[
    {
        "Chapter": "1",
        "sentence_range": "1-4",
        "Text": "vProofs are to Mathematics what calligraphy is to poetry Mathematical works do consist of proofs just as\npoems do consist of characters \u2014 VLADIMIR ARNOLD v\nA 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "2-5",
        "Text": "Mathematical works do consist of proofs just as\npoems do consist of characters \u2014 VLADIMIR ARNOLD v\nA 1 1  Introduction\nIn Classes IX, X and XI, we have learnt about the concepts of a statement, compound\nstatement, negation, converse and contrapositive of a statement; axioms, conjectures,\ntheorems and deductive reasoning"
    },
    {
        "Chapter": "1",
        "sentence_range": "3-6",
        "Text": "\u2014 VLADIMIR ARNOLD v\nA 1 1  Introduction\nIn Classes IX, X and XI, we have learnt about the concepts of a statement, compound\nstatement, negation, converse and contrapositive of a statement; axioms, conjectures,\ntheorems and deductive reasoning Here, we will discuss various methods of proving mathematical propositions"
    },
    {
        "Chapter": "1",
        "sentence_range": "4-7",
        "Text": "1 1  Introduction\nIn Classes IX, X and XI, we have learnt about the concepts of a statement, compound\nstatement, negation, converse and contrapositive of a statement; axioms, conjectures,\ntheorems and deductive reasoning Here, we will discuss various methods of proving mathematical propositions A"
    },
    {
        "Chapter": "1",
        "sentence_range": "5-8",
        "Text": "1  Introduction\nIn Classes IX, X and XI, we have learnt about the concepts of a statement, compound\nstatement, negation, converse and contrapositive of a statement; axioms, conjectures,\ntheorems and deductive reasoning Here, we will discuss various methods of proving mathematical propositions A 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "6-9",
        "Text": "Here, we will discuss various methods of proving mathematical propositions A 1 2  What is a Proof"
    },
    {
        "Chapter": "1",
        "sentence_range": "7-10",
        "Text": "A 1 2  What is a Proof Proof of a mathematical statement consists of sequence of statements, each statement\nbeing justified with a definition or an axiom or a proposition that is previously established\nby the  method of deduction using only the allowed logical rules"
    },
    {
        "Chapter": "1",
        "sentence_range": "8-11",
        "Text": "1 2  What is a Proof Proof of a mathematical statement consists of sequence of statements, each statement\nbeing justified with a definition or an axiom or a proposition that is previously established\nby the  method of deduction using only the allowed logical rules Thus, each proof is a chain of deductive arguments each of which has its premises\nand conclusions"
    },
    {
        "Chapter": "1",
        "sentence_range": "9-12",
        "Text": "2  What is a Proof Proof of a mathematical statement consists of sequence of statements, each statement\nbeing justified with a definition or an axiom or a proposition that is previously established\nby the  method of deduction using only the allowed logical rules Thus, each proof is a chain of deductive arguments each of which has its premises\nand conclusions Many a times, we prove a proposition directly from what is given in\nthe proposition"
    },
    {
        "Chapter": "1",
        "sentence_range": "10-13",
        "Text": "Proof of a mathematical statement consists of sequence of statements, each statement\nbeing justified with a definition or an axiom or a proposition that is previously established\nby the  method of deduction using only the allowed logical rules Thus, each proof is a chain of deductive arguments each of which has its premises\nand conclusions Many a times, we prove a proposition directly from what is given in\nthe proposition But some times it is easier to prove an equivalent proposition rather\nthan proving the proposition itself"
    },
    {
        "Chapter": "1",
        "sentence_range": "11-14",
        "Text": "Thus, each proof is a chain of deductive arguments each of which has its premises\nand conclusions Many a times, we prove a proposition directly from what is given in\nthe proposition But some times it is easier to prove an equivalent proposition rather\nthan proving the proposition itself This leads to, two ways of proving a proposition\ndirectly or indirectly and the proofs obtained are called direct proof and indirect proof\nand further each has three different ways of proving which is discussed below"
    },
    {
        "Chapter": "1",
        "sentence_range": "12-15",
        "Text": "Many a times, we prove a proposition directly from what is given in\nthe proposition But some times it is easier to prove an equivalent proposition rather\nthan proving the proposition itself This leads to, two ways of proving a proposition\ndirectly or indirectly and the proofs obtained are called direct proof and indirect proof\nand further each has three different ways of proving which is discussed below Direct Proof  It is the proof of a proposition in which we directly start the proof with\nwhat is given in the proposition"
    },
    {
        "Chapter": "1",
        "sentence_range": "13-16",
        "Text": "But some times it is easier to prove an equivalent proposition rather\nthan proving the proposition itself This leads to, two ways of proving a proposition\ndirectly or indirectly and the proofs obtained are called direct proof and indirect proof\nand further each has three different ways of proving which is discussed below Direct Proof  It is the proof of a proposition in which we directly start the proof with\nwhat is given in the proposition (i)\nStraight forward approach It is a chain of arguments which leads directly from\nwhat is given or assumed,  with the help of axioms, definitions or already proved\ntheorems, to what is to be proved using rules of logic"
    },
    {
        "Chapter": "1",
        "sentence_range": "14-17",
        "Text": "This leads to, two ways of proving a proposition\ndirectly or indirectly and the proofs obtained are called direct proof and indirect proof\nand further each has three different ways of proving which is discussed below Direct Proof  It is the proof of a proposition in which we directly start the proof with\nwhat is given in the proposition (i)\nStraight forward approach It is a chain of arguments which leads directly from\nwhat is given or assumed,  with the help of axioms, definitions or already proved\ntheorems, to what is to be proved using rules of logic Consider the following example:\nExample 1 Show that if x2 \u2013 5x + 6 = 0, then x = 3 or x = 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "15-18",
        "Text": "Direct Proof  It is the proof of a proposition in which we directly start the proof with\nwhat is given in the proposition (i)\nStraight forward approach It is a chain of arguments which leads directly from\nwhat is given or assumed,  with the help of axioms, definitions or already proved\ntheorems, to what is to be proved using rules of logic Consider the following example:\nExample 1 Show that if x2 \u2013 5x + 6 = 0, then x = 3 or x = 2 Solution x2 \u2013 5x + 6 = 0 (given)\nAppendix 1\nPROOFS IN MATHEMATICS\nRationalised 2023-24\n MATHEMATICS\n188\n\u21d2 (x \u2013 3) (x \u2013 2) = 0 (replacing an expression by an equal/equivalent expression)\n\u21d2 x \u2013 3 = 0 or x \u2013 2 = 0 (from the established theorem ab = 0 \u21d2 either a = 0 or\nb = 0, for a, b in R)\n\u21d2 x \u2013 3 + 3 = 0 + 3 or  x \u2013 2 + 2 = 0 + 2 (adding equal quantities on either side of the\nequation does not alter the nature of the\nequation)\n\u21d2 x + 0 = 3 or x + 0 = 2  (using the identity property of integers under addition)\n\u21d2 x = 3 or x = 2 (using the identity property of integers under addition)\nHence, x2 \u2013 5x + 6 = 0 implies x = 3 or x = 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "16-19",
        "Text": "(i)\nStraight forward approach It is a chain of arguments which leads directly from\nwhat is given or assumed,  with the help of axioms, definitions or already proved\ntheorems, to what is to be proved using rules of logic Consider the following example:\nExample 1 Show that if x2 \u2013 5x + 6 = 0, then x = 3 or x = 2 Solution x2 \u2013 5x + 6 = 0 (given)\nAppendix 1\nPROOFS IN MATHEMATICS\nRationalised 2023-24\n MATHEMATICS\n188\n\u21d2 (x \u2013 3) (x \u2013 2) = 0 (replacing an expression by an equal/equivalent expression)\n\u21d2 x \u2013 3 = 0 or x \u2013 2 = 0 (from the established theorem ab = 0 \u21d2 either a = 0 or\nb = 0, for a, b in R)\n\u21d2 x \u2013 3 + 3 = 0 + 3 or  x \u2013 2 + 2 = 0 + 2 (adding equal quantities on either side of the\nequation does not alter the nature of the\nequation)\n\u21d2 x + 0 = 3 or x + 0 = 2  (using the identity property of integers under addition)\n\u21d2 x = 3 or x = 2 (using the identity property of integers under addition)\nHence, x2 \u2013 5x + 6 = 0 implies x = 3 or x = 2 Explanation Let p be the given statement \u201cx2 \u2013 5x + 6 = 0\u201d and q be the conclusion\nstatement \u201cx = 3 or x = 2\u201d"
    },
    {
        "Chapter": "1",
        "sentence_range": "17-20",
        "Text": "Consider the following example:\nExample 1 Show that if x2 \u2013 5x + 6 = 0, then x = 3 or x = 2 Solution x2 \u2013 5x + 6 = 0 (given)\nAppendix 1\nPROOFS IN MATHEMATICS\nRationalised 2023-24\n MATHEMATICS\n188\n\u21d2 (x \u2013 3) (x \u2013 2) = 0 (replacing an expression by an equal/equivalent expression)\n\u21d2 x \u2013 3 = 0 or x \u2013 2 = 0 (from the established theorem ab = 0 \u21d2 either a = 0 or\nb = 0, for a, b in R)\n\u21d2 x \u2013 3 + 3 = 0 + 3 or  x \u2013 2 + 2 = 0 + 2 (adding equal quantities on either side of the\nequation does not alter the nature of the\nequation)\n\u21d2 x + 0 = 3 or x + 0 = 2  (using the identity property of integers under addition)\n\u21d2 x = 3 or x = 2 (using the identity property of integers under addition)\nHence, x2 \u2013 5x + 6 = 0 implies x = 3 or x = 2 Explanation Let p be the given statement \u201cx2 \u2013 5x + 6 = 0\u201d and q be the conclusion\nstatement \u201cx = 3 or x = 2\u201d From the statement p, we deduced the statement r : \u201c(x \u2013 3) (x \u2013 2) = 0\u201d by\nreplacing the expression x2 \u2013 5x + 6 in the statement p by another expression (x \u2013 3)\n(x \u2013 2) which is equal to x2 \u2013 5x + 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "18-21",
        "Text": "Solution x2 \u2013 5x + 6 = 0 (given)\nAppendix 1\nPROOFS IN MATHEMATICS\nRationalised 2023-24\n MATHEMATICS\n188\n\u21d2 (x \u2013 3) (x \u2013 2) = 0 (replacing an expression by an equal/equivalent expression)\n\u21d2 x \u2013 3 = 0 or x \u2013 2 = 0 (from the established theorem ab = 0 \u21d2 either a = 0 or\nb = 0, for a, b in R)\n\u21d2 x \u2013 3 + 3 = 0 + 3 or  x \u2013 2 + 2 = 0 + 2 (adding equal quantities on either side of the\nequation does not alter the nature of the\nequation)\n\u21d2 x + 0 = 3 or x + 0 = 2  (using the identity property of integers under addition)\n\u21d2 x = 3 or x = 2 (using the identity property of integers under addition)\nHence, x2 \u2013 5x + 6 = 0 implies x = 3 or x = 2 Explanation Let p be the given statement \u201cx2 \u2013 5x + 6 = 0\u201d and q be the conclusion\nstatement \u201cx = 3 or x = 2\u201d From the statement p, we deduced the statement r : \u201c(x \u2013 3) (x \u2013 2) = 0\u201d by\nreplacing the expression x2 \u2013 5x + 6 in the statement p by another expression (x \u2013 3)\n(x \u2013 2) which is equal to x2 \u2013 5x + 6 (i)There arise two questions:\nHow does the expression (x \u2013 3) (x \u2013 2) is equal to the expression x2 \u2013 5x + 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "19-22",
        "Text": "Explanation Let p be the given statement \u201cx2 \u2013 5x + 6 = 0\u201d and q be the conclusion\nstatement \u201cx = 3 or x = 2\u201d From the statement p, we deduced the statement r : \u201c(x \u2013 3) (x \u2013 2) = 0\u201d by\nreplacing the expression x2 \u2013 5x + 6 in the statement p by another expression (x \u2013 3)\n(x \u2013 2) which is equal to x2 \u2013 5x + 6 (i)There arise two questions:\nHow does the expression (x \u2013 3) (x \u2013 2) is equal to the expression x2 \u2013 5x + 6 (ii)\nHow can we replace an expression with another expression which is equal to\nthe former"
    },
    {
        "Chapter": "1",
        "sentence_range": "20-23",
        "Text": "From the statement p, we deduced the statement r : \u201c(x \u2013 3) (x \u2013 2) = 0\u201d by\nreplacing the expression x2 \u2013 5x + 6 in the statement p by another expression (x \u2013 3)\n(x \u2013 2) which is equal to x2 \u2013 5x + 6 (i)There arise two questions:\nHow does the expression (x \u2013 3) (x \u2013 2) is equal to the expression x2 \u2013 5x + 6 (ii)\nHow can we replace an expression with another expression which is equal to\nthe former The first one is proved in earlier classes by factorization, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "21-24",
        "Text": "(i)There arise two questions:\nHow does the expression (x \u2013 3) (x \u2013 2) is equal to the expression x2 \u2013 5x + 6 (ii)\nHow can we replace an expression with another expression which is equal to\nthe former The first one is proved in earlier classes by factorization, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "22-25",
        "Text": "(ii)\nHow can we replace an expression with another expression which is equal to\nthe former The first one is proved in earlier classes by factorization, i e ,\nx2 \u2013 5x + 6 = x2 \u2013 3x \u2013 2x + 6 = x (x \u2013 3) \u20132 (x \u2013 3) = (x \u2013 3) (x \u2013 2)"
    },
    {
        "Chapter": "1",
        "sentence_range": "23-26",
        "Text": "The first one is proved in earlier classes by factorization, i e ,\nx2 \u2013 5x + 6 = x2 \u2013 3x \u2013 2x + 6 = x (x \u2013 3) \u20132 (x \u2013 3) = (x \u2013 3) (x \u2013 2) The second one is by valid form of argumentation (rules of logic)\nNext this statement r becomes premises or given and deduce the statement s\n\u201c x \u2013 3 = 0 or x \u2013 2 = 0\u201d and the reasons are given in the brackets"
    },
    {
        "Chapter": "1",
        "sentence_range": "24-27",
        "Text": "e ,\nx2 \u2013 5x + 6 = x2 \u2013 3x \u2013 2x + 6 = x (x \u2013 3) \u20132 (x \u2013 3) = (x \u2013 3) (x \u2013 2) The second one is by valid form of argumentation (rules of logic)\nNext this statement r becomes premises or given and deduce the statement s\n\u201c x \u2013 3 = 0 or x \u2013 2 = 0\u201d and the reasons are given in the brackets This process continues till we reach the conclusion"
    },
    {
        "Chapter": "1",
        "sentence_range": "25-28",
        "Text": ",\nx2 \u2013 5x + 6 = x2 \u2013 3x \u2013 2x + 6 = x (x \u2013 3) \u20132 (x \u2013 3) = (x \u2013 3) (x \u2013 2) The second one is by valid form of argumentation (rules of logic)\nNext this statement r becomes premises or given and deduce the statement s\n\u201c x \u2013 3 = 0 or x \u2013 2 = 0\u201d and the reasons are given in the brackets This process continues till we reach the conclusion The symbolic equivalent of the argument is to prove by deduction that p \u21d2 q\nis true"
    },
    {
        "Chapter": "1",
        "sentence_range": "26-29",
        "Text": "The second one is by valid form of argumentation (rules of logic)\nNext this statement r becomes premises or given and deduce the statement s\n\u201c x \u2013 3 = 0 or x \u2013 2 = 0\u201d and the reasons are given in the brackets This process continues till we reach the conclusion The symbolic equivalent of the argument is to prove by deduction that p \u21d2 q\nis true Starting with p, we deduce p \u21d2 r \u21d2 s \u21d2 \u2026 \u21d2 q"
    },
    {
        "Chapter": "1",
        "sentence_range": "27-30",
        "Text": "This process continues till we reach the conclusion The symbolic equivalent of the argument is to prove by deduction that p \u21d2 q\nis true Starting with p, we deduce p \u21d2 r \u21d2 s \u21d2 \u2026 \u21d2 q This implies that \u201cp \u21d2 q\u201d is true"
    },
    {
        "Chapter": "1",
        "sentence_range": "28-31",
        "Text": "The symbolic equivalent of the argument is to prove by deduction that p \u21d2 q\nis true Starting with p, we deduce p \u21d2 r \u21d2 s \u21d2 \u2026 \u21d2 q This implies that \u201cp \u21d2 q\u201d is true Example 2 Prove that the function f : R \u2192\n\u2192\n\u2192\n\u2192\n\u2192 R\ndefined by f (x) = 2x + 5 is one-one"
    },
    {
        "Chapter": "1",
        "sentence_range": "29-32",
        "Text": "Starting with p, we deduce p \u21d2 r \u21d2 s \u21d2 \u2026 \u21d2 q This implies that \u201cp \u21d2 q\u201d is true Example 2 Prove that the function f : R \u2192\n\u2192\n\u2192\n\u2192\n\u2192 R\ndefined by f (x) = 2x + 5 is one-one Solution Note that a function f is one-one if\nf (x1) = f (x2) \u21d2 x1 = x2  (definition of one-one function)\nNow,  given that\nf (x1) = f (x2), i"
    },
    {
        "Chapter": "1",
        "sentence_range": "30-33",
        "Text": "This implies that \u201cp \u21d2 q\u201d is true Example 2 Prove that the function f : R \u2192\n\u2192\n\u2192\n\u2192\n\u2192 R\ndefined by f (x) = 2x + 5 is one-one Solution Note that a function f is one-one if\nf (x1) = f (x2) \u21d2 x1 = x2  (definition of one-one function)\nNow,  given that\nf (x1) = f (x2), i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "31-34",
        "Text": "Example 2 Prove that the function f : R \u2192\n\u2192\n\u2192\n\u2192\n\u2192 R\ndefined by f (x) = 2x + 5 is one-one Solution Note that a function f is one-one if\nf (x1) = f (x2) \u21d2 x1 = x2  (definition of one-one function)\nNow,  given that\nf (x1) = f (x2), i e , 2x1+ 5 = 2x2 + 5\n\u21d2\n2x1+ 5 \u2013 5 = 2x2 + 5 \u2013 5  (adding the same quantity on both sides)\nRationalised 2023-24\nPROOFS IN MATHEMATICS\n189\n\u21d2\n2x1+ 0 = 2x2 + 0\n\u21d2\n2x1 = 2x2 (using additive identity of real number)\n\u21d2\n1\n2\n2 x  =\n2\n2\n2 x  (dividing by the same non zero quantity)\n\u21d2\nx1 = x2\nHence, the given function is one-one"
    },
    {
        "Chapter": "1",
        "sentence_range": "32-35",
        "Text": "Solution Note that a function f is one-one if\nf (x1) = f (x2) \u21d2 x1 = x2  (definition of one-one function)\nNow,  given that\nf (x1) = f (x2), i e , 2x1+ 5 = 2x2 + 5\n\u21d2\n2x1+ 5 \u2013 5 = 2x2 + 5 \u2013 5  (adding the same quantity on both sides)\nRationalised 2023-24\nPROOFS IN MATHEMATICS\n189\n\u21d2\n2x1+ 0 = 2x2 + 0\n\u21d2\n2x1 = 2x2 (using additive identity of real number)\n\u21d2\n1\n2\n2 x  =\n2\n2\n2 x  (dividing by the same non zero quantity)\n\u21d2\nx1 = x2\nHence, the given function is one-one (ii) Mathematical Induction\nMathematical induction, is a strategy, of proving a proposition which is deductive in\nnature"
    },
    {
        "Chapter": "1",
        "sentence_range": "33-36",
        "Text": "e , 2x1+ 5 = 2x2 + 5\n\u21d2\n2x1+ 5 \u2013 5 = 2x2 + 5 \u2013 5  (adding the same quantity on both sides)\nRationalised 2023-24\nPROOFS IN MATHEMATICS\n189\n\u21d2\n2x1+ 0 = 2x2 + 0\n\u21d2\n2x1 = 2x2 (using additive identity of real number)\n\u21d2\n1\n2\n2 x  =\n2\n2\n2 x  (dividing by the same non zero quantity)\n\u21d2\nx1 = x2\nHence, the given function is one-one (ii) Mathematical Induction\nMathematical induction, is a strategy, of proving a proposition which is deductive in\nnature The whole basis of proof of this method depends on the following axiom:\nFor a given subset S of N, if\n(i)\nthe natural number 1 \u2208 S  and\n(ii)\nthe natural number k + 1 \u2208 S whenever k \u2208 S, then S = N"
    },
    {
        "Chapter": "1",
        "sentence_range": "34-37",
        "Text": ", 2x1+ 5 = 2x2 + 5\n\u21d2\n2x1+ 5 \u2013 5 = 2x2 + 5 \u2013 5  (adding the same quantity on both sides)\nRationalised 2023-24\nPROOFS IN MATHEMATICS\n189\n\u21d2\n2x1+ 0 = 2x2 + 0\n\u21d2\n2x1 = 2x2 (using additive identity of real number)\n\u21d2\n1\n2\n2 x  =\n2\n2\n2 x  (dividing by the same non zero quantity)\n\u21d2\nx1 = x2\nHence, the given function is one-one (ii) Mathematical Induction\nMathematical induction, is a strategy, of proving a proposition which is deductive in\nnature The whole basis of proof of this method depends on the following axiom:\nFor a given subset S of N, if\n(i)\nthe natural number 1 \u2208 S  and\n(ii)\nthe natural number k + 1 \u2208 S whenever k \u2208 S, then S = N According to the principle of mathematical induction, if a statement \u201cS(n) is true\nfor n = 1\u201d (or for some starting point j), and if \u201cS(n) is true for n = k\u201d implies that \u201cS(n)\nis true for n = k + 1\u201d (whatever integer k \u2265  j may be), then the statement is true for any\npositive integer n, for all n \u2265 j"
    },
    {
        "Chapter": "1",
        "sentence_range": "35-38",
        "Text": "(ii) Mathematical Induction\nMathematical induction, is a strategy, of proving a proposition which is deductive in\nnature The whole basis of proof of this method depends on the following axiom:\nFor a given subset S of N, if\n(i)\nthe natural number 1 \u2208 S  and\n(ii)\nthe natural number k + 1 \u2208 S whenever k \u2208 S, then S = N According to the principle of mathematical induction, if a statement \u201cS(n) is true\nfor n = 1\u201d (or for some starting point j), and if \u201cS(n) is true for n = k\u201d implies that \u201cS(n)\nis true for n = k + 1\u201d (whatever integer k \u2265  j may be), then the statement is true for any\npositive integer n, for all n \u2265 j We now consider some examples"
    },
    {
        "Chapter": "1",
        "sentence_range": "36-39",
        "Text": "The whole basis of proof of this method depends on the following axiom:\nFor a given subset S of N, if\n(i)\nthe natural number 1 \u2208 S  and\n(ii)\nthe natural number k + 1 \u2208 S whenever k \u2208 S, then S = N According to the principle of mathematical induction, if a statement \u201cS(n) is true\nfor n = 1\u201d (or for some starting point j), and if \u201cS(n) is true for n = k\u201d implies that \u201cS(n)\nis true for n = k + 1\u201d (whatever integer k \u2265  j may be), then the statement is true for any\npositive integer n, for all n \u2265 j We now consider some examples Example 3 Show that if\n A = \ncos\nsin\nsin\ncos\n\u03b8\n\u03b8\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n\u03b8\n\u03b8\n\uf8f0\n\uf8fb , then   An =\ncos\nsin\nsin\ncos\nn\nn\nn\nn\n\u03b8\n\u03b8\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n\u03b8\n\u03b8\n\uf8f0\n\uf8fb\nSolution We have\nP(n) : An =\ncos\nsin\nsin\ncos\nn\nn\nn\nn\n\u03b8\n\u03b8\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n\u03b8\n\u03b8\n\uf8f0\n\uf8fb\nWe note that\nP(1) : A1 =\ncos\nsin\nsin\ncos\n\u03b8\n\u03b8\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n\u03b8\n\u03b8\n\uf8f0\n\uf8fb\nTherefore, P(1) is true"
    },
    {
        "Chapter": "1",
        "sentence_range": "37-40",
        "Text": "According to the principle of mathematical induction, if a statement \u201cS(n) is true\nfor n = 1\u201d (or for some starting point j), and if \u201cS(n) is true for n = k\u201d implies that \u201cS(n)\nis true for n = k + 1\u201d (whatever integer k \u2265  j may be), then the statement is true for any\npositive integer n, for all n \u2265 j We now consider some examples Example 3 Show that if\n A = \ncos\nsin\nsin\ncos\n\u03b8\n\u03b8\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n\u03b8\n\u03b8\n\uf8f0\n\uf8fb , then   An =\ncos\nsin\nsin\ncos\nn\nn\nn\nn\n\u03b8\n\u03b8\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n\u03b8\n\u03b8\n\uf8f0\n\uf8fb\nSolution We have\nP(n) : An =\ncos\nsin\nsin\ncos\nn\nn\nn\nn\n\u03b8\n\u03b8\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n\u03b8\n\u03b8\n\uf8f0\n\uf8fb\nWe note that\nP(1) : A1 =\ncos\nsin\nsin\ncos\n\u03b8\n\u03b8\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n\u03b8\n\u03b8\n\uf8f0\n\uf8fb\nTherefore, P(1) is true Assume that P(k) is true, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "38-41",
        "Text": "We now consider some examples Example 3 Show that if\n A = \ncos\nsin\nsin\ncos\n\u03b8\n\u03b8\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n\u03b8\n\u03b8\n\uf8f0\n\uf8fb , then   An =\ncos\nsin\nsin\ncos\nn\nn\nn\nn\n\u03b8\n\u03b8\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n\u03b8\n\u03b8\n\uf8f0\n\uf8fb\nSolution We have\nP(n) : An =\ncos\nsin\nsin\ncos\nn\nn\nn\nn\n\u03b8\n\u03b8\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n\u03b8\n\u03b8\n\uf8f0\n\uf8fb\nWe note that\nP(1) : A1 =\ncos\nsin\nsin\ncos\n\u03b8\n\u03b8\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n\u03b8\n\u03b8\n\uf8f0\n\uf8fb\nTherefore, P(1) is true Assume that P(k) is true, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "39-42",
        "Text": "Example 3 Show that if\n A = \ncos\nsin\nsin\ncos\n\u03b8\n\u03b8\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n\u03b8\n\u03b8\n\uf8f0\n\uf8fb , then   An =\ncos\nsin\nsin\ncos\nn\nn\nn\nn\n\u03b8\n\u03b8\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n\u03b8\n\u03b8\n\uf8f0\n\uf8fb\nSolution We have\nP(n) : An =\ncos\nsin\nsin\ncos\nn\nn\nn\nn\n\u03b8\n\u03b8\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n\u03b8\n\u03b8\n\uf8f0\n\uf8fb\nWe note that\nP(1) : A1 =\ncos\nsin\nsin\ncos\n\u03b8\n\u03b8\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n\u03b8\n\u03b8\n\uf8f0\n\uf8fb\nTherefore, P(1) is true Assume that P(k) is true, i e ,\nP(k) : Ak =\ncos\nsin\nsin\ncos\nk\nk\nk\nk\n\u03b8\n\u03b8\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n\u03b8\n\u03b8\n\uf8f0\n\uf8fb\nRationalised 2023-24\n MATHEMATICS\n190\nWe want to prove that P(k + 1) is true whenever P(k) is true, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "40-43",
        "Text": "Assume that P(k) is true, i e ,\nP(k) : Ak =\ncos\nsin\nsin\ncos\nk\nk\nk\nk\n\u03b8\n\u03b8\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n\u03b8\n\u03b8\n\uf8f0\n\uf8fb\nRationalised 2023-24\n MATHEMATICS\n190\nWe want to prove that P(k + 1) is true whenever P(k) is true, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "41-44",
        "Text": "e ,\nP(k) : Ak =\ncos\nsin\nsin\ncos\nk\nk\nk\nk\n\u03b8\n\u03b8\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n\u03b8\n\u03b8\n\uf8f0\n\uf8fb\nRationalised 2023-24\n MATHEMATICS\n190\nWe want to prove that P(k + 1) is true whenever P(k) is true, i e ,\nP(k + 1) : Ak+1 =\ncos (\n1)\nsin (\n1)\nsin(\n1)\ncos (\n1 )\nk\nk\nk\nk\n+\n\u03b8\n+\n\u03b8\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n+\n\u03b8\n+\n\u03b8\n\uf8f0\n\uf8fb\nNow\nAk+1 = Ak"
    },
    {
        "Chapter": "1",
        "sentence_range": "42-45",
        "Text": ",\nP(k) : Ak =\ncos\nsin\nsin\ncos\nk\nk\nk\nk\n\u03b8\n\u03b8\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n\u03b8\n\u03b8\n\uf8f0\n\uf8fb\nRationalised 2023-24\n MATHEMATICS\n190\nWe want to prove that P(k + 1) is true whenever P(k) is true, i e ,\nP(k + 1) : Ak+1 =\ncos (\n1)\nsin (\n1)\nsin(\n1)\ncos (\n1 )\nk\nk\nk\nk\n+\n\u03b8\n+\n\u03b8\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n+\n\u03b8\n+\n\u03b8\n\uf8f0\n\uf8fb\nNow\nAk+1 = Ak A\nSince P(k) is true, we have\nAk+1 =\ncos\nsin\nsin\ncos\nk\nk\nk\nk\n\u03b8\n\u03b8\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n\u03b8\n\u03b8\n\uf8f0\n\uf8fb   \ncos\nsin\nsin\ncos\n\u03b8\n\u03b8\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n\u03b8\n\u03b8\n\uf8f0\n\uf8fb\n= \ncos\ncos\nsin\nsin\ncos\nsin\nsin\ncos\nsin\ncos\ncos\nsin\nsin\nsin\ncos\ncos\nk\nk\nk\nk\nk\nk\nk\nk\n\u03b8\n\u03b8 \u2212\n\u03b8\n\u03b8\n\u03b8\n\u03b8 +\n\u03b8\n\u03b8\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n\u03b8\n\u03b8 \u2212\n\u03b8\n\u03b8\n\u2212\n\u03b8\n\u03b8 +\n\u03b8\n\u03b8\n\uf8f0\n\uf8fb\n(by matrix multiplication)\n= \ncos (\n1)\nsin (\n1)\nsin(\n1)\ncos (\n1 )\nk\nk\nk\nk\n+\n\u03b8\n+\n\u03b8\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n+\n\u03b8\n+\n\u03b8\n\uf8f0\n\uf8fb\nThus, P(k + 1) is true whenever P(k) is true"
    },
    {
        "Chapter": "1",
        "sentence_range": "43-46",
        "Text": "e ,\nP(k + 1) : Ak+1 =\ncos (\n1)\nsin (\n1)\nsin(\n1)\ncos (\n1 )\nk\nk\nk\nk\n+\n\u03b8\n+\n\u03b8\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n+\n\u03b8\n+\n\u03b8\n\uf8f0\n\uf8fb\nNow\nAk+1 = Ak A\nSince P(k) is true, we have\nAk+1 =\ncos\nsin\nsin\ncos\nk\nk\nk\nk\n\u03b8\n\u03b8\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n\u03b8\n\u03b8\n\uf8f0\n\uf8fb   \ncos\nsin\nsin\ncos\n\u03b8\n\u03b8\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n\u03b8\n\u03b8\n\uf8f0\n\uf8fb\n= \ncos\ncos\nsin\nsin\ncos\nsin\nsin\ncos\nsin\ncos\ncos\nsin\nsin\nsin\ncos\ncos\nk\nk\nk\nk\nk\nk\nk\nk\n\u03b8\n\u03b8 \u2212\n\u03b8\n\u03b8\n\u03b8\n\u03b8 +\n\u03b8\n\u03b8\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n\u03b8\n\u03b8 \u2212\n\u03b8\n\u03b8\n\u2212\n\u03b8\n\u03b8 +\n\u03b8\n\u03b8\n\uf8f0\n\uf8fb\n(by matrix multiplication)\n= \ncos (\n1)\nsin (\n1)\nsin(\n1)\ncos (\n1 )\nk\nk\nk\nk\n+\n\u03b8\n+\n\u03b8\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n+\n\u03b8\n+\n\u03b8\n\uf8f0\n\uf8fb\nThus, P(k + 1) is true whenever P(k) is true Hence, P(n) is true for all n \u2265 1 (by the principle of mathematical induction)"
    },
    {
        "Chapter": "1",
        "sentence_range": "44-47",
        "Text": ",\nP(k + 1) : Ak+1 =\ncos (\n1)\nsin (\n1)\nsin(\n1)\ncos (\n1 )\nk\nk\nk\nk\n+\n\u03b8\n+\n\u03b8\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n+\n\u03b8\n+\n\u03b8\n\uf8f0\n\uf8fb\nNow\nAk+1 = Ak A\nSince P(k) is true, we have\nAk+1 =\ncos\nsin\nsin\ncos\nk\nk\nk\nk\n\u03b8\n\u03b8\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n\u03b8\n\u03b8\n\uf8f0\n\uf8fb   \ncos\nsin\nsin\ncos\n\u03b8\n\u03b8\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n\u03b8\n\u03b8\n\uf8f0\n\uf8fb\n= \ncos\ncos\nsin\nsin\ncos\nsin\nsin\ncos\nsin\ncos\ncos\nsin\nsin\nsin\ncos\ncos\nk\nk\nk\nk\nk\nk\nk\nk\n\u03b8\n\u03b8 \u2212\n\u03b8\n\u03b8\n\u03b8\n\u03b8 +\n\u03b8\n\u03b8\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n\u03b8\n\u03b8 \u2212\n\u03b8\n\u03b8\n\u2212\n\u03b8\n\u03b8 +\n\u03b8\n\u03b8\n\uf8f0\n\uf8fb\n(by matrix multiplication)\n= \ncos (\n1)\nsin (\n1)\nsin(\n1)\ncos (\n1 )\nk\nk\nk\nk\n+\n\u03b8\n+\n\u03b8\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n+\n\u03b8\n+\n\u03b8\n\uf8f0\n\uf8fb\nThus, P(k + 1) is true whenever P(k) is true Hence, P(n) is true for all n \u2265 1 (by the principle of mathematical induction) (iii) Proof by cases or by exhaustion\nThis method of proving a statement p \u21d2 q is possible only when p can be split into\nseveral cases, r, s, t  (say) so that  p = r \u2228  s \u2228  t (where \u201c\u2228 \u201d is the symbol for \u201cOR\u201d)"
    },
    {
        "Chapter": "1",
        "sentence_range": "45-48",
        "Text": "A\nSince P(k) is true, we have\nAk+1 =\ncos\nsin\nsin\ncos\nk\nk\nk\nk\n\u03b8\n\u03b8\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n\u03b8\n\u03b8\n\uf8f0\n\uf8fb   \ncos\nsin\nsin\ncos\n\u03b8\n\u03b8\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n\u03b8\n\u03b8\n\uf8f0\n\uf8fb\n= \ncos\ncos\nsin\nsin\ncos\nsin\nsin\ncos\nsin\ncos\ncos\nsin\nsin\nsin\ncos\ncos\nk\nk\nk\nk\nk\nk\nk\nk\n\u03b8\n\u03b8 \u2212\n\u03b8\n\u03b8\n\u03b8\n\u03b8 +\n\u03b8\n\u03b8\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n\u03b8\n\u03b8 \u2212\n\u03b8\n\u03b8\n\u2212\n\u03b8\n\u03b8 +\n\u03b8\n\u03b8\n\uf8f0\n\uf8fb\n(by matrix multiplication)\n= \ncos (\n1)\nsin (\n1)\nsin(\n1)\ncos (\n1 )\nk\nk\nk\nk\n+\n\u03b8\n+\n\u03b8\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n+\n\u03b8\n+\n\u03b8\n\uf8f0\n\uf8fb\nThus, P(k + 1) is true whenever P(k) is true Hence, P(n) is true for all n \u2265 1 (by the principle of mathematical induction) (iii) Proof by cases or by exhaustion\nThis method of proving a statement p \u21d2 q is possible only when p can be split into\nseveral cases, r, s, t  (say) so that  p = r \u2228  s \u2228  t (where \u201c\u2228 \u201d is the symbol for \u201cOR\u201d) If the conditionals\nr \u21d2 q;\ns \u21d2 q;\nand\nt \u21d2 q\nare proved, then (r \u2228  s \u2228  t) \u21d2 q, is proved and so p \u21d2 q is proved"
    },
    {
        "Chapter": "1",
        "sentence_range": "46-49",
        "Text": "Hence, P(n) is true for all n \u2265 1 (by the principle of mathematical induction) (iii) Proof by cases or by exhaustion\nThis method of proving a statement p \u21d2 q is possible only when p can be split into\nseveral cases, r, s, t  (say) so that  p = r \u2228  s \u2228  t (where \u201c\u2228 \u201d is the symbol for \u201cOR\u201d) If the conditionals\nr \u21d2 q;\ns \u21d2 q;\nand\nt \u21d2 q\nare proved, then (r \u2228  s \u2228  t) \u21d2 q, is proved and so p \u21d2 q is proved The method consists of examining every possible case of the hypothesis"
    },
    {
        "Chapter": "1",
        "sentence_range": "47-50",
        "Text": "(iii) Proof by cases or by exhaustion\nThis method of proving a statement p \u21d2 q is possible only when p can be split into\nseveral cases, r, s, t  (say) so that  p = r \u2228  s \u2228  t (where \u201c\u2228 \u201d is the symbol for \u201cOR\u201d) If the conditionals\nr \u21d2 q;\ns \u21d2 q;\nand\nt \u21d2 q\nare proved, then (r \u2228  s \u2228  t) \u21d2 q, is proved and so p \u21d2 q is proved The method consists of examining every possible case of the hypothesis It is\npractically convenient only when the number of possible cases are few"
    },
    {
        "Chapter": "1",
        "sentence_range": "48-51",
        "Text": "If the conditionals\nr \u21d2 q;\ns \u21d2 q;\nand\nt \u21d2 q\nare proved, then (r \u2228  s \u2228  t) \u21d2 q, is proved and so p \u21d2 q is proved The method consists of examining every possible case of the hypothesis It is\npractically convenient only when the number of possible cases are few Example 4 Show that in any triangle ABC,\na = b cos C + c cos B\nSolution Let p be the statement \u201cABC is any triangle\u201d and q be the statement\n\u201ca = b cos C + c cos B\u201d\nLet ABC be a triangle"
    },
    {
        "Chapter": "1",
        "sentence_range": "49-52",
        "Text": "The method consists of examining every possible case of the hypothesis It is\npractically convenient only when the number of possible cases are few Example 4 Show that in any triangle ABC,\na = b cos C + c cos B\nSolution Let p be the statement \u201cABC is any triangle\u201d and q be the statement\n\u201ca = b cos C + c cos B\u201d\nLet ABC be a triangle From A draw AD a perpendicular to BC (BC produced if\nnecessary)"
    },
    {
        "Chapter": "1",
        "sentence_range": "50-53",
        "Text": "It is\npractically convenient only when the number of possible cases are few Example 4 Show that in any triangle ABC,\na = b cos C + c cos B\nSolution Let p be the statement \u201cABC is any triangle\u201d and q be the statement\n\u201ca = b cos C + c cos B\u201d\nLet ABC be a triangle From A draw AD a perpendicular to BC (BC produced if\nnecessary) As we know that any triangle has to be either acute or obtuse or right angled, we\ncan split p into three statements r, s and t, where\nRationalised 2023-24\nPROOFS IN MATHEMATICS\n191\nr : ABC is an acute angled triangle with \u2220 C is acute"
    },
    {
        "Chapter": "1",
        "sentence_range": "51-54",
        "Text": "Example 4 Show that in any triangle ABC,\na = b cos C + c cos B\nSolution Let p be the statement \u201cABC is any triangle\u201d and q be the statement\n\u201ca = b cos C + c cos B\u201d\nLet ABC be a triangle From A draw AD a perpendicular to BC (BC produced if\nnecessary) As we know that any triangle has to be either acute or obtuse or right angled, we\ncan split p into three statements r, s and t, where\nRationalised 2023-24\nPROOFS IN MATHEMATICS\n191\nr : ABC is an acute angled triangle with \u2220 C is acute s : ABC is an obtuse angled triangle with \u2220 C is obtuse"
    },
    {
        "Chapter": "1",
        "sentence_range": "52-55",
        "Text": "From A draw AD a perpendicular to BC (BC produced if\nnecessary) As we know that any triangle has to be either acute or obtuse or right angled, we\ncan split p into three statements r, s and t, where\nRationalised 2023-24\nPROOFS IN MATHEMATICS\n191\nr : ABC is an acute angled triangle with \u2220 C is acute s : ABC is an obtuse angled triangle with \u2220 C is obtuse t : ABC is a right angled triangle with \u2220 C is right angle"
    },
    {
        "Chapter": "1",
        "sentence_range": "53-56",
        "Text": "As we know that any triangle has to be either acute or obtuse or right angled, we\ncan split p into three statements r, s and t, where\nRationalised 2023-24\nPROOFS IN MATHEMATICS\n191\nr : ABC is an acute angled triangle with \u2220 C is acute s : ABC is an obtuse angled triangle with \u2220 C is obtuse t : ABC is a right angled triangle with \u2220 C is right angle Hence, we prove the theorem by three cases"
    },
    {
        "Chapter": "1",
        "sentence_range": "54-57",
        "Text": "s : ABC is an obtuse angled triangle with \u2220 C is obtuse t : ABC is a right angled triangle with \u2220 C is right angle Hence, we prove the theorem by three cases Case (i)  When \u2220 C is acute (Fig"
    },
    {
        "Chapter": "1",
        "sentence_range": "55-58",
        "Text": "t : ABC is a right angled triangle with \u2220 C is right angle Hence, we prove the theorem by three cases Case (i)  When \u2220 C is acute (Fig A1"
    },
    {
        "Chapter": "1",
        "sentence_range": "56-59",
        "Text": "Hence, we prove the theorem by three cases Case (i)  When \u2220 C is acute (Fig A1 1)"
    },
    {
        "Chapter": "1",
        "sentence_range": "57-60",
        "Text": "Case (i)  When \u2220 C is acute (Fig A1 1) From the right angled triangle ADB,\nBD\nAB  = cos B\ni"
    },
    {
        "Chapter": "1",
        "sentence_range": "58-61",
        "Text": "A1 1) From the right angled triangle ADB,\nBD\nAB  = cos B\ni e"
    },
    {
        "Chapter": "1",
        "sentence_range": "59-62",
        "Text": "1) From the right angled triangle ADB,\nBD\nAB  = cos B\ni e BD = AB cos B\n= c cos B\nFrom the right angled triangle ADC,\nCD\nAC  = cos C\ni"
    },
    {
        "Chapter": "1",
        "sentence_range": "60-63",
        "Text": "From the right angled triangle ADB,\nBD\nAB  = cos B\ni e BD = AB cos B\n= c cos B\nFrom the right angled triangle ADC,\nCD\nAC  = cos C\ni e"
    },
    {
        "Chapter": "1",
        "sentence_range": "61-64",
        "Text": "e BD = AB cos B\n= c cos B\nFrom the right angled triangle ADC,\nCD\nAC  = cos C\ni e CD = AC cos C\n= b cos C\nNow\na = BD + CD\n= c cos B + b cos C"
    },
    {
        "Chapter": "1",
        "sentence_range": "62-65",
        "Text": "BD = AB cos B\n= c cos B\nFrom the right angled triangle ADC,\nCD\nAC  = cos C\ni e CD = AC cos C\n= b cos C\nNow\na = BD + CD\n= c cos B + b cos C (1)\nCase (ii)  When \u2220 C is obtuse (Fig A1"
    },
    {
        "Chapter": "1",
        "sentence_range": "63-66",
        "Text": "e CD = AC cos C\n= b cos C\nNow\na = BD + CD\n= c cos B + b cos C (1)\nCase (ii)  When \u2220 C is obtuse (Fig A1 2)"
    },
    {
        "Chapter": "1",
        "sentence_range": "64-67",
        "Text": "CD = AC cos C\n= b cos C\nNow\na = BD + CD\n= c cos B + b cos C (1)\nCase (ii)  When \u2220 C is obtuse (Fig A1 2) From the right angled triangle ADB,\nBD\nAB  = cos B\ni"
    },
    {
        "Chapter": "1",
        "sentence_range": "65-68",
        "Text": "(1)\nCase (ii)  When \u2220 C is obtuse (Fig A1 2) From the right angled triangle ADB,\nBD\nAB  = cos B\ni e"
    },
    {
        "Chapter": "1",
        "sentence_range": "66-69",
        "Text": "2) From the right angled triangle ADB,\nBD\nAB  = cos B\ni e BD = AB cos B\n= c cos B\nFrom the right angled triangle ADC,\nCD\nAC  = cos \u2220 ACD\n= cos (180\u00b0 \u2013 C)\n= \u2013 cos C\ni"
    },
    {
        "Chapter": "1",
        "sentence_range": "67-70",
        "Text": "From the right angled triangle ADB,\nBD\nAB  = cos B\ni e BD = AB cos B\n= c cos B\nFrom the right angled triangle ADC,\nCD\nAC  = cos \u2220 ACD\n= cos (180\u00b0 \u2013 C)\n= \u2013 cos C\ni e"
    },
    {
        "Chapter": "1",
        "sentence_range": "68-71",
        "Text": "e BD = AB cos B\n= c cos B\nFrom the right angled triangle ADC,\nCD\nAC  = cos \u2220 ACD\n= cos (180\u00b0 \u2013 C)\n= \u2013 cos C\ni e CD = \u2013 AC cos C\n= \u2013 b cos C\n Fig  A1"
    },
    {
        "Chapter": "1",
        "sentence_range": "69-72",
        "Text": "BD = AB cos B\n= c cos B\nFrom the right angled triangle ADC,\nCD\nAC  = cos \u2220 ACD\n= cos (180\u00b0 \u2013 C)\n= \u2013 cos C\ni e CD = \u2013 AC cos C\n= \u2013 b cos C\n Fig  A1 1\n Fig  A1"
    },
    {
        "Chapter": "1",
        "sentence_range": "70-73",
        "Text": "e CD = \u2013 AC cos C\n= \u2013 b cos C\n Fig  A1 1\n Fig  A1 2\nRationalised 2023-24\n MATHEMATICS\n192\nNow\na = BC = BD \u2013 CD\ni"
    },
    {
        "Chapter": "1",
        "sentence_range": "71-74",
        "Text": "CD = \u2013 AC cos C\n= \u2013 b cos C\n Fig  A1 1\n Fig  A1 2\nRationalised 2023-24\n MATHEMATICS\n192\nNow\na = BC = BD \u2013 CD\ni e"
    },
    {
        "Chapter": "1",
        "sentence_range": "72-75",
        "Text": "1\n Fig  A1 2\nRationalised 2023-24\n MATHEMATICS\n192\nNow\na = BC = BD \u2013 CD\ni e a = c cos B \u2013 ( \u2013 b cos C)\na = c cos B + b cos C"
    },
    {
        "Chapter": "1",
        "sentence_range": "73-76",
        "Text": "2\nRationalised 2023-24\n MATHEMATICS\n192\nNow\na = BC = BD \u2013 CD\ni e a = c cos B \u2013 ( \u2013 b cos C)\na = c cos B + b cos C (2)\nCase (iii) When \u2220 C is a right angle (Fig A1"
    },
    {
        "Chapter": "1",
        "sentence_range": "74-77",
        "Text": "e a = c cos B \u2013 ( \u2013 b cos C)\na = c cos B + b cos C (2)\nCase (iii) When \u2220 C is a right angle (Fig A1 3)"
    },
    {
        "Chapter": "1",
        "sentence_range": "75-78",
        "Text": "a = c cos B \u2013 ( \u2013 b cos C)\na = c cos B + b cos C (2)\nCase (iii) When \u2220 C is a right angle (Fig A1 3) From the right angled triangle ACB,\nBC\nAB  = cos B\ni"
    },
    {
        "Chapter": "1",
        "sentence_range": "76-79",
        "Text": "(2)\nCase (iii) When \u2220 C is a right angle (Fig A1 3) From the right angled triangle ACB,\nBC\nAB  = cos B\ni e"
    },
    {
        "Chapter": "1",
        "sentence_range": "77-80",
        "Text": "3) From the right angled triangle ACB,\nBC\nAB  = cos B\ni e BC = AB cos B\na = c cos B,\nand\nb cos C = b cos 900 = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "78-81",
        "Text": "From the right angled triangle ACB,\nBC\nAB  = cos B\ni e BC = AB cos B\na = c cos B,\nand\nb cos C = b cos 900 = 0 Thus, we may write\na = 0 + c cos B\n= b cos C + c cos B"
    },
    {
        "Chapter": "1",
        "sentence_range": "79-82",
        "Text": "e BC = AB cos B\na = c cos B,\nand\nb cos C = b cos 900 = 0 Thus, we may write\na = 0 + c cos B\n= b cos C + c cos B (3)\nFrom (1), (2) and (3)"
    },
    {
        "Chapter": "1",
        "sentence_range": "80-83",
        "Text": "BC = AB cos B\na = c cos B,\nand\nb cos C = b cos 900 = 0 Thus, we may write\na = 0 + c cos B\n= b cos C + c cos B (3)\nFrom (1), (2) and (3) We assert that for any triangle ABC,\na = b cos C + c cos B\nBy case (i), r \u21d2 q is proved"
    },
    {
        "Chapter": "1",
        "sentence_range": "81-84",
        "Text": "Thus, we may write\na = 0 + c cos B\n= b cos C + c cos B (3)\nFrom (1), (2) and (3) We assert that for any triangle ABC,\na = b cos C + c cos B\nBy case (i), r \u21d2 q is proved By case (ii), s \u21d2 q is proved"
    },
    {
        "Chapter": "1",
        "sentence_range": "82-85",
        "Text": "(3)\nFrom (1), (2) and (3) We assert that for any triangle ABC,\na = b cos C + c cos B\nBy case (i), r \u21d2 q is proved By case (ii), s \u21d2 q is proved By case (iii), t \u21d2 q is proved"
    },
    {
        "Chapter": "1",
        "sentence_range": "83-86",
        "Text": "We assert that for any triangle ABC,\na = b cos C + c cos B\nBy case (i), r \u21d2 q is proved By case (ii), s \u21d2 q is proved By case (iii), t \u21d2 q is proved Hence, from the proof by cases, (r  \u2228  s  \u2228  t) \u21d2 q is proved, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "84-87",
        "Text": "By case (ii), s \u21d2 q is proved By case (iii), t \u21d2 q is proved Hence, from the proof by cases, (r  \u2228  s  \u2228  t) \u21d2 q is proved, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "85-88",
        "Text": "By case (iii), t \u21d2 q is proved Hence, from the proof by cases, (r  \u2228  s  \u2228  t) \u21d2 q is proved, i e , p \u21d2 q is proved"
    },
    {
        "Chapter": "1",
        "sentence_range": "86-89",
        "Text": "Hence, from the proof by cases, (r  \u2228  s  \u2228  t) \u21d2 q is proved, i e , p \u21d2 q is proved Indirect Proof Instead of proving the given proposition directly, we establish the proof\nof the proposition through proving a proposition which is equivalent to the given\nproposition"
    },
    {
        "Chapter": "1",
        "sentence_range": "87-90",
        "Text": "e , p \u21d2 q is proved Indirect Proof Instead of proving the given proposition directly, we establish the proof\nof the proposition through proving a proposition which is equivalent to the given\nproposition (i)\nProof by contradiction (Reductio Ad Absurdum) : Here, we start with the\nassumption that the given statement is false"
    },
    {
        "Chapter": "1",
        "sentence_range": "88-91",
        "Text": ", p \u21d2 q is proved Indirect Proof Instead of proving the given proposition directly, we establish the proof\nof the proposition through proving a proposition which is equivalent to the given\nproposition (i)\nProof by contradiction (Reductio Ad Absurdum) : Here, we start with the\nassumption that the given statement is false By rules of logic, we arrive at a\nconclusion contradicting the assumption and hence it is inferred that the assumption\nis wrong and hence the given statement is true"
    },
    {
        "Chapter": "1",
        "sentence_range": "89-92",
        "Text": "Indirect Proof Instead of proving the given proposition directly, we establish the proof\nof the proposition through proving a proposition which is equivalent to the given\nproposition (i)\nProof by contradiction (Reductio Ad Absurdum) : Here, we start with the\nassumption that the given statement is false By rules of logic, we arrive at a\nconclusion contradicting the assumption and hence it is inferred that the assumption\nis wrong and hence the given statement is true Let us illustrate this method by an example"
    },
    {
        "Chapter": "1",
        "sentence_range": "90-93",
        "Text": "(i)\nProof by contradiction (Reductio Ad Absurdum) : Here, we start with the\nassumption that the given statement is false By rules of logic, we arrive at a\nconclusion contradicting the assumption and hence it is inferred that the assumption\nis wrong and hence the given statement is true Let us illustrate this method by an example Example 5 Show that the set of all prime numbers is infinite"
    },
    {
        "Chapter": "1",
        "sentence_range": "91-94",
        "Text": "By rules of logic, we arrive at a\nconclusion contradicting the assumption and hence it is inferred that the assumption\nis wrong and hence the given statement is true Let us illustrate this method by an example Example 5 Show that the set of all prime numbers is infinite Solution Let P be the set of all prime numbers"
    },
    {
        "Chapter": "1",
        "sentence_range": "92-95",
        "Text": "Let us illustrate this method by an example Example 5 Show that the set of all prime numbers is infinite Solution Let P be the set of all prime numbers We take the negation of the statement\n\u201cthe set of all prime numbers is infinite\u201d, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "93-96",
        "Text": "Example 5 Show that the set of all prime numbers is infinite Solution Let P be the set of all prime numbers We take the negation of the statement\n\u201cthe set of all prime numbers is infinite\u201d, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "94-97",
        "Text": "Solution Let P be the set of all prime numbers We take the negation of the statement\n\u201cthe set of all prime numbers is infinite\u201d, i e , we assume the set of all prime numbers\nto be finite"
    },
    {
        "Chapter": "1",
        "sentence_range": "95-98",
        "Text": "We take the negation of the statement\n\u201cthe set of all prime numbers is infinite\u201d, i e , we assume the set of all prime numbers\nto be finite Hence, we can list all the prime numbers as P1, P2, P3,"
    },
    {
        "Chapter": "1",
        "sentence_range": "96-99",
        "Text": "e , we assume the set of all prime numbers\nto be finite Hence, we can list all the prime numbers as P1, P2, P3, , Pk (say)"
    },
    {
        "Chapter": "1",
        "sentence_range": "97-100",
        "Text": ", we assume the set of all prime numbers\nto be finite Hence, we can list all the prime numbers as P1, P2, P3, , Pk (say) Note\nthat we have assumed that there is no prime number other than P1, P2, P3,"
    },
    {
        "Chapter": "1",
        "sentence_range": "98-101",
        "Text": "Hence, we can list all the prime numbers as P1, P2, P3, , Pk (say) Note\nthat we have assumed that there is no prime number other than P1, P2, P3, , Pk"
    },
    {
        "Chapter": "1",
        "sentence_range": "99-102",
        "Text": ", Pk (say) Note\nthat we have assumed that there is no prime number other than P1, P2, P3, , Pk Now consider N = (P1 P2 P3\u2026Pk)  + 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "100-103",
        "Text": "Note\nthat we have assumed that there is no prime number other than P1, P2, P3, , Pk Now consider N = (P1 P2 P3\u2026Pk)  + 1 (1)\nN is not in the list as N is larger than any of the numbers in the list"
    },
    {
        "Chapter": "1",
        "sentence_range": "101-104",
        "Text": ", Pk Now consider N = (P1 P2 P3\u2026Pk)  + 1 (1)\nN is not in the list as N is larger than any of the numbers in the list N is either prime or composite"
    },
    {
        "Chapter": "1",
        "sentence_range": "102-105",
        "Text": "Now consider N = (P1 P2 P3\u2026Pk)  + 1 (1)\nN is not in the list as N is larger than any of the numbers in the list N is either prime or composite Fig  A1"
    },
    {
        "Chapter": "1",
        "sentence_range": "103-106",
        "Text": "(1)\nN is not in the list as N is larger than any of the numbers in the list N is either prime or composite Fig  A1 3\nRationalised 2023-24\nPROOFS IN MATHEMATICS\n193\nIf N is a prime, then by (1), there exists a prime number which is not listed"
    },
    {
        "Chapter": "1",
        "sentence_range": "104-107",
        "Text": "N is either prime or composite Fig  A1 3\nRationalised 2023-24\nPROOFS IN MATHEMATICS\n193\nIf N is a prime, then by (1), there exists a prime number which is not listed On the other hand, if N is composite, it should have a prime divisor"
    },
    {
        "Chapter": "1",
        "sentence_range": "105-108",
        "Text": "Fig  A1 3\nRationalised 2023-24\nPROOFS IN MATHEMATICS\n193\nIf N is a prime, then by (1), there exists a prime number which is not listed On the other hand, if N is composite, it should have a prime divisor But none of the\nnumbers in the list can divide N, because they all leave the remainder 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "106-109",
        "Text": "3\nRationalised 2023-24\nPROOFS IN MATHEMATICS\n193\nIf N is a prime, then by (1), there exists a prime number which is not listed On the other hand, if N is composite, it should have a prime divisor But none of the\nnumbers in the list can divide N, because they all leave the remainder 1 Hence, the\nprime divisor should be other than the one in the list"
    },
    {
        "Chapter": "1",
        "sentence_range": "107-110",
        "Text": "On the other hand, if N is composite, it should have a prime divisor But none of the\nnumbers in the list can divide N, because they all leave the remainder 1 Hence, the\nprime divisor should be other than the one in the list Thus, in both the cases whether N is a prime or a composite, we ended up with\ncontradiction to the fact that we have listed all the prime numbers"
    },
    {
        "Chapter": "1",
        "sentence_range": "108-111",
        "Text": "But none of the\nnumbers in the list can divide N, because they all leave the remainder 1 Hence, the\nprime divisor should be other than the one in the list Thus, in both the cases whether N is a prime or a composite, we ended up with\ncontradiction to the fact that we have listed all the prime numbers Hence, our assumption that set of all prime numbers is finite is false"
    },
    {
        "Chapter": "1",
        "sentence_range": "109-112",
        "Text": "Hence, the\nprime divisor should be other than the one in the list Thus, in both the cases whether N is a prime or a composite, we ended up with\ncontradiction to the fact that we have listed all the prime numbers Hence, our assumption that set of all prime numbers is finite is false Thus, the set of all prime numbers is infinite"
    },
    {
        "Chapter": "1",
        "sentence_range": "110-113",
        "Text": "Thus, in both the cases whether N is a prime or a composite, we ended up with\ncontradiction to the fact that we have listed all the prime numbers Hence, our assumption that set of all prime numbers is finite is false Thus, the set of all prime numbers is infinite ANote  Observe that the above proof also uses the method of proof by cases"
    },
    {
        "Chapter": "1",
        "sentence_range": "111-114",
        "Text": "Hence, our assumption that set of all prime numbers is finite is false Thus, the set of all prime numbers is infinite ANote  Observe that the above proof also uses the method of proof by cases (ii)\nProof by using contrapositive statement of the given statement\nInstead of proving the conditional p \u21d2 q, we prove its equivalent, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "112-115",
        "Text": "Thus, the set of all prime numbers is infinite ANote  Observe that the above proof also uses the method of proof by cases (ii)\nProof by using contrapositive statement of the given statement\nInstead of proving the conditional p \u21d2 q, we prove its equivalent, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "113-116",
        "Text": "ANote  Observe that the above proof also uses the method of proof by cases (ii)\nProof by using contrapositive statement of the given statement\nInstead of proving the conditional p \u21d2 q, we prove its equivalent, i e ,\n~ q \u21d2 ~ p"
    },
    {
        "Chapter": "1",
        "sentence_range": "114-117",
        "Text": "(ii)\nProof by using contrapositive statement of the given statement\nInstead of proving the conditional p \u21d2 q, we prove its equivalent, i e ,\n~ q \u21d2 ~ p (students can verify)"
    },
    {
        "Chapter": "1",
        "sentence_range": "115-118",
        "Text": "e ,\n~ q \u21d2 ~ p (students can verify) The contrapositive of a conditional can be formed by interchanging the conclusion\nand the hypothesis and negating both"
    },
    {
        "Chapter": "1",
        "sentence_range": "116-119",
        "Text": ",\n~ q \u21d2 ~ p (students can verify) The contrapositive of a conditional can be formed by interchanging the conclusion\nand the hypothesis and negating both Example 6 Prove that the function f : R \u2192\n\u2192\n\u2192\n\u2192\n\u2192 R  defined by f (x) = 2x + 5 is one-one"
    },
    {
        "Chapter": "1",
        "sentence_range": "117-120",
        "Text": "(students can verify) The contrapositive of a conditional can be formed by interchanging the conclusion\nand the hypothesis and negating both Example 6 Prove that the function f : R \u2192\n\u2192\n\u2192\n\u2192\n\u2192 R  defined by f (x) = 2x + 5 is one-one Solution A function is one-one if f (x1) = f (x2) \u21d2 x1 = x2"
    },
    {
        "Chapter": "1",
        "sentence_range": "118-121",
        "Text": "The contrapositive of a conditional can be formed by interchanging the conclusion\nand the hypothesis and negating both Example 6 Prove that the function f : R \u2192\n\u2192\n\u2192\n\u2192\n\u2192 R  defined by f (x) = 2x + 5 is one-one Solution A function is one-one if f (x1) = f (x2) \u21d2 x1 = x2 Using this we have to show that \u201c2x1+ 5 = 2x2 + 5\u201d \u21d2 \u201cx1 = x2\u201d"
    },
    {
        "Chapter": "1",
        "sentence_range": "119-122",
        "Text": "Example 6 Prove that the function f : R \u2192\n\u2192\n\u2192\n\u2192\n\u2192 R  defined by f (x) = 2x + 5 is one-one Solution A function is one-one if f (x1) = f (x2) \u21d2 x1 = x2 Using this we have to show that \u201c2x1+ 5 = 2x2 + 5\u201d \u21d2 \u201cx1 = x2\u201d This is of the form\np \u21d2 q, where, p is 2x1+ 5 = 2x2 + 5 and q : x1  = x2"
    },
    {
        "Chapter": "1",
        "sentence_range": "120-123",
        "Text": "Solution A function is one-one if f (x1) = f (x2) \u21d2 x1 = x2 Using this we have to show that \u201c2x1+ 5 = 2x2 + 5\u201d \u21d2 \u201cx1 = x2\u201d This is of the form\np \u21d2 q, where, p is 2x1+ 5 = 2x2 + 5 and q : x1  = x2 We have proved this in Example 2\nof \u201cdirect method\u201d"
    },
    {
        "Chapter": "1",
        "sentence_range": "121-124",
        "Text": "Using this we have to show that \u201c2x1+ 5 = 2x2 + 5\u201d \u21d2 \u201cx1 = x2\u201d This is of the form\np \u21d2 q, where, p is 2x1+ 5 = 2x2 + 5 and q : x1  = x2 We have proved this in Example 2\nof \u201cdirect method\u201d We can also prove the same by using contrapositive of the statement"
    },
    {
        "Chapter": "1",
        "sentence_range": "122-125",
        "Text": "This is of the form\np \u21d2 q, where, p is 2x1+ 5 = 2x2 + 5 and q : x1  = x2 We have proved this in Example 2\nof \u201cdirect method\u201d We can also prove the same by using contrapositive of the statement Now\ncontrapositive of this statement is ~ q \u21d2 ~ p, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "123-126",
        "Text": "We have proved this in Example 2\nof \u201cdirect method\u201d We can also prove the same by using contrapositive of the statement Now\ncontrapositive of this statement is ~ q \u21d2 ~ p, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "124-127",
        "Text": "We can also prove the same by using contrapositive of the statement Now\ncontrapositive of this statement is ~ q \u21d2 ~ p, i e , contrapositive of \u201c if f (x1) = f (x2),\nthen x1 = x2\u201d is \u201cif x1 \u2260x2, then  f (x1) \u2260 f (x2)\u201d"
    },
    {
        "Chapter": "1",
        "sentence_range": "125-128",
        "Text": "Now\ncontrapositive of this statement is ~ q \u21d2 ~ p, i e , contrapositive of \u201c if f (x1) = f (x2),\nthen x1 = x2\u201d is \u201cif x1 \u2260x2, then  f (x1) \u2260 f (x2)\u201d Now\nx1 \u2260 x2\n\u21d2\n2x1 \u2260 2x2\n\u21d2\n2x1+ 5 \u2260 2x2 + 5\n\u21d2\nf (x1) \u2260 f (x2)"
    },
    {
        "Chapter": "1",
        "sentence_range": "126-129",
        "Text": "e , contrapositive of \u201c if f (x1) = f (x2),\nthen x1 = x2\u201d is \u201cif x1 \u2260x2, then  f (x1) \u2260 f (x2)\u201d Now\nx1 \u2260 x2\n\u21d2\n2x1 \u2260 2x2\n\u21d2\n2x1+ 5 \u2260 2x2 + 5\n\u21d2\nf (x1) \u2260 f (x2) Since \u201c~ q \u21d2 ~ p\u201d, is equivalent to \u201cp \u21d2 q\u201d the proof is complete"
    },
    {
        "Chapter": "1",
        "sentence_range": "127-130",
        "Text": ", contrapositive of \u201c if f (x1) = f (x2),\nthen x1 = x2\u201d is \u201cif x1 \u2260x2, then  f (x1) \u2260 f (x2)\u201d Now\nx1 \u2260 x2\n\u21d2\n2x1 \u2260 2x2\n\u21d2\n2x1+ 5 \u2260 2x2 + 5\n\u21d2\nf (x1) \u2260 f (x2) Since \u201c~ q \u21d2 ~ p\u201d, is equivalent to \u201cp \u21d2 q\u201d the proof is complete Example 7 Show that \u201cif a matrix A is invertible, then A is non singular\u201d"
    },
    {
        "Chapter": "1",
        "sentence_range": "128-131",
        "Text": "Now\nx1 \u2260 x2\n\u21d2\n2x1 \u2260 2x2\n\u21d2\n2x1+ 5 \u2260 2x2 + 5\n\u21d2\nf (x1) \u2260 f (x2) Since \u201c~ q \u21d2 ~ p\u201d, is equivalent to \u201cp \u21d2 q\u201d the proof is complete Example 7 Show that \u201cif a matrix A is invertible, then A is non singular\u201d Solution Writing the above statement in symbolic form, we have\np \u21d2 q, where, p is \u201cmatrix A is invertible\u201d and q is \u201cA is non singular\u201d\nInstead of proving the given statement, we prove its contrapositive statement, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "129-132",
        "Text": "Since \u201c~ q \u21d2 ~ p\u201d, is equivalent to \u201cp \u21d2 q\u201d the proof is complete Example 7 Show that \u201cif a matrix A is invertible, then A is non singular\u201d Solution Writing the above statement in symbolic form, we have\np \u21d2 q, where, p is \u201cmatrix A is invertible\u201d and q is \u201cA is non singular\u201d\nInstead of proving the given statement, we prove its contrapositive statement, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "130-133",
        "Text": "Example 7 Show that \u201cif a matrix A is invertible, then A is non singular\u201d Solution Writing the above statement in symbolic form, we have\np \u21d2 q, where, p is \u201cmatrix A is invertible\u201d and q is \u201cA is non singular\u201d\nInstead of proving the given statement, we prove its contrapositive statement, i e ,\nif A is not a non singular matrix, then the matrix A is not invertible"
    },
    {
        "Chapter": "1",
        "sentence_range": "131-134",
        "Text": "Solution Writing the above statement in symbolic form, we have\np \u21d2 q, where, p is \u201cmatrix A is invertible\u201d and q is \u201cA is non singular\u201d\nInstead of proving the given statement, we prove its contrapositive statement, i e ,\nif A is not a non singular matrix, then the matrix A is not invertible Rationalised 2023-24\n MATHEMATICS\n194\nIf A is not a non singular matrix, then it means the matrix A is singular, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "132-135",
        "Text": "e ,\nif A is not a non singular matrix, then the matrix A is not invertible Rationalised 2023-24\n MATHEMATICS\n194\nIf A is not a non singular matrix, then it means the matrix A is singular, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "133-136",
        "Text": ",\nif A is not a non singular matrix, then the matrix A is not invertible Rationalised 2023-24\n MATHEMATICS\n194\nIf A is not a non singular matrix, then it means the matrix A is singular, i e ,\n|A| = 0\nThen\nA\u20131 =\nA\nadj|A|\n does not exist as |A| = 0\nHence, A is not invertible"
    },
    {
        "Chapter": "1",
        "sentence_range": "134-137",
        "Text": "Rationalised 2023-24\n MATHEMATICS\n194\nIf A is not a non singular matrix, then it means the matrix A is singular, i e ,\n|A| = 0\nThen\nA\u20131 =\nA\nadj|A|\n does not exist as |A| = 0\nHence, A is not invertible Thus, we have proved that if A is not a non singular matrix, then A is not invertible"
    },
    {
        "Chapter": "1",
        "sentence_range": "135-138",
        "Text": "e ,\n|A| = 0\nThen\nA\u20131 =\nA\nadj|A|\n does not exist as |A| = 0\nHence, A is not invertible Thus, we have proved that if A is not a non singular matrix, then A is not invertible i"
    },
    {
        "Chapter": "1",
        "sentence_range": "136-139",
        "Text": ",\n|A| = 0\nThen\nA\u20131 =\nA\nadj|A|\n does not exist as |A| = 0\nHence, A is not invertible Thus, we have proved that if A is not a non singular matrix, then A is not invertible i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "137-140",
        "Text": "Thus, we have proved that if A is not a non singular matrix, then A is not invertible i e , ~ q \u21d2 ~ p"
    },
    {
        "Chapter": "1",
        "sentence_range": "138-141",
        "Text": "i e , ~ q \u21d2 ~ p Hence, if a matrix A is invertible, then A is non singular"
    },
    {
        "Chapter": "1",
        "sentence_range": "139-142",
        "Text": "e , ~ q \u21d2 ~ p Hence, if a matrix A is invertible, then A is non singular (iii)\nProof by a counter example\nIn the history of Mathematics, there are occasions when all attempts to find a\nvalid proof of a statement fail and the uncertainty of the truth value of the statement\nremains unresolved"
    },
    {
        "Chapter": "1",
        "sentence_range": "140-143",
        "Text": ", ~ q \u21d2 ~ p Hence, if a matrix A is invertible, then A is non singular (iii)\nProof by a counter example\nIn the history of Mathematics, there are occasions when all attempts to find a\nvalid proof of a statement fail and the uncertainty of the truth value of the statement\nremains unresolved In such a situation,  it is beneficial, if we find an example to falsify the statement"
    },
    {
        "Chapter": "1",
        "sentence_range": "141-144",
        "Text": "Hence, if a matrix A is invertible, then A is non singular (iii)\nProof by a counter example\nIn the history of Mathematics, there are occasions when all attempts to find a\nvalid proof of a statement fail and the uncertainty of the truth value of the statement\nremains unresolved In such a situation,  it is beneficial, if we find an example to falsify the statement The example to disprove the statement is called a counter example"
    },
    {
        "Chapter": "1",
        "sentence_range": "142-145",
        "Text": "(iii)\nProof by a counter example\nIn the history of Mathematics, there are occasions when all attempts to find a\nvalid proof of a statement fail and the uncertainty of the truth value of the statement\nremains unresolved In such a situation,  it is beneficial, if we find an example to falsify the statement The example to disprove the statement is called a counter example Since the disproof\nof a proposition p \u21d2 q is merely a proof of the proposition ~ (p \u21d2 q)"
    },
    {
        "Chapter": "1",
        "sentence_range": "143-146",
        "Text": "In such a situation,  it is beneficial, if we find an example to falsify the statement The example to disprove the statement is called a counter example Since the disproof\nof a proposition p \u21d2 q is merely a proof of the proposition ~ (p \u21d2 q) Hence, this is\nalso a method of proof"
    },
    {
        "Chapter": "1",
        "sentence_range": "144-147",
        "Text": "The example to disprove the statement is called a counter example Since the disproof\nof a proposition p \u21d2 q is merely a proof of the proposition ~ (p \u21d2 q) Hence, this is\nalso a method of proof Example 8 For each n,\n22\n1\nn + is a prime (n \u2208 N)"
    },
    {
        "Chapter": "1",
        "sentence_range": "145-148",
        "Text": "Since the disproof\nof a proposition p \u21d2 q is merely a proof of the proposition ~ (p \u21d2 q) Hence, this is\nalso a method of proof Example 8 For each n,\n22\n1\nn + is a prime (n \u2208 N) This was once thought to be true on the basis that\n212\n1\n+  = 22 +  1 = 5 is a prime"
    },
    {
        "Chapter": "1",
        "sentence_range": "146-149",
        "Text": "Hence, this is\nalso a method of proof Example 8 For each n,\n22\n1\nn + is a prime (n \u2208 N) This was once thought to be true on the basis that\n212\n1\n+  = 22 +  1 = 5 is a prime 222\n1\n+  = 24 + 1 = 17 is a prime"
    },
    {
        "Chapter": "1",
        "sentence_range": "147-150",
        "Text": "Example 8 For each n,\n22\n1\nn + is a prime (n \u2208 N) This was once thought to be true on the basis that\n212\n1\n+  = 22 +  1 = 5 is a prime 222\n1\n+  = 24 + 1 = 17 is a prime 223\n1\n+  = 28 + 1 = 257 is a prime"
    },
    {
        "Chapter": "1",
        "sentence_range": "148-151",
        "Text": "This was once thought to be true on the basis that\n212\n1\n+  = 22 +  1 = 5 is a prime 222\n1\n+  = 24 + 1 = 17 is a prime 223\n1\n+  = 28 + 1 = 257 is a prime However, at first sight the generalisation looks to be correct"
    },
    {
        "Chapter": "1",
        "sentence_range": "149-152",
        "Text": "222\n1\n+  = 24 + 1 = 17 is a prime 223\n1\n+  = 28 + 1 = 257 is a prime However, at first sight the generalisation looks to be correct But, eventually it was\nshown that\n225\n1\n+  = 232 + 1 = 4294967297\nwhich is not a prime since 4294967297  =  641 \u00d7 6700417 (a product of two numbers)"
    },
    {
        "Chapter": "1",
        "sentence_range": "150-153",
        "Text": "223\n1\n+  = 28 + 1 = 257 is a prime However, at first sight the generalisation looks to be correct But, eventually it was\nshown that\n225\n1\n+  = 232 + 1 = 4294967297\nwhich is not a prime since 4294967297  =  641 \u00d7 6700417 (a product of two numbers) So the generalisation \u201cFor each n, \n22\n1\nn +  is a prime (n \u2208 N)\u201d is false"
    },
    {
        "Chapter": "1",
        "sentence_range": "151-154",
        "Text": "However, at first sight the generalisation looks to be correct But, eventually it was\nshown that\n225\n1\n+  = 232 + 1 = 4294967297\nwhich is not a prime since 4294967297  =  641 \u00d7 6700417 (a product of two numbers) So the generalisation \u201cFor each n, \n22\n1\nn +  is a prime (n \u2208 N)\u201d is false Just this one example \n225\n1\n+  is sufficient to disprove the generalisation"
    },
    {
        "Chapter": "1",
        "sentence_range": "152-155",
        "Text": "But, eventually it was\nshown that\n225\n1\n+  = 232 + 1 = 4294967297\nwhich is not a prime since 4294967297  =  641 \u00d7 6700417 (a product of two numbers) So the generalisation \u201cFor each n, \n22\n1\nn +  is a prime (n \u2208 N)\u201d is false Just this one example \n225\n1\n+  is sufficient to disprove the generalisation This is the\ncounter example"
    },
    {
        "Chapter": "1",
        "sentence_range": "153-156",
        "Text": "So the generalisation \u201cFor each n, \n22\n1\nn +  is a prime (n \u2208 N)\u201d is false Just this one example \n225\n1\n+  is sufficient to disprove the generalisation This is the\ncounter example Thus, we have proved that the generalisation \u201cFor each n,\n22\n1\nn +  is a prime\n(n \u2208 N)\u201d is not true in general"
    },
    {
        "Chapter": "1",
        "sentence_range": "154-157",
        "Text": "Just this one example \n225\n1\n+  is sufficient to disprove the generalisation This is the\ncounter example Thus, we have proved that the generalisation \u201cFor each n,\n22\n1\nn +  is a prime\n(n \u2208 N)\u201d is not true in general Rationalised 2023-24\nPROOFS IN MATHEMATICS\n195\nExample 9 Every continuous function is differentiable"
    },
    {
        "Chapter": "1",
        "sentence_range": "155-158",
        "Text": "This is the\ncounter example Thus, we have proved that the generalisation \u201cFor each n,\n22\n1\nn +  is a prime\n(n \u2208 N)\u201d is not true in general Rationalised 2023-24\nPROOFS IN MATHEMATICS\n195\nExample 9 Every continuous function is differentiable Proof  We consider some functions given by\n(i)\nf (x) = x2\n(ii)\ng(x) = ex\n(iii)\nh(x) = sin x\nThese functions are continuous for all values of x"
    },
    {
        "Chapter": "1",
        "sentence_range": "156-159",
        "Text": "Thus, we have proved that the generalisation \u201cFor each n,\n22\n1\nn +  is a prime\n(n \u2208 N)\u201d is not true in general Rationalised 2023-24\nPROOFS IN MATHEMATICS\n195\nExample 9 Every continuous function is differentiable Proof  We consider some functions given by\n(i)\nf (x) = x2\n(ii)\ng(x) = ex\n(iii)\nh(x) = sin x\nThese functions are continuous for all values of x If we check for their\ndifferentiability, we find that they are all differentiable for all the values of x"
    },
    {
        "Chapter": "1",
        "sentence_range": "157-160",
        "Text": "Rationalised 2023-24\nPROOFS IN MATHEMATICS\n195\nExample 9 Every continuous function is differentiable Proof  We consider some functions given by\n(i)\nf (x) = x2\n(ii)\ng(x) = ex\n(iii)\nh(x) = sin x\nThese functions are continuous for all values of x If we check for their\ndifferentiability, we find that they are all differentiable for all the values of x This\nmakes us to believe that the generalisation \u201cEvery continuous function is differentiable\u201d\nmay be true"
    },
    {
        "Chapter": "1",
        "sentence_range": "158-161",
        "Text": "Proof  We consider some functions given by\n(i)\nf (x) = x2\n(ii)\ng(x) = ex\n(iii)\nh(x) = sin x\nThese functions are continuous for all values of x If we check for their\ndifferentiability, we find that they are all differentiable for all the values of x This\nmakes us to believe that the generalisation \u201cEvery continuous function is differentiable\u201d\nmay be true But if we check the differentiability of the function given by \u201c\u03c6(x) = | x|\u201d\nwhich is continuous, we find that it is not differentiable at x = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "159-162",
        "Text": "If we check for their\ndifferentiability, we find that they are all differentiable for all the values of x This\nmakes us to believe that the generalisation \u201cEvery continuous function is differentiable\u201d\nmay be true But if we check the differentiability of the function given by \u201c\u03c6(x) = | x|\u201d\nwhich is continuous, we find that it is not differentiable at x = 0 This means that the\nstatement \u201cEvery continuous function is differentiable\u201d is false, in general"
    },
    {
        "Chapter": "1",
        "sentence_range": "160-163",
        "Text": "This\nmakes us to believe that the generalisation \u201cEvery continuous function is differentiable\u201d\nmay be true But if we check the differentiability of the function given by \u201c\u03c6(x) = | x|\u201d\nwhich is continuous, we find that it is not differentiable at x = 0 This means that the\nstatement \u201cEvery continuous function is differentiable\u201d is false, in general Just this\none function \u201c\u03c6(x) = | x|\u201d is sufficient to disprove the statement"
    },
    {
        "Chapter": "1",
        "sentence_range": "161-164",
        "Text": "But if we check the differentiability of the function given by \u201c\u03c6(x) = | x|\u201d\nwhich is continuous, we find that it is not differentiable at x = 0 This means that the\nstatement \u201cEvery continuous function is differentiable\u201d is false, in general Just this\none function \u201c\u03c6(x) = | x|\u201d is sufficient to disprove the statement Hence,  \u201c\u03c6(x) = | x|\u201d\nis called a counter example to disprove \u201cEvery continuous function is differentiable\u201d"
    },
    {
        "Chapter": "1",
        "sentence_range": "162-165",
        "Text": "This means that the\nstatement \u201cEvery continuous function is differentiable\u201d is false, in general Just this\none function \u201c\u03c6(x) = | x|\u201d is sufficient to disprove the statement Hence,  \u201c\u03c6(x) = | x|\u201d\nis called a counter example to disprove \u201cEvery continuous function is differentiable\u201d \u2014v\nv\nv\nv\nv\u2014\nRationalised 2023-24\n196\nMATHEMATICS\nA"
    },
    {
        "Chapter": "1",
        "sentence_range": "163-166",
        "Text": "Just this\none function \u201c\u03c6(x) = | x|\u201d is sufficient to disprove the statement Hence,  \u201c\u03c6(x) = | x|\u201d\nis called a counter example to disprove \u201cEvery continuous function is differentiable\u201d \u2014v\nv\nv\nv\nv\u2014\nRationalised 2023-24\n196\nMATHEMATICS\nA 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "164-167",
        "Text": "Hence,  \u201c\u03c6(x) = | x|\u201d\nis called a counter example to disprove \u201cEvery continuous function is differentiable\u201d \u2014v\nv\nv\nv\nv\u2014\nRationalised 2023-24\n196\nMATHEMATICS\nA 2 1  Introduction\nIn class XI, we have learnt about mathematical modelling as an attempt to study some\npart (or form) of some real-life problems in mathematical terms, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "165-168",
        "Text": "\u2014v\nv\nv\nv\nv\u2014\nRationalised 2023-24\n196\nMATHEMATICS\nA 2 1  Introduction\nIn class XI, we have learnt about mathematical modelling as an attempt to study some\npart (or form) of some real-life problems in mathematical terms, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "166-169",
        "Text": "2 1  Introduction\nIn class XI, we have learnt about mathematical modelling as an attempt to study some\npart (or form) of some real-life problems in mathematical terms, i e , the conversion of\na physical situation into mathematics using some suitable conditions"
    },
    {
        "Chapter": "1",
        "sentence_range": "167-170",
        "Text": "1  Introduction\nIn class XI, we have learnt about mathematical modelling as an attempt to study some\npart (or form) of some real-life problems in mathematical terms, i e , the conversion of\na physical situation into mathematics using some suitable conditions Roughly speaking\nmathematical modelling is an activity in which we make models to describe the behaviour\nof various phenomenal activities of our interest in many ways using words, drawings or\nsketches, computer programs, mathematical formulae etc"
    },
    {
        "Chapter": "1",
        "sentence_range": "168-171",
        "Text": "e , the conversion of\na physical situation into mathematics using some suitable conditions Roughly speaking\nmathematical modelling is an activity in which we make models to describe the behaviour\nof various phenomenal activities of our interest in many ways using words, drawings or\nsketches, computer programs, mathematical formulae etc In earlier classes, we have observed that solutions to many problems, involving\napplications of various mathematical concepts, involve mathematical modelling in one\nway or the other"
    },
    {
        "Chapter": "1",
        "sentence_range": "169-172",
        "Text": ", the conversion of\na physical situation into mathematics using some suitable conditions Roughly speaking\nmathematical modelling is an activity in which we make models to describe the behaviour\nof various phenomenal activities of our interest in many ways using words, drawings or\nsketches, computer programs, mathematical formulae etc In earlier classes, we have observed that solutions to many problems, involving\napplications of various mathematical concepts, involve mathematical modelling in one\nway or the other Therefore, it is important to study mathematical modelling as a separate\ntopic"
    },
    {
        "Chapter": "1",
        "sentence_range": "170-173",
        "Text": "Roughly speaking\nmathematical modelling is an activity in which we make models to describe the behaviour\nof various phenomenal activities of our interest in many ways using words, drawings or\nsketches, computer programs, mathematical formulae etc In earlier classes, we have observed that solutions to many problems, involving\napplications of various mathematical concepts, involve mathematical modelling in one\nway or the other Therefore, it is important to study mathematical modelling as a separate\ntopic In this chapter, we shall further study mathematical modelling of some real-life\nproblems using techniques/results from matrix, calculus and linear programming"
    },
    {
        "Chapter": "1",
        "sentence_range": "171-174",
        "Text": "In earlier classes, we have observed that solutions to many problems, involving\napplications of various mathematical concepts, involve mathematical modelling in one\nway or the other Therefore, it is important to study mathematical modelling as a separate\ntopic In this chapter, we shall further study mathematical modelling of some real-life\nproblems using techniques/results from matrix, calculus and linear programming A"
    },
    {
        "Chapter": "1",
        "sentence_range": "172-175",
        "Text": "Therefore, it is important to study mathematical modelling as a separate\ntopic In this chapter, we shall further study mathematical modelling of some real-life\nproblems using techniques/results from matrix, calculus and linear programming A 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "173-176",
        "Text": "In this chapter, we shall further study mathematical modelling of some real-life\nproblems using techniques/results from matrix, calculus and linear programming A 2 2  Why Mathematical Modelling"
    },
    {
        "Chapter": "1",
        "sentence_range": "174-177",
        "Text": "A 2 2  Why Mathematical Modelling Students are aware of the solution of word problems in arithmetic, algebra, trigonometry\nand linear programming etc"
    },
    {
        "Chapter": "1",
        "sentence_range": "175-178",
        "Text": "2 2  Why Mathematical Modelling Students are aware of the solution of word problems in arithmetic, algebra, trigonometry\nand linear programming etc Sometimes we solve the problems without going into the\nphysical insight of the situational problems"
    },
    {
        "Chapter": "1",
        "sentence_range": "176-179",
        "Text": "2  Why Mathematical Modelling Students are aware of the solution of word problems in arithmetic, algebra, trigonometry\nand linear programming etc Sometimes we solve the problems without going into the\nphysical insight of the situational problems Situational problems need physical insight\nthat is introduction of physical laws and some symbols to compare the mathematical\nresults obtained with practical values"
    },
    {
        "Chapter": "1",
        "sentence_range": "177-180",
        "Text": "Students are aware of the solution of word problems in arithmetic, algebra, trigonometry\nand linear programming etc Sometimes we solve the problems without going into the\nphysical insight of the situational problems Situational problems need physical insight\nthat is introduction of physical laws and some symbols to compare the mathematical\nresults obtained with practical values To solve many problems faced by us, we need a\ntechnique and this is what is known as mathematical modelling"
    },
    {
        "Chapter": "1",
        "sentence_range": "178-181",
        "Text": "Sometimes we solve the problems without going into the\nphysical insight of the situational problems Situational problems need physical insight\nthat is introduction of physical laws and some symbols to compare the mathematical\nresults obtained with practical values To solve many problems faced by us, we need a\ntechnique and this is what is known as mathematical modelling Let us consider the\nfollowing problems:\n(i)\nTo find the width of a river (particularly, when it is difficult to cross the river)"
    },
    {
        "Chapter": "1",
        "sentence_range": "179-182",
        "Text": "Situational problems need physical insight\nthat is introduction of physical laws and some symbols to compare the mathematical\nresults obtained with practical values To solve many problems faced by us, we need a\ntechnique and this is what is known as mathematical modelling Let us consider the\nfollowing problems:\n(i)\nTo find the width of a river (particularly, when it is difficult to cross the river) (ii)\nTo find the optimal angle in case of shot-put (by considering the variables\nsuch as : the height of the thrower, resistance of the media, acceleration due to\ngravity etc"
    },
    {
        "Chapter": "1",
        "sentence_range": "180-183",
        "Text": "To solve many problems faced by us, we need a\ntechnique and this is what is known as mathematical modelling Let us consider the\nfollowing problems:\n(i)\nTo find the width of a river (particularly, when it is difficult to cross the river) (ii)\nTo find the optimal angle in case of shot-put (by considering the variables\nsuch as : the height of the thrower, resistance of the media, acceleration due to\ngravity etc )"
    },
    {
        "Chapter": "1",
        "sentence_range": "181-184",
        "Text": "Let us consider the\nfollowing problems:\n(i)\nTo find the width of a river (particularly, when it is difficult to cross the river) (ii)\nTo find the optimal angle in case of shot-put (by considering the variables\nsuch as : the height of the thrower, resistance of the media, acceleration due to\ngravity etc ) (iii)\nTo find the height of a tower (particularly, when it is not possible to reach the top\nof the tower)"
    },
    {
        "Chapter": "1",
        "sentence_range": "182-185",
        "Text": "(ii)\nTo find the optimal angle in case of shot-put (by considering the variables\nsuch as : the height of the thrower, resistance of the media, acceleration due to\ngravity etc ) (iii)\nTo find the height of a tower (particularly, when it is not possible to reach the top\nof the tower) (iv)\nTo find the temperature at the surface of the Sun"
    },
    {
        "Chapter": "1",
        "sentence_range": "183-186",
        "Text": ") (iii)\nTo find the height of a tower (particularly, when it is not possible to reach the top\nof the tower) (iv)\nTo find the temperature at the surface of the Sun Appendix 2\nMATHEMATICAL MODELLING\nRationalised 2023-24\nMATHEMATICAL MODELLING        197\n(v)\nWhy heart patients are not allowed to use lift"
    },
    {
        "Chapter": "1",
        "sentence_range": "184-187",
        "Text": "(iii)\nTo find the height of a tower (particularly, when it is not possible to reach the top\nof the tower) (iv)\nTo find the temperature at the surface of the Sun Appendix 2\nMATHEMATICAL MODELLING\nRationalised 2023-24\nMATHEMATICAL MODELLING        197\n(v)\nWhy heart patients are not allowed to use lift (without knowing the physiology\nof a human being)"
    },
    {
        "Chapter": "1",
        "sentence_range": "185-188",
        "Text": "(iv)\nTo find the temperature at the surface of the Sun Appendix 2\nMATHEMATICAL MODELLING\nRationalised 2023-24\nMATHEMATICAL MODELLING        197\n(v)\nWhy heart patients are not allowed to use lift (without knowing the physiology\nof a human being) (vi)\nTo find the mass of the Earth"
    },
    {
        "Chapter": "1",
        "sentence_range": "186-189",
        "Text": "Appendix 2\nMATHEMATICAL MODELLING\nRationalised 2023-24\nMATHEMATICAL MODELLING        197\n(v)\nWhy heart patients are not allowed to use lift (without knowing the physiology\nof a human being) (vi)\nTo find the mass of the Earth (vii)\nEstimate the yield of pulses in India from the standing crops (a person is not\nallowed to cut all of it)"
    },
    {
        "Chapter": "1",
        "sentence_range": "187-190",
        "Text": "(without knowing the physiology\nof a human being) (vi)\nTo find the mass of the Earth (vii)\nEstimate the yield of pulses in India from the standing crops (a person is not\nallowed to cut all of it) (viii) Find the volume of blood inside the body of a person (a person is not allowed to\nbleed completely)"
    },
    {
        "Chapter": "1",
        "sentence_range": "188-191",
        "Text": "(vi)\nTo find the mass of the Earth (vii)\nEstimate the yield of pulses in India from the standing crops (a person is not\nallowed to cut all of it) (viii) Find the volume of blood inside the body of a person (a person is not allowed to\nbleed completely) (ix)\nEstimate the population of India in the year 2020 (a person is not allowed to wait\ntill then)"
    },
    {
        "Chapter": "1",
        "sentence_range": "189-192",
        "Text": "(vii)\nEstimate the yield of pulses in India from the standing crops (a person is not\nallowed to cut all of it) (viii) Find the volume of blood inside the body of a person (a person is not allowed to\nbleed completely) (ix)\nEstimate the population of India in the year 2020 (a person is not allowed to wait\ntill then) All of these problems can be solved and infact have been solved with the help of\nMathematics using mathematical modelling"
    },
    {
        "Chapter": "1",
        "sentence_range": "190-193",
        "Text": "(viii) Find the volume of blood inside the body of a person (a person is not allowed to\nbleed completely) (ix)\nEstimate the population of India in the year 2020 (a person is not allowed to wait\ntill then) All of these problems can be solved and infact have been solved with the help of\nMathematics using mathematical modelling In fact, you might have studied the methods\nfor solving some of them in the present textbook itself"
    },
    {
        "Chapter": "1",
        "sentence_range": "191-194",
        "Text": "(ix)\nEstimate the population of India in the year 2020 (a person is not allowed to wait\ntill then) All of these problems can be solved and infact have been solved with the help of\nMathematics using mathematical modelling In fact, you might have studied the methods\nfor solving some of them in the present textbook itself However, it will be instructive if\nyou first try to solve them yourself and that too without the help of Mathematics, if\npossible, you will then appreciate the power of Mathematics and the need for\nmathematical modelling"
    },
    {
        "Chapter": "1",
        "sentence_range": "192-195",
        "Text": "All of these problems can be solved and infact have been solved with the help of\nMathematics using mathematical modelling In fact, you might have studied the methods\nfor solving some of them in the present textbook itself However, it will be instructive if\nyou first try to solve them yourself and that too without the help of Mathematics, if\npossible, you will then appreciate the power of Mathematics and the need for\nmathematical modelling A"
    },
    {
        "Chapter": "1",
        "sentence_range": "193-196",
        "Text": "In fact, you might have studied the methods\nfor solving some of them in the present textbook itself However, it will be instructive if\nyou first try to solve them yourself and that too without the help of Mathematics, if\npossible, you will then appreciate the power of Mathematics and the need for\nmathematical modelling A 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "194-197",
        "Text": "However, it will be instructive if\nyou first try to solve them yourself and that too without the help of Mathematics, if\npossible, you will then appreciate the power of Mathematics and the need for\nmathematical modelling A 2 3  Principles of Mathematical Modelling\nMathematical modelling is a principled activity and so it has some principles behind it"
    },
    {
        "Chapter": "1",
        "sentence_range": "195-198",
        "Text": "A 2 3  Principles of Mathematical Modelling\nMathematical modelling is a principled activity and so it has some principles behind it These principles are almost philosophical in nature"
    },
    {
        "Chapter": "1",
        "sentence_range": "196-199",
        "Text": "2 3  Principles of Mathematical Modelling\nMathematical modelling is a principled activity and so it has some principles behind it These principles are almost philosophical in nature Some of the basic principles of\nmathematical modelling are listed below in terms of instructions:\n(i)\nIdentify the need for the model"
    },
    {
        "Chapter": "1",
        "sentence_range": "197-200",
        "Text": "3  Principles of Mathematical Modelling\nMathematical modelling is a principled activity and so it has some principles behind it These principles are almost philosophical in nature Some of the basic principles of\nmathematical modelling are listed below in terms of instructions:\n(i)\nIdentify the need for the model (for what we are looking for)\n(ii)\nList the parameters/variables which are required for the model"
    },
    {
        "Chapter": "1",
        "sentence_range": "198-201",
        "Text": "These principles are almost philosophical in nature Some of the basic principles of\nmathematical modelling are listed below in terms of instructions:\n(i)\nIdentify the need for the model (for what we are looking for)\n(ii)\nList the parameters/variables which are required for the model (iii)\nIdentify the available relevent data"
    },
    {
        "Chapter": "1",
        "sentence_range": "199-202",
        "Text": "Some of the basic principles of\nmathematical modelling are listed below in terms of instructions:\n(i)\nIdentify the need for the model (for what we are looking for)\n(ii)\nList the parameters/variables which are required for the model (iii)\nIdentify the available relevent data (what is given"
    },
    {
        "Chapter": "1",
        "sentence_range": "200-203",
        "Text": "(for what we are looking for)\n(ii)\nList the parameters/variables which are required for the model (iii)\nIdentify the available relevent data (what is given )\n(iv)\nIdentify the circumstances that can be applied (assumptions)\n(v)\nIdentify the governing physical principles"
    },
    {
        "Chapter": "1",
        "sentence_range": "201-204",
        "Text": "(iii)\nIdentify the available relevent data (what is given )\n(iv)\nIdentify the circumstances that can be applied (assumptions)\n(v)\nIdentify the governing physical principles (vi)\nIdentify\n(a) the equations that will be used"
    },
    {
        "Chapter": "1",
        "sentence_range": "202-205",
        "Text": "(what is given )\n(iv)\nIdentify the circumstances that can be applied (assumptions)\n(v)\nIdentify the governing physical principles (vi)\nIdentify\n(a) the equations that will be used (b) the calculations that will be made"
    },
    {
        "Chapter": "1",
        "sentence_range": "203-206",
        "Text": ")\n(iv)\nIdentify the circumstances that can be applied (assumptions)\n(v)\nIdentify the governing physical principles (vi)\nIdentify\n(a) the equations that will be used (b) the calculations that will be made (c) the solution which will follow"
    },
    {
        "Chapter": "1",
        "sentence_range": "204-207",
        "Text": "(vi)\nIdentify\n(a) the equations that will be used (b) the calculations that will be made (c) the solution which will follow (vii)\nIdentify tests that can check the\n(a) consistency of the model"
    },
    {
        "Chapter": "1",
        "sentence_range": "205-208",
        "Text": "(b) the calculations that will be made (c) the solution which will follow (vii)\nIdentify tests that can check the\n(a) consistency of the model (b) utility of the model"
    },
    {
        "Chapter": "1",
        "sentence_range": "206-209",
        "Text": "(c) the solution which will follow (vii)\nIdentify tests that can check the\n(a) consistency of the model (b) utility of the model (viii) Identify the parameter values that can improve the model"
    },
    {
        "Chapter": "1",
        "sentence_range": "207-210",
        "Text": "(vii)\nIdentify tests that can check the\n(a) consistency of the model (b) utility of the model (viii) Identify the parameter values that can improve the model Rationalised 2023-24\n198\nMATHEMATICS\nThe above principles of mathematical modelling lead to the following: steps for\nmathematical modelling"
    },
    {
        "Chapter": "1",
        "sentence_range": "208-211",
        "Text": "(b) utility of the model (viii) Identify the parameter values that can improve the model Rationalised 2023-24\n198\nMATHEMATICS\nThe above principles of mathematical modelling lead to the following: steps for\nmathematical modelling Step 1: Identify the physical situation"
    },
    {
        "Chapter": "1",
        "sentence_range": "209-212",
        "Text": "(viii) Identify the parameter values that can improve the model Rationalised 2023-24\n198\nMATHEMATICS\nThe above principles of mathematical modelling lead to the following: steps for\nmathematical modelling Step 1: Identify the physical situation Step 2: Convert the physical situation into a mathematical model by introducing\nparameters / variables and using various known physical laws and symbols"
    },
    {
        "Chapter": "1",
        "sentence_range": "210-213",
        "Text": "Rationalised 2023-24\n198\nMATHEMATICS\nThe above principles of mathematical modelling lead to the following: steps for\nmathematical modelling Step 1: Identify the physical situation Step 2: Convert the physical situation into a mathematical model by introducing\nparameters / variables and using various known physical laws and symbols Step 3: Find the solution of the mathematical problem"
    },
    {
        "Chapter": "1",
        "sentence_range": "211-214",
        "Text": "Step 1: Identify the physical situation Step 2: Convert the physical situation into a mathematical model by introducing\nparameters / variables and using various known physical laws and symbols Step 3: Find the solution of the mathematical problem Step 4: Interpret the result in terms of the original problem and compare the result\nwith observations or experiments"
    },
    {
        "Chapter": "1",
        "sentence_range": "212-215",
        "Text": "Step 2: Convert the physical situation into a mathematical model by introducing\nparameters / variables and using various known physical laws and symbols Step 3: Find the solution of the mathematical problem Step 4: Interpret the result in terms of the original problem and compare the result\nwith observations or experiments Step 5: If the result is in good agreement, then accept the model"
    },
    {
        "Chapter": "1",
        "sentence_range": "213-216",
        "Text": "Step 3: Find the solution of the mathematical problem Step 4: Interpret the result in terms of the original problem and compare the result\nwith observations or experiments Step 5: If the result is in good agreement, then accept the model Otherwise modify\nthe hypotheses / assumptions according to the physical situation and go to\nStep 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "214-217",
        "Text": "Step 4: Interpret the result in terms of the original problem and compare the result\nwith observations or experiments Step 5: If the result is in good agreement, then accept the model Otherwise modify\nthe hypotheses / assumptions according to the physical situation and go to\nStep 2 The above steps can also be viewed through the following diagram:\nFig  A"
    },
    {
        "Chapter": "1",
        "sentence_range": "215-218",
        "Text": "Step 5: If the result is in good agreement, then accept the model Otherwise modify\nthe hypotheses / assumptions according to the physical situation and go to\nStep 2 The above steps can also be viewed through the following diagram:\nFig  A 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "216-219",
        "Text": "Otherwise modify\nthe hypotheses / assumptions according to the physical situation and go to\nStep 2 The above steps can also be viewed through the following diagram:\nFig  A 2 1\nExample 1 Find the height of a given tower using mathematical modelling"
    },
    {
        "Chapter": "1",
        "sentence_range": "217-220",
        "Text": "The above steps can also be viewed through the following diagram:\nFig  A 2 1\nExample 1 Find the height of a given tower using mathematical modelling Solution Step 1 Given physical situation is \u201cto find the height of a given tower\u201d"
    },
    {
        "Chapter": "1",
        "sentence_range": "218-221",
        "Text": "2 1\nExample 1 Find the height of a given tower using mathematical modelling Solution Step 1 Given physical situation is \u201cto find the height of a given tower\u201d Step 2 Let AB be the given tower (Fig A"
    },
    {
        "Chapter": "1",
        "sentence_range": "219-222",
        "Text": "1\nExample 1 Find the height of a given tower using mathematical modelling Solution Step 1 Given physical situation is \u201cto find the height of a given tower\u201d Step 2 Let AB be the given tower (Fig A 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "220-223",
        "Text": "Solution Step 1 Given physical situation is \u201cto find the height of a given tower\u201d Step 2 Let AB be the given tower (Fig A 2 2)"
    },
    {
        "Chapter": "1",
        "sentence_range": "221-224",
        "Text": "Step 2 Let AB be the given tower (Fig A 2 2) Let PQ be an observer measuring the\nheight of the tower with his eye at P"
    },
    {
        "Chapter": "1",
        "sentence_range": "222-225",
        "Text": "2 2) Let PQ be an observer measuring the\nheight of the tower with his eye at P Let  PQ = h  and let height of tower be H"
    },
    {
        "Chapter": "1",
        "sentence_range": "223-226",
        "Text": "2) Let PQ be an observer measuring the\nheight of the tower with his eye at P Let  PQ = h  and let height of tower be H Let \u03b1\nbe the angle of elevation from the eye of the observer to the top of the tower"
    },
    {
        "Chapter": "1",
        "sentence_range": "224-227",
        "Text": "Let PQ be an observer measuring the\nheight of the tower with his eye at P Let  PQ = h  and let height of tower be H Let \u03b1\nbe the angle of elevation from the eye of the observer to the top of the tower Fig A"
    },
    {
        "Chapter": "1",
        "sentence_range": "225-228",
        "Text": "Let  PQ = h  and let height of tower be H Let \u03b1\nbe the angle of elevation from the eye of the observer to the top of the tower Fig A 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "226-229",
        "Text": "Let \u03b1\nbe the angle of elevation from the eye of the observer to the top of the tower Fig A 2 2\nRationalised 2023-24\nMATHEMATICAL MODELLING        199\nLet\nl = PC = QB\nNow\ntan \u03b1 = AC\nH\nPC\nh\n\u2212l\n=\nor\nH = h + l tan \u03b1"
    },
    {
        "Chapter": "1",
        "sentence_range": "227-230",
        "Text": "Fig A 2 2\nRationalised 2023-24\nMATHEMATICAL MODELLING        199\nLet\nl = PC = QB\nNow\ntan \u03b1 = AC\nH\nPC\nh\n\u2212l\n=\nor\nH = h + l tan \u03b1 (1)\nStep 3 Note that the values of the parameters h, l and \u03b1 (using sextant) are known to\nthe observer and so (1) gives the solution of the problem"
    },
    {
        "Chapter": "1",
        "sentence_range": "228-231",
        "Text": "2 2\nRationalised 2023-24\nMATHEMATICAL MODELLING        199\nLet\nl = PC = QB\nNow\ntan \u03b1 = AC\nH\nPC\nh\n\u2212l\n=\nor\nH = h + l tan \u03b1 (1)\nStep 3 Note that the values of the parameters h, l and \u03b1 (using sextant) are known to\nthe observer and so (1) gives the solution of the problem Step 4 In case, if the foot of the tower is not  accessible, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "229-232",
        "Text": "2\nRationalised 2023-24\nMATHEMATICAL MODELLING        199\nLet\nl = PC = QB\nNow\ntan \u03b1 = AC\nH\nPC\nh\n\u2212l\n=\nor\nH = h + l tan \u03b1 (1)\nStep 3 Note that the values of the parameters h, l and \u03b1 (using sextant) are known to\nthe observer and so (1) gives the solution of the problem Step 4 In case, if the foot of the tower is not  accessible, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "230-233",
        "Text": "(1)\nStep 3 Note that the values of the parameters h, l and \u03b1 (using sextant) are known to\nthe observer and so (1) gives the solution of the problem Step 4 In case, if the foot of the tower is not  accessible, i e , when l is not known to the\nobserver, let \u03b2 be the angle of depression from P to the foot B of the tower"
    },
    {
        "Chapter": "1",
        "sentence_range": "231-234",
        "Text": "Step 4 In case, if the foot of the tower is not  accessible, i e , when l is not known to the\nobserver, let \u03b2 be the angle of depression from P to the foot B of the tower So from\n\u2206PQB, we have\nPQ\ntan\nQB\nlh\n\u03b2 =\n=\n or l = h cot \u03b2\nStep 5 is not required in this situation as exact values of the parameters h, l, \u03b1 and \u03b2\nare known"
    },
    {
        "Chapter": "1",
        "sentence_range": "232-235",
        "Text": "e , when l is not known to the\nobserver, let \u03b2 be the angle of depression from P to the foot B of the tower So from\n\u2206PQB, we have\nPQ\ntan\nQB\nlh\n\u03b2 =\n=\n or l = h cot \u03b2\nStep 5 is not required in this situation as exact values of the parameters h, l, \u03b1 and \u03b2\nare known Example 2 Let a business firm produces three types of products P1, P2 and P3 that\nuses three types of raw materials R1, R2 and R3"
    },
    {
        "Chapter": "1",
        "sentence_range": "233-236",
        "Text": ", when l is not known to the\nobserver, let \u03b2 be the angle of depression from P to the foot B of the tower So from\n\u2206PQB, we have\nPQ\ntan\nQB\nlh\n\u03b2 =\n=\n or l = h cot \u03b2\nStep 5 is not required in this situation as exact values of the parameters h, l, \u03b1 and \u03b2\nare known Example 2 Let a business firm produces three types of products P1, P2 and P3 that\nuses three types of raw materials R1, R2 and R3 Let the firm has purchase orders from\ntwo clients F1 and F2"
    },
    {
        "Chapter": "1",
        "sentence_range": "234-237",
        "Text": "So from\n\u2206PQB, we have\nPQ\ntan\nQB\nlh\n\u03b2 =\n=\n or l = h cot \u03b2\nStep 5 is not required in this situation as exact values of the parameters h, l, \u03b1 and \u03b2\nare known Example 2 Let a business firm produces three types of products P1, P2 and P3 that\nuses three types of raw materials R1, R2 and R3 Let the firm has purchase orders from\ntwo clients F1 and F2 Considering the situation that the firm has a limited quantity of\nR1, R2 and R3, respectively, prepare a model to determine the quantities of the raw\nmaterial R1, R2 and R3 required to meet the purchase orders"
    },
    {
        "Chapter": "1",
        "sentence_range": "235-238",
        "Text": "Example 2 Let a business firm produces three types of products P1, P2 and P3 that\nuses three types of raw materials R1, R2 and R3 Let the firm has purchase orders from\ntwo clients F1 and F2 Considering the situation that the firm has a limited quantity of\nR1, R2 and R3, respectively, prepare a model to determine the quantities of the raw\nmaterial R1, R2 and R3 required to meet the purchase orders Solution Step 1 The physical situation is well identified in the problem"
    },
    {
        "Chapter": "1",
        "sentence_range": "236-239",
        "Text": "Let the firm has purchase orders from\ntwo clients F1 and F2 Considering the situation that the firm has a limited quantity of\nR1, R2 and R3, respectively, prepare a model to determine the quantities of the raw\nmaterial R1, R2 and R3 required to meet the purchase orders Solution Step 1 The physical situation is well identified in the problem Step 2 Let A be a matrix that represents purchase orders from the two clients F1 and\nF2"
    },
    {
        "Chapter": "1",
        "sentence_range": "237-240",
        "Text": "Considering the situation that the firm has a limited quantity of\nR1, R2 and R3, respectively, prepare a model to determine the quantities of the raw\nmaterial R1, R2 and R3 required to meet the purchase orders Solution Step 1 The physical situation is well identified in the problem Step 2 Let A be a matrix that represents purchase orders from the two clients F1 and\nF2 Then, A is of the form\n1\n2\n3\n1\n2\nP P\nP\nF \u2022\n\u2022\n\u2022\nA\nF\n\u2022\n\u2022\n\u2022\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nLet B be the matrix that represents the amount of raw materials R1, R2 and R3,\nrequired to manufacture each unit of the products P1, P2 and P3"
    },
    {
        "Chapter": "1",
        "sentence_range": "238-241",
        "Text": "Solution Step 1 The physical situation is well identified in the problem Step 2 Let A be a matrix that represents purchase orders from the two clients F1 and\nF2 Then, A is of the form\n1\n2\n3\n1\n2\nP P\nP\nF \u2022\n\u2022\n\u2022\nA\nF\n\u2022\n\u2022\n\u2022\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nLet B be the matrix that represents the amount of raw materials R1, R2 and R3,\nrequired to manufacture each unit of the products P1, P2 and P3 Then, B is of the form\n1\n2\n3\n1\n2\n3\n\u2022R R R\n\u2022\n\u2022\nP\nB\nP\n\u2022\n\u2022\n\u2022\nP\n\u2022\n\u2022\n\u2022\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nRationalised 2023-24\n200\nMATHEMATICS\nStep 3 Note that the product (which in this case is well defined) of matrices A and B\nis given by the following matrix\n1\n2\n3\n1\n2\nR R R\nF \u2022\n\u2022\n\u2022\nAB F\n\u2022\n\u2022\n\u2022\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nwhich in fact gives the desired quantities of the raw materials R1, R2 and R3 to fulfill\nthe purchase orders of the two clients F1 and F2"
    },
    {
        "Chapter": "1",
        "sentence_range": "239-242",
        "Text": "Step 2 Let A be a matrix that represents purchase orders from the two clients F1 and\nF2 Then, A is of the form\n1\n2\n3\n1\n2\nP P\nP\nF \u2022\n\u2022\n\u2022\nA\nF\n\u2022\n\u2022\n\u2022\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nLet B be the matrix that represents the amount of raw materials R1, R2 and R3,\nrequired to manufacture each unit of the products P1, P2 and P3 Then, B is of the form\n1\n2\n3\n1\n2\n3\n\u2022R R R\n\u2022\n\u2022\nP\nB\nP\n\u2022\n\u2022\n\u2022\nP\n\u2022\n\u2022\n\u2022\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nRationalised 2023-24\n200\nMATHEMATICS\nStep 3 Note that the product (which in this case is well defined) of matrices A and B\nis given by the following matrix\n1\n2\n3\n1\n2\nR R R\nF \u2022\n\u2022\n\u2022\nAB F\n\u2022\n\u2022\n\u2022\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nwhich in fact gives the desired quantities of the raw materials R1, R2 and R3 to fulfill\nthe purchase orders of the two clients F1 and F2 Example 3 Interpret the model in Example 2, in case\n3\n4\n0\n10\n15\n6\nA =\n, B\n7\n9\n3\n10\n20\n0\n5\n12\n7\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nand the available raw materials are 330 units of R1, 455 units of R2 and 140 units of R3"
    },
    {
        "Chapter": "1",
        "sentence_range": "240-243",
        "Text": "Then, A is of the form\n1\n2\n3\n1\n2\nP P\nP\nF \u2022\n\u2022\n\u2022\nA\nF\n\u2022\n\u2022\n\u2022\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nLet B be the matrix that represents the amount of raw materials R1, R2 and R3,\nrequired to manufacture each unit of the products P1, P2 and P3 Then, B is of the form\n1\n2\n3\n1\n2\n3\n\u2022R R R\n\u2022\n\u2022\nP\nB\nP\n\u2022\n\u2022\n\u2022\nP\n\u2022\n\u2022\n\u2022\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nRationalised 2023-24\n200\nMATHEMATICS\nStep 3 Note that the product (which in this case is well defined) of matrices A and B\nis given by the following matrix\n1\n2\n3\n1\n2\nR R R\nF \u2022\n\u2022\n\u2022\nAB F\n\u2022\n\u2022\n\u2022\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nwhich in fact gives the desired quantities of the raw materials R1, R2 and R3 to fulfill\nthe purchase orders of the two clients F1 and F2 Example 3 Interpret the model in Example 2, in case\n3\n4\n0\n10\n15\n6\nA =\n, B\n7\n9\n3\n10\n20\n0\n5\n12\n7\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nand the available raw materials are 330 units of R1, 455 units of R2 and 140 units of R3 Solution Note that\nAB =\n3\n4\n0\n10\n15\n6\n7\n9\n3\n10\n20\n0\n5\n12\n7\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9 \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n=\nR\nR\nR\nF\nF\n1\n2\n3\n1\n2\n165\n247\n87\n170\n220\n60\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa\nThis clearly shows that to meet the purchase order of F1 and F2, the raw material\nrequired is 335 units of R1, 467 units of R2 and 147 units of R3 which is much more than\nthe available raw material"
    },
    {
        "Chapter": "1",
        "sentence_range": "241-244",
        "Text": "Then, B is of the form\n1\n2\n3\n1\n2\n3\n\u2022R R R\n\u2022\n\u2022\nP\nB\nP\n\u2022\n\u2022\n\u2022\nP\n\u2022\n\u2022\n\u2022\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nRationalised 2023-24\n200\nMATHEMATICS\nStep 3 Note that the product (which in this case is well defined) of matrices A and B\nis given by the following matrix\n1\n2\n3\n1\n2\nR R R\nF \u2022\n\u2022\n\u2022\nAB F\n\u2022\n\u2022\n\u2022\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nwhich in fact gives the desired quantities of the raw materials R1, R2 and R3 to fulfill\nthe purchase orders of the two clients F1 and F2 Example 3 Interpret the model in Example 2, in case\n3\n4\n0\n10\n15\n6\nA =\n, B\n7\n9\n3\n10\n20\n0\n5\n12\n7\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nand the available raw materials are 330 units of R1, 455 units of R2 and 140 units of R3 Solution Note that\nAB =\n3\n4\n0\n10\n15\n6\n7\n9\n3\n10\n20\n0\n5\n12\n7\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9 \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n=\nR\nR\nR\nF\nF\n1\n2\n3\n1\n2\n165\n247\n87\n170\n220\n60\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa\nThis clearly shows that to meet the purchase order of F1 and F2, the raw material\nrequired is 335 units of R1, 467 units of R2 and 147 units of R3 which is much more than\nthe available raw material Since the amount of raw material required to manufacture\neach unit of the three products is fixed, we can either ask for an increase in the\navailable raw material or we may ask the clients to reduce their orders"
    },
    {
        "Chapter": "1",
        "sentence_range": "242-245",
        "Text": "Example 3 Interpret the model in Example 2, in case\n3\n4\n0\n10\n15\n6\nA =\n, B\n7\n9\n3\n10\n20\n0\n5\n12\n7\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nand the available raw materials are 330 units of R1, 455 units of R2 and 140 units of R3 Solution Note that\nAB =\n3\n4\n0\n10\n15\n6\n7\n9\n3\n10\n20\n0\n5\n12\n7\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9 \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n=\nR\nR\nR\nF\nF\n1\n2\n3\n1\n2\n165\n247\n87\n170\n220\n60\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa\nThis clearly shows that to meet the purchase order of F1 and F2, the raw material\nrequired is 335 units of R1, 467 units of R2 and 147 units of R3 which is much more than\nthe available raw material Since the amount of raw material required to manufacture\neach unit of the three products is fixed, we can either ask for an increase in the\navailable raw material or we may ask the clients to reduce their orders Remark If we replace A in Example 3 by A1 given by\nA1 = 9\n12\n6\n10\n20\n0\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\ni"
    },
    {
        "Chapter": "1",
        "sentence_range": "243-246",
        "Text": "Solution Note that\nAB =\n3\n4\n0\n10\n15\n6\n7\n9\n3\n10\n20\n0\n5\n12\n7\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9 \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n=\nR\nR\nR\nF\nF\n1\n2\n3\n1\n2\n165\n247\n87\n170\n220\n60\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa\nThis clearly shows that to meet the purchase order of F1 and F2, the raw material\nrequired is 335 units of R1, 467 units of R2 and 147 units of R3 which is much more than\nthe available raw material Since the amount of raw material required to manufacture\neach unit of the three products is fixed, we can either ask for an increase in the\navailable raw material or we may ask the clients to reduce their orders Remark If we replace A in Example 3 by A1 given by\nA1 = 9\n12\n6\n10\n20\n0\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\ni e"
    },
    {
        "Chapter": "1",
        "sentence_range": "244-247",
        "Text": "Since the amount of raw material required to manufacture\neach unit of the three products is fixed, we can either ask for an increase in the\navailable raw material or we may ask the clients to reduce their orders Remark If we replace A in Example 3 by A1 given by\nA1 = 9\n12\n6\n10\n20\n0\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\ni e , if the clients agree to reduce their purchase orders, then\nA1 B = \n3\n4\n0\n9\n12\n6\n7\n9\n3\n10\n20\n0\n5\n12\n7\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9 \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n  = \uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa\n141\n216\n78\n170\n220\n60\nRationalised 2023-24\nMATHEMATICAL MODELLING        201\nThis requires 311 units of R1, 436 units of R2 and 138 units of R3 which are well\nbelow the available raw materials, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "245-248",
        "Text": "Remark If we replace A in Example 3 by A1 given by\nA1 = 9\n12\n6\n10\n20\n0\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\ni e , if the clients agree to reduce their purchase orders, then\nA1 B = \n3\n4\n0\n9\n12\n6\n7\n9\n3\n10\n20\n0\n5\n12\n7\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9 \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n  = \uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa\n141\n216\n78\n170\n220\n60\nRationalised 2023-24\nMATHEMATICAL MODELLING        201\nThis requires 311 units of R1, 436 units of R2 and 138 units of R3 which are well\nbelow the available raw materials, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "246-249",
        "Text": "e , if the clients agree to reduce their purchase orders, then\nA1 B = \n3\n4\n0\n9\n12\n6\n7\n9\n3\n10\n20\n0\n5\n12\n7\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9 \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n  = \uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa\n141\n216\n78\n170\n220\n60\nRationalised 2023-24\nMATHEMATICAL MODELLING        201\nThis requires 311 units of R1, 436 units of R2 and 138 units of R3 which are well\nbelow the available raw materials, i e , 330 units of R1, 455 units of R2 and 140 units of\nR3"
    },
    {
        "Chapter": "1",
        "sentence_range": "247-250",
        "Text": ", if the clients agree to reduce their purchase orders, then\nA1 B = \n3\n4\n0\n9\n12\n6\n7\n9\n3\n10\n20\n0\n5\n12\n7\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9 \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n  = \uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa\n141\n216\n78\n170\n220\n60\nRationalised 2023-24\nMATHEMATICAL MODELLING        201\nThis requires 311 units of R1, 436 units of R2 and 138 units of R3 which are well\nbelow the available raw materials, i e , 330 units of R1, 455 units of R2 and 140 units of\nR3 Thus, if the revised purchase orders of the clients are given by A1, then the firm\ncan easily supply the purchase orders of the two clients"
    },
    {
        "Chapter": "1",
        "sentence_range": "248-251",
        "Text": "e , 330 units of R1, 455 units of R2 and 140 units of\nR3 Thus, if the revised purchase orders of the clients are given by A1, then the firm\ncan easily supply the purchase orders of the two clients ANote One may further modify A so as to make full use of the available\nraw material"
    },
    {
        "Chapter": "1",
        "sentence_range": "249-252",
        "Text": ", 330 units of R1, 455 units of R2 and 140 units of\nR3 Thus, if the revised purchase orders of the clients are given by A1, then the firm\ncan easily supply the purchase orders of the two clients ANote One may further modify A so as to make full use of the available\nraw material Query Can we make a mathematical model with a given B and with fixed quantities of\nthe available raw material that can help the firm owner to ask the clients to modify their\norders in such a way that the firm makes the full use of its available raw material"
    },
    {
        "Chapter": "1",
        "sentence_range": "250-253",
        "Text": "Thus, if the revised purchase orders of the clients are given by A1, then the firm\ncan easily supply the purchase orders of the two clients ANote One may further modify A so as to make full use of the available\nraw material Query Can we make a mathematical model with a given B and with fixed quantities of\nthe available raw material that can help the firm owner to ask the clients to modify their\norders in such a way that the firm makes the full use of its available raw material The answer to this query is given in the following example:\nExample 4 Suppose P1, P2, P3 and R1, R2, R3 are as in Example 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "251-254",
        "Text": "ANote One may further modify A so as to make full use of the available\nraw material Query Can we make a mathematical model with a given B and with fixed quantities of\nthe available raw material that can help the firm owner to ask the clients to modify their\norders in such a way that the firm makes the full use of its available raw material The answer to this query is given in the following example:\nExample 4 Suppose P1, P2, P3 and R1, R2, R3 are as in Example 2 Let the firm has\n330 units of R1, 455 units of R2 and 140 units of R3 available with it and let the amount\nof raw materials R1, R2 and R3 required to manufacture each unit of the three products\nis given by\n1\n2\n3\n1\n2\n3\nR\nR\nR\n3\n4\n0\nP\nB\nP\n7\n9\n3\nP\n5\n12\n7\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nHow many units of each product is to be made so as to utilise the full available raw\nmaterial"
    },
    {
        "Chapter": "1",
        "sentence_range": "252-255",
        "Text": "Query Can we make a mathematical model with a given B and with fixed quantities of\nthe available raw material that can help the firm owner to ask the clients to modify their\norders in such a way that the firm makes the full use of its available raw material The answer to this query is given in the following example:\nExample 4 Suppose P1, P2, P3 and R1, R2, R3 are as in Example 2 Let the firm has\n330 units of R1, 455 units of R2 and 140 units of R3 available with it and let the amount\nof raw materials R1, R2 and R3 required to manufacture each unit of the three products\nis given by\n1\n2\n3\n1\n2\n3\nR\nR\nR\n3\n4\n0\nP\nB\nP\n7\n9\n3\nP\n5\n12\n7\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nHow many units of each product is to be made so as to utilise the full available raw\nmaterial Solution Step 1 The situation is easily identifiable"
    },
    {
        "Chapter": "1",
        "sentence_range": "253-256",
        "Text": "The answer to this query is given in the following example:\nExample 4 Suppose P1, P2, P3 and R1, R2, R3 are as in Example 2 Let the firm has\n330 units of R1, 455 units of R2 and 140 units of R3 available with it and let the amount\nof raw materials R1, R2 and R3 required to manufacture each unit of the three products\nis given by\n1\n2\n3\n1\n2\n3\nR\nR\nR\n3\n4\n0\nP\nB\nP\n7\n9\n3\nP\n5\n12\n7\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nHow many units of each product is to be made so as to utilise the full available raw\nmaterial Solution Step 1 The situation is easily identifiable Step 2 Suppose the firm produces x units of P1, y units of P2 and z units of P3"
    },
    {
        "Chapter": "1",
        "sentence_range": "254-257",
        "Text": "Let the firm has\n330 units of R1, 455 units of R2 and 140 units of R3 available with it and let the amount\nof raw materials R1, R2 and R3 required to manufacture each unit of the three products\nis given by\n1\n2\n3\n1\n2\n3\nR\nR\nR\n3\n4\n0\nP\nB\nP\n7\n9\n3\nP\n5\n12\n7\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nHow many units of each product is to be made so as to utilise the full available raw\nmaterial Solution Step 1 The situation is easily identifiable Step 2 Suppose the firm produces x units of P1, y units of P2 and z units of P3 Since\nproduct P1 requires 3 units of R1, P2 requires 7 units of R1 and P3 requires 5 units of R1\n(observe matrix B) and the total number of units, of R1, available is 330, we have\n3x + 7y + 5z = 330 (for raw material R1)\nSimilarly, we have\n4x + 9y + 12z = 455 (for raw material R2)\nand\n3y + 7z = 140 (for raw material R3)\nThis system of equations can be expressed in matrix form as\n3\n7\n5\n4\n9\n12\n0\n3\n7\n330\n455\n140\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\uf8fa\n=\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\nx\ny\nz\nRationalised 2023-24\n202\nMATHEMATICS\nStep 3 Using elementary row operations, we obtain\n1\n0\n0\n0\n1\n0\n0\n0\n1\n20\n35\n5\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\uf8fa\n=\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\nx\ny\nz\nThis gives x = 20, y = 35 and z = 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "255-258",
        "Text": "Solution Step 1 The situation is easily identifiable Step 2 Suppose the firm produces x units of P1, y units of P2 and z units of P3 Since\nproduct P1 requires 3 units of R1, P2 requires 7 units of R1 and P3 requires 5 units of R1\n(observe matrix B) and the total number of units, of R1, available is 330, we have\n3x + 7y + 5z = 330 (for raw material R1)\nSimilarly, we have\n4x + 9y + 12z = 455 (for raw material R2)\nand\n3y + 7z = 140 (for raw material R3)\nThis system of equations can be expressed in matrix form as\n3\n7\n5\n4\n9\n12\n0\n3\n7\n330\n455\n140\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\uf8fa\n=\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\nx\ny\nz\nRationalised 2023-24\n202\nMATHEMATICS\nStep 3 Using elementary row operations, we obtain\n1\n0\n0\n0\n1\n0\n0\n0\n1\n20\n35\n5\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\uf8fa\n=\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\nx\ny\nz\nThis gives x = 20, y = 35 and z = 5 Thus, the firm can produce 20 units of P1, 35\nunits of P2 and 5 units of P3 to make full use of its available raw material"
    },
    {
        "Chapter": "1",
        "sentence_range": "256-259",
        "Text": "Step 2 Suppose the firm produces x units of P1, y units of P2 and z units of P3 Since\nproduct P1 requires 3 units of R1, P2 requires 7 units of R1 and P3 requires 5 units of R1\n(observe matrix B) and the total number of units, of R1, available is 330, we have\n3x + 7y + 5z = 330 (for raw material R1)\nSimilarly, we have\n4x + 9y + 12z = 455 (for raw material R2)\nand\n3y + 7z = 140 (for raw material R3)\nThis system of equations can be expressed in matrix form as\n3\n7\n5\n4\n9\n12\n0\n3\n7\n330\n455\n140\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\uf8fa\n=\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\nx\ny\nz\nRationalised 2023-24\n202\nMATHEMATICS\nStep 3 Using elementary row operations, we obtain\n1\n0\n0\n0\n1\n0\n0\n0\n1\n20\n35\n5\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\uf8fa\n=\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\nx\ny\nz\nThis gives x = 20, y = 35 and z = 5 Thus, the firm can produce 20 units of P1, 35\nunits of P2 and 5 units of P3 to make full use of its available raw material Remark One may observe that if the manufacturer decides to manufacture according\nto the available raw material and not according to the purchase orders of the two\nclients F1 and F2 (as in Example 3), he/she is unable to meet these purchase orders as\nF1 demanded 6 units of P3 where as the manufacturer can make only 5 units of P3"
    },
    {
        "Chapter": "1",
        "sentence_range": "257-260",
        "Text": "Since\nproduct P1 requires 3 units of R1, P2 requires 7 units of R1 and P3 requires 5 units of R1\n(observe matrix B) and the total number of units, of R1, available is 330, we have\n3x + 7y + 5z = 330 (for raw material R1)\nSimilarly, we have\n4x + 9y + 12z = 455 (for raw material R2)\nand\n3y + 7z = 140 (for raw material R3)\nThis system of equations can be expressed in matrix form as\n3\n7\n5\n4\n9\n12\n0\n3\n7\n330\n455\n140\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\uf8fa\n=\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\nx\ny\nz\nRationalised 2023-24\n202\nMATHEMATICS\nStep 3 Using elementary row operations, we obtain\n1\n0\n0\n0\n1\n0\n0\n0\n1\n20\n35\n5\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\uf8fa\n=\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\nx\ny\nz\nThis gives x = 20, y = 35 and z = 5 Thus, the firm can produce 20 units of P1, 35\nunits of P2 and 5 units of P3 to make full use of its available raw material Remark One may observe that if the manufacturer decides to manufacture according\nto the available raw material and not according to the purchase orders of the two\nclients F1 and F2 (as in Example 3), he/she is unable to meet these purchase orders as\nF1 demanded 6 units of P3 where as the manufacturer can make only 5 units of P3 Example 5 A manufacturer of medicines is preparing a production plan of medicines\nM1 and M2"
    },
    {
        "Chapter": "1",
        "sentence_range": "258-261",
        "Text": "Thus, the firm can produce 20 units of P1, 35\nunits of P2 and 5 units of P3 to make full use of its available raw material Remark One may observe that if the manufacturer decides to manufacture according\nto the available raw material and not according to the purchase orders of the two\nclients F1 and F2 (as in Example 3), he/she is unable to meet these purchase orders as\nF1 demanded 6 units of P3 where as the manufacturer can make only 5 units of P3 Example 5 A manufacturer of medicines is preparing a production plan of medicines\nM1 and M2 There are sufficient raw materials available to make 20000 bottles of M1\nand 40000 bottles of M2, but there are only 45000 bottles into which either of the\nmedicines can be put"
    },
    {
        "Chapter": "1",
        "sentence_range": "259-262",
        "Text": "Remark One may observe that if the manufacturer decides to manufacture according\nto the available raw material and not according to the purchase orders of the two\nclients F1 and F2 (as in Example 3), he/she is unable to meet these purchase orders as\nF1 demanded 6 units of P3 where as the manufacturer can make only 5 units of P3 Example 5 A manufacturer of medicines is preparing a production plan of medicines\nM1 and M2 There are sufficient raw materials available to make 20000 bottles of M1\nand 40000 bottles of M2, but there are only 45000 bottles into which either of the\nmedicines can be put Further, it takes 3 hours to prepare enough material to fill 1000\nbottles of M1, it takes 1 hour to prepare enough material to fill 1000 bottles of M2 and\nthere are 66 hours available for this operation"
    },
    {
        "Chapter": "1",
        "sentence_range": "260-263",
        "Text": "Example 5 A manufacturer of medicines is preparing a production plan of medicines\nM1 and M2 There are sufficient raw materials available to make 20000 bottles of M1\nand 40000 bottles of M2, but there are only 45000 bottles into which either of the\nmedicines can be put Further, it takes 3 hours to prepare enough material to fill 1000\nbottles of M1, it takes 1 hour to prepare enough material to fill 1000 bottles of M2 and\nthere are 66 hours available for this operation The profit is Rs 8 per bottle for M1 and\nRs 7 per bottle for M2"
    },
    {
        "Chapter": "1",
        "sentence_range": "261-264",
        "Text": "There are sufficient raw materials available to make 20000 bottles of M1\nand 40000 bottles of M2, but there are only 45000 bottles into which either of the\nmedicines can be put Further, it takes 3 hours to prepare enough material to fill 1000\nbottles of M1, it takes 1 hour to prepare enough material to fill 1000 bottles of M2 and\nthere are 66 hours available for this operation The profit is Rs 8 per bottle for M1 and\nRs 7 per bottle for M2 How should the manufacturer schedule his/her production in\norder to maximise profit"
    },
    {
        "Chapter": "1",
        "sentence_range": "262-265",
        "Text": "Further, it takes 3 hours to prepare enough material to fill 1000\nbottles of M1, it takes 1 hour to prepare enough material to fill 1000 bottles of M2 and\nthere are 66 hours available for this operation The profit is Rs 8 per bottle for M1 and\nRs 7 per bottle for M2 How should the manufacturer schedule his/her production in\norder to maximise profit Solution Step 1 To find the number of bottles of M1 and M2 in order to maximise the\nprofit under the given hypotheses"
    },
    {
        "Chapter": "1",
        "sentence_range": "263-266",
        "Text": "The profit is Rs 8 per bottle for M1 and\nRs 7 per bottle for M2 How should the manufacturer schedule his/her production in\norder to maximise profit Solution Step 1 To find the number of bottles of M1 and M2 in order to maximise the\nprofit under the given hypotheses Step 2 Let x be the number of bottles of type M1 medicine and y be the number of\nbottles of type M2 medicine"
    },
    {
        "Chapter": "1",
        "sentence_range": "264-267",
        "Text": "How should the manufacturer schedule his/her production in\norder to maximise profit Solution Step 1 To find the number of bottles of M1 and M2 in order to maximise the\nprofit under the given hypotheses Step 2 Let x be the number of bottles of type M1 medicine and y be the number of\nbottles of type M2 medicine Since profit is Rs 8 per bottle for M1 and Rs 7 per bottle\nfor M2, therefore the objective function (which is to be maximised) is given by\nZ \u2261 Z (x, y) = 8x + 7y\nThe objective function is to be maximised subject to the constraints (Refer Chapter\n12 on Linear Programming)\nx\ny\nx\nxy\ny\nx\ny\n\u2264\n\u2264\n+\n+\u2264\n\u2264\n\u2265\n\u2265\n\uf8fc\n\uf8fd\n\uf8f4\n\uf8f4\uf8f4\n\uf8fe\n\uf8f4\uf8f4\n\uf8f4\n20000\n40000\n45000\n3\n66000\n0\n0\n,"
    },
    {
        "Chapter": "1",
        "sentence_range": "265-268",
        "Text": "Solution Step 1 To find the number of bottles of M1 and M2 in order to maximise the\nprofit under the given hypotheses Step 2 Let x be the number of bottles of type M1 medicine and y be the number of\nbottles of type M2 medicine Since profit is Rs 8 per bottle for M1 and Rs 7 per bottle\nfor M2, therefore the objective function (which is to be maximised) is given by\nZ \u2261 Z (x, y) = 8x + 7y\nThe objective function is to be maximised subject to the constraints (Refer Chapter\n12 on Linear Programming)\nx\ny\nx\nxy\ny\nx\ny\n\u2264\n\u2264\n+\n+\u2264\n\u2264\n\u2265\n\u2265\n\uf8fc\n\uf8fd\n\uf8f4\n\uf8f4\uf8f4\n\uf8fe\n\uf8f4\uf8f4\n\uf8f4\n20000\n40000\n45000\n3\n66000\n0\n0\n, (1)\nStep 3 The shaded region OPQRST is the feasible region for the constraints (1)\n(Fig A"
    },
    {
        "Chapter": "1",
        "sentence_range": "266-269",
        "Text": "Step 2 Let x be the number of bottles of type M1 medicine and y be the number of\nbottles of type M2 medicine Since profit is Rs 8 per bottle for M1 and Rs 7 per bottle\nfor M2, therefore the objective function (which is to be maximised) is given by\nZ \u2261 Z (x, y) = 8x + 7y\nThe objective function is to be maximised subject to the constraints (Refer Chapter\n12 on Linear Programming)\nx\ny\nx\nxy\ny\nx\ny\n\u2264\n\u2264\n+\n+\u2264\n\u2264\n\u2265\n\u2265\n\uf8fc\n\uf8fd\n\uf8f4\n\uf8f4\uf8f4\n\uf8fe\n\uf8f4\uf8f4\n\uf8f4\n20000\n40000\n45000\n3\n66000\n0\n0\n, (1)\nStep 3 The shaded region OPQRST is the feasible region for the constraints (1)\n(Fig A 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "267-270",
        "Text": "Since profit is Rs 8 per bottle for M1 and Rs 7 per bottle\nfor M2, therefore the objective function (which is to be maximised) is given by\nZ \u2261 Z (x, y) = 8x + 7y\nThe objective function is to be maximised subject to the constraints (Refer Chapter\n12 on Linear Programming)\nx\ny\nx\nxy\ny\nx\ny\n\u2264\n\u2264\n+\n+\u2264\n\u2264\n\u2265\n\u2265\n\uf8fc\n\uf8fd\n\uf8f4\n\uf8f4\uf8f4\n\uf8fe\n\uf8f4\uf8f4\n\uf8f4\n20000\n40000\n45000\n3\n66000\n0\n0\n, (1)\nStep 3 The shaded region OPQRST is the feasible region for the constraints (1)\n(Fig A 2 3)"
    },
    {
        "Chapter": "1",
        "sentence_range": "268-271",
        "Text": "(1)\nStep 3 The shaded region OPQRST is the feasible region for the constraints (1)\n(Fig A 2 3) The co-ordinates of vertices O, P, Q, R, S and T are (0, 0), (20000, 0),\n(20000, 6000), (10500, 34500), (5000, 40000) and (0, 40000), respectively"
    },
    {
        "Chapter": "1",
        "sentence_range": "269-272",
        "Text": "2 3) The co-ordinates of vertices O, P, Q, R, S and T are (0, 0), (20000, 0),\n(20000, 6000), (10500, 34500), (5000, 40000) and (0, 40000), respectively Rationalised 2023-24\nMATHEMATICAL MODELLING        203\nFig A"
    },
    {
        "Chapter": "1",
        "sentence_range": "270-273",
        "Text": "3) The co-ordinates of vertices O, P, Q, R, S and T are (0, 0), (20000, 0),\n(20000, 6000), (10500, 34500), (5000, 40000) and (0, 40000), respectively Rationalised 2023-24\nMATHEMATICAL MODELLING        203\nFig A 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "271-274",
        "Text": "The co-ordinates of vertices O, P, Q, R, S and T are (0, 0), (20000, 0),\n(20000, 6000), (10500, 34500), (5000, 40000) and (0, 40000), respectively Rationalised 2023-24\nMATHEMATICAL MODELLING        203\nFig A 2 3\nNote that\nZ at P (0, 0) = 0\nZ at P (20000, 0) = 8 \u00d7 20000 = 160000\nZ at Q (20000, 6000)  = 8 \u00d7 20000 + 7 \u00d7 6000 = 202000\nZ at R (10500, 34500)  = 8 \u00d7 10500 + 7 \u00d7 34500 = 325500\nZ at S = (5000, 40000) = 8 \u00d7 5000 + 7 \u00d7 40000 = 320000\nZ at T = (0, 40000)  = 7 \u00d7 40000 = 280000\nNow observe that the profit is maximum at x = 10500 and y = 34500 and the\nmaximum profit is ` 325500"
    },
    {
        "Chapter": "1",
        "sentence_range": "272-275",
        "Text": "Rationalised 2023-24\nMATHEMATICAL MODELLING        203\nFig A 2 3\nNote that\nZ at P (0, 0) = 0\nZ at P (20000, 0) = 8 \u00d7 20000 = 160000\nZ at Q (20000, 6000)  = 8 \u00d7 20000 + 7 \u00d7 6000 = 202000\nZ at R (10500, 34500)  = 8 \u00d7 10500 + 7 \u00d7 34500 = 325500\nZ at S = (5000, 40000) = 8 \u00d7 5000 + 7 \u00d7 40000 = 320000\nZ at T = (0, 40000)  = 7 \u00d7 40000 = 280000\nNow observe that the profit is maximum at x = 10500 and y = 34500 and the\nmaximum profit is ` 325500 Hence, the manufacturer should produce 10500 bottles of\nM1 medicine and 34500 bottles of M2 medicine in order to get maximum profit of\n` 325500"
    },
    {
        "Chapter": "1",
        "sentence_range": "273-276",
        "Text": "2 3\nNote that\nZ at P (0, 0) = 0\nZ at P (20000, 0) = 8 \u00d7 20000 = 160000\nZ at Q (20000, 6000)  = 8 \u00d7 20000 + 7 \u00d7 6000 = 202000\nZ at R (10500, 34500)  = 8 \u00d7 10500 + 7 \u00d7 34500 = 325500\nZ at S = (5000, 40000) = 8 \u00d7 5000 + 7 \u00d7 40000 = 320000\nZ at T = (0, 40000)  = 7 \u00d7 40000 = 280000\nNow observe that the profit is maximum at x = 10500 and y = 34500 and the\nmaximum profit is ` 325500 Hence, the manufacturer should produce 10500 bottles of\nM1 medicine and 34500 bottles of M2 medicine in order to get maximum profit of\n` 325500 Example 6 Suppose a company plans to produce a new product that incur some costs\n(fixed and variable) and let the company plans to sell the product at a fixed price"
    },
    {
        "Chapter": "1",
        "sentence_range": "274-277",
        "Text": "3\nNote that\nZ at P (0, 0) = 0\nZ at P (20000, 0) = 8 \u00d7 20000 = 160000\nZ at Q (20000, 6000)  = 8 \u00d7 20000 + 7 \u00d7 6000 = 202000\nZ at R (10500, 34500)  = 8 \u00d7 10500 + 7 \u00d7 34500 = 325500\nZ at S = (5000, 40000) = 8 \u00d7 5000 + 7 \u00d7 40000 = 320000\nZ at T = (0, 40000)  = 7 \u00d7 40000 = 280000\nNow observe that the profit is maximum at x = 10500 and y = 34500 and the\nmaximum profit is ` 325500 Hence, the manufacturer should produce 10500 bottles of\nM1 medicine and 34500 bottles of M2 medicine in order to get maximum profit of\n` 325500 Example 6 Suppose a company plans to produce a new product that incur some costs\n(fixed and variable) and let the company plans to sell the product at a fixed price Prepare a mathematical model to examine the profitability"
    },
    {
        "Chapter": "1",
        "sentence_range": "275-278",
        "Text": "Hence, the manufacturer should produce 10500 bottles of\nM1 medicine and 34500 bottles of M2 medicine in order to get maximum profit of\n` 325500 Example 6 Suppose a company plans to produce a new product that incur some costs\n(fixed and variable) and let the company plans to sell the product at a fixed price Prepare a mathematical model to examine the profitability Solution Step 1 Situation is clearly identifiable"
    },
    {
        "Chapter": "1",
        "sentence_range": "276-279",
        "Text": "Example 6 Suppose a company plans to produce a new product that incur some costs\n(fixed and variable) and let the company plans to sell the product at a fixed price Prepare a mathematical model to examine the profitability Solution Step 1 Situation is clearly identifiable Rationalised 2023-24\n204\nMATHEMATICS\nStep 2 Formulation: We are given that the costs are of two types: fixed and variable"
    },
    {
        "Chapter": "1",
        "sentence_range": "277-280",
        "Text": "Prepare a mathematical model to examine the profitability Solution Step 1 Situation is clearly identifiable Rationalised 2023-24\n204\nMATHEMATICS\nStep 2 Formulation: We are given that the costs are of two types: fixed and variable The fixed costs are independent of the number of units produced (e"
    },
    {
        "Chapter": "1",
        "sentence_range": "278-281",
        "Text": "Solution Step 1 Situation is clearly identifiable Rationalised 2023-24\n204\nMATHEMATICS\nStep 2 Formulation: We are given that the costs are of two types: fixed and variable The fixed costs are independent of the number of units produced (e g"
    },
    {
        "Chapter": "1",
        "sentence_range": "279-282",
        "Text": "Rationalised 2023-24\n204\nMATHEMATICS\nStep 2 Formulation: We are given that the costs are of two types: fixed and variable The fixed costs are independent of the number of units produced (e g , rent and rates),\nwhile the variable costs increase with the number of units produced (e"
    },
    {
        "Chapter": "1",
        "sentence_range": "280-283",
        "Text": "The fixed costs are independent of the number of units produced (e g , rent and rates),\nwhile the variable costs increase with the number of units produced (e g"
    },
    {
        "Chapter": "1",
        "sentence_range": "281-284",
        "Text": "g , rent and rates),\nwhile the variable costs increase with the number of units produced (e g , material)"
    },
    {
        "Chapter": "1",
        "sentence_range": "282-285",
        "Text": ", rent and rates),\nwhile the variable costs increase with the number of units produced (e g , material) Initially, we assume that the variable costs are directly proportional to the number of\nunits produced \u2014 this should simplify our model"
    },
    {
        "Chapter": "1",
        "sentence_range": "283-286",
        "Text": "g , material) Initially, we assume that the variable costs are directly proportional to the number of\nunits produced \u2014 this should simplify our model The company earn a certain amount\nof money by selling its products and wants to ensure that it is maximum"
    },
    {
        "Chapter": "1",
        "sentence_range": "284-287",
        "Text": ", material) Initially, we assume that the variable costs are directly proportional to the number of\nunits produced \u2014 this should simplify our model The company earn a certain amount\nof money by selling its products and wants to ensure that it is maximum For convenience,\nwe assume that all units produced are sold immediately"
    },
    {
        "Chapter": "1",
        "sentence_range": "285-288",
        "Text": "Initially, we assume that the variable costs are directly proportional to the number of\nunits produced \u2014 this should simplify our model The company earn a certain amount\nof money by selling its products and wants to ensure that it is maximum For convenience,\nwe assume that all units produced are sold immediately The mathematical model\nLet\nx = number of units produced and sold\nC = total cost of production (in rupees)\nI = income from sales (in rupees)\nP = profit (in rupees)\nOur assumptions above state that C consists of two parts:\n(i)\nfixed cost = a (in rupees),\n(ii)\nThenvariable cost = b (rupees/unit produced)"
    },
    {
        "Chapter": "1",
        "sentence_range": "286-289",
        "Text": "The company earn a certain amount\nof money by selling its products and wants to ensure that it is maximum For convenience,\nwe assume that all units produced are sold immediately The mathematical model\nLet\nx = number of units produced and sold\nC = total cost of production (in rupees)\nI = income from sales (in rupees)\nP = profit (in rupees)\nOur assumptions above state that C consists of two parts:\n(i)\nfixed cost = a (in rupees),\n(ii)\nThenvariable cost = b (rupees/unit produced) C = a + bx"
    },
    {
        "Chapter": "1",
        "sentence_range": "287-290",
        "Text": "For convenience,\nwe assume that all units produced are sold immediately The mathematical model\nLet\nx = number of units produced and sold\nC = total cost of production (in rupees)\nI = income from sales (in rupees)\nP = profit (in rupees)\nOur assumptions above state that C consists of two parts:\n(i)\nfixed cost = a (in rupees),\n(ii)\nThenvariable cost = b (rupees/unit produced) C = a + bx (1)\nAlso, income I depends on selling price s (rupees/unit)\nThus\nI = sx"
    },
    {
        "Chapter": "1",
        "sentence_range": "288-291",
        "Text": "The mathematical model\nLet\nx = number of units produced and sold\nC = total cost of production (in rupees)\nI = income from sales (in rupees)\nP = profit (in rupees)\nOur assumptions above state that C consists of two parts:\n(i)\nfixed cost = a (in rupees),\n(ii)\nThenvariable cost = b (rupees/unit produced) C = a + bx (1)\nAlso, income I depends on selling price s (rupees/unit)\nThus\nI = sx (2)\nThe profit P is then the difference between income and costs"
    },
    {
        "Chapter": "1",
        "sentence_range": "289-292",
        "Text": "C = a + bx (1)\nAlso, income I depends on selling price s (rupees/unit)\nThus\nI = sx (2)\nThe profit P is then the difference between income and costs So\nP = I \u2013 C\n= sx \u2013 (a + bx)\n= (s \u2013 b) x \u2013 a"
    },
    {
        "Chapter": "1",
        "sentence_range": "290-293",
        "Text": "(1)\nAlso, income I depends on selling price s (rupees/unit)\nThus\nI = sx (2)\nThe profit P is then the difference between income and costs So\nP = I \u2013 C\n= sx \u2013 (a + bx)\n= (s \u2013 b) x \u2013 a (3)\nWe now have a mathematical model of the relationships (1)  to  (3) between\nthe variables x, C, I, P, a, b, s"
    },
    {
        "Chapter": "1",
        "sentence_range": "291-294",
        "Text": "(2)\nThe profit P is then the difference between income and costs So\nP = I \u2013 C\n= sx \u2013 (a + bx)\n= (s \u2013 b) x \u2013 a (3)\nWe now have a mathematical model of the relationships (1)  to  (3) between\nthe variables x, C, I, P, a, b, s These variables may be classified as:\nindependent\nx\ndependent\nC, I, P\nparameters\na, b, s\nThe manufacturer, knowing x, a, b, s can determine P"
    },
    {
        "Chapter": "1",
        "sentence_range": "292-295",
        "Text": "So\nP = I \u2013 C\n= sx \u2013 (a + bx)\n= (s \u2013 b) x \u2013 a (3)\nWe now have a mathematical model of the relationships (1)  to  (3) between\nthe variables x, C, I, P, a, b, s These variables may be classified as:\nindependent\nx\ndependent\nC, I, P\nparameters\na, b, s\nThe manufacturer, knowing x, a, b, s can determine P Step 3 From (3), we can observe that for the break even point (i"
    },
    {
        "Chapter": "1",
        "sentence_range": "293-296",
        "Text": "(3)\nWe now have a mathematical model of the relationships (1)  to  (3) between\nthe variables x, C, I, P, a, b, s These variables may be classified as:\nindependent\nx\ndependent\nC, I, P\nparameters\na, b, s\nThe manufacturer, knowing x, a, b, s can determine P Step 3 From (3), we can observe that for the break even point (i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "294-297",
        "Text": "These variables may be classified as:\nindependent\nx\ndependent\nC, I, P\nparameters\na, b, s\nThe manufacturer, knowing x, a, b, s can determine P Step 3 From (3), we can observe that for the break even point (i e , make neither profit\nnor loss), he must have P = 0, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "295-298",
        "Text": "Step 3 From (3), we can observe that for the break even point (i e , make neither profit\nnor loss), he must have P = 0, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "296-299",
        "Text": "e , make neither profit\nnor loss), he must have P = 0, i e , \nunits"
    },
    {
        "Chapter": "1",
        "sentence_range": "297-300",
        "Text": ", make neither profit\nnor loss), he must have P = 0, i e , \nunits a\nx\ns\nb\n= \u2212\nSteps 4 and 5 In view of the break even point, one may conclude that if the company\nproduces few units, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "298-301",
        "Text": "e , \nunits a\nx\ns\nb\n= \u2212\nSteps 4 and 5 In view of the break even point, one may conclude that if the company\nproduces few units, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "299-302",
        "Text": ", \nunits a\nx\ns\nb\n= \u2212\nSteps 4 and 5 In view of the break even point, one may conclude that if the company\nproduces few units, i e , less than \nunits\na\nx\ns b\n= \u2212\n,  then the company will suffer loss\nRationalised 2023-24\nMATHEMATICAL MODELLING        205\nand if it produces large number of units, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "300-303",
        "Text": "a\nx\ns\nb\n= \u2212\nSteps 4 and 5 In view of the break even point, one may conclude that if the company\nproduces few units, i e , less than \nunits\na\nx\ns b\n= \u2212\n,  then the company will suffer loss\nRationalised 2023-24\nMATHEMATICAL MODELLING        205\nand if it produces large number of units, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "301-304",
        "Text": "e , less than \nunits\na\nx\ns b\n= \u2212\n,  then the company will suffer loss\nRationalised 2023-24\nMATHEMATICAL MODELLING        205\nand if it produces large number of units, i e , much more than \nunits\na\ns b\n\u2212\n, then it can\nmake huge profit"
    },
    {
        "Chapter": "1",
        "sentence_range": "302-305",
        "Text": ", less than \nunits\na\nx\ns b\n= \u2212\n,  then the company will suffer loss\nRationalised 2023-24\nMATHEMATICAL MODELLING        205\nand if it produces large number of units, i e , much more than \nunits\na\ns b\n\u2212\n, then it can\nmake huge profit Further, if the break even point proves to be unrealistic, then another\nmodel could be tried or the assumptions regarding cash flow may be modified"
    },
    {
        "Chapter": "1",
        "sentence_range": "303-306",
        "Text": "e , much more than \nunits\na\ns b\n\u2212\n, then it can\nmake huge profit Further, if the break even point proves to be unrealistic, then another\nmodel could be tried or the assumptions regarding cash flow may be modified Remark From (3), we also have\ndP\ns\nb\nd x =\n\u2212\nThis means that rate of change of P with respect to x depends on the quantity\ns \u2013 b, which is the difference of selling price and the variable cost of each product"
    },
    {
        "Chapter": "1",
        "sentence_range": "304-307",
        "Text": ", much more than \nunits\na\ns b\n\u2212\n, then it can\nmake huge profit Further, if the break even point proves to be unrealistic, then another\nmodel could be tried or the assumptions regarding cash flow may be modified Remark From (3), we also have\ndP\ns\nb\nd x =\n\u2212\nThis means that rate of change of P with respect to x depends on the quantity\ns \u2013 b, which is the difference of selling price and the variable cost of each product Thus, in order to gain profit, this should be positive and to get large gains, we need to\nproduce large quantity of the product and at the same time try to reduce the variable\ncost"
    },
    {
        "Chapter": "1",
        "sentence_range": "305-308",
        "Text": "Further, if the break even point proves to be unrealistic, then another\nmodel could be tried or the assumptions regarding cash flow may be modified Remark From (3), we also have\ndP\ns\nb\nd x =\n\u2212\nThis means that rate of change of P with respect to x depends on the quantity\ns \u2013 b, which is the difference of selling price and the variable cost of each product Thus, in order to gain profit, this should be positive and to get large gains, we need to\nproduce large quantity of the product and at the same time try to reduce the variable\ncost Example 7 Let a tank contains 1000 litres of brine which contains 250 g of salt per\nlitre"
    },
    {
        "Chapter": "1",
        "sentence_range": "306-309",
        "Text": "Remark From (3), we also have\ndP\ns\nb\nd x =\n\u2212\nThis means that rate of change of P with respect to x depends on the quantity\ns \u2013 b, which is the difference of selling price and the variable cost of each product Thus, in order to gain profit, this should be positive and to get large gains, we need to\nproduce large quantity of the product and at the same time try to reduce the variable\ncost Example 7 Let a tank contains 1000 litres of brine which contains 250 g of salt per\nlitre Brine containing 200 g of salt per litre flows into the tank at the rate of 25 litres per\nminute and the mixture flows out at the same rate"
    },
    {
        "Chapter": "1",
        "sentence_range": "307-310",
        "Text": "Thus, in order to gain profit, this should be positive and to get large gains, we need to\nproduce large quantity of the product and at the same time try to reduce the variable\ncost Example 7 Let a tank contains 1000 litres of brine which contains 250 g of salt per\nlitre Brine containing 200 g of salt per litre flows into the tank at the rate of 25 litres per\nminute and the mixture flows out at the same rate Assume that the mixture is kept\nuniform all the time by stirring"
    },
    {
        "Chapter": "1",
        "sentence_range": "308-311",
        "Text": "Example 7 Let a tank contains 1000 litres of brine which contains 250 g of salt per\nlitre Brine containing 200 g of salt per litre flows into the tank at the rate of 25 litres per\nminute and the mixture flows out at the same rate Assume that the mixture is kept\nuniform all the time by stirring What would be the amount of salt in the tank at\nany time t"
    },
    {
        "Chapter": "1",
        "sentence_range": "309-312",
        "Text": "Brine containing 200 g of salt per litre flows into the tank at the rate of 25 litres per\nminute and the mixture flows out at the same rate Assume that the mixture is kept\nuniform all the time by stirring What would be the amount of salt in the tank at\nany time t Solution Step 1 The situation is easily identifiable"
    },
    {
        "Chapter": "1",
        "sentence_range": "310-313",
        "Text": "Assume that the mixture is kept\nuniform all the time by stirring What would be the amount of salt in the tank at\nany time t Solution Step 1 The situation is easily identifiable Step 2 Let y = y (t) denote the amount of salt (in kg) in the tank at time t (in minutes)\nafter the inflow, outflow starts"
    },
    {
        "Chapter": "1",
        "sentence_range": "311-314",
        "Text": "What would be the amount of salt in the tank at\nany time t Solution Step 1 The situation is easily identifiable Step 2 Let y = y (t) denote the amount of salt (in kg) in the tank at time t (in minutes)\nafter the inflow, outflow starts Further assume that y is a differentiable function"
    },
    {
        "Chapter": "1",
        "sentence_range": "312-315",
        "Text": "Solution Step 1 The situation is easily identifiable Step 2 Let y = y (t) denote the amount of salt (in kg) in the tank at time t (in minutes)\nafter the inflow, outflow starts Further assume that y is a differentiable function When t = 0, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "313-316",
        "Text": "Step 2 Let y = y (t) denote the amount of salt (in kg) in the tank at time t (in minutes)\nafter the inflow, outflow starts Further assume that y is a differentiable function When t = 0, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "314-317",
        "Text": "Further assume that y is a differentiable function When t = 0, i e , before the inflow\u2013outflow of the brine starts,\n y = 250 g \u00d7 1000 = 250 kg\nNote that the change in y occurs due to the inflow, outflow of the mixture"
    },
    {
        "Chapter": "1",
        "sentence_range": "315-318",
        "Text": "When t = 0, i e , before the inflow\u2013outflow of the brine starts,\n y = 250 g \u00d7 1000 = 250 kg\nNote that the change in y occurs due to the inflow, outflow of the mixture Now the inflow of brine brings salt into the tank at the rate of 5 kg per minute\n(as 25 \u00d7 200 g = 5 kg) and the outflow of brine takes salt out of the tank at the rate of\n25 1000\n40\ny\ny\n\uf8ed\uf8ec\uf8eb\n\uf8f6\n\uf8f8\uf8f7 =\n kg per minute (as at time t, the salt in the tank is 1000\ny\nkg)"
    },
    {
        "Chapter": "1",
        "sentence_range": "316-319",
        "Text": "e , before the inflow\u2013outflow of the brine starts,\n y = 250 g \u00d7 1000 = 250 kg\nNote that the change in y occurs due to the inflow, outflow of the mixture Now the inflow of brine brings salt into the tank at the rate of 5 kg per minute\n(as 25 \u00d7 200 g = 5 kg) and the outflow of brine takes salt out of the tank at the rate of\n25 1000\n40\ny\ny\n\uf8ed\uf8ec\uf8eb\n\uf8f6\n\uf8f8\uf8f7 =\n kg per minute (as at time t, the salt in the tank is 1000\ny\nkg) Thus, the rate of change of salt with respect to t is given by\ndy\ndt  = 5\n\u2212y40\n(Why"
    },
    {
        "Chapter": "1",
        "sentence_range": "317-320",
        "Text": ", before the inflow\u2013outflow of the brine starts,\n y = 250 g \u00d7 1000 = 250 kg\nNote that the change in y occurs due to the inflow, outflow of the mixture Now the inflow of brine brings salt into the tank at the rate of 5 kg per minute\n(as 25 \u00d7 200 g = 5 kg) and the outflow of brine takes salt out of the tank at the rate of\n25 1000\n40\ny\ny\n\uf8ed\uf8ec\uf8eb\n\uf8f6\n\uf8f8\uf8f7 =\n kg per minute (as at time t, the salt in the tank is 1000\ny\nkg) Thus, the rate of change of salt with respect to t is given by\ndy\ndt  = 5\n\u2212y40\n(Why )\nor\n1\n40\ndy\ny\ndt +\n = 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "318-321",
        "Text": "Now the inflow of brine brings salt into the tank at the rate of 5 kg per minute\n(as 25 \u00d7 200 g = 5 kg) and the outflow of brine takes salt out of the tank at the rate of\n25 1000\n40\ny\ny\n\uf8ed\uf8ec\uf8eb\n\uf8f6\n\uf8f8\uf8f7 =\n kg per minute (as at time t, the salt in the tank is 1000\ny\nkg) Thus, the rate of change of salt with respect to t is given by\ndy\ndt  = 5\n\u2212y40\n(Why )\nor\n1\n40\ndy\ny\ndt +\n = 5 (1)\nRationalised 2023-24\n206\nMATHEMATICS\nThis gives a mathematical model for the given problem"
    },
    {
        "Chapter": "1",
        "sentence_range": "319-322",
        "Text": "Thus, the rate of change of salt with respect to t is given by\ndy\ndt  = 5\n\u2212y40\n(Why )\nor\n1\n40\ndy\ny\ndt +\n = 5 (1)\nRationalised 2023-24\n206\nMATHEMATICS\nThis gives a mathematical model for the given problem Step 3 Equation (1) is a linear equation and can be easily solved"
    },
    {
        "Chapter": "1",
        "sentence_range": "320-323",
        "Text": ")\nor\n1\n40\ndy\ny\ndt +\n = 5 (1)\nRationalised 2023-24\n206\nMATHEMATICS\nThis gives a mathematical model for the given problem Step 3 Equation (1) is a linear equation and can be easily solved The solution of (1) is\ngiven by\n40\n40\n200\nC\nt\nt\nye\ne\n=\n+\n or y (t) = 200 + C \n40\nt\ne\n\u2212"
    },
    {
        "Chapter": "1",
        "sentence_range": "321-324",
        "Text": "(1)\nRationalised 2023-24\n206\nMATHEMATICS\nThis gives a mathematical model for the given problem Step 3 Equation (1) is a linear equation and can be easily solved The solution of (1) is\ngiven by\n40\n40\n200\nC\nt\nt\nye\ne\n=\n+\n or y (t) = 200 + C \n40\nt\ne\n\u2212 (2)\nwhere, c is the constant of integration"
    },
    {
        "Chapter": "1",
        "sentence_range": "322-325",
        "Text": "Step 3 Equation (1) is a linear equation and can be easily solved The solution of (1) is\ngiven by\n40\n40\n200\nC\nt\nt\nye\ne\n=\n+\n or y (t) = 200 + C \n40\nt\ne\n\u2212 (2)\nwhere, c is the constant of integration Note that when t = 0, y = 250"
    },
    {
        "Chapter": "1",
        "sentence_range": "323-326",
        "Text": "The solution of (1) is\ngiven by\n40\n40\n200\nC\nt\nt\nye\ne\n=\n+\n or y (t) = 200 + C \n40\nt\ne\n\u2212 (2)\nwhere, c is the constant of integration Note that when t = 0, y = 250 Therefore, 250 = 200 + C\nor\nC = 50\nThen (2) reduces to\ny = 200 + 50 \n40\nt\ne\n\u2212"
    },
    {
        "Chapter": "1",
        "sentence_range": "324-327",
        "Text": "(2)\nwhere, c is the constant of integration Note that when t = 0, y = 250 Therefore, 250 = 200 + C\nor\nC = 50\nThen (2) reduces to\ny = 200 + 50 \n40\nt\ne\n\u2212 (3)\nor\n200\n50\ny\u2212\n = \n40\nt\ne\n\u2212\nor\n40\nt\ne\n = \n50\n200\ny\u2212\nTherefore\nt = 40\n50\n200\nloge\n\uf8ed\uf8ec\uf8eby \u2212\n\uf8f8\uf8f7\uf8f6"
    },
    {
        "Chapter": "1",
        "sentence_range": "325-328",
        "Text": "Note that when t = 0, y = 250 Therefore, 250 = 200 + C\nor\nC = 50\nThen (2) reduces to\ny = 200 + 50 \n40\nt\ne\n\u2212 (3)\nor\n200\n50\ny\u2212\n = \n40\nt\ne\n\u2212\nor\n40\nt\ne\n = \n50\n200\ny\u2212\nTherefore\nt = 40\n50\n200\nloge\n\uf8ed\uf8ec\uf8eby \u2212\n\uf8f8\uf8f7\uf8f6 (4)\nHere, the equation (4) gives the time t at which the salt in tank is y kg"
    },
    {
        "Chapter": "1",
        "sentence_range": "326-329",
        "Text": "Therefore, 250 = 200 + C\nor\nC = 50\nThen (2) reduces to\ny = 200 + 50 \n40\nt\ne\n\u2212 (3)\nor\n200\n50\ny\u2212\n = \n40\nt\ne\n\u2212\nor\n40\nt\ne\n = \n50\n200\ny\u2212\nTherefore\nt = 40\n50\n200\nloge\n\uf8ed\uf8ec\uf8eby \u2212\n\uf8f8\uf8f7\uf8f6 (4)\nHere, the equation (4) gives the time t at which the salt in tank is y kg Step 4 Since \n40\nt\ne\n\u2212\n is always positive, from (3), we conclude that y > 200 at all times\nThus, the minimum amount of salt content in the tank is 200 kg"
    },
    {
        "Chapter": "1",
        "sentence_range": "327-330",
        "Text": "(3)\nor\n200\n50\ny\u2212\n = \n40\nt\ne\n\u2212\nor\n40\nt\ne\n = \n50\n200\ny\u2212\nTherefore\nt = 40\n50\n200\nloge\n\uf8ed\uf8ec\uf8eby \u2212\n\uf8f8\uf8f7\uf8f6 (4)\nHere, the equation (4) gives the time t at which the salt in tank is y kg Step 4 Since \n40\nt\ne\n\u2212\n is always positive, from (3), we conclude that y > 200 at all times\nThus, the minimum amount of salt content in the tank is 200 kg Also, from (4), we conclude that t > 0 if and only if 0 < y \u2013 200 < 50 i"
    },
    {
        "Chapter": "1",
        "sentence_range": "328-331",
        "Text": "(4)\nHere, the equation (4) gives the time t at which the salt in tank is y kg Step 4 Since \n40\nt\ne\n\u2212\n is always positive, from (3), we conclude that y > 200 at all times\nThus, the minimum amount of salt content in the tank is 200 kg Also, from (4), we conclude that t > 0 if and only if 0 < y \u2013 200 < 50 i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "329-332",
        "Text": "Step 4 Since \n40\nt\ne\n\u2212\n is always positive, from (3), we conclude that y > 200 at all times\nThus, the minimum amount of salt content in the tank is 200 kg Also, from (4), we conclude that t > 0 if and only if 0 < y \u2013 200 < 50 i e , if and only\nif 200 < y < 250 i"
    },
    {
        "Chapter": "1",
        "sentence_range": "330-333",
        "Text": "Also, from (4), we conclude that t > 0 if and only if 0 < y \u2013 200 < 50 i e , if and only\nif 200 < y < 250 i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "331-334",
        "Text": "e , if and only\nif 200 < y < 250 i e , the amount of salt content in the tank after the start of inflow and\noutflow of the brine is between 200 kg and 250 kg"
    },
    {
        "Chapter": "1",
        "sentence_range": "332-335",
        "Text": ", if and only\nif 200 < y < 250 i e , the amount of salt content in the tank after the start of inflow and\noutflow of the brine is between 200 kg and 250 kg Limitations of Mathematical Modelling\nTill today many mathematical models have been developed and applied successfully\nto understand and get an insight into thousands of situations"
    },
    {
        "Chapter": "1",
        "sentence_range": "333-336",
        "Text": "e , the amount of salt content in the tank after the start of inflow and\noutflow of the brine is between 200 kg and 250 kg Limitations of Mathematical Modelling\nTill today many mathematical models have been developed and applied successfully\nto understand and get an insight into thousands of situations Some of the subjects like\nmathematical physics, mathematical economics, operations research, bio-mathematics\netc"
    },
    {
        "Chapter": "1",
        "sentence_range": "334-337",
        "Text": ", the amount of salt content in the tank after the start of inflow and\noutflow of the brine is between 200 kg and 250 kg Limitations of Mathematical Modelling\nTill today many mathematical models have been developed and applied successfully\nto understand and get an insight into thousands of situations Some of the subjects like\nmathematical physics, mathematical economics, operations research, bio-mathematics\netc are almost synonymous with mathematical modelling"
    },
    {
        "Chapter": "1",
        "sentence_range": "335-338",
        "Text": "Limitations of Mathematical Modelling\nTill today many mathematical models have been developed and applied successfully\nto understand and get an insight into thousands of situations Some of the subjects like\nmathematical physics, mathematical economics, operations research, bio-mathematics\netc are almost synonymous with mathematical modelling But there are still a large number of situations which are yet to be modelled"
    },
    {
        "Chapter": "1",
        "sentence_range": "336-339",
        "Text": "Some of the subjects like\nmathematical physics, mathematical economics, operations research, bio-mathematics\netc are almost synonymous with mathematical modelling But there are still a large number of situations which are yet to be modelled The\nreason behind this is that either the situation are found to be very complex or the\nmathematical models formed are mathematically intractable"
    },
    {
        "Chapter": "1",
        "sentence_range": "337-340",
        "Text": "are almost synonymous with mathematical modelling But there are still a large number of situations which are yet to be modelled The\nreason behind this is that either the situation are found to be very complex or the\nmathematical models formed are mathematically intractable Rationalised 2023-24\nMATHEMATICAL MODELLING        207\nThe development of  the powerful computers and super computers has enabled us\nto mathematically model a large number of situations (even complex situations)"
    },
    {
        "Chapter": "1",
        "sentence_range": "338-341",
        "Text": "But there are still a large number of situations which are yet to be modelled The\nreason behind this is that either the situation are found to be very complex or the\nmathematical models formed are mathematically intractable Rationalised 2023-24\nMATHEMATICAL MODELLING        207\nThe development of  the powerful computers and super computers has enabled us\nto mathematically model a large number of situations (even complex situations) Due\nto these fast and advanced computers, it has been possible to prepare more realistic\nmodels which can obtain better agreements with observations"
    },
    {
        "Chapter": "1",
        "sentence_range": "339-342",
        "Text": "The\nreason behind this is that either the situation are found to be very complex or the\nmathematical models formed are mathematically intractable Rationalised 2023-24\nMATHEMATICAL MODELLING        207\nThe development of  the powerful computers and super computers has enabled us\nto mathematically model a large number of situations (even complex situations) Due\nto these fast and advanced computers, it has been possible to prepare more realistic\nmodels which can obtain better agreements with observations However, we do not have good guidelines for choosing various parameters / variables\nand also for estimating the values of these parameters / variables used in a mathematical\nmodel"
    },
    {
        "Chapter": "1",
        "sentence_range": "340-343",
        "Text": "Rationalised 2023-24\nMATHEMATICAL MODELLING        207\nThe development of  the powerful computers and super computers has enabled us\nto mathematically model a large number of situations (even complex situations) Due\nto these fast and advanced computers, it has been possible to prepare more realistic\nmodels which can obtain better agreements with observations However, we do not have good guidelines for choosing various parameters / variables\nand also for estimating the values of these parameters / variables used in a mathematical\nmodel Infact, we can prepare reasonably accurate models to fit any data by choosing\nfive or six parameters / variables"
    },
    {
        "Chapter": "1",
        "sentence_range": "341-344",
        "Text": "Due\nto these fast and advanced computers, it has been possible to prepare more realistic\nmodels which can obtain better agreements with observations However, we do not have good guidelines for choosing various parameters / variables\nand also for estimating the values of these parameters / variables used in a mathematical\nmodel Infact, we can prepare reasonably accurate models to fit any data by choosing\nfive or six parameters / variables We require a minimal number of parameters / variables\nto be able to estimate them accurately"
    },
    {
        "Chapter": "1",
        "sentence_range": "342-345",
        "Text": "However, we do not have good guidelines for choosing various parameters / variables\nand also for estimating the values of these parameters / variables used in a mathematical\nmodel Infact, we can prepare reasonably accurate models to fit any data by choosing\nfive or six parameters / variables We require a minimal number of parameters / variables\nto be able to estimate them accurately Mathematical modelling of large or complex situations has its own special problems"
    },
    {
        "Chapter": "1",
        "sentence_range": "343-346",
        "Text": "Infact, we can prepare reasonably accurate models to fit any data by choosing\nfive or six parameters / variables We require a minimal number of parameters / variables\nto be able to estimate them accurately Mathematical modelling of large or complex situations has its own special problems These type of situations usually occur in the study of world models of environment,\noceanography, pollution control etc"
    },
    {
        "Chapter": "1",
        "sentence_range": "344-347",
        "Text": "We require a minimal number of parameters / variables\nto be able to estimate them accurately Mathematical modelling of large or complex situations has its own special problems These type of situations usually occur in the study of world models of environment,\noceanography, pollution control etc Mathematical modellers from all disciplines \u2014\nmathematics, computer science, physics, engineering, social sciences, etc"
    },
    {
        "Chapter": "1",
        "sentence_range": "345-348",
        "Text": "Mathematical modelling of large or complex situations has its own special problems These type of situations usually occur in the study of world models of environment,\noceanography, pollution control etc Mathematical modellers from all disciplines \u2014\nmathematics, computer science, physics, engineering, social sciences, etc , are involved\nin meeting these challenges with courage"
    },
    {
        "Chapter": "1",
        "sentence_range": "346-349",
        "Text": "These type of situations usually occur in the study of world models of environment,\noceanography, pollution control etc Mathematical modellers from all disciplines \u2014\nmathematics, computer science, physics, engineering, social sciences, etc , are involved\nin meeting these challenges with courage \u2014v\nv\nv\nv\nv\u2014\nRationalised 2023-24\nvThere is no permanent place in the world for ugly mathematics"
    },
    {
        "Chapter": "1",
        "sentence_range": "347-350",
        "Text": "Mathematical modellers from all disciplines \u2014\nmathematics, computer science, physics, engineering, social sciences, etc , are involved\nin meeting these challenges with courage \u2014v\nv\nv\nv\nv\u2014\nRationalised 2023-24\nvThere is no permanent place in the world for ugly mathematics It may\nbe very hard to define mathematical beauty but that is just as true of\nbeauty of any kind, we may not know quite what we mean by a\nbeautiful poem, but that does not prevent us from recognising\none when we read it"
    },
    {
        "Chapter": "1",
        "sentence_range": "348-351",
        "Text": ", are involved\nin meeting these challenges with courage \u2014v\nv\nv\nv\nv\u2014\nRationalised 2023-24\nvThere is no permanent place in the world for ugly mathematics It may\nbe very hard to define mathematical beauty but that is just as true of\nbeauty of any kind, we may not know quite what we mean by a\nbeautiful poem, but that does not prevent us from recognising\none when we read it \u2014 G"
    },
    {
        "Chapter": "1",
        "sentence_range": "349-352",
        "Text": "\u2014v\nv\nv\nv\nv\u2014\nRationalised 2023-24\nvThere is no permanent place in the world for ugly mathematics It may\nbe very hard to define mathematical beauty but that is just as true of\nbeauty of any kind, we may not know quite what we mean by a\nbeautiful poem, but that does not prevent us from recognising\none when we read it \u2014 G H"
    },
    {
        "Chapter": "1",
        "sentence_range": "350-353",
        "Text": "It may\nbe very hard to define mathematical beauty but that is just as true of\nbeauty of any kind, we may not know quite what we mean by a\nbeautiful poem, but that does not prevent us from recognising\none when we read it \u2014 G H HARDY v\n1"
    },
    {
        "Chapter": "1",
        "sentence_range": "351-354",
        "Text": "\u2014 G H HARDY v\n1 1  Introduction\nRecall that the notion of relations and functions, domain,\nco-domain and range have been introduced in Class XI\nalong with different types of specific real valued functions\nand their graphs"
    },
    {
        "Chapter": "1",
        "sentence_range": "352-355",
        "Text": "H HARDY v\n1 1  Introduction\nRecall that the notion of relations and functions, domain,\nco-domain and range have been introduced in Class XI\nalong with different types of specific real valued functions\nand their graphs The concept of the term \u2018relation\u2019 in\nmathematics has been drawn from the meaning of relation\nin English language, according to which two objects or\nquantities are related if there is a recognisable connection\nor link between the two objects or quantities"
    },
    {
        "Chapter": "1",
        "sentence_range": "353-356",
        "Text": "HARDY v\n1 1  Introduction\nRecall that the notion of relations and functions, domain,\nco-domain and range have been introduced in Class XI\nalong with different types of specific real valued functions\nand their graphs The concept of the term \u2018relation\u2019 in\nmathematics has been drawn from the meaning of relation\nin English language, according to which two objects or\nquantities are related if there is a recognisable connection\nor link between the two objects or quantities Let  A be\nthe set of students of Class XII of a school and B be the\nset of students of Class XI of the same school"
    },
    {
        "Chapter": "1",
        "sentence_range": "354-357",
        "Text": "1  Introduction\nRecall that the notion of relations and functions, domain,\nco-domain and range have been introduced in Class XI\nalong with different types of specific real valued functions\nand their graphs The concept of the term \u2018relation\u2019 in\nmathematics has been drawn from the meaning of relation\nin English language, according to which two objects or\nquantities are related if there is a recognisable connection\nor link between the two objects or quantities Let  A be\nthe set of students of Class XII of a school and B be the\nset of students of Class XI of the same school Then some\nof the examples of relations from A to B are\n(i)\n{(a, b) \u2208 A \u00d7 B: a is brother of b},\n(ii)\n{(a, b) \u2208 A \u00d7 B: a is sister of b},\n(iii)\n{(a, b) \u2208 A \u00d7 B: age of a is greater than age of b},\n(iv)\n{(a, b) \u2208 A \u00d7 B: total marks obtained by a in the final examination is less than\nthe total marks obtained by b in the final examination},\n(v)\n{(a, b) \u2208 A \u00d7 B: a lives in the same locality as b}"
    },
    {
        "Chapter": "1",
        "sentence_range": "355-358",
        "Text": "The concept of the term \u2018relation\u2019 in\nmathematics has been drawn from the meaning of relation\nin English language, according to which two objects or\nquantities are related if there is a recognisable connection\nor link between the two objects or quantities Let  A be\nthe set of students of Class XII of a school and B be the\nset of students of Class XI of the same school Then some\nof the examples of relations from A to B are\n(i)\n{(a, b) \u2208 A \u00d7 B: a is brother of b},\n(ii)\n{(a, b) \u2208 A \u00d7 B: a is sister of b},\n(iii)\n{(a, b) \u2208 A \u00d7 B: age of a is greater than age of b},\n(iv)\n{(a, b) \u2208 A \u00d7 B: total marks obtained by a in the final examination is less than\nthe total marks obtained by b in the final examination},\n(v)\n{(a, b) \u2208 A \u00d7 B: a lives in the same locality as b} However, abstracting from\nthis, we define mathematically a relation R from A to B as an arbitrary subset\nof A \u00d7 B"
    },
    {
        "Chapter": "1",
        "sentence_range": "356-359",
        "Text": "Let  A be\nthe set of students of Class XII of a school and B be the\nset of students of Class XI of the same school Then some\nof the examples of relations from A to B are\n(i)\n{(a, b) \u2208 A \u00d7 B: a is brother of b},\n(ii)\n{(a, b) \u2208 A \u00d7 B: a is sister of b},\n(iii)\n{(a, b) \u2208 A \u00d7 B: age of a is greater than age of b},\n(iv)\n{(a, b) \u2208 A \u00d7 B: total marks obtained by a in the final examination is less than\nthe total marks obtained by b in the final examination},\n(v)\n{(a, b) \u2208 A \u00d7 B: a lives in the same locality as b} However, abstracting from\nthis, we define mathematically a relation R from A to B as an arbitrary subset\nof A \u00d7 B If (a, b) \u2208 R, we say that a is related to b under the relation R and we write as\na R b"
    },
    {
        "Chapter": "1",
        "sentence_range": "357-360",
        "Text": "Then some\nof the examples of relations from A to B are\n(i)\n{(a, b) \u2208 A \u00d7 B: a is brother of b},\n(ii)\n{(a, b) \u2208 A \u00d7 B: a is sister of b},\n(iii)\n{(a, b) \u2208 A \u00d7 B: age of a is greater than age of b},\n(iv)\n{(a, b) \u2208 A \u00d7 B: total marks obtained by a in the final examination is less than\nthe total marks obtained by b in the final examination},\n(v)\n{(a, b) \u2208 A \u00d7 B: a lives in the same locality as b} However, abstracting from\nthis, we define mathematically a relation R from A to B as an arbitrary subset\nof A \u00d7 B If (a, b) \u2208 R, we say that a is related to b under the relation R and we write as\na R b In general, (a, b) \u2208 R, we do not bother whether there is a recognisable\nconnection or link between a and b"
    },
    {
        "Chapter": "1",
        "sentence_range": "358-361",
        "Text": "However, abstracting from\nthis, we define mathematically a relation R from A to B as an arbitrary subset\nof A \u00d7 B If (a, b) \u2208 R, we say that a is related to b under the relation R and we write as\na R b In general, (a, b) \u2208 R, we do not bother whether there is a recognisable\nconnection or link between a and b As seen in Class XI, functions are special kind of\nrelations"
    },
    {
        "Chapter": "1",
        "sentence_range": "359-362",
        "Text": "If (a, b) \u2208 R, we say that a is related to b under the relation R and we write as\na R b In general, (a, b) \u2208 R, we do not bother whether there is a recognisable\nconnection or link between a and b As seen in Class XI, functions are special kind of\nrelations In this chapter, we will study different types of relations and functions, composition\nof functions, invertible functions and binary operations"
    },
    {
        "Chapter": "1",
        "sentence_range": "360-363",
        "Text": "In general, (a, b) \u2208 R, we do not bother whether there is a recognisable\nconnection or link between a and b As seen in Class XI, functions are special kind of\nrelations In this chapter, we will study different types of relations and functions, composition\nof functions, invertible functions and binary operations Chapter 1\nRELATIONS AND FUNCTIONS\nLejeune Dirichlet\n (1805-1859)\nRationalised 2023-24\nMATHEMATICS\n2\n1"
    },
    {
        "Chapter": "1",
        "sentence_range": "361-364",
        "Text": "As seen in Class XI, functions are special kind of\nrelations In this chapter, we will study different types of relations and functions, composition\nof functions, invertible functions and binary operations Chapter 1\nRELATIONS AND FUNCTIONS\nLejeune Dirichlet\n (1805-1859)\nRationalised 2023-24\nMATHEMATICS\n2\n1 2  Types of Relations\nIn this section, we would like to study different types of relations"
    },
    {
        "Chapter": "1",
        "sentence_range": "362-365",
        "Text": "In this chapter, we will study different types of relations and functions, composition\nof functions, invertible functions and binary operations Chapter 1\nRELATIONS AND FUNCTIONS\nLejeune Dirichlet\n (1805-1859)\nRationalised 2023-24\nMATHEMATICS\n2\n1 2  Types of Relations\nIn this section, we would like to study different types of relations We know that a\nrelation in a set A is a subset of A \u00d7 A"
    },
    {
        "Chapter": "1",
        "sentence_range": "363-366",
        "Text": "Chapter 1\nRELATIONS AND FUNCTIONS\nLejeune Dirichlet\n (1805-1859)\nRationalised 2023-24\nMATHEMATICS\n2\n1 2  Types of Relations\nIn this section, we would like to study different types of relations We know that a\nrelation in a set A is a subset of A \u00d7 A Thus, the empty set \u03c6 and A \u00d7 A are two\nextreme relations"
    },
    {
        "Chapter": "1",
        "sentence_range": "364-367",
        "Text": "2  Types of Relations\nIn this section, we would like to study different types of relations We know that a\nrelation in a set A is a subset of A \u00d7 A Thus, the empty set \u03c6 and A \u00d7 A are two\nextreme relations For illustration, consider a relation R in the set A = {1, 2, 3, 4} given by\nR = {(a, b): a \u2013 b = 10}"
    },
    {
        "Chapter": "1",
        "sentence_range": "365-368",
        "Text": "We know that a\nrelation in a set A is a subset of A \u00d7 A Thus, the empty set \u03c6 and A \u00d7 A are two\nextreme relations For illustration, consider a relation R in the set A = {1, 2, 3, 4} given by\nR = {(a, b): a \u2013 b = 10} This is the empty set, as no pair (a, b) satisfies the condition\na \u2013 b = 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "366-369",
        "Text": "Thus, the empty set \u03c6 and A \u00d7 A are two\nextreme relations For illustration, consider a relation R in the set A = {1, 2, 3, 4} given by\nR = {(a, b): a \u2013 b = 10} This is the empty set, as no pair (a, b) satisfies the condition\na \u2013 b = 10 Similarly, R\u2032 = {(a, b) : | a \u2013 b | \u2265 0} is the whole set A \u00d7 A, as all pairs\n(a, b) in A \u00d7 A satisfy | a \u2013 b | \u2265 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "367-370",
        "Text": "For illustration, consider a relation R in the set A = {1, 2, 3, 4} given by\nR = {(a, b): a \u2013 b = 10} This is the empty set, as no pair (a, b) satisfies the condition\na \u2013 b = 10 Similarly, R\u2032 = {(a, b) : | a \u2013 b | \u2265 0} is the whole set A \u00d7 A, as all pairs\n(a, b) in A \u00d7 A satisfy | a \u2013 b | \u2265 0 These two extreme examples lead us to the\nfollowing definitions"
    },
    {
        "Chapter": "1",
        "sentence_range": "368-371",
        "Text": "This is the empty set, as no pair (a, b) satisfies the condition\na \u2013 b = 10 Similarly, R\u2032 = {(a, b) : | a \u2013 b | \u2265 0} is the whole set A \u00d7 A, as all pairs\n(a, b) in A \u00d7 A satisfy | a \u2013 b | \u2265 0 These two extreme examples lead us to the\nfollowing definitions Definition 1 A relation R in a set A is called empty relation, if no element of A is\nrelated to any element of A, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "369-372",
        "Text": "Similarly, R\u2032 = {(a, b) : | a \u2013 b | \u2265 0} is the whole set A \u00d7 A, as all pairs\n(a, b) in A \u00d7 A satisfy | a \u2013 b | \u2265 0 These two extreme examples lead us to the\nfollowing definitions Definition 1 A relation R in a set A is called empty relation, if no element of A is\nrelated to any element of A, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "370-373",
        "Text": "These two extreme examples lead us to the\nfollowing definitions Definition 1 A relation R in a set A is called empty relation, if no element of A is\nrelated to any element of A, i e , R = \u03c6 \u2282 A \u00d7 A"
    },
    {
        "Chapter": "1",
        "sentence_range": "371-374",
        "Text": "Definition 1 A relation R in a set A is called empty relation, if no element of A is\nrelated to any element of A, i e , R = \u03c6 \u2282 A \u00d7 A Definition 2  A relation R in a set A is called universal relation, if each element of A\nis related to every element of A, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "372-375",
        "Text": "e , R = \u03c6 \u2282 A \u00d7 A Definition 2  A relation R in a set A is called universal relation, if each element of A\nis related to every element of A, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "373-376",
        "Text": ", R = \u03c6 \u2282 A \u00d7 A Definition 2  A relation R in a set A is called universal relation, if each element of A\nis related to every element of A, i e , R = A \u00d7 A"
    },
    {
        "Chapter": "1",
        "sentence_range": "374-377",
        "Text": "Definition 2  A relation R in a set A is called universal relation, if each element of A\nis related to every element of A, i e , R = A \u00d7 A Both the empty relation and the universal relation are some times called trivial\nrelations"
    },
    {
        "Chapter": "1",
        "sentence_range": "375-378",
        "Text": "e , R = A \u00d7 A Both the empty relation and the universal relation are some times called trivial\nrelations Example 1 Let A be the set of all students of a boys school"
    },
    {
        "Chapter": "1",
        "sentence_range": "376-379",
        "Text": ", R = A \u00d7 A Both the empty relation and the universal relation are some times called trivial\nrelations Example 1 Let A be the set of all students of a boys school Show that the relation R\nin A given by R = {(a, b) : a is sister of b} is the empty relation and R\u2032 = {(a, b) : the\ndifference between heights of a and b is less than 3 meters} is the universal relation"
    },
    {
        "Chapter": "1",
        "sentence_range": "377-380",
        "Text": "Both the empty relation and the universal relation are some times called trivial\nrelations Example 1 Let A be the set of all students of a boys school Show that the relation R\nin A given by R = {(a, b) : a is sister of b} is the empty relation and R\u2032 = {(a, b) : the\ndifference between heights of a and b is less than 3 meters} is the universal relation Solution Since the school is boys school, no student of the school can be sister of any\nstudent of the school"
    },
    {
        "Chapter": "1",
        "sentence_range": "378-381",
        "Text": "Example 1 Let A be the set of all students of a boys school Show that the relation R\nin A given by R = {(a, b) : a is sister of b} is the empty relation and R\u2032 = {(a, b) : the\ndifference between heights of a and b is less than 3 meters} is the universal relation Solution Since the school is boys school, no student of the school can be sister of any\nstudent of the school Hence, R = \u03c6, showing that R is the empty relation"
    },
    {
        "Chapter": "1",
        "sentence_range": "379-382",
        "Text": "Show that the relation R\nin A given by R = {(a, b) : a is sister of b} is the empty relation and R\u2032 = {(a, b) : the\ndifference between heights of a and b is less than 3 meters} is the universal relation Solution Since the school is boys school, no student of the school can be sister of any\nstudent of the school Hence, R = \u03c6, showing that R is the empty relation It is also\nobvious that the difference between heights of any two students of the school has to be\nless than 3 meters"
    },
    {
        "Chapter": "1",
        "sentence_range": "380-383",
        "Text": "Solution Since the school is boys school, no student of the school can be sister of any\nstudent of the school Hence, R = \u03c6, showing that R is the empty relation It is also\nobvious that the difference between heights of any two students of the school has to be\nless than 3 meters This shows that R\u2032 = A \u00d7 A is the universal relation"
    },
    {
        "Chapter": "1",
        "sentence_range": "381-384",
        "Text": "Hence, R = \u03c6, showing that R is the empty relation It is also\nobvious that the difference between heights of any two students of the school has to be\nless than 3 meters This shows that R\u2032 = A \u00d7 A is the universal relation Remark In Class XI, we have seen two ways of representing a relation, namely raster\nmethod and set builder method"
    },
    {
        "Chapter": "1",
        "sentence_range": "382-385",
        "Text": "It is also\nobvious that the difference between heights of any two students of the school has to be\nless than 3 meters This shows that R\u2032 = A \u00d7 A is the universal relation Remark In Class XI, we have seen two ways of representing a relation, namely raster\nmethod and set builder method However, a relation R in the set {1, 2, 3, 4} defined by R\n= {(a, b) : b = a + 1} is also expressed as a R b if and only if\nb = a + 1 by many authors"
    },
    {
        "Chapter": "1",
        "sentence_range": "383-386",
        "Text": "This shows that R\u2032 = A \u00d7 A is the universal relation Remark In Class XI, we have seen two ways of representing a relation, namely raster\nmethod and set builder method However, a relation R in the set {1, 2, 3, 4} defined by R\n= {(a, b) : b = a + 1} is also expressed as a R b if and only if\nb = a + 1 by many authors We may also use this notation, as and when convenient"
    },
    {
        "Chapter": "1",
        "sentence_range": "384-387",
        "Text": "Remark In Class XI, we have seen two ways of representing a relation, namely raster\nmethod and set builder method However, a relation R in the set {1, 2, 3, 4} defined by R\n= {(a, b) : b = a + 1} is also expressed as a R b if and only if\nb = a + 1 by many authors We may also use this notation, as and when convenient If (a, b) \u2208 R, we say that a is related to b and we denote it as a R b"
    },
    {
        "Chapter": "1",
        "sentence_range": "385-388",
        "Text": "However, a relation R in the set {1, 2, 3, 4} defined by R\n= {(a, b) : b = a + 1} is also expressed as a R b if and only if\nb = a + 1 by many authors We may also use this notation, as and when convenient If (a, b) \u2208 R, we say that a is related to b and we denote it as a R b One of the most important relation, which plays a significant role in Mathematics,\nis an equivalence relation"
    },
    {
        "Chapter": "1",
        "sentence_range": "386-389",
        "Text": "We may also use this notation, as and when convenient If (a, b) \u2208 R, we say that a is related to b and we denote it as a R b One of the most important relation, which plays a significant role in Mathematics,\nis an equivalence relation To study equivalence relation, we first consider three\ntypes of relations, namely reflexive, symmetric and transitive"
    },
    {
        "Chapter": "1",
        "sentence_range": "387-390",
        "Text": "If (a, b) \u2208 R, we say that a is related to b and we denote it as a R b One of the most important relation, which plays a significant role in Mathematics,\nis an equivalence relation To study equivalence relation, we first consider three\ntypes of relations, namely reflexive, symmetric and transitive Definition 3 A relation R in a set A is called\n(i)\nreflexive, if (a, a) \u2208 R, for every a \u2208 A,\n(ii)\nsymmetric, if (a1, a2) \u2208 R implies that (a2, a1) \u2208 R, for all a1, a2 \u2208 A"
    },
    {
        "Chapter": "1",
        "sentence_range": "388-391",
        "Text": "One of the most important relation, which plays a significant role in Mathematics,\nis an equivalence relation To study equivalence relation, we first consider three\ntypes of relations, namely reflexive, symmetric and transitive Definition 3 A relation R in a set A is called\n(i)\nreflexive, if (a, a) \u2208 R, for every a \u2208 A,\n(ii)\nsymmetric, if (a1, a2) \u2208 R implies that (a2, a1) \u2208 R, for all a1, a2 \u2208 A (iii)\ntransitive, if (a1, a2) \u2208 R and (a2, a3) \u2208 R implies that (a1, a3) \u2208 R, for all a1, a2,\na3 \u2208 A"
    },
    {
        "Chapter": "1",
        "sentence_range": "389-392",
        "Text": "To study equivalence relation, we first consider three\ntypes of relations, namely reflexive, symmetric and transitive Definition 3 A relation R in a set A is called\n(i)\nreflexive, if (a, a) \u2208 R, for every a \u2208 A,\n(ii)\nsymmetric, if (a1, a2) \u2208 R implies that (a2, a1) \u2208 R, for all a1, a2 \u2208 A (iii)\ntransitive, if (a1, a2) \u2208 R and (a2, a3) \u2208 R implies that (a1, a3) \u2208 R, for all a1, a2,\na3 \u2208 A Rationalised 2023-24\nRELATIONS AND FUNCTIONS\n3\nDefinition 4 A relation R in a set A is said to be an equivalence relation if R is\nreflexive, symmetric and transitive"
    },
    {
        "Chapter": "1",
        "sentence_range": "390-393",
        "Text": "Definition 3 A relation R in a set A is called\n(i)\nreflexive, if (a, a) \u2208 R, for every a \u2208 A,\n(ii)\nsymmetric, if (a1, a2) \u2208 R implies that (a2, a1) \u2208 R, for all a1, a2 \u2208 A (iii)\ntransitive, if (a1, a2) \u2208 R and (a2, a3) \u2208 R implies that (a1, a3) \u2208 R, for all a1, a2,\na3 \u2208 A Rationalised 2023-24\nRELATIONS AND FUNCTIONS\n3\nDefinition 4 A relation R in a set A is said to be an equivalence relation if R is\nreflexive, symmetric and transitive Example 2 Let T be the set of all triangles in a plane with R a relation in T given by\nR = {(T1, T2) : T1 is congruent to T2}"
    },
    {
        "Chapter": "1",
        "sentence_range": "391-394",
        "Text": "(iii)\ntransitive, if (a1, a2) \u2208 R and (a2, a3) \u2208 R implies that (a1, a3) \u2208 R, for all a1, a2,\na3 \u2208 A Rationalised 2023-24\nRELATIONS AND FUNCTIONS\n3\nDefinition 4 A relation R in a set A is said to be an equivalence relation if R is\nreflexive, symmetric and transitive Example 2 Let T be the set of all triangles in a plane with R a relation in T given by\nR = {(T1, T2) : T1 is congruent to T2} Show that R is an equivalence relation"
    },
    {
        "Chapter": "1",
        "sentence_range": "392-395",
        "Text": "Rationalised 2023-24\nRELATIONS AND FUNCTIONS\n3\nDefinition 4 A relation R in a set A is said to be an equivalence relation if R is\nreflexive, symmetric and transitive Example 2 Let T be the set of all triangles in a plane with R a relation in T given by\nR = {(T1, T2) : T1 is congruent to T2} Show that R is an equivalence relation Solution R is reflexive, since every triangle is congruent to itself"
    },
    {
        "Chapter": "1",
        "sentence_range": "393-396",
        "Text": "Example 2 Let T be the set of all triangles in a plane with R a relation in T given by\nR = {(T1, T2) : T1 is congruent to T2} Show that R is an equivalence relation Solution R is reflexive, since every triangle is congruent to itself Further,\n(T1, T2) \u2208 R \u21d2 T1 is congruent to T2 \u21d2 T2 is congruent to T1 \u21d2 (T2, T1) \u2208 R"
    },
    {
        "Chapter": "1",
        "sentence_range": "394-397",
        "Text": "Show that R is an equivalence relation Solution R is reflexive, since every triangle is congruent to itself Further,\n(T1, T2) \u2208 R \u21d2 T1 is congruent to T2 \u21d2 T2 is congruent to T1 \u21d2 (T2, T1) \u2208 R Hence,\nR is symmetric"
    },
    {
        "Chapter": "1",
        "sentence_range": "395-398",
        "Text": "Solution R is reflexive, since every triangle is congruent to itself Further,\n(T1, T2) \u2208 R \u21d2 T1 is congruent to T2 \u21d2 T2 is congruent to T1 \u21d2 (T2, T1) \u2208 R Hence,\nR is symmetric Moreover, (T1, T2), (T2, T3) \u2208 R \u21d2 T1 is  congruent to T2 and T2 is\ncongruent to T3 \u21d2 T1 is congruent to T3 \u21d2 (T1, T3) \u2208 R"
    },
    {
        "Chapter": "1",
        "sentence_range": "396-399",
        "Text": "Further,\n(T1, T2) \u2208 R \u21d2 T1 is congruent to T2 \u21d2 T2 is congruent to T1 \u21d2 (T2, T1) \u2208 R Hence,\nR is symmetric Moreover, (T1, T2), (T2, T3) \u2208 R \u21d2 T1 is  congruent to T2 and T2 is\ncongruent to T3 \u21d2 T1 is congruent to T3 \u21d2 (T1, T3) \u2208 R Therefore, R is an equivalence\nrelation"
    },
    {
        "Chapter": "1",
        "sentence_range": "397-400",
        "Text": "Hence,\nR is symmetric Moreover, (T1, T2), (T2, T3) \u2208 R \u21d2 T1 is  congruent to T2 and T2 is\ncongruent to T3 \u21d2 T1 is congruent to T3 \u21d2 (T1, T3) \u2208 R Therefore, R is an equivalence\nrelation Example 3 Let L be the set of all lines in a plane and R be the relation in L defined as\nR = {(L1, L2) : L1 is perpendicular to L2}"
    },
    {
        "Chapter": "1",
        "sentence_range": "398-401",
        "Text": "Moreover, (T1, T2), (T2, T3) \u2208 R \u21d2 T1 is  congruent to T2 and T2 is\ncongruent to T3 \u21d2 T1 is congruent to T3 \u21d2 (T1, T3) \u2208 R Therefore, R is an equivalence\nrelation Example 3 Let L be the set of all lines in a plane and R be the relation in L defined as\nR = {(L1, L2) : L1 is perpendicular to L2} Show that R is symmetric but neither\nreflexive nor transitive"
    },
    {
        "Chapter": "1",
        "sentence_range": "399-402",
        "Text": "Therefore, R is an equivalence\nrelation Example 3 Let L be the set of all lines in a plane and R be the relation in L defined as\nR = {(L1, L2) : L1 is perpendicular to L2} Show that R is symmetric but neither\nreflexive nor transitive Solution R is not reflexive, as a line L1 can not be perpendicular to itself, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "400-403",
        "Text": "Example 3 Let L be the set of all lines in a plane and R be the relation in L defined as\nR = {(L1, L2) : L1 is perpendicular to L2} Show that R is symmetric but neither\nreflexive nor transitive Solution R is not reflexive, as a line L1 can not be perpendicular to itself, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "401-404",
        "Text": "Show that R is symmetric but neither\nreflexive nor transitive Solution R is not reflexive, as a line L1 can not be perpendicular to itself, i e , (L1, L1)\n\u2209 R"
    },
    {
        "Chapter": "1",
        "sentence_range": "402-405",
        "Text": "Solution R is not reflexive, as a line L1 can not be perpendicular to itself, i e , (L1, L1)\n\u2209 R R is symmetric as (L1, L2) \u2208 R\n\u21d2\nL1 is perpendicular to L2\n\u21d2\nL2 is perpendicular to L1\n\u21d2\n(L2, L1) \u2208 R"
    },
    {
        "Chapter": "1",
        "sentence_range": "403-406",
        "Text": "e , (L1, L1)\n\u2209 R R is symmetric as (L1, L2) \u2208 R\n\u21d2\nL1 is perpendicular to L2\n\u21d2\nL2 is perpendicular to L1\n\u21d2\n(L2, L1) \u2208 R R is not transitive"
    },
    {
        "Chapter": "1",
        "sentence_range": "404-407",
        "Text": ", (L1, L1)\n\u2209 R R is symmetric as (L1, L2) \u2208 R\n\u21d2\nL1 is perpendicular to L2\n\u21d2\nL2 is perpendicular to L1\n\u21d2\n(L2, L1) \u2208 R R is not transitive Indeed, if L1 is perpendicular to L2 and\nL2 is perpendicular to L3, then L1 can never be perpendicular to\nL3"
    },
    {
        "Chapter": "1",
        "sentence_range": "405-408",
        "Text": "R is symmetric as (L1, L2) \u2208 R\n\u21d2\nL1 is perpendicular to L2\n\u21d2\nL2 is perpendicular to L1\n\u21d2\n(L2, L1) \u2208 R R is not transitive Indeed, if L1 is perpendicular to L2 and\nL2 is perpendicular to L3, then L1 can never be perpendicular to\nL3 In fact, L1 is parallel to L3, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "406-409",
        "Text": "R is not transitive Indeed, if L1 is perpendicular to L2 and\nL2 is perpendicular to L3, then L1 can never be perpendicular to\nL3 In fact, L1 is parallel to L3, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "407-410",
        "Text": "Indeed, if L1 is perpendicular to L2 and\nL2 is perpendicular to L3, then L1 can never be perpendicular to\nL3 In fact, L1 is parallel to L3, i e , (L1, L2) \u2208 R, (L2, L3) \u2208 R but (L1, L3) \u2209 R"
    },
    {
        "Chapter": "1",
        "sentence_range": "408-411",
        "Text": "In fact, L1 is parallel to L3, i e , (L1, L2) \u2208 R, (L2, L3) \u2208 R but (L1, L3) \u2209 R Example 4 Show that the relation R in the set {1, 2, 3} given by R = {(1, 1), (2, 2),\n(3, 3), (1, 2), (2, 3)} is reflexive but neither symmetric nor transitive"
    },
    {
        "Chapter": "1",
        "sentence_range": "409-412",
        "Text": "e , (L1, L2) \u2208 R, (L2, L3) \u2208 R but (L1, L3) \u2209 R Example 4 Show that the relation R in the set {1, 2, 3} given by R = {(1, 1), (2, 2),\n(3, 3), (1, 2), (2, 3)} is reflexive but neither symmetric nor transitive Solution R is reflexive, since (1, 1), (2, 2) and (3, 3) lie in R"
    },
    {
        "Chapter": "1",
        "sentence_range": "410-413",
        "Text": ", (L1, L2) \u2208 R, (L2, L3) \u2208 R but (L1, L3) \u2209 R Example 4 Show that the relation R in the set {1, 2, 3} given by R = {(1, 1), (2, 2),\n(3, 3), (1, 2), (2, 3)} is reflexive but neither symmetric nor transitive Solution R is reflexive, since (1, 1), (2, 2) and (3, 3) lie in R Also, R is not symmetric,\nas (1, 2) \u2208 R but (2, 1) \u2209 R"
    },
    {
        "Chapter": "1",
        "sentence_range": "411-414",
        "Text": "Example 4 Show that the relation R in the set {1, 2, 3} given by R = {(1, 1), (2, 2),\n(3, 3), (1, 2), (2, 3)} is reflexive but neither symmetric nor transitive Solution R is reflexive, since (1, 1), (2, 2) and (3, 3) lie in R Also, R is not symmetric,\nas (1, 2) \u2208 R but (2, 1) \u2209 R Similarly, R is not transitive, as (1, 2) \u2208 R and (2, 3) \u2208 R\nbut (1, 3) \u2209 R"
    },
    {
        "Chapter": "1",
        "sentence_range": "412-415",
        "Text": "Solution R is reflexive, since (1, 1), (2, 2) and (3, 3) lie in R Also, R is not symmetric,\nas (1, 2) \u2208 R but (2, 1) \u2209 R Similarly, R is not transitive, as (1, 2) \u2208 R and (2, 3) \u2208 R\nbut (1, 3) \u2209 R Example 5 Show that the relation R in the set Z of integers given by\nR = {(a, b) : 2 divides a \u2013 b}\nis an equivalence relation"
    },
    {
        "Chapter": "1",
        "sentence_range": "413-416",
        "Text": "Also, R is not symmetric,\nas (1, 2) \u2208 R but (2, 1) \u2209 R Similarly, R is not transitive, as (1, 2) \u2208 R and (2, 3) \u2208 R\nbut (1, 3) \u2209 R Example 5 Show that the relation R in the set Z of integers given by\nR = {(a, b) : 2 divides a \u2013 b}\nis an equivalence relation Solution R is reflexive, as 2 divides (a \u2013 a) for all a \u2208 Z"
    },
    {
        "Chapter": "1",
        "sentence_range": "414-417",
        "Text": "Similarly, R is not transitive, as (1, 2) \u2208 R and (2, 3) \u2208 R\nbut (1, 3) \u2209 R Example 5 Show that the relation R in the set Z of integers given by\nR = {(a, b) : 2 divides a \u2013 b}\nis an equivalence relation Solution R is reflexive, as 2 divides (a \u2013 a) for all a \u2208 Z Further, if (a, b) \u2208 R, then\n2 divides a \u2013 b"
    },
    {
        "Chapter": "1",
        "sentence_range": "415-418",
        "Text": "Example 5 Show that the relation R in the set Z of integers given by\nR = {(a, b) : 2 divides a \u2013 b}\nis an equivalence relation Solution R is reflexive, as 2 divides (a \u2013 a) for all a \u2208 Z Further, if (a, b) \u2208 R, then\n2 divides a \u2013 b Therefore, 2 divides b \u2013 a"
    },
    {
        "Chapter": "1",
        "sentence_range": "416-419",
        "Text": "Solution R is reflexive, as 2 divides (a \u2013 a) for all a \u2208 Z Further, if (a, b) \u2208 R, then\n2 divides a \u2013 b Therefore, 2 divides b \u2013 a Hence, (b, a) \u2208 R, which shows that R is\nsymmetric"
    },
    {
        "Chapter": "1",
        "sentence_range": "417-420",
        "Text": "Further, if (a, b) \u2208 R, then\n2 divides a \u2013 b Therefore, 2 divides b \u2013 a Hence, (b, a) \u2208 R, which shows that R is\nsymmetric Similarly, if (a, b) \u2208 R and (b, c) \u2208 R, then a \u2013 b and b \u2013 c are divisible by\n2"
    },
    {
        "Chapter": "1",
        "sentence_range": "418-421",
        "Text": "Therefore, 2 divides b \u2013 a Hence, (b, a) \u2208 R, which shows that R is\nsymmetric Similarly, if (a, b) \u2208 R and (b, c) \u2208 R, then a \u2013 b and b \u2013 c are divisible by\n2 Now, a \u2013 c = (a \u2013 b) + (b \u2013 c) is even (Why"
    },
    {
        "Chapter": "1",
        "sentence_range": "419-422",
        "Text": "Hence, (b, a) \u2208 R, which shows that R is\nsymmetric Similarly, if (a, b) \u2208 R and (b, c) \u2208 R, then a \u2013 b and b \u2013 c are divisible by\n2 Now, a \u2013 c = (a \u2013 b) + (b \u2013 c) is even (Why )"
    },
    {
        "Chapter": "1",
        "sentence_range": "420-423",
        "Text": "Similarly, if (a, b) \u2208 R and (b, c) \u2208 R, then a \u2013 b and b \u2013 c are divisible by\n2 Now, a \u2013 c = (a \u2013 b) + (b \u2013 c) is even (Why ) So, (a \u2013 c) is divisible by 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "421-424",
        "Text": "Now, a \u2013 c = (a \u2013 b) + (b \u2013 c) is even (Why ) So, (a \u2013 c) is divisible by 2 This\nshows that R is transitive"
    },
    {
        "Chapter": "1",
        "sentence_range": "422-425",
        "Text": ") So, (a \u2013 c) is divisible by 2 This\nshows that R is transitive Thus, R is an equivalence relation in Z"
    },
    {
        "Chapter": "1",
        "sentence_range": "423-426",
        "Text": "So, (a \u2013 c) is divisible by 2 This\nshows that R is transitive Thus, R is an equivalence relation in Z Fig 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "424-427",
        "Text": "This\nshows that R is transitive Thus, R is an equivalence relation in Z Fig 1 1\nRationalised 2023-24\nMATHEMATICS\n4\nIn Example 5, note that all even integers are related to zero, as (0, \u00b1 2), (0, \u00b1 4)\netc"
    },
    {
        "Chapter": "1",
        "sentence_range": "425-428",
        "Text": "Thus, R is an equivalence relation in Z Fig 1 1\nRationalised 2023-24\nMATHEMATICS\n4\nIn Example 5, note that all even integers are related to zero, as (0, \u00b1 2), (0, \u00b1 4)\netc , lie in R and no odd integer is related to 0, as (0, \u00b1 1), (0, \u00b1 3) etc"
    },
    {
        "Chapter": "1",
        "sentence_range": "426-429",
        "Text": "Fig 1 1\nRationalised 2023-24\nMATHEMATICS\n4\nIn Example 5, note that all even integers are related to zero, as (0, \u00b1 2), (0, \u00b1 4)\netc , lie in R and no odd integer is related to 0, as (0, \u00b1 1), (0, \u00b1 3) etc , do not lie in R"
    },
    {
        "Chapter": "1",
        "sentence_range": "427-430",
        "Text": "1\nRationalised 2023-24\nMATHEMATICS\n4\nIn Example 5, note that all even integers are related to zero, as (0, \u00b1 2), (0, \u00b1 4)\netc , lie in R and no odd integer is related to 0, as (0, \u00b1 1), (0, \u00b1 3) etc , do not lie in R Similarly, all odd integers are related to one and no even integer is related to one"
    },
    {
        "Chapter": "1",
        "sentence_range": "428-431",
        "Text": ", lie in R and no odd integer is related to 0, as (0, \u00b1 1), (0, \u00b1 3) etc , do not lie in R Similarly, all odd integers are related to one and no even integer is related to one Therefore, the set E of all even integers and the set O of all odd integers are subsets of\nZ satisfying following conditions:\n(i)\nAll elements of E are related to each other and all elements of O are related to\neach other"
    },
    {
        "Chapter": "1",
        "sentence_range": "429-432",
        "Text": ", do not lie in R Similarly, all odd integers are related to one and no even integer is related to one Therefore, the set E of all even integers and the set O of all odd integers are subsets of\nZ satisfying following conditions:\n(i)\nAll elements of E are related to each other and all elements of O are related to\neach other (ii)\nNo element of E is related to any element of O and vice-versa"
    },
    {
        "Chapter": "1",
        "sentence_range": "430-433",
        "Text": "Similarly, all odd integers are related to one and no even integer is related to one Therefore, the set E of all even integers and the set O of all odd integers are subsets of\nZ satisfying following conditions:\n(i)\nAll elements of E are related to each other and all elements of O are related to\neach other (ii)\nNo element of E is related to any element of O and vice-versa (iii)\nE and O are disjoint and Z = E \u222a O"
    },
    {
        "Chapter": "1",
        "sentence_range": "431-434",
        "Text": "Therefore, the set E of all even integers and the set O of all odd integers are subsets of\nZ satisfying following conditions:\n(i)\nAll elements of E are related to each other and all elements of O are related to\neach other (ii)\nNo element of E is related to any element of O and vice-versa (iii)\nE and O are disjoint and Z = E \u222a O The subset E is called the equivalence class containing zero and is denoted by\n[0]"
    },
    {
        "Chapter": "1",
        "sentence_range": "432-435",
        "Text": "(ii)\nNo element of E is related to any element of O and vice-versa (iii)\nE and O are disjoint and Z = E \u222a O The subset E is called the equivalence class containing zero and is denoted by\n[0] Similarly, O is the equivalence class containing 1 and is denoted by [1]"
    },
    {
        "Chapter": "1",
        "sentence_range": "433-436",
        "Text": "(iii)\nE and O are disjoint and Z = E \u222a O The subset E is called the equivalence class containing zero and is denoted by\n[0] Similarly, O is the equivalence class containing 1 and is denoted by [1] Note that\n[0] \u2260 [1], [0] = [2r] and [1] = [2r + 1], r \u2208 Z"
    },
    {
        "Chapter": "1",
        "sentence_range": "434-437",
        "Text": "The subset E is called the equivalence class containing zero and is denoted by\n[0] Similarly, O is the equivalence class containing 1 and is denoted by [1] Note that\n[0] \u2260 [1], [0] = [2r] and [1] = [2r + 1], r \u2208 Z Infact, what we have seen above is true\nfor an arbitrary equivalence relation R in a set X"
    },
    {
        "Chapter": "1",
        "sentence_range": "435-438",
        "Text": "Similarly, O is the equivalence class containing 1 and is denoted by [1] Note that\n[0] \u2260 [1], [0] = [2r] and [1] = [2r + 1], r \u2208 Z Infact, what we have seen above is true\nfor an arbitrary equivalence relation R in a set X Given an arbitrary equivalence\nrelation R in an arbitrary set X, R divides X into mutually disjoint subsets Ai called\npartitions or subdivisions  of X satisfying:\n(i)\nall elements of Ai are related to each other, for all i"
    },
    {
        "Chapter": "1",
        "sentence_range": "436-439",
        "Text": "Note that\n[0] \u2260 [1], [0] = [2r] and [1] = [2r + 1], r \u2208 Z Infact, what we have seen above is true\nfor an arbitrary equivalence relation R in a set X Given an arbitrary equivalence\nrelation R in an arbitrary set X, R divides X into mutually disjoint subsets Ai called\npartitions or subdivisions  of X satisfying:\n(i)\nall elements of Ai are related to each other, for all i (ii)\nno element of Ai is related to any element of Aj , i \u2260 j"
    },
    {
        "Chapter": "1",
        "sentence_range": "437-440",
        "Text": "Infact, what we have seen above is true\nfor an arbitrary equivalence relation R in a set X Given an arbitrary equivalence\nrelation R in an arbitrary set X, R divides X into mutually disjoint subsets Ai called\npartitions or subdivisions  of X satisfying:\n(i)\nall elements of Ai are related to each other, for all i (ii)\nno element of Ai is related to any element of Aj , i \u2260 j (iii)\n\u222a Aj = X and Ai \u2229 Aj = \u03c6, i \u2260 j"
    },
    {
        "Chapter": "1",
        "sentence_range": "438-441",
        "Text": "Given an arbitrary equivalence\nrelation R in an arbitrary set X, R divides X into mutually disjoint subsets Ai called\npartitions or subdivisions  of X satisfying:\n(i)\nall elements of Ai are related to each other, for all i (ii)\nno element of Ai is related to any element of Aj , i \u2260 j (iii)\n\u222a Aj = X and Ai \u2229 Aj = \u03c6, i \u2260 j The subsets Ai are called equivalence classes"
    },
    {
        "Chapter": "1",
        "sentence_range": "439-442",
        "Text": "(ii)\nno element of Ai is related to any element of Aj , i \u2260 j (iii)\n\u222a Aj = X and Ai \u2229 Aj = \u03c6, i \u2260 j The subsets Ai are called equivalence classes The interesting part of the situation\nis that we can go reverse also"
    },
    {
        "Chapter": "1",
        "sentence_range": "440-443",
        "Text": "(iii)\n\u222a Aj = X and Ai \u2229 Aj = \u03c6, i \u2260 j The subsets Ai are called equivalence classes The interesting part of the situation\nis that we can go reverse also For example, consider a subdivision of the set Z given\nby three mutually disjoint subsets A1, A2 and A3 whose union is Z with\nA1 = {x \u2208 Z : x is a multiple of 3} = {"
    },
    {
        "Chapter": "1",
        "sentence_range": "441-444",
        "Text": "The subsets Ai are called equivalence classes The interesting part of the situation\nis that we can go reverse also For example, consider a subdivision of the set Z given\nby three mutually disjoint subsets A1, A2 and A3 whose union is Z with\nA1 = {x \u2208 Z : x is a multiple of 3} = { , \u2013 6, \u2013 3, 0, 3, 6,"
    },
    {
        "Chapter": "1",
        "sentence_range": "442-445",
        "Text": "The interesting part of the situation\nis that we can go reverse also For example, consider a subdivision of the set Z given\nby three mutually disjoint subsets A1, A2 and A3 whose union is Z with\nA1 = {x \u2208 Z : x is a multiple of 3} = { , \u2013 6, \u2013 3, 0, 3, 6, }\nA2 = {x \u2208 Z : x \u2013 1 is a multiple of 3} = {"
    },
    {
        "Chapter": "1",
        "sentence_range": "443-446",
        "Text": "For example, consider a subdivision of the set Z given\nby three mutually disjoint subsets A1, A2 and A3 whose union is Z with\nA1 = {x \u2208 Z : x is a multiple of 3} = { , \u2013 6, \u2013 3, 0, 3, 6, }\nA2 = {x \u2208 Z : x \u2013 1 is a multiple of 3} = { , \u2013 5, \u2013 2, 1, 4, 7,"
    },
    {
        "Chapter": "1",
        "sentence_range": "444-447",
        "Text": ", \u2013 6, \u2013 3, 0, 3, 6, }\nA2 = {x \u2208 Z : x \u2013 1 is a multiple of 3} = { , \u2013 5, \u2013 2, 1, 4, 7, }\nA3 = {x \u2208 Z : x \u2013 2 is a multiple of 3} = {"
    },
    {
        "Chapter": "1",
        "sentence_range": "445-448",
        "Text": "}\nA2 = {x \u2208 Z : x \u2013 1 is a multiple of 3} = { , \u2013 5, \u2013 2, 1, 4, 7, }\nA3 = {x \u2208 Z : x \u2013 2 is a multiple of 3} = { , \u2013 4, \u2013 1, 2, 5, 8,"
    },
    {
        "Chapter": "1",
        "sentence_range": "446-449",
        "Text": ", \u2013 5, \u2013 2, 1, 4, 7, }\nA3 = {x \u2208 Z : x \u2013 2 is a multiple of 3} = { , \u2013 4, \u2013 1, 2, 5, 8, }\nDefine a relation R in Z given by R = {(a, b) : 3 divides a \u2013 b}"
    },
    {
        "Chapter": "1",
        "sentence_range": "447-450",
        "Text": "}\nA3 = {x \u2208 Z : x \u2013 2 is a multiple of 3} = { , \u2013 4, \u2013 1, 2, 5, 8, }\nDefine a relation R in Z given by R = {(a, b) : 3 divides a \u2013 b} Following the\narguments similar to those used in Example 5, we can show that R is an equivalence\nrelation"
    },
    {
        "Chapter": "1",
        "sentence_range": "448-451",
        "Text": ", \u2013 4, \u2013 1, 2, 5, 8, }\nDefine a relation R in Z given by R = {(a, b) : 3 divides a \u2013 b} Following the\narguments similar to those used in Example 5, we can show that R is an equivalence\nrelation Also, A1 coincides with the set of all integers in Z which are related to zero, A2\ncoincides with the set of all integers which are related to 1 and A3 coincides with the\nset of all integers in Z which are related to 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "449-452",
        "Text": "}\nDefine a relation R in Z given by R = {(a, b) : 3 divides a \u2013 b} Following the\narguments similar to those used in Example 5, we can show that R is an equivalence\nrelation Also, A1 coincides with the set of all integers in Z which are related to zero, A2\ncoincides with the set of all integers which are related to 1 and A3 coincides with the\nset of all integers in Z which are related to 2 Thus, A1 = [0], A2 = [1] and A3 = [2]"
    },
    {
        "Chapter": "1",
        "sentence_range": "450-453",
        "Text": "Following the\narguments similar to those used in Example 5, we can show that R is an equivalence\nrelation Also, A1 coincides with the set of all integers in Z which are related to zero, A2\ncoincides with the set of all integers which are related to 1 and A3 coincides with the\nset of all integers in Z which are related to 2 Thus, A1 = [0], A2 = [1] and A3 = [2] In fact, A1 = [3r], A2 = [3r + 1] and A3 = [3r + 2], for all r \u2208 Z"
    },
    {
        "Chapter": "1",
        "sentence_range": "451-454",
        "Text": "Also, A1 coincides with the set of all integers in Z which are related to zero, A2\ncoincides with the set of all integers which are related to 1 and A3 coincides with the\nset of all integers in Z which are related to 2 Thus, A1 = [0], A2 = [1] and A3 = [2] In fact, A1 = [3r], A2 = [3r + 1] and A3 = [3r + 2], for all r \u2208 Z Example 6 Let R be the relation defined in the set A = {1, 2, 3, 4, 5, 6, 7} by\nR = {(a, b) : both a and b are either odd or even}"
    },
    {
        "Chapter": "1",
        "sentence_range": "452-455",
        "Text": "Thus, A1 = [0], A2 = [1] and A3 = [2] In fact, A1 = [3r], A2 = [3r + 1] and A3 = [3r + 2], for all r \u2208 Z Example 6 Let R be the relation defined in the set A = {1, 2, 3, 4, 5, 6, 7} by\nR = {(a, b) : both a and b are either odd or even} Show that R is an equivalence\nrelation"
    },
    {
        "Chapter": "1",
        "sentence_range": "453-456",
        "Text": "In fact, A1 = [3r], A2 = [3r + 1] and A3 = [3r + 2], for all r \u2208 Z Example 6 Let R be the relation defined in the set A = {1, 2, 3, 4, 5, 6, 7} by\nR = {(a, b) : both a and b are either odd or even} Show that R is an equivalence\nrelation Further, show that all the elements of the subset {1, 3, 5, 7} are related to each\nother and all the elements of the subset {2, 4, 6} are related to each other, but no\nelement of the subset {1, 3, 5, 7} is related to any element of the subset {2, 4, 6}"
    },
    {
        "Chapter": "1",
        "sentence_range": "454-457",
        "Text": "Example 6 Let R be the relation defined in the set A = {1, 2, 3, 4, 5, 6, 7} by\nR = {(a, b) : both a and b are either odd or even} Show that R is an equivalence\nrelation Further, show that all the elements of the subset {1, 3, 5, 7} are related to each\nother and all the elements of the subset {2, 4, 6} are related to each other, but no\nelement of the subset {1, 3, 5, 7} is related to any element of the subset {2, 4, 6} Rationalised 2023-24\nRELATIONS AND FUNCTIONS\n5\nSolution Given any element a in A, both a and a must be either odd or even, so\nthat (a, a) \u2208 R"
    },
    {
        "Chapter": "1",
        "sentence_range": "455-458",
        "Text": "Show that R is an equivalence\nrelation Further, show that all the elements of the subset {1, 3, 5, 7} are related to each\nother and all the elements of the subset {2, 4, 6} are related to each other, but no\nelement of the subset {1, 3, 5, 7} is related to any element of the subset {2, 4, 6} Rationalised 2023-24\nRELATIONS AND FUNCTIONS\n5\nSolution Given any element a in A, both a and a must be either odd or even, so\nthat (a, a) \u2208 R Further, (a, b) \u2208 R \u21d2 both a and b must be either odd or even\n\u21d2 (b, a) \u2208 R"
    },
    {
        "Chapter": "1",
        "sentence_range": "456-459",
        "Text": "Further, show that all the elements of the subset {1, 3, 5, 7} are related to each\nother and all the elements of the subset {2, 4, 6} are related to each other, but no\nelement of the subset {1, 3, 5, 7} is related to any element of the subset {2, 4, 6} Rationalised 2023-24\nRELATIONS AND FUNCTIONS\n5\nSolution Given any element a in A, both a and a must be either odd or even, so\nthat (a, a) \u2208 R Further, (a, b) \u2208 R \u21d2 both a and b must be either odd or even\n\u21d2 (b, a) \u2208 R Similarly, (a, b) \u2208 R and (b, c) \u2208 R \u21d2 all elements a, b, c, must be\neither even or odd simultaneously \u21d2 (a, c) \u2208 R"
    },
    {
        "Chapter": "1",
        "sentence_range": "457-460",
        "Text": "Rationalised 2023-24\nRELATIONS AND FUNCTIONS\n5\nSolution Given any element a in A, both a and a must be either odd or even, so\nthat (a, a) \u2208 R Further, (a, b) \u2208 R \u21d2 both a and b must be either odd or even\n\u21d2 (b, a) \u2208 R Similarly, (a, b) \u2208 R and (b, c) \u2208 R \u21d2 all elements a, b, c, must be\neither even or odd simultaneously \u21d2 (a, c) \u2208 R Hence, R is an equivalence relation"
    },
    {
        "Chapter": "1",
        "sentence_range": "458-461",
        "Text": "Further, (a, b) \u2208 R \u21d2 both a and b must be either odd or even\n\u21d2 (b, a) \u2208 R Similarly, (a, b) \u2208 R and (b, c) \u2208 R \u21d2 all elements a, b, c, must be\neither even or odd simultaneously \u21d2 (a, c) \u2208 R Hence, R is an equivalence relation Further, all the elements of {1, 3, 5, 7} are related to each other, as all the elements\nof this subset are odd"
    },
    {
        "Chapter": "1",
        "sentence_range": "459-462",
        "Text": "Similarly, (a, b) \u2208 R and (b, c) \u2208 R \u21d2 all elements a, b, c, must be\neither even or odd simultaneously \u21d2 (a, c) \u2208 R Hence, R is an equivalence relation Further, all the elements of {1, 3, 5, 7} are related to each other, as all the elements\nof this subset are odd Similarly, all the elements of the subset {2, 4, 6} are related to\neach other, as all of them are even"
    },
    {
        "Chapter": "1",
        "sentence_range": "460-463",
        "Text": "Hence, R is an equivalence relation Further, all the elements of {1, 3, 5, 7} are related to each other, as all the elements\nof this subset are odd Similarly, all the elements of the subset {2, 4, 6} are related to\neach other, as all of them are even Also, no element of the subset {1, 3, 5, 7} can be\nrelated to any element of {2, 4, 6}, as elements of {1, 3, 5, 7} are odd, while elements\nof {2, 4, 6} are even"
    },
    {
        "Chapter": "1",
        "sentence_range": "461-464",
        "Text": "Further, all the elements of {1, 3, 5, 7} are related to each other, as all the elements\nof this subset are odd Similarly, all the elements of the subset {2, 4, 6} are related to\neach other, as all of them are even Also, no element of the subset {1, 3, 5, 7} can be\nrelated to any element of {2, 4, 6}, as elements of {1, 3, 5, 7} are odd, while elements\nof {2, 4, 6} are even EXERCISE 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "462-465",
        "Text": "Similarly, all the elements of the subset {2, 4, 6} are related to\neach other, as all of them are even Also, no element of the subset {1, 3, 5, 7} can be\nrelated to any element of {2, 4, 6}, as elements of {1, 3, 5, 7} are odd, while elements\nof {2, 4, 6} are even EXERCISE 1 1\n1"
    },
    {
        "Chapter": "1",
        "sentence_range": "463-466",
        "Text": "Also, no element of the subset {1, 3, 5, 7} can be\nrelated to any element of {2, 4, 6}, as elements of {1, 3, 5, 7} are odd, while elements\nof {2, 4, 6} are even EXERCISE 1 1\n1 Determine whether each of the following relations are reflexive, symmetric and\ntransitive:\n(i) Relation R in the set A = {1, 2, 3,"
    },
    {
        "Chapter": "1",
        "sentence_range": "464-467",
        "Text": "EXERCISE 1 1\n1 Determine whether each of the following relations are reflexive, symmetric and\ntransitive:\n(i) Relation R in the set A = {1, 2, 3, , 13, 14} defined as\nR = {(x, y) : 3x \u2013 y = 0}\n(ii) Relation R in the set N of natural numbers defined as\nR = {(x, y) : y = x + 5 and x < 4}\n(iii) Relation R in the set A = {1, 2, 3, 4, 5, 6} as\nR = {(x, y) : y is divisible by x}\n(iv) Relation R in the set Z of all integers defined as\nR = {(x, y) : x \u2013 y is an integer}\n(v) Relation R in the set A of human beings in a town at a particular time given by\n(a)  R = {(x, y) : x and y work at the same place}\n(b)  R = {(x, y) : x and y live in the same locality}\n(c)  R = {(x, y) : x is exactly 7 cm taller than y}\n(d)  R = {(x, y) : x is wife of y}\n(e)  R = {(x, y) : x is father of y}\n2"
    },
    {
        "Chapter": "1",
        "sentence_range": "465-468",
        "Text": "1\n1 Determine whether each of the following relations are reflexive, symmetric and\ntransitive:\n(i) Relation R in the set A = {1, 2, 3, , 13, 14} defined as\nR = {(x, y) : 3x \u2013 y = 0}\n(ii) Relation R in the set N of natural numbers defined as\nR = {(x, y) : y = x + 5 and x < 4}\n(iii) Relation R in the set A = {1, 2, 3, 4, 5, 6} as\nR = {(x, y) : y is divisible by x}\n(iv) Relation R in the set Z of all integers defined as\nR = {(x, y) : x \u2013 y is an integer}\n(v) Relation R in the set A of human beings in a town at a particular time given by\n(a)  R = {(x, y) : x and y work at the same place}\n(b)  R = {(x, y) : x and y live in the same locality}\n(c)  R = {(x, y) : x is exactly 7 cm taller than y}\n(d)  R = {(x, y) : x is wife of y}\n(e)  R = {(x, y) : x is father of y}\n2 Show that the relation R in the set R of real numbers, defined as\nR = {(a, b) : a \u2264 b2} is neither reflexive nor symmetric nor transitive"
    },
    {
        "Chapter": "1",
        "sentence_range": "466-469",
        "Text": "Determine whether each of the following relations are reflexive, symmetric and\ntransitive:\n(i) Relation R in the set A = {1, 2, 3, , 13, 14} defined as\nR = {(x, y) : 3x \u2013 y = 0}\n(ii) Relation R in the set N of natural numbers defined as\nR = {(x, y) : y = x + 5 and x < 4}\n(iii) Relation R in the set A = {1, 2, 3, 4, 5, 6} as\nR = {(x, y) : y is divisible by x}\n(iv) Relation R in the set Z of all integers defined as\nR = {(x, y) : x \u2013 y is an integer}\n(v) Relation R in the set A of human beings in a town at a particular time given by\n(a)  R = {(x, y) : x and y work at the same place}\n(b)  R = {(x, y) : x and y live in the same locality}\n(c)  R = {(x, y) : x is exactly 7 cm taller than y}\n(d)  R = {(x, y) : x is wife of y}\n(e)  R = {(x, y) : x is father of y}\n2 Show that the relation R in the set R of real numbers, defined as\nR = {(a, b) : a \u2264 b2} is neither reflexive nor symmetric nor transitive 3"
    },
    {
        "Chapter": "1",
        "sentence_range": "467-470",
        "Text": ", 13, 14} defined as\nR = {(x, y) : 3x \u2013 y = 0}\n(ii) Relation R in the set N of natural numbers defined as\nR = {(x, y) : y = x + 5 and x < 4}\n(iii) Relation R in the set A = {1, 2, 3, 4, 5, 6} as\nR = {(x, y) : y is divisible by x}\n(iv) Relation R in the set Z of all integers defined as\nR = {(x, y) : x \u2013 y is an integer}\n(v) Relation R in the set A of human beings in a town at a particular time given by\n(a)  R = {(x, y) : x and y work at the same place}\n(b)  R = {(x, y) : x and y live in the same locality}\n(c)  R = {(x, y) : x is exactly 7 cm taller than y}\n(d)  R = {(x, y) : x is wife of y}\n(e)  R = {(x, y) : x is father of y}\n2 Show that the relation R in the set R of real numbers, defined as\nR = {(a, b) : a \u2264 b2} is neither reflexive nor symmetric nor transitive 3 Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as\nR = {(a, b) : b = a + 1} is reflexive, symmetric or transitive"
    },
    {
        "Chapter": "1",
        "sentence_range": "468-471",
        "Text": "Show that the relation R in the set R of real numbers, defined as\nR = {(a, b) : a \u2264 b2} is neither reflexive nor symmetric nor transitive 3 Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as\nR = {(a, b) : b = a + 1} is reflexive, symmetric or transitive 4"
    },
    {
        "Chapter": "1",
        "sentence_range": "469-472",
        "Text": "3 Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as\nR = {(a, b) : b = a + 1} is reflexive, symmetric or transitive 4 Show that the relation R in R defined as R = {(a, b) : a \u2264 b}, is reflexive and\ntransitive but not symmetric"
    },
    {
        "Chapter": "1",
        "sentence_range": "470-473",
        "Text": "Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as\nR = {(a, b) : b = a + 1} is reflexive, symmetric or transitive 4 Show that the relation R in R defined as R = {(a, b) : a \u2264 b}, is reflexive and\ntransitive but not symmetric 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "471-474",
        "Text": "4 Show that the relation R in R defined as R = {(a, b) : a \u2264 b}, is reflexive and\ntransitive but not symmetric 5 Check whether the relation R in R defined by R = {(a, b) : a \u2264 b3} is reflexive,\nsymmetric or transitive"
    },
    {
        "Chapter": "1",
        "sentence_range": "472-475",
        "Text": "Show that the relation R in R defined as R = {(a, b) : a \u2264 b}, is reflexive and\ntransitive but not symmetric 5 Check whether the relation R in R defined by R = {(a, b) : a \u2264 b3} is reflexive,\nsymmetric or transitive Rationalised 2023-24\nMATHEMATICS\n6\n6"
    },
    {
        "Chapter": "1",
        "sentence_range": "473-476",
        "Text": "5 Check whether the relation R in R defined by R = {(a, b) : a \u2264 b3} is reflexive,\nsymmetric or transitive Rationalised 2023-24\nMATHEMATICS\n6\n6 Show that the relation R in  the set {1, 2, 3} given by R = {(1, 2), (2, 1)} is\nsymmetric but neither reflexive nor transitive"
    },
    {
        "Chapter": "1",
        "sentence_range": "474-477",
        "Text": "Check whether the relation R in R defined by R = {(a, b) : a \u2264 b3} is reflexive,\nsymmetric or transitive Rationalised 2023-24\nMATHEMATICS\n6\n6 Show that the relation R in  the set {1, 2, 3} given by R = {(1, 2), (2, 1)} is\nsymmetric but neither reflexive nor transitive 7"
    },
    {
        "Chapter": "1",
        "sentence_range": "475-478",
        "Text": "Rationalised 2023-24\nMATHEMATICS\n6\n6 Show that the relation R in  the set {1, 2, 3} given by R = {(1, 2), (2, 1)} is\nsymmetric but neither reflexive nor transitive 7 Show that the relation R in the set A of all the books in a library of a college,\ngiven by R = {(x, y) : x and y have same number of pages} is an equivalence\nrelation"
    },
    {
        "Chapter": "1",
        "sentence_range": "476-479",
        "Text": "Show that the relation R in  the set {1, 2, 3} given by R = {(1, 2), (2, 1)} is\nsymmetric but neither reflexive nor transitive 7 Show that the relation R in the set A of all the books in a library of a college,\ngiven by R = {(x, y) : x and y have same number of pages} is an equivalence\nrelation 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "477-480",
        "Text": "7 Show that the relation R in the set A of all the books in a library of a college,\ngiven by R = {(x, y) : x and y have same number of pages} is an equivalence\nrelation 8 Show that the relation R in the set A = {1, 2, 3, 4, 5} given by\nR = {(a, b) : |a \u2013 b| is even}, is an equivalence relation"
    },
    {
        "Chapter": "1",
        "sentence_range": "478-481",
        "Text": "Show that the relation R in the set A of all the books in a library of a college,\ngiven by R = {(x, y) : x and y have same number of pages} is an equivalence\nrelation 8 Show that the relation R in the set A = {1, 2, 3, 4, 5} given by\nR = {(a, b) : |a \u2013 b| is even}, is an equivalence relation Show that all the\nelements of {1, 3, 5} are related to each other and all the elements of {2, 4} are\nrelated to each other"
    },
    {
        "Chapter": "1",
        "sentence_range": "479-482",
        "Text": "8 Show that the relation R in the set A = {1, 2, 3, 4, 5} given by\nR = {(a, b) : |a \u2013 b| is even}, is an equivalence relation Show that all the\nelements of {1, 3, 5} are related to each other and all the elements of {2, 4} are\nrelated to each other But  no element of {1, 3, 5} is related to any element of {2, 4}"
    },
    {
        "Chapter": "1",
        "sentence_range": "480-483",
        "Text": "Show that the relation R in the set A = {1, 2, 3, 4, 5} given by\nR = {(a, b) : |a \u2013 b| is even}, is an equivalence relation Show that all the\nelements of {1, 3, 5} are related to each other and all the elements of {2, 4} are\nrelated to each other But  no element of {1, 3, 5} is related to any element of {2, 4} 9"
    },
    {
        "Chapter": "1",
        "sentence_range": "481-484",
        "Text": "Show that all the\nelements of {1, 3, 5} are related to each other and all the elements of {2, 4} are\nrelated to each other But  no element of {1, 3, 5} is related to any element of {2, 4} 9 Show that each of the relation R in the set A = {x \u2208 Z : 0 \u2264 x \u2264 12}, given by\n(i) R = {(a, b) : |a \u2013 b| is a multiple of 4}\n(ii) R = {(a, b) : a = b}\nis an equivalence relation"
    },
    {
        "Chapter": "1",
        "sentence_range": "482-485",
        "Text": "But  no element of {1, 3, 5} is related to any element of {2, 4} 9 Show that each of the relation R in the set A = {x \u2208 Z : 0 \u2264 x \u2264 12}, given by\n(i) R = {(a, b) : |a \u2013 b| is a multiple of 4}\n(ii) R = {(a, b) : a = b}\nis an equivalence relation Find the set of all elements related to 1 in each case"
    },
    {
        "Chapter": "1",
        "sentence_range": "483-486",
        "Text": "9 Show that each of the relation R in the set A = {x \u2208 Z : 0 \u2264 x \u2264 12}, given by\n(i) R = {(a, b) : |a \u2013 b| is a multiple of 4}\n(ii) R = {(a, b) : a = b}\nis an equivalence relation Find the set of all elements related to 1 in each case 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "484-487",
        "Text": "Show that each of the relation R in the set A = {x \u2208 Z : 0 \u2264 x \u2264 12}, given by\n(i) R = {(a, b) : |a \u2013 b| is a multiple of 4}\n(ii) R = {(a, b) : a = b}\nis an equivalence relation Find the set of all elements related to 1 in each case 10 Give an example of a relation"
    },
    {
        "Chapter": "1",
        "sentence_range": "485-488",
        "Text": "Find the set of all elements related to 1 in each case 10 Give an example of a relation Which is\n(i) Symmetric but neither reflexive nor transitive"
    },
    {
        "Chapter": "1",
        "sentence_range": "486-489",
        "Text": "10 Give an example of a relation Which is\n(i) Symmetric but neither reflexive nor transitive (ii) Transitive but neither reflexive nor symmetric"
    },
    {
        "Chapter": "1",
        "sentence_range": "487-490",
        "Text": "Give an example of a relation Which is\n(i) Symmetric but neither reflexive nor transitive (ii) Transitive but neither reflexive nor symmetric (iii) Reflexive and symmetric but not transitive"
    },
    {
        "Chapter": "1",
        "sentence_range": "488-491",
        "Text": "Which is\n(i) Symmetric but neither reflexive nor transitive (ii) Transitive but neither reflexive nor symmetric (iii) Reflexive and symmetric but not transitive (iv) Reflexive and transitive but not symmetric"
    },
    {
        "Chapter": "1",
        "sentence_range": "489-492",
        "Text": "(ii) Transitive but neither reflexive nor symmetric (iii) Reflexive and symmetric but not transitive (iv) Reflexive and transitive but not symmetric (v) Symmetric and transitive but not reflexive"
    },
    {
        "Chapter": "1",
        "sentence_range": "490-493",
        "Text": "(iii) Reflexive and symmetric but not transitive (iv) Reflexive and transitive but not symmetric (v) Symmetric and transitive but not reflexive 11"
    },
    {
        "Chapter": "1",
        "sentence_range": "491-494",
        "Text": "(iv) Reflexive and transitive but not symmetric (v) Symmetric and transitive but not reflexive 11 Show that the relation R in the set A of points in a plane given by\nR = {(P, Q) : distance of the point P from the origin is same as the distance of the\npoint Q from the origin}, is an equivalence relation"
    },
    {
        "Chapter": "1",
        "sentence_range": "492-495",
        "Text": "(v) Symmetric and transitive but not reflexive 11 Show that the relation R in the set A of points in a plane given by\nR = {(P, Q) : distance of the point P from the origin is same as the distance of the\npoint Q from the origin}, is an equivalence relation Further, show that the set of\nall points related to a point P \u2260 (0, 0) is the circle passing through P with origin as\ncentre"
    },
    {
        "Chapter": "1",
        "sentence_range": "493-496",
        "Text": "11 Show that the relation R in the set A of points in a plane given by\nR = {(P, Q) : distance of the point P from the origin is same as the distance of the\npoint Q from the origin}, is an equivalence relation Further, show that the set of\nall points related to a point P \u2260 (0, 0) is the circle passing through P with origin as\ncentre 12"
    },
    {
        "Chapter": "1",
        "sentence_range": "494-497",
        "Text": "Show that the relation R in the set A of points in a plane given by\nR = {(P, Q) : distance of the point P from the origin is same as the distance of the\npoint Q from the origin}, is an equivalence relation Further, show that the set of\nall points related to a point P \u2260 (0, 0) is the circle passing through P with origin as\ncentre 12 Show that the relation R defined in the set A of all triangles as R = {(T1, T2) : T1\nis similar to T2}, is equivalence relation"
    },
    {
        "Chapter": "1",
        "sentence_range": "495-498",
        "Text": "Further, show that the set of\nall points related to a point P \u2260 (0, 0) is the circle passing through P with origin as\ncentre 12 Show that the relation R defined in the set A of all triangles as R = {(T1, T2) : T1\nis similar to T2}, is equivalence relation Consider three right angle triangles T1\nwith sides 3, 4, 5,  T2 with sides 5, 12, 13 and T3 with sides 6, 8, 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "496-499",
        "Text": "12 Show that the relation R defined in the set A of all triangles as R = {(T1, T2) : T1\nis similar to T2}, is equivalence relation Consider three right angle triangles T1\nwith sides 3, 4, 5,  T2 with sides 5, 12, 13 and T3 with sides 6, 8, 10 Which\ntriangles among T1, T2 and T3 are related"
    },
    {
        "Chapter": "1",
        "sentence_range": "497-500",
        "Text": "Show that the relation R defined in the set A of all triangles as R = {(T1, T2) : T1\nis similar to T2}, is equivalence relation Consider three right angle triangles T1\nwith sides 3, 4, 5,  T2 with sides 5, 12, 13 and T3 with sides 6, 8, 10 Which\ntriangles among T1, T2 and T3 are related 13"
    },
    {
        "Chapter": "1",
        "sentence_range": "498-501",
        "Text": "Consider three right angle triangles T1\nwith sides 3, 4, 5,  T2 with sides 5, 12, 13 and T3 with sides 6, 8, 10 Which\ntriangles among T1, T2 and T3 are related 13 Show that the relation R defined in the set A of all polygons as R = {(P1, P2) :\nP1 and P2 have same number of sides}, is an equivalence relation"
    },
    {
        "Chapter": "1",
        "sentence_range": "499-502",
        "Text": "Which\ntriangles among T1, T2 and T3 are related 13 Show that the relation R defined in the set A of all polygons as R = {(P1, P2) :\nP1 and P2 have same number of sides}, is an equivalence relation What is the\nset of all elements in A related to the right angle triangle T with sides 3, 4 and 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "500-503",
        "Text": "13 Show that the relation R defined in the set A of all polygons as R = {(P1, P2) :\nP1 and P2 have same number of sides}, is an equivalence relation What is the\nset of all elements in A related to the right angle triangle T with sides 3, 4 and 5 14"
    },
    {
        "Chapter": "1",
        "sentence_range": "501-504",
        "Text": "Show that the relation R defined in the set A of all polygons as R = {(P1, P2) :\nP1 and P2 have same number of sides}, is an equivalence relation What is the\nset of all elements in A related to the right angle triangle T with sides 3, 4 and 5 14 Let L be the set of all lines in XY plane and R be the relation in L defined as\nR = {(L1, L2) : L1 is parallel to L2}"
    },
    {
        "Chapter": "1",
        "sentence_range": "502-505",
        "Text": "What is the\nset of all elements in A related to the right angle triangle T with sides 3, 4 and 5 14 Let L be the set of all lines in XY plane and R be the relation in L defined as\nR = {(L1, L2) : L1 is parallel to L2} Show that R is an equivalence relation"
    },
    {
        "Chapter": "1",
        "sentence_range": "503-506",
        "Text": "14 Let L be the set of all lines in XY plane and R be the relation in L defined as\nR = {(L1, L2) : L1 is parallel to L2} Show that R is an equivalence relation Find\nthe set of all lines related to the line y = 2x + 4"
    },
    {
        "Chapter": "1",
        "sentence_range": "504-507",
        "Text": "Let L be the set of all lines in XY plane and R be the relation in L defined as\nR = {(L1, L2) : L1 is parallel to L2} Show that R is an equivalence relation Find\nthe set of all lines related to the line y = 2x + 4 Rationalised 2023-24\nRELATIONS AND FUNCTIONS\n7\n15"
    },
    {
        "Chapter": "1",
        "sentence_range": "505-508",
        "Text": "Show that R is an equivalence relation Find\nthe set of all lines related to the line y = 2x + 4 Rationalised 2023-24\nRELATIONS AND FUNCTIONS\n7\n15 Let R be the relation in the set {1, 2, 3, 4} given by R = {(1, 2), (2, 2), (1, 1), (4,4),\n(1, 3), (3, 3), (3, 2)}"
    },
    {
        "Chapter": "1",
        "sentence_range": "506-509",
        "Text": "Find\nthe set of all lines related to the line y = 2x + 4 Rationalised 2023-24\nRELATIONS AND FUNCTIONS\n7\n15 Let R be the relation in the set {1, 2, 3, 4} given by R = {(1, 2), (2, 2), (1, 1), (4,4),\n(1, 3), (3, 3), (3, 2)} Choose the correct answer"
    },
    {
        "Chapter": "1",
        "sentence_range": "507-510",
        "Text": "Rationalised 2023-24\nRELATIONS AND FUNCTIONS\n7\n15 Let R be the relation in the set {1, 2, 3, 4} given by R = {(1, 2), (2, 2), (1, 1), (4,4),\n(1, 3), (3, 3), (3, 2)} Choose the correct answer (A) R is reflexive and symmetric but not transitive"
    },
    {
        "Chapter": "1",
        "sentence_range": "508-511",
        "Text": "Let R be the relation in the set {1, 2, 3, 4} given by R = {(1, 2), (2, 2), (1, 1), (4,4),\n(1, 3), (3, 3), (3, 2)} Choose the correct answer (A) R is reflexive and symmetric but not transitive (B) R is reflexive and transitive but not symmetric"
    },
    {
        "Chapter": "1",
        "sentence_range": "509-512",
        "Text": "Choose the correct answer (A) R is reflexive and symmetric but not transitive (B) R is reflexive and transitive but not symmetric (C) R is symmetric and transitive but not reflexive"
    },
    {
        "Chapter": "1",
        "sentence_range": "510-513",
        "Text": "(A) R is reflexive and symmetric but not transitive (B) R is reflexive and transitive but not symmetric (C) R is symmetric and transitive but not reflexive (D) R is an equivalence relation"
    },
    {
        "Chapter": "1",
        "sentence_range": "511-514",
        "Text": "(B) R is reflexive and transitive but not symmetric (C) R is symmetric and transitive but not reflexive (D) R is an equivalence relation 16"
    },
    {
        "Chapter": "1",
        "sentence_range": "512-515",
        "Text": "(C) R is symmetric and transitive but not reflexive (D) R is an equivalence relation 16 Let R be the relation in the set N given by R = {(a, b) : a = b \u2013 2, b > 6}"
    },
    {
        "Chapter": "1",
        "sentence_range": "513-516",
        "Text": "(D) R is an equivalence relation 16 Let R be the relation in the set N given by R = {(a, b) : a = b \u2013 2, b > 6} Choose\nthe correct answer"
    },
    {
        "Chapter": "1",
        "sentence_range": "514-517",
        "Text": "16 Let R be the relation in the set N given by R = {(a, b) : a = b \u2013 2, b > 6} Choose\nthe correct answer (A) (2, 4) \u2208 R\n(B) (3, 8) \u2208 R\n(C) (6, 8) \u2208 R\n(D) (8, 7) \u2208 R\n1"
    },
    {
        "Chapter": "1",
        "sentence_range": "515-518",
        "Text": "Let R be the relation in the set N given by R = {(a, b) : a = b \u2013 2, b > 6} Choose\nthe correct answer (A) (2, 4) \u2208 R\n(B) (3, 8) \u2208 R\n(C) (6, 8) \u2208 R\n(D) (8, 7) \u2208 R\n1 3  Types of Functions\nThe notion of a function along with some special functions like identity function, constant\nfunction, polynomial function, rational function, modulus function, signum function etc"
    },
    {
        "Chapter": "1",
        "sentence_range": "516-519",
        "Text": "Choose\nthe correct answer (A) (2, 4) \u2208 R\n(B) (3, 8) \u2208 R\n(C) (6, 8) \u2208 R\n(D) (8, 7) \u2208 R\n1 3  Types of Functions\nThe notion of a function along with some special functions like identity function, constant\nfunction, polynomial function, rational function, modulus function, signum function etc along with their graphs have been given in Class XI"
    },
    {
        "Chapter": "1",
        "sentence_range": "517-520",
        "Text": "(A) (2, 4) \u2208 R\n(B) (3, 8) \u2208 R\n(C) (6, 8) \u2208 R\n(D) (8, 7) \u2208 R\n1 3  Types of Functions\nThe notion of a function along with some special functions like identity function, constant\nfunction, polynomial function, rational function, modulus function, signum function etc along with their graphs have been given in Class XI Addition, subtraction, multiplication and division of two functions have also been\nstudied"
    },
    {
        "Chapter": "1",
        "sentence_range": "518-521",
        "Text": "3  Types of Functions\nThe notion of a function along with some special functions like identity function, constant\nfunction, polynomial function, rational function, modulus function, signum function etc along with their graphs have been given in Class XI Addition, subtraction, multiplication and division of two functions have also been\nstudied As the concept of function is of paramount importance in mathematics and\namong other  disciplines as well, we would like to extend our study about function from\nwhere we finished earlier"
    },
    {
        "Chapter": "1",
        "sentence_range": "519-522",
        "Text": "along with their graphs have been given in Class XI Addition, subtraction, multiplication and division of two functions have also been\nstudied As the concept of function is of paramount importance in mathematics and\namong other  disciplines as well, we would like to extend our study about function from\nwhere we finished earlier In this section, we would like to study different types of\nfunctions"
    },
    {
        "Chapter": "1",
        "sentence_range": "520-523",
        "Text": "Addition, subtraction, multiplication and division of two functions have also been\nstudied As the concept of function is of paramount importance in mathematics and\namong other  disciplines as well, we would like to extend our study about function from\nwhere we finished earlier In this section, we would like to study different types of\nfunctions Consider the functions  f1, f2, f3 and f4 given by the following diagrams"
    },
    {
        "Chapter": "1",
        "sentence_range": "521-524",
        "Text": "As the concept of function is of paramount importance in mathematics and\namong other  disciplines as well, we would like to extend our study about function from\nwhere we finished earlier In this section, we would like to study different types of\nfunctions Consider the functions  f1, f2, f3 and f4 given by the following diagrams In Fig 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "522-525",
        "Text": "In this section, we would like to study different types of\nfunctions Consider the functions  f1, f2, f3 and f4 given by the following diagrams In Fig 1 2, we observe that the images of distinct elements of X1 under the function\nf1 are distinct, but the image of two distinct elements 1 and 2 of X1 under f2 is same,\nnamely b"
    },
    {
        "Chapter": "1",
        "sentence_range": "523-526",
        "Text": "Consider the functions  f1, f2, f3 and f4 given by the following diagrams In Fig 1 2, we observe that the images of distinct elements of X1 under the function\nf1 are distinct, but the image of two distinct elements 1 and 2 of X1 under f2 is same,\nnamely b Further, there are some elements like e and f in X2 which are not images of\nany element of X1 under f1, while all elements of X3 are images of some elements of X1\nunder f3"
    },
    {
        "Chapter": "1",
        "sentence_range": "524-527",
        "Text": "In Fig 1 2, we observe that the images of distinct elements of X1 under the function\nf1 are distinct, but the image of two distinct elements 1 and 2 of X1 under f2 is same,\nnamely b Further, there are some elements like e and f in X2 which are not images of\nany element of X1 under f1, while all elements of X3 are images of some elements of X1\nunder f3 The above observations lead to the following definitions:\nDefinition 5 A function f : X \u2192 Y is defined to be one-one (or injective), if the images\nof distinct elements of X under f are distinct, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "525-528",
        "Text": "2, we observe that the images of distinct elements of X1 under the function\nf1 are distinct, but the image of two distinct elements 1 and 2 of X1 under f2 is same,\nnamely b Further, there are some elements like e and f in X2 which are not images of\nany element of X1 under f1, while all elements of X3 are images of some elements of X1\nunder f3 The above observations lead to the following definitions:\nDefinition 5 A function f : X \u2192 Y is defined to be one-one (or injective), if the images\nof distinct elements of X under f are distinct, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "526-529",
        "Text": "Further, there are some elements like e and f in X2 which are not images of\nany element of X1 under f1, while all elements of X3 are images of some elements of X1\nunder f3 The above observations lead to the following definitions:\nDefinition 5 A function f : X \u2192 Y is defined to be one-one (or injective), if the images\nof distinct elements of X under f are distinct, i e , for every x1, x2 \u2208 X, f(x1) = f(x2)\nimplies x1 = x2"
    },
    {
        "Chapter": "1",
        "sentence_range": "527-530",
        "Text": "The above observations lead to the following definitions:\nDefinition 5 A function f : X \u2192 Y is defined to be one-one (or injective), if the images\nof distinct elements of X under f are distinct, i e , for every x1, x2 \u2208 X, f(x1) = f(x2)\nimplies x1 = x2 Otherwise, f is called many-one"
    },
    {
        "Chapter": "1",
        "sentence_range": "528-531",
        "Text": "e , for every x1, x2 \u2208 X, f(x1) = f(x2)\nimplies x1 = x2 Otherwise, f is called many-one The function f1 and f4 in Fig 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "529-532",
        "Text": ", for every x1, x2 \u2208 X, f(x1) = f(x2)\nimplies x1 = x2 Otherwise, f is called many-one The function f1 and f4 in Fig 1 2 (i) and (iv) are one-one and the function f2 and f3\nin Fig 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "530-533",
        "Text": "Otherwise, f is called many-one The function f1 and f4 in Fig 1 2 (i) and (iv) are one-one and the function f2 and f3\nin Fig 1 2 (ii) and (iii) are many-one"
    },
    {
        "Chapter": "1",
        "sentence_range": "531-534",
        "Text": "The function f1 and f4 in Fig 1 2 (i) and (iv) are one-one and the function f2 and f3\nin Fig 1 2 (ii) and (iii) are many-one Definition 6 A function f : X \u2192 Y is said to be onto (or surjective), if every element\nof Y is the image of some element of X under f, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "532-535",
        "Text": "2 (i) and (iv) are one-one and the function f2 and f3\nin Fig 1 2 (ii) and (iii) are many-one Definition 6 A function f : X \u2192 Y is said to be onto (or surjective), if every element\nof Y is the image of some element of X under f, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "533-536",
        "Text": "2 (ii) and (iii) are many-one Definition 6 A function f : X \u2192 Y is said to be onto (or surjective), if every element\nof Y is the image of some element of X under f, i e , for every y \u2208 Y, there exists an\nelement x in X such that f (x) = y"
    },
    {
        "Chapter": "1",
        "sentence_range": "534-537",
        "Text": "Definition 6 A function f : X \u2192 Y is said to be onto (or surjective), if every element\nof Y is the image of some element of X under f, i e , for every y \u2208 Y, there exists an\nelement x in X such that f (x) = y The function f3 and f4 in Fig 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "535-538",
        "Text": "e , for every y \u2208 Y, there exists an\nelement x in X such that f (x) = y The function f3 and f4 in Fig 1 2 (iii), (iv) are onto and the function f1 in Fig 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "536-539",
        "Text": ", for every y \u2208 Y, there exists an\nelement x in X such that f (x) = y The function f3 and f4 in Fig 1 2 (iii), (iv) are onto and the function f1 in Fig 1 2 (i) is\nnot onto as elements e, f in X2 are not the image of any element in X1 under f1"
    },
    {
        "Chapter": "1",
        "sentence_range": "537-540",
        "Text": "The function f3 and f4 in Fig 1 2 (iii), (iv) are onto and the function f1 in Fig 1 2 (i) is\nnot onto as elements e, f in X2 are not the image of any element in X1 under f1 Rationalised 2023-24\nMATHEMATICS\n8\nRemark  f : X \u2192 Y is onto if and only if Range of f = Y"
    },
    {
        "Chapter": "1",
        "sentence_range": "538-541",
        "Text": "2 (iii), (iv) are onto and the function f1 in Fig 1 2 (i) is\nnot onto as elements e, f in X2 are not the image of any element in X1 under f1 Rationalised 2023-24\nMATHEMATICS\n8\nRemark  f : X \u2192 Y is onto if and only if Range of f = Y Definition 7 A function f : X \u2192 Y is said to be one-one and onto (or bijective), if f is\nboth one-one and onto"
    },
    {
        "Chapter": "1",
        "sentence_range": "539-542",
        "Text": "2 (i) is\nnot onto as elements e, f in X2 are not the image of any element in X1 under f1 Rationalised 2023-24\nMATHEMATICS\n8\nRemark  f : X \u2192 Y is onto if and only if Range of f = Y Definition 7 A function f : X \u2192 Y is said to be one-one and onto (or bijective), if f is\nboth one-one and onto The function f4 in Fig 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "540-543",
        "Text": "Rationalised 2023-24\nMATHEMATICS\n8\nRemark  f : X \u2192 Y is onto if and only if Range of f = Y Definition 7 A function f : X \u2192 Y is said to be one-one and onto (or bijective), if f is\nboth one-one and onto The function f4 in Fig 1 2 (iv) is one-one and onto"
    },
    {
        "Chapter": "1",
        "sentence_range": "541-544",
        "Text": "Definition 7 A function f : X \u2192 Y is said to be one-one and onto (or bijective), if f is\nboth one-one and onto The function f4 in Fig 1 2 (iv) is one-one and onto Example 7 Let A be the set of all 50 students of Class X in a school"
    },
    {
        "Chapter": "1",
        "sentence_range": "542-545",
        "Text": "The function f4 in Fig 1 2 (iv) is one-one and onto Example 7 Let A be the set of all 50 students of Class X in a school Let f : A \u2192 N be\nfunction defined by f (x) = roll number of the student x"
    },
    {
        "Chapter": "1",
        "sentence_range": "543-546",
        "Text": "2 (iv) is one-one and onto Example 7 Let A be the set of all 50 students of Class X in a school Let f : A \u2192 N be\nfunction defined by f (x) = roll number of the student x Show that f is one-one\nbut not onto"
    },
    {
        "Chapter": "1",
        "sentence_range": "544-547",
        "Text": "Example 7 Let A be the set of all 50 students of Class X in a school Let f : A \u2192 N be\nfunction defined by f (x) = roll number of the student x Show that f is one-one\nbut not onto Solution No two different students of the class can have same roll number"
    },
    {
        "Chapter": "1",
        "sentence_range": "545-548",
        "Text": "Let f : A \u2192 N be\nfunction defined by f (x) = roll number of the student x Show that f is one-one\nbut not onto Solution No two different students of the class can have same roll number Therefore,\nf must be one-one"
    },
    {
        "Chapter": "1",
        "sentence_range": "546-549",
        "Text": "Show that f is one-one\nbut not onto Solution No two different students of the class can have same roll number Therefore,\nf must be one-one We can assume without any loss of generality that roll numbers of\nstudents are from 1 to 50"
    },
    {
        "Chapter": "1",
        "sentence_range": "547-550",
        "Text": "Solution No two different students of the class can have same roll number Therefore,\nf must be one-one We can assume without any loss of generality that roll numbers of\nstudents are from 1 to 50 This implies that 51 in N is not roll number of any student of\nthe class, so that 51 can not be image of any element of X under f"
    },
    {
        "Chapter": "1",
        "sentence_range": "548-551",
        "Text": "Therefore,\nf must be one-one We can assume without any loss of generality that roll numbers of\nstudents are from 1 to 50 This implies that 51 in N is not roll number of any student of\nthe class, so that 51 can not be image of any element of X under f Hence, f is not onto"
    },
    {
        "Chapter": "1",
        "sentence_range": "549-552",
        "Text": "We can assume without any loss of generality that roll numbers of\nstudents are from 1 to 50 This implies that 51 in N is not roll number of any student of\nthe class, so that 51 can not be image of any element of X under f Hence, f is not onto Example 8 Show that the function f : N \u2192 N, given by f(x) = 2x, is one-one but not\nonto"
    },
    {
        "Chapter": "1",
        "sentence_range": "550-553",
        "Text": "This implies that 51 in N is not roll number of any student of\nthe class, so that 51 can not be image of any element of X under f Hence, f is not onto Example 8 Show that the function f : N \u2192 N, given by f(x) = 2x, is one-one but not\nonto Solution The function f is one-one, for f (x1) = f(x2) \u21d2 2x1 = 2x2 \u21d2 x1 = x2"
    },
    {
        "Chapter": "1",
        "sentence_range": "551-554",
        "Text": "Hence, f is not onto Example 8 Show that the function f : N \u2192 N, given by f(x) = 2x, is one-one but not\nonto Solution The function f is one-one, for f (x1) = f(x2) \u21d2 2x1 = 2x2 \u21d2 x1 = x2 Further,\nf is not onto, as for 1 \u2208 N, there does not exist any x in N such that f(x) = 2x = 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "552-555",
        "Text": "Example 8 Show that the function f : N \u2192 N, given by f(x) = 2x, is one-one but not\nonto Solution The function f is one-one, for f (x1) = f(x2) \u21d2 2x1 = 2x2 \u21d2 x1 = x2 Further,\nf is not onto, as for 1 \u2208 N, there does not exist any x in N such that f(x) = 2x = 1 Fig 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "553-556",
        "Text": "Solution The function f is one-one, for f (x1) = f(x2) \u21d2 2x1 = 2x2 \u21d2 x1 = x2 Further,\nf is not onto, as for 1 \u2208 N, there does not exist any x in N such that f(x) = 2x = 1 Fig 1 2 (i) to (iv)\nRationalised 2023-24\nRELATIONS AND FUNCTIONS\n9\nExample 9  Prove that the function f : R \u2192 R, given by f (x) = 2x, is one-one and onto"
    },
    {
        "Chapter": "1",
        "sentence_range": "554-557",
        "Text": "Further,\nf is not onto, as for 1 \u2208 N, there does not exist any x in N such that f(x) = 2x = 1 Fig 1 2 (i) to (iv)\nRationalised 2023-24\nRELATIONS AND FUNCTIONS\n9\nExample 9  Prove that the function f : R \u2192 R, given by f (x) = 2x, is one-one and onto Solution  f is one-one, as f (x1) = f (x2) \u21d2 2x1 = 2x2 \u21d2 x1 = x2"
    },
    {
        "Chapter": "1",
        "sentence_range": "555-558",
        "Text": "Fig 1 2 (i) to (iv)\nRationalised 2023-24\nRELATIONS AND FUNCTIONS\n9\nExample 9  Prove that the function f : R \u2192 R, given by f (x) = 2x, is one-one and onto Solution  f is one-one, as f (x1) = f (x2) \u21d2 2x1 = 2x2 \u21d2 x1 = x2 Also, given any real\nnumber y in R, there exists 2\ny  in R such that f ( 2\ny ) = 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "556-559",
        "Text": "2 (i) to (iv)\nRationalised 2023-24\nRELATIONS AND FUNCTIONS\n9\nExample 9  Prove that the function f : R \u2192 R, given by f (x) = 2x, is one-one and onto Solution  f is one-one, as f (x1) = f (x2) \u21d2 2x1 = 2x2 \u21d2 x1 = x2 Also, given any real\nnumber y in R, there exists 2\ny  in R such that f ( 2\ny ) = 2 ( 2\ny ) = y"
    },
    {
        "Chapter": "1",
        "sentence_range": "557-560",
        "Text": "Solution  f is one-one, as f (x1) = f (x2) \u21d2 2x1 = 2x2 \u21d2 x1 = x2 Also, given any real\nnumber y in R, there exists 2\ny  in R such that f ( 2\ny ) = 2 ( 2\ny ) = y Hence, f is onto"
    },
    {
        "Chapter": "1",
        "sentence_range": "558-561",
        "Text": "Also, given any real\nnumber y in R, there exists 2\ny  in R such that f ( 2\ny ) = 2 ( 2\ny ) = y Hence, f is onto Fig 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "559-562",
        "Text": "( 2\ny ) = y Hence, f is onto Fig 1 3\nExample 10  Show that the function f : N \u2192 N, given by f (1) = f (2) = 1 and f(x) = x \u2013 1,\nfor every x > 2, is onto but not one-one"
    },
    {
        "Chapter": "1",
        "sentence_range": "560-563",
        "Text": "Hence, f is onto Fig 1 3\nExample 10  Show that the function f : N \u2192 N, given by f (1) = f (2) = 1 and f(x) = x \u2013 1,\nfor every x > 2, is onto but not one-one Solution  f is not one-one, as f (1) = f (2) = 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "561-564",
        "Text": "Fig 1 3\nExample 10  Show that the function f : N \u2192 N, given by f (1) = f (2) = 1 and f(x) = x \u2013 1,\nfor every x > 2, is onto but not one-one Solution  f is not one-one, as f (1) = f (2) = 1 But f is onto, as given any y \u2208 N, y \u2260 1,\nwe can choose x as y + 1 such that f (y + 1) = y + 1 \u2013 1 = y"
    },
    {
        "Chapter": "1",
        "sentence_range": "562-565",
        "Text": "3\nExample 10  Show that the function f : N \u2192 N, given by f (1) = f (2) = 1 and f(x) = x \u2013 1,\nfor every x > 2, is onto but not one-one Solution  f is not one-one, as f (1) = f (2) = 1 But f is onto, as given any y \u2208 N, y \u2260 1,\nwe can choose x as y + 1 such that f (y + 1) = y + 1 \u2013 1 = y Also for 1 \u2208 N, we\nhave f (1) = 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "563-566",
        "Text": "Solution  f is not one-one, as f (1) = f (2) = 1 But f is onto, as given any y \u2208 N, y \u2260 1,\nwe can choose x as y + 1 such that f (y + 1) = y + 1 \u2013 1 = y Also for 1 \u2208 N, we\nhave f (1) = 1 Example 11  Show that the function f : R \u2192 R,\ndefined as f (x) = x2, is neither one-one nor onto"
    },
    {
        "Chapter": "1",
        "sentence_range": "564-567",
        "Text": "But f is onto, as given any y \u2208 N, y \u2260 1,\nwe can choose x as y + 1 such that f (y + 1) = y + 1 \u2013 1 = y Also for 1 \u2208 N, we\nhave f (1) = 1 Example 11  Show that the function f : R \u2192 R,\ndefined as f (x) = x2, is neither one-one nor onto Solution  Since f (\u2013 1) = 1 = f (1), f is not one-\none"
    },
    {
        "Chapter": "1",
        "sentence_range": "565-568",
        "Text": "Also for 1 \u2208 N, we\nhave f (1) = 1 Example 11  Show that the function f : R \u2192 R,\ndefined as f (x) = x2, is neither one-one nor onto Solution  Since f (\u2013 1) = 1 = f (1), f is not one-\none Also, the element \u2013 2 in the co-domain R is\nnot image of any element x in the domain R\n(Why"
    },
    {
        "Chapter": "1",
        "sentence_range": "566-569",
        "Text": "Example 11  Show that the function f : R \u2192 R,\ndefined as f (x) = x2, is neither one-one nor onto Solution  Since f (\u2013 1) = 1 = f (1), f is not one-\none Also, the element \u2013 2 in the co-domain R is\nnot image of any element x in the domain R\n(Why )"
    },
    {
        "Chapter": "1",
        "sentence_range": "567-570",
        "Text": "Solution  Since f (\u2013 1) = 1 = f (1), f is not one-\none Also, the element \u2013 2 in the co-domain R is\nnot image of any element x in the domain R\n(Why ) Therefore f is not onto"
    },
    {
        "Chapter": "1",
        "sentence_range": "568-571",
        "Text": "Also, the element \u2013 2 in the co-domain R is\nnot image of any element x in the domain R\n(Why ) Therefore f is not onto Example 12  Show that f : N \u2192 N, given by\n1,if\nis odd,\n( )\n1,if\nis even\nx\nx\nf x\nx\nx\n+\n=   \n\u2212\n \nis both one-one and onto"
    },
    {
        "Chapter": "1",
        "sentence_range": "569-572",
        "Text": ") Therefore f is not onto Example 12  Show that f : N \u2192 N, given by\n1,if\nis odd,\n( )\n1,if\nis even\nx\nx\nf x\nx\nx\n+\n=   \n\u2212\n \nis both one-one and onto Fig 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "570-573",
        "Text": "Therefore f is not onto Example 12  Show that f : N \u2192 N, given by\n1,if\nis odd,\n( )\n1,if\nis even\nx\nx\nf x\nx\nx\n+\n=   \n\u2212\n \nis both one-one and onto Fig 1 4\nRationalised 2023-24\nMATHEMATICS\n10\nSolution  Suppose f (x1) = f (x2)"
    },
    {
        "Chapter": "1",
        "sentence_range": "571-574",
        "Text": "Example 12  Show that f : N \u2192 N, given by\n1,if\nis odd,\n( )\n1,if\nis even\nx\nx\nf x\nx\nx\n+\n=   \n\u2212\n \nis both one-one and onto Fig 1 4\nRationalised 2023-24\nMATHEMATICS\n10\nSolution  Suppose f (x1) = f (x2) Note that if x1 is odd and x2 is even, then we will have\nx1 + 1 = x2 \u2013 1, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "572-575",
        "Text": "Fig 1 4\nRationalised 2023-24\nMATHEMATICS\n10\nSolution  Suppose f (x1) = f (x2) Note that if x1 is odd and x2 is even, then we will have\nx1 + 1 = x2 \u2013 1, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "573-576",
        "Text": "4\nRationalised 2023-24\nMATHEMATICS\n10\nSolution  Suppose f (x1) = f (x2) Note that if x1 is odd and x2 is even, then we will have\nx1 + 1 = x2 \u2013 1, i e , x2 \u2013 x1 = 2 which is impossible"
    },
    {
        "Chapter": "1",
        "sentence_range": "574-577",
        "Text": "Note that if x1 is odd and x2 is even, then we will have\nx1 + 1 = x2 \u2013 1, i e , x2 \u2013 x1 = 2 which is impossible Similarly, the possibility of x1 being\neven and x2 being odd can also be ruled out, using the similar argument"
    },
    {
        "Chapter": "1",
        "sentence_range": "575-578",
        "Text": "e , x2 \u2013 x1 = 2 which is impossible Similarly, the possibility of x1 being\neven and x2 being odd can also be ruled out, using the similar argument Therefore,\nboth x1 and x2 must be either odd or even"
    },
    {
        "Chapter": "1",
        "sentence_range": "576-579",
        "Text": ", x2 \u2013 x1 = 2 which is impossible Similarly, the possibility of x1 being\neven and x2 being odd can also be ruled out, using the similar argument Therefore,\nboth x1 and x2 must be either odd or even Suppose both x1 and x2 are odd"
    },
    {
        "Chapter": "1",
        "sentence_range": "577-580",
        "Text": "Similarly, the possibility of x1 being\neven and x2 being odd can also be ruled out, using the similar argument Therefore,\nboth x1 and x2 must be either odd or even Suppose both x1 and x2 are odd Then\nf (x1) = f (x2) \u21d2 x1 + 1 = x2 + 1 \u21d2 x1 = x2"
    },
    {
        "Chapter": "1",
        "sentence_range": "578-581",
        "Text": "Therefore,\nboth x1 and x2 must be either odd or even Suppose both x1 and x2 are odd Then\nf (x1) = f (x2) \u21d2 x1 + 1 = x2 + 1 \u21d2 x1 = x2 Similarly, if both x1 and x2 are even, then also\nf (x1) = f (x2) \u21d2 x1 \u2013 1 = x2 \u2013 1 \u21d2 x1 = x2"
    },
    {
        "Chapter": "1",
        "sentence_range": "579-582",
        "Text": "Suppose both x1 and x2 are odd Then\nf (x1) = f (x2) \u21d2 x1 + 1 = x2 + 1 \u21d2 x1 = x2 Similarly, if both x1 and x2 are even, then also\nf (x1) = f (x2) \u21d2 x1 \u2013 1 = x2 \u2013 1 \u21d2 x1 = x2 Thus, f is one-one"
    },
    {
        "Chapter": "1",
        "sentence_range": "580-583",
        "Text": "Then\nf (x1) = f (x2) \u21d2 x1 + 1 = x2 + 1 \u21d2 x1 = x2 Similarly, if both x1 and x2 are even, then also\nf (x1) = f (x2) \u21d2 x1 \u2013 1 = x2 \u2013 1 \u21d2 x1 = x2 Thus, f is one-one Also, any odd number\n2r + 1 in the co-domain N is the image of 2r + 2 in the domain N and any even number\n2r in the co-domain N is the image of 2r \u2013 1 in the domain N"
    },
    {
        "Chapter": "1",
        "sentence_range": "581-584",
        "Text": "Similarly, if both x1 and x2 are even, then also\nf (x1) = f (x2) \u21d2 x1 \u2013 1 = x2 \u2013 1 \u21d2 x1 = x2 Thus, f is one-one Also, any odd number\n2r + 1 in the co-domain N is the image of 2r + 2 in the domain N and any even number\n2r in the co-domain N is the image of 2r \u2013 1 in the domain N Thus, f is onto"
    },
    {
        "Chapter": "1",
        "sentence_range": "582-585",
        "Text": "Thus, f is one-one Also, any odd number\n2r + 1 in the co-domain N is the image of 2r + 2 in the domain N and any even number\n2r in the co-domain N is the image of 2r \u2013 1 in the domain N Thus, f is onto Example 13 Show that an onto function f : {1, 2, 3} \u2192 {1, 2, 3} is always one-one"
    },
    {
        "Chapter": "1",
        "sentence_range": "583-586",
        "Text": "Also, any odd number\n2r + 1 in the co-domain N is the image of 2r + 2 in the domain N and any even number\n2r in the co-domain N is the image of 2r \u2013 1 in the domain N Thus, f is onto Example 13 Show that an onto function f : {1, 2, 3} \u2192 {1, 2, 3} is always one-one Solution Suppose f is not one-one"
    },
    {
        "Chapter": "1",
        "sentence_range": "584-587",
        "Text": "Thus, f is onto Example 13 Show that an onto function f : {1, 2, 3} \u2192 {1, 2, 3} is always one-one Solution Suppose f is not one-one Then there exists two elements, say 1 and 2 in the\ndomain whose image in the co-domain is same"
    },
    {
        "Chapter": "1",
        "sentence_range": "585-588",
        "Text": "Example 13 Show that an onto function f : {1, 2, 3} \u2192 {1, 2, 3} is always one-one Solution Suppose f is not one-one Then there exists two elements, say 1 and 2 in the\ndomain whose image in the co-domain is same Also, the image of 3 under f can be\nonly one element"
    },
    {
        "Chapter": "1",
        "sentence_range": "586-589",
        "Text": "Solution Suppose f is not one-one Then there exists two elements, say 1 and 2 in the\ndomain whose image in the co-domain is same Also, the image of 3 under f can be\nonly one element Therefore, the range set can have at the most two elements of the\nco-domain {1, 2, 3}, showing that f  is not onto, a contradiction"
    },
    {
        "Chapter": "1",
        "sentence_range": "587-590",
        "Text": "Then there exists two elements, say 1 and 2 in the\ndomain whose image in the co-domain is same Also, the image of 3 under f can be\nonly one element Therefore, the range set can have at the most two elements of the\nco-domain {1, 2, 3}, showing that f  is not onto, a contradiction Hence, f must be one-one"
    },
    {
        "Chapter": "1",
        "sentence_range": "588-591",
        "Text": "Also, the image of 3 under f can be\nonly one element Therefore, the range set can have at the most two elements of the\nco-domain {1, 2, 3}, showing that f  is not onto, a contradiction Hence, f must be one-one Example 14 Show that a one-one function f : {1, 2, 3} \u2192 {1, 2, 3} must be onto"
    },
    {
        "Chapter": "1",
        "sentence_range": "589-592",
        "Text": "Therefore, the range set can have at the most two elements of the\nco-domain {1, 2, 3}, showing that f  is not onto, a contradiction Hence, f must be one-one Example 14 Show that a one-one function f : {1, 2, 3} \u2192 {1, 2, 3} must be onto Solution Since f is one-one, three elements of {1, 2, 3} must be taken to 3 different\nelements of the co-domain {1, 2, 3} under f"
    },
    {
        "Chapter": "1",
        "sentence_range": "590-593",
        "Text": "Hence, f must be one-one Example 14 Show that a one-one function f : {1, 2, 3} \u2192 {1, 2, 3} must be onto Solution Since f is one-one, three elements of {1, 2, 3} must be taken to 3 different\nelements of the co-domain {1, 2, 3} under f Hence, f has to be onto"
    },
    {
        "Chapter": "1",
        "sentence_range": "591-594",
        "Text": "Example 14 Show that a one-one function f : {1, 2, 3} \u2192 {1, 2, 3} must be onto Solution Since f is one-one, three elements of {1, 2, 3} must be taken to 3 different\nelements of the co-domain {1, 2, 3} under f Hence, f has to be onto Remark The results mentioned in Examples 13 and 14 are also true for an arbitrary\nfinite set X, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "592-595",
        "Text": "Solution Since f is one-one, three elements of {1, 2, 3} must be taken to 3 different\nelements of the co-domain {1, 2, 3} under f Hence, f has to be onto Remark The results mentioned in Examples 13 and 14 are also true for an arbitrary\nfinite set X, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "593-596",
        "Text": "Hence, f has to be onto Remark The results mentioned in Examples 13 and 14 are also true for an arbitrary\nfinite set X, i e , a one-one function f : X \u2192 X is necessarily onto and an onto map\nf : X \u2192 X is necessarily one-one, for every finite set X"
    },
    {
        "Chapter": "1",
        "sentence_range": "594-597",
        "Text": "Remark The results mentioned in Examples 13 and 14 are also true for an arbitrary\nfinite set X, i e , a one-one function f : X \u2192 X is necessarily onto and an onto map\nf : X \u2192 X is necessarily one-one, for every finite set X In contrast to this, Examples 8\nand 10 show that for an infinite set, this may not be true"
    },
    {
        "Chapter": "1",
        "sentence_range": "595-598",
        "Text": "e , a one-one function f : X \u2192 X is necessarily onto and an onto map\nf : X \u2192 X is necessarily one-one, for every finite set X In contrast to this, Examples 8\nand 10 show that for an infinite set, this may not be true In fact, this is a characteristic\ndifference between a finite and an infinite set"
    },
    {
        "Chapter": "1",
        "sentence_range": "596-599",
        "Text": ", a one-one function f : X \u2192 X is necessarily onto and an onto map\nf : X \u2192 X is necessarily one-one, for every finite set X In contrast to this, Examples 8\nand 10 show that for an infinite set, this may not be true In fact, this is a characteristic\ndifference between a finite and an infinite set EXERCISE 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "597-600",
        "Text": "In contrast to this, Examples 8\nand 10 show that for an infinite set, this may not be true In fact, this is a characteristic\ndifference between a finite and an infinite set EXERCISE 1 2\n1"
    },
    {
        "Chapter": "1",
        "sentence_range": "598-601",
        "Text": "In fact, this is a characteristic\ndifference between a finite and an infinite set EXERCISE 1 2\n1 Show that the function f : R\u2217\u2217\u2217\u2217\u2217 \u2192 R\u2217\u2217\u2217\u2217\u2217 defined by f (x) = 1\nx  is one-one and onto,\nwhere R\u2217\u2217\u2217\u2217\u2217 is the set of all non-zero real numbers"
    },
    {
        "Chapter": "1",
        "sentence_range": "599-602",
        "Text": "EXERCISE 1 2\n1 Show that the function f : R\u2217\u2217\u2217\u2217\u2217 \u2192 R\u2217\u2217\u2217\u2217\u2217 defined by f (x) = 1\nx  is one-one and onto,\nwhere R\u2217\u2217\u2217\u2217\u2217 is the set of all non-zero real numbers Is the result true, if the domain\nR\u2217\u2217\u2217\u2217\u2217 is replaced by N with co-domain being same as R\u2217\u2217\u2217\u2217\u2217"
    },
    {
        "Chapter": "1",
        "sentence_range": "600-603",
        "Text": "2\n1 Show that the function f : R\u2217\u2217\u2217\u2217\u2217 \u2192 R\u2217\u2217\u2217\u2217\u2217 defined by f (x) = 1\nx  is one-one and onto,\nwhere R\u2217\u2217\u2217\u2217\u2217 is the set of all non-zero real numbers Is the result true, if the domain\nR\u2217\u2217\u2217\u2217\u2217 is replaced by N with co-domain being same as R\u2217\u2217\u2217\u2217\u2217 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "601-604",
        "Text": "Show that the function f : R\u2217\u2217\u2217\u2217\u2217 \u2192 R\u2217\u2217\u2217\u2217\u2217 defined by f (x) = 1\nx  is one-one and onto,\nwhere R\u2217\u2217\u2217\u2217\u2217 is the set of all non-zero real numbers Is the result true, if the domain\nR\u2217\u2217\u2217\u2217\u2217 is replaced by N with co-domain being same as R\u2217\u2217\u2217\u2217\u2217 2 Check the injectivity and surjectivity of the following functions:\n(i) f : N \u2192 N given by f(x) = x2\n(ii) f : Z \u2192 Z given by f(x) = x2\n(iii) f : R \u2192 R given by f(x) = x2\n(iv) f : N \u2192 N given by f(x) = x3\n(v) f : Z \u2192 Z given by f(x) = x3\n3"
    },
    {
        "Chapter": "1",
        "sentence_range": "602-605",
        "Text": "Is the result true, if the domain\nR\u2217\u2217\u2217\u2217\u2217 is replaced by N with co-domain being same as R\u2217\u2217\u2217\u2217\u2217 2 Check the injectivity and surjectivity of the following functions:\n(i) f : N \u2192 N given by f(x) = x2\n(ii) f : Z \u2192 Z given by f(x) = x2\n(iii) f : R \u2192 R given by f(x) = x2\n(iv) f : N \u2192 N given by f(x) = x3\n(v) f : Z \u2192 Z given by f(x) = x3\n3 Prove that the Greatest Integer Function f : R \u2192 R, given by f(x) = [x], is neither\none-one nor onto, where [x] denotes the greatest integer less than or equal to x"
    },
    {
        "Chapter": "1",
        "sentence_range": "603-606",
        "Text": "2 Check the injectivity and surjectivity of the following functions:\n(i) f : N \u2192 N given by f(x) = x2\n(ii) f : Z \u2192 Z given by f(x) = x2\n(iii) f : R \u2192 R given by f(x) = x2\n(iv) f : N \u2192 N given by f(x) = x3\n(v) f : Z \u2192 Z given by f(x) = x3\n3 Prove that the Greatest Integer Function f : R \u2192 R, given by f(x) = [x], is neither\none-one nor onto, where [x] denotes the greatest integer less than or equal to x Rationalised 2023-24\nRELATIONS AND FUNCTIONS\n11\n4"
    },
    {
        "Chapter": "1",
        "sentence_range": "604-607",
        "Text": "Check the injectivity and surjectivity of the following functions:\n(i) f : N \u2192 N given by f(x) = x2\n(ii) f : Z \u2192 Z given by f(x) = x2\n(iii) f : R \u2192 R given by f(x) = x2\n(iv) f : N \u2192 N given by f(x) = x3\n(v) f : Z \u2192 Z given by f(x) = x3\n3 Prove that the Greatest Integer Function f : R \u2192 R, given by f(x) = [x], is neither\none-one nor onto, where [x] denotes the greatest integer less than or equal to x Rationalised 2023-24\nRELATIONS AND FUNCTIONS\n11\n4 Show that the Modulus Function f : R \u2192 R, given by f (x) = | x|, is neither one-\none nor onto, where | x | is x, if x is positive or 0 and |x | is \u2013 x, if x is negative"
    },
    {
        "Chapter": "1",
        "sentence_range": "605-608",
        "Text": "Prove that the Greatest Integer Function f : R \u2192 R, given by f(x) = [x], is neither\none-one nor onto, where [x] denotes the greatest integer less than or equal to x Rationalised 2023-24\nRELATIONS AND FUNCTIONS\n11\n4 Show that the Modulus Function f : R \u2192 R, given by f (x) = | x|, is neither one-\none nor onto, where | x | is x, if x is positive or 0 and |x | is \u2013 x, if x is negative 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "606-609",
        "Text": "Rationalised 2023-24\nRELATIONS AND FUNCTIONS\n11\n4 Show that the Modulus Function f : R \u2192 R, given by f (x) = | x|, is neither one-\none nor onto, where | x | is x, if x is positive or 0 and |x | is \u2013 x, if x is negative 5 Show that the Signum Function f : R \u2192 R, given by\nf x\nx\nx\nx\n( )\n,\n,\n\ufffd ,\n=\n>\n=\n<\n\uf8f1\n\uf8f2\uf8f4\n\uf8f3\uf8f4\n1\n0\n0\n0\n1\n0\nif\nif\nif\nis neither one-one nor onto"
    },
    {
        "Chapter": "1",
        "sentence_range": "607-610",
        "Text": "Show that the Modulus Function f : R \u2192 R, given by f (x) = | x|, is neither one-\none nor onto, where | x | is x, if x is positive or 0 and |x | is \u2013 x, if x is negative 5 Show that the Signum Function f : R \u2192 R, given by\nf x\nx\nx\nx\n( )\n,\n,\n\ufffd ,\n=\n>\n=\n<\n\uf8f1\n\uf8f2\uf8f4\n\uf8f3\uf8f4\n1\n0\n0\n0\n1\n0\nif\nif\nif\nis neither one-one nor onto 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "608-611",
        "Text": "5 Show that the Signum Function f : R \u2192 R, given by\nf x\nx\nx\nx\n( )\n,\n,\n\ufffd ,\n=\n>\n=\n<\n\uf8f1\n\uf8f2\uf8f4\n\uf8f3\uf8f4\n1\n0\n0\n0\n1\n0\nif\nif\nif\nis neither one-one nor onto 6 Let A = {1, 2, 3}, B = {4, 5, 6, 7} and let f = {(1, 4), (2, 5), (3, 6)} be a function\nfrom A to B"
    },
    {
        "Chapter": "1",
        "sentence_range": "609-612",
        "Text": "Show that the Signum Function f : R \u2192 R, given by\nf x\nx\nx\nx\n( )\n,\n,\n\ufffd ,\n=\n>\n=\n<\n\uf8f1\n\uf8f2\uf8f4\n\uf8f3\uf8f4\n1\n0\n0\n0\n1\n0\nif\nif\nif\nis neither one-one nor onto 6 Let A = {1, 2, 3}, B = {4, 5, 6, 7} and let f = {(1, 4), (2, 5), (3, 6)} be a function\nfrom A to B Show that f is one-one"
    },
    {
        "Chapter": "1",
        "sentence_range": "610-613",
        "Text": "6 Let A = {1, 2, 3}, B = {4, 5, 6, 7} and let f = {(1, 4), (2, 5), (3, 6)} be a function\nfrom A to B Show that f is one-one 7"
    },
    {
        "Chapter": "1",
        "sentence_range": "611-614",
        "Text": "Let A = {1, 2, 3}, B = {4, 5, 6, 7} and let f = {(1, 4), (2, 5), (3, 6)} be a function\nfrom A to B Show that f is one-one 7 In each of the following cases, state whether the function is one-one, onto or\nbijective"
    },
    {
        "Chapter": "1",
        "sentence_range": "612-615",
        "Text": "Show that f is one-one 7 In each of the following cases, state whether the function is one-one, onto or\nbijective Justify your answer"
    },
    {
        "Chapter": "1",
        "sentence_range": "613-616",
        "Text": "7 In each of the following cases, state whether the function is one-one, onto or\nbijective Justify your answer (i) f : R \u2192 R defined by f (x) = 3 \u2013 4x\n(ii) f : R \u2192 R defined by f (x) = 1 + x2\n8"
    },
    {
        "Chapter": "1",
        "sentence_range": "614-617",
        "Text": "In each of the following cases, state whether the function is one-one, onto or\nbijective Justify your answer (i) f : R \u2192 R defined by f (x) = 3 \u2013 4x\n(ii) f : R \u2192 R defined by f (x) = 1 + x2\n8 Let A and B be sets"
    },
    {
        "Chapter": "1",
        "sentence_range": "615-618",
        "Text": "Justify your answer (i) f : R \u2192 R defined by f (x) = 3 \u2013 4x\n(ii) f : R \u2192 R defined by f (x) = 1 + x2\n8 Let A and B be sets Show that f : A \u00d7 B \u2192 B \u00d7 A such that f (a, b) = (b, a) is\nbijective function"
    },
    {
        "Chapter": "1",
        "sentence_range": "616-619",
        "Text": "(i) f : R \u2192 R defined by f (x) = 3 \u2013 4x\n(ii) f : R \u2192 R defined by f (x) = 1 + x2\n8 Let A and B be sets Show that f : A \u00d7 B \u2192 B \u00d7 A such that f (a, b) = (b, a) is\nbijective function 9"
    },
    {
        "Chapter": "1",
        "sentence_range": "617-620",
        "Text": "Let A and B be sets Show that f : A \u00d7 B \u2192 B \u00d7 A such that f (a, b) = (b, a) is\nbijective function 9 Let f : N \u2192 N be defined by f (n) = \nn\nn\nn\nn\n+\n\uf8f1\n\uf8f2\n\uf8f4\uf8f4\n\uf8f4\uf8f3\n\uf8f4\n1\n2\n2\n,\n,\nif\nis odd\nif\nis even\n for all n \u2208 N"
    },
    {
        "Chapter": "1",
        "sentence_range": "618-621",
        "Text": "Show that f : A \u00d7 B \u2192 B \u00d7 A such that f (a, b) = (b, a) is\nbijective function 9 Let f : N \u2192 N be defined by f (n) = \nn\nn\nn\nn\n+\n\uf8f1\n\uf8f2\n\uf8f4\uf8f4\n\uf8f4\uf8f3\n\uf8f4\n1\n2\n2\n,\n,\nif\nis odd\nif\nis even\n for all n \u2208 N State whether the function f is bijective"
    },
    {
        "Chapter": "1",
        "sentence_range": "619-622",
        "Text": "9 Let f : N \u2192 N be defined by f (n) = \nn\nn\nn\nn\n+\n\uf8f1\n\uf8f2\n\uf8f4\uf8f4\n\uf8f4\uf8f3\n\uf8f4\n1\n2\n2\n,\n,\nif\nis odd\nif\nis even\n for all n \u2208 N State whether the function f is bijective Justify your answer"
    },
    {
        "Chapter": "1",
        "sentence_range": "620-623",
        "Text": "Let f : N \u2192 N be defined by f (n) = \nn\nn\nn\nn\n+\n\uf8f1\n\uf8f2\n\uf8f4\uf8f4\n\uf8f4\uf8f3\n\uf8f4\n1\n2\n2\n,\n,\nif\nis odd\nif\nis even\n for all n \u2208 N State whether the function f is bijective Justify your answer 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "621-624",
        "Text": "State whether the function f is bijective Justify your answer 10 Let A = R \u2013 {3} and B = R \u2013 {1}"
    },
    {
        "Chapter": "1",
        "sentence_range": "622-625",
        "Text": "Justify your answer 10 Let A = R \u2013 {3} and B = R \u2013 {1} Consider the function  f : A \u2192 B defined by\nf (x) = \n2\n3\nxx\n\u2212\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\u2212\n\uf8ed\n\uf8f8"
    },
    {
        "Chapter": "1",
        "sentence_range": "623-626",
        "Text": "10 Let A = R \u2013 {3} and B = R \u2013 {1} Consider the function  f : A \u2192 B defined by\nf (x) = \n2\n3\nxx\n\u2212\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\u2212\n\uf8ed\n\uf8f8 Is f one-one and onto"
    },
    {
        "Chapter": "1",
        "sentence_range": "624-627",
        "Text": "Let A = R \u2013 {3} and B = R \u2013 {1} Consider the function  f : A \u2192 B defined by\nf (x) = \n2\n3\nxx\n\u2212\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\u2212\n\uf8ed\n\uf8f8 Is f one-one and onto Justify your answer"
    },
    {
        "Chapter": "1",
        "sentence_range": "625-628",
        "Text": "Consider the function  f : A \u2192 B defined by\nf (x) = \n2\n3\nxx\n\u2212\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\u2212\n\uf8ed\n\uf8f8 Is f one-one and onto Justify your answer 11"
    },
    {
        "Chapter": "1",
        "sentence_range": "626-629",
        "Text": "Is f one-one and onto Justify your answer 11 Let f : R \u2192 R be defined as f(x) = x4"
    },
    {
        "Chapter": "1",
        "sentence_range": "627-630",
        "Text": "Justify your answer 11 Let f : R \u2192 R be defined as f(x) = x4 Choose the correct answer"
    },
    {
        "Chapter": "1",
        "sentence_range": "628-631",
        "Text": "11 Let f : R \u2192 R be defined as f(x) = x4 Choose the correct answer (A) f is one-one onto\n(B) f is many-one onto\n(C) f is one-one but not onto\n(D) f is neither one-one nor onto"
    },
    {
        "Chapter": "1",
        "sentence_range": "629-632",
        "Text": "Let f : R \u2192 R be defined as f(x) = x4 Choose the correct answer (A) f is one-one onto\n(B) f is many-one onto\n(C) f is one-one but not onto\n(D) f is neither one-one nor onto 12"
    },
    {
        "Chapter": "1",
        "sentence_range": "630-633",
        "Text": "Choose the correct answer (A) f is one-one onto\n(B) f is many-one onto\n(C) f is one-one but not onto\n(D) f is neither one-one nor onto 12 Let f : R \u2192 R be defined as f (x) = 3x"
    },
    {
        "Chapter": "1",
        "sentence_range": "631-634",
        "Text": "(A) f is one-one onto\n(B) f is many-one onto\n(C) f is one-one but not onto\n(D) f is neither one-one nor onto 12 Let f : R \u2192 R be defined as f (x) = 3x Choose the correct answer"
    },
    {
        "Chapter": "1",
        "sentence_range": "632-635",
        "Text": "12 Let f : R \u2192 R be defined as f (x) = 3x Choose the correct answer (A) f is one-one onto\n(B) f is many-one onto\n(C) f is one-one but not onto\n(D) f is neither one-one nor onto"
    },
    {
        "Chapter": "1",
        "sentence_range": "633-636",
        "Text": "Let f : R \u2192 R be defined as f (x) = 3x Choose the correct answer (A) f is one-one onto\n(B) f is many-one onto\n(C) f is one-one but not onto\n(D) f is neither one-one nor onto Rationalised 2023-24\nMATHEMATICS\n12\n1"
    },
    {
        "Chapter": "1",
        "sentence_range": "634-637",
        "Text": "Choose the correct answer (A) f is one-one onto\n(B) f is many-one onto\n(C) f is one-one but not onto\n(D) f is neither one-one nor onto Rationalised 2023-24\nMATHEMATICS\n12\n1 4  Composition of Functions and Invertible Function\nDefinition 8 Let f : A \u2192 B and g : B \u2192 C be two functions"
    },
    {
        "Chapter": "1",
        "sentence_range": "635-638",
        "Text": "(A) f is one-one onto\n(B) f is many-one onto\n(C) f is one-one but not onto\n(D) f is neither one-one nor onto Rationalised 2023-24\nMATHEMATICS\n12\n1 4  Composition of Functions and Invertible Function\nDefinition 8 Let f : A \u2192 B and g : B \u2192 C be two functions Then the composition of\nf and g, denoted by gof, is defined as the function gof : A \u2192 C given by\ngof (x) = g(f (x)), \u2200  x \u2208 A"
    },
    {
        "Chapter": "1",
        "sentence_range": "636-639",
        "Text": "Rationalised 2023-24\nMATHEMATICS\n12\n1 4  Composition of Functions and Invertible Function\nDefinition 8 Let f : A \u2192 B and g : B \u2192 C be two functions Then the composition of\nf and g, denoted by gof, is defined as the function gof : A \u2192 C given by\ngof (x) = g(f (x)), \u2200  x \u2208 A Fig 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "637-640",
        "Text": "4  Composition of Functions and Invertible Function\nDefinition 8 Let f : A \u2192 B and g : B \u2192 C be two functions Then the composition of\nf and g, denoted by gof, is defined as the function gof : A \u2192 C given by\ngof (x) = g(f (x)), \u2200  x \u2208 A Fig 1 5\nExample 15 Let f : {2, 3, 4, 5} \u2192 {3, 4, 5, 9} and g : {3, 4, 5, 9} \u2192 {7, 11, 15} be\nfunctions defined as f (2) = 3, f(3) = 4, f(4) =  f(5) = 5 and g (3) = g (4) = 7 and\ng (5) =  g(9) = 11"
    },
    {
        "Chapter": "1",
        "sentence_range": "638-641",
        "Text": "Then the composition of\nf and g, denoted by gof, is defined as the function gof : A \u2192 C given by\ngof (x) = g(f (x)), \u2200  x \u2208 A Fig 1 5\nExample 15 Let f : {2, 3, 4, 5} \u2192 {3, 4, 5, 9} and g : {3, 4, 5, 9} \u2192 {7, 11, 15} be\nfunctions defined as f (2) = 3, f(3) = 4, f(4) =  f(5) = 5 and g (3) = g (4) = 7 and\ng (5) =  g(9) = 11 Find gof"
    },
    {
        "Chapter": "1",
        "sentence_range": "639-642",
        "Text": "Fig 1 5\nExample 15 Let f : {2, 3, 4, 5} \u2192 {3, 4, 5, 9} and g : {3, 4, 5, 9} \u2192 {7, 11, 15} be\nfunctions defined as f (2) = 3, f(3) = 4, f(4) =  f(5) = 5 and g (3) = g (4) = 7 and\ng (5) =  g(9) = 11 Find gof Solution We have gof (2) =  g (f (2)) = g (3) = 7, gof (3) =  g (f (3)) = g (4) = 7,\ngof(4) =  g (f(4)) = g (5) = 11 and gof(5) =  g (5) = 11"
    },
    {
        "Chapter": "1",
        "sentence_range": "640-643",
        "Text": "5\nExample 15 Let f : {2, 3, 4, 5} \u2192 {3, 4, 5, 9} and g : {3, 4, 5, 9} \u2192 {7, 11, 15} be\nfunctions defined as f (2) = 3, f(3) = 4, f(4) =  f(5) = 5 and g (3) = g (4) = 7 and\ng (5) =  g(9) = 11 Find gof Solution We have gof (2) =  g (f (2)) = g (3) = 7, gof (3) =  g (f (3)) = g (4) = 7,\ngof(4) =  g (f(4)) = g (5) = 11 and gof(5) =  g (5) = 11 Example 16 Find gof and fog, if f : R \u2192 R and g : R \u2192 R are given by f(x) = cos x\nand g(x) = 3x2"
    },
    {
        "Chapter": "1",
        "sentence_range": "641-644",
        "Text": "Find gof Solution We have gof (2) =  g (f (2)) = g (3) = 7, gof (3) =  g (f (3)) = g (4) = 7,\ngof(4) =  g (f(4)) = g (5) = 11 and gof(5) =  g (5) = 11 Example 16 Find gof and fog, if f : R \u2192 R and g : R \u2192 R are given by f(x) = cos x\nand g(x) = 3x2 Show that gof \u2260 fog"
    },
    {
        "Chapter": "1",
        "sentence_range": "642-645",
        "Text": "Solution We have gof (2) =  g (f (2)) = g (3) = 7, gof (3) =  g (f (3)) = g (4) = 7,\ngof(4) =  g (f(4)) = g (5) = 11 and gof(5) =  g (5) = 11 Example 16 Find gof and fog, if f : R \u2192 R and g : R \u2192 R are given by f(x) = cos x\nand g(x) = 3x2 Show that gof \u2260 fog Solution We have gof(x) =  g (f(x)) = g(cos x) = 3 (cos x)2 = 3 cos2 x"
    },
    {
        "Chapter": "1",
        "sentence_range": "643-646",
        "Text": "Example 16 Find gof and fog, if f : R \u2192 R and g : R \u2192 R are given by f(x) = cos x\nand g(x) = 3x2 Show that gof \u2260 fog Solution We have gof(x) =  g (f(x)) = g(cos x) = 3 (cos x)2 = 3 cos2 x Similarly,\nfog(x) =  f(g(x)) =  f (3x2) = cos (3x2)"
    },
    {
        "Chapter": "1",
        "sentence_range": "644-647",
        "Text": "Show that gof \u2260 fog Solution We have gof(x) =  g (f(x)) = g(cos x) = 3 (cos x)2 = 3 cos2 x Similarly,\nfog(x) =  f(g(x)) =  f (3x2) = cos (3x2) Note that 3cos2 x \u2260 cos 3x2, for x = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "645-648",
        "Text": "Solution We have gof(x) =  g (f(x)) = g(cos x) = 3 (cos x)2 = 3 cos2 x Similarly,\nfog(x) =  f(g(x)) =  f (3x2) = cos (3x2) Note that 3cos2 x \u2260 cos 3x2, for x = 0 Hence,\ngof \u2260 fog"
    },
    {
        "Chapter": "1",
        "sentence_range": "646-649",
        "Text": "Similarly,\nfog(x) =  f(g(x)) =  f (3x2) = cos (3x2) Note that 3cos2 x \u2260 cos 3x2, for x = 0 Hence,\ngof \u2260 fog Definition 9 A function f : X \u2192 Y is defined to be invertible, if there exists a function\ng : Y \u2192 X such that  gof = IX and fog = IY"
    },
    {
        "Chapter": "1",
        "sentence_range": "647-650",
        "Text": "Note that 3cos2 x \u2260 cos 3x2, for x = 0 Hence,\ngof \u2260 fog Definition 9 A function f : X \u2192 Y is defined to be invertible, if there exists a function\ng : Y \u2192 X such that  gof = IX and fog = IY The function g is called the inverse of f  and\nis denoted by f \u20131"
    },
    {
        "Chapter": "1",
        "sentence_range": "648-651",
        "Text": "Hence,\ngof \u2260 fog Definition 9 A function f : X \u2192 Y is defined to be invertible, if there exists a function\ng : Y \u2192 X such that  gof = IX and fog = IY The function g is called the inverse of f  and\nis denoted by f \u20131 Thus, if f is invertible, then f must be one-one and onto and conversely, if f is\none-one and onto, then f must be invertible"
    },
    {
        "Chapter": "1",
        "sentence_range": "649-652",
        "Text": "Definition 9 A function f : X \u2192 Y is defined to be invertible, if there exists a function\ng : Y \u2192 X such that  gof = IX and fog = IY The function g is called the inverse of f  and\nis denoted by f \u20131 Thus, if f is invertible, then f must be one-one and onto and conversely, if f is\none-one and onto, then f must be invertible This fact significantly helps for proving a\nfunction f to be invertible by showing that f is one-one and onto, specially when the\nactual inverse of f is not to be determined"
    },
    {
        "Chapter": "1",
        "sentence_range": "650-653",
        "Text": "The function g is called the inverse of f  and\nis denoted by f \u20131 Thus, if f is invertible, then f must be one-one and onto and conversely, if f is\none-one and onto, then f must be invertible This fact significantly helps for proving a\nfunction f to be invertible by showing that f is one-one and onto, specially when the\nactual inverse of f is not to be determined Example 17 Let f : N \u2192 Y be a function defined as f (x) = 4x + 3, where,\nY = {y \u2208 N: y = 4x + 3 for some x \u2208 N}"
    },
    {
        "Chapter": "1",
        "sentence_range": "651-654",
        "Text": "Thus, if f is invertible, then f must be one-one and onto and conversely, if f is\none-one and onto, then f must be invertible This fact significantly helps for proving a\nfunction f to be invertible by showing that f is one-one and onto, specially when the\nactual inverse of f is not to be determined Example 17 Let f : N \u2192 Y be a function defined as f (x) = 4x + 3, where,\nY = {y \u2208 N: y = 4x + 3 for some x \u2208 N} Show that f is invertible"
    },
    {
        "Chapter": "1",
        "sentence_range": "652-655",
        "Text": "This fact significantly helps for proving a\nfunction f to be invertible by showing that f is one-one and onto, specially when the\nactual inverse of f is not to be determined Example 17 Let f : N \u2192 Y be a function defined as f (x) = 4x + 3, where,\nY = {y \u2208 N: y = 4x + 3 for some x \u2208 N} Show that f is invertible Find the inverse"
    },
    {
        "Chapter": "1",
        "sentence_range": "653-656",
        "Text": "Example 17 Let f : N \u2192 Y be a function defined as f (x) = 4x + 3, where,\nY = {y \u2208 N: y = 4x + 3 for some x \u2208 N} Show that f is invertible Find the inverse Solution Consider an arbitrary element y of Y"
    },
    {
        "Chapter": "1",
        "sentence_range": "654-657",
        "Text": "Show that f is invertible Find the inverse Solution Consider an arbitrary element y of Y By the definition of Y, y = 4x + 3,\nfor some x in the domain N"
    },
    {
        "Chapter": "1",
        "sentence_range": "655-658",
        "Text": "Find the inverse Solution Consider an arbitrary element y of Y By the definition of Y, y = 4x + 3,\nfor some x in the domain N This shows that \n(\n3)\n4\ny\nx\n\u2212\n="
    },
    {
        "Chapter": "1",
        "sentence_range": "656-659",
        "Text": "Solution Consider an arbitrary element y of Y By the definition of Y, y = 4x + 3,\nfor some x in the domain N This shows that \n(\n3)\n4\ny\nx\n\u2212\n= Define g : Y \u2192 N by\nRationalised 2023-24\nRELATIONS AND FUNCTIONS\n13\n(\n3)\n( )\n4\ny\ng y\n\u2212\n="
    },
    {
        "Chapter": "1",
        "sentence_range": "657-660",
        "Text": "By the definition of Y, y = 4x + 3,\nfor some x in the domain N This shows that \n(\n3)\n4\ny\nx\n\u2212\n= Define g : Y \u2192 N by\nRationalised 2023-24\nRELATIONS AND FUNCTIONS\n13\n(\n3)\n( )\n4\ny\ng y\n\u2212\n= Now, gof(x) = g (f(x)) = g(4x + 3) = (4\n3\n3)\n4\nx\nx\n+\n\u2212\n=\n and\nfog(y) = f (g (y)) = f\n(\n3)\n4 (\n3)\n3\n4\n4\ny\ny\n\u2212\n\u2212\n\uf8eb\n\uf8f6 =\n+\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n = y \u2013 3 + 3 = y"
    },
    {
        "Chapter": "1",
        "sentence_range": "658-661",
        "Text": "This shows that \n(\n3)\n4\ny\nx\n\u2212\n= Define g : Y \u2192 N by\nRationalised 2023-24\nRELATIONS AND FUNCTIONS\n13\n(\n3)\n( )\n4\ny\ng y\n\u2212\n= Now, gof(x) = g (f(x)) = g(4x + 3) = (4\n3\n3)\n4\nx\nx\n+\n\u2212\n=\n and\nfog(y) = f (g (y)) = f\n(\n3)\n4 (\n3)\n3\n4\n4\ny\ny\n\u2212\n\u2212\n\uf8eb\n\uf8f6 =\n+\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n = y \u2013 3 + 3 = y This shows that gof = IN\nand fog = IY, which implies that f is invertible and g is the inverse of f"
    },
    {
        "Chapter": "1",
        "sentence_range": "659-662",
        "Text": "Define g : Y \u2192 N by\nRationalised 2023-24\nRELATIONS AND FUNCTIONS\n13\n(\n3)\n( )\n4\ny\ng y\n\u2212\n= Now, gof(x) = g (f(x)) = g(4x + 3) = (4\n3\n3)\n4\nx\nx\n+\n\u2212\n=\n and\nfog(y) = f (g (y)) = f\n(\n3)\n4 (\n3)\n3\n4\n4\ny\ny\n\u2212\n\u2212\n\uf8eb\n\uf8f6 =\n+\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n = y \u2013 3 + 3 = y This shows that gof = IN\nand fog = IY, which implies that f is invertible and g is the inverse of f Miscellaneous Examples\nExample 18 If R1\n and R2 are equivalence relations in a set A, show that R1 \u2229 R2 is\nalso an equivalence relation"
    },
    {
        "Chapter": "1",
        "sentence_range": "660-663",
        "Text": "Now, gof(x) = g (f(x)) = g(4x + 3) = (4\n3\n3)\n4\nx\nx\n+\n\u2212\n=\n and\nfog(y) = f (g (y)) = f\n(\n3)\n4 (\n3)\n3\n4\n4\ny\ny\n\u2212\n\u2212\n\uf8eb\n\uf8f6 =\n+\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n = y \u2013 3 + 3 = y This shows that gof = IN\nand fog = IY, which implies that f is invertible and g is the inverse of f Miscellaneous Examples\nExample 18 If R1\n and R2 are equivalence relations in a set A, show that R1 \u2229 R2 is\nalso an equivalence relation Solution Since R1\n and R2 are equivalence relations, (a, a) \u2208 R1, and (a, a) \u2208 R2 \u2200 a \u2208 A"
    },
    {
        "Chapter": "1",
        "sentence_range": "661-664",
        "Text": "This shows that gof = IN\nand fog = IY, which implies that f is invertible and g is the inverse of f Miscellaneous Examples\nExample 18 If R1\n and R2 are equivalence relations in a set A, show that R1 \u2229 R2 is\nalso an equivalence relation Solution Since R1\n and R2 are equivalence relations, (a, a) \u2208 R1, and (a, a) \u2208 R2 \u2200 a \u2208 A This implies that (a, a) \u2208 R1 \u2229 R2, \u2200 a, showing R1 \u2229 R2 is reflexive"
    },
    {
        "Chapter": "1",
        "sentence_range": "662-665",
        "Text": "Miscellaneous Examples\nExample 18 If R1\n and R2 are equivalence relations in a set A, show that R1 \u2229 R2 is\nalso an equivalence relation Solution Since R1\n and R2 are equivalence relations, (a, a) \u2208 R1, and (a, a) \u2208 R2 \u2200 a \u2208 A This implies that (a, a) \u2208 R1 \u2229 R2, \u2200 a, showing R1 \u2229 R2 is reflexive Further,\n(a, b) \u2208 R1 \u2229 R2 \u21d2 (a, b) \u2208 R1 and (a, b) \u2208 R2 \u21d2 (b, a) \u2208 R1 and (b, a) \u2208 R2 \u21d2\n(b, a) \u2208 R1 \u2229 R2, hence, R1 \u2229 R2 is symmetric"
    },
    {
        "Chapter": "1",
        "sentence_range": "663-666",
        "Text": "Solution Since R1\n and R2 are equivalence relations, (a, a) \u2208 R1, and (a, a) \u2208 R2 \u2200 a \u2208 A This implies that (a, a) \u2208 R1 \u2229 R2, \u2200 a, showing R1 \u2229 R2 is reflexive Further,\n(a, b) \u2208 R1 \u2229 R2 \u21d2 (a, b) \u2208 R1 and (a, b) \u2208 R2 \u21d2 (b, a) \u2208 R1 and (b, a) \u2208 R2 \u21d2\n(b, a) \u2208 R1 \u2229 R2, hence, R1 \u2229 R2 is symmetric Similarly, (a, b) \u2208 R1 \u2229 R2 and\n(b, c) \u2208 R1 \u2229 R2 \u21d2 (a, c) \u2208 R1 and (a, c) \u2208 R2 \u21d2 (a, c) \u2208 R1 \u2229 R2"
    },
    {
        "Chapter": "1",
        "sentence_range": "664-667",
        "Text": "This implies that (a, a) \u2208 R1 \u2229 R2, \u2200 a, showing R1 \u2229 R2 is reflexive Further,\n(a, b) \u2208 R1 \u2229 R2 \u21d2 (a, b) \u2208 R1 and (a, b) \u2208 R2 \u21d2 (b, a) \u2208 R1 and (b, a) \u2208 R2 \u21d2\n(b, a) \u2208 R1 \u2229 R2, hence, R1 \u2229 R2 is symmetric Similarly, (a, b) \u2208 R1 \u2229 R2 and\n(b, c) \u2208 R1 \u2229 R2 \u21d2 (a, c) \u2208 R1 and (a, c) \u2208 R2 \u21d2 (a, c) \u2208 R1 \u2229 R2 This shows that\nR1 \u2229 R2 is transitive"
    },
    {
        "Chapter": "1",
        "sentence_range": "665-668",
        "Text": "Further,\n(a, b) \u2208 R1 \u2229 R2 \u21d2 (a, b) \u2208 R1 and (a, b) \u2208 R2 \u21d2 (b, a) \u2208 R1 and (b, a) \u2208 R2 \u21d2\n(b, a) \u2208 R1 \u2229 R2, hence, R1 \u2229 R2 is symmetric Similarly, (a, b) \u2208 R1 \u2229 R2 and\n(b, c) \u2208 R1 \u2229 R2 \u21d2 (a, c) \u2208 R1 and (a, c) \u2208 R2 \u21d2 (a, c) \u2208 R1 \u2229 R2 This shows that\nR1 \u2229 R2 is transitive Thus, R1 \u2229 R2 is an equivalence relation"
    },
    {
        "Chapter": "1",
        "sentence_range": "666-669",
        "Text": "Similarly, (a, b) \u2208 R1 \u2229 R2 and\n(b, c) \u2208 R1 \u2229 R2 \u21d2 (a, c) \u2208 R1 and (a, c) \u2208 R2 \u21d2 (a, c) \u2208 R1 \u2229 R2 This shows that\nR1 \u2229 R2 is transitive Thus, R1 \u2229 R2 is an equivalence relation Example 19 Let R be a relation on the set A of ordered pairs of positive integers\ndefined by (x, y) R (u, v) if  and only if xv = yu"
    },
    {
        "Chapter": "1",
        "sentence_range": "667-670",
        "Text": "This shows that\nR1 \u2229 R2 is transitive Thus, R1 \u2229 R2 is an equivalence relation Example 19 Let R be a relation on the set A of ordered pairs of positive integers\ndefined by (x, y) R (u, v) if  and only if xv = yu Show that R is an equivalence relation"
    },
    {
        "Chapter": "1",
        "sentence_range": "668-671",
        "Text": "Thus, R1 \u2229 R2 is an equivalence relation Example 19 Let R be a relation on the set A of ordered pairs of positive integers\ndefined by (x, y) R (u, v) if  and only if xv = yu Show that R is an equivalence relation Solution Clearly, (x, y) R (x, y), \u2200 (x, y) \u2208 A, since xy = yx"
    },
    {
        "Chapter": "1",
        "sentence_range": "669-672",
        "Text": "Example 19 Let R be a relation on the set A of ordered pairs of positive integers\ndefined by (x, y) R (u, v) if  and only if xv = yu Show that R is an equivalence relation Solution Clearly, (x, y) R (x, y), \u2200 (x, y) \u2208 A, since xy = yx This shows that R is\nreflexive"
    },
    {
        "Chapter": "1",
        "sentence_range": "670-673",
        "Text": "Show that R is an equivalence relation Solution Clearly, (x, y) R (x, y), \u2200 (x, y) \u2208 A, since xy = yx This shows that R is\nreflexive Further, (x, y) R (u, v) \u21d2 xv = yu \u21d2 uy = vx and hence (u, v) R (x, y)"
    },
    {
        "Chapter": "1",
        "sentence_range": "671-674",
        "Text": "Solution Clearly, (x, y) R (x, y), \u2200 (x, y) \u2208 A, since xy = yx This shows that R is\nreflexive Further, (x, y) R (u, v) \u21d2 xv = yu \u21d2 uy = vx and hence (u, v) R (x, y) This\nshows that R is symmetric"
    },
    {
        "Chapter": "1",
        "sentence_range": "672-675",
        "Text": "This shows that R is\nreflexive Further, (x, y) R (u, v) \u21d2 xv = yu \u21d2 uy = vx and hence (u, v) R (x, y) This\nshows that R is symmetric Similarly, (x, y) R (u, v) and (u, v) R (a, b) \u21d2 xv = yu and\nub = va \u21d2 \na\na\nxv\nyu\nu\nu\n=\n\u21d2 \nb\na\nxv\nyu\nv\nu\n=\n \u21d2 xb = ya and hence (x, y) R (a, b)"
    },
    {
        "Chapter": "1",
        "sentence_range": "673-676",
        "Text": "Further, (x, y) R (u, v) \u21d2 xv = yu \u21d2 uy = vx and hence (u, v) R (x, y) This\nshows that R is symmetric Similarly, (x, y) R (u, v) and (u, v) R (a, b) \u21d2 xv = yu and\nub = va \u21d2 \na\na\nxv\nyu\nu\nu\n=\n\u21d2 \nb\na\nxv\nyu\nv\nu\n=\n \u21d2 xb = ya and hence (x, y) R (a, b) Thus, R\nis transitive"
    },
    {
        "Chapter": "1",
        "sentence_range": "674-677",
        "Text": "This\nshows that R is symmetric Similarly, (x, y) R (u, v) and (u, v) R (a, b) \u21d2 xv = yu and\nub = va \u21d2 \na\na\nxv\nyu\nu\nu\n=\n\u21d2 \nb\na\nxv\nyu\nv\nu\n=\n \u21d2 xb = ya and hence (x, y) R (a, b) Thus, R\nis transitive Thus, R is an equivalence relation"
    },
    {
        "Chapter": "1",
        "sentence_range": "675-678",
        "Text": "Similarly, (x, y) R (u, v) and (u, v) R (a, b) \u21d2 xv = yu and\nub = va \u21d2 \na\na\nxv\nyu\nu\nu\n=\n\u21d2 \nb\na\nxv\nyu\nv\nu\n=\n \u21d2 xb = ya and hence (x, y) R (a, b) Thus, R\nis transitive Thus, R is an equivalence relation Example 20 Let X = {1, 2, 3, 4, 5, 6, 7, 8, 9}"
    },
    {
        "Chapter": "1",
        "sentence_range": "676-679",
        "Text": "Thus, R\nis transitive Thus, R is an equivalence relation Example 20 Let X = {1, 2, 3, 4, 5, 6, 7, 8, 9} Let R1 be a relation in X given\nby R1 = {(x, y) : x \u2013 y is divisible by 3} and R2 be another relation on X given by\nR2 = {(x, y): {x, y} \u2282 {1, 4, 7}} or {x, y} \u2282 {2, 5, 8} or {x, y} \u2282 {3, 6, 9}}"
    },
    {
        "Chapter": "1",
        "sentence_range": "677-680",
        "Text": "Thus, R is an equivalence relation Example 20 Let X = {1, 2, 3, 4, 5, 6, 7, 8, 9} Let R1 be a relation in X given\nby R1 = {(x, y) : x \u2013 y is divisible by 3} and R2 be another relation on X given by\nR2 = {(x, y): {x, y} \u2282 {1, 4, 7}} or {x, y} \u2282 {2, 5, 8} or {x, y} \u2282 {3, 6, 9}} Show that\nR1 = R2"
    },
    {
        "Chapter": "1",
        "sentence_range": "678-681",
        "Text": "Example 20 Let X = {1, 2, 3, 4, 5, 6, 7, 8, 9} Let R1 be a relation in X given\nby R1 = {(x, y) : x \u2013 y is divisible by 3} and R2 be another relation on X given by\nR2 = {(x, y): {x, y} \u2282 {1, 4, 7}} or {x, y} \u2282 {2, 5, 8} or {x, y} \u2282 {3, 6, 9}} Show that\nR1 = R2 Solution Note that the characteristic of sets {1, 4, 7}, {2, 5, 8} and {3, 6, 9} is\nthat difference between any two elements of these sets is a multiple of 3"
    },
    {
        "Chapter": "1",
        "sentence_range": "679-682",
        "Text": "Let R1 be a relation in X given\nby R1 = {(x, y) : x \u2013 y is divisible by 3} and R2 be another relation on X given by\nR2 = {(x, y): {x, y} \u2282 {1, 4, 7}} or {x, y} \u2282 {2, 5, 8} or {x, y} \u2282 {3, 6, 9}} Show that\nR1 = R2 Solution Note that the characteristic of sets {1, 4, 7}, {2, 5, 8} and {3, 6, 9} is\nthat difference between any two elements of these sets is a multiple of 3 Therefore,\n(x, y) \u2208 R1 \u21d2 x \u2013 y is a multiple of 3 \u21d2 {x, y} \u2282 {1, 4, 7} or {x, y} \u2282 {2, 5, 8}\nor {x, y} \u2282 {3, 6, 9} \u21d2 (x, y) \u2208 R2"
    },
    {
        "Chapter": "1",
        "sentence_range": "680-683",
        "Text": "Show that\nR1 = R2 Solution Note that the characteristic of sets {1, 4, 7}, {2, 5, 8} and {3, 6, 9} is\nthat difference between any two elements of these sets is a multiple of 3 Therefore,\n(x, y) \u2208 R1 \u21d2 x \u2013 y is a multiple of 3 \u21d2 {x, y} \u2282 {1, 4, 7} or {x, y} \u2282 {2, 5, 8}\nor {x, y} \u2282 {3, 6, 9} \u21d2 (x, y) \u2208 R2 Hence, R1 \u2282 R2"
    },
    {
        "Chapter": "1",
        "sentence_range": "681-684",
        "Text": "Solution Note that the characteristic of sets {1, 4, 7}, {2, 5, 8} and {3, 6, 9} is\nthat difference between any two elements of these sets is a multiple of 3 Therefore,\n(x, y) \u2208 R1 \u21d2 x \u2013 y is a multiple of 3 \u21d2 {x, y} \u2282 {1, 4, 7} or {x, y} \u2282 {2, 5, 8}\nor {x, y} \u2282 {3, 6, 9} \u21d2 (x, y) \u2208 R2 Hence, R1 \u2282 R2 Similarly, {x, y} \u2208 R2 \u21d2 {x, y}\nRationalised 2023-24\nMATHEMATICS\n14\n\u2282 {1, 4, 7} or {x, y} \u2282 {2, 5, 8} or  {x, y} \u2282 {3, 6, 9} \u21d2 x \u2013 y is divisible by\n3 \u21d2 {x, y} \u2208 R1"
    },
    {
        "Chapter": "1",
        "sentence_range": "682-685",
        "Text": "Therefore,\n(x, y) \u2208 R1 \u21d2 x \u2013 y is a multiple of 3 \u21d2 {x, y} \u2282 {1, 4, 7} or {x, y} \u2282 {2, 5, 8}\nor {x, y} \u2282 {3, 6, 9} \u21d2 (x, y) \u2208 R2 Hence, R1 \u2282 R2 Similarly, {x, y} \u2208 R2 \u21d2 {x, y}\nRationalised 2023-24\nMATHEMATICS\n14\n\u2282 {1, 4, 7} or {x, y} \u2282 {2, 5, 8} or  {x, y} \u2282 {3, 6, 9} \u21d2 x \u2013 y is divisible by\n3 \u21d2 {x, y} \u2208 R1 This shows that R2 \u2282 R1"
    },
    {
        "Chapter": "1",
        "sentence_range": "683-686",
        "Text": "Hence, R1 \u2282 R2 Similarly, {x, y} \u2208 R2 \u21d2 {x, y}\nRationalised 2023-24\nMATHEMATICS\n14\n\u2282 {1, 4, 7} or {x, y} \u2282 {2, 5, 8} or  {x, y} \u2282 {3, 6, 9} \u21d2 x \u2013 y is divisible by\n3 \u21d2 {x, y} \u2208 R1 This shows that R2 \u2282 R1 Hence, R1 = R2"
    },
    {
        "Chapter": "1",
        "sentence_range": "684-687",
        "Text": "Similarly, {x, y} \u2208 R2 \u21d2 {x, y}\nRationalised 2023-24\nMATHEMATICS\n14\n\u2282 {1, 4, 7} or {x, y} \u2282 {2, 5, 8} or  {x, y} \u2282 {3, 6, 9} \u21d2 x \u2013 y is divisible by\n3 \u21d2 {x, y} \u2208 R1 This shows that R2 \u2282 R1 Hence, R1 = R2 Example 21 Let f : X \u2192 Y be a function"
    },
    {
        "Chapter": "1",
        "sentence_range": "685-688",
        "Text": "This shows that R2 \u2282 R1 Hence, R1 = R2 Example 21 Let f : X \u2192 Y be a function Define a relation R in X given by\nR = {(a, b): f(a) = f(b)}"
    },
    {
        "Chapter": "1",
        "sentence_range": "686-689",
        "Text": "Hence, R1 = R2 Example 21 Let f : X \u2192 Y be a function Define a relation R in X given by\nR = {(a, b): f(a) = f(b)} Examine whether R is an equivalence relation or not"
    },
    {
        "Chapter": "1",
        "sentence_range": "687-690",
        "Text": "Example 21 Let f : X \u2192 Y be a function Define a relation R in X given by\nR = {(a, b): f(a) = f(b)} Examine whether R is an equivalence relation or not Solution For every a \u2208 X, (a, a) \u2208 R, since f(a) = f (a), showing that R is reflexive"
    },
    {
        "Chapter": "1",
        "sentence_range": "688-691",
        "Text": "Define a relation R in X given by\nR = {(a, b): f(a) = f(b)} Examine whether R is an equivalence relation or not Solution For every a \u2208 X, (a, a) \u2208 R, since f(a) = f (a), showing that R is reflexive Similarly, (a, b) \u2208 R \u21d2 f (a) = f (b) \u21d2 f (b) = f (a) \u21d2 (b, a) \u2208 R"
    },
    {
        "Chapter": "1",
        "sentence_range": "689-692",
        "Text": "Examine whether R is an equivalence relation or not Solution For every a \u2208 X, (a, a) \u2208 R, since f(a) = f (a), showing that R is reflexive Similarly, (a, b) \u2208 R \u21d2 f (a) = f (b) \u21d2 f (b) = f (a) \u21d2 (b, a) \u2208 R Therefore, R is\nsymmetric"
    },
    {
        "Chapter": "1",
        "sentence_range": "690-693",
        "Text": "Solution For every a \u2208 X, (a, a) \u2208 R, since f(a) = f (a), showing that R is reflexive Similarly, (a, b) \u2208 R \u21d2 f (a) = f (b) \u21d2 f (b) = f (a) \u21d2 (b, a) \u2208 R Therefore, R is\nsymmetric Further, (a, b) \u2208 R and (b, c) \u2208 R \u21d2 f (a) = f (b) and f(b) = f (c) \u21d2 f (a)\n= f(c) \u21d2 (a, c) \u2208 R, which implies that R is transitive"
    },
    {
        "Chapter": "1",
        "sentence_range": "691-694",
        "Text": "Similarly, (a, b) \u2208 R \u21d2 f (a) = f (b) \u21d2 f (b) = f (a) \u21d2 (b, a) \u2208 R Therefore, R is\nsymmetric Further, (a, b) \u2208 R and (b, c) \u2208 R \u21d2 f (a) = f (b) and f(b) = f (c) \u21d2 f (a)\n= f(c) \u21d2 (a, c) \u2208 R, which implies that R is transitive Hence, R is an equivalence\nrelation"
    },
    {
        "Chapter": "1",
        "sentence_range": "692-695",
        "Text": "Therefore, R is\nsymmetric Further, (a, b) \u2208 R and (b, c) \u2208 R \u21d2 f (a) = f (b) and f(b) = f (c) \u21d2 f (a)\n= f(c) \u21d2 (a, c) \u2208 R, which implies that R is transitive Hence, R is an equivalence\nrelation Example 22 Find the number of all one-one functions from set A = {1, 2, 3} to itself"
    },
    {
        "Chapter": "1",
        "sentence_range": "693-696",
        "Text": "Further, (a, b) \u2208 R and (b, c) \u2208 R \u21d2 f (a) = f (b) and f(b) = f (c) \u21d2 f (a)\n= f(c) \u21d2 (a, c) \u2208 R, which implies that R is transitive Hence, R is an equivalence\nrelation Example 22 Find the number of all one-one functions from set A = {1, 2, 3} to itself Solution One-one function from {1, 2, 3} to itself is simply a permutation on three\nsymbols 1, 2, 3"
    },
    {
        "Chapter": "1",
        "sentence_range": "694-697",
        "Text": "Hence, R is an equivalence\nrelation Example 22 Find the number of all one-one functions from set A = {1, 2, 3} to itself Solution One-one function from {1, 2, 3} to itself is simply a permutation on three\nsymbols 1, 2, 3 Therefore, total number of one-one maps from {1, 2, 3} to itself is\nsame as total number of permutations on three symbols 1, 2, 3 which is 3"
    },
    {
        "Chapter": "1",
        "sentence_range": "695-698",
        "Text": "Example 22 Find the number of all one-one functions from set A = {1, 2, 3} to itself Solution One-one function from {1, 2, 3} to itself is simply a permutation on three\nsymbols 1, 2, 3 Therefore, total number of one-one maps from {1, 2, 3} to itself is\nsame as total number of permutations on three symbols 1, 2, 3 which is 3 = 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "696-699",
        "Text": "Solution One-one function from {1, 2, 3} to itself is simply a permutation on three\nsymbols 1, 2, 3 Therefore, total number of one-one maps from {1, 2, 3} to itself is\nsame as total number of permutations on three symbols 1, 2, 3 which is 3 = 6 Example 23 Let A = {1, 2, 3}"
    },
    {
        "Chapter": "1",
        "sentence_range": "697-700",
        "Text": "Therefore, total number of one-one maps from {1, 2, 3} to itself is\nsame as total number of permutations on three symbols 1, 2, 3 which is 3 = 6 Example 23 Let A = {1, 2, 3} Then show that  the number of relations containing (1, 2)\nand (2, 3) which are reflexive and transitive but not symmetric is three"
    },
    {
        "Chapter": "1",
        "sentence_range": "698-701",
        "Text": "= 6 Example 23 Let A = {1, 2, 3} Then show that  the number of relations containing (1, 2)\nand (2, 3) which are reflexive and transitive but not symmetric is three Solution The smallest relation R1 containing (1, 2) and (2, 3) which is reflexive and\ntransitive but not symmetric is {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (1, 3)}"
    },
    {
        "Chapter": "1",
        "sentence_range": "699-702",
        "Text": "Example 23 Let A = {1, 2, 3} Then show that  the number of relations containing (1, 2)\nand (2, 3) which are reflexive and transitive but not symmetric is three Solution The smallest relation R1 containing (1, 2) and (2, 3) which is reflexive and\ntransitive but not symmetric is {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (1, 3)} Now, if we add\nthe pair (2, 1) to R1 to get R2, then the relation R2 will be reflexive, transitive but not\nsymmetric"
    },
    {
        "Chapter": "1",
        "sentence_range": "700-703",
        "Text": "Then show that  the number of relations containing (1, 2)\nand (2, 3) which are reflexive and transitive but not symmetric is three Solution The smallest relation R1 containing (1, 2) and (2, 3) which is reflexive and\ntransitive but not symmetric is {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (1, 3)} Now, if we add\nthe pair (2, 1) to R1 to get R2, then the relation R2 will be reflexive, transitive but not\nsymmetric Similarly, we can obtain R3 by adding (3, 2) to R1 to get the desired relation"
    },
    {
        "Chapter": "1",
        "sentence_range": "701-704",
        "Text": "Solution The smallest relation R1 containing (1, 2) and (2, 3) which is reflexive and\ntransitive but not symmetric is {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (1, 3)} Now, if we add\nthe pair (2, 1) to R1 to get R2, then the relation R2 will be reflexive, transitive but not\nsymmetric Similarly, we can obtain R3 by adding (3, 2) to R1 to get the desired relation However, we can not add two pairs (2, 1), (3, 2) or single pair (3, 1) to R1 at a time, as\nby doing so, we will be forced to add the remaining pair in order to maintain transitivity\nand in the process, the relation will become symmetric also which is not required"
    },
    {
        "Chapter": "1",
        "sentence_range": "702-705",
        "Text": "Now, if we add\nthe pair (2, 1) to R1 to get R2, then the relation R2 will be reflexive, transitive but not\nsymmetric Similarly, we can obtain R3 by adding (3, 2) to R1 to get the desired relation However, we can not add two pairs (2, 1), (3, 2) or single pair (3, 1) to R1 at a time, as\nby doing so, we will be forced to add the remaining pair in order to maintain transitivity\nand in the process, the relation will become symmetric also which is not required Thus,\nthe total number of desired relations is three"
    },
    {
        "Chapter": "1",
        "sentence_range": "703-706",
        "Text": "Similarly, we can obtain R3 by adding (3, 2) to R1 to get the desired relation However, we can not add two pairs (2, 1), (3, 2) or single pair (3, 1) to R1 at a time, as\nby doing so, we will be forced to add the remaining pair in order to maintain transitivity\nand in the process, the relation will become symmetric also which is not required Thus,\nthe total number of desired relations is three Example 24  Show that the number of equivalence relation in the set {1, 2, 3} containing\n(1, 2) and (2, 1) is two"
    },
    {
        "Chapter": "1",
        "sentence_range": "704-707",
        "Text": "However, we can not add two pairs (2, 1), (3, 2) or single pair (3, 1) to R1 at a time, as\nby doing so, we will be forced to add the remaining pair in order to maintain transitivity\nand in the process, the relation will become symmetric also which is not required Thus,\nthe total number of desired relations is three Example 24  Show that the number of equivalence relation in the set {1, 2, 3} containing\n(1, 2) and (2, 1) is two Solution The smallest equivalence relation R1 containing (1, 2) and (2, 1) is {(1, 1),\n(2, 2), (3, 3), (1, 2), (2, 1)}"
    },
    {
        "Chapter": "1",
        "sentence_range": "705-708",
        "Text": "Thus,\nthe total number of desired relations is three Example 24  Show that the number of equivalence relation in the set {1, 2, 3} containing\n(1, 2) and (2, 1) is two Solution The smallest equivalence relation R1 containing (1, 2) and (2, 1) is {(1, 1),\n(2, 2), (3, 3), (1, 2), (2, 1)} Now we are left with only 4 pairs namely (2, 3), (3, 2),\n(1, 3) and (3, 1)"
    },
    {
        "Chapter": "1",
        "sentence_range": "706-709",
        "Text": "Example 24  Show that the number of equivalence relation in the set {1, 2, 3} containing\n(1, 2) and (2, 1) is two Solution The smallest equivalence relation R1 containing (1, 2) and (2, 1) is {(1, 1),\n(2, 2), (3, 3), (1, 2), (2, 1)} Now we are left with only 4 pairs namely (2, 3), (3, 2),\n(1, 3) and (3, 1) If we add any one, say (2, 3) to R1, then for symmetry we must add\n(3, 2) also and now for transitivity we are forced to add (1, 3) and (3, 1)"
    },
    {
        "Chapter": "1",
        "sentence_range": "707-710",
        "Text": "Solution The smallest equivalence relation R1 containing (1, 2) and (2, 1) is {(1, 1),\n(2, 2), (3, 3), (1, 2), (2, 1)} Now we are left with only 4 pairs namely (2, 3), (3, 2),\n(1, 3) and (3, 1) If we add any one, say (2, 3) to R1, then for symmetry we must add\n(3, 2) also and now for transitivity we are forced to add (1, 3) and (3, 1) Thus, the only\nequivalence relation bigger than R1 is the universal relation"
    },
    {
        "Chapter": "1",
        "sentence_range": "708-711",
        "Text": "Now we are left with only 4 pairs namely (2, 3), (3, 2),\n(1, 3) and (3, 1) If we add any one, say (2, 3) to R1, then for symmetry we must add\n(3, 2) also and now for transitivity we are forced to add (1, 3) and (3, 1) Thus, the only\nequivalence relation bigger than R1 is the universal relation This shows that the total\nnumber of equivalence relations containing (1, 2) and (2, 1) is two"
    },
    {
        "Chapter": "1",
        "sentence_range": "709-712",
        "Text": "If we add any one, say (2, 3) to R1, then for symmetry we must add\n(3, 2) also and now for transitivity we are forced to add (1, 3) and (3, 1) Thus, the only\nequivalence relation bigger than R1 is the universal relation This shows that the total\nnumber of equivalence relations containing (1, 2) and (2, 1) is two Example 25 Consider the identity function IN : N \u2192 N defined as IN (x) = x \u2200 x \u2208 N"
    },
    {
        "Chapter": "1",
        "sentence_range": "710-713",
        "Text": "Thus, the only\nequivalence relation bigger than R1 is the universal relation This shows that the total\nnumber of equivalence relations containing (1, 2) and (2, 1) is two Example 25 Consider the identity function IN : N \u2192 N defined as IN (x) = x \u2200 x \u2208 N Show that although IN is onto but IN + IN : N \u2192 N defined as\n(IN + IN) (x) = IN (x) + IN (x) = x + x = 2x is not onto"
    },
    {
        "Chapter": "1",
        "sentence_range": "711-714",
        "Text": "This shows that the total\nnumber of equivalence relations containing (1, 2) and (2, 1) is two Example 25 Consider the identity function IN : N \u2192 N defined as IN (x) = x \u2200 x \u2208 N Show that although IN is onto but IN + IN : N \u2192 N defined as\n(IN + IN) (x) = IN (x) + IN (x) = x + x = 2x is not onto Rationalised 2023-24\nRELATIONS AND FUNCTIONS\n15\nSolution Clearly IN is onto"
    },
    {
        "Chapter": "1",
        "sentence_range": "712-715",
        "Text": "Example 25 Consider the identity function IN : N \u2192 N defined as IN (x) = x \u2200 x \u2208 N Show that although IN is onto but IN + IN : N \u2192 N defined as\n(IN + IN) (x) = IN (x) + IN (x) = x + x = 2x is not onto Rationalised 2023-24\nRELATIONS AND FUNCTIONS\n15\nSolution Clearly IN is onto But IN + IN is not onto, as we can find an element 3\nin the co-domain N such that there does not exist any x in the domain N with\n(IN + IN) (x) = 2x = 3"
    },
    {
        "Chapter": "1",
        "sentence_range": "713-716",
        "Text": "Show that although IN is onto but IN + IN : N \u2192 N defined as\n(IN + IN) (x) = IN (x) + IN (x) = x + x = 2x is not onto Rationalised 2023-24\nRELATIONS AND FUNCTIONS\n15\nSolution Clearly IN is onto But IN + IN is not onto, as we can find an element 3\nin the co-domain N such that there does not exist any x in the domain N with\n(IN + IN) (x) = 2x = 3 Example 26 Consider a function f : 0, 2\n\u03c0\n\uf8ee\n\uf8f9\u2192\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nR given by f (x) = sin x and\ng : 0, 2\n\u03c0\n\uf8ee\n\uf8f9\u2192\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nR given by g(x) = cos x"
    },
    {
        "Chapter": "1",
        "sentence_range": "714-717",
        "Text": "Rationalised 2023-24\nRELATIONS AND FUNCTIONS\n15\nSolution Clearly IN is onto But IN + IN is not onto, as we can find an element 3\nin the co-domain N such that there does not exist any x in the domain N with\n(IN + IN) (x) = 2x = 3 Example 26 Consider a function f : 0, 2\n\u03c0\n\uf8ee\n\uf8f9\u2192\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nR given by f (x) = sin x and\ng : 0, 2\n\u03c0\n\uf8ee\n\uf8f9\u2192\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nR given by g(x) = cos x Show that f and g are one-one, but f + g is not\none-one"
    },
    {
        "Chapter": "1",
        "sentence_range": "715-718",
        "Text": "But IN + IN is not onto, as we can find an element 3\nin the co-domain N such that there does not exist any x in the domain N with\n(IN + IN) (x) = 2x = 3 Example 26 Consider a function f : 0, 2\n\u03c0\n\uf8ee\n\uf8f9\u2192\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nR given by f (x) = sin x and\ng : 0, 2\n\u03c0\n\uf8ee\n\uf8f9\u2192\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nR given by g(x) = cos x Show that f and g are one-one, but f + g is not\none-one Solution Since for any two distinct elements x1 and x2 in 0, 2\n\u03c0\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, sin x1 \u2260 sin x2 and\ncos x1 \u2260 cos x2, both f and g must be one-one"
    },
    {
        "Chapter": "1",
        "sentence_range": "716-719",
        "Text": "Example 26 Consider a function f : 0, 2\n\u03c0\n\uf8ee\n\uf8f9\u2192\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nR given by f (x) = sin x and\ng : 0, 2\n\u03c0\n\uf8ee\n\uf8f9\u2192\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nR given by g(x) = cos x Show that f and g are one-one, but f + g is not\none-one Solution Since for any two distinct elements x1 and x2 in 0, 2\n\u03c0\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, sin x1 \u2260 sin x2 and\ncos x1 \u2260 cos x2, both f and g must be one-one But (f + g) (0) = sin 0 + cos 0 = 1 and\n(f + g)\n\uf8eb\u03c02\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n = sin\ncos\n1\n2\n2\n\u03c0\n\u03c0\n+\n="
    },
    {
        "Chapter": "1",
        "sentence_range": "717-720",
        "Text": "Show that f and g are one-one, but f + g is not\none-one Solution Since for any two distinct elements x1 and x2 in 0, 2\n\u03c0\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, sin x1 \u2260 sin x2 and\ncos x1 \u2260 cos x2, both f and g must be one-one But (f + g) (0) = sin 0 + cos 0 = 1 and\n(f + g)\n\uf8eb\u03c02\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n = sin\ncos\n1\n2\n2\n\u03c0\n\u03c0\n+\n= Therefore, f + g is not one-one"
    },
    {
        "Chapter": "1",
        "sentence_range": "718-721",
        "Text": "Solution Since for any two distinct elements x1 and x2 in 0, 2\n\u03c0\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, sin x1 \u2260 sin x2 and\ncos x1 \u2260 cos x2, both f and g must be one-one But (f + g) (0) = sin 0 + cos 0 = 1 and\n(f + g)\n\uf8eb\u03c02\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n = sin\ncos\n1\n2\n2\n\u03c0\n\u03c0\n+\n= Therefore, f + g is not one-one Miscellaneous Exercise on Chapter 1\n1"
    },
    {
        "Chapter": "1",
        "sentence_range": "719-722",
        "Text": "But (f + g) (0) = sin 0 + cos 0 = 1 and\n(f + g)\n\uf8eb\u03c02\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n = sin\ncos\n1\n2\n2\n\u03c0\n\u03c0\n+\n= Therefore, f + g is not one-one Miscellaneous Exercise on Chapter 1\n1 Show that the function f : R \u2192 {x \u2208 R : \u2013 1 < x < 1} defined by \n( )\n1 |\n|\nx\nf x\n= +x\n,\nx \u2208 R is one one and onto function"
    },
    {
        "Chapter": "1",
        "sentence_range": "720-723",
        "Text": "Therefore, f + g is not one-one Miscellaneous Exercise on Chapter 1\n1 Show that the function f : R \u2192 {x \u2208 R : \u2013 1 < x < 1} defined by \n( )\n1 |\n|\nx\nf x\n= +x\n,\nx \u2208 R is one one and onto function 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "721-724",
        "Text": "Miscellaneous Exercise on Chapter 1\n1 Show that the function f : R \u2192 {x \u2208 R : \u2013 1 < x < 1} defined by \n( )\n1 |\n|\nx\nf x\n= +x\n,\nx \u2208 R is one one and onto function 2 Show that the function f : R \u2192 R given by f (x) = x3 is injective"
    },
    {
        "Chapter": "1",
        "sentence_range": "722-725",
        "Text": "Show that the function f : R \u2192 {x \u2208 R : \u2013 1 < x < 1} defined by \n( )\n1 |\n|\nx\nf x\n= +x\n,\nx \u2208 R is one one and onto function 2 Show that the function f : R \u2192 R given by f (x) = x3 is injective 3"
    },
    {
        "Chapter": "1",
        "sentence_range": "723-726",
        "Text": "2 Show that the function f : R \u2192 R given by f (x) = x3 is injective 3 Given a non empty set X, consider P(X) which is the set of all subsets of X"
    },
    {
        "Chapter": "1",
        "sentence_range": "724-727",
        "Text": "Show that the function f : R \u2192 R given by f (x) = x3 is injective 3 Given a non empty set X, consider P(X) which is the set of all subsets of X Define the relation R in P(X) as follows:\nFor subsets A, B in P(X), ARB if and only if A \u2282 B"
    },
    {
        "Chapter": "1",
        "sentence_range": "725-728",
        "Text": "3 Given a non empty set X, consider P(X) which is the set of all subsets of X Define the relation R in P(X) as follows:\nFor subsets A, B in P(X), ARB if and only if A \u2282 B Is R an equivalence relation\non P(X)"
    },
    {
        "Chapter": "1",
        "sentence_range": "726-729",
        "Text": "Given a non empty set X, consider P(X) which is the set of all subsets of X Define the relation R in P(X) as follows:\nFor subsets A, B in P(X), ARB if and only if A \u2282 B Is R an equivalence relation\non P(X) Justify your answer"
    },
    {
        "Chapter": "1",
        "sentence_range": "727-730",
        "Text": "Define the relation R in P(X) as follows:\nFor subsets A, B in P(X), ARB if and only if A \u2282 B Is R an equivalence relation\non P(X) Justify your answer 4"
    },
    {
        "Chapter": "1",
        "sentence_range": "728-731",
        "Text": "Is R an equivalence relation\non P(X) Justify your answer 4 Find the number of all onto functions from the set {1, 2, 3,"
    },
    {
        "Chapter": "1",
        "sentence_range": "729-732",
        "Text": "Justify your answer 4 Find the number of all onto functions from the set {1, 2, 3, , n} to itself"
    },
    {
        "Chapter": "1",
        "sentence_range": "730-733",
        "Text": "4 Find the number of all onto functions from the set {1, 2, 3, , n} to itself 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "731-734",
        "Text": "Find the number of all onto functions from the set {1, 2, 3, , n} to itself 5 Let A = {\u2013 1, 0, 1, 2}, B = {\u2013 4, \u2013 2, 0, 2} and f, g : A \u2192 B be functions defined\nby f (x) = x2 \u2013 x, x \u2208 A and \n1\n( )\n2\n1,\n2\ng x\nx\n=\n\u2212\n\u2212\n x \u2208 A"
    },
    {
        "Chapter": "1",
        "sentence_range": "732-735",
        "Text": ", n} to itself 5 Let A = {\u2013 1, 0, 1, 2}, B = {\u2013 4, \u2013 2, 0, 2} and f, g : A \u2192 B be functions defined\nby f (x) = x2 \u2013 x, x \u2208 A and \n1\n( )\n2\n1,\n2\ng x\nx\n=\n\u2212\n\u2212\n x \u2208 A Are f and g equal"
    },
    {
        "Chapter": "1",
        "sentence_range": "733-736",
        "Text": "5 Let A = {\u2013 1, 0, 1, 2}, B = {\u2013 4, \u2013 2, 0, 2} and f, g : A \u2192 B be functions defined\nby f (x) = x2 \u2013 x, x \u2208 A and \n1\n( )\n2\n1,\n2\ng x\nx\n=\n\u2212\n\u2212\n x \u2208 A Are f and g equal Justify your answer"
    },
    {
        "Chapter": "1",
        "sentence_range": "734-737",
        "Text": "Let A = {\u2013 1, 0, 1, 2}, B = {\u2013 4, \u2013 2, 0, 2} and f, g : A \u2192 B be functions defined\nby f (x) = x2 \u2013 x, x \u2208 A and \n1\n( )\n2\n1,\n2\ng x\nx\n=\n\u2212\n\u2212\n x \u2208 A Are f and g equal Justify your answer (Hint: One may note that two functions f : A \u2192 B and\ng : A \u2192 B such that f (a) = g(a) \u2200 a \u2208 A, are called equal functions)"
    },
    {
        "Chapter": "1",
        "sentence_range": "735-738",
        "Text": "Are f and g equal Justify your answer (Hint: One may note that two functions f : A \u2192 B and\ng : A \u2192 B such that f (a) = g(a) \u2200 a \u2208 A, are called equal functions) Rationalised 2023-24\nMATHEMATICS\n16\n6"
    },
    {
        "Chapter": "1",
        "sentence_range": "736-739",
        "Text": "Justify your answer (Hint: One may note that two functions f : A \u2192 B and\ng : A \u2192 B such that f (a) = g(a) \u2200 a \u2208 A, are called equal functions) Rationalised 2023-24\nMATHEMATICS\n16\n6 Let A = {1, 2, 3}"
    },
    {
        "Chapter": "1",
        "sentence_range": "737-740",
        "Text": "(Hint: One may note that two functions f : A \u2192 B and\ng : A \u2192 B such that f (a) = g(a) \u2200 a \u2208 A, are called equal functions) Rationalised 2023-24\nMATHEMATICS\n16\n6 Let A = {1, 2, 3} Then number of relations containing (1, 2) and (1, 3) which are\nreflexive and symmetric but not transitive is\n(A) 1\n(B) 2\n(C) 3\n(D) 4\n7"
    },
    {
        "Chapter": "1",
        "sentence_range": "738-741",
        "Text": "Rationalised 2023-24\nMATHEMATICS\n16\n6 Let A = {1, 2, 3} Then number of relations containing (1, 2) and (1, 3) which are\nreflexive and symmetric but not transitive is\n(A) 1\n(B) 2\n(C) 3\n(D) 4\n7 Let A = {1, 2, 3}"
    },
    {
        "Chapter": "1",
        "sentence_range": "739-742",
        "Text": "Let A = {1, 2, 3} Then number of relations containing (1, 2) and (1, 3) which are\nreflexive and symmetric but not transitive is\n(A) 1\n(B) 2\n(C) 3\n(D) 4\n7 Let A = {1, 2, 3} Then number of equivalence relations containing (1, 2) is\n(A) 1\n(B) 2\n(C) 3\n(D) 4\nSummary\nIn this chapter, we studied different types of relations and equivalence relation,\ncomposition of functions, invertible functions and binary operations"
    },
    {
        "Chapter": "1",
        "sentence_range": "740-743",
        "Text": "Then number of relations containing (1, 2) and (1, 3) which are\nreflexive and symmetric but not transitive is\n(A) 1\n(B) 2\n(C) 3\n(D) 4\n7 Let A = {1, 2, 3} Then number of equivalence relations containing (1, 2) is\n(A) 1\n(B) 2\n(C) 3\n(D) 4\nSummary\nIn this chapter, we studied different types of relations and equivalence relation,\ncomposition of functions, invertible functions and binary operations The main features\nof this chapter are as follows:\n\u00ae\nEmpty relation is the relation R in X given by R = \u03c6 \u2282 X \u00d7 X"
    },
    {
        "Chapter": "1",
        "sentence_range": "741-744",
        "Text": "Let A = {1, 2, 3} Then number of equivalence relations containing (1, 2) is\n(A) 1\n(B) 2\n(C) 3\n(D) 4\nSummary\nIn this chapter, we studied different types of relations and equivalence relation,\ncomposition of functions, invertible functions and binary operations The main features\nof this chapter are as follows:\n\u00ae\nEmpty relation is the relation R in X given by R = \u03c6 \u2282 X \u00d7 X \u00ae\nUniversal relation is the relation R in X given by R = X \u00d7 X"
    },
    {
        "Chapter": "1",
        "sentence_range": "742-745",
        "Text": "Then number of equivalence relations containing (1, 2) is\n(A) 1\n(B) 2\n(C) 3\n(D) 4\nSummary\nIn this chapter, we studied different types of relations and equivalence relation,\ncomposition of functions, invertible functions and binary operations The main features\nof this chapter are as follows:\n\u00ae\nEmpty relation is the relation R in X given by R = \u03c6 \u2282 X \u00d7 X \u00ae\nUniversal relation is the relation R in X given by R = X \u00d7 X \u00ae\nReflexive relation R in X is a relation with (a, a) \u2208 R \u2200 a \u2208 X"
    },
    {
        "Chapter": "1",
        "sentence_range": "743-746",
        "Text": "The main features\nof this chapter are as follows:\n\u00ae\nEmpty relation is the relation R in X given by R = \u03c6 \u2282 X \u00d7 X \u00ae\nUniversal relation is the relation R in X given by R = X \u00d7 X \u00ae\nReflexive relation R in X is a relation with (a, a) \u2208 R \u2200 a \u2208 X \u00ae\nSymmetric relation R in X is a relation satisfying (a, b) \u2208 R implies (b, a) \u2208 R"
    },
    {
        "Chapter": "1",
        "sentence_range": "744-747",
        "Text": "\u00ae\nUniversal relation is the relation R in X given by R = X \u00d7 X \u00ae\nReflexive relation R in X is a relation with (a, a) \u2208 R \u2200 a \u2208 X \u00ae\nSymmetric relation R in X is a relation satisfying (a, b) \u2208 R implies (b, a) \u2208 R \u00ae\nTransitive relation R in X is a relation satisfying (a, b) \u2208 R and (b, c) \u2208 R\nimplies that (a, c) \u2208 R"
    },
    {
        "Chapter": "1",
        "sentence_range": "745-748",
        "Text": "\u00ae\nReflexive relation R in X is a relation with (a, a) \u2208 R \u2200 a \u2208 X \u00ae\nSymmetric relation R in X is a relation satisfying (a, b) \u2208 R implies (b, a) \u2208 R \u00ae\nTransitive relation R in X is a relation satisfying (a, b) \u2208 R and (b, c) \u2208 R\nimplies that (a, c) \u2208 R \u00ae\nEquivalence relation R in X is a relation which is reflexive, symmetric and\ntransitive"
    },
    {
        "Chapter": "1",
        "sentence_range": "746-749",
        "Text": "\u00ae\nSymmetric relation R in X is a relation satisfying (a, b) \u2208 R implies (b, a) \u2208 R \u00ae\nTransitive relation R in X is a relation satisfying (a, b) \u2208 R and (b, c) \u2208 R\nimplies that (a, c) \u2208 R \u00ae\nEquivalence relation R in X is a relation which is reflexive, symmetric and\ntransitive \u00ae\nEquivalence class [a] containing a \u2208 X for an equivalence relation R in X is\nthe subset of X containing all elements b related to a"
    },
    {
        "Chapter": "1",
        "sentence_range": "747-750",
        "Text": "\u00ae\nTransitive relation R in X is a relation satisfying (a, b) \u2208 R and (b, c) \u2208 R\nimplies that (a, c) \u2208 R \u00ae\nEquivalence relation R in X is a relation which is reflexive, symmetric and\ntransitive \u00ae\nEquivalence class [a] containing a \u2208 X for an equivalence relation R in X is\nthe subset of X containing all elements b related to a \u00ae\nA function f : X \u2192 Y is one-one (or injective) if\nf (x1) = f(x2) \u21d2 x1 = x2 \u2200  x1, x2 \u2208 X"
    },
    {
        "Chapter": "1",
        "sentence_range": "748-751",
        "Text": "\u00ae\nEquivalence relation R in X is a relation which is reflexive, symmetric and\ntransitive \u00ae\nEquivalence class [a] containing a \u2208 X for an equivalence relation R in X is\nthe subset of X containing all elements b related to a \u00ae\nA function f : X \u2192 Y is one-one (or injective) if\nf (x1) = f(x2) \u21d2 x1 = x2 \u2200  x1, x2 \u2208 X \u00ae\nA function f : X \u2192 Y is onto (or surjective) if given any y \u2208 Y, \u2203 x \u2208 X such\nthat f (x) = y"
    },
    {
        "Chapter": "1",
        "sentence_range": "749-752",
        "Text": "\u00ae\nEquivalence class [a] containing a \u2208 X for an equivalence relation R in X is\nthe subset of X containing all elements b related to a \u00ae\nA function f : X \u2192 Y is one-one (or injective) if\nf (x1) = f(x2) \u21d2 x1 = x2 \u2200  x1, x2 \u2208 X \u00ae\nA function f : X \u2192 Y is onto (or surjective) if given any y \u2208 Y, \u2203 x \u2208 X such\nthat f (x) = y \u00ae\nA function f : X \u2192 Y is one-one and onto (or bijective), if f is both one-one\nand onto"
    },
    {
        "Chapter": "1",
        "sentence_range": "750-753",
        "Text": "\u00ae\nA function f : X \u2192 Y is one-one (or injective) if\nf (x1) = f(x2) \u21d2 x1 = x2 \u2200  x1, x2 \u2208 X \u00ae\nA function f : X \u2192 Y is onto (or surjective) if given any y \u2208 Y, \u2203 x \u2208 X such\nthat f (x) = y \u00ae\nA function f : X \u2192 Y is one-one and onto (or bijective), if f is both one-one\nand onto \u00ae\nGiven a finite set X, a function f : X \u2192 X is one-one (respectively onto) if and\nonly if f is onto (respectively one-one)"
    },
    {
        "Chapter": "1",
        "sentence_range": "751-754",
        "Text": "\u00ae\nA function f : X \u2192 Y is onto (or surjective) if given any y \u2208 Y, \u2203 x \u2208 X such\nthat f (x) = y \u00ae\nA function f : X \u2192 Y is one-one and onto (or bijective), if f is both one-one\nand onto \u00ae\nGiven a finite set X, a function f : X \u2192 X is one-one (respectively onto) if and\nonly if f is onto (respectively one-one) This is the characteristic property of a\nfinite set"
    },
    {
        "Chapter": "1",
        "sentence_range": "752-755",
        "Text": "\u00ae\nA function f : X \u2192 Y is one-one and onto (or bijective), if f is both one-one\nand onto \u00ae\nGiven a finite set X, a function f : X \u2192 X is one-one (respectively onto) if and\nonly if f is onto (respectively one-one) This is the characteristic property of a\nfinite set This is not true for infinite set\nRationalised 2023-24\nRELATIONS AND FUNCTIONS\n17\n\u2014v\nv\nv\nv\nv\u2014\nHistorical Note\nThe concept of function has evolved over a long period of time starting from\nR"
    },
    {
        "Chapter": "1",
        "sentence_range": "753-756",
        "Text": "\u00ae\nGiven a finite set X, a function f : X \u2192 X is one-one (respectively onto) if and\nonly if f is onto (respectively one-one) This is the characteristic property of a\nfinite set This is not true for infinite set\nRationalised 2023-24\nRELATIONS AND FUNCTIONS\n17\n\u2014v\nv\nv\nv\nv\u2014\nHistorical Note\nThe concept of function has evolved over a long period of time starting from\nR Descartes (1596-1650), who used the word \u2018function\u2019 in his manuscript\n\u201cGeometrie\u201d in 1637 to mean some positive integral power xn of a variable x\nwhile studying geometrical curves like hyperbola, parabola and ellipse"
    },
    {
        "Chapter": "1",
        "sentence_range": "754-757",
        "Text": "This is the characteristic property of a\nfinite set This is not true for infinite set\nRationalised 2023-24\nRELATIONS AND FUNCTIONS\n17\n\u2014v\nv\nv\nv\nv\u2014\nHistorical Note\nThe concept of function has evolved over a long period of time starting from\nR Descartes (1596-1650), who used the word \u2018function\u2019 in his manuscript\n\u201cGeometrie\u201d in 1637 to mean some positive integral power xn of a variable x\nwhile studying geometrical curves like hyperbola, parabola and ellipse James\nGregory (1636-1675) in his work \u201c Vera Circuli et Hyperbolae Quadratura\u201d\n(1667) considered function as a quantity obtained from other quantities by\nsuccessive use of algebraic operations or by any other operations"
    },
    {
        "Chapter": "1",
        "sentence_range": "755-758",
        "Text": "This is not true for infinite set\nRationalised 2023-24\nRELATIONS AND FUNCTIONS\n17\n\u2014v\nv\nv\nv\nv\u2014\nHistorical Note\nThe concept of function has evolved over a long period of time starting from\nR Descartes (1596-1650), who used the word \u2018function\u2019 in his manuscript\n\u201cGeometrie\u201d in 1637 to mean some positive integral power xn of a variable x\nwhile studying geometrical curves like hyperbola, parabola and ellipse James\nGregory (1636-1675) in his work \u201c Vera Circuli et Hyperbolae Quadratura\u201d\n(1667) considered function as a quantity obtained from other quantities by\nsuccessive use of algebraic operations or by any other operations Later G"
    },
    {
        "Chapter": "1",
        "sentence_range": "756-759",
        "Text": "Descartes (1596-1650), who used the word \u2018function\u2019 in his manuscript\n\u201cGeometrie\u201d in 1637 to mean some positive integral power xn of a variable x\nwhile studying geometrical curves like hyperbola, parabola and ellipse James\nGregory (1636-1675) in his work \u201c Vera Circuli et Hyperbolae Quadratura\u201d\n(1667) considered function as a quantity obtained from other quantities by\nsuccessive use of algebraic operations or by any other operations Later G W"
    },
    {
        "Chapter": "1",
        "sentence_range": "757-760",
        "Text": "James\nGregory (1636-1675) in his work \u201c Vera Circuli et Hyperbolae Quadratura\u201d\n(1667) considered function as a quantity obtained from other quantities by\nsuccessive use of algebraic operations or by any other operations Later G W Leibnitz (1646-1716) in his manuscript \u201cMethodus tangentium inversa, seu  de\nfunctionibus\u201d written in 1673 used the word \u2018function\u2019 to mean a quantity varying\nfrom point to point on a curve such as the coordinates of a point on the curve, the\nslope of the curve, the tangent and the normal to the curve at a point"
    },
    {
        "Chapter": "1",
        "sentence_range": "758-761",
        "Text": "Later G W Leibnitz (1646-1716) in his manuscript \u201cMethodus tangentium inversa, seu  de\nfunctionibus\u201d written in 1673 used the word \u2018function\u2019 to mean a quantity varying\nfrom point to point on a curve such as the coordinates of a point on the curve, the\nslope of the curve, the tangent and the normal to the curve at a point However,\nin his manuscript \u201cHistoria\u201d (1714), Leibnitz used the word \u2018function\u2019 to mean\nquantities that depend on a variable"
    },
    {
        "Chapter": "1",
        "sentence_range": "759-762",
        "Text": "W Leibnitz (1646-1716) in his manuscript \u201cMethodus tangentium inversa, seu  de\nfunctionibus\u201d written in 1673 used the word \u2018function\u2019 to mean a quantity varying\nfrom point to point on a curve such as the coordinates of a point on the curve, the\nslope of the curve, the tangent and the normal to the curve at a point However,\nin his manuscript \u201cHistoria\u201d (1714), Leibnitz used the word \u2018function\u2019 to mean\nquantities that depend on a variable He was the first to use the phrase \u2018function\nof x\u2019"
    },
    {
        "Chapter": "1",
        "sentence_range": "760-763",
        "Text": "Leibnitz (1646-1716) in his manuscript \u201cMethodus tangentium inversa, seu  de\nfunctionibus\u201d written in 1673 used the word \u2018function\u2019 to mean a quantity varying\nfrom point to point on a curve such as the coordinates of a point on the curve, the\nslope of the curve, the tangent and the normal to the curve at a point However,\nin his manuscript \u201cHistoria\u201d (1714), Leibnitz used the word \u2018function\u2019 to mean\nquantities that depend on a variable He was the first to use the phrase \u2018function\nof x\u2019 John Bernoulli (1667-1748) used the notation \u03c6x for the first time in 1718 to\nindicate a function of x"
    },
    {
        "Chapter": "1",
        "sentence_range": "761-764",
        "Text": "However,\nin his manuscript \u201cHistoria\u201d (1714), Leibnitz used the word \u2018function\u2019 to mean\nquantities that depend on a variable He was the first to use the phrase \u2018function\nof x\u2019 John Bernoulli (1667-1748) used the notation \u03c6x for the first time in 1718 to\nindicate a function of x But the general adoption of symbols like f, F, \u03c6, \u03c8"
    },
    {
        "Chapter": "1",
        "sentence_range": "762-765",
        "Text": "He was the first to use the phrase \u2018function\nof x\u2019 John Bernoulli (1667-1748) used the notation \u03c6x for the first time in 1718 to\nindicate a function of x But the general adoption of symbols like f, F, \u03c6, \u03c8 to\nrepresent functions was made by Leonhard Euler (1707-1783) in 1734 in the first\npart of his manuscript \u201cAnalysis Infinitorium\u201d"
    },
    {
        "Chapter": "1",
        "sentence_range": "763-766",
        "Text": "John Bernoulli (1667-1748) used the notation \u03c6x for the first time in 1718 to\nindicate a function of x But the general adoption of symbols like f, F, \u03c6, \u03c8 to\nrepresent functions was made by Leonhard Euler (1707-1783) in 1734 in the first\npart of his manuscript \u201cAnalysis Infinitorium\u201d Later on, Joeph Louis Lagrange\n(1736-1813) published his manuscripts \u201cTheorie des functions analytiques\u201d in\n1793, where he discussed  about analytic function and used the notion f (x), F(x),\n\u03c6(x) etc"
    },
    {
        "Chapter": "1",
        "sentence_range": "764-767",
        "Text": "But the general adoption of symbols like f, F, \u03c6, \u03c8 to\nrepresent functions was made by Leonhard Euler (1707-1783) in 1734 in the first\npart of his manuscript \u201cAnalysis Infinitorium\u201d Later on, Joeph Louis Lagrange\n(1736-1813) published his manuscripts \u201cTheorie des functions analytiques\u201d in\n1793, where he discussed  about analytic function and used the notion f (x), F(x),\n\u03c6(x) etc for different function of x"
    },
    {
        "Chapter": "1",
        "sentence_range": "765-768",
        "Text": "to\nrepresent functions was made by Leonhard Euler (1707-1783) in 1734 in the first\npart of his manuscript \u201cAnalysis Infinitorium\u201d Later on, Joeph Louis Lagrange\n(1736-1813) published his manuscripts \u201cTheorie des functions analytiques\u201d in\n1793, where he discussed  about analytic function and used the notion f (x), F(x),\n\u03c6(x) etc for different function of x Subsequently, Lejeunne Dirichlet\n(1805-1859) gave the definition of function which was being used till the set\ntheoretic definition of function presently used, was given after set theory was\ndeveloped by Georg Cantor (1845-1918)"
    },
    {
        "Chapter": "1",
        "sentence_range": "766-769",
        "Text": "Later on, Joeph Louis Lagrange\n(1736-1813) published his manuscripts \u201cTheorie des functions analytiques\u201d in\n1793, where he discussed  about analytic function and used the notion f (x), F(x),\n\u03c6(x) etc for different function of x Subsequently, Lejeunne Dirichlet\n(1805-1859) gave the definition of function which was being used till the set\ntheoretic definition of function presently used, was given after set theory was\ndeveloped by Georg Cantor (1845-1918) The set theoretic definition of function\nknown to us presently is simply an abstraction of the definition given by Dirichlet\nin a rigorous manner"
    },
    {
        "Chapter": "1",
        "sentence_range": "767-770",
        "Text": "for different function of x Subsequently, Lejeunne Dirichlet\n(1805-1859) gave the definition of function which was being used till the set\ntheoretic definition of function presently used, was given after set theory was\ndeveloped by Georg Cantor (1845-1918) The set theoretic definition of function\nknown to us presently is simply an abstraction of the definition given by Dirichlet\nin a rigorous manner Rationalised 2023-24\n 18\nMATHEMATICS\nvMathematics, in general, is fundamentally the science of\nself-evident things"
    },
    {
        "Chapter": "1",
        "sentence_range": "768-771",
        "Text": "Subsequently, Lejeunne Dirichlet\n(1805-1859) gave the definition of function which was being used till the set\ntheoretic definition of function presently used, was given after set theory was\ndeveloped by Georg Cantor (1845-1918) The set theoretic definition of function\nknown to us presently is simply an abstraction of the definition given by Dirichlet\nin a rigorous manner Rationalised 2023-24\n 18\nMATHEMATICS\nvMathematics, in general, is fundamentally the science of\nself-evident things \u2014 FELIX KLEIN v\n2"
    },
    {
        "Chapter": "1",
        "sentence_range": "769-772",
        "Text": "The set theoretic definition of function\nknown to us presently is simply an abstraction of the definition given by Dirichlet\nin a rigorous manner Rationalised 2023-24\n 18\nMATHEMATICS\nvMathematics, in general, is fundamentally the science of\nself-evident things \u2014 FELIX KLEIN v\n2 1  Introduction\nIn Chapter 1, we have studied that the inverse of a function\nf, denoted by f \u20131, exists if f is one-one and onto"
    },
    {
        "Chapter": "1",
        "sentence_range": "770-773",
        "Text": "Rationalised 2023-24\n 18\nMATHEMATICS\nvMathematics, in general, is fundamentally the science of\nself-evident things \u2014 FELIX KLEIN v\n2 1  Introduction\nIn Chapter 1, we have studied that the inverse of a function\nf, denoted by f \u20131, exists if f is one-one and onto There are\nmany functions which are not one-one, onto or both and\nhence we can not talk of their inverses"
    },
    {
        "Chapter": "1",
        "sentence_range": "771-774",
        "Text": "\u2014 FELIX KLEIN v\n2 1  Introduction\nIn Chapter 1, we have studied that the inverse of a function\nf, denoted by f \u20131, exists if f is one-one and onto There are\nmany functions which are not one-one, onto or both and\nhence we can not talk of their inverses In Class XI, we\nstudied that trigonometric functions are not one-one and\nonto over their natural domains and ranges and hence their\ninverses do not exist"
    },
    {
        "Chapter": "1",
        "sentence_range": "772-775",
        "Text": "1  Introduction\nIn Chapter 1, we have studied that the inverse of a function\nf, denoted by f \u20131, exists if f is one-one and onto There are\nmany functions which are not one-one, onto or both and\nhence we can not talk of their inverses In Class XI, we\nstudied that trigonometric functions are not one-one and\nonto over their natural domains and ranges and hence their\ninverses do not exist In this chapter, we shall study about\nthe restrictions on domains and ranges of trigonometric\nfunctions which ensure the existence of their inverses and\nobserve their behaviour through graphical representations"
    },
    {
        "Chapter": "1",
        "sentence_range": "773-776",
        "Text": "There are\nmany functions which are not one-one, onto or both and\nhence we can not talk of their inverses In Class XI, we\nstudied that trigonometric functions are not one-one and\nonto over their natural domains and ranges and hence their\ninverses do not exist In this chapter, we shall study about\nthe restrictions on domains and ranges of trigonometric\nfunctions which ensure the existence of their inverses and\nobserve their behaviour through graphical representations Besides, some elementary properties will also be discussed"
    },
    {
        "Chapter": "1",
        "sentence_range": "774-777",
        "Text": "In Class XI, we\nstudied that trigonometric functions are not one-one and\nonto over their natural domains and ranges and hence their\ninverses do not exist In this chapter, we shall study about\nthe restrictions on domains and ranges of trigonometric\nfunctions which ensure the existence of their inverses and\nobserve their behaviour through graphical representations Besides, some elementary properties will also be discussed The inverse trigonometric functions play an important\nrole in calculus for they serve to define many integrals"
    },
    {
        "Chapter": "1",
        "sentence_range": "775-778",
        "Text": "In this chapter, we shall study about\nthe restrictions on domains and ranges of trigonometric\nfunctions which ensure the existence of their inverses and\nobserve their behaviour through graphical representations Besides, some elementary properties will also be discussed The inverse trigonometric functions play an important\nrole in calculus for they serve to define many integrals The concepts of inverse trigonometric functions is also used in science and engineering"
    },
    {
        "Chapter": "1",
        "sentence_range": "776-779",
        "Text": "Besides, some elementary properties will also be discussed The inverse trigonometric functions play an important\nrole in calculus for they serve to define many integrals The concepts of inverse trigonometric functions is also used in science and engineering 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "777-780",
        "Text": "The inverse trigonometric functions play an important\nrole in calculus for they serve to define many integrals The concepts of inverse trigonometric functions is also used in science and engineering 2 2  Basic Concepts\nIn Class XI, we have studied trigonometric functions, which are defined as follows:\nsine function, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "778-781",
        "Text": "The concepts of inverse trigonometric functions is also used in science and engineering 2 2  Basic Concepts\nIn Class XI, we have studied trigonometric functions, which are defined as follows:\nsine function, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "779-782",
        "Text": "2 2  Basic Concepts\nIn Class XI, we have studied trigonometric functions, which are defined as follows:\nsine function, i e , sine : R \u2192 [\u2013 1, 1]\ncosine function, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "780-783",
        "Text": "2  Basic Concepts\nIn Class XI, we have studied trigonometric functions, which are defined as follows:\nsine function, i e , sine : R \u2192 [\u2013 1, 1]\ncosine function, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "781-784",
        "Text": "e , sine : R \u2192 [\u2013 1, 1]\ncosine function, i e , cos : R \u2192 [\u2013 1, 1]\ntangent function, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "782-785",
        "Text": ", sine : R \u2192 [\u2013 1, 1]\ncosine function, i e , cos : R \u2192 [\u2013 1, 1]\ntangent function, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "783-786",
        "Text": "e , cos : R \u2192 [\u2013 1, 1]\ntangent function, i e , tan : R \u2013 { x : x = (2n + 1) 2\n\u03c0 , n \u2208 Z} \u2192 R\ncotangent function, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "784-787",
        "Text": ", cos : R \u2192 [\u2013 1, 1]\ntangent function, i e , tan : R \u2013 { x : x = (2n + 1) 2\n\u03c0 , n \u2208 Z} \u2192 R\ncotangent function, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "785-788",
        "Text": "e , tan : R \u2013 { x : x = (2n + 1) 2\n\u03c0 , n \u2208 Z} \u2192 R\ncotangent function, i e , cot : R \u2013 { x : x = n\u03c0, n \u2208 Z} \u2192 R\nsecant function, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "786-789",
        "Text": ", tan : R \u2013 { x : x = (2n + 1) 2\n\u03c0 , n \u2208 Z} \u2192 R\ncotangent function, i e , cot : R \u2013 { x : x = n\u03c0, n \u2208 Z} \u2192 R\nsecant function, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "787-790",
        "Text": "e , cot : R \u2013 { x : x = n\u03c0, n \u2208 Z} \u2192 R\nsecant function, i e , sec : R \u2013 { x : x = (2n + 1) 2\n\u03c0 , n \u2208 Z} \u2192 R \u2013 (\u2013 1, 1)\ncosecant function, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "788-791",
        "Text": ", cot : R \u2013 { x : x = n\u03c0, n \u2208 Z} \u2192 R\nsecant function, i e , sec : R \u2013 { x : x = (2n + 1) 2\n\u03c0 , n \u2208 Z} \u2192 R \u2013 (\u2013 1, 1)\ncosecant function, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "789-792",
        "Text": "e , sec : R \u2013 { x : x = (2n + 1) 2\n\u03c0 , n \u2208 Z} \u2192 R \u2013 (\u2013 1, 1)\ncosecant function, i e , cosec : R \u2013 { x : x = n\u03c0,  n \u2208 Z} \u2192 R \u2013 (\u2013 1, 1)\nChapter 2\nINVERSE TRIGONOMETRIC\nFUNCTIONS\nAryabhata\n (476-550 A"
    },
    {
        "Chapter": "1",
        "sentence_range": "790-793",
        "Text": ", sec : R \u2013 { x : x = (2n + 1) 2\n\u03c0 , n \u2208 Z} \u2192 R \u2013 (\u2013 1, 1)\ncosecant function, i e , cosec : R \u2013 { x : x = n\u03c0,  n \u2208 Z} \u2192 R \u2013 (\u2013 1, 1)\nChapter 2\nINVERSE TRIGONOMETRIC\nFUNCTIONS\nAryabhata\n (476-550 A D"
    },
    {
        "Chapter": "1",
        "sentence_range": "791-794",
        "Text": "e , cosec : R \u2013 { x : x = n\u03c0,  n \u2208 Z} \u2192 R \u2013 (\u2013 1, 1)\nChapter 2\nINVERSE TRIGONOMETRIC\nFUNCTIONS\nAryabhata\n (476-550 A D )\nRationalised 2023-24\nINVERSE TRIGONOMETRIC FUNCTIONS     19\nWe have also learnt in Chapter 1 that if f : X\u2192Y such that f (x) = y is one-one and\nonto, then we can define a unique function g : Y\u2192X such that g(y) = x, where x \u2208 X\nand y = f (x), y \u2208 Y"
    },
    {
        "Chapter": "1",
        "sentence_range": "792-795",
        "Text": ", cosec : R \u2013 { x : x = n\u03c0,  n \u2208 Z} \u2192 R \u2013 (\u2013 1, 1)\nChapter 2\nINVERSE TRIGONOMETRIC\nFUNCTIONS\nAryabhata\n (476-550 A D )\nRationalised 2023-24\nINVERSE TRIGONOMETRIC FUNCTIONS     19\nWe have also learnt in Chapter 1 that if f : X\u2192Y such that f (x) = y is one-one and\nonto, then we can define a unique function g : Y\u2192X such that g(y) = x, where x \u2208 X\nand y = f (x), y \u2208 Y Here, the domain of g = range of f and the range of g =  domain\nof f"
    },
    {
        "Chapter": "1",
        "sentence_range": "793-796",
        "Text": "D )\nRationalised 2023-24\nINVERSE TRIGONOMETRIC FUNCTIONS     19\nWe have also learnt in Chapter 1 that if f : X\u2192Y such that f (x) = y is one-one and\nonto, then we can define a unique function g : Y\u2192X such that g(y) = x, where x \u2208 X\nand y = f (x), y \u2208 Y Here, the domain of g = range of f and the range of g =  domain\nof f The function g is called the inverse of f and is denoted by f \u20131"
    },
    {
        "Chapter": "1",
        "sentence_range": "794-797",
        "Text": ")\nRationalised 2023-24\nINVERSE TRIGONOMETRIC FUNCTIONS     19\nWe have also learnt in Chapter 1 that if f : X\u2192Y such that f (x) = y is one-one and\nonto, then we can define a unique function g : Y\u2192X such that g(y) = x, where x \u2208 X\nand y = f (x), y \u2208 Y Here, the domain of g = range of f and the range of g =  domain\nof f The function g is called the inverse of f and is denoted by f \u20131 Further, g is also\none-one and onto and inverse of g is f"
    },
    {
        "Chapter": "1",
        "sentence_range": "795-798",
        "Text": "Here, the domain of g = range of f and the range of g =  domain\nof f The function g is called the inverse of f and is denoted by f \u20131 Further, g is also\none-one and onto and inverse of g is f Thus, g\u20131 = (f \u20131)\u20131 = f"
    },
    {
        "Chapter": "1",
        "sentence_range": "796-799",
        "Text": "The function g is called the inverse of f and is denoted by f \u20131 Further, g is also\none-one and onto and inverse of g is f Thus, g\u20131 = (f \u20131)\u20131 = f We also have\n(f \u20131 o f ) (x) = f \u20131 (f (x)) = f \u20131(y) = x\nand\n(f o f \u20131) (y) = f (f \u20131(y))  = f (x) = y\nSince the domain of sine function is the set of all real numbers and range is the\nclosed interval [\u20131, 1]"
    },
    {
        "Chapter": "1",
        "sentence_range": "797-800",
        "Text": "Further, g is also\none-one and onto and inverse of g is f Thus, g\u20131 = (f \u20131)\u20131 = f We also have\n(f \u20131 o f ) (x) = f \u20131 (f (x)) = f \u20131(y) = x\nand\n(f o f \u20131) (y) = f (f \u20131(y))  = f (x) = y\nSince the domain of sine function is the set of all real numbers and range is the\nclosed interval [\u20131, 1] If we restrict its domain to\n2,\n2\n\uf8ee\u2212\u03c0 \u03c0\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, then it becomes one-one\nand onto with range [\u2013 1, 1]"
    },
    {
        "Chapter": "1",
        "sentence_range": "798-801",
        "Text": "Thus, g\u20131 = (f \u20131)\u20131 = f We also have\n(f \u20131 o f ) (x) = f \u20131 (f (x)) = f \u20131(y) = x\nand\n(f o f \u20131) (y) = f (f \u20131(y))  = f (x) = y\nSince the domain of sine function is the set of all real numbers and range is the\nclosed interval [\u20131, 1] If we restrict its domain to\n2,\n2\n\uf8ee\u2212\u03c0 \u03c0\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, then it becomes one-one\nand onto with range [\u2013 1, 1] Actually, sine function restricted to any of the intervals\n\uf8f0\uf8ef\uf8ee\u2212\n\uf8fb\uf8fa\uf8f9\n23\n2\n\u03c0\n, \ufffd\u03c0\n,\n2,\n2\n\uf8ee\u2212\u03c0 \u03c0\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, \n2,3\n2\n\u03c0\n\u03c0\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\netc"
    },
    {
        "Chapter": "1",
        "sentence_range": "799-802",
        "Text": "We also have\n(f \u20131 o f ) (x) = f \u20131 (f (x)) = f \u20131(y) = x\nand\n(f o f \u20131) (y) = f (f \u20131(y))  = f (x) = y\nSince the domain of sine function is the set of all real numbers and range is the\nclosed interval [\u20131, 1] If we restrict its domain to\n2,\n2\n\uf8ee\u2212\u03c0 \u03c0\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, then it becomes one-one\nand onto with range [\u2013 1, 1] Actually, sine function restricted to any of the intervals\n\uf8f0\uf8ef\uf8ee\u2212\n\uf8fb\uf8fa\uf8f9\n23\n2\n\u03c0\n, \ufffd\u03c0\n,\n2,\n2\n\uf8ee\u2212\u03c0 \u03c0\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, \n2,3\n2\n\u03c0\n\u03c0\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\netc , is one-one and its range is [\u20131, 1]"
    },
    {
        "Chapter": "1",
        "sentence_range": "800-803",
        "Text": "If we restrict its domain to\n2,\n2\n\uf8ee\u2212\u03c0 \u03c0\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, then it becomes one-one\nand onto with range [\u2013 1, 1] Actually, sine function restricted to any of the intervals\n\uf8f0\uf8ef\uf8ee\u2212\n\uf8fb\uf8fa\uf8f9\n23\n2\n\u03c0\n, \ufffd\u03c0\n,\n2,\n2\n\uf8ee\u2212\u03c0 \u03c0\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, \n2,3\n2\n\u03c0\n\u03c0\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\netc , is one-one and its range is [\u20131, 1] We can,\ntherefore, define the inverse of sine function in each of these intervals"
    },
    {
        "Chapter": "1",
        "sentence_range": "801-804",
        "Text": "Actually, sine function restricted to any of the intervals\n\uf8f0\uf8ef\uf8ee\u2212\n\uf8fb\uf8fa\uf8f9\n23\n2\n\u03c0\n, \ufffd\u03c0\n,\n2,\n2\n\uf8ee\u2212\u03c0 \u03c0\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, \n2,3\n2\n\u03c0\n\u03c0\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\netc , is one-one and its range is [\u20131, 1] We can,\ntherefore, define the inverse of sine function in each of these intervals We denote the\ninverse of sine function by sin\u20131 (arc sine function)"
    },
    {
        "Chapter": "1",
        "sentence_range": "802-805",
        "Text": ", is one-one and its range is [\u20131, 1] We can,\ntherefore, define the inverse of sine function in each of these intervals We denote the\ninverse of sine function by sin\u20131 (arc sine function) Thus, sin\u20131 is a function whose\ndomain is [\u2013 1, 1] and range could be any of the intervals \n23 ,\n2\n\uf8ee\u2212 \u03c0 \u2212\u03c0\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, \n2,\n2\n\uf8ee\u2212\u03c0 \u03c0\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n or\n2,3\n2\n\u03c0\n\u03c0\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, and so on"
    },
    {
        "Chapter": "1",
        "sentence_range": "803-806",
        "Text": "We can,\ntherefore, define the inverse of sine function in each of these intervals We denote the\ninverse of sine function by sin\u20131 (arc sine function) Thus, sin\u20131 is a function whose\ndomain is [\u2013 1, 1] and range could be any of the intervals \n23 ,\n2\n\uf8ee\u2212 \u03c0 \u2212\u03c0\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, \n2,\n2\n\uf8ee\u2212\u03c0 \u03c0\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n or\n2,3\n2\n\u03c0\n\u03c0\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, and so on Corresponding to each such interval, we get a branch of the\nfunction sin\u20131"
    },
    {
        "Chapter": "1",
        "sentence_range": "804-807",
        "Text": "We denote the\ninverse of sine function by sin\u20131 (arc sine function) Thus, sin\u20131 is a function whose\ndomain is [\u2013 1, 1] and range could be any of the intervals \n23 ,\n2\n\uf8ee\u2212 \u03c0 \u2212\u03c0\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, \n2,\n2\n\uf8ee\u2212\u03c0 \u03c0\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n or\n2,3\n2\n\u03c0\n\u03c0\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, and so on Corresponding to each such interval, we get a branch of the\nfunction sin\u20131 The branch with range \n2,\n2\n\uf8ee\u2212\u03c0 \u03c0\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb  is called the principal value branch,\nwhereas other intervals as range give different branches of sin\u20131"
    },
    {
        "Chapter": "1",
        "sentence_range": "805-808",
        "Text": "Thus, sin\u20131 is a function whose\ndomain is [\u2013 1, 1] and range could be any of the intervals \n23 ,\n2\n\uf8ee\u2212 \u03c0 \u2212\u03c0\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, \n2,\n2\n\uf8ee\u2212\u03c0 \u03c0\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n or\n2,3\n2\n\u03c0\n\u03c0\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, and so on Corresponding to each such interval, we get a branch of the\nfunction sin\u20131 The branch with range \n2,\n2\n\uf8ee\u2212\u03c0 \u03c0\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb  is called the principal value branch,\nwhereas other intervals as range give different branches of sin\u20131 When we refer\nto the function sin\u20131, we take it as the function whose domain is [\u20131, 1] and range is\n2,\n2\n\uf8ee\u2212\u03c0 \u03c0\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb"
    },
    {
        "Chapter": "1",
        "sentence_range": "806-809",
        "Text": "Corresponding to each such interval, we get a branch of the\nfunction sin\u20131 The branch with range \n2,\n2\n\uf8ee\u2212\u03c0 \u03c0\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb  is called the principal value branch,\nwhereas other intervals as range give different branches of sin\u20131 When we refer\nto the function sin\u20131, we take it as the function whose domain is [\u20131, 1] and range is\n2,\n2\n\uf8ee\u2212\u03c0 \u03c0\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb We write  sin\u20131 : [\u20131, 1] \u2192 \n2,\n2\n\uf8ee\u2212\u03c0 \u03c0\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nFrom the definition of the inverse functions, it follows that sin (sin\u20131 x) = x\nif  \u2013 1 \u2264 x \u2264 1 and sin\u20131 (sin x) = x if \n2\n2\nx\n\u03c0\n\u03c0\n\u2212\n\u2264\n\u2264"
    },
    {
        "Chapter": "1",
        "sentence_range": "807-810",
        "Text": "The branch with range \n2,\n2\n\uf8ee\u2212\u03c0 \u03c0\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb  is called the principal value branch,\nwhereas other intervals as range give different branches of sin\u20131 When we refer\nto the function sin\u20131, we take it as the function whose domain is [\u20131, 1] and range is\n2,\n2\n\uf8ee\u2212\u03c0 \u03c0\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb We write  sin\u20131 : [\u20131, 1] \u2192 \n2,\n2\n\uf8ee\u2212\u03c0 \u03c0\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nFrom the definition of the inverse functions, it follows that sin (sin\u20131 x) = x\nif  \u2013 1 \u2264 x \u2264 1 and sin\u20131 (sin x) = x if \n2\n2\nx\n\u03c0\n\u03c0\n\u2212\n\u2264\n\u2264 In other words, if y = sin\u20131 x, then\nsin y = x"
    },
    {
        "Chapter": "1",
        "sentence_range": "808-811",
        "Text": "When we refer\nto the function sin\u20131, we take it as the function whose domain is [\u20131, 1] and range is\n2,\n2\n\uf8ee\u2212\u03c0 \u03c0\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb We write  sin\u20131 : [\u20131, 1] \u2192 \n2,\n2\n\uf8ee\u2212\u03c0 \u03c0\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nFrom the definition of the inverse functions, it follows that sin (sin\u20131 x) = x\nif  \u2013 1 \u2264 x \u2264 1 and sin\u20131 (sin x) = x if \n2\n2\nx\n\u03c0\n\u03c0\n\u2212\n\u2264\n\u2264 In other words, if y = sin\u20131 x, then\nsin y = x Remarks\n(i)\nWe know from Chapter 1, that if y = f(x) is an invertible function, then  x = f \u20131 (y)"
    },
    {
        "Chapter": "1",
        "sentence_range": "809-812",
        "Text": "We write  sin\u20131 : [\u20131, 1] \u2192 \n2,\n2\n\uf8ee\u2212\u03c0 \u03c0\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nFrom the definition of the inverse functions, it follows that sin (sin\u20131 x) = x\nif  \u2013 1 \u2264 x \u2264 1 and sin\u20131 (sin x) = x if \n2\n2\nx\n\u03c0\n\u03c0\n\u2212\n\u2264\n\u2264 In other words, if y = sin\u20131 x, then\nsin y = x Remarks\n(i)\nWe know from Chapter 1, that if y = f(x) is an invertible function, then  x = f \u20131 (y) Thus, the graph of sin\u20131 function can be obtained from the graph of original\nfunction  by interchanging x and y axes, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "810-813",
        "Text": "In other words, if y = sin\u20131 x, then\nsin y = x Remarks\n(i)\nWe know from Chapter 1, that if y = f(x) is an invertible function, then  x = f \u20131 (y) Thus, the graph of sin\u20131 function can be obtained from the graph of original\nfunction  by interchanging x and y axes, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "811-814",
        "Text": "Remarks\n(i)\nWe know from Chapter 1, that if y = f(x) is an invertible function, then  x = f \u20131 (y) Thus, the graph of sin\u20131 function can be obtained from the graph of original\nfunction  by interchanging x and y axes, i e , if (a, b) is a point on the graph of\nsine function, then (b, a) becomes the corresponding point on the graph of inverse\nRationalised 2023-24\n 20\nMATHEMATICS\nof sine function"
    },
    {
        "Chapter": "1",
        "sentence_range": "812-815",
        "Text": "Thus, the graph of sin\u20131 function can be obtained from the graph of original\nfunction  by interchanging x and y axes, i e , if (a, b) is a point on the graph of\nsine function, then (b, a) becomes the corresponding point on the graph of inverse\nRationalised 2023-24\n 20\nMATHEMATICS\nof sine function Thus, the graph of the function y = sin\u20131 x can be obtained from\nthe graph of y = sin x by interchanging x and y axes"
    },
    {
        "Chapter": "1",
        "sentence_range": "813-816",
        "Text": "e , if (a, b) is a point on the graph of\nsine function, then (b, a) becomes the corresponding point on the graph of inverse\nRationalised 2023-24\n 20\nMATHEMATICS\nof sine function Thus, the graph of the function y = sin\u20131 x can be obtained from\nthe graph of y = sin x by interchanging x and y axes The graphs of y = sin x and\ny = sin\u20131 x are as given in Fig 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "814-817",
        "Text": ", if (a, b) is a point on the graph of\nsine function, then (b, a) becomes the corresponding point on the graph of inverse\nRationalised 2023-24\n 20\nMATHEMATICS\nof sine function Thus, the graph of the function y = sin\u20131 x can be obtained from\nthe graph of y = sin x by interchanging x and y axes The graphs of y = sin x and\ny = sin\u20131 x are as given in Fig 2 1 (i), (ii), (iii)"
    },
    {
        "Chapter": "1",
        "sentence_range": "815-818",
        "Text": "Thus, the graph of the function y = sin\u20131 x can be obtained from\nthe graph of y = sin x by interchanging x and y axes The graphs of y = sin x and\ny = sin\u20131 x are as given in Fig 2 1 (i), (ii), (iii) The dark portion of the graph of\ny = sin\u20131 x represent the principal value branch"
    },
    {
        "Chapter": "1",
        "sentence_range": "816-819",
        "Text": "The graphs of y = sin x and\ny = sin\u20131 x are as given in Fig 2 1 (i), (ii), (iii) The dark portion of the graph of\ny = sin\u20131 x represent the principal value branch (ii)\nIt can be shown that the graph of an inverse function can be obtained from the\ncorresponding graph of original function as a mirror image (i"
    },
    {
        "Chapter": "1",
        "sentence_range": "817-820",
        "Text": "1 (i), (ii), (iii) The dark portion of the graph of\ny = sin\u20131 x represent the principal value branch (ii)\nIt can be shown that the graph of an inverse function can be obtained from the\ncorresponding graph of original function as a mirror image (i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "818-821",
        "Text": "The dark portion of the graph of\ny = sin\u20131 x represent the principal value branch (ii)\nIt can be shown that the graph of an inverse function can be obtained from the\ncorresponding graph of original function as a mirror image (i e , reflection) along\nthe line y = x"
    },
    {
        "Chapter": "1",
        "sentence_range": "819-822",
        "Text": "(ii)\nIt can be shown that the graph of an inverse function can be obtained from the\ncorresponding graph of original function as a mirror image (i e , reflection) along\nthe line y = x This can be visualised by looking the graphs of y = sin x and\ny = sin\u20131 x as given in the same axes (Fig 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "820-823",
        "Text": "e , reflection) along\nthe line y = x This can be visualised by looking the graphs of y = sin x and\ny = sin\u20131 x as given in the same axes (Fig 2 1 (iii))"
    },
    {
        "Chapter": "1",
        "sentence_range": "821-824",
        "Text": ", reflection) along\nthe line y = x This can be visualised by looking the graphs of y = sin x and\ny = sin\u20131 x as given in the same axes (Fig 2 1 (iii)) Like sine function, the cosine function is a function whose domain is the set of all\nreal numbers and range is the set [\u20131, 1]"
    },
    {
        "Chapter": "1",
        "sentence_range": "822-825",
        "Text": "This can be visualised by looking the graphs of y = sin x and\ny = sin\u20131 x as given in the same axes (Fig 2 1 (iii)) Like sine function, the cosine function is a function whose domain is the set of all\nreal numbers and range is the set [\u20131, 1] If we restrict the domain of cosine function\nto [0, \u03c0], then it becomes one-one and onto with range [\u20131, 1]"
    },
    {
        "Chapter": "1",
        "sentence_range": "823-826",
        "Text": "1 (iii)) Like sine function, the cosine function is a function whose domain is the set of all\nreal numbers and range is the set [\u20131, 1] If we restrict the domain of cosine function\nto [0, \u03c0], then it becomes one-one and onto with range [\u20131, 1] Actually,  cosine function\nFig 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "824-827",
        "Text": "Like sine function, the cosine function is a function whose domain is the set of all\nreal numbers and range is the set [\u20131, 1] If we restrict the domain of cosine function\nto [0, \u03c0], then it becomes one-one and onto with range [\u20131, 1] Actually,  cosine function\nFig 2 1 (ii)\nFig 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "825-828",
        "Text": "If we restrict the domain of cosine function\nto [0, \u03c0], then it becomes one-one and onto with range [\u20131, 1] Actually,  cosine function\nFig 2 1 (ii)\nFig 2 1 (iii)\nFig 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "826-829",
        "Text": "Actually,  cosine function\nFig 2 1 (ii)\nFig 2 1 (iii)\nFig 2 1 (i)\nRationalised 2023-24\nINVERSE TRIGONOMETRIC FUNCTIONS     21\nrestricted to any of the intervals [\u2013 \u03c0, 0],  [0,\u03c0], [\u03c0, 2\u03c0] etc"
    },
    {
        "Chapter": "1",
        "sentence_range": "827-830",
        "Text": "1 (ii)\nFig 2 1 (iii)\nFig 2 1 (i)\nRationalised 2023-24\nINVERSE TRIGONOMETRIC FUNCTIONS     21\nrestricted to any of the intervals [\u2013 \u03c0, 0],  [0,\u03c0], [\u03c0, 2\u03c0] etc , is bijective with range as\n[\u20131, 1]"
    },
    {
        "Chapter": "1",
        "sentence_range": "828-831",
        "Text": "1 (iii)\nFig 2 1 (i)\nRationalised 2023-24\nINVERSE TRIGONOMETRIC FUNCTIONS     21\nrestricted to any of the intervals [\u2013 \u03c0, 0],  [0,\u03c0], [\u03c0, 2\u03c0] etc , is bijective with range as\n[\u20131, 1] We can, therefore, define the inverse of cosine function in each of these\nintervals"
    },
    {
        "Chapter": "1",
        "sentence_range": "829-832",
        "Text": "1 (i)\nRationalised 2023-24\nINVERSE TRIGONOMETRIC FUNCTIONS     21\nrestricted to any of the intervals [\u2013 \u03c0, 0],  [0,\u03c0], [\u03c0, 2\u03c0] etc , is bijective with range as\n[\u20131, 1] We can, therefore, define the inverse of cosine function in each of these\nintervals We denote the inverse of the cosine function by cos\u20131 (arc cosine function)"
    },
    {
        "Chapter": "1",
        "sentence_range": "830-833",
        "Text": ", is bijective with range as\n[\u20131, 1] We can, therefore, define the inverse of cosine function in each of these\nintervals We denote the inverse of the cosine function by cos\u20131 (arc cosine function) Thus, cos\u20131 is a function whose domain is [\u20131, 1] and range\ncould be any of the intervals [\u2013\u03c0, 0], [0, \u03c0], [\u03c0, 2\u03c0] etc"
    },
    {
        "Chapter": "1",
        "sentence_range": "831-834",
        "Text": "We can, therefore, define the inverse of cosine function in each of these\nintervals We denote the inverse of the cosine function by cos\u20131 (arc cosine function) Thus, cos\u20131 is a function whose domain is [\u20131, 1] and range\ncould be any of the intervals [\u2013\u03c0, 0], [0, \u03c0], [\u03c0, 2\u03c0] etc Corresponding to each such interval, we get a branch of the\nfunction cos\u20131"
    },
    {
        "Chapter": "1",
        "sentence_range": "832-835",
        "Text": "We denote the inverse of the cosine function by cos\u20131 (arc cosine function) Thus, cos\u20131 is a function whose domain is [\u20131, 1] and range\ncould be any of the intervals [\u2013\u03c0, 0], [0, \u03c0], [\u03c0, 2\u03c0] etc Corresponding to each such interval, we get a branch of the\nfunction cos\u20131 The branch with range [0, \u03c0] is called the principal\nvalue branch of the function cos\u20131"
    },
    {
        "Chapter": "1",
        "sentence_range": "833-836",
        "Text": "Thus, cos\u20131 is a function whose domain is [\u20131, 1] and range\ncould be any of the intervals [\u2013\u03c0, 0], [0, \u03c0], [\u03c0, 2\u03c0] etc Corresponding to each such interval, we get a branch of the\nfunction cos\u20131 The branch with range [0, \u03c0] is called the principal\nvalue branch of the function cos\u20131 We write\ncos\u20131 : [\u20131, 1] \u2192 [0, \u03c0]"
    },
    {
        "Chapter": "1",
        "sentence_range": "834-837",
        "Text": "Corresponding to each such interval, we get a branch of the\nfunction cos\u20131 The branch with range [0, \u03c0] is called the principal\nvalue branch of the function cos\u20131 We write\ncos\u20131 : [\u20131, 1] \u2192 [0, \u03c0] The graph of the function given by y = cos\u20131 x can be drawn\nin the same way as discussed about the graph of y = sin\u20131 x"
    },
    {
        "Chapter": "1",
        "sentence_range": "835-838",
        "Text": "The branch with range [0, \u03c0] is called the principal\nvalue branch of the function cos\u20131 We write\ncos\u20131 : [\u20131, 1] \u2192 [0, \u03c0] The graph of the function given by y = cos\u20131 x can be drawn\nin the same way as discussed about the graph of y = sin\u20131 x The\ngraphs of y = cos x and y = cos\u20131x are given in Fig 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "836-839",
        "Text": "We write\ncos\u20131 : [\u20131, 1] \u2192 [0, \u03c0] The graph of the function given by y = cos\u20131 x can be drawn\nin the same way as discussed about the graph of y = sin\u20131 x The\ngraphs of y = cos x and y = cos\u20131x are given in Fig 2 2 (i) and (ii)"
    },
    {
        "Chapter": "1",
        "sentence_range": "837-840",
        "Text": "The graph of the function given by y = cos\u20131 x can be drawn\nin the same way as discussed about the graph of y = sin\u20131 x The\ngraphs of y = cos x and y = cos\u20131x are given in Fig 2 2 (i) and (ii) Fig 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "838-841",
        "Text": "The\ngraphs of y = cos x and y = cos\u20131x are given in Fig 2 2 (i) and (ii) Fig 2 2 (ii)\nLet us now discuss cosec\u20131x and sec\u20131x as follows:\nSince, cosec x = \n1\nsin x , the domain of the cosec function is the set {x : x \u2208 R and\nx \u2260 n\u03c0, n \u2208 Z} and the range is the set {y : y \u2208 R, y \u2265 1 or y \u2264 \u20131} i"
    },
    {
        "Chapter": "1",
        "sentence_range": "839-842",
        "Text": "2 (i) and (ii) Fig 2 2 (ii)\nLet us now discuss cosec\u20131x and sec\u20131x as follows:\nSince, cosec x = \n1\nsin x , the domain of the cosec function is the set {x : x \u2208 R and\nx \u2260 n\u03c0, n \u2208 Z} and the range is the set {y : y \u2208 R, y \u2265 1 or y \u2264 \u20131} i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "840-843",
        "Text": "Fig 2 2 (ii)\nLet us now discuss cosec\u20131x and sec\u20131x as follows:\nSince, cosec x = \n1\nsin x , the domain of the cosec function is the set {x : x \u2208 R and\nx \u2260 n\u03c0, n \u2208 Z} and the range is the set {y : y \u2208 R, y \u2265 1 or y \u2264 \u20131} i e , the set\nR \u2013 (\u20131, 1)"
    },
    {
        "Chapter": "1",
        "sentence_range": "841-844",
        "Text": "2 (ii)\nLet us now discuss cosec\u20131x and sec\u20131x as follows:\nSince, cosec x = \n1\nsin x , the domain of the cosec function is the set {x : x \u2208 R and\nx \u2260 n\u03c0, n \u2208 Z} and the range is the set {y : y \u2208 R, y \u2265 1 or y \u2264 \u20131} i e , the set\nR \u2013 (\u20131, 1) It means that y = cosec x assumes all real values except \u20131 < y < 1 and is\nnot defined for integral multiple of \u03c0"
    },
    {
        "Chapter": "1",
        "sentence_range": "842-845",
        "Text": "e , the set\nR \u2013 (\u20131, 1) It means that y = cosec x assumes all real values except \u20131 < y < 1 and is\nnot defined for integral multiple of \u03c0 If we restrict the domain of cosec function to\n,\n2 2\n\uf8ee\u03c0 \u03c0\n\uf8f9\n\uf8ef\u2212\n\uf8fa\n\uf8f0\n\uf8fb \u2013 {0}, then it is one to one and onto with its range as the set R \u2013 (\u2013 1, 1)"
    },
    {
        "Chapter": "1",
        "sentence_range": "843-846",
        "Text": ", the set\nR \u2013 (\u20131, 1) It means that y = cosec x assumes all real values except \u20131 < y < 1 and is\nnot defined for integral multiple of \u03c0 If we restrict the domain of cosec function to\n,\n2 2\n\uf8ee\u03c0 \u03c0\n\uf8f9\n\uf8ef\u2212\n\uf8fa\n\uf8f0\n\uf8fb \u2013 {0}, then it is one to one and onto with its range as the set R \u2013 (\u2013 1, 1) Actually,\ncosec function restricted to any of the intervals \n3 ,\n{\n}\n2\n2\n\uf8ee\u2212 \u03c0 \u2212\u03c0\n\uf8f9 \u2212 \u2212\u03c0\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, \n2,\n2\n\uf8ee\u2212\u03c0 \u03c0\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb  \u2013 {0},\n,3\n{ }\n2\n2\n\u03c0\n\u03c0\n\uf8ee\n\uf8f9 \u2212 \u03c0\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n etc"
    },
    {
        "Chapter": "1",
        "sentence_range": "844-847",
        "Text": "It means that y = cosec x assumes all real values except \u20131 < y < 1 and is\nnot defined for integral multiple of \u03c0 If we restrict the domain of cosec function to\n,\n2 2\n\uf8ee\u03c0 \u03c0\n\uf8f9\n\uf8ef\u2212\n\uf8fa\n\uf8f0\n\uf8fb \u2013 {0}, then it is one to one and onto with its range as the set R \u2013 (\u2013 1, 1) Actually,\ncosec function restricted to any of the intervals \n3 ,\n{\n}\n2\n2\n\uf8ee\u2212 \u03c0 \u2212\u03c0\n\uf8f9 \u2212 \u2212\u03c0\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, \n2,\n2\n\uf8ee\u2212\u03c0 \u03c0\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb  \u2013 {0},\n,3\n{ }\n2\n2\n\u03c0\n\u03c0\n\uf8ee\n\uf8f9 \u2212 \u03c0\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n etc , is bijective and its range is the set of all real numbers R \u2013 (\u20131, 1)"
    },
    {
        "Chapter": "1",
        "sentence_range": "845-848",
        "Text": "If we restrict the domain of cosec function to\n,\n2 2\n\uf8ee\u03c0 \u03c0\n\uf8f9\n\uf8ef\u2212\n\uf8fa\n\uf8f0\n\uf8fb \u2013 {0}, then it is one to one and onto with its range as the set R \u2013 (\u2013 1, 1) Actually,\ncosec function restricted to any of the intervals \n3 ,\n{\n}\n2\n2\n\uf8ee\u2212 \u03c0 \u2212\u03c0\n\uf8f9 \u2212 \u2212\u03c0\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, \n2,\n2\n\uf8ee\u2212\u03c0 \u03c0\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb  \u2013 {0},\n,3\n{ }\n2\n2\n\u03c0\n\u03c0\n\uf8ee\n\uf8f9 \u2212 \u03c0\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n etc , is bijective and its range is the set of all real numbers R \u2013 (\u20131, 1) Fig 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "846-849",
        "Text": "Actually,\ncosec function restricted to any of the intervals \n3 ,\n{\n}\n2\n2\n\uf8ee\u2212 \u03c0 \u2212\u03c0\n\uf8f9 \u2212 \u2212\u03c0\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, \n2,\n2\n\uf8ee\u2212\u03c0 \u03c0\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb  \u2013 {0},\n,3\n{ }\n2\n2\n\u03c0\n\u03c0\n\uf8ee\n\uf8f9 \u2212 \u03c0\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n etc , is bijective and its range is the set of all real numbers R \u2013 (\u20131, 1) Fig 2 2 (i)\nRationalised 2023-24\n 22\nMATHEMATICS\nThus cosec\u20131 can be defined as a function whose domain is R \u2013 (\u20131, 1) and range could\nbe any of the intervals \n\u2212\n\u2212\n\uf8f0\uf8ef\uf8ee\n\uf8f9\n\uf8fb\uf8fa \u2212 \u2212\n23\n2\n\u03c0\n\u03c0\n\u03c0\n,\n{\n}, \n\uf8f0\uf8ef\uf8ee\u2212\n\uf8f9\n2\u03c0 \u03c0\uf8fb\uf8fa \u2212\n2\n0\n,\n{ }, \n,3\n{ }\n2\n2\n\u03c0\n\u03c0\n\uf8ee\n\uf8f9 \u2212 \u03c0\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\netc"
    },
    {
        "Chapter": "1",
        "sentence_range": "847-850",
        "Text": ", is bijective and its range is the set of all real numbers R \u2013 (\u20131, 1) Fig 2 2 (i)\nRationalised 2023-24\n 22\nMATHEMATICS\nThus cosec\u20131 can be defined as a function whose domain is R \u2013 (\u20131, 1) and range could\nbe any of the intervals \n\u2212\n\u2212\n\uf8f0\uf8ef\uf8ee\n\uf8f9\n\uf8fb\uf8fa \u2212 \u2212\n23\n2\n\u03c0\n\u03c0\n\u03c0\n,\n{\n}, \n\uf8f0\uf8ef\uf8ee\u2212\n\uf8f9\n2\u03c0 \u03c0\uf8fb\uf8fa \u2212\n2\n0\n,\n{ }, \n,3\n{ }\n2\n2\n\u03c0\n\u03c0\n\uf8ee\n\uf8f9 \u2212 \u03c0\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\netc The\nfunction corresponding to the range \n,\n{0}\n2\n\uf8ee\u2212\u03c0 \u03c02\n\uf8f9 \u2212\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nis called the principal value branch\nof cosec\u20131"
    },
    {
        "Chapter": "1",
        "sentence_range": "848-851",
        "Text": "Fig 2 2 (i)\nRationalised 2023-24\n 22\nMATHEMATICS\nThus cosec\u20131 can be defined as a function whose domain is R \u2013 (\u20131, 1) and range could\nbe any of the intervals \n\u2212\n\u2212\n\uf8f0\uf8ef\uf8ee\n\uf8f9\n\uf8fb\uf8fa \u2212 \u2212\n23\n2\n\u03c0\n\u03c0\n\u03c0\n,\n{\n}, \n\uf8f0\uf8ef\uf8ee\u2212\n\uf8f9\n2\u03c0 \u03c0\uf8fb\uf8fa \u2212\n2\n0\n,\n{ }, \n,3\n{ }\n2\n2\n\u03c0\n\u03c0\n\uf8ee\n\uf8f9 \u2212 \u03c0\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\netc The\nfunction corresponding to the range \n,\n{0}\n2\n\uf8ee\u2212\u03c0 \u03c02\n\uf8f9 \u2212\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nis called the principal value branch\nof cosec\u20131 We thus have principal branch as\ncosec\u20131  : R \u2013 (\u20131, 1) \u2192 \n,\n{0}\n2\n\uf8ee\u2212\u03c0 \u03c02\n\uf8f9 \u2212\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nThe graphs of y = cosec x and y = cosec\u20131 x are given in Fig 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "849-852",
        "Text": "2 (i)\nRationalised 2023-24\n 22\nMATHEMATICS\nThus cosec\u20131 can be defined as a function whose domain is R \u2013 (\u20131, 1) and range could\nbe any of the intervals \n\u2212\n\u2212\n\uf8f0\uf8ef\uf8ee\n\uf8f9\n\uf8fb\uf8fa \u2212 \u2212\n23\n2\n\u03c0\n\u03c0\n\u03c0\n,\n{\n}, \n\uf8f0\uf8ef\uf8ee\u2212\n\uf8f9\n2\u03c0 \u03c0\uf8fb\uf8fa \u2212\n2\n0\n,\n{ }, \n,3\n{ }\n2\n2\n\u03c0\n\u03c0\n\uf8ee\n\uf8f9 \u2212 \u03c0\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\netc The\nfunction corresponding to the range \n,\n{0}\n2\n\uf8ee\u2212\u03c0 \u03c02\n\uf8f9 \u2212\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nis called the principal value branch\nof cosec\u20131 We thus have principal branch as\ncosec\u20131  : R \u2013 (\u20131, 1) \u2192 \n,\n{0}\n2\n\uf8ee\u2212\u03c0 \u03c02\n\uf8f9 \u2212\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nThe graphs of y = cosec x and y = cosec\u20131 x are given in Fig 2 3 (i), (ii)"
    },
    {
        "Chapter": "1",
        "sentence_range": "850-853",
        "Text": "The\nfunction corresponding to the range \n,\n{0}\n2\n\uf8ee\u2212\u03c0 \u03c02\n\uf8f9 \u2212\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nis called the principal value branch\nof cosec\u20131 We thus have principal branch as\ncosec\u20131  : R \u2013 (\u20131, 1) \u2192 \n,\n{0}\n2\n\uf8ee\u2212\u03c0 \u03c02\n\uf8f9 \u2212\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nThe graphs of y = cosec x and y = cosec\u20131 x are given in Fig 2 3 (i), (ii) Also, since sec x = \n1\ncos x , the domain of y = sec x is the set R \u2013 {x : x = (2n + 1) 2\n\u03c0 ,\nn \u2208 Z} and range is the set R \u2013 (\u20131, 1)"
    },
    {
        "Chapter": "1",
        "sentence_range": "851-854",
        "Text": "We thus have principal branch as\ncosec\u20131  : R \u2013 (\u20131, 1) \u2192 \n,\n{0}\n2\n\uf8ee\u2212\u03c0 \u03c02\n\uf8f9 \u2212\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nThe graphs of y = cosec x and y = cosec\u20131 x are given in Fig 2 3 (i), (ii) Also, since sec x = \n1\ncos x , the domain of y = sec x is the set R \u2013 {x : x = (2n + 1) 2\n\u03c0 ,\nn \u2208 Z} and range is the set R \u2013 (\u20131, 1) It means that sec (secant function) assumes\nall real values except \u20131 < y < 1 and is not defined for odd multiples of 2\n\u03c0"
    },
    {
        "Chapter": "1",
        "sentence_range": "852-855",
        "Text": "3 (i), (ii) Also, since sec x = \n1\ncos x , the domain of y = sec x is the set R \u2013 {x : x = (2n + 1) 2\n\u03c0 ,\nn \u2208 Z} and range is the set R \u2013 (\u20131, 1) It means that sec (secant function) assumes\nall real values except \u20131 < y < 1 and is not defined for odd multiples of 2\n\u03c0 If we\nrestrict the domain of  secant function to [0, \u03c0] \u2013 { 2\n\u03c0 }, then it is one-one and onto with\nFig 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "853-856",
        "Text": "Also, since sec x = \n1\ncos x , the domain of y = sec x is the set R \u2013 {x : x = (2n + 1) 2\n\u03c0 ,\nn \u2208 Z} and range is the set R \u2013 (\u20131, 1) It means that sec (secant function) assumes\nall real values except \u20131 < y < 1 and is not defined for odd multiples of 2\n\u03c0 If we\nrestrict the domain of  secant function to [0, \u03c0] \u2013 { 2\n\u03c0 }, then it is one-one and onto with\nFig 2 3 (i)\nFig 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "854-857",
        "Text": "It means that sec (secant function) assumes\nall real values except \u20131 < y < 1 and is not defined for odd multiples of 2\n\u03c0 If we\nrestrict the domain of  secant function to [0, \u03c0] \u2013 { 2\n\u03c0 }, then it is one-one and onto with\nFig 2 3 (i)\nFig 2 3 (ii)\nRationalised 2023-24\nINVERSE TRIGONOMETRIC FUNCTIONS     23\nits range as the set R \u2013 (\u20131, 1)"
    },
    {
        "Chapter": "1",
        "sentence_range": "855-858",
        "Text": "If we\nrestrict the domain of  secant function to [0, \u03c0] \u2013 { 2\n\u03c0 }, then it is one-one and onto with\nFig 2 3 (i)\nFig 2 3 (ii)\nRationalised 2023-24\nINVERSE TRIGONOMETRIC FUNCTIONS     23\nits range as the set R \u2013 (\u20131, 1) Actually, secant function  restricted to any of the\nintervals [\u2013\u03c0, 0] \u2013 { 2\n\u2212\u03c0 }, [0, ] \u2013\n\u03c02\n\uf8f1 \uf8fc\n\u03c0\n\uf8f2 \uf8fd\n\uf8f3 \uf8fe\n, [\u03c0, 2\u03c0] \u2013 { 3\n2\n\u03c0 } etc"
    },
    {
        "Chapter": "1",
        "sentence_range": "856-859",
        "Text": "3 (i)\nFig 2 3 (ii)\nRationalised 2023-24\nINVERSE TRIGONOMETRIC FUNCTIONS     23\nits range as the set R \u2013 (\u20131, 1) Actually, secant function  restricted to any of the\nintervals [\u2013\u03c0, 0] \u2013 { 2\n\u2212\u03c0 }, [0, ] \u2013\n\u03c02\n\uf8f1 \uf8fc\n\u03c0\n\uf8f2 \uf8fd\n\uf8f3 \uf8fe\n, [\u03c0, 2\u03c0] \u2013 { 3\n2\n\u03c0 } etc , is bijective and its range\nis R \u2013 {\u20131, 1}"
    },
    {
        "Chapter": "1",
        "sentence_range": "857-860",
        "Text": "3 (ii)\nRationalised 2023-24\nINVERSE TRIGONOMETRIC FUNCTIONS     23\nits range as the set R \u2013 (\u20131, 1) Actually, secant function  restricted to any of the\nintervals [\u2013\u03c0, 0] \u2013 { 2\n\u2212\u03c0 }, [0, ] \u2013\n\u03c02\n\uf8f1 \uf8fc\n\u03c0\n\uf8f2 \uf8fd\n\uf8f3 \uf8fe\n, [\u03c0, 2\u03c0] \u2013 { 3\n2\n\u03c0 } etc , is bijective and its range\nis R \u2013 {\u20131, 1} Thus sec\u20131 can be defined as a function whose domain is R\u2013 (\u20131, 1) and\nrange could be any of the intervals [\u2013 \u03c0, 0] \u2013 { 2\n\u2212\u03c0 }, [0, \u03c0] \u2013 { 2\n\u03c0 }, [\u03c0, 2\u03c0] \u2013 { 3\n2\n\u03c0 } etc"
    },
    {
        "Chapter": "1",
        "sentence_range": "858-861",
        "Text": "Actually, secant function  restricted to any of the\nintervals [\u2013\u03c0, 0] \u2013 { 2\n\u2212\u03c0 }, [0, ] \u2013\n\u03c02\n\uf8f1 \uf8fc\n\u03c0\n\uf8f2 \uf8fd\n\uf8f3 \uf8fe\n, [\u03c0, 2\u03c0] \u2013 { 3\n2\n\u03c0 } etc , is bijective and its range\nis R \u2013 {\u20131, 1} Thus sec\u20131 can be defined as a function whose domain is R\u2013 (\u20131, 1) and\nrange could be any of the intervals [\u2013 \u03c0, 0] \u2013 { 2\n\u2212\u03c0 }, [0, \u03c0] \u2013 { 2\n\u03c0 }, [\u03c0, 2\u03c0] \u2013 { 3\n2\n\u03c0 } etc Corresponding to each of these intervals, we get different branches of the function sec\u20131"
    },
    {
        "Chapter": "1",
        "sentence_range": "859-862",
        "Text": ", is bijective and its range\nis R \u2013 {\u20131, 1} Thus sec\u20131 can be defined as a function whose domain is R\u2013 (\u20131, 1) and\nrange could be any of the intervals [\u2013 \u03c0, 0] \u2013 { 2\n\u2212\u03c0 }, [0, \u03c0] \u2013 { 2\n\u03c0 }, [\u03c0, 2\u03c0] \u2013 { 3\n2\n\u03c0 } etc Corresponding to each of these intervals, we get different branches of the function sec\u20131 The branch with range [0, \u03c0] \u2013 { 2\n\u03c0 } is called the principal value branch of the\nfunction sec\u20131"
    },
    {
        "Chapter": "1",
        "sentence_range": "860-863",
        "Text": "Thus sec\u20131 can be defined as a function whose domain is R\u2013 (\u20131, 1) and\nrange could be any of the intervals [\u2013 \u03c0, 0] \u2013 { 2\n\u2212\u03c0 }, [0, \u03c0] \u2013 { 2\n\u03c0 }, [\u03c0, 2\u03c0] \u2013 { 3\n2\n\u03c0 } etc Corresponding to each of these intervals, we get different branches of the function sec\u20131 The branch with range [0, \u03c0] \u2013 { 2\n\u03c0 } is called the principal value branch of the\nfunction sec\u20131 We thus have\nsec\u20131 : R \u2013 (\u20131,1) \u2192 [0, \u03c0] \u2013 { 2\n\u03c0 }\nThe graphs of the functions y = sec x and y = sec-1 x are given in Fig 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "861-864",
        "Text": "Corresponding to each of these intervals, we get different branches of the function sec\u20131 The branch with range [0, \u03c0] \u2013 { 2\n\u03c0 } is called the principal value branch of the\nfunction sec\u20131 We thus have\nsec\u20131 : R \u2013 (\u20131,1) \u2192 [0, \u03c0] \u2013 { 2\n\u03c0 }\nThe graphs of the functions y = sec x and y = sec-1 x are given in Fig 2 4 (i), (ii)"
    },
    {
        "Chapter": "1",
        "sentence_range": "862-865",
        "Text": "The branch with range [0, \u03c0] \u2013 { 2\n\u03c0 } is called the principal value branch of the\nfunction sec\u20131 We thus have\nsec\u20131 : R \u2013 (\u20131,1) \u2192 [0, \u03c0] \u2013 { 2\n\u03c0 }\nThe graphs of the functions y = sec x and y = sec-1 x are given in Fig 2 4 (i), (ii) Finally, we now discuss tan\u20131 and cot\u20131\nWe know that the domain of the tan function (tangent function) is the set\n{x : x \u2208 R and x \u2260 (2n +1) 2\n\u03c0 , n \u2208 Z} and the range is R"
    },
    {
        "Chapter": "1",
        "sentence_range": "863-866",
        "Text": "We thus have\nsec\u20131 : R \u2013 (\u20131,1) \u2192 [0, \u03c0] \u2013 { 2\n\u03c0 }\nThe graphs of the functions y = sec x and y = sec-1 x are given in Fig 2 4 (i), (ii) Finally, we now discuss tan\u20131 and cot\u20131\nWe know that the domain of the tan function (tangent function) is the set\n{x : x \u2208 R and x \u2260 (2n +1) 2\n\u03c0 , n \u2208 Z} and the range is R It means that tan function\nis not defined for odd multiples of  2\n\u03c0"
    },
    {
        "Chapter": "1",
        "sentence_range": "864-867",
        "Text": "4 (i), (ii) Finally, we now discuss tan\u20131 and cot\u20131\nWe know that the domain of the tan function (tangent function) is the set\n{x : x \u2208 R and x \u2260 (2n +1) 2\n\u03c0 , n \u2208 Z} and the range is R It means that tan function\nis not defined for odd multiples of  2\n\u03c0 If we restrict the domain of tangent function to\nFig 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "865-868",
        "Text": "Finally, we now discuss tan\u20131 and cot\u20131\nWe know that the domain of the tan function (tangent function) is the set\n{x : x \u2208 R and x \u2260 (2n +1) 2\n\u03c0 , n \u2208 Z} and the range is R It means that tan function\nis not defined for odd multiples of  2\n\u03c0 If we restrict the domain of tangent function to\nFig 2 4 (i)\nFig 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "866-869",
        "Text": "It means that tan function\nis not defined for odd multiples of  2\n\u03c0 If we restrict the domain of tangent function to\nFig 2 4 (i)\nFig 2 4 (ii)\nRationalised 2023-24\n 24\nMATHEMATICS\n2,\n2\n\uf8eb\u2212\u03c0 \u03c0\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n, then it is one-one and onto with its range as R"
    },
    {
        "Chapter": "1",
        "sentence_range": "867-870",
        "Text": "If we restrict the domain of tangent function to\nFig 2 4 (i)\nFig 2 4 (ii)\nRationalised 2023-24\n 24\nMATHEMATICS\n2,\n2\n\uf8eb\u2212\u03c0 \u03c0\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n, then it is one-one and onto with its range as R Actually, tangent function\nrestricted to any of the intervals \n23 ,\n2\n\uf8eb\u2212 \u03c0 \u2212\u03c0\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n, \n2,\n2\n\uf8eb\u2212\u03c0 \u03c0\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n, \n2,3\n2\n\u03c0\n\u03c0\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n etc"
    },
    {
        "Chapter": "1",
        "sentence_range": "868-871",
        "Text": "4 (i)\nFig 2 4 (ii)\nRationalised 2023-24\n 24\nMATHEMATICS\n2,\n2\n\uf8eb\u2212\u03c0 \u03c0\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n, then it is one-one and onto with its range as R Actually, tangent function\nrestricted to any of the intervals \n23 ,\n2\n\uf8eb\u2212 \u03c0 \u2212\u03c0\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n, \n2,\n2\n\uf8eb\u2212\u03c0 \u03c0\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n, \n2,3\n2\n\u03c0\n\u03c0\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n etc , is bijective\nand its range is R"
    },
    {
        "Chapter": "1",
        "sentence_range": "869-872",
        "Text": "4 (ii)\nRationalised 2023-24\n 24\nMATHEMATICS\n2,\n2\n\uf8eb\u2212\u03c0 \u03c0\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n, then it is one-one and onto with its range as R Actually, tangent function\nrestricted to any of the intervals \n23 ,\n2\n\uf8eb\u2212 \u03c0 \u2212\u03c0\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n, \n2,\n2\n\uf8eb\u2212\u03c0 \u03c0\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n, \n2,3\n2\n\u03c0\n\u03c0\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n etc , is bijective\nand its range is R Thus tan\u20131 can be defined as a function whose domain is R and\nrange could be any of the intervals \n23 ,\n2\n\uf8eb\u2212 \u03c0 \u2212\u03c0\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8 , \n2,\n2\n\uf8eb\u2212\u03c0 \u03c0\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8 , \n2,3\n2\n\u03c0\n\u03c0\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8  and so on"
    },
    {
        "Chapter": "1",
        "sentence_range": "870-873",
        "Text": "Actually, tangent function\nrestricted to any of the intervals \n23 ,\n2\n\uf8eb\u2212 \u03c0 \u2212\u03c0\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n, \n2,\n2\n\uf8eb\u2212\u03c0 \u03c0\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n, \n2,3\n2\n\u03c0\n\u03c0\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n etc , is bijective\nand its range is R Thus tan\u20131 can be defined as a function whose domain is R and\nrange could be any of the intervals \n23 ,\n2\n\uf8eb\u2212 \u03c0 \u2212\u03c0\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8 , \n2,\n2\n\uf8eb\u2212\u03c0 \u03c0\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8 , \n2,3\n2\n\u03c0\n\u03c0\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8  and so on These\nintervals give different branches of the function  tan\u20131"
    },
    {
        "Chapter": "1",
        "sentence_range": "871-874",
        "Text": ", is bijective\nand its range is R Thus tan\u20131 can be defined as a function whose domain is R and\nrange could be any of the intervals \n23 ,\n2\n\uf8eb\u2212 \u03c0 \u2212\u03c0\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8 , \n2,\n2\n\uf8eb\u2212\u03c0 \u03c0\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8 , \n2,3\n2\n\u03c0\n\u03c0\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8  and so on These\nintervals give different branches of the function  tan\u20131 The branch with range \n2,\n2\n\uf8eb\u2212\u03c0 \u03c0\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\nis called the principal value branch of the function tan\u20131"
    },
    {
        "Chapter": "1",
        "sentence_range": "872-875",
        "Text": "Thus tan\u20131 can be defined as a function whose domain is R and\nrange could be any of the intervals \n23 ,\n2\n\uf8eb\u2212 \u03c0 \u2212\u03c0\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8 , \n2,\n2\n\uf8eb\u2212\u03c0 \u03c0\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8 , \n2,3\n2\n\u03c0\n\u03c0\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8  and so on These\nintervals give different branches of the function  tan\u20131 The branch with range \n2,\n2\n\uf8eb\u2212\u03c0 \u03c0\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\nis called the principal value branch of the function tan\u20131 We thus have\ntan\u20131 : R \u2192 \n2,\n2\n\uf8eb\u2212\u03c0 \u03c0\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\nThe graphs of the function  y = tan x and y = tan\u20131x are given in Fig 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "873-876",
        "Text": "These\nintervals give different branches of the function  tan\u20131 The branch with range \n2,\n2\n\uf8eb\u2212\u03c0 \u03c0\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\nis called the principal value branch of the function tan\u20131 We thus have\ntan\u20131 : R \u2192 \n2,\n2\n\uf8eb\u2212\u03c0 \u03c0\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\nThe graphs of the function  y = tan x and y = tan\u20131x are given in Fig 2 5 (i), (ii)"
    },
    {
        "Chapter": "1",
        "sentence_range": "874-877",
        "Text": "The branch with range \n2,\n2\n\uf8eb\u2212\u03c0 \u03c0\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\nis called the principal value branch of the function tan\u20131 We thus have\ntan\u20131 : R \u2192 \n2,\n2\n\uf8eb\u2212\u03c0 \u03c0\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\nThe graphs of the function  y = tan x and y = tan\u20131x are given in Fig 2 5 (i), (ii) Fig 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "875-878",
        "Text": "We thus have\ntan\u20131 : R \u2192 \n2,\n2\n\uf8eb\u2212\u03c0 \u03c0\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\nThe graphs of the function  y = tan x and y = tan\u20131x are given in Fig 2 5 (i), (ii) Fig 2 5 (i)\nFig  2"
    },
    {
        "Chapter": "1",
        "sentence_range": "876-879",
        "Text": "5 (i), (ii) Fig 2 5 (i)\nFig  2 5 (ii)\nWe know that domain of the cot function (cotangent function) is the set\n{x : x \u2208 R and x \u2260 n\u03c0, n \u2208 Z} and range is R"
    },
    {
        "Chapter": "1",
        "sentence_range": "877-880",
        "Text": "Fig 2 5 (i)\nFig  2 5 (ii)\nWe know that domain of the cot function (cotangent function) is the set\n{x : x \u2208 R and x \u2260 n\u03c0, n \u2208 Z} and range is R It means that cotangent function is not\ndefined for integral multiples of \u03c0"
    },
    {
        "Chapter": "1",
        "sentence_range": "878-881",
        "Text": "5 (i)\nFig  2 5 (ii)\nWe know that domain of the cot function (cotangent function) is the set\n{x : x \u2208 R and x \u2260 n\u03c0, n \u2208 Z} and range is R It means that cotangent function is not\ndefined for integral multiples of \u03c0 If we restrict the domain of cotangent function to\n(0, \u03c0), then it is bijective with and its range as R"
    },
    {
        "Chapter": "1",
        "sentence_range": "879-882",
        "Text": "5 (ii)\nWe know that domain of the cot function (cotangent function) is the set\n{x : x \u2208 R and x \u2260 n\u03c0, n \u2208 Z} and range is R It means that cotangent function is not\ndefined for integral multiples of \u03c0 If we restrict the domain of cotangent function to\n(0, \u03c0), then it is bijective with and its range as R In fact, cotangent function restricted\nto any of the intervals (\u2013\u03c0, 0), (0, \u03c0), (\u03c0, 2\u03c0) etc"
    },
    {
        "Chapter": "1",
        "sentence_range": "880-883",
        "Text": "It means that cotangent function is not\ndefined for integral multiples of \u03c0 If we restrict the domain of cotangent function to\n(0, \u03c0), then it is bijective with and its range as R In fact, cotangent function restricted\nto any of the intervals (\u2013\u03c0, 0), (0, \u03c0), (\u03c0, 2\u03c0) etc , is bijective and its range is R"
    },
    {
        "Chapter": "1",
        "sentence_range": "881-884",
        "Text": "If we restrict the domain of cotangent function to\n(0, \u03c0), then it is bijective with and its range as R In fact, cotangent function restricted\nto any of the intervals (\u2013\u03c0, 0), (0, \u03c0), (\u03c0, 2\u03c0) etc , is bijective and its range is R Thus\ncot \u20131 can be defined as a function whose domain is the R and range as any of the\nRationalised 2023-24\nINVERSE TRIGONOMETRIC FUNCTIONS     25\nintervals (\u2013\u03c0, 0), (0, \u03c0), (\u03c0, 2\u03c0) etc"
    },
    {
        "Chapter": "1",
        "sentence_range": "882-885",
        "Text": "In fact, cotangent function restricted\nto any of the intervals (\u2013\u03c0, 0), (0, \u03c0), (\u03c0, 2\u03c0) etc , is bijective and its range is R Thus\ncot \u20131 can be defined as a function whose domain is the R and range as any of the\nRationalised 2023-24\nINVERSE TRIGONOMETRIC FUNCTIONS     25\nintervals (\u2013\u03c0, 0), (0, \u03c0), (\u03c0, 2\u03c0) etc These intervals give different branches of the\nfunction cot \u20131"
    },
    {
        "Chapter": "1",
        "sentence_range": "883-886",
        "Text": ", is bijective and its range is R Thus\ncot \u20131 can be defined as a function whose domain is the R and range as any of the\nRationalised 2023-24\nINVERSE TRIGONOMETRIC FUNCTIONS     25\nintervals (\u2013\u03c0, 0), (0, \u03c0), (\u03c0, 2\u03c0) etc These intervals give different branches of the\nfunction cot \u20131 The function with range (0, \u03c0) is called the principal value branch of\nthe function cot \u20131"
    },
    {
        "Chapter": "1",
        "sentence_range": "884-887",
        "Text": "Thus\ncot \u20131 can be defined as a function whose domain is the R and range as any of the\nRationalised 2023-24\nINVERSE TRIGONOMETRIC FUNCTIONS     25\nintervals (\u2013\u03c0, 0), (0, \u03c0), (\u03c0, 2\u03c0) etc These intervals give different branches of the\nfunction cot \u20131 The function with range (0, \u03c0) is called the principal value branch of\nthe function cot \u20131 We thus have\ncot\u20131 : R \u2192 (0, \u03c0)\nThe graphs of y = cot x and y = cot\u20131x are given in Fig 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "885-888",
        "Text": "These intervals give different branches of the\nfunction cot \u20131 The function with range (0, \u03c0) is called the principal value branch of\nthe function cot \u20131 We thus have\ncot\u20131 : R \u2192 (0, \u03c0)\nThe graphs of y = cot x and y = cot\u20131x are given in Fig 2 6 (i), (ii)"
    },
    {
        "Chapter": "1",
        "sentence_range": "886-889",
        "Text": "The function with range (0, \u03c0) is called the principal value branch of\nthe function cot \u20131 We thus have\ncot\u20131 : R \u2192 (0, \u03c0)\nThe graphs of y = cot x and y = cot\u20131x are given in Fig 2 6 (i), (ii) Fig 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "887-890",
        "Text": "We thus have\ncot\u20131 : R \u2192 (0, \u03c0)\nThe graphs of y = cot x and y = cot\u20131x are given in Fig 2 6 (i), (ii) Fig 2 6 (i)\nFig 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "888-891",
        "Text": "6 (i), (ii) Fig 2 6 (i)\nFig 2 6 (ii)\nThe following table gives the inverse trigonometric function (principal value\nbranches) along with their domains and ranges"
    },
    {
        "Chapter": "1",
        "sentence_range": "889-892",
        "Text": "Fig 2 6 (i)\nFig 2 6 (ii)\nThe following table gives the inverse trigonometric function (principal value\nbranches) along with their domains and ranges sin\u20131\n:\n[\u20131, 1]\n\u2192\n,\n2 2\n\uf8ee\u03c0 \u03c0\n\uf8f9\n\uf8ef\u2212\n\uf8fa\n\uf8f0\n\uf8fb\ncos\u20131\n:\n[\u20131, 1]\n\u2192\n[0, \u03c0]\ncosec\u20131\n:\nR \u2013 (\u20131,1)\n\u2192\n,\n2 2\n\uf8ee\u03c0 \u03c0\n\uf8f9\n\uf8ef\u2212\n\uf8fa\n\uf8f0\n\uf8fb \u2013 {0}\nsec\u20131\n:\nR \u2013 (\u20131, 1)\n\u2192\n[0, \u03c0] \u2013 { }\n2\n\u03c0\ntan\u20131\n:\nR\n\u2192\n2,\n2\n\uf8eb\u2212\u03c0 \u03c0\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\ncot\u20131\n:\nR\n\u2192\n(0, \u03c0)\nRationalised 2023-24\n 26\nMATHEMATICS\nANote\n1"
    },
    {
        "Chapter": "1",
        "sentence_range": "890-893",
        "Text": "6 (i)\nFig 2 6 (ii)\nThe following table gives the inverse trigonometric function (principal value\nbranches) along with their domains and ranges sin\u20131\n:\n[\u20131, 1]\n\u2192\n,\n2 2\n\uf8ee\u03c0 \u03c0\n\uf8f9\n\uf8ef\u2212\n\uf8fa\n\uf8f0\n\uf8fb\ncos\u20131\n:\n[\u20131, 1]\n\u2192\n[0, \u03c0]\ncosec\u20131\n:\nR \u2013 (\u20131,1)\n\u2192\n,\n2 2\n\uf8ee\u03c0 \u03c0\n\uf8f9\n\uf8ef\u2212\n\uf8fa\n\uf8f0\n\uf8fb \u2013 {0}\nsec\u20131\n:\nR \u2013 (\u20131, 1)\n\u2192\n[0, \u03c0] \u2013 { }\n2\n\u03c0\ntan\u20131\n:\nR\n\u2192\n2,\n2\n\uf8eb\u2212\u03c0 \u03c0\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\ncot\u20131\n:\nR\n\u2192\n(0, \u03c0)\nRationalised 2023-24\n 26\nMATHEMATICS\nANote\n1 sin\u20131x should not be confused with (sin x)\u20131"
    },
    {
        "Chapter": "1",
        "sentence_range": "891-894",
        "Text": "6 (ii)\nThe following table gives the inverse trigonometric function (principal value\nbranches) along with their domains and ranges sin\u20131\n:\n[\u20131, 1]\n\u2192\n,\n2 2\n\uf8ee\u03c0 \u03c0\n\uf8f9\n\uf8ef\u2212\n\uf8fa\n\uf8f0\n\uf8fb\ncos\u20131\n:\n[\u20131, 1]\n\u2192\n[0, \u03c0]\ncosec\u20131\n:\nR \u2013 (\u20131,1)\n\u2192\n,\n2 2\n\uf8ee\u03c0 \u03c0\n\uf8f9\n\uf8ef\u2212\n\uf8fa\n\uf8f0\n\uf8fb \u2013 {0}\nsec\u20131\n:\nR \u2013 (\u20131, 1)\n\u2192\n[0, \u03c0] \u2013 { }\n2\n\u03c0\ntan\u20131\n:\nR\n\u2192\n2,\n2\n\uf8eb\u2212\u03c0 \u03c0\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\ncot\u20131\n:\nR\n\u2192\n(0, \u03c0)\nRationalised 2023-24\n 26\nMATHEMATICS\nANote\n1 sin\u20131x should not be confused with (sin x)\u20131 In fact (sin x)\u20131 = \n1\nsin x  and\nsimilarly for other trigonometric functions"
    },
    {
        "Chapter": "1",
        "sentence_range": "892-895",
        "Text": "sin\u20131\n:\n[\u20131, 1]\n\u2192\n,\n2 2\n\uf8ee\u03c0 \u03c0\n\uf8f9\n\uf8ef\u2212\n\uf8fa\n\uf8f0\n\uf8fb\ncos\u20131\n:\n[\u20131, 1]\n\u2192\n[0, \u03c0]\ncosec\u20131\n:\nR \u2013 (\u20131,1)\n\u2192\n,\n2 2\n\uf8ee\u03c0 \u03c0\n\uf8f9\n\uf8ef\u2212\n\uf8fa\n\uf8f0\n\uf8fb \u2013 {0}\nsec\u20131\n:\nR \u2013 (\u20131, 1)\n\u2192\n[0, \u03c0] \u2013 { }\n2\n\u03c0\ntan\u20131\n:\nR\n\u2192\n2,\n2\n\uf8eb\u2212\u03c0 \u03c0\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\ncot\u20131\n:\nR\n\u2192\n(0, \u03c0)\nRationalised 2023-24\n 26\nMATHEMATICS\nANote\n1 sin\u20131x should not be confused with (sin x)\u20131 In fact (sin x)\u20131 = \n1\nsin x  and\nsimilarly for other trigonometric functions 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "893-896",
        "Text": "sin\u20131x should not be confused with (sin x)\u20131 In fact (sin x)\u20131 = \n1\nsin x  and\nsimilarly for other trigonometric functions 2 Whenever no branch of an inverse trigonometric functions is mentioned, we\nmean the principal value branch of that function"
    },
    {
        "Chapter": "1",
        "sentence_range": "894-897",
        "Text": "In fact (sin x)\u20131 = \n1\nsin x  and\nsimilarly for other trigonometric functions 2 Whenever no branch of an inverse trigonometric functions is mentioned, we\nmean the principal value branch of that function 3"
    },
    {
        "Chapter": "1",
        "sentence_range": "895-898",
        "Text": "2 Whenever no branch of an inverse trigonometric functions is mentioned, we\nmean the principal value branch of that function 3 The value of an inverse trigonometric functions which lies in the range of\nprincipal branch is called the principal value of that inverse trigonometric\nfunctions"
    },
    {
        "Chapter": "1",
        "sentence_range": "896-899",
        "Text": "Whenever no branch of an inverse trigonometric functions is mentioned, we\nmean the principal value branch of that function 3 The value of an inverse trigonometric functions which lies in the range of\nprincipal branch is called the principal value of that inverse trigonometric\nfunctions We now consider some examples:\nExample 1 Find the principal value of sin\u20131 \n1\n2\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8"
    },
    {
        "Chapter": "1",
        "sentence_range": "897-900",
        "Text": "3 The value of an inverse trigonometric functions which lies in the range of\nprincipal branch is called the principal value of that inverse trigonometric\nfunctions We now consider some examples:\nExample 1 Find the principal value of sin\u20131 \n1\n2\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8 Solution Let sin\u20131 \n1\n2\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n= y"
    },
    {
        "Chapter": "1",
        "sentence_range": "898-901",
        "Text": "The value of an inverse trigonometric functions which lies in the range of\nprincipal branch is called the principal value of that inverse trigonometric\nfunctions We now consider some examples:\nExample 1 Find the principal value of sin\u20131 \n1\n2\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8 Solution Let sin\u20131 \n1\n2\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n= y Then, sin y = 1\n2"
    },
    {
        "Chapter": "1",
        "sentence_range": "899-902",
        "Text": "We now consider some examples:\nExample 1 Find the principal value of sin\u20131 \n1\n2\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8 Solution Let sin\u20131 \n1\n2\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n= y Then, sin y = 1\n2 We know that the range of the principal value branch of sin\u20131 is \n\uf8ed\uf8ec\uf8eb\u2212\n2\u03c0 \u03c0\uf8f8\uf8f7\uf8f6\n,2\n and\nsin 4\n\uf8eb\u03c0\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n= \n1\n2"
    },
    {
        "Chapter": "1",
        "sentence_range": "900-903",
        "Text": "Solution Let sin\u20131 \n1\n2\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n= y Then, sin y = 1\n2 We know that the range of the principal value branch of sin\u20131 is \n\uf8ed\uf8ec\uf8eb\u2212\n2\u03c0 \u03c0\uf8f8\uf8f7\uf8f6\n,2\n and\nsin 4\n\uf8eb\u03c0\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n= \n1\n2 Therefore, principal value of sin\u20131 \n1\n2\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8  is 4\n\u03c0\nExample 2 Find the principal value of cot\u20131 \n1\n3\n\uf8eb\u2212\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\nSolution Let cot\u20131 \n1\n3\n\uf8eb\u2212\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n = y"
    },
    {
        "Chapter": "1",
        "sentence_range": "901-904",
        "Text": "Then, sin y = 1\n2 We know that the range of the principal value branch of sin\u20131 is \n\uf8ed\uf8ec\uf8eb\u2212\n2\u03c0 \u03c0\uf8f8\uf8f7\uf8f6\n,2\n and\nsin 4\n\uf8eb\u03c0\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n= \n1\n2 Therefore, principal value of sin\u20131 \n1\n2\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8  is 4\n\u03c0\nExample 2 Find the principal value of cot\u20131 \n1\n3\n\uf8eb\u2212\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\nSolution Let cot\u20131 \n1\n3\n\uf8eb\u2212\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n = y Then,\n1\ncot\ncot 3\n3\ny\n\u2212\n\uf8eb\u03c0\n\uf8f6\n=\n= \u2212\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8 = cot\n\u03c03\n\uf8eb\n\uf8f6\n\uf8ec\u03c0 \u2212\n\uf8f7\n\uf8ed\n\uf8f8  = \ncot2\n\uf8eb\u03c03\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\nWe know that the range of principal value branch of cot\u20131 is (0, \u03c0) and\ncot 2\n\uf8eb\u03c03\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n= \n1\n3\n\u2212"
    },
    {
        "Chapter": "1",
        "sentence_range": "902-905",
        "Text": "We know that the range of the principal value branch of sin\u20131 is \n\uf8ed\uf8ec\uf8eb\u2212\n2\u03c0 \u03c0\uf8f8\uf8f7\uf8f6\n,2\n and\nsin 4\n\uf8eb\u03c0\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n= \n1\n2 Therefore, principal value of sin\u20131 \n1\n2\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8  is 4\n\u03c0\nExample 2 Find the principal value of cot\u20131 \n1\n3\n\uf8eb\u2212\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\nSolution Let cot\u20131 \n1\n3\n\uf8eb\u2212\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n = y Then,\n1\ncot\ncot 3\n3\ny\n\u2212\n\uf8eb\u03c0\n\uf8f6\n=\n= \u2212\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8 = cot\n\u03c03\n\uf8eb\n\uf8f6\n\uf8ec\u03c0 \u2212\n\uf8f7\n\uf8ed\n\uf8f8  = \ncot2\n\uf8eb\u03c03\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\nWe know that the range of principal value branch of cot\u20131 is (0, \u03c0) and\ncot 2\n\uf8eb\u03c03\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n= \n1\n3\n\u2212 Hence, principal value of cot\u20131 \n1\n3\n\uf8eb\u2212\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8  is 2\n3\n\u03c0\nEXERCISE 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "903-906",
        "Text": "Therefore, principal value of sin\u20131 \n1\n2\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8  is 4\n\u03c0\nExample 2 Find the principal value of cot\u20131 \n1\n3\n\uf8eb\u2212\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\nSolution Let cot\u20131 \n1\n3\n\uf8eb\u2212\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n = y Then,\n1\ncot\ncot 3\n3\ny\n\u2212\n\uf8eb\u03c0\n\uf8f6\n=\n= \u2212\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8 = cot\n\u03c03\n\uf8eb\n\uf8f6\n\uf8ec\u03c0 \u2212\n\uf8f7\n\uf8ed\n\uf8f8  = \ncot2\n\uf8eb\u03c03\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\nWe know that the range of principal value branch of cot\u20131 is (0, \u03c0) and\ncot 2\n\uf8eb\u03c03\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n= \n1\n3\n\u2212 Hence, principal value of cot\u20131 \n1\n3\n\uf8eb\u2212\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8  is 2\n3\n\u03c0\nEXERCISE 2 1\nFind the principal values of the following:\n1"
    },
    {
        "Chapter": "1",
        "sentence_range": "904-907",
        "Text": "Then,\n1\ncot\ncot 3\n3\ny\n\u2212\n\uf8eb\u03c0\n\uf8f6\n=\n= \u2212\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8 = cot\n\u03c03\n\uf8eb\n\uf8f6\n\uf8ec\u03c0 \u2212\n\uf8f7\n\uf8ed\n\uf8f8  = \ncot2\n\uf8eb\u03c03\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\nWe know that the range of principal value branch of cot\u20131 is (0, \u03c0) and\ncot 2\n\uf8eb\u03c03\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n= \n1\n3\n\u2212 Hence, principal value of cot\u20131 \n1\n3\n\uf8eb\u2212\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8  is 2\n3\n\u03c0\nEXERCISE 2 1\nFind the principal values of the following:\n1 sin\u20131 \n1\n2\n\uf8eb\n\uf8f6\n\uf8ec\u2212\n\uf8f7\n\uf8ed\n\uf8f8\n2"
    },
    {
        "Chapter": "1",
        "sentence_range": "905-908",
        "Text": "Hence, principal value of cot\u20131 \n1\n3\n\uf8eb\u2212\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8  is 2\n3\n\u03c0\nEXERCISE 2 1\nFind the principal values of the following:\n1 sin\u20131 \n1\n2\n\uf8eb\n\uf8f6\n\uf8ec\u2212\n\uf8f7\n\uf8ed\n\uf8f8\n2 cos\u20131 \n3\n2\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n3"
    },
    {
        "Chapter": "1",
        "sentence_range": "906-909",
        "Text": "1\nFind the principal values of the following:\n1 sin\u20131 \n1\n2\n\uf8eb\n\uf8f6\n\uf8ec\u2212\n\uf8f7\n\uf8ed\n\uf8f8\n2 cos\u20131 \n3\n2\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n3 cosec\u20131 (2)\n4"
    },
    {
        "Chapter": "1",
        "sentence_range": "907-910",
        "Text": "sin\u20131 \n1\n2\n\uf8eb\n\uf8f6\n\uf8ec\u2212\n\uf8f7\n\uf8ed\n\uf8f8\n2 cos\u20131 \n3\n2\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n3 cosec\u20131 (2)\n4 tan\u20131 (\n\u22123)\n5"
    },
    {
        "Chapter": "1",
        "sentence_range": "908-911",
        "Text": "cos\u20131 \n3\n2\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n3 cosec\u20131 (2)\n4 tan\u20131 (\n\u22123)\n5 cos\u20131 \n1\n2\n\uf8eb\n\uf8f6\n\uf8ec\u2212\n\uf8f7\n\uf8ed\n\uf8f8\n6"
    },
    {
        "Chapter": "1",
        "sentence_range": "909-912",
        "Text": "cosec\u20131 (2)\n4 tan\u20131 (\n\u22123)\n5 cos\u20131 \n1\n2\n\uf8eb\n\uf8f6\n\uf8ec\u2212\n\uf8f7\n\uf8ed\n\uf8f8\n6 tan\u20131 (\u20131)\nRationalised 2023-24\nINVERSE TRIGONOMETRIC FUNCTIONS     27\n7"
    },
    {
        "Chapter": "1",
        "sentence_range": "910-913",
        "Text": "tan\u20131 (\n\u22123)\n5 cos\u20131 \n1\n2\n\uf8eb\n\uf8f6\n\uf8ec\u2212\n\uf8f7\n\uf8ed\n\uf8f8\n6 tan\u20131 (\u20131)\nRationalised 2023-24\nINVERSE TRIGONOMETRIC FUNCTIONS     27\n7 sec\u20131 \n2\n3\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n8"
    },
    {
        "Chapter": "1",
        "sentence_range": "911-914",
        "Text": "cos\u20131 \n1\n2\n\uf8eb\n\uf8f6\n\uf8ec\u2212\n\uf8f7\n\uf8ed\n\uf8f8\n6 tan\u20131 (\u20131)\nRationalised 2023-24\nINVERSE TRIGONOMETRIC FUNCTIONS     27\n7 sec\u20131 \n2\n3\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n8 cot\u20131 ( 3)\n9"
    },
    {
        "Chapter": "1",
        "sentence_range": "912-915",
        "Text": "tan\u20131 (\u20131)\nRationalised 2023-24\nINVERSE TRIGONOMETRIC FUNCTIONS     27\n7 sec\u20131 \n2\n3\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n8 cot\u20131 ( 3)\n9 cos\u20131 \n1\n2\n\uf8eb\n\uf8f6\n\uf8ec\u2212\n\uf8f7\n\uf8ed\n\uf8f8\n10"
    },
    {
        "Chapter": "1",
        "sentence_range": "913-916",
        "Text": "sec\u20131 \n2\n3\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n8 cot\u20131 ( 3)\n9 cos\u20131 \n1\n2\n\uf8eb\n\uf8f6\n\uf8ec\u2212\n\uf8f7\n\uf8ed\n\uf8f8\n10 cosec\u20131 (\n\u22122\n)\nFind the values of  the following:\n11"
    },
    {
        "Chapter": "1",
        "sentence_range": "914-917",
        "Text": "cot\u20131 ( 3)\n9 cos\u20131 \n1\n2\n\uf8eb\n\uf8f6\n\uf8ec\u2212\n\uf8f7\n\uf8ed\n\uf8f8\n10 cosec\u20131 (\n\u22122\n)\nFind the values of  the following:\n11 tan\u20131(1) + cos\u20131 \n1\n2\n \n \n \u2212\n \n \n + sin\u20131 \n1\n2\n \n \n \u2212\n \n \n \n12"
    },
    {
        "Chapter": "1",
        "sentence_range": "915-918",
        "Text": "cos\u20131 \n1\n2\n\uf8eb\n\uf8f6\n\uf8ec\u2212\n\uf8f7\n\uf8ed\n\uf8f8\n10 cosec\u20131 (\n\u22122\n)\nFind the values of  the following:\n11 tan\u20131(1) + cos\u20131 \n1\n2\n \n \n \u2212\n \n \n + sin\u20131 \n1\n2\n \n \n \u2212\n \n \n \n12 cos\u20131 1\n2\n \n \n \n \n \n + 2 sin\u20131 1\n2\n \n \n \n \n \n \n13"
    },
    {
        "Chapter": "1",
        "sentence_range": "916-919",
        "Text": "cosec\u20131 (\n\u22122\n)\nFind the values of  the following:\n11 tan\u20131(1) + cos\u20131 \n1\n2\n \n \n \u2212\n \n \n + sin\u20131 \n1\n2\n \n \n \u2212\n \n \n \n12 cos\u20131 1\n2\n \n \n \n \n \n + 2 sin\u20131 1\n2\n \n \n \n \n \n \n13 If sin\u20131 x = y, then\n(A) 0 \u2264 y \u2264 \u03c0\n(B)\n2\n2\ny\n\u03c0\n\u03c0\n\u2212\n\u2264\n\u2264\n(C) 0 <  y < \u03c0\n(D)\n2\n2\ny\n\u03c0\n\u03c0\n\u2212\n<\n<\n14"
    },
    {
        "Chapter": "1",
        "sentence_range": "917-920",
        "Text": "tan\u20131(1) + cos\u20131 \n1\n2\n \n \n \u2212\n \n \n + sin\u20131 \n1\n2\n \n \n \u2212\n \n \n \n12 cos\u20131 1\n2\n \n \n \n \n \n + 2 sin\u20131 1\n2\n \n \n \n \n \n \n13 If sin\u20131 x = y, then\n(A) 0 \u2264 y \u2264 \u03c0\n(B)\n2\n2\ny\n\u03c0\n\u03c0\n\u2212\n\u2264\n\u2264\n(C) 0 <  y < \u03c0\n(D)\n2\n2\ny\n\u03c0\n\u03c0\n\u2212\n<\n<\n14 tan\u20131 \n(\n)\n1\n3\nsec\n2\n\u2212\n\u2212\n\u2212\n is equal to\n(A) \u03c0\n(B)\n\u2212\u03c03\n(C)\n\u03c03\n(D) 2\n3\n\u03c0\n2"
    },
    {
        "Chapter": "1",
        "sentence_range": "918-921",
        "Text": "cos\u20131 1\n2\n \n \n \n \n \n + 2 sin\u20131 1\n2\n \n \n \n \n \n \n13 If sin\u20131 x = y, then\n(A) 0 \u2264 y \u2264 \u03c0\n(B)\n2\n2\ny\n\u03c0\n\u03c0\n\u2212\n\u2264\n\u2264\n(C) 0 <  y < \u03c0\n(D)\n2\n2\ny\n\u03c0\n\u03c0\n\u2212\n<\n<\n14 tan\u20131 \n(\n)\n1\n3\nsec\n2\n\u2212\n\u2212\n\u2212\n is equal to\n(A) \u03c0\n(B)\n\u2212\u03c03\n(C)\n\u03c03\n(D) 2\n3\n\u03c0\n2 3  Properties of Inverse Trigonometric Functions\nIn this section, we shall prove some important properties of inverse trigonometric\nfunctions"
    },
    {
        "Chapter": "1",
        "sentence_range": "919-922",
        "Text": "If sin\u20131 x = y, then\n(A) 0 \u2264 y \u2264 \u03c0\n(B)\n2\n2\ny\n\u03c0\n\u03c0\n\u2212\n\u2264\n\u2264\n(C) 0 <  y < \u03c0\n(D)\n2\n2\ny\n\u03c0\n\u03c0\n\u2212\n<\n<\n14 tan\u20131 \n(\n)\n1\n3\nsec\n2\n\u2212\n\u2212\n\u2212\n is equal to\n(A) \u03c0\n(B)\n\u2212\u03c03\n(C)\n\u03c03\n(D) 2\n3\n\u03c0\n2 3  Properties of Inverse Trigonometric Functions\nIn this section, we shall prove some important properties of inverse trigonometric\nfunctions It may be mentioned here that these results are valid within the principal\nvalue branches of the corresponding inverse trigonometric functions and wherever\nthey are defined"
    },
    {
        "Chapter": "1",
        "sentence_range": "920-923",
        "Text": "tan\u20131 \n(\n)\n1\n3\nsec\n2\n\u2212\n\u2212\n\u2212\n is equal to\n(A) \u03c0\n(B)\n\u2212\u03c03\n(C)\n\u03c03\n(D) 2\n3\n\u03c0\n2 3  Properties of Inverse Trigonometric Functions\nIn this section, we shall prove some important properties of inverse trigonometric\nfunctions It may be mentioned here that these results are valid within the principal\nvalue branches of the corresponding inverse trigonometric functions and wherever\nthey are defined Some results may not be valid for all values of the domains of inverse\ntrigonometric functions"
    },
    {
        "Chapter": "1",
        "sentence_range": "921-924",
        "Text": "3  Properties of Inverse Trigonometric Functions\nIn this section, we shall prove some important properties of inverse trigonometric\nfunctions It may be mentioned here that these results are valid within the principal\nvalue branches of the corresponding inverse trigonometric functions and wherever\nthey are defined Some results may not be valid for all values of the domains of inverse\ntrigonometric functions In fact, they will be valid only for some values of x for which\ninverse trigonometric functions are defined"
    },
    {
        "Chapter": "1",
        "sentence_range": "922-925",
        "Text": "It may be mentioned here that these results are valid within the principal\nvalue branches of the corresponding inverse trigonometric functions and wherever\nthey are defined Some results may not be valid for all values of the domains of inverse\ntrigonometric functions In fact, they will be valid only for some values of x for which\ninverse trigonometric functions are defined We will not go into the details of these\nvalues of x in the domain as this discussion goes beyond the scope of this textbook"
    },
    {
        "Chapter": "1",
        "sentence_range": "923-926",
        "Text": "Some results may not be valid for all values of the domains of inverse\ntrigonometric functions In fact, they will be valid only for some values of x for which\ninverse trigonometric functions are defined We will not go into the details of these\nvalues of x in the domain as this discussion goes beyond the scope of this textbook Let us recall that if y = sin\u20131x, then x = sin y and if x = sin y, then y = sin\u20131x"
    },
    {
        "Chapter": "1",
        "sentence_range": "924-927",
        "Text": "In fact, they will be valid only for some values of x for which\ninverse trigonometric functions are defined We will not go into the details of these\nvalues of x in the domain as this discussion goes beyond the scope of this textbook Let us recall that if y = sin\u20131x, then x = sin y and if x = sin y, then y = sin\u20131x This\nis equivalent to\nsin (sin\u20131 x) = x, x \u2208  [\u2013 1, 1] and sin\u20131 (sin x) = x, x \u2208 \n2,\n2\n\uf8ee\u2212\u03c0 \u03c0\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nFor suitable values of  domain similar results follow for remaining trigonometric\nfunctions"
    },
    {
        "Chapter": "1",
        "sentence_range": "925-928",
        "Text": "We will not go into the details of these\nvalues of x in the domain as this discussion goes beyond the scope of this textbook Let us recall that if y = sin\u20131x, then x = sin y and if x = sin y, then y = sin\u20131x This\nis equivalent to\nsin (sin\u20131 x) = x, x \u2208  [\u2013 1, 1] and sin\u20131 (sin x) = x, x \u2208 \n2,\n2\n\uf8ee\u2212\u03c0 \u03c0\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nFor suitable values of  domain similar results follow for remaining trigonometric\nfunctions Rationalised 2023-24\n 28\nMATHEMATICS\nWe now consider some examples"
    },
    {
        "Chapter": "1",
        "sentence_range": "926-929",
        "Text": "Let us recall that if y = sin\u20131x, then x = sin y and if x = sin y, then y = sin\u20131x This\nis equivalent to\nsin (sin\u20131 x) = x, x \u2208  [\u2013 1, 1] and sin\u20131 (sin x) = x, x \u2208 \n2,\n2\n\uf8ee\u2212\u03c0 \u03c0\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nFor suitable values of  domain similar results follow for remaining trigonometric\nfunctions Rationalised 2023-24\n 28\nMATHEMATICS\nWe now consider some examples Example 3 Show that\n(i)\nsin\u20131 (\n2)\n2\nx1\n\u2212x\n = 2 sin\u20131 x, \n1\n1\n2\n2\nx\n\u2212\n\u2264\n\u2264\n(ii)\nsin\u20131 (\n2)\n2\nx1\n\u2212x\n = 2 cos\u20131 x, \n1\n1\n2\n\u2264x\n\u2264\nSolution\n(i)\nLet x = sin \u03b8"
    },
    {
        "Chapter": "1",
        "sentence_range": "927-930",
        "Text": "This\nis equivalent to\nsin (sin\u20131 x) = x, x \u2208  [\u2013 1, 1] and sin\u20131 (sin x) = x, x \u2208 \n2,\n2\n\uf8ee\u2212\u03c0 \u03c0\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nFor suitable values of  domain similar results follow for remaining trigonometric\nfunctions Rationalised 2023-24\n 28\nMATHEMATICS\nWe now consider some examples Example 3 Show that\n(i)\nsin\u20131 (\n2)\n2\nx1\n\u2212x\n = 2 sin\u20131 x, \n1\n1\n2\n2\nx\n\u2212\n\u2264\n\u2264\n(ii)\nsin\u20131 (\n2)\n2\nx1\n\u2212x\n = 2 cos\u20131 x, \n1\n1\n2\n\u2264x\n\u2264\nSolution\n(i)\nLet x = sin \u03b8 Then sin\u20131 x = \u03b8"
    },
    {
        "Chapter": "1",
        "sentence_range": "928-931",
        "Text": "Rationalised 2023-24\n 28\nMATHEMATICS\nWe now consider some examples Example 3 Show that\n(i)\nsin\u20131 (\n2)\n2\nx1\n\u2212x\n = 2 sin\u20131 x, \n1\n1\n2\n2\nx\n\u2212\n\u2264\n\u2264\n(ii)\nsin\u20131 (\n2)\n2\nx1\n\u2212x\n = 2 cos\u20131 x, \n1\n1\n2\n\u2264x\n\u2264\nSolution\n(i)\nLet x = sin \u03b8 Then sin\u20131 x = \u03b8 We have\nsin\u20131 (\n2)\n2\nx1\n\u2212x\n = sin\u20131 (\n)\n2\n2sin\n1\nsin\n\u03b8\n\u2212\n\u03b8\n= sin\u20131 (2sin\u03b8 cos\u03b8) = sin\u20131 (sin2\u03b8) = 2\u03b8\n= 2 sin\u20131 x\n(ii)\nTake x = cos \u03b8, then proceeding as above, we get, sin\u20131 (\n2)\n2\nx1\n\u2212x\n= 2 cos\u20131 x\nExample 4 Express \n1\ncos\ntan\n1\nsin\nx\nx\n\u2212  \n \n \n \n \n \n\u2212\n, \n23\n2\n\u03c0\n\u03c0\n\u2212\n<\nx<\n in the simplest form"
    },
    {
        "Chapter": "1",
        "sentence_range": "929-932",
        "Text": "Example 3 Show that\n(i)\nsin\u20131 (\n2)\n2\nx1\n\u2212x\n = 2 sin\u20131 x, \n1\n1\n2\n2\nx\n\u2212\n\u2264\n\u2264\n(ii)\nsin\u20131 (\n2)\n2\nx1\n\u2212x\n = 2 cos\u20131 x, \n1\n1\n2\n\u2264x\n\u2264\nSolution\n(i)\nLet x = sin \u03b8 Then sin\u20131 x = \u03b8 We have\nsin\u20131 (\n2)\n2\nx1\n\u2212x\n = sin\u20131 (\n)\n2\n2sin\n1\nsin\n\u03b8\n\u2212\n\u03b8\n= sin\u20131 (2sin\u03b8 cos\u03b8) = sin\u20131 (sin2\u03b8) = 2\u03b8\n= 2 sin\u20131 x\n(ii)\nTake x = cos \u03b8, then proceeding as above, we get, sin\u20131 (\n2)\n2\nx1\n\u2212x\n= 2 cos\u20131 x\nExample 4 Express \n1\ncos\ntan\n1\nsin\nx\nx\n\u2212  \n \n \n \n \n \n\u2212\n, \n23\n2\n\u03c0\n\u03c0\n\u2212\n<\nx<\n in the simplest form Solution We write\n2\n2\n1\n\u20131\n2\n2\ncos\nsin\ncos\n2\n2\ntan\ntan\n1\nsin\ncos\nsin\n2sin\ncos\n2\n2\n2\n2\nx\nx\nx\nx\nx\nx\nx\nx\n\u2212\n\uf8ee\n\uf8f9\n\u2212\n\uf8ef\n\uf8fa\n\uf8eb\n\uf8f6 =\n\uf8ef\n\uf8fa\n\uf8ec\n\uf8f7\n\uf8ed\u2212\n\uf8f8\n\uf8ef\n\uf8fa\n+\n\u2212\n\uf8f0\n\uf8fb\n=\n\u20131\n2\ncos\nsin\ncos\nsin\n2\n2\n2\n2\ntan\ncos\nsin\n2\n2\nx\nx\nx\nx\nx\nx\n\uf8ee\n\uf8f9\n\uf8eb\n\uf8f6\uf8eb\n\uf8f6\n+\n\u2212\n\uf8ec\n\uf8f7\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8ed\n\uf8f8\uf8ed\n\uf8f8\n\uf8ef\n\uf8fa\n\uf8eb\n\uf8f6\n\uf8ef\n\uf8fa\n\u2212\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8ed\n\uf8f8\n\uf8f0\n\uf8fb\n=\n\u20131 cos\nsin\n2\n2\ntan\ncos\nsin\n2\n2\nx\nx\nx\nx\n\uf8ee\n\uf8f9\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n \n\u20131 1\ntan 2\ntan\n1\ntan 2\nx\nx\n\uf8ee\n\uf8f9\n\uf8ef+\n\uf8fa\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\u2212\n\uf8fb\n=\ntan\u20131\ntan 4\n2\n4\n2\nx\nx\n\uf8ee\n\uf8f9\n\u03c0\n\u03c0\n\uf8eb\n\uf8f6\n+\n=\n+\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8ed\n\uf8f8\n\uf8f0\n\uf8fb\nRationalised 2023-24\nINVERSE TRIGONOMETRIC FUNCTIONS     29\nExample 5 Write \n\u20131\n12\ncot\n1\nx\n\uf8eb\n\uf8f6\n\uf8ec\n\u2212\uf8f7\n\uf8ed\n\uf8f8\n,  x  > 1 in the simplest form"
    },
    {
        "Chapter": "1",
        "sentence_range": "930-933",
        "Text": "Then sin\u20131 x = \u03b8 We have\nsin\u20131 (\n2)\n2\nx1\n\u2212x\n = sin\u20131 (\n)\n2\n2sin\n1\nsin\n\u03b8\n\u2212\n\u03b8\n= sin\u20131 (2sin\u03b8 cos\u03b8) = sin\u20131 (sin2\u03b8) = 2\u03b8\n= 2 sin\u20131 x\n(ii)\nTake x = cos \u03b8, then proceeding as above, we get, sin\u20131 (\n2)\n2\nx1\n\u2212x\n= 2 cos\u20131 x\nExample 4 Express \n1\ncos\ntan\n1\nsin\nx\nx\n\u2212  \n \n \n \n \n \n\u2212\n, \n23\n2\n\u03c0\n\u03c0\n\u2212\n<\nx<\n in the simplest form Solution We write\n2\n2\n1\n\u20131\n2\n2\ncos\nsin\ncos\n2\n2\ntan\ntan\n1\nsin\ncos\nsin\n2sin\ncos\n2\n2\n2\n2\nx\nx\nx\nx\nx\nx\nx\nx\n\u2212\n\uf8ee\n\uf8f9\n\u2212\n\uf8ef\n\uf8fa\n\uf8eb\n\uf8f6 =\n\uf8ef\n\uf8fa\n\uf8ec\n\uf8f7\n\uf8ed\u2212\n\uf8f8\n\uf8ef\n\uf8fa\n+\n\u2212\n\uf8f0\n\uf8fb\n=\n\u20131\n2\ncos\nsin\ncos\nsin\n2\n2\n2\n2\ntan\ncos\nsin\n2\n2\nx\nx\nx\nx\nx\nx\n\uf8ee\n\uf8f9\n\uf8eb\n\uf8f6\uf8eb\n\uf8f6\n+\n\u2212\n\uf8ec\n\uf8f7\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8ed\n\uf8f8\uf8ed\n\uf8f8\n\uf8ef\n\uf8fa\n\uf8eb\n\uf8f6\n\uf8ef\n\uf8fa\n\u2212\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8ed\n\uf8f8\n\uf8f0\n\uf8fb\n=\n\u20131 cos\nsin\n2\n2\ntan\ncos\nsin\n2\n2\nx\nx\nx\nx\n\uf8ee\n\uf8f9\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n \n\u20131 1\ntan 2\ntan\n1\ntan 2\nx\nx\n\uf8ee\n\uf8f9\n\uf8ef+\n\uf8fa\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\u2212\n\uf8fb\n=\ntan\u20131\ntan 4\n2\n4\n2\nx\nx\n\uf8ee\n\uf8f9\n\u03c0\n\u03c0\n\uf8eb\n\uf8f6\n+\n=\n+\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8ed\n\uf8f8\n\uf8f0\n\uf8fb\nRationalised 2023-24\nINVERSE TRIGONOMETRIC FUNCTIONS     29\nExample 5 Write \n\u20131\n12\ncot\n1\nx\n\uf8eb\n\uf8f6\n\uf8ec\n\u2212\uf8f7\n\uf8ed\n\uf8f8\n,  x  > 1 in the simplest form Solution Let x = sec \u03b8, then \n2\n1\nx \u2212 = \nsec2\n1\n\u03b8 \u2212 =tan\n\u03b8\nTherefore, \n\u20131\n12\ncot\n1\nx \u2212\n= cot\u20131 (cot \u03b8) = \u03b8 = sec\u20131 x, which is the simplest form"
    },
    {
        "Chapter": "1",
        "sentence_range": "931-934",
        "Text": "We have\nsin\u20131 (\n2)\n2\nx1\n\u2212x\n = sin\u20131 (\n)\n2\n2sin\n1\nsin\n\u03b8\n\u2212\n\u03b8\n= sin\u20131 (2sin\u03b8 cos\u03b8) = sin\u20131 (sin2\u03b8) = 2\u03b8\n= 2 sin\u20131 x\n(ii)\nTake x = cos \u03b8, then proceeding as above, we get, sin\u20131 (\n2)\n2\nx1\n\u2212x\n= 2 cos\u20131 x\nExample 4 Express \n1\ncos\ntan\n1\nsin\nx\nx\n\u2212  \n \n \n \n \n \n\u2212\n, \n23\n2\n\u03c0\n\u03c0\n\u2212\n<\nx<\n in the simplest form Solution We write\n2\n2\n1\n\u20131\n2\n2\ncos\nsin\ncos\n2\n2\ntan\ntan\n1\nsin\ncos\nsin\n2sin\ncos\n2\n2\n2\n2\nx\nx\nx\nx\nx\nx\nx\nx\n\u2212\n\uf8ee\n\uf8f9\n\u2212\n\uf8ef\n\uf8fa\n\uf8eb\n\uf8f6 =\n\uf8ef\n\uf8fa\n\uf8ec\n\uf8f7\n\uf8ed\u2212\n\uf8f8\n\uf8ef\n\uf8fa\n+\n\u2212\n\uf8f0\n\uf8fb\n=\n\u20131\n2\ncos\nsin\ncos\nsin\n2\n2\n2\n2\ntan\ncos\nsin\n2\n2\nx\nx\nx\nx\nx\nx\n\uf8ee\n\uf8f9\n\uf8eb\n\uf8f6\uf8eb\n\uf8f6\n+\n\u2212\n\uf8ec\n\uf8f7\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8ed\n\uf8f8\uf8ed\n\uf8f8\n\uf8ef\n\uf8fa\n\uf8eb\n\uf8f6\n\uf8ef\n\uf8fa\n\u2212\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8ed\n\uf8f8\n\uf8f0\n\uf8fb\n=\n\u20131 cos\nsin\n2\n2\ntan\ncos\nsin\n2\n2\nx\nx\nx\nx\n\uf8ee\n\uf8f9\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n \n\u20131 1\ntan 2\ntan\n1\ntan 2\nx\nx\n\uf8ee\n\uf8f9\n\uf8ef+\n\uf8fa\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\u2212\n\uf8fb\n=\ntan\u20131\ntan 4\n2\n4\n2\nx\nx\n\uf8ee\n\uf8f9\n\u03c0\n\u03c0\n\uf8eb\n\uf8f6\n+\n=\n+\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8ed\n\uf8f8\n\uf8f0\n\uf8fb\nRationalised 2023-24\nINVERSE TRIGONOMETRIC FUNCTIONS     29\nExample 5 Write \n\u20131\n12\ncot\n1\nx\n\uf8eb\n\uf8f6\n\uf8ec\n\u2212\uf8f7\n\uf8ed\n\uf8f8\n,  x  > 1 in the simplest form Solution Let x = sec \u03b8, then \n2\n1\nx \u2212 = \nsec2\n1\n\u03b8 \u2212 =tan\n\u03b8\nTherefore, \n\u20131\n12\ncot\n1\nx \u2212\n= cot\u20131 (cot \u03b8) = \u03b8 = sec\u20131 x, which is the simplest form EXERCISE 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "932-935",
        "Text": "Solution We write\n2\n2\n1\n\u20131\n2\n2\ncos\nsin\ncos\n2\n2\ntan\ntan\n1\nsin\ncos\nsin\n2sin\ncos\n2\n2\n2\n2\nx\nx\nx\nx\nx\nx\nx\nx\n\u2212\n\uf8ee\n\uf8f9\n\u2212\n\uf8ef\n\uf8fa\n\uf8eb\n\uf8f6 =\n\uf8ef\n\uf8fa\n\uf8ec\n\uf8f7\n\uf8ed\u2212\n\uf8f8\n\uf8ef\n\uf8fa\n+\n\u2212\n\uf8f0\n\uf8fb\n=\n\u20131\n2\ncos\nsin\ncos\nsin\n2\n2\n2\n2\ntan\ncos\nsin\n2\n2\nx\nx\nx\nx\nx\nx\n\uf8ee\n\uf8f9\n\uf8eb\n\uf8f6\uf8eb\n\uf8f6\n+\n\u2212\n\uf8ec\n\uf8f7\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8ed\n\uf8f8\uf8ed\n\uf8f8\n\uf8ef\n\uf8fa\n\uf8eb\n\uf8f6\n\uf8ef\n\uf8fa\n\u2212\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8ed\n\uf8f8\n\uf8f0\n\uf8fb\n=\n\u20131 cos\nsin\n2\n2\ntan\ncos\nsin\n2\n2\nx\nx\nx\nx\n\uf8ee\n\uf8f9\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n \n\u20131 1\ntan 2\ntan\n1\ntan 2\nx\nx\n\uf8ee\n\uf8f9\n\uf8ef+\n\uf8fa\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\u2212\n\uf8fb\n=\ntan\u20131\ntan 4\n2\n4\n2\nx\nx\n\uf8ee\n\uf8f9\n\u03c0\n\u03c0\n\uf8eb\n\uf8f6\n+\n=\n+\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8ed\n\uf8f8\n\uf8f0\n\uf8fb\nRationalised 2023-24\nINVERSE TRIGONOMETRIC FUNCTIONS     29\nExample 5 Write \n\u20131\n12\ncot\n1\nx\n\uf8eb\n\uf8f6\n\uf8ec\n\u2212\uf8f7\n\uf8ed\n\uf8f8\n,  x  > 1 in the simplest form Solution Let x = sec \u03b8, then \n2\n1\nx \u2212 = \nsec2\n1\n\u03b8 \u2212 =tan\n\u03b8\nTherefore, \n\u20131\n12\ncot\n1\nx \u2212\n= cot\u20131 (cot \u03b8) = \u03b8 = sec\u20131 x, which is the simplest form EXERCISE 2 2\nProve the following:\n1"
    },
    {
        "Chapter": "1",
        "sentence_range": "933-936",
        "Text": "Solution Let x = sec \u03b8, then \n2\n1\nx \u2212 = \nsec2\n1\n\u03b8 \u2212 =tan\n\u03b8\nTherefore, \n\u20131\n12\ncot\n1\nx \u2212\n= cot\u20131 (cot \u03b8) = \u03b8 = sec\u20131 x, which is the simplest form EXERCISE 2 2\nProve the following:\n1 3sin\u20131 x = sin\u20131 (3x \u2013 4x3), \n1\n1\n\u2013\n2,\n2\nx\n\uf8ee\n\uf8f9\n\u2208\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n2"
    },
    {
        "Chapter": "1",
        "sentence_range": "934-937",
        "Text": "EXERCISE 2 2\nProve the following:\n1 3sin\u20131 x = sin\u20131 (3x \u2013 4x3), \n1\n1\n\u2013\n2,\n2\nx\n\uf8ee\n\uf8f9\n\u2208\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n2 3cos\u20131 x = cos\u20131 (4x3 \u2013 3x), \n21 , 1\nx\n\uf8ee\n\uf8f9\n\u2208\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nWrite the following functions in the simplest form:\n3"
    },
    {
        "Chapter": "1",
        "sentence_range": "935-938",
        "Text": "2\nProve the following:\n1 3sin\u20131 x = sin\u20131 (3x \u2013 4x3), \n1\n1\n\u2013\n2,\n2\nx\n\uf8ee\n\uf8f9\n\u2208\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n2 3cos\u20131 x = cos\u20131 (4x3 \u2013 3x), \n21 , 1\nx\n\uf8ee\n\uf8f9\n\u2208\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nWrite the following functions in the simplest form:\n3 2\n1\n1\n1\ntan\nx\nx\n\u2212\n+\n\u2212 , x \u2260 0\n4"
    },
    {
        "Chapter": "1",
        "sentence_range": "936-939",
        "Text": "3sin\u20131 x = sin\u20131 (3x \u2013 4x3), \n1\n1\n\u2013\n2,\n2\nx\n\uf8ee\n\uf8f9\n\u2208\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n2 3cos\u20131 x = cos\u20131 (4x3 \u2013 3x), \n21 , 1\nx\n\uf8ee\n\uf8f9\n\u2208\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nWrite the following functions in the simplest form:\n3 2\n1\n1\n1\ntan\nx\nx\n\u2212\n+\n\u2212 , x \u2260 0\n4 1\n1\ncos\ntan\n1\ncos\nx\nx\n\u2212 \uf8eb\n\uf8f6\n\u2212\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n+\n\uf8ed\n\uf8f8\n, 0 < x < \u03c0\n5"
    },
    {
        "Chapter": "1",
        "sentence_range": "937-940",
        "Text": "3cos\u20131 x = cos\u20131 (4x3 \u2013 3x), \n21 , 1\nx\n\uf8ee\n\uf8f9\n\u2208\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nWrite the following functions in the simplest form:\n3 2\n1\n1\n1\ntan\nx\nx\n\u2212\n+\n\u2212 , x \u2260 0\n4 1\n1\ncos\ntan\n1\ncos\nx\nx\n\u2212 \uf8eb\n\uf8f6\n\u2212\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n+\n\uf8ed\n\uf8f8\n, 0 < x < \u03c0\n5 1 cos\nsin\ntan\ncos\nsin\nx\nx\nx\nx\n\u2212 \uf8eb\n\uf8f6\n\u2212\n\uf8ec\n\uf8f7\n+\n\uf8ed\n\uf8f8\n, 4\n\u2212\u03c0  < x < 3\n4\n\u03c0\n6"
    },
    {
        "Chapter": "1",
        "sentence_range": "938-941",
        "Text": "2\n1\n1\n1\ntan\nx\nx\n\u2212\n+\n\u2212 , x \u2260 0\n4 1\n1\ncos\ntan\n1\ncos\nx\nx\n\u2212 \uf8eb\n\uf8f6\n\u2212\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n+\n\uf8ed\n\uf8f8\n, 0 < x < \u03c0\n5 1 cos\nsin\ntan\ncos\nsin\nx\nx\nx\nx\n\u2212 \uf8eb\n\uf8f6\n\u2212\n\uf8ec\n\uf8f7\n+\n\uf8ed\n\uf8f8\n, 4\n\u2212\u03c0  < x < 3\n4\n\u03c0\n6 1\n2\n2\ntan\nx\na\nx\n\u2212\n\u2212\n, |x| < a\n7"
    },
    {
        "Chapter": "1",
        "sentence_range": "939-942",
        "Text": "1\n1\ncos\ntan\n1\ncos\nx\nx\n\u2212 \uf8eb\n\uf8f6\n\u2212\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n+\n\uf8ed\n\uf8f8\n, 0 < x < \u03c0\n5 1 cos\nsin\ntan\ncos\nsin\nx\nx\nx\nx\n\u2212 \uf8eb\n\uf8f6\n\u2212\n\uf8ec\n\uf8f7\n+\n\uf8ed\n\uf8f8\n, 4\n\u2212\u03c0  < x < 3\n4\n\u03c0\n6 1\n2\n2\ntan\nx\na\nx\n\u2212\n\u2212\n, |x| < a\n7 2\n3\n1\n3\n2\n3\ntan\n3\na x\nx\na\nax\n\u2212 \uf8eb\n\uf8f6\n\u2212\n\uf8ec\n\uf8f7\n\u2212\n\uf8ed\n\uf8f8\n, a > 0; \n3\n3\n\u2212\n<\n<\na\na\nx\nFind the values of each of the following:\n8"
    },
    {
        "Chapter": "1",
        "sentence_range": "940-943",
        "Text": "1 cos\nsin\ntan\ncos\nsin\nx\nx\nx\nx\n\u2212 \uf8eb\n\uf8f6\n\u2212\n\uf8ec\n\uf8f7\n+\n\uf8ed\n\uf8f8\n, 4\n\u2212\u03c0  < x < 3\n4\n\u03c0\n6 1\n2\n2\ntan\nx\na\nx\n\u2212\n\u2212\n, |x| < a\n7 2\n3\n1\n3\n2\n3\ntan\n3\na x\nx\na\nax\n\u2212 \uf8eb\n\uf8f6\n\u2212\n\uf8ec\n\uf8f7\n\u2212\n\uf8ed\n\uf8f8\n, a > 0; \n3\n3\n\u2212\n<\n<\na\na\nx\nFind the values of each of the following:\n8 \u20131\n\u20131 1\ntan\n2cos 2sin\n2\n\uf8ee\n\uf8f9\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8ed\n\uf8f8\n\uf8f0\n\uf8fb\n9"
    },
    {
        "Chapter": "1",
        "sentence_range": "941-944",
        "Text": "1\n2\n2\ntan\nx\na\nx\n\u2212\n\u2212\n, |x| < a\n7 2\n3\n1\n3\n2\n3\ntan\n3\na x\nx\na\nax\n\u2212 \uf8eb\n\uf8f6\n\u2212\n\uf8ec\n\uf8f7\n\u2212\n\uf8ed\n\uf8f8\n, a > 0; \n3\n3\n\u2212\n<\n<\na\na\nx\nFind the values of each of the following:\n8 \u20131\n\u20131 1\ntan\n2cos 2sin\n2\n\uf8ee\n\uf8f9\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8ed\n\uf8f8\n\uf8f0\n\uf8fb\n9 2\n\u20131\n\u20131\n2\n2\n1\n2\n1\ntan\nsin\ncos\n2\n1\n1\nx\ny\nx\ny\n\uf8ee\n\uf8f9\n\u2212\n+\n\uf8ef\n\uf8fa\n+\n+\n\uf8f0\n\uf8fb\n, |x | < 1, y > 0 and xy < 1\nRationalised 2023-24\n 30\nMATHEMATICS\nFind the values of each of the expressions in Exercises 16 to 18"
    },
    {
        "Chapter": "1",
        "sentence_range": "942-945",
        "Text": "2\n3\n1\n3\n2\n3\ntan\n3\na x\nx\na\nax\n\u2212 \uf8eb\n\uf8f6\n\u2212\n\uf8ec\n\uf8f7\n\u2212\n\uf8ed\n\uf8f8\n, a > 0; \n3\n3\n\u2212\n<\n<\na\na\nx\nFind the values of each of the following:\n8 \u20131\n\u20131 1\ntan\n2cos 2sin\n2\n\uf8ee\n\uf8f9\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8ed\n\uf8f8\n\uf8f0\n\uf8fb\n9 2\n\u20131\n\u20131\n2\n2\n1\n2\n1\ntan\nsin\ncos\n2\n1\n1\nx\ny\nx\ny\n\uf8ee\n\uf8f9\n\u2212\n+\n\uf8ef\n\uf8fa\n+\n+\n\uf8f0\n\uf8fb\n, |x | < 1, y > 0 and xy < 1\nRationalised 2023-24\n 30\nMATHEMATICS\nFind the values of each of the expressions in Exercises 16 to 18 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "943-946",
        "Text": "\u20131\n\u20131 1\ntan\n2cos 2sin\n2\n\uf8ee\n\uf8f9\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8ed\n\uf8f8\n\uf8f0\n\uf8fb\n9 2\n\u20131\n\u20131\n2\n2\n1\n2\n1\ntan\nsin\ncos\n2\n1\n1\nx\ny\nx\ny\n\uf8ee\n\uf8f9\n\u2212\n+\n\uf8ef\n\uf8fa\n+\n+\n\uf8f0\n\uf8fb\n, |x | < 1, y > 0 and xy < 1\nRationalised 2023-24\n 30\nMATHEMATICS\nFind the values of each of the expressions in Exercises 16 to 18 10 \u20131\n2\nsin\nsin 3\n\u03c0\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n11"
    },
    {
        "Chapter": "1",
        "sentence_range": "944-947",
        "Text": "2\n\u20131\n\u20131\n2\n2\n1\n2\n1\ntan\nsin\ncos\n2\n1\n1\nx\ny\nx\ny\n\uf8ee\n\uf8f9\n\u2212\n+\n\uf8ef\n\uf8fa\n+\n+\n\uf8f0\n\uf8fb\n, |x | < 1, y > 0 and xy < 1\nRationalised 2023-24\n 30\nMATHEMATICS\nFind the values of each of the expressions in Exercises 16 to 18 10 \u20131\n2\nsin\nsin 3\n\u03c0\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n11 \u20131\n3\ntan\ntan 4\n\u03c0\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n12"
    },
    {
        "Chapter": "1",
        "sentence_range": "945-948",
        "Text": "10 \u20131\n2\nsin\nsin 3\n\u03c0\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n11 \u20131\n3\ntan\ntan 4\n\u03c0\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n12 \u20131\n\u20131\n3\n3\ntan sin\ncot\n5\n2\n\uf8eb\n\uf8f6\n+\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n13"
    },
    {
        "Chapter": "1",
        "sentence_range": "946-949",
        "Text": "\u20131\n2\nsin\nsin 3\n\u03c0\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n11 \u20131\n3\ntan\ntan 4\n\u03c0\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n12 \u20131\n\u20131\n3\n3\ntan sin\ncot\n5\n2\n\uf8eb\n\uf8f6\n+\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n13 1\n7\ncos\ncos\nis equal to\n6\n\u2212\n\u03c0\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n(A) 7\n\u03c06\n(B) 5\n\u03c06\n(C)\n\u03c03\n(D)\n6\n\u03c0\n14"
    },
    {
        "Chapter": "1",
        "sentence_range": "947-950",
        "Text": "\u20131\n3\ntan\ntan 4\n\u03c0\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n12 \u20131\n\u20131\n3\n3\ntan sin\ncot\n5\n2\n\uf8eb\n\uf8f6\n+\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n13 1\n7\ncos\ncos\nis equal to\n6\n\u2212\n\u03c0\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n(A) 7\n\u03c06\n(B) 5\n\u03c06\n(C)\n\u03c03\n(D)\n6\n\u03c0\n14 1\n1\nsin\nsin\n(\n)\n3\n2\n\u2212\n\uf8eb\u03c0\n\uf8f6\n\u2212\n\u2212\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8  is equal to\n(A) 1\n2\n(B) 1\n3\n(C) 1\n4\n(D) 1\n15"
    },
    {
        "Chapter": "1",
        "sentence_range": "948-951",
        "Text": "\u20131\n\u20131\n3\n3\ntan sin\ncot\n5\n2\n\uf8eb\n\uf8f6\n+\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n13 1\n7\ncos\ncos\nis equal to\n6\n\u2212\n\u03c0\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n(A) 7\n\u03c06\n(B) 5\n\u03c06\n(C)\n\u03c03\n(D)\n6\n\u03c0\n14 1\n1\nsin\nsin\n(\n)\n3\n2\n\u2212\n\uf8eb\u03c0\n\uf8f6\n\u2212\n\u2212\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8  is equal to\n(A) 1\n2\n(B) 1\n3\n(C) 1\n4\n(D) 1\n15 1\n1\ntan\n3\ncot\n(\n3)\n\u2212\n\u2212\n\u2212\n\u2212\n is equal to\n(A) \u03c0\n(B)\n\u2212\u03c02\n(C) 0\n(D) 2 3\nMiscellaneous Examples\nExample 6 Find the value of \n1\n3\nsin\n(sin\n)\n5\n\u2212\n\u03c0\nSolution We know that \nsin1\n(sin )\nx\nx\n\u2212\n="
    },
    {
        "Chapter": "1",
        "sentence_range": "949-952",
        "Text": "1\n7\ncos\ncos\nis equal to\n6\n\u2212\n\u03c0\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n(A) 7\n\u03c06\n(B) 5\n\u03c06\n(C)\n\u03c03\n(D)\n6\n\u03c0\n14 1\n1\nsin\nsin\n(\n)\n3\n2\n\u2212\n\uf8eb\u03c0\n\uf8f6\n\u2212\n\u2212\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8  is equal to\n(A) 1\n2\n(B) 1\n3\n(C) 1\n4\n(D) 1\n15 1\n1\ntan\n3\ncot\n(\n3)\n\u2212\n\u2212\n\u2212\n\u2212\n is equal to\n(A) \u03c0\n(B)\n\u2212\u03c02\n(C) 0\n(D) 2 3\nMiscellaneous Examples\nExample 6 Find the value of \n1\n3\nsin\n(sin\n)\n5\n\u2212\n\u03c0\nSolution We know that \nsin1\n(sin )\nx\nx\n\u2212\n= Therefore, \n1\n3\n3\nsin\n(sin\n5)\n5\n\u2212\n\u03c0\n\u03c0\n=\nBut\n3\n,\n5\n2 2\n\u03c0\n\u03c0 \u03c0\n\uf8ee\n\uf8f9\n\u2209 \u2212\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb , which is the principal branch of sin\u20131 x\nHowever\n3\n3\n2\nsin (\n)\nsin(\n)\nsin\n5\n5\n5\n\u03c0\n\u03c0\n\u03c0\n=\n\u03c0 \u2212\n=\n and 2\n,\n5\n2 2\n\u03c0\n\u03c0 \u03c0\n\uf8ee\n\uf8f9\n\u2208 \u2212\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nTherefore\n1\n1\n3\n2\n2\nsin\n(sin\n)\nsin\n(sin\n)\n5\n5\n5\n\u2212\n\u2212\n\u03c0\n\u03c0\n\u03c0\n=\n=\nRationalised 2023-24\nINVERSE TRIGONOMETRIC FUNCTIONS     31\nMiscellaneous Exercise on Chapter 2\nFind the value of the following:\n1"
    },
    {
        "Chapter": "1",
        "sentence_range": "950-953",
        "Text": "1\n1\nsin\nsin\n(\n)\n3\n2\n\u2212\n\uf8eb\u03c0\n\uf8f6\n\u2212\n\u2212\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8  is equal to\n(A) 1\n2\n(B) 1\n3\n(C) 1\n4\n(D) 1\n15 1\n1\ntan\n3\ncot\n(\n3)\n\u2212\n\u2212\n\u2212\n\u2212\n is equal to\n(A) \u03c0\n(B)\n\u2212\u03c02\n(C) 0\n(D) 2 3\nMiscellaneous Examples\nExample 6 Find the value of \n1\n3\nsin\n(sin\n)\n5\n\u2212\n\u03c0\nSolution We know that \nsin1\n(sin )\nx\nx\n\u2212\n= Therefore, \n1\n3\n3\nsin\n(sin\n5)\n5\n\u2212\n\u03c0\n\u03c0\n=\nBut\n3\n,\n5\n2 2\n\u03c0\n\u03c0 \u03c0\n\uf8ee\n\uf8f9\n\u2209 \u2212\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb , which is the principal branch of sin\u20131 x\nHowever\n3\n3\n2\nsin (\n)\nsin(\n)\nsin\n5\n5\n5\n\u03c0\n\u03c0\n\u03c0\n=\n\u03c0 \u2212\n=\n and 2\n,\n5\n2 2\n\u03c0\n\u03c0 \u03c0\n\uf8ee\n\uf8f9\n\u2208 \u2212\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nTherefore\n1\n1\n3\n2\n2\nsin\n(sin\n)\nsin\n(sin\n)\n5\n5\n5\n\u2212\n\u2212\n\u03c0\n\u03c0\n\u03c0\n=\n=\nRationalised 2023-24\nINVERSE TRIGONOMETRIC FUNCTIONS     31\nMiscellaneous Exercise on Chapter 2\nFind the value of the following:\n1 \u20131\n13\ncos\ncos 6\n\u03c0\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n2"
    },
    {
        "Chapter": "1",
        "sentence_range": "951-954",
        "Text": "1\n1\ntan\n3\ncot\n(\n3)\n\u2212\n\u2212\n\u2212\n\u2212\n is equal to\n(A) \u03c0\n(B)\n\u2212\u03c02\n(C) 0\n(D) 2 3\nMiscellaneous Examples\nExample 6 Find the value of \n1\n3\nsin\n(sin\n)\n5\n\u2212\n\u03c0\nSolution We know that \nsin1\n(sin )\nx\nx\n\u2212\n= Therefore, \n1\n3\n3\nsin\n(sin\n5)\n5\n\u2212\n\u03c0\n\u03c0\n=\nBut\n3\n,\n5\n2 2\n\u03c0\n\u03c0 \u03c0\n\uf8ee\n\uf8f9\n\u2209 \u2212\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb , which is the principal branch of sin\u20131 x\nHowever\n3\n3\n2\nsin (\n)\nsin(\n)\nsin\n5\n5\n5\n\u03c0\n\u03c0\n\u03c0\n=\n\u03c0 \u2212\n=\n and 2\n,\n5\n2 2\n\u03c0\n\u03c0 \u03c0\n\uf8ee\n\uf8f9\n\u2208 \u2212\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nTherefore\n1\n1\n3\n2\n2\nsin\n(sin\n)\nsin\n(sin\n)\n5\n5\n5\n\u2212\n\u2212\n\u03c0\n\u03c0\n\u03c0\n=\n=\nRationalised 2023-24\nINVERSE TRIGONOMETRIC FUNCTIONS     31\nMiscellaneous Exercise on Chapter 2\nFind the value of the following:\n1 \u20131\n13\ncos\ncos 6\n\u03c0\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n2 \u20131\n7\ntan\ntan 6\n\u03c0\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\nProve that\n3"
    },
    {
        "Chapter": "1",
        "sentence_range": "952-955",
        "Text": "Therefore, \n1\n3\n3\nsin\n(sin\n5)\n5\n\u2212\n\u03c0\n\u03c0\n=\nBut\n3\n,\n5\n2 2\n\u03c0\n\u03c0 \u03c0\n\uf8ee\n\uf8f9\n\u2209 \u2212\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb , which is the principal branch of sin\u20131 x\nHowever\n3\n3\n2\nsin (\n)\nsin(\n)\nsin\n5\n5\n5\n\u03c0\n\u03c0\n\u03c0\n=\n\u03c0 \u2212\n=\n and 2\n,\n5\n2 2\n\u03c0\n\u03c0 \u03c0\n\uf8ee\n\uf8f9\n\u2208 \u2212\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nTherefore\n1\n1\n3\n2\n2\nsin\n(sin\n)\nsin\n(sin\n)\n5\n5\n5\n\u2212\n\u2212\n\u03c0\n\u03c0\n\u03c0\n=\n=\nRationalised 2023-24\nINVERSE TRIGONOMETRIC FUNCTIONS     31\nMiscellaneous Exercise on Chapter 2\nFind the value of the following:\n1 \u20131\n13\ncos\ncos 6\n\u03c0\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n2 \u20131\n7\ntan\ntan 6\n\u03c0\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\nProve that\n3 \u20131\n\u20131\n3\n24\n2sin\ntan\n5\n7\n=\n4"
    },
    {
        "Chapter": "1",
        "sentence_range": "953-956",
        "Text": "\u20131\n13\ncos\ncos 6\n\u03c0\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n2 \u20131\n7\ntan\ntan 6\n\u03c0\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\nProve that\n3 \u20131\n\u20131\n3\n24\n2sin\ntan\n5\n7\n=\n4 \u20131\n\u20131\n\u20131\n8\n3\n77\nsin\nsin\ntan\n17\n5\n36\n+\n=\n5"
    },
    {
        "Chapter": "1",
        "sentence_range": "954-957",
        "Text": "\u20131\n7\ntan\ntan 6\n\u03c0\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\nProve that\n3 \u20131\n\u20131\n3\n24\n2sin\ntan\n5\n7\n=\n4 \u20131\n\u20131\n\u20131\n8\n3\n77\nsin\nsin\ntan\n17\n5\n36\n+\n=\n5 \u20131\n\u20131\n\u20131\n4\n12\n33\ncos\ncos\ncos\n5\n13\n65\n+\n=\n6"
    },
    {
        "Chapter": "1",
        "sentence_range": "955-958",
        "Text": "\u20131\n\u20131\n3\n24\n2sin\ntan\n5\n7\n=\n4 \u20131\n\u20131\n\u20131\n8\n3\n77\nsin\nsin\ntan\n17\n5\n36\n+\n=\n5 \u20131\n\u20131\n\u20131\n4\n12\n33\ncos\ncos\ncos\n5\n13\n65\n+\n=\n6 \u20131\n\u20131\n\u20131\n12\n3\n56\ncos\nsin\nsin\n13\n5\n65\n+\n=\n7"
    },
    {
        "Chapter": "1",
        "sentence_range": "956-959",
        "Text": "\u20131\n\u20131\n\u20131\n8\n3\n77\nsin\nsin\ntan\n17\n5\n36\n+\n=\n5 \u20131\n\u20131\n\u20131\n4\n12\n33\ncos\ncos\ncos\n5\n13\n65\n+\n=\n6 \u20131\n\u20131\n\u20131\n12\n3\n56\ncos\nsin\nsin\n13\n5\n65\n+\n=\n7 \u20131\n\u20131\n\u20131\n63\n5\n3\ntan\nsin\ncos\n16\n13\n5\n=\n+\nProve that\n8"
    },
    {
        "Chapter": "1",
        "sentence_range": "957-960",
        "Text": "\u20131\n\u20131\n\u20131\n4\n12\n33\ncos\ncos\ncos\n5\n13\n65\n+\n=\n6 \u20131\n\u20131\n\u20131\n12\n3\n56\ncos\nsin\nsin\n13\n5\n65\n+\n=\n7 \u20131\n\u20131\n\u20131\n63\n5\n3\ntan\nsin\ncos\n16\n13\n5\n=\n+\nProve that\n8 \u20131\n\u20131\n1\n1\ntan\n2cos\n1\nx\nx\nx\n\u2212\n \n \n=\n \n \n \n \n+\n, x \u2208 [0, 1]\n9"
    },
    {
        "Chapter": "1",
        "sentence_range": "958-961",
        "Text": "\u20131\n\u20131\n\u20131\n12\n3\n56\ncos\nsin\nsin\n13\n5\n65\n+\n=\n7 \u20131\n\u20131\n\u20131\n63\n5\n3\ntan\nsin\ncos\n16\n13\n5\n=\n+\nProve that\n8 \u20131\n\u20131\n1\n1\ntan\n2cos\n1\nx\nx\nx\n\u2212\n \n \n=\n \n \n \n \n+\n, x \u2208 [0, 1]\n9 \u20131\n1\nsin\n1\nsin\ncot\n2\n1\nsin\n1\nsin\nx\nx\nx\nx\nx\n\uf8eb\n\uf8f6\n+\n+\n\u2212\n=\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n+\n\u2212\n\u2212\n\uf8ed\n\uf8f8\n, \n0, 4\nx\n\u03c0\n\uf8eb\n\uf8f6\n\u2208\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n10"
    },
    {
        "Chapter": "1",
        "sentence_range": "959-962",
        "Text": "\u20131\n\u20131\n\u20131\n63\n5\n3\ntan\nsin\ncos\n16\n13\n5\n=\n+\nProve that\n8 \u20131\n\u20131\n1\n1\ntan\n2cos\n1\nx\nx\nx\n\u2212\n \n \n=\n \n \n \n \n+\n, x \u2208 [0, 1]\n9 \u20131\n1\nsin\n1\nsin\ncot\n2\n1\nsin\n1\nsin\nx\nx\nx\nx\nx\n\uf8eb\n\uf8f6\n+\n+\n\u2212\n=\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n+\n\u2212\n\u2212\n\uf8ed\n\uf8f8\n, \n0, 4\nx\n\u03c0\n\uf8eb\n\uf8f6\n\u2208\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n10 \u20131\n\u20131\n1\n1\n1\ntan\ncos\n4\n2\n1\n1\nx\nx\nx\nx\nx\n\uf8eb\n\uf8f6\n+\n\u2212\n\u2212\n=\u03c0\n\u2212\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n+\n+\n\u2212\n\uf8ed\n\uf8f8\n, \n1\n1\n2\nx\n\u2212\n\u2264\n\u2264  [Hint: Put x = cos 2\u03b8]\nSolve the following equations:\n11"
    },
    {
        "Chapter": "1",
        "sentence_range": "960-963",
        "Text": "\u20131\n\u20131\n1\n1\ntan\n2cos\n1\nx\nx\nx\n\u2212\n \n \n=\n \n \n \n \n+\n, x \u2208 [0, 1]\n9 \u20131\n1\nsin\n1\nsin\ncot\n2\n1\nsin\n1\nsin\nx\nx\nx\nx\nx\n\uf8eb\n\uf8f6\n+\n+\n\u2212\n=\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n+\n\u2212\n\u2212\n\uf8ed\n\uf8f8\n, \n0, 4\nx\n\u03c0\n\uf8eb\n\uf8f6\n\u2208\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n10 \u20131\n\u20131\n1\n1\n1\ntan\ncos\n4\n2\n1\n1\nx\nx\nx\nx\nx\n\uf8eb\n\uf8f6\n+\n\u2212\n\u2212\n=\u03c0\n\u2212\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n+\n+\n\u2212\n\uf8ed\n\uf8f8\n, \n1\n1\n2\nx\n\u2212\n\u2264\n\u2264  [Hint: Put x = cos 2\u03b8]\nSolve the following equations:\n11 2tan\u20131 (cos x) = tan\u20131 (2 cosec x)\n12"
    },
    {
        "Chapter": "1",
        "sentence_range": "961-964",
        "Text": "\u20131\n1\nsin\n1\nsin\ncot\n2\n1\nsin\n1\nsin\nx\nx\nx\nx\nx\n\uf8eb\n\uf8f6\n+\n+\n\u2212\n=\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n+\n\u2212\n\u2212\n\uf8ed\n\uf8f8\n, \n0, 4\nx\n\u03c0\n\uf8eb\n\uf8f6\n\u2208\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n10 \u20131\n\u20131\n1\n1\n1\ntan\ncos\n4\n2\n1\n1\nx\nx\nx\nx\nx\n\uf8eb\n\uf8f6\n+\n\u2212\n\u2212\n=\u03c0\n\u2212\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n+\n+\n\u2212\n\uf8ed\n\uf8f8\n, \n1\n1\n2\nx\n\u2212\n\u2264\n\u2264  [Hint: Put x = cos 2\u03b8]\nSolve the following equations:\n11 2tan\u20131 (cos x) = tan\u20131 (2 cosec x)\n12 \u20131\n\u20131\n1\n1\ntan\ntan\n,(\n0)\n1\n2\nx\nx x\nx\n\u2212\n=\n>\n+\n13"
    },
    {
        "Chapter": "1",
        "sentence_range": "962-965",
        "Text": "\u20131\n\u20131\n1\n1\n1\ntan\ncos\n4\n2\n1\n1\nx\nx\nx\nx\nx\n\uf8eb\n\uf8f6\n+\n\u2212\n\u2212\n=\u03c0\n\u2212\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n+\n+\n\u2212\n\uf8ed\n\uf8f8\n, \n1\n1\n2\nx\n\u2212\n\u2264\n\u2264  [Hint: Put x = cos 2\u03b8]\nSolve the following equations:\n11 2tan\u20131 (cos x) = tan\u20131 (2 cosec x)\n12 \u20131\n\u20131\n1\n1\ntan\ntan\n,(\n0)\n1\n2\nx\nx x\nx\n\u2212\n=\n>\n+\n13 sin (tan\u20131 x), |x| < 1 is equal to\n(A)\n2\n1\nx\nx\n\u2212\n(B)\n2\n1\n1\nx\n\u2212\n(C)\n2\n1\n1\nx\n+\n(D)\n2\n1\nx\nx\n+\n14"
    },
    {
        "Chapter": "1",
        "sentence_range": "963-966",
        "Text": "2tan\u20131 (cos x) = tan\u20131 (2 cosec x)\n12 \u20131\n\u20131\n1\n1\ntan\ntan\n,(\n0)\n1\n2\nx\nx x\nx\n\u2212\n=\n>\n+\n13 sin (tan\u20131 x), |x| < 1 is equal to\n(A)\n2\n1\nx\nx\n\u2212\n(B)\n2\n1\n1\nx\n\u2212\n(C)\n2\n1\n1\nx\n+\n(D)\n2\n1\nx\nx\n+\n14 sin\u20131 (1 \u2013 x) \u2013 2 sin\u20131 x = 2\n\u03c0 , then x is equal to\n(A) 0, 1\n2\n(B) 1, 1\n2\n(C) 0\n(D) 1\n2\nRationalised 2023-24\n 32\nMATHEMATICS\nSummary\n\u00ae The domains and ranges (principal value branches) of inverse trigonometric\nfunctions are given in the following table:\nFunctions\nDomain\nRange\n(Principal Value Branches)\ny = sin\u20131 x\n[\u20131, 1]\n2,\n2\n\uf8ee\u2212\u03c0 \u03c0\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\ny = cos\u20131 x\n[\u20131, 1]\n [0, \u03c0]\ny = cosec\u20131 x\nR \u2013 (\u20131,1)\n2,\n2\n\uf8ee\u2212\u03c0 \u03c0\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\u2013 {0}\ny = sec\u20131 x\nR \u2013 (\u20131, 1)\n[0, \u03c0] \u2013 { }\n2\n\u03c0\ny = tan\u20131 x\nR\n,\n2 2\n\u03c0 \u03c0\n\uf8eb\n\uf8f6\n\uf8ec\u2212\n\uf8f7\n\uf8ed\n\uf8f8\ny = cot\u20131 x\nR\n(0, \u03c0)\n\u00ae sin\u20131x should not be confused with (sin x)\u20131"
    },
    {
        "Chapter": "1",
        "sentence_range": "964-967",
        "Text": "\u20131\n\u20131\n1\n1\ntan\ntan\n,(\n0)\n1\n2\nx\nx x\nx\n\u2212\n=\n>\n+\n13 sin (tan\u20131 x), |x| < 1 is equal to\n(A)\n2\n1\nx\nx\n\u2212\n(B)\n2\n1\n1\nx\n\u2212\n(C)\n2\n1\n1\nx\n+\n(D)\n2\n1\nx\nx\n+\n14 sin\u20131 (1 \u2013 x) \u2013 2 sin\u20131 x = 2\n\u03c0 , then x is equal to\n(A) 0, 1\n2\n(B) 1, 1\n2\n(C) 0\n(D) 1\n2\nRationalised 2023-24\n 32\nMATHEMATICS\nSummary\n\u00ae The domains and ranges (principal value branches) of inverse trigonometric\nfunctions are given in the following table:\nFunctions\nDomain\nRange\n(Principal Value Branches)\ny = sin\u20131 x\n[\u20131, 1]\n2,\n2\n\uf8ee\u2212\u03c0 \u03c0\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\ny = cos\u20131 x\n[\u20131, 1]\n [0, \u03c0]\ny = cosec\u20131 x\nR \u2013 (\u20131,1)\n2,\n2\n\uf8ee\u2212\u03c0 \u03c0\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\u2013 {0}\ny = sec\u20131 x\nR \u2013 (\u20131, 1)\n[0, \u03c0] \u2013 { }\n2\n\u03c0\ny = tan\u20131 x\nR\n,\n2 2\n\u03c0 \u03c0\n\uf8eb\n\uf8f6\n\uf8ec\u2212\n\uf8f7\n\uf8ed\n\uf8f8\ny = cot\u20131 x\nR\n(0, \u03c0)\n\u00ae sin\u20131x should not be confused with (sin x)\u20131 In fact (sin x)\u20131 = \n1\nsin x  and\nsimilarly for other trigonometric functions"
    },
    {
        "Chapter": "1",
        "sentence_range": "965-968",
        "Text": "sin (tan\u20131 x), |x| < 1 is equal to\n(A)\n2\n1\nx\nx\n\u2212\n(B)\n2\n1\n1\nx\n\u2212\n(C)\n2\n1\n1\nx\n+\n(D)\n2\n1\nx\nx\n+\n14 sin\u20131 (1 \u2013 x) \u2013 2 sin\u20131 x = 2\n\u03c0 , then x is equal to\n(A) 0, 1\n2\n(B) 1, 1\n2\n(C) 0\n(D) 1\n2\nRationalised 2023-24\n 32\nMATHEMATICS\nSummary\n\u00ae The domains and ranges (principal value branches) of inverse trigonometric\nfunctions are given in the following table:\nFunctions\nDomain\nRange\n(Principal Value Branches)\ny = sin\u20131 x\n[\u20131, 1]\n2,\n2\n\uf8ee\u2212\u03c0 \u03c0\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\ny = cos\u20131 x\n[\u20131, 1]\n [0, \u03c0]\ny = cosec\u20131 x\nR \u2013 (\u20131,1)\n2,\n2\n\uf8ee\u2212\u03c0 \u03c0\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\u2013 {0}\ny = sec\u20131 x\nR \u2013 (\u20131, 1)\n[0, \u03c0] \u2013 { }\n2\n\u03c0\ny = tan\u20131 x\nR\n,\n2 2\n\u03c0 \u03c0\n\uf8eb\n\uf8f6\n\uf8ec\u2212\n\uf8f7\n\uf8ed\n\uf8f8\ny = cot\u20131 x\nR\n(0, \u03c0)\n\u00ae sin\u20131x should not be confused with (sin x)\u20131 In fact (sin x)\u20131 = \n1\nsin x  and\nsimilarly for other trigonometric functions \u00ae The value of an inverse trigonometric functions which lies in its principal\nvalue branch is called the principal value of that inverse trigonometric\nfunctions"
    },
    {
        "Chapter": "1",
        "sentence_range": "966-969",
        "Text": "sin\u20131 (1 \u2013 x) \u2013 2 sin\u20131 x = 2\n\u03c0 , then x is equal to\n(A) 0, 1\n2\n(B) 1, 1\n2\n(C) 0\n(D) 1\n2\nRationalised 2023-24\n 32\nMATHEMATICS\nSummary\n\u00ae The domains and ranges (principal value branches) of inverse trigonometric\nfunctions are given in the following table:\nFunctions\nDomain\nRange\n(Principal Value Branches)\ny = sin\u20131 x\n[\u20131, 1]\n2,\n2\n\uf8ee\u2212\u03c0 \u03c0\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\ny = cos\u20131 x\n[\u20131, 1]\n [0, \u03c0]\ny = cosec\u20131 x\nR \u2013 (\u20131,1)\n2,\n2\n\uf8ee\u2212\u03c0 \u03c0\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\u2013 {0}\ny = sec\u20131 x\nR \u2013 (\u20131, 1)\n[0, \u03c0] \u2013 { }\n2\n\u03c0\ny = tan\u20131 x\nR\n,\n2 2\n\u03c0 \u03c0\n\uf8eb\n\uf8f6\n\uf8ec\u2212\n\uf8f7\n\uf8ed\n\uf8f8\ny = cot\u20131 x\nR\n(0, \u03c0)\n\u00ae sin\u20131x should not be confused with (sin x)\u20131 In fact (sin x)\u20131 = \n1\nsin x  and\nsimilarly for other trigonometric functions \u00ae The value of an inverse trigonometric functions which lies in its principal\nvalue branch is called the principal value of that inverse trigonometric\nfunctions For suitable values of domain, we have\n\u00ae y = sin\u20131 x \u21d2 x = sin y\n\u00ae x = sin y  \u21d2 y = sin\u20131 x\n\u00ae sin (sin\u20131 x) = x\n\u00ae sin\u20131 (sin x) = x\nRationalised 2023-24\nINVERSE TRIGONOMETRIC FUNCTIONS     33\nHistorical Note\nThe study of trigonometry was first started in India"
    },
    {
        "Chapter": "1",
        "sentence_range": "967-970",
        "Text": "In fact (sin x)\u20131 = \n1\nsin x  and\nsimilarly for other trigonometric functions \u00ae The value of an inverse trigonometric functions which lies in its principal\nvalue branch is called the principal value of that inverse trigonometric\nfunctions For suitable values of domain, we have\n\u00ae y = sin\u20131 x \u21d2 x = sin y\n\u00ae x = sin y  \u21d2 y = sin\u20131 x\n\u00ae sin (sin\u20131 x) = x\n\u00ae sin\u20131 (sin x) = x\nRationalised 2023-24\nINVERSE TRIGONOMETRIC FUNCTIONS     33\nHistorical Note\nThe study of trigonometry was first started in India The ancient Indian\nMathematicians, Aryabhata (476A"
    },
    {
        "Chapter": "1",
        "sentence_range": "968-971",
        "Text": "\u00ae The value of an inverse trigonometric functions which lies in its principal\nvalue branch is called the principal value of that inverse trigonometric\nfunctions For suitable values of domain, we have\n\u00ae y = sin\u20131 x \u21d2 x = sin y\n\u00ae x = sin y  \u21d2 y = sin\u20131 x\n\u00ae sin (sin\u20131 x) = x\n\u00ae sin\u20131 (sin x) = x\nRationalised 2023-24\nINVERSE TRIGONOMETRIC FUNCTIONS     33\nHistorical Note\nThe study of trigonometry was first started in India The ancient Indian\nMathematicians, Aryabhata (476A D"
    },
    {
        "Chapter": "1",
        "sentence_range": "969-972",
        "Text": "For suitable values of domain, we have\n\u00ae y = sin\u20131 x \u21d2 x = sin y\n\u00ae x = sin y  \u21d2 y = sin\u20131 x\n\u00ae sin (sin\u20131 x) = x\n\u00ae sin\u20131 (sin x) = x\nRationalised 2023-24\nINVERSE TRIGONOMETRIC FUNCTIONS     33\nHistorical Note\nThe study of trigonometry was first started in India The ancient Indian\nMathematicians, Aryabhata (476A D ), Brahmagupta (598 A"
    },
    {
        "Chapter": "1",
        "sentence_range": "970-973",
        "Text": "The ancient Indian\nMathematicians, Aryabhata (476A D ), Brahmagupta (598 A D"
    },
    {
        "Chapter": "1",
        "sentence_range": "971-974",
        "Text": "D ), Brahmagupta (598 A D ), Bhaskara I\n(600 A"
    },
    {
        "Chapter": "1",
        "sentence_range": "972-975",
        "Text": "), Brahmagupta (598 A D ), Bhaskara I\n(600 A D"
    },
    {
        "Chapter": "1",
        "sentence_range": "973-976",
        "Text": "D ), Bhaskara I\n(600 A D ) and Bhaskara II (1114 A"
    },
    {
        "Chapter": "1",
        "sentence_range": "974-977",
        "Text": "), Bhaskara I\n(600 A D ) and Bhaskara II (1114 A D"
    },
    {
        "Chapter": "1",
        "sentence_range": "975-978",
        "Text": "D ) and Bhaskara II (1114 A D ) got important results of trigonometry"
    },
    {
        "Chapter": "1",
        "sentence_range": "976-979",
        "Text": ") and Bhaskara II (1114 A D ) got important results of trigonometry All\nthis knowledge went from India to Arabia and then from there to Europe"
    },
    {
        "Chapter": "1",
        "sentence_range": "977-980",
        "Text": "D ) got important results of trigonometry All\nthis knowledge went from India to Arabia and then from there to Europe The\nGreeks had also started the study of trigonometry but their approach was so\nclumsy that when the Indian approach became known, it was immediately adopted\nthroughout the world"
    },
    {
        "Chapter": "1",
        "sentence_range": "978-981",
        "Text": ") got important results of trigonometry All\nthis knowledge went from India to Arabia and then from there to Europe The\nGreeks had also started the study of trigonometry but their approach was so\nclumsy that when the Indian approach became known, it was immediately adopted\nthroughout the world In India, the predecessor of the modern trigonometric functions, known as\nthe sine of an angle, and the introduction of the sine function represents one of\nthe main contribution of the siddhantas (Sanskrit astronomical works) to\nmathematics"
    },
    {
        "Chapter": "1",
        "sentence_range": "979-982",
        "Text": "All\nthis knowledge went from India to Arabia and then from there to Europe The\nGreeks had also started the study of trigonometry but their approach was so\nclumsy that when the Indian approach became known, it was immediately adopted\nthroughout the world In India, the predecessor of the modern trigonometric functions, known as\nthe sine of an angle, and the introduction of the sine function represents one of\nthe main contribution of the siddhantas (Sanskrit astronomical works) to\nmathematics Bhaskara I (about 600 A"
    },
    {
        "Chapter": "1",
        "sentence_range": "980-983",
        "Text": "The\nGreeks had also started the study of trigonometry but their approach was so\nclumsy that when the Indian approach became known, it was immediately adopted\nthroughout the world In India, the predecessor of the modern trigonometric functions, known as\nthe sine of an angle, and the introduction of the sine function represents one of\nthe main contribution of the siddhantas (Sanskrit astronomical works) to\nmathematics Bhaskara I (about 600 A D"
    },
    {
        "Chapter": "1",
        "sentence_range": "981-984",
        "Text": "In India, the predecessor of the modern trigonometric functions, known as\nthe sine of an angle, and the introduction of the sine function represents one of\nthe main contribution of the siddhantas (Sanskrit astronomical works) to\nmathematics Bhaskara I (about 600 A D ) gave formulae to find the values of sine functions\nfor angles more than 90\u00b0"
    },
    {
        "Chapter": "1",
        "sentence_range": "982-985",
        "Text": "Bhaskara I (about 600 A D ) gave formulae to find the values of sine functions\nfor angles more than 90\u00b0 A sixteenth century Malayalam work Yuktibhasa\ncontains a proof for the expansion of sin (A + B)"
    },
    {
        "Chapter": "1",
        "sentence_range": "983-986",
        "Text": "D ) gave formulae to find the values of sine functions\nfor angles more than 90\u00b0 A sixteenth century Malayalam work Yuktibhasa\ncontains a proof for the expansion of sin (A + B) Exact expression for sines or\ncosines of 18\u00b0, 36\u00b0, 54\u00b0, 72\u00b0, etc"
    },
    {
        "Chapter": "1",
        "sentence_range": "984-987",
        "Text": ") gave formulae to find the values of sine functions\nfor angles more than 90\u00b0 A sixteenth century Malayalam work Yuktibhasa\ncontains a proof for the expansion of sin (A + B) Exact expression for sines or\ncosines of 18\u00b0, 36\u00b0, 54\u00b0, 72\u00b0, etc , were given by Bhaskara II"
    },
    {
        "Chapter": "1",
        "sentence_range": "985-988",
        "Text": "A sixteenth century Malayalam work Yuktibhasa\ncontains a proof for the expansion of sin (A + B) Exact expression for sines or\ncosines of 18\u00b0, 36\u00b0, 54\u00b0, 72\u00b0, etc , were given by Bhaskara II The symbols sin\u20131 x, cos\u20131 x, etc"
    },
    {
        "Chapter": "1",
        "sentence_range": "986-989",
        "Text": "Exact expression for sines or\ncosines of 18\u00b0, 36\u00b0, 54\u00b0, 72\u00b0, etc , were given by Bhaskara II The symbols sin\u20131 x, cos\u20131 x, etc , for arc sin x, arc cos x, etc"
    },
    {
        "Chapter": "1",
        "sentence_range": "987-990",
        "Text": ", were given by Bhaskara II The symbols sin\u20131 x, cos\u20131 x, etc , for arc sin x, arc cos x, etc , were suggested\nby the astronomer Sir John F"
    },
    {
        "Chapter": "1",
        "sentence_range": "988-991",
        "Text": "The symbols sin\u20131 x, cos\u20131 x, etc , for arc sin x, arc cos x, etc , were suggested\nby the astronomer Sir John F W"
    },
    {
        "Chapter": "1",
        "sentence_range": "989-992",
        "Text": ", for arc sin x, arc cos x, etc , were suggested\nby the astronomer Sir John F W Hersehel (1813) The name of Thales\n(about 600 B"
    },
    {
        "Chapter": "1",
        "sentence_range": "990-993",
        "Text": ", were suggested\nby the astronomer Sir John F W Hersehel (1813) The name of Thales\n(about 600 B C"
    },
    {
        "Chapter": "1",
        "sentence_range": "991-994",
        "Text": "W Hersehel (1813) The name of Thales\n(about 600 B C ) is invariably associated with height and distance problems"
    },
    {
        "Chapter": "1",
        "sentence_range": "992-995",
        "Text": "Hersehel (1813) The name of Thales\n(about 600 B C ) is invariably associated with height and distance problems He\nis credited with the determination of the height of a great pyramid in Egypt by\nmeasuring shadows of the pyramid and an auxiliary staff (or gnomon) of known\nheight, and comparing the ratios:\nH\nS\n=sh\n = tan (sun\u2019s altitude)\nThales is also said to have calculated the distance of a ship at sea through\nthe proportionality of sides of similar triangles"
    },
    {
        "Chapter": "1",
        "sentence_range": "993-996",
        "Text": "C ) is invariably associated with height and distance problems He\nis credited with the determination of the height of a great pyramid in Egypt by\nmeasuring shadows of the pyramid and an auxiliary staff (or gnomon) of known\nheight, and comparing the ratios:\nH\nS\n=sh\n = tan (sun\u2019s altitude)\nThales is also said to have calculated the distance of a ship at sea through\nthe proportionality of sides of similar triangles Problems on height and distance\nusing the similarity property are also found in ancient Indian works"
    },
    {
        "Chapter": "1",
        "sentence_range": "994-997",
        "Text": ") is invariably associated with height and distance problems He\nis credited with the determination of the height of a great pyramid in Egypt by\nmeasuring shadows of the pyramid and an auxiliary staff (or gnomon) of known\nheight, and comparing the ratios:\nH\nS\n=sh\n = tan (sun\u2019s altitude)\nThales is also said to have calculated the distance of a ship at sea through\nthe proportionality of sides of similar triangles Problems on height and distance\nusing the similarity property are also found in ancient Indian works \u2014v\nv\nv\nv\nv\u2014\nRationalised 2023-24\n 34\nMATHEMATICS\nvThe essence of Mathematics lies in its freedom"
    },
    {
        "Chapter": "1",
        "sentence_range": "995-998",
        "Text": "He\nis credited with the determination of the height of a great pyramid in Egypt by\nmeasuring shadows of the pyramid and an auxiliary staff (or gnomon) of known\nheight, and comparing the ratios:\nH\nS\n=sh\n = tan (sun\u2019s altitude)\nThales is also said to have calculated the distance of a ship at sea through\nthe proportionality of sides of similar triangles Problems on height and distance\nusing the similarity property are also found in ancient Indian works \u2014v\nv\nv\nv\nv\u2014\nRationalised 2023-24\n 34\nMATHEMATICS\nvThe essence of Mathematics lies in its freedom \u2014 CANTOR v\n3"
    },
    {
        "Chapter": "1",
        "sentence_range": "996-999",
        "Text": "Problems on height and distance\nusing the similarity property are also found in ancient Indian works \u2014v\nv\nv\nv\nv\u2014\nRationalised 2023-24\n 34\nMATHEMATICS\nvThe essence of Mathematics lies in its freedom \u2014 CANTOR v\n3 1  Introduction\nThe knowledge of matrices is necessary in various branches of mathematics"
    },
    {
        "Chapter": "1",
        "sentence_range": "997-1000",
        "Text": "\u2014v\nv\nv\nv\nv\u2014\nRationalised 2023-24\n 34\nMATHEMATICS\nvThe essence of Mathematics lies in its freedom \u2014 CANTOR v\n3 1  Introduction\nThe knowledge of matrices is necessary in various branches of mathematics Matrices\nare one of the most powerful tools in mathematics"
    },
    {
        "Chapter": "1",
        "sentence_range": "998-1001",
        "Text": "\u2014 CANTOR v\n3 1  Introduction\nThe knowledge of matrices is necessary in various branches of mathematics Matrices\nare one of the most powerful tools in mathematics This mathematical tool simplifies\nour work to a great extent when compared with other straight forward methods"
    },
    {
        "Chapter": "1",
        "sentence_range": "999-1002",
        "Text": "1  Introduction\nThe knowledge of matrices is necessary in various branches of mathematics Matrices\nare one of the most powerful tools in mathematics This mathematical tool simplifies\nour work to a great extent when compared with other straight forward methods The\nevolution of concept of matrices is the result of an attempt to obtain compact and\nsimple methods of solving system of linear equations"
    },
    {
        "Chapter": "1",
        "sentence_range": "1000-1003",
        "Text": "Matrices\nare one of the most powerful tools in mathematics This mathematical tool simplifies\nour work to a great extent when compared with other straight forward methods The\nevolution of concept of matrices is the result of an attempt to obtain compact and\nsimple methods of solving system of linear equations Matrices are not only used as a\nrepresentation of the coefficients in system of linear equations, but utility of matrices\nfar exceeds that use"
    },
    {
        "Chapter": "1",
        "sentence_range": "1001-1004",
        "Text": "This mathematical tool simplifies\nour work to a great extent when compared with other straight forward methods The\nevolution of concept of matrices is the result of an attempt to obtain compact and\nsimple methods of solving system of linear equations Matrices are not only used as a\nrepresentation of the coefficients in system of linear equations, but utility of matrices\nfar exceeds that use Matrix notation and operations are used in electronic spreadsheet\nprograms for personal computer, which in turn is used in different areas of business\nand science like budgeting, sales projection, cost estimation, analysing the results of an\nexperiment etc"
    },
    {
        "Chapter": "1",
        "sentence_range": "1002-1005",
        "Text": "The\nevolution of concept of matrices is the result of an attempt to obtain compact and\nsimple methods of solving system of linear equations Matrices are not only used as a\nrepresentation of the coefficients in system of linear equations, but utility of matrices\nfar exceeds that use Matrix notation and operations are used in electronic spreadsheet\nprograms for personal computer, which in turn is used in different areas of business\nand science like budgeting, sales projection, cost estimation, analysing the results of an\nexperiment etc Also, many physical operations such as magnification, rotation and\nreflection through a plane can be represented mathematically by matrices"
    },
    {
        "Chapter": "1",
        "sentence_range": "1003-1006",
        "Text": "Matrices are not only used as a\nrepresentation of the coefficients in system of linear equations, but utility of matrices\nfar exceeds that use Matrix notation and operations are used in electronic spreadsheet\nprograms for personal computer, which in turn is used in different areas of business\nand science like budgeting, sales projection, cost estimation, analysing the results of an\nexperiment etc Also, many physical operations such as magnification, rotation and\nreflection through a plane can be represented mathematically by matrices Matrices\nare also used in cryptography"
    },
    {
        "Chapter": "1",
        "sentence_range": "1004-1007",
        "Text": "Matrix notation and operations are used in electronic spreadsheet\nprograms for personal computer, which in turn is used in different areas of business\nand science like budgeting, sales projection, cost estimation, analysing the results of an\nexperiment etc Also, many physical operations such as magnification, rotation and\nreflection through a plane can be represented mathematically by matrices Matrices\nare also used in cryptography This mathematical tool is not only used in certain branches\nof sciences, but also in genetics, economics, sociology, modern psychology and industrial\nmanagement"
    },
    {
        "Chapter": "1",
        "sentence_range": "1005-1008",
        "Text": "Also, many physical operations such as magnification, rotation and\nreflection through a plane can be represented mathematically by matrices Matrices\nare also used in cryptography This mathematical tool is not only used in certain branches\nof sciences, but also in genetics, economics, sociology, modern psychology and industrial\nmanagement In this chapter, we shall find it interesting to become acquainted with the\nfundamentals of matrix and matrix algebra"
    },
    {
        "Chapter": "1",
        "sentence_range": "1006-1009",
        "Text": "Matrices\nare also used in cryptography This mathematical tool is not only used in certain branches\nof sciences, but also in genetics, economics, sociology, modern psychology and industrial\nmanagement In this chapter, we shall find it interesting to become acquainted with the\nfundamentals of matrix and matrix algebra 3"
    },
    {
        "Chapter": "1",
        "sentence_range": "1007-1010",
        "Text": "This mathematical tool is not only used in certain branches\nof sciences, but also in genetics, economics, sociology, modern psychology and industrial\nmanagement In this chapter, we shall find it interesting to become acquainted with the\nfundamentals of matrix and matrix algebra 3 2  Matrix\nSuppose we wish to express the information that Radha has 15 notebooks"
    },
    {
        "Chapter": "1",
        "sentence_range": "1008-1011",
        "Text": "In this chapter, we shall find it interesting to become acquainted with the\nfundamentals of matrix and matrix algebra 3 2  Matrix\nSuppose we wish to express the information that Radha has 15 notebooks We may\nexpress it as [15] with the understanding that the number inside [ ] is the number of\nnotebooks that Radha has"
    },
    {
        "Chapter": "1",
        "sentence_range": "1009-1012",
        "Text": "3 2  Matrix\nSuppose we wish to express the information that Radha has 15 notebooks We may\nexpress it as [15] with the understanding that the number inside [ ] is the number of\nnotebooks that Radha has Now, if we have to express that Radha has 15 notebooks\nand 6 pens"
    },
    {
        "Chapter": "1",
        "sentence_range": "1010-1013",
        "Text": "2  Matrix\nSuppose we wish to express the information that Radha has 15 notebooks We may\nexpress it as [15] with the understanding that the number inside [ ] is the number of\nnotebooks that Radha has Now, if we have to express that Radha has 15 notebooks\nand 6 pens We may express it as [15  6] with the understanding that first number\ninside [ ] is the number of notebooks while the other one is the number of pens possessed\nby Radha"
    },
    {
        "Chapter": "1",
        "sentence_range": "1011-1014",
        "Text": "We may\nexpress it as [15] with the understanding that the number inside [ ] is the number of\nnotebooks that Radha has Now, if we have to express that Radha has 15 notebooks\nand 6 pens We may express it as [15  6] with the understanding that first number\ninside [ ] is the number of notebooks while the other one is the number of pens possessed\nby Radha Let us now suppose that we wish to express the information of possession\nChapter 3\nMATRICES\nRationalised 2023-24\nMATRICES         35\nof notebooks and pens by Radha and her two friends Fauzia and Simran which\nis as follows:\nRadha\nhas\n15\nnotebooks\nand\n6 pens,\nFauzia\nhas\n10\nnotebooks\nand\n2 pens,\nSimran\nhas\n13\nnotebooks\nand\n5 pens"
    },
    {
        "Chapter": "1",
        "sentence_range": "1012-1015",
        "Text": "Now, if we have to express that Radha has 15 notebooks\nand 6 pens We may express it as [15  6] with the understanding that first number\ninside [ ] is the number of notebooks while the other one is the number of pens possessed\nby Radha Let us now suppose that we wish to express the information of possession\nChapter 3\nMATRICES\nRationalised 2023-24\nMATRICES         35\nof notebooks and pens by Radha and her two friends Fauzia and Simran which\nis as follows:\nRadha\nhas\n15\nnotebooks\nand\n6 pens,\nFauzia\nhas\n10\nnotebooks\nand\n2 pens,\nSimran\nhas\n13\nnotebooks\nand\n5 pens Now this could be arranged in the tabular form as follows:\nNotebooks\nPens\nRadha\n15\n6\nFauzia\n10\n2\nSimran\n13\n5\nand this can be expressed as\nor\nRadha\nFauzia\nSimran\nNotebooks\n15\n10\n13\nPens\n6\n2\n5\nwhich can be expressed as:\nIn the first arrangement the entries in the first column represent the number of\nnote books possessed by Radha, Fauzia and Simran, respectively and the entries in the\nsecond column represent the number of pens possessed by Radha, Fauzia and Simran,\nRationalised 2023-24\n 36\nMATHEMATICS\nrespectively"
    },
    {
        "Chapter": "1",
        "sentence_range": "1013-1016",
        "Text": "We may express it as [15  6] with the understanding that first number\ninside [ ] is the number of notebooks while the other one is the number of pens possessed\nby Radha Let us now suppose that we wish to express the information of possession\nChapter 3\nMATRICES\nRationalised 2023-24\nMATRICES         35\nof notebooks and pens by Radha and her two friends Fauzia and Simran which\nis as follows:\nRadha\nhas\n15\nnotebooks\nand\n6 pens,\nFauzia\nhas\n10\nnotebooks\nand\n2 pens,\nSimran\nhas\n13\nnotebooks\nand\n5 pens Now this could be arranged in the tabular form as follows:\nNotebooks\nPens\nRadha\n15\n6\nFauzia\n10\n2\nSimran\n13\n5\nand this can be expressed as\nor\nRadha\nFauzia\nSimran\nNotebooks\n15\n10\n13\nPens\n6\n2\n5\nwhich can be expressed as:\nIn the first arrangement the entries in the first column represent the number of\nnote books possessed by Radha, Fauzia and Simran, respectively and the entries in the\nsecond column represent the number of pens possessed by Radha, Fauzia and Simran,\nRationalised 2023-24\n 36\nMATHEMATICS\nrespectively Similarly, in the second arrangement, the entries in the first row represent\nthe number of notebooks possessed by Radha, Fauzia and Simran, respectively"
    },
    {
        "Chapter": "1",
        "sentence_range": "1014-1017",
        "Text": "Let us now suppose that we wish to express the information of possession\nChapter 3\nMATRICES\nRationalised 2023-24\nMATRICES         35\nof notebooks and pens by Radha and her two friends Fauzia and Simran which\nis as follows:\nRadha\nhas\n15\nnotebooks\nand\n6 pens,\nFauzia\nhas\n10\nnotebooks\nand\n2 pens,\nSimran\nhas\n13\nnotebooks\nand\n5 pens Now this could be arranged in the tabular form as follows:\nNotebooks\nPens\nRadha\n15\n6\nFauzia\n10\n2\nSimran\n13\n5\nand this can be expressed as\nor\nRadha\nFauzia\nSimran\nNotebooks\n15\n10\n13\nPens\n6\n2\n5\nwhich can be expressed as:\nIn the first arrangement the entries in the first column represent the number of\nnote books possessed by Radha, Fauzia and Simran, respectively and the entries in the\nsecond column represent the number of pens possessed by Radha, Fauzia and Simran,\nRationalised 2023-24\n 36\nMATHEMATICS\nrespectively Similarly, in the second arrangement, the entries in the first row represent\nthe number of notebooks possessed by Radha, Fauzia and Simran, respectively The\nentries in the second row represent the number of pens possessed by Radha, Fauzia\nand Simran, respectively"
    },
    {
        "Chapter": "1",
        "sentence_range": "1015-1018",
        "Text": "Now this could be arranged in the tabular form as follows:\nNotebooks\nPens\nRadha\n15\n6\nFauzia\n10\n2\nSimran\n13\n5\nand this can be expressed as\nor\nRadha\nFauzia\nSimran\nNotebooks\n15\n10\n13\nPens\n6\n2\n5\nwhich can be expressed as:\nIn the first arrangement the entries in the first column represent the number of\nnote books possessed by Radha, Fauzia and Simran, respectively and the entries in the\nsecond column represent the number of pens possessed by Radha, Fauzia and Simran,\nRationalised 2023-24\n 36\nMATHEMATICS\nrespectively Similarly, in the second arrangement, the entries in the first row represent\nthe number of notebooks possessed by Radha, Fauzia and Simran, respectively The\nentries in the second row represent the number of pens possessed by Radha, Fauzia\nand Simran, respectively An arrangement or display of the above kind is called a\nmatrix"
    },
    {
        "Chapter": "1",
        "sentence_range": "1016-1019",
        "Text": "Similarly, in the second arrangement, the entries in the first row represent\nthe number of notebooks possessed by Radha, Fauzia and Simran, respectively The\nentries in the second row represent the number of pens possessed by Radha, Fauzia\nand Simran, respectively An arrangement or display of the above kind is called a\nmatrix Formally, we define matrix as:\nDefinition 1  A matrix is an ordered rectangular array of numbers or functions"
    },
    {
        "Chapter": "1",
        "sentence_range": "1017-1020",
        "Text": "The\nentries in the second row represent the number of pens possessed by Radha, Fauzia\nand Simran, respectively An arrangement or display of the above kind is called a\nmatrix Formally, we define matrix as:\nDefinition 1  A matrix is an ordered rectangular array of numbers or functions The\nnumbers or functions are called the elements or the entries of the matrix"
    },
    {
        "Chapter": "1",
        "sentence_range": "1018-1021",
        "Text": "An arrangement or display of the above kind is called a\nmatrix Formally, we define matrix as:\nDefinition 1  A matrix is an ordered rectangular array of numbers or functions The\nnumbers or functions are called the elements or the entries of the matrix We denote matrices by capital letters"
    },
    {
        "Chapter": "1",
        "sentence_range": "1019-1022",
        "Text": "Formally, we define matrix as:\nDefinition 1  A matrix is an ordered rectangular array of numbers or functions The\nnumbers or functions are called the elements or the entries of the matrix We denote matrices by capital letters The following are some examples of matrices:\n5\n\u2013 2\nA\n0\n5\n3\n6\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n= \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, \n1\n2\n3\n2\nB\n3"
    },
    {
        "Chapter": "1",
        "sentence_range": "1020-1023",
        "Text": "The\nnumbers or functions are called the elements or the entries of the matrix We denote matrices by capital letters The following are some examples of matrices:\n5\n\u2013 2\nA\n0\n5\n3\n6\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n= \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, \n1\n2\n3\n2\nB\n3 5\n\u20131\n2\n5\n3\n5\n7\ni\n\uf8ee\n\uf8f9\n+\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n= \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n,  \n3\n1\n3\nC\ncos\ntan\nsin\n2\nx\nx\nx\nx\nx\n\uf8ee\n\uf8f9\n= \uf8ef+\n\uf8fa\n+\n\uf8f0\n\uf8fb\nIn the above examples, the horizontal lines of elements are said to constitute, rows\nof the matrix and the vertical lines of elements are said to constitute, columns of the\nmatrix"
    },
    {
        "Chapter": "1",
        "sentence_range": "1021-1024",
        "Text": "We denote matrices by capital letters The following are some examples of matrices:\n5\n\u2013 2\nA\n0\n5\n3\n6\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n= \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, \n1\n2\n3\n2\nB\n3 5\n\u20131\n2\n5\n3\n5\n7\ni\n\uf8ee\n\uf8f9\n+\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n= \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n,  \n3\n1\n3\nC\ncos\ntan\nsin\n2\nx\nx\nx\nx\nx\n\uf8ee\n\uf8f9\n= \uf8ef+\n\uf8fa\n+\n\uf8f0\n\uf8fb\nIn the above examples, the horizontal lines of elements are said to constitute, rows\nof the matrix and the vertical lines of elements are said to constitute, columns of the\nmatrix Thus A has 3 rows and 2 columns, B has 3 rows and 3 columns while C has 2\nrows and 3 columns"
    },
    {
        "Chapter": "1",
        "sentence_range": "1022-1025",
        "Text": "The following are some examples of matrices:\n5\n\u2013 2\nA\n0\n5\n3\n6\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n= \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, \n1\n2\n3\n2\nB\n3 5\n\u20131\n2\n5\n3\n5\n7\ni\n\uf8ee\n\uf8f9\n+\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n= \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n,  \n3\n1\n3\nC\ncos\ntan\nsin\n2\nx\nx\nx\nx\nx\n\uf8ee\n\uf8f9\n= \uf8ef+\n\uf8fa\n+\n\uf8f0\n\uf8fb\nIn the above examples, the horizontal lines of elements are said to constitute, rows\nof the matrix and the vertical lines of elements are said to constitute, columns of the\nmatrix Thus A has 3 rows and 2 columns, B has 3 rows and 3 columns while C has 2\nrows and 3 columns 3"
    },
    {
        "Chapter": "1",
        "sentence_range": "1023-1026",
        "Text": "5\n\u20131\n2\n5\n3\n5\n7\ni\n\uf8ee\n\uf8f9\n+\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n= \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n,  \n3\n1\n3\nC\ncos\ntan\nsin\n2\nx\nx\nx\nx\nx\n\uf8ee\n\uf8f9\n= \uf8ef+\n\uf8fa\n+\n\uf8f0\n\uf8fb\nIn the above examples, the horizontal lines of elements are said to constitute, rows\nof the matrix and the vertical lines of elements are said to constitute, columns of the\nmatrix Thus A has 3 rows and 2 columns, B has 3 rows and 3 columns while C has 2\nrows and 3 columns 3 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "1024-1027",
        "Text": "Thus A has 3 rows and 2 columns, B has 3 rows and 3 columns while C has 2\nrows and 3 columns 3 2 1  Order of a matrix\nA matrix having m rows and n columns is called a matrix of order m \u00d7 n or simply m \u00d7 n\nmatrix (read as an m by n matrix)"
    },
    {
        "Chapter": "1",
        "sentence_range": "1025-1028",
        "Text": "3 2 1  Order of a matrix\nA matrix having m rows and n columns is called a matrix of order m \u00d7 n or simply m \u00d7 n\nmatrix (read as an m by n matrix) So referring to the above examples of matrices, we\nhave A as 3 \u00d7 2 matrix, B as 3 \u00d7 3 matrix and C as 2 \u00d7 3 matrix"
    },
    {
        "Chapter": "1",
        "sentence_range": "1026-1029",
        "Text": "2 1  Order of a matrix\nA matrix having m rows and n columns is called a matrix of order m \u00d7 n or simply m \u00d7 n\nmatrix (read as an m by n matrix) So referring to the above examples of matrices, we\nhave A as 3 \u00d7 2 matrix, B as 3 \u00d7 3 matrix and C as 2 \u00d7 3 matrix We observe that A has\n3 \u00d7 2 = 6 elements, B and C have 9 and 6 elements, respectively"
    },
    {
        "Chapter": "1",
        "sentence_range": "1027-1030",
        "Text": "1  Order of a matrix\nA matrix having m rows and n columns is called a matrix of order m \u00d7 n or simply m \u00d7 n\nmatrix (read as an m by n matrix) So referring to the above examples of matrices, we\nhave A as 3 \u00d7 2 matrix, B as 3 \u00d7 3 matrix and C as 2 \u00d7 3 matrix We observe that A has\n3 \u00d7 2 = 6 elements, B and C have 9 and 6 elements, respectively In general, an m \u00d7 n matrix has the following rectangular array:\nor\nA = [aij]m \u00d7 n, 1\u2264 i \u2264 m, 1\u2264 j \u2264 n   i, j \u2208 N\nThus the ith  row consists of the elements ai1, ai2, ai3,"
    },
    {
        "Chapter": "1",
        "sentence_range": "1028-1031",
        "Text": "So referring to the above examples of matrices, we\nhave A as 3 \u00d7 2 matrix, B as 3 \u00d7 3 matrix and C as 2 \u00d7 3 matrix We observe that A has\n3 \u00d7 2 = 6 elements, B and C have 9 and 6 elements, respectively In general, an m \u00d7 n matrix has the following rectangular array:\nor\nA = [aij]m \u00d7 n, 1\u2264 i \u2264 m, 1\u2264 j \u2264 n   i, j \u2208 N\nThus the ith  row consists of the elements ai1, ai2, ai3, , ain, while the jth column\nconsists of the elements a1j, a2j, a3j,"
    },
    {
        "Chapter": "1",
        "sentence_range": "1029-1032",
        "Text": "We observe that A has\n3 \u00d7 2 = 6 elements, B and C have 9 and 6 elements, respectively In general, an m \u00d7 n matrix has the following rectangular array:\nor\nA = [aij]m \u00d7 n, 1\u2264 i \u2264 m, 1\u2264 j \u2264 n   i, j \u2208 N\nThus the ith  row consists of the elements ai1, ai2, ai3, , ain, while the jth column\nconsists of the elements a1j, a2j, a3j, , amj,\nIn general aij, is an element lying in the ith row and jth column"
    },
    {
        "Chapter": "1",
        "sentence_range": "1030-1033",
        "Text": "In general, an m \u00d7 n matrix has the following rectangular array:\nor\nA = [aij]m \u00d7 n, 1\u2264 i \u2264 m, 1\u2264 j \u2264 n   i, j \u2208 N\nThus the ith  row consists of the elements ai1, ai2, ai3, , ain, while the jth column\nconsists of the elements a1j, a2j, a3j, , amj,\nIn general aij, is an element lying in the ith row and jth column We can also call\nit as the (i, j)th element of A"
    },
    {
        "Chapter": "1",
        "sentence_range": "1031-1034",
        "Text": ", ain, while the jth column\nconsists of the elements a1j, a2j, a3j, , amj,\nIn general aij, is an element lying in the ith row and jth column We can also call\nit as the (i, j)th element of A The number of elements in an m \u00d7 n matrix will be\nequal to mn"
    },
    {
        "Chapter": "1",
        "sentence_range": "1032-1035",
        "Text": ", amj,\nIn general aij, is an element lying in the ith row and jth column We can also call\nit as the (i, j)th element of A The number of elements in an m \u00d7 n matrix will be\nequal to mn Rationalised 2023-24\nMATRICES         37\nANote  In this chapter\n1"
    },
    {
        "Chapter": "1",
        "sentence_range": "1033-1036",
        "Text": "We can also call\nit as the (i, j)th element of A The number of elements in an m \u00d7 n matrix will be\nequal to mn Rationalised 2023-24\nMATRICES         37\nANote  In this chapter\n1 We shall follow the notation, namely A = [aij]m \u00d7 n to indicate that A is a matrix\nof order m \u00d7 n"
    },
    {
        "Chapter": "1",
        "sentence_range": "1034-1037",
        "Text": "The number of elements in an m \u00d7 n matrix will be\nequal to mn Rationalised 2023-24\nMATRICES         37\nANote  In this chapter\n1 We shall follow the notation, namely A = [aij]m \u00d7 n to indicate that A is a matrix\nof order m \u00d7 n 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "1035-1038",
        "Text": "Rationalised 2023-24\nMATRICES         37\nANote  In this chapter\n1 We shall follow the notation, namely A = [aij]m \u00d7 n to indicate that A is a matrix\nof order m \u00d7 n 2 We shall consider only those matrices whose elements are real numbers or\nfunctions taking real values"
    },
    {
        "Chapter": "1",
        "sentence_range": "1036-1039",
        "Text": "We shall follow the notation, namely A = [aij]m \u00d7 n to indicate that A is a matrix\nof order m \u00d7 n 2 We shall consider only those matrices whose elements are real numbers or\nfunctions taking real values We can also represent any point (x, y) in a plane by a matrix (column or row) as\nx\ny\n\uf8ee \uf8f9\n\uf8ef \uf8fa\n\uf8f0 \uf8fb  (or [x, y])"
    },
    {
        "Chapter": "1",
        "sentence_range": "1037-1040",
        "Text": "2 We shall consider only those matrices whose elements are real numbers or\nfunctions taking real values We can also represent any point (x, y) in a plane by a matrix (column or row) as\nx\ny\n\uf8ee \uf8f9\n\uf8ef \uf8fa\n\uf8f0 \uf8fb  (or [x, y]) For example point P(0, 1) as a matrix representation may be given as\n0\nP\n1\n\uf8ee \uf8f9\n= \uf8ef \uf8fa\n\uf8f0 \uf8fb\n or [0 1]"
    },
    {
        "Chapter": "1",
        "sentence_range": "1038-1041",
        "Text": "We shall consider only those matrices whose elements are real numbers or\nfunctions taking real values We can also represent any point (x, y) in a plane by a matrix (column or row) as\nx\ny\n\uf8ee \uf8f9\n\uf8ef \uf8fa\n\uf8f0 \uf8fb  (or [x, y]) For example point P(0, 1) as a matrix representation may be given as\n0\nP\n1\n\uf8ee \uf8f9\n= \uf8ef \uf8fa\n\uf8f0 \uf8fb\n or [0 1] Observe that in this way we can also express the vertices of a closed rectilinear\nfigure in the form of a matrix"
    },
    {
        "Chapter": "1",
        "sentence_range": "1039-1042",
        "Text": "We can also represent any point (x, y) in a plane by a matrix (column or row) as\nx\ny\n\uf8ee \uf8f9\n\uf8ef \uf8fa\n\uf8f0 \uf8fb  (or [x, y]) For example point P(0, 1) as a matrix representation may be given as\n0\nP\n1\n\uf8ee \uf8f9\n= \uf8ef \uf8fa\n\uf8f0 \uf8fb\n or [0 1] Observe that in this way we can also express the vertices of a closed rectilinear\nfigure in the form of a matrix For example, consider a quadrilateral ABCD with vertices\nA (1, 0), B (3, 2), C (1, 3), D (\u20131, 2)"
    },
    {
        "Chapter": "1",
        "sentence_range": "1040-1043",
        "Text": "For example point P(0, 1) as a matrix representation may be given as\n0\nP\n1\n\uf8ee \uf8f9\n= \uf8ef \uf8fa\n\uf8f0 \uf8fb\n or [0 1] Observe that in this way we can also express the vertices of a closed rectilinear\nfigure in the form of a matrix For example, consider a quadrilateral ABCD with vertices\nA (1, 0), B (3, 2), C (1, 3), D (\u20131, 2) Now, quadrilateral ABCD in the matrix form, can be represented as\n2\n4\nA\nB\nC D\n1\n3\n1\n1\nX\n0 2\n3\n2\n\u00d7\n\u2212\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nor\n4 2\nA 1\n0\nB\n3\n2\nY\nC\n1\n3\nD\n1\n2\n\u00d7\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\uf8f0\n\uf8fb\nThus, matrices can be used as representation of vertices of geometrical figures in\na plane"
    },
    {
        "Chapter": "1",
        "sentence_range": "1041-1044",
        "Text": "Observe that in this way we can also express the vertices of a closed rectilinear\nfigure in the form of a matrix For example, consider a quadrilateral ABCD with vertices\nA (1, 0), B (3, 2), C (1, 3), D (\u20131, 2) Now, quadrilateral ABCD in the matrix form, can be represented as\n2\n4\nA\nB\nC D\n1\n3\n1\n1\nX\n0 2\n3\n2\n\u00d7\n\u2212\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nor\n4 2\nA 1\n0\nB\n3\n2\nY\nC\n1\n3\nD\n1\n2\n\u00d7\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\uf8f0\n\uf8fb\nThus, matrices can be used as representation of vertices of geometrical figures in\na plane Now, let us consider some examples"
    },
    {
        "Chapter": "1",
        "sentence_range": "1042-1045",
        "Text": "For example, consider a quadrilateral ABCD with vertices\nA (1, 0), B (3, 2), C (1, 3), D (\u20131, 2) Now, quadrilateral ABCD in the matrix form, can be represented as\n2\n4\nA\nB\nC D\n1\n3\n1\n1\nX\n0 2\n3\n2\n\u00d7\n\u2212\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nor\n4 2\nA 1\n0\nB\n3\n2\nY\nC\n1\n3\nD\n1\n2\n\u00d7\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\uf8f0\n\uf8fb\nThus, matrices can be used as representation of vertices of geometrical figures in\na plane Now, let us consider some examples Example 1 Consider the following information regarding the number of men and women\nworkers in three factories I, II and III\nMen workers\nWomen workers\nI\n30\n25\nII\n25\n31\nIII\n27\n26\nRepresent the above information in the form of a 3 \u00d7 2 matrix"
    },
    {
        "Chapter": "1",
        "sentence_range": "1043-1046",
        "Text": "Now, quadrilateral ABCD in the matrix form, can be represented as\n2\n4\nA\nB\nC D\n1\n3\n1\n1\nX\n0 2\n3\n2\n\u00d7\n\u2212\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nor\n4 2\nA 1\n0\nB\n3\n2\nY\nC\n1\n3\nD\n1\n2\n\u00d7\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\uf8f0\n\uf8fb\nThus, matrices can be used as representation of vertices of geometrical figures in\na plane Now, let us consider some examples Example 1 Consider the following information regarding the number of men and women\nworkers in three factories I, II and III\nMen workers\nWomen workers\nI\n30\n25\nII\n25\n31\nIII\n27\n26\nRepresent the above information in the form of a 3 \u00d7 2 matrix What does the entry\nin the third row and second column represent"
    },
    {
        "Chapter": "1",
        "sentence_range": "1044-1047",
        "Text": "Now, let us consider some examples Example 1 Consider the following information regarding the number of men and women\nworkers in three factories I, II and III\nMen workers\nWomen workers\nI\n30\n25\nII\n25\n31\nIII\n27\n26\nRepresent the above information in the form of a 3 \u00d7 2 matrix What does the entry\nin the third row and second column represent Rationalised 2023-24\n 38\nMATHEMATICS\nSolution The information is represented in the form of a 3 \u00d7 2 matrix as follows:\n30\n25\nA\n25\n31\n27\n26\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n= \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nThe entry in the third row and second column represents the number of women\nworkers in factory III"
    },
    {
        "Chapter": "1",
        "sentence_range": "1045-1048",
        "Text": "Example 1 Consider the following information regarding the number of men and women\nworkers in three factories I, II and III\nMen workers\nWomen workers\nI\n30\n25\nII\n25\n31\nIII\n27\n26\nRepresent the above information in the form of a 3 \u00d7 2 matrix What does the entry\nin the third row and second column represent Rationalised 2023-24\n 38\nMATHEMATICS\nSolution The information is represented in the form of a 3 \u00d7 2 matrix as follows:\n30\n25\nA\n25\n31\n27\n26\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n= \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nThe entry in the third row and second column represents the number of women\nworkers in factory III Example 2 If a matrix has 8 elements, what are the possible orders it can have"
    },
    {
        "Chapter": "1",
        "sentence_range": "1046-1049",
        "Text": "What does the entry\nin the third row and second column represent Rationalised 2023-24\n 38\nMATHEMATICS\nSolution The information is represented in the form of a 3 \u00d7 2 matrix as follows:\n30\n25\nA\n25\n31\n27\n26\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n= \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nThe entry in the third row and second column represents the number of women\nworkers in factory III Example 2 If a matrix has 8 elements, what are the possible orders it can have Solution We know that if a matrix is of order m \u00d7 n, it has mn elements"
    },
    {
        "Chapter": "1",
        "sentence_range": "1047-1050",
        "Text": "Rationalised 2023-24\n 38\nMATHEMATICS\nSolution The information is represented in the form of a 3 \u00d7 2 matrix as follows:\n30\n25\nA\n25\n31\n27\n26\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n= \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nThe entry in the third row and second column represents the number of women\nworkers in factory III Example 2 If a matrix has 8 elements, what are the possible orders it can have Solution We know that if a matrix is of order m \u00d7 n, it has mn elements Thus, to find\nall possible orders of a matrix with 8 elements, we will find all ordered pairs of natural\nnumbers, whose product is 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "1048-1051",
        "Text": "Example 2 If a matrix has 8 elements, what are the possible orders it can have Solution We know that if a matrix is of order m \u00d7 n, it has mn elements Thus, to find\nall possible orders of a matrix with 8 elements, we will find all ordered pairs of natural\nnumbers, whose product is 8 Thus, all possible ordered pairs are (1, 8), (8, 1), (4, 2), (2, 4)\nHence, possible orders are 1 \u00d7 8, 8 \u00d71, 4 \u00d7 2, 2 \u00d7 4\nExample 3 Construct a 3 \u00d7 2 matrix whose elements are given by \n1 |\n3 |\n2\naij\ni\nj\n=\n\u2212"
    },
    {
        "Chapter": "1",
        "sentence_range": "1049-1052",
        "Text": "Solution We know that if a matrix is of order m \u00d7 n, it has mn elements Thus, to find\nall possible orders of a matrix with 8 elements, we will find all ordered pairs of natural\nnumbers, whose product is 8 Thus, all possible ordered pairs are (1, 8), (8, 1), (4, 2), (2, 4)\nHence, possible orders are 1 \u00d7 8, 8 \u00d71, 4 \u00d7 2, 2 \u00d7 4\nExample 3 Construct a 3 \u00d7 2 matrix whose elements are given by \n1 |\n3 |\n2\naij\ni\nj\n=\n\u2212 Solution In general a 3 \u00d7 2 matrix is given by \n11\n12\n21\n22\n31\n32\nA\na\na\na\na\na\na\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n= \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb"
    },
    {
        "Chapter": "1",
        "sentence_range": "1050-1053",
        "Text": "Thus, to find\nall possible orders of a matrix with 8 elements, we will find all ordered pairs of natural\nnumbers, whose product is 8 Thus, all possible ordered pairs are (1, 8), (8, 1), (4, 2), (2, 4)\nHence, possible orders are 1 \u00d7 8, 8 \u00d71, 4 \u00d7 2, 2 \u00d7 4\nExample 3 Construct a 3 \u00d7 2 matrix whose elements are given by \n1 |\n3 |\n2\naij\ni\nj\n=\n\u2212 Solution In general a 3 \u00d7 2 matrix is given by \n11\n12\n21\n22\n31\n32\nA\na\na\na\na\na\na\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n= \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb Now\n1 |\n3 |\n2\naij\ni\nj\n=\n\u2212\n, i = 1, 2, 3 and  j = 1, 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "1051-1054",
        "Text": "Thus, all possible ordered pairs are (1, 8), (8, 1), (4, 2), (2, 4)\nHence, possible orders are 1 \u00d7 8, 8 \u00d71, 4 \u00d7 2, 2 \u00d7 4\nExample 3 Construct a 3 \u00d7 2 matrix whose elements are given by \n1 |\n3 |\n2\naij\ni\nj\n=\n\u2212 Solution In general a 3 \u00d7 2 matrix is given by \n11\n12\n21\n22\n31\n32\nA\na\na\na\na\na\na\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n= \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb Now\n1 |\n3 |\n2\naij\ni\nj\n=\n\u2212\n, i = 1, 2, 3 and  j = 1, 2 Therefore\n11\n1 |1 3 1|\n1\n2\na\n=\n\u2212 \u00d7\n=\n12\n1\n5\n2|1 3 2|\n2\na\n=\n\u2212 \u00d7\n=\n21\n1\n1\n| 2\n3 1|\n2\n2\na\n=\n\u2212 \u00d7\n=\n22\n1 | 2\n3 2|\n2\n2\na\n=\n\u2212 \u00d7\n=\n31\n1 |3\n3 1|\n0\n2\na\n=\n\u2212 \u00d7\n=\n32\n1\n3\n|3\n3 2 |\n2\n2\na\n=\n\u2212 \u00d7\n=\nHence the required matrix is given by \n5\n1\n2\n1\nA\n2\n2\n3\n0\n2\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n= \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb"
    },
    {
        "Chapter": "1",
        "sentence_range": "1052-1055",
        "Text": "Solution In general a 3 \u00d7 2 matrix is given by \n11\n12\n21\n22\n31\n32\nA\na\na\na\na\na\na\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n= \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb Now\n1 |\n3 |\n2\naij\ni\nj\n=\n\u2212\n, i = 1, 2, 3 and  j = 1, 2 Therefore\n11\n1 |1 3 1|\n1\n2\na\n=\n\u2212 \u00d7\n=\n12\n1\n5\n2|1 3 2|\n2\na\n=\n\u2212 \u00d7\n=\n21\n1\n1\n| 2\n3 1|\n2\n2\na\n=\n\u2212 \u00d7\n=\n22\n1 | 2\n3 2|\n2\n2\na\n=\n\u2212 \u00d7\n=\n31\n1 |3\n3 1|\n0\n2\na\n=\n\u2212 \u00d7\n=\n32\n1\n3\n|3\n3 2 |\n2\n2\na\n=\n\u2212 \u00d7\n=\nHence the required matrix is given by \n5\n1\n2\n1\nA\n2\n2\n3\n0\n2\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n= \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb Rationalised 2023-24\nMATRICES         39\n3"
    },
    {
        "Chapter": "1",
        "sentence_range": "1053-1056",
        "Text": "Now\n1 |\n3 |\n2\naij\ni\nj\n=\n\u2212\n, i = 1, 2, 3 and  j = 1, 2 Therefore\n11\n1 |1 3 1|\n1\n2\na\n=\n\u2212 \u00d7\n=\n12\n1\n5\n2|1 3 2|\n2\na\n=\n\u2212 \u00d7\n=\n21\n1\n1\n| 2\n3 1|\n2\n2\na\n=\n\u2212 \u00d7\n=\n22\n1 | 2\n3 2|\n2\n2\na\n=\n\u2212 \u00d7\n=\n31\n1 |3\n3 1|\n0\n2\na\n=\n\u2212 \u00d7\n=\n32\n1\n3\n|3\n3 2 |\n2\n2\na\n=\n\u2212 \u00d7\n=\nHence the required matrix is given by \n5\n1\n2\n1\nA\n2\n2\n3\n0\n2\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n= \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb Rationalised 2023-24\nMATRICES         39\n3 3  Types of Matrices\n In this section, we shall discuss different types of matrices"
    },
    {
        "Chapter": "1",
        "sentence_range": "1054-1057",
        "Text": "Therefore\n11\n1 |1 3 1|\n1\n2\na\n=\n\u2212 \u00d7\n=\n12\n1\n5\n2|1 3 2|\n2\na\n=\n\u2212 \u00d7\n=\n21\n1\n1\n| 2\n3 1|\n2\n2\na\n=\n\u2212 \u00d7\n=\n22\n1 | 2\n3 2|\n2\n2\na\n=\n\u2212 \u00d7\n=\n31\n1 |3\n3 1|\n0\n2\na\n=\n\u2212 \u00d7\n=\n32\n1\n3\n|3\n3 2 |\n2\n2\na\n=\n\u2212 \u00d7\n=\nHence the required matrix is given by \n5\n1\n2\n1\nA\n2\n2\n3\n0\n2\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n= \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb Rationalised 2023-24\nMATRICES         39\n3 3  Types of Matrices\n In this section, we shall discuss different types of matrices (i)\nColumn matrix\nA matrix is said to be a column matrix if it has only one column"
    },
    {
        "Chapter": "1",
        "sentence_range": "1055-1058",
        "Text": "Rationalised 2023-24\nMATRICES         39\n3 3  Types of Matrices\n In this section, we shall discuss different types of matrices (i)\nColumn matrix\nA matrix is said to be a column matrix if it has only one column For example,  \n0\n3\nA\n1\n1/ 2\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n=\n\uf8ef\u2212\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n is a column matrix of order 4 \u00d7 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "1056-1059",
        "Text": "3  Types of Matrices\n In this section, we shall discuss different types of matrices (i)\nColumn matrix\nA matrix is said to be a column matrix if it has only one column For example,  \n0\n3\nA\n1\n1/ 2\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n=\n\uf8ef\u2212\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n is a column matrix of order 4 \u00d7 1 In general, A = [aij] m \u00d7 1 is a column matrix of order m \u00d7 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "1057-1060",
        "Text": "(i)\nColumn matrix\nA matrix is said to be a column matrix if it has only one column For example,  \n0\n3\nA\n1\n1/ 2\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n=\n\uf8ef\u2212\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n is a column matrix of order 4 \u00d7 1 In general, A = [aij] m \u00d7 1 is a column matrix of order m \u00d7 1 (ii)\nRow matrix\nA matrix is said to be a row matrix if it has only one row"
    },
    {
        "Chapter": "1",
        "sentence_range": "1058-1061",
        "Text": "For example,  \n0\n3\nA\n1\n1/ 2\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n=\n\uf8ef\u2212\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n is a column matrix of order 4 \u00d7 1 In general, A = [aij] m \u00d7 1 is a column matrix of order m \u00d7 1 (ii)\nRow matrix\nA matrix is said to be a row matrix if it has only one row For example, \n1 4\n1\nB\n5 2 3\n2\n\u00d7\n\uf8ee\n\uf8f9\n= \u2212\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n is a row matrix"
    },
    {
        "Chapter": "1",
        "sentence_range": "1059-1062",
        "Text": "In general, A = [aij] m \u00d7 1 is a column matrix of order m \u00d7 1 (ii)\nRow matrix\nA matrix is said to be a row matrix if it has only one row For example, \n1 4\n1\nB\n5 2 3\n2\n\u00d7\n\uf8ee\n\uf8f9\n= \u2212\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n is a row matrix In general, B = [bij] 1 \u00d7 n is a row matrix of order 1 \u00d7 n"
    },
    {
        "Chapter": "1",
        "sentence_range": "1060-1063",
        "Text": "(ii)\nRow matrix\nA matrix is said to be a row matrix if it has only one row For example, \n1 4\n1\nB\n5 2 3\n2\n\u00d7\n\uf8ee\n\uf8f9\n= \u2212\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n is a row matrix In general, B = [bij] 1 \u00d7 n is a row matrix of order 1 \u00d7 n (iii)\nSquare matrix\nA matrix in which the number of rows are equal to the number of columns, is\nsaid to be a square matrix"
    },
    {
        "Chapter": "1",
        "sentence_range": "1061-1064",
        "Text": "For example, \n1 4\n1\nB\n5 2 3\n2\n\u00d7\n\uf8ee\n\uf8f9\n= \u2212\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n is a row matrix In general, B = [bij] 1 \u00d7 n is a row matrix of order 1 \u00d7 n (iii)\nSquare matrix\nA matrix in which the number of rows are equal to the number of columns, is\nsaid to be a square matrix Thus an m \u00d7 n matrix is said to be a square matrix if\nm = n and is known as a square matrix of order \u2018n\u2019"
    },
    {
        "Chapter": "1",
        "sentence_range": "1062-1065",
        "Text": "In general, B = [bij] 1 \u00d7 n is a row matrix of order 1 \u00d7 n (iii)\nSquare matrix\nA matrix in which the number of rows are equal to the number of columns, is\nsaid to be a square matrix Thus an m \u00d7 n matrix is said to be a square matrix if\nm = n and is known as a square matrix of order \u2018n\u2019 For example \n3\n1\n0\n3\nA\n3 2\n1\n42\n3\n1\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n= \uf8ef\n\uf8fa\n\uf8ef\n\u2212\uf8fa\n\uf8f0\n\uf8fb\n is a square matrix of order 3"
    },
    {
        "Chapter": "1",
        "sentence_range": "1063-1066",
        "Text": "(iii)\nSquare matrix\nA matrix in which the number of rows are equal to the number of columns, is\nsaid to be a square matrix Thus an m \u00d7 n matrix is said to be a square matrix if\nm = n and is known as a square matrix of order \u2018n\u2019 For example \n3\n1\n0\n3\nA\n3 2\n1\n42\n3\n1\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n= \uf8ef\n\uf8fa\n\uf8ef\n\u2212\uf8fa\n\uf8f0\n\uf8fb\n is a square matrix of order 3 In general, A = [aij] m \u00d7 m is a square matrix of order m"
    },
    {
        "Chapter": "1",
        "sentence_range": "1064-1067",
        "Text": "Thus an m \u00d7 n matrix is said to be a square matrix if\nm = n and is known as a square matrix of order \u2018n\u2019 For example \n3\n1\n0\n3\nA\n3 2\n1\n42\n3\n1\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n= \uf8ef\n\uf8fa\n\uf8ef\n\u2212\uf8fa\n\uf8f0\n\uf8fb\n is a square matrix of order 3 In general, A = [aij] m \u00d7 m is a square matrix of order m ANote  If A = [aij] is a square matrix of order n, then elements (entries) a11, a22,"
    },
    {
        "Chapter": "1",
        "sentence_range": "1065-1068",
        "Text": "For example \n3\n1\n0\n3\nA\n3 2\n1\n42\n3\n1\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n= \uf8ef\n\uf8fa\n\uf8ef\n\u2212\uf8fa\n\uf8f0\n\uf8fb\n is a square matrix of order 3 In general, A = [aij] m \u00d7 m is a square matrix of order m ANote  If A = [aij] is a square matrix of order n, then elements (entries) a11, a22, , ann\nare said to constitute the diagonal, of the matrix A"
    },
    {
        "Chapter": "1",
        "sentence_range": "1066-1069",
        "Text": "In general, A = [aij] m \u00d7 m is a square matrix of order m ANote  If A = [aij] is a square matrix of order n, then elements (entries) a11, a22, , ann\nare said to constitute the diagonal, of the matrix A Thus, if \n1\n3\n1\nA\n2\n4\n1\n3\n5\n6\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n=\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb"
    },
    {
        "Chapter": "1",
        "sentence_range": "1067-1070",
        "Text": "ANote  If A = [aij] is a square matrix of order n, then elements (entries) a11, a22, , ann\nare said to constitute the diagonal, of the matrix A Thus, if \n1\n3\n1\nA\n2\n4\n1\n3\n5\n6\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n=\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb Then the elements of the diagonal of A are 1, 4, 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "1068-1071",
        "Text": ", ann\nare said to constitute the diagonal, of the matrix A Thus, if \n1\n3\n1\nA\n2\n4\n1\n3\n5\n6\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n=\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb Then the elements of the diagonal of A are 1, 4, 6 Rationalised 2023-24\n 40\nMATHEMATICS\n(iv)\nDiagonal matrix\nA square matrix B = [bij] m \u00d7 m is said to be a diagonal matrix if all its non\ndiagonal elements are zero, that is a matrix B = [bij] m \u00d7 m is said to be a diagonal\nmatrix if bij = 0, when i \u2260 j"
    },
    {
        "Chapter": "1",
        "sentence_range": "1069-1072",
        "Text": "Thus, if \n1\n3\n1\nA\n2\n4\n1\n3\n5\n6\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n=\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb Then the elements of the diagonal of A are 1, 4, 6 Rationalised 2023-24\n 40\nMATHEMATICS\n(iv)\nDiagonal matrix\nA square matrix B = [bij] m \u00d7 m is said to be a diagonal matrix if all its non\ndiagonal elements are zero, that is a matrix B = [bij] m \u00d7 m is said to be a diagonal\nmatrix if bij = 0, when i \u2260 j For example,  A = [4], \n1\n0\nB\n0\n2\n\uf8ee\u2212\n\uf8f9\n= \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, \n1"
    },
    {
        "Chapter": "1",
        "sentence_range": "1070-1073",
        "Text": "Then the elements of the diagonal of A are 1, 4, 6 Rationalised 2023-24\n 40\nMATHEMATICS\n(iv)\nDiagonal matrix\nA square matrix B = [bij] m \u00d7 m is said to be a diagonal matrix if all its non\ndiagonal elements are zero, that is a matrix B = [bij] m \u00d7 m is said to be a diagonal\nmatrix if bij = 0, when i \u2260 j For example,  A = [4], \n1\n0\nB\n0\n2\n\uf8ee\u2212\n\uf8f9\n= \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, \n1 1\n0\n0\nC\n0\n2\n0\n0\n0\n3\n\uf8ee\u2212\n\uf8f9\n\uf8ef\n\uf8fa\n= \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, are diagonal matrices\nof order 1, 2, 3, respectively"
    },
    {
        "Chapter": "1",
        "sentence_range": "1071-1074",
        "Text": "Rationalised 2023-24\n 40\nMATHEMATICS\n(iv)\nDiagonal matrix\nA square matrix B = [bij] m \u00d7 m is said to be a diagonal matrix if all its non\ndiagonal elements are zero, that is a matrix B = [bij] m \u00d7 m is said to be a diagonal\nmatrix if bij = 0, when i \u2260 j For example,  A = [4], \n1\n0\nB\n0\n2\n\uf8ee\u2212\n\uf8f9\n= \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, \n1 1\n0\n0\nC\n0\n2\n0\n0\n0\n3\n\uf8ee\u2212\n\uf8f9\n\uf8ef\n\uf8fa\n= \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, are diagonal matrices\nof order 1, 2, 3, respectively (v)\nScalar matrix\nA diagonal matrix is said to be a scalar matrix if its diagonal elements are equal,\nthat is, a square matrix B = [bij] n \u00d7 n is said to be a scalar matrix if\nbij = 0,    when i  \u2260 j\nbij = k,    when i  = j, for some constant k"
    },
    {
        "Chapter": "1",
        "sentence_range": "1072-1075",
        "Text": "For example,  A = [4], \n1\n0\nB\n0\n2\n\uf8ee\u2212\n\uf8f9\n= \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, \n1 1\n0\n0\nC\n0\n2\n0\n0\n0\n3\n\uf8ee\u2212\n\uf8f9\n\uf8ef\n\uf8fa\n= \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, are diagonal matrices\nof order 1, 2, 3, respectively (v)\nScalar matrix\nA diagonal matrix is said to be a scalar matrix if its diagonal elements are equal,\nthat is, a square matrix B = [bij] n \u00d7 n is said to be a scalar matrix if\nbij = 0,    when i  \u2260 j\nbij = k,    when i  = j, for some constant k For example\nA = [3],     \n1\n0\nB\n0\n1\n\uf8ee\u2212\n\uf8f9\n= \uf8ef\n\u2212\uf8fa\n\uf8f0\n\uf8fb\n,      \n3\n0\n0\nC\n0\n3\n0\n0\n0\n3\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n= \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nare scalar matrices of order 1, 2 and 3, respectively"
    },
    {
        "Chapter": "1",
        "sentence_range": "1073-1076",
        "Text": "1\n0\n0\nC\n0\n2\n0\n0\n0\n3\n\uf8ee\u2212\n\uf8f9\n\uf8ef\n\uf8fa\n= \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, are diagonal matrices\nof order 1, 2, 3, respectively (v)\nScalar matrix\nA diagonal matrix is said to be a scalar matrix if its diagonal elements are equal,\nthat is, a square matrix B = [bij] n \u00d7 n is said to be a scalar matrix if\nbij = 0,    when i  \u2260 j\nbij = k,    when i  = j, for some constant k For example\nA = [3],     \n1\n0\nB\n0\n1\n\uf8ee\u2212\n\uf8f9\n= \uf8ef\n\u2212\uf8fa\n\uf8f0\n\uf8fb\n,      \n3\n0\n0\nC\n0\n3\n0\n0\n0\n3\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n= \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nare scalar matrices of order 1, 2 and 3, respectively (vi)\nIdentity matrix\nA square matrix in which elements in the diagonal are all 1 and rest are all zero\nis called an identity matrix"
    },
    {
        "Chapter": "1",
        "sentence_range": "1074-1077",
        "Text": "(v)\nScalar matrix\nA diagonal matrix is said to be a scalar matrix if its diagonal elements are equal,\nthat is, a square matrix B = [bij] n \u00d7 n is said to be a scalar matrix if\nbij = 0,    when i  \u2260 j\nbij = k,    when i  = j, for some constant k For example\nA = [3],     \n1\n0\nB\n0\n1\n\uf8ee\u2212\n\uf8f9\n= \uf8ef\n\u2212\uf8fa\n\uf8f0\n\uf8fb\n,      \n3\n0\n0\nC\n0\n3\n0\n0\n0\n3\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n= \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nare scalar matrices of order 1, 2 and 3, respectively (vi)\nIdentity matrix\nA square matrix in which elements in the diagonal are all 1 and rest are all zero\nis called an identity matrix In other words, the square matrix A = [aij] n \u00d7 n is an\nidentity matrix, if \n1\n0if\nif\nij\ni\nj\na\ni\nj\n=\n= \uf8f2\uf8f1\n\u2260\n\uf8f3"
    },
    {
        "Chapter": "1",
        "sentence_range": "1075-1078",
        "Text": "For example\nA = [3],     \n1\n0\nB\n0\n1\n\uf8ee\u2212\n\uf8f9\n= \uf8ef\n\u2212\uf8fa\n\uf8f0\n\uf8fb\n,      \n3\n0\n0\nC\n0\n3\n0\n0\n0\n3\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n= \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nare scalar matrices of order 1, 2 and 3, respectively (vi)\nIdentity matrix\nA square matrix in which elements in the diagonal are all 1 and rest are all zero\nis called an identity matrix In other words, the square matrix A = [aij] n \u00d7 n is an\nidentity matrix, if \n1\n0if\nif\nij\ni\nj\na\ni\nj\n=\n= \uf8f2\uf8f1\n\u2260\n\uf8f3 We denote the identity matrix of order n by In"
    },
    {
        "Chapter": "1",
        "sentence_range": "1076-1079",
        "Text": "(vi)\nIdentity matrix\nA square matrix in which elements in the diagonal are all 1 and rest are all zero\nis called an identity matrix In other words, the square matrix A = [aij] n \u00d7 n is an\nidentity matrix, if \n1\n0if\nif\nij\ni\nj\na\ni\nj\n=\n= \uf8f2\uf8f1\n\u2260\n\uf8f3 We denote the identity matrix of order n by In When order is clear from the\ncontext, we simply write it as I"
    },
    {
        "Chapter": "1",
        "sentence_range": "1077-1080",
        "Text": "In other words, the square matrix A = [aij] n \u00d7 n is an\nidentity matrix, if \n1\n0if\nif\nij\ni\nj\na\ni\nj\n=\n= \uf8f2\uf8f1\n\u2260\n\uf8f3 We denote the identity matrix of order n by In When order is clear from the\ncontext, we simply write it as I For example [1], \n1\n0\n0\n1\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb , \n1\n0\n0\n0\n1\n0\n0\n0\n1\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n are identity matrices of order 1, 2 and 3,\nrespectively"
    },
    {
        "Chapter": "1",
        "sentence_range": "1078-1081",
        "Text": "We denote the identity matrix of order n by In When order is clear from the\ncontext, we simply write it as I For example [1], \n1\n0\n0\n1\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb , \n1\n0\n0\n0\n1\n0\n0\n0\n1\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n are identity matrices of order 1, 2 and 3,\nrespectively Observe that a scalar matrix is an identity matrix when k = 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "1079-1082",
        "Text": "When order is clear from the\ncontext, we simply write it as I For example [1], \n1\n0\n0\n1\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb , \n1\n0\n0\n0\n1\n0\n0\n0\n1\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n are identity matrices of order 1, 2 and 3,\nrespectively Observe that a scalar matrix is an identity matrix when k = 1 But every identity\nmatrix is clearly a scalar matrix"
    },
    {
        "Chapter": "1",
        "sentence_range": "1080-1083",
        "Text": "For example [1], \n1\n0\n0\n1\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb , \n1\n0\n0\n0\n1\n0\n0\n0\n1\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n are identity matrices of order 1, 2 and 3,\nrespectively Observe that a scalar matrix is an identity matrix when k = 1 But every identity\nmatrix is clearly a scalar matrix Rationalised 2023-24\nMATRICES         41\n(vii)\nZero matrix\nA matrix is said to be zero matrix or null matrix if all its elements are zero"
    },
    {
        "Chapter": "1",
        "sentence_range": "1081-1084",
        "Text": "Observe that a scalar matrix is an identity matrix when k = 1 But every identity\nmatrix is clearly a scalar matrix Rationalised 2023-24\nMATRICES         41\n(vii)\nZero matrix\nA matrix is said to be zero matrix or null matrix if all its elements are zero For example, [0], \n0\n0\n0\n0\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb , \n0\n0\n0\n0\n0\n0\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb , [0, 0] are all zero matrices"
    },
    {
        "Chapter": "1",
        "sentence_range": "1082-1085",
        "Text": "But every identity\nmatrix is clearly a scalar matrix Rationalised 2023-24\nMATRICES         41\n(vii)\nZero matrix\nA matrix is said to be zero matrix or null matrix if all its elements are zero For example, [0], \n0\n0\n0\n0\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb , \n0\n0\n0\n0\n0\n0\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb , [0, 0] are all zero matrices We denote\nzero matrix by O"
    },
    {
        "Chapter": "1",
        "sentence_range": "1083-1086",
        "Text": "Rationalised 2023-24\nMATRICES         41\n(vii)\nZero matrix\nA matrix is said to be zero matrix or null matrix if all its elements are zero For example, [0], \n0\n0\n0\n0\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb , \n0\n0\n0\n0\n0\n0\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb , [0, 0] are all zero matrices We denote\nzero matrix by O Its order will be clear from the context"
    },
    {
        "Chapter": "1",
        "sentence_range": "1084-1087",
        "Text": "For example, [0], \n0\n0\n0\n0\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb , \n0\n0\n0\n0\n0\n0\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb , [0, 0] are all zero matrices We denote\nzero matrix by O Its order will be clear from the context 3"
    },
    {
        "Chapter": "1",
        "sentence_range": "1085-1088",
        "Text": "We denote\nzero matrix by O Its order will be clear from the context 3 3"
    },
    {
        "Chapter": "1",
        "sentence_range": "1086-1089",
        "Text": "Its order will be clear from the context 3 3 1  Equality of matrices\nDefinition 2 Two matrices A = [aij] and B = [bij] are said to be equal if\n(i)\nthey are of the same order\n(ii)\neach element of A is equal to the corresponding element of B, that is aij = bij for\nall i and j"
    },
    {
        "Chapter": "1",
        "sentence_range": "1087-1090",
        "Text": "3 3 1  Equality of matrices\nDefinition 2 Two matrices A = [aij] and B = [bij] are said to be equal if\n(i)\nthey are of the same order\n(ii)\neach element of A is equal to the corresponding element of B, that is aij = bij for\nall i and j For example, 2\n3\n2\n3\nand\n0\n1\n0\n1\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n are equal matrices but 3\n2\n2\n3\nand\n0\n1\n0\n1\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n are\nnot equal matrices"
    },
    {
        "Chapter": "1",
        "sentence_range": "1088-1091",
        "Text": "3 1  Equality of matrices\nDefinition 2 Two matrices A = [aij] and B = [bij] are said to be equal if\n(i)\nthey are of the same order\n(ii)\neach element of A is equal to the corresponding element of B, that is aij = bij for\nall i and j For example, 2\n3\n2\n3\nand\n0\n1\n0\n1\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n are equal matrices but 3\n2\n2\n3\nand\n0\n1\n0\n1\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n are\nnot equal matrices Symbolically, if two matrices A and B are equal, we write A = B"
    },
    {
        "Chapter": "1",
        "sentence_range": "1089-1092",
        "Text": "1  Equality of matrices\nDefinition 2 Two matrices A = [aij] and B = [bij] are said to be equal if\n(i)\nthey are of the same order\n(ii)\neach element of A is equal to the corresponding element of B, that is aij = bij for\nall i and j For example, 2\n3\n2\n3\nand\n0\n1\n0\n1\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n are equal matrices but 3\n2\n2\n3\nand\n0\n1\n0\n1\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n are\nnot equal matrices Symbolically, if two matrices A and B are equal, we write A = B If \n1"
    },
    {
        "Chapter": "1",
        "sentence_range": "1090-1093",
        "Text": "For example, 2\n3\n2\n3\nand\n0\n1\n0\n1\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n are equal matrices but 3\n2\n2\n3\nand\n0\n1\n0\n1\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n are\nnot equal matrices Symbolically, if two matrices A and B are equal, we write A = B If \n1 5\n0\n2\n6\n3\n2\nx\ny\nz\na\nb\nc\n\uf8ee\u2212\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa = \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n, then x = \u2013 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "1091-1094",
        "Text": "Symbolically, if two matrices A and B are equal, we write A = B If \n1 5\n0\n2\n6\n3\n2\nx\ny\nz\na\nb\nc\n\uf8ee\u2212\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa = \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n, then x = \u2013 1 5, y = 0, z = 2, a = \n6 , b = 3, c = 2\nExample 4 If  \n3\n4\n2\n7\n0\n6\n3\n2\n6\n1\n0\n6\n3\n2\n2\n3\n21\n0\n2\n4\n21\n0\nx\nz\ny\ny\na\nc\nb\nb\n+\n+\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n= \u2212\n\u2212\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n+\n\u2212\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nFind the values of a, b, c, x, y and z"
    },
    {
        "Chapter": "1",
        "sentence_range": "1092-1095",
        "Text": "If \n1 5\n0\n2\n6\n3\n2\nx\ny\nz\na\nb\nc\n\uf8ee\u2212\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa = \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n, then x = \u2013 1 5, y = 0, z = 2, a = \n6 , b = 3, c = 2\nExample 4 If  \n3\n4\n2\n7\n0\n6\n3\n2\n6\n1\n0\n6\n3\n2\n2\n3\n21\n0\n2\n4\n21\n0\nx\nz\ny\ny\na\nc\nb\nb\n+\n+\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n= \u2212\n\u2212\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n+\n\u2212\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nFind the values of a, b, c, x, y and z Solution As the given matrices are equal, therefore, their corresponding elements\nmust be equal"
    },
    {
        "Chapter": "1",
        "sentence_range": "1093-1096",
        "Text": "5\n0\n2\n6\n3\n2\nx\ny\nz\na\nb\nc\n\uf8ee\u2212\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa = \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n, then x = \u2013 1 5, y = 0, z = 2, a = \n6 , b = 3, c = 2\nExample 4 If  \n3\n4\n2\n7\n0\n6\n3\n2\n6\n1\n0\n6\n3\n2\n2\n3\n21\n0\n2\n4\n21\n0\nx\nz\ny\ny\na\nc\nb\nb\n+\n+\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n= \u2212\n\u2212\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n+\n\u2212\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nFind the values of a, b, c, x, y and z Solution As the given matrices are equal, therefore, their corresponding elements\nmust be equal Comparing the corresponding elements, we get\nx + 3 = 0,\nz + 4 = 6,\n2y \u2013 7 = 3y \u2013 2\na \u2013 1 = \u2013 3,\n0 = 2c + 2\nb \u2013 3 = 2b + 4,\nSimplifying, we get\na = \u2013 2, b = \u2013 7, c = \u2013 1, x = \u2013 3, y = \u20135, z = 2\nExample 5 Find the values of a, b, c, and d from the following equation:\n2\n2\n4\n3\n5\n4\n3\n11\n24\na\nb\na\nb\nc\nd\nc\nd\n+\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n+\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nRationalised 2023-24\n 42\nMATHEMATICS\nSolution By equality of two matrices, equating the corresponding elements, we get\n2a + b = 4\n5c \u2013 d = 11\na \u2013 2b = \u2013 3\n4c + 3d = 24\nSolving these equations, we get\na = 1, b = 2, c = 3 and d = 4\nEXERCISE 3"
    },
    {
        "Chapter": "1",
        "sentence_range": "1094-1097",
        "Text": "5, y = 0, z = 2, a = \n6 , b = 3, c = 2\nExample 4 If  \n3\n4\n2\n7\n0\n6\n3\n2\n6\n1\n0\n6\n3\n2\n2\n3\n21\n0\n2\n4\n21\n0\nx\nz\ny\ny\na\nc\nb\nb\n+\n+\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n= \u2212\n\u2212\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n+\n\u2212\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nFind the values of a, b, c, x, y and z Solution As the given matrices are equal, therefore, their corresponding elements\nmust be equal Comparing the corresponding elements, we get\nx + 3 = 0,\nz + 4 = 6,\n2y \u2013 7 = 3y \u2013 2\na \u2013 1 = \u2013 3,\n0 = 2c + 2\nb \u2013 3 = 2b + 4,\nSimplifying, we get\na = \u2013 2, b = \u2013 7, c = \u2013 1, x = \u2013 3, y = \u20135, z = 2\nExample 5 Find the values of a, b, c, and d from the following equation:\n2\n2\n4\n3\n5\n4\n3\n11\n24\na\nb\na\nb\nc\nd\nc\nd\n+\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n+\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nRationalised 2023-24\n 42\nMATHEMATICS\nSolution By equality of two matrices, equating the corresponding elements, we get\n2a + b = 4\n5c \u2013 d = 11\na \u2013 2b = \u2013 3\n4c + 3d = 24\nSolving these equations, we get\na = 1, b = 2, c = 3 and d = 4\nEXERCISE 3 1\n1"
    },
    {
        "Chapter": "1",
        "sentence_range": "1095-1098",
        "Text": "Solution As the given matrices are equal, therefore, their corresponding elements\nmust be equal Comparing the corresponding elements, we get\nx + 3 = 0,\nz + 4 = 6,\n2y \u2013 7 = 3y \u2013 2\na \u2013 1 = \u2013 3,\n0 = 2c + 2\nb \u2013 3 = 2b + 4,\nSimplifying, we get\na = \u2013 2, b = \u2013 7, c = \u2013 1, x = \u2013 3, y = \u20135, z = 2\nExample 5 Find the values of a, b, c, and d from the following equation:\n2\n2\n4\n3\n5\n4\n3\n11\n24\na\nb\na\nb\nc\nd\nc\nd\n+\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n+\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nRationalised 2023-24\n 42\nMATHEMATICS\nSolution By equality of two matrices, equating the corresponding elements, we get\n2a + b = 4\n5c \u2013 d = 11\na \u2013 2b = \u2013 3\n4c + 3d = 24\nSolving these equations, we get\na = 1, b = 2, c = 3 and d = 4\nEXERCISE 3 1\n1 In the matrix \n2\n5\n19\n7\n5\nA\n35\n2\n12\n2\n17\n3\n1\n5\n\uf8ee\n\uf8f9\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n=\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n, write:\n(i) The order of the matrix,\n(ii) The number of elements,\n(iii) Write the elements a13, a21, a33, a24, a23"
    },
    {
        "Chapter": "1",
        "sentence_range": "1096-1099",
        "Text": "Comparing the corresponding elements, we get\nx + 3 = 0,\nz + 4 = 6,\n2y \u2013 7 = 3y \u2013 2\na \u2013 1 = \u2013 3,\n0 = 2c + 2\nb \u2013 3 = 2b + 4,\nSimplifying, we get\na = \u2013 2, b = \u2013 7, c = \u2013 1, x = \u2013 3, y = \u20135, z = 2\nExample 5 Find the values of a, b, c, and d from the following equation:\n2\n2\n4\n3\n5\n4\n3\n11\n24\na\nb\na\nb\nc\nd\nc\nd\n+\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n+\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nRationalised 2023-24\n 42\nMATHEMATICS\nSolution By equality of two matrices, equating the corresponding elements, we get\n2a + b = 4\n5c \u2013 d = 11\na \u2013 2b = \u2013 3\n4c + 3d = 24\nSolving these equations, we get\na = 1, b = 2, c = 3 and d = 4\nEXERCISE 3 1\n1 In the matrix \n2\n5\n19\n7\n5\nA\n35\n2\n12\n2\n17\n3\n1\n5\n\uf8ee\n\uf8f9\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n=\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n, write:\n(i) The order of the matrix,\n(ii) The number of elements,\n(iii) Write the elements a13, a21, a33, a24, a23 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "1097-1100",
        "Text": "1\n1 In the matrix \n2\n5\n19\n7\n5\nA\n35\n2\n12\n2\n17\n3\n1\n5\n\uf8ee\n\uf8f9\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n=\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n, write:\n(i) The order of the matrix,\n(ii) The number of elements,\n(iii) Write the elements a13, a21, a33, a24, a23 2 If a matrix has 24 elements, what are the possible orders it can have"
    },
    {
        "Chapter": "1",
        "sentence_range": "1098-1101",
        "Text": "In the matrix \n2\n5\n19\n7\n5\nA\n35\n2\n12\n2\n17\n3\n1\n5\n\uf8ee\n\uf8f9\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n=\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n, write:\n(i) The order of the matrix,\n(ii) The number of elements,\n(iii) Write the elements a13, a21, a33, a24, a23 2 If a matrix has 24 elements, what are the possible orders it can have What, if it\nhas 13 elements"
    },
    {
        "Chapter": "1",
        "sentence_range": "1099-1102",
        "Text": "2 If a matrix has 24 elements, what are the possible orders it can have What, if it\nhas 13 elements 3"
    },
    {
        "Chapter": "1",
        "sentence_range": "1100-1103",
        "Text": "If a matrix has 24 elements, what are the possible orders it can have What, if it\nhas 13 elements 3 If a matrix has 18 elements, what are the possible orders it can have"
    },
    {
        "Chapter": "1",
        "sentence_range": "1101-1104",
        "Text": "What, if it\nhas 13 elements 3 If a matrix has 18 elements, what are the possible orders it can have What, if it\nhas 5 elements"
    },
    {
        "Chapter": "1",
        "sentence_range": "1102-1105",
        "Text": "3 If a matrix has 18 elements, what are the possible orders it can have What, if it\nhas 5 elements 4"
    },
    {
        "Chapter": "1",
        "sentence_range": "1103-1106",
        "Text": "If a matrix has 18 elements, what are the possible orders it can have What, if it\nhas 5 elements 4 Construct a 2 \u00d7 2 matrix, A = [aij], whose elements are given by:\n(i)\n2\n(\n)\n2\nij\ni\nj\na\n+\n=\n(ii)\nij\ni\na\nj\n=\n(iii)\n2\n(\n2 )\n2\nij\ni\nj\na\n+\n=\n5"
    },
    {
        "Chapter": "1",
        "sentence_range": "1104-1107",
        "Text": "What, if it\nhas 5 elements 4 Construct a 2 \u00d7 2 matrix, A = [aij], whose elements are given by:\n(i)\n2\n(\n)\n2\nij\ni\nj\na\n+\n=\n(ii)\nij\ni\na\nj\n=\n(iii)\n2\n(\n2 )\n2\nij\ni\nj\na\n+\n=\n5 Construct a 3 \u00d7 4 matrix, whose elements are given by:\n(i)\n1 | 3\n|\n2\naij\ni\nj\n=\n\u2212\n+\n(ii)\n2\naij\ni\nj\n=\n\u2212\n6"
    },
    {
        "Chapter": "1",
        "sentence_range": "1105-1108",
        "Text": "4 Construct a 2 \u00d7 2 matrix, A = [aij], whose elements are given by:\n(i)\n2\n(\n)\n2\nij\ni\nj\na\n+\n=\n(ii)\nij\ni\na\nj\n=\n(iii)\n2\n(\n2 )\n2\nij\ni\nj\na\n+\n=\n5 Construct a 3 \u00d7 4 matrix, whose elements are given by:\n(i)\n1 | 3\n|\n2\naij\ni\nj\n=\n\u2212\n+\n(ii)\n2\naij\ni\nj\n=\n\u2212\n6 Find the values of x, y and z from the following equations:\n(i)\n4\n3\n5\n1\n5\ny\nz\n\uf8eex\n\uf8f9\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n(ii)\n2\n6\n2\n5\n5\n8\nx\ny\nz\nxy\n\uf8ee+\n\uf8f9\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0+\n\uf8fb\n\uf8f0\n\uf8fb\n(iii)\n9\n5\n7\nx\ny\nz\nx\nz\ny\nz\n+\n+\n\uf8ee\n\uf8f9\n\uf8ee \uf8f9\n\uf8ef\n\uf8fa\n\uf8ef \uf8fa\n+\n=\n\uf8ef\n\uf8fa\n\uf8ef \uf8fa\n\uf8ef\n\uf8fa\n\uf8ef \uf8fa\n+\n\uf8f0\n\uf8fb\n\uf8f0 \uf8fb\n7"
    },
    {
        "Chapter": "1",
        "sentence_range": "1106-1109",
        "Text": "Construct a 2 \u00d7 2 matrix, A = [aij], whose elements are given by:\n(i)\n2\n(\n)\n2\nij\ni\nj\na\n+\n=\n(ii)\nij\ni\na\nj\n=\n(iii)\n2\n(\n2 )\n2\nij\ni\nj\na\n+\n=\n5 Construct a 3 \u00d7 4 matrix, whose elements are given by:\n(i)\n1 | 3\n|\n2\naij\ni\nj\n=\n\u2212\n+\n(ii)\n2\naij\ni\nj\n=\n\u2212\n6 Find the values of x, y and z from the following equations:\n(i)\n4\n3\n5\n1\n5\ny\nz\n\uf8eex\n\uf8f9\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n(ii)\n2\n6\n2\n5\n5\n8\nx\ny\nz\nxy\n\uf8ee+\n\uf8f9\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0+\n\uf8fb\n\uf8f0\n\uf8fb\n(iii)\n9\n5\n7\nx\ny\nz\nx\nz\ny\nz\n+\n+\n\uf8ee\n\uf8f9\n\uf8ee \uf8f9\n\uf8ef\n\uf8fa\n\uf8ef \uf8fa\n+\n=\n\uf8ef\n\uf8fa\n\uf8ef \uf8fa\n\uf8ef\n\uf8fa\n\uf8ef \uf8fa\n+\n\uf8f0\n\uf8fb\n\uf8f0 \uf8fb\n7 Find the value of a, b, c and d from the equation:\n2\n1\n5\n2\n3\n0\n13\na\nb\na\nc\na\nb\nc\nd\n\u2212\n+\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n+\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nRationalised 2023-24\nMATRICES         43\n8"
    },
    {
        "Chapter": "1",
        "sentence_range": "1107-1110",
        "Text": "Construct a 3 \u00d7 4 matrix, whose elements are given by:\n(i)\n1 | 3\n|\n2\naij\ni\nj\n=\n\u2212\n+\n(ii)\n2\naij\ni\nj\n=\n\u2212\n6 Find the values of x, y and z from the following equations:\n(i)\n4\n3\n5\n1\n5\ny\nz\n\uf8eex\n\uf8f9\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n(ii)\n2\n6\n2\n5\n5\n8\nx\ny\nz\nxy\n\uf8ee+\n\uf8f9\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0+\n\uf8fb\n\uf8f0\n\uf8fb\n(iii)\n9\n5\n7\nx\ny\nz\nx\nz\ny\nz\n+\n+\n\uf8ee\n\uf8f9\n\uf8ee \uf8f9\n\uf8ef\n\uf8fa\n\uf8ef \uf8fa\n+\n=\n\uf8ef\n\uf8fa\n\uf8ef \uf8fa\n\uf8ef\n\uf8fa\n\uf8ef \uf8fa\n+\n\uf8f0\n\uf8fb\n\uf8f0 \uf8fb\n7 Find the value of a, b, c and d from the equation:\n2\n1\n5\n2\n3\n0\n13\na\nb\na\nc\na\nb\nc\nd\n\u2212\n+\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n+\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nRationalised 2023-24\nMATRICES         43\n8 A = [aij]m \u00d7 n\\ is a square matrix, if\n(A) m < n\n(B) m > n\n(C) m = n\n(D) None of these\n9"
    },
    {
        "Chapter": "1",
        "sentence_range": "1108-1111",
        "Text": "Find the values of x, y and z from the following equations:\n(i)\n4\n3\n5\n1\n5\ny\nz\n\uf8eex\n\uf8f9\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n(ii)\n2\n6\n2\n5\n5\n8\nx\ny\nz\nxy\n\uf8ee+\n\uf8f9\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0+\n\uf8fb\n\uf8f0\n\uf8fb\n(iii)\n9\n5\n7\nx\ny\nz\nx\nz\ny\nz\n+\n+\n\uf8ee\n\uf8f9\n\uf8ee \uf8f9\n\uf8ef\n\uf8fa\n\uf8ef \uf8fa\n+\n=\n\uf8ef\n\uf8fa\n\uf8ef \uf8fa\n\uf8ef\n\uf8fa\n\uf8ef \uf8fa\n+\n\uf8f0\n\uf8fb\n\uf8f0 \uf8fb\n7 Find the value of a, b, c and d from the equation:\n2\n1\n5\n2\n3\n0\n13\na\nb\na\nc\na\nb\nc\nd\n\u2212\n+\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n+\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nRationalised 2023-24\nMATRICES         43\n8 A = [aij]m \u00d7 n\\ is a square matrix, if\n(A) m < n\n(B) m > n\n(C) m = n\n(D) None of these\n9 Which of the given values of x and y make the following pair of matrices equal\n3\n7\n5\n1\n2\n3\nx\ny\nx\n+\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n+\n\u2212\n\uf8f0\n\uf8fb  , \n0\n2\n8\n4\ny \u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n(A)\n1,\n7\n3\nx\ny\n=\u2212\n=\n(B) Not possible to find\n(C) y = 7,   \n32\nx\n=\u2212\n(D)\n1\n2\n3,\n3\nx\ny\n\u2212\n\u2212\n=\n=\n10"
    },
    {
        "Chapter": "1",
        "sentence_range": "1109-1112",
        "Text": "Find the value of a, b, c and d from the equation:\n2\n1\n5\n2\n3\n0\n13\na\nb\na\nc\na\nb\nc\nd\n\u2212\n+\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n+\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nRationalised 2023-24\nMATRICES         43\n8 A = [aij]m \u00d7 n\\ is a square matrix, if\n(A) m < n\n(B) m > n\n(C) m = n\n(D) None of these\n9 Which of the given values of x and y make the following pair of matrices equal\n3\n7\n5\n1\n2\n3\nx\ny\nx\n+\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n+\n\u2212\n\uf8f0\n\uf8fb  , \n0\n2\n8\n4\ny \u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n(A)\n1,\n7\n3\nx\ny\n=\u2212\n=\n(B) Not possible to find\n(C) y = 7,   \n32\nx\n=\u2212\n(D)\n1\n2\n3,\n3\nx\ny\n\u2212\n\u2212\n=\n=\n10 The number of all possible matrices of order 3 \u00d7 3 with each entry 0 or 1 is:\n(A) 27\n(B) 18\n(C) 81\n(D) 512\n3"
    },
    {
        "Chapter": "1",
        "sentence_range": "1110-1113",
        "Text": "A = [aij]m \u00d7 n\\ is a square matrix, if\n(A) m < n\n(B) m > n\n(C) m = n\n(D) None of these\n9 Which of the given values of x and y make the following pair of matrices equal\n3\n7\n5\n1\n2\n3\nx\ny\nx\n+\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n+\n\u2212\n\uf8f0\n\uf8fb  , \n0\n2\n8\n4\ny \u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n(A)\n1,\n7\n3\nx\ny\n=\u2212\n=\n(B) Not possible to find\n(C) y = 7,   \n32\nx\n=\u2212\n(D)\n1\n2\n3,\n3\nx\ny\n\u2212\n\u2212\n=\n=\n10 The number of all possible matrices of order 3 \u00d7 3 with each entry 0 or 1 is:\n(A) 27\n(B) 18\n(C) 81\n(D) 512\n3 4  Operations on Matrices\nIn this section, we shall introduce certain operations on matrices, namely, addition of\nmatrices, multiplication of a matrix by a scalar, difference and multiplication of matrices"
    },
    {
        "Chapter": "1",
        "sentence_range": "1111-1114",
        "Text": "Which of the given values of x and y make the following pair of matrices equal\n3\n7\n5\n1\n2\n3\nx\ny\nx\n+\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n+\n\u2212\n\uf8f0\n\uf8fb  , \n0\n2\n8\n4\ny \u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n(A)\n1,\n7\n3\nx\ny\n=\u2212\n=\n(B) Not possible to find\n(C) y = 7,   \n32\nx\n=\u2212\n(D)\n1\n2\n3,\n3\nx\ny\n\u2212\n\u2212\n=\n=\n10 The number of all possible matrices of order 3 \u00d7 3 with each entry 0 or 1 is:\n(A) 27\n(B) 18\n(C) 81\n(D) 512\n3 4  Operations on Matrices\nIn this section, we shall introduce certain operations on matrices, namely, addition of\nmatrices, multiplication of a matrix by a scalar, difference and multiplication of matrices 3"
    },
    {
        "Chapter": "1",
        "sentence_range": "1112-1115",
        "Text": "The number of all possible matrices of order 3 \u00d7 3 with each entry 0 or 1 is:\n(A) 27\n(B) 18\n(C) 81\n(D) 512\n3 4  Operations on Matrices\nIn this section, we shall introduce certain operations on matrices, namely, addition of\nmatrices, multiplication of a matrix by a scalar, difference and multiplication of matrices 3 4"
    },
    {
        "Chapter": "1",
        "sentence_range": "1113-1116",
        "Text": "4  Operations on Matrices\nIn this section, we shall introduce certain operations on matrices, namely, addition of\nmatrices, multiplication of a matrix by a scalar, difference and multiplication of matrices 3 4 1  Addition of matrices\nSuppose Fatima has two factories at places A and B"
    },
    {
        "Chapter": "1",
        "sentence_range": "1114-1117",
        "Text": "3 4 1  Addition of matrices\nSuppose Fatima has two factories at places A and B Each factory produces sport\nshoes for boys and girls in three different price categories labelled 1, 2 and 3"
    },
    {
        "Chapter": "1",
        "sentence_range": "1115-1118",
        "Text": "4 1  Addition of matrices\nSuppose Fatima has two factories at places A and B Each factory produces sport\nshoes for boys and girls in three different price categories labelled 1, 2 and 3 The\nquantities produced by each factory are represented as matrices given below:\nSuppose Fatima wants to know the total production of sport shoes in each price\ncategory"
    },
    {
        "Chapter": "1",
        "sentence_range": "1116-1119",
        "Text": "1  Addition of matrices\nSuppose Fatima has two factories at places A and B Each factory produces sport\nshoes for boys and girls in three different price categories labelled 1, 2 and 3 The\nquantities produced by each factory are represented as matrices given below:\nSuppose Fatima wants to know the total production of sport shoes in each price\ncategory Then the total production\nIn category 1 : for boys (80 + 90), for girls (60 + 50)\nIn category 2 : for boys (75 + 70), for girls (65 + 55)\nIn category 3 : for boys (90 + 75), for girls (85 + 75)\nThis can be represented in the matrix form as \n80\n90\n60\n50\n75\n70\n65\n55\n90\n75\n85\n75\n+\n+\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n+\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n+\n+\n\uf8f0\n\uf8fb"
    },
    {
        "Chapter": "1",
        "sentence_range": "1117-1120",
        "Text": "Each factory produces sport\nshoes for boys and girls in three different price categories labelled 1, 2 and 3 The\nquantities produced by each factory are represented as matrices given below:\nSuppose Fatima wants to know the total production of sport shoes in each price\ncategory Then the total production\nIn category 1 : for boys (80 + 90), for girls (60 + 50)\nIn category 2 : for boys (75 + 70), for girls (65 + 55)\nIn category 3 : for boys (90 + 75), for girls (85 + 75)\nThis can be represented in the matrix form as \n80\n90\n60\n50\n75\n70\n65\n55\n90\n75\n85\n75\n+\n+\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n+\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n+\n+\n\uf8f0\n\uf8fb Rationalised 2023-24\n 44\nMATHEMATICS\nThis new matrix is the sum of the above two matrices"
    },
    {
        "Chapter": "1",
        "sentence_range": "1118-1121",
        "Text": "The\nquantities produced by each factory are represented as matrices given below:\nSuppose Fatima wants to know the total production of sport shoes in each price\ncategory Then the total production\nIn category 1 : for boys (80 + 90), for girls (60 + 50)\nIn category 2 : for boys (75 + 70), for girls (65 + 55)\nIn category 3 : for boys (90 + 75), for girls (85 + 75)\nThis can be represented in the matrix form as \n80\n90\n60\n50\n75\n70\n65\n55\n90\n75\n85\n75\n+\n+\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n+\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n+\n+\n\uf8f0\n\uf8fb Rationalised 2023-24\n 44\nMATHEMATICS\nThis new matrix is the sum of the above two matrices We observe that the sum of\ntwo matrices is a matrix obtained by adding the corresponding elements of the given\nmatrices"
    },
    {
        "Chapter": "1",
        "sentence_range": "1119-1122",
        "Text": "Then the total production\nIn category 1 : for boys (80 + 90), for girls (60 + 50)\nIn category 2 : for boys (75 + 70), for girls (65 + 55)\nIn category 3 : for boys (90 + 75), for girls (85 + 75)\nThis can be represented in the matrix form as \n80\n90\n60\n50\n75\n70\n65\n55\n90\n75\n85\n75\n+\n+\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n+\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n+\n+\n\uf8f0\n\uf8fb Rationalised 2023-24\n 44\nMATHEMATICS\nThis new matrix is the sum of the above two matrices We observe that the sum of\ntwo matrices is a matrix obtained by adding the corresponding elements of the given\nmatrices Furthermore, the two matrices have to be of the same order"
    },
    {
        "Chapter": "1",
        "sentence_range": "1120-1123",
        "Text": "Rationalised 2023-24\n 44\nMATHEMATICS\nThis new matrix is the sum of the above two matrices We observe that the sum of\ntwo matrices is a matrix obtained by adding the corresponding elements of the given\nmatrices Furthermore, the two matrices have to be of the same order Thus, if  \n11\n12\n13\n21\n22\n23\nA\na\na\na\na\na\na\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n is a 2 \u00d7 3 matrix and \n11\n12\n13\n21\n22\n23\nB\nb\nb\nb\nb\nb\nb\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n is another\n2\u00d73 matrix"
    },
    {
        "Chapter": "1",
        "sentence_range": "1121-1124",
        "Text": "We observe that the sum of\ntwo matrices is a matrix obtained by adding the corresponding elements of the given\nmatrices Furthermore, the two matrices have to be of the same order Thus, if  \n11\n12\n13\n21\n22\n23\nA\na\na\na\na\na\na\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n is a 2 \u00d7 3 matrix and \n11\n12\n13\n21\n22\n23\nB\nb\nb\nb\nb\nb\nb\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n is another\n2\u00d73 matrix Then, we define \n11\n11\n12\n12\n13\n13\n21\n21\n22\n22\n23\n23\nA + B\na\nb\na\nb\na\nb\na\nb\na\nb\na\nb\n+\n+\n+\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n+\n+\n+\n\uf8f0\n\uf8fb"
    },
    {
        "Chapter": "1",
        "sentence_range": "1122-1125",
        "Text": "Furthermore, the two matrices have to be of the same order Thus, if  \n11\n12\n13\n21\n22\n23\nA\na\na\na\na\na\na\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n is a 2 \u00d7 3 matrix and \n11\n12\n13\n21\n22\n23\nB\nb\nb\nb\nb\nb\nb\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n is another\n2\u00d73 matrix Then, we define \n11\n11\n12\n12\n13\n13\n21\n21\n22\n22\n23\n23\nA + B\na\nb\na\nb\na\nb\na\nb\na\nb\na\nb\n+\n+\n+\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n+\n+\n+\n\uf8f0\n\uf8fb In general, if A = [aij] and B = [bij] are two matrices of the same order, say m \u00d7 n"
    },
    {
        "Chapter": "1",
        "sentence_range": "1123-1126",
        "Text": "Thus, if  \n11\n12\n13\n21\n22\n23\nA\na\na\na\na\na\na\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n is a 2 \u00d7 3 matrix and \n11\n12\n13\n21\n22\n23\nB\nb\nb\nb\nb\nb\nb\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n is another\n2\u00d73 matrix Then, we define \n11\n11\n12\n12\n13\n13\n21\n21\n22\n22\n23\n23\nA + B\na\nb\na\nb\na\nb\na\nb\na\nb\na\nb\n+\n+\n+\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n+\n+\n+\n\uf8f0\n\uf8fb In general, if A = [aij] and B = [bij] are two matrices of the same order, say m \u00d7 n Then, the sum of the two matrices A and B is defined as a matrix C = [cij]m \u00d7 n, where\ncij = aij + bij, for all possible values of i and j"
    },
    {
        "Chapter": "1",
        "sentence_range": "1124-1127",
        "Text": "Then, we define \n11\n11\n12\n12\n13\n13\n21\n21\n22\n22\n23\n23\nA + B\na\nb\na\nb\na\nb\na\nb\na\nb\na\nb\n+\n+\n+\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n+\n+\n+\n\uf8f0\n\uf8fb In general, if A = [aij] and B = [bij] are two matrices of the same order, say m \u00d7 n Then, the sum of the two matrices A and B is defined as a matrix C = [cij]m \u00d7 n, where\ncij = aij + bij, for all possible values of i and j Example 6  Given \n3\n1\n1\nA\n2\n3\n0\n\uf8ee\n\u2212\uf8f9\n= \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n and \n2\n5\n1\nB\n1\n2\n3\n2\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n= \uf8ef\n\uf8fa\n\uf8ef\u2212\n\uf8fa\n\uf8f0\n\uf8fb\n, find A + B\nSince A, B are of the same order 2 \u00d7 3"
    },
    {
        "Chapter": "1",
        "sentence_range": "1125-1128",
        "Text": "In general, if A = [aij] and B = [bij] are two matrices of the same order, say m \u00d7 n Then, the sum of the two matrices A and B is defined as a matrix C = [cij]m \u00d7 n, where\ncij = aij + bij, for all possible values of i and j Example 6  Given \n3\n1\n1\nA\n2\n3\n0\n\uf8ee\n\u2212\uf8f9\n= \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n and \n2\n5\n1\nB\n1\n2\n3\n2\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n= \uf8ef\n\uf8fa\n\uf8ef\u2212\n\uf8fa\n\uf8f0\n\uf8fb\n, find A + B\nSince A, B are of the same order 2 \u00d7 3 Therefore, addition of A and B is defined\nand is given by\n2\n3\n1\n5\n1 1\n2\n3\n1\n5\n0\nA+B\n1\n1\n2\n2\n3\n3\n0\n0\n6\n2\n2\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n+\n+\n\u2212\n+\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n=\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n+\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nANote\n1"
    },
    {
        "Chapter": "1",
        "sentence_range": "1126-1129",
        "Text": "Then, the sum of the two matrices A and B is defined as a matrix C = [cij]m \u00d7 n, where\ncij = aij + bij, for all possible values of i and j Example 6  Given \n3\n1\n1\nA\n2\n3\n0\n\uf8ee\n\u2212\uf8f9\n= \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n and \n2\n5\n1\nB\n1\n2\n3\n2\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n= \uf8ef\n\uf8fa\n\uf8ef\u2212\n\uf8fa\n\uf8f0\n\uf8fb\n, find A + B\nSince A, B are of the same order 2 \u00d7 3 Therefore, addition of A and B is defined\nand is given by\n2\n3\n1\n5\n1 1\n2\n3\n1\n5\n0\nA+B\n1\n1\n2\n2\n3\n3\n0\n0\n6\n2\n2\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n+\n+\n\u2212\n+\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n=\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n+\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nANote\n1 We emphasise that if A and B are not of the same order, then A + B is not\ndefined"
    },
    {
        "Chapter": "1",
        "sentence_range": "1127-1130",
        "Text": "Example 6  Given \n3\n1\n1\nA\n2\n3\n0\n\uf8ee\n\u2212\uf8f9\n= \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n and \n2\n5\n1\nB\n1\n2\n3\n2\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n= \uf8ef\n\uf8fa\n\uf8ef\u2212\n\uf8fa\n\uf8f0\n\uf8fb\n, find A + B\nSince A, B are of the same order 2 \u00d7 3 Therefore, addition of A and B is defined\nand is given by\n2\n3\n1\n5\n1 1\n2\n3\n1\n5\n0\nA+B\n1\n1\n2\n2\n3\n3\n0\n0\n6\n2\n2\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n+\n+\n\u2212\n+\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n=\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n+\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nANote\n1 We emphasise that if A and B are not of the same order, then A + B is not\ndefined For example if \n2\n3\nA\n1\n0\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, \n1\n2\n3\nB\n,\n1\n0\n1\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n then A + B is not defined"
    },
    {
        "Chapter": "1",
        "sentence_range": "1128-1131",
        "Text": "Therefore, addition of A and B is defined\nand is given by\n2\n3\n1\n5\n1 1\n2\n3\n1\n5\n0\nA+B\n1\n1\n2\n2\n3\n3\n0\n0\n6\n2\n2\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n+\n+\n\u2212\n+\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n=\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n+\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nANote\n1 We emphasise that if A and B are not of the same order, then A + B is not\ndefined For example if \n2\n3\nA\n1\n0\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, \n1\n2\n3\nB\n,\n1\n0\n1\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n then A + B is not defined 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "1129-1132",
        "Text": "We emphasise that if A and B are not of the same order, then A + B is not\ndefined For example if \n2\n3\nA\n1\n0\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, \n1\n2\n3\nB\n,\n1\n0\n1\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n then A + B is not defined 2 We may observe that addition of matrices is an example of binary operation\non the set of matrices of the same order"
    },
    {
        "Chapter": "1",
        "sentence_range": "1130-1133",
        "Text": "For example if \n2\n3\nA\n1\n0\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, \n1\n2\n3\nB\n,\n1\n0\n1\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n then A + B is not defined 2 We may observe that addition of matrices is an example of binary operation\non the set of matrices of the same order 3"
    },
    {
        "Chapter": "1",
        "sentence_range": "1131-1134",
        "Text": "2 We may observe that addition of matrices is an example of binary operation\non the set of matrices of the same order 3 4"
    },
    {
        "Chapter": "1",
        "sentence_range": "1132-1135",
        "Text": "We may observe that addition of matrices is an example of binary operation\non the set of matrices of the same order 3 4 2  Multiplication of a matrix by a scalar\nNow suppose that Fatima has doubled the production at a factory A in all categories\n(refer to 3"
    },
    {
        "Chapter": "1",
        "sentence_range": "1133-1136",
        "Text": "3 4 2  Multiplication of a matrix by a scalar\nNow suppose that Fatima has doubled the production at a factory A in all categories\n(refer to 3 4"
    },
    {
        "Chapter": "1",
        "sentence_range": "1134-1137",
        "Text": "4 2  Multiplication of a matrix by a scalar\nNow suppose that Fatima has doubled the production at a factory A in all categories\n(refer to 3 4 1)"
    },
    {
        "Chapter": "1",
        "sentence_range": "1135-1138",
        "Text": "2  Multiplication of a matrix by a scalar\nNow suppose that Fatima has doubled the production at a factory A in all categories\n(refer to 3 4 1) Rationalised 2023-24\nMATRICES         45\nPreviously quantities (in standard units) produced by factory A  were\nRevised quantities produced by factory A are as given below:\nBoys\nGirls\n2 80\n2\n60\n1\n2 2\n75\n2\n65\n3 2 90\n2 85\n\u00d7\n\u00d7\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u00d7\n\u00d7\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u00d7\n\u00d7\n\uf8f0\n\uf8fb\nThis can be represented in the matrix form as \n160\n120\n150\n130\n180\n170\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb"
    },
    {
        "Chapter": "1",
        "sentence_range": "1136-1139",
        "Text": "4 1) Rationalised 2023-24\nMATRICES         45\nPreviously quantities (in standard units) produced by factory A  were\nRevised quantities produced by factory A are as given below:\nBoys\nGirls\n2 80\n2\n60\n1\n2 2\n75\n2\n65\n3 2 90\n2 85\n\u00d7\n\u00d7\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u00d7\n\u00d7\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u00d7\n\u00d7\n\uf8f0\n\uf8fb\nThis can be represented in the matrix form as \n160\n120\n150\n130\n180\n170\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb We observe that\nthe new matrix is obtained by multiplying each element of the previous matrix by 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "1137-1140",
        "Text": "1) Rationalised 2023-24\nMATRICES         45\nPreviously quantities (in standard units) produced by factory A  were\nRevised quantities produced by factory A are as given below:\nBoys\nGirls\n2 80\n2\n60\n1\n2 2\n75\n2\n65\n3 2 90\n2 85\n\u00d7\n\u00d7\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u00d7\n\u00d7\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u00d7\n\u00d7\n\uf8f0\n\uf8fb\nThis can be represented in the matrix form as \n160\n120\n150\n130\n180\n170\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb We observe that\nthe new matrix is obtained by multiplying each element of the previous matrix by 2 In general, we may define multiplication of a matrix by a scalar as follows: if\nA = [aij] m \u00d7 n is a matrix and k is a scalar, then kA is another matrix which is obtained\nby multiplying each element of A by the scalar k"
    },
    {
        "Chapter": "1",
        "sentence_range": "1138-1141",
        "Text": "Rationalised 2023-24\nMATRICES         45\nPreviously quantities (in standard units) produced by factory A  were\nRevised quantities produced by factory A are as given below:\nBoys\nGirls\n2 80\n2\n60\n1\n2 2\n75\n2\n65\n3 2 90\n2 85\n\u00d7\n\u00d7\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u00d7\n\u00d7\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u00d7\n\u00d7\n\uf8f0\n\uf8fb\nThis can be represented in the matrix form as \n160\n120\n150\n130\n180\n170\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb We observe that\nthe new matrix is obtained by multiplying each element of the previous matrix by 2 In general, we may define multiplication of a matrix by a scalar as follows: if\nA = [aij] m \u00d7 n is a matrix and k is a scalar, then kA is another matrix which is obtained\nby multiplying each element of A by the scalar k In other words, kA = k[aij] m \u00d7 n = [k (aij)] m \u00d7 n, that is, (i, j)th element of kA is kaij\nfor all possible values of i and j"
    },
    {
        "Chapter": "1",
        "sentence_range": "1139-1142",
        "Text": "We observe that\nthe new matrix is obtained by multiplying each element of the previous matrix by 2 In general, we may define multiplication of a matrix by a scalar as follows: if\nA = [aij] m \u00d7 n is a matrix and k is a scalar, then kA is another matrix which is obtained\nby multiplying each element of A by the scalar k In other words, kA = k[aij] m \u00d7 n = [k (aij)] m \u00d7 n, that is, (i, j)th element of kA is kaij\nfor all possible values of i and j For example, if\nA =\n3\n1\n1"
    },
    {
        "Chapter": "1",
        "sentence_range": "1140-1143",
        "Text": "In general, we may define multiplication of a matrix by a scalar as follows: if\nA = [aij] m \u00d7 n is a matrix and k is a scalar, then kA is another matrix which is obtained\nby multiplying each element of A by the scalar k In other words, kA = k[aij] m \u00d7 n = [k (aij)] m \u00d7 n, that is, (i, j)th element of kA is kaij\nfor all possible values of i and j For example, if\nA =\n3\n1\n1 5\n5\n7\n3\n2\n0\n5\n\uf8ee\n\uf8f9\n\uf8ef\n\u2212\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, then\n3A =\n3\n1\n1"
    },
    {
        "Chapter": "1",
        "sentence_range": "1141-1144",
        "Text": "In other words, kA = k[aij] m \u00d7 n = [k (aij)] m \u00d7 n, that is, (i, j)th element of kA is kaij\nfor all possible values of i and j For example, if\nA =\n3\n1\n1 5\n5\n7\n3\n2\n0\n5\n\uf8ee\n\uf8f9\n\uf8ef\n\u2212\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, then\n3A =\n3\n1\n1 5\n9\n3\n4"
    },
    {
        "Chapter": "1",
        "sentence_range": "1142-1145",
        "Text": "For example, if\nA =\n3\n1\n1 5\n5\n7\n3\n2\n0\n5\n\uf8ee\n\uf8f9\n\uf8ef\n\u2212\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, then\n3A =\n3\n1\n1 5\n9\n3\n4 5\n3\n5\n7\n3\n3 5\n21\n9\n2\n0\n5\n6\n0\n15\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n=\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nNegative of a matrix The negative of a matrix is denoted by \u2013 A"
    },
    {
        "Chapter": "1",
        "sentence_range": "1143-1146",
        "Text": "5\n5\n7\n3\n2\n0\n5\n\uf8ee\n\uf8f9\n\uf8ef\n\u2212\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, then\n3A =\n3\n1\n1 5\n9\n3\n4 5\n3\n5\n7\n3\n3 5\n21\n9\n2\n0\n5\n6\n0\n15\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n=\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nNegative of a matrix The negative of a matrix is denoted by \u2013 A We define\n\u2013A = (\u2013 1) A"
    },
    {
        "Chapter": "1",
        "sentence_range": "1144-1147",
        "Text": "5\n9\n3\n4 5\n3\n5\n7\n3\n3 5\n21\n9\n2\n0\n5\n6\n0\n15\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n=\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nNegative of a matrix The negative of a matrix is denoted by \u2013 A We define\n\u2013A = (\u2013 1) A Rationalised 2023-24\n 46\nMATHEMATICS\nFor example, let\nA =\n3\n1\n5\nx\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\u2212\n\uf8fb\n, then \u2013 A is given by\n\u2013 A = (\u2013 1)\n3\n1\n3\n1\nA\n( 1)\n5\n5\nx\nx\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n= \u2212\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nDifference of matrices  If A = [aij], B = [bij] are two matrices of the same order,\nsay m \u00d7 n, then difference A \u2013 B is defined as a matrix D = [dij], where dij = aij \u2013 bij,\nfor all value of i and j"
    },
    {
        "Chapter": "1",
        "sentence_range": "1145-1148",
        "Text": "5\n3\n5\n7\n3\n3 5\n21\n9\n2\n0\n5\n6\n0\n15\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n=\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nNegative of a matrix The negative of a matrix is denoted by \u2013 A We define\n\u2013A = (\u2013 1) A Rationalised 2023-24\n 46\nMATHEMATICS\nFor example, let\nA =\n3\n1\n5\nx\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\u2212\n\uf8fb\n, then \u2013 A is given by\n\u2013 A = (\u2013 1)\n3\n1\n3\n1\nA\n( 1)\n5\n5\nx\nx\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n= \u2212\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nDifference of matrices  If A = [aij], B = [bij] are two matrices of the same order,\nsay m \u00d7 n, then difference A \u2013 B is defined as a matrix D = [dij], where dij = aij \u2013 bij,\nfor all value of i and j In other words, D = A \u2013 B = A + (\u20131) B, that is sum of the matrix\nA and the matrix \u2013 B"
    },
    {
        "Chapter": "1",
        "sentence_range": "1146-1149",
        "Text": "We define\n\u2013A = (\u2013 1) A Rationalised 2023-24\n 46\nMATHEMATICS\nFor example, let\nA =\n3\n1\n5\nx\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\u2212\n\uf8fb\n, then \u2013 A is given by\n\u2013 A = (\u2013 1)\n3\n1\n3\n1\nA\n( 1)\n5\n5\nx\nx\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n= \u2212\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nDifference of matrices  If A = [aij], B = [bij] are two matrices of the same order,\nsay m \u00d7 n, then difference A \u2013 B is defined as a matrix D = [dij], where dij = aij \u2013 bij,\nfor all value of i and j In other words, D = A \u2013 B = A + (\u20131) B, that is sum of the matrix\nA and the matrix \u2013 B Example 7 If \n1\n2\n3\n3\n1\n3\nA\nand B\n2\n3\n1\n1\n0\n2\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n, then find 2A \u2013 B"
    },
    {
        "Chapter": "1",
        "sentence_range": "1147-1150",
        "Text": "Rationalised 2023-24\n 46\nMATHEMATICS\nFor example, let\nA =\n3\n1\n5\nx\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\u2212\n\uf8fb\n, then \u2013 A is given by\n\u2013 A = (\u2013 1)\n3\n1\n3\n1\nA\n( 1)\n5\n5\nx\nx\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n= \u2212\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nDifference of matrices  If A = [aij], B = [bij] are two matrices of the same order,\nsay m \u00d7 n, then difference A \u2013 B is defined as a matrix D = [dij], where dij = aij \u2013 bij,\nfor all value of i and j In other words, D = A \u2013 B = A + (\u20131) B, that is sum of the matrix\nA and the matrix \u2013 B Example 7 If \n1\n2\n3\n3\n1\n3\nA\nand B\n2\n3\n1\n1\n0\n2\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n, then find 2A \u2013 B Solution We have\n2A \u2013 B = 2 1\n2\n3\n2\n3 1\n3\n1\n3\n1\n0\n2\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa \u2212\n\u2212\n\u2212\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa\n=\n2\n4\n6\n3 1\n3\n4\n6\n2\n1\n0\n2\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\u2212\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n=\n2\n3\n4\n1\n6\n3\n1\n5\n3\n4\n1\n6\n0\n2\n2\n5\n6\n0\n\u2212\n+\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n+\n+\n\u2212\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n3"
    },
    {
        "Chapter": "1",
        "sentence_range": "1148-1151",
        "Text": "In other words, D = A \u2013 B = A + (\u20131) B, that is sum of the matrix\nA and the matrix \u2013 B Example 7 If \n1\n2\n3\n3\n1\n3\nA\nand B\n2\n3\n1\n1\n0\n2\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n, then find 2A \u2013 B Solution We have\n2A \u2013 B = 2 1\n2\n3\n2\n3 1\n3\n1\n3\n1\n0\n2\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa \u2212\n\u2212\n\u2212\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa\n=\n2\n4\n6\n3 1\n3\n4\n6\n2\n1\n0\n2\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\u2212\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n=\n2\n3\n4\n1\n6\n3\n1\n5\n3\n4\n1\n6\n0\n2\n2\n5\n6\n0\n\u2212\n+\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n+\n+\n\u2212\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n3 4"
    },
    {
        "Chapter": "1",
        "sentence_range": "1149-1152",
        "Text": "Example 7 If \n1\n2\n3\n3\n1\n3\nA\nand B\n2\n3\n1\n1\n0\n2\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n, then find 2A \u2013 B Solution We have\n2A \u2013 B = 2 1\n2\n3\n2\n3 1\n3\n1\n3\n1\n0\n2\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa \u2212\n\u2212\n\u2212\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa\n=\n2\n4\n6\n3 1\n3\n4\n6\n2\n1\n0\n2\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\u2212\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n=\n2\n3\n4\n1\n6\n3\n1\n5\n3\n4\n1\n6\n0\n2\n2\n5\n6\n0\n\u2212\n+\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n+\n+\n\u2212\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n3 4 3  Properties of matrix addition\nThe addition of matrices satisfy the following properties:\n(i)\nCommutative Law  If A = [aij], B = [bij] are matrices of the same order, say\nm \u00d7 n, then A + B = B + A"
    },
    {
        "Chapter": "1",
        "sentence_range": "1150-1153",
        "Text": "Solution We have\n2A \u2013 B = 2 1\n2\n3\n2\n3 1\n3\n1\n3\n1\n0\n2\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa \u2212\n\u2212\n\u2212\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa\n=\n2\n4\n6\n3 1\n3\n4\n6\n2\n1\n0\n2\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\u2212\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n=\n2\n3\n4\n1\n6\n3\n1\n5\n3\n4\n1\n6\n0\n2\n2\n5\n6\n0\n\u2212\n+\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n+\n+\n\u2212\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n3 4 3  Properties of matrix addition\nThe addition of matrices satisfy the following properties:\n(i)\nCommutative Law  If A = [aij], B = [bij] are matrices of the same order, say\nm \u00d7 n, then A + B = B + A Now\nA + B = [aij] + [bij] = [aij + bij]\n= [bij + aij] (addition of numbers is commutative)\n= ([bij] + [aij]) = B + A\n(ii)\nAssociative Law  For any three matrices A = [aij], B = [bij], C = [cij] of the\nsame order, say m \u00d7 n, (A + B) + C = A + (B + C)"
    },
    {
        "Chapter": "1",
        "sentence_range": "1151-1154",
        "Text": "4 3  Properties of matrix addition\nThe addition of matrices satisfy the following properties:\n(i)\nCommutative Law  If A = [aij], B = [bij] are matrices of the same order, say\nm \u00d7 n, then A + B = B + A Now\nA + B = [aij] + [bij] = [aij + bij]\n= [bij + aij] (addition of numbers is commutative)\n= ([bij] + [aij]) = B + A\n(ii)\nAssociative Law  For any three matrices A = [aij], B = [bij], C = [cij] of the\nsame order, say m \u00d7 n, (A + B) + C = A + (B + C) Now\n(A + B) + C = ([aij] + [bij]) + [cij]\n= [aij + bij] + [cij] = [(aij + bij) + cij]\n= [aij + (bij + cij)]\n(Why"
    },
    {
        "Chapter": "1",
        "sentence_range": "1152-1155",
        "Text": "3  Properties of matrix addition\nThe addition of matrices satisfy the following properties:\n(i)\nCommutative Law  If A = [aij], B = [bij] are matrices of the same order, say\nm \u00d7 n, then A + B = B + A Now\nA + B = [aij] + [bij] = [aij + bij]\n= [bij + aij] (addition of numbers is commutative)\n= ([bij] + [aij]) = B + A\n(ii)\nAssociative Law  For any three matrices A = [aij], B = [bij], C = [cij] of the\nsame order, say m \u00d7 n, (A + B) + C = A + (B + C) Now\n(A + B) + C = ([aij] + [bij]) + [cij]\n= [aij + bij] + [cij] = [(aij + bij) + cij]\n= [aij + (bij + cij)]\n(Why )\n= [aij] + [(bij + cij)] = [aij] + ([bij] + [cij]) = A + (B + C)\nRationalised 2023-24\nMATRICES         47\n(iii)\nExistence of additive identity Let A = [aij] be an m \u00d7 n matrix and\nO be an m \u00d7 n zero matrix, then A + O = O  + A = A"
    },
    {
        "Chapter": "1",
        "sentence_range": "1153-1156",
        "Text": "Now\nA + B = [aij] + [bij] = [aij + bij]\n= [bij + aij] (addition of numbers is commutative)\n= ([bij] + [aij]) = B + A\n(ii)\nAssociative Law  For any three matrices A = [aij], B = [bij], C = [cij] of the\nsame order, say m \u00d7 n, (A + B) + C = A + (B + C) Now\n(A + B) + C = ([aij] + [bij]) + [cij]\n= [aij + bij] + [cij] = [(aij + bij) + cij]\n= [aij + (bij + cij)]\n(Why )\n= [aij] + [(bij + cij)] = [aij] + ([bij] + [cij]) = A + (B + C)\nRationalised 2023-24\nMATRICES         47\n(iii)\nExistence of additive identity Let A = [aij] be an m \u00d7 n matrix and\nO be an m \u00d7 n zero matrix, then A + O = O  + A = A In other words, O is the\nadditive identity for matrix addition"
    },
    {
        "Chapter": "1",
        "sentence_range": "1154-1157",
        "Text": "Now\n(A + B) + C = ([aij] + [bij]) + [cij]\n= [aij + bij] + [cij] = [(aij + bij) + cij]\n= [aij + (bij + cij)]\n(Why )\n= [aij] + [(bij + cij)] = [aij] + ([bij] + [cij]) = A + (B + C)\nRationalised 2023-24\nMATRICES         47\n(iii)\nExistence of additive identity Let A = [aij] be an m \u00d7 n matrix and\nO be an m \u00d7 n zero matrix, then A + O = O  + A = A In other words, O is the\nadditive identity for matrix addition (iv)\nThe existence of additive inverse Let A = [aij]m \u00d7 n be any matrix, then we\nhave another matrix as \u2013 A = [\u2013 aij]m \u00d7 n such that A + (\u2013 A) = (\u2013 A) + A= O"
    },
    {
        "Chapter": "1",
        "sentence_range": "1155-1158",
        "Text": ")\n= [aij] + [(bij + cij)] = [aij] + ([bij] + [cij]) = A + (B + C)\nRationalised 2023-24\nMATRICES         47\n(iii)\nExistence of additive identity Let A = [aij] be an m \u00d7 n matrix and\nO be an m \u00d7 n zero matrix, then A + O = O  + A = A In other words, O is the\nadditive identity for matrix addition (iv)\nThe existence of additive inverse Let A = [aij]m \u00d7 n be any matrix, then we\nhave another matrix as \u2013 A = [\u2013 aij]m \u00d7 n such that A + (\u2013 A) = (\u2013 A) + A= O So\n\u2013 A is the additive inverse of A or negative of A"
    },
    {
        "Chapter": "1",
        "sentence_range": "1156-1159",
        "Text": "In other words, O is the\nadditive identity for matrix addition (iv)\nThe existence of additive inverse Let A = [aij]m \u00d7 n be any matrix, then we\nhave another matrix as \u2013 A = [\u2013 aij]m \u00d7 n such that A + (\u2013 A) = (\u2013 A) + A= O So\n\u2013 A is the additive inverse of A or negative of A 3"
    },
    {
        "Chapter": "1",
        "sentence_range": "1157-1160",
        "Text": "(iv)\nThe existence of additive inverse Let A = [aij]m \u00d7 n be any matrix, then we\nhave another matrix as \u2013 A = [\u2013 aij]m \u00d7 n such that A + (\u2013 A) = (\u2013 A) + A= O So\n\u2013 A is the additive inverse of A or negative of A 3 4"
    },
    {
        "Chapter": "1",
        "sentence_range": "1158-1161",
        "Text": "So\n\u2013 A is the additive inverse of A or negative of A 3 4 4  Properties of scalar multiplication of a matrix\nIf A = [aij] and B = [bij] be two matrices of the same order, say m \u00d7 n, and k and l are\nscalars, then\n(i)\nk(A +B) = k A + kB, (ii) (k + l)A = k A + l A\n(ii)\nk (A + B) = k ([aij] + [bij])\n= k [aij + bij] = [k (aij + bij)] = [(k aij) + (k bij)]\n= [k aij] + [k bij] = k [aij] + k [bij] = kA + kB\n(iii)\n( k + l) A = (k + l) [aij]\n= [(k + l) aij] + [k aij] + [l aij] = k [aij] + l [aij] = k A + l A\nExample 8 If \n8\n0\n2\n2\nA\n4\n2 and B\n4\n2\n3\n6\n5 1\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n=\n\u2212\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n, then find the matrix X, such that\n2A + 3X = 5B"
    },
    {
        "Chapter": "1",
        "sentence_range": "1159-1162",
        "Text": "3 4 4  Properties of scalar multiplication of a matrix\nIf A = [aij] and B = [bij] be two matrices of the same order, say m \u00d7 n, and k and l are\nscalars, then\n(i)\nk(A +B) = k A + kB, (ii) (k + l)A = k A + l A\n(ii)\nk (A + B) = k ([aij] + [bij])\n= k [aij + bij] = [k (aij + bij)] = [(k aij) + (k bij)]\n= [k aij] + [k bij] = k [aij] + k [bij] = kA + kB\n(iii)\n( k + l) A = (k + l) [aij]\n= [(k + l) aij] + [k aij] + [l aij] = k [aij] + l [aij] = k A + l A\nExample 8 If \n8\n0\n2\n2\nA\n4\n2 and B\n4\n2\n3\n6\n5 1\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n=\n\u2212\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n, then find the matrix X, such that\n2A + 3X = 5B Solution We have 2A + 3X = 5B\nor\n2A + 3X \u2013 2A = 5B \u2013 2A\nor\n2A \u2013 2A + 3X = 5B \u2013 2A\n(Matrix addition is commutative)\nor\nO + 3X = 5B \u2013 2A\n(\u2013 2A is the additive inverse of 2A)\nor\n3X = 5B \u2013 2A\n(O is the additive identity)\nor\nX = 1\n3  (5B \u2013 2A)\nor\n2\n2\n8 0\n1\nX\n5\n4\n2\n2 4\n2\n3\n5 1\n3 6\n\uf8eb\n\uf8f6\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n=\n\u2212\n\u2212\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\u2212\n\uf8fb\n\uf8f0\n\uf8fb\n\uf8ed\n\uf8f8\n = \n10\n10\n16\n0\n1\n20\n10\n8\n4\n3\n25\n5\n6\n12\n\uf8eb\n\uf8f6\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n+\n\u2212\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\u2212\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\uf8ed\n\uf8f8\nRationalised 2023-24\n 48\nMATHEMATICS\n    = \n10\n16\n10\n0\n1\n20\n8\n10\n4\n3\n25\n6\n5 12\n\u2212\n\u2212\n+\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\u2212\n\uf8f0\n\uf8fb\n = \n6\n10\n1\n12\n14\n3\n31\n7\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8f0\n\uf8fb\n = \n10\n2\n143\n4\n3\n31\n7\n3\n3\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\u2212\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nExample 9 Find  X and Y, if \n5\n2\nX\nY\n0\n9\n\uf8ee\n\uf8f9\n+\n= \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n and \n3\n6\nX\nY\n0\n1\n\uf8ee\n\uf8f9\n\u2212\n= \uf8ef\n\u2212\uf8fa\n\uf8f0\n\uf8fb"
    },
    {
        "Chapter": "1",
        "sentence_range": "1160-1163",
        "Text": "4 4  Properties of scalar multiplication of a matrix\nIf A = [aij] and B = [bij] be two matrices of the same order, say m \u00d7 n, and k and l are\nscalars, then\n(i)\nk(A +B) = k A + kB, (ii) (k + l)A = k A + l A\n(ii)\nk (A + B) = k ([aij] + [bij])\n= k [aij + bij] = [k (aij + bij)] = [(k aij) + (k bij)]\n= [k aij] + [k bij] = k [aij] + k [bij] = kA + kB\n(iii)\n( k + l) A = (k + l) [aij]\n= [(k + l) aij] + [k aij] + [l aij] = k [aij] + l [aij] = k A + l A\nExample 8 If \n8\n0\n2\n2\nA\n4\n2 and B\n4\n2\n3\n6\n5 1\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n=\n\u2212\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n, then find the matrix X, such that\n2A + 3X = 5B Solution We have 2A + 3X = 5B\nor\n2A + 3X \u2013 2A = 5B \u2013 2A\nor\n2A \u2013 2A + 3X = 5B \u2013 2A\n(Matrix addition is commutative)\nor\nO + 3X = 5B \u2013 2A\n(\u2013 2A is the additive inverse of 2A)\nor\n3X = 5B \u2013 2A\n(O is the additive identity)\nor\nX = 1\n3  (5B \u2013 2A)\nor\n2\n2\n8 0\n1\nX\n5\n4\n2\n2 4\n2\n3\n5 1\n3 6\n\uf8eb\n\uf8f6\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n=\n\u2212\n\u2212\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\u2212\n\uf8fb\n\uf8f0\n\uf8fb\n\uf8ed\n\uf8f8\n = \n10\n10\n16\n0\n1\n20\n10\n8\n4\n3\n25\n5\n6\n12\n\uf8eb\n\uf8f6\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n+\n\u2212\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\u2212\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\uf8ed\n\uf8f8\nRationalised 2023-24\n 48\nMATHEMATICS\n    = \n10\n16\n10\n0\n1\n20\n8\n10\n4\n3\n25\n6\n5 12\n\u2212\n\u2212\n+\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\u2212\n\uf8f0\n\uf8fb\n = \n6\n10\n1\n12\n14\n3\n31\n7\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8f0\n\uf8fb\n = \n10\n2\n143\n4\n3\n31\n7\n3\n3\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\u2212\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nExample 9 Find  X and Y, if \n5\n2\nX\nY\n0\n9\n\uf8ee\n\uf8f9\n+\n= \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n and \n3\n6\nX\nY\n0\n1\n\uf8ee\n\uf8f9\n\u2212\n= \uf8ef\n\u2212\uf8fa\n\uf8f0\n\uf8fb Solution We have (\n)\n(\n)\n5\n2\n3\n6\nX\nY\nX\nY\n0\n9\n0\n1\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n+\n+\n\u2212\n=\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\u2212\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb"
    },
    {
        "Chapter": "1",
        "sentence_range": "1161-1164",
        "Text": "4  Properties of scalar multiplication of a matrix\nIf A = [aij] and B = [bij] be two matrices of the same order, say m \u00d7 n, and k and l are\nscalars, then\n(i)\nk(A +B) = k A + kB, (ii) (k + l)A = k A + l A\n(ii)\nk (A + B) = k ([aij] + [bij])\n= k [aij + bij] = [k (aij + bij)] = [(k aij) + (k bij)]\n= [k aij] + [k bij] = k [aij] + k [bij] = kA + kB\n(iii)\n( k + l) A = (k + l) [aij]\n= [(k + l) aij] + [k aij] + [l aij] = k [aij] + l [aij] = k A + l A\nExample 8 If \n8\n0\n2\n2\nA\n4\n2 and B\n4\n2\n3\n6\n5 1\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n=\n\u2212\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n, then find the matrix X, such that\n2A + 3X = 5B Solution We have 2A + 3X = 5B\nor\n2A + 3X \u2013 2A = 5B \u2013 2A\nor\n2A \u2013 2A + 3X = 5B \u2013 2A\n(Matrix addition is commutative)\nor\nO + 3X = 5B \u2013 2A\n(\u2013 2A is the additive inverse of 2A)\nor\n3X = 5B \u2013 2A\n(O is the additive identity)\nor\nX = 1\n3  (5B \u2013 2A)\nor\n2\n2\n8 0\n1\nX\n5\n4\n2\n2 4\n2\n3\n5 1\n3 6\n\uf8eb\n\uf8f6\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n=\n\u2212\n\u2212\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\u2212\n\uf8fb\n\uf8f0\n\uf8fb\n\uf8ed\n\uf8f8\n = \n10\n10\n16\n0\n1\n20\n10\n8\n4\n3\n25\n5\n6\n12\n\uf8eb\n\uf8f6\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n+\n\u2212\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\u2212\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\uf8ed\n\uf8f8\nRationalised 2023-24\n 48\nMATHEMATICS\n    = \n10\n16\n10\n0\n1\n20\n8\n10\n4\n3\n25\n6\n5 12\n\u2212\n\u2212\n+\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\u2212\n\uf8f0\n\uf8fb\n = \n6\n10\n1\n12\n14\n3\n31\n7\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8f0\n\uf8fb\n = \n10\n2\n143\n4\n3\n31\n7\n3\n3\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\u2212\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nExample 9 Find  X and Y, if \n5\n2\nX\nY\n0\n9\n\uf8ee\n\uf8f9\n+\n= \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n and \n3\n6\nX\nY\n0\n1\n\uf8ee\n\uf8f9\n\u2212\n= \uf8ef\n\u2212\uf8fa\n\uf8f0\n\uf8fb Solution We have (\n)\n(\n)\n5\n2\n3\n6\nX\nY\nX\nY\n0\n9\n0\n1\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n+\n+\n\u2212\n=\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\u2212\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb or\n(X + X) + (Y \u2013 Y) =\n8\n8\n0\n8\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n \u21d2 \n8\n8\n2X\n0\n8\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nor\nX =\n8\n8\n4\n4\n1\n0\n8\n0\n4\n2\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nAlso\n(X + Y) \u2013 (X \u2013 Y) =\n5\n2\n3\n6\n0\n9\n0\n1\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\u2212\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nor\n(X \u2013 X) + (Y + Y) =\n5\n3\n2\n6\n0\n9\n1\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n+\uf8fa\n\uf8f0\n\uf8fb\n \u21d2 \n2\n4\n2Y\n0\n10\n\u2212\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nor\nY =\n2\n4\n1\n2\n1\n0\n10\n0\n5\n2\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nExample 10 Find the values of x and y from the following equation:\n5\n3\n4\n2 7\n3\n1\n2\nx\ny\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb  = \n7\n6\n15\n14\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nSolution We have\n   \n5\n3\n4\n2 7\n3\n1\n2\nx\ny\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb  = \n7\n6\n15\n14\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n \u21d2 2\n10\n3\n4\n7\n6\n14\n2\n6\n1\n2\n15\n14\nx\ny\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n+\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nRationalised 2023-24\nMATRICES         49\nor\n2\n3\n10\n4\n14\n1\n2\n6\n2\nx\ny\n+\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n+\n\u2212\n+\n\uf8f0\n\uf8fb  =\n7\n6\n\uf8ee15 14\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n \u21d2 2\n3\n6\n7\n6\n15\n2\n4\n15\n14\nx\ny\n+\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nor\n2x + 3 = 7\nand\n2y \u2013 4 = 14\n(Why"
    },
    {
        "Chapter": "1",
        "sentence_range": "1162-1165",
        "Text": "Solution We have 2A + 3X = 5B\nor\n2A + 3X \u2013 2A = 5B \u2013 2A\nor\n2A \u2013 2A + 3X = 5B \u2013 2A\n(Matrix addition is commutative)\nor\nO + 3X = 5B \u2013 2A\n(\u2013 2A is the additive inverse of 2A)\nor\n3X = 5B \u2013 2A\n(O is the additive identity)\nor\nX = 1\n3  (5B \u2013 2A)\nor\n2\n2\n8 0\n1\nX\n5\n4\n2\n2 4\n2\n3\n5 1\n3 6\n\uf8eb\n\uf8f6\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n=\n\u2212\n\u2212\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\u2212\n\uf8fb\n\uf8f0\n\uf8fb\n\uf8ed\n\uf8f8\n = \n10\n10\n16\n0\n1\n20\n10\n8\n4\n3\n25\n5\n6\n12\n\uf8eb\n\uf8f6\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n+\n\u2212\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\u2212\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\uf8ed\n\uf8f8\nRationalised 2023-24\n 48\nMATHEMATICS\n    = \n10\n16\n10\n0\n1\n20\n8\n10\n4\n3\n25\n6\n5 12\n\u2212\n\u2212\n+\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\u2212\n\uf8f0\n\uf8fb\n = \n6\n10\n1\n12\n14\n3\n31\n7\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8f0\n\uf8fb\n = \n10\n2\n143\n4\n3\n31\n7\n3\n3\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\u2212\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nExample 9 Find  X and Y, if \n5\n2\nX\nY\n0\n9\n\uf8ee\n\uf8f9\n+\n= \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n and \n3\n6\nX\nY\n0\n1\n\uf8ee\n\uf8f9\n\u2212\n= \uf8ef\n\u2212\uf8fa\n\uf8f0\n\uf8fb Solution We have (\n)\n(\n)\n5\n2\n3\n6\nX\nY\nX\nY\n0\n9\n0\n1\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n+\n+\n\u2212\n=\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\u2212\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb or\n(X + X) + (Y \u2013 Y) =\n8\n8\n0\n8\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n \u21d2 \n8\n8\n2X\n0\n8\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nor\nX =\n8\n8\n4\n4\n1\n0\n8\n0\n4\n2\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nAlso\n(X + Y) \u2013 (X \u2013 Y) =\n5\n2\n3\n6\n0\n9\n0\n1\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\u2212\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nor\n(X \u2013 X) + (Y + Y) =\n5\n3\n2\n6\n0\n9\n1\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n+\uf8fa\n\uf8f0\n\uf8fb\n \u21d2 \n2\n4\n2Y\n0\n10\n\u2212\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nor\nY =\n2\n4\n1\n2\n1\n0\n10\n0\n5\n2\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nExample 10 Find the values of x and y from the following equation:\n5\n3\n4\n2 7\n3\n1\n2\nx\ny\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb  = \n7\n6\n15\n14\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nSolution We have\n   \n5\n3\n4\n2 7\n3\n1\n2\nx\ny\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb  = \n7\n6\n15\n14\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n \u21d2 2\n10\n3\n4\n7\n6\n14\n2\n6\n1\n2\n15\n14\nx\ny\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n+\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nRationalised 2023-24\nMATRICES         49\nor\n2\n3\n10\n4\n14\n1\n2\n6\n2\nx\ny\n+\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n+\n\u2212\n+\n\uf8f0\n\uf8fb  =\n7\n6\n\uf8ee15 14\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n \u21d2 2\n3\n6\n7\n6\n15\n2\n4\n15\n14\nx\ny\n+\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nor\n2x + 3 = 7\nand\n2y \u2013 4 = 14\n(Why )\nor\n2x = 7 \u2013 3\nand\n2y = 18\nor\nx = 4\n2\nand\ny = 18\n2\ni"
    },
    {
        "Chapter": "1",
        "sentence_range": "1163-1166",
        "Text": "Solution We have (\n)\n(\n)\n5\n2\n3\n6\nX\nY\nX\nY\n0\n9\n0\n1\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n+\n+\n\u2212\n=\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\u2212\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb or\n(X + X) + (Y \u2013 Y) =\n8\n8\n0\n8\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n \u21d2 \n8\n8\n2X\n0\n8\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nor\nX =\n8\n8\n4\n4\n1\n0\n8\n0\n4\n2\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nAlso\n(X + Y) \u2013 (X \u2013 Y) =\n5\n2\n3\n6\n0\n9\n0\n1\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\u2212\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nor\n(X \u2013 X) + (Y + Y) =\n5\n3\n2\n6\n0\n9\n1\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n+\uf8fa\n\uf8f0\n\uf8fb\n \u21d2 \n2\n4\n2Y\n0\n10\n\u2212\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nor\nY =\n2\n4\n1\n2\n1\n0\n10\n0\n5\n2\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nExample 10 Find the values of x and y from the following equation:\n5\n3\n4\n2 7\n3\n1\n2\nx\ny\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb  = \n7\n6\n15\n14\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nSolution We have\n   \n5\n3\n4\n2 7\n3\n1\n2\nx\ny\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb  = \n7\n6\n15\n14\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n \u21d2 2\n10\n3\n4\n7\n6\n14\n2\n6\n1\n2\n15\n14\nx\ny\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n+\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nRationalised 2023-24\nMATRICES         49\nor\n2\n3\n10\n4\n14\n1\n2\n6\n2\nx\ny\n+\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n+\n\u2212\n+\n\uf8f0\n\uf8fb  =\n7\n6\n\uf8ee15 14\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n \u21d2 2\n3\n6\n7\n6\n15\n2\n4\n15\n14\nx\ny\n+\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nor\n2x + 3 = 7\nand\n2y \u2013 4 = 14\n(Why )\nor\n2x = 7 \u2013 3\nand\n2y = 18\nor\nx = 4\n2\nand\ny = 18\n2\ni e"
    },
    {
        "Chapter": "1",
        "sentence_range": "1164-1167",
        "Text": "or\n(X + X) + (Y \u2013 Y) =\n8\n8\n0\n8\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n \u21d2 \n8\n8\n2X\n0\n8\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nor\nX =\n8\n8\n4\n4\n1\n0\n8\n0\n4\n2\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nAlso\n(X + Y) \u2013 (X \u2013 Y) =\n5\n2\n3\n6\n0\n9\n0\n1\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\u2212\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nor\n(X \u2013 X) + (Y + Y) =\n5\n3\n2\n6\n0\n9\n1\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n+\uf8fa\n\uf8f0\n\uf8fb\n \u21d2 \n2\n4\n2Y\n0\n10\n\u2212\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nor\nY =\n2\n4\n1\n2\n1\n0\n10\n0\n5\n2\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nExample 10 Find the values of x and y from the following equation:\n5\n3\n4\n2 7\n3\n1\n2\nx\ny\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb  = \n7\n6\n15\n14\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nSolution We have\n   \n5\n3\n4\n2 7\n3\n1\n2\nx\ny\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb  = \n7\n6\n15\n14\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n \u21d2 2\n10\n3\n4\n7\n6\n14\n2\n6\n1\n2\n15\n14\nx\ny\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n+\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nRationalised 2023-24\nMATRICES         49\nor\n2\n3\n10\n4\n14\n1\n2\n6\n2\nx\ny\n+\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n+\n\u2212\n+\n\uf8f0\n\uf8fb  =\n7\n6\n\uf8ee15 14\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n \u21d2 2\n3\n6\n7\n6\n15\n2\n4\n15\n14\nx\ny\n+\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nor\n2x + 3 = 7\nand\n2y \u2013 4 = 14\n(Why )\nor\n2x = 7 \u2013 3\nand\n2y = 18\nor\nx = 4\n2\nand\ny = 18\n2\ni e x = 2\nand\ny = 9"
    },
    {
        "Chapter": "1",
        "sentence_range": "1165-1168",
        "Text": ")\nor\n2x = 7 \u2013 3\nand\n2y = 18\nor\nx = 4\n2\nand\ny = 18\n2\ni e x = 2\nand\ny = 9 Example 11 Two farmers Ramkishan and Gurcharan Singh cultivates only three\nvarieties of rice namely Basmati, Permal and Naura"
    },
    {
        "Chapter": "1",
        "sentence_range": "1166-1169",
        "Text": "e x = 2\nand\ny = 9 Example 11 Two farmers Ramkishan and Gurcharan Singh cultivates only three\nvarieties of rice namely Basmati, Permal and Naura The sale (in Rupees) of these\nvarieties of rice by both the farmers in the month of September and October are given\nby the following matrices A and B"
    },
    {
        "Chapter": "1",
        "sentence_range": "1167-1170",
        "Text": "x = 2\nand\ny = 9 Example 11 Two farmers Ramkishan and Gurcharan Singh cultivates only three\nvarieties of rice namely Basmati, Permal and Naura The sale (in Rupees) of these\nvarieties of rice by both the farmers in the month of September and October are given\nby the following matrices A and B (i)\nFind the combined sales in September and October for each farmer in each\nvariety"
    },
    {
        "Chapter": "1",
        "sentence_range": "1168-1171",
        "Text": "Example 11 Two farmers Ramkishan and Gurcharan Singh cultivates only three\nvarieties of rice namely Basmati, Permal and Naura The sale (in Rupees) of these\nvarieties of rice by both the farmers in the month of September and October are given\nby the following matrices A and B (i)\nFind the combined sales in September and October for each farmer in each\nvariety (ii)\nFind the decrease in sales from September to October"
    },
    {
        "Chapter": "1",
        "sentence_range": "1169-1172",
        "Text": "The sale (in Rupees) of these\nvarieties of rice by both the farmers in the month of September and October are given\nby the following matrices A and B (i)\nFind the combined sales in September and October for each farmer in each\nvariety (ii)\nFind the decrease in sales from September to October (iii)\nIf both farmers receive 2%  profit on gross sales, compute the profit for each\nfarmer and for each variety sold in October"
    },
    {
        "Chapter": "1",
        "sentence_range": "1170-1173",
        "Text": "(i)\nFind the combined sales in September and October for each farmer in each\nvariety (ii)\nFind the decrease in sales from September to October (iii)\nIf both farmers receive 2%  profit on gross sales, compute the profit for each\nfarmer and for each variety sold in October Solution\n(i)\nCombined sales in September and October for each farmer in each variety is\ngiven by\nRationalised 2023-24\n 50\nMATHEMATICS\n(ii)\nChange in sales from September to October is given by\n(iii)\n2% of B = 2\n100 \u00d7B\n= 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "1171-1174",
        "Text": "(ii)\nFind the decrease in sales from September to October (iii)\nIf both farmers receive 2%  profit on gross sales, compute the profit for each\nfarmer and for each variety sold in October Solution\n(i)\nCombined sales in September and October for each farmer in each variety is\ngiven by\nRationalised 2023-24\n 50\nMATHEMATICS\n(ii)\nChange in sales from September to October is given by\n(iii)\n2% of B = 2\n100 \u00d7B\n= 0 02 \u00d7 B\n= 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "1172-1175",
        "Text": "(iii)\nIf both farmers receive 2%  profit on gross sales, compute the profit for each\nfarmer and for each variety sold in October Solution\n(i)\nCombined sales in September and October for each farmer in each variety is\ngiven by\nRationalised 2023-24\n 50\nMATHEMATICS\n(ii)\nChange in sales from September to October is given by\n(iii)\n2% of B = 2\n100 \u00d7B\n= 0 02 \u00d7 B\n= 0 02 \n=  \nThus, in October Ramkishan receives ` 100, ` 200 and ` 120 as profit in the\nsale of  each variety of rice, respectively, and Grucharan Singh receives profit of ` 400,\n` 200 and ` 200 in the sale of each variety of rice, respectively"
    },
    {
        "Chapter": "1",
        "sentence_range": "1173-1176",
        "Text": "Solution\n(i)\nCombined sales in September and October for each farmer in each variety is\ngiven by\nRationalised 2023-24\n 50\nMATHEMATICS\n(ii)\nChange in sales from September to October is given by\n(iii)\n2% of B = 2\n100 \u00d7B\n= 0 02 \u00d7 B\n= 0 02 \n=  \nThus, in October Ramkishan receives ` 100, ` 200 and ` 120 as profit in the\nsale of  each variety of rice, respectively, and Grucharan Singh receives profit of ` 400,\n` 200 and ` 200 in the sale of each variety of rice, respectively 3"
    },
    {
        "Chapter": "1",
        "sentence_range": "1174-1177",
        "Text": "02 \u00d7 B\n= 0 02 \n=  \nThus, in October Ramkishan receives ` 100, ` 200 and ` 120 as profit in the\nsale of  each variety of rice, respectively, and Grucharan Singh receives profit of ` 400,\n` 200 and ` 200 in the sale of each variety of rice, respectively 3 4"
    },
    {
        "Chapter": "1",
        "sentence_range": "1175-1178",
        "Text": "02 \n=  \nThus, in October Ramkishan receives ` 100, ` 200 and ` 120 as profit in the\nsale of  each variety of rice, respectively, and Grucharan Singh receives profit of ` 400,\n` 200 and ` 200 in the sale of each variety of rice, respectively 3 4 5  Multiplication of matrices\nSuppose Meera and Nadeem are two friends"
    },
    {
        "Chapter": "1",
        "sentence_range": "1176-1179",
        "Text": "3 4 5  Multiplication of matrices\nSuppose Meera and Nadeem are two friends Meera wants to buy 2 pens and 5 story\nbooks, while Nadeem needs 8 pens and 10 story books"
    },
    {
        "Chapter": "1",
        "sentence_range": "1177-1180",
        "Text": "4 5  Multiplication of matrices\nSuppose Meera and Nadeem are two friends Meera wants to buy 2 pens and 5 story\nbooks, while Nadeem needs 8 pens and 10 story books They both go to a shop to\nenquire about the rates which are quoted as follows:\nPen \u2013 ` 5 each, story book \u2013 ` 50 each"
    },
    {
        "Chapter": "1",
        "sentence_range": "1178-1181",
        "Text": "5  Multiplication of matrices\nSuppose Meera and Nadeem are two friends Meera wants to buy 2 pens and 5 story\nbooks, while Nadeem needs 8 pens and 10 story books They both go to a shop to\nenquire about the rates which are quoted as follows:\nPen \u2013 ` 5 each, story book \u2013 ` 50 each How much money does each need to spend"
    },
    {
        "Chapter": "1",
        "sentence_range": "1179-1182",
        "Text": "Meera wants to buy 2 pens and 5 story\nbooks, while Nadeem needs 8 pens and 10 story books They both go to a shop to\nenquire about the rates which are quoted as follows:\nPen \u2013 ` 5 each, story book \u2013 ` 50 each How much money does each need to spend Clearly, Meera needs ` (5 \u00d7 2 + 50 \u00d7 5)\nthat is ` 260, while Nadeem needs (8 \u00d7 5 + 50 \u00d7 10) `, that is ` 540"
    },
    {
        "Chapter": "1",
        "sentence_range": "1180-1183",
        "Text": "They both go to a shop to\nenquire about the rates which are quoted as follows:\nPen \u2013 ` 5 each, story book \u2013 ` 50 each How much money does each need to spend Clearly, Meera needs ` (5 \u00d7 2 + 50 \u00d7 5)\nthat is ` 260, while Nadeem needs (8 \u00d7 5 + 50 \u00d7 10) `, that is ` 540 In terms of matrix\nrepresentation, we can write the above information as follows:\nRequirements\nPrices per piece (in Rupees) Money needed (in Rupees)\n2\n5\n8\n10\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n5\n\uf8ee50\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n5\n2\n5 50\n260\n8 5\n10\n50\n540\n\u00d7\n+\n\u00d7\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u00d7\n+\n\u00d7\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nSuppose that they enquire about the rates from another shop, quoted as follows:\npen \u2013 ` 4 each,  story book \u2013 ` 40 each"
    },
    {
        "Chapter": "1",
        "sentence_range": "1181-1184",
        "Text": "How much money does each need to spend Clearly, Meera needs ` (5 \u00d7 2 + 50 \u00d7 5)\nthat is ` 260, while Nadeem needs (8 \u00d7 5 + 50 \u00d7 10) `, that is ` 540 In terms of matrix\nrepresentation, we can write the above information as follows:\nRequirements\nPrices per piece (in Rupees) Money needed (in Rupees)\n2\n5\n8\n10\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n5\n\uf8ee50\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n5\n2\n5 50\n260\n8 5\n10\n50\n540\n\u00d7\n+\n\u00d7\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u00d7\n+\n\u00d7\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nSuppose that they enquire about the rates from another shop, quoted as follows:\npen \u2013 ` 4 each,  story book \u2013 ` 40 each Now, the money required by Meera and Nadeem to make purchases will be\nrespectively ` (4 \u00d7 2 + 40 \u00d7 5) = ` 208 and ` (8 \u00d7 4 + 10 \u00d7 40) = ` 432\nRationalised 2023-24\nMATRICES         51\nAgain, the above information can be represented as follows:\nRequirements\nPrices per piece (in Rupees) Money needed (in Rupees)\n2\n5\n8\n10\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8ee404\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n4\n2\n40 5\n208\n8 4\n10 4 0\n432\n\u00d7\n+\n\u00d7\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u00d7\n+\n\u00d7\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nNow, the information in both the cases can be combined and expressed in terms of\nmatrices as follows:\nRequirements\nPrices per piece (in Rupees)\nMoney needed (in Rupees)\n2\n5\n8\n10\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n5\n4\n50\n40\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n5\n2\n5 50\n4\n2\n40 5\n8 5\n10 5 0 8 4\n10 4 0\n\u00d7\n+ \u00d7\n\u00d7\n+\n\u00d7\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u00d7\n+\n\u00d7\n\u00d7\n+\n\u00d7\n\uf8f0\n\uf8fb\n= \n260\n208\n540\n432\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nThe above is an example of multiplication of matrices"
    },
    {
        "Chapter": "1",
        "sentence_range": "1182-1185",
        "Text": "Clearly, Meera needs ` (5 \u00d7 2 + 50 \u00d7 5)\nthat is ` 260, while Nadeem needs (8 \u00d7 5 + 50 \u00d7 10) `, that is ` 540 In terms of matrix\nrepresentation, we can write the above information as follows:\nRequirements\nPrices per piece (in Rupees) Money needed (in Rupees)\n2\n5\n8\n10\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n5\n\uf8ee50\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n5\n2\n5 50\n260\n8 5\n10\n50\n540\n\u00d7\n+\n\u00d7\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u00d7\n+\n\u00d7\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nSuppose that they enquire about the rates from another shop, quoted as follows:\npen \u2013 ` 4 each,  story book \u2013 ` 40 each Now, the money required by Meera and Nadeem to make purchases will be\nrespectively ` (4 \u00d7 2 + 40 \u00d7 5) = ` 208 and ` (8 \u00d7 4 + 10 \u00d7 40) = ` 432\nRationalised 2023-24\nMATRICES         51\nAgain, the above information can be represented as follows:\nRequirements\nPrices per piece (in Rupees) Money needed (in Rupees)\n2\n5\n8\n10\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8ee404\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n4\n2\n40 5\n208\n8 4\n10 4 0\n432\n\u00d7\n+\n\u00d7\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u00d7\n+\n\u00d7\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nNow, the information in both the cases can be combined and expressed in terms of\nmatrices as follows:\nRequirements\nPrices per piece (in Rupees)\nMoney needed (in Rupees)\n2\n5\n8\n10\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n5\n4\n50\n40\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n5\n2\n5 50\n4\n2\n40 5\n8 5\n10 5 0 8 4\n10 4 0\n\u00d7\n+ \u00d7\n\u00d7\n+\n\u00d7\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u00d7\n+\n\u00d7\n\u00d7\n+\n\u00d7\n\uf8f0\n\uf8fb\n= \n260\n208\n540\n432\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nThe above is an example of multiplication of matrices We observe that, for\nmultiplication of two matrices A and B, the number of columns in A should be equal to\nthe number of rows in B"
    },
    {
        "Chapter": "1",
        "sentence_range": "1183-1186",
        "Text": "In terms of matrix\nrepresentation, we can write the above information as follows:\nRequirements\nPrices per piece (in Rupees) Money needed (in Rupees)\n2\n5\n8\n10\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n5\n\uf8ee50\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n5\n2\n5 50\n260\n8 5\n10\n50\n540\n\u00d7\n+\n\u00d7\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u00d7\n+\n\u00d7\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nSuppose that they enquire about the rates from another shop, quoted as follows:\npen \u2013 ` 4 each,  story book \u2013 ` 40 each Now, the money required by Meera and Nadeem to make purchases will be\nrespectively ` (4 \u00d7 2 + 40 \u00d7 5) = ` 208 and ` (8 \u00d7 4 + 10 \u00d7 40) = ` 432\nRationalised 2023-24\nMATRICES         51\nAgain, the above information can be represented as follows:\nRequirements\nPrices per piece (in Rupees) Money needed (in Rupees)\n2\n5\n8\n10\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8ee404\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n4\n2\n40 5\n208\n8 4\n10 4 0\n432\n\u00d7\n+\n\u00d7\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u00d7\n+\n\u00d7\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nNow, the information in both the cases can be combined and expressed in terms of\nmatrices as follows:\nRequirements\nPrices per piece (in Rupees)\nMoney needed (in Rupees)\n2\n5\n8\n10\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n5\n4\n50\n40\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n5\n2\n5 50\n4\n2\n40 5\n8 5\n10 5 0 8 4\n10 4 0\n\u00d7\n+ \u00d7\n\u00d7\n+\n\u00d7\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u00d7\n+\n\u00d7\n\u00d7\n+\n\u00d7\n\uf8f0\n\uf8fb\n= \n260\n208\n540\n432\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nThe above is an example of multiplication of matrices We observe that, for\nmultiplication of two matrices A and B, the number of columns in A should be equal to\nthe number of rows in B Furthermore for getting the elements of the product matrix,\nwe take rows of A and columns of B, multiply them element-wise and take the sum"
    },
    {
        "Chapter": "1",
        "sentence_range": "1184-1187",
        "Text": "Now, the money required by Meera and Nadeem to make purchases will be\nrespectively ` (4 \u00d7 2 + 40 \u00d7 5) = ` 208 and ` (8 \u00d7 4 + 10 \u00d7 40) = ` 432\nRationalised 2023-24\nMATRICES         51\nAgain, the above information can be represented as follows:\nRequirements\nPrices per piece (in Rupees) Money needed (in Rupees)\n2\n5\n8\n10\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8ee404\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n4\n2\n40 5\n208\n8 4\n10 4 0\n432\n\u00d7\n+\n\u00d7\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u00d7\n+\n\u00d7\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nNow, the information in both the cases can be combined and expressed in terms of\nmatrices as follows:\nRequirements\nPrices per piece (in Rupees)\nMoney needed (in Rupees)\n2\n5\n8\n10\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n5\n4\n50\n40\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n5\n2\n5 50\n4\n2\n40 5\n8 5\n10 5 0 8 4\n10 4 0\n\u00d7\n+ \u00d7\n\u00d7\n+\n\u00d7\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u00d7\n+\n\u00d7\n\u00d7\n+\n\u00d7\n\uf8f0\n\uf8fb\n= \n260\n208\n540\n432\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nThe above is an example of multiplication of matrices We observe that, for\nmultiplication of two matrices A and B, the number of columns in A should be equal to\nthe number of rows in B Furthermore for getting the elements of the product matrix,\nwe take rows of A and columns of B, multiply them element-wise and take the sum Formally, we define multiplication of matrices as follows:\nThe product of two matrices A and B is defined if the number of columns of A is\nequal to the number of rows of B"
    },
    {
        "Chapter": "1",
        "sentence_range": "1185-1188",
        "Text": "We observe that, for\nmultiplication of two matrices A and B, the number of columns in A should be equal to\nthe number of rows in B Furthermore for getting the elements of the product matrix,\nwe take rows of A and columns of B, multiply them element-wise and take the sum Formally, we define multiplication of matrices as follows:\nThe product of two matrices A and B is defined if the number of columns of A is\nequal to the number of rows of B Let A = [aij] be an m \u00d7 n matrix and B = [bjk] be an\nn \u00d7 p matrix"
    },
    {
        "Chapter": "1",
        "sentence_range": "1186-1189",
        "Text": "Furthermore for getting the elements of the product matrix,\nwe take rows of A and columns of B, multiply them element-wise and take the sum Formally, we define multiplication of matrices as follows:\nThe product of two matrices A and B is defined if the number of columns of A is\nequal to the number of rows of B Let A = [aij] be an m \u00d7 n matrix and B = [bjk] be an\nn \u00d7 p matrix Then the product of the matrices A and B is the matrix C of order m \u00d7 p"
    },
    {
        "Chapter": "1",
        "sentence_range": "1187-1190",
        "Text": "Formally, we define multiplication of matrices as follows:\nThe product of two matrices A and B is defined if the number of columns of A is\nequal to the number of rows of B Let A = [aij] be an m \u00d7 n matrix and B = [bjk] be an\nn \u00d7 p matrix Then the product of the matrices A and B is the matrix C of order m \u00d7 p To get the (i, k)th element cik of the matrix C, we take the ith row of A and kth column\nof B, multiply them elementwise and take the sum of all these products"
    },
    {
        "Chapter": "1",
        "sentence_range": "1188-1191",
        "Text": "Let A = [aij] be an m \u00d7 n matrix and B = [bjk] be an\nn \u00d7 p matrix Then the product of the matrices A and B is the matrix C of order m \u00d7 p To get the (i, k)th element cik of the matrix C, we take the ith row of A and kth column\nof B, multiply them elementwise and take the sum of all these products In other words,\nif A = [aij]m \u00d7 n, B = [bjk]n \u00d7 p, then the ith row of A is [ai1 ai2"
    },
    {
        "Chapter": "1",
        "sentence_range": "1189-1192",
        "Text": "Then the product of the matrices A and B is the matrix C of order m \u00d7 p To get the (i, k)th element cik of the matrix C, we take the ith row of A and kth column\nof B, multiply them elementwise and take the sum of all these products In other words,\nif A = [aij]m \u00d7 n, B = [bjk]n \u00d7 p, then the ith row of A is [ai1 ai2 ain] and the kth column of\nB is \n1"
    },
    {
        "Chapter": "1",
        "sentence_range": "1190-1193",
        "Text": "To get the (i, k)th element cik of the matrix C, we take the ith row of A and kth column\nof B, multiply them elementwise and take the sum of all these products In other words,\nif A = [aij]m \u00d7 n, B = [bjk]n \u00d7 p, then the ith row of A is [ai1 ai2 ain] and the kth column of\nB is \n1 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "1191-1194",
        "Text": "In other words,\nif A = [aij]m \u00d7 n, B = [bjk]n \u00d7 p, then the ith row of A is [ai1 ai2 ain] and the kth column of\nB is \n1 2 k\nk\nnk\nb\nb\nb\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n , then cik =  ai1 b1k + ai2 b2k + ai3 b3k +"
    },
    {
        "Chapter": "1",
        "sentence_range": "1192-1195",
        "Text": "ain] and the kth column of\nB is \n1 2 k\nk\nnk\nb\nb\nb\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n , then cik =  ai1 b1k + ai2 b2k + ai3 b3k + + ain bnk = \n1\nn\nij\njk\nj\na b\n=\u2211"
    },
    {
        "Chapter": "1",
        "sentence_range": "1193-1196",
        "Text": "2 k\nk\nnk\nb\nb\nb\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n , then cik =  ai1 b1k + ai2 b2k + ai3 b3k + + ain bnk = \n1\nn\nij\njk\nj\na b\n=\u2211 The matrix C = [cik]m \u00d7 p is the product of A and B"
    },
    {
        "Chapter": "1",
        "sentence_range": "1194-1197",
        "Text": "k\nk\nnk\nb\nb\nb\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n , then cik =  ai1 b1k + ai2 b2k + ai3 b3k + + ain bnk = \n1\nn\nij\njk\nj\na b\n=\u2211 The matrix C = [cik]m \u00d7 p is the product of A and B For example, if \n1\n1\n2\nC\n0\n\u22123 4\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n and \n2\n17\nD\n1\n5\n4\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n= \u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\u2212\uf8fa\n\uf8f0\n\uf8fb\n , then the product CD is defined\nRationalised 2023-24\n 52\nMATHEMATICS\nand is given by \n2\n7\n1\n1\n2\nCD\n1\n1\n0\n3\n4\n5\n4\n\uf8ee\n\uf8f9\n\u2212\n\uf8ee\n\uf8f9 \uf8ef\n\uf8fa\n=\n\u2212\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb \uf8ef\n\u2212\uf8fa\n\uf8f0\n\uf8fb"
    },
    {
        "Chapter": "1",
        "sentence_range": "1195-1198",
        "Text": "+ ain bnk = \n1\nn\nij\njk\nj\na b\n=\u2211 The matrix C = [cik]m \u00d7 p is the product of A and B For example, if \n1\n1\n2\nC\n0\n\u22123 4\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n and \n2\n17\nD\n1\n5\n4\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n= \u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\u2212\uf8fa\n\uf8f0\n\uf8fb\n , then the product CD is defined\nRationalised 2023-24\n 52\nMATHEMATICS\nand is given by \n2\n7\n1\n1\n2\nCD\n1\n1\n0\n3\n4\n5\n4\n\uf8ee\n\uf8f9\n\u2212\n\uf8ee\n\uf8f9 \uf8ef\n\uf8fa\n=\n\u2212\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb \uf8ef\n\u2212\uf8fa\n\uf8f0\n\uf8fb This is a 2 \u00d7 2  matrix in which each\nentry is the sum of the products across some row of C with the corresponding entries\ndown some column of D"
    },
    {
        "Chapter": "1",
        "sentence_range": "1196-1199",
        "Text": "The matrix C = [cik]m \u00d7 p is the product of A and B For example, if \n1\n1\n2\nC\n0\n\u22123 4\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n and \n2\n17\nD\n1\n5\n4\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n= \u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\u2212\uf8fa\n\uf8f0\n\uf8fb\n , then the product CD is defined\nRationalised 2023-24\n 52\nMATHEMATICS\nand is given by \n2\n7\n1\n1\n2\nCD\n1\n1\n0\n3\n4\n5\n4\n\uf8ee\n\uf8f9\n\u2212\n\uf8ee\n\uf8f9 \uf8ef\n\uf8fa\n=\n\u2212\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb \uf8ef\n\u2212\uf8fa\n\uf8f0\n\uf8fb This is a 2 \u00d7 2  matrix in which each\nentry is the sum of the products across some row of C with the corresponding entries\ndown some column of D These four computations are\nThus \n13\n2\nCD\n17\n13\n\u2212\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\nExample 12 Find AB, if \n6\n9\n2\n6\n0\nA\nand B\n2\n3\n7\n9\n8\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb"
    },
    {
        "Chapter": "1",
        "sentence_range": "1197-1200",
        "Text": "For example, if \n1\n1\n2\nC\n0\n\u22123 4\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n and \n2\n17\nD\n1\n5\n4\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n= \u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\u2212\uf8fa\n\uf8f0\n\uf8fb\n , then the product CD is defined\nRationalised 2023-24\n 52\nMATHEMATICS\nand is given by \n2\n7\n1\n1\n2\nCD\n1\n1\n0\n3\n4\n5\n4\n\uf8ee\n\uf8f9\n\u2212\n\uf8ee\n\uf8f9 \uf8ef\n\uf8fa\n=\n\u2212\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb \uf8ef\n\u2212\uf8fa\n\uf8f0\n\uf8fb This is a 2 \u00d7 2  matrix in which each\nentry is the sum of the products across some row of C with the corresponding entries\ndown some column of D These four computations are\nThus \n13\n2\nCD\n17\n13\n\u2212\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\nExample 12 Find AB, if \n6\n9\n2\n6\n0\nA\nand B\n2\n3\n7\n9\n8\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb Solution The matrix A has 2 columns which is equal to the number of rows of B"
    },
    {
        "Chapter": "1",
        "sentence_range": "1198-1201",
        "Text": "This is a 2 \u00d7 2  matrix in which each\nentry is the sum of the products across some row of C with the corresponding entries\ndown some column of D These four computations are\nThus \n13\n2\nCD\n17\n13\n\u2212\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\nExample 12 Find AB, if \n6\n9\n2\n6\n0\nA\nand B\n2\n3\n7\n9\n8\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb Solution The matrix A has 2 columns which is equal to the number of rows of B Hence AB is defined"
    },
    {
        "Chapter": "1",
        "sentence_range": "1199-1202",
        "Text": "These four computations are\nThus \n13\n2\nCD\n17\n13\n\u2212\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\nExample 12 Find AB, if \n6\n9\n2\n6\n0\nA\nand B\n2\n3\n7\n9\n8\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb Solution The matrix A has 2 columns which is equal to the number of rows of B Hence AB is defined Now\n6(2)\n9(7)\n6(6)\n9(9)\n6(0)\n9(8)\nAB\n2(2)\n3(7)\n2(6)\n3(9)\n2(0)\n3(8)\n+\n+\n+\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n+\n+\n+\n\uf8f0\n\uf8fb\n=\n12\n63\n36\n81\n0\n72\n4\n21\n12\n27\n0\n24\n+\n+\n+\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n+\n+\n+\n\uf8f0\n\uf8fb\n = \n75\n117\n72\n25\n39\n24\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nRationalised 2023-24\nMATRICES         53\nRemark  If AB is defined, then BA need not be defined"
    },
    {
        "Chapter": "1",
        "sentence_range": "1200-1203",
        "Text": "Solution The matrix A has 2 columns which is equal to the number of rows of B Hence AB is defined Now\n6(2)\n9(7)\n6(6)\n9(9)\n6(0)\n9(8)\nAB\n2(2)\n3(7)\n2(6)\n3(9)\n2(0)\n3(8)\n+\n+\n+\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n+\n+\n+\n\uf8f0\n\uf8fb\n=\n12\n63\n36\n81\n0\n72\n4\n21\n12\n27\n0\n24\n+\n+\n+\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n+\n+\n+\n\uf8f0\n\uf8fb\n = \n75\n117\n72\n25\n39\n24\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nRationalised 2023-24\nMATRICES         53\nRemark  If AB is defined, then BA need not be defined In the above example, AB is\ndefined but BA is not defined because B has 3 column while A has only 2 (and not 3)\nrows"
    },
    {
        "Chapter": "1",
        "sentence_range": "1201-1204",
        "Text": "Hence AB is defined Now\n6(2)\n9(7)\n6(6)\n9(9)\n6(0)\n9(8)\nAB\n2(2)\n3(7)\n2(6)\n3(9)\n2(0)\n3(8)\n+\n+\n+\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n+\n+\n+\n\uf8f0\n\uf8fb\n=\n12\n63\n36\n81\n0\n72\n4\n21\n12\n27\n0\n24\n+\n+\n+\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n+\n+\n+\n\uf8f0\n\uf8fb\n = \n75\n117\n72\n25\n39\n24\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nRationalised 2023-24\nMATRICES         53\nRemark  If AB is defined, then BA need not be defined In the above example, AB is\ndefined but BA is not defined because B has 3 column while A has only 2 (and not 3)\nrows If A, B are, respectively m \u00d7 n, k \u00d7 l matrices, then both AB and BA are defined\nif and only if n = k and l = m"
    },
    {
        "Chapter": "1",
        "sentence_range": "1202-1205",
        "Text": "Now\n6(2)\n9(7)\n6(6)\n9(9)\n6(0)\n9(8)\nAB\n2(2)\n3(7)\n2(6)\n3(9)\n2(0)\n3(8)\n+\n+\n+\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n+\n+\n+\n\uf8f0\n\uf8fb\n=\n12\n63\n36\n81\n0\n72\n4\n21\n12\n27\n0\n24\n+\n+\n+\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n+\n+\n+\n\uf8f0\n\uf8fb\n = \n75\n117\n72\n25\n39\n24\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nRationalised 2023-24\nMATRICES         53\nRemark  If AB is defined, then BA need not be defined In the above example, AB is\ndefined but BA is not defined because B has 3 column while A has only 2 (and not 3)\nrows If A, B are, respectively m \u00d7 n, k \u00d7 l matrices, then both AB and BA are defined\nif and only if n = k and l = m In particular, if both A and B are square matrices of the\nsame order, then both AB and BA are defined"
    },
    {
        "Chapter": "1",
        "sentence_range": "1203-1206",
        "Text": "In the above example, AB is\ndefined but BA is not defined because B has 3 column while A has only 2 (and not 3)\nrows If A, B are, respectively m \u00d7 n, k \u00d7 l matrices, then both AB and BA are defined\nif and only if n = k and l = m In particular, if both A and B are square matrices of the\nsame order, then both AB and BA are defined Non-commutativity of multiplication of matrices\nNow, we shall see by an example that even if AB and BA are both defined, it is not\nnecessary that AB = BA"
    },
    {
        "Chapter": "1",
        "sentence_range": "1204-1207",
        "Text": "If A, B are, respectively m \u00d7 n, k \u00d7 l matrices, then both AB and BA are defined\nif and only if n = k and l = m In particular, if both A and B are square matrices of the\nsame order, then both AB and BA are defined Non-commutativity of multiplication of matrices\nNow, we shall see by an example that even if AB and BA are both defined, it is not\nnecessary that AB = BA Example 13 If \n2 3\n1\n2\n3\nA\nand B\n4 5\n4\n2\n5\n2 1\n\uf8ee\n\uf8f9\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n=\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\u2212\n\uf8fb\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, then find AB, BA"
    },
    {
        "Chapter": "1",
        "sentence_range": "1205-1208",
        "Text": "In particular, if both A and B are square matrices of the\nsame order, then both AB and BA are defined Non-commutativity of multiplication of matrices\nNow, we shall see by an example that even if AB and BA are both defined, it is not\nnecessary that AB = BA Example 13 If \n2 3\n1\n2\n3\nA\nand B\n4 5\n4\n2\n5\n2 1\n\uf8ee\n\uf8f9\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n=\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\u2212\n\uf8fb\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, then find AB, BA Show that\nAB \u2260 BA"
    },
    {
        "Chapter": "1",
        "sentence_range": "1206-1209",
        "Text": "Non-commutativity of multiplication of matrices\nNow, we shall see by an example that even if AB and BA are both defined, it is not\nnecessary that AB = BA Example 13 If \n2 3\n1\n2\n3\nA\nand B\n4 5\n4\n2\n5\n2 1\n\uf8ee\n\uf8f9\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n=\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\u2212\n\uf8fb\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, then find AB, BA Show that\nAB \u2260 BA Solution Since A is a 2 \u00d7 3 matrix and B is 3 \u00d7 2 matrix"
    },
    {
        "Chapter": "1",
        "sentence_range": "1207-1210",
        "Text": "Example 13 If \n2 3\n1\n2\n3\nA\nand B\n4 5\n4\n2\n5\n2 1\n\uf8ee\n\uf8f9\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n=\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\u2212\n\uf8fb\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, then find AB, BA Show that\nAB \u2260 BA Solution Since A is a 2 \u00d7 3 matrix and B is 3 \u00d7 2 matrix Hence AB and BA are both\ndefined and are matrices of order 2 \u00d7 2 and 3 \u00d7 3, respectively"
    },
    {
        "Chapter": "1",
        "sentence_range": "1208-1211",
        "Text": "Show that\nAB \u2260 BA Solution Since A is a 2 \u00d7 3 matrix and B is 3 \u00d7 2 matrix Hence AB and BA are both\ndefined and are matrices of order 2 \u00d7 2 and 3 \u00d7 3, respectively Note that\n2 3\n1\n2\n3\nAB\n4 5\n4\n2\n5\n2 1\n\uf8ee\n\uf8f9\n\u2212\n\uf8ee\n\uf8f9 \uf8ef\n\uf8fa\n= \uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\u2212\uf8f0\n\uf8fb \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n =  \n2\n8\n6\n3 10\n3\n0\n4\n8\n8\n10\n12\n10\n5\n10\n3\n\u2212\n+\n\u2212\n+\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212 +\n+\n\u2212\n+\n+\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nand\n2 3\n2 12\n4\n6\n6\n15\n1\n2\n3\nBA\n4 5\n4\n20\n8 10\n12\n25\n4\n2\n5\n2 1\n2\n4\n4\n2\n6\n5\n\u2212\n\u2212 +\n+\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n=\n=\n\u2212\n\u2212 +\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\uf8f0\n\uf8fb\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212 +\n+\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n \n10\n2\n21\n16\n2\n37\n2\n2\n11\n\uf8ee\u2212\n\uf8f9\n\uf8ef\n\uf8fa\n= \u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8f0\n\uf8fb\nClearly AB \u2260 BA\nIn the above example both AB and BA are of different order and so AB \u2260 BA"
    },
    {
        "Chapter": "1",
        "sentence_range": "1209-1212",
        "Text": "Solution Since A is a 2 \u00d7 3 matrix and B is 3 \u00d7 2 matrix Hence AB and BA are both\ndefined and are matrices of order 2 \u00d7 2 and 3 \u00d7 3, respectively Note that\n2 3\n1\n2\n3\nAB\n4 5\n4\n2\n5\n2 1\n\uf8ee\n\uf8f9\n\u2212\n\uf8ee\n\uf8f9 \uf8ef\n\uf8fa\n= \uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\u2212\uf8f0\n\uf8fb \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n =  \n2\n8\n6\n3 10\n3\n0\n4\n8\n8\n10\n12\n10\n5\n10\n3\n\u2212\n+\n\u2212\n+\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212 +\n+\n\u2212\n+\n+\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nand\n2 3\n2 12\n4\n6\n6\n15\n1\n2\n3\nBA\n4 5\n4\n20\n8 10\n12\n25\n4\n2\n5\n2 1\n2\n4\n4\n2\n6\n5\n\u2212\n\u2212 +\n+\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n=\n=\n\u2212\n\u2212 +\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\uf8f0\n\uf8fb\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212 +\n+\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n \n10\n2\n21\n16\n2\n37\n2\n2\n11\n\uf8ee\u2212\n\uf8f9\n\uf8ef\n\uf8fa\n= \u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8f0\n\uf8fb\nClearly AB \u2260 BA\nIn the above example both AB and BA are of different order and so AB \u2260 BA But\none may think that perhaps AB and BA could be the same if they were of the same\norder"
    },
    {
        "Chapter": "1",
        "sentence_range": "1210-1213",
        "Text": "Hence AB and BA are both\ndefined and are matrices of order 2 \u00d7 2 and 3 \u00d7 3, respectively Note that\n2 3\n1\n2\n3\nAB\n4 5\n4\n2\n5\n2 1\n\uf8ee\n\uf8f9\n\u2212\n\uf8ee\n\uf8f9 \uf8ef\n\uf8fa\n= \uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\u2212\uf8f0\n\uf8fb \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n =  \n2\n8\n6\n3 10\n3\n0\n4\n8\n8\n10\n12\n10\n5\n10\n3\n\u2212\n+\n\u2212\n+\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212 +\n+\n\u2212\n+\n+\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nand\n2 3\n2 12\n4\n6\n6\n15\n1\n2\n3\nBA\n4 5\n4\n20\n8 10\n12\n25\n4\n2\n5\n2 1\n2\n4\n4\n2\n6\n5\n\u2212\n\u2212 +\n+\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n=\n=\n\u2212\n\u2212 +\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\uf8f0\n\uf8fb\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212 +\n+\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n \n10\n2\n21\n16\n2\n37\n2\n2\n11\n\uf8ee\u2212\n\uf8f9\n\uf8ef\n\uf8fa\n= \u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8f0\n\uf8fb\nClearly AB \u2260 BA\nIn the above example both AB and BA are of different order and so AB \u2260 BA But\none may think that perhaps AB and BA could be the same if they were of the same\norder But it is not so, here we give an example to show that even if AB and BA are of\nsame order they may not be same"
    },
    {
        "Chapter": "1",
        "sentence_range": "1211-1214",
        "Text": "Note that\n2 3\n1\n2\n3\nAB\n4 5\n4\n2\n5\n2 1\n\uf8ee\n\uf8f9\n\u2212\n\uf8ee\n\uf8f9 \uf8ef\n\uf8fa\n= \uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\u2212\uf8f0\n\uf8fb \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n =  \n2\n8\n6\n3 10\n3\n0\n4\n8\n8\n10\n12\n10\n5\n10\n3\n\u2212\n+\n\u2212\n+\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212 +\n+\n\u2212\n+\n+\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nand\n2 3\n2 12\n4\n6\n6\n15\n1\n2\n3\nBA\n4 5\n4\n20\n8 10\n12\n25\n4\n2\n5\n2 1\n2\n4\n4\n2\n6\n5\n\u2212\n\u2212 +\n+\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n=\n=\n\u2212\n\u2212 +\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\uf8f0\n\uf8fb\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212 +\n+\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n \n10\n2\n21\n16\n2\n37\n2\n2\n11\n\uf8ee\u2212\n\uf8f9\n\uf8ef\n\uf8fa\n= \u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8f0\n\uf8fb\nClearly AB \u2260 BA\nIn the above example both AB and BA are of different order and so AB \u2260 BA But\none may think that perhaps AB and BA could be the same if they were of the same\norder But it is not so, here we give an example to show that even if AB and BA are of\nsame order they may not be same Example 14 If \n1\n0\nA\n0\n1\n\uf8ee\n\uf8f9\n= \uf8ef\n\u2212\uf8fa\n\uf8f0\n\uf8fb\n and \n0\n1\nB\n1\n0\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, then \n0\n1\nAB\n1\n0\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\uf8f0\u2212\n\uf8fb"
    },
    {
        "Chapter": "1",
        "sentence_range": "1212-1215",
        "Text": "But\none may think that perhaps AB and BA could be the same if they were of the same\norder But it is not so, here we give an example to show that even if AB and BA are of\nsame order they may not be same Example 14 If \n1\n0\nA\n0\n1\n\uf8ee\n\uf8f9\n= \uf8ef\n\u2212\uf8fa\n\uf8f0\n\uf8fb\n and \n0\n1\nB\n1\n0\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, then \n0\n1\nAB\n1\n0\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\uf8f0\u2212\n\uf8fb and\n0\n1\nBA\n1\n\u22120\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb"
    },
    {
        "Chapter": "1",
        "sentence_range": "1213-1216",
        "Text": "But it is not so, here we give an example to show that even if AB and BA are of\nsame order they may not be same Example 14 If \n1\n0\nA\n0\n1\n\uf8ee\n\uf8f9\n= \uf8ef\n\u2212\uf8fa\n\uf8f0\n\uf8fb\n and \n0\n1\nB\n1\n0\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, then \n0\n1\nAB\n1\n0\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\uf8f0\u2212\n\uf8fb and\n0\n1\nBA\n1\n\u22120\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb Clearly AB \u2260 BA"
    },
    {
        "Chapter": "1",
        "sentence_range": "1214-1217",
        "Text": "Example 14 If \n1\n0\nA\n0\n1\n\uf8ee\n\uf8f9\n= \uf8ef\n\u2212\uf8fa\n\uf8f0\n\uf8fb\n and \n0\n1\nB\n1\n0\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, then \n0\n1\nAB\n1\n0\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\uf8f0\u2212\n\uf8fb and\n0\n1\nBA\n1\n\u22120\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb Clearly AB \u2260 BA Thus matrix multiplication is not commutative"
    },
    {
        "Chapter": "1",
        "sentence_range": "1215-1218",
        "Text": "and\n0\n1\nBA\n1\n\u22120\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb Clearly AB \u2260 BA Thus matrix multiplication is not commutative Rationalised 2023-24\n 54\nMATHEMATICS\nANote  This does not mean that AB \u2260 BA for every pair of matrices A, B for\nwhich AB and BA, are defined"
    },
    {
        "Chapter": "1",
        "sentence_range": "1216-1219",
        "Text": "Clearly AB \u2260 BA Thus matrix multiplication is not commutative Rationalised 2023-24\n 54\nMATHEMATICS\nANote  This does not mean that AB \u2260 BA for every pair of matrices A, B for\nwhich AB and BA, are defined For instance,\nIf  \n1\n0\n3\n0\nA\n, B\n0\n2\n0\n4\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n, then AB = BA = \n3\n0\n0\n8\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nObserve that multiplication of diagonal matrices of same order will be commutative"
    },
    {
        "Chapter": "1",
        "sentence_range": "1217-1220",
        "Text": "Thus matrix multiplication is not commutative Rationalised 2023-24\n 54\nMATHEMATICS\nANote  This does not mean that AB \u2260 BA for every pair of matrices A, B for\nwhich AB and BA, are defined For instance,\nIf  \n1\n0\n3\n0\nA\n, B\n0\n2\n0\n4\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n, then AB = BA = \n3\n0\n0\n8\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nObserve that multiplication of diagonal matrices of same order will be commutative Zero matrix as the product of two non zero matrices\nWe know that, for real numbers a, b if ab = 0, then either a = 0 or b = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "1218-1221",
        "Text": "Rationalised 2023-24\n 54\nMATHEMATICS\nANote  This does not mean that AB \u2260 BA for every pair of matrices A, B for\nwhich AB and BA, are defined For instance,\nIf  \n1\n0\n3\n0\nA\n, B\n0\n2\n0\n4\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n, then AB = BA = \n3\n0\n0\n8\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nObserve that multiplication of diagonal matrices of same order will be commutative Zero matrix as the product of two non zero matrices\nWe know that, for real numbers a, b if ab = 0, then either a = 0 or b = 0 This need\nnot be true for matrices, we will observe this through an example"
    },
    {
        "Chapter": "1",
        "sentence_range": "1219-1222",
        "Text": "For instance,\nIf  \n1\n0\n3\n0\nA\n, B\n0\n2\n0\n4\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n, then AB = BA = \n3\n0\n0\n8\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nObserve that multiplication of diagonal matrices of same order will be commutative Zero matrix as the product of two non zero matrices\nWe know that, for real numbers a, b if ab = 0, then either a = 0 or b = 0 This need\nnot be true for matrices, we will observe this through an example Example 15 Find AB, if \n0\n1\nA\n0\n\u22122\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n and \n3\n5\nB\n0\n0\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb"
    },
    {
        "Chapter": "1",
        "sentence_range": "1220-1223",
        "Text": "Zero matrix as the product of two non zero matrices\nWe know that, for real numbers a, b if ab = 0, then either a = 0 or b = 0 This need\nnot be true for matrices, we will observe this through an example Example 15 Find AB, if \n0\n1\nA\n0\n\u22122\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n and \n3\n5\nB\n0\n0\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb Solution We have \n0\n1\n3\n5\n0\n0\nAB\n0\n2\n0\n0\n0\n0\n\u2212\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n=\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n\uf8f0\n\uf8fb"
    },
    {
        "Chapter": "1",
        "sentence_range": "1221-1224",
        "Text": "This need\nnot be true for matrices, we will observe this through an example Example 15 Find AB, if \n0\n1\nA\n0\n\u22122\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n and \n3\n5\nB\n0\n0\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb Solution We have \n0\n1\n3\n5\n0\n0\nAB\n0\n2\n0\n0\n0\n0\n\u2212\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n=\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n\uf8f0\n\uf8fb Thus, if the product of two matrices is a zero matrix, it is not necessary that one of\nthe matrices is a zero matrix"
    },
    {
        "Chapter": "1",
        "sentence_range": "1222-1225",
        "Text": "Example 15 Find AB, if \n0\n1\nA\n0\n\u22122\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n and \n3\n5\nB\n0\n0\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb Solution We have \n0\n1\n3\n5\n0\n0\nAB\n0\n2\n0\n0\n0\n0\n\u2212\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n=\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n\uf8f0\n\uf8fb Thus, if the product of two matrices is a zero matrix, it is not necessary that one of\nthe matrices is a zero matrix 3"
    },
    {
        "Chapter": "1",
        "sentence_range": "1223-1226",
        "Text": "Solution We have \n0\n1\n3\n5\n0\n0\nAB\n0\n2\n0\n0\n0\n0\n\u2212\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n=\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n\uf8f0\n\uf8fb Thus, if the product of two matrices is a zero matrix, it is not necessary that one of\nthe matrices is a zero matrix 3 4"
    },
    {
        "Chapter": "1",
        "sentence_range": "1224-1227",
        "Text": "Thus, if the product of two matrices is a zero matrix, it is not necessary that one of\nthe matrices is a zero matrix 3 4 6  Properties of multiplication of matrices\nThe multiplication of matrices possesses the following properties, which we state without\nproof"
    },
    {
        "Chapter": "1",
        "sentence_range": "1225-1228",
        "Text": "3 4 6  Properties of multiplication of matrices\nThe multiplication of matrices possesses the following properties, which we state without\nproof 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "1226-1229",
        "Text": "4 6  Properties of multiplication of matrices\nThe multiplication of matrices possesses the following properties, which we state without\nproof 1 The associative law For any three matrices A, B and C"
    },
    {
        "Chapter": "1",
        "sentence_range": "1227-1230",
        "Text": "6  Properties of multiplication of matrices\nThe multiplication of matrices possesses the following properties, which we state without\nproof 1 The associative law For any three matrices A, B and C We have\n(AB) C = A (BC), whenever both sides of the equality are defined"
    },
    {
        "Chapter": "1",
        "sentence_range": "1228-1231",
        "Text": "1 The associative law For any three matrices A, B and C We have\n(AB) C = A (BC), whenever both sides of the equality are defined 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "1229-1232",
        "Text": "The associative law For any three matrices A, B and C We have\n(AB) C = A (BC), whenever both sides of the equality are defined 2 The distributive law For three matrices A, B and C"
    },
    {
        "Chapter": "1",
        "sentence_range": "1230-1233",
        "Text": "We have\n(AB) C = A (BC), whenever both sides of the equality are defined 2 The distributive law For three matrices A, B and C (i) A (B+C) = AB + AC\n(ii) (A+B) C = AC + BC, whenever both sides of equality are defined"
    },
    {
        "Chapter": "1",
        "sentence_range": "1231-1234",
        "Text": "2 The distributive law For three matrices A, B and C (i) A (B+C) = AB + AC\n(ii) (A+B) C = AC + BC, whenever both sides of equality are defined 3"
    },
    {
        "Chapter": "1",
        "sentence_range": "1232-1235",
        "Text": "The distributive law For three matrices A, B and C (i) A (B+C) = AB + AC\n(ii) (A+B) C = AC + BC, whenever both sides of equality are defined 3 The existence of multiplicative identity For every square matrix A, there\nexist an identity matrix of same order such that IA = AI = A"
    },
    {
        "Chapter": "1",
        "sentence_range": "1233-1236",
        "Text": "(i) A (B+C) = AB + AC\n(ii) (A+B) C = AC + BC, whenever both sides of equality are defined 3 The existence of multiplicative identity For every square matrix A, there\nexist an identity matrix of same order such that IA = AI = A Now, we shall verify these properties by examples"
    },
    {
        "Chapter": "1",
        "sentence_range": "1234-1237",
        "Text": "3 The existence of multiplicative identity For every square matrix A, there\nexist an identity matrix of same order such that IA = AI = A Now, we shall verify these properties by examples Example 16 If \n1\n1\n1\n1 3\n1\n2\n3\n4\nA\n2\n0\n3 , B\n0 2\nand C\n2\n0\n2\n1\n3\n1\n2\n1 4\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n=\n=\n= \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n, find\nA(BC), (AB)C and show that (AB)C = A(BC)"
    },
    {
        "Chapter": "1",
        "sentence_range": "1235-1238",
        "Text": "The existence of multiplicative identity For every square matrix A, there\nexist an identity matrix of same order such that IA = AI = A Now, we shall verify these properties by examples Example 16 If \n1\n1\n1\n1 3\n1\n2\n3\n4\nA\n2\n0\n3 , B\n0 2\nand C\n2\n0\n2\n1\n3\n1\n2\n1 4\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n=\n=\n= \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n, find\nA(BC), (AB)C and show that (AB)C = A(BC) Rationalised 2023-24\nMATRICES         55\nSolution We have \n1\n1\n1\n1 3\n1\n0 1 3\n2\n4\n2\n1\nAB\n2\n0\n3\n0 2\n2\n0\n3 6\n0 12\n1 18\n3\n1\n2\n1 4\n3\n0\n2 9\n2\n8\n1 15\n\u2212\n+\n+\n+\n\u2212\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n=\n=\n+\n\u2212\n+\n+\n= \u2212\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n+\n\u2212\n\u2212\n+\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n(AB) (C)\n2\n2\n4\n0\n6\n2\n8\n1\n2\n1\n1\n2\n3\n4\n1 18\n1\n36\n2\n0\n3\n36\n4\n18\n2\n0\n2\n1\n1 15\n1\n30\n2\n0\n3\n30\n4\n15\n+\n+\n\u2212\n\u2212\n+\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n= \u2212\n= \u2212 +\n\u2212 +\n\u2212 \u2212\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n+\n+\n\u2212\n\u2212\n+\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n=\n4\n4\n4\n7\n35\n2\n39\n22\n31\n2\n27\n11\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\nNow\nBC =\n1\n6\n2\n0\n3\n6\n4\n3\n1 3\n1\n2\n3\n4\n0 2\n0\n4\n0\n0\n0\n4\n0\n2\n2\n0\n2\n1\n1 4\n1\n8\n2\n0\n3\n8\n4\n4\n+\n+\n\u2212\n\u2212 +\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n=\n+\n+\n\u2212\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212 +\n\u2212 +\n\u2212 \u2212\n+\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n=\n7\n2\n3\n1\n4\n0\n4\n2\n7\n2\n11\n8\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8f0\n\uf8fb\nTherefore\nA(BC) =\n7\n2\n3\n1\n1\n1\n1\n2\n0\n3\n4\n0\n4\n2\n3\n1\n2\n7\n2\n11\n8\n\u2212\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\u2212\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n=\n7\n4\n7\n2\n0\n2\n3\n4\n11\n1\n2\n8\n14\n0\n21\n4\n0\n6\n6\n0\n33\n2\n0\n24\n21\n4\n14\n6\n0\n4\n9\n4\n22\n3\n2\n16\n+\n\u2212\n+\n+\n\u2212 \u2212\n+\n\u2212 +\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n+\n+\n+\n\u2212\n\u2212 +\n\u2212\n\u2212 +\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n+\n+\n\u2212\n\u2212 +\n\u2212\n\u2212 \u2212\n+\n\uf8f0\n\uf8fb\n=\n4\n4\n4\n7\n35\n2\n39\n22\n31\n2\n27\n11\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb"
    },
    {
        "Chapter": "1",
        "sentence_range": "1236-1239",
        "Text": "Now, we shall verify these properties by examples Example 16 If \n1\n1\n1\n1 3\n1\n2\n3\n4\nA\n2\n0\n3 , B\n0 2\nand C\n2\n0\n2\n1\n3\n1\n2\n1 4\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n=\n=\n= \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n, find\nA(BC), (AB)C and show that (AB)C = A(BC) Rationalised 2023-24\nMATRICES         55\nSolution We have \n1\n1\n1\n1 3\n1\n0 1 3\n2\n4\n2\n1\nAB\n2\n0\n3\n0 2\n2\n0\n3 6\n0 12\n1 18\n3\n1\n2\n1 4\n3\n0\n2 9\n2\n8\n1 15\n\u2212\n+\n+\n+\n\u2212\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n=\n=\n+\n\u2212\n+\n+\n= \u2212\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n+\n\u2212\n\u2212\n+\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n(AB) (C)\n2\n2\n4\n0\n6\n2\n8\n1\n2\n1\n1\n2\n3\n4\n1 18\n1\n36\n2\n0\n3\n36\n4\n18\n2\n0\n2\n1\n1 15\n1\n30\n2\n0\n3\n30\n4\n15\n+\n+\n\u2212\n\u2212\n+\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n= \u2212\n= \u2212 +\n\u2212 +\n\u2212 \u2212\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n+\n+\n\u2212\n\u2212\n+\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n=\n4\n4\n4\n7\n35\n2\n39\n22\n31\n2\n27\n11\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\nNow\nBC =\n1\n6\n2\n0\n3\n6\n4\n3\n1 3\n1\n2\n3\n4\n0 2\n0\n4\n0\n0\n0\n4\n0\n2\n2\n0\n2\n1\n1 4\n1\n8\n2\n0\n3\n8\n4\n4\n+\n+\n\u2212\n\u2212 +\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n=\n+\n+\n\u2212\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212 +\n\u2212 +\n\u2212 \u2212\n+\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n=\n7\n2\n3\n1\n4\n0\n4\n2\n7\n2\n11\n8\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8f0\n\uf8fb\nTherefore\nA(BC) =\n7\n2\n3\n1\n1\n1\n1\n2\n0\n3\n4\n0\n4\n2\n3\n1\n2\n7\n2\n11\n8\n\u2212\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\u2212\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n=\n7\n4\n7\n2\n0\n2\n3\n4\n11\n1\n2\n8\n14\n0\n21\n4\n0\n6\n6\n0\n33\n2\n0\n24\n21\n4\n14\n6\n0\n4\n9\n4\n22\n3\n2\n16\n+\n\u2212\n+\n+\n\u2212 \u2212\n+\n\u2212 +\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n+\n+\n+\n\u2212\n\u2212 +\n\u2212\n\u2212 +\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n+\n+\n\u2212\n\u2212 +\n\u2212\n\u2212 \u2212\n+\n\uf8f0\n\uf8fb\n=\n4\n4\n4\n7\n35\n2\n39\n22\n31\n2\n27\n11\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb Clearly, (AB) C = A  (BC)\nRationalised 2023-24\n 56\nMATHEMATICS\nExample 17 If \n0\n6\n7\n0\n1\n1\n2\nA\n6\n0\n8 , B\n1\n0\n2 , C\n2\n7\n8\n0\n1\n2\n0\n3\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n= \u2212\n=\n= \u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nCalculate AC, BC and (A + B)C"
    },
    {
        "Chapter": "1",
        "sentence_range": "1237-1240",
        "Text": "Example 16 If \n1\n1\n1\n1 3\n1\n2\n3\n4\nA\n2\n0\n3 , B\n0 2\nand C\n2\n0\n2\n1\n3\n1\n2\n1 4\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n=\n=\n= \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n, find\nA(BC), (AB)C and show that (AB)C = A(BC) Rationalised 2023-24\nMATRICES         55\nSolution We have \n1\n1\n1\n1 3\n1\n0 1 3\n2\n4\n2\n1\nAB\n2\n0\n3\n0 2\n2\n0\n3 6\n0 12\n1 18\n3\n1\n2\n1 4\n3\n0\n2 9\n2\n8\n1 15\n\u2212\n+\n+\n+\n\u2212\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n=\n=\n+\n\u2212\n+\n+\n= \u2212\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n+\n\u2212\n\u2212\n+\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n(AB) (C)\n2\n2\n4\n0\n6\n2\n8\n1\n2\n1\n1\n2\n3\n4\n1 18\n1\n36\n2\n0\n3\n36\n4\n18\n2\n0\n2\n1\n1 15\n1\n30\n2\n0\n3\n30\n4\n15\n+\n+\n\u2212\n\u2212\n+\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n= \u2212\n= \u2212 +\n\u2212 +\n\u2212 \u2212\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n+\n+\n\u2212\n\u2212\n+\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n=\n4\n4\n4\n7\n35\n2\n39\n22\n31\n2\n27\n11\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\nNow\nBC =\n1\n6\n2\n0\n3\n6\n4\n3\n1 3\n1\n2\n3\n4\n0 2\n0\n4\n0\n0\n0\n4\n0\n2\n2\n0\n2\n1\n1 4\n1\n8\n2\n0\n3\n8\n4\n4\n+\n+\n\u2212\n\u2212 +\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n=\n+\n+\n\u2212\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212 +\n\u2212 +\n\u2212 \u2212\n+\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n=\n7\n2\n3\n1\n4\n0\n4\n2\n7\n2\n11\n8\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8f0\n\uf8fb\nTherefore\nA(BC) =\n7\n2\n3\n1\n1\n1\n1\n2\n0\n3\n4\n0\n4\n2\n3\n1\n2\n7\n2\n11\n8\n\u2212\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\u2212\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n=\n7\n4\n7\n2\n0\n2\n3\n4\n11\n1\n2\n8\n14\n0\n21\n4\n0\n6\n6\n0\n33\n2\n0\n24\n21\n4\n14\n6\n0\n4\n9\n4\n22\n3\n2\n16\n+\n\u2212\n+\n+\n\u2212 \u2212\n+\n\u2212 +\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n+\n+\n+\n\u2212\n\u2212 +\n\u2212\n\u2212 +\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n+\n+\n\u2212\n\u2212 +\n\u2212\n\u2212 \u2212\n+\n\uf8f0\n\uf8fb\n=\n4\n4\n4\n7\n35\n2\n39\n22\n31\n2\n27\n11\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb Clearly, (AB) C = A  (BC)\nRationalised 2023-24\n 56\nMATHEMATICS\nExample 17 If \n0\n6\n7\n0\n1\n1\n2\nA\n6\n0\n8 , B\n1\n0\n2 , C\n2\n7\n8\n0\n1\n2\n0\n3\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n= \u2212\n=\n= \u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nCalculate AC, BC and (A + B)C Also, verify that (A + B)C = AC + BC\nSolution Now, \n0\n7\n8\nA+B\n5\n0\n10\n8\n6\n0\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n= \u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\nSo\n(A + B) C =\n0\n7\n8\n2\n0\n14\n24\n10\n5\n0\n10\n2\n10\n0\n30\n20\n8\n6\n0\n3\n16\n12\n0\n28\n\u2212\n+\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n= \u2212\n+\n+\n=\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n+\n+\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nFurther\nAC =\n0\n6\n7\n2\n0 12\n21\n9\n6\n0\n8\n2\n12\n0\n24\n12\n7\n8\n0\n3\n14\n16\n0\n30\n\u2212\n+\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n= \u2212\n+\n+\n=\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n+\n+\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nand\nBC =\n0\n1\n1\n2\n0\n2\n3\n1\n1\n0\n2\n2\n2\n0\n6\n8\n1\n2\n0\n3\n2\n4\n0\n2\n\u2212\n+\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n=\n+\n+\n=\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n+\n\u2212\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nSo\nAC + BC =\n9\n1\n10\n12\n8\n20\n30\n2\n28\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n+\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nClearly,\n(A + B) C = AC + BC\nExample 18 If \n1\n2\n3\nA\n3\n2\n1\n4\n2\n1\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n=\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, then show that A3 \u2013 23A \u2013 40 I = O\nSolution We have \n2\n1\n2\n3\n1\n2\n3\n19\n4\n8\nA\nA"
    },
    {
        "Chapter": "1",
        "sentence_range": "1238-1241",
        "Text": "Rationalised 2023-24\nMATRICES         55\nSolution We have \n1\n1\n1\n1 3\n1\n0 1 3\n2\n4\n2\n1\nAB\n2\n0\n3\n0 2\n2\n0\n3 6\n0 12\n1 18\n3\n1\n2\n1 4\n3\n0\n2 9\n2\n8\n1 15\n\u2212\n+\n+\n+\n\u2212\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n=\n=\n+\n\u2212\n+\n+\n= \u2212\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n+\n\u2212\n\u2212\n+\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n(AB) (C)\n2\n2\n4\n0\n6\n2\n8\n1\n2\n1\n1\n2\n3\n4\n1 18\n1\n36\n2\n0\n3\n36\n4\n18\n2\n0\n2\n1\n1 15\n1\n30\n2\n0\n3\n30\n4\n15\n+\n+\n\u2212\n\u2212\n+\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n= \u2212\n= \u2212 +\n\u2212 +\n\u2212 \u2212\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n+\n+\n\u2212\n\u2212\n+\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n=\n4\n4\n4\n7\n35\n2\n39\n22\n31\n2\n27\n11\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\nNow\nBC =\n1\n6\n2\n0\n3\n6\n4\n3\n1 3\n1\n2\n3\n4\n0 2\n0\n4\n0\n0\n0\n4\n0\n2\n2\n0\n2\n1\n1 4\n1\n8\n2\n0\n3\n8\n4\n4\n+\n+\n\u2212\n\u2212 +\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n=\n+\n+\n\u2212\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212 +\n\u2212 +\n\u2212 \u2212\n+\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n=\n7\n2\n3\n1\n4\n0\n4\n2\n7\n2\n11\n8\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8f0\n\uf8fb\nTherefore\nA(BC) =\n7\n2\n3\n1\n1\n1\n1\n2\n0\n3\n4\n0\n4\n2\n3\n1\n2\n7\n2\n11\n8\n\u2212\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\u2212\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n=\n7\n4\n7\n2\n0\n2\n3\n4\n11\n1\n2\n8\n14\n0\n21\n4\n0\n6\n6\n0\n33\n2\n0\n24\n21\n4\n14\n6\n0\n4\n9\n4\n22\n3\n2\n16\n+\n\u2212\n+\n+\n\u2212 \u2212\n+\n\u2212 +\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n+\n+\n+\n\u2212\n\u2212 +\n\u2212\n\u2212 +\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n+\n+\n\u2212\n\u2212 +\n\u2212\n\u2212 \u2212\n+\n\uf8f0\n\uf8fb\n=\n4\n4\n4\n7\n35\n2\n39\n22\n31\n2\n27\n11\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb Clearly, (AB) C = A  (BC)\nRationalised 2023-24\n 56\nMATHEMATICS\nExample 17 If \n0\n6\n7\n0\n1\n1\n2\nA\n6\n0\n8 , B\n1\n0\n2 , C\n2\n7\n8\n0\n1\n2\n0\n3\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n= \u2212\n=\n= \u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nCalculate AC, BC and (A + B)C Also, verify that (A + B)C = AC + BC\nSolution Now, \n0\n7\n8\nA+B\n5\n0\n10\n8\n6\n0\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n= \u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\nSo\n(A + B) C =\n0\n7\n8\n2\n0\n14\n24\n10\n5\n0\n10\n2\n10\n0\n30\n20\n8\n6\n0\n3\n16\n12\n0\n28\n\u2212\n+\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n= \u2212\n+\n+\n=\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n+\n+\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nFurther\nAC =\n0\n6\n7\n2\n0 12\n21\n9\n6\n0\n8\n2\n12\n0\n24\n12\n7\n8\n0\n3\n14\n16\n0\n30\n\u2212\n+\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n= \u2212\n+\n+\n=\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n+\n+\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nand\nBC =\n0\n1\n1\n2\n0\n2\n3\n1\n1\n0\n2\n2\n2\n0\n6\n8\n1\n2\n0\n3\n2\n4\n0\n2\n\u2212\n+\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n=\n+\n+\n=\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n+\n\u2212\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nSo\nAC + BC =\n9\n1\n10\n12\n8\n20\n30\n2\n28\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n+\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nClearly,\n(A + B) C = AC + BC\nExample 18 If \n1\n2\n3\nA\n3\n2\n1\n4\n2\n1\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n=\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, then show that A3 \u2013 23A \u2013 40 I = O\nSolution We have \n2\n1\n2\n3\n1\n2\n3\n19\n4\n8\nA\nA A\n3\n2\n1\n3\n2\n1\n1\n12\n8\n4\n2\n1\n4\n2\n1\n14\n6\n15\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n=\n=\n\u2212\n\u2212\n=\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nRationalised 2023-24\nMATRICES         57\nSo\nA3 = A A2 = \n1\n2\n3\n19\n4\n8\n63\n46\n69\n3\n2\n1\n1\n12\n8\n69\n6\n23\n4\n2\n1\n14\n6\n15\n92\n46\n63\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n=\n\u2212\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nNow\nA3 \u2013 23A \u2013 40I = \n63\n46\n69\n1\n2\n3\n1\n0\n0\n69\n6\n23 \u2013 23 3\n2\n1 \u2013 40 0\n1\n0\n92\n46\n63\n4\n2\n1\n0\n0 1\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n=\n63\n46\n69\n23\n46\n69\n40\n0\n0\n69\n6\n23\n69\n46\n23\n0\n40\n0\n92\n46\n63\n92\n46\n23\n0\n0\n40\n\u2212\n\u2212\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n+ \u2212\n\u2212\n+\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\u2212\n\u2212\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n=\n63\n23\n40\n46\n46\n0\n69\n69\n0\n69\n69\n0\n6\n46\n40\n23\n23\n0\n92\n92\n0\n46\n46\n0\n63\n23\n40\n\u2212\n\u2212\n\u2212\n+\n\u2212\n+\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n+\n\u2212 +\n\u2212\n\u2212\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n+\n\u2212\n+\n\u2212\n\u2212\n\uf8f0\n\uf8fb\n=\n0\n0\n0\n0\n0\n0\nO\n0\n0\n0\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa =\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nExample 19 In a legislative assembly election, a political group hired a public relations\nfirm to promote its candidate in three ways: telephone, house calls, and letters"
    },
    {
        "Chapter": "1",
        "sentence_range": "1239-1242",
        "Text": "Clearly, (AB) C = A  (BC)\nRationalised 2023-24\n 56\nMATHEMATICS\nExample 17 If \n0\n6\n7\n0\n1\n1\n2\nA\n6\n0\n8 , B\n1\n0\n2 , C\n2\n7\n8\n0\n1\n2\n0\n3\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n= \u2212\n=\n= \u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nCalculate AC, BC and (A + B)C Also, verify that (A + B)C = AC + BC\nSolution Now, \n0\n7\n8\nA+B\n5\n0\n10\n8\n6\n0\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n= \u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\nSo\n(A + B) C =\n0\n7\n8\n2\n0\n14\n24\n10\n5\n0\n10\n2\n10\n0\n30\n20\n8\n6\n0\n3\n16\n12\n0\n28\n\u2212\n+\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n= \u2212\n+\n+\n=\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n+\n+\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nFurther\nAC =\n0\n6\n7\n2\n0 12\n21\n9\n6\n0\n8\n2\n12\n0\n24\n12\n7\n8\n0\n3\n14\n16\n0\n30\n\u2212\n+\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n= \u2212\n+\n+\n=\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n+\n+\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nand\nBC =\n0\n1\n1\n2\n0\n2\n3\n1\n1\n0\n2\n2\n2\n0\n6\n8\n1\n2\n0\n3\n2\n4\n0\n2\n\u2212\n+\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n=\n+\n+\n=\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n+\n\u2212\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nSo\nAC + BC =\n9\n1\n10\n12\n8\n20\n30\n2\n28\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n+\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nClearly,\n(A + B) C = AC + BC\nExample 18 If \n1\n2\n3\nA\n3\n2\n1\n4\n2\n1\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n=\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, then show that A3 \u2013 23A \u2013 40 I = O\nSolution We have \n2\n1\n2\n3\n1\n2\n3\n19\n4\n8\nA\nA A\n3\n2\n1\n3\n2\n1\n1\n12\n8\n4\n2\n1\n4\n2\n1\n14\n6\n15\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n=\n=\n\u2212\n\u2212\n=\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nRationalised 2023-24\nMATRICES         57\nSo\nA3 = A A2 = \n1\n2\n3\n19\n4\n8\n63\n46\n69\n3\n2\n1\n1\n12\n8\n69\n6\n23\n4\n2\n1\n14\n6\n15\n92\n46\n63\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n=\n\u2212\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nNow\nA3 \u2013 23A \u2013 40I = \n63\n46\n69\n1\n2\n3\n1\n0\n0\n69\n6\n23 \u2013 23 3\n2\n1 \u2013 40 0\n1\n0\n92\n46\n63\n4\n2\n1\n0\n0 1\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n=\n63\n46\n69\n23\n46\n69\n40\n0\n0\n69\n6\n23\n69\n46\n23\n0\n40\n0\n92\n46\n63\n92\n46\n23\n0\n0\n40\n\u2212\n\u2212\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n+ \u2212\n\u2212\n+\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\u2212\n\u2212\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n=\n63\n23\n40\n46\n46\n0\n69\n69\n0\n69\n69\n0\n6\n46\n40\n23\n23\n0\n92\n92\n0\n46\n46\n0\n63\n23\n40\n\u2212\n\u2212\n\u2212\n+\n\u2212\n+\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n+\n\u2212 +\n\u2212\n\u2212\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n+\n\u2212\n+\n\u2212\n\u2212\n\uf8f0\n\uf8fb\n=\n0\n0\n0\n0\n0\n0\nO\n0\n0\n0\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa =\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nExample 19 In a legislative assembly election, a political group hired a public relations\nfirm to promote its candidate in three ways: telephone, house calls, and letters The\ncost per contact (in paise) is given in matrix A as\nA = \n40\nTelephone\n100\nHousecall\n50\nLetter\n\uf8eeCost per contact\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nThe number of contacts of each type made in two cities X and Y is given by\nTelephone\nHousecall\nLetter\n1000\n500\n5000\nX\nB\nY\n3000\n1000 10,000\n\u2192\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\u2192\n\uf8f0\n\uf8fb"
    },
    {
        "Chapter": "1",
        "sentence_range": "1240-1243",
        "Text": "Also, verify that (A + B)C = AC + BC\nSolution Now, \n0\n7\n8\nA+B\n5\n0\n10\n8\n6\n0\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n= \u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\nSo\n(A + B) C =\n0\n7\n8\n2\n0\n14\n24\n10\n5\n0\n10\n2\n10\n0\n30\n20\n8\n6\n0\n3\n16\n12\n0\n28\n\u2212\n+\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n= \u2212\n+\n+\n=\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n+\n+\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nFurther\nAC =\n0\n6\n7\n2\n0 12\n21\n9\n6\n0\n8\n2\n12\n0\n24\n12\n7\n8\n0\n3\n14\n16\n0\n30\n\u2212\n+\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n= \u2212\n+\n+\n=\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n+\n+\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nand\nBC =\n0\n1\n1\n2\n0\n2\n3\n1\n1\n0\n2\n2\n2\n0\n6\n8\n1\n2\n0\n3\n2\n4\n0\n2\n\u2212\n+\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n=\n+\n+\n=\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n+\n\u2212\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nSo\nAC + BC =\n9\n1\n10\n12\n8\n20\n30\n2\n28\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n+\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nClearly,\n(A + B) C = AC + BC\nExample 18 If \n1\n2\n3\nA\n3\n2\n1\n4\n2\n1\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n=\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, then show that A3 \u2013 23A \u2013 40 I = O\nSolution We have \n2\n1\n2\n3\n1\n2\n3\n19\n4\n8\nA\nA A\n3\n2\n1\n3\n2\n1\n1\n12\n8\n4\n2\n1\n4\n2\n1\n14\n6\n15\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n=\n=\n\u2212\n\u2212\n=\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nRationalised 2023-24\nMATRICES         57\nSo\nA3 = A A2 = \n1\n2\n3\n19\n4\n8\n63\n46\n69\n3\n2\n1\n1\n12\n8\n69\n6\n23\n4\n2\n1\n14\n6\n15\n92\n46\n63\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n=\n\u2212\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nNow\nA3 \u2013 23A \u2013 40I = \n63\n46\n69\n1\n2\n3\n1\n0\n0\n69\n6\n23 \u2013 23 3\n2\n1 \u2013 40 0\n1\n0\n92\n46\n63\n4\n2\n1\n0\n0 1\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n=\n63\n46\n69\n23\n46\n69\n40\n0\n0\n69\n6\n23\n69\n46\n23\n0\n40\n0\n92\n46\n63\n92\n46\n23\n0\n0\n40\n\u2212\n\u2212\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n+ \u2212\n\u2212\n+\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\u2212\n\u2212\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n=\n63\n23\n40\n46\n46\n0\n69\n69\n0\n69\n69\n0\n6\n46\n40\n23\n23\n0\n92\n92\n0\n46\n46\n0\n63\n23\n40\n\u2212\n\u2212\n\u2212\n+\n\u2212\n+\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n+\n\u2212 +\n\u2212\n\u2212\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n+\n\u2212\n+\n\u2212\n\u2212\n\uf8f0\n\uf8fb\n=\n0\n0\n0\n0\n0\n0\nO\n0\n0\n0\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa =\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nExample 19 In a legislative assembly election, a political group hired a public relations\nfirm to promote its candidate in three ways: telephone, house calls, and letters The\ncost per contact (in paise) is given in matrix A as\nA = \n40\nTelephone\n100\nHousecall\n50\nLetter\n\uf8eeCost per contact\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nThe number of contacts of each type made in two cities X and Y is given by\nTelephone\nHousecall\nLetter\n1000\n500\n5000\nX\nB\nY\n3000\n1000 10,000\n\u2192\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\u2192\n\uf8f0\n\uf8fb Find the total amount spent by the group in the two\ncities X and Y"
    },
    {
        "Chapter": "1",
        "sentence_range": "1241-1244",
        "Text": "A\n3\n2\n1\n3\n2\n1\n1\n12\n8\n4\n2\n1\n4\n2\n1\n14\n6\n15\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n=\n=\n\u2212\n\u2212\n=\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nRationalised 2023-24\nMATRICES         57\nSo\nA3 = A A2 = \n1\n2\n3\n19\n4\n8\n63\n46\n69\n3\n2\n1\n1\n12\n8\n69\n6\n23\n4\n2\n1\n14\n6\n15\n92\n46\n63\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n=\n\u2212\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nNow\nA3 \u2013 23A \u2013 40I = \n63\n46\n69\n1\n2\n3\n1\n0\n0\n69\n6\n23 \u2013 23 3\n2\n1 \u2013 40 0\n1\n0\n92\n46\n63\n4\n2\n1\n0\n0 1\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n=\n63\n46\n69\n23\n46\n69\n40\n0\n0\n69\n6\n23\n69\n46\n23\n0\n40\n0\n92\n46\n63\n92\n46\n23\n0\n0\n40\n\u2212\n\u2212\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n+ \u2212\n\u2212\n+\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\u2212\n\u2212\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n=\n63\n23\n40\n46\n46\n0\n69\n69\n0\n69\n69\n0\n6\n46\n40\n23\n23\n0\n92\n92\n0\n46\n46\n0\n63\n23\n40\n\u2212\n\u2212\n\u2212\n+\n\u2212\n+\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n+\n\u2212 +\n\u2212\n\u2212\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n+\n\u2212\n+\n\u2212\n\u2212\n\uf8f0\n\uf8fb\n=\n0\n0\n0\n0\n0\n0\nO\n0\n0\n0\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa =\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nExample 19 In a legislative assembly election, a political group hired a public relations\nfirm to promote its candidate in three ways: telephone, house calls, and letters The\ncost per contact (in paise) is given in matrix A as\nA = \n40\nTelephone\n100\nHousecall\n50\nLetter\n\uf8eeCost per contact\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nThe number of contacts of each type made in two cities X and Y is given by\nTelephone\nHousecall\nLetter\n1000\n500\n5000\nX\nB\nY\n3000\n1000 10,000\n\u2192\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\u2192\n\uf8f0\n\uf8fb Find the total amount spent by the group in the two\ncities X and Y Rationalised 2023-24\n 58\nMATHEMATICS\nSolution We have\nBA =\n40,000\n50,000\n250,000\nX\nY\n120,000 +100,000 +500,000\n+\n+\n\u2192\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa \u2192\n\uf8f0\n\uf8fb\n=\n340,000\nX\nY\n720,000\n\u2192\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa \u2192\n\uf8f0\n\uf8fb\nSo the total amount spent by the group in the two cities is 340,000 paise and\n720,000 paise, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "1242-1245",
        "Text": "The\ncost per contact (in paise) is given in matrix A as\nA = \n40\nTelephone\n100\nHousecall\n50\nLetter\n\uf8eeCost per contact\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nThe number of contacts of each type made in two cities X and Y is given by\nTelephone\nHousecall\nLetter\n1000\n500\n5000\nX\nB\nY\n3000\n1000 10,000\n\u2192\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\u2192\n\uf8f0\n\uf8fb Find the total amount spent by the group in the two\ncities X and Y Rationalised 2023-24\n 58\nMATHEMATICS\nSolution We have\nBA =\n40,000\n50,000\n250,000\nX\nY\n120,000 +100,000 +500,000\n+\n+\n\u2192\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa \u2192\n\uf8f0\n\uf8fb\n=\n340,000\nX\nY\n720,000\n\u2192\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa \u2192\n\uf8f0\n\uf8fb\nSo the total amount spent by the group in the two cities is 340,000 paise and\n720,000 paise, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "1243-1246",
        "Text": "Find the total amount spent by the group in the two\ncities X and Y Rationalised 2023-24\n 58\nMATHEMATICS\nSolution We have\nBA =\n40,000\n50,000\n250,000\nX\nY\n120,000 +100,000 +500,000\n+\n+\n\u2192\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa \u2192\n\uf8f0\n\uf8fb\n=\n340,000\nX\nY\n720,000\n\u2192\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa \u2192\n\uf8f0\n\uf8fb\nSo the total amount spent by the group in the two cities is 340,000 paise and\n720,000 paise, i e , ` 3400 and ` 7200, respectively"
    },
    {
        "Chapter": "1",
        "sentence_range": "1244-1247",
        "Text": "Rationalised 2023-24\n 58\nMATHEMATICS\nSolution We have\nBA =\n40,000\n50,000\n250,000\nX\nY\n120,000 +100,000 +500,000\n+\n+\n\u2192\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa \u2192\n\uf8f0\n\uf8fb\n=\n340,000\nX\nY\n720,000\n\u2192\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa \u2192\n\uf8f0\n\uf8fb\nSo the total amount spent by the group in the two cities is 340,000 paise and\n720,000 paise, i e , ` 3400 and ` 7200, respectively EXERCISE 3"
    },
    {
        "Chapter": "1",
        "sentence_range": "1245-1248",
        "Text": "e , ` 3400 and ` 7200, respectively EXERCISE 3 2\n1"
    },
    {
        "Chapter": "1",
        "sentence_range": "1246-1249",
        "Text": ", ` 3400 and ` 7200, respectively EXERCISE 3 2\n1 Let \n2\n4\n1\n3\n2\n5\nA\n, B\n, C\n3\n2\n2\n5\n3\n4\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n=\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nFind each of the following:\n(i) A + B\n(ii) A \u2013 B\n(iii) 3A \u2013 C\n(iv) AB\n(v) BA\n2"
    },
    {
        "Chapter": "1",
        "sentence_range": "1247-1250",
        "Text": "EXERCISE 3 2\n1 Let \n2\n4\n1\n3\n2\n5\nA\n, B\n, C\n3\n2\n2\n5\n3\n4\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n=\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nFind each of the following:\n(i) A + B\n(ii) A \u2013 B\n(iii) 3A \u2013 C\n(iv) AB\n(v) BA\n2 Compute the following:\n(i)\na\nb\na\nb\nb\na\nb\na\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n(ii)\n2\n2\n2\n2\n2\n2\n2\n2\n2\n2\n2\n2\na\nb\nb\nc\nab\nbc\nac\nab\na\nc\na\nb\n\uf8ee\n\uf8f9\n+\n+\n\uf8ee\n\uf8f9\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n+\n+\n\uf8f0\n\uf8fb\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n(iii)\n1\n4\n6\n12\n7\n6\n8\n5\n16\n8\n0\n5\n2\n8\n5\n3\n2\n4\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n(iv)\n2\n2\n2\n2\n2\n2\n2\n2\ncos\nsin\nsin\ncos\nsin\ncos\ncos\nsin\nx\nx\nx\nx\nx\nx\nx\nx\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n3"
    },
    {
        "Chapter": "1",
        "sentence_range": "1248-1251",
        "Text": "2\n1 Let \n2\n4\n1\n3\n2\n5\nA\n, B\n, C\n3\n2\n2\n5\n3\n4\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n=\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nFind each of the following:\n(i) A + B\n(ii) A \u2013 B\n(iii) 3A \u2013 C\n(iv) AB\n(v) BA\n2 Compute the following:\n(i)\na\nb\na\nb\nb\na\nb\na\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n(ii)\n2\n2\n2\n2\n2\n2\n2\n2\n2\n2\n2\n2\na\nb\nb\nc\nab\nbc\nac\nab\na\nc\na\nb\n\uf8ee\n\uf8f9\n+\n+\n\uf8ee\n\uf8f9\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n+\n+\n\uf8f0\n\uf8fb\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n(iii)\n1\n4\n6\n12\n7\n6\n8\n5\n16\n8\n0\n5\n2\n8\n5\n3\n2\n4\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n(iv)\n2\n2\n2\n2\n2\n2\n2\n2\ncos\nsin\nsin\ncos\nsin\ncos\ncos\nsin\nx\nx\nx\nx\nx\nx\nx\nx\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n3 Compute the indicated products"
    },
    {
        "Chapter": "1",
        "sentence_range": "1249-1252",
        "Text": "Let \n2\n4\n1\n3\n2\n5\nA\n, B\n, C\n3\n2\n2\n5\n3\n4\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n=\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nFind each of the following:\n(i) A + B\n(ii) A \u2013 B\n(iii) 3A \u2013 C\n(iv) AB\n(v) BA\n2 Compute the following:\n(i)\na\nb\na\nb\nb\na\nb\na\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n(ii)\n2\n2\n2\n2\n2\n2\n2\n2\n2\n2\n2\n2\na\nb\nb\nc\nab\nbc\nac\nab\na\nc\na\nb\n\uf8ee\n\uf8f9\n+\n+\n\uf8ee\n\uf8f9\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n+\n+\n\uf8f0\n\uf8fb\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n(iii)\n1\n4\n6\n12\n7\n6\n8\n5\n16\n8\n0\n5\n2\n8\n5\n3\n2\n4\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n(iv)\n2\n2\n2\n2\n2\n2\n2\n2\ncos\nsin\nsin\ncos\nsin\ncos\ncos\nsin\nx\nx\nx\nx\nx\nx\nx\nx\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n3 Compute the indicated products (i)\na\nb\na\nb\nb\na\nb\na\n\u2212\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\u2212\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n(ii)\n1\n2\n3\n\uf8ee \uf8f9\n\uf8ef \uf8fa\n\uf8ef \uf8fa\n\uf8ef \uf8fa\n\uf8f0 \uf8fb\n[2   3   4]\n(iii)\n1\n2\n1\n2\n3\n2\n3\n2\n3\n1\n\u2212\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n(iv)\n2\n3\n4\n1\n3\n5\n3\n4\n5\n0\n2\n4\n4\n5\n6\n3\n0\n5\n\u2212\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n(v)\n2 1\n1\n0\n1\n3 2\n1\n2\n1\n1 1\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa \u2212\uf8f0\n\uf8fb\n\uf8ef\n\uf8fa\n\u2212\uf8f0\n\uf8fb\n(vi)\n2\n3\n3\n1\n3\n1\n0\n1\n0\n2\n3\n1\n\u2212\n\uf8ee\n\uf8f9\n\u2212\n\uf8ee\n\uf8f9 \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\u2212\uf8f0\n\uf8fb \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nRationalised 2023-24\nMATRICES         59\n4"
    },
    {
        "Chapter": "1",
        "sentence_range": "1250-1253",
        "Text": "Compute the following:\n(i)\na\nb\na\nb\nb\na\nb\na\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n(ii)\n2\n2\n2\n2\n2\n2\n2\n2\n2\n2\n2\n2\na\nb\nb\nc\nab\nbc\nac\nab\na\nc\na\nb\n\uf8ee\n\uf8f9\n+\n+\n\uf8ee\n\uf8f9\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n+\n+\n\uf8f0\n\uf8fb\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n(iii)\n1\n4\n6\n12\n7\n6\n8\n5\n16\n8\n0\n5\n2\n8\n5\n3\n2\n4\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n(iv)\n2\n2\n2\n2\n2\n2\n2\n2\ncos\nsin\nsin\ncos\nsin\ncos\ncos\nsin\nx\nx\nx\nx\nx\nx\nx\nx\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n3 Compute the indicated products (i)\na\nb\na\nb\nb\na\nb\na\n\u2212\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\u2212\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n(ii)\n1\n2\n3\n\uf8ee \uf8f9\n\uf8ef \uf8fa\n\uf8ef \uf8fa\n\uf8ef \uf8fa\n\uf8f0 \uf8fb\n[2   3   4]\n(iii)\n1\n2\n1\n2\n3\n2\n3\n2\n3\n1\n\u2212\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n(iv)\n2\n3\n4\n1\n3\n5\n3\n4\n5\n0\n2\n4\n4\n5\n6\n3\n0\n5\n\u2212\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n(v)\n2 1\n1\n0\n1\n3 2\n1\n2\n1\n1 1\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa \u2212\uf8f0\n\uf8fb\n\uf8ef\n\uf8fa\n\u2212\uf8f0\n\uf8fb\n(vi)\n2\n3\n3\n1\n3\n1\n0\n1\n0\n2\n3\n1\n\u2212\n\uf8ee\n\uf8f9\n\u2212\n\uf8ee\n\uf8f9 \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\u2212\uf8f0\n\uf8fb \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nRationalised 2023-24\nMATRICES         59\n4 If \n1\n2\n3\n3\n1\n2\n4\n1\n2\nA\n5\n0\n2 , B\n4\n2\n5\nand C\n0\n3\n2\n1\n1\n1\n2\n0\n3\n1\n2\n3\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n=\n=\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n, then compute\n(A+B) and (B \u2013 C)"
    },
    {
        "Chapter": "1",
        "sentence_range": "1251-1254",
        "Text": "Compute the indicated products (i)\na\nb\na\nb\nb\na\nb\na\n\u2212\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\u2212\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n(ii)\n1\n2\n3\n\uf8ee \uf8f9\n\uf8ef \uf8fa\n\uf8ef \uf8fa\n\uf8ef \uf8fa\n\uf8f0 \uf8fb\n[2   3   4]\n(iii)\n1\n2\n1\n2\n3\n2\n3\n2\n3\n1\n\u2212\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n(iv)\n2\n3\n4\n1\n3\n5\n3\n4\n5\n0\n2\n4\n4\n5\n6\n3\n0\n5\n\u2212\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n(v)\n2 1\n1\n0\n1\n3 2\n1\n2\n1\n1 1\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa \u2212\uf8f0\n\uf8fb\n\uf8ef\n\uf8fa\n\u2212\uf8f0\n\uf8fb\n(vi)\n2\n3\n3\n1\n3\n1\n0\n1\n0\n2\n3\n1\n\u2212\n\uf8ee\n\uf8f9\n\u2212\n\uf8ee\n\uf8f9 \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\u2212\uf8f0\n\uf8fb \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nRationalised 2023-24\nMATRICES         59\n4 If \n1\n2\n3\n3\n1\n2\n4\n1\n2\nA\n5\n0\n2 , B\n4\n2\n5\nand C\n0\n3\n2\n1\n1\n1\n2\n0\n3\n1\n2\n3\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n=\n=\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n, then compute\n(A+B) and (B \u2013 C) Also, verify that A + (B \u2013 C) = (A + B) \u2013 C"
    },
    {
        "Chapter": "1",
        "sentence_range": "1252-1255",
        "Text": "(i)\na\nb\na\nb\nb\na\nb\na\n\u2212\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\u2212\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n(ii)\n1\n2\n3\n\uf8ee \uf8f9\n\uf8ef \uf8fa\n\uf8ef \uf8fa\n\uf8ef \uf8fa\n\uf8f0 \uf8fb\n[2   3   4]\n(iii)\n1\n2\n1\n2\n3\n2\n3\n2\n3\n1\n\u2212\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n(iv)\n2\n3\n4\n1\n3\n5\n3\n4\n5\n0\n2\n4\n4\n5\n6\n3\n0\n5\n\u2212\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n(v)\n2 1\n1\n0\n1\n3 2\n1\n2\n1\n1 1\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa \u2212\uf8f0\n\uf8fb\n\uf8ef\n\uf8fa\n\u2212\uf8f0\n\uf8fb\n(vi)\n2\n3\n3\n1\n3\n1\n0\n1\n0\n2\n3\n1\n\u2212\n\uf8ee\n\uf8f9\n\u2212\n\uf8ee\n\uf8f9 \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\u2212\uf8f0\n\uf8fb \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nRationalised 2023-24\nMATRICES         59\n4 If \n1\n2\n3\n3\n1\n2\n4\n1\n2\nA\n5\n0\n2 , B\n4\n2\n5\nand C\n0\n3\n2\n1\n1\n1\n2\n0\n3\n1\n2\n3\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n=\n=\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n, then compute\n(A+B) and (B \u2013 C) Also, verify that A + (B \u2013 C) = (A + B) \u2013 C 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "1253-1256",
        "Text": "If \n1\n2\n3\n3\n1\n2\n4\n1\n2\nA\n5\n0\n2 , B\n4\n2\n5\nand C\n0\n3\n2\n1\n1\n1\n2\n0\n3\n1\n2\n3\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n=\n=\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n, then compute\n(A+B) and (B \u2013 C) Also, verify that A + (B \u2013 C) = (A + B) \u2013 C 5 If \n2\n5\n2\n3\n1\n1\n3\n3\n5\n5\n1\n2\n4\n1\n2\n4\nA\nand B\n3\n3\n3\n5\n5\n5\n7\n2\n7\n6\n2\n2\n3\n3\n5\n5\n5\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n=\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n, then compute 3A \u2013 5B"
    },
    {
        "Chapter": "1",
        "sentence_range": "1254-1257",
        "Text": "Also, verify that A + (B \u2013 C) = (A + B) \u2013 C 5 If \n2\n5\n2\n3\n1\n1\n3\n3\n5\n5\n1\n2\n4\n1\n2\n4\nA\nand B\n3\n3\n3\n5\n5\n5\n7\n2\n7\n6\n2\n2\n3\n3\n5\n5\n5\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n=\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n, then compute 3A \u2013 5B 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "1255-1258",
        "Text": "5 If \n2\n5\n2\n3\n1\n1\n3\n3\n5\n5\n1\n2\n4\n1\n2\n4\nA\nand B\n3\n3\n3\n5\n5\n5\n7\n2\n7\n6\n2\n2\n3\n3\n5\n5\n5\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n=\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n, then compute 3A \u2013 5B 6 Simplify \ncos\nsin\nsin\ncos\ncos\n+ sin\nsin\ncos\ncos\nsin\n\u03b8\n\u03b8\n\u03b8\n\u2212\n\u03b8\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\u03b8\n\u03b8\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u03b8\n\u03b8\n\u03b8\n\u03b8\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n7"
    },
    {
        "Chapter": "1",
        "sentence_range": "1256-1259",
        "Text": "If \n2\n5\n2\n3\n1\n1\n3\n3\n5\n5\n1\n2\n4\n1\n2\n4\nA\nand B\n3\n3\n3\n5\n5\n5\n7\n2\n7\n6\n2\n2\n3\n3\n5\n5\n5\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n=\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n, then compute 3A \u2013 5B 6 Simplify \ncos\nsin\nsin\ncos\ncos\n+ sin\nsin\ncos\ncos\nsin\n\u03b8\n\u03b8\n\u03b8\n\u2212\n\u03b8\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\u03b8\n\u03b8\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u03b8\n\u03b8\n\u03b8\n\u03b8\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n7 Find X and Y, if\n(i)\n7\n0\n3\n0\nX + Y\nand X \u2013 Y\n2\n5\n0\n3\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n(ii)\n2\n3\n2\n2\n2X + 3Y\nand 3X\n2Y\n4\n0\n1\n\u22125\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n+\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n8"
    },
    {
        "Chapter": "1",
        "sentence_range": "1257-1260",
        "Text": "6 Simplify \ncos\nsin\nsin\ncos\ncos\n+ sin\nsin\ncos\ncos\nsin\n\u03b8\n\u03b8\n\u03b8\n\u2212\n\u03b8\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\u03b8\n\u03b8\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u03b8\n\u03b8\n\u03b8\n\u03b8\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n7 Find X and Y, if\n(i)\n7\n0\n3\n0\nX + Y\nand X \u2013 Y\n2\n5\n0\n3\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n(ii)\n2\n3\n2\n2\n2X + 3Y\nand 3X\n2Y\n4\n0\n1\n\u22125\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n+\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n8 Find X, if Y = \n3\n2\n1\n4\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n and 2X + Y = \n1\n0\n3\n2\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\u2212\n\uf8fb\n9"
    },
    {
        "Chapter": "1",
        "sentence_range": "1258-1261",
        "Text": "Simplify \ncos\nsin\nsin\ncos\ncos\n+ sin\nsin\ncos\ncos\nsin\n\u03b8\n\u03b8\n\u03b8\n\u2212\n\u03b8\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\u03b8\n\u03b8\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u03b8\n\u03b8\n\u03b8\n\u03b8\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n7 Find X and Y, if\n(i)\n7\n0\n3\n0\nX + Y\nand X \u2013 Y\n2\n5\n0\n3\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n(ii)\n2\n3\n2\n2\n2X + 3Y\nand 3X\n2Y\n4\n0\n1\n\u22125\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n+\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n8 Find X, if Y = \n3\n2\n1\n4\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n and 2X + Y = \n1\n0\n3\n2\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\u2212\n\uf8fb\n9 Find x and y, if \n1\n3\n0\n5\n6\n2 0\n1\n2\n1\n8\ny\nx\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n+\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n10"
    },
    {
        "Chapter": "1",
        "sentence_range": "1259-1262",
        "Text": "Find X and Y, if\n(i)\n7\n0\n3\n0\nX + Y\nand X \u2013 Y\n2\n5\n0\n3\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n(ii)\n2\n3\n2\n2\n2X + 3Y\nand 3X\n2Y\n4\n0\n1\n\u22125\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n+\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n8 Find X, if Y = \n3\n2\n1\n4\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n and 2X + Y = \n1\n0\n3\n2\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\u2212\n\uf8fb\n9 Find x and y, if \n1\n3\n0\n5\n6\n2 0\n1\n2\n1\n8\ny\nx\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n+\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n10 Solve the equation for x, y, z and t, if \n1\n1\n3\n5\n2\n3\n3\n0\n2\n4\n6\nx\nz\ny\nt\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n+\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n11"
    },
    {
        "Chapter": "1",
        "sentence_range": "1260-1263",
        "Text": "Find X, if Y = \n3\n2\n1\n4\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n and 2X + Y = \n1\n0\n3\n2\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\u2212\n\uf8fb\n9 Find x and y, if \n1\n3\n0\n5\n6\n2 0\n1\n2\n1\n8\ny\nx\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n+\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n10 Solve the equation for x, y, z and t, if \n1\n1\n3\n5\n2\n3\n3\n0\n2\n4\n6\nx\nz\ny\nt\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n+\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n11 If \n2\n1\n10\n3\n1\n5\nx\ny \u2212\n\uf8ee \uf8f9\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n+\n=\n\uf8ef \uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0 \uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n, find the values of x and y"
    },
    {
        "Chapter": "1",
        "sentence_range": "1261-1264",
        "Text": "Find x and y, if \n1\n3\n0\n5\n6\n2 0\n1\n2\n1\n8\ny\nx\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n+\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n10 Solve the equation for x, y, z and t, if \n1\n1\n3\n5\n2\n3\n3\n0\n2\n4\n6\nx\nz\ny\nt\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n+\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n11 If \n2\n1\n10\n3\n1\n5\nx\ny \u2212\n\uf8ee \uf8f9\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n+\n=\n\uf8ef \uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0 \uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n, find the values of x and y 12"
    },
    {
        "Chapter": "1",
        "sentence_range": "1262-1265",
        "Text": "Solve the equation for x, y, z and t, if \n1\n1\n3\n5\n2\n3\n3\n0\n2\n4\n6\nx\nz\ny\nt\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n+\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n11 If \n2\n1\n10\n3\n1\n5\nx\ny \u2212\n\uf8ee \uf8f9\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n+\n=\n\uf8ef \uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0 \uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n, find the values of x and y 12 Given \n6\n4\n3\n1\n2\n3\nx\ny\nx\nx\ny\nz\nw\nw\nz\nw\n+\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n+\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n, find the values of x, y, z and w"
    },
    {
        "Chapter": "1",
        "sentence_range": "1263-1266",
        "Text": "If \n2\n1\n10\n3\n1\n5\nx\ny \u2212\n\uf8ee \uf8f9\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n+\n=\n\uf8ef \uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0 \uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n, find the values of x and y 12 Given \n6\n4\n3\n1\n2\n3\nx\ny\nx\nx\ny\nz\nw\nw\nz\nw\n+\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n+\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n, find the values of x, y, z and w Rationalised 2023-24\n 60\nMATHEMATICS\n13"
    },
    {
        "Chapter": "1",
        "sentence_range": "1264-1267",
        "Text": "12 Given \n6\n4\n3\n1\n2\n3\nx\ny\nx\nx\ny\nz\nw\nw\nz\nw\n+\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n+\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n, find the values of x, y, z and w Rationalised 2023-24\n 60\nMATHEMATICS\n13 If \ncos\nsin\n0\nF ( )\nsin\ncos\n0\n0\n0\n1\nx\nx\nx\nx\nx\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n= \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, show that F(x) F(y) = F(x + y)"
    },
    {
        "Chapter": "1",
        "sentence_range": "1265-1268",
        "Text": "Given \n6\n4\n3\n1\n2\n3\nx\ny\nx\nx\ny\nz\nw\nw\nz\nw\n+\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n+\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n, find the values of x, y, z and w Rationalised 2023-24\n 60\nMATHEMATICS\n13 If \ncos\nsin\n0\nF ( )\nsin\ncos\n0\n0\n0\n1\nx\nx\nx\nx\nx\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n= \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, show that F(x) F(y) = F(x + y) 14"
    },
    {
        "Chapter": "1",
        "sentence_range": "1266-1269",
        "Text": "Rationalised 2023-24\n 60\nMATHEMATICS\n13 If \ncos\nsin\n0\nF ( )\nsin\ncos\n0\n0\n0\n1\nx\nx\nx\nx\nx\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n= \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, show that F(x) F(y) = F(x + y) 14 Show that\n(i)\n5\n1\n2\n1\n2\n1\n5\n1\n6\n7\n3\n4\n3\n4\n6\n7\n\u2212\n\u2212\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\u2260\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n(ii)\n1\n2\n3\n1\n1\n0\n1\n1\n0\n1\n2\n3\n0\n1\n0\n0\n1\n1\n0\n1\n1\n0\n1\n0\n1\n1\n0\n2\n3\n4\n2\n3\n4\n1\n1\n0\n\u2212\n\u2212\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\u2212\n\u2260\n\u2212\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n15"
    },
    {
        "Chapter": "1",
        "sentence_range": "1267-1270",
        "Text": "If \ncos\nsin\n0\nF ( )\nsin\ncos\n0\n0\n0\n1\nx\nx\nx\nx\nx\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n= \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, show that F(x) F(y) = F(x + y) 14 Show that\n(i)\n5\n1\n2\n1\n2\n1\n5\n1\n6\n7\n3\n4\n3\n4\n6\n7\n\u2212\n\u2212\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\u2260\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n(ii)\n1\n2\n3\n1\n1\n0\n1\n1\n0\n1\n2\n3\n0\n1\n0\n0\n1\n1\n0\n1\n1\n0\n1\n0\n1\n1\n0\n2\n3\n4\n2\n3\n4\n1\n1\n0\n\u2212\n\u2212\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\u2212\n\u2260\n\u2212\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n15 Find A2 \u2013 5A + 6I, if \n2\n0\n1\nA\n2\n1\n3\n1\n1\n0\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n= \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n16"
    },
    {
        "Chapter": "1",
        "sentence_range": "1268-1271",
        "Text": "14 Show that\n(i)\n5\n1\n2\n1\n2\n1\n5\n1\n6\n7\n3\n4\n3\n4\n6\n7\n\u2212\n\u2212\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\u2260\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n(ii)\n1\n2\n3\n1\n1\n0\n1\n1\n0\n1\n2\n3\n0\n1\n0\n0\n1\n1\n0\n1\n1\n0\n1\n0\n1\n1\n0\n2\n3\n4\n2\n3\n4\n1\n1\n0\n\u2212\n\u2212\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\u2212\n\u2260\n\u2212\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n15 Find A2 \u2013 5A + 6I, if \n2\n0\n1\nA\n2\n1\n3\n1\n1\n0\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n= \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n16 If \n1\n0\n2\nA\n0\n2\n1\n2\n0\n3\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n= \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, prove that A3 \u2013 6A2 + 7A + 2I = 0\n17"
    },
    {
        "Chapter": "1",
        "sentence_range": "1269-1272",
        "Text": "Show that\n(i)\n5\n1\n2\n1\n2\n1\n5\n1\n6\n7\n3\n4\n3\n4\n6\n7\n\u2212\n\u2212\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\u2260\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n(ii)\n1\n2\n3\n1\n1\n0\n1\n1\n0\n1\n2\n3\n0\n1\n0\n0\n1\n1\n0\n1\n1\n0\n1\n0\n1\n1\n0\n2\n3\n4\n2\n3\n4\n1\n1\n0\n\u2212\n\u2212\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\u2212\n\u2260\n\u2212\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n15 Find A2 \u2013 5A + 6I, if \n2\n0\n1\nA\n2\n1\n3\n1\n1\n0\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n= \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n16 If \n1\n0\n2\nA\n0\n2\n1\n2\n0\n3\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n= \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, prove that A3 \u2013 6A2 + 7A + 2I = 0\n17 If \n3\n2\n1\n0\nA\nand I=\n4\n2\n0\n1\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n, find k so that A2 = kA \u2013 2I\n18"
    },
    {
        "Chapter": "1",
        "sentence_range": "1270-1273",
        "Text": "Find A2 \u2013 5A + 6I, if \n2\n0\n1\nA\n2\n1\n3\n1\n1\n0\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n= \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n16 If \n1\n0\n2\nA\n0\n2\n1\n2\n0\n3\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n= \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, prove that A3 \u2013 6A2 + 7A + 2I = 0\n17 If \n3\n2\n1\n0\nA\nand I=\n4\n2\n0\n1\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n, find k so that A2 = kA \u2013 2I\n18 If \n0\ntan 2\nA\ntan\n0\n2\n\u03b1\n\uf8ee\n\uf8f9\n\u2212\n\uf8ef\n\uf8fa\n= \uf8ef\n\uf8fa\n\u03b1\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n and I is the identity matrix of order 2, show that\nI + A = (I \u2013 A) \ncos\nsin\nsin\ncos\n\u03b1\n\u2212\n\u03b1\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u03b1\n\u03b1\n\uf8f0\n\uf8fb\n19"
    },
    {
        "Chapter": "1",
        "sentence_range": "1271-1274",
        "Text": "If \n1\n0\n2\nA\n0\n2\n1\n2\n0\n3\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n= \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, prove that A3 \u2013 6A2 + 7A + 2I = 0\n17 If \n3\n2\n1\n0\nA\nand I=\n4\n2\n0\n1\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n, find k so that A2 = kA \u2013 2I\n18 If \n0\ntan 2\nA\ntan\n0\n2\n\u03b1\n\uf8ee\n\uf8f9\n\u2212\n\uf8ef\n\uf8fa\n= \uf8ef\n\uf8fa\n\u03b1\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n and I is the identity matrix of order 2, show that\nI + A = (I \u2013 A) \ncos\nsin\nsin\ncos\n\u03b1\n\u2212\n\u03b1\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u03b1\n\u03b1\n\uf8f0\n\uf8fb\n19 A trust fund has ` 30,000 that must be invested in two different types of bonds"
    },
    {
        "Chapter": "1",
        "sentence_range": "1272-1275",
        "Text": "If \n3\n2\n1\n0\nA\nand I=\n4\n2\n0\n1\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n, find k so that A2 = kA \u2013 2I\n18 If \n0\ntan 2\nA\ntan\n0\n2\n\u03b1\n\uf8ee\n\uf8f9\n\u2212\n\uf8ef\n\uf8fa\n= \uf8ef\n\uf8fa\n\u03b1\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n and I is the identity matrix of order 2, show that\nI + A = (I \u2013 A) \ncos\nsin\nsin\ncos\n\u03b1\n\u2212\n\u03b1\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u03b1\n\u03b1\n\uf8f0\n\uf8fb\n19 A trust fund has ` 30,000 that must be invested in two different types of bonds The first bond pays 5% interest per year, and the second bond pays 7% interest\nper year"
    },
    {
        "Chapter": "1",
        "sentence_range": "1273-1276",
        "Text": "If \n0\ntan 2\nA\ntan\n0\n2\n\u03b1\n\uf8ee\n\uf8f9\n\u2212\n\uf8ef\n\uf8fa\n= \uf8ef\n\uf8fa\n\u03b1\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n and I is the identity matrix of order 2, show that\nI + A = (I \u2013 A) \ncos\nsin\nsin\ncos\n\u03b1\n\u2212\n\u03b1\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u03b1\n\u03b1\n\uf8f0\n\uf8fb\n19 A trust fund has ` 30,000 that must be invested in two different types of bonds The first bond pays 5% interest per year, and the second bond pays 7% interest\nper year Using matrix multiplication, determine how to divide ` 30,000 among\nthe two types of bonds"
    },
    {
        "Chapter": "1",
        "sentence_range": "1274-1277",
        "Text": "A trust fund has ` 30,000 that must be invested in two different types of bonds The first bond pays 5% interest per year, and the second bond pays 7% interest\nper year Using matrix multiplication, determine how to divide ` 30,000 among\nthe two types of bonds If the trust fund must obtain an annual total interest of:\n(a)\n` 1800\n(b)\n` 2000\nRationalised 2023-24\nMATRICES         61\n20"
    },
    {
        "Chapter": "1",
        "sentence_range": "1275-1278",
        "Text": "The first bond pays 5% interest per year, and the second bond pays 7% interest\nper year Using matrix multiplication, determine how to divide ` 30,000 among\nthe two types of bonds If the trust fund must obtain an annual total interest of:\n(a)\n` 1800\n(b)\n` 2000\nRationalised 2023-24\nMATRICES         61\n20 The bookshop of a particular school has 10 dozen chemistry books, 8 dozen\nphysics books, 10 dozen economics books"
    },
    {
        "Chapter": "1",
        "sentence_range": "1276-1279",
        "Text": "Using matrix multiplication, determine how to divide ` 30,000 among\nthe two types of bonds If the trust fund must obtain an annual total interest of:\n(a)\n` 1800\n(b)\n` 2000\nRationalised 2023-24\nMATRICES         61\n20 The bookshop of a particular school has 10 dozen chemistry books, 8 dozen\nphysics books, 10 dozen economics books Their selling prices are ` 80, ` 60 and\n` 40 each respectively"
    },
    {
        "Chapter": "1",
        "sentence_range": "1277-1280",
        "Text": "If the trust fund must obtain an annual total interest of:\n(a)\n` 1800\n(b)\n` 2000\nRationalised 2023-24\nMATRICES         61\n20 The bookshop of a particular school has 10 dozen chemistry books, 8 dozen\nphysics books, 10 dozen economics books Their selling prices are ` 80, ` 60 and\n` 40 each respectively Find the total amount the bookshop will receive from\nselling all the books using matrix algebra"
    },
    {
        "Chapter": "1",
        "sentence_range": "1278-1281",
        "Text": "The bookshop of a particular school has 10 dozen chemistry books, 8 dozen\nphysics books, 10 dozen economics books Their selling prices are ` 80, ` 60 and\n` 40 each respectively Find the total amount the bookshop will receive from\nselling all the books using matrix algebra Assume X, Y, Z, W and P are matrices of order 2 \u00d7 n, 3 \u00d7 k, 2 \u00d7 p, n \u00d7 3 and p \u00d7 k,\nrespectively"
    },
    {
        "Chapter": "1",
        "sentence_range": "1279-1282",
        "Text": "Their selling prices are ` 80, ` 60 and\n` 40 each respectively Find the total amount the bookshop will receive from\nselling all the books using matrix algebra Assume X, Y, Z, W and P are matrices of order 2 \u00d7 n, 3 \u00d7 k, 2 \u00d7 p, n \u00d7 3 and p \u00d7 k,\nrespectively Choose the correct answer in Exercises 21 and 22"
    },
    {
        "Chapter": "1",
        "sentence_range": "1280-1283",
        "Text": "Find the total amount the bookshop will receive from\nselling all the books using matrix algebra Assume X, Y, Z, W and P are matrices of order 2 \u00d7 n, 3 \u00d7 k, 2 \u00d7 p, n \u00d7 3 and p \u00d7 k,\nrespectively Choose the correct answer in Exercises 21 and 22 21"
    },
    {
        "Chapter": "1",
        "sentence_range": "1281-1284",
        "Text": "Assume X, Y, Z, W and P are matrices of order 2 \u00d7 n, 3 \u00d7 k, 2 \u00d7 p, n \u00d7 3 and p \u00d7 k,\nrespectively Choose the correct answer in Exercises 21 and 22 21 The restriction on n, k and p so that PY + WY will be defined are:\n(A) k = 3, p = n\n(B) k is arbitrary, p = 2\n(C) p is arbitrary, k = 3\n(D) k = 2, p = 3\n22"
    },
    {
        "Chapter": "1",
        "sentence_range": "1282-1285",
        "Text": "Choose the correct answer in Exercises 21 and 22 21 The restriction on n, k and p so that PY + WY will be defined are:\n(A) k = 3, p = n\n(B) k is arbitrary, p = 2\n(C) p is arbitrary, k = 3\n(D) k = 2, p = 3\n22 If n = p, then the order of the matrix 7X \u2013 5Z is:\n(A) p \u00d7 2\n(B) 2 \u00d7 n\n(C) n \u00d7 3\n(D) p \u00d7 n\n3"
    },
    {
        "Chapter": "1",
        "sentence_range": "1283-1286",
        "Text": "21 The restriction on n, k and p so that PY + WY will be defined are:\n(A) k = 3, p = n\n(B) k is arbitrary, p = 2\n(C) p is arbitrary, k = 3\n(D) k = 2, p = 3\n22 If n = p, then the order of the matrix 7X \u2013 5Z is:\n(A) p \u00d7 2\n(B) 2 \u00d7 n\n(C) n \u00d7 3\n(D) p \u00d7 n\n3 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "1284-1287",
        "Text": "The restriction on n, k and p so that PY + WY will be defined are:\n(A) k = 3, p = n\n(B) k is arbitrary, p = 2\n(C) p is arbitrary, k = 3\n(D) k = 2, p = 3\n22 If n = p, then the order of the matrix 7X \u2013 5Z is:\n(A) p \u00d7 2\n(B) 2 \u00d7 n\n(C) n \u00d7 3\n(D) p \u00d7 n\n3 5 Transpose of a Matrix\nIn this section, we shall learn about transpose of a matrix and special types of matrices\nsuch as symmetric and skew symmetric matrices"
    },
    {
        "Chapter": "1",
        "sentence_range": "1285-1288",
        "Text": "If n = p, then the order of the matrix 7X \u2013 5Z is:\n(A) p \u00d7 2\n(B) 2 \u00d7 n\n(C) n \u00d7 3\n(D) p \u00d7 n\n3 5 Transpose of a Matrix\nIn this section, we shall learn about transpose of a matrix and special types of matrices\nsuch as symmetric and skew symmetric matrices Definition 3  If A = [aij] be an m \u00d7 n matrix, then the matrix obtained by interchanging\nthe rows and columns of A is called the transpose of A"
    },
    {
        "Chapter": "1",
        "sentence_range": "1286-1289",
        "Text": "5 Transpose of a Matrix\nIn this section, we shall learn about transpose of a matrix and special types of matrices\nsuch as symmetric and skew symmetric matrices Definition 3  If A = [aij] be an m \u00d7 n matrix, then the matrix obtained by interchanging\nthe rows and columns of A is called the transpose of A Transpose of the matrix A is\ndenoted by A\u2032 or (AT)"
    },
    {
        "Chapter": "1",
        "sentence_range": "1287-1290",
        "Text": "Transpose of a Matrix\nIn this section, we shall learn about transpose of a matrix and special types of matrices\nsuch as symmetric and skew symmetric matrices Definition 3  If A = [aij] be an m \u00d7 n matrix, then the matrix obtained by interchanging\nthe rows and columns of A is called the transpose of A Transpose of the matrix A is\ndenoted by A\u2032 or (AT) In other words, if A = [aij]m \u00d7 n, then A\u2032 = [aji]n \u00d7 m"
    },
    {
        "Chapter": "1",
        "sentence_range": "1288-1291",
        "Text": "Definition 3  If A = [aij] be an m \u00d7 n matrix, then the matrix obtained by interchanging\nthe rows and columns of A is called the transpose of A Transpose of the matrix A is\ndenoted by A\u2032 or (AT) In other words, if A = [aij]m \u00d7 n, then A\u2032 = [aji]n \u00d7 m For example,\nif \n2\n3\n3\n2\n3\n5\n3\n3\n0\nA\n3\n1\n, then A\n1\n5\n1\n0\n1\n5\n5\n\u00d7\n\u00d7\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n=\n\u2032 =\n\uf8ef\n\uf8fa\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n3"
    },
    {
        "Chapter": "1",
        "sentence_range": "1289-1292",
        "Text": "Transpose of the matrix A is\ndenoted by A\u2032 or (AT) In other words, if A = [aij]m \u00d7 n, then A\u2032 = [aji]n \u00d7 m For example,\nif \n2\n3\n3\n2\n3\n5\n3\n3\n0\nA\n3\n1\n, then A\n1\n5\n1\n0\n1\n5\n5\n\u00d7\n\u00d7\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n=\n\u2032 =\n\uf8ef\n\uf8fa\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n3 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "1290-1293",
        "Text": "In other words, if A = [aij]m \u00d7 n, then A\u2032 = [aji]n \u00d7 m For example,\nif \n2\n3\n3\n2\n3\n5\n3\n3\n0\nA\n3\n1\n, then A\n1\n5\n1\n0\n1\n5\n5\n\u00d7\n\u00d7\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n=\n\u2032 =\n\uf8ef\n\uf8fa\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n3 5 1  Properties of transpose of the matrices\nWe now state the following properties of transpose of matrices without proof"
    },
    {
        "Chapter": "1",
        "sentence_range": "1291-1294",
        "Text": "For example,\nif \n2\n3\n3\n2\n3\n5\n3\n3\n0\nA\n3\n1\n, then A\n1\n5\n1\n0\n1\n5\n5\n\u00d7\n\u00d7\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n=\n\u2032 =\n\uf8ef\n\uf8fa\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n3 5 1  Properties of transpose of the matrices\nWe now state the following properties of transpose of matrices without proof These\nmay be verified by taking suitable examples"
    },
    {
        "Chapter": "1",
        "sentence_range": "1292-1295",
        "Text": "5 1  Properties of transpose of the matrices\nWe now state the following properties of transpose of matrices without proof These\nmay be verified by taking suitable examples (i)For any matrices A and B of suitable orders, we have\n(A\u2032)\u2032 = A,\n(ii)\n(kA)\u2032 = kA\u2032 (where k is any constant)\n(iii)\n(A + B)\u2032 = A\u2032 + B\u2032\n(iv)\n(A B)\u2032 = B\u2032 A\u2032\nExample 20 If \n2\n1\n2\n3\n3\n2\nA\nand B\n1\n2\n4\n4\n2\n0\n\uf8ee\n\uf8f9\n\u2212\n\uf8ee\n\uf8f9\n=\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n, verify that\n(i)\n(A\u2032)\u2032 = A,\n(ii)\n(A + B)\u2032 = A\u2032 + B\u2032,\n(iii)\n(kB)\u2032 = kB\u2032, where k is any constant"
    },
    {
        "Chapter": "1",
        "sentence_range": "1293-1296",
        "Text": "1  Properties of transpose of the matrices\nWe now state the following properties of transpose of matrices without proof These\nmay be verified by taking suitable examples (i)For any matrices A and B of suitable orders, we have\n(A\u2032)\u2032 = A,\n(ii)\n(kA)\u2032 = kA\u2032 (where k is any constant)\n(iii)\n(A + B)\u2032 = A\u2032 + B\u2032\n(iv)\n(A B)\u2032 = B\u2032 A\u2032\nExample 20 If \n2\n1\n2\n3\n3\n2\nA\nand B\n1\n2\n4\n4\n2\n0\n\uf8ee\n\uf8f9\n\u2212\n\uf8ee\n\uf8f9\n=\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n, verify that\n(i)\n(A\u2032)\u2032 = A,\n(ii)\n(A + B)\u2032 = A\u2032 + B\u2032,\n(iii)\n(kB)\u2032 = kB\u2032, where k is any constant Rationalised 2023-24\n 62\nMATHEMATICS\nSolution\n(i)\nWe have\nA =\n(\n)\n3\n4\n3\n3\n2\n3\n3\n2\nA\n3 2\nA\nA\n4\n2\n0\n4\n2\n0\n2\n0\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2032\n\u2032\n\u2032\n\u21d2\n=\n\u21d2\n=\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nThus\n(A\u2032)\u2032 = A\n(ii)\nWe have\nA = 3\n3\n2 ,\n4\n2\n0\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n B =\n2\n1\n2\n5\n3\n1\n4\nA\nB\n1\n2 4\n5\n4\n4\n\uf8ee\n\uf8f9\n\u2212\n\uf8ee\n\uf8f9\n\u2212\n\u21d2\n+\n= \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nTherefore\n(A + B)\u2032 =\n5\n5\n3\n1\n4\n4\n4\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nNow\nA\u2032 =\n3\n4\n2 1\n3 2 , B\n1 2 ,\n2\n0\n2 4\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2032 = \u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nSo\nA\u2032 + B\u2032 =\n5\n5\n43 1 4\n4\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nThus\n(A + B)\u2032 = A\u2032 + B\u2032\n(iii)\nWe have\nkB = k 2\n1\n2\n2\n2\n1\n2\n4\n2\n4\nk\nk\nk\nk\nk\nk\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nThen\n(kB)\u2032 =\n2\n2 1\n2\n1 2\nB\n2\n4\n2 4\nk\nk\nk\nk\nk\nk\nk\nk\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2032\n\u2212\n=\n\u2212\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nThus\n(kB)\u2032 = kB\u2032\nRationalised 2023-24\nMATRICES         63\nExample 21 If \n[\n]\n2\nA\n4 , B\n1\n3\n6\n5\n\uf8ee\u2212\n\uf8f9\n\uf8ef\n\uf8fa\n=\n=\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, verify that (AB)\u2032 = B\u2032A\u2032"
    },
    {
        "Chapter": "1",
        "sentence_range": "1294-1297",
        "Text": "These\nmay be verified by taking suitable examples (i)For any matrices A and B of suitable orders, we have\n(A\u2032)\u2032 = A,\n(ii)\n(kA)\u2032 = kA\u2032 (where k is any constant)\n(iii)\n(A + B)\u2032 = A\u2032 + B\u2032\n(iv)\n(A B)\u2032 = B\u2032 A\u2032\nExample 20 If \n2\n1\n2\n3\n3\n2\nA\nand B\n1\n2\n4\n4\n2\n0\n\uf8ee\n\uf8f9\n\u2212\n\uf8ee\n\uf8f9\n=\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n, verify that\n(i)\n(A\u2032)\u2032 = A,\n(ii)\n(A + B)\u2032 = A\u2032 + B\u2032,\n(iii)\n(kB)\u2032 = kB\u2032, where k is any constant Rationalised 2023-24\n 62\nMATHEMATICS\nSolution\n(i)\nWe have\nA =\n(\n)\n3\n4\n3\n3\n2\n3\n3\n2\nA\n3 2\nA\nA\n4\n2\n0\n4\n2\n0\n2\n0\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2032\n\u2032\n\u2032\n\u21d2\n=\n\u21d2\n=\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nThus\n(A\u2032)\u2032 = A\n(ii)\nWe have\nA = 3\n3\n2 ,\n4\n2\n0\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n B =\n2\n1\n2\n5\n3\n1\n4\nA\nB\n1\n2 4\n5\n4\n4\n\uf8ee\n\uf8f9\n\u2212\n\uf8ee\n\uf8f9\n\u2212\n\u21d2\n+\n= \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nTherefore\n(A + B)\u2032 =\n5\n5\n3\n1\n4\n4\n4\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nNow\nA\u2032 =\n3\n4\n2 1\n3 2 , B\n1 2 ,\n2\n0\n2 4\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2032 = \u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nSo\nA\u2032 + B\u2032 =\n5\n5\n43 1 4\n4\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nThus\n(A + B)\u2032 = A\u2032 + B\u2032\n(iii)\nWe have\nkB = k 2\n1\n2\n2\n2\n1\n2\n4\n2\n4\nk\nk\nk\nk\nk\nk\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nThen\n(kB)\u2032 =\n2\n2 1\n2\n1 2\nB\n2\n4\n2 4\nk\nk\nk\nk\nk\nk\nk\nk\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2032\n\u2212\n=\n\u2212\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nThus\n(kB)\u2032 = kB\u2032\nRationalised 2023-24\nMATRICES         63\nExample 21 If \n[\n]\n2\nA\n4 , B\n1\n3\n6\n5\n\uf8ee\u2212\n\uf8f9\n\uf8ef\n\uf8fa\n=\n=\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, verify that (AB)\u2032 = B\u2032A\u2032 Solution We have\nA =\n[\n]\n2\n4 , B\n1\n3\n6\n5\n\uf8ee\u2212\n\uf8f9\n\uf8ef\n\uf8fa\n=\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nthen\nAB =\n[\n]\n2\n4\n1\n3\n6\n5\n\uf8ee\u2212\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n = \n2\n6\n12\n4\n12\n24\n5\n15\n30\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\nNow\nA\u2032 = [\u20132 4 5] , \n1\nB\n3\n6\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2032 = \uf8ef\n\uf8fa\n\uf8ef\n\uf8f0\u2212\uf8fa\n\uf8fb\nB\u2032A\u2032 =\n[\n]\n1\n2\n4\n5\n3\n2\n4\n5\n6\n12\n15\n(AB)\n6\n12\n24\n30\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2032\n\u2212\n= \u2212\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\u2212\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nClearly\n(AB)\u2032 = B\u2032A\u2032\n3"
    },
    {
        "Chapter": "1",
        "sentence_range": "1295-1298",
        "Text": "(i)For any matrices A and B of suitable orders, we have\n(A\u2032)\u2032 = A,\n(ii)\n(kA)\u2032 = kA\u2032 (where k is any constant)\n(iii)\n(A + B)\u2032 = A\u2032 + B\u2032\n(iv)\n(A B)\u2032 = B\u2032 A\u2032\nExample 20 If \n2\n1\n2\n3\n3\n2\nA\nand B\n1\n2\n4\n4\n2\n0\n\uf8ee\n\uf8f9\n\u2212\n\uf8ee\n\uf8f9\n=\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n, verify that\n(i)\n(A\u2032)\u2032 = A,\n(ii)\n(A + B)\u2032 = A\u2032 + B\u2032,\n(iii)\n(kB)\u2032 = kB\u2032, where k is any constant Rationalised 2023-24\n 62\nMATHEMATICS\nSolution\n(i)\nWe have\nA =\n(\n)\n3\n4\n3\n3\n2\n3\n3\n2\nA\n3 2\nA\nA\n4\n2\n0\n4\n2\n0\n2\n0\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2032\n\u2032\n\u2032\n\u21d2\n=\n\u21d2\n=\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nThus\n(A\u2032)\u2032 = A\n(ii)\nWe have\nA = 3\n3\n2 ,\n4\n2\n0\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n B =\n2\n1\n2\n5\n3\n1\n4\nA\nB\n1\n2 4\n5\n4\n4\n\uf8ee\n\uf8f9\n\u2212\n\uf8ee\n\uf8f9\n\u2212\n\u21d2\n+\n= \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nTherefore\n(A + B)\u2032 =\n5\n5\n3\n1\n4\n4\n4\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nNow\nA\u2032 =\n3\n4\n2 1\n3 2 , B\n1 2 ,\n2\n0\n2 4\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2032 = \u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nSo\nA\u2032 + B\u2032 =\n5\n5\n43 1 4\n4\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nThus\n(A + B)\u2032 = A\u2032 + B\u2032\n(iii)\nWe have\nkB = k 2\n1\n2\n2\n2\n1\n2\n4\n2\n4\nk\nk\nk\nk\nk\nk\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nThen\n(kB)\u2032 =\n2\n2 1\n2\n1 2\nB\n2\n4\n2 4\nk\nk\nk\nk\nk\nk\nk\nk\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2032\n\u2212\n=\n\u2212\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nThus\n(kB)\u2032 = kB\u2032\nRationalised 2023-24\nMATRICES         63\nExample 21 If \n[\n]\n2\nA\n4 , B\n1\n3\n6\n5\n\uf8ee\u2212\n\uf8f9\n\uf8ef\n\uf8fa\n=\n=\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, verify that (AB)\u2032 = B\u2032A\u2032 Solution We have\nA =\n[\n]\n2\n4 , B\n1\n3\n6\n5\n\uf8ee\u2212\n\uf8f9\n\uf8ef\n\uf8fa\n=\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nthen\nAB =\n[\n]\n2\n4\n1\n3\n6\n5\n\uf8ee\u2212\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n = \n2\n6\n12\n4\n12\n24\n5\n15\n30\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\nNow\nA\u2032 = [\u20132 4 5] , \n1\nB\n3\n6\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2032 = \uf8ef\n\uf8fa\n\uf8ef\n\uf8f0\u2212\uf8fa\n\uf8fb\nB\u2032A\u2032 =\n[\n]\n1\n2\n4\n5\n3\n2\n4\n5\n6\n12\n15\n(AB)\n6\n12\n24\n30\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2032\n\u2212\n= \u2212\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\u2212\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nClearly\n(AB)\u2032 = B\u2032A\u2032\n3 6  Symmetric and Skew Symmetric Matrices\nDefinition 4 A square matrix A = [aij] is said to be symmetric if A\u2032 = A, that is,\n[aij] = [aji] for all possible values of i and j"
    },
    {
        "Chapter": "1",
        "sentence_range": "1296-1299",
        "Text": "Rationalised 2023-24\n 62\nMATHEMATICS\nSolution\n(i)\nWe have\nA =\n(\n)\n3\n4\n3\n3\n2\n3\n3\n2\nA\n3 2\nA\nA\n4\n2\n0\n4\n2\n0\n2\n0\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2032\n\u2032\n\u2032\n\u21d2\n=\n\u21d2\n=\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nThus\n(A\u2032)\u2032 = A\n(ii)\nWe have\nA = 3\n3\n2 ,\n4\n2\n0\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n B =\n2\n1\n2\n5\n3\n1\n4\nA\nB\n1\n2 4\n5\n4\n4\n\uf8ee\n\uf8f9\n\u2212\n\uf8ee\n\uf8f9\n\u2212\n\u21d2\n+\n= \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nTherefore\n(A + B)\u2032 =\n5\n5\n3\n1\n4\n4\n4\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nNow\nA\u2032 =\n3\n4\n2 1\n3 2 , B\n1 2 ,\n2\n0\n2 4\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2032 = \u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nSo\nA\u2032 + B\u2032 =\n5\n5\n43 1 4\n4\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nThus\n(A + B)\u2032 = A\u2032 + B\u2032\n(iii)\nWe have\nkB = k 2\n1\n2\n2\n2\n1\n2\n4\n2\n4\nk\nk\nk\nk\nk\nk\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nThen\n(kB)\u2032 =\n2\n2 1\n2\n1 2\nB\n2\n4\n2 4\nk\nk\nk\nk\nk\nk\nk\nk\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2032\n\u2212\n=\n\u2212\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nThus\n(kB)\u2032 = kB\u2032\nRationalised 2023-24\nMATRICES         63\nExample 21 If \n[\n]\n2\nA\n4 , B\n1\n3\n6\n5\n\uf8ee\u2212\n\uf8f9\n\uf8ef\n\uf8fa\n=\n=\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, verify that (AB)\u2032 = B\u2032A\u2032 Solution We have\nA =\n[\n]\n2\n4 , B\n1\n3\n6\n5\n\uf8ee\u2212\n\uf8f9\n\uf8ef\n\uf8fa\n=\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nthen\nAB =\n[\n]\n2\n4\n1\n3\n6\n5\n\uf8ee\u2212\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n = \n2\n6\n12\n4\n12\n24\n5\n15\n30\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\nNow\nA\u2032 = [\u20132 4 5] , \n1\nB\n3\n6\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2032 = \uf8ef\n\uf8fa\n\uf8ef\n\uf8f0\u2212\uf8fa\n\uf8fb\nB\u2032A\u2032 =\n[\n]\n1\n2\n4\n5\n3\n2\n4\n5\n6\n12\n15\n(AB)\n6\n12\n24\n30\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2032\n\u2212\n= \u2212\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\u2212\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nClearly\n(AB)\u2032 = B\u2032A\u2032\n3 6  Symmetric and Skew Symmetric Matrices\nDefinition 4 A square matrix A = [aij] is said to be symmetric if A\u2032 = A, that is,\n[aij] = [aji] for all possible values of i and j For example \n3\n2\n3\nA\n2\n1"
    },
    {
        "Chapter": "1",
        "sentence_range": "1297-1300",
        "Text": "Solution We have\nA =\n[\n]\n2\n4 , B\n1\n3\n6\n5\n\uf8ee\u2212\n\uf8f9\n\uf8ef\n\uf8fa\n=\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nthen\nAB =\n[\n]\n2\n4\n1\n3\n6\n5\n\uf8ee\u2212\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n = \n2\n6\n12\n4\n12\n24\n5\n15\n30\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\nNow\nA\u2032 = [\u20132 4 5] , \n1\nB\n3\n6\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2032 = \uf8ef\n\uf8fa\n\uf8ef\n\uf8f0\u2212\uf8fa\n\uf8fb\nB\u2032A\u2032 =\n[\n]\n1\n2\n4\n5\n3\n2\n4\n5\n6\n12\n15\n(AB)\n6\n12\n24\n30\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2032\n\u2212\n= \u2212\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\u2212\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nClearly\n(AB)\u2032 = B\u2032A\u2032\n3 6  Symmetric and Skew Symmetric Matrices\nDefinition 4 A square matrix A = [aij] is said to be symmetric if A\u2032 = A, that is,\n[aij] = [aji] for all possible values of i and j For example \n3\n2\n3\nA\n2\n1 5\n1\n3\n1\n1\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n=\n\u2212\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n is a symmetric matrix as A\u2032 = A\nDefinition 5 A square matrix A = [aij] is said to be skew symmetric matrix if\nA\u2032 = \u2013 A, that is aji = \u2013 aij for all possible values of i  and j"
    },
    {
        "Chapter": "1",
        "sentence_range": "1298-1301",
        "Text": "6  Symmetric and Skew Symmetric Matrices\nDefinition 4 A square matrix A = [aij] is said to be symmetric if A\u2032 = A, that is,\n[aij] = [aji] for all possible values of i and j For example \n3\n2\n3\nA\n2\n1 5\n1\n3\n1\n1\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n=\n\u2212\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n is a symmetric matrix as A\u2032 = A\nDefinition 5 A square matrix A = [aij] is said to be skew symmetric matrix if\nA\u2032 = \u2013 A, that is aji = \u2013 aij for all possible values of i  and j Now, if we put i = j, we\nhave aii = \u2013 aii"
    },
    {
        "Chapter": "1",
        "sentence_range": "1299-1302",
        "Text": "For example \n3\n2\n3\nA\n2\n1 5\n1\n3\n1\n1\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n=\n\u2212\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n is a symmetric matrix as A\u2032 = A\nDefinition 5 A square matrix A = [aij] is said to be skew symmetric matrix if\nA\u2032 = \u2013 A, that is aji = \u2013 aij for all possible values of i  and j Now, if we put i = j, we\nhave aii = \u2013 aii Therefore 2aii = 0 or aii = 0 for all i\u2019s"
    },
    {
        "Chapter": "1",
        "sentence_range": "1300-1303",
        "Text": "5\n1\n3\n1\n1\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n=\n\u2212\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n is a symmetric matrix as A\u2032 = A\nDefinition 5 A square matrix A = [aij] is said to be skew symmetric matrix if\nA\u2032 = \u2013 A, that is aji = \u2013 aij for all possible values of i  and j Now, if we put i = j, we\nhave aii = \u2013 aii Therefore 2aii = 0 or aii = 0 for all i\u2019s This means that all the diagonal elements of a skew symmetric matrix are zero"
    },
    {
        "Chapter": "1",
        "sentence_range": "1301-1304",
        "Text": "Now, if we put i = j, we\nhave aii = \u2013 aii Therefore 2aii = 0 or aii = 0 for all i\u2019s This means that all the diagonal elements of a skew symmetric matrix are zero Rationalised 2023-24\n 64\nMATHEMATICS\nFor example, the matrix \n0\nB\n0\n0\ne\nf\ne\ng\nf\ng\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n=\n\uf8ef\u2212\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8f0\n\uf8fb\n is a skew symmetric matrix as B\u2032= \u2013B\nNow, we are going to prove some results of symmetric and skew-symmetric\nmatrices"
    },
    {
        "Chapter": "1",
        "sentence_range": "1302-1305",
        "Text": "Therefore 2aii = 0 or aii = 0 for all i\u2019s This means that all the diagonal elements of a skew symmetric matrix are zero Rationalised 2023-24\n 64\nMATHEMATICS\nFor example, the matrix \n0\nB\n0\n0\ne\nf\ne\ng\nf\ng\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n=\n\uf8ef\u2212\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8f0\n\uf8fb\n is a skew symmetric matrix as B\u2032= \u2013B\nNow, we are going to prove some results of symmetric and skew-symmetric\nmatrices Theorem 1 For any square matrix A with real number entries, A + A\u2032 is a symmetric\nmatrix and A \u2013 A\u2032 is a skew symmetric matrix"
    },
    {
        "Chapter": "1",
        "sentence_range": "1303-1306",
        "Text": "This means that all the diagonal elements of a skew symmetric matrix are zero Rationalised 2023-24\n 64\nMATHEMATICS\nFor example, the matrix \n0\nB\n0\n0\ne\nf\ne\ng\nf\ng\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n=\n\uf8ef\u2212\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8f0\n\uf8fb\n is a skew symmetric matrix as B\u2032= \u2013B\nNow, we are going to prove some results of symmetric and skew-symmetric\nmatrices Theorem 1 For any square matrix A with real number entries, A + A\u2032 is a symmetric\nmatrix and A \u2013 A\u2032 is a skew symmetric matrix Proof Let B = A + A\u2032, then\nB\u2032 = (A + A\u2032)\u2032\n= A\u2032 + (A\u2032)\u2032 (as (A + B)\u2032 = A\u2032 + B\u2032)\n= A\u2032 + A (as (A\u2032)\u2032 = A)\n= A + A\u2032 (as A + B = B + A)\n= B\nTherefore\nB = A + A\u2032 is a symmetric matrix\nNow let\nC = A \u2013 A\u2032\nC\u2032 = (A \u2013 A\u2032)\u2032 = A\u2032 \u2013 (A\u2032)\u2032     (Why"
    },
    {
        "Chapter": "1",
        "sentence_range": "1304-1307",
        "Text": "Rationalised 2023-24\n 64\nMATHEMATICS\nFor example, the matrix \n0\nB\n0\n0\ne\nf\ne\ng\nf\ng\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n=\n\uf8ef\u2212\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8f0\n\uf8fb\n is a skew symmetric matrix as B\u2032= \u2013B\nNow, we are going to prove some results of symmetric and skew-symmetric\nmatrices Theorem 1 For any square matrix A with real number entries, A + A\u2032 is a symmetric\nmatrix and A \u2013 A\u2032 is a skew symmetric matrix Proof Let B = A + A\u2032, then\nB\u2032 = (A + A\u2032)\u2032\n= A\u2032 + (A\u2032)\u2032 (as (A + B)\u2032 = A\u2032 + B\u2032)\n= A\u2032 + A (as (A\u2032)\u2032 = A)\n= A + A\u2032 (as A + B = B + A)\n= B\nTherefore\nB = A + A\u2032 is a symmetric matrix\nNow let\nC = A \u2013 A\u2032\nC\u2032 = (A \u2013 A\u2032)\u2032 = A\u2032 \u2013 (A\u2032)\u2032     (Why )\n= A\u2032 \u2013 A    (Why"
    },
    {
        "Chapter": "1",
        "sentence_range": "1305-1308",
        "Text": "Theorem 1 For any square matrix A with real number entries, A + A\u2032 is a symmetric\nmatrix and A \u2013 A\u2032 is a skew symmetric matrix Proof Let B = A + A\u2032, then\nB\u2032 = (A + A\u2032)\u2032\n= A\u2032 + (A\u2032)\u2032 (as (A + B)\u2032 = A\u2032 + B\u2032)\n= A\u2032 + A (as (A\u2032)\u2032 = A)\n= A + A\u2032 (as A + B = B + A)\n= B\nTherefore\nB = A + A\u2032 is a symmetric matrix\nNow let\nC = A \u2013 A\u2032\nC\u2032 = (A \u2013 A\u2032)\u2032 = A\u2032 \u2013 (A\u2032)\u2032     (Why )\n= A\u2032 \u2013 A    (Why )\n= \u2013 (A \u2013 A\u2032) = \u2013 C\nTherefore\nC = A \u2013 A\u2032 is a skew symmetric matrix"
    },
    {
        "Chapter": "1",
        "sentence_range": "1306-1309",
        "Text": "Proof Let B = A + A\u2032, then\nB\u2032 = (A + A\u2032)\u2032\n= A\u2032 + (A\u2032)\u2032 (as (A + B)\u2032 = A\u2032 + B\u2032)\n= A\u2032 + A (as (A\u2032)\u2032 = A)\n= A + A\u2032 (as A + B = B + A)\n= B\nTherefore\nB = A + A\u2032 is a symmetric matrix\nNow let\nC = A \u2013 A\u2032\nC\u2032 = (A \u2013 A\u2032)\u2032 = A\u2032 \u2013 (A\u2032)\u2032     (Why )\n= A\u2032 \u2013 A    (Why )\n= \u2013 (A \u2013 A\u2032) = \u2013 C\nTherefore\nC = A \u2013 A\u2032 is a skew symmetric matrix Theorem 2 Any square matrix can be expressed as the sum of a symmetric and a\nskew symmetric matrix"
    },
    {
        "Chapter": "1",
        "sentence_range": "1307-1310",
        "Text": ")\n= A\u2032 \u2013 A    (Why )\n= \u2013 (A \u2013 A\u2032) = \u2013 C\nTherefore\nC = A \u2013 A\u2032 is a skew symmetric matrix Theorem 2 Any square matrix can be expressed as the sum of a symmetric and a\nskew symmetric matrix Proof Let A be a square matrix, then we can write\n1\n1\nA\n(A\nA )\n(A\nA )\n2\n2\n\u2032\n\u2032\n=\n+\n+\n\u2212\nFrom the Theorem 1, we know that (A + A\u2032) is a symmetric matrix and (A \u2013 A\u2032) is\na skew symmetric matrix"
    },
    {
        "Chapter": "1",
        "sentence_range": "1308-1311",
        "Text": ")\n= \u2013 (A \u2013 A\u2032) = \u2013 C\nTherefore\nC = A \u2013 A\u2032 is a skew symmetric matrix Theorem 2 Any square matrix can be expressed as the sum of a symmetric and a\nskew symmetric matrix Proof Let A be a square matrix, then we can write\n1\n1\nA\n(A\nA )\n(A\nA )\n2\n2\n\u2032\n\u2032\n=\n+\n+\n\u2212\nFrom the Theorem 1, we know that (A + A\u2032) is a symmetric matrix and (A \u2013 A\u2032) is\na skew symmetric matrix Since for any matrix A, (kA)\u2032 = kA\u2032, it follows that 1 (A\nA )\n2\n\u2032\n+\nRationalised 2023-24\nMATRICES         65\nis symmetric matrix and 1 (A\nA )\n2\n\u2032\n\u2212\n is skew symmetric matrix"
    },
    {
        "Chapter": "1",
        "sentence_range": "1309-1312",
        "Text": "Theorem 2 Any square matrix can be expressed as the sum of a symmetric and a\nskew symmetric matrix Proof Let A be a square matrix, then we can write\n1\n1\nA\n(A\nA )\n(A\nA )\n2\n2\n\u2032\n\u2032\n=\n+\n+\n\u2212\nFrom the Theorem 1, we know that (A + A\u2032) is a symmetric matrix and (A \u2013 A\u2032) is\na skew symmetric matrix Since for any matrix A, (kA)\u2032 = kA\u2032, it follows that 1 (A\nA )\n2\n\u2032\n+\nRationalised 2023-24\nMATRICES         65\nis symmetric matrix and 1 (A\nA )\n2\n\u2032\n\u2212\n is skew symmetric matrix Thus, any square\nmatrix can be expressed as the sum of a symmetric and a skew symmetric matrix"
    },
    {
        "Chapter": "1",
        "sentence_range": "1310-1313",
        "Text": "Proof Let A be a square matrix, then we can write\n1\n1\nA\n(A\nA )\n(A\nA )\n2\n2\n\u2032\n\u2032\n=\n+\n+\n\u2212\nFrom the Theorem 1, we know that (A + A\u2032) is a symmetric matrix and (A \u2013 A\u2032) is\na skew symmetric matrix Since for any matrix A, (kA)\u2032 = kA\u2032, it follows that 1 (A\nA )\n2\n\u2032\n+\nRationalised 2023-24\nMATRICES         65\nis symmetric matrix and 1 (A\nA )\n2\n\u2032\n\u2212\n is skew symmetric matrix Thus, any square\nmatrix can be expressed as the sum of a symmetric and a skew symmetric matrix Example 22 Express the matrix \n2\n2\n4\nB\n1\n3\n4\n1\n2\n3\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n= \u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8f0\n\uf8fb\n as the sum of a symmetric and a\nskew symmetric matrix"
    },
    {
        "Chapter": "1",
        "sentence_range": "1311-1314",
        "Text": "Since for any matrix A, (kA)\u2032 = kA\u2032, it follows that 1 (A\nA )\n2\n\u2032\n+\nRationalised 2023-24\nMATRICES         65\nis symmetric matrix and 1 (A\nA )\n2\n\u2032\n\u2212\n is skew symmetric matrix Thus, any square\nmatrix can be expressed as the sum of a symmetric and a skew symmetric matrix Example 22 Express the matrix \n2\n2\n4\nB\n1\n3\n4\n1\n2\n3\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n= \u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8f0\n\uf8fb\n as the sum of a symmetric and a\nskew symmetric matrix Solution Here\nB\u2032 =\n2\n1\n1\n2\n3\n2\n4\n4\n3\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8f0\n\uf8fb\nLet\nP =\n4\n3\n3\n1\n1\n(B + B )\n3\n6\n2\n2\n2\n3\n2\n6\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2032 =\n\uf8ef\u2212\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8f0\n\uf8fb\n = \n3\n3\n2\n2\n2\n3\n3\n1\n32\n1\n3\n2\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\u2212\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\u2212\n\u2212\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n,\nNow\nP\u2032 =\n3\n3\n2\n2\n2\n3\n3\n1\n2\n3\n1\n3\n2\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\u2212\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\u2212\n\u2212\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n= P\nThus\nP = 1 (B + B )\n2\n\u2032  is a symmetric matrix"
    },
    {
        "Chapter": "1",
        "sentence_range": "1312-1315",
        "Text": "Thus, any square\nmatrix can be expressed as the sum of a symmetric and a skew symmetric matrix Example 22 Express the matrix \n2\n2\n4\nB\n1\n3\n4\n1\n2\n3\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n= \u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8f0\n\uf8fb\n as the sum of a symmetric and a\nskew symmetric matrix Solution Here\nB\u2032 =\n2\n1\n1\n2\n3\n2\n4\n4\n3\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8f0\n\uf8fb\nLet\nP =\n4\n3\n3\n1\n1\n(B + B )\n3\n6\n2\n2\n2\n3\n2\n6\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2032 =\n\uf8ef\u2212\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8f0\n\uf8fb\n = \n3\n3\n2\n2\n2\n3\n3\n1\n32\n1\n3\n2\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\u2212\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\u2212\n\u2212\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n,\nNow\nP\u2032 =\n3\n3\n2\n2\n2\n3\n3\n1\n2\n3\n1\n3\n2\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\u2212\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\u2212\n\u2212\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n= P\nThus\nP = 1 (B + B )\n2\n\u2032  is a symmetric matrix Also, let\nQ =\n1\n5\n0\n2\n2\n0\n1\n5\n1\n1\n1\n(B \u2013 B )\n1\n0\n6\n0\n3\n2\n2\n2\n5\n6\n0\n5\n3\n0\n2\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2032 =\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8ef\n\uf8fa\n\u2212\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nRationalised 2023-24\n 66\nMATHEMATICS\nThen\nQ\u2032 =\n1\n5\n0\n2\n3\n1\n0\n3\nQ\n52\n3\n0\n2\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\u2212\n\u2212\uf8fa\n= \u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\u2212\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nThus\nQ = 1 (B \u2013 B )\n2\n\u2032  is a skew symmetric matrix"
    },
    {
        "Chapter": "1",
        "sentence_range": "1313-1316",
        "Text": "Example 22 Express the matrix \n2\n2\n4\nB\n1\n3\n4\n1\n2\n3\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n= \u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8f0\n\uf8fb\n as the sum of a symmetric and a\nskew symmetric matrix Solution Here\nB\u2032 =\n2\n1\n1\n2\n3\n2\n4\n4\n3\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8f0\n\uf8fb\nLet\nP =\n4\n3\n3\n1\n1\n(B + B )\n3\n6\n2\n2\n2\n3\n2\n6\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2032 =\n\uf8ef\u2212\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8f0\n\uf8fb\n = \n3\n3\n2\n2\n2\n3\n3\n1\n32\n1\n3\n2\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\u2212\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\u2212\n\u2212\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n,\nNow\nP\u2032 =\n3\n3\n2\n2\n2\n3\n3\n1\n2\n3\n1\n3\n2\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\u2212\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\u2212\n\u2212\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n= P\nThus\nP = 1 (B + B )\n2\n\u2032  is a symmetric matrix Also, let\nQ =\n1\n5\n0\n2\n2\n0\n1\n5\n1\n1\n1\n(B \u2013 B )\n1\n0\n6\n0\n3\n2\n2\n2\n5\n6\n0\n5\n3\n0\n2\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2032 =\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8ef\n\uf8fa\n\u2212\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nRationalised 2023-24\n 66\nMATHEMATICS\nThen\nQ\u2032 =\n1\n5\n0\n2\n3\n1\n0\n3\nQ\n52\n3\n0\n2\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\u2212\n\u2212\uf8fa\n= \u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\u2212\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nThus\nQ = 1 (B \u2013 B )\n2\n\u2032  is a skew symmetric matrix Now\n3\n3\n1\n5\n2\n0\n2\n2\n2\n2\n2\n2\n4\n3\n1\nP + Q\n3\n1\n0\n3\n1\n3\n4\nB\n2\n2\n1\n2\n3\n3\n5\n1\n3\n3\n0\n2\n2\n\u2212\n\u2212\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n=\n+\n= \u2212\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8ef\u2212\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nThus, B is represented as the sum of a symmetric and a skew symmetric matrix"
    },
    {
        "Chapter": "1",
        "sentence_range": "1314-1317",
        "Text": "Solution Here\nB\u2032 =\n2\n1\n1\n2\n3\n2\n4\n4\n3\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8f0\n\uf8fb\nLet\nP =\n4\n3\n3\n1\n1\n(B + B )\n3\n6\n2\n2\n2\n3\n2\n6\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2032 =\n\uf8ef\u2212\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8f0\n\uf8fb\n = \n3\n3\n2\n2\n2\n3\n3\n1\n32\n1\n3\n2\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\u2212\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\u2212\n\u2212\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n,\nNow\nP\u2032 =\n3\n3\n2\n2\n2\n3\n3\n1\n2\n3\n1\n3\n2\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\u2212\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\u2212\n\u2212\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n= P\nThus\nP = 1 (B + B )\n2\n\u2032  is a symmetric matrix Also, let\nQ =\n1\n5\n0\n2\n2\n0\n1\n5\n1\n1\n1\n(B \u2013 B )\n1\n0\n6\n0\n3\n2\n2\n2\n5\n6\n0\n5\n3\n0\n2\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2032 =\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8ef\n\uf8fa\n\u2212\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nRationalised 2023-24\n 66\nMATHEMATICS\nThen\nQ\u2032 =\n1\n5\n0\n2\n3\n1\n0\n3\nQ\n52\n3\n0\n2\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\u2212\n\u2212\uf8fa\n= \u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\u2212\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nThus\nQ = 1 (B \u2013 B )\n2\n\u2032  is a skew symmetric matrix Now\n3\n3\n1\n5\n2\n0\n2\n2\n2\n2\n2\n2\n4\n3\n1\nP + Q\n3\n1\n0\n3\n1\n3\n4\nB\n2\n2\n1\n2\n3\n3\n5\n1\n3\n3\n0\n2\n2\n\u2212\n\u2212\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n=\n+\n= \u2212\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8ef\u2212\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nThus, B is represented as the sum of a symmetric and a skew symmetric matrix EXERCISE 3"
    },
    {
        "Chapter": "1",
        "sentence_range": "1315-1318",
        "Text": "Also, let\nQ =\n1\n5\n0\n2\n2\n0\n1\n5\n1\n1\n1\n(B \u2013 B )\n1\n0\n6\n0\n3\n2\n2\n2\n5\n6\n0\n5\n3\n0\n2\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2032 =\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8ef\n\uf8fa\n\u2212\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nRationalised 2023-24\n 66\nMATHEMATICS\nThen\nQ\u2032 =\n1\n5\n0\n2\n3\n1\n0\n3\nQ\n52\n3\n0\n2\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\u2212\n\u2212\uf8fa\n= \u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\u2212\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nThus\nQ = 1 (B \u2013 B )\n2\n\u2032  is a skew symmetric matrix Now\n3\n3\n1\n5\n2\n0\n2\n2\n2\n2\n2\n2\n4\n3\n1\nP + Q\n3\n1\n0\n3\n1\n3\n4\nB\n2\n2\n1\n2\n3\n3\n5\n1\n3\n3\n0\n2\n2\n\u2212\n\u2212\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n=\n+\n= \u2212\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8ef\u2212\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nThus, B is represented as the sum of a symmetric and a skew symmetric matrix EXERCISE 3 3\n1"
    },
    {
        "Chapter": "1",
        "sentence_range": "1316-1319",
        "Text": "Now\n3\n3\n1\n5\n2\n0\n2\n2\n2\n2\n2\n2\n4\n3\n1\nP + Q\n3\n1\n0\n3\n1\n3\n4\nB\n2\n2\n1\n2\n3\n3\n5\n1\n3\n3\n0\n2\n2\n\u2212\n\u2212\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n=\n+\n= \u2212\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8ef\u2212\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nThus, B is represented as the sum of a symmetric and a skew symmetric matrix EXERCISE 3 3\n1 Find the transpose of each of the following matrices:\n(i)\n5\n21\n1\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8f0\u2212\uf8fa\n\uf8fb\n(ii)\n1\n1\n2\n3\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n(iii)\n1\n5\n6\n3\n5\n6\n2\n3\n1\n\uf8ee\u2212\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\u2212\uf8fa\n\uf8f0\n\uf8fb\n2"
    },
    {
        "Chapter": "1",
        "sentence_range": "1317-1320",
        "Text": "EXERCISE 3 3\n1 Find the transpose of each of the following matrices:\n(i)\n5\n21\n1\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8f0\u2212\uf8fa\n\uf8fb\n(ii)\n1\n1\n2\n3\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n(iii)\n1\n5\n6\n3\n5\n6\n2\n3\n1\n\uf8ee\u2212\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\u2212\uf8fa\n\uf8f0\n\uf8fb\n2 If \n1\n2\n3\n4\n1\n5\nA\n5\n7\n9\nand B\n1\n2\n0\n2\n1\n1\n1\n3\n1\n\u2212\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n=\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\u2212\n\uf8fb\n\uf8f0\n\uf8fb\n, then verify that\n(i) (A + B)\u2032 = A\u2032 + B\u2032,\n(ii) (A \u2013 B)\u2032 = A\u2032 \u2013 B\u2032\n3"
    },
    {
        "Chapter": "1",
        "sentence_range": "1318-1321",
        "Text": "3\n1 Find the transpose of each of the following matrices:\n(i)\n5\n21\n1\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8f0\u2212\uf8fa\n\uf8fb\n(ii)\n1\n1\n2\n3\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n(iii)\n1\n5\n6\n3\n5\n6\n2\n3\n1\n\uf8ee\u2212\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\u2212\uf8fa\n\uf8f0\n\uf8fb\n2 If \n1\n2\n3\n4\n1\n5\nA\n5\n7\n9\nand B\n1\n2\n0\n2\n1\n1\n1\n3\n1\n\u2212\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n=\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\u2212\n\uf8fb\n\uf8f0\n\uf8fb\n, then verify that\n(i) (A + B)\u2032 = A\u2032 + B\u2032,\n(ii) (A \u2013 B)\u2032 = A\u2032 \u2013 B\u2032\n3 If \n3 4\n1\n2\n1\nA\n1 2\nand B\n1\n2 3\n0 1\n\uf8ee\n\uf8f9\n\uf8ee\u2212\n\uf8f9\n\uf8ef\n\uf8fa\n\u2032 = \u2212\n= \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, then verify that\n(i) (A + B)\u2032 = A\u2032 + B\u2032\n(ii) (A \u2013 B)\u2032 = A\u2032 \u2013 B\u2032\nRationalised 2023-24\nMATRICES         67\n4"
    },
    {
        "Chapter": "1",
        "sentence_range": "1319-1322",
        "Text": "Find the transpose of each of the following matrices:\n(i)\n5\n21\n1\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8f0\u2212\uf8fa\n\uf8fb\n(ii)\n1\n1\n2\n3\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n(iii)\n1\n5\n6\n3\n5\n6\n2\n3\n1\n\uf8ee\u2212\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\u2212\uf8fa\n\uf8f0\n\uf8fb\n2 If \n1\n2\n3\n4\n1\n5\nA\n5\n7\n9\nand B\n1\n2\n0\n2\n1\n1\n1\n3\n1\n\u2212\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n=\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\u2212\n\uf8fb\n\uf8f0\n\uf8fb\n, then verify that\n(i) (A + B)\u2032 = A\u2032 + B\u2032,\n(ii) (A \u2013 B)\u2032 = A\u2032 \u2013 B\u2032\n3 If \n3 4\n1\n2\n1\nA\n1 2\nand B\n1\n2 3\n0 1\n\uf8ee\n\uf8f9\n\uf8ee\u2212\n\uf8f9\n\uf8ef\n\uf8fa\n\u2032 = \u2212\n= \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, then verify that\n(i) (A + B)\u2032 = A\u2032 + B\u2032\n(ii) (A \u2013 B)\u2032 = A\u2032 \u2013 B\u2032\nRationalised 2023-24\nMATRICES         67\n4 If \n2\n3\n1\n0\nA\nand B\n1\n2\n1\n2\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\u2032 =\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n, then find (A + 2B)\u2032\n5"
    },
    {
        "Chapter": "1",
        "sentence_range": "1320-1323",
        "Text": "If \n1\n2\n3\n4\n1\n5\nA\n5\n7\n9\nand B\n1\n2\n0\n2\n1\n1\n1\n3\n1\n\u2212\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n=\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\u2212\n\uf8fb\n\uf8f0\n\uf8fb\n, then verify that\n(i) (A + B)\u2032 = A\u2032 + B\u2032,\n(ii) (A \u2013 B)\u2032 = A\u2032 \u2013 B\u2032\n3 If \n3 4\n1\n2\n1\nA\n1 2\nand B\n1\n2 3\n0 1\n\uf8ee\n\uf8f9\n\uf8ee\u2212\n\uf8f9\n\uf8ef\n\uf8fa\n\u2032 = \u2212\n= \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, then verify that\n(i) (A + B)\u2032 = A\u2032 + B\u2032\n(ii) (A \u2013 B)\u2032 = A\u2032 \u2013 B\u2032\nRationalised 2023-24\nMATRICES         67\n4 If \n2\n3\n1\n0\nA\nand B\n1\n2\n1\n2\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\u2032 =\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n, then find (A + 2B)\u2032\n5 For the matrices A and B, verify that (AB)\u2032 = B\u2032A\u2032, where\n(i)\n[\n]\n1\nA\n4\n, B\n1\n2\n1\n3\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n= \u2212\n= \u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n(ii)\n[\n]\n0\nA\n1\n, B\n1\n5\n7\n2\n\uf8ee \uf8f9\n=\uf8ef \uf8fa\n=\n\uf8ef \uf8fa\n\uf8ef \uf8fa\n\uf8f0 \uf8fb\n6"
    },
    {
        "Chapter": "1",
        "sentence_range": "1321-1324",
        "Text": "If \n3 4\n1\n2\n1\nA\n1 2\nand B\n1\n2 3\n0 1\n\uf8ee\n\uf8f9\n\uf8ee\u2212\n\uf8f9\n\uf8ef\n\uf8fa\n\u2032 = \u2212\n= \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, then verify that\n(i) (A + B)\u2032 = A\u2032 + B\u2032\n(ii) (A \u2013 B)\u2032 = A\u2032 \u2013 B\u2032\nRationalised 2023-24\nMATRICES         67\n4 If \n2\n3\n1\n0\nA\nand B\n1\n2\n1\n2\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\u2032 =\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n, then find (A + 2B)\u2032\n5 For the matrices A and B, verify that (AB)\u2032 = B\u2032A\u2032, where\n(i)\n[\n]\n1\nA\n4\n, B\n1\n2\n1\n3\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n= \u2212\n= \u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n(ii)\n[\n]\n0\nA\n1\n, B\n1\n5\n7\n2\n\uf8ee \uf8f9\n=\uf8ef \uf8fa\n=\n\uf8ef \uf8fa\n\uf8ef \uf8fa\n\uf8f0 \uf8fb\n6 If (i)\ncos\nsin\nA\nsin\ncos\n\u03b1\n\u03b1\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\u2212\n\u03b1\n\u03b1\n\uf8f0\n\uf8fb\n, then verify that A\u2032 A = I\n(ii) If \nsin\ncos\nA\ncos\nsin\n\u03b1\n\u03b1\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\u2212\n\u03b1\n\u03b1\n\uf8f0\n\uf8fb\n, then verify that A\u2032 A = I\n7"
    },
    {
        "Chapter": "1",
        "sentence_range": "1322-1325",
        "Text": "If \n2\n3\n1\n0\nA\nand B\n1\n2\n1\n2\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\u2032 =\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n, then find (A + 2B)\u2032\n5 For the matrices A and B, verify that (AB)\u2032 = B\u2032A\u2032, where\n(i)\n[\n]\n1\nA\n4\n, B\n1\n2\n1\n3\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n= \u2212\n= \u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n(ii)\n[\n]\n0\nA\n1\n, B\n1\n5\n7\n2\n\uf8ee \uf8f9\n=\uf8ef \uf8fa\n=\n\uf8ef \uf8fa\n\uf8ef \uf8fa\n\uf8f0 \uf8fb\n6 If (i)\ncos\nsin\nA\nsin\ncos\n\u03b1\n\u03b1\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\u2212\n\u03b1\n\u03b1\n\uf8f0\n\uf8fb\n, then verify that A\u2032 A = I\n(ii) If \nsin\ncos\nA\ncos\nsin\n\u03b1\n\u03b1\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\u2212\n\u03b1\n\u03b1\n\uf8f0\n\uf8fb\n, then verify that A\u2032 A = I\n7 (i) Show that the matrix \n1\n1\n5\nA\n1\n2\n1\n5\n1\n3\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n= \u2212\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n is a symmetric matrix"
    },
    {
        "Chapter": "1",
        "sentence_range": "1323-1326",
        "Text": "For the matrices A and B, verify that (AB)\u2032 = B\u2032A\u2032, where\n(i)\n[\n]\n1\nA\n4\n, B\n1\n2\n1\n3\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n= \u2212\n= \u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n(ii)\n[\n]\n0\nA\n1\n, B\n1\n5\n7\n2\n\uf8ee \uf8f9\n=\uf8ef \uf8fa\n=\n\uf8ef \uf8fa\n\uf8ef \uf8fa\n\uf8f0 \uf8fb\n6 If (i)\ncos\nsin\nA\nsin\ncos\n\u03b1\n\u03b1\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\u2212\n\u03b1\n\u03b1\n\uf8f0\n\uf8fb\n, then verify that A\u2032 A = I\n(ii) If \nsin\ncos\nA\ncos\nsin\n\u03b1\n\u03b1\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\u2212\n\u03b1\n\u03b1\n\uf8f0\n\uf8fb\n, then verify that A\u2032 A = I\n7 (i) Show that the matrix \n1\n1\n5\nA\n1\n2\n1\n5\n1\n3\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n= \u2212\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n is a symmetric matrix (ii)  Show that the matrix \n0\n1\n1\nA\n1\n0\n1\n1\n1\n0\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n= \u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n is a skew symmetric matrix"
    },
    {
        "Chapter": "1",
        "sentence_range": "1324-1327",
        "Text": "If (i)\ncos\nsin\nA\nsin\ncos\n\u03b1\n\u03b1\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\u2212\n\u03b1\n\u03b1\n\uf8f0\n\uf8fb\n, then verify that A\u2032 A = I\n(ii) If \nsin\ncos\nA\ncos\nsin\n\u03b1\n\u03b1\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\u2212\n\u03b1\n\u03b1\n\uf8f0\n\uf8fb\n, then verify that A\u2032 A = I\n7 (i) Show that the matrix \n1\n1\n5\nA\n1\n2\n1\n5\n1\n3\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n= \u2212\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n is a symmetric matrix (ii)  Show that the matrix \n0\n1\n1\nA\n1\n0\n1\n1\n1\n0\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n= \u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n is a skew symmetric matrix 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "1325-1328",
        "Text": "(i) Show that the matrix \n1\n1\n5\nA\n1\n2\n1\n5\n1\n3\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n= \u2212\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n is a symmetric matrix (ii)  Show that the matrix \n0\n1\n1\nA\n1\n0\n1\n1\n1\n0\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n= \u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n is a skew symmetric matrix 8 For the matrix \n1\n5\nA\n6\n7\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, verify that\n(i) (A + A\u2032) is a symmetric matrix\n(ii) (A \u2013 A\u2032) is a skew symmetric matrix\n9"
    },
    {
        "Chapter": "1",
        "sentence_range": "1326-1329",
        "Text": "(ii)  Show that the matrix \n0\n1\n1\nA\n1\n0\n1\n1\n1\n0\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n= \u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n is a skew symmetric matrix 8 For the matrix \n1\n5\nA\n6\n7\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, verify that\n(i) (A + A\u2032) is a symmetric matrix\n(ii) (A \u2013 A\u2032) is a skew symmetric matrix\n9 Find \n(\n)\n1 A\nA\n2\n\u2032\n+\n and \n(\n)\n1 A\nA\n2\n\u2032\n\u2212\n, when \n0\nA\n0\n0\na\nb\na\nc\nb\nc\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n= \u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8f0\n\uf8fb\n10"
    },
    {
        "Chapter": "1",
        "sentence_range": "1327-1330",
        "Text": "8 For the matrix \n1\n5\nA\n6\n7\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, verify that\n(i) (A + A\u2032) is a symmetric matrix\n(ii) (A \u2013 A\u2032) is a skew symmetric matrix\n9 Find \n(\n)\n1 A\nA\n2\n\u2032\n+\n and \n(\n)\n1 A\nA\n2\n\u2032\n\u2212\n, when \n0\nA\n0\n0\na\nb\na\nc\nb\nc\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n= \u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8f0\n\uf8fb\n10 Express the following matrices as the sum of a symmetric and a skew symmetric\nmatrix:\nRationalised 2023-24\n 68\nMATHEMATICS\n(i)\n3\n5\n1\n1\n\uf8ee\n\uf8f9\n\uf8ef\n\u2212\uf8fa\n\uf8f0\n\uf8fb\n(ii)\n6\n2\n2\n2\n3\n1\n2\n1\n3\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n(iii)\n3\n3\n1\n2\n2\n1\n4\n5\n2\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8f0\n\uf8fb\n(iv)\n1\n5\n1\n2\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\u2212\n\uf8fb\nChoose the correct answer in the Exercises 11 and 12"
    },
    {
        "Chapter": "1",
        "sentence_range": "1328-1331",
        "Text": "For the matrix \n1\n5\nA\n6\n7\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, verify that\n(i) (A + A\u2032) is a symmetric matrix\n(ii) (A \u2013 A\u2032) is a skew symmetric matrix\n9 Find \n(\n)\n1 A\nA\n2\n\u2032\n+\n and \n(\n)\n1 A\nA\n2\n\u2032\n\u2212\n, when \n0\nA\n0\n0\na\nb\na\nc\nb\nc\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n= \u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8f0\n\uf8fb\n10 Express the following matrices as the sum of a symmetric and a skew symmetric\nmatrix:\nRationalised 2023-24\n 68\nMATHEMATICS\n(i)\n3\n5\n1\n1\n\uf8ee\n\uf8f9\n\uf8ef\n\u2212\uf8fa\n\uf8f0\n\uf8fb\n(ii)\n6\n2\n2\n2\n3\n1\n2\n1\n3\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n(iii)\n3\n3\n1\n2\n2\n1\n4\n5\n2\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8f0\n\uf8fb\n(iv)\n1\n5\n1\n2\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\u2212\n\uf8fb\nChoose the correct answer in the Exercises 11 and 12 11"
    },
    {
        "Chapter": "1",
        "sentence_range": "1329-1332",
        "Text": "Find \n(\n)\n1 A\nA\n2\n\u2032\n+\n and \n(\n)\n1 A\nA\n2\n\u2032\n\u2212\n, when \n0\nA\n0\n0\na\nb\na\nc\nb\nc\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n= \u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8f0\n\uf8fb\n10 Express the following matrices as the sum of a symmetric and a skew symmetric\nmatrix:\nRationalised 2023-24\n 68\nMATHEMATICS\n(i)\n3\n5\n1\n1\n\uf8ee\n\uf8f9\n\uf8ef\n\u2212\uf8fa\n\uf8f0\n\uf8fb\n(ii)\n6\n2\n2\n2\n3\n1\n2\n1\n3\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n(iii)\n3\n3\n1\n2\n2\n1\n4\n5\n2\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8f0\n\uf8fb\n(iv)\n1\n5\n1\n2\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\u2212\n\uf8fb\nChoose the correct answer in the Exercises 11 and 12 11 If A, B are symmetric matrices of same order, then AB \u2013 BA is a\n(A) Skew symmetric matrix\n(B) Symmetric matrix\n(C) Zero matrix\n(D) Identity matrix\n12"
    },
    {
        "Chapter": "1",
        "sentence_range": "1330-1333",
        "Text": "Express the following matrices as the sum of a symmetric and a skew symmetric\nmatrix:\nRationalised 2023-24\n 68\nMATHEMATICS\n(i)\n3\n5\n1\n1\n\uf8ee\n\uf8f9\n\uf8ef\n\u2212\uf8fa\n\uf8f0\n\uf8fb\n(ii)\n6\n2\n2\n2\n3\n1\n2\n1\n3\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n(iii)\n3\n3\n1\n2\n2\n1\n4\n5\n2\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8f0\n\uf8fb\n(iv)\n1\n5\n1\n2\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\u2212\n\uf8fb\nChoose the correct answer in the Exercises 11 and 12 11 If A, B are symmetric matrices of same order, then AB \u2013 BA is a\n(A) Skew symmetric matrix\n(B) Symmetric matrix\n(C) Zero matrix\n(D) Identity matrix\n12 If \ncos\nsin\nA\n,\nsin\ncos\n\u03b1\n\u2212\n\u03b1\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\u03b1\n\u03b1\n\uf8f0\n\uf8fb\nand  A + A\u2032 = I, then the value of \u03b1 is\n(A) 6\n\u03c0\n(B) 3\n\u03c0\n(C) \u03c0\n(D) 3\n2\n\u03c0\n3"
    },
    {
        "Chapter": "1",
        "sentence_range": "1331-1334",
        "Text": "11 If A, B are symmetric matrices of same order, then AB \u2013 BA is a\n(A) Skew symmetric matrix\n(B) Symmetric matrix\n(C) Zero matrix\n(D) Identity matrix\n12 If \ncos\nsin\nA\n,\nsin\ncos\n\u03b1\n\u2212\n\u03b1\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\u03b1\n\u03b1\n\uf8f0\n\uf8fb\nand  A + A\u2032 = I, then the value of \u03b1 is\n(A) 6\n\u03c0\n(B) 3\n\u03c0\n(C) \u03c0\n(D) 3\n2\n\u03c0\n3 7  Invertible Matrices\nDefinition 6 If A is a square matrix of order m, and if there exists another square\nmatrix B of the same order m, such that AB = BA = I, then B is called the inverse\nmatrix of A and it is denoted by A\u2013 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "1332-1335",
        "Text": "If A, B are symmetric matrices of same order, then AB \u2013 BA is a\n(A) Skew symmetric matrix\n(B) Symmetric matrix\n(C) Zero matrix\n(D) Identity matrix\n12 If \ncos\nsin\nA\n,\nsin\ncos\n\u03b1\n\u2212\n\u03b1\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\u03b1\n\u03b1\n\uf8f0\n\uf8fb\nand  A + A\u2032 = I, then the value of \u03b1 is\n(A) 6\n\u03c0\n(B) 3\n\u03c0\n(C) \u03c0\n(D) 3\n2\n\u03c0\n3 7  Invertible Matrices\nDefinition 6 If A is a square matrix of order m, and if there exists another square\nmatrix B of the same order m, such that AB = BA = I, then B is called the inverse\nmatrix of A and it is denoted by A\u2013 1 In that case A is said to be invertible"
    },
    {
        "Chapter": "1",
        "sentence_range": "1333-1336",
        "Text": "If \ncos\nsin\nA\n,\nsin\ncos\n\u03b1\n\u2212\n\u03b1\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\u03b1\n\u03b1\n\uf8f0\n\uf8fb\nand  A + A\u2032 = I, then the value of \u03b1 is\n(A) 6\n\u03c0\n(B) 3\n\u03c0\n(C) \u03c0\n(D) 3\n2\n\u03c0\n3 7  Invertible Matrices\nDefinition 6 If A is a square matrix of order m, and if there exists another square\nmatrix B of the same order m, such that AB = BA = I, then B is called the inverse\nmatrix of A and it is denoted by A\u2013 1 In that case A is said to be invertible For example, let\nA =\n2\n3\n1\n2\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nand B = \n2\n3\n1\n2\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\u2212\n\uf8fb\nbe two matrices"
    },
    {
        "Chapter": "1",
        "sentence_range": "1334-1337",
        "Text": "7  Invertible Matrices\nDefinition 6 If A is a square matrix of order m, and if there exists another square\nmatrix B of the same order m, such that AB = BA = I, then B is called the inverse\nmatrix of A and it is denoted by A\u2013 1 In that case A is said to be invertible For example, let\nA =\n2\n3\n1\n2\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nand B = \n2\n3\n1\n2\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\u2212\n\uf8fb\nbe two matrices Now\nAB =\n2\n3\n2\n3\n1\n2\n1\n2\n\u2212\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n=\n4\n3\n6\n6\n1\n0\nI\n2\n2\n3\n4\n0\n1\n\u2212\n\u2212 +\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212 +\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nAlso\nBA =\n1\n0\nI\n0\n1\n\uf8ee\n\uf8f9 =\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb"
    },
    {
        "Chapter": "1",
        "sentence_range": "1335-1338",
        "Text": "In that case A is said to be invertible For example, let\nA =\n2\n3\n1\n2\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nand B = \n2\n3\n1\n2\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\u2212\n\uf8fb\nbe two matrices Now\nAB =\n2\n3\n2\n3\n1\n2\n1\n2\n\u2212\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n=\n4\n3\n6\n6\n1\n0\nI\n2\n2\n3\n4\n0\n1\n\u2212\n\u2212 +\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212 +\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nAlso\nBA =\n1\n0\nI\n0\n1\n\uf8ee\n\uf8f9 =\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb Thus B is the inverse of A, in other\nwords B = A\u2013 1 and A is inverse of B, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "1336-1339",
        "Text": "For example, let\nA =\n2\n3\n1\n2\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nand B = \n2\n3\n1\n2\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\u2212\n\uf8fb\nbe two matrices Now\nAB =\n2\n3\n2\n3\n1\n2\n1\n2\n\u2212\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n=\n4\n3\n6\n6\n1\n0\nI\n2\n2\n3\n4\n0\n1\n\u2212\n\u2212 +\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212 +\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nAlso\nBA =\n1\n0\nI\n0\n1\n\uf8ee\n\uf8f9 =\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb Thus B is the inverse of A, in other\nwords B = A\u2013 1 and A is inverse of B, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "1337-1340",
        "Text": "Now\nAB =\n2\n3\n2\n3\n1\n2\n1\n2\n\u2212\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n=\n4\n3\n6\n6\n1\n0\nI\n2\n2\n3\n4\n0\n1\n\u2212\n\u2212 +\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212 +\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nAlso\nBA =\n1\n0\nI\n0\n1\n\uf8ee\n\uf8f9 =\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb Thus B is the inverse of A, in other\nwords B = A\u2013 1 and A is inverse of B, i e , A = B\u20131\nRationalised 2023-24\nMATRICES         69\nANote\n1"
    },
    {
        "Chapter": "1",
        "sentence_range": "1338-1341",
        "Text": "Thus B is the inverse of A, in other\nwords B = A\u2013 1 and A is inverse of B, i e , A = B\u20131\nRationalised 2023-24\nMATRICES         69\nANote\n1 A rectangular matrix does not possess inverse matrix, since for products BA\nand AB to be defined and to be equal, it is necessary that matrices A and B\nshould be square matrices of the same order"
    },
    {
        "Chapter": "1",
        "sentence_range": "1339-1342",
        "Text": "e , A = B\u20131\nRationalised 2023-24\nMATRICES         69\nANote\n1 A rectangular matrix does not possess inverse matrix, since for products BA\nand AB to be defined and to be equal, it is necessary that matrices A and B\nshould be square matrices of the same order 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "1340-1343",
        "Text": ", A = B\u20131\nRationalised 2023-24\nMATRICES         69\nANote\n1 A rectangular matrix does not possess inverse matrix, since for products BA\nand AB to be defined and to be equal, it is necessary that matrices A and B\nshould be square matrices of the same order 2 If B is the inverse of A, then A is also the inverse of B"
    },
    {
        "Chapter": "1",
        "sentence_range": "1341-1344",
        "Text": "A rectangular matrix does not possess inverse matrix, since for products BA\nand AB to be defined and to be equal, it is necessary that matrices A and B\nshould be square matrices of the same order 2 If B is the inverse of A, then A is also the inverse of B Theorem 3 (Uniqueness of inverse) Inverse of a square matrix, if it exists, is unique"
    },
    {
        "Chapter": "1",
        "sentence_range": "1342-1345",
        "Text": "2 If B is the inverse of A, then A is also the inverse of B Theorem 3 (Uniqueness of inverse) Inverse of a square matrix, if it exists, is unique Proof Let A = [aij] be a square matrix of order m"
    },
    {
        "Chapter": "1",
        "sentence_range": "1343-1346",
        "Text": "If B is the inverse of A, then A is also the inverse of B Theorem 3 (Uniqueness of inverse) Inverse of a square matrix, if it exists, is unique Proof Let A = [aij] be a square matrix of order m If possible, let B and C be two\ninverses of A"
    },
    {
        "Chapter": "1",
        "sentence_range": "1344-1347",
        "Text": "Theorem 3 (Uniqueness of inverse) Inverse of a square matrix, if it exists, is unique Proof Let A = [aij] be a square matrix of order m If possible, let B and C be two\ninverses of A We shall show that B = C"
    },
    {
        "Chapter": "1",
        "sentence_range": "1345-1348",
        "Text": "Proof Let A = [aij] be a square matrix of order m If possible, let B and C be two\ninverses of A We shall show that B = C Since B is the inverse of A\nAB = BA = I"
    },
    {
        "Chapter": "1",
        "sentence_range": "1346-1349",
        "Text": "If possible, let B and C be two\ninverses of A We shall show that B = C Since B is the inverse of A\nAB = BA = I (1)\nSince C is also the inverse of A\nAC = CA = I"
    },
    {
        "Chapter": "1",
        "sentence_range": "1347-1350",
        "Text": "We shall show that B = C Since B is the inverse of A\nAB = BA = I (1)\nSince C is also the inverse of A\nAC = CA = I (2)\nThus\nB = BI = B (AC) = (BA) C = IC = C\nTheorem 4 If A and B are invertible matrices of the same order, then (AB)\u20131 = B\u20131 A\u20131"
    },
    {
        "Chapter": "1",
        "sentence_range": "1348-1351",
        "Text": "Since B is the inverse of A\nAB = BA = I (1)\nSince C is also the inverse of A\nAC = CA = I (2)\nThus\nB = BI = B (AC) = (BA) C = IC = C\nTheorem 4 If A and B are invertible matrices of the same order, then (AB)\u20131 = B\u20131 A\u20131 Proof From the definition of inverse of a matrix, we have\n(AB) (AB)\u20131 = 1\nor\nA\u20131 (AB) (AB)\u20131 = A\u20131I\n(Pre multiplying both sides by A\u20131)\nor\n(A\u20131A) B (AB)\u20131 = A\u20131\n(Since A\u20131 I = A\u20131)\nor\nIB (AB)\u20131 = A\u20131\nor\nB (AB)\u20131 = A\u20131\nor\nB\u20131 B (AB)\u20131 = B\u20131 A\u20131\nor\nI (AB)\u20131 = B\u20131 A\u20131\nHence\n(AB)\u20131 = B\u20131 A\u20131\n1"
    },
    {
        "Chapter": "1",
        "sentence_range": "1349-1352",
        "Text": "(1)\nSince C is also the inverse of A\nAC = CA = I (2)\nThus\nB = BI = B (AC) = (BA) C = IC = C\nTheorem 4 If A and B are invertible matrices of the same order, then (AB)\u20131 = B\u20131 A\u20131 Proof From the definition of inverse of a matrix, we have\n(AB) (AB)\u20131 = 1\nor\nA\u20131 (AB) (AB)\u20131 = A\u20131I\n(Pre multiplying both sides by A\u20131)\nor\n(A\u20131A) B (AB)\u20131 = A\u20131\n(Since A\u20131 I = A\u20131)\nor\nIB (AB)\u20131 = A\u20131\nor\nB (AB)\u20131 = A\u20131\nor\nB\u20131 B (AB)\u20131 = B\u20131 A\u20131\nor\nI (AB)\u20131 = B\u20131 A\u20131\nHence\n(AB)\u20131 = B\u20131 A\u20131\n1 Matrices A and B will be inverse of each other only if\n     (A) AB = BA (B) AB = BA = 0\n     (C) AB = 0, BA = I (D) AB = BA = I\nEXERCISE 3"
    },
    {
        "Chapter": "1",
        "sentence_range": "1350-1353",
        "Text": "(2)\nThus\nB = BI = B (AC) = (BA) C = IC = C\nTheorem 4 If A and B are invertible matrices of the same order, then (AB)\u20131 = B\u20131 A\u20131 Proof From the definition of inverse of a matrix, we have\n(AB) (AB)\u20131 = 1\nor\nA\u20131 (AB) (AB)\u20131 = A\u20131I\n(Pre multiplying both sides by A\u20131)\nor\n(A\u20131A) B (AB)\u20131 = A\u20131\n(Since A\u20131 I = A\u20131)\nor\nIB (AB)\u20131 = A\u20131\nor\nB (AB)\u20131 = A\u20131\nor\nB\u20131 B (AB)\u20131 = B\u20131 A\u20131\nor\nI (AB)\u20131 = B\u20131 A\u20131\nHence\n(AB)\u20131 = B\u20131 A\u20131\n1 Matrices A and B will be inverse of each other only if\n     (A) AB = BA (B) AB = BA = 0\n     (C) AB = 0, BA = I (D) AB = BA = I\nEXERCISE 3 4\nRationalised 2023-24\n 70\nMATHEMATICS\nMiscellaneous Examples\nExample 23 If \ncos\nsin\nA\nsin\ncos\n\u03b8\n\u03b8\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\u2212\n\u03b8\n\u03b8\n\uf8f0\n\uf8fb\n, then prove that \ncos\nsin\nA\nsin\ncos\nn\nn\nn\nn\nn\n\u03b8\n\u03b8\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\u2212\n\u03b8\n\u03b8\n\uf8f0\n\uf8fb\n, n \u2208 N"
    },
    {
        "Chapter": "1",
        "sentence_range": "1351-1354",
        "Text": "Proof From the definition of inverse of a matrix, we have\n(AB) (AB)\u20131 = 1\nor\nA\u20131 (AB) (AB)\u20131 = A\u20131I\n(Pre multiplying both sides by A\u20131)\nor\n(A\u20131A) B (AB)\u20131 = A\u20131\n(Since A\u20131 I = A\u20131)\nor\nIB (AB)\u20131 = A\u20131\nor\nB (AB)\u20131 = A\u20131\nor\nB\u20131 B (AB)\u20131 = B\u20131 A\u20131\nor\nI (AB)\u20131 = B\u20131 A\u20131\nHence\n(AB)\u20131 = B\u20131 A\u20131\n1 Matrices A and B will be inverse of each other only if\n     (A) AB = BA (B) AB = BA = 0\n     (C) AB = 0, BA = I (D) AB = BA = I\nEXERCISE 3 4\nRationalised 2023-24\n 70\nMATHEMATICS\nMiscellaneous Examples\nExample 23 If \ncos\nsin\nA\nsin\ncos\n\u03b8\n\u03b8\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\u2212\n\u03b8\n\u03b8\n\uf8f0\n\uf8fb\n, then prove that \ncos\nsin\nA\nsin\ncos\nn\nn\nn\nn\nn\n\u03b8\n\u03b8\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\u2212\n\u03b8\n\u03b8\n\uf8f0\n\uf8fb\n, n \u2208 N Solution We shall prove the result by using principle of mathematical induction"
    },
    {
        "Chapter": "1",
        "sentence_range": "1352-1355",
        "Text": "Matrices A and B will be inverse of each other only if\n     (A) AB = BA (B) AB = BA = 0\n     (C) AB = 0, BA = I (D) AB = BA = I\nEXERCISE 3 4\nRationalised 2023-24\n 70\nMATHEMATICS\nMiscellaneous Examples\nExample 23 If \ncos\nsin\nA\nsin\ncos\n\u03b8\n\u03b8\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\u2212\n\u03b8\n\u03b8\n\uf8f0\n\uf8fb\n, then prove that \ncos\nsin\nA\nsin\ncos\nn\nn\nn\nn\nn\n\u03b8\n\u03b8\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\u2212\n\u03b8\n\u03b8\n\uf8f0\n\uf8fb\n, n \u2208 N Solution We shall prove the result by using principle of mathematical induction We have\nP(n) : If \ncos\nsin\nA\nsin\ncos\n\u03b8\n\u03b8\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\u2212\n\u03b8\n\u03b8\n\uf8f0\n\uf8fb\n, then \ncos\nsin\nA\nsin\ncos\nn\nn\nn\nn\nn\n\u03b8\n\u03b8\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\u2212\n\u03b8\n\u03b8\n\uf8f0\n\uf8fb\n, n \u2208 N\nP(1) : \ncos\nsin\nA\nsin\ncos\n\u03b8\n\u03b8\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\u2212\n\u03b8\n\u03b8\n\uf8f0\n\uf8fb\n, so \n1\ncos\nsin\nA\nsin\ncos\n\u03b8\n\u03b8\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\u2212\n\u03b8\n\u03b8\n\uf8f0\n\uf8fb\nTherefore,\nthe result is true for n = 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "1353-1356",
        "Text": "4\nRationalised 2023-24\n 70\nMATHEMATICS\nMiscellaneous Examples\nExample 23 If \ncos\nsin\nA\nsin\ncos\n\u03b8\n\u03b8\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\u2212\n\u03b8\n\u03b8\n\uf8f0\n\uf8fb\n, then prove that \ncos\nsin\nA\nsin\ncos\nn\nn\nn\nn\nn\n\u03b8\n\u03b8\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\u2212\n\u03b8\n\u03b8\n\uf8f0\n\uf8fb\n, n \u2208 N Solution We shall prove the result by using principle of mathematical induction We have\nP(n) : If \ncos\nsin\nA\nsin\ncos\n\u03b8\n\u03b8\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\u2212\n\u03b8\n\u03b8\n\uf8f0\n\uf8fb\n, then \ncos\nsin\nA\nsin\ncos\nn\nn\nn\nn\nn\n\u03b8\n\u03b8\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\u2212\n\u03b8\n\u03b8\n\uf8f0\n\uf8fb\n, n \u2208 N\nP(1) : \ncos\nsin\nA\nsin\ncos\n\u03b8\n\u03b8\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\u2212\n\u03b8\n\u03b8\n\uf8f0\n\uf8fb\n, so \n1\ncos\nsin\nA\nsin\ncos\n\u03b8\n\u03b8\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\u2212\n\u03b8\n\u03b8\n\uf8f0\n\uf8fb\nTherefore,\nthe result is true for n = 1 Let the result be true for n = k"
    },
    {
        "Chapter": "1",
        "sentence_range": "1354-1357",
        "Text": "Solution We shall prove the result by using principle of mathematical induction We have\nP(n) : If \ncos\nsin\nA\nsin\ncos\n\u03b8\n\u03b8\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\u2212\n\u03b8\n\u03b8\n\uf8f0\n\uf8fb\n, then \ncos\nsin\nA\nsin\ncos\nn\nn\nn\nn\nn\n\u03b8\n\u03b8\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\u2212\n\u03b8\n\u03b8\n\uf8f0\n\uf8fb\n, n \u2208 N\nP(1) : \ncos\nsin\nA\nsin\ncos\n\u03b8\n\u03b8\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\u2212\n\u03b8\n\u03b8\n\uf8f0\n\uf8fb\n, so \n1\ncos\nsin\nA\nsin\ncos\n\u03b8\n\u03b8\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\u2212\n\u03b8\n\u03b8\n\uf8f0\n\uf8fb\nTherefore,\nthe result is true for n = 1 Let the result be true for n = k So\nP(k) : \ncos\nsin\nA\nsin\ncos\n\u03b8\n\u03b8\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\u2212\n\u03b8\n\u03b8\n\uf8f0\n\uf8fb\n, then \ncos\nsin\nA\nsin\ncos\nk\nk\nk\nk\nk\n\u03b8\n\u03b8\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\u2212\n\u03b8\n\u03b8\n\uf8f0\n\uf8fb\nNow, we prove that the result holds for n = k +1\nNow\nAk + 1 =\ncos\nsin\ncos\nsin\nA A\nsin\ncos\nsin\ncos\nk\nk\nk\nk\nk\n\u03b8\n\u03b8\n\u03b8\n\u03b8\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\u22c5\n= \uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\u2212\n\u03b8\n\u03b8\n\u2212\n\u03b8\n\u03b8\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n=\ncos cos\n\u2013 sin sin\ncos sin\nsin cos\nsin cos\ncos sin\nsin sin\ncos cos\nk\nk\nk\nk\nk\nk\nk\nk\n\u03b8\n\u03b8\n\u03b8\n\u03b8\n\u03b8\n\u03b8 +\n\u03b8\n\u03b8\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n\u03b8\n\u03b8 +\n\u03b8\n\u03b8\n\u2212\n\u03b8\n\u03b8 +\n\u03b8\n\u03b8\n\uf8f0\n\uf8fb\n=\ncos(\n)\nsin(\n)\ncos(\n1)\nsin (\n1)\nsin(\n)\ncos(\n)\nsin (\n1)\ncos(\n1)\nk\nk\nk\nk\nk\nk\nk\nk\n\u03b8 + \u03b8\n\u03b8 + \u03b8\n+\n\u03b8\n+\n\u03b8\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u03b8 + \u03b8\n\u03b8 + \u03b8\n\u2212\n+\n\u03b8\n+\n\u03b8\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nTherefore, the result is true for n = k  + 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "1355-1358",
        "Text": "We have\nP(n) : If \ncos\nsin\nA\nsin\ncos\n\u03b8\n\u03b8\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\u2212\n\u03b8\n\u03b8\n\uf8f0\n\uf8fb\n, then \ncos\nsin\nA\nsin\ncos\nn\nn\nn\nn\nn\n\u03b8\n\u03b8\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\u2212\n\u03b8\n\u03b8\n\uf8f0\n\uf8fb\n, n \u2208 N\nP(1) : \ncos\nsin\nA\nsin\ncos\n\u03b8\n\u03b8\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\u2212\n\u03b8\n\u03b8\n\uf8f0\n\uf8fb\n, so \n1\ncos\nsin\nA\nsin\ncos\n\u03b8\n\u03b8\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\u2212\n\u03b8\n\u03b8\n\uf8f0\n\uf8fb\nTherefore,\nthe result is true for n = 1 Let the result be true for n = k So\nP(k) : \ncos\nsin\nA\nsin\ncos\n\u03b8\n\u03b8\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\u2212\n\u03b8\n\u03b8\n\uf8f0\n\uf8fb\n, then \ncos\nsin\nA\nsin\ncos\nk\nk\nk\nk\nk\n\u03b8\n\u03b8\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\u2212\n\u03b8\n\u03b8\n\uf8f0\n\uf8fb\nNow, we prove that the result holds for n = k +1\nNow\nAk + 1 =\ncos\nsin\ncos\nsin\nA A\nsin\ncos\nsin\ncos\nk\nk\nk\nk\nk\n\u03b8\n\u03b8\n\u03b8\n\u03b8\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\u22c5\n= \uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\u2212\n\u03b8\n\u03b8\n\u2212\n\u03b8\n\u03b8\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n=\ncos cos\n\u2013 sin sin\ncos sin\nsin cos\nsin cos\ncos sin\nsin sin\ncos cos\nk\nk\nk\nk\nk\nk\nk\nk\n\u03b8\n\u03b8\n\u03b8\n\u03b8\n\u03b8\n\u03b8 +\n\u03b8\n\u03b8\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n\u03b8\n\u03b8 +\n\u03b8\n\u03b8\n\u2212\n\u03b8\n\u03b8 +\n\u03b8\n\u03b8\n\uf8f0\n\uf8fb\n=\ncos(\n)\nsin(\n)\ncos(\n1)\nsin (\n1)\nsin(\n)\ncos(\n)\nsin (\n1)\ncos(\n1)\nk\nk\nk\nk\nk\nk\nk\nk\n\u03b8 + \u03b8\n\u03b8 + \u03b8\n+\n\u03b8\n+\n\u03b8\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u03b8 + \u03b8\n\u03b8 + \u03b8\n\u2212\n+\n\u03b8\n+\n\u03b8\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nTherefore, the result is true for n = k  + 1 Thus by principle of mathematical induction,\nwe have \ncos\nsin\nA\nsin\ncos\nn\nn\nn\nn\nn\n\u03b8\n\u03b8\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\u2212\n\u03b8\n\u03b8\n\uf8f0\n\uf8fb\n, holds for all natural numbers"
    },
    {
        "Chapter": "1",
        "sentence_range": "1356-1359",
        "Text": "Let the result be true for n = k So\nP(k) : \ncos\nsin\nA\nsin\ncos\n\u03b8\n\u03b8\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\u2212\n\u03b8\n\u03b8\n\uf8f0\n\uf8fb\n, then \ncos\nsin\nA\nsin\ncos\nk\nk\nk\nk\nk\n\u03b8\n\u03b8\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\u2212\n\u03b8\n\u03b8\n\uf8f0\n\uf8fb\nNow, we prove that the result holds for n = k +1\nNow\nAk + 1 =\ncos\nsin\ncos\nsin\nA A\nsin\ncos\nsin\ncos\nk\nk\nk\nk\nk\n\u03b8\n\u03b8\n\u03b8\n\u03b8\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\u22c5\n= \uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\u2212\n\u03b8\n\u03b8\n\u2212\n\u03b8\n\u03b8\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n=\ncos cos\n\u2013 sin sin\ncos sin\nsin cos\nsin cos\ncos sin\nsin sin\ncos cos\nk\nk\nk\nk\nk\nk\nk\nk\n\u03b8\n\u03b8\n\u03b8\n\u03b8\n\u03b8\n\u03b8 +\n\u03b8\n\u03b8\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n\u03b8\n\u03b8 +\n\u03b8\n\u03b8\n\u2212\n\u03b8\n\u03b8 +\n\u03b8\n\u03b8\n\uf8f0\n\uf8fb\n=\ncos(\n)\nsin(\n)\ncos(\n1)\nsin (\n1)\nsin(\n)\ncos(\n)\nsin (\n1)\ncos(\n1)\nk\nk\nk\nk\nk\nk\nk\nk\n\u03b8 + \u03b8\n\u03b8 + \u03b8\n+\n\u03b8\n+\n\u03b8\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u03b8 + \u03b8\n\u03b8 + \u03b8\n\u2212\n+\n\u03b8\n+\n\u03b8\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nTherefore, the result is true for n = k  + 1 Thus by principle of mathematical induction,\nwe have \ncos\nsin\nA\nsin\ncos\nn\nn\nn\nn\nn\n\u03b8\n\u03b8\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\u2212\n\u03b8\n\u03b8\n\uf8f0\n\uf8fb\n, holds for all natural numbers Example 24 If A and B are symmetric matrices of the same order, then show that AB\nis symmetric if and only if A and B commute, that is AB = BA"
    },
    {
        "Chapter": "1",
        "sentence_range": "1357-1360",
        "Text": "So\nP(k) : \ncos\nsin\nA\nsin\ncos\n\u03b8\n\u03b8\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\u2212\n\u03b8\n\u03b8\n\uf8f0\n\uf8fb\n, then \ncos\nsin\nA\nsin\ncos\nk\nk\nk\nk\nk\n\u03b8\n\u03b8\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\u2212\n\u03b8\n\u03b8\n\uf8f0\n\uf8fb\nNow, we prove that the result holds for n = k +1\nNow\nAk + 1 =\ncos\nsin\ncos\nsin\nA A\nsin\ncos\nsin\ncos\nk\nk\nk\nk\nk\n\u03b8\n\u03b8\n\u03b8\n\u03b8\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\u22c5\n= \uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\u2212\n\u03b8\n\u03b8\n\u2212\n\u03b8\n\u03b8\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n=\ncos cos\n\u2013 sin sin\ncos sin\nsin cos\nsin cos\ncos sin\nsin sin\ncos cos\nk\nk\nk\nk\nk\nk\nk\nk\n\u03b8\n\u03b8\n\u03b8\n\u03b8\n\u03b8\n\u03b8 +\n\u03b8\n\u03b8\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n\u03b8\n\u03b8 +\n\u03b8\n\u03b8\n\u2212\n\u03b8\n\u03b8 +\n\u03b8\n\u03b8\n\uf8f0\n\uf8fb\n=\ncos(\n)\nsin(\n)\ncos(\n1)\nsin (\n1)\nsin(\n)\ncos(\n)\nsin (\n1)\ncos(\n1)\nk\nk\nk\nk\nk\nk\nk\nk\n\u03b8 + \u03b8\n\u03b8 + \u03b8\n+\n\u03b8\n+\n\u03b8\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u03b8 + \u03b8\n\u03b8 + \u03b8\n\u2212\n+\n\u03b8\n+\n\u03b8\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nTherefore, the result is true for n = k  + 1 Thus by principle of mathematical induction,\nwe have \ncos\nsin\nA\nsin\ncos\nn\nn\nn\nn\nn\n\u03b8\n\u03b8\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\u2212\n\u03b8\n\u03b8\n\uf8f0\n\uf8fb\n, holds for all natural numbers Example 24 If A and B are symmetric matrices of the same order, then show that AB\nis symmetric if and only if A and B commute, that is AB = BA Solution Since A and B are both symmetric matrices, therefore A\u2032 = A and B\u2032 = B"
    },
    {
        "Chapter": "1",
        "sentence_range": "1358-1361",
        "Text": "Thus by principle of mathematical induction,\nwe have \ncos\nsin\nA\nsin\ncos\nn\nn\nn\nn\nn\n\u03b8\n\u03b8\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\u2212\n\u03b8\n\u03b8\n\uf8f0\n\uf8fb\n, holds for all natural numbers Example 24 If A and B are symmetric matrices of the same order, then show that AB\nis symmetric if and only if A and B commute, that is AB = BA Solution Since A and B are both symmetric matrices, therefore A\u2032 = A and B\u2032 = B Rationalised 2023-24\nMATRICES         71\nLet\nAB be symmetric, then (AB)\u2032 = AB\nBut\n(AB)\u2032 = B\u2032A\u2032= BA (Why"
    },
    {
        "Chapter": "1",
        "sentence_range": "1359-1362",
        "Text": "Example 24 If A and B are symmetric matrices of the same order, then show that AB\nis symmetric if and only if A and B commute, that is AB = BA Solution Since A and B are both symmetric matrices, therefore A\u2032 = A and B\u2032 = B Rationalised 2023-24\nMATRICES         71\nLet\nAB be symmetric, then (AB)\u2032 = AB\nBut\n(AB)\u2032 = B\u2032A\u2032= BA (Why )\nTherefore\nBA = AB\nConversely, if AB = BA, then we shall show that AB is symmetric"
    },
    {
        "Chapter": "1",
        "sentence_range": "1360-1363",
        "Text": "Solution Since A and B are both symmetric matrices, therefore A\u2032 = A and B\u2032 = B Rationalised 2023-24\nMATRICES         71\nLet\nAB be symmetric, then (AB)\u2032 = AB\nBut\n(AB)\u2032 = B\u2032A\u2032= BA (Why )\nTherefore\nBA = AB\nConversely, if AB = BA, then we shall show that AB is symmetric Now\n(AB)\u2032 = B\u2032A\u2032\n= B A (as A and B are symmetric)\n= AB\nHence AB is symmetric"
    },
    {
        "Chapter": "1",
        "sentence_range": "1361-1364",
        "Text": "Rationalised 2023-24\nMATRICES         71\nLet\nAB be symmetric, then (AB)\u2032 = AB\nBut\n(AB)\u2032 = B\u2032A\u2032= BA (Why )\nTherefore\nBA = AB\nConversely, if AB = BA, then we shall show that AB is symmetric Now\n(AB)\u2032 = B\u2032A\u2032\n= B A (as A and B are symmetric)\n= AB\nHence AB is symmetric Example 25 Let \n2\n1\n5\n2\n2\n5\nA\n, B\n, C\n3\n4\n7\n4\n3\n8\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n=\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb"
    },
    {
        "Chapter": "1",
        "sentence_range": "1362-1365",
        "Text": ")\nTherefore\nBA = AB\nConversely, if AB = BA, then we shall show that AB is symmetric Now\n(AB)\u2032 = B\u2032A\u2032\n= B A (as A and B are symmetric)\n= AB\nHence AB is symmetric Example 25 Let \n2\n1\n5\n2\n2\n5\nA\n, B\n, C\n3\n4\n7\n4\n3\n8\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n=\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb Find a matrix D such that\nCD \u2013 AB = O"
    },
    {
        "Chapter": "1",
        "sentence_range": "1363-1366",
        "Text": "Now\n(AB)\u2032 = B\u2032A\u2032\n= B A (as A and B are symmetric)\n= AB\nHence AB is symmetric Example 25 Let \n2\n1\n5\n2\n2\n5\nA\n, B\n, C\n3\n4\n7\n4\n3\n8\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n=\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb Find a matrix D such that\nCD \u2013 AB = O Solution Since A, B, C are all square matrices of order 2, and CD \u2013 AB is well\ndefined, D must be a square matrix of order 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "1364-1367",
        "Text": "Example 25 Let \n2\n1\n5\n2\n2\n5\nA\n, B\n, C\n3\n4\n7\n4\n3\n8\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n=\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb Find a matrix D such that\nCD \u2013 AB = O Solution Since A, B, C are all square matrices of order 2, and CD \u2013 AB is well\ndefined, D must be a square matrix of order 2 Let\nD = a\nb\nc\nd\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb"
    },
    {
        "Chapter": "1",
        "sentence_range": "1365-1368",
        "Text": "Find a matrix D such that\nCD \u2013 AB = O Solution Since A, B, C are all square matrices of order 2, and CD \u2013 AB is well\ndefined, D must be a square matrix of order 2 Let\nD = a\nb\nc\nd\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb Then CD \u2013 AB = 0 gives\n2\n5\n2\n1\n5\n2\n3\n8\n3\n4\n7\n4\na\nb\nc\nd\n\u2212\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\u2212\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n = O\nor\n2\n5\n2\n5\n3\n0\n3\n8\n3\n8\n43\n22\na\nc\nb\nd\na\nc\nb\nd\n+\n+\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n+\n+\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n = 0\n0\n0\n0\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nor\n2\n5\n3\n2\n5\n3\n8\n43\n3\n8\n22\na\nc\nb\nd\na\nc\nb\nd\n+\n\u2212\n+\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n+\n\u2212\n+\n\u2212\n\uf8f0\n\uf8fb\n = 0\n0\n0\n0\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nBy equality of matrices, we get\n2a + 5c \u2013 3 = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "1366-1369",
        "Text": "Solution Since A, B, C are all square matrices of order 2, and CD \u2013 AB is well\ndefined, D must be a square matrix of order 2 Let\nD = a\nb\nc\nd\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb Then CD \u2013 AB = 0 gives\n2\n5\n2\n1\n5\n2\n3\n8\n3\n4\n7\n4\na\nb\nc\nd\n\u2212\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\u2212\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n = O\nor\n2\n5\n2\n5\n3\n0\n3\n8\n3\n8\n43\n22\na\nc\nb\nd\na\nc\nb\nd\n+\n+\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n+\n+\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n = 0\n0\n0\n0\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nor\n2\n5\n3\n2\n5\n3\n8\n43\n3\n8\n22\na\nc\nb\nd\na\nc\nb\nd\n+\n\u2212\n+\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n+\n\u2212\n+\n\u2212\n\uf8f0\n\uf8fb\n = 0\n0\n0\n0\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nBy equality of matrices, we get\n2a + 5c \u2013 3 = 0 (1)\n3a + 8c \u2013 43 = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "1367-1370",
        "Text": "Let\nD = a\nb\nc\nd\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb Then CD \u2013 AB = 0 gives\n2\n5\n2\n1\n5\n2\n3\n8\n3\n4\n7\n4\na\nb\nc\nd\n\u2212\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\u2212\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n = O\nor\n2\n5\n2\n5\n3\n0\n3\n8\n3\n8\n43\n22\na\nc\nb\nd\na\nc\nb\nd\n+\n+\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n+\n+\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n = 0\n0\n0\n0\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nor\n2\n5\n3\n2\n5\n3\n8\n43\n3\n8\n22\na\nc\nb\nd\na\nc\nb\nd\n+\n\u2212\n+\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n+\n\u2212\n+\n\u2212\n\uf8f0\n\uf8fb\n = 0\n0\n0\n0\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nBy equality of matrices, we get\n2a + 5c \u2013 3 = 0 (1)\n3a + 8c \u2013 43 = 0 (2)\n2b + 5d = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "1368-1371",
        "Text": "Then CD \u2013 AB = 0 gives\n2\n5\n2\n1\n5\n2\n3\n8\n3\n4\n7\n4\na\nb\nc\nd\n\u2212\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\u2212\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n = O\nor\n2\n5\n2\n5\n3\n0\n3\n8\n3\n8\n43\n22\na\nc\nb\nd\na\nc\nb\nd\n+\n+\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n+\n+\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n = 0\n0\n0\n0\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nor\n2\n5\n3\n2\n5\n3\n8\n43\n3\n8\n22\na\nc\nb\nd\na\nc\nb\nd\n+\n\u2212\n+\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n+\n\u2212\n+\n\u2212\n\uf8f0\n\uf8fb\n = 0\n0\n0\n0\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nBy equality of matrices, we get\n2a + 5c \u2013 3 = 0 (1)\n3a + 8c \u2013 43 = 0 (2)\n2b + 5d = 0 (3)\nand\n3b + 8d \u2013 22 = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "1369-1372",
        "Text": "(1)\n3a + 8c \u2013 43 = 0 (2)\n2b + 5d = 0 (3)\nand\n3b + 8d \u2013 22 = 0 (4)\nSolving (1) and (2), we get a = \u2013191, c = 77"
    },
    {
        "Chapter": "1",
        "sentence_range": "1370-1373",
        "Text": "(2)\n2b + 5d = 0 (3)\nand\n3b + 8d \u2013 22 = 0 (4)\nSolving (1) and (2), we get a = \u2013191, c = 77 Solving (3) and (4), we get b = \u2013 110,\nd = 44"
    },
    {
        "Chapter": "1",
        "sentence_range": "1371-1374",
        "Text": "(3)\nand\n3b + 8d \u2013 22 = 0 (4)\nSolving (1) and (2), we get a = \u2013191, c = 77 Solving (3) and (4), we get b = \u2013 110,\nd = 44 Rationalised 2023-24\n 72\nMATHEMATICS\nTherefore\nD =\n191\n110\n77\n44\na\nb\nc\nd\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nMiscellaneous Exercise on Chapter 3\n1"
    },
    {
        "Chapter": "1",
        "sentence_range": "1372-1375",
        "Text": "(4)\nSolving (1) and (2), we get a = \u2013191, c = 77 Solving (3) and (4), we get b = \u2013 110,\nd = 44 Rationalised 2023-24\n 72\nMATHEMATICS\nTherefore\nD =\n191\n110\n77\n44\na\nb\nc\nd\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nMiscellaneous Exercise on Chapter 3\n1 If A and B are symmetric matrices, prove that AB \u2013 BA is a skew symmetric\nmatrix"
    },
    {
        "Chapter": "1",
        "sentence_range": "1373-1376",
        "Text": "Solving (3) and (4), we get b = \u2013 110,\nd = 44 Rationalised 2023-24\n 72\nMATHEMATICS\nTherefore\nD =\n191\n110\n77\n44\na\nb\nc\nd\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nMiscellaneous Exercise on Chapter 3\n1 If A and B are symmetric matrices, prove that AB \u2013 BA is a skew symmetric\nmatrix 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "1374-1377",
        "Text": "Rationalised 2023-24\n 72\nMATHEMATICS\nTherefore\nD =\n191\n110\n77\n44\na\nb\nc\nd\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nMiscellaneous Exercise on Chapter 3\n1 If A and B are symmetric matrices, prove that AB \u2013 BA is a skew symmetric\nmatrix 2 Show that the matrix B\u2032AB is symmetric or skew symmetric according as A is\nsymmetric or skew symmetric"
    },
    {
        "Chapter": "1",
        "sentence_range": "1375-1378",
        "Text": "If A and B are symmetric matrices, prove that AB \u2013 BA is a skew symmetric\nmatrix 2 Show that the matrix B\u2032AB is symmetric or skew symmetric according as A is\nsymmetric or skew symmetric 3"
    },
    {
        "Chapter": "1",
        "sentence_range": "1376-1379",
        "Text": "2 Show that the matrix B\u2032AB is symmetric or skew symmetric according as A is\nsymmetric or skew symmetric 3 Find the values of x, y, z if the matrix \n0\n2\nA\ny\nz\nx\ny\nz\nx\ny\nz\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n=\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n satisfy the equation\nA\u2032A  = I"
    },
    {
        "Chapter": "1",
        "sentence_range": "1377-1380",
        "Text": "Show that the matrix B\u2032AB is symmetric or skew symmetric according as A is\nsymmetric or skew symmetric 3 Find the values of x, y, z if the matrix \n0\n2\nA\ny\nz\nx\ny\nz\nx\ny\nz\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n=\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n satisfy the equation\nA\u2032A  = I 4"
    },
    {
        "Chapter": "1",
        "sentence_range": "1378-1381",
        "Text": "3 Find the values of x, y, z if the matrix \n0\n2\nA\ny\nz\nx\ny\nz\nx\ny\nz\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n=\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n satisfy the equation\nA\u2032A  = I 4 For what values of x : [\n]\n1\n2\n0\n0\n1\n2\n1\n2\n0\n1\n2\n1\n0\n2\nx\n\uf8ee\n\uf8f9 \uf8ee \uf8f9\n\uf8ef\n\uf8fa \uf8ef \uf8fa\n\uf8ef\n\uf8fa \uf8ef \uf8fa\n\uf8ef\n\uf8fa \uf8ef \uf8fa\n\uf8f0\n\uf8fb \uf8f0 \uf8fb\n = O"
    },
    {
        "Chapter": "1",
        "sentence_range": "1379-1382",
        "Text": "Find the values of x, y, z if the matrix \n0\n2\nA\ny\nz\nx\ny\nz\nx\ny\nz\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n=\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n satisfy the equation\nA\u2032A  = I 4 For what values of x : [\n]\n1\n2\n0\n0\n1\n2\n1\n2\n0\n1\n2\n1\n0\n2\nx\n\uf8ee\n\uf8f9 \uf8ee \uf8f9\n\uf8ef\n\uf8fa \uf8ef \uf8fa\n\uf8ef\n\uf8fa \uf8ef \uf8fa\n\uf8ef\n\uf8fa \uf8ef \uf8fa\n\uf8f0\n\uf8fb \uf8f0 \uf8fb\n = O 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "1380-1383",
        "Text": "4 For what values of x : [\n]\n1\n2\n0\n0\n1\n2\n1\n2\n0\n1\n2\n1\n0\n2\nx\n\uf8ee\n\uf8f9 \uf8ee \uf8f9\n\uf8ef\n\uf8fa \uf8ef \uf8fa\n\uf8ef\n\uf8fa \uf8ef \uf8fa\n\uf8ef\n\uf8fa \uf8ef \uf8fa\n\uf8f0\n\uf8fb \uf8f0 \uf8fb\n = O 5 If \n3\n1\nA\n1\n2\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\uf8f0\u2212\n\uf8fb\n, show that A2 \u2013 5A + 7I = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "1381-1384",
        "Text": "For what values of x : [\n]\n1\n2\n0\n0\n1\n2\n1\n2\n0\n1\n2\n1\n0\n2\nx\n\uf8ee\n\uf8f9 \uf8ee \uf8f9\n\uf8ef\n\uf8fa \uf8ef \uf8fa\n\uf8ef\n\uf8fa \uf8ef \uf8fa\n\uf8ef\n\uf8fa \uf8ef \uf8fa\n\uf8f0\n\uf8fb \uf8f0 \uf8fb\n = O 5 If \n3\n1\nA\n1\n2\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\uf8f0\u2212\n\uf8fb\n, show that A2 \u2013 5A + 7I = 0 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "1382-1385",
        "Text": "5 If \n3\n1\nA\n1\n2\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\uf8f0\u2212\n\uf8fb\n, show that A2 \u2013 5A + 7I = 0 6 Find x, if [\n]\n1\n0\n2\n5\n1\n0\n2\n1\n4\nO\n2\n0\n3\n1\nx\nx\n\uf8ee\n\uf8f9 \uf8ee \uf8f9\n\uf8ef\n\uf8fa \uf8ef \uf8fa\n\u2212\n\u2212\n=\n\uf8ef\n\uf8fa \uf8ef \uf8fa\n\uf8ef\n\uf8fa \uf8ef \uf8fa\n\uf8f0\n\uf8fb \uf8f0 \uf8fb\n7"
    },
    {
        "Chapter": "1",
        "sentence_range": "1383-1386",
        "Text": "If \n3\n1\nA\n1\n2\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\uf8f0\u2212\n\uf8fb\n, show that A2 \u2013 5A + 7I = 0 6 Find x, if [\n]\n1\n0\n2\n5\n1\n0\n2\n1\n4\nO\n2\n0\n3\n1\nx\nx\n\uf8ee\n\uf8f9 \uf8ee \uf8f9\n\uf8ef\n\uf8fa \uf8ef \uf8fa\n\u2212\n\u2212\n=\n\uf8ef\n\uf8fa \uf8ef \uf8fa\n\uf8ef\n\uf8fa \uf8ef \uf8fa\n\uf8f0\n\uf8fb \uf8f0 \uf8fb\n7 A manufacturer produces three products x, y, z which he sells in two markets"
    },
    {
        "Chapter": "1",
        "sentence_range": "1384-1387",
        "Text": "6 Find x, if [\n]\n1\n0\n2\n5\n1\n0\n2\n1\n4\nO\n2\n0\n3\n1\nx\nx\n\uf8ee\n\uf8f9 \uf8ee \uf8f9\n\uf8ef\n\uf8fa \uf8ef \uf8fa\n\u2212\n\u2212\n=\n\uf8ef\n\uf8fa \uf8ef \uf8fa\n\uf8ef\n\uf8fa \uf8ef \uf8fa\n\uf8f0\n\uf8fb \uf8f0 \uf8fb\n7 A manufacturer produces three products x, y, z which he sells in two markets Annual sales are indicated below:\nMarket\nProducts\nI\n10,000\n2,000\n18,000\nII\n6,000\n20,000\n8,000\nRationalised 2023-24\nMATRICES         73\n(a) If unit sale prices of x, y and z are ` 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "1385-1388",
        "Text": "Find x, if [\n]\n1\n0\n2\n5\n1\n0\n2\n1\n4\nO\n2\n0\n3\n1\nx\nx\n\uf8ee\n\uf8f9 \uf8ee \uf8f9\n\uf8ef\n\uf8fa \uf8ef \uf8fa\n\u2212\n\u2212\n=\n\uf8ef\n\uf8fa \uf8ef \uf8fa\n\uf8ef\n\uf8fa \uf8ef \uf8fa\n\uf8f0\n\uf8fb \uf8f0 \uf8fb\n7 A manufacturer produces three products x, y, z which he sells in two markets Annual sales are indicated below:\nMarket\nProducts\nI\n10,000\n2,000\n18,000\nII\n6,000\n20,000\n8,000\nRationalised 2023-24\nMATRICES         73\n(a) If unit sale prices of x, y and z are ` 2 50, ` 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "1386-1389",
        "Text": "A manufacturer produces three products x, y, z which he sells in two markets Annual sales are indicated below:\nMarket\nProducts\nI\n10,000\n2,000\n18,000\nII\n6,000\n20,000\n8,000\nRationalised 2023-24\nMATRICES         73\n(a) If unit sale prices of x, y and z are ` 2 50, ` 1 50 and ` 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "1387-1390",
        "Text": "Annual sales are indicated below:\nMarket\nProducts\nI\n10,000\n2,000\n18,000\nII\n6,000\n20,000\n8,000\nRationalised 2023-24\nMATRICES         73\n(a) If unit sale prices of x, y and z are ` 2 50, ` 1 50 and ` 1 00, respectively,\nfind the total revenue in each market with the help of matrix algebra"
    },
    {
        "Chapter": "1",
        "sentence_range": "1388-1391",
        "Text": "50, ` 1 50 and ` 1 00, respectively,\nfind the total revenue in each market with the help of matrix algebra (b) If the unit costs of the above three commodities are ` 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "1389-1392",
        "Text": "50 and ` 1 00, respectively,\nfind the total revenue in each market with the help of matrix algebra (b) If the unit costs of the above three commodities are ` 2 00, ` 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "1390-1393",
        "Text": "00, respectively,\nfind the total revenue in each market with the help of matrix algebra (b) If the unit costs of the above three commodities are ` 2 00, ` 1 00 and\n50 paise respectively"
    },
    {
        "Chapter": "1",
        "sentence_range": "1391-1394",
        "Text": "(b) If the unit costs of the above three commodities are ` 2 00, ` 1 00 and\n50 paise respectively Find the gross profit"
    },
    {
        "Chapter": "1",
        "sentence_range": "1392-1395",
        "Text": "00, ` 1 00 and\n50 paise respectively Find the gross profit 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "1393-1396",
        "Text": "00 and\n50 paise respectively Find the gross profit 8 Find the matrix X so that \n1\n2\n3\n7\n8\n9\nX 4\n5\n6\n2\n4\n6\n\u2212\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nChoose the correct answer in the following questions:\n9"
    },
    {
        "Chapter": "1",
        "sentence_range": "1394-1397",
        "Text": "Find the gross profit 8 Find the matrix X so that \n1\n2\n3\n7\n8\n9\nX 4\n5\n6\n2\n4\n6\n\u2212\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nChoose the correct answer in the following questions:\n9 If A =                 is such that A\u00b2 = I, then\n(A) 1 + \u03b1\u00b2 + \u03b2\u03b3 = 0\n(B) 1 \u2013 \u03b1\u00b2 + \u03b2\u03b3 = 0\n(C) 1 \u2013 \u03b1\u00b2 \u2013 \u03b2\u03b3 = 0\n(D) 1 + \u03b1\u00b2 \u2013 \u03b2\u03b3 = 0\n10"
    },
    {
        "Chapter": "1",
        "sentence_range": "1395-1398",
        "Text": "8 Find the matrix X so that \n1\n2\n3\n7\n8\n9\nX 4\n5\n6\n2\n4\n6\n\u2212\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nChoose the correct answer in the following questions:\n9 If A =                 is such that A\u00b2 = I, then\n(A) 1 + \u03b1\u00b2 + \u03b2\u03b3 = 0\n(B) 1 \u2013 \u03b1\u00b2 + \u03b2\u03b3 = 0\n(C) 1 \u2013 \u03b1\u00b2 \u2013 \u03b2\u03b3 = 0\n(D) 1 + \u03b1\u00b2 \u2013 \u03b2\u03b3 = 0\n10 If the matrix A is both symmetric and skew symmetric, then\n(A) A is a diagonal matrix\n(B) A is a zero matrix\n(C) A is a square matrix\n(D) None of these\n11"
    },
    {
        "Chapter": "1",
        "sentence_range": "1396-1399",
        "Text": "Find the matrix X so that \n1\n2\n3\n7\n8\n9\nX 4\n5\n6\n2\n4\n6\n\u2212\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nChoose the correct answer in the following questions:\n9 If A =                 is such that A\u00b2 = I, then\n(A) 1 + \u03b1\u00b2 + \u03b2\u03b3 = 0\n(B) 1 \u2013 \u03b1\u00b2 + \u03b2\u03b3 = 0\n(C) 1 \u2013 \u03b1\u00b2 \u2013 \u03b2\u03b3 = 0\n(D) 1 + \u03b1\u00b2 \u2013 \u03b2\u03b3 = 0\n10 If the matrix A is both symmetric and skew symmetric, then\n(A) A is a diagonal matrix\n(B) A is a zero matrix\n(C) A is a square matrix\n(D) None of these\n11 If A is square matrix such that A2 = A, then (I + A)\u00b3 \u2013 7 A is equal to\n(A) A\n(B) I \u2013 A\n(C) I\n(D) 3A\nSummary\n\u00ae A matrix is an ordered rectangular array of numbers or functions"
    },
    {
        "Chapter": "1",
        "sentence_range": "1397-1400",
        "Text": "If A =                 is such that A\u00b2 = I, then\n(A) 1 + \u03b1\u00b2 + \u03b2\u03b3 = 0\n(B) 1 \u2013 \u03b1\u00b2 + \u03b2\u03b3 = 0\n(C) 1 \u2013 \u03b1\u00b2 \u2013 \u03b2\u03b3 = 0\n(D) 1 + \u03b1\u00b2 \u2013 \u03b2\u03b3 = 0\n10 If the matrix A is both symmetric and skew symmetric, then\n(A) A is a diagonal matrix\n(B) A is a zero matrix\n(C) A is a square matrix\n(D) None of these\n11 If A is square matrix such that A2 = A, then (I + A)\u00b3 \u2013 7 A is equal to\n(A) A\n(B) I \u2013 A\n(C) I\n(D) 3A\nSummary\n\u00ae A matrix is an ordered rectangular array of numbers or functions \u00ae A matrix having m rows and n columns is called a matrix of order m \u00d7 n"
    },
    {
        "Chapter": "1",
        "sentence_range": "1398-1401",
        "Text": "If the matrix A is both symmetric and skew symmetric, then\n(A) A is a diagonal matrix\n(B) A is a zero matrix\n(C) A is a square matrix\n(D) None of these\n11 If A is square matrix such that A2 = A, then (I + A)\u00b3 \u2013 7 A is equal to\n(A) A\n(B) I \u2013 A\n(C) I\n(D) 3A\nSummary\n\u00ae A matrix is an ordered rectangular array of numbers or functions \u00ae A matrix having m rows and n columns is called a matrix of order m \u00d7 n \u00ae [aij]m \u00d7 1 is a column matrix"
    },
    {
        "Chapter": "1",
        "sentence_range": "1399-1402",
        "Text": "If A is square matrix such that A2 = A, then (I + A)\u00b3 \u2013 7 A is equal to\n(A) A\n(B) I \u2013 A\n(C) I\n(D) 3A\nSummary\n\u00ae A matrix is an ordered rectangular array of numbers or functions \u00ae A matrix having m rows and n columns is called a matrix of order m \u00d7 n \u00ae [aij]m \u00d7 1 is a column matrix \u00ae [aij]1 \u00d7 n is a row matrix"
    },
    {
        "Chapter": "1",
        "sentence_range": "1400-1403",
        "Text": "\u00ae A matrix having m rows and n columns is called a matrix of order m \u00d7 n \u00ae [aij]m \u00d7 1 is a column matrix \u00ae [aij]1 \u00d7 n is a row matrix \u00ae An m \u00d7 n matrix is a square matrix if m = n"
    },
    {
        "Chapter": "1",
        "sentence_range": "1401-1404",
        "Text": "\u00ae [aij]m \u00d7 1 is a column matrix \u00ae [aij]1 \u00d7 n is a row matrix \u00ae An m \u00d7 n matrix is a square matrix if m = n \u00ae A = [aij]m \u00d7 m is a diagonal matrix if aij = 0, when i \u2260 j"
    },
    {
        "Chapter": "1",
        "sentence_range": "1402-1405",
        "Text": "\u00ae [aij]1 \u00d7 n is a row matrix \u00ae An m \u00d7 n matrix is a square matrix if m = n \u00ae A = [aij]m \u00d7 m is a diagonal matrix if aij = 0, when i \u2260 j \u00ae A = [aij]n \u00d7 n is a scalar matrix if aij = 0, when i \u2260 j, aij = k, (k is some\nconstant), when i = j"
    },
    {
        "Chapter": "1",
        "sentence_range": "1403-1406",
        "Text": "\u00ae An m \u00d7 n matrix is a square matrix if m = n \u00ae A = [aij]m \u00d7 m is a diagonal matrix if aij = 0, when i \u2260 j \u00ae A = [aij]n \u00d7 n is a scalar matrix if aij = 0, when i \u2260 j, aij = k, (k is some\nconstant), when i = j \u00ae A = [aij]n \u00d7 n is  an identity matrix, if aij = 1, when i = j, aij = 0, when i \u2260 j"
    },
    {
        "Chapter": "1",
        "sentence_range": "1404-1407",
        "Text": "\u00ae A = [aij]m \u00d7 m is a diagonal matrix if aij = 0, when i \u2260 j \u00ae A = [aij]n \u00d7 n is a scalar matrix if aij = 0, when i \u2260 j, aij = k, (k is some\nconstant), when i = j \u00ae A = [aij]n \u00d7 n is  an identity matrix, if aij = 1, when i = j, aij = 0, when i \u2260 j \u00ae A zero matrix has all its elements as zero"
    },
    {
        "Chapter": "1",
        "sentence_range": "1405-1408",
        "Text": "\u00ae A = [aij]n \u00d7 n is a scalar matrix if aij = 0, when i \u2260 j, aij = k, (k is some\nconstant), when i = j \u00ae A = [aij]n \u00d7 n is  an identity matrix, if aij = 1, when i = j, aij = 0, when i \u2260 j \u00ae A zero matrix has all its elements as zero \u00ae A = [aij] = [bij] = B if (i) A and B are of same order, (ii)  aij = bij for all\npossible values of i and j"
    },
    {
        "Chapter": "1",
        "sentence_range": "1406-1409",
        "Text": "\u00ae A = [aij]n \u00d7 n is  an identity matrix, if aij = 1, when i = j, aij = 0, when i \u2260 j \u00ae A zero matrix has all its elements as zero \u00ae A = [aij] = [bij] = B if (i) A and B are of same order, (ii)  aij = bij for all\npossible values of i and j \u03b1\n\u03b2\n\u03b3\n\u2212\u03b1\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa\nRationalised 2023-24\n 74\nMATHEMATICS\n\u00ae kA = k[aij]m \u00d7 n = [k(aij)]m \u00d7 n\n\u00ae \u2013 A = (\u20131)A\n\u00ae A \u2013 B = A + (\u20131) B\n\u00ae A + B = B + A\n\u00ae (A + B) + C = A + (B + C), where A, B and C are of same order"
    },
    {
        "Chapter": "1",
        "sentence_range": "1407-1410",
        "Text": "\u00ae A zero matrix has all its elements as zero \u00ae A = [aij] = [bij] = B if (i) A and B are of same order, (ii)  aij = bij for all\npossible values of i and j \u03b1\n\u03b2\n\u03b3\n\u2212\u03b1\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa\nRationalised 2023-24\n 74\nMATHEMATICS\n\u00ae kA = k[aij]m \u00d7 n = [k(aij)]m \u00d7 n\n\u00ae \u2013 A = (\u20131)A\n\u00ae A \u2013 B = A + (\u20131) B\n\u00ae A + B = B + A\n\u00ae (A + B) + C = A + (B + C), where A, B and C are of same order \u00ae k(A + B) = kA + kB, where A and B are of same order, k is constant"
    },
    {
        "Chapter": "1",
        "sentence_range": "1408-1411",
        "Text": "\u00ae A = [aij] = [bij] = B if (i) A and B are of same order, (ii)  aij = bij for all\npossible values of i and j \u03b1\n\u03b2\n\u03b3\n\u2212\u03b1\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa\nRationalised 2023-24\n 74\nMATHEMATICS\n\u00ae kA = k[aij]m \u00d7 n = [k(aij)]m \u00d7 n\n\u00ae \u2013 A = (\u20131)A\n\u00ae A \u2013 B = A + (\u20131) B\n\u00ae A + B = B + A\n\u00ae (A + B) + C = A + (B + C), where A, B and C are of same order \u00ae k(A + B) = kA + kB, where A and B are of same order, k is constant \u00ae (k + l ) A = kA + lA, where k and l are constant"
    },
    {
        "Chapter": "1",
        "sentence_range": "1409-1412",
        "Text": "\u03b1\n\u03b2\n\u03b3\n\u2212\u03b1\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa\nRationalised 2023-24\n 74\nMATHEMATICS\n\u00ae kA = k[aij]m \u00d7 n = [k(aij)]m \u00d7 n\n\u00ae \u2013 A = (\u20131)A\n\u00ae A \u2013 B = A + (\u20131) B\n\u00ae A + B = B + A\n\u00ae (A + B) + C = A + (B + C), where A, B and C are of same order \u00ae k(A + B) = kA + kB, where A and B are of same order, k is constant \u00ae (k + l ) A = kA + lA, where k and l are constant \u00ae\u00ae If A = [aij]m \u00d7 n and B = [bjk]n \u00d7 p, then AB = C = [cik]m \u00d7 p, where      =\n(i) A(BC) = (AB)C,  (ii)   A(B + C) = AB + AC,  (iii)   (A + B)C = AC + BC\n\u00ae\u00ae  If A = [aij]m \u00d7 n, then A\u2032 or AT = [aji]n \u00d7 m\n(i) (A\u2032)\u2032 = A,  (ii)   (kA)\u2032 = kA\u2032,  (iii)   (A + B)\u2032 = A\u2032 + B\u2032,  (iv)  (AB)\u2032 = B\u2032A\u2032\n\u00ae A is a symmetric matrix if A\u2032 = A"
    },
    {
        "Chapter": "1",
        "sentence_range": "1410-1413",
        "Text": "\u00ae k(A + B) = kA + kB, where A and B are of same order, k is constant \u00ae (k + l ) A = kA + lA, where k and l are constant \u00ae\u00ae If A = [aij]m \u00d7 n and B = [bjk]n \u00d7 p, then AB = C = [cik]m \u00d7 p, where      =\n(i) A(BC) = (AB)C,  (ii)   A(B + C) = AB + AC,  (iii)   (A + B)C = AC + BC\n\u00ae\u00ae  If A = [aij]m \u00d7 n, then A\u2032 or AT = [aji]n \u00d7 m\n(i) (A\u2032)\u2032 = A,  (ii)   (kA)\u2032 = kA\u2032,  (iii)   (A + B)\u2032 = A\u2032 + B\u2032,  (iv)  (AB)\u2032 = B\u2032A\u2032\n\u00ae A is a symmetric matrix if A\u2032 = A \u00ae A is a skew symmetric matrix if A\u2032 = \u2013A"
    },
    {
        "Chapter": "1",
        "sentence_range": "1411-1414",
        "Text": "\u00ae (k + l ) A = kA + lA, where k and l are constant \u00ae\u00ae If A = [aij]m \u00d7 n and B = [bjk]n \u00d7 p, then AB = C = [cik]m \u00d7 p, where      =\n(i) A(BC) = (AB)C,  (ii)   A(B + C) = AB + AC,  (iii)   (A + B)C = AC + BC\n\u00ae\u00ae  If A = [aij]m \u00d7 n, then A\u2032 or AT = [aji]n \u00d7 m\n(i) (A\u2032)\u2032 = A,  (ii)   (kA)\u2032 = kA\u2032,  (iii)   (A + B)\u2032 = A\u2032 + B\u2032,  (iv)  (AB)\u2032 = B\u2032A\u2032\n\u00ae A is a symmetric matrix if A\u2032 = A \u00ae A is a skew symmetric matrix if A\u2032 = \u2013A \u00ae Any square matrix can be represented as the sum of a symmetric and a\nskew symmetric matrix"
    },
    {
        "Chapter": "1",
        "sentence_range": "1412-1415",
        "Text": "\u00ae\u00ae If A = [aij]m \u00d7 n and B = [bjk]n \u00d7 p, then AB = C = [cik]m \u00d7 p, where      =\n(i) A(BC) = (AB)C,  (ii)   A(B + C) = AB + AC,  (iii)   (A + B)C = AC + BC\n\u00ae\u00ae  If A = [aij]m \u00d7 n, then A\u2032 or AT = [aji]n \u00d7 m\n(i) (A\u2032)\u2032 = A,  (ii)   (kA)\u2032 = kA\u2032,  (iii)   (A + B)\u2032 = A\u2032 + B\u2032,  (iv)  (AB)\u2032 = B\u2032A\u2032\n\u00ae A is a symmetric matrix if A\u2032 = A \u00ae A is a skew symmetric matrix if A\u2032 = \u2013A \u00ae Any square matrix can be represented as the sum of a symmetric and a\nskew symmetric matrix \u00ae If A and B are two square matrices such that AB = BA = I, then B is the\ninverse matrix of A and is denoted by A\u20131 and A is the inverse of B"
    },
    {
        "Chapter": "1",
        "sentence_range": "1413-1416",
        "Text": "\u00ae A is a skew symmetric matrix if A\u2032 = \u2013A \u00ae Any square matrix can be represented as the sum of a symmetric and a\nskew symmetric matrix \u00ae If A and B are two square matrices such that AB = BA = I, then B is the\ninverse matrix of A and is denoted by A\u20131 and A is the inverse of B \u00ae Inverse of a square matrix, if it exists, is unique"
    },
    {
        "Chapter": "1",
        "sentence_range": "1414-1417",
        "Text": "\u00ae Any square matrix can be represented as the sum of a symmetric and a\nskew symmetric matrix \u00ae If A and B are two square matrices such that AB = BA = I, then B is the\ninverse matrix of A and is denoted by A\u20131 and A is the inverse of B \u00ae Inverse of a square matrix, if it exists, is unique \u2014v\nv\nv\nv\nv\u2014\n1\n= \nn\nik\nij\njk\nj\nc\na b\n\u2211\nRationalised 2023-24\nMATRICES         75\nNOTES\nRationalised 2023-24\n 76\nMATHEMATICS\nv All Mathematical truths are relative and conditional"
    },
    {
        "Chapter": "1",
        "sentence_range": "1415-1418",
        "Text": "\u00ae If A and B are two square matrices such that AB = BA = I, then B is the\ninverse matrix of A and is denoted by A\u20131 and A is the inverse of B \u00ae Inverse of a square matrix, if it exists, is unique \u2014v\nv\nv\nv\nv\u2014\n1\n= \nn\nik\nij\njk\nj\nc\na b\n\u2211\nRationalised 2023-24\nMATRICES         75\nNOTES\nRationalised 2023-24\n 76\nMATHEMATICS\nv All Mathematical truths are relative and conditional \u2014 C"
    },
    {
        "Chapter": "1",
        "sentence_range": "1416-1419",
        "Text": "\u00ae Inverse of a square matrix, if it exists, is unique \u2014v\nv\nv\nv\nv\u2014\n1\n= \nn\nik\nij\njk\nj\nc\na b\n\u2211\nRationalised 2023-24\nMATRICES         75\nNOTES\nRationalised 2023-24\n 76\nMATHEMATICS\nv All Mathematical truths are relative and conditional \u2014 C P"
    },
    {
        "Chapter": "1",
        "sentence_range": "1417-1420",
        "Text": "\u2014v\nv\nv\nv\nv\u2014\n1\n= \nn\nik\nij\njk\nj\nc\na b\n\u2211\nRationalised 2023-24\nMATRICES         75\nNOTES\nRationalised 2023-24\n 76\nMATHEMATICS\nv All Mathematical truths are relative and conditional \u2014 C P STEINMETZ v\n4"
    },
    {
        "Chapter": "1",
        "sentence_range": "1418-1421",
        "Text": "\u2014 C P STEINMETZ v\n4 1  Introduction\nIn the previous chapter, we have studied about matrices\nand algebra of matrices"
    },
    {
        "Chapter": "1",
        "sentence_range": "1419-1422",
        "Text": "P STEINMETZ v\n4 1  Introduction\nIn the previous chapter, we have studied about matrices\nand algebra of matrices We have also learnt that a system\nof algebraic equations can be expressed in the form of\nmatrices"
    },
    {
        "Chapter": "1",
        "sentence_range": "1420-1423",
        "Text": "STEINMETZ v\n4 1  Introduction\nIn the previous chapter, we have studied about matrices\nand algebra of matrices We have also learnt that a system\nof algebraic equations can be expressed in the form of\nmatrices This means, a system of linear equations like\na1 x + b1 y = c1\na2 x + b2 y = c2\ncan be represented as \n1\n1\n1\n2\n2\n2\na\nb\nc\nx\na\nb\nc\ny\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee \uf8f9 =\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef \uf8fa\n\uf8f0 \uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb"
    },
    {
        "Chapter": "1",
        "sentence_range": "1421-1424",
        "Text": "1  Introduction\nIn the previous chapter, we have studied about matrices\nand algebra of matrices We have also learnt that a system\nof algebraic equations can be expressed in the form of\nmatrices This means, a system of linear equations like\na1 x + b1 y = c1\na2 x + b2 y = c2\ncan be represented as \n1\n1\n1\n2\n2\n2\na\nb\nc\nx\na\nb\nc\ny\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee \uf8f9 =\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef \uf8fa\n\uf8f0 \uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb Now, this\nsystem of equations has a unique solution or not, is\ndetermined by the number a1 b2 \u2013 a2 b1"
    },
    {
        "Chapter": "1",
        "sentence_range": "1422-1425",
        "Text": "We have also learnt that a system\nof algebraic equations can be expressed in the form of\nmatrices This means, a system of linear equations like\na1 x + b1 y = c1\na2 x + b2 y = c2\ncan be represented as \n1\n1\n1\n2\n2\n2\na\nb\nc\nx\na\nb\nc\ny\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee \uf8f9 =\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef \uf8fa\n\uf8f0 \uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb Now, this\nsystem of equations has a unique solution or not, is\ndetermined by the number a1 b2 \u2013 a2 b1 (Recall that if\n1\n1\n2\n2\na\nb\na\n\u2260b\n or, a1 b2 \u2013 a2 b1 \u2260 0,  then the system of linear\nequations has a unique solution)"
    },
    {
        "Chapter": "1",
        "sentence_range": "1423-1426",
        "Text": "This means, a system of linear equations like\na1 x + b1 y = c1\na2 x + b2 y = c2\ncan be represented as \n1\n1\n1\n2\n2\n2\na\nb\nc\nx\na\nb\nc\ny\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee \uf8f9 =\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef \uf8fa\n\uf8f0 \uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb Now, this\nsystem of equations has a unique solution or not, is\ndetermined by the number a1 b2 \u2013 a2 b1 (Recall that if\n1\n1\n2\n2\na\nb\na\n\u2260b\n or, a1 b2 \u2013 a2 b1 \u2260 0,  then the system of linear\nequations has a unique solution) The number a1 b2 \u2013 a2 b1\nwhich determines uniqueness of solution is associated with the matrix \n1\n1\n2\n2\nA\na\nb\na\nb\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nand is called the determinant of A or det A"
    },
    {
        "Chapter": "1",
        "sentence_range": "1424-1427",
        "Text": "Now, this\nsystem of equations has a unique solution or not, is\ndetermined by the number a1 b2 \u2013 a2 b1 (Recall that if\n1\n1\n2\n2\na\nb\na\n\u2260b\n or, a1 b2 \u2013 a2 b1 \u2260 0,  then the system of linear\nequations has a unique solution) The number a1 b2 \u2013 a2 b1\nwhich determines uniqueness of solution is associated with the matrix \n1\n1\n2\n2\nA\na\nb\na\nb\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nand is called the determinant of A or det A Determinants have wide applications in\nEngineering, Science, Economics, Social Science, etc"
    },
    {
        "Chapter": "1",
        "sentence_range": "1425-1428",
        "Text": "(Recall that if\n1\n1\n2\n2\na\nb\na\n\u2260b\n or, a1 b2 \u2013 a2 b1 \u2260 0,  then the system of linear\nequations has a unique solution) The number a1 b2 \u2013 a2 b1\nwhich determines uniqueness of solution is associated with the matrix \n1\n1\n2\n2\nA\na\nb\na\nb\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nand is called the determinant of A or det A Determinants have wide applications in\nEngineering, Science, Economics, Social Science, etc In this chapter, we shall study determinants up to order three only with real entries"
    },
    {
        "Chapter": "1",
        "sentence_range": "1426-1429",
        "Text": "The number a1 b2 \u2013 a2 b1\nwhich determines uniqueness of solution is associated with the matrix \n1\n1\n2\n2\nA\na\nb\na\nb\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nand is called the determinant of A or det A Determinants have wide applications in\nEngineering, Science, Economics, Social Science, etc In this chapter, we shall study determinants up to order three only with real entries Also, we will study various properties of determinants, minors, cofactors and applications\nof determinants in finding the area of a triangle, adjoint and inverse of a square matrix,\nconsistency and inconsistency of system of linear equations and solution of linear\nequations in two or three variables using inverse of a matrix"
    },
    {
        "Chapter": "1",
        "sentence_range": "1427-1430",
        "Text": "Determinants have wide applications in\nEngineering, Science, Economics, Social Science, etc In this chapter, we shall study determinants up to order three only with real entries Also, we will study various properties of determinants, minors, cofactors and applications\nof determinants in finding the area of a triangle, adjoint and inverse of a square matrix,\nconsistency and inconsistency of system of linear equations and solution of linear\nequations in two or three variables using inverse of a matrix 4"
    },
    {
        "Chapter": "1",
        "sentence_range": "1428-1431",
        "Text": "In this chapter, we shall study determinants up to order three only with real entries Also, we will study various properties of determinants, minors, cofactors and applications\nof determinants in finding the area of a triangle, adjoint and inverse of a square matrix,\nconsistency and inconsistency of system of linear equations and solution of linear\nequations in two or three variables using inverse of a matrix 4 2  Determinant\nTo every square matrix A = [aij] of order n, we can associate a number (real or\ncomplex) called determinant of the square matrix A, where aij = (i, j)th element of A"
    },
    {
        "Chapter": "1",
        "sentence_range": "1429-1432",
        "Text": "Also, we will study various properties of determinants, minors, cofactors and applications\nof determinants in finding the area of a triangle, adjoint and inverse of a square matrix,\nconsistency and inconsistency of system of linear equations and solution of linear\nequations in two or three variables using inverse of a matrix 4 2  Determinant\nTo every square matrix A = [aij] of order n, we can associate a number (real or\ncomplex) called determinant of the square matrix A, where aij = (i, j)th element of A Chapter 4\nDETERMINANTS\nP"
    },
    {
        "Chapter": "1",
        "sentence_range": "1430-1433",
        "Text": "4 2  Determinant\nTo every square matrix A = [aij] of order n, we can associate a number (real or\ncomplex) called determinant of the square matrix A, where aij = (i, j)th element of A Chapter 4\nDETERMINANTS\nP S"
    },
    {
        "Chapter": "1",
        "sentence_range": "1431-1434",
        "Text": "2  Determinant\nTo every square matrix A = [aij] of order n, we can associate a number (real or\ncomplex) called determinant of the square matrix A, where aij = (i, j)th element of A Chapter 4\nDETERMINANTS\nP S Laplace\n(1749-1827)\nRationalised 2023-24\nDETERMINANTS     77\nThis may be thought of as a function which associates each square matrix with a\nunique number (real or complex)"
    },
    {
        "Chapter": "1",
        "sentence_range": "1432-1435",
        "Text": "Chapter 4\nDETERMINANTS\nP S Laplace\n(1749-1827)\nRationalised 2023-24\nDETERMINANTS     77\nThis may be thought of as a function which associates each square matrix with a\nunique number (real or complex) If M is the set of square matrices, K is the set of\nnumbers (real or complex) and f : M \u2192 K is defined by f (A) = k, where A \u2208 M and\nk \u2208 K, then f (A) is called the determinant of A"
    },
    {
        "Chapter": "1",
        "sentence_range": "1433-1436",
        "Text": "S Laplace\n(1749-1827)\nRationalised 2023-24\nDETERMINANTS     77\nThis may be thought of as a function which associates each square matrix with a\nunique number (real or complex) If M is the set of square matrices, K is the set of\nnumbers (real or complex) and f : M \u2192 K is defined by f (A) = k, where A \u2208 M and\nk \u2208 K, then f (A) is called the determinant of A It is also denoted by |A| or det A or \u2206"
    },
    {
        "Chapter": "1",
        "sentence_range": "1434-1437",
        "Text": "Laplace\n(1749-1827)\nRationalised 2023-24\nDETERMINANTS     77\nThis may be thought of as a function which associates each square matrix with a\nunique number (real or complex) If M is the set of square matrices, K is the set of\nnumbers (real or complex) and f : M \u2192 K is defined by f (A) = k, where A \u2208 M and\nk \u2208 K, then f (A) is called the determinant of A It is also denoted by |A| or det A or \u2206 If A = \na\nb\nc\nd\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, then determinant of A is written as |A| = \na\nb\nc\nd  = det (A)\nRemarks\n(i)\nFor matrix A, |A| is read as determinant of A and not modulus of A"
    },
    {
        "Chapter": "1",
        "sentence_range": "1435-1438",
        "Text": "If M is the set of square matrices, K is the set of\nnumbers (real or complex) and f : M \u2192 K is defined by f (A) = k, where A \u2208 M and\nk \u2208 K, then f (A) is called the determinant of A It is also denoted by |A| or det A or \u2206 If A = \na\nb\nc\nd\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, then determinant of A is written as |A| = \na\nb\nc\nd  = det (A)\nRemarks\n(i)\nFor matrix A, |A| is read as determinant of A and not modulus of A (ii)\nOnly square matrices have determinants"
    },
    {
        "Chapter": "1",
        "sentence_range": "1436-1439",
        "Text": "It is also denoted by |A| or det A or \u2206 If A = \na\nb\nc\nd\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, then determinant of A is written as |A| = \na\nb\nc\nd  = det (A)\nRemarks\n(i)\nFor matrix A, |A| is read as determinant of A and not modulus of A (ii)\nOnly square matrices have determinants 4"
    },
    {
        "Chapter": "1",
        "sentence_range": "1437-1440",
        "Text": "If A = \na\nb\nc\nd\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, then determinant of A is written as |A| = \na\nb\nc\nd  = det (A)\nRemarks\n(i)\nFor matrix A, |A| is read as determinant of A and not modulus of A (ii)\nOnly square matrices have determinants 4 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "1438-1441",
        "Text": "(ii)\nOnly square matrices have determinants 4 2 1  Determinant of a matrix of order one\nLet A = [a] be the matrix of order 1, then determinant of A is defined to be equal to a\n4"
    },
    {
        "Chapter": "1",
        "sentence_range": "1439-1442",
        "Text": "4 2 1  Determinant of a matrix of order one\nLet A = [a] be the matrix of order 1, then determinant of A is defined to be equal to a\n4 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "1440-1443",
        "Text": "2 1  Determinant of a matrix of order one\nLet A = [a] be the matrix of order 1, then determinant of A is defined to be equal to a\n4 2 2  Determinant of a matrix of order two\nLet\nA = \n11\n12\n21\n22\na\na\na\na\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n be a matrix of order 2 \u00d7 2,\nthen the determinant of A is defined as:\ndet (A) = |A| = \u2206 = \n = a11a22 \u2013 a21a12\nExample 1 Evaluate \n2\n4\n\u20131\n2"
    },
    {
        "Chapter": "1",
        "sentence_range": "1441-1444",
        "Text": "1  Determinant of a matrix of order one\nLet A = [a] be the matrix of order 1, then determinant of A is defined to be equal to a\n4 2 2  Determinant of a matrix of order two\nLet\nA = \n11\n12\n21\n22\na\na\na\na\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n be a matrix of order 2 \u00d7 2,\nthen the determinant of A is defined as:\ndet (A) = |A| = \u2206 = \n = a11a22 \u2013 a21a12\nExample 1 Evaluate \n2\n4\n\u20131\n2 Solution We have \n2\n4\n\u20131\n2  = 2(2) \u2013 4(\u20131) = 4 + 4 = 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "1442-1445",
        "Text": "2 2  Determinant of a matrix of order two\nLet\nA = \n11\n12\n21\n22\na\na\na\na\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n be a matrix of order 2 \u00d7 2,\nthen the determinant of A is defined as:\ndet (A) = |A| = \u2206 = \n = a11a22 \u2013 a21a12\nExample 1 Evaluate \n2\n4\n\u20131\n2 Solution We have \n2\n4\n\u20131\n2  = 2(2) \u2013 4(\u20131) = 4 + 4 = 8 Example 2 Evaluate \n1\nx\u2013 1\nx\nx\nx\n+\nSolution We have\n1\nx\u2013 1\nx\nx\nx\n+\n = x (x) \u2013 (x + 1) (x \u2013 1)  = x2 \u2013 (x2 \u2013 1) = x2 \u2013 x2 + 1 = 1\n4"
    },
    {
        "Chapter": "1",
        "sentence_range": "1443-1446",
        "Text": "2  Determinant of a matrix of order two\nLet\nA = \n11\n12\n21\n22\na\na\na\na\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n be a matrix of order 2 \u00d7 2,\nthen the determinant of A is defined as:\ndet (A) = |A| = \u2206 = \n = a11a22 \u2013 a21a12\nExample 1 Evaluate \n2\n4\n\u20131\n2 Solution We have \n2\n4\n\u20131\n2  = 2(2) \u2013 4(\u20131) = 4 + 4 = 8 Example 2 Evaluate \n1\nx\u2013 1\nx\nx\nx\n+\nSolution We have\n1\nx\u2013 1\nx\nx\nx\n+\n = x (x) \u2013 (x + 1) (x \u2013 1)  = x2 \u2013 (x2 \u2013 1) = x2 \u2013 x2 + 1 = 1\n4 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "1444-1447",
        "Text": "Solution We have \n2\n4\n\u20131\n2  = 2(2) \u2013 4(\u20131) = 4 + 4 = 8 Example 2 Evaluate \n1\nx\u2013 1\nx\nx\nx\n+\nSolution We have\n1\nx\u2013 1\nx\nx\nx\n+\n = x (x) \u2013 (x + 1) (x \u2013 1)  = x2 \u2013 (x2 \u2013 1) = x2 \u2013 x2 + 1 = 1\n4 2 3  Determinant of a matrix of order 3 \u00d7 3\nDeterminant of a matrix of order three can be determined by expressing it in terms of\nsecond order determinants"
    },
    {
        "Chapter": "1",
        "sentence_range": "1445-1448",
        "Text": "Example 2 Evaluate \n1\nx\u2013 1\nx\nx\nx\n+\nSolution We have\n1\nx\u2013 1\nx\nx\nx\n+\n = x (x) \u2013 (x + 1) (x \u2013 1)  = x2 \u2013 (x2 \u2013 1) = x2 \u2013 x2 + 1 = 1\n4 2 3  Determinant of a matrix of order 3 \u00d7 3\nDeterminant of a matrix of order three can be determined by expressing it in terms of\nsecond order determinants This is known as expansion of a determinant along\na row (or a column)"
    },
    {
        "Chapter": "1",
        "sentence_range": "1446-1449",
        "Text": "2 3  Determinant of a matrix of order 3 \u00d7 3\nDeterminant of a matrix of order three can be determined by expressing it in terms of\nsecond order determinants This is known as expansion of a determinant along\na row (or a column) There are six ways of expanding a determinant of order\nRationalised 2023-24\n 78\nMATHEMATICS\n3 corresponding to each of three rows (R1, R2 and R3) and three columns (C1, C2 and\nC3) giving the same value as shown below"
    },
    {
        "Chapter": "1",
        "sentence_range": "1447-1450",
        "Text": "3  Determinant of a matrix of order 3 \u00d7 3\nDeterminant of a matrix of order three can be determined by expressing it in terms of\nsecond order determinants This is known as expansion of a determinant along\na row (or a column) There are six ways of expanding a determinant of order\nRationalised 2023-24\n 78\nMATHEMATICS\n3 corresponding to each of three rows (R1, R2 and R3) and three columns (C1, C2 and\nC3) giving the same value as shown below Consider the determinant of square matrix A = [aij]3 \u00d7 3\ni"
    },
    {
        "Chapter": "1",
        "sentence_range": "1448-1451",
        "Text": "This is known as expansion of a determinant along\na row (or a column) There are six ways of expanding a determinant of order\nRationalised 2023-24\n 78\nMATHEMATICS\n3 corresponding to each of three rows (R1, R2 and R3) and three columns (C1, C2 and\nC3) giving the same value as shown below Consider the determinant of square matrix A = [aij]3 \u00d7 3\ni e"
    },
    {
        "Chapter": "1",
        "sentence_range": "1449-1452",
        "Text": "There are six ways of expanding a determinant of order\nRationalised 2023-24\n 78\nMATHEMATICS\n3 corresponding to each of three rows (R1, R2 and R3) and three columns (C1, C2 and\nC3) giving the same value as shown below Consider the determinant of square matrix A = [aij]3 \u00d7 3\ni e ,\n| A | =\n21\n22\n23\n31\n32\n33\na\na\na\na\na\na\n11\n12\n13\na\na\na\nExpansion along first Row (R1)\nStep 1 Multiply first element a11 of R1 by (\u20131)(1 + 1) [(\u20131)sum of suffixes in a11] and with the\nsecond order determinant obtained by deleting the elements of first row (R1) and first\ncolumn (C1) of | A | as a11 lies in R1 and C1,\ni"
    },
    {
        "Chapter": "1",
        "sentence_range": "1450-1453",
        "Text": "Consider the determinant of square matrix A = [aij]3 \u00d7 3\ni e ,\n| A | =\n21\n22\n23\n31\n32\n33\na\na\na\na\na\na\n11\n12\n13\na\na\na\nExpansion along first Row (R1)\nStep 1 Multiply first element a11 of R1 by (\u20131)(1 + 1) [(\u20131)sum of suffixes in a11] and with the\nsecond order determinant obtained by deleting the elements of first row (R1) and first\ncolumn (C1) of | A | as a11 lies in R1 and C1,\ni e"
    },
    {
        "Chapter": "1",
        "sentence_range": "1451-1454",
        "Text": "e ,\n| A | =\n21\n22\n23\n31\n32\n33\na\na\na\na\na\na\n11\n12\n13\na\na\na\nExpansion along first Row (R1)\nStep 1 Multiply first element a11 of R1 by (\u20131)(1 + 1) [(\u20131)sum of suffixes in a11] and with the\nsecond order determinant obtained by deleting the elements of first row (R1) and first\ncolumn (C1) of | A | as a11 lies in R1 and C1,\ni e ,\n(\u20131)1 + 1 a11 \n22\n23\n32\n33\na\na\na\na\nStep 2 Multiply 2nd element a12 of R1 by (\u20131)1 + 2 [(\u20131)sum of suffixes in a12] and the second\norder determinant obtained by deleting elements of first row (R1) and 2nd column (C2)\nof | A | as a12 lies in R1 and C2,\ni"
    },
    {
        "Chapter": "1",
        "sentence_range": "1452-1455",
        "Text": ",\n| A | =\n21\n22\n23\n31\n32\n33\na\na\na\na\na\na\n11\n12\n13\na\na\na\nExpansion along first Row (R1)\nStep 1 Multiply first element a11 of R1 by (\u20131)(1 + 1) [(\u20131)sum of suffixes in a11] and with the\nsecond order determinant obtained by deleting the elements of first row (R1) and first\ncolumn (C1) of | A | as a11 lies in R1 and C1,\ni e ,\n(\u20131)1 + 1 a11 \n22\n23\n32\n33\na\na\na\na\nStep 2 Multiply 2nd element a12 of R1 by (\u20131)1 + 2 [(\u20131)sum of suffixes in a12] and the second\norder determinant obtained by deleting elements of first row (R1) and 2nd column (C2)\nof | A | as a12 lies in R1 and C2,\ni e"
    },
    {
        "Chapter": "1",
        "sentence_range": "1453-1456",
        "Text": "e ,\n(\u20131)1 + 1 a11 \n22\n23\n32\n33\na\na\na\na\nStep 2 Multiply 2nd element a12 of R1 by (\u20131)1 + 2 [(\u20131)sum of suffixes in a12] and the second\norder determinant obtained by deleting elements of first row (R1) and 2nd column (C2)\nof | A | as a12 lies in R1 and C2,\ni e ,\n(\u20131)1 + 2 a12 \n21\n23\n31\n33\na\na\na\na\nStep 3 Multiply third element a13 of R1 by (\u20131)1 + 3 [(\u20131)sum of suffixes in a13] and the second\norder determinant obtained by deleting elements of first row (R1) and third column (C3)\nof | A | as a13 lies in R1 and C3,\ni"
    },
    {
        "Chapter": "1",
        "sentence_range": "1454-1457",
        "Text": ",\n(\u20131)1 + 1 a11 \n22\n23\n32\n33\na\na\na\na\nStep 2 Multiply 2nd element a12 of R1 by (\u20131)1 + 2 [(\u20131)sum of suffixes in a12] and the second\norder determinant obtained by deleting elements of first row (R1) and 2nd column (C2)\nof | A | as a12 lies in R1 and C2,\ni e ,\n(\u20131)1 + 2 a12 \n21\n23\n31\n33\na\na\na\na\nStep 3 Multiply third element a13 of R1 by (\u20131)1 + 3 [(\u20131)sum of suffixes in a13] and the second\norder determinant obtained by deleting elements of first row (R1) and third column (C3)\nof | A | as a13 lies in R1 and C3,\ni e"
    },
    {
        "Chapter": "1",
        "sentence_range": "1455-1458",
        "Text": "e ,\n(\u20131)1 + 2 a12 \n21\n23\n31\n33\na\na\na\na\nStep 3 Multiply third element a13 of R1 by (\u20131)1 + 3 [(\u20131)sum of suffixes in a13] and the second\norder determinant obtained by deleting elements of first row (R1) and third column (C3)\nof | A | as a13 lies in R1 and C3,\ni e ,\n(\u20131)1 + 3 a13 \n21\n22\n31\n32\na\na\na\na\nStep 4 Now the expansion of determinant of A, that is, | A | written as sum of all three\nterms obtained in steps 1, 2 and 3 above is given by\ndet A = |A| = (\u20131)1 + 1 a11 \n22\n23\n21\n23\n1\n2\n12\n32\n33\n31\n33\n(\u20131)\na\na\na\na\na\na\na\na\na\n+\n+\n         + \n21\n22\n1\n3\n13\n31\n32\n(\u20131)\na\na\na\na\na\n+\nor\n|A| = a11 (a22 a33 \u2013 a32 a23) \u2013 a12 (a21 a33 \u2013 a31 a23)\n+ a13 (a21 a32 \u2013 a31 a22)\nRationalised 2023-24\nDETERMINANTS     79\n= a11 a22 a33 \u2013 a11 a32 a23 \u2013 a12 a21 a33 + a12 a31 a23 + a13 a21 a32\n\u2013 a13 a31 a22"
    },
    {
        "Chapter": "1",
        "sentence_range": "1456-1459",
        "Text": ",\n(\u20131)1 + 2 a12 \n21\n23\n31\n33\na\na\na\na\nStep 3 Multiply third element a13 of R1 by (\u20131)1 + 3 [(\u20131)sum of suffixes in a13] and the second\norder determinant obtained by deleting elements of first row (R1) and third column (C3)\nof | A | as a13 lies in R1 and C3,\ni e ,\n(\u20131)1 + 3 a13 \n21\n22\n31\n32\na\na\na\na\nStep 4 Now the expansion of determinant of A, that is, | A | written as sum of all three\nterms obtained in steps 1, 2 and 3 above is given by\ndet A = |A| = (\u20131)1 + 1 a11 \n22\n23\n21\n23\n1\n2\n12\n32\n33\n31\n33\n(\u20131)\na\na\na\na\na\na\na\na\na\n+\n+\n         + \n21\n22\n1\n3\n13\n31\n32\n(\u20131)\na\na\na\na\na\n+\nor\n|A| = a11 (a22 a33 \u2013 a32 a23) \u2013 a12 (a21 a33 \u2013 a31 a23)\n+ a13 (a21 a32 \u2013 a31 a22)\nRationalised 2023-24\nDETERMINANTS     79\n= a11 a22 a33 \u2013 a11 a32 a23 \u2013 a12 a21 a33 + a12 a31 a23 + a13 a21 a32\n\u2013 a13 a31 a22 (1)\nANote  We shall apply all four steps together"
    },
    {
        "Chapter": "1",
        "sentence_range": "1457-1460",
        "Text": "e ,\n(\u20131)1 + 3 a13 \n21\n22\n31\n32\na\na\na\na\nStep 4 Now the expansion of determinant of A, that is, | A | written as sum of all three\nterms obtained in steps 1, 2 and 3 above is given by\ndet A = |A| = (\u20131)1 + 1 a11 \n22\n23\n21\n23\n1\n2\n12\n32\n33\n31\n33\n(\u20131)\na\na\na\na\na\na\na\na\na\n+\n+\n         + \n21\n22\n1\n3\n13\n31\n32\n(\u20131)\na\na\na\na\na\n+\nor\n|A| = a11 (a22 a33 \u2013 a32 a23) \u2013 a12 (a21 a33 \u2013 a31 a23)\n+ a13 (a21 a32 \u2013 a31 a22)\nRationalised 2023-24\nDETERMINANTS     79\n= a11 a22 a33 \u2013 a11 a32 a23 \u2013 a12 a21 a33 + a12 a31 a23 + a13 a21 a32\n\u2013 a13 a31 a22 (1)\nANote  We shall apply all four steps together Expansion along second row (R2)\n| A | =\n11\n12\n13\n31\n32\n33\na\na\na\na\na\na\n21\n22\n23\na\na\na\nExpanding along R2, we get\n| A | =\n12\n13\n11\n13\n2\n1\n2\n2\n21\n22\n32\n33\n31\n33\n(\u20131)\n(\u20131)\na\na\na\na\na\na\na\na\na\na\n+\n+\n+\n11\n12\n2\n3\n23\n31\n32\n(\u20131)\na\na\na\na\na\n+\n+\n= \u2013 a21 (a12 a33 \u2013 a32 a13) + a22 (a11 a33 \u2013 a31 a13)\n\u2013 a23 (a11 a32 \u2013 a31 a12)\n| A | = \u2013 a21 a12 a33 + a21 a32 a13 + a22 a11 a33 \u2013 a22 a31 a13 \u2013 a23 a11 a32\n \n+ a23 a31 a12\n= a11 a22 a33 \u2013 a11 a23 a32 \u2013 a12 a21 a33 + a12 a23 a31 + a13 a21 a32\n\u2013 a13 a31 a22"
    },
    {
        "Chapter": "1",
        "sentence_range": "1458-1461",
        "Text": ",\n(\u20131)1 + 3 a13 \n21\n22\n31\n32\na\na\na\na\nStep 4 Now the expansion of determinant of A, that is, | A | written as sum of all three\nterms obtained in steps 1, 2 and 3 above is given by\ndet A = |A| = (\u20131)1 + 1 a11 \n22\n23\n21\n23\n1\n2\n12\n32\n33\n31\n33\n(\u20131)\na\na\na\na\na\na\na\na\na\n+\n+\n         + \n21\n22\n1\n3\n13\n31\n32\n(\u20131)\na\na\na\na\na\n+\nor\n|A| = a11 (a22 a33 \u2013 a32 a23) \u2013 a12 (a21 a33 \u2013 a31 a23)\n+ a13 (a21 a32 \u2013 a31 a22)\nRationalised 2023-24\nDETERMINANTS     79\n= a11 a22 a33 \u2013 a11 a32 a23 \u2013 a12 a21 a33 + a12 a31 a23 + a13 a21 a32\n\u2013 a13 a31 a22 (1)\nANote  We shall apply all four steps together Expansion along second row (R2)\n| A | =\n11\n12\n13\n31\n32\n33\na\na\na\na\na\na\n21\n22\n23\na\na\na\nExpanding along R2, we get\n| A | =\n12\n13\n11\n13\n2\n1\n2\n2\n21\n22\n32\n33\n31\n33\n(\u20131)\n(\u20131)\na\na\na\na\na\na\na\na\na\na\n+\n+\n+\n11\n12\n2\n3\n23\n31\n32\n(\u20131)\na\na\na\na\na\n+\n+\n= \u2013 a21 (a12 a33 \u2013 a32 a13) + a22 (a11 a33 \u2013 a31 a13)\n\u2013 a23 (a11 a32 \u2013 a31 a12)\n| A | = \u2013 a21 a12 a33 + a21 a32 a13 + a22 a11 a33 \u2013 a22 a31 a13 \u2013 a23 a11 a32\n \n+ a23 a31 a12\n= a11 a22 a33 \u2013 a11 a23 a32 \u2013 a12 a21 a33 + a12 a23 a31 + a13 a21 a32\n\u2013 a13 a31 a22 (2)\nExpansion along first Column (C1)\n| A | =\n12\n13\n22\n23\n32\n33\n11\n21\n31\na\na\na\na\na\na\na\na\na\nBy expanding along C1, we get\n| A | =\n22\n23\n12\n13\n1\n1\n2\n1\n11\n21\n32\n33\n32\n33\n(\u20131)\n( 1)\na\na\na\na\na\na\na\na\na\na\n+\n+\n+\n\u2212\n+ \n12\n13\n3\n1\n31\n22\n23\n(\u20131)\na\na\na\na\na\n+\n= a11 (a22 a33 \u2013 a23 a32) \u2013 a21 (a12 a33 \u2013 a13 a32) + a31 (a12 a23 \u2013 a13 a22)\nRationalised 2023-24\n 80\nMATHEMATICS\n| A | = a11 a22 a33 \u2013 a11 a23 a32 \u2013 a21 a12 a33 + a21 a13 a32 + a31 a12 a23\n\u2013 a31 a13 a22\n= a11 a22 a33 \u2013 a11 a23 a32 \u2013 a12 a21 a33 + a12 a23 a31 + a13 a21 a32\n\u2013 a13 a31 a22"
    },
    {
        "Chapter": "1",
        "sentence_range": "1459-1462",
        "Text": "(1)\nANote  We shall apply all four steps together Expansion along second row (R2)\n| A | =\n11\n12\n13\n31\n32\n33\na\na\na\na\na\na\n21\n22\n23\na\na\na\nExpanding along R2, we get\n| A | =\n12\n13\n11\n13\n2\n1\n2\n2\n21\n22\n32\n33\n31\n33\n(\u20131)\n(\u20131)\na\na\na\na\na\na\na\na\na\na\n+\n+\n+\n11\n12\n2\n3\n23\n31\n32\n(\u20131)\na\na\na\na\na\n+\n+\n= \u2013 a21 (a12 a33 \u2013 a32 a13) + a22 (a11 a33 \u2013 a31 a13)\n\u2013 a23 (a11 a32 \u2013 a31 a12)\n| A | = \u2013 a21 a12 a33 + a21 a32 a13 + a22 a11 a33 \u2013 a22 a31 a13 \u2013 a23 a11 a32\n \n+ a23 a31 a12\n= a11 a22 a33 \u2013 a11 a23 a32 \u2013 a12 a21 a33 + a12 a23 a31 + a13 a21 a32\n\u2013 a13 a31 a22 (2)\nExpansion along first Column (C1)\n| A | =\n12\n13\n22\n23\n32\n33\n11\n21\n31\na\na\na\na\na\na\na\na\na\nBy expanding along C1, we get\n| A | =\n22\n23\n12\n13\n1\n1\n2\n1\n11\n21\n32\n33\n32\n33\n(\u20131)\n( 1)\na\na\na\na\na\na\na\na\na\na\n+\n+\n+\n\u2212\n+ \n12\n13\n3\n1\n31\n22\n23\n(\u20131)\na\na\na\na\na\n+\n= a11 (a22 a33 \u2013 a23 a32) \u2013 a21 (a12 a33 \u2013 a13 a32) + a31 (a12 a23 \u2013 a13 a22)\nRationalised 2023-24\n 80\nMATHEMATICS\n| A | = a11 a22 a33 \u2013 a11 a23 a32 \u2013 a21 a12 a33 + a21 a13 a32 + a31 a12 a23\n\u2013 a31 a13 a22\n= a11 a22 a33 \u2013 a11 a23 a32 \u2013 a12 a21 a33 + a12 a23 a31 + a13 a21 a32\n\u2013 a13 a31 a22 (3)\nClearly, values of |A| in (1), (2) and (3) are equal"
    },
    {
        "Chapter": "1",
        "sentence_range": "1460-1463",
        "Text": "Expansion along second row (R2)\n| A | =\n11\n12\n13\n31\n32\n33\na\na\na\na\na\na\n21\n22\n23\na\na\na\nExpanding along R2, we get\n| A | =\n12\n13\n11\n13\n2\n1\n2\n2\n21\n22\n32\n33\n31\n33\n(\u20131)\n(\u20131)\na\na\na\na\na\na\na\na\na\na\n+\n+\n+\n11\n12\n2\n3\n23\n31\n32\n(\u20131)\na\na\na\na\na\n+\n+\n= \u2013 a21 (a12 a33 \u2013 a32 a13) + a22 (a11 a33 \u2013 a31 a13)\n\u2013 a23 (a11 a32 \u2013 a31 a12)\n| A | = \u2013 a21 a12 a33 + a21 a32 a13 + a22 a11 a33 \u2013 a22 a31 a13 \u2013 a23 a11 a32\n \n+ a23 a31 a12\n= a11 a22 a33 \u2013 a11 a23 a32 \u2013 a12 a21 a33 + a12 a23 a31 + a13 a21 a32\n\u2013 a13 a31 a22 (2)\nExpansion along first Column (C1)\n| A | =\n12\n13\n22\n23\n32\n33\n11\n21\n31\na\na\na\na\na\na\na\na\na\nBy expanding along C1, we get\n| A | =\n22\n23\n12\n13\n1\n1\n2\n1\n11\n21\n32\n33\n32\n33\n(\u20131)\n( 1)\na\na\na\na\na\na\na\na\na\na\n+\n+\n+\n\u2212\n+ \n12\n13\n3\n1\n31\n22\n23\n(\u20131)\na\na\na\na\na\n+\n= a11 (a22 a33 \u2013 a23 a32) \u2013 a21 (a12 a33 \u2013 a13 a32) + a31 (a12 a23 \u2013 a13 a22)\nRationalised 2023-24\n 80\nMATHEMATICS\n| A | = a11 a22 a33 \u2013 a11 a23 a32 \u2013 a21 a12 a33 + a21 a13 a32 + a31 a12 a23\n\u2013 a31 a13 a22\n= a11 a22 a33 \u2013 a11 a23 a32 \u2013 a12 a21 a33 + a12 a23 a31 + a13 a21 a32\n\u2013 a13 a31 a22 (3)\nClearly, values of |A| in (1), (2) and (3) are equal It is left as an exercise to the\nreader to verify that the values of |A| by expanding along R3, C2 and C3 are equal to the\nvalue of |A| obtained in (1), (2) or (3)"
    },
    {
        "Chapter": "1",
        "sentence_range": "1461-1464",
        "Text": "(2)\nExpansion along first Column (C1)\n| A | =\n12\n13\n22\n23\n32\n33\n11\n21\n31\na\na\na\na\na\na\na\na\na\nBy expanding along C1, we get\n| A | =\n22\n23\n12\n13\n1\n1\n2\n1\n11\n21\n32\n33\n32\n33\n(\u20131)\n( 1)\na\na\na\na\na\na\na\na\na\na\n+\n+\n+\n\u2212\n+ \n12\n13\n3\n1\n31\n22\n23\n(\u20131)\na\na\na\na\na\n+\n= a11 (a22 a33 \u2013 a23 a32) \u2013 a21 (a12 a33 \u2013 a13 a32) + a31 (a12 a23 \u2013 a13 a22)\nRationalised 2023-24\n 80\nMATHEMATICS\n| A | = a11 a22 a33 \u2013 a11 a23 a32 \u2013 a21 a12 a33 + a21 a13 a32 + a31 a12 a23\n\u2013 a31 a13 a22\n= a11 a22 a33 \u2013 a11 a23 a32 \u2013 a12 a21 a33 + a12 a23 a31 + a13 a21 a32\n\u2013 a13 a31 a22 (3)\nClearly, values of |A| in (1), (2) and (3) are equal It is left as an exercise to the\nreader to verify that the values of |A| by expanding along R3, C2 and C3 are equal to the\nvalue of |A| obtained in (1), (2) or (3) Hence, expanding a determinant along any row or column gives same value"
    },
    {
        "Chapter": "1",
        "sentence_range": "1462-1465",
        "Text": "(3)\nClearly, values of |A| in (1), (2) and (3) are equal It is left as an exercise to the\nreader to verify that the values of |A| by expanding along R3, C2 and C3 are equal to the\nvalue of |A| obtained in (1), (2) or (3) Hence, expanding a determinant along any row or column gives same value Remarks\n(i)\nFor easier calculations, we shall expand the determinant along that row or column\nwhich contains maximum number of zeros"
    },
    {
        "Chapter": "1",
        "sentence_range": "1463-1466",
        "Text": "It is left as an exercise to the\nreader to verify that the values of |A| by expanding along R3, C2 and C3 are equal to the\nvalue of |A| obtained in (1), (2) or (3) Hence, expanding a determinant along any row or column gives same value Remarks\n(i)\nFor easier calculations, we shall expand the determinant along that row or column\nwhich contains maximum number of zeros (ii)\nWhile expanding, instead of multiplying by (\u20131)i + j, we can multiply by +1 or \u20131\naccording as (i + j) is even or odd"
    },
    {
        "Chapter": "1",
        "sentence_range": "1464-1467",
        "Text": "Hence, expanding a determinant along any row or column gives same value Remarks\n(i)\nFor easier calculations, we shall expand the determinant along that row or column\nwhich contains maximum number of zeros (ii)\nWhile expanding, instead of multiplying by (\u20131)i + j, we can multiply by +1 or \u20131\naccording as (i + j) is even or odd (iii)\nLet A = 2\n2\n4\n0\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n and B = \n1\n1\n2\n0\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb"
    },
    {
        "Chapter": "1",
        "sentence_range": "1465-1468",
        "Text": "Remarks\n(i)\nFor easier calculations, we shall expand the determinant along that row or column\nwhich contains maximum number of zeros (ii)\nWhile expanding, instead of multiplying by (\u20131)i + j, we can multiply by +1 or \u20131\naccording as (i + j) is even or odd (iii)\nLet A = 2\n2\n4\n0\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n and B = \n1\n1\n2\n0\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb Then, it is easy to verify that A = 2B"
    },
    {
        "Chapter": "1",
        "sentence_range": "1466-1469",
        "Text": "(ii)\nWhile expanding, instead of multiplying by (\u20131)i + j, we can multiply by +1 or \u20131\naccording as (i + j) is even or odd (iii)\nLet A = 2\n2\n4\n0\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n and B = \n1\n1\n2\n0\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb Then, it is easy to verify that A = 2B Also\n|A| = 0 \u2013 8 = \u2013 8 and |B| = 0 \u2013 2 = \u2013 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "1467-1470",
        "Text": "(iii)\nLet A = 2\n2\n4\n0\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n and B = \n1\n1\n2\n0\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb Then, it is easy to verify that A = 2B Also\n|A| = 0 \u2013 8 = \u2013 8 and |B| = 0 \u2013 2 = \u2013 2 Observe that, |A| = 4(\u2013 2) = 22|B| or |A| = 2n|B|, where n = 2 is the order of\nsquare matrices A and B"
    },
    {
        "Chapter": "1",
        "sentence_range": "1468-1471",
        "Text": "Then, it is easy to verify that A = 2B Also\n|A| = 0 \u2013 8 = \u2013 8 and |B| = 0 \u2013 2 = \u2013 2 Observe that, |A| = 4(\u2013 2) = 22|B| or |A| = 2n|B|, where n = 2 is the order of\nsquare matrices A and B In general, if A = kB where A and B are square matrices of order n, then | A| = kn\n| B |, where n = 1, 2, 3\nExample 3 Evaluate the determinant  \u2206 = \n1\n2\n4\n\u20131\n3\n0\n4\n1\n0"
    },
    {
        "Chapter": "1",
        "sentence_range": "1469-1472",
        "Text": "Also\n|A| = 0 \u2013 8 = \u2013 8 and |B| = 0 \u2013 2 = \u2013 2 Observe that, |A| = 4(\u2013 2) = 22|B| or |A| = 2n|B|, where n = 2 is the order of\nsquare matrices A and B In general, if A = kB where A and B are square matrices of order n, then | A| = kn\n| B |, where n = 1, 2, 3\nExample 3 Evaluate the determinant  \u2206 = \n1\n2\n4\n\u20131\n3\n0\n4\n1\n0 Solution Note that in the third column, two entries are zero"
    },
    {
        "Chapter": "1",
        "sentence_range": "1470-1473",
        "Text": "Observe that, |A| = 4(\u2013 2) = 22|B| or |A| = 2n|B|, where n = 2 is the order of\nsquare matrices A and B In general, if A = kB where A and B are square matrices of order n, then | A| = kn\n| B |, where n = 1, 2, 3\nExample 3 Evaluate the determinant  \u2206 = \n1\n2\n4\n\u20131\n3\n0\n4\n1\n0 Solution Note that in the third column, two entries are zero So expanding along third\ncolumn (C3), we get\n\u2206 =\n\u20131\n3\n1\n2\n1\n2\n4\n\u2013 0\n0\n4\n1\n4\n1\n\u20131\n3\n+\n= 4 (\u20131 \u2013 12) \u2013 0 + 0  = \u2013 52\nExample 4 Evaluate \u2206 = \n0\nsin\n\u2013cos\n\u2013sin\n0\nsin\ncos\n\u2013sin\n0\n\u03b1\n\u03b1\n\u03b1\n\u03b2\n\u03b1\n\u03b2"
    },
    {
        "Chapter": "1",
        "sentence_range": "1471-1474",
        "Text": "In general, if A = kB where A and B are square matrices of order n, then | A| = kn\n| B |, where n = 1, 2, 3\nExample 3 Evaluate the determinant  \u2206 = \n1\n2\n4\n\u20131\n3\n0\n4\n1\n0 Solution Note that in the third column, two entries are zero So expanding along third\ncolumn (C3), we get\n\u2206 =\n\u20131\n3\n1\n2\n1\n2\n4\n\u2013 0\n0\n4\n1\n4\n1\n\u20131\n3\n+\n= 4 (\u20131 \u2013 12) \u2013 0 + 0  = \u2013 52\nExample 4 Evaluate \u2206 = \n0\nsin\n\u2013cos\n\u2013sin\n0\nsin\ncos\n\u2013sin\n0\n\u03b1\n\u03b1\n\u03b1\n\u03b2\n\u03b1\n\u03b2 Rationalised 2023-24\nDETERMINANTS     81\nSolution Expanding along R1, we get\n\u2206 =\n0\nsin\n\u2013sin\nsin\n\u2013sin\n0\n0\n\u2013 sin\n\u2013 cos\n\u2013sin\n0\ncos\n0\ncos\n\u2013sin\n\u03b2\n\u03b1\n\u03b2\n\u03b1\n\u03b1\n\u03b1\n\u03b2\n\u03b1\n\u03b1\n\u03b2\n= 0 \u2013 sin \u03b1 (0 \u2013 sin \u03b2 cos \u03b1) \u2013 cos \u03b1 (sin \u03b1 sin \u03b2 \u2013 0)\n= sin \u03b1 sin \u03b2 cos \u03b1 \u2013 cos \u03b1 sin \u03b1 sin \u03b2 = 0\nExample 5 Find values of x for which 3\n3\n2\n1\n4\n1\nx\nx\n="
    },
    {
        "Chapter": "1",
        "sentence_range": "1472-1475",
        "Text": "Solution Note that in the third column, two entries are zero So expanding along third\ncolumn (C3), we get\n\u2206 =\n\u20131\n3\n1\n2\n1\n2\n4\n\u2013 0\n0\n4\n1\n4\n1\n\u20131\n3\n+\n= 4 (\u20131 \u2013 12) \u2013 0 + 0  = \u2013 52\nExample 4 Evaluate \u2206 = \n0\nsin\n\u2013cos\n\u2013sin\n0\nsin\ncos\n\u2013sin\n0\n\u03b1\n\u03b1\n\u03b1\n\u03b2\n\u03b1\n\u03b2 Rationalised 2023-24\nDETERMINANTS     81\nSolution Expanding along R1, we get\n\u2206 =\n0\nsin\n\u2013sin\nsin\n\u2013sin\n0\n0\n\u2013 sin\n\u2013 cos\n\u2013sin\n0\ncos\n0\ncos\n\u2013sin\n\u03b2\n\u03b1\n\u03b2\n\u03b1\n\u03b1\n\u03b1\n\u03b2\n\u03b1\n\u03b1\n\u03b2\n= 0 \u2013 sin \u03b1 (0 \u2013 sin \u03b2 cos \u03b1) \u2013 cos \u03b1 (sin \u03b1 sin \u03b2 \u2013 0)\n= sin \u03b1 sin \u03b2 cos \u03b1 \u2013 cos \u03b1 sin \u03b1 sin \u03b2 = 0\nExample 5 Find values of x for which 3\n3\n2\n1\n4\n1\nx\nx\n= Solution We have 3\n3\n2\n1\n4\n1\nx\nx\n=\ni"
    },
    {
        "Chapter": "1",
        "sentence_range": "1473-1476",
        "Text": "So expanding along third\ncolumn (C3), we get\n\u2206 =\n\u20131\n3\n1\n2\n1\n2\n4\n\u2013 0\n0\n4\n1\n4\n1\n\u20131\n3\n+\n= 4 (\u20131 \u2013 12) \u2013 0 + 0  = \u2013 52\nExample 4 Evaluate \u2206 = \n0\nsin\n\u2013cos\n\u2013sin\n0\nsin\ncos\n\u2013sin\n0\n\u03b1\n\u03b1\n\u03b1\n\u03b2\n\u03b1\n\u03b2 Rationalised 2023-24\nDETERMINANTS     81\nSolution Expanding along R1, we get\n\u2206 =\n0\nsin\n\u2013sin\nsin\n\u2013sin\n0\n0\n\u2013 sin\n\u2013 cos\n\u2013sin\n0\ncos\n0\ncos\n\u2013sin\n\u03b2\n\u03b1\n\u03b2\n\u03b1\n\u03b1\n\u03b1\n\u03b2\n\u03b1\n\u03b1\n\u03b2\n= 0 \u2013 sin \u03b1 (0 \u2013 sin \u03b2 cos \u03b1) \u2013 cos \u03b1 (sin \u03b1 sin \u03b2 \u2013 0)\n= sin \u03b1 sin \u03b2 cos \u03b1 \u2013 cos \u03b1 sin \u03b1 sin \u03b2 = 0\nExample 5 Find values of x for which 3\n3\n2\n1\n4\n1\nx\nx\n= Solution We have 3\n3\n2\n1\n4\n1\nx\nx\n=\ni e"
    },
    {
        "Chapter": "1",
        "sentence_range": "1474-1477",
        "Text": "Rationalised 2023-24\nDETERMINANTS     81\nSolution Expanding along R1, we get\n\u2206 =\n0\nsin\n\u2013sin\nsin\n\u2013sin\n0\n0\n\u2013 sin\n\u2013 cos\n\u2013sin\n0\ncos\n0\ncos\n\u2013sin\n\u03b2\n\u03b1\n\u03b2\n\u03b1\n\u03b1\n\u03b1\n\u03b2\n\u03b1\n\u03b1\n\u03b2\n= 0 \u2013 sin \u03b1 (0 \u2013 sin \u03b2 cos \u03b1) \u2013 cos \u03b1 (sin \u03b1 sin \u03b2 \u2013 0)\n= sin \u03b1 sin \u03b2 cos \u03b1 \u2013 cos \u03b1 sin \u03b1 sin \u03b2 = 0\nExample 5 Find values of x for which 3\n3\n2\n1\n4\n1\nx\nx\n= Solution We have 3\n3\n2\n1\n4\n1\nx\nx\n=\ni e 3 \u2013 x2 = 3 \u2013 8\ni"
    },
    {
        "Chapter": "1",
        "sentence_range": "1475-1478",
        "Text": "Solution We have 3\n3\n2\n1\n4\n1\nx\nx\n=\ni e 3 \u2013 x2 = 3 \u2013 8\ni e"
    },
    {
        "Chapter": "1",
        "sentence_range": "1476-1479",
        "Text": "e 3 \u2013 x2 = 3 \u2013 8\ni e x2 = 8\nHence\nx =\n2 2\n\u00b1\nEXERCISE 4"
    },
    {
        "Chapter": "1",
        "sentence_range": "1477-1480",
        "Text": "3 \u2013 x2 = 3 \u2013 8\ni e x2 = 8\nHence\nx =\n2 2\n\u00b1\nEXERCISE 4 1\nEvaluate the determinants in Exercises 1 and 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "1478-1481",
        "Text": "e x2 = 8\nHence\nx =\n2 2\n\u00b1\nEXERCISE 4 1\nEvaluate the determinants in Exercises 1 and 2 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "1479-1482",
        "Text": "x2 = 8\nHence\nx =\n2 2\n\u00b1\nEXERCISE 4 1\nEvaluate the determinants in Exercises 1 and 2 1 2\n4\n\u20135\n\u20131\n2"
    },
    {
        "Chapter": "1",
        "sentence_range": "1480-1483",
        "Text": "1\nEvaluate the determinants in Exercises 1 and 2 1 2\n4\n\u20135\n\u20131\n2 (i)\ncos\n\u2013sin\nsin\ncos\n\u03b8\n\u03b8\n\u03b8\n\u03b8\n(ii)\n2 \u2013\n1\n\u2013 1\n1\n1\nx\nx\nx\nx\nx\n+\n+\n+\n3"
    },
    {
        "Chapter": "1",
        "sentence_range": "1481-1484",
        "Text": "1 2\n4\n\u20135\n\u20131\n2 (i)\ncos\n\u2013sin\nsin\ncos\n\u03b8\n\u03b8\n\u03b8\n\u03b8\n(ii)\n2 \u2013\n1\n\u2013 1\n1\n1\nx\nx\nx\nx\nx\n+\n+\n+\n3 If\nA = \n1\n2\n4\n2\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, then show that | 2A | = 4 | A |\n4"
    },
    {
        "Chapter": "1",
        "sentence_range": "1482-1485",
        "Text": "2\n4\n\u20135\n\u20131\n2 (i)\ncos\n\u2013sin\nsin\ncos\n\u03b8\n\u03b8\n\u03b8\n\u03b8\n(ii)\n2 \u2013\n1\n\u2013 1\n1\n1\nx\nx\nx\nx\nx\n+\n+\n+\n3 If\nA = \n1\n2\n4\n2\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, then show that | 2A | = 4 | A |\n4 If\nA = \n1\n0\n1\n0\n1\n2\n0\n0\n4\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, then show that | 3 A | = 27 | A |\n5"
    },
    {
        "Chapter": "1",
        "sentence_range": "1483-1486",
        "Text": "(i)\ncos\n\u2013sin\nsin\ncos\n\u03b8\n\u03b8\n\u03b8\n\u03b8\n(ii)\n2 \u2013\n1\n\u2013 1\n1\n1\nx\nx\nx\nx\nx\n+\n+\n+\n3 If\nA = \n1\n2\n4\n2\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, then show that | 2A | = 4 | A |\n4 If\nA = \n1\n0\n1\n0\n1\n2\n0\n0\n4\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, then show that | 3 A | = 27 | A |\n5 Evaluate the determinants\n(i)\n3\n\u20131\n\u20132\n0\n0\n\u20131\n3\n\u20135\n0\n(ii)\n3\n\u2013 4\n5\n1\n1\n\u20132\n2\n3\n1\nRationalised 2023-24\n 82\nMATHEMATICS\n(iii)\n0\n1\n2\n\u20131\n0\n\u20133\n\u20132\n3\n0\n(iv)\n2\n\u20131\n\u20132\n0\n2\n\u20131\n3\n\u20135\n0\n6"
    },
    {
        "Chapter": "1",
        "sentence_range": "1484-1487",
        "Text": "If\nA = \n1\n2\n4\n2\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, then show that | 2A | = 4 | A |\n4 If\nA = \n1\n0\n1\n0\n1\n2\n0\n0\n4\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, then show that | 3 A | = 27 | A |\n5 Evaluate the determinants\n(i)\n3\n\u20131\n\u20132\n0\n0\n\u20131\n3\n\u20135\n0\n(ii)\n3\n\u2013 4\n5\n1\n1\n\u20132\n2\n3\n1\nRationalised 2023-24\n 82\nMATHEMATICS\n(iii)\n0\n1\n2\n\u20131\n0\n\u20133\n\u20132\n3\n0\n(iv)\n2\n\u20131\n\u20132\n0\n2\n\u20131\n3\n\u20135\n0\n6 If A = \n1\n1\n\u20132\n2\n1\n\u20133\n5\n4\n\u20139\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, find | A |\n7"
    },
    {
        "Chapter": "1",
        "sentence_range": "1485-1488",
        "Text": "If\nA = \n1\n0\n1\n0\n1\n2\n0\n0\n4\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, then show that | 3 A | = 27 | A |\n5 Evaluate the determinants\n(i)\n3\n\u20131\n\u20132\n0\n0\n\u20131\n3\n\u20135\n0\n(ii)\n3\n\u2013 4\n5\n1\n1\n\u20132\n2\n3\n1\nRationalised 2023-24\n 82\nMATHEMATICS\n(iii)\n0\n1\n2\n\u20131\n0\n\u20133\n\u20132\n3\n0\n(iv)\n2\n\u20131\n\u20132\n0\n2\n\u20131\n3\n\u20135\n0\n6 If A = \n1\n1\n\u20132\n2\n1\n\u20133\n5\n4\n\u20139\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, find | A |\n7 Find values of x, if\n(i)\n2\n4\n2\n4\n5\n1\n6\nx\nx\n=\n(ii)\n2\n3\n3\n4\n5\n2\n5\nx\nx\n=\n8"
    },
    {
        "Chapter": "1",
        "sentence_range": "1486-1489",
        "Text": "Evaluate the determinants\n(i)\n3\n\u20131\n\u20132\n0\n0\n\u20131\n3\n\u20135\n0\n(ii)\n3\n\u2013 4\n5\n1\n1\n\u20132\n2\n3\n1\nRationalised 2023-24\n 82\nMATHEMATICS\n(iii)\n0\n1\n2\n\u20131\n0\n\u20133\n\u20132\n3\n0\n(iv)\n2\n\u20131\n\u20132\n0\n2\n\u20131\n3\n\u20135\n0\n6 If A = \n1\n1\n\u20132\n2\n1\n\u20133\n5\n4\n\u20139\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, find | A |\n7 Find values of x, if\n(i)\n2\n4\n2\n4\n5\n1\n6\nx\nx\n=\n(ii)\n2\n3\n3\n4\n5\n2\n5\nx\nx\n=\n8 If \n2\n6\n2\n18\n18\n6\nx\nx =\n, then x is equal to\n(A) 6\n(B) \u00b1 6\n(C) \u2013 6\n(D) 0\n4"
    },
    {
        "Chapter": "1",
        "sentence_range": "1487-1490",
        "Text": "If A = \n1\n1\n\u20132\n2\n1\n\u20133\n5\n4\n\u20139\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, find | A |\n7 Find values of x, if\n(i)\n2\n4\n2\n4\n5\n1\n6\nx\nx\n=\n(ii)\n2\n3\n3\n4\n5\n2\n5\nx\nx\n=\n8 If \n2\n6\n2\n18\n18\n6\nx\nx =\n, then x is equal to\n(A) 6\n(B) \u00b1 6\n(C) \u2013 6\n(D) 0\n4 3  Area of a Triangle\nIn earlier classes, we have studied that the area of a triangle whose vertices are\n(x1, y1), (x2, y2) and (x3, y3), is given by the expression 1\n2 [x1(y2\u2013y3) + x2 (y3\u2013y1) +\nx3 (y1\u2013y2)]"
    },
    {
        "Chapter": "1",
        "sentence_range": "1488-1491",
        "Text": "Find values of x, if\n(i)\n2\n4\n2\n4\n5\n1\n6\nx\nx\n=\n(ii)\n2\n3\n3\n4\n5\n2\n5\nx\nx\n=\n8 If \n2\n6\n2\n18\n18\n6\nx\nx =\n, then x is equal to\n(A) 6\n(B) \u00b1 6\n(C) \u2013 6\n(D) 0\n4 3  Area of a Triangle\nIn earlier classes, we have studied that the area of a triangle whose vertices are\n(x1, y1), (x2, y2) and (x3, y3), is given by the expression 1\n2 [x1(y2\u2013y3) + x2 (y3\u2013y1) +\nx3 (y1\u2013y2)] Now this expression can be written in the form of a determinant as\n\u2206 =\n1\n1\n2\n2\n3\n3\n1\n1\n1\n2\n1\nx\ny\nx\ny\nx\ny"
    },
    {
        "Chapter": "1",
        "sentence_range": "1489-1492",
        "Text": "If \n2\n6\n2\n18\n18\n6\nx\nx =\n, then x is equal to\n(A) 6\n(B) \u00b1 6\n(C) \u2013 6\n(D) 0\n4 3  Area of a Triangle\nIn earlier classes, we have studied that the area of a triangle whose vertices are\n(x1, y1), (x2, y2) and (x3, y3), is given by the expression 1\n2 [x1(y2\u2013y3) + x2 (y3\u2013y1) +\nx3 (y1\u2013y2)] Now this expression can be written in the form of a determinant as\n\u2206 =\n1\n1\n2\n2\n3\n3\n1\n1\n1\n2\n1\nx\ny\nx\ny\nx\ny (1)\nRemarks\n(i)\nSince area is a positive quantity, we always take the absolute value of the\ndeterminant in (1)"
    },
    {
        "Chapter": "1",
        "sentence_range": "1490-1493",
        "Text": "3  Area of a Triangle\nIn earlier classes, we have studied that the area of a triangle whose vertices are\n(x1, y1), (x2, y2) and (x3, y3), is given by the expression 1\n2 [x1(y2\u2013y3) + x2 (y3\u2013y1) +\nx3 (y1\u2013y2)] Now this expression can be written in the form of a determinant as\n\u2206 =\n1\n1\n2\n2\n3\n3\n1\n1\n1\n2\n1\nx\ny\nx\ny\nx\ny (1)\nRemarks\n(i)\nSince area is a positive quantity, we always take the absolute value of the\ndeterminant in (1) (ii)\nIf area is given, use both positive and negative values of the determinant for\ncalculation"
    },
    {
        "Chapter": "1",
        "sentence_range": "1491-1494",
        "Text": "Now this expression can be written in the form of a determinant as\n\u2206 =\n1\n1\n2\n2\n3\n3\n1\n1\n1\n2\n1\nx\ny\nx\ny\nx\ny (1)\nRemarks\n(i)\nSince area is a positive quantity, we always take the absolute value of the\ndeterminant in (1) (ii)\nIf area is given, use both positive and negative values of the determinant for\ncalculation (iii)\nThe area of the triangle formed by three collinear points is zero"
    },
    {
        "Chapter": "1",
        "sentence_range": "1492-1495",
        "Text": "(1)\nRemarks\n(i)\nSince area is a positive quantity, we always take the absolute value of the\ndeterminant in (1) (ii)\nIf area is given, use both positive and negative values of the determinant for\ncalculation (iii)\nThe area of the triangle formed by three collinear points is zero Example 6 Find the area of the triangle whose vertices are (3, 8), (\u2013 4, 2) and (5, 1)"
    },
    {
        "Chapter": "1",
        "sentence_range": "1493-1496",
        "Text": "(ii)\nIf area is given, use both positive and negative values of the determinant for\ncalculation (iii)\nThe area of the triangle formed by three collinear points is zero Example 6 Find the area of the triangle whose vertices are (3, 8), (\u2013 4, 2) and (5, 1) Solution The area of triangle is given by\n\u2206 =\n3\n8\n1\n1\n4\n2\n1\n2\n5\n1\n1\n\u2013\nRationalised 2023-24\nDETERMINANTS     83\n=\n(\n)\n(\n)\n(\n)\n1 3 2 \u20131 \u2013 8 \u20134 \u2013 5\n1 \u20134 \u201310\n2 \uf8ee\n+\n\uf8f9\n\uf8f0\n\uf8fb\n \n=\n(\n)\n1\n61\n3\n72\n14\n2\n2\n\u2013\n+\n=\nExample 7 Find the equation of the line joining A(1, 3) and B (0, 0) using determinants\nand find k if D(k, 0) is a point such that area of triangle ABD is 3sq units"
    },
    {
        "Chapter": "1",
        "sentence_range": "1494-1497",
        "Text": "(iii)\nThe area of the triangle formed by three collinear points is zero Example 6 Find the area of the triangle whose vertices are (3, 8), (\u2013 4, 2) and (5, 1) Solution The area of triangle is given by\n\u2206 =\n3\n8\n1\n1\n4\n2\n1\n2\n5\n1\n1\n\u2013\nRationalised 2023-24\nDETERMINANTS     83\n=\n(\n)\n(\n)\n(\n)\n1 3 2 \u20131 \u2013 8 \u20134 \u2013 5\n1 \u20134 \u201310\n2 \uf8ee\n+\n\uf8f9\n\uf8f0\n\uf8fb\n \n=\n(\n)\n1\n61\n3\n72\n14\n2\n2\n\u2013\n+\n=\nExample 7 Find the equation of the line joining A(1, 3) and B (0, 0) using determinants\nand find k if D(k, 0) is a point such that area of triangle ABD is 3sq units Solution Let P (x, y) be any point on AB"
    },
    {
        "Chapter": "1",
        "sentence_range": "1495-1498",
        "Text": "Example 6 Find the area of the triangle whose vertices are (3, 8), (\u2013 4, 2) and (5, 1) Solution The area of triangle is given by\n\u2206 =\n3\n8\n1\n1\n4\n2\n1\n2\n5\n1\n1\n\u2013\nRationalised 2023-24\nDETERMINANTS     83\n=\n(\n)\n(\n)\n(\n)\n1 3 2 \u20131 \u2013 8 \u20134 \u2013 5\n1 \u20134 \u201310\n2 \uf8ee\n+\n\uf8f9\n\uf8f0\n\uf8fb\n \n=\n(\n)\n1\n61\n3\n72\n14\n2\n2\n\u2013\n+\n=\nExample 7 Find the equation of the line joining A(1, 3) and B (0, 0) using determinants\nand find k if D(k, 0) is a point such that area of triangle ABD is 3sq units Solution Let P (x, y) be any point on AB Then, area of triangle ABP is zero (Why"
    },
    {
        "Chapter": "1",
        "sentence_range": "1496-1499",
        "Text": "Solution The area of triangle is given by\n\u2206 =\n3\n8\n1\n1\n4\n2\n1\n2\n5\n1\n1\n\u2013\nRationalised 2023-24\nDETERMINANTS     83\n=\n(\n)\n(\n)\n(\n)\n1 3 2 \u20131 \u2013 8 \u20134 \u2013 5\n1 \u20134 \u201310\n2 \uf8ee\n+\n\uf8f9\n\uf8f0\n\uf8fb\n \n=\n(\n)\n1\n61\n3\n72\n14\n2\n2\n\u2013\n+\n=\nExample 7 Find the equation of the line joining A(1, 3) and B (0, 0) using determinants\nand find k if D(k, 0) is a point such that area of triangle ABD is 3sq units Solution Let P (x, y) be any point on AB Then, area of triangle ABP is zero (Why )"
    },
    {
        "Chapter": "1",
        "sentence_range": "1497-1500",
        "Text": "Solution Let P (x, y) be any point on AB Then, area of triangle ABP is zero (Why ) So\n0\n0\n1\n1 1\n3\n1\n2\n1\nx\ny\n = 0\nThis gives\n(\n)\n1\n2 y \u20133\nx  = 0 or y = 3x,\nwhich is the equation of required line AB"
    },
    {
        "Chapter": "1",
        "sentence_range": "1498-1501",
        "Text": "Then, area of triangle ABP is zero (Why ) So\n0\n0\n1\n1 1\n3\n1\n2\n1\nx\ny\n = 0\nThis gives\n(\n)\n1\n2 y \u20133\nx  = 0 or y = 3x,\nwhich is the equation of required line AB Also, since the area of the triangle ABD is 3 sq"
    },
    {
        "Chapter": "1",
        "sentence_range": "1499-1502",
        "Text": ") So\n0\n0\n1\n1 1\n3\n1\n2\n1\nx\ny\n = 0\nThis gives\n(\n)\n1\n2 y \u20133\nx  = 0 or y = 3x,\nwhich is the equation of required line AB Also, since the area of the triangle ABD is 3 sq units, we have\n1\n3\n1\n1 0\n0\n1\n2\n0\n1\nk\n = \u00b1 3\nThis gives, \n3\n3\n2\n\u2212k\n= \u00b1 , i"
    },
    {
        "Chapter": "1",
        "sentence_range": "1500-1503",
        "Text": "So\n0\n0\n1\n1 1\n3\n1\n2\n1\nx\ny\n = 0\nThis gives\n(\n)\n1\n2 y \u20133\nx  = 0 or y = 3x,\nwhich is the equation of required line AB Also, since the area of the triangle ABD is 3 sq units, we have\n1\n3\n1\n1 0\n0\n1\n2\n0\n1\nk\n = \u00b1 3\nThis gives, \n3\n3\n2\n\u2212k\n= \u00b1 , i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "1501-1504",
        "Text": "Also, since the area of the triangle ABD is 3 sq units, we have\n1\n3\n1\n1 0\n0\n1\n2\n0\n1\nk\n = \u00b1 3\nThis gives, \n3\n3\n2\n\u2212k\n= \u00b1 , i e , k = \u2213  2"
    },
    {
        "Chapter": "1",
        "sentence_range": "1502-1505",
        "Text": "units, we have\n1\n3\n1\n1 0\n0\n1\n2\n0\n1\nk\n = \u00b1 3\nThis gives, \n3\n3\n2\n\u2212k\n= \u00b1 , i e , k = \u2213  2 EXERCISE 4"
    },
    {
        "Chapter": "1",
        "sentence_range": "1503-1506",
        "Text": "e , k = \u2213  2 EXERCISE 4 2\n1"
    },
    {
        "Chapter": "1",
        "sentence_range": "1504-1507",
        "Text": ", k = \u2213  2 EXERCISE 4 2\n1 Find area of the triangle with vertices at the point given in each of the following :\n(i) (1, 0), (6, 0), (4, 3)\n(ii) (2, 7), (1, 1), (10, 8)\n(iii) (\u20132, \u20133), (3, 2), (\u20131, \u20138)\n2"
    },
    {
        "Chapter": "1",
        "sentence_range": "1505-1508",
        "Text": "EXERCISE 4 2\n1 Find area of the triangle with vertices at the point given in each of the following :\n(i) (1, 0), (6, 0), (4, 3)\n(ii) (2, 7), (1, 1), (10, 8)\n(iii) (\u20132, \u20133), (3, 2), (\u20131, \u20138)\n2 Show that points\nA (a, b + c), B (b, c + a), C (c, a + b) are collinear"
    },
    {
        "Chapter": "1",
        "sentence_range": "1506-1509",
        "Text": "2\n1 Find area of the triangle with vertices at the point given in each of the following :\n(i) (1, 0), (6, 0), (4, 3)\n(ii) (2, 7), (1, 1), (10, 8)\n(iii) (\u20132, \u20133), (3, 2), (\u20131, \u20138)\n2 Show that points\nA (a, b + c), B (b, c + a), C (c, a + b) are collinear 3"
    },
    {
        "Chapter": "1",
        "sentence_range": "1507-1510",
        "Text": "Find area of the triangle with vertices at the point given in each of the following :\n(i) (1, 0), (6, 0), (4, 3)\n(ii) (2, 7), (1, 1), (10, 8)\n(iii) (\u20132, \u20133), (3, 2), (\u20131, \u20138)\n2 Show that points\nA (a, b + c), B (b, c + a), C (c, a + b) are collinear 3 Find values of k if area of triangle is 4 sq"
    },
    {
        "Chapter": "1",
        "sentence_range": "1508-1511",
        "Text": "Show that points\nA (a, b + c), B (b, c + a), C (c, a + b) are collinear 3 Find values of k if area of triangle is 4 sq units and vertices are\n(i) (k, 0), (4, 0), (0, 2)\n(ii) (\u20132, 0), (0, 4), (0, k)\n4"
    },
    {
        "Chapter": "1",
        "sentence_range": "1509-1512",
        "Text": "3 Find values of k if area of triangle is 4 sq units and vertices are\n(i) (k, 0), (4, 0), (0, 2)\n(ii) (\u20132, 0), (0, 4), (0, k)\n4 (i) Find equation of line joining (1, 2) and (3, 6) using determinants"
    },
    {
        "Chapter": "1",
        "sentence_range": "1510-1513",
        "Text": "Find values of k if area of triangle is 4 sq units and vertices are\n(i) (k, 0), (4, 0), (0, 2)\n(ii) (\u20132, 0), (0, 4), (0, k)\n4 (i) Find equation of line joining (1, 2) and (3, 6) using determinants (ii) Find equation of line joining (3, 1) and (9, 3) using determinants"
    },
    {
        "Chapter": "1",
        "sentence_range": "1511-1514",
        "Text": "units and vertices are\n(i) (k, 0), (4, 0), (0, 2)\n(ii) (\u20132, 0), (0, 4), (0, k)\n4 (i) Find equation of line joining (1, 2) and (3, 6) using determinants (ii) Find equation of line joining (3, 1) and (9, 3) using determinants 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "1512-1515",
        "Text": "(i) Find equation of line joining (1, 2) and (3, 6) using determinants (ii) Find equation of line joining (3, 1) and (9, 3) using determinants 5 If area of triangle is 35 sq units with vertices (2, \u2013 6), (5, 4) and (k, 4)"
    },
    {
        "Chapter": "1",
        "sentence_range": "1513-1516",
        "Text": "(ii) Find equation of line joining (3, 1) and (9, 3) using determinants 5 If area of triangle is 35 sq units with vertices (2, \u2013 6), (5, 4) and (k, 4) Then k is\n(A) 12\n(B) \u20132\n(C) \u201312, \u20132\n(D) 12, \u20132\nRationalised 2023-24\n 84\nMATHEMATICS\n4"
    },
    {
        "Chapter": "1",
        "sentence_range": "1514-1517",
        "Text": "5 If area of triangle is 35 sq units with vertices (2, \u2013 6), (5, 4) and (k, 4) Then k is\n(A) 12\n(B) \u20132\n(C) \u201312, \u20132\n(D) 12, \u20132\nRationalised 2023-24\n 84\nMATHEMATICS\n4 4  Minors and Cofactors\nIn this section, we will learn to write the expansion of a determinant in compact form\nusing minors and cofactors"
    },
    {
        "Chapter": "1",
        "sentence_range": "1515-1518",
        "Text": "If area of triangle is 35 sq units with vertices (2, \u2013 6), (5, 4) and (k, 4) Then k is\n(A) 12\n(B) \u20132\n(C) \u201312, \u20132\n(D) 12, \u20132\nRationalised 2023-24\n 84\nMATHEMATICS\n4 4  Minors and Cofactors\nIn this section, we will learn to write the expansion of a determinant in compact form\nusing minors and cofactors Definition 1 Minor of an element aij of a determinant is the determinant obtained by\ndeleting its ith row and jth column in which element aij lies"
    },
    {
        "Chapter": "1",
        "sentence_range": "1516-1519",
        "Text": "Then k is\n(A) 12\n(B) \u20132\n(C) \u201312, \u20132\n(D) 12, \u20132\nRationalised 2023-24\n 84\nMATHEMATICS\n4 4  Minors and Cofactors\nIn this section, we will learn to write the expansion of a determinant in compact form\nusing minors and cofactors Definition 1 Minor of an element aij of a determinant is the determinant obtained by\ndeleting its ith row and jth column in which element aij lies Minor of an element aij is\ndenoted by Mij"
    },
    {
        "Chapter": "1",
        "sentence_range": "1517-1520",
        "Text": "4  Minors and Cofactors\nIn this section, we will learn to write the expansion of a determinant in compact form\nusing minors and cofactors Definition 1 Minor of an element aij of a determinant is the determinant obtained by\ndeleting its ith row and jth column in which element aij lies Minor of an element aij is\ndenoted by Mij Remark Minor of an element of a determinant of order n(n \u2265 2) is a determinant of\norder n \u2013 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "1518-1521",
        "Text": "Definition 1 Minor of an element aij of a determinant is the determinant obtained by\ndeleting its ith row and jth column in which element aij lies Minor of an element aij is\ndenoted by Mij Remark Minor of an element of a determinant of order n(n \u2265 2) is a determinant of\norder n \u2013 1 Example 8 Find the minor of element 6 in the determinant \n1\n2\n3\n4\n5\n6\n7\n8\n9\n\u2206 =\nSolution Since 6 lies in the second row and third column, its minor M23 is given by\nM23 =\n1\n2\n7\n8  = 8 \u2013 14 = \u2013 6 (obtained by deleting R2 and C3 in \u2206)"
    },
    {
        "Chapter": "1",
        "sentence_range": "1519-1522",
        "Text": "Minor of an element aij is\ndenoted by Mij Remark Minor of an element of a determinant of order n(n \u2265 2) is a determinant of\norder n \u2013 1 Example 8 Find the minor of element 6 in the determinant \n1\n2\n3\n4\n5\n6\n7\n8\n9\n\u2206 =\nSolution Since 6 lies in the second row and third column, its minor M23 is given by\nM23 =\n1\n2\n7\n8  = 8 \u2013 14 = \u2013 6 (obtained by deleting R2 and C3 in \u2206) Definition 2 Cofactor of an element aij, denoted by Aij is defined by\nAij = (\u20131)i + j Mij, where Mij is minor of aij"
    },
    {
        "Chapter": "1",
        "sentence_range": "1520-1523",
        "Text": "Remark Minor of an element of a determinant of order n(n \u2265 2) is a determinant of\norder n \u2013 1 Example 8 Find the minor of element 6 in the determinant \n1\n2\n3\n4\n5\n6\n7\n8\n9\n\u2206 =\nSolution Since 6 lies in the second row and third column, its minor M23 is given by\nM23 =\n1\n2\n7\n8  = 8 \u2013 14 = \u2013 6 (obtained by deleting R2 and C3 in \u2206) Definition 2 Cofactor of an element aij, denoted by Aij is defined by\nAij = (\u20131)i + j Mij, where Mij is minor of aij Example 9 Find minors and cofactors of all the elements of the determinant \n1\n\u20132\n4\n3\nSolution Minor of the element aij is Mij\nHere a11 = 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "1521-1524",
        "Text": "Example 8 Find the minor of element 6 in the determinant \n1\n2\n3\n4\n5\n6\n7\n8\n9\n\u2206 =\nSolution Since 6 lies in the second row and third column, its minor M23 is given by\nM23 =\n1\n2\n7\n8  = 8 \u2013 14 = \u2013 6 (obtained by deleting R2 and C3 in \u2206) Definition 2 Cofactor of an element aij, denoted by Aij is defined by\nAij = (\u20131)i + j Mij, where Mij is minor of aij Example 9 Find minors and cofactors of all the elements of the determinant \n1\n\u20132\n4\n3\nSolution Minor of the element aij is Mij\nHere a11 = 1 So M11 = Minor of a11= 3\nM12\n = Minor of the element a12 = 4\nM21 = Minor of the element a21 = \u20132\nM22 = Minor of the element a22 = 1\nNow, cofactor of aij is Aij"
    },
    {
        "Chapter": "1",
        "sentence_range": "1522-1525",
        "Text": "Definition 2 Cofactor of an element aij, denoted by Aij is defined by\nAij = (\u20131)i + j Mij, where Mij is minor of aij Example 9 Find minors and cofactors of all the elements of the determinant \n1\n\u20132\n4\n3\nSolution Minor of the element aij is Mij\nHere a11 = 1 So M11 = Minor of a11= 3\nM12\n = Minor of the element a12 = 4\nM21 = Minor of the element a21 = \u20132\nM22 = Minor of the element a22 = 1\nNow, cofactor of aij is Aij So\nA11 = (\u20131)1 + 1  M11 = (\u20131)2 (3) = 3\nA12 = (\u20131)1 + 2  M12 = (\u20131)3 (4) = \u2013 4\nA21 = (\u20131)2 + 1  M21 = (\u20131)3 (\u20132) = 2\nA22 = (\u20131)2 + 2  M22 = (\u20131)4 (1) = 1\nRationalised 2023-24\nDETERMINANTS     85\nExample 10 Find minors and cofactors of the elements a11, a21 in the determinant\n\u2206 =\n11\n12\n13\n21\n22\n23\n31\n32\n33\na\na\na\na\na\na\na\na\na\nSolution By definition of minors and cofactors, we have\nMinor of a11 = M11 = \n22\n23\n32\n33\na\na\na\na\n = a22 a33\u2013 a23 a32\nCofactor of a11 = A11 = (\u20131)1+1  M11 = a22 a33 \u2013 a23 a32\nMinor of a21 = M21 = \n12\n13\n32\n33\na\na\na\na\n = a12 a33 \u2013 a13 a32\nCofactor of a21 = A21 = (\u20131)2+1  M21 = (\u20131) (a12 a33 \u2013 a13 a32) = \u2013 a12 a33 + a13 a32\nRemark Expanding the determinant \u2206, in Example 21, along R1, we have\n\u2206 = (\u20131)1+1 a11 \n22\n23\n32\n33\na\na\na\na\n+ (\u20131)1+2 a12 \n21\n23\n31\n33\na\na\na\na\n + (\u20131)1+3  a13 \n21\n22\n31\n32\na\na\na\na\n   = a11 A11 + a12 A12 + a13 A13, where Aij is cofactor of aij\n    = sum of product of elements of R1 with their corresponding cofactors\nSimilarly, \u2206 can be calculated by other five ways of expansion that is along R2, R3,\nC1, C2 and C3"
    },
    {
        "Chapter": "1",
        "sentence_range": "1523-1526",
        "Text": "Example 9 Find minors and cofactors of all the elements of the determinant \n1\n\u20132\n4\n3\nSolution Minor of the element aij is Mij\nHere a11 = 1 So M11 = Minor of a11= 3\nM12\n = Minor of the element a12 = 4\nM21 = Minor of the element a21 = \u20132\nM22 = Minor of the element a22 = 1\nNow, cofactor of aij is Aij So\nA11 = (\u20131)1 + 1  M11 = (\u20131)2 (3) = 3\nA12 = (\u20131)1 + 2  M12 = (\u20131)3 (4) = \u2013 4\nA21 = (\u20131)2 + 1  M21 = (\u20131)3 (\u20132) = 2\nA22 = (\u20131)2 + 2  M22 = (\u20131)4 (1) = 1\nRationalised 2023-24\nDETERMINANTS     85\nExample 10 Find minors and cofactors of the elements a11, a21 in the determinant\n\u2206 =\n11\n12\n13\n21\n22\n23\n31\n32\n33\na\na\na\na\na\na\na\na\na\nSolution By definition of minors and cofactors, we have\nMinor of a11 = M11 = \n22\n23\n32\n33\na\na\na\na\n = a22 a33\u2013 a23 a32\nCofactor of a11 = A11 = (\u20131)1+1  M11 = a22 a33 \u2013 a23 a32\nMinor of a21 = M21 = \n12\n13\n32\n33\na\na\na\na\n = a12 a33 \u2013 a13 a32\nCofactor of a21 = A21 = (\u20131)2+1  M21 = (\u20131) (a12 a33 \u2013 a13 a32) = \u2013 a12 a33 + a13 a32\nRemark Expanding the determinant \u2206, in Example 21, along R1, we have\n\u2206 = (\u20131)1+1 a11 \n22\n23\n32\n33\na\na\na\na\n+ (\u20131)1+2 a12 \n21\n23\n31\n33\na\na\na\na\n + (\u20131)1+3  a13 \n21\n22\n31\n32\na\na\na\na\n   = a11 A11 + a12 A12 + a13 A13, where Aij is cofactor of aij\n    = sum of product of elements of R1 with their corresponding cofactors\nSimilarly, \u2206 can be calculated by other five ways of expansion that is along R2, R3,\nC1, C2 and C3 Hence \u2206 = sum of the product of elements of any row (or column) with their\ncorresponding cofactors"
    },
    {
        "Chapter": "1",
        "sentence_range": "1524-1527",
        "Text": "So M11 = Minor of a11= 3\nM12\n = Minor of the element a12 = 4\nM21 = Minor of the element a21 = \u20132\nM22 = Minor of the element a22 = 1\nNow, cofactor of aij is Aij So\nA11 = (\u20131)1 + 1  M11 = (\u20131)2 (3) = 3\nA12 = (\u20131)1 + 2  M12 = (\u20131)3 (4) = \u2013 4\nA21 = (\u20131)2 + 1  M21 = (\u20131)3 (\u20132) = 2\nA22 = (\u20131)2 + 2  M22 = (\u20131)4 (1) = 1\nRationalised 2023-24\nDETERMINANTS     85\nExample 10 Find minors and cofactors of the elements a11, a21 in the determinant\n\u2206 =\n11\n12\n13\n21\n22\n23\n31\n32\n33\na\na\na\na\na\na\na\na\na\nSolution By definition of minors and cofactors, we have\nMinor of a11 = M11 = \n22\n23\n32\n33\na\na\na\na\n = a22 a33\u2013 a23 a32\nCofactor of a11 = A11 = (\u20131)1+1  M11 = a22 a33 \u2013 a23 a32\nMinor of a21 = M21 = \n12\n13\n32\n33\na\na\na\na\n = a12 a33 \u2013 a13 a32\nCofactor of a21 = A21 = (\u20131)2+1  M21 = (\u20131) (a12 a33 \u2013 a13 a32) = \u2013 a12 a33 + a13 a32\nRemark Expanding the determinant \u2206, in Example 21, along R1, we have\n\u2206 = (\u20131)1+1 a11 \n22\n23\n32\n33\na\na\na\na\n+ (\u20131)1+2 a12 \n21\n23\n31\n33\na\na\na\na\n + (\u20131)1+3  a13 \n21\n22\n31\n32\na\na\na\na\n   = a11 A11 + a12 A12 + a13 A13, where Aij is cofactor of aij\n    = sum of product of elements of R1 with their corresponding cofactors\nSimilarly, \u2206 can be calculated by other five ways of expansion that is along R2, R3,\nC1, C2 and C3 Hence \u2206 = sum of the product of elements of any row (or column) with their\ncorresponding cofactors ANote  If elements of a row (or column) are multiplied with cofactors of any\nother row (or column), then their sum is zero"
    },
    {
        "Chapter": "1",
        "sentence_range": "1525-1528",
        "Text": "So\nA11 = (\u20131)1 + 1  M11 = (\u20131)2 (3) = 3\nA12 = (\u20131)1 + 2  M12 = (\u20131)3 (4) = \u2013 4\nA21 = (\u20131)2 + 1  M21 = (\u20131)3 (\u20132) = 2\nA22 = (\u20131)2 + 2  M22 = (\u20131)4 (1) = 1\nRationalised 2023-24\nDETERMINANTS     85\nExample 10 Find minors and cofactors of the elements a11, a21 in the determinant\n\u2206 =\n11\n12\n13\n21\n22\n23\n31\n32\n33\na\na\na\na\na\na\na\na\na\nSolution By definition of minors and cofactors, we have\nMinor of a11 = M11 = \n22\n23\n32\n33\na\na\na\na\n = a22 a33\u2013 a23 a32\nCofactor of a11 = A11 = (\u20131)1+1  M11 = a22 a33 \u2013 a23 a32\nMinor of a21 = M21 = \n12\n13\n32\n33\na\na\na\na\n = a12 a33 \u2013 a13 a32\nCofactor of a21 = A21 = (\u20131)2+1  M21 = (\u20131) (a12 a33 \u2013 a13 a32) = \u2013 a12 a33 + a13 a32\nRemark Expanding the determinant \u2206, in Example 21, along R1, we have\n\u2206 = (\u20131)1+1 a11 \n22\n23\n32\n33\na\na\na\na\n+ (\u20131)1+2 a12 \n21\n23\n31\n33\na\na\na\na\n + (\u20131)1+3  a13 \n21\n22\n31\n32\na\na\na\na\n   = a11 A11 + a12 A12 + a13 A13, where Aij is cofactor of aij\n    = sum of product of elements of R1 with their corresponding cofactors\nSimilarly, \u2206 can be calculated by other five ways of expansion that is along R2, R3,\nC1, C2 and C3 Hence \u2206 = sum of the product of elements of any row (or column) with their\ncorresponding cofactors ANote  If elements of a row (or column) are multiplied with cofactors of any\nother row (or column), then their sum is zero For example,\n\u2206 = a11 A21 + a12 A22 + a13 A23\n= a11 (\u20131)1+1 \n12\n13\n32\n33\na\na\na\na\n+ a12 (\u20131)1+2 \n11\n13\n31\n33\na\na\na\na\n+ a13 (\u20131)1+3 \n11\n12\n31\n32\na\na\na\na\n= \n11\n12\n13\n11\n12\n13\n31\n32\n33\na\na\na\na\na\na\na\na\na\n = 0 (since  R1 and R2 are identical)\nSimilarly, we can try for other rows and columns"
    },
    {
        "Chapter": "1",
        "sentence_range": "1526-1529",
        "Text": "Hence \u2206 = sum of the product of elements of any row (or column) with their\ncorresponding cofactors ANote  If elements of a row (or column) are multiplied with cofactors of any\nother row (or column), then their sum is zero For example,\n\u2206 = a11 A21 + a12 A22 + a13 A23\n= a11 (\u20131)1+1 \n12\n13\n32\n33\na\na\na\na\n+ a12 (\u20131)1+2 \n11\n13\n31\n33\na\na\na\na\n+ a13 (\u20131)1+3 \n11\n12\n31\n32\na\na\na\na\n= \n11\n12\n13\n11\n12\n13\n31\n32\n33\na\na\na\na\na\na\na\na\na\n = 0 (since  R1 and R2 are identical)\nSimilarly, we can try for other rows and columns Rationalised 2023-24\n 86\nMATHEMATICS\nExample 11 Find minors and cofactors of the elements of the determinant\n2\n3\n5\n6\n0\n4\n1\n5\n7\n\u2013\n\u2013\nand verify that a11 A31 + a12 A32 + a13 A33= 0\nSolution We have M11 = \n0\n4\n5\n\u20137\n = 0 \u201320 = \u201320;  A11 = (\u20131)1+1 (\u201320) = \u201320\nM12 = \n6\n4\n1\n\u20137\n = \u2013 42 \u2013 4 = \u2013 46;\n A12 = (\u20131)1+2 (\u2013 46) = 46\nM13 = \n6\n0\n1\n5  = 30 \u2013 0 = 30;\n A13 = (\u20131)1+3 (30) = 30\nM21 = \n3\n5\n5\n7\n\u2013\n\u2013  = 21 \u2013 25 = \u2013 4;\n A21 = (\u20131)2+1 (\u2013 4) = 4\nM22 = \n2\n5\n1\n\u20137\n = \u201314 \u2013 5 = \u201319;\n A22 = (\u20131)2+2 (\u201319) = \u201319\nM23 = \n2\n3\n1\n5\n\u2013\n = 10 + 3 = 13;\n A23 = (\u20131)2+3 (13) = \u201313\nM31 = \n3\n5\n0\n4\n\u2013\n = \u201312 \u2013 0 = \u201312;\n A31 = (\u20131)3+1 (\u201312) = \u201312\nM32 = \n2\n5\n6\n4  = 8 \u2013 30 = \u201322;\n A32 = (\u20131)3+2 (\u201322) = 22\nand\nM33 = \n2\n3\n6\n0\n\u2013\n = 0 + 18 = 18;\n A33 = (\u20131)3+3 (18) = 18\nNow\na11 = 2, a12 = \u20133, a13 = 5;  A31 = \u201312, A32 = 22, A33 = 18\nSo\na11 A31 + a12 A32 + a13 A33\n         = 2 (\u201312) + (\u20133) (22) + 5 (18) = \u201324 \u2013 66 + 90 = 0\nRationalised 2023-24\nDETERMINANTS     87\nEXERCISE 4"
    },
    {
        "Chapter": "1",
        "sentence_range": "1527-1530",
        "Text": "ANote  If elements of a row (or column) are multiplied with cofactors of any\nother row (or column), then their sum is zero For example,\n\u2206 = a11 A21 + a12 A22 + a13 A23\n= a11 (\u20131)1+1 \n12\n13\n32\n33\na\na\na\na\n+ a12 (\u20131)1+2 \n11\n13\n31\n33\na\na\na\na\n+ a13 (\u20131)1+3 \n11\n12\n31\n32\na\na\na\na\n= \n11\n12\n13\n11\n12\n13\n31\n32\n33\na\na\na\na\na\na\na\na\na\n = 0 (since  R1 and R2 are identical)\nSimilarly, we can try for other rows and columns Rationalised 2023-24\n 86\nMATHEMATICS\nExample 11 Find minors and cofactors of the elements of the determinant\n2\n3\n5\n6\n0\n4\n1\n5\n7\n\u2013\n\u2013\nand verify that a11 A31 + a12 A32 + a13 A33= 0\nSolution We have M11 = \n0\n4\n5\n\u20137\n = 0 \u201320 = \u201320;  A11 = (\u20131)1+1 (\u201320) = \u201320\nM12 = \n6\n4\n1\n\u20137\n = \u2013 42 \u2013 4 = \u2013 46;\n A12 = (\u20131)1+2 (\u2013 46) = 46\nM13 = \n6\n0\n1\n5  = 30 \u2013 0 = 30;\n A13 = (\u20131)1+3 (30) = 30\nM21 = \n3\n5\n5\n7\n\u2013\n\u2013  = 21 \u2013 25 = \u2013 4;\n A21 = (\u20131)2+1 (\u2013 4) = 4\nM22 = \n2\n5\n1\n\u20137\n = \u201314 \u2013 5 = \u201319;\n A22 = (\u20131)2+2 (\u201319) = \u201319\nM23 = \n2\n3\n1\n5\n\u2013\n = 10 + 3 = 13;\n A23 = (\u20131)2+3 (13) = \u201313\nM31 = \n3\n5\n0\n4\n\u2013\n = \u201312 \u2013 0 = \u201312;\n A31 = (\u20131)3+1 (\u201312) = \u201312\nM32 = \n2\n5\n6\n4  = 8 \u2013 30 = \u201322;\n A32 = (\u20131)3+2 (\u201322) = 22\nand\nM33 = \n2\n3\n6\n0\n\u2013\n = 0 + 18 = 18;\n A33 = (\u20131)3+3 (18) = 18\nNow\na11 = 2, a12 = \u20133, a13 = 5;  A31 = \u201312, A32 = 22, A33 = 18\nSo\na11 A31 + a12 A32 + a13 A33\n         = 2 (\u201312) + (\u20133) (22) + 5 (18) = \u201324 \u2013 66 + 90 = 0\nRationalised 2023-24\nDETERMINANTS     87\nEXERCISE 4 3\nWrite Minors and Cofactors of the elements of following determinants:\n1"
    },
    {
        "Chapter": "1",
        "sentence_range": "1528-1531",
        "Text": "For example,\n\u2206 = a11 A21 + a12 A22 + a13 A23\n= a11 (\u20131)1+1 \n12\n13\n32\n33\na\na\na\na\n+ a12 (\u20131)1+2 \n11\n13\n31\n33\na\na\na\na\n+ a13 (\u20131)1+3 \n11\n12\n31\n32\na\na\na\na\n= \n11\n12\n13\n11\n12\n13\n31\n32\n33\na\na\na\na\na\na\na\na\na\n = 0 (since  R1 and R2 are identical)\nSimilarly, we can try for other rows and columns Rationalised 2023-24\n 86\nMATHEMATICS\nExample 11 Find minors and cofactors of the elements of the determinant\n2\n3\n5\n6\n0\n4\n1\n5\n7\n\u2013\n\u2013\nand verify that a11 A31 + a12 A32 + a13 A33= 0\nSolution We have M11 = \n0\n4\n5\n\u20137\n = 0 \u201320 = \u201320;  A11 = (\u20131)1+1 (\u201320) = \u201320\nM12 = \n6\n4\n1\n\u20137\n = \u2013 42 \u2013 4 = \u2013 46;\n A12 = (\u20131)1+2 (\u2013 46) = 46\nM13 = \n6\n0\n1\n5  = 30 \u2013 0 = 30;\n A13 = (\u20131)1+3 (30) = 30\nM21 = \n3\n5\n5\n7\n\u2013\n\u2013  = 21 \u2013 25 = \u2013 4;\n A21 = (\u20131)2+1 (\u2013 4) = 4\nM22 = \n2\n5\n1\n\u20137\n = \u201314 \u2013 5 = \u201319;\n A22 = (\u20131)2+2 (\u201319) = \u201319\nM23 = \n2\n3\n1\n5\n\u2013\n = 10 + 3 = 13;\n A23 = (\u20131)2+3 (13) = \u201313\nM31 = \n3\n5\n0\n4\n\u2013\n = \u201312 \u2013 0 = \u201312;\n A31 = (\u20131)3+1 (\u201312) = \u201312\nM32 = \n2\n5\n6\n4  = 8 \u2013 30 = \u201322;\n A32 = (\u20131)3+2 (\u201322) = 22\nand\nM33 = \n2\n3\n6\n0\n\u2013\n = 0 + 18 = 18;\n A33 = (\u20131)3+3 (18) = 18\nNow\na11 = 2, a12 = \u20133, a13 = 5;  A31 = \u201312, A32 = 22, A33 = 18\nSo\na11 A31 + a12 A32 + a13 A33\n         = 2 (\u201312) + (\u20133) (22) + 5 (18) = \u201324 \u2013 66 + 90 = 0\nRationalised 2023-24\nDETERMINANTS     87\nEXERCISE 4 3\nWrite Minors and Cofactors of the elements of following determinants:\n1 (i)  2\n4\n0\n3\n\u2013\n(ii)\na\nc\nb\nd\n2"
    },
    {
        "Chapter": "1",
        "sentence_range": "1529-1532",
        "Text": "Rationalised 2023-24\n 86\nMATHEMATICS\nExample 11 Find minors and cofactors of the elements of the determinant\n2\n3\n5\n6\n0\n4\n1\n5\n7\n\u2013\n\u2013\nand verify that a11 A31 + a12 A32 + a13 A33= 0\nSolution We have M11 = \n0\n4\n5\n\u20137\n = 0 \u201320 = \u201320;  A11 = (\u20131)1+1 (\u201320) = \u201320\nM12 = \n6\n4\n1\n\u20137\n = \u2013 42 \u2013 4 = \u2013 46;\n A12 = (\u20131)1+2 (\u2013 46) = 46\nM13 = \n6\n0\n1\n5  = 30 \u2013 0 = 30;\n A13 = (\u20131)1+3 (30) = 30\nM21 = \n3\n5\n5\n7\n\u2013\n\u2013  = 21 \u2013 25 = \u2013 4;\n A21 = (\u20131)2+1 (\u2013 4) = 4\nM22 = \n2\n5\n1\n\u20137\n = \u201314 \u2013 5 = \u201319;\n A22 = (\u20131)2+2 (\u201319) = \u201319\nM23 = \n2\n3\n1\n5\n\u2013\n = 10 + 3 = 13;\n A23 = (\u20131)2+3 (13) = \u201313\nM31 = \n3\n5\n0\n4\n\u2013\n = \u201312 \u2013 0 = \u201312;\n A31 = (\u20131)3+1 (\u201312) = \u201312\nM32 = \n2\n5\n6\n4  = 8 \u2013 30 = \u201322;\n A32 = (\u20131)3+2 (\u201322) = 22\nand\nM33 = \n2\n3\n6\n0\n\u2013\n = 0 + 18 = 18;\n A33 = (\u20131)3+3 (18) = 18\nNow\na11 = 2, a12 = \u20133, a13 = 5;  A31 = \u201312, A32 = 22, A33 = 18\nSo\na11 A31 + a12 A32 + a13 A33\n         = 2 (\u201312) + (\u20133) (22) + 5 (18) = \u201324 \u2013 66 + 90 = 0\nRationalised 2023-24\nDETERMINANTS     87\nEXERCISE 4 3\nWrite Minors and Cofactors of the elements of following determinants:\n1 (i)  2\n4\n0\n3\n\u2013\n(ii)\na\nc\nb\nd\n2 (i)  \n1\n0\n0\n0\n1\n0\n0\n0\n1\n(ii)\n1\n0\n4\n3\n5\n1\n0\n1\n2\n\u2013\n3"
    },
    {
        "Chapter": "1",
        "sentence_range": "1530-1533",
        "Text": "3\nWrite Minors and Cofactors of the elements of following determinants:\n1 (i)  2\n4\n0\n3\n\u2013\n(ii)\na\nc\nb\nd\n2 (i)  \n1\n0\n0\n0\n1\n0\n0\n0\n1\n(ii)\n1\n0\n4\n3\n5\n1\n0\n1\n2\n\u2013\n3 Using Cofactors of elements of second row, evaluate \u2206 = \n5\n3\n8\n2\n0\n1\n1\n2\n3"
    },
    {
        "Chapter": "1",
        "sentence_range": "1531-1534",
        "Text": "(i)  2\n4\n0\n3\n\u2013\n(ii)\na\nc\nb\nd\n2 (i)  \n1\n0\n0\n0\n1\n0\n0\n0\n1\n(ii)\n1\n0\n4\n3\n5\n1\n0\n1\n2\n\u2013\n3 Using Cofactors of elements of second row, evaluate \u2206 = \n5\n3\n8\n2\n0\n1\n1\n2\n3 4"
    },
    {
        "Chapter": "1",
        "sentence_range": "1532-1535",
        "Text": "(i)  \n1\n0\n0\n0\n1\n0\n0\n0\n1\n(ii)\n1\n0\n4\n3\n5\n1\n0\n1\n2\n\u2013\n3 Using Cofactors of elements of second row, evaluate \u2206 = \n5\n3\n8\n2\n0\n1\n1\n2\n3 4 Using Cofactors of elements of third column, evaluate \u2206 = \n1\n1\n1\nx\nyz\ny\nzx\nz\nxy"
    },
    {
        "Chapter": "1",
        "sentence_range": "1533-1536",
        "Text": "Using Cofactors of elements of second row, evaluate \u2206 = \n5\n3\n8\n2\n0\n1\n1\n2\n3 4 Using Cofactors of elements of third column, evaluate \u2206 = \n1\n1\n1\nx\nyz\ny\nzx\nz\nxy 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "1534-1537",
        "Text": "4 Using Cofactors of elements of third column, evaluate \u2206 = \n1\n1\n1\nx\nyz\ny\nzx\nz\nxy 5 If \u2206 = \n11\n12\n13\n21\n22\n23\n31\n32\n33\na\na\na\na\na\na\na\na\na\n and Aij is Cofactors of aij, then value of \u2206 is given by\n(A)\na11 A31+ a12 A32 + a13 A33\n(B)\na11 A11+ a12 A21 + a13 A31\n(C)\na21 A11+ a22 A12 + a23 A13\n(D)\na11 A11+ a21 A21 + a31 A31\n4"
    },
    {
        "Chapter": "1",
        "sentence_range": "1535-1538",
        "Text": "Using Cofactors of elements of third column, evaluate \u2206 = \n1\n1\n1\nx\nyz\ny\nzx\nz\nxy 5 If \u2206 = \n11\n12\n13\n21\n22\n23\n31\n32\n33\na\na\na\na\na\na\na\na\na\n and Aij is Cofactors of aij, then value of \u2206 is given by\n(A)\na11 A31+ a12 A32 + a13 A33\n(B)\na11 A11+ a12 A21 + a13 A31\n(C)\na21 A11+ a22 A12 + a23 A13\n(D)\na11 A11+ a21 A21 + a31 A31\n4 5  Adjoint and Inverse of a Matrix\nIn the previous chapter, we have studied inverse of a matrix"
    },
    {
        "Chapter": "1",
        "sentence_range": "1536-1539",
        "Text": "5 If \u2206 = \n11\n12\n13\n21\n22\n23\n31\n32\n33\na\na\na\na\na\na\na\na\na\n and Aij is Cofactors of aij, then value of \u2206 is given by\n(A)\na11 A31+ a12 A32 + a13 A33\n(B)\na11 A11+ a12 A21 + a13 A31\n(C)\na21 A11+ a22 A12 + a23 A13\n(D)\na11 A11+ a21 A21 + a31 A31\n4 5  Adjoint and Inverse of a Matrix\nIn the previous chapter, we have studied inverse of a matrix In this section, we shall\ndiscuss the condition for existence of inverse of a matrix"
    },
    {
        "Chapter": "1",
        "sentence_range": "1537-1540",
        "Text": "If \u2206 = \n11\n12\n13\n21\n22\n23\n31\n32\n33\na\na\na\na\na\na\na\na\na\n and Aij is Cofactors of aij, then value of \u2206 is given by\n(A)\na11 A31+ a12 A32 + a13 A33\n(B)\na11 A11+ a12 A21 + a13 A31\n(C)\na21 A11+ a22 A12 + a23 A13\n(D)\na11 A11+ a21 A21 + a31 A31\n4 5  Adjoint and Inverse of a Matrix\nIn the previous chapter, we have studied inverse of a matrix In this section, we shall\ndiscuss the condition for existence of inverse of a matrix To find inverse of a matrix A, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "1538-1541",
        "Text": "5  Adjoint and Inverse of a Matrix\nIn the previous chapter, we have studied inverse of a matrix In this section, we shall\ndiscuss the condition for existence of inverse of a matrix To find inverse of a matrix A, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "1539-1542",
        "Text": "In this section, we shall\ndiscuss the condition for existence of inverse of a matrix To find inverse of a matrix A, i e , A\u20131 we shall first define adjoint of a matrix"
    },
    {
        "Chapter": "1",
        "sentence_range": "1540-1543",
        "Text": "To find inverse of a matrix A, i e , A\u20131 we shall first define adjoint of a matrix 4"
    },
    {
        "Chapter": "1",
        "sentence_range": "1541-1544",
        "Text": "e , A\u20131 we shall first define adjoint of a matrix 4 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "1542-1545",
        "Text": ", A\u20131 we shall first define adjoint of a matrix 4 5 1  Adjoint of a matrix\nDefinition 3 The adjoint of a square matrix A = [aij]n \u00d7 n is defined as the transpose of\nthe matrix [Aij]n \u00d7 n, where Aij is the cofactor of the element aij"
    },
    {
        "Chapter": "1",
        "sentence_range": "1543-1546",
        "Text": "4 5 1  Adjoint of a matrix\nDefinition 3 The adjoint of a square matrix A = [aij]n \u00d7 n is defined as the transpose of\nthe matrix [Aij]n \u00d7 n, where Aij is the cofactor of the element aij Adjoint of the matrix A\nis denoted by adj A"
    },
    {
        "Chapter": "1",
        "sentence_range": "1544-1547",
        "Text": "5 1  Adjoint of a matrix\nDefinition 3 The adjoint of a square matrix A = [aij]n \u00d7 n is defined as the transpose of\nthe matrix [Aij]n \u00d7 n, where Aij is the cofactor of the element aij Adjoint of the matrix A\nis denoted by adj A Let\n11\n12\n13\n21\n22\n23\n31\n32\n33\nA =\na\na\na\na\na\na\na\na\na\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nRationalised 2023-24\n 88\nMATHEMATICS\nThen\n11\n12\n13\n21\n22\n23\n31\n32\n33\nA\nA\nA\nA =Transposeof A\nA\nA\nA\nA\nA\nadj\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n11\n21\n31\n12\n22\n32\n13\n23\n33\nA\nA\nA\n= A\nA\nA\nA\nA\nA\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nExample 12 \n2\n3\nFind \n A for A = 1\n4\nadj\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nSolution We have  A11 = 4, A12 = \u20131, A21 = \u20133, A22 = 2\nHence\nadj A =\n11\n21\n12\n22\nA\nA\n4\n\u20133\n =\nA\nA\n\u20131\n2\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nRemark For a square matrix of order 2, given by\nA =\n11\n12\n21\n22\na\na\na\na\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nThe adj A can also be obtained by interchanging a11 and a22 and by changing signs\nof a12 and a21, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "1545-1548",
        "Text": "1  Adjoint of a matrix\nDefinition 3 The adjoint of a square matrix A = [aij]n \u00d7 n is defined as the transpose of\nthe matrix [Aij]n \u00d7 n, where Aij is the cofactor of the element aij Adjoint of the matrix A\nis denoted by adj A Let\n11\n12\n13\n21\n22\n23\n31\n32\n33\nA =\na\na\na\na\na\na\na\na\na\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nRationalised 2023-24\n 88\nMATHEMATICS\nThen\n11\n12\n13\n21\n22\n23\n31\n32\n33\nA\nA\nA\nA =Transposeof A\nA\nA\nA\nA\nA\nadj\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n11\n21\n31\n12\n22\n32\n13\n23\n33\nA\nA\nA\n= A\nA\nA\nA\nA\nA\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nExample 12 \n2\n3\nFind \n A for A = 1\n4\nadj\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nSolution We have  A11 = 4, A12 = \u20131, A21 = \u20133, A22 = 2\nHence\nadj A =\n11\n21\n12\n22\nA\nA\n4\n\u20133\n =\nA\nA\n\u20131\n2\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nRemark For a square matrix of order 2, given by\nA =\n11\n12\n21\n22\na\na\na\na\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nThe adj A can also be obtained by interchanging a11 and a22 and by changing signs\nof a12 and a21, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "1546-1549",
        "Text": "Adjoint of the matrix A\nis denoted by adj A Let\n11\n12\n13\n21\n22\n23\n31\n32\n33\nA =\na\na\na\na\na\na\na\na\na\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nRationalised 2023-24\n 88\nMATHEMATICS\nThen\n11\n12\n13\n21\n22\n23\n31\n32\n33\nA\nA\nA\nA =Transposeof A\nA\nA\nA\nA\nA\nadj\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n11\n21\n31\n12\n22\n32\n13\n23\n33\nA\nA\nA\n= A\nA\nA\nA\nA\nA\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nExample 12 \n2\n3\nFind \n A for A = 1\n4\nadj\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nSolution We have  A11 = 4, A12 = \u20131, A21 = \u20133, A22 = 2\nHence\nadj A =\n11\n21\n12\n22\nA\nA\n4\n\u20133\n =\nA\nA\n\u20131\n2\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nRemark For a square matrix of order 2, given by\nA =\n11\n12\n21\n22\na\na\na\na\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nThe adj A can also be obtained by interchanging a11 and a22 and by changing signs\nof a12 and a21, i e ,\nWe state the following theorem without proof"
    },
    {
        "Chapter": "1",
        "sentence_range": "1547-1550",
        "Text": "Let\n11\n12\n13\n21\n22\n23\n31\n32\n33\nA =\na\na\na\na\na\na\na\na\na\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nRationalised 2023-24\n 88\nMATHEMATICS\nThen\n11\n12\n13\n21\n22\n23\n31\n32\n33\nA\nA\nA\nA =Transposeof A\nA\nA\nA\nA\nA\nadj\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n11\n21\n31\n12\n22\n32\n13\n23\n33\nA\nA\nA\n= A\nA\nA\nA\nA\nA\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nExample 12 \n2\n3\nFind \n A for A = 1\n4\nadj\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nSolution We have  A11 = 4, A12 = \u20131, A21 = \u20133, A22 = 2\nHence\nadj A =\n11\n21\n12\n22\nA\nA\n4\n\u20133\n =\nA\nA\n\u20131\n2\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nRemark For a square matrix of order 2, given by\nA =\n11\n12\n21\n22\na\na\na\na\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nThe adj A can also be obtained by interchanging a11 and a22 and by changing signs\nof a12 and a21, i e ,\nWe state the following theorem without proof Theorem 1 If A be any given square matrix of order n, then\nA(adj A) = (adj A) A = A I ,\nwhere I is the identity matrix of order n\nVerification\nLet\nA = \n11\n12\n13\n21\n22\n23\n31\n32\n33\na\na\na\na\na\na\na\na\na\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n,  then adj  A = \n11\n21\n31\n12\n22\n32\n13\n23\n33\nA\nA\nA\nA\nA\nA\nA\nA\nA\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nSince sum of product of elements of a row (or a column) with corresponding\ncofactors is equal to |A| and otherwise zero, we have\nRationalised 2023-24\nDETERMINANTS     89\nA (adj A) = \nA\n0\n0\n0\nA\n0\n0\n0\nA\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n = A  \n1\n0\n0\n0\n1\n0\n0\n0\n1\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n = A  I\nSimilarly, we can show  (adj A) A = A  I\nHence A (adj A) = (adj A) A = A  I\nDefinition 4 A square matrix A is said to be singular if A  = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "1548-1551",
        "Text": "e ,\nWe state the following theorem without proof Theorem 1 If A be any given square matrix of order n, then\nA(adj A) = (adj A) A = A I ,\nwhere I is the identity matrix of order n\nVerification\nLet\nA = \n11\n12\n13\n21\n22\n23\n31\n32\n33\na\na\na\na\na\na\na\na\na\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n,  then adj  A = \n11\n21\n31\n12\n22\n32\n13\n23\n33\nA\nA\nA\nA\nA\nA\nA\nA\nA\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nSince sum of product of elements of a row (or a column) with corresponding\ncofactors is equal to |A| and otherwise zero, we have\nRationalised 2023-24\nDETERMINANTS     89\nA (adj A) = \nA\n0\n0\n0\nA\n0\n0\n0\nA\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n = A  \n1\n0\n0\n0\n1\n0\n0\n0\n1\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n = A  I\nSimilarly, we can show  (adj A) A = A  I\nHence A (adj A) = (adj A) A = A  I\nDefinition 4 A square matrix A is said to be singular if A  = 0 For example, the determinant of matrix A = \n1\n2\n4\n8\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa  is zero\nHence A is a singular matrix"
    },
    {
        "Chapter": "1",
        "sentence_range": "1549-1552",
        "Text": ",\nWe state the following theorem without proof Theorem 1 If A be any given square matrix of order n, then\nA(adj A) = (adj A) A = A I ,\nwhere I is the identity matrix of order n\nVerification\nLet\nA = \n11\n12\n13\n21\n22\n23\n31\n32\n33\na\na\na\na\na\na\na\na\na\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n,  then adj  A = \n11\n21\n31\n12\n22\n32\n13\n23\n33\nA\nA\nA\nA\nA\nA\nA\nA\nA\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nSince sum of product of elements of a row (or a column) with corresponding\ncofactors is equal to |A| and otherwise zero, we have\nRationalised 2023-24\nDETERMINANTS     89\nA (adj A) = \nA\n0\n0\n0\nA\n0\n0\n0\nA\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n = A  \n1\n0\n0\n0\n1\n0\n0\n0\n1\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n = A  I\nSimilarly, we can show  (adj A) A = A  I\nHence A (adj A) = (adj A) A = A  I\nDefinition 4 A square matrix A is said to be singular if A  = 0 For example, the determinant of matrix A = \n1\n2\n4\n8\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa  is zero\nHence A is a singular matrix Definition 5 A square matrix A is said to be non-singular if A  \u2260 0\nLet\nA = \n1\n2\n3\n4\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb"
    },
    {
        "Chapter": "1",
        "sentence_range": "1550-1553",
        "Text": "Theorem 1 If A be any given square matrix of order n, then\nA(adj A) = (adj A) A = A I ,\nwhere I is the identity matrix of order n\nVerification\nLet\nA = \n11\n12\n13\n21\n22\n23\n31\n32\n33\na\na\na\na\na\na\na\na\na\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n,  then adj  A = \n11\n21\n31\n12\n22\n32\n13\n23\n33\nA\nA\nA\nA\nA\nA\nA\nA\nA\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nSince sum of product of elements of a row (or a column) with corresponding\ncofactors is equal to |A| and otherwise zero, we have\nRationalised 2023-24\nDETERMINANTS     89\nA (adj A) = \nA\n0\n0\n0\nA\n0\n0\n0\nA\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n = A  \n1\n0\n0\n0\n1\n0\n0\n0\n1\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n = A  I\nSimilarly, we can show  (adj A) A = A  I\nHence A (adj A) = (adj A) A = A  I\nDefinition 4 A square matrix A is said to be singular if A  = 0 For example, the determinant of matrix A = \n1\n2\n4\n8\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa  is zero\nHence A is a singular matrix Definition 5 A square matrix A is said to be non-singular if A  \u2260 0\nLet\nA = \n1\n2\n3\n4\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb Then A = \n1\n2\n3\n4  = 4 \u2013 6 = \u2013 2 \u2260 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "1551-1554",
        "Text": "For example, the determinant of matrix A = \n1\n2\n4\n8\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa  is zero\nHence A is a singular matrix Definition 5 A square matrix A is said to be non-singular if A  \u2260 0\nLet\nA = \n1\n2\n3\n4\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb Then A = \n1\n2\n3\n4  = 4 \u2013 6 = \u2013 2 \u2260 0 Hence A is a nonsingular matrix\nWe state the following theorems without proof"
    },
    {
        "Chapter": "1",
        "sentence_range": "1552-1555",
        "Text": "Definition 5 A square matrix A is said to be non-singular if A  \u2260 0\nLet\nA = \n1\n2\n3\n4\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb Then A = \n1\n2\n3\n4  = 4 \u2013 6 = \u2013 2 \u2260 0 Hence A is a nonsingular matrix\nWe state the following theorems without proof Theorem 2 If A and B are nonsingular matrices of the same order, then AB and BA\nare also nonsingular matrices of the same order"
    },
    {
        "Chapter": "1",
        "sentence_range": "1553-1556",
        "Text": "Then A = \n1\n2\n3\n4  = 4 \u2013 6 = \u2013 2 \u2260 0 Hence A is a nonsingular matrix\nWe state the following theorems without proof Theorem 2 If A and B are nonsingular matrices of the same order, then AB and BA\nare also nonsingular matrices of the same order Theorem 3 The determinant of the product of matrices is equal to product of their\nrespective determinants, that is, AB  = A  B , where A and B are square matrices of\nthe same order\nRemark We know that (adj A) A = A  I = \nA\nA\nA\nA\n0\n0\n0\n0\n0\n0\n0\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\uf8fa\n\u2260\n,\nWriting determinants of matrices on both sides, we have\n(\nadjA)A\n =\nA\n0\n0\n0\nA\n0\n0\n0\nA\nRationalised 2023-24\n 90\nMATHEMATICS\ni"
    },
    {
        "Chapter": "1",
        "sentence_range": "1554-1557",
        "Text": "Hence A is a nonsingular matrix\nWe state the following theorems without proof Theorem 2 If A and B are nonsingular matrices of the same order, then AB and BA\nare also nonsingular matrices of the same order Theorem 3 The determinant of the product of matrices is equal to product of their\nrespective determinants, that is, AB  = A  B , where A and B are square matrices of\nthe same order\nRemark We know that (adj A) A = A  I = \nA\nA\nA\nA\n0\n0\n0\n0\n0\n0\n0\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\uf8fa\n\u2260\n,\nWriting determinants of matrices on both sides, we have\n(\nadjA)A\n =\nA\n0\n0\n0\nA\n0\n0\n0\nA\nRationalised 2023-24\n 90\nMATHEMATICS\ni e"
    },
    {
        "Chapter": "1",
        "sentence_range": "1555-1558",
        "Text": "Theorem 2 If A and B are nonsingular matrices of the same order, then AB and BA\nare also nonsingular matrices of the same order Theorem 3 The determinant of the product of matrices is equal to product of their\nrespective determinants, that is, AB  = A  B , where A and B are square matrices of\nthe same order\nRemark We know that (adj A) A = A  I = \nA\nA\nA\nA\n0\n0\n0\n0\n0\n0\n0\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\uf8fa\n\u2260\n,\nWriting determinants of matrices on both sides, we have\n(\nadjA)A\n =\nA\n0\n0\n0\nA\n0\n0\n0\nA\nRationalised 2023-24\n 90\nMATHEMATICS\ni e |(adj A)| |A| =\n3\n1\n0\n0\nA\n0\n1\n0\n0\n0\n1\n(Why"
    },
    {
        "Chapter": "1",
        "sentence_range": "1556-1559",
        "Text": "Theorem 3 The determinant of the product of matrices is equal to product of their\nrespective determinants, that is, AB  = A  B , where A and B are square matrices of\nthe same order\nRemark We know that (adj A) A = A  I = \nA\nA\nA\nA\n0\n0\n0\n0\n0\n0\n0\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\uf8fa\n\u2260\n,\nWriting determinants of matrices on both sides, we have\n(\nadjA)A\n =\nA\n0\n0\n0\nA\n0\n0\n0\nA\nRationalised 2023-24\n 90\nMATHEMATICS\ni e |(adj A)| |A| =\n3\n1\n0\n0\nA\n0\n1\n0\n0\n0\n1\n(Why )\ni"
    },
    {
        "Chapter": "1",
        "sentence_range": "1557-1560",
        "Text": "e |(adj A)| |A| =\n3\n1\n0\n0\nA\n0\n1\n0\n0\n0\n1\n(Why )\ni e"
    },
    {
        "Chapter": "1",
        "sentence_range": "1558-1561",
        "Text": "|(adj A)| |A| =\n3\n1\n0\n0\nA\n0\n1\n0\n0\n0\n1\n(Why )\ni e |(adj A)| |A| = |A|3 (1)\ni"
    },
    {
        "Chapter": "1",
        "sentence_range": "1559-1562",
        "Text": ")\ni e |(adj A)| |A| = |A|3 (1)\ni e"
    },
    {
        "Chapter": "1",
        "sentence_range": "1560-1563",
        "Text": "e |(adj A)| |A| = |A|3 (1)\ni e |(adj A)| = | A |2\nIn general, if A is a square matrix of order n, then |adj(A)| = |A|n \u2013 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "1561-1564",
        "Text": "|(adj A)| |A| = |A|3 (1)\ni e |(adj A)| = | A |2\nIn general, if A is a square matrix of order n, then |adj(A)| = |A|n \u2013 1 Theorem 4 A square matrix A is invertible if and only if A is nonsingular matrix"
    },
    {
        "Chapter": "1",
        "sentence_range": "1562-1565",
        "Text": "e |(adj A)| = | A |2\nIn general, if A is a square matrix of order n, then |adj(A)| = |A|n \u2013 1 Theorem 4 A square matrix A is invertible if and only if A is nonsingular matrix Proof Let A be invertible matrix of order n and I be the identity matrix of order n"
    },
    {
        "Chapter": "1",
        "sentence_range": "1563-1566",
        "Text": "|(adj A)| = | A |2\nIn general, if A is a square matrix of order n, then |adj(A)| = |A|n \u2013 1 Theorem 4 A square matrix A is invertible if and only if A is nonsingular matrix Proof Let A be invertible matrix of order n and I be the identity matrix of order n Then, there exists a square matrix B of order n such that AB = BA = I\nNow\nAB = I"
    },
    {
        "Chapter": "1",
        "sentence_range": "1564-1567",
        "Text": "Theorem 4 A square matrix A is invertible if and only if A is nonsingular matrix Proof Let A be invertible matrix of order n and I be the identity matrix of order n Then, there exists a square matrix B of order n such that AB = BA = I\nNow\nAB = I So AB  = I     or   A  B  = 1    (since I\n1, AB\nA B )\n=\n=\nThis gives\nA \u2260 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "1565-1568",
        "Text": "Proof Let A be invertible matrix of order n and I be the identity matrix of order n Then, there exists a square matrix B of order n such that AB = BA = I\nNow\nAB = I So AB  = I     or   A  B  = 1    (since I\n1, AB\nA B )\n=\n=\nThis gives\nA \u2260 0 Hence A is nonsingular"
    },
    {
        "Chapter": "1",
        "sentence_range": "1566-1569",
        "Text": "Then, there exists a square matrix B of order n such that AB = BA = I\nNow\nAB = I So AB  = I     or   A  B  = 1    (since I\n1, AB\nA B )\n=\n=\nThis gives\nA \u2260 0 Hence A is nonsingular Conversely, let A be nonsingular"
    },
    {
        "Chapter": "1",
        "sentence_range": "1567-1570",
        "Text": "So AB  = I     or   A  B  = 1    (since I\n1, AB\nA B )\n=\n=\nThis gives\nA \u2260 0 Hence A is nonsingular Conversely, let A be nonsingular Then A  \u2260 0\nNow\nA (adj A) = (adj A) A = A I\n(Theorem 1)\nor\nA \n1\n1\nA\nA A\nI\n| A |\n| A |\nadj\nadj\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n=\n=\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\nor\nAB = BA = I, where B = \n1\nA\n| A | adj\nThus\nA is invertible and A\u20131 = \n1\nA\n| A | adj\nExample 13 If A = \n1\n3\n3\n1\n4\n3\n1\n3\n4\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\n, then verify that A adj A = |A| I"
    },
    {
        "Chapter": "1",
        "sentence_range": "1568-1571",
        "Text": "Hence A is nonsingular Conversely, let A be nonsingular Then A  \u2260 0\nNow\nA (adj A) = (adj A) A = A I\n(Theorem 1)\nor\nA \n1\n1\nA\nA A\nI\n| A |\n| A |\nadj\nadj\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n=\n=\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\nor\nAB = BA = I, where B = \n1\nA\n| A | adj\nThus\nA is invertible and A\u20131 = \n1\nA\n| A | adj\nExample 13 If A = \n1\n3\n3\n1\n4\n3\n1\n3\n4\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\n, then verify that A adj A = |A| I Also find A\u20131"
    },
    {
        "Chapter": "1",
        "sentence_range": "1569-1572",
        "Text": "Conversely, let A be nonsingular Then A  \u2260 0\nNow\nA (adj A) = (adj A) A = A I\n(Theorem 1)\nor\nA \n1\n1\nA\nA A\nI\n| A |\n| A |\nadj\nadj\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n=\n=\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\nor\nAB = BA = I, where B = \n1\nA\n| A | adj\nThus\nA is invertible and A\u20131 = \n1\nA\n| A | adj\nExample 13 If A = \n1\n3\n3\n1\n4\n3\n1\n3\n4\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\n, then verify that A adj A = |A| I Also find A\u20131 Solution We have A  = 1 (16 \u2013 9) \u20133 (4 \u2013 3) + 3 (3 \u2013 4) = 1 \u2260 0\nNow A11 = 7,  A12 = \u20131, A13 = \u20131, A21 = \u20133, A22 = 1,A23 = 0, A31 = \u20133, A32 = 0,\nA33 = 1\nTherefore\nadj A =\n7\n3\n3\n1\n1\n0\n1\n0\n1\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\u2212\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\u2212\n\uf8fb\nRationalised 2023-24\nDETERMINANTS     91\nNow\nA (adj A) =\n1\n3\n3\n7\n3\n3\n1\n4\n3\n1\n1\n0\n1\n3\n4\n1\n0\n1\n\u2212\n\u2212\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\u2212\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n=\n7\n3\n3\n3\n3\n0\n3\n0\n3\n7\n4\n3\n3\n4\n0\n3\n0\n3\n7\n3\n4\n3\n3\n0\n3\n0\n4\n\u2212\n\u2212\n\u2212 +\n+\n\u2212 +\n+\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\u2212 +\n+\n\u2212 +\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\u2212 +\n+\n\u2212 +\n+\n\uf8f0\n\uf8fb\n=\n1\n0\n0\n0\n1\n0\n0\n0\n1\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n = (1)   \n1\n0\n0\n0\n1\n0\n0\n0\n1\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n = A"
    },
    {
        "Chapter": "1",
        "sentence_range": "1570-1573",
        "Text": "Then A  \u2260 0\nNow\nA (adj A) = (adj A) A = A I\n(Theorem 1)\nor\nA \n1\n1\nA\nA A\nI\n| A |\n| A |\nadj\nadj\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n=\n=\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\nor\nAB = BA = I, where B = \n1\nA\n| A | adj\nThus\nA is invertible and A\u20131 = \n1\nA\n| A | adj\nExample 13 If A = \n1\n3\n3\n1\n4\n3\n1\n3\n4\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\n, then verify that A adj A = |A| I Also find A\u20131 Solution We have A  = 1 (16 \u2013 9) \u20133 (4 \u2013 3) + 3 (3 \u2013 4) = 1 \u2260 0\nNow A11 = 7,  A12 = \u20131, A13 = \u20131, A21 = \u20133, A22 = 1,A23 = 0, A31 = \u20133, A32 = 0,\nA33 = 1\nTherefore\nadj A =\n7\n3\n3\n1\n1\n0\n1\n0\n1\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\u2212\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\u2212\n\uf8fb\nRationalised 2023-24\nDETERMINANTS     91\nNow\nA (adj A) =\n1\n3\n3\n7\n3\n3\n1\n4\n3\n1\n1\n0\n1\n3\n4\n1\n0\n1\n\u2212\n\u2212\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\u2212\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n=\n7\n3\n3\n3\n3\n0\n3\n0\n3\n7\n4\n3\n3\n4\n0\n3\n0\n3\n7\n3\n4\n3\n3\n0\n3\n0\n4\n\u2212\n\u2212\n\u2212 +\n+\n\u2212 +\n+\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\u2212 +\n+\n\u2212 +\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\u2212 +\n+\n\u2212 +\n+\n\uf8f0\n\uf8fb\n=\n1\n0\n0\n0\n1\n0\n0\n0\n1\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n = (1)   \n1\n0\n0\n0\n1\n0\n0\n0\n1\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n = A I\nAlso\nA\u20131\n1\nA\nA\na d j\n=\n =\n7\n3\n3\n1\n1\n1\n0\n1\n1\n0\n1\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\u2212\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\u2212\n\uf8fb\n = \n7\n3\n3\n1\n1\n0\n1\n0\n1\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\u2212\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\u2212\n\uf8fb\nExample 14 If A = \n2\n3\n1\n2\nand B\n1\n4\n1\n\u22123\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n, then verify that (AB)\u20131 = B\u20131A\u20131"
    },
    {
        "Chapter": "1",
        "sentence_range": "1571-1574",
        "Text": "Also find A\u20131 Solution We have A  = 1 (16 \u2013 9) \u20133 (4 \u2013 3) + 3 (3 \u2013 4) = 1 \u2260 0\nNow A11 = 7,  A12 = \u20131, A13 = \u20131, A21 = \u20133, A22 = 1,A23 = 0, A31 = \u20133, A32 = 0,\nA33 = 1\nTherefore\nadj A =\n7\n3\n3\n1\n1\n0\n1\n0\n1\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\u2212\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\u2212\n\uf8fb\nRationalised 2023-24\nDETERMINANTS     91\nNow\nA (adj A) =\n1\n3\n3\n7\n3\n3\n1\n4\n3\n1\n1\n0\n1\n3\n4\n1\n0\n1\n\u2212\n\u2212\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\u2212\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n=\n7\n3\n3\n3\n3\n0\n3\n0\n3\n7\n4\n3\n3\n4\n0\n3\n0\n3\n7\n3\n4\n3\n3\n0\n3\n0\n4\n\u2212\n\u2212\n\u2212 +\n+\n\u2212 +\n+\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\u2212 +\n+\n\u2212 +\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\u2212 +\n+\n\u2212 +\n+\n\uf8f0\n\uf8fb\n=\n1\n0\n0\n0\n1\n0\n0\n0\n1\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n = (1)   \n1\n0\n0\n0\n1\n0\n0\n0\n1\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n = A I\nAlso\nA\u20131\n1\nA\nA\na d j\n=\n =\n7\n3\n3\n1\n1\n1\n0\n1\n1\n0\n1\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\u2212\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\u2212\n\uf8fb\n = \n7\n3\n3\n1\n1\n0\n1\n0\n1\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\u2212\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\u2212\n\uf8fb\nExample 14 If A = \n2\n3\n1\n2\nand B\n1\n4\n1\n\u22123\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n, then verify that (AB)\u20131 = B\u20131A\u20131 Solution We have AB = \n2\n3\n1\n2\n1\n5\n1\n4\n1\n3\n5\n14\n\u2212\n\u2212\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\u2212\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nSince,\nAB  = \u201311 \u2260 0, (AB)\u20131 exists and is given by\n(AB)\u20131 = \n14\n5\n1\n1\n(AB)\n5\n1\nAB\n11\nadj\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n=\u2212\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8f0\n\uf8fb\n \n14\n5\n1\n5\n1\n11\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nFurther, A  = \u201311 \u2260 0 and B  = 1 \u2260 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "1572-1575",
        "Text": "Solution We have A  = 1 (16 \u2013 9) \u20133 (4 \u2013 3) + 3 (3 \u2013 4) = 1 \u2260 0\nNow A11 = 7,  A12 = \u20131, A13 = \u20131, A21 = \u20133, A22 = 1,A23 = 0, A31 = \u20133, A32 = 0,\nA33 = 1\nTherefore\nadj A =\n7\n3\n3\n1\n1\n0\n1\n0\n1\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\u2212\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\u2212\n\uf8fb\nRationalised 2023-24\nDETERMINANTS     91\nNow\nA (adj A) =\n1\n3\n3\n7\n3\n3\n1\n4\n3\n1\n1\n0\n1\n3\n4\n1\n0\n1\n\u2212\n\u2212\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\u2212\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n=\n7\n3\n3\n3\n3\n0\n3\n0\n3\n7\n4\n3\n3\n4\n0\n3\n0\n3\n7\n3\n4\n3\n3\n0\n3\n0\n4\n\u2212\n\u2212\n\u2212 +\n+\n\u2212 +\n+\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\u2212 +\n+\n\u2212 +\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\u2212 +\n+\n\u2212 +\n+\n\uf8f0\n\uf8fb\n=\n1\n0\n0\n0\n1\n0\n0\n0\n1\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n = (1)   \n1\n0\n0\n0\n1\n0\n0\n0\n1\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n = A I\nAlso\nA\u20131\n1\nA\nA\na d j\n=\n =\n7\n3\n3\n1\n1\n1\n0\n1\n1\n0\n1\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\u2212\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\u2212\n\uf8fb\n = \n7\n3\n3\n1\n1\n0\n1\n0\n1\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\u2212\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\u2212\n\uf8fb\nExample 14 If A = \n2\n3\n1\n2\nand B\n1\n4\n1\n\u22123\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n, then verify that (AB)\u20131 = B\u20131A\u20131 Solution We have AB = \n2\n3\n1\n2\n1\n5\n1\n4\n1\n3\n5\n14\n\u2212\n\u2212\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\u2212\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nSince,\nAB  = \u201311 \u2260 0, (AB)\u20131 exists and is given by\n(AB)\u20131 = \n14\n5\n1\n1\n(AB)\n5\n1\nAB\n11\nadj\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n=\u2212\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8f0\n\uf8fb\n \n14\n5\n1\n5\n1\n11\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nFurther, A  = \u201311 \u2260 0 and B  = 1 \u2260 0 Therefore, A\u20131 and B\u20131 both exist and are given by\nA\u20131 = \u2212\n\u2212\n\u2212\n\u2212\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa\n= \uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa\n\u2212\n1\n11\n4\n3\n1\n2\n3\n2\n1\n1\n1\n,B\nTherefore\nB A\n\u2212\n\u2212 = \u2212\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa\n\u2212\n\u2212\n\u2212\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa\n1\n1\n1\n11\n3\n2\n1\n1\n4\n3\n1\n2\n = \u2212\n\u2212\n\u2212\n\u2212\n\u2212\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa\n1\n11\n14\n5\n5\n1  \n14\n5\n1\n5\n1\n11\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nHence (AB)\u20131 = B\u20131 A\u20131\nRationalised 2023-24\n 92\nMATHEMATICS\nExample 15 Show that the matrix A = \n2\n3\n1\n2\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa  satisfies the equation A2 \u2013 4A + I = O,\nwhere I is 2 \u00d7 2 identity matrix and O is 2 \u00d7 2 zero matrix"
    },
    {
        "Chapter": "1",
        "sentence_range": "1573-1576",
        "Text": "I\nAlso\nA\u20131\n1\nA\nA\na d j\n=\n =\n7\n3\n3\n1\n1\n1\n0\n1\n1\n0\n1\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\u2212\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\u2212\n\uf8fb\n = \n7\n3\n3\n1\n1\n0\n1\n0\n1\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\u2212\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\u2212\n\uf8fb\nExample 14 If A = \n2\n3\n1\n2\nand B\n1\n4\n1\n\u22123\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n, then verify that (AB)\u20131 = B\u20131A\u20131 Solution We have AB = \n2\n3\n1\n2\n1\n5\n1\n4\n1\n3\n5\n14\n\u2212\n\u2212\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\u2212\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nSince,\nAB  = \u201311 \u2260 0, (AB)\u20131 exists and is given by\n(AB)\u20131 = \n14\n5\n1\n1\n(AB)\n5\n1\nAB\n11\nadj\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n=\u2212\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8f0\n\uf8fb\n \n14\n5\n1\n5\n1\n11\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nFurther, A  = \u201311 \u2260 0 and B  = 1 \u2260 0 Therefore, A\u20131 and B\u20131 both exist and are given by\nA\u20131 = \u2212\n\u2212\n\u2212\n\u2212\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa\n= \uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa\n\u2212\n1\n11\n4\n3\n1\n2\n3\n2\n1\n1\n1\n,B\nTherefore\nB A\n\u2212\n\u2212 = \u2212\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa\n\u2212\n\u2212\n\u2212\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa\n1\n1\n1\n11\n3\n2\n1\n1\n4\n3\n1\n2\n = \u2212\n\u2212\n\u2212\n\u2212\n\u2212\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa\n1\n11\n14\n5\n5\n1  \n14\n5\n1\n5\n1\n11\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nHence (AB)\u20131 = B\u20131 A\u20131\nRationalised 2023-24\n 92\nMATHEMATICS\nExample 15 Show that the matrix A = \n2\n3\n1\n2\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa  satisfies the equation A2 \u2013 4A + I = O,\nwhere I is 2 \u00d7 2 identity matrix and O is 2 \u00d7 2 zero matrix Using this equation, find A\u20131"
    },
    {
        "Chapter": "1",
        "sentence_range": "1574-1577",
        "Text": "Solution We have AB = \n2\n3\n1\n2\n1\n5\n1\n4\n1\n3\n5\n14\n\u2212\n\u2212\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\u2212\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nSince,\nAB  = \u201311 \u2260 0, (AB)\u20131 exists and is given by\n(AB)\u20131 = \n14\n5\n1\n1\n(AB)\n5\n1\nAB\n11\nadj\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n=\u2212\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\uf8f0\n\uf8fb\n \n14\n5\n1\n5\n1\n11\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nFurther, A  = \u201311 \u2260 0 and B  = 1 \u2260 0 Therefore, A\u20131 and B\u20131 both exist and are given by\nA\u20131 = \u2212\n\u2212\n\u2212\n\u2212\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa\n= \uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa\n\u2212\n1\n11\n4\n3\n1\n2\n3\n2\n1\n1\n1\n,B\nTherefore\nB A\n\u2212\n\u2212 = \u2212\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa\n\u2212\n\u2212\n\u2212\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa\n1\n1\n1\n11\n3\n2\n1\n1\n4\n3\n1\n2\n = \u2212\n\u2212\n\u2212\n\u2212\n\u2212\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa\n1\n11\n14\n5\n5\n1  \n14\n5\n1\n5\n1\n11\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nHence (AB)\u20131 = B\u20131 A\u20131\nRationalised 2023-24\n 92\nMATHEMATICS\nExample 15 Show that the matrix A = \n2\n3\n1\n2\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa  satisfies the equation A2 \u2013 4A + I = O,\nwhere I is 2 \u00d7 2 identity matrix and O is 2 \u00d7 2 zero matrix Using this equation, find A\u20131 Solution We have \n2\n2\n3\n2\n3\n7\n12\nA\nA"
    },
    {
        "Chapter": "1",
        "sentence_range": "1575-1578",
        "Text": "Therefore, A\u20131 and B\u20131 both exist and are given by\nA\u20131 = \u2212\n\u2212\n\u2212\n\u2212\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa\n= \uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa\n\u2212\n1\n11\n4\n3\n1\n2\n3\n2\n1\n1\n1\n,B\nTherefore\nB A\n\u2212\n\u2212 = \u2212\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa\n\u2212\n\u2212\n\u2212\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa\n1\n1\n1\n11\n3\n2\n1\n1\n4\n3\n1\n2\n = \u2212\n\u2212\n\u2212\n\u2212\n\u2212\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa\n1\n11\n14\n5\n5\n1  \n14\n5\n1\n5\n1\n11\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nHence (AB)\u20131 = B\u20131 A\u20131\nRationalised 2023-24\n 92\nMATHEMATICS\nExample 15 Show that the matrix A = \n2\n3\n1\n2\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa  satisfies the equation A2 \u2013 4A + I = O,\nwhere I is 2 \u00d7 2 identity matrix and O is 2 \u00d7 2 zero matrix Using this equation, find A\u20131 Solution We have \n2\n2\n3\n2\n3\n7\n12\nA\nA A\n1\n2\n1\n2\n4\n7\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n=\n=\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nHence\n2\n7\n12\n8\n12\n1\n0\nA\n4A\nI\n4\n7\n4\n8\n0\n1\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\u2212\n+ =\n\u2212\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n0\n0\nO\n0\n0\n\uf8ee\n\uf8f9\n=\n=\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nNow\nA2 \u2013 4A + I = O\nTherefore\nA A \u2013 4A = \u2013 I\nor\nA  A (A\u20131) \u2013 4 A A\u20131 = \u2013 I A\u20131 (Post multiplying by A\u20131 because |A| \u2260 0)\nor\nA (A A\u20131) \u2013 4I = \u2013 A\u20131\nor\nAI \u2013 4I = \u2013 A\u20131\nor\nA\u20131  =  4I \u2013 A  = \n4\n0\n2\n3\n2\n3\n0\n4\n1\n2\n1\n2\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\u2212\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nHence\n1\n2\n3\nA\n1\n2\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\uf8f0\u2212\n\uf8fb\nEXERCISE 4"
    },
    {
        "Chapter": "1",
        "sentence_range": "1576-1579",
        "Text": "Using this equation, find A\u20131 Solution We have \n2\n2\n3\n2\n3\n7\n12\nA\nA A\n1\n2\n1\n2\n4\n7\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n=\n=\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nHence\n2\n7\n12\n8\n12\n1\n0\nA\n4A\nI\n4\n7\n4\n8\n0\n1\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\u2212\n+ =\n\u2212\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n0\n0\nO\n0\n0\n\uf8ee\n\uf8f9\n=\n=\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nNow\nA2 \u2013 4A + I = O\nTherefore\nA A \u2013 4A = \u2013 I\nor\nA  A (A\u20131) \u2013 4 A A\u20131 = \u2013 I A\u20131 (Post multiplying by A\u20131 because |A| \u2260 0)\nor\nA (A A\u20131) \u2013 4I = \u2013 A\u20131\nor\nAI \u2013 4I = \u2013 A\u20131\nor\nA\u20131  =  4I \u2013 A  = \n4\n0\n2\n3\n2\n3\n0\n4\n1\n2\n1\n2\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\u2212\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nHence\n1\n2\n3\nA\n1\n2\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\uf8f0\u2212\n\uf8fb\nEXERCISE 4 4\nFind adjoint of each of the matrices in Exercises 1 and 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "1577-1580",
        "Text": "Solution We have \n2\n2\n3\n2\n3\n7\n12\nA\nA A\n1\n2\n1\n2\n4\n7\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n=\n=\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nHence\n2\n7\n12\n8\n12\n1\n0\nA\n4A\nI\n4\n7\n4\n8\n0\n1\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\u2212\n+ =\n\u2212\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n0\n0\nO\n0\n0\n\uf8ee\n\uf8f9\n=\n=\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nNow\nA2 \u2013 4A + I = O\nTherefore\nA A \u2013 4A = \u2013 I\nor\nA  A (A\u20131) \u2013 4 A A\u20131 = \u2013 I A\u20131 (Post multiplying by A\u20131 because |A| \u2260 0)\nor\nA (A A\u20131) \u2013 4I = \u2013 A\u20131\nor\nAI \u2013 4I = \u2013 A\u20131\nor\nA\u20131  =  4I \u2013 A  = \n4\n0\n2\n3\n2\n3\n0\n4\n1\n2\n1\n2\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\u2212\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nHence\n1\n2\n3\nA\n1\n2\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\uf8f0\u2212\n\uf8fb\nEXERCISE 4 4\nFind adjoint of each of the matrices in Exercises 1 and 2 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "1578-1581",
        "Text": "A\n1\n2\n1\n2\n4\n7\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n=\n=\n=\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nHence\n2\n7\n12\n8\n12\n1\n0\nA\n4A\nI\n4\n7\n4\n8\n0\n1\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\u2212\n+ =\n\u2212\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n0\n0\nO\n0\n0\n\uf8ee\n\uf8f9\n=\n=\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nNow\nA2 \u2013 4A + I = O\nTherefore\nA A \u2013 4A = \u2013 I\nor\nA  A (A\u20131) \u2013 4 A A\u20131 = \u2013 I A\u20131 (Post multiplying by A\u20131 because |A| \u2260 0)\nor\nA (A A\u20131) \u2013 4I = \u2013 A\u20131\nor\nAI \u2013 4I = \u2013 A\u20131\nor\nA\u20131  =  4I \u2013 A  = \n4\n0\n2\n3\n2\n3\n0\n4\n1\n2\n1\n2\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\u2212\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nHence\n1\n2\n3\nA\n1\n2\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n= \uf8ef\n\uf8fa\n\uf8f0\u2212\n\uf8fb\nEXERCISE 4 4\nFind adjoint of each of the matrices in Exercises 1 and 2 1 1\n2\n3\n4\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa\n2"
    },
    {
        "Chapter": "1",
        "sentence_range": "1579-1582",
        "Text": "4\nFind adjoint of each of the matrices in Exercises 1 and 2 1 1\n2\n3\n4\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa\n2 1\n1\n2\n2\n3\n5\n2\n0\n1\n\u2212\n\u2212\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\nVerify A (adj A) = (adj A) A = |A| I in Exercises 3 and 4\n3"
    },
    {
        "Chapter": "1",
        "sentence_range": "1580-1583",
        "Text": "1 1\n2\n3\n4\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa\n2 1\n1\n2\n2\n3\n5\n2\n0\n1\n\u2212\n\u2212\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\nVerify A (adj A) = (adj A) A = |A| I in Exercises 3 and 4\n3 2\n3\n4\n6\n\u2212\n\u2212\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa\n4"
    },
    {
        "Chapter": "1",
        "sentence_range": "1581-1584",
        "Text": "1\n2\n3\n4\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa\n2 1\n1\n2\n2\n3\n5\n2\n0\n1\n\u2212\n\u2212\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\nVerify A (adj A) = (adj A) A = |A| I in Exercises 3 and 4\n3 2\n3\n4\n6\n\u2212\n\u2212\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa\n4 1\n1\n2\n3\n0\n2\n1\n0\n3\n\u2212\n\u2212\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\nFind the inverse of each of the matrices (if it exists) given in Exercises 5 to 11"
    },
    {
        "Chapter": "1",
        "sentence_range": "1582-1585",
        "Text": "1\n1\n2\n2\n3\n5\n2\n0\n1\n\u2212\n\u2212\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\nVerify A (adj A) = (adj A) A = |A| I in Exercises 3 and 4\n3 2\n3\n4\n6\n\u2212\n\u2212\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa\n4 1\n1\n2\n3\n0\n2\n1\n0\n3\n\u2212\n\u2212\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\nFind the inverse of each of the matrices (if it exists) given in Exercises 5 to 11 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "1583-1586",
        "Text": "2\n3\n4\n6\n\u2212\n\u2212\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa\n4 1\n1\n2\n3\n0\n2\n1\n0\n3\n\u2212\n\u2212\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\nFind the inverse of each of the matrices (if it exists) given in Exercises 5 to 11 5 2\n2\n4\n\u22123\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa\n6"
    },
    {
        "Chapter": "1",
        "sentence_range": "1584-1587",
        "Text": "1\n1\n2\n3\n0\n2\n1\n0\n3\n\u2212\n\u2212\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\nFind the inverse of each of the matrices (if it exists) given in Exercises 5 to 11 5 2\n2\n4\n\u22123\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa\n6 \u2212\u2212\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa\n1\n5\n3\n2\n7"
    },
    {
        "Chapter": "1",
        "sentence_range": "1585-1588",
        "Text": "5 2\n2\n4\n\u22123\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa\n6 \u2212\u2212\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa\n1\n5\n3\n2\n7 1\n2\n3\n0\n2\n4\n0\n0\n5\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\nRationalised 2023-24\nDETERMINANTS     93\n8"
    },
    {
        "Chapter": "1",
        "sentence_range": "1586-1589",
        "Text": "2\n2\n4\n\u22123\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa\n6 \u2212\u2212\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa\n1\n5\n3\n2\n7 1\n2\n3\n0\n2\n4\n0\n0\n5\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\nRationalised 2023-24\nDETERMINANTS     93\n8 1\n0\n0\n3\n3\n0\n5\n2\n1\n\u2212\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\n9"
    },
    {
        "Chapter": "1",
        "sentence_range": "1587-1590",
        "Text": "\u2212\u2212\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa\n1\n5\n3\n2\n7 1\n2\n3\n0\n2\n4\n0\n0\n5\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\nRationalised 2023-24\nDETERMINANTS     93\n8 1\n0\n0\n3\n3\n0\n5\n2\n1\n\u2212\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\n9 2\n1\n3\n4\n1\n0\n7\n2\n1\n\u2212\n\u2212\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\n10"
    },
    {
        "Chapter": "1",
        "sentence_range": "1588-1591",
        "Text": "1\n2\n3\n0\n2\n4\n0\n0\n5\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\nRationalised 2023-24\nDETERMINANTS     93\n8 1\n0\n0\n3\n3\n0\n5\n2\n1\n\u2212\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\n9 2\n1\n3\n4\n1\n0\n7\n2\n1\n\u2212\n\u2212\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\n10 1\n1\n2\n0\n2\n3\n3\n2\n4\n\u2212\n\u2212\n\u2212\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\n11"
    },
    {
        "Chapter": "1",
        "sentence_range": "1589-1592",
        "Text": "1\n0\n0\n3\n3\n0\n5\n2\n1\n\u2212\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\n9 2\n1\n3\n4\n1\n0\n7\n2\n1\n\u2212\n\u2212\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\n10 1\n1\n2\n0\n2\n3\n3\n2\n4\n\u2212\n\u2212\n\u2212\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\n11 1\n0\n0\n0\ncos\nsin\n0\nsin\ncos\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u03b1\n\u03b1\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u03b1\n\u2212\n\u03b1\n\uf8f0\n\uf8fb\n12"
    },
    {
        "Chapter": "1",
        "sentence_range": "1590-1593",
        "Text": "2\n1\n3\n4\n1\n0\n7\n2\n1\n\u2212\n\u2212\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\n10 1\n1\n2\n0\n2\n3\n3\n2\n4\n\u2212\n\u2212\n\u2212\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\n11 1\n0\n0\n0\ncos\nsin\n0\nsin\ncos\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u03b1\n\u03b1\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u03b1\n\u2212\n\u03b1\n\uf8f0\n\uf8fb\n12 Let A = \n3\n7\n2\n5\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa  and B = 6\n8\n7\n9\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa"
    },
    {
        "Chapter": "1",
        "sentence_range": "1591-1594",
        "Text": "1\n1\n2\n0\n2\n3\n3\n2\n4\n\u2212\n\u2212\n\u2212\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\n11 1\n0\n0\n0\ncos\nsin\n0\nsin\ncos\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u03b1\n\u03b1\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u03b1\n\u2212\n\u03b1\n\uf8f0\n\uf8fb\n12 Let A = \n3\n7\n2\n5\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa  and B = 6\n8\n7\n9\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa Verify that (AB)\u20131 = B\u20131 A\u20131"
    },
    {
        "Chapter": "1",
        "sentence_range": "1592-1595",
        "Text": "1\n0\n0\n0\ncos\nsin\n0\nsin\ncos\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u03b1\n\u03b1\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u03b1\n\u2212\n\u03b1\n\uf8f0\n\uf8fb\n12 Let A = \n3\n7\n2\n5\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa  and B = 6\n8\n7\n9\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa Verify that (AB)\u20131 = B\u20131 A\u20131 13"
    },
    {
        "Chapter": "1",
        "sentence_range": "1593-1596",
        "Text": "Let A = \n3\n7\n2\n5\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa  and B = 6\n8\n7\n9\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa Verify that (AB)\u20131 = B\u20131 A\u20131 13 If A = \n3\n1\n1\n2\n\u2212\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa , show that A2 \u2013 5A + 7I = O"
    },
    {
        "Chapter": "1",
        "sentence_range": "1594-1597",
        "Text": "Verify that (AB)\u20131 = B\u20131 A\u20131 13 If A = \n3\n1\n1\n2\n\u2212\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa , show that A2 \u2013 5A + 7I = O Hence find A\u20131"
    },
    {
        "Chapter": "1",
        "sentence_range": "1595-1598",
        "Text": "13 If A = \n3\n1\n1\n2\n\u2212\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa , show that A2 \u2013 5A + 7I = O Hence find A\u20131 14"
    },
    {
        "Chapter": "1",
        "sentence_range": "1596-1599",
        "Text": "If A = \n3\n1\n1\n2\n\u2212\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa , show that A2 \u2013 5A + 7I = O Hence find A\u20131 14 For the matrix A = \n3\n2\n1\n1\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa , find the numbers a and b such that A2 + aA + bI = O"
    },
    {
        "Chapter": "1",
        "sentence_range": "1597-1600",
        "Text": "Hence find A\u20131 14 For the matrix A = \n3\n2\n1\n1\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa , find the numbers a and b such that A2 + aA + bI = O 15"
    },
    {
        "Chapter": "1",
        "sentence_range": "1598-1601",
        "Text": "14 For the matrix A = \n3\n2\n1\n1\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa , find the numbers a and b such that A2 + aA + bI = O 15 For the matrix A = \n1\n1\n1\n1\n2\n3\n2\n1\n\u22123\n\u2212\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\nShow that  A3\u2013 6A2 + 5A + 11 I = O"
    },
    {
        "Chapter": "1",
        "sentence_range": "1599-1602",
        "Text": "For the matrix A = \n3\n2\n1\n1\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa , find the numbers a and b such that A2 + aA + bI = O 15 For the matrix A = \n1\n1\n1\n1\n2\n3\n2\n1\n\u22123\n\u2212\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\nShow that  A3\u2013 6A2 + 5A + 11 I = O Hence, find A\u20131"
    },
    {
        "Chapter": "1",
        "sentence_range": "1600-1603",
        "Text": "15 For the matrix A = \n1\n1\n1\n1\n2\n3\n2\n1\n\u22123\n\u2212\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\nShow that  A3\u2013 6A2 + 5A + 11 I = O Hence, find A\u20131 16"
    },
    {
        "Chapter": "1",
        "sentence_range": "1601-1604",
        "Text": "For the matrix A = \n1\n1\n1\n1\n2\n3\n2\n1\n\u22123\n\u2212\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\nShow that  A3\u2013 6A2 + 5A + 11 I = O Hence, find A\u20131 16 If A = \n2\n1\n1\n1\n2\n1\n1\n1\n2\n\u2212\n\u2212\n\u2212\n\u2212\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\nVerify that A3 \u2013 6A2 + 9A \u2013 4I = O and hence find A\u20131\n17"
    },
    {
        "Chapter": "1",
        "sentence_range": "1602-1605",
        "Text": "Hence, find A\u20131 16 If A = \n2\n1\n1\n1\n2\n1\n1\n1\n2\n\u2212\n\u2212\n\u2212\n\u2212\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\nVerify that A3 \u2013 6A2 + 9A \u2013 4I = O and hence find A\u20131\n17 Let A be a nonsingular square matrix of order 3 \u00d7 3"
    },
    {
        "Chapter": "1",
        "sentence_range": "1603-1606",
        "Text": "16 If A = \n2\n1\n1\n1\n2\n1\n1\n1\n2\n\u2212\n\u2212\n\u2212\n\u2212\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\nVerify that A3 \u2013 6A2 + 9A \u2013 4I = O and hence find A\u20131\n17 Let A be a nonsingular square matrix of order 3 \u00d7 3 Then |adj A| is equal to\n(A) | A |\n(B) | A |2\n(C) | A |3\n(D) 3|A|\n18"
    },
    {
        "Chapter": "1",
        "sentence_range": "1604-1607",
        "Text": "If A = \n2\n1\n1\n1\n2\n1\n1\n1\n2\n\u2212\n\u2212\n\u2212\n\u2212\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\nVerify that A3 \u2013 6A2 + 9A \u2013 4I = O and hence find A\u20131\n17 Let A be a nonsingular square matrix of order 3 \u00d7 3 Then |adj A| is equal to\n(A) | A |\n(B) | A |2\n(C) | A |3\n(D) 3|A|\n18 If A is an invertible matrix of order 2, then det (A\u20131) is equal to\n(A) det (A)\n(B)\n1\ndet (A)\n(C) 1\n(D) 0\n4"
    },
    {
        "Chapter": "1",
        "sentence_range": "1605-1608",
        "Text": "Let A be a nonsingular square matrix of order 3 \u00d7 3 Then |adj A| is equal to\n(A) | A |\n(B) | A |2\n(C) | A |3\n(D) 3|A|\n18 If A is an invertible matrix of order 2, then det (A\u20131) is equal to\n(A) det (A)\n(B)\n1\ndet (A)\n(C) 1\n(D) 0\n4 6  Applications of Determinants and Matrices\nIn this section, we shall discuss application of determinants and matrices for solving the\nsystem of linear equations in two or three variables and for checking the consistency of\nthe system of linear equations"
    },
    {
        "Chapter": "1",
        "sentence_range": "1606-1609",
        "Text": "Then |adj A| is equal to\n(A) | A |\n(B) | A |2\n(C) | A |3\n(D) 3|A|\n18 If A is an invertible matrix of order 2, then det (A\u20131) is equal to\n(A) det (A)\n(B)\n1\ndet (A)\n(C) 1\n(D) 0\n4 6  Applications of Determinants and Matrices\nIn this section, we shall discuss application of determinants and matrices for solving the\nsystem of linear equations in two or three variables and for checking the consistency of\nthe system of linear equations Rationalised 2023-24\n 94\nMATHEMATICS\nConsistent system A system of equations is said to be consistent if its solution (one\nor more) exists"
    },
    {
        "Chapter": "1",
        "sentence_range": "1607-1610",
        "Text": "If A is an invertible matrix of order 2, then det (A\u20131) is equal to\n(A) det (A)\n(B)\n1\ndet (A)\n(C) 1\n(D) 0\n4 6  Applications of Determinants and Matrices\nIn this section, we shall discuss application of determinants and matrices for solving the\nsystem of linear equations in two or three variables and for checking the consistency of\nthe system of linear equations Rationalised 2023-24\n 94\nMATHEMATICS\nConsistent system A system of equations is said to be consistent if its solution (one\nor more) exists Inconsistent system A system of equations is said to be inconsistent if its solution\ndoes not exist"
    },
    {
        "Chapter": "1",
        "sentence_range": "1608-1611",
        "Text": "6  Applications of Determinants and Matrices\nIn this section, we shall discuss application of determinants and matrices for solving the\nsystem of linear equations in two or three variables and for checking the consistency of\nthe system of linear equations Rationalised 2023-24\n 94\nMATHEMATICS\nConsistent system A system of equations is said to be consistent if its solution (one\nor more) exists Inconsistent system A system of equations is said to be inconsistent if its solution\ndoes not exist ANote In this chapter, we restrict ourselves to the system of linear equations\nhaving unique solutions only"
    },
    {
        "Chapter": "1",
        "sentence_range": "1609-1612",
        "Text": "Rationalised 2023-24\n 94\nMATHEMATICS\nConsistent system A system of equations is said to be consistent if its solution (one\nor more) exists Inconsistent system A system of equations is said to be inconsistent if its solution\ndoes not exist ANote In this chapter, we restrict ourselves to the system of linear equations\nhaving unique solutions only 4"
    },
    {
        "Chapter": "1",
        "sentence_range": "1610-1613",
        "Text": "Inconsistent system A system of equations is said to be inconsistent if its solution\ndoes not exist ANote In this chapter, we restrict ourselves to the system of linear equations\nhaving unique solutions only 4 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "1611-1614",
        "Text": "ANote In this chapter, we restrict ourselves to the system of linear equations\nhaving unique solutions only 4 6 1  Solution of system of linear equations using inverse of a matrix\nLet us express the system of linear equations as matrix equations and solve them using\ninverse of the coefficient matrix"
    },
    {
        "Chapter": "1",
        "sentence_range": "1612-1615",
        "Text": "4 6 1  Solution of system of linear equations using inverse of a matrix\nLet us express the system of linear equations as matrix equations and solve them using\ninverse of the coefficient matrix Consider the system of equations\na1 x + b1 y + c1 z =  d1\na2 x + b2 y + c2 z = d 2\na3 x + b3 y + c3 z = d 3\nLet\nA =\n1\n1\n1\n1\n2\n2\n2\n2\n3\n3\n3\n3\n, X\nand B\na\nb\nc\nx\nd\na\nb\nc\ny\nd\na\nb\nc\nz\nd\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee \uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n=\uf8ef \uf8fa\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef \uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef \uf8fa\n\uf8f0 \uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nThen, the system of equations can be written as, AX = B, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "1613-1616",
        "Text": "6 1  Solution of system of linear equations using inverse of a matrix\nLet us express the system of linear equations as matrix equations and solve them using\ninverse of the coefficient matrix Consider the system of equations\na1 x + b1 y + c1 z =  d1\na2 x + b2 y + c2 z = d 2\na3 x + b3 y + c3 z = d 3\nLet\nA =\n1\n1\n1\n1\n2\n2\n2\n2\n3\n3\n3\n3\n, X\nand B\na\nb\nc\nx\nd\na\nb\nc\ny\nd\na\nb\nc\nz\nd\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee \uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n=\uf8ef \uf8fa\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef \uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef \uf8fa\n\uf8f0 \uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nThen, the system of equations can be written as, AX = B, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "1614-1617",
        "Text": "1  Solution of system of linear equations using inverse of a matrix\nLet us express the system of linear equations as matrix equations and solve them using\ninverse of the coefficient matrix Consider the system of equations\na1 x + b1 y + c1 z =  d1\na2 x + b2 y + c2 z = d 2\na3 x + b3 y + c3 z = d 3\nLet\nA =\n1\n1\n1\n1\n2\n2\n2\n2\n3\n3\n3\n3\n, X\nand B\na\nb\nc\nx\nd\na\nb\nc\ny\nd\na\nb\nc\nz\nd\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee \uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n=\uf8ef \uf8fa\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef \uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef \uf8fa\n\uf8f0 \uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nThen, the system of equations can be written as, AX = B, i e ,\n \n1\n1\n1\n2\n2\n2\n3\n3\n3\na\nb\nc\nx\na\nb\nc\ny\na\nb\nc\nz\n\uf8ee\n\uf8f9 \uf8ee \uf8f9\n\uf8ef\n\uf8fa \uf8ef \uf8fa\n\uf8ef\n\uf8fa \uf8ef \uf8fa\n\uf8ef\n\uf8fa \uf8ef \uf8fa\n\uf8f0 \uf8fb\n\uf8f0\n\uf8fb\n =\n1\n2\n3\nd\nd\nd\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nCase I If A is a nonsingular matrix, then its inverse exists"
    },
    {
        "Chapter": "1",
        "sentence_range": "1615-1618",
        "Text": "Consider the system of equations\na1 x + b1 y + c1 z =  d1\na2 x + b2 y + c2 z = d 2\na3 x + b3 y + c3 z = d 3\nLet\nA =\n1\n1\n1\n1\n2\n2\n2\n2\n3\n3\n3\n3\n, X\nand B\na\nb\nc\nx\nd\na\nb\nc\ny\nd\na\nb\nc\nz\nd\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee \uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n=\uf8ef \uf8fa\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef \uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef \uf8fa\n\uf8f0 \uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nThen, the system of equations can be written as, AX = B, i e ,\n \n1\n1\n1\n2\n2\n2\n3\n3\n3\na\nb\nc\nx\na\nb\nc\ny\na\nb\nc\nz\n\uf8ee\n\uf8f9 \uf8ee \uf8f9\n\uf8ef\n\uf8fa \uf8ef \uf8fa\n\uf8ef\n\uf8fa \uf8ef \uf8fa\n\uf8ef\n\uf8fa \uf8ef \uf8fa\n\uf8f0 \uf8fb\n\uf8f0\n\uf8fb\n =\n1\n2\n3\nd\nd\nd\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nCase I If A is a nonsingular matrix, then its inverse exists Now\nAX = B\nor\nA\u20131 (AX) = A\u20131 B\n(premultiplying by A\u20131)\nor\n(A\u20131A) X = A\u20131 B\n(by associative property)\nor\nI X = A\u20131 B\nor\nX = A\u20131 B\nThis matrix equation provides unique solution for the given system of equations as\ninverse of a matrix is unique"
    },
    {
        "Chapter": "1",
        "sentence_range": "1616-1619",
        "Text": "e ,\n \n1\n1\n1\n2\n2\n2\n3\n3\n3\na\nb\nc\nx\na\nb\nc\ny\na\nb\nc\nz\n\uf8ee\n\uf8f9 \uf8ee \uf8f9\n\uf8ef\n\uf8fa \uf8ef \uf8fa\n\uf8ef\n\uf8fa \uf8ef \uf8fa\n\uf8ef\n\uf8fa \uf8ef \uf8fa\n\uf8f0 \uf8fb\n\uf8f0\n\uf8fb\n =\n1\n2\n3\nd\nd\nd\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nCase I If A is a nonsingular matrix, then its inverse exists Now\nAX = B\nor\nA\u20131 (AX) = A\u20131 B\n(premultiplying by A\u20131)\nor\n(A\u20131A) X = A\u20131 B\n(by associative property)\nor\nI X = A\u20131 B\nor\nX = A\u20131 B\nThis matrix equation provides unique solution for the given system of equations as\ninverse of a matrix is unique This method of solving system of equations is known as\nMatrix Method"
    },
    {
        "Chapter": "1",
        "sentence_range": "1617-1620",
        "Text": ",\n \n1\n1\n1\n2\n2\n2\n3\n3\n3\na\nb\nc\nx\na\nb\nc\ny\na\nb\nc\nz\n\uf8ee\n\uf8f9 \uf8ee \uf8f9\n\uf8ef\n\uf8fa \uf8ef \uf8fa\n\uf8ef\n\uf8fa \uf8ef \uf8fa\n\uf8ef\n\uf8fa \uf8ef \uf8fa\n\uf8f0 \uf8fb\n\uf8f0\n\uf8fb\n =\n1\n2\n3\nd\nd\nd\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nCase I If A is a nonsingular matrix, then its inverse exists Now\nAX = B\nor\nA\u20131 (AX) = A\u20131 B\n(premultiplying by A\u20131)\nor\n(A\u20131A) X = A\u20131 B\n(by associative property)\nor\nI X = A\u20131 B\nor\nX = A\u20131 B\nThis matrix equation provides unique solution for the given system of equations as\ninverse of a matrix is unique This method of solving system of equations is known as\nMatrix Method Case II If A is a singular matrix, then |A| = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "1618-1621",
        "Text": "Now\nAX = B\nor\nA\u20131 (AX) = A\u20131 B\n(premultiplying by A\u20131)\nor\n(A\u20131A) X = A\u20131 B\n(by associative property)\nor\nI X = A\u20131 B\nor\nX = A\u20131 B\nThis matrix equation provides unique solution for the given system of equations as\ninverse of a matrix is unique This method of solving system of equations is known as\nMatrix Method Case II If A is a singular matrix, then |A| = 0 In this case, we calculate (adj A) B"
    },
    {
        "Chapter": "1",
        "sentence_range": "1619-1622",
        "Text": "This method of solving system of equations is known as\nMatrix Method Case II If A is a singular matrix, then |A| = 0 In this case, we calculate (adj A) B If (adj A) B \u2260 O, (O being zero matrix), then solution does not exist and the\nsystem of equations is called inconsistent"
    },
    {
        "Chapter": "1",
        "sentence_range": "1620-1623",
        "Text": "Case II If A is a singular matrix, then |A| = 0 In this case, we calculate (adj A) B If (adj A) B \u2260 O, (O being zero matrix), then solution does not exist and the\nsystem of equations is called inconsistent Rationalised 2023-24\nDETERMINANTS     95\nIf (adj A) B = O, then system may be either consistent or inconsistent according\nas the system have either infinitely many solutions or no solution"
    },
    {
        "Chapter": "1",
        "sentence_range": "1621-1624",
        "Text": "In this case, we calculate (adj A) B If (adj A) B \u2260 O, (O being zero matrix), then solution does not exist and the\nsystem of equations is called inconsistent Rationalised 2023-24\nDETERMINANTS     95\nIf (adj A) B = O, then system may be either consistent or inconsistent according\nas the system have either infinitely many solutions or no solution Example 16 Solve the system of equations\n2x + 5y = 1\n3x + 2y = 7\nSolution The system of equations can be written in the form AX = B, where\nA =\n2\n5\n1\n,X\nand B\n3\n2\n7\nx\ny\n\uf8ee\n\uf8f9\n\uf8ee \uf8f9\n\uf8ee \uf8f9\n=\n=\n\uf8ef\n\uf8fa\n\uf8ef \uf8fa\n\uf8ef \uf8fa\n\uf8f0\n\uf8fb\n\uf8f0 \uf8fb\n\uf8f0 \uf8fb\nNow, A  = \u201311 \u2260 0, Hence, A is nonsingular matrix and so has a unique solution"
    },
    {
        "Chapter": "1",
        "sentence_range": "1622-1625",
        "Text": "If (adj A) B \u2260 O, (O being zero matrix), then solution does not exist and the\nsystem of equations is called inconsistent Rationalised 2023-24\nDETERMINANTS     95\nIf (adj A) B = O, then system may be either consistent or inconsistent according\nas the system have either infinitely many solutions or no solution Example 16 Solve the system of equations\n2x + 5y = 1\n3x + 2y = 7\nSolution The system of equations can be written in the form AX = B, where\nA =\n2\n5\n1\n,X\nand B\n3\n2\n7\nx\ny\n\uf8ee\n\uf8f9\n\uf8ee \uf8f9\n\uf8ee \uf8f9\n=\n=\n\uf8ef\n\uf8fa\n\uf8ef \uf8fa\n\uf8ef \uf8fa\n\uf8f0\n\uf8fb\n\uf8f0 \uf8fb\n\uf8f0 \uf8fb\nNow, A  = \u201311 \u2260 0, Hence, A is nonsingular matrix and so has a unique solution Note that\nA\u20131 = \u2212\n\u2212\n\u2212\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa\n1\n11\n2\n5\n3\n2\nTherefore\nX = A\u20131B = \u2013 1\n11\n2\n5\n3\n2\n71\n\u2212\n\u2212\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa\ni"
    },
    {
        "Chapter": "1",
        "sentence_range": "1623-1626",
        "Text": "Rationalised 2023-24\nDETERMINANTS     95\nIf (adj A) B = O, then system may be either consistent or inconsistent according\nas the system have either infinitely many solutions or no solution Example 16 Solve the system of equations\n2x + 5y = 1\n3x + 2y = 7\nSolution The system of equations can be written in the form AX = B, where\nA =\n2\n5\n1\n,X\nand B\n3\n2\n7\nx\ny\n\uf8ee\n\uf8f9\n\uf8ee \uf8f9\n\uf8ee \uf8f9\n=\n=\n\uf8ef\n\uf8fa\n\uf8ef \uf8fa\n\uf8ef \uf8fa\n\uf8f0\n\uf8fb\n\uf8f0 \uf8fb\n\uf8f0 \uf8fb\nNow, A  = \u201311 \u2260 0, Hence, A is nonsingular matrix and so has a unique solution Note that\nA\u20131 = \u2212\n\u2212\n\u2212\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa\n1\n11\n2\n5\n3\n2\nTherefore\nX = A\u20131B = \u2013 1\n11\n2\n5\n3\n2\n71\n\u2212\n\u2212\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa\ni e"
    },
    {
        "Chapter": "1",
        "sentence_range": "1624-1627",
        "Text": "Example 16 Solve the system of equations\n2x + 5y = 1\n3x + 2y = 7\nSolution The system of equations can be written in the form AX = B, where\nA =\n2\n5\n1\n,X\nand B\n3\n2\n7\nx\ny\n\uf8ee\n\uf8f9\n\uf8ee \uf8f9\n\uf8ee \uf8f9\n=\n=\n\uf8ef\n\uf8fa\n\uf8ef \uf8fa\n\uf8ef \uf8fa\n\uf8f0\n\uf8fb\n\uf8f0 \uf8fb\n\uf8f0 \uf8fb\nNow, A  = \u201311 \u2260 0, Hence, A is nonsingular matrix and so has a unique solution Note that\nA\u20131 = \u2212\n\u2212\n\u2212\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa\n1\n11\n2\n5\n3\n2\nTherefore\nX = A\u20131B = \u2013 1\n11\n2\n5\n3\n2\n71\n\u2212\n\u2212\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa\ni e x\ny\n\uf8ee \uf8f9\n\uf8ef \uf8fa\n\uf8f0 \uf8fb  = \u2212\n\u2212\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa= \u2212\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa\n1\n11\n33\n11\n3\n1\nHence\nx = 3, y = \u2013 1\nExample 17 Solve the following system of equations by matrix method"
    },
    {
        "Chapter": "1",
        "sentence_range": "1625-1628",
        "Text": "Note that\nA\u20131 = \u2212\n\u2212\n\u2212\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa\n1\n11\n2\n5\n3\n2\nTherefore\nX = A\u20131B = \u2013 1\n11\n2\n5\n3\n2\n71\n\u2212\n\u2212\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa\ni e x\ny\n\uf8ee \uf8f9\n\uf8ef \uf8fa\n\uf8f0 \uf8fb  = \u2212\n\u2212\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa= \u2212\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa\n1\n11\n33\n11\n3\n1\nHence\nx = 3, y = \u2013 1\nExample 17 Solve the following system of equations by matrix method 3x \u2013 2y + 3z = 8\n2x + y \u2013 z = 1\n4x \u2013 3y + 2z = 4\nSolution The system of equations can be written in the form AX = B, where\n3\n2\n3\n8\nA\n2\n1\n1 , X\nand B\n1\n4\n3\n2\n4\nx\ny\nz\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee \uf8f9\n\uf8ee \uf8f9\n\uf8ef\n\uf8fa\n\uf8ef \uf8fa\n\uf8ef \uf8fa\n=\n\u2212\n=\n=\n\uf8ef\n\uf8fa\n\uf8ef \uf8fa\n\uf8ef \uf8fa\n\uf8ef\n\uf8fa\n\uf8ef \uf8fa\n\uf8ef \uf8fa\n\u2212\n\uf8f0\n\uf8fb\n\uf8f0 \uf8fb\n\uf8f0 \uf8fb\nWe see that\nA  = 3 (2 \u2013 3) + 2(4 + 4) + 3 (\u2013 6 \u2013 4) = \u2013 17 \u2260 0\nHence, A is nonsingular and so its inverse exists"
    },
    {
        "Chapter": "1",
        "sentence_range": "1626-1629",
        "Text": "e x\ny\n\uf8ee \uf8f9\n\uf8ef \uf8fa\n\uf8f0 \uf8fb  = \u2212\n\u2212\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa= \u2212\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa\n1\n11\n33\n11\n3\n1\nHence\nx = 3, y = \u2013 1\nExample 17 Solve the following system of equations by matrix method 3x \u2013 2y + 3z = 8\n2x + y \u2013 z = 1\n4x \u2013 3y + 2z = 4\nSolution The system of equations can be written in the form AX = B, where\n3\n2\n3\n8\nA\n2\n1\n1 , X\nand B\n1\n4\n3\n2\n4\nx\ny\nz\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee \uf8f9\n\uf8ee \uf8f9\n\uf8ef\n\uf8fa\n\uf8ef \uf8fa\n\uf8ef \uf8fa\n=\n\u2212\n=\n=\n\uf8ef\n\uf8fa\n\uf8ef \uf8fa\n\uf8ef \uf8fa\n\uf8ef\n\uf8fa\n\uf8ef \uf8fa\n\uf8ef \uf8fa\n\u2212\n\uf8f0\n\uf8fb\n\uf8f0 \uf8fb\n\uf8f0 \uf8fb\nWe see that\nA  = 3 (2 \u2013 3) + 2(4 + 4) + 3 (\u2013 6 \u2013 4) = \u2013 17 \u2260 0\nHence, A is nonsingular and so its inverse exists Now\nA11 = \u20131,\nA12 = \u2013 8,\nA13 = \u201310\nA21 = \u20135,\nA22 = \u2013 6,\nA23 = 1\nA31 = \u20131,\nA32 = 9,\nA33 = 7\nRationalised 2023-24\n 96\nMATHEMATICS\nTherefore\nA\u20131 =\n1\n5\n1\n1\n8\n6\n9\n17\n10\n1\n7\n\u2212\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\u2212\n\uf8fb\nSo\nX =\n\u20131\n1\n5\n1\n8\n1\nA B =\n8\n6\n9\n1\n17\n10\n1\n7\n4\n\u2212\n\u2212\n\u2212\n\uf8ee\n\uf8f9 \uf8ee \uf8f9\n\uf8ef\n\uf8fa \uf8ef \uf8fa\n\u2212\n\u2212\n\u2212\n\uf8ef\n\uf8fa \uf8ef \uf8fa\n\uf8ef\n\uf8fa \uf8ef \uf8fa\n\uf8f0\u2212\n\uf8fb \uf8f0 \uf8fb\ni"
    },
    {
        "Chapter": "1",
        "sentence_range": "1627-1630",
        "Text": "x\ny\n\uf8ee \uf8f9\n\uf8ef \uf8fa\n\uf8f0 \uf8fb  = \u2212\n\u2212\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa= \u2212\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa\n1\n11\n33\n11\n3\n1\nHence\nx = 3, y = \u2013 1\nExample 17 Solve the following system of equations by matrix method 3x \u2013 2y + 3z = 8\n2x + y \u2013 z = 1\n4x \u2013 3y + 2z = 4\nSolution The system of equations can be written in the form AX = B, where\n3\n2\n3\n8\nA\n2\n1\n1 , X\nand B\n1\n4\n3\n2\n4\nx\ny\nz\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee \uf8f9\n\uf8ee \uf8f9\n\uf8ef\n\uf8fa\n\uf8ef \uf8fa\n\uf8ef \uf8fa\n=\n\u2212\n=\n=\n\uf8ef\n\uf8fa\n\uf8ef \uf8fa\n\uf8ef \uf8fa\n\uf8ef\n\uf8fa\n\uf8ef \uf8fa\n\uf8ef \uf8fa\n\u2212\n\uf8f0\n\uf8fb\n\uf8f0 \uf8fb\n\uf8f0 \uf8fb\nWe see that\nA  = 3 (2 \u2013 3) + 2(4 + 4) + 3 (\u2013 6 \u2013 4) = \u2013 17 \u2260 0\nHence, A is nonsingular and so its inverse exists Now\nA11 = \u20131,\nA12 = \u2013 8,\nA13 = \u201310\nA21 = \u20135,\nA22 = \u2013 6,\nA23 = 1\nA31 = \u20131,\nA32 = 9,\nA33 = 7\nRationalised 2023-24\n 96\nMATHEMATICS\nTherefore\nA\u20131 =\n1\n5\n1\n1\n8\n6\n9\n17\n10\n1\n7\n\u2212\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\u2212\n\uf8fb\nSo\nX =\n\u20131\n1\n5\n1\n8\n1\nA B =\n8\n6\n9\n1\n17\n10\n1\n7\n4\n\u2212\n\u2212\n\u2212\n\uf8ee\n\uf8f9 \uf8ee \uf8f9\n\uf8ef\n\uf8fa \uf8ef \uf8fa\n\u2212\n\u2212\n\u2212\n\uf8ef\n\uf8fa \uf8ef \uf8fa\n\uf8ef\n\uf8fa \uf8ef \uf8fa\n\uf8f0\u2212\n\uf8fb \uf8f0 \uf8fb\ni e"
    },
    {
        "Chapter": "1",
        "sentence_range": "1628-1631",
        "Text": "3x \u2013 2y + 3z = 8\n2x + y \u2013 z = 1\n4x \u2013 3y + 2z = 4\nSolution The system of equations can be written in the form AX = B, where\n3\n2\n3\n8\nA\n2\n1\n1 , X\nand B\n1\n4\n3\n2\n4\nx\ny\nz\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee \uf8f9\n\uf8ee \uf8f9\n\uf8ef\n\uf8fa\n\uf8ef \uf8fa\n\uf8ef \uf8fa\n=\n\u2212\n=\n=\n\uf8ef\n\uf8fa\n\uf8ef \uf8fa\n\uf8ef \uf8fa\n\uf8ef\n\uf8fa\n\uf8ef \uf8fa\n\uf8ef \uf8fa\n\u2212\n\uf8f0\n\uf8fb\n\uf8f0 \uf8fb\n\uf8f0 \uf8fb\nWe see that\nA  = 3 (2 \u2013 3) + 2(4 + 4) + 3 (\u2013 6 \u2013 4) = \u2013 17 \u2260 0\nHence, A is nonsingular and so its inverse exists Now\nA11 = \u20131,\nA12 = \u2013 8,\nA13 = \u201310\nA21 = \u20135,\nA22 = \u2013 6,\nA23 = 1\nA31 = \u20131,\nA32 = 9,\nA33 = 7\nRationalised 2023-24\n 96\nMATHEMATICS\nTherefore\nA\u20131 =\n1\n5\n1\n1\n8\n6\n9\n17\n10\n1\n7\n\u2212\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\u2212\n\uf8fb\nSo\nX =\n\u20131\n1\n5\n1\n8\n1\nA B =\n8\n6\n9\n1\n17\n10\n1\n7\n4\n\u2212\n\u2212\n\u2212\n\uf8ee\n\uf8f9 \uf8ee \uf8f9\n\uf8ef\n\uf8fa \uf8ef \uf8fa\n\u2212\n\u2212\n\u2212\n\uf8ef\n\uf8fa \uf8ef \uf8fa\n\uf8ef\n\uf8fa \uf8ef \uf8fa\n\uf8f0\u2212\n\uf8fb \uf8f0 \uf8fb\ni e x\ny\nz\n\uf8ee \uf8f9\n\uf8ef \uf8fa\n\uf8ef \uf8fa\n\uf8ef \uf8fa\n\uf8f0 \uf8fb\n =\n17\n1\n1\n34\n2\n17\n51\n3\n\uf8ee\u2212\n\uf8f9\n\uf8ee \uf8f9\n\uf8ef\n\uf8fa\n\uf8ef \uf8fa\n\u2212\n\u2212\n=\n\uf8ef\n\uf8fa\n\uf8ef \uf8fa\n\uf8ef\n\uf8fa\n\uf8ef \uf8fa\n\uf8f0\u2212\n\uf8fb\n\uf8f0 \uf8fb\nHence\nx = 1, y = 2 and z = 3"
    },
    {
        "Chapter": "1",
        "sentence_range": "1629-1632",
        "Text": "Now\nA11 = \u20131,\nA12 = \u2013 8,\nA13 = \u201310\nA21 = \u20135,\nA22 = \u2013 6,\nA23 = 1\nA31 = \u20131,\nA32 = 9,\nA33 = 7\nRationalised 2023-24\n 96\nMATHEMATICS\nTherefore\nA\u20131 =\n1\n5\n1\n1\n8\n6\n9\n17\n10\n1\n7\n\u2212\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\u2212\n\uf8fb\nSo\nX =\n\u20131\n1\n5\n1\n8\n1\nA B =\n8\n6\n9\n1\n17\n10\n1\n7\n4\n\u2212\n\u2212\n\u2212\n\uf8ee\n\uf8f9 \uf8ee \uf8f9\n\uf8ef\n\uf8fa \uf8ef \uf8fa\n\u2212\n\u2212\n\u2212\n\uf8ef\n\uf8fa \uf8ef \uf8fa\n\uf8ef\n\uf8fa \uf8ef \uf8fa\n\uf8f0\u2212\n\uf8fb \uf8f0 \uf8fb\ni e x\ny\nz\n\uf8ee \uf8f9\n\uf8ef \uf8fa\n\uf8ef \uf8fa\n\uf8ef \uf8fa\n\uf8f0 \uf8fb\n =\n17\n1\n1\n34\n2\n17\n51\n3\n\uf8ee\u2212\n\uf8f9\n\uf8ee \uf8f9\n\uf8ef\n\uf8fa\n\uf8ef \uf8fa\n\u2212\n\u2212\n=\n\uf8ef\n\uf8fa\n\uf8ef \uf8fa\n\uf8ef\n\uf8fa\n\uf8ef \uf8fa\n\uf8f0\u2212\n\uf8fb\n\uf8f0 \uf8fb\nHence\nx = 1, y = 2 and z = 3 Example 18 The sum of three numbers is 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "1630-1633",
        "Text": "e x\ny\nz\n\uf8ee \uf8f9\n\uf8ef \uf8fa\n\uf8ef \uf8fa\n\uf8ef \uf8fa\n\uf8f0 \uf8fb\n =\n17\n1\n1\n34\n2\n17\n51\n3\n\uf8ee\u2212\n\uf8f9\n\uf8ee \uf8f9\n\uf8ef\n\uf8fa\n\uf8ef \uf8fa\n\u2212\n\u2212\n=\n\uf8ef\n\uf8fa\n\uf8ef \uf8fa\n\uf8ef\n\uf8fa\n\uf8ef \uf8fa\n\uf8f0\u2212\n\uf8fb\n\uf8f0 \uf8fb\nHence\nx = 1, y = 2 and z = 3 Example 18 The sum of three numbers is 6 If we multiply third number by 3 and add\nsecond number to it, we get 11"
    },
    {
        "Chapter": "1",
        "sentence_range": "1631-1634",
        "Text": "x\ny\nz\n\uf8ee \uf8f9\n\uf8ef \uf8fa\n\uf8ef \uf8fa\n\uf8ef \uf8fa\n\uf8f0 \uf8fb\n =\n17\n1\n1\n34\n2\n17\n51\n3\n\uf8ee\u2212\n\uf8f9\n\uf8ee \uf8f9\n\uf8ef\n\uf8fa\n\uf8ef \uf8fa\n\u2212\n\u2212\n=\n\uf8ef\n\uf8fa\n\uf8ef \uf8fa\n\uf8ef\n\uf8fa\n\uf8ef \uf8fa\n\uf8f0\u2212\n\uf8fb\n\uf8f0 \uf8fb\nHence\nx = 1, y = 2 and z = 3 Example 18 The sum of three numbers is 6 If we multiply third number by 3 and add\nsecond number to it, we get 11 By adding first and third numbers, we get double of the\nsecond number"
    },
    {
        "Chapter": "1",
        "sentence_range": "1632-1635",
        "Text": "Example 18 The sum of three numbers is 6 If we multiply third number by 3 and add\nsecond number to it, we get 11 By adding first and third numbers, we get double of the\nsecond number Represent it algebraically and find the numbers using matrix method"
    },
    {
        "Chapter": "1",
        "sentence_range": "1633-1636",
        "Text": "If we multiply third number by 3 and add\nsecond number to it, we get 11 By adding first and third numbers, we get double of the\nsecond number Represent it algebraically and find the numbers using matrix method Solution Let first, second and third numbers be denoted by x, y and z, respectively"
    },
    {
        "Chapter": "1",
        "sentence_range": "1634-1637",
        "Text": "By adding first and third numbers, we get double of the\nsecond number Represent it algebraically and find the numbers using matrix method Solution Let first, second and third numbers be denoted by x, y and z, respectively Then, according to given conditions, we have\nx + y + z = 6\ny + 3z = 11\nx + z = 2y  or x \u2013 2y + z = 0\nThis system can be written as A X = B, where\nA =  \n1\n1\n1\n0\n1\n3\n1\n2\n1\n\ufffd\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\n, X = \nx\ny\nz\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\n and B = \n6\n11\n0\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\nHere \n(\n)\n(\n)\nA\n1 1\n6 \u2013 (0 \u2013 3)\n0 \u20131\n9\n0\n=\n+\n+\n=\n\u2260"
    },
    {
        "Chapter": "1",
        "sentence_range": "1635-1638",
        "Text": "Represent it algebraically and find the numbers using matrix method Solution Let first, second and third numbers be denoted by x, y and z, respectively Then, according to given conditions, we have\nx + y + z = 6\ny + 3z = 11\nx + z = 2y  or x \u2013 2y + z = 0\nThis system can be written as A X = B, where\nA =  \n1\n1\n1\n0\n1\n3\n1\n2\n1\n\ufffd\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\n, X = \nx\ny\nz\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\n and B = \n6\n11\n0\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\nHere \n(\n)\n(\n)\nA\n1 1\n6 \u2013 (0 \u2013 3)\n0 \u20131\n9\n0\n=\n+\n+\n=\n\u2260 Now we find adj A\nA11 = 1 (1 + 6) = 7,\nA12 = \u2013 (0 \u2013 3) = 3,\nA13 = \u2013 1\nA21 = \u2013 (1 + 2) = \u2013 3,\nA22 = 0,\nA23 = \u2013 (\u2013 2 \u2013 1) = 3\nA31 = (3 \u2013 1) = 2,\nA32 = \u2013 (3 \u2013 0) = \u2013 3,\nA33 = (1 \u2013 0) = 1\nHence\nadj A =\n7\n\u20133\n2\n3\n0\n\u20133\n\u20131\n3\n1\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nRationalised 2023-24\nDETERMINANTS     97\nThus\nA \u20131 =\n1\nA  adj (A) = \n7\n3\n2\n1\n3\n0\n3\n9\n1\n3\n1\n\u2013\n\u2013\n\u2013\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nSince\nX = A\u20131 B\nX =\n7\n3\n2\n6\n1\n3\n0\n3\n11\n9\n1\n3\n1\n0\n\u2013\n\u2013\n\u2013\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\nor\nx\ny\nz\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\n =  1\n9  \n42\n33\n0\n18\n0\n0\n6\n33\n0\n\u2212\n+\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n+\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212 +\n+\n\uf8f0\n\uf8fb\n = 1\n9  \n9\n18\n27\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\n = \n1\n2\n3\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\nThus\nx = 1, y = 2, z = 3\nEXERCISE 4"
    },
    {
        "Chapter": "1",
        "sentence_range": "1636-1639",
        "Text": "Solution Let first, second and third numbers be denoted by x, y and z, respectively Then, according to given conditions, we have\nx + y + z = 6\ny + 3z = 11\nx + z = 2y  or x \u2013 2y + z = 0\nThis system can be written as A X = B, where\nA =  \n1\n1\n1\n0\n1\n3\n1\n2\n1\n\ufffd\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\n, X = \nx\ny\nz\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\n and B = \n6\n11\n0\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\nHere \n(\n)\n(\n)\nA\n1 1\n6 \u2013 (0 \u2013 3)\n0 \u20131\n9\n0\n=\n+\n+\n=\n\u2260 Now we find adj A\nA11 = 1 (1 + 6) = 7,\nA12 = \u2013 (0 \u2013 3) = 3,\nA13 = \u2013 1\nA21 = \u2013 (1 + 2) = \u2013 3,\nA22 = 0,\nA23 = \u2013 (\u2013 2 \u2013 1) = 3\nA31 = (3 \u2013 1) = 2,\nA32 = \u2013 (3 \u2013 0) = \u2013 3,\nA33 = (1 \u2013 0) = 1\nHence\nadj A =\n7\n\u20133\n2\n3\n0\n\u20133\n\u20131\n3\n1\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nRationalised 2023-24\nDETERMINANTS     97\nThus\nA \u20131 =\n1\nA  adj (A) = \n7\n3\n2\n1\n3\n0\n3\n9\n1\n3\n1\n\u2013\n\u2013\n\u2013\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nSince\nX = A\u20131 B\nX =\n7\n3\n2\n6\n1\n3\n0\n3\n11\n9\n1\n3\n1\n0\n\u2013\n\u2013\n\u2013\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\nor\nx\ny\nz\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\n =  1\n9  \n42\n33\n0\n18\n0\n0\n6\n33\n0\n\u2212\n+\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n+\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212 +\n+\n\uf8f0\n\uf8fb\n = 1\n9  \n9\n18\n27\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\n = \n1\n2\n3\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\nThus\nx = 1, y = 2, z = 3\nEXERCISE 4 5\nExamine the consistency of the system of equations in Exercises 1 to 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "1637-1640",
        "Text": "Then, according to given conditions, we have\nx + y + z = 6\ny + 3z = 11\nx + z = 2y  or x \u2013 2y + z = 0\nThis system can be written as A X = B, where\nA =  \n1\n1\n1\n0\n1\n3\n1\n2\n1\n\ufffd\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\n, X = \nx\ny\nz\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\n and B = \n6\n11\n0\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\nHere \n(\n)\n(\n)\nA\n1 1\n6 \u2013 (0 \u2013 3)\n0 \u20131\n9\n0\n=\n+\n+\n=\n\u2260 Now we find adj A\nA11 = 1 (1 + 6) = 7,\nA12 = \u2013 (0 \u2013 3) = 3,\nA13 = \u2013 1\nA21 = \u2013 (1 + 2) = \u2013 3,\nA22 = 0,\nA23 = \u2013 (\u2013 2 \u2013 1) = 3\nA31 = (3 \u2013 1) = 2,\nA32 = \u2013 (3 \u2013 0) = \u2013 3,\nA33 = (1 \u2013 0) = 1\nHence\nadj A =\n7\n\u20133\n2\n3\n0\n\u20133\n\u20131\n3\n1\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nRationalised 2023-24\nDETERMINANTS     97\nThus\nA \u20131 =\n1\nA  adj (A) = \n7\n3\n2\n1\n3\n0\n3\n9\n1\n3\n1\n\u2013\n\u2013\n\u2013\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nSince\nX = A\u20131 B\nX =\n7\n3\n2\n6\n1\n3\n0\n3\n11\n9\n1\n3\n1\n0\n\u2013\n\u2013\n\u2013\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\nor\nx\ny\nz\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\n =  1\n9  \n42\n33\n0\n18\n0\n0\n6\n33\n0\n\u2212\n+\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n+\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212 +\n+\n\uf8f0\n\uf8fb\n = 1\n9  \n9\n18\n27\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\n = \n1\n2\n3\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\nThus\nx = 1, y = 2, z = 3\nEXERCISE 4 5\nExamine the consistency of the system of equations in Exercises 1 to 6 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "1638-1641",
        "Text": "Now we find adj A\nA11 = 1 (1 + 6) = 7,\nA12 = \u2013 (0 \u2013 3) = 3,\nA13 = \u2013 1\nA21 = \u2013 (1 + 2) = \u2013 3,\nA22 = 0,\nA23 = \u2013 (\u2013 2 \u2013 1) = 3\nA31 = (3 \u2013 1) = 2,\nA32 = \u2013 (3 \u2013 0) = \u2013 3,\nA33 = (1 \u2013 0) = 1\nHence\nadj A =\n7\n\u20133\n2\n3\n0\n\u20133\n\u20131\n3\n1\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nRationalised 2023-24\nDETERMINANTS     97\nThus\nA \u20131 =\n1\nA  adj (A) = \n7\n3\n2\n1\n3\n0\n3\n9\n1\n3\n1\n\u2013\n\u2013\n\u2013\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nSince\nX = A\u20131 B\nX =\n7\n3\n2\n6\n1\n3\n0\n3\n11\n9\n1\n3\n1\n0\n\u2013\n\u2013\n\u2013\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\nor\nx\ny\nz\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\n =  1\n9  \n42\n33\n0\n18\n0\n0\n6\n33\n0\n\u2212\n+\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n+\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212 +\n+\n\uf8f0\n\uf8fb\n = 1\n9  \n9\n18\n27\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\n = \n1\n2\n3\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\nThus\nx = 1, y = 2, z = 3\nEXERCISE 4 5\nExamine the consistency of the system of equations in Exercises 1 to 6 1 x + 2y = 2\n2"
    },
    {
        "Chapter": "1",
        "sentence_range": "1639-1642",
        "Text": "5\nExamine the consistency of the system of equations in Exercises 1 to 6 1 x + 2y = 2\n2 2x \u2013 y = 5\n3"
    },
    {
        "Chapter": "1",
        "sentence_range": "1640-1643",
        "Text": "1 x + 2y = 2\n2 2x \u2013 y = 5\n3 x + 3y = 5\n2x + 3y = 3\nx + y = 4\n2x + 6y = 8\n4"
    },
    {
        "Chapter": "1",
        "sentence_range": "1641-1644",
        "Text": "x + 2y = 2\n2 2x \u2013 y = 5\n3 x + 3y = 5\n2x + 3y = 3\nx + y = 4\n2x + 6y = 8\n4 x + y + z = 1\n5"
    },
    {
        "Chapter": "1",
        "sentence_range": "1642-1645",
        "Text": "2x \u2013 y = 5\n3 x + 3y = 5\n2x + 3y = 3\nx + y = 4\n2x + 6y = 8\n4 x + y + z = 1\n5 3x\u2013y \u2013 2z = 2\n6"
    },
    {
        "Chapter": "1",
        "sentence_range": "1643-1646",
        "Text": "x + 3y = 5\n2x + 3y = 3\nx + y = 4\n2x + 6y = 8\n4 x + y + z = 1\n5 3x\u2013y \u2013 2z = 2\n6 5x \u2013 y + 4z = 5\n2x + 3y + 2z = 2\n2y \u2013 z = \u20131\n2x + 3y + 5z = 2\nax + ay + 2az = 4\n3x \u2013 5y = 3\n5x \u2013 2y + 6z = \u20131\nSolve system of linear equations, using matrix method, in Exercises 7 to 14"
    },
    {
        "Chapter": "1",
        "sentence_range": "1644-1647",
        "Text": "x + y + z = 1\n5 3x\u2013y \u2013 2z = 2\n6 5x \u2013 y + 4z = 5\n2x + 3y + 2z = 2\n2y \u2013 z = \u20131\n2x + 3y + 5z = 2\nax + ay + 2az = 4\n3x \u2013 5y = 3\n5x \u2013 2y + 6z = \u20131\nSolve system of linear equations, using matrix method, in Exercises 7 to 14 7"
    },
    {
        "Chapter": "1",
        "sentence_range": "1645-1648",
        "Text": "3x\u2013y \u2013 2z = 2\n6 5x \u2013 y + 4z = 5\n2x + 3y + 2z = 2\n2y \u2013 z = \u20131\n2x + 3y + 5z = 2\nax + ay + 2az = 4\n3x \u2013 5y = 3\n5x \u2013 2y + 6z = \u20131\nSolve system of linear equations, using matrix method, in Exercises 7 to 14 7 5x + 2y = 4\n8"
    },
    {
        "Chapter": "1",
        "sentence_range": "1646-1649",
        "Text": "5x \u2013 y + 4z = 5\n2x + 3y + 2z = 2\n2y \u2013 z = \u20131\n2x + 3y + 5z = 2\nax + ay + 2az = 4\n3x \u2013 5y = 3\n5x \u2013 2y + 6z = \u20131\nSolve system of linear equations, using matrix method, in Exercises 7 to 14 7 5x + 2y = 4\n8 2x \u2013 y = \u20132\n9"
    },
    {
        "Chapter": "1",
        "sentence_range": "1647-1650",
        "Text": "7 5x + 2y = 4\n8 2x \u2013 y = \u20132\n9 4x \u2013 3y = 3\n7x + 3y = 5\n3x + 4y = 3\n3x \u2013 5y = 7\n10"
    },
    {
        "Chapter": "1",
        "sentence_range": "1648-1651",
        "Text": "5x + 2y = 4\n8 2x \u2013 y = \u20132\n9 4x \u2013 3y = 3\n7x + 3y = 5\n3x + 4y = 3\n3x \u2013 5y = 7\n10 5x + 2y = 3\n11"
    },
    {
        "Chapter": "1",
        "sentence_range": "1649-1652",
        "Text": "2x \u2013 y = \u20132\n9 4x \u2013 3y = 3\n7x + 3y = 5\n3x + 4y = 3\n3x \u2013 5y = 7\n10 5x + 2y = 3\n11 2x + y + z = 1\n12"
    },
    {
        "Chapter": "1",
        "sentence_range": "1650-1653",
        "Text": "4x \u2013 3y = 3\n7x + 3y = 5\n3x + 4y = 3\n3x \u2013 5y = 7\n10 5x + 2y = 3\n11 2x + y + z = 1\n12 x \u2013 y + z = 4\n3x + 2y = 5\nx \u2013 2y \u2013 z = 3\n2\n2x + y \u2013 3z = 0\n3y \u2013 5z = 9\nx + y + z = 2\n13"
    },
    {
        "Chapter": "1",
        "sentence_range": "1651-1654",
        "Text": "5x + 2y = 3\n11 2x + y + z = 1\n12 x \u2013 y + z = 4\n3x + 2y = 5\nx \u2013 2y \u2013 z = 3\n2\n2x + y \u2013 3z = 0\n3y \u2013 5z = 9\nx + y + z = 2\n13 2x + 3y +3 z = 5\n14"
    },
    {
        "Chapter": "1",
        "sentence_range": "1652-1655",
        "Text": "2x + y + z = 1\n12 x \u2013 y + z = 4\n3x + 2y = 5\nx \u2013 2y \u2013 z = 3\n2\n2x + y \u2013 3z = 0\n3y \u2013 5z = 9\nx + y + z = 2\n13 2x + 3y +3 z = 5\n14 x \u2013 y + 2z = 7\nx \u2013 2y + z = \u2013 4\n3x + 4y \u2013 5z = \u2013 5\n3x \u2013 y \u2013 2z = 3\n2x \u2013 y + 3z = 12\nRationalised 2023-24\n 98\nMATHEMATICS\n15"
    },
    {
        "Chapter": "1",
        "sentence_range": "1653-1656",
        "Text": "x \u2013 y + z = 4\n3x + 2y = 5\nx \u2013 2y \u2013 z = 3\n2\n2x + y \u2013 3z = 0\n3y \u2013 5z = 9\nx + y + z = 2\n13 2x + 3y +3 z = 5\n14 x \u2013 y + 2z = 7\nx \u2013 2y + z = \u2013 4\n3x + 4y \u2013 5z = \u2013 5\n3x \u2013 y \u2013 2z = 3\n2x \u2013 y + 3z = 12\nRationalised 2023-24\n 98\nMATHEMATICS\n15 If  A = \n2\n\u20133\n5\n3\n2\n\u2013 4\n1\n1\n\u20132\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, find A\u20131"
    },
    {
        "Chapter": "1",
        "sentence_range": "1654-1657",
        "Text": "2x + 3y +3 z = 5\n14 x \u2013 y + 2z = 7\nx \u2013 2y + z = \u2013 4\n3x + 4y \u2013 5z = \u2013 5\n3x \u2013 y \u2013 2z = 3\n2x \u2013 y + 3z = 12\nRationalised 2023-24\n 98\nMATHEMATICS\n15 If  A = \n2\n\u20133\n5\n3\n2\n\u2013 4\n1\n1\n\u20132\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, find A\u20131 Using A\u20131 solve the system of equations\n2x \u2013 3y + 5z = 11\n3x + 2y \u2013 4z =  \u2013 5\nx + y \u2013 2z =  \u2013 3\n16"
    },
    {
        "Chapter": "1",
        "sentence_range": "1655-1658",
        "Text": "x \u2013 y + 2z = 7\nx \u2013 2y + z = \u2013 4\n3x + 4y \u2013 5z = \u2013 5\n3x \u2013 y \u2013 2z = 3\n2x \u2013 y + 3z = 12\nRationalised 2023-24\n 98\nMATHEMATICS\n15 If  A = \n2\n\u20133\n5\n3\n2\n\u2013 4\n1\n1\n\u20132\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, find A\u20131 Using A\u20131 solve the system of equations\n2x \u2013 3y + 5z = 11\n3x + 2y \u2013 4z =  \u2013 5\nx + y \u2013 2z =  \u2013 3\n16 The cost of 4 kg onion, 3 kg wheat and 2 kg rice is ` 60"
    },
    {
        "Chapter": "1",
        "sentence_range": "1656-1659",
        "Text": "If  A = \n2\n\u20133\n5\n3\n2\n\u2013 4\n1\n1\n\u20132\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, find A\u20131 Using A\u20131 solve the system of equations\n2x \u2013 3y + 5z = 11\n3x + 2y \u2013 4z =  \u2013 5\nx + y \u2013 2z =  \u2013 3\n16 The cost of 4 kg onion, 3 kg wheat and 2 kg rice is ` 60 The cost of 2 kg onion,\n4 kg wheat and 6 kg rice is ` 90"
    },
    {
        "Chapter": "1",
        "sentence_range": "1657-1660",
        "Text": "Using A\u20131 solve the system of equations\n2x \u2013 3y + 5z = 11\n3x + 2y \u2013 4z =  \u2013 5\nx + y \u2013 2z =  \u2013 3\n16 The cost of 4 kg onion, 3 kg wheat and 2 kg rice is ` 60 The cost of 2 kg onion,\n4 kg wheat and 6 kg rice is ` 90 The cost of 6 kg onion 2 kg wheat and 3 kg rice\nis ` 70"
    },
    {
        "Chapter": "1",
        "sentence_range": "1658-1661",
        "Text": "The cost of 4 kg onion, 3 kg wheat and 2 kg rice is ` 60 The cost of 2 kg onion,\n4 kg wheat and 6 kg rice is ` 90 The cost of 6 kg onion 2 kg wheat and 3 kg rice\nis ` 70 Find cost of each item per kg by matrix method"
    },
    {
        "Chapter": "1",
        "sentence_range": "1659-1662",
        "Text": "The cost of 2 kg onion,\n4 kg wheat and 6 kg rice is ` 90 The cost of 6 kg onion 2 kg wheat and 3 kg rice\nis ` 70 Find cost of each item per kg by matrix method Miscellaneous Examples\nExample 19 Use product \n1\n1\n2\n0\n2\n3\n3\n2\n4\n2\n0\n1\n9\n2\n3\n6\n1\n2\n\ufffd\n\ufffd\n\ufffd\n\ufffd\n\ufffd\n\ufffd\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\n to solve the system of equations\nx \u2013 y + 2z = 1\n2y \u2013 3z = 1\n3x \u2013 2y + 4z = 2\nSolution Consider the product \n1\n1\n2\n2\n0\n1\n0\n2\n3\n9\n2\n3\n3\n2\n4\n6\n1\n2\n\u2013\n\u2013\n\u2013\n\u2013\n\u2013\n\u2013\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n =\n2\n9\n12\n0\n2\n2\n1\n3\n4\n0\n18 18\n0\n4\n3\n0\n6\n6\n6 18\n24\n0\n4\n4\n3\n6\n8\n\u2212 \u2212\n+\n\u2212\n+\n+\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n+\n\u2212\n+\n\u2212\n\u2212\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212 \u2212\n+\n\u2212\n+\n+\n\u2212\n\uf8f0\n\uf8fb\n = \n1\n0\n0\n0\n1\n0\n0\n0\n1\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\nHence\n  \n1\n1\n2\n0\n2\n3\n3\n2\n4\n2\n0\n1\n9\n2\n3\n6\n1\n2\n1\n\u2013\n\u2013\n\u2013\n\u2013\n\u2013\n\u2013\n\u2013\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\uf8fa\n=\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\nNow, given system of equations can be written, in matrix form, as follows\n1\n\u20131\n2\n0\n2\n\u20133\n3\n\u20132\n4\nx\ny\nz\n\uf8ee\n\uf8f9 \uf8ee \uf8f9\n\uf8ef\n\uf8fa \uf8ef \uf8fa\n\uf8ef\n\uf8fa \uf8ef \uf8fa\n\uf8ef\n\uf8fa \uf8ef \uf8fa\n\uf8f0 \uf8fb\n\uf8f0\n\uf8fb\n =\n1\n1\n2\n\uf8ee \uf8f9\n\uf8ef \uf8fa\n\uf8ef \uf8fa\n\uf8ef \uf8fa\n\uf8f0 \uf8fb\nRationalised 2023-24\nDETERMINANTS     99\nor\n  \n  \n  \n  \n  \nx\ny\nz\n =\n1\n1\n1\n2\n1\n0\n2\n3\n1\n3\n2\n4\n2\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee \uf8f9\n\uf8ef\n\uf8fa\n\uf8ef \uf8fa\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef \uf8fa\n\uf8ef\n\uf8fa\n\uf8ef \uf8fa\n\u2212\n\uf8f0\n\uf8fb\n\uf8f0 \uf8fb\n = \n\ufffd\n\ufffd\n\ufffd\n2\n0\n1\n9\n2\n3\n6\n1\n2\n1\n1\n2\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\n=\n2\n0\n2\n0\n9\n2\n6\n5\n6\n1\n4\n3\n\u2212 +\n+\n\uf8ee\n\uf8f9\n\uf8ee \uf8f9\n\uf8ef\n\uf8fa\n\uf8ef \uf8fa\n+\n\u2212\n=\n\uf8ef\n\uf8fa\n\uf8ef \uf8fa\n\uf8ef\n\uf8fa\n\uf8ef \uf8fa\n+ \u2212\n\uf8f0\n\uf8fb\n\uf8f0 \uf8fb\nHence\nx = 0, y = 5 and z = 3\nMiscellaneous Exercises on Chapter 4\n1"
    },
    {
        "Chapter": "1",
        "sentence_range": "1660-1663",
        "Text": "The cost of 6 kg onion 2 kg wheat and 3 kg rice\nis ` 70 Find cost of each item per kg by matrix method Miscellaneous Examples\nExample 19 Use product \n1\n1\n2\n0\n2\n3\n3\n2\n4\n2\n0\n1\n9\n2\n3\n6\n1\n2\n\ufffd\n\ufffd\n\ufffd\n\ufffd\n\ufffd\n\ufffd\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\n to solve the system of equations\nx \u2013 y + 2z = 1\n2y \u2013 3z = 1\n3x \u2013 2y + 4z = 2\nSolution Consider the product \n1\n1\n2\n2\n0\n1\n0\n2\n3\n9\n2\n3\n3\n2\n4\n6\n1\n2\n\u2013\n\u2013\n\u2013\n\u2013\n\u2013\n\u2013\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n =\n2\n9\n12\n0\n2\n2\n1\n3\n4\n0\n18 18\n0\n4\n3\n0\n6\n6\n6 18\n24\n0\n4\n4\n3\n6\n8\n\u2212 \u2212\n+\n\u2212\n+\n+\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n+\n\u2212\n+\n\u2212\n\u2212\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212 \u2212\n+\n\u2212\n+\n+\n\u2212\n\uf8f0\n\uf8fb\n = \n1\n0\n0\n0\n1\n0\n0\n0\n1\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\nHence\n  \n1\n1\n2\n0\n2\n3\n3\n2\n4\n2\n0\n1\n9\n2\n3\n6\n1\n2\n1\n\u2013\n\u2013\n\u2013\n\u2013\n\u2013\n\u2013\n\u2013\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\uf8fa\n=\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\nNow, given system of equations can be written, in matrix form, as follows\n1\n\u20131\n2\n0\n2\n\u20133\n3\n\u20132\n4\nx\ny\nz\n\uf8ee\n\uf8f9 \uf8ee \uf8f9\n\uf8ef\n\uf8fa \uf8ef \uf8fa\n\uf8ef\n\uf8fa \uf8ef \uf8fa\n\uf8ef\n\uf8fa \uf8ef \uf8fa\n\uf8f0 \uf8fb\n\uf8f0\n\uf8fb\n =\n1\n1\n2\n\uf8ee \uf8f9\n\uf8ef \uf8fa\n\uf8ef \uf8fa\n\uf8ef \uf8fa\n\uf8f0 \uf8fb\nRationalised 2023-24\nDETERMINANTS     99\nor\n  \n  \n  \n  \n  \nx\ny\nz\n =\n1\n1\n1\n2\n1\n0\n2\n3\n1\n3\n2\n4\n2\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee \uf8f9\n\uf8ef\n\uf8fa\n\uf8ef \uf8fa\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef \uf8fa\n\uf8ef\n\uf8fa\n\uf8ef \uf8fa\n\u2212\n\uf8f0\n\uf8fb\n\uf8f0 \uf8fb\n = \n\ufffd\n\ufffd\n\ufffd\n2\n0\n1\n9\n2\n3\n6\n1\n2\n1\n1\n2\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\n=\n2\n0\n2\n0\n9\n2\n6\n5\n6\n1\n4\n3\n\u2212 +\n+\n\uf8ee\n\uf8f9\n\uf8ee \uf8f9\n\uf8ef\n\uf8fa\n\uf8ef \uf8fa\n+\n\u2212\n=\n\uf8ef\n\uf8fa\n\uf8ef \uf8fa\n\uf8ef\n\uf8fa\n\uf8ef \uf8fa\n+ \u2212\n\uf8f0\n\uf8fb\n\uf8f0 \uf8fb\nHence\nx = 0, y = 5 and z = 3\nMiscellaneous Exercises on Chapter 4\n1 Prove that the determinant\nsin\ncos\n\u2013sin\n\u2013\n1\ncos\n1\nx\nx\nx\n\u03b8\n\u03b8\n\u03b8\n\u03b8\n is independent of \u03b8"
    },
    {
        "Chapter": "1",
        "sentence_range": "1661-1664",
        "Text": "Find cost of each item per kg by matrix method Miscellaneous Examples\nExample 19 Use product \n1\n1\n2\n0\n2\n3\n3\n2\n4\n2\n0\n1\n9\n2\n3\n6\n1\n2\n\ufffd\n\ufffd\n\ufffd\n\ufffd\n\ufffd\n\ufffd\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\n to solve the system of equations\nx \u2013 y + 2z = 1\n2y \u2013 3z = 1\n3x \u2013 2y + 4z = 2\nSolution Consider the product \n1\n1\n2\n2\n0\n1\n0\n2\n3\n9\n2\n3\n3\n2\n4\n6\n1\n2\n\u2013\n\u2013\n\u2013\n\u2013\n\u2013\n\u2013\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n =\n2\n9\n12\n0\n2\n2\n1\n3\n4\n0\n18 18\n0\n4\n3\n0\n6\n6\n6 18\n24\n0\n4\n4\n3\n6\n8\n\u2212 \u2212\n+\n\u2212\n+\n+\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n+\n\u2212\n+\n\u2212\n\u2212\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212 \u2212\n+\n\u2212\n+\n+\n\u2212\n\uf8f0\n\uf8fb\n = \n1\n0\n0\n0\n1\n0\n0\n0\n1\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\nHence\n  \n1\n1\n2\n0\n2\n3\n3\n2\n4\n2\n0\n1\n9\n2\n3\n6\n1\n2\n1\n\u2013\n\u2013\n\u2013\n\u2013\n\u2013\n\u2013\n\u2013\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\uf8fa\n=\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\nNow, given system of equations can be written, in matrix form, as follows\n1\n\u20131\n2\n0\n2\n\u20133\n3\n\u20132\n4\nx\ny\nz\n\uf8ee\n\uf8f9 \uf8ee \uf8f9\n\uf8ef\n\uf8fa \uf8ef \uf8fa\n\uf8ef\n\uf8fa \uf8ef \uf8fa\n\uf8ef\n\uf8fa \uf8ef \uf8fa\n\uf8f0 \uf8fb\n\uf8f0\n\uf8fb\n =\n1\n1\n2\n\uf8ee \uf8f9\n\uf8ef \uf8fa\n\uf8ef \uf8fa\n\uf8ef \uf8fa\n\uf8f0 \uf8fb\nRationalised 2023-24\nDETERMINANTS     99\nor\n  \n  \n  \n  \n  \nx\ny\nz\n =\n1\n1\n1\n2\n1\n0\n2\n3\n1\n3\n2\n4\n2\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee \uf8f9\n\uf8ef\n\uf8fa\n\uf8ef \uf8fa\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef \uf8fa\n\uf8ef\n\uf8fa\n\uf8ef \uf8fa\n\u2212\n\uf8f0\n\uf8fb\n\uf8f0 \uf8fb\n = \n\ufffd\n\ufffd\n\ufffd\n2\n0\n1\n9\n2\n3\n6\n1\n2\n1\n1\n2\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\n=\n2\n0\n2\n0\n9\n2\n6\n5\n6\n1\n4\n3\n\u2212 +\n+\n\uf8ee\n\uf8f9\n\uf8ee \uf8f9\n\uf8ef\n\uf8fa\n\uf8ef \uf8fa\n+\n\u2212\n=\n\uf8ef\n\uf8fa\n\uf8ef \uf8fa\n\uf8ef\n\uf8fa\n\uf8ef \uf8fa\n+ \u2212\n\uf8f0\n\uf8fb\n\uf8f0 \uf8fb\nHence\nx = 0, y = 5 and z = 3\nMiscellaneous Exercises on Chapter 4\n1 Prove that the determinant\nsin\ncos\n\u2013sin\n\u2013\n1\ncos\n1\nx\nx\nx\n\u03b8\n\u03b8\n\u03b8\n\u03b8\n is independent of \u03b8 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "1662-1665",
        "Text": "Miscellaneous Examples\nExample 19 Use product \n1\n1\n2\n0\n2\n3\n3\n2\n4\n2\n0\n1\n9\n2\n3\n6\n1\n2\n\ufffd\n\ufffd\n\ufffd\n\ufffd\n\ufffd\n\ufffd\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\n to solve the system of equations\nx \u2013 y + 2z = 1\n2y \u2013 3z = 1\n3x \u2013 2y + 4z = 2\nSolution Consider the product \n1\n1\n2\n2\n0\n1\n0\n2\n3\n9\n2\n3\n3\n2\n4\n6\n1\n2\n\u2013\n\u2013\n\u2013\n\u2013\n\u2013\n\u2013\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n =\n2\n9\n12\n0\n2\n2\n1\n3\n4\n0\n18 18\n0\n4\n3\n0\n6\n6\n6 18\n24\n0\n4\n4\n3\n6\n8\n\u2212 \u2212\n+\n\u2212\n+\n+\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n+\n\u2212\n+\n\u2212\n\u2212\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212 \u2212\n+\n\u2212\n+\n+\n\u2212\n\uf8f0\n\uf8fb\n = \n1\n0\n0\n0\n1\n0\n0\n0\n1\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\nHence\n  \n1\n1\n2\n0\n2\n3\n3\n2\n4\n2\n0\n1\n9\n2\n3\n6\n1\n2\n1\n\u2013\n\u2013\n\u2013\n\u2013\n\u2013\n\u2013\n\u2013\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\uf8fa\n=\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\nNow, given system of equations can be written, in matrix form, as follows\n1\n\u20131\n2\n0\n2\n\u20133\n3\n\u20132\n4\nx\ny\nz\n\uf8ee\n\uf8f9 \uf8ee \uf8f9\n\uf8ef\n\uf8fa \uf8ef \uf8fa\n\uf8ef\n\uf8fa \uf8ef \uf8fa\n\uf8ef\n\uf8fa \uf8ef \uf8fa\n\uf8f0 \uf8fb\n\uf8f0\n\uf8fb\n =\n1\n1\n2\n\uf8ee \uf8f9\n\uf8ef \uf8fa\n\uf8ef \uf8fa\n\uf8ef \uf8fa\n\uf8f0 \uf8fb\nRationalised 2023-24\nDETERMINANTS     99\nor\n  \n  \n  \n  \n  \nx\ny\nz\n =\n1\n1\n1\n2\n1\n0\n2\n3\n1\n3\n2\n4\n2\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee \uf8f9\n\uf8ef\n\uf8fa\n\uf8ef \uf8fa\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef \uf8fa\n\uf8ef\n\uf8fa\n\uf8ef \uf8fa\n\u2212\n\uf8f0\n\uf8fb\n\uf8f0 \uf8fb\n = \n\ufffd\n\ufffd\n\ufffd\n2\n0\n1\n9\n2\n3\n6\n1\n2\n1\n1\n2\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\n=\n2\n0\n2\n0\n9\n2\n6\n5\n6\n1\n4\n3\n\u2212 +\n+\n\uf8ee\n\uf8f9\n\uf8ee \uf8f9\n\uf8ef\n\uf8fa\n\uf8ef \uf8fa\n+\n\u2212\n=\n\uf8ef\n\uf8fa\n\uf8ef \uf8fa\n\uf8ef\n\uf8fa\n\uf8ef \uf8fa\n+ \u2212\n\uf8f0\n\uf8fb\n\uf8f0 \uf8fb\nHence\nx = 0, y = 5 and z = 3\nMiscellaneous Exercises on Chapter 4\n1 Prove that the determinant\nsin\ncos\n\u2013sin\n\u2013\n1\ncos\n1\nx\nx\nx\n\u03b8\n\u03b8\n\u03b8\n\u03b8\n is independent of \u03b8 2 Evaluate \ncos\ncos\ncos\nsin\n\u2013sin\n\u2013sin\ncos\n0\nsin\ncos\nsin\nsin\ncos\n\u03b1\n\u03b2\n\u03b1\n\u03b2\n\u03b1\n\u03b2\n\u03b2\n\u03b1\n\u03b2\n\u03b1\n\u03b2\n\u03b1"
    },
    {
        "Chapter": "1",
        "sentence_range": "1663-1666",
        "Text": "Prove that the determinant\nsin\ncos\n\u2013sin\n\u2013\n1\ncos\n1\nx\nx\nx\n\u03b8\n\u03b8\n\u03b8\n\u03b8\n is independent of \u03b8 2 Evaluate \ncos\ncos\ncos\nsin\n\u2013sin\n\u2013sin\ncos\n0\nsin\ncos\nsin\nsin\ncos\n\u03b1\n\u03b2\n\u03b1\n\u03b2\n\u03b1\n\u03b2\n\u03b2\n\u03b1\n\u03b2\n\u03b1\n\u03b2\n\u03b1 3"
    },
    {
        "Chapter": "1",
        "sentence_range": "1664-1667",
        "Text": "2 Evaluate \ncos\ncos\ncos\nsin\n\u2013sin\n\u2013sin\ncos\n0\nsin\ncos\nsin\nsin\ncos\n\u03b1\n\u03b2\n\u03b1\n\u03b2\n\u03b1\n\u03b2\n\u03b2\n\u03b1\n\u03b2\n\u03b1\n\u03b2\n\u03b1 3 If A\u20131 = \n(\n)\n1\n3\n1\n1\n1\n2\n2\n15\n6\n5 and B\n1\n3\n0\n, find AB\n5\n2\n2\n0\n2\n1\n\u2013\n\u2013\n\u2013\n\u2013\n\u2013\n\u2013\n\u2013\n\u2013\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n4"
    },
    {
        "Chapter": "1",
        "sentence_range": "1665-1668",
        "Text": "Evaluate \ncos\ncos\ncos\nsin\n\u2013sin\n\u2013sin\ncos\n0\nsin\ncos\nsin\nsin\ncos\n\u03b1\n\u03b2\n\u03b1\n\u03b2\n\u03b1\n\u03b2\n\u03b2\n\u03b1\n\u03b2\n\u03b1\n\u03b2\n\u03b1 3 If A\u20131 = \n(\n)\n1\n3\n1\n1\n1\n2\n2\n15\n6\n5 and B\n1\n3\n0\n, find AB\n5\n2\n2\n0\n2\n1\n\u2013\n\u2013\n\u2013\n\u2013\n\u2013\n\u2013\n\u2013\n\u2013\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n4 Let A = \n1\n2\n1\n2\n3\n1\n1\n1\n5\n\ufffd\n\ufffd\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa"
    },
    {
        "Chapter": "1",
        "sentence_range": "1666-1669",
        "Text": "3 If A\u20131 = \n(\n)\n1\n3\n1\n1\n1\n2\n2\n15\n6\n5 and B\n1\n3\n0\n, find AB\n5\n2\n2\n0\n2\n1\n\u2013\n\u2013\n\u2013\n\u2013\n\u2013\n\u2013\n\u2013\n\u2013\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n4 Let A = \n1\n2\n1\n2\n3\n1\n1\n1\n5\n\ufffd\n\ufffd\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa Verify that\n(i) [adj A]\u20131 = adj (A\u20131)\n(ii) (A\u20131)\u20131 = A\n5"
    },
    {
        "Chapter": "1",
        "sentence_range": "1667-1670",
        "Text": "If A\u20131 = \n(\n)\n1\n3\n1\n1\n1\n2\n2\n15\n6\n5 and B\n1\n3\n0\n, find AB\n5\n2\n2\n0\n2\n1\n\u2013\n\u2013\n\u2013\n\u2013\n\u2013\n\u2013\n\u2013\n\u2013\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n4 Let A = \n1\n2\n1\n2\n3\n1\n1\n1\n5\n\ufffd\n\ufffd\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa Verify that\n(i) [adj A]\u20131 = adj (A\u20131)\n(ii) (A\u20131)\u20131 = A\n5 Evaluate \nx\ny\nx\ny\ny\nx\ny\nx\nx\ny\nx\ny\n+\n+\n+\n6"
    },
    {
        "Chapter": "1",
        "sentence_range": "1668-1671",
        "Text": "Let A = \n1\n2\n1\n2\n3\n1\n1\n1\n5\n\ufffd\n\ufffd\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa Verify that\n(i) [adj A]\u20131 = adj (A\u20131)\n(ii) (A\u20131)\u20131 = A\n5 Evaluate \nx\ny\nx\ny\ny\nx\ny\nx\nx\ny\nx\ny\n+\n+\n+\n6 Evaluate \n1\n1\n1\nx\ny\nx\ny\ny\nx\nx+ y\n+\nRationalised 2023-24\n 100\nMATHEMATICS\nUsing properties  of determinants in Exercises 11 to 15, prove that:\n7"
    },
    {
        "Chapter": "1",
        "sentence_range": "1669-1672",
        "Text": "Verify that\n(i) [adj A]\u20131 = adj (A\u20131)\n(ii) (A\u20131)\u20131 = A\n5 Evaluate \nx\ny\nx\ny\ny\nx\ny\nx\nx\ny\nx\ny\n+\n+\n+\n6 Evaluate \n1\n1\n1\nx\ny\nx\ny\ny\nx\nx+ y\n+\nRationalised 2023-24\n 100\nMATHEMATICS\nUsing properties  of determinants in Exercises 11 to 15, prove that:\n7 Solve the system of equations\n2\n3\n10\n4\n+\n+\n=\nx\ny\nz\n4\n6\n5\n1\n+\n=\nx\u2013\ny\nz\n6\n9\n20\n2\n+\n=\n\u2013\nx\ny\nz\nChoose the correct answer in Exercise 17 to 19"
    },
    {
        "Chapter": "1",
        "sentence_range": "1670-1673",
        "Text": "Evaluate \nx\ny\nx\ny\ny\nx\ny\nx\nx\ny\nx\ny\n+\n+\n+\n6 Evaluate \n1\n1\n1\nx\ny\nx\ny\ny\nx\nx+ y\n+\nRationalised 2023-24\n 100\nMATHEMATICS\nUsing properties  of determinants in Exercises 11 to 15, prove that:\n7 Solve the system of equations\n2\n3\n10\n4\n+\n+\n=\nx\ny\nz\n4\n6\n5\n1\n+\n=\nx\u2013\ny\nz\n6\n9\n20\n2\n+\n=\n\u2013\nx\ny\nz\nChoose the correct answer in Exercise 17 to 19 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "1671-1674",
        "Text": "Evaluate \n1\n1\n1\nx\ny\nx\ny\ny\nx\nx+ y\n+\nRationalised 2023-24\n 100\nMATHEMATICS\nUsing properties  of determinants in Exercises 11 to 15, prove that:\n7 Solve the system of equations\n2\n3\n10\n4\n+\n+\n=\nx\ny\nz\n4\n6\n5\n1\n+\n=\nx\u2013\ny\nz\n6\n9\n20\n2\n+\n=\n\u2013\nx\ny\nz\nChoose the correct answer in Exercise 17 to 19 8 If  x, y, z are nonzero real numbers, then the inverse of matrix \n0\n0\nA\n0\n0\n0\n0\nx\ny\nz\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n= \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nis\n(A)\n1\n1\n1\n0\n0\n0\n0\n0\n0\nx\ny\nz\n\u2212\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n(B)\n1\n1\n1\n0\n0\n0\n0\n0\n0\nx\nxyz\ny\nz\n\u2212\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n(C)\n0\n0\n1\n0\n0\n0\n0\nx\ny\nxyz\nz\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n(D)\n1\n0\n0\n1\n0\n1\n0\n0\n0\n1\nxyz\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n9"
    },
    {
        "Chapter": "1",
        "sentence_range": "1672-1675",
        "Text": "Solve the system of equations\n2\n3\n10\n4\n+\n+\n=\nx\ny\nz\n4\n6\n5\n1\n+\n=\nx\u2013\ny\nz\n6\n9\n20\n2\n+\n=\n\u2013\nx\ny\nz\nChoose the correct answer in Exercise 17 to 19 8 If  x, y, z are nonzero real numbers, then the inverse of matrix \n0\n0\nA\n0\n0\n0\n0\nx\ny\nz\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n= \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nis\n(A)\n1\n1\n1\n0\n0\n0\n0\n0\n0\nx\ny\nz\n\u2212\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n(B)\n1\n1\n1\n0\n0\n0\n0\n0\n0\nx\nxyz\ny\nz\n\u2212\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n(C)\n0\n0\n1\n0\n0\n0\n0\nx\ny\nxyz\nz\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n(D)\n1\n0\n0\n1\n0\n1\n0\n0\n0\n1\nxyz\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n9 Let A = \n1\nsin\n1\nsin\n1\nsin\n1\nsin\n1\n\u03b8\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n\u03b8\n\u03b8\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\u03b8\n\uf8f0\n\uf8fb\n, where 0 \u2264 \u03b8 \u2264 2\u03c0"
    },
    {
        "Chapter": "1",
        "sentence_range": "1673-1676",
        "Text": "8 If  x, y, z are nonzero real numbers, then the inverse of matrix \n0\n0\nA\n0\n0\n0\n0\nx\ny\nz\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n= \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nis\n(A)\n1\n1\n1\n0\n0\n0\n0\n0\n0\nx\ny\nz\n\u2212\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n(B)\n1\n1\n1\n0\n0\n0\n0\n0\n0\nx\nxyz\ny\nz\n\u2212\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n(C)\n0\n0\n1\n0\n0\n0\n0\nx\ny\nxyz\nz\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n(D)\n1\n0\n0\n1\n0\n1\n0\n0\n0\n1\nxyz\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n9 Let A = \n1\nsin\n1\nsin\n1\nsin\n1\nsin\n1\n\u03b8\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n\u03b8\n\u03b8\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\u03b8\n\uf8f0\n\uf8fb\n, where 0 \u2264 \u03b8 \u2264 2\u03c0 Then\n(A) Det (A) = 0\n(B) Det (A) \u2208 (2, \u221e)\n(C) Det (A) \u2208 (2, 4)\n(D) Det (A) \u2208 [2, 4]\nRationalised 2023-24\nDETERMINANTS     101\nSummary\n\u00ae Determinant of a matrix A = [a11]1\u00d71 is given by |a11| = a11\n\u00ae Determinant of a matrix A = \uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa\na\na\na\na\n11\n12\n21\n22\n is given by\n11\n12\n21\n22\nA\na\na\na\na\n=\n= a11 a22 \u2013 a12 a21\n\u00ae Determinant of a matrix A =\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\na\nb\nc\na\nb\nc\na\nb\nc\n1\n1\n1\n2\n2\n2\n3\n3\n3\n is given by (expanding along R1)\n1\n1\n1\n2\n2\n2\n2\n2\n2\n2\n2\n2\n1\n1\n1\n3\n3\n3\n3\n3\n3\n3\n3\n3\nA\na\nb\nc\nb\nc\na\nc\na\nb\na\nb\nc\na\nb\nc\nb\nc\na\nc\na\nb\na\nb\nc\n=\n=\n\u2212\n+\nFor any square matrix A, the |A| satisfy following properties"
    },
    {
        "Chapter": "1",
        "sentence_range": "1674-1677",
        "Text": "If  x, y, z are nonzero real numbers, then the inverse of matrix \n0\n0\nA\n0\n0\n0\n0\nx\ny\nz\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n= \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nis\n(A)\n1\n1\n1\n0\n0\n0\n0\n0\n0\nx\ny\nz\n\u2212\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n(B)\n1\n1\n1\n0\n0\n0\n0\n0\n0\nx\nxyz\ny\nz\n\u2212\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n(C)\n0\n0\n1\n0\n0\n0\n0\nx\ny\nxyz\nz\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n(D)\n1\n0\n0\n1\n0\n1\n0\n0\n0\n1\nxyz\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n9 Let A = \n1\nsin\n1\nsin\n1\nsin\n1\nsin\n1\n\u03b8\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n\u03b8\n\u03b8\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\u03b8\n\uf8f0\n\uf8fb\n, where 0 \u2264 \u03b8 \u2264 2\u03c0 Then\n(A) Det (A) = 0\n(B) Det (A) \u2208 (2, \u221e)\n(C) Det (A) \u2208 (2, 4)\n(D) Det (A) \u2208 [2, 4]\nRationalised 2023-24\nDETERMINANTS     101\nSummary\n\u00ae Determinant of a matrix A = [a11]1\u00d71 is given by |a11| = a11\n\u00ae Determinant of a matrix A = \uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa\na\na\na\na\n11\n12\n21\n22\n is given by\n11\n12\n21\n22\nA\na\na\na\na\n=\n= a11 a22 \u2013 a12 a21\n\u00ae Determinant of a matrix A =\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\na\nb\nc\na\nb\nc\na\nb\nc\n1\n1\n1\n2\n2\n2\n3\n3\n3\n is given by (expanding along R1)\n1\n1\n1\n2\n2\n2\n2\n2\n2\n2\n2\n2\n1\n1\n1\n3\n3\n3\n3\n3\n3\n3\n3\n3\nA\na\nb\nc\nb\nc\na\nc\na\nb\na\nb\nc\na\nb\nc\nb\nc\na\nc\na\nb\na\nb\nc\n=\n=\n\u2212\n+\nFor any square matrix A, the |A| satisfy following properties \u00ae Area of a triangle with vertices (x1, y1), (x2, y2) and (x3, y3) is given by\n1\n1\n2\n2\n3\n3\n1\n1\n1\n2\n1\nx\ny\nx\ny\nx\ny\n\u2206=\n\u00ae Minor of an element aij of the determinant of matrix A is the determinant\nobtained by deleting ith row and jth column and denoted by Mij"
    },
    {
        "Chapter": "1",
        "sentence_range": "1675-1678",
        "Text": "Let A = \n1\nsin\n1\nsin\n1\nsin\n1\nsin\n1\n\u03b8\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u2212\n\u03b8\n\u03b8\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u2212\n\u2212\n\u03b8\n\uf8f0\n\uf8fb\n, where 0 \u2264 \u03b8 \u2264 2\u03c0 Then\n(A) Det (A) = 0\n(B) Det (A) \u2208 (2, \u221e)\n(C) Det (A) \u2208 (2, 4)\n(D) Det (A) \u2208 [2, 4]\nRationalised 2023-24\nDETERMINANTS     101\nSummary\n\u00ae Determinant of a matrix A = [a11]1\u00d71 is given by |a11| = a11\n\u00ae Determinant of a matrix A = \uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa\na\na\na\na\n11\n12\n21\n22\n is given by\n11\n12\n21\n22\nA\na\na\na\na\n=\n= a11 a22 \u2013 a12 a21\n\u00ae Determinant of a matrix A =\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\na\nb\nc\na\nb\nc\na\nb\nc\n1\n1\n1\n2\n2\n2\n3\n3\n3\n is given by (expanding along R1)\n1\n1\n1\n2\n2\n2\n2\n2\n2\n2\n2\n2\n1\n1\n1\n3\n3\n3\n3\n3\n3\n3\n3\n3\nA\na\nb\nc\nb\nc\na\nc\na\nb\na\nb\nc\na\nb\nc\nb\nc\na\nc\na\nb\na\nb\nc\n=\n=\n\u2212\n+\nFor any square matrix A, the |A| satisfy following properties \u00ae Area of a triangle with vertices (x1, y1), (x2, y2) and (x3, y3) is given by\n1\n1\n2\n2\n3\n3\n1\n1\n1\n2\n1\nx\ny\nx\ny\nx\ny\n\u2206=\n\u00ae Minor of an element aij of the determinant of matrix A is the determinant\nobtained by deleting ith row and jth column and denoted by Mij \u00ae Cofactor of aij of given by Aij = (\u2013 1)i+ j Mij\n\u00ae Value of determinant of a matrix A is obtained by sum of product of elements\nof a row (or a column) with corresponding cofactors"
    },
    {
        "Chapter": "1",
        "sentence_range": "1676-1679",
        "Text": "Then\n(A) Det (A) = 0\n(B) Det (A) \u2208 (2, \u221e)\n(C) Det (A) \u2208 (2, 4)\n(D) Det (A) \u2208 [2, 4]\nRationalised 2023-24\nDETERMINANTS     101\nSummary\n\u00ae Determinant of a matrix A = [a11]1\u00d71 is given by |a11| = a11\n\u00ae Determinant of a matrix A = \uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa\na\na\na\na\n11\n12\n21\n22\n is given by\n11\n12\n21\n22\nA\na\na\na\na\n=\n= a11 a22 \u2013 a12 a21\n\u00ae Determinant of a matrix A =\n\uf8ee\n\uf8f0\n\uf8ef\uf8ef\n\uf8ef\n\uf8f9\n\uf8fb\n\uf8fa\uf8fa\n\uf8fa\na\nb\nc\na\nb\nc\na\nb\nc\n1\n1\n1\n2\n2\n2\n3\n3\n3\n is given by (expanding along R1)\n1\n1\n1\n2\n2\n2\n2\n2\n2\n2\n2\n2\n1\n1\n1\n3\n3\n3\n3\n3\n3\n3\n3\n3\nA\na\nb\nc\nb\nc\na\nc\na\nb\na\nb\nc\na\nb\nc\nb\nc\na\nc\na\nb\na\nb\nc\n=\n=\n\u2212\n+\nFor any square matrix A, the |A| satisfy following properties \u00ae Area of a triangle with vertices (x1, y1), (x2, y2) and (x3, y3) is given by\n1\n1\n2\n2\n3\n3\n1\n1\n1\n2\n1\nx\ny\nx\ny\nx\ny\n\u2206=\n\u00ae Minor of an element aij of the determinant of matrix A is the determinant\nobtained by deleting ith row and jth column and denoted by Mij \u00ae Cofactor of aij of given by Aij = (\u2013 1)i+ j Mij\n\u00ae Value of determinant of a matrix A is obtained by sum of product of elements\nof a row (or a column) with corresponding cofactors For example,\nA = a11 A11 + a12 A12 + a13 A13"
    },
    {
        "Chapter": "1",
        "sentence_range": "1677-1680",
        "Text": "\u00ae Area of a triangle with vertices (x1, y1), (x2, y2) and (x3, y3) is given by\n1\n1\n2\n2\n3\n3\n1\n1\n1\n2\n1\nx\ny\nx\ny\nx\ny\n\u2206=\n\u00ae Minor of an element aij of the determinant of matrix A is the determinant\nobtained by deleting ith row and jth column and denoted by Mij \u00ae Cofactor of aij of given by Aij = (\u2013 1)i+ j Mij\n\u00ae Value of determinant of a matrix A is obtained by sum of product of elements\nof a row (or a column) with corresponding cofactors For example,\nA = a11 A11 + a12 A12 + a13 A13 \u00ae If elements of one row (or column) are multiplied with cofactors of elements\nof any other row (or column), then their sum is zero"
    },
    {
        "Chapter": "1",
        "sentence_range": "1678-1681",
        "Text": "\u00ae Cofactor of aij of given by Aij = (\u2013 1)i+ j Mij\n\u00ae Value of determinant of a matrix A is obtained by sum of product of elements\nof a row (or a column) with corresponding cofactors For example,\nA = a11 A11 + a12 A12 + a13 A13 \u00ae If elements of one row (or column) are multiplied with cofactors of elements\nof any other row (or column), then their sum is zero For example, a11 A21 + a12\nA22 + a13 A23 = 0\nRationalised 2023-24\n 102\nMATHEMATICS\n\u00ae If \n11\n12\n13\n21\n22\n23\n31\n32\n33\nA\n,\na\na\na\na\na\na\na\na\na\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n= \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n then \n11\n21\n31\n12\n22\n32\n13\n23\n33\nA\nA\nA\nA\nA\nA\nA\nA\nA\nA\nadj\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n= \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, where Aij is\ncofactor of aij\n\u00ae A (adj A) = (adj A) A = |A| I, where A is square matrix of order n"
    },
    {
        "Chapter": "1",
        "sentence_range": "1679-1682",
        "Text": "For example,\nA = a11 A11 + a12 A12 + a13 A13 \u00ae If elements of one row (or column) are multiplied with cofactors of elements\nof any other row (or column), then their sum is zero For example, a11 A21 + a12\nA22 + a13 A23 = 0\nRationalised 2023-24\n 102\nMATHEMATICS\n\u00ae If \n11\n12\n13\n21\n22\n23\n31\n32\n33\nA\n,\na\na\na\na\na\na\na\na\na\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n= \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n then \n11\n21\n31\n12\n22\n32\n13\n23\n33\nA\nA\nA\nA\nA\nA\nA\nA\nA\nA\nadj\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n= \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, where Aij is\ncofactor of aij\n\u00ae A (adj A) = (adj A) A = |A| I, where A is square matrix of order n \u00ae A square matrix A is said to be singular or non-singular according as\n|A| = 0 or |A| \u2260 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "1680-1683",
        "Text": "\u00ae If elements of one row (or column) are multiplied with cofactors of elements\nof any other row (or column), then their sum is zero For example, a11 A21 + a12\nA22 + a13 A23 = 0\nRationalised 2023-24\n 102\nMATHEMATICS\n\u00ae If \n11\n12\n13\n21\n22\n23\n31\n32\n33\nA\n,\na\na\na\na\na\na\na\na\na\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n= \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n then \n11\n21\n31\n12\n22\n32\n13\n23\n33\nA\nA\nA\nA\nA\nA\nA\nA\nA\nA\nadj\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n= \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, where Aij is\ncofactor of aij\n\u00ae A (adj A) = (adj A) A = |A| I, where A is square matrix of order n \u00ae A square matrix A is said to be singular or non-singular according as\n|A| = 0 or |A| \u2260 0 \u00ae If AB = BA = I, where B is square matrix, then B is called inverse of A"
    },
    {
        "Chapter": "1",
        "sentence_range": "1681-1684",
        "Text": "For example, a11 A21 + a12\nA22 + a13 A23 = 0\nRationalised 2023-24\n 102\nMATHEMATICS\n\u00ae If \n11\n12\n13\n21\n22\n23\n31\n32\n33\nA\n,\na\na\na\na\na\na\na\na\na\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n= \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n then \n11\n21\n31\n12\n22\n32\n13\n23\n33\nA\nA\nA\nA\nA\nA\nA\nA\nA\nA\nadj\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n= \uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n, where Aij is\ncofactor of aij\n\u00ae A (adj A) = (adj A) A = |A| I, where A is square matrix of order n \u00ae A square matrix A is said to be singular or non-singular according as\n|A| = 0 or |A| \u2260 0 \u00ae If AB = BA = I, where B is square matrix, then B is called inverse of A Also A\u20131 = B or B\u20131 = A and hence (A\u20131)\u20131 = A"
    },
    {
        "Chapter": "1",
        "sentence_range": "1682-1685",
        "Text": "\u00ae A square matrix A is said to be singular or non-singular according as\n|A| = 0 or |A| \u2260 0 \u00ae If AB = BA = I, where B is square matrix, then B is called inverse of A Also A\u20131 = B or B\u20131 = A and hence (A\u20131)\u20131 = A \u00ae A square matrix A has inverse if and only if A is non-singular"
    },
    {
        "Chapter": "1",
        "sentence_range": "1683-1686",
        "Text": "\u00ae If AB = BA = I, where B is square matrix, then B is called inverse of A Also A\u20131 = B or B\u20131 = A and hence (A\u20131)\u20131 = A \u00ae A square matrix A has inverse if and only if A is non-singular \u00ae\n\u20131\n1\nA\n(\nA adjA)\n=\n\u00ae If\na1 x + b1 y + c1 z = d1\na2 x + b2 y + c2 z = d2\na3 x + b3 y + c3 z = d3,\nthen these equations can be written as A X = B, where\n1\n1\n1\n1\n2\n2\n2\n2\n3\n3\n3\n3\nA\n,X=\nand B=\na\nb\nc\nx\nd\na\nb\nc\ny\nd\na\nb\nc\nz\nd\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee \uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef \uf8fa\n=\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef \uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef \uf8fa\n\uf8f0 \uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\u00ae Unique solution of equation AX = B is given by X = A\u20131 B, where A\n\u22600"
    },
    {
        "Chapter": "1",
        "sentence_range": "1684-1687",
        "Text": "Also A\u20131 = B or B\u20131 = A and hence (A\u20131)\u20131 = A \u00ae A square matrix A has inverse if and only if A is non-singular \u00ae\n\u20131\n1\nA\n(\nA adjA)\n=\n\u00ae If\na1 x + b1 y + c1 z = d1\na2 x + b2 y + c2 z = d2\na3 x + b3 y + c3 z = d3,\nthen these equations can be written as A X = B, where\n1\n1\n1\n1\n2\n2\n2\n2\n3\n3\n3\n3\nA\n,X=\nand B=\na\nb\nc\nx\nd\na\nb\nc\ny\nd\na\nb\nc\nz\nd\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee \uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef \uf8fa\n=\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef \uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef \uf8fa\n\uf8f0 \uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\u00ae Unique solution of equation AX = B is given by X = A\u20131 B, where A\n\u22600 \u00ae A system of equation is consistent or inconsistent according as its solution\nexists or not"
    },
    {
        "Chapter": "1",
        "sentence_range": "1685-1688",
        "Text": "\u00ae A square matrix A has inverse if and only if A is non-singular \u00ae\n\u20131\n1\nA\n(\nA adjA)\n=\n\u00ae If\na1 x + b1 y + c1 z = d1\na2 x + b2 y + c2 z = d2\na3 x + b3 y + c3 z = d3,\nthen these equations can be written as A X = B, where\n1\n1\n1\n1\n2\n2\n2\n2\n3\n3\n3\n3\nA\n,X=\nand B=\na\nb\nc\nx\nd\na\nb\nc\ny\nd\na\nb\nc\nz\nd\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee \uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef \uf8fa\n=\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef \uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef \uf8fa\n\uf8f0 \uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\u00ae Unique solution of equation AX = B is given by X = A\u20131 B, where A\n\u22600 \u00ae A system of equation is consistent or inconsistent according as its solution\nexists or not \u00ae For a square matrix A in matrix equation AX = B\n(i) |A| \u2260 0, there exists unique solution\n(ii) |A| = 0 and (adj A) B \u2260 0, then there exists no solution\n(iii) |A| = 0 and (adj A) B = 0, then system may or may not be consistent"
    },
    {
        "Chapter": "1",
        "sentence_range": "1686-1689",
        "Text": "\u00ae\n\u20131\n1\nA\n(\nA adjA)\n=\n\u00ae If\na1 x + b1 y + c1 z = d1\na2 x + b2 y + c2 z = d2\na3 x + b3 y + c3 z = d3,\nthen these equations can be written as A X = B, where\n1\n1\n1\n1\n2\n2\n2\n2\n3\n3\n3\n3\nA\n,X=\nand B=\na\nb\nc\nx\nd\na\nb\nc\ny\nd\na\nb\nc\nz\nd\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee \uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef \uf8fa\n=\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef \uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef \uf8fa\n\uf8f0 \uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\u00ae Unique solution of equation AX = B is given by X = A\u20131 B, where A\n\u22600 \u00ae A system of equation is consistent or inconsistent according as its solution\nexists or not \u00ae For a square matrix A in matrix equation AX = B\n(i) |A| \u2260 0, there exists unique solution\n(ii) |A| = 0 and (adj A) B \u2260 0, then there exists no solution\n(iii) |A| = 0 and (adj A) B = 0, then system may or may not be consistent Rationalised 2023-24\nDETERMINANTS     103\nHistorical Note\nThe Chinese method of representing the coefficients of the unknowns of\nseveral linear equations by using rods on a calculating board naturally led to the\ndiscovery of simple method of elimination"
    },
    {
        "Chapter": "1",
        "sentence_range": "1687-1690",
        "Text": "\u00ae A system of equation is consistent or inconsistent according as its solution\nexists or not \u00ae For a square matrix A in matrix equation AX = B\n(i) |A| \u2260 0, there exists unique solution\n(ii) |A| = 0 and (adj A) B \u2260 0, then there exists no solution\n(iii) |A| = 0 and (adj A) B = 0, then system may or may not be consistent Rationalised 2023-24\nDETERMINANTS     103\nHistorical Note\nThe Chinese method of representing the coefficients of the unknowns of\nseveral linear equations by using rods on a calculating board naturally led to the\ndiscovery of simple method of elimination The arrangement of rods was precisely\nthat of the numbers in a determinant"
    },
    {
        "Chapter": "1",
        "sentence_range": "1688-1691",
        "Text": "\u00ae For a square matrix A in matrix equation AX = B\n(i) |A| \u2260 0, there exists unique solution\n(ii) |A| = 0 and (adj A) B \u2260 0, then there exists no solution\n(iii) |A| = 0 and (adj A) B = 0, then system may or may not be consistent Rationalised 2023-24\nDETERMINANTS     103\nHistorical Note\nThe Chinese method of representing the coefficients of the unknowns of\nseveral linear equations by using rods on a calculating board naturally led to the\ndiscovery of simple method of elimination The arrangement of rods was precisely\nthat of the numbers in a determinant The Chinese, therefore, early developed the\nidea of subtracting columns and rows as in simplification of a determinant\nMikami, China, pp 30, 93"
    },
    {
        "Chapter": "1",
        "sentence_range": "1689-1692",
        "Text": "Rationalised 2023-24\nDETERMINANTS     103\nHistorical Note\nThe Chinese method of representing the coefficients of the unknowns of\nseveral linear equations by using rods on a calculating board naturally led to the\ndiscovery of simple method of elimination The arrangement of rods was precisely\nthat of the numbers in a determinant The Chinese, therefore, early developed the\nidea of subtracting columns and rows as in simplification of a determinant\nMikami, China, pp 30, 93 Seki Kowa, the greatest of the Japanese Mathematicians of seventeenth\ncentury in his work \u2018Kai Fukudai no Ho\u2019 in 1683 showed that he had the idea of\ndeterminants and of their expansion"
    },
    {
        "Chapter": "1",
        "sentence_range": "1690-1693",
        "Text": "The arrangement of rods was precisely\nthat of the numbers in a determinant The Chinese, therefore, early developed the\nidea of subtracting columns and rows as in simplification of a determinant\nMikami, China, pp 30, 93 Seki Kowa, the greatest of the Japanese Mathematicians of seventeenth\ncentury in his work \u2018Kai Fukudai no Ho\u2019 in 1683 showed that he had the idea of\ndeterminants and of their expansion But he used this device only in eliminating a\nquantity from two equations and not directly in the solution of a set of simultaneous\nlinear equations"
    },
    {
        "Chapter": "1",
        "sentence_range": "1691-1694",
        "Text": "The Chinese, therefore, early developed the\nidea of subtracting columns and rows as in simplification of a determinant\nMikami, China, pp 30, 93 Seki Kowa, the greatest of the Japanese Mathematicians of seventeenth\ncentury in his work \u2018Kai Fukudai no Ho\u2019 in 1683 showed that he had the idea of\ndeterminants and of their expansion But he used this device only in eliminating a\nquantity from two equations and not directly in the solution of a set of simultaneous\nlinear equations T"
    },
    {
        "Chapter": "1",
        "sentence_range": "1692-1695",
        "Text": "Seki Kowa, the greatest of the Japanese Mathematicians of seventeenth\ncentury in his work \u2018Kai Fukudai no Ho\u2019 in 1683 showed that he had the idea of\ndeterminants and of their expansion But he used this device only in eliminating a\nquantity from two equations and not directly in the solution of a set of simultaneous\nlinear equations T Hayashi, \u201cThe Fakudoi and Determinants in Japanese\nMathematics,\u201d in the proc"
    },
    {
        "Chapter": "1",
        "sentence_range": "1693-1696",
        "Text": "But he used this device only in eliminating a\nquantity from two equations and not directly in the solution of a set of simultaneous\nlinear equations T Hayashi, \u201cThe Fakudoi and Determinants in Japanese\nMathematics,\u201d in the proc of the Tokyo Math"
    },
    {
        "Chapter": "1",
        "sentence_range": "1694-1697",
        "Text": "T Hayashi, \u201cThe Fakudoi and Determinants in Japanese\nMathematics,\u201d in the proc of the Tokyo Math Soc"
    },
    {
        "Chapter": "1",
        "sentence_range": "1695-1698",
        "Text": "Hayashi, \u201cThe Fakudoi and Determinants in Japanese\nMathematics,\u201d in the proc of the Tokyo Math Soc , V"
    },
    {
        "Chapter": "1",
        "sentence_range": "1696-1699",
        "Text": "of the Tokyo Math Soc , V Vendermonde was the first to recognise determinants as independent functions"
    },
    {
        "Chapter": "1",
        "sentence_range": "1697-1700",
        "Text": "Soc , V Vendermonde was the first to recognise determinants as independent functions He may be called the formal founder"
    },
    {
        "Chapter": "1",
        "sentence_range": "1698-1701",
        "Text": ", V Vendermonde was the first to recognise determinants as independent functions He may be called the formal founder Laplace (1772), gave general method of\nexpanding a determinant in terms of its complementary minors"
    },
    {
        "Chapter": "1",
        "sentence_range": "1699-1702",
        "Text": "Vendermonde was the first to recognise determinants as independent functions He may be called the formal founder Laplace (1772), gave general method of\nexpanding a determinant in terms of its complementary minors In 1773 Lagrange\ntreated determinants of the second and third orders and used them for purpose\nother than the solution of equations"
    },
    {
        "Chapter": "1",
        "sentence_range": "1700-1703",
        "Text": "He may be called the formal founder Laplace (1772), gave general method of\nexpanding a determinant in terms of its complementary minors In 1773 Lagrange\ntreated determinants of the second and third orders and used them for purpose\nother than the solution of equations In 1801, Gauss used determinants in his\ntheory of numbers"
    },
    {
        "Chapter": "1",
        "sentence_range": "1701-1704",
        "Text": "Laplace (1772), gave general method of\nexpanding a determinant in terms of its complementary minors In 1773 Lagrange\ntreated determinants of the second and third orders and used them for purpose\nother than the solution of equations In 1801, Gauss used determinants in his\ntheory of numbers The next great contributor was Jacques - Philippe - Marie Binet, (1812) who\nstated the theorem relating to the product of two matrices of m-columns and n-\nrows, which for the special case of m = n reduces to the multiplication theorem"
    },
    {
        "Chapter": "1",
        "sentence_range": "1702-1705",
        "Text": "In 1773 Lagrange\ntreated determinants of the second and third orders and used them for purpose\nother than the solution of equations In 1801, Gauss used determinants in his\ntheory of numbers The next great contributor was Jacques - Philippe - Marie Binet, (1812) who\nstated the theorem relating to the product of two matrices of m-columns and n-\nrows, which for the special case of m = n reduces to the multiplication theorem Also on the same day, Cauchy (1812) presented one on the same subject"
    },
    {
        "Chapter": "1",
        "sentence_range": "1703-1706",
        "Text": "In 1801, Gauss used determinants in his\ntheory of numbers The next great contributor was Jacques - Philippe - Marie Binet, (1812) who\nstated the theorem relating to the product of two matrices of m-columns and n-\nrows, which for the special case of m = n reduces to the multiplication theorem Also on the same day, Cauchy (1812) presented one on the same subject He\nused the word \u2018determinant\u2019 in its present sense"
    },
    {
        "Chapter": "1",
        "sentence_range": "1704-1707",
        "Text": "The next great contributor was Jacques - Philippe - Marie Binet, (1812) who\nstated the theorem relating to the product of two matrices of m-columns and n-\nrows, which for the special case of m = n reduces to the multiplication theorem Also on the same day, Cauchy (1812) presented one on the same subject He\nused the word \u2018determinant\u2019 in its present sense He gave the proof of multiplication\ntheorem more satisfactory than Binet\u2019s"
    },
    {
        "Chapter": "1",
        "sentence_range": "1705-1708",
        "Text": "Also on the same day, Cauchy (1812) presented one on the same subject He\nused the word \u2018determinant\u2019 in its present sense He gave the proof of multiplication\ntheorem more satisfactory than Binet\u2019s The greatest contributor to the theory was Carl Gustav Jacob Jacobi, after\nthis the word determinant received its final acceptance"
    },
    {
        "Chapter": "1",
        "sentence_range": "1706-1709",
        "Text": "He\nused the word \u2018determinant\u2019 in its present sense He gave the proof of multiplication\ntheorem more satisfactory than Binet\u2019s The greatest contributor to the theory was Carl Gustav Jacob Jacobi, after\nthis the word determinant received its final acceptance Rationalised 2023-24\n MATHEMATICS\n104\nvThe whole of science is nothing more than a refinement\nof everyday thinking"
    },
    {
        "Chapter": "1",
        "sentence_range": "1707-1710",
        "Text": "He gave the proof of multiplication\ntheorem more satisfactory than Binet\u2019s The greatest contributor to the theory was Carl Gustav Jacob Jacobi, after\nthis the word determinant received its final acceptance Rationalised 2023-24\n MATHEMATICS\n104\nvThe whole of science is nothing more than a refinement\nof everyday thinking \u201d \u2014 ALBERT EINSTEIN v\n5"
    },
    {
        "Chapter": "1",
        "sentence_range": "1708-1711",
        "Text": "The greatest contributor to the theory was Carl Gustav Jacob Jacobi, after\nthis the word determinant received its final acceptance Rationalised 2023-24\n MATHEMATICS\n104\nvThe whole of science is nothing more than a refinement\nof everyday thinking \u201d \u2014 ALBERT EINSTEIN v\n5 1  Introduction\nThis chapter is essentially a continuation of our study of\ndifferentiation of functions in Class XI"
    },
    {
        "Chapter": "1",
        "sentence_range": "1709-1712",
        "Text": "Rationalised 2023-24\n MATHEMATICS\n104\nvThe whole of science is nothing more than a refinement\nof everyday thinking \u201d \u2014 ALBERT EINSTEIN v\n5 1  Introduction\nThis chapter is essentially a continuation of our study of\ndifferentiation of functions in Class XI We had learnt to\ndifferentiate certain functions like polynomial functions and\ntrigonometric functions"
    },
    {
        "Chapter": "1",
        "sentence_range": "1710-1713",
        "Text": "\u201d \u2014 ALBERT EINSTEIN v\n5 1  Introduction\nThis chapter is essentially a continuation of our study of\ndifferentiation of functions in Class XI We had learnt to\ndifferentiate certain functions like polynomial functions and\ntrigonometric functions In this chapter, we introduce the\nvery important concepts of continuity, differentiability and\nrelations between them"
    },
    {
        "Chapter": "1",
        "sentence_range": "1711-1714",
        "Text": "1  Introduction\nThis chapter is essentially a continuation of our study of\ndifferentiation of functions in Class XI We had learnt to\ndifferentiate certain functions like polynomial functions and\ntrigonometric functions In this chapter, we introduce the\nvery important concepts of continuity, differentiability and\nrelations between them We will also learn differentiation\nof inverse trigonometric functions"
    },
    {
        "Chapter": "1",
        "sentence_range": "1712-1715",
        "Text": "We had learnt to\ndifferentiate certain functions like polynomial functions and\ntrigonometric functions In this chapter, we introduce the\nvery important concepts of continuity, differentiability and\nrelations between them We will also learn differentiation\nof inverse trigonometric functions Further, we introduce a\nnew class of functions called exponential and logarithmic\nfunctions"
    },
    {
        "Chapter": "1",
        "sentence_range": "1713-1716",
        "Text": "In this chapter, we introduce the\nvery important concepts of continuity, differentiability and\nrelations between them We will also learn differentiation\nof inverse trigonometric functions Further, we introduce a\nnew class of functions called exponential and logarithmic\nfunctions These functions lead to powerful techniques of\ndifferentiation"
    },
    {
        "Chapter": "1",
        "sentence_range": "1714-1717",
        "Text": "We will also learn differentiation\nof inverse trigonometric functions Further, we introduce a\nnew class of functions called exponential and logarithmic\nfunctions These functions lead to powerful techniques of\ndifferentiation We illustrate certain geometrically obvious\nconditions through differential calculus"
    },
    {
        "Chapter": "1",
        "sentence_range": "1715-1718",
        "Text": "Further, we introduce a\nnew class of functions called exponential and logarithmic\nfunctions These functions lead to powerful techniques of\ndifferentiation We illustrate certain geometrically obvious\nconditions through differential calculus In the process, we\nwill learn some fundamental theorems in this area"
    },
    {
        "Chapter": "1",
        "sentence_range": "1716-1719",
        "Text": "These functions lead to powerful techniques of\ndifferentiation We illustrate certain geometrically obvious\nconditions through differential calculus In the process, we\nwill learn some fundamental theorems in this area 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "1717-1720",
        "Text": "We illustrate certain geometrically obvious\nconditions through differential calculus In the process, we\nwill learn some fundamental theorems in this area 5 2  Continuity\nWe start the section with two informal examples to get a feel of continuity"
    },
    {
        "Chapter": "1",
        "sentence_range": "1718-1721",
        "Text": "In the process, we\nwill learn some fundamental theorems in this area 5 2  Continuity\nWe start the section with two informal examples to get a feel of continuity Consider\nthe function\n1, if\n0\n( )\n2, if\n0\nx\nf x\nx\n\u2264\n= \uf8f2\uf8f1\n>\n\uf8f3\nThis function is of course defined at every\npoint of the real line"
    },
    {
        "Chapter": "1",
        "sentence_range": "1719-1722",
        "Text": "5 2  Continuity\nWe start the section with two informal examples to get a feel of continuity Consider\nthe function\n1, if\n0\n( )\n2, if\n0\nx\nf x\nx\n\u2264\n= \uf8f2\uf8f1\n>\n\uf8f3\nThis function is of course defined at every\npoint of the real line Graph of this function is\ngiven in the Fig 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "1720-1723",
        "Text": "2  Continuity\nWe start the section with two informal examples to get a feel of continuity Consider\nthe function\n1, if\n0\n( )\n2, if\n0\nx\nf x\nx\n\u2264\n= \uf8f2\uf8f1\n>\n\uf8f3\nThis function is of course defined at every\npoint of the real line Graph of this function is\ngiven in the Fig 5 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "1721-1724",
        "Text": "Consider\nthe function\n1, if\n0\n( )\n2, if\n0\nx\nf x\nx\n\u2264\n= \uf8f2\uf8f1\n>\n\uf8f3\nThis function is of course defined at every\npoint of the real line Graph of this function is\ngiven in the Fig 5 1 One can deduce from the\ngraph that the value of the function at nearby\npoints on x-axis remain close to each other\nexcept at x = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "1722-1725",
        "Text": "Graph of this function is\ngiven in the Fig 5 1 One can deduce from the\ngraph that the value of the function at nearby\npoints on x-axis remain close to each other\nexcept at x = 0 At the points near and to the\nleft of 0, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "1723-1726",
        "Text": "1 One can deduce from the\ngraph that the value of the function at nearby\npoints on x-axis remain close to each other\nexcept at x = 0 At the points near and to the\nleft of 0, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "1724-1727",
        "Text": "One can deduce from the\ngraph that the value of the function at nearby\npoints on x-axis remain close to each other\nexcept at x = 0 At the points near and to the\nleft of 0, i e , at points like \u2013 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "1725-1728",
        "Text": "At the points near and to the\nleft of 0, i e , at points like \u2013 0 1, \u2013 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "1726-1729",
        "Text": "e , at points like \u2013 0 1, \u2013 0 01, \u2013 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "1727-1730",
        "Text": ", at points like \u2013 0 1, \u2013 0 01, \u2013 0 001,\nthe value of the function is 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "1728-1731",
        "Text": "1, \u2013 0 01, \u2013 0 001,\nthe value of the function is 1 At the points near\nand to the right of 0, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "1729-1732",
        "Text": "01, \u2013 0 001,\nthe value of the function is 1 At the points near\nand to the right of 0, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "1730-1733",
        "Text": "001,\nthe value of the function is 1 At the points near\nand to the right of 0, i e , at points like 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "1731-1734",
        "Text": "At the points near\nand to the right of 0, i e , at points like 0 1, 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "1732-1735",
        "Text": "e , at points like 0 1, 0 01,\nChapter 5\nCONTINUITY AND\nDIFFERENTIABILITY\nSir Issac Newton\n(1642-1727)\nFig 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "1733-1736",
        "Text": ", at points like 0 1, 0 01,\nChapter 5\nCONTINUITY AND\nDIFFERENTIABILITY\nSir Issac Newton\n(1642-1727)\nFig 5 1\nRationalised 2023-24\nCONTINUITY AND DIFFERENTIABILITY\n105\n0"
    },
    {
        "Chapter": "1",
        "sentence_range": "1734-1737",
        "Text": "1, 0 01,\nChapter 5\nCONTINUITY AND\nDIFFERENTIABILITY\nSir Issac Newton\n(1642-1727)\nFig 5 1\nRationalised 2023-24\nCONTINUITY AND DIFFERENTIABILITY\n105\n0 001, the value of the function is 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "1735-1738",
        "Text": "01,\nChapter 5\nCONTINUITY AND\nDIFFERENTIABILITY\nSir Issac Newton\n(1642-1727)\nFig 5 1\nRationalised 2023-24\nCONTINUITY AND DIFFERENTIABILITY\n105\n0 001, the value of the function is 2 Using the language of left and right hand limits, we\nmay say that the left (respectively right) hand limit of f at 0 is 1 (respectively 2)"
    },
    {
        "Chapter": "1",
        "sentence_range": "1736-1739",
        "Text": "1\nRationalised 2023-24\nCONTINUITY AND DIFFERENTIABILITY\n105\n0 001, the value of the function is 2 Using the language of left and right hand limits, we\nmay say that the left (respectively right) hand limit of f at 0 is 1 (respectively 2) In\nparticular the left and right hand limits do not coincide"
    },
    {
        "Chapter": "1",
        "sentence_range": "1737-1740",
        "Text": "001, the value of the function is 2 Using the language of left and right hand limits, we\nmay say that the left (respectively right) hand limit of f at 0 is 1 (respectively 2) In\nparticular the left and right hand limits do not coincide We also observe that the value\nof the function at x = 0 concides with the left hand limit"
    },
    {
        "Chapter": "1",
        "sentence_range": "1738-1741",
        "Text": "Using the language of left and right hand limits, we\nmay say that the left (respectively right) hand limit of f at 0 is 1 (respectively 2) In\nparticular the left and right hand limits do not coincide We also observe that the value\nof the function at x = 0 concides with the left hand limit Note that when we try to draw\nthe graph, we cannot draw it in one stroke, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "1739-1742",
        "Text": "In\nparticular the left and right hand limits do not coincide We also observe that the value\nof the function at x = 0 concides with the left hand limit Note that when we try to draw\nthe graph, we cannot draw it in one stroke, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "1740-1743",
        "Text": "We also observe that the value\nof the function at x = 0 concides with the left hand limit Note that when we try to draw\nthe graph, we cannot draw it in one stroke, i e , without lifting pen from the plane of the\npaper, we can not draw the graph of this function"
    },
    {
        "Chapter": "1",
        "sentence_range": "1741-1744",
        "Text": "Note that when we try to draw\nthe graph, we cannot draw it in one stroke, i e , without lifting pen from the plane of the\npaper, we can not draw the graph of this function In fact, we need to lift the pen when\nwe come to 0 from left"
    },
    {
        "Chapter": "1",
        "sentence_range": "1742-1745",
        "Text": "e , without lifting pen from the plane of the\npaper, we can not draw the graph of this function In fact, we need to lift the pen when\nwe come to 0 from left This is one instance of function being not continuous at x = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "1743-1746",
        "Text": ", without lifting pen from the plane of the\npaper, we can not draw the graph of this function In fact, we need to lift the pen when\nwe come to 0 from left This is one instance of function being not continuous at x = 0 Now, consider the function defined as\nf x\nxx\n( )\n,,\n=\n=\u2260\n\uf8f2\uf8f1\n\uf8f3\n1\n0\n2\n0\nif\nif\nThis function is also defined at every point"
    },
    {
        "Chapter": "1",
        "sentence_range": "1744-1747",
        "Text": "In fact, we need to lift the pen when\nwe come to 0 from left This is one instance of function being not continuous at x = 0 Now, consider the function defined as\nf x\nxx\n( )\n,,\n=\n=\u2260\n\uf8f2\uf8f1\n\uf8f3\n1\n0\n2\n0\nif\nif\nThis function is also defined at every point Left and the right hand limits at x = 0\nare both equal to 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "1745-1748",
        "Text": "This is one instance of function being not continuous at x = 0 Now, consider the function defined as\nf x\nxx\n( )\n,,\n=\n=\u2260\n\uf8f2\uf8f1\n\uf8f3\n1\n0\n2\n0\nif\nif\nThis function is also defined at every point Left and the right hand limits at x = 0\nare both equal to 1 But the value of the\nfunction at x = 0 equals 2 which does not\ncoincide with the common value of the left\nand right hand limits"
    },
    {
        "Chapter": "1",
        "sentence_range": "1746-1749",
        "Text": "Now, consider the function defined as\nf x\nxx\n( )\n,,\n=\n=\u2260\n\uf8f2\uf8f1\n\uf8f3\n1\n0\n2\n0\nif\nif\nThis function is also defined at every point Left and the right hand limits at x = 0\nare both equal to 1 But the value of the\nfunction at x = 0 equals 2 which does not\ncoincide with the common value of the left\nand right hand limits Again, we note that we\ncannot draw the graph of the function without\nlifting the pen"
    },
    {
        "Chapter": "1",
        "sentence_range": "1747-1750",
        "Text": "Left and the right hand limits at x = 0\nare both equal to 1 But the value of the\nfunction at x = 0 equals 2 which does not\ncoincide with the common value of the left\nand right hand limits Again, we note that we\ncannot draw the graph of the function without\nlifting the pen This is yet another instance of\na function being not continuous at x = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "1748-1751",
        "Text": "But the value of the\nfunction at x = 0 equals 2 which does not\ncoincide with the common value of the left\nand right hand limits Again, we note that we\ncannot draw the graph of the function without\nlifting the pen This is yet another instance of\na function being not continuous at x = 0 Naively, we may say that a function is\ncontinuous at a fixed point if we can draw the\ngraph of the function around that point without\nlifting the pen from the plane of the paper"
    },
    {
        "Chapter": "1",
        "sentence_range": "1749-1752",
        "Text": "Again, we note that we\ncannot draw the graph of the function without\nlifting the pen This is yet another instance of\na function being not continuous at x = 0 Naively, we may say that a function is\ncontinuous at a fixed point if we can draw the\ngraph of the function around that point without\nlifting the pen from the plane of the paper Mathematically, it may be phrased precisely as follows:\nDefinition 1 Suppose f is a real function on a subset of the real numbers and let  c be\na point in the domain of f"
    },
    {
        "Chapter": "1",
        "sentence_range": "1750-1753",
        "Text": "This is yet another instance of\na function being not continuous at x = 0 Naively, we may say that a function is\ncontinuous at a fixed point if we can draw the\ngraph of the function around that point without\nlifting the pen from the plane of the paper Mathematically, it may be phrased precisely as follows:\nDefinition 1 Suppose f is a real function on a subset of the real numbers and let  c be\na point in the domain of f Then f is continuous at c if\nlim\n( )\n( )\nx\nc f x\nf c\n\u2192\n=\nMore elaborately, if the left hand limit, right hand limit and the value of the function\nat x = c exist and equal to each other, then f is said to be continuous at x = c"
    },
    {
        "Chapter": "1",
        "sentence_range": "1751-1754",
        "Text": "Naively, we may say that a function is\ncontinuous at a fixed point if we can draw the\ngraph of the function around that point without\nlifting the pen from the plane of the paper Mathematically, it may be phrased precisely as follows:\nDefinition 1 Suppose f is a real function on a subset of the real numbers and let  c be\na point in the domain of f Then f is continuous at c if\nlim\n( )\n( )\nx\nc f x\nf c\n\u2192\n=\nMore elaborately, if the left hand limit, right hand limit and the value of the function\nat x = c exist and equal to each other, then f is said to be continuous at x = c Recall that\nif the right hand and left hand limits at x = c coincide, then we say that the common\nvalue is the limit of the function at x = c"
    },
    {
        "Chapter": "1",
        "sentence_range": "1752-1755",
        "Text": "Mathematically, it may be phrased precisely as follows:\nDefinition 1 Suppose f is a real function on a subset of the real numbers and let  c be\na point in the domain of f Then f is continuous at c if\nlim\n( )\n( )\nx\nc f x\nf c\n\u2192\n=\nMore elaborately, if the left hand limit, right hand limit and the value of the function\nat x = c exist and equal to each other, then f is said to be continuous at x = c Recall that\nif the right hand and left hand limits at x = c coincide, then we say that the common\nvalue is the limit of the function at x = c Hence we may also rephrase  the definition of\ncontinuity as follows: a function is continuous at x = c if the function is defined at\nx = c and if the value of the function at x = c equals the limit of the function at\nx = c"
    },
    {
        "Chapter": "1",
        "sentence_range": "1753-1756",
        "Text": "Then f is continuous at c if\nlim\n( )\n( )\nx\nc f x\nf c\n\u2192\n=\nMore elaborately, if the left hand limit, right hand limit and the value of the function\nat x = c exist and equal to each other, then f is said to be continuous at x = c Recall that\nif the right hand and left hand limits at x = c coincide, then we say that the common\nvalue is the limit of the function at x = c Hence we may also rephrase  the definition of\ncontinuity as follows: a function is continuous at x = c if the function is defined at\nx = c and if the value of the function at x = c equals the limit of the function at\nx = c If f is not continuous at c, we say f is discontinuous at c and c is called a point\nof discontinuity of f"
    },
    {
        "Chapter": "1",
        "sentence_range": "1754-1757",
        "Text": "Recall that\nif the right hand and left hand limits at x = c coincide, then we say that the common\nvalue is the limit of the function at x = c Hence we may also rephrase  the definition of\ncontinuity as follows: a function is continuous at x = c if the function is defined at\nx = c and if the value of the function at x = c equals the limit of the function at\nx = c If f is not continuous at c, we say f is discontinuous at c and c is called a point\nof discontinuity of f Fig 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "1755-1758",
        "Text": "Hence we may also rephrase  the definition of\ncontinuity as follows: a function is continuous at x = c if the function is defined at\nx = c and if the value of the function at x = c equals the limit of the function at\nx = c If f is not continuous at c, we say f is discontinuous at c and c is called a point\nof discontinuity of f Fig 5 2\nRationalised 2023-24\n MATHEMATICS\n106\nExample 1 Check the continuity of the function f given by f(x) = 2x + 3 at x = 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "1756-1759",
        "Text": "If f is not continuous at c, we say f is discontinuous at c and c is called a point\nof discontinuity of f Fig 5 2\nRationalised 2023-24\n MATHEMATICS\n106\nExample 1 Check the continuity of the function f given by f(x) = 2x + 3 at x = 1 Solution First note that the function is defined at the given point x = 1 and its value is 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "1757-1760",
        "Text": "Fig 5 2\nRationalised 2023-24\n MATHEMATICS\n106\nExample 1 Check the continuity of the function f given by f(x) = 2x + 3 at x = 1 Solution First note that the function is defined at the given point x = 1 and its value is 5 Then find the limit of the function at x = 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "1758-1761",
        "Text": "2\nRationalised 2023-24\n MATHEMATICS\n106\nExample 1 Check the continuity of the function f given by f(x) = 2x + 3 at x = 1 Solution First note that the function is defined at the given point x = 1 and its value is 5 Then find the limit of the function at x = 1 Clearly\n1\n1\nlim\n( )\nlim(2\n3)\n2(1)\n3\n5\nx\nx\nf x\nx\n\u2192\n=\u2192\n+\n=\n+\n=\nThus\nlim1\n( )\n5\n(1)\nx\nf x\nf\n\u2192\n=\n=\nHence, f is continuous at x = 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "1759-1762",
        "Text": "Solution First note that the function is defined at the given point x = 1 and its value is 5 Then find the limit of the function at x = 1 Clearly\n1\n1\nlim\n( )\nlim(2\n3)\n2(1)\n3\n5\nx\nx\nf x\nx\n\u2192\n=\u2192\n+\n=\n+\n=\nThus\nlim1\n( )\n5\n(1)\nx\nf x\nf\n\u2192\n=\n=\nHence, f is continuous at x = 1 Example 2 Examine whether the function f given by f(x) = x2 is continuous at x = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "1760-1763",
        "Text": "Then find the limit of the function at x = 1 Clearly\n1\n1\nlim\n( )\nlim(2\n3)\n2(1)\n3\n5\nx\nx\nf x\nx\n\u2192\n=\u2192\n+\n=\n+\n=\nThus\nlim1\n( )\n5\n(1)\nx\nf x\nf\n\u2192\n=\n=\nHence, f is continuous at x = 1 Example 2 Examine whether the function f given by f(x) = x2 is continuous at x = 0 Solution First note that the function is defined at the given point x = 0 and its value is 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "1761-1764",
        "Text": "Clearly\n1\n1\nlim\n( )\nlim(2\n3)\n2(1)\n3\n5\nx\nx\nf x\nx\n\u2192\n=\u2192\n+\n=\n+\n=\nThus\nlim1\n( )\n5\n(1)\nx\nf x\nf\n\u2192\n=\n=\nHence, f is continuous at x = 1 Example 2 Examine whether the function f given by f(x) = x2 is continuous at x = 0 Solution First note that the function is defined at the given point x = 0 and its value is 0 Then find the limit of the function at x = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "1762-1765",
        "Text": "Example 2 Examine whether the function f given by f(x) = x2 is continuous at x = 0 Solution First note that the function is defined at the given point x = 0 and its value is 0 Then find the limit of the function at x = 0 Clearly\n2\n2\n0\n0\nlim\n( )\nlim\n0\n0\nx\nx\nf x\nx\n\u2192\n=\u2192\n=\n=\nThus\nlim0\n( )\n0\n(0)\nx\nf x\nf\n\u2192\n=\n=\nHence, f is continuous at x = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "1763-1766",
        "Text": "Solution First note that the function is defined at the given point x = 0 and its value is 0 Then find the limit of the function at x = 0 Clearly\n2\n2\n0\n0\nlim\n( )\nlim\n0\n0\nx\nx\nf x\nx\n\u2192\n=\u2192\n=\n=\nThus\nlim0\n( )\n0\n(0)\nx\nf x\nf\n\u2192\n=\n=\nHence, f is continuous at x = 0 Example 3 Discuss the continuity of the function f given by f(x) = | x | at x = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "1764-1767",
        "Text": "Then find the limit of the function at x = 0 Clearly\n2\n2\n0\n0\nlim\n( )\nlim\n0\n0\nx\nx\nf x\nx\n\u2192\n=\u2192\n=\n=\nThus\nlim0\n( )\n0\n(0)\nx\nf x\nf\n\u2192\n=\n=\nHence, f is continuous at x = 0 Example 3 Discuss the continuity of the function f given by f(x) = | x | at x = 0 Solution By definition\nf(x) =\n, if\n0\n, if\n0\nx\nx\nx\nx\n\u2212\n<\n\uf8f2\uf8f1\n\u2265\n\uf8f3\nClearly the function is defined at 0 and f(0) = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "1765-1768",
        "Text": "Clearly\n2\n2\n0\n0\nlim\n( )\nlim\n0\n0\nx\nx\nf x\nx\n\u2192\n=\u2192\n=\n=\nThus\nlim0\n( )\n0\n(0)\nx\nf x\nf\n\u2192\n=\n=\nHence, f is continuous at x = 0 Example 3 Discuss the continuity of the function f given by f(x) = | x | at x = 0 Solution By definition\nf(x) =\n, if\n0\n, if\n0\nx\nx\nx\nx\n\u2212\n<\n\uf8f2\uf8f1\n\u2265\n\uf8f3\nClearly the function is defined at 0 and f(0) = 0 Left hand limit of f at 0 is\n0\n0\nlim\n( )\nlim (\u2013 )\n0\nx\nx\nf x\nx\n\u2212\n\u2212\n\u2192\n=\u2192\n=\nSimilarly, the right hand limit of f at 0 is\n0\n0\nlim\n( )\nlim\n0\nx\nx\nf x\nx\n+\n+\n\u2192\n=\u2192\n=\nThus, the left hand limit, right hand limit and the value of the function coincide at\nx = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "1766-1769",
        "Text": "Example 3 Discuss the continuity of the function f given by f(x) = | x | at x = 0 Solution By definition\nf(x) =\n, if\n0\n, if\n0\nx\nx\nx\nx\n\u2212\n<\n\uf8f2\uf8f1\n\u2265\n\uf8f3\nClearly the function is defined at 0 and f(0) = 0 Left hand limit of f at 0 is\n0\n0\nlim\n( )\nlim (\u2013 )\n0\nx\nx\nf x\nx\n\u2212\n\u2212\n\u2192\n=\u2192\n=\nSimilarly, the right hand limit of f at 0 is\n0\n0\nlim\n( )\nlim\n0\nx\nx\nf x\nx\n+\n+\n\u2192\n=\u2192\n=\nThus, the left hand limit, right hand limit and the value of the function coincide at\nx = 0 Hence, f is continuous at x = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "1767-1770",
        "Text": "Solution By definition\nf(x) =\n, if\n0\n, if\n0\nx\nx\nx\nx\n\u2212\n<\n\uf8f2\uf8f1\n\u2265\n\uf8f3\nClearly the function is defined at 0 and f(0) = 0 Left hand limit of f at 0 is\n0\n0\nlim\n( )\nlim (\u2013 )\n0\nx\nx\nf x\nx\n\u2212\n\u2212\n\u2192\n=\u2192\n=\nSimilarly, the right hand limit of f at 0 is\n0\n0\nlim\n( )\nlim\n0\nx\nx\nf x\nx\n+\n+\n\u2192\n=\u2192\n=\nThus, the left hand limit, right hand limit and the value of the function coincide at\nx = 0 Hence, f is continuous at x = 0 Example 4 Show that the function f given by\nf(x) =\n3\n3, if\n0\n1,\nif\n0\nx\nx\nx\n\uf8f1\n+\n\u2260\n\uf8f4\uf8f2\n=\n\uf8f4\uf8f3\nis not continuous at x = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "1768-1771",
        "Text": "Left hand limit of f at 0 is\n0\n0\nlim\n( )\nlim (\u2013 )\n0\nx\nx\nf x\nx\n\u2212\n\u2212\n\u2192\n=\u2192\n=\nSimilarly, the right hand limit of f at 0 is\n0\n0\nlim\n( )\nlim\n0\nx\nx\nf x\nx\n+\n+\n\u2192\n=\u2192\n=\nThus, the left hand limit, right hand limit and the value of the function coincide at\nx = 0 Hence, f is continuous at x = 0 Example 4 Show that the function f given by\nf(x) =\n3\n3, if\n0\n1,\nif\n0\nx\nx\nx\n\uf8f1\n+\n\u2260\n\uf8f4\uf8f2\n=\n\uf8f4\uf8f3\nis not continuous at x = 0 Rationalised 2023-24\nCONTINUITY AND DIFFERENTIABILITY\n107\nSolution The function is defined at x = 0 and its value at x = 0 is 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "1769-1772",
        "Text": "Hence, f is continuous at x = 0 Example 4 Show that the function f given by\nf(x) =\n3\n3, if\n0\n1,\nif\n0\nx\nx\nx\n\uf8f1\n+\n\u2260\n\uf8f4\uf8f2\n=\n\uf8f4\uf8f3\nis not continuous at x = 0 Rationalised 2023-24\nCONTINUITY AND DIFFERENTIABILITY\n107\nSolution The function is defined at x = 0 and its value at x = 0 is 1 When x \u2260 0, the\nfunction is given by a polynomial"
    },
    {
        "Chapter": "1",
        "sentence_range": "1770-1773",
        "Text": "Example 4 Show that the function f given by\nf(x) =\n3\n3, if\n0\n1,\nif\n0\nx\nx\nx\n\uf8f1\n+\n\u2260\n\uf8f4\uf8f2\n=\n\uf8f4\uf8f3\nis not continuous at x = 0 Rationalised 2023-24\nCONTINUITY AND DIFFERENTIABILITY\n107\nSolution The function is defined at x = 0 and its value at x = 0 is 1 When x \u2260 0, the\nfunction is given by a polynomial Hence,\nlim0\n( )\nx\nf x\n\u2192\n =\n3\n3\n0\nlim (\n3)\n0\n3\n3\nx\nx\n\u2192\n+\n=\n+\n=\nSince the limit of f at x = 0 does not coincide with f(0), the function is not continuous\nat x = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "1771-1774",
        "Text": "Rationalised 2023-24\nCONTINUITY AND DIFFERENTIABILITY\n107\nSolution The function is defined at x = 0 and its value at x = 0 is 1 When x \u2260 0, the\nfunction is given by a polynomial Hence,\nlim0\n( )\nx\nf x\n\u2192\n =\n3\n3\n0\nlim (\n3)\n0\n3\n3\nx\nx\n\u2192\n+\n=\n+\n=\nSince the limit of f at x = 0 does not coincide with f(0), the function is not continuous\nat x = 0 It may be noted that x = 0 is the only point of discontinuity for this function"
    },
    {
        "Chapter": "1",
        "sentence_range": "1772-1775",
        "Text": "When x \u2260 0, the\nfunction is given by a polynomial Hence,\nlim0\n( )\nx\nf x\n\u2192\n =\n3\n3\n0\nlim (\n3)\n0\n3\n3\nx\nx\n\u2192\n+\n=\n+\n=\nSince the limit of f at x = 0 does not coincide with f(0), the function is not continuous\nat x = 0 It may be noted that x = 0 is the only point of discontinuity for this function Example 5 Check the points where the constant function f(x) = k is continuous"
    },
    {
        "Chapter": "1",
        "sentence_range": "1773-1776",
        "Text": "Hence,\nlim0\n( )\nx\nf x\n\u2192\n =\n3\n3\n0\nlim (\n3)\n0\n3\n3\nx\nx\n\u2192\n+\n=\n+\n=\nSince the limit of f at x = 0 does not coincide with f(0), the function is not continuous\nat x = 0 It may be noted that x = 0 is the only point of discontinuity for this function Example 5 Check the points where the constant function f(x) = k is continuous Solution The function is defined at all real numbers and by definition, its value at any\nreal number equals k"
    },
    {
        "Chapter": "1",
        "sentence_range": "1774-1777",
        "Text": "It may be noted that x = 0 is the only point of discontinuity for this function Example 5 Check the points where the constant function f(x) = k is continuous Solution The function is defined at all real numbers and by definition, its value at any\nreal number equals k Let c be any real number"
    },
    {
        "Chapter": "1",
        "sentence_range": "1775-1778",
        "Text": "Example 5 Check the points where the constant function f(x) = k is continuous Solution The function is defined at all real numbers and by definition, its value at any\nreal number equals k Let c be any real number Then\nlim\n( )\nx\nc f x\n\u2192\n = lim\nx\nc k\nk\n\u2192\n=\nSince f(c) = k = lim\nx\nc\n\u2192  f(x) for any real number c, the function f is continuous at\nevery real number"
    },
    {
        "Chapter": "1",
        "sentence_range": "1776-1779",
        "Text": "Solution The function is defined at all real numbers and by definition, its value at any\nreal number equals k Let c be any real number Then\nlim\n( )\nx\nc f x\n\u2192\n = lim\nx\nc k\nk\n\u2192\n=\nSince f(c) = k = lim\nx\nc\n\u2192  f(x) for any real number c, the function f is continuous at\nevery real number Example 6 Prove that the identity function on real numbers given by f(x) = x is\ncontinuous at every real number"
    },
    {
        "Chapter": "1",
        "sentence_range": "1777-1780",
        "Text": "Let c be any real number Then\nlim\n( )\nx\nc f x\n\u2192\n = lim\nx\nc k\nk\n\u2192\n=\nSince f(c) = k = lim\nx\nc\n\u2192  f(x) for any real number c, the function f is continuous at\nevery real number Example 6 Prove that the identity function on real numbers given by f(x) = x is\ncontinuous at every real number Solution The function is clearly defined at every point and f (c) = c for every real\nnumber c"
    },
    {
        "Chapter": "1",
        "sentence_range": "1778-1781",
        "Text": "Then\nlim\n( )\nx\nc f x\n\u2192\n = lim\nx\nc k\nk\n\u2192\n=\nSince f(c) = k = lim\nx\nc\n\u2192  f(x) for any real number c, the function f is continuous at\nevery real number Example 6 Prove that the identity function on real numbers given by f(x) = x is\ncontinuous at every real number Solution The function is clearly defined at every point and f (c) = c for every real\nnumber c Also,\nlim\n( )\nx\nc f x\n\u2192\n = lim\nx\nc x\nc\n\u2192\n=\nThus, lim\nx\nc\n\u2192 f(x) = c = f(c) and hence the function is continuous at every real number"
    },
    {
        "Chapter": "1",
        "sentence_range": "1779-1782",
        "Text": "Example 6 Prove that the identity function on real numbers given by f(x) = x is\ncontinuous at every real number Solution The function is clearly defined at every point and f (c) = c for every real\nnumber c Also,\nlim\n( )\nx\nc f x\n\u2192\n = lim\nx\nc x\nc\n\u2192\n=\nThus, lim\nx\nc\n\u2192 f(x) = c = f(c) and hence the function is continuous at every real number Having defined continuity of a function at a given point, now we make a natural\nextension of this definition to discuss continuity of a function"
    },
    {
        "Chapter": "1",
        "sentence_range": "1780-1783",
        "Text": "Solution The function is clearly defined at every point and f (c) = c for every real\nnumber c Also,\nlim\n( )\nx\nc f x\n\u2192\n = lim\nx\nc x\nc\n\u2192\n=\nThus, lim\nx\nc\n\u2192 f(x) = c = f(c) and hence the function is continuous at every real number Having defined continuity of a function at a given point, now we make a natural\nextension of this definition to discuss continuity of a function Definition 2 A real function f is said to be continuous if it is continuous at every point\nin the domain of f"
    },
    {
        "Chapter": "1",
        "sentence_range": "1781-1784",
        "Text": "Also,\nlim\n( )\nx\nc f x\n\u2192\n = lim\nx\nc x\nc\n\u2192\n=\nThus, lim\nx\nc\n\u2192 f(x) = c = f(c) and hence the function is continuous at every real number Having defined continuity of a function at a given point, now we make a natural\nextension of this definition to discuss continuity of a function Definition 2 A real function f is said to be continuous if it is continuous at every point\nin the domain of f This definition requires a bit of elaboration"
    },
    {
        "Chapter": "1",
        "sentence_range": "1782-1785",
        "Text": "Having defined continuity of a function at a given point, now we make a natural\nextension of this definition to discuss continuity of a function Definition 2 A real function f is said to be continuous if it is continuous at every point\nin the domain of f This definition requires a bit of elaboration Suppose f is a function defined on a\nclosed interval [a, b], then for f to be continuous, it needs to be continuous at every\npoint in [a, b] including the end points a and b"
    },
    {
        "Chapter": "1",
        "sentence_range": "1783-1786",
        "Text": "Definition 2 A real function f is said to be continuous if it is continuous at every point\nin the domain of f This definition requires a bit of elaboration Suppose f is a function defined on a\nclosed interval [a, b], then for f to be continuous, it needs to be continuous at every\npoint in [a, b] including the end points a and b Continuity of f at a means\nlim\n( )\nx\na f x\n\u2192+\n= f (a)\nand continuity of f  at b means\nlim\u2013\n( )\nx\nb f x\n\u2192\n= f(b)\nObserve that lim\n( )\nx\na f x\n\u2192\u2212\n and lim\n( )\nx\nb f x\n\u2192+\ndo not make sense"
    },
    {
        "Chapter": "1",
        "sentence_range": "1784-1787",
        "Text": "This definition requires a bit of elaboration Suppose f is a function defined on a\nclosed interval [a, b], then for f to be continuous, it needs to be continuous at every\npoint in [a, b] including the end points a and b Continuity of f at a means\nlim\n( )\nx\na f x\n\u2192+\n= f (a)\nand continuity of f  at b means\nlim\u2013\n( )\nx\nb f x\n\u2192\n= f(b)\nObserve that lim\n( )\nx\na f x\n\u2192\u2212\n and lim\n( )\nx\nb f x\n\u2192+\ndo not make sense As a consequence\nof this definition, if f is defined only at one point, it is continuous there, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "1785-1788",
        "Text": "Suppose f is a function defined on a\nclosed interval [a, b], then for f to be continuous, it needs to be continuous at every\npoint in [a, b] including the end points a and b Continuity of f at a means\nlim\n( )\nx\na f x\n\u2192+\n= f (a)\nand continuity of f  at b means\nlim\u2013\n( )\nx\nb f x\n\u2192\n= f(b)\nObserve that lim\n( )\nx\na f x\n\u2192\u2212\n and lim\n( )\nx\nb f x\n\u2192+\ndo not make sense As a consequence\nof this definition, if f is defined only at one point, it is continuous there, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "1786-1789",
        "Text": "Continuity of f at a means\nlim\n( )\nx\na f x\n\u2192+\n= f (a)\nand continuity of f  at b means\nlim\u2013\n( )\nx\nb f x\n\u2192\n= f(b)\nObserve that lim\n( )\nx\na f x\n\u2192\u2212\n and lim\n( )\nx\nb f x\n\u2192+\ndo not make sense As a consequence\nof this definition, if f is defined only at one point, it is continuous there, i e , if the\ndomain of f is a singleton, f is a continuous function"
    },
    {
        "Chapter": "1",
        "sentence_range": "1787-1790",
        "Text": "As a consequence\nof this definition, if f is defined only at one point, it is continuous there, i e , if the\ndomain of f is a singleton, f is a continuous function Rationalised 2023-24\n MATHEMATICS\n108\nExample 7 Is the function defined by f(x) = | x |, a continuous function"
    },
    {
        "Chapter": "1",
        "sentence_range": "1788-1791",
        "Text": "e , if the\ndomain of f is a singleton, f is a continuous function Rationalised 2023-24\n MATHEMATICS\n108\nExample 7 Is the function defined by f(x) = | x |, a continuous function Solution We may rewrite f as\nf (x) =\n, if\n0\n, if\n0\nx\nx\nx\nx\n\u2212\n<\n\uf8f2\uf8f1\n\u2265\n\uf8f3\nBy Example 3, we know that f is continuous at x = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "1789-1792",
        "Text": ", if the\ndomain of f is a singleton, f is a continuous function Rationalised 2023-24\n MATHEMATICS\n108\nExample 7 Is the function defined by f(x) = | x |, a continuous function Solution We may rewrite f as\nf (x) =\n, if\n0\n, if\n0\nx\nx\nx\nx\n\u2212\n<\n\uf8f2\uf8f1\n\u2265\n\uf8f3\nBy Example 3, we know that f is continuous at x = 0 Let c be a real number such that c < 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "1790-1793",
        "Text": "Rationalised 2023-24\n MATHEMATICS\n108\nExample 7 Is the function defined by f(x) = | x |, a continuous function Solution We may rewrite f as\nf (x) =\n, if\n0\n, if\n0\nx\nx\nx\nx\n\u2212\n<\n\uf8f2\uf8f1\n\u2265\n\uf8f3\nBy Example 3, we know that f is continuous at x = 0 Let c be a real number such that c < 0 Then f(c) = \u2013 c"
    },
    {
        "Chapter": "1",
        "sentence_range": "1791-1794",
        "Text": "Solution We may rewrite f as\nf (x) =\n, if\n0\n, if\n0\nx\nx\nx\nx\n\u2212\n<\n\uf8f2\uf8f1\n\u2265\n\uf8f3\nBy Example 3, we know that f is continuous at x = 0 Let c be a real number such that c < 0 Then f(c) = \u2013 c Also\nlim\n( )\nx\nc f x\n\u2192\n = lim (\n)\n\u2013\nx\nc\nx\nc\n\u2192\n\u2212\n=\n          (Why"
    },
    {
        "Chapter": "1",
        "sentence_range": "1792-1795",
        "Text": "Let c be a real number such that c < 0 Then f(c) = \u2013 c Also\nlim\n( )\nx\nc f x\n\u2192\n = lim (\n)\n\u2013\nx\nc\nx\nc\n\u2192\n\u2212\n=\n          (Why )\nSince lim\n( )\n( )\nx\nc f x\nf c\n\u2192\n=\n,  f  is continuous at all negative real numbers"
    },
    {
        "Chapter": "1",
        "sentence_range": "1793-1796",
        "Text": "Then f(c) = \u2013 c Also\nlim\n( )\nx\nc f x\n\u2192\n = lim (\n)\n\u2013\nx\nc\nx\nc\n\u2192\n\u2212\n=\n          (Why )\nSince lim\n( )\n( )\nx\nc f x\nf c\n\u2192\n=\n,  f  is continuous at all negative real numbers Now, let c be a real number such that c > 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "1794-1797",
        "Text": "Also\nlim\n( )\nx\nc f x\n\u2192\n = lim (\n)\n\u2013\nx\nc\nx\nc\n\u2192\n\u2212\n=\n          (Why )\nSince lim\n( )\n( )\nx\nc f x\nf c\n\u2192\n=\n,  f  is continuous at all negative real numbers Now, let c be a real number such that c > 0 Then f (c) = c"
    },
    {
        "Chapter": "1",
        "sentence_range": "1795-1798",
        "Text": ")\nSince lim\n( )\n( )\nx\nc f x\nf c\n\u2192\n=\n,  f  is continuous at all negative real numbers Now, let c be a real number such that c > 0 Then f (c) = c Also\nlim\n( )\nx\nc f x\n\u2192\n = lim\nx\nc x\nc\n\u2192\n=                  (Why"
    },
    {
        "Chapter": "1",
        "sentence_range": "1796-1799",
        "Text": "Now, let c be a real number such that c > 0 Then f (c) = c Also\nlim\n( )\nx\nc f x\n\u2192\n = lim\nx\nc x\nc\n\u2192\n=                  (Why )\nSince lim\n( )\n( )\nx\nc f x\nf c\n\u2192\n=\n, f is continuous at all positive real numbers"
    },
    {
        "Chapter": "1",
        "sentence_range": "1797-1800",
        "Text": "Then f (c) = c Also\nlim\n( )\nx\nc f x\n\u2192\n = lim\nx\nc x\nc\n\u2192\n=                  (Why )\nSince lim\n( )\n( )\nx\nc f x\nf c\n\u2192\n=\n, f is continuous at all positive real numbers Hence, f\nis continuous at all points"
    },
    {
        "Chapter": "1",
        "sentence_range": "1798-1801",
        "Text": "Also\nlim\n( )\nx\nc f x\n\u2192\n = lim\nx\nc x\nc\n\u2192\n=                  (Why )\nSince lim\n( )\n( )\nx\nc f x\nf c\n\u2192\n=\n, f is continuous at all positive real numbers Hence, f\nis continuous at all points Example 8 Discuss the continuity of the function f given by f (x) = x3 + x2 \u2013 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "1799-1802",
        "Text": ")\nSince lim\n( )\n( )\nx\nc f x\nf c\n\u2192\n=\n, f is continuous at all positive real numbers Hence, f\nis continuous at all points Example 8 Discuss the continuity of the function f given by f (x) = x3 + x2 \u2013 1 Solution Clearly f is defined at every real number c and its value at c is c3 + c2 \u2013 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "1800-1803",
        "Text": "Hence, f\nis continuous at all points Example 8 Discuss the continuity of the function f given by f (x) = x3 + x2 \u2013 1 Solution Clearly f is defined at every real number c and its value at c is c3 + c2 \u2013 1 We\nalso know that\nlim\n( )\nx\nc f x\n\u2192\n =\n3\n2\n3\n2\nlim (\n1)\n1\nx\nc x\nx\nc\nc\n\u2192\n+\n\u2212\n=\n+\n\u2212\nThus lim\n( )\n( )\nx\nc f x\nf c\n\u2192\n=\n, and hence f is continuous at every real number"
    },
    {
        "Chapter": "1",
        "sentence_range": "1801-1804",
        "Text": "Example 8 Discuss the continuity of the function f given by f (x) = x3 + x2 \u2013 1 Solution Clearly f is defined at every real number c and its value at c is c3 + c2 \u2013 1 We\nalso know that\nlim\n( )\nx\nc f x\n\u2192\n =\n3\n2\n3\n2\nlim (\n1)\n1\nx\nc x\nx\nc\nc\n\u2192\n+\n\u2212\n=\n+\n\u2212\nThus lim\n( )\n( )\nx\nc f x\nf c\n\u2192\n=\n, and hence f is continuous at every real number This means\nf is a continuous function"
    },
    {
        "Chapter": "1",
        "sentence_range": "1802-1805",
        "Text": "Solution Clearly f is defined at every real number c and its value at c is c3 + c2 \u2013 1 We\nalso know that\nlim\n( )\nx\nc f x\n\u2192\n =\n3\n2\n3\n2\nlim (\n1)\n1\nx\nc x\nx\nc\nc\n\u2192\n+\n\u2212\n=\n+\n\u2212\nThus lim\n( )\n( )\nx\nc f x\nf c\n\u2192\n=\n, and hence f is continuous at every real number This means\nf is a continuous function Example 9 Discuss the continuity of the function f defined by f (x) = 1\nx , x \u2260 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "1803-1806",
        "Text": "We\nalso know that\nlim\n( )\nx\nc f x\n\u2192\n =\n3\n2\n3\n2\nlim (\n1)\n1\nx\nc x\nx\nc\nc\n\u2192\n+\n\u2212\n=\n+\n\u2212\nThus lim\n( )\n( )\nx\nc f x\nf c\n\u2192\n=\n, and hence f is continuous at every real number This means\nf is a continuous function Example 9 Discuss the continuity of the function f defined by f (x) = 1\nx , x \u2260 0 Solution Fix any non zero real number c, we have\n1\n1\nlim\n( )\nlim\nx\nc\nx\nc\nf x\nx\nc\n\u2192\n=\u2192\n=\nAlso, since for c \u2260 0, \n1\nf c( )\n=c\n, we have lim\n( )\n( )\nx\nc f x\nf c\n\u2192\n=\n and hence, f is continuous\nat every point in the domain of f"
    },
    {
        "Chapter": "1",
        "sentence_range": "1804-1807",
        "Text": "This means\nf is a continuous function Example 9 Discuss the continuity of the function f defined by f (x) = 1\nx , x \u2260 0 Solution Fix any non zero real number c, we have\n1\n1\nlim\n( )\nlim\nx\nc\nx\nc\nf x\nx\nc\n\u2192\n=\u2192\n=\nAlso, since for c \u2260 0, \n1\nf c( )\n=c\n, we have lim\n( )\n( )\nx\nc f x\nf c\n\u2192\n=\n and hence, f is continuous\nat every point in the domain of f Thus f is a continuous function"
    },
    {
        "Chapter": "1",
        "sentence_range": "1805-1808",
        "Text": "Example 9 Discuss the continuity of the function f defined by f (x) = 1\nx , x \u2260 0 Solution Fix any non zero real number c, we have\n1\n1\nlim\n( )\nlim\nx\nc\nx\nc\nf x\nx\nc\n\u2192\n=\u2192\n=\nAlso, since for c \u2260 0, \n1\nf c( )\n=c\n, we have lim\n( )\n( )\nx\nc f x\nf c\n\u2192\n=\n and hence, f is continuous\nat every point in the domain of f Thus f is a continuous function Rationalised 2023-24\nCONTINUITY AND DIFFERENTIABILITY\n109\nWe take this opportunity to explain the concept of infinity"
    },
    {
        "Chapter": "1",
        "sentence_range": "1806-1809",
        "Text": "Solution Fix any non zero real number c, we have\n1\n1\nlim\n( )\nlim\nx\nc\nx\nc\nf x\nx\nc\n\u2192\n=\u2192\n=\nAlso, since for c \u2260 0, \n1\nf c( )\n=c\n, we have lim\n( )\n( )\nx\nc f x\nf c\n\u2192\n=\n and hence, f is continuous\nat every point in the domain of f Thus f is a continuous function Rationalised 2023-24\nCONTINUITY AND DIFFERENTIABILITY\n109\nWe take this opportunity to explain the concept of infinity This we do by analysing\nthe function f (x) = 1\nx  near x = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "1807-1810",
        "Text": "Thus f is a continuous function Rationalised 2023-24\nCONTINUITY AND DIFFERENTIABILITY\n109\nWe take this opportunity to explain the concept of infinity This we do by analysing\nthe function f (x) = 1\nx  near x = 0 To carry out this analysis we follow the usual trick of\nfinding the value of the function at real numbers close to 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "1808-1811",
        "Text": "Rationalised 2023-24\nCONTINUITY AND DIFFERENTIABILITY\n109\nWe take this opportunity to explain the concept of infinity This we do by analysing\nthe function f (x) = 1\nx  near x = 0 To carry out this analysis we follow the usual trick of\nfinding the value of the function at real numbers close to 0 Essentially we are trying to\nfind the right hand limit of f at 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "1809-1812",
        "Text": "This we do by analysing\nthe function f (x) = 1\nx  near x = 0 To carry out this analysis we follow the usual trick of\nfinding the value of the function at real numbers close to 0 Essentially we are trying to\nfind the right hand limit of f at 0 We tabulate this in the following (Table 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "1810-1813",
        "Text": "To carry out this analysis we follow the usual trick of\nfinding the value of the function at real numbers close to 0 Essentially we are trying to\nfind the right hand limit of f at 0 We tabulate this in the following (Table 5 1)"
    },
    {
        "Chapter": "1",
        "sentence_range": "1811-1814",
        "Text": "Essentially we are trying to\nfind the right hand limit of f at 0 We tabulate this in the following (Table 5 1) Table 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "1812-1815",
        "Text": "We tabulate this in the following (Table 5 1) Table 5 1\nx\n1\n0"
    },
    {
        "Chapter": "1",
        "sentence_range": "1813-1816",
        "Text": "1) Table 5 1\nx\n1\n0 3\n0"
    },
    {
        "Chapter": "1",
        "sentence_range": "1814-1817",
        "Text": "Table 5 1\nx\n1\n0 3\n0 2\n0"
    },
    {
        "Chapter": "1",
        "sentence_range": "1815-1818",
        "Text": "1\nx\n1\n0 3\n0 2\n0 1 = 10\u20131\n0"
    },
    {
        "Chapter": "1",
        "sentence_range": "1816-1819",
        "Text": "3\n0 2\n0 1 = 10\u20131\n0 01 = 10\u20132\n0"
    },
    {
        "Chapter": "1",
        "sentence_range": "1817-1820",
        "Text": "2\n0 1 = 10\u20131\n0 01 = 10\u20132\n0 001 = 10\u20133\n10\u2013n\nf (x)\n1\n3"
    },
    {
        "Chapter": "1",
        "sentence_range": "1818-1821",
        "Text": "1 = 10\u20131\n0 01 = 10\u20132\n0 001 = 10\u20133\n10\u2013n\nf (x)\n1\n3 333"
    },
    {
        "Chapter": "1",
        "sentence_range": "1819-1822",
        "Text": "01 = 10\u20132\n0 001 = 10\u20133\n10\u2013n\nf (x)\n1\n3 333 5\n10\n100 = 102\n1000 = 103\n10n\nWe observe that as x gets closer to 0 from the right, the value of f (x) shoots up\nhigher"
    },
    {
        "Chapter": "1",
        "sentence_range": "1820-1823",
        "Text": "001 = 10\u20133\n10\u2013n\nf (x)\n1\n3 333 5\n10\n100 = 102\n1000 = 103\n10n\nWe observe that as x gets closer to 0 from the right, the value of f (x) shoots up\nhigher This may be rephrased as: the value of f (x) may be made larger than any given\nnumber by choosing a positive real number very close to 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "1821-1824",
        "Text": "333 5\n10\n100 = 102\n1000 = 103\n10n\nWe observe that as x gets closer to 0 from the right, the value of f (x) shoots up\nhigher This may be rephrased as: the value of f (x) may be made larger than any given\nnumber by choosing a positive real number very close to 0 In symbols, we write\nlim0\n( )\nx\nf x\n\u2192+\n= + \u221e\n(to be read as: the right hand limit of f (x) at 0 is plus infinity)"
    },
    {
        "Chapter": "1",
        "sentence_range": "1822-1825",
        "Text": "5\n10\n100 = 102\n1000 = 103\n10n\nWe observe that as x gets closer to 0 from the right, the value of f (x) shoots up\nhigher This may be rephrased as: the value of f (x) may be made larger than any given\nnumber by choosing a positive real number very close to 0 In symbols, we write\nlim0\n( )\nx\nf x\n\u2192+\n= + \u221e\n(to be read as: the right hand limit of f (x) at 0 is plus infinity) We wish to emphasise\nthat + \u221e is NOT a real number and hence the right hand limit of f at 0 does not exist (as\na real number)"
    },
    {
        "Chapter": "1",
        "sentence_range": "1823-1826",
        "Text": "This may be rephrased as: the value of f (x) may be made larger than any given\nnumber by choosing a positive real number very close to 0 In symbols, we write\nlim0\n( )\nx\nf x\n\u2192+\n= + \u221e\n(to be read as: the right hand limit of f (x) at 0 is plus infinity) We wish to emphasise\nthat + \u221e is NOT a real number and hence the right hand limit of f at 0 does not exist (as\na real number) Similarly, the left hand limit of f  at 0 may be found"
    },
    {
        "Chapter": "1",
        "sentence_range": "1824-1827",
        "Text": "In symbols, we write\nlim0\n( )\nx\nf x\n\u2192+\n= + \u221e\n(to be read as: the right hand limit of f (x) at 0 is plus infinity) We wish to emphasise\nthat + \u221e is NOT a real number and hence the right hand limit of f at 0 does not exist (as\na real number) Similarly, the left hand limit of f  at 0 may be found The following table is self\nexplanatory"
    },
    {
        "Chapter": "1",
        "sentence_range": "1825-1828",
        "Text": "We wish to emphasise\nthat + \u221e is NOT a real number and hence the right hand limit of f at 0 does not exist (as\na real number) Similarly, the left hand limit of f  at 0 may be found The following table is self\nexplanatory Table 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "1826-1829",
        "Text": "Similarly, the left hand limit of f  at 0 may be found The following table is self\nexplanatory Table 5 2\nx\n\u2013 1\n\u2013 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "1827-1830",
        "Text": "The following table is self\nexplanatory Table 5 2\nx\n\u2013 1\n\u2013 0 3\n\u2013 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "1828-1831",
        "Text": "Table 5 2\nx\n\u2013 1\n\u2013 0 3\n\u2013 0 2\n\u2013 10\u20131\n\u2013 10\u20132\n\u2013 10\u20133\n\u2013 10\u2013n\nf (x)\n\u2013 1\n\u2013 3"
    },
    {
        "Chapter": "1",
        "sentence_range": "1829-1832",
        "Text": "2\nx\n\u2013 1\n\u2013 0 3\n\u2013 0 2\n\u2013 10\u20131\n\u2013 10\u20132\n\u2013 10\u20133\n\u2013 10\u2013n\nf (x)\n\u2013 1\n\u2013 3 333"
    },
    {
        "Chapter": "1",
        "sentence_range": "1830-1833",
        "Text": "3\n\u2013 0 2\n\u2013 10\u20131\n\u2013 10\u20132\n\u2013 10\u20133\n\u2013 10\u2013n\nf (x)\n\u2013 1\n\u2013 3 333 \u2013 5\n\u2013 10\n\u2013 102\n\u2013 103\n\u2013 10n\nFrom the Table 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "1831-1834",
        "Text": "2\n\u2013 10\u20131\n\u2013 10\u20132\n\u2013 10\u20133\n\u2013 10\u2013n\nf (x)\n\u2013 1\n\u2013 3 333 \u2013 5\n\u2013 10\n\u2013 102\n\u2013 103\n\u2013 10n\nFrom the Table 5 2, we deduce that the\nvalue of f(x) may be made smaller than any\ngiven number by choosing a negative real\nnumber very close to 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "1832-1835",
        "Text": "333 \u2013 5\n\u2013 10\n\u2013 102\n\u2013 103\n\u2013 10n\nFrom the Table 5 2, we deduce that the\nvalue of f(x) may be made smaller than any\ngiven number by choosing a negative real\nnumber very close to 0 In symbols,\nwe write\nlim0\n( )\nx\nf x\n\u2192\u2212\n= \u2212 \u221e\n(to be read as: the left hand limit of f (x) at 0 is\nminus infinity)"
    },
    {
        "Chapter": "1",
        "sentence_range": "1833-1836",
        "Text": "\u2013 5\n\u2013 10\n\u2013 102\n\u2013 103\n\u2013 10n\nFrom the Table 5 2, we deduce that the\nvalue of f(x) may be made smaller than any\ngiven number by choosing a negative real\nnumber very close to 0 In symbols,\nwe write\nlim0\n( )\nx\nf x\n\u2192\u2212\n= \u2212 \u221e\n(to be read as: the left hand limit of f (x) at 0 is\nminus infinity) Again, we wish to emphasise\nthat \u2013 \u221e is NOT a real number and hence the\nleft hand limit of f at 0 does not exist (as a real\nnumber)"
    },
    {
        "Chapter": "1",
        "sentence_range": "1834-1837",
        "Text": "2, we deduce that the\nvalue of f(x) may be made smaller than any\ngiven number by choosing a negative real\nnumber very close to 0 In symbols,\nwe write\nlim0\n( )\nx\nf x\n\u2192\u2212\n= \u2212 \u221e\n(to be read as: the left hand limit of f (x) at 0 is\nminus infinity) Again, we wish to emphasise\nthat \u2013 \u221e is NOT a real number and hence the\nleft hand limit of f at 0 does not exist (as a real\nnumber) The graph of the reciprocal function\ngiven in Fig 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "1835-1838",
        "Text": "In symbols,\nwe write\nlim0\n( )\nx\nf x\n\u2192\u2212\n= \u2212 \u221e\n(to be read as: the left hand limit of f (x) at 0 is\nminus infinity) Again, we wish to emphasise\nthat \u2013 \u221e is NOT a real number and hence the\nleft hand limit of f at 0 does not exist (as a real\nnumber) The graph of the reciprocal function\ngiven in Fig 5 3 is a geometric representation\nof the above mentioned facts"
    },
    {
        "Chapter": "1",
        "sentence_range": "1836-1839",
        "Text": "Again, we wish to emphasise\nthat \u2013 \u221e is NOT a real number and hence the\nleft hand limit of f at 0 does not exist (as a real\nnumber) The graph of the reciprocal function\ngiven in Fig 5 3 is a geometric representation\nof the above mentioned facts Fig 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "1837-1840",
        "Text": "The graph of the reciprocal function\ngiven in Fig 5 3 is a geometric representation\nof the above mentioned facts Fig 5 3\nRationalised 2023-24\n MATHEMATICS\n110\nExample 10 Discuss the continuity of the function f defined by\nf (x) =\n2, if\n1\n2, if\n1\nx\nx\nx\nx\n+\n\u2264\n\uf8f1\n\uf8f2 \u2212\n>\n\uf8f3\nSolution The function f is defined at all points of the real line"
    },
    {
        "Chapter": "1",
        "sentence_range": "1838-1841",
        "Text": "3 is a geometric representation\nof the above mentioned facts Fig 5 3\nRationalised 2023-24\n MATHEMATICS\n110\nExample 10 Discuss the continuity of the function f defined by\nf (x) =\n2, if\n1\n2, if\n1\nx\nx\nx\nx\n+\n\u2264\n\uf8f1\n\uf8f2 \u2212\n>\n\uf8f3\nSolution The function f is defined at all points of the real line Case 1 If c < 1, then f(c) = c + 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "1839-1842",
        "Text": "Fig 5 3\nRationalised 2023-24\n MATHEMATICS\n110\nExample 10 Discuss the continuity of the function f defined by\nf (x) =\n2, if\n1\n2, if\n1\nx\nx\nx\nx\n+\n\u2264\n\uf8f1\n\uf8f2 \u2212\n>\n\uf8f3\nSolution The function f is defined at all points of the real line Case 1 If c < 1, then f(c) = c + 2 Therefore, lim\n( )\nlim(\n2)\n2\nx\nc\nx\nc\nf x\nx\nc\n\u2192\n=\u2192\n+\n=\n+\nThus, f is continuous at all real numbers less than 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "1840-1843",
        "Text": "3\nRationalised 2023-24\n MATHEMATICS\n110\nExample 10 Discuss the continuity of the function f defined by\nf (x) =\n2, if\n1\n2, if\n1\nx\nx\nx\nx\n+\n\u2264\n\uf8f1\n\uf8f2 \u2212\n>\n\uf8f3\nSolution The function f is defined at all points of the real line Case 1 If c < 1, then f(c) = c + 2 Therefore, lim\n( )\nlim(\n2)\n2\nx\nc\nx\nc\nf x\nx\nc\n\u2192\n=\u2192\n+\n=\n+\nThus, f is continuous at all real numbers less than 1 Case 2 If c > 1, then f (c) = c \u2013 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "1841-1844",
        "Text": "Case 1 If c < 1, then f(c) = c + 2 Therefore, lim\n( )\nlim(\n2)\n2\nx\nc\nx\nc\nf x\nx\nc\n\u2192\n=\u2192\n+\n=\n+\nThus, f is continuous at all real numbers less than 1 Case 2 If c > 1, then f (c) = c \u2013 2 Therefore,\nlim\n( )\nlim\nx\nc\nx\nc\nf x\n\u2192\n=\u2192\n(x \u2013 2) = c \u2013 2 = f (c)\nThus, f is continuous at all points x > 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "1842-1845",
        "Text": "Therefore, lim\n( )\nlim(\n2)\n2\nx\nc\nx\nc\nf x\nx\nc\n\u2192\n=\u2192\n+\n=\n+\nThus, f is continuous at all real numbers less than 1 Case 2 If c > 1, then f (c) = c \u2013 2 Therefore,\nlim\n( )\nlim\nx\nc\nx\nc\nf x\n\u2192\n=\u2192\n(x \u2013 2) = c \u2013 2 = f (c)\nThus, f is continuous at all points x > 1 Case 3 If c = 1, then the left hand limit of f at\nx = 1 is\n\u2013\n\u2013\n1\n1\nlim\n( )\nlim (\n2)\n1\n2\n3\nx\nx\nf x\nx\n\u2192\n=\u2192\n+\n= +\n=\nThe right hand limit of f at x = 1 is\n1\n1\nlim\n( )\nlim (\n2)\n1\n2\n1\nx\nx\nf x\nx\n+\n+\n\u2192\n=\u2192\n\u2212\n= \u2212\n= \u2212\nSince the left and right hand limits of f at x = 1\ndo not coincide, f  is not continuous at x = 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "1843-1846",
        "Text": "Case 2 If c > 1, then f (c) = c \u2013 2 Therefore,\nlim\n( )\nlim\nx\nc\nx\nc\nf x\n\u2192\n=\u2192\n(x \u2013 2) = c \u2013 2 = f (c)\nThus, f is continuous at all points x > 1 Case 3 If c = 1, then the left hand limit of f at\nx = 1 is\n\u2013\n\u2013\n1\n1\nlim\n( )\nlim (\n2)\n1\n2\n3\nx\nx\nf x\nx\n\u2192\n=\u2192\n+\n= +\n=\nThe right hand limit of f at x = 1 is\n1\n1\nlim\n( )\nlim (\n2)\n1\n2\n1\nx\nx\nf x\nx\n+\n+\n\u2192\n=\u2192\n\u2212\n= \u2212\n= \u2212\nSince the left and right hand limits of f at x = 1\ndo not coincide, f  is not continuous at x = 1 Hence\nx = 1 is the only point of discontinuity of f"
    },
    {
        "Chapter": "1",
        "sentence_range": "1844-1847",
        "Text": "Therefore,\nlim\n( )\nlim\nx\nc\nx\nc\nf x\n\u2192\n=\u2192\n(x \u2013 2) = c \u2013 2 = f (c)\nThus, f is continuous at all points x > 1 Case 3 If c = 1, then the left hand limit of f at\nx = 1 is\n\u2013\n\u2013\n1\n1\nlim\n( )\nlim (\n2)\n1\n2\n3\nx\nx\nf x\nx\n\u2192\n=\u2192\n+\n= +\n=\nThe right hand limit of f at x = 1 is\n1\n1\nlim\n( )\nlim (\n2)\n1\n2\n1\nx\nx\nf x\nx\n+\n+\n\u2192\n=\u2192\n\u2212\n= \u2212\n= \u2212\nSince the left and right hand limits of f at x = 1\ndo not coincide, f  is not continuous at x = 1 Hence\nx = 1 is the only point of discontinuity of f The graph of the function is given in Fig 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "1845-1848",
        "Text": "Case 3 If c = 1, then the left hand limit of f at\nx = 1 is\n\u2013\n\u2013\n1\n1\nlim\n( )\nlim (\n2)\n1\n2\n3\nx\nx\nf x\nx\n\u2192\n=\u2192\n+\n= +\n=\nThe right hand limit of f at x = 1 is\n1\n1\nlim\n( )\nlim (\n2)\n1\n2\n1\nx\nx\nf x\nx\n+\n+\n\u2192\n=\u2192\n\u2212\n= \u2212\n= \u2212\nSince the left and right hand limits of f at x = 1\ndo not coincide, f  is not continuous at x = 1 Hence\nx = 1 is the only point of discontinuity of f The graph of the function is given in Fig 5 4"
    },
    {
        "Chapter": "1",
        "sentence_range": "1846-1849",
        "Text": "Hence\nx = 1 is the only point of discontinuity of f The graph of the function is given in Fig 5 4 Example 11 Find all the points of discontinuity of the function f defined by\nf (x) = \n2, if\n1\n0,\nif\n1\n2, if\n1\nx\nx\nx\nx\nx\n+\n<\n\uf8f4\uf8f1\n=\n\uf8f2\n\uf8f4 \u2212\n>\n\uf8f3\nSolution As in the previous example we find that f\nis continuous at all real numbers x \u2260 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "1847-1850",
        "Text": "The graph of the function is given in Fig 5 4 Example 11 Find all the points of discontinuity of the function f defined by\nf (x) = \n2, if\n1\n0,\nif\n1\n2, if\n1\nx\nx\nx\nx\nx\n+\n<\n\uf8f4\uf8f1\n=\n\uf8f2\n\uf8f4 \u2212\n>\n\uf8f3\nSolution As in the previous example we find that f\nis continuous at all real numbers x \u2260 1 The left\nhand limit of f at x = 1 is\n\u2013\n1\n1\nlim\n( )\nlim (\n2)\n1\n2\n3\nx\nx\nf x\nx\n\u2192\u2212\n=\u2192\n+\n= +\n=\nThe right hand limit of f at x = 1 is\n1\n1\nlim\n( )\nlim (\n2)\n1\n2\n1\nx\nx\nf x\nx\n+\n+\n\u2192\n=\u2192\n\u2212\n= \u2212\n= \u2212\nSince, the left and right hand limits of f at x = 1\ndo not coincide, f is not continuous at x = 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "1848-1851",
        "Text": "4 Example 11 Find all the points of discontinuity of the function f defined by\nf (x) = \n2, if\n1\n0,\nif\n1\n2, if\n1\nx\nx\nx\nx\nx\n+\n<\n\uf8f4\uf8f1\n=\n\uf8f2\n\uf8f4 \u2212\n>\n\uf8f3\nSolution As in the previous example we find that f\nis continuous at all real numbers x \u2260 1 The left\nhand limit of f at x = 1 is\n\u2013\n1\n1\nlim\n( )\nlim (\n2)\n1\n2\n3\nx\nx\nf x\nx\n\u2192\u2212\n=\u2192\n+\n= +\n=\nThe right hand limit of f at x = 1 is\n1\n1\nlim\n( )\nlim (\n2)\n1\n2\n1\nx\nx\nf x\nx\n+\n+\n\u2192\n=\u2192\n\u2212\n= \u2212\n= \u2212\nSince, the left and right hand limits of f at x = 1\ndo not coincide, f is not continuous at x = 1 Hence\nx = 1 is the only point of discontinuity of f"
    },
    {
        "Chapter": "1",
        "sentence_range": "1849-1852",
        "Text": "Example 11 Find all the points of discontinuity of the function f defined by\nf (x) = \n2, if\n1\n0,\nif\n1\n2, if\n1\nx\nx\nx\nx\nx\n+\n<\n\uf8f4\uf8f1\n=\n\uf8f2\n\uf8f4 \u2212\n>\n\uf8f3\nSolution As in the previous example we find that f\nis continuous at all real numbers x \u2260 1 The left\nhand limit of f at x = 1 is\n\u2013\n1\n1\nlim\n( )\nlim (\n2)\n1\n2\n3\nx\nx\nf x\nx\n\u2192\u2212\n=\u2192\n+\n= +\n=\nThe right hand limit of f at x = 1 is\n1\n1\nlim\n( )\nlim (\n2)\n1\n2\n1\nx\nx\nf x\nx\n+\n+\n\u2192\n=\u2192\n\u2212\n= \u2212\n= \u2212\nSince, the left and right hand limits of f at x = 1\ndo not coincide, f is not continuous at x = 1 Hence\nx = 1 is the only point of discontinuity of f The\ngraph of the function is given in the Fig 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "1850-1853",
        "Text": "The left\nhand limit of f at x = 1 is\n\u2013\n1\n1\nlim\n( )\nlim (\n2)\n1\n2\n3\nx\nx\nf x\nx\n\u2192\u2212\n=\u2192\n+\n= +\n=\nThe right hand limit of f at x = 1 is\n1\n1\nlim\n( )\nlim (\n2)\n1\n2\n1\nx\nx\nf x\nx\n+\n+\n\u2192\n=\u2192\n\u2212\n= \u2212\n= \u2212\nSince, the left and right hand limits of f at x = 1\ndo not coincide, f is not continuous at x = 1 Hence\nx = 1 is the only point of discontinuity of f The\ngraph of the function is given in the Fig 5 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "1851-1854",
        "Text": "Hence\nx = 1 is the only point of discontinuity of f The\ngraph of the function is given in the Fig 5 5 Fig  5"
    },
    {
        "Chapter": "1",
        "sentence_range": "1852-1855",
        "Text": "The\ngraph of the function is given in the Fig 5 5 Fig  5 4\nFig  5"
    },
    {
        "Chapter": "1",
        "sentence_range": "1853-1856",
        "Text": "5 Fig  5 4\nFig  5 5\nRationalised 2023-24\nCONTINUITY AND DIFFERENTIABILITY\n111\nExample 12 Discuss the continuity of the function defined by\nf(x) =\n2, if\n0\n2, if\n0\nx\nx\nx\nx\n+\n<\n\uf8f1\n\uf8f2\u2212 +\n>\n\uf8f3\nSolution Observe that the function is defined at all real numbers except at 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "1854-1857",
        "Text": "Fig  5 4\nFig  5 5\nRationalised 2023-24\nCONTINUITY AND DIFFERENTIABILITY\n111\nExample 12 Discuss the continuity of the function defined by\nf(x) =\n2, if\n0\n2, if\n0\nx\nx\nx\nx\n+\n<\n\uf8f1\n\uf8f2\u2212 +\n>\n\uf8f3\nSolution Observe that the function is defined at all real numbers except at 0 Domain\nof definition of this function is\nD1 \u222a D2 where  D1 = {x \u2208 R : x < 0} and\n      D2 = {x \u2208 R : x > 0}\nCase 1 If c \u2208 D1, then lim\n( )\nlim\nx\nc\nx\nc\nf x\n\u2192\n=\u2192\n (x + 2)\n= c + 2 = f (c) and hence f is continuous in D1"
    },
    {
        "Chapter": "1",
        "sentence_range": "1855-1858",
        "Text": "4\nFig  5 5\nRationalised 2023-24\nCONTINUITY AND DIFFERENTIABILITY\n111\nExample 12 Discuss the continuity of the function defined by\nf(x) =\n2, if\n0\n2, if\n0\nx\nx\nx\nx\n+\n<\n\uf8f1\n\uf8f2\u2212 +\n>\n\uf8f3\nSolution Observe that the function is defined at all real numbers except at 0 Domain\nof definition of this function is\nD1 \u222a D2 where  D1 = {x \u2208 R : x < 0} and\n      D2 = {x \u2208 R : x > 0}\nCase 1 If c \u2208 D1, then lim\n( )\nlim\nx\nc\nx\nc\nf x\n\u2192\n=\u2192\n (x + 2)\n= c + 2 = f (c) and hence f is continuous in D1 Case 2 If c \u2208 D2, then lim\n( )\nlim\nx\nc\nx\nc\nf x\n\u2192\n=\u2192\n (\u2013 x + 2)\n= \u2013 c + 2 = f (c) and hence f is continuous in D2"
    },
    {
        "Chapter": "1",
        "sentence_range": "1856-1859",
        "Text": "5\nRationalised 2023-24\nCONTINUITY AND DIFFERENTIABILITY\n111\nExample 12 Discuss the continuity of the function defined by\nf(x) =\n2, if\n0\n2, if\n0\nx\nx\nx\nx\n+\n<\n\uf8f1\n\uf8f2\u2212 +\n>\n\uf8f3\nSolution Observe that the function is defined at all real numbers except at 0 Domain\nof definition of this function is\nD1 \u222a D2 where  D1 = {x \u2208 R : x < 0} and\n      D2 = {x \u2208 R : x > 0}\nCase 1 If c \u2208 D1, then lim\n( )\nlim\nx\nc\nx\nc\nf x\n\u2192\n=\u2192\n (x + 2)\n= c + 2 = f (c) and hence f is continuous in D1 Case 2 If c \u2208 D2, then lim\n( )\nlim\nx\nc\nx\nc\nf x\n\u2192\n=\u2192\n (\u2013 x + 2)\n= \u2013 c + 2 = f (c) and hence f is continuous in D2 Since f is continuous at all points in the domain of f,\nwe deduce that f is continuous"
    },
    {
        "Chapter": "1",
        "sentence_range": "1857-1860",
        "Text": "Domain\nof definition of this function is\nD1 \u222a D2 where  D1 = {x \u2208 R : x < 0} and\n      D2 = {x \u2208 R : x > 0}\nCase 1 If c \u2208 D1, then lim\n( )\nlim\nx\nc\nx\nc\nf x\n\u2192\n=\u2192\n (x + 2)\n= c + 2 = f (c) and hence f is continuous in D1 Case 2 If c \u2208 D2, then lim\n( )\nlim\nx\nc\nx\nc\nf x\n\u2192\n=\u2192\n (\u2013 x + 2)\n= \u2013 c + 2 = f (c) and hence f is continuous in D2 Since f is continuous at all points in the domain of f,\nwe deduce that f is continuous Graph of this\nfunction is given in the Fig 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "1858-1861",
        "Text": "Case 2 If c \u2208 D2, then lim\n( )\nlim\nx\nc\nx\nc\nf x\n\u2192\n=\u2192\n (\u2013 x + 2)\n= \u2013 c + 2 = f (c) and hence f is continuous in D2 Since f is continuous at all points in the domain of f,\nwe deduce that f is continuous Graph of this\nfunction is given in the Fig 5 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "1859-1862",
        "Text": "Since f is continuous at all points in the domain of f,\nwe deduce that f is continuous Graph of this\nfunction is given in the Fig 5 6 Note that to graph\nthis function we need to lift the pen from the plane\nof the paper, but we need to do that only for those points where the function is not\ndefined"
    },
    {
        "Chapter": "1",
        "sentence_range": "1860-1863",
        "Text": "Graph of this\nfunction is given in the Fig 5 6 Note that to graph\nthis function we need to lift the pen from the plane\nof the paper, but we need to do that only for those points where the function is not\ndefined Example 13 Discuss the continuity of the function f given by\nf (x) = \n,2\nif\n0\n, if\n0\nx\nx\nx\nx\n\u2265\n\uf8f1\uf8f4\uf8f2\n<\n\uf8f4\uf8f3\nSolution Clearly the function is defined at\nevery real number"
    },
    {
        "Chapter": "1",
        "sentence_range": "1861-1864",
        "Text": "6 Note that to graph\nthis function we need to lift the pen from the plane\nof the paper, but we need to do that only for those points where the function is not\ndefined Example 13 Discuss the continuity of the function f given by\nf (x) = \n,2\nif\n0\n, if\n0\nx\nx\nx\nx\n\u2265\n\uf8f1\uf8f4\uf8f2\n<\n\uf8f4\uf8f3\nSolution Clearly the function is defined at\nevery real number Graph of the function is\ngiven in Fig 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "1862-1865",
        "Text": "Note that to graph\nthis function we need to lift the pen from the plane\nof the paper, but we need to do that only for those points where the function is not\ndefined Example 13 Discuss the continuity of the function f given by\nf (x) = \n,2\nif\n0\n, if\n0\nx\nx\nx\nx\n\u2265\n\uf8f1\uf8f4\uf8f2\n<\n\uf8f4\uf8f3\nSolution Clearly the function is defined at\nevery real number Graph of the function is\ngiven in Fig 5 7"
    },
    {
        "Chapter": "1",
        "sentence_range": "1863-1866",
        "Text": "Example 13 Discuss the continuity of the function f given by\nf (x) = \n,2\nif\n0\n, if\n0\nx\nx\nx\nx\n\u2265\n\uf8f1\uf8f4\uf8f2\n<\n\uf8f4\uf8f3\nSolution Clearly the function is defined at\nevery real number Graph of the function is\ngiven in Fig 5 7 By inspection, it seems prudent\nto partition the domain of definition of f into\nthree disjoint subsets of the real line"
    },
    {
        "Chapter": "1",
        "sentence_range": "1864-1867",
        "Text": "Graph of the function is\ngiven in Fig 5 7 By inspection, it seems prudent\nto partition the domain of definition of f into\nthree disjoint subsets of the real line Let\nD1 = {x \u2208 R : x < 0}, D2 = {0} and\nD3 = {x \u2208 R : x > 0}\nCase 1 At any point in D1, we have f(x) = x2 and it is easy to see that it is continuous\nthere (see Example 2)"
    },
    {
        "Chapter": "1",
        "sentence_range": "1865-1868",
        "Text": "7 By inspection, it seems prudent\nto partition the domain of definition of f into\nthree disjoint subsets of the real line Let\nD1 = {x \u2208 R : x < 0}, D2 = {0} and\nD3 = {x \u2208 R : x > 0}\nCase 1 At any point in D1, we have f(x) = x2 and it is easy to see that it is continuous\nthere (see Example 2) Case 2 At any point in D3, we have f(x) = x and it is easy to see that it is continuous\nthere (see Example 6)"
    },
    {
        "Chapter": "1",
        "sentence_range": "1866-1869",
        "Text": "By inspection, it seems prudent\nto partition the domain of definition of f into\nthree disjoint subsets of the real line Let\nD1 = {x \u2208 R : x < 0}, D2 = {0} and\nD3 = {x \u2208 R : x > 0}\nCase 1 At any point in D1, we have f(x) = x2 and it is easy to see that it is continuous\nthere (see Example 2) Case 2 At any point in D3, we have f(x) = x and it is easy to see that it is continuous\nthere (see Example 6) Fig 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "1867-1870",
        "Text": "Let\nD1 = {x \u2208 R : x < 0}, D2 = {0} and\nD3 = {x \u2208 R : x > 0}\nCase 1 At any point in D1, we have f(x) = x2 and it is easy to see that it is continuous\nthere (see Example 2) Case 2 At any point in D3, we have f(x) = x and it is easy to see that it is continuous\nthere (see Example 6) Fig 5 6\n  Fig 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "1868-1871",
        "Text": "Case 2 At any point in D3, we have f(x) = x and it is easy to see that it is continuous\nthere (see Example 6) Fig 5 6\n  Fig 5 7\nRationalised 2023-24\n MATHEMATICS\n112\nCase 3 Now we analyse the function at x = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "1869-1872",
        "Text": "Fig 5 6\n  Fig 5 7\nRationalised 2023-24\n MATHEMATICS\n112\nCase 3 Now we analyse the function at x = 0 The value of the function at 0 is f(0) = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "1870-1873",
        "Text": "6\n  Fig 5 7\nRationalised 2023-24\n MATHEMATICS\n112\nCase 3 Now we analyse the function at x = 0 The value of the function at 0 is f(0) = 0 The left hand limit of  f at 0 is\n\u2013\n2\n2\n0\n0\nlim\n( )\nlim\n0\n0\nx\nx\nf x\n\u2212x\n\u2192\n=\u2192\n=\n=\nThe right hand limit of f at 0 is\n0\n0\nlim\n( )\nlim\n0\nx\nx\nf x\nx\n+\n+\n\u2192\n=\u2192\n=\nThus \nlim0\n( )\n0\nx\nf x\n\u2192\n=\n= f(0) and hence f is continuous at 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "1871-1874",
        "Text": "7\nRationalised 2023-24\n MATHEMATICS\n112\nCase 3 Now we analyse the function at x = 0 The value of the function at 0 is f(0) = 0 The left hand limit of  f at 0 is\n\u2013\n2\n2\n0\n0\nlim\n( )\nlim\n0\n0\nx\nx\nf x\n\u2212x\n\u2192\n=\u2192\n=\n=\nThe right hand limit of f at 0 is\n0\n0\nlim\n( )\nlim\n0\nx\nx\nf x\nx\n+\n+\n\u2192\n=\u2192\n=\nThus \nlim0\n( )\n0\nx\nf x\n\u2192\n=\n= f(0) and hence f is continuous at 0 This means that f is\ncontinuous at every point in its domain and hence, f is a continuous function"
    },
    {
        "Chapter": "1",
        "sentence_range": "1872-1875",
        "Text": "The value of the function at 0 is f(0) = 0 The left hand limit of  f at 0 is\n\u2013\n2\n2\n0\n0\nlim\n( )\nlim\n0\n0\nx\nx\nf x\n\u2212x\n\u2192\n=\u2192\n=\n=\nThe right hand limit of f at 0 is\n0\n0\nlim\n( )\nlim\n0\nx\nx\nf x\nx\n+\n+\n\u2192\n=\u2192\n=\nThus \nlim0\n( )\n0\nx\nf x\n\u2192\n=\n= f(0) and hence f is continuous at 0 This means that f is\ncontinuous at every point in its domain and hence, f is a continuous function Example 14 Show that every polynomial function is continuous"
    },
    {
        "Chapter": "1",
        "sentence_range": "1873-1876",
        "Text": "The left hand limit of  f at 0 is\n\u2013\n2\n2\n0\n0\nlim\n( )\nlim\n0\n0\nx\nx\nf x\n\u2212x\n\u2192\n=\u2192\n=\n=\nThe right hand limit of f at 0 is\n0\n0\nlim\n( )\nlim\n0\nx\nx\nf x\nx\n+\n+\n\u2192\n=\u2192\n=\nThus \nlim0\n( )\n0\nx\nf x\n\u2192\n=\n= f(0) and hence f is continuous at 0 This means that f is\ncontinuous at every point in its domain and hence, f is a continuous function Example 14 Show that every polynomial function is continuous Solution Recall that a function p is a polynomial function if it is defined by\np(x) = a0 + a1 x +"
    },
    {
        "Chapter": "1",
        "sentence_range": "1874-1877",
        "Text": "This means that f is\ncontinuous at every point in its domain and hence, f is a continuous function Example 14 Show that every polynomial function is continuous Solution Recall that a function p is a polynomial function if it is defined by\np(x) = a0 + a1 x + + an xn for some natural number n, an \u2260 0 and ai \u2208 R"
    },
    {
        "Chapter": "1",
        "sentence_range": "1875-1878",
        "Text": "Example 14 Show that every polynomial function is continuous Solution Recall that a function p is a polynomial function if it is defined by\np(x) = a0 + a1 x + + an xn for some natural number n, an \u2260 0 and ai \u2208 R Clearly this\nfunction is defined for every real number"
    },
    {
        "Chapter": "1",
        "sentence_range": "1876-1879",
        "Text": "Solution Recall that a function p is a polynomial function if it is defined by\np(x) = a0 + a1 x + + an xn for some natural number n, an \u2260 0 and ai \u2208 R Clearly this\nfunction is defined for every real number For a fixed real number c, we have\nlim\n( )\n( )\nx\nc p x\np c\n\u2192\n=\nBy definition, p is continuous at c"
    },
    {
        "Chapter": "1",
        "sentence_range": "1877-1880",
        "Text": "+ an xn for some natural number n, an \u2260 0 and ai \u2208 R Clearly this\nfunction is defined for every real number For a fixed real number c, we have\nlim\n( )\n( )\nx\nc p x\np c\n\u2192\n=\nBy definition, p is continuous at c Since c is any real number, p is continuous at\nevery real number and hence p is a continuous function"
    },
    {
        "Chapter": "1",
        "sentence_range": "1878-1881",
        "Text": "Clearly this\nfunction is defined for every real number For a fixed real number c, we have\nlim\n( )\n( )\nx\nc p x\np c\n\u2192\n=\nBy definition, p is continuous at c Since c is any real number, p is continuous at\nevery real number and hence p is a continuous function Example 15  Find all the points of discontinuity of the greatest integer function defined\nby f (x) = [x], where [x] denotes the greatest integer less than or equal to x"
    },
    {
        "Chapter": "1",
        "sentence_range": "1879-1882",
        "Text": "For a fixed real number c, we have\nlim\n( )\n( )\nx\nc p x\np c\n\u2192\n=\nBy definition, p is continuous at c Since c is any real number, p is continuous at\nevery real number and hence p is a continuous function Example 15  Find all the points of discontinuity of the greatest integer function defined\nby f (x) = [x], where [x] denotes the greatest integer less than or equal to x Solution First observe that f is defined for all real numbers"
    },
    {
        "Chapter": "1",
        "sentence_range": "1880-1883",
        "Text": "Since c is any real number, p is continuous at\nevery real number and hence p is a continuous function Example 15  Find all the points of discontinuity of the greatest integer function defined\nby f (x) = [x], where [x] denotes the greatest integer less than or equal to x Solution First observe that f is defined for all real numbers Graph of the function is\ngiven in Fig 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "1881-1884",
        "Text": "Example 15  Find all the points of discontinuity of the greatest integer function defined\nby f (x) = [x], where [x] denotes the greatest integer less than or equal to x Solution First observe that f is defined for all real numbers Graph of the function is\ngiven in Fig 5 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "1882-1885",
        "Text": "Solution First observe that f is defined for all real numbers Graph of the function is\ngiven in Fig 5 8 From the graph it looks like that f is discontinuous at every integral\npoint"
    },
    {
        "Chapter": "1",
        "sentence_range": "1883-1886",
        "Text": "Graph of the function is\ngiven in Fig 5 8 From the graph it looks like that f is discontinuous at every integral\npoint Below we explore, if this is true"
    },
    {
        "Chapter": "1",
        "sentence_range": "1884-1887",
        "Text": "8 From the graph it looks like that f is discontinuous at every integral\npoint Below we explore, if this is true Fig 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "1885-1888",
        "Text": "From the graph it looks like that f is discontinuous at every integral\npoint Below we explore, if this is true Fig 5 8\nRationalised 2023-24\nCONTINUITY AND DIFFERENTIABILITY\n113\nCase 1 Let c be a real number which is not equal to any integer"
    },
    {
        "Chapter": "1",
        "sentence_range": "1886-1889",
        "Text": "Below we explore, if this is true Fig 5 8\nRationalised 2023-24\nCONTINUITY AND DIFFERENTIABILITY\n113\nCase 1 Let c be a real number which is not equal to any integer It is evident from the\ngraph that for all real numbers close to c the value of the function is equal to [c]; i"
    },
    {
        "Chapter": "1",
        "sentence_range": "1887-1890",
        "Text": "Fig 5 8\nRationalised 2023-24\nCONTINUITY AND DIFFERENTIABILITY\n113\nCase 1 Let c be a real number which is not equal to any integer It is evident from the\ngraph that for all real numbers close to c the value of the function is equal to [c]; i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "1888-1891",
        "Text": "8\nRationalised 2023-24\nCONTINUITY AND DIFFERENTIABILITY\n113\nCase 1 Let c be a real number which is not equal to any integer It is evident from the\ngraph that for all real numbers close to c the value of the function is equal to [c]; i e ,\nlim\n( )\nlim [ ]\n[ ]\nx\nc\nx\nc\nf x\nx\nc\n\u2192\n=\u2192\n="
    },
    {
        "Chapter": "1",
        "sentence_range": "1889-1892",
        "Text": "It is evident from the\ngraph that for all real numbers close to c the value of the function is equal to [c]; i e ,\nlim\n( )\nlim [ ]\n[ ]\nx\nc\nx\nc\nf x\nx\nc\n\u2192\n=\u2192\n= Also f(c) = [c] and hence the function is continuous at all real\nnumbers not equal to integers"
    },
    {
        "Chapter": "1",
        "sentence_range": "1890-1893",
        "Text": "e ,\nlim\n( )\nlim [ ]\n[ ]\nx\nc\nx\nc\nf x\nx\nc\n\u2192\n=\u2192\n= Also f(c) = [c] and hence the function is continuous at all real\nnumbers not equal to integers Case 2 Let c be an integer"
    },
    {
        "Chapter": "1",
        "sentence_range": "1891-1894",
        "Text": ",\nlim\n( )\nlim [ ]\n[ ]\nx\nc\nx\nc\nf x\nx\nc\n\u2192\n=\u2192\n= Also f(c) = [c] and hence the function is continuous at all real\nnumbers not equal to integers Case 2 Let c be an integer Then we can find a sufficiently small real number\nr > 0 such that [c \u2013 r] = c \u2013 1 whereas [c + r] = c"
    },
    {
        "Chapter": "1",
        "sentence_range": "1892-1895",
        "Text": "Also f(c) = [c] and hence the function is continuous at all real\nnumbers not equal to integers Case 2 Let c be an integer Then we can find a sufficiently small real number\nr > 0 such that [c \u2013 r] = c \u2013 1 whereas [c + r] = c This, in terms of limits mean that\nxlim\n\u2192c\u2212\nf (x) = c \u2013 1, lim\nx\n\u2192c+\nf (x) = c\nSince these limits cannot be equal to each other for any c, the function is\ndiscontinuous at every integral point"
    },
    {
        "Chapter": "1",
        "sentence_range": "1893-1896",
        "Text": "Case 2 Let c be an integer Then we can find a sufficiently small real number\nr > 0 such that [c \u2013 r] = c \u2013 1 whereas [c + r] = c This, in terms of limits mean that\nxlim\n\u2192c\u2212\nf (x) = c \u2013 1, lim\nx\n\u2192c+\nf (x) = c\nSince these limits cannot be equal to each other for any c, the function is\ndiscontinuous at every integral point 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "1894-1897",
        "Text": "Then we can find a sufficiently small real number\nr > 0 such that [c \u2013 r] = c \u2013 1 whereas [c + r] = c This, in terms of limits mean that\nxlim\n\u2192c\u2212\nf (x) = c \u2013 1, lim\nx\n\u2192c+\nf (x) = c\nSince these limits cannot be equal to each other for any c, the function is\ndiscontinuous at every integral point 5 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "1895-1898",
        "Text": "This, in terms of limits mean that\nxlim\n\u2192c\u2212\nf (x) = c \u2013 1, lim\nx\n\u2192c+\nf (x) = c\nSince these limits cannot be equal to each other for any c, the function is\ndiscontinuous at every integral point 5 2 1  Algebra of continuous functions\nIn the previous class, after having understood the concept of limits, we learnt some\nalgebra of limits"
    },
    {
        "Chapter": "1",
        "sentence_range": "1896-1899",
        "Text": "5 2 1  Algebra of continuous functions\nIn the previous class, after having understood the concept of limits, we learnt some\nalgebra of limits Analogously, now we will study some algebra of continuous functions"
    },
    {
        "Chapter": "1",
        "sentence_range": "1897-1900",
        "Text": "2 1  Algebra of continuous functions\nIn the previous class, after having understood the concept of limits, we learnt some\nalgebra of limits Analogously, now we will study some algebra of continuous functions Since continuity of a function at a point is entirely dictated by the limit of the function at\nthat point, it is reasonable to expect results analogous to the case of limits"
    },
    {
        "Chapter": "1",
        "sentence_range": "1898-1901",
        "Text": "1  Algebra of continuous functions\nIn the previous class, after having understood the concept of limits, we learnt some\nalgebra of limits Analogously, now we will study some algebra of continuous functions Since continuity of a function at a point is entirely dictated by the limit of the function at\nthat point, it is reasonable to expect results analogous to the case of limits Theorem 1 Suppose f and g be two real functions continuous at a real number c"
    },
    {
        "Chapter": "1",
        "sentence_range": "1899-1902",
        "Text": "Analogously, now we will study some algebra of continuous functions Since continuity of a function at a point is entirely dictated by the limit of the function at\nthat point, it is reasonable to expect results analogous to the case of limits Theorem 1 Suppose f and g be two real functions continuous at a real number c Then\n(1)\nf + g is continuous at x = c"
    },
    {
        "Chapter": "1",
        "sentence_range": "1900-1903",
        "Text": "Since continuity of a function at a point is entirely dictated by the limit of the function at\nthat point, it is reasonable to expect results analogous to the case of limits Theorem 1 Suppose f and g be two real functions continuous at a real number c Then\n(1)\nf + g is continuous at x = c (2)\nf \u2013 g is continuous at x = c"
    },
    {
        "Chapter": "1",
        "sentence_range": "1901-1904",
        "Text": "Theorem 1 Suppose f and g be two real functions continuous at a real number c Then\n(1)\nf + g is continuous at x = c (2)\nf \u2013 g is continuous at x = c (3)\nf"
    },
    {
        "Chapter": "1",
        "sentence_range": "1902-1905",
        "Text": "Then\n(1)\nf + g is continuous at x = c (2)\nf \u2013 g is continuous at x = c (3)\nf g is continuous at x = c"
    },
    {
        "Chapter": "1",
        "sentence_range": "1903-1906",
        "Text": "(2)\nf \u2013 g is continuous at x = c (3)\nf g is continuous at x = c (4)\nf\ng\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n is continuous at x = c, (provided g(c) \u2260 0)"
    },
    {
        "Chapter": "1",
        "sentence_range": "1904-1907",
        "Text": "(3)\nf g is continuous at x = c (4)\nf\ng\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n is continuous at x = c, (provided g(c) \u2260 0) Proof We are investigating continuity of (f + g) at x = c"
    },
    {
        "Chapter": "1",
        "sentence_range": "1905-1908",
        "Text": "g is continuous at x = c (4)\nf\ng\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n is continuous at x = c, (provided g(c) \u2260 0) Proof We are investigating continuity of (f + g) at x = c Clearly it is defined at\nx = c"
    },
    {
        "Chapter": "1",
        "sentence_range": "1906-1909",
        "Text": "(4)\nf\ng\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n is continuous at x = c, (provided g(c) \u2260 0) Proof We are investigating continuity of (f + g) at x = c Clearly it is defined at\nx = c We have\nlim(\n)( )\nx\nc f\ng\nx\n\u2192\n+\n = lim[ ( )\n( )]\nx\nc f x\ng x\n\u2192\n+\n(by definition of f + g)\n= lim\n( )\nlim ( )\nx\nc\nx\nc\nf x\ng x\n\u2192\n+\u2192\n(by the theorem on limits)\n= f (c) + g(c)\n(as f and g are continuous)\n= (f + g) (c)\n(by definition of f + g)\nHence, f + g is continuous at x = c"
    },
    {
        "Chapter": "1",
        "sentence_range": "1907-1910",
        "Text": "Proof We are investigating continuity of (f + g) at x = c Clearly it is defined at\nx = c We have\nlim(\n)( )\nx\nc f\ng\nx\n\u2192\n+\n = lim[ ( )\n( )]\nx\nc f x\ng x\n\u2192\n+\n(by definition of f + g)\n= lim\n( )\nlim ( )\nx\nc\nx\nc\nf x\ng x\n\u2192\n+\u2192\n(by the theorem on limits)\n= f (c) + g(c)\n(as f and g are continuous)\n= (f + g) (c)\n(by definition of f + g)\nHence, f + g is continuous at x = c Proofs for the remaining parts are similar and left as an exercise to the reader"
    },
    {
        "Chapter": "1",
        "sentence_range": "1908-1911",
        "Text": "Clearly it is defined at\nx = c We have\nlim(\n)( )\nx\nc f\ng\nx\n\u2192\n+\n = lim[ ( )\n( )]\nx\nc f x\ng x\n\u2192\n+\n(by definition of f + g)\n= lim\n( )\nlim ( )\nx\nc\nx\nc\nf x\ng x\n\u2192\n+\u2192\n(by the theorem on limits)\n= f (c) + g(c)\n(as f and g are continuous)\n= (f + g) (c)\n(by definition of f + g)\nHence, f + g is continuous at x = c Proofs for the remaining parts are similar and left as an exercise to the reader Rationalised 2023-24\n MATHEMATICS\n114\nRemarks\n(i)\nAs a special case of (3) above, if f is a constant function, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "1909-1912",
        "Text": "We have\nlim(\n)( )\nx\nc f\ng\nx\n\u2192\n+\n = lim[ ( )\n( )]\nx\nc f x\ng x\n\u2192\n+\n(by definition of f + g)\n= lim\n( )\nlim ( )\nx\nc\nx\nc\nf x\ng x\n\u2192\n+\u2192\n(by the theorem on limits)\n= f (c) + g(c)\n(as f and g are continuous)\n= (f + g) (c)\n(by definition of f + g)\nHence, f + g is continuous at x = c Proofs for the remaining parts are similar and left as an exercise to the reader Rationalised 2023-24\n MATHEMATICS\n114\nRemarks\n(i)\nAs a special case of (3) above, if f is a constant function, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "1910-1913",
        "Text": "Proofs for the remaining parts are similar and left as an exercise to the reader Rationalised 2023-24\n MATHEMATICS\n114\nRemarks\n(i)\nAs a special case of (3) above, if f is a constant function, i e , f (x) = \u03bb for some\nreal number \u03bb, then the function (\u03bb"
    },
    {
        "Chapter": "1",
        "sentence_range": "1911-1914",
        "Text": "Rationalised 2023-24\n MATHEMATICS\n114\nRemarks\n(i)\nAs a special case of (3) above, if f is a constant function, i e , f (x) = \u03bb for some\nreal number \u03bb, then the function (\u03bb g) defined by (\u03bb"
    },
    {
        "Chapter": "1",
        "sentence_range": "1912-1915",
        "Text": "e , f (x) = \u03bb for some\nreal number \u03bb, then the function (\u03bb g) defined by (\u03bb g) (x) = \u03bb"
    },
    {
        "Chapter": "1",
        "sentence_range": "1913-1916",
        "Text": ", f (x) = \u03bb for some\nreal number \u03bb, then the function (\u03bb g) defined by (\u03bb g) (x) = \u03bb g(x) is also\ncontinuous"
    },
    {
        "Chapter": "1",
        "sentence_range": "1914-1917",
        "Text": "g) defined by (\u03bb g) (x) = \u03bb g(x) is also\ncontinuous In particular if \u03bb = \u2013 1, the continuity of f implies continuity of \u2013 f"
    },
    {
        "Chapter": "1",
        "sentence_range": "1915-1918",
        "Text": "g) (x) = \u03bb g(x) is also\ncontinuous In particular if \u03bb = \u2013 1, the continuity of f implies continuity of \u2013 f (ii)\nAs a special case of (4) above, if f is the constant function f (x) = \u03bb, then the\nfunction g\n\u03bb  defined by \n( )\n( )\ngx\ng x\n\u03bb\n\u03bb\n=\nis also continuous wherever g(x) \u2260 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "1916-1919",
        "Text": "g(x) is also\ncontinuous In particular if \u03bb = \u2013 1, the continuity of f implies continuity of \u2013 f (ii)\nAs a special case of (4) above, if f is the constant function f (x) = \u03bb, then the\nfunction g\n\u03bb  defined by \n( )\n( )\ngx\ng x\n\u03bb\n\u03bb\n=\nis also continuous wherever g(x) \u2260 0 In\nparticular, the continuity of g implies continuity of 1\ng"
    },
    {
        "Chapter": "1",
        "sentence_range": "1917-1920",
        "Text": "In particular if \u03bb = \u2013 1, the continuity of f implies continuity of \u2013 f (ii)\nAs a special case of (4) above, if f is the constant function f (x) = \u03bb, then the\nfunction g\n\u03bb  defined by \n( )\n( )\ngx\ng x\n\u03bb\n\u03bb\n=\nis also continuous wherever g(x) \u2260 0 In\nparticular, the continuity of g implies continuity of 1\ng The above theorem can be exploited to generate many continuous functions"
    },
    {
        "Chapter": "1",
        "sentence_range": "1918-1921",
        "Text": "(ii)\nAs a special case of (4) above, if f is the constant function f (x) = \u03bb, then the\nfunction g\n\u03bb  defined by \n( )\n( )\ngx\ng x\n\u03bb\n\u03bb\n=\nis also continuous wherever g(x) \u2260 0 In\nparticular, the continuity of g implies continuity of 1\ng The above theorem can be exploited to generate many continuous functions They\nalso aid in deciding if certain functions are continuous or not"
    },
    {
        "Chapter": "1",
        "sentence_range": "1919-1922",
        "Text": "In\nparticular, the continuity of g implies continuity of 1\ng The above theorem can be exploited to generate many continuous functions They\nalso aid in deciding if certain functions are continuous or not The following examples\nillustrate this:\nExample 16 Prove that every rational function is continuous"
    },
    {
        "Chapter": "1",
        "sentence_range": "1920-1923",
        "Text": "The above theorem can be exploited to generate many continuous functions They\nalso aid in deciding if certain functions are continuous or not The following examples\nillustrate this:\nExample 16 Prove that every rational function is continuous Solution Recall that every rational function f is given by\n( )\n( )\n,\n( )\n0\n( )\np x\nf x\nq x\n=q x\n\u2260\nwhere p and q are polynomial functions"
    },
    {
        "Chapter": "1",
        "sentence_range": "1921-1924",
        "Text": "They\nalso aid in deciding if certain functions are continuous or not The following examples\nillustrate this:\nExample 16 Prove that every rational function is continuous Solution Recall that every rational function f is given by\n( )\n( )\n,\n( )\n0\n( )\np x\nf x\nq x\n=q x\n\u2260\nwhere p and q are polynomial functions The domain of f is all real numbers except\npoints at which q is zero"
    },
    {
        "Chapter": "1",
        "sentence_range": "1922-1925",
        "Text": "The following examples\nillustrate this:\nExample 16 Prove that every rational function is continuous Solution Recall that every rational function f is given by\n( )\n( )\n,\n( )\n0\n( )\np x\nf x\nq x\n=q x\n\u2260\nwhere p and q are polynomial functions The domain of f is all real numbers except\npoints at which q is zero Since polynomial functions are continuous (Example 14), f is\ncontinuous by (4) of  Theorem 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "1923-1926",
        "Text": "Solution Recall that every rational function f is given by\n( )\n( )\n,\n( )\n0\n( )\np x\nf x\nq x\n=q x\n\u2260\nwhere p and q are polynomial functions The domain of f is all real numbers except\npoints at which q is zero Since polynomial functions are continuous (Example 14), f is\ncontinuous by (4) of  Theorem 1 Example 17 Discuss the continuity of sine function"
    },
    {
        "Chapter": "1",
        "sentence_range": "1924-1927",
        "Text": "The domain of f is all real numbers except\npoints at which q is zero Since polynomial functions are continuous (Example 14), f is\ncontinuous by (4) of  Theorem 1 Example 17 Discuss the continuity of sine function Solution To see this we use the following facts\n0\nlim sin\n0\nx\nx\n\u2192\n=\nWe have not proved it, but is intuitively clear from the graph of sin x near 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "1925-1928",
        "Text": "Since polynomial functions are continuous (Example 14), f is\ncontinuous by (4) of  Theorem 1 Example 17 Discuss the continuity of sine function Solution To see this we use the following facts\n0\nlim sin\n0\nx\nx\n\u2192\n=\nWe have not proved it, but is intuitively clear from the graph of sin x near 0 Now, observe that f (x) = sin x is defined for every real number"
    },
    {
        "Chapter": "1",
        "sentence_range": "1926-1929",
        "Text": "Example 17 Discuss the continuity of sine function Solution To see this we use the following facts\n0\nlim sin\n0\nx\nx\n\u2192\n=\nWe have not proved it, but is intuitively clear from the graph of sin x near 0 Now, observe that f (x) = sin x is defined for every real number Let c be a real\nnumber"
    },
    {
        "Chapter": "1",
        "sentence_range": "1927-1930",
        "Text": "Solution To see this we use the following facts\n0\nlim sin\n0\nx\nx\n\u2192\n=\nWe have not proved it, but is intuitively clear from the graph of sin x near 0 Now, observe that f (x) = sin x is defined for every real number Let c be a real\nnumber Put x = c + h"
    },
    {
        "Chapter": "1",
        "sentence_range": "1928-1931",
        "Text": "Now, observe that f (x) = sin x is defined for every real number Let c be a real\nnumber Put x = c + h If x \u2192 c we know that h \u2192 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "1929-1932",
        "Text": "Let c be a real\nnumber Put x = c + h If x \u2192 c we know that h \u2192 0 Therefore\nlim\n( )\nx\nc f x\n\u2192\n = lim sin\nx\nc\nx\n\u2192\n=\n0\nlim sin(\n)\nh\nc\nh\n\u2192\n+\n=\n0\nlim [sin cos\ncos sin ]\nh\nc\nh\nc\nh\n\u2192\n+\n=\n0\n0\nlim [sin cos ]\nlim [cos sin ]\nh\nh\nc\nh\nc\nh\n\u2192\n\u2192\n+\n= sin c + 0 = sin c = f (c)\nThus lim\nx\nc\n\u2192  f (x) = f(c) and hence f is a continuous function"
    },
    {
        "Chapter": "1",
        "sentence_range": "1930-1933",
        "Text": "Put x = c + h If x \u2192 c we know that h \u2192 0 Therefore\nlim\n( )\nx\nc f x\n\u2192\n = lim sin\nx\nc\nx\n\u2192\n=\n0\nlim sin(\n)\nh\nc\nh\n\u2192\n+\n=\n0\nlim [sin cos\ncos sin ]\nh\nc\nh\nc\nh\n\u2192\n+\n=\n0\n0\nlim [sin cos ]\nlim [cos sin ]\nh\nh\nc\nh\nc\nh\n\u2192\n\u2192\n+\n= sin c + 0 = sin c = f (c)\nThus lim\nx\nc\n\u2192  f (x) = f(c) and hence f is a continuous function Rationalised 2023-24\nCONTINUITY AND DIFFERENTIABILITY\n115\nRemark A similar proof may be given for the continuity of cosine function"
    },
    {
        "Chapter": "1",
        "sentence_range": "1931-1934",
        "Text": "If x \u2192 c we know that h \u2192 0 Therefore\nlim\n( )\nx\nc f x\n\u2192\n = lim sin\nx\nc\nx\n\u2192\n=\n0\nlim sin(\n)\nh\nc\nh\n\u2192\n+\n=\n0\nlim [sin cos\ncos sin ]\nh\nc\nh\nc\nh\n\u2192\n+\n=\n0\n0\nlim [sin cos ]\nlim [cos sin ]\nh\nh\nc\nh\nc\nh\n\u2192\n\u2192\n+\n= sin c + 0 = sin c = f (c)\nThus lim\nx\nc\n\u2192  f (x) = f(c) and hence f is a continuous function Rationalised 2023-24\nCONTINUITY AND DIFFERENTIABILITY\n115\nRemark A similar proof may be given for the continuity of cosine function Example 18 Prove that the function defined by f(x) = tan x is a continuous function"
    },
    {
        "Chapter": "1",
        "sentence_range": "1932-1935",
        "Text": "Therefore\nlim\n( )\nx\nc f x\n\u2192\n = lim sin\nx\nc\nx\n\u2192\n=\n0\nlim sin(\n)\nh\nc\nh\n\u2192\n+\n=\n0\nlim [sin cos\ncos sin ]\nh\nc\nh\nc\nh\n\u2192\n+\n=\n0\n0\nlim [sin cos ]\nlim [cos sin ]\nh\nh\nc\nh\nc\nh\n\u2192\n\u2192\n+\n= sin c + 0 = sin c = f (c)\nThus lim\nx\nc\n\u2192  f (x) = f(c) and hence f is a continuous function Rationalised 2023-24\nCONTINUITY AND DIFFERENTIABILITY\n115\nRemark A similar proof may be given for the continuity of cosine function Example 18 Prove that the function defined by f(x) = tan x is a continuous function Solution The function f (x) = tan x = sin\ncos\nx\nx"
    },
    {
        "Chapter": "1",
        "sentence_range": "1933-1936",
        "Text": "Rationalised 2023-24\nCONTINUITY AND DIFFERENTIABILITY\n115\nRemark A similar proof may be given for the continuity of cosine function Example 18 Prove that the function defined by f(x) = tan x is a continuous function Solution The function f (x) = tan x = sin\ncos\nx\nx This is defined for all real numbers such\nthat cos x \u2260 0, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "1934-1937",
        "Text": "Example 18 Prove that the function defined by f(x) = tan x is a continuous function Solution The function f (x) = tan x = sin\ncos\nx\nx This is defined for all real numbers such\nthat cos x \u2260 0, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "1935-1938",
        "Text": "Solution The function f (x) = tan x = sin\ncos\nx\nx This is defined for all real numbers such\nthat cos x \u2260 0, i e , x \u2260 (2n +1) 2\n\u03c0"
    },
    {
        "Chapter": "1",
        "sentence_range": "1936-1939",
        "Text": "This is defined for all real numbers such\nthat cos x \u2260 0, i e , x \u2260 (2n +1) 2\n\u03c0 We have just proved that both sine and cosine\nfunctions are continuous"
    },
    {
        "Chapter": "1",
        "sentence_range": "1937-1940",
        "Text": "e , x \u2260 (2n +1) 2\n\u03c0 We have just proved that both sine and cosine\nfunctions are continuous Thus tan x being a quotient of two continuous functions is\ncontinuous wherever it is defined"
    },
    {
        "Chapter": "1",
        "sentence_range": "1938-1941",
        "Text": ", x \u2260 (2n +1) 2\n\u03c0 We have just proved that both sine and cosine\nfunctions are continuous Thus tan x being a quotient of two continuous functions is\ncontinuous wherever it is defined An interesting fact is the behaviour of continuous functions with respect to\ncomposition of functions"
    },
    {
        "Chapter": "1",
        "sentence_range": "1939-1942",
        "Text": "We have just proved that both sine and cosine\nfunctions are continuous Thus tan x being a quotient of two continuous functions is\ncontinuous wherever it is defined An interesting fact is the behaviour of continuous functions with respect to\ncomposition of functions Recall that if f and g are two real functions, then\n(f o g) (x) = f(g (x))\nis defined whenever the range of g is a subset of domain of f"
    },
    {
        "Chapter": "1",
        "sentence_range": "1940-1943",
        "Text": "Thus tan x being a quotient of two continuous functions is\ncontinuous wherever it is defined An interesting fact is the behaviour of continuous functions with respect to\ncomposition of functions Recall that if f and g are two real functions, then\n(f o g) (x) = f(g (x))\nis defined whenever the range of g is a subset of domain of f The following theorem\n(stated without proof) captures the continuity of composite functions"
    },
    {
        "Chapter": "1",
        "sentence_range": "1941-1944",
        "Text": "An interesting fact is the behaviour of continuous functions with respect to\ncomposition of functions Recall that if f and g are two real functions, then\n(f o g) (x) = f(g (x))\nis defined whenever the range of g is a subset of domain of f The following theorem\n(stated without proof) captures the continuity of composite functions Theorem 2 Suppose f and g are real valued functions such that (f o g) is defined at c"
    },
    {
        "Chapter": "1",
        "sentence_range": "1942-1945",
        "Text": "Recall that if f and g are two real functions, then\n(f o g) (x) = f(g (x))\nis defined whenever the range of g is a subset of domain of f The following theorem\n(stated without proof) captures the continuity of composite functions Theorem 2 Suppose f and g are real valued functions such that (f o g) is defined at c If g is continuous at c and if f is continuous at g(c), then (f o g) is continuous at c"
    },
    {
        "Chapter": "1",
        "sentence_range": "1943-1946",
        "Text": "The following theorem\n(stated without proof) captures the continuity of composite functions Theorem 2 Suppose f and g are real valued functions such that (f o g) is defined at c If g is continuous at c and if f is continuous at g(c), then (f o g) is continuous at c The following examples illustrate this theorem"
    },
    {
        "Chapter": "1",
        "sentence_range": "1944-1947",
        "Text": "Theorem 2 Suppose f and g are real valued functions such that (f o g) is defined at c If g is continuous at c and if f is continuous at g(c), then (f o g) is continuous at c The following examples illustrate this theorem Example 19 Show that the function defined by f(x) = sin (x2) is a continuous function"
    },
    {
        "Chapter": "1",
        "sentence_range": "1945-1948",
        "Text": "If g is continuous at c and if f is continuous at g(c), then (f o g) is continuous at c The following examples illustrate this theorem Example 19 Show that the function defined by f(x) = sin (x2) is a continuous function Solution Observe that the function is defined for every real number"
    },
    {
        "Chapter": "1",
        "sentence_range": "1946-1949",
        "Text": "The following examples illustrate this theorem Example 19 Show that the function defined by f(x) = sin (x2) is a continuous function Solution Observe that the function is defined for every real number The function\nf may be thought of as a composition g o h of the two functions g and h, where\ng (x) = sin x and h(x) = x2"
    },
    {
        "Chapter": "1",
        "sentence_range": "1947-1950",
        "Text": "Example 19 Show that the function defined by f(x) = sin (x2) is a continuous function Solution Observe that the function is defined for every real number The function\nf may be thought of as a composition g o h of the two functions g and h, where\ng (x) = sin x and h(x) = x2 Since both g and h are continuous functions, by Theorem 2,\nit can be deduced that f is a continuous function"
    },
    {
        "Chapter": "1",
        "sentence_range": "1948-1951",
        "Text": "Solution Observe that the function is defined for every real number The function\nf may be thought of as a composition g o h of the two functions g and h, where\ng (x) = sin x and h(x) = x2 Since both g and h are continuous functions, by Theorem 2,\nit can be deduced that f is a continuous function Example 20 Show that the function f defined by\nf (x) = |1 \u2013 x + | x||,\nwhere x is any real number, is a continuous function"
    },
    {
        "Chapter": "1",
        "sentence_range": "1949-1952",
        "Text": "The function\nf may be thought of as a composition g o h of the two functions g and h, where\ng (x) = sin x and h(x) = x2 Since both g and h are continuous functions, by Theorem 2,\nit can be deduced that f is a continuous function Example 20 Show that the function f defined by\nf (x) = |1 \u2013 x + | x||,\nwhere x is any real number, is a continuous function Solution Define g by g(x) = 1 \u2013 x + |x| and  h by h(x) = |x| for all real x"
    },
    {
        "Chapter": "1",
        "sentence_range": "1950-1953",
        "Text": "Since both g and h are continuous functions, by Theorem 2,\nit can be deduced that f is a continuous function Example 20 Show that the function f defined by\nf (x) = |1 \u2013 x + | x||,\nwhere x is any real number, is a continuous function Solution Define g by g(x) = 1 \u2013 x + |x| and  h by h(x) = |x| for all real x Then\n(h o g) (x) = h (g (x))\n= h (1\u2013 x + |x |)\n= |1\u2013 x + | x|| = f (x)\nIn Example 7, we have seen that h is a continuous function"
    },
    {
        "Chapter": "1",
        "sentence_range": "1951-1954",
        "Text": "Example 20 Show that the function f defined by\nf (x) = |1 \u2013 x + | x||,\nwhere x is any real number, is a continuous function Solution Define g by g(x) = 1 \u2013 x + |x| and  h by h(x) = |x| for all real x Then\n(h o g) (x) = h (g (x))\n= h (1\u2013 x + |x |)\n= |1\u2013 x + | x|| = f (x)\nIn Example 7, we have seen that h is a continuous function Hence g being a sum\nof a polynomial function and the modulus function is continuous"
    },
    {
        "Chapter": "1",
        "sentence_range": "1952-1955",
        "Text": "Solution Define g by g(x) = 1 \u2013 x + |x| and  h by h(x) = |x| for all real x Then\n(h o g) (x) = h (g (x))\n= h (1\u2013 x + |x |)\n= |1\u2013 x + | x|| = f (x)\nIn Example 7, we have seen that h is a continuous function Hence g being a sum\nof a polynomial function and the modulus function is continuous But then f  being a\ncomposite of two continuous functions is continuous"
    },
    {
        "Chapter": "1",
        "sentence_range": "1953-1956",
        "Text": "Then\n(h o g) (x) = h (g (x))\n= h (1\u2013 x + |x |)\n= |1\u2013 x + | x|| = f (x)\nIn Example 7, we have seen that h is a continuous function Hence g being a sum\nof a polynomial function and the modulus function is continuous But then f  being a\ncomposite of two continuous functions is continuous Rationalised 2023-24\n MATHEMATICS\n116\nEXERCISE 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "1954-1957",
        "Text": "Hence g being a sum\nof a polynomial function and the modulus function is continuous But then f  being a\ncomposite of two continuous functions is continuous Rationalised 2023-24\n MATHEMATICS\n116\nEXERCISE 5 1\n1"
    },
    {
        "Chapter": "1",
        "sentence_range": "1955-1958",
        "Text": "But then f  being a\ncomposite of two continuous functions is continuous Rationalised 2023-24\n MATHEMATICS\n116\nEXERCISE 5 1\n1 Prove that the function f(x) = 5x \u2013 3 is continuous at x = 0, at x = \u2013 3 and at x = 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "1956-1959",
        "Text": "Rationalised 2023-24\n MATHEMATICS\n116\nEXERCISE 5 1\n1 Prove that the function f(x) = 5x \u2013 3 is continuous at x = 0, at x = \u2013 3 and at x = 5 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "1957-1960",
        "Text": "1\n1 Prove that the function f(x) = 5x \u2013 3 is continuous at x = 0, at x = \u2013 3 and at x = 5 2 Examine the continuity of the function f(x) = 2x2 \u2013 1 at x = 3"
    },
    {
        "Chapter": "1",
        "sentence_range": "1958-1961",
        "Text": "Prove that the function f(x) = 5x \u2013 3 is continuous at x = 0, at x = \u2013 3 and at x = 5 2 Examine the continuity of the function f(x) = 2x2 \u2013 1 at x = 3 3"
    },
    {
        "Chapter": "1",
        "sentence_range": "1959-1962",
        "Text": "2 Examine the continuity of the function f(x) = 2x2 \u2013 1 at x = 3 3 Examine the following functions for continuity"
    },
    {
        "Chapter": "1",
        "sentence_range": "1960-1963",
        "Text": "Examine the continuity of the function f(x) = 2x2 \u2013 1 at x = 3 3 Examine the following functions for continuity (a)\nf(x) = x \u2013 5\n(b)\nf(x) = \n1\nx \u22125\n, x \u2260 5\n(c)\nf(x) = \n2\n25\n5\nx\nx\n+\u2212\n, x \u2260 \u20135\n(d)\nf(x) = |x \u2013 5 |\n4"
    },
    {
        "Chapter": "1",
        "sentence_range": "1961-1964",
        "Text": "3 Examine the following functions for continuity (a)\nf(x) = x \u2013 5\n(b)\nf(x) = \n1\nx \u22125\n, x \u2260 5\n(c)\nf(x) = \n2\n25\n5\nx\nx\n+\u2212\n, x \u2260 \u20135\n(d)\nf(x) = |x \u2013 5 |\n4 Prove that the function f(x) = xn is continuous at x = n, where n is a positive\ninteger"
    },
    {
        "Chapter": "1",
        "sentence_range": "1962-1965",
        "Text": "Examine the following functions for continuity (a)\nf(x) = x \u2013 5\n(b)\nf(x) = \n1\nx \u22125\n, x \u2260 5\n(c)\nf(x) = \n2\n25\n5\nx\nx\n+\u2212\n, x \u2260 \u20135\n(d)\nf(x) = |x \u2013 5 |\n4 Prove that the function f(x) = xn is continuous at x = n, where n is a positive\ninteger 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "1963-1966",
        "Text": "(a)\nf(x) = x \u2013 5\n(b)\nf(x) = \n1\nx \u22125\n, x \u2260 5\n(c)\nf(x) = \n2\n25\n5\nx\nx\n+\u2212\n, x \u2260 \u20135\n(d)\nf(x) = |x \u2013 5 |\n4 Prove that the function f(x) = xn is continuous at x = n, where n is a positive\ninteger 5 Is the function f defined by\n, if\n1\n( )\n5, if\n>1\nx\nx\nf x\nx\n\u2264\n= \uf8f2\uf8f1\n\uf8f3\ncontinuous at x = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "1964-1967",
        "Text": "Prove that the function f(x) = xn is continuous at x = n, where n is a positive\ninteger 5 Is the function f defined by\n, if\n1\n( )\n5, if\n>1\nx\nx\nf x\nx\n\u2264\n= \uf8f2\uf8f1\n\uf8f3\ncontinuous at x = 0 At x = 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "1965-1968",
        "Text": "5 Is the function f defined by\n, if\n1\n( )\n5, if\n>1\nx\nx\nf x\nx\n\u2264\n= \uf8f2\uf8f1\n\uf8f3\ncontinuous at x = 0 At x = 1 At x = 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "1966-1969",
        "Text": "Is the function f defined by\n, if\n1\n( )\n5, if\n>1\nx\nx\nf x\nx\n\u2264\n= \uf8f2\uf8f1\n\uf8f3\ncontinuous at x = 0 At x = 1 At x = 2 Find all points of discontinuity of f, where f is defined by\n6"
    },
    {
        "Chapter": "1",
        "sentence_range": "1967-1970",
        "Text": "At x = 1 At x = 2 Find all points of discontinuity of f, where f is defined by\n6 2\n3, if\n2\n( )\n2\n3, if\n> 2\nx\nx\nf x\nx\nx\n+\n\u2264\n= \uf8f2\uf8f1\n\u2212\n\uf8f3\n7"
    },
    {
        "Chapter": "1",
        "sentence_range": "1968-1971",
        "Text": "At x = 2 Find all points of discontinuity of f, where f is defined by\n6 2\n3, if\n2\n( )\n2\n3, if\n> 2\nx\nx\nf x\nx\nx\n+\n\u2264\n= \uf8f2\uf8f1\n\u2212\n\uf8f3\n7 |\n| 3, if\n3\n( )\n2 , if\n3\n< 3\n6\n2, if\n3\nx\nx\nf x\nx\nx\nx\nx\n+\n\u2264 \u2212\n=\uf8f4\uf8f1\n\u2212\n\u2212\n<\n\uf8f4\uf8f2\n+\n\u2265\n\uf8f3\n8"
    },
    {
        "Chapter": "1",
        "sentence_range": "1969-1972",
        "Text": "Find all points of discontinuity of f, where f is defined by\n6 2\n3, if\n2\n( )\n2\n3, if\n> 2\nx\nx\nf x\nx\nx\n+\n\u2264\n= \uf8f2\uf8f1\n\u2212\n\uf8f3\n7 |\n| 3, if\n3\n( )\n2 , if\n3\n< 3\n6\n2, if\n3\nx\nx\nf x\nx\nx\nx\nx\n+\n\u2264 \u2212\n=\uf8f4\uf8f1\n\u2212\n\u2212\n<\n\uf8f4\uf8f2\n+\n\u2265\n\uf8f3\n8 |\n|, if\n0\n( )\n0,\nif\n0\nx\nx\nf x\nx\nx\n\uf8f1\n\u2260\n= \uf8f2\uf8f4\n\uf8f4\n=\n\uf8f3\n9"
    },
    {
        "Chapter": "1",
        "sentence_range": "1970-1973",
        "Text": "2\n3, if\n2\n( )\n2\n3, if\n> 2\nx\nx\nf x\nx\nx\n+\n\u2264\n= \uf8f2\uf8f1\n\u2212\n\uf8f3\n7 |\n| 3, if\n3\n( )\n2 , if\n3\n< 3\n6\n2, if\n3\nx\nx\nf x\nx\nx\nx\nx\n+\n\u2264 \u2212\n=\uf8f4\uf8f1\n\u2212\n\u2212\n<\n\uf8f4\uf8f2\n+\n\u2265\n\uf8f3\n8 |\n|, if\n0\n( )\n0,\nif\n0\nx\nx\nf x\nx\nx\n\uf8f1\n\u2260\n= \uf8f2\uf8f4\n\uf8f4\n=\n\uf8f3\n9 , if\n0\n|\n|\n( )\n1,\nif\n0\nx\nx\nx\nf x\nx\n\uf8f1\n<\n= \uf8f2\uf8f4\n\uf8f4\u2212\n\u2265\n\uf8f3\n10"
    },
    {
        "Chapter": "1",
        "sentence_range": "1971-1974",
        "Text": "|\n| 3, if\n3\n( )\n2 , if\n3\n< 3\n6\n2, if\n3\nx\nx\nf x\nx\nx\nx\nx\n+\n\u2264 \u2212\n=\uf8f4\uf8f1\n\u2212\n\u2212\n<\n\uf8f4\uf8f2\n+\n\u2265\n\uf8f3\n8 |\n|, if\n0\n( )\n0,\nif\n0\nx\nx\nf x\nx\nx\n\uf8f1\n\u2260\n= \uf8f2\uf8f4\n\uf8f4\n=\n\uf8f3\n9 , if\n0\n|\n|\n( )\n1,\nif\n0\nx\nx\nx\nf x\nx\n\uf8f1\n<\n= \uf8f2\uf8f4\n\uf8f4\u2212\n\u2265\n\uf8f3\n10 2\n1, if\n1\n( )\n1, if\n1\nx\nx\nf x\nx\nx\n+\n\u2265\n= \uf8f2\uf8f1\uf8f4\n+\n<\n\uf8f4\uf8f3\n11"
    },
    {
        "Chapter": "1",
        "sentence_range": "1972-1975",
        "Text": "|\n|, if\n0\n( )\n0,\nif\n0\nx\nx\nf x\nx\nx\n\uf8f1\n\u2260\n= \uf8f2\uf8f4\n\uf8f4\n=\n\uf8f3\n9 , if\n0\n|\n|\n( )\n1,\nif\n0\nx\nx\nx\nf x\nx\n\uf8f1\n<\n= \uf8f2\uf8f4\n\uf8f4\u2212\n\u2265\n\uf8f3\n10 2\n1, if\n1\n( )\n1, if\n1\nx\nx\nf x\nx\nx\n+\n\u2265\n= \uf8f2\uf8f1\uf8f4\n+\n<\n\uf8f4\uf8f3\n11 3\n2\n3, if\n2\n( )\n1,\nif\n2\nx\nx\nf x\nx\nx\n\uf8f1\n\u2212\n\u2264\n= \uf8f2\uf8f4\n+\n>\n\uf8f4\uf8f3\n12"
    },
    {
        "Chapter": "1",
        "sentence_range": "1973-1976",
        "Text": ", if\n0\n|\n|\n( )\n1,\nif\n0\nx\nx\nx\nf x\nx\n\uf8f1\n<\n= \uf8f2\uf8f4\n\uf8f4\u2212\n\u2265\n\uf8f3\n10 2\n1, if\n1\n( )\n1, if\n1\nx\nx\nf x\nx\nx\n+\n\u2265\n= \uf8f2\uf8f1\uf8f4\n+\n<\n\uf8f4\uf8f3\n11 3\n2\n3, if\n2\n( )\n1,\nif\n2\nx\nx\nf x\nx\nx\n\uf8f1\n\u2212\n\u2264\n= \uf8f2\uf8f4\n+\n>\n\uf8f4\uf8f3\n12 10\n2\n1, if\n1\n( )\n,\nif\n1\nx\nx\nf x\nx\nx\n\uf8f1\n\u2212\n\u2264\n= \uf8f2\uf8f4\n>\n\uf8f4\uf8f3\n13"
    },
    {
        "Chapter": "1",
        "sentence_range": "1974-1977",
        "Text": "2\n1, if\n1\n( )\n1, if\n1\nx\nx\nf x\nx\nx\n+\n\u2265\n= \uf8f2\uf8f1\uf8f4\n+\n<\n\uf8f4\uf8f3\n11 3\n2\n3, if\n2\n( )\n1,\nif\n2\nx\nx\nf x\nx\nx\n\uf8f1\n\u2212\n\u2264\n= \uf8f2\uf8f4\n+\n>\n\uf8f4\uf8f3\n12 10\n2\n1, if\n1\n( )\n,\nif\n1\nx\nx\nf x\nx\nx\n\uf8f1\n\u2212\n\u2264\n= \uf8f2\uf8f4\n>\n\uf8f4\uf8f3\n13 Is the function defined by\n5, if\n1\n( )\n5, if\n1\nx\nx\nf x\nx\nx\n+\n\u2264\n\uf8f1\n= \uf8f2 \u2212\n>\n\uf8f3\na continuous function"
    },
    {
        "Chapter": "1",
        "sentence_range": "1975-1978",
        "Text": "3\n2\n3, if\n2\n( )\n1,\nif\n2\nx\nx\nf x\nx\nx\n\uf8f1\n\u2212\n\u2264\n= \uf8f2\uf8f4\n+\n>\n\uf8f4\uf8f3\n12 10\n2\n1, if\n1\n( )\n,\nif\n1\nx\nx\nf x\nx\nx\n\uf8f1\n\u2212\n\u2264\n= \uf8f2\uf8f4\n>\n\uf8f4\uf8f3\n13 Is the function defined by\n5, if\n1\n( )\n5, if\n1\nx\nx\nf x\nx\nx\n+\n\u2264\n\uf8f1\n= \uf8f2 \u2212\n>\n\uf8f3\na continuous function Rationalised 2023-24\nCONTINUITY AND DIFFERENTIABILITY\n117\nDiscuss the continuity of the function f, where f is defined by\n14"
    },
    {
        "Chapter": "1",
        "sentence_range": "1976-1979",
        "Text": "10\n2\n1, if\n1\n( )\n,\nif\n1\nx\nx\nf x\nx\nx\n\uf8f1\n\u2212\n\u2264\n= \uf8f2\uf8f4\n>\n\uf8f4\uf8f3\n13 Is the function defined by\n5, if\n1\n( )\n5, if\n1\nx\nx\nf x\nx\nx\n+\n\u2264\n\uf8f1\n= \uf8f2 \u2212\n>\n\uf8f3\na continuous function Rationalised 2023-24\nCONTINUITY AND DIFFERENTIABILITY\n117\nDiscuss the continuity of the function f, where f is defined by\n14 3, if 0\n1\n( )\n4, if 1\n3\n5, if 3\n10\nx\nf x\nx\nx\n\u2264\n\u2264\n=\uf8f4\uf8f1\n<\n<\n\uf8f4\uf8f2\n\u2264\n\u2264\n\uf8f3\n15"
    },
    {
        "Chapter": "1",
        "sentence_range": "1977-1980",
        "Text": "Is the function defined by\n5, if\n1\n( )\n5, if\n1\nx\nx\nf x\nx\nx\n+\n\u2264\n\uf8f1\n= \uf8f2 \u2212\n>\n\uf8f3\na continuous function Rationalised 2023-24\nCONTINUITY AND DIFFERENTIABILITY\n117\nDiscuss the continuity of the function f, where f is defined by\n14 3, if 0\n1\n( )\n4, if 1\n3\n5, if 3\n10\nx\nf x\nx\nx\n\u2264\n\u2264\n=\uf8f4\uf8f1\n<\n<\n\uf8f4\uf8f2\n\u2264\n\u2264\n\uf8f3\n15 2 , if\n0\n( )\n0,\nif 0\n1\n4 , if\n>1\nx\nx\nf x\nx\nx\nx\n<\n=\uf8f4\uf8f1\n\u2264\n\u2264\n\uf8f2\n\uf8f4\uf8f3\n16"
    },
    {
        "Chapter": "1",
        "sentence_range": "1978-1981",
        "Text": "Rationalised 2023-24\nCONTINUITY AND DIFFERENTIABILITY\n117\nDiscuss the continuity of the function f, where f is defined by\n14 3, if 0\n1\n( )\n4, if 1\n3\n5, if 3\n10\nx\nf x\nx\nx\n\u2264\n\u2264\n=\uf8f4\uf8f1\n<\n<\n\uf8f4\uf8f2\n\u2264\n\u2264\n\uf8f3\n15 2 , if\n0\n( )\n0,\nif 0\n1\n4 , if\n>1\nx\nx\nf x\nx\nx\nx\n<\n=\uf8f4\uf8f1\n\u2264\n\u2264\n\uf8f2\n\uf8f4\uf8f3\n16 2, if\n1\n( )\n2 , if\n1\n1\n2,\nif\n1\nx\nf x\nx\nx\nx\n\u2212\n\u2264 \u2212\n=\uf8f4\uf8f1\n\u2212 <\n\u2264\n\uf8f4\uf8f2\n>\n\uf8f3\n17"
    },
    {
        "Chapter": "1",
        "sentence_range": "1979-1982",
        "Text": "3, if 0\n1\n( )\n4, if 1\n3\n5, if 3\n10\nx\nf x\nx\nx\n\u2264\n\u2264\n=\uf8f4\uf8f1\n<\n<\n\uf8f4\uf8f2\n\u2264\n\u2264\n\uf8f3\n15 2 , if\n0\n( )\n0,\nif 0\n1\n4 , if\n>1\nx\nx\nf x\nx\nx\nx\n<\n=\uf8f4\uf8f1\n\u2264\n\u2264\n\uf8f2\n\uf8f4\uf8f3\n16 2, if\n1\n( )\n2 , if\n1\n1\n2,\nif\n1\nx\nf x\nx\nx\nx\n\u2212\n\u2264 \u2212\n=\uf8f4\uf8f1\n\u2212 <\n\u2264\n\uf8f4\uf8f2\n>\n\uf8f3\n17 Find the relationship between a and b so that the function f defined by\n1, if\n3\n( )\n3, if\n3\nax\nx\nf x\nbx\nx\n+\n\u2264\n= \uf8f2\uf8f1\n+\n>\n\uf8f3\nis continuous at x = 3"
    },
    {
        "Chapter": "1",
        "sentence_range": "1980-1983",
        "Text": "2 , if\n0\n( )\n0,\nif 0\n1\n4 , if\n>1\nx\nx\nf x\nx\nx\nx\n<\n=\uf8f4\uf8f1\n\u2264\n\u2264\n\uf8f2\n\uf8f4\uf8f3\n16 2, if\n1\n( )\n2 , if\n1\n1\n2,\nif\n1\nx\nf x\nx\nx\nx\n\u2212\n\u2264 \u2212\n=\uf8f4\uf8f1\n\u2212 <\n\u2264\n\uf8f4\uf8f2\n>\n\uf8f3\n17 Find the relationship between a and b so that the function f defined by\n1, if\n3\n( )\n3, if\n3\nax\nx\nf x\nbx\nx\n+\n\u2264\n= \uf8f2\uf8f1\n+\n>\n\uf8f3\nis continuous at x = 3 18"
    },
    {
        "Chapter": "1",
        "sentence_range": "1981-1984",
        "Text": "2, if\n1\n( )\n2 , if\n1\n1\n2,\nif\n1\nx\nf x\nx\nx\nx\n\u2212\n\u2264 \u2212\n=\uf8f4\uf8f1\n\u2212 <\n\u2264\n\uf8f4\uf8f2\n>\n\uf8f3\n17 Find the relationship between a and b so that the function f defined by\n1, if\n3\n( )\n3, if\n3\nax\nx\nf x\nbx\nx\n+\n\u2264\n= \uf8f2\uf8f1\n+\n>\n\uf8f3\nis continuous at x = 3 18 For what value of \u03bb is the function defined by\n(2\n2 ), if\n0\n( )\n4\n1,\nif\n0\nx\nx\nx\nf x\nx\nx\n\uf8f1\u03bb\n\u2212\n\u2264\n= \uf8f2\uf8f4\n+\n>\n\uf8f4\uf8f3\ncontinuous at x = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "1982-1985",
        "Text": "Find the relationship between a and b so that the function f defined by\n1, if\n3\n( )\n3, if\n3\nax\nx\nf x\nbx\nx\n+\n\u2264\n= \uf8f2\uf8f1\n+\n>\n\uf8f3\nis continuous at x = 3 18 For what value of \u03bb is the function defined by\n(2\n2 ), if\n0\n( )\n4\n1,\nif\n0\nx\nx\nx\nf x\nx\nx\n\uf8f1\u03bb\n\u2212\n\u2264\n= \uf8f2\uf8f4\n+\n>\n\uf8f4\uf8f3\ncontinuous at x = 0 What about continuity at x = 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "1983-1986",
        "Text": "18 For what value of \u03bb is the function defined by\n(2\n2 ), if\n0\n( )\n4\n1,\nif\n0\nx\nx\nx\nf x\nx\nx\n\uf8f1\u03bb\n\u2212\n\u2264\n= \uf8f2\uf8f4\n+\n>\n\uf8f4\uf8f3\ncontinuous at x = 0 What about continuity at x = 1 19"
    },
    {
        "Chapter": "1",
        "sentence_range": "1984-1987",
        "Text": "For what value of \u03bb is the function defined by\n(2\n2 ), if\n0\n( )\n4\n1,\nif\n0\nx\nx\nx\nf x\nx\nx\n\uf8f1\u03bb\n\u2212\n\u2264\n= \uf8f2\uf8f4\n+\n>\n\uf8f4\uf8f3\ncontinuous at x = 0 What about continuity at x = 1 19 Show that the function defined by g (x) = x \u2013 [x] is discontinuous at all integral\npoints"
    },
    {
        "Chapter": "1",
        "sentence_range": "1985-1988",
        "Text": "What about continuity at x = 1 19 Show that the function defined by g (x) = x \u2013 [x] is discontinuous at all integral\npoints Here [x] denotes the greatest integer less than or equal to x"
    },
    {
        "Chapter": "1",
        "sentence_range": "1986-1989",
        "Text": "19 Show that the function defined by g (x) = x \u2013 [x] is discontinuous at all integral\npoints Here [x] denotes the greatest integer less than or equal to x 20"
    },
    {
        "Chapter": "1",
        "sentence_range": "1987-1990",
        "Text": "Show that the function defined by g (x) = x \u2013 [x] is discontinuous at all integral\npoints Here [x] denotes the greatest integer less than or equal to x 20 Is the function defined by f(x) = x2 \u2013 sin x + 5 continuous at x = \u03c0"
    },
    {
        "Chapter": "1",
        "sentence_range": "1988-1991",
        "Text": "Here [x] denotes the greatest integer less than or equal to x 20 Is the function defined by f(x) = x2 \u2013 sin x + 5 continuous at x = \u03c0 21"
    },
    {
        "Chapter": "1",
        "sentence_range": "1989-1992",
        "Text": "20 Is the function defined by f(x) = x2 \u2013 sin x + 5 continuous at x = \u03c0 21 Discuss the continuity of the following functions:\n(a)\nf(x) = sin x + cos x\n(b)\nf(x) = sin x \u2013 cos x\n(c)\nf(x) = sin x"
    },
    {
        "Chapter": "1",
        "sentence_range": "1990-1993",
        "Text": "Is the function defined by f(x) = x2 \u2013 sin x + 5 continuous at x = \u03c0 21 Discuss the continuity of the following functions:\n(a)\nf(x) = sin x + cos x\n(b)\nf(x) = sin x \u2013 cos x\n(c)\nf(x) = sin x cos x\n22"
    },
    {
        "Chapter": "1",
        "sentence_range": "1991-1994",
        "Text": "21 Discuss the continuity of the following functions:\n(a)\nf(x) = sin x + cos x\n(b)\nf(x) = sin x \u2013 cos x\n(c)\nf(x) = sin x cos x\n22 Discuss the continuity of the cosine, cosecant, secant and cotangent functions"
    },
    {
        "Chapter": "1",
        "sentence_range": "1992-1995",
        "Text": "Discuss the continuity of the following functions:\n(a)\nf(x) = sin x + cos x\n(b)\nf(x) = sin x \u2013 cos x\n(c)\nf(x) = sin x cos x\n22 Discuss the continuity of the cosine, cosecant, secant and cotangent functions 23"
    },
    {
        "Chapter": "1",
        "sentence_range": "1993-1996",
        "Text": "cos x\n22 Discuss the continuity of the cosine, cosecant, secant and cotangent functions 23 Find all points of discontinuity of f, where\nsin\n, if\n0\n( )\n1,\nif\n0\nx\nx\nf x\nxx\nx\n\uf8f1\n<\n= \uf8f2\uf8f4\n\uf8f4 +\n\u2265\n\uf8f3\n24"
    },
    {
        "Chapter": "1",
        "sentence_range": "1994-1997",
        "Text": "Discuss the continuity of the cosine, cosecant, secant and cotangent functions 23 Find all points of discontinuity of f, where\nsin\n, if\n0\n( )\n1,\nif\n0\nx\nx\nf x\nxx\nx\n\uf8f1\n<\n= \uf8f2\uf8f4\n\uf8f4 +\n\u2265\n\uf8f3\n24 Determine if f defined by\n2\nsin1\n, if\n0\n( )\n0,\nif\n0\nx\nx\nf x\nx\nx\n\uf8f1\n\u2260\n= \uf8f2\uf8f4\n\uf8f4\n=\n\uf8f3\nis a continuous function"
    },
    {
        "Chapter": "1",
        "sentence_range": "1995-1998",
        "Text": "23 Find all points of discontinuity of f, where\nsin\n, if\n0\n( )\n1,\nif\n0\nx\nx\nf x\nxx\nx\n\uf8f1\n<\n= \uf8f2\uf8f4\n\uf8f4 +\n\u2265\n\uf8f3\n24 Determine if f defined by\n2\nsin1\n, if\n0\n( )\n0,\nif\n0\nx\nx\nf x\nx\nx\n\uf8f1\n\u2260\n= \uf8f2\uf8f4\n\uf8f4\n=\n\uf8f3\nis a continuous function Rationalised 2023-24\n MATHEMATICS\n118\n25"
    },
    {
        "Chapter": "1",
        "sentence_range": "1996-1999",
        "Text": "Find all points of discontinuity of f, where\nsin\n, if\n0\n( )\n1,\nif\n0\nx\nx\nf x\nxx\nx\n\uf8f1\n<\n= \uf8f2\uf8f4\n\uf8f4 +\n\u2265\n\uf8f3\n24 Determine if f defined by\n2\nsin1\n, if\n0\n( )\n0,\nif\n0\nx\nx\nf x\nx\nx\n\uf8f1\n\u2260\n= \uf8f2\uf8f4\n\uf8f4\n=\n\uf8f3\nis a continuous function Rationalised 2023-24\n MATHEMATICS\n118\n25 Examine the continuity of f, where f is defined by\nsin\ncos , if\n0\n( )\n1,\nif\n0\nx\nx\nx\nf x\nx\n\u2212\n\u2260\n\uf8f1\n= \uf8f2\u2212\n=\n\uf8f3\nFind the values of k so that the function f is continuous at the indicated point in Exercises\n26 to 29"
    },
    {
        "Chapter": "1",
        "sentence_range": "1997-2000",
        "Text": "Determine if f defined by\n2\nsin1\n, if\n0\n( )\n0,\nif\n0\nx\nx\nf x\nx\nx\n\uf8f1\n\u2260\n= \uf8f2\uf8f4\n\uf8f4\n=\n\uf8f3\nis a continuous function Rationalised 2023-24\n MATHEMATICS\n118\n25 Examine the continuity of f, where f is defined by\nsin\ncos , if\n0\n( )\n1,\nif\n0\nx\nx\nx\nf x\nx\n\u2212\n\u2260\n\uf8f1\n= \uf8f2\u2212\n=\n\uf8f3\nFind the values of k so that the function f is continuous at the indicated point in Exercises\n26 to 29 26"
    },
    {
        "Chapter": "1",
        "sentence_range": "1998-2001",
        "Text": "Rationalised 2023-24\n MATHEMATICS\n118\n25 Examine the continuity of f, where f is defined by\nsin\ncos , if\n0\n( )\n1,\nif\n0\nx\nx\nx\nf x\nx\n\u2212\n\u2260\n\uf8f1\n= \uf8f2\u2212\n=\n\uf8f3\nFind the values of k so that the function f is continuous at the indicated point in Exercises\n26 to 29 26 cos , if\n2\n2\n( )\n3,\nif\n2\nk\nx\nx\nx\nf x\nx\n\u03c0\n\uf8f1\n\u2260\n= \uf8f2\uf8f4\uf8f4\u03c0 \u2212\n\u03c0\n\uf8f4\n=\n\uf8f4\uf8f3\n       at x = 2\n\u03c0\n27"
    },
    {
        "Chapter": "1",
        "sentence_range": "1999-2002",
        "Text": "Examine the continuity of f, where f is defined by\nsin\ncos , if\n0\n( )\n1,\nif\n0\nx\nx\nx\nf x\nx\n\u2212\n\u2260\n\uf8f1\n= \uf8f2\u2212\n=\n\uf8f3\nFind the values of k so that the function f is continuous at the indicated point in Exercises\n26 to 29 26 cos , if\n2\n2\n( )\n3,\nif\n2\nk\nx\nx\nx\nf x\nx\n\u03c0\n\uf8f1\n\u2260\n= \uf8f2\uf8f4\uf8f4\u03c0 \u2212\n\u03c0\n\uf8f4\n=\n\uf8f4\uf8f3\n       at x = 2\n\u03c0\n27 2, if\n2\n( )\n3,\nif\n2\nkx\nx\nf x\nx\n\uf8f1\n\u2264\n= \uf8f2\uf8f4\n>\n\uf8f4\uf8f3\n            at x = 2\n28"
    },
    {
        "Chapter": "1",
        "sentence_range": "2000-2003",
        "Text": "26 cos , if\n2\n2\n( )\n3,\nif\n2\nk\nx\nx\nx\nf x\nx\n\u03c0\n\uf8f1\n\u2260\n= \uf8f2\uf8f4\uf8f4\u03c0 \u2212\n\u03c0\n\uf8f4\n=\n\uf8f4\uf8f3\n       at x = 2\n\u03c0\n27 2, if\n2\n( )\n3,\nif\n2\nkx\nx\nf x\nx\n\uf8f1\n\u2264\n= \uf8f2\uf8f4\n>\n\uf8f4\uf8f3\n            at x = 2\n28 1, if\n( )\ncos ,\nif\nkx\nx\nf x\nx\nx\n+\n\u2264 \u03c0\n= \uf8f2\uf8f1\n> \u03c0\n\uf8f3\n         at x = \u03c0\n29"
    },
    {
        "Chapter": "1",
        "sentence_range": "2001-2004",
        "Text": "cos , if\n2\n2\n( )\n3,\nif\n2\nk\nx\nx\nx\nf x\nx\n\u03c0\n\uf8f1\n\u2260\n= \uf8f2\uf8f4\uf8f4\u03c0 \u2212\n\u03c0\n\uf8f4\n=\n\uf8f4\uf8f3\n       at x = 2\n\u03c0\n27 2, if\n2\n( )\n3,\nif\n2\nkx\nx\nf x\nx\n\uf8f1\n\u2264\n= \uf8f2\uf8f4\n>\n\uf8f4\uf8f3\n            at x = 2\n28 1, if\n( )\ncos ,\nif\nkx\nx\nf x\nx\nx\n+\n\u2264 \u03c0\n= \uf8f2\uf8f1\n> \u03c0\n\uf8f3\n         at x = \u03c0\n29 1, if\n5\n( )\n3\n5, if\n5\nkx\nx\nf x\nx\nx\n+\n\u2264\n= \uf8f2\uf8f1\n\u2212\n>\n\uf8f3\n         at x = 5\n30"
    },
    {
        "Chapter": "1",
        "sentence_range": "2002-2005",
        "Text": "2, if\n2\n( )\n3,\nif\n2\nkx\nx\nf x\nx\n\uf8f1\n\u2264\n= \uf8f2\uf8f4\n>\n\uf8f4\uf8f3\n            at x = 2\n28 1, if\n( )\ncos ,\nif\nkx\nx\nf x\nx\nx\n+\n\u2264 \u03c0\n= \uf8f2\uf8f1\n> \u03c0\n\uf8f3\n         at x = \u03c0\n29 1, if\n5\n( )\n3\n5, if\n5\nkx\nx\nf x\nx\nx\n+\n\u2264\n= \uf8f2\uf8f1\n\u2212\n>\n\uf8f3\n         at x = 5\n30 Find the values of a and b such that the function defined by\n5,\nif\n2\n( )\n, if 2\n10\n21,\nif\n10\nx\nf x\nax\nb\nx\nx\n\u2264\n=\uf8f4\uf8f1\n+\n<\n<\n\uf8f4\uf8f2\n\u2265\n\uf8f3\nis a continuous function"
    },
    {
        "Chapter": "1",
        "sentence_range": "2003-2006",
        "Text": "1, if\n( )\ncos ,\nif\nkx\nx\nf x\nx\nx\n+\n\u2264 \u03c0\n= \uf8f2\uf8f1\n> \u03c0\n\uf8f3\n         at x = \u03c0\n29 1, if\n5\n( )\n3\n5, if\n5\nkx\nx\nf x\nx\nx\n+\n\u2264\n= \uf8f2\uf8f1\n\u2212\n>\n\uf8f3\n         at x = 5\n30 Find the values of a and b such that the function defined by\n5,\nif\n2\n( )\n, if 2\n10\n21,\nif\n10\nx\nf x\nax\nb\nx\nx\n\u2264\n=\uf8f4\uf8f1\n+\n<\n<\n\uf8f4\uf8f2\n\u2265\n\uf8f3\nis a continuous function 31"
    },
    {
        "Chapter": "1",
        "sentence_range": "2004-2007",
        "Text": "1, if\n5\n( )\n3\n5, if\n5\nkx\nx\nf x\nx\nx\n+\n\u2264\n= \uf8f2\uf8f1\n\u2212\n>\n\uf8f3\n         at x = 5\n30 Find the values of a and b such that the function defined by\n5,\nif\n2\n( )\n, if 2\n10\n21,\nif\n10\nx\nf x\nax\nb\nx\nx\n\u2264\n=\uf8f4\uf8f1\n+\n<\n<\n\uf8f4\uf8f2\n\u2265\n\uf8f3\nis a continuous function 31 Show that the function defined by f(x) = cos (x2) is a continuous function"
    },
    {
        "Chapter": "1",
        "sentence_range": "2005-2008",
        "Text": "Find the values of a and b such that the function defined by\n5,\nif\n2\n( )\n, if 2\n10\n21,\nif\n10\nx\nf x\nax\nb\nx\nx\n\u2264\n=\uf8f4\uf8f1\n+\n<\n<\n\uf8f4\uf8f2\n\u2265\n\uf8f3\nis a continuous function 31 Show that the function defined by f(x) = cos (x2) is a continuous function 32"
    },
    {
        "Chapter": "1",
        "sentence_range": "2006-2009",
        "Text": "31 Show that the function defined by f(x) = cos (x2) is a continuous function 32 Show that the function defined by f(x) = |cos x | is a continuous function"
    },
    {
        "Chapter": "1",
        "sentence_range": "2007-2010",
        "Text": "Show that the function defined by f(x) = cos (x2) is a continuous function 32 Show that the function defined by f(x) = |cos x | is a continuous function 33"
    },
    {
        "Chapter": "1",
        "sentence_range": "2008-2011",
        "Text": "32 Show that the function defined by f(x) = |cos x | is a continuous function 33 Examine that sin |x| is a continuous function"
    },
    {
        "Chapter": "1",
        "sentence_range": "2009-2012",
        "Text": "Show that the function defined by f(x) = |cos x | is a continuous function 33 Examine that sin |x| is a continuous function 34"
    },
    {
        "Chapter": "1",
        "sentence_range": "2010-2013",
        "Text": "33 Examine that sin |x| is a continuous function 34 Find all the points of discontinuity of f defined by f(x) = |x | \u2013 |x + 1 |"
    },
    {
        "Chapter": "1",
        "sentence_range": "2011-2014",
        "Text": "Examine that sin |x| is a continuous function 34 Find all the points of discontinuity of f defined by f(x) = |x | \u2013 |x + 1 | 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "2012-2015",
        "Text": "34 Find all the points of discontinuity of f defined by f(x) = |x | \u2013 |x + 1 | 5 3"
    },
    {
        "Chapter": "1",
        "sentence_range": "2013-2016",
        "Text": "Find all the points of discontinuity of f defined by f(x) = |x | \u2013 |x + 1 | 5 3 Differentiability\nRecall the following facts from previous class"
    },
    {
        "Chapter": "1",
        "sentence_range": "2014-2017",
        "Text": "5 3 Differentiability\nRecall the following facts from previous class We had defined the derivative of a real\nfunction as follows:\nSuppose f is a real function and c is a point in its domain"
    },
    {
        "Chapter": "1",
        "sentence_range": "2015-2018",
        "Text": "3 Differentiability\nRecall the following facts from previous class We had defined the derivative of a real\nfunction as follows:\nSuppose f is a real function and c is a point in its domain The derivative of f at c is\ndefined by\n0\n(\n)\n( )\nlim\nh\nf c\nh\nf c\nh\n\u2192\n+\n\u2212\nRationalised 2023-24\nCONTINUITY AND DIFFERENTIABILITY\n119\nf (x)\nxn\nsin x\ncos x\ntan x\nf \u2032(x)\nnxn \u2013 1\ncos x\n\u2013 sin x\nsec2 x\nprovided this limit exists"
    },
    {
        "Chapter": "1",
        "sentence_range": "2016-2019",
        "Text": "Differentiability\nRecall the following facts from previous class We had defined the derivative of a real\nfunction as follows:\nSuppose f is a real function and c is a point in its domain The derivative of f at c is\ndefined by\n0\n(\n)\n( )\nlim\nh\nf c\nh\nf c\nh\n\u2192\n+\n\u2212\nRationalised 2023-24\nCONTINUITY AND DIFFERENTIABILITY\n119\nf (x)\nxn\nsin x\ncos x\ntan x\nf \u2032(x)\nnxn \u2013 1\ncos x\n\u2013 sin x\nsec2 x\nprovided this limit exists Derivative of f at c is denoted by f \u2032(c) or \n( ( )) |c\nd\ndxf x"
    },
    {
        "Chapter": "1",
        "sentence_range": "2017-2020",
        "Text": "We had defined the derivative of a real\nfunction as follows:\nSuppose f is a real function and c is a point in its domain The derivative of f at c is\ndefined by\n0\n(\n)\n( )\nlim\nh\nf c\nh\nf c\nh\n\u2192\n+\n\u2212\nRationalised 2023-24\nCONTINUITY AND DIFFERENTIABILITY\n119\nf (x)\nxn\nsin x\ncos x\ntan x\nf \u2032(x)\nnxn \u2013 1\ncos x\n\u2013 sin x\nsec2 x\nprovided this limit exists Derivative of f at c is denoted by f \u2032(c) or \n( ( )) |c\nd\ndxf x The\nfunction defined by\n0\n(\n)\n( )\n( )\nlim\nh\nf x\nh\nf x\nf\nx\nh\n\u2192\n+\n\u2212\n\u2032\n=\nwherever the limit exists is defined to be the derivative of f"
    },
    {
        "Chapter": "1",
        "sentence_range": "2018-2021",
        "Text": "The derivative of f at c is\ndefined by\n0\n(\n)\n( )\nlim\nh\nf c\nh\nf c\nh\n\u2192\n+\n\u2212\nRationalised 2023-24\nCONTINUITY AND DIFFERENTIABILITY\n119\nf (x)\nxn\nsin x\ncos x\ntan x\nf \u2032(x)\nnxn \u2013 1\ncos x\n\u2013 sin x\nsec2 x\nprovided this limit exists Derivative of f at c is denoted by f \u2032(c) or \n( ( )) |c\nd\ndxf x The\nfunction defined by\n0\n(\n)\n( )\n( )\nlim\nh\nf x\nh\nf x\nf\nx\nh\n\u2192\n+\n\u2212\n\u2032\n=\nwherever the limit exists is defined to be the derivative of f The derivative of f is\ndenoted by f \u2032(x) or \nd( ( ))\ndxf x\nor if y = f (x) by dy\ndx or y\u2032"
    },
    {
        "Chapter": "1",
        "sentence_range": "2019-2022",
        "Text": "Derivative of f at c is denoted by f \u2032(c) or \n( ( )) |c\nd\ndxf x The\nfunction defined by\n0\n(\n)\n( )\n( )\nlim\nh\nf x\nh\nf x\nf\nx\nh\n\u2192\n+\n\u2212\n\u2032\n=\nwherever the limit exists is defined to be the derivative of f The derivative of f is\ndenoted by f \u2032(x) or \nd( ( ))\ndxf x\nor if y = f (x) by dy\ndx or y\u2032 The process of finding\nderivative of a function is called differentiation"
    },
    {
        "Chapter": "1",
        "sentence_range": "2020-2023",
        "Text": "The\nfunction defined by\n0\n(\n)\n( )\n( )\nlim\nh\nf x\nh\nf x\nf\nx\nh\n\u2192\n+\n\u2212\n\u2032\n=\nwherever the limit exists is defined to be the derivative of f The derivative of f is\ndenoted by f \u2032(x) or \nd( ( ))\ndxf x\nor if y = f (x) by dy\ndx or y\u2032 The process of finding\nderivative of a function is called differentiation We also use the phrase differentiate\nf(x) with respect to x to mean find f \u2032(x)"
    },
    {
        "Chapter": "1",
        "sentence_range": "2021-2024",
        "Text": "The derivative of f is\ndenoted by f \u2032(x) or \nd( ( ))\ndxf x\nor if y = f (x) by dy\ndx or y\u2032 The process of finding\nderivative of a function is called differentiation We also use the phrase differentiate\nf(x) with respect to x to mean find f \u2032(x) (1)The following rules were established as a part of algebra of derivatives:\n(u \u00b1 v)\u2032 = u\u2032 \u00b1 v\u2032\n(2)\n(uv)\u2032 = u\u2032v + uv\u2032 (Leibnitz or product rule)\n(3)\n2\nu\nu v\nuv\nv\nv\n\u2032\n\u2032 \u2212\n\u2032\n\uf8eb\n\uf8f6 =\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n, wherever v \u2260 0 (Quotient rule)"
    },
    {
        "Chapter": "1",
        "sentence_range": "2022-2025",
        "Text": "The process of finding\nderivative of a function is called differentiation We also use the phrase differentiate\nf(x) with respect to x to mean find f \u2032(x) (1)The following rules were established as a part of algebra of derivatives:\n(u \u00b1 v)\u2032 = u\u2032 \u00b1 v\u2032\n(2)\n(uv)\u2032 = u\u2032v + uv\u2032 (Leibnitz or product rule)\n(3)\n2\nu\nu v\nuv\nv\nv\n\u2032\n\u2032 \u2212\n\u2032\n\uf8eb\n\uf8f6 =\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n, wherever v \u2260 0 (Quotient rule) The following table gives a list of derivatives of certain standard functions:\nTable 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "2023-2026",
        "Text": "We also use the phrase differentiate\nf(x) with respect to x to mean find f \u2032(x) (1)The following rules were established as a part of algebra of derivatives:\n(u \u00b1 v)\u2032 = u\u2032 \u00b1 v\u2032\n(2)\n(uv)\u2032 = u\u2032v + uv\u2032 (Leibnitz or product rule)\n(3)\n2\nu\nu v\nuv\nv\nv\n\u2032\n\u2032 \u2212\n\u2032\n\uf8eb\n\uf8f6 =\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n, wherever v \u2260 0 (Quotient rule) The following table gives a list of derivatives of certain standard functions:\nTable 5 3\nWhenever we defined derivative, we had put a caution provided the limit exists"
    },
    {
        "Chapter": "1",
        "sentence_range": "2024-2027",
        "Text": "(1)The following rules were established as a part of algebra of derivatives:\n(u \u00b1 v)\u2032 = u\u2032 \u00b1 v\u2032\n(2)\n(uv)\u2032 = u\u2032v + uv\u2032 (Leibnitz or product rule)\n(3)\n2\nu\nu v\nuv\nv\nv\n\u2032\n\u2032 \u2212\n\u2032\n\uf8eb\n\uf8f6 =\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n, wherever v \u2260 0 (Quotient rule) The following table gives a list of derivatives of certain standard functions:\nTable 5 3\nWhenever we defined derivative, we had put a caution provided the limit exists Now the natural question is; what if it doesn\u2019t"
    },
    {
        "Chapter": "1",
        "sentence_range": "2025-2028",
        "Text": "The following table gives a list of derivatives of certain standard functions:\nTable 5 3\nWhenever we defined derivative, we had put a caution provided the limit exists Now the natural question is; what if it doesn\u2019t The question is quite pertinent and so is\nits answer"
    },
    {
        "Chapter": "1",
        "sentence_range": "2026-2029",
        "Text": "3\nWhenever we defined derivative, we had put a caution provided the limit exists Now the natural question is; what if it doesn\u2019t The question is quite pertinent and so is\nits answer If  \n0\n(\n)\n( )\nlim\nh\nf c\nh\nf c\nh\n\u2192\n+\n\u2212\n does not exist, we say that f is not differentiable at c"
    },
    {
        "Chapter": "1",
        "sentence_range": "2027-2030",
        "Text": "Now the natural question is; what if it doesn\u2019t The question is quite pertinent and so is\nits answer If  \n0\n(\n)\n( )\nlim\nh\nf c\nh\nf c\nh\n\u2192\n+\n\u2212\n does not exist, we say that f is not differentiable at c In other words, we say that a function f is differentiable at a point c in its domain if both\n\u2013\n0\n(\n)\n( )\nlim\nh\nf c\nh\nf c\nh\n\u2192\n+\n\u2212\n and \n0\n(\n)\n( )\nlim\nh\nf c\nh\nf c\nh\n+\n\u2192\n+\n\u2212\n are finite and equal"
    },
    {
        "Chapter": "1",
        "sentence_range": "2028-2031",
        "Text": "The question is quite pertinent and so is\nits answer If  \n0\n(\n)\n( )\nlim\nh\nf c\nh\nf c\nh\n\u2192\n+\n\u2212\n does not exist, we say that f is not differentiable at c In other words, we say that a function f is differentiable at a point c in its domain if both\n\u2013\n0\n(\n)\n( )\nlim\nh\nf c\nh\nf c\nh\n\u2192\n+\n\u2212\n and \n0\n(\n)\n( )\nlim\nh\nf c\nh\nf c\nh\n+\n\u2192\n+\n\u2212\n are finite and equal A function is said\nto be differentiable in an interval [a, b] if it is differentiable at every point of [a, b]"
    },
    {
        "Chapter": "1",
        "sentence_range": "2029-2032",
        "Text": "If  \n0\n(\n)\n( )\nlim\nh\nf c\nh\nf c\nh\n\u2192\n+\n\u2212\n does not exist, we say that f is not differentiable at c In other words, we say that a function f is differentiable at a point c in its domain if both\n\u2013\n0\n(\n)\n( )\nlim\nh\nf c\nh\nf c\nh\n\u2192\n+\n\u2212\n and \n0\n(\n)\n( )\nlim\nh\nf c\nh\nf c\nh\n+\n\u2192\n+\n\u2212\n are finite and equal A function is said\nto be differentiable in an interval [a, b] if it is differentiable at every point of [a, b] As\nin case of continuity, at the end points a and b, we take the right hand limit and left hand\nlimit, which are nothing but left hand derivative and right hand derivative of the function\nat a and b respectively"
    },
    {
        "Chapter": "1",
        "sentence_range": "2030-2033",
        "Text": "In other words, we say that a function f is differentiable at a point c in its domain if both\n\u2013\n0\n(\n)\n( )\nlim\nh\nf c\nh\nf c\nh\n\u2192\n+\n\u2212\n and \n0\n(\n)\n( )\nlim\nh\nf c\nh\nf c\nh\n+\n\u2192\n+\n\u2212\n are finite and equal A function is said\nto be differentiable in an interval [a, b] if it is differentiable at every point of [a, b] As\nin case of continuity, at the end points a and b, we take the right hand limit and left hand\nlimit, which are nothing but left hand derivative and right hand derivative of the function\nat a and b respectively Similarly, a function is said to be differentiable in an interval\n(a, b) if it is differentiable at every point of (a, b)"
    },
    {
        "Chapter": "1",
        "sentence_range": "2031-2034",
        "Text": "A function is said\nto be differentiable in an interval [a, b] if it is differentiable at every point of [a, b] As\nin case of continuity, at the end points a and b, we take the right hand limit and left hand\nlimit, which are nothing but left hand derivative and right hand derivative of the function\nat a and b respectively Similarly, a function is said to be differentiable in an interval\n(a, b) if it is differentiable at every point of (a, b) Rationalised 2023-24\n MATHEMATICS\n120\nTheorem 3 If a function f is differentiable at a point c, then it is also continuous at that\npoint"
    },
    {
        "Chapter": "1",
        "sentence_range": "2032-2035",
        "Text": "As\nin case of continuity, at the end points a and b, we take the right hand limit and left hand\nlimit, which are nothing but left hand derivative and right hand derivative of the function\nat a and b respectively Similarly, a function is said to be differentiable in an interval\n(a, b) if it is differentiable at every point of (a, b) Rationalised 2023-24\n MATHEMATICS\n120\nTheorem 3 If a function f is differentiable at a point c, then it is also continuous at that\npoint Proof Since f is differentiable at c, we have\n( )\n( )\nlim\n( )\nx\nc\nf x\nf c\nf c\nx\nc\n\u2192\n\u2212\n=\n\u2032\n\u2212\nBut for x \u2260 c, we have\nf (x) \u2013 f (c) =\n( )\n( )"
    },
    {
        "Chapter": "1",
        "sentence_range": "2033-2036",
        "Text": "Similarly, a function is said to be differentiable in an interval\n(a, b) if it is differentiable at every point of (a, b) Rationalised 2023-24\n MATHEMATICS\n120\nTheorem 3 If a function f is differentiable at a point c, then it is also continuous at that\npoint Proof Since f is differentiable at c, we have\n( )\n( )\nlim\n( )\nx\nc\nf x\nf c\nf c\nx\nc\n\u2192\n\u2212\n=\n\u2032\n\u2212\nBut for x \u2260 c, we have\nf (x) \u2013 f (c) =\n( )\n( ) (\n)\nf x\nf c\nx\nc\nx\nc\n\u2212\n\u2212\n\u2212\nTherefore\nlim[ ( )\n( )]\nx\nc f x\nf c\n\u2192\n\u2212\n =\n( )\n( )\nlim"
    },
    {
        "Chapter": "1",
        "sentence_range": "2034-2037",
        "Text": "Rationalised 2023-24\n MATHEMATICS\n120\nTheorem 3 If a function f is differentiable at a point c, then it is also continuous at that\npoint Proof Since f is differentiable at c, we have\n( )\n( )\nlim\n( )\nx\nc\nf x\nf c\nf c\nx\nc\n\u2192\n\u2212\n=\n\u2032\n\u2212\nBut for x \u2260 c, we have\nf (x) \u2013 f (c) =\n( )\n( ) (\n)\nf x\nf c\nx\nc\nx\nc\n\u2212\n\u2212\n\u2212\nTherefore\nlim[ ( )\n( )]\nx\nc f x\nf c\n\u2192\n\u2212\n =\n( )\n( )\nlim (\n)\nx\nc\nf x\nf c\nx\nc\nx\nc\n\u2192\n\u2212\n\uf8ee\n\uf8f9\n\u2212\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\nor\nlim[ ( )]\nlim[ ( )]\nx\nc\nx\nc\nf x\nf c\n\u2192\n\u2212\u2192\n =\n( )\n( )\nlim"
    },
    {
        "Chapter": "1",
        "sentence_range": "2035-2038",
        "Text": "Proof Since f is differentiable at c, we have\n( )\n( )\nlim\n( )\nx\nc\nf x\nf c\nf c\nx\nc\n\u2192\n\u2212\n=\n\u2032\n\u2212\nBut for x \u2260 c, we have\nf (x) \u2013 f (c) =\n( )\n( ) (\n)\nf x\nf c\nx\nc\nx\nc\n\u2212\n\u2212\n\u2212\nTherefore\nlim[ ( )\n( )]\nx\nc f x\nf c\n\u2192\n\u2212\n =\n( )\n( )\nlim (\n)\nx\nc\nf x\nf c\nx\nc\nx\nc\n\u2192\n\u2212\n\uf8ee\n\uf8f9\n\u2212\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\nor\nlim[ ( )]\nlim[ ( )]\nx\nc\nx\nc\nf x\nf c\n\u2192\n\u2212\u2192\n =\n( )\n( )\nlim lim [(\n)]\nx\nc\nx\nc\nf x\nf c\nx\nc\nx\nc\n\u2192\n\u2192\n\u2212\n\uf8ee\n\uf8f9\n\u2212\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n= f\u2032(c)"
    },
    {
        "Chapter": "1",
        "sentence_range": "2036-2039",
        "Text": "(\n)\nf x\nf c\nx\nc\nx\nc\n\u2212\n\u2212\n\u2212\nTherefore\nlim[ ( )\n( )]\nx\nc f x\nf c\n\u2192\n\u2212\n =\n( )\n( )\nlim (\n)\nx\nc\nf x\nf c\nx\nc\nx\nc\n\u2192\n\u2212\n\uf8ee\n\uf8f9\n\u2212\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\nor\nlim[ ( )]\nlim[ ( )]\nx\nc\nx\nc\nf x\nf c\n\u2192\n\u2212\u2192\n =\n( )\n( )\nlim lim [(\n)]\nx\nc\nx\nc\nf x\nf c\nx\nc\nx\nc\n\u2192\n\u2192\n\u2212\n\uf8ee\n\uf8f9\n\u2212\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n= f\u2032(c) 0 = 0\nor\nlim\n( )\nx\nc f x\n\u2192\n = f (c)\nHence f is continuous at x = c"
    },
    {
        "Chapter": "1",
        "sentence_range": "2037-2040",
        "Text": "(\n)\nx\nc\nf x\nf c\nx\nc\nx\nc\n\u2192\n\u2212\n\uf8ee\n\uf8f9\n\u2212\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\nor\nlim[ ( )]\nlim[ ( )]\nx\nc\nx\nc\nf x\nf c\n\u2192\n\u2212\u2192\n =\n( )\n( )\nlim lim [(\n)]\nx\nc\nx\nc\nf x\nf c\nx\nc\nx\nc\n\u2192\n\u2192\n\u2212\n\uf8ee\n\uf8f9\n\u2212\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n= f\u2032(c) 0 = 0\nor\nlim\n( )\nx\nc f x\n\u2192\n = f (c)\nHence f is continuous at x = c Corollary 1 Every differentiable function is continuous"
    },
    {
        "Chapter": "1",
        "sentence_range": "2038-2041",
        "Text": "lim [(\n)]\nx\nc\nx\nc\nf x\nf c\nx\nc\nx\nc\n\u2192\n\u2192\n\u2212\n\uf8ee\n\uf8f9\n\u2212\n\uf8ef\n\uf8fa\n\u2212\n\uf8f0\n\uf8fb\n= f\u2032(c) 0 = 0\nor\nlim\n( )\nx\nc f x\n\u2192\n = f (c)\nHence f is continuous at x = c Corollary 1 Every differentiable function is continuous We remark that the converse of the above statement is not true"
    },
    {
        "Chapter": "1",
        "sentence_range": "2039-2042",
        "Text": "0 = 0\nor\nlim\n( )\nx\nc f x\n\u2192\n = f (c)\nHence f is continuous at x = c Corollary 1 Every differentiable function is continuous We remark that the converse of the above statement is not true Indeed we have\nseen that the function defined by f(x) = |x | is a continuous function"
    },
    {
        "Chapter": "1",
        "sentence_range": "2040-2043",
        "Text": "Corollary 1 Every differentiable function is continuous We remark that the converse of the above statement is not true Indeed we have\nseen that the function defined by f(x) = |x | is a continuous function Consider the left\nhand limit\n0\u2013\n(0\n)\n(0)\nlim\n1\nh\nf\nh\nf\nh\nh\nh\n\u2192\n+\n\u2212\n=\u2212\n= \u2212\nThe right hand limit\n0\n(0\n)\n(0)\nlim\n1\nh\nf\nh\nf\nh\nh\nh\n+\n\u2192\n+\n\u2212\n=\n=\nSince the above left and right hand limits at 0 are not equal, \n0\n(0\n)\n(0)\nlim\nh\nf\nh\nf\nh\n\u2192\n+\n\u2212\ndoes not exist and hence f is not differentiable at 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "2041-2044",
        "Text": "We remark that the converse of the above statement is not true Indeed we have\nseen that the function defined by f(x) = |x | is a continuous function Consider the left\nhand limit\n0\u2013\n(0\n)\n(0)\nlim\n1\nh\nf\nh\nf\nh\nh\nh\n\u2192\n+\n\u2212\n=\u2212\n= \u2212\nThe right hand limit\n0\n(0\n)\n(0)\nlim\n1\nh\nf\nh\nf\nh\nh\nh\n+\n\u2192\n+\n\u2212\n=\n=\nSince the above left and right hand limits at 0 are not equal, \n0\n(0\n)\n(0)\nlim\nh\nf\nh\nf\nh\n\u2192\n+\n\u2212\ndoes not exist and hence f is not differentiable at 0 Thus f is not a differentiable\nfunction"
    },
    {
        "Chapter": "1",
        "sentence_range": "2042-2045",
        "Text": "Indeed we have\nseen that the function defined by f(x) = |x | is a continuous function Consider the left\nhand limit\n0\u2013\n(0\n)\n(0)\nlim\n1\nh\nf\nh\nf\nh\nh\nh\n\u2192\n+\n\u2212\n=\u2212\n= \u2212\nThe right hand limit\n0\n(0\n)\n(0)\nlim\n1\nh\nf\nh\nf\nh\nh\nh\n+\n\u2192\n+\n\u2212\n=\n=\nSince the above left and right hand limits at 0 are not equal, \n0\n(0\n)\n(0)\nlim\nh\nf\nh\nf\nh\n\u2192\n+\n\u2212\ndoes not exist and hence f is not differentiable at 0 Thus f is not a differentiable\nfunction 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "2043-2046",
        "Text": "Consider the left\nhand limit\n0\u2013\n(0\n)\n(0)\nlim\n1\nh\nf\nh\nf\nh\nh\nh\n\u2192\n+\n\u2212\n=\u2212\n= \u2212\nThe right hand limit\n0\n(0\n)\n(0)\nlim\n1\nh\nf\nh\nf\nh\nh\nh\n+\n\u2192\n+\n\u2212\n=\n=\nSince the above left and right hand limits at 0 are not equal, \n0\n(0\n)\n(0)\nlim\nh\nf\nh\nf\nh\n\u2192\n+\n\u2212\ndoes not exist and hence f is not differentiable at 0 Thus f is not a differentiable\nfunction 5 3"
    },
    {
        "Chapter": "1",
        "sentence_range": "2044-2047",
        "Text": "Thus f is not a differentiable\nfunction 5 3 1  Derivatives of composite functions\nTo study derivative of composite functions, we start with an illustrative example"
    },
    {
        "Chapter": "1",
        "sentence_range": "2045-2048",
        "Text": "5 3 1  Derivatives of composite functions\nTo study derivative of composite functions, we start with an illustrative example Say,\nwe want to find the derivative of f, where\nf (x) = (2x + 1)3\nRationalised 2023-24\nCONTINUITY AND DIFFERENTIABILITY\n121\nOne way is to expand (2x + 1)3 using binomial theorem and find the derivative as\na polynomial function as illustrated below"
    },
    {
        "Chapter": "1",
        "sentence_range": "2046-2049",
        "Text": "3 1  Derivatives of composite functions\nTo study derivative of composite functions, we start with an illustrative example Say,\nwe want to find the derivative of f, where\nf (x) = (2x + 1)3\nRationalised 2023-24\nCONTINUITY AND DIFFERENTIABILITY\n121\nOne way is to expand (2x + 1)3 using binomial theorem and find the derivative as\na polynomial function as illustrated below ( )\ndxd f x\n =\n3\n(2\n1)\nd\nx\ndx\n\uf8ee\n\uf8f9\n+\n\uf8f0\n\uf8fb\n=\n3\n2\n(8\n12\n6\n1)\nd\nx\nx\nx\ndx\n+\n+\n+\n= 24x2 + 24x + 6\n= 6 (2x + 1)2\nNow, observe that\nf (x) = (h o g) (x)\nwhere g(x) = 2x + 1 and h(x) = x3"
    },
    {
        "Chapter": "1",
        "sentence_range": "2047-2050",
        "Text": "1  Derivatives of composite functions\nTo study derivative of composite functions, we start with an illustrative example Say,\nwe want to find the derivative of f, where\nf (x) = (2x + 1)3\nRationalised 2023-24\nCONTINUITY AND DIFFERENTIABILITY\n121\nOne way is to expand (2x + 1)3 using binomial theorem and find the derivative as\na polynomial function as illustrated below ( )\ndxd f x\n =\n3\n(2\n1)\nd\nx\ndx\n\uf8ee\n\uf8f9\n+\n\uf8f0\n\uf8fb\n=\n3\n2\n(8\n12\n6\n1)\nd\nx\nx\nx\ndx\n+\n+\n+\n= 24x2 + 24x + 6\n= 6 (2x + 1)2\nNow, observe that\nf (x) = (h o g) (x)\nwhere g(x) = 2x + 1 and h(x) = x3 Put t = g(x) = 2x + 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "2048-2051",
        "Text": "Say,\nwe want to find the derivative of f, where\nf (x) = (2x + 1)3\nRationalised 2023-24\nCONTINUITY AND DIFFERENTIABILITY\n121\nOne way is to expand (2x + 1)3 using binomial theorem and find the derivative as\na polynomial function as illustrated below ( )\ndxd f x\n =\n3\n(2\n1)\nd\nx\ndx\n\uf8ee\n\uf8f9\n+\n\uf8f0\n\uf8fb\n=\n3\n2\n(8\n12\n6\n1)\nd\nx\nx\nx\ndx\n+\n+\n+\n= 24x2 + 24x + 6\n= 6 (2x + 1)2\nNow, observe that\nf (x) = (h o g) (x)\nwhere g(x) = 2x + 1 and h(x) = x3 Put t = g(x) = 2x + 1 Then f(x) = h(t) = t3"
    },
    {
        "Chapter": "1",
        "sentence_range": "2049-2052",
        "Text": "( )\ndxd f x\n =\n3\n(2\n1)\nd\nx\ndx\n\uf8ee\n\uf8f9\n+\n\uf8f0\n\uf8fb\n=\n3\n2\n(8\n12\n6\n1)\nd\nx\nx\nx\ndx\n+\n+\n+\n= 24x2 + 24x + 6\n= 6 (2x + 1)2\nNow, observe that\nf (x) = (h o g) (x)\nwhere g(x) = 2x + 1 and h(x) = x3 Put t = g(x) = 2x + 1 Then f(x) = h(t) = t3 Thus\ndf\ndx  =\n6 (2x + 1)2 = 3(2x + 1)2"
    },
    {
        "Chapter": "1",
        "sentence_range": "2050-2053",
        "Text": "Put t = g(x) = 2x + 1 Then f(x) = h(t) = t3 Thus\ndf\ndx  =\n6 (2x + 1)2 = 3(2x + 1)2 2 = 3t2"
    },
    {
        "Chapter": "1",
        "sentence_range": "2051-2054",
        "Text": "Then f(x) = h(t) = t3 Thus\ndf\ndx  =\n6 (2x + 1)2 = 3(2x + 1)2 2 = 3t2 2 = dh dt\ndt\ndx\n\u22c5\nThe advantage with such observation is that it simplifies the calculation in finding\nthe derivative of, say, (2x + 1)100"
    },
    {
        "Chapter": "1",
        "sentence_range": "2052-2055",
        "Text": "Thus\ndf\ndx  =\n6 (2x + 1)2 = 3(2x + 1)2 2 = 3t2 2 = dh dt\ndt\ndx\n\u22c5\nThe advantage with such observation is that it simplifies the calculation in finding\nthe derivative of, say, (2x + 1)100 We may formalise this observation in the following\ntheorem called the chain rule"
    },
    {
        "Chapter": "1",
        "sentence_range": "2053-2056",
        "Text": "2 = 3t2 2 = dh dt\ndt\ndx\n\u22c5\nThe advantage with such observation is that it simplifies the calculation in finding\nthe derivative of, say, (2x + 1)100 We may formalise this observation in the following\ntheorem called the chain rule Theorem 4 (Chain Rule) Let f be a real valued function which is a composite of two\nfunctions u and v; i"
    },
    {
        "Chapter": "1",
        "sentence_range": "2054-2057",
        "Text": "2 = dh dt\ndt\ndx\n\u22c5\nThe advantage with such observation is that it simplifies the calculation in finding\nthe derivative of, say, (2x + 1)100 We may formalise this observation in the following\ntheorem called the chain rule Theorem 4 (Chain Rule) Let f be a real valued function which is a composite of two\nfunctions u and v; i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "2055-2058",
        "Text": "We may formalise this observation in the following\ntheorem called the chain rule Theorem 4 (Chain Rule) Let f be a real valued function which is a composite of two\nfunctions u and v; i e , f = v o u"
    },
    {
        "Chapter": "1",
        "sentence_range": "2056-2059",
        "Text": "Theorem 4 (Chain Rule) Let f be a real valued function which is a composite of two\nfunctions u and v; i e , f = v o u Suppose t = u(x) and if both dt\ndx  and dv\ndt  exist, we have\ndf\ndv dt\ndx\n=dt dx\n\u22c5\nWe skip the proof of this theorem"
    },
    {
        "Chapter": "1",
        "sentence_range": "2057-2060",
        "Text": "e , f = v o u Suppose t = u(x) and if both dt\ndx  and dv\ndt  exist, we have\ndf\ndv dt\ndx\n=dt dx\n\u22c5\nWe skip the proof of this theorem Chain rule may be extended as follows"
    },
    {
        "Chapter": "1",
        "sentence_range": "2058-2061",
        "Text": ", f = v o u Suppose t = u(x) and if both dt\ndx  and dv\ndt  exist, we have\ndf\ndv dt\ndx\n=dt dx\n\u22c5\nWe skip the proof of this theorem Chain rule may be extended as follows Suppose\nf is a real valued function which is a composite of three functions u, v and w; i"
    },
    {
        "Chapter": "1",
        "sentence_range": "2059-2062",
        "Text": "Suppose t = u(x) and if both dt\ndx  and dv\ndt  exist, we have\ndf\ndv dt\ndx\n=dt dx\n\u22c5\nWe skip the proof of this theorem Chain rule may be extended as follows Suppose\nf is a real valued function which is a composite of three functions u, v and w; i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "2060-2063",
        "Text": "Chain rule may be extended as follows Suppose\nf is a real valued function which is a composite of three functions u, v and w; i e ,\nf = (w o u) o v"
    },
    {
        "Chapter": "1",
        "sentence_range": "2061-2064",
        "Text": "Suppose\nf is a real valued function which is a composite of three functions u, v and w; i e ,\nf = (w o u) o v If t = v (x) and s = u (t), then\n( o )\ndf\nd w u\ndt\ndw ds\ndt\ndx\ndt\ndx\nds\ndt\ndx\n=\n\u22c5\n=\n\u22c5\n\u22c5\nprovided all the derivatives in the statement exist"
    },
    {
        "Chapter": "1",
        "sentence_range": "2062-2065",
        "Text": "e ,\nf = (w o u) o v If t = v (x) and s = u (t), then\n( o )\ndf\nd w u\ndt\ndw ds\ndt\ndx\ndt\ndx\nds\ndt\ndx\n=\n\u22c5\n=\n\u22c5\n\u22c5\nprovided all the derivatives in the statement exist Reader is invited to formulate chain\nrule for composite of more functions"
    },
    {
        "Chapter": "1",
        "sentence_range": "2063-2066",
        "Text": ",\nf = (w o u) o v If t = v (x) and s = u (t), then\n( o )\ndf\nd w u\ndt\ndw ds\ndt\ndx\ndt\ndx\nds\ndt\ndx\n=\n\u22c5\n=\n\u22c5\n\u22c5\nprovided all the derivatives in the statement exist Reader is invited to formulate chain\nrule for composite of more functions Example 21 Find the derivative of the function given by f(x) = sin (x2)"
    },
    {
        "Chapter": "1",
        "sentence_range": "2064-2067",
        "Text": "If t = v (x) and s = u (t), then\n( o )\ndf\nd w u\ndt\ndw ds\ndt\ndx\ndt\ndx\nds\ndt\ndx\n=\n\u22c5\n=\n\u22c5\n\u22c5\nprovided all the derivatives in the statement exist Reader is invited to formulate chain\nrule for composite of more functions Example 21 Find the derivative of the function given by f(x) = sin (x2) Solution Observe that the given function is a composite of two functions"
    },
    {
        "Chapter": "1",
        "sentence_range": "2065-2068",
        "Text": "Reader is invited to formulate chain\nrule for composite of more functions Example 21 Find the derivative of the function given by f(x) = sin (x2) Solution Observe that the given function is a composite of two functions Indeed, if\nt = u(x) = x2 and v(t) = sin t, then\nf (x) = (v o u) (x) = v(u(x)) = v(x2) = sin x2\nRationalised 2023-24\n MATHEMATICS\n122\nPut t = u(x) = x2"
    },
    {
        "Chapter": "1",
        "sentence_range": "2066-2069",
        "Text": "Example 21 Find the derivative of the function given by f(x) = sin (x2) Solution Observe that the given function is a composite of two functions Indeed, if\nt = u(x) = x2 and v(t) = sin t, then\nf (x) = (v o u) (x) = v(u(x)) = v(x2) = sin x2\nRationalised 2023-24\n MATHEMATICS\n122\nPut t = u(x) = x2 Observe that \ncos\ndv\nt\ndt =\n and \n2\ndt\ndx =x\n exist"
    },
    {
        "Chapter": "1",
        "sentence_range": "2067-2070",
        "Text": "Solution Observe that the given function is a composite of two functions Indeed, if\nt = u(x) = x2 and v(t) = sin t, then\nf (x) = (v o u) (x) = v(u(x)) = v(x2) = sin x2\nRationalised 2023-24\n MATHEMATICS\n122\nPut t = u(x) = x2 Observe that \ncos\ndv\nt\ndt =\n and \n2\ndt\ndx =x\n exist Hence, by chain rule\ndf\ndx  =\ncos\n2\ndv\ndt\nt\nx\ndt\n\u22c5dx\n=\n\u22c5\nIt is normal practice to express the final result only in terms of x"
    },
    {
        "Chapter": "1",
        "sentence_range": "2068-2071",
        "Text": "Indeed, if\nt = u(x) = x2 and v(t) = sin t, then\nf (x) = (v o u) (x) = v(u(x)) = v(x2) = sin x2\nRationalised 2023-24\n MATHEMATICS\n122\nPut t = u(x) = x2 Observe that \ncos\ndv\nt\ndt =\n and \n2\ndt\ndx =x\n exist Hence, by chain rule\ndf\ndx  =\ncos\n2\ndv\ndt\nt\nx\ndt\n\u22c5dx\n=\n\u22c5\nIt is normal practice to express the final result only in terms of x Thus\ndf\ndx  =\n2\ncos\n2\n2 cos\nt\nx\nx\nx\n\u22c5\n=\nEXERCISE 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "2069-2072",
        "Text": "Observe that \ncos\ndv\nt\ndt =\n and \n2\ndt\ndx =x\n exist Hence, by chain rule\ndf\ndx  =\ncos\n2\ndv\ndt\nt\nx\ndt\n\u22c5dx\n=\n\u22c5\nIt is normal practice to express the final result only in terms of x Thus\ndf\ndx  =\n2\ncos\n2\n2 cos\nt\nx\nx\nx\n\u22c5\n=\nEXERCISE 5 2\nDifferentiate the functions with respect to x in Exercises 1 to 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "2070-2073",
        "Text": "Hence, by chain rule\ndf\ndx  =\ncos\n2\ndv\ndt\nt\nx\ndt\n\u22c5dx\n=\n\u22c5\nIt is normal practice to express the final result only in terms of x Thus\ndf\ndx  =\n2\ncos\n2\n2 cos\nt\nx\nx\nx\n\u22c5\n=\nEXERCISE 5 2\nDifferentiate the functions with respect to x in Exercises 1 to 8 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "2071-2074",
        "Text": "Thus\ndf\ndx  =\n2\ncos\n2\n2 cos\nt\nx\nx\nx\n\u22c5\n=\nEXERCISE 5 2\nDifferentiate the functions with respect to x in Exercises 1 to 8 1 sin (x2 + 5)\n2"
    },
    {
        "Chapter": "1",
        "sentence_range": "2072-2075",
        "Text": "2\nDifferentiate the functions with respect to x in Exercises 1 to 8 1 sin (x2 + 5)\n2 cos (sin x)\n3"
    },
    {
        "Chapter": "1",
        "sentence_range": "2073-2076",
        "Text": "1 sin (x2 + 5)\n2 cos (sin x)\n3 sin (ax + b)\n4"
    },
    {
        "Chapter": "1",
        "sentence_range": "2074-2077",
        "Text": "sin (x2 + 5)\n2 cos (sin x)\n3 sin (ax + b)\n4 sec (tan (\nx ))\n5"
    },
    {
        "Chapter": "1",
        "sentence_range": "2075-2078",
        "Text": "cos (sin x)\n3 sin (ax + b)\n4 sec (tan (\nx ))\n5 sin (\n)\ncos (\n)\nax\nb\ncx\nd\n++\n6"
    },
    {
        "Chapter": "1",
        "sentence_range": "2076-2079",
        "Text": "sin (ax + b)\n4 sec (tan (\nx ))\n5 sin (\n)\ncos (\n)\nax\nb\ncx\nd\n++\n6 cos x3"
    },
    {
        "Chapter": "1",
        "sentence_range": "2077-2080",
        "Text": "sec (tan (\nx ))\n5 sin (\n)\ncos (\n)\nax\nb\ncx\nd\n++\n6 cos x3 sin2 (x5)\n7"
    },
    {
        "Chapter": "1",
        "sentence_range": "2078-2081",
        "Text": "sin (\n)\ncos (\n)\nax\nb\ncx\nd\n++\n6 cos x3 sin2 (x5)\n7 (\n2 cot x2)\n8"
    },
    {
        "Chapter": "1",
        "sentence_range": "2079-2082",
        "Text": "cos x3 sin2 (x5)\n7 (\n2 cot x2)\n8 (\n)\ncos\nx\n9"
    },
    {
        "Chapter": "1",
        "sentence_range": "2080-2083",
        "Text": "sin2 (x5)\n7 (\n2 cot x2)\n8 (\n)\ncos\nx\n9 Prove that the function f given by\nf (x) = |x \u2013 1|, x \u2208 R\nis not differentiable at x = 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "2081-2084",
        "Text": "(\n2 cot x2)\n8 (\n)\ncos\nx\n9 Prove that the function f given by\nf (x) = |x \u2013 1|, x \u2208 R\nis not differentiable at x = 1 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "2082-2085",
        "Text": "(\n)\ncos\nx\n9 Prove that the function f given by\nf (x) = |x \u2013 1|, x \u2208 R\nis not differentiable at x = 1 10 Prove that the greatest integer function defined by\nf (x) = [x], 0 < x < 3\nis not differentiable at x = 1 and x = 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "2083-2086",
        "Text": "Prove that the function f given by\nf (x) = |x \u2013 1|, x \u2208 R\nis not differentiable at x = 1 10 Prove that the greatest integer function defined by\nf (x) = [x], 0 < x < 3\nis not differentiable at x = 1 and x = 2 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "2084-2087",
        "Text": "10 Prove that the greatest integer function defined by\nf (x) = [x], 0 < x < 3\nis not differentiable at x = 1 and x = 2 5 3"
    },
    {
        "Chapter": "1",
        "sentence_range": "2085-2088",
        "Text": "Prove that the greatest integer function defined by\nf (x) = [x], 0 < x < 3\nis not differentiable at x = 1 and x = 2 5 3 2  Derivatives of implicit functions\nUntil now we have been differentiating various functions given in the form y = f(x)"
    },
    {
        "Chapter": "1",
        "sentence_range": "2086-2089",
        "Text": "5 3 2  Derivatives of implicit functions\nUntil now we have been differentiating various functions given in the form y = f(x) But it is not necessary that functions are always expressed in this form"
    },
    {
        "Chapter": "1",
        "sentence_range": "2087-2090",
        "Text": "3 2  Derivatives of implicit functions\nUntil now we have been differentiating various functions given in the form y = f(x) But it is not necessary that functions are always expressed in this form For example,\nconsider one of the following relationships between x and y:\nx \u2013 y \u2013 \u03c0 = 0\nx + sin xy \u2013 y = 0\nIn the first case, we can solve for y and rewrite the relationship as y = x \u2013 \u03c0"
    },
    {
        "Chapter": "1",
        "sentence_range": "2088-2091",
        "Text": "2  Derivatives of implicit functions\nUntil now we have been differentiating various functions given in the form y = f(x) But it is not necessary that functions are always expressed in this form For example,\nconsider one of the following relationships between x and y:\nx \u2013 y \u2013 \u03c0 = 0\nx + sin xy \u2013 y = 0\nIn the first case, we can solve for y and rewrite the relationship as y = x \u2013 \u03c0 In\nthe second case, it does not seem that there is an easy way to solve for y"
    },
    {
        "Chapter": "1",
        "sentence_range": "2089-2092",
        "Text": "But it is not necessary that functions are always expressed in this form For example,\nconsider one of the following relationships between x and y:\nx \u2013 y \u2013 \u03c0 = 0\nx + sin xy \u2013 y = 0\nIn the first case, we can solve for y and rewrite the relationship as y = x \u2013 \u03c0 In\nthe second case, it does not seem that there is an easy way to solve for y Nevertheless,\nthere is no doubt about the dependence of y on x in either of the cases"
    },
    {
        "Chapter": "1",
        "sentence_range": "2090-2093",
        "Text": "For example,\nconsider one of the following relationships between x and y:\nx \u2013 y \u2013 \u03c0 = 0\nx + sin xy \u2013 y = 0\nIn the first case, we can solve for y and rewrite the relationship as y = x \u2013 \u03c0 In\nthe second case, it does not seem that there is an easy way to solve for y Nevertheless,\nthere is no doubt about the dependence of y on x in either of the cases When a\nrelationship between x and y is expressed in a way that it is easy to solve for y and\nwrite y = f(x), we say that y is given as an explicit function of x"
    },
    {
        "Chapter": "1",
        "sentence_range": "2091-2094",
        "Text": "In\nthe second case, it does not seem that there is an easy way to solve for y Nevertheless,\nthere is no doubt about the dependence of y on x in either of the cases When a\nrelationship between x and y is expressed in a way that it is easy to solve for y and\nwrite y = f(x), we say that y is given as an explicit function of x In the latter case it\nRationalised 2023-24\nCONTINUITY AND DIFFERENTIABILITY\n123\nis implicit that y is a function of x and we say that the relationship of the second type,\nabove, gives function implicitly"
    },
    {
        "Chapter": "1",
        "sentence_range": "2092-2095",
        "Text": "Nevertheless,\nthere is no doubt about the dependence of y on x in either of the cases When a\nrelationship between x and y is expressed in a way that it is easy to solve for y and\nwrite y = f(x), we say that y is given as an explicit function of x In the latter case it\nRationalised 2023-24\nCONTINUITY AND DIFFERENTIABILITY\n123\nis implicit that y is a function of x and we say that the relationship of the second type,\nabove, gives function implicitly In this subsection, we learn to differentiate implicit\nfunctions"
    },
    {
        "Chapter": "1",
        "sentence_range": "2093-2096",
        "Text": "When a\nrelationship between x and y is expressed in a way that it is easy to solve for y and\nwrite y = f(x), we say that y is given as an explicit function of x In the latter case it\nRationalised 2023-24\nCONTINUITY AND DIFFERENTIABILITY\n123\nis implicit that y is a function of x and we say that the relationship of the second type,\nabove, gives function implicitly In this subsection, we learn to differentiate implicit\nfunctions Example 22 Find dy\ndx  if x \u2013 y = \u03c0"
    },
    {
        "Chapter": "1",
        "sentence_range": "2094-2097",
        "Text": "In the latter case it\nRationalised 2023-24\nCONTINUITY AND DIFFERENTIABILITY\n123\nis implicit that y is a function of x and we say that the relationship of the second type,\nabove, gives function implicitly In this subsection, we learn to differentiate implicit\nfunctions Example 22 Find dy\ndx  if x \u2013 y = \u03c0 Solution One way is to solve for y and rewrite the above as\ny = x \u2013 \u03c0\nBut then\ndy\ndx  = 1\nAlternatively, directly differentiating the relationship w"
    },
    {
        "Chapter": "1",
        "sentence_range": "2095-2098",
        "Text": "In this subsection, we learn to differentiate implicit\nfunctions Example 22 Find dy\ndx  if x \u2013 y = \u03c0 Solution One way is to solve for y and rewrite the above as\ny = x \u2013 \u03c0\nBut then\ndy\ndx  = 1\nAlternatively, directly differentiating the relationship w r"
    },
    {
        "Chapter": "1",
        "sentence_range": "2096-2099",
        "Text": "Example 22 Find dy\ndx  if x \u2013 y = \u03c0 Solution One way is to solve for y and rewrite the above as\ny = x \u2013 \u03c0\nBut then\ndy\ndx  = 1\nAlternatively, directly differentiating the relationship w r t"
    },
    {
        "Chapter": "1",
        "sentence_range": "2097-2100",
        "Text": "Solution One way is to solve for y and rewrite the above as\ny = x \u2013 \u03c0\nBut then\ndy\ndx  = 1\nAlternatively, directly differentiating the relationship w r t , x, we have\n(\n)\nd\nx\ny\ndx\n\u2212\n =  d\ndx\n\u03c0\nRecall that d\ndx\n\u03c0  means to differentiate the constant function taking value \u03c0\neverywhere w"
    },
    {
        "Chapter": "1",
        "sentence_range": "2098-2101",
        "Text": "r t , x, we have\n(\n)\nd\nx\ny\ndx\n\u2212\n =  d\ndx\n\u03c0\nRecall that d\ndx\n\u03c0  means to differentiate the constant function taking value \u03c0\neverywhere w r"
    },
    {
        "Chapter": "1",
        "sentence_range": "2099-2102",
        "Text": "t , x, we have\n(\n)\nd\nx\ny\ndx\n\u2212\n =  d\ndx\n\u03c0\nRecall that d\ndx\n\u03c0  means to differentiate the constant function taking value \u03c0\neverywhere w r t"
    },
    {
        "Chapter": "1",
        "sentence_range": "2100-2103",
        "Text": ", x, we have\n(\n)\nd\nx\ny\ndx\n\u2212\n =  d\ndx\n\u03c0\nRecall that d\ndx\n\u03c0  means to differentiate the constant function taking value \u03c0\neverywhere w r t , x"
    },
    {
        "Chapter": "1",
        "sentence_range": "2101-2104",
        "Text": "r t , x Thus\n( )\n( )\nd\nd\nx\ny\ndx\n\u2212dx\n = 0\nwhich implies that\ndy\ndx  =\n1\ndx\ndx =\nExample 23 Find dy\ndx\n, if y + sin y = cos x"
    },
    {
        "Chapter": "1",
        "sentence_range": "2102-2105",
        "Text": "t , x Thus\n( )\n( )\nd\nd\nx\ny\ndx\n\u2212dx\n = 0\nwhich implies that\ndy\ndx  =\n1\ndx\ndx =\nExample 23 Find dy\ndx\n, if y + sin y = cos x Solution We differentiate the relationship directly with respect to x, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "2103-2106",
        "Text": ", x Thus\n( )\n( )\nd\nd\nx\ny\ndx\n\u2212dx\n = 0\nwhich implies that\ndy\ndx  =\n1\ndx\ndx =\nExample 23 Find dy\ndx\n, if y + sin y = cos x Solution We differentiate the relationship directly with respect to x, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "2104-2107",
        "Text": "Thus\n( )\n( )\nd\nd\nx\ny\ndx\n\u2212dx\n = 0\nwhich implies that\ndy\ndx  =\n1\ndx\ndx =\nExample 23 Find dy\ndx\n, if y + sin y = cos x Solution We differentiate the relationship directly with respect to x, i e ,\n(sin )\ndy\nd\ny\ndx\n+dx\n =\nd(cos )\nx\ndx\nwhich implies using chain rule\ncos\ndy\nydy\ndx\ndx\n+\n\u22c5\n = \u2013 sin x\nThis gives\ndy\ndx  =\nsin\n1\ncos\nx\ny\n\u2212 +\nwhere\ny \u2260 (2n + 1) \u03c0\nRationalised 2023-24\n MATHEMATICS\n124\n5"
    },
    {
        "Chapter": "1",
        "sentence_range": "2105-2108",
        "Text": "Solution We differentiate the relationship directly with respect to x, i e ,\n(sin )\ndy\nd\ny\ndx\n+dx\n =\nd(cos )\nx\ndx\nwhich implies using chain rule\ncos\ndy\nydy\ndx\ndx\n+\n\u22c5\n = \u2013 sin x\nThis gives\ndy\ndx  =\nsin\n1\ncos\nx\ny\n\u2212 +\nwhere\ny \u2260 (2n + 1) \u03c0\nRationalised 2023-24\n MATHEMATICS\n124\n5 3"
    },
    {
        "Chapter": "1",
        "sentence_range": "2106-2109",
        "Text": "e ,\n(sin )\ndy\nd\ny\ndx\n+dx\n =\nd(cos )\nx\ndx\nwhich implies using chain rule\ncos\ndy\nydy\ndx\ndx\n+\n\u22c5\n = \u2013 sin x\nThis gives\ndy\ndx  =\nsin\n1\ncos\nx\ny\n\u2212 +\nwhere\ny \u2260 (2n + 1) \u03c0\nRationalised 2023-24\n MATHEMATICS\n124\n5 3 3  Derivatives of inverse trigonometric functions\nWe remark that inverse trigonometric functions are continuous functions, but we will\nnot prove this"
    },
    {
        "Chapter": "1",
        "sentence_range": "2107-2110",
        "Text": ",\n(sin )\ndy\nd\ny\ndx\n+dx\n =\nd(cos )\nx\ndx\nwhich implies using chain rule\ncos\ndy\nydy\ndx\ndx\n+\n\u22c5\n = \u2013 sin x\nThis gives\ndy\ndx  =\nsin\n1\ncos\nx\ny\n\u2212 +\nwhere\ny \u2260 (2n + 1) \u03c0\nRationalised 2023-24\n MATHEMATICS\n124\n5 3 3  Derivatives of inverse trigonometric functions\nWe remark that inverse trigonometric functions are continuous functions, but we will\nnot prove this Now we use chain rule to find derivatives of these functions"
    },
    {
        "Chapter": "1",
        "sentence_range": "2108-2111",
        "Text": "3 3  Derivatives of inverse trigonometric functions\nWe remark that inverse trigonometric functions are continuous functions, but we will\nnot prove this Now we use chain rule to find derivatives of these functions Example 24 Find the derivative of f given by f(x) = sin\u20131 x assuming it exists"
    },
    {
        "Chapter": "1",
        "sentence_range": "2109-2112",
        "Text": "3  Derivatives of inverse trigonometric functions\nWe remark that inverse trigonometric functions are continuous functions, but we will\nnot prove this Now we use chain rule to find derivatives of these functions Example 24 Find the derivative of f given by f(x) = sin\u20131 x assuming it exists Solution Let y = sin\u20131 x"
    },
    {
        "Chapter": "1",
        "sentence_range": "2110-2113",
        "Text": "Now we use chain rule to find derivatives of these functions Example 24 Find the derivative of f given by f(x) = sin\u20131 x assuming it exists Solution Let y = sin\u20131 x Then, x = sin y"
    },
    {
        "Chapter": "1",
        "sentence_range": "2111-2114",
        "Text": "Example 24 Find the derivative of f given by f(x) = sin\u20131 x assuming it exists Solution Let y = sin\u20131 x Then, x = sin y Differentiating both sides w"
    },
    {
        "Chapter": "1",
        "sentence_range": "2112-2115",
        "Text": "Solution Let y = sin\u20131 x Then, x = sin y Differentiating both sides w r"
    },
    {
        "Chapter": "1",
        "sentence_range": "2113-2116",
        "Text": "Then, x = sin y Differentiating both sides w r t"
    },
    {
        "Chapter": "1",
        "sentence_range": "2114-2117",
        "Text": "Differentiating both sides w r t x, we get\n1 = cos y dy\ndx\nwhich implies that\ndy\ndx  =\n1\n1\n1\ncos\ncos(sin\n)\ny\nx\n\u2212\n=\nObserve that this is defined only for cos y \u2260 0, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "2115-2118",
        "Text": "r t x, we get\n1 = cos y dy\ndx\nwhich implies that\ndy\ndx  =\n1\n1\n1\ncos\ncos(sin\n)\ny\nx\n\u2212\n=\nObserve that this is defined only for cos y \u2260 0, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "2116-2119",
        "Text": "t x, we get\n1 = cos y dy\ndx\nwhich implies that\ndy\ndx  =\n1\n1\n1\ncos\ncos(sin\n)\ny\nx\n\u2212\n=\nObserve that this is defined only for cos y \u2260 0, i e , sin\u20131 x \u2260 \n,\n2 2\n\u2212\u03c0 \u03c0\n, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "2117-2120",
        "Text": "x, we get\n1 = cos y dy\ndx\nwhich implies that\ndy\ndx  =\n1\n1\n1\ncos\ncos(sin\n)\ny\nx\n\u2212\n=\nObserve that this is defined only for cos y \u2260 0, i e , sin\u20131 x \u2260 \n,\n2 2\n\u2212\u03c0 \u03c0\n, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "2118-2121",
        "Text": "e , sin\u20131 x \u2260 \n,\n2 2\n\u2212\u03c0 \u03c0\n, i e , x \u2260 \u2013 1, 1,\ni"
    },
    {
        "Chapter": "1",
        "sentence_range": "2119-2122",
        "Text": ", sin\u20131 x \u2260 \n,\n2 2\n\u2212\u03c0 \u03c0\n, i e , x \u2260 \u2013 1, 1,\ni e"
    },
    {
        "Chapter": "1",
        "sentence_range": "2120-2123",
        "Text": "e , x \u2260 \u2013 1, 1,\ni e , x \u2208 (\u2013 1, 1)"
    },
    {
        "Chapter": "1",
        "sentence_range": "2121-2124",
        "Text": ", x \u2260 \u2013 1, 1,\ni e , x \u2208 (\u2013 1, 1) To make this result a bit more attractive, we carry out the following manipulation"
    },
    {
        "Chapter": "1",
        "sentence_range": "2122-2125",
        "Text": "e , x \u2208 (\u2013 1, 1) To make this result a bit more attractive, we carry out the following manipulation Recall that for x \u2208 (\u2013 1, 1), sin (sin\u20131 x) = x and hence\ncos2 y = 1 \u2013 (sin y)2 = 1 \u2013 (sin (sin\u20131 x))2 = 1 \u2013 x2\nAlso, since y \u2208 \n,\n2 2\n\u03c0 \u03c0\n\uf8eb\n\uf8f6\n\uf8ec\u2212\n\uf8f7\n\uf8ed\n\uf8f8, cos y is positive and hence cos y = \n2\n1\nx\n\u2212\nThus, for x \u2208 (\u2013 1, 1),\n2\n1\n1\ncos\n1\ndy\ndx\ny\nx\n=\n=\n\u2212\n2\n1\n1\nx\n\u2212\n2\n1\n1\nx\n\u2212\n\u2212\n2\n1\n1\nx\n+\nf(x)                    sin\u20131 x                      cos-1 x                   tan-1x\nDomain off               (-1, 1)                     (-1, 1)                        R\nf 1(x)\nRationalised 2023-24\nCONTINUITY AND DIFFERENTIABILITY\n125\nEXERCISE 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "2123-2126",
        "Text": ", x \u2208 (\u2013 1, 1) To make this result a bit more attractive, we carry out the following manipulation Recall that for x \u2208 (\u2013 1, 1), sin (sin\u20131 x) = x and hence\ncos2 y = 1 \u2013 (sin y)2 = 1 \u2013 (sin (sin\u20131 x))2 = 1 \u2013 x2\nAlso, since y \u2208 \n,\n2 2\n\u03c0 \u03c0\n\uf8eb\n\uf8f6\n\uf8ec\u2212\n\uf8f7\n\uf8ed\n\uf8f8, cos y is positive and hence cos y = \n2\n1\nx\n\u2212\nThus, for x \u2208 (\u2013 1, 1),\n2\n1\n1\ncos\n1\ndy\ndx\ny\nx\n=\n=\n\u2212\n2\n1\n1\nx\n\u2212\n2\n1\n1\nx\n\u2212\n\u2212\n2\n1\n1\nx\n+\nf(x)                    sin\u20131 x                      cos-1 x                   tan-1x\nDomain off               (-1, 1)                     (-1, 1)                        R\nf 1(x)\nRationalised 2023-24\nCONTINUITY AND DIFFERENTIABILITY\n125\nEXERCISE 5 3\nFind dy\ndx  in the following:\n1"
    },
    {
        "Chapter": "1",
        "sentence_range": "2124-2127",
        "Text": "To make this result a bit more attractive, we carry out the following manipulation Recall that for x \u2208 (\u2013 1, 1), sin (sin\u20131 x) = x and hence\ncos2 y = 1 \u2013 (sin y)2 = 1 \u2013 (sin (sin\u20131 x))2 = 1 \u2013 x2\nAlso, since y \u2208 \n,\n2 2\n\u03c0 \u03c0\n\uf8eb\n\uf8f6\n\uf8ec\u2212\n\uf8f7\n\uf8ed\n\uf8f8, cos y is positive and hence cos y = \n2\n1\nx\n\u2212\nThus, for x \u2208 (\u2013 1, 1),\n2\n1\n1\ncos\n1\ndy\ndx\ny\nx\n=\n=\n\u2212\n2\n1\n1\nx\n\u2212\n2\n1\n1\nx\n\u2212\n\u2212\n2\n1\n1\nx\n+\nf(x)                    sin\u20131 x                      cos-1 x                   tan-1x\nDomain off               (-1, 1)                     (-1, 1)                        R\nf 1(x)\nRationalised 2023-24\nCONTINUITY AND DIFFERENTIABILITY\n125\nEXERCISE 5 3\nFind dy\ndx  in the following:\n1 2x + 3y = sin x\n2"
    },
    {
        "Chapter": "1",
        "sentence_range": "2125-2128",
        "Text": "Recall that for x \u2208 (\u2013 1, 1), sin (sin\u20131 x) = x and hence\ncos2 y = 1 \u2013 (sin y)2 = 1 \u2013 (sin (sin\u20131 x))2 = 1 \u2013 x2\nAlso, since y \u2208 \n,\n2 2\n\u03c0 \u03c0\n\uf8eb\n\uf8f6\n\uf8ec\u2212\n\uf8f7\n\uf8ed\n\uf8f8, cos y is positive and hence cos y = \n2\n1\nx\n\u2212\nThus, for x \u2208 (\u2013 1, 1),\n2\n1\n1\ncos\n1\ndy\ndx\ny\nx\n=\n=\n\u2212\n2\n1\n1\nx\n\u2212\n2\n1\n1\nx\n\u2212\n\u2212\n2\n1\n1\nx\n+\nf(x)                    sin\u20131 x                      cos-1 x                   tan-1x\nDomain off               (-1, 1)                     (-1, 1)                        R\nf 1(x)\nRationalised 2023-24\nCONTINUITY AND DIFFERENTIABILITY\n125\nEXERCISE 5 3\nFind dy\ndx  in the following:\n1 2x + 3y = sin x\n2 2x + 3y = sin y\n3"
    },
    {
        "Chapter": "1",
        "sentence_range": "2126-2129",
        "Text": "3\nFind dy\ndx  in the following:\n1 2x + 3y = sin x\n2 2x + 3y = sin y\n3 ax + by2 = cos y\n4"
    },
    {
        "Chapter": "1",
        "sentence_range": "2127-2130",
        "Text": "2x + 3y = sin x\n2 2x + 3y = sin y\n3 ax + by2 = cos y\n4 xy + y2 = tan x + y\n5"
    },
    {
        "Chapter": "1",
        "sentence_range": "2128-2131",
        "Text": "2x + 3y = sin y\n3 ax + by2 = cos y\n4 xy + y2 = tan x + y\n5 x2 + xy + y2 = 100\n6"
    },
    {
        "Chapter": "1",
        "sentence_range": "2129-2132",
        "Text": "ax + by2 = cos y\n4 xy + y2 = tan x + y\n5 x2 + xy + y2 = 100\n6 x3 + x2y + xy2 + y3 = 81\n7"
    },
    {
        "Chapter": "1",
        "sentence_range": "2130-2133",
        "Text": "xy + y2 = tan x + y\n5 x2 + xy + y2 = 100\n6 x3 + x2y + xy2 + y3 = 81\n7 sin2 y + cos xy = \u03ba\n8"
    },
    {
        "Chapter": "1",
        "sentence_range": "2131-2134",
        "Text": "x2 + xy + y2 = 100\n6 x3 + x2y + xy2 + y3 = 81\n7 sin2 y + cos xy = \u03ba\n8 sin2 x + cos2 y = 1\n9"
    },
    {
        "Chapter": "1",
        "sentence_range": "2132-2135",
        "Text": "x3 + x2y + xy2 + y3 = 81\n7 sin2 y + cos xy = \u03ba\n8 sin2 x + cos2 y = 1\n9 y = sin\u20131 \n2\n2\n1\nx\nx\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed+\n\uf8f8\n10"
    },
    {
        "Chapter": "1",
        "sentence_range": "2133-2136",
        "Text": "sin2 y + cos xy = \u03ba\n8 sin2 x + cos2 y = 1\n9 y = sin\u20131 \n2\n2\n1\nx\nx\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed+\n\uf8f8\n10 y = tan\u20131\n3\n2\n3\n,\n1\n3\nx\nx\nx\n\uf8eb\n\uf8f6\n\u2212\n\uf8ec\n\uf8f7\n\uf8ed\u2212\n\uf8f8\n \n1\n1\n3\n3\nx\n\u2212\n<\n<\n11"
    },
    {
        "Chapter": "1",
        "sentence_range": "2134-2137",
        "Text": "sin2 x + cos2 y = 1\n9 y = sin\u20131 \n2\n2\n1\nx\nx\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed+\n\uf8f8\n10 y = tan\u20131\n3\n2\n3\n,\n1\n3\nx\nx\nx\n\uf8eb\n\uf8f6\n\u2212\n\uf8ec\n\uf8f7\n\uf8ed\u2212\n\uf8f8\n \n1\n1\n3\n3\nx\n\u2212\n<\n<\n11 2\n1\n2\n1\n,\ncos\n0\n1\n1\nx\ny\nx\nx\n\u2212 \uf8eb\n\uf8f6\n\u2212\n=\n<\n<\n\uf8ec\n\uf8f7\n\uf8ed+\n\uf8f8\n12"
    },
    {
        "Chapter": "1",
        "sentence_range": "2135-2138",
        "Text": "y = sin\u20131 \n2\n2\n1\nx\nx\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed+\n\uf8f8\n10 y = tan\u20131\n3\n2\n3\n,\n1\n3\nx\nx\nx\n\uf8eb\n\uf8f6\n\u2212\n\uf8ec\n\uf8f7\n\uf8ed\u2212\n\uf8f8\n \n1\n1\n3\n3\nx\n\u2212\n<\n<\n11 2\n1\n2\n1\n,\ncos\n0\n1\n1\nx\ny\nx\nx\n\u2212 \uf8eb\n\uf8f6\n\u2212\n=\n<\n<\n\uf8ec\n\uf8f7\n\uf8ed+\n\uf8f8\n12 2\n1\n2\n1\n,\nsin\n0\n1\n1\nx\ny\nx\nx\n\u2212 \uf8eb\n\uf8f6\n\u2212\n=\n<\n<\n\uf8ec\n\uf8f7\n\uf8ed+\n\uf8f8\n13"
    },
    {
        "Chapter": "1",
        "sentence_range": "2136-2139",
        "Text": "y = tan\u20131\n3\n2\n3\n,\n1\n3\nx\nx\nx\n\uf8eb\n\uf8f6\n\u2212\n\uf8ec\n\uf8f7\n\uf8ed\u2212\n\uf8f8\n \n1\n1\n3\n3\nx\n\u2212\n<\n<\n11 2\n1\n2\n1\n,\ncos\n0\n1\n1\nx\ny\nx\nx\n\u2212 \uf8eb\n\uf8f6\n\u2212\n=\n<\n<\n\uf8ec\n\uf8f7\n\uf8ed+\n\uf8f8\n12 2\n1\n2\n1\n,\nsin\n0\n1\n1\nx\ny\nx\nx\n\u2212 \uf8eb\n\uf8f6\n\u2212\n=\n<\n<\n\uf8ec\n\uf8f7\n\uf8ed+\n\uf8f8\n13 1\n22\n,\ncos\n1\n1\n1\nx\ny\nx\nx\n\u2212 \uf8eb\n\uf8f6\n=\n\u2212 <\n<\n\uf8ec\n\uf8f7\n\uf8ed+\n\uf8f8\n14"
    },
    {
        "Chapter": "1",
        "sentence_range": "2137-2140",
        "Text": "2\n1\n2\n1\n,\ncos\n0\n1\n1\nx\ny\nx\nx\n\u2212 \uf8eb\n\uf8f6\n\u2212\n=\n<\n<\n\uf8ec\n\uf8f7\n\uf8ed+\n\uf8f8\n12 2\n1\n2\n1\n,\nsin\n0\n1\n1\nx\ny\nx\nx\n\u2212 \uf8eb\n\uf8f6\n\u2212\n=\n<\n<\n\uf8ec\n\uf8f7\n\uf8ed+\n\uf8f8\n13 1\n22\n,\ncos\n1\n1\n1\nx\ny\nx\nx\n\u2212 \uf8eb\n\uf8f6\n=\n\u2212 <\n<\n\uf8ec\n\uf8f7\n\uf8ed+\n\uf8f8\n14 (\n)\n1\n2\n1\n1\n,\nsin\n2\n1\n2\n2\ny\nx\nx\nx\n\u2212\n=\n\u2212\n\u2212\n<\n<\n15"
    },
    {
        "Chapter": "1",
        "sentence_range": "2138-2141",
        "Text": "2\n1\n2\n1\n,\nsin\n0\n1\n1\nx\ny\nx\nx\n\u2212 \uf8eb\n\uf8f6\n\u2212\n=\n<\n<\n\uf8ec\n\uf8f7\n\uf8ed+\n\uf8f8\n13 1\n22\n,\ncos\n1\n1\n1\nx\ny\nx\nx\n\u2212 \uf8eb\n\uf8f6\n=\n\u2212 <\n<\n\uf8ec\n\uf8f7\n\uf8ed+\n\uf8f8\n14 (\n)\n1\n2\n1\n1\n,\nsin\n2\n1\n2\n2\ny\nx\nx\nx\n\u2212\n=\n\u2212\n\u2212\n<\n<\n15 1\n12\n1\n,\nsec\n0\n2\n1\n2\ny\nx\nx\n\u2212 \uf8eb\n\uf8f6\n=\n<\n<\n\uf8ec\n\u2212\uf8f7\n\uf8ed\n\uf8f8\n5"
    },
    {
        "Chapter": "1",
        "sentence_range": "2139-2142",
        "Text": "1\n22\n,\ncos\n1\n1\n1\nx\ny\nx\nx\n\u2212 \uf8eb\n\uf8f6\n=\n\u2212 <\n<\n\uf8ec\n\uf8f7\n\uf8ed+\n\uf8f8\n14 (\n)\n1\n2\n1\n1\n,\nsin\n2\n1\n2\n2\ny\nx\nx\nx\n\u2212\n=\n\u2212\n\u2212\n<\n<\n15 1\n12\n1\n,\nsec\n0\n2\n1\n2\ny\nx\nx\n\u2212 \uf8eb\n\uf8f6\n=\n<\n<\n\uf8ec\n\u2212\uf8f7\n\uf8ed\n\uf8f8\n5 4  Exponential and Logarithmic Functions\nTill now we have learnt some aspects of different classes of functions like polynomial\nfunctions, rational functions and trigonometric functions"
    },
    {
        "Chapter": "1",
        "sentence_range": "2140-2143",
        "Text": "(\n)\n1\n2\n1\n1\n,\nsin\n2\n1\n2\n2\ny\nx\nx\nx\n\u2212\n=\n\u2212\n\u2212\n<\n<\n15 1\n12\n1\n,\nsec\n0\n2\n1\n2\ny\nx\nx\n\u2212 \uf8eb\n\uf8f6\n=\n<\n<\n\uf8ec\n\u2212\uf8f7\n\uf8ed\n\uf8f8\n5 4  Exponential and Logarithmic Functions\nTill now we have learnt some aspects of different classes of functions like polynomial\nfunctions, rational functions and trigonometric functions In this section, we shall\nlearn about a new class of (related) functions called exponential functions and logarithmic\nfunctions"
    },
    {
        "Chapter": "1",
        "sentence_range": "2141-2144",
        "Text": "1\n12\n1\n,\nsec\n0\n2\n1\n2\ny\nx\nx\n\u2212 \uf8eb\n\uf8f6\n=\n<\n<\n\uf8ec\n\u2212\uf8f7\n\uf8ed\n\uf8f8\n5 4  Exponential and Logarithmic Functions\nTill now we have learnt some aspects of different classes of functions like polynomial\nfunctions, rational functions and trigonometric functions In this section, we shall\nlearn about a new class of (related) functions called exponential functions and logarithmic\nfunctions It needs to be emphasized that many statements made in this section are\nmotivational and precise proofs of these are well beyond the scope of this text"
    },
    {
        "Chapter": "1",
        "sentence_range": "2142-2145",
        "Text": "4  Exponential and Logarithmic Functions\nTill now we have learnt some aspects of different classes of functions like polynomial\nfunctions, rational functions and trigonometric functions In this section, we shall\nlearn about a new class of (related) functions called exponential functions and logarithmic\nfunctions It needs to be emphasized that many statements made in this section are\nmotivational and precise proofs of these are well beyond the scope of this text The Fig 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "2143-2146",
        "Text": "In this section, we shall\nlearn about a new class of (related) functions called exponential functions and logarithmic\nfunctions It needs to be emphasized that many statements made in this section are\nmotivational and precise proofs of these are well beyond the scope of this text The Fig 5 9 gives a sketch of y = f1(x) = x, y = f2(x) = x2, y = f3(x) = x3 and y = f4(x)\n= x4"
    },
    {
        "Chapter": "1",
        "sentence_range": "2144-2147",
        "Text": "It needs to be emphasized that many statements made in this section are\nmotivational and precise proofs of these are well beyond the scope of this text The Fig 5 9 gives a sketch of y = f1(x) = x, y = f2(x) = x2, y = f3(x) = x3 and y = f4(x)\n= x4 Observe that the curves get steeper as the power of x increases"
    },
    {
        "Chapter": "1",
        "sentence_range": "2145-2148",
        "Text": "The Fig 5 9 gives a sketch of y = f1(x) = x, y = f2(x) = x2, y = f3(x) = x3 and y = f4(x)\n= x4 Observe that the curves get steeper as the power of x increases Steeper the\ncurve, faster is the rate of growth"
    },
    {
        "Chapter": "1",
        "sentence_range": "2146-2149",
        "Text": "9 gives a sketch of y = f1(x) = x, y = f2(x) = x2, y = f3(x) = x3 and y = f4(x)\n= x4 Observe that the curves get steeper as the power of x increases Steeper the\ncurve, faster is the rate of growth What this means is that for a fixed increment in the\nvalue of x (> 1), the increment in the value of y = fn (x) increases as n increases for n\n= 1, 2, 3, 4"
    },
    {
        "Chapter": "1",
        "sentence_range": "2147-2150",
        "Text": "Observe that the curves get steeper as the power of x increases Steeper the\ncurve, faster is the rate of growth What this means is that for a fixed increment in the\nvalue of x (> 1), the increment in the value of y = fn (x) increases as n increases for n\n= 1, 2, 3, 4 It is conceivable that such a statement is true for all positive values of n,\nRationalised 2023-24\n MATHEMATICS\n126\nwhere fn (x) = xn"
    },
    {
        "Chapter": "1",
        "sentence_range": "2148-2151",
        "Text": "Steeper the\ncurve, faster is the rate of growth What this means is that for a fixed increment in the\nvalue of x (> 1), the increment in the value of y = fn (x) increases as n increases for n\n= 1, 2, 3, 4 It is conceivable that such a statement is true for all positive values of n,\nRationalised 2023-24\n MATHEMATICS\n126\nwhere fn (x) = xn Essentially, this means\nthat the graph of y = fn (x) leans more\ntowards the y-axis as n increases"
    },
    {
        "Chapter": "1",
        "sentence_range": "2149-2152",
        "Text": "What this means is that for a fixed increment in the\nvalue of x (> 1), the increment in the value of y = fn (x) increases as n increases for n\n= 1, 2, 3, 4 It is conceivable that such a statement is true for all positive values of n,\nRationalised 2023-24\n MATHEMATICS\n126\nwhere fn (x) = xn Essentially, this means\nthat the graph of y = fn (x) leans more\ntowards the y-axis as n increases For\nexample, consider f10(x) = x10 and f15(x)\n= x15"
    },
    {
        "Chapter": "1",
        "sentence_range": "2150-2153",
        "Text": "It is conceivable that such a statement is true for all positive values of n,\nRationalised 2023-24\n MATHEMATICS\n126\nwhere fn (x) = xn Essentially, this means\nthat the graph of y = fn (x) leans more\ntowards the y-axis as n increases For\nexample, consider f10(x) = x10 and f15(x)\n= x15 If x increases from 1 to 2, f10\nincreases from 1 to 210 whereas f15\nincreases from 1 to 215"
    },
    {
        "Chapter": "1",
        "sentence_range": "2151-2154",
        "Text": "Essentially, this means\nthat the graph of y = fn (x) leans more\ntowards the y-axis as n increases For\nexample, consider f10(x) = x10 and f15(x)\n= x15 If x increases from 1 to 2, f10\nincreases from 1 to 210 whereas f15\nincreases from 1 to 215 Thus, for the same\nincrement in x, f15 grow faster than f10"
    },
    {
        "Chapter": "1",
        "sentence_range": "2152-2155",
        "Text": "For\nexample, consider f10(x) = x10 and f15(x)\n= x15 If x increases from 1 to 2, f10\nincreases from 1 to 210 whereas f15\nincreases from 1 to 215 Thus, for the same\nincrement in x, f15 grow faster than f10 Upshot of the above discussion is that\nthe growth of polynomial functions is\ndependent  on the degree of the polynomial\nfunction \u2013 higher the degree, greater is\nthe growth"
    },
    {
        "Chapter": "1",
        "sentence_range": "2153-2156",
        "Text": "If x increases from 1 to 2, f10\nincreases from 1 to 210 whereas f15\nincreases from 1 to 215 Thus, for the same\nincrement in x, f15 grow faster than f10 Upshot of the above discussion is that\nthe growth of polynomial functions is\ndependent  on the degree of the polynomial\nfunction \u2013 higher the degree, greater is\nthe growth The next natural question is:\nIs there a function which grows faster than any polynomial function"
    },
    {
        "Chapter": "1",
        "sentence_range": "2154-2157",
        "Text": "Thus, for the same\nincrement in x, f15 grow faster than f10 Upshot of the above discussion is that\nthe growth of polynomial functions is\ndependent  on the degree of the polynomial\nfunction \u2013 higher the degree, greater is\nthe growth The next natural question is:\nIs there a function which grows faster than any polynomial function The answer is in\naffirmative and an example of such a function is\ny  = f(x) = 10x"
    },
    {
        "Chapter": "1",
        "sentence_range": "2155-2158",
        "Text": "Upshot of the above discussion is that\nthe growth of polynomial functions is\ndependent  on the degree of the polynomial\nfunction \u2013 higher the degree, greater is\nthe growth The next natural question is:\nIs there a function which grows faster than any polynomial function The answer is in\naffirmative and an example of such a function is\ny  = f(x) = 10x Our claim is that this function f grows faster than fn (x) = xn for any positive integer n"
    },
    {
        "Chapter": "1",
        "sentence_range": "2156-2159",
        "Text": "The next natural question is:\nIs there a function which grows faster than any polynomial function The answer is in\naffirmative and an example of such a function is\ny  = f(x) = 10x Our claim is that this function f grows faster than fn (x) = xn for any positive integer n For example, we can prove that 10x grows faster than f100 (x) = x100"
    },
    {
        "Chapter": "1",
        "sentence_range": "2157-2160",
        "Text": "The answer is in\naffirmative and an example of such a function is\ny  = f(x) = 10x Our claim is that this function f grows faster than fn (x) = xn for any positive integer n For example, we can prove that 10x grows faster than f100 (x) = x100 For large values\nof x like x = 103, note that f100 (x) = (103)100 = 10300 whereas f(103) = \n10103\n= 101000"
    },
    {
        "Chapter": "1",
        "sentence_range": "2158-2161",
        "Text": "Our claim is that this function f grows faster than fn (x) = xn for any positive integer n For example, we can prove that 10x grows faster than f100 (x) = x100 For large values\nof x like x = 103, note that f100 (x) = (103)100 = 10300 whereas f(103) = \n10103\n= 101000 Clearly f (x) is much greater than f100 (x)"
    },
    {
        "Chapter": "1",
        "sentence_range": "2159-2162",
        "Text": "For example, we can prove that 10x grows faster than f100 (x) = x100 For large values\nof x like x = 103, note that f100 (x) = (103)100 = 10300 whereas f(103) = \n10103\n= 101000 Clearly f (x) is much greater than f100 (x) It is not difficult to prove that for all\nx > 103, f (x) > f100 (x)"
    },
    {
        "Chapter": "1",
        "sentence_range": "2160-2163",
        "Text": "For large values\nof x like x = 103, note that f100 (x) = (103)100 = 10300 whereas f(103) = \n10103\n= 101000 Clearly f (x) is much greater than f100 (x) It is not difficult to prove that for all\nx > 103, f (x) > f100 (x) But we will not attempt to give a proof of this here"
    },
    {
        "Chapter": "1",
        "sentence_range": "2161-2164",
        "Text": "Clearly f (x) is much greater than f100 (x) It is not difficult to prove that for all\nx > 103, f (x) > f100 (x) But we will not attempt to give a proof of this here Similarly, by\nchoosing large values of x, one can verify that f(x) grows faster than fn (x) for any\npositive integer n"
    },
    {
        "Chapter": "1",
        "sentence_range": "2162-2165",
        "Text": "It is not difficult to prove that for all\nx > 103, f (x) > f100 (x) But we will not attempt to give a proof of this here Similarly, by\nchoosing large values of x, one can verify that f(x) grows faster than fn (x) for any\npositive integer n Definition 3 The exponential function with positive base b > 1 is the function\ny = f (x) = bx\nThe graph of y = 10x is given in the Fig 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "2163-2166",
        "Text": "But we will not attempt to give a proof of this here Similarly, by\nchoosing large values of x, one can verify that f(x) grows faster than fn (x) for any\npositive integer n Definition 3 The exponential function with positive base b > 1 is the function\ny = f (x) = bx\nThe graph of y = 10x is given in the Fig 5 9"
    },
    {
        "Chapter": "1",
        "sentence_range": "2164-2167",
        "Text": "Similarly, by\nchoosing large values of x, one can verify that f(x) grows faster than fn (x) for any\npositive integer n Definition 3 The exponential function with positive base b > 1 is the function\ny = f (x) = bx\nThe graph of y = 10x is given in the Fig 5 9 It is advised that the reader plots this graph for particular values of b like 2, 3 and 4"
    },
    {
        "Chapter": "1",
        "sentence_range": "2165-2168",
        "Text": "Definition 3 The exponential function with positive base b > 1 is the function\ny = f (x) = bx\nThe graph of y = 10x is given in the Fig 5 9 It is advised that the reader plots this graph for particular values of b like 2, 3 and 4 Following are some of the salient features of the exponential functions:\n(1)\nDomain of the exponential function is R, the set of all real numbers"
    },
    {
        "Chapter": "1",
        "sentence_range": "2166-2169",
        "Text": "9 It is advised that the reader plots this graph for particular values of b like 2, 3 and 4 Following are some of the salient features of the exponential functions:\n(1)\nDomain of the exponential function is R, the set of all real numbers (2)\nRange of the exponential function is the set of all positive real numbers"
    },
    {
        "Chapter": "1",
        "sentence_range": "2167-2170",
        "Text": "It is advised that the reader plots this graph for particular values of b like 2, 3 and 4 Following are some of the salient features of the exponential functions:\n(1)\nDomain of the exponential function is R, the set of all real numbers (2)\nRange of the exponential function is the set of all positive real numbers (3)\nThe point (0, 1) is always on the graph of the exponential function (this is a\nrestatement of the fact that b0 = 1 for any real b > 1)"
    },
    {
        "Chapter": "1",
        "sentence_range": "2168-2171",
        "Text": "Following are some of the salient features of the exponential functions:\n(1)\nDomain of the exponential function is R, the set of all real numbers (2)\nRange of the exponential function is the set of all positive real numbers (3)\nThe point (0, 1) is always on the graph of the exponential function (this is a\nrestatement of the fact that b0 = 1 for any real b > 1) (4)\nExponential function is ever increasing; i"
    },
    {
        "Chapter": "1",
        "sentence_range": "2169-2172",
        "Text": "(2)\nRange of the exponential function is the set of all positive real numbers (3)\nThe point (0, 1) is always on the graph of the exponential function (this is a\nrestatement of the fact that b0 = 1 for any real b > 1) (4)\nExponential function is ever increasing; i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "2170-2173",
        "Text": "(3)\nThe point (0, 1) is always on the graph of the exponential function (this is a\nrestatement of the fact that b0 = 1 for any real b > 1) (4)\nExponential function is ever increasing; i e , as we move from left to right, the\ngraph rises above"
    },
    {
        "Chapter": "1",
        "sentence_range": "2171-2174",
        "Text": "(4)\nExponential function is ever increasing; i e , as we move from left to right, the\ngraph rises above Fig  5"
    },
    {
        "Chapter": "1",
        "sentence_range": "2172-2175",
        "Text": "e , as we move from left to right, the\ngraph rises above Fig  5 9\nRationalised 2023-24\nCONTINUITY AND DIFFERENTIABILITY\n127\n(5)\nFor very large negative values of x, the exponential function is very close to 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "2173-2176",
        "Text": ", as we move from left to right, the\ngraph rises above Fig  5 9\nRationalised 2023-24\nCONTINUITY AND DIFFERENTIABILITY\n127\n(5)\nFor very large negative values of x, the exponential function is very close to 0 In\nother words, in the second quadrant, the graph approaches x-axis (but never\nmeets it)"
    },
    {
        "Chapter": "1",
        "sentence_range": "2174-2177",
        "Text": "Fig  5 9\nRationalised 2023-24\nCONTINUITY AND DIFFERENTIABILITY\n127\n(5)\nFor very large negative values of x, the exponential function is very close to 0 In\nother words, in the second quadrant, the graph approaches x-axis (but never\nmeets it) Exponential function with base 10 is called the common exponential function"
    },
    {
        "Chapter": "1",
        "sentence_range": "2175-2178",
        "Text": "9\nRationalised 2023-24\nCONTINUITY AND DIFFERENTIABILITY\n127\n(5)\nFor very large negative values of x, the exponential function is very close to 0 In\nother words, in the second quadrant, the graph approaches x-axis (but never\nmeets it) Exponential function with base 10 is called the common exponential function In\nthe Appendix A"
    },
    {
        "Chapter": "1",
        "sentence_range": "2176-2179",
        "Text": "In\nother words, in the second quadrant, the graph approaches x-axis (but never\nmeets it) Exponential function with base 10 is called the common exponential function In\nthe Appendix A 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "2177-2180",
        "Text": "Exponential function with base 10 is called the common exponential function In\nthe Appendix A 1 4 of Class XI, it was observed that the sum of the series\n1\n1\n1"
    },
    {
        "Chapter": "1",
        "sentence_range": "2178-2181",
        "Text": "In\nthe Appendix A 1 4 of Class XI, it was observed that the sum of the series\n1\n1\n1 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "2179-2182",
        "Text": "1 4 of Class XI, it was observed that the sum of the series\n1\n1\n1 1 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "2180-2183",
        "Text": "4 of Class XI, it was observed that the sum of the series\n1\n1\n1 1 2 +\n+\n+\nis a number between 2 and 3 and is denoted by e"
    },
    {
        "Chapter": "1",
        "sentence_range": "2181-2184",
        "Text": "1 2 +\n+\n+\nis a number between 2 and 3 and is denoted by e Using this e as the base we obtain an\nextremely important exponential function y = ex"
    },
    {
        "Chapter": "1",
        "sentence_range": "2182-2185",
        "Text": "2 +\n+\n+\nis a number between 2 and 3 and is denoted by e Using this e as the base we obtain an\nextremely important exponential function y = ex This is called natural exponential function"
    },
    {
        "Chapter": "1",
        "sentence_range": "2183-2186",
        "Text": "+\n+\n+\nis a number between 2 and 3 and is denoted by e Using this e as the base we obtain an\nextremely important exponential function y = ex This is called natural exponential function It would be interesting to know if the inverse of the exponential function exists and\nhas nice interpretation"
    },
    {
        "Chapter": "1",
        "sentence_range": "2184-2187",
        "Text": "Using this e as the base we obtain an\nextremely important exponential function y = ex This is called natural exponential function It would be interesting to know if the inverse of the exponential function exists and\nhas nice interpretation This search motivates the following definition"
    },
    {
        "Chapter": "1",
        "sentence_range": "2185-2188",
        "Text": "This is called natural exponential function It would be interesting to know if the inverse of the exponential function exists and\nhas nice interpretation This search motivates the following definition Definition 4 Let b > 1 be a real number"
    },
    {
        "Chapter": "1",
        "sentence_range": "2186-2189",
        "Text": "It would be interesting to know if the inverse of the exponential function exists and\nhas nice interpretation This search motivates the following definition Definition 4 Let b > 1 be a real number Then we say logarithm of a to base b is x if\nbx = a"
    },
    {
        "Chapter": "1",
        "sentence_range": "2187-2190",
        "Text": "This search motivates the following definition Definition 4 Let b > 1 be a real number Then we say logarithm of a to base b is x if\nbx = a Logarithm of a to base b is denoted by logb a"
    },
    {
        "Chapter": "1",
        "sentence_range": "2188-2191",
        "Text": "Definition 4 Let b > 1 be a real number Then we say logarithm of a to base b is x if\nbx = a Logarithm of a to base b is denoted by logb a Thus logb a = x if bx = a"
    },
    {
        "Chapter": "1",
        "sentence_range": "2189-2192",
        "Text": "Then we say logarithm of a to base b is x if\nbx = a Logarithm of a to base b is denoted by logb a Thus logb a = x if bx = a Let us\nwork with a few explicit examples to get a feel for this"
    },
    {
        "Chapter": "1",
        "sentence_range": "2190-2193",
        "Text": "Logarithm of a to base b is denoted by logb a Thus logb a = x if bx = a Let us\nwork with a few explicit examples to get a feel for this We know 23 = 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "2191-2194",
        "Text": "Thus logb a = x if bx = a Let us\nwork with a few explicit examples to get a feel for this We know 23 = 8 In terms of\nlogarithms, we may rewrite this as log2 8 = 3"
    },
    {
        "Chapter": "1",
        "sentence_range": "2192-2195",
        "Text": "Let us\nwork with a few explicit examples to get a feel for this We know 23 = 8 In terms of\nlogarithms, we may rewrite this as log2 8 = 3 Similarly, 104 = 10000 is equivalent to\nsaying log10 10000 = 4"
    },
    {
        "Chapter": "1",
        "sentence_range": "2193-2196",
        "Text": "We know 23 = 8 In terms of\nlogarithms, we may rewrite this as log2 8 = 3 Similarly, 104 = 10000 is equivalent to\nsaying log10 10000 = 4 Also, 625 = 54 = 252 is equivalent to saying log5 625 = 4 or\nlog25 625 = 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "2194-2197",
        "Text": "In terms of\nlogarithms, we may rewrite this as log2 8 = 3 Similarly, 104 = 10000 is equivalent to\nsaying log10 10000 = 4 Also, 625 = 54 = 252 is equivalent to saying log5 625 = 4 or\nlog25 625 = 2 On a slightly more mature note, fixing a  base b > 1, we may look at logarithm as\na function from positive real numbers to all real numbers"
    },
    {
        "Chapter": "1",
        "sentence_range": "2195-2198",
        "Text": "Similarly, 104 = 10000 is equivalent to\nsaying log10 10000 = 4 Also, 625 = 54 = 252 is equivalent to saying log5 625 = 4 or\nlog25 625 = 2 On a slightly more mature note, fixing a  base b > 1, we may look at logarithm as\na function from positive real numbers to all real numbers This function, called the\nlogarithmic function, is defined by\nlogb : R+ \u2192 R\nx \u2192 logb x = y   if by = x\nAs before if the base b = 10, we say it\nis common logarithms and if b = e, then\nwe say it is natural logarithms"
    },
    {
        "Chapter": "1",
        "sentence_range": "2196-2199",
        "Text": "Also, 625 = 54 = 252 is equivalent to saying log5 625 = 4 or\nlog25 625 = 2 On a slightly more mature note, fixing a  base b > 1, we may look at logarithm as\na function from positive real numbers to all real numbers This function, called the\nlogarithmic function, is defined by\nlogb : R+ \u2192 R\nx \u2192 logb x = y   if by = x\nAs before if the base b = 10, we say it\nis common logarithms and if b = e, then\nwe say it is natural logarithms Often\nnatural logarithm is denoted by ln"
    },
    {
        "Chapter": "1",
        "sentence_range": "2197-2200",
        "Text": "On a slightly more mature note, fixing a  base b > 1, we may look at logarithm as\na function from positive real numbers to all real numbers This function, called the\nlogarithmic function, is defined by\nlogb : R+ \u2192 R\nx \u2192 logb x = y   if by = x\nAs before if the base b = 10, we say it\nis common logarithms and if b = e, then\nwe say it is natural logarithms Often\nnatural logarithm is denoted by ln In this\nchapter, log x denotes the logarithm\nfunction to base e, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "2198-2201",
        "Text": "This function, called the\nlogarithmic function, is defined by\nlogb : R+ \u2192 R\nx \u2192 logb x = y   if by = x\nAs before if the base b = 10, we say it\nis common logarithms and if b = e, then\nwe say it is natural logarithms Often\nnatural logarithm is denoted by ln In this\nchapter, log x denotes the logarithm\nfunction to base e, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "2199-2202",
        "Text": "Often\nnatural logarithm is denoted by ln In this\nchapter, log x denotes the logarithm\nfunction to base e, i e , ln x will be written\nas simply log x"
    },
    {
        "Chapter": "1",
        "sentence_range": "2200-2203",
        "Text": "In this\nchapter, log x denotes the logarithm\nfunction to base e, i e , ln x will be written\nas simply log x The Fig 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "2201-2204",
        "Text": "e , ln x will be written\nas simply log x The Fig 5 10 gives the plots\nof logarithm function to base 2, e and 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "2202-2205",
        "Text": ", ln x will be written\nas simply log x The Fig 5 10 gives the plots\nof logarithm function to base 2, e and 10 Some of the important observations\nabout the logarithm function to any base\nb > 1 are listed below:\n Fig 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "2203-2206",
        "Text": "The Fig 5 10 gives the plots\nof logarithm function to base 2, e and 10 Some of the important observations\nabout the logarithm function to any base\nb > 1 are listed below:\n Fig 5 10\nRationalised 2023-24\n MATHEMATICS\n128\n Fig 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "2204-2207",
        "Text": "10 gives the plots\nof logarithm function to base 2, e and 10 Some of the important observations\nabout the logarithm function to any base\nb > 1 are listed below:\n Fig 5 10\nRationalised 2023-24\n MATHEMATICS\n128\n Fig 5 11\n(1)\nWe cannot make a meaningful definition of logarithm of non-positive numbers\nand hence the domain of log function is R+"
    },
    {
        "Chapter": "1",
        "sentence_range": "2205-2208",
        "Text": "Some of the important observations\nabout the logarithm function to any base\nb > 1 are listed below:\n Fig 5 10\nRationalised 2023-24\n MATHEMATICS\n128\n Fig 5 11\n(1)\nWe cannot make a meaningful definition of logarithm of non-positive numbers\nand hence the domain of log function is R+ (2)\nThe range of log function is the set of all real numbers"
    },
    {
        "Chapter": "1",
        "sentence_range": "2206-2209",
        "Text": "10\nRationalised 2023-24\n MATHEMATICS\n128\n Fig 5 11\n(1)\nWe cannot make a meaningful definition of logarithm of non-positive numbers\nand hence the domain of log function is R+ (2)\nThe range of log function is the set of all real numbers (3)\nThe point (1, 0) is always on the graph of the log function"
    },
    {
        "Chapter": "1",
        "sentence_range": "2207-2210",
        "Text": "11\n(1)\nWe cannot make a meaningful definition of logarithm of non-positive numbers\nand hence the domain of log function is R+ (2)\nThe range of log function is the set of all real numbers (3)\nThe point (1, 0) is always on the graph of the log function (4)\nThe log function is ever increasing,\ni"
    },
    {
        "Chapter": "1",
        "sentence_range": "2208-2211",
        "Text": "(2)\nThe range of log function is the set of all real numbers (3)\nThe point (1, 0) is always on the graph of the log function (4)\nThe log function is ever increasing,\ni e"
    },
    {
        "Chapter": "1",
        "sentence_range": "2209-2212",
        "Text": "(3)\nThe point (1, 0) is always on the graph of the log function (4)\nThe log function is ever increasing,\ni e , as we move from left to right\nthe graph rises above"
    },
    {
        "Chapter": "1",
        "sentence_range": "2210-2213",
        "Text": "(4)\nThe log function is ever increasing,\ni e , as we move from left to right\nthe graph rises above (5)\nFor x very near to zero, the value\nof log x can be made lesser than\nany given real number"
    },
    {
        "Chapter": "1",
        "sentence_range": "2211-2214",
        "Text": "e , as we move from left to right\nthe graph rises above (5)\nFor x very near to zero, the value\nof log x can be made lesser than\nany given real number In other\nwords in the fourth quadrant the\ngraph approaches y-axis (but\nnever meets it)"
    },
    {
        "Chapter": "1",
        "sentence_range": "2212-2215",
        "Text": ", as we move from left to right\nthe graph rises above (5)\nFor x very near to zero, the value\nof log x can be made lesser than\nany given real number In other\nwords in the fourth quadrant the\ngraph approaches y-axis (but\nnever meets it) (6)\nFig 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "2213-2216",
        "Text": "(5)\nFor x very near to zero, the value\nof log x can be made lesser than\nany given real number In other\nwords in the fourth quadrant the\ngraph approaches y-axis (but\nnever meets it) (6)\nFig 5 11 gives the plot of y = ex and\ny = ln x"
    },
    {
        "Chapter": "1",
        "sentence_range": "2214-2217",
        "Text": "In other\nwords in the fourth quadrant the\ngraph approaches y-axis (but\nnever meets it) (6)\nFig 5 11 gives the plot of y = ex and\ny = ln x It is of interest to observe\nthat the two curves are the mirror\nimages of each other reflected in the line y = x"
    },
    {
        "Chapter": "1",
        "sentence_range": "2215-2218",
        "Text": "(6)\nFig 5 11 gives the plot of y = ex and\ny = ln x It is of interest to observe\nthat the two curves are the mirror\nimages of each other reflected in the line y = x Two  properties of \u2018log\u2019 functions are proved below:\n(1)\nThere is a standard change of base rule to obtain loga p in terms of logb p"
    },
    {
        "Chapter": "1",
        "sentence_range": "2216-2219",
        "Text": "11 gives the plot of y = ex and\ny = ln x It is of interest to observe\nthat the two curves are the mirror\nimages of each other reflected in the line y = x Two  properties of \u2018log\u2019 functions are proved below:\n(1)\nThere is a standard change of base rule to obtain loga p in terms of logb p Let\nloga p = \u03b1, logb p = \u03b2 and logb a = \u03b3"
    },
    {
        "Chapter": "1",
        "sentence_range": "2217-2220",
        "Text": "It is of interest to observe\nthat the two curves are the mirror\nimages of each other reflected in the line y = x Two  properties of \u2018log\u2019 functions are proved below:\n(1)\nThere is a standard change of base rule to obtain loga p in terms of logb p Let\nloga p = \u03b1, logb p = \u03b2 and logb a = \u03b3 This means a\u03b1 = p, b\u03b2 = p and b\u03b3 = a"
    },
    {
        "Chapter": "1",
        "sentence_range": "2218-2221",
        "Text": "Two  properties of \u2018log\u2019 functions are proved below:\n(1)\nThere is a standard change of base rule to obtain loga p in terms of logb p Let\nloga p = \u03b1, logb p = \u03b2 and logb a = \u03b3 This means a\u03b1 = p, b\u03b2 = p and b\u03b3 = a Substituting the third equation in the first one, we have\n(b\u03b3)\u03b1 = b\u03b3\u03b1 = p\nUsing this in the second equation, we get\nb\u03b2 = p = b\u03b3\u03b1\nwhich implies\n\u03b2 = \u03b1\u03b3 or \u03b1 = \u03b2\n\u03b3"
    },
    {
        "Chapter": "1",
        "sentence_range": "2219-2222",
        "Text": "Let\nloga p = \u03b1, logb p = \u03b2 and logb a = \u03b3 This means a\u03b1 = p, b\u03b2 = p and b\u03b3 = a Substituting the third equation in the first one, we have\n(b\u03b3)\u03b1 = b\u03b3\u03b1 = p\nUsing this in the second equation, we get\nb\u03b2 = p = b\u03b3\u03b1\nwhich implies\n\u03b2 = \u03b1\u03b3 or \u03b1 = \u03b2\n\u03b3 But then\nloga p = log\nlog\nb\nb\np\na\n(2)\nAnother interesting property of the log function is its effect on products"
    },
    {
        "Chapter": "1",
        "sentence_range": "2220-2223",
        "Text": "This means a\u03b1 = p, b\u03b2 = p and b\u03b3 = a Substituting the third equation in the first one, we have\n(b\u03b3)\u03b1 = b\u03b3\u03b1 = p\nUsing this in the second equation, we get\nb\u03b2 = p = b\u03b3\u03b1\nwhich implies\n\u03b2 = \u03b1\u03b3 or \u03b1 = \u03b2\n\u03b3 But then\nloga p = log\nlog\nb\nb\np\na\n(2)\nAnother interesting property of the log function is its effect on products Let\nlogb pq = \u03b1"
    },
    {
        "Chapter": "1",
        "sentence_range": "2221-2224",
        "Text": "Substituting the third equation in the first one, we have\n(b\u03b3)\u03b1 = b\u03b3\u03b1 = p\nUsing this in the second equation, we get\nb\u03b2 = p = b\u03b3\u03b1\nwhich implies\n\u03b2 = \u03b1\u03b3 or \u03b1 = \u03b2\n\u03b3 But then\nloga p = log\nlog\nb\nb\np\na\n(2)\nAnother interesting property of the log function is its effect on products Let\nlogb pq = \u03b1 Then b\u03b1 = pq"
    },
    {
        "Chapter": "1",
        "sentence_range": "2222-2225",
        "Text": "But then\nloga p = log\nlog\nb\nb\np\na\n(2)\nAnother interesting property of the log function is its effect on products Let\nlogb pq = \u03b1 Then b\u03b1 = pq If logb p = \u03b2 and logb q = \u03b3, then b\u03b2 = p and b\u03b3 = q"
    },
    {
        "Chapter": "1",
        "sentence_range": "2223-2226",
        "Text": "Let\nlogb pq = \u03b1 Then b\u03b1 = pq If logb p = \u03b2 and logb q = \u03b3, then b\u03b2 = p and b\u03b3 = q But then b\u03b1 = pq = b\u03b2b\u03b3 = b\u03b2 + \u03b3\nwhich implies \u03b1 = \u03b2 + \u03b3, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "2224-2227",
        "Text": "Then b\u03b1 = pq If logb p = \u03b2 and logb q = \u03b3, then b\u03b2 = p and b\u03b3 = q But then b\u03b1 = pq = b\u03b2b\u03b3 = b\u03b2 + \u03b3\nwhich implies \u03b1 = \u03b2 + \u03b3, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "2225-2228",
        "Text": "If logb p = \u03b2 and logb q = \u03b3, then b\u03b2 = p and b\u03b3 = q But then b\u03b1 = pq = b\u03b2b\u03b3 = b\u03b2 + \u03b3\nwhich implies \u03b1 = \u03b2 + \u03b3, i e ,\nlogb pq = logb p + logb q\nRationalised 2023-24\nCONTINUITY AND DIFFERENTIABILITY\n129\nA particularly interesting and important consequence of this is when p = q"
    },
    {
        "Chapter": "1",
        "sentence_range": "2226-2229",
        "Text": "But then b\u03b1 = pq = b\u03b2b\u03b3 = b\u03b2 + \u03b3\nwhich implies \u03b1 = \u03b2 + \u03b3, i e ,\nlogb pq = logb p + logb q\nRationalised 2023-24\nCONTINUITY AND DIFFERENTIABILITY\n129\nA particularly interesting and important consequence of this is when p = q In\nthis case the above may be rewritten as\nlogb p2 =  logb p + logb p = 2 log p\nAn easy generalisation of this (left as an exercise"
    },
    {
        "Chapter": "1",
        "sentence_range": "2227-2230",
        "Text": "e ,\nlogb pq = logb p + logb q\nRationalised 2023-24\nCONTINUITY AND DIFFERENTIABILITY\n129\nA particularly interesting and important consequence of this is when p = q In\nthis case the above may be rewritten as\nlogb p2 =  logb p + logb p = 2 log p\nAn easy generalisation of this (left as an exercise ) is\nlogb pn = n log p\nfor any positive integer n"
    },
    {
        "Chapter": "1",
        "sentence_range": "2228-2231",
        "Text": ",\nlogb pq = logb p + logb q\nRationalised 2023-24\nCONTINUITY AND DIFFERENTIABILITY\n129\nA particularly interesting and important consequence of this is when p = q In\nthis case the above may be rewritten as\nlogb p2 =  logb p + logb p = 2 log p\nAn easy generalisation of this (left as an exercise ) is\nlogb pn = n log p\nfor any positive integer n In fact this is true for any real number n, but we will\nnot attempt to prove this"
    },
    {
        "Chapter": "1",
        "sentence_range": "2229-2232",
        "Text": "In\nthis case the above may be rewritten as\nlogb p2 =  logb p + logb p = 2 log p\nAn easy generalisation of this (left as an exercise ) is\nlogb pn = n log p\nfor any positive integer n In fact this is true for any real number n, but we will\nnot attempt to prove this On the similar lines the reader is invited to verify\nlogb\nx\ny  = logb x \u2013 logb y\nExample 25 Is it true that x = elog x for all real x"
    },
    {
        "Chapter": "1",
        "sentence_range": "2230-2233",
        "Text": ") is\nlogb pn = n log p\nfor any positive integer n In fact this is true for any real number n, but we will\nnot attempt to prove this On the similar lines the reader is invited to verify\nlogb\nx\ny  = logb x \u2013 logb y\nExample 25 Is it true that x = elog x for all real x Solution First, observe that the domain of log function is set of all positive real numbers"
    },
    {
        "Chapter": "1",
        "sentence_range": "2231-2234",
        "Text": "In fact this is true for any real number n, but we will\nnot attempt to prove this On the similar lines the reader is invited to verify\nlogb\nx\ny  = logb x \u2013 logb y\nExample 25 Is it true that x = elog x for all real x Solution First, observe that the domain of log function is set of all positive real numbers So the above equation is not true for non-positive real numbers"
    },
    {
        "Chapter": "1",
        "sentence_range": "2232-2235",
        "Text": "On the similar lines the reader is invited to verify\nlogb\nx\ny  = logb x \u2013 logb y\nExample 25 Is it true that x = elog x for all real x Solution First, observe that the domain of log function is set of all positive real numbers So the above equation is not true for non-positive real numbers Now, let y = elog x"
    },
    {
        "Chapter": "1",
        "sentence_range": "2233-2236",
        "Text": "Solution First, observe that the domain of log function is set of all positive real numbers So the above equation is not true for non-positive real numbers Now, let y = elog x If\ny > 0, we may take logarithm which gives us log y = log (elog x) = log x"
    },
    {
        "Chapter": "1",
        "sentence_range": "2234-2237",
        "Text": "So the above equation is not true for non-positive real numbers Now, let y = elog x If\ny > 0, we may take logarithm which gives us log y = log (elog x) = log x log e = log x"
    },
    {
        "Chapter": "1",
        "sentence_range": "2235-2238",
        "Text": "Now, let y = elog x If\ny > 0, we may take logarithm which gives us log y = log (elog x) = log x log e = log x Thus\ny = x"
    },
    {
        "Chapter": "1",
        "sentence_range": "2236-2239",
        "Text": "If\ny > 0, we may take logarithm which gives us log y = log (elog x) = log x log e = log x Thus\ny = x Hence x = elog x is true only for positive values of x"
    },
    {
        "Chapter": "1",
        "sentence_range": "2237-2240",
        "Text": "log e = log x Thus\ny = x Hence x = elog x is true only for positive values of x One of the striking properties of the natural exponential function in differential\ncalculus is that it doesn\u2019t change during the process of differentiation"
    },
    {
        "Chapter": "1",
        "sentence_range": "2238-2241",
        "Text": "Thus\ny = x Hence x = elog x is true only for positive values of x One of the striking properties of the natural exponential function in differential\ncalculus is that it doesn\u2019t change during the process of differentiation This is captured\nin the following theorem whose proof we skip"
    },
    {
        "Chapter": "1",
        "sentence_range": "2239-2242",
        "Text": "Hence x = elog x is true only for positive values of x One of the striking properties of the natural exponential function in differential\ncalculus is that it doesn\u2019t change during the process of differentiation This is captured\nin the following theorem whose proof we skip Theorem 5*\n(1)\nThe derivative of ex w"
    },
    {
        "Chapter": "1",
        "sentence_range": "2240-2243",
        "Text": "One of the striking properties of the natural exponential function in differential\ncalculus is that it doesn\u2019t change during the process of differentiation This is captured\nin the following theorem whose proof we skip Theorem 5*\n(1)\nThe derivative of ex w r"
    },
    {
        "Chapter": "1",
        "sentence_range": "2241-2244",
        "Text": "This is captured\nin the following theorem whose proof we skip Theorem 5*\n(1)\nThe derivative of ex w r t"
    },
    {
        "Chapter": "1",
        "sentence_range": "2242-2245",
        "Text": "Theorem 5*\n(1)\nThe derivative of ex w r t , x is ex; i"
    },
    {
        "Chapter": "1",
        "sentence_range": "2243-2246",
        "Text": "r t , x is ex; i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "2244-2247",
        "Text": "t , x is ex; i e , d\ndx (ex) = ex"
    },
    {
        "Chapter": "1",
        "sentence_range": "2245-2248",
        "Text": ", x is ex; i e , d\ndx (ex) = ex (2)\nThe derivative of log x w"
    },
    {
        "Chapter": "1",
        "sentence_range": "2246-2249",
        "Text": "e , d\ndx (ex) = ex (2)\nThe derivative of log x w r"
    },
    {
        "Chapter": "1",
        "sentence_range": "2247-2250",
        "Text": ", d\ndx (ex) = ex (2)\nThe derivative of log x w r t"
    },
    {
        "Chapter": "1",
        "sentence_range": "2248-2251",
        "Text": "(2)\nThe derivative of log x w r t , x is 1\nx ; i"
    },
    {
        "Chapter": "1",
        "sentence_range": "2249-2252",
        "Text": "r t , x is 1\nx ; i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "2250-2253",
        "Text": "t , x is 1\nx ; i e , d\ndx (log x) = 1\nx"
    },
    {
        "Chapter": "1",
        "sentence_range": "2251-2254",
        "Text": ", x is 1\nx ; i e , d\ndx (log x) = 1\nx Example 26 Differentiate the following w"
    },
    {
        "Chapter": "1",
        "sentence_range": "2252-2255",
        "Text": "e , d\ndx (log x) = 1\nx Example 26 Differentiate the following w r"
    },
    {
        "Chapter": "1",
        "sentence_range": "2253-2256",
        "Text": ", d\ndx (log x) = 1\nx Example 26 Differentiate the following w r t"
    },
    {
        "Chapter": "1",
        "sentence_range": "2254-2257",
        "Text": "Example 26 Differentiate the following w r t x:\n(i)\ne\u2013x\n(ii)   sin (log x), x > 0\n    (iii)   cos\u20131 (ex)\n (iv)    ecos x\nSolution\n(i)\nLet y = e\u2013 x"
    },
    {
        "Chapter": "1",
        "sentence_range": "2255-2258",
        "Text": "r t x:\n(i)\ne\u2013x\n(ii)   sin (log x), x > 0\n    (iii)   cos\u20131 (ex)\n (iv)    ecos x\nSolution\n(i)\nLet y = e\u2013 x Using chain rule, we have\ndy\ndx  =\nx\nd\ne\ndx\n\u2212 \u22c5\n (\u2013 x) = \u2013 e\u2013 x\n(ii)\nLet y = sin (log x)"
    },
    {
        "Chapter": "1",
        "sentence_range": "2256-2259",
        "Text": "t x:\n(i)\ne\u2013x\n(ii)   sin (log x), x > 0\n    (iii)   cos\u20131 (ex)\n (iv)    ecos x\nSolution\n(i)\nLet y = e\u2013 x Using chain rule, we have\ndy\ndx  =\nx\nd\ne\ndx\n\u2212 \u22c5\n (\u2013 x) = \u2013 e\u2013 x\n(ii)\nLet y = sin (log x) Using chain rule, we have\ndy\ndx  =\ncos (log )\ncos (log )\nd(log )\nx\nx\nx\ndx\nx\n\u22c5\n=\n* Please see supplementary material on Page 222"
    },
    {
        "Chapter": "1",
        "sentence_range": "2257-2260",
        "Text": "x:\n(i)\ne\u2013x\n(ii)   sin (log x), x > 0\n    (iii)   cos\u20131 (ex)\n (iv)    ecos x\nSolution\n(i)\nLet y = e\u2013 x Using chain rule, we have\ndy\ndx  =\nx\nd\ne\ndx\n\u2212 \u22c5\n (\u2013 x) = \u2013 e\u2013 x\n(ii)\nLet y = sin (log x) Using chain rule, we have\ndy\ndx  =\ncos (log )\ncos (log )\nd(log )\nx\nx\nx\ndx\nx\n\u22c5\n=\n* Please see supplementary material on Page 222 Rationalised 2023-24\n MATHEMATICS\n130\n(iii)\nLet y = cos\u20131 (ex)"
    },
    {
        "Chapter": "1",
        "sentence_range": "2258-2261",
        "Text": "Using chain rule, we have\ndy\ndx  =\nx\nd\ne\ndx\n\u2212 \u22c5\n (\u2013 x) = \u2013 e\u2013 x\n(ii)\nLet y = sin (log x) Using chain rule, we have\ndy\ndx  =\ncos (log )\ncos (log )\nd(log )\nx\nx\nx\ndx\nx\n\u22c5\n=\n* Please see supplementary material on Page 222 Rationalised 2023-24\n MATHEMATICS\n130\n(iii)\nLet y = cos\u20131 (ex) Using chain rule, we have\ndy\ndx  =\n2\n2\n1\n(\n)\n1\n(\n)\n1\nx\nx\nx\nx\nd\ne\ndxe\ne\ne\n\u2212\n\u2212\n\u22c5\n=\n\u2212\n\u2212\n(iv)\nLet y = ecos x"
    },
    {
        "Chapter": "1",
        "sentence_range": "2259-2262",
        "Text": "Using chain rule, we have\ndy\ndx  =\ncos (log )\ncos (log )\nd(log )\nx\nx\nx\ndx\nx\n\u22c5\n=\n* Please see supplementary material on Page 222 Rationalised 2023-24\n MATHEMATICS\n130\n(iii)\nLet y = cos\u20131 (ex) Using chain rule, we have\ndy\ndx  =\n2\n2\n1\n(\n)\n1\n(\n)\n1\nx\nx\nx\nx\nd\ne\ndxe\ne\ne\n\u2212\n\u2212\n\u22c5\n=\n\u2212\n\u2212\n(iv)\nLet y = ecos x Using chain rule, we have\ndy\ndx  =\ncos\ncos\n( sin )\n(sin )\nx\nx\ne\nx\nx e\n\u22c5 \u2212\n= \u2212\nEXERCISE 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "2260-2263",
        "Text": "Rationalised 2023-24\n MATHEMATICS\n130\n(iii)\nLet y = cos\u20131 (ex) Using chain rule, we have\ndy\ndx  =\n2\n2\n1\n(\n)\n1\n(\n)\n1\nx\nx\nx\nx\nd\ne\ndxe\ne\ne\n\u2212\n\u2212\n\u22c5\n=\n\u2212\n\u2212\n(iv)\nLet y = ecos x Using chain rule, we have\ndy\ndx  =\ncos\ncos\n( sin )\n(sin )\nx\nx\ne\nx\nx e\n\u22c5 \u2212\n= \u2212\nEXERCISE 5 4\nDifferentiate the following w"
    },
    {
        "Chapter": "1",
        "sentence_range": "2261-2264",
        "Text": "Using chain rule, we have\ndy\ndx  =\n2\n2\n1\n(\n)\n1\n(\n)\n1\nx\nx\nx\nx\nd\ne\ndxe\ne\ne\n\u2212\n\u2212\n\u22c5\n=\n\u2212\n\u2212\n(iv)\nLet y = ecos x Using chain rule, we have\ndy\ndx  =\ncos\ncos\n( sin )\n(sin )\nx\nx\ne\nx\nx e\n\u22c5 \u2212\n= \u2212\nEXERCISE 5 4\nDifferentiate the following w r"
    },
    {
        "Chapter": "1",
        "sentence_range": "2262-2265",
        "Text": "Using chain rule, we have\ndy\ndx  =\ncos\ncos\n( sin )\n(sin )\nx\nx\ne\nx\nx e\n\u22c5 \u2212\n= \u2212\nEXERCISE 5 4\nDifferentiate the following w r t"
    },
    {
        "Chapter": "1",
        "sentence_range": "2263-2266",
        "Text": "4\nDifferentiate the following w r t x:\n1"
    },
    {
        "Chapter": "1",
        "sentence_range": "2264-2267",
        "Text": "r t x:\n1 sin\nxe\nx\n2"
    },
    {
        "Chapter": "1",
        "sentence_range": "2265-2268",
        "Text": "t x:\n1 sin\nxe\nx\n2 sin1\nx\ne\n\u2212\n3"
    },
    {
        "Chapter": "1",
        "sentence_range": "2266-2269",
        "Text": "x:\n1 sin\nxe\nx\n2 sin1\nx\ne\n\u2212\n3 3xe\n4"
    },
    {
        "Chapter": "1",
        "sentence_range": "2267-2270",
        "Text": "sin\nxe\nx\n2 sin1\nx\ne\n\u2212\n3 3xe\n4 sin (tan\u20131 e\u2013x)\n5"
    },
    {
        "Chapter": "1",
        "sentence_range": "2268-2271",
        "Text": "sin1\nx\ne\n\u2212\n3 3xe\n4 sin (tan\u20131 e\u2013x)\n5 log (cos ex)\n6"
    },
    {
        "Chapter": "1",
        "sentence_range": "2269-2272",
        "Text": "3xe\n4 sin (tan\u20131 e\u2013x)\n5 log (cos ex)\n6 2\n5"
    },
    {
        "Chapter": "1",
        "sentence_range": "2270-2273",
        "Text": "sin (tan\u20131 e\u2013x)\n5 log (cos ex)\n6 2\n5 x\nx\nx\ne\ne\ne\n+\n+\n+\n7"
    },
    {
        "Chapter": "1",
        "sentence_range": "2271-2274",
        "Text": "log (cos ex)\n6 2\n5 x\nx\nx\ne\ne\ne\n+\n+\n+\n7 ,\n0\nex\nx >\n8"
    },
    {
        "Chapter": "1",
        "sentence_range": "2272-2275",
        "Text": "2\n5 x\nx\nx\ne\ne\ne\n+\n+\n+\n7 ,\n0\nex\nx >\n8 log (log x), x > 1\n9"
    },
    {
        "Chapter": "1",
        "sentence_range": "2273-2276",
        "Text": "x\nx\nx\ne\ne\ne\n+\n+\n+\n7 ,\n0\nex\nx >\n8 log (log x), x > 1\n9 cos ,\n0\nlog\nx\nx\nx\n>\n10"
    },
    {
        "Chapter": "1",
        "sentence_range": "2274-2277",
        "Text": ",\n0\nex\nx >\n8 log (log x), x > 1\n9 cos ,\n0\nlog\nx\nx\nx\n>\n10 cos (log x + ex), x > 0\n5"
    },
    {
        "Chapter": "1",
        "sentence_range": "2275-2278",
        "Text": "log (log x), x > 1\n9 cos ,\n0\nlog\nx\nx\nx\n>\n10 cos (log x + ex), x > 0\n5 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "2276-2279",
        "Text": "cos ,\n0\nlog\nx\nx\nx\n>\n10 cos (log x + ex), x > 0\n5 5 Logarithmic Differentiation\nIn this section, we will learn to differentiate certain special class of functions given in\nthe form\ny = f (x) = [u(x)]v (x)\nBy taking logarithm (to base e) the above may be rewritten as\nlog y = v(x) log [u(x)]\nUsing chain rule we may differentiate this to get\n1\n1\n( )\n( )\ndy\nv x\ny dx\nu x\n\u22c5\n=\n\u22c5"
    },
    {
        "Chapter": "1",
        "sentence_range": "2277-2280",
        "Text": "cos (log x + ex), x > 0\n5 5 Logarithmic Differentiation\nIn this section, we will learn to differentiate certain special class of functions given in\nthe form\ny = f (x) = [u(x)]v (x)\nBy taking logarithm (to base e) the above may be rewritten as\nlog y = v(x) log [u(x)]\nUsing chain rule we may differentiate this to get\n1\n1\n( )\n( )\ndy\nv x\ny dx\nu x\n\u22c5\n=\n\u22c5 u\u2032(x) + v\u2032(x)"
    },
    {
        "Chapter": "1",
        "sentence_range": "2278-2281",
        "Text": "5 Logarithmic Differentiation\nIn this section, we will learn to differentiate certain special class of functions given in\nthe form\ny = f (x) = [u(x)]v (x)\nBy taking logarithm (to base e) the above may be rewritten as\nlog y = v(x) log [u(x)]\nUsing chain rule we may differentiate this to get\n1\n1\n( )\n( )\ndy\nv x\ny dx\nu x\n\u22c5\n=\n\u22c5 u\u2032(x) + v\u2032(x) log [u(x)]\nwhich implies that\n[\n]\n( )\n( )\n( ) log\n( )\n( )\ndy\nyv x\nu x\nv x\nu x\ndx\n\uf8eeu x\n\uf8f9\n=\n\u22c5 \u2032\n+ \u2032\n\u22c5\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nThe main point to be noted in this method is that f(x) and u(x) must always be\npositive as otherwise their logarithms are not defined"
    },
    {
        "Chapter": "1",
        "sentence_range": "2279-2282",
        "Text": "Logarithmic Differentiation\nIn this section, we will learn to differentiate certain special class of functions given in\nthe form\ny = f (x) = [u(x)]v (x)\nBy taking logarithm (to base e) the above may be rewritten as\nlog y = v(x) log [u(x)]\nUsing chain rule we may differentiate this to get\n1\n1\n( )\n( )\ndy\nv x\ny dx\nu x\n\u22c5\n=\n\u22c5 u\u2032(x) + v\u2032(x) log [u(x)]\nwhich implies that\n[\n]\n( )\n( )\n( ) log\n( )\n( )\ndy\nyv x\nu x\nv x\nu x\ndx\n\uf8eeu x\n\uf8f9\n=\n\u22c5 \u2032\n+ \u2032\n\u22c5\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nThe main point to be noted in this method is that f(x) and u(x) must always be\npositive as otherwise their logarithms are not defined This process of differentiation is\nknown as logarithms differentiation and is illustrated by the following examples:\nRationalised 2023-24\nCONTINUITY AND DIFFERENTIABILITY\n131\nExample 27 Differentiate \n2\n2\n(\n3) (\n4)\n3\n4\n5\nx\nx\nx\nx\n\u2212\n+\n+\n+\n w"
    },
    {
        "Chapter": "1",
        "sentence_range": "2280-2283",
        "Text": "u\u2032(x) + v\u2032(x) log [u(x)]\nwhich implies that\n[\n]\n( )\n( )\n( ) log\n( )\n( )\ndy\nyv x\nu x\nv x\nu x\ndx\n\uf8eeu x\n\uf8f9\n=\n\u22c5 \u2032\n+ \u2032\n\u22c5\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nThe main point to be noted in this method is that f(x) and u(x) must always be\npositive as otherwise their logarithms are not defined This process of differentiation is\nknown as logarithms differentiation and is illustrated by the following examples:\nRationalised 2023-24\nCONTINUITY AND DIFFERENTIABILITY\n131\nExample 27 Differentiate \n2\n2\n(\n3) (\n4)\n3\n4\n5\nx\nx\nx\nx\n\u2212\n+\n+\n+\n w r"
    },
    {
        "Chapter": "1",
        "sentence_range": "2281-2284",
        "Text": "log [u(x)]\nwhich implies that\n[\n]\n( )\n( )\n( ) log\n( )\n( )\ndy\nyv x\nu x\nv x\nu x\ndx\n\uf8eeu x\n\uf8f9\n=\n\u22c5 \u2032\n+ \u2032\n\u22c5\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nThe main point to be noted in this method is that f(x) and u(x) must always be\npositive as otherwise their logarithms are not defined This process of differentiation is\nknown as logarithms differentiation and is illustrated by the following examples:\nRationalised 2023-24\nCONTINUITY AND DIFFERENTIABILITY\n131\nExample 27 Differentiate \n2\n2\n(\n3) (\n4)\n3\n4\n5\nx\nx\nx\nx\n\u2212\n+\n+\n+\n w r t"
    },
    {
        "Chapter": "1",
        "sentence_range": "2282-2285",
        "Text": "This process of differentiation is\nknown as logarithms differentiation and is illustrated by the following examples:\nRationalised 2023-24\nCONTINUITY AND DIFFERENTIABILITY\n131\nExample 27 Differentiate \n2\n2\n(\n3) (\n4)\n3\n4\n5\nx\nx\nx\nx\n\u2212\n+\n+\n+\n w r t x"
    },
    {
        "Chapter": "1",
        "sentence_range": "2283-2286",
        "Text": "r t x Solution Let \n2\n2\n(\n3) (\n4)\n(3\n4\n5)\nx\nx\ny\nx\nx\n\u2212\n+\n=\n+\n+\nTaking logarithm on both sides, we have\nlog y = 1\n2 [log (x \u2013 3) + log (x2 + 4) \u2013 log (3x2 + 4x + 5)]\nNow, differentiating both sides w"
    },
    {
        "Chapter": "1",
        "sentence_range": "2284-2287",
        "Text": "t x Solution Let \n2\n2\n(\n3) (\n4)\n(3\n4\n5)\nx\nx\ny\nx\nx\n\u2212\n+\n=\n+\n+\nTaking logarithm on both sides, we have\nlog y = 1\n2 [log (x \u2013 3) + log (x2 + 4) \u2013 log (3x2 + 4x + 5)]\nNow, differentiating both sides w r"
    },
    {
        "Chapter": "1",
        "sentence_range": "2285-2288",
        "Text": "x Solution Let \n2\n2\n(\n3) (\n4)\n(3\n4\n5)\nx\nx\ny\nx\nx\n\u2212\n+\n=\n+\n+\nTaking logarithm on both sides, we have\nlog y = 1\n2 [log (x \u2013 3) + log (x2 + 4) \u2013 log (3x2 + 4x + 5)]\nNow, differentiating both sides w r t"
    },
    {
        "Chapter": "1",
        "sentence_range": "2286-2289",
        "Text": "Solution Let \n2\n2\n(\n3) (\n4)\n(3\n4\n5)\nx\nx\ny\nx\nx\n\u2212\n+\n=\n+\n+\nTaking logarithm on both sides, we have\nlog y = 1\n2 [log (x \u2013 3) + log (x2 + 4) \u2013 log (3x2 + 4x + 5)]\nNow, differentiating both sides w r t x, we get\n1 dy\ny dx\n\u22c5\n =\n2\n2\n1\n1\n2\n6\n4\n2 (\n3)\n4\n3\n4\n5\nx\nx\nx\nx\nx\n+x\n\uf8ee\n\uf8f9\n+\n\u2212\n\uf8ef\n\uf8fa\n\u2212\n+\n+\n+\n\uf8f0\n\uf8fb\nor\ndy\ndx  =\n2\n2\n1\n2\n6\n4\n2\n(\n3)\n4\n3\n4\n5\ny\nx\nx\nx\nx\nx\n+x\n\uf8ee\n\uf8f9\n+\n\u2212\n\uf8ef\n\uf8fa\n\u2212\n+\n+\n+\n\uf8f0\n\uf8fb\n=\n2\n2\n2\n2\n1\n(\n3)(\n4)\n1\n2\n6\n4\n2\n(\n3)\n3\n4\n5\n4\n3\n4\n5\nx\nx\nx\nx\nx\nx\nx\nx\nx\nx\n\u2212\n+\n+\n\uf8ee\n\uf8f9\n+\n\u2212\n\uf8ef\n\uf8fa\n\u2212\n+\n+\n+\n+\n+\n\uf8f0\n\uf8fb\nExample 28 Differentiate ax w"
    },
    {
        "Chapter": "1",
        "sentence_range": "2287-2290",
        "Text": "r t x, we get\n1 dy\ny dx\n\u22c5\n =\n2\n2\n1\n1\n2\n6\n4\n2 (\n3)\n4\n3\n4\n5\nx\nx\nx\nx\nx\n+x\n\uf8ee\n\uf8f9\n+\n\u2212\n\uf8ef\n\uf8fa\n\u2212\n+\n+\n+\n\uf8f0\n\uf8fb\nor\ndy\ndx  =\n2\n2\n1\n2\n6\n4\n2\n(\n3)\n4\n3\n4\n5\ny\nx\nx\nx\nx\nx\n+x\n\uf8ee\n\uf8f9\n+\n\u2212\n\uf8ef\n\uf8fa\n\u2212\n+\n+\n+\n\uf8f0\n\uf8fb\n=\n2\n2\n2\n2\n1\n(\n3)(\n4)\n1\n2\n6\n4\n2\n(\n3)\n3\n4\n5\n4\n3\n4\n5\nx\nx\nx\nx\nx\nx\nx\nx\nx\nx\n\u2212\n+\n+\n\uf8ee\n\uf8f9\n+\n\u2212\n\uf8ef\n\uf8fa\n\u2212\n+\n+\n+\n+\n+\n\uf8f0\n\uf8fb\nExample 28 Differentiate ax w r"
    },
    {
        "Chapter": "1",
        "sentence_range": "2288-2291",
        "Text": "t x, we get\n1 dy\ny dx\n\u22c5\n =\n2\n2\n1\n1\n2\n6\n4\n2 (\n3)\n4\n3\n4\n5\nx\nx\nx\nx\nx\n+x\n\uf8ee\n\uf8f9\n+\n\u2212\n\uf8ef\n\uf8fa\n\u2212\n+\n+\n+\n\uf8f0\n\uf8fb\nor\ndy\ndx  =\n2\n2\n1\n2\n6\n4\n2\n(\n3)\n4\n3\n4\n5\ny\nx\nx\nx\nx\nx\n+x\n\uf8ee\n\uf8f9\n+\n\u2212\n\uf8ef\n\uf8fa\n\u2212\n+\n+\n+\n\uf8f0\n\uf8fb\n=\n2\n2\n2\n2\n1\n(\n3)(\n4)\n1\n2\n6\n4\n2\n(\n3)\n3\n4\n5\n4\n3\n4\n5\nx\nx\nx\nx\nx\nx\nx\nx\nx\nx\n\u2212\n+\n+\n\uf8ee\n\uf8f9\n+\n\u2212\n\uf8ef\n\uf8fa\n\u2212\n+\n+\n+\n+\n+\n\uf8f0\n\uf8fb\nExample 28 Differentiate ax w r t"
    },
    {
        "Chapter": "1",
        "sentence_range": "2289-2292",
        "Text": "x, we get\n1 dy\ny dx\n\u22c5\n =\n2\n2\n1\n1\n2\n6\n4\n2 (\n3)\n4\n3\n4\n5\nx\nx\nx\nx\nx\n+x\n\uf8ee\n\uf8f9\n+\n\u2212\n\uf8ef\n\uf8fa\n\u2212\n+\n+\n+\n\uf8f0\n\uf8fb\nor\ndy\ndx  =\n2\n2\n1\n2\n6\n4\n2\n(\n3)\n4\n3\n4\n5\ny\nx\nx\nx\nx\nx\n+x\n\uf8ee\n\uf8f9\n+\n\u2212\n\uf8ef\n\uf8fa\n\u2212\n+\n+\n+\n\uf8f0\n\uf8fb\n=\n2\n2\n2\n2\n1\n(\n3)(\n4)\n1\n2\n6\n4\n2\n(\n3)\n3\n4\n5\n4\n3\n4\n5\nx\nx\nx\nx\nx\nx\nx\nx\nx\nx\n\u2212\n+\n+\n\uf8ee\n\uf8f9\n+\n\u2212\n\uf8ef\n\uf8fa\n\u2212\n+\n+\n+\n+\n+\n\uf8f0\n\uf8fb\nExample 28 Differentiate ax w r t x, where a is a positive constant"
    },
    {
        "Chapter": "1",
        "sentence_range": "2290-2293",
        "Text": "r t x, where a is a positive constant Solution Let y = ax"
    },
    {
        "Chapter": "1",
        "sentence_range": "2291-2294",
        "Text": "t x, where a is a positive constant Solution Let y = ax Then\nlog y = x log a\nDifferentiating both sides w"
    },
    {
        "Chapter": "1",
        "sentence_range": "2292-2295",
        "Text": "x, where a is a positive constant Solution Let y = ax Then\nlog y = x log a\nDifferentiating both sides w r"
    },
    {
        "Chapter": "1",
        "sentence_range": "2293-2296",
        "Text": "Solution Let y = ax Then\nlog y = x log a\nDifferentiating both sides w r t"
    },
    {
        "Chapter": "1",
        "sentence_range": "2294-2297",
        "Text": "Then\nlog y = x log a\nDifferentiating both sides w r t x, we have\n1 dy\ny dx  = log a\nor\ndy\ndx  = y log a\nThus\n(\nx)\nd\ndxa\n = ax log a\nAlternatively\n(\nx)\nd\ndxa\n =\nlog\nlog\n(\n)\n( log )\nx\na\nx\na\nd\nd\ne\ne\nx\na\ndx\ndx\n=\n= ex log a"
    },
    {
        "Chapter": "1",
        "sentence_range": "2295-2298",
        "Text": "r t x, we have\n1 dy\ny dx  = log a\nor\ndy\ndx  = y log a\nThus\n(\nx)\nd\ndxa\n = ax log a\nAlternatively\n(\nx)\nd\ndxa\n =\nlog\nlog\n(\n)\n( log )\nx\na\nx\na\nd\nd\ne\ne\nx\na\ndx\ndx\n=\n= ex log a log a = ax log a"
    },
    {
        "Chapter": "1",
        "sentence_range": "2296-2299",
        "Text": "t x, we have\n1 dy\ny dx  = log a\nor\ndy\ndx  = y log a\nThus\n(\nx)\nd\ndxa\n = ax log a\nAlternatively\n(\nx)\nd\ndxa\n =\nlog\nlog\n(\n)\n( log )\nx\na\nx\na\nd\nd\ne\ne\nx\na\ndx\ndx\n=\n= ex log a log a = ax log a Rationalised 2023-24\n MATHEMATICS\n132\nExample 29 Differentiate xsin x, x > 0 w"
    },
    {
        "Chapter": "1",
        "sentence_range": "2297-2300",
        "Text": "x, we have\n1 dy\ny dx  = log a\nor\ndy\ndx  = y log a\nThus\n(\nx)\nd\ndxa\n = ax log a\nAlternatively\n(\nx)\nd\ndxa\n =\nlog\nlog\n(\n)\n( log )\nx\na\nx\na\nd\nd\ne\ne\nx\na\ndx\ndx\n=\n= ex log a log a = ax log a Rationalised 2023-24\n MATHEMATICS\n132\nExample 29 Differentiate xsin x, x > 0 w r"
    },
    {
        "Chapter": "1",
        "sentence_range": "2298-2301",
        "Text": "log a = ax log a Rationalised 2023-24\n MATHEMATICS\n132\nExample 29 Differentiate xsin x, x > 0 w r t"
    },
    {
        "Chapter": "1",
        "sentence_range": "2299-2302",
        "Text": "Rationalised 2023-24\n MATHEMATICS\n132\nExample 29 Differentiate xsin x, x > 0 w r t x"
    },
    {
        "Chapter": "1",
        "sentence_range": "2300-2303",
        "Text": "r t x Solution Let y = xsin x"
    },
    {
        "Chapter": "1",
        "sentence_range": "2301-2304",
        "Text": "t x Solution Let y = xsin x Taking logarithm on both sides, we have\nlog y = sin x log x\nTherefore\n1"
    },
    {
        "Chapter": "1",
        "sentence_range": "2302-2305",
        "Text": "x Solution Let y = xsin x Taking logarithm on both sides, we have\nlog y = sin x log x\nTherefore\n1 dy\ny dx  = sin\n(log )\nlog\n(sin )\nd\nd\nx\nx\nx\nx\ndx\ndx\n+\nor\n1 dy\ny dx  =\n(sin )1\nlog\ncos\nx\nx\nx\nx +\nor\ndy\ndx  =\nsin\ncos\nlog\nx\ny\nx\nx\nx\n\uf8ee\n\uf8f9\n+\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n=\nsin\nsin\ncos\nlog\nx\nx\nx\nx\nx\nx\n\uf8ee\n\uf8f9\n+\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n=\nsin\n1\nsin\nsin\ncos\nlog\nx\nx\nx\nx\nx\nx\nx\n\u2212 \u22c5\n+\n\u22c5\nExample 30 Find dy\ndx , if yx + xy + xx = ab"
    },
    {
        "Chapter": "1",
        "sentence_range": "2303-2306",
        "Text": "Solution Let y = xsin x Taking logarithm on both sides, we have\nlog y = sin x log x\nTherefore\n1 dy\ny dx  = sin\n(log )\nlog\n(sin )\nd\nd\nx\nx\nx\nx\ndx\ndx\n+\nor\n1 dy\ny dx  =\n(sin )1\nlog\ncos\nx\nx\nx\nx +\nor\ndy\ndx  =\nsin\ncos\nlog\nx\ny\nx\nx\nx\n\uf8ee\n\uf8f9\n+\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n=\nsin\nsin\ncos\nlog\nx\nx\nx\nx\nx\nx\n\uf8ee\n\uf8f9\n+\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n=\nsin\n1\nsin\nsin\ncos\nlog\nx\nx\nx\nx\nx\nx\nx\n\u2212 \u22c5\n+\n\u22c5\nExample 30 Find dy\ndx , if yx + xy + xx = ab Solution Given that yx + xy + xx = ab"
    },
    {
        "Chapter": "1",
        "sentence_range": "2304-2307",
        "Text": "Taking logarithm on both sides, we have\nlog y = sin x log x\nTherefore\n1 dy\ny dx  = sin\n(log )\nlog\n(sin )\nd\nd\nx\nx\nx\nx\ndx\ndx\n+\nor\n1 dy\ny dx  =\n(sin )1\nlog\ncos\nx\nx\nx\nx +\nor\ndy\ndx  =\nsin\ncos\nlog\nx\ny\nx\nx\nx\n\uf8ee\n\uf8f9\n+\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n=\nsin\nsin\ncos\nlog\nx\nx\nx\nx\nx\nx\n\uf8ee\n\uf8f9\n+\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n=\nsin\n1\nsin\nsin\ncos\nlog\nx\nx\nx\nx\nx\nx\nx\n\u2212 \u22c5\n+\n\u22c5\nExample 30 Find dy\ndx , if yx + xy + xx = ab Solution Given that yx + xy + xx = ab Putting u = yx, v = xy and w = xx, we get u + v + w = ab\nTherefore\n0\ndu\ndv\ndw\ndx\ndx\ndx\n+\n+\n="
    },
    {
        "Chapter": "1",
        "sentence_range": "2305-2308",
        "Text": "dy\ny dx  = sin\n(log )\nlog\n(sin )\nd\nd\nx\nx\nx\nx\ndx\ndx\n+\nor\n1 dy\ny dx  =\n(sin )1\nlog\ncos\nx\nx\nx\nx +\nor\ndy\ndx  =\nsin\ncos\nlog\nx\ny\nx\nx\nx\n\uf8ee\n\uf8f9\n+\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n=\nsin\nsin\ncos\nlog\nx\nx\nx\nx\nx\nx\n\uf8ee\n\uf8f9\n+\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n=\nsin\n1\nsin\nsin\ncos\nlog\nx\nx\nx\nx\nx\nx\nx\n\u2212 \u22c5\n+\n\u22c5\nExample 30 Find dy\ndx , if yx + xy + xx = ab Solution Given that yx + xy + xx = ab Putting u = yx, v = xy and w = xx, we get u + v + w = ab\nTherefore\n0\ndu\ndv\ndw\ndx\ndx\ndx\n+\n+\n= (1)\nNow, u = yx"
    },
    {
        "Chapter": "1",
        "sentence_range": "2306-2309",
        "Text": "Solution Given that yx + xy + xx = ab Putting u = yx, v = xy and w = xx, we get u + v + w = ab\nTherefore\n0\ndu\ndv\ndw\ndx\ndx\ndx\n+\n+\n= (1)\nNow, u = yx Taking logarithm on both sides, we have\nlog u = x log y\nDifferentiating both sides w"
    },
    {
        "Chapter": "1",
        "sentence_range": "2307-2310",
        "Text": "Putting u = yx, v = xy and w = xx, we get u + v + w = ab\nTherefore\n0\ndu\ndv\ndw\ndx\ndx\ndx\n+\n+\n= (1)\nNow, u = yx Taking logarithm on both sides, we have\nlog u = x log y\nDifferentiating both sides w r"
    },
    {
        "Chapter": "1",
        "sentence_range": "2308-2311",
        "Text": "(1)\nNow, u = yx Taking logarithm on both sides, we have\nlog u = x log y\nDifferentiating both sides w r t"
    },
    {
        "Chapter": "1",
        "sentence_range": "2309-2312",
        "Text": "Taking logarithm on both sides, we have\nlog u = x log y\nDifferentiating both sides w r t x, we have\n1 du\nu dx\n\u22c5\n =\n(log )\nlog\n( )\nd\nd\nx\ny\ny\nx\ndx\ndx\n+\n=\n1\nlog\n1\ndy\nx\ny\ny dx\n\u22c5\n+\n\u22c5\nSo\ndu\ndx  =\nlog\nlog\nx\nx dy\nx dy\nu\ny\ny\ny\ny dx\ny dx\n\uf8eb\n\uf8f6\n\uf8ee\n\uf8f9\n+\n=\n+\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8ed\n\uf8f8\n\uf8f0\n\uf8fb"
    },
    {
        "Chapter": "1",
        "sentence_range": "2310-2313",
        "Text": "r t x, we have\n1 du\nu dx\n\u22c5\n =\n(log )\nlog\n( )\nd\nd\nx\ny\ny\nx\ndx\ndx\n+\n=\n1\nlog\n1\ndy\nx\ny\ny dx\n\u22c5\n+\n\u22c5\nSo\ndu\ndx  =\nlog\nlog\nx\nx dy\nx dy\nu\ny\ny\ny\ny dx\ny dx\n\uf8eb\n\uf8f6\n\uf8ee\n\uf8f9\n+\n=\n+\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8ed\n\uf8f8\n\uf8f0\n\uf8fb (2)\nAlso v = xy\nRationalised 2023-24\nCONTINUITY AND DIFFERENTIABILITY\n133\nTaking logarithm on both sides, we have\nlog v = y log x\nDifferentiating both sides w"
    },
    {
        "Chapter": "1",
        "sentence_range": "2311-2314",
        "Text": "t x, we have\n1 du\nu dx\n\u22c5\n =\n(log )\nlog\n( )\nd\nd\nx\ny\ny\nx\ndx\ndx\n+\n=\n1\nlog\n1\ndy\nx\ny\ny dx\n\u22c5\n+\n\u22c5\nSo\ndu\ndx  =\nlog\nlog\nx\nx dy\nx dy\nu\ny\ny\ny\ny dx\ny dx\n\uf8eb\n\uf8f6\n\uf8ee\n\uf8f9\n+\n=\n+\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8ed\n\uf8f8\n\uf8f0\n\uf8fb (2)\nAlso v = xy\nRationalised 2023-24\nCONTINUITY AND DIFFERENTIABILITY\n133\nTaking logarithm on both sides, we have\nlog v = y log x\nDifferentiating both sides w r"
    },
    {
        "Chapter": "1",
        "sentence_range": "2312-2315",
        "Text": "x, we have\n1 du\nu dx\n\u22c5\n =\n(log )\nlog\n( )\nd\nd\nx\ny\ny\nx\ndx\ndx\n+\n=\n1\nlog\n1\ndy\nx\ny\ny dx\n\u22c5\n+\n\u22c5\nSo\ndu\ndx  =\nlog\nlog\nx\nx dy\nx dy\nu\ny\ny\ny\ny dx\ny dx\n\uf8eb\n\uf8f6\n\uf8ee\n\uf8f9\n+\n=\n+\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8ed\n\uf8f8\n\uf8f0\n\uf8fb (2)\nAlso v = xy\nRationalised 2023-24\nCONTINUITY AND DIFFERENTIABILITY\n133\nTaking logarithm on both sides, we have\nlog v = y log x\nDifferentiating both sides w r t"
    },
    {
        "Chapter": "1",
        "sentence_range": "2313-2316",
        "Text": "(2)\nAlso v = xy\nRationalised 2023-24\nCONTINUITY AND DIFFERENTIABILITY\n133\nTaking logarithm on both sides, we have\nlog v = y log x\nDifferentiating both sides w r t x, we have\n1 dv\nv dx\n\u22c5\n =\n(log )\nlog\nd\ndy\ny\nx\nx\ndx\ndx\n+\n=\n1\nlog\ndy\ny\nx\nx\ndx\n\u22c5\n+\n\u22c5\nSo\ndv\ndx  =\nlog\ny\ndy\nv\nx\nx\ndx\n\uf8ee\n\uf8f9\n+\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n=\nlog\ny\ny\ndy\nx\nx\nx\ndx\n\uf8ee\n\uf8f9\n+\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb"
    },
    {
        "Chapter": "1",
        "sentence_range": "2314-2317",
        "Text": "r t x, we have\n1 dv\nv dx\n\u22c5\n =\n(log )\nlog\nd\ndy\ny\nx\nx\ndx\ndx\n+\n=\n1\nlog\ndy\ny\nx\nx\ndx\n\u22c5\n+\n\u22c5\nSo\ndv\ndx  =\nlog\ny\ndy\nv\nx\nx\ndx\n\uf8ee\n\uf8f9\n+\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n=\nlog\ny\ny\ndy\nx\nx\nx\ndx\n\uf8ee\n\uf8f9\n+\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb (3)\nAgain\nw = xx\nTaking logarithm on both sides, we have\nlog w = x log x"
    },
    {
        "Chapter": "1",
        "sentence_range": "2315-2318",
        "Text": "t x, we have\n1 dv\nv dx\n\u22c5\n =\n(log )\nlog\nd\ndy\ny\nx\nx\ndx\ndx\n+\n=\n1\nlog\ndy\ny\nx\nx\ndx\n\u22c5\n+\n\u22c5\nSo\ndv\ndx  =\nlog\ny\ndy\nv\nx\nx\ndx\n\uf8ee\n\uf8f9\n+\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n=\nlog\ny\ny\ndy\nx\nx\nx\ndx\n\uf8ee\n\uf8f9\n+\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb (3)\nAgain\nw = xx\nTaking logarithm on both sides, we have\nlog w = x log x Differentiating both sides w"
    },
    {
        "Chapter": "1",
        "sentence_range": "2316-2319",
        "Text": "x, we have\n1 dv\nv dx\n\u22c5\n =\n(log )\nlog\nd\ndy\ny\nx\nx\ndx\ndx\n+\n=\n1\nlog\ndy\ny\nx\nx\ndx\n\u22c5\n+\n\u22c5\nSo\ndv\ndx  =\nlog\ny\ndy\nv\nx\nx\ndx\n\uf8ee\n\uf8f9\n+\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n=\nlog\ny\ny\ndy\nx\nx\nx\ndx\n\uf8ee\n\uf8f9\n+\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb (3)\nAgain\nw = xx\nTaking logarithm on both sides, we have\nlog w = x log x Differentiating both sides w r"
    },
    {
        "Chapter": "1",
        "sentence_range": "2317-2320",
        "Text": "(3)\nAgain\nw = xx\nTaking logarithm on both sides, we have\nlog w = x log x Differentiating both sides w r t"
    },
    {
        "Chapter": "1",
        "sentence_range": "2318-2321",
        "Text": "Differentiating both sides w r t x, we have\n1 dw\nw dx\n\u22c5\n =\n(log )\nlog\n( )\nd\nd\nx\nx\nx\nx\ndx\ndx\n+\n\u22c5\n=\n1\nlog\n1\nx\nx\n\u22c5x\n+\n\u22c5\ni"
    },
    {
        "Chapter": "1",
        "sentence_range": "2319-2322",
        "Text": "r t x, we have\n1 dw\nw dx\n\u22c5\n =\n(log )\nlog\n( )\nd\nd\nx\nx\nx\nx\ndx\ndx\n+\n\u22c5\n=\n1\nlog\n1\nx\nx\n\u22c5x\n+\n\u22c5\ni e"
    },
    {
        "Chapter": "1",
        "sentence_range": "2320-2323",
        "Text": "t x, we have\n1 dw\nw dx\n\u22c5\n =\n(log )\nlog\n( )\nd\nd\nx\nx\nx\nx\ndx\ndx\n+\n\u22c5\n=\n1\nlog\n1\nx\nx\n\u22c5x\n+\n\u22c5\ni e dw\ndx  = w (1 + log x)\n= xx (1 + log x)"
    },
    {
        "Chapter": "1",
        "sentence_range": "2321-2324",
        "Text": "x, we have\n1 dw\nw dx\n\u22c5\n =\n(log )\nlog\n( )\nd\nd\nx\nx\nx\nx\ndx\ndx\n+\n\u22c5\n=\n1\nlog\n1\nx\nx\n\u22c5x\n+\n\u22c5\ni e dw\ndx  = w (1 + log x)\n= xx (1 + log x) (4)\nFrom (1), (2), (3), (4), we have\nlog\nlog\nx\ny\nx dy\ny\ndy\ny\ny\nx\nx\ny dx\nx\ndx\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n+\n+\n+\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n + xx (1 + log x) = 0\nor\n(x"
    },
    {
        "Chapter": "1",
        "sentence_range": "2322-2325",
        "Text": "e dw\ndx  = w (1 + log x)\n= xx (1 + log x) (4)\nFrom (1), (2), (3), (4), we have\nlog\nlog\nx\ny\nx dy\ny\ndy\ny\ny\nx\nx\ny dx\nx\ndx\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n+\n+\n+\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n + xx (1 + log x) = 0\nor\n(x yx \u2013 1 + xy"
    },
    {
        "Chapter": "1",
        "sentence_range": "2323-2326",
        "Text": "dw\ndx  = w (1 + log x)\n= xx (1 + log x) (4)\nFrom (1), (2), (3), (4), we have\nlog\nlog\nx\ny\nx dy\ny\ndy\ny\ny\nx\nx\ny dx\nx\ndx\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n+\n+\n+\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n + xx (1 + log x) = 0\nor\n(x yx \u2013 1 + xy log x) dy\ndx  = \u2013 xx (1 + log x) \u2013 y"
    },
    {
        "Chapter": "1",
        "sentence_range": "2324-2327",
        "Text": "(4)\nFrom (1), (2), (3), (4), we have\nlog\nlog\nx\ny\nx dy\ny\ndy\ny\ny\nx\nx\ny dx\nx\ndx\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n+\n+\n+\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n + xx (1 + log x) = 0\nor\n(x yx \u2013 1 + xy log x) dy\ndx  = \u2013 xx (1 + log x) \u2013 y xy\u20131 \u2013 yx log y\nTherefore\ndy\ndx  =\n1\n1\n[\nlog"
    },
    {
        "Chapter": "1",
        "sentence_range": "2325-2328",
        "Text": "yx \u2013 1 + xy log x) dy\ndx  = \u2013 xx (1 + log x) \u2013 y xy\u20131 \u2013 yx log y\nTherefore\ndy\ndx  =\n1\n1\n[\nlog (1\nlog )]"
    },
    {
        "Chapter": "1",
        "sentence_range": "2326-2329",
        "Text": "log x) dy\ndx  = \u2013 xx (1 + log x) \u2013 y xy\u20131 \u2013 yx log y\nTherefore\ndy\ndx  =\n1\n1\n[\nlog (1\nlog )] log\nx\ny\nx\nx\ny\ny\ny\ny x\nx\nx\nx y\nx\nx\n\u2212\n\u2212\n\u2212\n+\n+\n+\n+\nRationalised 2023-24\n MATHEMATICS\n134\nEXERCISE 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "2327-2330",
        "Text": "xy\u20131 \u2013 yx log y\nTherefore\ndy\ndx  =\n1\n1\n[\nlog (1\nlog )] log\nx\ny\nx\nx\ny\ny\ny\ny x\nx\nx\nx y\nx\nx\n\u2212\n\u2212\n\u2212\n+\n+\n+\n+\nRationalised 2023-24\n MATHEMATICS\n134\nEXERCISE 5 5\nDifferentiate the functions given in Exercises 1 to 11 w"
    },
    {
        "Chapter": "1",
        "sentence_range": "2328-2331",
        "Text": "(1\nlog )] log\nx\ny\nx\nx\ny\ny\ny\ny x\nx\nx\nx y\nx\nx\n\u2212\n\u2212\n\u2212\n+\n+\n+\n+\nRationalised 2023-24\n MATHEMATICS\n134\nEXERCISE 5 5\nDifferentiate the functions given in Exercises 1 to 11 w r"
    },
    {
        "Chapter": "1",
        "sentence_range": "2329-2332",
        "Text": "log\nx\ny\nx\nx\ny\ny\ny\ny x\nx\nx\nx y\nx\nx\n\u2212\n\u2212\n\u2212\n+\n+\n+\n+\nRationalised 2023-24\n MATHEMATICS\n134\nEXERCISE 5 5\nDifferentiate the functions given in Exercises 1 to 11 w r t"
    },
    {
        "Chapter": "1",
        "sentence_range": "2330-2333",
        "Text": "5\nDifferentiate the functions given in Exercises 1 to 11 w r t x"
    },
    {
        "Chapter": "1",
        "sentence_range": "2331-2334",
        "Text": "r t x 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "2332-2335",
        "Text": "t x 1 cos x"
    },
    {
        "Chapter": "1",
        "sentence_range": "2333-2336",
        "Text": "x 1 cos x cos 2x"
    },
    {
        "Chapter": "1",
        "sentence_range": "2334-2337",
        "Text": "1 cos x cos 2x cos 3x\n2"
    },
    {
        "Chapter": "1",
        "sentence_range": "2335-2338",
        "Text": "cos x cos 2x cos 3x\n2 (\n1) (\n2)\n(\n3) (\n4) (\n5)\nx\nx\nx\nx\nx\n\u2212\n\u2212\n\u2212\n\u2212\n\u2212\n3"
    },
    {
        "Chapter": "1",
        "sentence_range": "2336-2339",
        "Text": "cos 2x cos 3x\n2 (\n1) (\n2)\n(\n3) (\n4) (\n5)\nx\nx\nx\nx\nx\n\u2212\n\u2212\n\u2212\n\u2212\n\u2212\n3 (log x)cos x\n4"
    },
    {
        "Chapter": "1",
        "sentence_range": "2337-2340",
        "Text": "cos 3x\n2 (\n1) (\n2)\n(\n3) (\n4) (\n5)\nx\nx\nx\nx\nx\n\u2212\n\u2212\n\u2212\n\u2212\n\u2212\n3 (log x)cos x\n4 xx \u2013 2sin x\n5"
    },
    {
        "Chapter": "1",
        "sentence_range": "2338-2341",
        "Text": "(\n1) (\n2)\n(\n3) (\n4) (\n5)\nx\nx\nx\nx\nx\n\u2212\n\u2212\n\u2212\n\u2212\n\u2212\n3 (log x)cos x\n4 xx \u2013 2sin x\n5 (x + 3)2"
    },
    {
        "Chapter": "1",
        "sentence_range": "2339-2342",
        "Text": "(log x)cos x\n4 xx \u2013 2sin x\n5 (x + 3)2 (x + 4)3"
    },
    {
        "Chapter": "1",
        "sentence_range": "2340-2343",
        "Text": "xx \u2013 2sin x\n5 (x + 3)2 (x + 4)3 (x + 5)4\n6"
    },
    {
        "Chapter": "1",
        "sentence_range": "2341-2344",
        "Text": "(x + 3)2 (x + 4)3 (x + 5)4\n6 11\n1\nx\nx\nx\nx\nx\n\uf8eb\n\uf8f6\n\uf8ec+\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8eb\n\uf8f6\n+\n+\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n7"
    },
    {
        "Chapter": "1",
        "sentence_range": "2342-2345",
        "Text": "(x + 4)3 (x + 5)4\n6 11\n1\nx\nx\nx\nx\nx\n\uf8eb\n\uf8f6\n\uf8ec+\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8eb\n\uf8f6\n+\n+\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n7 (log x)x + xlog x\n8"
    },
    {
        "Chapter": "1",
        "sentence_range": "2343-2346",
        "Text": "(x + 5)4\n6 11\n1\nx\nx\nx\nx\nx\n\uf8eb\n\uf8f6\n\uf8ec+\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8eb\n\uf8f6\n+\n+\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n7 (log x)x + xlog x\n8 (sin x)x + sin\u20131 \nx\n9"
    },
    {
        "Chapter": "1",
        "sentence_range": "2344-2347",
        "Text": "11\n1\nx\nx\nx\nx\nx\n\uf8eb\n\uf8f6\n\uf8ec+\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8eb\n\uf8f6\n+\n+\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n7 (log x)x + xlog x\n8 (sin x)x + sin\u20131 \nx\n9 xsin x + (sin x)cos x\n10"
    },
    {
        "Chapter": "1",
        "sentence_range": "2345-2348",
        "Text": "(log x)x + xlog x\n8 (sin x)x + sin\u20131 \nx\n9 xsin x + (sin x)cos x\n10 2\ncos\n2\n1\n1\nx\nx\nx\nx\nx\n+\n+\n\u2212\n11"
    },
    {
        "Chapter": "1",
        "sentence_range": "2346-2349",
        "Text": "(sin x)x + sin\u20131 \nx\n9 xsin x + (sin x)cos x\n10 2\ncos\n2\n1\n1\nx\nx\nx\nx\nx\n+\n+\n\u2212\n11 (x cos x)x + \n1\n( sin ) x\nx\nx\nFind dy\ndx of the functions given in Exercises 12 to 15"
    },
    {
        "Chapter": "1",
        "sentence_range": "2347-2350",
        "Text": "xsin x + (sin x)cos x\n10 2\ncos\n2\n1\n1\nx\nx\nx\nx\nx\n+\n+\n\u2212\n11 (x cos x)x + \n1\n( sin ) x\nx\nx\nFind dy\ndx of the functions given in Exercises 12 to 15 12"
    },
    {
        "Chapter": "1",
        "sentence_range": "2348-2351",
        "Text": "2\ncos\n2\n1\n1\nx\nx\nx\nx\nx\n+\n+\n\u2212\n11 (x cos x)x + \n1\n( sin ) x\nx\nx\nFind dy\ndx of the functions given in Exercises 12 to 15 12 xy + yx = 1\n13"
    },
    {
        "Chapter": "1",
        "sentence_range": "2349-2352",
        "Text": "(x cos x)x + \n1\n( sin ) x\nx\nx\nFind dy\ndx of the functions given in Exercises 12 to 15 12 xy + yx = 1\n13 yx = xy\n14"
    },
    {
        "Chapter": "1",
        "sentence_range": "2350-2353",
        "Text": "12 xy + yx = 1\n13 yx = xy\n14 (cos x)y = (cos y)x\n15"
    },
    {
        "Chapter": "1",
        "sentence_range": "2351-2354",
        "Text": "xy + yx = 1\n13 yx = xy\n14 (cos x)y = (cos y)x\n15 xy = e(x \u2013 y)\n16"
    },
    {
        "Chapter": "1",
        "sentence_range": "2352-2355",
        "Text": "yx = xy\n14 (cos x)y = (cos y)x\n15 xy = e(x \u2013 y)\n16 Find the derivative of the function given by f(x) = (1 + x) (1 + x2) (1 + x4) (1 + x8)\nand hence find f\u2032(1)"
    },
    {
        "Chapter": "1",
        "sentence_range": "2353-2356",
        "Text": "(cos x)y = (cos y)x\n15 xy = e(x \u2013 y)\n16 Find the derivative of the function given by f(x) = (1 + x) (1 + x2) (1 + x4) (1 + x8)\nand hence find f\u2032(1) 17"
    },
    {
        "Chapter": "1",
        "sentence_range": "2354-2357",
        "Text": "xy = e(x \u2013 y)\n16 Find the derivative of the function given by f(x) = (1 + x) (1 + x2) (1 + x4) (1 + x8)\nand hence find f\u2032(1) 17 Differentiate (x2 \u2013 5x + 8) (x3 + 7x + 9) in three ways mentioned below:\n(i) by using product rule\n(ii) by expanding the product to obtain a single polynomial"
    },
    {
        "Chapter": "1",
        "sentence_range": "2355-2358",
        "Text": "Find the derivative of the function given by f(x) = (1 + x) (1 + x2) (1 + x4) (1 + x8)\nand hence find f\u2032(1) 17 Differentiate (x2 \u2013 5x + 8) (x3 + 7x + 9) in three ways mentioned below:\n(i) by using product rule\n(ii) by expanding the product to obtain a single polynomial (iii) by logarithmic differentiation"
    },
    {
        "Chapter": "1",
        "sentence_range": "2356-2359",
        "Text": "17 Differentiate (x2 \u2013 5x + 8) (x3 + 7x + 9) in three ways mentioned below:\n(i) by using product rule\n(ii) by expanding the product to obtain a single polynomial (iii) by logarithmic differentiation Do they all give the same answer"
    },
    {
        "Chapter": "1",
        "sentence_range": "2357-2360",
        "Text": "Differentiate (x2 \u2013 5x + 8) (x3 + 7x + 9) in three ways mentioned below:\n(i) by using product rule\n(ii) by expanding the product to obtain a single polynomial (iii) by logarithmic differentiation Do they all give the same answer 18"
    },
    {
        "Chapter": "1",
        "sentence_range": "2358-2361",
        "Text": "(iii) by logarithmic differentiation Do they all give the same answer 18 If u, v and w are functions of x, then show that\nd\ndx  (u"
    },
    {
        "Chapter": "1",
        "sentence_range": "2359-2362",
        "Text": "Do they all give the same answer 18 If u, v and w are functions of x, then show that\nd\ndx  (u v"
    },
    {
        "Chapter": "1",
        "sentence_range": "2360-2363",
        "Text": "18 If u, v and w are functions of x, then show that\nd\ndx  (u v w) = du\ndx  v"
    },
    {
        "Chapter": "1",
        "sentence_range": "2361-2364",
        "Text": "If u, v and w are functions of x, then show that\nd\ndx  (u v w) = du\ndx  v w + u"
    },
    {
        "Chapter": "1",
        "sentence_range": "2362-2365",
        "Text": "v w) = du\ndx  v w + u dv\ndx"
    },
    {
        "Chapter": "1",
        "sentence_range": "2363-2366",
        "Text": "w) = du\ndx  v w + u dv\ndx w + u"
    },
    {
        "Chapter": "1",
        "sentence_range": "2364-2367",
        "Text": "w + u dv\ndx w + u v dw\ndx\nin two ways - first by repeated application of product rule, second by logarithmic\ndifferentiation"
    },
    {
        "Chapter": "1",
        "sentence_range": "2365-2368",
        "Text": "dv\ndx w + u v dw\ndx\nin two ways - first by repeated application of product rule, second by logarithmic\ndifferentiation 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "2366-2369",
        "Text": "w + u v dw\ndx\nin two ways - first by repeated application of product rule, second by logarithmic\ndifferentiation 5 6  Derivatives of Functions in Parametric Forms\nSometimes the relation between two variables is neither explicit nor implicit, but some\nlink of a third variable with each of the two variables, separately, establishes a relation\nbetween the first two variables"
    },
    {
        "Chapter": "1",
        "sentence_range": "2367-2370",
        "Text": "v dw\ndx\nin two ways - first by repeated application of product rule, second by logarithmic\ndifferentiation 5 6  Derivatives of Functions in Parametric Forms\nSometimes the relation between two variables is neither explicit nor implicit, but some\nlink of a third variable with each of the two variables, separately, establishes a relation\nbetween the first two variables In such a situation, we say that the relation between\nRationalised 2023-24\nCONTINUITY AND DIFFERENTIABILITY\n135\nthem is expressed via a third variable"
    },
    {
        "Chapter": "1",
        "sentence_range": "2368-2371",
        "Text": "5 6  Derivatives of Functions in Parametric Forms\nSometimes the relation between two variables is neither explicit nor implicit, but some\nlink of a third variable with each of the two variables, separately, establishes a relation\nbetween the first two variables In such a situation, we say that the relation between\nRationalised 2023-24\nCONTINUITY AND DIFFERENTIABILITY\n135\nthem is expressed via a third variable The third variable is called the parameter"
    },
    {
        "Chapter": "1",
        "sentence_range": "2369-2372",
        "Text": "6  Derivatives of Functions in Parametric Forms\nSometimes the relation between two variables is neither explicit nor implicit, but some\nlink of a third variable with each of the two variables, separately, establishes a relation\nbetween the first two variables In such a situation, we say that the relation between\nRationalised 2023-24\nCONTINUITY AND DIFFERENTIABILITY\n135\nthem is expressed via a third variable The third variable is called the parameter More\nprecisely, a relation expressed between two variables x and y in the form\nx = f(t), y = g (t) is said to be parametric form with t as a parameter"
    },
    {
        "Chapter": "1",
        "sentence_range": "2370-2373",
        "Text": "In such a situation, we say that the relation between\nRationalised 2023-24\nCONTINUITY AND DIFFERENTIABILITY\n135\nthem is expressed via a third variable The third variable is called the parameter More\nprecisely, a relation expressed between two variables x and y in the form\nx = f(t), y = g (t) is said to be parametric form with t as a parameter In order to find derivative of function in such form, we have by chain rule"
    },
    {
        "Chapter": "1",
        "sentence_range": "2371-2374",
        "Text": "The third variable is called the parameter More\nprecisely, a relation expressed between two variables x and y in the form\nx = f(t), y = g (t) is said to be parametric form with t as a parameter In order to find derivative of function in such form, we have by chain rule dy\ndt  = dy dx\ndx dt\n\u22c5\nor\ndy\ndx  =\nwhenever\n0\ndy\ndx\ndt\ndx\ndt\ndt\n\uf8eb\n\uf8f6\n\u2260\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\nThus\ndy\ndx  =\n( )\nas\n( ) and\n( )\ng t( )\ndy\ndx\ng t\nf t\nf t\ndt\ndt\n\u2032\n\uf8eb\n\uf8f6\n=\n\u2032\n=\n\u2032\n\uf8ec\n\uf8f7\n\u2032\n\uf8ed\n\uf8f8  [provided f\u2032(t) \u2260 0]\nExample 31 Find dy\ndx , if x = a cos \u03b8, y = a sin \u03b8"
    },
    {
        "Chapter": "1",
        "sentence_range": "2372-2375",
        "Text": "More\nprecisely, a relation expressed between two variables x and y in the form\nx = f(t), y = g (t) is said to be parametric form with t as a parameter In order to find derivative of function in such form, we have by chain rule dy\ndt  = dy dx\ndx dt\n\u22c5\nor\ndy\ndx  =\nwhenever\n0\ndy\ndx\ndt\ndx\ndt\ndt\n\uf8eb\n\uf8f6\n\u2260\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\nThus\ndy\ndx  =\n( )\nas\n( ) and\n( )\ng t( )\ndy\ndx\ng t\nf t\nf t\ndt\ndt\n\u2032\n\uf8eb\n\uf8f6\n=\n\u2032\n=\n\u2032\n\uf8ec\n\uf8f7\n\u2032\n\uf8ed\n\uf8f8  [provided f\u2032(t) \u2260 0]\nExample 31 Find dy\ndx , if x = a cos \u03b8, y = a sin \u03b8 Solution Given that\nx = a cos \u03b8, y = a sin \u03b8\nTherefore\ndx\nd\u03b8  = \u2013 a sin \u03b8, dy\nd\u03b8  = a cos \u03b8\nHence\ndy\ndx  =\ncos\ncot\nsin\ndy\na\nd\ndx\na\nd\n\u03b8\n\u03b8 =\n= \u2212\n\u03b8\n\u2212\n\u03b8\n\u03b8\nExample 32 Find dy\ndx\n, if x = at2, y = 2at"
    },
    {
        "Chapter": "1",
        "sentence_range": "2373-2376",
        "Text": "In order to find derivative of function in such form, we have by chain rule dy\ndt  = dy dx\ndx dt\n\u22c5\nor\ndy\ndx  =\nwhenever\n0\ndy\ndx\ndt\ndx\ndt\ndt\n\uf8eb\n\uf8f6\n\u2260\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\nThus\ndy\ndx  =\n( )\nas\n( ) and\n( )\ng t( )\ndy\ndx\ng t\nf t\nf t\ndt\ndt\n\u2032\n\uf8eb\n\uf8f6\n=\n\u2032\n=\n\u2032\n\uf8ec\n\uf8f7\n\u2032\n\uf8ed\n\uf8f8  [provided f\u2032(t) \u2260 0]\nExample 31 Find dy\ndx , if x = a cos \u03b8, y = a sin \u03b8 Solution Given that\nx = a cos \u03b8, y = a sin \u03b8\nTherefore\ndx\nd\u03b8  = \u2013 a sin \u03b8, dy\nd\u03b8  = a cos \u03b8\nHence\ndy\ndx  =\ncos\ncot\nsin\ndy\na\nd\ndx\na\nd\n\u03b8\n\u03b8 =\n= \u2212\n\u03b8\n\u2212\n\u03b8\n\u03b8\nExample 32 Find dy\ndx\n, if x = at2, y = 2at Solution Given that x = at2, y = 2at\nSo\ndx\ndt  = 2at   and   dy\ndt = 2a\nTherefore\ndy\ndx  =\n2\n1\n2\ndy\na\ndt\ndx\nat\nt\ndt\n=\n=\nRationalised 2023-24\n MATHEMATICS\n136\nExample 33 Find dy\ndx\n, if x = a (\u03b8 + sin \u03b8), y = a (1 \u2013 cos \u03b8)"
    },
    {
        "Chapter": "1",
        "sentence_range": "2374-2377",
        "Text": "dy\ndt  = dy dx\ndx dt\n\u22c5\nor\ndy\ndx  =\nwhenever\n0\ndy\ndx\ndt\ndx\ndt\ndt\n\uf8eb\n\uf8f6\n\u2260\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\nThus\ndy\ndx  =\n( )\nas\n( ) and\n( )\ng t( )\ndy\ndx\ng t\nf t\nf t\ndt\ndt\n\u2032\n\uf8eb\n\uf8f6\n=\n\u2032\n=\n\u2032\n\uf8ec\n\uf8f7\n\u2032\n\uf8ed\n\uf8f8  [provided f\u2032(t) \u2260 0]\nExample 31 Find dy\ndx , if x = a cos \u03b8, y = a sin \u03b8 Solution Given that\nx = a cos \u03b8, y = a sin \u03b8\nTherefore\ndx\nd\u03b8  = \u2013 a sin \u03b8, dy\nd\u03b8  = a cos \u03b8\nHence\ndy\ndx  =\ncos\ncot\nsin\ndy\na\nd\ndx\na\nd\n\u03b8\n\u03b8 =\n= \u2212\n\u03b8\n\u2212\n\u03b8\n\u03b8\nExample 32 Find dy\ndx\n, if x = at2, y = 2at Solution Given that x = at2, y = 2at\nSo\ndx\ndt  = 2at   and   dy\ndt = 2a\nTherefore\ndy\ndx  =\n2\n1\n2\ndy\na\ndt\ndx\nat\nt\ndt\n=\n=\nRationalised 2023-24\n MATHEMATICS\n136\nExample 33 Find dy\ndx\n, if x = a (\u03b8 + sin \u03b8), y = a (1 \u2013 cos \u03b8) Solution We have dx\nd\u03b8 = a(1 + cos \u03b8), dy\nd\u03b8  = a (sin \u03b8)\nTherefore\ndy\ndx  =\nsin\ntan\n(1\ncos )\n2\ndy\na\nd\ndx\na\nd\n\u03b8\n\u03b8\n\u03b8 =\n=\n+\n\u03b8\n\u03b8\nANote  It may be noted here that dy\ndx  is expressed in terms of parameter only\nwithout directly involving the main variables x and y"
    },
    {
        "Chapter": "1",
        "sentence_range": "2375-2378",
        "Text": "Solution Given that\nx = a cos \u03b8, y = a sin \u03b8\nTherefore\ndx\nd\u03b8  = \u2013 a sin \u03b8, dy\nd\u03b8  = a cos \u03b8\nHence\ndy\ndx  =\ncos\ncot\nsin\ndy\na\nd\ndx\na\nd\n\u03b8\n\u03b8 =\n= \u2212\n\u03b8\n\u2212\n\u03b8\n\u03b8\nExample 32 Find dy\ndx\n, if x = at2, y = 2at Solution Given that x = at2, y = 2at\nSo\ndx\ndt  = 2at   and   dy\ndt = 2a\nTherefore\ndy\ndx  =\n2\n1\n2\ndy\na\ndt\ndx\nat\nt\ndt\n=\n=\nRationalised 2023-24\n MATHEMATICS\n136\nExample 33 Find dy\ndx\n, if x = a (\u03b8 + sin \u03b8), y = a (1 \u2013 cos \u03b8) Solution We have dx\nd\u03b8 = a(1 + cos \u03b8), dy\nd\u03b8  = a (sin \u03b8)\nTherefore\ndy\ndx  =\nsin\ntan\n(1\ncos )\n2\ndy\na\nd\ndx\na\nd\n\u03b8\n\u03b8\n\u03b8 =\n=\n+\n\u03b8\n\u03b8\nANote  It may be noted here that dy\ndx  is expressed in terms of parameter only\nwithout directly involving the main variables x and y Example 34 Find \n2\n2\n2\n3\n3\n3\ndy, if\nx\ny\na\ndx\n+\n="
    },
    {
        "Chapter": "1",
        "sentence_range": "2376-2379",
        "Text": "Solution Given that x = at2, y = 2at\nSo\ndx\ndt  = 2at   and   dy\ndt = 2a\nTherefore\ndy\ndx  =\n2\n1\n2\ndy\na\ndt\ndx\nat\nt\ndt\n=\n=\nRationalised 2023-24\n MATHEMATICS\n136\nExample 33 Find dy\ndx\n, if x = a (\u03b8 + sin \u03b8), y = a (1 \u2013 cos \u03b8) Solution We have dx\nd\u03b8 = a(1 + cos \u03b8), dy\nd\u03b8  = a (sin \u03b8)\nTherefore\ndy\ndx  =\nsin\ntan\n(1\ncos )\n2\ndy\na\nd\ndx\na\nd\n\u03b8\n\u03b8\n\u03b8 =\n=\n+\n\u03b8\n\u03b8\nANote  It may be noted here that dy\ndx  is expressed in terms of parameter only\nwithout directly involving the main variables x and y Example 34 Find \n2\n2\n2\n3\n3\n3\ndy, if\nx\ny\na\ndx\n+\n= Solution Let x = a cos3 \u03b8, y = a sin3 \u03b8"
    },
    {
        "Chapter": "1",
        "sentence_range": "2377-2380",
        "Text": "Solution We have dx\nd\u03b8 = a(1 + cos \u03b8), dy\nd\u03b8  = a (sin \u03b8)\nTherefore\ndy\ndx  =\nsin\ntan\n(1\ncos )\n2\ndy\na\nd\ndx\na\nd\n\u03b8\n\u03b8\n\u03b8 =\n=\n+\n\u03b8\n\u03b8\nANote  It may be noted here that dy\ndx  is expressed in terms of parameter only\nwithout directly involving the main variables x and y Example 34 Find \n2\n2\n2\n3\n3\n3\ndy, if\nx\ny\na\ndx\n+\n= Solution Let x = a cos3 \u03b8, y = a sin3 \u03b8 Then\n2\n2\n3\n3\nx\n+y\n =\n2\n2\n3\n3\n3\n3\n( cos\n)\n( sin\n)\na\na\n\u03b8\n+\n\u03b8\n=\n2\n2\n2\n2\n3\n3\n(cos\n(sin\n)\na\na\n\u03b8 +\n\u03b8 =\nHence, x = a cos3\u03b8, y = a sin3\u03b8 is parametric equation of \n2\n2\n2\n3\n3\n3\nx\ny\na\n+\n=\nNow\ndx\nd\u03b8  = \u2013 3a cos2 \u03b8 sin \u03b8 and dy\nd\u03b8  = 3a sin2 \u03b8 cos \u03b8\nTherefore\ndy\ndx  =\n2\n3\n3 sin2\ncos\ntan\n3 cos\nsin\ndy\na\ny\nd\ndx\nx\na\nd\n\u03b8\n\u03b8\n\u03b8 =\n= \u2212\n\u03b8 = \u2212\n\u2212\n\u03b8\n\u03b8\n\u03b8\nRationalised 2023-24\nCONTINUITY AND DIFFERENTIABILITY\n137\nEXERCISE 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "2378-2381",
        "Text": "Example 34 Find \n2\n2\n2\n3\n3\n3\ndy, if\nx\ny\na\ndx\n+\n= Solution Let x = a cos3 \u03b8, y = a sin3 \u03b8 Then\n2\n2\n3\n3\nx\n+y\n =\n2\n2\n3\n3\n3\n3\n( cos\n)\n( sin\n)\na\na\n\u03b8\n+\n\u03b8\n=\n2\n2\n2\n2\n3\n3\n(cos\n(sin\n)\na\na\n\u03b8 +\n\u03b8 =\nHence, x = a cos3\u03b8, y = a sin3\u03b8 is parametric equation of \n2\n2\n2\n3\n3\n3\nx\ny\na\n+\n=\nNow\ndx\nd\u03b8  = \u2013 3a cos2 \u03b8 sin \u03b8 and dy\nd\u03b8  = 3a sin2 \u03b8 cos \u03b8\nTherefore\ndy\ndx  =\n2\n3\n3 sin2\ncos\ntan\n3 cos\nsin\ndy\na\ny\nd\ndx\nx\na\nd\n\u03b8\n\u03b8\n\u03b8 =\n= \u2212\n\u03b8 = \u2212\n\u2212\n\u03b8\n\u03b8\n\u03b8\nRationalised 2023-24\nCONTINUITY AND DIFFERENTIABILITY\n137\nEXERCISE 5 6\nIf  x and y are connected parametrically by the equations given in Exercises 1 to 10,\nwithout eliminating the parameter, Find dy\ndx"
    },
    {
        "Chapter": "1",
        "sentence_range": "2379-2382",
        "Text": "Solution Let x = a cos3 \u03b8, y = a sin3 \u03b8 Then\n2\n2\n3\n3\nx\n+y\n =\n2\n2\n3\n3\n3\n3\n( cos\n)\n( sin\n)\na\na\n\u03b8\n+\n\u03b8\n=\n2\n2\n2\n2\n3\n3\n(cos\n(sin\n)\na\na\n\u03b8 +\n\u03b8 =\nHence, x = a cos3\u03b8, y = a sin3\u03b8 is parametric equation of \n2\n2\n2\n3\n3\n3\nx\ny\na\n+\n=\nNow\ndx\nd\u03b8  = \u2013 3a cos2 \u03b8 sin \u03b8 and dy\nd\u03b8  = 3a sin2 \u03b8 cos \u03b8\nTherefore\ndy\ndx  =\n2\n3\n3 sin2\ncos\ntan\n3 cos\nsin\ndy\na\ny\nd\ndx\nx\na\nd\n\u03b8\n\u03b8\n\u03b8 =\n= \u2212\n\u03b8 = \u2212\n\u2212\n\u03b8\n\u03b8\n\u03b8\nRationalised 2023-24\nCONTINUITY AND DIFFERENTIABILITY\n137\nEXERCISE 5 6\nIf  x and y are connected parametrically by the equations given in Exercises 1 to 10,\nwithout eliminating the parameter, Find dy\ndx 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "2380-2383",
        "Text": "Then\n2\n2\n3\n3\nx\n+y\n =\n2\n2\n3\n3\n3\n3\n( cos\n)\n( sin\n)\na\na\n\u03b8\n+\n\u03b8\n=\n2\n2\n2\n2\n3\n3\n(cos\n(sin\n)\na\na\n\u03b8 +\n\u03b8 =\nHence, x = a cos3\u03b8, y = a sin3\u03b8 is parametric equation of \n2\n2\n2\n3\n3\n3\nx\ny\na\n+\n=\nNow\ndx\nd\u03b8  = \u2013 3a cos2 \u03b8 sin \u03b8 and dy\nd\u03b8  = 3a sin2 \u03b8 cos \u03b8\nTherefore\ndy\ndx  =\n2\n3\n3 sin2\ncos\ntan\n3 cos\nsin\ndy\na\ny\nd\ndx\nx\na\nd\n\u03b8\n\u03b8\n\u03b8 =\n= \u2212\n\u03b8 = \u2212\n\u2212\n\u03b8\n\u03b8\n\u03b8\nRationalised 2023-24\nCONTINUITY AND DIFFERENTIABILITY\n137\nEXERCISE 5 6\nIf  x and y are connected parametrically by the equations given in Exercises 1 to 10,\nwithout eliminating the parameter, Find dy\ndx 1 x = 2at2, y = at4\n2"
    },
    {
        "Chapter": "1",
        "sentence_range": "2381-2384",
        "Text": "6\nIf  x and y are connected parametrically by the equations given in Exercises 1 to 10,\nwithout eliminating the parameter, Find dy\ndx 1 x = 2at2, y = at4\n2 x = a cos \u03b8, y = b cos \u03b8\n3"
    },
    {
        "Chapter": "1",
        "sentence_range": "2382-2385",
        "Text": "1 x = 2at2, y = at4\n2 x = a cos \u03b8, y = b cos \u03b8\n3 x = sin t, y = cos 2t\n4"
    },
    {
        "Chapter": "1",
        "sentence_range": "2383-2386",
        "Text": "x = 2at2, y = at4\n2 x = a cos \u03b8, y = b cos \u03b8\n3 x = sin t, y = cos 2t\n4 x = 4t, y = 4\nt\n5"
    },
    {
        "Chapter": "1",
        "sentence_range": "2384-2387",
        "Text": "x = a cos \u03b8, y = b cos \u03b8\n3 x = sin t, y = cos 2t\n4 x = 4t, y = 4\nt\n5 x = cos \u03b8 \u2013 cos 2\u03b8, y = sin \u03b8 \u2013 sin 2\u03b8\n6"
    },
    {
        "Chapter": "1",
        "sentence_range": "2385-2388",
        "Text": "x = sin t, y = cos 2t\n4 x = 4t, y = 4\nt\n5 x = cos \u03b8 \u2013 cos 2\u03b8, y = sin \u03b8 \u2013 sin 2\u03b8\n6 x = a (\u03b8 \u2013 sin \u03b8), y = a (1 + cos \u03b8)\n7"
    },
    {
        "Chapter": "1",
        "sentence_range": "2386-2389",
        "Text": "x = 4t, y = 4\nt\n5 x = cos \u03b8 \u2013 cos 2\u03b8, y = sin \u03b8 \u2013 sin 2\u03b8\n6 x = a (\u03b8 \u2013 sin \u03b8), y = a (1 + cos \u03b8)\n7 x = \nsin3\ncos2\nt\nt , \ncos3\ncos2\nt\ny\nt\n=\n8"
    },
    {
        "Chapter": "1",
        "sentence_range": "2387-2390",
        "Text": "x = cos \u03b8 \u2013 cos 2\u03b8, y = sin \u03b8 \u2013 sin 2\u03b8\n6 x = a (\u03b8 \u2013 sin \u03b8), y = a (1 + cos \u03b8)\n7 x = \nsin3\ncos2\nt\nt , \ncos3\ncos2\nt\ny\nt\n=\n8 cos\nlog tan 2\nt\nx\na\nt\n\uf8eb\n\uf8f6\n=\n+\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8  y = a sin t\n9"
    },
    {
        "Chapter": "1",
        "sentence_range": "2388-2391",
        "Text": "x = a (\u03b8 \u2013 sin \u03b8), y = a (1 + cos \u03b8)\n7 x = \nsin3\ncos2\nt\nt , \ncos3\ncos2\nt\ny\nt\n=\n8 cos\nlog tan 2\nt\nx\na\nt\n\uf8eb\n\uf8f6\n=\n+\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8  y = a sin t\n9 x = a sec \u03b8, y = b tan \u03b8\n10"
    },
    {
        "Chapter": "1",
        "sentence_range": "2389-2392",
        "Text": "x = \nsin3\ncos2\nt\nt , \ncos3\ncos2\nt\ny\nt\n=\n8 cos\nlog tan 2\nt\nx\na\nt\n\uf8eb\n\uf8f6\n=\n+\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8  y = a sin t\n9 x = a sec \u03b8, y = b tan \u03b8\n10 x = a (cos \u03b8 + \u03b8 sin \u03b8),  y = a (sin \u03b8 \u2013 \u03b8 cos \u03b8)\n11"
    },
    {
        "Chapter": "1",
        "sentence_range": "2390-2393",
        "Text": "cos\nlog tan 2\nt\nx\na\nt\n\uf8eb\n\uf8f6\n=\n+\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8  y = a sin t\n9 x = a sec \u03b8, y = b tan \u03b8\n10 x = a (cos \u03b8 + \u03b8 sin \u03b8),  y = a (sin \u03b8 \u2013 \u03b8 cos \u03b8)\n11 If \n1\n1\nsin\ncos\n,\n, show that\nt\nt\ndy\ny\nx\na\ny\na\ndx\nx\n\u2212\n\u2212\n=\n=\n= \u2212\n5"
    },
    {
        "Chapter": "1",
        "sentence_range": "2391-2394",
        "Text": "x = a sec \u03b8, y = b tan \u03b8\n10 x = a (cos \u03b8 + \u03b8 sin \u03b8),  y = a (sin \u03b8 \u2013 \u03b8 cos \u03b8)\n11 If \n1\n1\nsin\ncos\n,\n, show that\nt\nt\ndy\ny\nx\na\ny\na\ndx\nx\n\u2212\n\u2212\n=\n=\n= \u2212\n5 7  Second Order Derivative\nLet\ny = f (x)"
    },
    {
        "Chapter": "1",
        "sentence_range": "2392-2395",
        "Text": "x = a (cos \u03b8 + \u03b8 sin \u03b8),  y = a (sin \u03b8 \u2013 \u03b8 cos \u03b8)\n11 If \n1\n1\nsin\ncos\n,\n, show that\nt\nt\ndy\ny\nx\na\ny\na\ndx\nx\n\u2212\n\u2212\n=\n=\n= \u2212\n5 7  Second Order Derivative\nLet\ny = f (x) Then\ndy\ndx  = f \u2032(x)"
    },
    {
        "Chapter": "1",
        "sentence_range": "2393-2396",
        "Text": "If \n1\n1\nsin\ncos\n,\n, show that\nt\nt\ndy\ny\nx\na\ny\na\ndx\nx\n\u2212\n\u2212\n=\n=\n= \u2212\n5 7  Second Order Derivative\nLet\ny = f (x) Then\ndy\ndx  = f \u2032(x) (1)\nIf f\u2032(x) is differentiable, we may differentiate (1) again w"
    },
    {
        "Chapter": "1",
        "sentence_range": "2394-2397",
        "Text": "7  Second Order Derivative\nLet\ny = f (x) Then\ndy\ndx  = f \u2032(x) (1)\nIf f\u2032(x) is differentiable, we may differentiate (1) again w r"
    },
    {
        "Chapter": "1",
        "sentence_range": "2395-2398",
        "Text": "Then\ndy\ndx  = f \u2032(x) (1)\nIf f\u2032(x) is differentiable, we may differentiate (1) again w r t"
    },
    {
        "Chapter": "1",
        "sentence_range": "2396-2399",
        "Text": "(1)\nIf f\u2032(x) is differentiable, we may differentiate (1) again w r t x"
    },
    {
        "Chapter": "1",
        "sentence_range": "2397-2400",
        "Text": "r t x Then, the left hand\nside becomes d\ndy\ndx\ndx\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8  which is called the second order derivative of y w"
    },
    {
        "Chapter": "1",
        "sentence_range": "2398-2401",
        "Text": "t x Then, the left hand\nside becomes d\ndy\ndx\ndx\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8  which is called the second order derivative of y w r"
    },
    {
        "Chapter": "1",
        "sentence_range": "2399-2402",
        "Text": "x Then, the left hand\nside becomes d\ndy\ndx\ndx\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8  which is called the second order derivative of y w r t"
    },
    {
        "Chapter": "1",
        "sentence_range": "2400-2403",
        "Text": "Then, the left hand\nside becomes d\ndy\ndx\ndx\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8  which is called the second order derivative of y w r t x and\nis denoted by \n2\n2\nd y\ndx"
    },
    {
        "Chapter": "1",
        "sentence_range": "2401-2404",
        "Text": "r t x and\nis denoted by \n2\n2\nd y\ndx The second order derivative of f(x) is denoted by f \u2033(x)"
    },
    {
        "Chapter": "1",
        "sentence_range": "2402-2405",
        "Text": "t x and\nis denoted by \n2\n2\nd y\ndx The second order derivative of f(x) is denoted by f \u2033(x) It is also\ndenoted by D2 y or y\u2033 or y2 if y = f(x)"
    },
    {
        "Chapter": "1",
        "sentence_range": "2403-2406",
        "Text": "x and\nis denoted by \n2\n2\nd y\ndx The second order derivative of f(x) is denoted by f \u2033(x) It is also\ndenoted by D2 y or y\u2033 or y2 if y = f(x) We remark that higher order derivatives may be\ndefined similarly"
    },
    {
        "Chapter": "1",
        "sentence_range": "2404-2407",
        "Text": "The second order derivative of f(x) is denoted by f \u2033(x) It is also\ndenoted by D2 y or y\u2033 or y2 if y = f(x) We remark that higher order derivatives may be\ndefined similarly Rationalised 2023-24\n MATHEMATICS\n138\nExample 35 Find \n2\n2\nd y\ndx\n, if  y = x3 + tan x"
    },
    {
        "Chapter": "1",
        "sentence_range": "2405-2408",
        "Text": "It is also\ndenoted by D2 y or y\u2033 or y2 if y = f(x) We remark that higher order derivatives may be\ndefined similarly Rationalised 2023-24\n MATHEMATICS\n138\nExample 35 Find \n2\n2\nd y\ndx\n, if  y = x3 + tan x Solution Given that y = x3 + tan x"
    },
    {
        "Chapter": "1",
        "sentence_range": "2406-2409",
        "Text": "We remark that higher order derivatives may be\ndefined similarly Rationalised 2023-24\n MATHEMATICS\n138\nExample 35 Find \n2\n2\nd y\ndx\n, if  y = x3 + tan x Solution Given that y = x3 + tan x Then\ndy\ndx  = 3x2 + sec2 x\nTherefore\n2\n2\nd y\ndx\n =\n(\n)\n2\n2\n3\nsec\nd\nx\nx\ndx\n+\n= 6x + 2 sec x"
    },
    {
        "Chapter": "1",
        "sentence_range": "2407-2410",
        "Text": "Rationalised 2023-24\n MATHEMATICS\n138\nExample 35 Find \n2\n2\nd y\ndx\n, if  y = x3 + tan x Solution Given that y = x3 + tan x Then\ndy\ndx  = 3x2 + sec2 x\nTherefore\n2\n2\nd y\ndx\n =\n(\n)\n2\n2\n3\nsec\nd\nx\nx\ndx\n+\n= 6x + 2 sec x sec x tan x = 6x + 2 sec2 x tan x\nExample 36 If y = A sin x + B cos x, then prove that \n2\n2\n0\nd y\ny\ndx\n+\n="
    },
    {
        "Chapter": "1",
        "sentence_range": "2408-2411",
        "Text": "Solution Given that y = x3 + tan x Then\ndy\ndx  = 3x2 + sec2 x\nTherefore\n2\n2\nd y\ndx\n =\n(\n)\n2\n2\n3\nsec\nd\nx\nx\ndx\n+\n= 6x + 2 sec x sec x tan x = 6x + 2 sec2 x tan x\nExample 36 If y = A sin x + B cos x, then prove that \n2\n2\n0\nd y\ny\ndx\n+\n= Solution We have\ndy\ndx  = A cos x \u2013 B sin x\nand\n2\n2\nd y\ndx\n = d\ndx  (A cos x \u2013 B sin x)\n= \u2013 A sin x \u2013 B cos x = \u2013 y\nHence\n2\n2\nd y\ndx\n + y = 0\nExample 37 If  y = 3e2x + 2e3x, prove that \n2\n2\n5\n6\n0\nd y\ndy\ny\ndx\ndx\n\u2212\n+\n="
    },
    {
        "Chapter": "1",
        "sentence_range": "2409-2412",
        "Text": "Then\ndy\ndx  = 3x2 + sec2 x\nTherefore\n2\n2\nd y\ndx\n =\n(\n)\n2\n2\n3\nsec\nd\nx\nx\ndx\n+\n= 6x + 2 sec x sec x tan x = 6x + 2 sec2 x tan x\nExample 36 If y = A sin x + B cos x, then prove that \n2\n2\n0\nd y\ny\ndx\n+\n= Solution We have\ndy\ndx  = A cos x \u2013 B sin x\nand\n2\n2\nd y\ndx\n = d\ndx  (A cos x \u2013 B sin x)\n= \u2013 A sin x \u2013 B cos x = \u2013 y\nHence\n2\n2\nd y\ndx\n + y = 0\nExample 37 If  y = 3e2x + 2e3x, prove that \n2\n2\n5\n6\n0\nd y\ndy\ny\ndx\ndx\n\u2212\n+\n= Solution Given that y = 3e2x + 2e3x"
    },
    {
        "Chapter": "1",
        "sentence_range": "2410-2413",
        "Text": "sec x tan x = 6x + 2 sec2 x tan x\nExample 36 If y = A sin x + B cos x, then prove that \n2\n2\n0\nd y\ny\ndx\n+\n= Solution We have\ndy\ndx  = A cos x \u2013 B sin x\nand\n2\n2\nd y\ndx\n = d\ndx  (A cos x \u2013 B sin x)\n= \u2013 A sin x \u2013 B cos x = \u2013 y\nHence\n2\n2\nd y\ndx\n + y = 0\nExample 37 If  y = 3e2x + 2e3x, prove that \n2\n2\n5\n6\n0\nd y\ndy\ny\ndx\ndx\n\u2212\n+\n= Solution Given that y = 3e2x + 2e3x Then\ndy\ndx  = 6e2x + 6e3x = 6 (e2x + e3x)\nTherefore\n2\n2\nd y\ndx\n = 12e2x + 18e3x = 6 (2e2x + 3e3x)\nHence\n2\n2\n5\nd y\ndy\ndx\ndx\n\u2212\n + 6y = 6 (2e2x + 3e3x)\n\u2013 30 (e2x + e3x) + 6 (3e2x + 2e3x) = 0\nRationalised 2023-24\nCONTINUITY AND DIFFERENTIABILITY\n139\nExample 38 If y = sin\u20131 x, show that (1 \u2013 x2) \n2\n2\n0\nd y\ndy\nx dx\ndx\n\u2212\n="
    },
    {
        "Chapter": "1",
        "sentence_range": "2411-2414",
        "Text": "Solution We have\ndy\ndx  = A cos x \u2013 B sin x\nand\n2\n2\nd y\ndx\n = d\ndx  (A cos x \u2013 B sin x)\n= \u2013 A sin x \u2013 B cos x = \u2013 y\nHence\n2\n2\nd y\ndx\n + y = 0\nExample 37 If  y = 3e2x + 2e3x, prove that \n2\n2\n5\n6\n0\nd y\ndy\ny\ndx\ndx\n\u2212\n+\n= Solution Given that y = 3e2x + 2e3x Then\ndy\ndx  = 6e2x + 6e3x = 6 (e2x + e3x)\nTherefore\n2\n2\nd y\ndx\n = 12e2x + 18e3x = 6 (2e2x + 3e3x)\nHence\n2\n2\n5\nd y\ndy\ndx\ndx\n\u2212\n + 6y = 6 (2e2x + 3e3x)\n\u2013 30 (e2x + e3x) + 6 (3e2x + 2e3x) = 0\nRationalised 2023-24\nCONTINUITY AND DIFFERENTIABILITY\n139\nExample 38 If y = sin\u20131 x, show that (1 \u2013 x2) \n2\n2\n0\nd y\ndy\nx dx\ndx\n\u2212\n= Solution We have y = sin\u20131x"
    },
    {
        "Chapter": "1",
        "sentence_range": "2412-2415",
        "Text": "Solution Given that y = 3e2x + 2e3x Then\ndy\ndx  = 6e2x + 6e3x = 6 (e2x + e3x)\nTherefore\n2\n2\nd y\ndx\n = 12e2x + 18e3x = 6 (2e2x + 3e3x)\nHence\n2\n2\n5\nd y\ndy\ndx\ndx\n\u2212\n + 6y = 6 (2e2x + 3e3x)\n\u2013 30 (e2x + e3x) + 6 (3e2x + 2e3x) = 0\nRationalised 2023-24\nCONTINUITY AND DIFFERENTIABILITY\n139\nExample 38 If y = sin\u20131 x, show that (1 \u2013 x2) \n2\n2\n0\nd y\ndy\nx dx\ndx\n\u2212\n= Solution We have y = sin\u20131x Then\ndy\ndx  = \n2\n1\n(1\nx)\n\u2212\nor\n2\n(1\n)\n1\ndy\nx\ndx\n\u2212\n=\nSo\n2\n(1\n)"
    },
    {
        "Chapter": "1",
        "sentence_range": "2413-2416",
        "Text": "Then\ndy\ndx  = 6e2x + 6e3x = 6 (e2x + e3x)\nTherefore\n2\n2\nd y\ndx\n = 12e2x + 18e3x = 6 (2e2x + 3e3x)\nHence\n2\n2\n5\nd y\ndy\ndx\ndx\n\u2212\n + 6y = 6 (2e2x + 3e3x)\n\u2013 30 (e2x + e3x) + 6 (3e2x + 2e3x) = 0\nRationalised 2023-24\nCONTINUITY AND DIFFERENTIABILITY\n139\nExample 38 If y = sin\u20131 x, show that (1 \u2013 x2) \n2\n2\n0\nd y\ndy\nx dx\ndx\n\u2212\n= Solution We have y = sin\u20131x Then\ndy\ndx  = \n2\n1\n(1\nx)\n\u2212\nor\n2\n(1\n)\n1\ndy\nx\ndx\n\u2212\n=\nSo\n2\n(1\n) 0\nd\ndy\nx\ndx\ndx\n\uf8eb\n\uf8f6\n\u2212\n=\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\nor\n(\n)\n2\n2\n2\n2\n(1\n)\n(1\n)\n0\nd y\ndy\nd\nx\nx\ndx dx\ndx\n\u2212\n\u22c5\n+\n\u22c5\n\u2212\n=\nor\n2\n2\n2\n2\n2\n(1\n)\n0\n2 1\nd y\ndy\nx\nx\ndx\ndx\nx\n\u2212\n\u22c5\n\u2212\n\u22c5\n=\n\u2212\nHence\n2\n2\n2\n(1\n)\n0\nd y\ndy\nx\nx dx\ndx\n\u2212\n\u2212\n=\nAlternatively, Given that y = sin\u20131 x, we have\n1\n2\n1\n1\ny\nx\n=\n\u2212\n, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "2414-2417",
        "Text": "Solution We have y = sin\u20131x Then\ndy\ndx  = \n2\n1\n(1\nx)\n\u2212\nor\n2\n(1\n)\n1\ndy\nx\ndx\n\u2212\n=\nSo\n2\n(1\n) 0\nd\ndy\nx\ndx\ndx\n\uf8eb\n\uf8f6\n\u2212\n=\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\nor\n(\n)\n2\n2\n2\n2\n(1\n)\n(1\n)\n0\nd y\ndy\nd\nx\nx\ndx dx\ndx\n\u2212\n\u22c5\n+\n\u22c5\n\u2212\n=\nor\n2\n2\n2\n2\n2\n(1\n)\n0\n2 1\nd y\ndy\nx\nx\ndx\ndx\nx\n\u2212\n\u22c5\n\u2212\n\u22c5\n=\n\u2212\nHence\n2\n2\n2\n(1\n)\n0\nd y\ndy\nx\nx dx\ndx\n\u2212\n\u2212\n=\nAlternatively, Given that y = sin\u20131 x, we have\n1\n2\n1\n1\ny\nx\n=\n\u2212\n, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "2415-2418",
        "Text": "Then\ndy\ndx  = \n2\n1\n(1\nx)\n\u2212\nor\n2\n(1\n)\n1\ndy\nx\ndx\n\u2212\n=\nSo\n2\n(1\n) 0\nd\ndy\nx\ndx\ndx\n\uf8eb\n\uf8f6\n\u2212\n=\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\nor\n(\n)\n2\n2\n2\n2\n(1\n)\n(1\n)\n0\nd y\ndy\nd\nx\nx\ndx dx\ndx\n\u2212\n\u22c5\n+\n\u22c5\n\u2212\n=\nor\n2\n2\n2\n2\n2\n(1\n)\n0\n2 1\nd y\ndy\nx\nx\ndx\ndx\nx\n\u2212\n\u22c5\n\u2212\n\u22c5\n=\n\u2212\nHence\n2\n2\n2\n(1\n)\n0\nd y\ndy\nx\nx dx\ndx\n\u2212\n\u2212\n=\nAlternatively, Given that y = sin\u20131 x, we have\n1\n2\n1\n1\ny\nx\n=\n\u2212\n, i e , (\n2)\n12\n1\n1\nx\ny\n\u2212\n=\nSo\n2\n2\n1\n2\n1\n(1\n)"
    },
    {
        "Chapter": "1",
        "sentence_range": "2416-2419",
        "Text": "0\nd\ndy\nx\ndx\ndx\n\uf8eb\n\uf8f6\n\u2212\n=\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\nor\n(\n)\n2\n2\n2\n2\n(1\n)\n(1\n)\n0\nd y\ndy\nd\nx\nx\ndx dx\ndx\n\u2212\n\u22c5\n+\n\u22c5\n\u2212\n=\nor\n2\n2\n2\n2\n2\n(1\n)\n0\n2 1\nd y\ndy\nx\nx\ndx\ndx\nx\n\u2212\n\u22c5\n\u2212\n\u22c5\n=\n\u2212\nHence\n2\n2\n2\n(1\n)\n0\nd y\ndy\nx\nx dx\ndx\n\u2212\n\u2212\n=\nAlternatively, Given that y = sin\u20131 x, we have\n1\n2\n1\n1\ny\nx\n=\n\u2212\n, i e , (\n2)\n12\n1\n1\nx\ny\n\u2212\n=\nSo\n2\n2\n1\n2\n1\n(1\n) 2\n(0\n2 )\n0\nx\ny y\ny\nx\n\u2212\n+\n\u2212\n=\nHence\n(1 \u2013 x2) y2 \u2013 xy1 = 0\nEXERCISE 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "2417-2420",
        "Text": "e , (\n2)\n12\n1\n1\nx\ny\n\u2212\n=\nSo\n2\n2\n1\n2\n1\n(1\n) 2\n(0\n2 )\n0\nx\ny y\ny\nx\n\u2212\n+\n\u2212\n=\nHence\n(1 \u2013 x2) y2 \u2013 xy1 = 0\nEXERCISE 5 7\nFind the second order derivatives of the functions given in Exercises 1 to 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "2418-2421",
        "Text": ", (\n2)\n12\n1\n1\nx\ny\n\u2212\n=\nSo\n2\n2\n1\n2\n1\n(1\n) 2\n(0\n2 )\n0\nx\ny y\ny\nx\n\u2212\n+\n\u2212\n=\nHence\n(1 \u2013 x2) y2 \u2013 xy1 = 0\nEXERCISE 5 7\nFind the second order derivatives of the functions given in Exercises 1 to 10 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "2419-2422",
        "Text": "2\n(0\n2 )\n0\nx\ny y\ny\nx\n\u2212\n+\n\u2212\n=\nHence\n(1 \u2013 x2) y2 \u2013 xy1 = 0\nEXERCISE 5 7\nFind the second order derivatives of the functions given in Exercises 1 to 10 1 x2 + 3x + 2\n2"
    },
    {
        "Chapter": "1",
        "sentence_range": "2420-2423",
        "Text": "7\nFind the second order derivatives of the functions given in Exercises 1 to 10 1 x2 + 3x + 2\n2 x20\n3"
    },
    {
        "Chapter": "1",
        "sentence_range": "2421-2424",
        "Text": "1 x2 + 3x + 2\n2 x20\n3 x"
    },
    {
        "Chapter": "1",
        "sentence_range": "2422-2425",
        "Text": "x2 + 3x + 2\n2 x20\n3 x cos x\n4"
    },
    {
        "Chapter": "1",
        "sentence_range": "2423-2426",
        "Text": "x20\n3 x cos x\n4 log x\n5"
    },
    {
        "Chapter": "1",
        "sentence_range": "2424-2427",
        "Text": "x cos x\n4 log x\n5 x3 log x\n6"
    },
    {
        "Chapter": "1",
        "sentence_range": "2425-2428",
        "Text": "cos x\n4 log x\n5 x3 log x\n6 ex sin 5x\n7"
    },
    {
        "Chapter": "1",
        "sentence_range": "2426-2429",
        "Text": "log x\n5 x3 log x\n6 ex sin 5x\n7 e6x cos 3x\n8"
    },
    {
        "Chapter": "1",
        "sentence_range": "2427-2430",
        "Text": "x3 log x\n6 ex sin 5x\n7 e6x cos 3x\n8 tan\u20131 x\n9"
    },
    {
        "Chapter": "1",
        "sentence_range": "2428-2431",
        "Text": "ex sin 5x\n7 e6x cos 3x\n8 tan\u20131 x\n9 log (log x)\n10"
    },
    {
        "Chapter": "1",
        "sentence_range": "2429-2432",
        "Text": "e6x cos 3x\n8 tan\u20131 x\n9 log (log x)\n10 sin (log x)\n11"
    },
    {
        "Chapter": "1",
        "sentence_range": "2430-2433",
        "Text": "tan\u20131 x\n9 log (log x)\n10 sin (log x)\n11 If y = 5 cos x \u2013 3 sin x, prove that \n2\n2\n0\nd y\ny\ndx\n+\n=\nRationalised 2023-24\n MATHEMATICS\n140\n12"
    },
    {
        "Chapter": "1",
        "sentence_range": "2431-2434",
        "Text": "log (log x)\n10 sin (log x)\n11 If y = 5 cos x \u2013 3 sin x, prove that \n2\n2\n0\nd y\ny\ndx\n+\n=\nRationalised 2023-24\n MATHEMATICS\n140\n12 If y = cos\u20131 x, Find  \n2\n2\nd y\ndx\nin terms of y alone"
    },
    {
        "Chapter": "1",
        "sentence_range": "2432-2435",
        "Text": "sin (log x)\n11 If y = 5 cos x \u2013 3 sin x, prove that \n2\n2\n0\nd y\ny\ndx\n+\n=\nRationalised 2023-24\n MATHEMATICS\n140\n12 If y = cos\u20131 x, Find  \n2\n2\nd y\ndx\nin terms of y alone 13"
    },
    {
        "Chapter": "1",
        "sentence_range": "2433-2436",
        "Text": "If y = 5 cos x \u2013 3 sin x, prove that \n2\n2\n0\nd y\ny\ndx\n+\n=\nRationalised 2023-24\n MATHEMATICS\n140\n12 If y = cos\u20131 x, Find  \n2\n2\nd y\ndx\nin terms of y alone 13 If y = 3 cos (log x) + 4 sin (log x), show that x2 y2 + xy1 + y = 0\n14"
    },
    {
        "Chapter": "1",
        "sentence_range": "2434-2437",
        "Text": "If y = cos\u20131 x, Find  \n2\n2\nd y\ndx\nin terms of y alone 13 If y = 3 cos (log x) + 4 sin (log x), show that x2 y2 + xy1 + y = 0\n14 If y = Aemx + Benx, show that \n2\n2\n(\n)\n0\nd y\ndy\nm\nn\nmny\ndx\ndx\n\u2212\n+\n+\n=\n15"
    },
    {
        "Chapter": "1",
        "sentence_range": "2435-2438",
        "Text": "13 If y = 3 cos (log x) + 4 sin (log x), show that x2 y2 + xy1 + y = 0\n14 If y = Aemx + Benx, show that \n2\n2\n(\n)\n0\nd y\ndy\nm\nn\nmny\ndx\ndx\n\u2212\n+\n+\n=\n15 If y = 500e7x + 600e\u20137x, show that \n2\n2\n49\nd y\ny\ndx\n=\n16"
    },
    {
        "Chapter": "1",
        "sentence_range": "2436-2439",
        "Text": "If y = 3 cos (log x) + 4 sin (log x), show that x2 y2 + xy1 + y = 0\n14 If y = Aemx + Benx, show that \n2\n2\n(\n)\n0\nd y\ndy\nm\nn\nmny\ndx\ndx\n\u2212\n+\n+\n=\n15 If y = 500e7x + 600e\u20137x, show that \n2\n2\n49\nd y\ny\ndx\n=\n16 If ey\n (x + 1) = 1, show that \n2\n2\n2\nd y\ndy\ndx\ndx\n\uf8eb\n\uf8f6\n= \uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n17"
    },
    {
        "Chapter": "1",
        "sentence_range": "2437-2440",
        "Text": "If y = Aemx + Benx, show that \n2\n2\n(\n)\n0\nd y\ndy\nm\nn\nmny\ndx\ndx\n\u2212\n+\n+\n=\n15 If y = 500e7x + 600e\u20137x, show that \n2\n2\n49\nd y\ny\ndx\n=\n16 If ey\n (x + 1) = 1, show that \n2\n2\n2\nd y\ndy\ndx\ndx\n\uf8eb\n\uf8f6\n= \uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n17 If y = (tan\u20131 x)2, show that (x2 + 1)2 y2 + 2x (x2 + 1) y1 = 2\nMiscellaneous Examples\nExample 39 Differentiate w"
    },
    {
        "Chapter": "1",
        "sentence_range": "2438-2441",
        "Text": "If y = 500e7x + 600e\u20137x, show that \n2\n2\n49\nd y\ny\ndx\n=\n16 If ey\n (x + 1) = 1, show that \n2\n2\n2\nd y\ndy\ndx\ndx\n\uf8eb\n\uf8f6\n= \uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n17 If y = (tan\u20131 x)2, show that (x2 + 1)2 y2 + 2x (x2 + 1) y1 = 2\nMiscellaneous Examples\nExample 39 Differentiate w r"
    },
    {
        "Chapter": "1",
        "sentence_range": "2439-2442",
        "Text": "If ey\n (x + 1) = 1, show that \n2\n2\n2\nd y\ndy\ndx\ndx\n\uf8eb\n\uf8f6\n= \uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n17 If y = (tan\u20131 x)2, show that (x2 + 1)2 y2 + 2x (x2 + 1) y1 = 2\nMiscellaneous Examples\nExample 39 Differentiate w r t"
    },
    {
        "Chapter": "1",
        "sentence_range": "2440-2443",
        "Text": "If y = (tan\u20131 x)2, show that (x2 + 1)2 y2 + 2x (x2 + 1) y1 = 2\nMiscellaneous Examples\nExample 39 Differentiate w r t x, the following function:\n(i)\n12\n3\n2\n2\n4\nx\nx\n+\n+\n+\n(ii)    log7 (log x)\nSolution\n(i)\nLet y = \n12\n3\n2\n2\n4\nx\nx\n+\n+\n+\n= \n1\n1\n2\n2\n2\n(3\n2)\n(2\n4)\nx\nx\n\u2212\n+\n+\n+\nNote that this function is defined at all real numbers \nx > \u221232"
    },
    {
        "Chapter": "1",
        "sentence_range": "2441-2444",
        "Text": "r t x, the following function:\n(i)\n12\n3\n2\n2\n4\nx\nx\n+\n+\n+\n(ii)    log7 (log x)\nSolution\n(i)\nLet y = \n12\n3\n2\n2\n4\nx\nx\n+\n+\n+\n= \n1\n1\n2\n2\n2\n(3\n2)\n(2\n4)\nx\nx\n\u2212\n+\n+\n+\nNote that this function is defined at all real numbers \nx > \u221232 Therefore\ndy\ndx  =\n1\n1\n1\n1\n2\n2\n2\n2\n1\n1\n(3\n2)\n(3\n2)\n(2\n4)\n(2\n4)\n2\n2\nd\nd\nx\nx\nx\nx\ndx\ndx\n\u2212\n\u2212 \u2212\n\uf8eb\n\uf8f6\n+\n\u22c5\n+\n+ \u2212\n+\n\u22c5\n+\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n= 1\n2 3\n2\n3\n21\n2\n4\n4\n21\n2\n23\n(\n)\n( )\n(\n)\nx\nx\nx\n+\n\u22c5\n\u2212 \uf8eb\n\uf8ed\uf8ec \uf8f6\n\uf8f8\uf8f7\n+\n\u22c5\n\u2212\n\u2212\n=\n(\n)\n3\n2\n2\n3\n2\n2 3\n2\n2\n4\nx\nx\nx\n\u2212\n+\n+\nThis is defined for all real numbers \nx > \u221232"
    },
    {
        "Chapter": "1",
        "sentence_range": "2442-2445",
        "Text": "t x, the following function:\n(i)\n12\n3\n2\n2\n4\nx\nx\n+\n+\n+\n(ii)    log7 (log x)\nSolution\n(i)\nLet y = \n12\n3\n2\n2\n4\nx\nx\n+\n+\n+\n= \n1\n1\n2\n2\n2\n(3\n2)\n(2\n4)\nx\nx\n\u2212\n+\n+\n+\nNote that this function is defined at all real numbers \nx > \u221232 Therefore\ndy\ndx  =\n1\n1\n1\n1\n2\n2\n2\n2\n1\n1\n(3\n2)\n(3\n2)\n(2\n4)\n(2\n4)\n2\n2\nd\nd\nx\nx\nx\nx\ndx\ndx\n\u2212\n\u2212 \u2212\n\uf8eb\n\uf8f6\n+\n\u22c5\n+\n+ \u2212\n+\n\u22c5\n+\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n= 1\n2 3\n2\n3\n21\n2\n4\n4\n21\n2\n23\n(\n)\n( )\n(\n)\nx\nx\nx\n+\n\u22c5\n\u2212 \uf8eb\n\uf8ed\uf8ec \uf8f6\n\uf8f8\uf8f7\n+\n\u22c5\n\u2212\n\u2212\n=\n(\n)\n3\n2\n2\n3\n2\n2 3\n2\n2\n4\nx\nx\nx\n\u2212\n+\n+\nThis is defined for all real numbers \nx > \u221232 Rationalised 2023-24\nCONTINUITY AND DIFFERENTIABILITY\n141\n(ii)\nLet y = log7 (log x) = log (log )\nlog7\nx  (by change of base formula)"
    },
    {
        "Chapter": "1",
        "sentence_range": "2443-2446",
        "Text": "x, the following function:\n(i)\n12\n3\n2\n2\n4\nx\nx\n+\n+\n+\n(ii)    log7 (log x)\nSolution\n(i)\nLet y = \n12\n3\n2\n2\n4\nx\nx\n+\n+\n+\n= \n1\n1\n2\n2\n2\n(3\n2)\n(2\n4)\nx\nx\n\u2212\n+\n+\n+\nNote that this function is defined at all real numbers \nx > \u221232 Therefore\ndy\ndx  =\n1\n1\n1\n1\n2\n2\n2\n2\n1\n1\n(3\n2)\n(3\n2)\n(2\n4)\n(2\n4)\n2\n2\nd\nd\nx\nx\nx\nx\ndx\ndx\n\u2212\n\u2212 \u2212\n\uf8eb\n\uf8f6\n+\n\u22c5\n+\n+ \u2212\n+\n\u22c5\n+\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n= 1\n2 3\n2\n3\n21\n2\n4\n4\n21\n2\n23\n(\n)\n( )\n(\n)\nx\nx\nx\n+\n\u22c5\n\u2212 \uf8eb\n\uf8ed\uf8ec \uf8f6\n\uf8f8\uf8f7\n+\n\u22c5\n\u2212\n\u2212\n=\n(\n)\n3\n2\n2\n3\n2\n2 3\n2\n2\n4\nx\nx\nx\n\u2212\n+\n+\nThis is defined for all real numbers \nx > \u221232 Rationalised 2023-24\nCONTINUITY AND DIFFERENTIABILITY\n141\n(ii)\nLet y = log7 (log x) = log (log )\nlog7\nx  (by change of base formula) The function is defined for all real numbers x > 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "2444-2447",
        "Text": "Therefore\ndy\ndx  =\n1\n1\n1\n1\n2\n2\n2\n2\n1\n1\n(3\n2)\n(3\n2)\n(2\n4)\n(2\n4)\n2\n2\nd\nd\nx\nx\nx\nx\ndx\ndx\n\u2212\n\u2212 \u2212\n\uf8eb\n\uf8f6\n+\n\u22c5\n+\n+ \u2212\n+\n\u22c5\n+\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n= 1\n2 3\n2\n3\n21\n2\n4\n4\n21\n2\n23\n(\n)\n( )\n(\n)\nx\nx\nx\n+\n\u22c5\n\u2212 \uf8eb\n\uf8ed\uf8ec \uf8f6\n\uf8f8\uf8f7\n+\n\u22c5\n\u2212\n\u2212\n=\n(\n)\n3\n2\n2\n3\n2\n2 3\n2\n2\n4\nx\nx\nx\n\u2212\n+\n+\nThis is defined for all real numbers \nx > \u221232 Rationalised 2023-24\nCONTINUITY AND DIFFERENTIABILITY\n141\n(ii)\nLet y = log7 (log x) = log (log )\nlog7\nx  (by change of base formula) The function is defined for all real numbers x > 1 Therefore\ndy\ndx  =\n1\n(log (log ))\nlog7\nd\nx\ndx\n=\n1\n1\n(log )\nlog7 log\nd\nx\nx dx\n\u22c5\n=\n1\nxlog7 log\nx\nExample 40 Differentiate the following w"
    },
    {
        "Chapter": "1",
        "sentence_range": "2445-2448",
        "Text": "Rationalised 2023-24\nCONTINUITY AND DIFFERENTIABILITY\n141\n(ii)\nLet y = log7 (log x) = log (log )\nlog7\nx  (by change of base formula) The function is defined for all real numbers x > 1 Therefore\ndy\ndx  =\n1\n(log (log ))\nlog7\nd\nx\ndx\n=\n1\n1\n(log )\nlog7 log\nd\nx\nx dx\n\u22c5\n=\n1\nxlog7 log\nx\nExample 40 Differentiate the following w r"
    },
    {
        "Chapter": "1",
        "sentence_range": "2446-2449",
        "Text": "The function is defined for all real numbers x > 1 Therefore\ndy\ndx  =\n1\n(log (log ))\nlog7\nd\nx\ndx\n=\n1\n1\n(log )\nlog7 log\nd\nx\nx dx\n\u22c5\n=\n1\nxlog7 log\nx\nExample 40 Differentiate the following w r t"
    },
    {
        "Chapter": "1",
        "sentence_range": "2447-2450",
        "Text": "Therefore\ndy\ndx  =\n1\n(log (log ))\nlog7\nd\nx\ndx\n=\n1\n1\n(log )\nlog7 log\nd\nx\nx dx\n\u22c5\n=\n1\nxlog7 log\nx\nExample 40 Differentiate the following w r t x"
    },
    {
        "Chapter": "1",
        "sentence_range": "2448-2451",
        "Text": "r t x (i)\ncos \u20131 (sin x)\n(ii)   \n1\nsin\ntan\n1\ncos\nx\nx\n\u2212 \uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed+\n\uf8f8\n (iii)   \n1\n1\n2\nsin\n1\n4\nx\nx\n+\n\u2212 \uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed+\n\uf8f8\nSolution\n(i)\nLet f (x) = cos \u20131 (sin x)"
    },
    {
        "Chapter": "1",
        "sentence_range": "2449-2452",
        "Text": "t x (i)\ncos \u20131 (sin x)\n(ii)   \n1\nsin\ntan\n1\ncos\nx\nx\n\u2212 \uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed+\n\uf8f8\n (iii)   \n1\n1\n2\nsin\n1\n4\nx\nx\n+\n\u2212 \uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed+\n\uf8f8\nSolution\n(i)\nLet f (x) = cos \u20131 (sin x) Observe that this function is defined for all real numbers"
    },
    {
        "Chapter": "1",
        "sentence_range": "2450-2453",
        "Text": "x (i)\ncos \u20131 (sin x)\n(ii)   \n1\nsin\ntan\n1\ncos\nx\nx\n\u2212 \uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed+\n\uf8f8\n (iii)   \n1\n1\n2\nsin\n1\n4\nx\nx\n+\n\u2212 \uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed+\n\uf8f8\nSolution\n(i)\nLet f (x) = cos \u20131 (sin x) Observe that this function is defined for all real numbers We may rewrite this function as\nf(x) = cos \u20131 (sin x)\n= cos\ncos\n\u2212\n\u2212\n\uf8ed\uf8ec\uf8eb\n\uf8f8\uf8f7\uf8f6\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa\n1\n\u03c02\nx\n= 2\nx\n\u03c0 \u2212\nThus\nf \u2032(x) = \u2013 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "2451-2454",
        "Text": "(i)\ncos \u20131 (sin x)\n(ii)   \n1\nsin\ntan\n1\ncos\nx\nx\n\u2212 \uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed+\n\uf8f8\n (iii)   \n1\n1\n2\nsin\n1\n4\nx\nx\n+\n\u2212 \uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed+\n\uf8f8\nSolution\n(i)\nLet f (x) = cos \u20131 (sin x) Observe that this function is defined for all real numbers We may rewrite this function as\nf(x) = cos \u20131 (sin x)\n= cos\ncos\n\u2212\n\u2212\n\uf8ed\uf8ec\uf8eb\n\uf8f8\uf8f7\uf8f6\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa\n1\n\u03c02\nx\n= 2\nx\n\u03c0 \u2212\nThus\nf \u2032(x) = \u2013 1 (ii)\nLet f(x) = tan \u20131 \nsin\n1\ncos\nx\nx\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed+\n\uf8f8"
    },
    {
        "Chapter": "1",
        "sentence_range": "2452-2455",
        "Text": "Observe that this function is defined for all real numbers We may rewrite this function as\nf(x) = cos \u20131 (sin x)\n= cos\ncos\n\u2212\n\u2212\n\uf8ed\uf8ec\uf8eb\n\uf8f8\uf8f7\uf8f6\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa\n1\n\u03c02\nx\n= 2\nx\n\u03c0 \u2212\nThus\nf \u2032(x) = \u2013 1 (ii)\nLet f(x) = tan \u20131 \nsin\n1\ncos\nx\nx\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed+\n\uf8f8 Observe that this function is defined for all real\nnumbers, where cos x \u2260 \u2013 1; i"
    },
    {
        "Chapter": "1",
        "sentence_range": "2453-2456",
        "Text": "We may rewrite this function as\nf(x) = cos \u20131 (sin x)\n= cos\ncos\n\u2212\n\u2212\n\uf8ed\uf8ec\uf8eb\n\uf8f8\uf8f7\uf8f6\n\uf8ee\n\uf8f0\uf8ef\n\uf8f9\n\uf8fb\uf8fa\n1\n\u03c02\nx\n= 2\nx\n\u03c0 \u2212\nThus\nf \u2032(x) = \u2013 1 (ii)\nLet f(x) = tan \u20131 \nsin\n1\ncos\nx\nx\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed+\n\uf8f8 Observe that this function is defined for all real\nnumbers, where cos x \u2260 \u2013 1; i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "2454-2457",
        "Text": "(ii)\nLet f(x) = tan \u20131 \nsin\n1\ncos\nx\nx\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed+\n\uf8f8 Observe that this function is defined for all real\nnumbers, where cos x \u2260 \u2013 1; i e , at all odd multiplies of \u03c0"
    },
    {
        "Chapter": "1",
        "sentence_range": "2455-2458",
        "Text": "Observe that this function is defined for all real\nnumbers, where cos x \u2260 \u2013 1; i e , at all odd multiplies of \u03c0 We may rewrite this\nfunction as\nf(x) =\n1\nsin\ntan\n1\ncos\nx\nx\n\u2212 \uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed+\n\uf8f8\n=\n1\n2\n2 sin\n2cos\n2\ntan\n2cos 2\nx\nx\nx\n\u2212\n\uf8ee\n\uf8f9\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nRationalised 2023-24\n MATHEMATICS\n142\n=\ntan1\ntan 2\n2\nx\nx\n\u2212 \uf8ee\n\uf8f9\n\uf8eb\n\uf8f6 =\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8ed\n\uf8f8\n\uf8f0\n\uf8fb\nObserve that we could cancel cos\n\uf8ebx2\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8 in both numerator and denominator as it\nis not equal to zero"
    },
    {
        "Chapter": "1",
        "sentence_range": "2456-2459",
        "Text": "e , at all odd multiplies of \u03c0 We may rewrite this\nfunction as\nf(x) =\n1\nsin\ntan\n1\ncos\nx\nx\n\u2212 \uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed+\n\uf8f8\n=\n1\n2\n2 sin\n2cos\n2\ntan\n2cos 2\nx\nx\nx\n\u2212\n\uf8ee\n\uf8f9\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nRationalised 2023-24\n MATHEMATICS\n142\n=\ntan1\ntan 2\n2\nx\nx\n\u2212 \uf8ee\n\uf8f9\n\uf8eb\n\uf8f6 =\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8ed\n\uf8f8\n\uf8f0\n\uf8fb\nObserve that we could cancel cos\n\uf8ebx2\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8 in both numerator and denominator as it\nis not equal to zero Thus f \u2032(x) = 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "2457-2460",
        "Text": ", at all odd multiplies of \u03c0 We may rewrite this\nfunction as\nf(x) =\n1\nsin\ntan\n1\ncos\nx\nx\n\u2212 \uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed+\n\uf8f8\n=\n1\n2\n2 sin\n2cos\n2\ntan\n2cos 2\nx\nx\nx\n\u2212\n\uf8ee\n\uf8f9\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nRationalised 2023-24\n MATHEMATICS\n142\n=\ntan1\ntan 2\n2\nx\nx\n\u2212 \uf8ee\n\uf8f9\n\uf8eb\n\uf8f6 =\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8ed\n\uf8f8\n\uf8f0\n\uf8fb\nObserve that we could cancel cos\n\uf8ebx2\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8 in both numerator and denominator as it\nis not equal to zero Thus f \u2032(x) = 1 2\n(iii)\nLet f(x) = sin\u20131 \n1\n12\n4\nx\nx\n+\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed+\n\uf8f8"
    },
    {
        "Chapter": "1",
        "sentence_range": "2458-2461",
        "Text": "We may rewrite this\nfunction as\nf(x) =\n1\nsin\ntan\n1\ncos\nx\nx\n\u2212 \uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed+\n\uf8f8\n=\n1\n2\n2 sin\n2cos\n2\ntan\n2cos 2\nx\nx\nx\n\u2212\n\uf8ee\n\uf8f9\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nRationalised 2023-24\n MATHEMATICS\n142\n=\ntan1\ntan 2\n2\nx\nx\n\u2212 \uf8ee\n\uf8f9\n\uf8eb\n\uf8f6 =\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8ed\n\uf8f8\n\uf8f0\n\uf8fb\nObserve that we could cancel cos\n\uf8ebx2\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8 in both numerator and denominator as it\nis not equal to zero Thus f \u2032(x) = 1 2\n(iii)\nLet f(x) = sin\u20131 \n1\n12\n4\nx\nx\n+\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed+\n\uf8f8 To find the domain of this function we need to find all\nx such that \n21\n1\n1\n1\n4\nx\nx\n+\n\u2212 \u2264\n\u2264\n+"
    },
    {
        "Chapter": "1",
        "sentence_range": "2459-2462",
        "Text": "Thus f \u2032(x) = 1 2\n(iii)\nLet f(x) = sin\u20131 \n1\n12\n4\nx\nx\n+\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed+\n\uf8f8 To find the domain of this function we need to find all\nx such that \n21\n1\n1\n1\n4\nx\nx\n+\n\u2212 \u2264\n\u2264\n+ Since the quantity in the middle is always positive,\nwe need to find all x such that \n1\n2\n1\n1\n4\nx\nx\n+\n\u2264\n+\n, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "2460-2463",
        "Text": "2\n(iii)\nLet f(x) = sin\u20131 \n1\n12\n4\nx\nx\n+\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed+\n\uf8f8 To find the domain of this function we need to find all\nx such that \n21\n1\n1\n1\n4\nx\nx\n+\n\u2212 \u2264\n\u2264\n+ Since the quantity in the middle is always positive,\nwe need to find all x such that \n1\n2\n1\n1\n4\nx\nx\n+\n\u2264\n+\n, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "2461-2464",
        "Text": "To find the domain of this function we need to find all\nx such that \n21\n1\n1\n1\n4\nx\nx\n+\n\u2212 \u2264\n\u2264\n+ Since the quantity in the middle is always positive,\nwe need to find all x such that \n1\n2\n1\n1\n4\nx\nx\n+\n\u2264\n+\n, i e , all x such that 2x + 1 \u2264 1 + 4x"
    },
    {
        "Chapter": "1",
        "sentence_range": "2462-2465",
        "Text": "Since the quantity in the middle is always positive,\nwe need to find all x such that \n1\n2\n1\n1\n4\nx\nx\n+\n\u2264\n+\n, i e , all x such that 2x + 1 \u2264 1 + 4x We\nmay rewrite this as 2 \u2264 1\n2x  + 2x which is true for all x"
    },
    {
        "Chapter": "1",
        "sentence_range": "2463-2466",
        "Text": "e , all x such that 2x + 1 \u2264 1 + 4x We\nmay rewrite this as 2 \u2264 1\n2x  + 2x which is true for all x Hence the function\nis defined at every real number"
    },
    {
        "Chapter": "1",
        "sentence_range": "2464-2467",
        "Text": ", all x such that 2x + 1 \u2264 1 + 4x We\nmay rewrite this as 2 \u2264 1\n2x  + 2x which is true for all x Hence the function\nis defined at every real number By putting 2x = tan \u03b8, this function may be\nrewritten as\nf(x) =\n1\n1\n2\nsin\n1\n4\nx\nx\n+\n\u2212 \uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0+\n\uf8fb\n= sin\u2212\n\u22c5\n+ (\n)\n\uf8ee\n\uf8ef\uf8f0\n\uf8ef\n\uf8f9\n\uf8fa\uf8fb\n\uf8fa\n1\n2\n2\n2\n1\n2\nx\nx\n=\n1\n2tan2\nsin\n1\ntan\n\u2212\n\u03b8\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n+\n\u03b8\n\uf8f0\n\uf8fb\n= sin \u20131 [sin 2\u03b8]\n= 2\u03b8 = 2 tan \u2013 1 (2x)\nThus\nf \u2032(x) =\n(\n)\n2\n1\n2\n(2 )\n1\n2\nx\nx\ndxd\n\u22c5\n\u22c5\n+\n=\n2\n(2 )log2\n1\n4\nx\nx \u22c5\n+\n=\n1\n2\nlog2\n1\n4\nx\nx\n+\n+\nRationalised 2023-24\nCONTINUITY AND DIFFERENTIABILITY\n143\nExample 41 Find  f \u2032(x) if f(x) = (sin x)sin x for all 0 < x < \u03c0"
    },
    {
        "Chapter": "1",
        "sentence_range": "2465-2468",
        "Text": "We\nmay rewrite this as 2 \u2264 1\n2x  + 2x which is true for all x Hence the function\nis defined at every real number By putting 2x = tan \u03b8, this function may be\nrewritten as\nf(x) =\n1\n1\n2\nsin\n1\n4\nx\nx\n+\n\u2212 \uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0+\n\uf8fb\n= sin\u2212\n\u22c5\n+ (\n)\n\uf8ee\n\uf8ef\uf8f0\n\uf8ef\n\uf8f9\n\uf8fa\uf8fb\n\uf8fa\n1\n2\n2\n2\n1\n2\nx\nx\n=\n1\n2tan2\nsin\n1\ntan\n\u2212\n\u03b8\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n+\n\u03b8\n\uf8f0\n\uf8fb\n= sin \u20131 [sin 2\u03b8]\n= 2\u03b8 = 2 tan \u2013 1 (2x)\nThus\nf \u2032(x) =\n(\n)\n2\n1\n2\n(2 )\n1\n2\nx\nx\ndxd\n\u22c5\n\u22c5\n+\n=\n2\n(2 )log2\n1\n4\nx\nx \u22c5\n+\n=\n1\n2\nlog2\n1\n4\nx\nx\n+\n+\nRationalised 2023-24\nCONTINUITY AND DIFFERENTIABILITY\n143\nExample 41 Find  f \u2032(x) if f(x) = (sin x)sin x for all 0 < x < \u03c0 Solution The function y = (sin x)sin x is defined for all positive real numbers"
    },
    {
        "Chapter": "1",
        "sentence_range": "2466-2469",
        "Text": "Hence the function\nis defined at every real number By putting 2x = tan \u03b8, this function may be\nrewritten as\nf(x) =\n1\n1\n2\nsin\n1\n4\nx\nx\n+\n\u2212 \uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0+\n\uf8fb\n= sin\u2212\n\u22c5\n+ (\n)\n\uf8ee\n\uf8ef\uf8f0\n\uf8ef\n\uf8f9\n\uf8fa\uf8fb\n\uf8fa\n1\n2\n2\n2\n1\n2\nx\nx\n=\n1\n2tan2\nsin\n1\ntan\n\u2212\n\u03b8\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n+\n\u03b8\n\uf8f0\n\uf8fb\n= sin \u20131 [sin 2\u03b8]\n= 2\u03b8 = 2 tan \u2013 1 (2x)\nThus\nf \u2032(x) =\n(\n)\n2\n1\n2\n(2 )\n1\n2\nx\nx\ndxd\n\u22c5\n\u22c5\n+\n=\n2\n(2 )log2\n1\n4\nx\nx \u22c5\n+\n=\n1\n2\nlog2\n1\n4\nx\nx\n+\n+\nRationalised 2023-24\nCONTINUITY AND DIFFERENTIABILITY\n143\nExample 41 Find  f \u2032(x) if f(x) = (sin x)sin x for all 0 < x < \u03c0 Solution The function y = (sin x)sin x is defined for all positive real numbers Taking\nlogarithms, we have\nlog y = log (sin x)sin x = sin x log (sin x)\nThen\n1 dy\ny dx  = d\ndx  (sin x log (sin x))\n= cos x log (sin x) + sin x"
    },
    {
        "Chapter": "1",
        "sentence_range": "2467-2470",
        "Text": "By putting 2x = tan \u03b8, this function may be\nrewritten as\nf(x) =\n1\n1\n2\nsin\n1\n4\nx\nx\n+\n\u2212 \uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0+\n\uf8fb\n= sin\u2212\n\u22c5\n+ (\n)\n\uf8ee\n\uf8ef\uf8f0\n\uf8ef\n\uf8f9\n\uf8fa\uf8fb\n\uf8fa\n1\n2\n2\n2\n1\n2\nx\nx\n=\n1\n2tan2\nsin\n1\ntan\n\u2212\n\u03b8\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n+\n\u03b8\n\uf8f0\n\uf8fb\n= sin \u20131 [sin 2\u03b8]\n= 2\u03b8 = 2 tan \u2013 1 (2x)\nThus\nf \u2032(x) =\n(\n)\n2\n1\n2\n(2 )\n1\n2\nx\nx\ndxd\n\u22c5\n\u22c5\n+\n=\n2\n(2 )log2\n1\n4\nx\nx \u22c5\n+\n=\n1\n2\nlog2\n1\n4\nx\nx\n+\n+\nRationalised 2023-24\nCONTINUITY AND DIFFERENTIABILITY\n143\nExample 41 Find  f \u2032(x) if f(x) = (sin x)sin x for all 0 < x < \u03c0 Solution The function y = (sin x)sin x is defined for all positive real numbers Taking\nlogarithms, we have\nlog y = log (sin x)sin x = sin x log (sin x)\nThen\n1 dy\ny dx  = d\ndx  (sin x log (sin x))\n= cos x log (sin x) + sin x 1\n(sin )\nsin\nd\nx\nx dx\n\u22c5\n= cos x log (sin x) + cos x\n= (1 + log (sin x)) cos x\nThus\ndy\ndx  = y((1 + log (sin x)) cos x) = (1 + log (sin x)) ( sin x)sin x cos x\nExample 42 For a positive constant a find dy\ndx , where\n1\n1\n, and\na\nt t\ny\na\nx\nt\nt\n+\n\uf8eb\n\uf8f6\n=\n=\n\uf8ec+\n\uf8f7\n\uf8ed\n\uf8f8\nSolution Observe that both y and x are defined for all real t \u2260 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "2468-2471",
        "Text": "Solution The function y = (sin x)sin x is defined for all positive real numbers Taking\nlogarithms, we have\nlog y = log (sin x)sin x = sin x log (sin x)\nThen\n1 dy\ny dx  = d\ndx  (sin x log (sin x))\n= cos x log (sin x) + sin x 1\n(sin )\nsin\nd\nx\nx dx\n\u22c5\n= cos x log (sin x) + cos x\n= (1 + log (sin x)) cos x\nThus\ndy\ndx  = y((1 + log (sin x)) cos x) = (1 + log (sin x)) ( sin x)sin x cos x\nExample 42 For a positive constant a find dy\ndx , where\n1\n1\n, and\na\nt t\ny\na\nx\nt\nt\n+\n\uf8eb\n\uf8f6\n=\n=\n\uf8ec+\n\uf8f7\n\uf8ed\n\uf8f8\nSolution Observe that both y and x are defined for all real t \u2260 0 Clearly\ndy\ndt  = \n(\n)\nt t1\nd\na\ndt\n+\n =\n1\n1\nlog\nat t d\nt\na\ndt\nt\n+\n\uf8eb\n+\uf8f6\n\u22c5\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n=\n1\n12\n1\nlog\nat t\na\nt\n+ \uf8eb\n\uf8f6\n\uf8ec\u2212\n\uf8f7\n\uf8ed\n\uf8f8\nSimilarly\ndx\ndt  =\n1\n1\n1\na\nd\na t\nt\nt\ndt\nt\n\u2212\n\uf8ee\n\uf8f9\n\uf8eb\n\uf8f6\n+\n\u22c5\n\uf8ec+\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8ed\n\uf8f8\n=\n1\n2\n1\n1\n1\na\na t\nt\nt\n\u2212\n\uf8ee\n\uf8f9\n\uf8eb\n\uf8f6\n+\n\u22c5\n\uf8ec\u2212\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8ed\n\uf8f8\ndx\ndt  \u2260 0 only if t \u2260 \u00b1 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "2469-2472",
        "Text": "Taking\nlogarithms, we have\nlog y = log (sin x)sin x = sin x log (sin x)\nThen\n1 dy\ny dx  = d\ndx  (sin x log (sin x))\n= cos x log (sin x) + sin x 1\n(sin )\nsin\nd\nx\nx dx\n\u22c5\n= cos x log (sin x) + cos x\n= (1 + log (sin x)) cos x\nThus\ndy\ndx  = y((1 + log (sin x)) cos x) = (1 + log (sin x)) ( sin x)sin x cos x\nExample 42 For a positive constant a find dy\ndx , where\n1\n1\n, and\na\nt t\ny\na\nx\nt\nt\n+\n\uf8eb\n\uf8f6\n=\n=\n\uf8ec+\n\uf8f7\n\uf8ed\n\uf8f8\nSolution Observe that both y and x are defined for all real t \u2260 0 Clearly\ndy\ndt  = \n(\n)\nt t1\nd\na\ndt\n+\n =\n1\n1\nlog\nat t d\nt\na\ndt\nt\n+\n\uf8eb\n+\uf8f6\n\u22c5\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n=\n1\n12\n1\nlog\nat t\na\nt\n+ \uf8eb\n\uf8f6\n\uf8ec\u2212\n\uf8f7\n\uf8ed\n\uf8f8\nSimilarly\ndx\ndt  =\n1\n1\n1\na\nd\na t\nt\nt\ndt\nt\n\u2212\n\uf8ee\n\uf8f9\n\uf8eb\n\uf8f6\n+\n\u22c5\n\uf8ec+\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8ed\n\uf8f8\n=\n1\n2\n1\n1\n1\na\na t\nt\nt\n\u2212\n\uf8ee\n\uf8f9\n\uf8eb\n\uf8f6\n+\n\u22c5\n\uf8ec\u2212\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8ed\n\uf8f8\ndx\ndt  \u2260 0 only if t \u2260 \u00b1 1 Thus for t \u2260 \u00b1 1,\nRationalised 2023-24\n MATHEMATICS\n144\ndy\ndy\ndt\ndx\ndx\ndt\n=\n =\na\nt\na\na t\nt\nt\nt t\na\n+\n\u2212\n\uf8ed\uf8ec\uf8eb\u2212\n\uf8f6\n\uf8f8\uf8f7\n\uf8f0\uf8ef\uf8ee+\n\uf8fb\uf8fa\uf8f9\n\u22c5\n\uf8ed\uf8ec\uf8eb\u2212\n\uf8f6\n\uf8f8\uf8f7\n1\n2\n1\n2\n1\n1\n1\n1\n1\nlog\n=\n1\n1\nlog\n1\nt t\na\na\na\na t\nt\n+\n\u2212\n\uf8eb\n\uf8ec+\uf8f6\n\uf8f7\n\uf8ed\n\uf8f8\nExample 43 Differentiate sin2 x w"
    },
    {
        "Chapter": "1",
        "sentence_range": "2470-2473",
        "Text": "1\n(sin )\nsin\nd\nx\nx dx\n\u22c5\n= cos x log (sin x) + cos x\n= (1 + log (sin x)) cos x\nThus\ndy\ndx  = y((1 + log (sin x)) cos x) = (1 + log (sin x)) ( sin x)sin x cos x\nExample 42 For a positive constant a find dy\ndx , where\n1\n1\n, and\na\nt t\ny\na\nx\nt\nt\n+\n\uf8eb\n\uf8f6\n=\n=\n\uf8ec+\n\uf8f7\n\uf8ed\n\uf8f8\nSolution Observe that both y and x are defined for all real t \u2260 0 Clearly\ndy\ndt  = \n(\n)\nt t1\nd\na\ndt\n+\n =\n1\n1\nlog\nat t d\nt\na\ndt\nt\n+\n\uf8eb\n+\uf8f6\n\u22c5\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n=\n1\n12\n1\nlog\nat t\na\nt\n+ \uf8eb\n\uf8f6\n\uf8ec\u2212\n\uf8f7\n\uf8ed\n\uf8f8\nSimilarly\ndx\ndt  =\n1\n1\n1\na\nd\na t\nt\nt\ndt\nt\n\u2212\n\uf8ee\n\uf8f9\n\uf8eb\n\uf8f6\n+\n\u22c5\n\uf8ec+\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8ed\n\uf8f8\n=\n1\n2\n1\n1\n1\na\na t\nt\nt\n\u2212\n\uf8ee\n\uf8f9\n\uf8eb\n\uf8f6\n+\n\u22c5\n\uf8ec\u2212\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8ed\n\uf8f8\ndx\ndt  \u2260 0 only if t \u2260 \u00b1 1 Thus for t \u2260 \u00b1 1,\nRationalised 2023-24\n MATHEMATICS\n144\ndy\ndy\ndt\ndx\ndx\ndt\n=\n =\na\nt\na\na t\nt\nt\nt t\na\n+\n\u2212\n\uf8ed\uf8ec\uf8eb\u2212\n\uf8f6\n\uf8f8\uf8f7\n\uf8f0\uf8ef\uf8ee+\n\uf8fb\uf8fa\uf8f9\n\u22c5\n\uf8ed\uf8ec\uf8eb\u2212\n\uf8f6\n\uf8f8\uf8f7\n1\n2\n1\n2\n1\n1\n1\n1\n1\nlog\n=\n1\n1\nlog\n1\nt t\na\na\na\na t\nt\n+\n\u2212\n\uf8eb\n\uf8ec+\uf8f6\n\uf8f7\n\uf8ed\n\uf8f8\nExample 43 Differentiate sin2 x w r"
    },
    {
        "Chapter": "1",
        "sentence_range": "2471-2474",
        "Text": "Clearly\ndy\ndt  = \n(\n)\nt t1\nd\na\ndt\n+\n =\n1\n1\nlog\nat t d\nt\na\ndt\nt\n+\n\uf8eb\n+\uf8f6\n\u22c5\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n=\n1\n12\n1\nlog\nat t\na\nt\n+ \uf8eb\n\uf8f6\n\uf8ec\u2212\n\uf8f7\n\uf8ed\n\uf8f8\nSimilarly\ndx\ndt  =\n1\n1\n1\na\nd\na t\nt\nt\ndt\nt\n\u2212\n\uf8ee\n\uf8f9\n\uf8eb\n\uf8f6\n+\n\u22c5\n\uf8ec+\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8ed\n\uf8f8\n=\n1\n2\n1\n1\n1\na\na t\nt\nt\n\u2212\n\uf8ee\n\uf8f9\n\uf8eb\n\uf8f6\n+\n\u22c5\n\uf8ec\u2212\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8ed\n\uf8f8\ndx\ndt  \u2260 0 only if t \u2260 \u00b1 1 Thus for t \u2260 \u00b1 1,\nRationalised 2023-24\n MATHEMATICS\n144\ndy\ndy\ndt\ndx\ndx\ndt\n=\n =\na\nt\na\na t\nt\nt\nt t\na\n+\n\u2212\n\uf8ed\uf8ec\uf8eb\u2212\n\uf8f6\n\uf8f8\uf8f7\n\uf8f0\uf8ef\uf8ee+\n\uf8fb\uf8fa\uf8f9\n\u22c5\n\uf8ed\uf8ec\uf8eb\u2212\n\uf8f6\n\uf8f8\uf8f7\n1\n2\n1\n2\n1\n1\n1\n1\n1\nlog\n=\n1\n1\nlog\n1\nt t\na\na\na\na t\nt\n+\n\u2212\n\uf8eb\n\uf8ec+\uf8f6\n\uf8f7\n\uf8ed\n\uf8f8\nExample 43 Differentiate sin2 x w r t"
    },
    {
        "Chapter": "1",
        "sentence_range": "2472-2475",
        "Text": "Thus for t \u2260 \u00b1 1,\nRationalised 2023-24\n MATHEMATICS\n144\ndy\ndy\ndt\ndx\ndx\ndt\n=\n =\na\nt\na\na t\nt\nt\nt t\na\n+\n\u2212\n\uf8ed\uf8ec\uf8eb\u2212\n\uf8f6\n\uf8f8\uf8f7\n\uf8f0\uf8ef\uf8ee+\n\uf8fb\uf8fa\uf8f9\n\u22c5\n\uf8ed\uf8ec\uf8eb\u2212\n\uf8f6\n\uf8f8\uf8f7\n1\n2\n1\n2\n1\n1\n1\n1\n1\nlog\n=\n1\n1\nlog\n1\nt t\na\na\na\na t\nt\n+\n\u2212\n\uf8eb\n\uf8ec+\uf8f6\n\uf8f7\n\uf8ed\n\uf8f8\nExample 43 Differentiate sin2 x w r t e cos x"
    },
    {
        "Chapter": "1",
        "sentence_range": "2473-2476",
        "Text": "r t e cos x Solution Let u (x) = sin2 x and v (x) = e cos x"
    },
    {
        "Chapter": "1",
        "sentence_range": "2474-2477",
        "Text": "t e cos x Solution Let u (x) = sin2 x and v (x) = e cos x We want to find \n/\n/\ndu\ndu dx\ndv\n=dv dx"
    },
    {
        "Chapter": "1",
        "sentence_range": "2475-2478",
        "Text": "e cos x Solution Let u (x) = sin2 x and v (x) = e cos x We want to find \n/\n/\ndu\ndu dx\ndv\n=dv dx Clearly\ndu\ndx  = 2 sin x cos x and dv\ndx  = e cos x (\u2013 sin x) = \u2013 (sin x) e cos x\nThus\ndu\ndv  =\ncos\ncos\n2sin\ncos\n2cos\nsin\nx\nx\nx\nx\nx\nx e\n= \u2212e\n\u2212\nMiscellaneous Exercise on Chapter 5\nDifferentiate w"
    },
    {
        "Chapter": "1",
        "sentence_range": "2476-2479",
        "Text": "Solution Let u (x) = sin2 x and v (x) = e cos x We want to find \n/\n/\ndu\ndu dx\ndv\n=dv dx Clearly\ndu\ndx  = 2 sin x cos x and dv\ndx  = e cos x (\u2013 sin x) = \u2013 (sin x) e cos x\nThus\ndu\ndv  =\ncos\ncos\n2sin\ncos\n2cos\nsin\nx\nx\nx\nx\nx\nx e\n= \u2212e\n\u2212\nMiscellaneous Exercise on Chapter 5\nDifferentiate w r"
    },
    {
        "Chapter": "1",
        "sentence_range": "2477-2480",
        "Text": "We want to find \n/\n/\ndu\ndu dx\ndv\n=dv dx Clearly\ndu\ndx  = 2 sin x cos x and dv\ndx  = e cos x (\u2013 sin x) = \u2013 (sin x) e cos x\nThus\ndu\ndv  =\ncos\ncos\n2sin\ncos\n2cos\nsin\nx\nx\nx\nx\nx\nx e\n= \u2212e\n\u2212\nMiscellaneous Exercise on Chapter 5\nDifferentiate w r t"
    },
    {
        "Chapter": "1",
        "sentence_range": "2478-2481",
        "Text": "Clearly\ndu\ndx  = 2 sin x cos x and dv\ndx  = e cos x (\u2013 sin x) = \u2013 (sin x) e cos x\nThus\ndu\ndv  =\ncos\ncos\n2sin\ncos\n2cos\nsin\nx\nx\nx\nx\nx\nx e\n= \u2212e\n\u2212\nMiscellaneous Exercise on Chapter 5\nDifferentiate w r t x the function in Exercises 1 to 11"
    },
    {
        "Chapter": "1",
        "sentence_range": "2479-2482",
        "Text": "r t x the function in Exercises 1 to 11 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "2480-2483",
        "Text": "t x the function in Exercises 1 to 11 1 (3x2 \u2013 9x + 5)9\n2"
    },
    {
        "Chapter": "1",
        "sentence_range": "2481-2484",
        "Text": "x the function in Exercises 1 to 11 1 (3x2 \u2013 9x + 5)9\n2 sin3 x + cos6 x\n3"
    },
    {
        "Chapter": "1",
        "sentence_range": "2482-2485",
        "Text": "1 (3x2 \u2013 9x + 5)9\n2 sin3 x + cos6 x\n3 (5x)3 cos 2x\n4"
    },
    {
        "Chapter": "1",
        "sentence_range": "2483-2486",
        "Text": "(3x2 \u2013 9x + 5)9\n2 sin3 x + cos6 x\n3 (5x)3 cos 2x\n4 sin\u20131(x \nx ), 0 \u2264 x \u2264 1\n5"
    },
    {
        "Chapter": "1",
        "sentence_range": "2484-2487",
        "Text": "sin3 x + cos6 x\n3 (5x)3 cos 2x\n4 sin\u20131(x \nx ), 0 \u2264 x \u2264 1\n5 cos1\n2\n2\n7\nx\nx\n\u2212\n+\n, \u2013 2 < x < 2\n6"
    },
    {
        "Chapter": "1",
        "sentence_range": "2485-2488",
        "Text": "(5x)3 cos 2x\n4 sin\u20131(x \nx ), 0 \u2264 x \u2264 1\n5 cos1\n2\n2\n7\nx\nx\n\u2212\n+\n, \u2013 2 < x < 2\n6 1\n1\nsin\n1 sin\ncot\n1\nsin\n1 sin\nx\nx\nx\nx\n\u2212 \uf8ee\n\uf8f9\n+\n+\n\u2212\n\uf8ef\n\uf8fa\n+\n\u2212\n\u2212\n\uf8f0\n\uf8fb , 0 < x < 2\n\u03c0\n7"
    },
    {
        "Chapter": "1",
        "sentence_range": "2486-2489",
        "Text": "sin\u20131(x \nx ), 0 \u2264 x \u2264 1\n5 cos1\n2\n2\n7\nx\nx\n\u2212\n+\n, \u2013 2 < x < 2\n6 1\n1\nsin\n1 sin\ncot\n1\nsin\n1 sin\nx\nx\nx\nx\n\u2212 \uf8ee\n\uf8f9\n+\n+\n\u2212\n\uf8ef\n\uf8fa\n+\n\u2212\n\u2212\n\uf8f0\n\uf8fb , 0 < x < 2\n\u03c0\n7 (log x)log x, x > 1\n8"
    },
    {
        "Chapter": "1",
        "sentence_range": "2487-2490",
        "Text": "cos1\n2\n2\n7\nx\nx\n\u2212\n+\n, \u2013 2 < x < 2\n6 1\n1\nsin\n1 sin\ncot\n1\nsin\n1 sin\nx\nx\nx\nx\n\u2212 \uf8ee\n\uf8f9\n+\n+\n\u2212\n\uf8ef\n\uf8fa\n+\n\u2212\n\u2212\n\uf8f0\n\uf8fb , 0 < x < 2\n\u03c0\n7 (log x)log x, x > 1\n8 cos (a cos x + b sin x), for some constant a and b"
    },
    {
        "Chapter": "1",
        "sentence_range": "2488-2491",
        "Text": "1\n1\nsin\n1 sin\ncot\n1\nsin\n1 sin\nx\nx\nx\nx\n\u2212 \uf8ee\n\uf8f9\n+\n+\n\u2212\n\uf8ef\n\uf8fa\n+\n\u2212\n\u2212\n\uf8f0\n\uf8fb , 0 < x < 2\n\u03c0\n7 (log x)log x, x > 1\n8 cos (a cos x + b sin x), for some constant a and b 9"
    },
    {
        "Chapter": "1",
        "sentence_range": "2489-2492",
        "Text": "(log x)log x, x > 1\n8 cos (a cos x + b sin x), for some constant a and b 9 (sin x \u2013 cos x) (sin x \u2013 cos x), \n3\n4\n4\nx\n\u03c0\n\u03c0\n<\n<\n10"
    },
    {
        "Chapter": "1",
        "sentence_range": "2490-2493",
        "Text": "cos (a cos x + b sin x), for some constant a and b 9 (sin x \u2013 cos x) (sin x \u2013 cos x), \n3\n4\n4\nx\n\u03c0\n\u03c0\n<\n<\n10 xx + xa + ax + aa, for some fixed a > 0 and x > 0\nRationalised 2023-24\nCONTINUITY AND DIFFERENTIABILITY\n145\n11"
    },
    {
        "Chapter": "1",
        "sentence_range": "2491-2494",
        "Text": "9 (sin x \u2013 cos x) (sin x \u2013 cos x), \n3\n4\n4\nx\n\u03c0\n\u03c0\n<\n<\n10 xx + xa + ax + aa, for some fixed a > 0 and x > 0\nRationalised 2023-24\nCONTINUITY AND DIFFERENTIABILITY\n145\n11 (\n)\n2\n2 3\n3\nx\nxx\n\u2212 +x\n\u2212\n, for x > 3\n12"
    },
    {
        "Chapter": "1",
        "sentence_range": "2492-2495",
        "Text": "(sin x \u2013 cos x) (sin x \u2013 cos x), \n3\n4\n4\nx\n\u03c0\n\u03c0\n<\n<\n10 xx + xa + ax + aa, for some fixed a > 0 and x > 0\nRationalised 2023-24\nCONTINUITY AND DIFFERENTIABILITY\n145\n11 (\n)\n2\n2 3\n3\nx\nxx\n\u2212 +x\n\u2212\n, for x > 3\n12 Find dy\ndx\n, if y = 12 (1 \u2013 cos t), x = 10 (t \u2013 sin t), \n2\n2\nt\n\u03c0\n\u03c0\n\u2212\n< <\n13"
    },
    {
        "Chapter": "1",
        "sentence_range": "2493-2496",
        "Text": "xx + xa + ax + aa, for some fixed a > 0 and x > 0\nRationalised 2023-24\nCONTINUITY AND DIFFERENTIABILITY\n145\n11 (\n)\n2\n2 3\n3\nx\nxx\n\u2212 +x\n\u2212\n, for x > 3\n12 Find dy\ndx\n, if y = 12 (1 \u2013 cos t), x = 10 (t \u2013 sin t), \n2\n2\nt\n\u03c0\n\u03c0\n\u2212\n< <\n13 Find dy\ndx , if y = sin\u20131 x + sin\u20131 \n2\n1\n\u2212x\n, 0 < x < 1\n14"
    },
    {
        "Chapter": "1",
        "sentence_range": "2494-2497",
        "Text": "(\n)\n2\n2 3\n3\nx\nxx\n\u2212 +x\n\u2212\n, for x > 3\n12 Find dy\ndx\n, if y = 12 (1 \u2013 cos t), x = 10 (t \u2013 sin t), \n2\n2\nt\n\u03c0\n\u03c0\n\u2212\n< <\n13 Find dy\ndx , if y = sin\u20131 x + sin\u20131 \n2\n1\n\u2212x\n, 0 < x < 1\n14 If \n1\n1\n0\nx\ny\ny\nx\n+\n+\n+\n=\n, for , \u2013 1 < x < 1, prove that\n(\n)2\n1\n1\ndy\ndx\nx\n= \u2212\n+\n15"
    },
    {
        "Chapter": "1",
        "sentence_range": "2495-2498",
        "Text": "Find dy\ndx\n, if y = 12 (1 \u2013 cos t), x = 10 (t \u2013 sin t), \n2\n2\nt\n\u03c0\n\u03c0\n\u2212\n< <\n13 Find dy\ndx , if y = sin\u20131 x + sin\u20131 \n2\n1\n\u2212x\n, 0 < x < 1\n14 If \n1\n1\n0\nx\ny\ny\nx\n+\n+\n+\n=\n, for , \u2013 1 < x < 1, prove that\n(\n)2\n1\n1\ndy\ndx\nx\n= \u2212\n+\n15 If (x \u2013 a)2 + (y \u2013 b)2 = c2, for some c > 0, prove that\n3\n2\n2\n2\n2\n1\ndy\ndx\nd y\ndx\n\uf8ee\n\uf8f9\n\uf8eb\n\uf8f6\n\uf8ef+\n\uf8fa\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8f0\n\uf8fb\nis a constant independent of a and b"
    },
    {
        "Chapter": "1",
        "sentence_range": "2496-2499",
        "Text": "Find dy\ndx , if y = sin\u20131 x + sin\u20131 \n2\n1\n\u2212x\n, 0 < x < 1\n14 If \n1\n1\n0\nx\ny\ny\nx\n+\n+\n+\n=\n, for , \u2013 1 < x < 1, prove that\n(\n)2\n1\n1\ndy\ndx\nx\n= \u2212\n+\n15 If (x \u2013 a)2 + (y \u2013 b)2 = c2, for some c > 0, prove that\n3\n2\n2\n2\n2\n1\ndy\ndx\nd y\ndx\n\uf8ee\n\uf8f9\n\uf8eb\n\uf8f6\n\uf8ef+\n\uf8fa\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8f0\n\uf8fb\nis a constant independent of a and b 16"
    },
    {
        "Chapter": "1",
        "sentence_range": "2497-2500",
        "Text": "If \n1\n1\n0\nx\ny\ny\nx\n+\n+\n+\n=\n, for , \u2013 1 < x < 1, prove that\n(\n)2\n1\n1\ndy\ndx\nx\n= \u2212\n+\n15 If (x \u2013 a)2 + (y \u2013 b)2 = c2, for some c > 0, prove that\n3\n2\n2\n2\n2\n1\ndy\ndx\nd y\ndx\n\uf8ee\n\uf8f9\n\uf8eb\n\uf8f6\n\uf8ef+\n\uf8fa\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8f0\n\uf8fb\nis a constant independent of a and b 16 If cos y = x cos (a + y), with cos a \u2260 \u00b1 1, prove that \ncos (2\n)\nsin\ndy\na\ny\ndx\n+a\n="
    },
    {
        "Chapter": "1",
        "sentence_range": "2498-2501",
        "Text": "If (x \u2013 a)2 + (y \u2013 b)2 = c2, for some c > 0, prove that\n3\n2\n2\n2\n2\n1\ndy\ndx\nd y\ndx\n\uf8ee\n\uf8f9\n\uf8eb\n\uf8f6\n\uf8ef+\n\uf8fa\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8f0\n\uf8fb\nis a constant independent of a and b 16 If cos y = x cos (a + y), with cos a \u2260 \u00b1 1, prove that \ncos (2\n)\nsin\ndy\na\ny\ndx\n+a\n= 17"
    },
    {
        "Chapter": "1",
        "sentence_range": "2499-2502",
        "Text": "16 If cos y = x cos (a + y), with cos a \u2260 \u00b1 1, prove that \ncos (2\n)\nsin\ndy\na\ny\ndx\n+a\n= 17 If x = a (cos t + t sin t) and y = a (sin t \u2013 t cos t), find \n2\n2\nd y\ndx"
    },
    {
        "Chapter": "1",
        "sentence_range": "2500-2503",
        "Text": "If cos y = x cos (a + y), with cos a \u2260 \u00b1 1, prove that \ncos (2\n)\nsin\ndy\na\ny\ndx\n+a\n= 17 If x = a (cos t + t sin t) and y = a (sin t \u2013 t cos t), find \n2\n2\nd y\ndx 18"
    },
    {
        "Chapter": "1",
        "sentence_range": "2501-2504",
        "Text": "17 If x = a (cos t + t sin t) and y = a (sin t \u2013 t cos t), find \n2\n2\nd y\ndx 18 If f(x) = | x |3, show that f \u2033(x) exists for all real x and find it"
    },
    {
        "Chapter": "1",
        "sentence_range": "2502-2505",
        "Text": "If x = a (cos t + t sin t) and y = a (sin t \u2013 t cos t), find \n2\n2\nd y\ndx 18 If f(x) = | x |3, show that f \u2033(x) exists for all real x and find it 19"
    },
    {
        "Chapter": "1",
        "sentence_range": "2503-2506",
        "Text": "18 If f(x) = | x |3, show that f \u2033(x) exists for all real x and find it 19 Using the fact that sin (A + B) = sin A cos B + cos A sin B and the differentiation,\nobtain the sum formula for cosines"
    },
    {
        "Chapter": "1",
        "sentence_range": "2504-2507",
        "Text": "If f(x) = | x |3, show that f \u2033(x) exists for all real x and find it 19 Using the fact that sin (A + B) = sin A cos B + cos A sin B and the differentiation,\nobtain the sum formula for cosines 20"
    },
    {
        "Chapter": "1",
        "sentence_range": "2505-2508",
        "Text": "19 Using the fact that sin (A + B) = sin A cos B + cos A sin B and the differentiation,\nobtain the sum formula for cosines 20 Does there exist a function which is continuous everywhere but not differentiable\nat exactly two points"
    },
    {
        "Chapter": "1",
        "sentence_range": "2506-2509",
        "Text": "Using the fact that sin (A + B) = sin A cos B + cos A sin B and the differentiation,\nobtain the sum formula for cosines 20 Does there exist a function which is continuous everywhere but not differentiable\nat exactly two points Justify your answer"
    },
    {
        "Chapter": "1",
        "sentence_range": "2507-2510",
        "Text": "20 Does there exist a function which is continuous everywhere but not differentiable\nat exactly two points Justify your answer 21"
    },
    {
        "Chapter": "1",
        "sentence_range": "2508-2511",
        "Text": "Does there exist a function which is continuous everywhere but not differentiable\nat exactly two points Justify your answer 21 If \n( )\n( )\n( )\nf x\ng x\nh x\ny\nl\nm\nn\na\nb\nc\n=\n, prove that \n( )\n( )\n( )\nf\nx\ng x\nh x\ndy\nl\nm\nn\ndx\na\nb\nc\n\u2032\n\u2032\n\u2032\n=\n22"
    },
    {
        "Chapter": "1",
        "sentence_range": "2509-2512",
        "Text": "Justify your answer 21 If \n( )\n( )\n( )\nf x\ng x\nh x\ny\nl\nm\nn\na\nb\nc\n=\n, prove that \n( )\n( )\n( )\nf\nx\ng x\nh x\ndy\nl\nm\nn\ndx\na\nb\nc\n\u2032\n\u2032\n\u2032\n=\n22 If y = \nacos1\nx\ne\n\u2212\n, \u2013 1 \u2264 x \u2264 1, show that (\n)\n2\n2\n2\n2\n1\n0\nd y\ndy\nx\nx\na y\ndx\ndx\n\u2212\n\u2212\n\u2212\n="
    },
    {
        "Chapter": "1",
        "sentence_range": "2510-2513",
        "Text": "21 If \n( )\n( )\n( )\nf x\ng x\nh x\ny\nl\nm\nn\na\nb\nc\n=\n, prove that \n( )\n( )\n( )\nf\nx\ng x\nh x\ndy\nl\nm\nn\ndx\na\nb\nc\n\u2032\n\u2032\n\u2032\n=\n22 If y = \nacos1\nx\ne\n\u2212\n, \u2013 1 \u2264 x \u2264 1, show that (\n)\n2\n2\n2\n2\n1\n0\nd y\ndy\nx\nx\na y\ndx\ndx\n\u2212\n\u2212\n\u2212\n= Rationalised 2023-24\n MATHEMATICS\n146\nSummary\n\u00ae A real valued function is continuous at a point in its domain if the limit of the\nfunction at that point equals the value of the function at that point"
    },
    {
        "Chapter": "1",
        "sentence_range": "2511-2514",
        "Text": "If \n( )\n( )\n( )\nf x\ng x\nh x\ny\nl\nm\nn\na\nb\nc\n=\n, prove that \n( )\n( )\n( )\nf\nx\ng x\nh x\ndy\nl\nm\nn\ndx\na\nb\nc\n\u2032\n\u2032\n\u2032\n=\n22 If y = \nacos1\nx\ne\n\u2212\n, \u2013 1 \u2264 x \u2264 1, show that (\n)\n2\n2\n2\n2\n1\n0\nd y\ndy\nx\nx\na y\ndx\ndx\n\u2212\n\u2212\n\u2212\n= Rationalised 2023-24\n MATHEMATICS\n146\nSummary\n\u00ae A real valued function is continuous at a point in its domain if the limit of the\nfunction at that point equals the value of the function at that point A function\nis continuous if it is continuous on the whole of its domain"
    },
    {
        "Chapter": "1",
        "sentence_range": "2512-2515",
        "Text": "If y = \nacos1\nx\ne\n\u2212\n, \u2013 1 \u2264 x \u2264 1, show that (\n)\n2\n2\n2\n2\n1\n0\nd y\ndy\nx\nx\na y\ndx\ndx\n\u2212\n\u2212\n\u2212\n= Rationalised 2023-24\n MATHEMATICS\n146\nSummary\n\u00ae A real valued function is continuous at a point in its domain if the limit of the\nfunction at that point equals the value of the function at that point A function\nis continuous if it is continuous on the whole of its domain \u00ae Sum, difference, product and quotient of continuous functions are continuous"
    },
    {
        "Chapter": "1",
        "sentence_range": "2513-2516",
        "Text": "Rationalised 2023-24\n MATHEMATICS\n146\nSummary\n\u00ae A real valued function is continuous at a point in its domain if the limit of the\nfunction at that point equals the value of the function at that point A function\nis continuous if it is continuous on the whole of its domain \u00ae Sum, difference, product and quotient of continuous functions are continuous i"
    },
    {
        "Chapter": "1",
        "sentence_range": "2514-2517",
        "Text": "A function\nis continuous if it is continuous on the whole of its domain \u00ae Sum, difference, product and quotient of continuous functions are continuous i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "2515-2518",
        "Text": "\u00ae Sum, difference, product and quotient of continuous functions are continuous i e , if f and g are continuous functions, then\n(f \u00b1 g) (x) = f (x) \u00b1 g(x) is continuous"
    },
    {
        "Chapter": "1",
        "sentence_range": "2516-2519",
        "Text": "i e , if f and g are continuous functions, then\n(f \u00b1 g) (x) = f (x) \u00b1 g(x) is continuous (f"
    },
    {
        "Chapter": "1",
        "sentence_range": "2517-2520",
        "Text": "e , if f and g are continuous functions, then\n(f \u00b1 g) (x) = f (x) \u00b1 g(x) is continuous (f g) (x) = f (x)"
    },
    {
        "Chapter": "1",
        "sentence_range": "2518-2521",
        "Text": ", if f and g are continuous functions, then\n(f \u00b1 g) (x) = f (x) \u00b1 g(x) is continuous (f g) (x) = f (x) g(x) is continuous"
    },
    {
        "Chapter": "1",
        "sentence_range": "2519-2522",
        "Text": "(f g) (x) = f (x) g(x) is continuous ( )\n( )\n( )\nf\nf x\nx\ng\ng x\n\uf8eb\n\uf8f6\n=\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n (wherever g(x) \u2260 0) is continuous"
    },
    {
        "Chapter": "1",
        "sentence_range": "2520-2523",
        "Text": "g) (x) = f (x) g(x) is continuous ( )\n( )\n( )\nf\nf x\nx\ng\ng x\n\uf8eb\n\uf8f6\n=\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n (wherever g(x) \u2260 0) is continuous \u00ae Every differentiable function is continuous, but the converse is not true"
    },
    {
        "Chapter": "1",
        "sentence_range": "2521-2524",
        "Text": "g(x) is continuous ( )\n( )\n( )\nf\nf x\nx\ng\ng x\n\uf8eb\n\uf8f6\n=\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n (wherever g(x) \u2260 0) is continuous \u00ae Every differentiable function is continuous, but the converse is not true \u00ae Chain rule is rule to differentiate composites of functions"
    },
    {
        "Chapter": "1",
        "sentence_range": "2522-2525",
        "Text": "( )\n( )\n( )\nf\nf x\nx\ng\ng x\n\uf8eb\n\uf8f6\n=\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n (wherever g(x) \u2260 0) is continuous \u00ae Every differentiable function is continuous, but the converse is not true \u00ae Chain rule is rule to differentiate composites of functions If f = v o u, t = u (x)\nand if both dt\ndx  and dv\ndt  exist then\ndf\ndv dt\ndx\n=dt dx\n\u22c5\n\u00ae Following are some of the standard derivatives (in appropriate domains):\n(\n)\n1\n2\n1\nsin\n1\nd\nx\ndx\nx\n\u2212\n=\n\u2212\n(\n)\n1\n2\n1\ncos\n1\nd\nx\ndx\nx\n\u2212\n= \u2212\n\u2212\n(\n)\n1\n2\n1\ntan\n1\nd\nx\ndx\nx\n\u2212\n=\n+\n(\nx)\nx\nd\ne\ne\ndx\n=\n(\n)\n1\nlog\nd\nx\ndx\nx\n=\n\u00ae Logarithmic differentiation is a powerful technique to differentiate functions\nof the form f (x) = [u (x)]v (x)"
    },
    {
        "Chapter": "1",
        "sentence_range": "2523-2526",
        "Text": "\u00ae Every differentiable function is continuous, but the converse is not true \u00ae Chain rule is rule to differentiate composites of functions If f = v o u, t = u (x)\nand if both dt\ndx  and dv\ndt  exist then\ndf\ndv dt\ndx\n=dt dx\n\u22c5\n\u00ae Following are some of the standard derivatives (in appropriate domains):\n(\n)\n1\n2\n1\nsin\n1\nd\nx\ndx\nx\n\u2212\n=\n\u2212\n(\n)\n1\n2\n1\ncos\n1\nd\nx\ndx\nx\n\u2212\n= \u2212\n\u2212\n(\n)\n1\n2\n1\ntan\n1\nd\nx\ndx\nx\n\u2212\n=\n+\n(\nx)\nx\nd\ne\ne\ndx\n=\n(\n)\n1\nlog\nd\nx\ndx\nx\n=\n\u00ae Logarithmic differentiation is a powerful technique to differentiate functions\nof the form f (x) = [u (x)]v (x) Here both f (x) and u(x) need to be positive for\nthis technique to make sense"
    },
    {
        "Chapter": "1",
        "sentence_range": "2524-2527",
        "Text": "\u00ae Chain rule is rule to differentiate composites of functions If f = v o u, t = u (x)\nand if both dt\ndx  and dv\ndt  exist then\ndf\ndv dt\ndx\n=dt dx\n\u22c5\n\u00ae Following are some of the standard derivatives (in appropriate domains):\n(\n)\n1\n2\n1\nsin\n1\nd\nx\ndx\nx\n\u2212\n=\n\u2212\n(\n)\n1\n2\n1\ncos\n1\nd\nx\ndx\nx\n\u2212\n= \u2212\n\u2212\n(\n)\n1\n2\n1\ntan\n1\nd\nx\ndx\nx\n\u2212\n=\n+\n(\nx)\nx\nd\ne\ne\ndx\n=\n(\n)\n1\nlog\nd\nx\ndx\nx\n=\n\u00ae Logarithmic differentiation is a powerful technique to differentiate functions\nof the form f (x) = [u (x)]v (x) Here both f (x) and u(x) need to be positive for\nthis technique to make sense \u2014v\nv\nv\nv\nv\u2014\nRationalised 2023-24\nv With the Calculus as a key, Mathematics can be successfully applied\nto the explanation of the course of Nature"
    },
    {
        "Chapter": "1",
        "sentence_range": "2525-2528",
        "Text": "If f = v o u, t = u (x)\nand if both dt\ndx  and dv\ndt  exist then\ndf\ndv dt\ndx\n=dt dx\n\u22c5\n\u00ae Following are some of the standard derivatives (in appropriate domains):\n(\n)\n1\n2\n1\nsin\n1\nd\nx\ndx\nx\n\u2212\n=\n\u2212\n(\n)\n1\n2\n1\ncos\n1\nd\nx\ndx\nx\n\u2212\n= \u2212\n\u2212\n(\n)\n1\n2\n1\ntan\n1\nd\nx\ndx\nx\n\u2212\n=\n+\n(\nx)\nx\nd\ne\ne\ndx\n=\n(\n)\n1\nlog\nd\nx\ndx\nx\n=\n\u00ae Logarithmic differentiation is a powerful technique to differentiate functions\nof the form f (x) = [u (x)]v (x) Here both f (x) and u(x) need to be positive for\nthis technique to make sense \u2014v\nv\nv\nv\nv\u2014\nRationalised 2023-24\nv With the Calculus as a key, Mathematics can be successfully applied\nto the explanation of the course of Nature \u201d \u2014 WHITEHEAD v\n6"
    },
    {
        "Chapter": "1",
        "sentence_range": "2526-2529",
        "Text": "Here both f (x) and u(x) need to be positive for\nthis technique to make sense \u2014v\nv\nv\nv\nv\u2014\nRationalised 2023-24\nv With the Calculus as a key, Mathematics can be successfully applied\nto the explanation of the course of Nature \u201d \u2014 WHITEHEAD v\n6 1 Introduction\nIn Chapter 5, we have learnt how to find derivative of composite functions, inverse\ntrigonometric functions, implicit functions, exponential functions and logarithmic functions"
    },
    {
        "Chapter": "1",
        "sentence_range": "2527-2530",
        "Text": "\u2014v\nv\nv\nv\nv\u2014\nRationalised 2023-24\nv With the Calculus as a key, Mathematics can be successfully applied\nto the explanation of the course of Nature \u201d \u2014 WHITEHEAD v\n6 1 Introduction\nIn Chapter 5, we have learnt how to find derivative of composite functions, inverse\ntrigonometric functions, implicit functions, exponential functions and logarithmic functions In this chapter, we will study applications of the derivative in various disciplines, e"
    },
    {
        "Chapter": "1",
        "sentence_range": "2528-2531",
        "Text": "\u201d \u2014 WHITEHEAD v\n6 1 Introduction\nIn Chapter 5, we have learnt how to find derivative of composite functions, inverse\ntrigonometric functions, implicit functions, exponential functions and logarithmic functions In this chapter, we will study applications of the derivative in various disciplines, e g"
    },
    {
        "Chapter": "1",
        "sentence_range": "2529-2532",
        "Text": "1 Introduction\nIn Chapter 5, we have learnt how to find derivative of composite functions, inverse\ntrigonometric functions, implicit functions, exponential functions and logarithmic functions In this chapter, we will study applications of the derivative in various disciplines, e g , in\nengineering, science, social science, and many  other fields"
    },
    {
        "Chapter": "1",
        "sentence_range": "2530-2533",
        "Text": "In this chapter, we will study applications of the derivative in various disciplines, e g , in\nengineering, science, social science, and many  other fields For instance, we will learn\nhow the derivative can be used (i) to determine rate of change of quantities, (ii) to find\nthe equations of tangent and normal to a curve at a point, (iii) to find turning points on\nthe graph of a function which in turn will help us to locate points at which largest or\nsmallest value (locally) of a function occurs"
    },
    {
        "Chapter": "1",
        "sentence_range": "2531-2534",
        "Text": "g , in\nengineering, science, social science, and many  other fields For instance, we will learn\nhow the derivative can be used (i) to determine rate of change of quantities, (ii) to find\nthe equations of tangent and normal to a curve at a point, (iii) to find turning points on\nthe graph of a function which in turn will help us to locate points at which largest or\nsmallest value (locally) of a function occurs We will also use derivative to find intervals\non which a function is increasing or decreasing"
    },
    {
        "Chapter": "1",
        "sentence_range": "2532-2535",
        "Text": ", in\nengineering, science, social science, and many  other fields For instance, we will learn\nhow the derivative can be used (i) to determine rate of change of quantities, (ii) to find\nthe equations of tangent and normal to a curve at a point, (iii) to find turning points on\nthe graph of a function which in turn will help us to locate points at which largest or\nsmallest value (locally) of a function occurs We will also use derivative to find intervals\non which a function is increasing or decreasing Finally, we use the derivative to find\napproximate value of certain quantities"
    },
    {
        "Chapter": "1",
        "sentence_range": "2533-2536",
        "Text": "For instance, we will learn\nhow the derivative can be used (i) to determine rate of change of quantities, (ii) to find\nthe equations of tangent and normal to a curve at a point, (iii) to find turning points on\nthe graph of a function which in turn will help us to locate points at which largest or\nsmallest value (locally) of a function occurs We will also use derivative to find intervals\non which a function is increasing or decreasing Finally, we use the derivative to find\napproximate value of certain quantities 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "2534-2537",
        "Text": "We will also use derivative to find intervals\non which a function is increasing or decreasing Finally, we use the derivative to find\napproximate value of certain quantities 6 2 Rate of Change of Quantities\nRecall that by the derivative ds\ndt , we mean the rate of change of distance s with\nrespect to the time t"
    },
    {
        "Chapter": "1",
        "sentence_range": "2535-2538",
        "Text": "Finally, we use the derivative to find\napproximate value of certain quantities 6 2 Rate of Change of Quantities\nRecall that by the derivative ds\ndt , we mean the rate of change of distance s with\nrespect to the time t In a similar fashion, whenever one quantity y varies with another\nquantity x, satisfying some rule \n( )\ny\n=f x\n, then dy\ndx  (or f\u2032(x)) represents the rate of\nchange of y with respect to x and  \ndy\ndx\nx x\n\uf8fb\uf8fa\uf8f9\n= 0\n (or f\u2032(x0)) represents the  rate of change\nof y with respect to x at \n0\nx\n=x"
    },
    {
        "Chapter": "1",
        "sentence_range": "2536-2539",
        "Text": "6 2 Rate of Change of Quantities\nRecall that by the derivative ds\ndt , we mean the rate of change of distance s with\nrespect to the time t In a similar fashion, whenever one quantity y varies with another\nquantity x, satisfying some rule \n( )\ny\n=f x\n, then dy\ndx  (or f\u2032(x)) represents the rate of\nchange of y with respect to x and  \ndy\ndx\nx x\n\uf8fb\uf8fa\uf8f9\n= 0\n (or f\u2032(x0)) represents the  rate of change\nof y with respect to x at \n0\nx\n=x if Further, if two variables x and y are varying with respect to another variable t, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "2537-2540",
        "Text": "2 Rate of Change of Quantities\nRecall that by the derivative ds\ndt , we mean the rate of change of distance s with\nrespect to the time t In a similar fashion, whenever one quantity y varies with another\nquantity x, satisfying some rule \n( )\ny\n=f x\n, then dy\ndx  (or f\u2032(x)) represents the rate of\nchange of y with respect to x and  \ndy\ndx\nx x\n\uf8fb\uf8fa\uf8f9\n= 0\n (or f\u2032(x0)) represents the  rate of change\nof y with respect to x at \n0\nx\n=x if Further, if two variables x and y are varying with respect to another variable t, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "2538-2541",
        "Text": "In a similar fashion, whenever one quantity y varies with another\nquantity x, satisfying some rule \n( )\ny\n=f x\n, then dy\ndx  (or f\u2032(x)) represents the rate of\nchange of y with respect to x and  \ndy\ndx\nx x\n\uf8fb\uf8fa\uf8f9\n= 0\n (or f\u2032(x0)) represents the  rate of change\nof y with respect to x at \n0\nx\n=x if Further, if two variables x and y are varying with respect to another variable t, i e ,\n( )\nx\n=f t\nand \n( )\ny\n=g t\n, then by Chain Rule\ndy\ndx  = dy\ndx\ndt\ndt ,  if  \n0\ndx\ndt \u2260\nChapter 6\nAPPLICATION OF\nDERIVATIVES\nRationalised 2023-24\n MATHEMATICS\n148\nThus, the rate of change of y with respect to x can be calculated using the rate of\nchange of y and that of x both with respect to t"
    },
    {
        "Chapter": "1",
        "sentence_range": "2539-2542",
        "Text": "if Further, if two variables x and y are varying with respect to another variable t, i e ,\n( )\nx\n=f t\nand \n( )\ny\n=g t\n, then by Chain Rule\ndy\ndx  = dy\ndx\ndt\ndt ,  if  \n0\ndx\ndt \u2260\nChapter 6\nAPPLICATION OF\nDERIVATIVES\nRationalised 2023-24\n MATHEMATICS\n148\nThus, the rate of change of y with respect to x can be calculated using the rate of\nchange of y and that of x both with respect to t Let us consider some examples"
    },
    {
        "Chapter": "1",
        "sentence_range": "2540-2543",
        "Text": "e ,\n( )\nx\n=f t\nand \n( )\ny\n=g t\n, then by Chain Rule\ndy\ndx  = dy\ndx\ndt\ndt ,  if  \n0\ndx\ndt \u2260\nChapter 6\nAPPLICATION OF\nDERIVATIVES\nRationalised 2023-24\n MATHEMATICS\n148\nThus, the rate of change of y with respect to x can be calculated using the rate of\nchange of y and that of x both with respect to t Let us consider some examples Example 1 Find the rate of change of the area of a circle per second with respect to\nits radius r when r = 5 cm"
    },
    {
        "Chapter": "1",
        "sentence_range": "2541-2544",
        "Text": ",\n( )\nx\n=f t\nand \n( )\ny\n=g t\n, then by Chain Rule\ndy\ndx  = dy\ndx\ndt\ndt ,  if  \n0\ndx\ndt \u2260\nChapter 6\nAPPLICATION OF\nDERIVATIVES\nRationalised 2023-24\n MATHEMATICS\n148\nThus, the rate of change of y with respect to x can be calculated using the rate of\nchange of y and that of x both with respect to t Let us consider some examples Example 1 Find the rate of change of the area of a circle per second with respect to\nits radius r when r = 5 cm Solution The area A of a circle with radius r is given by A = \u03c0r2"
    },
    {
        "Chapter": "1",
        "sentence_range": "2542-2545",
        "Text": "Let us consider some examples Example 1 Find the rate of change of the area of a circle per second with respect to\nits radius r when r = 5 cm Solution The area A of a circle with radius r is given by A = \u03c0r2 Therefore, the rate\nof change of the area A with respect to its radius r is given by \n2\nA\n(\n)\n2\nd\nd\nr\nr\ndr\n=dr\n\u03c0\n= \u03c0"
    },
    {
        "Chapter": "1",
        "sentence_range": "2543-2546",
        "Text": "Example 1 Find the rate of change of the area of a circle per second with respect to\nits radius r when r = 5 cm Solution The area A of a circle with radius r is given by A = \u03c0r2 Therefore, the rate\nof change of the area A with respect to its radius r is given by \n2\nA\n(\n)\n2\nd\nd\nr\nr\ndr\n=dr\n\u03c0\n= \u03c0 When r = 5 cm, \nA\n10\nd\ndr =\n\u03c0"
    },
    {
        "Chapter": "1",
        "sentence_range": "2544-2547",
        "Text": "Solution The area A of a circle with radius r is given by A = \u03c0r2 Therefore, the rate\nof change of the area A with respect to its radius r is given by \n2\nA\n(\n)\n2\nd\nd\nr\nr\ndr\n=dr\n\u03c0\n= \u03c0 When r = 5 cm, \nA\n10\nd\ndr =\n\u03c0 Thus, the area of the circle is changing at the rate of\n10\u03c0 cm2/s"
    },
    {
        "Chapter": "1",
        "sentence_range": "2545-2548",
        "Text": "Therefore, the rate\nof change of the area A with respect to its radius r is given by \n2\nA\n(\n)\n2\nd\nd\nr\nr\ndr\n=dr\n\u03c0\n= \u03c0 When r = 5 cm, \nA\n10\nd\ndr =\n\u03c0 Thus, the area of the circle is changing at the rate of\n10\u03c0 cm2/s Example 2 The volume of a cube is increasing at a rate of 9 cubic centimetres per\nsecond"
    },
    {
        "Chapter": "1",
        "sentence_range": "2546-2549",
        "Text": "When r = 5 cm, \nA\n10\nd\ndr =\n\u03c0 Thus, the area of the circle is changing at the rate of\n10\u03c0 cm2/s Example 2 The volume of a cube is increasing at a rate of 9 cubic centimetres per\nsecond How fast is the surface area increasing when the length of an edge is 10\ncentimetres"
    },
    {
        "Chapter": "1",
        "sentence_range": "2547-2550",
        "Text": "Thus, the area of the circle is changing at the rate of\n10\u03c0 cm2/s Example 2 The volume of a cube is increasing at a rate of 9 cubic centimetres per\nsecond How fast is the surface area increasing when the length of an edge is 10\ncentimetres Solution Let x be the length of a side, V be the volume and S be the surface area of\nthe cube"
    },
    {
        "Chapter": "1",
        "sentence_range": "2548-2551",
        "Text": "Example 2 The volume of a cube is increasing at a rate of 9 cubic centimetres per\nsecond How fast is the surface area increasing when the length of an edge is 10\ncentimetres Solution Let x be the length of a side, V be the volume and S be the surface area of\nthe cube Then, V = x3 and S = 6x2, where x is a function of time t"
    },
    {
        "Chapter": "1",
        "sentence_range": "2549-2552",
        "Text": "How fast is the surface area increasing when the length of an edge is 10\ncentimetres Solution Let x be the length of a side, V be the volume and S be the surface area of\nthe cube Then, V = x3 and S = 6x2, where x is a function of time t Now\ndV\ndt  = 9cm3/s (Given)\nTherefore\n9 =\n3\n3\nV\n(\n)\n(\n)\nd\nd\nd\ndx\nx\nx\ndt\ndt\ndx\ndt\n=\n=\n\u22c5\n(By Chain Rule)\n=\n32\ndx\nx\ndt\n\u22c5\nor\ndx\ndt  =\nx32"
    },
    {
        "Chapter": "1",
        "sentence_range": "2550-2553",
        "Text": "Solution Let x be the length of a side, V be the volume and S be the surface area of\nthe cube Then, V = x3 and S = 6x2, where x is a function of time t Now\ndV\ndt  = 9cm3/s (Given)\nTherefore\n9 =\n3\n3\nV\n(\n)\n(\n)\nd\nd\nd\ndx\nx\nx\ndt\ndt\ndx\ndt\n=\n=\n\u22c5\n(By Chain Rule)\n=\n32\ndx\nx\ndt\n\u22c5\nor\ndx\ndt  =\nx32 (1)\nNow\ndS\ndt  =\n2\n2\n(6\n)\n(6\n)\nd\nd\ndx\nx\nx\ndt\ndx\ndt\n=\n\u22c5\n(By Chain Rule)\n=\n32\n36\n12x\nx\n\uf8ebx\n\uf8f6\n\u22c5\n=\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n(Using (1))\nHence, when\nx = 10 cm, \n2\n3"
    },
    {
        "Chapter": "1",
        "sentence_range": "2551-2554",
        "Text": "Then, V = x3 and S = 6x2, where x is a function of time t Now\ndV\ndt  = 9cm3/s (Given)\nTherefore\n9 =\n3\n3\nV\n(\n)\n(\n)\nd\nd\nd\ndx\nx\nx\ndt\ndt\ndx\ndt\n=\n=\n\u22c5\n(By Chain Rule)\n=\n32\ndx\nx\ndt\n\u22c5\nor\ndx\ndt  =\nx32 (1)\nNow\ndS\ndt  =\n2\n2\n(6\n)\n(6\n)\nd\nd\ndx\nx\nx\ndt\ndx\ndt\n=\n\u22c5\n(By Chain Rule)\n=\n32\n36\n12x\nx\n\uf8ebx\n\uf8f6\n\u22c5\n=\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n(Using (1))\nHence, when\nx = 10 cm, \n2\n3 6 cm /s\ndS\ndt =\nRationalised 2023-24\nAPPLICATION OF DERIVATIVES\n149\nExample 3 A stone is dropped into a quiet lake and waves move in circles at a speed\nof 4cm per second"
    },
    {
        "Chapter": "1",
        "sentence_range": "2552-2555",
        "Text": "Now\ndV\ndt  = 9cm3/s (Given)\nTherefore\n9 =\n3\n3\nV\n(\n)\n(\n)\nd\nd\nd\ndx\nx\nx\ndt\ndt\ndx\ndt\n=\n=\n\u22c5\n(By Chain Rule)\n=\n32\ndx\nx\ndt\n\u22c5\nor\ndx\ndt  =\nx32 (1)\nNow\ndS\ndt  =\n2\n2\n(6\n)\n(6\n)\nd\nd\ndx\nx\nx\ndt\ndx\ndt\n=\n\u22c5\n(By Chain Rule)\n=\n32\n36\n12x\nx\n\uf8ebx\n\uf8f6\n\u22c5\n=\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n(Using (1))\nHence, when\nx = 10 cm, \n2\n3 6 cm /s\ndS\ndt =\nRationalised 2023-24\nAPPLICATION OF DERIVATIVES\n149\nExample 3 A stone is dropped into a quiet lake and waves move in circles at a speed\nof 4cm per second At the instant, when the radius of the circular wave is 10 cm, how\nfast is the enclosed area increasing"
    },
    {
        "Chapter": "1",
        "sentence_range": "2553-2556",
        "Text": "(1)\nNow\ndS\ndt  =\n2\n2\n(6\n)\n(6\n)\nd\nd\ndx\nx\nx\ndt\ndx\ndt\n=\n\u22c5\n(By Chain Rule)\n=\n32\n36\n12x\nx\n\uf8ebx\n\uf8f6\n\u22c5\n=\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n(Using (1))\nHence, when\nx = 10 cm, \n2\n3 6 cm /s\ndS\ndt =\nRationalised 2023-24\nAPPLICATION OF DERIVATIVES\n149\nExample 3 A stone is dropped into a quiet lake and waves move in circles at a speed\nof 4cm per second At the instant, when the radius of the circular wave is 10 cm, how\nfast is the enclosed area increasing Solution The area A of a circle with radius r is given by A = \u03c0r2"
    },
    {
        "Chapter": "1",
        "sentence_range": "2554-2557",
        "Text": "6 cm /s\ndS\ndt =\nRationalised 2023-24\nAPPLICATION OF DERIVATIVES\n149\nExample 3 A stone is dropped into a quiet lake and waves move in circles at a speed\nof 4cm per second At the instant, when the radius of the circular wave is 10 cm, how\nfast is the enclosed area increasing Solution The area A of a circle with radius r is given by A = \u03c0r2 Therefore, the rate\nof change of area A with respect to time t is\ndA\ndt  =\n2\n2\n(\n)\n(\n)\nd\nd\ndr\nr\nr\ndt\ndr\ndt\n\u03c0\n=\n\u03c0\n\u22c5\n = 2\u03c0 r dr\ndt\n(By Chain Rule)\nIt is given that\ndr\ndt  = 4cm/s\nTherefore, when r = 10 cm,\ndA\ndt  = 2\u03c0(10) (4) = 80\u03c0\nThus, the enclosed area is increasing at the rate of 80\u03c0 cm2/s, when r = 10 cm"
    },
    {
        "Chapter": "1",
        "sentence_range": "2555-2558",
        "Text": "At the instant, when the radius of the circular wave is 10 cm, how\nfast is the enclosed area increasing Solution The area A of a circle with radius r is given by A = \u03c0r2 Therefore, the rate\nof change of area A with respect to time t is\ndA\ndt  =\n2\n2\n(\n)\n(\n)\nd\nd\ndr\nr\nr\ndt\ndr\ndt\n\u03c0\n=\n\u03c0\n\u22c5\n = 2\u03c0 r dr\ndt\n(By Chain Rule)\nIt is given that\ndr\ndt  = 4cm/s\nTherefore, when r = 10 cm,\ndA\ndt  = 2\u03c0(10) (4) = 80\u03c0\nThus, the enclosed area is increasing at the rate of 80\u03c0 cm2/s, when r = 10 cm ANote  dy\ndx  is positive if y increases as x increases and is negative if y decreases\nas x increases"
    },
    {
        "Chapter": "1",
        "sentence_range": "2556-2559",
        "Text": "Solution The area A of a circle with radius r is given by A = \u03c0r2 Therefore, the rate\nof change of area A with respect to time t is\ndA\ndt  =\n2\n2\n(\n)\n(\n)\nd\nd\ndr\nr\nr\ndt\ndr\ndt\n\u03c0\n=\n\u03c0\n\u22c5\n = 2\u03c0 r dr\ndt\n(By Chain Rule)\nIt is given that\ndr\ndt  = 4cm/s\nTherefore, when r = 10 cm,\ndA\ndt  = 2\u03c0(10) (4) = 80\u03c0\nThus, the enclosed area is increasing at the rate of 80\u03c0 cm2/s, when r = 10 cm ANote  dy\ndx  is positive if y increases as x increases and is negative if y decreases\nas x increases Example 4 The length x of a rectangle is decreasing at the rate of 3 cm/minute and\nthe width y is increasing at the rate of 2cm/minute"
    },
    {
        "Chapter": "1",
        "sentence_range": "2557-2560",
        "Text": "Therefore, the rate\nof change of area A with respect to time t is\ndA\ndt  =\n2\n2\n(\n)\n(\n)\nd\nd\ndr\nr\nr\ndt\ndr\ndt\n\u03c0\n=\n\u03c0\n\u22c5\n = 2\u03c0 r dr\ndt\n(By Chain Rule)\nIt is given that\ndr\ndt  = 4cm/s\nTherefore, when r = 10 cm,\ndA\ndt  = 2\u03c0(10) (4) = 80\u03c0\nThus, the enclosed area is increasing at the rate of 80\u03c0 cm2/s, when r = 10 cm ANote  dy\ndx  is positive if y increases as x increases and is negative if y decreases\nas x increases Example 4 The length x of a rectangle is decreasing at the rate of 3 cm/minute and\nthe width y is increasing at the rate of 2cm/minute When x =10cm and y = 6cm, find\nthe rates of change of (a) the perimeter and (b) the area of the rectangle"
    },
    {
        "Chapter": "1",
        "sentence_range": "2558-2561",
        "Text": "ANote  dy\ndx  is positive if y increases as x increases and is negative if y decreases\nas x increases Example 4 The length x of a rectangle is decreasing at the rate of 3 cm/minute and\nthe width y is increasing at the rate of 2cm/minute When x =10cm and y = 6cm, find\nthe rates of change of (a) the perimeter and (b) the area of the rectangle Solution Since the length x is decreasing and the width y is increasing with respect to\ntime, we have\n3 cm/min\ndx\ndt = \u2212\nand\n2 cm/min\ndy\ndt =\n(a)\nThe perimeter P of a rectangle is given by\nP = 2 (x + y)\nTherefore\ndP\ndt  = 2\n2\n3\n2\n2\ndx\ndt\ndy\n+dt\n\uf8ed\uf8ec\uf8eb\n\uf8f6\n\uf8f8\uf8f7 =\n\u2212 +\n= \u2212\n(\n)\ncm/min\n(b)\nThe area A of the rectangle is given by\nA = x"
    },
    {
        "Chapter": "1",
        "sentence_range": "2559-2562",
        "Text": "Example 4 The length x of a rectangle is decreasing at the rate of 3 cm/minute and\nthe width y is increasing at the rate of 2cm/minute When x =10cm and y = 6cm, find\nthe rates of change of (a) the perimeter and (b) the area of the rectangle Solution Since the length x is decreasing and the width y is increasing with respect to\ntime, we have\n3 cm/min\ndx\ndt = \u2212\nand\n2 cm/min\ndy\ndt =\n(a)\nThe perimeter P of a rectangle is given by\nP = 2 (x + y)\nTherefore\ndP\ndt  = 2\n2\n3\n2\n2\ndx\ndt\ndy\n+dt\n\uf8ed\uf8ec\uf8eb\n\uf8f6\n\uf8f8\uf8f7 =\n\u2212 +\n= \u2212\n(\n)\ncm/min\n(b)\nThe area A of the rectangle is given by\nA = x y\nTherefore\ndA\ndt  = dx\ndy\ny\nx\ndt\ndt\n\u22c5\n+\n\u22c5\n= \u2013 3(6) + 10(2)\n(as x = 10 cm and y = 6 cm)\n= 2 cm2/min\nRationalised 2023-24\n MATHEMATICS\n150\nExample 5 The total cost C(x) in Rupees, associated with the production of x units of\nan item is given by\nC(x) = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "2560-2563",
        "Text": "When x =10cm and y = 6cm, find\nthe rates of change of (a) the perimeter and (b) the area of the rectangle Solution Since the length x is decreasing and the width y is increasing with respect to\ntime, we have\n3 cm/min\ndx\ndt = \u2212\nand\n2 cm/min\ndy\ndt =\n(a)\nThe perimeter P of a rectangle is given by\nP = 2 (x + y)\nTherefore\ndP\ndt  = 2\n2\n3\n2\n2\ndx\ndt\ndy\n+dt\n\uf8ed\uf8ec\uf8eb\n\uf8f6\n\uf8f8\uf8f7 =\n\u2212 +\n= \u2212\n(\n)\ncm/min\n(b)\nThe area A of the rectangle is given by\nA = x y\nTherefore\ndA\ndt  = dx\ndy\ny\nx\ndt\ndt\n\u22c5\n+\n\u22c5\n= \u2013 3(6) + 10(2)\n(as x = 10 cm and y = 6 cm)\n= 2 cm2/min\nRationalised 2023-24\n MATHEMATICS\n150\nExample 5 The total cost C(x) in Rupees, associated with the production of x units of\nan item is given by\nC(x) = 0 005 x3 \u2013 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "2561-2564",
        "Text": "Solution Since the length x is decreasing and the width y is increasing with respect to\ntime, we have\n3 cm/min\ndx\ndt = \u2212\nand\n2 cm/min\ndy\ndt =\n(a)\nThe perimeter P of a rectangle is given by\nP = 2 (x + y)\nTherefore\ndP\ndt  = 2\n2\n3\n2\n2\ndx\ndt\ndy\n+dt\n\uf8ed\uf8ec\uf8eb\n\uf8f6\n\uf8f8\uf8f7 =\n\u2212 +\n= \u2212\n(\n)\ncm/min\n(b)\nThe area A of the rectangle is given by\nA = x y\nTherefore\ndA\ndt  = dx\ndy\ny\nx\ndt\ndt\n\u22c5\n+\n\u22c5\n= \u2013 3(6) + 10(2)\n(as x = 10 cm and y = 6 cm)\n= 2 cm2/min\nRationalised 2023-24\n MATHEMATICS\n150\nExample 5 The total cost C(x) in Rupees, associated with the production of x units of\nan item is given by\nC(x) = 0 005 x3 \u2013 0 02 x2 + 30x + 5000\nFind the marginal cost when 3 units are produced, where by marginal cost we\nmean the instantaneous rate of change of total cost at any level of output"
    },
    {
        "Chapter": "1",
        "sentence_range": "2562-2565",
        "Text": "y\nTherefore\ndA\ndt  = dx\ndy\ny\nx\ndt\ndt\n\u22c5\n+\n\u22c5\n= \u2013 3(6) + 10(2)\n(as x = 10 cm and y = 6 cm)\n= 2 cm2/min\nRationalised 2023-24\n MATHEMATICS\n150\nExample 5 The total cost C(x) in Rupees, associated with the production of x units of\nan item is given by\nC(x) = 0 005 x3 \u2013 0 02 x2 + 30x + 5000\nFind the marginal cost when 3 units are produced, where by marginal cost we\nmean the instantaneous rate of change of total cost at any level of output Solution Since marginal cost is the rate of change of total cost with respect to the\noutput, we have\nMarginal\ncost (MC) =\n0"
    },
    {
        "Chapter": "1",
        "sentence_range": "2563-2566",
        "Text": "005 x3 \u2013 0 02 x2 + 30x + 5000\nFind the marginal cost when 3 units are produced, where by marginal cost we\nmean the instantaneous rate of change of total cost at any level of output Solution Since marginal cost is the rate of change of total cost with respect to the\noutput, we have\nMarginal\ncost (MC) =\n0 005(32\n)\n0"
    },
    {
        "Chapter": "1",
        "sentence_range": "2564-2567",
        "Text": "02 x2 + 30x + 5000\nFind the marginal cost when 3 units are produced, where by marginal cost we\nmean the instantaneous rate of change of total cost at any level of output Solution Since marginal cost is the rate of change of total cost with respect to the\noutput, we have\nMarginal\ncost (MC) =\n0 005(32\n)\n0 02(2 )\n30\ndC\nx\nx\ndx =\n\u2212\n+\nWhen\nx = 3, MC =\n0"
    },
    {
        "Chapter": "1",
        "sentence_range": "2565-2568",
        "Text": "Solution Since marginal cost is the rate of change of total cost with respect to the\noutput, we have\nMarginal\ncost (MC) =\n0 005(32\n)\n0 02(2 )\n30\ndC\nx\nx\ndx =\n\u2212\n+\nWhen\nx = 3, MC =\n0 015(3 )2\n0"
    },
    {
        "Chapter": "1",
        "sentence_range": "2566-2569",
        "Text": "005(32\n)\n0 02(2 )\n30\ndC\nx\nx\ndx =\n\u2212\n+\nWhen\nx = 3, MC =\n0 015(3 )2\n0 04(3)\n30\n\u2212\n+\n= 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "2567-2570",
        "Text": "02(2 )\n30\ndC\nx\nx\ndx =\n\u2212\n+\nWhen\nx = 3, MC =\n0 015(3 )2\n0 04(3)\n30\n\u2212\n+\n= 0 135 \u2013 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "2568-2571",
        "Text": "015(3 )2\n0 04(3)\n30\n\u2212\n+\n= 0 135 \u2013 0 12 + 30 = 30"
    },
    {
        "Chapter": "1",
        "sentence_range": "2569-2572",
        "Text": "04(3)\n30\n\u2212\n+\n= 0 135 \u2013 0 12 + 30 = 30 015\nHence, the required marginal cost is ` 30"
    },
    {
        "Chapter": "1",
        "sentence_range": "2570-2573",
        "Text": "135 \u2013 0 12 + 30 = 30 015\nHence, the required marginal cost is ` 30 02 (nearly)"
    },
    {
        "Chapter": "1",
        "sentence_range": "2571-2574",
        "Text": "12 + 30 = 30 015\nHence, the required marginal cost is ` 30 02 (nearly) Example 6 The total revenue in Rupees received from the sale of x units of a product\nis given by R(x) = 3x2 + 36x + 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "2572-2575",
        "Text": "015\nHence, the required marginal cost is ` 30 02 (nearly) Example 6 The total revenue in Rupees received from the sale of x units of a product\nis given by R(x) = 3x2 + 36x + 5 Find the marginal revenue, when x = 5, where by\nmarginal revenue we mean the rate of change of total revenue with respect to the\nnumber of items sold at an instant"
    },
    {
        "Chapter": "1",
        "sentence_range": "2573-2576",
        "Text": "02 (nearly) Example 6 The total revenue in Rupees received from the sale of x units of a product\nis given by R(x) = 3x2 + 36x + 5 Find the marginal revenue, when x = 5, where by\nmarginal revenue we mean the rate of change of total revenue with respect to the\nnumber of items sold at an instant Solution Since marginal revenue is the rate of change of total revenue with respect to\nthe number of units sold, we have\nMarginal Revenue\n(MR) =\nR\n6\n36\nd\ndx =x\n+\nWhen\nx = 5, MR = 6(5) + 36 = 66\nHence, the required marginal revenue is ` 66"
    },
    {
        "Chapter": "1",
        "sentence_range": "2574-2577",
        "Text": "Example 6 The total revenue in Rupees received from the sale of x units of a product\nis given by R(x) = 3x2 + 36x + 5 Find the marginal revenue, when x = 5, where by\nmarginal revenue we mean the rate of change of total revenue with respect to the\nnumber of items sold at an instant Solution Since marginal revenue is the rate of change of total revenue with respect to\nthe number of units sold, we have\nMarginal Revenue\n(MR) =\nR\n6\n36\nd\ndx =x\n+\nWhen\nx = 5, MR = 6(5) + 36 = 66\nHence, the required marginal revenue is ` 66 EXERCISE 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "2575-2578",
        "Text": "Find the marginal revenue, when x = 5, where by\nmarginal revenue we mean the rate of change of total revenue with respect to the\nnumber of items sold at an instant Solution Since marginal revenue is the rate of change of total revenue with respect to\nthe number of units sold, we have\nMarginal Revenue\n(MR) =\nR\n6\n36\nd\ndx =x\n+\nWhen\nx = 5, MR = 6(5) + 36 = 66\nHence, the required marginal revenue is ` 66 EXERCISE 6 1\n1"
    },
    {
        "Chapter": "1",
        "sentence_range": "2576-2579",
        "Text": "Solution Since marginal revenue is the rate of change of total revenue with respect to\nthe number of units sold, we have\nMarginal Revenue\n(MR) =\nR\n6\n36\nd\ndx =x\n+\nWhen\nx = 5, MR = 6(5) + 36 = 66\nHence, the required marginal revenue is ` 66 EXERCISE 6 1\n1 Find the rate of change of the area of a circle with respect to its radius r when\n(a)\nr = 3 cm\n(b)\nr = 4 cm\n2"
    },
    {
        "Chapter": "1",
        "sentence_range": "2577-2580",
        "Text": "EXERCISE 6 1\n1 Find the rate of change of the area of a circle with respect to its radius r when\n(a)\nr = 3 cm\n(b)\nr = 4 cm\n2 The volume of a cube is increasing at the rate of 8 cm3/s"
    },
    {
        "Chapter": "1",
        "sentence_range": "2578-2581",
        "Text": "1\n1 Find the rate of change of the area of a circle with respect to its radius r when\n(a)\nr = 3 cm\n(b)\nr = 4 cm\n2 The volume of a cube is increasing at the rate of 8 cm3/s How fast is the\nsurface area increasing when the length of an edge is 12 cm"
    },
    {
        "Chapter": "1",
        "sentence_range": "2579-2582",
        "Text": "Find the rate of change of the area of a circle with respect to its radius r when\n(a)\nr = 3 cm\n(b)\nr = 4 cm\n2 The volume of a cube is increasing at the rate of 8 cm3/s How fast is the\nsurface area increasing when the length of an edge is 12 cm 3"
    },
    {
        "Chapter": "1",
        "sentence_range": "2580-2583",
        "Text": "The volume of a cube is increasing at the rate of 8 cm3/s How fast is the\nsurface area increasing when the length of an edge is 12 cm 3 The radius of a circle is increasing uniformly at the rate of 3 cm/s"
    },
    {
        "Chapter": "1",
        "sentence_range": "2581-2584",
        "Text": "How fast is the\nsurface area increasing when the length of an edge is 12 cm 3 The radius of a circle is increasing uniformly at the rate of 3 cm/s Find the rate\nat which the area of the circle is increasing when the radius is 10 cm"
    },
    {
        "Chapter": "1",
        "sentence_range": "2582-2585",
        "Text": "3 The radius of a circle is increasing uniformly at the rate of 3 cm/s Find the rate\nat which the area of the circle is increasing when the radius is 10 cm 4"
    },
    {
        "Chapter": "1",
        "sentence_range": "2583-2586",
        "Text": "The radius of a circle is increasing uniformly at the rate of 3 cm/s Find the rate\nat which the area of the circle is increasing when the radius is 10 cm 4 An edge of a variable cube is increasing at the rate of 3 cm/s"
    },
    {
        "Chapter": "1",
        "sentence_range": "2584-2587",
        "Text": "Find the rate\nat which the area of the circle is increasing when the radius is 10 cm 4 An edge of a variable cube is increasing at the rate of 3 cm/s How fast is the\nvolume of the cube increasing when the edge is 10 cm long"
    },
    {
        "Chapter": "1",
        "sentence_range": "2585-2588",
        "Text": "4 An edge of a variable cube is increasing at the rate of 3 cm/s How fast is the\nvolume of the cube increasing when the edge is 10 cm long 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "2586-2589",
        "Text": "An edge of a variable cube is increasing at the rate of 3 cm/s How fast is the\nvolume of the cube increasing when the edge is 10 cm long 5 A stone is dropped into a quiet lake and waves move in circles at the speed of\n5 cm/s"
    },
    {
        "Chapter": "1",
        "sentence_range": "2587-2590",
        "Text": "How fast is the\nvolume of the cube increasing when the edge is 10 cm long 5 A stone is dropped into a quiet lake and waves move in circles at the speed of\n5 cm/s At the instant when the radius of the circular wave is 8 cm, how fast is\nthe enclosed area increasing"
    },
    {
        "Chapter": "1",
        "sentence_range": "2588-2591",
        "Text": "5 A stone is dropped into a quiet lake and waves move in circles at the speed of\n5 cm/s At the instant when the radius of the circular wave is 8 cm, how fast is\nthe enclosed area increasing Rationalised 2023-24\nAPPLICATION OF DERIVATIVES\n151\n6"
    },
    {
        "Chapter": "1",
        "sentence_range": "2589-2592",
        "Text": "A stone is dropped into a quiet lake and waves move in circles at the speed of\n5 cm/s At the instant when the radius of the circular wave is 8 cm, how fast is\nthe enclosed area increasing Rationalised 2023-24\nAPPLICATION OF DERIVATIVES\n151\n6 The radius of a circle is increasing at the rate of 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "2590-2593",
        "Text": "At the instant when the radius of the circular wave is 8 cm, how fast is\nthe enclosed area increasing Rationalised 2023-24\nAPPLICATION OF DERIVATIVES\n151\n6 The radius of a circle is increasing at the rate of 0 7 cm/s"
    },
    {
        "Chapter": "1",
        "sentence_range": "2591-2594",
        "Text": "Rationalised 2023-24\nAPPLICATION OF DERIVATIVES\n151\n6 The radius of a circle is increasing at the rate of 0 7 cm/s What is the rate of\nincrease of its circumference"
    },
    {
        "Chapter": "1",
        "sentence_range": "2592-2595",
        "Text": "The radius of a circle is increasing at the rate of 0 7 cm/s What is the rate of\nincrease of its circumference 7"
    },
    {
        "Chapter": "1",
        "sentence_range": "2593-2596",
        "Text": "7 cm/s What is the rate of\nincrease of its circumference 7 The length x of a rectangle is decreasing at the rate of 5 cm/minute and the\nwidth y is increasing at the rate of 4 cm/minute"
    },
    {
        "Chapter": "1",
        "sentence_range": "2594-2597",
        "Text": "What is the rate of\nincrease of its circumference 7 The length x of a rectangle is decreasing at the rate of 5 cm/minute and the\nwidth y is increasing at the rate of 4 cm/minute When x = 8cm and y = 6cm, find\nthe rates of change  of (a) the perimeter, and (b) the area of the rectangle"
    },
    {
        "Chapter": "1",
        "sentence_range": "2595-2598",
        "Text": "7 The length x of a rectangle is decreasing at the rate of 5 cm/minute and the\nwidth y is increasing at the rate of 4 cm/minute When x = 8cm and y = 6cm, find\nthe rates of change  of (a) the perimeter, and (b) the area of the rectangle 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "2596-2599",
        "Text": "The length x of a rectangle is decreasing at the rate of 5 cm/minute and the\nwidth y is increasing at the rate of 4 cm/minute When x = 8cm and y = 6cm, find\nthe rates of change  of (a) the perimeter, and (b) the area of the rectangle 8 A balloon, which always remains spherical on inflation, is being inflated by pumping\nin 900 cubic centimetres of gas per second"
    },
    {
        "Chapter": "1",
        "sentence_range": "2597-2600",
        "Text": "When x = 8cm and y = 6cm, find\nthe rates of change  of (a) the perimeter, and (b) the area of the rectangle 8 A balloon, which always remains spherical on inflation, is being inflated by pumping\nin 900 cubic centimetres of gas per second Find the rate at which the radius of\nthe balloon increases when the radius is 15 cm"
    },
    {
        "Chapter": "1",
        "sentence_range": "2598-2601",
        "Text": "8 A balloon, which always remains spherical on inflation, is being inflated by pumping\nin 900 cubic centimetres of gas per second Find the rate at which the radius of\nthe balloon increases when the radius is 15 cm 9"
    },
    {
        "Chapter": "1",
        "sentence_range": "2599-2602",
        "Text": "A balloon, which always remains spherical on inflation, is being inflated by pumping\nin 900 cubic centimetres of gas per second Find the rate at which the radius of\nthe balloon increases when the radius is 15 cm 9 A balloon, which always remains spherical has a variable radius"
    },
    {
        "Chapter": "1",
        "sentence_range": "2600-2603",
        "Text": "Find the rate at which the radius of\nthe balloon increases when the radius is 15 cm 9 A balloon, which always remains spherical has a variable radius Find the rate at\nwhich its volume is increasing with the radius when the later is 10 cm"
    },
    {
        "Chapter": "1",
        "sentence_range": "2601-2604",
        "Text": "9 A balloon, which always remains spherical has a variable radius Find the rate at\nwhich its volume is increasing with the radius when the later is 10 cm 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "2602-2605",
        "Text": "A balloon, which always remains spherical has a variable radius Find the rate at\nwhich its volume is increasing with the radius when the later is 10 cm 10 A ladder 5 m long is leaning against a wall"
    },
    {
        "Chapter": "1",
        "sentence_range": "2603-2606",
        "Text": "Find the rate at\nwhich its volume is increasing with the radius when the later is 10 cm 10 A ladder 5 m long is leaning against a wall The bottom of the ladder is pulled\nalong the ground, away from the wall, at the rate of 2cm/s"
    },
    {
        "Chapter": "1",
        "sentence_range": "2604-2607",
        "Text": "10 A ladder 5 m long is leaning against a wall The bottom of the ladder is pulled\nalong the ground, away from the wall, at the rate of 2cm/s How fast is its height\non the wall decreasing when the foot of the ladder is 4 m away from the wall"
    },
    {
        "Chapter": "1",
        "sentence_range": "2605-2608",
        "Text": "A ladder 5 m long is leaning against a wall The bottom of the ladder is pulled\nalong the ground, away from the wall, at the rate of 2cm/s How fast is its height\non the wall decreasing when the foot of the ladder is 4 m away from the wall 11"
    },
    {
        "Chapter": "1",
        "sentence_range": "2606-2609",
        "Text": "The bottom of the ladder is pulled\nalong the ground, away from the wall, at the rate of 2cm/s How fast is its height\non the wall decreasing when the foot of the ladder is 4 m away from the wall 11 A particle moves along the curve 6y = x3 +2"
    },
    {
        "Chapter": "1",
        "sentence_range": "2607-2610",
        "Text": "How fast is its height\non the wall decreasing when the foot of the ladder is 4 m away from the wall 11 A particle moves along the curve 6y = x3 +2 Find the points on the curve at\nwhich the y-coordinate is changing 8 times as fast as the x-coordinate"
    },
    {
        "Chapter": "1",
        "sentence_range": "2608-2611",
        "Text": "11 A particle moves along the curve 6y = x3 +2 Find the points on the curve at\nwhich the y-coordinate is changing 8 times as fast as the x-coordinate 12"
    },
    {
        "Chapter": "1",
        "sentence_range": "2609-2612",
        "Text": "A particle moves along the curve 6y = x3 +2 Find the points on the curve at\nwhich the y-coordinate is changing 8 times as fast as the x-coordinate 12 The radius of an air bubble is increasing at the rate of 1\n2 cm/s"
    },
    {
        "Chapter": "1",
        "sentence_range": "2610-2613",
        "Text": "Find the points on the curve at\nwhich the y-coordinate is changing 8 times as fast as the x-coordinate 12 The radius of an air bubble is increasing at the rate of 1\n2 cm/s At what rate is the\nvolume of the bubble increasing when the radius is 1 cm"
    },
    {
        "Chapter": "1",
        "sentence_range": "2611-2614",
        "Text": "12 The radius of an air bubble is increasing at the rate of 1\n2 cm/s At what rate is the\nvolume of the bubble increasing when the radius is 1 cm 13"
    },
    {
        "Chapter": "1",
        "sentence_range": "2612-2615",
        "Text": "The radius of an air bubble is increasing at the rate of 1\n2 cm/s At what rate is the\nvolume of the bubble increasing when the radius is 1 cm 13 A balloon, which always remains spherical, has a variable diameter 3 (2\n1)\n2\nx +"
    },
    {
        "Chapter": "1",
        "sentence_range": "2613-2616",
        "Text": "At what rate is the\nvolume of the bubble increasing when the radius is 1 cm 13 A balloon, which always remains spherical, has a variable diameter 3 (2\n1)\n2\nx + Find the rate of change of its volume with respect to x"
    },
    {
        "Chapter": "1",
        "sentence_range": "2614-2617",
        "Text": "13 A balloon, which always remains spherical, has a variable diameter 3 (2\n1)\n2\nx + Find the rate of change of its volume with respect to x 14"
    },
    {
        "Chapter": "1",
        "sentence_range": "2615-2618",
        "Text": "A balloon, which always remains spherical, has a variable diameter 3 (2\n1)\n2\nx + Find the rate of change of its volume with respect to x 14 Sand is pouring from a pipe at the rate of 12 cm3/s"
    },
    {
        "Chapter": "1",
        "sentence_range": "2616-2619",
        "Text": "Find the rate of change of its volume with respect to x 14 Sand is pouring from a pipe at the rate of 12 cm3/s The falling sand forms a cone\non the ground in such a way that the height of the cone is always one-sixth of the\nradius of the base"
    },
    {
        "Chapter": "1",
        "sentence_range": "2617-2620",
        "Text": "14 Sand is pouring from a pipe at the rate of 12 cm3/s The falling sand forms a cone\non the ground in such a way that the height of the cone is always one-sixth of the\nradius of the base How fast is the height of the sand cone increasing when the\nheight is 4 cm"
    },
    {
        "Chapter": "1",
        "sentence_range": "2618-2621",
        "Text": "Sand is pouring from a pipe at the rate of 12 cm3/s The falling sand forms a cone\non the ground in such a way that the height of the cone is always one-sixth of the\nradius of the base How fast is the height of the sand cone increasing when the\nheight is 4 cm 15"
    },
    {
        "Chapter": "1",
        "sentence_range": "2619-2622",
        "Text": "The falling sand forms a cone\non the ground in such a way that the height of the cone is always one-sixth of the\nradius of the base How fast is the height of the sand cone increasing when the\nheight is 4 cm 15 The total cost C (x) in Rupees associated with the production of x units of an\nitem is given by\nC (x) = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "2620-2623",
        "Text": "How fast is the height of the sand cone increasing when the\nheight is 4 cm 15 The total cost C (x) in Rupees associated with the production of x units of an\nitem is given by\nC (x) = 0 007x3 \u2013 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "2621-2624",
        "Text": "15 The total cost C (x) in Rupees associated with the production of x units of an\nitem is given by\nC (x) = 0 007x3 \u2013 0 003x2 + 15x + 4000"
    },
    {
        "Chapter": "1",
        "sentence_range": "2622-2625",
        "Text": "The total cost C (x) in Rupees associated with the production of x units of an\nitem is given by\nC (x) = 0 007x3 \u2013 0 003x2 + 15x + 4000 Find the marginal cost when 17 units are produced"
    },
    {
        "Chapter": "1",
        "sentence_range": "2623-2626",
        "Text": "007x3 \u2013 0 003x2 + 15x + 4000 Find the marginal cost when 17 units are produced 16"
    },
    {
        "Chapter": "1",
        "sentence_range": "2624-2627",
        "Text": "003x2 + 15x + 4000 Find the marginal cost when 17 units are produced 16 The total revenue in Rupees received from the sale of x units of a product is\ngiven by\nR (x) = 13x2 + 26x + 15"
    },
    {
        "Chapter": "1",
        "sentence_range": "2625-2628",
        "Text": "Find the marginal cost when 17 units are produced 16 The total revenue in Rupees received from the sale of x units of a product is\ngiven by\nR (x) = 13x2 + 26x + 15 Find the marginal revenue when  x = 7"
    },
    {
        "Chapter": "1",
        "sentence_range": "2626-2629",
        "Text": "16 The total revenue in Rupees received from the sale of x units of a product is\ngiven by\nR (x) = 13x2 + 26x + 15 Find the marginal revenue when  x = 7 Choose the correct answer for questions 17 and 18"
    },
    {
        "Chapter": "1",
        "sentence_range": "2627-2630",
        "Text": "The total revenue in Rupees received from the sale of x units of a product is\ngiven by\nR (x) = 13x2 + 26x + 15 Find the marginal revenue when  x = 7 Choose the correct answer for questions 17 and 18 17"
    },
    {
        "Chapter": "1",
        "sentence_range": "2628-2631",
        "Text": "Find the marginal revenue when  x = 7 Choose the correct answer for questions 17 and 18 17 The rate of change of the area of a circle with respect to its radius r at r = 6 cm is\n(A) 10\u03c0\n(B) 12\u03c0\n(C) 8\u03c0\n(D) 11\u03c0\nRationalised 2023-24\n MATHEMATICS\n152\n18"
    },
    {
        "Chapter": "1",
        "sentence_range": "2629-2632",
        "Text": "Choose the correct answer for questions 17 and 18 17 The rate of change of the area of a circle with respect to its radius r at r = 6 cm is\n(A) 10\u03c0\n(B) 12\u03c0\n(C) 8\u03c0\n(D) 11\u03c0\nRationalised 2023-24\n MATHEMATICS\n152\n18 The total revenue in Rupees received from the sale of x units of a product is\ngiven by\nR(x) = 3x2 + 36x + 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "2630-2633",
        "Text": "17 The rate of change of the area of a circle with respect to its radius r at r = 6 cm is\n(A) 10\u03c0\n(B) 12\u03c0\n(C) 8\u03c0\n(D) 11\u03c0\nRationalised 2023-24\n MATHEMATICS\n152\n18 The total revenue in Rupees received from the sale of x units of a product is\ngiven by\nR(x) = 3x2 + 36x + 5 The marginal revenue, when x = 15 is\n(A) 116\n(B) 96\n(C) 90\n(D) 126\n6"
    },
    {
        "Chapter": "1",
        "sentence_range": "2631-2634",
        "Text": "The rate of change of the area of a circle with respect to its radius r at r = 6 cm is\n(A) 10\u03c0\n(B) 12\u03c0\n(C) 8\u03c0\n(D) 11\u03c0\nRationalised 2023-24\n MATHEMATICS\n152\n18 The total revenue in Rupees received from the sale of x units of a product is\ngiven by\nR(x) = 3x2 + 36x + 5 The marginal revenue, when x = 15 is\n(A) 116\n(B) 96\n(C) 90\n(D) 126\n6 3  Increasing and Decreasing Functions\nIn this section, we will use differentiation to find out whether a function is increasing or\ndecreasing or none"
    },
    {
        "Chapter": "1",
        "sentence_range": "2632-2635",
        "Text": "The total revenue in Rupees received from the sale of x units of a product is\ngiven by\nR(x) = 3x2 + 36x + 5 The marginal revenue, when x = 15 is\n(A) 116\n(B) 96\n(C) 90\n(D) 126\n6 3  Increasing and Decreasing Functions\nIn this section, we will use differentiation to find out whether a function is increasing or\ndecreasing or none Consider the function f given by f (x) = x2, x \u2208 R"
    },
    {
        "Chapter": "1",
        "sentence_range": "2633-2636",
        "Text": "The marginal revenue, when x = 15 is\n(A) 116\n(B) 96\n(C) 90\n(D) 126\n6 3  Increasing and Decreasing Functions\nIn this section, we will use differentiation to find out whether a function is increasing or\ndecreasing or none Consider the function f given by f (x) = x2, x \u2208 R The graph of this function is a\nparabola  as given in Fig 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "2634-2637",
        "Text": "3  Increasing and Decreasing Functions\nIn this section, we will use differentiation to find out whether a function is increasing or\ndecreasing or none Consider the function f given by f (x) = x2, x \u2208 R The graph of this function is a\nparabola  as given in Fig 6 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "2635-2638",
        "Text": "Consider the function f given by f (x) = x2, x \u2208 R The graph of this function is a\nparabola  as given in Fig 6 1 Fig 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "2636-2639",
        "Text": "The graph of this function is a\nparabola  as given in Fig 6 1 Fig 6 1\nFirst consider the graph (Fig 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "2637-2640",
        "Text": "1 Fig 6 1\nFirst consider the graph (Fig 6 1) to the right of the origin"
    },
    {
        "Chapter": "1",
        "sentence_range": "2638-2641",
        "Text": "Fig 6 1\nFirst consider the graph (Fig 6 1) to the right of the origin Observe that as we\nmove from left to right along the graph, the height of the graph continuously increases"
    },
    {
        "Chapter": "1",
        "sentence_range": "2639-2642",
        "Text": "1\nFirst consider the graph (Fig 6 1) to the right of the origin Observe that as we\nmove from left to right along the graph, the height of the graph continuously increases For this reason, the function is said to be increasing for the real numbers x > 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "2640-2643",
        "Text": "1) to the right of the origin Observe that as we\nmove from left to right along the graph, the height of the graph continuously increases For this reason, the function is said to be increasing for the real numbers x > 0 Now consider the graph to the left of the origin and observe here that as we move\nfrom left to right along the graph, the height of the graph continuously decreases"
    },
    {
        "Chapter": "1",
        "sentence_range": "2641-2644",
        "Text": "Observe that as we\nmove from left to right along the graph, the height of the graph continuously increases For this reason, the function is said to be increasing for the real numbers x > 0 Now consider the graph to the left of the origin and observe here that as we move\nfrom left to right along the graph, the height of the graph continuously decreases Consequently, the function is said to be decreasing for the real numbers  x < 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "2642-2645",
        "Text": "For this reason, the function is said to be increasing for the real numbers x > 0 Now consider the graph to the left of the origin and observe here that as we move\nfrom left to right along the graph, the height of the graph continuously decreases Consequently, the function is said to be decreasing for the real numbers  x < 0 We shall now give the following analytical definitions for a function which is\nincreasing or decreasing on an interval"
    },
    {
        "Chapter": "1",
        "sentence_range": "2643-2646",
        "Text": "Now consider the graph to the left of the origin and observe here that as we move\nfrom left to right along the graph, the height of the graph continuously decreases Consequently, the function is said to be decreasing for the real numbers  x < 0 We shall now give the following analytical definitions for a function which is\nincreasing or decreasing on an interval Definition 1 Let I be an interval contained in the domain of a real valued function f"
    },
    {
        "Chapter": "1",
        "sentence_range": "2644-2647",
        "Text": "Consequently, the function is said to be decreasing for the real numbers  x < 0 We shall now give the following analytical definitions for a function which is\nincreasing or decreasing on an interval Definition 1 Let I be an interval contained in the domain of a real valued function f Then f is said to be\n(i)\nincreasing on I if x1 < x2 in I \u21d2 f (x1) < f (x2) for all x1, x2 \u2208 I"
    },
    {
        "Chapter": "1",
        "sentence_range": "2645-2648",
        "Text": "We shall now give the following analytical definitions for a function which is\nincreasing or decreasing on an interval Definition 1 Let I be an interval contained in the domain of a real valued function f Then f is said to be\n(i)\nincreasing on I if x1 < x2 in I \u21d2 f (x1) < f (x2) for all x1, x2 \u2208 I (ii)\ndecreasing on I, if x1, x2 in I \u21d2 f (x1) < f (x2) for all x1, x2 \u2208 I"
    },
    {
        "Chapter": "1",
        "sentence_range": "2646-2649",
        "Text": "Definition 1 Let I be an interval contained in the domain of a real valued function f Then f is said to be\n(i)\nincreasing on I if x1 < x2 in I \u21d2 f (x1) < f (x2) for all x1, x2 \u2208 I (ii)\ndecreasing on I, if x1, x2 in I \u21d2 f (x1) < f (x2) for all x1, x2 \u2208 I (iii)\nconstant on I, if f(x) = c for all x \u2208 I, where c is a constant"
    },
    {
        "Chapter": "1",
        "sentence_range": "2647-2650",
        "Text": "Then f is said to be\n(i)\nincreasing on I if x1 < x2 in I \u21d2 f (x1) < f (x2) for all x1, x2 \u2208 I (ii)\ndecreasing on I, if x1, x2 in I \u21d2 f (x1) < f (x2) for all x1, x2 \u2208 I (iii)\nconstant on I, if f(x) = c for all x \u2208 I, where c is a constant x\nf (x) = x2\n \u20132\n4\n\u221223\n9\n4\n \u20131\n1\n\u221221\n1\n4\n  0\n0\nValues left to origin\nas we move from left to right, the\nheight of the graph decreases\nx\nf (x) = x2\n0\n0\n1\n2\n1\n4\n 1\n1\n3\n2\n9\n4\n 2\n4\nValues right to origin\nas we move from left to right, the\nheight of the graph increases\nRationalised 2023-24\nAPPLICATION OF DERIVATIVES\n153\n(iv)\ndecreasing on I if x1 < x2 in I \u21d2 f (x1) \u2265 f (x2) for all x1, x2 \u2208 I"
    },
    {
        "Chapter": "1",
        "sentence_range": "2648-2651",
        "Text": "(ii)\ndecreasing on I, if x1, x2 in I \u21d2 f (x1) < f (x2) for all x1, x2 \u2208 I (iii)\nconstant on I, if f(x) = c for all x \u2208 I, where c is a constant x\nf (x) = x2\n \u20132\n4\n\u221223\n9\n4\n \u20131\n1\n\u221221\n1\n4\n  0\n0\nValues left to origin\nas we move from left to right, the\nheight of the graph decreases\nx\nf (x) = x2\n0\n0\n1\n2\n1\n4\n 1\n1\n3\n2\n9\n4\n 2\n4\nValues right to origin\nas we move from left to right, the\nheight of the graph increases\nRationalised 2023-24\nAPPLICATION OF DERIVATIVES\n153\n(iv)\ndecreasing on I if x1 < x2 in I \u21d2 f (x1) \u2265 f (x2) for all x1, x2 \u2208 I (v)\nstrictly decreasing on I if x1 < x2 in I \u21d2 f (x1) > f (x2) for all x1, x2 \u2208 I"
    },
    {
        "Chapter": "1",
        "sentence_range": "2649-2652",
        "Text": "(iii)\nconstant on I, if f(x) = c for all x \u2208 I, where c is a constant x\nf (x) = x2\n \u20132\n4\n\u221223\n9\n4\n \u20131\n1\n\u221221\n1\n4\n  0\n0\nValues left to origin\nas we move from left to right, the\nheight of the graph decreases\nx\nf (x) = x2\n0\n0\n1\n2\n1\n4\n 1\n1\n3\n2\n9\n4\n 2\n4\nValues right to origin\nas we move from left to right, the\nheight of the graph increases\nRationalised 2023-24\nAPPLICATION OF DERIVATIVES\n153\n(iv)\ndecreasing on I if x1 < x2 in I \u21d2 f (x1) \u2265 f (x2) for all x1, x2 \u2208 I (v)\nstrictly decreasing on I if x1 < x2 in I \u21d2 f (x1) > f (x2) for all x1, x2 \u2208 I For graphical representation of such functions see Fig 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "2650-2653",
        "Text": "x\nf (x) = x2\n \u20132\n4\n\u221223\n9\n4\n \u20131\n1\n\u221221\n1\n4\n  0\n0\nValues left to origin\nas we move from left to right, the\nheight of the graph decreases\nx\nf (x) = x2\n0\n0\n1\n2\n1\n4\n 1\n1\n3\n2\n9\n4\n 2\n4\nValues right to origin\nas we move from left to right, the\nheight of the graph increases\nRationalised 2023-24\nAPPLICATION OF DERIVATIVES\n153\n(iv)\ndecreasing on I if x1 < x2 in I \u21d2 f (x1) \u2265 f (x2) for all x1, x2 \u2208 I (v)\nstrictly decreasing on I if x1 < x2 in I \u21d2 f (x1) > f (x2) for all x1, x2 \u2208 I For graphical representation of such functions see Fig 6 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "2651-2654",
        "Text": "(v)\nstrictly decreasing on I if x1 < x2 in I \u21d2 f (x1) > f (x2) for all x1, x2 \u2208 I For graphical representation of such functions see Fig 6 2 Fig 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "2652-2655",
        "Text": "For graphical representation of such functions see Fig 6 2 Fig 6 2\nWe shall now define when a function is increasing or decreasing at a point"
    },
    {
        "Chapter": "1",
        "sentence_range": "2653-2656",
        "Text": "2 Fig 6 2\nWe shall now define when a function is increasing or decreasing at a point Definition 2 Let x0 be a point in the domain of definition of a real valued function  f"
    },
    {
        "Chapter": "1",
        "sentence_range": "2654-2657",
        "Text": "Fig 6 2\nWe shall now define when a function is increasing or decreasing at a point Definition 2 Let x0 be a point in the domain of definition of a real valued function  f Then  f  is said to be increasing, decreasing at x0 if  there exists an open interval I\ncontaining x0 such that f is increasing, decreasing, respectively, in I"
    },
    {
        "Chapter": "1",
        "sentence_range": "2655-2658",
        "Text": "2\nWe shall now define when a function is increasing or decreasing at a point Definition 2 Let x0 be a point in the domain of definition of a real valued function  f Then  f  is said to be increasing, decreasing at x0 if  there exists an open interval I\ncontaining x0 such that f is increasing, decreasing, respectively, in I Let us clarify this definition for the case of increasing function"
    },
    {
        "Chapter": "1",
        "sentence_range": "2656-2659",
        "Text": "Definition 2 Let x0 be a point in the domain of definition of a real valued function  f Then  f  is said to be increasing, decreasing at x0 if  there exists an open interval I\ncontaining x0 such that f is increasing, decreasing, respectively, in I Let us clarify this definition for the case of increasing function Example 7 Show that the function given by f(x) = 7x \u2013 3 is increasing on R"
    },
    {
        "Chapter": "1",
        "sentence_range": "2657-2660",
        "Text": "Then  f  is said to be increasing, decreasing at x0 if  there exists an open interval I\ncontaining x0 such that f is increasing, decreasing, respectively, in I Let us clarify this definition for the case of increasing function Example 7 Show that the function given by f(x) = 7x \u2013 3 is increasing on R Solution Let x1 and x2 be any two numbers in R"
    },
    {
        "Chapter": "1",
        "sentence_range": "2658-2661",
        "Text": "Let us clarify this definition for the case of increasing function Example 7 Show that the function given by f(x) = 7x \u2013 3 is increasing on R Solution Let x1 and x2 be any two numbers in R Then\nx1 < x2 \u21d2 7x1 < 7x2  \u21d2 7x1 \u2013 3 < 7x2 \u2013 3 \u21d2 f (x1) < f (x2)\nThus, by Definition 1, it follows that f is strictly increasing on R"
    },
    {
        "Chapter": "1",
        "sentence_range": "2659-2662",
        "Text": "Example 7 Show that the function given by f(x) = 7x \u2013 3 is increasing on R Solution Let x1 and x2 be any two numbers in R Then\nx1 < x2 \u21d2 7x1 < 7x2  \u21d2 7x1 \u2013 3 < 7x2 \u2013 3 \u21d2 f (x1) < f (x2)\nThus, by Definition 1, it follows that f is strictly increasing on R We shall now give the first derivative test for increasing and decreasing functions"
    },
    {
        "Chapter": "1",
        "sentence_range": "2660-2663",
        "Text": "Solution Let x1 and x2 be any two numbers in R Then\nx1 < x2 \u21d2 7x1 < 7x2  \u21d2 7x1 \u2013 3 < 7x2 \u2013 3 \u21d2 f (x1) < f (x2)\nThus, by Definition 1, it follows that f is strictly increasing on R We shall now give the first derivative test for increasing and decreasing functions The proof of this test requires the Mean Value Theorem studied in Chapter 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "2661-2664",
        "Text": "Then\nx1 < x2 \u21d2 7x1 < 7x2  \u21d2 7x1 \u2013 3 < 7x2 \u2013 3 \u21d2 f (x1) < f (x2)\nThus, by Definition 1, it follows that f is strictly increasing on R We shall now give the first derivative test for increasing and decreasing functions The proof of this test requires the Mean Value Theorem studied in Chapter 5 Theorem 1 Let  f  be continuous on [a, b] and differentiable on the open interval\n(a,b)"
    },
    {
        "Chapter": "1",
        "sentence_range": "2662-2665",
        "Text": "We shall now give the first derivative test for increasing and decreasing functions The proof of this test requires the Mean Value Theorem studied in Chapter 5 Theorem 1 Let  f  be continuous on [a, b] and differentiable on the open interval\n(a,b) Then\n(a)\nf  is increasing in [a,b] if f \u2032(x) > 0 for each x \u2208 (a, b)\n(b)\nf  is decreasing in [a,b] if f \u2032(x) < 0 for each x \u2208 (a, b)\n(c)\nf  is a constant function in [a,b] if f \u2032(x) = 0 for each x \u2208 (a, b)\nStrictly Increasing function\n(i)\nNeither Increasing nor\nDecreasing function\n(iii)\nStrictly Decreasing function\n(ii)\nRationalised 2023-24\n MATHEMATICS\n154\nProof (a) Let x1, x2 \u2208 [a, b] be such that x1 < x2"
    },
    {
        "Chapter": "1",
        "sentence_range": "2663-2666",
        "Text": "The proof of this test requires the Mean Value Theorem studied in Chapter 5 Theorem 1 Let  f  be continuous on [a, b] and differentiable on the open interval\n(a,b) Then\n(a)\nf  is increasing in [a,b] if f \u2032(x) > 0 for each x \u2208 (a, b)\n(b)\nf  is decreasing in [a,b] if f \u2032(x) < 0 for each x \u2208 (a, b)\n(c)\nf  is a constant function in [a,b] if f \u2032(x) = 0 for each x \u2208 (a, b)\nStrictly Increasing function\n(i)\nNeither Increasing nor\nDecreasing function\n(iii)\nStrictly Decreasing function\n(ii)\nRationalised 2023-24\n MATHEMATICS\n154\nProof (a) Let x1, x2 \u2208 [a, b] be such that x1 < x2 Then, by Mean Value Theorem (Theorem 8 in Chapter 5), there exists a point c\nbetween x1 and x2 such that\nf(x2) \u2013 f(x1) = f \u2032(c) (x2 \u2013 x1)\ni"
    },
    {
        "Chapter": "1",
        "sentence_range": "2664-2667",
        "Text": "Theorem 1 Let  f  be continuous on [a, b] and differentiable on the open interval\n(a,b) Then\n(a)\nf  is increasing in [a,b] if f \u2032(x) > 0 for each x \u2208 (a, b)\n(b)\nf  is decreasing in [a,b] if f \u2032(x) < 0 for each x \u2208 (a, b)\n(c)\nf  is a constant function in [a,b] if f \u2032(x) = 0 for each x \u2208 (a, b)\nStrictly Increasing function\n(i)\nNeither Increasing nor\nDecreasing function\n(iii)\nStrictly Decreasing function\n(ii)\nRationalised 2023-24\n MATHEMATICS\n154\nProof (a) Let x1, x2 \u2208 [a, b] be such that x1 < x2 Then, by Mean Value Theorem (Theorem 8 in Chapter 5), there exists a point c\nbetween x1 and x2 such that\nf(x2) \u2013 f(x1) = f \u2032(c) (x2 \u2013 x1)\ni e"
    },
    {
        "Chapter": "1",
        "sentence_range": "2665-2668",
        "Text": "Then\n(a)\nf  is increasing in [a,b] if f \u2032(x) > 0 for each x \u2208 (a, b)\n(b)\nf  is decreasing in [a,b] if f \u2032(x) < 0 for each x \u2208 (a, b)\n(c)\nf  is a constant function in [a,b] if f \u2032(x) = 0 for each x \u2208 (a, b)\nStrictly Increasing function\n(i)\nNeither Increasing nor\nDecreasing function\n(iii)\nStrictly Decreasing function\n(ii)\nRationalised 2023-24\n MATHEMATICS\n154\nProof (a) Let x1, x2 \u2208 [a, b] be such that x1 < x2 Then, by Mean Value Theorem (Theorem 8 in Chapter 5), there exists a point c\nbetween x1 and x2 such that\nf(x2) \u2013 f(x1) = f \u2032(c) (x2 \u2013 x1)\ni e f(x2) \u2013 f(x1) > 0\n(as f \u2032(c) > 0 (given))\ni"
    },
    {
        "Chapter": "1",
        "sentence_range": "2666-2669",
        "Text": "Then, by Mean Value Theorem (Theorem 8 in Chapter 5), there exists a point c\nbetween x1 and x2 such that\nf(x2) \u2013 f(x1) = f \u2032(c) (x2 \u2013 x1)\ni e f(x2) \u2013 f(x1) > 0\n(as f \u2032(c) > 0 (given))\ni e"
    },
    {
        "Chapter": "1",
        "sentence_range": "2667-2670",
        "Text": "e f(x2) \u2013 f(x1) > 0\n(as f \u2032(c) > 0 (given))\ni e f(x2) > f (x1)\nThus, we have\n1\n2\n1\n2\n1\n2\n(\n)\n(\n), for all\n,\n[ , ]\nx\nx\nf x\nf x\nx x\na b\n<\n \n \n \n \n \nHence, f  is an increasing function in [a,b]"
    },
    {
        "Chapter": "1",
        "sentence_range": "2668-2671",
        "Text": "f(x2) \u2013 f(x1) > 0\n(as f \u2032(c) > 0 (given))\ni e f(x2) > f (x1)\nThus, we have\n1\n2\n1\n2\n1\n2\n(\n)\n(\n), for all\n,\n[ , ]\nx\nx\nf x\nf x\nx x\na b\n<\n \n \n \n \n \nHence, f  is an increasing function in [a,b] The proofs of part (b) and (c) are similar"
    },
    {
        "Chapter": "1",
        "sentence_range": "2669-2672",
        "Text": "e f(x2) > f (x1)\nThus, we have\n1\n2\n1\n2\n1\n2\n(\n)\n(\n), for all\n,\n[ , ]\nx\nx\nf x\nf x\nx x\na b\n<\n \n \n \n \n \nHence, f  is an increasing function in [a,b] The proofs of part (b) and (c) are similar It is left as an exercise to the reader"
    },
    {
        "Chapter": "1",
        "sentence_range": "2670-2673",
        "Text": "f(x2) > f (x1)\nThus, we have\n1\n2\n1\n2\n1\n2\n(\n)\n(\n), for all\n,\n[ , ]\nx\nx\nf x\nf x\nx x\na b\n<\n \n \n \n \n \nHence, f  is an increasing function in [a,b] The proofs of part (b) and (c) are similar It is left as an exercise to the reader Remarks\nThere is a more generalised theorem, which states that if f\u00a2(x) > 0 for x in an interval\nexcluding the end points and f is continuous in the interval, then f is increasing"
    },
    {
        "Chapter": "1",
        "sentence_range": "2671-2674",
        "Text": "The proofs of part (b) and (c) are similar It is left as an exercise to the reader Remarks\nThere is a more generalised theorem, which states that if f\u00a2(x) > 0 for x in an interval\nexcluding the end points and f is continuous in the interval, then f is increasing Similarly,\nif f\u00a2(x) < 0 for x in an interval excluding the end points and f is continuous in the\ninterval, then f is decreasing"
    },
    {
        "Chapter": "1",
        "sentence_range": "2672-2675",
        "Text": "It is left as an exercise to the reader Remarks\nThere is a more generalised theorem, which states that if f\u00a2(x) > 0 for x in an interval\nexcluding the end points and f is continuous in the interval, then f is increasing Similarly,\nif f\u00a2(x) < 0 for x in an interval excluding the end points and f is continuous in the\ninterval, then f is decreasing Example 8 Show that the function  f  given by\nf (x) = x3 \u2013 3x2 + 4x, x \u2208 R\nis increasing on R"
    },
    {
        "Chapter": "1",
        "sentence_range": "2673-2676",
        "Text": "Remarks\nThere is a more generalised theorem, which states that if f\u00a2(x) > 0 for x in an interval\nexcluding the end points and f is continuous in the interval, then f is increasing Similarly,\nif f\u00a2(x) < 0 for x in an interval excluding the end points and f is continuous in the\ninterval, then f is decreasing Example 8 Show that the function  f  given by\nf (x) = x3 \u2013 3x2 + 4x, x \u2208 R\nis increasing on R Solution Note that\nf \u2032(x) = 3x2 \u2013 6x + 4\n= 3(x2 \u2013 2x + 1) + 1\n= 3(x \u2013 1)2 + 1 > 0, in every interval of R\nTherefore, the function  f is increasing on R"
    },
    {
        "Chapter": "1",
        "sentence_range": "2674-2677",
        "Text": "Similarly,\nif f\u00a2(x) < 0 for x in an interval excluding the end points and f is continuous in the\ninterval, then f is decreasing Example 8 Show that the function  f  given by\nf (x) = x3 \u2013 3x2 + 4x, x \u2208 R\nis increasing on R Solution Note that\nf \u2032(x) = 3x2 \u2013 6x + 4\n= 3(x2 \u2013 2x + 1) + 1\n= 3(x \u2013 1)2 + 1 > 0, in every interval of R\nTherefore, the function  f is increasing on R Example 9 Prove that the function given by f (x) = cos x is\n(a)\ndecreasing in (0, \u03c0)\n(b)\nincreasing in (\u03c0, 2\u03c0), and\n(c)\nneither increasing nor decreasing in (0, 2\u03c0)"
    },
    {
        "Chapter": "1",
        "sentence_range": "2675-2678",
        "Text": "Example 8 Show that the function  f  given by\nf (x) = x3 \u2013 3x2 + 4x, x \u2208 R\nis increasing on R Solution Note that\nf \u2032(x) = 3x2 \u2013 6x + 4\n= 3(x2 \u2013 2x + 1) + 1\n= 3(x \u2013 1)2 + 1 > 0, in every interval of R\nTherefore, the function  f is increasing on R Example 9 Prove that the function given by f (x) = cos x is\n(a)\ndecreasing in (0, \u03c0)\n(b)\nincreasing in (\u03c0, 2\u03c0), and\n(c)\nneither increasing nor decreasing in (0, 2\u03c0) Rationalised 2023-24\nAPPLICATION OF DERIVATIVES\n155\nFig 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "2676-2679",
        "Text": "Solution Note that\nf \u2032(x) = 3x2 \u2013 6x + 4\n= 3(x2 \u2013 2x + 1) + 1\n= 3(x \u2013 1)2 + 1 > 0, in every interval of R\nTherefore, the function  f is increasing on R Example 9 Prove that the function given by f (x) = cos x is\n(a)\ndecreasing in (0, \u03c0)\n(b)\nincreasing in (\u03c0, 2\u03c0), and\n(c)\nneither increasing nor decreasing in (0, 2\u03c0) Rationalised 2023-24\nAPPLICATION OF DERIVATIVES\n155\nFig 6 4\nSolution Note that f \u2032(x) = \u2013 sin x\n(a)\nSince for each x \u2208 (0, \u03c0), sin x > 0, we have f \u2032(x) < 0 and so f is decreasing in\n(0, \u03c0)"
    },
    {
        "Chapter": "1",
        "sentence_range": "2677-2680",
        "Text": "Example 9 Prove that the function given by f (x) = cos x is\n(a)\ndecreasing in (0, \u03c0)\n(b)\nincreasing in (\u03c0, 2\u03c0), and\n(c)\nneither increasing nor decreasing in (0, 2\u03c0) Rationalised 2023-24\nAPPLICATION OF DERIVATIVES\n155\nFig 6 4\nSolution Note that f \u2032(x) = \u2013 sin x\n(a)\nSince for each x \u2208 (0, \u03c0), sin x > 0, we have f \u2032(x) < 0 and so f is decreasing in\n(0, \u03c0) (b)\nSince for each x \u2208 (\u03c0, 2\u03c0), sin x < 0, we have f \u2032(x) > 0 and so f is increasing in\n(\u03c0, 2\u03c0)"
    },
    {
        "Chapter": "1",
        "sentence_range": "2678-2681",
        "Text": "Rationalised 2023-24\nAPPLICATION OF DERIVATIVES\n155\nFig 6 4\nSolution Note that f \u2032(x) = \u2013 sin x\n(a)\nSince for each x \u2208 (0, \u03c0), sin x > 0, we have f \u2032(x) < 0 and so f is decreasing in\n(0, \u03c0) (b)\nSince for each x \u2208 (\u03c0, 2\u03c0), sin x < 0, we have f \u2032(x) > 0 and so f is increasing in\n(\u03c0, 2\u03c0) (c)\nClearly by (a) and (b) above, f is neither increasing nor decreasing in (0, 2\u03c0)"
    },
    {
        "Chapter": "1",
        "sentence_range": "2679-2682",
        "Text": "4\nSolution Note that f \u2032(x) = \u2013 sin x\n(a)\nSince for each x \u2208 (0, \u03c0), sin x > 0, we have f \u2032(x) < 0 and so f is decreasing in\n(0, \u03c0) (b)\nSince for each x \u2208 (\u03c0, 2\u03c0), sin x < 0, we have f \u2032(x) > 0 and so f is increasing in\n(\u03c0, 2\u03c0) (c)\nClearly by (a) and (b) above, f is neither increasing nor decreasing in (0, 2\u03c0) Example 10 Find the intervals in which the function f given by f(x) = x2 \u2013 4x + 6 is\n(a) increasing\n(b)  decreasing\nSolution We have\nf (x)\n= x2 \u2013 4x + 6\nor\nf \u2032(x) = 2x \u2013 4\nTherefore, f \u2032(x) = 0 gives x = 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "2680-2683",
        "Text": "(b)\nSince for each x \u2208 (\u03c0, 2\u03c0), sin x < 0, we have f \u2032(x) > 0 and so f is increasing in\n(\u03c0, 2\u03c0) (c)\nClearly by (a) and (b) above, f is neither increasing nor decreasing in (0, 2\u03c0) Example 10 Find the intervals in which the function f given by f(x) = x2 \u2013 4x + 6 is\n(a) increasing\n(b)  decreasing\nSolution We have\nf (x)\n= x2 \u2013 4x + 6\nor\nf \u2032(x) = 2x \u2013 4\nTherefore, f \u2032(x) = 0 gives x = 2 Now the point x = 2 divides the real line into two\ndisjoint intervals namely, (\u2013 \u221e, 2) and (2, \u221e) (Fig 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "2681-2684",
        "Text": "(c)\nClearly by (a) and (b) above, f is neither increasing nor decreasing in (0, 2\u03c0) Example 10 Find the intervals in which the function f given by f(x) = x2 \u2013 4x + 6 is\n(a) increasing\n(b)  decreasing\nSolution We have\nf (x)\n= x2 \u2013 4x + 6\nor\nf \u2032(x) = 2x \u2013 4\nTherefore, f \u2032(x) = 0 gives x = 2 Now the point x = 2 divides the real line into two\ndisjoint intervals namely, (\u2013 \u221e, 2) and (2, \u221e) (Fig 6 3)"
    },
    {
        "Chapter": "1",
        "sentence_range": "2682-2685",
        "Text": "Example 10 Find the intervals in which the function f given by f(x) = x2 \u2013 4x + 6 is\n(a) increasing\n(b)  decreasing\nSolution We have\nf (x)\n= x2 \u2013 4x + 6\nor\nf \u2032(x) = 2x \u2013 4\nTherefore, f \u2032(x) = 0 gives x = 2 Now the point x = 2 divides the real line into two\ndisjoint intervals namely, (\u2013 \u221e, 2) and (2, \u221e) (Fig 6 3) In the interval (\u2013 \u221e, 2), f \u2032(x) = 2x\n\u2013 4 < 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "2683-2686",
        "Text": "Now the point x = 2 divides the real line into two\ndisjoint intervals namely, (\u2013 \u221e, 2) and (2, \u221e) (Fig 6 3) In the interval (\u2013 \u221e, 2), f \u2032(x) = 2x\n\u2013 4 < 0 Therefore, f is decreasing in this interval"
    },
    {
        "Chapter": "1",
        "sentence_range": "2684-2687",
        "Text": "3) In the interval (\u2013 \u221e, 2), f \u2032(x) = 2x\n\u2013 4 < 0 Therefore, f is decreasing in this interval Also, in the interval (2,\n)\n\u221e , \n( )\n0\nf\nx >\n\u2032\nand so the function  f  is increasing in this interval"
    },
    {
        "Chapter": "1",
        "sentence_range": "2685-2688",
        "Text": "In the interval (\u2013 \u221e, 2), f \u2032(x) = 2x\n\u2013 4 < 0 Therefore, f is decreasing in this interval Also, in the interval (2,\n)\n\u221e , \n( )\n0\nf\nx >\n\u2032\nand so the function  f  is increasing in this interval Example 11 Find the intervals in which the function f given by f (x) = 4x3 \u2013 6x2 \u2013 72x\n+ 30 is (a) increasing (b) decreasing"
    },
    {
        "Chapter": "1",
        "sentence_range": "2686-2689",
        "Text": "Therefore, f is decreasing in this interval Also, in the interval (2,\n)\n\u221e , \n( )\n0\nf\nx >\n\u2032\nand so the function  f  is increasing in this interval Example 11 Find the intervals in which the function f given by f (x) = 4x3 \u2013 6x2 \u2013 72x\n+ 30 is (a) increasing (b) decreasing Solution We have\nf (x)\n= 4x3 \u2013 6x2 \u2013 72x + 30\nor\nf \u2032(x) = 12x2 \u2013 12x \u2013 72\n= 12(x2 \u2013 x \u2013 6)\n= 12(x \u2013 3) (x + 2)\nTherefore, f \u2032(x) = 0 gives x = \u2013 2, 3"
    },
    {
        "Chapter": "1",
        "sentence_range": "2687-2690",
        "Text": "Also, in the interval (2,\n)\n\u221e , \n( )\n0\nf\nx >\n\u2032\nand so the function  f  is increasing in this interval Example 11 Find the intervals in which the function f given by f (x) = 4x3 \u2013 6x2 \u2013 72x\n+ 30 is (a) increasing (b) decreasing Solution We have\nf (x)\n= 4x3 \u2013 6x2 \u2013 72x + 30\nor\nf \u2032(x) = 12x2 \u2013 12x \u2013 72\n= 12(x2 \u2013 x \u2013 6)\n= 12(x \u2013 3) (x + 2)\nTherefore, f \u2032(x) = 0 gives x = \u2013 2, 3 The\npoints x = \u2013 2 and  x = 3 divides the real line into\nthree disjoint intervals, namely, (\u2013 \u221e, \u2013 2), (\u2013 2, 3)\nand (3, \u221e)"
    },
    {
        "Chapter": "1",
        "sentence_range": "2688-2691",
        "Text": "Example 11 Find the intervals in which the function f given by f (x) = 4x3 \u2013 6x2 \u2013 72x\n+ 30 is (a) increasing (b) decreasing Solution We have\nf (x)\n= 4x3 \u2013 6x2 \u2013 72x + 30\nor\nf \u2032(x) = 12x2 \u2013 12x \u2013 72\n= 12(x2 \u2013 x \u2013 6)\n= 12(x \u2013 3) (x + 2)\nTherefore, f \u2032(x) = 0 gives x = \u2013 2, 3 The\npoints x = \u2013 2 and  x = 3 divides the real line into\nthree disjoint intervals, namely, (\u2013 \u221e, \u2013 2), (\u2013 2, 3)\nand (3, \u221e) Fig 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "2689-2692",
        "Text": "Solution We have\nf (x)\n= 4x3 \u2013 6x2 \u2013 72x + 30\nor\nf \u2032(x) = 12x2 \u2013 12x \u2013 72\n= 12(x2 \u2013 x \u2013 6)\n= 12(x \u2013 3) (x + 2)\nTherefore, f \u2032(x) = 0 gives x = \u2013 2, 3 The\npoints x = \u2013 2 and  x = 3 divides the real line into\nthree disjoint intervals, namely, (\u2013 \u221e, \u2013 2), (\u2013 2, 3)\nand (3, \u221e) Fig 6 3\nRationalised 2023-24\n MATHEMATICS\n156\nIn the intervals (\u2013 \u221e, \u2013 2) and (3, \u221e), f \u2032(x) is positive while in the interval (\u2013 2, 3),\nf \u2032(x) is negative"
    },
    {
        "Chapter": "1",
        "sentence_range": "2690-2693",
        "Text": "The\npoints x = \u2013 2 and  x = 3 divides the real line into\nthree disjoint intervals, namely, (\u2013 \u221e, \u2013 2), (\u2013 2, 3)\nand (3, \u221e) Fig 6 3\nRationalised 2023-24\n MATHEMATICS\n156\nIn the intervals (\u2013 \u221e, \u2013 2) and (3, \u221e), f \u2032(x) is positive while in the interval (\u2013 2, 3),\nf \u2032(x) is negative Consequently, the function f is increasing in the intervals\n(\u2013 \u221e, \u2013 2) and (3, \u221e) while the function is decreasing in the interval (\u2013 2, 3)"
    },
    {
        "Chapter": "1",
        "sentence_range": "2691-2694",
        "Text": "Fig 6 3\nRationalised 2023-24\n MATHEMATICS\n156\nIn the intervals (\u2013 \u221e, \u2013 2) and (3, \u221e), f \u2032(x) is positive while in the interval (\u2013 2, 3),\nf \u2032(x) is negative Consequently, the function f is increasing in the intervals\n(\u2013 \u221e, \u2013 2) and (3, \u221e) while the function is decreasing in the interval (\u2013 2, 3) However,\nf is neither increasing nor decreasing in R"
    },
    {
        "Chapter": "1",
        "sentence_range": "2692-2695",
        "Text": "3\nRationalised 2023-24\n MATHEMATICS\n156\nIn the intervals (\u2013 \u221e, \u2013 2) and (3, \u221e), f \u2032(x) is positive while in the interval (\u2013 2, 3),\nf \u2032(x) is negative Consequently, the function f is increasing in the intervals\n(\u2013 \u221e, \u2013 2) and (3, \u221e) while the function is decreasing in the interval (\u2013 2, 3) However,\nf is neither increasing nor decreasing in R Interval\nSign of f \u2032(x)\nNature of function f\n(\u2013 \u221e, \u2013 2)\n(\u2013) (\u2013) > 0\nf is increasing\n(\u2013 2, 3)\n(\u2013) (+) < 0\nf is decreasing\n(3, \u221e)\n(+) (+) > 0\nf is increasing\nExample 12 Find intervals in which the function given by f (x) = sin 3x, x \u2208\uf8ee\n\uf8f0\uf8ef\n0 2\uf8fb\uf8fa\uf8f9\n, \u03c0  is\n(a) increasing (b) decreasing"
    },
    {
        "Chapter": "1",
        "sentence_range": "2693-2696",
        "Text": "Consequently, the function f is increasing in the intervals\n(\u2013 \u221e, \u2013 2) and (3, \u221e) while the function is decreasing in the interval (\u2013 2, 3) However,\nf is neither increasing nor decreasing in R Interval\nSign of f \u2032(x)\nNature of function f\n(\u2013 \u221e, \u2013 2)\n(\u2013) (\u2013) > 0\nf is increasing\n(\u2013 2, 3)\n(\u2013) (+) < 0\nf is decreasing\n(3, \u221e)\n(+) (+) > 0\nf is increasing\nExample 12 Find intervals in which the function given by f (x) = sin 3x, x \u2208\uf8ee\n\uf8f0\uf8ef\n0 2\uf8fb\uf8fa\uf8f9\n, \u03c0  is\n(a) increasing (b) decreasing Solution We have\nf (x) = sin 3x\nor\nf \u2032(x) = 3cos 3x\nTherefore, f\u2032(x) = 0 gives cos 3x = 0 which in turn gives \n3\n3\n2,\n2\nx\n\u03c0\n\u03c0\n=\n (as x \u2208\uf8ee\n\uf8f0\uf8ef\n0 2\uf8fb\uf8fa\uf8f9\n, \u03c0\nimplies \n3\n3\n0, 2\nx\n\u03c0\n\uf8ee\n\uf8f9\n\u2208 \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n)"
    },
    {
        "Chapter": "1",
        "sentence_range": "2694-2697",
        "Text": "However,\nf is neither increasing nor decreasing in R Interval\nSign of f \u2032(x)\nNature of function f\n(\u2013 \u221e, \u2013 2)\n(\u2013) (\u2013) > 0\nf is increasing\n(\u2013 2, 3)\n(\u2013) (+) < 0\nf is decreasing\n(3, \u221e)\n(+) (+) > 0\nf is increasing\nExample 12 Find intervals in which the function given by f (x) = sin 3x, x \u2208\uf8ee\n\uf8f0\uf8ef\n0 2\uf8fb\uf8fa\uf8f9\n, \u03c0  is\n(a) increasing (b) decreasing Solution We have\nf (x) = sin 3x\nor\nf \u2032(x) = 3cos 3x\nTherefore, f\u2032(x) = 0 gives cos 3x = 0 which in turn gives \n3\n3\n2,\n2\nx\n\u03c0\n\u03c0\n=\n (as x \u2208\uf8ee\n\uf8f0\uf8ef\n0 2\uf8fb\uf8fa\uf8f9\n, \u03c0\nimplies \n3\n3\n0, 2\nx\n\u03c0\n\uf8ee\n\uf8f9\n\u2208 \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n) So \n6\nx\n=\u03c0\n and 2\n\u03c0"
    },
    {
        "Chapter": "1",
        "sentence_range": "2695-2698",
        "Text": "Interval\nSign of f \u2032(x)\nNature of function f\n(\u2013 \u221e, \u2013 2)\n(\u2013) (\u2013) > 0\nf is increasing\n(\u2013 2, 3)\n(\u2013) (+) < 0\nf is decreasing\n(3, \u221e)\n(+) (+) > 0\nf is increasing\nExample 12 Find intervals in which the function given by f (x) = sin 3x, x \u2208\uf8ee\n\uf8f0\uf8ef\n0 2\uf8fb\uf8fa\uf8f9\n, \u03c0  is\n(a) increasing (b) decreasing Solution We have\nf (x) = sin 3x\nor\nf \u2032(x) = 3cos 3x\nTherefore, f\u2032(x) = 0 gives cos 3x = 0 which in turn gives \n3\n3\n2,\n2\nx\n\u03c0\n\u03c0\n=\n (as x \u2208\uf8ee\n\uf8f0\uf8ef\n0 2\uf8fb\uf8fa\uf8f9\n, \u03c0\nimplies \n3\n3\n0, 2\nx\n\u03c0\n\uf8ee\n\uf8f9\n\u2208 \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n) So \n6\nx\n=\u03c0\n and 2\n\u03c0 The point \n6\nx\n=\u03c0\n divides the interval 0 2\n\uf8f0\uf8ef\uf8ee, \u03c0\n\uf8f9\n\uf8fb\uf8fa\ninto two disjoint intervals 0, 6\n\u03c0\n\uf8ee\n\uf8f7\uf8f6\n\uf8ef\uf8f0\n\uf8f8  and \u03c0 \u03c0\n6 2\n,\n\uf8ed\uf8ec\uf8eb\n\uf8f9\n\uf8fb\uf8fa"
    },
    {
        "Chapter": "1",
        "sentence_range": "2696-2699",
        "Text": "Solution We have\nf (x) = sin 3x\nor\nf \u2032(x) = 3cos 3x\nTherefore, f\u2032(x) = 0 gives cos 3x = 0 which in turn gives \n3\n3\n2,\n2\nx\n\u03c0\n\u03c0\n=\n (as x \u2208\uf8ee\n\uf8f0\uf8ef\n0 2\uf8fb\uf8fa\uf8f9\n, \u03c0\nimplies \n3\n3\n0, 2\nx\n\u03c0\n\uf8ee\n\uf8f9\n\u2208 \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n) So \n6\nx\n=\u03c0\n and 2\n\u03c0 The point \n6\nx\n=\u03c0\n divides the interval 0 2\n\uf8f0\uf8ef\uf8ee, \u03c0\n\uf8f9\n\uf8fb\uf8fa\ninto two disjoint intervals 0, 6\n\u03c0\n\uf8ee\n\uf8f7\uf8f6\n\uf8ef\uf8f0\n\uf8f8  and \u03c0 \u03c0\n6 2\n,\n\uf8ed\uf8ec\uf8eb\n\uf8f9\n\uf8fb\uf8fa Now, \n( )\n0\nf\n\u2032x >\n for all \n0, 6\nx\n\u03c0\n\uf8ee\n\uf8f6\n\u2208\n\uf8f7\n\uf8ef\uf8f0\n\uf8f8  as 0\n0\n3\n6\n2\nx\nx\n\u03c0\n\u03c0\n\u2264\n<\n\u21d2\n\u2264\n<\n and \n( )\n0\nf\n\u2032x <\n for\nall \n,\n6 2\nx\n\uf8eb\u03c0 \u03c0\n\uf8f6\n\u2208\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8  as \n3\n3\n6\n2\n2\n2\nx\nx\n\u03c0\n\u03c0\n\u03c0\n\u03c0\n<\n<\n\u21d2\n<\n<"
    },
    {
        "Chapter": "1",
        "sentence_range": "2697-2700",
        "Text": "So \n6\nx\n=\u03c0\n and 2\n\u03c0 The point \n6\nx\n=\u03c0\n divides the interval 0 2\n\uf8f0\uf8ef\uf8ee, \u03c0\n\uf8f9\n\uf8fb\uf8fa\ninto two disjoint intervals 0, 6\n\u03c0\n\uf8ee\n\uf8f7\uf8f6\n\uf8ef\uf8f0\n\uf8f8  and \u03c0 \u03c0\n6 2\n,\n\uf8ed\uf8ec\uf8eb\n\uf8f9\n\uf8fb\uf8fa Now, \n( )\n0\nf\n\u2032x >\n for all \n0, 6\nx\n\u03c0\n\uf8ee\n\uf8f6\n\u2208\n\uf8f7\n\uf8ef\uf8f0\n\uf8f8  as 0\n0\n3\n6\n2\nx\nx\n\u03c0\n\u03c0\n\u2264\n<\n\u21d2\n\u2264\n<\n and \n( )\n0\nf\n\u2032x <\n for\nall \n,\n6 2\nx\n\uf8eb\u03c0 \u03c0\n\uf8f6\n\u2208\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8  as \n3\n3\n6\n2\n2\n2\nx\nx\n\u03c0\n\u03c0\n\u03c0\n\u03c0\n<\n<\n\u21d2\n<\n< Therefore, f is increasing in 0, 6\n\u03c0\n\uf8ee\n\uf8f7\uf8f6\n\uf8ef\uf8f0\n\uf8f8  and decreasing in \n,\n6 2\n\uf8eb\u03c0 \u03c0\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8"
    },
    {
        "Chapter": "1",
        "sentence_range": "2698-2701",
        "Text": "The point \n6\nx\n=\u03c0\n divides the interval 0 2\n\uf8f0\uf8ef\uf8ee, \u03c0\n\uf8f9\n\uf8fb\uf8fa\ninto two disjoint intervals 0, 6\n\u03c0\n\uf8ee\n\uf8f7\uf8f6\n\uf8ef\uf8f0\n\uf8f8  and \u03c0 \u03c0\n6 2\n,\n\uf8ed\uf8ec\uf8eb\n\uf8f9\n\uf8fb\uf8fa Now, \n( )\n0\nf\n\u2032x >\n for all \n0, 6\nx\n\u03c0\n\uf8ee\n\uf8f6\n\u2208\n\uf8f7\n\uf8ef\uf8f0\n\uf8f8  as 0\n0\n3\n6\n2\nx\nx\n\u03c0\n\u03c0\n\u2264\n<\n\u21d2\n\u2264\n<\n and \n( )\n0\nf\n\u2032x <\n for\nall \n,\n6 2\nx\n\uf8eb\u03c0 \u03c0\n\uf8f6\n\u2208\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8  as \n3\n3\n6\n2\n2\n2\nx\nx\n\u03c0\n\u03c0\n\u03c0\n\u03c0\n<\n<\n\u21d2\n<\n< Therefore, f is increasing in 0, 6\n\u03c0\n\uf8ee\n\uf8f7\uf8f6\n\uf8ef\uf8f0\n\uf8f8  and decreasing in \n,\n6 2\n\uf8eb\u03c0 \u03c0\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8 Fig 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "2699-2702",
        "Text": "Now, \n( )\n0\nf\n\u2032x >\n for all \n0, 6\nx\n\u03c0\n\uf8ee\n\uf8f6\n\u2208\n\uf8f7\n\uf8ef\uf8f0\n\uf8f8  as 0\n0\n3\n6\n2\nx\nx\n\u03c0\n\u03c0\n\u2264\n<\n\u21d2\n\u2264\n<\n and \n( )\n0\nf\n\u2032x <\n for\nall \n,\n6 2\nx\n\uf8eb\u03c0 \u03c0\n\uf8f6\n\u2208\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8  as \n3\n3\n6\n2\n2\n2\nx\nx\n\u03c0\n\u03c0\n\u03c0\n\u03c0\n<\n<\n\u21d2\n<\n< Therefore, f is increasing in 0, 6\n\u03c0\n\uf8ee\n\uf8f7\uf8f6\n\uf8ef\uf8f0\n\uf8f8  and decreasing in \n,\n6 2\n\uf8eb\u03c0 \u03c0\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8 Fig 6 5\nRationalised 2023-24\nAPPLICATION OF DERIVATIVES\n157\nAlso, the given function is continuous at x = 0 and \n6\nx\n=\u03c0"
    },
    {
        "Chapter": "1",
        "sentence_range": "2700-2703",
        "Text": "Therefore, f is increasing in 0, 6\n\u03c0\n\uf8ee\n\uf8f7\uf8f6\n\uf8ef\uf8f0\n\uf8f8  and decreasing in \n,\n6 2\n\uf8eb\u03c0 \u03c0\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8 Fig 6 5\nRationalised 2023-24\nAPPLICATION OF DERIVATIVES\n157\nAlso, the given function is continuous at x = 0 and \n6\nx\n=\u03c0 Therefore, by Theorem 1,\nf is increasing on 0 6\n\uf8f0\uf8ef\uf8ee, \u03c0\n\uf8f9\n\uf8fb\uf8fa  and decreasing on \n\u03c0 \u03c0\n6 2\n,\n\uf8f0\uf8ef\uf8ee\n\uf8f9\n\uf8fb\uf8fa"
    },
    {
        "Chapter": "1",
        "sentence_range": "2701-2704",
        "Text": "Fig 6 5\nRationalised 2023-24\nAPPLICATION OF DERIVATIVES\n157\nAlso, the given function is continuous at x = 0 and \n6\nx\n=\u03c0 Therefore, by Theorem 1,\nf is increasing on 0 6\n\uf8f0\uf8ef\uf8ee, \u03c0\n\uf8f9\n\uf8fb\uf8fa  and decreasing on \n\u03c0 \u03c0\n6 2\n,\n\uf8f0\uf8ef\uf8ee\n\uf8f9\n\uf8fb\uf8fa Example 13 Find the intervals in which the function f given by\n f (x) = sin x + cos x, 0 \u2264 x \u2264 2\u03c0\nis increasing or decreasing"
    },
    {
        "Chapter": "1",
        "sentence_range": "2702-2705",
        "Text": "5\nRationalised 2023-24\nAPPLICATION OF DERIVATIVES\n157\nAlso, the given function is continuous at x = 0 and \n6\nx\n=\u03c0 Therefore, by Theorem 1,\nf is increasing on 0 6\n\uf8f0\uf8ef\uf8ee, \u03c0\n\uf8f9\n\uf8fb\uf8fa  and decreasing on \n\u03c0 \u03c0\n6 2\n,\n\uf8f0\uf8ef\uf8ee\n\uf8f9\n\uf8fb\uf8fa Example 13 Find the intervals in which the function f given by\n f (x) = sin x + cos x, 0 \u2264 x \u2264 2\u03c0\nis increasing or decreasing Solution We have\nf(x) = sin x + cos x,\nor\nf \u2032(x) = cos x \u2013 sin x\nNow \n( )\n0\nf\n\u2032x =\n gives sin x = cos x which gives  that \n4\nx\n=\u03c0\n, 5\n4\n\u03c0  as 0\n2\n\u2264x\n\u2264 \u03c0\nThe points \n4\nx\n=\u03c0\n and \n5\n4\nx\n=\u03c0\n divide the interval [0, 2\u03c0] into three disjoint intervals,\nnamely, 0, 4\n\u03c0\n\uf8ee\n\uf8f7\uf8f6\n\uf8ef\uf8f0\n\uf8f8 , \u03c0\n\u03c0\n4\n5\n,4\n\uf8ed\uf8ec\uf8eb\n\uf8f6\n\uf8f8\uf8f7  and 5 ,2\n\uf8eb\u03c04\n\u03c0\uf8f9\n\uf8ec\n\uf8fa\n\uf8ed\n\uf8fb"
    },
    {
        "Chapter": "1",
        "sentence_range": "2703-2706",
        "Text": "Therefore, by Theorem 1,\nf is increasing on 0 6\n\uf8f0\uf8ef\uf8ee, \u03c0\n\uf8f9\n\uf8fb\uf8fa  and decreasing on \n\u03c0 \u03c0\n6 2\n,\n\uf8f0\uf8ef\uf8ee\n\uf8f9\n\uf8fb\uf8fa Example 13 Find the intervals in which the function f given by\n f (x) = sin x + cos x, 0 \u2264 x \u2264 2\u03c0\nis increasing or decreasing Solution We have\nf(x) = sin x + cos x,\nor\nf \u2032(x) = cos x \u2013 sin x\nNow \n( )\n0\nf\n\u2032x =\n gives sin x = cos x which gives  that \n4\nx\n=\u03c0\n, 5\n4\n\u03c0  as 0\n2\n\u2264x\n\u2264 \u03c0\nThe points \n4\nx\n=\u03c0\n and \n5\n4\nx\n=\u03c0\n divide the interval [0, 2\u03c0] into three disjoint intervals,\nnamely, 0, 4\n\u03c0\n\uf8ee\n\uf8f7\uf8f6\n\uf8ef\uf8f0\n\uf8f8 , \u03c0\n\u03c0\n4\n5\n,4\n\uf8ed\uf8ec\uf8eb\n\uf8f6\n\uf8f8\uf8f7  and 5 ,2\n\uf8eb\u03c04\n\u03c0\uf8f9\n\uf8ec\n\uf8fa\n\uf8ed\n\uf8fb Note that\n5\n( )\n0 if\n0,\n,2\n4\n4\nf\nx\nx\n\u03c0\n\u03c0\n\uf8ee\n\uf8f6\n\uf8eb\n\uf8f9\n\u2032\n>\n\u2208\n\u222a\n\u03c0\n\uf8f7\n\uf8ec\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8f8\n\uf8ed\n\uf8fb\nor\nf  is increasing in the intervals 0, 4\n\u03c0\n\uf8ee\n\uf8f7\uf8f6\n\uf8ef\uf8f0\n\uf8f8  and \n\uf8eb\u03c045 ,2\n\u03c0\uf8f9\n\uf8ec\n\uf8fa\n\uf8ed\n\uf8fb\nAlso\n\u2032\n<\n\u2208\uf8eb\n\uf8ed\uf8ec\n\uf8f8\uf8f7\uf8f6\nf\nx\nx\n( )\n,\n0\n4\n5\n4\nif\n\u03c0\n\u03c0\nor\n f  is decreasing in \n\u03c0\n\u03c0\n4\n5\n,4\n\uf8ed\uf8ec\uf8eb\n\uf8f6\n\uf8f8\uf8f7\nFig 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "2704-2707",
        "Text": "Example 13 Find the intervals in which the function f given by\n f (x) = sin x + cos x, 0 \u2264 x \u2264 2\u03c0\nis increasing or decreasing Solution We have\nf(x) = sin x + cos x,\nor\nf \u2032(x) = cos x \u2013 sin x\nNow \n( )\n0\nf\n\u2032x =\n gives sin x = cos x which gives  that \n4\nx\n=\u03c0\n, 5\n4\n\u03c0  as 0\n2\n\u2264x\n\u2264 \u03c0\nThe points \n4\nx\n=\u03c0\n and \n5\n4\nx\n=\u03c0\n divide the interval [0, 2\u03c0] into three disjoint intervals,\nnamely, 0, 4\n\u03c0\n\uf8ee\n\uf8f7\uf8f6\n\uf8ef\uf8f0\n\uf8f8 , \u03c0\n\u03c0\n4\n5\n,4\n\uf8ed\uf8ec\uf8eb\n\uf8f6\n\uf8f8\uf8f7  and 5 ,2\n\uf8eb\u03c04\n\u03c0\uf8f9\n\uf8ec\n\uf8fa\n\uf8ed\n\uf8fb Note that\n5\n( )\n0 if\n0,\n,2\n4\n4\nf\nx\nx\n\u03c0\n\u03c0\n\uf8ee\n\uf8f6\n\uf8eb\n\uf8f9\n\u2032\n>\n\u2208\n\u222a\n\u03c0\n\uf8f7\n\uf8ec\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8f8\n\uf8ed\n\uf8fb\nor\nf  is increasing in the intervals 0, 4\n\u03c0\n\uf8ee\n\uf8f7\uf8f6\n\uf8ef\uf8f0\n\uf8f8  and \n\uf8eb\u03c045 ,2\n\u03c0\uf8f9\n\uf8ec\n\uf8fa\n\uf8ed\n\uf8fb\nAlso\n\u2032\n<\n\u2208\uf8eb\n\uf8ed\uf8ec\n\uf8f8\uf8f7\uf8f6\nf\nx\nx\n( )\n,\n0\n4\n5\n4\nif\n\u03c0\n\u03c0\nor\n f  is decreasing in \n\u03c0\n\u03c0\n4\n5\n,4\n\uf8ed\uf8ec\uf8eb\n\uf8f6\n\uf8f8\uf8f7\nFig 6 6\nRationalised 2023-24\n MATHEMATICS\n158\nInterval\nSign of \nf( )\n\u2032x\nNature of function\n0, 4\n\u03c0\n\uf8ee\n\uf8f7\uf8f6\n\uf8ef\uf8f0\n\uf8f8\n> 0\nf  is increasing\n\u03c0\n\u03c0\n4\n5\n,4\n\uf8ed\uf8ec\uf8eb\n\uf8f8\uf8f7\uf8f6\n< 0\nf  is decreasing\n\uf8eb\u03c045 ,2\n\u03c0\uf8f9\n\uf8ec\n\uf8fa\n\uf8ed\n\uf8fb\n> 0\nf  is increasing\nEXERCISE 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "2705-2708",
        "Text": "Solution We have\nf(x) = sin x + cos x,\nor\nf \u2032(x) = cos x \u2013 sin x\nNow \n( )\n0\nf\n\u2032x =\n gives sin x = cos x which gives  that \n4\nx\n=\u03c0\n, 5\n4\n\u03c0  as 0\n2\n\u2264x\n\u2264 \u03c0\nThe points \n4\nx\n=\u03c0\n and \n5\n4\nx\n=\u03c0\n divide the interval [0, 2\u03c0] into three disjoint intervals,\nnamely, 0, 4\n\u03c0\n\uf8ee\n\uf8f7\uf8f6\n\uf8ef\uf8f0\n\uf8f8 , \u03c0\n\u03c0\n4\n5\n,4\n\uf8ed\uf8ec\uf8eb\n\uf8f6\n\uf8f8\uf8f7  and 5 ,2\n\uf8eb\u03c04\n\u03c0\uf8f9\n\uf8ec\n\uf8fa\n\uf8ed\n\uf8fb Note that\n5\n( )\n0 if\n0,\n,2\n4\n4\nf\nx\nx\n\u03c0\n\u03c0\n\uf8ee\n\uf8f6\n\uf8eb\n\uf8f9\n\u2032\n>\n\u2208\n\u222a\n\u03c0\n\uf8f7\n\uf8ec\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8f8\n\uf8ed\n\uf8fb\nor\nf  is increasing in the intervals 0, 4\n\u03c0\n\uf8ee\n\uf8f7\uf8f6\n\uf8ef\uf8f0\n\uf8f8  and \n\uf8eb\u03c045 ,2\n\u03c0\uf8f9\n\uf8ec\n\uf8fa\n\uf8ed\n\uf8fb\nAlso\n\u2032\n<\n\u2208\uf8eb\n\uf8ed\uf8ec\n\uf8f8\uf8f7\uf8f6\nf\nx\nx\n( )\n,\n0\n4\n5\n4\nif\n\u03c0\n\u03c0\nor\n f  is decreasing in \n\u03c0\n\u03c0\n4\n5\n,4\n\uf8ed\uf8ec\uf8eb\n\uf8f6\n\uf8f8\uf8f7\nFig 6 6\nRationalised 2023-24\n MATHEMATICS\n158\nInterval\nSign of \nf( )\n\u2032x\nNature of function\n0, 4\n\u03c0\n\uf8ee\n\uf8f7\uf8f6\n\uf8ef\uf8f0\n\uf8f8\n> 0\nf  is increasing\n\u03c0\n\u03c0\n4\n5\n,4\n\uf8ed\uf8ec\uf8eb\n\uf8f8\uf8f7\uf8f6\n< 0\nf  is decreasing\n\uf8eb\u03c045 ,2\n\u03c0\uf8f9\n\uf8ec\n\uf8fa\n\uf8ed\n\uf8fb\n> 0\nf  is increasing\nEXERCISE 6 2\n1"
    },
    {
        "Chapter": "1",
        "sentence_range": "2706-2709",
        "Text": "Note that\n5\n( )\n0 if\n0,\n,2\n4\n4\nf\nx\nx\n\u03c0\n\u03c0\n\uf8ee\n\uf8f6\n\uf8eb\n\uf8f9\n\u2032\n>\n\u2208\n\u222a\n\u03c0\n\uf8f7\n\uf8ec\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8f8\n\uf8ed\n\uf8fb\nor\nf  is increasing in the intervals 0, 4\n\u03c0\n\uf8ee\n\uf8f7\uf8f6\n\uf8ef\uf8f0\n\uf8f8  and \n\uf8eb\u03c045 ,2\n\u03c0\uf8f9\n\uf8ec\n\uf8fa\n\uf8ed\n\uf8fb\nAlso\n\u2032\n<\n\u2208\uf8eb\n\uf8ed\uf8ec\n\uf8f8\uf8f7\uf8f6\nf\nx\nx\n( )\n,\n0\n4\n5\n4\nif\n\u03c0\n\u03c0\nor\n f  is decreasing in \n\u03c0\n\u03c0\n4\n5\n,4\n\uf8ed\uf8ec\uf8eb\n\uf8f6\n\uf8f8\uf8f7\nFig 6 6\nRationalised 2023-24\n MATHEMATICS\n158\nInterval\nSign of \nf( )\n\u2032x\nNature of function\n0, 4\n\u03c0\n\uf8ee\n\uf8f7\uf8f6\n\uf8ef\uf8f0\n\uf8f8\n> 0\nf  is increasing\n\u03c0\n\u03c0\n4\n5\n,4\n\uf8ed\uf8ec\uf8eb\n\uf8f8\uf8f7\uf8f6\n< 0\nf  is decreasing\n\uf8eb\u03c045 ,2\n\u03c0\uf8f9\n\uf8ec\n\uf8fa\n\uf8ed\n\uf8fb\n> 0\nf  is increasing\nEXERCISE 6 2\n1 Show that the function given by f (x) = 3x + 17 is increasing on R"
    },
    {
        "Chapter": "1",
        "sentence_range": "2707-2710",
        "Text": "6\nRationalised 2023-24\n MATHEMATICS\n158\nInterval\nSign of \nf( )\n\u2032x\nNature of function\n0, 4\n\u03c0\n\uf8ee\n\uf8f7\uf8f6\n\uf8ef\uf8f0\n\uf8f8\n> 0\nf  is increasing\n\u03c0\n\u03c0\n4\n5\n,4\n\uf8ed\uf8ec\uf8eb\n\uf8f8\uf8f7\uf8f6\n< 0\nf  is decreasing\n\uf8eb\u03c045 ,2\n\u03c0\uf8f9\n\uf8ec\n\uf8fa\n\uf8ed\n\uf8fb\n> 0\nf  is increasing\nEXERCISE 6 2\n1 Show that the function given by f (x) = 3x + 17 is increasing on R 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "2708-2711",
        "Text": "2\n1 Show that the function given by f (x) = 3x + 17 is increasing on R 2 Show that the function given by f (x) = e2x is increasing on R"
    },
    {
        "Chapter": "1",
        "sentence_range": "2709-2712",
        "Text": "Show that the function given by f (x) = 3x + 17 is increasing on R 2 Show that the function given by f (x) = e2x is increasing on R 3"
    },
    {
        "Chapter": "1",
        "sentence_range": "2710-2713",
        "Text": "2 Show that the function given by f (x) = e2x is increasing on R 3 Show that the function given by f (x) = sin x is\n(a)\nincreasing in 0, 2\n\u03c0\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n(b)\ndecreasing in \n\uf8eb\u03c02,\n\u03c0\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n(c) neither increasing nor decreasing in (0, \u03c0)\n4"
    },
    {
        "Chapter": "1",
        "sentence_range": "2711-2714",
        "Text": "Show that the function given by f (x) = e2x is increasing on R 3 Show that the function given by f (x) = sin x is\n(a)\nincreasing in 0, 2\n\u03c0\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n(b)\ndecreasing in \n\uf8eb\u03c02,\n\u03c0\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n(c) neither increasing nor decreasing in (0, \u03c0)\n4 Find the intervals in which the function f given by  f (x) = 2x2 \u2013 3x is\n(a)\nincreasing\n(b)\ndecreasing\n5"
    },
    {
        "Chapter": "1",
        "sentence_range": "2712-2715",
        "Text": "3 Show that the function given by f (x) = sin x is\n(a)\nincreasing in 0, 2\n\u03c0\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n(b)\ndecreasing in \n\uf8eb\u03c02,\n\u03c0\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n(c) neither increasing nor decreasing in (0, \u03c0)\n4 Find the intervals in which the function f given by  f (x) = 2x2 \u2013 3x is\n(a)\nincreasing\n(b)\ndecreasing\n5 Find the intervals in which the function f given by f(x) = 2x3 \u2013 3x2 \u2013 36x + 7 is\n(a)\nincreasing\n(b)\ndecreasing\n6"
    },
    {
        "Chapter": "1",
        "sentence_range": "2713-2716",
        "Text": "Show that the function given by f (x) = sin x is\n(a)\nincreasing in 0, 2\n\u03c0\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n(b)\ndecreasing in \n\uf8eb\u03c02,\n\u03c0\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n(c) neither increasing nor decreasing in (0, \u03c0)\n4 Find the intervals in which the function f given by  f (x) = 2x2 \u2013 3x is\n(a)\nincreasing\n(b)\ndecreasing\n5 Find the intervals in which the function f given by f(x) = 2x3 \u2013 3x2 \u2013 36x + 7 is\n(a)\nincreasing\n(b)\ndecreasing\n6 Find the intervals in which the following functions are strictly increasing or\ndecreasing:\n(a)\nx2 + 2x \u2013 5\n(b)\n10 \u2013 6x \u2013 2x2\n(c)\n\u20132x3 \u2013 9x2 \u2013 12x + 1\n(d)\n6 \u2013 9x \u2013 x2\n(e)\n(x + 1)3 (x \u2013 3)3\n7"
    },
    {
        "Chapter": "1",
        "sentence_range": "2714-2717",
        "Text": "Find the intervals in which the function f given by  f (x) = 2x2 \u2013 3x is\n(a)\nincreasing\n(b)\ndecreasing\n5 Find the intervals in which the function f given by f(x) = 2x3 \u2013 3x2 \u2013 36x + 7 is\n(a)\nincreasing\n(b)\ndecreasing\n6 Find the intervals in which the following functions are strictly increasing or\ndecreasing:\n(a)\nx2 + 2x \u2013 5\n(b)\n10 \u2013 6x \u2013 2x2\n(c)\n\u20132x3 \u2013 9x2 \u2013 12x + 1\n(d)\n6 \u2013 9x \u2013 x2\n(e)\n(x + 1)3 (x \u2013 3)3\n7 Show that \n2\nlog(1\n)\n2\nx\ny\nx\nx\n=\n+\n\u2212\n+\n, x > \u2013 1, is  an increasing function of x\nthroughout its domain"
    },
    {
        "Chapter": "1",
        "sentence_range": "2715-2718",
        "Text": "Find the intervals in which the function f given by f(x) = 2x3 \u2013 3x2 \u2013 36x + 7 is\n(a)\nincreasing\n(b)\ndecreasing\n6 Find the intervals in which the following functions are strictly increasing or\ndecreasing:\n(a)\nx2 + 2x \u2013 5\n(b)\n10 \u2013 6x \u2013 2x2\n(c)\n\u20132x3 \u2013 9x2 \u2013 12x + 1\n(d)\n6 \u2013 9x \u2013 x2\n(e)\n(x + 1)3 (x \u2013 3)3\n7 Show that \n2\nlog(1\n)\n2\nx\ny\nx\nx\n=\n+\n\u2212\n+\n, x > \u2013 1, is  an increasing function of x\nthroughout its domain 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "2716-2719",
        "Text": "Find the intervals in which the following functions are strictly increasing or\ndecreasing:\n(a)\nx2 + 2x \u2013 5\n(b)\n10 \u2013 6x \u2013 2x2\n(c)\n\u20132x3 \u2013 9x2 \u2013 12x + 1\n(d)\n6 \u2013 9x \u2013 x2\n(e)\n(x + 1)3 (x \u2013 3)3\n7 Show that \n2\nlog(1\n)\n2\nx\ny\nx\nx\n=\n+\n\u2212\n+\n, x > \u2013 1, is  an increasing function of x\nthroughout its domain 8 Find the values of x for which y = [x(x \u2013 2)]2 is an increasing function"
    },
    {
        "Chapter": "1",
        "sentence_range": "2717-2720",
        "Text": "Show that \n2\nlog(1\n)\n2\nx\ny\nx\nx\n=\n+\n\u2212\n+\n, x > \u2013 1, is  an increasing function of x\nthroughout its domain 8 Find the values of x for which y = [x(x \u2013 2)]2 is an increasing function 9"
    },
    {
        "Chapter": "1",
        "sentence_range": "2718-2721",
        "Text": "8 Find the values of x for which y = [x(x \u2013 2)]2 is an increasing function 9 Prove that \n(24sin\ncos )\ny\n\u03b8\n=\n\u2212 \u03b8\n+\n\u03b8\n is an increasing function of \u03b8  in 0 2\n\uf8f0\uf8ef\uf8ee, \u03c0\n\uf8f9\n\uf8fb\uf8fa"
    },
    {
        "Chapter": "1",
        "sentence_range": "2719-2722",
        "Text": "Find the values of x for which y = [x(x \u2013 2)]2 is an increasing function 9 Prove that \n(24sin\ncos )\ny\n\u03b8\n=\n\u2212 \u03b8\n+\n\u03b8\n is an increasing function of \u03b8  in 0 2\n\uf8f0\uf8ef\uf8ee, \u03c0\n\uf8f9\n\uf8fb\uf8fa Rationalised 2023-24\nAPPLICATION OF DERIVATIVES\n159\n10"
    },
    {
        "Chapter": "1",
        "sentence_range": "2720-2723",
        "Text": "9 Prove that \n(24sin\ncos )\ny\n\u03b8\n=\n\u2212 \u03b8\n+\n\u03b8\n is an increasing function of \u03b8  in 0 2\n\uf8f0\uf8ef\uf8ee, \u03c0\n\uf8f9\n\uf8fb\uf8fa Rationalised 2023-24\nAPPLICATION OF DERIVATIVES\n159\n10 Prove that the logarithmic function is increasing on (0, \u221e)"
    },
    {
        "Chapter": "1",
        "sentence_range": "2721-2724",
        "Text": "Prove that \n(24sin\ncos )\ny\n\u03b8\n=\n\u2212 \u03b8\n+\n\u03b8\n is an increasing function of \u03b8  in 0 2\n\uf8f0\uf8ef\uf8ee, \u03c0\n\uf8f9\n\uf8fb\uf8fa Rationalised 2023-24\nAPPLICATION OF DERIVATIVES\n159\n10 Prove that the logarithmic function is increasing on (0, \u221e) 11"
    },
    {
        "Chapter": "1",
        "sentence_range": "2722-2725",
        "Text": "Rationalised 2023-24\nAPPLICATION OF DERIVATIVES\n159\n10 Prove that the logarithmic function is increasing on (0, \u221e) 11 Prove that the function f given by f(x) = x2 \u2013 x + 1 is neither strictly increasing\nnor decreasing on (\u2013 1, 1)"
    },
    {
        "Chapter": "1",
        "sentence_range": "2723-2726",
        "Text": "Prove that the logarithmic function is increasing on (0, \u221e) 11 Prove that the function f given by f(x) = x2 \u2013 x + 1 is neither strictly increasing\nnor decreasing on (\u2013 1, 1) 12"
    },
    {
        "Chapter": "1",
        "sentence_range": "2724-2727",
        "Text": "11 Prove that the function f given by f(x) = x2 \u2013 x + 1 is neither strictly increasing\nnor decreasing on (\u2013 1, 1) 12 Which of the following functions are decreasing on 0, 2\n\u03c0"
    },
    {
        "Chapter": "1",
        "sentence_range": "2725-2728",
        "Text": "Prove that the function f given by f(x) = x2 \u2013 x + 1 is neither strictly increasing\nnor decreasing on (\u2013 1, 1) 12 Which of the following functions are decreasing on 0, 2\n\u03c0 (A) cos x\n(B) cos 2x\n(C) cos 3x\n(D) tan x\n13"
    },
    {
        "Chapter": "1",
        "sentence_range": "2726-2729",
        "Text": "12 Which of the following functions are decreasing on 0, 2\n\u03c0 (A) cos x\n(B) cos 2x\n(C) cos 3x\n(D) tan x\n13 On which of the following intervals is the function f given by f (x) = x100 + sin x \u20131\ndecreasing"
    },
    {
        "Chapter": "1",
        "sentence_range": "2727-2730",
        "Text": "Which of the following functions are decreasing on 0, 2\n\u03c0 (A) cos x\n(B) cos 2x\n(C) cos 3x\n(D) tan x\n13 On which of the following intervals is the function f given by f (x) = x100 + sin x \u20131\ndecreasing (A) (0,1)\n(B)\n \u03c02,\n\u03c0 \n \n \n \n \n(C)\n0, 2\n\u03c0\n \n \n \n \n \n \n(D) None of these\n14"
    },
    {
        "Chapter": "1",
        "sentence_range": "2728-2731",
        "Text": "(A) cos x\n(B) cos 2x\n(C) cos 3x\n(D) tan x\n13 On which of the following intervals is the function f given by f (x) = x100 + sin x \u20131\ndecreasing (A) (0,1)\n(B)\n \u03c02,\n\u03c0 \n \n \n \n \n(C)\n0, 2\n\u03c0\n \n \n \n \n \n \n(D) None of these\n14 For what values of a the function f given by f (x) = x2 + ax + 1 is increasing on\n[1, 2]"
    },
    {
        "Chapter": "1",
        "sentence_range": "2729-2732",
        "Text": "On which of the following intervals is the function f given by f (x) = x100 + sin x \u20131\ndecreasing (A) (0,1)\n(B)\n \u03c02,\n\u03c0 \n \n \n \n \n(C)\n0, 2\n\u03c0\n \n \n \n \n \n \n(D) None of these\n14 For what values of a the function f given by f (x) = x2 + ax + 1 is increasing on\n[1, 2] 15"
    },
    {
        "Chapter": "1",
        "sentence_range": "2730-2733",
        "Text": "(A) (0,1)\n(B)\n \u03c02,\n\u03c0 \n \n \n \n \n(C)\n0, 2\n\u03c0\n \n \n \n \n \n \n(D) None of these\n14 For what values of a the function f given by f (x) = x2 + ax + 1 is increasing on\n[1, 2] 15 Let I be any interval disjoint from [\u20131, 1]"
    },
    {
        "Chapter": "1",
        "sentence_range": "2731-2734",
        "Text": "For what values of a the function f given by f (x) = x2 + ax + 1 is increasing on\n[1, 2] 15 Let I be any interval disjoint from [\u20131, 1] Prove that the function f  given by\n1\n( )\nf x\nx\nx\n=\n+\n is increasing on I"
    },
    {
        "Chapter": "1",
        "sentence_range": "2732-2735",
        "Text": "15 Let I be any interval disjoint from [\u20131, 1] Prove that the function f  given by\n1\n( )\nf x\nx\nx\n=\n+\n is increasing on I 16"
    },
    {
        "Chapter": "1",
        "sentence_range": "2733-2736",
        "Text": "Let I be any interval disjoint from [\u20131, 1] Prove that the function f  given by\n1\n( )\nf x\nx\nx\n=\n+\n is increasing on I 16 Prove that the function f given by f (x) = log sin x  is increasing on 0 2\n\uf8ed\uf8ec\uf8eb, \u03c0\n\uf8f6\n\uf8f8\uf8f7  and\ndecreasing on \u03c0 \u03c0\n\uf8ed\uf8ec\uf8eb2 ,\n\uf8f6\n\uf8f8\uf8f7"
    },
    {
        "Chapter": "1",
        "sentence_range": "2734-2737",
        "Text": "Prove that the function f  given by\n1\n( )\nf x\nx\nx\n=\n+\n is increasing on I 16 Prove that the function f given by f (x) = log sin x  is increasing on 0 2\n\uf8ed\uf8ec\uf8eb, \u03c0\n\uf8f6\n\uf8f8\uf8f7  and\ndecreasing on \u03c0 \u03c0\n\uf8ed\uf8ec\uf8eb2 ,\n\uf8f6\n\uf8f8\uf8f7 17"
    },
    {
        "Chapter": "1",
        "sentence_range": "2735-2738",
        "Text": "16 Prove that the function f given by f (x) = log sin x  is increasing on 0 2\n\uf8ed\uf8ec\uf8eb, \u03c0\n\uf8f6\n\uf8f8\uf8f7  and\ndecreasing on \u03c0 \u03c0\n\uf8ed\uf8ec\uf8eb2 ,\n\uf8f6\n\uf8f8\uf8f7 17 Prove that the function f given by f (x) = log |cos x| is decreasing on 0, 2\n\u03c0\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n and\nincreasing on 3 , 2\n\uf8eb\u03c02\n\u03c0\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8"
    },
    {
        "Chapter": "1",
        "sentence_range": "2736-2739",
        "Text": "Prove that the function f given by f (x) = log sin x  is increasing on 0 2\n\uf8ed\uf8ec\uf8eb, \u03c0\n\uf8f6\n\uf8f8\uf8f7  and\ndecreasing on \u03c0 \u03c0\n\uf8ed\uf8ec\uf8eb2 ,\n\uf8f6\n\uf8f8\uf8f7 17 Prove that the function f given by f (x) = log |cos x| is decreasing on 0, 2\n\u03c0\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n and\nincreasing on 3 , 2\n\uf8eb\u03c02\n\u03c0\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8 18"
    },
    {
        "Chapter": "1",
        "sentence_range": "2737-2740",
        "Text": "17 Prove that the function f given by f (x) = log |cos x| is decreasing on 0, 2\n\u03c0\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n and\nincreasing on 3 , 2\n\uf8eb\u03c02\n\u03c0\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8 18 Prove that the function given by f (x) = x3 \u2013 3x2 + 3x \u2013 100 is increasing in R"
    },
    {
        "Chapter": "1",
        "sentence_range": "2738-2741",
        "Text": "Prove that the function f given by f (x) = log |cos x| is decreasing on 0, 2\n\u03c0\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n and\nincreasing on 3 , 2\n\uf8eb\u03c02\n\u03c0\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8 18 Prove that the function given by f (x) = x3 \u2013 3x2 + 3x \u2013 100 is increasing in R 19"
    },
    {
        "Chapter": "1",
        "sentence_range": "2739-2742",
        "Text": "18 Prove that the function given by f (x) = x3 \u2013 3x2 + 3x \u2013 100 is increasing in R 19 The interval in which y = x2 e\u2013x is increasing is\n(A) (\u2013 \u221e, \u221e)\n(B) (\u2013 2, 0)\n(C) (2, \u221e)\n(D) (0, 2)\n6"
    },
    {
        "Chapter": "1",
        "sentence_range": "2740-2743",
        "Text": "Prove that the function given by f (x) = x3 \u2013 3x2 + 3x \u2013 100 is increasing in R 19 The interval in which y = x2 e\u2013x is increasing is\n(A) (\u2013 \u221e, \u221e)\n(B) (\u2013 2, 0)\n(C) (2, \u221e)\n(D) (0, 2)\n6 4  Maxima and Minima\nIn this section, we will use the concept of derivatives to calculate the maximum or\nminimum values of  various functions"
    },
    {
        "Chapter": "1",
        "sentence_range": "2741-2744",
        "Text": "19 The interval in which y = x2 e\u2013x is increasing is\n(A) (\u2013 \u221e, \u221e)\n(B) (\u2013 2, 0)\n(C) (2, \u221e)\n(D) (0, 2)\n6 4  Maxima and Minima\nIn this section, we will use the concept of derivatives to calculate the maximum or\nminimum values of  various functions In fact, we will find the \u2018turning points\u2019 of the\ngraph of a function and thus find points at which the graph reaches its highest (or\nRationalised 2023-24\n MATHEMATICS\n160\nlowest) locally"
    },
    {
        "Chapter": "1",
        "sentence_range": "2742-2745",
        "Text": "The interval in which y = x2 e\u2013x is increasing is\n(A) (\u2013 \u221e, \u221e)\n(B) (\u2013 2, 0)\n(C) (2, \u221e)\n(D) (0, 2)\n6 4  Maxima and Minima\nIn this section, we will use the concept of derivatives to calculate the maximum or\nminimum values of  various functions In fact, we will find the \u2018turning points\u2019 of the\ngraph of a function and thus find points at which the graph reaches its highest (or\nRationalised 2023-24\n MATHEMATICS\n160\nlowest) locally The knowledge of such points is very useful in sketching the graph of\na given function"
    },
    {
        "Chapter": "1",
        "sentence_range": "2743-2746",
        "Text": "4  Maxima and Minima\nIn this section, we will use the concept of derivatives to calculate the maximum or\nminimum values of  various functions In fact, we will find the \u2018turning points\u2019 of the\ngraph of a function and thus find points at which the graph reaches its highest (or\nRationalised 2023-24\n MATHEMATICS\n160\nlowest) locally The knowledge of such points is very useful in sketching the graph of\na given function Further, we will also find the absolute maximum and absolute minimum\nof a function that are necessary for the solution of many applied problems"
    },
    {
        "Chapter": "1",
        "sentence_range": "2744-2747",
        "Text": "In fact, we will find the \u2018turning points\u2019 of the\ngraph of a function and thus find points at which the graph reaches its highest (or\nRationalised 2023-24\n MATHEMATICS\n160\nlowest) locally The knowledge of such points is very useful in sketching the graph of\na given function Further, we will also find the absolute maximum and absolute minimum\nof a function that are necessary for the solution of many applied problems (i)Let us consider the following problems that arise in day to day life"
    },
    {
        "Chapter": "1",
        "sentence_range": "2745-2748",
        "Text": "The knowledge of such points is very useful in sketching the graph of\na given function Further, we will also find the absolute maximum and absolute minimum\nof a function that are necessary for the solution of many applied problems (i)Let us consider the following problems that arise in day to day life The profit from a grove of orange trees is given by P(x) = ax + bx2, where a,b\nare constants and x is the number of orange trees per acre"
    },
    {
        "Chapter": "1",
        "sentence_range": "2746-2749",
        "Text": "Further, we will also find the absolute maximum and absolute minimum\nof a function that are necessary for the solution of many applied problems (i)Let us consider the following problems that arise in day to day life The profit from a grove of orange trees is given by P(x) = ax + bx2, where a,b\nare constants and x is the number of orange trees per acre How many trees per\nacre will maximise the profit"
    },
    {
        "Chapter": "1",
        "sentence_range": "2747-2750",
        "Text": "(i)Let us consider the following problems that arise in day to day life The profit from a grove of orange trees is given by P(x) = ax + bx2, where a,b\nare constants and x is the number of orange trees per acre How many trees per\nacre will maximise the profit (ii)\nA ball, thrown into the air from a building 60 metres high, travels along a path\ngiven by \n2\n( )\n60\nx60\nh x\nx\n=\n+\n\u2212\n, where x is the horizontal distance from the building\nand h(x) is the height of the ball"
    },
    {
        "Chapter": "1",
        "sentence_range": "2748-2751",
        "Text": "The profit from a grove of orange trees is given by P(x) = ax + bx2, where a,b\nare constants and x is the number of orange trees per acre How many trees per\nacre will maximise the profit (ii)\nA ball, thrown into the air from a building 60 metres high, travels along a path\ngiven by \n2\n( )\n60\nx60\nh x\nx\n=\n+\n\u2212\n, where x is the horizontal distance from the building\nand h(x) is the height of the ball What is the maximum height the ball will\nreach"
    },
    {
        "Chapter": "1",
        "sentence_range": "2749-2752",
        "Text": "How many trees per\nacre will maximise the profit (ii)\nA ball, thrown into the air from a building 60 metres high, travels along a path\ngiven by \n2\n( )\n60\nx60\nh x\nx\n=\n+\n\u2212\n, where x is the horizontal distance from the building\nand h(x) is the height of the ball What is the maximum height the ball will\nreach (iii)\nAn Apache helicopter of enemy is flying along the path given by the curve\nf (x) = x2 + 7"
    },
    {
        "Chapter": "1",
        "sentence_range": "2750-2753",
        "Text": "(ii)\nA ball, thrown into the air from a building 60 metres high, travels along a path\ngiven by \n2\n( )\n60\nx60\nh x\nx\n=\n+\n\u2212\n, where x is the horizontal distance from the building\nand h(x) is the height of the ball What is the maximum height the ball will\nreach (iii)\nAn Apache helicopter of enemy is flying along the path given by the curve\nf (x) = x2 + 7 A soldier, placed at the point (1, 2), wants to shoot the helicopter\nwhen it is nearest to him"
    },
    {
        "Chapter": "1",
        "sentence_range": "2751-2754",
        "Text": "What is the maximum height the ball will\nreach (iii)\nAn Apache helicopter of enemy is flying along the path given by the curve\nf (x) = x2 + 7 A soldier, placed at the point (1, 2), wants to shoot the helicopter\nwhen it is nearest to him What is the nearest distance"
    },
    {
        "Chapter": "1",
        "sentence_range": "2752-2755",
        "Text": "(iii)\nAn Apache helicopter of enemy is flying along the path given by the curve\nf (x) = x2 + 7 A soldier, placed at the point (1, 2), wants to shoot the helicopter\nwhen it is nearest to him What is the nearest distance In each of the above problem, there is something common, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "2753-2756",
        "Text": "A soldier, placed at the point (1, 2), wants to shoot the helicopter\nwhen it is nearest to him What is the nearest distance In each of the above problem, there is something common, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "2754-2757",
        "Text": "What is the nearest distance In each of the above problem, there is something common, i e , we wish to find out\nthe maximum or minimum values of the given functions"
    },
    {
        "Chapter": "1",
        "sentence_range": "2755-2758",
        "Text": "In each of the above problem, there is something common, i e , we wish to find out\nthe maximum or minimum values of the given functions In order to tackle such problems,\nwe first formally define maximum or minimum values of a function, points of local\nmaxima and minima and test for determining such points"
    },
    {
        "Chapter": "1",
        "sentence_range": "2756-2759",
        "Text": "e , we wish to find out\nthe maximum or minimum values of the given functions In order to tackle such problems,\nwe first formally define maximum or minimum values of a function, points of local\nmaxima and minima and test for determining such points Definition 3 Let f be a function defined on an interval I"
    },
    {
        "Chapter": "1",
        "sentence_range": "2757-2760",
        "Text": ", we wish to find out\nthe maximum or minimum values of the given functions In order to tackle such problems,\nwe first formally define maximum or minimum values of a function, points of local\nmaxima and minima and test for determining such points Definition 3 Let f be a function defined on an interval I Then\n(a)\n f is said to have a maximum value in I, if there exists a point c in I such that\n( )\n( )\nf c>\nf x , for all x \u2208 I"
    },
    {
        "Chapter": "1",
        "sentence_range": "2758-2761",
        "Text": "In order to tackle such problems,\nwe first formally define maximum or minimum values of a function, points of local\nmaxima and minima and test for determining such points Definition 3 Let f be a function defined on an interval I Then\n(a)\n f is said to have a maximum value in I, if there exists a point c in I such that\n( )\n( )\nf c>\nf x , for all x \u2208 I The number f (c) is called the maximum value of f in I and the point c is called a\npoint of maximum value of f in I"
    },
    {
        "Chapter": "1",
        "sentence_range": "2759-2762",
        "Text": "Definition 3 Let f be a function defined on an interval I Then\n(a)\n f is said to have a maximum value in I, if there exists a point c in I such that\n( )\n( )\nf c>\nf x , for all x \u2208 I The number f (c) is called the maximum value of f in I and the point c is called a\npoint of maximum value of f in I (b)\n f  is said to have a minimum value in I, if there exists a point c in I such that\nf (c) < f (x), for all x \u2208 I"
    },
    {
        "Chapter": "1",
        "sentence_range": "2760-2763",
        "Text": "Then\n(a)\n f is said to have a maximum value in I, if there exists a point c in I such that\n( )\n( )\nf c>\nf x , for all x \u2208 I The number f (c) is called the maximum value of f in I and the point c is called a\npoint of maximum value of f in I (b)\n f  is said to have a minimum value in I, if there exists a point c in I such that\nf (c) < f (x), for all x \u2208 I The number f (c), in this case, is called the minimum value of f in I and the point\nc, in this case, is called a point of minimum value of  f  in I"
    },
    {
        "Chapter": "1",
        "sentence_range": "2761-2764",
        "Text": "The number f (c) is called the maximum value of f in I and the point c is called a\npoint of maximum value of f in I (b)\n f  is said to have a minimum value in I, if there exists a point c in I such that\nf (c) < f (x), for all x \u2208 I The number f (c), in this case, is called the minimum value of f in I and the point\nc, in this case, is called a point of minimum value of  f  in I (c)\nf is said to have an extreme value in I if there exists a point c in I such that\nf (c) is either a maximum value or a minimum value of f  in I"
    },
    {
        "Chapter": "1",
        "sentence_range": "2762-2765",
        "Text": "(b)\n f  is said to have a minimum value in I, if there exists a point c in I such that\nf (c) < f (x), for all x \u2208 I The number f (c), in this case, is called the minimum value of f in I and the point\nc, in this case, is called a point of minimum value of  f  in I (c)\nf is said to have an extreme value in I if there exists a point c in I such that\nf (c) is either a maximum value or a minimum value of f  in I The number  f (c), in this case, is called an extreme value of  f  in I and the point c\nis called an extreme point"
    },
    {
        "Chapter": "1",
        "sentence_range": "2763-2766",
        "Text": "The number f (c), in this case, is called the minimum value of f in I and the point\nc, in this case, is called a point of minimum value of  f  in I (c)\nf is said to have an extreme value in I if there exists a point c in I such that\nf (c) is either a maximum value or a minimum value of f  in I The number  f (c), in this case, is called an extreme value of  f  in I and the point c\nis called an extreme point Remark In Fig 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "2764-2767",
        "Text": "(c)\nf is said to have an extreme value in I if there exists a point c in I such that\nf (c) is either a maximum value or a minimum value of f  in I The number  f (c), in this case, is called an extreme value of  f  in I and the point c\nis called an extreme point Remark In Fig 6 7(a), (b) and (c), we have exhibited that graphs of certain particular\nfunctions  help us to find maximum value and minimum value at a point"
    },
    {
        "Chapter": "1",
        "sentence_range": "2765-2768",
        "Text": "The number  f (c), in this case, is called an extreme value of  f  in I and the point c\nis called an extreme point Remark In Fig 6 7(a), (b) and (c), we have exhibited that graphs of certain particular\nfunctions  help us to find maximum value and minimum value at a point Infact, through\ngraphs, we can even find maximum/minimum value of a function at a point at which it\nis not even differentiable (Example 15)"
    },
    {
        "Chapter": "1",
        "sentence_range": "2766-2769",
        "Text": "Remark In Fig 6 7(a), (b) and (c), we have exhibited that graphs of certain particular\nfunctions  help us to find maximum value and minimum value at a point Infact, through\ngraphs, we can even find maximum/minimum value of a function at a point at which it\nis not even differentiable (Example 15) Rationalised 2023-24\nAPPLICATION OF DERIVATIVES\n161\nFig 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "2767-2770",
        "Text": "7(a), (b) and (c), we have exhibited that graphs of certain particular\nfunctions  help us to find maximum value and minimum value at a point Infact, through\ngraphs, we can even find maximum/minimum value of a function at a point at which it\nis not even differentiable (Example 15) Rationalised 2023-24\nAPPLICATION OF DERIVATIVES\n161\nFig 6 7\nExample 14 Find the maximum and the minimum values, if\nany, of the function f  given by\nf (x) = x2, x \u2208 R"
    },
    {
        "Chapter": "1",
        "sentence_range": "2768-2771",
        "Text": "Infact, through\ngraphs, we can even find maximum/minimum value of a function at a point at which it\nis not even differentiable (Example 15) Rationalised 2023-24\nAPPLICATION OF DERIVATIVES\n161\nFig 6 7\nExample 14 Find the maximum and the minimum values, if\nany, of the function f  given by\nf (x) = x2, x \u2208 R Solution From the graph of the given function (Fig 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "2769-2772",
        "Text": "Rationalised 2023-24\nAPPLICATION OF DERIVATIVES\n161\nFig 6 7\nExample 14 Find the maximum and the minimum values, if\nany, of the function f  given by\nf (x) = x2, x \u2208 R Solution From the graph of the given function (Fig 6 8), we\nhave  f (x) = 0 if x = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "2770-2773",
        "Text": "7\nExample 14 Find the maximum and the minimum values, if\nany, of the function f  given by\nf (x) = x2, x \u2208 R Solution From the graph of the given function (Fig 6 8), we\nhave  f (x) = 0 if x = 0 Also\nf (x) \u2265 0, for all x \u2208 R"
    },
    {
        "Chapter": "1",
        "sentence_range": "2771-2774",
        "Text": "Solution From the graph of the given function (Fig 6 8), we\nhave  f (x) = 0 if x = 0 Also\nf (x) \u2265 0, for all x \u2208 R Therefore, the minimum value of  f  is 0 and the point of\nminimum value of f is x = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "2772-2775",
        "Text": "8), we\nhave  f (x) = 0 if x = 0 Also\nf (x) \u2265 0, for all x \u2208 R Therefore, the minimum value of  f  is 0 and the point of\nminimum value of f is x = 0 Further, it may be observed\nfrom the graph of the function that f  has no maximum value\nand hence no point of maximum value of  f  in R"
    },
    {
        "Chapter": "1",
        "sentence_range": "2773-2776",
        "Text": "Also\nf (x) \u2265 0, for all x \u2208 R Therefore, the minimum value of  f  is 0 and the point of\nminimum value of f is x = 0 Further, it may be observed\nfrom the graph of the function that f  has no maximum value\nand hence no point of maximum value of  f  in R ANote  If we restrict the domain of  f to [\u2013 2, 1] only,\nthen f will have maximum value(\u2013 2)2 = 4 at x = \u2013 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "2774-2777",
        "Text": "Therefore, the minimum value of  f  is 0 and the point of\nminimum value of f is x = 0 Further, it may be observed\nfrom the graph of the function that f  has no maximum value\nand hence no point of maximum value of  f  in R ANote  If we restrict the domain of  f to [\u2013 2, 1] only,\nthen f will have maximum value(\u2013 2)2 = 4 at x = \u2013 2 Example 15 Find the maximum and minimum values\nof  f , if any, of the function given by f(x) = |x|, x \u2208 R"
    },
    {
        "Chapter": "1",
        "sentence_range": "2775-2778",
        "Text": "Further, it may be observed\nfrom the graph of the function that f  has no maximum value\nand hence no point of maximum value of  f  in R ANote  If we restrict the domain of  f to [\u2013 2, 1] only,\nthen f will have maximum value(\u2013 2)2 = 4 at x = \u2013 2 Example 15 Find the maximum and minimum values\nof  f , if any, of the function given by f(x) = |x|, x \u2208 R Solution From the graph of the given function\n(Fig 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "2776-2779",
        "Text": "ANote  If we restrict the domain of  f to [\u2013 2, 1] only,\nthen f will have maximum value(\u2013 2)2 = 4 at x = \u2013 2 Example 15 Find the maximum and minimum values\nof  f , if any, of the function given by f(x) = |x|, x \u2208 R Solution From the graph of the given function\n(Fig 6 9) , note that\nf (x) \u2265 0, for all x \u2208 R and  f (x) = 0  if  x = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "2777-2780",
        "Text": "Example 15 Find the maximum and minimum values\nof  f , if any, of the function given by f(x) = |x|, x \u2208 R Solution From the graph of the given function\n(Fig 6 9) , note that\nf (x) \u2265 0, for all x \u2208 R and  f (x) = 0  if  x = 0 Therefore, the function f has a minimum value 0\nand the point of minimum value of f is x = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "2778-2781",
        "Text": "Solution From the graph of the given function\n(Fig 6 9) , note that\nf (x) \u2265 0, for all x \u2208 R and  f (x) = 0  if  x = 0 Therefore, the function f has a minimum value 0\nand the point of minimum value of f is x = 0 Also, the\ngraph clearly shows that f has no maximum value in R\nand hence no point of maximum value in R"
    },
    {
        "Chapter": "1",
        "sentence_range": "2779-2782",
        "Text": "9) , note that\nf (x) \u2265 0, for all x \u2208 R and  f (x) = 0  if  x = 0 Therefore, the function f has a minimum value 0\nand the point of minimum value of f is x = 0 Also, the\ngraph clearly shows that f has no maximum value in R\nand hence no point of maximum value in R ANote\n(i)\nIf we restrict the domain of f to [\u2013 2, 1] only, then f will have maximum value\n|\u2013 2| = 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "2780-2783",
        "Text": "Therefore, the function f has a minimum value 0\nand the point of minimum value of f is x = 0 Also, the\ngraph clearly shows that f has no maximum value in R\nand hence no point of maximum value in R ANote\n(i)\nIf we restrict the domain of f to [\u2013 2, 1] only, then f will have maximum value\n|\u2013 2| = 2 Fig 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "2781-2784",
        "Text": "Also, the\ngraph clearly shows that f has no maximum value in R\nand hence no point of maximum value in R ANote\n(i)\nIf we restrict the domain of f to [\u2013 2, 1] only, then f will have maximum value\n|\u2013 2| = 2 Fig 6 8\nFig 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "2782-2785",
        "Text": "ANote\n(i)\nIf we restrict the domain of f to [\u2013 2, 1] only, then f will have maximum value\n|\u2013 2| = 2 Fig 6 8\nFig 6 9\nRationalised 2023-24\n MATHEMATICS\n162\nFig 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "2783-2786",
        "Text": "Fig 6 8\nFig 6 9\nRationalised 2023-24\n MATHEMATICS\n162\nFig 6 10\n(ii)\nOne may note that the function f in Example 27 is not differentiable at\nx = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "2784-2787",
        "Text": "8\nFig 6 9\nRationalised 2023-24\n MATHEMATICS\n162\nFig 6 10\n(ii)\nOne may note that the function f in Example 27 is not differentiable at\nx = 0 Example 16 Find the maximum and the minimum values, if any, of the function\ngiven by\nf (x) = x, x \u2208 (0, 1)"
    },
    {
        "Chapter": "1",
        "sentence_range": "2785-2788",
        "Text": "9\nRationalised 2023-24\n MATHEMATICS\n162\nFig 6 10\n(ii)\nOne may note that the function f in Example 27 is not differentiable at\nx = 0 Example 16 Find the maximum and the minimum values, if any, of the function\ngiven by\nf (x) = x, x \u2208 (0, 1) Solution The given function is an increasing (strictly) function in the given interval\n(0, 1)"
    },
    {
        "Chapter": "1",
        "sentence_range": "2786-2789",
        "Text": "10\n(ii)\nOne may note that the function f in Example 27 is not differentiable at\nx = 0 Example 16 Find the maximum and the minimum values, if any, of the function\ngiven by\nf (x) = x, x \u2208 (0, 1) Solution The given function is an increasing (strictly) function in the given interval\n(0, 1) From the graph (Fig 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "2787-2790",
        "Text": "Example 16 Find the maximum and the minimum values, if any, of the function\ngiven by\nf (x) = x, x \u2208 (0, 1) Solution The given function is an increasing (strictly) function in the given interval\n(0, 1) From the graph (Fig 6 10) of the function f , it\nseems that, it should have the minimum value at a\npoint closest to 0 on its right and the maximum value\nat a point closest to 1 on its left"
    },
    {
        "Chapter": "1",
        "sentence_range": "2788-2791",
        "Text": "Solution The given function is an increasing (strictly) function in the given interval\n(0, 1) From the graph (Fig 6 10) of the function f , it\nseems that, it should have the minimum value at a\npoint closest to 0 on its right and the maximum value\nat a point closest to 1 on its left Are such points\navailable"
    },
    {
        "Chapter": "1",
        "sentence_range": "2789-2792",
        "Text": "From the graph (Fig 6 10) of the function f , it\nseems that, it should have the minimum value at a\npoint closest to 0 on its right and the maximum value\nat a point closest to 1 on its left Are such points\navailable Of course, not"
    },
    {
        "Chapter": "1",
        "sentence_range": "2790-2793",
        "Text": "10) of the function f , it\nseems that, it should have the minimum value at a\npoint closest to 0 on its right and the maximum value\nat a point closest to 1 on its left Are such points\navailable Of course, not It is not possible to locate\nsuch points"
    },
    {
        "Chapter": "1",
        "sentence_range": "2791-2794",
        "Text": "Are such points\navailable Of course, not It is not possible to locate\nsuch points Infact, if a point x0 is closest to 0, then\nwe find \n0\n0\nx2\n<x\n for all \n0\nx \u2208(0,1)"
    },
    {
        "Chapter": "1",
        "sentence_range": "2792-2795",
        "Text": "Of course, not It is not possible to locate\nsuch points Infact, if a point x0 is closest to 0, then\nwe find \n0\n0\nx2\n<x\n for all \n0\nx \u2208(0,1) Also, if x1 is closest\nto 1, then \n1\n1\nx21\nx\n+\n>\n for all \n1\nx \u2208(0,1)"
    },
    {
        "Chapter": "1",
        "sentence_range": "2793-2796",
        "Text": "It is not possible to locate\nsuch points Infact, if a point x0 is closest to 0, then\nwe find \n0\n0\nx2\n<x\n for all \n0\nx \u2208(0,1) Also, if x1 is closest\nto 1, then \n1\n1\nx21\nx\n+\n>\n for all \n1\nx \u2208(0,1) Therefore, the given function has neither the\nmaximum value nor the minimum value in the interval (0,1)"
    },
    {
        "Chapter": "1",
        "sentence_range": "2794-2797",
        "Text": "Infact, if a point x0 is closest to 0, then\nwe find \n0\n0\nx2\n<x\n for all \n0\nx \u2208(0,1) Also, if x1 is closest\nto 1, then \n1\n1\nx21\nx\n+\n>\n for all \n1\nx \u2208(0,1) Therefore, the given function has neither the\nmaximum value nor the minimum value in the interval (0,1) Remark The reader may observe that in Example 16, if we include the points 0 and 1\nin the domain of  f , i"
    },
    {
        "Chapter": "1",
        "sentence_range": "2795-2798",
        "Text": "Also, if x1 is closest\nto 1, then \n1\n1\nx21\nx\n+\n>\n for all \n1\nx \u2208(0,1) Therefore, the given function has neither the\nmaximum value nor the minimum value in the interval (0,1) Remark The reader may observe that in Example 16, if we include the points 0 and 1\nin the domain of  f , i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "2796-2799",
        "Text": "Therefore, the given function has neither the\nmaximum value nor the minimum value in the interval (0,1) Remark The reader may observe that in Example 16, if we include the points 0 and 1\nin the domain of  f , i e , if we extend the domain of  f  to [0,1], then the function  f  has\nminimum value 0 at x = 0 and maximum value 1 at x = 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "2797-2800",
        "Text": "Remark The reader may observe that in Example 16, if we include the points 0 and 1\nin the domain of  f , i e , if we extend the domain of  f  to [0,1], then the function  f  has\nminimum value 0 at x = 0 and maximum value 1 at x = 1 Infact, we have the following\nresults (The proof of these results are beyond the scope of the present text)\nEvery monotonic function assumes its maximum/minimum value at the end\npoints of the domain of definition of the function"
    },
    {
        "Chapter": "1",
        "sentence_range": "2798-2801",
        "Text": "e , if we extend the domain of  f  to [0,1], then the function  f  has\nminimum value 0 at x = 0 and maximum value 1 at x = 1 Infact, we have the following\nresults (The proof of these results are beyond the scope of the present text)\nEvery monotonic function assumes its maximum/minimum value at the end\npoints of the domain of definition of the function A more general result is\nEvery continuous function on a closed interval has a maximum and a minimum\nvalue"
    },
    {
        "Chapter": "1",
        "sentence_range": "2799-2802",
        "Text": ", if we extend the domain of  f  to [0,1], then the function  f  has\nminimum value 0 at x = 0 and maximum value 1 at x = 1 Infact, we have the following\nresults (The proof of these results are beyond the scope of the present text)\nEvery monotonic function assumes its maximum/minimum value at the end\npoints of the domain of definition of the function A more general result is\nEvery continuous function on a closed interval has a maximum and a minimum\nvalue ANote  By a monotonic function f  in an interval I, we mean that f  is either\nincreasing in I or decreasing in I"
    },
    {
        "Chapter": "1",
        "sentence_range": "2800-2803",
        "Text": "Infact, we have the following\nresults (The proof of these results are beyond the scope of the present text)\nEvery monotonic function assumes its maximum/minimum value at the end\npoints of the domain of definition of the function A more general result is\nEvery continuous function on a closed interval has a maximum and a minimum\nvalue ANote  By a monotonic function f  in an interval I, we mean that f  is either\nincreasing in I or decreasing in I Maximum and minimum values of a function defined on a closed interval will be\ndiscussed later in this section"
    },
    {
        "Chapter": "1",
        "sentence_range": "2801-2804",
        "Text": "A more general result is\nEvery continuous function on a closed interval has a maximum and a minimum\nvalue ANote  By a monotonic function f  in an interval I, we mean that f  is either\nincreasing in I or decreasing in I Maximum and minimum values of a function defined on a closed interval will be\ndiscussed later in this section Let us now examine the graph of a function as shown in Fig 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "2802-2805",
        "Text": "ANote  By a monotonic function f  in an interval I, we mean that f  is either\nincreasing in I or decreasing in I Maximum and minimum values of a function defined on a closed interval will be\ndiscussed later in this section Let us now examine the graph of a function as shown in Fig 6 11"
    },
    {
        "Chapter": "1",
        "sentence_range": "2803-2806",
        "Text": "Maximum and minimum values of a function defined on a closed interval will be\ndiscussed later in this section Let us now examine the graph of a function as shown in Fig 6 11 Observe that at\npoints A, B, C and D on the graph, the function changes its nature from decreasing to\nincreasing or vice-versa"
    },
    {
        "Chapter": "1",
        "sentence_range": "2804-2807",
        "Text": "Let us now examine the graph of a function as shown in Fig 6 11 Observe that at\npoints A, B, C and D on the graph, the function changes its nature from decreasing to\nincreasing or vice-versa These points may be called turning points of the given\nfunction"
    },
    {
        "Chapter": "1",
        "sentence_range": "2805-2808",
        "Text": "11 Observe that at\npoints A, B, C and D on the graph, the function changes its nature from decreasing to\nincreasing or vice-versa These points may be called turning points of the given\nfunction Further, observe that at turning points, the graph has either a little hill or a little\nvalley"
    },
    {
        "Chapter": "1",
        "sentence_range": "2806-2809",
        "Text": "Observe that at\npoints A, B, C and D on the graph, the function changes its nature from decreasing to\nincreasing or vice-versa These points may be called turning points of the given\nfunction Further, observe that at turning points, the graph has either a little hill or a little\nvalley Roughly speaking, the function has minimum value in some neighbourhood\n(interval) of each of the points A and C which are at the bottom of their respective\nRationalised 2023-24\nAPPLICATION OF DERIVATIVES\n163\nvalleys"
    },
    {
        "Chapter": "1",
        "sentence_range": "2807-2810",
        "Text": "These points may be called turning points of the given\nfunction Further, observe that at turning points, the graph has either a little hill or a little\nvalley Roughly speaking, the function has minimum value in some neighbourhood\n(interval) of each of the points A and C which are at the bottom of their respective\nRationalised 2023-24\nAPPLICATION OF DERIVATIVES\n163\nvalleys Similarly, the function has maximum value in some neighbourhood of points B\nand D which are at the top of their respective hills"
    },
    {
        "Chapter": "1",
        "sentence_range": "2808-2811",
        "Text": "Further, observe that at turning points, the graph has either a little hill or a little\nvalley Roughly speaking, the function has minimum value in some neighbourhood\n(interval) of each of the points A and C which are at the bottom of their respective\nRationalised 2023-24\nAPPLICATION OF DERIVATIVES\n163\nvalleys Similarly, the function has maximum value in some neighbourhood of points B\nand D which are at the top of their respective hills For this reason, the points A and C\nmay be regarded as points of local minimum value (or relative minimum value) and\npoints B and D may be regarded as points of local maximum value (or relative maximum\nvalue) for the function"
    },
    {
        "Chapter": "1",
        "sentence_range": "2809-2812",
        "Text": "Roughly speaking, the function has minimum value in some neighbourhood\n(interval) of each of the points A and C which are at the bottom of their respective\nRationalised 2023-24\nAPPLICATION OF DERIVATIVES\n163\nvalleys Similarly, the function has maximum value in some neighbourhood of points B\nand D which are at the top of their respective hills For this reason, the points A and C\nmay be regarded as points of local minimum value (or relative minimum value) and\npoints B and D may be regarded as points of local maximum value (or relative maximum\nvalue) for the function The local maximum value and local minimum value of the\nfunction are referred to as local maxima and local minima, respectively, of the function"
    },
    {
        "Chapter": "1",
        "sentence_range": "2810-2813",
        "Text": "Similarly, the function has maximum value in some neighbourhood of points B\nand D which are at the top of their respective hills For this reason, the points A and C\nmay be regarded as points of local minimum value (or relative minimum value) and\npoints B and D may be regarded as points of local maximum value (or relative maximum\nvalue) for the function The local maximum value and local minimum value of the\nfunction are referred to as local maxima and local minima, respectively, of the function We now formally give the following definition\nDefinition 4 Let  f  be a real valued function and let c be an interior point in the domain\nof f"
    },
    {
        "Chapter": "1",
        "sentence_range": "2811-2814",
        "Text": "For this reason, the points A and C\nmay be regarded as points of local minimum value (or relative minimum value) and\npoints B and D may be regarded as points of local maximum value (or relative maximum\nvalue) for the function The local maximum value and local minimum value of the\nfunction are referred to as local maxima and local minima, respectively, of the function We now formally give the following definition\nDefinition 4 Let  f  be a real valued function and let c be an interior point in the domain\nof f Then\n(a)\nc is called a point of local maxima if there is an h > 0 such that\nf (c) \u2265 f (x), for all x in (c \u2013 h, c + h), x \u2260 c\nThe value f (c) is called the local maximum value of f"
    },
    {
        "Chapter": "1",
        "sentence_range": "2812-2815",
        "Text": "The local maximum value and local minimum value of the\nfunction are referred to as local maxima and local minima, respectively, of the function We now formally give the following definition\nDefinition 4 Let  f  be a real valued function and let c be an interior point in the domain\nof f Then\n(a)\nc is called a point of local maxima if there is an h > 0 such that\nf (c) \u2265 f (x), for all x in (c \u2013 h, c + h), x \u2260 c\nThe value f (c) is called the local maximum value of f (b)\nc is called a point of local minima if there is an h > 0 such that\nf (c) \u2264 f (x), for all  x in (c \u2013 h, c + h)\nThe value f (c) is called the local minimum value of f"
    },
    {
        "Chapter": "1",
        "sentence_range": "2813-2816",
        "Text": "We now formally give the following definition\nDefinition 4 Let  f  be a real valued function and let c be an interior point in the domain\nof f Then\n(a)\nc is called a point of local maxima if there is an h > 0 such that\nf (c) \u2265 f (x), for all x in (c \u2013 h, c + h), x \u2260 c\nThe value f (c) is called the local maximum value of f (b)\nc is called a point of local minima if there is an h > 0 such that\nf (c) \u2264 f (x), for all  x in (c \u2013 h, c + h)\nThe value f (c) is called the local minimum value of f Geometrically, the above definition states that if  x = c is a point of local maxima of  f,\nthen the graph of f around c will be as shown in Fig 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "2814-2817",
        "Text": "Then\n(a)\nc is called a point of local maxima if there is an h > 0 such that\nf (c) \u2265 f (x), for all x in (c \u2013 h, c + h), x \u2260 c\nThe value f (c) is called the local maximum value of f (b)\nc is called a point of local minima if there is an h > 0 such that\nf (c) \u2264 f (x), for all  x in (c \u2013 h, c + h)\nThe value f (c) is called the local minimum value of f Geometrically, the above definition states that if  x = c is a point of local maxima of  f,\nthen the graph of f around c will be as shown in Fig 6 12(a)"
    },
    {
        "Chapter": "1",
        "sentence_range": "2815-2818",
        "Text": "(b)\nc is called a point of local minima if there is an h > 0 such that\nf (c) \u2264 f (x), for all  x in (c \u2013 h, c + h)\nThe value f (c) is called the local minimum value of f Geometrically, the above definition states that if  x = c is a point of local maxima of  f,\nthen the graph of f around c will be as shown in Fig 6 12(a) Note that the function f  is\nincreasing (i"
    },
    {
        "Chapter": "1",
        "sentence_range": "2816-2819",
        "Text": "Geometrically, the above definition states that if  x = c is a point of local maxima of  f,\nthen the graph of f around c will be as shown in Fig 6 12(a) Note that the function f  is\nincreasing (i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "2817-2820",
        "Text": "12(a) Note that the function f  is\nincreasing (i e , f \u2032(x) > 0) in the interval (c \u2013 h, c) and decreasing (i"
    },
    {
        "Chapter": "1",
        "sentence_range": "2818-2821",
        "Text": "Note that the function f  is\nincreasing (i e , f \u2032(x) > 0) in the interval (c \u2013 h, c) and decreasing (i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "2819-2822",
        "Text": "e , f \u2032(x) > 0) in the interval (c \u2013 h, c) and decreasing (i e , f \u2032(x) < 0) in the\ninterval (c, c + h)"
    },
    {
        "Chapter": "1",
        "sentence_range": "2820-2823",
        "Text": ", f \u2032(x) > 0) in the interval (c \u2013 h, c) and decreasing (i e , f \u2032(x) < 0) in the\ninterval (c, c + h) This suggests that f \u2032(c) must be zero"
    },
    {
        "Chapter": "1",
        "sentence_range": "2821-2824",
        "Text": "e , f \u2032(x) < 0) in the\ninterval (c, c + h) This suggests that f \u2032(c) must be zero Fig 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "2822-2825",
        "Text": ", f \u2032(x) < 0) in the\ninterval (c, c + h) This suggests that f \u2032(c) must be zero Fig 6 11\nFig 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "2823-2826",
        "Text": "This suggests that f \u2032(c) must be zero Fig 6 11\nFig 6 12\nRationalised 2023-24\n MATHEMATICS\n164\nFig 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "2824-2827",
        "Text": "Fig 6 11\nFig 6 12\nRationalised 2023-24\n MATHEMATICS\n164\nFig 6 13\nSimilarly, if c is a point of local minima of f , then the graph of f around c will be as\nshown in Fig 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "2825-2828",
        "Text": "11\nFig 6 12\nRationalised 2023-24\n MATHEMATICS\n164\nFig 6 13\nSimilarly, if c is a point of local minima of f , then the graph of f around c will be as\nshown in Fig 6 14(b)"
    },
    {
        "Chapter": "1",
        "sentence_range": "2826-2829",
        "Text": "12\nRationalised 2023-24\n MATHEMATICS\n164\nFig 6 13\nSimilarly, if c is a point of local minima of f , then the graph of f around c will be as\nshown in Fig 6 14(b) Here f  is decreasing (i"
    },
    {
        "Chapter": "1",
        "sentence_range": "2827-2830",
        "Text": "13\nSimilarly, if c is a point of local minima of f , then the graph of f around c will be as\nshown in Fig 6 14(b) Here f  is decreasing (i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "2828-2831",
        "Text": "14(b) Here f  is decreasing (i e , f \u2032(x) < 0) in the interval (c \u2013 h, c) and\nincreasing (i"
    },
    {
        "Chapter": "1",
        "sentence_range": "2829-2832",
        "Text": "Here f  is decreasing (i e , f \u2032(x) < 0) in the interval (c \u2013 h, c) and\nincreasing (i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "2830-2833",
        "Text": "e , f \u2032(x) < 0) in the interval (c \u2013 h, c) and\nincreasing (i e , f \u2032(x) > 0) in the interval (c, c + h)"
    },
    {
        "Chapter": "1",
        "sentence_range": "2831-2834",
        "Text": ", f \u2032(x) < 0) in the interval (c \u2013 h, c) and\nincreasing (i e , f \u2032(x) > 0) in the interval (c, c + h) This again suggest that f \u2032(c) must\nbe zero"
    },
    {
        "Chapter": "1",
        "sentence_range": "2832-2835",
        "Text": "e , f \u2032(x) > 0) in the interval (c, c + h) This again suggest that f \u2032(c) must\nbe zero The above discussion lead us to the following theorem (without proof)"
    },
    {
        "Chapter": "1",
        "sentence_range": "2833-2836",
        "Text": ", f \u2032(x) > 0) in the interval (c, c + h) This again suggest that f \u2032(c) must\nbe zero The above discussion lead us to the following theorem (without proof) Theorem 2 Let  f  be a function defined on an open interval I"
    },
    {
        "Chapter": "1",
        "sentence_range": "2834-2837",
        "Text": "This again suggest that f \u2032(c) must\nbe zero The above discussion lead us to the following theorem (without proof) Theorem 2 Let  f  be a function defined on an open interval I Suppose c \u2208 I be  any\npoint"
    },
    {
        "Chapter": "1",
        "sentence_range": "2835-2838",
        "Text": "The above discussion lead us to the following theorem (without proof) Theorem 2 Let  f  be a function defined on an open interval I Suppose c \u2208 I be  any\npoint If f  has a local maxima or a local minima at x = c, then either f \u2032(c) = 0 or f  is not\ndifferentiable at c"
    },
    {
        "Chapter": "1",
        "sentence_range": "2836-2839",
        "Text": "Theorem 2 Let  f  be a function defined on an open interval I Suppose c \u2208 I be  any\npoint If f  has a local maxima or a local minima at x = c, then either f \u2032(c) = 0 or f  is not\ndifferentiable at c Remark The converse of above theorem need not\nbe true, that is, a point at which the derivative vanishes\nneed not be a point of local maxima or local minima"
    },
    {
        "Chapter": "1",
        "sentence_range": "2837-2840",
        "Text": "Suppose c \u2208 I be  any\npoint If f  has a local maxima or a local minima at x = c, then either f \u2032(c) = 0 or f  is not\ndifferentiable at c Remark The converse of above theorem need not\nbe true, that is, a point at which the derivative vanishes\nneed not be a point of local maxima or local minima For example, if f (x) = x3, then f \u2032(x) = 3x2 and so\nf \u2032(0) = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "2838-2841",
        "Text": "If f  has a local maxima or a local minima at x = c, then either f \u2032(c) = 0 or f  is not\ndifferentiable at c Remark The converse of above theorem need not\nbe true, that is, a point at which the derivative vanishes\nneed not be a point of local maxima or local minima For example, if f (x) = x3, then f \u2032(x) = 3x2 and so\nf \u2032(0) = 0 But 0 is neither a point of local maxima nor\na point of local minima (Fig 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "2839-2842",
        "Text": "Remark The converse of above theorem need not\nbe true, that is, a point at which the derivative vanishes\nneed not be a point of local maxima or local minima For example, if f (x) = x3, then f \u2032(x) = 3x2 and so\nf \u2032(0) = 0 But 0 is neither a point of local maxima nor\na point of local minima (Fig 6 13)"
    },
    {
        "Chapter": "1",
        "sentence_range": "2840-2843",
        "Text": "For example, if f (x) = x3, then f \u2032(x) = 3x2 and so\nf \u2032(0) = 0 But 0 is neither a point of local maxima nor\na point of local minima (Fig 6 13) ANote   A point c in the domain of a function f at\nwhich either f \u2032(c) = 0 or f is not differentiable is\ncalled a critical point of f"
    },
    {
        "Chapter": "1",
        "sentence_range": "2841-2844",
        "Text": "But 0 is neither a point of local maxima nor\na point of local minima (Fig 6 13) ANote   A point c in the domain of a function f at\nwhich either f \u2032(c) = 0 or f is not differentiable is\ncalled a critical point of f Note that if f is continuous\nat c and f \u2032(c) = 0, then there exists an h > 0 such\nthat f is differentiable in the interval\n(c \u2013 h, c + h)"
    },
    {
        "Chapter": "1",
        "sentence_range": "2842-2845",
        "Text": "13) ANote   A point c in the domain of a function f at\nwhich either f \u2032(c) = 0 or f is not differentiable is\ncalled a critical point of f Note that if f is continuous\nat c and f \u2032(c) = 0, then there exists an h > 0 such\nthat f is differentiable in the interval\n(c \u2013 h, c + h) We shall now give a working rule for finding points of local maxima or points of\nlocal minima using only the first order derivatives"
    },
    {
        "Chapter": "1",
        "sentence_range": "2843-2846",
        "Text": "ANote   A point c in the domain of a function f at\nwhich either f \u2032(c) = 0 or f is not differentiable is\ncalled a critical point of f Note that if f is continuous\nat c and f \u2032(c) = 0, then there exists an h > 0 such\nthat f is differentiable in the interval\n(c \u2013 h, c + h) We shall now give a working rule for finding points of local maxima or points of\nlocal minima using only the first order derivatives Theorem 3 (First Derivative Test) Let f be a function defined on an open interval I"
    },
    {
        "Chapter": "1",
        "sentence_range": "2844-2847",
        "Text": "Note that if f is continuous\nat c and f \u2032(c) = 0, then there exists an h > 0 such\nthat f is differentiable in the interval\n(c \u2013 h, c + h) We shall now give a working rule for finding points of local maxima or points of\nlocal minima using only the first order derivatives Theorem 3 (First Derivative Test) Let f be a function defined on an open interval I Let f be continuous at a critical point c in I"
    },
    {
        "Chapter": "1",
        "sentence_range": "2845-2848",
        "Text": "We shall now give a working rule for finding points of local maxima or points of\nlocal minima using only the first order derivatives Theorem 3 (First Derivative Test) Let f be a function defined on an open interval I Let f be continuous at a critical point c in I Then\n(i)\nIf f \u2032(x) changes sign from positive to negative as x increases through c, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "2846-2849",
        "Text": "Theorem 3 (First Derivative Test) Let f be a function defined on an open interval I Let f be continuous at a critical point c in I Then\n(i)\nIf f \u2032(x) changes sign from positive to negative as x increases through c, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "2847-2850",
        "Text": "Let f be continuous at a critical point c in I Then\n(i)\nIf f \u2032(x) changes sign from positive to negative as x increases through c, i e , if\nf \u2032(x) > 0 at every point sufficiently close to and to the left of c, and f \u2032(x) < 0 at\nevery point sufficiently close to and to the right of c, then c is a point of local\nmaxima"
    },
    {
        "Chapter": "1",
        "sentence_range": "2848-2851",
        "Text": "Then\n(i)\nIf f \u2032(x) changes sign from positive to negative as x increases through c, i e , if\nf \u2032(x) > 0 at every point sufficiently close to and to the left of c, and f \u2032(x) < 0 at\nevery point sufficiently close to and to the right of c, then c is a point of local\nmaxima (ii)\nIf f \u2032(x) changes sign from negative to positive as x increases through c, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "2849-2852",
        "Text": "e , if\nf \u2032(x) > 0 at every point sufficiently close to and to the left of c, and f \u2032(x) < 0 at\nevery point sufficiently close to and to the right of c, then c is a point of local\nmaxima (ii)\nIf f \u2032(x) changes sign from negative to positive as x increases through c, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "2850-2853",
        "Text": ", if\nf \u2032(x) > 0 at every point sufficiently close to and to the left of c, and f \u2032(x) < 0 at\nevery point sufficiently close to and to the right of c, then c is a point of local\nmaxima (ii)\nIf f \u2032(x) changes sign from negative to positive as x increases through c, i e , if\nf \u2032(x) < 0 at every point sufficiently close to and to the left of c, and f \u2032(x) > 0 at\nevery point sufficiently close to and to the right of c, then c is a point of local\nminima"
    },
    {
        "Chapter": "1",
        "sentence_range": "2851-2854",
        "Text": "(ii)\nIf f \u2032(x) changes sign from negative to positive as x increases through c, i e , if\nf \u2032(x) < 0 at every point sufficiently close to and to the left of c, and f \u2032(x) > 0 at\nevery point sufficiently close to and to the right of c, then c is a point of local\nminima (iii)\nIf f \u2032(x) does not change sign as x increases through c, then c is neither a point of\nlocal maxima nor a point of local minima"
    },
    {
        "Chapter": "1",
        "sentence_range": "2852-2855",
        "Text": "e , if\nf \u2032(x) < 0 at every point sufficiently close to and to the left of c, and f \u2032(x) > 0 at\nevery point sufficiently close to and to the right of c, then c is a point of local\nminima (iii)\nIf f \u2032(x) does not change sign as x increases through c, then c is neither a point of\nlocal maxima nor a point of local minima Infact, such a point is called point of\ninflection (Fig 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "2853-2856",
        "Text": ", if\nf \u2032(x) < 0 at every point sufficiently close to and to the left of c, and f \u2032(x) > 0 at\nevery point sufficiently close to and to the right of c, then c is a point of local\nminima (iii)\nIf f \u2032(x) does not change sign as x increases through c, then c is neither a point of\nlocal maxima nor a point of local minima Infact, such a point is called point of\ninflection (Fig 6 13)"
    },
    {
        "Chapter": "1",
        "sentence_range": "2854-2857",
        "Text": "(iii)\nIf f \u2032(x) does not change sign as x increases through c, then c is neither a point of\nlocal maxima nor a point of local minima Infact, such a point is called point of\ninflection (Fig 6 13) Rationalised 2023-24\nAPPLICATION OF DERIVATIVES\n165\nANote  If c is a point of local maxima of f , then f (c) is a local maximum value of\nf"
    },
    {
        "Chapter": "1",
        "sentence_range": "2855-2858",
        "Text": "Infact, such a point is called point of\ninflection (Fig 6 13) Rationalised 2023-24\nAPPLICATION OF DERIVATIVES\n165\nANote  If c is a point of local maxima of f , then f (c) is a local maximum value of\nf Similarly, if c is a point of local minima of f , then  f(c) is a local minimum value of f"
    },
    {
        "Chapter": "1",
        "sentence_range": "2856-2859",
        "Text": "13) Rationalised 2023-24\nAPPLICATION OF DERIVATIVES\n165\nANote  If c is a point of local maxima of f , then f (c) is a local maximum value of\nf Similarly, if c is a point of local minima of f , then  f(c) is a local minimum value of f Figures 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "2857-2860",
        "Text": "Rationalised 2023-24\nAPPLICATION OF DERIVATIVES\n165\nANote  If c is a point of local maxima of f , then f (c) is a local maximum value of\nf Similarly, if c is a point of local minima of f , then  f(c) is a local minimum value of f Figures 6 13 and 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "2858-2861",
        "Text": "Similarly, if c is a point of local minima of f , then  f(c) is a local minimum value of f Figures 6 13 and 6 14, geometrically explain Theorem 3"
    },
    {
        "Chapter": "1",
        "sentence_range": "2859-2862",
        "Text": "Figures 6 13 and 6 14, geometrically explain Theorem 3 Fig 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "2860-2863",
        "Text": "13 and 6 14, geometrically explain Theorem 3 Fig 6 14\nExample 17 Find all points of local maxima and local minima of the function f\ngiven by\nf (x) = x3 \u2013 3x + 3"
    },
    {
        "Chapter": "1",
        "sentence_range": "2861-2864",
        "Text": "14, geometrically explain Theorem 3 Fig 6 14\nExample 17 Find all points of local maxima and local minima of the function f\ngiven by\nf (x) = x3 \u2013 3x + 3 Solution We have\nf (x) = x3 \u2013 3x + 3\nor\nf \u2032(x) = 3x2 \u2013 3 = 3(x \u2013 1) (x + 1)\nor\nf \u2032(x) = 0 at x = 1 and x = \u2013 1\nThus, x = \u00b1 1 are the only critical points which could possibly be the points of local\nmaxima and/or local minima of f"
    },
    {
        "Chapter": "1",
        "sentence_range": "2862-2865",
        "Text": "Fig 6 14\nExample 17 Find all points of local maxima and local minima of the function f\ngiven by\nf (x) = x3 \u2013 3x + 3 Solution We have\nf (x) = x3 \u2013 3x + 3\nor\nf \u2032(x) = 3x2 \u2013 3 = 3(x \u2013 1) (x + 1)\nor\nf \u2032(x) = 0 at x = 1 and x = \u2013 1\nThus, x = \u00b1 1 are the only critical points which could possibly be the points of local\nmaxima and/or local minima of f Let us first examine the point x = 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "2863-2866",
        "Text": "14\nExample 17 Find all points of local maxima and local minima of the function f\ngiven by\nf (x) = x3 \u2013 3x + 3 Solution We have\nf (x) = x3 \u2013 3x + 3\nor\nf \u2032(x) = 3x2 \u2013 3 = 3(x \u2013 1) (x + 1)\nor\nf \u2032(x) = 0 at x = 1 and x = \u2013 1\nThus, x = \u00b1 1 are the only critical points which could possibly be the points of local\nmaxima and/or local minima of f Let us first examine the point x = 1 Note that for values close to 1 and to the right of 1, f \u2032(x) > 0 and  for values close\nto 1 and to the left of 1, f \u2032(x) < 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "2864-2867",
        "Text": "Solution We have\nf (x) = x3 \u2013 3x + 3\nor\nf \u2032(x) = 3x2 \u2013 3 = 3(x \u2013 1) (x + 1)\nor\nf \u2032(x) = 0 at x = 1 and x = \u2013 1\nThus, x = \u00b1 1 are the only critical points which could possibly be the points of local\nmaxima and/or local minima of f Let us first examine the point x = 1 Note that for values close to 1 and to the right of 1, f \u2032(x) > 0 and  for values close\nto 1 and to the left of 1, f \u2032(x) < 0 Therefore, by first derivative test, x  = 1 is a point\nof local minima and local minimum value is f (1) = 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "2865-2868",
        "Text": "Let us first examine the point x = 1 Note that for values close to 1 and to the right of 1, f \u2032(x) > 0 and  for values close\nto 1 and to the left of 1, f \u2032(x) < 0 Therefore, by first derivative test, x  = 1 is a point\nof local minima and local minimum value is f (1) = 1 In the case of x = \u20131, note that\nf \u2032(x) > 0, for values close to and to the left of \u20131 and f \u2032(x) < 0, for values close to and\nto the right of \u2013 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "2866-2869",
        "Text": "Note that for values close to 1 and to the right of 1, f \u2032(x) > 0 and  for values close\nto 1 and to the left of 1, f \u2032(x) < 0 Therefore, by first derivative test, x  = 1 is a point\nof local minima and local minimum value is f (1) = 1 In the case of x = \u20131, note that\nf \u2032(x) > 0, for values close to and to the left of \u20131 and f \u2032(x) < 0, for values close to and\nto the right of \u2013 1 Therefore, by first derivative test, x = \u2013 1 is a point of local maxima\nand local maximum value is  f (\u20131) = 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "2867-2870",
        "Text": "Therefore, by first derivative test, x  = 1 is a point\nof local minima and local minimum value is f (1) = 1 In the case of x = \u20131, note that\nf \u2032(x) > 0, for values close to and to the left of \u20131 and f \u2032(x) < 0, for values close to and\nto the right of \u2013 1 Therefore, by first derivative test, x = \u2013 1 is a point of local maxima\nand local maximum value is  f (\u20131) = 5 Values of  x\nSign of f \u2032\u2032\u2032\u2032\u2032(x) = 3(x \u2013 1) (x + 1)\nClose to 1     \nto the right (say 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "2868-2871",
        "Text": "In the case of x = \u20131, note that\nf \u2032(x) > 0, for values close to and to the left of \u20131 and f \u2032(x) < 0, for values close to and\nto the right of \u2013 1 Therefore, by first derivative test, x = \u2013 1 is a point of local maxima\nand local maximum value is  f (\u20131) = 5 Values of  x\nSign of f \u2032\u2032\u2032\u2032\u2032(x) = 3(x \u2013 1) (x + 1)\nClose to 1     \nto the right (say 1 1 etc"
    },
    {
        "Chapter": "1",
        "sentence_range": "2869-2872",
        "Text": "Therefore, by first derivative test, x = \u2013 1 is a point of local maxima\nand local maximum value is  f (\u20131) = 5 Values of  x\nSign of f \u2032\u2032\u2032\u2032\u2032(x) = 3(x \u2013 1) (x + 1)\nClose to 1     \nto the right (say 1 1 etc )\n>0\nto the left (say 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "2870-2873",
        "Text": "Values of  x\nSign of f \u2032\u2032\u2032\u2032\u2032(x) = 3(x \u2013 1) (x + 1)\nClose to 1     \nto the right (say 1 1 etc )\n>0\nto the left (say 0 9 etc"
    },
    {
        "Chapter": "1",
        "sentence_range": "2871-2874",
        "Text": "1 etc )\n>0\nto the left (say 0 9 etc )\n<0\nClose to \u20131  \nto the right (say \n0"
    },
    {
        "Chapter": "1",
        "sentence_range": "2872-2875",
        "Text": ")\n>0\nto the left (say 0 9 etc )\n<0\nClose to \u20131  \nto the right (say \n0 9 etc"
    },
    {
        "Chapter": "1",
        "sentence_range": "2873-2876",
        "Text": "9 etc )\n<0\nClose to \u20131  \nto the right (say \n0 9 etc )\n0\nto the left (say \n1"
    },
    {
        "Chapter": "1",
        "sentence_range": "2874-2877",
        "Text": ")\n<0\nClose to \u20131  \nto the right (say \n0 9 etc )\n0\nto the left (say \n1 1 etc"
    },
    {
        "Chapter": "1",
        "sentence_range": "2875-2878",
        "Text": "9 etc )\n0\nto the left (say \n1 1 etc )\n0\n\u2212\n<\n\u2212\n>\nRationalised 2023-24\n MATHEMATICS\n166\nFig 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "2876-2879",
        "Text": ")\n0\nto the left (say \n1 1 etc )\n0\n\u2212\n<\n\u2212\n>\nRationalised 2023-24\n MATHEMATICS\n166\nFig 6 15\nExample 18 Find all the points of local maxima and local minima of the function f\ngiven by\nf (x) = 2x3 \u2013 6x2 + 6x +5"
    },
    {
        "Chapter": "1",
        "sentence_range": "2877-2880",
        "Text": "1 etc )\n0\n\u2212\n<\n\u2212\n>\nRationalised 2023-24\n MATHEMATICS\n166\nFig 6 15\nExample 18 Find all the points of local maxima and local minima of the function f\ngiven by\nf (x) = 2x3 \u2013 6x2 + 6x +5 Solution We have\nf (x) = 2x3 \u2013 6x2 + 6x + 5\nor\nf \u2032(x) = 6x2 \u2013 12x + 6 = 6(x \u2013 1)2\nor\nf \u2032(x) = 0    at   x = 1\nThus, x = 1 is the only critical point of  f"
    },
    {
        "Chapter": "1",
        "sentence_range": "2878-2881",
        "Text": ")\n0\n\u2212\n<\n\u2212\n>\nRationalised 2023-24\n MATHEMATICS\n166\nFig 6 15\nExample 18 Find all the points of local maxima and local minima of the function f\ngiven by\nf (x) = 2x3 \u2013 6x2 + 6x +5 Solution We have\nf (x) = 2x3 \u2013 6x2 + 6x + 5\nor\nf \u2032(x) = 6x2 \u2013 12x + 6 = 6(x \u2013 1)2\nor\nf \u2032(x) = 0    at   x = 1\nThus, x = 1 is the only critical point of  f We shall now examine this point for local\nmaxima and/or local minima of f"
    },
    {
        "Chapter": "1",
        "sentence_range": "2879-2882",
        "Text": "15\nExample 18 Find all the points of local maxima and local minima of the function f\ngiven by\nf (x) = 2x3 \u2013 6x2 + 6x +5 Solution We have\nf (x) = 2x3 \u2013 6x2 + 6x + 5\nor\nf \u2032(x) = 6x2 \u2013 12x + 6 = 6(x \u2013 1)2\nor\nf \u2032(x) = 0    at   x = 1\nThus, x = 1 is the only critical point of  f We shall now examine this point for local\nmaxima and/or local minima of f Observe that f \u2032(x) \u2265 0, for all x \u2208 R and in particular\nf \u2032(x) > 0, for values close to 1 and to the left and to the right of 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "2880-2883",
        "Text": "Solution We have\nf (x) = 2x3 \u2013 6x2 + 6x + 5\nor\nf \u2032(x) = 6x2 \u2013 12x + 6 = 6(x \u2013 1)2\nor\nf \u2032(x) = 0    at   x = 1\nThus, x = 1 is the only critical point of  f We shall now examine this point for local\nmaxima and/or local minima of f Observe that f \u2032(x) \u2265 0, for all x \u2208 R and in particular\nf \u2032(x) > 0, for values close to 1 and to the left and to the right of 1 Therefore, by first\nderivative test, the point x = 1 is neither a point of local maxima nor a point of local\nminima"
    },
    {
        "Chapter": "1",
        "sentence_range": "2881-2884",
        "Text": "We shall now examine this point for local\nmaxima and/or local minima of f Observe that f \u2032(x) \u2265 0, for all x \u2208 R and in particular\nf \u2032(x) > 0, for values close to 1 and to the left and to the right of 1 Therefore, by first\nderivative test, the point x = 1 is neither a point of local maxima nor a point of local\nminima Hence x = 1 is a point of inflexion"
    },
    {
        "Chapter": "1",
        "sentence_range": "2882-2885",
        "Text": "Observe that f \u2032(x) \u2265 0, for all x \u2208 R and in particular\nf \u2032(x) > 0, for values close to 1 and to the left and to the right of 1 Therefore, by first\nderivative test, the point x = 1 is neither a point of local maxima nor a point of local\nminima Hence x = 1 is a point of inflexion Remark One may note that since f \u2032(x), in Example 30, never changes its sign on R,\ngraph of f has no turning points and hence no point of local maxima or local minima"
    },
    {
        "Chapter": "1",
        "sentence_range": "2883-2886",
        "Text": "Therefore, by first\nderivative test, the point x = 1 is neither a point of local maxima nor a point of local\nminima Hence x = 1 is a point of inflexion Remark One may note that since f \u2032(x), in Example 30, never changes its sign on R,\ngraph of f has no turning points and hence no point of local maxima or local minima We shall now give another test to examine local maxima and local minima of a\ngiven function"
    },
    {
        "Chapter": "1",
        "sentence_range": "2884-2887",
        "Text": "Hence x = 1 is a point of inflexion Remark One may note that since f \u2032(x), in Example 30, never changes its sign on R,\ngraph of f has no turning points and hence no point of local maxima or local minima We shall now give another test to examine local maxima and local minima of a\ngiven function This test is often easier to apply than the first derivative test"
    },
    {
        "Chapter": "1",
        "sentence_range": "2885-2888",
        "Text": "Remark One may note that since f \u2032(x), in Example 30, never changes its sign on R,\ngraph of f has no turning points and hence no point of local maxima or local minima We shall now give another test to examine local maxima and local minima of a\ngiven function This test is often easier to apply than the first derivative test Theorem 4 (Second Derivative Test) Let  f  be a function defined on an interval I\nand c \u2208 I"
    },
    {
        "Chapter": "1",
        "sentence_range": "2886-2889",
        "Text": "We shall now give another test to examine local maxima and local minima of a\ngiven function This test is often easier to apply than the first derivative test Theorem 4 (Second Derivative Test) Let  f  be a function defined on an interval I\nand c \u2208 I Let f  be twice differentiable at c"
    },
    {
        "Chapter": "1",
        "sentence_range": "2887-2890",
        "Text": "This test is often easier to apply than the first derivative test Theorem 4 (Second Derivative Test) Let  f  be a function defined on an interval I\nand c \u2208 I Let f  be twice differentiable at c Then\n(i)\nx = c is a point of local maxima if f \u2032(c) = 0 and f \u2033(c) < 0\nThe value f (c) is local maximum value of  f"
    },
    {
        "Chapter": "1",
        "sentence_range": "2888-2891",
        "Text": "Theorem 4 (Second Derivative Test) Let  f  be a function defined on an interval I\nand c \u2208 I Let f  be twice differentiable at c Then\n(i)\nx = c is a point of local maxima if f \u2032(c) = 0 and f \u2033(c) < 0\nThe value f (c) is local maximum value of  f (ii)\nx = c is a point of local minima if \n( )\n0\nf\n\u2032c =\n and f \u2033(c) > 0\nIn this case, f (c) is local minimum value of  f"
    },
    {
        "Chapter": "1",
        "sentence_range": "2889-2892",
        "Text": "Let f  be twice differentiable at c Then\n(i)\nx = c is a point of local maxima if f \u2032(c) = 0 and f \u2033(c) < 0\nThe value f (c) is local maximum value of  f (ii)\nx = c is a point of local minima if \n( )\n0\nf\n\u2032c =\n and f \u2033(c) > 0\nIn this case, f (c) is local minimum value of  f (iii)\nThe test fails if f \u2032(c) = 0 and f \u2033(c) = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "2890-2893",
        "Text": "Then\n(i)\nx = c is a point of local maxima if f \u2032(c) = 0 and f \u2033(c) < 0\nThe value f (c) is local maximum value of  f (ii)\nx = c is a point of local minima if \n( )\n0\nf\n\u2032c =\n and f \u2033(c) > 0\nIn this case, f (c) is local minimum value of  f (iii)\nThe test fails if f \u2032(c) = 0 and f \u2033(c) = 0 In this case, we go back to the first derivative test and find whether c is a point of\nlocal maxima, local minima or a point of inflexion"
    },
    {
        "Chapter": "1",
        "sentence_range": "2891-2894",
        "Text": "(ii)\nx = c is a point of local minima if \n( )\n0\nf\n\u2032c =\n and f \u2033(c) > 0\nIn this case, f (c) is local minimum value of  f (iii)\nThe test fails if f \u2032(c) = 0 and f \u2033(c) = 0 In this case, we go back to the first derivative test and find whether c is a point of\nlocal maxima, local minima or a point of inflexion ANote  As f is twice differentiable at c, we mean\nsecond order derivative of f exists at c"
    },
    {
        "Chapter": "1",
        "sentence_range": "2892-2895",
        "Text": "(iii)\nThe test fails if f \u2032(c) = 0 and f \u2033(c) = 0 In this case, we go back to the first derivative test and find whether c is a point of\nlocal maxima, local minima or a point of inflexion ANote  As f is twice differentiable at c, we mean\nsecond order derivative of f exists at c Example 19 Find local minimum value of the function f\ngiven by  f (x) = 3 + |x|, x \u2208 R"
    },
    {
        "Chapter": "1",
        "sentence_range": "2893-2896",
        "Text": "In this case, we go back to the first derivative test and find whether c is a point of\nlocal maxima, local minima or a point of inflexion ANote  As f is twice differentiable at c, we mean\nsecond order derivative of f exists at c Example 19 Find local minimum value of the function f\ngiven by  f (x) = 3 + |x|, x \u2208 R Solution Note that the given function is not differentiable\nat x  = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "2894-2897",
        "Text": "ANote  As f is twice differentiable at c, we mean\nsecond order derivative of f exists at c Example 19 Find local minimum value of the function f\ngiven by  f (x) = 3 + |x|, x \u2208 R Solution Note that the given function is not differentiable\nat x  = 0 So, second derivative test fails"
    },
    {
        "Chapter": "1",
        "sentence_range": "2895-2898",
        "Text": "Example 19 Find local minimum value of the function f\ngiven by  f (x) = 3 + |x|, x \u2208 R Solution Note that the given function is not differentiable\nat x  = 0 So, second derivative test fails Let us try first\nderivative test"
    },
    {
        "Chapter": "1",
        "sentence_range": "2896-2899",
        "Text": "Solution Note that the given function is not differentiable\nat x  = 0 So, second derivative test fails Let us try first\nderivative test Note that 0 is a critical point of f"
    },
    {
        "Chapter": "1",
        "sentence_range": "2897-2900",
        "Text": "So, second derivative test fails Let us try first\nderivative test Note that 0 is a critical point of f Now to\nthe left of 0, f (x) = 3 \u2013 x and so f \u2032(x) = \u2013 1 < 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "2898-2901",
        "Text": "Let us try first\nderivative test Note that 0 is a critical point of f Now to\nthe left of 0, f (x) = 3 \u2013 x and so f \u2032(x) = \u2013 1 < 0 Also to\nRationalised 2023-24\nAPPLICATION OF DERIVATIVES\n167\nthe right of 0, f (x) = 3 + x and so  f \u2032(x) = 1 > 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "2899-2902",
        "Text": "Note that 0 is a critical point of f Now to\nthe left of 0, f (x) = 3 \u2013 x and so f \u2032(x) = \u2013 1 < 0 Also to\nRationalised 2023-24\nAPPLICATION OF DERIVATIVES\n167\nthe right of 0, f (x) = 3 + x and so  f \u2032(x) = 1 > 0 Therefore, by first derivative test, x =\n0 is a point of  local minima of  f  and local minimum value of  f  is f (0) = 3"
    },
    {
        "Chapter": "1",
        "sentence_range": "2900-2903",
        "Text": "Now to\nthe left of 0, f (x) = 3 \u2013 x and so f \u2032(x) = \u2013 1 < 0 Also to\nRationalised 2023-24\nAPPLICATION OF DERIVATIVES\n167\nthe right of 0, f (x) = 3 + x and so  f \u2032(x) = 1 > 0 Therefore, by first derivative test, x =\n0 is a point of  local minima of  f  and local minimum value of  f  is f (0) = 3 Example 20 Find local maximum and local minimum values of the function  f  given by\nf (x) = 3x4 + 4x3 \u2013 12x2 + 12\nSolution We have\nf (x) = 3x4 + 4x3 \u2013 12x2 + 12\nor\nf \u2032(x) = 12x3 + 12x2 \u2013 24x = 12x (x \u2013 1) (x + 2)\nor\nf \u2032(x) = 0 at x = 0, x = 1 and x = \u2013 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "2901-2904",
        "Text": "Also to\nRationalised 2023-24\nAPPLICATION OF DERIVATIVES\n167\nthe right of 0, f (x) = 3 + x and so  f \u2032(x) = 1 > 0 Therefore, by first derivative test, x =\n0 is a point of  local minima of  f  and local minimum value of  f  is f (0) = 3 Example 20 Find local maximum and local minimum values of the function  f  given by\nf (x) = 3x4 + 4x3 \u2013 12x2 + 12\nSolution We have\nf (x) = 3x4 + 4x3 \u2013 12x2 + 12\nor\nf \u2032(x) = 12x3 + 12x2 \u2013 24x = 12x (x \u2013 1) (x + 2)\nor\nf \u2032(x) = 0 at x = 0, x = 1 and x = \u2013 2 Now\nf \u2033(x) = 36x2 + 24x \u2013 24 = 12 (3x2 + 2x \u2013 2)\nor\n\u2032\u2032\n= \u2212\n<\n\u2032\u2032\n=\n>\n\u2032\u2032 \u2212\n=\n>\n\uf8f1\n\uf8f2\uf8f4\n\uf8f3\uf8f4\nf\nf\nf\n( )\n( )\n(\n)\n0\n24\n0\n1\n36\n0\n2\n72\n0\nTherefore, by second derivative test, x  = 0 is a point of local maxima and local\nmaximum value of f at x = 0 is f (0) = 12 while x = 1 and x = \u2013 2 are the points of local\nminima and local minimum values of f at x = \u2013 1 and \u2013 2 are f (1) = 7 and f (\u20132) = \u201320,\nrespectively"
    },
    {
        "Chapter": "1",
        "sentence_range": "2902-2905",
        "Text": "Therefore, by first derivative test, x =\n0 is a point of  local minima of  f  and local minimum value of  f  is f (0) = 3 Example 20 Find local maximum and local minimum values of the function  f  given by\nf (x) = 3x4 + 4x3 \u2013 12x2 + 12\nSolution We have\nf (x) = 3x4 + 4x3 \u2013 12x2 + 12\nor\nf \u2032(x) = 12x3 + 12x2 \u2013 24x = 12x (x \u2013 1) (x + 2)\nor\nf \u2032(x) = 0 at x = 0, x = 1 and x = \u2013 2 Now\nf \u2033(x) = 36x2 + 24x \u2013 24 = 12 (3x2 + 2x \u2013 2)\nor\n\u2032\u2032\n= \u2212\n<\n\u2032\u2032\n=\n>\n\u2032\u2032 \u2212\n=\n>\n\uf8f1\n\uf8f2\uf8f4\n\uf8f3\uf8f4\nf\nf\nf\n( )\n( )\n(\n)\n0\n24\n0\n1\n36\n0\n2\n72\n0\nTherefore, by second derivative test, x  = 0 is a point of local maxima and local\nmaximum value of f at x = 0 is f (0) = 12 while x = 1 and x = \u2013 2 are the points of local\nminima and local minimum values of f at x = \u2013 1 and \u2013 2 are f (1) = 7 and f (\u20132) = \u201320,\nrespectively Example 21 Find all the points of local maxima and local minima of the function f\ngiven by\nf(x) = 2x3 \u2013 6x2 + 6x +5"
    },
    {
        "Chapter": "1",
        "sentence_range": "2903-2906",
        "Text": "Example 20 Find local maximum and local minimum values of the function  f  given by\nf (x) = 3x4 + 4x3 \u2013 12x2 + 12\nSolution We have\nf (x) = 3x4 + 4x3 \u2013 12x2 + 12\nor\nf \u2032(x) = 12x3 + 12x2 \u2013 24x = 12x (x \u2013 1) (x + 2)\nor\nf \u2032(x) = 0 at x = 0, x = 1 and x = \u2013 2 Now\nf \u2033(x) = 36x2 + 24x \u2013 24 = 12 (3x2 + 2x \u2013 2)\nor\n\u2032\u2032\n= \u2212\n<\n\u2032\u2032\n=\n>\n\u2032\u2032 \u2212\n=\n>\n\uf8f1\n\uf8f2\uf8f4\n\uf8f3\uf8f4\nf\nf\nf\n( )\n( )\n(\n)\n0\n24\n0\n1\n36\n0\n2\n72\n0\nTherefore, by second derivative test, x  = 0 is a point of local maxima and local\nmaximum value of f at x = 0 is f (0) = 12 while x = 1 and x = \u2013 2 are the points of local\nminima and local minimum values of f at x = \u2013 1 and \u2013 2 are f (1) = 7 and f (\u20132) = \u201320,\nrespectively Example 21 Find all the points of local maxima and local minima of the function f\ngiven by\nf(x) = 2x3 \u2013 6x2 + 6x +5 Solution We have\nf(x) = 2x3 \u2013 6x2 + 6x +5\nor\n2\n2\n( )\n6\n12\n6\n6(\n1)\n( )\n12(\n1)\nf\nx\nx\nx\nx\nf\nx\nx\n\uf8f1 \u2032\n=\n\u2212\n+\n=\n\u2212\n\uf8f4\uf8f2 \u2032\u2032\n=\n\u2212\n\uf8f4\uf8f3\nNow f \u2032(x) = 0 gives x =1"
    },
    {
        "Chapter": "1",
        "sentence_range": "2904-2907",
        "Text": "Now\nf \u2033(x) = 36x2 + 24x \u2013 24 = 12 (3x2 + 2x \u2013 2)\nor\n\u2032\u2032\n= \u2212\n<\n\u2032\u2032\n=\n>\n\u2032\u2032 \u2212\n=\n>\n\uf8f1\n\uf8f2\uf8f4\n\uf8f3\uf8f4\nf\nf\nf\n( )\n( )\n(\n)\n0\n24\n0\n1\n36\n0\n2\n72\n0\nTherefore, by second derivative test, x  = 0 is a point of local maxima and local\nmaximum value of f at x = 0 is f (0) = 12 while x = 1 and x = \u2013 2 are the points of local\nminima and local minimum values of f at x = \u2013 1 and \u2013 2 are f (1) = 7 and f (\u20132) = \u201320,\nrespectively Example 21 Find all the points of local maxima and local minima of the function f\ngiven by\nf(x) = 2x3 \u2013 6x2 + 6x +5 Solution We have\nf(x) = 2x3 \u2013 6x2 + 6x +5\nor\n2\n2\n( )\n6\n12\n6\n6(\n1)\n( )\n12(\n1)\nf\nx\nx\nx\nx\nf\nx\nx\n\uf8f1 \u2032\n=\n\u2212\n+\n=\n\u2212\n\uf8f4\uf8f2 \u2032\u2032\n=\n\u2212\n\uf8f4\uf8f3\nNow f \u2032(x) = 0 gives x =1 Also f \u2033(1) = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "2905-2908",
        "Text": "Example 21 Find all the points of local maxima and local minima of the function f\ngiven by\nf(x) = 2x3 \u2013 6x2 + 6x +5 Solution We have\nf(x) = 2x3 \u2013 6x2 + 6x +5\nor\n2\n2\n( )\n6\n12\n6\n6(\n1)\n( )\n12(\n1)\nf\nx\nx\nx\nx\nf\nx\nx\n\uf8f1 \u2032\n=\n\u2212\n+\n=\n\u2212\n\uf8f4\uf8f2 \u2032\u2032\n=\n\u2212\n\uf8f4\uf8f3\nNow f \u2032(x) = 0 gives x =1 Also f \u2033(1) = 0 Therefore, the second derivative test\nfails in this case"
    },
    {
        "Chapter": "1",
        "sentence_range": "2906-2909",
        "Text": "Solution We have\nf(x) = 2x3 \u2013 6x2 + 6x +5\nor\n2\n2\n( )\n6\n12\n6\n6(\n1)\n( )\n12(\n1)\nf\nx\nx\nx\nx\nf\nx\nx\n\uf8f1 \u2032\n=\n\u2212\n+\n=\n\u2212\n\uf8f4\uf8f2 \u2032\u2032\n=\n\u2212\n\uf8f4\uf8f3\nNow f \u2032(x) = 0 gives x =1 Also f \u2033(1) = 0 Therefore, the second derivative test\nfails in this case So, we shall go back to the first derivative test"
    },
    {
        "Chapter": "1",
        "sentence_range": "2907-2910",
        "Text": "Also f \u2033(1) = 0 Therefore, the second derivative test\nfails in this case So, we shall go back to the first derivative test We have already seen (Example 18) that, using first derivative test, x =1 is neither\na point of local maxima nor a point of local minima and so it is a point of inflexion"
    },
    {
        "Chapter": "1",
        "sentence_range": "2908-2911",
        "Text": "Therefore, the second derivative test\nfails in this case So, we shall go back to the first derivative test We have already seen (Example 18) that, using first derivative test, x =1 is neither\na point of local maxima nor a point of local minima and so it is a point of inflexion Example 22 Find two positive numbers whose sum is 15 and the sum of whose\nsquares is minimum"
    },
    {
        "Chapter": "1",
        "sentence_range": "2909-2912",
        "Text": "So, we shall go back to the first derivative test We have already seen (Example 18) that, using first derivative test, x =1 is neither\na point of local maxima nor a point of local minima and so it is a point of inflexion Example 22 Find two positive numbers whose sum is 15 and the sum of whose\nsquares is minimum Solution Let one of the numbers be x"
    },
    {
        "Chapter": "1",
        "sentence_range": "2910-2913",
        "Text": "We have already seen (Example 18) that, using first derivative test, x =1 is neither\na point of local maxima nor a point of local minima and so it is a point of inflexion Example 22 Find two positive numbers whose sum is 15 and the sum of whose\nsquares is minimum Solution Let one of the numbers be x Then the other number is (15 \u2013 x)"
    },
    {
        "Chapter": "1",
        "sentence_range": "2911-2914",
        "Text": "Example 22 Find two positive numbers whose sum is 15 and the sum of whose\nsquares is minimum Solution Let one of the numbers be x Then the other number is (15 \u2013 x) Let S(x)\ndenote the sum of the squares of these numbers"
    },
    {
        "Chapter": "1",
        "sentence_range": "2912-2915",
        "Text": "Solution Let one of the numbers be x Then the other number is (15 \u2013 x) Let S(x)\ndenote the sum of the squares of these numbers Then\nRationalised 2023-24\n MATHEMATICS\n168\nS(x) = x2 + (15 \u2013 x)2 = 2x2 \u2013 30x + 225\nor\nS ( )\n4\n30\nS ( )\n4\nx\nx\nx\n\u2032\n=\n\u2212\n\uf8f1\n\uf8f2 \u2032\u2032\n=\n\uf8f3\nNow S\u2032(x) = 0 gives \nx =215"
    },
    {
        "Chapter": "1",
        "sentence_range": "2913-2916",
        "Text": "Then the other number is (15 \u2013 x) Let S(x)\ndenote the sum of the squares of these numbers Then\nRationalised 2023-24\n MATHEMATICS\n168\nS(x) = x2 + (15 \u2013 x)2 = 2x2 \u2013 30x + 225\nor\nS ( )\n4\n30\nS ( )\n4\nx\nx\nx\n\u2032\n=\n\u2212\n\uf8f1\n\uf8f2 \u2032\u2032\n=\n\uf8f3\nNow S\u2032(x) = 0 gives \nx =215 Also \n15\nS\n4\n0\n\uf8eb2\n\uf8f6\n\u2032\u2032\n=\n>\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8"
    },
    {
        "Chapter": "1",
        "sentence_range": "2914-2917",
        "Text": "Let S(x)\ndenote the sum of the squares of these numbers Then\nRationalised 2023-24\n MATHEMATICS\n168\nS(x) = x2 + (15 \u2013 x)2 = 2x2 \u2013 30x + 225\nor\nS ( )\n4\n30\nS ( )\n4\nx\nx\nx\n\u2032\n=\n\u2212\n\uf8f1\n\uf8f2 \u2032\u2032\n=\n\uf8f3\nNow S\u2032(x) = 0 gives \nx =215 Also \n15\nS\n4\n0\n\uf8eb2\n\uf8f6\n\u2032\u2032\n=\n>\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8 Therefore, by second derivative\ntest, \nx =215\n is the point of local minima of S"
    },
    {
        "Chapter": "1",
        "sentence_range": "2915-2918",
        "Text": "Then\nRationalised 2023-24\n MATHEMATICS\n168\nS(x) = x2 + (15 \u2013 x)2 = 2x2 \u2013 30x + 225\nor\nS ( )\n4\n30\nS ( )\n4\nx\nx\nx\n\u2032\n=\n\u2212\n\uf8f1\n\uf8f2 \u2032\u2032\n=\n\uf8f3\nNow S\u2032(x) = 0 gives \nx =215 Also \n15\nS\n4\n0\n\uf8eb2\n\uf8f6\n\u2032\u2032\n=\n>\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8 Therefore, by second derivative\ntest, \nx =215\n is the point of local minima of S Hence the sum of squares of numbers is\nminimum when the numbers are 15\n2  and \n15\n15\n15\n2\n2\n\u2212\n="
    },
    {
        "Chapter": "1",
        "sentence_range": "2916-2919",
        "Text": "Also \n15\nS\n4\n0\n\uf8eb2\n\uf8f6\n\u2032\u2032\n=\n>\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8 Therefore, by second derivative\ntest, \nx =215\n is the point of local minima of S Hence the sum of squares of numbers is\nminimum when the numbers are 15\n2  and \n15\n15\n15\n2\n2\n\u2212\n= Remark Proceeding as in Example 34 one may prove that the two positive numbers,\nwhose sum is k and the sum of whose squares is minimum, are \n2and\n2\nk\nk"
    },
    {
        "Chapter": "1",
        "sentence_range": "2917-2920",
        "Text": "Therefore, by second derivative\ntest, \nx =215\n is the point of local minima of S Hence the sum of squares of numbers is\nminimum when the numbers are 15\n2  and \n15\n15\n15\n2\n2\n\u2212\n= Remark Proceeding as in Example 34 one may prove that the two positive numbers,\nwhose sum is k and the sum of whose squares is minimum, are \n2and\n2\nk\nk Example 23 Find the shortest distance of the point (0, c) from the parabola y = x2,\nwhere 1\n2  \u2264 c \u2264 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "2918-2921",
        "Text": "Hence the sum of squares of numbers is\nminimum when the numbers are 15\n2  and \n15\n15\n15\n2\n2\n\u2212\n= Remark Proceeding as in Example 34 one may prove that the two positive numbers,\nwhose sum is k and the sum of whose squares is minimum, are \n2and\n2\nk\nk Example 23 Find the shortest distance of the point (0, c) from the parabola y = x2,\nwhere 1\n2  \u2264 c \u2264 5 Solution Let (h, k) be any point on the parabola y = x2"
    },
    {
        "Chapter": "1",
        "sentence_range": "2919-2922",
        "Text": "Remark Proceeding as in Example 34 one may prove that the two positive numbers,\nwhose sum is k and the sum of whose squares is minimum, are \n2and\n2\nk\nk Example 23 Find the shortest distance of the point (0, c) from the parabola y = x2,\nwhere 1\n2  \u2264 c \u2264 5 Solution Let (h, k) be any point on the parabola y = x2 Let D be the required distance\nbetween (h, k) and (0, c)"
    },
    {
        "Chapter": "1",
        "sentence_range": "2920-2923",
        "Text": "Example 23 Find the shortest distance of the point (0, c) from the parabola y = x2,\nwhere 1\n2  \u2264 c \u2264 5 Solution Let (h, k) be any point on the parabola y = x2 Let D be the required distance\nbetween (h, k) and (0, c) Then\n                      \n2\n2\n2\n2\nD\n(\n0)\n(\n)\n(\n)\nh\nk\nc\nh\nk\nc\n=\n\u2212\n+\n\u2212\n=\n+\n\u2212"
    },
    {
        "Chapter": "1",
        "sentence_range": "2921-2924",
        "Text": "Solution Let (h, k) be any point on the parabola y = x2 Let D be the required distance\nbetween (h, k) and (0, c) Then\n                      \n2\n2\n2\n2\nD\n(\n0)\n(\n)\n(\n)\nh\nk\nc\nh\nk\nc\n=\n\u2212\n+\n\u2212\n=\n+\n\u2212 (1)\nSince (h, k) lies on the parabola y = x2, we have k = h2"
    },
    {
        "Chapter": "1",
        "sentence_range": "2922-2925",
        "Text": "Let D be the required distance\nbetween (h, k) and (0, c) Then\n                      \n2\n2\n2\n2\nD\n(\n0)\n(\n)\n(\n)\nh\nk\nc\nh\nk\nc\n=\n\u2212\n+\n\u2212\n=\n+\n\u2212 (1)\nSince (h, k) lies on the parabola y = x2, we have k = h2 So (1) gives\nD \u2261 D(k) = \n2\n(\n)\nk\nk\nc\n+\n\u2212\nor\nD\u2032(k) =\n2\n1\n2(\n)\n2\n(\n)\nk\nc\nk\nk\nc\n+\n\u2212\n+\n\u2212\nNow\nD\u2032(k) = 0 gives \n2\n1\nc2\nk\n\u2212\n=\nObserve that when \n2\n1\nc2\nk\n\u2212\n<\n, then 2(\n)\n1\n0\nk\n\u2212c\n+ <\n, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "2923-2926",
        "Text": "Then\n                      \n2\n2\n2\n2\nD\n(\n0)\n(\n)\n(\n)\nh\nk\nc\nh\nk\nc\n=\n\u2212\n+\n\u2212\n=\n+\n\u2212 (1)\nSince (h, k) lies on the parabola y = x2, we have k = h2 So (1) gives\nD \u2261 D(k) = \n2\n(\n)\nk\nk\nc\n+\n\u2212\nor\nD\u2032(k) =\n2\n1\n2(\n)\n2\n(\n)\nk\nc\nk\nk\nc\n+\n\u2212\n+\n\u2212\nNow\nD\u2032(k) = 0 gives \n2\n1\nc2\nk\n\u2212\n=\nObserve that when \n2\n1\nc2\nk\n\u2212\n<\n, then 2(\n)\n1\n0\nk\n\u2212c\n+ <\n, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "2924-2927",
        "Text": "(1)\nSince (h, k) lies on the parabola y = x2, we have k = h2 So (1) gives\nD \u2261 D(k) = \n2\n(\n)\nk\nk\nc\n+\n\u2212\nor\nD\u2032(k) =\n2\n1\n2(\n)\n2\n(\n)\nk\nc\nk\nk\nc\n+\n\u2212\n+\n\u2212\nNow\nD\u2032(k) = 0 gives \n2\n1\nc2\nk\n\u2212\n=\nObserve that when \n2\n1\nc2\nk\n\u2212\n<\n, then 2(\n)\n1\n0\nk\n\u2212c\n+ <\n, i e , D ( )\n0\n\u2032k\n<"
    },
    {
        "Chapter": "1",
        "sentence_range": "2925-2928",
        "Text": "So (1) gives\nD \u2261 D(k) = \n2\n(\n)\nk\nk\nc\n+\n\u2212\nor\nD\u2032(k) =\n2\n1\n2(\n)\n2\n(\n)\nk\nc\nk\nk\nc\n+\n\u2212\n+\n\u2212\nNow\nD\u2032(k) = 0 gives \n2\n1\nc2\nk\n\u2212\n=\nObserve that when \n2\n1\nc2\nk\n\u2212\n<\n, then 2(\n)\n1\n0\nk\n\u2212c\n+ <\n, i e , D ( )\n0\n\u2032k\n< Also when\n2\n1\nc2\nk\n\u2212\n>\n, then D ( )\n0\n\u2032k\n>"
    },
    {
        "Chapter": "1",
        "sentence_range": "2926-2929",
        "Text": "e , D ( )\n0\n\u2032k\n< Also when\n2\n1\nc2\nk\n\u2212\n>\n, then D ( )\n0\n\u2032k\n> So, by first derivative test, D (k) is minimum at \n2\n1\nc2\nk\n\u2212\n="
    },
    {
        "Chapter": "1",
        "sentence_range": "2927-2930",
        "Text": ", D ( )\n0\n\u2032k\n< Also when\n2\n1\nc2\nk\n\u2212\n>\n, then D ( )\n0\n\u2032k\n> So, by first derivative test, D (k) is minimum at \n2\n1\nc2\nk\n\u2212\n= Rationalised 2023-24\nAPPLICATION OF DERIVATIVES\n169\nHence, the required shortest distance is given by\n2\n2\n1\n2\n1\n2\n1\n4\n1\nD\n2\n2\n2\n2\nc\nc\nc\nc\nc\n\u2212\n\u2212\n\u2212\n\u2212\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n=\n+\n\u2212\n=\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\nANote  The reader may note that in Example 35, we have used first derivative\ntest instead of the second derivative test as the former is easy and short"
    },
    {
        "Chapter": "1",
        "sentence_range": "2928-2931",
        "Text": "Also when\n2\n1\nc2\nk\n\u2212\n>\n, then D ( )\n0\n\u2032k\n> So, by first derivative test, D (k) is minimum at \n2\n1\nc2\nk\n\u2212\n= Rationalised 2023-24\nAPPLICATION OF DERIVATIVES\n169\nHence, the required shortest distance is given by\n2\n2\n1\n2\n1\n2\n1\n4\n1\nD\n2\n2\n2\n2\nc\nc\nc\nc\nc\n\u2212\n\u2212\n\u2212\n\u2212\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n=\n+\n\u2212\n=\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\nANote  The reader may note that in Example 35, we have used first derivative\ntest instead of the second derivative test as the former is easy and short Example 24 Let AP and BQ be two vertical poles at\npoints A and B, respectively"
    },
    {
        "Chapter": "1",
        "sentence_range": "2929-2932",
        "Text": "So, by first derivative test, D (k) is minimum at \n2\n1\nc2\nk\n\u2212\n= Rationalised 2023-24\nAPPLICATION OF DERIVATIVES\n169\nHence, the required shortest distance is given by\n2\n2\n1\n2\n1\n2\n1\n4\n1\nD\n2\n2\n2\n2\nc\nc\nc\nc\nc\n\u2212\n\u2212\n\u2212\n\u2212\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n=\n+\n\u2212\n=\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\nANote  The reader may note that in Example 35, we have used first derivative\ntest instead of the second derivative test as the former is easy and short Example 24 Let AP and BQ be two vertical poles at\npoints A and B, respectively If AP = 16 m, BQ = 22 m\nand AB = 20 m, then find the distance of a point R on\nAB from the point A such that RP2 + RQ2 is minimum"
    },
    {
        "Chapter": "1",
        "sentence_range": "2930-2933",
        "Text": "Rationalised 2023-24\nAPPLICATION OF DERIVATIVES\n169\nHence, the required shortest distance is given by\n2\n2\n1\n2\n1\n2\n1\n4\n1\nD\n2\n2\n2\n2\nc\nc\nc\nc\nc\n\u2212\n\u2212\n\u2212\n\u2212\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n=\n+\n\u2212\n=\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\nANote  The reader may note that in Example 35, we have used first derivative\ntest instead of the second derivative test as the former is easy and short Example 24 Let AP and BQ be two vertical poles at\npoints A and B, respectively If AP = 16 m, BQ = 22 m\nand AB = 20 m, then find the distance of a point R on\nAB from the point A such that RP2 + RQ2 is minimum Solution Let R be a point on AB such that AR = x m"
    },
    {
        "Chapter": "1",
        "sentence_range": "2931-2934",
        "Text": "Example 24 Let AP and BQ be two vertical poles at\npoints A and B, respectively If AP = 16 m, BQ = 22 m\nand AB = 20 m, then find the distance of a point R on\nAB from the point A such that RP2 + RQ2 is minimum Solution Let R be a point on AB such that AR = x m Then RB = (20 \u2013 x) m (as AB = 20 m)"
    },
    {
        "Chapter": "1",
        "sentence_range": "2932-2935",
        "Text": "If AP = 16 m, BQ = 22 m\nand AB = 20 m, then find the distance of a point R on\nAB from the point A such that RP2 + RQ2 is minimum Solution Let R be a point on AB such that AR = x m Then RB = (20 \u2013 x) m (as AB = 20 m) From Fig 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "2933-2936",
        "Text": "Solution Let R be a point on AB such that AR = x m Then RB = (20 \u2013 x) m (as AB = 20 m) From Fig 6 16,\nwe have\nRP2 = AR2 + AP2\nand\nRQ2 = RB2 + BQ2\nTherefore\nRP2 + RQ2 = AR2 + AP2 + RB2 + BQ2\n= x2 + (16)2 + (20 \u2013 x)2 + (22)2\n= 2x2 \u2013 40x + 1140\nLet\nS \u2261 S(x) = RP2 + RQ2 = 2x2 \u2013 40x + 1140"
    },
    {
        "Chapter": "1",
        "sentence_range": "2934-2937",
        "Text": "Then RB = (20 \u2013 x) m (as AB = 20 m) From Fig 6 16,\nwe have\nRP2 = AR2 + AP2\nand\nRQ2 = RB2 + BQ2\nTherefore\nRP2 + RQ2 = AR2 + AP2 + RB2 + BQ2\n= x2 + (16)2 + (20 \u2013 x)2 + (22)2\n= 2x2 \u2013 40x + 1140\nLet\nS \u2261 S(x) = RP2 + RQ2 = 2x2 \u2013 40x + 1140 Therefore\nS\u2032(x) = 4x \u2013 40"
    },
    {
        "Chapter": "1",
        "sentence_range": "2935-2938",
        "Text": "From Fig 6 16,\nwe have\nRP2 = AR2 + AP2\nand\nRQ2 = RB2 + BQ2\nTherefore\nRP2 + RQ2 = AR2 + AP2 + RB2 + BQ2\n= x2 + (16)2 + (20 \u2013 x)2 + (22)2\n= 2x2 \u2013 40x + 1140\nLet\nS \u2261 S(x) = RP2 + RQ2 = 2x2 \u2013 40x + 1140 Therefore\nS\u2032(x) = 4x \u2013 40 Now S\u2032(x) = 0 gives x = 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "2936-2939",
        "Text": "16,\nwe have\nRP2 = AR2 + AP2\nand\nRQ2 = RB2 + BQ2\nTherefore\nRP2 + RQ2 = AR2 + AP2 + RB2 + BQ2\n= x2 + (16)2 + (20 \u2013 x)2 + (22)2\n= 2x2 \u2013 40x + 1140\nLet\nS \u2261 S(x) = RP2 + RQ2 = 2x2 \u2013 40x + 1140 Therefore\nS\u2032(x) = 4x \u2013 40 Now S\u2032(x) = 0 gives x = 10 Also S\u2033(x) = 4 > 0, for all x and so S\u2033(10) > 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "2937-2940",
        "Text": "Therefore\nS\u2032(x) = 4x \u2013 40 Now S\u2032(x) = 0 gives x = 10 Also S\u2033(x) = 4 > 0, for all x and so S\u2033(10) > 0 Therefore, by second derivative test, x = 10 is the point of local minima of S"
    },
    {
        "Chapter": "1",
        "sentence_range": "2938-2941",
        "Text": "Now S\u2032(x) = 0 gives x = 10 Also S\u2033(x) = 4 > 0, for all x and so S\u2033(10) > 0 Therefore, by second derivative test, x = 10 is the point of local minima of S Thus, the\ndistance of R from A on AB is AR = x =10 m"
    },
    {
        "Chapter": "1",
        "sentence_range": "2939-2942",
        "Text": "Also S\u2033(x) = 4 > 0, for all x and so S\u2033(10) > 0 Therefore, by second derivative test, x = 10 is the point of local minima of S Thus, the\ndistance of R from A on AB is AR = x =10 m Example 25 If length of three sides of a trapezium other than base are equal to 10cm,\nthen find the area of the trapezium when it is maximum"
    },
    {
        "Chapter": "1",
        "sentence_range": "2940-2943",
        "Text": "Therefore, by second derivative test, x = 10 is the point of local minima of S Thus, the\ndistance of R from A on AB is AR = x =10 m Example 25 If length of three sides of a trapezium other than base are equal to 10cm,\nthen find the area of the trapezium when it is maximum Solution The required trapezium is as given in Fig 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "2941-2944",
        "Text": "Thus, the\ndistance of R from A on AB is AR = x =10 m Example 25 If length of three sides of a trapezium other than base are equal to 10cm,\nthen find the area of the trapezium when it is maximum Solution The required trapezium is as given in Fig 6 17"
    },
    {
        "Chapter": "1",
        "sentence_range": "2942-2945",
        "Text": "Example 25 If length of three sides of a trapezium other than base are equal to 10cm,\nthen find the area of the trapezium when it is maximum Solution The required trapezium is as given in Fig 6 17 Draw perpendiculars DP and\nFig 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "2943-2946",
        "Text": "Solution The required trapezium is as given in Fig 6 17 Draw perpendiculars DP and\nFig 6 16\nFig 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "2944-2947",
        "Text": "17 Draw perpendiculars DP and\nFig 6 16\nFig 6 17\nRationalised 2023-24\n MATHEMATICS\n170\nCQ on AB"
    },
    {
        "Chapter": "1",
        "sentence_range": "2945-2948",
        "Text": "Draw perpendiculars DP and\nFig 6 16\nFig 6 17\nRationalised 2023-24\n MATHEMATICS\n170\nCQ on AB Let AP = x cm"
    },
    {
        "Chapter": "1",
        "sentence_range": "2946-2949",
        "Text": "16\nFig 6 17\nRationalised 2023-24\n MATHEMATICS\n170\nCQ on AB Let AP = x cm Note that \u2206APD ~ \u2206BQC"
    },
    {
        "Chapter": "1",
        "sentence_range": "2947-2950",
        "Text": "17\nRationalised 2023-24\n MATHEMATICS\n170\nCQ on AB Let AP = x cm Note that \u2206APD ~ \u2206BQC Therefore, QB = x cm"
    },
    {
        "Chapter": "1",
        "sentence_range": "2948-2951",
        "Text": "Let AP = x cm Note that \u2206APD ~ \u2206BQC Therefore, QB = x cm Also, by\nPythagoras theorem, DP = QC = \n2\n100\n\u2212x"
    },
    {
        "Chapter": "1",
        "sentence_range": "2949-2952",
        "Text": "Note that \u2206APD ~ \u2206BQC Therefore, QB = x cm Also, by\nPythagoras theorem, DP = QC = \n2\n100\n\u2212x Let A be the area of the trapezium"
    },
    {
        "Chapter": "1",
        "sentence_range": "2950-2953",
        "Text": "Therefore, QB = x cm Also, by\nPythagoras theorem, DP = QC = \n2\n100\n\u2212x Let A be the area of the trapezium Then\nA \u2261 A(x) = 1\n2 (sum of parallel sides) (height)\n=\n(\n2)\n1 (2\n10 10)\n100\n2\nx\nx\n+\n+\n\u2212\n=\n(\n2)\n(\n10)\n100\nx\nx\n+\n\u2212\nor\nA\u2032(x) =\n(\n)\n2\n( 2 )2\n(\n10)\n100\n2 100\nx\nx\nx\nx\n\u2212\n+\n+\n\u2212\n\u2212\n=\n2\n2\n2\n10\n100\n100\nx\nx\nx\n\u2212\n\u2212\n+\n\u2212\nNow\nA\u2032(x) = 0 gives 2x2 + 10x \u2013 100 = 0, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "2951-2954",
        "Text": "Also, by\nPythagoras theorem, DP = QC = \n2\n100\n\u2212x Let A be the area of the trapezium Then\nA \u2261 A(x) = 1\n2 (sum of parallel sides) (height)\n=\n(\n2)\n1 (2\n10 10)\n100\n2\nx\nx\n+\n+\n\u2212\n=\n(\n2)\n(\n10)\n100\nx\nx\n+\n\u2212\nor\nA\u2032(x) =\n(\n)\n2\n( 2 )2\n(\n10)\n100\n2 100\nx\nx\nx\nx\n\u2212\n+\n+\n\u2212\n\u2212\n=\n2\n2\n2\n10\n100\n100\nx\nx\nx\n\u2212\n\u2212\n+\n\u2212\nNow\nA\u2032(x) = 0 gives 2x2 + 10x \u2013 100 = 0, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "2952-2955",
        "Text": "Let A be the area of the trapezium Then\nA \u2261 A(x) = 1\n2 (sum of parallel sides) (height)\n=\n(\n2)\n1 (2\n10 10)\n100\n2\nx\nx\n+\n+\n\u2212\n=\n(\n2)\n(\n10)\n100\nx\nx\n+\n\u2212\nor\nA\u2032(x) =\n(\n)\n2\n( 2 )2\n(\n10)\n100\n2 100\nx\nx\nx\nx\n\u2212\n+\n+\n\u2212\n\u2212\n=\n2\n2\n2\n10\n100\n100\nx\nx\nx\n\u2212\n\u2212\n+\n\u2212\nNow\nA\u2032(x) = 0 gives 2x2 + 10x \u2013 100 = 0, i e , x  = 5 and x = \u201310"
    },
    {
        "Chapter": "1",
        "sentence_range": "2953-2956",
        "Text": "Then\nA \u2261 A(x) = 1\n2 (sum of parallel sides) (height)\n=\n(\n2)\n1 (2\n10 10)\n100\n2\nx\nx\n+\n+\n\u2212\n=\n(\n2)\n(\n10)\n100\nx\nx\n+\n\u2212\nor\nA\u2032(x) =\n(\n)\n2\n( 2 )2\n(\n10)\n100\n2 100\nx\nx\nx\nx\n\u2212\n+\n+\n\u2212\n\u2212\n=\n2\n2\n2\n10\n100\n100\nx\nx\nx\n\u2212\n\u2212\n+\n\u2212\nNow\nA\u2032(x) = 0 gives 2x2 + 10x \u2013 100 = 0, i e , x  = 5 and x = \u201310 Since x represents distance, it can not be negative"
    },
    {
        "Chapter": "1",
        "sentence_range": "2954-2957",
        "Text": "e , x  = 5 and x = \u201310 Since x represents distance, it can not be negative So,\nx = 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "2955-2958",
        "Text": ", x  = 5 and x = \u201310 Since x represents distance, it can not be negative So,\nx = 5 Now\nA\u2033(x) =\n2\n2\n2\n2\n( 2 )\n100\n( 4\n10)\n( 2\n10\n100)\n2 100\n100\nx\nx\nx\nx\nx\nx\nx\n\u2212\n\u2212\n\u2212\n\u2212\n\u2212 \u2212\n\u2212\n+\n\u2212\n\u2212\n=\n3\n3\n2\n2\n2\n300\n1000\n(100\n)\nx\nx\nx\n\u2212\n\u2212\n\u2212\n (on simplification)\nor\nA\u2033(5) =\n3\n3\n2\n2\n2(5)\n300(5)\n1000\n2250\n30\n0\n75 75\n75\n(100\n(5) )\n\u2212\n\u2212\n\u2212\n\u2212\n=\n=\n<\n\u2212\nThus, area of trapezium is maximum at x = 5  and the area is given by\nA(5) =\n2\n2\n(5\n10) 100\n(5)\n15 75\n75 3 cm\n+\n\u2212\n=\n=\nExample 26 Prove that the radius of the right circular cylinder of greatest curved\nsurface area which can be inscribed in a given cone is half of that of the cone"
    },
    {
        "Chapter": "1",
        "sentence_range": "2956-2959",
        "Text": "Since x represents distance, it can not be negative So,\nx = 5 Now\nA\u2033(x) =\n2\n2\n2\n2\n( 2 )\n100\n( 4\n10)\n( 2\n10\n100)\n2 100\n100\nx\nx\nx\nx\nx\nx\nx\n\u2212\n\u2212\n\u2212\n\u2212\n\u2212 \u2212\n\u2212\n+\n\u2212\n\u2212\n=\n3\n3\n2\n2\n2\n300\n1000\n(100\n)\nx\nx\nx\n\u2212\n\u2212\n\u2212\n (on simplification)\nor\nA\u2033(5) =\n3\n3\n2\n2\n2(5)\n300(5)\n1000\n2250\n30\n0\n75 75\n75\n(100\n(5) )\n\u2212\n\u2212\n\u2212\n\u2212\n=\n=\n<\n\u2212\nThus, area of trapezium is maximum at x = 5  and the area is given by\nA(5) =\n2\n2\n(5\n10) 100\n(5)\n15 75\n75 3 cm\n+\n\u2212\n=\n=\nExample 26 Prove that the radius of the right circular cylinder of greatest curved\nsurface area which can be inscribed in a given cone is half of that of the cone Solution Let OC = r be the radius of the cone and OA = h be its height"
    },
    {
        "Chapter": "1",
        "sentence_range": "2957-2960",
        "Text": "So,\nx = 5 Now\nA\u2033(x) =\n2\n2\n2\n2\n( 2 )\n100\n( 4\n10)\n( 2\n10\n100)\n2 100\n100\nx\nx\nx\nx\nx\nx\nx\n\u2212\n\u2212\n\u2212\n\u2212\n\u2212 \u2212\n\u2212\n+\n\u2212\n\u2212\n=\n3\n3\n2\n2\n2\n300\n1000\n(100\n)\nx\nx\nx\n\u2212\n\u2212\n\u2212\n (on simplification)\nor\nA\u2033(5) =\n3\n3\n2\n2\n2(5)\n300(5)\n1000\n2250\n30\n0\n75 75\n75\n(100\n(5) )\n\u2212\n\u2212\n\u2212\n\u2212\n=\n=\n<\n\u2212\nThus, area of trapezium is maximum at x = 5  and the area is given by\nA(5) =\n2\n2\n(5\n10) 100\n(5)\n15 75\n75 3 cm\n+\n\u2212\n=\n=\nExample 26 Prove that the radius of the right circular cylinder of greatest curved\nsurface area which can be inscribed in a given cone is half of that of the cone Solution Let OC = r be the radius of the cone and OA = h be its height Let a cylinder\nwith radius OE = x inscribed in the given cone (Fig 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "2958-2961",
        "Text": "Now\nA\u2033(x) =\n2\n2\n2\n2\n( 2 )\n100\n( 4\n10)\n( 2\n10\n100)\n2 100\n100\nx\nx\nx\nx\nx\nx\nx\n\u2212\n\u2212\n\u2212\n\u2212\n\u2212 \u2212\n\u2212\n+\n\u2212\n\u2212\n=\n3\n3\n2\n2\n2\n300\n1000\n(100\n)\nx\nx\nx\n\u2212\n\u2212\n\u2212\n (on simplification)\nor\nA\u2033(5) =\n3\n3\n2\n2\n2(5)\n300(5)\n1000\n2250\n30\n0\n75 75\n75\n(100\n(5) )\n\u2212\n\u2212\n\u2212\n\u2212\n=\n=\n<\n\u2212\nThus, area of trapezium is maximum at x = 5  and the area is given by\nA(5) =\n2\n2\n(5\n10) 100\n(5)\n15 75\n75 3 cm\n+\n\u2212\n=\n=\nExample 26 Prove that the radius of the right circular cylinder of greatest curved\nsurface area which can be inscribed in a given cone is half of that of the cone Solution Let OC = r be the radius of the cone and OA = h be its height Let a cylinder\nwith radius OE = x inscribed in the given cone (Fig 6 18)"
    },
    {
        "Chapter": "1",
        "sentence_range": "2959-2962",
        "Text": "Solution Let OC = r be the radius of the cone and OA = h be its height Let a cylinder\nwith radius OE = x inscribed in the given cone (Fig 6 18) The height QE of the cylinder\nis given by\nRationalised 2023-24\nAPPLICATION OF DERIVATIVES\n171\nQE\nOA  = EC\nOC    (since \u2206QEC ~ \u2206AOC)\nor\nQE\nh  = r\nx\nr\n\u2212\nor\nQE =\n(\n)\nh r\nx\nr\n\u2212\nLet S be the curved surface area of the given\ncylinder"
    },
    {
        "Chapter": "1",
        "sentence_range": "2960-2963",
        "Text": "Let a cylinder\nwith radius OE = x inscribed in the given cone (Fig 6 18) The height QE of the cylinder\nis given by\nRationalised 2023-24\nAPPLICATION OF DERIVATIVES\n171\nQE\nOA  = EC\nOC    (since \u2206QEC ~ \u2206AOC)\nor\nQE\nh  = r\nx\nr\n\u2212\nor\nQE =\n(\n)\nh r\nx\nr\n\u2212\nLet S be the curved surface area of the given\ncylinder Then\nS  \u2261 S(x) = 2\n(\n)\nxh r\nx\nr\n\u03c0\n\u2212\n = \n2\n2\n(\n)\nh rx\nx\n\u03c0r\n\u2212\nor\n2\nS ( )\n(\n2 )\n4\nS ( )\nh\nx\nr\nx\nr\nh\nx\nr\n\u03c0\n\uf8f1 \u2032\n=\n\u2212\n\uf8f4\uf8f4\uf8f2\n\u2212 \u03c0\n\uf8f4 \u2032\u2032\n=\n\uf8f4\uf8f3\nNow S\u2032(x) = 0 gives \nx =r2"
    },
    {
        "Chapter": "1",
        "sentence_range": "2961-2964",
        "Text": "18) The height QE of the cylinder\nis given by\nRationalised 2023-24\nAPPLICATION OF DERIVATIVES\n171\nQE\nOA  = EC\nOC    (since \u2206QEC ~ \u2206AOC)\nor\nQE\nh  = r\nx\nr\n\u2212\nor\nQE =\n(\n)\nh r\nx\nr\n\u2212\nLet S be the curved surface area of the given\ncylinder Then\nS  \u2261 S(x) = 2\n(\n)\nxh r\nx\nr\n\u03c0\n\u2212\n = \n2\n2\n(\n)\nh rx\nx\n\u03c0r\n\u2212\nor\n2\nS ( )\n(\n2 )\n4\nS ( )\nh\nx\nr\nx\nr\nh\nx\nr\n\u03c0\n\uf8f1 \u2032\n=\n\u2212\n\uf8f4\uf8f4\uf8f2\n\u2212 \u03c0\n\uf8f4 \u2032\u2032\n=\n\uf8f4\uf8f3\nNow S\u2032(x) = 0 gives \nx =r2 Since S\u2033(x) < 0 for all x, S\n0\n\uf8ebr2\n\uf8f6\n\u2032\u2032\n<\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8"
    },
    {
        "Chapter": "1",
        "sentence_range": "2962-2965",
        "Text": "The height QE of the cylinder\nis given by\nRationalised 2023-24\nAPPLICATION OF DERIVATIVES\n171\nQE\nOA  = EC\nOC    (since \u2206QEC ~ \u2206AOC)\nor\nQE\nh  = r\nx\nr\n\u2212\nor\nQE =\n(\n)\nh r\nx\nr\n\u2212\nLet S be the curved surface area of the given\ncylinder Then\nS  \u2261 S(x) = 2\n(\n)\nxh r\nx\nr\n\u03c0\n\u2212\n = \n2\n2\n(\n)\nh rx\nx\n\u03c0r\n\u2212\nor\n2\nS ( )\n(\n2 )\n4\nS ( )\nh\nx\nr\nx\nr\nh\nx\nr\n\u03c0\n\uf8f1 \u2032\n=\n\u2212\n\uf8f4\uf8f4\uf8f2\n\u2212 \u03c0\n\uf8f4 \u2032\u2032\n=\n\uf8f4\uf8f3\nNow S\u2032(x) = 0 gives \nx =r2 Since S\u2033(x) < 0 for all x, S\n0\n\uf8ebr2\n\uf8f6\n\u2032\u2032\n<\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8 So \nx =r2\n is a\npoint of maxima of S"
    },
    {
        "Chapter": "1",
        "sentence_range": "2963-2966",
        "Text": "Then\nS  \u2261 S(x) = 2\n(\n)\nxh r\nx\nr\n\u03c0\n\u2212\n = \n2\n2\n(\n)\nh rx\nx\n\u03c0r\n\u2212\nor\n2\nS ( )\n(\n2 )\n4\nS ( )\nh\nx\nr\nx\nr\nh\nx\nr\n\u03c0\n\uf8f1 \u2032\n=\n\u2212\n\uf8f4\uf8f4\uf8f2\n\u2212 \u03c0\n\uf8f4 \u2032\u2032\n=\n\uf8f4\uf8f3\nNow S\u2032(x) = 0 gives \nx =r2 Since S\u2033(x) < 0 for all x, S\n0\n\uf8ebr2\n\uf8f6\n\u2032\u2032\n<\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8 So \nx =r2\n is a\npoint of maxima of S Hence, the radius of the cylinder of greatest curved surface area\nwhich can be inscribed in a given cone is half of that of the cone"
    },
    {
        "Chapter": "1",
        "sentence_range": "2964-2967",
        "Text": "Since S\u2033(x) < 0 for all x, S\n0\n\uf8ebr2\n\uf8f6\n\u2032\u2032\n<\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8 So \nx =r2\n is a\npoint of maxima of S Hence, the radius of the cylinder of greatest curved surface area\nwhich can be inscribed in a given cone is half of that of the cone 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "2965-2968",
        "Text": "So \nx =r2\n is a\npoint of maxima of S Hence, the radius of the cylinder of greatest curved surface area\nwhich can be inscribed in a given cone is half of that of the cone 6 4"
    },
    {
        "Chapter": "1",
        "sentence_range": "2966-2969",
        "Text": "Hence, the radius of the cylinder of greatest curved surface area\nwhich can be inscribed in a given cone is half of that of the cone 6 4 1  Maximum and Minimum Values of a Function in a Closed Interval\nLet us consider a function  f  given by\nf (x) = x + 2, x \u2208 (0, 1)\nObserve that the function is continuous on (0, 1) and neither has a maximum value\nnor has a minimum value"
    },
    {
        "Chapter": "1",
        "sentence_range": "2967-2970",
        "Text": "6 4 1  Maximum and Minimum Values of a Function in a Closed Interval\nLet us consider a function  f  given by\nf (x) = x + 2, x \u2208 (0, 1)\nObserve that the function is continuous on (0, 1) and neither has a maximum value\nnor has a minimum value Further, we may note that the function even has neither a\nlocal maximum value nor a local minimum value"
    },
    {
        "Chapter": "1",
        "sentence_range": "2968-2971",
        "Text": "4 1  Maximum and Minimum Values of a Function in a Closed Interval\nLet us consider a function  f  given by\nf (x) = x + 2, x \u2208 (0, 1)\nObserve that the function is continuous on (0, 1) and neither has a maximum value\nnor has a minimum value Further, we may note that the function even has neither a\nlocal maximum value nor a local minimum value However, if we extend the domain of  f  to the closed interval [0, 1], then f still may\nnot have a local maximum (minimum) values but it certainly does have maximum value\n3 = f (1) and minimum value 2 =  f (0)"
    },
    {
        "Chapter": "1",
        "sentence_range": "2969-2972",
        "Text": "1  Maximum and Minimum Values of a Function in a Closed Interval\nLet us consider a function  f  given by\nf (x) = x + 2, x \u2208 (0, 1)\nObserve that the function is continuous on (0, 1) and neither has a maximum value\nnor has a minimum value Further, we may note that the function even has neither a\nlocal maximum value nor a local minimum value However, if we extend the domain of  f  to the closed interval [0, 1], then f still may\nnot have a local maximum (minimum) values but it certainly does have maximum value\n3 = f (1) and minimum value 2 =  f (0) The maximum value 3 of f at x = 1 is called\nabsolute maximum value (global maximum or greatest value) of f on the interval\n[0, 1]"
    },
    {
        "Chapter": "1",
        "sentence_range": "2970-2973",
        "Text": "Further, we may note that the function even has neither a\nlocal maximum value nor a local minimum value However, if we extend the domain of  f  to the closed interval [0, 1], then f still may\nnot have a local maximum (minimum) values but it certainly does have maximum value\n3 = f (1) and minimum value 2 =  f (0) The maximum value 3 of f at x = 1 is called\nabsolute maximum value (global maximum or greatest value) of f on the interval\n[0, 1] Similarly, the minimum value 2 of f  at x  = 0 is called the absolute minimum\nvalue (global minimum or least value) of f on [0, 1]"
    },
    {
        "Chapter": "1",
        "sentence_range": "2971-2974",
        "Text": "However, if we extend the domain of  f  to the closed interval [0, 1], then f still may\nnot have a local maximum (minimum) values but it certainly does have maximum value\n3 = f (1) and minimum value 2 =  f (0) The maximum value 3 of f at x = 1 is called\nabsolute maximum value (global maximum or greatest value) of f on the interval\n[0, 1] Similarly, the minimum value 2 of f  at x  = 0 is called the absolute minimum\nvalue (global minimum or least value) of f on [0, 1] Consider the graph given in Fig 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "2972-2975",
        "Text": "The maximum value 3 of f at x = 1 is called\nabsolute maximum value (global maximum or greatest value) of f on the interval\n[0, 1] Similarly, the minimum value 2 of f  at x  = 0 is called the absolute minimum\nvalue (global minimum or least value) of f on [0, 1] Consider the graph given in Fig 6 19 of a continuous function defined on a closed\ninterval [a, d]"
    },
    {
        "Chapter": "1",
        "sentence_range": "2973-2976",
        "Text": "Similarly, the minimum value 2 of f  at x  = 0 is called the absolute minimum\nvalue (global minimum or least value) of f on [0, 1] Consider the graph given in Fig 6 19 of a continuous function defined on a closed\ninterval [a, d] Observe that the function f has a local minima at x = b and local\nFig 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "2974-2977",
        "Text": "Consider the graph given in Fig 6 19 of a continuous function defined on a closed\ninterval [a, d] Observe that the function f has a local minima at x = b and local\nFig 6 18\nRationalised 2023-24\n MATHEMATICS\n172\nminimum value is f(b)"
    },
    {
        "Chapter": "1",
        "sentence_range": "2975-2978",
        "Text": "19 of a continuous function defined on a closed\ninterval [a, d] Observe that the function f has a local minima at x = b and local\nFig 6 18\nRationalised 2023-24\n MATHEMATICS\n172\nminimum value is f(b) The function also has a local maxima at x = c and local maximum\nvalue is f (c)"
    },
    {
        "Chapter": "1",
        "sentence_range": "2976-2979",
        "Text": "Observe that the function f has a local minima at x = b and local\nFig 6 18\nRationalised 2023-24\n MATHEMATICS\n172\nminimum value is f(b) The function also has a local maxima at x = c and local maximum\nvalue is f (c) Also from the graph, it is evident that f has  absolute maximum value f (a) and\nabsolute minimum value f (d)"
    },
    {
        "Chapter": "1",
        "sentence_range": "2977-2980",
        "Text": "18\nRationalised 2023-24\n MATHEMATICS\n172\nminimum value is f(b) The function also has a local maxima at x = c and local maximum\nvalue is f (c) Also from the graph, it is evident that f has  absolute maximum value f (a) and\nabsolute minimum value f (d) Further note that the absolute maximum (minimum)\nvalue of f is different from local maximum (minimum) value of f"
    },
    {
        "Chapter": "1",
        "sentence_range": "2978-2981",
        "Text": "The function also has a local maxima at x = c and local maximum\nvalue is f (c) Also from the graph, it is evident that f has  absolute maximum value f (a) and\nabsolute minimum value f (d) Further note that the absolute maximum (minimum)\nvalue of f is different from local maximum (minimum) value of f We will now state two results (without proof) regarding absolute maximum and\nabsolute minimum values of a function on a closed interval I"
    },
    {
        "Chapter": "1",
        "sentence_range": "2979-2982",
        "Text": "Also from the graph, it is evident that f has  absolute maximum value f (a) and\nabsolute minimum value f (d) Further note that the absolute maximum (minimum)\nvalue of f is different from local maximum (minimum) value of f We will now state two results (without proof) regarding absolute maximum and\nabsolute minimum values of a function on a closed interval I Theorem 5 Let f  be a continuous function on an interval I = [a, b]"
    },
    {
        "Chapter": "1",
        "sentence_range": "2980-2983",
        "Text": "Further note that the absolute maximum (minimum)\nvalue of f is different from local maximum (minimum) value of f We will now state two results (without proof) regarding absolute maximum and\nabsolute minimum values of a function on a closed interval I Theorem 5 Let f  be a continuous function on an interval I = [a, b] Then f has the\nabsolute maximum value and f attains it at least once in I"
    },
    {
        "Chapter": "1",
        "sentence_range": "2981-2984",
        "Text": "We will now state two results (without proof) regarding absolute maximum and\nabsolute minimum values of a function on a closed interval I Theorem 5 Let f  be a continuous function on an interval I = [a, b] Then f has the\nabsolute maximum value and f attains it at least once in I Also, f has the absolute\nminimum value and attains it at least once in I"
    },
    {
        "Chapter": "1",
        "sentence_range": "2982-2985",
        "Text": "Theorem 5 Let f  be a continuous function on an interval I = [a, b] Then f has the\nabsolute maximum value and f attains it at least once in I Also, f has the absolute\nminimum value and attains it at least once in I Theorem 6 Let f be a differentiable function on a closed interval I and let c be any\ninterior point of I"
    },
    {
        "Chapter": "1",
        "sentence_range": "2983-2986",
        "Text": "Then f has the\nabsolute maximum value and f attains it at least once in I Also, f has the absolute\nminimum value and attains it at least once in I Theorem 6 Let f be a differentiable function on a closed interval I and let c be any\ninterior point of I Then\n(i) f \u2032(c) = 0 if  f attains its absolute maximum value at c"
    },
    {
        "Chapter": "1",
        "sentence_range": "2984-2987",
        "Text": "Also, f has the absolute\nminimum value and attains it at least once in I Theorem 6 Let f be a differentiable function on a closed interval I and let c be any\ninterior point of I Then\n(i) f \u2032(c) = 0 if  f attains its absolute maximum value at c (ii) f \u2032(c) = 0 if  f attains its absolute minimum value at c"
    },
    {
        "Chapter": "1",
        "sentence_range": "2985-2988",
        "Text": "Theorem 6 Let f be a differentiable function on a closed interval I and let c be any\ninterior point of I Then\n(i) f \u2032(c) = 0 if  f attains its absolute maximum value at c (ii) f \u2032(c) = 0 if  f attains its absolute minimum value at c In view of the above results, we have the following working rule for finding absolute\nmaximum and/or absolute minimum values of a function in a given closed interval\n[a, b]"
    },
    {
        "Chapter": "1",
        "sentence_range": "2986-2989",
        "Text": "Then\n(i) f \u2032(c) = 0 if  f attains its absolute maximum value at c (ii) f \u2032(c) = 0 if  f attains its absolute minimum value at c In view of the above results, we have the following working rule for finding absolute\nmaximum and/or absolute minimum values of a function in a given closed interval\n[a, b] Working Rule\nStep 1: Find all critical points of  f  in the interval, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "2987-2990",
        "Text": "(ii) f \u2032(c) = 0 if  f attains its absolute minimum value at c In view of the above results, we have the following working rule for finding absolute\nmaximum and/or absolute minimum values of a function in a given closed interval\n[a, b] Working Rule\nStep 1: Find all critical points of  f  in the interval, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "2988-2991",
        "Text": "In view of the above results, we have the following working rule for finding absolute\nmaximum and/or absolute minimum values of a function in a given closed interval\n[a, b] Working Rule\nStep 1: Find all critical points of  f  in the interval, i e , find points x where either\n( )\n0\nf\n\u2032x =\n or f  is not differentiable"
    },
    {
        "Chapter": "1",
        "sentence_range": "2989-2992",
        "Text": "Working Rule\nStep 1: Find all critical points of  f  in the interval, i e , find points x where either\n( )\n0\nf\n\u2032x =\n or f  is not differentiable Step 2: Take the end points of the interval"
    },
    {
        "Chapter": "1",
        "sentence_range": "2990-2993",
        "Text": "e , find points x where either\n( )\n0\nf\n\u2032x =\n or f  is not differentiable Step 2: Take the end points of the interval Step 3: At all these points (listed in Step 1 and 2), calculate the values of  f"
    },
    {
        "Chapter": "1",
        "sentence_range": "2991-2994",
        "Text": ", find points x where either\n( )\n0\nf\n\u2032x =\n or f  is not differentiable Step 2: Take the end points of the interval Step 3: At all these points (listed in Step 1 and 2), calculate the values of  f Step 4: Identify the maximum and minimum values of  f  out of the values calculated in\nStep 3"
    },
    {
        "Chapter": "1",
        "sentence_range": "2992-2995",
        "Text": "Step 2: Take the end points of the interval Step 3: At all these points (listed in Step 1 and 2), calculate the values of  f Step 4: Identify the maximum and minimum values of  f  out of the values calculated in\nStep 3 This maximum value will be the absolute maximum (greatest) value of\nf  and the minimum value will be the absolute minimum (least) value of  f"
    },
    {
        "Chapter": "1",
        "sentence_range": "2993-2996",
        "Text": "Step 3: At all these points (listed in Step 1 and 2), calculate the values of  f Step 4: Identify the maximum and minimum values of  f  out of the values calculated in\nStep 3 This maximum value will be the absolute maximum (greatest) value of\nf  and the minimum value will be the absolute minimum (least) value of  f Fig 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "2994-2997",
        "Text": "Step 4: Identify the maximum and minimum values of  f  out of the values calculated in\nStep 3 This maximum value will be the absolute maximum (greatest) value of\nf  and the minimum value will be the absolute minimum (least) value of  f Fig 6 19\nRationalised 2023-24\nAPPLICATION OF DERIVATIVES\n173\nExample 27 Find the absolute maximum and minimum values of a function f given by\nf (x) = 2x3 \u2013 15x2 + 36x +1 on the interval [1, 5]"
    },
    {
        "Chapter": "1",
        "sentence_range": "2995-2998",
        "Text": "This maximum value will be the absolute maximum (greatest) value of\nf  and the minimum value will be the absolute minimum (least) value of  f Fig 6 19\nRationalised 2023-24\nAPPLICATION OF DERIVATIVES\n173\nExample 27 Find the absolute maximum and minimum values of a function f given by\nf (x) = 2x3 \u2013 15x2 + 36x +1 on the interval [1, 5] Solution We have\nf (x) = 2x3 \u2013 15x2 + 36x + 1\nor\nf \u2032(x) = 6x2 \u2013 30x + 36 = 6 (x \u2013 3) (x \u2013 2)\nNote that f \u2032(x) = 0 gives  x = 2 and x = 3"
    },
    {
        "Chapter": "1",
        "sentence_range": "2996-2999",
        "Text": "Fig 6 19\nRationalised 2023-24\nAPPLICATION OF DERIVATIVES\n173\nExample 27 Find the absolute maximum and minimum values of a function f given by\nf (x) = 2x3 \u2013 15x2 + 36x +1 on the interval [1, 5] Solution We have\nf (x) = 2x3 \u2013 15x2 + 36x + 1\nor\nf \u2032(x) = 6x2 \u2013 30x + 36 = 6 (x \u2013 3) (x \u2013 2)\nNote that f \u2032(x) = 0 gives  x = 2 and x = 3 We shall now evaluate the value of f at these points and at the end points of the\ninterval [1, 5], i"
    },
    {
        "Chapter": "1",
        "sentence_range": "2997-3000",
        "Text": "19\nRationalised 2023-24\nAPPLICATION OF DERIVATIVES\n173\nExample 27 Find the absolute maximum and minimum values of a function f given by\nf (x) = 2x3 \u2013 15x2 + 36x +1 on the interval [1, 5] Solution We have\nf (x) = 2x3 \u2013 15x2 + 36x + 1\nor\nf \u2032(x) = 6x2 \u2013 30x + 36 = 6 (x \u2013 3) (x \u2013 2)\nNote that f \u2032(x) = 0 gives  x = 2 and x = 3 We shall now evaluate the value of f at these points and at the end points of the\ninterval [1, 5], i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "2998-3001",
        "Text": "Solution We have\nf (x) = 2x3 \u2013 15x2 + 36x + 1\nor\nf \u2032(x) = 6x2 \u2013 30x + 36 = 6 (x \u2013 3) (x \u2013 2)\nNote that f \u2032(x) = 0 gives  x = 2 and x = 3 We shall now evaluate the value of f at these points and at the end points of the\ninterval [1, 5], i e , at x = 1, x = 2, x = 3 and at x = 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "2999-3002",
        "Text": "We shall now evaluate the value of f at these points and at the end points of the\ninterval [1, 5], i e , at x = 1, x = 2, x = 3 and at x = 5 So\nf (1) = 2(13) \u2013 15(12) + 36 (1) + 1 = 24\nf (2) = 2(23) \u2013 15(22) + 36 (2) + 1 = 29\nf (3) = 2(33) \u2013 15(32) + 36 (3) + 1 = 28\nf (5) = 2(53) \u2013 15(52) + 36 (5) + 1 = 56\nThus, we conclude that absolute maximum value of f on [1, 5] is 56, occurring at\nx =5, and absolute  minimum value of f on [1, 5] is 24 which occurs at x = 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "3000-3003",
        "Text": "e , at x = 1, x = 2, x = 3 and at x = 5 So\nf (1) = 2(13) \u2013 15(12) + 36 (1) + 1 = 24\nf (2) = 2(23) \u2013 15(22) + 36 (2) + 1 = 29\nf (3) = 2(33) \u2013 15(32) + 36 (3) + 1 = 28\nf (5) = 2(53) \u2013 15(52) + 36 (5) + 1 = 56\nThus, we conclude that absolute maximum value of f on [1, 5] is 56, occurring at\nx =5, and absolute  minimum value of f on [1, 5] is 24 which occurs at x = 1 Example 28 Find absolute maximum and minimum values of a function f given by\n4\n1\n3\n3\n( )\n12\n6\n,\n[ 1, 1]\nf x\nx\nx\nx\n=\n\u2212\n\u2208 \u2212\nSolution We have\nf (x) =\n4\n1\n3\n3\n12\n6\nx\nx\n\u2212\nor\nf \u2032(x) =\n31\n2\n2\n3\n3\n2\n2(8\n1)\n16\nx\nx\nx\nx\n\u2212\n\u2212\n=\nThus, f \u2032(x) = 0 gives \nx =81"
    },
    {
        "Chapter": "1",
        "sentence_range": "3001-3004",
        "Text": ", at x = 1, x = 2, x = 3 and at x = 5 So\nf (1) = 2(13) \u2013 15(12) + 36 (1) + 1 = 24\nf (2) = 2(23) \u2013 15(22) + 36 (2) + 1 = 29\nf (3) = 2(33) \u2013 15(32) + 36 (3) + 1 = 28\nf (5) = 2(53) \u2013 15(52) + 36 (5) + 1 = 56\nThus, we conclude that absolute maximum value of f on [1, 5] is 56, occurring at\nx =5, and absolute  minimum value of f on [1, 5] is 24 which occurs at x = 1 Example 28 Find absolute maximum and minimum values of a function f given by\n4\n1\n3\n3\n( )\n12\n6\n,\n[ 1, 1]\nf x\nx\nx\nx\n=\n\u2212\n\u2208 \u2212\nSolution We have\nf (x) =\n4\n1\n3\n3\n12\n6\nx\nx\n\u2212\nor\nf \u2032(x) =\n31\n2\n2\n3\n3\n2\n2(8\n1)\n16\nx\nx\nx\nx\n\u2212\n\u2212\n=\nThus, f \u2032(x) = 0 gives \nx =81 Further note that f \u2032(x) is not defined at x = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "3002-3005",
        "Text": "So\nf (1) = 2(13) \u2013 15(12) + 36 (1) + 1 = 24\nf (2) = 2(23) \u2013 15(22) + 36 (2) + 1 = 29\nf (3) = 2(33) \u2013 15(32) + 36 (3) + 1 = 28\nf (5) = 2(53) \u2013 15(52) + 36 (5) + 1 = 56\nThus, we conclude that absolute maximum value of f on [1, 5] is 56, occurring at\nx =5, and absolute  minimum value of f on [1, 5] is 24 which occurs at x = 1 Example 28 Find absolute maximum and minimum values of a function f given by\n4\n1\n3\n3\n( )\n12\n6\n,\n[ 1, 1]\nf x\nx\nx\nx\n=\n\u2212\n\u2208 \u2212\nSolution We have\nf (x) =\n4\n1\n3\n3\n12\n6\nx\nx\n\u2212\nor\nf \u2032(x) =\n31\n2\n2\n3\n3\n2\n2(8\n1)\n16\nx\nx\nx\nx\n\u2212\n\u2212\n=\nThus, f \u2032(x) = 0 gives \nx =81 Further note that f \u2032(x) is not defined at x = 0 So the\ncritical points are  x = 0 and \nx =81"
    },
    {
        "Chapter": "1",
        "sentence_range": "3003-3006",
        "Text": "Example 28 Find absolute maximum and minimum values of a function f given by\n4\n1\n3\n3\n( )\n12\n6\n,\n[ 1, 1]\nf x\nx\nx\nx\n=\n\u2212\n\u2208 \u2212\nSolution We have\nf (x) =\n4\n1\n3\n3\n12\n6\nx\nx\n\u2212\nor\nf \u2032(x) =\n31\n2\n2\n3\n3\n2\n2(8\n1)\n16\nx\nx\nx\nx\n\u2212\n\u2212\n=\nThus, f \u2032(x) = 0 gives \nx =81 Further note that f \u2032(x) is not defined at x = 0 So the\ncritical points are  x = 0 and \nx =81 Now evaluating the value of f at critical points\nx = 0, 1\n8  and at end points of the interval x = \u20131 and x = 1, we have\nf (\u20131) =\n4\n1\n3\n3\n12( 1)\n6( 1)\n18\n\u2212\n\u2212\n\u2212\n=\nf (0) = 12 (0) \u2013 6(0) = 0\nRationalised 2023-24\n MATHEMATICS\n174\nf \uf8eb81\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8  =\n4\n1\n3\n3\n1\n1\n9\n12\n6\n8\n8\n\u22124\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n\u2212\n=\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\nf (1) =\n4\n1\n3\n3\n12(1)\n6(1)\n6\n\u2212\n=\nHence, we conclude that absolute maximum value of  f  is 18 that occurs at x = \u20131\nand absolute minimum value of f is \n49\n\u2212  that occurs at \nx =81"
    },
    {
        "Chapter": "1",
        "sentence_range": "3004-3007",
        "Text": "Further note that f \u2032(x) is not defined at x = 0 So the\ncritical points are  x = 0 and \nx =81 Now evaluating the value of f at critical points\nx = 0, 1\n8  and at end points of the interval x = \u20131 and x = 1, we have\nf (\u20131) =\n4\n1\n3\n3\n12( 1)\n6( 1)\n18\n\u2212\n\u2212\n\u2212\n=\nf (0) = 12 (0) \u2013 6(0) = 0\nRationalised 2023-24\n MATHEMATICS\n174\nf \uf8eb81\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8  =\n4\n1\n3\n3\n1\n1\n9\n12\n6\n8\n8\n\u22124\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n\u2212\n=\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\nf (1) =\n4\n1\n3\n3\n12(1)\n6(1)\n6\n\u2212\n=\nHence, we conclude that absolute maximum value of  f  is 18 that occurs at x = \u20131\nand absolute minimum value of f is \n49\n\u2212  that occurs at \nx =81 Example 29 An Apache helicopter of enemy is flying along the curve given by\ny = x2 + 7"
    },
    {
        "Chapter": "1",
        "sentence_range": "3005-3008",
        "Text": "So the\ncritical points are  x = 0 and \nx =81 Now evaluating the value of f at critical points\nx = 0, 1\n8  and at end points of the interval x = \u20131 and x = 1, we have\nf (\u20131) =\n4\n1\n3\n3\n12( 1)\n6( 1)\n18\n\u2212\n\u2212\n\u2212\n=\nf (0) = 12 (0) \u2013 6(0) = 0\nRationalised 2023-24\n MATHEMATICS\n174\nf \uf8eb81\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8  =\n4\n1\n3\n3\n1\n1\n9\n12\n6\n8\n8\n\u22124\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n\u2212\n=\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\nf (1) =\n4\n1\n3\n3\n12(1)\n6(1)\n6\n\u2212\n=\nHence, we conclude that absolute maximum value of  f  is 18 that occurs at x = \u20131\nand absolute minimum value of f is \n49\n\u2212  that occurs at \nx =81 Example 29 An Apache helicopter of enemy is flying along the curve given by\ny = x2 + 7 A soldier, placed at (3, 7), wants to shoot down the helicopter when it is\nnearest to him"
    },
    {
        "Chapter": "1",
        "sentence_range": "3006-3009",
        "Text": "Now evaluating the value of f at critical points\nx = 0, 1\n8  and at end points of the interval x = \u20131 and x = 1, we have\nf (\u20131) =\n4\n1\n3\n3\n12( 1)\n6( 1)\n18\n\u2212\n\u2212\n\u2212\n=\nf (0) = 12 (0) \u2013 6(0) = 0\nRationalised 2023-24\n MATHEMATICS\n174\nf \uf8eb81\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8  =\n4\n1\n3\n3\n1\n1\n9\n12\n6\n8\n8\n\u22124\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n\u2212\n=\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\nf (1) =\n4\n1\n3\n3\n12(1)\n6(1)\n6\n\u2212\n=\nHence, we conclude that absolute maximum value of  f  is 18 that occurs at x = \u20131\nand absolute minimum value of f is \n49\n\u2212  that occurs at \nx =81 Example 29 An Apache helicopter of enemy is flying along the curve given by\ny = x2 + 7 A soldier, placed at (3, 7), wants to shoot down the helicopter when it is\nnearest to him Find the nearest distance"
    },
    {
        "Chapter": "1",
        "sentence_range": "3007-3010",
        "Text": "Example 29 An Apache helicopter of enemy is flying along the curve given by\ny = x2 + 7 A soldier, placed at (3, 7), wants to shoot down the helicopter when it is\nnearest to him Find the nearest distance Solution For each value of x, the helicopter\u2019s position is at point (x, x2 + 7)"
    },
    {
        "Chapter": "1",
        "sentence_range": "3008-3011",
        "Text": "A soldier, placed at (3, 7), wants to shoot down the helicopter when it is\nnearest to him Find the nearest distance Solution For each value of x, the helicopter\u2019s position is at point (x, x2 + 7) Therefore, the distance between the helicopter and the soldier placed at (3,7) is\n2\n2\n2\n(\n3)\n(\n7\n7)\nx\nx\n\u2212\n+\n+\n\u2212\n, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "3009-3012",
        "Text": "Find the nearest distance Solution For each value of x, the helicopter\u2019s position is at point (x, x2 + 7) Therefore, the distance between the helicopter and the soldier placed at (3,7) is\n2\n2\n2\n(\n3)\n(\n7\n7)\nx\nx\n\u2212\n+\n+\n\u2212\n, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "3010-3013",
        "Text": "Solution For each value of x, the helicopter\u2019s position is at point (x, x2 + 7) Therefore, the distance between the helicopter and the soldier placed at (3,7) is\n2\n2\n2\n(\n3)\n(\n7\n7)\nx\nx\n\u2212\n+\n+\n\u2212\n, i e , \n2\n4\n(\n3)\nx\nx\n\u2212\n+"
    },
    {
        "Chapter": "1",
        "sentence_range": "3011-3014",
        "Text": "Therefore, the distance between the helicopter and the soldier placed at (3,7) is\n2\n2\n2\n(\n3)\n(\n7\n7)\nx\nx\n\u2212\n+\n+\n\u2212\n, i e , \n2\n4\n(\n3)\nx\nx\n\u2212\n+ Let\nf (x) = (x \u2013 3)2 + x4\nor\nf \u2032(x) = 2(x \u2013 3) + 4x3 = 2(x \u2013 1) (2x2 + 2x + 3)\nThus, f \u2032(x) = 0 gives x = 1 or 2x2 + 2x + 3 = 0 for which there are  no real roots"
    },
    {
        "Chapter": "1",
        "sentence_range": "3012-3015",
        "Text": "e , \n2\n4\n(\n3)\nx\nx\n\u2212\n+ Let\nf (x) = (x \u2013 3)2 + x4\nor\nf \u2032(x) = 2(x \u2013 3) + 4x3 = 2(x \u2013 1) (2x2 + 2x + 3)\nThus, f \u2032(x) = 0 gives x = 1 or 2x2 + 2x + 3 = 0 for which there are  no real roots Also, there are no end points of the interval to be added to the set for which f \u2032 is zero,\ni"
    },
    {
        "Chapter": "1",
        "sentence_range": "3013-3016",
        "Text": ", \n2\n4\n(\n3)\nx\nx\n\u2212\n+ Let\nf (x) = (x \u2013 3)2 + x4\nor\nf \u2032(x) = 2(x \u2013 3) + 4x3 = 2(x \u2013 1) (2x2 + 2x + 3)\nThus, f \u2032(x) = 0 gives x = 1 or 2x2 + 2x + 3 = 0 for which there are  no real roots Also, there are no end points of the interval to be added to the set for which f \u2032 is zero,\ni e"
    },
    {
        "Chapter": "1",
        "sentence_range": "3014-3017",
        "Text": "Let\nf (x) = (x \u2013 3)2 + x4\nor\nf \u2032(x) = 2(x \u2013 3) + 4x3 = 2(x \u2013 1) (2x2 + 2x + 3)\nThus, f \u2032(x) = 0 gives x = 1 or 2x2 + 2x + 3 = 0 for which there are  no real roots Also, there are no end points of the interval to be added to the set for which f \u2032 is zero,\ni e , there is only one point, namely, x = 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "3015-3018",
        "Text": "Also, there are no end points of the interval to be added to the set for which f \u2032 is zero,\ni e , there is only one point, namely, x = 1 The value of f at this point is given by\nf (1) = (1 \u2013 3)2 + (1)4  = 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "3016-3019",
        "Text": "e , there is only one point, namely, x = 1 The value of f at this point is given by\nf (1) = (1 \u2013 3)2 + (1)4  = 5 Thus, the distance between the solider and the helicopter is\n(1)\n5\nf\n="
    },
    {
        "Chapter": "1",
        "sentence_range": "3017-3020",
        "Text": ", there is only one point, namely, x = 1 The value of f at this point is given by\nf (1) = (1 \u2013 3)2 + (1)4  = 5 Thus, the distance between the solider and the helicopter is\n(1)\n5\nf\n= Note that 5  is either a maximum value or a minimum value"
    },
    {
        "Chapter": "1",
        "sentence_range": "3018-3021",
        "Text": "The value of f at this point is given by\nf (1) = (1 \u2013 3)2 + (1)4  = 5 Thus, the distance between the solider and the helicopter is\n(1)\n5\nf\n= Note that 5  is either a maximum value or a minimum value Since\nf(0)\n =\n2\n4\n(0\n3)\n(0)\n3\n5\n\u2212\n+\n=\n>\n,\nit follows that \n5  is the minimum value of  \n( )\nf x"
    },
    {
        "Chapter": "1",
        "sentence_range": "3019-3022",
        "Text": "Thus, the distance between the solider and the helicopter is\n(1)\n5\nf\n= Note that 5  is either a maximum value or a minimum value Since\nf(0)\n =\n2\n4\n(0\n3)\n(0)\n3\n5\n\u2212\n+\n=\n>\n,\nit follows that \n5  is the minimum value of  \n( )\nf x Hence, \n5  is the minimum\ndistance between the soldier and the helicopter"
    },
    {
        "Chapter": "1",
        "sentence_range": "3020-3023",
        "Text": "Note that 5  is either a maximum value or a minimum value Since\nf(0)\n =\n2\n4\n(0\n3)\n(0)\n3\n5\n\u2212\n+\n=\n>\n,\nit follows that \n5  is the minimum value of  \n( )\nf x Hence, \n5  is the minimum\ndistance between the soldier and the helicopter EXERCISE 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "3021-3024",
        "Text": "Since\nf(0)\n =\n2\n4\n(0\n3)\n(0)\n3\n5\n\u2212\n+\n=\n>\n,\nit follows that \n5  is the minimum value of  \n( )\nf x Hence, \n5  is the minimum\ndistance between the soldier and the helicopter EXERCISE 6 3\n1"
    },
    {
        "Chapter": "1",
        "sentence_range": "3022-3025",
        "Text": "Hence, \n5  is the minimum\ndistance between the soldier and the helicopter EXERCISE 6 3\n1 Find the maximum and minimum values, if any, of the following functions\ngiven by\n(i) f (x) = (2x \u2013 1)2 + 3\n(ii) f (x) = 9x2 + 12x + 2\n(iii) f (x) = \u2013 (x \u2013 1)2 + 10\n(iv) g(x) = x3 + 1\nRationalised 2023-24\nAPPLICATION OF DERIVATIVES\n175\n2"
    },
    {
        "Chapter": "1",
        "sentence_range": "3023-3026",
        "Text": "EXERCISE 6 3\n1 Find the maximum and minimum values, if any, of the following functions\ngiven by\n(i) f (x) = (2x \u2013 1)2 + 3\n(ii) f (x) = 9x2 + 12x + 2\n(iii) f (x) = \u2013 (x \u2013 1)2 + 10\n(iv) g(x) = x3 + 1\nRationalised 2023-24\nAPPLICATION OF DERIVATIVES\n175\n2 Find the maximum and minimum values, if any, of the following functions\ngiven by\n(i) f (x) = |x + 2| \u2013 1\n(ii) g(x) = \u2013 |x + 1| + 3\n(iii) h(x) = sin(2x) + 5\n(iv) f (x) = |sin 4x + 3|\n(v) h(x) = x + 1, x \u2208 (\u2013 1, 1)\n3"
    },
    {
        "Chapter": "1",
        "sentence_range": "3024-3027",
        "Text": "3\n1 Find the maximum and minimum values, if any, of the following functions\ngiven by\n(i) f (x) = (2x \u2013 1)2 + 3\n(ii) f (x) = 9x2 + 12x + 2\n(iii) f (x) = \u2013 (x \u2013 1)2 + 10\n(iv) g(x) = x3 + 1\nRationalised 2023-24\nAPPLICATION OF DERIVATIVES\n175\n2 Find the maximum and minimum values, if any, of the following functions\ngiven by\n(i) f (x) = |x + 2| \u2013 1\n(ii) g(x) = \u2013 |x + 1| + 3\n(iii) h(x) = sin(2x) + 5\n(iv) f (x) = |sin 4x + 3|\n(v) h(x) = x + 1, x \u2208 (\u2013 1, 1)\n3 Find the local maxima and local minima, if any, of the following functions"
    },
    {
        "Chapter": "1",
        "sentence_range": "3025-3028",
        "Text": "Find the maximum and minimum values, if any, of the following functions\ngiven by\n(i) f (x) = (2x \u2013 1)2 + 3\n(ii) f (x) = 9x2 + 12x + 2\n(iii) f (x) = \u2013 (x \u2013 1)2 + 10\n(iv) g(x) = x3 + 1\nRationalised 2023-24\nAPPLICATION OF DERIVATIVES\n175\n2 Find the maximum and minimum values, if any, of the following functions\ngiven by\n(i) f (x) = |x + 2| \u2013 1\n(ii) g(x) = \u2013 |x + 1| + 3\n(iii) h(x) = sin(2x) + 5\n(iv) f (x) = |sin 4x + 3|\n(v) h(x) = x + 1, x \u2208 (\u2013 1, 1)\n3 Find the local maxima and local minima, if any, of the following functions Find\nalso the local maximum and the local minimum values, as the case may be:\n(i) f (x) = x2\n(ii) g(x) = x3 \u2013 3x\n(iii) h(x) = sin x + cos x, 0\n2\nx\n\u03c0\n<\n<\n(iv) f (x) = sin x \u2013 cos x, 0\n2\n<x\n< \u03c0\n(v) f (x) = x3 \u2013 6x2 + 9x + 15\n(vi)\n2\n( )\n,\n0\nx2\ng x\nx\nx\n=\n+\n>\n(vii)\n2\n1\n( )\n2\ng x\nx\n=\n+\n(viii)\n( )\n1\n, 0\n1\n=\n\u2212\n<\n<\nf x\nx\nx\nx\n4"
    },
    {
        "Chapter": "1",
        "sentence_range": "3026-3029",
        "Text": "Find the maximum and minimum values, if any, of the following functions\ngiven by\n(i) f (x) = |x + 2| \u2013 1\n(ii) g(x) = \u2013 |x + 1| + 3\n(iii) h(x) = sin(2x) + 5\n(iv) f (x) = |sin 4x + 3|\n(v) h(x) = x + 1, x \u2208 (\u2013 1, 1)\n3 Find the local maxima and local minima, if any, of the following functions Find\nalso the local maximum and the local minimum values, as the case may be:\n(i) f (x) = x2\n(ii) g(x) = x3 \u2013 3x\n(iii) h(x) = sin x + cos x, 0\n2\nx\n\u03c0\n<\n<\n(iv) f (x) = sin x \u2013 cos x, 0\n2\n<x\n< \u03c0\n(v) f (x) = x3 \u2013 6x2 + 9x + 15\n(vi)\n2\n( )\n,\n0\nx2\ng x\nx\nx\n=\n+\n>\n(vii)\n2\n1\n( )\n2\ng x\nx\n=\n+\n(viii)\n( )\n1\n, 0\n1\n=\n\u2212\n<\n<\nf x\nx\nx\nx\n4 Prove that the following  functions do not have maxima or minima:\n(i) f (x) = ex\n(ii) g(x) = log x\n(iii) h (x) = x3 + x2 + x +1\n5"
    },
    {
        "Chapter": "1",
        "sentence_range": "3027-3030",
        "Text": "Find the local maxima and local minima, if any, of the following functions Find\nalso the local maximum and the local minimum values, as the case may be:\n(i) f (x) = x2\n(ii) g(x) = x3 \u2013 3x\n(iii) h(x) = sin x + cos x, 0\n2\nx\n\u03c0\n<\n<\n(iv) f (x) = sin x \u2013 cos x, 0\n2\n<x\n< \u03c0\n(v) f (x) = x3 \u2013 6x2 + 9x + 15\n(vi)\n2\n( )\n,\n0\nx2\ng x\nx\nx\n=\n+\n>\n(vii)\n2\n1\n( )\n2\ng x\nx\n=\n+\n(viii)\n( )\n1\n, 0\n1\n=\n\u2212\n<\n<\nf x\nx\nx\nx\n4 Prove that the following  functions do not have maxima or minima:\n(i) f (x) = ex\n(ii) g(x) = log x\n(iii) h (x) = x3 + x2 + x +1\n5 Find the absolute maximum value and the absolute minimum value of the following\nfunctions in the given intervals:\n(i) f (x) = x3, x \u2208 [\u2013 2, 2]\n(ii) f (x) = sin x + cos x , x \u2208 [0, \u03c0]\n(iii) f (x) =\n12\n9\n4\n,\n2,\n2\n2\nx\nx\nx\n\uf8ee\n\uf8f9\n\u2212\n\u2208 \u2212\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n(iv)\n2\n( )\n(\n1)\n3,\n[ 3,1]\nf x\nx\nx\n=\n\u2212\n+\n\u2208 \u2212\n6"
    },
    {
        "Chapter": "1",
        "sentence_range": "3028-3031",
        "Text": "Find\nalso the local maximum and the local minimum values, as the case may be:\n(i) f (x) = x2\n(ii) g(x) = x3 \u2013 3x\n(iii) h(x) = sin x + cos x, 0\n2\nx\n\u03c0\n<\n<\n(iv) f (x) = sin x \u2013 cos x, 0\n2\n<x\n< \u03c0\n(v) f (x) = x3 \u2013 6x2 + 9x + 15\n(vi)\n2\n( )\n,\n0\nx2\ng x\nx\nx\n=\n+\n>\n(vii)\n2\n1\n( )\n2\ng x\nx\n=\n+\n(viii)\n( )\n1\n, 0\n1\n=\n\u2212\n<\n<\nf x\nx\nx\nx\n4 Prove that the following  functions do not have maxima or minima:\n(i) f (x) = ex\n(ii) g(x) = log x\n(iii) h (x) = x3 + x2 + x +1\n5 Find the absolute maximum value and the absolute minimum value of the following\nfunctions in the given intervals:\n(i) f (x) = x3, x \u2208 [\u2013 2, 2]\n(ii) f (x) = sin x + cos x , x \u2208 [0, \u03c0]\n(iii) f (x) =\n12\n9\n4\n,\n2,\n2\n2\nx\nx\nx\n\uf8ee\n\uf8f9\n\u2212\n\u2208 \u2212\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n(iv)\n2\n( )\n(\n1)\n3,\n[ 3,1]\nf x\nx\nx\n=\n\u2212\n+\n\u2208 \u2212\n6 Find the maximum profit that a company can make, if the profit function is\ngiven by\np(x) = 41 \u2013 72x \u2013 18x2\n7"
    },
    {
        "Chapter": "1",
        "sentence_range": "3029-3032",
        "Text": "Prove that the following  functions do not have maxima or minima:\n(i) f (x) = ex\n(ii) g(x) = log x\n(iii) h (x) = x3 + x2 + x +1\n5 Find the absolute maximum value and the absolute minimum value of the following\nfunctions in the given intervals:\n(i) f (x) = x3, x \u2208 [\u2013 2, 2]\n(ii) f (x) = sin x + cos x , x \u2208 [0, \u03c0]\n(iii) f (x) =\n12\n9\n4\n,\n2,\n2\n2\nx\nx\nx\n\uf8ee\n\uf8f9\n\u2212\n\u2208 \u2212\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n(iv)\n2\n( )\n(\n1)\n3,\n[ 3,1]\nf x\nx\nx\n=\n\u2212\n+\n\u2208 \u2212\n6 Find the maximum profit that a company can make, if the profit function is\ngiven by\np(x) = 41 \u2013 72x \u2013 18x2\n7 Find both the maximum value and the minimum value of\n3x4 \u2013 8x3 + 12x2  \u2013 48x + 25 on the interval [0, 3]"
    },
    {
        "Chapter": "1",
        "sentence_range": "3030-3033",
        "Text": "Find the absolute maximum value and the absolute minimum value of the following\nfunctions in the given intervals:\n(i) f (x) = x3, x \u2208 [\u2013 2, 2]\n(ii) f (x) = sin x + cos x , x \u2208 [0, \u03c0]\n(iii) f (x) =\n12\n9\n4\n,\n2,\n2\n2\nx\nx\nx\n\uf8ee\n\uf8f9\n\u2212\n\u2208 \u2212\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n(iv)\n2\n( )\n(\n1)\n3,\n[ 3,1]\nf x\nx\nx\n=\n\u2212\n+\n\u2208 \u2212\n6 Find the maximum profit that a company can make, if the profit function is\ngiven by\np(x) = 41 \u2013 72x \u2013 18x2\n7 Find both the maximum value and the minimum value of\n3x4 \u2013 8x3 + 12x2  \u2013 48x + 25 on the interval [0, 3] 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "3031-3034",
        "Text": "Find the maximum profit that a company can make, if the profit function is\ngiven by\np(x) = 41 \u2013 72x \u2013 18x2\n7 Find both the maximum value and the minimum value of\n3x4 \u2013 8x3 + 12x2  \u2013 48x + 25 on the interval [0, 3] 8 At what points in the interval [0, 2\u03c0], does the function sin 2x attain its maximum\nvalue"
    },
    {
        "Chapter": "1",
        "sentence_range": "3032-3035",
        "Text": "Find both the maximum value and the minimum value of\n3x4 \u2013 8x3 + 12x2  \u2013 48x + 25 on the interval [0, 3] 8 At what points in the interval [0, 2\u03c0], does the function sin 2x attain its maximum\nvalue 9"
    },
    {
        "Chapter": "1",
        "sentence_range": "3033-3036",
        "Text": "8 At what points in the interval [0, 2\u03c0], does the function sin 2x attain its maximum\nvalue 9 What is the maximum value of the function sin x + cos x"
    },
    {
        "Chapter": "1",
        "sentence_range": "3034-3037",
        "Text": "At what points in the interval [0, 2\u03c0], does the function sin 2x attain its maximum\nvalue 9 What is the maximum value of the function sin x + cos x 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "3035-3038",
        "Text": "9 What is the maximum value of the function sin x + cos x 10 Find the maximum value of 2x3 \u2013 24x + 107 in the interval [1, 3]"
    },
    {
        "Chapter": "1",
        "sentence_range": "3036-3039",
        "Text": "What is the maximum value of the function sin x + cos x 10 Find the maximum value of 2x3 \u2013 24x + 107 in the interval [1, 3] Find the\nmaximum value of the same function in [\u20133, \u20131]"
    },
    {
        "Chapter": "1",
        "sentence_range": "3037-3040",
        "Text": "10 Find the maximum value of 2x3 \u2013 24x + 107 in the interval [1, 3] Find the\nmaximum value of the same function in [\u20133, \u20131] Rationalised 2023-24\n MATHEMATICS\n176\n11"
    },
    {
        "Chapter": "1",
        "sentence_range": "3038-3041",
        "Text": "Find the maximum value of 2x3 \u2013 24x + 107 in the interval [1, 3] Find the\nmaximum value of the same function in [\u20133, \u20131] Rationalised 2023-24\n MATHEMATICS\n176\n11 It is given that at x = 1, the function x4 \u2013 62x2 + ax + 9 attains its maximum value,\non the interval [0, 2]"
    },
    {
        "Chapter": "1",
        "sentence_range": "3039-3042",
        "Text": "Find the\nmaximum value of the same function in [\u20133, \u20131] Rationalised 2023-24\n MATHEMATICS\n176\n11 It is given that at x = 1, the function x4 \u2013 62x2 + ax + 9 attains its maximum value,\non the interval [0, 2] Find the value of a"
    },
    {
        "Chapter": "1",
        "sentence_range": "3040-3043",
        "Text": "Rationalised 2023-24\n MATHEMATICS\n176\n11 It is given that at x = 1, the function x4 \u2013 62x2 + ax + 9 attains its maximum value,\non the interval [0, 2] Find the value of a 12"
    },
    {
        "Chapter": "1",
        "sentence_range": "3041-3044",
        "Text": "It is given that at x = 1, the function x4 \u2013 62x2 + ax + 9 attains its maximum value,\non the interval [0, 2] Find the value of a 12 Find the maximum and minimum values of  x + sin 2x on [0, 2\u03c0]"
    },
    {
        "Chapter": "1",
        "sentence_range": "3042-3045",
        "Text": "Find the value of a 12 Find the maximum and minimum values of  x + sin 2x on [0, 2\u03c0] 13"
    },
    {
        "Chapter": "1",
        "sentence_range": "3043-3046",
        "Text": "12 Find the maximum and minimum values of  x + sin 2x on [0, 2\u03c0] 13 Find two numbers whose sum is 24 and whose product is as large as possible"
    },
    {
        "Chapter": "1",
        "sentence_range": "3044-3047",
        "Text": "Find the maximum and minimum values of  x + sin 2x on [0, 2\u03c0] 13 Find two numbers whose sum is 24 and whose product is as large as possible 14"
    },
    {
        "Chapter": "1",
        "sentence_range": "3045-3048",
        "Text": "13 Find two numbers whose sum is 24 and whose product is as large as possible 14 Find two positive numbers x and y such that x + y = 60 and xy3 is maximum"
    },
    {
        "Chapter": "1",
        "sentence_range": "3046-3049",
        "Text": "Find two numbers whose sum is 24 and whose product is as large as possible 14 Find two positive numbers x and y such that x + y = 60 and xy3 is maximum 15"
    },
    {
        "Chapter": "1",
        "sentence_range": "3047-3050",
        "Text": "14 Find two positive numbers x and y such that x + y = 60 and xy3 is maximum 15 Find two positive numbers x and y such that their sum is 35 and the product x2 y5\nis a maximum"
    },
    {
        "Chapter": "1",
        "sentence_range": "3048-3051",
        "Text": "Find two positive numbers x and y such that x + y = 60 and xy3 is maximum 15 Find two positive numbers x and y such that their sum is 35 and the product x2 y5\nis a maximum 16"
    },
    {
        "Chapter": "1",
        "sentence_range": "3049-3052",
        "Text": "15 Find two positive numbers x and y such that their sum is 35 and the product x2 y5\nis a maximum 16 Find two positive numbers whose sum is 16 and the sum of whose cubes is\nminimum"
    },
    {
        "Chapter": "1",
        "sentence_range": "3050-3053",
        "Text": "Find two positive numbers x and y such that their sum is 35 and the product x2 y5\nis a maximum 16 Find two positive numbers whose sum is 16 and the sum of whose cubes is\nminimum 17"
    },
    {
        "Chapter": "1",
        "sentence_range": "3051-3054",
        "Text": "16 Find two positive numbers whose sum is 16 and the sum of whose cubes is\nminimum 17 A square piece of tin of side 18 cm is to be made into a box without top, by\ncutting a square from each corner and folding up the flaps to form the box"
    },
    {
        "Chapter": "1",
        "sentence_range": "3052-3055",
        "Text": "Find two positive numbers whose sum is 16 and the sum of whose cubes is\nminimum 17 A square piece of tin of side 18 cm is to be made into a box without top, by\ncutting a square from each corner and folding up the flaps to form the box What\nshould be the side of the square to be cut off so that the volume of the box is the\nmaximum possible"
    },
    {
        "Chapter": "1",
        "sentence_range": "3053-3056",
        "Text": "17 A square piece of tin of side 18 cm is to be made into a box without top, by\ncutting a square from each corner and folding up the flaps to form the box What\nshould be the side of the square to be cut off so that the volume of the box is the\nmaximum possible 18"
    },
    {
        "Chapter": "1",
        "sentence_range": "3054-3057",
        "Text": "A square piece of tin of side 18 cm is to be made into a box without top, by\ncutting a square from each corner and folding up the flaps to form the box What\nshould be the side of the square to be cut off so that the volume of the box is the\nmaximum possible 18 A rectangular sheet of tin 45 cm by 24 cm is to be made into a box without top,\nby cutting off square from each corner and folding up the flaps"
    },
    {
        "Chapter": "1",
        "sentence_range": "3055-3058",
        "Text": "What\nshould be the side of the square to be cut off so that the volume of the box is the\nmaximum possible 18 A rectangular sheet of tin 45 cm by 24 cm is to be made into a box without top,\nby cutting off square from each corner and folding up the flaps What should be\nthe side of the square to be cut off so that the volume of the box is maximum"
    },
    {
        "Chapter": "1",
        "sentence_range": "3056-3059",
        "Text": "18 A rectangular sheet of tin 45 cm by 24 cm is to be made into a box without top,\nby cutting off square from each corner and folding up the flaps What should be\nthe side of the square to be cut off so that the volume of the box is maximum 19"
    },
    {
        "Chapter": "1",
        "sentence_range": "3057-3060",
        "Text": "A rectangular sheet of tin 45 cm by 24 cm is to be made into a box without top,\nby cutting off square from each corner and folding up the flaps What should be\nthe side of the square to be cut off so that the volume of the box is maximum 19 Show that of all the rectangles inscribed  in a given fixed circle, the square has\nthe maximum area"
    },
    {
        "Chapter": "1",
        "sentence_range": "3058-3061",
        "Text": "What should be\nthe side of the square to be cut off so that the volume of the box is maximum 19 Show that of all the rectangles inscribed  in a given fixed circle, the square has\nthe maximum area 20"
    },
    {
        "Chapter": "1",
        "sentence_range": "3059-3062",
        "Text": "19 Show that of all the rectangles inscribed  in a given fixed circle, the square has\nthe maximum area 20 Show that the right circular cylinder of given surface and maximum volume is\nsuch that its height is equal to the diameter of the base"
    },
    {
        "Chapter": "1",
        "sentence_range": "3060-3063",
        "Text": "Show that of all the rectangles inscribed  in a given fixed circle, the square has\nthe maximum area 20 Show that the right circular cylinder of given surface and maximum volume is\nsuch that its height is equal to the diameter of the base 21"
    },
    {
        "Chapter": "1",
        "sentence_range": "3061-3064",
        "Text": "20 Show that the right circular cylinder of given surface and maximum volume is\nsuch that its height is equal to the diameter of the base 21 Of all the closed cylindrical cans (right circular), of a given volume of 100 cubic\ncentimetres, find the dimensions of the can which has the minimum surface\narea"
    },
    {
        "Chapter": "1",
        "sentence_range": "3062-3065",
        "Text": "Show that the right circular cylinder of given surface and maximum volume is\nsuch that its height is equal to the diameter of the base 21 Of all the closed cylindrical cans (right circular), of a given volume of 100 cubic\ncentimetres, find the dimensions of the can which has the minimum surface\narea 22"
    },
    {
        "Chapter": "1",
        "sentence_range": "3063-3066",
        "Text": "21 Of all the closed cylindrical cans (right circular), of a given volume of 100 cubic\ncentimetres, find the dimensions of the can which has the minimum surface\narea 22 A wire of length 28 m is to be cut into two pieces"
    },
    {
        "Chapter": "1",
        "sentence_range": "3064-3067",
        "Text": "Of all the closed cylindrical cans (right circular), of a given volume of 100 cubic\ncentimetres, find the dimensions of the can which has the minimum surface\narea 22 A wire of length 28 m is to be cut into two pieces One of the pieces is to be\nmade into a square and the other into a circle"
    },
    {
        "Chapter": "1",
        "sentence_range": "3065-3068",
        "Text": "22 A wire of length 28 m is to be cut into two pieces One of the pieces is to be\nmade into a square and the other into a circle What should be the length of the\ntwo pieces so that the combined area of the square and the circle is minimum"
    },
    {
        "Chapter": "1",
        "sentence_range": "3066-3069",
        "Text": "A wire of length 28 m is to be cut into two pieces One of the pieces is to be\nmade into a square and the other into a circle What should be the length of the\ntwo pieces so that the combined area of the square and the circle is minimum 23"
    },
    {
        "Chapter": "1",
        "sentence_range": "3067-3070",
        "Text": "One of the pieces is to be\nmade into a square and the other into a circle What should be the length of the\ntwo pieces so that the combined area of the square and the circle is minimum 23 Prove that the volume of the largest cone that can be inscribed in a sphere of\nradius R is 8\n27  of the volume of the sphere"
    },
    {
        "Chapter": "1",
        "sentence_range": "3068-3071",
        "Text": "What should be the length of the\ntwo pieces so that the combined area of the square and the circle is minimum 23 Prove that the volume of the largest cone that can be inscribed in a sphere of\nradius R is 8\n27  of the volume of the sphere 24"
    },
    {
        "Chapter": "1",
        "sentence_range": "3069-3072",
        "Text": "23 Prove that the volume of the largest cone that can be inscribed in a sphere of\nradius R is 8\n27  of the volume of the sphere 24 Show that the right circular cone of least curved surface and given volume has\nan altitude equal to \n2  time the radius of the base"
    },
    {
        "Chapter": "1",
        "sentence_range": "3070-3073",
        "Text": "Prove that the volume of the largest cone that can be inscribed in a sphere of\nradius R is 8\n27  of the volume of the sphere 24 Show that the right circular cone of least curved surface and given volume has\nan altitude equal to \n2  time the radius of the base 25"
    },
    {
        "Chapter": "1",
        "sentence_range": "3071-3074",
        "Text": "24 Show that the right circular cone of least curved surface and given volume has\nan altitude equal to \n2  time the radius of the base 25 Show that the semi-vertical angle of the cone of the maximum volume and of\ngiven slant height is \ntan1\n2\n\u2212"
    },
    {
        "Chapter": "1",
        "sentence_range": "3072-3075",
        "Text": "Show that the right circular cone of least curved surface and given volume has\nan altitude equal to \n2  time the radius of the base 25 Show that the semi-vertical angle of the cone of the maximum volume and of\ngiven slant height is \ntan1\n2\n\u2212 26"
    },
    {
        "Chapter": "1",
        "sentence_range": "3073-3076",
        "Text": "25 Show that the semi-vertical angle of the cone of the maximum volume and of\ngiven slant height is \ntan1\n2\n\u2212 26 Show that semi-vertical angle of right circular cone of given surface area and\nmaximum volume is sin\u2212 \uf8eb\n\uf8ed\uf8ec\n1 1\uf8f8\uf8f7\uf8f6\n3"
    },
    {
        "Chapter": "1",
        "sentence_range": "3074-3077",
        "Text": "Show that the semi-vertical angle of the cone of the maximum volume and of\ngiven slant height is \ntan1\n2\n\u2212 26 Show that semi-vertical angle of right circular cone of given surface area and\nmaximum volume is sin\u2212 \uf8eb\n\uf8ed\uf8ec\n1 1\uf8f8\uf8f7\uf8f6\n3 Rationalised 2023-24\nAPPLICATION OF DERIVATIVES\n177\nChoose the correct answer in Questions 27 and 29"
    },
    {
        "Chapter": "1",
        "sentence_range": "3075-3078",
        "Text": "26 Show that semi-vertical angle of right circular cone of given surface area and\nmaximum volume is sin\u2212 \uf8eb\n\uf8ed\uf8ec\n1 1\uf8f8\uf8f7\uf8f6\n3 Rationalised 2023-24\nAPPLICATION OF DERIVATIVES\n177\nChoose the correct answer in Questions 27 and 29 27"
    },
    {
        "Chapter": "1",
        "sentence_range": "3076-3079",
        "Text": "Show that semi-vertical angle of right circular cone of given surface area and\nmaximum volume is sin\u2212 \uf8eb\n\uf8ed\uf8ec\n1 1\uf8f8\uf8f7\uf8f6\n3 Rationalised 2023-24\nAPPLICATION OF DERIVATIVES\n177\nChoose the correct answer in Questions 27 and 29 27 The point on the curve x2 = 2y which is nearest to the point (0, 5) is\n(A) (2 2,4)\n(B) (2 2,0)\n(C) (0, 0)\n(D) (2, 2)\n28"
    },
    {
        "Chapter": "1",
        "sentence_range": "3077-3080",
        "Text": "Rationalised 2023-24\nAPPLICATION OF DERIVATIVES\n177\nChoose the correct answer in Questions 27 and 29 27 The point on the curve x2 = 2y which is nearest to the point (0, 5) is\n(A) (2 2,4)\n(B) (2 2,0)\n(C) (0, 0)\n(D) (2, 2)\n28 For all real values of x, the minimum value of \n2\n2\n1\n1\nx\nx\nx\nx\n\u2212\n+\n+\n+\n is\n(A) 0\n(B) 1\n(C) 3\n(D) 1\n3\n29"
    },
    {
        "Chapter": "1",
        "sentence_range": "3078-3081",
        "Text": "27 The point on the curve x2 = 2y which is nearest to the point (0, 5) is\n(A) (2 2,4)\n(B) (2 2,0)\n(C) (0, 0)\n(D) (2, 2)\n28 For all real values of x, the minimum value of \n2\n2\n1\n1\nx\nx\nx\nx\n\u2212\n+\n+\n+\n is\n(A) 0\n(B) 1\n(C) 3\n(D) 1\n3\n29 The maximum value of \n31\n[ (\n1)\n1]\nx x \u2212\n+\n, 0\n1\n\u2264x\n\u2264  is\n(A)\n1\n3\n31\n\uf8ed\uf8ec\uf8eb\n\uf8f8\uf8f7\uf8f6\n(B) 1\n2\n(C) 1\n(D) 0\nMiscellaneous Examples\nExample 30 A car starts from a point P at time t = 0 seconds and stops at point Q"
    },
    {
        "Chapter": "1",
        "sentence_range": "3079-3082",
        "Text": "The point on the curve x2 = 2y which is nearest to the point (0, 5) is\n(A) (2 2,4)\n(B) (2 2,0)\n(C) (0, 0)\n(D) (2, 2)\n28 For all real values of x, the minimum value of \n2\n2\n1\n1\nx\nx\nx\nx\n\u2212\n+\n+\n+\n is\n(A) 0\n(B) 1\n(C) 3\n(D) 1\n3\n29 The maximum value of \n31\n[ (\n1)\n1]\nx x \u2212\n+\n, 0\n1\n\u2264x\n\u2264  is\n(A)\n1\n3\n31\n\uf8ed\uf8ec\uf8eb\n\uf8f8\uf8f7\uf8f6\n(B) 1\n2\n(C) 1\n(D) 0\nMiscellaneous Examples\nExample 30 A car starts from a point P at time t = 0 seconds and stops at point Q The\ndistance x, in metres, covered by it, in t seconds is given by\nx\nt\nt\n=\n\u2212\n\uf8ed\uf8ec\uf8eb\n\uf8f8\uf8f7\uf8f6\n2 2\n3\nFind the time taken by it to reach Q and also find distance between P and Q"
    },
    {
        "Chapter": "1",
        "sentence_range": "3080-3083",
        "Text": "For all real values of x, the minimum value of \n2\n2\n1\n1\nx\nx\nx\nx\n\u2212\n+\n+\n+\n is\n(A) 0\n(B) 1\n(C) 3\n(D) 1\n3\n29 The maximum value of \n31\n[ (\n1)\n1]\nx x \u2212\n+\n, 0\n1\n\u2264x\n\u2264  is\n(A)\n1\n3\n31\n\uf8ed\uf8ec\uf8eb\n\uf8f8\uf8f7\uf8f6\n(B) 1\n2\n(C) 1\n(D) 0\nMiscellaneous Examples\nExample 30 A car starts from a point P at time t = 0 seconds and stops at point Q The\ndistance x, in metres, covered by it, in t seconds is given by\nx\nt\nt\n=\n\u2212\n\uf8ed\uf8ec\uf8eb\n\uf8f8\uf8f7\uf8f6\n2 2\n3\nFind the time taken by it to reach Q and also find distance between P and Q Solution Let v be the velocity of the car at t seconds"
    },
    {
        "Chapter": "1",
        "sentence_range": "3081-3084",
        "Text": "The maximum value of \n31\n[ (\n1)\n1]\nx x \u2212\n+\n, 0\n1\n\u2264x\n\u2264  is\n(A)\n1\n3\n31\n\uf8ed\uf8ec\uf8eb\n\uf8f8\uf8f7\uf8f6\n(B) 1\n2\n(C) 1\n(D) 0\nMiscellaneous Examples\nExample 30 A car starts from a point P at time t = 0 seconds and stops at point Q The\ndistance x, in metres, covered by it, in t seconds is given by\nx\nt\nt\n=\n\u2212\n\uf8ed\uf8ec\uf8eb\n\uf8f8\uf8f7\uf8f6\n2 2\n3\nFind the time taken by it to reach Q and also find distance between P and Q Solution Let v be the velocity of the car at t seconds Now\nx =\n2 2\nt3\nt \uf8eb\n\uf8f6\n\u2212\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\nTherefore\nv = dx\ndt  = 4t \u2013 t2 = t(4 \u2013 t)\nThus, v = 0 gives t = 0 and/or t = 4"
    },
    {
        "Chapter": "1",
        "sentence_range": "3082-3085",
        "Text": "The\ndistance x, in metres, covered by it, in t seconds is given by\nx\nt\nt\n=\n\u2212\n\uf8ed\uf8ec\uf8eb\n\uf8f8\uf8f7\uf8f6\n2 2\n3\nFind the time taken by it to reach Q and also find distance between P and Q Solution Let v be the velocity of the car at t seconds Now\nx =\n2 2\nt3\nt \uf8eb\n\uf8f6\n\u2212\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\nTherefore\nv = dx\ndt  = 4t \u2013 t2 = t(4 \u2013 t)\nThus, v = 0 gives t = 0 and/or t = 4 Now v = 0 at P as well as at Q and at P, t = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "3083-3086",
        "Text": "Solution Let v be the velocity of the car at t seconds Now\nx =\n2 2\nt3\nt \uf8eb\n\uf8f6\n\u2212\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\nTherefore\nv = dx\ndt  = 4t \u2013 t2 = t(4 \u2013 t)\nThus, v = 0 gives t = 0 and/or t = 4 Now v = 0 at P as well as at Q and at P, t = 0 So, at Q, t = 4"
    },
    {
        "Chapter": "1",
        "sentence_range": "3084-3087",
        "Text": "Now\nx =\n2 2\nt3\nt \uf8eb\n\uf8f6\n\u2212\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\nTherefore\nv = dx\ndt  = 4t \u2013 t2 = t(4 \u2013 t)\nThus, v = 0 gives t = 0 and/or t = 4 Now v = 0 at P as well as at Q and at P, t = 0 So, at Q, t = 4 Thus, the car will\nreach the point Q after 4 seconds"
    },
    {
        "Chapter": "1",
        "sentence_range": "3085-3088",
        "Text": "Now v = 0 at P as well as at Q and at P, t = 0 So, at Q, t = 4 Thus, the car will\nreach the point Q after 4 seconds Also the distance travelled in 4 seconds is given by\nx]t = 4 =\n2\n4\n2\n32\n4\n2\n16\nm\n3\n3\n3\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n\u2212\n=\n=\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\nRationalised 2023-24\n MATHEMATICS\n178\nExample 31 A water tank has the shape of an inverted right circular cone with its axis\nvertical and vertex lowermost"
    },
    {
        "Chapter": "1",
        "sentence_range": "3086-3089",
        "Text": "So, at Q, t = 4 Thus, the car will\nreach the point Q after 4 seconds Also the distance travelled in 4 seconds is given by\nx]t = 4 =\n2\n4\n2\n32\n4\n2\n16\nm\n3\n3\n3\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n\u2212\n=\n=\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\nRationalised 2023-24\n MATHEMATICS\n178\nExample 31 A water tank has the shape of an inverted right circular cone with its axis\nvertical and vertex lowermost Its semi-vertical angle is tan\u20131(0"
    },
    {
        "Chapter": "1",
        "sentence_range": "3087-3090",
        "Text": "Thus, the car will\nreach the point Q after 4 seconds Also the distance travelled in 4 seconds is given by\nx]t = 4 =\n2\n4\n2\n32\n4\n2\n16\nm\n3\n3\n3\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n\u2212\n=\n=\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\nRationalised 2023-24\n MATHEMATICS\n178\nExample 31 A water tank has the shape of an inverted right circular cone with its axis\nvertical and vertex lowermost Its semi-vertical angle is tan\u20131(0 5)"
    },
    {
        "Chapter": "1",
        "sentence_range": "3088-3091",
        "Text": "Also the distance travelled in 4 seconds is given by\nx]t = 4 =\n2\n4\n2\n32\n4\n2\n16\nm\n3\n3\n3\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n\u2212\n=\n=\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\nRationalised 2023-24\n MATHEMATICS\n178\nExample 31 A water tank has the shape of an inverted right circular cone with its axis\nvertical and vertex lowermost Its semi-vertical angle is tan\u20131(0 5) Water is poured\ninto it at a constant rate of 5 cubic metre per hour"
    },
    {
        "Chapter": "1",
        "sentence_range": "3089-3092",
        "Text": "Its semi-vertical angle is tan\u20131(0 5) Water is poured\ninto it at a constant rate of 5 cubic metre per hour Find the rate at which the level of\nthe water is rising at the instant when the depth of water in the tank is 4 m"
    },
    {
        "Chapter": "1",
        "sentence_range": "3090-3093",
        "Text": "5) Water is poured\ninto it at a constant rate of 5 cubic metre per hour Find the rate at which the level of\nthe water is rising at the instant when the depth of water in the tank is 4 m Solution Let r, h and \u03b1 be as in Fig 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "3091-3094",
        "Text": "Water is poured\ninto it at a constant rate of 5 cubic metre per hour Find the rate at which the level of\nthe water is rising at the instant when the depth of water in the tank is 4 m Solution Let r, h and \u03b1 be as in Fig 6 20"
    },
    {
        "Chapter": "1",
        "sentence_range": "3092-3095",
        "Text": "Find the rate at which the level of\nthe water is rising at the instant when the depth of water in the tank is 4 m Solution Let r, h and \u03b1 be as in Fig 6 20 Then"
    },
    {
        "Chapter": "1",
        "sentence_range": "3093-3096",
        "Text": "Solution Let r, h and \u03b1 be as in Fig 6 20 Then tan\nhr\n\u03b1 =\nSo\n\u03b1 =\ntan1\nr\nh\n\u2212 \uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8"
    },
    {
        "Chapter": "1",
        "sentence_range": "3094-3097",
        "Text": "20 Then tan\nhr\n\u03b1 =\nSo\n\u03b1 =\ntan1\nr\nh\n\u2212 \uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8 But\n\u03b1 = tan\u20131(0"
    },
    {
        "Chapter": "1",
        "sentence_range": "3095-3098",
        "Text": "Then tan\nhr\n\u03b1 =\nSo\n\u03b1 =\ntan1\nr\nh\n\u2212 \uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8 But\n\u03b1 = tan\u20131(0 5)     (given)\nor\nr\nh  = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "3096-3099",
        "Text": "tan\nhr\n\u03b1 =\nSo\n\u03b1 =\ntan1\nr\nh\n\u2212 \uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8 But\n\u03b1 = tan\u20131(0 5)     (given)\nor\nr\nh  = 0 5\nor\nr = 2\nh\nLet V be the  volume of the cone"
    },
    {
        "Chapter": "1",
        "sentence_range": "3097-3100",
        "Text": "But\n\u03b1 = tan\u20131(0 5)     (given)\nor\nr\nh  = 0 5\nor\nr = 2\nh\nLet V be the  volume of the cone Then\nV =\n2\n3\n2\n1\n1\n3\n3\n2\n12\nh\nh\nr h\nh\n\u03c0\n\uf8eb\n\uf8f6\n\u03c0\n=\n\u03c0\n=\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\nTherefore\ndV\ndt  =\n3\n12\nd\nh\ndh\ndh\ndt\n\uf8eb\n\uf8f6\n\u03c0\n\u22c5\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n(by Chain Rule)\n=\n2\n4\ndh\nh dt\n\u03c0\nNow rate of change of volume, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "3098-3101",
        "Text": "5)     (given)\nor\nr\nh  = 0 5\nor\nr = 2\nh\nLet V be the  volume of the cone Then\nV =\n2\n3\n2\n1\n1\n3\n3\n2\n12\nh\nh\nr h\nh\n\u03c0\n\uf8eb\n\uf8f6\n\u03c0\n=\n\u03c0\n=\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\nTherefore\ndV\ndt  =\n3\n12\nd\nh\ndh\ndh\ndt\n\uf8eb\n\uf8f6\n\u03c0\n\u22c5\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n(by Chain Rule)\n=\n2\n4\ndh\nh dt\n\u03c0\nNow rate of change of volume, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "3099-3102",
        "Text": "5\nor\nr = 2\nh\nLet V be the  volume of the cone Then\nV =\n2\n3\n2\n1\n1\n3\n3\n2\n12\nh\nh\nr h\nh\n\u03c0\n\uf8eb\n\uf8f6\n\u03c0\n=\n\u03c0\n=\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\nTherefore\ndV\ndt  =\n3\n12\nd\nh\ndh\ndh\ndt\n\uf8eb\n\uf8f6\n\u03c0\n\u22c5\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n(by Chain Rule)\n=\n2\n4\ndh\nh dt\n\u03c0\nNow rate of change of volume, i e , V\n5\nd\ndt =\nm3/h and h = 4 m"
    },
    {
        "Chapter": "1",
        "sentence_range": "3100-3103",
        "Text": "Then\nV =\n2\n3\n2\n1\n1\n3\n3\n2\n12\nh\nh\nr h\nh\n\u03c0\n\uf8eb\n\uf8f6\n\u03c0\n=\n\u03c0\n=\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\nTherefore\ndV\ndt  =\n3\n12\nd\nh\ndh\ndh\ndt\n\uf8eb\n\uf8f6\n\u03c0\n\u22c5\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n(by Chain Rule)\n=\n2\n4\ndh\nh dt\n\u03c0\nNow rate of change of volume, i e , V\n5\nd\ndt =\nm3/h and h = 4 m Therefore\n5 =\n(4)2\n4\ndh\ndt\n\u03c0\n\u22c5\nor\ndh\ndt  = 5\n35\n22\nm/h\n4\n88\n7\n\uf8eb\n\uf8f6\n=\n\uf8ec\u03c0 =\n\uf8f7\n\u03c0\n\uf8ed\n\uf8f8\nThus, the rate of change of water level is 35 m/h\n88"
    },
    {
        "Chapter": "1",
        "sentence_range": "3101-3104",
        "Text": "e , V\n5\nd\ndt =\nm3/h and h = 4 m Therefore\n5 =\n(4)2\n4\ndh\ndt\n\u03c0\n\u22c5\nor\ndh\ndt  = 5\n35\n22\nm/h\n4\n88\n7\n\uf8eb\n\uf8f6\n=\n\uf8ec\u03c0 =\n\uf8f7\n\u03c0\n\uf8ed\n\uf8f8\nThus, the rate of change of water level is 35 m/h\n88 Example 32 A man of height 2 metres walks at a uniform speed of 5 km/h away from\na lamp post which is 6 metres high"
    },
    {
        "Chapter": "1",
        "sentence_range": "3102-3105",
        "Text": ", V\n5\nd\ndt =\nm3/h and h = 4 m Therefore\n5 =\n(4)2\n4\ndh\ndt\n\u03c0\n\u22c5\nor\ndh\ndt  = 5\n35\n22\nm/h\n4\n88\n7\n\uf8eb\n\uf8f6\n=\n\uf8ec\u03c0 =\n\uf8f7\n\u03c0\n\uf8ed\n\uf8f8\nThus, the rate of change of water level is 35 m/h\n88 Example 32 A man of height 2 metres walks at a uniform speed of 5 km/h away from\na lamp post which is 6 metres high Find the rate at which the length of his shadow\nincreases"
    },
    {
        "Chapter": "1",
        "sentence_range": "3103-3106",
        "Text": "Therefore\n5 =\n(4)2\n4\ndh\ndt\n\u03c0\n\u22c5\nor\ndh\ndt  = 5\n35\n22\nm/h\n4\n88\n7\n\uf8eb\n\uf8f6\n=\n\uf8ec\u03c0 =\n\uf8f7\n\u03c0\n\uf8ed\n\uf8f8\nThus, the rate of change of water level is 35 m/h\n88 Example 32 A man of height 2 metres walks at a uniform speed of 5 km/h away from\na lamp post which is 6 metres high Find the rate at which the length of his shadow\nincreases Fig 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "3104-3107",
        "Text": "Example 32 A man of height 2 metres walks at a uniform speed of 5 km/h away from\na lamp post which is 6 metres high Find the rate at which the length of his shadow\nincreases Fig 6 20\nRationalised 2023-24\nAPPLICATION OF DERIVATIVES\n179\nSolution In Fig 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "3105-3108",
        "Text": "Find the rate at which the length of his shadow\nincreases Fig 6 20\nRationalised 2023-24\nAPPLICATION OF DERIVATIVES\n179\nSolution In Fig 6 21, Let AB be the lamp-post, the\nlamp being at the position B and let MN be the man at\na particular time t and let AM = l metres"
    },
    {
        "Chapter": "1",
        "sentence_range": "3106-3109",
        "Text": "Fig 6 20\nRationalised 2023-24\nAPPLICATION OF DERIVATIVES\n179\nSolution In Fig 6 21, Let AB be the lamp-post, the\nlamp being at the position B and let MN be the man at\na particular time t and let AM = l metres Then, MS is\nthe shadow of the man"
    },
    {
        "Chapter": "1",
        "sentence_range": "3107-3110",
        "Text": "20\nRationalised 2023-24\nAPPLICATION OF DERIVATIVES\n179\nSolution In Fig 6 21, Let AB be the lamp-post, the\nlamp being at the position B and let MN be the man at\na particular time t and let AM = l metres Then, MS is\nthe shadow of the man Let MS = s metres"
    },
    {
        "Chapter": "1",
        "sentence_range": "3108-3111",
        "Text": "21, Let AB be the lamp-post, the\nlamp being at the position B and let MN be the man at\na particular time t and let AM = l metres Then, MS is\nthe shadow of the man Let MS = s metres Note that\n\u2206MSN ~ \u2206ASB\nor\nMS\nAS  = MN\nAB\nor\nAS = 3s (as MN =\n2 and AB = 6 (given))\nThus\nAM = 3s \u2013 s = 2s"
    },
    {
        "Chapter": "1",
        "sentence_range": "3109-3112",
        "Text": "Then, MS is\nthe shadow of the man Let MS = s metres Note that\n\u2206MSN ~ \u2206ASB\nor\nMS\nAS  = MN\nAB\nor\nAS = 3s (as MN =\n2 and AB = 6 (given))\nThus\nAM = 3s \u2013 s = 2s But AM = l\nSo\nl = 2s\nTherefore\ndl\ndt  = 2 ds\ndt\nSince \n5\ndl\ndt =\nkm/h"
    },
    {
        "Chapter": "1",
        "sentence_range": "3110-3113",
        "Text": "Let MS = s metres Note that\n\u2206MSN ~ \u2206ASB\nor\nMS\nAS  = MN\nAB\nor\nAS = 3s (as MN =\n2 and AB = 6 (given))\nThus\nAM = 3s \u2013 s = 2s But AM = l\nSo\nl = 2s\nTherefore\ndl\ndt  = 2 ds\ndt\nSince \n5\ndl\ndt =\nkm/h Hence, the length of the shadow increases at the rate 5\n2 km/h"
    },
    {
        "Chapter": "1",
        "sentence_range": "3111-3114",
        "Text": "Note that\n\u2206MSN ~ \u2206ASB\nor\nMS\nAS  = MN\nAB\nor\nAS = 3s (as MN =\n2 and AB = 6 (given))\nThus\nAM = 3s \u2013 s = 2s But AM = l\nSo\nl = 2s\nTherefore\ndl\ndt  = 2 ds\ndt\nSince \n5\ndl\ndt =\nkm/h Hence, the length of the shadow increases at the rate 5\n2 km/h Example 33 Find intervals in which the function given by\nf (x) =\n4\n3\n2\n3\n4\n36\n3\n11\n10\n5\n5\nx\nx\nx\nx\n\u2212\n\u2212\n+\n+\nis (a) increasing (b) decreasing"
    },
    {
        "Chapter": "1",
        "sentence_range": "3112-3115",
        "Text": "But AM = l\nSo\nl = 2s\nTherefore\ndl\ndt  = 2 ds\ndt\nSince \n5\ndl\ndt =\nkm/h Hence, the length of the shadow increases at the rate 5\n2 km/h Example 33 Find intervals in which the function given by\nf (x) =\n4\n3\n2\n3\n4\n36\n3\n11\n10\n5\n5\nx\nx\nx\nx\n\u2212\n\u2212\n+\n+\nis (a) increasing (b) decreasing Solution We have\nf (x) =\n4\n3\n2\n3\n4\n36\n3\n11\n10\n5\n5\nx\nx\nx\nx\n\u2212\n\u2212\n+\n+\nTherefore\nf \u2032(x) =\n3\n2\n3\n4\n36\n(4\n)\n(3\n)\n3(2 )\n10\n5\n5\nx\nx\nx\n\u2212\n\u2212\n+\n= 6(\n1)(\n2)(\n3)\n5 x\nx\nx\n\u2212\n+\n\u2212\n(on simplification)\nFig 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "3113-3116",
        "Text": "Hence, the length of the shadow increases at the rate 5\n2 km/h Example 33 Find intervals in which the function given by\nf (x) =\n4\n3\n2\n3\n4\n36\n3\n11\n10\n5\n5\nx\nx\nx\nx\n\u2212\n\u2212\n+\n+\nis (a) increasing (b) decreasing Solution We have\nf (x) =\n4\n3\n2\n3\n4\n36\n3\n11\n10\n5\n5\nx\nx\nx\nx\n\u2212\n\u2212\n+\n+\nTherefore\nf \u2032(x) =\n3\n2\n3\n4\n36\n(4\n)\n(3\n)\n3(2 )\n10\n5\n5\nx\nx\nx\n\u2212\n\u2212\n+\n= 6(\n1)(\n2)(\n3)\n5 x\nx\nx\n\u2212\n+\n\u2212\n(on simplification)\nFig 6 21\nRationalised 2023-24\n MATHEMATICS\n180\nNow f \u2032(x) = 0 gives x = 1, x = \u2013 2, or x = 3"
    },
    {
        "Chapter": "1",
        "sentence_range": "3114-3117",
        "Text": "Example 33 Find intervals in which the function given by\nf (x) =\n4\n3\n2\n3\n4\n36\n3\n11\n10\n5\n5\nx\nx\nx\nx\n\u2212\n\u2212\n+\n+\nis (a) increasing (b) decreasing Solution We have\nf (x) =\n4\n3\n2\n3\n4\n36\n3\n11\n10\n5\n5\nx\nx\nx\nx\n\u2212\n\u2212\n+\n+\nTherefore\nf \u2032(x) =\n3\n2\n3\n4\n36\n(4\n)\n(3\n)\n3(2 )\n10\n5\n5\nx\nx\nx\n\u2212\n\u2212\n+\n= 6(\n1)(\n2)(\n3)\n5 x\nx\nx\n\u2212\n+\n\u2212\n(on simplification)\nFig 6 21\nRationalised 2023-24\n MATHEMATICS\n180\nNow f \u2032(x) = 0 gives x = 1, x = \u2013 2, or x = 3 The\npoints x = 1, \u2013 2, and 3 divide the real line into four\ndisjoint intervals namely, (\u2013 \u221e, \u2013 2), (\u2013 2, 1), (1, 3)\nand (3, \u221e) (Fig 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "3115-3118",
        "Text": "Solution We have\nf (x) =\n4\n3\n2\n3\n4\n36\n3\n11\n10\n5\n5\nx\nx\nx\nx\n\u2212\n\u2212\n+\n+\nTherefore\nf \u2032(x) =\n3\n2\n3\n4\n36\n(4\n)\n(3\n)\n3(2 )\n10\n5\n5\nx\nx\nx\n\u2212\n\u2212\n+\n= 6(\n1)(\n2)(\n3)\n5 x\nx\nx\n\u2212\n+\n\u2212\n(on simplification)\nFig 6 21\nRationalised 2023-24\n MATHEMATICS\n180\nNow f \u2032(x) = 0 gives x = 1, x = \u2013 2, or x = 3 The\npoints x = 1, \u2013 2, and 3 divide the real line into four\ndisjoint intervals namely, (\u2013 \u221e, \u2013 2), (\u2013 2, 1), (1, 3)\nand (3, \u221e) (Fig 6 22)"
    },
    {
        "Chapter": "1",
        "sentence_range": "3116-3119",
        "Text": "21\nRationalised 2023-24\n MATHEMATICS\n180\nNow f \u2032(x) = 0 gives x = 1, x = \u2013 2, or x = 3 The\npoints x = 1, \u2013 2, and 3 divide the real line into four\ndisjoint intervals namely, (\u2013 \u221e, \u2013 2), (\u2013 2, 1), (1, 3)\nand (3, \u221e) (Fig 6 22) Consider the interval (\u2013 \u221e, \u2013 2), i"
    },
    {
        "Chapter": "1",
        "sentence_range": "3117-3120",
        "Text": "The\npoints x = 1, \u2013 2, and 3 divide the real line into four\ndisjoint intervals namely, (\u2013 \u221e, \u2013 2), (\u2013 2, 1), (1, 3)\nand (3, \u221e) (Fig 6 22) Consider the interval (\u2013 \u221e, \u2013 2), i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "3118-3121",
        "Text": "22) Consider the interval (\u2013 \u221e, \u2013 2), i e , when \u2013 \u221e < x < \u2013 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "3119-3122",
        "Text": "Consider the interval (\u2013 \u221e, \u2013 2), i e , when \u2013 \u221e < x < \u2013 2 In this case, we have x \u2013 1 < 0, x + 2 < 0 and x \u2013 3 < 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "3120-3123",
        "Text": "e , when \u2013 \u221e < x < \u2013 2 In this case, we have x \u2013 1 < 0, x + 2 < 0 and x \u2013 3 < 0 (In particular, observe that for x = \u20133, f \u2032(x) = (x \u2013 1) (x + 2) (x \u2013 3) = (\u2013 4) (\u2013 1)\n(\u2013 6) < 0)\nTherefore,\nf \u2032(x) < 0 when \u2013 \u221e < x < \u2013 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "3121-3124",
        "Text": ", when \u2013 \u221e < x < \u2013 2 In this case, we have x \u2013 1 < 0, x + 2 < 0 and x \u2013 3 < 0 (In particular, observe that for x = \u20133, f \u2032(x) = (x \u2013 1) (x + 2) (x \u2013 3) = (\u2013 4) (\u2013 1)\n(\u2013 6) < 0)\nTherefore,\nf \u2032(x) < 0 when \u2013 \u221e < x < \u2013 2 Thus, the function f is decreasing in (\u2013 \u221e, \u2013 2)"
    },
    {
        "Chapter": "1",
        "sentence_range": "3122-3125",
        "Text": "In this case, we have x \u2013 1 < 0, x + 2 < 0 and x \u2013 3 < 0 (In particular, observe that for x = \u20133, f \u2032(x) = (x \u2013 1) (x + 2) (x \u2013 3) = (\u2013 4) (\u2013 1)\n(\u2013 6) < 0)\nTherefore,\nf \u2032(x) < 0 when \u2013 \u221e < x < \u2013 2 Thus, the function f is decreasing in (\u2013 \u221e, \u2013 2) Consider the interval (\u2013 2, 1), i"
    },
    {
        "Chapter": "1",
        "sentence_range": "3123-3126",
        "Text": "(In particular, observe that for x = \u20133, f \u2032(x) = (x \u2013 1) (x + 2) (x \u2013 3) = (\u2013 4) (\u2013 1)\n(\u2013 6) < 0)\nTherefore,\nf \u2032(x) < 0 when \u2013 \u221e < x < \u2013 2 Thus, the function f is decreasing in (\u2013 \u221e, \u2013 2) Consider the interval (\u2013 2, 1), i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "3124-3127",
        "Text": "Thus, the function f is decreasing in (\u2013 \u221e, \u2013 2) Consider the interval (\u2013 2, 1), i e , when  \u2013 2 < x < 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "3125-3128",
        "Text": "Consider the interval (\u2013 2, 1), i e , when  \u2013 2 < x < 1 In this case, we have x \u2013 1 < 0, x + 2 > 0 and x \u2013 3 < 0\n(In particular, observe that for x = 0, f \u2032(x) = (x \u2013 1) (x + 2) (x \u2013 3) = (\u20131) (2) (\u20133)\n= 6 > 0)\nSo\nf \u2032(x) > 0 when \u2013 2 < x < 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "3126-3129",
        "Text": "e , when  \u2013 2 < x < 1 In this case, we have x \u2013 1 < 0, x + 2 > 0 and x \u2013 3 < 0\n(In particular, observe that for x = 0, f \u2032(x) = (x \u2013 1) (x + 2) (x \u2013 3) = (\u20131) (2) (\u20133)\n= 6 > 0)\nSo\nf \u2032(x) > 0 when \u2013 2 < x < 1 Thus,\nf is increasing in (\u2013 2, 1)"
    },
    {
        "Chapter": "1",
        "sentence_range": "3127-3130",
        "Text": ", when  \u2013 2 < x < 1 In this case, we have x \u2013 1 < 0, x + 2 > 0 and x \u2013 3 < 0\n(In particular, observe that for x = 0, f \u2032(x) = (x \u2013 1) (x + 2) (x \u2013 3) = (\u20131) (2) (\u20133)\n= 6 > 0)\nSo\nf \u2032(x) > 0 when \u2013 2 < x < 1 Thus,\nf is increasing in (\u2013 2, 1) Now consider the interval (1, 3), i"
    },
    {
        "Chapter": "1",
        "sentence_range": "3128-3131",
        "Text": "In this case, we have x \u2013 1 < 0, x + 2 > 0 and x \u2013 3 < 0\n(In particular, observe that for x = 0, f \u2032(x) = (x \u2013 1) (x + 2) (x \u2013 3) = (\u20131) (2) (\u20133)\n= 6 > 0)\nSo\nf \u2032(x) > 0 when \u2013 2 < x < 1 Thus,\nf is increasing in (\u2013 2, 1) Now consider the interval (1, 3), i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "3129-3132",
        "Text": "Thus,\nf is increasing in (\u2013 2, 1) Now consider the interval (1, 3), i e , when 1 < x < 3"
    },
    {
        "Chapter": "1",
        "sentence_range": "3130-3133",
        "Text": "Now consider the interval (1, 3), i e , when 1 < x < 3 In this case, we have\nx \u2013 1 > 0, x  + 2 > 0 and x \u2013 3 < 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "3131-3134",
        "Text": "e , when 1 < x < 3 In this case, we have\nx \u2013 1 > 0, x  + 2 > 0 and x \u2013 3 < 0 So,\nf \u2032(x) < 0 when 1 < x < 3"
    },
    {
        "Chapter": "1",
        "sentence_range": "3132-3135",
        "Text": ", when 1 < x < 3 In this case, we have\nx \u2013 1 > 0, x  + 2 > 0 and x \u2013 3 < 0 So,\nf \u2032(x) < 0 when 1 < x < 3 Thus,\n f is decreasing in (1, 3)"
    },
    {
        "Chapter": "1",
        "sentence_range": "3133-3136",
        "Text": "In this case, we have\nx \u2013 1 > 0, x  + 2 > 0 and x \u2013 3 < 0 So,\nf \u2032(x) < 0 when 1 < x < 3 Thus,\n f is decreasing in (1, 3) Finally, consider the interval (3, \u221e), i"
    },
    {
        "Chapter": "1",
        "sentence_range": "3134-3137",
        "Text": "So,\nf \u2032(x) < 0 when 1 < x < 3 Thus,\n f is decreasing in (1, 3) Finally, consider the interval (3, \u221e), i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "3135-3138",
        "Text": "Thus,\n f is decreasing in (1, 3) Finally, consider the interval (3, \u221e), i e , when  x > 3"
    },
    {
        "Chapter": "1",
        "sentence_range": "3136-3139",
        "Text": "Finally, consider the interval (3, \u221e), i e , when  x > 3 In this case, we have x \u2013 1 > 0,\nx + 2 > 0 and  x \u2013 3 > 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "3137-3140",
        "Text": "e , when  x > 3 In this case, we have x \u2013 1 > 0,\nx + 2 > 0 and  x \u2013 3 > 0 So f \u2032(x) > 0 when x > 3"
    },
    {
        "Chapter": "1",
        "sentence_range": "3138-3141",
        "Text": ", when  x > 3 In this case, we have x \u2013 1 > 0,\nx + 2 > 0 and  x \u2013 3 > 0 So f \u2032(x) > 0 when x > 3 Thus,  f  is increasing in the interval (3, \u221e)"
    },
    {
        "Chapter": "1",
        "sentence_range": "3139-3142",
        "Text": "In this case, we have x \u2013 1 > 0,\nx + 2 > 0 and  x \u2013 3 > 0 So f \u2032(x) > 0 when x > 3 Thus,  f  is increasing in the interval (3, \u221e) Example 34 Show that the function f given by\nf (x) = tan\u20131(sin x + cos x), x > 0\nis always an increasing function in 0 4\n\uf8ed\uf8ec\uf8eb, \u03c0\n\uf8f6\n\uf8f8\uf8f7"
    },
    {
        "Chapter": "1",
        "sentence_range": "3140-3143",
        "Text": "So f \u2032(x) > 0 when x > 3 Thus,  f  is increasing in the interval (3, \u221e) Example 34 Show that the function f given by\nf (x) = tan\u20131(sin x + cos x), x > 0\nis always an increasing function in 0 4\n\uf8ed\uf8ec\uf8eb, \u03c0\n\uf8f6\n\uf8f8\uf8f7 Solution We have\nf (x) = tan\u20131(sin x + cos x), x > 0\nTherefore\nf \u2032(x) =\n2\n1\n(cos\nsin )\n1\n(sin\ncos )\nx\nx\nx\nx\n\u2212\n+\n+\nFig 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "3141-3144",
        "Text": "Thus,  f  is increasing in the interval (3, \u221e) Example 34 Show that the function f given by\nf (x) = tan\u20131(sin x + cos x), x > 0\nis always an increasing function in 0 4\n\uf8ed\uf8ec\uf8eb, \u03c0\n\uf8f6\n\uf8f8\uf8f7 Solution We have\nf (x) = tan\u20131(sin x + cos x), x > 0\nTherefore\nf \u2032(x) =\n2\n1\n(cos\nsin )\n1\n(sin\ncos )\nx\nx\nx\nx\n\u2212\n+\n+\nFig 6 22\nRationalised 2023-24\nAPPLICATION OF DERIVATIVES\n181\n= cos\nsin\n2\nxsin 2\nx\nx\n\u2212\n+\n(on simplification)\nNote that 2 + sin 2x > 0 for all x in 0, 4\n\u03c0"
    },
    {
        "Chapter": "1",
        "sentence_range": "3142-3145",
        "Text": "Example 34 Show that the function f given by\nf (x) = tan\u20131(sin x + cos x), x > 0\nis always an increasing function in 0 4\n\uf8ed\uf8ec\uf8eb, \u03c0\n\uf8f6\n\uf8f8\uf8f7 Solution We have\nf (x) = tan\u20131(sin x + cos x), x > 0\nTherefore\nf \u2032(x) =\n2\n1\n(cos\nsin )\n1\n(sin\ncos )\nx\nx\nx\nx\n\u2212\n+\n+\nFig 6 22\nRationalised 2023-24\nAPPLICATION OF DERIVATIVES\n181\n= cos\nsin\n2\nxsin 2\nx\nx\n\u2212\n+\n(on simplification)\nNote that 2 + sin 2x > 0 for all x in 0, 4\n\u03c0 Therefore\nf \u2032(x) > 0  if cos x \u2013 sin x > 0\nor\nf \u2032(x) > 0  if cos x > sin x  or cot x > 1\nNow\ncot x > 1 if tan x < 1, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "3143-3146",
        "Text": "Solution We have\nf (x) = tan\u20131(sin x + cos x), x > 0\nTherefore\nf \u2032(x) =\n2\n1\n(cos\nsin )\n1\n(sin\ncos )\nx\nx\nx\nx\n\u2212\n+\n+\nFig 6 22\nRationalised 2023-24\nAPPLICATION OF DERIVATIVES\n181\n= cos\nsin\n2\nxsin 2\nx\nx\n\u2212\n+\n(on simplification)\nNote that 2 + sin 2x > 0 for all x in 0, 4\n\u03c0 Therefore\nf \u2032(x) > 0  if cos x \u2013 sin x > 0\nor\nf \u2032(x) > 0  if cos x > sin x  or cot x > 1\nNow\ncot x > 1 if tan x < 1, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "3144-3147",
        "Text": "22\nRationalised 2023-24\nAPPLICATION OF DERIVATIVES\n181\n= cos\nsin\n2\nxsin 2\nx\nx\n\u2212\n+\n(on simplification)\nNote that 2 + sin 2x > 0 for all x in 0, 4\n\u03c0 Therefore\nf \u2032(x) > 0  if cos x \u2013 sin x > 0\nor\nf \u2032(x) > 0  if cos x > sin x  or cot x > 1\nNow\ncot x > 1 if tan x < 1, i e , if 0\n4\nx\n\u03c0\n<\n<\nThus\nf \u2032(x) > 0  in 0 4\n\uf8ed\uf8ec\uf8eb, \u03c0\n\uf8f6\n\uf8f8\uf8f7\nHence f is increasing function in 0, 4\n\u03c0\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8"
    },
    {
        "Chapter": "1",
        "sentence_range": "3145-3148",
        "Text": "Therefore\nf \u2032(x) > 0  if cos x \u2013 sin x > 0\nor\nf \u2032(x) > 0  if cos x > sin x  or cot x > 1\nNow\ncot x > 1 if tan x < 1, i e , if 0\n4\nx\n\u03c0\n<\n<\nThus\nf \u2032(x) > 0  in 0 4\n\uf8ed\uf8ec\uf8eb, \u03c0\n\uf8f6\n\uf8f8\uf8f7\nHence f is increasing function in 0, 4\n\u03c0\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8 Example 35 A circular disc of radius 3 cm is being heated"
    },
    {
        "Chapter": "1",
        "sentence_range": "3146-3149",
        "Text": "e , if 0\n4\nx\n\u03c0\n<\n<\nThus\nf \u2032(x) > 0  in 0 4\n\uf8ed\uf8ec\uf8eb, \u03c0\n\uf8f6\n\uf8f8\uf8f7\nHence f is increasing function in 0, 4\n\u03c0\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8 Example 35 A circular disc of radius 3 cm is being heated Due to expansion, its\nradius increases at the rate of 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "3147-3150",
        "Text": ", if 0\n4\nx\n\u03c0\n<\n<\nThus\nf \u2032(x) > 0  in 0 4\n\uf8ed\uf8ec\uf8eb, \u03c0\n\uf8f6\n\uf8f8\uf8f7\nHence f is increasing function in 0, 4\n\u03c0\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8 Example 35 A circular disc of radius 3 cm is being heated Due to expansion, its\nradius increases at the rate of 0 05 cm/s"
    },
    {
        "Chapter": "1",
        "sentence_range": "3148-3151",
        "Text": "Example 35 A circular disc of radius 3 cm is being heated Due to expansion, its\nradius increases at the rate of 0 05 cm/s Find the rate at which its area is increasing\nwhen radius is 3"
    },
    {
        "Chapter": "1",
        "sentence_range": "3149-3152",
        "Text": "Due to expansion, its\nradius increases at the rate of 0 05 cm/s Find the rate at which its area is increasing\nwhen radius is 3 2 cm"
    },
    {
        "Chapter": "1",
        "sentence_range": "3150-3153",
        "Text": "05 cm/s Find the rate at which its area is increasing\nwhen radius is 3 2 cm Solution Let r be the radius of the given disc and A be its area"
    },
    {
        "Chapter": "1",
        "sentence_range": "3151-3154",
        "Text": "Find the rate at which its area is increasing\nwhen radius is 3 2 cm Solution Let r be the radius of the given disc and A be its area Then\nA = \u03c0r2\nor\ndA\ndt  = 2\ndr\n\u03c0r dt\n(by Chain Rule)\nNow approximate rate of increase of radius = dr = \n0"
    },
    {
        "Chapter": "1",
        "sentence_range": "3152-3155",
        "Text": "2 cm Solution Let r be the radius of the given disc and A be its area Then\nA = \u03c0r2\nor\ndA\ndt  = 2\ndr\n\u03c0r dt\n(by Chain Rule)\nNow approximate rate of increase of radius = dr = \n0 05\ndr\nt\ndt \u2206 =\ncm/s"
    },
    {
        "Chapter": "1",
        "sentence_range": "3153-3156",
        "Text": "Solution Let r be the radius of the given disc and A be its area Then\nA = \u03c0r2\nor\ndA\ndt  = 2\ndr\n\u03c0r dt\n(by Chain Rule)\nNow approximate rate of increase of radius = dr = \n0 05\ndr\nt\ndt \u2206 =\ncm/s Therefore, the approximate rate of increase in area is given by\ndA =\nA (\n)\nd\ndt \u2206t\n = 2\nrdr\nt\ndt\n\uf8eb\n\uf8f6\n\u03c0\n\u2206\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n= 2\u03c0 (3"
    },
    {
        "Chapter": "1",
        "sentence_range": "3154-3157",
        "Text": "Then\nA = \u03c0r2\nor\ndA\ndt  = 2\ndr\n\u03c0r dt\n(by Chain Rule)\nNow approximate rate of increase of radius = dr = \n0 05\ndr\nt\ndt \u2206 =\ncm/s Therefore, the approximate rate of increase in area is given by\ndA =\nA (\n)\nd\ndt \u2206t\n = 2\nrdr\nt\ndt\n\uf8eb\n\uf8f6\n\u03c0\n\u2206\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n= 2\u03c0 (3 2) (0"
    },
    {
        "Chapter": "1",
        "sentence_range": "3155-3158",
        "Text": "05\ndr\nt\ndt \u2206 =\ncm/s Therefore, the approximate rate of increase in area is given by\ndA =\nA (\n)\nd\ndt \u2206t\n = 2\nrdr\nt\ndt\n\uf8eb\n\uf8f6\n\u03c0\n\u2206\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n= 2\u03c0 (3 2) (0 05) = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "3156-3159",
        "Text": "Therefore, the approximate rate of increase in area is given by\ndA =\nA (\n)\nd\ndt \u2206t\n = 2\nrdr\nt\ndt\n\uf8eb\n\uf8f6\n\u03c0\n\u2206\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n= 2\u03c0 (3 2) (0 05) = 0 320\u03c0 cm2/s\n(r = 3"
    },
    {
        "Chapter": "1",
        "sentence_range": "3157-3160",
        "Text": "2) (0 05) = 0 320\u03c0 cm2/s\n(r = 3 2 cm)\nExample 36 An open topped box is to be constructed by removing equal squares from\neach corner of a 3 metre by 8 metre rectangular sheet of aluminium and folding up the\nsides"
    },
    {
        "Chapter": "1",
        "sentence_range": "3158-3161",
        "Text": "05) = 0 320\u03c0 cm2/s\n(r = 3 2 cm)\nExample 36 An open topped box is to be constructed by removing equal squares from\neach corner of a 3 metre by 8 metre rectangular sheet of aluminium and folding up the\nsides Find the volume of the largest such box"
    },
    {
        "Chapter": "1",
        "sentence_range": "3159-3162",
        "Text": "320\u03c0 cm2/s\n(r = 3 2 cm)\nExample 36 An open topped box is to be constructed by removing equal squares from\neach corner of a 3 metre by 8 metre rectangular sheet of aluminium and folding up the\nsides Find the volume of the largest such box Rationalised 2023-24\n MATHEMATICS\n182\nSolution Let x metre be the length of a side of the removed squares"
    },
    {
        "Chapter": "1",
        "sentence_range": "3160-3163",
        "Text": "2 cm)\nExample 36 An open topped box is to be constructed by removing equal squares from\neach corner of a 3 metre by 8 metre rectangular sheet of aluminium and folding up the\nsides Find the volume of the largest such box Rationalised 2023-24\n MATHEMATICS\n182\nSolution Let x metre be the length of a side of the removed squares Then, the height\nof the box is x, length is 8 \u2013 2x and breadth is 3 \u2013 2x (Fig 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "3161-3164",
        "Text": "Find the volume of the largest such box Rationalised 2023-24\n MATHEMATICS\n182\nSolution Let x metre be the length of a side of the removed squares Then, the height\nof the box is x, length is 8 \u2013 2x and breadth is 3 \u2013 2x (Fig 6 23)"
    },
    {
        "Chapter": "1",
        "sentence_range": "3162-3165",
        "Text": "Rationalised 2023-24\n MATHEMATICS\n182\nSolution Let x metre be the length of a side of the removed squares Then, the height\nof the box is x, length is 8 \u2013 2x and breadth is 3 \u2013 2x (Fig 6 23) If V(x) is the volume\nof the box, then\nFig 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "3163-3166",
        "Text": "Then, the height\nof the box is x, length is 8 \u2013 2x and breadth is 3 \u2013 2x (Fig 6 23) If V(x) is the volume\nof the box, then\nFig 6 23\nV(x) = x(3 \u2013 2x) (8 \u2013 2x)\n= 4x3 \u2013 22x2 + 24x\nTherefore\n2\nV ( )\n12\n44\n24\n4(\n3)(3\n2)\nV ( )\n24\n44\nx\nx\nx\nx\nx\nx\nx\n\uf8f1 \u2032\n=\n\u2212\n+\n=\n\u2212\n\u2212\n\uf8f4\uf8f2 \u2032\u2032\n=\n\u2212\n\uf8f4\uf8f3\nNow\nV\u2032(x) = 0 gives \nx =3, 32"
    },
    {
        "Chapter": "1",
        "sentence_range": "3164-3167",
        "Text": "23) If V(x) is the volume\nof the box, then\nFig 6 23\nV(x) = x(3 \u2013 2x) (8 \u2013 2x)\n= 4x3 \u2013 22x2 + 24x\nTherefore\n2\nV ( )\n12\n44\n24\n4(\n3)(3\n2)\nV ( )\n24\n44\nx\nx\nx\nx\nx\nx\nx\n\uf8f1 \u2032\n=\n\u2212\n+\n=\n\u2212\n\u2212\n\uf8f4\uf8f2 \u2032\u2032\n=\n\u2212\n\uf8f4\uf8f3\nNow\nV\u2032(x) = 0 gives \nx =3, 32 But x \u2260 3 (Why"
    },
    {
        "Chapter": "1",
        "sentence_range": "3165-3168",
        "Text": "If V(x) is the volume\nof the box, then\nFig 6 23\nV(x) = x(3 \u2013 2x) (8 \u2013 2x)\n= 4x3 \u2013 22x2 + 24x\nTherefore\n2\nV ( )\n12\n44\n24\n4(\n3)(3\n2)\nV ( )\n24\n44\nx\nx\nx\nx\nx\nx\nx\n\uf8f1 \u2032\n=\n\u2212\n+\n=\n\u2212\n\u2212\n\uf8f4\uf8f2 \u2032\u2032\n=\n\u2212\n\uf8f4\uf8f3\nNow\nV\u2032(x) = 0 gives \nx =3, 32 But x \u2260 3 (Why )\nThus, we have\nx =32"
    },
    {
        "Chapter": "1",
        "sentence_range": "3166-3169",
        "Text": "23\nV(x) = x(3 \u2013 2x) (8 \u2013 2x)\n= 4x3 \u2013 22x2 + 24x\nTherefore\n2\nV ( )\n12\n44\n24\n4(\n3)(3\n2)\nV ( )\n24\n44\nx\nx\nx\nx\nx\nx\nx\n\uf8f1 \u2032\n=\n\u2212\n+\n=\n\u2212\n\u2212\n\uf8f4\uf8f2 \u2032\u2032\n=\n\u2212\n\uf8f4\uf8f3\nNow\nV\u2032(x) = 0 gives \nx =3, 32 But x \u2260 3 (Why )\nThus, we have\nx =32 Now \n2\n2\nV\n24\n44\n28\n0\n3\n3\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n\u2032\u2032\n=\n\u2212\n= \u2212\n<\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8"
    },
    {
        "Chapter": "1",
        "sentence_range": "3167-3170",
        "Text": "But x \u2260 3 (Why )\nThus, we have\nx =32 Now \n2\n2\nV\n24\n44\n28\n0\n3\n3\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n\u2032\u2032\n=\n\u2212\n= \u2212\n<\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8 Therefore, \nx =32\n is the point of maxima, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "3168-3171",
        "Text": ")\nThus, we have\nx =32 Now \n2\n2\nV\n24\n44\n28\n0\n3\n3\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n\u2032\u2032\n=\n\u2212\n= \u2212\n<\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8 Therefore, \nx =32\n is the point of maxima, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "3169-3172",
        "Text": "Now \n2\n2\nV\n24\n44\n28\n0\n3\n3\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n\u2032\u2032\n=\n\u2212\n= \u2212\n<\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8 Therefore, \nx =32\n is the point of maxima, i e , if we remove a square of side 2\n3\nmetre from each corner of the sheet and make a box from the remaining sheet, then\nthe volume of the box such obtained will be the largest and it is given by\n2\nV 3\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8  =\n3\n2\n2\n2\n2\n4\n22\n24\n3\n3\n3\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n\u2212\n+\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n=\n200 m3\n27\nExample 37 Manufacturer can sell x items at a price of rupees 5\n\uf8ed\uf8ec\uf8eb\u2212100\n\uf8f8\uf8f7\uf8f6\nx\n each"
    },
    {
        "Chapter": "1",
        "sentence_range": "3170-3173",
        "Text": "Therefore, \nx =32\n is the point of maxima, i e , if we remove a square of side 2\n3\nmetre from each corner of the sheet and make a box from the remaining sheet, then\nthe volume of the box such obtained will be the largest and it is given by\n2\nV 3\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8  =\n3\n2\n2\n2\n2\n4\n22\n24\n3\n3\n3\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n\u2212\n+\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n=\n200 m3\n27\nExample 37 Manufacturer can sell x items at a price of rupees 5\n\uf8ed\uf8ec\uf8eb\u2212100\n\uf8f8\uf8f7\uf8f6\nx\n each The\ncost price of x items is Rs \n5x\n+500\n\uf8ed\uf8ec\uf8eb\n\uf8f6\n\uf8f8\uf8f7"
    },
    {
        "Chapter": "1",
        "sentence_range": "3171-3174",
        "Text": "e , if we remove a square of side 2\n3\nmetre from each corner of the sheet and make a box from the remaining sheet, then\nthe volume of the box such obtained will be the largest and it is given by\n2\nV 3\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8  =\n3\n2\n2\n2\n2\n4\n22\n24\n3\n3\n3\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n\u2212\n+\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n=\n200 m3\n27\nExample 37 Manufacturer can sell x items at a price of rupees 5\n\uf8ed\uf8ec\uf8eb\u2212100\n\uf8f8\uf8f7\uf8f6\nx\n each The\ncost price of x items is Rs \n5x\n+500\n\uf8ed\uf8ec\uf8eb\n\uf8f6\n\uf8f8\uf8f7 Find the number of items he should sell to earn\nmaximum profit"
    },
    {
        "Chapter": "1",
        "sentence_range": "3172-3175",
        "Text": ", if we remove a square of side 2\n3\nmetre from each corner of the sheet and make a box from the remaining sheet, then\nthe volume of the box such obtained will be the largest and it is given by\n2\nV 3\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8  =\n3\n2\n2\n2\n2\n4\n22\n24\n3\n3\n3\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n\u2212\n+\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n=\n200 m3\n27\nExample 37 Manufacturer can sell x items at a price of rupees 5\n\uf8ed\uf8ec\uf8eb\u2212100\n\uf8f8\uf8f7\uf8f6\nx\n each The\ncost price of x items is Rs \n5x\n+500\n\uf8ed\uf8ec\uf8eb\n\uf8f6\n\uf8f8\uf8f7 Find the number of items he should sell to earn\nmaximum profit Rationalised 2023-24\nAPPLICATION OF DERIVATIVES\n183\nSolution Let S(x) be the selling price of x items and let C(x) be the cost price of x\nitems"
    },
    {
        "Chapter": "1",
        "sentence_range": "3173-3176",
        "Text": "The\ncost price of x items is Rs \n5x\n+500\n\uf8ed\uf8ec\uf8eb\n\uf8f6\n\uf8f8\uf8f7 Find the number of items he should sell to earn\nmaximum profit Rationalised 2023-24\nAPPLICATION OF DERIVATIVES\n183\nSolution Let S(x) be the selling price of x items and let C(x) be the cost price of x\nitems Then, we have\nS(x) =\n2\n5\n5\n100\n100\nx\nx\nx\nx\n\uf8eb\n\uf8f6\n\u2212\n=\n\u2212\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\nand\nC(x) =\n500\n5\nx +\nThus, the profit function P(x) is given by\nP(x) =\n2\nS( )\nC( )\n5\n500\n100\n5\nx\nx\nx\nx\nx\n\u2212\n=\n\u2212\n\u2212\n\u2212\ni"
    },
    {
        "Chapter": "1",
        "sentence_range": "3174-3177",
        "Text": "Find the number of items he should sell to earn\nmaximum profit Rationalised 2023-24\nAPPLICATION OF DERIVATIVES\n183\nSolution Let S(x) be the selling price of x items and let C(x) be the cost price of x\nitems Then, we have\nS(x) =\n2\n5\n5\n100\n100\nx\nx\nx\nx\n\uf8eb\n\uf8f6\n\u2212\n=\n\u2212\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\nand\nC(x) =\n500\n5\nx +\nThus, the profit function P(x) is given by\nP(x) =\n2\nS( )\nC( )\n5\n500\n100\n5\nx\nx\nx\nx\nx\n\u2212\n=\n\u2212\n\u2212\n\u2212\ni e"
    },
    {
        "Chapter": "1",
        "sentence_range": "3175-3178",
        "Text": "Rationalised 2023-24\nAPPLICATION OF DERIVATIVES\n183\nSolution Let S(x) be the selling price of x items and let C(x) be the cost price of x\nitems Then, we have\nS(x) =\n2\n5\n5\n100\n100\nx\nx\nx\nx\n\uf8eb\n\uf8f6\n\u2212\n=\n\u2212\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\nand\nC(x) =\n500\n5\nx +\nThus, the profit function P(x) is given by\nP(x) =\n2\nS( )\nC( )\n5\n500\n100\n5\nx\nx\nx\nx\nx\n\u2212\n=\n\u2212\n\u2212\n\u2212\ni e P(x) =\n2\n24\n500\n5\n100\nx \u2212x\n\u2212\nor\nP\u2032(x) = 24\n5\nx50\n\u2212\nNow P\u2032(x) = 0 gives x = 240"
    },
    {
        "Chapter": "1",
        "sentence_range": "3176-3179",
        "Text": "Then, we have\nS(x) =\n2\n5\n5\n100\n100\nx\nx\nx\nx\n\uf8eb\n\uf8f6\n\u2212\n=\n\u2212\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\nand\nC(x) =\n500\n5\nx +\nThus, the profit function P(x) is given by\nP(x) =\n2\nS( )\nC( )\n5\n500\n100\n5\nx\nx\nx\nx\nx\n\u2212\n=\n\u2212\n\u2212\n\u2212\ni e P(x) =\n2\n24\n500\n5\n100\nx \u2212x\n\u2212\nor\nP\u2032(x) = 24\n5\nx50\n\u2212\nNow P\u2032(x) = 0 gives x = 240 Also \n1\nP ( )\n50\nx\n\u2212\n\u2032\u2032\n="
    },
    {
        "Chapter": "1",
        "sentence_range": "3177-3180",
        "Text": "e P(x) =\n2\n24\n500\n5\n100\nx \u2212x\n\u2212\nor\nP\u2032(x) = 24\n5\nx50\n\u2212\nNow P\u2032(x) = 0 gives x = 240 Also \n1\nP ( )\n50\nx\n\u2212\n\u2032\u2032\n= So \n1\nP (240)\n0\n\u221250\n\u2032\u2032\n=\n<\nThus, x = 240 is a point of maxima"
    },
    {
        "Chapter": "1",
        "sentence_range": "3178-3181",
        "Text": "P(x) =\n2\n24\n500\n5\n100\nx \u2212x\n\u2212\nor\nP\u2032(x) = 24\n5\nx50\n\u2212\nNow P\u2032(x) = 0 gives x = 240 Also \n1\nP ( )\n50\nx\n\u2212\n\u2032\u2032\n= So \n1\nP (240)\n0\n\u221250\n\u2032\u2032\n=\n<\nThus, x = 240 is a point of maxima Hence, the manufacturer can earn maximum\nprofit, if he sells 240 items"
    },
    {
        "Chapter": "1",
        "sentence_range": "3179-3182",
        "Text": "Also \n1\nP ( )\n50\nx\n\u2212\n\u2032\u2032\n= So \n1\nP (240)\n0\n\u221250\n\u2032\u2032\n=\n<\nThus, x = 240 is a point of maxima Hence, the manufacturer can earn maximum\nprofit, if he sells 240 items Miscellaneous Exercise on Chapter 6\n1"
    },
    {
        "Chapter": "1",
        "sentence_range": "3180-3183",
        "Text": "So \n1\nP (240)\n0\n\u221250\n\u2032\u2032\n=\n<\nThus, x = 240 is a point of maxima Hence, the manufacturer can earn maximum\nprofit, if he sells 240 items Miscellaneous Exercise on Chapter 6\n1 Show that the function given by \nlog\n( )\nx\nf x\nx\n=\n has maximum at x = e"
    },
    {
        "Chapter": "1",
        "sentence_range": "3181-3184",
        "Text": "Hence, the manufacturer can earn maximum\nprofit, if he sells 240 items Miscellaneous Exercise on Chapter 6\n1 Show that the function given by \nlog\n( )\nx\nf x\nx\n=\n has maximum at x = e 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "3182-3185",
        "Text": "Miscellaneous Exercise on Chapter 6\n1 Show that the function given by \nlog\n( )\nx\nf x\nx\n=\n has maximum at x = e 2 The two equal sides of an isosceles triangle with fixed base b are decreasing at\nthe rate of 3 cm per second"
    },
    {
        "Chapter": "1",
        "sentence_range": "3183-3186",
        "Text": "Show that the function given by \nlog\n( )\nx\nf x\nx\n=\n has maximum at x = e 2 The two equal sides of an isosceles triangle with fixed base b are decreasing at\nthe rate of 3 cm per second How fast is the area decreasing when the two equal\nsides are equal to the base"
    },
    {
        "Chapter": "1",
        "sentence_range": "3184-3187",
        "Text": "2 The two equal sides of an isosceles triangle with fixed base b are decreasing at\nthe rate of 3 cm per second How fast is the area decreasing when the two equal\nsides are equal to the base 3"
    },
    {
        "Chapter": "1",
        "sentence_range": "3185-3188",
        "Text": "The two equal sides of an isosceles triangle with fixed base b are decreasing at\nthe rate of 3 cm per second How fast is the area decreasing when the two equal\nsides are equal to the base 3 Find the intervals in which the function f given by\n4sin\n2\ncos\n( )\n2\ncos\nx\nx\nx\nx\nf x\nx\n\u2212\n\u2212\n=\n+\nis (i) increasing (ii) decreasing"
    },
    {
        "Chapter": "1",
        "sentence_range": "3186-3189",
        "Text": "How fast is the area decreasing when the two equal\nsides are equal to the base 3 Find the intervals in which the function f given by\n4sin\n2\ncos\n( )\n2\ncos\nx\nx\nx\nx\nf x\nx\n\u2212\n\u2212\n=\n+\nis (i) increasing (ii) decreasing 4"
    },
    {
        "Chapter": "1",
        "sentence_range": "3187-3190",
        "Text": "3 Find the intervals in which the function f given by\n4sin\n2\ncos\n( )\n2\ncos\nx\nx\nx\nx\nf x\nx\n\u2212\n\u2212\n=\n+\nis (i) increasing (ii) decreasing 4 Find the intervals in which the function f given by\n3\n13\n( )\n,\n0\nf x\nx\nx\nx\n=\n+\n\u2260\n is\n(i) increasing\n(ii) decreasing"
    },
    {
        "Chapter": "1",
        "sentence_range": "3188-3191",
        "Text": "Find the intervals in which the function f given by\n4sin\n2\ncos\n( )\n2\ncos\nx\nx\nx\nx\nf x\nx\n\u2212\n\u2212\n=\n+\nis (i) increasing (ii) decreasing 4 Find the intervals in which the function f given by\n3\n13\n( )\n,\n0\nf x\nx\nx\nx\n=\n+\n\u2260\n is\n(i) increasing\n(ii) decreasing Rationalised 2023-24\n MATHEMATICS\n184\n5"
    },
    {
        "Chapter": "1",
        "sentence_range": "3189-3192",
        "Text": "4 Find the intervals in which the function f given by\n3\n13\n( )\n,\n0\nf x\nx\nx\nx\n=\n+\n\u2260\n is\n(i) increasing\n(ii) decreasing Rationalised 2023-24\n MATHEMATICS\n184\n5 Find the maximum area of an isosceles triangle inscribed in the ellipse \n2\n2\n2\n2\n1\nx\ny\na\nb\n+\n=\nwith its vertex at one end of the major axis"
    },
    {
        "Chapter": "1",
        "sentence_range": "3190-3193",
        "Text": "Find the intervals in which the function f given by\n3\n13\n( )\n,\n0\nf x\nx\nx\nx\n=\n+\n\u2260\n is\n(i) increasing\n(ii) decreasing Rationalised 2023-24\n MATHEMATICS\n184\n5 Find the maximum area of an isosceles triangle inscribed in the ellipse \n2\n2\n2\n2\n1\nx\ny\na\nb\n+\n=\nwith its vertex at one end of the major axis 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "3191-3194",
        "Text": "Rationalised 2023-24\n MATHEMATICS\n184\n5 Find the maximum area of an isosceles triangle inscribed in the ellipse \n2\n2\n2\n2\n1\nx\ny\na\nb\n+\n=\nwith its vertex at one end of the major axis 6 A tank with rectangular base and rectangular sides, open at the top is to be\nconstructed so that its depth is 2 m and volume is 8 m3"
    },
    {
        "Chapter": "1",
        "sentence_range": "3192-3195",
        "Text": "Find the maximum area of an isosceles triangle inscribed in the ellipse \n2\n2\n2\n2\n1\nx\ny\na\nb\n+\n=\nwith its vertex at one end of the major axis 6 A tank with rectangular base and rectangular sides, open at the top is to be\nconstructed so that its depth is 2 m and volume is 8 m3 If building of tank costs\nRs 70 per sq metres for the base and Rs 45 per square metre for sides"
    },
    {
        "Chapter": "1",
        "sentence_range": "3193-3196",
        "Text": "6 A tank with rectangular base and rectangular sides, open at the top is to be\nconstructed so that its depth is 2 m and volume is 8 m3 If building of tank costs\nRs 70 per sq metres for the base and Rs 45 per square metre for sides What is\nthe cost of least expensive tank"
    },
    {
        "Chapter": "1",
        "sentence_range": "3194-3197",
        "Text": "A tank with rectangular base and rectangular sides, open at the top is to be\nconstructed so that its depth is 2 m and volume is 8 m3 If building of tank costs\nRs 70 per sq metres for the base and Rs 45 per square metre for sides What is\nthe cost of least expensive tank 7"
    },
    {
        "Chapter": "1",
        "sentence_range": "3195-3198",
        "Text": "If building of tank costs\nRs 70 per sq metres for the base and Rs 45 per square metre for sides What is\nthe cost of least expensive tank 7 The sum of the perimeter of a circle and square is k, where k is some constant"
    },
    {
        "Chapter": "1",
        "sentence_range": "3196-3199",
        "Text": "What is\nthe cost of least expensive tank 7 The sum of the perimeter of a circle and square is k, where k is some constant Prove that the sum of their areas is least when the side of square is double the\nradius of the circle"
    },
    {
        "Chapter": "1",
        "sentence_range": "3197-3200",
        "Text": "7 The sum of the perimeter of a circle and square is k, where k is some constant Prove that the sum of their areas is least when the side of square is double the\nradius of the circle 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "3198-3201",
        "Text": "The sum of the perimeter of a circle and square is k, where k is some constant Prove that the sum of their areas is least when the side of square is double the\nradius of the circle 8 A window is in the form of a rectangle surmounted by a semicircular opening"
    },
    {
        "Chapter": "1",
        "sentence_range": "3199-3202",
        "Text": "Prove that the sum of their areas is least when the side of square is double the\nradius of the circle 8 A window is in the form of a rectangle surmounted by a semicircular opening The total perimeter of the window is 10 m"
    },
    {
        "Chapter": "1",
        "sentence_range": "3200-3203",
        "Text": "8 A window is in the form of a rectangle surmounted by a semicircular opening The total perimeter of the window is 10 m Find the dimensions of the window to\nadmit maximum light through the whole opening"
    },
    {
        "Chapter": "1",
        "sentence_range": "3201-3204",
        "Text": "A window is in the form of a rectangle surmounted by a semicircular opening The total perimeter of the window is 10 m Find the dimensions of the window to\nadmit maximum light through the whole opening 9"
    },
    {
        "Chapter": "1",
        "sentence_range": "3202-3205",
        "Text": "The total perimeter of the window is 10 m Find the dimensions of the window to\nadmit maximum light through the whole opening 9 A point on the hypotenuse of a triangle is at distance a and b from the sides of\nthe triangle"
    },
    {
        "Chapter": "1",
        "sentence_range": "3203-3206",
        "Text": "Find the dimensions of the window to\nadmit maximum light through the whole opening 9 A point on the hypotenuse of a triangle is at distance a and b from the sides of\nthe triangle Show that the minimum length of the hypotenuse is \n2\n2\n3\n3\n3\n2\n(\n)\na\n+b"
    },
    {
        "Chapter": "1",
        "sentence_range": "3204-3207",
        "Text": "9 A point on the hypotenuse of a triangle is at distance a and b from the sides of\nthe triangle Show that the minimum length of the hypotenuse is \n2\n2\n3\n3\n3\n2\n(\n)\na\n+b 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "3205-3208",
        "Text": "A point on the hypotenuse of a triangle is at distance a and b from the sides of\nthe triangle Show that the minimum length of the hypotenuse is \n2\n2\n3\n3\n3\n2\n(\n)\na\n+b 10 Find the points at which the function  f given by  f (x) = (x \u2013 2)4 (x + 1)3 has\n(i) local maxima\n(ii) local minima\n(iii) point of inflexion\n11"
    },
    {
        "Chapter": "1",
        "sentence_range": "3206-3209",
        "Text": "Show that the minimum length of the hypotenuse is \n2\n2\n3\n3\n3\n2\n(\n)\na\n+b 10 Find the points at which the function  f given by  f (x) = (x \u2013 2)4 (x + 1)3 has\n(i) local maxima\n(ii) local minima\n(iii) point of inflexion\n11 Find the absolute maximum and minimum values of the function f given by\nf (x) = cos2 x + sin x, x \u2208 [0, \u03c0]\n12"
    },
    {
        "Chapter": "1",
        "sentence_range": "3207-3210",
        "Text": "10 Find the points at which the function  f given by  f (x) = (x \u2013 2)4 (x + 1)3 has\n(i) local maxima\n(ii) local minima\n(iii) point of inflexion\n11 Find the absolute maximum and minimum values of the function f given by\nf (x) = cos2 x + sin x, x \u2208 [0, \u03c0]\n12 Show that the altitude of the right circular cone of maximum volume that can be\ninscribed in a sphere of radius r is 4\n3\nr"
    },
    {
        "Chapter": "1",
        "sentence_range": "3208-3211",
        "Text": "Find the points at which the function  f given by  f (x) = (x \u2013 2)4 (x + 1)3 has\n(i) local maxima\n(ii) local minima\n(iii) point of inflexion\n11 Find the absolute maximum and minimum values of the function f given by\nf (x) = cos2 x + sin x, x \u2208 [0, \u03c0]\n12 Show that the altitude of the right circular cone of maximum volume that can be\ninscribed in a sphere of radius r is 4\n3\nr 13"
    },
    {
        "Chapter": "1",
        "sentence_range": "3209-3212",
        "Text": "Find the absolute maximum and minimum values of the function f given by\nf (x) = cos2 x + sin x, x \u2208 [0, \u03c0]\n12 Show that the altitude of the right circular cone of maximum volume that can be\ninscribed in a sphere of radius r is 4\n3\nr 13 Let f be a function defined on [a, b] such that f \u2032(x) > 0, for all x \u2208 (a, b)"
    },
    {
        "Chapter": "1",
        "sentence_range": "3210-3213",
        "Text": "Show that the altitude of the right circular cone of maximum volume that can be\ninscribed in a sphere of radius r is 4\n3\nr 13 Let f be a function defined on [a, b] such that f \u2032(x) > 0, for all x \u2208 (a, b) Then\nprove that f  is an increasing function on (a, b)"
    },
    {
        "Chapter": "1",
        "sentence_range": "3211-3214",
        "Text": "13 Let f be a function defined on [a, b] such that f \u2032(x) > 0, for all x \u2208 (a, b) Then\nprove that f  is an increasing function on (a, b) 14"
    },
    {
        "Chapter": "1",
        "sentence_range": "3212-3215",
        "Text": "Let f be a function defined on [a, b] such that f \u2032(x) > 0, for all x \u2208 (a, b) Then\nprove that f  is an increasing function on (a, b) 14 Show that the height of the cylinder of maximum volume that can be inscribed in\na sphere of radius R is 2R\n3"
    },
    {
        "Chapter": "1",
        "sentence_range": "3213-3216",
        "Text": "Then\nprove that f  is an increasing function on (a, b) 14 Show that the height of the cylinder of maximum volume that can be inscribed in\na sphere of radius R is 2R\n3 Also find the maximum volume"
    },
    {
        "Chapter": "1",
        "sentence_range": "3214-3217",
        "Text": "14 Show that the height of the cylinder of maximum volume that can be inscribed in\na sphere of radius R is 2R\n3 Also find the maximum volume 15"
    },
    {
        "Chapter": "1",
        "sentence_range": "3215-3218",
        "Text": "Show that the height of the cylinder of maximum volume that can be inscribed in\na sphere of radius R is 2R\n3 Also find the maximum volume 15 Show that height of the cylinder of greatest volume which can be inscribed in a\nright circular cone of height h and semi vertical angle \u03b1 is one-third that of the\ncone and the greatest volume of cylinder is \n3\n2\n4\n27 htan\n\u03c0\n\u03b1"
    },
    {
        "Chapter": "1",
        "sentence_range": "3216-3219",
        "Text": "Also find the maximum volume 15 Show that height of the cylinder of greatest volume which can be inscribed in a\nright circular cone of height h and semi vertical angle \u03b1 is one-third that of the\ncone and the greatest volume of cylinder is \n3\n2\n4\n27 htan\n\u03c0\n\u03b1 Rationalised 2023-24\nAPPLICATION OF DERIVATIVES\n185\n16"
    },
    {
        "Chapter": "1",
        "sentence_range": "3217-3220",
        "Text": "15 Show that height of the cylinder of greatest volume which can be inscribed in a\nright circular cone of height h and semi vertical angle \u03b1 is one-third that of the\ncone and the greatest volume of cylinder is \n3\n2\n4\n27 htan\n\u03c0\n\u03b1 Rationalised 2023-24\nAPPLICATION OF DERIVATIVES\n185\n16 A cylindrical tank of radius 10 m is being filled with wheat at the rate of 314\ncubic metre per hour"
    },
    {
        "Chapter": "1",
        "sentence_range": "3218-3221",
        "Text": "Show that height of the cylinder of greatest volume which can be inscribed in a\nright circular cone of height h and semi vertical angle \u03b1 is one-third that of the\ncone and the greatest volume of cylinder is \n3\n2\n4\n27 htan\n\u03c0\n\u03b1 Rationalised 2023-24\nAPPLICATION OF DERIVATIVES\n185\n16 A cylindrical tank of radius 10 m is being filled with wheat at the rate of 314\ncubic metre per hour Then the depth of the wheat is increasing at the rate of\n(A) 1 m/h\n(B) 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "3219-3222",
        "Text": "Rationalised 2023-24\nAPPLICATION OF DERIVATIVES\n185\n16 A cylindrical tank of radius 10 m is being filled with wheat at the rate of 314\ncubic metre per hour Then the depth of the wheat is increasing at the rate of\n(A) 1 m/h\n(B) 0 1 m/h\n(C)  1"
    },
    {
        "Chapter": "1",
        "sentence_range": "3220-3223",
        "Text": "A cylindrical tank of radius 10 m is being filled with wheat at the rate of 314\ncubic metre per hour Then the depth of the wheat is increasing at the rate of\n(A) 1 m/h\n(B) 0 1 m/h\n(C)  1 1 m/h\n(D) 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "3221-3224",
        "Text": "Then the depth of the wheat is increasing at the rate of\n(A) 1 m/h\n(B) 0 1 m/h\n(C)  1 1 m/h\n(D) 0 5 m/h\nSummary\n\u00ae If a quantity y varies with another quantity x, satisfying some rule \n( )\ny\n=f x\n,\nthen dy\ndx  (or \nf( )\n\u2032x\n) represents the rate of change of y with respect to x and\n0\nx x\ndy\ndx\n=\n   \n (or \n( )0\n\u2032f x\n) represents the  rate of change of y with respect to x at\n0\nx\n=x"
    },
    {
        "Chapter": "1",
        "sentence_range": "3222-3225",
        "Text": "1 m/h\n(C)  1 1 m/h\n(D) 0 5 m/h\nSummary\n\u00ae If a quantity y varies with another quantity x, satisfying some rule \n( )\ny\n=f x\n,\nthen dy\ndx  (or \nf( )\n\u2032x\n) represents the rate of change of y with respect to x and\n0\nx x\ndy\ndx\n=\n   \n (or \n( )0\n\u2032f x\n) represents the  rate of change of y with respect to x at\n0\nx\n=x \u00ae If two variables x and y are varying with respect to another variable t, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "3223-3226",
        "Text": "1 m/h\n(D) 0 5 m/h\nSummary\n\u00ae If a quantity y varies with another quantity x, satisfying some rule \n( )\ny\n=f x\n,\nthen dy\ndx  (or \nf( )\n\u2032x\n) represents the rate of change of y with respect to x and\n0\nx x\ndy\ndx\n=\n   \n (or \n( )0\n\u2032f x\n) represents the  rate of change of y with respect to x at\n0\nx\n=x \u00ae If two variables x and y are varying with respect to another variable t, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "3224-3227",
        "Text": "5 m/h\nSummary\n\u00ae If a quantity y varies with another quantity x, satisfying some rule \n( )\ny\n=f x\n,\nthen dy\ndx  (or \nf( )\n\u2032x\n) represents the rate of change of y with respect to x and\n0\nx x\ndy\ndx\n=\n   \n (or \n( )0\n\u2032f x\n) represents the  rate of change of y with respect to x at\n0\nx\n=x \u00ae If two variables x and y are varying with respect to another variable t, i e , if\n( )\nx\n=f t\nand \n( )\ny\n=g t\n, then by Chain Rule\ndy\ndy\ndx\ndt\ndt\ndx =\n,  if  \n0\ndx\ndt \u2260"
    },
    {
        "Chapter": "1",
        "sentence_range": "3225-3228",
        "Text": "\u00ae If two variables x and y are varying with respect to another variable t, i e , if\n( )\nx\n=f t\nand \n( )\ny\n=g t\n, then by Chain Rule\ndy\ndy\ndx\ndt\ndt\ndx =\n,  if  \n0\ndx\ndt \u2260 \u00ae A function  f  is said to be\n(a) increasing on an interval (a, b) if\nx1 < x2  in (a, b) \u21d2 f (x1) < f (x2) for all x1, x2 \u2208 (a, b)"
    },
    {
        "Chapter": "1",
        "sentence_range": "3226-3229",
        "Text": "e , if\n( )\nx\n=f t\nand \n( )\ny\n=g t\n, then by Chain Rule\ndy\ndy\ndx\ndt\ndt\ndx =\n,  if  \n0\ndx\ndt \u2260 \u00ae A function  f  is said to be\n(a) increasing on an interval (a, b) if\nx1 < x2  in (a, b) \u21d2 f (x1) < f (x2) for all x1, x2 \u2208 (a, b) Alternatively, if f \u2032(x) \u2265 0 for each x in (a, b)\n(b) decreasing on (a,b) if\nx1 < x2  in (a, b) \u21d2 f (x1) > f (x2) for all x1, x2 \u2208 (a, b)"
    },
    {
        "Chapter": "1",
        "sentence_range": "3227-3230",
        "Text": ", if\n( )\nx\n=f t\nand \n( )\ny\n=g t\n, then by Chain Rule\ndy\ndy\ndx\ndt\ndt\ndx =\n,  if  \n0\ndx\ndt \u2260 \u00ae A function  f  is said to be\n(a) increasing on an interval (a, b) if\nx1 < x2  in (a, b) \u21d2 f (x1) < f (x2) for all x1, x2 \u2208 (a, b) Alternatively, if f \u2032(x) \u2265 0 for each x in (a, b)\n(b) decreasing on (a,b) if\nx1 < x2  in (a, b) \u21d2 f (x1) > f (x2) for all x1, x2 \u2208 (a, b) (c) constant in (a, b), if f (x) = c for all x \u2208 (a, b), where c is a constant"
    },
    {
        "Chapter": "1",
        "sentence_range": "3228-3231",
        "Text": "\u00ae A function  f  is said to be\n(a) increasing on an interval (a, b) if\nx1 < x2  in (a, b) \u21d2 f (x1) < f (x2) for all x1, x2 \u2208 (a, b) Alternatively, if f \u2032(x) \u2265 0 for each x in (a, b)\n(b) decreasing on (a,b) if\nx1 < x2  in (a, b) \u21d2 f (x1) > f (x2) for all x1, x2 \u2208 (a, b) (c) constant in (a, b), if f (x) = c for all x \u2208 (a, b), where c is a constant \u00ae A point c in the domain of a function f at which either f \u2032(c) = 0 or f is not\ndifferentiable is called a critical point of f"
    },
    {
        "Chapter": "1",
        "sentence_range": "3229-3232",
        "Text": "Alternatively, if f \u2032(x) \u2265 0 for each x in (a, b)\n(b) decreasing on (a,b) if\nx1 < x2  in (a, b) \u21d2 f (x1) > f (x2) for all x1, x2 \u2208 (a, b) (c) constant in (a, b), if f (x) = c for all x \u2208 (a, b), where c is a constant \u00ae A point c in the domain of a function f at which either f \u2032(c) = 0 or f is not\ndifferentiable is called a critical point of f \u00ae First Derivative Test Let f  be a function defined on an open interval I"
    },
    {
        "Chapter": "1",
        "sentence_range": "3230-3233",
        "Text": "(c) constant in (a, b), if f (x) = c for all x \u2208 (a, b), where c is a constant \u00ae A point c in the domain of a function f at which either f \u2032(c) = 0 or f is not\ndifferentiable is called a critical point of f \u00ae First Derivative Test Let f  be a function defined on an open interval I Let\nf   be continuous at a critical point c in I"
    },
    {
        "Chapter": "1",
        "sentence_range": "3231-3234",
        "Text": "\u00ae A point c in the domain of a function f at which either f \u2032(c) = 0 or f is not\ndifferentiable is called a critical point of f \u00ae First Derivative Test Let f  be a function defined on an open interval I Let\nf   be continuous at a critical point c in I Then\n (i) If f \u2032(x) changes sign from positive to negative as x increases through c,\ni"
    },
    {
        "Chapter": "1",
        "sentence_range": "3232-3235",
        "Text": "\u00ae First Derivative Test Let f  be a function defined on an open interval I Let\nf   be continuous at a critical point c in I Then\n (i) If f \u2032(x) changes sign from positive to negative as x increases through c,\ni e"
    },
    {
        "Chapter": "1",
        "sentence_range": "3233-3236",
        "Text": "Let\nf   be continuous at a critical point c in I Then\n (i) If f \u2032(x) changes sign from positive to negative as x increases through c,\ni e , if f \u2032(x) > 0 at every point sufficiently close to and to the left of c,\nand f \u2032(x) < 0 at every point sufficiently close to and to the right of c,\nthen c is a point of local maxima"
    },
    {
        "Chapter": "1",
        "sentence_range": "3234-3237",
        "Text": "Then\n (i) If f \u2032(x) changes sign from positive to negative as x increases through c,\ni e , if f \u2032(x) > 0 at every point sufficiently close to and to the left of c,\nand f \u2032(x) < 0 at every point sufficiently close to and to the right of c,\nthen c is a point of local maxima Rationalised 2023-24\n MATHEMATICS\n186\n(ii)\nIf f \u2032(x) changes sign from negative to positive as x increases through c,\ni"
    },
    {
        "Chapter": "1",
        "sentence_range": "3235-3238",
        "Text": "e , if f \u2032(x) > 0 at every point sufficiently close to and to the left of c,\nand f \u2032(x) < 0 at every point sufficiently close to and to the right of c,\nthen c is a point of local maxima Rationalised 2023-24\n MATHEMATICS\n186\n(ii)\nIf f \u2032(x) changes sign from negative to positive as x increases through c,\ni e"
    },
    {
        "Chapter": "1",
        "sentence_range": "3236-3239",
        "Text": ", if f \u2032(x) > 0 at every point sufficiently close to and to the left of c,\nand f \u2032(x) < 0 at every point sufficiently close to and to the right of c,\nthen c is a point of local maxima Rationalised 2023-24\n MATHEMATICS\n186\n(ii)\nIf f \u2032(x) changes sign from negative to positive as x increases through c,\ni e , if f \u2032(x) < 0 at every point sufficiently close to and to the left of c,\nand f \u2032(x) > 0 at every point sufficiently close to and to the right of c,\nthen c is a point of local minima"
    },
    {
        "Chapter": "1",
        "sentence_range": "3237-3240",
        "Text": "Rationalised 2023-24\n MATHEMATICS\n186\n(ii)\nIf f \u2032(x) changes sign from negative to positive as x increases through c,\ni e , if f \u2032(x) < 0 at every point sufficiently close to and to the left of c,\nand f \u2032(x) > 0 at every point sufficiently close to and to the right of c,\nthen c is a point of local minima (iii) If f \u2032(x) does not change sign as x increases through c, then c is neither\na point of local maxima nor a point of local minima"
    },
    {
        "Chapter": "1",
        "sentence_range": "3238-3241",
        "Text": "e , if f \u2032(x) < 0 at every point sufficiently close to and to the left of c,\nand f \u2032(x) > 0 at every point sufficiently close to and to the right of c,\nthen c is a point of local minima (iii) If f \u2032(x) does not change sign as x increases through c, then c is neither\na point of local maxima nor a point of local minima Infact, such a point\nis called point of inflexion"
    },
    {
        "Chapter": "1",
        "sentence_range": "3239-3242",
        "Text": ", if f \u2032(x) < 0 at every point sufficiently close to and to the left of c,\nand f \u2032(x) > 0 at every point sufficiently close to and to the right of c,\nthen c is a point of local minima (iii) If f \u2032(x) does not change sign as x increases through c, then c is neither\na point of local maxima nor a point of local minima Infact, such a point\nis called point of inflexion \u00ae Second Derivative Test Let f be a function defined on an interval I and\nc \u2208 I"
    },
    {
        "Chapter": "1",
        "sentence_range": "3240-3243",
        "Text": "(iii) If f \u2032(x) does not change sign as x increases through c, then c is neither\na point of local maxima nor a point of local minima Infact, such a point\nis called point of inflexion \u00ae Second Derivative Test Let f be a function defined on an interval I and\nc \u2208 I Let f  be twice differentiable at c"
    },
    {
        "Chapter": "1",
        "sentence_range": "3241-3244",
        "Text": "Infact, such a point\nis called point of inflexion \u00ae Second Derivative Test Let f be a function defined on an interval I and\nc \u2208 I Let f  be twice differentiable at c Then\n (i) x = c is a point of local maxima if f \u2032(c) = 0 and f \u2033(c) < 0\nThe values f (c) is local maximum value of  f"
    },
    {
        "Chapter": "1",
        "sentence_range": "3242-3245",
        "Text": "\u00ae Second Derivative Test Let f be a function defined on an interval I and\nc \u2208 I Let f  be twice differentiable at c Then\n (i) x = c is a point of local maxima if f \u2032(c) = 0 and f \u2033(c) < 0\nThe values f (c) is local maximum value of  f (ii)\nx = c is a point of local minima if f \u2032(c) = 0 and f \u2033(c) > 0\nIn this case, f (c) is local minimum value of  f"
    },
    {
        "Chapter": "1",
        "sentence_range": "3243-3246",
        "Text": "Let f  be twice differentiable at c Then\n (i) x = c is a point of local maxima if f \u2032(c) = 0 and f \u2033(c) < 0\nThe values f (c) is local maximum value of  f (ii)\nx = c is a point of local minima if f \u2032(c) = 0 and f \u2033(c) > 0\nIn this case, f (c) is local minimum value of  f (iii) The test fails if f \u2032(c) = 0 and f \u2033(c) = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "3244-3247",
        "Text": "Then\n (i) x = c is a point of local maxima if f \u2032(c) = 0 and f \u2033(c) < 0\nThe values f (c) is local maximum value of  f (ii)\nx = c is a point of local minima if f \u2032(c) = 0 and f \u2033(c) > 0\nIn this case, f (c) is local minimum value of  f (iii) The test fails if f \u2032(c) = 0 and f \u2033(c) = 0 In this case, we go back to the first derivative test and find whether c is\na point of maxima, minima or a point of inflexion"
    },
    {
        "Chapter": "1",
        "sentence_range": "3245-3248",
        "Text": "(ii)\nx = c is a point of local minima if f \u2032(c) = 0 and f \u2033(c) > 0\nIn this case, f (c) is local minimum value of  f (iii) The test fails if f \u2032(c) = 0 and f \u2033(c) = 0 In this case, we go back to the first derivative test and find whether c is\na point of maxima, minima or a point of inflexion \u00ae Working rule for finding absolute maxima and/or absolute minima\nStep 1: Find all critical points of f in the interval, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "3246-3249",
        "Text": "(iii) The test fails if f \u2032(c) = 0 and f \u2033(c) = 0 In this case, we go back to the first derivative test and find whether c is\na point of maxima, minima or a point of inflexion \u00ae Working rule for finding absolute maxima and/or absolute minima\nStep 1: Find all critical points of f in the interval, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "3247-3250",
        "Text": "In this case, we go back to the first derivative test and find whether c is\na point of maxima, minima or a point of inflexion \u00ae Working rule for finding absolute maxima and/or absolute minima\nStep 1: Find all critical points of f in the interval, i e , find points x where\neither f \u2032(x) = 0 or f is not differentiable"
    },
    {
        "Chapter": "1",
        "sentence_range": "3248-3251",
        "Text": "\u00ae Working rule for finding absolute maxima and/or absolute minima\nStep 1: Find all critical points of f in the interval, i e , find points x where\neither f \u2032(x) = 0 or f is not differentiable Step 2:Take the end points of the interval"
    },
    {
        "Chapter": "1",
        "sentence_range": "3249-3252",
        "Text": "e , find points x where\neither f \u2032(x) = 0 or f is not differentiable Step 2:Take the end points of the interval Step 3: At all these points (listed in Step 1 and 2), calculate the values of  f"
    },
    {
        "Chapter": "1",
        "sentence_range": "3250-3253",
        "Text": ", find points x where\neither f \u2032(x) = 0 or f is not differentiable Step 2:Take the end points of the interval Step 3: At all these points (listed in Step 1 and 2), calculate the values of  f Step 4: Identify the maximum and minimum values of f out of the values\ncalculated in Step 3"
    },
    {
        "Chapter": "1",
        "sentence_range": "3251-3254",
        "Text": "Step 2:Take the end points of the interval Step 3: At all these points (listed in Step 1 and 2), calculate the values of  f Step 4: Identify the maximum and minimum values of f out of the values\ncalculated in Step 3 This maximum value will be the absolute maximum\nvalue of  f  and the minimum value will be the absolute minimum value of  f"
    },
    {
        "Chapter": "1",
        "sentence_range": "3252-3255",
        "Text": "Step 3: At all these points (listed in Step 1 and 2), calculate the values of  f Step 4: Identify the maximum and minimum values of f out of the values\ncalculated in Step 3 This maximum value will be the absolute maximum\nvalue of  f  and the minimum value will be the absolute minimum value of  f \u2014v\nv\nv\nv\nv\u2014\nRationalised 2023-24\nINTEGRALS         287\n\ufffdJust as a mountaineer climbs a mountain \u2013 because it is there, so\na good mathematics student studies new material because\nit is there"
    },
    {
        "Chapter": "1",
        "sentence_range": "3253-3256",
        "Text": "Step 4: Identify the maximum and minimum values of f out of the values\ncalculated in Step 3 This maximum value will be the absolute maximum\nvalue of  f  and the minimum value will be the absolute minimum value of  f \u2014v\nv\nv\nv\nv\u2014\nRationalised 2023-24\nINTEGRALS         287\n\ufffdJust as a mountaineer climbs a mountain \u2013 because it is there, so\na good mathematics student studies new material because\nit is there \u2014 JAMES B"
    },
    {
        "Chapter": "1",
        "sentence_range": "3254-3257",
        "Text": "This maximum value will be the absolute maximum\nvalue of  f  and the minimum value will be the absolute minimum value of  f \u2014v\nv\nv\nv\nv\u2014\nRationalised 2023-24\nINTEGRALS         287\n\ufffdJust as a mountaineer climbs a mountain \u2013 because it is there, so\na good mathematics student studies new material because\nit is there \u2014 JAMES B BRISTOL \ufffd\n7"
    },
    {
        "Chapter": "1",
        "sentence_range": "3255-3258",
        "Text": "\u2014v\nv\nv\nv\nv\u2014\nRationalised 2023-24\nINTEGRALS         287\n\ufffdJust as a mountaineer climbs a mountain \u2013 because it is there, so\na good mathematics student studies new material because\nit is there \u2014 JAMES B BRISTOL \ufffd\n7 1  Introduction\nDifferential Calculus is centred on the concept of the\nderivative"
    },
    {
        "Chapter": "1",
        "sentence_range": "3256-3259",
        "Text": "\u2014 JAMES B BRISTOL \ufffd\n7 1  Introduction\nDifferential Calculus is centred on the concept of the\nderivative The original motivation for the derivative was\nthe problem of defining tangent lines to the graphs of\nfunctions and calculating the slope of such lines"
    },
    {
        "Chapter": "1",
        "sentence_range": "3257-3260",
        "Text": "BRISTOL \ufffd\n7 1  Introduction\nDifferential Calculus is centred on the concept of the\nderivative The original motivation for the derivative was\nthe problem of defining tangent lines to the graphs of\nfunctions and calculating the slope of such lines Integral\nCalculus is motivated by the problem of defining and\ncalculating the area of the region bounded by the graph of\nthe functions"
    },
    {
        "Chapter": "1",
        "sentence_range": "3258-3261",
        "Text": "1  Introduction\nDifferential Calculus is centred on the concept of the\nderivative The original motivation for the derivative was\nthe problem of defining tangent lines to the graphs of\nfunctions and calculating the slope of such lines Integral\nCalculus is motivated by the problem of defining and\ncalculating the area of the region bounded by the graph of\nthe functions If a function f  is differentiable in an interval I, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "3259-3262",
        "Text": "The original motivation for the derivative was\nthe problem of defining tangent lines to the graphs of\nfunctions and calculating the slope of such lines Integral\nCalculus is motivated by the problem of defining and\ncalculating the area of the region bounded by the graph of\nthe functions If a function f  is differentiable in an interval I, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "3260-3263",
        "Text": "Integral\nCalculus is motivated by the problem of defining and\ncalculating the area of the region bounded by the graph of\nthe functions If a function f  is differentiable in an interval I, i e , its\nderivative f \u2032exists at each point of I, then a natural question\narises that given f \u2032at each point of I, can we determine\nthe function"
    },
    {
        "Chapter": "1",
        "sentence_range": "3261-3264",
        "Text": "If a function f  is differentiable in an interval I, i e , its\nderivative f \u2032exists at each point of I, then a natural question\narises that given f \u2032at each point of I, can we determine\nthe function The functions that could possibly have given\nfunction as a derivative are called anti derivatives (or\nprimitive) of the function"
    },
    {
        "Chapter": "1",
        "sentence_range": "3262-3265",
        "Text": "e , its\nderivative f \u2032exists at each point of I, then a natural question\narises that given f \u2032at each point of I, can we determine\nthe function The functions that could possibly have given\nfunction as a derivative are called anti derivatives (or\nprimitive) of the function Further, the formula that gives\nall these anti derivatives is called the indefinite integral of the function and such\nprocess of finding anti derivatives is called integration"
    },
    {
        "Chapter": "1",
        "sentence_range": "3263-3266",
        "Text": ", its\nderivative f \u2032exists at each point of I, then a natural question\narises that given f \u2032at each point of I, can we determine\nthe function The functions that could possibly have given\nfunction as a derivative are called anti derivatives (or\nprimitive) of the function Further, the formula that gives\nall these anti derivatives is called the indefinite integral of the function and such\nprocess of finding anti derivatives is called integration Such type of problems arise in\nmany practical situations"
    },
    {
        "Chapter": "1",
        "sentence_range": "3264-3267",
        "Text": "The functions that could possibly have given\nfunction as a derivative are called anti derivatives (or\nprimitive) of the function Further, the formula that gives\nall these anti derivatives is called the indefinite integral of the function and such\nprocess of finding anti derivatives is called integration Such type of problems arise in\nmany practical situations For instance, if we know the instantaneous velocity of an\nobject at any instant, then there arises a natural question, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "3265-3268",
        "Text": "Further, the formula that gives\nall these anti derivatives is called the indefinite integral of the function and such\nprocess of finding anti derivatives is called integration Such type of problems arise in\nmany practical situations For instance, if we know the instantaneous velocity of an\nobject at any instant, then there arises a natural question, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "3266-3269",
        "Text": "Such type of problems arise in\nmany practical situations For instance, if we know the instantaneous velocity of an\nobject at any instant, then there arises a natural question, i e , can we determine the\nposition of the object at any instant"
    },
    {
        "Chapter": "1",
        "sentence_range": "3267-3270",
        "Text": "For instance, if we know the instantaneous velocity of an\nobject at any instant, then there arises a natural question, i e , can we determine the\nposition of the object at any instant There are several such practical and theoretical\nsituations where the process of integration is involved"
    },
    {
        "Chapter": "1",
        "sentence_range": "3268-3271",
        "Text": "e , can we determine the\nposition of the object at any instant There are several such practical and theoretical\nsituations where the process of integration is involved The development of integral\ncalculus arises out of the efforts of solving the problems of the following types:\n(a)\nthe problem of finding a function whenever its derivative is given,\n(b)\nthe problem of finding the area bounded by the graph of a function under certain\nconditions"
    },
    {
        "Chapter": "1",
        "sentence_range": "3269-3272",
        "Text": ", can we determine the\nposition of the object at any instant There are several such practical and theoretical\nsituations where the process of integration is involved The development of integral\ncalculus arises out of the efforts of solving the problems of the following types:\n(a)\nthe problem of finding a function whenever its derivative is given,\n(b)\nthe problem of finding the area bounded by the graph of a function under certain\nconditions These  two problems lead to the two forms of the integrals, e"
    },
    {
        "Chapter": "1",
        "sentence_range": "3270-3273",
        "Text": "There are several such practical and theoretical\nsituations where the process of integration is involved The development of integral\ncalculus arises out of the efforts of solving the problems of the following types:\n(a)\nthe problem of finding a function whenever its derivative is given,\n(b)\nthe problem of finding the area bounded by the graph of a function under certain\nconditions These  two problems lead to the two forms of the integrals, e g"
    },
    {
        "Chapter": "1",
        "sentence_range": "3271-3274",
        "Text": "The development of integral\ncalculus arises out of the efforts of solving the problems of the following types:\n(a)\nthe problem of finding a function whenever its derivative is given,\n(b)\nthe problem of finding the area bounded by the graph of a function under certain\nconditions These  two problems lead to the two forms of the integrals, e g , indefinite and\ndefinite integrals, which together constitute the Integral Calculus"
    },
    {
        "Chapter": "1",
        "sentence_range": "3272-3275",
        "Text": "These  two problems lead to the two forms of the integrals, e g , indefinite and\ndefinite integrals, which together constitute the Integral Calculus Chapter 7\nINTEGRALS\nG"
    },
    {
        "Chapter": "1",
        "sentence_range": "3273-3276",
        "Text": "g , indefinite and\ndefinite integrals, which together constitute the Integral Calculus Chapter 7\nINTEGRALS\nG W"
    },
    {
        "Chapter": "1",
        "sentence_range": "3274-3277",
        "Text": ", indefinite and\ndefinite integrals, which together constitute the Integral Calculus Chapter 7\nINTEGRALS\nG W Leibnitz\n(1646 -1716)\n288\nMATHEMATICS\nThere is a connection, known as the Fundamental Theorem of Calculus, between\nindefinite integral and definite integral which makes the definite integral as a practical\ntool for science and engineering"
    },
    {
        "Chapter": "1",
        "sentence_range": "3275-3278",
        "Text": "Chapter 7\nINTEGRALS\nG W Leibnitz\n(1646 -1716)\n288\nMATHEMATICS\nThere is a connection, known as the Fundamental Theorem of Calculus, between\nindefinite integral and definite integral which makes the definite integral as a practical\ntool for science and engineering The definite integral is also used to solve many interesting\nproblems from various disciplines like economics, finance and probability"
    },
    {
        "Chapter": "1",
        "sentence_range": "3276-3279",
        "Text": "W Leibnitz\n(1646 -1716)\n288\nMATHEMATICS\nThere is a connection, known as the Fundamental Theorem of Calculus, between\nindefinite integral and definite integral which makes the definite integral as a practical\ntool for science and engineering The definite integral is also used to solve many interesting\nproblems from various disciplines like economics, finance and probability In this Chapter, we shall confine ourselves to the study of indefinite and definite\nintegrals and their elementary properties including some techniques of integration"
    },
    {
        "Chapter": "1",
        "sentence_range": "3277-3280",
        "Text": "Leibnitz\n(1646 -1716)\n288\nMATHEMATICS\nThere is a connection, known as the Fundamental Theorem of Calculus, between\nindefinite integral and definite integral which makes the definite integral as a practical\ntool for science and engineering The definite integral is also used to solve many interesting\nproblems from various disciplines like economics, finance and probability In this Chapter, we shall confine ourselves to the study of indefinite and definite\nintegrals and their elementary properties including some techniques of integration 7"
    },
    {
        "Chapter": "1",
        "sentence_range": "3278-3281",
        "Text": "The definite integral is also used to solve many interesting\nproblems from various disciplines like economics, finance and probability In this Chapter, we shall confine ourselves to the study of indefinite and definite\nintegrals and their elementary properties including some techniques of integration 7 2  Integration as an Inverse Process of Differentiation\nIntegration is the inverse process of differentiation"
    },
    {
        "Chapter": "1",
        "sentence_range": "3279-3282",
        "Text": "In this Chapter, we shall confine ourselves to the study of indefinite and definite\nintegrals and their elementary properties including some techniques of integration 7 2  Integration as an Inverse Process of Differentiation\nIntegration is the inverse process of differentiation Instead of differentiating a function,\nwe are given the derivative of a function and asked to find its primitive, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "3280-3283",
        "Text": "7 2  Integration as an Inverse Process of Differentiation\nIntegration is the inverse process of differentiation Instead of differentiating a function,\nwe are given the derivative of a function and asked to find its primitive, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "3281-3284",
        "Text": "2  Integration as an Inverse Process of Differentiation\nIntegration is the inverse process of differentiation Instead of differentiating a function,\nwe are given the derivative of a function and asked to find its primitive, i e , the original\nfunction"
    },
    {
        "Chapter": "1",
        "sentence_range": "3282-3285",
        "Text": "Instead of differentiating a function,\nwe are given the derivative of a function and asked to find its primitive, i e , the original\nfunction Such a process is called integration or anti differentiation"
    },
    {
        "Chapter": "1",
        "sentence_range": "3283-3286",
        "Text": "e , the original\nfunction Such a process is called integration or anti differentiation Let us consider the following examples:\nWe know that\nd(sin )\nx\ndx\n = cos x"
    },
    {
        "Chapter": "1",
        "sentence_range": "3284-3287",
        "Text": ", the original\nfunction Such a process is called integration or anti differentiation Let us consider the following examples:\nWe know that\nd(sin )\nx\ndx\n = cos x (1)\n3\n(\n)\n3\nd\nx\ndx\n = x 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "3285-3288",
        "Text": "Such a process is called integration or anti differentiation Let us consider the following examples:\nWe know that\nd(sin )\nx\ndx\n = cos x (1)\n3\n(\n)\n3\nd\nx\ndx\n = x 2 (2)\nand\n(\nx)\nd\ne\ndx\n= ex"
    },
    {
        "Chapter": "1",
        "sentence_range": "3286-3289",
        "Text": "Let us consider the following examples:\nWe know that\nd(sin )\nx\ndx\n = cos x (1)\n3\n(\n)\n3\nd\nx\ndx\n = x 2 (2)\nand\n(\nx)\nd\ne\ndx\n= ex (3)\nWe observe that in (1), the function cos x is the derived function of sin x"
    },
    {
        "Chapter": "1",
        "sentence_range": "3287-3290",
        "Text": "(1)\n3\n(\n)\n3\nd\nx\ndx\n = x 2 (2)\nand\n(\nx)\nd\ne\ndx\n= ex (3)\nWe observe that in (1), the function cos x is the derived function of sin x We say\nthat sin x is an anti derivative (or an integral) of cos x"
    },
    {
        "Chapter": "1",
        "sentence_range": "3288-3291",
        "Text": "(2)\nand\n(\nx)\nd\ne\ndx\n= ex (3)\nWe observe that in (1), the function cos x is the derived function of sin x We say\nthat sin x is an anti derivative (or an integral) of cos x Similarly, in (2) and (3), \n3\n3\nx  and\nex are the anti derivatives (or integrals) of x2 and ex, respectively"
    },
    {
        "Chapter": "1",
        "sentence_range": "3289-3292",
        "Text": "(3)\nWe observe that in (1), the function cos x is the derived function of sin x We say\nthat sin x is an anti derivative (or an integral) of cos x Similarly, in (2) and (3), \n3\n3\nx  and\nex are the anti derivatives (or integrals) of x2 and ex, respectively Again, we note that\nfor any real number C, treated as constant function, its derivative is zero and hence, we\ncan write (1), (2) and (3) as follows :\n(sin\n+ C)\n=cos\nd\nx\nx\ndx\n, \n3\n2\n(\n+ C)\n3\n=\nd\nx\nx\ndx\nand \n(\nx+ C) =\nx\nd\ne\ne\ndx\nThus, anti derivatives (or integrals) of the above cited functions are not unique"
    },
    {
        "Chapter": "1",
        "sentence_range": "3290-3293",
        "Text": "We say\nthat sin x is an anti derivative (or an integral) of cos x Similarly, in (2) and (3), \n3\n3\nx  and\nex are the anti derivatives (or integrals) of x2 and ex, respectively Again, we note that\nfor any real number C, treated as constant function, its derivative is zero and hence, we\ncan write (1), (2) and (3) as follows :\n(sin\n+ C)\n=cos\nd\nx\nx\ndx\n, \n3\n2\n(\n+ C)\n3\n=\nd\nx\nx\ndx\nand \n(\nx+ C) =\nx\nd\ne\ne\ndx\nThus, anti derivatives (or integrals) of the above cited functions are not unique Actually, there exist infinitely many anti derivatives of each of these functions which\ncan be obtained by choosing C arbitrarily from the set of real numbers"
    },
    {
        "Chapter": "1",
        "sentence_range": "3291-3294",
        "Text": "Similarly, in (2) and (3), \n3\n3\nx  and\nex are the anti derivatives (or integrals) of x2 and ex, respectively Again, we note that\nfor any real number C, treated as constant function, its derivative is zero and hence, we\ncan write (1), (2) and (3) as follows :\n(sin\n+ C)\n=cos\nd\nx\nx\ndx\n, \n3\n2\n(\n+ C)\n3\n=\nd\nx\nx\ndx\nand \n(\nx+ C) =\nx\nd\ne\ne\ndx\nThus, anti derivatives (or integrals) of the above cited functions are not unique Actually, there exist infinitely many anti derivatives of each of these functions which\ncan be obtained by choosing C arbitrarily from the set of real numbers For this reason\nC is customarily referred to as arbitrary constant"
    },
    {
        "Chapter": "1",
        "sentence_range": "3292-3295",
        "Text": "Again, we note that\nfor any real number C, treated as constant function, its derivative is zero and hence, we\ncan write (1), (2) and (3) as follows :\n(sin\n+ C)\n=cos\nd\nx\nx\ndx\n, \n3\n2\n(\n+ C)\n3\n=\nd\nx\nx\ndx\nand \n(\nx+ C) =\nx\nd\ne\ne\ndx\nThus, anti derivatives (or integrals) of the above cited functions are not unique Actually, there exist infinitely many anti derivatives of each of these functions which\ncan be obtained by choosing C arbitrarily from the set of real numbers For this reason\nC is customarily referred to as arbitrary constant In fact, C is the parameter by\nvarying which one gets different anti derivatives (or integrals) of the given function"
    },
    {
        "Chapter": "1",
        "sentence_range": "3293-3296",
        "Text": "Actually, there exist infinitely many anti derivatives of each of these functions which\ncan be obtained by choosing C arbitrarily from the set of real numbers For this reason\nC is customarily referred to as arbitrary constant In fact, C is the parameter by\nvarying which one gets different anti derivatives (or integrals) of the given function More generally, if there is a function F such that \nF ( ) =\n( )\nd\nx\nf\nx\ndx\n, \u2200 x \u2208 I (interval),\nthen for any arbitrary real number C, (also called constant of integration)\n[\nF ( ) + C]\nd\nx\ndx\n = f (x), x \u2208 I\nINTEGRALS         289\nThus,\n{F + C, C \u2208 R} denotes a family of anti derivatives of f"
    },
    {
        "Chapter": "1",
        "sentence_range": "3294-3297",
        "Text": "For this reason\nC is customarily referred to as arbitrary constant In fact, C is the parameter by\nvarying which one gets different anti derivatives (or integrals) of the given function More generally, if there is a function F such that \nF ( ) =\n( )\nd\nx\nf\nx\ndx\n, \u2200 x \u2208 I (interval),\nthen for any arbitrary real number C, (also called constant of integration)\n[\nF ( ) + C]\nd\nx\ndx\n = f (x), x \u2208 I\nINTEGRALS         289\nThus,\n{F + C, C \u2208 R} denotes a family of anti derivatives of f Remark  Functions with same derivatives differ by a constant"
    },
    {
        "Chapter": "1",
        "sentence_range": "3295-3298",
        "Text": "In fact, C is the parameter by\nvarying which one gets different anti derivatives (or integrals) of the given function More generally, if there is a function F such that \nF ( ) =\n( )\nd\nx\nf\nx\ndx\n, \u2200 x \u2208 I (interval),\nthen for any arbitrary real number C, (also called constant of integration)\n[\nF ( ) + C]\nd\nx\ndx\n = f (x), x \u2208 I\nINTEGRALS         289\nThus,\n{F + C, C \u2208 R} denotes a family of anti derivatives of f Remark  Functions with same derivatives differ by a constant To show this, let g and h\nbe two functions having the same derivatives on an interval I"
    },
    {
        "Chapter": "1",
        "sentence_range": "3296-3299",
        "Text": "More generally, if there is a function F such that \nF ( ) =\n( )\nd\nx\nf\nx\ndx\n, \u2200 x \u2208 I (interval),\nthen for any arbitrary real number C, (also called constant of integration)\n[\nF ( ) + C]\nd\nx\ndx\n = f (x), x \u2208 I\nINTEGRALS         289\nThus,\n{F + C, C \u2208 R} denotes a family of anti derivatives of f Remark  Functions with same derivatives differ by a constant To show this, let g and h\nbe two functions having the same derivatives on an interval I Consider the function f = g \u2013 h defined by f (x) = g(x) \u2013 h(x), \u2200 x \u2208 I\nThen\ndf\ndx = f\u2032 = g\u2032  \u2013 h\u2032 giving  f\u2032 (x) = g\u2032 (x) \u2013 h\u2032 (x) \u2200 x \u2208 I\nor\nf\u2032 (x) = 0, \u2200x \u2208 I by hypothesis,\ni"
    },
    {
        "Chapter": "1",
        "sentence_range": "3297-3300",
        "Text": "Remark  Functions with same derivatives differ by a constant To show this, let g and h\nbe two functions having the same derivatives on an interval I Consider the function f = g \u2013 h defined by f (x) = g(x) \u2013 h(x), \u2200 x \u2208 I\nThen\ndf\ndx = f\u2032 = g\u2032  \u2013 h\u2032 giving  f\u2032 (x) = g\u2032 (x) \u2013 h\u2032 (x) \u2200 x \u2208 I\nor\nf\u2032 (x) = 0, \u2200x \u2208 I by hypothesis,\ni e"
    },
    {
        "Chapter": "1",
        "sentence_range": "3298-3301",
        "Text": "To show this, let g and h\nbe two functions having the same derivatives on an interval I Consider the function f = g \u2013 h defined by f (x) = g(x) \u2013 h(x), \u2200 x \u2208 I\nThen\ndf\ndx = f\u2032 = g\u2032  \u2013 h\u2032 giving  f\u2032 (x) = g\u2032 (x) \u2013 h\u2032 (x) \u2200 x \u2208 I\nor\nf\u2032 (x) = 0, \u2200x \u2208 I by hypothesis,\ni e , the rate of change of f with respect to x is zero on I and hence f is constant"
    },
    {
        "Chapter": "1",
        "sentence_range": "3299-3302",
        "Text": "Consider the function f = g \u2013 h defined by f (x) = g(x) \u2013 h(x), \u2200 x \u2208 I\nThen\ndf\ndx = f\u2032 = g\u2032  \u2013 h\u2032 giving  f\u2032 (x) = g\u2032 (x) \u2013 h\u2032 (x) \u2200 x \u2208 I\nor\nf\u2032 (x) = 0, \u2200x \u2208 I by hypothesis,\ni e , the rate of change of f with respect to x is zero on I and hence f is constant In view of the above remark, it is justified to infer that the family {F + C, C \u2208 R}\nprovides all possible anti derivatives of f"
    },
    {
        "Chapter": "1",
        "sentence_range": "3300-3303",
        "Text": "e , the rate of change of f with respect to x is zero on I and hence f is constant In view of the above remark, it is justified to infer that the family {F + C, C \u2208 R}\nprovides all possible anti derivatives of f We introduce a new symbol, namely, \n( )\n\u222bf x dx\n which will represent the entire\nclass of anti derivatives read as the indefinite integral of f with respect to x"
    },
    {
        "Chapter": "1",
        "sentence_range": "3301-3304",
        "Text": ", the rate of change of f with respect to x is zero on I and hence f is constant In view of the above remark, it is justified to infer that the family {F + C, C \u2208 R}\nprovides all possible anti derivatives of f We introduce a new symbol, namely, \n( )\n\u222bf x dx\n which will represent the entire\nclass of anti derivatives read as the indefinite integral of f with respect to x Symbolically, we write \n( )\nf x dx= F ( ) + C\nx\n\u222b"
    },
    {
        "Chapter": "1",
        "sentence_range": "3302-3305",
        "Text": "In view of the above remark, it is justified to infer that the family {F + C, C \u2208 R}\nprovides all possible anti derivatives of f We introduce a new symbol, namely, \n( )\n\u222bf x dx\n which will represent the entire\nclass of anti derivatives read as the indefinite integral of f with respect to x Symbolically, we write \n( )\nf x dx= F ( ) + C\nx\n\u222b Notation Given that  \n( )\ndy\nf x\ndx\n=\n, we write y = \n( )\n\u222bf x dx"
    },
    {
        "Chapter": "1",
        "sentence_range": "3303-3306",
        "Text": "We introduce a new symbol, namely, \n( )\n\u222bf x dx\n which will represent the entire\nclass of anti derivatives read as the indefinite integral of f with respect to x Symbolically, we write \n( )\nf x dx= F ( ) + C\nx\n\u222b Notation Given that  \n( )\ndy\nf x\ndx\n=\n, we write y = \n( )\n\u222bf x dx For the sake of convenience, we mention below the following symbols/terms/phrases\nwith their meanings as given in the Table (7"
    },
    {
        "Chapter": "1",
        "sentence_range": "3304-3307",
        "Text": "Symbolically, we write \n( )\nf x dx= F ( ) + C\nx\n\u222b Notation Given that  \n( )\ndy\nf x\ndx\n=\n, we write y = \n( )\n\u222bf x dx For the sake of convenience, we mention below the following symbols/terms/phrases\nwith their meanings as given in the Table (7 1)"
    },
    {
        "Chapter": "1",
        "sentence_range": "3305-3308",
        "Text": "Notation Given that  \n( )\ndy\nf x\ndx\n=\n, we write y = \n( )\n\u222bf x dx For the sake of convenience, we mention below the following symbols/terms/phrases\nwith their meanings as given in the Table (7 1) Table 7"
    },
    {
        "Chapter": "1",
        "sentence_range": "3306-3309",
        "Text": "For the sake of convenience, we mention below the following symbols/terms/phrases\nwith their meanings as given in the Table (7 1) Table 7 1\nSymbols/Terms/Phrases\nMeaning\n( )\n\u222bf x dx\nIntegral of f with respect to x\nf (x) in \n( )\n\u222bf x dx\nIntegrand\nx in  \n( )\n\u222bf x dx\nVariable of integration\nIntegrate\nFind the  integral\nAn integral of f\nA function F such that\nF\u2032(x) = f (x)\nIntegration\nThe process of finding the integral\nConstant of Integration\nAny real number C, considered as\nconstant function\n290\nMATHEMATICS\nWe already know the formulae for the derivatives of many important functions"
    },
    {
        "Chapter": "1",
        "sentence_range": "3307-3310",
        "Text": "1) Table 7 1\nSymbols/Terms/Phrases\nMeaning\n( )\n\u222bf x dx\nIntegral of f with respect to x\nf (x) in \n( )\n\u222bf x dx\nIntegrand\nx in  \n( )\n\u222bf x dx\nVariable of integration\nIntegrate\nFind the  integral\nAn integral of f\nA function F such that\nF\u2032(x) = f (x)\nIntegration\nThe process of finding the integral\nConstant of Integration\nAny real number C, considered as\nconstant function\n290\nMATHEMATICS\nWe already know the formulae for the derivatives of many important functions From these formulae, we can write down immediately the corresponding formulae\n(referred to as standard formulae) for the integrals of these functions, as listed below\nwhich will be used to find integrals of other functions"
    },
    {
        "Chapter": "1",
        "sentence_range": "3308-3311",
        "Text": "Table 7 1\nSymbols/Terms/Phrases\nMeaning\n( )\n\u222bf x dx\nIntegral of f with respect to x\nf (x) in \n( )\n\u222bf x dx\nIntegrand\nx in  \n( )\n\u222bf x dx\nVariable of integration\nIntegrate\nFind the  integral\nAn integral of f\nA function F such that\nF\u2032(x) = f (x)\nIntegration\nThe process of finding the integral\nConstant of Integration\nAny real number C, considered as\nconstant function\n290\nMATHEMATICS\nWe already know the formulae for the derivatives of many important functions From these formulae, we can write down immediately the corresponding formulae\n(referred to as standard formulae) for the integrals of these functions, as listed below\nwhich will be used to find integrals of other functions Derivatives\nIntegrals (Anti derivatives)\n(i)\n1\n1\nn\nn\nd\nx\nx\ndx\nn\n+\n\uf8eb\n\uf8f6 =\n\uf8ec\n\uf8f7\n+\n\uf8ed\n\uf8f8\n ;\n1\nC\n1\nn\nn\nx\nx dx\nn\n+\n=\n+\n+\n\u222b\n, n \u2260 \u20131\nParticularly, we note that\n( )\n1\nd\ndxx\n=  ;\n      \nC\ndx\n=x\n+\n\u222b\n(ii)\n(\nsin)\ncos\nd\nx\nx\ndx\n=\n ;\ncos\nsin\nC\nx dx\nx\n=\n+\n\u222b\n(iii)\n(\n\u2013 cos)\nsin\nd\nx\nx\ndx\n=\n ;\nsin\ncos\nC\nx dx\n\u2013\nx\n=\n+\n\u222b\n(iv)\n(\n)\n2\ntan\nsec\nd\nx\nx\ndx\n=\n ;\nsec2\ntan\nC\nx dx\nx\n=\n+\n\u222b\n(v)\n(\n)\n2\n\u2013 cot\ncosec\nd\nx\nx\ndx\n=\n ;\ncosec2\ncot\nC\nx dx\n\u2013\nx\n=\n+\n\u222b\n(vi)\n(\nsec)\nsec\ntan\nd\nx\nx\nx\ndx\n=\n ;\nsec\ntan\nsec\nC\nx\nx dx\nx\n=\n+\n\u222b\n(vii)\n(\n\u2013 cosec)\ncosec\ncot\nd\nx\nx\nx\ndx\n=\n ;\ncosec\ncot\n\u2013 cosec\nC\nx\nx dx\nx\n=\n+\n\u222b\n(viii)\n(\n)\n\u2013 1\n2\n1\nsin\n1\nd\nx\ndx\n\u2013 x\n=\n ;\n\u2013 1\n2\nsin\nC\n1\ndx\nx\n\u2013 x\n=\n+\n\u222b\n(ix)\n(\n)\n\u2013 1\n2\n1\n\u2013 cos\n1\nd\nx\ndx\n\u2013 x\n=\n ;\n\u2013 1\n2\ncos\nC\n1\ndx\n\u2013\nx\n\u2013 x\n=\n+\n\u222b\n(x)\n(\n)\n\u2013 1\n2\n1\ntan\n1\nd\nx\ndx\nx\n=\n+\n ;\n\u2013 1\n2\ntan\nC\n1\ndx\nx\nx\n=\n+\n+\n\u222b\n(xi)\n(\n)\n\u2013 1\n2\n1\n\u2013 cot\n1\nd\nx\ndx\nx\n=\n+\n ;\n\u2013 1\n2\ncot\nC\n1\ndx\n\u2013\nx\nx\n=\n+\n\u222b+\nINTEGRALS         291\n(xii)\n(\n)\n\u2013 1\n12\nsec\n1\nd\nx\ndx\nx\nx \u2013\n=\n ;\n\u2013 1\n2\nsec\nC\n1\ndx\nx\nx\nx \u2013\n=\n+\n\u222b\n(xiii)\n(\n)\n\u2013 1\n12\n\u2013 cosec\n1\nd\nx\ndx\nx\nx \u2013\n=\n ;\n\u2013 1\n2\ncosec\nC\n1\ndx\n\u2013\nx\nx\nx \u2013\n=\n+\n\u222b\n(xiv)\n(\nx)\nx\nd\ne\ne\ndx\n=\n ;\nC\nx\nx\ne dx\n=e\n+\n\u222b\n(xv)\n(\n)\n1\nlog|\n|\nd\nx\ndx\n=x\n;\n1\nlog|\n| C\ndx\nx\nx\n=\n+\n\u222b\n(xvi)\nx\nx\nd\na\na\ndx\nlog a\n\uf8eb\n\uf8f6 =\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n ;\nC\nx\nx\na\na dx\nlog a\n=\n+\n\u222b\n\ufffdNote  In practice, we normally do not mention the interval over which the various\nfunctions are defined"
    },
    {
        "Chapter": "1",
        "sentence_range": "3309-3312",
        "Text": "1\nSymbols/Terms/Phrases\nMeaning\n( )\n\u222bf x dx\nIntegral of f with respect to x\nf (x) in \n( )\n\u222bf x dx\nIntegrand\nx in  \n( )\n\u222bf x dx\nVariable of integration\nIntegrate\nFind the  integral\nAn integral of f\nA function F such that\nF\u2032(x) = f (x)\nIntegration\nThe process of finding the integral\nConstant of Integration\nAny real number C, considered as\nconstant function\n290\nMATHEMATICS\nWe already know the formulae for the derivatives of many important functions From these formulae, we can write down immediately the corresponding formulae\n(referred to as standard formulae) for the integrals of these functions, as listed below\nwhich will be used to find integrals of other functions Derivatives\nIntegrals (Anti derivatives)\n(i)\n1\n1\nn\nn\nd\nx\nx\ndx\nn\n+\n\uf8eb\n\uf8f6 =\n\uf8ec\n\uf8f7\n+\n\uf8ed\n\uf8f8\n ;\n1\nC\n1\nn\nn\nx\nx dx\nn\n+\n=\n+\n+\n\u222b\n, n \u2260 \u20131\nParticularly, we note that\n( )\n1\nd\ndxx\n=  ;\n      \nC\ndx\n=x\n+\n\u222b\n(ii)\n(\nsin)\ncos\nd\nx\nx\ndx\n=\n ;\ncos\nsin\nC\nx dx\nx\n=\n+\n\u222b\n(iii)\n(\n\u2013 cos)\nsin\nd\nx\nx\ndx\n=\n ;\nsin\ncos\nC\nx dx\n\u2013\nx\n=\n+\n\u222b\n(iv)\n(\n)\n2\ntan\nsec\nd\nx\nx\ndx\n=\n ;\nsec2\ntan\nC\nx dx\nx\n=\n+\n\u222b\n(v)\n(\n)\n2\n\u2013 cot\ncosec\nd\nx\nx\ndx\n=\n ;\ncosec2\ncot\nC\nx dx\n\u2013\nx\n=\n+\n\u222b\n(vi)\n(\nsec)\nsec\ntan\nd\nx\nx\nx\ndx\n=\n ;\nsec\ntan\nsec\nC\nx\nx dx\nx\n=\n+\n\u222b\n(vii)\n(\n\u2013 cosec)\ncosec\ncot\nd\nx\nx\nx\ndx\n=\n ;\ncosec\ncot\n\u2013 cosec\nC\nx\nx dx\nx\n=\n+\n\u222b\n(viii)\n(\n)\n\u2013 1\n2\n1\nsin\n1\nd\nx\ndx\n\u2013 x\n=\n ;\n\u2013 1\n2\nsin\nC\n1\ndx\nx\n\u2013 x\n=\n+\n\u222b\n(ix)\n(\n)\n\u2013 1\n2\n1\n\u2013 cos\n1\nd\nx\ndx\n\u2013 x\n=\n ;\n\u2013 1\n2\ncos\nC\n1\ndx\n\u2013\nx\n\u2013 x\n=\n+\n\u222b\n(x)\n(\n)\n\u2013 1\n2\n1\ntan\n1\nd\nx\ndx\nx\n=\n+\n ;\n\u2013 1\n2\ntan\nC\n1\ndx\nx\nx\n=\n+\n+\n\u222b\n(xi)\n(\n)\n\u2013 1\n2\n1\n\u2013 cot\n1\nd\nx\ndx\nx\n=\n+\n ;\n\u2013 1\n2\ncot\nC\n1\ndx\n\u2013\nx\nx\n=\n+\n\u222b+\nINTEGRALS         291\n(xii)\n(\n)\n\u2013 1\n12\nsec\n1\nd\nx\ndx\nx\nx \u2013\n=\n ;\n\u2013 1\n2\nsec\nC\n1\ndx\nx\nx\nx \u2013\n=\n+\n\u222b\n(xiii)\n(\n)\n\u2013 1\n12\n\u2013 cosec\n1\nd\nx\ndx\nx\nx \u2013\n=\n ;\n\u2013 1\n2\ncosec\nC\n1\ndx\n\u2013\nx\nx\nx \u2013\n=\n+\n\u222b\n(xiv)\n(\nx)\nx\nd\ne\ne\ndx\n=\n ;\nC\nx\nx\ne dx\n=e\n+\n\u222b\n(xv)\n(\n)\n1\nlog|\n|\nd\nx\ndx\n=x\n;\n1\nlog|\n| C\ndx\nx\nx\n=\n+\n\u222b\n(xvi)\nx\nx\nd\na\na\ndx\nlog a\n\uf8eb\n\uf8f6 =\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n ;\nC\nx\nx\na\na dx\nlog a\n=\n+\n\u222b\n\ufffdNote  In practice, we normally do not mention the interval over which the various\nfunctions are defined However, in any specific problem one has to keep it in mind"
    },
    {
        "Chapter": "1",
        "sentence_range": "3310-3313",
        "Text": "From these formulae, we can write down immediately the corresponding formulae\n(referred to as standard formulae) for the integrals of these functions, as listed below\nwhich will be used to find integrals of other functions Derivatives\nIntegrals (Anti derivatives)\n(i)\n1\n1\nn\nn\nd\nx\nx\ndx\nn\n+\n\uf8eb\n\uf8f6 =\n\uf8ec\n\uf8f7\n+\n\uf8ed\n\uf8f8\n ;\n1\nC\n1\nn\nn\nx\nx dx\nn\n+\n=\n+\n+\n\u222b\n, n \u2260 \u20131\nParticularly, we note that\n( )\n1\nd\ndxx\n=  ;\n      \nC\ndx\n=x\n+\n\u222b\n(ii)\n(\nsin)\ncos\nd\nx\nx\ndx\n=\n ;\ncos\nsin\nC\nx dx\nx\n=\n+\n\u222b\n(iii)\n(\n\u2013 cos)\nsin\nd\nx\nx\ndx\n=\n ;\nsin\ncos\nC\nx dx\n\u2013\nx\n=\n+\n\u222b\n(iv)\n(\n)\n2\ntan\nsec\nd\nx\nx\ndx\n=\n ;\nsec2\ntan\nC\nx dx\nx\n=\n+\n\u222b\n(v)\n(\n)\n2\n\u2013 cot\ncosec\nd\nx\nx\ndx\n=\n ;\ncosec2\ncot\nC\nx dx\n\u2013\nx\n=\n+\n\u222b\n(vi)\n(\nsec)\nsec\ntan\nd\nx\nx\nx\ndx\n=\n ;\nsec\ntan\nsec\nC\nx\nx dx\nx\n=\n+\n\u222b\n(vii)\n(\n\u2013 cosec)\ncosec\ncot\nd\nx\nx\nx\ndx\n=\n ;\ncosec\ncot\n\u2013 cosec\nC\nx\nx dx\nx\n=\n+\n\u222b\n(viii)\n(\n)\n\u2013 1\n2\n1\nsin\n1\nd\nx\ndx\n\u2013 x\n=\n ;\n\u2013 1\n2\nsin\nC\n1\ndx\nx\n\u2013 x\n=\n+\n\u222b\n(ix)\n(\n)\n\u2013 1\n2\n1\n\u2013 cos\n1\nd\nx\ndx\n\u2013 x\n=\n ;\n\u2013 1\n2\ncos\nC\n1\ndx\n\u2013\nx\n\u2013 x\n=\n+\n\u222b\n(x)\n(\n)\n\u2013 1\n2\n1\ntan\n1\nd\nx\ndx\nx\n=\n+\n ;\n\u2013 1\n2\ntan\nC\n1\ndx\nx\nx\n=\n+\n+\n\u222b\n(xi)\n(\n)\n\u2013 1\n2\n1\n\u2013 cot\n1\nd\nx\ndx\nx\n=\n+\n ;\n\u2013 1\n2\ncot\nC\n1\ndx\n\u2013\nx\nx\n=\n+\n\u222b+\nINTEGRALS         291\n(xii)\n(\n)\n\u2013 1\n12\nsec\n1\nd\nx\ndx\nx\nx \u2013\n=\n ;\n\u2013 1\n2\nsec\nC\n1\ndx\nx\nx\nx \u2013\n=\n+\n\u222b\n(xiii)\n(\n)\n\u2013 1\n12\n\u2013 cosec\n1\nd\nx\ndx\nx\nx \u2013\n=\n ;\n\u2013 1\n2\ncosec\nC\n1\ndx\n\u2013\nx\nx\nx \u2013\n=\n+\n\u222b\n(xiv)\n(\nx)\nx\nd\ne\ne\ndx\n=\n ;\nC\nx\nx\ne dx\n=e\n+\n\u222b\n(xv)\n(\n)\n1\nlog|\n|\nd\nx\ndx\n=x\n;\n1\nlog|\n| C\ndx\nx\nx\n=\n+\n\u222b\n(xvi)\nx\nx\nd\na\na\ndx\nlog a\n\uf8eb\n\uf8f6 =\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n ;\nC\nx\nx\na\na dx\nlog a\n=\n+\n\u222b\n\ufffdNote  In practice, we normally do not mention the interval over which the various\nfunctions are defined However, in any specific problem one has to keep it in mind 7"
    },
    {
        "Chapter": "1",
        "sentence_range": "3311-3314",
        "Text": "Derivatives\nIntegrals (Anti derivatives)\n(i)\n1\n1\nn\nn\nd\nx\nx\ndx\nn\n+\n\uf8eb\n\uf8f6 =\n\uf8ec\n\uf8f7\n+\n\uf8ed\n\uf8f8\n ;\n1\nC\n1\nn\nn\nx\nx dx\nn\n+\n=\n+\n+\n\u222b\n, n \u2260 \u20131\nParticularly, we note that\n( )\n1\nd\ndxx\n=  ;\n      \nC\ndx\n=x\n+\n\u222b\n(ii)\n(\nsin)\ncos\nd\nx\nx\ndx\n=\n ;\ncos\nsin\nC\nx dx\nx\n=\n+\n\u222b\n(iii)\n(\n\u2013 cos)\nsin\nd\nx\nx\ndx\n=\n ;\nsin\ncos\nC\nx dx\n\u2013\nx\n=\n+\n\u222b\n(iv)\n(\n)\n2\ntan\nsec\nd\nx\nx\ndx\n=\n ;\nsec2\ntan\nC\nx dx\nx\n=\n+\n\u222b\n(v)\n(\n)\n2\n\u2013 cot\ncosec\nd\nx\nx\ndx\n=\n ;\ncosec2\ncot\nC\nx dx\n\u2013\nx\n=\n+\n\u222b\n(vi)\n(\nsec)\nsec\ntan\nd\nx\nx\nx\ndx\n=\n ;\nsec\ntan\nsec\nC\nx\nx dx\nx\n=\n+\n\u222b\n(vii)\n(\n\u2013 cosec)\ncosec\ncot\nd\nx\nx\nx\ndx\n=\n ;\ncosec\ncot\n\u2013 cosec\nC\nx\nx dx\nx\n=\n+\n\u222b\n(viii)\n(\n)\n\u2013 1\n2\n1\nsin\n1\nd\nx\ndx\n\u2013 x\n=\n ;\n\u2013 1\n2\nsin\nC\n1\ndx\nx\n\u2013 x\n=\n+\n\u222b\n(ix)\n(\n)\n\u2013 1\n2\n1\n\u2013 cos\n1\nd\nx\ndx\n\u2013 x\n=\n ;\n\u2013 1\n2\ncos\nC\n1\ndx\n\u2013\nx\n\u2013 x\n=\n+\n\u222b\n(x)\n(\n)\n\u2013 1\n2\n1\ntan\n1\nd\nx\ndx\nx\n=\n+\n ;\n\u2013 1\n2\ntan\nC\n1\ndx\nx\nx\n=\n+\n+\n\u222b\n(xi)\n(\n)\n\u2013 1\n2\n1\n\u2013 cot\n1\nd\nx\ndx\nx\n=\n+\n ;\n\u2013 1\n2\ncot\nC\n1\ndx\n\u2013\nx\nx\n=\n+\n\u222b+\nINTEGRALS         291\n(xii)\n(\n)\n\u2013 1\n12\nsec\n1\nd\nx\ndx\nx\nx \u2013\n=\n ;\n\u2013 1\n2\nsec\nC\n1\ndx\nx\nx\nx \u2013\n=\n+\n\u222b\n(xiii)\n(\n)\n\u2013 1\n12\n\u2013 cosec\n1\nd\nx\ndx\nx\nx \u2013\n=\n ;\n\u2013 1\n2\ncosec\nC\n1\ndx\n\u2013\nx\nx\nx \u2013\n=\n+\n\u222b\n(xiv)\n(\nx)\nx\nd\ne\ne\ndx\n=\n ;\nC\nx\nx\ne dx\n=e\n+\n\u222b\n(xv)\n(\n)\n1\nlog|\n|\nd\nx\ndx\n=x\n;\n1\nlog|\n| C\ndx\nx\nx\n=\n+\n\u222b\n(xvi)\nx\nx\nd\na\na\ndx\nlog a\n\uf8eb\n\uf8f6 =\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n ;\nC\nx\nx\na\na dx\nlog a\n=\n+\n\u222b\n\ufffdNote  In practice, we normally do not mention the interval over which the various\nfunctions are defined However, in any specific problem one has to keep it in mind 7 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "3312-3315",
        "Text": "However, in any specific problem one has to keep it in mind 7 2 1 Geometrical interpretation of indefinite integral\nLet f (x) = 2x"
    },
    {
        "Chapter": "1",
        "sentence_range": "3313-3316",
        "Text": "7 2 1 Geometrical interpretation of indefinite integral\nLet f (x) = 2x Then \n2\n( )\nC\nf x dx\n=x\n+\n\u222b"
    },
    {
        "Chapter": "1",
        "sentence_range": "3314-3317",
        "Text": "2 1 Geometrical interpretation of indefinite integral\nLet f (x) = 2x Then \n2\n( )\nC\nf x dx\n=x\n+\n\u222b For different values of C, we get different\nintegrals"
    },
    {
        "Chapter": "1",
        "sentence_range": "3315-3318",
        "Text": "1 Geometrical interpretation of indefinite integral\nLet f (x) = 2x Then \n2\n( )\nC\nf x dx\n=x\n+\n\u222b For different values of C, we get different\nintegrals But these integrals are very similar geometrically"
    },
    {
        "Chapter": "1",
        "sentence_range": "3316-3319",
        "Text": "Then \n2\n( )\nC\nf x dx\n=x\n+\n\u222b For different values of C, we get different\nintegrals But these integrals are very similar geometrically Thus, y = x2 + C, where C is arbitrary constant, represents a family of integrals"
    },
    {
        "Chapter": "1",
        "sentence_range": "3317-3320",
        "Text": "For different values of C, we get different\nintegrals But these integrals are very similar geometrically Thus, y = x2 + C, where C is arbitrary constant, represents a family of integrals By\nassigning different values to C, we get different members of the family"
    },
    {
        "Chapter": "1",
        "sentence_range": "3318-3321",
        "Text": "But these integrals are very similar geometrically Thus, y = x2 + C, where C is arbitrary constant, represents a family of integrals By\nassigning different values to C, we get different members of the family These together\nconstitute the indefinite integral"
    },
    {
        "Chapter": "1",
        "sentence_range": "3319-3322",
        "Text": "Thus, y = x2 + C, where C is arbitrary constant, represents a family of integrals By\nassigning different values to C, we get different members of the family These together\nconstitute the indefinite integral In this case, each integral represents a parabola with\nits axis along y-axis"
    },
    {
        "Chapter": "1",
        "sentence_range": "3320-3323",
        "Text": "By\nassigning different values to C, we get different members of the family These together\nconstitute the indefinite integral In this case, each integral represents a parabola with\nits axis along y-axis Clearly, for C = 0, we obtain y = x2, a parabola with its vertex on the origin"
    },
    {
        "Chapter": "1",
        "sentence_range": "3321-3324",
        "Text": "These together\nconstitute the indefinite integral In this case, each integral represents a parabola with\nits axis along y-axis Clearly, for C = 0, we obtain y = x2, a parabola with its vertex on the origin The\ncurve y = x2 + 1 for C = 1 is obtained by shifting the parabola y = x2 one unit along\ny-axis in positive direction"
    },
    {
        "Chapter": "1",
        "sentence_range": "3322-3325",
        "Text": "In this case, each integral represents a parabola with\nits axis along y-axis Clearly, for C = 0, we obtain y = x2, a parabola with its vertex on the origin The\ncurve y = x2 + 1 for C = 1 is obtained by shifting the parabola y = x2 one unit along\ny-axis in positive direction For C = \u2013 1, y = x2 \u2013 1 is obtained by shifting the parabola\ny = x2 one unit along y-axis in the negative direction"
    },
    {
        "Chapter": "1",
        "sentence_range": "3323-3326",
        "Text": "Clearly, for C = 0, we obtain y = x2, a parabola with its vertex on the origin The\ncurve y = x2 + 1 for C = 1 is obtained by shifting the parabola y = x2 one unit along\ny-axis in positive direction For C = \u2013 1, y = x2 \u2013 1 is obtained by shifting the parabola\ny = x2 one unit along y-axis in the negative direction Thus, for each positive value of C,\neach parabola of the family has its vertex on the positive side of the y-axis and for\nnegative values of C, each has its vertex along the negative side of the y-axis"
    },
    {
        "Chapter": "1",
        "sentence_range": "3324-3327",
        "Text": "The\ncurve y = x2 + 1 for C = 1 is obtained by shifting the parabola y = x2 one unit along\ny-axis in positive direction For C = \u2013 1, y = x2 \u2013 1 is obtained by shifting the parabola\ny = x2 one unit along y-axis in the negative direction Thus, for each positive value of C,\neach parabola of the family has its vertex on the positive side of the y-axis and for\nnegative values of C, each has its vertex along the negative side of the y-axis Some of\nthese have been shown in the Fig 7"
    },
    {
        "Chapter": "1",
        "sentence_range": "3325-3328",
        "Text": "For C = \u2013 1, y = x2 \u2013 1 is obtained by shifting the parabola\ny = x2 one unit along y-axis in the negative direction Thus, for each positive value of C,\neach parabola of the family has its vertex on the positive side of the y-axis and for\nnegative values of C, each has its vertex along the negative side of the y-axis Some of\nthese have been shown in the Fig 7 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "3326-3329",
        "Text": "Thus, for each positive value of C,\neach parabola of the family has its vertex on the positive side of the y-axis and for\nnegative values of C, each has its vertex along the negative side of the y-axis Some of\nthese have been shown in the Fig 7 1 Let us consider the intersection of all these parabolas by a line x = a"
    },
    {
        "Chapter": "1",
        "sentence_range": "3327-3330",
        "Text": "Some of\nthese have been shown in the Fig 7 1 Let us consider the intersection of all these parabolas by a line x = a In the Fig 7"
    },
    {
        "Chapter": "1",
        "sentence_range": "3328-3331",
        "Text": "1 Let us consider the intersection of all these parabolas by a line x = a In the Fig 7 1,\nwe have taken a > 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "3329-3332",
        "Text": "Let us consider the intersection of all these parabolas by a line x = a In the Fig 7 1,\nwe have taken a > 0 The same is true when a < 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "3330-3333",
        "Text": "In the Fig 7 1,\nwe have taken a > 0 The same is true when a < 0 If the line x = a intersects the\nparabolas y = x2, y = x2 + 1, y = x2 + 2, y = x2 \u2013 1, y = x2 \u2013 2 at P0, P1, P2, P\u20131, P\u20132 etc"
    },
    {
        "Chapter": "1",
        "sentence_range": "3331-3334",
        "Text": "1,\nwe have taken a > 0 The same is true when a < 0 If the line x = a intersects the\nparabolas y = x2, y = x2 + 1, y = x2 + 2, y = x2 \u2013 1, y = x2 \u2013 2 at P0, P1, P2, P\u20131, P\u20132 etc ,\nrespectively, then dy\ndx  at these points equals 2a"
    },
    {
        "Chapter": "1",
        "sentence_range": "3332-3335",
        "Text": "The same is true when a < 0 If the line x = a intersects the\nparabolas y = x2, y = x2 + 1, y = x2 + 2, y = x2 \u2013 1, y = x2 \u2013 2 at P0, P1, P2, P\u20131, P\u20132 etc ,\nrespectively, then dy\ndx  at these points equals 2a This indicates that the tangents to the\ncurves at these points are parallel"
    },
    {
        "Chapter": "1",
        "sentence_range": "3333-3336",
        "Text": "If the line x = a intersects the\nparabolas y = x2, y = x2 + 1, y = x2 + 2, y = x2 \u2013 1, y = x2 \u2013 2 at P0, P1, P2, P\u20131, P\u20132 etc ,\nrespectively, then dy\ndx  at these points equals 2a This indicates that the tangents to the\ncurves at these points are parallel Thus, \n2\nC\n2\nC\nF ( )\nx dx\nx\nx\n=\n+\n=\n\u222b\n(say), implies that\n292\nMATHEMATICS\nthe tangents to all the curves y = F C (x), C \u2208 R, at the points of intersection of the\ncurves by the line x = a, (a \u2208 R), are parallel"
    },
    {
        "Chapter": "1",
        "sentence_range": "3334-3337",
        "Text": ",\nrespectively, then dy\ndx  at these points equals 2a This indicates that the tangents to the\ncurves at these points are parallel Thus, \n2\nC\n2\nC\nF ( )\nx dx\nx\nx\n=\n+\n=\n\u222b\n(say), implies that\n292\nMATHEMATICS\nthe tangents to all the curves y = F C (x), C \u2208 R, at the points of intersection of the\ncurves by the line x = a, (a \u2208 R), are parallel Further, the following equation (statement)\n( )\nF ( )\nC\n(say)\nf x dx\nx\ny\n=\n+\n=\n\u222b\n,\nrepresents a family of curves"
    },
    {
        "Chapter": "1",
        "sentence_range": "3335-3338",
        "Text": "This indicates that the tangents to the\ncurves at these points are parallel Thus, \n2\nC\n2\nC\nF ( )\nx dx\nx\nx\n=\n+\n=\n\u222b\n(say), implies that\n292\nMATHEMATICS\nthe tangents to all the curves y = F C (x), C \u2208 R, at the points of intersection of the\ncurves by the line x = a, (a \u2208 R), are parallel Further, the following equation (statement)\n( )\nF ( )\nC\n(say)\nf x dx\nx\ny\n=\n+\n=\n\u222b\n,\nrepresents a family of curves The different values of C will correspond to different\nmembers of this family and these members can be obtained by shifting any one of the\ncurves parallel to itself"
    },
    {
        "Chapter": "1",
        "sentence_range": "3336-3339",
        "Text": "Thus, \n2\nC\n2\nC\nF ( )\nx dx\nx\nx\n=\n+\n=\n\u222b\n(say), implies that\n292\nMATHEMATICS\nthe tangents to all the curves y = F C (x), C \u2208 R, at the points of intersection of the\ncurves by the line x = a, (a \u2208 R), are parallel Further, the following equation (statement)\n( )\nF ( )\nC\n(say)\nf x dx\nx\ny\n=\n+\n=\n\u222b\n,\nrepresents a family of curves The different values of C will correspond to different\nmembers of this family and these members can be obtained by shifting any one of the\ncurves parallel to itself This is the geometrical interpretation of indefinite integral"
    },
    {
        "Chapter": "1",
        "sentence_range": "3337-3340",
        "Text": "Further, the following equation (statement)\n( )\nF ( )\nC\n(say)\nf x dx\nx\ny\n=\n+\n=\n\u222b\n,\nrepresents a family of curves The different values of C will correspond to different\nmembers of this family and these members can be obtained by shifting any one of the\ncurves parallel to itself This is the geometrical interpretation of indefinite integral 7"
    },
    {
        "Chapter": "1",
        "sentence_range": "3338-3341",
        "Text": "The different values of C will correspond to different\nmembers of this family and these members can be obtained by shifting any one of the\ncurves parallel to itself This is the geometrical interpretation of indefinite integral 7 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "3339-3342",
        "Text": "This is the geometrical interpretation of indefinite integral 7 2 2 Some properties of indefinite integral\nIn this sub section, we shall derive some properties of indefinite integrals"
    },
    {
        "Chapter": "1",
        "sentence_range": "3340-3343",
        "Text": "7 2 2 Some properties of indefinite integral\nIn this sub section, we shall derive some properties of indefinite integrals (I)\nThe process of differentiation and integration are inverses of each other in the\nsense of the following results :\n( )\nd\nf x dx\ndx \u222b\n = f (x)\nand\nf( )\n\u222b\u2032x dx\n = f (x) + C, where C is any arbitrary constant"
    },
    {
        "Chapter": "1",
        "sentence_range": "3341-3344",
        "Text": "2 2 Some properties of indefinite integral\nIn this sub section, we shall derive some properties of indefinite integrals (I)\nThe process of differentiation and integration are inverses of each other in the\nsense of the following results :\n( )\nd\nf x dx\ndx \u222b\n = f (x)\nand\nf( )\n\u222b\u2032x dx\n = f (x) + C, where C is any arbitrary constant Fig 7"
    },
    {
        "Chapter": "1",
        "sentence_range": "3342-3345",
        "Text": "2 Some properties of indefinite integral\nIn this sub section, we shall derive some properties of indefinite integrals (I)\nThe process of differentiation and integration are inverses of each other in the\nsense of the following results :\n( )\nd\nf x dx\ndx \u222b\n = f (x)\nand\nf( )\n\u222b\u2032x dx\n = f (x) + C, where C is any arbitrary constant Fig 7 1\nINTEGRALS         293\nProof Let F be any anti derivative of f, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "3343-3346",
        "Text": "(I)\nThe process of differentiation and integration are inverses of each other in the\nsense of the following results :\n( )\nd\nf x dx\ndx \u222b\n = f (x)\nand\nf( )\n\u222b\u2032x dx\n = f (x) + C, where C is any arbitrary constant Fig 7 1\nINTEGRALS         293\nProof Let F be any anti derivative of f, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "3344-3347",
        "Text": "Fig 7 1\nINTEGRALS         293\nProof Let F be any anti derivative of f, i e ,\ndF( )\nx\ndx\n = f (x)\nThen\n( )\n\u222bf x dx\n = F(x) + C\nTherefore\n( )\nd\ndx \u222bf x dx\n =\n(\nF ( ) + C)\nd\nx\ndx\n=\nF ( ) =\n( )\nd\nx\nf x\ndx\nSimilarly, we note that\nf \u2032(x) =\n( )\nd f x\ndx\nand hence\nf( )\n\u222b\u2032x dx\n = f (x) + C\nwhere C is arbitrary constant called constant of integration"
    },
    {
        "Chapter": "1",
        "sentence_range": "3345-3348",
        "Text": "1\nINTEGRALS         293\nProof Let F be any anti derivative of f, i e ,\ndF( )\nx\ndx\n = f (x)\nThen\n( )\n\u222bf x dx\n = F(x) + C\nTherefore\n( )\nd\ndx \u222bf x dx\n =\n(\nF ( ) + C)\nd\nx\ndx\n=\nF ( ) =\n( )\nd\nx\nf x\ndx\nSimilarly, we note that\nf \u2032(x) =\n( )\nd f x\ndx\nand hence\nf( )\n\u222b\u2032x dx\n = f (x) + C\nwhere C is arbitrary constant called constant of integration (II)\nTwo indefinite integrals with the same derivative lead to the same family of\ncurves and so they are equivalent"
    },
    {
        "Chapter": "1",
        "sentence_range": "3346-3349",
        "Text": "e ,\ndF( )\nx\ndx\n = f (x)\nThen\n( )\n\u222bf x dx\n = F(x) + C\nTherefore\n( )\nd\ndx \u222bf x dx\n =\n(\nF ( ) + C)\nd\nx\ndx\n=\nF ( ) =\n( )\nd\nx\nf x\ndx\nSimilarly, we note that\nf \u2032(x) =\n( )\nd f x\ndx\nand hence\nf( )\n\u222b\u2032x dx\n = f (x) + C\nwhere C is arbitrary constant called constant of integration (II)\nTwo indefinite integrals with the same derivative lead to the same family of\ncurves and so they are equivalent Proof Let f and g be two functions such that\n( )\nd\nf x dx\ndx \u222b\n =\n( )\nd\ng x dx\ndx \u222b\nor\n( )\n( )\nd\nf x dx \u2013\ng x dx\ndx\n\uf8ee\n\uf8f9\n\uf8f0\n\uf8fb\n\u222b\n\u222b\n = 0\nHence\n( )\n( )\nf x dx \u2013\ng x dx\n\u222b\n\u222b\n= C, where C is any real number\n  (Why"
    },
    {
        "Chapter": "1",
        "sentence_range": "3347-3350",
        "Text": ",\ndF( )\nx\ndx\n = f (x)\nThen\n( )\n\u222bf x dx\n = F(x) + C\nTherefore\n( )\nd\ndx \u222bf x dx\n =\n(\nF ( ) + C)\nd\nx\ndx\n=\nF ( ) =\n( )\nd\nx\nf x\ndx\nSimilarly, we note that\nf \u2032(x) =\n( )\nd f x\ndx\nand hence\nf( )\n\u222b\u2032x dx\n = f (x) + C\nwhere C is arbitrary constant called constant of integration (II)\nTwo indefinite integrals with the same derivative lead to the same family of\ncurves and so they are equivalent Proof Let f and g be two functions such that\n( )\nd\nf x dx\ndx \u222b\n =\n( )\nd\ng x dx\ndx \u222b\nor\n( )\n( )\nd\nf x dx \u2013\ng x dx\ndx\n\uf8ee\n\uf8f9\n\uf8f0\n\uf8fb\n\u222b\n\u222b\n = 0\nHence\n( )\n( )\nf x dx \u2013\ng x dx\n\u222b\n\u222b\n= C, where C is any real number\n  (Why )\nor\n( )\n\u222bf x dx\n = \n( )\ng x dx +C\n\u222b\nSo the families of curves {\n}\n1\n1\n( )\nC ,C\nR\nf x dx +\n\u2208\n\u222b\nand\n{\n}\n2\n2\n( )\nC , C\nR\ng x dx +\n\u2208\n\u222b\n are identical"
    },
    {
        "Chapter": "1",
        "sentence_range": "3348-3351",
        "Text": "(II)\nTwo indefinite integrals with the same derivative lead to the same family of\ncurves and so they are equivalent Proof Let f and g be two functions such that\n( )\nd\nf x dx\ndx \u222b\n =\n( )\nd\ng x dx\ndx \u222b\nor\n( )\n( )\nd\nf x dx \u2013\ng x dx\ndx\n\uf8ee\n\uf8f9\n\uf8f0\n\uf8fb\n\u222b\n\u222b\n = 0\nHence\n( )\n( )\nf x dx \u2013\ng x dx\n\u222b\n\u222b\n= C, where C is any real number\n  (Why )\nor\n( )\n\u222bf x dx\n = \n( )\ng x dx +C\n\u222b\nSo the families of curves {\n}\n1\n1\n( )\nC ,C\nR\nf x dx +\n\u2208\n\u222b\nand\n{\n}\n2\n2\n( )\nC , C\nR\ng x dx +\n\u2208\n\u222b\n are identical Hence, in this sense, \n( )\nand\n( )\nf x dx\ng x dx\n\u222b\n\u222b\n are equivalent"
    },
    {
        "Chapter": "1",
        "sentence_range": "3349-3352",
        "Text": "Proof Let f and g be two functions such that\n( )\nd\nf x dx\ndx \u222b\n =\n( )\nd\ng x dx\ndx \u222b\nor\n( )\n( )\nd\nf x dx \u2013\ng x dx\ndx\n\uf8ee\n\uf8f9\n\uf8f0\n\uf8fb\n\u222b\n\u222b\n = 0\nHence\n( )\n( )\nf x dx \u2013\ng x dx\n\u222b\n\u222b\n= C, where C is any real number\n  (Why )\nor\n( )\n\u222bf x dx\n = \n( )\ng x dx +C\n\u222b\nSo the families of curves {\n}\n1\n1\n( )\nC ,C\nR\nf x dx +\n\u2208\n\u222b\nand\n{\n}\n2\n2\n( )\nC , C\nR\ng x dx +\n\u2208\n\u222b\n are identical Hence, in this sense, \n( )\nand\n( )\nf x dx\ng x dx\n\u222b\n\u222b\n are equivalent 294\nMATHEMATICS\n\ufffd Note The equivalence of the families {\n}\n1\n1\n( )\nf x dx+ C ,C\n\u2208\n\u222b\nR  and\n{\n}\n2\n2\n( )\ng x dx+ C ,C\n\u2208\n\u222b\nR  is customarily expressed by writing \n( )\n=\n( )\nf x dx\ng x dx\n\u222b\n\u222b\n,\nwithout mentioning the parameter"
    },
    {
        "Chapter": "1",
        "sentence_range": "3350-3353",
        "Text": ")\nor\n( )\n\u222bf x dx\n = \n( )\ng x dx +C\n\u222b\nSo the families of curves {\n}\n1\n1\n( )\nC ,C\nR\nf x dx +\n\u2208\n\u222b\nand\n{\n}\n2\n2\n( )\nC , C\nR\ng x dx +\n\u2208\n\u222b\n are identical Hence, in this sense, \n( )\nand\n( )\nf x dx\ng x dx\n\u222b\n\u222b\n are equivalent 294\nMATHEMATICS\n\ufffd Note The equivalence of the families {\n}\n1\n1\n( )\nf x dx+ C ,C\n\u2208\n\u222b\nR  and\n{\n}\n2\n2\n( )\ng x dx+ C ,C\n\u2208\n\u222b\nR  is customarily expressed by writing \n( )\n=\n( )\nf x dx\ng x dx\n\u222b\n\u222b\n,\nwithout mentioning the parameter (III)\n[\n]\n( ) +\n( )\n( )\n+\n( )\nf x\ng x\ndx\nf x dx\ng x dx\n=\n\u222b\n\u222b\n\u222b\nProof\nBy Property (I), we have\n[ ( ) +\n( )]\nd\nf x\ng x\ndx\ndx\n\uf8ee\n\uf8f9\n\uf8f0\n\uf8fb\n\u222b\n = f (x) + g(x)"
    },
    {
        "Chapter": "1",
        "sentence_range": "3351-3354",
        "Text": "Hence, in this sense, \n( )\nand\n( )\nf x dx\ng x dx\n\u222b\n\u222b\n are equivalent 294\nMATHEMATICS\n\ufffd Note The equivalence of the families {\n}\n1\n1\n( )\nf x dx+ C ,C\n\u2208\n\u222b\nR  and\n{\n}\n2\n2\n( )\ng x dx+ C ,C\n\u2208\n\u222b\nR  is customarily expressed by writing \n( )\n=\n( )\nf x dx\ng x dx\n\u222b\n\u222b\n,\nwithout mentioning the parameter (III)\n[\n]\n( ) +\n( )\n( )\n+\n( )\nf x\ng x\ndx\nf x dx\ng x dx\n=\n\u222b\n\u222b\n\u222b\nProof\nBy Property (I), we have\n[ ( ) +\n( )]\nd\nf x\ng x\ndx\ndx\n\uf8ee\n\uf8f9\n\uf8f0\n\uf8fb\n\u222b\n = f (x) + g(x) (1)\n On the otherhand, we find that\n( )\n+\n( )\nd\nf x dx\ng x dx\ndx\n\uf8ee\n\uf8f9\n\uf8f0\n\uf8fb\n\u222b\n\u222b\n =\n( )\n+\n( )\nd\nd\nf x dx\ng x dx\ndx\ndx\n\u222b\n\u222b\n= f (x) + g(x)"
    },
    {
        "Chapter": "1",
        "sentence_range": "3352-3355",
        "Text": "294\nMATHEMATICS\n\ufffd Note The equivalence of the families {\n}\n1\n1\n( )\nf x dx+ C ,C\n\u2208\n\u222b\nR  and\n{\n}\n2\n2\n( )\ng x dx+ C ,C\n\u2208\n\u222b\nR  is customarily expressed by writing \n( )\n=\n( )\nf x dx\ng x dx\n\u222b\n\u222b\n,\nwithout mentioning the parameter (III)\n[\n]\n( ) +\n( )\n( )\n+\n( )\nf x\ng x\ndx\nf x dx\ng x dx\n=\n\u222b\n\u222b\n\u222b\nProof\nBy Property (I), we have\n[ ( ) +\n( )]\nd\nf x\ng x\ndx\ndx\n\uf8ee\n\uf8f9\n\uf8f0\n\uf8fb\n\u222b\n = f (x) + g(x) (1)\n On the otherhand, we find that\n( )\n+\n( )\nd\nf x dx\ng x dx\ndx\n\uf8ee\n\uf8f9\n\uf8f0\n\uf8fb\n\u222b\n\u222b\n =\n( )\n+\n( )\nd\nd\nf x dx\ng x dx\ndx\ndx\n\u222b\n\u222b\n= f (x) + g(x) (2)\n  Thus, in view of Property (II), it follows by (1) and (2)  that\n(\n)\n( )\n( )\nf x\ng x\ndx\n+\n\u222b\n=\n( )\n( )\nf x dx\ng x dx\n+\n\u222b\n\u222b"
    },
    {
        "Chapter": "1",
        "sentence_range": "3353-3356",
        "Text": "(III)\n[\n]\n( ) +\n( )\n( )\n+\n( )\nf x\ng x\ndx\nf x dx\ng x dx\n=\n\u222b\n\u222b\n\u222b\nProof\nBy Property (I), we have\n[ ( ) +\n( )]\nd\nf x\ng x\ndx\ndx\n\uf8ee\n\uf8f9\n\uf8f0\n\uf8fb\n\u222b\n = f (x) + g(x) (1)\n On the otherhand, we find that\n( )\n+\n( )\nd\nf x dx\ng x dx\ndx\n\uf8ee\n\uf8f9\n\uf8f0\n\uf8fb\n\u222b\n\u222b\n =\n( )\n+\n( )\nd\nd\nf x dx\ng x dx\ndx\ndx\n\u222b\n\u222b\n= f (x) + g(x) (2)\n  Thus, in view of Property (II), it follows by (1) and (2)  that\n(\n)\n( )\n( )\nf x\ng x\ndx\n+\n\u222b\n=\n( )\n( )\nf x dx\ng x dx\n+\n\u222b\n\u222b (IV)  For any real number k, \n( )\n( )\nk f x dx\nk\nf x dx\n=\n\u222b\n\u222b\nProof By the Property (I), \n( )\n( )\nd\nk f x dx\nk f x\ndx\n=\n\u222b"
    },
    {
        "Chapter": "1",
        "sentence_range": "3354-3357",
        "Text": "(1)\n On the otherhand, we find that\n( )\n+\n( )\nd\nf x dx\ng x dx\ndx\n\uf8ee\n\uf8f9\n\uf8f0\n\uf8fb\n\u222b\n\u222b\n =\n( )\n+\n( )\nd\nd\nf x dx\ng x dx\ndx\ndx\n\u222b\n\u222b\n= f (x) + g(x) (2)\n  Thus, in view of Property (II), it follows by (1) and (2)  that\n(\n)\n( )\n( )\nf x\ng x\ndx\n+\n\u222b\n=\n( )\n( )\nf x dx\ng x dx\n+\n\u222b\n\u222b (IV)  For any real number k, \n( )\n( )\nk f x dx\nk\nf x dx\n=\n\u222b\n\u222b\nProof By the Property (I), \n( )\n( )\nd\nk f x dx\nk f x\ndx\n=\n\u222b Also\n( )\nd\nk\nf x dx\ndx\n\uf8ee\n\uf8f9\n\uf8f0\n\uf8fb\n\u222b\n =  \n( )\n=\n( )\nkd\nf x dx\nk f x\ndx \u222b\n Therefore, using the Property (II), we have \n( )\n( )\nk f x dx\nk\nf x dx\n=\n\u222b\n\u222b"
    },
    {
        "Chapter": "1",
        "sentence_range": "3355-3358",
        "Text": "(2)\n  Thus, in view of Property (II), it follows by (1) and (2)  that\n(\n)\n( )\n( )\nf x\ng x\ndx\n+\n\u222b\n=\n( )\n( )\nf x dx\ng x dx\n+\n\u222b\n\u222b (IV)  For any real number k, \n( )\n( )\nk f x dx\nk\nf x dx\n=\n\u222b\n\u222b\nProof By the Property (I), \n( )\n( )\nd\nk f x dx\nk f x\ndx\n=\n\u222b Also\n( )\nd\nk\nf x dx\ndx\n\uf8ee\n\uf8f9\n\uf8f0\n\uf8fb\n\u222b\n =  \n( )\n=\n( )\nkd\nf x dx\nk f x\ndx \u222b\n Therefore, using the Property (II), we have \n( )\n( )\nk f x dx\nk\nf x dx\n=\n\u222b\n\u222b (V)\nProperties (III) and (IV) can be generalised to a finite number of functions\nf1, f2,"
    },
    {
        "Chapter": "1",
        "sentence_range": "3356-3359",
        "Text": "(IV)  For any real number k, \n( )\n( )\nk f x dx\nk\nf x dx\n=\n\u222b\n\u222b\nProof By the Property (I), \n( )\n( )\nd\nk f x dx\nk f x\ndx\n=\n\u222b Also\n( )\nd\nk\nf x dx\ndx\n\uf8ee\n\uf8f9\n\uf8f0\n\uf8fb\n\u222b\n =  \n( )\n=\n( )\nkd\nf x dx\nk f x\ndx \u222b\n Therefore, using the Property (II), we have \n( )\n( )\nk f x dx\nk\nf x dx\n=\n\u222b\n\u222b (V)\nProperties (III) and (IV) can be generalised to a finite number of functions\nf1, f2, , fn and the real numbers, k1, k2,"
    },
    {
        "Chapter": "1",
        "sentence_range": "3357-3360",
        "Text": "Also\n( )\nd\nk\nf x dx\ndx\n\uf8ee\n\uf8f9\n\uf8f0\n\uf8fb\n\u222b\n =  \n( )\n=\n( )\nkd\nf x dx\nk f x\ndx \u222b\n Therefore, using the Property (II), we have \n( )\n( )\nk f x dx\nk\nf x dx\n=\n\u222b\n\u222b (V)\nProperties (III) and (IV) can be generalised to a finite number of functions\nf1, f2, , fn and the real numbers, k1, k2, , kn giving\n[\n]\n1 1\n2 2\n( )\n( )\n( )\nn\nn\nk f x\nk f\nx"
    },
    {
        "Chapter": "1",
        "sentence_range": "3358-3361",
        "Text": "(V)\nProperties (III) and (IV) can be generalised to a finite number of functions\nf1, f2, , fn and the real numbers, k1, k2, , kn giving\n[\n]\n1 1\n2 2\n( )\n( )\n( )\nn\nn\nk f x\nk f\nx k f\nx\ndx\n+\n+\n+\n\u222b\n= \n1\n1\n2\n2\n( )\n( )\n( )\nn\nn\nk\nf x dx\nk\nf\nx dx"
    },
    {
        "Chapter": "1",
        "sentence_range": "3359-3362",
        "Text": ", fn and the real numbers, k1, k2, , kn giving\n[\n]\n1 1\n2 2\n( )\n( )\n( )\nn\nn\nk f x\nk f\nx k f\nx\ndx\n+\n+\n+\n\u222b\n= \n1\n1\n2\n2\n( )\n( )\n( )\nn\nn\nk\nf x dx\nk\nf\nx dx k\nf\nx dx\n+\n+\n+\n\u222b\n\u222b\n\u222b"
    },
    {
        "Chapter": "1",
        "sentence_range": "3360-3363",
        "Text": ", kn giving\n[\n]\n1 1\n2 2\n( )\n( )\n( )\nn\nn\nk f x\nk f\nx k f\nx\ndx\n+\n+\n+\n\u222b\n= \n1\n1\n2\n2\n( )\n( )\n( )\nn\nn\nk\nf x dx\nk\nf\nx dx k\nf\nx dx\n+\n+\n+\n\u222b\n\u222b\n\u222b To find an anti derivative of a given function, we search intuitively for a function\nwhose derivative is the given function"
    },
    {
        "Chapter": "1",
        "sentence_range": "3361-3364",
        "Text": "k f\nx\ndx\n+\n+\n+\n\u222b\n= \n1\n1\n2\n2\n( )\n( )\n( )\nn\nn\nk\nf x dx\nk\nf\nx dx k\nf\nx dx\n+\n+\n+\n\u222b\n\u222b\n\u222b To find an anti derivative of a given function, we search intuitively for a function\nwhose derivative is the given function The search for the requisite function for finding\nan anti derivative is known as integration by the method of inspection"
    },
    {
        "Chapter": "1",
        "sentence_range": "3362-3365",
        "Text": "k\nf\nx dx\n+\n+\n+\n\u222b\n\u222b\n\u222b To find an anti derivative of a given function, we search intuitively for a function\nwhose derivative is the given function The search for the requisite function for finding\nan anti derivative is known as integration by the method of inspection We illustrate it\nthrough some examples"
    },
    {
        "Chapter": "1",
        "sentence_range": "3363-3366",
        "Text": "To find an anti derivative of a given function, we search intuitively for a function\nwhose derivative is the given function The search for the requisite function for finding\nan anti derivative is known as integration by the method of inspection We illustrate it\nthrough some examples INTEGRALS         295\nExample 1 Write an anti derivative for each of the following functions using the\nmethod of inspection:\n(i)\ncos 2x\n(ii)\n3x2 + 4x3\n(iii)\n1\nx , x \u2260 0\nSolution\n(i)\nWe look for a function whose derivative is cos 2x"
    },
    {
        "Chapter": "1",
        "sentence_range": "3364-3367",
        "Text": "The search for the requisite function for finding\nan anti derivative is known as integration by the method of inspection We illustrate it\nthrough some examples INTEGRALS         295\nExample 1 Write an anti derivative for each of the following functions using the\nmethod of inspection:\n(i)\ncos 2x\n(ii)\n3x2 + 4x3\n(iii)\n1\nx , x \u2260 0\nSolution\n(i)\nWe look for a function whose derivative is cos 2x Recall that\nd\ndx  sin 2x = 2 cos 2x\nor\ncos 2x = 1\n2\nd\ndx  (sin 2x) =\n21 sin 2\nd\nx\ndx\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\nTherefore, an anti derivative of cos 2x is \n1 sin 2\n2\nx"
    },
    {
        "Chapter": "1",
        "sentence_range": "3365-3368",
        "Text": "We illustrate it\nthrough some examples INTEGRALS         295\nExample 1 Write an anti derivative for each of the following functions using the\nmethod of inspection:\n(i)\ncos 2x\n(ii)\n3x2 + 4x3\n(iii)\n1\nx , x \u2260 0\nSolution\n(i)\nWe look for a function whose derivative is cos 2x Recall that\nd\ndx  sin 2x = 2 cos 2x\nor\ncos 2x = 1\n2\nd\ndx  (sin 2x) =\n21 sin 2\nd\nx\ndx\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\nTherefore, an anti derivative of cos 2x is \n1 sin 2\n2\nx (ii)\nWe look for a function whose derivative is 3x2 + 4x3"
    },
    {
        "Chapter": "1",
        "sentence_range": "3366-3369",
        "Text": "INTEGRALS         295\nExample 1 Write an anti derivative for each of the following functions using the\nmethod of inspection:\n(i)\ncos 2x\n(ii)\n3x2 + 4x3\n(iii)\n1\nx , x \u2260 0\nSolution\n(i)\nWe look for a function whose derivative is cos 2x Recall that\nd\ndx  sin 2x = 2 cos 2x\nor\ncos 2x = 1\n2\nd\ndx  (sin 2x) =\n21 sin 2\nd\nx\ndx\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\nTherefore, an anti derivative of cos 2x is \n1 sin 2\n2\nx (ii)\nWe look for a function whose derivative is 3x2 + 4x3 Note that\n(\n)\n3\n4\nd\nx\nx\ndx\n+\n= 3x2 + 4x3"
    },
    {
        "Chapter": "1",
        "sentence_range": "3367-3370",
        "Text": "Recall that\nd\ndx  sin 2x = 2 cos 2x\nor\ncos 2x = 1\n2\nd\ndx  (sin 2x) =\n21 sin 2\nd\nx\ndx\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\nTherefore, an anti derivative of cos 2x is \n1 sin 2\n2\nx (ii)\nWe look for a function whose derivative is 3x2 + 4x3 Note that\n(\n)\n3\n4\nd\nx\nx\ndx\n+\n= 3x2 + 4x3 Therefore, an anti derivative of 3x2 + 4x3  is  x3 + x4"
    },
    {
        "Chapter": "1",
        "sentence_range": "3368-3371",
        "Text": "(ii)\nWe look for a function whose derivative is 3x2 + 4x3 Note that\n(\n)\n3\n4\nd\nx\nx\ndx\n+\n= 3x2 + 4x3 Therefore, an anti derivative of 3x2 + 4x3  is  x3 + x4 (iii)\nWe know that\n1\n1\n1\n(log )\n0 and\n[log (\n)]\n(\n1)\n0\nd\nd\nx\n,x\n\u2013 x\n\u2013\n,x\ndx\nx\ndx\n\u2013 x\nx\n=\n>\n=\n=\n<\nCombining above, we get \n(\n)\n1\nlog\n0\nd\nx\n, x\ndx\n=x\n\u2260\nTherefore, \n1\nlog\ndx\nx\nx\n=\n\u222b\n is one of the anti derivatives of 1\nx"
    },
    {
        "Chapter": "1",
        "sentence_range": "3369-3372",
        "Text": "Note that\n(\n)\n3\n4\nd\nx\nx\ndx\n+\n= 3x2 + 4x3 Therefore, an anti derivative of 3x2 + 4x3  is  x3 + x4 (iii)\nWe know that\n1\n1\n1\n(log )\n0 and\n[log (\n)]\n(\n1)\n0\nd\nd\nx\n,x\n\u2013 x\n\u2013\n,x\ndx\nx\ndx\n\u2013 x\nx\n=\n>\n=\n=\n<\nCombining above, we get \n(\n)\n1\nlog\n0\nd\nx\n, x\ndx\n=x\n\u2260\nTherefore, \n1\nlog\ndx\nx\nx\n=\n\u222b\n is one of the anti derivatives of 1\nx Example 2 Find the following integrals:\n(i)\n3\n2\n1\nx \u2013 dx\n\u222bx\n(ii)   \n(32\n1)\nx\ndx\n+\n\u222b\n(iii)   \u222b\n23\n1\n(\n2\n\u2013\n)\n+\n\u222b\nx\nx\ne\ndx\nx\nSolution\n(i)\nWe have\n3\n2\n2\n1\n\u2013\nx \u2013\ndx\nx dx \u2013\nx\ndx\nx\n=\n\u222b\n\u222b\n\u222b\n(by Property V)\n296\nMATHEMATICS\n= \n1\n1\n2\n1\n1\n2\nC\nC\n1 1\n2\n1\n\u2013\nx\nx\n\u2013\n\u2013\n+\n+\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n+\n+\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n+\n+\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n;  C1, C2 are constants of integration\n= \n2\n1\n1\n2\nC\nC\n2\n1\n\u2013\nx\n\u2013x\n\u2013\n\u2013\n+\n = \n2\n1\n2\n1 + C\nC\nx2\n\u2013\nx\n+\n= \n2\n1 + C\n2\nx\nx\n+\n, where C = C 1 \u2013 C2 is another constant of integration"
    },
    {
        "Chapter": "1",
        "sentence_range": "3370-3373",
        "Text": "Therefore, an anti derivative of 3x2 + 4x3  is  x3 + x4 (iii)\nWe know that\n1\n1\n1\n(log )\n0 and\n[log (\n)]\n(\n1)\n0\nd\nd\nx\n,x\n\u2013 x\n\u2013\n,x\ndx\nx\ndx\n\u2013 x\nx\n=\n>\n=\n=\n<\nCombining above, we get \n(\n)\n1\nlog\n0\nd\nx\n, x\ndx\n=x\n\u2260\nTherefore, \n1\nlog\ndx\nx\nx\n=\n\u222b\n is one of the anti derivatives of 1\nx Example 2 Find the following integrals:\n(i)\n3\n2\n1\nx \u2013 dx\n\u222bx\n(ii)   \n(32\n1)\nx\ndx\n+\n\u222b\n(iii)   \u222b\n23\n1\n(\n2\n\u2013\n)\n+\n\u222b\nx\nx\ne\ndx\nx\nSolution\n(i)\nWe have\n3\n2\n2\n1\n\u2013\nx \u2013\ndx\nx dx \u2013\nx\ndx\nx\n=\n\u222b\n\u222b\n\u222b\n(by Property V)\n296\nMATHEMATICS\n= \n1\n1\n2\n1\n1\n2\nC\nC\n1 1\n2\n1\n\u2013\nx\nx\n\u2013\n\u2013\n+\n+\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n+\n+\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n+\n+\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n;  C1, C2 are constants of integration\n= \n2\n1\n1\n2\nC\nC\n2\n1\n\u2013\nx\n\u2013x\n\u2013\n\u2013\n+\n = \n2\n1\n2\n1 + C\nC\nx2\n\u2013\nx\n+\n= \n2\n1 + C\n2\nx\nx\n+\n, where C = C 1 \u2013 C2 is another constant of integration \ufffdNote  From now onwards, we shall write only one constant of integration in the\nfinal answer"
    },
    {
        "Chapter": "1",
        "sentence_range": "3371-3374",
        "Text": "(iii)\nWe know that\n1\n1\n1\n(log )\n0 and\n[log (\n)]\n(\n1)\n0\nd\nd\nx\n,x\n\u2013 x\n\u2013\n,x\ndx\nx\ndx\n\u2013 x\nx\n=\n>\n=\n=\n<\nCombining above, we get \n(\n)\n1\nlog\n0\nd\nx\n, x\ndx\n=x\n\u2260\nTherefore, \n1\nlog\ndx\nx\nx\n=\n\u222b\n is one of the anti derivatives of 1\nx Example 2 Find the following integrals:\n(i)\n3\n2\n1\nx \u2013 dx\n\u222bx\n(ii)   \n(32\n1)\nx\ndx\n+\n\u222b\n(iii)   \u222b\n23\n1\n(\n2\n\u2013\n)\n+\n\u222b\nx\nx\ne\ndx\nx\nSolution\n(i)\nWe have\n3\n2\n2\n1\n\u2013\nx \u2013\ndx\nx dx \u2013\nx\ndx\nx\n=\n\u222b\n\u222b\n\u222b\n(by Property V)\n296\nMATHEMATICS\n= \n1\n1\n2\n1\n1\n2\nC\nC\n1 1\n2\n1\n\u2013\nx\nx\n\u2013\n\u2013\n+\n+\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n+\n+\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n+\n+\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n;  C1, C2 are constants of integration\n= \n2\n1\n1\n2\nC\nC\n2\n1\n\u2013\nx\n\u2013x\n\u2013\n\u2013\n+\n = \n2\n1\n2\n1 + C\nC\nx2\n\u2013\nx\n+\n= \n2\n1 + C\n2\nx\nx\n+\n, where C = C 1 \u2013 C2 is another constant of integration \ufffdNote  From now onwards, we shall write only one constant of integration in the\nfinal answer (ii) We have\n2\n2\n3\n3\n(\n1)\nx\ndx\nx dx\ndx\n+\n=\n+\n\u222b\n\u222b\n\u222b\n=\n2\n1\n3\nC\n2\n1\n3\nx\nx\n+\n+\n+\n+\n = \n335\nC\n5\nx\n+x\n+\n(iii) We have \n3\n3\n2\n2\n1\n1\n(\n2\n)\n2\nx\nx\nx\ne \u2013\ndx\nx\ndx\ne dx \u2013\ndx\nx\nx\n+\n=\n+\n\u222b\n\u222b\n\u222b\n\u222b\n=\n3\n1\n2\n2\n\u2013 log\n+ C\n3\n1\n2\nx\nx\ne\nx\n+\n+\n+\n=\n25\n2\n2\n\u2013 log\n+ C\n5\nx\nx\ne\nx\n+\nExample 3 Find the following integrals:\n(i)\n(sin\ncos )\nx\nx dx\n+\n\u222b\n(ii) cosec\n(cosec \ncot\n)\nx\nx\nx dx\n+\n\u222b\n(iii)\n2\n1\n\u2013cossin\nxx dx\n\u222b\nSolution\n(i)\nWe have\n(sin\ncos )\nsin\ncos\nx\nx dx\nx dx\nx dx\n+\n=\n+\n\u222b\n\u222b\n\u222b\n= \u2013 cos\nsin\nC\nx\nx\n+\n+\nINTEGRALS         297\n(ii)\nWe have\n2\n(cosec\n(cosec\n+ cot\n)\ncosec\ncosec\ncot\nx\nx\nx dx\nx dx\nx\nx dx\n=\n+\n\u222b\n\u222b\n\u222b\n= \u2013 cot\ncosec\nC\nx \u2013\nx +\n(iii)\nWe have\n2\n2\n2\n1 sin\n1\nsin\ncos\ncos\ncos\n\u2013\nx\nx\ndx\ndx \u2013\ndx\nx\nx\nx\n=\n\u222b\n\u222b\n\u222b\n= \nsec2\ntan\nsec\nx dx \u2013\nx\nx dx\n\u222b\n\u222b\n= tan\nsec\nC\nx \u2013\nx +\nExample 4 Find the anti derivative F of  f defined by f (x) = 4x3 \u2013 6, where F (0) = 3\nSolution One anti derivative of f (x) is x4 \u2013 6x since\n(4\n6 )\nd\nx \u2013 x\ndx\n = 4x3 \u2013 6\nTherefore, the anti derivative F is given by\nF(x) = x4 \u2013 6x + C, where C is constant"
    },
    {
        "Chapter": "1",
        "sentence_range": "3372-3375",
        "Text": "Example 2 Find the following integrals:\n(i)\n3\n2\n1\nx \u2013 dx\n\u222bx\n(ii)   \n(32\n1)\nx\ndx\n+\n\u222b\n(iii)   \u222b\n23\n1\n(\n2\n\u2013\n)\n+\n\u222b\nx\nx\ne\ndx\nx\nSolution\n(i)\nWe have\n3\n2\n2\n1\n\u2013\nx \u2013\ndx\nx dx \u2013\nx\ndx\nx\n=\n\u222b\n\u222b\n\u222b\n(by Property V)\n296\nMATHEMATICS\n= \n1\n1\n2\n1\n1\n2\nC\nC\n1 1\n2\n1\n\u2013\nx\nx\n\u2013\n\u2013\n+\n+\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n+\n+\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n+\n+\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n;  C1, C2 are constants of integration\n= \n2\n1\n1\n2\nC\nC\n2\n1\n\u2013\nx\n\u2013x\n\u2013\n\u2013\n+\n = \n2\n1\n2\n1 + C\nC\nx2\n\u2013\nx\n+\n= \n2\n1 + C\n2\nx\nx\n+\n, where C = C 1 \u2013 C2 is another constant of integration \ufffdNote  From now onwards, we shall write only one constant of integration in the\nfinal answer (ii) We have\n2\n2\n3\n3\n(\n1)\nx\ndx\nx dx\ndx\n+\n=\n+\n\u222b\n\u222b\n\u222b\n=\n2\n1\n3\nC\n2\n1\n3\nx\nx\n+\n+\n+\n+\n = \n335\nC\n5\nx\n+x\n+\n(iii) We have \n3\n3\n2\n2\n1\n1\n(\n2\n)\n2\nx\nx\nx\ne \u2013\ndx\nx\ndx\ne dx \u2013\ndx\nx\nx\n+\n=\n+\n\u222b\n\u222b\n\u222b\n\u222b\n=\n3\n1\n2\n2\n\u2013 log\n+ C\n3\n1\n2\nx\nx\ne\nx\n+\n+\n+\n=\n25\n2\n2\n\u2013 log\n+ C\n5\nx\nx\ne\nx\n+\nExample 3 Find the following integrals:\n(i)\n(sin\ncos )\nx\nx dx\n+\n\u222b\n(ii) cosec\n(cosec \ncot\n)\nx\nx\nx dx\n+\n\u222b\n(iii)\n2\n1\n\u2013cossin\nxx dx\n\u222b\nSolution\n(i)\nWe have\n(sin\ncos )\nsin\ncos\nx\nx dx\nx dx\nx dx\n+\n=\n+\n\u222b\n\u222b\n\u222b\n= \u2013 cos\nsin\nC\nx\nx\n+\n+\nINTEGRALS         297\n(ii)\nWe have\n2\n(cosec\n(cosec\n+ cot\n)\ncosec\ncosec\ncot\nx\nx\nx dx\nx dx\nx\nx dx\n=\n+\n\u222b\n\u222b\n\u222b\n= \u2013 cot\ncosec\nC\nx \u2013\nx +\n(iii)\nWe have\n2\n2\n2\n1 sin\n1\nsin\ncos\ncos\ncos\n\u2013\nx\nx\ndx\ndx \u2013\ndx\nx\nx\nx\n=\n\u222b\n\u222b\n\u222b\n= \nsec2\ntan\nsec\nx dx \u2013\nx\nx dx\n\u222b\n\u222b\n= tan\nsec\nC\nx \u2013\nx +\nExample 4 Find the anti derivative F of  f defined by f (x) = 4x3 \u2013 6, where F (0) = 3\nSolution One anti derivative of f (x) is x4 \u2013 6x since\n(4\n6 )\nd\nx \u2013 x\ndx\n = 4x3 \u2013 6\nTherefore, the anti derivative F is given by\nF(x) = x4 \u2013 6x + C, where C is constant Given that\nF(0) = 3, which gives,\n3 = 0 \u2013 6 \u00d7 0 + C    or    C = 3\nHence, the required anti derivative is the unique function F defined by\nF(x) = x4 \u2013 6x + 3"
    },
    {
        "Chapter": "1",
        "sentence_range": "3373-3376",
        "Text": "\ufffdNote  From now onwards, we shall write only one constant of integration in the\nfinal answer (ii) We have\n2\n2\n3\n3\n(\n1)\nx\ndx\nx dx\ndx\n+\n=\n+\n\u222b\n\u222b\n\u222b\n=\n2\n1\n3\nC\n2\n1\n3\nx\nx\n+\n+\n+\n+\n = \n335\nC\n5\nx\n+x\n+\n(iii) We have \n3\n3\n2\n2\n1\n1\n(\n2\n)\n2\nx\nx\nx\ne \u2013\ndx\nx\ndx\ne dx \u2013\ndx\nx\nx\n+\n=\n+\n\u222b\n\u222b\n\u222b\n\u222b\n=\n3\n1\n2\n2\n\u2013 log\n+ C\n3\n1\n2\nx\nx\ne\nx\n+\n+\n+\n=\n25\n2\n2\n\u2013 log\n+ C\n5\nx\nx\ne\nx\n+\nExample 3 Find the following integrals:\n(i)\n(sin\ncos )\nx\nx dx\n+\n\u222b\n(ii) cosec\n(cosec \ncot\n)\nx\nx\nx dx\n+\n\u222b\n(iii)\n2\n1\n\u2013cossin\nxx dx\n\u222b\nSolution\n(i)\nWe have\n(sin\ncos )\nsin\ncos\nx\nx dx\nx dx\nx dx\n+\n=\n+\n\u222b\n\u222b\n\u222b\n= \u2013 cos\nsin\nC\nx\nx\n+\n+\nINTEGRALS         297\n(ii)\nWe have\n2\n(cosec\n(cosec\n+ cot\n)\ncosec\ncosec\ncot\nx\nx\nx dx\nx dx\nx\nx dx\n=\n+\n\u222b\n\u222b\n\u222b\n= \u2013 cot\ncosec\nC\nx \u2013\nx +\n(iii)\nWe have\n2\n2\n2\n1 sin\n1\nsin\ncos\ncos\ncos\n\u2013\nx\nx\ndx\ndx \u2013\ndx\nx\nx\nx\n=\n\u222b\n\u222b\n\u222b\n= \nsec2\ntan\nsec\nx dx \u2013\nx\nx dx\n\u222b\n\u222b\n= tan\nsec\nC\nx \u2013\nx +\nExample 4 Find the anti derivative F of  f defined by f (x) = 4x3 \u2013 6, where F (0) = 3\nSolution One anti derivative of f (x) is x4 \u2013 6x since\n(4\n6 )\nd\nx \u2013 x\ndx\n = 4x3 \u2013 6\nTherefore, the anti derivative F is given by\nF(x) = x4 \u2013 6x + C, where C is constant Given that\nF(0) = 3, which gives,\n3 = 0 \u2013 6 \u00d7 0 + C    or    C = 3\nHence, the required anti derivative is the unique function F defined by\nF(x) = x4 \u2013 6x + 3 Remarks\n(i)\nWe see that if F is an anti derivative of f, then so is F + C, where C is any\nconstant"
    },
    {
        "Chapter": "1",
        "sentence_range": "3374-3377",
        "Text": "(ii) We have\n2\n2\n3\n3\n(\n1)\nx\ndx\nx dx\ndx\n+\n=\n+\n\u222b\n\u222b\n\u222b\n=\n2\n1\n3\nC\n2\n1\n3\nx\nx\n+\n+\n+\n+\n = \n335\nC\n5\nx\n+x\n+\n(iii) We have \n3\n3\n2\n2\n1\n1\n(\n2\n)\n2\nx\nx\nx\ne \u2013\ndx\nx\ndx\ne dx \u2013\ndx\nx\nx\n+\n=\n+\n\u222b\n\u222b\n\u222b\n\u222b\n=\n3\n1\n2\n2\n\u2013 log\n+ C\n3\n1\n2\nx\nx\ne\nx\n+\n+\n+\n=\n25\n2\n2\n\u2013 log\n+ C\n5\nx\nx\ne\nx\n+\nExample 3 Find the following integrals:\n(i)\n(sin\ncos )\nx\nx dx\n+\n\u222b\n(ii) cosec\n(cosec \ncot\n)\nx\nx\nx dx\n+\n\u222b\n(iii)\n2\n1\n\u2013cossin\nxx dx\n\u222b\nSolution\n(i)\nWe have\n(sin\ncos )\nsin\ncos\nx\nx dx\nx dx\nx dx\n+\n=\n+\n\u222b\n\u222b\n\u222b\n= \u2013 cos\nsin\nC\nx\nx\n+\n+\nINTEGRALS         297\n(ii)\nWe have\n2\n(cosec\n(cosec\n+ cot\n)\ncosec\ncosec\ncot\nx\nx\nx dx\nx dx\nx\nx dx\n=\n+\n\u222b\n\u222b\n\u222b\n= \u2013 cot\ncosec\nC\nx \u2013\nx +\n(iii)\nWe have\n2\n2\n2\n1 sin\n1\nsin\ncos\ncos\ncos\n\u2013\nx\nx\ndx\ndx \u2013\ndx\nx\nx\nx\n=\n\u222b\n\u222b\n\u222b\n= \nsec2\ntan\nsec\nx dx \u2013\nx\nx dx\n\u222b\n\u222b\n= tan\nsec\nC\nx \u2013\nx +\nExample 4 Find the anti derivative F of  f defined by f (x) = 4x3 \u2013 6, where F (0) = 3\nSolution One anti derivative of f (x) is x4 \u2013 6x since\n(4\n6 )\nd\nx \u2013 x\ndx\n = 4x3 \u2013 6\nTherefore, the anti derivative F is given by\nF(x) = x4 \u2013 6x + C, where C is constant Given that\nF(0) = 3, which gives,\n3 = 0 \u2013 6 \u00d7 0 + C    or    C = 3\nHence, the required anti derivative is the unique function F defined by\nF(x) = x4 \u2013 6x + 3 Remarks\n(i)\nWe see that if F is an anti derivative of f, then so is F + C, where C is any\nconstant Thus, if we know one anti derivative F of a function f, we can write\ndown an infinite number of anti derivatives of f by adding any constant to F\nexpressed by F(x)  + C, C \u2208 R"
    },
    {
        "Chapter": "1",
        "sentence_range": "3375-3378",
        "Text": "Given that\nF(0) = 3, which gives,\n3 = 0 \u2013 6 \u00d7 0 + C    or    C = 3\nHence, the required anti derivative is the unique function F defined by\nF(x) = x4 \u2013 6x + 3 Remarks\n(i)\nWe see that if F is an anti derivative of f, then so is F + C, where C is any\nconstant Thus, if we know one anti derivative F of a function f, we can write\ndown an infinite number of anti derivatives of f by adding any constant to F\nexpressed by F(x)  + C, C \u2208 R In applications, it is often necessary to satisfy an\nadditional condition which then determines a specific value of C giving unique\nanti derivative of the given function"
    },
    {
        "Chapter": "1",
        "sentence_range": "3376-3379",
        "Text": "Remarks\n(i)\nWe see that if F is an anti derivative of f, then so is F + C, where C is any\nconstant Thus, if we know one anti derivative F of a function f, we can write\ndown an infinite number of anti derivatives of f by adding any constant to F\nexpressed by F(x)  + C, C \u2208 R In applications, it is often necessary to satisfy an\nadditional condition which then determines a specific value of C giving unique\nanti derivative of the given function (ii)\nSometimes, F is not expressible in terms of elementary functions viz"
    },
    {
        "Chapter": "1",
        "sentence_range": "3377-3380",
        "Text": "Thus, if we know one anti derivative F of a function f, we can write\ndown an infinite number of anti derivatives of f by adding any constant to F\nexpressed by F(x)  + C, C \u2208 R In applications, it is often necessary to satisfy an\nadditional condition which then determines a specific value of C giving unique\nanti derivative of the given function (ii)\nSometimes, F is not expressible in terms of elementary functions viz , polynomial,\nlogarithmic, exponential, trigonometric functions and their inverses etc"
    },
    {
        "Chapter": "1",
        "sentence_range": "3378-3381",
        "Text": "In applications, it is often necessary to satisfy an\nadditional condition which then determines a specific value of C giving unique\nanti derivative of the given function (ii)\nSometimes, F is not expressible in terms of elementary functions viz , polynomial,\nlogarithmic, exponential, trigonometric functions and their inverses etc We are\ntherefore blocked for finding \n( )\n\u222bf x dx"
    },
    {
        "Chapter": "1",
        "sentence_range": "3379-3382",
        "Text": "(ii)\nSometimes, F is not expressible in terms of elementary functions viz , polynomial,\nlogarithmic, exponential, trigonometric functions and their inverses etc We are\ntherefore blocked for finding \n( )\n\u222bf x dx For example, it is not possible to find\ne\u2013 x2\ndx\n\u222b\n by inspection since we can not find a function whose derivative is \ne\u2013 x2\n298\nMATHEMATICS\n(iii)\nWhen the variable of integration is denoted by a variable other than x, the integral\nformulae are modified accordingly"
    },
    {
        "Chapter": "1",
        "sentence_range": "3380-3383",
        "Text": ", polynomial,\nlogarithmic, exponential, trigonometric functions and their inverses etc We are\ntherefore blocked for finding \n( )\n\u222bf x dx For example, it is not possible to find\ne\u2013 x2\ndx\n\u222b\n by inspection since we can not find a function whose derivative is \ne\u2013 x2\n298\nMATHEMATICS\n(iii)\nWhen the variable of integration is denoted by a variable other than x, the integral\nformulae are modified accordingly For instance\n4\n1\n4\n5\n1\nC\nC\n4\n1\n5\ny\ny dy\ny\n+\n=\n+\n=\n+\n+\n\u222b\n7"
    },
    {
        "Chapter": "1",
        "sentence_range": "3381-3384",
        "Text": "We are\ntherefore blocked for finding \n( )\n\u222bf x dx For example, it is not possible to find\ne\u2013 x2\ndx\n\u222b\n by inspection since we can not find a function whose derivative is \ne\u2013 x2\n298\nMATHEMATICS\n(iii)\nWhen the variable of integration is denoted by a variable other than x, the integral\nformulae are modified accordingly For instance\n4\n1\n4\n5\n1\nC\nC\n4\n1\n5\ny\ny dy\ny\n+\n=\n+\n=\n+\n+\n\u222b\n7 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "3382-3385",
        "Text": "For example, it is not possible to find\ne\u2013 x2\ndx\n\u222b\n by inspection since we can not find a function whose derivative is \ne\u2013 x2\n298\nMATHEMATICS\n(iii)\nWhen the variable of integration is denoted by a variable other than x, the integral\nformulae are modified accordingly For instance\n4\n1\n4\n5\n1\nC\nC\n4\n1\n5\ny\ny dy\ny\n+\n=\n+\n=\n+\n+\n\u222b\n7 2 3 Comparison between differentiation and integration\n1"
    },
    {
        "Chapter": "1",
        "sentence_range": "3383-3386",
        "Text": "For instance\n4\n1\n4\n5\n1\nC\nC\n4\n1\n5\ny\ny dy\ny\n+\n=\n+\n=\n+\n+\n\u222b\n7 2 3 Comparison between differentiation and integration\n1 Both are operations on functions"
    },
    {
        "Chapter": "1",
        "sentence_range": "3384-3387",
        "Text": "2 3 Comparison between differentiation and integration\n1 Both are operations on functions 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "3385-3388",
        "Text": "3 Comparison between differentiation and integration\n1 Both are operations on functions 2 Both satisfy the property of linearity, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "3386-3389",
        "Text": "Both are operations on functions 2 Both satisfy the property of linearity, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "3387-3390",
        "Text": "2 Both satisfy the property of linearity, i e ,\n(i)\n[\n]\n1\n1\n2\n2\n1\n1\n2\n2\n( )\n( )\n( )\n( )\nd\nd\nd\nk f\nx\nk\nf\nx\nk\nf\nx\nk\nf\nx\ndx\ndx\ndx\n+\n=\n+\n(ii)\n[\n]\n1\n1\n2\n2\n1\n1\n2\n2\n( )\n( )\n( )\n( )\nk f\nx\nk\nf\nx\ndx\nk\nf\nx dx\nk\nf\nx dx\n+\n=\n+\n\u222b\n\u222b\n\u222b\nHere k1 and k2 are constants"
    },
    {
        "Chapter": "1",
        "sentence_range": "3388-3391",
        "Text": "Both satisfy the property of linearity, i e ,\n(i)\n[\n]\n1\n1\n2\n2\n1\n1\n2\n2\n( )\n( )\n( )\n( )\nd\nd\nd\nk f\nx\nk\nf\nx\nk\nf\nx\nk\nf\nx\ndx\ndx\ndx\n+\n=\n+\n(ii)\n[\n]\n1\n1\n2\n2\n1\n1\n2\n2\n( )\n( )\n( )\n( )\nk f\nx\nk\nf\nx\ndx\nk\nf\nx dx\nk\nf\nx dx\n+\n=\n+\n\u222b\n\u222b\n\u222b\nHere k1 and k2 are constants 3"
    },
    {
        "Chapter": "1",
        "sentence_range": "3389-3392",
        "Text": "e ,\n(i)\n[\n]\n1\n1\n2\n2\n1\n1\n2\n2\n( )\n( )\n( )\n( )\nd\nd\nd\nk f\nx\nk\nf\nx\nk\nf\nx\nk\nf\nx\ndx\ndx\ndx\n+\n=\n+\n(ii)\n[\n]\n1\n1\n2\n2\n1\n1\n2\n2\n( )\n( )\n( )\n( )\nk f\nx\nk\nf\nx\ndx\nk\nf\nx dx\nk\nf\nx dx\n+\n=\n+\n\u222b\n\u222b\n\u222b\nHere k1 and k2 are constants 3 We have already seen that all functions are not differentiable"
    },
    {
        "Chapter": "1",
        "sentence_range": "3390-3393",
        "Text": ",\n(i)\n[\n]\n1\n1\n2\n2\n1\n1\n2\n2\n( )\n( )\n( )\n( )\nd\nd\nd\nk f\nx\nk\nf\nx\nk\nf\nx\nk\nf\nx\ndx\ndx\ndx\n+\n=\n+\n(ii)\n[\n]\n1\n1\n2\n2\n1\n1\n2\n2\n( )\n( )\n( )\n( )\nk f\nx\nk\nf\nx\ndx\nk\nf\nx dx\nk\nf\nx dx\n+\n=\n+\n\u222b\n\u222b\n\u222b\nHere k1 and k2 are constants 3 We have already seen that all functions are not differentiable Similarly, all functions\nare not integrable"
    },
    {
        "Chapter": "1",
        "sentence_range": "3391-3394",
        "Text": "3 We have already seen that all functions are not differentiable Similarly, all functions\nare not integrable We will learn more about nondifferentiable functions and\nnonintegrable functions in higher classes"
    },
    {
        "Chapter": "1",
        "sentence_range": "3392-3395",
        "Text": "We have already seen that all functions are not differentiable Similarly, all functions\nare not integrable We will learn more about nondifferentiable functions and\nnonintegrable functions in higher classes 4"
    },
    {
        "Chapter": "1",
        "sentence_range": "3393-3396",
        "Text": "Similarly, all functions\nare not integrable We will learn more about nondifferentiable functions and\nnonintegrable functions in higher classes 4 The derivative of a function, when it exists, is a unique function"
    },
    {
        "Chapter": "1",
        "sentence_range": "3394-3397",
        "Text": "We will learn more about nondifferentiable functions and\nnonintegrable functions in higher classes 4 The derivative of a function, when it exists, is a unique function The integral of\na function is not so"
    },
    {
        "Chapter": "1",
        "sentence_range": "3395-3398",
        "Text": "4 The derivative of a function, when it exists, is a unique function The integral of\na function is not so However, they are unique upto an additive constant, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "3396-3399",
        "Text": "The derivative of a function, when it exists, is a unique function The integral of\na function is not so However, they are unique upto an additive constant, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "3397-3400",
        "Text": "The integral of\na function is not so However, they are unique upto an additive constant, i e , any\ntwo integrals of a function differ by a constant"
    },
    {
        "Chapter": "1",
        "sentence_range": "3398-3401",
        "Text": "However, they are unique upto an additive constant, i e , any\ntwo integrals of a function differ by a constant 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "3399-3402",
        "Text": "e , any\ntwo integrals of a function differ by a constant 5 When a polynomial function P is differentiated, the result is a polynomial whose\ndegree is 1 less than the degree of  P"
    },
    {
        "Chapter": "1",
        "sentence_range": "3400-3403",
        "Text": ", any\ntwo integrals of a function differ by a constant 5 When a polynomial function P is differentiated, the result is a polynomial whose\ndegree is 1 less than the degree of  P When a polynomial function P is integrated,\nthe result is a polynomial whose degree is 1 more than that of P"
    },
    {
        "Chapter": "1",
        "sentence_range": "3401-3404",
        "Text": "5 When a polynomial function P is differentiated, the result is a polynomial whose\ndegree is 1 less than the degree of  P When a polynomial function P is integrated,\nthe result is a polynomial whose degree is 1 more than that of P 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "3402-3405",
        "Text": "When a polynomial function P is differentiated, the result is a polynomial whose\ndegree is 1 less than the degree of  P When a polynomial function P is integrated,\nthe result is a polynomial whose degree is 1 more than that of P 6 We can speak of the derivative at a point"
    },
    {
        "Chapter": "1",
        "sentence_range": "3403-3406",
        "Text": "When a polynomial function P is integrated,\nthe result is a polynomial whose degree is 1 more than that of P 6 We can speak of the derivative at a point We never speak of the integral at a\npoint, we speak of the integral of a function over an interval on which the integral\nis defined as will be seen in Section 7"
    },
    {
        "Chapter": "1",
        "sentence_range": "3404-3407",
        "Text": "6 We can speak of the derivative at a point We never speak of the integral at a\npoint, we speak of the integral of a function over an interval on which the integral\nis defined as will be seen in Section 7 7"
    },
    {
        "Chapter": "1",
        "sentence_range": "3405-3408",
        "Text": "We can speak of the derivative at a point We never speak of the integral at a\npoint, we speak of the integral of a function over an interval on which the integral\nis defined as will be seen in Section 7 7 7"
    },
    {
        "Chapter": "1",
        "sentence_range": "3406-3409",
        "Text": "We never speak of the integral at a\npoint, we speak of the integral of a function over an interval on which the integral\nis defined as will be seen in Section 7 7 7 The derivative of a function has a geometrical meaning, namely, the slope of the\ntangent to the corresponding curve at a point"
    },
    {
        "Chapter": "1",
        "sentence_range": "3407-3410",
        "Text": "7 7 The derivative of a function has a geometrical meaning, namely, the slope of the\ntangent to the corresponding curve at a point Similarly, the indefinite integral of\na function represents geometrically, a family of curves placed parallel to each\nother having parallel tangents at the points of intersection of the curves of the\nfamily with the lines orthogonal (perpendicular) to the axis representing the variable\nof integration"
    },
    {
        "Chapter": "1",
        "sentence_range": "3408-3411",
        "Text": "7 The derivative of a function has a geometrical meaning, namely, the slope of the\ntangent to the corresponding curve at a point Similarly, the indefinite integral of\na function represents geometrically, a family of curves placed parallel to each\nother having parallel tangents at the points of intersection of the curves of the\nfamily with the lines orthogonal (perpendicular) to the axis representing the variable\nof integration 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "3409-3412",
        "Text": "The derivative of a function has a geometrical meaning, namely, the slope of the\ntangent to the corresponding curve at a point Similarly, the indefinite integral of\na function represents geometrically, a family of curves placed parallel to each\nother having parallel tangents at the points of intersection of the curves of the\nfamily with the lines orthogonal (perpendicular) to the axis representing the variable\nof integration 8 The derivative is used for finding some physical quantities like the velocity of a\nmoving particle, when the distance traversed at any time t is known"
    },
    {
        "Chapter": "1",
        "sentence_range": "3410-3413",
        "Text": "Similarly, the indefinite integral of\na function represents geometrically, a family of curves placed parallel to each\nother having parallel tangents at the points of intersection of the curves of the\nfamily with the lines orthogonal (perpendicular) to the axis representing the variable\nof integration 8 The derivative is used for finding some physical quantities like the velocity of a\nmoving particle, when the distance traversed at any time t is known Similarly,\nthe integral is used in calculating the distance traversed when the velocity at time\nt is known"
    },
    {
        "Chapter": "1",
        "sentence_range": "3411-3414",
        "Text": "8 The derivative is used for finding some physical quantities like the velocity of a\nmoving particle, when the distance traversed at any time t is known Similarly,\nthe integral is used in calculating the distance traversed when the velocity at time\nt is known 9"
    },
    {
        "Chapter": "1",
        "sentence_range": "3412-3415",
        "Text": "The derivative is used for finding some physical quantities like the velocity of a\nmoving particle, when the distance traversed at any time t is known Similarly,\nthe integral is used in calculating the distance traversed when the velocity at time\nt is known 9 Differentiation is a process involving limits"
    },
    {
        "Chapter": "1",
        "sentence_range": "3413-3416",
        "Text": "Similarly,\nthe integral is used in calculating the distance traversed when the velocity at time\nt is known 9 Differentiation is a process involving limits So is integration, as will be seen in\nSection 7"
    },
    {
        "Chapter": "1",
        "sentence_range": "3414-3417",
        "Text": "9 Differentiation is a process involving limits So is integration, as will be seen in\nSection 7 7"
    },
    {
        "Chapter": "1",
        "sentence_range": "3415-3418",
        "Text": "Differentiation is a process involving limits So is integration, as will be seen in\nSection 7 7 INTEGRALS         299\n10"
    },
    {
        "Chapter": "1",
        "sentence_range": "3416-3419",
        "Text": "So is integration, as will be seen in\nSection 7 7 INTEGRALS         299\n10 The process of differentiation and integration are inverses of each other as\ndiscussed in Section 7"
    },
    {
        "Chapter": "1",
        "sentence_range": "3417-3420",
        "Text": "7 INTEGRALS         299\n10 The process of differentiation and integration are inverses of each other as\ndiscussed in Section 7 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "3418-3421",
        "Text": "INTEGRALS         299\n10 The process of differentiation and integration are inverses of each other as\ndiscussed in Section 7 2 2 (i)"
    },
    {
        "Chapter": "1",
        "sentence_range": "3419-3422",
        "Text": "The process of differentiation and integration are inverses of each other as\ndiscussed in Section 7 2 2 (i) EXERCISE 7"
    },
    {
        "Chapter": "1",
        "sentence_range": "3420-3423",
        "Text": "2 2 (i) EXERCISE 7 1\nFind an anti derivative (or integral) of the following functions by the method of inspection"
    },
    {
        "Chapter": "1",
        "sentence_range": "3421-3424",
        "Text": "2 (i) EXERCISE 7 1\nFind an anti derivative (or integral) of the following functions by the method of inspection 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "3422-3425",
        "Text": "EXERCISE 7 1\nFind an anti derivative (or integral) of the following functions by the method of inspection 1 sin 2x\n2"
    },
    {
        "Chapter": "1",
        "sentence_range": "3423-3426",
        "Text": "1\nFind an anti derivative (or integral) of the following functions by the method of inspection 1 sin 2x\n2 cos 3x\n3"
    },
    {
        "Chapter": "1",
        "sentence_range": "3424-3427",
        "Text": "1 sin 2x\n2 cos 3x\n3 e 2x\n4"
    },
    {
        "Chapter": "1",
        "sentence_range": "3425-3428",
        "Text": "sin 2x\n2 cos 3x\n3 e 2x\n4 (ax + b)2\n5"
    },
    {
        "Chapter": "1",
        "sentence_range": "3426-3429",
        "Text": "cos 3x\n3 e 2x\n4 (ax + b)2\n5 sin 2x \u2013 4 e3x\nFind the following integrals in Exercises 6 to 20:\n6"
    },
    {
        "Chapter": "1",
        "sentence_range": "3427-3430",
        "Text": "e 2x\n4 (ax + b)2\n5 sin 2x \u2013 4 e3x\nFind the following integrals in Exercises 6 to 20:\n6 (4 3\nex+ 1) \ndx\n\u222b\n7"
    },
    {
        "Chapter": "1",
        "sentence_range": "3428-3431",
        "Text": "(ax + b)2\n5 sin 2x \u2013 4 e3x\nFind the following integrals in Exercises 6 to 20:\n6 (4 3\nex+ 1) \ndx\n\u222b\n7 2\n(1\u201312\n)\nx\ndx\nx\n\u222b\n8"
    },
    {
        "Chapter": "1",
        "sentence_range": "3429-3432",
        "Text": "sin 2x \u2013 4 e3x\nFind the following integrals in Exercises 6 to 20:\n6 (4 3\nex+ 1) \ndx\n\u222b\n7 2\n(1\u201312\n)\nx\ndx\nx\n\u222b\n8 2\n(\n)\nax\nbx\nc dx\n+\n+\n\u222b\n9"
    },
    {
        "Chapter": "1",
        "sentence_range": "3430-3433",
        "Text": "(4 3\nex+ 1) \ndx\n\u222b\n7 2\n(1\u201312\n)\nx\ndx\nx\n\u222b\n8 2\n(\n)\nax\nbx\nc dx\n+\n+\n\u222b\n9 (22\nx)\nx\ne\ndx\n+\n\u222b\n10"
    },
    {
        "Chapter": "1",
        "sentence_range": "3431-3434",
        "Text": "2\n(1\u201312\n)\nx\ndx\nx\n\u222b\n8 2\n(\n)\nax\nbx\nc dx\n+\n+\n\u222b\n9 (22\nx)\nx\ne\ndx\n+\n\u222b\n10 2\nx \u20131\ndx\nx\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\u222b\n11"
    },
    {
        "Chapter": "1",
        "sentence_range": "3432-3435",
        "Text": "2\n(\n)\nax\nbx\nc dx\n+\n+\n\u222b\n9 (22\nx)\nx\ne\ndx\n+\n\u222b\n10 2\nx \u20131\ndx\nx\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\u222b\n11 3\n2\n2\n5\n4\nx\nx \u2013\ndx\n+x\n\u222b\n12"
    },
    {
        "Chapter": "1",
        "sentence_range": "3433-3436",
        "Text": "(22\nx)\nx\ne\ndx\n+\n\u222b\n10 2\nx \u20131\ndx\nx\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\u222b\n11 3\n2\n2\n5\n4\nx\nx \u2013\ndx\n+x\n\u222b\n12 3\n3\n4\nx\nx\ndx\nx\n+\n+\n\u222b\n13"
    },
    {
        "Chapter": "1",
        "sentence_range": "3434-3437",
        "Text": "2\nx \u20131\ndx\nx\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\u222b\n11 3\n2\n2\n5\n4\nx\nx \u2013\ndx\n+x\n\u222b\n12 3\n3\n4\nx\nx\ndx\nx\n+\n+\n\u222b\n13 3\n2\n1\n1\nx\nx\nx \u2013\ndx\n\u2212x \u2013\n+\n\u222b\n14"
    },
    {
        "Chapter": "1",
        "sentence_range": "3435-3438",
        "Text": "3\n2\n2\n5\n4\nx\nx \u2013\ndx\n+x\n\u222b\n12 3\n3\n4\nx\nx\ndx\nx\n+\n+\n\u222b\n13 3\n2\n1\n1\nx\nx\nx \u2013\ndx\n\u2212x \u2013\n+\n\u222b\n14 (1\n\u2013 x)\nx dx\n\u222b\n15"
    },
    {
        "Chapter": "1",
        "sentence_range": "3436-3439",
        "Text": "3\n3\n4\nx\nx\ndx\nx\n+\n+\n\u222b\n13 3\n2\n1\n1\nx\nx\nx \u2013\ndx\n\u2212x \u2013\n+\n\u222b\n14 (1\n\u2013 x)\nx dx\n\u222b\n15 ( 32\n2\n3)\nx\nx\nx\ndx\n+\n+\n\u222b\n16"
    },
    {
        "Chapter": "1",
        "sentence_range": "3437-3440",
        "Text": "3\n2\n1\n1\nx\nx\nx \u2013\ndx\n\u2212x \u2013\n+\n\u222b\n14 (1\n\u2013 x)\nx dx\n\u222b\n15 ( 32\n2\n3)\nx\nx\nx\ndx\n+\n+\n\u222b\n16 (2\n3cos\nx)\nx \u2013\nx\ne\ndx\n+\n\u222b\n17"
    },
    {
        "Chapter": "1",
        "sentence_range": "3438-3441",
        "Text": "(1\n\u2013 x)\nx dx\n\u222b\n15 ( 32\n2\n3)\nx\nx\nx\ndx\n+\n+\n\u222b\n16 (2\n3cos\nx)\nx \u2013\nx\ne\ndx\n+\n\u222b\n17 (22\n3sin\n5\n)\nx \u2013\nx\nx dx\n+\n\u222b\n18"
    },
    {
        "Chapter": "1",
        "sentence_range": "3439-3442",
        "Text": "( 32\n2\n3)\nx\nx\nx\ndx\n+\n+\n\u222b\n16 (2\n3cos\nx)\nx \u2013\nx\ne\ndx\n+\n\u222b\n17 (22\n3sin\n5\n)\nx \u2013\nx\nx dx\n+\n\u222b\n18 sec\n(sec\ntan )\nx\nx\nx dx\n+\n\u222b\n19"
    },
    {
        "Chapter": "1",
        "sentence_range": "3440-3443",
        "Text": "(2\n3cos\nx)\nx \u2013\nx\ne\ndx\n+\n\u222b\n17 (22\n3sin\n5\n)\nx \u2013\nx\nx dx\n+\n\u222b\n18 sec\n(sec\ntan )\nx\nx\nx dx\n+\n\u222b\n19 2\nsec2\ncosec\nx\ndx\nx\n\u222b\n20"
    },
    {
        "Chapter": "1",
        "sentence_range": "3441-3444",
        "Text": "(22\n3sin\n5\n)\nx \u2013\nx\nx dx\n+\n\u222b\n18 sec\n(sec\ntan )\nx\nx\nx dx\n+\n\u222b\n19 2\nsec2\ncosec\nx\ndx\nx\n\u222b\n20 2\n2 \u2013 3sin\ncos\nx\nx\n\u222b\ndx"
    },
    {
        "Chapter": "1",
        "sentence_range": "3442-3445",
        "Text": "sec\n(sec\ntan )\nx\nx\nx dx\n+\n\u222b\n19 2\nsec2\ncosec\nx\ndx\nx\n\u222b\n20 2\n2 \u2013 3sin\ncos\nx\nx\n\u222b\ndx Choose the correct answer in Exercises 21 and 22"
    },
    {
        "Chapter": "1",
        "sentence_range": "3443-3446",
        "Text": "2\nsec2\ncosec\nx\ndx\nx\n\u222b\n20 2\n2 \u2013 3sin\ncos\nx\nx\n\u222b\ndx Choose the correct answer in Exercises 21 and 22 21"
    },
    {
        "Chapter": "1",
        "sentence_range": "3444-3447",
        "Text": "2\n2 \u2013 3sin\ncos\nx\nx\n\u222b\ndx Choose the correct answer in Exercises 21 and 22 21 The anti derivative of \n1\nx\nx\n\uf8eb\n\uf8f6\n+\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n equals\n(A)\n1\n1\n3\n2\n1\n2\nC\n3\nx\nx\n+\n+\n(B)\n2\n2\n23\n1\nC\n3\n2\nx\nx\n+\n+\n(C)\n3\n1\n2\n2\n2\n2\nC\n3\nx\nx\n+\n+\n(D)\n3\n1\n2\n2\n3\n1\nC\n2\n2\nx\nx\n+\n+\n22"
    },
    {
        "Chapter": "1",
        "sentence_range": "3445-3448",
        "Text": "Choose the correct answer in Exercises 21 and 22 21 The anti derivative of \n1\nx\nx\n\uf8eb\n\uf8f6\n+\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n equals\n(A)\n1\n1\n3\n2\n1\n2\nC\n3\nx\nx\n+\n+\n(B)\n2\n2\n23\n1\nC\n3\n2\nx\nx\n+\n+\n(C)\n3\n1\n2\n2\n2\n2\nC\n3\nx\nx\n+\n+\n(D)\n3\n1\n2\n2\n3\n1\nC\n2\n2\nx\nx\n+\n+\n22 If \n3\n34\n( )\n4\nd f x\nx\ndx\nx\n=\n\u2212\n such that f (2) = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "3446-3449",
        "Text": "21 The anti derivative of \n1\nx\nx\n\uf8eb\n\uf8f6\n+\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n equals\n(A)\n1\n1\n3\n2\n1\n2\nC\n3\nx\nx\n+\n+\n(B)\n2\n2\n23\n1\nC\n3\n2\nx\nx\n+\n+\n(C)\n3\n1\n2\n2\n2\n2\nC\n3\nx\nx\n+\n+\n(D)\n3\n1\n2\n2\n3\n1\nC\n2\n2\nx\nx\n+\n+\n22 If \n3\n34\n( )\n4\nd f x\nx\ndx\nx\n=\n\u2212\n such that f (2) = 0 Then f (x) is\n(A)\n4\n13\n129\n8\nx\n+x\n\u2212\n(B)\n3\n14\n129\n8\nx\n+x\n+\n(C)\n4\n13\n129\n8\nx\n+x\n+\n(D)\n3\n14\n129\n8\nx\n+x\n\u2212\n300\nMATHEMATICS\n7"
    },
    {
        "Chapter": "1",
        "sentence_range": "3447-3450",
        "Text": "The anti derivative of \n1\nx\nx\n\uf8eb\n\uf8f6\n+\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n equals\n(A)\n1\n1\n3\n2\n1\n2\nC\n3\nx\nx\n+\n+\n(B)\n2\n2\n23\n1\nC\n3\n2\nx\nx\n+\n+\n(C)\n3\n1\n2\n2\n2\n2\nC\n3\nx\nx\n+\n+\n(D)\n3\n1\n2\n2\n3\n1\nC\n2\n2\nx\nx\n+\n+\n22 If \n3\n34\n( )\n4\nd f x\nx\ndx\nx\n=\n\u2212\n such that f (2) = 0 Then f (x) is\n(A)\n4\n13\n129\n8\nx\n+x\n\u2212\n(B)\n3\n14\n129\n8\nx\n+x\n+\n(C)\n4\n13\n129\n8\nx\n+x\n+\n(D)\n3\n14\n129\n8\nx\n+x\n\u2212\n300\nMATHEMATICS\n7 3  Methods of Integration\nIn previous section, we discussed integrals of those functions which were readily\nobtainable from derivatives of some functions"
    },
    {
        "Chapter": "1",
        "sentence_range": "3448-3451",
        "Text": "If \n3\n34\n( )\n4\nd f x\nx\ndx\nx\n=\n\u2212\n such that f (2) = 0 Then f (x) is\n(A)\n4\n13\n129\n8\nx\n+x\n\u2212\n(B)\n3\n14\n129\n8\nx\n+x\n+\n(C)\n4\n13\n129\n8\nx\n+x\n+\n(D)\n3\n14\n129\n8\nx\n+x\n\u2212\n300\nMATHEMATICS\n7 3  Methods of Integration\nIn previous section, we discussed integrals of those functions which were readily\nobtainable from derivatives of some functions It was based on inspection, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "3449-3452",
        "Text": "Then f (x) is\n(A)\n4\n13\n129\n8\nx\n+x\n\u2212\n(B)\n3\n14\n129\n8\nx\n+x\n+\n(C)\n4\n13\n129\n8\nx\n+x\n+\n(D)\n3\n14\n129\n8\nx\n+x\n\u2212\n300\nMATHEMATICS\n7 3  Methods of Integration\nIn previous section, we discussed integrals of those functions which were readily\nobtainable from derivatives of some functions It was based on inspection, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "3450-3453",
        "Text": "3  Methods of Integration\nIn previous section, we discussed integrals of those functions which were readily\nobtainable from derivatives of some functions It was based on inspection, i e , on the\nsearch of a function F whose derivative is f which led us to the integral of f"
    },
    {
        "Chapter": "1",
        "sentence_range": "3451-3454",
        "Text": "It was based on inspection, i e , on the\nsearch of a function F whose derivative is f which led us to the integral of f However,\nthis method, which depends on inspection, is not very suitable for many functions"
    },
    {
        "Chapter": "1",
        "sentence_range": "3452-3455",
        "Text": "e , on the\nsearch of a function F whose derivative is f which led us to the integral of f However,\nthis method, which depends on inspection, is not very suitable for many functions Hence, we need to develop additional techniques or methods for finding the integrals\nby reducing them into standard forms"
    },
    {
        "Chapter": "1",
        "sentence_range": "3453-3456",
        "Text": ", on the\nsearch of a function F whose derivative is f which led us to the integral of f However,\nthis method, which depends on inspection, is not very suitable for many functions Hence, we need to develop additional techniques or methods for finding the integrals\nby reducing them into standard forms Prominent among them are methods based on:\n1"
    },
    {
        "Chapter": "1",
        "sentence_range": "3454-3457",
        "Text": "However,\nthis method, which depends on inspection, is not very suitable for many functions Hence, we need to develop additional techniques or methods for finding the integrals\nby reducing them into standard forms Prominent among them are methods based on:\n1 Integration by Substitution\n2"
    },
    {
        "Chapter": "1",
        "sentence_range": "3455-3458",
        "Text": "Hence, we need to develop additional techniques or methods for finding the integrals\nby reducing them into standard forms Prominent among them are methods based on:\n1 Integration by Substitution\n2 Integration using Partial Fractions\n3"
    },
    {
        "Chapter": "1",
        "sentence_range": "3456-3459",
        "Text": "Prominent among them are methods based on:\n1 Integration by Substitution\n2 Integration using Partial Fractions\n3 Integration by Parts\n7"
    },
    {
        "Chapter": "1",
        "sentence_range": "3457-3460",
        "Text": "Integration by Substitution\n2 Integration using Partial Fractions\n3 Integration by Parts\n7 3"
    },
    {
        "Chapter": "1",
        "sentence_range": "3458-3461",
        "Text": "Integration using Partial Fractions\n3 Integration by Parts\n7 3 1 Integration by substitution\nIn this section, we consider the method of integration by substitution"
    },
    {
        "Chapter": "1",
        "sentence_range": "3459-3462",
        "Text": "Integration by Parts\n7 3 1 Integration by substitution\nIn this section, we consider the method of integration by substitution The given integral \n( )\n\u222bf x dx\n can be transformed into another form by changing\nthe independent variable x to t by substituting x = g (t)"
    },
    {
        "Chapter": "1",
        "sentence_range": "3460-3463",
        "Text": "3 1 Integration by substitution\nIn this section, we consider the method of integration by substitution The given integral \n( )\n\u222bf x dx\n can be transformed into another form by changing\nthe independent variable x to t by substituting x = g (t) Consider\nI =\n( )\nf x dx\n\u222b\nPut x = g(t) so that dx\ndt  = g\u2032(t)"
    },
    {
        "Chapter": "1",
        "sentence_range": "3461-3464",
        "Text": "1 Integration by substitution\nIn this section, we consider the method of integration by substitution The given integral \n( )\n\u222bf x dx\n can be transformed into another form by changing\nthe independent variable x to t by substituting x = g (t) Consider\nI =\n( )\nf x dx\n\u222b\nPut x = g(t) so that dx\ndt  = g\u2032(t) We write\ndx = g\u2032(t) dt\nThus\nI =\n( )\n( ( ))\n( )\nf x dx\nf g t\ng t dt\n=\n\u2032\n\u222b\n\u222b\nThis change of variable formula is one of the important tools available to us in the\nname of integration by substitution"
    },
    {
        "Chapter": "1",
        "sentence_range": "3462-3465",
        "Text": "The given integral \n( )\n\u222bf x dx\n can be transformed into another form by changing\nthe independent variable x to t by substituting x = g (t) Consider\nI =\n( )\nf x dx\n\u222b\nPut x = g(t) so that dx\ndt  = g\u2032(t) We write\ndx = g\u2032(t) dt\nThus\nI =\n( )\n( ( ))\n( )\nf x dx\nf g t\ng t dt\n=\n\u2032\n\u222b\n\u222b\nThis change of variable formula is one of the important tools available to us in the\nname of integration by substitution It is often important to guess what will be the useful\nsubstitution"
    },
    {
        "Chapter": "1",
        "sentence_range": "3463-3466",
        "Text": "Consider\nI =\n( )\nf x dx\n\u222b\nPut x = g(t) so that dx\ndt  = g\u2032(t) We write\ndx = g\u2032(t) dt\nThus\nI =\n( )\n( ( ))\n( )\nf x dx\nf g t\ng t dt\n=\n\u2032\n\u222b\n\u222b\nThis change of variable formula is one of the important tools available to us in the\nname of integration by substitution It is often important to guess what will be the useful\nsubstitution Usually, we make a substitution for a function whose derivative also occurs\nin the integrand as illustrated in the following examples"
    },
    {
        "Chapter": "1",
        "sentence_range": "3464-3467",
        "Text": "We write\ndx = g\u2032(t) dt\nThus\nI =\n( )\n( ( ))\n( )\nf x dx\nf g t\ng t dt\n=\n\u2032\n\u222b\n\u222b\nThis change of variable formula is one of the important tools available to us in the\nname of integration by substitution It is often important to guess what will be the useful\nsubstitution Usually, we make a substitution for a function whose derivative also occurs\nin the integrand as illustrated in the following examples Example 5 Integrate the following functions w"
    },
    {
        "Chapter": "1",
        "sentence_range": "3465-3468",
        "Text": "It is often important to guess what will be the useful\nsubstitution Usually, we make a substitution for a function whose derivative also occurs\nin the integrand as illustrated in the following examples Example 5 Integrate the following functions w r"
    },
    {
        "Chapter": "1",
        "sentence_range": "3466-3469",
        "Text": "Usually, we make a substitution for a function whose derivative also occurs\nin the integrand as illustrated in the following examples Example 5 Integrate the following functions w r t"
    },
    {
        "Chapter": "1",
        "sentence_range": "3467-3470",
        "Text": "Example 5 Integrate the following functions w r t x:\n(i)\nsin mx\n(ii)\n2x sin (x2 + 1)\n(iii)\n4\n2\ntan\nxsec\nx\nx\n(iv)\n1\n2\nsin (tan\n)\n1\nx\u2013 x\n+\nSolution\n(i)\nWe know that derivative of mx is m"
    },
    {
        "Chapter": "1",
        "sentence_range": "3468-3471",
        "Text": "r t x:\n(i)\nsin mx\n(ii)\n2x sin (x2 + 1)\n(iii)\n4\n2\ntan\nxsec\nx\nx\n(iv)\n1\n2\nsin (tan\n)\n1\nx\u2013 x\n+\nSolution\n(i)\nWe know that derivative of mx is m Thus, we make the substitution\nmx = t so that mdx = dt"
    },
    {
        "Chapter": "1",
        "sentence_range": "3469-3472",
        "Text": "t x:\n(i)\nsin mx\n(ii)\n2x sin (x2 + 1)\n(iii)\n4\n2\ntan\nxsec\nx\nx\n(iv)\n1\n2\nsin (tan\n)\n1\nx\u2013 x\n+\nSolution\n(i)\nWe know that derivative of mx is m Thus, we make the substitution\nmx = t so that mdx = dt Therefore,      \n1\nsin\nsin\nmx dx\nt dt\nm\n=\n\u222b\n\u222b\n =  \u2013 1\nm\ncos t + C  = \u2013 1\nm cos mx + C\nINTEGRALS         301\n(ii)\nDerivative of x2 + 1 is 2x"
    },
    {
        "Chapter": "1",
        "sentence_range": "3470-3473",
        "Text": "x:\n(i)\nsin mx\n(ii)\n2x sin (x2 + 1)\n(iii)\n4\n2\ntan\nxsec\nx\nx\n(iv)\n1\n2\nsin (tan\n)\n1\nx\u2013 x\n+\nSolution\n(i)\nWe know that derivative of mx is m Thus, we make the substitution\nmx = t so that mdx = dt Therefore,      \n1\nsin\nsin\nmx dx\nt dt\nm\n=\n\u222b\n\u222b\n =  \u2013 1\nm\ncos t + C  = \u2013 1\nm cos mx + C\nINTEGRALS         301\n(ii)\nDerivative of x2 + 1 is 2x Thus, we use the substitution x2 + 1 = t so that\n2x dx = dt"
    },
    {
        "Chapter": "1",
        "sentence_range": "3471-3474",
        "Text": "Thus, we make the substitution\nmx = t so that mdx = dt Therefore,      \n1\nsin\nsin\nmx dx\nt dt\nm\n=\n\u222b\n\u222b\n =  \u2013 1\nm\ncos t + C  = \u2013 1\nm cos mx + C\nINTEGRALS         301\n(ii)\nDerivative of x2 + 1 is 2x Thus, we use the substitution x2 + 1 = t so that\n2x dx = dt Therefore,  \n2 sin (2\n1)\nsin\nx\nx\ndx\nt dt\n+\n=\n\u222b\n\u222b\n =  \u2013 cos t + C  = \u2013 cos (x2 + 1) + C\n(iii)\nDerivative of \nx  is \n21\n1\n1\n2\n2\n\u2013\nx\nx\n="
    },
    {
        "Chapter": "1",
        "sentence_range": "3472-3475",
        "Text": "Therefore,      \n1\nsin\nsin\nmx dx\nt dt\nm\n=\n\u222b\n\u222b\n =  \u2013 1\nm\ncos t + C  = \u2013 1\nm cos mx + C\nINTEGRALS         301\n(ii)\nDerivative of x2 + 1 is 2x Thus, we use the substitution x2 + 1 = t so that\n2x dx = dt Therefore,  \n2 sin (2\n1)\nsin\nx\nx\ndx\nt dt\n+\n=\n\u222b\n\u222b\n =  \u2013 cos t + C  = \u2013 cos (x2 + 1) + C\n(iii)\nDerivative of \nx  is \n21\n1\n1\n2\n2\n\u2013\nx\nx\n= Thus, we use the substitution\n1\nso that\ngiving\n2\nx\nt\ndx\ndt\nx\n=\n=\n dx = 2t dt"
    },
    {
        "Chapter": "1",
        "sentence_range": "3473-3476",
        "Text": "Thus, we use the substitution x2 + 1 = t so that\n2x dx = dt Therefore,  \n2 sin (2\n1)\nsin\nx\nx\ndx\nt dt\n+\n=\n\u222b\n\u222b\n =  \u2013 cos t + C  = \u2013 cos (x2 + 1) + C\n(iii)\nDerivative of \nx  is \n21\n1\n1\n2\n2\n\u2013\nx\nx\n= Thus, we use the substitution\n1\nso that\ngiving\n2\nx\nt\ndx\ndt\nx\n=\n=\n dx = 2t dt Thus,\n4\n2\n4\n2\ntan\nsec\n2 tan\nsec\nx\nx\nt\nt\nt dt\ndx\nt\nx\n=\n\u222b\n\u222b\n = \n4\n2\n2 tan\ntsec\nt dt\n\u222b\nAgain, we make another substitution tan t = u so that\nsec2 t dt = du\nTherefore,\n4\n2\n4\n2 tan\nsec\n2\nt\nt dt\nu du\n=\n\u222b\n\u222b\n = \n5\n2\nC\n5\nu +\n=\n2 tan5\nC\n5\nt +\n (since u = tan t)\n=\n2 tan5\nC (since\n)\n5\nx\nt\nx\n+\n=\nHence,\n4\n2\ntan\nxsec\nx dx\nx\n\u222b\n =\n2 tan5\nC\n5\nx +\nAlternatively, make the substitution tan\nx\nt\n=\n(iv)\nDerivative of  \n1\n2\n1\ntan\n1\n\u2013 x\nx\n= +"
    },
    {
        "Chapter": "1",
        "sentence_range": "3474-3477",
        "Text": "Therefore,  \n2 sin (2\n1)\nsin\nx\nx\ndx\nt dt\n+\n=\n\u222b\n\u222b\n =  \u2013 cos t + C  = \u2013 cos (x2 + 1) + C\n(iii)\nDerivative of \nx  is \n21\n1\n1\n2\n2\n\u2013\nx\nx\n= Thus, we use the substitution\n1\nso that\ngiving\n2\nx\nt\ndx\ndt\nx\n=\n=\n dx = 2t dt Thus,\n4\n2\n4\n2\ntan\nsec\n2 tan\nsec\nx\nx\nt\nt\nt dt\ndx\nt\nx\n=\n\u222b\n\u222b\n = \n4\n2\n2 tan\ntsec\nt dt\n\u222b\nAgain, we make another substitution tan t = u so that\nsec2 t dt = du\nTherefore,\n4\n2\n4\n2 tan\nsec\n2\nt\nt dt\nu du\n=\n\u222b\n\u222b\n = \n5\n2\nC\n5\nu +\n=\n2 tan5\nC\n5\nt +\n (since u = tan t)\n=\n2 tan5\nC (since\n)\n5\nx\nt\nx\n+\n=\nHence,\n4\n2\ntan\nxsec\nx dx\nx\n\u222b\n =\n2 tan5\nC\n5\nx +\nAlternatively, make the substitution tan\nx\nt\n=\n(iv)\nDerivative of  \n1\n2\n1\ntan\n1\n\u2013 x\nx\n= + Thus, we use the substitution\ntan\u20131 x = t so that \n2\n1\ndx\nx\n+\n = dt"
    },
    {
        "Chapter": "1",
        "sentence_range": "3475-3478",
        "Text": "Thus, we use the substitution\n1\nso that\ngiving\n2\nx\nt\ndx\ndt\nx\n=\n=\n dx = 2t dt Thus,\n4\n2\n4\n2\ntan\nsec\n2 tan\nsec\nx\nx\nt\nt\nt dt\ndx\nt\nx\n=\n\u222b\n\u222b\n = \n4\n2\n2 tan\ntsec\nt dt\n\u222b\nAgain, we make another substitution tan t = u so that\nsec2 t dt = du\nTherefore,\n4\n2\n4\n2 tan\nsec\n2\nt\nt dt\nu du\n=\n\u222b\n\u222b\n = \n5\n2\nC\n5\nu +\n=\n2 tan5\nC\n5\nt +\n (since u = tan t)\n=\n2 tan5\nC (since\n)\n5\nx\nt\nx\n+\n=\nHence,\n4\n2\ntan\nxsec\nx dx\nx\n\u222b\n =\n2 tan5\nC\n5\nx +\nAlternatively, make the substitution tan\nx\nt\n=\n(iv)\nDerivative of  \n1\n2\n1\ntan\n1\n\u2013 x\nx\n= + Thus, we use the substitution\ntan\u20131 x = t so that \n2\n1\ndx\nx\n+\n = dt Therefore ,  \n1\nsin (tan2\n)\nsin\n1\n\u2013 x dx\nt dt\nx\n=\n+\n\u222b\n\u222b\n =  \u2013 cos t + C = \u2013 cos (tan \u20131x) + C\nNow, we discuss some important integrals involving trigonometric functions and\ntheir standard integrals using substitution technique"
    },
    {
        "Chapter": "1",
        "sentence_range": "3476-3479",
        "Text": "Thus,\n4\n2\n4\n2\ntan\nsec\n2 tan\nsec\nx\nx\nt\nt\nt dt\ndx\nt\nx\n=\n\u222b\n\u222b\n = \n4\n2\n2 tan\ntsec\nt dt\n\u222b\nAgain, we make another substitution tan t = u so that\nsec2 t dt = du\nTherefore,\n4\n2\n4\n2 tan\nsec\n2\nt\nt dt\nu du\n=\n\u222b\n\u222b\n = \n5\n2\nC\n5\nu +\n=\n2 tan5\nC\n5\nt +\n (since u = tan t)\n=\n2 tan5\nC (since\n)\n5\nx\nt\nx\n+\n=\nHence,\n4\n2\ntan\nxsec\nx dx\nx\n\u222b\n =\n2 tan5\nC\n5\nx +\nAlternatively, make the substitution tan\nx\nt\n=\n(iv)\nDerivative of  \n1\n2\n1\ntan\n1\n\u2013 x\nx\n= + Thus, we use the substitution\ntan\u20131 x = t so that \n2\n1\ndx\nx\n+\n = dt Therefore ,  \n1\nsin (tan2\n)\nsin\n1\n\u2013 x dx\nt dt\nx\n=\n+\n\u222b\n\u222b\n =  \u2013 cos t + C = \u2013 cos (tan \u20131x) + C\nNow, we discuss some important integrals involving trigonometric functions and\ntheir standard integrals using substitution technique These will be used later without\nreference"
    },
    {
        "Chapter": "1",
        "sentence_range": "3477-3480",
        "Text": "Thus, we use the substitution\ntan\u20131 x = t so that \n2\n1\ndx\nx\n+\n = dt Therefore ,  \n1\nsin (tan2\n)\nsin\n1\n\u2013 x dx\nt dt\nx\n=\n+\n\u222b\n\u222b\n =  \u2013 cos t + C = \u2013 cos (tan \u20131x) + C\nNow, we discuss some important integrals involving trigonometric functions and\ntheir standard integrals using substitution technique These will be used later without\nreference (i) \u222btan\n= log sec\n+ C\nx dx\nx\nWe have\nsin\ntan\ncos\nx\nx dx\ndx\nx\n=\n\u222b\n\u222b\n302\nMATHEMATICS\nPut  cos x = t so that sin x dx = \u2013 dt\nThen\ntan\nlog\nC\nlog cos\nC\ndt\nx dx\n\u2013\n\u2013\nt\n\u2013\nx\nt\n=\n=\n+\n=\n+\n\u222b\n\u222b\nor\ntan\nlog sec\nC\nx dx\nx\n=\n+\n\u222b\n(ii) \u222bcot\n= log sin\n+ C\nx dx\nx\nWe have\ncos\ncot\nsin\nx\nx dx\ndx\nx\n=\n\u222b\n\u222b\nPut  sin x = t so that cos x dx = dt\nThen\ncot\ndt\nx dx\nt\n=\n\u222b\n\u222b\n = log\nt +C\n = log sin\nC\nx +\n(iii) \u222bsec\n= log sec\n+ tan\n+ C\nx dx\nx\nx\nWe have\nsec\n(sec\ntan )\nsec\nsec\n+ tan\nx\nx\nx\nx dx\ndx\nx\n+x\n=\n\u222b\n\u222b\nPut sec x + tan x = t so that sec x (tan x + sec x) dx = dt\nTherefore, sec\nlog\n+ C = log sec\ntan\nC\ndt\nx dx\nt\nx\nx\nt\n=\n=\n+\n+\n\u222b\n\u222b\n(iv) \u222bcosec\n= log cosec\n\u2013 cot\n+ C\nx dx\nx\nx\nWe have\ncosec\n(cosec\ncot )\ncosec\n(cosec\ncot )\nx\nx\nx\nx dx\ndx\nx\n+x\n=\n+\n\u222b\n\u222b\nPut cosec x + cot x = t so that \u2013 cosec x (cosec x + cot x) dx = dt\nSo\ncosec\n\u2013\n\u2013log| |\n\u2013 log|cosec\ncot\n|\nC\ndt\nx dx\nt\nx\nx\nt\n=\n=\n=\n+\n+\n\u222b\n\u222b\n=\n2\n2\ncosec\ncot\n\u2013 log\nC\ncosec\ncot\nx\nx\nx\nx\n\u2212\n+\n\u2212\n= log cosec\ncot\nC\nx \u2013\nx +\nExample 6 Find the following integrals:\n(i)\n3\n2\nsin\nxcos\nx dx\n\u222b\n(ii)    \nsin\nsin (\n)\nx\ndx\nx\n+a\n\u222b\n    (iii)  \n1\n1\ntan\ndx\nx\n\u222b+\nINTEGRALS         303\nSolution\n(i)\nWe have\n3\n2\n2\n2\nsin\ncos\nsin\ncos\n(sin )\nx\nx dx\nx\nx\nx dx\n=\n\u222b\n\u222b\n= \n2\n2\n(1\u2013 cos\n) cos\n(sin )\nx\nx\nx dx\n\u222b\nPut t = cos x so that dt = \u2013 sin x dx\nTherefore,    \n2\n2\nsin\ncos\n(sin )\nx\nx\nx dx\n\u222b\n = \n2\n2\n(1\u2013\nt)\nt dt\n\u2212\u222b\n= \n3\n5\n2\n4\n(\n\u2013\n)\nC\n3\n5\nt\nt\n\u2013\nt\nt\ndt\n\u2013\n\u2013\n\uf8eb\n\uf8f6\n=\n+\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\u222b\n= \n3\n5\n1\n1\ncos\ncos\nC\n3\n5\n\u2013\nx\nx\n+\n+\n(ii)\nPut x + a = t"
    },
    {
        "Chapter": "1",
        "sentence_range": "3478-3481",
        "Text": "Therefore ,  \n1\nsin (tan2\n)\nsin\n1\n\u2013 x dx\nt dt\nx\n=\n+\n\u222b\n\u222b\n =  \u2013 cos t + C = \u2013 cos (tan \u20131x) + C\nNow, we discuss some important integrals involving trigonometric functions and\ntheir standard integrals using substitution technique These will be used later without\nreference (i) \u222btan\n= log sec\n+ C\nx dx\nx\nWe have\nsin\ntan\ncos\nx\nx dx\ndx\nx\n=\n\u222b\n\u222b\n302\nMATHEMATICS\nPut  cos x = t so that sin x dx = \u2013 dt\nThen\ntan\nlog\nC\nlog cos\nC\ndt\nx dx\n\u2013\n\u2013\nt\n\u2013\nx\nt\n=\n=\n+\n=\n+\n\u222b\n\u222b\nor\ntan\nlog sec\nC\nx dx\nx\n=\n+\n\u222b\n(ii) \u222bcot\n= log sin\n+ C\nx dx\nx\nWe have\ncos\ncot\nsin\nx\nx dx\ndx\nx\n=\n\u222b\n\u222b\nPut  sin x = t so that cos x dx = dt\nThen\ncot\ndt\nx dx\nt\n=\n\u222b\n\u222b\n = log\nt +C\n = log sin\nC\nx +\n(iii) \u222bsec\n= log sec\n+ tan\n+ C\nx dx\nx\nx\nWe have\nsec\n(sec\ntan )\nsec\nsec\n+ tan\nx\nx\nx\nx dx\ndx\nx\n+x\n=\n\u222b\n\u222b\nPut sec x + tan x = t so that sec x (tan x + sec x) dx = dt\nTherefore, sec\nlog\n+ C = log sec\ntan\nC\ndt\nx dx\nt\nx\nx\nt\n=\n=\n+\n+\n\u222b\n\u222b\n(iv) \u222bcosec\n= log cosec\n\u2013 cot\n+ C\nx dx\nx\nx\nWe have\ncosec\n(cosec\ncot )\ncosec\n(cosec\ncot )\nx\nx\nx\nx dx\ndx\nx\n+x\n=\n+\n\u222b\n\u222b\nPut cosec x + cot x = t so that \u2013 cosec x (cosec x + cot x) dx = dt\nSo\ncosec\n\u2013\n\u2013log| |\n\u2013 log|cosec\ncot\n|\nC\ndt\nx dx\nt\nx\nx\nt\n=\n=\n=\n+\n+\n\u222b\n\u222b\n=\n2\n2\ncosec\ncot\n\u2013 log\nC\ncosec\ncot\nx\nx\nx\nx\n\u2212\n+\n\u2212\n= log cosec\ncot\nC\nx \u2013\nx +\nExample 6 Find the following integrals:\n(i)\n3\n2\nsin\nxcos\nx dx\n\u222b\n(ii)    \nsin\nsin (\n)\nx\ndx\nx\n+a\n\u222b\n    (iii)  \n1\n1\ntan\ndx\nx\n\u222b+\nINTEGRALS         303\nSolution\n(i)\nWe have\n3\n2\n2\n2\nsin\ncos\nsin\ncos\n(sin )\nx\nx dx\nx\nx\nx dx\n=\n\u222b\n\u222b\n= \n2\n2\n(1\u2013 cos\n) cos\n(sin )\nx\nx\nx dx\n\u222b\nPut t = cos x so that dt = \u2013 sin x dx\nTherefore,    \n2\n2\nsin\ncos\n(sin )\nx\nx\nx dx\n\u222b\n = \n2\n2\n(1\u2013\nt)\nt dt\n\u2212\u222b\n= \n3\n5\n2\n4\n(\n\u2013\n)\nC\n3\n5\nt\nt\n\u2013\nt\nt\ndt\n\u2013\n\u2013\n\uf8eb\n\uf8f6\n=\n+\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\u222b\n= \n3\n5\n1\n1\ncos\ncos\nC\n3\n5\n\u2013\nx\nx\n+\n+\n(ii)\nPut x + a = t Then dx = dt"
    },
    {
        "Chapter": "1",
        "sentence_range": "3479-3482",
        "Text": "These will be used later without\nreference (i) \u222btan\n= log sec\n+ C\nx dx\nx\nWe have\nsin\ntan\ncos\nx\nx dx\ndx\nx\n=\n\u222b\n\u222b\n302\nMATHEMATICS\nPut  cos x = t so that sin x dx = \u2013 dt\nThen\ntan\nlog\nC\nlog cos\nC\ndt\nx dx\n\u2013\n\u2013\nt\n\u2013\nx\nt\n=\n=\n+\n=\n+\n\u222b\n\u222b\nor\ntan\nlog sec\nC\nx dx\nx\n=\n+\n\u222b\n(ii) \u222bcot\n= log sin\n+ C\nx dx\nx\nWe have\ncos\ncot\nsin\nx\nx dx\ndx\nx\n=\n\u222b\n\u222b\nPut  sin x = t so that cos x dx = dt\nThen\ncot\ndt\nx dx\nt\n=\n\u222b\n\u222b\n = log\nt +C\n = log sin\nC\nx +\n(iii) \u222bsec\n= log sec\n+ tan\n+ C\nx dx\nx\nx\nWe have\nsec\n(sec\ntan )\nsec\nsec\n+ tan\nx\nx\nx\nx dx\ndx\nx\n+x\n=\n\u222b\n\u222b\nPut sec x + tan x = t so that sec x (tan x + sec x) dx = dt\nTherefore, sec\nlog\n+ C = log sec\ntan\nC\ndt\nx dx\nt\nx\nx\nt\n=\n=\n+\n+\n\u222b\n\u222b\n(iv) \u222bcosec\n= log cosec\n\u2013 cot\n+ C\nx dx\nx\nx\nWe have\ncosec\n(cosec\ncot )\ncosec\n(cosec\ncot )\nx\nx\nx\nx dx\ndx\nx\n+x\n=\n+\n\u222b\n\u222b\nPut cosec x + cot x = t so that \u2013 cosec x (cosec x + cot x) dx = dt\nSo\ncosec\n\u2013\n\u2013log| |\n\u2013 log|cosec\ncot\n|\nC\ndt\nx dx\nt\nx\nx\nt\n=\n=\n=\n+\n+\n\u222b\n\u222b\n=\n2\n2\ncosec\ncot\n\u2013 log\nC\ncosec\ncot\nx\nx\nx\nx\n\u2212\n+\n\u2212\n= log cosec\ncot\nC\nx \u2013\nx +\nExample 6 Find the following integrals:\n(i)\n3\n2\nsin\nxcos\nx dx\n\u222b\n(ii)    \nsin\nsin (\n)\nx\ndx\nx\n+a\n\u222b\n    (iii)  \n1\n1\ntan\ndx\nx\n\u222b+\nINTEGRALS         303\nSolution\n(i)\nWe have\n3\n2\n2\n2\nsin\ncos\nsin\ncos\n(sin )\nx\nx dx\nx\nx\nx dx\n=\n\u222b\n\u222b\n= \n2\n2\n(1\u2013 cos\n) cos\n(sin )\nx\nx\nx dx\n\u222b\nPut t = cos x so that dt = \u2013 sin x dx\nTherefore,    \n2\n2\nsin\ncos\n(sin )\nx\nx\nx dx\n\u222b\n = \n2\n2\n(1\u2013\nt)\nt dt\n\u2212\u222b\n= \n3\n5\n2\n4\n(\n\u2013\n)\nC\n3\n5\nt\nt\n\u2013\nt\nt\ndt\n\u2013\n\u2013\n\uf8eb\n\uf8f6\n=\n+\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\u222b\n= \n3\n5\n1\n1\ncos\ncos\nC\n3\n5\n\u2013\nx\nx\n+\n+\n(ii)\nPut x + a = t Then dx = dt Therefore\nsin\nsin (\n)\nsin (\n)\nsin\nx\nt \u2013 a\ndx\ndt\nx\na\nt\n=\n+\n\u222b\n\u222b\n= \nsin cos\ncos sin\nsin\nt\na \u2013\nt\na dt\nt\n\u222b\n= cos\n\u2013 sin\ncot\na dt\na\nt dt\n\u222b\n\u222b\n= \n1\n(cos )\n(sin ) log sin\nC\na t \u2013\na\nt\n\uf8ee\n\uf8f9\n+\n\uf8f0\n\uf8fb\n= \n1\n(cos ) (\n)\n(sin ) log sin (\n)\nC\na\nx\na \u2013\na\nx\na\n\uf8ee\n\uf8f9\n+\n+\n+\n\uf8f0\n\uf8fb\n= \n1\ncos\ncos\n(sin ) log sin (\n)\nC sin\nx\na\na\na \u2013\na\nx\na \u2013\na\n+\n+\nHence, \nsin\nsin (\n)\nx\ndx\nx\n+a\n\u222b\n = x cos a \u2013 sin a log |sin (x + a)| + C,\nwhere,  C = \u2013 C1 sin a + a cos a, is another arbitrary constant"
    },
    {
        "Chapter": "1",
        "sentence_range": "3480-3483",
        "Text": "(i) \u222btan\n= log sec\n+ C\nx dx\nx\nWe have\nsin\ntan\ncos\nx\nx dx\ndx\nx\n=\n\u222b\n\u222b\n302\nMATHEMATICS\nPut  cos x = t so that sin x dx = \u2013 dt\nThen\ntan\nlog\nC\nlog cos\nC\ndt\nx dx\n\u2013\n\u2013\nt\n\u2013\nx\nt\n=\n=\n+\n=\n+\n\u222b\n\u222b\nor\ntan\nlog sec\nC\nx dx\nx\n=\n+\n\u222b\n(ii) \u222bcot\n= log sin\n+ C\nx dx\nx\nWe have\ncos\ncot\nsin\nx\nx dx\ndx\nx\n=\n\u222b\n\u222b\nPut  sin x = t so that cos x dx = dt\nThen\ncot\ndt\nx dx\nt\n=\n\u222b\n\u222b\n = log\nt +C\n = log sin\nC\nx +\n(iii) \u222bsec\n= log sec\n+ tan\n+ C\nx dx\nx\nx\nWe have\nsec\n(sec\ntan )\nsec\nsec\n+ tan\nx\nx\nx\nx dx\ndx\nx\n+x\n=\n\u222b\n\u222b\nPut sec x + tan x = t so that sec x (tan x + sec x) dx = dt\nTherefore, sec\nlog\n+ C = log sec\ntan\nC\ndt\nx dx\nt\nx\nx\nt\n=\n=\n+\n+\n\u222b\n\u222b\n(iv) \u222bcosec\n= log cosec\n\u2013 cot\n+ C\nx dx\nx\nx\nWe have\ncosec\n(cosec\ncot )\ncosec\n(cosec\ncot )\nx\nx\nx\nx dx\ndx\nx\n+x\n=\n+\n\u222b\n\u222b\nPut cosec x + cot x = t so that \u2013 cosec x (cosec x + cot x) dx = dt\nSo\ncosec\n\u2013\n\u2013log| |\n\u2013 log|cosec\ncot\n|\nC\ndt\nx dx\nt\nx\nx\nt\n=\n=\n=\n+\n+\n\u222b\n\u222b\n=\n2\n2\ncosec\ncot\n\u2013 log\nC\ncosec\ncot\nx\nx\nx\nx\n\u2212\n+\n\u2212\n= log cosec\ncot\nC\nx \u2013\nx +\nExample 6 Find the following integrals:\n(i)\n3\n2\nsin\nxcos\nx dx\n\u222b\n(ii)    \nsin\nsin (\n)\nx\ndx\nx\n+a\n\u222b\n    (iii)  \n1\n1\ntan\ndx\nx\n\u222b+\nINTEGRALS         303\nSolution\n(i)\nWe have\n3\n2\n2\n2\nsin\ncos\nsin\ncos\n(sin )\nx\nx dx\nx\nx\nx dx\n=\n\u222b\n\u222b\n= \n2\n2\n(1\u2013 cos\n) cos\n(sin )\nx\nx\nx dx\n\u222b\nPut t = cos x so that dt = \u2013 sin x dx\nTherefore,    \n2\n2\nsin\ncos\n(sin )\nx\nx\nx dx\n\u222b\n = \n2\n2\n(1\u2013\nt)\nt dt\n\u2212\u222b\n= \n3\n5\n2\n4\n(\n\u2013\n)\nC\n3\n5\nt\nt\n\u2013\nt\nt\ndt\n\u2013\n\u2013\n\uf8eb\n\uf8f6\n=\n+\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\u222b\n= \n3\n5\n1\n1\ncos\ncos\nC\n3\n5\n\u2013\nx\nx\n+\n+\n(ii)\nPut x + a = t Then dx = dt Therefore\nsin\nsin (\n)\nsin (\n)\nsin\nx\nt \u2013 a\ndx\ndt\nx\na\nt\n=\n+\n\u222b\n\u222b\n= \nsin cos\ncos sin\nsin\nt\na \u2013\nt\na dt\nt\n\u222b\n= cos\n\u2013 sin\ncot\na dt\na\nt dt\n\u222b\n\u222b\n= \n1\n(cos )\n(sin ) log sin\nC\na t \u2013\na\nt\n\uf8ee\n\uf8f9\n+\n\uf8f0\n\uf8fb\n= \n1\n(cos ) (\n)\n(sin ) log sin (\n)\nC\na\nx\na \u2013\na\nx\na\n\uf8ee\n\uf8f9\n+\n+\n+\n\uf8f0\n\uf8fb\n= \n1\ncos\ncos\n(sin ) log sin (\n)\nC sin\nx\na\na\na \u2013\na\nx\na \u2013\na\n+\n+\nHence, \nsin\nsin (\n)\nx\ndx\nx\n+a\n\u222b\n = x cos a \u2013 sin a log |sin (x + a)| + C,\nwhere,  C = \u2013 C1 sin a + a cos a, is another arbitrary constant (iii)\ncos\n1\ntan\ncos\nsin\ndx\nx dx\nx\nx\nx\n=\n+\n+\n\u222b\n\u222b\n= \n1\n(cos\n+ sin\n+ cos\n\u2013 sin )\n2\ncos\nsin\nx\nx\nx\nx dx\nx\nx\n+\n\u222b\n304\nMATHEMATICS\n= \n1\n1\ncos\n\u2013 sin\n2\n2\ncos\nsin\nx\nx\ndx\ndx\nx\nx\n+\n+\n\u222b\n\u222b\n= \nC1\n1\ncos\nsin\n2\n2\n2\ncos\nsin\nx\nx \u2013\nx dx\nx\nx\n+\n+\n+\n\u222b"
    },
    {
        "Chapter": "1",
        "sentence_range": "3481-3484",
        "Text": "Then dx = dt Therefore\nsin\nsin (\n)\nsin (\n)\nsin\nx\nt \u2013 a\ndx\ndt\nx\na\nt\n=\n+\n\u222b\n\u222b\n= \nsin cos\ncos sin\nsin\nt\na \u2013\nt\na dt\nt\n\u222b\n= cos\n\u2013 sin\ncot\na dt\na\nt dt\n\u222b\n\u222b\n= \n1\n(cos )\n(sin ) log sin\nC\na t \u2013\na\nt\n\uf8ee\n\uf8f9\n+\n\uf8f0\n\uf8fb\n= \n1\n(cos ) (\n)\n(sin ) log sin (\n)\nC\na\nx\na \u2013\na\nx\na\n\uf8ee\n\uf8f9\n+\n+\n+\n\uf8f0\n\uf8fb\n= \n1\ncos\ncos\n(sin ) log sin (\n)\nC sin\nx\na\na\na \u2013\na\nx\na \u2013\na\n+\n+\nHence, \nsin\nsin (\n)\nx\ndx\nx\n+a\n\u222b\n = x cos a \u2013 sin a log |sin (x + a)| + C,\nwhere,  C = \u2013 C1 sin a + a cos a, is another arbitrary constant (iii)\ncos\n1\ntan\ncos\nsin\ndx\nx dx\nx\nx\nx\n=\n+\n+\n\u222b\n\u222b\n= \n1\n(cos\n+ sin\n+ cos\n\u2013 sin )\n2\ncos\nsin\nx\nx\nx\nx dx\nx\nx\n+\n\u222b\n304\nMATHEMATICS\n= \n1\n1\ncos\n\u2013 sin\n2\n2\ncos\nsin\nx\nx\ndx\ndx\nx\nx\n+\n+\n\u222b\n\u222b\n= \nC1\n1\ncos\nsin\n2\n2\n2\ncos\nsin\nx\nx \u2013\nx dx\nx\nx\n+\n+\n+\n\u222b (1)\nNow, consider \ncos\nsin\nI\ncos\nsin\nx \u2013\nx dx\nx\nx\n=\n+\n\u222b\nPut cos x + sin x = t so that (cos x \u2013 sin x) dx = dt\nTherefore       \n2\nI\nlog\nC\ndt\nt\nt\n=\n=\n+\n\u222b\n= \n2\nlog cos\nsin\nC\nx\nx\n+\n+\nPutting it in (1), we get\n1\n2\nC\nC\n1\n+\n+\nlog cos\nsin\n1\ntan\n2\n2\n2\n2\ndx\nx\nx\nx\nx\n=\n+\n+\n+\n\u222b\n= \n1\n2\nC\nC\n+1\nlog cos\nsin\n2\n2\n2\n2\nx\nx\nx\n+\n+\n+\n= \n1\n2\nC\nC\n+1\nlog cos\nsin\nC\nC\n2\n2\n2\n2\nx\nx\nx\n,\uf8eb\n\uf8f6\n+\n+\n=\n+\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\nEXERCISE 7"
    },
    {
        "Chapter": "1",
        "sentence_range": "3482-3485",
        "Text": "Therefore\nsin\nsin (\n)\nsin (\n)\nsin\nx\nt \u2013 a\ndx\ndt\nx\na\nt\n=\n+\n\u222b\n\u222b\n= \nsin cos\ncos sin\nsin\nt\na \u2013\nt\na dt\nt\n\u222b\n= cos\n\u2013 sin\ncot\na dt\na\nt dt\n\u222b\n\u222b\n= \n1\n(cos )\n(sin ) log sin\nC\na t \u2013\na\nt\n\uf8ee\n\uf8f9\n+\n\uf8f0\n\uf8fb\n= \n1\n(cos ) (\n)\n(sin ) log sin (\n)\nC\na\nx\na \u2013\na\nx\na\n\uf8ee\n\uf8f9\n+\n+\n+\n\uf8f0\n\uf8fb\n= \n1\ncos\ncos\n(sin ) log sin (\n)\nC sin\nx\na\na\na \u2013\na\nx\na \u2013\na\n+\n+\nHence, \nsin\nsin (\n)\nx\ndx\nx\n+a\n\u222b\n = x cos a \u2013 sin a log |sin (x + a)| + C,\nwhere,  C = \u2013 C1 sin a + a cos a, is another arbitrary constant (iii)\ncos\n1\ntan\ncos\nsin\ndx\nx dx\nx\nx\nx\n=\n+\n+\n\u222b\n\u222b\n= \n1\n(cos\n+ sin\n+ cos\n\u2013 sin )\n2\ncos\nsin\nx\nx\nx\nx dx\nx\nx\n+\n\u222b\n304\nMATHEMATICS\n= \n1\n1\ncos\n\u2013 sin\n2\n2\ncos\nsin\nx\nx\ndx\ndx\nx\nx\n+\n+\n\u222b\n\u222b\n= \nC1\n1\ncos\nsin\n2\n2\n2\ncos\nsin\nx\nx \u2013\nx dx\nx\nx\n+\n+\n+\n\u222b (1)\nNow, consider \ncos\nsin\nI\ncos\nsin\nx \u2013\nx dx\nx\nx\n=\n+\n\u222b\nPut cos x + sin x = t so that (cos x \u2013 sin x) dx = dt\nTherefore       \n2\nI\nlog\nC\ndt\nt\nt\n=\n=\n+\n\u222b\n= \n2\nlog cos\nsin\nC\nx\nx\n+\n+\nPutting it in (1), we get\n1\n2\nC\nC\n1\n+\n+\nlog cos\nsin\n1\ntan\n2\n2\n2\n2\ndx\nx\nx\nx\nx\n=\n+\n+\n+\n\u222b\n= \n1\n2\nC\nC\n+1\nlog cos\nsin\n2\n2\n2\n2\nx\nx\nx\n+\n+\n+\n= \n1\n2\nC\nC\n+1\nlog cos\nsin\nC\nC\n2\n2\n2\n2\nx\nx\nx\n,\uf8eb\n\uf8f6\n+\n+\n=\n+\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\nEXERCISE 7 2\nIntegrate the functions in Exercises 1 to 37:\n1"
    },
    {
        "Chapter": "1",
        "sentence_range": "3483-3486",
        "Text": "(iii)\ncos\n1\ntan\ncos\nsin\ndx\nx dx\nx\nx\nx\n=\n+\n+\n\u222b\n\u222b\n= \n1\n(cos\n+ sin\n+ cos\n\u2013 sin )\n2\ncos\nsin\nx\nx\nx\nx dx\nx\nx\n+\n\u222b\n304\nMATHEMATICS\n= \n1\n1\ncos\n\u2013 sin\n2\n2\ncos\nsin\nx\nx\ndx\ndx\nx\nx\n+\n+\n\u222b\n\u222b\n= \nC1\n1\ncos\nsin\n2\n2\n2\ncos\nsin\nx\nx \u2013\nx dx\nx\nx\n+\n+\n+\n\u222b (1)\nNow, consider \ncos\nsin\nI\ncos\nsin\nx \u2013\nx dx\nx\nx\n=\n+\n\u222b\nPut cos x + sin x = t so that (cos x \u2013 sin x) dx = dt\nTherefore       \n2\nI\nlog\nC\ndt\nt\nt\n=\n=\n+\n\u222b\n= \n2\nlog cos\nsin\nC\nx\nx\n+\n+\nPutting it in (1), we get\n1\n2\nC\nC\n1\n+\n+\nlog cos\nsin\n1\ntan\n2\n2\n2\n2\ndx\nx\nx\nx\nx\n=\n+\n+\n+\n\u222b\n= \n1\n2\nC\nC\n+1\nlog cos\nsin\n2\n2\n2\n2\nx\nx\nx\n+\n+\n+\n= \n1\n2\nC\nC\n+1\nlog cos\nsin\nC\nC\n2\n2\n2\n2\nx\nx\nx\n,\uf8eb\n\uf8f6\n+\n+\n=\n+\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\nEXERCISE 7 2\nIntegrate the functions in Exercises 1 to 37:\n1 2\n2\n1\n+xx\n2"
    },
    {
        "Chapter": "1",
        "sentence_range": "3484-3487",
        "Text": "(1)\nNow, consider \ncos\nsin\nI\ncos\nsin\nx \u2013\nx dx\nx\nx\n=\n+\n\u222b\nPut cos x + sin x = t so that (cos x \u2013 sin x) dx = dt\nTherefore       \n2\nI\nlog\nC\ndt\nt\nt\n=\n=\n+\n\u222b\n= \n2\nlog cos\nsin\nC\nx\nx\n+\n+\nPutting it in (1), we get\n1\n2\nC\nC\n1\n+\n+\nlog cos\nsin\n1\ntan\n2\n2\n2\n2\ndx\nx\nx\nx\nx\n=\n+\n+\n+\n\u222b\n= \n1\n2\nC\nC\n+1\nlog cos\nsin\n2\n2\n2\n2\nx\nx\nx\n+\n+\n+\n= \n1\n2\nC\nC\n+1\nlog cos\nsin\nC\nC\n2\n2\n2\n2\nx\nx\nx\n,\uf8eb\n\uf8f6\n+\n+\n=\n+\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\nEXERCISE 7 2\nIntegrate the functions in Exercises 1 to 37:\n1 2\n2\n1\n+xx\n2 (\n)\n2\nlog x\nx\n3"
    },
    {
        "Chapter": "1",
        "sentence_range": "3485-3488",
        "Text": "2\nIntegrate the functions in Exercises 1 to 37:\n1 2\n2\n1\n+xx\n2 (\n)\n2\nlog x\nx\n3 1\nlog\nx\nx\nx\n+\n4"
    },
    {
        "Chapter": "1",
        "sentence_range": "3486-3489",
        "Text": "2\n2\n1\n+xx\n2 (\n)\n2\nlog x\nx\n3 1\nlog\nx\nx\nx\n+\n4 sin\nxsin (cos )\nx\n5"
    },
    {
        "Chapter": "1",
        "sentence_range": "3487-3490",
        "Text": "(\n)\n2\nlog x\nx\n3 1\nlog\nx\nx\nx\n+\n4 sin\nxsin (cos )\nx\n5 sin (\n) cos (\n)\nax\nb\nax\nb\n+\n+\n6"
    },
    {
        "Chapter": "1",
        "sentence_range": "3488-3491",
        "Text": "1\nlog\nx\nx\nx\n+\n4 sin\nxsin (cos )\nx\n5 sin (\n) cos (\n)\nax\nb\nax\nb\n+\n+\n6 ax\n+b\n7"
    },
    {
        "Chapter": "1",
        "sentence_range": "3489-3492",
        "Text": "sin\nxsin (cos )\nx\n5 sin (\n) cos (\n)\nax\nb\nax\nb\n+\n+\n6 ax\n+b\n7 2\nx\nx +\n8"
    },
    {
        "Chapter": "1",
        "sentence_range": "3490-3493",
        "Text": "sin (\n) cos (\n)\nax\nb\nax\nb\n+\n+\n6 ax\n+b\n7 2\nx\nx +\n8 2\n1\n2\nx\nx\n+\n9"
    },
    {
        "Chapter": "1",
        "sentence_range": "3491-3494",
        "Text": "ax\n+b\n7 2\nx\nx +\n8 2\n1\n2\nx\nx\n+\n9 2\n(4\n2)\n1\nx\nx\nx\n+\n+\n+\n10"
    },
    {
        "Chapter": "1",
        "sentence_range": "3492-3495",
        "Text": "2\nx\nx +\n8 2\n1\n2\nx\nx\n+\n9 2\n(4\n2)\n1\nx\nx\nx\n+\n+\n+\n10 1\nx \u2013\nx\n11"
    },
    {
        "Chapter": "1",
        "sentence_range": "3493-3496",
        "Text": "2\n1\n2\nx\nx\n+\n9 2\n(4\n2)\n1\nx\nx\nx\n+\n+\n+\n10 1\nx \u2013\nx\n11 4\nx\nx +\n, x > 0\n12"
    },
    {
        "Chapter": "1",
        "sentence_range": "3494-3497",
        "Text": "2\n(4\n2)\n1\nx\nx\nx\n+\n+\n+\n10 1\nx \u2013\nx\n11 4\nx\nx +\n, x > 0\n12 1\n3\n5\n3\n(\nx \u20131)\nx\n13"
    },
    {
        "Chapter": "1",
        "sentence_range": "3495-3498",
        "Text": "1\nx \u2013\nx\n11 4\nx\nx +\n, x > 0\n12 1\n3\n5\n3\n(\nx \u20131)\nx\n13 2\n3 3\n(2\n3\n)\nx\n+x\n14"
    },
    {
        "Chapter": "1",
        "sentence_range": "3496-3499",
        "Text": "4\nx\nx +\n, x > 0\n12 1\n3\n5\n3\n(\nx \u20131)\nx\n13 2\n3 3\n(2\n3\n)\nx\n+x\n14 1\nx(log )m\nx\n, x > 0, \n\u22601\nm\n15"
    },
    {
        "Chapter": "1",
        "sentence_range": "3497-3500",
        "Text": "1\n3\n5\n3\n(\nx \u20131)\nx\n13 2\n3 3\n(2\n3\n)\nx\n+x\n14 1\nx(log )m\nx\n, x > 0, \n\u22601\nm\n15 2\n9\n4\nx\n\u2013 x\n16"
    },
    {
        "Chapter": "1",
        "sentence_range": "3498-3501",
        "Text": "2\n3 3\n(2\n3\n)\nx\n+x\n14 1\nx(log )m\nx\n, x > 0, \n\u22601\nm\n15 2\n9\n4\nx\n\u2013 x\n16 2\n3\nex\n+\n17"
    },
    {
        "Chapter": "1",
        "sentence_range": "3499-3502",
        "Text": "1\nx(log )m\nx\n, x > 0, \n\u22601\nm\n15 2\n9\n4\nx\n\u2013 x\n16 2\n3\nex\n+\n17 x2\nx\ne\nINTEGRALS         305\n18"
    },
    {
        "Chapter": "1",
        "sentence_range": "3500-3503",
        "Text": "2\n9\n4\nx\n\u2013 x\n16 2\n3\nex\n+\n17 x2\nx\ne\nINTEGRALS         305\n18 1\n2\n1\ntan\u2013\nx\ne\n+x\n19"
    },
    {
        "Chapter": "1",
        "sentence_range": "3501-3504",
        "Text": "2\n3\nex\n+\n17 x2\nx\ne\nINTEGRALS         305\n18 1\n2\n1\ntan\u2013\nx\ne\n+x\n19 2\n2\n1\n1\nx\nx\ne\n\u2013\ne\n+\n20"
    },
    {
        "Chapter": "1",
        "sentence_range": "3502-3505",
        "Text": "x2\nx\ne\nINTEGRALS         305\n18 1\n2\n1\ntan\u2013\nx\ne\n+x\n19 2\n2\n1\n1\nx\nx\ne\n\u2013\ne\n+\n20 2\n2\n2\n2\nx\n\u2013\nx\nx\n\u2013\nx\ne\n\u2013 e\ne\ne\n+\n21"
    },
    {
        "Chapter": "1",
        "sentence_range": "3503-3506",
        "Text": "1\n2\n1\ntan\u2013\nx\ne\n+x\n19 2\n2\n1\n1\nx\nx\ne\n\u2013\ne\n+\n20 2\n2\n2\n2\nx\n\u2013\nx\nx\n\u2013\nx\ne\n\u2013 e\ne\ne\n+\n21 tan2 (2x \u2013 3)\n22"
    },
    {
        "Chapter": "1",
        "sentence_range": "3504-3507",
        "Text": "2\n2\n1\n1\nx\nx\ne\n\u2013\ne\n+\n20 2\n2\n2\n2\nx\n\u2013\nx\nx\n\u2013\nx\ne\n\u2013 e\ne\ne\n+\n21 tan2 (2x \u2013 3)\n22 sec2 (7 \u2013 4x)\n23"
    },
    {
        "Chapter": "1",
        "sentence_range": "3505-3508",
        "Text": "2\n2\n2\n2\nx\n\u2013\nx\nx\n\u2013\nx\ne\n\u2013 e\ne\ne\n+\n21 tan2 (2x \u2013 3)\n22 sec2 (7 \u2013 4x)\n23 1\n2\nsin\n1\n\u2013 x\n\u2013 x\n24"
    },
    {
        "Chapter": "1",
        "sentence_range": "3506-3509",
        "Text": "tan2 (2x \u2013 3)\n22 sec2 (7 \u2013 4x)\n23 1\n2\nsin\n1\n\u2013 x\n\u2013 x\n24 2cos\n3sin\n6cos\n4sin\nx \u2013\nx\nx\nx\n+\n25"
    },
    {
        "Chapter": "1",
        "sentence_range": "3507-3510",
        "Text": "sec2 (7 \u2013 4x)\n23 1\n2\nsin\n1\n\u2013 x\n\u2013 x\n24 2cos\n3sin\n6cos\n4sin\nx \u2013\nx\nx\nx\n+\n25 2\n2\n1\ncos\n(1\ntan )\nx\n\u2013\nx\n26"
    },
    {
        "Chapter": "1",
        "sentence_range": "3508-3511",
        "Text": "1\n2\nsin\n1\n\u2013 x\n\u2013 x\n24 2cos\n3sin\n6cos\n4sin\nx \u2013\nx\nx\nx\n+\n25 2\n2\n1\ncos\n(1\ntan )\nx\n\u2013\nx\n26 cos\nx\nx\n27"
    },
    {
        "Chapter": "1",
        "sentence_range": "3509-3512",
        "Text": "2cos\n3sin\n6cos\n4sin\nx \u2013\nx\nx\nx\n+\n25 2\n2\n1\ncos\n(1\ntan )\nx\n\u2013\nx\n26 cos\nx\nx\n27 sin 2 cos 2\nx\nx\n28"
    },
    {
        "Chapter": "1",
        "sentence_range": "3510-3513",
        "Text": "2\n2\n1\ncos\n(1\ntan )\nx\n\u2013\nx\n26 cos\nx\nx\n27 sin 2 cos 2\nx\nx\n28 cos\n1\nsin\nx\nx\n+\n29"
    },
    {
        "Chapter": "1",
        "sentence_range": "3511-3514",
        "Text": "cos\nx\nx\n27 sin 2 cos 2\nx\nx\n28 cos\n1\nsin\nx\nx\n+\n29 cot x log sin x\n30"
    },
    {
        "Chapter": "1",
        "sentence_range": "3512-3515",
        "Text": "sin 2 cos 2\nx\nx\n28 cos\n1\nsin\nx\nx\n+\n29 cot x log sin x\n30 sin\n1\ncos\nx\nx\n+\n31"
    },
    {
        "Chapter": "1",
        "sentence_range": "3513-3516",
        "Text": "cos\n1\nsin\nx\nx\n+\n29 cot x log sin x\n30 sin\n1\ncos\nx\nx\n+\n31 (\n)\n2\nsin\n1\ncos\nx\nx\n+\n32"
    },
    {
        "Chapter": "1",
        "sentence_range": "3514-3517",
        "Text": "cot x log sin x\n30 sin\n1\ncos\nx\nx\n+\n31 (\n)\n2\nsin\n1\ncos\nx\nx\n+\n32 1\n1\ncot x\n+\n33"
    },
    {
        "Chapter": "1",
        "sentence_range": "3515-3518",
        "Text": "sin\n1\ncos\nx\nx\n+\n31 (\n)\n2\nsin\n1\ncos\nx\nx\n+\n32 1\n1\ncot x\n+\n33 1\n1\n\u2013tan\nx\n34"
    },
    {
        "Chapter": "1",
        "sentence_range": "3516-3519",
        "Text": "(\n)\n2\nsin\n1\ncos\nx\nx\n+\n32 1\n1\ncot x\n+\n33 1\n1\n\u2013tan\nx\n34 tan\nsin\ncos\nx\nx\nx\n35"
    },
    {
        "Chapter": "1",
        "sentence_range": "3517-3520",
        "Text": "1\n1\ncot x\n+\n33 1\n1\n\u2013tan\nx\n34 tan\nsin\ncos\nx\nx\nx\n35 (\n)\n2\n1\nlog x\nx\n+\n36"
    },
    {
        "Chapter": "1",
        "sentence_range": "3518-3521",
        "Text": "1\n1\n\u2013tan\nx\n34 tan\nsin\ncos\nx\nx\nx\n35 (\n)\n2\n1\nlog x\nx\n+\n36 (\n)\n2\n(\n1)\nlog\nx\nx\nx\nx\n+\n+\n37"
    },
    {
        "Chapter": "1",
        "sentence_range": "3519-3522",
        "Text": "tan\nsin\ncos\nx\nx\nx\n35 (\n)\n2\n1\nlog x\nx\n+\n36 (\n)\n2\n(\n1)\nlog\nx\nx\nx\nx\n+\n+\n37 (\n)\n3\n1 4\nsin tan\n1\n\u2013\nx\nx\nx8\n+\nChoose the correct answer in Exercises 38 and 39"
    },
    {
        "Chapter": "1",
        "sentence_range": "3520-3523",
        "Text": "(\n)\n2\n1\nlog x\nx\n+\n36 (\n)\n2\n(\n1)\nlog\nx\nx\nx\nx\n+\n+\n37 (\n)\n3\n1 4\nsin tan\n1\n\u2013\nx\nx\nx8\n+\nChoose the correct answer in Exercises 38 and 39 38"
    },
    {
        "Chapter": "1",
        "sentence_range": "3521-3524",
        "Text": "(\n)\n2\n(\n1)\nlog\nx\nx\nx\nx\n+\n+\n37 (\n)\n3\n1 4\nsin tan\n1\n\u2013\nx\nx\nx8\n+\nChoose the correct answer in Exercises 38 and 39 38 10\n9\n10\n10\n10 log\n10\nx\ne\nx\nx\ndx\nx\n+\n+\n\u222b\n equals\n(A) 10x \u2013 x10 + C\n(B) 10x + x10 + C\n(C) (10x \u2013 x10)\u20131 + C\n(D) log (10x + x10) + C\n39"
    },
    {
        "Chapter": "1",
        "sentence_range": "3522-3525",
        "Text": "(\n)\n3\n1 4\nsin tan\n1\n\u2013\nx\nx\nx8\n+\nChoose the correct answer in Exercises 38 and 39 38 10\n9\n10\n10\n10 log\n10\nx\ne\nx\nx\ndx\nx\n+\n+\n\u222b\n equals\n(A) 10x \u2013 x10 + C\n(B) 10x + x10 + C\n(C) (10x \u2013 x10)\u20131 + C\n(D) log (10x + x10) + C\n39 2\n2\nequals\nsin\nxdxcos\nx\n\u222b\n(A)\ntan x + cot x + C\n(B)  tan x \u2013 cot x + C\n(C)\ntan x cot x + C\n(D)  tan x \u2013 cot 2x + C\n7"
    },
    {
        "Chapter": "1",
        "sentence_range": "3523-3526",
        "Text": "38 10\n9\n10\n10\n10 log\n10\nx\ne\nx\nx\ndx\nx\n+\n+\n\u222b\n equals\n(A) 10x \u2013 x10 + C\n(B) 10x + x10 + C\n(C) (10x \u2013 x10)\u20131 + C\n(D) log (10x + x10) + C\n39 2\n2\nequals\nsin\nxdxcos\nx\n\u222b\n(A)\ntan x + cot x + C\n(B)  tan x \u2013 cot x + C\n(C)\ntan x cot x + C\n(D)  tan x \u2013 cot 2x + C\n7 3"
    },
    {
        "Chapter": "1",
        "sentence_range": "3524-3527",
        "Text": "10\n9\n10\n10\n10 log\n10\nx\ne\nx\nx\ndx\nx\n+\n+\n\u222b\n equals\n(A) 10x \u2013 x10 + C\n(B) 10x + x10 + C\n(C) (10x \u2013 x10)\u20131 + C\n(D) log (10x + x10) + C\n39 2\n2\nequals\nsin\nxdxcos\nx\n\u222b\n(A)\ntan x + cot x + C\n(B)  tan x \u2013 cot x + C\n(C)\ntan x cot x + C\n(D)  tan x \u2013 cot 2x + C\n7 3 2  Integration using trigonometric identities\nWhen the integrand involves some trigonometric functions, we use some known identities\nto find the integral as illustrated through the following example"
    },
    {
        "Chapter": "1",
        "sentence_range": "3525-3528",
        "Text": "2\n2\nequals\nsin\nxdxcos\nx\n\u222b\n(A)\ntan x + cot x + C\n(B)  tan x \u2013 cot x + C\n(C)\ntan x cot x + C\n(D)  tan x \u2013 cot 2x + C\n7 3 2  Integration using trigonometric identities\nWhen the integrand involves some trigonometric functions, we use some known identities\nto find the integral as illustrated through the following example Example 7 Find (i) \n2\n\u222bcos x dx\n (ii) sin 2 cos 3\nx\nx dx\n\u222b\n (iii) \n3\n\u222bsin x dx\n306\nMATHEMATICS\nSolution\n(i)\nRecall the identity cos 2x = 2 cos2 x \u2013 1, which gives\ncos2x = 1\ncos 2\n2\nx\n+\nTherefore,      \n\u222bcos2\nx dx  = 1 (1+ cos 2 )\n2\nx dx\n\u222b\n= 1\n1\ncos 2\n2\n2\ndx\nx dx\n+\n\u222b\n\u222b\n= \n1 sin 2\nC\n2\n4\nx\nx\n+\n+\n(ii)\nRecall the identity sin x cos y = 1\n2 [sin (x + y) + sin (x \u2013 y)]\n(Why"
    },
    {
        "Chapter": "1",
        "sentence_range": "3526-3529",
        "Text": "3 2  Integration using trigonometric identities\nWhen the integrand involves some trigonometric functions, we use some known identities\nto find the integral as illustrated through the following example Example 7 Find (i) \n2\n\u222bcos x dx\n (ii) sin 2 cos 3\nx\nx dx\n\u222b\n (iii) \n3\n\u222bsin x dx\n306\nMATHEMATICS\nSolution\n(i)\nRecall the identity cos 2x = 2 cos2 x \u2013 1, which gives\ncos2x = 1\ncos 2\n2\nx\n+\nTherefore,      \n\u222bcos2\nx dx  = 1 (1+ cos 2 )\n2\nx dx\n\u222b\n= 1\n1\ncos 2\n2\n2\ndx\nx dx\n+\n\u222b\n\u222b\n= \n1 sin 2\nC\n2\n4\nx\nx\n+\n+\n(ii)\nRecall the identity sin x cos y = 1\n2 [sin (x + y) + sin (x \u2013 y)]\n(Why )\nThen   sin 2 cos 3\n\u222b\nx\nxdx  = \n1\nsin 5\nsin\n2\n\u2022\n\uf8ee\n\uf8f9\n\uf8f0\n\uf8fb\n\u222b\n\u222b\nx dx\nx dx\n= \n1\n1 cos 5\ncos\nC\n2\n\u20135\nx\nx\n\uf8ee\n\uf8f9\n+\n+\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n= \n1\n1\ncos 5\ncos\nC\n10\n2\n\u2013\nx\nx\n+\n+\n(iii)\nFrom the identity sin 3x = 3 sin x \u2013 4 sin3 x, we find that\nsin3x = 3sin\nsin 3\n4\nx \u2013\nx\nTherefore,      \n3\n\u222bsin x dx\n = 3\n1\nsin\nsin 3\n4\n4\nx dx \u2013\nx dx\n\u222b\n\u222b\n                                      = \n3\n1\n\u2013\ncos\ncos 3\nC\n4\n12\nx\nx\n+\n+\nAlternatively, \n3\n2\nsin\nsin\nsin\nx dx\nx\nx dx\n=\n\u222b\n\u222b\n = \n(1\u2013 cos2\nx) sin\nx dx\n\u222b\nPut cos x = t so that \u2013 sin x dx = dt\nTherefore,     \n3\n\u222bsin x dx\n = \n(\n)\n1 \u2013 t2\ndt\n\u2212\u222b\n = \n3\n2\nC\nt3\n\u2013\ndt\nt dt\n\u2013 t\n+\n=\n+\n+\n\u222b\n\u222b\n= \n3\n1\ncos\ncos\nC\n3\n\u2013\nx\nx\n+\n+\nRemark It can be shown using trigonometric identities that both answers are equivalent"
    },
    {
        "Chapter": "1",
        "sentence_range": "3527-3530",
        "Text": "2  Integration using trigonometric identities\nWhen the integrand involves some trigonometric functions, we use some known identities\nto find the integral as illustrated through the following example Example 7 Find (i) \n2\n\u222bcos x dx\n (ii) sin 2 cos 3\nx\nx dx\n\u222b\n (iii) \n3\n\u222bsin x dx\n306\nMATHEMATICS\nSolution\n(i)\nRecall the identity cos 2x = 2 cos2 x \u2013 1, which gives\ncos2x = 1\ncos 2\n2\nx\n+\nTherefore,      \n\u222bcos2\nx dx  = 1 (1+ cos 2 )\n2\nx dx\n\u222b\n= 1\n1\ncos 2\n2\n2\ndx\nx dx\n+\n\u222b\n\u222b\n= \n1 sin 2\nC\n2\n4\nx\nx\n+\n+\n(ii)\nRecall the identity sin x cos y = 1\n2 [sin (x + y) + sin (x \u2013 y)]\n(Why )\nThen   sin 2 cos 3\n\u222b\nx\nxdx  = \n1\nsin 5\nsin\n2\n\u2022\n\uf8ee\n\uf8f9\n\uf8f0\n\uf8fb\n\u222b\n\u222b\nx dx\nx dx\n= \n1\n1 cos 5\ncos\nC\n2\n\u20135\nx\nx\n\uf8ee\n\uf8f9\n+\n+\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n= \n1\n1\ncos 5\ncos\nC\n10\n2\n\u2013\nx\nx\n+\n+\n(iii)\nFrom the identity sin 3x = 3 sin x \u2013 4 sin3 x, we find that\nsin3x = 3sin\nsin 3\n4\nx \u2013\nx\nTherefore,      \n3\n\u222bsin x dx\n = 3\n1\nsin\nsin 3\n4\n4\nx dx \u2013\nx dx\n\u222b\n\u222b\n                                      = \n3\n1\n\u2013\ncos\ncos 3\nC\n4\n12\nx\nx\n+\n+\nAlternatively, \n3\n2\nsin\nsin\nsin\nx dx\nx\nx dx\n=\n\u222b\n\u222b\n = \n(1\u2013 cos2\nx) sin\nx dx\n\u222b\nPut cos x = t so that \u2013 sin x dx = dt\nTherefore,     \n3\n\u222bsin x dx\n = \n(\n)\n1 \u2013 t2\ndt\n\u2212\u222b\n = \n3\n2\nC\nt3\n\u2013\ndt\nt dt\n\u2013 t\n+\n=\n+\n+\n\u222b\n\u222b\n= \n3\n1\ncos\ncos\nC\n3\n\u2013\nx\nx\n+\n+\nRemark It can be shown using trigonometric identities that both answers are equivalent INTEGRALS         307\nEXERCISE 7"
    },
    {
        "Chapter": "1",
        "sentence_range": "3528-3531",
        "Text": "Example 7 Find (i) \n2\n\u222bcos x dx\n (ii) sin 2 cos 3\nx\nx dx\n\u222b\n (iii) \n3\n\u222bsin x dx\n306\nMATHEMATICS\nSolution\n(i)\nRecall the identity cos 2x = 2 cos2 x \u2013 1, which gives\ncos2x = 1\ncos 2\n2\nx\n+\nTherefore,      \n\u222bcos2\nx dx  = 1 (1+ cos 2 )\n2\nx dx\n\u222b\n= 1\n1\ncos 2\n2\n2\ndx\nx dx\n+\n\u222b\n\u222b\n= \n1 sin 2\nC\n2\n4\nx\nx\n+\n+\n(ii)\nRecall the identity sin x cos y = 1\n2 [sin (x + y) + sin (x \u2013 y)]\n(Why )\nThen   sin 2 cos 3\n\u222b\nx\nxdx  = \n1\nsin 5\nsin\n2\n\u2022\n\uf8ee\n\uf8f9\n\uf8f0\n\uf8fb\n\u222b\n\u222b\nx dx\nx dx\n= \n1\n1 cos 5\ncos\nC\n2\n\u20135\nx\nx\n\uf8ee\n\uf8f9\n+\n+\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n= \n1\n1\ncos 5\ncos\nC\n10\n2\n\u2013\nx\nx\n+\n+\n(iii)\nFrom the identity sin 3x = 3 sin x \u2013 4 sin3 x, we find that\nsin3x = 3sin\nsin 3\n4\nx \u2013\nx\nTherefore,      \n3\n\u222bsin x dx\n = 3\n1\nsin\nsin 3\n4\n4\nx dx \u2013\nx dx\n\u222b\n\u222b\n                                      = \n3\n1\n\u2013\ncos\ncos 3\nC\n4\n12\nx\nx\n+\n+\nAlternatively, \n3\n2\nsin\nsin\nsin\nx dx\nx\nx dx\n=\n\u222b\n\u222b\n = \n(1\u2013 cos2\nx) sin\nx dx\n\u222b\nPut cos x = t so that \u2013 sin x dx = dt\nTherefore,     \n3\n\u222bsin x dx\n = \n(\n)\n1 \u2013 t2\ndt\n\u2212\u222b\n = \n3\n2\nC\nt3\n\u2013\ndt\nt dt\n\u2013 t\n+\n=\n+\n+\n\u222b\n\u222b\n= \n3\n1\ncos\ncos\nC\n3\n\u2013\nx\nx\n+\n+\nRemark It can be shown using trigonometric identities that both answers are equivalent INTEGRALS         307\nEXERCISE 7 3\nFind the integrals of the functions in Exercises 1 to 22:\n1"
    },
    {
        "Chapter": "1",
        "sentence_range": "3529-3532",
        "Text": ")\nThen   sin 2 cos 3\n\u222b\nx\nxdx  = \n1\nsin 5\nsin\n2\n\u2022\n\uf8ee\n\uf8f9\n\uf8f0\n\uf8fb\n\u222b\n\u222b\nx dx\nx dx\n= \n1\n1 cos 5\ncos\nC\n2\n\u20135\nx\nx\n\uf8ee\n\uf8f9\n+\n+\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n= \n1\n1\ncos 5\ncos\nC\n10\n2\n\u2013\nx\nx\n+\n+\n(iii)\nFrom the identity sin 3x = 3 sin x \u2013 4 sin3 x, we find that\nsin3x = 3sin\nsin 3\n4\nx \u2013\nx\nTherefore,      \n3\n\u222bsin x dx\n = 3\n1\nsin\nsin 3\n4\n4\nx dx \u2013\nx dx\n\u222b\n\u222b\n                                      = \n3\n1\n\u2013\ncos\ncos 3\nC\n4\n12\nx\nx\n+\n+\nAlternatively, \n3\n2\nsin\nsin\nsin\nx dx\nx\nx dx\n=\n\u222b\n\u222b\n = \n(1\u2013 cos2\nx) sin\nx dx\n\u222b\nPut cos x = t so that \u2013 sin x dx = dt\nTherefore,     \n3\n\u222bsin x dx\n = \n(\n)\n1 \u2013 t2\ndt\n\u2212\u222b\n = \n3\n2\nC\nt3\n\u2013\ndt\nt dt\n\u2013 t\n+\n=\n+\n+\n\u222b\n\u222b\n= \n3\n1\ncos\ncos\nC\n3\n\u2013\nx\nx\n+\n+\nRemark It can be shown using trigonometric identities that both answers are equivalent INTEGRALS         307\nEXERCISE 7 3\nFind the integrals of the functions in Exercises 1 to 22:\n1 sin2 (2x + 5)\n2"
    },
    {
        "Chapter": "1",
        "sentence_range": "3530-3533",
        "Text": "INTEGRALS         307\nEXERCISE 7 3\nFind the integrals of the functions in Exercises 1 to 22:\n1 sin2 (2x + 5)\n2 sin 3x cos 4x\n3"
    },
    {
        "Chapter": "1",
        "sentence_range": "3531-3534",
        "Text": "3\nFind the integrals of the functions in Exercises 1 to 22:\n1 sin2 (2x + 5)\n2 sin 3x cos 4x\n3 cos 2x cos 4x cos 6x\n4"
    },
    {
        "Chapter": "1",
        "sentence_range": "3532-3535",
        "Text": "sin2 (2x + 5)\n2 sin 3x cos 4x\n3 cos 2x cos 4x cos 6x\n4 sin3 (2x + 1)\n5"
    },
    {
        "Chapter": "1",
        "sentence_range": "3533-3536",
        "Text": "sin 3x cos 4x\n3 cos 2x cos 4x cos 6x\n4 sin3 (2x + 1)\n5 sin3 x cos3 x\n6"
    },
    {
        "Chapter": "1",
        "sentence_range": "3534-3537",
        "Text": "cos 2x cos 4x cos 6x\n4 sin3 (2x + 1)\n5 sin3 x cos3 x\n6 sin x sin 2x sin 3x\n7"
    },
    {
        "Chapter": "1",
        "sentence_range": "3535-3538",
        "Text": "sin3 (2x + 1)\n5 sin3 x cos3 x\n6 sin x sin 2x sin 3x\n7 sin 4x sin 8x\n8"
    },
    {
        "Chapter": "1",
        "sentence_range": "3536-3539",
        "Text": "sin3 x cos3 x\n6 sin x sin 2x sin 3x\n7 sin 4x sin 8x\n8 1\ncos\n1\ncos\n\u2013\nxx\n+\n9"
    },
    {
        "Chapter": "1",
        "sentence_range": "3537-3540",
        "Text": "sin x sin 2x sin 3x\n7 sin 4x sin 8x\n8 1\ncos\n1\ncos\n\u2013\nxx\n+\n9 cos\n1\ncos\nx\nx\n+\n10"
    },
    {
        "Chapter": "1",
        "sentence_range": "3538-3541",
        "Text": "sin 4x sin 8x\n8 1\ncos\n1\ncos\n\u2013\nxx\n+\n9 cos\n1\ncos\nx\nx\n+\n10 sin4 x\n11"
    },
    {
        "Chapter": "1",
        "sentence_range": "3539-3542",
        "Text": "1\ncos\n1\ncos\n\u2013\nxx\n+\n9 cos\n1\ncos\nx\nx\n+\n10 sin4 x\n11 cos4 2x\n12"
    },
    {
        "Chapter": "1",
        "sentence_range": "3540-3543",
        "Text": "cos\n1\ncos\nx\nx\n+\n10 sin4 x\n11 cos4 2x\n12 2\nsin\n1\ncos\nx\nx\n+\n13"
    },
    {
        "Chapter": "1",
        "sentence_range": "3541-3544",
        "Text": "sin4 x\n11 cos4 2x\n12 2\nsin\n1\ncos\nx\nx\n+\n13 cos 2\ncos 2\ncos\nx \u2013cos\nx \u2013\n\u03b1\n\u03b1\n14"
    },
    {
        "Chapter": "1",
        "sentence_range": "3542-3545",
        "Text": "cos4 2x\n12 2\nsin\n1\ncos\nx\nx\n+\n13 cos 2\ncos 2\ncos\nx \u2013cos\nx \u2013\n\u03b1\n\u03b1\n14 cos\nsin\n1\nsin 2\nx \u2013\nx\nx\n+\n15"
    },
    {
        "Chapter": "1",
        "sentence_range": "3543-3546",
        "Text": "2\nsin\n1\ncos\nx\nx\n+\n13 cos 2\ncos 2\ncos\nx \u2013cos\nx \u2013\n\u03b1\n\u03b1\n14 cos\nsin\n1\nsin 2\nx \u2013\nx\nx\n+\n15 tan3 2x sec 2x\n16"
    },
    {
        "Chapter": "1",
        "sentence_range": "3544-3547",
        "Text": "cos 2\ncos 2\ncos\nx \u2013cos\nx \u2013\n\u03b1\n\u03b1\n14 cos\nsin\n1\nsin 2\nx \u2013\nx\nx\n+\n15 tan3 2x sec 2x\n16 tan4x\n17"
    },
    {
        "Chapter": "1",
        "sentence_range": "3545-3548",
        "Text": "cos\nsin\n1\nsin 2\nx \u2013\nx\nx\n+\n15 tan3 2x sec 2x\n16 tan4x\n17 3\n3\n2\n2\nsin\ncos\nsin\ncos\nx\nx\nx\nx\n+\n18"
    },
    {
        "Chapter": "1",
        "sentence_range": "3546-3549",
        "Text": "tan3 2x sec 2x\n16 tan4x\n17 3\n3\n2\n2\nsin\ncos\nsin\ncos\nx\nx\nx\nx\n+\n18 2\n2\ncos 2\n2sin\nxcos\nx\nx\n+\n19"
    },
    {
        "Chapter": "1",
        "sentence_range": "3547-3550",
        "Text": "tan4x\n17 3\n3\n2\n2\nsin\ncos\nsin\ncos\nx\nx\nx\nx\n+\n18 2\n2\ncos 2\n2sin\nxcos\nx\nx\n+\n19 3\n1\nsin\nxcos\nx\n20"
    },
    {
        "Chapter": "1",
        "sentence_range": "3548-3551",
        "Text": "3\n3\n2\n2\nsin\ncos\nsin\ncos\nx\nx\nx\nx\n+\n18 2\n2\ncos 2\n2sin\nxcos\nx\nx\n+\n19 3\n1\nsin\nxcos\nx\n20 (\n)\n2\ncos 2\ncos\nsin\nx\nx\nx\n+\n21"
    },
    {
        "Chapter": "1",
        "sentence_range": "3549-3552",
        "Text": "2\n2\ncos 2\n2sin\nxcos\nx\nx\n+\n19 3\n1\nsin\nxcos\nx\n20 (\n)\n2\ncos 2\ncos\nsin\nx\nx\nx\n+\n21 sin \u2013 1 (cos x)\n22"
    },
    {
        "Chapter": "1",
        "sentence_range": "3550-3553",
        "Text": "3\n1\nsin\nxcos\nx\n20 (\n)\n2\ncos 2\ncos\nsin\nx\nx\nx\n+\n21 sin \u2013 1 (cos x)\n22 1\ncos (\n) cos (\n)\nx \u2013 a\nx \u2013 b\nChoose the correct answer in Exercises 23 and 24"
    },
    {
        "Chapter": "1",
        "sentence_range": "3551-3554",
        "Text": "(\n)\n2\ncos 2\ncos\nsin\nx\nx\nx\n+\n21 sin \u2013 1 (cos x)\n22 1\ncos (\n) cos (\n)\nx \u2013 a\nx \u2013 b\nChoose the correct answer in Exercises 23 and 24 23"
    },
    {
        "Chapter": "1",
        "sentence_range": "3552-3555",
        "Text": "sin \u2013 1 (cos x)\n22 1\ncos (\n) cos (\n)\nx \u2013 a\nx \u2013 b\nChoose the correct answer in Exercises 23 and 24 23 2\n2\n2\n2\nsin\ncos\nis equal to\nsin\nxcos\nx dx\nx\nx\n\u2212\n\u222b\n(A) tan x + cot x + C\n(B) tan x + cosec x + C\n(C) \u2013 tan x + cot x + C\n(D) tan x + sec x + C\n24"
    },
    {
        "Chapter": "1",
        "sentence_range": "3553-3556",
        "Text": "1\ncos (\n) cos (\n)\nx \u2013 a\nx \u2013 b\nChoose the correct answer in Exercises 23 and 24 23 2\n2\n2\n2\nsin\ncos\nis equal to\nsin\nxcos\nx dx\nx\nx\n\u2212\n\u222b\n(A) tan x + cot x + C\n(B) tan x + cosec x + C\n(C) \u2013 tan x + cot x + C\n(D) tan x + sec x + C\n24 (12\n)\nequals\ncos (\n)\nx\nx\ne\nx\n+e xdx\n\u222b\n(A) \u2013 cot (exx) + C\n(B) tan (xex) + C\n(C) tan (ex) + C\n(D) cot (ex) + C\n7"
    },
    {
        "Chapter": "1",
        "sentence_range": "3554-3557",
        "Text": "23 2\n2\n2\n2\nsin\ncos\nis equal to\nsin\nxcos\nx dx\nx\nx\n\u2212\n\u222b\n(A) tan x + cot x + C\n(B) tan x + cosec x + C\n(C) \u2013 tan x + cot x + C\n(D) tan x + sec x + C\n24 (12\n)\nequals\ncos (\n)\nx\nx\ne\nx\n+e xdx\n\u222b\n(A) \u2013 cot (exx) + C\n(B) tan (xex) + C\n(C) tan (ex) + C\n(D) cot (ex) + C\n7 4  Integrals of Some Particular Functions\nIn this section, we mention below some important formulae of integrals and apply them\nfor integrating many other related standard integrals:\n(1) \u222b\n2\n2\n1\n\u2013\n=\nlog\n+ C\n2\n+\ndx\u2013\nx\na\na\nx\na\nx\na\n308\nMATHEMATICS\n(2) \u222b\n2\n2\n1\n+\n=\nlog\n+ C\n2\n\u2013\ndx\u2013\na\nx\na\na\nx\na\nx\n(3) \u222b\n\u2013 1\n2\n2\n1 tan\nC\ndx\nx\n=\n+\na\na\nx + a\n(4) \u222b\n2\n2\n2\n2 = log\n+\n\u2013\n+C\n\u2013\ndx\nx\nx\na\nx\na\n(5) \u222b\n\u2013 1\n2\n2 = sin\n+ C\n\u2013\ndx\nax\na\nx\n(6) \u222b\n2\n2\n2\n2 = log\n+\n+\n+ C\n+\ndx\nx\nx\na\nx\na\nWe now prove the above results:\n(1)\nWe have  \n2\n12\n1\n(\n) (\n)\nx \u2013 a\nx\na\nx \u2013 a\n=\n+\n= \n1\n(\n) \u2013 (\n)\n1\n1\n1\n2\n(\n) (\n)\n2\nx\na\nx \u2013 a\n\u2013\na\nx \u2013 a\nx\na\na\nx \u2013 a\nx\na\n\uf8ee\n\uf8f9\n+\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n+\n+\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nTherefore,  \n2\n2\n1\n2\ndx\ndx\ndx\n\u2013\na\nx \u2013 a\nx\na\nx \u2013 a\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa\n+\n\uf8f0\n\uf8fb\n\u222b\n\u222b\n\u222b\n= \n[\n]\n1\nlog (\n)|\nlog (\n)|\nC\n2\n| x \u2013 a \u2013\n| x\na\na\n+\n+\n= 1 log\nC\n2\nx \u2013 a\na\nx\na +\n+\n(2)\nIn view of (1) above, we have\n2\n2\n1\n1\n(\n)\n(\n)\n2\n(\n) (\n)\n\u2013\na\nx\na\nx\na\na\nx\na\nx\na\nx\n\uf8ee\n\uf8f9\n+\n+\n\u2212\n=\n\uf8ef\n\uf8fa\n+\n\u2212\n\uf8f0\n\uf8fb  = 1\n1\n1\n2a\na\nx\na\nx\n\uf8ee\n\uf8f9\n+\n\uf8ef\n\uf8fa\n\u2212\n+\n\uf8f0\n\uf8fb\nINTEGRALS         309\n      Therefore,\n2\n\u20132\ndx\na\nx\n\u222b\n = \n1\n2\ndx\ndx\na\na\nx\na\nx\n\uf8ee\n\uf8f9\n+\n\uf8ef\n\uf8fa\n\u2212\n+\n\uf8f0\n\uf8fb\n\u222b\n\u222b\n= 1 [ log |\n|\nlog |\n|]\nC\n2\na\nx\na\nx\na \u2212\n\u2212\n+\n+\n+\n= \n1 log\nC\n2\na\nx\na\na\n+x\n+\n\u2212\n\ufffdNote  The technique used in (1) will be explained in Section 7"
    },
    {
        "Chapter": "1",
        "sentence_range": "3555-3558",
        "Text": "2\n2\n2\n2\nsin\ncos\nis equal to\nsin\nxcos\nx dx\nx\nx\n\u2212\n\u222b\n(A) tan x + cot x + C\n(B) tan x + cosec x + C\n(C) \u2013 tan x + cot x + C\n(D) tan x + sec x + C\n24 (12\n)\nequals\ncos (\n)\nx\nx\ne\nx\n+e xdx\n\u222b\n(A) \u2013 cot (exx) + C\n(B) tan (xex) + C\n(C) tan (ex) + C\n(D) cot (ex) + C\n7 4  Integrals of Some Particular Functions\nIn this section, we mention below some important formulae of integrals and apply them\nfor integrating many other related standard integrals:\n(1) \u222b\n2\n2\n1\n\u2013\n=\nlog\n+ C\n2\n+\ndx\u2013\nx\na\na\nx\na\nx\na\n308\nMATHEMATICS\n(2) \u222b\n2\n2\n1\n+\n=\nlog\n+ C\n2\n\u2013\ndx\u2013\na\nx\na\na\nx\na\nx\n(3) \u222b\n\u2013 1\n2\n2\n1 tan\nC\ndx\nx\n=\n+\na\na\nx + a\n(4) \u222b\n2\n2\n2\n2 = log\n+\n\u2013\n+C\n\u2013\ndx\nx\nx\na\nx\na\n(5) \u222b\n\u2013 1\n2\n2 = sin\n+ C\n\u2013\ndx\nax\na\nx\n(6) \u222b\n2\n2\n2\n2 = log\n+\n+\n+ C\n+\ndx\nx\nx\na\nx\na\nWe now prove the above results:\n(1)\nWe have  \n2\n12\n1\n(\n) (\n)\nx \u2013 a\nx\na\nx \u2013 a\n=\n+\n= \n1\n(\n) \u2013 (\n)\n1\n1\n1\n2\n(\n) (\n)\n2\nx\na\nx \u2013 a\n\u2013\na\nx \u2013 a\nx\na\na\nx \u2013 a\nx\na\n\uf8ee\n\uf8f9\n+\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n+\n+\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nTherefore,  \n2\n2\n1\n2\ndx\ndx\ndx\n\u2013\na\nx \u2013 a\nx\na\nx \u2013 a\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa\n+\n\uf8f0\n\uf8fb\n\u222b\n\u222b\n\u222b\n= \n[\n]\n1\nlog (\n)|\nlog (\n)|\nC\n2\n| x \u2013 a \u2013\n| x\na\na\n+\n+\n= 1 log\nC\n2\nx \u2013 a\na\nx\na +\n+\n(2)\nIn view of (1) above, we have\n2\n2\n1\n1\n(\n)\n(\n)\n2\n(\n) (\n)\n\u2013\na\nx\na\nx\na\na\nx\na\nx\na\nx\n\uf8ee\n\uf8f9\n+\n+\n\u2212\n=\n\uf8ef\n\uf8fa\n+\n\u2212\n\uf8f0\n\uf8fb  = 1\n1\n1\n2a\na\nx\na\nx\n\uf8ee\n\uf8f9\n+\n\uf8ef\n\uf8fa\n\u2212\n+\n\uf8f0\n\uf8fb\nINTEGRALS         309\n      Therefore,\n2\n\u20132\ndx\na\nx\n\u222b\n = \n1\n2\ndx\ndx\na\na\nx\na\nx\n\uf8ee\n\uf8f9\n+\n\uf8ef\n\uf8fa\n\u2212\n+\n\uf8f0\n\uf8fb\n\u222b\n\u222b\n= 1 [ log |\n|\nlog |\n|]\nC\n2\na\nx\na\nx\na \u2212\n\u2212\n+\n+\n+\n= \n1 log\nC\n2\na\nx\na\na\n+x\n+\n\u2212\n\ufffdNote  The technique used in (1) will be explained in Section 7 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "3556-3559",
        "Text": "(12\n)\nequals\ncos (\n)\nx\nx\ne\nx\n+e xdx\n\u222b\n(A) \u2013 cot (exx) + C\n(B) tan (xex) + C\n(C) tan (ex) + C\n(D) cot (ex) + C\n7 4  Integrals of Some Particular Functions\nIn this section, we mention below some important formulae of integrals and apply them\nfor integrating many other related standard integrals:\n(1) \u222b\n2\n2\n1\n\u2013\n=\nlog\n+ C\n2\n+\ndx\u2013\nx\na\na\nx\na\nx\na\n308\nMATHEMATICS\n(2) \u222b\n2\n2\n1\n+\n=\nlog\n+ C\n2\n\u2013\ndx\u2013\na\nx\na\na\nx\na\nx\n(3) \u222b\n\u2013 1\n2\n2\n1 tan\nC\ndx\nx\n=\n+\na\na\nx + a\n(4) \u222b\n2\n2\n2\n2 = log\n+\n\u2013\n+C\n\u2013\ndx\nx\nx\na\nx\na\n(5) \u222b\n\u2013 1\n2\n2 = sin\n+ C\n\u2013\ndx\nax\na\nx\n(6) \u222b\n2\n2\n2\n2 = log\n+\n+\n+ C\n+\ndx\nx\nx\na\nx\na\nWe now prove the above results:\n(1)\nWe have  \n2\n12\n1\n(\n) (\n)\nx \u2013 a\nx\na\nx \u2013 a\n=\n+\n= \n1\n(\n) \u2013 (\n)\n1\n1\n1\n2\n(\n) (\n)\n2\nx\na\nx \u2013 a\n\u2013\na\nx \u2013 a\nx\na\na\nx \u2013 a\nx\na\n\uf8ee\n\uf8f9\n+\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n+\n+\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nTherefore,  \n2\n2\n1\n2\ndx\ndx\ndx\n\u2013\na\nx \u2013 a\nx\na\nx \u2013 a\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa\n+\n\uf8f0\n\uf8fb\n\u222b\n\u222b\n\u222b\n= \n[\n]\n1\nlog (\n)|\nlog (\n)|\nC\n2\n| x \u2013 a \u2013\n| x\na\na\n+\n+\n= 1 log\nC\n2\nx \u2013 a\na\nx\na +\n+\n(2)\nIn view of (1) above, we have\n2\n2\n1\n1\n(\n)\n(\n)\n2\n(\n) (\n)\n\u2013\na\nx\na\nx\na\na\nx\na\nx\na\nx\n\uf8ee\n\uf8f9\n+\n+\n\u2212\n=\n\uf8ef\n\uf8fa\n+\n\u2212\n\uf8f0\n\uf8fb  = 1\n1\n1\n2a\na\nx\na\nx\n\uf8ee\n\uf8f9\n+\n\uf8ef\n\uf8fa\n\u2212\n+\n\uf8f0\n\uf8fb\nINTEGRALS         309\n      Therefore,\n2\n\u20132\ndx\na\nx\n\u222b\n = \n1\n2\ndx\ndx\na\na\nx\na\nx\n\uf8ee\n\uf8f9\n+\n\uf8ef\n\uf8fa\n\u2212\n+\n\uf8f0\n\uf8fb\n\u222b\n\u222b\n= 1 [ log |\n|\nlog |\n|]\nC\n2\na\nx\na\nx\na \u2212\n\u2212\n+\n+\n+\n= \n1 log\nC\n2\na\nx\na\na\n+x\n+\n\u2212\n\ufffdNote  The technique used in (1) will be explained in Section 7 5 (3) Put x = a tan \u03b8"
    },
    {
        "Chapter": "1",
        "sentence_range": "3557-3560",
        "Text": "4  Integrals of Some Particular Functions\nIn this section, we mention below some important formulae of integrals and apply them\nfor integrating many other related standard integrals:\n(1) \u222b\n2\n2\n1\n\u2013\n=\nlog\n+ C\n2\n+\ndx\u2013\nx\na\na\nx\na\nx\na\n308\nMATHEMATICS\n(2) \u222b\n2\n2\n1\n+\n=\nlog\n+ C\n2\n\u2013\ndx\u2013\na\nx\na\na\nx\na\nx\n(3) \u222b\n\u2013 1\n2\n2\n1 tan\nC\ndx\nx\n=\n+\na\na\nx + a\n(4) \u222b\n2\n2\n2\n2 = log\n+\n\u2013\n+C\n\u2013\ndx\nx\nx\na\nx\na\n(5) \u222b\n\u2013 1\n2\n2 = sin\n+ C\n\u2013\ndx\nax\na\nx\n(6) \u222b\n2\n2\n2\n2 = log\n+\n+\n+ C\n+\ndx\nx\nx\na\nx\na\nWe now prove the above results:\n(1)\nWe have  \n2\n12\n1\n(\n) (\n)\nx \u2013 a\nx\na\nx \u2013 a\n=\n+\n= \n1\n(\n) \u2013 (\n)\n1\n1\n1\n2\n(\n) (\n)\n2\nx\na\nx \u2013 a\n\u2013\na\nx \u2013 a\nx\na\na\nx \u2013 a\nx\na\n\uf8ee\n\uf8f9\n+\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n+\n+\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nTherefore,  \n2\n2\n1\n2\ndx\ndx\ndx\n\u2013\na\nx \u2013 a\nx\na\nx \u2013 a\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa\n+\n\uf8f0\n\uf8fb\n\u222b\n\u222b\n\u222b\n= \n[\n]\n1\nlog (\n)|\nlog (\n)|\nC\n2\n| x \u2013 a \u2013\n| x\na\na\n+\n+\n= 1 log\nC\n2\nx \u2013 a\na\nx\na +\n+\n(2)\nIn view of (1) above, we have\n2\n2\n1\n1\n(\n)\n(\n)\n2\n(\n) (\n)\n\u2013\na\nx\na\nx\na\na\nx\na\nx\na\nx\n\uf8ee\n\uf8f9\n+\n+\n\u2212\n=\n\uf8ef\n\uf8fa\n+\n\u2212\n\uf8f0\n\uf8fb  = 1\n1\n1\n2a\na\nx\na\nx\n\uf8ee\n\uf8f9\n+\n\uf8ef\n\uf8fa\n\u2212\n+\n\uf8f0\n\uf8fb\nINTEGRALS         309\n      Therefore,\n2\n\u20132\ndx\na\nx\n\u222b\n = \n1\n2\ndx\ndx\na\na\nx\na\nx\n\uf8ee\n\uf8f9\n+\n\uf8ef\n\uf8fa\n\u2212\n+\n\uf8f0\n\uf8fb\n\u222b\n\u222b\n= 1 [ log |\n|\nlog |\n|]\nC\n2\na\nx\na\nx\na \u2212\n\u2212\n+\n+\n+\n= \n1 log\nC\n2\na\nx\na\na\n+x\n+\n\u2212\n\ufffdNote  The technique used in (1) will be explained in Section 7 5 (3) Put x = a tan \u03b8 Then dx = a sec2 \u03b8 d\u03b8"
    },
    {
        "Chapter": "1",
        "sentence_range": "3558-3561",
        "Text": "5 (3) Put x = a tan \u03b8 Then dx = a sec2 \u03b8 d\u03b8 Therefore,      \n2\n2\ndx\nx\n+a\n\u222b\n = \n2\n2\n2\n2\n\u03b8\n\u03b8\nsec\u03b8\ntan\na\nd\na\n+a\n\u222b\n=\n1\n1\n1\n1\n\u03b8\n\u03b8\nC\ntan\nC\n\u2013 x\nd\na\na\na\na\n=\n+\n=\n+\n\u222b\n(4) Let x = a sec\u03b8"
    },
    {
        "Chapter": "1",
        "sentence_range": "3559-3562",
        "Text": "(3) Put x = a tan \u03b8 Then dx = a sec2 \u03b8 d\u03b8 Therefore,      \n2\n2\ndx\nx\n+a\n\u222b\n = \n2\n2\n2\n2\n\u03b8\n\u03b8\nsec\u03b8\ntan\na\nd\na\n+a\n\u222b\n=\n1\n1\n1\n1\n\u03b8\n\u03b8\nC\ntan\nC\n\u2013 x\nd\na\na\na\na\n=\n+\n=\n+\n\u222b\n(4) Let x = a sec\u03b8 Then dx = a sec\u03b8 tan \u03b8 d \u03b8"
    },
    {
        "Chapter": "1",
        "sentence_range": "3560-3563",
        "Text": "Then dx = a sec2 \u03b8 d\u03b8 Therefore,      \n2\n2\ndx\nx\n+a\n\u222b\n = \n2\n2\n2\n2\n\u03b8\n\u03b8\nsec\u03b8\ntan\na\nd\na\n+a\n\u222b\n=\n1\n1\n1\n1\n\u03b8\n\u03b8\nC\ntan\nC\n\u2013 x\nd\na\na\na\na\n=\n+\n=\n+\n\u222b\n(4) Let x = a sec\u03b8 Then dx = a sec\u03b8 tan \u03b8 d \u03b8 Therefore,\n2\n2\ndx\nx\n\u2212a\n\u222b\n =\n2\n2\n2\nsec\u03b8 tan\u03b8 \u03b8\nsec \u03b8\na\nd\na\n\u2212a\n\u222b\n=\n1\nsec\u03b8 \u03b8\nd =log sec\u03b8 + tan\u03b8 + C\n\u222b\n=\n2\n1\n2\nlog\n1\nC\nx\nx \u2013\na\na\n+\n+\n=\n2\n2\n1\nlog\nlog\nC\nx\nx \u2013 a\na\n+\n\u2212\n+\n=\n2\n2\nlog\n+ C\nx\nx \u2013 a\n+\n, where C = C1 \u2013 log |a|\n(5) Let x = a sin\u03b8"
    },
    {
        "Chapter": "1",
        "sentence_range": "3561-3564",
        "Text": "Therefore,      \n2\n2\ndx\nx\n+a\n\u222b\n = \n2\n2\n2\n2\n\u03b8\n\u03b8\nsec\u03b8\ntan\na\nd\na\n+a\n\u222b\n=\n1\n1\n1\n1\n\u03b8\n\u03b8\nC\ntan\nC\n\u2013 x\nd\na\na\na\na\n=\n+\n=\n+\n\u222b\n(4) Let x = a sec\u03b8 Then dx = a sec\u03b8 tan \u03b8 d \u03b8 Therefore,\n2\n2\ndx\nx\n\u2212a\n\u222b\n =\n2\n2\n2\nsec\u03b8 tan\u03b8 \u03b8\nsec \u03b8\na\nd\na\n\u2212a\n\u222b\n=\n1\nsec\u03b8 \u03b8\nd =log sec\u03b8 + tan\u03b8 + C\n\u222b\n=\n2\n1\n2\nlog\n1\nC\nx\nx \u2013\na\na\n+\n+\n=\n2\n2\n1\nlog\nlog\nC\nx\nx \u2013 a\na\n+\n\u2212\n+\n=\n2\n2\nlog\n+ C\nx\nx \u2013 a\n+\n, where C = C1 \u2013 log |a|\n(5) Let x = a sin\u03b8 Then dx = a cos\u03b8 d\u03b8"
    },
    {
        "Chapter": "1",
        "sentence_range": "3562-3565",
        "Text": "Then dx = a sec\u03b8 tan \u03b8 d \u03b8 Therefore,\n2\n2\ndx\nx\n\u2212a\n\u222b\n =\n2\n2\n2\nsec\u03b8 tan\u03b8 \u03b8\nsec \u03b8\na\nd\na\n\u2212a\n\u222b\n=\n1\nsec\u03b8 \u03b8\nd =log sec\u03b8 + tan\u03b8 + C\n\u222b\n=\n2\n1\n2\nlog\n1\nC\nx\nx \u2013\na\na\n+\n+\n=\n2\n2\n1\nlog\nlog\nC\nx\nx \u2013 a\na\n+\n\u2212\n+\n=\n2\n2\nlog\n+ C\nx\nx \u2013 a\n+\n, where C = C1 \u2013 log |a|\n(5) Let x = a sin\u03b8 Then dx = a cos\u03b8 d\u03b8 Therefore,\n \n2\n2\ndx\na\n\u2212x\n\u222b\n =\n2\n2\n2\n\u03b8 \u03b8\n\u03b8\ncos\nsin\na\nd\na \u2013 a\n\u222b\n=\n\u03b8 = \u03b8 + C = sin1\nC\n\u2013\nx\nd\na\n+\n\u222b\n(6) Let x = a tan\u03b8"
    },
    {
        "Chapter": "1",
        "sentence_range": "3563-3566",
        "Text": "Therefore,\n2\n2\ndx\nx\n\u2212a\n\u222b\n =\n2\n2\n2\nsec\u03b8 tan\u03b8 \u03b8\nsec \u03b8\na\nd\na\n\u2212a\n\u222b\n=\n1\nsec\u03b8 \u03b8\nd =log sec\u03b8 + tan\u03b8 + C\n\u222b\n=\n2\n1\n2\nlog\n1\nC\nx\nx \u2013\na\na\n+\n+\n=\n2\n2\n1\nlog\nlog\nC\nx\nx \u2013 a\na\n+\n\u2212\n+\n=\n2\n2\nlog\n+ C\nx\nx \u2013 a\n+\n, where C = C1 \u2013 log |a|\n(5) Let x = a sin\u03b8 Then dx = a cos\u03b8 d\u03b8 Therefore,\n \n2\n2\ndx\na\n\u2212x\n\u222b\n =\n2\n2\n2\n\u03b8 \u03b8\n\u03b8\ncos\nsin\na\nd\na \u2013 a\n\u222b\n=\n\u03b8 = \u03b8 + C = sin1\nC\n\u2013\nx\nd\na\n+\n\u222b\n(6) Let x = a tan\u03b8 Then dx = a sec2\u03b8 d\u03b8"
    },
    {
        "Chapter": "1",
        "sentence_range": "3564-3567",
        "Text": "Then dx = a cos\u03b8 d\u03b8 Therefore,\n \n2\n2\ndx\na\n\u2212x\n\u222b\n =\n2\n2\n2\n\u03b8 \u03b8\n\u03b8\ncos\nsin\na\nd\na \u2013 a\n\u222b\n=\n\u03b8 = \u03b8 + C = sin1\nC\n\u2013\nx\nd\na\n+\n\u222b\n(6) Let x = a tan\u03b8 Then dx = a sec2\u03b8 d\u03b8 Therefore,\n2\n2\ndx\nx\n+a\n\u222b\n =\n2\n2\n2\n2\n\u03b8\n\u03b8\n\u03b8\nsec\ntan\na\nd\na\n+a\n\u222b\n  =\n1\n\u03b8\n\u03b8\nsec\u03b8 \u03b8 = log (sec\ntan )\nC\nd\n+\n+\n\u222b\n310\nMATHEMATICS\n=\n2\n1\n2\nlog\n1\nC\nx\nx\na\na\n+\n+\n+\n=\n2\n1\nlog\nlog\nC\nx\nx\na\n|a|\n2\n+\n+\n\u2212\n+\n=\n2\nlog\nC\nx\nx\na2\n+\n+\n+\n, where C = C1 \u2013 log |a|\nApplying these standard formulae, we now obtain some more formulae which\nare useful from applications point of view and can be applied directly to evaluate\nother integrals"
    },
    {
        "Chapter": "1",
        "sentence_range": "3565-3568",
        "Text": "Therefore,\n \n2\n2\ndx\na\n\u2212x\n\u222b\n =\n2\n2\n2\n\u03b8 \u03b8\n\u03b8\ncos\nsin\na\nd\na \u2013 a\n\u222b\n=\n\u03b8 = \u03b8 + C = sin1\nC\n\u2013\nx\nd\na\n+\n\u222b\n(6) Let x = a tan\u03b8 Then dx = a sec2\u03b8 d\u03b8 Therefore,\n2\n2\ndx\nx\n+a\n\u222b\n =\n2\n2\n2\n2\n\u03b8\n\u03b8\n\u03b8\nsec\ntan\na\nd\na\n+a\n\u222b\n  =\n1\n\u03b8\n\u03b8\nsec\u03b8 \u03b8 = log (sec\ntan )\nC\nd\n+\n+\n\u222b\n310\nMATHEMATICS\n=\n2\n1\n2\nlog\n1\nC\nx\nx\na\na\n+\n+\n+\n=\n2\n1\nlog\nlog\nC\nx\nx\na\n|a|\n2\n+\n+\n\u2212\n+\n=\n2\nlog\nC\nx\nx\na2\n+\n+\n+\n, where C = C1 \u2013 log |a|\nApplying these standard formulae, we now obtain some more formulae which\nare useful from applications point of view and can be applied directly to evaluate\nother integrals (7)\nTo find the integral \n2\ndx\nax\nbx\nc\n+\n+\n\u222b\n, we write\nax 2 + bx + c = \n2\n2\n2\n2\n2\n4\nb\nc\nb\nc\nb\na x\nx\na\nx\n\u2013\na\na\na\na\na\n\uf8ee\n\uf8f9\n\uf8eb\n\uf8f6\n\uf8ee\n\uf8f9\n\uf8eb\n\uf8f6\n+\n+\n=\n+\n+\n\uf8ef\n\uf8fa\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8ed\n\uf8f8\n\uf8ef\n\uf8fa\n\uf8ed\n\uf8f8\n\uf8f0\n\uf8fb\nNow, put \nb2\nx\nt\na\n+\n= so that dx = dt and writing \n2\n2\n42\nc\n\u2013b\nk\na\na\n= \u00b1"
    },
    {
        "Chapter": "1",
        "sentence_range": "3566-3569",
        "Text": "Then dx = a sec2\u03b8 d\u03b8 Therefore,\n2\n2\ndx\nx\n+a\n\u222b\n =\n2\n2\n2\n2\n\u03b8\n\u03b8\n\u03b8\nsec\ntan\na\nd\na\n+a\n\u222b\n  =\n1\n\u03b8\n\u03b8\nsec\u03b8 \u03b8 = log (sec\ntan )\nC\nd\n+\n+\n\u222b\n310\nMATHEMATICS\n=\n2\n1\n2\nlog\n1\nC\nx\nx\na\na\n+\n+\n+\n=\n2\n1\nlog\nlog\nC\nx\nx\na\n|a|\n2\n+\n+\n\u2212\n+\n=\n2\nlog\nC\nx\nx\na2\n+\n+\n+\n, where C = C1 \u2013 log |a|\nApplying these standard formulae, we now obtain some more formulae which\nare useful from applications point of view and can be applied directly to evaluate\nother integrals (7)\nTo find the integral \n2\ndx\nax\nbx\nc\n+\n+\n\u222b\n, we write\nax 2 + bx + c = \n2\n2\n2\n2\n2\n4\nb\nc\nb\nc\nb\na x\nx\na\nx\n\u2013\na\na\na\na\na\n\uf8ee\n\uf8f9\n\uf8eb\n\uf8f6\n\uf8ee\n\uf8f9\n\uf8eb\n\uf8f6\n+\n+\n=\n+\n+\n\uf8ef\n\uf8fa\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8ed\n\uf8f8\n\uf8ef\n\uf8fa\n\uf8ed\n\uf8f8\n\uf8f0\n\uf8fb\nNow, put \nb2\nx\nt\na\n+\n= so that dx = dt and writing \n2\n2\n42\nc\n\u2013b\nk\na\na\n= \u00b1 We find the\nintegral reduced to the form \n2\n2\n1\ndt\na\nt\n\u222b\u00b1k\n depending upon the sign of \n2\n42\nc\na\u2013b\na\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\nand hence can be evaluated"
    },
    {
        "Chapter": "1",
        "sentence_range": "3567-3570",
        "Text": "Therefore,\n2\n2\ndx\nx\n+a\n\u222b\n =\n2\n2\n2\n2\n\u03b8\n\u03b8\n\u03b8\nsec\ntan\na\nd\na\n+a\n\u222b\n  =\n1\n\u03b8\n\u03b8\nsec\u03b8 \u03b8 = log (sec\ntan )\nC\nd\n+\n+\n\u222b\n310\nMATHEMATICS\n=\n2\n1\n2\nlog\n1\nC\nx\nx\na\na\n+\n+\n+\n=\n2\n1\nlog\nlog\nC\nx\nx\na\n|a|\n2\n+\n+\n\u2212\n+\n=\n2\nlog\nC\nx\nx\na2\n+\n+\n+\n, where C = C1 \u2013 log |a|\nApplying these standard formulae, we now obtain some more formulae which\nare useful from applications point of view and can be applied directly to evaluate\nother integrals (7)\nTo find the integral \n2\ndx\nax\nbx\nc\n+\n+\n\u222b\n, we write\nax 2 + bx + c = \n2\n2\n2\n2\n2\n4\nb\nc\nb\nc\nb\na x\nx\na\nx\n\u2013\na\na\na\na\na\n\uf8ee\n\uf8f9\n\uf8eb\n\uf8f6\n\uf8ee\n\uf8f9\n\uf8eb\n\uf8f6\n+\n+\n=\n+\n+\n\uf8ef\n\uf8fa\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8ed\n\uf8f8\n\uf8ef\n\uf8fa\n\uf8ed\n\uf8f8\n\uf8f0\n\uf8fb\nNow, put \nb2\nx\nt\na\n+\n= so that dx = dt and writing \n2\n2\n42\nc\n\u2013b\nk\na\na\n= \u00b1 We find the\nintegral reduced to the form \n2\n2\n1\ndt\na\nt\n\u222b\u00b1k\n depending upon the sign of \n2\n42\nc\na\u2013b\na\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\nand hence can be evaluated (8)\nTo find the integral of the type \n2\ndx\nax\nbx\nc\n+\n+\n\u222b\n, proceeding as in (7), we\nobtain the integral using the standard formulae"
    },
    {
        "Chapter": "1",
        "sentence_range": "3568-3571",
        "Text": "(7)\nTo find the integral \n2\ndx\nax\nbx\nc\n+\n+\n\u222b\n, we write\nax 2 + bx + c = \n2\n2\n2\n2\n2\n4\nb\nc\nb\nc\nb\na x\nx\na\nx\n\u2013\na\na\na\na\na\n\uf8ee\n\uf8f9\n\uf8eb\n\uf8f6\n\uf8ee\n\uf8f9\n\uf8eb\n\uf8f6\n+\n+\n=\n+\n+\n\uf8ef\n\uf8fa\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8ed\n\uf8f8\n\uf8ef\n\uf8fa\n\uf8ed\n\uf8f8\n\uf8f0\n\uf8fb\nNow, put \nb2\nx\nt\na\n+\n= so that dx = dt and writing \n2\n2\n42\nc\n\u2013b\nk\na\na\n= \u00b1 We find the\nintegral reduced to the form \n2\n2\n1\ndt\na\nt\n\u222b\u00b1k\n depending upon the sign of \n2\n42\nc\na\u2013b\na\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\nand hence can be evaluated (8)\nTo find the integral of the type \n2\ndx\nax\nbx\nc\n+\n+\n\u222b\n, proceeding as in (7), we\nobtain the integral using the standard formulae (9)\nTo find the integral of the type \n2\npx\nq\ndx\nax\nbx\nc\n++\n+\n\u222b\n, where p, q, a, b, c are\nconstants, we are to find real numbers A, B such that\n2\n+\n= A\n(\n) + B = A (2\n) + B\nd\npx\nq\nax\nbx\nc\nax\nb\ndx\n+\n+\n+\nTo determine A and B, we equate from both sides the coefficients of x and the\nconstant terms"
    },
    {
        "Chapter": "1",
        "sentence_range": "3569-3572",
        "Text": "We find the\nintegral reduced to the form \n2\n2\n1\ndt\na\nt\n\u222b\u00b1k\n depending upon the sign of \n2\n42\nc\na\u2013b\na\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\nand hence can be evaluated (8)\nTo find the integral of the type \n2\ndx\nax\nbx\nc\n+\n+\n\u222b\n, proceeding as in (7), we\nobtain the integral using the standard formulae (9)\nTo find the integral of the type \n2\npx\nq\ndx\nax\nbx\nc\n++\n+\n\u222b\n, where p, q, a, b, c are\nconstants, we are to find real numbers A, B such that\n2\n+\n= A\n(\n) + B = A (2\n) + B\nd\npx\nq\nax\nbx\nc\nax\nb\ndx\n+\n+\n+\nTo determine A and B, we equate from both sides the coefficients of x and the\nconstant terms A and B are thus obtained and hence the integral is reduced to\none of the known forms"
    },
    {
        "Chapter": "1",
        "sentence_range": "3570-3573",
        "Text": "(8)\nTo find the integral of the type \n2\ndx\nax\nbx\nc\n+\n+\n\u222b\n, proceeding as in (7), we\nobtain the integral using the standard formulae (9)\nTo find the integral of the type \n2\npx\nq\ndx\nax\nbx\nc\n++\n+\n\u222b\n, where p, q, a, b, c are\nconstants, we are to find real numbers A, B such that\n2\n+\n= A\n(\n) + B = A (2\n) + B\nd\npx\nq\nax\nbx\nc\nax\nb\ndx\n+\n+\n+\nTo determine A and B, we equate from both sides the coefficients of x and the\nconstant terms A and B are thus obtained and hence the integral is reduced to\none of the known forms INTEGRALS         311\n(10) For the evaluation of the integral of the type \n2\n(\n)\npx\nq dx\nax\nbx\nc\n+\n+\n+\n\u222b\n, we proceed\nas in (9) and transform the integral into known standard forms"
    },
    {
        "Chapter": "1",
        "sentence_range": "3571-3574",
        "Text": "(9)\nTo find the integral of the type \n2\npx\nq\ndx\nax\nbx\nc\n++\n+\n\u222b\n, where p, q, a, b, c are\nconstants, we are to find real numbers A, B such that\n2\n+\n= A\n(\n) + B = A (2\n) + B\nd\npx\nq\nax\nbx\nc\nax\nb\ndx\n+\n+\n+\nTo determine A and B, we equate from both sides the coefficients of x and the\nconstant terms A and B are thus obtained and hence the integral is reduced to\none of the known forms INTEGRALS         311\n(10) For the evaluation of the integral of the type \n2\n(\n)\npx\nq dx\nax\nbx\nc\n+\n+\n+\n\u222b\n, we proceed\nas in (9) and transform the integral into known standard forms Let us illustrate the above methods by some examples"
    },
    {
        "Chapter": "1",
        "sentence_range": "3572-3575",
        "Text": "A and B are thus obtained and hence the integral is reduced to\none of the known forms INTEGRALS         311\n(10) For the evaluation of the integral of the type \n2\n(\n)\npx\nq dx\nax\nbx\nc\n+\n+\n+\n\u222b\n, we proceed\nas in (9) and transform the integral into known standard forms Let us illustrate the above methods by some examples Example 8 Find the following integrals:\n(i)\n2\n16\ndx\n\u222bx \u2212\n(ii)\n2\n2\ndx\nx\n\u2212x\n\u222b\nSolution\n(i)\nWe have \n2\n2\n2\n16\n4\ndx\ndx\nx\nx \u2013\n=\n\u2212\n\u222b\n\u222b\n = \n4\nlog\nC\n8\nxx \u20134\n1\n+\n+\n[by 7"
    },
    {
        "Chapter": "1",
        "sentence_range": "3573-3576",
        "Text": "INTEGRALS         311\n(10) For the evaluation of the integral of the type \n2\n(\n)\npx\nq dx\nax\nbx\nc\n+\n+\n+\n\u222b\n, we proceed\nas in (9) and transform the integral into known standard forms Let us illustrate the above methods by some examples Example 8 Find the following integrals:\n(i)\n2\n16\ndx\n\u222bx \u2212\n(ii)\n2\n2\ndx\nx\n\u2212x\n\u222b\nSolution\n(i)\nWe have \n2\n2\n2\n16\n4\ndx\ndx\nx\nx \u2013\n=\n\u2212\n\u222b\n\u222b\n = \n4\nlog\nC\n8\nxx \u20134\n1\n+\n+\n[by 7 4 (1)]\n(ii)\n(\n)\n2\n2\n2\n1\n1\n=\n\u2212\n\u222b\n\u222b\ndx\ndx\nx\nx\n\u2013 x \u2013\nPut x \u2013 1 = t"
    },
    {
        "Chapter": "1",
        "sentence_range": "3574-3577",
        "Text": "Let us illustrate the above methods by some examples Example 8 Find the following integrals:\n(i)\n2\n16\ndx\n\u222bx \u2212\n(ii)\n2\n2\ndx\nx\n\u2212x\n\u222b\nSolution\n(i)\nWe have \n2\n2\n2\n16\n4\ndx\ndx\nx\nx \u2013\n=\n\u2212\n\u222b\n\u222b\n = \n4\nlog\nC\n8\nxx \u20134\n1\n+\n+\n[by 7 4 (1)]\n(ii)\n(\n)\n2\n2\n2\n1\n1\n=\n\u2212\n\u222b\n\u222b\ndx\ndx\nx\nx\n\u2013 x \u2013\nPut x \u2013 1 = t Then dx = dt"
    },
    {
        "Chapter": "1",
        "sentence_range": "3575-3578",
        "Text": "Example 8 Find the following integrals:\n(i)\n2\n16\ndx\n\u222bx \u2212\n(ii)\n2\n2\ndx\nx\n\u2212x\n\u222b\nSolution\n(i)\nWe have \n2\n2\n2\n16\n4\ndx\ndx\nx\nx \u2013\n=\n\u2212\n\u222b\n\u222b\n = \n4\nlog\nC\n8\nxx \u20134\n1\n+\n+\n[by 7 4 (1)]\n(ii)\n(\n)\n2\n2\n2\n1\n1\n=\n\u2212\n\u222b\n\u222b\ndx\ndx\nx\nx\n\u2013 x \u2013\nPut x \u2013 1 = t Then dx = dt Therefore,\n2\n2\ndx\nx\n\u2212x\n\u222b\n =\n2\n1\ndt\n\u2013 t\n\u222b\n = \nsin1\n( )\nC\n\u2013\nt +\n[by 7"
    },
    {
        "Chapter": "1",
        "sentence_range": "3576-3579",
        "Text": "4 (1)]\n(ii)\n(\n)\n2\n2\n2\n1\n1\n=\n\u2212\n\u222b\n\u222b\ndx\ndx\nx\nx\n\u2013 x \u2013\nPut x \u2013 1 = t Then dx = dt Therefore,\n2\n2\ndx\nx\n\u2212x\n\u222b\n =\n2\n1\ndt\n\u2013 t\n\u222b\n = \nsin1\n( )\nC\n\u2013\nt +\n[by 7 4 (5)]\n=\nsin1\n( \u20131)\nC\n\u2013\nx\n+\nExample 9 Find the following integrals :\n(i)\n2\n6\n13\ndx\nx\n\u2212x\n+\n\u222b\n(ii)\n32\n13\n10\ndx\nx\nx\n+\n\u2212\n\u222b\n(iii)\n52\n2\ndx\nx\nx\n\u2212\n\u222b\nSolution\n(i)\nWe have  x2 \u2013 6x + 13 = x2 \u2013 6x + 32 \u2013 32 + 13 = (x \u2013 3)2 + 4\nSo,\n6\n13\ndx\nx\n2 \u2212x\n+\n\u222b\n =\n(\n)\n2\n2\n1\n3\n2\ndx\nx \u2013\n+\n\u222b\nLet\nx \u2013 3 = t"
    },
    {
        "Chapter": "1",
        "sentence_range": "3577-3580",
        "Text": "Then dx = dt Therefore,\n2\n2\ndx\nx\n\u2212x\n\u222b\n =\n2\n1\ndt\n\u2013 t\n\u222b\n = \nsin1\n( )\nC\n\u2013\nt +\n[by 7 4 (5)]\n=\nsin1\n( \u20131)\nC\n\u2013\nx\n+\nExample 9 Find the following integrals :\n(i)\n2\n6\n13\ndx\nx\n\u2212x\n+\n\u222b\n(ii)\n32\n13\n10\ndx\nx\nx\n+\n\u2212\n\u222b\n(iii)\n52\n2\ndx\nx\nx\n\u2212\n\u222b\nSolution\n(i)\nWe have  x2 \u2013 6x + 13 = x2 \u2013 6x + 32 \u2013 32 + 13 = (x \u2013 3)2 + 4\nSo,\n6\n13\ndx\nx\n2 \u2212x\n+\n\u222b\n =\n(\n)\n2\n2\n1\n3\n2\ndx\nx \u2013\n+\n\u222b\nLet\nx \u2013 3 = t Then dx = dt\nTherefore,\n6\n13\ndx\nx\n2 \u2212x\n+\n\u222b\n =  \n1\n2\n2\n1 tan\nC\n2\n2\n2\n\u2013\ndt\nt\nt\n=\n+\n+\n\u222b\n[by 7"
    },
    {
        "Chapter": "1",
        "sentence_range": "3578-3581",
        "Text": "Therefore,\n2\n2\ndx\nx\n\u2212x\n\u222b\n =\n2\n1\ndt\n\u2013 t\n\u222b\n = \nsin1\n( )\nC\n\u2013\nt +\n[by 7 4 (5)]\n=\nsin1\n( \u20131)\nC\n\u2013\nx\n+\nExample 9 Find the following integrals :\n(i)\n2\n6\n13\ndx\nx\n\u2212x\n+\n\u222b\n(ii)\n32\n13\n10\ndx\nx\nx\n+\n\u2212\n\u222b\n(iii)\n52\n2\ndx\nx\nx\n\u2212\n\u222b\nSolution\n(i)\nWe have  x2 \u2013 6x + 13 = x2 \u2013 6x + 32 \u2013 32 + 13 = (x \u2013 3)2 + 4\nSo,\n6\n13\ndx\nx\n2 \u2212x\n+\n\u222b\n =\n(\n)\n2\n2\n1\n3\n2\ndx\nx \u2013\n+\n\u222b\nLet\nx \u2013 3 = t Then dx = dt\nTherefore,\n6\n13\ndx\nx\n2 \u2212x\n+\n\u222b\n =  \n1\n2\n2\n1 tan\nC\n2\n2\n2\n\u2013\ndt\nt\nt\n=\n+\n+\n\u222b\n[by 7 4 (3)]\n=\n1\n1\n3\ntan\nC\n2\n2\n\u2013\nx \u2013\n+\n312\nMATHEMATICS\n(ii)\nThe given integral is of the form 7"
    },
    {
        "Chapter": "1",
        "sentence_range": "3579-3582",
        "Text": "4 (5)]\n=\nsin1\n( \u20131)\nC\n\u2013\nx\n+\nExample 9 Find the following integrals :\n(i)\n2\n6\n13\ndx\nx\n\u2212x\n+\n\u222b\n(ii)\n32\n13\n10\ndx\nx\nx\n+\n\u2212\n\u222b\n(iii)\n52\n2\ndx\nx\nx\n\u2212\n\u222b\nSolution\n(i)\nWe have  x2 \u2013 6x + 13 = x2 \u2013 6x + 32 \u2013 32 + 13 = (x \u2013 3)2 + 4\nSo,\n6\n13\ndx\nx\n2 \u2212x\n+\n\u222b\n =\n(\n)\n2\n2\n1\n3\n2\ndx\nx \u2013\n+\n\u222b\nLet\nx \u2013 3 = t Then dx = dt\nTherefore,\n6\n13\ndx\nx\n2 \u2212x\n+\n\u222b\n =  \n1\n2\n2\n1 tan\nC\n2\n2\n2\n\u2013\ndt\nt\nt\n=\n+\n+\n\u222b\n[by 7 4 (3)]\n=\n1\n1\n3\ntan\nC\n2\n2\n\u2013\nx \u2013\n+\n312\nMATHEMATICS\n(ii)\nThe given integral is of the form 7 4 (7)"
    },
    {
        "Chapter": "1",
        "sentence_range": "3580-3583",
        "Text": "Then dx = dt\nTherefore,\n6\n13\ndx\nx\n2 \u2212x\n+\n\u222b\n =  \n1\n2\n2\n1 tan\nC\n2\n2\n2\n\u2013\ndt\nt\nt\n=\n+\n+\n\u222b\n[by 7 4 (3)]\n=\n1\n1\n3\ntan\nC\n2\n2\n\u2013\nx \u2013\n+\n312\nMATHEMATICS\n(ii)\nThe given integral is of the form 7 4 (7) We write the denominator of the integrand,\n32\n13\n10\nx\nx \u2013\n+\n =\n2\n13\n10\n3\n3\n3\nx\nx\n\u2013\n\uf8eb\n\uf8f6\n+\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n=\n2\n2\n13\n17\n3\n6\n6\nx\n\u2013\n\uf8ee\n\uf8f9\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n+\n\uf8ef\n\uf8fa\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n(completing the square)\nThus\n3\n13\n10\ndx\nx\nx\n2 +\n\u2212\n\u222b\n =\n2\n2\n31\n13\n17\n6\n6\ndx\n\uf8ebx\n\uf8f6\n\uf8eb\n\uf8f6\n+\n\u2212\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n\u222b\nPut \n13\n6\nx\nt\n+\n="
    },
    {
        "Chapter": "1",
        "sentence_range": "3581-3584",
        "Text": "4 (3)]\n=\n1\n1\n3\ntan\nC\n2\n2\n\u2013\nx \u2013\n+\n312\nMATHEMATICS\n(ii)\nThe given integral is of the form 7 4 (7) We write the denominator of the integrand,\n32\n13\n10\nx\nx \u2013\n+\n =\n2\n13\n10\n3\n3\n3\nx\nx\n\u2013\n\uf8eb\n\uf8f6\n+\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n=\n2\n2\n13\n17\n3\n6\n6\nx\n\u2013\n\uf8ee\n\uf8f9\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n+\n\uf8ef\n\uf8fa\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n(completing the square)\nThus\n3\n13\n10\ndx\nx\nx\n2 +\n\u2212\n\u222b\n =\n2\n2\n31\n13\n17\n6\n6\ndx\n\uf8ebx\n\uf8f6\n\uf8eb\n\uf8f6\n+\n\u2212\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n\u222b\nPut \n13\n6\nx\nt\n+\n= Then dx = dt"
    },
    {
        "Chapter": "1",
        "sentence_range": "3582-3585",
        "Text": "4 (7) We write the denominator of the integrand,\n32\n13\n10\nx\nx \u2013\n+\n =\n2\n13\n10\n3\n3\n3\nx\nx\n\u2013\n\uf8eb\n\uf8f6\n+\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n=\n2\n2\n13\n17\n3\n6\n6\nx\n\u2013\n\uf8ee\n\uf8f9\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n+\n\uf8ef\n\uf8fa\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n(completing the square)\nThus\n3\n13\n10\ndx\nx\nx\n2 +\n\u2212\n\u222b\n =\n2\n2\n31\n13\n17\n6\n6\ndx\n\uf8ebx\n\uf8f6\n\uf8eb\n\uf8f6\n+\n\u2212\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n\u222b\nPut \n13\n6\nx\nt\n+\n= Then dx = dt Therefore,\n3\n13\n10\ndx\nx\nx\n2 +\n\u2212\n\u222b\n =\n2\n2\n31\n17\n6\ndt\nt\n\uf8eb\n\uf8f6\n\u2212 \uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\u222b\n=\n1\n17\n1\n6\nlog\nC\n17\n17\n3\n2\n6\n6\nt \u2013\nt\n+\n\u00d7 \u00d7\n+\n[by 7"
    },
    {
        "Chapter": "1",
        "sentence_range": "3583-3586",
        "Text": "We write the denominator of the integrand,\n32\n13\n10\nx\nx \u2013\n+\n =\n2\n13\n10\n3\n3\n3\nx\nx\n\u2013\n\uf8eb\n\uf8f6\n+\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n=\n2\n2\n13\n17\n3\n6\n6\nx\n\u2013\n\uf8ee\n\uf8f9\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n+\n\uf8ef\n\uf8fa\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n(completing the square)\nThus\n3\n13\n10\ndx\nx\nx\n2 +\n\u2212\n\u222b\n =\n2\n2\n31\n13\n17\n6\n6\ndx\n\uf8ebx\n\uf8f6\n\uf8eb\n\uf8f6\n+\n\u2212\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n\u222b\nPut \n13\n6\nx\nt\n+\n= Then dx = dt Therefore,\n3\n13\n10\ndx\nx\nx\n2 +\n\u2212\n\u222b\n =\n2\n2\n31\n17\n6\ndt\nt\n\uf8eb\n\uf8f6\n\u2212 \uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\u222b\n=\n1\n17\n1\n6\nlog\nC\n17\n17\n3\n2\n6\n6\nt \u2013\nt\n+\n\u00d7 \u00d7\n+\n[by 7 4 (i)]\n=\n1\n13\n17\n1\n6\n6\nlog\nC\n13\n17\n17\n6\n6\nx\n\u2013\nx\n+\n+\n+\n+\n=\n1\n1\n6\n4\nlog\nC\n17\n6\nxx30\n\u2212\n+\n+\n=\n1\n1\n3\n2\n1\n1\nlog\nC\nlog\n17\n5\n17\n3\nxx\n\u2212\n+\n+\n+\n=\n1\n3\n2\nlog\nC\n17\nxx5\n\u2212\n+\n+\n, where C = \n1\n1\n1\nC\n17log\n3\n+\nINTEGRALS         313\n(iii)\nWe have \n2\n2\n5\n2\n5\n5\ndx\ndx\nx\nx\nx\nx \u2013\n2\n=\n\uf8eb\n\uf8f6\n\u2212\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\u222b\n\u222b\n=\n2\n2\n1\n5\n1\n1\n5\n5\ndx\nx \u2013\n\u2013\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n\u222b\n (completing the square)\nPut \n1\n5\nx \u2013\nt\n="
    },
    {
        "Chapter": "1",
        "sentence_range": "3584-3587",
        "Text": "Then dx = dt Therefore,\n3\n13\n10\ndx\nx\nx\n2 +\n\u2212\n\u222b\n =\n2\n2\n31\n17\n6\ndt\nt\n\uf8eb\n\uf8f6\n\u2212 \uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\u222b\n=\n1\n17\n1\n6\nlog\nC\n17\n17\n3\n2\n6\n6\nt \u2013\nt\n+\n\u00d7 \u00d7\n+\n[by 7 4 (i)]\n=\n1\n13\n17\n1\n6\n6\nlog\nC\n13\n17\n17\n6\n6\nx\n\u2013\nx\n+\n+\n+\n+\n=\n1\n1\n6\n4\nlog\nC\n17\n6\nxx30\n\u2212\n+\n+\n=\n1\n1\n3\n2\n1\n1\nlog\nC\nlog\n17\n5\n17\n3\nxx\n\u2212\n+\n+\n+\n=\n1\n3\n2\nlog\nC\n17\nxx5\n\u2212\n+\n+\n, where C = \n1\n1\n1\nC\n17log\n3\n+\nINTEGRALS         313\n(iii)\nWe have \n2\n2\n5\n2\n5\n5\ndx\ndx\nx\nx\nx\nx \u2013\n2\n=\n\uf8eb\n\uf8f6\n\u2212\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\u222b\n\u222b\n=\n2\n2\n1\n5\n1\n1\n5\n5\ndx\nx \u2013\n\u2013\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n\u222b\n (completing the square)\nPut \n1\n5\nx \u2013\nt\n= Then dx = dt"
    },
    {
        "Chapter": "1",
        "sentence_range": "3585-3588",
        "Text": "Therefore,\n3\n13\n10\ndx\nx\nx\n2 +\n\u2212\n\u222b\n =\n2\n2\n31\n17\n6\ndt\nt\n\uf8eb\n\uf8f6\n\u2212 \uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\u222b\n=\n1\n17\n1\n6\nlog\nC\n17\n17\n3\n2\n6\n6\nt \u2013\nt\n+\n\u00d7 \u00d7\n+\n[by 7 4 (i)]\n=\n1\n13\n17\n1\n6\n6\nlog\nC\n13\n17\n17\n6\n6\nx\n\u2013\nx\n+\n+\n+\n+\n=\n1\n1\n6\n4\nlog\nC\n17\n6\nxx30\n\u2212\n+\n+\n=\n1\n1\n3\n2\n1\n1\nlog\nC\nlog\n17\n5\n17\n3\nxx\n\u2212\n+\n+\n+\n=\n1\n3\n2\nlog\nC\n17\nxx5\n\u2212\n+\n+\n, where C = \n1\n1\n1\nC\n17log\n3\n+\nINTEGRALS         313\n(iii)\nWe have \n2\n2\n5\n2\n5\n5\ndx\ndx\nx\nx\nx\nx \u2013\n2\n=\n\uf8eb\n\uf8f6\n\u2212\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\u222b\n\u222b\n=\n2\n2\n1\n5\n1\n1\n5\n5\ndx\nx \u2013\n\u2013\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n\u222b\n (completing the square)\nPut \n1\n5\nx \u2013\nt\n= Then dx = dt Therefore,\n5\n2\ndx\nx\nx\n2 \u2212\n\u222b\n =\n2\n2\n1\n5\n1\n5\ndt\nt \u2013\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\u222b\n=\n2\n2\n1\n1\nlog\nC\n5\n5\nt\nt \u2013 \uf8eb\n\uf8f6\n+\n+\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n[by 7"
    },
    {
        "Chapter": "1",
        "sentence_range": "3586-3589",
        "Text": "4 (i)]\n=\n1\n13\n17\n1\n6\n6\nlog\nC\n13\n17\n17\n6\n6\nx\n\u2013\nx\n+\n+\n+\n+\n=\n1\n1\n6\n4\nlog\nC\n17\n6\nxx30\n\u2212\n+\n+\n=\n1\n1\n3\n2\n1\n1\nlog\nC\nlog\n17\n5\n17\n3\nxx\n\u2212\n+\n+\n+\n=\n1\n3\n2\nlog\nC\n17\nxx5\n\u2212\n+\n+\n, where C = \n1\n1\n1\nC\n17log\n3\n+\nINTEGRALS         313\n(iii)\nWe have \n2\n2\n5\n2\n5\n5\ndx\ndx\nx\nx\nx\nx \u2013\n2\n=\n\uf8eb\n\uf8f6\n\u2212\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\u222b\n\u222b\n=\n2\n2\n1\n5\n1\n1\n5\n5\ndx\nx \u2013\n\u2013\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n\u222b\n (completing the square)\nPut \n1\n5\nx \u2013\nt\n= Then dx = dt Therefore,\n5\n2\ndx\nx\nx\n2 \u2212\n\u222b\n =\n2\n2\n1\n5\n1\n5\ndt\nt \u2013\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\u222b\n=\n2\n2\n1\n1\nlog\nC\n5\n5\nt\nt \u2013 \uf8eb\n\uf8f6\n+\n+\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n[by 7 4 (4)]\n=\n2\n1\n1\n2\nlog\nC\n5\n5\n5\nx\nx \u2013\nx \u2013\n+\n+\nExample 10 Find the following integrals:\n(i)\n2\n2\n6\n5\nx\ndx\nx\nx\n2\n++\n+\n\u222b\n(ii)\n2\n3\n5\n4\nx\ndx\nx\nx\n+\n\u2212\n+\n\u222b\nSolution\n(i)\nUsing the formula 7"
    },
    {
        "Chapter": "1",
        "sentence_range": "3587-3590",
        "Text": "Then dx = dt Therefore,\n5\n2\ndx\nx\nx\n2 \u2212\n\u222b\n =\n2\n2\n1\n5\n1\n5\ndt\nt \u2013\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\u222b\n=\n2\n2\n1\n1\nlog\nC\n5\n5\nt\nt \u2013 \uf8eb\n\uf8f6\n+\n+\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n[by 7 4 (4)]\n=\n2\n1\n1\n2\nlog\nC\n5\n5\n5\nx\nx \u2013\nx \u2013\n+\n+\nExample 10 Find the following integrals:\n(i)\n2\n2\n6\n5\nx\ndx\nx\nx\n2\n++\n+\n\u222b\n(ii)\n2\n3\n5\n4\nx\ndx\nx\nx\n+\n\u2212\n+\n\u222b\nSolution\n(i)\nUsing the formula 7 4 (9), we express\nx + 2 = \n(\n)\n2\nA\n2\n6\n5\nB\nd\nx\nx\ndx\n+\n+\n+\n = A (4\n6)\nB\nx +\n+\nEquating the coefficients of x and the constant terms from both sides, we get\n4A = 1 and 6A + B = 2   or    A = 1\n4  and B = 1\n2"
    },
    {
        "Chapter": "1",
        "sentence_range": "3588-3591",
        "Text": "Therefore,\n5\n2\ndx\nx\nx\n2 \u2212\n\u222b\n =\n2\n2\n1\n5\n1\n5\ndt\nt \u2013\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\u222b\n=\n2\n2\n1\n1\nlog\nC\n5\n5\nt\nt \u2013 \uf8eb\n\uf8f6\n+\n+\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n[by 7 4 (4)]\n=\n2\n1\n1\n2\nlog\nC\n5\n5\n5\nx\nx \u2013\nx \u2013\n+\n+\nExample 10 Find the following integrals:\n(i)\n2\n2\n6\n5\nx\ndx\nx\nx\n2\n++\n+\n\u222b\n(ii)\n2\n3\n5\n4\nx\ndx\nx\nx\n+\n\u2212\n+\n\u222b\nSolution\n(i)\nUsing the formula 7 4 (9), we express\nx + 2 = \n(\n)\n2\nA\n2\n6\n5\nB\nd\nx\nx\ndx\n+\n+\n+\n = A (4\n6)\nB\nx +\n+\nEquating the coefficients of x and the constant terms from both sides, we get\n4A = 1 and 6A + B = 2   or    A = 1\n4  and B = 1\n2 Therefore,\n2\n2\n6\n5\nxx\nx\n2\n++\n+\n\u222b\n =\n1\n4\n6\n1\n4\n2\n2\n6\n5\n2\n6\n5\nx\ndx\ndx\nx\nx\nx\nx\n2\n2\n+\n+\n+\n+\n+\n+\n\u222b\n\u222b\n=\n1\n2\n1\n1\nI\nI\n4\n2\n+\n    (say)"
    },
    {
        "Chapter": "1",
        "sentence_range": "3589-3592",
        "Text": "4 (4)]\n=\n2\n1\n1\n2\nlog\nC\n5\n5\n5\nx\nx \u2013\nx \u2013\n+\n+\nExample 10 Find the following integrals:\n(i)\n2\n2\n6\n5\nx\ndx\nx\nx\n2\n++\n+\n\u222b\n(ii)\n2\n3\n5\n4\nx\ndx\nx\nx\n+\n\u2212\n+\n\u222b\nSolution\n(i)\nUsing the formula 7 4 (9), we express\nx + 2 = \n(\n)\n2\nA\n2\n6\n5\nB\nd\nx\nx\ndx\n+\n+\n+\n = A (4\n6)\nB\nx +\n+\nEquating the coefficients of x and the constant terms from both sides, we get\n4A = 1 and 6A + B = 2   or    A = 1\n4  and B = 1\n2 Therefore,\n2\n2\n6\n5\nxx\nx\n2\n++\n+\n\u222b\n =\n1\n4\n6\n1\n4\n2\n2\n6\n5\n2\n6\n5\nx\ndx\ndx\nx\nx\nx\nx\n2\n2\n+\n+\n+\n+\n+\n+\n\u222b\n\u222b\n=\n1\n2\n1\n1\nI\nI\n4\n2\n+\n    (say) (1)\n314\nMATHEMATICS\nIn I1, put 2x2 + 6x + 5 = t, so that (4x + 6) dx = dt\nTherefore,\nI1 =\n1\nlog\nC\ndt\nt\nt\n=\n+\n\u222b\n=\n2\n1\nlog | 2\n6\n5|\nC\nx\nx\n+\n+\n+"
    },
    {
        "Chapter": "1",
        "sentence_range": "3590-3593",
        "Text": "4 (9), we express\nx + 2 = \n(\n)\n2\nA\n2\n6\n5\nB\nd\nx\nx\ndx\n+\n+\n+\n = A (4\n6)\nB\nx +\n+\nEquating the coefficients of x and the constant terms from both sides, we get\n4A = 1 and 6A + B = 2   or    A = 1\n4  and B = 1\n2 Therefore,\n2\n2\n6\n5\nxx\nx\n2\n++\n+\n\u222b\n =\n1\n4\n6\n1\n4\n2\n2\n6\n5\n2\n6\n5\nx\ndx\ndx\nx\nx\nx\nx\n2\n2\n+\n+\n+\n+\n+\n+\n\u222b\n\u222b\n=\n1\n2\n1\n1\nI\nI\n4\n2\n+\n    (say) (1)\n314\nMATHEMATICS\nIn I1, put 2x2 + 6x + 5 = t, so that (4x + 6) dx = dt\nTherefore,\nI1 =\n1\nlog\nC\ndt\nt\nt\n=\n+\n\u222b\n=\n2\n1\nlog | 2\n6\n5|\nC\nx\nx\n+\n+\n+ (2)\nand\nI2 =\n2\n2\n1\n5\n2\n2\n6\n5\n3\n2\ndx\ndx\nx\nx\nx\nx\n=\n+\n+\n+\n+\n\u222b\n\u222b\n=\n2\n2\n21\n3\n1\n2\n2\ndx\n\uf8ebx\n\uf8f6\n\uf8eb\n\uf8f6\n+\n+\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n\u222b\nPut \n3\n2\nx\nt\n+\n= , so that dx = dt, we get\nI2 =\n2\n2\n21\n1\n2\ndt\nt\n\uf8eb\n\uf8f6\n+ \uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\u222b\n = \n1\n2\n1\ntan 2\nC\n1\n2\n2\n\u2013\nt +\n\u00d7\n[by 7"
    },
    {
        "Chapter": "1",
        "sentence_range": "3591-3594",
        "Text": "Therefore,\n2\n2\n6\n5\nxx\nx\n2\n++\n+\n\u222b\n =\n1\n4\n6\n1\n4\n2\n2\n6\n5\n2\n6\n5\nx\ndx\ndx\nx\nx\nx\nx\n2\n2\n+\n+\n+\n+\n+\n+\n\u222b\n\u222b\n=\n1\n2\n1\n1\nI\nI\n4\n2\n+\n    (say) (1)\n314\nMATHEMATICS\nIn I1, put 2x2 + 6x + 5 = t, so that (4x + 6) dx = dt\nTherefore,\nI1 =\n1\nlog\nC\ndt\nt\nt\n=\n+\n\u222b\n=\n2\n1\nlog | 2\n6\n5|\nC\nx\nx\n+\n+\n+ (2)\nand\nI2 =\n2\n2\n1\n5\n2\n2\n6\n5\n3\n2\ndx\ndx\nx\nx\nx\nx\n=\n+\n+\n+\n+\n\u222b\n\u222b\n=\n2\n2\n21\n3\n1\n2\n2\ndx\n\uf8ebx\n\uf8f6\n\uf8eb\n\uf8f6\n+\n+\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n\u222b\nPut \n3\n2\nx\nt\n+\n= , so that dx = dt, we get\nI2 =\n2\n2\n21\n1\n2\ndt\nt\n\uf8eb\n\uf8f6\n+ \uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\u222b\n = \n1\n2\n1\ntan 2\nC\n1\n2\n2\n\u2013\nt +\n\u00d7\n[by 7 4 (3)]\n=\n1\n2\n3\ntan 2\n+ C\n2\n\u2013\n\uf8ebx\n\uf8f6\n\uf8ec+\n\uf8f7\n\uf8ed\n\uf8f8\n = \n(\n)\n1\n2\ntan\n2\n3 + C\n\u2013\nx +"
    },
    {
        "Chapter": "1",
        "sentence_range": "3592-3595",
        "Text": "(1)\n314\nMATHEMATICS\nIn I1, put 2x2 + 6x + 5 = t, so that (4x + 6) dx = dt\nTherefore,\nI1 =\n1\nlog\nC\ndt\nt\nt\n=\n+\n\u222b\n=\n2\n1\nlog | 2\n6\n5|\nC\nx\nx\n+\n+\n+ (2)\nand\nI2 =\n2\n2\n1\n5\n2\n2\n6\n5\n3\n2\ndx\ndx\nx\nx\nx\nx\n=\n+\n+\n+\n+\n\u222b\n\u222b\n=\n2\n2\n21\n3\n1\n2\n2\ndx\n\uf8ebx\n\uf8f6\n\uf8eb\n\uf8f6\n+\n+\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n\u222b\nPut \n3\n2\nx\nt\n+\n= , so that dx = dt, we get\nI2 =\n2\n2\n21\n1\n2\ndt\nt\n\uf8eb\n\uf8f6\n+ \uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\u222b\n = \n1\n2\n1\ntan 2\nC\n1\n2\n2\n\u2013\nt +\n\u00d7\n[by 7 4 (3)]\n=\n1\n2\n3\ntan 2\n+ C\n2\n\u2013\n\uf8ebx\n\uf8f6\n\uf8ec+\n\uf8f7\n\uf8ed\n\uf8f8\n = \n(\n)\n1\n2\ntan\n2\n3 + C\n\u2013\nx + (3)\nUsing (2) and (3) in (1), we get\n(\n)\n2\n1\n2\n1\n1\nlog 2\n6\n5\ntan\n2\n3\nC\n4\n2\n2\n6\n5\n\u2013\nx\ndx\nx\nx\nx\nx\nx\n2\n+\n=\n+\n+\n+\n+\n+\n+\n+\n\u222b\nwhere,\nC =\n1\n2\nC\nC\n4\n2\n+\n(ii)\nThis integral is of the form given in 7"
    },
    {
        "Chapter": "1",
        "sentence_range": "3593-3596",
        "Text": "(2)\nand\nI2 =\n2\n2\n1\n5\n2\n2\n6\n5\n3\n2\ndx\ndx\nx\nx\nx\nx\n=\n+\n+\n+\n+\n\u222b\n\u222b\n=\n2\n2\n21\n3\n1\n2\n2\ndx\n\uf8ebx\n\uf8f6\n\uf8eb\n\uf8f6\n+\n+\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n\u222b\nPut \n3\n2\nx\nt\n+\n= , so that dx = dt, we get\nI2 =\n2\n2\n21\n1\n2\ndt\nt\n\uf8eb\n\uf8f6\n+ \uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\u222b\n = \n1\n2\n1\ntan 2\nC\n1\n2\n2\n\u2013\nt +\n\u00d7\n[by 7 4 (3)]\n=\n1\n2\n3\ntan 2\n+ C\n2\n\u2013\n\uf8ebx\n\uf8f6\n\uf8ec+\n\uf8f7\n\uf8ed\n\uf8f8\n = \n(\n)\n1\n2\ntan\n2\n3 + C\n\u2013\nx + (3)\nUsing (2) and (3) in (1), we get\n(\n)\n2\n1\n2\n1\n1\nlog 2\n6\n5\ntan\n2\n3\nC\n4\n2\n2\n6\n5\n\u2013\nx\ndx\nx\nx\nx\nx\nx\n2\n+\n=\n+\n+\n+\n+\n+\n+\n+\n\u222b\nwhere,\nC =\n1\n2\nC\nC\n4\n2\n+\n(ii)\nThis integral is of the form given in 7 4 (10)"
    },
    {
        "Chapter": "1",
        "sentence_range": "3594-3597",
        "Text": "4 (3)]\n=\n1\n2\n3\ntan 2\n+ C\n2\n\u2013\n\uf8ebx\n\uf8f6\n\uf8ec+\n\uf8f7\n\uf8ed\n\uf8f8\n = \n(\n)\n1\n2\ntan\n2\n3 + C\n\u2013\nx + (3)\nUsing (2) and (3) in (1), we get\n(\n)\n2\n1\n2\n1\n1\nlog 2\n6\n5\ntan\n2\n3\nC\n4\n2\n2\n6\n5\n\u2013\nx\ndx\nx\nx\nx\nx\nx\n2\n+\n=\n+\n+\n+\n+\n+\n+\n+\n\u222b\nwhere,\nC =\n1\n2\nC\nC\n4\n2\n+\n(ii)\nThis integral is of the form given in 7 4 (10) Let us express\nx + 3 = \n2\nA\n(5\n4\n) + B\nd\n\u2013 x \u2013 x\ndx\n= A (\u2013 4 \u2013 2x) + B\nEquating the coefficients of x and the constant terms from both sides, we get\n\u2013 2A = 1 and \u2013 4 A + B = 3, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "3595-3598",
        "Text": "(3)\nUsing (2) and (3) in (1), we get\n(\n)\n2\n1\n2\n1\n1\nlog 2\n6\n5\ntan\n2\n3\nC\n4\n2\n2\n6\n5\n\u2013\nx\ndx\nx\nx\nx\nx\nx\n2\n+\n=\n+\n+\n+\n+\n+\n+\n+\n\u222b\nwhere,\nC =\n1\n2\nC\nC\n4\n2\n+\n(ii)\nThis integral is of the form given in 7 4 (10) Let us express\nx + 3 = \n2\nA\n(5\n4\n) + B\nd\n\u2013 x \u2013 x\ndx\n= A (\u2013 4 \u2013 2x) + B\nEquating the coefficients of x and the constant terms from both sides, we get\n\u2013 2A = 1 and \u2013 4 A + B = 3, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "3596-3599",
        "Text": "4 (10) Let us express\nx + 3 = \n2\nA\n(5\n4\n) + B\nd\n\u2013 x \u2013 x\ndx\n= A (\u2013 4 \u2013 2x) + B\nEquating the coefficients of x and the constant terms from both sides, we get\n\u2013 2A = 1 and \u2013 4 A + B = 3, i e , A = \n1\n2\n\u2013\n and B = 1\nINTEGRALS         315\nTherefore,\n2\n3\n5\n4\nx\ndx\nx\nx\n+\n\u2212\n\u2212\n\u222b\n =\n(\n)\n2\n2\n4\n2\n21\n5\n4\n5\n4\n\u2013\n\u2013 x dx\ndx\n\u2013\nx\nx\nx\nx\n+\n\u2212\n\u2212\n\u2212\n\u2212\n\u222b\n\u222b\n=\n\u201321\n I1 + I2"
    },
    {
        "Chapter": "1",
        "sentence_range": "3597-3600",
        "Text": "Let us express\nx + 3 = \n2\nA\n(5\n4\n) + B\nd\n\u2013 x \u2013 x\ndx\n= A (\u2013 4 \u2013 2x) + B\nEquating the coefficients of x and the constant terms from both sides, we get\n\u2013 2A = 1 and \u2013 4 A + B = 3, i e , A = \n1\n2\n\u2013\n and B = 1\nINTEGRALS         315\nTherefore,\n2\n3\n5\n4\nx\ndx\nx\nx\n+\n\u2212\n\u2212\n\u222b\n =\n(\n)\n2\n2\n4\n2\n21\n5\n4\n5\n4\n\u2013\n\u2013 x dx\ndx\n\u2013\nx\nx\nx\nx\n+\n\u2212\n\u2212\n\u2212\n\u2212\n\u222b\n\u222b\n=\n\u201321\n I1 + I2 (1)\nIn I1, put 5 \u2013 4x \u2013 x2 = t, so that (\u2013 4 \u2013 2x) dx = dt"
    },
    {
        "Chapter": "1",
        "sentence_range": "3598-3601",
        "Text": "e , A = \n1\n2\n\u2013\n and B = 1\nINTEGRALS         315\nTherefore,\n2\n3\n5\n4\nx\ndx\nx\nx\n+\n\u2212\n\u2212\n\u222b\n =\n(\n)\n2\n2\n4\n2\n21\n5\n4\n5\n4\n\u2013\n\u2013 x dx\ndx\n\u2013\nx\nx\nx\nx\n+\n\u2212\n\u2212\n\u2212\n\u2212\n\u222b\n\u222b\n=\n\u201321\n I1 + I2 (1)\nIn I1, put 5 \u2013 4x \u2013 x2 = t, so that (\u2013 4 \u2013 2x) dx = dt Therefore,\nI1=  (\n)\n2\n4\n2\n5\n4\n\u2013\nx dx\ndt\nt\nx\nx\n\u2212\n=\n\u2212\n\u2212\n\u222b\n\u222b\n = \n1\n2\nC\nt +\n=\n2\n1\n2 5 4\n\u2013 x \u2013 x +C"
    },
    {
        "Chapter": "1",
        "sentence_range": "3599-3602",
        "Text": ", A = \n1\n2\n\u2013\n and B = 1\nINTEGRALS         315\nTherefore,\n2\n3\n5\n4\nx\ndx\nx\nx\n+\n\u2212\n\u2212\n\u222b\n =\n(\n)\n2\n2\n4\n2\n21\n5\n4\n5\n4\n\u2013\n\u2013 x dx\ndx\n\u2013\nx\nx\nx\nx\n+\n\u2212\n\u2212\n\u2212\n\u2212\n\u222b\n\u222b\n=\n\u201321\n I1 + I2 (1)\nIn I1, put 5 \u2013 4x \u2013 x2 = t, so that (\u2013 4 \u2013 2x) dx = dt Therefore,\nI1=  (\n)\n2\n4\n2\n5\n4\n\u2013\nx dx\ndt\nt\nx\nx\n\u2212\n=\n\u2212\n\u2212\n\u222b\n\u222b\n = \n1\n2\nC\nt +\n=\n2\n1\n2 5 4\n\u2013 x \u2013 x +C (2)\nNow consider\nI2 =\n2\n2\n5\n4\n9\n(\n2)\ndx\ndx\nx\nx\n\u2013 x\n=\n\u2212\n\u2212\n+\n\u222b\n\u222b\nPut x + 2 = t, so that dx = dt"
    },
    {
        "Chapter": "1",
        "sentence_range": "3600-3603",
        "Text": "(1)\nIn I1, put 5 \u2013 4x \u2013 x2 = t, so that (\u2013 4 \u2013 2x) dx = dt Therefore,\nI1=  (\n)\n2\n4\n2\n5\n4\n\u2013\nx dx\ndt\nt\nx\nx\n\u2212\n=\n\u2212\n\u2212\n\u222b\n\u222b\n = \n1\n2\nC\nt +\n=\n2\n1\n2 5 4\n\u2013 x \u2013 x +C (2)\nNow consider\nI2 =\n2\n2\n5\n4\n9\n(\n2)\ndx\ndx\nx\nx\n\u2013 x\n=\n\u2212\n\u2212\n+\n\u222b\n\u222b\nPut x + 2 = t, so that dx = dt Therefore,\nI2 =\n1\n2\n2\n2\nsin\n3+ C\n3\n\u2013\ndt\nt\nt\n=\n\u2212\n\u222b\n[by 7"
    },
    {
        "Chapter": "1",
        "sentence_range": "3601-3604",
        "Text": "Therefore,\nI1=  (\n)\n2\n4\n2\n5\n4\n\u2013\nx dx\ndt\nt\nx\nx\n\u2212\n=\n\u2212\n\u2212\n\u222b\n\u222b\n = \n1\n2\nC\nt +\n=\n2\n1\n2 5 4\n\u2013 x \u2013 x +C (2)\nNow consider\nI2 =\n2\n2\n5\n4\n9\n(\n2)\ndx\ndx\nx\nx\n\u2013 x\n=\n\u2212\n\u2212\n+\n\u222b\n\u222b\nPut x + 2 = t, so that dx = dt Therefore,\nI2 =\n1\n2\n2\n2\nsin\n3+ C\n3\n\u2013\ndt\nt\nt\n=\n\u2212\n\u222b\n[by 7 4 (5)]\n=\n1\n2\n2\nsin\nC\n3\n\u2013 x +\n+"
    },
    {
        "Chapter": "1",
        "sentence_range": "3602-3605",
        "Text": "(2)\nNow consider\nI2 =\n2\n2\n5\n4\n9\n(\n2)\ndx\ndx\nx\nx\n\u2013 x\n=\n\u2212\n\u2212\n+\n\u222b\n\u222b\nPut x + 2 = t, so that dx = dt Therefore,\nI2 =\n1\n2\n2\n2\nsin\n3+ C\n3\n\u2013\ndt\nt\nt\n=\n\u2212\n\u222b\n[by 7 4 (5)]\n=\n1\n2\n2\nsin\nC\n3\n\u2013 x +\n+ (3)\nSubstituting (2) and (3) in (1), we obtain\n2\n1\n2\n3\n2\n5 \u2013 4 \u2013\n+ sin\nC\n3\n5\n4\n\u2013\nx\nx\n\u2013\nx\nx\n\u2013 x \u2013 x\n+\n+\n=\n+\n\u222b\n, where \n1\n2\nC\nC\nC\n2\n\u2013\n=\nEXERCISE 7"
    },
    {
        "Chapter": "1",
        "sentence_range": "3603-3606",
        "Text": "Therefore,\nI2 =\n1\n2\n2\n2\nsin\n3+ C\n3\n\u2013\ndt\nt\nt\n=\n\u2212\n\u222b\n[by 7 4 (5)]\n=\n1\n2\n2\nsin\nC\n3\n\u2013 x +\n+ (3)\nSubstituting (2) and (3) in (1), we obtain\n2\n1\n2\n3\n2\n5 \u2013 4 \u2013\n+ sin\nC\n3\n5\n4\n\u2013\nx\nx\n\u2013\nx\nx\n\u2013 x \u2013 x\n+\n+\n=\n+\n\u222b\n, where \n1\n2\nC\nC\nC\n2\n\u2013\n=\nEXERCISE 7 4\nIntegrate the functions in Exercises 1 to 23"
    },
    {
        "Chapter": "1",
        "sentence_range": "3604-3607",
        "Text": "4 (5)]\n=\n1\n2\n2\nsin\nC\n3\n\u2013 x +\n+ (3)\nSubstituting (2) and (3) in (1), we obtain\n2\n1\n2\n3\n2\n5 \u2013 4 \u2013\n+ sin\nC\n3\n5\n4\n\u2013\nx\nx\n\u2013\nx\nx\n\u2013 x \u2013 x\n+\n+\n=\n+\n\u222b\n, where \n1\n2\nC\nC\nC\n2\n\u2013\n=\nEXERCISE 7 4\nIntegrate the functions in Exercises 1 to 23 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "3605-3608",
        "Text": "(3)\nSubstituting (2) and (3) in (1), we obtain\n2\n1\n2\n3\n2\n5 \u2013 4 \u2013\n+ sin\nC\n3\n5\n4\n\u2013\nx\nx\n\u2013\nx\nx\n\u2013 x \u2013 x\n+\n+\n=\n+\n\u222b\n, where \n1\n2\nC\nC\nC\n2\n\u2013\n=\nEXERCISE 7 4\nIntegrate the functions in Exercises 1 to 23 1 2\n6\n3\n1\nx\nx +\n2"
    },
    {
        "Chapter": "1",
        "sentence_range": "3606-3609",
        "Text": "4\nIntegrate the functions in Exercises 1 to 23 1 2\n6\n3\n1\nx\nx +\n2 2\n1\n1\n4x\n+\n3"
    },
    {
        "Chapter": "1",
        "sentence_range": "3607-3610",
        "Text": "1 2\n6\n3\n1\nx\nx +\n2 2\n1\n1\n4x\n+\n3 (\n)\n2\n1\n2\n1\n\u2013 x\n+\n4"
    },
    {
        "Chapter": "1",
        "sentence_range": "3608-3611",
        "Text": "2\n6\n3\n1\nx\nx +\n2 2\n1\n1\n4x\n+\n3 (\n)\n2\n1\n2\n1\n\u2013 x\n+\n4 2\n1\n9\n\u201325\nx\n5"
    },
    {
        "Chapter": "1",
        "sentence_range": "3609-3612",
        "Text": "2\n1\n1\n4x\n+\n3 (\n)\n2\n1\n2\n1\n\u2013 x\n+\n4 2\n1\n9\n\u201325\nx\n5 4\n3\n1 2\nx\nx\n+\n6"
    },
    {
        "Chapter": "1",
        "sentence_range": "3610-3613",
        "Text": "(\n)\n2\n1\n2\n1\n\u2013 x\n+\n4 2\n1\n9\n\u201325\nx\n5 4\n3\n1 2\nx\nx\n+\n6 2\n6\n1\nx\nx\n\u2212\n7"
    },
    {
        "Chapter": "1",
        "sentence_range": "3611-3614",
        "Text": "2\n1\n9\n\u201325\nx\n5 4\n3\n1 2\nx\nx\n+\n6 2\n6\n1\nx\nx\n\u2212\n7 2\n1\n1\nx \u2013\nx \u2013\n8"
    },
    {
        "Chapter": "1",
        "sentence_range": "3612-3615",
        "Text": "4\n3\n1 2\nx\nx\n+\n6 2\n6\n1\nx\nx\n\u2212\n7 2\n1\n1\nx \u2013\nx \u2013\n8 2\n6\n6\nx\nx\na\n+\n9"
    },
    {
        "Chapter": "1",
        "sentence_range": "3613-3616",
        "Text": "2\n6\n1\nx\nx\n\u2212\n7 2\n1\n1\nx \u2013\nx \u2013\n8 2\n6\n6\nx\nx\na\n+\n9 2\n2\nsec\ntan\n4\nx\nx +\n316\nMATHEMATICS\n10"
    },
    {
        "Chapter": "1",
        "sentence_range": "3614-3617",
        "Text": "2\n1\n1\nx \u2013\nx \u2013\n8 2\n6\n6\nx\nx\na\n+\n9 2\n2\nsec\ntan\n4\nx\nx +\n316\nMATHEMATICS\n10 2\n1\n2\n2\nx\n+x\n+\n11"
    },
    {
        "Chapter": "1",
        "sentence_range": "3615-3618",
        "Text": "2\n6\n6\nx\nx\na\n+\n9 2\n2\nsec\ntan\n4\nx\nx +\n316\nMATHEMATICS\n10 2\n1\n2\n2\nx\n+x\n+\n11 2\n1\n9\n6\n5\nx\n+x\n+\n12"
    },
    {
        "Chapter": "1",
        "sentence_range": "3616-3619",
        "Text": "2\n2\nsec\ntan\n4\nx\nx +\n316\nMATHEMATICS\n10 2\n1\n2\n2\nx\n+x\n+\n11 2\n1\n9\n6\n5\nx\n+x\n+\n12 2\n1\n7\n\u20136\nx \u2013 x\n13"
    },
    {
        "Chapter": "1",
        "sentence_range": "3617-3620",
        "Text": "2\n1\n2\n2\nx\n+x\n+\n11 2\n1\n9\n6\n5\nx\n+x\n+\n12 2\n1\n7\n\u20136\nx \u2013 x\n13 (\n)(\n)\n1\n1\n2\nx \u2013\nx \u2013\n14"
    },
    {
        "Chapter": "1",
        "sentence_range": "3618-3621",
        "Text": "2\n1\n9\n6\n5\nx\n+x\n+\n12 2\n1\n7\n\u20136\nx \u2013 x\n13 (\n)(\n)\n1\n1\n2\nx \u2013\nx \u2013\n14 2\n1\n8\n3x \u2013 x\n+\n15"
    },
    {
        "Chapter": "1",
        "sentence_range": "3619-3622",
        "Text": "2\n1\n7\n\u20136\nx \u2013 x\n13 (\n)(\n)\n1\n1\n2\nx \u2013\nx \u2013\n14 2\n1\n8\n3x \u2013 x\n+\n15 (\n)(\n)\n1\nx \u2013 a\nx \u2013b\n16"
    },
    {
        "Chapter": "1",
        "sentence_range": "3620-3623",
        "Text": "(\n)(\n)\n1\n1\n2\nx \u2013\nx \u2013\n14 2\n1\n8\n3x \u2013 x\n+\n15 (\n)(\n)\n1\nx \u2013 a\nx \u2013b\n16 42\n1\n2\n3\nx\nx\nx \u2013\n+\n+\n17"
    },
    {
        "Chapter": "1",
        "sentence_range": "3621-3624",
        "Text": "2\n1\n8\n3x \u2013 x\n+\n15 (\n)(\n)\n1\nx \u2013 a\nx \u2013b\n16 42\n1\n2\n3\nx\nx\nx \u2013\n+\n+\n17 2\n2\n1\nx\nx \u2013\n+\n18"
    },
    {
        "Chapter": "1",
        "sentence_range": "3622-3625",
        "Text": "(\n)(\n)\n1\nx \u2013 a\nx \u2013b\n16 42\n1\n2\n3\nx\nx\nx \u2013\n+\n+\n17 2\n2\n1\nx\nx \u2013\n+\n18 2\n5\n2\n1 2\n3\nxx\nx\n\u2212\n+\n+\n19"
    },
    {
        "Chapter": "1",
        "sentence_range": "3623-3626",
        "Text": "42\n1\n2\n3\nx\nx\nx \u2013\n+\n+\n17 2\n2\n1\nx\nx \u2013\n+\n18 2\n5\n2\n1 2\n3\nxx\nx\n\u2212\n+\n+\n19 (\n)(\n)\n6\n7\n5\n4\nx\nx \u2013\nx \u2013\n+\n20"
    },
    {
        "Chapter": "1",
        "sentence_range": "3624-3627",
        "Text": "2\n2\n1\nx\nx \u2013\n+\n18 2\n5\n2\n1 2\n3\nxx\nx\n\u2212\n+\n+\n19 (\n)(\n)\n6\n7\n5\n4\nx\nx \u2013\nx \u2013\n+\n20 2\n2\n4\nx\nx \u2013 x\n+\n21"
    },
    {
        "Chapter": "1",
        "sentence_range": "3625-3628",
        "Text": "2\n5\n2\n1 2\n3\nxx\nx\n\u2212\n+\n+\n19 (\n)(\n)\n6\n7\n5\n4\nx\nx \u2013\nx \u2013\n+\n20 2\n2\n4\nx\nx \u2013 x\n+\n21 2\n2\n2\n3\nx\nx\nx\n+\n+\n+\n22"
    },
    {
        "Chapter": "1",
        "sentence_range": "3626-3629",
        "Text": "(\n)(\n)\n6\n7\n5\n4\nx\nx \u2013\nx \u2013\n+\n20 2\n2\n4\nx\nx \u2013 x\n+\n21 2\n2\n2\n3\nx\nx\nx\n+\n+\n+\n22 2\n3\n2\n5\nx\nx \u2013 x\n+\n\u2212\n23"
    },
    {
        "Chapter": "1",
        "sentence_range": "3627-3630",
        "Text": "2\n2\n4\nx\nx \u2013 x\n+\n21 2\n2\n2\n3\nx\nx\nx\n+\n+\n+\n22 2\n3\n2\n5\nx\nx \u2013 x\n+\n\u2212\n23 2\n5\n3\n4\n10\nx\nx\nx\n+\n+\n+"
    },
    {
        "Chapter": "1",
        "sentence_range": "3628-3631",
        "Text": "2\n2\n2\n3\nx\nx\nx\n+\n+\n+\n22 2\n3\n2\n5\nx\nx \u2013 x\n+\n\u2212\n23 2\n5\n3\n4\n10\nx\nx\nx\n+\n+\n+ Choose the correct answer in Exercises 24 and 25"
    },
    {
        "Chapter": "1",
        "sentence_range": "3629-3632",
        "Text": "2\n3\n2\n5\nx\nx \u2013 x\n+\n\u2212\n23 2\n5\n3\n4\n10\nx\nx\nx\n+\n+\n+ Choose the correct answer in Exercises 24 and 25 24"
    },
    {
        "Chapter": "1",
        "sentence_range": "3630-3633",
        "Text": "2\n5\n3\n4\n10\nx\nx\nx\n+\n+\n+ Choose the correct answer in Exercises 24 and 25 24 2\nequals\n2\n2\ndx\nx\n+x\n+\n\u222b\n(A)\nx tan\u20131 (x + 1) + C\n(B)\ntan\u20131 (x + 1) + C\n(C)\n(x + 1) tan\u20131x + C\n(D)\ntan\u20131x + C\n25"
    },
    {
        "Chapter": "1",
        "sentence_range": "3631-3634",
        "Text": "Choose the correct answer in Exercises 24 and 25 24 2\nequals\n2\n2\ndx\nx\n+x\n+\n\u222b\n(A)\nx tan\u20131 (x + 1) + C\n(B)\ntan\u20131 (x + 1) + C\n(C)\n(x + 1) tan\u20131x + C\n(D)\ntan\u20131x + C\n25 2 equals\n9\n4\ndx\nx\nx\n\u2212\n\u222b\n(A)\n\u20131\n1\n9\n8\nsin\nC\n9\n8\n\uf8ebx \u2212\n\uf8f6 +\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n(B)\n\u20131\n1\n8\n9\nsin\nC\n2\n9\n\uf8ebx \u2212\n\uf8f6 +\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n(C)\n\u20131\n1\n9\n8\nsin\nC\n3\n8\n\uf8ebx \u2212\n\uf8f6 +\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n(D)\n\u20131\n1\n9\n8\nsin\nC\n2\n9\n\uf8ebx \u2212\n\uf8f6 +\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n7"
    },
    {
        "Chapter": "1",
        "sentence_range": "3632-3635",
        "Text": "24 2\nequals\n2\n2\ndx\nx\n+x\n+\n\u222b\n(A)\nx tan\u20131 (x + 1) + C\n(B)\ntan\u20131 (x + 1) + C\n(C)\n(x + 1) tan\u20131x + C\n(D)\ntan\u20131x + C\n25 2 equals\n9\n4\ndx\nx\nx\n\u2212\n\u222b\n(A)\n\u20131\n1\n9\n8\nsin\nC\n9\n8\n\uf8ebx \u2212\n\uf8f6 +\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n(B)\n\u20131\n1\n8\n9\nsin\nC\n2\n9\n\uf8ebx \u2212\n\uf8f6 +\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n(C)\n\u20131\n1\n9\n8\nsin\nC\n3\n8\n\uf8ebx \u2212\n\uf8f6 +\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n(D)\n\u20131\n1\n9\n8\nsin\nC\n2\n9\n\uf8ebx \u2212\n\uf8f6 +\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n7 5  Integration by Partial Fractions\nRecall that a rational function is defined as the ratio of two polynomials in the form\nP( )\nQ( )\nx\nx\n, where P (x) and Q(x) are polynomials in x and Q(x) \u2260 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "3633-3636",
        "Text": "2\nequals\n2\n2\ndx\nx\n+x\n+\n\u222b\n(A)\nx tan\u20131 (x + 1) + C\n(B)\ntan\u20131 (x + 1) + C\n(C)\n(x + 1) tan\u20131x + C\n(D)\ntan\u20131x + C\n25 2 equals\n9\n4\ndx\nx\nx\n\u2212\n\u222b\n(A)\n\u20131\n1\n9\n8\nsin\nC\n9\n8\n\uf8ebx \u2212\n\uf8f6 +\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n(B)\n\u20131\n1\n8\n9\nsin\nC\n2\n9\n\uf8ebx \u2212\n\uf8f6 +\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n(C)\n\u20131\n1\n9\n8\nsin\nC\n3\n8\n\uf8ebx \u2212\n\uf8f6 +\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n(D)\n\u20131\n1\n9\n8\nsin\nC\n2\n9\n\uf8ebx \u2212\n\uf8f6 +\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n7 5  Integration by Partial Fractions\nRecall that a rational function is defined as the ratio of two polynomials in the form\nP( )\nQ( )\nx\nx\n, where P (x) and Q(x) are polynomials in x and Q(x) \u2260 0 If the degree of P(x)\nis less than the degree of Q(x), then the rational function is called proper, otherwise, it\nis called improper"
    },
    {
        "Chapter": "1",
        "sentence_range": "3634-3637",
        "Text": "2 equals\n9\n4\ndx\nx\nx\n\u2212\n\u222b\n(A)\n\u20131\n1\n9\n8\nsin\nC\n9\n8\n\uf8ebx \u2212\n\uf8f6 +\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n(B)\n\u20131\n1\n8\n9\nsin\nC\n2\n9\n\uf8ebx \u2212\n\uf8f6 +\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n(C)\n\u20131\n1\n9\n8\nsin\nC\n3\n8\n\uf8ebx \u2212\n\uf8f6 +\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n(D)\n\u20131\n1\n9\n8\nsin\nC\n2\n9\n\uf8ebx \u2212\n\uf8f6 +\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n7 5  Integration by Partial Fractions\nRecall that a rational function is defined as the ratio of two polynomials in the form\nP( )\nQ( )\nx\nx\n, where P (x) and Q(x) are polynomials in x and Q(x) \u2260 0 If the degree of P(x)\nis less than the degree of Q(x), then the rational function is called proper, otherwise, it\nis called improper The improper rational functions can be reduced to the proper rational\nINTEGRALS         317\nfunctions by long division process"
    },
    {
        "Chapter": "1",
        "sentence_range": "3635-3638",
        "Text": "5  Integration by Partial Fractions\nRecall that a rational function is defined as the ratio of two polynomials in the form\nP( )\nQ( )\nx\nx\n, where P (x) and Q(x) are polynomials in x and Q(x) \u2260 0 If the degree of P(x)\nis less than the degree of Q(x), then the rational function is called proper, otherwise, it\nis called improper The improper rational functions can be reduced to the proper rational\nINTEGRALS         317\nfunctions by long division process Thus, if P( )\nQ( )\nx\nx  is improper, then \n1P ( )\nP( )\nT( )\nQ( )\nQ( )\nx\nx\nx\nx\nx\n=\n+\n,\nwhere T(x) is a polynomial in x and \n1P ( )\nQ( )\nx\nx is a proper rational function"
    },
    {
        "Chapter": "1",
        "sentence_range": "3636-3639",
        "Text": "If the degree of P(x)\nis less than the degree of Q(x), then the rational function is called proper, otherwise, it\nis called improper The improper rational functions can be reduced to the proper rational\nINTEGRALS         317\nfunctions by long division process Thus, if P( )\nQ( )\nx\nx  is improper, then \n1P ( )\nP( )\nT( )\nQ( )\nQ( )\nx\nx\nx\nx\nx\n=\n+\n,\nwhere T(x) is a polynomial in x and \n1P ( )\nQ( )\nx\nx is a proper rational function As we know\nhow to integrate polynomials, the integration of any rational function is reduced to the\nintegration of a proper rational function"
    },
    {
        "Chapter": "1",
        "sentence_range": "3637-3640",
        "Text": "The improper rational functions can be reduced to the proper rational\nINTEGRALS         317\nfunctions by long division process Thus, if P( )\nQ( )\nx\nx  is improper, then \n1P ( )\nP( )\nT( )\nQ( )\nQ( )\nx\nx\nx\nx\nx\n=\n+\n,\nwhere T(x) is a polynomial in x and \n1P ( )\nQ( )\nx\nx is a proper rational function As we know\nhow to integrate polynomials, the integration of any rational function is reduced to the\nintegration of a proper rational function The rational functions which we shall consider\nhere for integration purposes will be those whose denominators can be factorised into\nlinear and quadratic factors"
    },
    {
        "Chapter": "1",
        "sentence_range": "3638-3641",
        "Text": "Thus, if P( )\nQ( )\nx\nx  is improper, then \n1P ( )\nP( )\nT( )\nQ( )\nQ( )\nx\nx\nx\nx\nx\n=\n+\n,\nwhere T(x) is a polynomial in x and \n1P ( )\nQ( )\nx\nx is a proper rational function As we know\nhow to integrate polynomials, the integration of any rational function is reduced to the\nintegration of a proper rational function The rational functions which we shall consider\nhere for integration purposes will be those whose denominators can be factorised into\nlinear and quadratic factors Assume that we want to evaluate \nP( )\nQ( )\n\u222bxx dx\n, where P( )\nQ( )\nx\nx\nis proper rational function"
    },
    {
        "Chapter": "1",
        "sentence_range": "3639-3642",
        "Text": "As we know\nhow to integrate polynomials, the integration of any rational function is reduced to the\nintegration of a proper rational function The rational functions which we shall consider\nhere for integration purposes will be those whose denominators can be factorised into\nlinear and quadratic factors Assume that we want to evaluate \nP( )\nQ( )\n\u222bxx dx\n, where P( )\nQ( )\nx\nx\nis proper rational function It is always possible to write the integrand as a sum of\nsimpler rational functions by a method called partial fraction decomposition"
    },
    {
        "Chapter": "1",
        "sentence_range": "3640-3643",
        "Text": "The rational functions which we shall consider\nhere for integration purposes will be those whose denominators can be factorised into\nlinear and quadratic factors Assume that we want to evaluate \nP( )\nQ( )\n\u222bxx dx\n, where P( )\nQ( )\nx\nx\nis proper rational function It is always possible to write the integrand as a sum of\nsimpler rational functions by a method called partial fraction decomposition After this,\nthe integration can be carried out easily using the already known methods"
    },
    {
        "Chapter": "1",
        "sentence_range": "3641-3644",
        "Text": "Assume that we want to evaluate \nP( )\nQ( )\n\u222bxx dx\n, where P( )\nQ( )\nx\nx\nis proper rational function It is always possible to write the integrand as a sum of\nsimpler rational functions by a method called partial fraction decomposition After this,\nthe integration can be carried out easily using the already known methods The following\nTable 7"
    },
    {
        "Chapter": "1",
        "sentence_range": "3642-3645",
        "Text": "It is always possible to write the integrand as a sum of\nsimpler rational functions by a method called partial fraction decomposition After this,\nthe integration can be carried out easily using the already known methods The following\nTable 7 2 indicates the types of simpler partial fractions that are to be associated with\nvarious kind of rational functions"
    },
    {
        "Chapter": "1",
        "sentence_range": "3643-3646",
        "Text": "After this,\nthe integration can be carried out easily using the already known methods The following\nTable 7 2 indicates the types of simpler partial fractions that are to be associated with\nvarious kind of rational functions Table 7"
    },
    {
        "Chapter": "1",
        "sentence_range": "3644-3647",
        "Text": "The following\nTable 7 2 indicates the types of simpler partial fractions that are to be associated with\nvarious kind of rational functions Table 7 2\n S"
    },
    {
        "Chapter": "1",
        "sentence_range": "3645-3648",
        "Text": "2 indicates the types of simpler partial fractions that are to be associated with\nvarious kind of rational functions Table 7 2\n S No"
    },
    {
        "Chapter": "1",
        "sentence_range": "3646-3649",
        "Text": "Table 7 2\n S No Form of the rational function\nForm of the partial fraction\n1"
    },
    {
        "Chapter": "1",
        "sentence_range": "3647-3650",
        "Text": "2\n S No Form of the rational function\nForm of the partial fraction\n1 ( \u2013 ) ( \u2013 )\npx\nq\nx a\n+x b\n, a \u2260 b\nA\nB\nx \u2013 a\nx \u2013 b\n+\n2"
    },
    {
        "Chapter": "1",
        "sentence_range": "3648-3651",
        "Text": "No Form of the rational function\nForm of the partial fraction\n1 ( \u2013 ) ( \u2013 )\npx\nq\nx a\n+x b\n, a \u2260 b\nA\nB\nx \u2013 a\nx \u2013 b\n+\n2 ( \u2013 )2\npx\nq\nx\na\n+\n(\n)\n2\nA\nB\nx \u2013 a\nx \u2013 a\n+\n3"
    },
    {
        "Chapter": "1",
        "sentence_range": "3649-3652",
        "Text": "Form of the rational function\nForm of the partial fraction\n1 ( \u2013 ) ( \u2013 )\npx\nq\nx a\n+x b\n, a \u2260 b\nA\nB\nx \u2013 a\nx \u2013 b\n+\n2 ( \u2013 )2\npx\nq\nx\na\n+\n(\n)\n2\nA\nB\nx \u2013 a\nx \u2013 a\n+\n3 2\n( \u2013 ) (\n) (\n)\npx\nqx\nr\nx\na\nx \u2013 b\nx \u2013 c\n+\n+\nA\nB\nC\nx \u2013 a\nx \u2013 b\nx \u2013c\n+\n+\n4"
    },
    {
        "Chapter": "1",
        "sentence_range": "3650-3653",
        "Text": "( \u2013 ) ( \u2013 )\npx\nq\nx a\n+x b\n, a \u2260 b\nA\nB\nx \u2013 a\nx \u2013 b\n+\n2 ( \u2013 )2\npx\nq\nx\na\n+\n(\n)\n2\nA\nB\nx \u2013 a\nx \u2013 a\n+\n3 2\n( \u2013 ) (\n) (\n)\npx\nqx\nr\nx\na\nx \u2013 b\nx \u2013 c\n+\n+\nA\nB\nC\nx \u2013 a\nx \u2013 b\nx \u2013c\n+\n+\n4 2\n2\n( \u2013 ) (\n)\npx\nqx\nr\nx\na\nx \u2013 b\n+\n+\n2\nA\nB\nC\n(\n)\nx \u2013 a\nx \u2013 b\n+x \u2013 a\n+\n5"
    },
    {
        "Chapter": "1",
        "sentence_range": "3651-3654",
        "Text": "( \u2013 )2\npx\nq\nx\na\n+\n(\n)\n2\nA\nB\nx \u2013 a\nx \u2013 a\n+\n3 2\n( \u2013 ) (\n) (\n)\npx\nqx\nr\nx\na\nx \u2013 b\nx \u2013 c\n+\n+\nA\nB\nC\nx \u2013 a\nx \u2013 b\nx \u2013c\n+\n+\n4 2\n2\n( \u2013 ) (\n)\npx\nqx\nr\nx\na\nx \u2013 b\n+\n+\n2\nA\nB\nC\n(\n)\nx \u2013 a\nx \u2013 b\n+x \u2013 a\n+\n5 2\n2\n( \u2013\n) (\n)\npx\nqx\nr\nx\na\nx\nbx\nc\n+\n+\n+\n+\n2\nA\nB + C\nx\nx \u2013 a\nx\nbx\nc\n+\n+\n+\n,\nwhere x2 + bx + c cannot be factorised further\nIn the above table, A, B and C are real numbers to be determined suitably"
    },
    {
        "Chapter": "1",
        "sentence_range": "3652-3655",
        "Text": "2\n( \u2013 ) (\n) (\n)\npx\nqx\nr\nx\na\nx \u2013 b\nx \u2013 c\n+\n+\nA\nB\nC\nx \u2013 a\nx \u2013 b\nx \u2013c\n+\n+\n4 2\n2\n( \u2013 ) (\n)\npx\nqx\nr\nx\na\nx \u2013 b\n+\n+\n2\nA\nB\nC\n(\n)\nx \u2013 a\nx \u2013 b\n+x \u2013 a\n+\n5 2\n2\n( \u2013\n) (\n)\npx\nqx\nr\nx\na\nx\nbx\nc\n+\n+\n+\n+\n2\nA\nB + C\nx\nx \u2013 a\nx\nbx\nc\n+\n+\n+\n,\nwhere x2 + bx + c cannot be factorised further\nIn the above table, A, B and C are real numbers to be determined suitably 318\nMATHEMATICS\nExample 11 Find \n(\n1) (\n2)\ndx\nx\nx\n+\n+\n\u222b\nSolution The integrand is a proper rational function"
    },
    {
        "Chapter": "1",
        "sentence_range": "3653-3656",
        "Text": "2\n2\n( \u2013 ) (\n)\npx\nqx\nr\nx\na\nx \u2013 b\n+\n+\n2\nA\nB\nC\n(\n)\nx \u2013 a\nx \u2013 b\n+x \u2013 a\n+\n5 2\n2\n( \u2013\n) (\n)\npx\nqx\nr\nx\na\nx\nbx\nc\n+\n+\n+\n+\n2\nA\nB + C\nx\nx \u2013 a\nx\nbx\nc\n+\n+\n+\n,\nwhere x2 + bx + c cannot be factorised further\nIn the above table, A, B and C are real numbers to be determined suitably 318\nMATHEMATICS\nExample 11 Find \n(\n1) (\n2)\ndx\nx\nx\n+\n+\n\u222b\nSolution The integrand is a proper rational function Therefore, by using the form of\npartial fraction [Table 7"
    },
    {
        "Chapter": "1",
        "sentence_range": "3654-3657",
        "Text": "2\n2\n( \u2013\n) (\n)\npx\nqx\nr\nx\na\nx\nbx\nc\n+\n+\n+\n+\n2\nA\nB + C\nx\nx \u2013 a\nx\nbx\nc\n+\n+\n+\n,\nwhere x2 + bx + c cannot be factorised further\nIn the above table, A, B and C are real numbers to be determined suitably 318\nMATHEMATICS\nExample 11 Find \n(\n1) (\n2)\ndx\nx\nx\n+\n+\n\u222b\nSolution The integrand is a proper rational function Therefore, by using the form of\npartial fraction [Table 7 2 (i)], we write\n1\n(\n1) (\n2)\nx\nx\n+\n+\n =\nA\nB\n1\n2\nx\nx\n+\n+\n+"
    },
    {
        "Chapter": "1",
        "sentence_range": "3655-3658",
        "Text": "318\nMATHEMATICS\nExample 11 Find \n(\n1) (\n2)\ndx\nx\nx\n+\n+\n\u222b\nSolution The integrand is a proper rational function Therefore, by using the form of\npartial fraction [Table 7 2 (i)], we write\n1\n(\n1) (\n2)\nx\nx\n+\n+\n =\nA\nB\n1\n2\nx\nx\n+\n+\n+ (1)\nwhere, real numbers A and B are to be determined suitably"
    },
    {
        "Chapter": "1",
        "sentence_range": "3656-3659",
        "Text": "Therefore, by using the form of\npartial fraction [Table 7 2 (i)], we write\n1\n(\n1) (\n2)\nx\nx\n+\n+\n =\nA\nB\n1\n2\nx\nx\n+\n+\n+ (1)\nwhere, real numbers A and B are to be determined suitably This gives\n1 = A (x + 2) + B (x + 1)"
    },
    {
        "Chapter": "1",
        "sentence_range": "3657-3660",
        "Text": "2 (i)], we write\n1\n(\n1) (\n2)\nx\nx\n+\n+\n =\nA\nB\n1\n2\nx\nx\n+\n+\n+ (1)\nwhere, real numbers A and B are to be determined suitably This gives\n1 = A (x + 2) + B (x + 1) Equating the coefficients of x and the constant term, we get\nA + B = 0\nand\n2A + B = 1\nSolving these equations, we get A =1 and B = \u2013 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "3658-3661",
        "Text": "(1)\nwhere, real numbers A and B are to be determined suitably This gives\n1 = A (x + 2) + B (x + 1) Equating the coefficients of x and the constant term, we get\nA + B = 0\nand\n2A + B = 1\nSolving these equations, we get A =1 and B = \u2013 1 Thus, the integrand is given by\n1\n(\n1) (\n2)\nx\nx\n+\n+\n =\n1\n\u20131\n1\n2\nx\nx\n+\n+\n+\nTherefore,\n(\n1) (\n2)\ndx\nx\nx\n+\n+\n\u222b\n =\n1\n2\ndx\ndx\n\u2013\nx\nx\n+\n+\n\u222b\n\u222b\n= log\n1\nlog\n2\nC\nx\nx\n+\n\u2212\n+\n+\n=\n1\nlog\nC\n2\nxx\n+\n+\n+\nRemark The equation (1) above is an identity, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "3659-3662",
        "Text": "This gives\n1 = A (x + 2) + B (x + 1) Equating the coefficients of x and the constant term, we get\nA + B = 0\nand\n2A + B = 1\nSolving these equations, we get A =1 and B = \u2013 1 Thus, the integrand is given by\n1\n(\n1) (\n2)\nx\nx\n+\n+\n =\n1\n\u20131\n1\n2\nx\nx\n+\n+\n+\nTherefore,\n(\n1) (\n2)\ndx\nx\nx\n+\n+\n\u222b\n =\n1\n2\ndx\ndx\n\u2013\nx\nx\n+\n+\n\u222b\n\u222b\n= log\n1\nlog\n2\nC\nx\nx\n+\n\u2212\n+\n+\n=\n1\nlog\nC\n2\nxx\n+\n+\n+\nRemark The equation (1) above is an identity, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "3660-3663",
        "Text": "Equating the coefficients of x and the constant term, we get\nA + B = 0\nand\n2A + B = 1\nSolving these equations, we get A =1 and B = \u2013 1 Thus, the integrand is given by\n1\n(\n1) (\n2)\nx\nx\n+\n+\n =\n1\n\u20131\n1\n2\nx\nx\n+\n+\n+\nTherefore,\n(\n1) (\n2)\ndx\nx\nx\n+\n+\n\u222b\n =\n1\n2\ndx\ndx\n\u2013\nx\nx\n+\n+\n\u222b\n\u222b\n= log\n1\nlog\n2\nC\nx\nx\n+\n\u2212\n+\n+\n=\n1\nlog\nC\n2\nxx\n+\n+\n+\nRemark The equation (1) above is an identity, i e a statement true for all (permissible)\nvalues of x"
    },
    {
        "Chapter": "1",
        "sentence_range": "3661-3664",
        "Text": "Thus, the integrand is given by\n1\n(\n1) (\n2)\nx\nx\n+\n+\n =\n1\n\u20131\n1\n2\nx\nx\n+\n+\n+\nTherefore,\n(\n1) (\n2)\ndx\nx\nx\n+\n+\n\u222b\n =\n1\n2\ndx\ndx\n\u2013\nx\nx\n+\n+\n\u222b\n\u222b\n= log\n1\nlog\n2\nC\nx\nx\n+\n\u2212\n+\n+\n=\n1\nlog\nC\n2\nxx\n+\n+\n+\nRemark The equation (1) above is an identity, i e a statement true for all (permissible)\nvalues of x Some authors use the symbol \u2018\u2261\u2019 to indicate that the statement is an\nidentity and use the symbol \u2018=\u2019 to indicate that the statement is an equation, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "3662-3665",
        "Text": "e a statement true for all (permissible)\nvalues of x Some authors use the symbol \u2018\u2261\u2019 to indicate that the statement is an\nidentity and use the symbol \u2018=\u2019 to indicate that the statement is an equation, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "3663-3666",
        "Text": "a statement true for all (permissible)\nvalues of x Some authors use the symbol \u2018\u2261\u2019 to indicate that the statement is an\nidentity and use the symbol \u2018=\u2019 to indicate that the statement is an equation, i e , to\nindicate that the statement is true only for certain values of x"
    },
    {
        "Chapter": "1",
        "sentence_range": "3664-3667",
        "Text": "Some authors use the symbol \u2018\u2261\u2019 to indicate that the statement is an\nidentity and use the symbol \u2018=\u2019 to indicate that the statement is an equation, i e , to\nindicate that the statement is true only for certain values of x Example 12 Find \n2\n2\n1\n5\n6\nx\ndx\nx\n\u2212+x\n+\n\u222b\nSolution Here the integrand \n2\n2\n1\n5\n6\nx\nx \u2013 x\n+\n+\n is not proper rational function, so we divide\nx2 + 1 by x2 \u2013 5x + 6 and find that\nINTEGRALS         319\n2\n2\n1\n5\n6\nx\nx \u2013 x\n+\n+\n =\n2\n5\n5\n5\n5\n1\n1\n(\n2) (\n3)\n5\n6\nx \u2013\nx \u2013\nx \u2013\nx \u2013\n+x \u2013 x\n= +\n+\nLet\n5\n5\n(\n2) (\n3)\nx \u2013\nx \u2013\nx \u2013\n =\nA\nB\n2\n3\nx \u2013\nx \u2013\n+\nSo that\n5x \u2013 5 = A (x \u2013 3) + B (x \u2013 2)\nEquating the coefficients of x and constant terms on both sides, we get A + B = 5\nand 3A + 2B = 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "3665-3668",
        "Text": "e , to\nindicate that the statement is true only for certain values of x Example 12 Find \n2\n2\n1\n5\n6\nx\ndx\nx\n\u2212+x\n+\n\u222b\nSolution Here the integrand \n2\n2\n1\n5\n6\nx\nx \u2013 x\n+\n+\n is not proper rational function, so we divide\nx2 + 1 by x2 \u2013 5x + 6 and find that\nINTEGRALS         319\n2\n2\n1\n5\n6\nx\nx \u2013 x\n+\n+\n =\n2\n5\n5\n5\n5\n1\n1\n(\n2) (\n3)\n5\n6\nx \u2013\nx \u2013\nx \u2013\nx \u2013\n+x \u2013 x\n= +\n+\nLet\n5\n5\n(\n2) (\n3)\nx \u2013\nx \u2013\nx \u2013\n =\nA\nB\n2\n3\nx \u2013\nx \u2013\n+\nSo that\n5x \u2013 5 = A (x \u2013 3) + B (x \u2013 2)\nEquating the coefficients of x and constant terms on both sides, we get A + B = 5\nand 3A + 2B = 5 Solving these equations, we get A = \u2013 5  and B = 10\nThus,\n2\n2\n1\n5\n6\nx\nx \u2013 x\n+\n+\n =\n5\n10\n1\n2\n3\nx \u2013\nx \u2013\n\u2212\n+\nTherefore,\n2\n2\n1\n5\n6\nx\ndx\nx \u2013 x\n+\n+\n\u222b\n =\n1\n5\n10\n2\n3\ndx\ndx\ndx\nx \u2013\nx \u2013\n\u2212\n+\n\u222b\n\u222b\n\u222b\n= x \u2013 5 log |x \u2013 2| + 10 log |x \u2013 3| + C"
    },
    {
        "Chapter": "1",
        "sentence_range": "3666-3669",
        "Text": ", to\nindicate that the statement is true only for certain values of x Example 12 Find \n2\n2\n1\n5\n6\nx\ndx\nx\n\u2212+x\n+\n\u222b\nSolution Here the integrand \n2\n2\n1\n5\n6\nx\nx \u2013 x\n+\n+\n is not proper rational function, so we divide\nx2 + 1 by x2 \u2013 5x + 6 and find that\nINTEGRALS         319\n2\n2\n1\n5\n6\nx\nx \u2013 x\n+\n+\n =\n2\n5\n5\n5\n5\n1\n1\n(\n2) (\n3)\n5\n6\nx \u2013\nx \u2013\nx \u2013\nx \u2013\n+x \u2013 x\n= +\n+\nLet\n5\n5\n(\n2) (\n3)\nx \u2013\nx \u2013\nx \u2013\n =\nA\nB\n2\n3\nx \u2013\nx \u2013\n+\nSo that\n5x \u2013 5 = A (x \u2013 3) + B (x \u2013 2)\nEquating the coefficients of x and constant terms on both sides, we get A + B = 5\nand 3A + 2B = 5 Solving these equations, we get A = \u2013 5  and B = 10\nThus,\n2\n2\n1\n5\n6\nx\nx \u2013 x\n+\n+\n =\n5\n10\n1\n2\n3\nx \u2013\nx \u2013\n\u2212\n+\nTherefore,\n2\n2\n1\n5\n6\nx\ndx\nx \u2013 x\n+\n+\n\u222b\n =\n1\n5\n10\n2\n3\ndx\ndx\ndx\nx \u2013\nx \u2013\n\u2212\n+\n\u222b\n\u222b\n\u222b\n= x \u2013 5 log |x \u2013 2| + 10 log |x \u2013 3| + C Example 13 Find \n32\n2\n(\n1) (\n3)\nx\ndx\nx\n\u2212x\n+\n+\n\u222b\nSolution The integrand is of the type as given in Table 7"
    },
    {
        "Chapter": "1",
        "sentence_range": "3667-3670",
        "Text": "Example 12 Find \n2\n2\n1\n5\n6\nx\ndx\nx\n\u2212+x\n+\n\u222b\nSolution Here the integrand \n2\n2\n1\n5\n6\nx\nx \u2013 x\n+\n+\n is not proper rational function, so we divide\nx2 + 1 by x2 \u2013 5x + 6 and find that\nINTEGRALS         319\n2\n2\n1\n5\n6\nx\nx \u2013 x\n+\n+\n =\n2\n5\n5\n5\n5\n1\n1\n(\n2) (\n3)\n5\n6\nx \u2013\nx \u2013\nx \u2013\nx \u2013\n+x \u2013 x\n= +\n+\nLet\n5\n5\n(\n2) (\n3)\nx \u2013\nx \u2013\nx \u2013\n =\nA\nB\n2\n3\nx \u2013\nx \u2013\n+\nSo that\n5x \u2013 5 = A (x \u2013 3) + B (x \u2013 2)\nEquating the coefficients of x and constant terms on both sides, we get A + B = 5\nand 3A + 2B = 5 Solving these equations, we get A = \u2013 5  and B = 10\nThus,\n2\n2\n1\n5\n6\nx\nx \u2013 x\n+\n+\n =\n5\n10\n1\n2\n3\nx \u2013\nx \u2013\n\u2212\n+\nTherefore,\n2\n2\n1\n5\n6\nx\ndx\nx \u2013 x\n+\n+\n\u222b\n =\n1\n5\n10\n2\n3\ndx\ndx\ndx\nx \u2013\nx \u2013\n\u2212\n+\n\u222b\n\u222b\n\u222b\n= x \u2013 5 log |x \u2013 2| + 10 log |x \u2013 3| + C Example 13 Find \n32\n2\n(\n1) (\n3)\nx\ndx\nx\n\u2212x\n+\n+\n\u222b\nSolution The integrand is of the type as given in Table 7 2 (4)"
    },
    {
        "Chapter": "1",
        "sentence_range": "3668-3671",
        "Text": "Solving these equations, we get A = \u2013 5  and B = 10\nThus,\n2\n2\n1\n5\n6\nx\nx \u2013 x\n+\n+\n =\n5\n10\n1\n2\n3\nx \u2013\nx \u2013\n\u2212\n+\nTherefore,\n2\n2\n1\n5\n6\nx\ndx\nx \u2013 x\n+\n+\n\u222b\n =\n1\n5\n10\n2\n3\ndx\ndx\ndx\nx \u2013\nx \u2013\n\u2212\n+\n\u222b\n\u222b\n\u222b\n= x \u2013 5 log |x \u2013 2| + 10 log |x \u2013 3| + C Example 13 Find \n32\n2\n(\n1) (\n3)\nx\ndx\nx\n\u2212x\n+\n+\n\u222b\nSolution The integrand is of the type as given in Table 7 2 (4) We write\n32\n2\n(\n1) (\n3)\nx \u2013\nx\nx\n+\n+\n =\n2\nA\nB\nC\n1\n3\n(\n1)\nx\nx\n+x\n+\n+\n+\n+\nSo that\n3x \u2013 2 = A (x + 1) (x + 3) + B (x + 3) + C (x + 1)2\n= A (x2 + 4x + 3) + B (x + 3) + C (x2 + 2x + 1 )\nComparing coefficient of x 2, x and constant term on both sides, we get\nA + C = 0, 4A + B + 2C = 3 and 3A + 3B + C = \u2013 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "3669-3672",
        "Text": "Example 13 Find \n32\n2\n(\n1) (\n3)\nx\ndx\nx\n\u2212x\n+\n+\n\u222b\nSolution The integrand is of the type as given in Table 7 2 (4) We write\n32\n2\n(\n1) (\n3)\nx \u2013\nx\nx\n+\n+\n =\n2\nA\nB\nC\n1\n3\n(\n1)\nx\nx\n+x\n+\n+\n+\n+\nSo that\n3x \u2013 2 = A (x + 1) (x + 3) + B (x + 3) + C (x + 1)2\n= A (x2 + 4x + 3) + B (x + 3) + C (x2 + 2x + 1 )\nComparing coefficient of x 2, x and constant term on both sides, we get\nA + C = 0, 4A + B + 2C = 3 and 3A + 3B + C = \u2013 2 Solving these equations, we get\n11\n5\n11\nA\nB\nand C\n4\n2\n4\n\u2013\n\u2013\n,\n=\n=\n="
    },
    {
        "Chapter": "1",
        "sentence_range": "3670-3673",
        "Text": "2 (4) We write\n32\n2\n(\n1) (\n3)\nx \u2013\nx\nx\n+\n+\n =\n2\nA\nB\nC\n1\n3\n(\n1)\nx\nx\n+x\n+\n+\n+\n+\nSo that\n3x \u2013 2 = A (x + 1) (x + 3) + B (x + 3) + C (x + 1)2\n= A (x2 + 4x + 3) + B (x + 3) + C (x2 + 2x + 1 )\nComparing coefficient of x 2, x and constant term on both sides, we get\nA + C = 0, 4A + B + 2C = 3 and 3A + 3B + C = \u2013 2 Solving these equations, we get\n11\n5\n11\nA\nB\nand C\n4\n2\n4\n\u2013\n\u2013\n,\n=\n=\n= Thus the integrand is given by\n32\n2\n(\n1) (\n3)\nx\nx\nx\n\u2212\n+\n+\n =\n2\n11\n5\n11\n4(\n1)\n4(\n3)\n2(\n1)\n\u2013\n\u2013\nx\nx\nx\n+\n+\n+\nTherefore,\n32\n2\n(\n1) (\n3)\nx\nx\nx\n\u2212\n+\n+\n\u222b\n =\n2\n11\n5\n11\n4\n1\n2\n4\n3\n(\n1)\ndx\ndx\ndx\n\u2013\nx\nx\nx\n\u2212\n+\n+\n+\n\u222b\n\u222b\n\u222b\n=\n11\n5\n11\nlog\n+1\nlog\n3\nC\n4\n2( + 1)\n4\nx\nx\nx\n+\n\u2212\n+\n+\n= 11\n+1\n5\nlog\n+C\n4\n+3\n2 ( +1)\nxx\nx\n+\n320\nMATHEMATICS\nExample 14 Find \n2\n2\n2\n(\n1) (\n4)\nx\ndx\nx\nx\n+\n+\n\u222b\nSolution  Consider \n2\n2\n2\n(\n1) (\n4)\nx\nx\nx\n+\n+\n and put x2 = y"
    },
    {
        "Chapter": "1",
        "sentence_range": "3671-3674",
        "Text": "We write\n32\n2\n(\n1) (\n3)\nx \u2013\nx\nx\n+\n+\n =\n2\nA\nB\nC\n1\n3\n(\n1)\nx\nx\n+x\n+\n+\n+\n+\nSo that\n3x \u2013 2 = A (x + 1) (x + 3) + B (x + 3) + C (x + 1)2\n= A (x2 + 4x + 3) + B (x + 3) + C (x2 + 2x + 1 )\nComparing coefficient of x 2, x and constant term on both sides, we get\nA + C = 0, 4A + B + 2C = 3 and 3A + 3B + C = \u2013 2 Solving these equations, we get\n11\n5\n11\nA\nB\nand C\n4\n2\n4\n\u2013\n\u2013\n,\n=\n=\n= Thus the integrand is given by\n32\n2\n(\n1) (\n3)\nx\nx\nx\n\u2212\n+\n+\n =\n2\n11\n5\n11\n4(\n1)\n4(\n3)\n2(\n1)\n\u2013\n\u2013\nx\nx\nx\n+\n+\n+\nTherefore,\n32\n2\n(\n1) (\n3)\nx\nx\nx\n\u2212\n+\n+\n\u222b\n =\n2\n11\n5\n11\n4\n1\n2\n4\n3\n(\n1)\ndx\ndx\ndx\n\u2013\nx\nx\nx\n\u2212\n+\n+\n+\n\u222b\n\u222b\n\u222b\n=\n11\n5\n11\nlog\n+1\nlog\n3\nC\n4\n2( + 1)\n4\nx\nx\nx\n+\n\u2212\n+\n+\n= 11\n+1\n5\nlog\n+C\n4\n+3\n2 ( +1)\nxx\nx\n+\n320\nMATHEMATICS\nExample 14 Find \n2\n2\n2\n(\n1) (\n4)\nx\ndx\nx\nx\n+\n+\n\u222b\nSolution  Consider \n2\n2\n2\n(\n1) (\n4)\nx\nx\nx\n+\n+\n and put x2 = y Then\n2\n2\n2\n(\n1) (\n4)\nx\nx\nx\n+\n+\n = (\n1) (\n4)\ny\ny\ny\n+\n+\nWrite\n(\n1) (\n4)\ny\ny\ny\n+\n+\n =\nA\nB\n1\n4\ny\ny\n+\n+\n+\nSo that\ny =  A (y + 4) + B (y + 1)\nComparing coefficients of y and constant terms on both sides, we get A + B = 1\nand 4A + B = 0, which give\nA =\n1\n4\nand B\n3\n3\n\u2212\n=\nThus,\n2\n2\n2\n(\n1) (\n4)\nx\nx\nx\n+\n+\n =\n2\n2\n1\n4\n3(\n1)\n3 (\n4)\n\u2013\nx\nx\n+\n+\n+\nTherefore,\n2\n2\n2\n(\n1) (\n4)\nx dx\nx\nx\n+\n+\n\u222b\n =\n2\n2\n1\n4\n3\n3\n1\n4\ndx\ndx\n\u2013\nx\nx\n+\n+\n+\n\u222b\n\u222b\n=\n1\n1\n1\n4\n1\ntan\ntan\nC\n3\n3\n2\n2\n\u2013\n\u2013 x\n\u2013\nx +\n\u00d7\n+\n=\n1\n1\n1\n2\ntan\ntan\nC\n3\n3\n2\n\u2013\n\u2013 x\n\u2013\nx +\n+\nIn the above example, the substitution was made only for the partial fraction part\nand not for the integration part"
    },
    {
        "Chapter": "1",
        "sentence_range": "3672-3675",
        "Text": "Solving these equations, we get\n11\n5\n11\nA\nB\nand C\n4\n2\n4\n\u2013\n\u2013\n,\n=\n=\n= Thus the integrand is given by\n32\n2\n(\n1) (\n3)\nx\nx\nx\n\u2212\n+\n+\n =\n2\n11\n5\n11\n4(\n1)\n4(\n3)\n2(\n1)\n\u2013\n\u2013\nx\nx\nx\n+\n+\n+\nTherefore,\n32\n2\n(\n1) (\n3)\nx\nx\nx\n\u2212\n+\n+\n\u222b\n =\n2\n11\n5\n11\n4\n1\n2\n4\n3\n(\n1)\ndx\ndx\ndx\n\u2013\nx\nx\nx\n\u2212\n+\n+\n+\n\u222b\n\u222b\n\u222b\n=\n11\n5\n11\nlog\n+1\nlog\n3\nC\n4\n2( + 1)\n4\nx\nx\nx\n+\n\u2212\n+\n+\n= 11\n+1\n5\nlog\n+C\n4\n+3\n2 ( +1)\nxx\nx\n+\n320\nMATHEMATICS\nExample 14 Find \n2\n2\n2\n(\n1) (\n4)\nx\ndx\nx\nx\n+\n+\n\u222b\nSolution  Consider \n2\n2\n2\n(\n1) (\n4)\nx\nx\nx\n+\n+\n and put x2 = y Then\n2\n2\n2\n(\n1) (\n4)\nx\nx\nx\n+\n+\n = (\n1) (\n4)\ny\ny\ny\n+\n+\nWrite\n(\n1) (\n4)\ny\ny\ny\n+\n+\n =\nA\nB\n1\n4\ny\ny\n+\n+\n+\nSo that\ny =  A (y + 4) + B (y + 1)\nComparing coefficients of y and constant terms on both sides, we get A + B = 1\nand 4A + B = 0, which give\nA =\n1\n4\nand B\n3\n3\n\u2212\n=\nThus,\n2\n2\n2\n(\n1) (\n4)\nx\nx\nx\n+\n+\n =\n2\n2\n1\n4\n3(\n1)\n3 (\n4)\n\u2013\nx\nx\n+\n+\n+\nTherefore,\n2\n2\n2\n(\n1) (\n4)\nx dx\nx\nx\n+\n+\n\u222b\n =\n2\n2\n1\n4\n3\n3\n1\n4\ndx\ndx\n\u2013\nx\nx\n+\n+\n+\n\u222b\n\u222b\n=\n1\n1\n1\n4\n1\ntan\ntan\nC\n3\n3\n2\n2\n\u2013\n\u2013 x\n\u2013\nx +\n\u00d7\n+\n=\n1\n1\n1\n2\ntan\ntan\nC\n3\n3\n2\n\u2013\n\u2013 x\n\u2013\nx +\n+\nIn the above example, the substitution was made only for the partial fraction part\nand not for the integration part Now, we consider an example, where the integration\ninvolves a combination of the substitution method and the partial fraction method"
    },
    {
        "Chapter": "1",
        "sentence_range": "3673-3676",
        "Text": "Thus the integrand is given by\n32\n2\n(\n1) (\n3)\nx\nx\nx\n\u2212\n+\n+\n =\n2\n11\n5\n11\n4(\n1)\n4(\n3)\n2(\n1)\n\u2013\n\u2013\nx\nx\nx\n+\n+\n+\nTherefore,\n32\n2\n(\n1) (\n3)\nx\nx\nx\n\u2212\n+\n+\n\u222b\n =\n2\n11\n5\n11\n4\n1\n2\n4\n3\n(\n1)\ndx\ndx\ndx\n\u2013\nx\nx\nx\n\u2212\n+\n+\n+\n\u222b\n\u222b\n\u222b\n=\n11\n5\n11\nlog\n+1\nlog\n3\nC\n4\n2( + 1)\n4\nx\nx\nx\n+\n\u2212\n+\n+\n= 11\n+1\n5\nlog\n+C\n4\n+3\n2 ( +1)\nxx\nx\n+\n320\nMATHEMATICS\nExample 14 Find \n2\n2\n2\n(\n1) (\n4)\nx\ndx\nx\nx\n+\n+\n\u222b\nSolution  Consider \n2\n2\n2\n(\n1) (\n4)\nx\nx\nx\n+\n+\n and put x2 = y Then\n2\n2\n2\n(\n1) (\n4)\nx\nx\nx\n+\n+\n = (\n1) (\n4)\ny\ny\ny\n+\n+\nWrite\n(\n1) (\n4)\ny\ny\ny\n+\n+\n =\nA\nB\n1\n4\ny\ny\n+\n+\n+\nSo that\ny =  A (y + 4) + B (y + 1)\nComparing coefficients of y and constant terms on both sides, we get A + B = 1\nand 4A + B = 0, which give\nA =\n1\n4\nand B\n3\n3\n\u2212\n=\nThus,\n2\n2\n2\n(\n1) (\n4)\nx\nx\nx\n+\n+\n =\n2\n2\n1\n4\n3(\n1)\n3 (\n4)\n\u2013\nx\nx\n+\n+\n+\nTherefore,\n2\n2\n2\n(\n1) (\n4)\nx dx\nx\nx\n+\n+\n\u222b\n =\n2\n2\n1\n4\n3\n3\n1\n4\ndx\ndx\n\u2013\nx\nx\n+\n+\n+\n\u222b\n\u222b\n=\n1\n1\n1\n4\n1\ntan\ntan\nC\n3\n3\n2\n2\n\u2013\n\u2013 x\n\u2013\nx +\n\u00d7\n+\n=\n1\n1\n1\n2\ntan\ntan\nC\n3\n3\n2\n\u2013\n\u2013 x\n\u2013\nx +\n+\nIn the above example, the substitution was made only for the partial fraction part\nand not for the integration part Now, we consider an example, where the integration\ninvolves a combination of the substitution method and the partial fraction method Example 15 Find (\n)\n3 sin2\n2 cos\n5\ncos\n4sin\n\u2013\nd\n\u2013\n\u2013\n\u03c6\n\u03c6\n\u03c6\n\u03c6\n\u03c6\n\u222b\nSolution Let y = sin\u03c6\nThen\ndy = cos\u03c6 d\u03c6\nINTEGRALS         321\nTherefore,\n(\n)\n3sin2\n2 cos\n5\ncos\n4sin\n\u2013\nd\n\u2013\n\u2013\n\u03c6\n\u03c6\n\u03c6\n\u03c6\n\u03c6\n\u222b\n =\n2\n(3 \u2013 2)\n5\n(1\n)\n4\ny\ndy\n\u2013\n\u2013 y\n\u2013\ny\n\u222b\n=\n32\n2\n4\n4\ny \u2013\ndy\ny \u2013 y +\n\u222b\n=\n(\n)\n2\n3\n2\nI (say)\n2\ny \u2013\ny \u2013\n=\n\u222b\nNow, we write\n(\n)\n2\n3\n2\n2\ny \u2013\ny \u2013\n =\n2\nA\nB\n2\n(\n2)\ny\n+y\n\u2212\n\u2212\n[by Table 7"
    },
    {
        "Chapter": "1",
        "sentence_range": "3674-3677",
        "Text": "Then\n2\n2\n2\n(\n1) (\n4)\nx\nx\nx\n+\n+\n = (\n1) (\n4)\ny\ny\ny\n+\n+\nWrite\n(\n1) (\n4)\ny\ny\ny\n+\n+\n =\nA\nB\n1\n4\ny\ny\n+\n+\n+\nSo that\ny =  A (y + 4) + B (y + 1)\nComparing coefficients of y and constant terms on both sides, we get A + B = 1\nand 4A + B = 0, which give\nA =\n1\n4\nand B\n3\n3\n\u2212\n=\nThus,\n2\n2\n2\n(\n1) (\n4)\nx\nx\nx\n+\n+\n =\n2\n2\n1\n4\n3(\n1)\n3 (\n4)\n\u2013\nx\nx\n+\n+\n+\nTherefore,\n2\n2\n2\n(\n1) (\n4)\nx dx\nx\nx\n+\n+\n\u222b\n =\n2\n2\n1\n4\n3\n3\n1\n4\ndx\ndx\n\u2013\nx\nx\n+\n+\n+\n\u222b\n\u222b\n=\n1\n1\n1\n4\n1\ntan\ntan\nC\n3\n3\n2\n2\n\u2013\n\u2013 x\n\u2013\nx +\n\u00d7\n+\n=\n1\n1\n1\n2\ntan\ntan\nC\n3\n3\n2\n\u2013\n\u2013 x\n\u2013\nx +\n+\nIn the above example, the substitution was made only for the partial fraction part\nand not for the integration part Now, we consider an example, where the integration\ninvolves a combination of the substitution method and the partial fraction method Example 15 Find (\n)\n3 sin2\n2 cos\n5\ncos\n4sin\n\u2013\nd\n\u2013\n\u2013\n\u03c6\n\u03c6\n\u03c6\n\u03c6\n\u03c6\n\u222b\nSolution Let y = sin\u03c6\nThen\ndy = cos\u03c6 d\u03c6\nINTEGRALS         321\nTherefore,\n(\n)\n3sin2\n2 cos\n5\ncos\n4sin\n\u2013\nd\n\u2013\n\u2013\n\u03c6\n\u03c6\n\u03c6\n\u03c6\n\u03c6\n\u222b\n =\n2\n(3 \u2013 2)\n5\n(1\n)\n4\ny\ndy\n\u2013\n\u2013 y\n\u2013\ny\n\u222b\n=\n32\n2\n4\n4\ny \u2013\ndy\ny \u2013 y +\n\u222b\n=\n(\n)\n2\n3\n2\nI (say)\n2\ny \u2013\ny \u2013\n=\n\u222b\nNow, we write\n(\n)\n2\n3\n2\n2\ny \u2013\ny \u2013\n =\n2\nA\nB\n2\n(\n2)\ny\n+y\n\u2212\n\u2212\n[by Table 7 2 (2)]\nTherefore,\n3y \u2013 2 = A (y \u2013 2) + B\nComparing the coefficients of y and constant term, we get A = 3 and B \u2013 2A = \u2013 2,\nwhich gives A = 3 and B = 4"
    },
    {
        "Chapter": "1",
        "sentence_range": "3675-3678",
        "Text": "Now, we consider an example, where the integration\ninvolves a combination of the substitution method and the partial fraction method Example 15 Find (\n)\n3 sin2\n2 cos\n5\ncos\n4sin\n\u2013\nd\n\u2013\n\u2013\n\u03c6\n\u03c6\n\u03c6\n\u03c6\n\u03c6\n\u222b\nSolution Let y = sin\u03c6\nThen\ndy = cos\u03c6 d\u03c6\nINTEGRALS         321\nTherefore,\n(\n)\n3sin2\n2 cos\n5\ncos\n4sin\n\u2013\nd\n\u2013\n\u2013\n\u03c6\n\u03c6\n\u03c6\n\u03c6\n\u03c6\n\u222b\n =\n2\n(3 \u2013 2)\n5\n(1\n)\n4\ny\ndy\n\u2013\n\u2013 y\n\u2013\ny\n\u222b\n=\n32\n2\n4\n4\ny \u2013\ndy\ny \u2013 y +\n\u222b\n=\n(\n)\n2\n3\n2\nI (say)\n2\ny \u2013\ny \u2013\n=\n\u222b\nNow, we write\n(\n)\n2\n3\n2\n2\ny \u2013\ny \u2013\n =\n2\nA\nB\n2\n(\n2)\ny\n+y\n\u2212\n\u2212\n[by Table 7 2 (2)]\nTherefore,\n3y \u2013 2 = A (y \u2013 2) + B\nComparing the coefficients of y and constant term, we get A = 3 and B \u2013 2A = \u2013 2,\nwhich gives A = 3 and B = 4 Therefore, the required integral is given by\nI =\n2\n3\n4\n[\n+\n]\n2\n(\n2)\ndy\ny \u2013\ny \u2013\n\u222b\n = \n2\n3\n2+ 4\n(\n2)\ndy\ndy\ny \u2013\ny \u2013\n\u222b\n\u222b\n=\n1\n3log\n2\n4\nC\n2\ny\n\u2013\ny\n\uf8eb\n\uf8f6\n\u2212\n+\n+\n\uf8ec\n\uf8f7\n\u2212\n\uf8ed\n\uf8f8\n=\n4\n3log sin\n2\nC\n2\nsin\n\u2013\n\u03c6 \u2212\n+\n+\n\u03c6\n=\n4\n3log (2\nsin )\n+ C\n2\nsin\n\u2212\n\u03c6 +\n\u2212\n\u03c6\n (since, 2 \u2013 sin\u03c6 is always positive)\nExample 16 Find \n2\n2\n1\n(\n2) (\n1)\nx\nx\ndx\nx\nx\n+\n+\n+\n+\n\u222b\nSolution The integrand is a proper rational function"
    },
    {
        "Chapter": "1",
        "sentence_range": "3676-3679",
        "Text": "Example 15 Find (\n)\n3 sin2\n2 cos\n5\ncos\n4sin\n\u2013\nd\n\u2013\n\u2013\n\u03c6\n\u03c6\n\u03c6\n\u03c6\n\u03c6\n\u222b\nSolution Let y = sin\u03c6\nThen\ndy = cos\u03c6 d\u03c6\nINTEGRALS         321\nTherefore,\n(\n)\n3sin2\n2 cos\n5\ncos\n4sin\n\u2013\nd\n\u2013\n\u2013\n\u03c6\n\u03c6\n\u03c6\n\u03c6\n\u03c6\n\u222b\n =\n2\n(3 \u2013 2)\n5\n(1\n)\n4\ny\ndy\n\u2013\n\u2013 y\n\u2013\ny\n\u222b\n=\n32\n2\n4\n4\ny \u2013\ndy\ny \u2013 y +\n\u222b\n=\n(\n)\n2\n3\n2\nI (say)\n2\ny \u2013\ny \u2013\n=\n\u222b\nNow, we write\n(\n)\n2\n3\n2\n2\ny \u2013\ny \u2013\n =\n2\nA\nB\n2\n(\n2)\ny\n+y\n\u2212\n\u2212\n[by Table 7 2 (2)]\nTherefore,\n3y \u2013 2 = A (y \u2013 2) + B\nComparing the coefficients of y and constant term, we get A = 3 and B \u2013 2A = \u2013 2,\nwhich gives A = 3 and B = 4 Therefore, the required integral is given by\nI =\n2\n3\n4\n[\n+\n]\n2\n(\n2)\ndy\ny \u2013\ny \u2013\n\u222b\n = \n2\n3\n2+ 4\n(\n2)\ndy\ndy\ny \u2013\ny \u2013\n\u222b\n\u222b\n=\n1\n3log\n2\n4\nC\n2\ny\n\u2013\ny\n\uf8eb\n\uf8f6\n\u2212\n+\n+\n\uf8ec\n\uf8f7\n\u2212\n\uf8ed\n\uf8f8\n=\n4\n3log sin\n2\nC\n2\nsin\n\u2013\n\u03c6 \u2212\n+\n+\n\u03c6\n=\n4\n3log (2\nsin )\n+ C\n2\nsin\n\u2212\n\u03c6 +\n\u2212\n\u03c6\n (since, 2 \u2013 sin\u03c6 is always positive)\nExample 16 Find \n2\n2\n1\n(\n2) (\n1)\nx\nx\ndx\nx\nx\n+\n+\n+\n+\n\u222b\nSolution The integrand is a proper rational function Decompose the rational function\ninto partial fraction [Table 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "3677-3680",
        "Text": "2 (2)]\nTherefore,\n3y \u2013 2 = A (y \u2013 2) + B\nComparing the coefficients of y and constant term, we get A = 3 and B \u2013 2A = \u2013 2,\nwhich gives A = 3 and B = 4 Therefore, the required integral is given by\nI =\n2\n3\n4\n[\n+\n]\n2\n(\n2)\ndy\ny \u2013\ny \u2013\n\u222b\n = \n2\n3\n2+ 4\n(\n2)\ndy\ndy\ny \u2013\ny \u2013\n\u222b\n\u222b\n=\n1\n3log\n2\n4\nC\n2\ny\n\u2013\ny\n\uf8eb\n\uf8f6\n\u2212\n+\n+\n\uf8ec\n\uf8f7\n\u2212\n\uf8ed\n\uf8f8\n=\n4\n3log sin\n2\nC\n2\nsin\n\u2013\n\u03c6 \u2212\n+\n+\n\u03c6\n=\n4\n3log (2\nsin )\n+ C\n2\nsin\n\u2212\n\u03c6 +\n\u2212\n\u03c6\n (since, 2 \u2013 sin\u03c6 is always positive)\nExample 16 Find \n2\n2\n1\n(\n2) (\n1)\nx\nx\ndx\nx\nx\n+\n+\n+\n+\n\u222b\nSolution The integrand is a proper rational function Decompose the rational function\ninto partial fraction [Table 2 2(5)]"
    },
    {
        "Chapter": "1",
        "sentence_range": "3678-3681",
        "Text": "Therefore, the required integral is given by\nI =\n2\n3\n4\n[\n+\n]\n2\n(\n2)\ndy\ny \u2013\ny \u2013\n\u222b\n = \n2\n3\n2+ 4\n(\n2)\ndy\ndy\ny \u2013\ny \u2013\n\u222b\n\u222b\n=\n1\n3log\n2\n4\nC\n2\ny\n\u2013\ny\n\uf8eb\n\uf8f6\n\u2212\n+\n+\n\uf8ec\n\uf8f7\n\u2212\n\uf8ed\n\uf8f8\n=\n4\n3log sin\n2\nC\n2\nsin\n\u2013\n\u03c6 \u2212\n+\n+\n\u03c6\n=\n4\n3log (2\nsin )\n+ C\n2\nsin\n\u2212\n\u03c6 +\n\u2212\n\u03c6\n (since, 2 \u2013 sin\u03c6 is always positive)\nExample 16 Find \n2\n2\n1\n(\n2) (\n1)\nx\nx\ndx\nx\nx\n+\n+\n+\n+\n\u222b\nSolution The integrand is a proper rational function Decompose the rational function\ninto partial fraction [Table 2 2(5)] Write\n2\n2\n1\n(\n1) (\n2)\nx\nx\nx\n+x\n+\n+\n+\n =\n2\nA\nB + C\n2\n(\n1)\nx\nx\n+x\n+\n+\nTherefore,\nx2 + x + 1 = A (x2 + 1) + (Bx + C) (x + 2)\n322\nMATHEMATICS\nEquating the coefficients of x2, x and of constant term of both sides, we get\nA + B =1, 2B + C = 1 and A + 2C = 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "3679-3682",
        "Text": "Decompose the rational function\ninto partial fraction [Table 2 2(5)] Write\n2\n2\n1\n(\n1) (\n2)\nx\nx\nx\n+x\n+\n+\n+\n =\n2\nA\nB + C\n2\n(\n1)\nx\nx\n+x\n+\n+\nTherefore,\nx2 + x + 1 = A (x2 + 1) + (Bx + C) (x + 2)\n322\nMATHEMATICS\nEquating the coefficients of x2, x and of constant term of both sides, we get\nA + B =1, 2B + C = 1 and A + 2C = 1 Solving these equations, we get\n3\n2\n1\nA\n, B\nand C\n5\n5\n5\n=\n=\n=\nThus, the integrand is given by\n2\n2\n1\n(\n1) (\n2)\nx\nx\nx\n+x\n+\n+\n+\n =\n2\n2\n1\n3\n5\n5\n5 (\n2)\n1\nx\nx\nx\n+\n+\n+\n+\n = \n2\n3\n1\n2\n1\n5 (\n2)\n5\n1\nx\nx\nx\n+\n\uf8eb\n\uf8f6\n+\n\uf8ec\n\uf8f7\n+\n+\n\uf8ed\n\uf8f8\nTherefore,\n2\n2\n1\n(\n+1) (\n2)\nx\nx\ndx\nx\n+x\n+\n+\n\u222b\n =\n2\n2\n3\n1\n2\n1\n1\n5\n2\n5\n5\n1\n1\ndx\nx\ndx\ndx\nx\nx\nx\n+\n+\n+\n+\n+\n\u222b\n\u222b\n\u222b\n=\n2\n1\n3\n1\n1\nlog\n2\nlog\n1\ntan\nC\n5\n5\n5\n\u2013\nx\nx\nx\n+\n+\n+\n+\n+\nEXERCISE 7"
    },
    {
        "Chapter": "1",
        "sentence_range": "3680-3683",
        "Text": "2(5)] Write\n2\n2\n1\n(\n1) (\n2)\nx\nx\nx\n+x\n+\n+\n+\n =\n2\nA\nB + C\n2\n(\n1)\nx\nx\n+x\n+\n+\nTherefore,\nx2 + x + 1 = A (x2 + 1) + (Bx + C) (x + 2)\n322\nMATHEMATICS\nEquating the coefficients of x2, x and of constant term of both sides, we get\nA + B =1, 2B + C = 1 and A + 2C = 1 Solving these equations, we get\n3\n2\n1\nA\n, B\nand C\n5\n5\n5\n=\n=\n=\nThus, the integrand is given by\n2\n2\n1\n(\n1) (\n2)\nx\nx\nx\n+x\n+\n+\n+\n =\n2\n2\n1\n3\n5\n5\n5 (\n2)\n1\nx\nx\nx\n+\n+\n+\n+\n = \n2\n3\n1\n2\n1\n5 (\n2)\n5\n1\nx\nx\nx\n+\n\uf8eb\n\uf8f6\n+\n\uf8ec\n\uf8f7\n+\n+\n\uf8ed\n\uf8f8\nTherefore,\n2\n2\n1\n(\n+1) (\n2)\nx\nx\ndx\nx\n+x\n+\n+\n\u222b\n =\n2\n2\n3\n1\n2\n1\n1\n5\n2\n5\n5\n1\n1\ndx\nx\ndx\ndx\nx\nx\nx\n+\n+\n+\n+\n+\n\u222b\n\u222b\n\u222b\n=\n2\n1\n3\n1\n1\nlog\n2\nlog\n1\ntan\nC\n5\n5\n5\n\u2013\nx\nx\nx\n+\n+\n+\n+\n+\nEXERCISE 7 5\nIntegrate the rational functions in Exercises 1 to 21"
    },
    {
        "Chapter": "1",
        "sentence_range": "3681-3684",
        "Text": "Write\n2\n2\n1\n(\n1) (\n2)\nx\nx\nx\n+x\n+\n+\n+\n =\n2\nA\nB + C\n2\n(\n1)\nx\nx\n+x\n+\n+\nTherefore,\nx2 + x + 1 = A (x2 + 1) + (Bx + C) (x + 2)\n322\nMATHEMATICS\nEquating the coefficients of x2, x and of constant term of both sides, we get\nA + B =1, 2B + C = 1 and A + 2C = 1 Solving these equations, we get\n3\n2\n1\nA\n, B\nand C\n5\n5\n5\n=\n=\n=\nThus, the integrand is given by\n2\n2\n1\n(\n1) (\n2)\nx\nx\nx\n+x\n+\n+\n+\n =\n2\n2\n1\n3\n5\n5\n5 (\n2)\n1\nx\nx\nx\n+\n+\n+\n+\n = \n2\n3\n1\n2\n1\n5 (\n2)\n5\n1\nx\nx\nx\n+\n\uf8eb\n\uf8f6\n+\n\uf8ec\n\uf8f7\n+\n+\n\uf8ed\n\uf8f8\nTherefore,\n2\n2\n1\n(\n+1) (\n2)\nx\nx\ndx\nx\n+x\n+\n+\n\u222b\n =\n2\n2\n3\n1\n2\n1\n1\n5\n2\n5\n5\n1\n1\ndx\nx\ndx\ndx\nx\nx\nx\n+\n+\n+\n+\n+\n\u222b\n\u222b\n\u222b\n=\n2\n1\n3\n1\n1\nlog\n2\nlog\n1\ntan\nC\n5\n5\n5\n\u2013\nx\nx\nx\n+\n+\n+\n+\n+\nEXERCISE 7 5\nIntegrate the rational functions in Exercises 1 to 21 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "3682-3685",
        "Text": "Solving these equations, we get\n3\n2\n1\nA\n, B\nand C\n5\n5\n5\n=\n=\n=\nThus, the integrand is given by\n2\n2\n1\n(\n1) (\n2)\nx\nx\nx\n+x\n+\n+\n+\n =\n2\n2\n1\n3\n5\n5\n5 (\n2)\n1\nx\nx\nx\n+\n+\n+\n+\n = \n2\n3\n1\n2\n1\n5 (\n2)\n5\n1\nx\nx\nx\n+\n\uf8eb\n\uf8f6\n+\n\uf8ec\n\uf8f7\n+\n+\n\uf8ed\n\uf8f8\nTherefore,\n2\n2\n1\n(\n+1) (\n2)\nx\nx\ndx\nx\n+x\n+\n+\n\u222b\n =\n2\n2\n3\n1\n2\n1\n1\n5\n2\n5\n5\n1\n1\ndx\nx\ndx\ndx\nx\nx\nx\n+\n+\n+\n+\n+\n\u222b\n\u222b\n\u222b\n=\n2\n1\n3\n1\n1\nlog\n2\nlog\n1\ntan\nC\n5\n5\n5\n\u2013\nx\nx\nx\n+\n+\n+\n+\n+\nEXERCISE 7 5\nIntegrate the rational functions in Exercises 1 to 21 1 (\n1) (\n2)\nx\nx\nx\n+\n+\n2"
    },
    {
        "Chapter": "1",
        "sentence_range": "3683-3686",
        "Text": "5\nIntegrate the rational functions in Exercises 1 to 21 1 (\n1) (\n2)\nx\nx\nx\n+\n+\n2 2\n1\nx \u20139\n3"
    },
    {
        "Chapter": "1",
        "sentence_range": "3684-3687",
        "Text": "1 (\n1) (\n2)\nx\nx\nx\n+\n+\n2 2\n1\nx \u20139\n3 3\n1\n(\n1) (\n2) (\n3)\nx \u2013\nx \u2013\nx \u2013\nx \u2013\n4"
    },
    {
        "Chapter": "1",
        "sentence_range": "3685-3688",
        "Text": "(\n1) (\n2)\nx\nx\nx\n+\n+\n2 2\n1\nx \u20139\n3 3\n1\n(\n1) (\n2) (\n3)\nx \u2013\nx \u2013\nx \u2013\nx \u2013\n4 (\n1) (\n2) (\n3)\nx\nx \u2013\nx \u2013\nx \u2013\n5"
    },
    {
        "Chapter": "1",
        "sentence_range": "3686-3689",
        "Text": "2\n1\nx \u20139\n3 3\n1\n(\n1) (\n2) (\n3)\nx \u2013\nx \u2013\nx \u2013\nx \u2013\n4 (\n1) (\n2) (\n3)\nx\nx \u2013\nx \u2013\nx \u2013\n5 2\n32\n2\nx\nx\n+x\n+\n6"
    },
    {
        "Chapter": "1",
        "sentence_range": "3687-3690",
        "Text": "3\n1\n(\n1) (\n2) (\n3)\nx \u2013\nx \u2013\nx \u2013\nx \u2013\n4 (\n1) (\n2) (\n3)\nx\nx \u2013\nx \u2013\nx \u2013\n5 2\n32\n2\nx\nx\n+x\n+\n6 2\n(11\n2 )\n\u2013 x\nx\n\u2013 x\n7"
    },
    {
        "Chapter": "1",
        "sentence_range": "3688-3691",
        "Text": "(\n1) (\n2) (\n3)\nx\nx \u2013\nx \u2013\nx \u2013\n5 2\n32\n2\nx\nx\n+x\n+\n6 2\n(11\n2 )\n\u2013 x\nx\n\u2013 x\n7 (2\n1) ( \u2013 1)\nx\nx\nx\n+\n8"
    },
    {
        "Chapter": "1",
        "sentence_range": "3689-3692",
        "Text": "2\n32\n2\nx\nx\n+x\n+\n6 2\n(11\n2 )\n\u2013 x\nx\n\u2013 x\n7 (2\n1) ( \u2013 1)\nx\nx\nx\n+\n8 2\n(\n1) (\n2)\nx\nx \u2013\nx +\n9"
    },
    {
        "Chapter": "1",
        "sentence_range": "3690-3693",
        "Text": "2\n(11\n2 )\n\u2013 x\nx\n\u2013 x\n7 (2\n1) ( \u2013 1)\nx\nx\nx\n+\n8 2\n(\n1) (\n2)\nx\nx \u2013\nx +\n9 3\n32\n5\n1\nx \u2013 xx\n+x\n\u2212\n+\n10"
    },
    {
        "Chapter": "1",
        "sentence_range": "3691-3694",
        "Text": "(2\n1) ( \u2013 1)\nx\nx\nx\n+\n8 2\n(\n1) (\n2)\nx\nx \u2013\nx +\n9 3\n32\n5\n1\nx \u2013 xx\n+x\n\u2212\n+\n10 2\n2\n3\n(\n1) (2\n3)\nx \u2013x\nx\n\u2212\n+\n11"
    },
    {
        "Chapter": "1",
        "sentence_range": "3692-3695",
        "Text": "2\n(\n1) (\n2)\nx\nx \u2013\nx +\n9 3\n32\n5\n1\nx \u2013 xx\n+x\n\u2212\n+\n10 2\n2\n3\n(\n1) (2\n3)\nx \u2013x\nx\n\u2212\n+\n11 2\n5\n(\n1) (\n4)\nx\nx\nx\n+\n\u2212\n12"
    },
    {
        "Chapter": "1",
        "sentence_range": "3693-3696",
        "Text": "3\n32\n5\n1\nx \u2013 xx\n+x\n\u2212\n+\n10 2\n2\n3\n(\n1) (2\n3)\nx \u2013x\nx\n\u2212\n+\n11 2\n5\n(\n1) (\n4)\nx\nx\nx\n+\n\u2212\n12 3\n2\n1\n1\nx\nx\nx\n+\n+\n\u2212\n13"
    },
    {
        "Chapter": "1",
        "sentence_range": "3694-3697",
        "Text": "2\n2\n3\n(\n1) (2\n3)\nx \u2013x\nx\n\u2212\n+\n11 2\n5\n(\n1) (\n4)\nx\nx\nx\n+\n\u2212\n12 3\n2\n1\n1\nx\nx\nx\n+\n+\n\u2212\n13 2\n2\n(1\n) (1\n)\nx\nx\n\u2212\n+\n14"
    },
    {
        "Chapter": "1",
        "sentence_range": "3695-3698",
        "Text": "2\n5\n(\n1) (\n4)\nx\nx\nx\n+\n\u2212\n12 3\n2\n1\n1\nx\nx\nx\n+\n+\n\u2212\n13 2\n2\n(1\n) (1\n)\nx\nx\n\u2212\n+\n14 2\n3\n1\n(\n2)\nx \u2013\nx +\n15"
    },
    {
        "Chapter": "1",
        "sentence_range": "3696-3699",
        "Text": "3\n2\n1\n1\nx\nx\nx\n+\n+\n\u2212\n13 2\n2\n(1\n) (1\n)\nx\nx\n\u2212\n+\n14 2\n3\n1\n(\n2)\nx \u2013\nx +\n15 4\n1\n1\nx \u2212\n16"
    },
    {
        "Chapter": "1",
        "sentence_range": "3697-3700",
        "Text": "2\n2\n(1\n) (1\n)\nx\nx\n\u2212\n+\n14 2\n3\n1\n(\n2)\nx \u2013\nx +\n15 4\n1\n1\nx \u2212\n16 1\n(\n1)\nn\nx x +\n [Hint:  multiply numerator and denominator by x n \u2013 1 and put xn = t ]\n17"
    },
    {
        "Chapter": "1",
        "sentence_range": "3698-3701",
        "Text": "2\n3\n1\n(\n2)\nx \u2013\nx +\n15 4\n1\n1\nx \u2212\n16 1\n(\n1)\nn\nx x +\n [Hint:  multiply numerator and denominator by x n \u2013 1 and put xn = t ]\n17 cos\n(1\u2013 sin ) (2 \u2013 sin )\nx\nx\nx\n[Hint : Put sin x = t]\nINTEGRALS         323\n18"
    },
    {
        "Chapter": "1",
        "sentence_range": "3699-3702",
        "Text": "4\n1\n1\nx \u2212\n16 1\n(\n1)\nn\nx x +\n [Hint:  multiply numerator and denominator by x n \u2013 1 and put xn = t ]\n17 cos\n(1\u2013 sin ) (2 \u2013 sin )\nx\nx\nx\n[Hint : Put sin x = t]\nINTEGRALS         323\n18 2\n2\n2\n2\n(\n1) (\n2)\n(\n3)(\n4)\nx\nx\nx\nx\n+\n+\n+\n+\n19"
    },
    {
        "Chapter": "1",
        "sentence_range": "3700-3703",
        "Text": "1\n(\n1)\nn\nx x +\n [Hint:  multiply numerator and denominator by x n \u2013 1 and put xn = t ]\n17 cos\n(1\u2013 sin ) (2 \u2013 sin )\nx\nx\nx\n[Hint : Put sin x = t]\nINTEGRALS         323\n18 2\n2\n2\n2\n(\n1) (\n2)\n(\n3)(\n4)\nx\nx\nx\nx\n+\n+\n+\n+\n19 2\n2\n2\n(\n1) (\n3)\nx\nx\nx\n+\n+\n20"
    },
    {
        "Chapter": "1",
        "sentence_range": "3701-3704",
        "Text": "cos\n(1\u2013 sin ) (2 \u2013 sin )\nx\nx\nx\n[Hint : Put sin x = t]\nINTEGRALS         323\n18 2\n2\n2\n2\n(\n1) (\n2)\n(\n3)(\n4)\nx\nx\nx\nx\n+\n+\n+\n+\n19 2\n2\n2\n(\n1) (\n3)\nx\nx\nx\n+\n+\n20 4\n1\n(\n1)\nx x \u2013\n21"
    },
    {
        "Chapter": "1",
        "sentence_range": "3702-3705",
        "Text": "2\n2\n2\n2\n(\n1) (\n2)\n(\n3)(\n4)\nx\nx\nx\nx\n+\n+\n+\n+\n19 2\n2\n2\n(\n1) (\n3)\nx\nx\nx\n+\n+\n20 4\n1\n(\n1)\nx x \u2013\n21 1\n(\n1)\nx\ne \u2013\n[Hint : Put ex = t]\nChoose the correct answer in each of the Exercises 22 and 23"
    },
    {
        "Chapter": "1",
        "sentence_range": "3703-3706",
        "Text": "2\n2\n2\n(\n1) (\n3)\nx\nx\nx\n+\n+\n20 4\n1\n(\n1)\nx x \u2013\n21 1\n(\n1)\nx\ne \u2013\n[Hint : Put ex = t]\nChoose the correct answer in each of the Exercises 22 and 23 22"
    },
    {
        "Chapter": "1",
        "sentence_range": "3704-3707",
        "Text": "4\n1\n(\n1)\nx x \u2013\n21 1\n(\n1)\nx\ne \u2013\n[Hint : Put ex = t]\nChoose the correct answer in each of the Exercises 22 and 23 22 (\n1) (\n2)\nx dx\nx\nx\n\u2212\n\u2212\n\u222b\n equals\n(A)\n2\n(\n1)\nlog\nC\n2\nxx\n\u2212\n+\n\u2212\n(B)\n2\n(\n2)\nlog\nC\n1\nx\nx\n\u2212\n+\n\u2212\n(C)\n12\nlog\nC\n2\nxx\n\u2212\n\uf8eb\n\uf8f6\n+\n\uf8ec\n\uf8f7\n\u2212\n\uf8ed\n\uf8f8\n(D) log (\n1) (\n2)\nC\nx\nx\n\u2212\n\u2212\n+\n23"
    },
    {
        "Chapter": "1",
        "sentence_range": "3705-3708",
        "Text": "1\n(\n1)\nx\ne \u2013\n[Hint : Put ex = t]\nChoose the correct answer in each of the Exercises 22 and 23 22 (\n1) (\n2)\nx dx\nx\nx\n\u2212\n\u2212\n\u222b\n equals\n(A)\n2\n(\n1)\nlog\nC\n2\nxx\n\u2212\n+\n\u2212\n(B)\n2\n(\n2)\nlog\nC\n1\nx\nx\n\u2212\n+\n\u2212\n(C)\n12\nlog\nC\n2\nxx\n\u2212\n\uf8eb\n\uf8f6\n+\n\uf8ec\n\uf8f7\n\u2212\n\uf8ed\n\uf8f8\n(D) log (\n1) (\n2)\nC\nx\nx\n\u2212\n\u2212\n+\n23 (2\n1)\ndx\n\u222bx x +\nequals\n(A)\n2\n1\nlog\nlog (\n+1) + C\n2\nx\nx\n\u2212\n(B)\n2\n1\nlog\nlog (\n+1) + C\n2\nx\nx\n+\n(C)\n2\n1\nlog\nlog (\n+1) +C\n2\nx\nx\n\u2212\n+\n(D)\n2\n1log\nlog (\n+1) + C\n2\nx\nx\n+\n7"
    },
    {
        "Chapter": "1",
        "sentence_range": "3706-3709",
        "Text": "22 (\n1) (\n2)\nx dx\nx\nx\n\u2212\n\u2212\n\u222b\n equals\n(A)\n2\n(\n1)\nlog\nC\n2\nxx\n\u2212\n+\n\u2212\n(B)\n2\n(\n2)\nlog\nC\n1\nx\nx\n\u2212\n+\n\u2212\n(C)\n12\nlog\nC\n2\nxx\n\u2212\n\uf8eb\n\uf8f6\n+\n\uf8ec\n\uf8f7\n\u2212\n\uf8ed\n\uf8f8\n(D) log (\n1) (\n2)\nC\nx\nx\n\u2212\n\u2212\n+\n23 (2\n1)\ndx\n\u222bx x +\nequals\n(A)\n2\n1\nlog\nlog (\n+1) + C\n2\nx\nx\n\u2212\n(B)\n2\n1\nlog\nlog (\n+1) + C\n2\nx\nx\n+\n(C)\n2\n1\nlog\nlog (\n+1) +C\n2\nx\nx\n\u2212\n+\n(D)\n2\n1log\nlog (\n+1) + C\n2\nx\nx\n+\n7 6  Integration by Parts\nIn this section, we describe one more method of integration, that is found quite useful in\nintegrating products of functions"
    },
    {
        "Chapter": "1",
        "sentence_range": "3707-3710",
        "Text": "(\n1) (\n2)\nx dx\nx\nx\n\u2212\n\u2212\n\u222b\n equals\n(A)\n2\n(\n1)\nlog\nC\n2\nxx\n\u2212\n+\n\u2212\n(B)\n2\n(\n2)\nlog\nC\n1\nx\nx\n\u2212\n+\n\u2212\n(C)\n12\nlog\nC\n2\nxx\n\u2212\n\uf8eb\n\uf8f6\n+\n\uf8ec\n\uf8f7\n\u2212\n\uf8ed\n\uf8f8\n(D) log (\n1) (\n2)\nC\nx\nx\n\u2212\n\u2212\n+\n23 (2\n1)\ndx\n\u222bx x +\nequals\n(A)\n2\n1\nlog\nlog (\n+1) + C\n2\nx\nx\n\u2212\n(B)\n2\n1\nlog\nlog (\n+1) + C\n2\nx\nx\n+\n(C)\n2\n1\nlog\nlog (\n+1) +C\n2\nx\nx\n\u2212\n+\n(D)\n2\n1log\nlog (\n+1) + C\n2\nx\nx\n+\n7 6  Integration by Parts\nIn this section, we describe one more method of integration, that is found quite useful in\nintegrating products of functions If u and v are any two differentiable functions of a single variable x (say)"
    },
    {
        "Chapter": "1",
        "sentence_range": "3708-3711",
        "Text": "(2\n1)\ndx\n\u222bx x +\nequals\n(A)\n2\n1\nlog\nlog (\n+1) + C\n2\nx\nx\n\u2212\n(B)\n2\n1\nlog\nlog (\n+1) + C\n2\nx\nx\n+\n(C)\n2\n1\nlog\nlog (\n+1) +C\n2\nx\nx\n\u2212\n+\n(D)\n2\n1log\nlog (\n+1) + C\n2\nx\nx\n+\n7 6  Integration by Parts\nIn this section, we describe one more method of integration, that is found quite useful in\nintegrating products of functions If u and v are any two differentiable functions of a single variable x (say) Then, by\nthe product rule of differentiation, we have\n(\n)\nd\nuv\ndx\n =\ndv\ndu\nu\nv\ndx\ndx\n+\nIntegrating both sides, we get\nuv =\ndv\ndu\nu\ndx\nv\ndx\ndx\ndx\n+\n\u222b\n\u222b\nor\nudv\ndx\n\u222bdx\n =\nuv \u2013 vdu\ndx\n\u222bdx"
    },
    {
        "Chapter": "1",
        "sentence_range": "3709-3712",
        "Text": "6  Integration by Parts\nIn this section, we describe one more method of integration, that is found quite useful in\nintegrating products of functions If u and v are any two differentiable functions of a single variable x (say) Then, by\nthe product rule of differentiation, we have\n(\n)\nd\nuv\ndx\n =\ndv\ndu\nu\nv\ndx\ndx\n+\nIntegrating both sides, we get\nuv =\ndv\ndu\nu\ndx\nv\ndx\ndx\ndx\n+\n\u222b\n\u222b\nor\nudv\ndx\n\u222bdx\n =\nuv \u2013 vdu\ndx\n\u222bdx (1)\nLet\nu = f (x) and dv\ndx = g(x)"
    },
    {
        "Chapter": "1",
        "sentence_range": "3710-3713",
        "Text": "If u and v are any two differentiable functions of a single variable x (say) Then, by\nthe product rule of differentiation, we have\n(\n)\nd\nuv\ndx\n =\ndv\ndu\nu\nv\ndx\ndx\n+\nIntegrating both sides, we get\nuv =\ndv\ndu\nu\ndx\nv\ndx\ndx\ndx\n+\n\u222b\n\u222b\nor\nudv\ndx\n\u222bdx\n =\nuv \u2013 vdu\ndx\n\u222bdx (1)\nLet\nu = f (x) and dv\ndx = g(x) Then\ndu\ndx = f \u2032(x) and v = \n( )\n\u222bg x dx\n324\nMATHEMATICS\nTherefore, expression (1) can be rewritten as\n( ) ( )\n\u222bf x g x dx\n =\n( )\n( )\n[\n( )\n]\n( )\nf x\ng x dx \u2013\ng x dx f\n\u2032x dx\n\u222b\n\u222b \u222b\ni"
    },
    {
        "Chapter": "1",
        "sentence_range": "3711-3714",
        "Text": "Then, by\nthe product rule of differentiation, we have\n(\n)\nd\nuv\ndx\n =\ndv\ndu\nu\nv\ndx\ndx\n+\nIntegrating both sides, we get\nuv =\ndv\ndu\nu\ndx\nv\ndx\ndx\ndx\n+\n\u222b\n\u222b\nor\nudv\ndx\n\u222bdx\n =\nuv \u2013 vdu\ndx\n\u222bdx (1)\nLet\nu = f (x) and dv\ndx = g(x) Then\ndu\ndx = f \u2032(x) and v = \n( )\n\u222bg x dx\n324\nMATHEMATICS\nTherefore, expression (1) can be rewritten as\n( ) ( )\n\u222bf x g x dx\n =\n( )\n( )\n[\n( )\n]\n( )\nf x\ng x dx \u2013\ng x dx f\n\u2032x dx\n\u222b\n\u222b \u222b\ni e"
    },
    {
        "Chapter": "1",
        "sentence_range": "3712-3715",
        "Text": "(1)\nLet\nu = f (x) and dv\ndx = g(x) Then\ndu\ndx = f \u2032(x) and v = \n( )\n\u222bg x dx\n324\nMATHEMATICS\nTherefore, expression (1) can be rewritten as\n( ) ( )\n\u222bf x g x dx\n =\n( )\n( )\n[\n( )\n]\n( )\nf x\ng x dx \u2013\ng x dx f\n\u2032x dx\n\u222b\n\u222b \u222b\ni e ,\n( ) ( )\n\u222bf x g x dx\n =\n( )\n( )\n[\n( )\n( )\n]\nf x\ng x dx \u2013\nf\nx\ng x dx dx\n\u2032\n\u222b\n\u222b\n\u222b\nIf we take f as the first function and g as the second function, then this formula\nmay be stated as follows:\n\u201cThe integral of the product of two functions = (first function) \u00d7 (integral\nof the second function) \u2013 Integral of [(differential coefficient of the first function)\n\u00d7 (integral of the second function)]\u201d\nExample 17 Find \nxcos\nx dx\n\u222b\nSolution Put f (x) = x (first function) and g (x) = cos x (second function)"
    },
    {
        "Chapter": "1",
        "sentence_range": "3713-3716",
        "Text": "Then\ndu\ndx = f \u2032(x) and v = \n( )\n\u222bg x dx\n324\nMATHEMATICS\nTherefore, expression (1) can be rewritten as\n( ) ( )\n\u222bf x g x dx\n =\n( )\n( )\n[\n( )\n]\n( )\nf x\ng x dx \u2013\ng x dx f\n\u2032x dx\n\u222b\n\u222b \u222b\ni e ,\n( ) ( )\n\u222bf x g x dx\n =\n( )\n( )\n[\n( )\n( )\n]\nf x\ng x dx \u2013\nf\nx\ng x dx dx\n\u2032\n\u222b\n\u222b\n\u222b\nIf we take f as the first function and g as the second function, then this formula\nmay be stated as follows:\n\u201cThe integral of the product of two functions = (first function) \u00d7 (integral\nof the second function) \u2013 Integral of [(differential coefficient of the first function)\n\u00d7 (integral of the second function)]\u201d\nExample 17 Find \nxcos\nx dx\n\u222b\nSolution Put f (x) = x (first function) and g (x) = cos x (second function) Then, integration by parts gives\nxcos\nx dx\n\u222b\n =\ncos\n[\n( ) cos\n]\nd\nx\nx dx \u2013\nx\nx dx dx\ndx\n\u222b\n\u222b\n\u222b\n=\nsin\nsin\nx\nx \u2013\nx dx\n\u222b\n = x sin x + cos x + C\nSuppose, we take\nf (x) = cos x and g(x) = x"
    },
    {
        "Chapter": "1",
        "sentence_range": "3714-3717",
        "Text": "e ,\n( ) ( )\n\u222bf x g x dx\n =\n( )\n( )\n[\n( )\n( )\n]\nf x\ng x dx \u2013\nf\nx\ng x dx dx\n\u2032\n\u222b\n\u222b\n\u222b\nIf we take f as the first function and g as the second function, then this formula\nmay be stated as follows:\n\u201cThe integral of the product of two functions = (first function) \u00d7 (integral\nof the second function) \u2013 Integral of [(differential coefficient of the first function)\n\u00d7 (integral of the second function)]\u201d\nExample 17 Find \nxcos\nx dx\n\u222b\nSolution Put f (x) = x (first function) and g (x) = cos x (second function) Then, integration by parts gives\nxcos\nx dx\n\u222b\n =\ncos\n[\n( ) cos\n]\nd\nx\nx dx \u2013\nx\nx dx dx\ndx\n\u222b\n\u222b\n\u222b\n=\nsin\nsin\nx\nx \u2013\nx dx\n\u222b\n = x sin x + cos x + C\nSuppose, we take\nf (x) = cos x and g(x) = x Then\nxcos\nx dx\n\u222b\n = cos\n[\n(cos )\n]\nd\nx\nx dx \u2013\nx\nx dx dx\ndx\n\u222b\n\u222b\n\u222b\n= (\n)\n2\n2\ncos\nsin\n2\n2\nx\nx\nx\nx\ndx\n+\u222b\nThus, it shows that the integral \nxcos\nx dx\n\u222b\n is reduced to the comparatively more\ncomplicated integral having more power of x"
    },
    {
        "Chapter": "1",
        "sentence_range": "3715-3718",
        "Text": ",\n( ) ( )\n\u222bf x g x dx\n =\n( )\n( )\n[\n( )\n( )\n]\nf x\ng x dx \u2013\nf\nx\ng x dx dx\n\u2032\n\u222b\n\u222b\n\u222b\nIf we take f as the first function and g as the second function, then this formula\nmay be stated as follows:\n\u201cThe integral of the product of two functions = (first function) \u00d7 (integral\nof the second function) \u2013 Integral of [(differential coefficient of the first function)\n\u00d7 (integral of the second function)]\u201d\nExample 17 Find \nxcos\nx dx\n\u222b\nSolution Put f (x) = x (first function) and g (x) = cos x (second function) Then, integration by parts gives\nxcos\nx dx\n\u222b\n =\ncos\n[\n( ) cos\n]\nd\nx\nx dx \u2013\nx\nx dx dx\ndx\n\u222b\n\u222b\n\u222b\n=\nsin\nsin\nx\nx \u2013\nx dx\n\u222b\n = x sin x + cos x + C\nSuppose, we take\nf (x) = cos x and g(x) = x Then\nxcos\nx dx\n\u222b\n = cos\n[\n(cos )\n]\nd\nx\nx dx \u2013\nx\nx dx dx\ndx\n\u222b\n\u222b\n\u222b\n= (\n)\n2\n2\ncos\nsin\n2\n2\nx\nx\nx\nx\ndx\n+\u222b\nThus, it shows that the integral \nxcos\nx dx\n\u222b\n is reduced to the comparatively more\ncomplicated integral having more power of x Therefore, the proper choice of the first\nfunction and the second function is significant"
    },
    {
        "Chapter": "1",
        "sentence_range": "3716-3719",
        "Text": "Then, integration by parts gives\nxcos\nx dx\n\u222b\n =\ncos\n[\n( ) cos\n]\nd\nx\nx dx \u2013\nx\nx dx dx\ndx\n\u222b\n\u222b\n\u222b\n=\nsin\nsin\nx\nx \u2013\nx dx\n\u222b\n = x sin x + cos x + C\nSuppose, we take\nf (x) = cos x and g(x) = x Then\nxcos\nx dx\n\u222b\n = cos\n[\n(cos )\n]\nd\nx\nx dx \u2013\nx\nx dx dx\ndx\n\u222b\n\u222b\n\u222b\n= (\n)\n2\n2\ncos\nsin\n2\n2\nx\nx\nx\nx\ndx\n+\u222b\nThus, it shows that the integral \nxcos\nx dx\n\u222b\n is reduced to the comparatively more\ncomplicated integral having more power of x Therefore, the proper choice of the first\nfunction and the second function is significant Remarks\n(i)\nIt is worth mentioning that integration by parts is not applicable to product of\nfunctions in all cases"
    },
    {
        "Chapter": "1",
        "sentence_range": "3717-3720",
        "Text": "Then\nxcos\nx dx\n\u222b\n = cos\n[\n(cos )\n]\nd\nx\nx dx \u2013\nx\nx dx dx\ndx\n\u222b\n\u222b\n\u222b\n= (\n)\n2\n2\ncos\nsin\n2\n2\nx\nx\nx\nx\ndx\n+\u222b\nThus, it shows that the integral \nxcos\nx dx\n\u222b\n is reduced to the comparatively more\ncomplicated integral having more power of x Therefore, the proper choice of the first\nfunction and the second function is significant Remarks\n(i)\nIt is worth mentioning that integration by parts is not applicable to product of\nfunctions in all cases For instance, the method does not work for \nxsin\nx dx\n\u222b"
    },
    {
        "Chapter": "1",
        "sentence_range": "3718-3721",
        "Text": "Therefore, the proper choice of the first\nfunction and the second function is significant Remarks\n(i)\nIt is worth mentioning that integration by parts is not applicable to product of\nfunctions in all cases For instance, the method does not work for \nxsin\nx dx\n\u222b The reason is that there does not exist any function whose derivative is\nx  sin x"
    },
    {
        "Chapter": "1",
        "sentence_range": "3719-3722",
        "Text": "Remarks\n(i)\nIt is worth mentioning that integration by parts is not applicable to product of\nfunctions in all cases For instance, the method does not work for \nxsin\nx dx\n\u222b The reason is that there does not exist any function whose derivative is\nx  sin x (ii)\nObserve that while finding the integral of the second function, we did not add\nany constant of integration"
    },
    {
        "Chapter": "1",
        "sentence_range": "3720-3723",
        "Text": "For instance, the method does not work for \nxsin\nx dx\n\u222b The reason is that there does not exist any function whose derivative is\nx  sin x (ii)\nObserve that while finding the integral of the second function, we did not add\nany constant of integration If we write the integral of the second function cos x\nINTEGRALS         325\nas sin x + k, where k is any constant, then\nxcos\nx dx\n\u222b\n =\n(sin\n)\n(sin\n)\nx\nx\nk\nx\nk dx\n+\n\u2212\n+\n\u222b\n=\n(sin\n)\n(sin\nx\nx\nk\nx dx\nk dx\n+\n\u2212\n\u2212\n\u222b\n\u222b\n=\n(sin\n)\ncos\nC\nx\nx\nk\nx \u2013 kx\n+\n\u2212\n+\n = \nsin\ncos\nC\nx\nx\nx\n+\n+\nThis shows that adding a constant to the integral of the second function is\nsuperfluous so far as the final result is concerned while applying the method of\nintegration by parts"
    },
    {
        "Chapter": "1",
        "sentence_range": "3721-3724",
        "Text": "The reason is that there does not exist any function whose derivative is\nx  sin x (ii)\nObserve that while finding the integral of the second function, we did not add\nany constant of integration If we write the integral of the second function cos x\nINTEGRALS         325\nas sin x + k, where k is any constant, then\nxcos\nx dx\n\u222b\n =\n(sin\n)\n(sin\n)\nx\nx\nk\nx\nk dx\n+\n\u2212\n+\n\u222b\n=\n(sin\n)\n(sin\nx\nx\nk\nx dx\nk dx\n+\n\u2212\n\u2212\n\u222b\n\u222b\n=\n(sin\n)\ncos\nC\nx\nx\nk\nx \u2013 kx\n+\n\u2212\n+\n = \nsin\ncos\nC\nx\nx\nx\n+\n+\nThis shows that adding a constant to the integral of the second function is\nsuperfluous so far as the final result is concerned while applying the method of\nintegration by parts (iii)\nUsually, if any function is a power of x or a polynomial in x, then we take it as the\nfirst function"
    },
    {
        "Chapter": "1",
        "sentence_range": "3722-3725",
        "Text": "(ii)\nObserve that while finding the integral of the second function, we did not add\nany constant of integration If we write the integral of the second function cos x\nINTEGRALS         325\nas sin x + k, where k is any constant, then\nxcos\nx dx\n\u222b\n =\n(sin\n)\n(sin\n)\nx\nx\nk\nx\nk dx\n+\n\u2212\n+\n\u222b\n=\n(sin\n)\n(sin\nx\nx\nk\nx dx\nk dx\n+\n\u2212\n\u2212\n\u222b\n\u222b\n=\n(sin\n)\ncos\nC\nx\nx\nk\nx \u2013 kx\n+\n\u2212\n+\n = \nsin\ncos\nC\nx\nx\nx\n+\n+\nThis shows that adding a constant to the integral of the second function is\nsuperfluous so far as the final result is concerned while applying the method of\nintegration by parts (iii)\nUsually, if any function is a power of x or a polynomial in x, then we take it as the\nfirst function However, in cases where other function is inverse trigonometric\nfunction or logarithmic function, then we take them as first function"
    },
    {
        "Chapter": "1",
        "sentence_range": "3723-3726",
        "Text": "If we write the integral of the second function cos x\nINTEGRALS         325\nas sin x + k, where k is any constant, then\nxcos\nx dx\n\u222b\n =\n(sin\n)\n(sin\n)\nx\nx\nk\nx\nk dx\n+\n\u2212\n+\n\u222b\n=\n(sin\n)\n(sin\nx\nx\nk\nx dx\nk dx\n+\n\u2212\n\u2212\n\u222b\n\u222b\n=\n(sin\n)\ncos\nC\nx\nx\nk\nx \u2013 kx\n+\n\u2212\n+\n = \nsin\ncos\nC\nx\nx\nx\n+\n+\nThis shows that adding a constant to the integral of the second function is\nsuperfluous so far as the final result is concerned while applying the method of\nintegration by parts (iii)\nUsually, if any function is a power of x or a polynomial in x, then we take it as the\nfirst function However, in cases where other function is inverse trigonometric\nfunction or logarithmic function, then we take them as first function Example 18 Find log x dx\n\u222b\nSolution To start with, we are unable to guess a function whose derivative is log x"
    },
    {
        "Chapter": "1",
        "sentence_range": "3724-3727",
        "Text": "(iii)\nUsually, if any function is a power of x or a polynomial in x, then we take it as the\nfirst function However, in cases where other function is inverse trigonometric\nfunction or logarithmic function, then we take them as first function Example 18 Find log x dx\n\u222b\nSolution To start with, we are unable to guess a function whose derivative is log x We\ntake log x as the first function and the constant function 1 as the second function"
    },
    {
        "Chapter": "1",
        "sentence_range": "3725-3728",
        "Text": "However, in cases where other function is inverse trigonometric\nfunction or logarithmic function, then we take them as first function Example 18 Find log x dx\n\u222b\nSolution To start with, we are unable to guess a function whose derivative is log x We\ntake log x as the first function and the constant function 1 as the second function Then,\nthe integral of the second function is x"
    },
    {
        "Chapter": "1",
        "sentence_range": "3726-3729",
        "Text": "Example 18 Find log x dx\n\u222b\nSolution To start with, we are unable to guess a function whose derivative is log x We\ntake log x as the first function and the constant function 1 as the second function Then,\nthe integral of the second function is x Hence,\n(log"
    },
    {
        "Chapter": "1",
        "sentence_range": "3727-3730",
        "Text": "We\ntake log x as the first function and the constant function 1 as the second function Then,\nthe integral of the second function is x Hence,\n(log 1)\nx\ndx\n\u222b\n = log\n1\n[\n(log ) 1\n]\nd\nx\ndx\nx\ndx dx\ndx\n\u2212\n\u222b\n\u222b\n\u222b\n=\n1\n(log )\n\u2013\nlog\nC\nx\nx\nx dx\nx\nx \u2013 x\nx\n\u22c5\n=\n+\n\u222b"
    },
    {
        "Chapter": "1",
        "sentence_range": "3728-3731",
        "Text": "Then,\nthe integral of the second function is x Hence,\n(log 1)\nx\ndx\n\u222b\n = log\n1\n[\n(log ) 1\n]\nd\nx\ndx\nx\ndx dx\ndx\n\u2212\n\u222b\n\u222b\n\u222b\n=\n1\n(log )\n\u2013\nlog\nC\nx\nx\nx dx\nx\nx \u2013 x\nx\n\u22c5\n=\n+\n\u222b Example 19 Find \nx\nx e dx\n\u222b\nSolution Take first function as x and second function as ex"
    },
    {
        "Chapter": "1",
        "sentence_range": "3729-3732",
        "Text": "Hence,\n(log 1)\nx\ndx\n\u222b\n = log\n1\n[\n(log ) 1\n]\nd\nx\ndx\nx\ndx dx\ndx\n\u2212\n\u222b\n\u222b\n\u222b\n=\n1\n(log )\n\u2013\nlog\nC\nx\nx\nx dx\nx\nx \u2013 x\nx\n\u22c5\n=\n+\n\u222b Example 19 Find \nx\nx e dx\n\u222b\nSolution Take first function as x and second function as ex The integral of the second\nfunction is ex"
    },
    {
        "Chapter": "1",
        "sentence_range": "3730-3733",
        "Text": "1)\nx\ndx\n\u222b\n = log\n1\n[\n(log ) 1\n]\nd\nx\ndx\nx\ndx dx\ndx\n\u2212\n\u222b\n\u222b\n\u222b\n=\n1\n(log )\n\u2013\nlog\nC\nx\nx\nx dx\nx\nx \u2013 x\nx\n\u22c5\n=\n+\n\u222b Example 19 Find \nx\nx e dx\n\u222b\nSolution Take first function as x and second function as ex The integral of the second\nfunction is ex Therefore,\nx\n\u222bx e dx\n =\n1\nx\nx\nx e\ne dx\n\u2212\n\u222b\u22c5\n = xex \u2013 ex + C"
    },
    {
        "Chapter": "1",
        "sentence_range": "3731-3734",
        "Text": "Example 19 Find \nx\nx e dx\n\u222b\nSolution Take first function as x and second function as ex The integral of the second\nfunction is ex Therefore,\nx\n\u222bx e dx\n =\n1\nx\nx\nx e\ne dx\n\u2212\n\u222b\u22c5\n = xex \u2013 ex + C Example 20 Find \n1\n2\nsin\n1\n\u2013\nx\nx dx\n\u2212x\n\u222b\nSolution Let first function be sin \u2013 1x and second function be \n2\n1\nx\nx\n\u2212"
    },
    {
        "Chapter": "1",
        "sentence_range": "3732-3735",
        "Text": "The integral of the second\nfunction is ex Therefore,\nx\n\u222bx e dx\n =\n1\nx\nx\nx e\ne dx\n\u2212\n\u222b\u22c5\n = xex \u2013 ex + C Example 20 Find \n1\n2\nsin\n1\n\u2013\nx\nx dx\n\u2212x\n\u222b\nSolution Let first function be sin \u2013 1x and second function be \n2\n1\nx\nx\n\u2212 First we find the integral of the second function, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "3733-3736",
        "Text": "Therefore,\nx\n\u222bx e dx\n =\n1\nx\nx\nx e\ne dx\n\u2212\n\u222b\u22c5\n = xex \u2013 ex + C Example 20 Find \n1\n2\nsin\n1\n\u2013\nx\nx dx\n\u2212x\n\u222b\nSolution Let first function be sin \u2013 1x and second function be \n2\n1\nx\nx\n\u2212 First we find the integral of the second function, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "3734-3737",
        "Text": "Example 20 Find \n1\n2\nsin\n1\n\u2013\nx\nx dx\n\u2212x\n\u222b\nSolution Let first function be sin \u2013 1x and second function be \n2\n1\nx\nx\n\u2212 First we find the integral of the second function, i e , \n2\n1\nx dx\n\u2212x\n\u222b"
    },
    {
        "Chapter": "1",
        "sentence_range": "3735-3738",
        "Text": "First we find the integral of the second function, i e , \n2\n1\nx dx\n\u2212x\n\u222b Put t =1 \u2013 x2"
    },
    {
        "Chapter": "1",
        "sentence_range": "3736-3739",
        "Text": "e , \n2\n1\nx dx\n\u2212x\n\u222b Put t =1 \u2013 x2 Then dt = \u2013 2x dx\n326\nMATHEMATICS\nTherefore,\n2\n1\nx dx\n\u2212x\n\u222b\n =\n1\n2\ndt\n\u2013\n\u222bt\n = \n2\n\u2013\n1\nt\nx\n= \u2212\n\u2212\nHence,\n1\n2\nsin\n1\n\u2013\nx\nx dx\n\u2212x\n\u222b\n =\n(\n)\n1\n2\n2\n12\n(sin\n)\n1\n(\n1\n)\n1\n\u2013 x\n\u2013\nx\n\u2013\nx\ndx\nx\n\u2212\n\u2212\n\u2212\n\u2212\n\u222b\n=\n2\n1\n1\nsin\nC\n\u2013\nx\nx\nx\n\u2212\n\u2212\n+\n+\n = \n2\n1\n1\nsin\nC\nx \u2013\nx\nx\n\u2212\n\u2212\n+\nAlternatively, this integral can also be worked out by making substitution sin\u20131 x  = \u03b8 and\nthen integrating by parts"
    },
    {
        "Chapter": "1",
        "sentence_range": "3737-3740",
        "Text": ", \n2\n1\nx dx\n\u2212x\n\u222b Put t =1 \u2013 x2 Then dt = \u2013 2x dx\n326\nMATHEMATICS\nTherefore,\n2\n1\nx dx\n\u2212x\n\u222b\n =\n1\n2\ndt\n\u2013\n\u222bt\n = \n2\n\u2013\n1\nt\nx\n= \u2212\n\u2212\nHence,\n1\n2\nsin\n1\n\u2013\nx\nx dx\n\u2212x\n\u222b\n =\n(\n)\n1\n2\n2\n12\n(sin\n)\n1\n(\n1\n)\n1\n\u2013 x\n\u2013\nx\n\u2013\nx\ndx\nx\n\u2212\n\u2212\n\u2212\n\u2212\n\u222b\n=\n2\n1\n1\nsin\nC\n\u2013\nx\nx\nx\n\u2212\n\u2212\n+\n+\n = \n2\n1\n1\nsin\nC\nx \u2013\nx\nx\n\u2212\n\u2212\n+\nAlternatively, this integral can also be worked out by making substitution sin\u20131 x  = \u03b8 and\nthen integrating by parts Example 21  Find \nexsin\nx dx\n\u222b\nSolution  Take ex as the first function and sin x as second function"
    },
    {
        "Chapter": "1",
        "sentence_range": "3738-3741",
        "Text": "Put t =1 \u2013 x2 Then dt = \u2013 2x dx\n326\nMATHEMATICS\nTherefore,\n2\n1\nx dx\n\u2212x\n\u222b\n =\n1\n2\ndt\n\u2013\n\u222bt\n = \n2\n\u2013\n1\nt\nx\n= \u2212\n\u2212\nHence,\n1\n2\nsin\n1\n\u2013\nx\nx dx\n\u2212x\n\u222b\n =\n(\n)\n1\n2\n2\n12\n(sin\n)\n1\n(\n1\n)\n1\n\u2013 x\n\u2013\nx\n\u2013\nx\ndx\nx\n\u2212\n\u2212\n\u2212\n\u2212\n\u222b\n=\n2\n1\n1\nsin\nC\n\u2013\nx\nx\nx\n\u2212\n\u2212\n+\n+\n = \n2\n1\n1\nsin\nC\nx \u2013\nx\nx\n\u2212\n\u2212\n+\nAlternatively, this integral can also be worked out by making substitution sin\u20131 x  = \u03b8 and\nthen integrating by parts Example 21  Find \nexsin\nx dx\n\u222b\nSolution  Take ex as the first function and sin x as second function Then, integrating\nby parts, we have\nI\nsin\n(\ncos )\ncos\nx\nx\nx\ne\nx dx\ne\n\u2013\nx\ne\nx dx\n=\n=\n+\n\u222b\n\u222b\n= \u2013 e x cos x + I1 (say)"
    },
    {
        "Chapter": "1",
        "sentence_range": "3739-3742",
        "Text": "Then dt = \u2013 2x dx\n326\nMATHEMATICS\nTherefore,\n2\n1\nx dx\n\u2212x\n\u222b\n =\n1\n2\ndt\n\u2013\n\u222bt\n = \n2\n\u2013\n1\nt\nx\n= \u2212\n\u2212\nHence,\n1\n2\nsin\n1\n\u2013\nx\nx dx\n\u2212x\n\u222b\n =\n(\n)\n1\n2\n2\n12\n(sin\n)\n1\n(\n1\n)\n1\n\u2013 x\n\u2013\nx\n\u2013\nx\ndx\nx\n\u2212\n\u2212\n\u2212\n\u2212\n\u222b\n=\n2\n1\n1\nsin\nC\n\u2013\nx\nx\nx\n\u2212\n\u2212\n+\n+\n = \n2\n1\n1\nsin\nC\nx \u2013\nx\nx\n\u2212\n\u2212\n+\nAlternatively, this integral can also be worked out by making substitution sin\u20131 x  = \u03b8 and\nthen integrating by parts Example 21  Find \nexsin\nx dx\n\u222b\nSolution  Take ex as the first function and sin x as second function Then, integrating\nby parts, we have\nI\nsin\n(\ncos )\ncos\nx\nx\nx\ne\nx dx\ne\n\u2013\nx\ne\nx dx\n=\n=\n+\n\u222b\n\u222b\n= \u2013 e x cos x + I1 (say) (1)\nTaking ex\n and cos x as the first and second functions, respectively, in I1, we get\nI1 =\nsin\nsin\nx\nx\ne\nx \u2013 e\nx dx\n\u222b\nSubstituting the value of I1 in (1), we get\nI = \u2013 ex cos x + ex sin x \u2013 I  or  2I = ex (sin x \u2013 cos x)\nHence,\nI =\nsin\n(sin\ncos ) + C\n2\nx\nx\ne\ne\nx dx\nx \u2013\nx\n=\n\u222b\nAlternatively, above integral can also be determined by taking sin x as the first function\nand ex the second function"
    },
    {
        "Chapter": "1",
        "sentence_range": "3740-3743",
        "Text": "Example 21  Find \nexsin\nx dx\n\u222b\nSolution  Take ex as the first function and sin x as second function Then, integrating\nby parts, we have\nI\nsin\n(\ncos )\ncos\nx\nx\nx\ne\nx dx\ne\n\u2013\nx\ne\nx dx\n=\n=\n+\n\u222b\n\u222b\n= \u2013 e x cos x + I1 (say) (1)\nTaking ex\n and cos x as the first and second functions, respectively, in I1, we get\nI1 =\nsin\nsin\nx\nx\ne\nx \u2013 e\nx dx\n\u222b\nSubstituting the value of I1 in (1), we get\nI = \u2013 ex cos x + ex sin x \u2013 I  or  2I = ex (sin x \u2013 cos x)\nHence,\nI =\nsin\n(sin\ncos ) + C\n2\nx\nx\ne\ne\nx dx\nx \u2013\nx\n=\n\u222b\nAlternatively, above integral can also be determined by taking sin x as the first function\nand ex the second function 7"
    },
    {
        "Chapter": "1",
        "sentence_range": "3741-3744",
        "Text": "Then, integrating\nby parts, we have\nI\nsin\n(\ncos )\ncos\nx\nx\nx\ne\nx dx\ne\n\u2013\nx\ne\nx dx\n=\n=\n+\n\u222b\n\u222b\n= \u2013 e x cos x + I1 (say) (1)\nTaking ex\n and cos x as the first and second functions, respectively, in I1, we get\nI1 =\nsin\nsin\nx\nx\ne\nx \u2013 e\nx dx\n\u222b\nSubstituting the value of I1 in (1), we get\nI = \u2013 ex cos x + ex sin x \u2013 I  or  2I = ex (sin x \u2013 cos x)\nHence,\nI =\nsin\n(sin\ncos ) + C\n2\nx\nx\ne\ne\nx dx\nx \u2013\nx\n=\n\u222b\nAlternatively, above integral can also be determined by taking sin x as the first function\nand ex the second function 7 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "3742-3745",
        "Text": "(1)\nTaking ex\n and cos x as the first and second functions, respectively, in I1, we get\nI1 =\nsin\nsin\nx\nx\ne\nx \u2013 e\nx dx\n\u222b\nSubstituting the value of I1 in (1), we get\nI = \u2013 ex cos x + ex sin x \u2013 I  or  2I = ex (sin x \u2013 cos x)\nHence,\nI =\nsin\n(sin\ncos ) + C\n2\nx\nx\ne\ne\nx dx\nx \u2013\nx\n=\n\u222b\nAlternatively, above integral can also be determined by taking sin x as the first function\nand ex the second function 7 6 1 Integral of the type \n[\n( ) +\n( )]\nex\nf x\nf\nx\ndx\n\u2032\n\u222b\nWe have\nI =\n[\n( ) +\n( )]\nex\nf x\nf\nx\ndx\n\u2032\n\u222b\n = \n( )\n+\n( )\nx\nx\ne f x dx\ne f\n\u2032x dx\n\u222b\n\u222b\n=\n1\n1\nI\n( )\n, where I =\n( )\nx\nx\ne f\nx dx\ne f x dx\n\u2032\n+ \u222b\n\u222b"
    },
    {
        "Chapter": "1",
        "sentence_range": "3743-3746",
        "Text": "7 6 1 Integral of the type \n[\n( ) +\n( )]\nex\nf x\nf\nx\ndx\n\u2032\n\u222b\nWe have\nI =\n[\n( ) +\n( )]\nex\nf x\nf\nx\ndx\n\u2032\n\u222b\n = \n( )\n+\n( )\nx\nx\ne f x dx\ne f\n\u2032x dx\n\u222b\n\u222b\n=\n1\n1\nI\n( )\n, where I =\n( )\nx\nx\ne f\nx dx\ne f x dx\n\u2032\n+ \u222b\n\u222b (1)\nTaking f(x) and ex as the first function and second function, respectively, in I 1 and\nintegrating it by parts, we have I1 = f (x) ex \u2013 \n( )\nC\nx\nf\n\u2032x e dx\n+\n\u222b\nSubstituting I1 in (1), we get\nI =\n( )\n( )\n( )\nC\nx\nx\nx\ne f x\nf\nx e dx\ne f\nx dx\n\u2032\n\u2032\n\u2212\n+\n+\n\u222b\n\u222b\n = ex f (x) + C\nINTEGRALS         327\nThus,\n\u2032\n\u222b\n[\n( )\n( )]\nex\n f x + f\nx\ndx  = \n( )\nC\nx\ne f x +\nExample 22 Find (i) \n1\n2\n1\n(tan\n)\n1\nx\n\u2013\ne\nx\nx\n+\n+\n\u222b\ndx   (ii) \n2\n2\n(\n+ 1)\n( + 1)\nx\nx\ne\n\u222bx\n dx\nSolution\n(i) We have I =\n1\n12\n(tan\n)\n1\nx\n\u2013\ne\nx\ndx\nx\n+\n+\n\u222b\nConsider f (x) = tan\u2013 1x, then  f \u2032(x) = \n2\n1\n1\nx\n+\nThus, the given integrand is of the form ex [ f (x) + f \u2032(x)]"
    },
    {
        "Chapter": "1",
        "sentence_range": "3744-3747",
        "Text": "6 1 Integral of the type \n[\n( ) +\n( )]\nex\nf x\nf\nx\ndx\n\u2032\n\u222b\nWe have\nI =\n[\n( ) +\n( )]\nex\nf x\nf\nx\ndx\n\u2032\n\u222b\n = \n( )\n+\n( )\nx\nx\ne f x dx\ne f\n\u2032x dx\n\u222b\n\u222b\n=\n1\n1\nI\n( )\n, where I =\n( )\nx\nx\ne f\nx dx\ne f x dx\n\u2032\n+ \u222b\n\u222b (1)\nTaking f(x) and ex as the first function and second function, respectively, in I 1 and\nintegrating it by parts, we have I1 = f (x) ex \u2013 \n( )\nC\nx\nf\n\u2032x e dx\n+\n\u222b\nSubstituting I1 in (1), we get\nI =\n( )\n( )\n( )\nC\nx\nx\nx\ne f x\nf\nx e dx\ne f\nx dx\n\u2032\n\u2032\n\u2212\n+\n+\n\u222b\n\u222b\n = ex f (x) + C\nINTEGRALS         327\nThus,\n\u2032\n\u222b\n[\n( )\n( )]\nex\n f x + f\nx\ndx  = \n( )\nC\nx\ne f x +\nExample 22 Find (i) \n1\n2\n1\n(tan\n)\n1\nx\n\u2013\ne\nx\nx\n+\n+\n\u222b\ndx   (ii) \n2\n2\n(\n+ 1)\n( + 1)\nx\nx\ne\n\u222bx\n dx\nSolution\n(i) We have I =\n1\n12\n(tan\n)\n1\nx\n\u2013\ne\nx\ndx\nx\n+\n+\n\u222b\nConsider f (x) = tan\u2013 1x, then  f \u2032(x) = \n2\n1\n1\nx\n+\nThus, the given integrand is of the form ex [ f (x) + f \u2032(x)] Therefore, \n1\n12\nI\n(tan\n)\n1\nx\n\u2013\ne\nx\ndx\nx\n=\n+\n+\n\u222b\n = ex tan\u2013 1x + C\n(ii) We have \n2\n2\n(\n+1)\nI\n( +1)\nx\nx\ne\n= \u222bx\ndx\n2\n2\n1 +1+1)\n[\n]\n( +1)\nx\nx \u2013\ne\ndx\nx\n= \u222b\n2\n2\n2\n1\n2\n[\n]\n( +1)\n( +1)\nx\nx \u2013\ne\ndx\nx\nx\n=\n+\n\u222b\n \n2\n1\n2\n[\n+\n]\n+1\n( +1)\nx\nx \u2013\ne\ndx\nx\nx\n= \u222b\nConsider \n1\n( )\n1\nx\nf x\nx\n\u2212\n=\n+\n, then  \n2\n2\n( )\n(\n1)\nf\nx\nx\n\u2032\n=\n+\nThus, the given integrand is of the form ex [f (x) + f \u2032(x)]"
    },
    {
        "Chapter": "1",
        "sentence_range": "3745-3748",
        "Text": "1 Integral of the type \n[\n( ) +\n( )]\nex\nf x\nf\nx\ndx\n\u2032\n\u222b\nWe have\nI =\n[\n( ) +\n( )]\nex\nf x\nf\nx\ndx\n\u2032\n\u222b\n = \n( )\n+\n( )\nx\nx\ne f x dx\ne f\n\u2032x dx\n\u222b\n\u222b\n=\n1\n1\nI\n( )\n, where I =\n( )\nx\nx\ne f\nx dx\ne f x dx\n\u2032\n+ \u222b\n\u222b (1)\nTaking f(x) and ex as the first function and second function, respectively, in I 1 and\nintegrating it by parts, we have I1 = f (x) ex \u2013 \n( )\nC\nx\nf\n\u2032x e dx\n+\n\u222b\nSubstituting I1 in (1), we get\nI =\n( )\n( )\n( )\nC\nx\nx\nx\ne f x\nf\nx e dx\ne f\nx dx\n\u2032\n\u2032\n\u2212\n+\n+\n\u222b\n\u222b\n = ex f (x) + C\nINTEGRALS         327\nThus,\n\u2032\n\u222b\n[\n( )\n( )]\nex\n f x + f\nx\ndx  = \n( )\nC\nx\ne f x +\nExample 22 Find (i) \n1\n2\n1\n(tan\n)\n1\nx\n\u2013\ne\nx\nx\n+\n+\n\u222b\ndx   (ii) \n2\n2\n(\n+ 1)\n( + 1)\nx\nx\ne\n\u222bx\n dx\nSolution\n(i) We have I =\n1\n12\n(tan\n)\n1\nx\n\u2013\ne\nx\ndx\nx\n+\n+\n\u222b\nConsider f (x) = tan\u2013 1x, then  f \u2032(x) = \n2\n1\n1\nx\n+\nThus, the given integrand is of the form ex [ f (x) + f \u2032(x)] Therefore, \n1\n12\nI\n(tan\n)\n1\nx\n\u2013\ne\nx\ndx\nx\n=\n+\n+\n\u222b\n = ex tan\u2013 1x + C\n(ii) We have \n2\n2\n(\n+1)\nI\n( +1)\nx\nx\ne\n= \u222bx\ndx\n2\n2\n1 +1+1)\n[\n]\n( +1)\nx\nx \u2013\ne\ndx\nx\n= \u222b\n2\n2\n2\n1\n2\n[\n]\n( +1)\n( +1)\nx\nx \u2013\ne\ndx\nx\nx\n=\n+\n\u222b\n \n2\n1\n2\n[\n+\n]\n+1\n( +1)\nx\nx \u2013\ne\ndx\nx\nx\n= \u222b\nConsider \n1\n( )\n1\nx\nf x\nx\n\u2212\n=\n+\n, then  \n2\n2\n( )\n(\n1)\nf\nx\nx\n\u2032\n=\n+\nThus, the given integrand is of the form ex [f (x) + f \u2032(x)] Therefore,\n2\n12\n1\nC\n1\n(\n1)\nx\nx\nx\nx\ne dx\ne\nx\nx\n+\n\u2212\n=\n+\n+\n+\n\u222b\nEXERCISE 7"
    },
    {
        "Chapter": "1",
        "sentence_range": "3746-3749",
        "Text": "(1)\nTaking f(x) and ex as the first function and second function, respectively, in I 1 and\nintegrating it by parts, we have I1 = f (x) ex \u2013 \n( )\nC\nx\nf\n\u2032x e dx\n+\n\u222b\nSubstituting I1 in (1), we get\nI =\n( )\n( )\n( )\nC\nx\nx\nx\ne f x\nf\nx e dx\ne f\nx dx\n\u2032\n\u2032\n\u2212\n+\n+\n\u222b\n\u222b\n = ex f (x) + C\nINTEGRALS         327\nThus,\n\u2032\n\u222b\n[\n( )\n( )]\nex\n f x + f\nx\ndx  = \n( )\nC\nx\ne f x +\nExample 22 Find (i) \n1\n2\n1\n(tan\n)\n1\nx\n\u2013\ne\nx\nx\n+\n+\n\u222b\ndx   (ii) \n2\n2\n(\n+ 1)\n( + 1)\nx\nx\ne\n\u222bx\n dx\nSolution\n(i) We have I =\n1\n12\n(tan\n)\n1\nx\n\u2013\ne\nx\ndx\nx\n+\n+\n\u222b\nConsider f (x) = tan\u2013 1x, then  f \u2032(x) = \n2\n1\n1\nx\n+\nThus, the given integrand is of the form ex [ f (x) + f \u2032(x)] Therefore, \n1\n12\nI\n(tan\n)\n1\nx\n\u2013\ne\nx\ndx\nx\n=\n+\n+\n\u222b\n = ex tan\u2013 1x + C\n(ii) We have \n2\n2\n(\n+1)\nI\n( +1)\nx\nx\ne\n= \u222bx\ndx\n2\n2\n1 +1+1)\n[\n]\n( +1)\nx\nx \u2013\ne\ndx\nx\n= \u222b\n2\n2\n2\n1\n2\n[\n]\n( +1)\n( +1)\nx\nx \u2013\ne\ndx\nx\nx\n=\n+\n\u222b\n \n2\n1\n2\n[\n+\n]\n+1\n( +1)\nx\nx \u2013\ne\ndx\nx\nx\n= \u222b\nConsider \n1\n( )\n1\nx\nf x\nx\n\u2212\n=\n+\n, then  \n2\n2\n( )\n(\n1)\nf\nx\nx\n\u2032\n=\n+\nThus, the given integrand is of the form ex [f (x) + f \u2032(x)] Therefore,\n2\n12\n1\nC\n1\n(\n1)\nx\nx\nx\nx\ne dx\ne\nx\nx\n+\n\u2212\n=\n+\n+\n+\n\u222b\nEXERCISE 7 6\nIntegrate the functions in Exercises 1 to 22"
    },
    {
        "Chapter": "1",
        "sentence_range": "3747-3750",
        "Text": "Therefore, \n1\n12\nI\n(tan\n)\n1\nx\n\u2013\ne\nx\ndx\nx\n=\n+\n+\n\u222b\n = ex tan\u2013 1x + C\n(ii) We have \n2\n2\n(\n+1)\nI\n( +1)\nx\nx\ne\n= \u222bx\ndx\n2\n2\n1 +1+1)\n[\n]\n( +1)\nx\nx \u2013\ne\ndx\nx\n= \u222b\n2\n2\n2\n1\n2\n[\n]\n( +1)\n( +1)\nx\nx \u2013\ne\ndx\nx\nx\n=\n+\n\u222b\n \n2\n1\n2\n[\n+\n]\n+1\n( +1)\nx\nx \u2013\ne\ndx\nx\nx\n= \u222b\nConsider \n1\n( )\n1\nx\nf x\nx\n\u2212\n=\n+\n, then  \n2\n2\n( )\n(\n1)\nf\nx\nx\n\u2032\n=\n+\nThus, the given integrand is of the form ex [f (x) + f \u2032(x)] Therefore,\n2\n12\n1\nC\n1\n(\n1)\nx\nx\nx\nx\ne dx\ne\nx\nx\n+\n\u2212\n=\n+\n+\n+\n\u222b\nEXERCISE 7 6\nIntegrate the functions in Exercises 1 to 22 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "3748-3751",
        "Text": "Therefore,\n2\n12\n1\nC\n1\n(\n1)\nx\nx\nx\nx\ne dx\ne\nx\nx\n+\n\u2212\n=\n+\n+\n+\n\u222b\nEXERCISE 7 6\nIntegrate the functions in Exercises 1 to 22 1 x sin x\n2"
    },
    {
        "Chapter": "1",
        "sentence_range": "3749-3752",
        "Text": "6\nIntegrate the functions in Exercises 1 to 22 1 x sin x\n2 x sin 3x\n3"
    },
    {
        "Chapter": "1",
        "sentence_range": "3750-3753",
        "Text": "1 x sin x\n2 x sin 3x\n3 x2 ex\n4"
    },
    {
        "Chapter": "1",
        "sentence_range": "3751-3754",
        "Text": "x sin x\n2 x sin 3x\n3 x2 ex\n4 x log x\n5"
    },
    {
        "Chapter": "1",
        "sentence_range": "3752-3755",
        "Text": "x sin 3x\n3 x2 ex\n4 x log x\n5 x log 2x\n6"
    },
    {
        "Chapter": "1",
        "sentence_range": "3753-3756",
        "Text": "x2 ex\n4 x log x\n5 x log 2x\n6 x2 log x\n7"
    },
    {
        "Chapter": "1",
        "sentence_range": "3754-3757",
        "Text": "x log x\n5 x log 2x\n6 x2 log x\n7 x sin\u2013 1x\n8"
    },
    {
        "Chapter": "1",
        "sentence_range": "3755-3758",
        "Text": "x log 2x\n6 x2 log x\n7 x sin\u2013 1x\n8 x tan\u20131 x\n9"
    },
    {
        "Chapter": "1",
        "sentence_range": "3756-3759",
        "Text": "x2 log x\n7 x sin\u2013 1x\n8 x tan\u20131 x\n9 x cos\u20131 x\n10"
    },
    {
        "Chapter": "1",
        "sentence_range": "3757-3760",
        "Text": "x sin\u2013 1x\n8 x tan\u20131 x\n9 x cos\u20131 x\n10 (sin\u20131x)2\n11"
    },
    {
        "Chapter": "1",
        "sentence_range": "3758-3761",
        "Text": "x tan\u20131 x\n9 x cos\u20131 x\n10 (sin\u20131x)2\n11 1\n2\ncos\n1\nx\nx\nx\n\u2212\n\u2212\n12"
    },
    {
        "Chapter": "1",
        "sentence_range": "3759-3762",
        "Text": "x cos\u20131 x\n10 (sin\u20131x)2\n11 1\n2\ncos\n1\nx\nx\nx\n\u2212\n\u2212\n12 x sec2 x\n13"
    },
    {
        "Chapter": "1",
        "sentence_range": "3760-3763",
        "Text": "(sin\u20131x)2\n11 1\n2\ncos\n1\nx\nx\nx\n\u2212\n\u2212\n12 x sec2 x\n13 tan \u20131x\n14"
    },
    {
        "Chapter": "1",
        "sentence_range": "3761-3764",
        "Text": "1\n2\ncos\n1\nx\nx\nx\n\u2212\n\u2212\n12 x sec2 x\n13 tan \u20131x\n14 x (log x)2\n15"
    },
    {
        "Chapter": "1",
        "sentence_range": "3762-3765",
        "Text": "x sec2 x\n13 tan \u20131x\n14 x (log x)2\n15 (x2 + 1) log x\n328\nMATHEMATICS\n16"
    },
    {
        "Chapter": "1",
        "sentence_range": "3763-3766",
        "Text": "tan \u20131x\n14 x (log x)2\n15 (x2 + 1) log x\n328\nMATHEMATICS\n16 ex (sinx + cosx) 17"
    },
    {
        "Chapter": "1",
        "sentence_range": "3764-3767",
        "Text": "x (log x)2\n15 (x2 + 1) log x\n328\nMATHEMATICS\n16 ex (sinx + cosx) 17 2\n(1\n)\nx\nx e\n+x\n18"
    },
    {
        "Chapter": "1",
        "sentence_range": "3765-3768",
        "Text": "(x2 + 1) log x\n328\nMATHEMATICS\n16 ex (sinx + cosx) 17 2\n(1\n)\nx\nx e\n+x\n18 1\nsin\n1\ncos\nx\nx\ne\nx\n\uf8eb\n\uf8f6\n+\n\uf8ec\n\uf8f7\n\uf8ed+\n\uf8f8\n19"
    },
    {
        "Chapter": "1",
        "sentence_range": "3766-3769",
        "Text": "ex (sinx + cosx) 17 2\n(1\n)\nx\nx e\n+x\n18 1\nsin\n1\ncos\nx\nx\ne\nx\n\uf8eb\n\uf8f6\n+\n\uf8ec\n\uf8f7\n\uf8ed+\n\uf8f8\n19 2\n1\n\u20131\nxe\nx\nx\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n20"
    },
    {
        "Chapter": "1",
        "sentence_range": "3767-3770",
        "Text": "2\n(1\n)\nx\nx e\n+x\n18 1\nsin\n1\ncos\nx\nx\ne\nx\n\uf8eb\n\uf8f6\n+\n\uf8ec\n\uf8f7\n\uf8ed+\n\uf8f8\n19 2\n1\n\u20131\nxe\nx\nx\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n20 3\n(\n3)\n(\n1)\nx\nx\ne\n\u2212x\n\u2212\n21"
    },
    {
        "Chapter": "1",
        "sentence_range": "3768-3771",
        "Text": "1\nsin\n1\ncos\nx\nx\ne\nx\n\uf8eb\n\uf8f6\n+\n\uf8ec\n\uf8f7\n\uf8ed+\n\uf8f8\n19 2\n1\n\u20131\nxe\nx\nx\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n20 3\n(\n3)\n(\n1)\nx\nx\ne\n\u2212x\n\u2212\n21 e2x sin x\n22"
    },
    {
        "Chapter": "1",
        "sentence_range": "3769-3772",
        "Text": "2\n1\n\u20131\nxe\nx\nx\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n20 3\n(\n3)\n(\n1)\nx\nx\ne\n\u2212x\n\u2212\n21 e2x sin x\n22 1\n2\n2\nsin\n1\n\u2013\nx\nx\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed+\n\uf8f8\nChoose the correct answer in Exercises 23 and 24"
    },
    {
        "Chapter": "1",
        "sentence_range": "3770-3773",
        "Text": "3\n(\n3)\n(\n1)\nx\nx\ne\n\u2212x\n\u2212\n21 e2x sin x\n22 1\n2\n2\nsin\n1\n\u2013\nx\nx\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed+\n\uf8f8\nChoose the correct answer in Exercises 23 and 24 23"
    },
    {
        "Chapter": "1",
        "sentence_range": "3771-3774",
        "Text": "e2x sin x\n22 1\n2\n2\nsin\n1\n\u2013\nx\nx\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed+\n\uf8f8\nChoose the correct answer in Exercises 23 and 24 23 3\n2\nx\n\u222bx e dx\n equals\n(A)\n3\n1\nC\n3\nex\n+\n(B)\n2\n1\nC\n3\nex\n+\n(C)\n3\n1\nC\n2\nex\n+\n(D)\n2\n1\nC\n2\nex\n+\n24"
    },
    {
        "Chapter": "1",
        "sentence_range": "3772-3775",
        "Text": "1\n2\n2\nsin\n1\n\u2013\nx\nx\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed+\n\uf8f8\nChoose the correct answer in Exercises 23 and 24 23 3\n2\nx\n\u222bx e dx\n equals\n(A)\n3\n1\nC\n3\nex\n+\n(B)\n2\n1\nC\n3\nex\n+\n(C)\n3\n1\nC\n2\nex\n+\n(D)\n2\n1\nC\n2\nex\n+\n24 sec\n(1\ntan )\nex\nx\nx dx\n+\n\u222b\n equals\n(A) ex cos x + C\n(B) ex sec x + C\n(C) ex sin x + C\n(D) ex tan x + C\n7"
    },
    {
        "Chapter": "1",
        "sentence_range": "3773-3776",
        "Text": "23 3\n2\nx\n\u222bx e dx\n equals\n(A)\n3\n1\nC\n3\nex\n+\n(B)\n2\n1\nC\n3\nex\n+\n(C)\n3\n1\nC\n2\nex\n+\n(D)\n2\n1\nC\n2\nex\n+\n24 sec\n(1\ntan )\nex\nx\nx dx\n+\n\u222b\n equals\n(A) ex cos x + C\n(B) ex sec x + C\n(C) ex sin x + C\n(D) ex tan x + C\n7 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "3774-3777",
        "Text": "3\n2\nx\n\u222bx e dx\n equals\n(A)\n3\n1\nC\n3\nex\n+\n(B)\n2\n1\nC\n3\nex\n+\n(C)\n3\n1\nC\n2\nex\n+\n(D)\n2\n1\nC\n2\nex\n+\n24 sec\n(1\ntan )\nex\nx\nx dx\n+\n\u222b\n equals\n(A) ex cos x + C\n(B) ex sec x + C\n(C) ex sin x + C\n(D) ex tan x + C\n7 6 2 Integrals of some more types\nHere, we discuss some special types of standard integrals based on the technique of\nintegration  by parts :\n(i)\n2\n2\nx\na\ndx\n\u2212\n\u222b\n(ii)\n2\n2\nx\n+a dx\n\u222b\n(iii)\n2\n2\na\n\u2212x dx\n\u222b\n(i)\n Let \n2\n2\nI\nx\na dx\n=\n\u2212\n\u222b\nTaking constant function 1 as the second function and integrating by parts, we\nhave\nI =\n2\n2\n2\n2\n1\n2\n2\nx\nx\nx\na\nx dx\nx\na\n\u2212\n\u2212\n\u2212\n\u222b\n=\n2\n2\n2\n2\n2\nx\nx\nx\na\ndx\nx\na\n\u2212\n\u2212\n\u2212\n\u222b\n = \n2\n2\n2\n2\n2\n2\n2\nx\na\na\nx\nx\na\ndx\nx\na\n\u2212\n+\n\u2212\n\u2212\n\u2212\n\u222b\nINTEGRALS         329\n=\n2\n2\n2\n2\n2\n2\n2\ndx\nx\nx\na\nx\na dx\na\nx\na\n\u2212\n\u2212\n\u2212\n\u2212\n\u2212\n\u222b\n\u222b\n=\n2\n2\n2\n2\n2\nI\ndx\nx\nx\na\na\nx\na\n\u2212\n\u2212 \u2212\n\u2212\n\u222b\nor\n2I =\n2\n2\n2\n2\n2\ndx\nx\nx\na\na\nx\na\n\u2212\n\u2212\n\u2212\n\u222b\nor\nI = \u222b\n2\n2\nx \u2013 a dx = \n2\n2\n2\n2\n2\n\u2013\n\u2013\nlog\n+\n\u2013\n+ C\n2\n2\nx\na\nx\na\nx\nx\na\nSimilarly, integrating other two integrals by parts, taking constant function 1 as the\nsecond function, we get\n(ii) \u222b\n2\n2\n2\n2\n2\n2\n2\n1\n+\n=\n+\n+\nlog\n+\n+\n+ C\n2\na2\nx\na dx\nx\nx\na\nx\nx\na\n(iii) \u222b\n2\n2\n2\n2\n2\n\u20131\n1\n\u2013\n=\n\u2013\n+\nsin\n+ C\n2\na2\nx\na\nx dx\nx a\nx\na\nAlternatively, integrals (i), (ii) and (iii) can also be found by making trigonometric\nsubstitution x = a sec\u03b8 in (i), x = a tan\u03b8 in (ii) and x = a sin\u03b8 in (iii) respectively"
    },
    {
        "Chapter": "1",
        "sentence_range": "3775-3778",
        "Text": "sec\n(1\ntan )\nex\nx\nx dx\n+\n\u222b\n equals\n(A) ex cos x + C\n(B) ex sec x + C\n(C) ex sin x + C\n(D) ex tan x + C\n7 6 2 Integrals of some more types\nHere, we discuss some special types of standard integrals based on the technique of\nintegration  by parts :\n(i)\n2\n2\nx\na\ndx\n\u2212\n\u222b\n(ii)\n2\n2\nx\n+a dx\n\u222b\n(iii)\n2\n2\na\n\u2212x dx\n\u222b\n(i)\n Let \n2\n2\nI\nx\na dx\n=\n\u2212\n\u222b\nTaking constant function 1 as the second function and integrating by parts, we\nhave\nI =\n2\n2\n2\n2\n1\n2\n2\nx\nx\nx\na\nx dx\nx\na\n\u2212\n\u2212\n\u2212\n\u222b\n=\n2\n2\n2\n2\n2\nx\nx\nx\na\ndx\nx\na\n\u2212\n\u2212\n\u2212\n\u222b\n = \n2\n2\n2\n2\n2\n2\n2\nx\na\na\nx\nx\na\ndx\nx\na\n\u2212\n+\n\u2212\n\u2212\n\u2212\n\u222b\nINTEGRALS         329\n=\n2\n2\n2\n2\n2\n2\n2\ndx\nx\nx\na\nx\na dx\na\nx\na\n\u2212\n\u2212\n\u2212\n\u2212\n\u2212\n\u222b\n\u222b\n=\n2\n2\n2\n2\n2\nI\ndx\nx\nx\na\na\nx\na\n\u2212\n\u2212 \u2212\n\u2212\n\u222b\nor\n2I =\n2\n2\n2\n2\n2\ndx\nx\nx\na\na\nx\na\n\u2212\n\u2212\n\u2212\n\u222b\nor\nI = \u222b\n2\n2\nx \u2013 a dx = \n2\n2\n2\n2\n2\n\u2013\n\u2013\nlog\n+\n\u2013\n+ C\n2\n2\nx\na\nx\na\nx\nx\na\nSimilarly, integrating other two integrals by parts, taking constant function 1 as the\nsecond function, we get\n(ii) \u222b\n2\n2\n2\n2\n2\n2\n2\n1\n+\n=\n+\n+\nlog\n+\n+\n+ C\n2\na2\nx\na dx\nx\nx\na\nx\nx\na\n(iii) \u222b\n2\n2\n2\n2\n2\n\u20131\n1\n\u2013\n=\n\u2013\n+\nsin\n+ C\n2\na2\nx\na\nx dx\nx a\nx\na\nAlternatively, integrals (i), (ii) and (iii) can also be found by making trigonometric\nsubstitution x = a sec\u03b8 in (i), x = a tan\u03b8 in (ii) and x = a sin\u03b8 in (iii) respectively Example 23 Find \n2\n2\n5\nx\nx\ndx\n+\n+\n\u222b\nSolution Note that\n2\n2\n5\nx\nx\ndx\n+\n+\n\u222b\n =\n2\n(\n1)\n4\nx\ndx\n+\n+\n\u222b\nPut  x + 1 = y, so that dx = dy"
    },
    {
        "Chapter": "1",
        "sentence_range": "3776-3779",
        "Text": "6 2 Integrals of some more types\nHere, we discuss some special types of standard integrals based on the technique of\nintegration  by parts :\n(i)\n2\n2\nx\na\ndx\n\u2212\n\u222b\n(ii)\n2\n2\nx\n+a dx\n\u222b\n(iii)\n2\n2\na\n\u2212x dx\n\u222b\n(i)\n Let \n2\n2\nI\nx\na dx\n=\n\u2212\n\u222b\nTaking constant function 1 as the second function and integrating by parts, we\nhave\nI =\n2\n2\n2\n2\n1\n2\n2\nx\nx\nx\na\nx dx\nx\na\n\u2212\n\u2212\n\u2212\n\u222b\n=\n2\n2\n2\n2\n2\nx\nx\nx\na\ndx\nx\na\n\u2212\n\u2212\n\u2212\n\u222b\n = \n2\n2\n2\n2\n2\n2\n2\nx\na\na\nx\nx\na\ndx\nx\na\n\u2212\n+\n\u2212\n\u2212\n\u2212\n\u222b\nINTEGRALS         329\n=\n2\n2\n2\n2\n2\n2\n2\ndx\nx\nx\na\nx\na dx\na\nx\na\n\u2212\n\u2212\n\u2212\n\u2212\n\u2212\n\u222b\n\u222b\n=\n2\n2\n2\n2\n2\nI\ndx\nx\nx\na\na\nx\na\n\u2212\n\u2212 \u2212\n\u2212\n\u222b\nor\n2I =\n2\n2\n2\n2\n2\ndx\nx\nx\na\na\nx\na\n\u2212\n\u2212\n\u2212\n\u222b\nor\nI = \u222b\n2\n2\nx \u2013 a dx = \n2\n2\n2\n2\n2\n\u2013\n\u2013\nlog\n+\n\u2013\n+ C\n2\n2\nx\na\nx\na\nx\nx\na\nSimilarly, integrating other two integrals by parts, taking constant function 1 as the\nsecond function, we get\n(ii) \u222b\n2\n2\n2\n2\n2\n2\n2\n1\n+\n=\n+\n+\nlog\n+\n+\n+ C\n2\na2\nx\na dx\nx\nx\na\nx\nx\na\n(iii) \u222b\n2\n2\n2\n2\n2\n\u20131\n1\n\u2013\n=\n\u2013\n+\nsin\n+ C\n2\na2\nx\na\nx dx\nx a\nx\na\nAlternatively, integrals (i), (ii) and (iii) can also be found by making trigonometric\nsubstitution x = a sec\u03b8 in (i), x = a tan\u03b8 in (ii) and x = a sin\u03b8 in (iii) respectively Example 23 Find \n2\n2\n5\nx\nx\ndx\n+\n+\n\u222b\nSolution Note that\n2\n2\n5\nx\nx\ndx\n+\n+\n\u222b\n =\n2\n(\n1)\n4\nx\ndx\n+\n+\n\u222b\nPut  x + 1 = y, so that dx = dy Then\n2\n2\n5\nx\nx\ndx\n+\n+\n\u222b\n =\n2\n22\ny\ndy\n+\n\u222b\n=\n2\n2\n1\n4\n4\nlog\n4\nC\n2\n2\ny\ny\ny\ny\n+\n+\n+\n+\n+\n        [using 7"
    },
    {
        "Chapter": "1",
        "sentence_range": "3777-3780",
        "Text": "2 Integrals of some more types\nHere, we discuss some special types of standard integrals based on the technique of\nintegration  by parts :\n(i)\n2\n2\nx\na\ndx\n\u2212\n\u222b\n(ii)\n2\n2\nx\n+a dx\n\u222b\n(iii)\n2\n2\na\n\u2212x dx\n\u222b\n(i)\n Let \n2\n2\nI\nx\na dx\n=\n\u2212\n\u222b\nTaking constant function 1 as the second function and integrating by parts, we\nhave\nI =\n2\n2\n2\n2\n1\n2\n2\nx\nx\nx\na\nx dx\nx\na\n\u2212\n\u2212\n\u2212\n\u222b\n=\n2\n2\n2\n2\n2\nx\nx\nx\na\ndx\nx\na\n\u2212\n\u2212\n\u2212\n\u222b\n = \n2\n2\n2\n2\n2\n2\n2\nx\na\na\nx\nx\na\ndx\nx\na\n\u2212\n+\n\u2212\n\u2212\n\u2212\n\u222b\nINTEGRALS         329\n=\n2\n2\n2\n2\n2\n2\n2\ndx\nx\nx\na\nx\na dx\na\nx\na\n\u2212\n\u2212\n\u2212\n\u2212\n\u2212\n\u222b\n\u222b\n=\n2\n2\n2\n2\n2\nI\ndx\nx\nx\na\na\nx\na\n\u2212\n\u2212 \u2212\n\u2212\n\u222b\nor\n2I =\n2\n2\n2\n2\n2\ndx\nx\nx\na\na\nx\na\n\u2212\n\u2212\n\u2212\n\u222b\nor\nI = \u222b\n2\n2\nx \u2013 a dx = \n2\n2\n2\n2\n2\n\u2013\n\u2013\nlog\n+\n\u2013\n+ C\n2\n2\nx\na\nx\na\nx\nx\na\nSimilarly, integrating other two integrals by parts, taking constant function 1 as the\nsecond function, we get\n(ii) \u222b\n2\n2\n2\n2\n2\n2\n2\n1\n+\n=\n+\n+\nlog\n+\n+\n+ C\n2\na2\nx\na dx\nx\nx\na\nx\nx\na\n(iii) \u222b\n2\n2\n2\n2\n2\n\u20131\n1\n\u2013\n=\n\u2013\n+\nsin\n+ C\n2\na2\nx\na\nx dx\nx a\nx\na\nAlternatively, integrals (i), (ii) and (iii) can also be found by making trigonometric\nsubstitution x = a sec\u03b8 in (i), x = a tan\u03b8 in (ii) and x = a sin\u03b8 in (iii) respectively Example 23 Find \n2\n2\n5\nx\nx\ndx\n+\n+\n\u222b\nSolution Note that\n2\n2\n5\nx\nx\ndx\n+\n+\n\u222b\n =\n2\n(\n1)\n4\nx\ndx\n+\n+\n\u222b\nPut  x + 1 = y, so that dx = dy Then\n2\n2\n5\nx\nx\ndx\n+\n+\n\u222b\n =\n2\n22\ny\ndy\n+\n\u222b\n=\n2\n2\n1\n4\n4\nlog\n4\nC\n2\n2\ny\ny\ny\ny\n+\n+\n+\n+\n+\n        [using 7 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "3778-3781",
        "Text": "Example 23 Find \n2\n2\n5\nx\nx\ndx\n+\n+\n\u222b\nSolution Note that\n2\n2\n5\nx\nx\ndx\n+\n+\n\u222b\n =\n2\n(\n1)\n4\nx\ndx\n+\n+\n\u222b\nPut  x + 1 = y, so that dx = dy Then\n2\n2\n5\nx\nx\ndx\n+\n+\n\u222b\n =\n2\n22\ny\ndy\n+\n\u222b\n=\n2\n2\n1\n4\n4\nlog\n4\nC\n2\n2\ny\ny\ny\ny\n+\n+\n+\n+\n+\n        [using 7 6 2 (ii)]\n=\n2\n2\n1 (\n1)\n2\n5\n2 log\n1\n2\n5\nC\n2 x\nx\nx\nx\nx\nx\n+\n+\n+\n+\n+ +\n+\n+\n+\nExample 24 Find \n2\n3\n2x\nx dx\n\u2212\n\u2212\n\u222b\nSolution Note that \n2\n2\n3\n2\n4\n(\n1)\nx\nx\ndx\nx\ndx\n\u2212\n\u2212\n=\n\u2212\n+\n\u222b\n\u222b\n330\nMATHEMATICS\nPut x + 1 = y so that dx = dy"
    },
    {
        "Chapter": "1",
        "sentence_range": "3779-3782",
        "Text": "Then\n2\n2\n5\nx\nx\ndx\n+\n+\n\u222b\n =\n2\n22\ny\ndy\n+\n\u222b\n=\n2\n2\n1\n4\n4\nlog\n4\nC\n2\n2\ny\ny\ny\ny\n+\n+\n+\n+\n+\n        [using 7 6 2 (ii)]\n=\n2\n2\n1 (\n1)\n2\n5\n2 log\n1\n2\n5\nC\n2 x\nx\nx\nx\nx\nx\n+\n+\n+\n+\n+ +\n+\n+\n+\nExample 24 Find \n2\n3\n2x\nx dx\n\u2212\n\u2212\n\u222b\nSolution Note that \n2\n2\n3\n2\n4\n(\n1)\nx\nx\ndx\nx\ndx\n\u2212\n\u2212\n=\n\u2212\n+\n\u222b\n\u222b\n330\nMATHEMATICS\nPut x + 1 = y so that dx = dy Thus\n2\n3\n2x\nx dx\n\u2212\n\u2212\n\u222b\n =\n2\n4\n\u2212y dy\n\u222b\n=\n2\n1\n1\n4\n4\nsin\nC\n2\n2\n2\n\u2013 y\ny\n\u2212y\n+\n+\n[using 7"
    },
    {
        "Chapter": "1",
        "sentence_range": "3780-3783",
        "Text": "6 2 (ii)]\n=\n2\n2\n1 (\n1)\n2\n5\n2 log\n1\n2\n5\nC\n2 x\nx\nx\nx\nx\nx\n+\n+\n+\n+\n+ +\n+\n+\n+\nExample 24 Find \n2\n3\n2x\nx dx\n\u2212\n\u2212\n\u222b\nSolution Note that \n2\n2\n3\n2\n4\n(\n1)\nx\nx\ndx\nx\ndx\n\u2212\n\u2212\n=\n\u2212\n+\n\u222b\n\u222b\n330\nMATHEMATICS\nPut x + 1 = y so that dx = dy Thus\n2\n3\n2x\nx dx\n\u2212\n\u2212\n\u222b\n =\n2\n4\n\u2212y dy\n\u222b\n=\n2\n1\n1\n4\n4\nsin\nC\n2\n2\n2\n\u2013 y\ny\n\u2212y\n+\n+\n[using 7 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "3781-3784",
        "Text": "2 (ii)]\n=\n2\n2\n1 (\n1)\n2\n5\n2 log\n1\n2\n5\nC\n2 x\nx\nx\nx\nx\nx\n+\n+\n+\n+\n+ +\n+\n+\n+\nExample 24 Find \n2\n3\n2x\nx dx\n\u2212\n\u2212\n\u222b\nSolution Note that \n2\n2\n3\n2\n4\n(\n1)\nx\nx\ndx\nx\ndx\n\u2212\n\u2212\n=\n\u2212\n+\n\u222b\n\u222b\n330\nMATHEMATICS\nPut x + 1 = y so that dx = dy Thus\n2\n3\n2x\nx dx\n\u2212\n\u2212\n\u222b\n =\n2\n4\n\u2212y dy\n\u222b\n=\n2\n1\n1\n4\n4\nsin\nC\n2\n2\n2\n\u2013 y\ny\n\u2212y\n+\n+\n[using 7 6 2 (iii)]\n=\n2\n1\n1\n1\n(\n1)\n3\n2\n2 sin\nC\n2\n2\n\u2013\nx\nx\nx\nx\n+\n\uf8eb\n\uf8f6\n+\n\u2212\n\u2212\n+\n+\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\nEXERCISE 7"
    },
    {
        "Chapter": "1",
        "sentence_range": "3782-3785",
        "Text": "Thus\n2\n3\n2x\nx dx\n\u2212\n\u2212\n\u222b\n =\n2\n4\n\u2212y dy\n\u222b\n=\n2\n1\n1\n4\n4\nsin\nC\n2\n2\n2\n\u2013 y\ny\n\u2212y\n+\n+\n[using 7 6 2 (iii)]\n=\n2\n1\n1\n1\n(\n1)\n3\n2\n2 sin\nC\n2\n2\n\u2013\nx\nx\nx\nx\n+\n\uf8eb\n\uf8f6\n+\n\u2212\n\u2212\n+\n+\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\nEXERCISE 7 7\nIntegrate the functions in Exercises 1 to 9"
    },
    {
        "Chapter": "1",
        "sentence_range": "3783-3786",
        "Text": "6 2 (iii)]\n=\n2\n1\n1\n1\n(\n1)\n3\n2\n2 sin\nC\n2\n2\n\u2013\nx\nx\nx\nx\n+\n\uf8eb\n\uf8f6\n+\n\u2212\n\u2212\n+\n+\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\nEXERCISE 7 7\nIntegrate the functions in Exercises 1 to 9 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "3784-3787",
        "Text": "2 (iii)]\n=\n2\n1\n1\n1\n(\n1)\n3\n2\n2 sin\nC\n2\n2\n\u2013\nx\nx\nx\nx\n+\n\uf8eb\n\uf8f6\n+\n\u2212\n\u2212\n+\n+\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\nEXERCISE 7 7\nIntegrate the functions in Exercises 1 to 9 1 2\n4\n\u2212x\n2"
    },
    {
        "Chapter": "1",
        "sentence_range": "3785-3788",
        "Text": "7\nIntegrate the functions in Exercises 1 to 9 1 2\n4\n\u2212x\n2 2\n1\n\u22124x\n3"
    },
    {
        "Chapter": "1",
        "sentence_range": "3786-3789",
        "Text": "1 2\n4\n\u2212x\n2 2\n1\n\u22124x\n3 2\n4\n6\nx\nx\n+\n+\n4"
    },
    {
        "Chapter": "1",
        "sentence_range": "3787-3790",
        "Text": "2\n4\n\u2212x\n2 2\n1\n\u22124x\n3 2\n4\n6\nx\nx\n+\n+\n4 2\n4\n1\nx\nx\n+\n+\n5"
    },
    {
        "Chapter": "1",
        "sentence_range": "3788-3791",
        "Text": "2\n1\n\u22124x\n3 2\n4\n6\nx\nx\n+\n+\n4 2\n4\n1\nx\nx\n+\n+\n5 2\n1\n4x\nx\n\u2212\n\u2212\n6"
    },
    {
        "Chapter": "1",
        "sentence_range": "3789-3792",
        "Text": "2\n4\n6\nx\nx\n+\n+\n4 2\n4\n1\nx\nx\n+\n+\n5 2\n1\n4x\nx\n\u2212\n\u2212\n6 2\n4\n5\nx\nx\n+\n\u2212\n7"
    },
    {
        "Chapter": "1",
        "sentence_range": "3790-3793",
        "Text": "2\n4\n1\nx\nx\n+\n+\n5 2\n1\n4x\nx\n\u2212\n\u2212\n6 2\n4\n5\nx\nx\n+\n\u2212\n7 2\n1\n3x\nx\n+\n\u2212\n8"
    },
    {
        "Chapter": "1",
        "sentence_range": "3791-3794",
        "Text": "2\n1\n4x\nx\n\u2212\n\u2212\n6 2\n4\n5\nx\nx\n+\n\u2212\n7 2\n1\n3x\nx\n+\n\u2212\n8 2\n3\nx\n+x\n9"
    },
    {
        "Chapter": "1",
        "sentence_range": "3792-3795",
        "Text": "2\n4\n5\nx\nx\n+\n\u2212\n7 2\n1\n3x\nx\n+\n\u2212\n8 2\n3\nx\n+x\n9 2\n1\nx9\n+\nChoose the correct answer in Exercises 10 to 11"
    },
    {
        "Chapter": "1",
        "sentence_range": "3793-3796",
        "Text": "2\n1\n3x\nx\n+\n\u2212\n8 2\n3\nx\n+x\n9 2\n1\nx9\n+\nChoose the correct answer in Exercises 10 to 11 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "3794-3797",
        "Text": "2\n3\nx\n+x\n9 2\n1\nx9\n+\nChoose the correct answer in Exercises 10 to 11 10 2\n1\n+x dx\n\u222b\nis equal to\n(A)\n(\n)\n2\n2\n1\n1\nlog\n1\nC\n2\n2\nx\nx\nx\nx\n+\n+\n+\n+\n+\n(B)\n2 23\n2(1\n)\nC\n3\n+x\n+\n(C)\n2 23\n2\n(1\n)\nC\n3 x\n+x\n+\n(D)\n2\n2\n2\n2\n1\n1\nlog\n1\nC\n2\n2\nx\nx\nx\nx\nx\n+\n+\n+\n+\n+\n11"
    },
    {
        "Chapter": "1",
        "sentence_range": "3795-3798",
        "Text": "2\n1\nx9\n+\nChoose the correct answer in Exercises 10 to 11 10 2\n1\n+x dx\n\u222b\nis equal to\n(A)\n(\n)\n2\n2\n1\n1\nlog\n1\nC\n2\n2\nx\nx\nx\nx\n+\n+\n+\n+\n+\n(B)\n2 23\n2(1\n)\nC\n3\n+x\n+\n(C)\n2 23\n2\n(1\n)\nC\n3 x\n+x\n+\n(D)\n2\n2\n2\n2\n1\n1\nlog\n1\nC\n2\n2\nx\nx\nx\nx\nx\n+\n+\n+\n+\n+\n11 2\n8\n7\nx\nx\ndx\n\u2212\n+\n\u222b\n is equal to\n(A)\n2\n2\n1 (\n4)\n8\n7\n9log\n4\n8\n7\nC\n2 x\nx\nx\nx\nx\nx\n\u2212\n\u2212\n+\n+\n\u2212\n+\n\u2212\n+\n+\n(B)\n2\n2\n1 (\n4)\n8\n7\n9log\n4\n8\n7\nC\n2 x\nx\nx\nx\nx\nx\n+\n\u2212\n+\n+\n+ +\n\u2212\n+\n+\n(C)\n2\n2\n1 (\n4)\n8\n7\n3\n2log\n4\n8\n7\nC\n2 x\nx\nx\nx\nx\nx\n\u2212\n\u2212\n+\n\u2212\n\u2212\n+\n\u2212\n+\n+\n(D)\n2\n2\n1\n9\n(\n4)\n8\n7\nlog\n4\n8\n7\nC\n2\n2\nx\nx\nx\nx\nx\nx\n\u2212\n\u2212\n+\n\u2212\n\u2212\n+\n\u2212\n+\n+\nINTEGRALS         331\n7"
    },
    {
        "Chapter": "1",
        "sentence_range": "3796-3799",
        "Text": "10 2\n1\n+x dx\n\u222b\nis equal to\n(A)\n(\n)\n2\n2\n1\n1\nlog\n1\nC\n2\n2\nx\nx\nx\nx\n+\n+\n+\n+\n+\n(B)\n2 23\n2(1\n)\nC\n3\n+x\n+\n(C)\n2 23\n2\n(1\n)\nC\n3 x\n+x\n+\n(D)\n2\n2\n2\n2\n1\n1\nlog\n1\nC\n2\n2\nx\nx\nx\nx\nx\n+\n+\n+\n+\n+\n11 2\n8\n7\nx\nx\ndx\n\u2212\n+\n\u222b\n is equal to\n(A)\n2\n2\n1 (\n4)\n8\n7\n9log\n4\n8\n7\nC\n2 x\nx\nx\nx\nx\nx\n\u2212\n\u2212\n+\n+\n\u2212\n+\n\u2212\n+\n+\n(B)\n2\n2\n1 (\n4)\n8\n7\n9log\n4\n8\n7\nC\n2 x\nx\nx\nx\nx\nx\n+\n\u2212\n+\n+\n+ +\n\u2212\n+\n+\n(C)\n2\n2\n1 (\n4)\n8\n7\n3\n2log\n4\n8\n7\nC\n2 x\nx\nx\nx\nx\nx\n\u2212\n\u2212\n+\n\u2212\n\u2212\n+\n\u2212\n+\n+\n(D)\n2\n2\n1\n9\n(\n4)\n8\n7\nlog\n4\n8\n7\nC\n2\n2\nx\nx\nx\nx\nx\nx\n\u2212\n\u2212\n+\n\u2212\n\u2212\n+\n\u2212\n+\n+\nINTEGRALS         331\n7 7   Definite Integral\nIn the previous sections, we have studied about the indefinite integrals and discussed\nfew methods of finding them including integrals of some special functions"
    },
    {
        "Chapter": "1",
        "sentence_range": "3797-3800",
        "Text": "2\n1\n+x dx\n\u222b\nis equal to\n(A)\n(\n)\n2\n2\n1\n1\nlog\n1\nC\n2\n2\nx\nx\nx\nx\n+\n+\n+\n+\n+\n(B)\n2 23\n2(1\n)\nC\n3\n+x\n+\n(C)\n2 23\n2\n(1\n)\nC\n3 x\n+x\n+\n(D)\n2\n2\n2\n2\n1\n1\nlog\n1\nC\n2\n2\nx\nx\nx\nx\nx\n+\n+\n+\n+\n+\n11 2\n8\n7\nx\nx\ndx\n\u2212\n+\n\u222b\n is equal to\n(A)\n2\n2\n1 (\n4)\n8\n7\n9log\n4\n8\n7\nC\n2 x\nx\nx\nx\nx\nx\n\u2212\n\u2212\n+\n+\n\u2212\n+\n\u2212\n+\n+\n(B)\n2\n2\n1 (\n4)\n8\n7\n9log\n4\n8\n7\nC\n2 x\nx\nx\nx\nx\nx\n+\n\u2212\n+\n+\n+ +\n\u2212\n+\n+\n(C)\n2\n2\n1 (\n4)\n8\n7\n3\n2log\n4\n8\n7\nC\n2 x\nx\nx\nx\nx\nx\n\u2212\n\u2212\n+\n\u2212\n\u2212\n+\n\u2212\n+\n+\n(D)\n2\n2\n1\n9\n(\n4)\n8\n7\nlog\n4\n8\n7\nC\n2\n2\nx\nx\nx\nx\nx\nx\n\u2212\n\u2212\n+\n\u2212\n\u2212\n+\n\u2212\n+\n+\nINTEGRALS         331\n7 7   Definite Integral\nIn the previous sections, we have studied about the indefinite integrals and discussed\nfew methods of finding them including integrals of some special functions In this\nsection, we shall study what is called definite integral of a function"
    },
    {
        "Chapter": "1",
        "sentence_range": "3798-3801",
        "Text": "2\n8\n7\nx\nx\ndx\n\u2212\n+\n\u222b\n is equal to\n(A)\n2\n2\n1 (\n4)\n8\n7\n9log\n4\n8\n7\nC\n2 x\nx\nx\nx\nx\nx\n\u2212\n\u2212\n+\n+\n\u2212\n+\n\u2212\n+\n+\n(B)\n2\n2\n1 (\n4)\n8\n7\n9log\n4\n8\n7\nC\n2 x\nx\nx\nx\nx\nx\n+\n\u2212\n+\n+\n+ +\n\u2212\n+\n+\n(C)\n2\n2\n1 (\n4)\n8\n7\n3\n2log\n4\n8\n7\nC\n2 x\nx\nx\nx\nx\nx\n\u2212\n\u2212\n+\n\u2212\n\u2212\n+\n\u2212\n+\n+\n(D)\n2\n2\n1\n9\n(\n4)\n8\n7\nlog\n4\n8\n7\nC\n2\n2\nx\nx\nx\nx\nx\nx\n\u2212\n\u2212\n+\n\u2212\n\u2212\n+\n\u2212\n+\n+\nINTEGRALS         331\n7 7   Definite Integral\nIn the previous sections, we have studied about the indefinite integrals and discussed\nfew methods of finding them including integrals of some special functions In this\nsection, we shall study what is called definite integral of a function The definite integral\nhas a unique value"
    },
    {
        "Chapter": "1",
        "sentence_range": "3799-3802",
        "Text": "7   Definite Integral\nIn the previous sections, we have studied about the indefinite integrals and discussed\nfew methods of finding them including integrals of some special functions In this\nsection, we shall study what is called definite integral of a function The definite integral\nhas a unique value A definite integral is denoted by \n( )\nb\n\u222ba f x dx\n, where a is called the\nlower limit of the integral and b is called the upper limit of the integral"
    },
    {
        "Chapter": "1",
        "sentence_range": "3800-3803",
        "Text": "In this\nsection, we shall study what is called definite integral of a function The definite integral\nhas a unique value A definite integral is denoted by \n( )\nb\n\u222ba f x dx\n, where a is called the\nlower limit of the integral and b is called the upper limit of the integral The definite\nintegral is introduced either as the limit of a sum or if it has an anti derivative F in the\ninterval [a, b], then its value is the difference  between the values of F at the end\npoints, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "3801-3804",
        "Text": "The definite integral\nhas a unique value A definite integral is denoted by \n( )\nb\n\u222ba f x dx\n, where a is called the\nlower limit of the integral and b is called the upper limit of the integral The definite\nintegral is introduced either as the limit of a sum or if it has an anti derivative F in the\ninterval [a, b], then its value is the difference  between the values of F at the end\npoints, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "3802-3805",
        "Text": "A definite integral is denoted by \n( )\nb\n\u222ba f x dx\n, where a is called the\nlower limit of the integral and b is called the upper limit of the integral The definite\nintegral is introduced either as the limit of a sum or if it has an anti derivative F in the\ninterval [a, b], then its value is the difference  between the values of F at the end\npoints, i e , F(b) \u2013 F(a)"
    },
    {
        "Chapter": "1",
        "sentence_range": "3803-3806",
        "Text": "The definite\nintegral is introduced either as the limit of a sum or if it has an anti derivative F in the\ninterval [a, b], then its value is the difference  between the values of F at the end\npoints, i e , F(b) \u2013 F(a) Here, we shall consider these two cases separately as discussed\nbelow:\n7"
    },
    {
        "Chapter": "1",
        "sentence_range": "3804-3807",
        "Text": "e , F(b) \u2013 F(a) Here, we shall consider these two cases separately as discussed\nbelow:\n7 7"
    },
    {
        "Chapter": "1",
        "sentence_range": "3805-3808",
        "Text": ", F(b) \u2013 F(a) Here, we shall consider these two cases separately as discussed\nbelow:\n7 7 1  Definite integral as the limit of a sum\nLet f be a continuous function defined on close interval [a, b]"
    },
    {
        "Chapter": "1",
        "sentence_range": "3806-3809",
        "Text": "Here, we shall consider these two cases separately as discussed\nbelow:\n7 7 1  Definite integral as the limit of a sum\nLet f be a continuous function defined on close interval [a, b] Assume that all the\nvalues taken by the function are non negative, so the graph of the function is a curve\nabove the x-axis"
    },
    {
        "Chapter": "1",
        "sentence_range": "3807-3810",
        "Text": "7 1  Definite integral as the limit of a sum\nLet f be a continuous function defined on close interval [a, b] Assume that all the\nvalues taken by the function are non negative, so the graph of the function is a curve\nabove the x-axis The definite integral \n( )\nb\n\u222ba f x dx\n is the area bounded by the curve y = f (x), the\nordinates x = a, x = b and the x-axis"
    },
    {
        "Chapter": "1",
        "sentence_range": "3808-3811",
        "Text": "1  Definite integral as the limit of a sum\nLet f be a continuous function defined on close interval [a, b] Assume that all the\nvalues taken by the function are non negative, so the graph of the function is a curve\nabove the x-axis The definite integral \n( )\nb\n\u222ba f x dx\n is the area bounded by the curve y = f (x), the\nordinates x = a, x = b and the x-axis To evaluate this area, consider the region PRSQP\nbetween this curve, x-axis and the ordinates x = a and x = b (Fig 7"
    },
    {
        "Chapter": "1",
        "sentence_range": "3809-3812",
        "Text": "Assume that all the\nvalues taken by the function are non negative, so the graph of the function is a curve\nabove the x-axis The definite integral \n( )\nb\n\u222ba f x dx\n is the area bounded by the curve y = f (x), the\nordinates x = a, x = b and the x-axis To evaluate this area, consider the region PRSQP\nbetween this curve, x-axis and the ordinates x = a and x = b (Fig 7 2)"
    },
    {
        "Chapter": "1",
        "sentence_range": "3810-3813",
        "Text": "The definite integral \n( )\nb\n\u222ba f x dx\n is the area bounded by the curve y = f (x), the\nordinates x = a, x = b and the x-axis To evaluate this area, consider the region PRSQP\nbetween this curve, x-axis and the ordinates x = a and x = b (Fig 7 2) Divide the interval [a, b] into n equal subintervals denoted by [x0, x1], [x1, x2] ,"
    },
    {
        "Chapter": "1",
        "sentence_range": "3811-3814",
        "Text": "To evaluate this area, consider the region PRSQP\nbetween this curve, x-axis and the ordinates x = a and x = b (Fig 7 2) Divide the interval [a, b] into n equal subintervals denoted by [x0, x1], [x1, x2] , ,\n[xr \u2013 1, xr],"
    },
    {
        "Chapter": "1",
        "sentence_range": "3812-3815",
        "Text": "2) Divide the interval [a, b] into n equal subintervals denoted by [x0, x1], [x1, x2] , ,\n[xr \u2013 1, xr], , [xn \u2013 1, xn], where x0 = a, x1 = a + h, x2 = a + 2h,"
    },
    {
        "Chapter": "1",
        "sentence_range": "3813-3816",
        "Text": "Divide the interval [a, b] into n equal subintervals denoted by [x0, x1], [x1, x2] , ,\n[xr \u2013 1, xr], , [xn \u2013 1, xn], where x0 = a, x1 = a + h, x2 = a + 2h, , xr = a + rh and\nxn = b = a + nh or"
    },
    {
        "Chapter": "1",
        "sentence_range": "3814-3817",
        "Text": ",\n[xr \u2013 1, xr], , [xn \u2013 1, xn], where x0 = a, x1 = a + h, x2 = a + 2h, , xr = a + rh and\nxn = b = a + nh or b\na\nn\n\u2212h\n=\n We note that as n \u2192 \u221e, h \u2192 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "3815-3818",
        "Text": ", [xn \u2013 1, xn], where x0 = a, x1 = a + h, x2 = a + 2h, , xr = a + rh and\nxn = b = a + nh or b\na\nn\n\u2212h\n=\n We note that as n \u2192 \u221e, h \u2192 0 Fig 7"
    },
    {
        "Chapter": "1",
        "sentence_range": "3816-3819",
        "Text": ", xr = a + rh and\nxn = b = a + nh or b\na\nn\n\u2212h\n=\n We note that as n \u2192 \u221e, h \u2192 0 Fig 7 2\nO\nY\nX\nX'\nY'\nQ\nP\nC\nM\nD\nL\nS\nA\nB\nR\na = x0 x1 x2\nxr-1 xr\nx =b\nn\ny\nf x\n = ( )\n332\nMATHEMATICS\nThe region PRSQP under consideration is the sum of n subregions, where each\nsubregion is defined on subintervals [xr \u2013 1, xr], r = 1, 2, 3, \u2026, n"
    },
    {
        "Chapter": "1",
        "sentence_range": "3817-3820",
        "Text": "b\na\nn\n\u2212h\n=\n We note that as n \u2192 \u221e, h \u2192 0 Fig 7 2\nO\nY\nX\nX'\nY'\nQ\nP\nC\nM\nD\nL\nS\nA\nB\nR\na = x0 x1 x2\nxr-1 xr\nx =b\nn\ny\nf x\n = ( )\n332\nMATHEMATICS\nThe region PRSQP under consideration is the sum of n subregions, where each\nsubregion is defined on subintervals [xr \u2013 1, xr], r = 1, 2, 3, \u2026, n From Fig 7"
    },
    {
        "Chapter": "1",
        "sentence_range": "3818-3821",
        "Text": "Fig 7 2\nO\nY\nX\nX'\nY'\nQ\nP\nC\nM\nD\nL\nS\nA\nB\nR\na = x0 x1 x2\nxr-1 xr\nx =b\nn\ny\nf x\n = ( )\n332\nMATHEMATICS\nThe region PRSQP under consideration is the sum of n subregions, where each\nsubregion is defined on subintervals [xr \u2013 1, xr], r = 1, 2, 3, \u2026, n From Fig 7 2, we have\narea of the rectangle (ABLC) < area of the region (ABDCA) < area of the rectangle\n(ABDM)"
    },
    {
        "Chapter": "1",
        "sentence_range": "3819-3822",
        "Text": "2\nO\nY\nX\nX'\nY'\nQ\nP\nC\nM\nD\nL\nS\nA\nB\nR\na = x0 x1 x2\nxr-1 xr\nx =b\nn\ny\nf x\n = ( )\n332\nMATHEMATICS\nThe region PRSQP under consideration is the sum of n subregions, where each\nsubregion is defined on subintervals [xr \u2013 1, xr], r = 1, 2, 3, \u2026, n From Fig 7 2, we have\narea of the rectangle (ABLC) < area of the region (ABDCA) < area of the rectangle\n(ABDM) (1)\nEvidently as xr \u2013 xr\u20131 \u2192 0, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "3820-3823",
        "Text": "From Fig 7 2, we have\narea of the rectangle (ABLC) < area of the region (ABDCA) < area of the rectangle\n(ABDM) (1)\nEvidently as xr \u2013 xr\u20131 \u2192 0, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "3821-3824",
        "Text": "2, we have\narea of the rectangle (ABLC) < area of the region (ABDCA) < area of the rectangle\n(ABDM) (1)\nEvidently as xr \u2013 xr\u20131 \u2192 0, i e , h \u2192 0 all the three areas shown in (1) become\nnearly equal to each other"
    },
    {
        "Chapter": "1",
        "sentence_range": "3822-3825",
        "Text": "(1)\nEvidently as xr \u2013 xr\u20131 \u2192 0, i e , h \u2192 0 all the three areas shown in (1) become\nnearly equal to each other Now we form the following sums"
    },
    {
        "Chapter": "1",
        "sentence_range": "3823-3826",
        "Text": "e , h \u2192 0 all the three areas shown in (1) become\nnearly equal to each other Now we form the following sums sn = h [f(x0) + \u2026 + f (xn - 1)] = \n1\n0\n(\n)\nn\nr\nr\nh\nf x\n\u2212\n=\u2211"
    },
    {
        "Chapter": "1",
        "sentence_range": "3824-3827",
        "Text": ", h \u2192 0 all the three areas shown in (1) become\nnearly equal to each other Now we form the following sums sn = h [f(x0) + \u2026 + f (xn - 1)] = \n1\n0\n(\n)\nn\nr\nr\nh\nf x\n\u2212\n=\u2211 (2)\nand\n Sn =\n1\n2\n1\n[ ( )\n(\n)\n(\n)]\n(\n)\nn\nn\nr\nr\nh f x\nf x\nf x\nh\nf x\n=\n+\n+\u2026+\n= \u2211"
    },
    {
        "Chapter": "1",
        "sentence_range": "3825-3828",
        "Text": "Now we form the following sums sn = h [f(x0) + \u2026 + f (xn - 1)] = \n1\n0\n(\n)\nn\nr\nr\nh\nf x\n\u2212\n=\u2211 (2)\nand\n Sn =\n1\n2\n1\n[ ( )\n(\n)\n(\n)]\n(\n)\nn\nn\nr\nr\nh f x\nf x\nf x\nh\nf x\n=\n+\n+\u2026+\n= \u2211 (3)\nHere, sn and Sn denote the sum of areas of all lower rectangles and upper rectangles\nraised over subintervals [xr\u20131, xr] for r = 1, 2, 3, \u2026, n, respectively"
    },
    {
        "Chapter": "1",
        "sentence_range": "3826-3829",
        "Text": "sn = h [f(x0) + \u2026 + f (xn - 1)] = \n1\n0\n(\n)\nn\nr\nr\nh\nf x\n\u2212\n=\u2211 (2)\nand\n Sn =\n1\n2\n1\n[ ( )\n(\n)\n(\n)]\n(\n)\nn\nn\nr\nr\nh f x\nf x\nf x\nh\nf x\n=\n+\n+\u2026+\n= \u2211 (3)\nHere, sn and Sn denote the sum of areas of all lower rectangles and upper rectangles\nraised over subintervals [xr\u20131, xr] for r = 1, 2, 3, \u2026, n, respectively In view of the inequality (1) for an arbitrary subinterval [xr\u20131, xr], we have\nsn < area of the region PRSQP < Sn"
    },
    {
        "Chapter": "1",
        "sentence_range": "3827-3830",
        "Text": "(2)\nand\n Sn =\n1\n2\n1\n[ ( )\n(\n)\n(\n)]\n(\n)\nn\nn\nr\nr\nh f x\nf x\nf x\nh\nf x\n=\n+\n+\u2026+\n= \u2211 (3)\nHere, sn and Sn denote the sum of areas of all lower rectangles and upper rectangles\nraised over subintervals [xr\u20131, xr] for r = 1, 2, 3, \u2026, n, respectively In view of the inequality (1) for an arbitrary subinterval [xr\u20131, xr], we have\nsn < area of the region PRSQP < Sn (4)\nAs n \u2192 \u221e strips become narrower and narrower, it is assumed that the limiting\nvalues of (2) and (3) are the same in both cases and the common limiting value is the\nrequired area under the curve"
    },
    {
        "Chapter": "1",
        "sentence_range": "3828-3831",
        "Text": "(3)\nHere, sn and Sn denote the sum of areas of all lower rectangles and upper rectangles\nraised over subintervals [xr\u20131, xr] for r = 1, 2, 3, \u2026, n, respectively In view of the inequality (1) for an arbitrary subinterval [xr\u20131, xr], we have\nsn < area of the region PRSQP < Sn (4)\nAs n \u2192 \u221e strips become narrower and narrower, it is assumed that the limiting\nvalues of (2) and (3) are the same in both cases and the common limiting value is the\nrequired area under the curve Symbolically, we write\nlimSn\nn\u2192\u221e\n = lim\nn\nn\ns\n\u2192\u221e\n = area of the region PRSQP = \n( )\nb\n\u222ba f x dx"
    },
    {
        "Chapter": "1",
        "sentence_range": "3829-3832",
        "Text": "In view of the inequality (1) for an arbitrary subinterval [xr\u20131, xr], we have\nsn < area of the region PRSQP < Sn (4)\nAs n \u2192 \u221e strips become narrower and narrower, it is assumed that the limiting\nvalues of (2) and (3) are the same in both cases and the common limiting value is the\nrequired area under the curve Symbolically, we write\nlimSn\nn\u2192\u221e\n = lim\nn\nn\ns\n\u2192\u221e\n = area of the region PRSQP = \n( )\nb\n\u222ba f x dx (5)\nIt follows that this area is also the limiting value of any area which is between that\nof the rectangles below the curve and that of the rectangles above the curve"
    },
    {
        "Chapter": "1",
        "sentence_range": "3830-3833",
        "Text": "(4)\nAs n \u2192 \u221e strips become narrower and narrower, it is assumed that the limiting\nvalues of (2) and (3) are the same in both cases and the common limiting value is the\nrequired area under the curve Symbolically, we write\nlimSn\nn\u2192\u221e\n = lim\nn\nn\ns\n\u2192\u221e\n = area of the region PRSQP = \n( )\nb\n\u222ba f x dx (5)\nIt follows that this area is also the limiting value of any area which is between that\nof the rectangles below the curve and that of the rectangles above the curve For\nthe sake of convenience, we shall take rectangles with height equal to that of the\ncurve at the left hand edge of each subinterval"
    },
    {
        "Chapter": "1",
        "sentence_range": "3831-3834",
        "Text": "Symbolically, we write\nlimSn\nn\u2192\u221e\n = lim\nn\nn\ns\n\u2192\u221e\n = area of the region PRSQP = \n( )\nb\n\u222ba f x dx (5)\nIt follows that this area is also the limiting value of any area which is between that\nof the rectangles below the curve and that of the rectangles above the curve For\nthe sake of convenience, we shall take rectangles with height equal to that of the\ncurve at the left hand edge of each subinterval Thus, we rewrite (5) as\n( )\nb\n\u222ba f x dx\n =\nlim0\n[ ( )\n(\n)"
    },
    {
        "Chapter": "1",
        "sentence_range": "3832-3835",
        "Text": "(5)\nIt follows that this area is also the limiting value of any area which is between that\nof the rectangles below the curve and that of the rectangles above the curve For\nthe sake of convenience, we shall take rectangles with height equal to that of the\ncurve at the left hand edge of each subinterval Thus, we rewrite (5) as\n( )\nb\n\u222ba f x dx\n =\nlim0\n[ ( )\n(\n) (\n( \u2013 1) ]\nh\nh f a\nf a\nh\nf a\nn\nh\n\u2192\n+\n+\n+\n+\n+\nor\n( )\nb\n\u222ba f x dx\n =\n( \u2013 ) lim1\n[ ( )\n(\n)"
    },
    {
        "Chapter": "1",
        "sentence_range": "3833-3836",
        "Text": "For\nthe sake of convenience, we shall take rectangles with height equal to that of the\ncurve at the left hand edge of each subinterval Thus, we rewrite (5) as\n( )\nb\n\u222ba f x dx\n =\nlim0\n[ ( )\n(\n) (\n( \u2013 1) ]\nh\nh f a\nf a\nh\nf a\nn\nh\n\u2192\n+\n+\n+\n+\n+\nor\n( )\nb\n\u222ba f x dx\n =\n( \u2013 ) lim1\n[ ( )\n(\n) (\n( \u20131) ]\nn\nb\na\nf a\nf a\nh\nf a\nn\nh\n\u2192\u221en\n+\n+\n+\n+\n+"
    },
    {
        "Chapter": "1",
        "sentence_range": "3834-3837",
        "Text": "Thus, we rewrite (5) as\n( )\nb\n\u222ba f x dx\n =\nlim0\n[ ( )\n(\n) (\n( \u2013 1) ]\nh\nh f a\nf a\nh\nf a\nn\nh\n\u2192\n+\n+\n+\n+\n+\nor\n( )\nb\n\u222ba f x dx\n =\n( \u2013 ) lim1\n[ ( )\n(\n) (\n( \u20131) ]\nn\nb\na\nf a\nf a\nh\nf a\nn\nh\n\u2192\u221en\n+\n+\n+\n+\n+ (6)\nwhere\nh =\n\u2013\n0\nb\na\nas n\nn\n\u2192\n\u2192 \u221e\nThe above expression (6) is known as the definition of definite integral as the limit\nof sum"
    },
    {
        "Chapter": "1",
        "sentence_range": "3835-3838",
        "Text": "(\n( \u2013 1) ]\nh\nh f a\nf a\nh\nf a\nn\nh\n\u2192\n+\n+\n+\n+\n+\nor\n( )\nb\n\u222ba f x dx\n =\n( \u2013 ) lim1\n[ ( )\n(\n) (\n( \u20131) ]\nn\nb\na\nf a\nf a\nh\nf a\nn\nh\n\u2192\u221en\n+\n+\n+\n+\n+ (6)\nwhere\nh =\n\u2013\n0\nb\na\nas n\nn\n\u2192\n\u2192 \u221e\nThe above expression (6) is known as the definition of definite integral as the limit\nof sum Remark The value of the definite integral of a function over any particular interval\ndepends on the function and the interval, but not on the variable of integration that we\nINTEGRALS         333\nchoose to represent the independent variable"
    },
    {
        "Chapter": "1",
        "sentence_range": "3836-3839",
        "Text": "(\n( \u20131) ]\nn\nb\na\nf a\nf a\nh\nf a\nn\nh\n\u2192\u221en\n+\n+\n+\n+\n+ (6)\nwhere\nh =\n\u2013\n0\nb\na\nas n\nn\n\u2192\n\u2192 \u221e\nThe above expression (6) is known as the definition of definite integral as the limit\nof sum Remark The value of the definite integral of a function over any particular interval\ndepends on the function and the interval, but not on the variable of integration that we\nINTEGRALS         333\nchoose to represent the independent variable If the independent variable is denoted by\nt or u instead of x, we simply write the integral as \n( )\nb\n\u222ba f t dt\n or \n( )\nb\n\u222ba f u du\ninstead of\n( )\nb\n\u222ba f x dx"
    },
    {
        "Chapter": "1",
        "sentence_range": "3837-3840",
        "Text": "(6)\nwhere\nh =\n\u2013\n0\nb\na\nas n\nn\n\u2192\n\u2192 \u221e\nThe above expression (6) is known as the definition of definite integral as the limit\nof sum Remark The value of the definite integral of a function over any particular interval\ndepends on the function and the interval, but not on the variable of integration that we\nINTEGRALS         333\nchoose to represent the independent variable If the independent variable is denoted by\nt or u instead of x, we simply write the integral as \n( )\nb\n\u222ba f t dt\n or \n( )\nb\n\u222ba f u du\ninstead of\n( )\nb\n\u222ba f x dx Hence, the variable of integration is called a dummy variable"
    },
    {
        "Chapter": "1",
        "sentence_range": "3838-3841",
        "Text": "Remark The value of the definite integral of a function over any particular interval\ndepends on the function and the interval, but not on the variable of integration that we\nINTEGRALS         333\nchoose to represent the independent variable If the independent variable is denoted by\nt or u instead of x, we simply write the integral as \n( )\nb\n\u222ba f t dt\n or \n( )\nb\n\u222ba f u du\ninstead of\n( )\nb\n\u222ba f x dx Hence, the variable of integration is called a dummy variable Example 25 Find \n2\n0 (2\n1)\nx\ndx\n+\n\u222b\n as the limit of a sum"
    },
    {
        "Chapter": "1",
        "sentence_range": "3839-3842",
        "Text": "If the independent variable is denoted by\nt or u instead of x, we simply write the integral as \n( )\nb\n\u222ba f t dt\n or \n( )\nb\n\u222ba f u du\ninstead of\n( )\nb\n\u222ba f x dx Hence, the variable of integration is called a dummy variable Example 25 Find \n2\n0 (2\n1)\nx\ndx\n+\n\u222b\n as the limit of a sum Solution By definition\n( )\nb\n\u222ba f x dx\n =\n( \u2013 ) lim1\n[ ( )\n(\n)"
    },
    {
        "Chapter": "1",
        "sentence_range": "3840-3843",
        "Text": "Hence, the variable of integration is called a dummy variable Example 25 Find \n2\n0 (2\n1)\nx\ndx\n+\n\u222b\n as the limit of a sum Solution By definition\n( )\nb\n\u222ba f x dx\n =\n( \u2013 ) lim1\n[ ( )\n(\n) (\n( \u20131) ],\nn\nb\na\nf a\nf a\nh\nf a\nn\nh\n\u2192\u221en\n+\n+\n+\n+\n+\nwhere,\nh =\nb\u2013\na\nn\nIn this example, a = 0, b = 2, f (x) = x2 + 1, \n2 \u2013 0\n2\nh\nn\nn\n=\n=\nTherefore,\n2\n2\n0 (\n1)\nx\ndx\n+\n\u222b\n =  \n1\n2\n4\n2 ( \u20131)\n2 lim\n[ (0)\n( )\n( )"
    },
    {
        "Chapter": "1",
        "sentence_range": "3841-3844",
        "Text": "Example 25 Find \n2\n0 (2\n1)\nx\ndx\n+\n\u222b\n as the limit of a sum Solution By definition\n( )\nb\n\u222ba f x dx\n =\n( \u2013 ) lim1\n[ ( )\n(\n) (\n( \u20131) ],\nn\nb\na\nf a\nf a\nh\nf a\nn\nh\n\u2192\u221en\n+\n+\n+\n+\n+\nwhere,\nh =\nb\u2013\na\nn\nIn this example, a = 0, b = 2, f (x) = x2 + 1, \n2 \u2013 0\n2\nh\nn\nn\n=\n=\nTherefore,\n2\n2\n0 (\n1)\nx\ndx\n+\n\u222b\n =  \n1\n2\n4\n2 ( \u20131)\n2 lim\n[ (0)\n( )\n( ) (\n)]\nn\nn\nf\nf\nf\nf\nn\nn\nn\nn\n\u2192\u221e\n+\n+\n+\n+\n=\n2\n2\n2\n2\n2\n2\n1\n2\n4\n(2 \u2013 2)\n2 lim\n[1\n(\n1)\n(\n1)"
    },
    {
        "Chapter": "1",
        "sentence_range": "3842-3845",
        "Text": "Solution By definition\n( )\nb\n\u222ba f x dx\n =\n( \u2013 ) lim1\n[ ( )\n(\n) (\n( \u20131) ],\nn\nb\na\nf a\nf a\nh\nf a\nn\nh\n\u2192\u221en\n+\n+\n+\n+\n+\nwhere,\nh =\nb\u2013\na\nn\nIn this example, a = 0, b = 2, f (x) = x2 + 1, \n2 \u2013 0\n2\nh\nn\nn\n=\n=\nTherefore,\n2\n2\n0 (\n1)\nx\ndx\n+\n\u222b\n =  \n1\n2\n4\n2 ( \u20131)\n2 lim\n[ (0)\n( )\n( ) (\n)]\nn\nn\nf\nf\nf\nf\nn\nn\nn\nn\n\u2192\u221e\n+\n+\n+\n+\n=\n2\n2\n2\n2\n2\n2\n1\n2\n4\n(2 \u2013 2)\n2 lim\n[1\n(\n1)\n(\n1) 1 ]\nn\nn\nn\nn\nn\nn\n\u2192\u221e\n\uf8eb\n\uf8f6\n+\n+\n+\n+\n+\n+\n+\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n=\n2\n2\n2\n-\n1\n1\n2 lim\n[(1 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "3843-3846",
        "Text": "(\n( \u20131) ],\nn\nb\na\nf a\nf a\nh\nf a\nn\nh\n\u2192\u221en\n+\n+\n+\n+\n+\nwhere,\nh =\nb\u2013\na\nn\nIn this example, a = 0, b = 2, f (x) = x2 + 1, \n2 \u2013 0\n2\nh\nn\nn\n=\n=\nTherefore,\n2\n2\n0 (\n1)\nx\ndx\n+\n\u222b\n =  \n1\n2\n4\n2 ( \u20131)\n2 lim\n[ (0)\n( )\n( ) (\n)]\nn\nn\nf\nf\nf\nf\nn\nn\nn\nn\n\u2192\u221e\n+\n+\n+\n+\n=\n2\n2\n2\n2\n2\n2\n1\n2\n4\n(2 \u2013 2)\n2 lim\n[1\n(\n1)\n(\n1) 1 ]\nn\nn\nn\nn\nn\nn\n\u2192\u221e\n\uf8eb\n\uf8f6\n+\n+\n+\n+\n+\n+\n+\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n=\n2\n2\n2\n-\n1\n1\n2 lim\n[(1 1 1)\n(2\n4"
    },
    {
        "Chapter": "1",
        "sentence_range": "3844-3847",
        "Text": "(\n)]\nn\nn\nf\nf\nf\nf\nn\nn\nn\nn\n\u2192\u221e\n+\n+\n+\n+\n=\n2\n2\n2\n2\n2\n2\n1\n2\n4\n(2 \u2013 2)\n2 lim\n[1\n(\n1)\n(\n1) 1 ]\nn\nn\nn\nn\nn\nn\n\u2192\u221e\n\uf8eb\n\uf8f6\n+\n+\n+\n+\n+\n+\n+\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n=\n2\n2\n2\n-\n1\n1\n2 lim\n[(1 1 1)\n(2\n4 (2 \u2013 2) ]\n2\n\u2192\u221e\n+ +\n+\n+\n+\n+\n+\n144244\n3\nn\nn terms\nn\nn\nn\n=\n2\n2\n2\n2\n1\n2\n2 lim\n[\n(1\n2"
    },
    {
        "Chapter": "1",
        "sentence_range": "3845-3848",
        "Text": "1 ]\nn\nn\nn\nn\nn\nn\n\u2192\u221e\n\uf8eb\n\uf8f6\n+\n+\n+\n+\n+\n+\n+\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n=\n2\n2\n2\n-\n1\n1\n2 lim\n[(1 1 1)\n(2\n4 (2 \u2013 2) ]\n2\n\u2192\u221e\n+ +\n+\n+\n+\n+\n+\n144244\n3\nn\nn terms\nn\nn\nn\n=\n2\n2\n2\n2\n1\n2\n2 lim\n[\n(1\n2 ( \u20131) ]\nn\nn\nn\nn\nn2\n\u2192\u221e\n+\n+\n+\n+\n=\n1\n4 (\n1)\n(2 \u20131)\n2 lim\n[\n]\n6\nn\nn\nn\nn\nnn\nn2\n\u2192\u221e\n\u2212\n+\n=\n1\n2 (\n1) (2 \u20131)\n2 lim\n[\n]\n3\nn\nn\nn\nnn\nn\n\u2192\u221e\n\u2212\n+\n=\n2\n1\n1\n2 lim [1\n(1\n) (2 \u2013\n)]\n3\nn\nn\nn\n\u2192\u221e\n+\n\u2212\n = \n4\n2 [1\n]\n3\n+\n = 14\n3\n334\nMATHEMATICS\nExample 26 Evaluate \n2\n0\nxe dx\n\u222b\nas the limit of a sum"
    },
    {
        "Chapter": "1",
        "sentence_range": "3846-3849",
        "Text": "1)\n(2\n4 (2 \u2013 2) ]\n2\n\u2192\u221e\n+ +\n+\n+\n+\n+\n+\n144244\n3\nn\nn terms\nn\nn\nn\n=\n2\n2\n2\n2\n1\n2\n2 lim\n[\n(1\n2 ( \u20131) ]\nn\nn\nn\nn\nn2\n\u2192\u221e\n+\n+\n+\n+\n=\n1\n4 (\n1)\n(2 \u20131)\n2 lim\n[\n]\n6\nn\nn\nn\nn\nnn\nn2\n\u2192\u221e\n\u2212\n+\n=\n1\n2 (\n1) (2 \u20131)\n2 lim\n[\n]\n3\nn\nn\nn\nnn\nn\n\u2192\u221e\n\u2212\n+\n=\n2\n1\n1\n2 lim [1\n(1\n) (2 \u2013\n)]\n3\nn\nn\nn\n\u2192\u221e\n+\n\u2212\n = \n4\n2 [1\n]\n3\n+\n = 14\n3\n334\nMATHEMATICS\nExample 26 Evaluate \n2\n0\nxe dx\n\u222b\nas the limit of a sum Solution By definition\n2\n0\n\u222bxe dx\n =\n2\n4\n2\n\u2013 2\n0\n(2 \u2013 0) lim1"
    },
    {
        "Chapter": "1",
        "sentence_range": "3847-3850",
        "Text": "(2 \u2013 2) ]\n2\n\u2192\u221e\n+ +\n+\n+\n+\n+\n+\n144244\n3\nn\nn terms\nn\nn\nn\n=\n2\n2\n2\n2\n1\n2\n2 lim\n[\n(1\n2 ( \u20131) ]\nn\nn\nn\nn\nn2\n\u2192\u221e\n+\n+\n+\n+\n=\n1\n4 (\n1)\n(2 \u20131)\n2 lim\n[\n]\n6\nn\nn\nn\nn\nnn\nn2\n\u2192\u221e\n\u2212\n+\n=\n1\n2 (\n1) (2 \u20131)\n2 lim\n[\n]\n3\nn\nn\nn\nnn\nn\n\u2192\u221e\n\u2212\n+\n=\n2\n1\n1\n2 lim [1\n(1\n) (2 \u2013\n)]\n3\nn\nn\nn\n\u2192\u221e\n+\n\u2212\n = \n4\n2 [1\n]\n3\n+\n = 14\n3\n334\nMATHEMATICS\nExample 26 Evaluate \n2\n0\nxe dx\n\u222b\nas the limit of a sum Solution By definition\n2\n0\n\u222bxe dx\n =\n2\n4\n2\n\u2013 2\n0\n(2 \u2013 0) lim1 n\nn\nn\nn\nn\ne\ne\ne\ne\nn\n\u2192\u221e\n\uf8ee\n\uf8f9\n+\n+\n+\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nUsing the sum to n terms of a G"
    },
    {
        "Chapter": "1",
        "sentence_range": "3848-3851",
        "Text": "( \u20131) ]\nn\nn\nn\nn\nn2\n\u2192\u221e\n+\n+\n+\n+\n=\n1\n4 (\n1)\n(2 \u20131)\n2 lim\n[\n]\n6\nn\nn\nn\nn\nnn\nn2\n\u2192\u221e\n\u2212\n+\n=\n1\n2 (\n1) (2 \u20131)\n2 lim\n[\n]\n3\nn\nn\nn\nnn\nn\n\u2192\u221e\n\u2212\n+\n=\n2\n1\n1\n2 lim [1\n(1\n) (2 \u2013\n)]\n3\nn\nn\nn\n\u2192\u221e\n+\n\u2212\n = \n4\n2 [1\n]\n3\n+\n = 14\n3\n334\nMATHEMATICS\nExample 26 Evaluate \n2\n0\nxe dx\n\u222b\nas the limit of a sum Solution By definition\n2\n0\n\u222bxe dx\n =\n2\n4\n2\n\u2013 2\n0\n(2 \u2013 0) lim1 n\nn\nn\nn\nn\ne\ne\ne\ne\nn\n\u2192\u221e\n\uf8ee\n\uf8f9\n+\n+\n+\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nUsing the sum to n terms of a G P"
    },
    {
        "Chapter": "1",
        "sentence_range": "3849-3852",
        "Text": "Solution By definition\n2\n0\n\u222bxe dx\n =\n2\n4\n2\n\u2013 2\n0\n(2 \u2013 0) lim1 n\nn\nn\nn\nn\ne\ne\ne\ne\nn\n\u2192\u221e\n\uf8ee\n\uf8f9\n+\n+\n+\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nUsing the sum to n terms of a G P , where a = 1, \nn2\nr\n=e\n, we have\n2\n0\n\u222bxe dx\n=\n2\n2\n1\n\u20131\n2 lim\n[\n]\n1\nn\nn\nn\nn\nne\ne\n\u2192\u221e\n\u2212\n = \n2\n2\n1\n\u20131\n2 lim\n\u20131\nn\nn\nn ee\n\u2192\u221e\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n=\n2\n2\n2 (\n\u20131)\n\u20131\nlim\n2\n2\nn\nn\ne\ne\nn\n\u2192\u221e\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u22c5\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n = e2 \u2013 1\n[using \n0\n(\n1)\nlim\n1\nh\nh\ne\nh\n\u2192\n\u2212\n= ]\nEXERCISE 7"
    },
    {
        "Chapter": "1",
        "sentence_range": "3850-3853",
        "Text": "n\nn\nn\nn\nn\ne\ne\ne\ne\nn\n\u2192\u221e\n\uf8ee\n\uf8f9\n+\n+\n+\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nUsing the sum to n terms of a G P , where a = 1, \nn2\nr\n=e\n, we have\n2\n0\n\u222bxe dx\n=\n2\n2\n1\n\u20131\n2 lim\n[\n]\n1\nn\nn\nn\nn\nne\ne\n\u2192\u221e\n\u2212\n = \n2\n2\n1\n\u20131\n2 lim\n\u20131\nn\nn\nn ee\n\u2192\u221e\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n=\n2\n2\n2 (\n\u20131)\n\u20131\nlim\n2\n2\nn\nn\ne\ne\nn\n\u2192\u221e\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u22c5\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n = e2 \u2013 1\n[using \n0\n(\n1)\nlim\n1\nh\nh\ne\nh\n\u2192\n\u2212\n= ]\nEXERCISE 7 8\nEvaluate the following definite integrals as limit of sums"
    },
    {
        "Chapter": "1",
        "sentence_range": "3851-3854",
        "Text": "P , where a = 1, \nn2\nr\n=e\n, we have\n2\n0\n\u222bxe dx\n=\n2\n2\n1\n\u20131\n2 lim\n[\n]\n1\nn\nn\nn\nn\nne\ne\n\u2192\u221e\n\u2212\n = \n2\n2\n1\n\u20131\n2 lim\n\u20131\nn\nn\nn ee\n\u2192\u221e\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n=\n2\n2\n2 (\n\u20131)\n\u20131\nlim\n2\n2\nn\nn\ne\ne\nn\n\u2192\u221e\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u22c5\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n = e2 \u2013 1\n[using \n0\n(\n1)\nlim\n1\nh\nh\ne\nh\n\u2192\n\u2212\n= ]\nEXERCISE 7 8\nEvaluate the following definite integrals as limit of sums 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "3852-3855",
        "Text": ", where a = 1, \nn2\nr\n=e\n, we have\n2\n0\n\u222bxe dx\n=\n2\n2\n1\n\u20131\n2 lim\n[\n]\n1\nn\nn\nn\nn\nne\ne\n\u2192\u221e\n\u2212\n = \n2\n2\n1\n\u20131\n2 lim\n\u20131\nn\nn\nn ee\n\u2192\u221e\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n=\n2\n2\n2 (\n\u20131)\n\u20131\nlim\n2\n2\nn\nn\ne\ne\nn\n\u2192\u221e\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\u22c5\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n = e2 \u2013 1\n[using \n0\n(\n1)\nlim\n1\nh\nh\ne\nh\n\u2192\n\u2212\n= ]\nEXERCISE 7 8\nEvaluate the following definite integrals as limit of sums 1 b\n\u222ba x dx\n2"
    },
    {
        "Chapter": "1",
        "sentence_range": "3853-3856",
        "Text": "8\nEvaluate the following definite integrals as limit of sums 1 b\n\u222ba x dx\n2 5\n0 (\n1)\nx\ndx\n+\n\u222b\n3"
    },
    {
        "Chapter": "1",
        "sentence_range": "3854-3857",
        "Text": "1 b\n\u222ba x dx\n2 5\n0 (\n1)\nx\ndx\n+\n\u222b\n3 3\n2\n2 x dx\n\u222b\n4"
    },
    {
        "Chapter": "1",
        "sentence_range": "3855-3858",
        "Text": "b\n\u222ba x dx\n2 5\n0 (\n1)\nx\ndx\n+\n\u222b\n3 3\n2\n2 x dx\n\u222b\n4 4\n2\n1 (\n)\nx\n\u2212x dx\n\u222b\n5"
    },
    {
        "Chapter": "1",
        "sentence_range": "3856-3859",
        "Text": "5\n0 (\n1)\nx\ndx\n+\n\u222b\n3 3\n2\n2 x dx\n\u222b\n4 4\n2\n1 (\n)\nx\n\u2212x dx\n\u222b\n5 1\n1\nx\ne dx\n\u2212\u222b\n6"
    },
    {
        "Chapter": "1",
        "sentence_range": "3857-3860",
        "Text": "3\n2\n2 x dx\n\u222b\n4 4\n2\n1 (\n)\nx\n\u2212x dx\n\u222b\n5 1\n1\nx\ne dx\n\u2212\u222b\n6 4\n2\n0 (\nx)\nx\ne\ndx\n+\n\u222b\n7"
    },
    {
        "Chapter": "1",
        "sentence_range": "3858-3861",
        "Text": "4\n2\n1 (\n)\nx\n\u2212x dx\n\u222b\n5 1\n1\nx\ne dx\n\u2212\u222b\n6 4\n2\n0 (\nx)\nx\ne\ndx\n+\n\u222b\n7 8  Fundamental Theorem of Calculus\n7"
    },
    {
        "Chapter": "1",
        "sentence_range": "3859-3862",
        "Text": "1\n1\nx\ne dx\n\u2212\u222b\n6 4\n2\n0 (\nx)\nx\ne\ndx\n+\n\u222b\n7 8  Fundamental Theorem of Calculus\n7 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "3860-3863",
        "Text": "4\n2\n0 (\nx)\nx\ne\ndx\n+\n\u222b\n7 8  Fundamental Theorem of Calculus\n7 8 1  Area function\nWe have defined \n( )\nb\n\u222ba f x dx\n as the area of\nthe region bounded by the curve y = f(x),\nthe ordinates x = a and x = b and x-axis"
    },
    {
        "Chapter": "1",
        "sentence_range": "3861-3864",
        "Text": "8  Fundamental Theorem of Calculus\n7 8 1  Area function\nWe have defined \n( )\nb\n\u222ba f x dx\n as the area of\nthe region bounded by the curve y = f(x),\nthe ordinates x = a and x = b and x-axis Let x\nbe a given point in [a, b]"
    },
    {
        "Chapter": "1",
        "sentence_range": "3862-3865",
        "Text": "8 1  Area function\nWe have defined \n( )\nb\n\u222ba f x dx\n as the area of\nthe region bounded by the curve y = f(x),\nthe ordinates x = a and x = b and x-axis Let x\nbe a given point in [a, b] Then \n( )\nx\na f x dx\n\u222b\nrepresents the area of the light shaded region\nFig 7"
    },
    {
        "Chapter": "1",
        "sentence_range": "3863-3866",
        "Text": "1  Area function\nWe have defined \n( )\nb\n\u222ba f x dx\n as the area of\nthe region bounded by the curve y = f(x),\nthe ordinates x = a and x = b and x-axis Let x\nbe a given point in [a, b] Then \n( )\nx\na f x dx\n\u222b\nrepresents the area of the light shaded region\nFig 7 3\nINTEGRALS         335\nin Fig 7"
    },
    {
        "Chapter": "1",
        "sentence_range": "3864-3867",
        "Text": "Let x\nbe a given point in [a, b] Then \n( )\nx\na f x dx\n\u222b\nrepresents the area of the light shaded region\nFig 7 3\nINTEGRALS         335\nin Fig 7 3 [Here it is assumed that f (x) > 0 for x \u2208 [a, b], the assertion made below is\nequally true for other functions as well]"
    },
    {
        "Chapter": "1",
        "sentence_range": "3865-3868",
        "Text": "Then \n( )\nx\na f x dx\n\u222b\nrepresents the area of the light shaded region\nFig 7 3\nINTEGRALS         335\nin Fig 7 3 [Here it is assumed that f (x) > 0 for x \u2208 [a, b], the assertion made below is\nequally true for other functions as well] The area of this shaded region depends upon\nthe value of x"
    },
    {
        "Chapter": "1",
        "sentence_range": "3866-3869",
        "Text": "3\nINTEGRALS         335\nin Fig 7 3 [Here it is assumed that f (x) > 0 for x \u2208 [a, b], the assertion made below is\nequally true for other functions as well] The area of this shaded region depends upon\nthe value of x In other words, the area of this shaded region is a function of x"
    },
    {
        "Chapter": "1",
        "sentence_range": "3867-3870",
        "Text": "3 [Here it is assumed that f (x) > 0 for x \u2208 [a, b], the assertion made below is\nequally true for other functions as well] The area of this shaded region depends upon\nthe value of x In other words, the area of this shaded region is a function of x We denote this\nfunction of x by A(x)"
    },
    {
        "Chapter": "1",
        "sentence_range": "3868-3871",
        "Text": "The area of this shaded region depends upon\nthe value of x In other words, the area of this shaded region is a function of x We denote this\nfunction of x by A(x) We call the function A(x) as Area function and is given by\nA (x) = \u222b\n( )\nx\na f x dx"
    },
    {
        "Chapter": "1",
        "sentence_range": "3869-3872",
        "Text": "In other words, the area of this shaded region is a function of x We denote this\nfunction of x by A(x) We call the function A(x) as Area function and is given by\nA (x) = \u222b\n( )\nx\na f x dx (1)\nBased on this definition, the two basic fundamental theorems have been given"
    },
    {
        "Chapter": "1",
        "sentence_range": "3870-3873",
        "Text": "We denote this\nfunction of x by A(x) We call the function A(x) as Area function and is given by\nA (x) = \u222b\n( )\nx\na f x dx (1)\nBased on this definition, the two basic fundamental theorems have been given However, we only state them as their proofs are beyond the scope of this text book"
    },
    {
        "Chapter": "1",
        "sentence_range": "3871-3874",
        "Text": "We call the function A(x) as Area function and is given by\nA (x) = \u222b\n( )\nx\na f x dx (1)\nBased on this definition, the two basic fundamental theorems have been given However, we only state them as their proofs are beyond the scope of this text book 7"
    },
    {
        "Chapter": "1",
        "sentence_range": "3872-3875",
        "Text": "(1)\nBased on this definition, the two basic fundamental theorems have been given However, we only state them as their proofs are beyond the scope of this text book 7 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "3873-3876",
        "Text": "However, we only state them as their proofs are beyond the scope of this text book 7 8 2  First fundamental theorem of integral calculus\nTheorem 1 Let f be a continuous function on the closed interval [a, b] and let A (x) be\nthe area function"
    },
    {
        "Chapter": "1",
        "sentence_range": "3874-3877",
        "Text": "7 8 2  First fundamental theorem of integral calculus\nTheorem 1 Let f be a continuous function on the closed interval [a, b] and let A (x) be\nthe area function Then A\u2032\u2032\u2032\u2032\u2032(x) = f (x), for all x \u2208\u2208\u2208\u2208\u2208 [a, b]"
    },
    {
        "Chapter": "1",
        "sentence_range": "3875-3878",
        "Text": "8 2  First fundamental theorem of integral calculus\nTheorem 1 Let f be a continuous function on the closed interval [a, b] and let A (x) be\nthe area function Then A\u2032\u2032\u2032\u2032\u2032(x) = f (x), for all x \u2208\u2208\u2208\u2208\u2208 [a, b] 7"
    },
    {
        "Chapter": "1",
        "sentence_range": "3876-3879",
        "Text": "2  First fundamental theorem of integral calculus\nTheorem 1 Let f be a continuous function on the closed interval [a, b] and let A (x) be\nthe area function Then A\u2032\u2032\u2032\u2032\u2032(x) = f (x), for all x \u2208\u2208\u2208\u2208\u2208 [a, b] 7 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "3877-3880",
        "Text": "Then A\u2032\u2032\u2032\u2032\u2032(x) = f (x), for all x \u2208\u2208\u2208\u2208\u2208 [a, b] 7 8 3  Second fundamental theorem of integral calculus\nWe state below an important theorem which enables us to evaluate definite integrals\nby making use of anti derivative"
    },
    {
        "Chapter": "1",
        "sentence_range": "3878-3881",
        "Text": "7 8 3  Second fundamental theorem of integral calculus\nWe state below an important theorem which enables us to evaluate definite integrals\nby making use of anti derivative Theorem 2 Let f  be continuous function defined on the closed interval [a, b] and F be\nan anti derivative of f"
    },
    {
        "Chapter": "1",
        "sentence_range": "3879-3882",
        "Text": "8 3  Second fundamental theorem of integral calculus\nWe state below an important theorem which enables us to evaluate definite integrals\nby making use of anti derivative Theorem 2 Let f  be continuous function defined on the closed interval [a, b] and F be\nan anti derivative of f Then \u222b\n( )\nb\na f x dx = [F( )] =\nab\nx\n F (b) \u2013 F(a)"
    },
    {
        "Chapter": "1",
        "sentence_range": "3880-3883",
        "Text": "3  Second fundamental theorem of integral calculus\nWe state below an important theorem which enables us to evaluate definite integrals\nby making use of anti derivative Theorem 2 Let f  be continuous function defined on the closed interval [a, b] and F be\nan anti derivative of f Then \u222b\n( )\nb\na f x dx = [F( )] =\nab\nx\n F (b) \u2013 F(a) Remarks\n(i)\nIn words, the Theorem 2 tells us that \n( )\nb\n\u222ba f x dx\n= (value of the anti derivative F\nof f at the upper limit b \u2013 value of the same anti derivative at the lower limit a)"
    },
    {
        "Chapter": "1",
        "sentence_range": "3881-3884",
        "Text": "Theorem 2 Let f  be continuous function defined on the closed interval [a, b] and F be\nan anti derivative of f Then \u222b\n( )\nb\na f x dx = [F( )] =\nab\nx\n F (b) \u2013 F(a) Remarks\n(i)\nIn words, the Theorem 2 tells us that \n( )\nb\n\u222ba f x dx\n= (value of the anti derivative F\nof f at the upper limit b \u2013 value of the same anti derivative at the lower limit a) (ii)\nThis theorem is very useful, because it gives us a method of calculating the\ndefinite integral more easily, without calculating the limit of a sum"
    },
    {
        "Chapter": "1",
        "sentence_range": "3882-3885",
        "Text": "Then \u222b\n( )\nb\na f x dx = [F( )] =\nab\nx\n F (b) \u2013 F(a) Remarks\n(i)\nIn words, the Theorem 2 tells us that \n( )\nb\n\u222ba f x dx\n= (value of the anti derivative F\nof f at the upper limit b \u2013 value of the same anti derivative at the lower limit a) (ii)\nThis theorem is very useful, because it gives us a method of calculating the\ndefinite integral more easily, without calculating the limit of a sum (iii)\nThe crucial operation in evaluating a definite integral is that of finding a function\nwhose derivative is equal to the integrand"
    },
    {
        "Chapter": "1",
        "sentence_range": "3883-3886",
        "Text": "Remarks\n(i)\nIn words, the Theorem 2 tells us that \n( )\nb\n\u222ba f x dx\n= (value of the anti derivative F\nof f at the upper limit b \u2013 value of the same anti derivative at the lower limit a) (ii)\nThis theorem is very useful, because it gives us a method of calculating the\ndefinite integral more easily, without calculating the limit of a sum (iii)\nThe crucial operation in evaluating a definite integral is that of finding a function\nwhose derivative is equal to the integrand This strengthens the relationship\nbetween differentiation and integration"
    },
    {
        "Chapter": "1",
        "sentence_range": "3884-3887",
        "Text": "(ii)\nThis theorem is very useful, because it gives us a method of calculating the\ndefinite integral more easily, without calculating the limit of a sum (iii)\nThe crucial operation in evaluating a definite integral is that of finding a function\nwhose derivative is equal to the integrand This strengthens the relationship\nbetween differentiation and integration (iv)\nIn \n( )\nb\n\u222ba f x dx\n, the function f needs to be well defined and continuous in [a, b]"
    },
    {
        "Chapter": "1",
        "sentence_range": "3885-3888",
        "Text": "(iii)\nThe crucial operation in evaluating a definite integral is that of finding a function\nwhose derivative is equal to the integrand This strengthens the relationship\nbetween differentiation and integration (iv)\nIn \n( )\nb\n\u222ba f x dx\n, the function f needs to be well defined and continuous in [a, b] For instance, the consideration of definite integral \n1\n3\n2\n2\n2 (\nx x\u20131)\ndx\n\u2212\u222b\n is erroneous\nsince the function f expressed by f(x) = \n1\n2\n2\n(\nx x\u20131)\n is not defined in a portion\n\u2013 1 < x < 1 of the closed interval [\u2013 2, 3]"
    },
    {
        "Chapter": "1",
        "sentence_range": "3886-3889",
        "Text": "This strengthens the relationship\nbetween differentiation and integration (iv)\nIn \n( )\nb\n\u222ba f x dx\n, the function f needs to be well defined and continuous in [a, b] For instance, the consideration of definite integral \n1\n3\n2\n2\n2 (\nx x\u20131)\ndx\n\u2212\u222b\n is erroneous\nsince the function f expressed by f(x) = \n1\n2\n2\n(\nx x\u20131)\n is not defined in a portion\n\u2013 1 < x < 1 of the closed interval [\u2013 2, 3] 336\nMATHEMATICS\nSteps for calculating \n( )\nb\n\u222ba f x dx"
    },
    {
        "Chapter": "1",
        "sentence_range": "3887-3890",
        "Text": "(iv)\nIn \n( )\nb\n\u222ba f x dx\n, the function f needs to be well defined and continuous in [a, b] For instance, the consideration of definite integral \n1\n3\n2\n2\n2 (\nx x\u20131)\ndx\n\u2212\u222b\n is erroneous\nsince the function f expressed by f(x) = \n1\n2\n2\n(\nx x\u20131)\n is not defined in a portion\n\u2013 1 < x < 1 of the closed interval [\u2013 2, 3] 336\nMATHEMATICS\nSteps for calculating \n( )\nb\n\u222ba f x dx (i)\nFind the indefinite integral\n( )\n\u222bf x dx"
    },
    {
        "Chapter": "1",
        "sentence_range": "3888-3891",
        "Text": "For instance, the consideration of definite integral \n1\n3\n2\n2\n2 (\nx x\u20131)\ndx\n\u2212\u222b\n is erroneous\nsince the function f expressed by f(x) = \n1\n2\n2\n(\nx x\u20131)\n is not defined in a portion\n\u2013 1 < x < 1 of the closed interval [\u2013 2, 3] 336\nMATHEMATICS\nSteps for calculating \n( )\nb\n\u222ba f x dx (i)\nFind the indefinite integral\n( )\n\u222bf x dx Let this be F(x)"
    },
    {
        "Chapter": "1",
        "sentence_range": "3889-3892",
        "Text": "336\nMATHEMATICS\nSteps for calculating \n( )\nb\n\u222ba f x dx (i)\nFind the indefinite integral\n( )\n\u222bf x dx Let this be F(x) There is no need to keep\nintegration constant C because if we consider F(x) + C instead of F(x), we get\n( )\n[F ( )\nC]\n[F( )\nC]\u2013 [F( )\nC]\nF( ) \u2013 F( )\nb\nab\na f x dx\nx\nb\na\nb\na\n=\n+\n=\n+\n+\n=\n\u222b"
    },
    {
        "Chapter": "1",
        "sentence_range": "3890-3893",
        "Text": "(i)\nFind the indefinite integral\n( )\n\u222bf x dx Let this be F(x) There is no need to keep\nintegration constant C because if we consider F(x) + C instead of F(x), we get\n( )\n[F ( )\nC]\n[F( )\nC]\u2013 [F( )\nC]\nF( ) \u2013 F( )\nb\nab\na f x dx\nx\nb\na\nb\na\n=\n+\n=\n+\n+\n=\n\u222b Thus, the arbitrary constant disappears in evaluating the value of the definite\nintegral"
    },
    {
        "Chapter": "1",
        "sentence_range": "3891-3894",
        "Text": "Let this be F(x) There is no need to keep\nintegration constant C because if we consider F(x) + C instead of F(x), we get\n( )\n[F ( )\nC]\n[F( )\nC]\u2013 [F( )\nC]\nF( ) \u2013 F( )\nb\nab\na f x dx\nx\nb\na\nb\na\n=\n+\n=\n+\n+\n=\n\u222b Thus, the arbitrary constant disappears in evaluating the value of the definite\nintegral (ii)\nEvaluate F(b) \u2013 F(a) = [F ( )]b\nxa\n, which is the value of  \n( )\nb\n\u222ba f x dx"
    },
    {
        "Chapter": "1",
        "sentence_range": "3892-3895",
        "Text": "There is no need to keep\nintegration constant C because if we consider F(x) + C instead of F(x), we get\n( )\n[F ( )\nC]\n[F( )\nC]\u2013 [F( )\nC]\nF( ) \u2013 F( )\nb\nab\na f x dx\nx\nb\na\nb\na\n=\n+\n=\n+\n+\n=\n\u222b Thus, the arbitrary constant disappears in evaluating the value of the definite\nintegral (ii)\nEvaluate F(b) \u2013 F(a) = [F ( )]b\nxa\n, which is the value of  \n( )\nb\n\u222ba f x dx We now consider some examples\nExample 27 Evaluate the following integrals:\n(i)\n3\n2\n\u222b2 x dx\n(ii)\n9\n3\n4\n22\n(30\u2013\n)\nx\ndx\nx\n\u222b\n(iii)\n2\n1 (\n1) (\n2)\nx dx\nx\nx\n+\n+\n\u222b\n(iv)   \n3\n4\n0 sin 2 cos 2\nt\nt dt\n\u03c0\n\u222b\nSolution\n(i)\nLet \n3\n2\n2\nI\n= \u222bx dx"
    },
    {
        "Chapter": "1",
        "sentence_range": "3893-3896",
        "Text": "Thus, the arbitrary constant disappears in evaluating the value of the definite\nintegral (ii)\nEvaluate F(b) \u2013 F(a) = [F ( )]b\nxa\n, which is the value of  \n( )\nb\n\u222ba f x dx We now consider some examples\nExample 27 Evaluate the following integrals:\n(i)\n3\n2\n\u222b2 x dx\n(ii)\n9\n3\n4\n22\n(30\u2013\n)\nx\ndx\nx\n\u222b\n(iii)\n2\n1 (\n1) (\n2)\nx dx\nx\nx\n+\n+\n\u222b\n(iv)   \n3\n4\n0 sin 2 cos 2\nt\nt dt\n\u03c0\n\u222b\nSolution\n(i)\nLet \n3\n2\n2\nI\n= \u222bx dx Since \n3\n2\nF ( )\nx3\nx dx\nx\n=\n=\n\u222b\n,\nTherefore, by the second fundamental theorem, we get\nI = \n27\n8\n19\nF (3) \u2013 F (2)\n\u2013\n3\n3\n3\n=\n=\n(ii)\nLet \n9\n3\n4\n2\n2\nI\n(30 \u2013\n)\nx\ndx\nx\n= \u222b"
    },
    {
        "Chapter": "1",
        "sentence_range": "3894-3897",
        "Text": "(ii)\nEvaluate F(b) \u2013 F(a) = [F ( )]b\nxa\n, which is the value of  \n( )\nb\n\u222ba f x dx We now consider some examples\nExample 27 Evaluate the following integrals:\n(i)\n3\n2\n\u222b2 x dx\n(ii)\n9\n3\n4\n22\n(30\u2013\n)\nx\ndx\nx\n\u222b\n(iii)\n2\n1 (\n1) (\n2)\nx dx\nx\nx\n+\n+\n\u222b\n(iv)   \n3\n4\n0 sin 2 cos 2\nt\nt dt\n\u03c0\n\u222b\nSolution\n(i)\nLet \n3\n2\n2\nI\n= \u222bx dx Since \n3\n2\nF ( )\nx3\nx dx\nx\n=\n=\n\u222b\n,\nTherefore, by the second fundamental theorem, we get\nI = \n27\n8\n19\nF (3) \u2013 F (2)\n\u2013\n3\n3\n3\n=\n=\n(ii)\nLet \n9\n3\n4\n2\n2\nI\n(30 \u2013\n)\nx\ndx\nx\n= \u222b We first find the anti derivative of the integrand"
    },
    {
        "Chapter": "1",
        "sentence_range": "3895-3898",
        "Text": "We now consider some examples\nExample 27 Evaluate the following integrals:\n(i)\n3\n2\n\u222b2 x dx\n(ii)\n9\n3\n4\n22\n(30\u2013\n)\nx\ndx\nx\n\u222b\n(iii)\n2\n1 (\n1) (\n2)\nx dx\nx\nx\n+\n+\n\u222b\n(iv)   \n3\n4\n0 sin 2 cos 2\nt\nt dt\n\u03c0\n\u222b\nSolution\n(i)\nLet \n3\n2\n2\nI\n= \u222bx dx Since \n3\n2\nF ( )\nx3\nx dx\nx\n=\n=\n\u222b\n,\nTherefore, by the second fundamental theorem, we get\nI = \n27\n8\n19\nF (3) \u2013 F (2)\n\u2013\n3\n3\n3\n=\n=\n(ii)\nLet \n9\n3\n4\n2\n2\nI\n(30 \u2013\n)\nx\ndx\nx\n= \u222b We first find the anti derivative of the integrand Put \n23\n3\n30 \u2013"
    },
    {
        "Chapter": "1",
        "sentence_range": "3896-3899",
        "Text": "Since \n3\n2\nF ( )\nx3\nx dx\nx\n=\n=\n\u222b\n,\nTherefore, by the second fundamental theorem, we get\nI = \n27\n8\n19\nF (3) \u2013 F (2)\n\u2013\n3\n3\n3\n=\n=\n(ii)\nLet \n9\n3\n4\n2\n2\nI\n(30 \u2013\n)\nx\ndx\nx\n= \u222b We first find the anti derivative of the integrand Put \n23\n3\n30 \u2013 Then \u2013\n2\nx\nt\nx dx\ndt\n=\n=\n or \n2\n\u2013\n3\nx dx\ndt\n=\nThus,  \n3\n2\n2\n2\n\u20132\n3\n(30 \u2013\n)\nx\ndt\ndx\nt\nx\n=\n\u222b\n\u222b\n = \n2 1\n3 t\n\uf8ee \uf8f9\n\uf8ef \uf8fa\n\uf8f0 \uf8fb = \n3\n2\n2\n1\nF ( )\n3 (30 \u2013\n)\nx\nx\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa =\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nINTEGRALS         337\nTherefore, by the second fundamental theorem of calculus, we have\nI =\n9\n3\n2\n4\n2\n1\nF(9) \u2013 F(4)\n3\n(30 \u2013\n)\nx\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n=\n2\n1\n1\n3 (30 \u2013 27)\n30 \u20138\n\uf8ee\n\uf8f9\n\u2212\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n = \n2 1\n1\n19\n3 3\n22\n99\n\uf8ee\n\uf8f9\n\u2212\n=\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n(iii)\nLet \n2\n1\nI\n(\n1) (\n2)\nx dx\nx\nx\n=\n+\n+\n\u222b\nUsing partial fraction, we get  \n\u20131\n2\n(\n1) (\n2)\n1\n2\nx\nx\nx\nx\nx\n=\n+\n+\n+\n+\n+\nSo\n(\n1) (\n2)\nx dx\nx\nx\n+\n+\n\u222b\n = \u2013 log\n1\n2log\n2\nF( )\nx\nx\nx\n+\n+\n+\n=\nTherefore, by the second fundamental theorem of calculus, we have\nI = F(2) \u2013 F(1) = [\u2013 log 3 + 2 log 4] \u2013 [\u2013 log 2 + 2 log 3]\n= \u2013 3 log 3 + log 2 + 2 log 4 = \n32\nlog\n27\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n(iv)\nLet \n3\n04\nI\nsin 2 cos2\nt\nt dt\n= \u222b\u03c0"
    },
    {
        "Chapter": "1",
        "sentence_range": "3897-3900",
        "Text": "We first find the anti derivative of the integrand Put \n23\n3\n30 \u2013 Then \u2013\n2\nx\nt\nx dx\ndt\n=\n=\n or \n2\n\u2013\n3\nx dx\ndt\n=\nThus,  \n3\n2\n2\n2\n\u20132\n3\n(30 \u2013\n)\nx\ndt\ndx\nt\nx\n=\n\u222b\n\u222b\n = \n2 1\n3 t\n\uf8ee \uf8f9\n\uf8ef \uf8fa\n\uf8f0 \uf8fb = \n3\n2\n2\n1\nF ( )\n3 (30 \u2013\n)\nx\nx\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa =\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nINTEGRALS         337\nTherefore, by the second fundamental theorem of calculus, we have\nI =\n9\n3\n2\n4\n2\n1\nF(9) \u2013 F(4)\n3\n(30 \u2013\n)\nx\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n=\n2\n1\n1\n3 (30 \u2013 27)\n30 \u20138\n\uf8ee\n\uf8f9\n\u2212\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n = \n2 1\n1\n19\n3 3\n22\n99\n\uf8ee\n\uf8f9\n\u2212\n=\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n(iii)\nLet \n2\n1\nI\n(\n1) (\n2)\nx dx\nx\nx\n=\n+\n+\n\u222b\nUsing partial fraction, we get  \n\u20131\n2\n(\n1) (\n2)\n1\n2\nx\nx\nx\nx\nx\n=\n+\n+\n+\n+\n+\nSo\n(\n1) (\n2)\nx dx\nx\nx\n+\n+\n\u222b\n = \u2013 log\n1\n2log\n2\nF( )\nx\nx\nx\n+\n+\n+\n=\nTherefore, by the second fundamental theorem of calculus, we have\nI = F(2) \u2013 F(1) = [\u2013 log 3 + 2 log 4] \u2013 [\u2013 log 2 + 2 log 3]\n= \u2013 3 log 3 + log 2 + 2 log 4 = \n32\nlog\n27\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n(iv)\nLet \n3\n04\nI\nsin 2 cos2\nt\nt dt\n= \u222b\u03c0 Consider \n3\nsin 2 cos2\nt\nt dt\n\u222b\nPut sin 2t = u so that 2 cos 2t dt = du or cos 2t dt = 1\n2  du\nSo\n3\nsin 2 cos2\nt\nt dt\n\u222b\n =\n3\n1\n2\nu du\n\u222b\n=\n4\n4\n1\n1\n[\n]\nsin 2\nF ( )say\n8\n8\nu\nt\nt\n=\n=\nTherefore, by the second fundamental theorem of integral calculus\nI =\n4\n4\n1\n1\nF ( ) \u2013 F (0)\n[sin\n\u2013sin 0]\n4\n8\n2\n8\n\u03c0\n\u03c0\n=\n=\n338\nMATHEMATICS\nEXERCISE 7"
    },
    {
        "Chapter": "1",
        "sentence_range": "3898-3901",
        "Text": "Put \n23\n3\n30 \u2013 Then \u2013\n2\nx\nt\nx dx\ndt\n=\n=\n or \n2\n\u2013\n3\nx dx\ndt\n=\nThus,  \n3\n2\n2\n2\n\u20132\n3\n(30 \u2013\n)\nx\ndt\ndx\nt\nx\n=\n\u222b\n\u222b\n = \n2 1\n3 t\n\uf8ee \uf8f9\n\uf8ef \uf8fa\n\uf8f0 \uf8fb = \n3\n2\n2\n1\nF ( )\n3 (30 \u2013\n)\nx\nx\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa =\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nINTEGRALS         337\nTherefore, by the second fundamental theorem of calculus, we have\nI =\n9\n3\n2\n4\n2\n1\nF(9) \u2013 F(4)\n3\n(30 \u2013\n)\nx\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n=\n2\n1\n1\n3 (30 \u2013 27)\n30 \u20138\n\uf8ee\n\uf8f9\n\u2212\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n = \n2 1\n1\n19\n3 3\n22\n99\n\uf8ee\n\uf8f9\n\u2212\n=\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n(iii)\nLet \n2\n1\nI\n(\n1) (\n2)\nx dx\nx\nx\n=\n+\n+\n\u222b\nUsing partial fraction, we get  \n\u20131\n2\n(\n1) (\n2)\n1\n2\nx\nx\nx\nx\nx\n=\n+\n+\n+\n+\n+\nSo\n(\n1) (\n2)\nx dx\nx\nx\n+\n+\n\u222b\n = \u2013 log\n1\n2log\n2\nF( )\nx\nx\nx\n+\n+\n+\n=\nTherefore, by the second fundamental theorem of calculus, we have\nI = F(2) \u2013 F(1) = [\u2013 log 3 + 2 log 4] \u2013 [\u2013 log 2 + 2 log 3]\n= \u2013 3 log 3 + log 2 + 2 log 4 = \n32\nlog\n27\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n(iv)\nLet \n3\n04\nI\nsin 2 cos2\nt\nt dt\n= \u222b\u03c0 Consider \n3\nsin 2 cos2\nt\nt dt\n\u222b\nPut sin 2t = u so that 2 cos 2t dt = du or cos 2t dt = 1\n2  du\nSo\n3\nsin 2 cos2\nt\nt dt\n\u222b\n =\n3\n1\n2\nu du\n\u222b\n=\n4\n4\n1\n1\n[\n]\nsin 2\nF ( )say\n8\n8\nu\nt\nt\n=\n=\nTherefore, by the second fundamental theorem of integral calculus\nI =\n4\n4\n1\n1\nF ( ) \u2013 F (0)\n[sin\n\u2013sin 0]\n4\n8\n2\n8\n\u03c0\n\u03c0\n=\n=\n338\nMATHEMATICS\nEXERCISE 7 9\nEvaluate the definite integrals in Exercises 1 to 20"
    },
    {
        "Chapter": "1",
        "sentence_range": "3899-3902",
        "Text": "Then \u2013\n2\nx\nt\nx dx\ndt\n=\n=\n or \n2\n\u2013\n3\nx dx\ndt\n=\nThus,  \n3\n2\n2\n2\n\u20132\n3\n(30 \u2013\n)\nx\ndt\ndx\nt\nx\n=\n\u222b\n\u222b\n = \n2 1\n3 t\n\uf8ee \uf8f9\n\uf8ef \uf8fa\n\uf8f0 \uf8fb = \n3\n2\n2\n1\nF ( )\n3 (30 \u2013\n)\nx\nx\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa =\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nINTEGRALS         337\nTherefore, by the second fundamental theorem of calculus, we have\nI =\n9\n3\n2\n4\n2\n1\nF(9) \u2013 F(4)\n3\n(30 \u2013\n)\nx\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n=\n2\n1\n1\n3 (30 \u2013 27)\n30 \u20138\n\uf8ee\n\uf8f9\n\u2212\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n = \n2 1\n1\n19\n3 3\n22\n99\n\uf8ee\n\uf8f9\n\u2212\n=\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n(iii)\nLet \n2\n1\nI\n(\n1) (\n2)\nx dx\nx\nx\n=\n+\n+\n\u222b\nUsing partial fraction, we get  \n\u20131\n2\n(\n1) (\n2)\n1\n2\nx\nx\nx\nx\nx\n=\n+\n+\n+\n+\n+\nSo\n(\n1) (\n2)\nx dx\nx\nx\n+\n+\n\u222b\n = \u2013 log\n1\n2log\n2\nF( )\nx\nx\nx\n+\n+\n+\n=\nTherefore, by the second fundamental theorem of calculus, we have\nI = F(2) \u2013 F(1) = [\u2013 log 3 + 2 log 4] \u2013 [\u2013 log 2 + 2 log 3]\n= \u2013 3 log 3 + log 2 + 2 log 4 = \n32\nlog\n27\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n(iv)\nLet \n3\n04\nI\nsin 2 cos2\nt\nt dt\n= \u222b\u03c0 Consider \n3\nsin 2 cos2\nt\nt dt\n\u222b\nPut sin 2t = u so that 2 cos 2t dt = du or cos 2t dt = 1\n2  du\nSo\n3\nsin 2 cos2\nt\nt dt\n\u222b\n =\n3\n1\n2\nu du\n\u222b\n=\n4\n4\n1\n1\n[\n]\nsin 2\nF ( )say\n8\n8\nu\nt\nt\n=\n=\nTherefore, by the second fundamental theorem of integral calculus\nI =\n4\n4\n1\n1\nF ( ) \u2013 F (0)\n[sin\n\u2013sin 0]\n4\n8\n2\n8\n\u03c0\n\u03c0\n=\n=\n338\nMATHEMATICS\nEXERCISE 7 9\nEvaluate the definite integrals in Exercises 1 to 20 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "3900-3903",
        "Text": "Consider \n3\nsin 2 cos2\nt\nt dt\n\u222b\nPut sin 2t = u so that 2 cos 2t dt = du or cos 2t dt = 1\n2  du\nSo\n3\nsin 2 cos2\nt\nt dt\n\u222b\n =\n3\n1\n2\nu du\n\u222b\n=\n4\n4\n1\n1\n[\n]\nsin 2\nF ( )say\n8\n8\nu\nt\nt\n=\n=\nTherefore, by the second fundamental theorem of integral calculus\nI =\n4\n4\n1\n1\nF ( ) \u2013 F (0)\n[sin\n\u2013sin 0]\n4\n8\n2\n8\n\u03c0\n\u03c0\n=\n=\n338\nMATHEMATICS\nEXERCISE 7 9\nEvaluate the definite integrals in Exercises 1 to 20 1 1\n1(\n1)\nx\ndx\n\u2212\n+\n\u222b\n2"
    },
    {
        "Chapter": "1",
        "sentence_range": "3901-3904",
        "Text": "9\nEvaluate the definite integrals in Exercises 1 to 20 1 1\n1(\n1)\nx\ndx\n\u2212\n+\n\u222b\n2 3\n2\n\u222bx1 dx\n3"
    },
    {
        "Chapter": "1",
        "sentence_range": "3902-3905",
        "Text": "1 1\n1(\n1)\nx\ndx\n\u2212\n+\n\u222b\n2 3\n2\n\u222bx1 dx\n3 2\n3\n2\n1 (4\n\u2013 5\n6\n9)\nx\nx\nx\ndx\n+\n+\n\u222b\n4"
    },
    {
        "Chapter": "1",
        "sentence_range": "3903-3906",
        "Text": "1\n1(\n1)\nx\ndx\n\u2212\n+\n\u222b\n2 3\n2\n\u222bx1 dx\n3 2\n3\n2\n1 (4\n\u2013 5\n6\n9)\nx\nx\nx\ndx\n+\n+\n\u222b\n4 4\n0 sin 2x dx\n\u222b\u03c0\n5"
    },
    {
        "Chapter": "1",
        "sentence_range": "3904-3907",
        "Text": "3\n2\n\u222bx1 dx\n3 2\n3\n2\n1 (4\n\u2013 5\n6\n9)\nx\nx\nx\ndx\n+\n+\n\u222b\n4 4\n0 sin 2x dx\n\u222b\u03c0\n5 2\n0 cos 2x dx\n\u222b\u03c0\n6"
    },
    {
        "Chapter": "1",
        "sentence_range": "3905-3908",
        "Text": "2\n3\n2\n1 (4\n\u2013 5\n6\n9)\nx\nx\nx\ndx\n+\n+\n\u222b\n4 4\n0 sin 2x dx\n\u222b\u03c0\n5 2\n0 cos 2x dx\n\u222b\u03c0\n6 5\n4\nx\n\u222be dx\n7"
    },
    {
        "Chapter": "1",
        "sentence_range": "3906-3909",
        "Text": "4\n0 sin 2x dx\n\u222b\u03c0\n5 2\n0 cos 2x dx\n\u222b\u03c0\n6 5\n4\nx\n\u222be dx\n7 4\n0 tanx dx\n\u03c0\n\u222b\n8"
    },
    {
        "Chapter": "1",
        "sentence_range": "3907-3910",
        "Text": "2\n0 cos 2x dx\n\u222b\u03c0\n6 5\n4\nx\n\u222be dx\n7 4\n0 tanx dx\n\u03c0\n\u222b\n8 4\n6\ncosec x dx\n\u03c0\n\u03c0\u222b\n9"
    },
    {
        "Chapter": "1",
        "sentence_range": "3908-3911",
        "Text": "5\n4\nx\n\u222be dx\n7 4\n0 tanx dx\n\u03c0\n\u222b\n8 4\n6\ncosec x dx\n\u03c0\n\u03c0\u222b\n9 1\n0\n2\n1\u2013\ndx\nx\n\u222b\n10"
    },
    {
        "Chapter": "1",
        "sentence_range": "3909-3912",
        "Text": "4\n0 tanx dx\n\u03c0\n\u222b\n8 4\n6\ncosec x dx\n\u03c0\n\u03c0\u222b\n9 1\n0\n2\n1\u2013\ndx\nx\n\u222b\n10 1\n2\n01\ndx\n+x\n\u222b\n11"
    },
    {
        "Chapter": "1",
        "sentence_range": "3910-3913",
        "Text": "4\n6\ncosec x dx\n\u03c0\n\u03c0\u222b\n9 1\n0\n2\n1\u2013\ndx\nx\n\u222b\n10 1\n2\n01\ndx\n+x\n\u222b\n11 3\n2\n2\n1\ndx\nx \u2212\n\u222b\n12"
    },
    {
        "Chapter": "1",
        "sentence_range": "3911-3914",
        "Text": "1\n0\n2\n1\u2013\ndx\nx\n\u222b\n10 1\n2\n01\ndx\n+x\n\u222b\n11 3\n2\n2\n1\ndx\nx \u2212\n\u222b\n12 2\n2\n0 cos x dx\n\u222b\u03c0\n13"
    },
    {
        "Chapter": "1",
        "sentence_range": "3912-3915",
        "Text": "1\n2\n01\ndx\n+x\n\u222b\n11 3\n2\n2\n1\ndx\nx \u2212\n\u222b\n12 2\n2\n0 cos x dx\n\u222b\u03c0\n13 3\n2\n2\n1\nx dx\n\u222bx +\n14"
    },
    {
        "Chapter": "1",
        "sentence_range": "3913-3916",
        "Text": "3\n2\n2\n1\ndx\nx \u2212\n\u222b\n12 2\n2\n0 cos x dx\n\u222b\u03c0\n13 3\n2\n2\n1\nx dx\n\u222bx +\n14 1\n2\n0\n2\n3\n5\n1\nx\ndx\nx\n+\n+\n\u222b\n15"
    },
    {
        "Chapter": "1",
        "sentence_range": "3914-3917",
        "Text": "2\n2\n0 cos x dx\n\u222b\u03c0\n13 3\n2\n2\n1\nx dx\n\u222bx +\n14 1\n2\n0\n2\n3\n5\n1\nx\ndx\nx\n+\n+\n\u222b\n15 2\n1\n0\nx\nx e dx\n\u222b\n16"
    },
    {
        "Chapter": "1",
        "sentence_range": "3915-3918",
        "Text": "3\n2\n2\n1\nx dx\n\u222bx +\n14 1\n2\n0\n2\n3\n5\n1\nx\ndx\nx\n+\n+\n\u222b\n15 2\n1\n0\nx\nx e dx\n\u222b\n16 2\n2\n2\n1\n5\n4\n3\nx\nx\n+x\n+\n\u222b\n17"
    },
    {
        "Chapter": "1",
        "sentence_range": "3916-3919",
        "Text": "1\n2\n0\n2\n3\n5\n1\nx\ndx\nx\n+\n+\n\u222b\n15 2\n1\n0\nx\nx e dx\n\u222b\n16 2\n2\n2\n1\n5\n4\n3\nx\nx\n+x\n+\n\u222b\n17 2\n3\n4\n0 (2sec\n2)\nx\nx\ndx\n\u03c0\n+\n+\n\u222b\n18"
    },
    {
        "Chapter": "1",
        "sentence_range": "3917-3920",
        "Text": "2\n1\n0\nx\nx e dx\n\u222b\n16 2\n2\n2\n1\n5\n4\n3\nx\nx\n+x\n+\n\u222b\n17 2\n3\n4\n0 (2sec\n2)\nx\nx\ndx\n\u03c0\n+\n+\n\u222b\n18 2\n2\n0 (sin\n\u2013cos\n)\n2\n2\nx\nx dx\n\u03c0\u222b\n19"
    },
    {
        "Chapter": "1",
        "sentence_range": "3918-3921",
        "Text": "2\n2\n2\n1\n5\n4\n3\nx\nx\n+x\n+\n\u222b\n17 2\n3\n4\n0 (2sec\n2)\nx\nx\ndx\n\u03c0\n+\n+\n\u222b\n18 2\n2\n0 (sin\n\u2013cos\n)\n2\n2\nx\nx dx\n\u03c0\u222b\n19 2\n2\n0\n6\n3\n4\nx\ndx\nx\n++\n\u222b\n20"
    },
    {
        "Chapter": "1",
        "sentence_range": "3919-3922",
        "Text": "2\n3\n4\n0 (2sec\n2)\nx\nx\ndx\n\u03c0\n+\n+\n\u222b\n18 2\n2\n0 (sin\n\u2013cos\n)\n2\n2\nx\nx dx\n\u03c0\u222b\n19 2\n2\n0\n6\n3\n4\nx\ndx\nx\n++\n\u222b\n20 1\n0(\nsin\n)\n4\nx\nx\nx e\ndx\n\u03c0\n+\n\u222b\nChoose the correct answer in Exercises 21 and 22"
    },
    {
        "Chapter": "1",
        "sentence_range": "3920-3923",
        "Text": "2\n2\n0 (sin\n\u2013cos\n)\n2\n2\nx\nx dx\n\u03c0\u222b\n19 2\n2\n0\n6\n3\n4\nx\ndx\nx\n++\n\u222b\n20 1\n0(\nsin\n)\n4\nx\nx\nx e\ndx\n\u03c0\n+\n\u222b\nChoose the correct answer in Exercises 21 and 22 21"
    },
    {
        "Chapter": "1",
        "sentence_range": "3921-3924",
        "Text": "2\n2\n0\n6\n3\n4\nx\ndx\nx\n++\n\u222b\n20 1\n0(\nsin\n)\n4\nx\nx\nx e\ndx\n\u03c0\n+\n\u222b\nChoose the correct answer in Exercises 21 and 22 21 3\n2\n1 1\ndx\n+x\n\u222b\n equals\n(A)\n\u03c03\n(B)\n\u03c032\n(C)\n\u03c06\n(D) 12\n\u03c0\n22"
    },
    {
        "Chapter": "1",
        "sentence_range": "3922-3925",
        "Text": "1\n0(\nsin\n)\n4\nx\nx\nx e\ndx\n\u03c0\n+\n\u222b\nChoose the correct answer in Exercises 21 and 22 21 3\n2\n1 1\ndx\n+x\n\u222b\n equals\n(A)\n\u03c03\n(B)\n\u03c032\n(C)\n\u03c06\n(D) 12\n\u03c0\n22 2\n3\n2\n0 4\n9\ndx\nx\n+\n\u222b\n equals\n(A) 6\n\u03c0\n(B) 12\n\u03c0\n(C) 24\n\u03c0\n(D) 4\n\u03c0\n7"
    },
    {
        "Chapter": "1",
        "sentence_range": "3923-3926",
        "Text": "21 3\n2\n1 1\ndx\n+x\n\u222b\n equals\n(A)\n\u03c03\n(B)\n\u03c032\n(C)\n\u03c06\n(D) 12\n\u03c0\n22 2\n3\n2\n0 4\n9\ndx\nx\n+\n\u222b\n equals\n(A) 6\n\u03c0\n(B) 12\n\u03c0\n(C) 24\n\u03c0\n(D) 4\n\u03c0\n7 9  Evaluation of Definite Integrals by Substitution\nIn the previous sections, we have discussed several methods for finding the indefinite\nintegral"
    },
    {
        "Chapter": "1",
        "sentence_range": "3924-3927",
        "Text": "3\n2\n1 1\ndx\n+x\n\u222b\n equals\n(A)\n\u03c03\n(B)\n\u03c032\n(C)\n\u03c06\n(D) 12\n\u03c0\n22 2\n3\n2\n0 4\n9\ndx\nx\n+\n\u222b\n equals\n(A) 6\n\u03c0\n(B) 12\n\u03c0\n(C) 24\n\u03c0\n(D) 4\n\u03c0\n7 9  Evaluation of Definite Integrals by Substitution\nIn the previous sections, we have discussed several methods for finding the indefinite\nintegral One of the important methods for finding the indefinite integral is the method\nof substitution"
    },
    {
        "Chapter": "1",
        "sentence_range": "3925-3928",
        "Text": "2\n3\n2\n0 4\n9\ndx\nx\n+\n\u222b\n equals\n(A) 6\n\u03c0\n(B) 12\n\u03c0\n(C) 24\n\u03c0\n(D) 4\n\u03c0\n7 9  Evaluation of Definite Integrals by Substitution\nIn the previous sections, we have discussed several methods for finding the indefinite\nintegral One of the important methods for finding the indefinite integral is the method\nof substitution INTEGRALS         339\nTo evaluate \n( )\nb\n\u222ba f x dx\n, by substitution, the steps could be as follows:\n1"
    },
    {
        "Chapter": "1",
        "sentence_range": "3926-3929",
        "Text": "9  Evaluation of Definite Integrals by Substitution\nIn the previous sections, we have discussed several methods for finding the indefinite\nintegral One of the important methods for finding the indefinite integral is the method\nof substitution INTEGRALS         339\nTo evaluate \n( )\nb\n\u222ba f x dx\n, by substitution, the steps could be as follows:\n1 Consider the integral without limits and substitute, y = f (x) or x = g(y) to reduce\nthe given integral to a known form"
    },
    {
        "Chapter": "1",
        "sentence_range": "3927-3930",
        "Text": "One of the important methods for finding the indefinite integral is the method\nof substitution INTEGRALS         339\nTo evaluate \n( )\nb\n\u222ba f x dx\n, by substitution, the steps could be as follows:\n1 Consider the integral without limits and substitute, y = f (x) or x = g(y) to reduce\nthe given integral to a known form 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "3928-3931",
        "Text": "INTEGRALS         339\nTo evaluate \n( )\nb\n\u222ba f x dx\n, by substitution, the steps could be as follows:\n1 Consider the integral without limits and substitute, y = f (x) or x = g(y) to reduce\nthe given integral to a known form 2 Integrate the new integrand with respect to the new variable without mentioning\nthe constant of integration"
    },
    {
        "Chapter": "1",
        "sentence_range": "3929-3932",
        "Text": "Consider the integral without limits and substitute, y = f (x) or x = g(y) to reduce\nthe given integral to a known form 2 Integrate the new integrand with respect to the new variable without mentioning\nthe constant of integration 3"
    },
    {
        "Chapter": "1",
        "sentence_range": "3930-3933",
        "Text": "2 Integrate the new integrand with respect to the new variable without mentioning\nthe constant of integration 3 Resubstitute for the new variable and write the answer in terms of the original\nvariable"
    },
    {
        "Chapter": "1",
        "sentence_range": "3931-3934",
        "Text": "Integrate the new integrand with respect to the new variable without mentioning\nthe constant of integration 3 Resubstitute for the new variable and write the answer in terms of the original\nvariable 4"
    },
    {
        "Chapter": "1",
        "sentence_range": "3932-3935",
        "Text": "3 Resubstitute for the new variable and write the answer in terms of the original\nvariable 4 Find the values of answers obtained in (3) at the given limits of integral and find\nthe difference of the values at the upper and lower limits"
    },
    {
        "Chapter": "1",
        "sentence_range": "3933-3936",
        "Text": "Resubstitute for the new variable and write the answer in terms of the original\nvariable 4 Find the values of answers obtained in (3) at the given limits of integral and find\nthe difference of the values at the upper and lower limits \ufffdNote In order to quicken this method, we can proceed as follows: After\nperforming steps 1, and 2, there is no need of step 3"
    },
    {
        "Chapter": "1",
        "sentence_range": "3934-3937",
        "Text": "4 Find the values of answers obtained in (3) at the given limits of integral and find\nthe difference of the values at the upper and lower limits \ufffdNote In order to quicken this method, we can proceed as follows: After\nperforming steps 1, and 2, there is no need of step 3 Here, the integral will be kept\nin the new variable itself, and the limits of the integral will accordingly be changed,\nso that we can perform the last step"
    },
    {
        "Chapter": "1",
        "sentence_range": "3935-3938",
        "Text": "Find the values of answers obtained in (3) at the given limits of integral and find\nthe difference of the values at the upper and lower limits \ufffdNote In order to quicken this method, we can proceed as follows: After\nperforming steps 1, and 2, there is no need of step 3 Here, the integral will be kept\nin the new variable itself, and the limits of the integral will accordingly be changed,\nso that we can perform the last step Let us illustrate this by examples"
    },
    {
        "Chapter": "1",
        "sentence_range": "3936-3939",
        "Text": "\ufffdNote In order to quicken this method, we can proceed as follows: After\nperforming steps 1, and 2, there is no need of step 3 Here, the integral will be kept\nin the new variable itself, and the limits of the integral will accordingly be changed,\nso that we can perform the last step Let us illustrate this by examples Example 28 Evaluate \n1\n4\n5\n15\n1\nx\nx\ndx\n\u2212\n+\n\u222b"
    },
    {
        "Chapter": "1",
        "sentence_range": "3937-3940",
        "Text": "Here, the integral will be kept\nin the new variable itself, and the limits of the integral will accordingly be changed,\nso that we can perform the last step Let us illustrate this by examples Example 28 Evaluate \n1\n4\n5\n15\n1\nx\nx\ndx\n\u2212\n+\n\u222b Solution Put  t = x5 + 1, then dt = 5x4 dx"
    },
    {
        "Chapter": "1",
        "sentence_range": "3938-3941",
        "Text": "Let us illustrate this by examples Example 28 Evaluate \n1\n4\n5\n15\n1\nx\nx\ndx\n\u2212\n+\n\u222b Solution Put  t = x5 + 1, then dt = 5x4 dx Therefore,\n4\n5\n5\n1\nx\nx\ndx\n+\n\u222b\n =\n\u222bt dt\n = \n223\n3\nt  = \n3\n5\n2\n2 (\n1)\n3\nx +\nHence,\n1\n4\n5\n15\n1\nx\nx\ndx\n\u2212\n+\n\u222b\n =\n1\n3\n5\n2\n\u2013 1\n2 (\n1)\n3\nx\n\uf8ee\n\uf8f9\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n=\n(\n)\n3\n3\n5\n5\n2\n2\n2 (1\n1) \u2013 (\u20131)\n1\n3\n\uf8ee\n\uf8f9\n+\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n=\n3\n3\n2\n2\n2 2\n0\n3\n\uf8ee\n\uf8f9\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n = \n2\n4\n2\n(2\n2)\n3\n3\n=\nAlternatively, first we transform the integral and then evaluate the transformed integral\nwith new limits"
    },
    {
        "Chapter": "1",
        "sentence_range": "3939-3942",
        "Text": "Example 28 Evaluate \n1\n4\n5\n15\n1\nx\nx\ndx\n\u2212\n+\n\u222b Solution Put  t = x5 + 1, then dt = 5x4 dx Therefore,\n4\n5\n5\n1\nx\nx\ndx\n+\n\u222b\n =\n\u222bt dt\n = \n223\n3\nt  = \n3\n5\n2\n2 (\n1)\n3\nx +\nHence,\n1\n4\n5\n15\n1\nx\nx\ndx\n\u2212\n+\n\u222b\n =\n1\n3\n5\n2\n\u2013 1\n2 (\n1)\n3\nx\n\uf8ee\n\uf8f9\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n=\n(\n)\n3\n3\n5\n5\n2\n2\n2 (1\n1) \u2013 (\u20131)\n1\n3\n\uf8ee\n\uf8f9\n+\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n=\n3\n3\n2\n2\n2 2\n0\n3\n\uf8ee\n\uf8f9\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n = \n2\n4\n2\n(2\n2)\n3\n3\n=\nAlternatively, first we transform the integral and then evaluate the transformed integral\nwith new limits 340\nMATHEMATICS\nLet\nt = x5 + 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "3940-3943",
        "Text": "Solution Put  t = x5 + 1, then dt = 5x4 dx Therefore,\n4\n5\n5\n1\nx\nx\ndx\n+\n\u222b\n =\n\u222bt dt\n = \n223\n3\nt  = \n3\n5\n2\n2 (\n1)\n3\nx +\nHence,\n1\n4\n5\n15\n1\nx\nx\ndx\n\u2212\n+\n\u222b\n =\n1\n3\n5\n2\n\u2013 1\n2 (\n1)\n3\nx\n\uf8ee\n\uf8f9\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n=\n(\n)\n3\n3\n5\n5\n2\n2\n2 (1\n1) \u2013 (\u20131)\n1\n3\n\uf8ee\n\uf8f9\n+\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n=\n3\n3\n2\n2\n2 2\n0\n3\n\uf8ee\n\uf8f9\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n = \n2\n4\n2\n(2\n2)\n3\n3\n=\nAlternatively, first we transform the integral and then evaluate the transformed integral\nwith new limits 340\nMATHEMATICS\nLet\nt = x5 + 1 Then dt = 5 x4 dx"
    },
    {
        "Chapter": "1",
        "sentence_range": "3941-3944",
        "Text": "Therefore,\n4\n5\n5\n1\nx\nx\ndx\n+\n\u222b\n =\n\u222bt dt\n = \n223\n3\nt  = \n3\n5\n2\n2 (\n1)\n3\nx +\nHence,\n1\n4\n5\n15\n1\nx\nx\ndx\n\u2212\n+\n\u222b\n =\n1\n3\n5\n2\n\u2013 1\n2 (\n1)\n3\nx\n\uf8ee\n\uf8f9\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n=\n(\n)\n3\n3\n5\n5\n2\n2\n2 (1\n1) \u2013 (\u20131)\n1\n3\n\uf8ee\n\uf8f9\n+\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n=\n3\n3\n2\n2\n2 2\n0\n3\n\uf8ee\n\uf8f9\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n = \n2\n4\n2\n(2\n2)\n3\n3\n=\nAlternatively, first we transform the integral and then evaluate the transformed integral\nwith new limits 340\nMATHEMATICS\nLet\nt = x5 + 1 Then dt = 5 x4 dx Note that, when\nx = \u2013 1, t = 0 and when x = 1, t = 2\nThus,  as x varies from \u2013 1 to 1, t varies from 0 to 2\nTherefore\n1\n4\n5\n15\n1\nx\nx\ndx\n\u2212\n+\n\u222b\n =\n2\n0\nt dt\n\u222b\n=\n2\n3\n3\n3\n2\n2\n2\n0\n2\n2 2 \u2013 0\n3\n3\n\uf8eet\n\uf8f9\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n = \n2\n4\n2\n(2\n2)\n3\n3\n=\nExample 29 Evaluate \n\u2013 1\n1\n2\n0\ntan\n1\nx dx\n+x\n\u222b\nSolution Let t = tan \u2013 1x, then \n12\n1\ndt\ndx\nx\n=\n+"
    },
    {
        "Chapter": "1",
        "sentence_range": "3942-3945",
        "Text": "340\nMATHEMATICS\nLet\nt = x5 + 1 Then dt = 5 x4 dx Note that, when\nx = \u2013 1, t = 0 and when x = 1, t = 2\nThus,  as x varies from \u2013 1 to 1, t varies from 0 to 2\nTherefore\n1\n4\n5\n15\n1\nx\nx\ndx\n\u2212\n+\n\u222b\n =\n2\n0\nt dt\n\u222b\n=\n2\n3\n3\n3\n2\n2\n2\n0\n2\n2 2 \u2013 0\n3\n3\n\uf8eet\n\uf8f9\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n = \n2\n4\n2\n(2\n2)\n3\n3\n=\nExample 29 Evaluate \n\u2013 1\n1\n2\n0\ntan\n1\nx dx\n+x\n\u222b\nSolution Let t = tan \u2013 1x, then \n12\n1\ndt\ndx\nx\n=\n+ The new limits are, when x = 0, t = 0 and\nwhen x = 1, \n4\nt\n=\u03c0"
    },
    {
        "Chapter": "1",
        "sentence_range": "3943-3946",
        "Text": "Then dt = 5 x4 dx Note that, when\nx = \u2013 1, t = 0 and when x = 1, t = 2\nThus,  as x varies from \u2013 1 to 1, t varies from 0 to 2\nTherefore\n1\n4\n5\n15\n1\nx\nx\ndx\n\u2212\n+\n\u222b\n =\n2\n0\nt dt\n\u222b\n=\n2\n3\n3\n3\n2\n2\n2\n0\n2\n2 2 \u2013 0\n3\n3\n\uf8eet\n\uf8f9\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n = \n2\n4\n2\n(2\n2)\n3\n3\n=\nExample 29 Evaluate \n\u2013 1\n1\n2\n0\ntan\n1\nx dx\n+x\n\u222b\nSolution Let t = tan \u2013 1x, then \n12\n1\ndt\ndx\nx\n=\n+ The new limits are, when x = 0, t = 0 and\nwhen x = 1, \n4\nt\n=\u03c0 Thus, as x varies from 0 to 1, t varies from 0 to \n4\n\u03c0"
    },
    {
        "Chapter": "1",
        "sentence_range": "3944-3947",
        "Text": "Note that, when\nx = \u2013 1, t = 0 and when x = 1, t = 2\nThus,  as x varies from \u2013 1 to 1, t varies from 0 to 2\nTherefore\n1\n4\n5\n15\n1\nx\nx\ndx\n\u2212\n+\n\u222b\n =\n2\n0\nt dt\n\u222b\n=\n2\n3\n3\n3\n2\n2\n2\n0\n2\n2 2 \u2013 0\n3\n3\n\uf8eet\n\uf8f9\n\uf8ee\n\uf8f9\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n = \n2\n4\n2\n(2\n2)\n3\n3\n=\nExample 29 Evaluate \n\u2013 1\n1\n2\n0\ntan\n1\nx dx\n+x\n\u222b\nSolution Let t = tan \u2013 1x, then \n12\n1\ndt\ndx\nx\n=\n+ The new limits are, when x = 0, t = 0 and\nwhen x = 1, \n4\nt\n=\u03c0 Thus, as x varies from 0 to 1, t varies from 0 to \n4\n\u03c0 Therefore\n\u20131\n1\n2\n0\ntan\n1\nx dx\n+x\n\u222b\n=\n2\n4\n4\n0\nt20\nt dt\n\u03c0\n\u03c0\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\u222b\n = \n2\n2\n1\n\u2013 0\n2 16\n32\n\uf8ee\n\uf8f9\n\u03c0\n=\u03c0\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nEXERCISE 7"
    },
    {
        "Chapter": "1",
        "sentence_range": "3945-3948",
        "Text": "The new limits are, when x = 0, t = 0 and\nwhen x = 1, \n4\nt\n=\u03c0 Thus, as x varies from 0 to 1, t varies from 0 to \n4\n\u03c0 Therefore\n\u20131\n1\n2\n0\ntan\n1\nx dx\n+x\n\u222b\n=\n2\n4\n4\n0\nt20\nt dt\n\u03c0\n\u03c0\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\u222b\n = \n2\n2\n1\n\u2013 0\n2 16\n32\n\uf8ee\n\uf8f9\n\u03c0\n=\u03c0\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nEXERCISE 7 10\nEvaluate the integrals in Exercises 1 to 8 using substitution"
    },
    {
        "Chapter": "1",
        "sentence_range": "3946-3949",
        "Text": "Thus, as x varies from 0 to 1, t varies from 0 to \n4\n\u03c0 Therefore\n\u20131\n1\n2\n0\ntan\n1\nx dx\n+x\n\u222b\n=\n2\n4\n4\n0\nt20\nt dt\n\u03c0\n\u03c0\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\u222b\n = \n2\n2\n1\n\u2013 0\n2 16\n32\n\uf8ee\n\uf8f9\n\u03c0\n=\u03c0\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nEXERCISE 7 10\nEvaluate the integrals in Exercises 1 to 8 using substitution 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "3947-3950",
        "Text": "Therefore\n\u20131\n1\n2\n0\ntan\n1\nx dx\n+x\n\u222b\n=\n2\n4\n4\n0\nt20\nt dt\n\u03c0\n\u03c0\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\u222b\n = \n2\n2\n1\n\u2013 0\n2 16\n32\n\uf8ee\n\uf8f9\n\u03c0\n=\u03c0\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nEXERCISE 7 10\nEvaluate the integrals in Exercises 1 to 8 using substitution 1 1\n2\n0\n1\nx\ndx\n\u222bx +\n2"
    },
    {
        "Chapter": "1",
        "sentence_range": "3948-3951",
        "Text": "10\nEvaluate the integrals in Exercises 1 to 8 using substitution 1 1\n2\n0\n1\nx\ndx\n\u222bx +\n2 5\n02\nsin\ncos\nd\n\u03c0\n\u03c6\n\u03c6\n\u03c6\n\u222b\n3"
    },
    {
        "Chapter": "1",
        "sentence_range": "3949-3952",
        "Text": "1 1\n2\n0\n1\nx\ndx\n\u222bx +\n2 5\n02\nsin\ncos\nd\n\u03c0\n\u03c6\n\u03c6\n\u03c6\n\u222b\n3 1\n\u2013 1\n2\n0\n2\nsin\n1\nx\ndx\nx\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed+\n\uf8f8\n\u222b\n4"
    },
    {
        "Chapter": "1",
        "sentence_range": "3950-3953",
        "Text": "1\n2\n0\n1\nx\ndx\n\u222bx +\n2 5\n02\nsin\ncos\nd\n\u03c0\n\u03c6\n\u03c6\n\u03c6\n\u222b\n3 1\n\u2013 1\n2\n0\n2\nsin\n1\nx\ndx\nx\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed+\n\uf8f8\n\u222b\n4 2\n0\n2\nx\nx +\n\u222b\n (Put x + 2 = t2)\n5"
    },
    {
        "Chapter": "1",
        "sentence_range": "3951-3954",
        "Text": "5\n02\nsin\ncos\nd\n\u03c0\n\u03c6\n\u03c6\n\u03c6\n\u222b\n3 1\n\u2013 1\n2\n0\n2\nsin\n1\nx\ndx\nx\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed+\n\uf8f8\n\u222b\n4 2\n0\n2\nx\nx +\n\u222b\n (Put x + 2 = t2)\n5 2\n2\n0\n1sin\ncos\nx\ndx\nx\n\u03c0\n+\n\u222b\n6"
    },
    {
        "Chapter": "1",
        "sentence_range": "3952-3955",
        "Text": "1\n\u2013 1\n2\n0\n2\nsin\n1\nx\ndx\nx\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed+\n\uf8f8\n\u222b\n4 2\n0\n2\nx\nx +\n\u222b\n (Put x + 2 = t2)\n5 2\n2\n0\n1sin\ncos\nx\ndx\nx\n\u03c0\n+\n\u222b\n6 2\n2\n0\n4 \u2013\ndx\nx\nx\n+\n\u222b\n7"
    },
    {
        "Chapter": "1",
        "sentence_range": "3953-3956",
        "Text": "2\n0\n2\nx\nx +\n\u222b\n (Put x + 2 = t2)\n5 2\n2\n0\n1sin\ncos\nx\ndx\nx\n\u03c0\n+\n\u222b\n6 2\n2\n0\n4 \u2013\ndx\nx\nx\n+\n\u222b\n7 1\n2\n1\n2\n5\ndx\nx\nx\n\u2212\n+\n+\n\u222b\n8"
    },
    {
        "Chapter": "1",
        "sentence_range": "3954-3957",
        "Text": "2\n2\n0\n1sin\ncos\nx\ndx\nx\n\u03c0\n+\n\u222b\n6 2\n2\n0\n4 \u2013\ndx\nx\nx\n+\n\u222b\n7 1\n2\n1\n2\n5\ndx\nx\nx\n\u2212\n+\n+\n\u222b\n8 2\n2\n2\n1\n1\n\u20131\n2\nx\ne dx\nx\nx\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\u222b\nChoose the correct answer in Exercises 9 and 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "3955-3958",
        "Text": "2\n2\n0\n4 \u2013\ndx\nx\nx\n+\n\u222b\n7 1\n2\n1\n2\n5\ndx\nx\nx\n\u2212\n+\n+\n\u222b\n8 2\n2\n2\n1\n1\n\u20131\n2\nx\ne dx\nx\nx\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\u222b\nChoose the correct answer in Exercises 9 and 10 9"
    },
    {
        "Chapter": "1",
        "sentence_range": "3956-3959",
        "Text": "1\n2\n1\n2\n5\ndx\nx\nx\n\u2212\n+\n+\n\u222b\n8 2\n2\n2\n1\n1\n\u20131\n2\nx\ne dx\nx\nx\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\u222b\nChoose the correct answer in Exercises 9 and 10 9 The value of the integral \n3 31\n1\n1\n4\n3\n(\n)\nx\nx\ndx\n\u2212x\n\u222b\n is\n(A) 6\n(B) 0\n(C) 3\n(D) 4\n10"
    },
    {
        "Chapter": "1",
        "sentence_range": "3957-3960",
        "Text": "2\n2\n2\n1\n1\n\u20131\n2\nx\ne dx\nx\nx\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\u222b\nChoose the correct answer in Exercises 9 and 10 9 The value of the integral \n3 31\n1\n1\n4\n3\n(\n)\nx\nx\ndx\n\u2212x\n\u222b\n is\n(A) 6\n(B) 0\n(C) 3\n(D) 4\n10 If f (x) = \n0\nx tsin\nt dt\n\u222b\n, then f\u2032(x) is\n(A) cosx + x sin x\n(B) x sinx\n(C) x cosx\n(D) sinx + x cosx\nINTEGRALS         341\n7"
    },
    {
        "Chapter": "1",
        "sentence_range": "3958-3961",
        "Text": "9 The value of the integral \n3 31\n1\n1\n4\n3\n(\n)\nx\nx\ndx\n\u2212x\n\u222b\n is\n(A) 6\n(B) 0\n(C) 3\n(D) 4\n10 If f (x) = \n0\nx tsin\nt dt\n\u222b\n, then f\u2032(x) is\n(A) cosx + x sin x\n(B) x sinx\n(C) x cosx\n(D) sinx + x cosx\nINTEGRALS         341\n7 10  Some Properties of Definite Integrals\nWe list below some important properties of definite integrals"
    },
    {
        "Chapter": "1",
        "sentence_range": "3959-3962",
        "Text": "The value of the integral \n3 31\n1\n1\n4\n3\n(\n)\nx\nx\ndx\n\u2212x\n\u222b\n is\n(A) 6\n(B) 0\n(C) 3\n(D) 4\n10 If f (x) = \n0\nx tsin\nt dt\n\u222b\n, then f\u2032(x) is\n(A) cosx + x sin x\n(B) x sinx\n(C) x cosx\n(D) sinx + x cosx\nINTEGRALS         341\n7 10  Some Properties of Definite Integrals\nWe list below some important properties of definite integrals These will be useful in\nevaluating the definite integrals more easily"
    },
    {
        "Chapter": "1",
        "sentence_range": "3960-3963",
        "Text": "If f (x) = \n0\nx tsin\nt dt\n\u222b\n, then f\u2032(x) is\n(A) cosx + x sin x\n(B) x sinx\n(C) x cosx\n(D) sinx + x cosx\nINTEGRALS         341\n7 10  Some Properties of Definite Integrals\nWe list below some important properties of definite integrals These will be useful in\nevaluating the definite integrals more easily P0 :\n( )\n( )\nb\nb\na\na\nf x dx\nf t dt\n=\n\u222b\n\u222b\nP1 :\n( )\n\u2013\n( )\nb\na\na\nb\nf x dx\nf x dx\n=\n\u222b\n\u222b"
    },
    {
        "Chapter": "1",
        "sentence_range": "3961-3964",
        "Text": "10  Some Properties of Definite Integrals\nWe list below some important properties of definite integrals These will be useful in\nevaluating the definite integrals more easily P0 :\n( )\n( )\nb\nb\na\na\nf x dx\nf t dt\n=\n\u222b\n\u222b\nP1 :\n( )\n\u2013\n( )\nb\na\na\nb\nf x dx\nf x dx\n=\n\u222b\n\u222b In particular, \n( )\n0\na\na f x dx =\n\u222b\nP2 :\n( )\n( )\n( )\nb\nc\nb\na\na\nc\nf x dx\nf x dx\nf x dx\n=\n+\n\u222b\n\u222b\n\u222b\nP3 :\n( )\n(\n)\nb\nb\na\na\nf x dx\nf a\nb\nx dx\n=\n+\n\u2212\n\u222b\n\u222b\nP4 :\n0\n0\n( )\n(\n)\na\na\nf x dx\nf a\nx dx\n=\n\u2212\n\u222b\n\u222b\n(Note that P4 is a particular case of P3)\nP5 :\n2\n0\n0\n0\n( )\n( )\n(2\n)\na\na\na\nf x dx\nf x dx\nf\na\nx dx\n=\n+\n\u2212\n\u222b\n\u222b\n\u222b\nP6 :\n2\n0\n0\n( )\n2\n( )\n, if\n(2\n)\n( )\na\na\nf x dx\nf x dx\nf\na\nx\nf x\n=\n\u2212\n=\n\u222b\n\u222b\n  and\n                 0 if f (2a \u2013 x) = \u2013 f (x)\nP7 :\n(i)  \n0\n( )\n2\n( )\na\na\na f x dx\nf x dx\n\u2212\n=\n\u222b\n\u222b\n, if f is an even function, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "3962-3965",
        "Text": "These will be useful in\nevaluating the definite integrals more easily P0 :\n( )\n( )\nb\nb\na\na\nf x dx\nf t dt\n=\n\u222b\n\u222b\nP1 :\n( )\n\u2013\n( )\nb\na\na\nb\nf x dx\nf x dx\n=\n\u222b\n\u222b In particular, \n( )\n0\na\na f x dx =\n\u222b\nP2 :\n( )\n( )\n( )\nb\nc\nb\na\na\nc\nf x dx\nf x dx\nf x dx\n=\n+\n\u222b\n\u222b\n\u222b\nP3 :\n( )\n(\n)\nb\nb\na\na\nf x dx\nf a\nb\nx dx\n=\n+\n\u2212\n\u222b\n\u222b\nP4 :\n0\n0\n( )\n(\n)\na\na\nf x dx\nf a\nx dx\n=\n\u2212\n\u222b\n\u222b\n(Note that P4 is a particular case of P3)\nP5 :\n2\n0\n0\n0\n( )\n( )\n(2\n)\na\na\na\nf x dx\nf x dx\nf\na\nx dx\n=\n+\n\u2212\n\u222b\n\u222b\n\u222b\nP6 :\n2\n0\n0\n( )\n2\n( )\n, if\n(2\n)\n( )\na\na\nf x dx\nf x dx\nf\na\nx\nf x\n=\n\u2212\n=\n\u222b\n\u222b\n  and\n                 0 if f (2a \u2013 x) = \u2013 f (x)\nP7 :\n(i)  \n0\n( )\n2\n( )\na\na\na f x dx\nf x dx\n\u2212\n=\n\u222b\n\u222b\n, if f is an even function, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "3963-3966",
        "Text": "P0 :\n( )\n( )\nb\nb\na\na\nf x dx\nf t dt\n=\n\u222b\n\u222b\nP1 :\n( )\n\u2013\n( )\nb\na\na\nb\nf x dx\nf x dx\n=\n\u222b\n\u222b In particular, \n( )\n0\na\na f x dx =\n\u222b\nP2 :\n( )\n( )\n( )\nb\nc\nb\na\na\nc\nf x dx\nf x dx\nf x dx\n=\n+\n\u222b\n\u222b\n\u222b\nP3 :\n( )\n(\n)\nb\nb\na\na\nf x dx\nf a\nb\nx dx\n=\n+\n\u2212\n\u222b\n\u222b\nP4 :\n0\n0\n( )\n(\n)\na\na\nf x dx\nf a\nx dx\n=\n\u2212\n\u222b\n\u222b\n(Note that P4 is a particular case of P3)\nP5 :\n2\n0\n0\n0\n( )\n( )\n(2\n)\na\na\na\nf x dx\nf x dx\nf\na\nx dx\n=\n+\n\u2212\n\u222b\n\u222b\n\u222b\nP6 :\n2\n0\n0\n( )\n2\n( )\n, if\n(2\n)\n( )\na\na\nf x dx\nf x dx\nf\na\nx\nf x\n=\n\u2212\n=\n\u222b\n\u222b\n  and\n                 0 if f (2a \u2013 x) = \u2013 f (x)\nP7 :\n(i)  \n0\n( )\n2\n( )\na\na\na f x dx\nf x dx\n\u2212\n=\n\u222b\n\u222b\n, if f is an even function, i e , if f (\u2013 x) = f (x)"
    },
    {
        "Chapter": "1",
        "sentence_range": "3964-3967",
        "Text": "In particular, \n( )\n0\na\na f x dx =\n\u222b\nP2 :\n( )\n( )\n( )\nb\nc\nb\na\na\nc\nf x dx\nf x dx\nf x dx\n=\n+\n\u222b\n\u222b\n\u222b\nP3 :\n( )\n(\n)\nb\nb\na\na\nf x dx\nf a\nb\nx dx\n=\n+\n\u2212\n\u222b\n\u222b\nP4 :\n0\n0\n( )\n(\n)\na\na\nf x dx\nf a\nx dx\n=\n\u2212\n\u222b\n\u222b\n(Note that P4 is a particular case of P3)\nP5 :\n2\n0\n0\n0\n( )\n( )\n(2\n)\na\na\na\nf x dx\nf x dx\nf\na\nx dx\n=\n+\n\u2212\n\u222b\n\u222b\n\u222b\nP6 :\n2\n0\n0\n( )\n2\n( )\n, if\n(2\n)\n( )\na\na\nf x dx\nf x dx\nf\na\nx\nf x\n=\n\u2212\n=\n\u222b\n\u222b\n  and\n                 0 if f (2a \u2013 x) = \u2013 f (x)\nP7 :\n(i)  \n0\n( )\n2\n( )\na\na\na f x dx\nf x dx\n\u2212\n=\n\u222b\n\u222b\n, if f is an even function, i e , if f (\u2013 x) = f (x) (ii)  \n( )\n0\na\n\u2212a f x dx\n=\n\u222b\n, if f is an odd function, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "3965-3968",
        "Text": "e , if f (\u2013 x) = f (x) (ii)  \n( )\n0\na\n\u2212a f x dx\n=\n\u222b\n, if f is an odd function, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "3966-3969",
        "Text": ", if f (\u2013 x) = f (x) (ii)  \n( )\n0\na\n\u2212a f x dx\n=\n\u222b\n, if f is an odd function, i e , if f (\u2013 x) = \u2013 f (x)"
    },
    {
        "Chapter": "1",
        "sentence_range": "3967-3970",
        "Text": "(ii)  \n( )\n0\na\n\u2212a f x dx\n=\n\u222b\n, if f is an odd function, i e , if f (\u2013 x) = \u2013 f (x) We give the proofs of these properties one by one"
    },
    {
        "Chapter": "1",
        "sentence_range": "3968-3971",
        "Text": "e , if f (\u2013 x) = \u2013 f (x) We give the proofs of these properties one by one Proof of P0 It follows directly by making the substitution x = t"
    },
    {
        "Chapter": "1",
        "sentence_range": "3969-3972",
        "Text": ", if f (\u2013 x) = \u2013 f (x) We give the proofs of these properties one by one Proof of P0 It follows directly by making the substitution x = t Proof of P1 Let F be anti derivative of f"
    },
    {
        "Chapter": "1",
        "sentence_range": "3970-3973",
        "Text": "We give the proofs of these properties one by one Proof of P0 It follows directly by making the substitution x = t Proof of P1 Let F be anti derivative of f Then, by the second fundamental theorem of\ncalculus, we have \n( )\nF ( ) \u2013 F ( )\n\u2013[F ( )\nF ( )]\n( )\nb\na\na\nb\nf x dx\nb\na\na\nb\nf x dx\n=\n=\n\u2212\n= \u2212\n\u222b\n\u222b\nHere, we observe that, if a = b, then \n( )\n0\na\n\u222ba f x dx ="
    },
    {
        "Chapter": "1",
        "sentence_range": "3971-3974",
        "Text": "Proof of P0 It follows directly by making the substitution x = t Proof of P1 Let F be anti derivative of f Then, by the second fundamental theorem of\ncalculus, we have \n( )\nF ( ) \u2013 F ( )\n\u2013[F ( )\nF ( )]\n( )\nb\na\na\nb\nf x dx\nb\na\na\nb\nf x dx\n=\n=\n\u2212\n= \u2212\n\u222b\n\u222b\nHere, we observe that, if a = b, then \n( )\n0\na\n\u222ba f x dx = Proof of P2 Let F be anti derivative of f"
    },
    {
        "Chapter": "1",
        "sentence_range": "3972-3975",
        "Text": "Proof of P1 Let F be anti derivative of f Then, by the second fundamental theorem of\ncalculus, we have \n( )\nF ( ) \u2013 F ( )\n\u2013[F ( )\nF ( )]\n( )\nb\na\na\nb\nf x dx\nb\na\na\nb\nf x dx\n=\n=\n\u2212\n= \u2212\n\u222b\n\u222b\nHere, we observe that, if a = b, then \n( )\n0\na\n\u222ba f x dx = Proof of P2 Let F be anti derivative of f Then\n( )\nb\n\u222ba f x dx\n = F(b) \u2013 F(a)"
    },
    {
        "Chapter": "1",
        "sentence_range": "3973-3976",
        "Text": "Then, by the second fundamental theorem of\ncalculus, we have \n( )\nF ( ) \u2013 F ( )\n\u2013[F ( )\nF ( )]\n( )\nb\na\na\nb\nf x dx\nb\na\na\nb\nf x dx\n=\n=\n\u2212\n= \u2212\n\u222b\n\u222b\nHere, we observe that, if a = b, then \n( )\n0\na\n\u222ba f x dx = Proof of P2 Let F be anti derivative of f Then\n( )\nb\n\u222ba f x dx\n = F(b) \u2013 F(a) (1)\n( )\nc\n\u222ba f x dx\n = F(c) \u2013 F(a)"
    },
    {
        "Chapter": "1",
        "sentence_range": "3974-3977",
        "Text": "Proof of P2 Let F be anti derivative of f Then\n( )\nb\n\u222ba f x dx\n = F(b) \u2013 F(a) (1)\n( )\nc\n\u222ba f x dx\n = F(c) \u2013 F(a) (2)\nand\n( )\nb\n\u222bc f x dx\n = F(b) \u2013 F(c)"
    },
    {
        "Chapter": "1",
        "sentence_range": "3975-3978",
        "Text": "Then\n( )\nb\n\u222ba f x dx\n = F(b) \u2013 F(a) (1)\n( )\nc\n\u222ba f x dx\n = F(c) \u2013 F(a) (2)\nand\n( )\nb\n\u222bc f x dx\n = F(b) \u2013 F(c) (3)\n342\nMATHEMATICS\nAdding (2) and (3), we get \n( )\n( )\nF( ) \u2013 F( )\n( )\nc\nb\nb\na\nc\na\nf x dx\nf x dx\nb\na\nf x dx\n+\n=\n=\n\u222b\n\u222b\n\u222b\nThis proves the property P2"
    },
    {
        "Chapter": "1",
        "sentence_range": "3976-3979",
        "Text": "(1)\n( )\nc\n\u222ba f x dx\n = F(c) \u2013 F(a) (2)\nand\n( )\nb\n\u222bc f x dx\n = F(b) \u2013 F(c) (3)\n342\nMATHEMATICS\nAdding (2) and (3), we get \n( )\n( )\nF( ) \u2013 F( )\n( )\nc\nb\nb\na\nc\na\nf x dx\nf x dx\nb\na\nf x dx\n+\n=\n=\n\u222b\n\u222b\n\u222b\nThis proves the property P2 Proof of P3  Let t = a + b \u2013 x"
    },
    {
        "Chapter": "1",
        "sentence_range": "3977-3980",
        "Text": "(2)\nand\n( )\nb\n\u222bc f x dx\n = F(b) \u2013 F(c) (3)\n342\nMATHEMATICS\nAdding (2) and (3), we get \n( )\n( )\nF( ) \u2013 F( )\n( )\nc\nb\nb\na\nc\na\nf x dx\nf x dx\nb\na\nf x dx\n+\n=\n=\n\u222b\n\u222b\n\u222b\nThis proves the property P2 Proof of P3  Let t = a + b \u2013 x Then dt = \u2013 dx"
    },
    {
        "Chapter": "1",
        "sentence_range": "3978-3981",
        "Text": "(3)\n342\nMATHEMATICS\nAdding (2) and (3), we get \n( )\n( )\nF( ) \u2013 F( )\n( )\nc\nb\nb\na\nc\na\nf x dx\nf x dx\nb\na\nf x dx\n+\n=\n=\n\u222b\n\u222b\n\u222b\nThis proves the property P2 Proof of P3  Let t = a + b \u2013 x Then dt = \u2013 dx When x = a, t = b and when x = b, t = a"
    },
    {
        "Chapter": "1",
        "sentence_range": "3979-3982",
        "Text": "Proof of P3  Let t = a + b \u2013 x Then dt = \u2013 dx When x = a, t = b and when x = b, t = a Therefore\n( )\nb\n\u222ba f x dx\n =\n(\n\u2013 )\na\nb f a\nb\nt dt\n\u2212\n+\n\u222b\n=\n(\n\u2013 )\nb\na f a\nb\nt dt\n+\n\u222b\n (by P1)\n=\n(\n\u2013 )\nb\na f a\nb\nx\n+\n\u222b\ndx by P0\nProof of P4 Put t = a \u2013 x"
    },
    {
        "Chapter": "1",
        "sentence_range": "3980-3983",
        "Text": "Then dt = \u2013 dx When x = a, t = b and when x = b, t = a Therefore\n( )\nb\n\u222ba f x dx\n =\n(\n\u2013 )\na\nb f a\nb\nt dt\n\u2212\n+\n\u222b\n=\n(\n\u2013 )\nb\na f a\nb\nt dt\n+\n\u222b\n (by P1)\n=\n(\n\u2013 )\nb\na f a\nb\nx\n+\n\u222b\ndx by P0\nProof of P4 Put t = a \u2013 x Then dt = \u2013 dx"
    },
    {
        "Chapter": "1",
        "sentence_range": "3981-3984",
        "Text": "When x = a, t = b and when x = b, t = a Therefore\n( )\nb\n\u222ba f x dx\n =\n(\n\u2013 )\na\nb f a\nb\nt dt\n\u2212\n+\n\u222b\n=\n(\n\u2013 )\nb\na f a\nb\nt dt\n+\n\u222b\n (by P1)\n=\n(\n\u2013 )\nb\na f a\nb\nx\n+\n\u222b\ndx by P0\nProof of P4 Put t = a \u2013 x Then dt = \u2013 dx When x = 0, t = a and when x = a, t = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "3982-3985",
        "Text": "Therefore\n( )\nb\n\u222ba f x dx\n =\n(\n\u2013 )\na\nb f a\nb\nt dt\n\u2212\n+\n\u222b\n=\n(\n\u2013 )\nb\na f a\nb\nt dt\n+\n\u222b\n (by P1)\n=\n(\n\u2013 )\nb\na f a\nb\nx\n+\n\u222b\ndx by P0\nProof of P4 Put t = a \u2013 x Then dt = \u2013 dx When x = 0, t = a and when x = a, t = 0 Now\nproceed as in P3"
    },
    {
        "Chapter": "1",
        "sentence_range": "3983-3986",
        "Text": "Then dt = \u2013 dx When x = 0, t = a and when x = a, t = 0 Now\nproceed as in P3 Proof of P5 Using P2, we have \n2\n2\n0\n0\n( )\n( )\n( )\na\na\na\na\nf x dx\nf x dx\nf x dx\n=\n+\n\u222b\n\u222b\n\u222b"
    },
    {
        "Chapter": "1",
        "sentence_range": "3984-3987",
        "Text": "When x = 0, t = a and when x = a, t = 0 Now\nproceed as in P3 Proof of P5 Using P2, we have \n2\n2\n0\n0\n( )\n( )\n( )\na\na\na\na\nf x dx\nf x dx\nf x dx\n=\n+\n\u222b\n\u222b\n\u222b Let\nt = 2a \u2013 x in the second integral on the right hand side"
    },
    {
        "Chapter": "1",
        "sentence_range": "3985-3988",
        "Text": "Now\nproceed as in P3 Proof of P5 Using P2, we have \n2\n2\n0\n0\n( )\n( )\n( )\na\na\na\na\nf x dx\nf x dx\nf x dx\n=\n+\n\u222b\n\u222b\n\u222b Let\nt = 2a \u2013 x in the second integral on the right hand side Then\ndt = \u2013 dx"
    },
    {
        "Chapter": "1",
        "sentence_range": "3986-3989",
        "Text": "Proof of P5 Using P2, we have \n2\n2\n0\n0\n( )\n( )\n( )\na\na\na\na\nf x dx\nf x dx\nf x dx\n=\n+\n\u222b\n\u222b\n\u222b Let\nt = 2a \u2013 x in the second integral on the right hand side Then\ndt = \u2013 dx When x = a, t = a and when x = 2a, t = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "3987-3990",
        "Text": "Let\nt = 2a \u2013 x in the second integral on the right hand side Then\ndt = \u2013 dx When x = a, t = a and when x = 2a, t = 0 Also x = 2a \u2013 t"
    },
    {
        "Chapter": "1",
        "sentence_range": "3988-3991",
        "Text": "Then\ndt = \u2013 dx When x = a, t = a and when x = 2a, t = 0 Also x = 2a \u2013 t Therefore, the second integral becomes\n2\n( )\na\na\nf x dx\n\u222b\n =\n\u20130\n(2 \u2013 )\na f\na\nt dt\n\u222b\n = \n0\n(2 \u2013 )\na f\na\nt dt\n\u222b\n = \n0\n(2 \u2013 )\na f\na\nx dx\n\u222b\nHence\n2\n0\n( )\n\u222ba f x dx\n =\n0\n0\n( )\n(2\n)\na\na\nf x dx\nf\na\nx dx\n+\n\u2212\n\u222b\n\u222b\nProof of P6 Using P5, we have \n2\n0\n0\n0\n( )\n( )\n(2\n)\na\na\na\nf x dx\nf x dx\nf\na\nx dx\n=\n+\n\u2212\n\u222b\n\u222b\n\u222b"
    },
    {
        "Chapter": "1",
        "sentence_range": "3989-3992",
        "Text": "When x = a, t = a and when x = 2a, t = 0 Also x = 2a \u2013 t Therefore, the second integral becomes\n2\n( )\na\na\nf x dx\n\u222b\n =\n\u20130\n(2 \u2013 )\na f\na\nt dt\n\u222b\n = \n0\n(2 \u2013 )\na f\na\nt dt\n\u222b\n = \n0\n(2 \u2013 )\na f\na\nx dx\n\u222b\nHence\n2\n0\n( )\n\u222ba f x dx\n =\n0\n0\n( )\n(2\n)\na\na\nf x dx\nf\na\nx dx\n+\n\u2212\n\u222b\n\u222b\nProof of P6 Using P5, we have \n2\n0\n0\n0\n( )\n( )\n(2\n)\na\na\na\nf x dx\nf x dx\nf\na\nx dx\n=\n+\n\u2212\n\u222b\n\u222b\n\u222b (1)\nNow, if\nf (2a \u2013 x) = f (x), then (1) becomes\n2\n0\n( )\n\u222ba f x dx\n =\n0\n0\n0\n( )\n( )\n2\n( )\n,\na\na\na\nf x dx\nf x dx\nf x dx\n+\n=\n\u222b\n\u222b\n\u222b\nand if\nf(2a \u2013 x) = \u2013 f (x), then (1) becomes\n2\n0\n( )\n\u222ba f x dx\n =  \n0\n0\n( )\n( )\n0\na\na\nf x dx\nf x dx\n\u2212\n=\n\u222b\n\u222b\nProof of P7 Using P2, we have\n( )\na\na f x dx\n\u2212\u222b\n =\n0\n0\n( )\n( )\na\na f x dx\nf x dx\n\u2212\n+\n\u222b\n\u222b"
    },
    {
        "Chapter": "1",
        "sentence_range": "3990-3993",
        "Text": "Also x = 2a \u2013 t Therefore, the second integral becomes\n2\n( )\na\na\nf x dx\n\u222b\n =\n\u20130\n(2 \u2013 )\na f\na\nt dt\n\u222b\n = \n0\n(2 \u2013 )\na f\na\nt dt\n\u222b\n = \n0\n(2 \u2013 )\na f\na\nx dx\n\u222b\nHence\n2\n0\n( )\n\u222ba f x dx\n =\n0\n0\n( )\n(2\n)\na\na\nf x dx\nf\na\nx dx\n+\n\u2212\n\u222b\n\u222b\nProof of P6 Using P5, we have \n2\n0\n0\n0\n( )\n( )\n(2\n)\na\na\na\nf x dx\nf x dx\nf\na\nx dx\n=\n+\n\u2212\n\u222b\n\u222b\n\u222b (1)\nNow, if\nf (2a \u2013 x) = f (x), then (1) becomes\n2\n0\n( )\n\u222ba f x dx\n =\n0\n0\n0\n( )\n( )\n2\n( )\n,\na\na\na\nf x dx\nf x dx\nf x dx\n+\n=\n\u222b\n\u222b\n\u222b\nand if\nf(2a \u2013 x) = \u2013 f (x), then (1) becomes\n2\n0\n( )\n\u222ba f x dx\n =  \n0\n0\n( )\n( )\n0\na\na\nf x dx\nf x dx\n\u2212\n=\n\u222b\n\u222b\nProof of P7 Using P2, we have\n( )\na\na f x dx\n\u2212\u222b\n =\n0\n0\n( )\n( )\na\na f x dx\nf x dx\n\u2212\n+\n\u222b\n\u222b Then\nLet\nt = \u2013 x in the first integral on the right hand side"
    },
    {
        "Chapter": "1",
        "sentence_range": "3991-3994",
        "Text": "Therefore, the second integral becomes\n2\n( )\na\na\nf x dx\n\u222b\n =\n\u20130\n(2 \u2013 )\na f\na\nt dt\n\u222b\n = \n0\n(2 \u2013 )\na f\na\nt dt\n\u222b\n = \n0\n(2 \u2013 )\na f\na\nx dx\n\u222b\nHence\n2\n0\n( )\n\u222ba f x dx\n =\n0\n0\n( )\n(2\n)\na\na\nf x dx\nf\na\nx dx\n+\n\u2212\n\u222b\n\u222b\nProof of P6 Using P5, we have \n2\n0\n0\n0\n( )\n( )\n(2\n)\na\na\na\nf x dx\nf x dx\nf\na\nx dx\n=\n+\n\u2212\n\u222b\n\u222b\n\u222b (1)\nNow, if\nf (2a \u2013 x) = f (x), then (1) becomes\n2\n0\n( )\n\u222ba f x dx\n =\n0\n0\n0\n( )\n( )\n2\n( )\n,\na\na\na\nf x dx\nf x dx\nf x dx\n+\n=\n\u222b\n\u222b\n\u222b\nand if\nf(2a \u2013 x) = \u2013 f (x), then (1) becomes\n2\n0\n( )\n\u222ba f x dx\n =  \n0\n0\n( )\n( )\n0\na\na\nf x dx\nf x dx\n\u2212\n=\n\u222b\n\u222b\nProof of P7 Using P2, we have\n( )\na\na f x dx\n\u2212\u222b\n =\n0\n0\n( )\n( )\na\na f x dx\nf x dx\n\u2212\n+\n\u222b\n\u222b Then\nLet\nt = \u2013 x in the first integral on the right hand side dt = \u2013 dx"
    },
    {
        "Chapter": "1",
        "sentence_range": "3992-3995",
        "Text": "(1)\nNow, if\nf (2a \u2013 x) = f (x), then (1) becomes\n2\n0\n( )\n\u222ba f x dx\n =\n0\n0\n0\n( )\n( )\n2\n( )\n,\na\na\na\nf x dx\nf x dx\nf x dx\n+\n=\n\u222b\n\u222b\n\u222b\nand if\nf(2a \u2013 x) = \u2013 f (x), then (1) becomes\n2\n0\n( )\n\u222ba f x dx\n =  \n0\n0\n( )\n( )\n0\na\na\nf x dx\nf x dx\n\u2212\n=\n\u222b\n\u222b\nProof of P7 Using P2, we have\n( )\na\na f x dx\n\u2212\u222b\n =\n0\n0\n( )\n( )\na\na f x dx\nf x dx\n\u2212\n+\n\u222b\n\u222b Then\nLet\nt = \u2013 x in the first integral on the right hand side dt = \u2013 dx When x = \u2013 a, t = a and when\nx = 0, t = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "3993-3996",
        "Text": "Then\nLet\nt = \u2013 x in the first integral on the right hand side dt = \u2013 dx When x = \u2013 a, t = a and when\nx = 0, t = 0 Also x = \u2013 t"
    },
    {
        "Chapter": "1",
        "sentence_range": "3994-3997",
        "Text": "dt = \u2013 dx When x = \u2013 a, t = a and when\nx = 0, t = 0 Also x = \u2013 t INTEGRALS         343\nTherefore\n( )\na\na f x dx\n\u2212\u222b\n =\n0\n0\n\u2013\n(\u2013 )\n( )\na\na f\nt dt\nf x dx\n+\n\u222b\n\u222b\n=\n0\n0\n(\u2013 )\n( )\na\na\nf\nx dx\nf x dx\n+\n\u222b\n\u222b\n       (by P0)"
    },
    {
        "Chapter": "1",
        "sentence_range": "3995-3998",
        "Text": "When x = \u2013 a, t = a and when\nx = 0, t = 0 Also x = \u2013 t INTEGRALS         343\nTherefore\n( )\na\na f x dx\n\u2212\u222b\n =\n0\n0\n\u2013\n(\u2013 )\n( )\na\na f\nt dt\nf x dx\n+\n\u222b\n\u222b\n=\n0\n0\n(\u2013 )\n( )\na\na\nf\nx dx\nf x dx\n+\n\u222b\n\u222b\n       (by P0) (1)\n(i) Now, if f is an even function, then f (\u2013x) = f (x) and so (1) becomes\n0\n0\n0\n( )\n( )\n( )\n2\n( )\na\na\na\na\na f x dx\nf x dx\nf x dx\nf x dx\n\u2212\n=\n+\n=\n\u222b\n\u222b\n\u222b\n\u222b\n(ii) If f is an odd function, then f (\u2013x) = \u2013 f(x) and so (1) becomes\n0\n0\n( )\n( )\n( )\n0\na\na\na\na f x dx\nf x dx\nf x dx\n\u2212\n= \u2212\n+\n=\n\u222b\n\u222b\n\u222b\nExample 30 Evaluate \n2\n3\n1\nx\u2013\nx dx\n\u2212\u222b\nSolution We note that x3 \u2013 x \u2265 0 on [\u2013 1, 0] and x3 \u2013 x \u2264 0 on [0, 1] and that\nx3 \u2013 x \u2265 0 on [1, 2]"
    },
    {
        "Chapter": "1",
        "sentence_range": "3996-3999",
        "Text": "Also x = \u2013 t INTEGRALS         343\nTherefore\n( )\na\na f x dx\n\u2212\u222b\n =\n0\n0\n\u2013\n(\u2013 )\n( )\na\na f\nt dt\nf x dx\n+\n\u222b\n\u222b\n=\n0\n0\n(\u2013 )\n( )\na\na\nf\nx dx\nf x dx\n+\n\u222b\n\u222b\n       (by P0) (1)\n(i) Now, if f is an even function, then f (\u2013x) = f (x) and so (1) becomes\n0\n0\n0\n( )\n( )\n( )\n2\n( )\na\na\na\na\na f x dx\nf x dx\nf x dx\nf x dx\n\u2212\n=\n+\n=\n\u222b\n\u222b\n\u222b\n\u222b\n(ii) If f is an odd function, then f (\u2013x) = \u2013 f(x) and so (1) becomes\n0\n0\n( )\n( )\n( )\n0\na\na\na\na f x dx\nf x dx\nf x dx\n\u2212\n= \u2212\n+\n=\n\u222b\n\u222b\n\u222b\nExample 30 Evaluate \n2\n3\n1\nx\u2013\nx dx\n\u2212\u222b\nSolution We note that x3 \u2013 x \u2265 0 on [\u2013 1, 0] and x3 \u2013 x \u2264 0 on [0, 1] and that\nx3 \u2013 x \u2265 0 on [1, 2] So by P2 we write\n2\n3\n1\nx\u2013\nx dx\n\u2212\u222b\n =\n0\n1\n2\n3\n3\n3\n1\n0\n1\n(\n\u2013 )\n\u2013 (\n\u2013 )\n(\n\u2013 )\nx\nx dx\nx\nx dx\nx\nx dx\n\u2212\n+\n+\n\u222b\n\u222b\n\u222b\n=\n0\n1\n2\n3\n3\n3\n1\n0\n1\n(\n\u2013 )\n( \u2013\n)\n(\n\u2013 )\nx\nx dx\nx\nx\ndx\nx\nx dx\n\u2212\n+\n+\n\u222b\n\u222b\n\u222b\n=\n0\n1\n2\n4\n2\n2\n4\n4\n2\n\u20131\n0\n1\n\u2013\n\u2013\n\u2013\n4\n2\n2\n4\n4\n2\nx\nx\nx\nx\nx\nx\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n+\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n=\n(\n)\n1\n1\n1\n1\n1\n1\n\u2013\n\u2013\n\u2013\n4 \u2013 2 \u2013\n\u2013\n4\n2\n2\n4\n4\n2\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n+\n+\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n=\n1\n1\n1\n1\n1\n1\n\u2013\n2\n4\n2\n2\n4\n4\n2\n+\n+\n\u2212\n+\n\u2212\n+\n = 3\n3\n11\n2\n2\n4\n4\n\u2212\n+\n=\nExample 31 Evaluate \n2\n\u20134\n4\nsin x dx\n\u03c0\n\u03c0\n\u222b\nSolution We observe that sin2 x is an even function"
    },
    {
        "Chapter": "1",
        "sentence_range": "3997-4000",
        "Text": "INTEGRALS         343\nTherefore\n( )\na\na f x dx\n\u2212\u222b\n =\n0\n0\n\u2013\n(\u2013 )\n( )\na\na f\nt dt\nf x dx\n+\n\u222b\n\u222b\n=\n0\n0\n(\u2013 )\n( )\na\na\nf\nx dx\nf x dx\n+\n\u222b\n\u222b\n       (by P0) (1)\n(i) Now, if f is an even function, then f (\u2013x) = f (x) and so (1) becomes\n0\n0\n0\n( )\n( )\n( )\n2\n( )\na\na\na\na\na f x dx\nf x dx\nf x dx\nf x dx\n\u2212\n=\n+\n=\n\u222b\n\u222b\n\u222b\n\u222b\n(ii) If f is an odd function, then f (\u2013x) = \u2013 f(x) and so (1) becomes\n0\n0\n( )\n( )\n( )\n0\na\na\na\na f x dx\nf x dx\nf x dx\n\u2212\n= \u2212\n+\n=\n\u222b\n\u222b\n\u222b\nExample 30 Evaluate \n2\n3\n1\nx\u2013\nx dx\n\u2212\u222b\nSolution We note that x3 \u2013 x \u2265 0 on [\u2013 1, 0] and x3 \u2013 x \u2264 0 on [0, 1] and that\nx3 \u2013 x \u2265 0 on [1, 2] So by P2 we write\n2\n3\n1\nx\u2013\nx dx\n\u2212\u222b\n =\n0\n1\n2\n3\n3\n3\n1\n0\n1\n(\n\u2013 )\n\u2013 (\n\u2013 )\n(\n\u2013 )\nx\nx dx\nx\nx dx\nx\nx dx\n\u2212\n+\n+\n\u222b\n\u222b\n\u222b\n=\n0\n1\n2\n3\n3\n3\n1\n0\n1\n(\n\u2013 )\n( \u2013\n)\n(\n\u2013 )\nx\nx dx\nx\nx\ndx\nx\nx dx\n\u2212\n+\n+\n\u222b\n\u222b\n\u222b\n=\n0\n1\n2\n4\n2\n2\n4\n4\n2\n\u20131\n0\n1\n\u2013\n\u2013\n\u2013\n4\n2\n2\n4\n4\n2\nx\nx\nx\nx\nx\nx\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n+\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n=\n(\n)\n1\n1\n1\n1\n1\n1\n\u2013\n\u2013\n\u2013\n4 \u2013 2 \u2013\n\u2013\n4\n2\n2\n4\n4\n2\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n+\n+\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n=\n1\n1\n1\n1\n1\n1\n\u2013\n2\n4\n2\n2\n4\n4\n2\n+\n+\n\u2212\n+\n\u2212\n+\n = 3\n3\n11\n2\n2\n4\n4\n\u2212\n+\n=\nExample 31 Evaluate \n2\n\u20134\n4\nsin x dx\n\u03c0\n\u03c0\n\u222b\nSolution We observe that sin2 x is an even function Therefore, by P7 (i), we get\n2\n\u20134\n4\nsin x dx\n\u03c0\n\u222b\u03c0\n =\n2\n204\nsin x dx\n\u222b\u03c0\n344\nMATHEMATICS\n=\n4\n0\n(1 cos 2 )\n2\n2\nx dx\n\u03c0\n\u2212\n\u222b\n = \n4\n0 (1\ncos 2 )x dx\n\u03c0\n\u2212\n\u222b\n=\n4\n0\n\u20131\nsin 2\n2\nx\nx\n\u03c0\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n = \n1\n1\n\u2013\nsin\n\u20130\n\u2013\n4\n2\n2\n4\n2\n\u03c0\n\u03c0\n\u03c0\n\uf8eb\n\uf8f6\n=\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\nExample 32 Evaluate \n2\n0\nsin\n1\nxcos\nx\ndx\nx\n\u03c0\n+\n\u222b\nSolution Let I = \n2\n0\n1sin\nxcos\nx\ndx\nx\n\u03c0\n+\n\u222b"
    },
    {
        "Chapter": "1",
        "sentence_range": "3998-4001",
        "Text": "(1)\n(i) Now, if f is an even function, then f (\u2013x) = f (x) and so (1) becomes\n0\n0\n0\n( )\n( )\n( )\n2\n( )\na\na\na\na\na f x dx\nf x dx\nf x dx\nf x dx\n\u2212\n=\n+\n=\n\u222b\n\u222b\n\u222b\n\u222b\n(ii) If f is an odd function, then f (\u2013x) = \u2013 f(x) and so (1) becomes\n0\n0\n( )\n( )\n( )\n0\na\na\na\na f x dx\nf x dx\nf x dx\n\u2212\n= \u2212\n+\n=\n\u222b\n\u222b\n\u222b\nExample 30 Evaluate \n2\n3\n1\nx\u2013\nx dx\n\u2212\u222b\nSolution We note that x3 \u2013 x \u2265 0 on [\u2013 1, 0] and x3 \u2013 x \u2264 0 on [0, 1] and that\nx3 \u2013 x \u2265 0 on [1, 2] So by P2 we write\n2\n3\n1\nx\u2013\nx dx\n\u2212\u222b\n =\n0\n1\n2\n3\n3\n3\n1\n0\n1\n(\n\u2013 )\n\u2013 (\n\u2013 )\n(\n\u2013 )\nx\nx dx\nx\nx dx\nx\nx dx\n\u2212\n+\n+\n\u222b\n\u222b\n\u222b\n=\n0\n1\n2\n3\n3\n3\n1\n0\n1\n(\n\u2013 )\n( \u2013\n)\n(\n\u2013 )\nx\nx dx\nx\nx\ndx\nx\nx dx\n\u2212\n+\n+\n\u222b\n\u222b\n\u222b\n=\n0\n1\n2\n4\n2\n2\n4\n4\n2\n\u20131\n0\n1\n\u2013\n\u2013\n\u2013\n4\n2\n2\n4\n4\n2\nx\nx\nx\nx\nx\nx\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n+\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n=\n(\n)\n1\n1\n1\n1\n1\n1\n\u2013\n\u2013\n\u2013\n4 \u2013 2 \u2013\n\u2013\n4\n2\n2\n4\n4\n2\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n+\n+\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n=\n1\n1\n1\n1\n1\n1\n\u2013\n2\n4\n2\n2\n4\n4\n2\n+\n+\n\u2212\n+\n\u2212\n+\n = 3\n3\n11\n2\n2\n4\n4\n\u2212\n+\n=\nExample 31 Evaluate \n2\n\u20134\n4\nsin x dx\n\u03c0\n\u03c0\n\u222b\nSolution We observe that sin2 x is an even function Therefore, by P7 (i), we get\n2\n\u20134\n4\nsin x dx\n\u03c0\n\u222b\u03c0\n =\n2\n204\nsin x dx\n\u222b\u03c0\n344\nMATHEMATICS\n=\n4\n0\n(1 cos 2 )\n2\n2\nx dx\n\u03c0\n\u2212\n\u222b\n = \n4\n0 (1\ncos 2 )x dx\n\u03c0\n\u2212\n\u222b\n=\n4\n0\n\u20131\nsin 2\n2\nx\nx\n\u03c0\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n = \n1\n1\n\u2013\nsin\n\u20130\n\u2013\n4\n2\n2\n4\n2\n\u03c0\n\u03c0\n\u03c0\n\uf8eb\n\uf8f6\n=\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\nExample 32 Evaluate \n2\n0\nsin\n1\nxcos\nx\ndx\nx\n\u03c0\n+\n\u222b\nSolution Let I = \n2\n0\n1sin\nxcos\nx\ndx\nx\n\u03c0\n+\n\u222b Then, by P4, we have\nI =  \n2\n0\n(\n)sin (\n)\n1\ncos (\n)\nx\nx dx\nx\n\u03c0 \u03c0 \u2212\n\u03c0 \u2212\n+\n\u03c0 \u2212\n\u222b\n=\n2\n0\n(\n)sin\n1 cos\nx\nx dx\nx\n\u03c0 \u03c0 \u2212\n+\n\u222b\n = \n2\n0\nsin\nI\n1\ncos\nx dx\nx\n\u03c0\n\u03c0\n\u2212\n+\n\u222b\nor\n2 I =\n2\n0\n1sin\ncos\nx dx\nx\n\u03c0\n\u03c0\n+\n\u222b\nor\nI =\n2\n0\nsin\n2\n1\ncos\nx dx\nx\n\u03c0\n\u03c0\n+\n\u222b\nPut cos x = t so that \u2013 sin x dx = dt"
    },
    {
        "Chapter": "1",
        "sentence_range": "3999-4002",
        "Text": "So by P2 we write\n2\n3\n1\nx\u2013\nx dx\n\u2212\u222b\n =\n0\n1\n2\n3\n3\n3\n1\n0\n1\n(\n\u2013 )\n\u2013 (\n\u2013 )\n(\n\u2013 )\nx\nx dx\nx\nx dx\nx\nx dx\n\u2212\n+\n+\n\u222b\n\u222b\n\u222b\n=\n0\n1\n2\n3\n3\n3\n1\n0\n1\n(\n\u2013 )\n( \u2013\n)\n(\n\u2013 )\nx\nx dx\nx\nx\ndx\nx\nx dx\n\u2212\n+\n+\n\u222b\n\u222b\n\u222b\n=\n0\n1\n2\n4\n2\n2\n4\n4\n2\n\u20131\n0\n1\n\u2013\n\u2013\n\u2013\n4\n2\n2\n4\n4\n2\nx\nx\nx\nx\nx\nx\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n+\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n=\n(\n)\n1\n1\n1\n1\n1\n1\n\u2013\n\u2013\n\u2013\n4 \u2013 2 \u2013\n\u2013\n4\n2\n2\n4\n4\n2\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n+\n+\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n=\n1\n1\n1\n1\n1\n1\n\u2013\n2\n4\n2\n2\n4\n4\n2\n+\n+\n\u2212\n+\n\u2212\n+\n = 3\n3\n11\n2\n2\n4\n4\n\u2212\n+\n=\nExample 31 Evaluate \n2\n\u20134\n4\nsin x dx\n\u03c0\n\u03c0\n\u222b\nSolution We observe that sin2 x is an even function Therefore, by P7 (i), we get\n2\n\u20134\n4\nsin x dx\n\u03c0\n\u222b\u03c0\n =\n2\n204\nsin x dx\n\u222b\u03c0\n344\nMATHEMATICS\n=\n4\n0\n(1 cos 2 )\n2\n2\nx dx\n\u03c0\n\u2212\n\u222b\n = \n4\n0 (1\ncos 2 )x dx\n\u03c0\n\u2212\n\u222b\n=\n4\n0\n\u20131\nsin 2\n2\nx\nx\n\u03c0\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n = \n1\n1\n\u2013\nsin\n\u20130\n\u2013\n4\n2\n2\n4\n2\n\u03c0\n\u03c0\n\u03c0\n\uf8eb\n\uf8f6\n=\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\nExample 32 Evaluate \n2\n0\nsin\n1\nxcos\nx\ndx\nx\n\u03c0\n+\n\u222b\nSolution Let I = \n2\n0\n1sin\nxcos\nx\ndx\nx\n\u03c0\n+\n\u222b Then, by P4, we have\nI =  \n2\n0\n(\n)sin (\n)\n1\ncos (\n)\nx\nx dx\nx\n\u03c0 \u03c0 \u2212\n\u03c0 \u2212\n+\n\u03c0 \u2212\n\u222b\n=\n2\n0\n(\n)sin\n1 cos\nx\nx dx\nx\n\u03c0 \u03c0 \u2212\n+\n\u222b\n = \n2\n0\nsin\nI\n1\ncos\nx dx\nx\n\u03c0\n\u03c0\n\u2212\n+\n\u222b\nor\n2 I =\n2\n0\n1sin\ncos\nx dx\nx\n\u03c0\n\u03c0\n+\n\u222b\nor\nI =\n2\n0\nsin\n2\n1\ncos\nx dx\nx\n\u03c0\n\u03c0\n+\n\u222b\nPut cos x = t so that \u2013 sin x dx = dt When x = 0, t = 1 and when x = \u03c0, t = \u2013 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "4000-4003",
        "Text": "Therefore, by P7 (i), we get\n2\n\u20134\n4\nsin x dx\n\u03c0\n\u222b\u03c0\n =\n2\n204\nsin x dx\n\u222b\u03c0\n344\nMATHEMATICS\n=\n4\n0\n(1 cos 2 )\n2\n2\nx dx\n\u03c0\n\u2212\n\u222b\n = \n4\n0 (1\ncos 2 )x dx\n\u03c0\n\u2212\n\u222b\n=\n4\n0\n\u20131\nsin 2\n2\nx\nx\n\u03c0\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n = \n1\n1\n\u2013\nsin\n\u20130\n\u2013\n4\n2\n2\n4\n2\n\u03c0\n\u03c0\n\u03c0\n\uf8eb\n\uf8f6\n=\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\nExample 32 Evaluate \n2\n0\nsin\n1\nxcos\nx\ndx\nx\n\u03c0\n+\n\u222b\nSolution Let I = \n2\n0\n1sin\nxcos\nx\ndx\nx\n\u03c0\n+\n\u222b Then, by P4, we have\nI =  \n2\n0\n(\n)sin (\n)\n1\ncos (\n)\nx\nx dx\nx\n\u03c0 \u03c0 \u2212\n\u03c0 \u2212\n+\n\u03c0 \u2212\n\u222b\n=\n2\n0\n(\n)sin\n1 cos\nx\nx dx\nx\n\u03c0 \u03c0 \u2212\n+\n\u222b\n = \n2\n0\nsin\nI\n1\ncos\nx dx\nx\n\u03c0\n\u03c0\n\u2212\n+\n\u222b\nor\n2 I =\n2\n0\n1sin\ncos\nx dx\nx\n\u03c0\n\u03c0\n+\n\u222b\nor\nI =\n2\n0\nsin\n2\n1\ncos\nx dx\nx\n\u03c0\n\u03c0\n+\n\u222b\nPut cos x = t so that \u2013 sin x dx = dt When x = 0, t = 1 and when x = \u03c0, t = \u2013 1 Therefore, (by P1) we get\nI =\n1\n2\n1\n\u2013\n2\n1\ndt\nt\n\u2212\n\u03c0\n+\n\u222b\n= \n1\n2\n1\n2\n1\ndt\nt\n\u2212\n\u03c0\n+\n\u222b\n=\n1\n2\n0 1\ndt\nt\n\u03c0\n+\n\u222b\n (by P7, \n2\n1\nsince\n1 t\n+\n is even function)\n=\n2\n1\n\u2013 1\n\u2013 1\n1\n0\ntan\ntan\n1\u2013 tan\n0\n\u2013 0\n4\n4\nt\n\u2212\n\u03c0\n\u03c0\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\u03c0\n= \u03c0\n= \u03c0\n=\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nExample 33 Evaluate \n1\n5\n4\n1 sin\nxcos\nx dx\n\u2212\u222b\nSolution Let I = \n1\n5\n4\n1sin\nxcos\nx dx\n\u2212\u222b"
    },
    {
        "Chapter": "1",
        "sentence_range": "4001-4004",
        "Text": "Then, by P4, we have\nI =  \n2\n0\n(\n)sin (\n)\n1\ncos (\n)\nx\nx dx\nx\n\u03c0 \u03c0 \u2212\n\u03c0 \u2212\n+\n\u03c0 \u2212\n\u222b\n=\n2\n0\n(\n)sin\n1 cos\nx\nx dx\nx\n\u03c0 \u03c0 \u2212\n+\n\u222b\n = \n2\n0\nsin\nI\n1\ncos\nx dx\nx\n\u03c0\n\u03c0\n\u2212\n+\n\u222b\nor\n2 I =\n2\n0\n1sin\ncos\nx dx\nx\n\u03c0\n\u03c0\n+\n\u222b\nor\nI =\n2\n0\nsin\n2\n1\ncos\nx dx\nx\n\u03c0\n\u03c0\n+\n\u222b\nPut cos x = t so that \u2013 sin x dx = dt When x = 0, t = 1 and when x = \u03c0, t = \u2013 1 Therefore, (by P1) we get\nI =\n1\n2\n1\n\u2013\n2\n1\ndt\nt\n\u2212\n\u03c0\n+\n\u222b\n= \n1\n2\n1\n2\n1\ndt\nt\n\u2212\n\u03c0\n+\n\u222b\n=\n1\n2\n0 1\ndt\nt\n\u03c0\n+\n\u222b\n (by P7, \n2\n1\nsince\n1 t\n+\n is even function)\n=\n2\n1\n\u2013 1\n\u2013 1\n1\n0\ntan\ntan\n1\u2013 tan\n0\n\u2013 0\n4\n4\nt\n\u2212\n\u03c0\n\u03c0\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\u03c0\n= \u03c0\n= \u03c0\n=\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nExample 33 Evaluate \n1\n5\n4\n1 sin\nxcos\nx dx\n\u2212\u222b\nSolution Let I = \n1\n5\n4\n1sin\nxcos\nx dx\n\u2212\u222b Let f(x) = sin5 x cos4 x"
    },
    {
        "Chapter": "1",
        "sentence_range": "4002-4005",
        "Text": "When x = 0, t = 1 and when x = \u03c0, t = \u2013 1 Therefore, (by P1) we get\nI =\n1\n2\n1\n\u2013\n2\n1\ndt\nt\n\u2212\n\u03c0\n+\n\u222b\n= \n1\n2\n1\n2\n1\ndt\nt\n\u2212\n\u03c0\n+\n\u222b\n=\n1\n2\n0 1\ndt\nt\n\u03c0\n+\n\u222b\n (by P7, \n2\n1\nsince\n1 t\n+\n is even function)\n=\n2\n1\n\u2013 1\n\u2013 1\n1\n0\ntan\ntan\n1\u2013 tan\n0\n\u2013 0\n4\n4\nt\n\u2212\n\u03c0\n\u03c0\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\u03c0\n= \u03c0\n= \u03c0\n=\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nExample 33 Evaluate \n1\n5\n4\n1 sin\nxcos\nx dx\n\u2212\u222b\nSolution Let I = \n1\n5\n4\n1sin\nxcos\nx dx\n\u2212\u222b Let f(x) = sin5 x cos4 x Then\nf (\u2013 x) = sin5 (\u2013 x) cos4 (\u2013 x) = \u2013 sin5 x cos4 x = \u2013 f (x), i"
    },
    {
        "Chapter": "1",
        "sentence_range": "4003-4006",
        "Text": "Therefore, (by P1) we get\nI =\n1\n2\n1\n\u2013\n2\n1\ndt\nt\n\u2212\n\u03c0\n+\n\u222b\n= \n1\n2\n1\n2\n1\ndt\nt\n\u2212\n\u03c0\n+\n\u222b\n=\n1\n2\n0 1\ndt\nt\n\u03c0\n+\n\u222b\n (by P7, \n2\n1\nsince\n1 t\n+\n is even function)\n=\n2\n1\n\u2013 1\n\u2013 1\n1\n0\ntan\ntan\n1\u2013 tan\n0\n\u2013 0\n4\n4\nt\n\u2212\n\u03c0\n\u03c0\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\u03c0\n= \u03c0\n= \u03c0\n=\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nExample 33 Evaluate \n1\n5\n4\n1 sin\nxcos\nx dx\n\u2212\u222b\nSolution Let I = \n1\n5\n4\n1sin\nxcos\nx dx\n\u2212\u222b Let f(x) = sin5 x cos4 x Then\nf (\u2013 x) = sin5 (\u2013 x) cos4 (\u2013 x) = \u2013 sin5 x cos4 x = \u2013 f (x), i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "4004-4007",
        "Text": "Let f(x) = sin5 x cos4 x Then\nf (\u2013 x) = sin5 (\u2013 x) cos4 (\u2013 x) = \u2013 sin5 x cos4 x = \u2013 f (x), i e , f is an odd function"
    },
    {
        "Chapter": "1",
        "sentence_range": "4005-4008",
        "Text": "Then\nf (\u2013 x) = sin5 (\u2013 x) cos4 (\u2013 x) = \u2013 sin5 x cos4 x = \u2013 f (x), i e , f is an odd function Therefore, by P7 (ii), I = 0\nINTEGRALS         345\nExample 34 Evaluate \n4\n2\n4\n4\n0\nsinsin\ncos\nx\ndx\nx\nx\n\u03c0\n+\n\u222b\nSolution Let I = \n4\n2\n4\n4\n0\nsinsin\ncos\nx\ndx\nx\nx\n\u03c0\n+\n\u222b"
    },
    {
        "Chapter": "1",
        "sentence_range": "4006-4009",
        "Text": "e , f is an odd function Therefore, by P7 (ii), I = 0\nINTEGRALS         345\nExample 34 Evaluate \n4\n2\n4\n4\n0\nsinsin\ncos\nx\ndx\nx\nx\n\u03c0\n+\n\u222b\nSolution Let I = \n4\n2\n4\n4\n0\nsinsin\ncos\nx\ndx\nx\nx\n\u03c0\n+\n\u222b (1)\nThen, by P4\nI =\n4\n2\n0\n4\n4\nsin (\n)\n2\nsin (\n)\ncos (\n)\n2\n2\nx\ndx\nx\nx\n\u03c0\n\u03c0 \u2212\n\u03c0\n\u03c0\n\u2212\n+\n\u2212\n\u222b\n = \n4\n2\n4\n4\n0\ncoscos\nsin\nx\ndx\nx\nx\n\u03c0\n+\n\u222b"
    },
    {
        "Chapter": "1",
        "sentence_range": "4007-4010",
        "Text": ", f is an odd function Therefore, by P7 (ii), I = 0\nINTEGRALS         345\nExample 34 Evaluate \n4\n2\n4\n4\n0\nsinsin\ncos\nx\ndx\nx\nx\n\u03c0\n+\n\u222b\nSolution Let I = \n4\n2\n4\n4\n0\nsinsin\ncos\nx\ndx\nx\nx\n\u03c0\n+\n\u222b (1)\nThen, by P4\nI =\n4\n2\n0\n4\n4\nsin (\n)\n2\nsin (\n)\ncos (\n)\n2\n2\nx\ndx\nx\nx\n\u03c0\n\u03c0 \u2212\n\u03c0\n\u03c0\n\u2212\n+\n\u2212\n\u222b\n = \n4\n2\n4\n4\n0\ncoscos\nsin\nx\ndx\nx\nx\n\u03c0\n+\n\u222b (2)\nAdding (1) and (2), we get\n2I =\n4\n4\n2\n2\n2\n4\n4\n0\n0\n0\nsin\ncos\n[ ]\n2\nsin\nxcos\nx dx\ndx\nx\nx\nx\n\u03c0\n\u03c0\n\u03c0\n+\n\u03c0\n=\n=\n=\n+\n\u222b\n\u222b\nHence\nI =\n4\n\u03c0\nExample 35 Evaluate \n3\n6 1\ntan\ndx\nx\n\u03c0\n\u03c0\n+\n\u222b\nSolution  Let I = \n3\n3\n6\n6\ncos\n1\ntan\ncos\nsin\nx dx\ndx\nx\nx\nx\n\u03c0\n\u03c0\n\u03c0\n\u03c0\n=\n+\n+\n\u222b\n\u222b"
    },
    {
        "Chapter": "1",
        "sentence_range": "4008-4011",
        "Text": "Therefore, by P7 (ii), I = 0\nINTEGRALS         345\nExample 34 Evaluate \n4\n2\n4\n4\n0\nsinsin\ncos\nx\ndx\nx\nx\n\u03c0\n+\n\u222b\nSolution Let I = \n4\n2\n4\n4\n0\nsinsin\ncos\nx\ndx\nx\nx\n\u03c0\n+\n\u222b (1)\nThen, by P4\nI =\n4\n2\n0\n4\n4\nsin (\n)\n2\nsin (\n)\ncos (\n)\n2\n2\nx\ndx\nx\nx\n\u03c0\n\u03c0 \u2212\n\u03c0\n\u03c0\n\u2212\n+\n\u2212\n\u222b\n = \n4\n2\n4\n4\n0\ncoscos\nsin\nx\ndx\nx\nx\n\u03c0\n+\n\u222b (2)\nAdding (1) and (2), we get\n2I =\n4\n4\n2\n2\n2\n4\n4\n0\n0\n0\nsin\ncos\n[ ]\n2\nsin\nxcos\nx dx\ndx\nx\nx\nx\n\u03c0\n\u03c0\n\u03c0\n+\n\u03c0\n=\n=\n=\n+\n\u222b\n\u222b\nHence\nI =\n4\n\u03c0\nExample 35 Evaluate \n3\n6 1\ntan\ndx\nx\n\u03c0\n\u03c0\n+\n\u222b\nSolution  Let I = \n3\n3\n6\n6\ncos\n1\ntan\ncos\nsin\nx dx\ndx\nx\nx\nx\n\u03c0\n\u03c0\n\u03c0\n\u03c0\n=\n+\n+\n\u222b\n\u222b (1)\nThen, by P3\nI =\n3\n6\ncos\n3\n6\ncos\nsin\n3\n6\n3\n6\nx\ndx\nx\nx\n\u03c0\n\u03c0\n\u03c0\n\u03c0\n\uf8eb\n\uf8f6\n+\n\u2212\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\u03c0\n\u03c0\n\u03c0\n\u03c0\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n+\n\u2212\n+\n+\n\u2212\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n\u222b\n=\n3\n6\nsin\nsin\ncos\nx\ndx\nx\nx\n\u03c0\n\u03c0\n+\n\u222b"
    },
    {
        "Chapter": "1",
        "sentence_range": "4009-4012",
        "Text": "(1)\nThen, by P4\nI =\n4\n2\n0\n4\n4\nsin (\n)\n2\nsin (\n)\ncos (\n)\n2\n2\nx\ndx\nx\nx\n\u03c0\n\u03c0 \u2212\n\u03c0\n\u03c0\n\u2212\n+\n\u2212\n\u222b\n = \n4\n2\n4\n4\n0\ncoscos\nsin\nx\ndx\nx\nx\n\u03c0\n+\n\u222b (2)\nAdding (1) and (2), we get\n2I =\n4\n4\n2\n2\n2\n4\n4\n0\n0\n0\nsin\ncos\n[ ]\n2\nsin\nxcos\nx dx\ndx\nx\nx\nx\n\u03c0\n\u03c0\n\u03c0\n+\n\u03c0\n=\n=\n=\n+\n\u222b\n\u222b\nHence\nI =\n4\n\u03c0\nExample 35 Evaluate \n3\n6 1\ntan\ndx\nx\n\u03c0\n\u03c0\n+\n\u222b\nSolution  Let I = \n3\n3\n6\n6\ncos\n1\ntan\ncos\nsin\nx dx\ndx\nx\nx\nx\n\u03c0\n\u03c0\n\u03c0\n\u03c0\n=\n+\n+\n\u222b\n\u222b (1)\nThen, by P3\nI =\n3\n6\ncos\n3\n6\ncos\nsin\n3\n6\n3\n6\nx\ndx\nx\nx\n\u03c0\n\u03c0\n\u03c0\n\u03c0\n\uf8eb\n\uf8f6\n+\n\u2212\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\u03c0\n\u03c0\n\u03c0\n\u03c0\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n+\n\u2212\n+\n+\n\u2212\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n\u222b\n=\n3\n6\nsin\nsin\ncos\nx\ndx\nx\nx\n\u03c0\n\u03c0\n+\n\u222b (2)\nAdding (1) and (2), we get\n2I =\n[ ]\n3\n3\n6\n6\n3\n6\n6\ndx\nx\n\u03c0\n\u03c0\n\u03c0\n\u03c0\n\u03c0\n\u03c0\n\u03c0\n=\n=\n\u2212\n=\n\u222b"
    },
    {
        "Chapter": "1",
        "sentence_range": "4010-4013",
        "Text": "(2)\nAdding (1) and (2), we get\n2I =\n4\n4\n2\n2\n2\n4\n4\n0\n0\n0\nsin\ncos\n[ ]\n2\nsin\nxcos\nx dx\ndx\nx\nx\nx\n\u03c0\n\u03c0\n\u03c0\n+\n\u03c0\n=\n=\n=\n+\n\u222b\n\u222b\nHence\nI =\n4\n\u03c0\nExample 35 Evaluate \n3\n6 1\ntan\ndx\nx\n\u03c0\n\u03c0\n+\n\u222b\nSolution  Let I = \n3\n3\n6\n6\ncos\n1\ntan\ncos\nsin\nx dx\ndx\nx\nx\nx\n\u03c0\n\u03c0\n\u03c0\n\u03c0\n=\n+\n+\n\u222b\n\u222b (1)\nThen, by P3\nI =\n3\n6\ncos\n3\n6\ncos\nsin\n3\n6\n3\n6\nx\ndx\nx\nx\n\u03c0\n\u03c0\n\u03c0\n\u03c0\n\uf8eb\n\uf8f6\n+\n\u2212\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\u03c0\n\u03c0\n\u03c0\n\u03c0\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n+\n\u2212\n+\n+\n\u2212\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n\u222b\n=\n3\n6\nsin\nsin\ncos\nx\ndx\nx\nx\n\u03c0\n\u03c0\n+\n\u222b (2)\nAdding (1) and (2), we get\n2I =\n[ ]\n3\n3\n6\n6\n3\n6\n6\ndx\nx\n\u03c0\n\u03c0\n\u03c0\n\u03c0\n\u03c0\n\u03c0\n\u03c0\n=\n=\n\u2212\n=\n\u222b Hence I\n12\n=\u03c0\n346\nMATHEMATICS\nExample 36 Evaluate \n2\n0 log sin x dx\n\u03c0\n\u222b\nSolution Let I =\n2\n0 logsinx dx\n\u03c0\n\u222b\nThen, by P4\nI =\n2\n2\n0\n0\nlog sin\nlog cos\n2\nx dx\nx dx\n\u03c0\n\u03c0\n\uf8eb\u03c0\n\uf8f6\n\u2212\n=\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\u222b\n\u222b\nAdding the two values of I, we get\n2I =\n(\n)\n02\nlog sin\nlogcos\nx\nx dx\n\u03c0\n+\n\u222b\n=\n(\n)\n02\nlog sin\ncos\nlog2\nlog2\nx\nx\ndx\n\u03c0\n+\n\u2212\n\u222b\n(by adding and subtracting log2)\n=\n2\n2\n0\n0\nlog sin2\nlog2\nx dx\ndx\n\u03c0\n\u03c0\n\u2212\n\u222b\n\u222b\n(Why"
    },
    {
        "Chapter": "1",
        "sentence_range": "4011-4014",
        "Text": "(1)\nThen, by P3\nI =\n3\n6\ncos\n3\n6\ncos\nsin\n3\n6\n3\n6\nx\ndx\nx\nx\n\u03c0\n\u03c0\n\u03c0\n\u03c0\n\uf8eb\n\uf8f6\n+\n\u2212\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\u03c0\n\u03c0\n\u03c0\n\u03c0\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n+\n\u2212\n+\n+\n\u2212\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n\u222b\n=\n3\n6\nsin\nsin\ncos\nx\ndx\nx\nx\n\u03c0\n\u03c0\n+\n\u222b (2)\nAdding (1) and (2), we get\n2I =\n[ ]\n3\n3\n6\n6\n3\n6\n6\ndx\nx\n\u03c0\n\u03c0\n\u03c0\n\u03c0\n\u03c0\n\u03c0\n\u03c0\n=\n=\n\u2212\n=\n\u222b Hence I\n12\n=\u03c0\n346\nMATHEMATICS\nExample 36 Evaluate \n2\n0 log sin x dx\n\u03c0\n\u222b\nSolution Let I =\n2\n0 logsinx dx\n\u03c0\n\u222b\nThen, by P4\nI =\n2\n2\n0\n0\nlog sin\nlog cos\n2\nx dx\nx dx\n\u03c0\n\u03c0\n\uf8eb\u03c0\n\uf8f6\n\u2212\n=\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\u222b\n\u222b\nAdding the two values of I, we get\n2I =\n(\n)\n02\nlog sin\nlogcos\nx\nx dx\n\u03c0\n+\n\u222b\n=\n(\n)\n02\nlog sin\ncos\nlog2\nlog2\nx\nx\ndx\n\u03c0\n+\n\u2212\n\u222b\n(by adding and subtracting log2)\n=\n2\n2\n0\n0\nlog sin2\nlog2\nx dx\ndx\n\u03c0\n\u03c0\n\u2212\n\u222b\n\u222b\n(Why )\nPut 2x = t in the first integral"
    },
    {
        "Chapter": "1",
        "sentence_range": "4012-4015",
        "Text": "(2)\nAdding (1) and (2), we get\n2I =\n[ ]\n3\n3\n6\n6\n3\n6\n6\ndx\nx\n\u03c0\n\u03c0\n\u03c0\n\u03c0\n\u03c0\n\u03c0\n\u03c0\n=\n=\n\u2212\n=\n\u222b Hence I\n12\n=\u03c0\n346\nMATHEMATICS\nExample 36 Evaluate \n2\n0 log sin x dx\n\u03c0\n\u222b\nSolution Let I =\n2\n0 logsinx dx\n\u03c0\n\u222b\nThen, by P4\nI =\n2\n2\n0\n0\nlog sin\nlog cos\n2\nx dx\nx dx\n\u03c0\n\u03c0\n\uf8eb\u03c0\n\uf8f6\n\u2212\n=\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\u222b\n\u222b\nAdding the two values of I, we get\n2I =\n(\n)\n02\nlog sin\nlogcos\nx\nx dx\n\u03c0\n+\n\u222b\n=\n(\n)\n02\nlog sin\ncos\nlog2\nlog2\nx\nx\ndx\n\u03c0\n+\n\u2212\n\u222b\n(by adding and subtracting log2)\n=\n2\n2\n0\n0\nlog sin2\nlog2\nx dx\ndx\n\u03c0\n\u03c0\n\u2212\n\u222b\n\u222b\n(Why )\nPut 2x = t in the first integral Then 2 dx = dt, when x = 0, t = 0 and when \n2\nx\n=\u03c0\n,\nt = \u03c0"
    },
    {
        "Chapter": "1",
        "sentence_range": "4013-4016",
        "Text": "Hence I\n12\n=\u03c0\n346\nMATHEMATICS\nExample 36 Evaluate \n2\n0 log sin x dx\n\u03c0\n\u222b\nSolution Let I =\n2\n0 logsinx dx\n\u03c0\n\u222b\nThen, by P4\nI =\n2\n2\n0\n0\nlog sin\nlog cos\n2\nx dx\nx dx\n\u03c0\n\u03c0\n\uf8eb\u03c0\n\uf8f6\n\u2212\n=\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\u222b\n\u222b\nAdding the two values of I, we get\n2I =\n(\n)\n02\nlog sin\nlogcos\nx\nx dx\n\u03c0\n+\n\u222b\n=\n(\n)\n02\nlog sin\ncos\nlog2\nlog2\nx\nx\ndx\n\u03c0\n+\n\u2212\n\u222b\n(by adding and subtracting log2)\n=\n2\n2\n0\n0\nlog sin2\nlog2\nx dx\ndx\n\u03c0\n\u03c0\n\u2212\n\u222b\n\u222b\n(Why )\nPut 2x = t in the first integral Then 2 dx = dt, when x = 0, t = 0 and when \n2\nx\n=\u03c0\n,\nt = \u03c0 Therefore\n2I =\n0\n1\nlog sin\nlog2\n2\n2\nt dt\n\u03c0\n\u2212\u03c0\n\u222b\n=\n02\n2\nlog sin\nlog2\n2\n2\nt dt\n\u03c0\n\u2212\u03c0\n\u222b\n [by P6 as sin (\u03c0 \u2013 t) = sin t)\n=\n2\n0 log sin\nlog2\n2\nx dx\n\u03c0\n\u2212\u03c0\n\u222b\n (by changing variable t to x)\n= I\nlog2\n\u03c02\n\u2212\nHence\n2\n0 log sin x dx\n\u222b\u03c0\n = \u2013\n2log2\n\u03c0"
    },
    {
        "Chapter": "1",
        "sentence_range": "4014-4017",
        "Text": ")\nPut 2x = t in the first integral Then 2 dx = dt, when x = 0, t = 0 and when \n2\nx\n=\u03c0\n,\nt = \u03c0 Therefore\n2I =\n0\n1\nlog sin\nlog2\n2\n2\nt dt\n\u03c0\n\u2212\u03c0\n\u222b\n=\n02\n2\nlog sin\nlog2\n2\n2\nt dt\n\u03c0\n\u2212\u03c0\n\u222b\n [by P6 as sin (\u03c0 \u2013 t) = sin t)\n=\n2\n0 log sin\nlog2\n2\nx dx\n\u03c0\n\u2212\u03c0\n\u222b\n (by changing variable t to x)\n= I\nlog2\n\u03c02\n\u2212\nHence\n2\n0 log sin x dx\n\u222b\u03c0\n = \u2013\n2log2\n\u03c0 INTEGRALS         347\nEXERCISE 7"
    },
    {
        "Chapter": "1",
        "sentence_range": "4015-4018",
        "Text": "Then 2 dx = dt, when x = 0, t = 0 and when \n2\nx\n=\u03c0\n,\nt = \u03c0 Therefore\n2I =\n0\n1\nlog sin\nlog2\n2\n2\nt dt\n\u03c0\n\u2212\u03c0\n\u222b\n=\n02\n2\nlog sin\nlog2\n2\n2\nt dt\n\u03c0\n\u2212\u03c0\n\u222b\n [by P6 as sin (\u03c0 \u2013 t) = sin t)\n=\n2\n0 log sin\nlog2\n2\nx dx\n\u03c0\n\u2212\u03c0\n\u222b\n (by changing variable t to x)\n= I\nlog2\n\u03c02\n\u2212\nHence\n2\n0 log sin x dx\n\u222b\u03c0\n = \u2013\n2log2\n\u03c0 INTEGRALS         347\nEXERCISE 7 11\nBy using the properties of definite integrals, evaluate the integrals in Exercises 1 to 19"
    },
    {
        "Chapter": "1",
        "sentence_range": "4016-4019",
        "Text": "Therefore\n2I =\n0\n1\nlog sin\nlog2\n2\n2\nt dt\n\u03c0\n\u2212\u03c0\n\u222b\n=\n02\n2\nlog sin\nlog2\n2\n2\nt dt\n\u03c0\n\u2212\u03c0\n\u222b\n [by P6 as sin (\u03c0 \u2013 t) = sin t)\n=\n2\n0 log sin\nlog2\n2\nx dx\n\u03c0\n\u2212\u03c0\n\u222b\n (by changing variable t to x)\n= I\nlog2\n\u03c02\n\u2212\nHence\n2\n0 log sin x dx\n\u222b\u03c0\n = \u2013\n2log2\n\u03c0 INTEGRALS         347\nEXERCISE 7 11\nBy using the properties of definite integrals, evaluate the integrals in Exercises 1 to 19 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "4017-4020",
        "Text": "INTEGRALS         347\nEXERCISE 7 11\nBy using the properties of definite integrals, evaluate the integrals in Exercises 1 to 19 1 2\n2\n0 cos x dx\n\u222b\u03c0\n2"
    },
    {
        "Chapter": "1",
        "sentence_range": "4018-4021",
        "Text": "11\nBy using the properties of definite integrals, evaluate the integrals in Exercises 1 to 19 1 2\n2\n0 cos x dx\n\u222b\u03c0\n2 02\nsin\nsin\ncos\nx\ndx\nx\nx\n\u03c0\n+\n\u222b\n3"
    },
    {
        "Chapter": "1",
        "sentence_range": "4019-4022",
        "Text": "1 2\n2\n0 cos x dx\n\u222b\u03c0\n2 02\nsin\nsin\ncos\nx\ndx\nx\nx\n\u03c0\n+\n\u222b\n3 23\n2\n3\n3\n0\n2\n2\nsin\nsin\ncos\nx dx\nx\nx\n\u03c0\n+\n\u222b\n4"
    },
    {
        "Chapter": "1",
        "sentence_range": "4020-4023",
        "Text": "2\n2\n0 cos x dx\n\u222b\u03c0\n2 02\nsin\nsin\ncos\nx\ndx\nx\nx\n\u03c0\n+\n\u222b\n3 23\n2\n3\n3\n0\n2\n2\nsin\nsin\ncos\nx dx\nx\nx\n\u03c0\n+\n\u222b\n4 5\n2\n5\n5\n0\ncos\nsin\ncos\nx dx\nx\nx\n\u03c0\n+\n\u222b\n5"
    },
    {
        "Chapter": "1",
        "sentence_range": "4021-4024",
        "Text": "02\nsin\nsin\ncos\nx\ndx\nx\nx\n\u03c0\n+\n\u222b\n3 23\n2\n3\n3\n0\n2\n2\nsin\nsin\ncos\nx dx\nx\nx\n\u03c0\n+\n\u222b\n4 5\n2\n5\n5\n0\ncos\nsin\ncos\nx dx\nx\nx\n\u03c0\n+\n\u222b\n5 5\n5 |\n2|\nx\ndx\n\u2212\n+\n\u222b\n6"
    },
    {
        "Chapter": "1",
        "sentence_range": "4022-4025",
        "Text": "23\n2\n3\n3\n0\n2\n2\nsin\nsin\ncos\nx dx\nx\nx\n\u03c0\n+\n\u222b\n4 5\n2\n5\n5\n0\ncos\nsin\ncos\nx dx\nx\nx\n\u03c0\n+\n\u222b\n5 5\n5 |\n2|\nx\ndx\n\u2212\n+\n\u222b\n6 8\n2\n5\nx\ndx\n\u2212\n\u222b\n7"
    },
    {
        "Chapter": "1",
        "sentence_range": "4023-4026",
        "Text": "5\n2\n5\n5\n0\ncos\nsin\ncos\nx dx\nx\nx\n\u03c0\n+\n\u222b\n5 5\n5 |\n2|\nx\ndx\n\u2212\n+\n\u222b\n6 8\n2\n5\nx\ndx\n\u2212\n\u222b\n7 1\n0\n(1\n)n\nx\nx\ndx\n\u2212\n\u222b\n8"
    },
    {
        "Chapter": "1",
        "sentence_range": "4024-4027",
        "Text": "5\n5 |\n2|\nx\ndx\n\u2212\n+\n\u222b\n6 8\n2\n5\nx\ndx\n\u2212\n\u222b\n7 1\n0\n(1\n)n\nx\nx\ndx\n\u2212\n\u222b\n8 4\n0 log (1\ntan )x dx\n\u03c0\n+\n\u222b\n9"
    },
    {
        "Chapter": "1",
        "sentence_range": "4025-4028",
        "Text": "8\n2\n5\nx\ndx\n\u2212\n\u222b\n7 1\n0\n(1\n)n\nx\nx\ndx\n\u2212\n\u222b\n8 4\n0 log (1\ntan )x dx\n\u03c0\n+\n\u222b\n9 2\n0\n2\nx\n\u2212x dx\n\u222b\n10"
    },
    {
        "Chapter": "1",
        "sentence_range": "4026-4029",
        "Text": "1\n0\n(1\n)n\nx\nx\ndx\n\u2212\n\u222b\n8 4\n0 log (1\ntan )x dx\n\u03c0\n+\n\u222b\n9 2\n0\n2\nx\n\u2212x dx\n\u222b\n10 2\n0 (2log sin\nlogsin2 )\nx\nx dx\n\u03c0\n\u2212\n\u222b\n11"
    },
    {
        "Chapter": "1",
        "sentence_range": "4027-4030",
        "Text": "4\n0 log (1\ntan )x dx\n\u03c0\n+\n\u222b\n9 2\n0\n2\nx\n\u2212x dx\n\u222b\n10 2\n0 (2log sin\nlogsin2 )\nx\nx dx\n\u03c0\n\u2212\n\u222b\n11 2\n\u20132\n2\nsin x dx\n\u03c0\n\u03c0\n\u222b\n12"
    },
    {
        "Chapter": "1",
        "sentence_range": "4028-4031",
        "Text": "2\n0\n2\nx\n\u2212x dx\n\u222b\n10 2\n0 (2log sin\nlogsin2 )\nx\nx dx\n\u03c0\n\u2212\n\u222b\n11 2\n\u20132\n2\nsin x dx\n\u03c0\n\u03c0\n\u222b\n12 0 1\nsin\nx dx\nx\n\u03c0\n+\n\u222b\n13"
    },
    {
        "Chapter": "1",
        "sentence_range": "4029-4032",
        "Text": "2\n0 (2log sin\nlogsin2 )\nx\nx dx\n\u03c0\n\u2212\n\u222b\n11 2\n\u20132\n2\nsin x dx\n\u03c0\n\u03c0\n\u222b\n12 0 1\nsin\nx dx\nx\n\u03c0\n+\n\u222b\n13 7\n\u20132\n2\nsin x dx\n\u03c0\n\u222b\u03c0\n14"
    },
    {
        "Chapter": "1",
        "sentence_range": "4030-4033",
        "Text": "2\n\u20132\n2\nsin x dx\n\u03c0\n\u03c0\n\u222b\n12 0 1\nsin\nx dx\nx\n\u03c0\n+\n\u222b\n13 7\n\u20132\n2\nsin x dx\n\u03c0\n\u222b\u03c0\n14 2\n5\n0 cos x dx\n\u03c0\n\u222b\n15"
    },
    {
        "Chapter": "1",
        "sentence_range": "4031-4034",
        "Text": "0 1\nsin\nx dx\nx\n\u03c0\n+\n\u222b\n13 7\n\u20132\n2\nsin x dx\n\u03c0\n\u222b\u03c0\n14 2\n5\n0 cos x dx\n\u03c0\n\u222b\n15 02\nsin\ncos\n1\nsin\ncos\nx\nx dx\nx\nx\n\u03c0\n\u2212\n+\n\u222b\n16"
    },
    {
        "Chapter": "1",
        "sentence_range": "4032-4035",
        "Text": "7\n\u20132\n2\nsin x dx\n\u03c0\n\u222b\u03c0\n14 2\n5\n0 cos x dx\n\u03c0\n\u222b\n15 02\nsin\ncos\n1\nsin\ncos\nx\nx dx\nx\nx\n\u03c0\n\u2212\n+\n\u222b\n16 0 log (1\ncos )\nx dx\n\u03c0\n+\n\u222b\n17"
    },
    {
        "Chapter": "1",
        "sentence_range": "4033-4036",
        "Text": "2\n5\n0 cos x dx\n\u03c0\n\u222b\n15 02\nsin\ncos\n1\nsin\ncos\nx\nx dx\nx\nx\n\u03c0\n\u2212\n+\n\u222b\n16 0 log (1\ncos )\nx dx\n\u03c0\n+\n\u222b\n17 0\na\nx\ndx\nx\na\nx\n+\n\u2212\n\u222b\n18"
    },
    {
        "Chapter": "1",
        "sentence_range": "4034-4037",
        "Text": "02\nsin\ncos\n1\nsin\ncos\nx\nx dx\nx\nx\n\u03c0\n\u2212\n+\n\u222b\n16 0 log (1\ncos )\nx dx\n\u03c0\n+\n\u222b\n17 0\na\nx\ndx\nx\na\nx\n+\n\u2212\n\u222b\n18 4\n0\n1\nx\ndx\n\u2212\n\u222b\n19"
    },
    {
        "Chapter": "1",
        "sentence_range": "4035-4038",
        "Text": "0 log (1\ncos )\nx dx\n\u03c0\n+\n\u222b\n17 0\na\nx\ndx\nx\na\nx\n+\n\u2212\n\u222b\n18 4\n0\n1\nx\ndx\n\u2212\n\u222b\n19 Show that \n0\n0\n( ) ( )\n2\n( )\na\na\nf x g x dx\nf x dx\n=\n\u222b\n\u222b\n, if f and g are defined as f(x) = f(a \u2013 x)\nand g(x) + g(a \u2013 x) = 4\nChoose the correct answer in Exercises 20 and 21"
    },
    {
        "Chapter": "1",
        "sentence_range": "4036-4039",
        "Text": "0\na\nx\ndx\nx\na\nx\n+\n\u2212\n\u222b\n18 4\n0\n1\nx\ndx\n\u2212\n\u222b\n19 Show that \n0\n0\n( ) ( )\n2\n( )\na\na\nf x g x dx\nf x dx\n=\n\u222b\n\u222b\n, if f and g are defined as f(x) = f(a \u2013 x)\nand g(x) + g(a \u2013 x) = 4\nChoose the correct answer in Exercises 20 and 21 20"
    },
    {
        "Chapter": "1",
        "sentence_range": "4037-4040",
        "Text": "4\n0\n1\nx\ndx\n\u2212\n\u222b\n19 Show that \n0\n0\n( ) ( )\n2\n( )\na\na\nf x g x dx\nf x dx\n=\n\u222b\n\u222b\n, if f and g are defined as f(x) = f(a \u2013 x)\nand g(x) + g(a \u2013 x) = 4\nChoose the correct answer in Exercises 20 and 21 20 The value of \n3\n5\n2\n2\n(\ncos\ntan\n1)\nx\nx\nx\nx\ndx\n\u03c0\n\u2212\u03c0\n+\n+\n+\n\u222b\n is\n(A) 0\n(B) 2\n(C) \u03c0\n(D) 1\n21"
    },
    {
        "Chapter": "1",
        "sentence_range": "4038-4041",
        "Text": "Show that \n0\n0\n( ) ( )\n2\n( )\na\na\nf x g x dx\nf x dx\n=\n\u222b\n\u222b\n, if f and g are defined as f(x) = f(a \u2013 x)\nand g(x) + g(a \u2013 x) = 4\nChoose the correct answer in Exercises 20 and 21 20 The value of \n3\n5\n2\n2\n(\ncos\ntan\n1)\nx\nx\nx\nx\ndx\n\u03c0\n\u2212\u03c0\n+\n+\n+\n\u222b\n is\n(A) 0\n(B) 2\n(C) \u03c0\n(D) 1\n21 The value of \n02\n4\n3sin\nlog\n4\n3cos\nx\ndx\nx\n\u03c0\n\uf8eb\n\uf8f6\n+\n\uf8ec\n\uf8f7\n+\n\uf8ed\n\uf8f8\n\u222b\n is\n(A) 2\n(B)\n43\n(C) 0\n(D) \u20132\n348\nMATHEMATICS\nMiscellaneous Examples\nExample 37 Find cos 6\n1 sin 6\nx\nx dx\n+\n\u222b\nSolution Put t = 1 + sin 6x, so that dt = 6 cos 6x dx\nTherefore\n21\n1\ncos 6\n1 sin 6\n6\nx\nx dx\nt dt\n+\n=\n\u222b\n\u222b\n=\n3\n3\n2\n2\n1\n2\n1\n( )\nC = \n(1 sin 6 )\nC\n6\n3\n9\nt\nx\n\u00d7\n+\n+\n+\nExample 38 Find \n1\n4\n4\n5\n(\n)\nx\nx\ndx\n\u2212x\n\u222b\nSolution We have \n1\n1\n4\n4\n4\n3\n5\n4\n1\n(1\n)\n(\n)\nx\nx\nx\ndx\ndx\nx\nx\n\u2212\n\u2212\n=\n\u222b\n\u222b\nPut \n\u2013 3\n3\n4\n1\n3\n1\n1\u2013\n, so that\nx\nt\ndx\ndt\nx\nx\n\u2212\n=\n=\n=\nTherefore \n1\n1\n4\n4\n4\n5\n(\n)\n31\nx\nx\ndx\nt\ndt\n\u2212x\n=\n\u222b\n\u222b\n = \n5\n5\n4\n4\n3\n1\n4\n4\n1\nC =\n1\nC\n3\n5\n15\nt\nx\n\uf8eb\n\uf8f6\n\u00d7\n+\n\u2212\n+\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\nExample 39 Find \n4\n2\n(\n1) (\n1)\nx dx\nx\nx\n\u2212\n+\n\u222b\nSolution We have\n4\n2\n(\n1)(\n1)\nx\nx\nx\n\u2212\n+\n =\n3\n12\n(\n1)\n1\nx\nx\nx\nx\n+\n+\n\u2212\n+\n\u2212\n=\n12\n(\n1)\n(\n1) (\n1)\nx\nx\nx\n+\n+\n\u2212\n+"
    },
    {
        "Chapter": "1",
        "sentence_range": "4039-4042",
        "Text": "20 The value of \n3\n5\n2\n2\n(\ncos\ntan\n1)\nx\nx\nx\nx\ndx\n\u03c0\n\u2212\u03c0\n+\n+\n+\n\u222b\n is\n(A) 0\n(B) 2\n(C) \u03c0\n(D) 1\n21 The value of \n02\n4\n3sin\nlog\n4\n3cos\nx\ndx\nx\n\u03c0\n\uf8eb\n\uf8f6\n+\n\uf8ec\n\uf8f7\n+\n\uf8ed\n\uf8f8\n\u222b\n is\n(A) 2\n(B)\n43\n(C) 0\n(D) \u20132\n348\nMATHEMATICS\nMiscellaneous Examples\nExample 37 Find cos 6\n1 sin 6\nx\nx dx\n+\n\u222b\nSolution Put t = 1 + sin 6x, so that dt = 6 cos 6x dx\nTherefore\n21\n1\ncos 6\n1 sin 6\n6\nx\nx dx\nt dt\n+\n=\n\u222b\n\u222b\n=\n3\n3\n2\n2\n1\n2\n1\n( )\nC = \n(1 sin 6 )\nC\n6\n3\n9\nt\nx\n\u00d7\n+\n+\n+\nExample 38 Find \n1\n4\n4\n5\n(\n)\nx\nx\ndx\n\u2212x\n\u222b\nSolution We have \n1\n1\n4\n4\n4\n3\n5\n4\n1\n(1\n)\n(\n)\nx\nx\nx\ndx\ndx\nx\nx\n\u2212\n\u2212\n=\n\u222b\n\u222b\nPut \n\u2013 3\n3\n4\n1\n3\n1\n1\u2013\n, so that\nx\nt\ndx\ndt\nx\nx\n\u2212\n=\n=\n=\nTherefore \n1\n1\n4\n4\n4\n5\n(\n)\n31\nx\nx\ndx\nt\ndt\n\u2212x\n=\n\u222b\n\u222b\n = \n5\n5\n4\n4\n3\n1\n4\n4\n1\nC =\n1\nC\n3\n5\n15\nt\nx\n\uf8eb\n\uf8f6\n\u00d7\n+\n\u2212\n+\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\nExample 39 Find \n4\n2\n(\n1) (\n1)\nx dx\nx\nx\n\u2212\n+\n\u222b\nSolution We have\n4\n2\n(\n1)(\n1)\nx\nx\nx\n\u2212\n+\n =\n3\n12\n(\n1)\n1\nx\nx\nx\nx\n+\n+\n\u2212\n+\n\u2212\n=\n12\n(\n1)\n(\n1) (\n1)\nx\nx\nx\n+\n+\n\u2212\n+ (1)\nNow express\n2\n1\n(\n1)(\n1)\nx\nx\n\u2212\n+\n =\n2\nA\nB\nC\n(\n1)\n(\n1)\nx\nx\nx\n+\n+\n\u2212\n+"
    },
    {
        "Chapter": "1",
        "sentence_range": "4040-4043",
        "Text": "The value of \n3\n5\n2\n2\n(\ncos\ntan\n1)\nx\nx\nx\nx\ndx\n\u03c0\n\u2212\u03c0\n+\n+\n+\n\u222b\n is\n(A) 0\n(B) 2\n(C) \u03c0\n(D) 1\n21 The value of \n02\n4\n3sin\nlog\n4\n3cos\nx\ndx\nx\n\u03c0\n\uf8eb\n\uf8f6\n+\n\uf8ec\n\uf8f7\n+\n\uf8ed\n\uf8f8\n\u222b\n is\n(A) 2\n(B)\n43\n(C) 0\n(D) \u20132\n348\nMATHEMATICS\nMiscellaneous Examples\nExample 37 Find cos 6\n1 sin 6\nx\nx dx\n+\n\u222b\nSolution Put t = 1 + sin 6x, so that dt = 6 cos 6x dx\nTherefore\n21\n1\ncos 6\n1 sin 6\n6\nx\nx dx\nt dt\n+\n=\n\u222b\n\u222b\n=\n3\n3\n2\n2\n1\n2\n1\n( )\nC = \n(1 sin 6 )\nC\n6\n3\n9\nt\nx\n\u00d7\n+\n+\n+\nExample 38 Find \n1\n4\n4\n5\n(\n)\nx\nx\ndx\n\u2212x\n\u222b\nSolution We have \n1\n1\n4\n4\n4\n3\n5\n4\n1\n(1\n)\n(\n)\nx\nx\nx\ndx\ndx\nx\nx\n\u2212\n\u2212\n=\n\u222b\n\u222b\nPut \n\u2013 3\n3\n4\n1\n3\n1\n1\u2013\n, so that\nx\nt\ndx\ndt\nx\nx\n\u2212\n=\n=\n=\nTherefore \n1\n1\n4\n4\n4\n5\n(\n)\n31\nx\nx\ndx\nt\ndt\n\u2212x\n=\n\u222b\n\u222b\n = \n5\n5\n4\n4\n3\n1\n4\n4\n1\nC =\n1\nC\n3\n5\n15\nt\nx\n\uf8eb\n\uf8f6\n\u00d7\n+\n\u2212\n+\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\nExample 39 Find \n4\n2\n(\n1) (\n1)\nx dx\nx\nx\n\u2212\n+\n\u222b\nSolution We have\n4\n2\n(\n1)(\n1)\nx\nx\nx\n\u2212\n+\n =\n3\n12\n(\n1)\n1\nx\nx\nx\nx\n+\n+\n\u2212\n+\n\u2212\n=\n12\n(\n1)\n(\n1) (\n1)\nx\nx\nx\n+\n+\n\u2212\n+ (1)\nNow express\n2\n1\n(\n1)(\n1)\nx\nx\n\u2212\n+\n =\n2\nA\nB\nC\n(\n1)\n(\n1)\nx\nx\nx\n+\n+\n\u2212\n+ (2)\nINTEGRALS         349\nSo\n1 = A (x2 + 1) + (Bx + C) (x \u2013 1)\n= (A + B) x2 + (C \u2013 B) x + A \u2013 C\nEquating coefficients on both sides, we get A + B = 0, C \u2013 B = 0 and A \u2013 C = 1,\nwhich give \n1\n1\nA\n, B\nC\n\u2013\n2\n2\n=\n=\n="
    },
    {
        "Chapter": "1",
        "sentence_range": "4041-4044",
        "Text": "The value of \n02\n4\n3sin\nlog\n4\n3cos\nx\ndx\nx\n\u03c0\n\uf8eb\n\uf8f6\n+\n\uf8ec\n\uf8f7\n+\n\uf8ed\n\uf8f8\n\u222b\n is\n(A) 2\n(B)\n43\n(C) 0\n(D) \u20132\n348\nMATHEMATICS\nMiscellaneous Examples\nExample 37 Find cos 6\n1 sin 6\nx\nx dx\n+\n\u222b\nSolution Put t = 1 + sin 6x, so that dt = 6 cos 6x dx\nTherefore\n21\n1\ncos 6\n1 sin 6\n6\nx\nx dx\nt dt\n+\n=\n\u222b\n\u222b\n=\n3\n3\n2\n2\n1\n2\n1\n( )\nC = \n(1 sin 6 )\nC\n6\n3\n9\nt\nx\n\u00d7\n+\n+\n+\nExample 38 Find \n1\n4\n4\n5\n(\n)\nx\nx\ndx\n\u2212x\n\u222b\nSolution We have \n1\n1\n4\n4\n4\n3\n5\n4\n1\n(1\n)\n(\n)\nx\nx\nx\ndx\ndx\nx\nx\n\u2212\n\u2212\n=\n\u222b\n\u222b\nPut \n\u2013 3\n3\n4\n1\n3\n1\n1\u2013\n, so that\nx\nt\ndx\ndt\nx\nx\n\u2212\n=\n=\n=\nTherefore \n1\n1\n4\n4\n4\n5\n(\n)\n31\nx\nx\ndx\nt\ndt\n\u2212x\n=\n\u222b\n\u222b\n = \n5\n5\n4\n4\n3\n1\n4\n4\n1\nC =\n1\nC\n3\n5\n15\nt\nx\n\uf8eb\n\uf8f6\n\u00d7\n+\n\u2212\n+\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\nExample 39 Find \n4\n2\n(\n1) (\n1)\nx dx\nx\nx\n\u2212\n+\n\u222b\nSolution We have\n4\n2\n(\n1)(\n1)\nx\nx\nx\n\u2212\n+\n =\n3\n12\n(\n1)\n1\nx\nx\nx\nx\n+\n+\n\u2212\n+\n\u2212\n=\n12\n(\n1)\n(\n1) (\n1)\nx\nx\nx\n+\n+\n\u2212\n+ (1)\nNow express\n2\n1\n(\n1)(\n1)\nx\nx\n\u2212\n+\n =\n2\nA\nB\nC\n(\n1)\n(\n1)\nx\nx\nx\n+\n+\n\u2212\n+ (2)\nINTEGRALS         349\nSo\n1 = A (x2 + 1) + (Bx + C) (x \u2013 1)\n= (A + B) x2 + (C \u2013 B) x + A \u2013 C\nEquating coefficients on both sides, we get A + B = 0, C \u2013 B = 0 and A \u2013 C = 1,\nwhich give \n1\n1\nA\n, B\nC\n\u2013\n2\n2\n=\n=\n= Substituting values of A, B and C in (2), we get\n2\n1\n(\n1) (\n1)\nx\nx\n\u2212\n+\n =\n2\n2\n1\n1\n1\n2(\n1)\n2 (\n1)\n2(\n1)\nx\nx\nx\nx\n\u2212\n\u2212\n\u2212\n+\n+"
    },
    {
        "Chapter": "1",
        "sentence_range": "4042-4045",
        "Text": "(1)\nNow express\n2\n1\n(\n1)(\n1)\nx\nx\n\u2212\n+\n =\n2\nA\nB\nC\n(\n1)\n(\n1)\nx\nx\nx\n+\n+\n\u2212\n+ (2)\nINTEGRALS         349\nSo\n1 = A (x2 + 1) + (Bx + C) (x \u2013 1)\n= (A + B) x2 + (C \u2013 B) x + A \u2013 C\nEquating coefficients on both sides, we get A + B = 0, C \u2013 B = 0 and A \u2013 C = 1,\nwhich give \n1\n1\nA\n, B\nC\n\u2013\n2\n2\n=\n=\n= Substituting values of A, B and C in (2), we get\n2\n1\n(\n1) (\n1)\nx\nx\n\u2212\n+\n =\n2\n2\n1\n1\n1\n2(\n1)\n2 (\n1)\n2(\n1)\nx\nx\nx\nx\n\u2212\n\u2212\n\u2212\n+\n+ (3)\nAgain, substituting (3) in (1), we have\n4\n2\n(\n1) (\n1)\nx\nx\nx\nx\n\u2212\n+\n+\n =\n2\n2\n1\n1\n1\n(\n1)\n2(\n1)\n2 (\n1)\n2(\n1)\nx\nx\nx\nx\nx\n+\n+\n\u2212\n\u2212\n\u2212\n+\n+\nTherefore\n4\n2\n2\n\u2013 1\n2\n1\n1\n1\nlog\n1 \u2013\nlog (\n1) \u2013\ntan\nC\n2\n2\n4\n2\n(\n1) (\n1)\nx\nx\ndx\nx\nx\nx\nx\nx\nx\nx\n=\n+\n+\n\u2212\n+\n+\n\u2212\n+\n+\n\u222b\nExample 40 Find \n2\n1\nlog (log )\n(log )\nx\ndx\nx\n\uf8ee\n\uf8f9\n+\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\u222b\nSolution Let \n2\n1\nI\nlog (log )\n(log )\nx\ndx\nx\n\uf8ee\n\uf8f9\n=\n+\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\u222b\n=\n2\n1\nlog (log )\n(log )\nx dx\ndx\nx\n+\n\u222b\n\u222b\nIn the first integral, let us take 1 as the second function"
    },
    {
        "Chapter": "1",
        "sentence_range": "4043-4046",
        "Text": "(2)\nINTEGRALS         349\nSo\n1 = A (x2 + 1) + (Bx + C) (x \u2013 1)\n= (A + B) x2 + (C \u2013 B) x + A \u2013 C\nEquating coefficients on both sides, we get A + B = 0, C \u2013 B = 0 and A \u2013 C = 1,\nwhich give \n1\n1\nA\n, B\nC\n\u2013\n2\n2\n=\n=\n= Substituting values of A, B and C in (2), we get\n2\n1\n(\n1) (\n1)\nx\nx\n\u2212\n+\n =\n2\n2\n1\n1\n1\n2(\n1)\n2 (\n1)\n2(\n1)\nx\nx\nx\nx\n\u2212\n\u2212\n\u2212\n+\n+ (3)\nAgain, substituting (3) in (1), we have\n4\n2\n(\n1) (\n1)\nx\nx\nx\nx\n\u2212\n+\n+\n =\n2\n2\n1\n1\n1\n(\n1)\n2(\n1)\n2 (\n1)\n2(\n1)\nx\nx\nx\nx\nx\n+\n+\n\u2212\n\u2212\n\u2212\n+\n+\nTherefore\n4\n2\n2\n\u2013 1\n2\n1\n1\n1\nlog\n1 \u2013\nlog (\n1) \u2013\ntan\nC\n2\n2\n4\n2\n(\n1) (\n1)\nx\nx\ndx\nx\nx\nx\nx\nx\nx\nx\n=\n+\n+\n\u2212\n+\n+\n\u2212\n+\n+\n\u222b\nExample 40 Find \n2\n1\nlog (log )\n(log )\nx\ndx\nx\n\uf8ee\n\uf8f9\n+\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\u222b\nSolution Let \n2\n1\nI\nlog (log )\n(log )\nx\ndx\nx\n\uf8ee\n\uf8f9\n=\n+\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\u222b\n=\n2\n1\nlog (log )\n(log )\nx dx\ndx\nx\n+\n\u222b\n\u222b\nIn the first integral, let us take 1 as the second function Then integrating it by\nparts, we get\nI =\n2\n1\nlog (log )\nlog\n(log )\ndx\nx\nx\nx dx\nx\nx\nx\n\u2212\n+\n\u222b\n\u222b\n=\n2\nlog (log )\nlog\n(log )\ndx\ndx\nx\nx\nx\nx\n\u2212\n+\n\u222b\n\u222b"
    },
    {
        "Chapter": "1",
        "sentence_range": "4044-4047",
        "Text": "Substituting values of A, B and C in (2), we get\n2\n1\n(\n1) (\n1)\nx\nx\n\u2212\n+\n =\n2\n2\n1\n1\n1\n2(\n1)\n2 (\n1)\n2(\n1)\nx\nx\nx\nx\n\u2212\n\u2212\n\u2212\n+\n+ (3)\nAgain, substituting (3) in (1), we have\n4\n2\n(\n1) (\n1)\nx\nx\nx\nx\n\u2212\n+\n+\n =\n2\n2\n1\n1\n1\n(\n1)\n2(\n1)\n2 (\n1)\n2(\n1)\nx\nx\nx\nx\nx\n+\n+\n\u2212\n\u2212\n\u2212\n+\n+\nTherefore\n4\n2\n2\n\u2013 1\n2\n1\n1\n1\nlog\n1 \u2013\nlog (\n1) \u2013\ntan\nC\n2\n2\n4\n2\n(\n1) (\n1)\nx\nx\ndx\nx\nx\nx\nx\nx\nx\nx\n=\n+\n+\n\u2212\n+\n+\n\u2212\n+\n+\n\u222b\nExample 40 Find \n2\n1\nlog (log )\n(log )\nx\ndx\nx\n\uf8ee\n\uf8f9\n+\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\u222b\nSolution Let \n2\n1\nI\nlog (log )\n(log )\nx\ndx\nx\n\uf8ee\n\uf8f9\n=\n+\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\u222b\n=\n2\n1\nlog (log )\n(log )\nx dx\ndx\nx\n+\n\u222b\n\u222b\nIn the first integral, let us take 1 as the second function Then integrating it by\nparts, we get\nI =\n2\n1\nlog (log )\nlog\n(log )\ndx\nx\nx\nx dx\nx\nx\nx\n\u2212\n+\n\u222b\n\u222b\n=\n2\nlog (log )\nlog\n(log )\ndx\ndx\nx\nx\nx\nx\n\u2212\n+\n\u222b\n\u222b (1)\nAgain, consider \nlog\ndx\nx\n\u222b\n, take 1 as the second function and integrate it by parts,\nwe have \n2\n1\n1\n\u2013\n\u2013\nlog\nlog\n(log )\ndx\nx\nx\ndx\nx\nx\nx\nx\n\uf8ee\n\uf8f9\n\uf8f1\n\uf8fc\n\uf8eb\n\uf8f6\n= \uf8ef\n\uf8fa\n\uf8f2\n\uf8fd\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8ef\n\uf8fa\n\uf8f3\n\uf8fe\n\uf8f0\n\uf8fb\n\u222b\n\u222b"
    },
    {
        "Chapter": "1",
        "sentence_range": "4045-4048",
        "Text": "(3)\nAgain, substituting (3) in (1), we have\n4\n2\n(\n1) (\n1)\nx\nx\nx\nx\n\u2212\n+\n+\n =\n2\n2\n1\n1\n1\n(\n1)\n2(\n1)\n2 (\n1)\n2(\n1)\nx\nx\nx\nx\nx\n+\n+\n\u2212\n\u2212\n\u2212\n+\n+\nTherefore\n4\n2\n2\n\u2013 1\n2\n1\n1\n1\nlog\n1 \u2013\nlog (\n1) \u2013\ntan\nC\n2\n2\n4\n2\n(\n1) (\n1)\nx\nx\ndx\nx\nx\nx\nx\nx\nx\nx\n=\n+\n+\n\u2212\n+\n+\n\u2212\n+\n+\n\u222b\nExample 40 Find \n2\n1\nlog (log )\n(log )\nx\ndx\nx\n\uf8ee\n\uf8f9\n+\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\u222b\nSolution Let \n2\n1\nI\nlog (log )\n(log )\nx\ndx\nx\n\uf8ee\n\uf8f9\n=\n+\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\u222b\n=\n2\n1\nlog (log )\n(log )\nx dx\ndx\nx\n+\n\u222b\n\u222b\nIn the first integral, let us take 1 as the second function Then integrating it by\nparts, we get\nI =\n2\n1\nlog (log )\nlog\n(log )\ndx\nx\nx\nx dx\nx\nx\nx\n\u2212\n+\n\u222b\n\u222b\n=\n2\nlog (log )\nlog\n(log )\ndx\ndx\nx\nx\nx\nx\n\u2212\n+\n\u222b\n\u222b (1)\nAgain, consider \nlog\ndx\nx\n\u222b\n, take 1 as the second function and integrate it by parts,\nwe have \n2\n1\n1\n\u2013\n\u2013\nlog\nlog\n(log )\ndx\nx\nx\ndx\nx\nx\nx\nx\n\uf8ee\n\uf8f9\n\uf8f1\n\uf8fc\n\uf8eb\n\uf8f6\n= \uf8ef\n\uf8fa\n\uf8f2\n\uf8fd\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8ef\n\uf8fa\n\uf8f3\n\uf8fe\n\uf8f0\n\uf8fb\n\u222b\n\u222b (2)\n350\nMATHEMATICS\nPutting (2) in (1), we get\n2\n2\nI\nlog (log )\nlog\n(log )\n(log )\nx\ndx\ndx\nx\nx\nx\nx\nx\n=\n\u2212\n\u2212\n+\n\u222b\n\u222b\n = \nlog (log )\nC\nlog\nx\nx\nx\nx\n\u2212\n+\nExample 41 Find \ncot\ntan\nx\nx dx\n\uf8ee\n\uf8f9\n+\n\uf8f0\n\uf8fb\n\u222b\nSolution We have\nI =\ncot\ntan\nx\nx\ndx\n\uf8ee\n\uf8f9\n+\n\uf8f0\n\uf8fb\n\u222b\ntan (1\ncot )\nx\nx dx\n=\n+\n\u222b\nPut tan x = t2, so that sec2 x dx = 2t dt\nor\ndx =  \n4\n2\n1\nt dt\nt\n+\nThen\nI =\n2\n4\n1\n2\n1\n(1\n)\nt\nt\ndt\nt\nt\n\uf8eb\n\uf8f6\n\uf8ec+\n\uf8f7\n+\n\uf8ed\n\uf8f8\n\u222b\n=\n2\n2\n2\n4\n2\n2\n2\n1\n1\n1\n1\n(\n1)\n2\n= 2\n= 2\n1\n1\n1\n2\ndt\ndt\nt\nt\nt\ndt\nt\nt\nt\nt\nt\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n+\n+\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n+\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n\uf8eb\n\uf8f6\n+\n\uf8eb\n\uf8f6\n+\n\u2212\n+\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n\u222b\n\u222b\n\u222b\nPut \n1\nt\nt\n\u2212  = y, so that \n12\n1\nt\n\uf8eb\n\uf8f6\n\uf8ec+\n\uf8f7\n\uf8ed\n\uf8f8  dt = dy"
    },
    {
        "Chapter": "1",
        "sentence_range": "4046-4049",
        "Text": "Then integrating it by\nparts, we get\nI =\n2\n1\nlog (log )\nlog\n(log )\ndx\nx\nx\nx dx\nx\nx\nx\n\u2212\n+\n\u222b\n\u222b\n=\n2\nlog (log )\nlog\n(log )\ndx\ndx\nx\nx\nx\nx\n\u2212\n+\n\u222b\n\u222b (1)\nAgain, consider \nlog\ndx\nx\n\u222b\n, take 1 as the second function and integrate it by parts,\nwe have \n2\n1\n1\n\u2013\n\u2013\nlog\nlog\n(log )\ndx\nx\nx\ndx\nx\nx\nx\nx\n\uf8ee\n\uf8f9\n\uf8f1\n\uf8fc\n\uf8eb\n\uf8f6\n= \uf8ef\n\uf8fa\n\uf8f2\n\uf8fd\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8ef\n\uf8fa\n\uf8f3\n\uf8fe\n\uf8f0\n\uf8fb\n\u222b\n\u222b (2)\n350\nMATHEMATICS\nPutting (2) in (1), we get\n2\n2\nI\nlog (log )\nlog\n(log )\n(log )\nx\ndx\ndx\nx\nx\nx\nx\nx\n=\n\u2212\n\u2212\n+\n\u222b\n\u222b\n = \nlog (log )\nC\nlog\nx\nx\nx\nx\n\u2212\n+\nExample 41 Find \ncot\ntan\nx\nx dx\n\uf8ee\n\uf8f9\n+\n\uf8f0\n\uf8fb\n\u222b\nSolution We have\nI =\ncot\ntan\nx\nx\ndx\n\uf8ee\n\uf8f9\n+\n\uf8f0\n\uf8fb\n\u222b\ntan (1\ncot )\nx\nx dx\n=\n+\n\u222b\nPut tan x = t2, so that sec2 x dx = 2t dt\nor\ndx =  \n4\n2\n1\nt dt\nt\n+\nThen\nI =\n2\n4\n1\n2\n1\n(1\n)\nt\nt\ndt\nt\nt\n\uf8eb\n\uf8f6\n\uf8ec+\n\uf8f7\n+\n\uf8ed\n\uf8f8\n\u222b\n=\n2\n2\n2\n4\n2\n2\n2\n1\n1\n1\n1\n(\n1)\n2\n= 2\n= 2\n1\n1\n1\n2\ndt\ndt\nt\nt\nt\ndt\nt\nt\nt\nt\nt\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n+\n+\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n+\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n\uf8eb\n\uf8f6\n+\n\uf8eb\n\uf8f6\n+\n\u2212\n+\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n\u222b\n\u222b\n\u222b\nPut \n1\nt\nt\n\u2212  = y, so that \n12\n1\nt\n\uf8eb\n\uf8f6\n\uf8ec+\n\uf8f7\n\uf8ed\n\uf8f8  dt = dy Then\nI =\n(\n)\n\u2013 1\n\u2013 1\n2\n2\n1\n2\n2 tan\nC =\n2 tan\nC\n2\n2\n2\nt\ndy\ny\nt\ny\n\uf8eb\n\uf8f6\n\uf8ec\u2212\n\uf8f7\n\uf8ed\n\uf8f8\n=\n+\n+\n+\n\u222b\n=\n2\n\u2013 1\n\u2013 1\n1\ntan\n1\n2 tan\nC =\n2 tan\nC\n2\n2tan\nt\nx\nt\nx\n\uf8eb\n\u2212\uf8f6\n\u2212\n\uf8eb\n\uf8f6\n+\n+\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\nExample 42 Find \n4\nsin 2 cos 2\n9\u2013 cos (2 )\nx\nx dx\nx\n\u222b\nSolution Let \n4\nsin 2 cos 2\nI\n9 \u2013 cos 2\nx\nx dx\nx\n=\u222b\nINTEGRALS         351\nPut cos2 (2x) = t so that 4 sin 2x cos 2x dx = \u2013 dt\nTherefore\n\u20131\n1\n2\n2\n1\n1\n1\n1\nI\n\u2013\n\u2013\nsin\nC\nsin\ncos 2\nC\n4\n4\n3\n4\n3\n9 \u2013\ndt\nt\nx\nt\n\u2212\n\uf8eb\n\uf8f6\n\uf8ee\n\uf8f9\n=\n=\n+\n= \u2212\n+\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8ed\n\uf8f8\n\uf8f0\n\uf8fb\n\u222b\nExample 43 Evaluate \n123\nsin (\n)\nx\nx dx\n\u2212\n\u03c0\n\u222b\nSolution Here f (x) = | x sin \u03c0x | = \nsin\nfor\n1\n1\n3\nsin\nfor 1\n2\nx\nx\nx\nx\nx\nx\n\u03c0\n\u2212 \u2264\n\u2264\n\uf8f1\uf8f4\uf8f2\u2212\n\u03c0\n\u2264\n\u2264\n\uf8f4\uf8f3\nTherefore\n23\n1 |\nsin\n|\nx\nx dx\n\u2212\n\u03c0\n\u222b\n =\n3\n1\n2\n1\n1\nsin\nsin\nx\nx dx\nx\nx dx\n\u2212\n\u03c0\n+\n\u2212\n\u03c0\n\u222b\n\u222b\n=\n3\n1\n2\n1\n1\nsin\nsin\nx\nx dx\nx\nx dx\n\u2212\n\u03c0\n\u2212\n\u03c0\n\u222b\n\u222b\nIntegrating both integrals on righthand side, we get\n23\n1 |\nsin\n|\nx\nx dx\n\u2212\n\u03c0\n\u222b\n =\n3\n1\n2\n2\n2\n1\n1\n\u2013\ncos\nsin\ncos\nsin\nx\nx\nx\nx\nx\nx\n\u2212\n\u03c0\n\u03c0\n\u2212\n\u03c0\n\u03c0\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n+\n\u2212\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u03c0\n\u03c0\n\u03c0\n\u03c0\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n=\n2\n2\n2\n1\n1\n3\n1\n\uf8ee\n\uf8f9\n\u2212 \u2212\n\u2212\n=\n+\n\uf8ef\n\uf8fa\n\u03c0\n\u03c0\n\u03c0\n\u03c0\n\u03c0\n\uf8f0\n\uf8fb\nExample 44 Evaluate \n2\n2\n2\n2\n0\ncos\nsin\nx dx\na\nx\nb\nx\n\u03c0\n+\n\u222b\nSolution Let I = \n2\n2\n2\n2\n2\n2\n2\n2\n0\n0\n(\n)\ncos\nsin\ncos (\n)\nsin (\n)\nx dx\nx dx\na\nx\nb\nx\na\nx\nb\nx\n\u03c0\n\u03c0\n\u03c0 \u2212\n=\n+\n\u03c0 \u2212\n+\n\u03c0 \u2212\n\u222b\n\u222b\n(using P4)\n=\n2\n2\n2\n2\n2\n2\n2\n2\n0\n0\ncos\nsin\ncos\nsin\ndx\nx dx\na\nx\nb\nx\na\nx\nb\nx\n\u03c0\n\u03c0\n\u03c0\n\u2212\n+\n+\n\u222b\n\u222b\n=\n2\n2\n2\n2\n0\nI\ncos\nsin\ndx\na\nx\nb\nx\n\u03c0\n\u03c0\n\u2212\n+\n\u222b\nThus\n2I =\n2\n2\n2\n2\n0\ncos\nsin\ndx\na\nx\nb\nx\n\u03c0\n\u03c0\n+\n\u222b\n352\nMATHEMATICS\nor\nI =\n2\n2\n2\n2\n2\n2\n2\n2\n2\n0\n20\n2\n2\ncos\nsin\ncos\nsin\ndx\ndx\na\nx\nb\nx\na\nx\nb\nx\n\u03c0\n\u03c0\n\u03c0\n=\u03c0\n\u22c5\n+\n+\n\u222b\n\u222b\n(using P6)\n=\n2\n4\n2\n2\n2\n2\n2\n2\n2\n2\n0\n4\ncos\nsin\ncos\nsin\n\u03c0\n\u03c0\n\u03c0\n\uf8ee\n\uf8f9\n\u03c0\n+\n\uf8ef\n\uf8fa\n+\n+\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\u222b\n\u222b\ndx\ndx\na\nx\nb\nx\na\nx\nb\nx\n= \n2\n2\n2\n4\n2\n2\n2\n2\n2\n2\n0\n4\nsec\ncosec\ntan\ncot\n\u03c0\n\u03c0\n\u03c0\n\uf8ee\n\uf8f9\n\u03c0\n+\n\uf8ef\n\uf8fa\n+\n+\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\u222b\n\u222b\nxdx\nxdx\na\nb\nx\na\nx\nb\n= \n(\n)\n0\n1\n2\n2 2\n2\n2\n2\n0\n1\ntan\nt\ncot\n\uf8ee\n\uf8f9\n\u03c0\n\u2212\n=\n=\n\uf8ef\n\uf8fa\n+\n+\n\uf8f0\n\uf8fb\n\u222b\n\u222b\ndt\ndu\nput\nx\nand\nx\nu\na\nb t\na u\nb\n= \n1\n0\n\u20131\n\u20131\n0\n1\ntan\n\u2013\ntan\n\u03c0\n\u03c0\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nbt\nau\nab\na\nab\nb\n =  \n\u20131\n\u20131\ntan\ntan\n\u03c0 \uf8ee\n\uf8f9\n+\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nb\na\nab\na\nb = \n2\n2\n\u03c0\nab\nMiscellaneous Exercise on Chapter 7\nIntegrate the functions in Exercises 1 to 24"
    },
    {
        "Chapter": "1",
        "sentence_range": "4047-4050",
        "Text": "(1)\nAgain, consider \nlog\ndx\nx\n\u222b\n, take 1 as the second function and integrate it by parts,\nwe have \n2\n1\n1\n\u2013\n\u2013\nlog\nlog\n(log )\ndx\nx\nx\ndx\nx\nx\nx\nx\n\uf8ee\n\uf8f9\n\uf8f1\n\uf8fc\n\uf8eb\n\uf8f6\n= \uf8ef\n\uf8fa\n\uf8f2\n\uf8fd\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8ef\n\uf8fa\n\uf8f3\n\uf8fe\n\uf8f0\n\uf8fb\n\u222b\n\u222b (2)\n350\nMATHEMATICS\nPutting (2) in (1), we get\n2\n2\nI\nlog (log )\nlog\n(log )\n(log )\nx\ndx\ndx\nx\nx\nx\nx\nx\n=\n\u2212\n\u2212\n+\n\u222b\n\u222b\n = \nlog (log )\nC\nlog\nx\nx\nx\nx\n\u2212\n+\nExample 41 Find \ncot\ntan\nx\nx dx\n\uf8ee\n\uf8f9\n+\n\uf8f0\n\uf8fb\n\u222b\nSolution We have\nI =\ncot\ntan\nx\nx\ndx\n\uf8ee\n\uf8f9\n+\n\uf8f0\n\uf8fb\n\u222b\ntan (1\ncot )\nx\nx dx\n=\n+\n\u222b\nPut tan x = t2, so that sec2 x dx = 2t dt\nor\ndx =  \n4\n2\n1\nt dt\nt\n+\nThen\nI =\n2\n4\n1\n2\n1\n(1\n)\nt\nt\ndt\nt\nt\n\uf8eb\n\uf8f6\n\uf8ec+\n\uf8f7\n+\n\uf8ed\n\uf8f8\n\u222b\n=\n2\n2\n2\n4\n2\n2\n2\n1\n1\n1\n1\n(\n1)\n2\n= 2\n= 2\n1\n1\n1\n2\ndt\ndt\nt\nt\nt\ndt\nt\nt\nt\nt\nt\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n+\n+\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n+\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n\uf8eb\n\uf8f6\n+\n\uf8eb\n\uf8f6\n+\n\u2212\n+\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n\u222b\n\u222b\n\u222b\nPut \n1\nt\nt\n\u2212  = y, so that \n12\n1\nt\n\uf8eb\n\uf8f6\n\uf8ec+\n\uf8f7\n\uf8ed\n\uf8f8  dt = dy Then\nI =\n(\n)\n\u2013 1\n\u2013 1\n2\n2\n1\n2\n2 tan\nC =\n2 tan\nC\n2\n2\n2\nt\ndy\ny\nt\ny\n\uf8eb\n\uf8f6\n\uf8ec\u2212\n\uf8f7\n\uf8ed\n\uf8f8\n=\n+\n+\n+\n\u222b\n=\n2\n\u2013 1\n\u2013 1\n1\ntan\n1\n2 tan\nC =\n2 tan\nC\n2\n2tan\nt\nx\nt\nx\n\uf8eb\n\u2212\uf8f6\n\u2212\n\uf8eb\n\uf8f6\n+\n+\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\nExample 42 Find \n4\nsin 2 cos 2\n9\u2013 cos (2 )\nx\nx dx\nx\n\u222b\nSolution Let \n4\nsin 2 cos 2\nI\n9 \u2013 cos 2\nx\nx dx\nx\n=\u222b\nINTEGRALS         351\nPut cos2 (2x) = t so that 4 sin 2x cos 2x dx = \u2013 dt\nTherefore\n\u20131\n1\n2\n2\n1\n1\n1\n1\nI\n\u2013\n\u2013\nsin\nC\nsin\ncos 2\nC\n4\n4\n3\n4\n3\n9 \u2013\ndt\nt\nx\nt\n\u2212\n\uf8eb\n\uf8f6\n\uf8ee\n\uf8f9\n=\n=\n+\n= \u2212\n+\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8ed\n\uf8f8\n\uf8f0\n\uf8fb\n\u222b\nExample 43 Evaluate \n123\nsin (\n)\nx\nx dx\n\u2212\n\u03c0\n\u222b\nSolution Here f (x) = | x sin \u03c0x | = \nsin\nfor\n1\n1\n3\nsin\nfor 1\n2\nx\nx\nx\nx\nx\nx\n\u03c0\n\u2212 \u2264\n\u2264\n\uf8f1\uf8f4\uf8f2\u2212\n\u03c0\n\u2264\n\u2264\n\uf8f4\uf8f3\nTherefore\n23\n1 |\nsin\n|\nx\nx dx\n\u2212\n\u03c0\n\u222b\n =\n3\n1\n2\n1\n1\nsin\nsin\nx\nx dx\nx\nx dx\n\u2212\n\u03c0\n+\n\u2212\n\u03c0\n\u222b\n\u222b\n=\n3\n1\n2\n1\n1\nsin\nsin\nx\nx dx\nx\nx dx\n\u2212\n\u03c0\n\u2212\n\u03c0\n\u222b\n\u222b\nIntegrating both integrals on righthand side, we get\n23\n1 |\nsin\n|\nx\nx dx\n\u2212\n\u03c0\n\u222b\n =\n3\n1\n2\n2\n2\n1\n1\n\u2013\ncos\nsin\ncos\nsin\nx\nx\nx\nx\nx\nx\n\u2212\n\u03c0\n\u03c0\n\u2212\n\u03c0\n\u03c0\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n+\n\u2212\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u03c0\n\u03c0\n\u03c0\n\u03c0\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n=\n2\n2\n2\n1\n1\n3\n1\n\uf8ee\n\uf8f9\n\u2212 \u2212\n\u2212\n=\n+\n\uf8ef\n\uf8fa\n\u03c0\n\u03c0\n\u03c0\n\u03c0\n\u03c0\n\uf8f0\n\uf8fb\nExample 44 Evaluate \n2\n2\n2\n2\n0\ncos\nsin\nx dx\na\nx\nb\nx\n\u03c0\n+\n\u222b\nSolution Let I = \n2\n2\n2\n2\n2\n2\n2\n2\n0\n0\n(\n)\ncos\nsin\ncos (\n)\nsin (\n)\nx dx\nx dx\na\nx\nb\nx\na\nx\nb\nx\n\u03c0\n\u03c0\n\u03c0 \u2212\n=\n+\n\u03c0 \u2212\n+\n\u03c0 \u2212\n\u222b\n\u222b\n(using P4)\n=\n2\n2\n2\n2\n2\n2\n2\n2\n0\n0\ncos\nsin\ncos\nsin\ndx\nx dx\na\nx\nb\nx\na\nx\nb\nx\n\u03c0\n\u03c0\n\u03c0\n\u2212\n+\n+\n\u222b\n\u222b\n=\n2\n2\n2\n2\n0\nI\ncos\nsin\ndx\na\nx\nb\nx\n\u03c0\n\u03c0\n\u2212\n+\n\u222b\nThus\n2I =\n2\n2\n2\n2\n0\ncos\nsin\ndx\na\nx\nb\nx\n\u03c0\n\u03c0\n+\n\u222b\n352\nMATHEMATICS\nor\nI =\n2\n2\n2\n2\n2\n2\n2\n2\n2\n0\n20\n2\n2\ncos\nsin\ncos\nsin\ndx\ndx\na\nx\nb\nx\na\nx\nb\nx\n\u03c0\n\u03c0\n\u03c0\n=\u03c0\n\u22c5\n+\n+\n\u222b\n\u222b\n(using P6)\n=\n2\n4\n2\n2\n2\n2\n2\n2\n2\n2\n0\n4\ncos\nsin\ncos\nsin\n\u03c0\n\u03c0\n\u03c0\n\uf8ee\n\uf8f9\n\u03c0\n+\n\uf8ef\n\uf8fa\n+\n+\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\u222b\n\u222b\ndx\ndx\na\nx\nb\nx\na\nx\nb\nx\n= \n2\n2\n2\n4\n2\n2\n2\n2\n2\n2\n0\n4\nsec\ncosec\ntan\ncot\n\u03c0\n\u03c0\n\u03c0\n\uf8ee\n\uf8f9\n\u03c0\n+\n\uf8ef\n\uf8fa\n+\n+\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\u222b\n\u222b\nxdx\nxdx\na\nb\nx\na\nx\nb\n= \n(\n)\n0\n1\n2\n2 2\n2\n2\n2\n0\n1\ntan\nt\ncot\n\uf8ee\n\uf8f9\n\u03c0\n\u2212\n=\n=\n\uf8ef\n\uf8fa\n+\n+\n\uf8f0\n\uf8fb\n\u222b\n\u222b\ndt\ndu\nput\nx\nand\nx\nu\na\nb t\na u\nb\n= \n1\n0\n\u20131\n\u20131\n0\n1\ntan\n\u2013\ntan\n\u03c0\n\u03c0\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nbt\nau\nab\na\nab\nb\n =  \n\u20131\n\u20131\ntan\ntan\n\u03c0 \uf8ee\n\uf8f9\n+\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nb\na\nab\na\nb = \n2\n2\n\u03c0\nab\nMiscellaneous Exercise on Chapter 7\nIntegrate the functions in Exercises 1 to 24 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "4048-4051",
        "Text": "(2)\n350\nMATHEMATICS\nPutting (2) in (1), we get\n2\n2\nI\nlog (log )\nlog\n(log )\n(log )\nx\ndx\ndx\nx\nx\nx\nx\nx\n=\n\u2212\n\u2212\n+\n\u222b\n\u222b\n = \nlog (log )\nC\nlog\nx\nx\nx\nx\n\u2212\n+\nExample 41 Find \ncot\ntan\nx\nx dx\n\uf8ee\n\uf8f9\n+\n\uf8f0\n\uf8fb\n\u222b\nSolution We have\nI =\ncot\ntan\nx\nx\ndx\n\uf8ee\n\uf8f9\n+\n\uf8f0\n\uf8fb\n\u222b\ntan (1\ncot )\nx\nx dx\n=\n+\n\u222b\nPut tan x = t2, so that sec2 x dx = 2t dt\nor\ndx =  \n4\n2\n1\nt dt\nt\n+\nThen\nI =\n2\n4\n1\n2\n1\n(1\n)\nt\nt\ndt\nt\nt\n\uf8eb\n\uf8f6\n\uf8ec+\n\uf8f7\n+\n\uf8ed\n\uf8f8\n\u222b\n=\n2\n2\n2\n4\n2\n2\n2\n1\n1\n1\n1\n(\n1)\n2\n= 2\n= 2\n1\n1\n1\n2\ndt\ndt\nt\nt\nt\ndt\nt\nt\nt\nt\nt\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n+\n+\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n+\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n\uf8eb\n\uf8f6\n+\n\uf8eb\n\uf8f6\n+\n\u2212\n+\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n\u222b\n\u222b\n\u222b\nPut \n1\nt\nt\n\u2212  = y, so that \n12\n1\nt\n\uf8eb\n\uf8f6\n\uf8ec+\n\uf8f7\n\uf8ed\n\uf8f8  dt = dy Then\nI =\n(\n)\n\u2013 1\n\u2013 1\n2\n2\n1\n2\n2 tan\nC =\n2 tan\nC\n2\n2\n2\nt\ndy\ny\nt\ny\n\uf8eb\n\uf8f6\n\uf8ec\u2212\n\uf8f7\n\uf8ed\n\uf8f8\n=\n+\n+\n+\n\u222b\n=\n2\n\u2013 1\n\u2013 1\n1\ntan\n1\n2 tan\nC =\n2 tan\nC\n2\n2tan\nt\nx\nt\nx\n\uf8eb\n\u2212\uf8f6\n\u2212\n\uf8eb\n\uf8f6\n+\n+\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\nExample 42 Find \n4\nsin 2 cos 2\n9\u2013 cos (2 )\nx\nx dx\nx\n\u222b\nSolution Let \n4\nsin 2 cos 2\nI\n9 \u2013 cos 2\nx\nx dx\nx\n=\u222b\nINTEGRALS         351\nPut cos2 (2x) = t so that 4 sin 2x cos 2x dx = \u2013 dt\nTherefore\n\u20131\n1\n2\n2\n1\n1\n1\n1\nI\n\u2013\n\u2013\nsin\nC\nsin\ncos 2\nC\n4\n4\n3\n4\n3\n9 \u2013\ndt\nt\nx\nt\n\u2212\n\uf8eb\n\uf8f6\n\uf8ee\n\uf8f9\n=\n=\n+\n= \u2212\n+\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8ed\n\uf8f8\n\uf8f0\n\uf8fb\n\u222b\nExample 43 Evaluate \n123\nsin (\n)\nx\nx dx\n\u2212\n\u03c0\n\u222b\nSolution Here f (x) = | x sin \u03c0x | = \nsin\nfor\n1\n1\n3\nsin\nfor 1\n2\nx\nx\nx\nx\nx\nx\n\u03c0\n\u2212 \u2264\n\u2264\n\uf8f1\uf8f4\uf8f2\u2212\n\u03c0\n\u2264\n\u2264\n\uf8f4\uf8f3\nTherefore\n23\n1 |\nsin\n|\nx\nx dx\n\u2212\n\u03c0\n\u222b\n =\n3\n1\n2\n1\n1\nsin\nsin\nx\nx dx\nx\nx dx\n\u2212\n\u03c0\n+\n\u2212\n\u03c0\n\u222b\n\u222b\n=\n3\n1\n2\n1\n1\nsin\nsin\nx\nx dx\nx\nx dx\n\u2212\n\u03c0\n\u2212\n\u03c0\n\u222b\n\u222b\nIntegrating both integrals on righthand side, we get\n23\n1 |\nsin\n|\nx\nx dx\n\u2212\n\u03c0\n\u222b\n =\n3\n1\n2\n2\n2\n1\n1\n\u2013\ncos\nsin\ncos\nsin\nx\nx\nx\nx\nx\nx\n\u2212\n\u03c0\n\u03c0\n\u2212\n\u03c0\n\u03c0\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n+\n\u2212\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u03c0\n\u03c0\n\u03c0\n\u03c0\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n=\n2\n2\n2\n1\n1\n3\n1\n\uf8ee\n\uf8f9\n\u2212 \u2212\n\u2212\n=\n+\n\uf8ef\n\uf8fa\n\u03c0\n\u03c0\n\u03c0\n\u03c0\n\u03c0\n\uf8f0\n\uf8fb\nExample 44 Evaluate \n2\n2\n2\n2\n0\ncos\nsin\nx dx\na\nx\nb\nx\n\u03c0\n+\n\u222b\nSolution Let I = \n2\n2\n2\n2\n2\n2\n2\n2\n0\n0\n(\n)\ncos\nsin\ncos (\n)\nsin (\n)\nx dx\nx dx\na\nx\nb\nx\na\nx\nb\nx\n\u03c0\n\u03c0\n\u03c0 \u2212\n=\n+\n\u03c0 \u2212\n+\n\u03c0 \u2212\n\u222b\n\u222b\n(using P4)\n=\n2\n2\n2\n2\n2\n2\n2\n2\n0\n0\ncos\nsin\ncos\nsin\ndx\nx dx\na\nx\nb\nx\na\nx\nb\nx\n\u03c0\n\u03c0\n\u03c0\n\u2212\n+\n+\n\u222b\n\u222b\n=\n2\n2\n2\n2\n0\nI\ncos\nsin\ndx\na\nx\nb\nx\n\u03c0\n\u03c0\n\u2212\n+\n\u222b\nThus\n2I =\n2\n2\n2\n2\n0\ncos\nsin\ndx\na\nx\nb\nx\n\u03c0\n\u03c0\n+\n\u222b\n352\nMATHEMATICS\nor\nI =\n2\n2\n2\n2\n2\n2\n2\n2\n2\n0\n20\n2\n2\ncos\nsin\ncos\nsin\ndx\ndx\na\nx\nb\nx\na\nx\nb\nx\n\u03c0\n\u03c0\n\u03c0\n=\u03c0\n\u22c5\n+\n+\n\u222b\n\u222b\n(using P6)\n=\n2\n4\n2\n2\n2\n2\n2\n2\n2\n2\n0\n4\ncos\nsin\ncos\nsin\n\u03c0\n\u03c0\n\u03c0\n\uf8ee\n\uf8f9\n\u03c0\n+\n\uf8ef\n\uf8fa\n+\n+\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\u222b\n\u222b\ndx\ndx\na\nx\nb\nx\na\nx\nb\nx\n= \n2\n2\n2\n4\n2\n2\n2\n2\n2\n2\n0\n4\nsec\ncosec\ntan\ncot\n\u03c0\n\u03c0\n\u03c0\n\uf8ee\n\uf8f9\n\u03c0\n+\n\uf8ef\n\uf8fa\n+\n+\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\u222b\n\u222b\nxdx\nxdx\na\nb\nx\na\nx\nb\n= \n(\n)\n0\n1\n2\n2 2\n2\n2\n2\n0\n1\ntan\nt\ncot\n\uf8ee\n\uf8f9\n\u03c0\n\u2212\n=\n=\n\uf8ef\n\uf8fa\n+\n+\n\uf8f0\n\uf8fb\n\u222b\n\u222b\ndt\ndu\nput\nx\nand\nx\nu\na\nb t\na u\nb\n= \n1\n0\n\u20131\n\u20131\n0\n1\ntan\n\u2013\ntan\n\u03c0\n\u03c0\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nbt\nau\nab\na\nab\nb\n =  \n\u20131\n\u20131\ntan\ntan\n\u03c0 \uf8ee\n\uf8f9\n+\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nb\na\nab\na\nb = \n2\n2\n\u03c0\nab\nMiscellaneous Exercise on Chapter 7\nIntegrate the functions in Exercises 1 to 24 1 3\n1\nx\n\u2212x\n2"
    },
    {
        "Chapter": "1",
        "sentence_range": "4049-4052",
        "Text": "Then\nI =\n(\n)\n\u2013 1\n\u2013 1\n2\n2\n1\n2\n2 tan\nC =\n2 tan\nC\n2\n2\n2\nt\ndy\ny\nt\ny\n\uf8eb\n\uf8f6\n\uf8ec\u2212\n\uf8f7\n\uf8ed\n\uf8f8\n=\n+\n+\n+\n\u222b\n=\n2\n\u2013 1\n\u2013 1\n1\ntan\n1\n2 tan\nC =\n2 tan\nC\n2\n2tan\nt\nx\nt\nx\n\uf8eb\n\u2212\uf8f6\n\u2212\n\uf8eb\n\uf8f6\n+\n+\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\nExample 42 Find \n4\nsin 2 cos 2\n9\u2013 cos (2 )\nx\nx dx\nx\n\u222b\nSolution Let \n4\nsin 2 cos 2\nI\n9 \u2013 cos 2\nx\nx dx\nx\n=\u222b\nINTEGRALS         351\nPut cos2 (2x) = t so that 4 sin 2x cos 2x dx = \u2013 dt\nTherefore\n\u20131\n1\n2\n2\n1\n1\n1\n1\nI\n\u2013\n\u2013\nsin\nC\nsin\ncos 2\nC\n4\n4\n3\n4\n3\n9 \u2013\ndt\nt\nx\nt\n\u2212\n\uf8eb\n\uf8f6\n\uf8ee\n\uf8f9\n=\n=\n+\n= \u2212\n+\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8ed\n\uf8f8\n\uf8f0\n\uf8fb\n\u222b\nExample 43 Evaluate \n123\nsin (\n)\nx\nx dx\n\u2212\n\u03c0\n\u222b\nSolution Here f (x) = | x sin \u03c0x | = \nsin\nfor\n1\n1\n3\nsin\nfor 1\n2\nx\nx\nx\nx\nx\nx\n\u03c0\n\u2212 \u2264\n\u2264\n\uf8f1\uf8f4\uf8f2\u2212\n\u03c0\n\u2264\n\u2264\n\uf8f4\uf8f3\nTherefore\n23\n1 |\nsin\n|\nx\nx dx\n\u2212\n\u03c0\n\u222b\n =\n3\n1\n2\n1\n1\nsin\nsin\nx\nx dx\nx\nx dx\n\u2212\n\u03c0\n+\n\u2212\n\u03c0\n\u222b\n\u222b\n=\n3\n1\n2\n1\n1\nsin\nsin\nx\nx dx\nx\nx dx\n\u2212\n\u03c0\n\u2212\n\u03c0\n\u222b\n\u222b\nIntegrating both integrals on righthand side, we get\n23\n1 |\nsin\n|\nx\nx dx\n\u2212\n\u03c0\n\u222b\n =\n3\n1\n2\n2\n2\n1\n1\n\u2013\ncos\nsin\ncos\nsin\nx\nx\nx\nx\nx\nx\n\u2212\n\u03c0\n\u03c0\n\u2212\n\u03c0\n\u03c0\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n+\n\u2212\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\u03c0\n\u03c0\n\u03c0\n\u03c0\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n=\n2\n2\n2\n1\n1\n3\n1\n\uf8ee\n\uf8f9\n\u2212 \u2212\n\u2212\n=\n+\n\uf8ef\n\uf8fa\n\u03c0\n\u03c0\n\u03c0\n\u03c0\n\u03c0\n\uf8f0\n\uf8fb\nExample 44 Evaluate \n2\n2\n2\n2\n0\ncos\nsin\nx dx\na\nx\nb\nx\n\u03c0\n+\n\u222b\nSolution Let I = \n2\n2\n2\n2\n2\n2\n2\n2\n0\n0\n(\n)\ncos\nsin\ncos (\n)\nsin (\n)\nx dx\nx dx\na\nx\nb\nx\na\nx\nb\nx\n\u03c0\n\u03c0\n\u03c0 \u2212\n=\n+\n\u03c0 \u2212\n+\n\u03c0 \u2212\n\u222b\n\u222b\n(using P4)\n=\n2\n2\n2\n2\n2\n2\n2\n2\n0\n0\ncos\nsin\ncos\nsin\ndx\nx dx\na\nx\nb\nx\na\nx\nb\nx\n\u03c0\n\u03c0\n\u03c0\n\u2212\n+\n+\n\u222b\n\u222b\n=\n2\n2\n2\n2\n0\nI\ncos\nsin\ndx\na\nx\nb\nx\n\u03c0\n\u03c0\n\u2212\n+\n\u222b\nThus\n2I =\n2\n2\n2\n2\n0\ncos\nsin\ndx\na\nx\nb\nx\n\u03c0\n\u03c0\n+\n\u222b\n352\nMATHEMATICS\nor\nI =\n2\n2\n2\n2\n2\n2\n2\n2\n2\n0\n20\n2\n2\ncos\nsin\ncos\nsin\ndx\ndx\na\nx\nb\nx\na\nx\nb\nx\n\u03c0\n\u03c0\n\u03c0\n=\u03c0\n\u22c5\n+\n+\n\u222b\n\u222b\n(using P6)\n=\n2\n4\n2\n2\n2\n2\n2\n2\n2\n2\n0\n4\ncos\nsin\ncos\nsin\n\u03c0\n\u03c0\n\u03c0\n\uf8ee\n\uf8f9\n\u03c0\n+\n\uf8ef\n\uf8fa\n+\n+\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\u222b\n\u222b\ndx\ndx\na\nx\nb\nx\na\nx\nb\nx\n= \n2\n2\n2\n4\n2\n2\n2\n2\n2\n2\n0\n4\nsec\ncosec\ntan\ncot\n\u03c0\n\u03c0\n\u03c0\n\uf8ee\n\uf8f9\n\u03c0\n+\n\uf8ef\n\uf8fa\n+\n+\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\u222b\n\u222b\nxdx\nxdx\na\nb\nx\na\nx\nb\n= \n(\n)\n0\n1\n2\n2 2\n2\n2\n2\n0\n1\ntan\nt\ncot\n\uf8ee\n\uf8f9\n\u03c0\n\u2212\n=\n=\n\uf8ef\n\uf8fa\n+\n+\n\uf8f0\n\uf8fb\n\u222b\n\u222b\ndt\ndu\nput\nx\nand\nx\nu\na\nb t\na u\nb\n= \n1\n0\n\u20131\n\u20131\n0\n1\ntan\n\u2013\ntan\n\u03c0\n\u03c0\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\nbt\nau\nab\na\nab\nb\n =  \n\u20131\n\u20131\ntan\ntan\n\u03c0 \uf8ee\n\uf8f9\n+\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nb\na\nab\na\nb = \n2\n2\n\u03c0\nab\nMiscellaneous Exercise on Chapter 7\nIntegrate the functions in Exercises 1 to 24 1 3\n1\nx\n\u2212x\n2 1\nx\na\nx\nb\n+\n+\n+\n3"
    },
    {
        "Chapter": "1",
        "sentence_range": "4050-4053",
        "Text": "1 3\n1\nx\n\u2212x\n2 1\nx\na\nx\nb\n+\n+\n+\n3 2\n1\nx\nax\nx\n\u2212\n [Hint: Put x = a\nt ]\n4"
    },
    {
        "Chapter": "1",
        "sentence_range": "4051-4054",
        "Text": "3\n1\nx\n\u2212x\n2 1\nx\na\nx\nb\n+\n+\n+\n3 2\n1\nx\nax\nx\n\u2212\n [Hint: Put x = a\nt ]\n4 3\n2\n4\n4\n1\n(\n1)\nx\nx +\n5"
    },
    {
        "Chapter": "1",
        "sentence_range": "4052-4055",
        "Text": "1\nx\na\nx\nb\n+\n+\n+\n3 2\n1\nx\nax\nx\n\u2212\n [Hint: Put x = a\nt ]\n4 3\n2\n4\n4\n1\n(\n1)\nx\nx +\n5 1\n1\n3\n2\n1\nx\nx\n+\n      [Hint:\n1\n1\n1\n1\n3\n2\n3\n6\n1\n1\n1\nx\nx\nx\nx\n=\n\uf8eb\n\uf8f6\n+\n\uf8ec+\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n, put x = t6]\n6"
    },
    {
        "Chapter": "1",
        "sentence_range": "4053-4056",
        "Text": "2\n1\nx\nax\nx\n\u2212\n [Hint: Put x = a\nt ]\n4 3\n2\n4\n4\n1\n(\n1)\nx\nx +\n5 1\n1\n3\n2\n1\nx\nx\n+\n      [Hint:\n1\n1\n1\n1\n3\n2\n3\n6\n1\n1\n1\nx\nx\nx\nx\n=\n\uf8eb\n\uf8f6\n+\n\uf8ec+\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n, put x = t6]\n6 2\n5\n(\n1) (\n9)\nx\nx\nx\n+\n+\n7"
    },
    {
        "Chapter": "1",
        "sentence_range": "4054-4057",
        "Text": "3\n2\n4\n4\n1\n(\n1)\nx\nx +\n5 1\n1\n3\n2\n1\nx\nx\n+\n      [Hint:\n1\n1\n1\n1\n3\n2\n3\n6\n1\n1\n1\nx\nx\nx\nx\n=\n\uf8eb\n\uf8f6\n+\n\uf8ec+\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n, put x = t6]\n6 2\n5\n(\n1) (\n9)\nx\nx\nx\n+\n+\n7 sin\nsin (\n)\nx\nx\n\u2212a\n8"
    },
    {
        "Chapter": "1",
        "sentence_range": "4055-4058",
        "Text": "1\n1\n3\n2\n1\nx\nx\n+\n      [Hint:\n1\n1\n1\n1\n3\n2\n3\n6\n1\n1\n1\nx\nx\nx\nx\n=\n\uf8eb\n\uf8f6\n+\n\uf8ec+\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n, put x = t6]\n6 2\n5\n(\n1) (\n9)\nx\nx\nx\n+\n+\n7 sin\nsin (\n)\nx\nx\n\u2212a\n8 5 log\n4 log\n3 log\n2 log\nx\nx\nx\nx\ne\ne\ne\n\u2212e\n\u2212\n9"
    },
    {
        "Chapter": "1",
        "sentence_range": "4056-4059",
        "Text": "2\n5\n(\n1) (\n9)\nx\nx\nx\n+\n+\n7 sin\nsin (\n)\nx\nx\n\u2212a\n8 5 log\n4 log\n3 log\n2 log\nx\nx\nx\nx\ne\ne\ne\n\u2212e\n\u2212\n9 2\ncos\n4\nsin\nx\nx\n\u2212\n10"
    },
    {
        "Chapter": "1",
        "sentence_range": "4057-4060",
        "Text": "sin\nsin (\n)\nx\nx\n\u2212a\n8 5 log\n4 log\n3 log\n2 log\nx\nx\nx\nx\ne\ne\ne\n\u2212e\n\u2212\n9 2\ncos\n4\nsin\nx\nx\n\u2212\n10 8\n8\n2\n2\nsin\ncos\n1 2sin\ncos\nx\nx\nx\n\u2212\n\u2212\n11"
    },
    {
        "Chapter": "1",
        "sentence_range": "4058-4061",
        "Text": "5 log\n4 log\n3 log\n2 log\nx\nx\nx\nx\ne\ne\ne\n\u2212e\n\u2212\n9 2\ncos\n4\nsin\nx\nx\n\u2212\n10 8\n8\n2\n2\nsin\ncos\n1 2sin\ncos\nx\nx\nx\n\u2212\n\u2212\n11 1\ncos (\n) cos (\n)\nx\na\nx\nb\n+\n+\n12"
    },
    {
        "Chapter": "1",
        "sentence_range": "4059-4062",
        "Text": "2\ncos\n4\nsin\nx\nx\n\u2212\n10 8\n8\n2\n2\nsin\ncos\n1 2sin\ncos\nx\nx\nx\n\u2212\n\u2212\n11 1\ncos (\n) cos (\n)\nx\na\nx\nb\n+\n+\n12 3\n8\n1\nx\nx\n\u2212\n13"
    },
    {
        "Chapter": "1",
        "sentence_range": "4060-4063",
        "Text": "8\n8\n2\n2\nsin\ncos\n1 2sin\ncos\nx\nx\nx\n\u2212\n\u2212\n11 1\ncos (\n) cos (\n)\nx\na\nx\nb\n+\n+\n12 3\n8\n1\nx\nx\n\u2212\n13 (1\n) (2\n)\nx\nx\nx\ne\ne\ne\n+\n+\n14"
    },
    {
        "Chapter": "1",
        "sentence_range": "4061-4064",
        "Text": "1\ncos (\n) cos (\n)\nx\na\nx\nb\n+\n+\n12 3\n8\n1\nx\nx\n\u2212\n13 (1\n) (2\n)\nx\nx\nx\ne\ne\ne\n+\n+\n14 2\n2\n1\n(\n1) (\n4)\nx\nx\n+\n+\n15"
    },
    {
        "Chapter": "1",
        "sentence_range": "4062-4065",
        "Text": "3\n8\n1\nx\nx\n\u2212\n13 (1\n) (2\n)\nx\nx\nx\ne\ne\ne\n+\n+\n14 2\n2\n1\n(\n1) (\n4)\nx\nx\n+\n+\n15 cos 3x elog sinx\n16"
    },
    {
        "Chapter": "1",
        "sentence_range": "4063-4066",
        "Text": "(1\n) (2\n)\nx\nx\nx\ne\ne\ne\n+\n+\n14 2\n2\n1\n(\n1) (\n4)\nx\nx\n+\n+\n15 cos 3x elog sinx\n16 e3 logx (x4 + 1)\u2013 1\n17"
    },
    {
        "Chapter": "1",
        "sentence_range": "4064-4067",
        "Text": "2\n2\n1\n(\n1) (\n4)\nx\nx\n+\n+\n15 cos 3x elog sinx\n16 e3 logx (x4 + 1)\u2013 1\n17 f \u2032 (ax + b) [f (ax + b)] n\n18"
    },
    {
        "Chapter": "1",
        "sentence_range": "4065-4068",
        "Text": "cos 3x elog sinx\n16 e3 logx (x4 + 1)\u2013 1\n17 f \u2032 (ax + b) [f (ax + b)] n\n18 3\n1\nsin\nsin (\n)\nx\nx + \u03b1\n19"
    },
    {
        "Chapter": "1",
        "sentence_range": "4066-4069",
        "Text": "e3 logx (x4 + 1)\u2013 1\n17 f \u2032 (ax + b) [f (ax + b)] n\n18 3\n1\nsin\nsin (\n)\nx\nx + \u03b1\n19 1\n1\n1\n1\nsin\ncos\nsin\ncos\nx\nx\nx\nx\n\u2212\n\u2212\n\u2212\n\u2212\n\u2212\n+\n, x \u2208 [0, 1]\nINTEGRALS         353\n20"
    },
    {
        "Chapter": "1",
        "sentence_range": "4067-4070",
        "Text": "f \u2032 (ax + b) [f (ax + b)] n\n18 3\n1\nsin\nsin (\n)\nx\nx + \u03b1\n19 1\n1\n1\n1\nsin\ncos\nsin\ncos\nx\nx\nx\nx\n\u2212\n\u2212\n\u2212\n\u2212\n\u2212\n+\n, x \u2208 [0, 1]\nINTEGRALS         353\n20 1\n1\nx\nx\n\u2212\n+\n21"
    },
    {
        "Chapter": "1",
        "sentence_range": "4068-4071",
        "Text": "3\n1\nsin\nsin (\n)\nx\nx + \u03b1\n19 1\n1\n1\n1\nsin\ncos\nsin\ncos\nx\nx\nx\nx\n\u2212\n\u2212\n\u2212\n\u2212\n\u2212\n+\n, x \u2208 [0, 1]\nINTEGRALS         353\n20 1\n1\nx\nx\n\u2212\n+\n21 2\nsin 2\n1\ncos2\nx ex\nx\n++\n22"
    },
    {
        "Chapter": "1",
        "sentence_range": "4069-4072",
        "Text": "1\n1\n1\n1\nsin\ncos\nsin\ncos\nx\nx\nx\nx\n\u2212\n\u2212\n\u2212\n\u2212\n\u2212\n+\n, x \u2208 [0, 1]\nINTEGRALS         353\n20 1\n1\nx\nx\n\u2212\n+\n21 2\nsin 2\n1\ncos2\nx ex\nx\n++\n22 2\n2\n1\n(\n1) (\n2)\nx\nx\nx\n+x\n+\n+\n+\n23"
    },
    {
        "Chapter": "1",
        "sentence_range": "4070-4073",
        "Text": "1\n1\nx\nx\n\u2212\n+\n21 2\nsin 2\n1\ncos2\nx ex\nx\n++\n22 2\n2\n1\n(\n1) (\n2)\nx\nx\nx\n+x\n+\n+\n+\n23 \u2013 1\n1\ntan\n1\nx\nx\n+\u2212\n24"
    },
    {
        "Chapter": "1",
        "sentence_range": "4071-4074",
        "Text": "2\nsin 2\n1\ncos2\nx ex\nx\n++\n22 2\n2\n1\n(\n1) (\n2)\nx\nx\nx\n+x\n+\n+\n+\n23 \u2013 1\n1\ntan\n1\nx\nx\n+\u2212\n24 2\n2\n4\n1 log (\n1)\n2 log\nx\nx\nx\nx\n\uf8ee\n\uf8f9\n+\n+\n\u2212\n\uf8f0\n\uf8fb\nEvaluate the definite integrals in Exercises 25 to 33"
    },
    {
        "Chapter": "1",
        "sentence_range": "4072-4075",
        "Text": "2\n2\n1\n(\n1) (\n2)\nx\nx\nx\n+x\n+\n+\n+\n23 \u2013 1\n1\ntan\n1\nx\nx\n+\u2212\n24 2\n2\n4\n1 log (\n1)\n2 log\nx\nx\nx\nx\n\uf8ee\n\uf8f9\n+\n+\n\u2212\n\uf8f0\n\uf8fb\nEvaluate the definite integrals in Exercises 25 to 33 25"
    },
    {
        "Chapter": "1",
        "sentence_range": "4073-4076",
        "Text": "\u2013 1\n1\ntan\n1\nx\nx\n+\u2212\n24 2\n2\n4\n1 log (\n1)\n2 log\nx\nx\nx\nx\n\uf8ee\n\uf8f9\n+\n+\n\u2212\n\uf8f0\n\uf8fb\nEvaluate the definite integrals in Exercises 25 to 33 25 2\n1\nsin\n1 cos\n\u03c0\n\u03c0\n\uf8eb\u2212\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\u2212\n\uf8f8\n\u222b\nx\nx\ne\nxdx\n26"
    },
    {
        "Chapter": "1",
        "sentence_range": "4074-4077",
        "Text": "2\n2\n4\n1 log (\n1)\n2 log\nx\nx\nx\nx\n\uf8ee\n\uf8f9\n+\n+\n\u2212\n\uf8f0\n\uf8fb\nEvaluate the definite integrals in Exercises 25 to 33 25 2\n1\nsin\n1 cos\n\u03c0\n\u03c0\n\uf8eb\u2212\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\u2212\n\uf8f8\n\u222b\nx\nx\ne\nxdx\n26 4\n4\n4\n0\nsin\ncos\ncos\nsin\nx\nx\ndx\nx\nx\n\u03c0\n+\n\u222b\n27"
    },
    {
        "Chapter": "1",
        "sentence_range": "4075-4078",
        "Text": "25 2\n1\nsin\n1 cos\n\u03c0\n\u03c0\n\uf8eb\u2212\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\u2212\n\uf8f8\n\u222b\nx\nx\ne\nxdx\n26 4\n4\n4\n0\nsin\ncos\ncos\nsin\nx\nx\ndx\nx\nx\n\u03c0\n+\n\u222b\n27 2\n2\n2\n2\n0\ncos\ncos\n4 sin\nx dx\nx\nx\n\u03c0\n+\n\u222b\n28"
    },
    {
        "Chapter": "1",
        "sentence_range": "4076-4079",
        "Text": "2\n1\nsin\n1 cos\n\u03c0\n\u03c0\n\uf8eb\u2212\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\u2212\n\uf8f8\n\u222b\nx\nx\ne\nxdx\n26 4\n4\n4\n0\nsin\ncos\ncos\nsin\nx\nx\ndx\nx\nx\n\u03c0\n+\n\u222b\n27 2\n2\n2\n2\n0\ncos\ncos\n4 sin\nx dx\nx\nx\n\u03c0\n+\n\u222b\n28 3\n6\nsin\ncos\nsin 2\nx\nx dx\nx\n\u03c0\n\u03c0\n+\n\u222b\n29"
    },
    {
        "Chapter": "1",
        "sentence_range": "4077-4080",
        "Text": "4\n4\n4\n0\nsin\ncos\ncos\nsin\nx\nx\ndx\nx\nx\n\u03c0\n+\n\u222b\n27 2\n2\n2\n2\n0\ncos\ncos\n4 sin\nx dx\nx\nx\n\u03c0\n+\n\u222b\n28 3\n6\nsin\ncos\nsin 2\nx\nx dx\nx\n\u03c0\n\u03c0\n+\n\u222b\n29 1\n0\n1\ndx\nx\nx\n+\n\u2212\n\u222b\n30"
    },
    {
        "Chapter": "1",
        "sentence_range": "4078-4081",
        "Text": "2\n2\n2\n2\n0\ncos\ncos\n4 sin\nx dx\nx\nx\n\u03c0\n+\n\u222b\n28 3\n6\nsin\ncos\nsin 2\nx\nx dx\nx\n\u03c0\n\u03c0\n+\n\u222b\n29 1\n0\n1\ndx\nx\nx\n+\n\u2212\n\u222b\n30 04\nsin\ncos\n9\nx16 sin 2\nx dx\nx\n\u03c0\n+\n+\n\u222b\n31"
    },
    {
        "Chapter": "1",
        "sentence_range": "4079-4082",
        "Text": "3\n6\nsin\ncos\nsin 2\nx\nx dx\nx\n\u03c0\n\u03c0\n+\n\u222b\n29 1\n0\n1\ndx\nx\nx\n+\n\u2212\n\u222b\n30 04\nsin\ncos\n9\nx16 sin 2\nx dx\nx\n\u03c0\n+\n+\n\u222b\n31 1\n2\n0 sin 2 tan\n(sin )\nx\nx dx\n\u03c0\n\u2212\n\u222b\n32"
    },
    {
        "Chapter": "1",
        "sentence_range": "4080-4083",
        "Text": "1\n0\n1\ndx\nx\nx\n+\n\u2212\n\u222b\n30 04\nsin\ncos\n9\nx16 sin 2\nx dx\nx\n\u03c0\n+\n+\n\u222b\n31 1\n2\n0 sin 2 tan\n(sin )\nx\nx dx\n\u03c0\n\u2212\n\u222b\n32 0\ntan\nsec\ntan\nx\nx\ndx\nx\nx\n\u03c0\n+\n\u222b\n33"
    },
    {
        "Chapter": "1",
        "sentence_range": "4081-4084",
        "Text": "04\nsin\ncos\n9\nx16 sin 2\nx dx\nx\n\u03c0\n+\n+\n\u222b\n31 1\n2\n0 sin 2 tan\n(sin )\nx\nx dx\n\u03c0\n\u2212\n\u222b\n32 0\ntan\nsec\ntan\nx\nx\ndx\nx\nx\n\u03c0\n+\n\u222b\n33 4\n1 [\n1|\n|\n2|\n|\n3|]\nx\nx\nx\ndx\n\u2212\n+\n\u2212\n+\n\u2212\n\u222b\nProve the following (Exercises 34 to 39)\n34"
    },
    {
        "Chapter": "1",
        "sentence_range": "4082-4085",
        "Text": "1\n2\n0 sin 2 tan\n(sin )\nx\nx dx\n\u03c0\n\u2212\n\u222b\n32 0\ntan\nsec\ntan\nx\nx\ndx\nx\nx\n\u03c0\n+\n\u222b\n33 4\n1 [\n1|\n|\n2|\n|\n3|]\nx\nx\nx\ndx\n\u2212\n+\n\u2212\n+\n\u2212\n\u222b\nProve the following (Exercises 34 to 39)\n34 3\n2\n1\n2\n2\nlog\n3\n3\n(\nxdx1)\nx\n=\n+\n+\n\u222b\n35"
    },
    {
        "Chapter": "1",
        "sentence_range": "4083-4086",
        "Text": "0\ntan\nsec\ntan\nx\nx\ndx\nx\nx\n\u03c0\n+\n\u222b\n33 4\n1 [\n1|\n|\n2|\n|\n3|]\nx\nx\nx\ndx\n\u2212\n+\n\u2212\n+\n\u2212\n\u222b\nProve the following (Exercises 34 to 39)\n34 3\n2\n1\n2\n2\nlog\n3\n3\n(\nxdx1)\nx\n=\n+\n+\n\u222b\n35 1\n0\n1\nx\nx e dx =\n\u222b\n36"
    },
    {
        "Chapter": "1",
        "sentence_range": "4084-4087",
        "Text": "4\n1 [\n1|\n|\n2|\n|\n3|]\nx\nx\nx\ndx\n\u2212\n+\n\u2212\n+\n\u2212\n\u222b\nProve the following (Exercises 34 to 39)\n34 3\n2\n1\n2\n2\nlog\n3\n3\n(\nxdx1)\nx\n=\n+\n+\n\u222b\n35 1\n0\n1\nx\nx e dx =\n\u222b\n36 1\n17\n4\n1\ncos\n0\nx\nx dx\n\u2212\n=\n\u222b\n37"
    },
    {
        "Chapter": "1",
        "sentence_range": "4085-4088",
        "Text": "3\n2\n1\n2\n2\nlog\n3\n3\n(\nxdx1)\nx\n=\n+\n+\n\u222b\n35 1\n0\n1\nx\nx e dx =\n\u222b\n36 1\n17\n4\n1\ncos\n0\nx\nx dx\n\u2212\n=\n\u222b\n37 3\n2\n0\n2\nsin\n3\nx dx\n\u03c0\n=\n\u222b\n38"
    },
    {
        "Chapter": "1",
        "sentence_range": "4086-4089",
        "Text": "1\n0\n1\nx\nx e dx =\n\u222b\n36 1\n17\n4\n1\ncos\n0\nx\nx dx\n\u2212\n=\n\u222b\n37 3\n2\n0\n2\nsin\n3\nx dx\n\u03c0\n=\n\u222b\n38 3\n4\n0 2 tan\n1\nlog2\nx dx\n\u03c0\n= \u2212\n\u222b\n39"
    },
    {
        "Chapter": "1",
        "sentence_range": "4087-4090",
        "Text": "1\n17\n4\n1\ncos\n0\nx\nx dx\n\u2212\n=\n\u222b\n37 3\n2\n0\n2\nsin\n3\nx dx\n\u03c0\n=\n\u222b\n38 3\n4\n0 2 tan\n1\nlog2\nx dx\n\u03c0\n= \u2212\n\u222b\n39 1\n1\n0sin\n1\n2\nx dx\n\u2212\n=\u03c0\n\u2212\n\u222b\n40"
    },
    {
        "Chapter": "1",
        "sentence_range": "4088-4091",
        "Text": "3\n2\n0\n2\nsin\n3\nx dx\n\u03c0\n=\n\u222b\n38 3\n4\n0 2 tan\n1\nlog2\nx dx\n\u03c0\n= \u2212\n\u222b\n39 1\n1\n0sin\n1\n2\nx dx\n\u2212\n=\u03c0\n\u2212\n\u222b\n40 Evaluate \n1\n2 3\n0\nx\ne\ndx\n\u2212\n\u222b\n as a limit of a sum"
    },
    {
        "Chapter": "1",
        "sentence_range": "4089-4092",
        "Text": "3\n4\n0 2 tan\n1\nlog2\nx dx\n\u03c0\n= \u2212\n\u222b\n39 1\n1\n0sin\n1\n2\nx dx\n\u2212\n=\u03c0\n\u2212\n\u222b\n40 Evaluate \n1\n2 3\n0\nx\ne\ndx\n\u2212\n\u222b\n as a limit of a sum Choose the correct answers in Exercises 41 to 44"
    },
    {
        "Chapter": "1",
        "sentence_range": "4090-4093",
        "Text": "1\n1\n0sin\n1\n2\nx dx\n\u2212\n=\u03c0\n\u2212\n\u222b\n40 Evaluate \n1\n2 3\n0\nx\ne\ndx\n\u2212\n\u222b\n as a limit of a sum Choose the correct answers in Exercises 41 to 44 41"
    },
    {
        "Chapter": "1",
        "sentence_range": "4091-4094",
        "Text": "Evaluate \n1\n2 3\n0\nx\ne\ndx\n\u2212\n\u222b\n as a limit of a sum Choose the correct answers in Exercises 41 to 44 41 x\nx\ndx\ne\n+e\u2212\n\u222b\n is equal to\n(A) tan\u20131 (ex) + C\n(B) tan\u20131 (e\u2013x) + C\n(C) log (ex \u2013 e\u2013x) + C\n(D) log (ex + e\u2013x) + C\n42"
    },
    {
        "Chapter": "1",
        "sentence_range": "4092-4095",
        "Text": "Choose the correct answers in Exercises 41 to 44 41 x\nx\ndx\ne\n+e\u2212\n\u222b\n is equal to\n(A) tan\u20131 (ex) + C\n(B) tan\u20131 (e\u2013x) + C\n(C) log (ex \u2013 e\u2013x) + C\n(D) log (ex + e\u2013x) + C\n42 2\n(sincos2\ncos )\nx\ndx\nx\nx\n+\n\u222b\n is equal to\n(A)\n\u20131\nC\nsin\ncos\nx\nx\n+\n+\n(B) log |sin\ncos |\nC\nx\nx\n+\n+\n(C) log |sin\ncos |\nC\nx\nx\n\u2212\n+\n(D)\n2\n1\n(sin\ncos )\nx\nx\n+\n354\nMATHEMATICS\n43"
    },
    {
        "Chapter": "1",
        "sentence_range": "4093-4096",
        "Text": "41 x\nx\ndx\ne\n+e\u2212\n\u222b\n is equal to\n(A) tan\u20131 (ex) + C\n(B) tan\u20131 (e\u2013x) + C\n(C) log (ex \u2013 e\u2013x) + C\n(D) log (ex + e\u2013x) + C\n42 2\n(sincos2\ncos )\nx\ndx\nx\nx\n+\n\u222b\n is equal to\n(A)\n\u20131\nC\nsin\ncos\nx\nx\n+\n+\n(B) log |sin\ncos |\nC\nx\nx\n+\n+\n(C) log |sin\ncos |\nC\nx\nx\n\u2212\n+\n(D)\n2\n1\n(sin\ncos )\nx\nx\n+\n354\nMATHEMATICS\n43 If f (a + b \u2013 x) = f (x), then \n( )\nb\n\u222ba x f x dx\n is equal to\n(A)\n(\n)\n2\nb\na\na\nb\nf b\nx dx\n+\n\u2212\n\u222b\n(B)\n(\n)\n2\nb\na\na\nb\nf b\nx dx\n+\n+\n\u222b\n(C)\n( )\n2\nb\na\nb\na\n\u2212 \u222bf x dx\n(D)\n( )\n2\nb\na\na\nb\nf x dx\n+ \u222b\n44"
    },
    {
        "Chapter": "1",
        "sentence_range": "4094-4097",
        "Text": "x\nx\ndx\ne\n+e\u2212\n\u222b\n is equal to\n(A) tan\u20131 (ex) + C\n(B) tan\u20131 (e\u2013x) + C\n(C) log (ex \u2013 e\u2013x) + C\n(D) log (ex + e\u2013x) + C\n42 2\n(sincos2\ncos )\nx\ndx\nx\nx\n+\n\u222b\n is equal to\n(A)\n\u20131\nC\nsin\ncos\nx\nx\n+\n+\n(B) log |sin\ncos |\nC\nx\nx\n+\n+\n(C) log |sin\ncos |\nC\nx\nx\n\u2212\n+\n(D)\n2\n1\n(sin\ncos )\nx\nx\n+\n354\nMATHEMATICS\n43 If f (a + b \u2013 x) = f (x), then \n( )\nb\n\u222ba x f x dx\n is equal to\n(A)\n(\n)\n2\nb\na\na\nb\nf b\nx dx\n+\n\u2212\n\u222b\n(B)\n(\n)\n2\nb\na\na\nb\nf b\nx dx\n+\n+\n\u222b\n(C)\n( )\n2\nb\na\nb\na\n\u2212 \u222bf x dx\n(D)\n( )\n2\nb\na\na\nb\nf x dx\n+ \u222b\n44 The value of \n1\n1\n2\n0\n2\n1\ntan\n1\nx\ndx\nx\nx\n\u2212\n\u2212\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n+\n\u2212\n\uf8ed\n\uf8f8\n\u222b\n is\n(A) 1\n(B) 0\n(C) \u20131\n(D)\n4\n\u03c0\nSummary\n\ufffd Integration is the inverse process of differentiation"
    },
    {
        "Chapter": "1",
        "sentence_range": "4095-4098",
        "Text": "2\n(sincos2\ncos )\nx\ndx\nx\nx\n+\n\u222b\n is equal to\n(A)\n\u20131\nC\nsin\ncos\nx\nx\n+\n+\n(B) log |sin\ncos |\nC\nx\nx\n+\n+\n(C) log |sin\ncos |\nC\nx\nx\n\u2212\n+\n(D)\n2\n1\n(sin\ncos )\nx\nx\n+\n354\nMATHEMATICS\n43 If f (a + b \u2013 x) = f (x), then \n( )\nb\n\u222ba x f x dx\n is equal to\n(A)\n(\n)\n2\nb\na\na\nb\nf b\nx dx\n+\n\u2212\n\u222b\n(B)\n(\n)\n2\nb\na\na\nb\nf b\nx dx\n+\n+\n\u222b\n(C)\n( )\n2\nb\na\nb\na\n\u2212 \u222bf x dx\n(D)\n( )\n2\nb\na\na\nb\nf x dx\n+ \u222b\n44 The value of \n1\n1\n2\n0\n2\n1\ntan\n1\nx\ndx\nx\nx\n\u2212\n\u2212\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n+\n\u2212\n\uf8ed\n\uf8f8\n\u222b\n is\n(A) 1\n(B) 0\n(C) \u20131\n(D)\n4\n\u03c0\nSummary\n\ufffd Integration is the inverse process of differentiation In the differential calculus,\nwe are given a function and we have to find the derivative or differential of\nthis function, but in the integral calculus, we are to find a function whose\ndifferential is given"
    },
    {
        "Chapter": "1",
        "sentence_range": "4096-4099",
        "Text": "If f (a + b \u2013 x) = f (x), then \n( )\nb\n\u222ba x f x dx\n is equal to\n(A)\n(\n)\n2\nb\na\na\nb\nf b\nx dx\n+\n\u2212\n\u222b\n(B)\n(\n)\n2\nb\na\na\nb\nf b\nx dx\n+\n+\n\u222b\n(C)\n( )\n2\nb\na\nb\na\n\u2212 \u222bf x dx\n(D)\n( )\n2\nb\na\na\nb\nf x dx\n+ \u222b\n44 The value of \n1\n1\n2\n0\n2\n1\ntan\n1\nx\ndx\nx\nx\n\u2212\n\u2212\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n+\n\u2212\n\uf8ed\n\uf8f8\n\u222b\n is\n(A) 1\n(B) 0\n(C) \u20131\n(D)\n4\n\u03c0\nSummary\n\ufffd Integration is the inverse process of differentiation In the differential calculus,\nwe are given a function and we have to find the derivative or differential of\nthis function, but in the integral calculus, we are to find a function whose\ndifferential is given Thus, integration is a process which is the inverse of\ndifferentiation"
    },
    {
        "Chapter": "1",
        "sentence_range": "4097-4100",
        "Text": "The value of \n1\n1\n2\n0\n2\n1\ntan\n1\nx\ndx\nx\nx\n\u2212\n\u2212\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n+\n\u2212\n\uf8ed\n\uf8f8\n\u222b\n is\n(A) 1\n(B) 0\n(C) \u20131\n(D)\n4\n\u03c0\nSummary\n\ufffd Integration is the inverse process of differentiation In the differential calculus,\nwe are given a function and we have to find the derivative or differential of\nthis function, but in the integral calculus, we are to find a function whose\ndifferential is given Thus, integration is a process which is the inverse of\ndifferentiation Let \nF( )\n( )\nd\nx\nf x\ndx\n="
    },
    {
        "Chapter": "1",
        "sentence_range": "4098-4101",
        "Text": "In the differential calculus,\nwe are given a function and we have to find the derivative or differential of\nthis function, but in the integral calculus, we are to find a function whose\ndifferential is given Thus, integration is a process which is the inverse of\ndifferentiation Let \nF( )\n( )\nd\nx\nf x\ndx\n= Then we write \n( )\nF ( )\nC\nf x dx\nx\n=\n+\n\u222b"
    },
    {
        "Chapter": "1",
        "sentence_range": "4099-4102",
        "Text": "Thus, integration is a process which is the inverse of\ndifferentiation Let \nF( )\n( )\nd\nx\nf x\ndx\n= Then we write \n( )\nF ( )\nC\nf x dx\nx\n=\n+\n\u222b These integrals\nare called indefinite integrals or general integrals, C is called constant of\nintegration"
    },
    {
        "Chapter": "1",
        "sentence_range": "4100-4103",
        "Text": "Let \nF( )\n( )\nd\nx\nf x\ndx\n= Then we write \n( )\nF ( )\nC\nf x dx\nx\n=\n+\n\u222b These integrals\nare called indefinite integrals or general integrals, C is called constant of\nintegration All these integrals differ by a constant"
    },
    {
        "Chapter": "1",
        "sentence_range": "4101-4104",
        "Text": "Then we write \n( )\nF ( )\nC\nf x dx\nx\n=\n+\n\u222b These integrals\nare called indefinite integrals or general integrals, C is called constant of\nintegration All these integrals differ by a constant \ufffd From the geometric point of view, an indefinite integral is collection of family\nof curves, each of which is obtained by translating one of the curves parallel\nto itself upwards or downwards along the y-axis"
    },
    {
        "Chapter": "1",
        "sentence_range": "4102-4105",
        "Text": "These integrals\nare called indefinite integrals or general integrals, C is called constant of\nintegration All these integrals differ by a constant \ufffd From the geometric point of view, an indefinite integral is collection of family\nof curves, each of which is obtained by translating one of the curves parallel\nto itself upwards or downwards along the y-axis \ufffd Some properties of indefinite integrals are as follows:\n1"
    },
    {
        "Chapter": "1",
        "sentence_range": "4103-4106",
        "Text": "All these integrals differ by a constant \ufffd From the geometric point of view, an indefinite integral is collection of family\nof curves, each of which is obtained by translating one of the curves parallel\nto itself upwards or downwards along the y-axis \ufffd Some properties of indefinite integrals are as follows:\n1 [ ( )\n( )]\n( )\n( )\nf x\ng x\ndx\nf x dx\ng x dx\n+\n=\n+\n\u222b\n\u222b\n\u222b\n2"
    },
    {
        "Chapter": "1",
        "sentence_range": "4104-4107",
        "Text": "\ufffd From the geometric point of view, an indefinite integral is collection of family\nof curves, each of which is obtained by translating one of the curves parallel\nto itself upwards or downwards along the y-axis \ufffd Some properties of indefinite integrals are as follows:\n1 [ ( )\n( )]\n( )\n( )\nf x\ng x\ndx\nf x dx\ng x dx\n+\n=\n+\n\u222b\n\u222b\n\u222b\n2 For any real number k, \n( )\n( )\nk f x dx\nk\nf x dx\n=\n\u222b\n\u222b\nMore generally, if f1, f2, f3,"
    },
    {
        "Chapter": "1",
        "sentence_range": "4105-4108",
        "Text": "\ufffd Some properties of indefinite integrals are as follows:\n1 [ ( )\n( )]\n( )\n( )\nf x\ng x\ndx\nf x dx\ng x dx\n+\n=\n+\n\u222b\n\u222b\n\u222b\n2 For any real number k, \n( )\n( )\nk f x dx\nk\nf x dx\n=\n\u222b\n\u222b\nMore generally, if f1, f2, f3, , fn are functions and k1, k2,"
    },
    {
        "Chapter": "1",
        "sentence_range": "4106-4109",
        "Text": "[ ( )\n( )]\n( )\n( )\nf x\ng x\ndx\nf x dx\ng x dx\n+\n=\n+\n\u222b\n\u222b\n\u222b\n2 For any real number k, \n( )\n( )\nk f x dx\nk\nf x dx\n=\n\u222b\n\u222b\nMore generally, if f1, f2, f3, , fn are functions and k1, k2, ,kn are real\nnumbers"
    },
    {
        "Chapter": "1",
        "sentence_range": "4107-4110",
        "Text": "For any real number k, \n( )\n( )\nk f x dx\nk\nf x dx\n=\n\u222b\n\u222b\nMore generally, if f1, f2, f3, , fn are functions and k1, k2, ,kn are real\nnumbers Then\n1\n1\n2\n2\n[\n( )\n( )"
    },
    {
        "Chapter": "1",
        "sentence_range": "4108-4111",
        "Text": ", fn are functions and k1, k2, ,kn are real\nnumbers Then\n1\n1\n2\n2\n[\n( )\n( ) ( )]\nn\nn\nk f\nx\nk f\nx\nk f\nx\ndx\n+\n+\n+\n\u222b\n= \n1\n1\n2\n2\n( )\n( )"
    },
    {
        "Chapter": "1",
        "sentence_range": "4109-4112",
        "Text": ",kn are real\nnumbers Then\n1\n1\n2\n2\n[\n( )\n( ) ( )]\nn\nn\nk f\nx\nk f\nx\nk f\nx\ndx\n+\n+\n+\n\u222b\n= \n1\n1\n2\n2\n( )\n( ) ( )\nn\nn\nk\nf\nx dx\nk\nf\nx dx\nk\nf\nx dx\n+\n+\n+\n\u222b\n\u222b\n\u222b\nINTEGRALS         355\n\ufffd Some standard integrals\n(i)\n1\nC\n1\nn\nn\nx\nx dx\nn\n+\n=\n+\n+\n\u222b\n, n \u2260 \u2013 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "4110-4113",
        "Text": "Then\n1\n1\n2\n2\n[\n( )\n( ) ( )]\nn\nn\nk f\nx\nk f\nx\nk f\nx\ndx\n+\n+\n+\n\u222b\n= \n1\n1\n2\n2\n( )\n( ) ( )\nn\nn\nk\nf\nx dx\nk\nf\nx dx\nk\nf\nx dx\n+\n+\n+\n\u222b\n\u222b\n\u222b\nINTEGRALS         355\n\ufffd Some standard integrals\n(i)\n1\nC\n1\nn\nn\nx\nx dx\nn\n+\n=\n+\n+\n\u222b\n, n \u2260 \u2013 1 Particularly, \nC\ndx\n=x\n+\n\u222b\n(ii)\ncos\nsin\nC\nx dx\nx\n=\n+\n\u222b\n(iii)\nsin\n\u2013 cos\nC\nx dx\nx\n=\n+\n\u222b\n(iv)\nsec2\ntan\nC\nx dx\nx\n=\n+\n\u222b\n(v)\ncosec2\n\u2013 cot\nC\nx dx\nx\n=\n+\n\u222b\n(vi)\nsec\ntan\nsec\nC\nx\nx dx\nx\n=\n+\n\u222b\n(vii)\ncosec\ncot\n\u2013 cosec\nC\nx\nx dx\nx\n=\n+\n\u222b\n(viii)\n1\n2\nsin\nC\n1\ndx\nx\nx\n\u2212\n=\n+\n\u2212\n\u222b\n(ix)\n1\n2\ncos\nC\n1\ndx\nx\nx\n\u2212\n= \u2212\n+\n\u2212\n\u222b\n(x)\n1\n2\ntan\nC\n1\ndx\nx\nx\n\u2212\n=\n+\n+\n\u222b\n(xi)\n1\n2\ncot\nC\n1\ndx\nx\nx\n\u2212\n= \u2212\n+\n\u222b+\n(xii)\nC\nx\nx\ne dx\n=e\n+\n\u222b\n(xiii)\nC\nlog\nx\nx\na\na dx\na\n=\n+\n\u222b\n(xiv)\n1\n2\nsec\nC\n1\ndx\nx\nx x\n\u2212\n=\n+\n\u2212\n\u222b\n(xv)\n1\n2\ncosec\nC\n1\ndx\nx\nx x\n\u2212\n= \u2212\n+\n\u2212\n\u222b\n(xvi)\n1\nlog|\n|\nC\ndx\nx\nx\n=\n+\n\u222b\n\ufffd Integration by partial fractions\nRecall that a rational function is ratio of two polynomials of the form P( )\nQ( )\nx\nx\n,\nwhere P(x) and Q (x) are polynomials in x and Q (x) \u2260 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "4111-4114",
        "Text": "( )]\nn\nn\nk f\nx\nk f\nx\nk f\nx\ndx\n+\n+\n+\n\u222b\n= \n1\n1\n2\n2\n( )\n( ) ( )\nn\nn\nk\nf\nx dx\nk\nf\nx dx\nk\nf\nx dx\n+\n+\n+\n\u222b\n\u222b\n\u222b\nINTEGRALS         355\n\ufffd Some standard integrals\n(i)\n1\nC\n1\nn\nn\nx\nx dx\nn\n+\n=\n+\n+\n\u222b\n, n \u2260 \u2013 1 Particularly, \nC\ndx\n=x\n+\n\u222b\n(ii)\ncos\nsin\nC\nx dx\nx\n=\n+\n\u222b\n(iii)\nsin\n\u2013 cos\nC\nx dx\nx\n=\n+\n\u222b\n(iv)\nsec2\ntan\nC\nx dx\nx\n=\n+\n\u222b\n(v)\ncosec2\n\u2013 cot\nC\nx dx\nx\n=\n+\n\u222b\n(vi)\nsec\ntan\nsec\nC\nx\nx dx\nx\n=\n+\n\u222b\n(vii)\ncosec\ncot\n\u2013 cosec\nC\nx\nx dx\nx\n=\n+\n\u222b\n(viii)\n1\n2\nsin\nC\n1\ndx\nx\nx\n\u2212\n=\n+\n\u2212\n\u222b\n(ix)\n1\n2\ncos\nC\n1\ndx\nx\nx\n\u2212\n= \u2212\n+\n\u2212\n\u222b\n(x)\n1\n2\ntan\nC\n1\ndx\nx\nx\n\u2212\n=\n+\n+\n\u222b\n(xi)\n1\n2\ncot\nC\n1\ndx\nx\nx\n\u2212\n= \u2212\n+\n\u222b+\n(xii)\nC\nx\nx\ne dx\n=e\n+\n\u222b\n(xiii)\nC\nlog\nx\nx\na\na dx\na\n=\n+\n\u222b\n(xiv)\n1\n2\nsec\nC\n1\ndx\nx\nx x\n\u2212\n=\n+\n\u2212\n\u222b\n(xv)\n1\n2\ncosec\nC\n1\ndx\nx\nx x\n\u2212\n= \u2212\n+\n\u2212\n\u222b\n(xvi)\n1\nlog|\n|\nC\ndx\nx\nx\n=\n+\n\u222b\n\ufffd Integration by partial fractions\nRecall that a rational function is ratio of two polynomials of the form P( )\nQ( )\nx\nx\n,\nwhere P(x) and Q (x) are polynomials in x and Q (x) \u2260 0 If degree of the\npolynomial P (x) is greater than the degree of the polynomial Q (x), then we\nmay divide P (x) by Q (x) so that \n1P ( )\nP( )\nT ( )\nQ( )\nQ( )\nx\nx\nx\nx\nx\n=\n+\n, where T(x) is a\npolynomial in x and degree of P1(x) is less than the degree of Q(x)"
    },
    {
        "Chapter": "1",
        "sentence_range": "4112-4115",
        "Text": "( )\nn\nn\nk\nf\nx dx\nk\nf\nx dx\nk\nf\nx dx\n+\n+\n+\n\u222b\n\u222b\n\u222b\nINTEGRALS         355\n\ufffd Some standard integrals\n(i)\n1\nC\n1\nn\nn\nx\nx dx\nn\n+\n=\n+\n+\n\u222b\n, n \u2260 \u2013 1 Particularly, \nC\ndx\n=x\n+\n\u222b\n(ii)\ncos\nsin\nC\nx dx\nx\n=\n+\n\u222b\n(iii)\nsin\n\u2013 cos\nC\nx dx\nx\n=\n+\n\u222b\n(iv)\nsec2\ntan\nC\nx dx\nx\n=\n+\n\u222b\n(v)\ncosec2\n\u2013 cot\nC\nx dx\nx\n=\n+\n\u222b\n(vi)\nsec\ntan\nsec\nC\nx\nx dx\nx\n=\n+\n\u222b\n(vii)\ncosec\ncot\n\u2013 cosec\nC\nx\nx dx\nx\n=\n+\n\u222b\n(viii)\n1\n2\nsin\nC\n1\ndx\nx\nx\n\u2212\n=\n+\n\u2212\n\u222b\n(ix)\n1\n2\ncos\nC\n1\ndx\nx\nx\n\u2212\n= \u2212\n+\n\u2212\n\u222b\n(x)\n1\n2\ntan\nC\n1\ndx\nx\nx\n\u2212\n=\n+\n+\n\u222b\n(xi)\n1\n2\ncot\nC\n1\ndx\nx\nx\n\u2212\n= \u2212\n+\n\u222b+\n(xii)\nC\nx\nx\ne dx\n=e\n+\n\u222b\n(xiii)\nC\nlog\nx\nx\na\na dx\na\n=\n+\n\u222b\n(xiv)\n1\n2\nsec\nC\n1\ndx\nx\nx x\n\u2212\n=\n+\n\u2212\n\u222b\n(xv)\n1\n2\ncosec\nC\n1\ndx\nx\nx x\n\u2212\n= \u2212\n+\n\u2212\n\u222b\n(xvi)\n1\nlog|\n|\nC\ndx\nx\nx\n=\n+\n\u222b\n\ufffd Integration by partial fractions\nRecall that a rational function is ratio of two polynomials of the form P( )\nQ( )\nx\nx\n,\nwhere P(x) and Q (x) are polynomials in x and Q (x) \u2260 0 If degree of the\npolynomial P (x) is greater than the degree of the polynomial Q (x), then we\nmay divide P (x) by Q (x) so that \n1P ( )\nP( )\nT ( )\nQ( )\nQ( )\nx\nx\nx\nx\nx\n=\n+\n, where T(x) is a\npolynomial in x and degree of P1(x) is less than the degree of Q(x) T(x)\nbeing polynomial can be easily integrated"
    },
    {
        "Chapter": "1",
        "sentence_range": "4113-4116",
        "Text": "Particularly, \nC\ndx\n=x\n+\n\u222b\n(ii)\ncos\nsin\nC\nx dx\nx\n=\n+\n\u222b\n(iii)\nsin\n\u2013 cos\nC\nx dx\nx\n=\n+\n\u222b\n(iv)\nsec2\ntan\nC\nx dx\nx\n=\n+\n\u222b\n(v)\ncosec2\n\u2013 cot\nC\nx dx\nx\n=\n+\n\u222b\n(vi)\nsec\ntan\nsec\nC\nx\nx dx\nx\n=\n+\n\u222b\n(vii)\ncosec\ncot\n\u2013 cosec\nC\nx\nx dx\nx\n=\n+\n\u222b\n(viii)\n1\n2\nsin\nC\n1\ndx\nx\nx\n\u2212\n=\n+\n\u2212\n\u222b\n(ix)\n1\n2\ncos\nC\n1\ndx\nx\nx\n\u2212\n= \u2212\n+\n\u2212\n\u222b\n(x)\n1\n2\ntan\nC\n1\ndx\nx\nx\n\u2212\n=\n+\n+\n\u222b\n(xi)\n1\n2\ncot\nC\n1\ndx\nx\nx\n\u2212\n= \u2212\n+\n\u222b+\n(xii)\nC\nx\nx\ne dx\n=e\n+\n\u222b\n(xiii)\nC\nlog\nx\nx\na\na dx\na\n=\n+\n\u222b\n(xiv)\n1\n2\nsec\nC\n1\ndx\nx\nx x\n\u2212\n=\n+\n\u2212\n\u222b\n(xv)\n1\n2\ncosec\nC\n1\ndx\nx\nx x\n\u2212\n= \u2212\n+\n\u2212\n\u222b\n(xvi)\n1\nlog|\n|\nC\ndx\nx\nx\n=\n+\n\u222b\n\ufffd Integration by partial fractions\nRecall that a rational function is ratio of two polynomials of the form P( )\nQ( )\nx\nx\n,\nwhere P(x) and Q (x) are polynomials in x and Q (x) \u2260 0 If degree of the\npolynomial P (x) is greater than the degree of the polynomial Q (x), then we\nmay divide P (x) by Q (x) so that \n1P ( )\nP( )\nT ( )\nQ( )\nQ( )\nx\nx\nx\nx\nx\n=\n+\n, where T(x) is a\npolynomial in x and degree of P1(x) is less than the degree of Q(x) T(x)\nbeing polynomial can be easily integrated 1P ( )\nQ( )\nx\nx  can be integrated by\n356\nMATHEMATICS\nexpressing \n1P ( )\nQ( )\nx\nx  as the sum of partial fractions of the following type:\n1"
    },
    {
        "Chapter": "1",
        "sentence_range": "4114-4117",
        "Text": "If degree of the\npolynomial P (x) is greater than the degree of the polynomial Q (x), then we\nmay divide P (x) by Q (x) so that \n1P ( )\nP( )\nT ( )\nQ( )\nQ( )\nx\nx\nx\nx\nx\n=\n+\n, where T(x) is a\npolynomial in x and degree of P1(x) is less than the degree of Q(x) T(x)\nbeing polynomial can be easily integrated 1P ( )\nQ( )\nx\nx  can be integrated by\n356\nMATHEMATICS\nexpressing \n1P ( )\nQ( )\nx\nx  as the sum of partial fractions of the following type:\n1 (\n) (\n)\npx\nq\nx\na\nx\nb\n+\n\u2212\n\u2212\n=\nA\nB\nx\na\nx\nb\n+\n\u2212\n\u2212\n, a \u2260 b\n2"
    },
    {
        "Chapter": "1",
        "sentence_range": "4115-4118",
        "Text": "T(x)\nbeing polynomial can be easily integrated 1P ( )\nQ( )\nx\nx  can be integrated by\n356\nMATHEMATICS\nexpressing \n1P ( )\nQ( )\nx\nx  as the sum of partial fractions of the following type:\n1 (\n) (\n)\npx\nq\nx\na\nx\nb\n+\n\u2212\n\u2212\n=\nA\nB\nx\na\nx\nb\n+\n\u2212\n\u2212\n, a \u2260 b\n2 2\n(\n)\npx\nq\nx\na\n+\n\u2212\n=\n2\nA\nB\n(\n)\nx\na\nx\na\n+\n\u2212\n\u2212\n3"
    },
    {
        "Chapter": "1",
        "sentence_range": "4116-4119",
        "Text": "1P ( )\nQ( )\nx\nx  can be integrated by\n356\nMATHEMATICS\nexpressing \n1P ( )\nQ( )\nx\nx  as the sum of partial fractions of the following type:\n1 (\n) (\n)\npx\nq\nx\na\nx\nb\n+\n\u2212\n\u2212\n=\nA\nB\nx\na\nx\nb\n+\n\u2212\n\u2212\n, a \u2260 b\n2 2\n(\n)\npx\nq\nx\na\n+\n\u2212\n=\n2\nA\nB\n(\n)\nx\na\nx\na\n+\n\u2212\n\u2212\n3 2\n(\n) (\n) (\n)\npx\nqx\nr\nx\na\nx\nb\nx\nc\n+\n+\n\u2212\n\u2212\n\u2212\n=\nA\nB\nC\nx\na\nx\nb\nx\nc\n+\n+\n\u2212\n\u2212\n\u2212\n4"
    },
    {
        "Chapter": "1",
        "sentence_range": "4117-4120",
        "Text": "(\n) (\n)\npx\nq\nx\na\nx\nb\n+\n\u2212\n\u2212\n=\nA\nB\nx\na\nx\nb\n+\n\u2212\n\u2212\n, a \u2260 b\n2 2\n(\n)\npx\nq\nx\na\n+\n\u2212\n=\n2\nA\nB\n(\n)\nx\na\nx\na\n+\n\u2212\n\u2212\n3 2\n(\n) (\n) (\n)\npx\nqx\nr\nx\na\nx\nb\nx\nc\n+\n+\n\u2212\n\u2212\n\u2212\n=\nA\nB\nC\nx\na\nx\nb\nx\nc\n+\n+\n\u2212\n\u2212\n\u2212\n4 2\n2\n(\n) (\n)\npx\nqx\nr\nx\na\nx\nb\n+\n+\n\u2212\n\u2212\n=\n2\nA\nB\nC\n(\n)\nx\na\nx\nb\nx\na\n+\n+\n\u2212\n\u2212\n\u2212\n5"
    },
    {
        "Chapter": "1",
        "sentence_range": "4118-4121",
        "Text": "2\n(\n)\npx\nq\nx\na\n+\n\u2212\n=\n2\nA\nB\n(\n)\nx\na\nx\na\n+\n\u2212\n\u2212\n3 2\n(\n) (\n) (\n)\npx\nqx\nr\nx\na\nx\nb\nx\nc\n+\n+\n\u2212\n\u2212\n\u2212\n=\nA\nB\nC\nx\na\nx\nb\nx\nc\n+\n+\n\u2212\n\u2212\n\u2212\n4 2\n2\n(\n) (\n)\npx\nqx\nr\nx\na\nx\nb\n+\n+\n\u2212\n\u2212\n=\n2\nA\nB\nC\n(\n)\nx\na\nx\nb\nx\na\n+\n+\n\u2212\n\u2212\n\u2212\n5 2\n2\n(\n) (\n)\npx\nqx\nr\nx\na\nx\nbx\nc\n+\n+\n\u2212\n+\n+\n=\n2\nA\nB + C\nx\nx\na\nx\nbx\nc\n+\n\u2212\n+\n+\nwhere x2 + bx + c can not be factorised further"
    },
    {
        "Chapter": "1",
        "sentence_range": "4119-4122",
        "Text": "2\n(\n) (\n) (\n)\npx\nqx\nr\nx\na\nx\nb\nx\nc\n+\n+\n\u2212\n\u2212\n\u2212\n=\nA\nB\nC\nx\na\nx\nb\nx\nc\n+\n+\n\u2212\n\u2212\n\u2212\n4 2\n2\n(\n) (\n)\npx\nqx\nr\nx\na\nx\nb\n+\n+\n\u2212\n\u2212\n=\n2\nA\nB\nC\n(\n)\nx\na\nx\nb\nx\na\n+\n+\n\u2212\n\u2212\n\u2212\n5 2\n2\n(\n) (\n)\npx\nqx\nr\nx\na\nx\nbx\nc\n+\n+\n\u2212\n+\n+\n=\n2\nA\nB + C\nx\nx\na\nx\nbx\nc\n+\n\u2212\n+\n+\nwhere x2 + bx + c can not be factorised further \ufffd Integration by substitution\nA change in the variable of integration often reduces an integral to one of the\nfundamental integrals"
    },
    {
        "Chapter": "1",
        "sentence_range": "4120-4123",
        "Text": "2\n2\n(\n) (\n)\npx\nqx\nr\nx\na\nx\nb\n+\n+\n\u2212\n\u2212\n=\n2\nA\nB\nC\n(\n)\nx\na\nx\nb\nx\na\n+\n+\n\u2212\n\u2212\n\u2212\n5 2\n2\n(\n) (\n)\npx\nqx\nr\nx\na\nx\nbx\nc\n+\n+\n\u2212\n+\n+\n=\n2\nA\nB + C\nx\nx\na\nx\nbx\nc\n+\n\u2212\n+\n+\nwhere x2 + bx + c can not be factorised further \ufffd Integration by substitution\nA change in the variable of integration often reduces an integral to one of the\nfundamental integrals The method in which we change the variable to some\nother variable is called the method of substitution"
    },
    {
        "Chapter": "1",
        "sentence_range": "4121-4124",
        "Text": "2\n2\n(\n) (\n)\npx\nqx\nr\nx\na\nx\nbx\nc\n+\n+\n\u2212\n+\n+\n=\n2\nA\nB + C\nx\nx\na\nx\nbx\nc\n+\n\u2212\n+\n+\nwhere x2 + bx + c can not be factorised further \ufffd Integration by substitution\nA change in the variable of integration often reduces an integral to one of the\nfundamental integrals The method in which we change the variable to some\nother variable is called the method of substitution When the integrand involves\nsome trigonometric functions, we use some well known identities to find the\nintegrals"
    },
    {
        "Chapter": "1",
        "sentence_range": "4122-4125",
        "Text": "\ufffd Integration by substitution\nA change in the variable of integration often reduces an integral to one of the\nfundamental integrals The method in which we change the variable to some\nother variable is called the method of substitution When the integrand involves\nsome trigonometric functions, we use some well known identities to find the\nintegrals Using substitution technique, we obtain the following standard\nintegrals"
    },
    {
        "Chapter": "1",
        "sentence_range": "4123-4126",
        "Text": "The method in which we change the variable to some\nother variable is called the method of substitution When the integrand involves\nsome trigonometric functions, we use some well known identities to find the\nintegrals Using substitution technique, we obtain the following standard\nintegrals (i)\ntan\nlog sec\nC\nx dx\nx\n=\n+\n\u222b\n(ii)\ncot\nlog sin\nC\nx dx\nx\n=\n+\n\u222b\n(iii)\nsec\nlog sec\ntan\nC\nx dx\nx\nx\n=\n+\n+\n\u222b\n(iv)\ncosec\nlog cosec\ncot\nC\nx dx\nx\nx\n=\n\u2212\n+\n\u222b\n\ufffd Integrals of some special functions\n(i)\n2\n2\n1 log\nC\n2\ndx\nx\na\na\nx\na\nx\na\n\u2212\n=\n+\n+\n\u2212\n\u222b\n(ii)\n2\n2\n1 log\nC\n2\ndx\na\nx\na\na\nx\na\nx\n+\n=\n+\n\u2212\n\u2212\n\u222b\n(iii)\n1\n2\n2\n1 tan\nC\ndx\nx\na\na\nx\na\n\u2212\n=\n+\n+\n\u222b\nINTEGRALS         357\n(iv)\n2\n2\n2\n2\nlog\nC\ndx\nx\nx\na\nx\na\n=\n+\n\u2212\n+\n\u2212\n\u222b\n(v)\n1\n2\n2\nsin\nC\ndx\nax\na\nx\n\u2212\n=\n+\n\u2212\n\u222b\n(vi)\n2\n2\n2\n2\nlog |\n|\nC\ndx\nx\nx\na\nx\na\n=\n+\n+\n+\n+\n\u222b\n\ufffd Integration by parts\nFor given functions f1 and  f2, we have\n1\n2\n1\n2\n1\n2\n( )\n( )\n( )\n( )\n( )\n( )\nd\nf\nx\nf\nx dx\nf\nx\nf\nx dx\nf x\nf\nx dx dx\ndx\n\uf8ee\n\uf8f9\n\u22c5\n=\n\u2212\n\u22c5\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\u222b\n\u222b\n\u222b\n\u222b\n, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "4124-4127",
        "Text": "When the integrand involves\nsome trigonometric functions, we use some well known identities to find the\nintegrals Using substitution technique, we obtain the following standard\nintegrals (i)\ntan\nlog sec\nC\nx dx\nx\n=\n+\n\u222b\n(ii)\ncot\nlog sin\nC\nx dx\nx\n=\n+\n\u222b\n(iii)\nsec\nlog sec\ntan\nC\nx dx\nx\nx\n=\n+\n+\n\u222b\n(iv)\ncosec\nlog cosec\ncot\nC\nx dx\nx\nx\n=\n\u2212\n+\n\u222b\n\ufffd Integrals of some special functions\n(i)\n2\n2\n1 log\nC\n2\ndx\nx\na\na\nx\na\nx\na\n\u2212\n=\n+\n+\n\u2212\n\u222b\n(ii)\n2\n2\n1 log\nC\n2\ndx\na\nx\na\na\nx\na\nx\n+\n=\n+\n\u2212\n\u2212\n\u222b\n(iii)\n1\n2\n2\n1 tan\nC\ndx\nx\na\na\nx\na\n\u2212\n=\n+\n+\n\u222b\nINTEGRALS         357\n(iv)\n2\n2\n2\n2\nlog\nC\ndx\nx\nx\na\nx\na\n=\n+\n\u2212\n+\n\u2212\n\u222b\n(v)\n1\n2\n2\nsin\nC\ndx\nax\na\nx\n\u2212\n=\n+\n\u2212\n\u222b\n(vi)\n2\n2\n2\n2\nlog |\n|\nC\ndx\nx\nx\na\nx\na\n=\n+\n+\n+\n+\n\u222b\n\ufffd Integration by parts\nFor given functions f1 and  f2, we have\n1\n2\n1\n2\n1\n2\n( )\n( )\n( )\n( )\n( )\n( )\nd\nf\nx\nf\nx dx\nf\nx\nf\nx dx\nf x\nf\nx dx dx\ndx\n\uf8ee\n\uf8f9\n\u22c5\n=\n\u2212\n\u22c5\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\u222b\n\u222b\n\u222b\n\u222b\n, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "4125-4128",
        "Text": "Using substitution technique, we obtain the following standard\nintegrals (i)\ntan\nlog sec\nC\nx dx\nx\n=\n+\n\u222b\n(ii)\ncot\nlog sin\nC\nx dx\nx\n=\n+\n\u222b\n(iii)\nsec\nlog sec\ntan\nC\nx dx\nx\nx\n=\n+\n+\n\u222b\n(iv)\ncosec\nlog cosec\ncot\nC\nx dx\nx\nx\n=\n\u2212\n+\n\u222b\n\ufffd Integrals of some special functions\n(i)\n2\n2\n1 log\nC\n2\ndx\nx\na\na\nx\na\nx\na\n\u2212\n=\n+\n+\n\u2212\n\u222b\n(ii)\n2\n2\n1 log\nC\n2\ndx\na\nx\na\na\nx\na\nx\n+\n=\n+\n\u2212\n\u2212\n\u222b\n(iii)\n1\n2\n2\n1 tan\nC\ndx\nx\na\na\nx\na\n\u2212\n=\n+\n+\n\u222b\nINTEGRALS         357\n(iv)\n2\n2\n2\n2\nlog\nC\ndx\nx\nx\na\nx\na\n=\n+\n\u2212\n+\n\u2212\n\u222b\n(v)\n1\n2\n2\nsin\nC\ndx\nax\na\nx\n\u2212\n=\n+\n\u2212\n\u222b\n(vi)\n2\n2\n2\n2\nlog |\n|\nC\ndx\nx\nx\na\nx\na\n=\n+\n+\n+\n+\n\u222b\n\ufffd Integration by parts\nFor given functions f1 and  f2, we have\n1\n2\n1\n2\n1\n2\n( )\n( )\n( )\n( )\n( )\n( )\nd\nf\nx\nf\nx dx\nf\nx\nf\nx dx\nf x\nf\nx dx dx\ndx\n\uf8ee\n\uf8f9\n\u22c5\n=\n\u2212\n\u22c5\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\u222b\n\u222b\n\u222b\n\u222b\n, i e , the\nintegral of the product of two functions = first function \u00d7 integral of the\nsecond function \u2013 integral of {differential coefficient of the first function \u00d7\nintegral of the second function}"
    },
    {
        "Chapter": "1",
        "sentence_range": "4126-4129",
        "Text": "(i)\ntan\nlog sec\nC\nx dx\nx\n=\n+\n\u222b\n(ii)\ncot\nlog sin\nC\nx dx\nx\n=\n+\n\u222b\n(iii)\nsec\nlog sec\ntan\nC\nx dx\nx\nx\n=\n+\n+\n\u222b\n(iv)\ncosec\nlog cosec\ncot\nC\nx dx\nx\nx\n=\n\u2212\n+\n\u222b\n\ufffd Integrals of some special functions\n(i)\n2\n2\n1 log\nC\n2\ndx\nx\na\na\nx\na\nx\na\n\u2212\n=\n+\n+\n\u2212\n\u222b\n(ii)\n2\n2\n1 log\nC\n2\ndx\na\nx\na\na\nx\na\nx\n+\n=\n+\n\u2212\n\u2212\n\u222b\n(iii)\n1\n2\n2\n1 tan\nC\ndx\nx\na\na\nx\na\n\u2212\n=\n+\n+\n\u222b\nINTEGRALS         357\n(iv)\n2\n2\n2\n2\nlog\nC\ndx\nx\nx\na\nx\na\n=\n+\n\u2212\n+\n\u2212\n\u222b\n(v)\n1\n2\n2\nsin\nC\ndx\nax\na\nx\n\u2212\n=\n+\n\u2212\n\u222b\n(vi)\n2\n2\n2\n2\nlog |\n|\nC\ndx\nx\nx\na\nx\na\n=\n+\n+\n+\n+\n\u222b\n\ufffd Integration by parts\nFor given functions f1 and  f2, we have\n1\n2\n1\n2\n1\n2\n( )\n( )\n( )\n( )\n( )\n( )\nd\nf\nx\nf\nx dx\nf\nx\nf\nx dx\nf x\nf\nx dx dx\ndx\n\uf8ee\n\uf8f9\n\u22c5\n=\n\u2212\n\u22c5\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\u222b\n\u222b\n\u222b\n\u222b\n, i e , the\nintegral of the product of two functions = first function \u00d7 integral of the\nsecond function \u2013 integral of {differential coefficient of the first function \u00d7\nintegral of the second function} Care must be taken in choosing the first\nfunction and the second function"
    },
    {
        "Chapter": "1",
        "sentence_range": "4127-4130",
        "Text": "e , the\nintegral of the product of two functions = first function \u00d7 integral of the\nsecond function \u2013 integral of {differential coefficient of the first function \u00d7\nintegral of the second function} Care must be taken in choosing the first\nfunction and the second function Obviously, we must take that function as\nthe second function whose integral is well known to us"
    },
    {
        "Chapter": "1",
        "sentence_range": "4128-4131",
        "Text": ", the\nintegral of the product of two functions = first function \u00d7 integral of the\nsecond function \u2013 integral of {differential coefficient of the first function \u00d7\nintegral of the second function} Care must be taken in choosing the first\nfunction and the second function Obviously, we must take that function as\nthe second function whose integral is well known to us \ufffd\n[ ( )\n( )]\n( )\nC\nx\nx\ne\nf x\nf\nx\ndx\ne f x dx\n\u2032\n+\n=\n+\n\u222b\n\u222b\n\ufffd Some special types of integrals\n(i)\n2\n2\n2\n2\n2\n2\n2\nlog\nC\n2\n2\nx\na\nx\na\ndx\nx\na\nx\nx\na\n\u2212\n=\n\u2212\n\u2212\n+\n\u2212\n+\n\u222b\n(ii)\n2\n2\n2\n2\n2\n2\n2\nlog\nC\n2\n2\nx\na\nx\na\ndx\nx\na\nx\nx\na\n+\n=\n+\n+\n+\n+\n+\n\u222b\n(iii)\n2\n2\n2\n2\n2\nsin1\nC\n2\n2\nx\na\nx\na\nx\ndx\na\nx\na\n\u2212\n\u2212\n=\n\u2212\n+\n+\n\u222b\n(iv) Integrals of the types \n2\n2\nor\ndx\ndx\nax\nbx\nc\nax\nbx\nc\n+\n+\n+\n+\n\u222b\n\u222b\ncan be\ntransformed into standard form by expressing\nax2 + bx + c = \n2\n2\n2\n2\n2\n4\nb\nc\nb\nc\nb\na x\nx\na\nx\na\na\na\na\na\n\uf8ee\n\uf8f9\n\uf8eb\n\uf8f6\n\uf8ee\n\uf8f9\n\uf8eb\n\uf8f6\n+\n+\n=\n+\n+\n\u2212\n\uf8ef\n\uf8fa\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8ed\n\uf8f8\n\uf8ef\n\uf8fa\n\uf8ed\n\uf8f8\n\uf8f0\n\uf8fb\n(v) Integrals of the types \n2\n2\nor\npx\nq dx\npx\nq dx\nax\nbx\nc\nax\nbx\nc\n+\n+\n+\n+\n+\n+\n\u222b\n\u222b\ncan be\n358\nMATHEMATICS\ntransformed into standard form by expressing\n2\nA\n(\n)\nB\nA (2\n)\nB\nd\npx\nq\nax\nbx\nc\nax\nb\ndx\n+\n=\n+\n+\n+\n=\n+\n+\n, where A and B are\ndetermined by comparing coefficients on both sides"
    },
    {
        "Chapter": "1",
        "sentence_range": "4129-4132",
        "Text": "Care must be taken in choosing the first\nfunction and the second function Obviously, we must take that function as\nthe second function whose integral is well known to us \ufffd\n[ ( )\n( )]\n( )\nC\nx\nx\ne\nf x\nf\nx\ndx\ne f x dx\n\u2032\n+\n=\n+\n\u222b\n\u222b\n\ufffd Some special types of integrals\n(i)\n2\n2\n2\n2\n2\n2\n2\nlog\nC\n2\n2\nx\na\nx\na\ndx\nx\na\nx\nx\na\n\u2212\n=\n\u2212\n\u2212\n+\n\u2212\n+\n\u222b\n(ii)\n2\n2\n2\n2\n2\n2\n2\nlog\nC\n2\n2\nx\na\nx\na\ndx\nx\na\nx\nx\na\n+\n=\n+\n+\n+\n+\n+\n\u222b\n(iii)\n2\n2\n2\n2\n2\nsin1\nC\n2\n2\nx\na\nx\na\nx\ndx\na\nx\na\n\u2212\n\u2212\n=\n\u2212\n+\n+\n\u222b\n(iv) Integrals of the types \n2\n2\nor\ndx\ndx\nax\nbx\nc\nax\nbx\nc\n+\n+\n+\n+\n\u222b\n\u222b\ncan be\ntransformed into standard form by expressing\nax2 + bx + c = \n2\n2\n2\n2\n2\n4\nb\nc\nb\nc\nb\na x\nx\na\nx\na\na\na\na\na\n\uf8ee\n\uf8f9\n\uf8eb\n\uf8f6\n\uf8ee\n\uf8f9\n\uf8eb\n\uf8f6\n+\n+\n=\n+\n+\n\u2212\n\uf8ef\n\uf8fa\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8ed\n\uf8f8\n\uf8ef\n\uf8fa\n\uf8ed\n\uf8f8\n\uf8f0\n\uf8fb\n(v) Integrals of the types \n2\n2\nor\npx\nq dx\npx\nq dx\nax\nbx\nc\nax\nbx\nc\n+\n+\n+\n+\n+\n+\n\u222b\n\u222b\ncan be\n358\nMATHEMATICS\ntransformed into standard form by expressing\n2\nA\n(\n)\nB\nA (2\n)\nB\nd\npx\nq\nax\nbx\nc\nax\nb\ndx\n+\n=\n+\n+\n+\n=\n+\n+\n, where A and B are\ndetermined by comparing coefficients on both sides \ufffd We have defined\n( )\nb\n\u222ba f x dx\n as the area of the region bounded by the curve\ny = f (x), a \u2264 x \u2264 b, the x-axis and the ordinates x = a and x = b"
    },
    {
        "Chapter": "1",
        "sentence_range": "4130-4133",
        "Text": "Obviously, we must take that function as\nthe second function whose integral is well known to us \ufffd\n[ ( )\n( )]\n( )\nC\nx\nx\ne\nf x\nf\nx\ndx\ne f x dx\n\u2032\n+\n=\n+\n\u222b\n\u222b\n\ufffd Some special types of integrals\n(i)\n2\n2\n2\n2\n2\n2\n2\nlog\nC\n2\n2\nx\na\nx\na\ndx\nx\na\nx\nx\na\n\u2212\n=\n\u2212\n\u2212\n+\n\u2212\n+\n\u222b\n(ii)\n2\n2\n2\n2\n2\n2\n2\nlog\nC\n2\n2\nx\na\nx\na\ndx\nx\na\nx\nx\na\n+\n=\n+\n+\n+\n+\n+\n\u222b\n(iii)\n2\n2\n2\n2\n2\nsin1\nC\n2\n2\nx\na\nx\na\nx\ndx\na\nx\na\n\u2212\n\u2212\n=\n\u2212\n+\n+\n\u222b\n(iv) Integrals of the types \n2\n2\nor\ndx\ndx\nax\nbx\nc\nax\nbx\nc\n+\n+\n+\n+\n\u222b\n\u222b\ncan be\ntransformed into standard form by expressing\nax2 + bx + c = \n2\n2\n2\n2\n2\n4\nb\nc\nb\nc\nb\na x\nx\na\nx\na\na\na\na\na\n\uf8ee\n\uf8f9\n\uf8eb\n\uf8f6\n\uf8ee\n\uf8f9\n\uf8eb\n\uf8f6\n+\n+\n=\n+\n+\n\u2212\n\uf8ef\n\uf8fa\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8ed\n\uf8f8\n\uf8ef\n\uf8fa\n\uf8ed\n\uf8f8\n\uf8f0\n\uf8fb\n(v) Integrals of the types \n2\n2\nor\npx\nq dx\npx\nq dx\nax\nbx\nc\nax\nbx\nc\n+\n+\n+\n+\n+\n+\n\u222b\n\u222b\ncan be\n358\nMATHEMATICS\ntransformed into standard form by expressing\n2\nA\n(\n)\nB\nA (2\n)\nB\nd\npx\nq\nax\nbx\nc\nax\nb\ndx\n+\n=\n+\n+\n+\n=\n+\n+\n, where A and B are\ndetermined by comparing coefficients on both sides \ufffd We have defined\n( )\nb\n\u222ba f x dx\n as the area of the region bounded by the curve\ny = f (x), a \u2264 x \u2264 b, the x-axis and the ordinates x = a and x = b Let x be a\ngiven point in [a, b]"
    },
    {
        "Chapter": "1",
        "sentence_range": "4131-4134",
        "Text": "\ufffd\n[ ( )\n( )]\n( )\nC\nx\nx\ne\nf x\nf\nx\ndx\ne f x dx\n\u2032\n+\n=\n+\n\u222b\n\u222b\n\ufffd Some special types of integrals\n(i)\n2\n2\n2\n2\n2\n2\n2\nlog\nC\n2\n2\nx\na\nx\na\ndx\nx\na\nx\nx\na\n\u2212\n=\n\u2212\n\u2212\n+\n\u2212\n+\n\u222b\n(ii)\n2\n2\n2\n2\n2\n2\n2\nlog\nC\n2\n2\nx\na\nx\na\ndx\nx\na\nx\nx\na\n+\n=\n+\n+\n+\n+\n+\n\u222b\n(iii)\n2\n2\n2\n2\n2\nsin1\nC\n2\n2\nx\na\nx\na\nx\ndx\na\nx\na\n\u2212\n\u2212\n=\n\u2212\n+\n+\n\u222b\n(iv) Integrals of the types \n2\n2\nor\ndx\ndx\nax\nbx\nc\nax\nbx\nc\n+\n+\n+\n+\n\u222b\n\u222b\ncan be\ntransformed into standard form by expressing\nax2 + bx + c = \n2\n2\n2\n2\n2\n4\nb\nc\nb\nc\nb\na x\nx\na\nx\na\na\na\na\na\n\uf8ee\n\uf8f9\n\uf8eb\n\uf8f6\n\uf8ee\n\uf8f9\n\uf8eb\n\uf8f6\n+\n+\n=\n+\n+\n\u2212\n\uf8ef\n\uf8fa\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8ed\n\uf8f8\n\uf8ef\n\uf8fa\n\uf8ed\n\uf8f8\n\uf8f0\n\uf8fb\n(v) Integrals of the types \n2\n2\nor\npx\nq dx\npx\nq dx\nax\nbx\nc\nax\nbx\nc\n+\n+\n+\n+\n+\n+\n\u222b\n\u222b\ncan be\n358\nMATHEMATICS\ntransformed into standard form by expressing\n2\nA\n(\n)\nB\nA (2\n)\nB\nd\npx\nq\nax\nbx\nc\nax\nb\ndx\n+\n=\n+\n+\n+\n=\n+\n+\n, where A and B are\ndetermined by comparing coefficients on both sides \ufffd We have defined\n( )\nb\n\u222ba f x dx\n as the area of the region bounded by the curve\ny = f (x), a \u2264 x \u2264 b, the x-axis and the ordinates x = a and x = b Let x be a\ngiven point in [a, b] Then \n( )\nx\n\u222ba f x dx\n represents the Area function A (x)"
    },
    {
        "Chapter": "1",
        "sentence_range": "4132-4135",
        "Text": "\ufffd We have defined\n( )\nb\n\u222ba f x dx\n as the area of the region bounded by the curve\ny = f (x), a \u2264 x \u2264 b, the x-axis and the ordinates x = a and x = b Let x be a\ngiven point in [a, b] Then \n( )\nx\n\u222ba f x dx\n represents the Area function A (x) This concept of area function leads to the Fundamental Theorems of Integral\nCalculus"
    },
    {
        "Chapter": "1",
        "sentence_range": "4133-4136",
        "Text": "Let x be a\ngiven point in [a, b] Then \n( )\nx\n\u222ba f x dx\n represents the Area function A (x) This concept of area function leads to the Fundamental Theorems of Integral\nCalculus \ufffd First fundamental theorem of integral calculus\nLet the area function be defined by A(x) = \n( )\nx\n\u222ba f x dx\n for all x \u2265 a, where\nthe function f is assumed to be continuous on [a, b]"
    },
    {
        "Chapter": "1",
        "sentence_range": "4134-4137",
        "Text": "Then \n( )\nx\n\u222ba f x dx\n represents the Area function A (x) This concept of area function leads to the Fundamental Theorems of Integral\nCalculus \ufffd First fundamental theorem of integral calculus\nLet the area function be defined by A(x) = \n( )\nx\n\u222ba f x dx\n for all x \u2265 a, where\nthe function f is assumed to be continuous on [a, b] Then A\u2032(x) = f (x) for all\nx \u2208 [a, b]"
    },
    {
        "Chapter": "1",
        "sentence_range": "4135-4138",
        "Text": "This concept of area function leads to the Fundamental Theorems of Integral\nCalculus \ufffd First fundamental theorem of integral calculus\nLet the area function be defined by A(x) = \n( )\nx\n\u222ba f x dx\n for all x \u2265 a, where\nthe function f is assumed to be continuous on [a, b] Then A\u2032(x) = f (x) for all\nx \u2208 [a, b] \ufffd Second fundamental theorem of integral calculus\nLet f be a continuous function of x defined on the closed interval [a, b] and\nlet F be another function such that \nF( )\n( )\nd\nx\nf x\ndx\n=\n for all x in the domain of\nf, then \n[\n]\n( )\nF( )\nC\nF ( )\nF ( )\nb\nb\na\na f x dx\nx\nb\na\n=\n+\n=\n\u2212\n\u222b"
    },
    {
        "Chapter": "1",
        "sentence_range": "4136-4139",
        "Text": "\ufffd First fundamental theorem of integral calculus\nLet the area function be defined by A(x) = \n( )\nx\n\u222ba f x dx\n for all x \u2265 a, where\nthe function f is assumed to be continuous on [a, b] Then A\u2032(x) = f (x) for all\nx \u2208 [a, b] \ufffd Second fundamental theorem of integral calculus\nLet f be a continuous function of x defined on the closed interval [a, b] and\nlet F be another function such that \nF( )\n( )\nd\nx\nf x\ndx\n=\n for all x in the domain of\nf, then \n[\n]\n( )\nF( )\nC\nF ( )\nF ( )\nb\nb\na\na f x dx\nx\nb\na\n=\n+\n=\n\u2212\n\u222b This is called the definite integral of f over the range [a, b], where a and b\nare called the limits of integration, a being the lower limit and b the\nupper limit"
    },
    {
        "Chapter": "1",
        "sentence_range": "4137-4140",
        "Text": "Then A\u2032(x) = f (x) for all\nx \u2208 [a, b] \ufffd Second fundamental theorem of integral calculus\nLet f be a continuous function of x defined on the closed interval [a, b] and\nlet F be another function such that \nF( )\n( )\nd\nx\nf x\ndx\n=\n for all x in the domain of\nf, then \n[\n]\n( )\nF( )\nC\nF ( )\nF ( )\nb\nb\na\na f x dx\nx\nb\na\n=\n+\n=\n\u2212\n\u222b This is called the definite integral of f over the range [a, b], where a and b\nare called the limits of integration, a being the lower limit and b the\nupper limit \u2014\ufffd\n\ufffd\n\ufffd\n\ufffd\n\ufffd\u2014\nAPPLICATION OF INTEGRALS         359\nFig 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "4138-4141",
        "Text": "\ufffd Second fundamental theorem of integral calculus\nLet f be a continuous function of x defined on the closed interval [a, b] and\nlet F be another function such that \nF( )\n( )\nd\nx\nf x\ndx\n=\n for all x in the domain of\nf, then \n[\n]\n( )\nF( )\nC\nF ( )\nF ( )\nb\nb\na\na f x dx\nx\nb\na\n=\n+\n=\n\u2212\n\u222b This is called the definite integral of f over the range [a, b], where a and b\nare called the limits of integration, a being the lower limit and b the\nupper limit \u2014\ufffd\n\ufffd\n\ufffd\n\ufffd\n\ufffd\u2014\nAPPLICATION OF INTEGRALS         359\nFig 8 1\n\ufffd One should study Mathematics because it is only through Mathematics that\nnature can be conceived in harmonious form"
    },
    {
        "Chapter": "1",
        "sentence_range": "4139-4142",
        "Text": "This is called the definite integral of f over the range [a, b], where a and b\nare called the limits of integration, a being the lower limit and b the\nupper limit \u2014\ufffd\n\ufffd\n\ufffd\n\ufffd\n\ufffd\u2014\nAPPLICATION OF INTEGRALS         359\nFig 8 1\n\ufffd One should study Mathematics because it is only through Mathematics that\nnature can be conceived in harmonious form \u2013 BIRKHOFF \ufffd\n8"
    },
    {
        "Chapter": "1",
        "sentence_range": "4140-4143",
        "Text": "\u2014\ufffd\n\ufffd\n\ufffd\n\ufffd\n\ufffd\u2014\nAPPLICATION OF INTEGRALS         359\nFig 8 1\n\ufffd One should study Mathematics because it is only through Mathematics that\nnature can be conceived in harmonious form \u2013 BIRKHOFF \ufffd\n8 1  Introduction\nIn geometry, we have learnt formulae to calculate areas\nof various geometrical figures including triangles,\nrectangles, trapezias and circles"
    },
    {
        "Chapter": "1",
        "sentence_range": "4141-4144",
        "Text": "1\n\ufffd One should study Mathematics because it is only through Mathematics that\nnature can be conceived in harmonious form \u2013 BIRKHOFF \ufffd\n8 1  Introduction\nIn geometry, we have learnt formulae to calculate areas\nof various geometrical figures including triangles,\nrectangles, trapezias and circles Such formulae are\nfundamental in the applications of mathematics to many\nreal life problems"
    },
    {
        "Chapter": "1",
        "sentence_range": "4142-4145",
        "Text": "\u2013 BIRKHOFF \ufffd\n8 1  Introduction\nIn geometry, we have learnt formulae to calculate areas\nof various geometrical figures including triangles,\nrectangles, trapezias and circles Such formulae are\nfundamental in the applications of mathematics to many\nreal life problems The formulae of elementary geometry\nallow us to calculate areas of many simple figures"
    },
    {
        "Chapter": "1",
        "sentence_range": "4143-4146",
        "Text": "1  Introduction\nIn geometry, we have learnt formulae to calculate areas\nof various geometrical figures including triangles,\nrectangles, trapezias and circles Such formulae are\nfundamental in the applications of mathematics to many\nreal life problems The formulae of elementary geometry\nallow us to calculate areas of many simple figures However, they are inadequate for calculating the areas\nenclosed by curves"
    },
    {
        "Chapter": "1",
        "sentence_range": "4144-4147",
        "Text": "Such formulae are\nfundamental in the applications of mathematics to many\nreal life problems The formulae of elementary geometry\nallow us to calculate areas of many simple figures However, they are inadequate for calculating the areas\nenclosed by curves For that we shall need some concepts\nof Integral Calculus"
    },
    {
        "Chapter": "1",
        "sentence_range": "4145-4148",
        "Text": "The formulae of elementary geometry\nallow us to calculate areas of many simple figures However, they are inadequate for calculating the areas\nenclosed by curves For that we shall need some concepts\nof Integral Calculus In the previous chapter, we have studied to find the\narea bounded by the curve y = f (x), the ordinates x = a,\nx = b and x-axis, while calculating definite integral as the\nlimit of a sum"
    },
    {
        "Chapter": "1",
        "sentence_range": "4146-4149",
        "Text": "However, they are inadequate for calculating the areas\nenclosed by curves For that we shall need some concepts\nof Integral Calculus In the previous chapter, we have studied to find the\narea bounded by the curve y = f (x), the ordinates x = a,\nx = b and x-axis, while calculating definite integral as the\nlimit of a sum Here, in this chapter, we shall study a specific\napplication of integrals to find the area under simple curves,\narea between lines and arcs of circles, parabolas and\nellipses (standard forms only)"
    },
    {
        "Chapter": "1",
        "sentence_range": "4147-4150",
        "Text": "For that we shall need some concepts\nof Integral Calculus In the previous chapter, we have studied to find the\narea bounded by the curve y = f (x), the ordinates x = a,\nx = b and x-axis, while calculating definite integral as the\nlimit of a sum Here, in this chapter, we shall study a specific\napplication of integrals to find the area under simple curves,\narea between lines and arcs of circles, parabolas and\nellipses (standard forms only) We shall also deal with finding\nthe area bounded by the above said curves"
    },
    {
        "Chapter": "1",
        "sentence_range": "4148-4151",
        "Text": "In the previous chapter, we have studied to find the\narea bounded by the curve y = f (x), the ordinates x = a,\nx = b and x-axis, while calculating definite integral as the\nlimit of a sum Here, in this chapter, we shall study a specific\napplication of integrals to find the area under simple curves,\narea between lines and arcs of circles, parabolas and\nellipses (standard forms only) We shall also deal with finding\nthe area bounded by the above said curves 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "4149-4152",
        "Text": "Here, in this chapter, we shall study a specific\napplication of integrals to find the area under simple curves,\narea between lines and arcs of circles, parabolas and\nellipses (standard forms only) We shall also deal with finding\nthe area bounded by the above said curves 8 2 Area under Simple Curves\nIn the previous chapter, we have studied\ndefinite integral as the limit of a sum and\nhow to evaluate definite integral using\nFundamental Theorem of Calculus"
    },
    {
        "Chapter": "1",
        "sentence_range": "4150-4153",
        "Text": "We shall also deal with finding\nthe area bounded by the above said curves 8 2 Area under Simple Curves\nIn the previous chapter, we have studied\ndefinite integral as the limit of a sum and\nhow to evaluate definite integral using\nFundamental Theorem of Calculus Now,\nwe consider the easy and intuitive way of\nfinding the area bounded by the curve\ny = f (x), x-axis and the ordinates x = a and\nx = b"
    },
    {
        "Chapter": "1",
        "sentence_range": "4151-4154",
        "Text": "8 2 Area under Simple Curves\nIn the previous chapter, we have studied\ndefinite integral as the limit of a sum and\nhow to evaluate definite integral using\nFundamental Theorem of Calculus Now,\nwe consider the easy and intuitive way of\nfinding the area bounded by the curve\ny = f (x), x-axis and the ordinates x = a and\nx = b From Fig 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "4152-4155",
        "Text": "2 Area under Simple Curves\nIn the previous chapter, we have studied\ndefinite integral as the limit of a sum and\nhow to evaluate definite integral using\nFundamental Theorem of Calculus Now,\nwe consider the easy and intuitive way of\nfinding the area bounded by the curve\ny = f (x), x-axis and the ordinates x = a and\nx = b From Fig 8 1, we can think of area\nunder the curve as composed of large\nnumber of very thin vertical strips"
    },
    {
        "Chapter": "1",
        "sentence_range": "4153-4156",
        "Text": "Now,\nwe consider the easy and intuitive way of\nfinding the area bounded by the curve\ny = f (x), x-axis and the ordinates x = a and\nx = b From Fig 8 1, we can think of area\nunder the curve as composed of large\nnumber of very thin vertical strips Consider\nan arbitrary strip of height y and width dx,\nthen dA (area of the elementary strip)= ydx,\nwhere, y = f(x)"
    },
    {
        "Chapter": "1",
        "sentence_range": "4154-4157",
        "Text": "From Fig 8 1, we can think of area\nunder the curve as composed of large\nnumber of very thin vertical strips Consider\nan arbitrary strip of height y and width dx,\nthen dA (area of the elementary strip)= ydx,\nwhere, y = f(x) Chapter 8\nAPPLICATION OF INTEGRALS\nA"
    },
    {
        "Chapter": "1",
        "sentence_range": "4155-4158",
        "Text": "1, we can think of area\nunder the curve as composed of large\nnumber of very thin vertical strips Consider\nan arbitrary strip of height y and width dx,\nthen dA (area of the elementary strip)= ydx,\nwhere, y = f(x) Chapter 8\nAPPLICATION OF INTEGRALS\nA L"
    },
    {
        "Chapter": "1",
        "sentence_range": "4156-4159",
        "Text": "Consider\nan arbitrary strip of height y and width dx,\nthen dA (area of the elementary strip)= ydx,\nwhere, y = f(x) Chapter 8\nAPPLICATION OF INTEGRALS\nA L Cauchy\n(1789-1857)\n360\nMATHEMATICS\nFig  8"
    },
    {
        "Chapter": "1",
        "sentence_range": "4157-4160",
        "Text": "Chapter 8\nAPPLICATION OF INTEGRALS\nA L Cauchy\n(1789-1857)\n360\nMATHEMATICS\nFig  8 2\nThis area is called the elementary area which is located at an arbitrary position\nwithin the region which is specified by some value of x between a and b"
    },
    {
        "Chapter": "1",
        "sentence_range": "4158-4161",
        "Text": "L Cauchy\n(1789-1857)\n360\nMATHEMATICS\nFig  8 2\nThis area is called the elementary area which is located at an arbitrary position\nwithin the region which is specified by some value of x between a and b We can think\nof the total area A of the region between x-axis, ordinates x = a, x = b and the curve\ny = f (x) as the result of adding up the elementary areas of thin strips across the region\nPQRSP"
    },
    {
        "Chapter": "1",
        "sentence_range": "4159-4162",
        "Text": "Cauchy\n(1789-1857)\n360\nMATHEMATICS\nFig  8 2\nThis area is called the elementary area which is located at an arbitrary position\nwithin the region which is specified by some value of x between a and b We can think\nof the total area A of the region between x-axis, ordinates x = a, x = b and the curve\ny = f (x) as the result of adding up the elementary areas of thin strips across the region\nPQRSP Symbolically, we express\nA = \nA\n( )\nb\nb\nb\na\na\na\nd\nydx\nf x dx\n=\n=\n\u222b\n\u222b\n\u222b\nThe area A of the region bounded by\nthe curve x = g (y), y-axis and the lines y  =  c,\ny = d is given by\nA = \n( )\nd\nd\nc\nc\nxdy\ng y dy\n=\n\u222b\n\u222b\nHere, we consider horizontal strips as shown in\nthe Fig 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "4160-4163",
        "Text": "2\nThis area is called the elementary area which is located at an arbitrary position\nwithin the region which is specified by some value of x between a and b We can think\nof the total area A of the region between x-axis, ordinates x = a, x = b and the curve\ny = f (x) as the result of adding up the elementary areas of thin strips across the region\nPQRSP Symbolically, we express\nA = \nA\n( )\nb\nb\nb\na\na\na\nd\nydx\nf x dx\n=\n=\n\u222b\n\u222b\n\u222b\nThe area A of the region bounded by\nthe curve x = g (y), y-axis and the lines y  =  c,\ny = d is given by\nA = \n( )\nd\nd\nc\nc\nxdy\ng y dy\n=\n\u222b\n\u222b\nHere, we consider horizontal strips as shown in\nthe Fig 8 2\nRemark If the position of the curve under consideration is below the x-axis, then since\nf (x) < 0 from x = a to x = b, as shown in Fig 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "4161-4164",
        "Text": "We can think\nof the total area A of the region between x-axis, ordinates x = a, x = b and the curve\ny = f (x) as the result of adding up the elementary areas of thin strips across the region\nPQRSP Symbolically, we express\nA = \nA\n( )\nb\nb\nb\na\na\na\nd\nydx\nf x dx\n=\n=\n\u222b\n\u222b\n\u222b\nThe area A of the region bounded by\nthe curve x = g (y), y-axis and the lines y  =  c,\ny = d is given by\nA = \n( )\nd\nd\nc\nc\nxdy\ng y dy\n=\n\u222b\n\u222b\nHere, we consider horizontal strips as shown in\nthe Fig 8 2\nRemark If the position of the curve under consideration is below the x-axis, then since\nf (x) < 0 from x = a to x = b, as shown in Fig 8 3, the area bounded by the curve, x-axis\nand the ordinates x = a, x = b come out to be negative"
    },
    {
        "Chapter": "1",
        "sentence_range": "4162-4165",
        "Text": "Symbolically, we express\nA = \nA\n( )\nb\nb\nb\na\na\na\nd\nydx\nf x dx\n=\n=\n\u222b\n\u222b\n\u222b\nThe area A of the region bounded by\nthe curve x = g (y), y-axis and the lines y  =  c,\ny = d is given by\nA = \n( )\nd\nd\nc\nc\nxdy\ng y dy\n=\n\u222b\n\u222b\nHere, we consider horizontal strips as shown in\nthe Fig 8 2\nRemark If the position of the curve under consideration is below the x-axis, then since\nf (x) < 0 from x = a to x = b, as shown in Fig 8 3, the area bounded by the curve, x-axis\nand the ordinates x = a, x = b come out to be negative But, it is only the numerical\nvalue of the area which is taken into consideration"
    },
    {
        "Chapter": "1",
        "sentence_range": "4163-4166",
        "Text": "2\nRemark If the position of the curve under consideration is below the x-axis, then since\nf (x) < 0 from x = a to x = b, as shown in Fig 8 3, the area bounded by the curve, x-axis\nand the ordinates x = a, x = b come out to be negative But, it is only the numerical\nvalue of the area which is taken into consideration Thus, if the area is negative, we\ntake its absolute value, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "4164-4167",
        "Text": "3, the area bounded by the curve, x-axis\nand the ordinates x = a, x = b come out to be negative But, it is only the numerical\nvalue of the area which is taken into consideration Thus, if the area is negative, we\ntake its absolute value, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "4165-4168",
        "Text": "But, it is only the numerical\nvalue of the area which is taken into consideration Thus, if the area is negative, we\ntake its absolute value, i e , \n( )\nb\n\u222ba f x dx"
    },
    {
        "Chapter": "1",
        "sentence_range": "4166-4169",
        "Text": "Thus, if the area is negative, we\ntake its absolute value, i e , \n( )\nb\n\u222ba f x dx Fig 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "4167-4170",
        "Text": "e , \n( )\nb\n\u222ba f x dx Fig 8 3\nGenerally, it may happen that some portion of the curve is above x-axis and some is\nbelow the x-axis as shown in the Fig 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "4168-4171",
        "Text": ", \n( )\nb\n\u222ba f x dx Fig 8 3\nGenerally, it may happen that some portion of the curve is above x-axis and some is\nbelow the x-axis as shown in the Fig 8 4"
    },
    {
        "Chapter": "1",
        "sentence_range": "4169-4172",
        "Text": "Fig 8 3\nGenerally, it may happen that some portion of the curve is above x-axis and some is\nbelow the x-axis as shown in the Fig 8 4 Here, A1 < 0 and A2 > 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "4170-4173",
        "Text": "3\nGenerally, it may happen that some portion of the curve is above x-axis and some is\nbelow the x-axis as shown in the Fig 8 4 Here, A1 < 0 and A2 > 0 Therefore, the area\nA bounded by the curve y = f (x), x-axis and the ordinates x = a and x = b is given\nby A = |A1| + A2"
    },
    {
        "Chapter": "1",
        "sentence_range": "4171-4174",
        "Text": "4 Here, A1 < 0 and A2 > 0 Therefore, the area\nA bounded by the curve y = f (x), x-axis and the ordinates x = a and x = b is given\nby A = |A1| + A2 APPLICATION OF INTEGRALS         361\nExample 1 Find the area enclosed by the circle x2 + y2 = a2"
    },
    {
        "Chapter": "1",
        "sentence_range": "4172-4175",
        "Text": "Here, A1 < 0 and A2 > 0 Therefore, the area\nA bounded by the curve y = f (x), x-axis and the ordinates x = a and x = b is given\nby A = |A1| + A2 APPLICATION OF INTEGRALS         361\nExample 1 Find the area enclosed by the circle x2 + y2 = a2 Solution From Fig 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "4173-4176",
        "Text": "Therefore, the area\nA bounded by the curve y = f (x), x-axis and the ordinates x = a and x = b is given\nby A = |A1| + A2 APPLICATION OF INTEGRALS         361\nExample 1 Find the area enclosed by the circle x2 + y2 = a2 Solution From Fig 8 5, the whole area enclosed\nby the given circle\n= 4 (area of the region AOBA bounded by\nthe curve, x-axis and the ordinates x = 0 and\nx = a) [as the circle is symmetrical about both\nx-axis and y-axis]\n= \n40\n\u222ba ydx\n (taking vertical strips)\n= \n2\n2\n40\na\na\n\u2212x dx\n\u222b\nSince x2 + y2 = a2 gives     y = \n2\n2\na\nx\n\u00b1\n\u2212\nAs the region AOBA lies in the first quadrant, y is taken as positive"
    },
    {
        "Chapter": "1",
        "sentence_range": "4174-4177",
        "Text": "APPLICATION OF INTEGRALS         361\nExample 1 Find the area enclosed by the circle x2 + y2 = a2 Solution From Fig 8 5, the whole area enclosed\nby the given circle\n= 4 (area of the region AOBA bounded by\nthe curve, x-axis and the ordinates x = 0 and\nx = a) [as the circle is symmetrical about both\nx-axis and y-axis]\n= \n40\n\u222ba ydx\n (taking vertical strips)\n= \n2\n2\n40\na\na\n\u2212x dx\n\u222b\nSince x2 + y2 = a2 gives     y = \n2\n2\na\nx\n\u00b1\n\u2212\nAs the region AOBA lies in the first quadrant, y is taken as positive Integrating, we get\nthe whole area enclosed by the given circle\n= \n2\n2\n2\n\u20131\n0\n4\nsin\n2\n2\na\nx\na\nx\na\nx\na\n\uf8ee\n\uf8f9\n\u2212\n+\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n= \n2\n1\n4\n0\nsin 1\n0\n2\n2\na\na\n\u2212\n\uf8ee\n\uf8f9\n\uf8eb\n\uf8f6\n\u00d7 +\n\u2212\n\uf8ef\n\uf8fa\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8f0\n\uf8fb\n =  \n2\n2\n4\n2\n2\na\na\n\uf8eb\n\uf8f6\n\uf8eb\u03c0\n\uf8f6 =\u03c0\n\uf8ec\n\uf8f7 \uf8ec\n\uf8f7\n\uf8ed\n\uf8f8 \uf8ed\n\uf8f8\nFig 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "4175-4178",
        "Text": "Solution From Fig 8 5, the whole area enclosed\nby the given circle\n= 4 (area of the region AOBA bounded by\nthe curve, x-axis and the ordinates x = 0 and\nx = a) [as the circle is symmetrical about both\nx-axis and y-axis]\n= \n40\n\u222ba ydx\n (taking vertical strips)\n= \n2\n2\n40\na\na\n\u2212x dx\n\u222b\nSince x2 + y2 = a2 gives     y = \n2\n2\na\nx\n\u00b1\n\u2212\nAs the region AOBA lies in the first quadrant, y is taken as positive Integrating, we get\nthe whole area enclosed by the given circle\n= \n2\n2\n2\n\u20131\n0\n4\nsin\n2\n2\na\nx\na\nx\na\nx\na\n\uf8ee\n\uf8f9\n\u2212\n+\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n= \n2\n1\n4\n0\nsin 1\n0\n2\n2\na\na\n\u2212\n\uf8ee\n\uf8f9\n\uf8eb\n\uf8f6\n\u00d7 +\n\u2212\n\uf8ef\n\uf8fa\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8f0\n\uf8fb\n =  \n2\n2\n4\n2\n2\na\na\n\uf8eb\n\uf8f6\n\uf8eb\u03c0\n\uf8f6 =\u03c0\n\uf8ec\n\uf8f7 \uf8ec\n\uf8f7\n\uf8ed\n\uf8f8 \uf8ed\n\uf8f8\nFig 8 5\nFig  8"
    },
    {
        "Chapter": "1",
        "sentence_range": "4176-4179",
        "Text": "5, the whole area enclosed\nby the given circle\n= 4 (area of the region AOBA bounded by\nthe curve, x-axis and the ordinates x = 0 and\nx = a) [as the circle is symmetrical about both\nx-axis and y-axis]\n= \n40\n\u222ba ydx\n (taking vertical strips)\n= \n2\n2\n40\na\na\n\u2212x dx\n\u222b\nSince x2 + y2 = a2 gives     y = \n2\n2\na\nx\n\u00b1\n\u2212\nAs the region AOBA lies in the first quadrant, y is taken as positive Integrating, we get\nthe whole area enclosed by the given circle\n= \n2\n2\n2\n\u20131\n0\n4\nsin\n2\n2\na\nx\na\nx\na\nx\na\n\uf8ee\n\uf8f9\n\u2212\n+\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n= \n2\n1\n4\n0\nsin 1\n0\n2\n2\na\na\n\u2212\n\uf8ee\n\uf8f9\n\uf8eb\n\uf8f6\n\u00d7 +\n\u2212\n\uf8ef\n\uf8fa\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8f0\n\uf8fb\n =  \n2\n2\n4\n2\n2\na\na\n\uf8eb\n\uf8f6\n\uf8eb\u03c0\n\uf8f6 =\u03c0\n\uf8ec\n\uf8f7 \uf8ec\n\uf8f7\n\uf8ed\n\uf8f8 \uf8ed\n\uf8f8\nFig 8 5\nFig  8 4\n362\nMATHEMATICS\nAlternatively, considering horizontal strips as shown in Fig 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "4177-4180",
        "Text": "Integrating, we get\nthe whole area enclosed by the given circle\n= \n2\n2\n2\n\u20131\n0\n4\nsin\n2\n2\na\nx\na\nx\na\nx\na\n\uf8ee\n\uf8f9\n\u2212\n+\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n= \n2\n1\n4\n0\nsin 1\n0\n2\n2\na\na\n\u2212\n\uf8ee\n\uf8f9\n\uf8eb\n\uf8f6\n\u00d7 +\n\u2212\n\uf8ef\n\uf8fa\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8f0\n\uf8fb\n =  \n2\n2\n4\n2\n2\na\na\n\uf8eb\n\uf8f6\n\uf8eb\u03c0\n\uf8f6 =\u03c0\n\uf8ec\n\uf8f7 \uf8ec\n\uf8f7\n\uf8ed\n\uf8f8 \uf8ed\n\uf8f8\nFig 8 5\nFig  8 4\n362\nMATHEMATICS\nAlternatively, considering horizontal strips as shown in Fig 8 6, the whole area of the\nregion enclosed by circle\n= \n40\n\u222ba xdy\n = \n2\n2\n40\na\na\ny\ndy\n\u2212\n\u222b\n(Why"
    },
    {
        "Chapter": "1",
        "sentence_range": "4178-4181",
        "Text": "5\nFig  8 4\n362\nMATHEMATICS\nAlternatively, considering horizontal strips as shown in Fig 8 6, the whole area of the\nregion enclosed by circle\n= \n40\n\u222ba xdy\n = \n2\n2\n40\na\na\ny\ndy\n\u2212\n\u222b\n(Why )\n= \n2\n2\n2\n1\n0\n4\nsin\n2\n2\na\na\ny\ny\na\ny\na\n\u2212\n\uf8ee\n\uf8f9\n\u2212\n+\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n= \n2\n1\n4\n0\nsin 1\n0\n2\na2\na\n\u2212\n\uf8ee\n\uf8f9\n\uf8eb\n\uf8f6\n\u00d7 +\n\u2212\n\uf8ef\n\uf8fa\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8f0\n\uf8fb\n= \n2\n2\n4\na2 2\na\n\u03c0 = \u03c0\nExample 2 Find the area enclosed by the ellipse \n2\n2\n2\n2\n1\nx\ny\na\nb\n+\n=\nSolution From Fig 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "4179-4182",
        "Text": "4\n362\nMATHEMATICS\nAlternatively, considering horizontal strips as shown in Fig 8 6, the whole area of the\nregion enclosed by circle\n= \n40\n\u222ba xdy\n = \n2\n2\n40\na\na\ny\ndy\n\u2212\n\u222b\n(Why )\n= \n2\n2\n2\n1\n0\n4\nsin\n2\n2\na\na\ny\ny\na\ny\na\n\u2212\n\uf8ee\n\uf8f9\n\u2212\n+\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n= \n2\n1\n4\n0\nsin 1\n0\n2\na2\na\n\u2212\n\uf8ee\n\uf8f9\n\uf8eb\n\uf8f6\n\u00d7 +\n\u2212\n\uf8ef\n\uf8fa\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8f0\n\uf8fb\n= \n2\n2\n4\na2 2\na\n\u03c0 = \u03c0\nExample 2 Find the area enclosed by the ellipse \n2\n2\n2\n2\n1\nx\ny\na\nb\n+\n=\nSolution From Fig 8 7, the area of the region ABA\u2032B\u2032A bounded by the ellipse\n= \nin\n4\n,\n0,\narea of theregion AOBA\nthe first quadrant bounded\nbythecurve x\naxis and theordinates x\nx\na\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\u2212\n=\n=\n\uf8ed\n\uf8f8\n(as the ellipse is symmetrical about both x-axis and y-axis)\n= \n40\n(takingverticalstrips)\na ydx\n\u222b\nNow \n2\n2\n2\n2\nx\ny\na\n+b\n = 1 gives \n2\n2\nb\ny\na\nx\n=\u00b1a\n\u2212\n, but as the region AOBA lies in the first\nquadrant, y is taken as positive"
    },
    {
        "Chapter": "1",
        "sentence_range": "4180-4183",
        "Text": "6, the whole area of the\nregion enclosed by circle\n= \n40\n\u222ba xdy\n = \n2\n2\n40\na\na\ny\ndy\n\u2212\n\u222b\n(Why )\n= \n2\n2\n2\n1\n0\n4\nsin\n2\n2\na\na\ny\ny\na\ny\na\n\u2212\n\uf8ee\n\uf8f9\n\u2212\n+\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n= \n2\n1\n4\n0\nsin 1\n0\n2\na2\na\n\u2212\n\uf8ee\n\uf8f9\n\uf8eb\n\uf8f6\n\u00d7 +\n\u2212\n\uf8ef\n\uf8fa\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8f0\n\uf8fb\n= \n2\n2\n4\na2 2\na\n\u03c0 = \u03c0\nExample 2 Find the area enclosed by the ellipse \n2\n2\n2\n2\n1\nx\ny\na\nb\n+\n=\nSolution From Fig 8 7, the area of the region ABA\u2032B\u2032A bounded by the ellipse\n= \nin\n4\n,\n0,\narea of theregion AOBA\nthe first quadrant bounded\nbythecurve x\naxis and theordinates x\nx\na\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\u2212\n=\n=\n\uf8ed\n\uf8f8\n(as the ellipse is symmetrical about both x-axis and y-axis)\n= \n40\n(takingverticalstrips)\na ydx\n\u222b\nNow \n2\n2\n2\n2\nx\ny\na\n+b\n = 1 gives \n2\n2\nb\ny\na\nx\n=\u00b1a\n\u2212\n, but as the region AOBA lies in the first\nquadrant, y is taken as positive So, the required area is\n= \n2\n2\n40\na b\na\nx dx\na\n\u2212\n\u222b\n= \n2\n2\n2\n\u20131\n0\n4\nsin\n2\n2\na\nb x\na\nx\na\nx\na\na\n\uf8ee\n\uf8f9\n\u2212\n+\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n (Why"
    },
    {
        "Chapter": "1",
        "sentence_range": "4181-4184",
        "Text": ")\n= \n2\n2\n2\n1\n0\n4\nsin\n2\n2\na\na\ny\ny\na\ny\na\n\u2212\n\uf8ee\n\uf8f9\n\u2212\n+\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n= \n2\n1\n4\n0\nsin 1\n0\n2\na2\na\n\u2212\n\uf8ee\n\uf8f9\n\uf8eb\n\uf8f6\n\u00d7 +\n\u2212\n\uf8ef\n\uf8fa\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8f0\n\uf8fb\n= \n2\n2\n4\na2 2\na\n\u03c0 = \u03c0\nExample 2 Find the area enclosed by the ellipse \n2\n2\n2\n2\n1\nx\ny\na\nb\n+\n=\nSolution From Fig 8 7, the area of the region ABA\u2032B\u2032A bounded by the ellipse\n= \nin\n4\n,\n0,\narea of theregion AOBA\nthe first quadrant bounded\nbythecurve x\naxis and theordinates x\nx\na\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\u2212\n=\n=\n\uf8ed\n\uf8f8\n(as the ellipse is symmetrical about both x-axis and y-axis)\n= \n40\n(takingverticalstrips)\na ydx\n\u222b\nNow \n2\n2\n2\n2\nx\ny\na\n+b\n = 1 gives \n2\n2\nb\ny\na\nx\n=\u00b1a\n\u2212\n, but as the region AOBA lies in the first\nquadrant, y is taken as positive So, the required area is\n= \n2\n2\n40\na b\na\nx dx\na\n\u2212\n\u222b\n= \n2\n2\n2\n\u20131\n0\n4\nsin\n2\n2\na\nb x\na\nx\na\nx\na\na\n\uf8ee\n\uf8f9\n\u2212\n+\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n (Why )\n= \n2\n1\n4\n0\nsin\n1\n0\n2\n2\nb\na\na\na\n\u2212\n\uf8ee\n\uf8f9\n\uf8eb\n\uf8f6\n\u00d7 +\n\u2212\n\uf8ef\n\uf8fa\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8ed\n\uf8f8\n\uf8f0\n\uf8fb\n= \n2\n4\n2 2\nb a\nab\na\n\u03c0 =\u03c0\nFig 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "4182-4185",
        "Text": "7, the area of the region ABA\u2032B\u2032A bounded by the ellipse\n= \nin\n4\n,\n0,\narea of theregion AOBA\nthe first quadrant bounded\nbythecurve x\naxis and theordinates x\nx\na\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\u2212\n=\n=\n\uf8ed\n\uf8f8\n(as the ellipse is symmetrical about both x-axis and y-axis)\n= \n40\n(takingverticalstrips)\na ydx\n\u222b\nNow \n2\n2\n2\n2\nx\ny\na\n+b\n = 1 gives \n2\n2\nb\ny\na\nx\n=\u00b1a\n\u2212\n, but as the region AOBA lies in the first\nquadrant, y is taken as positive So, the required area is\n= \n2\n2\n40\na b\na\nx dx\na\n\u2212\n\u222b\n= \n2\n2\n2\n\u20131\n0\n4\nsin\n2\n2\na\nb x\na\nx\na\nx\na\na\n\uf8ee\n\uf8f9\n\u2212\n+\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n (Why )\n= \n2\n1\n4\n0\nsin\n1\n0\n2\n2\nb\na\na\na\n\u2212\n\uf8ee\n\uf8f9\n\uf8eb\n\uf8f6\n\u00d7 +\n\u2212\n\uf8ef\n\uf8fa\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8ed\n\uf8f8\n\uf8f0\n\uf8fb\n= \n2\n4\n2 2\nb a\nab\na\n\u03c0 =\u03c0\nFig 8 6\nFig 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "4183-4186",
        "Text": "So, the required area is\n= \n2\n2\n40\na b\na\nx dx\na\n\u2212\n\u222b\n= \n2\n2\n2\n\u20131\n0\n4\nsin\n2\n2\na\nb x\na\nx\na\nx\na\na\n\uf8ee\n\uf8f9\n\u2212\n+\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n (Why )\n= \n2\n1\n4\n0\nsin\n1\n0\n2\n2\nb\na\na\na\n\u2212\n\uf8ee\n\uf8f9\n\uf8eb\n\uf8f6\n\u00d7 +\n\u2212\n\uf8ef\n\uf8fa\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8ed\n\uf8f8\n\uf8f0\n\uf8fb\n= \n2\n4\n2 2\nb a\nab\na\n\u03c0 =\u03c0\nFig 8 6\nFig 8 7\nAPPLICATION OF INTEGRALS         363\nAlternatively, considering horizontal strips as\nshown in the Fig 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "4184-4187",
        "Text": ")\n= \n2\n1\n4\n0\nsin\n1\n0\n2\n2\nb\na\na\na\n\u2212\n\uf8ee\n\uf8f9\n\uf8eb\n\uf8f6\n\u00d7 +\n\u2212\n\uf8ef\n\uf8fa\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8ed\n\uf8f8\n\uf8f0\n\uf8fb\n= \n2\n4\n2 2\nb a\nab\na\n\u03c0 =\u03c0\nFig 8 6\nFig 8 7\nAPPLICATION OF INTEGRALS         363\nAlternatively, considering horizontal strips as\nshown in the Fig 8 8, the area of the ellipse is\n= \n0\n4\u222b\nb xdy  = \n2\n2\n0\n4\n\u2212\n\u222b\nab\nb\ny dy\nb\n (Why"
    },
    {
        "Chapter": "1",
        "sentence_range": "4185-4188",
        "Text": "6\nFig 8 7\nAPPLICATION OF INTEGRALS         363\nAlternatively, considering horizontal strips as\nshown in the Fig 8 8, the area of the ellipse is\n= \n0\n4\u222b\nb xdy  = \n2\n2\n0\n4\n\u2212\n\u222b\nab\nb\ny dy\nb\n (Why )\n= \n2\n2\n2\n\u20131\n0\n4\nsin\n2\n2\nb\na\ny\nb\ny\nb\ny\nb\nb\n\uf8ee\n\uf8f9\n\u2212\n+\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n= \n2\n\u20131\n4\n0\nsin 1\n0\n2\n2\na\nb\nb\nb\n\uf8ee\n\uf8f9\n\uf8eb\n\uf8f6\n\u00d7 +\n\u2212\n\uf8ef\n\uf8fa\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8ed\n\uf8f8\n\uf8f0\n\uf8fb\n= \n2\n4\n2 2\na b\nab\nb\n\u03c0 =\u03c0\n8"
    },
    {
        "Chapter": "1",
        "sentence_range": "4186-4189",
        "Text": "7\nAPPLICATION OF INTEGRALS         363\nAlternatively, considering horizontal strips as\nshown in the Fig 8 8, the area of the ellipse is\n= \n0\n4\u222b\nb xdy  = \n2\n2\n0\n4\n\u2212\n\u222b\nab\nb\ny dy\nb\n (Why )\n= \n2\n2\n2\n\u20131\n0\n4\nsin\n2\n2\nb\na\ny\nb\ny\nb\ny\nb\nb\n\uf8ee\n\uf8f9\n\u2212\n+\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n= \n2\n\u20131\n4\n0\nsin 1\n0\n2\n2\na\nb\nb\nb\n\uf8ee\n\uf8f9\n\uf8eb\n\uf8f6\n\u00d7 +\n\u2212\n\uf8ef\n\uf8fa\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8ed\n\uf8f8\n\uf8f0\n\uf8fb\n= \n2\n4\n2 2\na b\nab\nb\n\u03c0 =\u03c0\n8 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "4187-4190",
        "Text": "8, the area of the ellipse is\n= \n0\n4\u222b\nb xdy  = \n2\n2\n0\n4\n\u2212\n\u222b\nab\nb\ny dy\nb\n (Why )\n= \n2\n2\n2\n\u20131\n0\n4\nsin\n2\n2\nb\na\ny\nb\ny\nb\ny\nb\nb\n\uf8ee\n\uf8f9\n\u2212\n+\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n= \n2\n\u20131\n4\n0\nsin 1\n0\n2\n2\na\nb\nb\nb\n\uf8ee\n\uf8f9\n\uf8eb\n\uf8f6\n\u00d7 +\n\u2212\n\uf8ef\n\uf8fa\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8ed\n\uf8f8\n\uf8f0\n\uf8fb\n= \n2\n4\n2 2\na b\nab\nb\n\u03c0 =\u03c0\n8 2 1  The area of the region bounded by a curve and a line\nIn this subsection, we will find the area of the region bounded by a line and a circle,\na line and a parabola, a line and an ellipse"
    },
    {
        "Chapter": "1",
        "sentence_range": "4188-4191",
        "Text": ")\n= \n2\n2\n2\n\u20131\n0\n4\nsin\n2\n2\nb\na\ny\nb\ny\nb\ny\nb\nb\n\uf8ee\n\uf8f9\n\u2212\n+\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n= \n2\n\u20131\n4\n0\nsin 1\n0\n2\n2\na\nb\nb\nb\n\uf8ee\n\uf8f9\n\uf8eb\n\uf8f6\n\u00d7 +\n\u2212\n\uf8ef\n\uf8fa\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8ed\n\uf8f8\n\uf8f0\n\uf8fb\n= \n2\n4\n2 2\na b\nab\nb\n\u03c0 =\u03c0\n8 2 1  The area of the region bounded by a curve and a line\nIn this subsection, we will find the area of the region bounded by a line and a circle,\na line and a parabola, a line and an ellipse Equations of above mentioned curves will be\nin their standard forms only as the cases in other forms go beyond the scope of this\ntextbook"
    },
    {
        "Chapter": "1",
        "sentence_range": "4189-4192",
        "Text": "2 1  The area of the region bounded by a curve and a line\nIn this subsection, we will find the area of the region bounded by a line and a circle,\na line and a parabola, a line and an ellipse Equations of above mentioned curves will be\nin their standard forms only as the cases in other forms go beyond the scope of this\ntextbook Example 3 Find the area of the region bounded\nby the curve y = x2 and the line y = 4"
    },
    {
        "Chapter": "1",
        "sentence_range": "4190-4193",
        "Text": "1  The area of the region bounded by a curve and a line\nIn this subsection, we will find the area of the region bounded by a line and a circle,\na line and a parabola, a line and an ellipse Equations of above mentioned curves will be\nin their standard forms only as the cases in other forms go beyond the scope of this\ntextbook Example 3 Find the area of the region bounded\nby the curve y = x2 and the line y = 4 Solution Since the given curve represented by\nthe equation y = x2 is a parabola symmetrical\nabout y-axis only, therefore, from Fig 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "4191-4194",
        "Text": "Equations of above mentioned curves will be\nin their standard forms only as the cases in other forms go beyond the scope of this\ntextbook Example 3 Find the area of the region bounded\nby the curve y = x2 and the line y = 4 Solution Since the given curve represented by\nthe equation y = x2 is a parabola symmetrical\nabout y-axis only, therefore, from Fig 8 9, the\nrequired area of the region AOBA is given by\n4\n20\n\u222bxdy\n =\nareaof theregionBONB boundedbycurve,\naxis\n2 andthelines\n0and\n= 4\ny\ny\ny\n\u2212\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n=\n\uf8ed\n\uf8f8\n= \n4\n20\nydy\n\u222b\n = \n4\n3\n2\n0\n2\n2\n3\ny\n\uf8ee\n\uf8f9\n\u00d7\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n \n4\n32\n8\n3\n3\n=\n\u00d7\n=\n(Why"
    },
    {
        "Chapter": "1",
        "sentence_range": "4192-4195",
        "Text": "Example 3 Find the area of the region bounded\nby the curve y = x2 and the line y = 4 Solution Since the given curve represented by\nthe equation y = x2 is a parabola symmetrical\nabout y-axis only, therefore, from Fig 8 9, the\nrequired area of the region AOBA is given by\n4\n20\n\u222bxdy\n =\nareaof theregionBONB boundedbycurve,\naxis\n2 andthelines\n0and\n= 4\ny\ny\ny\n\u2212\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n=\n\uf8ed\n\uf8f8\n= \n4\n20\nydy\n\u222b\n = \n4\n3\n2\n0\n2\n2\n3\ny\n\uf8ee\n\uf8f9\n\u00d7\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n \n4\n32\n8\n3\n3\n=\n\u00d7\n=\n(Why )\nHere, we have taken horizontal strips as indicated in the Fig 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "4193-4196",
        "Text": "Solution Since the given curve represented by\nthe equation y = x2 is a parabola symmetrical\nabout y-axis only, therefore, from Fig 8 9, the\nrequired area of the region AOBA is given by\n4\n20\n\u222bxdy\n =\nareaof theregionBONB boundedbycurve,\naxis\n2 andthelines\n0and\n= 4\ny\ny\ny\n\u2212\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n=\n\uf8ed\n\uf8f8\n= \n4\n20\nydy\n\u222b\n = \n4\n3\n2\n0\n2\n2\n3\ny\n\uf8ee\n\uf8f9\n\u00d7\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n \n4\n32\n8\n3\n3\n=\n\u00d7\n=\n(Why )\nHere, we have taken horizontal strips as indicated in the Fig 8 9"
    },
    {
        "Chapter": "1",
        "sentence_range": "4194-4197",
        "Text": "9, the\nrequired area of the region AOBA is given by\n4\n20\n\u222bxdy\n =\nareaof theregionBONB boundedbycurve,\naxis\n2 andthelines\n0and\n= 4\ny\ny\ny\n\u2212\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n=\n\uf8ed\n\uf8f8\n= \n4\n20\nydy\n\u222b\n = \n4\n3\n2\n0\n2\n2\n3\ny\n\uf8ee\n\uf8f9\n\u00d7\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n \n4\n32\n8\n3\n3\n=\n\u00d7\n=\n(Why )\nHere, we have taken horizontal strips as indicated in the Fig 8 9 Fig 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "4195-4198",
        "Text": ")\nHere, we have taken horizontal strips as indicated in the Fig 8 9 Fig 8 8\nFig 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "4196-4199",
        "Text": "9 Fig 8 8\nFig 8 9\n364\nMATHEMATICS\nAlternatively, we may consider the vertical\nstrips like PQ as shown in the Fig 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "4197-4200",
        "Text": "Fig 8 8\nFig 8 9\n364\nMATHEMATICS\nAlternatively, we may consider the vertical\nstrips like PQ as shown in the Fig 8 10 to\nobtain the area of the region AOBA"
    },
    {
        "Chapter": "1",
        "sentence_range": "4198-4201",
        "Text": "8\nFig 8 9\n364\nMATHEMATICS\nAlternatively, we may consider the vertical\nstrips like PQ as shown in the Fig 8 10 to\nobtain the area of the region AOBA To this\nend, we solve the equations x2 = y and y = 4\nwhich gives x = \u20132 and x = 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "4199-4202",
        "Text": "9\n364\nMATHEMATICS\nAlternatively, we may consider the vertical\nstrips like PQ as shown in the Fig 8 10 to\nobtain the area of the region AOBA To this\nend, we solve the equations x2 = y and y = 4\nwhich gives x = \u20132 and x = 2 Thus, the region AOBA may be stated as\nthe region bounded by the curve y = x 2, y = 4\nand the ordinates x = \u20132 and x = 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "4200-4203",
        "Text": "10 to\nobtain the area of the region AOBA To this\nend, we solve the equations x2 = y and y = 4\nwhich gives x = \u20132 and x = 2 Thus, the region AOBA may be stated as\nthe region bounded by the curve y = x 2, y = 4\nand the ordinates x = \u20132 and x = 2 Therefore, the area of the region AOBA\n= \n2\n2 ydx\n\u2212\u222b\n  [ y = (y-coordinate of Q) \u2013 (y-coordinate of P) = 4 \u2013 x 2]\n= \n(\n)\n2\n2\n20\n4\nx\ndx\n\u2212\n\u222b\n(Why"
    },
    {
        "Chapter": "1",
        "sentence_range": "4201-4204",
        "Text": "To this\nend, we solve the equations x2 = y and y = 4\nwhich gives x = \u20132 and x = 2 Thus, the region AOBA may be stated as\nthe region bounded by the curve y = x 2, y = 4\nand the ordinates x = \u20132 and x = 2 Therefore, the area of the region AOBA\n= \n2\n2 ydx\n\u2212\u222b\n  [ y = (y-coordinate of Q) \u2013 (y-coordinate of P) = 4 \u2013 x 2]\n= \n(\n)\n2\n2\n20\n4\nx\ndx\n\u2212\n\u222b\n(Why )\n=\n2\n3\n0\n2 4\nx3\n\uf8eex\n\uf8f9\n\u2212\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n \n8\n2 4 2\n3\n\uf8ee\n\uf8f9\n=\n\uf8ef\u00d7 \u2212\n\uf8fa\n\uf8f0\n\uf8fb\n32\n3\n=\nRemark From the above examples, it is inferred that we can consider either vertical\nstrips or horizontal strips for calculating the area of the region"
    },
    {
        "Chapter": "1",
        "sentence_range": "4202-4205",
        "Text": "Thus, the region AOBA may be stated as\nthe region bounded by the curve y = x 2, y = 4\nand the ordinates x = \u20132 and x = 2 Therefore, the area of the region AOBA\n= \n2\n2 ydx\n\u2212\u222b\n  [ y = (y-coordinate of Q) \u2013 (y-coordinate of P) = 4 \u2013 x 2]\n= \n(\n)\n2\n2\n20\n4\nx\ndx\n\u2212\n\u222b\n(Why )\n=\n2\n3\n0\n2 4\nx3\n\uf8eex\n\uf8f9\n\u2212\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n \n8\n2 4 2\n3\n\uf8ee\n\uf8f9\n=\n\uf8ef\u00d7 \u2212\n\uf8fa\n\uf8f0\n\uf8fb\n32\n3\n=\nRemark From the above examples, it is inferred that we can consider either vertical\nstrips or horizontal strips for calculating the area of the region Henceforth, we shall\nconsider either of these two, most preferably vertical strips"
    },
    {
        "Chapter": "1",
        "sentence_range": "4203-4206",
        "Text": "Therefore, the area of the region AOBA\n= \n2\n2 ydx\n\u2212\u222b\n  [ y = (y-coordinate of Q) \u2013 (y-coordinate of P) = 4 \u2013 x 2]\n= \n(\n)\n2\n2\n20\n4\nx\ndx\n\u2212\n\u222b\n(Why )\n=\n2\n3\n0\n2 4\nx3\n\uf8eex\n\uf8f9\n\u2212\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n \n8\n2 4 2\n3\n\uf8ee\n\uf8f9\n=\n\uf8ef\u00d7 \u2212\n\uf8fa\n\uf8f0\n\uf8fb\n32\n3\n=\nRemark From the above examples, it is inferred that we can consider either vertical\nstrips or horizontal strips for calculating the area of the region Henceforth, we shall\nconsider either of these two, most preferably vertical strips Example 4 Find the area of the region in the first quadrant enclosed by the x-axis,\nthe line y = x, and the circle x2 + y2 = 32"
    },
    {
        "Chapter": "1",
        "sentence_range": "4204-4207",
        "Text": ")\n=\n2\n3\n0\n2 4\nx3\n\uf8eex\n\uf8f9\n\u2212\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n \n8\n2 4 2\n3\n\uf8ee\n\uf8f9\n=\n\uf8ef\u00d7 \u2212\n\uf8fa\n\uf8f0\n\uf8fb\n32\n3\n=\nRemark From the above examples, it is inferred that we can consider either vertical\nstrips or horizontal strips for calculating the area of the region Henceforth, we shall\nconsider either of these two, most preferably vertical strips Example 4 Find the area of the region in the first quadrant enclosed by the x-axis,\nthe line y = x, and the circle x2 + y2 = 32 Solution The given equations are\ny = x"
    },
    {
        "Chapter": "1",
        "sentence_range": "4205-4208",
        "Text": "Henceforth, we shall\nconsider either of these two, most preferably vertical strips Example 4 Find the area of the region in the first quadrant enclosed by the x-axis,\nthe line y = x, and the circle x2 + y2 = 32 Solution The given equations are\ny = x (1)\nand\nx2 + y2 = 32"
    },
    {
        "Chapter": "1",
        "sentence_range": "4206-4209",
        "Text": "Example 4 Find the area of the region in the first quadrant enclosed by the x-axis,\nthe line y = x, and the circle x2 + y2 = 32 Solution The given equations are\ny = x (1)\nand\nx2 + y2 = 32 (2)\nSolving (1) and (2), we find that the line\nand the circle meet at B(4, 4) in the first\nquadrant (Fig 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "4207-4210",
        "Text": "Solution The given equations are\ny = x (1)\nand\nx2 + y2 = 32 (2)\nSolving (1) and (2), we find that the line\nand the circle meet at B(4, 4) in the first\nquadrant (Fig 8 11)"
    },
    {
        "Chapter": "1",
        "sentence_range": "4208-4211",
        "Text": "(1)\nand\nx2 + y2 = 32 (2)\nSolving (1) and (2), we find that the line\nand the circle meet at B(4, 4) in the first\nquadrant (Fig 8 11) Draw perpendicular\nBM to the x-axis"
    },
    {
        "Chapter": "1",
        "sentence_range": "4209-4212",
        "Text": "(2)\nSolving (1) and (2), we find that the line\nand the circle meet at B(4, 4) in the first\nquadrant (Fig 8 11) Draw perpendicular\nBM to the x-axis Therefore, the required area = area of\nthe region OBMO + area of the region\nBMAB"
    },
    {
        "Chapter": "1",
        "sentence_range": "4210-4213",
        "Text": "11) Draw perpendicular\nBM to the x-axis Therefore, the required area = area of\nthe region OBMO + area of the region\nBMAB Now, the area of the region OBMO\n= \n4\n4\n0\n0\nydx\nxdx\n=\n\u222b\n\u222b"
    },
    {
        "Chapter": "1",
        "sentence_range": "4211-4214",
        "Text": "Draw perpendicular\nBM to the x-axis Therefore, the required area = area of\nthe region OBMO + area of the region\nBMAB Now, the area of the region OBMO\n= \n4\n4\n0\n0\nydx\nxdx\n=\n\u222b\n\u222b (3)\n= \n4\n2\n0\n1\n2 x\n\uf8ee\n\uf8f9\n\uf8f0\n\uf8fb\n= 8\nFig 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "4212-4215",
        "Text": "Therefore, the required area = area of\nthe region OBMO + area of the region\nBMAB Now, the area of the region OBMO\n= \n4\n4\n0\n0\nydx\nxdx\n=\n\u222b\n\u222b (3)\n= \n4\n2\n0\n1\n2 x\n\uf8ee\n\uf8f9\n\uf8f0\n\uf8fb\n= 8\nFig 8 10\nFig 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "4213-4216",
        "Text": "Now, the area of the region OBMO\n= \n4\n4\n0\n0\nydx\nxdx\n=\n\u222b\n\u222b (3)\n= \n4\n2\n0\n1\n2 x\n\uf8ee\n\uf8f9\n\uf8f0\n\uf8fb\n= 8\nFig 8 10\nFig 8 11\nY\nO\nA\ny\nx\n=\nY'\nB\nM\n(4\ufffd4)\n,\nX\nX'\n(4\n2 0)\n,\nAPPLICATION OF INTEGRALS         365\nO\nF (\no)\nae,\nB\nY\nY\u2032\nB'\nS\nR\nX\nX\u2032\nx\nae\n=\nAgain, the area of the region BMAB\n= \n4 2\n4\nydx\n\u222b\n= \n4 2\n2\n4\n32\n\u2212x dx\n\u222b\n= \n4 2\n2\n\u20131\n4\n1\n1\n32\n32\nsin\n2\n2\n4 2\nx\nx\nx\n\uf8ee\n\uf8f9\n\u2212\n+\n\u00d7\n\u00d7\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n= \n\u20131\n\u20131\n1\n1\n4\n1\n1\n4 2\n0\n32\nsin 1\n32\n16\n32\nsin\n2\n2\n2\n2\n2\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n\u00d7\n+\n\u00d7\n\u00d7\n\u2212\n\u2212\n+\n\u00d7\n\u00d7\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n= 8 \u03c0 \u2013 (8 + 4\u03c0) = 4\u03c0 \u2013 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "4214-4217",
        "Text": "(3)\n= \n4\n2\n0\n1\n2 x\n\uf8ee\n\uf8f9\n\uf8f0\n\uf8fb\n= 8\nFig 8 10\nFig 8 11\nY\nO\nA\ny\nx\n=\nY'\nB\nM\n(4\ufffd4)\n,\nX\nX'\n(4\n2 0)\n,\nAPPLICATION OF INTEGRALS         365\nO\nF (\no)\nae,\nB\nY\nY\u2032\nB'\nS\nR\nX\nX\u2032\nx\nae\n=\nAgain, the area of the region BMAB\n= \n4 2\n4\nydx\n\u222b\n= \n4 2\n2\n4\n32\n\u2212x dx\n\u222b\n= \n4 2\n2\n\u20131\n4\n1\n1\n32\n32\nsin\n2\n2\n4 2\nx\nx\nx\n\uf8ee\n\uf8f9\n\u2212\n+\n\u00d7\n\u00d7\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n= \n\u20131\n\u20131\n1\n1\n4\n1\n1\n4 2\n0\n32\nsin 1\n32\n16\n32\nsin\n2\n2\n2\n2\n2\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n\u00d7\n+\n\u00d7\n\u00d7\n\u2212\n\u2212\n+\n\u00d7\n\u00d7\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n= 8 \u03c0 \u2013 (8 + 4\u03c0) = 4\u03c0 \u2013 8 (4)\nAdding (3) and (4), we get, the required area = 4\u03c0"
    },
    {
        "Chapter": "1",
        "sentence_range": "4215-4218",
        "Text": "10\nFig 8 11\nY\nO\nA\ny\nx\n=\nY'\nB\nM\n(4\ufffd4)\n,\nX\nX'\n(4\n2 0)\n,\nAPPLICATION OF INTEGRALS         365\nO\nF (\no)\nae,\nB\nY\nY\u2032\nB'\nS\nR\nX\nX\u2032\nx\nae\n=\nAgain, the area of the region BMAB\n= \n4 2\n4\nydx\n\u222b\n= \n4 2\n2\n4\n32\n\u2212x dx\n\u222b\n= \n4 2\n2\n\u20131\n4\n1\n1\n32\n32\nsin\n2\n2\n4 2\nx\nx\nx\n\uf8ee\n\uf8f9\n\u2212\n+\n\u00d7\n\u00d7\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n= \n\u20131\n\u20131\n1\n1\n4\n1\n1\n4 2\n0\n32\nsin 1\n32\n16\n32\nsin\n2\n2\n2\n2\n2\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n\u00d7\n+\n\u00d7\n\u00d7\n\u2212\n\u2212\n+\n\u00d7\n\u00d7\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n= 8 \u03c0 \u2013 (8 + 4\u03c0) = 4\u03c0 \u2013 8 (4)\nAdding (3) and (4), we get, the required area = 4\u03c0 Example 5 Find the area bounded by the ellipse \n2\n2\n2\n2\n1\nx\ny\na\nb\n+\n=  and the ordinates x = 0\nand x = ae, where, b2 = a2 (1 \u2013 e2) and e < 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "4216-4219",
        "Text": "11\nY\nO\nA\ny\nx\n=\nY'\nB\nM\n(4\ufffd4)\n,\nX\nX'\n(4\n2 0)\n,\nAPPLICATION OF INTEGRALS         365\nO\nF (\no)\nae,\nB\nY\nY\u2032\nB'\nS\nR\nX\nX\u2032\nx\nae\n=\nAgain, the area of the region BMAB\n= \n4 2\n4\nydx\n\u222b\n= \n4 2\n2\n4\n32\n\u2212x dx\n\u222b\n= \n4 2\n2\n\u20131\n4\n1\n1\n32\n32\nsin\n2\n2\n4 2\nx\nx\nx\n\uf8ee\n\uf8f9\n\u2212\n+\n\u00d7\n\u00d7\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n= \n\u20131\n\u20131\n1\n1\n4\n1\n1\n4 2\n0\n32\nsin 1\n32\n16\n32\nsin\n2\n2\n2\n2\n2\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n\u00d7\n+\n\u00d7\n\u00d7\n\u2212\n\u2212\n+\n\u00d7\n\u00d7\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n= 8 \u03c0 \u2013 (8 + 4\u03c0) = 4\u03c0 \u2013 8 (4)\nAdding (3) and (4), we get, the required area = 4\u03c0 Example 5 Find the area bounded by the ellipse \n2\n2\n2\n2\n1\nx\ny\na\nb\n+\n=  and the ordinates x = 0\nand x = ae, where, b2 = a2 (1 \u2013 e2) and e < 1 Solution The required area (Fig 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "4217-4220",
        "Text": "(4)\nAdding (3) and (4), we get, the required area = 4\u03c0 Example 5 Find the area bounded by the ellipse \n2\n2\n2\n2\n1\nx\ny\na\nb\n+\n=  and the ordinates x = 0\nand x = ae, where, b2 = a2 (1 \u2013 e2) and e < 1 Solution The required area (Fig 8 12) of the region BOB\u2032RFSB is enclosed by the\nellipse and the lines x = 0 and x = ae"
    },
    {
        "Chapter": "1",
        "sentence_range": "4218-4221",
        "Text": "Example 5 Find the area bounded by the ellipse \n2\n2\n2\n2\n1\nx\ny\na\nb\n+\n=  and the ordinates x = 0\nand x = ae, where, b2 = a2 (1 \u2013 e2) and e < 1 Solution The required area (Fig 8 12) of the region BOB\u2032RFSB is enclosed by the\nellipse and the lines x = 0 and x = ae Note that the area of the region BOB\u2032RFSB\n= \n20\n\u222bae ydx\n = \n2\n2\n0\n2\nae\nb\na\nx dx\na\n\u2212\n\u222b\n= \n2\n2\n2\n\u20131\n0\n2\nsin\n2\n2\nae\nb x\na\nx\na\nx\na\na\n\uf8ee\n\uf8f9\n\u2212\n+\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n= \n2\n2 2\n2\n\u20131\n2\nsin\n2\nb ae a\na e\na\ne\na\n\uf8ee\n\uf8f9\n\u2212\n+\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n= \n2\n\u20131\n1\nsin\nab e\ne\ne\n\uf8ee\n\uf8f9\n\u2212\n+\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nEXERCISE 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "4219-4222",
        "Text": "Solution The required area (Fig 8 12) of the region BOB\u2032RFSB is enclosed by the\nellipse and the lines x = 0 and x = ae Note that the area of the region BOB\u2032RFSB\n= \n20\n\u222bae ydx\n = \n2\n2\n0\n2\nae\nb\na\nx dx\na\n\u2212\n\u222b\n= \n2\n2\n2\n\u20131\n0\n2\nsin\n2\n2\nae\nb x\na\nx\na\nx\na\na\n\uf8ee\n\uf8f9\n\u2212\n+\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n= \n2\n2 2\n2\n\u20131\n2\nsin\n2\nb ae a\na e\na\ne\na\n\uf8ee\n\uf8f9\n\u2212\n+\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n= \n2\n\u20131\n1\nsin\nab e\ne\ne\n\uf8ee\n\uf8f9\n\u2212\n+\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nEXERCISE 8 1\n1"
    },
    {
        "Chapter": "1",
        "sentence_range": "4220-4223",
        "Text": "12) of the region BOB\u2032RFSB is enclosed by the\nellipse and the lines x = 0 and x = ae Note that the area of the region BOB\u2032RFSB\n= \n20\n\u222bae ydx\n = \n2\n2\n0\n2\nae\nb\na\nx dx\na\n\u2212\n\u222b\n= \n2\n2\n2\n\u20131\n0\n2\nsin\n2\n2\nae\nb x\na\nx\na\nx\na\na\n\uf8ee\n\uf8f9\n\u2212\n+\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n= \n2\n2 2\n2\n\u20131\n2\nsin\n2\nb ae a\na e\na\ne\na\n\uf8ee\n\uf8f9\n\u2212\n+\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n= \n2\n\u20131\n1\nsin\nab e\ne\ne\n\uf8ee\n\uf8f9\n\u2212\n+\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nEXERCISE 8 1\n1 Find the area of the region bounded by the curve y2 = x and the lines x = 1,\nx = 4 and the x-axis in the first quadrant"
    },
    {
        "Chapter": "1",
        "sentence_range": "4221-4224",
        "Text": "Note that the area of the region BOB\u2032RFSB\n= \n20\n\u222bae ydx\n = \n2\n2\n0\n2\nae\nb\na\nx dx\na\n\u2212\n\u222b\n= \n2\n2\n2\n\u20131\n0\n2\nsin\n2\n2\nae\nb x\na\nx\na\nx\na\na\n\uf8ee\n\uf8f9\n\u2212\n+\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n= \n2\n2 2\n2\n\u20131\n2\nsin\n2\nb ae a\na e\na\ne\na\n\uf8ee\n\uf8f9\n\u2212\n+\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n= \n2\n\u20131\n1\nsin\nab e\ne\ne\n\uf8ee\n\uf8f9\n\u2212\n+\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\nEXERCISE 8 1\n1 Find the area of the region bounded by the curve y2 = x and the lines x = 1,\nx = 4 and the x-axis in the first quadrant 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "4222-4225",
        "Text": "1\n1 Find the area of the region bounded by the curve y2 = x and the lines x = 1,\nx = 4 and the x-axis in the first quadrant 2 Find the area of the region bounded by y2 = 9x, x = 2, x = 4 and the x-axis in the\nfirst quadrant"
    },
    {
        "Chapter": "1",
        "sentence_range": "4223-4226",
        "Text": "Find the area of the region bounded by the curve y2 = x and the lines x = 1,\nx = 4 and the x-axis in the first quadrant 2 Find the area of the region bounded by y2 = 9x, x = 2, x = 4 and the x-axis in the\nfirst quadrant Fig 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "4224-4227",
        "Text": "2 Find the area of the region bounded by y2 = 9x, x = 2, x = 4 and the x-axis in the\nfirst quadrant Fig 8 12\n366\nMATHEMATICS\n3"
    },
    {
        "Chapter": "1",
        "sentence_range": "4225-4228",
        "Text": "Find the area of the region bounded by y2 = 9x, x = 2, x = 4 and the x-axis in the\nfirst quadrant Fig 8 12\n366\nMATHEMATICS\n3 Find the area of the region bounded by x2 = 4y, y = 2, y = 4 and the y-axis in the\nfirst quadrant"
    },
    {
        "Chapter": "1",
        "sentence_range": "4226-4229",
        "Text": "Fig 8 12\n366\nMATHEMATICS\n3 Find the area of the region bounded by x2 = 4y, y = 2, y = 4 and the y-axis in the\nfirst quadrant 4"
    },
    {
        "Chapter": "1",
        "sentence_range": "4227-4230",
        "Text": "12\n366\nMATHEMATICS\n3 Find the area of the region bounded by x2 = 4y, y = 2, y = 4 and the y-axis in the\nfirst quadrant 4 Find the area of the region bounded by the ellipse \n2\n2\n1\n16\n9\nx\n+y\n="
    },
    {
        "Chapter": "1",
        "sentence_range": "4228-4231",
        "Text": "Find the area of the region bounded by x2 = 4y, y = 2, y = 4 and the y-axis in the\nfirst quadrant 4 Find the area of the region bounded by the ellipse \n2\n2\n1\n16\n9\nx\n+y\n= 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "4229-4232",
        "Text": "4 Find the area of the region bounded by the ellipse \n2\n2\n1\n16\n9\nx\n+y\n= 5 Find the area of the region bounded by the ellipse \n2\n2\n1\n4\n9\nx\n+y\n="
    },
    {
        "Chapter": "1",
        "sentence_range": "4230-4233",
        "Text": "Find the area of the region bounded by the ellipse \n2\n2\n1\n16\n9\nx\n+y\n= 5 Find the area of the region bounded by the ellipse \n2\n2\n1\n4\n9\nx\n+y\n= 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "4231-4234",
        "Text": "5 Find the area of the region bounded by the ellipse \n2\n2\n1\n4\n9\nx\n+y\n= 6 Find the area of the region in the first quadrant enclosed by x-axis, line x = 3 y\nand the circle x2 + y2 = 4"
    },
    {
        "Chapter": "1",
        "sentence_range": "4232-4235",
        "Text": "Find the area of the region bounded by the ellipse \n2\n2\n1\n4\n9\nx\n+y\n= 6 Find the area of the region in the first quadrant enclosed by x-axis, line x = 3 y\nand the circle x2 + y2 = 4 7"
    },
    {
        "Chapter": "1",
        "sentence_range": "4233-4236",
        "Text": "6 Find the area of the region in the first quadrant enclosed by x-axis, line x = 3 y\nand the circle x2 + y2 = 4 7 Find the area of the smaller part of the circle x2 + y2 = a2 cut off by the line \n2\nx=a"
    },
    {
        "Chapter": "1",
        "sentence_range": "4234-4237",
        "Text": "Find the area of the region in the first quadrant enclosed by x-axis, line x = 3 y\nand the circle x2 + y2 = 4 7 Find the area of the smaller part of the circle x2 + y2 = a2 cut off by the line \n2\nx=a 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "4235-4238",
        "Text": "7 Find the area of the smaller part of the circle x2 + y2 = a2 cut off by the line \n2\nx=a 8 The area between x = y2 and x = 4 is divided into two equal parts by the line\nx = a, find the value of a"
    },
    {
        "Chapter": "1",
        "sentence_range": "4236-4239",
        "Text": "Find the area of the smaller part of the circle x2 + y2 = a2 cut off by the line \n2\nx=a 8 The area between x = y2 and x = 4 is divided into two equal parts by the line\nx = a, find the value of a 9"
    },
    {
        "Chapter": "1",
        "sentence_range": "4237-4240",
        "Text": "8 The area between x = y2 and x = 4 is divided into two equal parts by the line\nx = a, find the value of a 9 Find the area of the region bounded by the parabola y = x2 and y = x"
    },
    {
        "Chapter": "1",
        "sentence_range": "4238-4241",
        "Text": "The area between x = y2 and x = 4 is divided into two equal parts by the line\nx = a, find the value of a 9 Find the area of the region bounded by the parabola y = x2 and y = x 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "4239-4242",
        "Text": "9 Find the area of the region bounded by the parabola y = x2 and y = x 10 Find the area bounded by the curve x2 = 4y and the line x = 4y \u2013 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "4240-4243",
        "Text": "Find the area of the region bounded by the parabola y = x2 and y = x 10 Find the area bounded by the curve x2 = 4y and the line x = 4y \u2013 2 11"
    },
    {
        "Chapter": "1",
        "sentence_range": "4241-4244",
        "Text": "10 Find the area bounded by the curve x2 = 4y and the line x = 4y \u2013 2 11 Find the area of the region bounded by the curve y2 = 4x and the line x = 3"
    },
    {
        "Chapter": "1",
        "sentence_range": "4242-4245",
        "Text": "Find the area bounded by the curve x2 = 4y and the line x = 4y \u2013 2 11 Find the area of the region bounded by the curve y2 = 4x and the line x = 3 Choose the correct answer in the following Exercises 12 and 13"
    },
    {
        "Chapter": "1",
        "sentence_range": "4243-4246",
        "Text": "11 Find the area of the region bounded by the curve y2 = 4x and the line x = 3 Choose the correct answer in the following Exercises 12 and 13 12"
    },
    {
        "Chapter": "1",
        "sentence_range": "4244-4247",
        "Text": "Find the area of the region bounded by the curve y2 = 4x and the line x = 3 Choose the correct answer in the following Exercises 12 and 13 12 Area lying in the first quadrant and bounded by the circle x2 + y2 = 4 and the lines\nx = 0 and x = 2 is\n(A) \u03c0\n(B) 2\n\u03c0\n(C) 3\n\u03c0\n(D) 4\n\u03c0\n13"
    },
    {
        "Chapter": "1",
        "sentence_range": "4245-4248",
        "Text": "Choose the correct answer in the following Exercises 12 and 13 12 Area lying in the first quadrant and bounded by the circle x2 + y2 = 4 and the lines\nx = 0 and x = 2 is\n(A) \u03c0\n(B) 2\n\u03c0\n(C) 3\n\u03c0\n(D) 4\n\u03c0\n13 Area of the region bounded by the curve y2 = 4x, y-axis and the line y = 3 is\n(A) 2\n(B)\n49\n(C)\n39\n(D)\n9\n2\n8"
    },
    {
        "Chapter": "1",
        "sentence_range": "4246-4249",
        "Text": "12 Area lying in the first quadrant and bounded by the circle x2 + y2 = 4 and the lines\nx = 0 and x = 2 is\n(A) \u03c0\n(B) 2\n\u03c0\n(C) 3\n\u03c0\n(D) 4\n\u03c0\n13 Area of the region bounded by the curve y2 = 4x, y-axis and the line y = 3 is\n(A) 2\n(B)\n49\n(C)\n39\n(D)\n9\n2\n8 3  Area between Two Curves\nIntuitively, true in the sense of Leibnitz, integration is the act of calculating the area by\ncutting the region into a large number of small strips of elementary area and then\nadding up these elementary areas"
    },
    {
        "Chapter": "1",
        "sentence_range": "4247-4250",
        "Text": "Area lying in the first quadrant and bounded by the circle x2 + y2 = 4 and the lines\nx = 0 and x = 2 is\n(A) \u03c0\n(B) 2\n\u03c0\n(C) 3\n\u03c0\n(D) 4\n\u03c0\n13 Area of the region bounded by the curve y2 = 4x, y-axis and the line y = 3 is\n(A) 2\n(B)\n49\n(C)\n39\n(D)\n9\n2\n8 3  Area between Two Curves\nIntuitively, true in the sense of Leibnitz, integration is the act of calculating the area by\ncutting the region into a large number of small strips of elementary area and then\nadding up these elementary areas Suppose we are given two curves represented by\ny = f (x), y = g (x), where f (x) \u2265 g(x) in [a, b] as shown in Fig 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "4248-4251",
        "Text": "Area of the region bounded by the curve y2 = 4x, y-axis and the line y = 3 is\n(A) 2\n(B)\n49\n(C)\n39\n(D)\n9\n2\n8 3  Area between Two Curves\nIntuitively, true in the sense of Leibnitz, integration is the act of calculating the area by\ncutting the region into a large number of small strips of elementary area and then\nadding up these elementary areas Suppose we are given two curves represented by\ny = f (x), y = g (x), where f (x) \u2265 g(x) in [a, b] as shown in Fig 8 13"
    },
    {
        "Chapter": "1",
        "sentence_range": "4249-4252",
        "Text": "3  Area between Two Curves\nIntuitively, true in the sense of Leibnitz, integration is the act of calculating the area by\ncutting the region into a large number of small strips of elementary area and then\nadding up these elementary areas Suppose we are given two curves represented by\ny = f (x), y = g (x), where f (x) \u2265 g(x) in [a, b] as shown in Fig 8 13 Here the points of\nintersection of these two curves are given by x = a and x = b obtained by taking\ncommon values of y from the given equation of two curves"
    },
    {
        "Chapter": "1",
        "sentence_range": "4250-4253",
        "Text": "Suppose we are given two curves represented by\ny = f (x), y = g (x), where f (x) \u2265 g(x) in [a, b] as shown in Fig 8 13 Here the points of\nintersection of these two curves are given by x = a and x = b obtained by taking\ncommon values of y from the given equation of two curves For setting up a formula for the integral, it is convenient to take elementary area in\nthe form of vertical strips"
    },
    {
        "Chapter": "1",
        "sentence_range": "4251-4254",
        "Text": "13 Here the points of\nintersection of these two curves are given by x = a and x = b obtained by taking\ncommon values of y from the given equation of two curves For setting up a formula for the integral, it is convenient to take elementary area in\nthe form of vertical strips As indicated in the Fig 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "4252-4255",
        "Text": "Here the points of\nintersection of these two curves are given by x = a and x = b obtained by taking\ncommon values of y from the given equation of two curves For setting up a formula for the integral, it is convenient to take elementary area in\nthe form of vertical strips As indicated in the Fig 8 13, elementary strip has height\nAPPLICATION OF INTEGRALS         367\ny\n=f x\n(\ufffd\ufffd)\nX\nY\ny\ng x\n=\ufffd\ufffd\ufffd\ufffd(\ufffd\ufffd)\nx\n=a\nx\nc\n=\ny\ng x\n=\ufffd\ufffd\ufffd\ufffd(\ufffd\ufffd)\ny\n=f x\n(\ufffd\ufffd)\nx\nb\n=\nA\nB\nR\nC\nD\nQ\nO\nP\nX\u2032\nY\u2032\nf(x) \u2013 g(x) and width dx so that the elementary area\nFig 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "4253-4256",
        "Text": "For setting up a formula for the integral, it is convenient to take elementary area in\nthe form of vertical strips As indicated in the Fig 8 13, elementary strip has height\nAPPLICATION OF INTEGRALS         367\ny\n=f x\n(\ufffd\ufffd)\nX\nY\ny\ng x\n=\ufffd\ufffd\ufffd\ufffd(\ufffd\ufffd)\nx\n=a\nx\nc\n=\ny\ng x\n=\ufffd\ufffd\ufffd\ufffd(\ufffd\ufffd)\ny\n=f x\n(\ufffd\ufffd)\nx\nb\n=\nA\nB\nR\nC\nD\nQ\nO\nP\nX\u2032\nY\u2032\nf(x) \u2013 g(x) and width dx so that the elementary area\nFig 8 13\nFig 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "4254-4257",
        "Text": "As indicated in the Fig 8 13, elementary strip has height\nAPPLICATION OF INTEGRALS         367\ny\n=f x\n(\ufffd\ufffd)\nX\nY\ny\ng x\n=\ufffd\ufffd\ufffd\ufffd(\ufffd\ufffd)\nx\n=a\nx\nc\n=\ny\ng x\n=\ufffd\ufffd\ufffd\ufffd(\ufffd\ufffd)\ny\n=f x\n(\ufffd\ufffd)\nx\nb\n=\nA\nB\nR\nC\nD\nQ\nO\nP\nX\u2032\nY\u2032\nf(x) \u2013 g(x) and width dx so that the elementary area\nFig 8 13\nFig 8 14\ndA = [f (x) \u2013 g(x)] dx, and the total area A can be taken as\nA = \n[ ( )\n( )]\nb\na f x\n\u2212g x dx\n\u222b\nAlternatively,\nA = [area bounded by y = f (x), x-axis and the lines x = a, x = b]\n\u2013 [area bounded by y = g (x), x-axis and the lines x = a, x = b]\n=\n( )\n( )\nb\nb\na\na\nf x dx\ng x dx\n\u2212\n\u222b\n\u222b\n =\n[\n]\n( )\n( )\n,\nb\na f x\ng x\ndx\n\u2212\n\u222b\nwhere f (x) \u2265 g (x) in [a, b]\nIf f (x) \u2265 g (x) in [a, c] and f (x) \u2264 g (x) in [c, b], where a < c < b as shown in the\nFig 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "4255-4258",
        "Text": "13, elementary strip has height\nAPPLICATION OF INTEGRALS         367\ny\n=f x\n(\ufffd\ufffd)\nX\nY\ny\ng x\n=\ufffd\ufffd\ufffd\ufffd(\ufffd\ufffd)\nx\n=a\nx\nc\n=\ny\ng x\n=\ufffd\ufffd\ufffd\ufffd(\ufffd\ufffd)\ny\n=f x\n(\ufffd\ufffd)\nx\nb\n=\nA\nB\nR\nC\nD\nQ\nO\nP\nX\u2032\nY\u2032\nf(x) \u2013 g(x) and width dx so that the elementary area\nFig 8 13\nFig 8 14\ndA = [f (x) \u2013 g(x)] dx, and the total area A can be taken as\nA = \n[ ( )\n( )]\nb\na f x\n\u2212g x dx\n\u222b\nAlternatively,\nA = [area bounded by y = f (x), x-axis and the lines x = a, x = b]\n\u2013 [area bounded by y = g (x), x-axis and the lines x = a, x = b]\n=\n( )\n( )\nb\nb\na\na\nf x dx\ng x dx\n\u2212\n\u222b\n\u222b\n =\n[\n]\n( )\n( )\n,\nb\na f x\ng x\ndx\n\u2212\n\u222b\nwhere f (x) \u2265 g (x) in [a, b]\nIf f (x) \u2265 g (x) in [a, c] and f (x) \u2264 g (x) in [c, b], where a < c < b as shown in the\nFig 8 14, then the area of the regions bounded by curves can be written as\nTotal Area = Area of the region ACBDA + Area of the region BPRQB\n =\n[\n]\n[\n]\n( )\n( )\n( )\n( )\nc\nb\na\nc\nf x\ng x\ndx\ng x\nf x dx\n\u2212\n+\n\u2212\n\u222b\n\u222b\n368\nMATHEMATICS\nY\nO\nP\ufffd(4,\ufffd4)\nC\ufffd(4,\ufffd0)\nY\ufffd\nX\ufffd\nX\nQ\ufffd(8,\ufffd0)\nFig 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "4256-4259",
        "Text": "13\nFig 8 14\ndA = [f (x) \u2013 g(x)] dx, and the total area A can be taken as\nA = \n[ ( )\n( )]\nb\na f x\n\u2212g x dx\n\u222b\nAlternatively,\nA = [area bounded by y = f (x), x-axis and the lines x = a, x = b]\n\u2013 [area bounded by y = g (x), x-axis and the lines x = a, x = b]\n=\n( )\n( )\nb\nb\na\na\nf x dx\ng x dx\n\u2212\n\u222b\n\u222b\n =\n[\n]\n( )\n( )\n,\nb\na f x\ng x\ndx\n\u2212\n\u222b\nwhere f (x) \u2265 g (x) in [a, b]\nIf f (x) \u2265 g (x) in [a, c] and f (x) \u2264 g (x) in [c, b], where a < c < b as shown in the\nFig 8 14, then the area of the regions bounded by curves can be written as\nTotal Area = Area of the region ACBDA + Area of the region BPRQB\n =\n[\n]\n[\n]\n( )\n( )\n( )\n( )\nc\nb\na\nc\nf x\ng x\ndx\ng x\nf x dx\n\u2212\n+\n\u2212\n\u222b\n\u222b\n368\nMATHEMATICS\nY\nO\nP\ufffd(4,\ufffd4)\nC\ufffd(4,\ufffd0)\nY\ufffd\nX\ufffd\nX\nQ\ufffd(8,\ufffd0)\nFig 8 16\nExample 6 Find the area of the region bounded by the two parabolas y = x2 and y2 = x"
    },
    {
        "Chapter": "1",
        "sentence_range": "4257-4260",
        "Text": "14\ndA = [f (x) \u2013 g(x)] dx, and the total area A can be taken as\nA = \n[ ( )\n( )]\nb\na f x\n\u2212g x dx\n\u222b\nAlternatively,\nA = [area bounded by y = f (x), x-axis and the lines x = a, x = b]\n\u2013 [area bounded by y = g (x), x-axis and the lines x = a, x = b]\n=\n( )\n( )\nb\nb\na\na\nf x dx\ng x dx\n\u2212\n\u222b\n\u222b\n =\n[\n]\n( )\n( )\n,\nb\na f x\ng x\ndx\n\u2212\n\u222b\nwhere f (x) \u2265 g (x) in [a, b]\nIf f (x) \u2265 g (x) in [a, c] and f (x) \u2264 g (x) in [c, b], where a < c < b as shown in the\nFig 8 14, then the area of the regions bounded by curves can be written as\nTotal Area = Area of the region ACBDA + Area of the region BPRQB\n =\n[\n]\n[\n]\n( )\n( )\n( )\n( )\nc\nb\na\nc\nf x\ng x\ndx\ng x\nf x dx\n\u2212\n+\n\u2212\n\u222b\n\u222b\n368\nMATHEMATICS\nY\nO\nP\ufffd(4,\ufffd4)\nC\ufffd(4,\ufffd0)\nY\ufffd\nX\ufffd\nX\nQ\ufffd(8,\ufffd0)\nFig 8 16\nExample 6 Find the area of the region bounded by the two parabolas y = x2 and y2 = x Solution The point of intersection of these two\nparabolas are O (0, 0) and A (1, 1) as shown in\nthe Fig 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "4258-4261",
        "Text": "14, then the area of the regions bounded by curves can be written as\nTotal Area = Area of the region ACBDA + Area of the region BPRQB\n =\n[\n]\n[\n]\n( )\n( )\n( )\n( )\nc\nb\na\nc\nf x\ng x\ndx\ng x\nf x dx\n\u2212\n+\n\u2212\n\u222b\n\u222b\n368\nMATHEMATICS\nY\nO\nP\ufffd(4,\ufffd4)\nC\ufffd(4,\ufffd0)\nY\ufffd\nX\ufffd\nX\nQ\ufffd(8,\ufffd0)\nFig 8 16\nExample 6 Find the area of the region bounded by the two parabolas y = x2 and y2 = x Solution The point of intersection of these two\nparabolas are O (0, 0) and A (1, 1) as shown in\nthe Fig 8 15"
    },
    {
        "Chapter": "1",
        "sentence_range": "4259-4262",
        "Text": "16\nExample 6 Find the area of the region bounded by the two parabolas y = x2 and y2 = x Solution The point of intersection of these two\nparabolas are O (0, 0) and A (1, 1) as shown in\nthe Fig 8 15 Here, we can set y 2 = x or y =\nx = f(x) and y = x2\n= g(x), where, f (x) \u2265 g (x) in [0, 1]"
    },
    {
        "Chapter": "1",
        "sentence_range": "4260-4263",
        "Text": "Solution The point of intersection of these two\nparabolas are O (0, 0) and A (1, 1) as shown in\nthe Fig 8 15 Here, we can set y 2 = x or y =\nx = f(x) and y = x2\n= g(x), where, f (x) \u2265 g (x) in [0, 1] Therefore, the required area of the shaded region\n= \n[\n]\n1\n0\n( )\n( )\nf x\ng x\ndx\n\u2212\n\u222b\n= \n1\n2\n0\nx\nx\ndx\n\uf8ee\n\uf8f9\n\u2212\n\uf8f0\n\uf8fb\n\u222b\n1\n3\n3\n2\n0\n32\nx3\nx\n\uf8ee\n\uf8f9\n=\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n= \n2\n1\n1\n3\n3\n3\n\u2212\n=\nExample 7 Find the area lying above x-axis and included between the circle\nx2 + y2 = 8x and inside of the parabola y2 = 4x"
    },
    {
        "Chapter": "1",
        "sentence_range": "4261-4264",
        "Text": "15 Here, we can set y 2 = x or y =\nx = f(x) and y = x2\n= g(x), where, f (x) \u2265 g (x) in [0, 1] Therefore, the required area of the shaded region\n= \n[\n]\n1\n0\n( )\n( )\nf x\ng x\ndx\n\u2212\n\u222b\n= \n1\n2\n0\nx\nx\ndx\n\uf8ee\n\uf8f9\n\u2212\n\uf8f0\n\uf8fb\n\u222b\n1\n3\n3\n2\n0\n32\nx3\nx\n\uf8ee\n\uf8f9\n=\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n= \n2\n1\n1\n3\n3\n3\n\u2212\n=\nExample 7 Find the area lying above x-axis and included between the circle\nx2 + y2 = 8x and inside of the parabola y2 = 4x Solution The given equation of the circle x2 + y 2 = 8x can be expressed as\n(x \u2013 4)2 + y2 = 16"
    },
    {
        "Chapter": "1",
        "sentence_range": "4262-4265",
        "Text": "Here, we can set y 2 = x or y =\nx = f(x) and y = x2\n= g(x), where, f (x) \u2265 g (x) in [0, 1] Therefore, the required area of the shaded region\n= \n[\n]\n1\n0\n( )\n( )\nf x\ng x\ndx\n\u2212\n\u222b\n= \n1\n2\n0\nx\nx\ndx\n\uf8ee\n\uf8f9\n\u2212\n\uf8f0\n\uf8fb\n\u222b\n1\n3\n3\n2\n0\n32\nx3\nx\n\uf8ee\n\uf8f9\n=\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n= \n2\n1\n1\n3\n3\n3\n\u2212\n=\nExample 7 Find the area lying above x-axis and included between the circle\nx2 + y2 = 8x and inside of the parabola y2 = 4x Solution The given equation of the circle x2 + y 2 = 8x can be expressed as\n(x \u2013 4)2 + y2 = 16 Thus, the centre of the\ncircle is (4, 0) and radius is 4"
    },
    {
        "Chapter": "1",
        "sentence_range": "4263-4266",
        "Text": "Therefore, the required area of the shaded region\n= \n[\n]\n1\n0\n( )\n( )\nf x\ng x\ndx\n\u2212\n\u222b\n= \n1\n2\n0\nx\nx\ndx\n\uf8ee\n\uf8f9\n\u2212\n\uf8f0\n\uf8fb\n\u222b\n1\n3\n3\n2\n0\n32\nx3\nx\n\uf8ee\n\uf8f9\n=\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n= \n2\n1\n1\n3\n3\n3\n\u2212\n=\nExample 7 Find the area lying above x-axis and included between the circle\nx2 + y2 = 8x and inside of the parabola y2 = 4x Solution The given equation of the circle x2 + y 2 = 8x can be expressed as\n(x \u2013 4)2 + y2 = 16 Thus, the centre of the\ncircle is (4, 0) and radius is 4 Its intersection\nwith the parabola y2 = 4x gives\nx2 + 4x = 8x\nor\nx2 \u2013 4x = 0\nor\nx (x \u2013 4) = 0\nor\nx = 0, x = 4\nThus, the points of intersection of these\ntwo curves are O(0,0) and P(4,4) above the\nx-axis"
    },
    {
        "Chapter": "1",
        "sentence_range": "4264-4267",
        "Text": "Solution The given equation of the circle x2 + y 2 = 8x can be expressed as\n(x \u2013 4)2 + y2 = 16 Thus, the centre of the\ncircle is (4, 0) and radius is 4 Its intersection\nwith the parabola y2 = 4x gives\nx2 + 4x = 8x\nor\nx2 \u2013 4x = 0\nor\nx (x \u2013 4) = 0\nor\nx = 0, x = 4\nThus, the points of intersection of these\ntwo curves are O(0,0) and P(4,4) above the\nx-axis From the Fig 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "4265-4268",
        "Text": "Thus, the centre of the\ncircle is (4, 0) and radius is 4 Its intersection\nwith the parabola y2 = 4x gives\nx2 + 4x = 8x\nor\nx2 \u2013 4x = 0\nor\nx (x \u2013 4) = 0\nor\nx = 0, x = 4\nThus, the points of intersection of these\ntwo curves are O(0,0) and P(4,4) above the\nx-axis From the Fig 8 16, the required area of\nthe region OPQCO included between these\ntwo curves above x-axis is\n= (area of the region OCPO) + (area of the region PCQP)\n= \n4\n8\n0\n4\nydx\nydx\n+\n\u222b\n\u222b\n= \n4\n8\n2\n2\n0\n4\n2\n4\n(\n4)\nx dx\nx\ndx\n+\n\u2212\n\u2212\n\u222b\n\u222b\n   (Why"
    },
    {
        "Chapter": "1",
        "sentence_range": "4266-4269",
        "Text": "Its intersection\nwith the parabola y2 = 4x gives\nx2 + 4x = 8x\nor\nx2 \u2013 4x = 0\nor\nx (x \u2013 4) = 0\nor\nx = 0, x = 4\nThus, the points of intersection of these\ntwo curves are O(0,0) and P(4,4) above the\nx-axis From the Fig 8 16, the required area of\nthe region OPQCO included between these\ntwo curves above x-axis is\n= (area of the region OCPO) + (area of the region PCQP)\n= \n4\n8\n0\n4\nydx\nydx\n+\n\u222b\n\u222b\n= \n4\n8\n2\n2\n0\n4\n2\n4\n(\n4)\nx dx\nx\ndx\n+\n\u2212\n\u2212\n\u222b\n\u222b\n   (Why )\nFig 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "4267-4270",
        "Text": "From the Fig 8 16, the required area of\nthe region OPQCO included between these\ntwo curves above x-axis is\n= (area of the region OCPO) + (area of the region PCQP)\n= \n4\n8\n0\n4\nydx\nydx\n+\n\u222b\n\u222b\n= \n4\n8\n2\n2\n0\n4\n2\n4\n(\n4)\nx dx\nx\ndx\n+\n\u2212\n\u2212\n\u222b\n\u222b\n   (Why )\nFig 8 15\nAPPLICATION OF INTEGRALS         369\n= \n4\n3\n4\n2\n2\n2\n0\n0\n2\n2\n4\n, where,\n4\n3\nx\nt dt\nx\nt\n\uf8ee\n\uf8f9\n\u00d7\n+\n\u2212\n\u2212\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\u222b\n(Why"
    },
    {
        "Chapter": "1",
        "sentence_range": "4268-4271",
        "Text": "16, the required area of\nthe region OPQCO included between these\ntwo curves above x-axis is\n= (area of the region OCPO) + (area of the region PCQP)\n= \n4\n8\n0\n4\nydx\nydx\n+\n\u222b\n\u222b\n= \n4\n8\n2\n2\n0\n4\n2\n4\n(\n4)\nx dx\nx\ndx\n+\n\u2212\n\u2212\n\u222b\n\u222b\n   (Why )\nFig 8 15\nAPPLICATION OF INTEGRALS         369\n= \n4\n3\n4\n2\n2\n2\n0\n0\n2\n2\n4\n, where,\n4\n3\nx\nt dt\nx\nt\n\uf8ee\n\uf8f9\n\u00d7\n+\n\u2212\n\u2212\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\u222b\n(Why )\n= \n4\n2\n2\n2\n\u20131\n0\n32\n1\n4\n4\nsin\n3\n2\n2\n4\nt\nt\nt\n\uf8ee\n\uf8f9\n+\n\u2212\n+\n\u00d7\n\u00d7\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n= \n2\n\u20131\n32\n4\n1\n0\n4\nsin 1\n3\n2\n2\n\uf8ee\n\uf8f9\n+\n\u00d7\n+\n\u00d7\n\u00d7\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n32\n32\n0\n8\n4\n3\n2\n3\n\u03c0\n\uf8ee\n\uf8f9\n=\n+\n+\n\u00d7\n=\n+ \u03c0\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n = 4(8 3 )\n3\n+ \u03c0\nExample 8 In Fig 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "4269-4272",
        "Text": ")\nFig 8 15\nAPPLICATION OF INTEGRALS         369\n= \n4\n3\n4\n2\n2\n2\n0\n0\n2\n2\n4\n, where,\n4\n3\nx\nt dt\nx\nt\n\uf8ee\n\uf8f9\n\u00d7\n+\n\u2212\n\u2212\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\u222b\n(Why )\n= \n4\n2\n2\n2\n\u20131\n0\n32\n1\n4\n4\nsin\n3\n2\n2\n4\nt\nt\nt\n\uf8ee\n\uf8f9\n+\n\u2212\n+\n\u00d7\n\u00d7\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n= \n2\n\u20131\n32\n4\n1\n0\n4\nsin 1\n3\n2\n2\n\uf8ee\n\uf8f9\n+\n\u00d7\n+\n\u00d7\n\u00d7\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n32\n32\n0\n8\n4\n3\n2\n3\n\u03c0\n\uf8ee\n\uf8f9\n=\n+\n+\n\u00d7\n=\n+ \u03c0\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n = 4(8 3 )\n3\n+ \u03c0\nExample 8 In Fig 8 17, AOBA is the part of the ellipse 9x2 + y2 = 36 in the first\nquadrant such that OA = 2 and OB = 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "4270-4273",
        "Text": "15\nAPPLICATION OF INTEGRALS         369\n= \n4\n3\n4\n2\n2\n2\n0\n0\n2\n2\n4\n, where,\n4\n3\nx\nt dt\nx\nt\n\uf8ee\n\uf8f9\n\u00d7\n+\n\u2212\n\u2212\n=\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\u222b\n(Why )\n= \n4\n2\n2\n2\n\u20131\n0\n32\n1\n4\n4\nsin\n3\n2\n2\n4\nt\nt\nt\n\uf8ee\n\uf8f9\n+\n\u2212\n+\n\u00d7\n\u00d7\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n= \n2\n\u20131\n32\n4\n1\n0\n4\nsin 1\n3\n2\n2\n\uf8ee\n\uf8f9\n+\n\u00d7\n+\n\u00d7\n\u00d7\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n32\n32\n0\n8\n4\n3\n2\n3\n\u03c0\n\uf8ee\n\uf8f9\n=\n+\n+\n\u00d7\n=\n+ \u03c0\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n = 4(8 3 )\n3\n+ \u03c0\nExample 8 In Fig 8 17, AOBA is the part of the ellipse 9x2 + y2 = 36 in the first\nquadrant such that OA = 2 and OB = 6 Find the area between the arc AB and the\nchord AB"
    },
    {
        "Chapter": "1",
        "sentence_range": "4271-4274",
        "Text": ")\n= \n4\n2\n2\n2\n\u20131\n0\n32\n1\n4\n4\nsin\n3\n2\n2\n4\nt\nt\nt\n\uf8ee\n\uf8f9\n+\n\u2212\n+\n\u00d7\n\u00d7\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n= \n2\n\u20131\n32\n4\n1\n0\n4\nsin 1\n3\n2\n2\n\uf8ee\n\uf8f9\n+\n\u00d7\n+\n\u00d7\n\u00d7\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n32\n32\n0\n8\n4\n3\n2\n3\n\u03c0\n\uf8ee\n\uf8f9\n=\n+\n+\n\u00d7\n=\n+ \u03c0\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n = 4(8 3 )\n3\n+ \u03c0\nExample 8 In Fig 8 17, AOBA is the part of the ellipse 9x2 + y2 = 36 in the first\nquadrant such that OA = 2 and OB = 6 Find the area between the arc AB and the\nchord AB Solution Given equation of the ellipse 9x2 + y2 = 36 can be expressed as \n2\n2\n1\n4\n36\nx\n+y\n=  or\n2\n2\n2\n2\n1\n2\n6\nx\n+y\n= and hence, its shape is as given in Fig 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "4272-4275",
        "Text": "17, AOBA is the part of the ellipse 9x2 + y2 = 36 in the first\nquadrant such that OA = 2 and OB = 6 Find the area between the arc AB and the\nchord AB Solution Given equation of the ellipse 9x2 + y2 = 36 can be expressed as \n2\n2\n1\n4\n36\nx\n+y\n=  or\n2\n2\n2\n2\n1\n2\n6\nx\n+y\n= and hence, its shape is as given in Fig 8 17"
    },
    {
        "Chapter": "1",
        "sentence_range": "4273-4276",
        "Text": "Find the area between the arc AB and the\nchord AB Solution Given equation of the ellipse 9x2 + y2 = 36 can be expressed as \n2\n2\n1\n4\n36\nx\n+y\n=  or\n2\n2\n2\n2\n1\n2\n6\nx\n+y\n= and hence, its shape is as given in Fig 8 17 Accordingly, the equation of the chord AB is\ny \u2013 0 = 6\n0 (\n2)\n0\n\u22122 x\n\u2212\n\u2212\nor\ny = \u2013 3(x \u2013 2)\nor\ny = \u2013 3x + 6\nArea of the shaded region as shown in the Fig 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "4274-4277",
        "Text": "Solution Given equation of the ellipse 9x2 + y2 = 36 can be expressed as \n2\n2\n1\n4\n36\nx\n+y\n=  or\n2\n2\n2\n2\n1\n2\n6\nx\n+y\n= and hence, its shape is as given in Fig 8 17 Accordingly, the equation of the chord AB is\ny \u2013 0 = 6\n0 (\n2)\n0\n\u22122 x\n\u2212\n\u2212\nor\ny = \u2013 3(x \u2013 2)\nor\ny = \u2013 3x + 6\nArea of the shaded region as shown in the Fig 8 17"
    },
    {
        "Chapter": "1",
        "sentence_range": "4275-4278",
        "Text": "17 Accordingly, the equation of the chord AB is\ny \u2013 0 = 6\n0 (\n2)\n0\n\u22122 x\n\u2212\n\u2212\nor\ny = \u2013 3(x \u2013 2)\nor\ny = \u2013 3x + 6\nArea of the shaded region as shown in the Fig 8 17 = \n2\n2\n2\n0\n0\n3\n4\n(6\n3 )\nx dx\nx dx\n\u2212\n\u2212\n\u2212\n\u222b\n\u222b\n(Why"
    },
    {
        "Chapter": "1",
        "sentence_range": "4276-4279",
        "Text": "Accordingly, the equation of the chord AB is\ny \u2013 0 = 6\n0 (\n2)\n0\n\u22122 x\n\u2212\n\u2212\nor\ny = \u2013 3(x \u2013 2)\nor\ny = \u2013 3x + 6\nArea of the shaded region as shown in the Fig 8 17 = \n2\n2\n2\n0\n0\n3\n4\n(6\n3 )\nx dx\nx dx\n\u2212\n\u2212\n\u2212\n\u222b\n\u222b\n(Why )\n= \n2\n2\n2\n2\n\u20131\n0\n0\n4\n3\n3\n4\nsin\n6\n2\n2\n2\n2\nx\nx\nx\nx\n\uf8eex\n\uf8f9\n\uf8ee\n\uf8f9\n\u2212\n+\n\u2212\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n= \n\u20131\n2\n12\n3\n0\n2sin\n(1)\n12\n2\n2\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\u00d7\n+\n\u2212\n\u2212\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n\u03c0\n3\n2\n6\n2\n=\n\u00d7\n\u00d7\n\u2212\n = 3\u03c0 \u2013 6\nFig 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "4277-4280",
        "Text": "17 = \n2\n2\n2\n0\n0\n3\n4\n(6\n3 )\nx dx\nx dx\n\u2212\n\u2212\n\u2212\n\u222b\n\u222b\n(Why )\n= \n2\n2\n2\n2\n\u20131\n0\n0\n4\n3\n3\n4\nsin\n6\n2\n2\n2\n2\nx\nx\nx\nx\n\uf8eex\n\uf8f9\n\uf8ee\n\uf8f9\n\u2212\n+\n\u2212\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n= \n\u20131\n2\n12\n3\n0\n2sin\n(1)\n12\n2\n2\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\u00d7\n+\n\u2212\n\u2212\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n\u03c0\n3\n2\n6\n2\n=\n\u00d7\n\u00d7\n\u2212\n = 3\u03c0 \u2013 6\nFig 8 17\n370\nMATHEMATICS\nExample 9 Using integration find the area of region bounded by the triangle whose\nvertices are (1, 0), (2, 2) and (3, 1)"
    },
    {
        "Chapter": "1",
        "sentence_range": "4278-4281",
        "Text": "= \n2\n2\n2\n0\n0\n3\n4\n(6\n3 )\nx dx\nx dx\n\u2212\n\u2212\n\u2212\n\u222b\n\u222b\n(Why )\n= \n2\n2\n2\n2\n\u20131\n0\n0\n4\n3\n3\n4\nsin\n6\n2\n2\n2\n2\nx\nx\nx\nx\n\uf8eex\n\uf8f9\n\uf8ee\n\uf8f9\n\u2212\n+\n\u2212\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n= \n\u20131\n2\n12\n3\n0\n2sin\n(1)\n12\n2\n2\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\u00d7\n+\n\u2212\n\u2212\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n\u03c0\n3\n2\n6\n2\n=\n\u00d7\n\u00d7\n\u2212\n = 3\u03c0 \u2013 6\nFig 8 17\n370\nMATHEMATICS\nExample 9 Using integration find the area of region bounded by the triangle whose\nvertices are (1, 0), (2, 2) and (3, 1) Solution Let A(1, 0), B(2, 2) and C (3, 1) be\nthe vertices of a triangle ABC (Fig 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "4279-4282",
        "Text": ")\n= \n2\n2\n2\n2\n\u20131\n0\n0\n4\n3\n3\n4\nsin\n6\n2\n2\n2\n2\nx\nx\nx\nx\n\uf8eex\n\uf8f9\n\uf8ee\n\uf8f9\n\u2212\n+\n\u2212\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n= \n\u20131\n2\n12\n3\n0\n2sin\n(1)\n12\n2\n2\n\uf8ee\n\uf8f9 \uf8ee\n\uf8f9\n\u00d7\n+\n\u2212\n\u2212\n\uf8ef\n\uf8fa \uf8ef\n\uf8fa\n\uf8f0\n\uf8fb \uf8f0\n\uf8fb\n\u03c0\n3\n2\n6\n2\n=\n\u00d7\n\u00d7\n\u2212\n = 3\u03c0 \u2013 6\nFig 8 17\n370\nMATHEMATICS\nExample 9 Using integration find the area of region bounded by the triangle whose\nvertices are (1, 0), (2, 2) and (3, 1) Solution Let A(1, 0), B(2, 2) and C (3, 1) be\nthe vertices of a triangle ABC (Fig 8 18)"
    },
    {
        "Chapter": "1",
        "sentence_range": "4280-4283",
        "Text": "17\n370\nMATHEMATICS\nExample 9 Using integration find the area of region bounded by the triangle whose\nvertices are (1, 0), (2, 2) and (3, 1) Solution Let A(1, 0), B(2, 2) and C (3, 1) be\nthe vertices of a triangle ABC (Fig 8 18) Area of \u2206ABC\n= Area of \u2206ABD  + Area of trapezium\n    BDEC \u2013 Area of \u2206AEC\nNow equation of the sides AB, BC and\nCA are given by\ny = 2(x \u2013 1), y = 4 \u2013 x, y = 1\n2  (x \u2013 1), respectively"
    },
    {
        "Chapter": "1",
        "sentence_range": "4281-4284",
        "Text": "Solution Let A(1, 0), B(2, 2) and C (3, 1) be\nthe vertices of a triangle ABC (Fig 8 18) Area of \u2206ABC\n= Area of \u2206ABD  + Area of trapezium\n    BDEC \u2013 Area of \u2206AEC\nNow equation of the sides AB, BC and\nCA are given by\ny = 2(x \u2013 1), y = 4 \u2013 x, y = 1\n2  (x \u2013 1), respectively Hence,\narea of \u2206 ABC =\n2\n3\n3\n1\n2\n1\n1\n2 (\n1)\n(4\n)\n2\nx\nx\ndx\nx dx\n\u2212dx\n\u2212\n+\n\u2212\n\u2212\n\u222b\n\u222b\n\u222b\n=\n2\n3\n3\n2\n2\n2\n1\n2\n1\n1\n2\n4\n2\n2\n2\n2\nx\nx\nx\nx\nx\nx\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\u2212\n+\n\u2212\n\u2212\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n=\n2\n2\n2\n2\n1\n3\n2\n2\n2\n1\n4 3\n4 2\n2\n2\n2\n2\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n\u2212\n\u2212\n\u2212\n+\n\u00d7 \u2212\n\u2212\n\u00d7\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\u2013\n2\n1\n3\n1\n3\n1\n2\n2\n2\n\uf8ee\n\uf8f9\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n\u2212\n\u2212\n\u2212\n\uf8ef\n\uf8fa\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n\uf8f0\n\uf8fb\n= 3\n2\nExample 10 Find the area of the region enclosed between the two circles: x2 + y2  = 4\nand (x \u2013 2)2 + y2 = 4"
    },
    {
        "Chapter": "1",
        "sentence_range": "4282-4285",
        "Text": "18) Area of \u2206ABC\n= Area of \u2206ABD  + Area of trapezium\n    BDEC \u2013 Area of \u2206AEC\nNow equation of the sides AB, BC and\nCA are given by\ny = 2(x \u2013 1), y = 4 \u2013 x, y = 1\n2  (x \u2013 1), respectively Hence,\narea of \u2206 ABC =\n2\n3\n3\n1\n2\n1\n1\n2 (\n1)\n(4\n)\n2\nx\nx\ndx\nx dx\n\u2212dx\n\u2212\n+\n\u2212\n\u2212\n\u222b\n\u222b\n\u222b\n=\n2\n3\n3\n2\n2\n2\n1\n2\n1\n1\n2\n4\n2\n2\n2\n2\nx\nx\nx\nx\nx\nx\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\u2212\n+\n\u2212\n\u2212\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n=\n2\n2\n2\n2\n1\n3\n2\n2\n2\n1\n4 3\n4 2\n2\n2\n2\n2\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n\u2212\n\u2212\n\u2212\n+\n\u00d7 \u2212\n\u2212\n\u00d7\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\u2013\n2\n1\n3\n1\n3\n1\n2\n2\n2\n\uf8ee\n\uf8f9\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n\u2212\n\u2212\n\u2212\n\uf8ef\n\uf8fa\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n\uf8f0\n\uf8fb\n= 3\n2\nExample 10 Find the area of the region enclosed between the two circles: x2 + y2  = 4\nand (x \u2013 2)2 + y2 = 4 Solution Equations of the given circles are\nx2 + y2 = 4"
    },
    {
        "Chapter": "1",
        "sentence_range": "4283-4286",
        "Text": "Area of \u2206ABC\n= Area of \u2206ABD  + Area of trapezium\n    BDEC \u2013 Area of \u2206AEC\nNow equation of the sides AB, BC and\nCA are given by\ny = 2(x \u2013 1), y = 4 \u2013 x, y = 1\n2  (x \u2013 1), respectively Hence,\narea of \u2206 ABC =\n2\n3\n3\n1\n2\n1\n1\n2 (\n1)\n(4\n)\n2\nx\nx\ndx\nx dx\n\u2212dx\n\u2212\n+\n\u2212\n\u2212\n\u222b\n\u222b\n\u222b\n=\n2\n3\n3\n2\n2\n2\n1\n2\n1\n1\n2\n4\n2\n2\n2\n2\nx\nx\nx\nx\nx\nx\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\u2212\n+\n\u2212\n\u2212\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n=\n2\n2\n2\n2\n1\n3\n2\n2\n2\n1\n4 3\n4 2\n2\n2\n2\n2\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n\u2212\n\u2212\n\u2212\n+\n\u00d7 \u2212\n\u2212\n\u00d7\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\u2013\n2\n1\n3\n1\n3\n1\n2\n2\n2\n\uf8ee\n\uf8f9\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n\u2212\n\u2212\n\u2212\n\uf8ef\n\uf8fa\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n\uf8f0\n\uf8fb\n= 3\n2\nExample 10 Find the area of the region enclosed between the two circles: x2 + y2  = 4\nand (x \u2013 2)2 + y2 = 4 Solution Equations of the given circles are\nx2 + y2 = 4 (1)\nand\n(x \u2013 2)2 + y2 = 4"
    },
    {
        "Chapter": "1",
        "sentence_range": "4284-4287",
        "Text": "Hence,\narea of \u2206 ABC =\n2\n3\n3\n1\n2\n1\n1\n2 (\n1)\n(4\n)\n2\nx\nx\ndx\nx dx\n\u2212dx\n\u2212\n+\n\u2212\n\u2212\n\u222b\n\u222b\n\u222b\n=\n2\n3\n3\n2\n2\n2\n1\n2\n1\n1\n2\n4\n2\n2\n2\n2\nx\nx\nx\nx\nx\nx\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\u2212\n+\n\u2212\n\u2212\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n=\n2\n2\n2\n2\n1\n3\n2\n2\n2\n1\n4 3\n4 2\n2\n2\n2\n2\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n\u2212\n\u2212\n\u2212\n+\n\u00d7 \u2212\n\u2212\n\u00d7\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n\u2013\n2\n1\n3\n1\n3\n1\n2\n2\n2\n\uf8ee\n\uf8f9\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n\u2212\n\u2212\n\u2212\n\uf8ef\n\uf8fa\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n\uf8f0\n\uf8fb\n= 3\n2\nExample 10 Find the area of the region enclosed between the two circles: x2 + y2  = 4\nand (x \u2013 2)2 + y2 = 4 Solution Equations of the given circles are\nx2 + y2 = 4 (1)\nand\n(x \u2013 2)2 + y2 = 4 (2)\nEquation (1) is a circle with centre O at the\norigin and radius 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "4285-4288",
        "Text": "Solution Equations of the given circles are\nx2 + y2 = 4 (1)\nand\n(x \u2013 2)2 + y2 = 4 (2)\nEquation (1) is a circle with centre O at the\norigin and radius 2 Equation (2) is a circle with\ncentre C (2, 0) and radius 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "4286-4289",
        "Text": "(1)\nand\n(x \u2013 2)2 + y2 = 4 (2)\nEquation (1) is a circle with centre O at the\norigin and radius 2 Equation (2) is a circle with\ncentre C (2, 0) and radius 2 Solving equations\n(1) and (2), we have\n(x \u20132)2 + y2 = x2 + y2\nor\nx2 \u2013 4x + 4 + y2 = x2 + y2\nor\nx = 1 which gives y =\n3\n\u00b1\nThus, the points of intersection of the given\ncircles are A(1, \n3 ) and A\u2032(1, \u2013\n3 ) as shown in\nthe Fig 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "4287-4290",
        "Text": "(2)\nEquation (1) is a circle with centre O at the\norigin and radius 2 Equation (2) is a circle with\ncentre C (2, 0) and radius 2 Solving equations\n(1) and (2), we have\n(x \u20132)2 + y2 = x2 + y2\nor\nx2 \u2013 4x + 4 + y2 = x2 + y2\nor\nx = 1 which gives y =\n3\n\u00b1\nThus, the points of intersection of the given\ncircles are A(1, \n3 ) and A\u2032(1, \u2013\n3 ) as shown in\nthe Fig 8 19"
    },
    {
        "Chapter": "1",
        "sentence_range": "4288-4291",
        "Text": "Equation (2) is a circle with\ncentre C (2, 0) and radius 2 Solving equations\n(1) and (2), we have\n(x \u20132)2 + y2 = x2 + y2\nor\nx2 \u2013 4x + 4 + y2 = x2 + y2\nor\nx = 1 which gives y =\n3\n\u00b1\nThus, the points of intersection of the given\ncircles are A(1, \n3 ) and A\u2032(1, \u2013\n3 ) as shown in\nthe Fig 8 19 Fig 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "4289-4292",
        "Text": "Solving equations\n(1) and (2), we have\n(x \u20132)2 + y2 = x2 + y2\nor\nx2 \u2013 4x + 4 + y2 = x2 + y2\nor\nx = 1 which gives y =\n3\n\u00b1\nThus, the points of intersection of the given\ncircles are A(1, \n3 ) and A\u2032(1, \u2013\n3 ) as shown in\nthe Fig 8 19 Fig 8 18\nFig 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "4290-4293",
        "Text": "19 Fig 8 18\nFig 8 19\nAPPLICATION OF INTEGRALS         371\nRequired area of the enclosed region OACA\u2032O between circles\n= 2 [area of the region ODCAO]\n(Why"
    },
    {
        "Chapter": "1",
        "sentence_range": "4291-4294",
        "Text": "Fig 8 18\nFig 8 19\nAPPLICATION OF INTEGRALS         371\nRequired area of the enclosed region OACA\u2032O between circles\n= 2 [area of the region ODCAO]\n(Why )\n= 2 [area of the region ODAO + area of the region DCAD]\n= \n1\n2\n0\n1\n2\ny dx\ny dx\n\uf8ee\n\uf8f9\n+\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\u222b\n\u222b\n= \n1\n2\n2\n2\n0\n1\n2\n4\n(\n2)\n4\nx\ndx\nx dx\n\uf8ee\n\uf8f9\n\u2212\n\u2212\n+\n\u2212\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\u222b\n\u222b\n(Why"
    },
    {
        "Chapter": "1",
        "sentence_range": "4292-4295",
        "Text": "18\nFig 8 19\nAPPLICATION OF INTEGRALS         371\nRequired area of the enclosed region OACA\u2032O between circles\n= 2 [area of the region ODCAO]\n(Why )\n= 2 [area of the region ODAO + area of the region DCAD]\n= \n1\n2\n0\n1\n2\ny dx\ny dx\n\uf8ee\n\uf8f9\n+\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\u222b\n\u222b\n= \n1\n2\n2\n2\n0\n1\n2\n4\n(\n2)\n4\nx\ndx\nx dx\n\uf8ee\n\uf8f9\n\u2212\n\u2212\n+\n\u2212\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\u222b\n\u222b\n(Why )\n= \n1\n2\n\u20131\n0\n1\n1\n2\n2\n(\n2) 4\n(\n2)\n4sin\n2\n2\nx2\nx\nx\n\uf8ee\n\uf8f9\n\u2212\n\uf8eb\n\uf8f6\n\u2212\n\u2212\n\u2212\n+\n\u00d7\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8ed\n\uf8f8\n\uf8f0\n\uf8fb\n    + \n2\n2\n\u20131\n1\n1\n1\n2\n4\n4sin\n2\n2\nx2\nx\nx\n\uf8ee\n\uf8f9\n\u2212\n+\n\u00d7\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n= \n1\n2\n2\n\u20131\n2\n\u20131\n1\n0\n2\n(\n2) 4\n(\n2)\n4sin\n4\n4sin\n2\n2\nx\nx\nx\nx\nx\nx\n\uf8ee\n\uf8f9\n\u2212\n\uf8eb\n\uf8f6\n\uf8ee\n\uf8f9\n\u2212\n\u2212\n\u2212\n+\n+\n\u2212\n+\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ed\n\uf8f8\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n= \n\u20131\n1\n\u20131\n1\n1\n1\n3\n4sin\n4sin ( 1)\n4sin 1\n3\n4sin\n2\n2\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8eb\n\uf8f6\n\uf8eb\u2212\n\uf8f6\n\uf8ee\n\uf8f9\n\u2212\n+\n\u2212\n\u2212\n+\n\u2212\n\u2212\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8ed\n\uf8f8\n\uf8f0\n\uf8fb\n\uf8ed\n\uf8f8\n\uf8f0\n\uf8fb\n= \n3\n4\n4\n4\n3\n4\n6\n2\n2\n6\n\uf8ee\n\uf8f9\n\u03c0\n\u03c0\n\u03c0\n\u03c0\n\uf8eb\n\uf8f6\n\uf8ee\n\uf8f9\n\u2212\n\u2212\n\u00d7\n+\n\u00d7\n+\n\u00d7\n\u2212\n\u2212\n\u00d7\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ed\n\uf8f8\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n= \n2\n2\n3\n2\n2\n3\n3\n3\n\u03c0\n\u03c0\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n\u2212\n\u2212\n+ \u03c0 +\n\u03c0 \u2212\n\u2212\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n= 8\n2 3\n3\n\u03c0 \u2212\nEXERCISE 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "4293-4296",
        "Text": "19\nAPPLICATION OF INTEGRALS         371\nRequired area of the enclosed region OACA\u2032O between circles\n= 2 [area of the region ODCAO]\n(Why )\n= 2 [area of the region ODAO + area of the region DCAD]\n= \n1\n2\n0\n1\n2\ny dx\ny dx\n\uf8ee\n\uf8f9\n+\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\u222b\n\u222b\n= \n1\n2\n2\n2\n0\n1\n2\n4\n(\n2)\n4\nx\ndx\nx dx\n\uf8ee\n\uf8f9\n\u2212\n\u2212\n+\n\u2212\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\u222b\n\u222b\n(Why )\n= \n1\n2\n\u20131\n0\n1\n1\n2\n2\n(\n2) 4\n(\n2)\n4sin\n2\n2\nx2\nx\nx\n\uf8ee\n\uf8f9\n\u2212\n\uf8eb\n\uf8f6\n\u2212\n\u2212\n\u2212\n+\n\u00d7\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8ed\n\uf8f8\n\uf8f0\n\uf8fb\n    + \n2\n2\n\u20131\n1\n1\n1\n2\n4\n4sin\n2\n2\nx2\nx\nx\n\uf8ee\n\uf8f9\n\u2212\n+\n\u00d7\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n= \n1\n2\n2\n\u20131\n2\n\u20131\n1\n0\n2\n(\n2) 4\n(\n2)\n4sin\n4\n4sin\n2\n2\nx\nx\nx\nx\nx\nx\n\uf8ee\n\uf8f9\n\u2212\n\uf8eb\n\uf8f6\n\uf8ee\n\uf8f9\n\u2212\n\u2212\n\u2212\n+\n+\n\u2212\n+\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ed\n\uf8f8\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n= \n\u20131\n1\n\u20131\n1\n1\n1\n3\n4sin\n4sin ( 1)\n4sin 1\n3\n4sin\n2\n2\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8eb\n\uf8f6\n\uf8eb\u2212\n\uf8f6\n\uf8ee\n\uf8f9\n\u2212\n+\n\u2212\n\u2212\n+\n\u2212\n\u2212\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8ed\n\uf8f8\n\uf8f0\n\uf8fb\n\uf8ed\n\uf8f8\n\uf8f0\n\uf8fb\n= \n3\n4\n4\n4\n3\n4\n6\n2\n2\n6\n\uf8ee\n\uf8f9\n\u03c0\n\u03c0\n\u03c0\n\u03c0\n\uf8eb\n\uf8f6\n\uf8ee\n\uf8f9\n\u2212\n\u2212\n\u00d7\n+\n\u00d7\n+\n\u00d7\n\u2212\n\u2212\n\u00d7\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ed\n\uf8f8\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n= \n2\n2\n3\n2\n2\n3\n3\n3\n\u03c0\n\u03c0\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n\u2212\n\u2212\n+ \u03c0 +\n\u03c0 \u2212\n\u2212\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n= 8\n2 3\n3\n\u03c0 \u2212\nEXERCISE 8 2\n1"
    },
    {
        "Chapter": "1",
        "sentence_range": "4294-4297",
        "Text": ")\n= 2 [area of the region ODAO + area of the region DCAD]\n= \n1\n2\n0\n1\n2\ny dx\ny dx\n\uf8ee\n\uf8f9\n+\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\u222b\n\u222b\n= \n1\n2\n2\n2\n0\n1\n2\n4\n(\n2)\n4\nx\ndx\nx dx\n\uf8ee\n\uf8f9\n\u2212\n\u2212\n+\n\u2212\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\u222b\n\u222b\n(Why )\n= \n1\n2\n\u20131\n0\n1\n1\n2\n2\n(\n2) 4\n(\n2)\n4sin\n2\n2\nx2\nx\nx\n\uf8ee\n\uf8f9\n\u2212\n\uf8eb\n\uf8f6\n\u2212\n\u2212\n\u2212\n+\n\u00d7\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8ed\n\uf8f8\n\uf8f0\n\uf8fb\n    + \n2\n2\n\u20131\n1\n1\n1\n2\n4\n4sin\n2\n2\nx2\nx\nx\n\uf8ee\n\uf8f9\n\u2212\n+\n\u00d7\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n= \n1\n2\n2\n\u20131\n2\n\u20131\n1\n0\n2\n(\n2) 4\n(\n2)\n4sin\n4\n4sin\n2\n2\nx\nx\nx\nx\nx\nx\n\uf8ee\n\uf8f9\n\u2212\n\uf8eb\n\uf8f6\n\uf8ee\n\uf8f9\n\u2212\n\u2212\n\u2212\n+\n+\n\u2212\n+\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ed\n\uf8f8\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n= \n\u20131\n1\n\u20131\n1\n1\n1\n3\n4sin\n4sin ( 1)\n4sin 1\n3\n4sin\n2\n2\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8eb\n\uf8f6\n\uf8eb\u2212\n\uf8f6\n\uf8ee\n\uf8f9\n\u2212\n+\n\u2212\n\u2212\n+\n\u2212\n\u2212\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8ed\n\uf8f8\n\uf8f0\n\uf8fb\n\uf8ed\n\uf8f8\n\uf8f0\n\uf8fb\n= \n3\n4\n4\n4\n3\n4\n6\n2\n2\n6\n\uf8ee\n\uf8f9\n\u03c0\n\u03c0\n\u03c0\n\u03c0\n\uf8eb\n\uf8f6\n\uf8ee\n\uf8f9\n\u2212\n\u2212\n\u00d7\n+\n\u00d7\n+\n\u00d7\n\u2212\n\u2212\n\u00d7\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ed\n\uf8f8\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n= \n2\n2\n3\n2\n2\n3\n3\n3\n\u03c0\n\u03c0\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n\u2212\n\u2212\n+ \u03c0 +\n\u03c0 \u2212\n\u2212\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n= 8\n2 3\n3\n\u03c0 \u2212\nEXERCISE 8 2\n1 Find the area of the circle 4x2 + 4y2 = 9 which is interior to the parabola x2 = 4y"
    },
    {
        "Chapter": "1",
        "sentence_range": "4295-4298",
        "Text": ")\n= \n1\n2\n\u20131\n0\n1\n1\n2\n2\n(\n2) 4\n(\n2)\n4sin\n2\n2\nx2\nx\nx\n\uf8ee\n\uf8f9\n\u2212\n\uf8eb\n\uf8f6\n\u2212\n\u2212\n\u2212\n+\n\u00d7\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8ed\n\uf8f8\n\uf8f0\n\uf8fb\n    + \n2\n2\n\u20131\n1\n1\n1\n2\n4\n4sin\n2\n2\nx2\nx\nx\n\uf8ee\n\uf8f9\n\u2212\n+\n\u00d7\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n= \n1\n2\n2\n\u20131\n2\n\u20131\n1\n0\n2\n(\n2) 4\n(\n2)\n4sin\n4\n4sin\n2\n2\nx\nx\nx\nx\nx\nx\n\uf8ee\n\uf8f9\n\u2212\n\uf8eb\n\uf8f6\n\uf8ee\n\uf8f9\n\u2212\n\u2212\n\u2212\n+\n+\n\u2212\n+\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ed\n\uf8f8\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n= \n\u20131\n1\n\u20131\n1\n1\n1\n3\n4sin\n4sin ( 1)\n4sin 1\n3\n4sin\n2\n2\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8eb\n\uf8f6\n\uf8eb\u2212\n\uf8f6\n\uf8ee\n\uf8f9\n\u2212\n+\n\u2212\n\u2212\n+\n\u2212\n\u2212\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8ed\n\uf8f8\n\uf8f0\n\uf8fb\n\uf8ed\n\uf8f8\n\uf8f0\n\uf8fb\n= \n3\n4\n4\n4\n3\n4\n6\n2\n2\n6\n\uf8ee\n\uf8f9\n\u03c0\n\u03c0\n\u03c0\n\u03c0\n\uf8eb\n\uf8f6\n\uf8ee\n\uf8f9\n\u2212\n\u2212\n\u00d7\n+\n\u00d7\n+\n\u00d7\n\u2212\n\u2212\n\u00d7\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ed\n\uf8f8\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n= \n2\n2\n3\n2\n2\n3\n3\n3\n\u03c0\n\u03c0\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n\u2212\n\u2212\n+ \u03c0 +\n\u03c0 \u2212\n\u2212\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n= 8\n2 3\n3\n\u03c0 \u2212\nEXERCISE 8 2\n1 Find the area of the circle 4x2 + 4y2 = 9 which is interior to the parabola x2 = 4y 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "4296-4299",
        "Text": "2\n1 Find the area of the circle 4x2 + 4y2 = 9 which is interior to the parabola x2 = 4y 2 Find the area bounded by curves (x  \u2013 1)2 + y2 = 1 and  x2 + y2 = 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "4297-4300",
        "Text": "Find the area of the circle 4x2 + 4y2 = 9 which is interior to the parabola x2 = 4y 2 Find the area bounded by curves (x  \u2013 1)2 + y2 = 1 and  x2 + y2 = 1 3"
    },
    {
        "Chapter": "1",
        "sentence_range": "4298-4301",
        "Text": "2 Find the area bounded by curves (x  \u2013 1)2 + y2 = 1 and  x2 + y2 = 1 3 Find the area of the region bounded by the curves y = x2 + 2,  y = x, x = 0 and\nx = 3"
    },
    {
        "Chapter": "1",
        "sentence_range": "4299-4302",
        "Text": "Find the area bounded by curves (x  \u2013 1)2 + y2 = 1 and  x2 + y2 = 1 3 Find the area of the region bounded by the curves y = x2 + 2,  y = x, x = 0 and\nx = 3 4"
    },
    {
        "Chapter": "1",
        "sentence_range": "4300-4303",
        "Text": "3 Find the area of the region bounded by the curves y = x2 + 2,  y = x, x = 0 and\nx = 3 4 Using integration find the area of region bounded by the triangle whose vertices\nare (\u2013 1, 0), (1, 3) and (3, 2)"
    },
    {
        "Chapter": "1",
        "sentence_range": "4301-4304",
        "Text": "Find the area of the region bounded by the curves y = x2 + 2,  y = x, x = 0 and\nx = 3 4 Using integration find the area of region bounded by the triangle whose vertices\nare (\u2013 1, 0), (1, 3) and (3, 2) 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "4302-4305",
        "Text": "4 Using integration find the area of region bounded by the triangle whose vertices\nare (\u2013 1, 0), (1, 3) and (3, 2) 5 Using integration find the area of the triangular region whose sides have the\nequations y = 2x + 1, y = 3x + 1 and x = 4"
    },
    {
        "Chapter": "1",
        "sentence_range": "4303-4306",
        "Text": "Using integration find the area of region bounded by the triangle whose vertices\nare (\u2013 1, 0), (1, 3) and (3, 2) 5 Using integration find the area of the triangular region whose sides have the\nequations y = 2x + 1, y = 3x + 1 and x = 4 372\nMATHEMATICS\nChoose the correct answer in the following exercises 6 and 7"
    },
    {
        "Chapter": "1",
        "sentence_range": "4304-4307",
        "Text": "5 Using integration find the area of the triangular region whose sides have the\nequations y = 2x + 1, y = 3x + 1 and x = 4 372\nMATHEMATICS\nChoose the correct answer in the following exercises 6 and 7 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "4305-4308",
        "Text": "Using integration find the area of the triangular region whose sides have the\nequations y = 2x + 1, y = 3x + 1 and x = 4 372\nMATHEMATICS\nChoose the correct answer in the following exercises 6 and 7 6 Smaller area enclosed by the circle x2 + y2 = 4 and the line x + y = 2 is\n (A) 2 (\u03c0 \u2013 2)\n(B) \u03c0 \u2013 2\n(C) 2\u03c0 \u2013 1\n(D) 2 (\u03c0 + 2)\n7"
    },
    {
        "Chapter": "1",
        "sentence_range": "4306-4309",
        "Text": "372\nMATHEMATICS\nChoose the correct answer in the following exercises 6 and 7 6 Smaller area enclosed by the circle x2 + y2 = 4 and the line x + y = 2 is\n (A) 2 (\u03c0 \u2013 2)\n(B) \u03c0 \u2013 2\n(C) 2\u03c0 \u2013 1\n(D) 2 (\u03c0 + 2)\n7 Area lying between the curves y2 = 4x and y = 2x is\n(A)\n32\n(B) 1\n3\n(C)\n41\n(D)\n3\n4\nMiscellaneous Examples\nExample 11 Find the area of the parabola y2 = 4ax bounded by its latus rectum"
    },
    {
        "Chapter": "1",
        "sentence_range": "4307-4310",
        "Text": "6 Smaller area enclosed by the circle x2 + y2 = 4 and the line x + y = 2 is\n (A) 2 (\u03c0 \u2013 2)\n(B) \u03c0 \u2013 2\n(C) 2\u03c0 \u2013 1\n(D) 2 (\u03c0 + 2)\n7 Area lying between the curves y2 = 4x and y = 2x is\n(A)\n32\n(B) 1\n3\n(C)\n41\n(D)\n3\n4\nMiscellaneous Examples\nExample 11 Find the area of the parabola y2 = 4ax bounded by its latus rectum Solution From Fig 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "4308-4311",
        "Text": "Smaller area enclosed by the circle x2 + y2 = 4 and the line x + y = 2 is\n (A) 2 (\u03c0 \u2013 2)\n(B) \u03c0 \u2013 2\n(C) 2\u03c0 \u2013 1\n(D) 2 (\u03c0 + 2)\n7 Area lying between the curves y2 = 4x and y = 2x is\n(A)\n32\n(B) 1\n3\n(C)\n41\n(D)\n3\n4\nMiscellaneous Examples\nExample 11 Find the area of the parabola y2 = 4ax bounded by its latus rectum Solution From Fig 8 20, the vertex of the parabola\ny2 = 4ax is at origin (0, 0)"
    },
    {
        "Chapter": "1",
        "sentence_range": "4309-4312",
        "Text": "Area lying between the curves y2 = 4x and y = 2x is\n(A)\n32\n(B) 1\n3\n(C)\n41\n(D)\n3\n4\nMiscellaneous Examples\nExample 11 Find the area of the parabola y2 = 4ax bounded by its latus rectum Solution From Fig 8 20, the vertex of the parabola\ny2 = 4ax is at origin (0, 0) The equation of the\nlatus rectum LSL\u2032 is x = a"
    },
    {
        "Chapter": "1",
        "sentence_range": "4310-4313",
        "Text": "Solution From Fig 8 20, the vertex of the parabola\ny2 = 4ax is at origin (0, 0) The equation of the\nlatus rectum LSL\u2032 is x = a Also, parabola is\nsymmetrical about the x-axis"
    },
    {
        "Chapter": "1",
        "sentence_range": "4311-4314",
        "Text": "20, the vertex of the parabola\ny2 = 4ax is at origin (0, 0) The equation of the\nlatus rectum LSL\u2032 is x = a Also, parabola is\nsymmetrical about the x-axis The required area of the region OLL\u2032O\n= 2(area of the region OLSO)\n= \n20\n\u222ba ydx\n = \n20\n4\na\nax dx\n\u222b\n= \n0\n2\n2\na\na\nxdx\n\u00d7\n\u222b\n= \n3\n2\n0\n2\n4\n3\na\na\n\uf8eex\n\uf8f9\n\u00d7\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n= \n23\n8\n3\na a\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n = \n82\n3\na\nExample 12 Find the area of the region bounded\nby the line y = 3x + 2, the x-axis and the ordinates\nx = \u20131 and x = 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "4312-4315",
        "Text": "The equation of the\nlatus rectum LSL\u2032 is x = a Also, parabola is\nsymmetrical about the x-axis The required area of the region OLL\u2032O\n= 2(area of the region OLSO)\n= \n20\n\u222ba ydx\n = \n20\n4\na\nax dx\n\u222b\n= \n0\n2\n2\na\na\nxdx\n\u00d7\n\u222b\n= \n3\n2\n0\n2\n4\n3\na\na\n\uf8eex\n\uf8f9\n\u00d7\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n= \n23\n8\n3\na a\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n = \n82\n3\na\nExample 12 Find the area of the region bounded\nby the line y = 3x + 2, the x-axis and the ordinates\nx = \u20131 and x = 1 Solution As shown in the Fig 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "4313-4316",
        "Text": "Also, parabola is\nsymmetrical about the x-axis The required area of the region OLL\u2032O\n= 2(area of the region OLSO)\n= \n20\n\u222ba ydx\n = \n20\n4\na\nax dx\n\u222b\n= \n0\n2\n2\na\na\nxdx\n\u00d7\n\u222b\n= \n3\n2\n0\n2\n4\n3\na\na\n\uf8eex\n\uf8f9\n\u00d7\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n= \n23\n8\n3\na a\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n = \n82\n3\na\nExample 12 Find the area of the region bounded\nby the line y = 3x + 2, the x-axis and the ordinates\nx = \u20131 and x = 1 Solution As shown in the Fig 8 21, the line\ny = 3x + 2 meets x-axis at x = \n32\n\u2212\n and its graph\nlies below x-axis for\n1, 32\nx\n\u2212\n\uf8eb\n\uf8f6\n\u2208 \u2212\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8 and above\nx-axis for \n32 ,1\nx\n\uf8eb\u2212\n\uf8f6\n\u2208\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8"
    },
    {
        "Chapter": "1",
        "sentence_range": "4314-4317",
        "Text": "The required area of the region OLL\u2032O\n= 2(area of the region OLSO)\n= \n20\n\u222ba ydx\n = \n20\n4\na\nax dx\n\u222b\n= \n0\n2\n2\na\na\nxdx\n\u00d7\n\u222b\n= \n3\n2\n0\n2\n4\n3\na\na\n\uf8eex\n\uf8f9\n\u00d7\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n= \n23\n8\n3\na a\n\uf8ee\n\uf8f9\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n = \n82\n3\na\nExample 12 Find the area of the region bounded\nby the line y = 3x + 2, the x-axis and the ordinates\nx = \u20131 and x = 1 Solution As shown in the Fig 8 21, the line\ny = 3x + 2 meets x-axis at x = \n32\n\u2212\n and its graph\nlies below x-axis for\n1, 32\nx\n\u2212\n\uf8eb\n\uf8f6\n\u2208 \u2212\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8 and above\nx-axis for \n32 ,1\nx\n\uf8eb\u2212\n\uf8f6\n\u2208\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8 Fig 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "4315-4318",
        "Text": "Solution As shown in the Fig 8 21, the line\ny = 3x + 2 meets x-axis at x = \n32\n\u2212\n and its graph\nlies below x-axis for\n1, 32\nx\n\u2212\n\uf8eb\n\uf8f6\n\u2208 \u2212\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8 and above\nx-axis for \n32 ,1\nx\n\uf8eb\u2212\n\uf8f6\n\u2208\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8 Fig 8 21\nX\u2032\nO\nY\u2032\nX\nY\nS\nL\nL'\n(\ufffd\ufffd,\ufffd0)\na\nFig 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "4316-4319",
        "Text": "21, the line\ny = 3x + 2 meets x-axis at x = \n32\n\u2212\n and its graph\nlies below x-axis for\n1, 32\nx\n\u2212\n\uf8eb\n\uf8f6\n\u2208 \u2212\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8 and above\nx-axis for \n32 ,1\nx\n\uf8eb\u2212\n\uf8f6\n\u2208\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8 Fig 8 21\nX\u2032\nO\nY\u2032\nX\nY\nS\nL\nL'\n(\ufffd\ufffd,\ufffd0)\na\nFig 8 20\nAPPLICATION OF INTEGRALS         373\nFig 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "4317-4320",
        "Text": "Fig 8 21\nX\u2032\nO\nY\u2032\nX\nY\nS\nL\nL'\n(\ufffd\ufffd,\ufffd0)\na\nFig 8 20\nAPPLICATION OF INTEGRALS         373\nFig 8 23\nThe required area = Area of the region ACBA + Area of the region ADEA\n=\n2\n1\n3\n2\n1\n3\n(3\n2)\n(3\n2)\nx\ndx\nx\ndx\n\u2212\n\u2212\n\u2212\n+\n+\n+\n\u222b\n\u222b\n=\n2\n1\n2\n2\n3\n2\n1\n3\n3\n3\n2\n2\n2\n2\nx\nx\nx\nx\n\u2212\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n+\n+\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n = 1\n25\n13\n6\n6\n3\n+\n=\nExample 13 Find the area bounded by\nthe curve y = cos x between x = 0 and\nx = 2\u03c0"
    },
    {
        "Chapter": "1",
        "sentence_range": "4318-4321",
        "Text": "21\nX\u2032\nO\nY\u2032\nX\nY\nS\nL\nL'\n(\ufffd\ufffd,\ufffd0)\na\nFig 8 20\nAPPLICATION OF INTEGRALS         373\nFig 8 23\nThe required area = Area of the region ACBA + Area of the region ADEA\n=\n2\n1\n3\n2\n1\n3\n(3\n2)\n(3\n2)\nx\ndx\nx\ndx\n\u2212\n\u2212\n\u2212\n+\n+\n+\n\u222b\n\u222b\n=\n2\n1\n2\n2\n3\n2\n1\n3\n3\n3\n2\n2\n2\n2\nx\nx\nx\nx\n\u2212\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n+\n+\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n = 1\n25\n13\n6\n6\n3\n+\n=\nExample 13 Find the area bounded by\nthe curve y = cos x between x = 0 and\nx = 2\u03c0 Solution From the Fig 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "4319-4322",
        "Text": "20\nAPPLICATION OF INTEGRALS         373\nFig 8 23\nThe required area = Area of the region ACBA + Area of the region ADEA\n=\n2\n1\n3\n2\n1\n3\n(3\n2)\n(3\n2)\nx\ndx\nx\ndx\n\u2212\n\u2212\n\u2212\n+\n+\n+\n\u222b\n\u222b\n=\n2\n1\n2\n2\n3\n2\n1\n3\n3\n3\n2\n2\n2\n2\nx\nx\nx\nx\n\u2212\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n+\n+\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n = 1\n25\n13\n6\n6\n3\n+\n=\nExample 13 Find the area bounded by\nthe curve y = cos x between x = 0 and\nx = 2\u03c0 Solution From the Fig 8 22, the required\narea = area of the region OABO + area\nof the region BCDB + area of the region\nDEFD"
    },
    {
        "Chapter": "1",
        "sentence_range": "4320-4323",
        "Text": "23\nThe required area = Area of the region ACBA + Area of the region ADEA\n=\n2\n1\n3\n2\n1\n3\n(3\n2)\n(3\n2)\nx\ndx\nx\ndx\n\u2212\n\u2212\n\u2212\n+\n+\n+\n\u222b\n\u222b\n=\n2\n1\n2\n2\n3\n2\n1\n3\n3\n3\n2\n2\n2\n2\nx\nx\nx\nx\n\u2212\n\u2212\n\u2212\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n+\n+\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n = 1\n25\n13\n6\n6\n3\n+\n=\nExample 13 Find the area bounded by\nthe curve y = cos x between x = 0 and\nx = 2\u03c0 Solution From the Fig 8 22, the required\narea = area of the region OABO + area\nof the region BCDB + area of the region\nDEFD Thus, we have the required area\n= \n3\u03c0\n\u03c0\n2\u03c0\n2\n2\n3\u03c0\n\u03c0\n0\n2\n2\ncos\ncos\ncos\nxdx\nxdx\nx dx\n+\n+\n\u222b\n\u222b\n\u222b\n= [\n]\n[\n]\n[\n]\n3\n2\n2\n2\n3\n0\n2\n2\nsin\nsin\nsin\nx\nx\nx\n\u03c0\n\u03c0\n\u03c0\n\u03c0\n\u03c0\n+\n+\n= 1 + 2 + 1 = 4\nExample 13 Prove that the curves y2 = 4x and x2 = 4y\ndivide the area of the square bounded by x = 0, x = 4,\ny = 4 and y = 0 into three equal parts"
    },
    {
        "Chapter": "1",
        "sentence_range": "4321-4324",
        "Text": "Solution From the Fig 8 22, the required\narea = area of the region OABO + area\nof the region BCDB + area of the region\nDEFD Thus, we have the required area\n= \n3\u03c0\n\u03c0\n2\u03c0\n2\n2\n3\u03c0\n\u03c0\n0\n2\n2\ncos\ncos\ncos\nxdx\nxdx\nx dx\n+\n+\n\u222b\n\u222b\n\u222b\n= [\n]\n[\n]\n[\n]\n3\n2\n2\n2\n3\n0\n2\n2\nsin\nsin\nsin\nx\nx\nx\n\u03c0\n\u03c0\n\u03c0\n\u03c0\n\u03c0\n+\n+\n= 1 + 2 + 1 = 4\nExample 13 Prove that the curves y2 = 4x and x2 = 4y\ndivide the area of the square bounded by x = 0, x = 4,\ny = 4 and y = 0 into three equal parts Solution Note that the point of intersection of the\nparabolas y2 = 4x and x2 = 4y are (0, 0) and (4, 4) as\nFig 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "4322-4325",
        "Text": "22, the required\narea = area of the region OABO + area\nof the region BCDB + area of the region\nDEFD Thus, we have the required area\n= \n3\u03c0\n\u03c0\n2\u03c0\n2\n2\n3\u03c0\n\u03c0\n0\n2\n2\ncos\ncos\ncos\nxdx\nxdx\nx dx\n+\n+\n\u222b\n\u222b\n\u222b\n= [\n]\n[\n]\n[\n]\n3\n2\n2\n2\n3\n0\n2\n2\nsin\nsin\nsin\nx\nx\nx\n\u03c0\n\u03c0\n\u03c0\n\u03c0\n\u03c0\n+\n+\n= 1 + 2 + 1 = 4\nExample 13 Prove that the curves y2 = 4x and x2 = 4y\ndivide the area of the square bounded by x = 0, x = 4,\ny = 4 and y = 0 into three equal parts Solution Note that the point of intersection of the\nparabolas y2 = 4x and x2 = 4y are (0, 0) and (4, 4) as\nFig 8 22\n374\nMATHEMATICS\nY'\nR\nO\nX\nX'\nx =\ufffd2\nT\nS\nP\ufffd(0,1) x =\ufffd1\nY\nQ(1,2)\nshown in the Fig 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "4323-4326",
        "Text": "Thus, we have the required area\n= \n3\u03c0\n\u03c0\n2\u03c0\n2\n2\n3\u03c0\n\u03c0\n0\n2\n2\ncos\ncos\ncos\nxdx\nxdx\nx dx\n+\n+\n\u222b\n\u222b\n\u222b\n= [\n]\n[\n]\n[\n]\n3\n2\n2\n2\n3\n0\n2\n2\nsin\nsin\nsin\nx\nx\nx\n\u03c0\n\u03c0\n\u03c0\n\u03c0\n\u03c0\n+\n+\n= 1 + 2 + 1 = 4\nExample 13 Prove that the curves y2 = 4x and x2 = 4y\ndivide the area of the square bounded by x = 0, x = 4,\ny = 4 and y = 0 into three equal parts Solution Note that the point of intersection of the\nparabolas y2 = 4x and x2 = 4y are (0, 0) and (4, 4) as\nFig 8 22\n374\nMATHEMATICS\nY'\nR\nO\nX\nX'\nx =\ufffd2\nT\nS\nP\ufffd(0,1) x =\ufffd1\nY\nQ(1,2)\nshown in the Fig 8 23"
    },
    {
        "Chapter": "1",
        "sentence_range": "4324-4327",
        "Text": "Solution Note that the point of intersection of the\nparabolas y2 = 4x and x2 = 4y are (0, 0) and (4, 4) as\nFig 8 22\n374\nMATHEMATICS\nY'\nR\nO\nX\nX'\nx =\ufffd2\nT\nS\nP\ufffd(0,1) x =\ufffd1\nY\nQ(1,2)\nshown in the Fig 8 23 Now, the area of the region OAQBO bounded by curves y2 = 4x and x2 = 4y"
    },
    {
        "Chapter": "1",
        "sentence_range": "4325-4328",
        "Text": "22\n374\nMATHEMATICS\nY'\nR\nO\nX\nX'\nx =\ufffd2\nT\nS\nP\ufffd(0,1) x =\ufffd1\nY\nQ(1,2)\nshown in the Fig 8 23 Now, the area of the region OAQBO bounded by curves y2 = 4x and x2 = 4y =\n2\n4\n0\n2\nx4\nx\ndx\n\uf8eb\n\uf8f6\n\u2212\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\u222b\n = \n4\n3\n3\n2\n0\n2\n2\n3\nx12\nx\n\uf8ee\n\uf8f9\n\u00d7\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n= 32\n16\n16\n3\n3\n3\n\u2212\n="
    },
    {
        "Chapter": "1",
        "sentence_range": "4326-4329",
        "Text": "23 Now, the area of the region OAQBO bounded by curves y2 = 4x and x2 = 4y =\n2\n4\n0\n2\nx4\nx\ndx\n\uf8eb\n\uf8f6\n\u2212\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\u222b\n = \n4\n3\n3\n2\n0\n2\n2\n3\nx12\nx\n\uf8ee\n\uf8f9\n\u00d7\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n= 32\n16\n16\n3\n3\n3\n\u2212\n= (1)\nAgain, the area of the region OPQAO bounded by the curves x2 = 4y, x = 0, x = 4\nand x-axis\n=\n2\n4\n4\n3\n0\n0\n1\n16\n4\n12\n3\nx\ndx\n\uf8eex\n\uf8f9\n=\n=\n\uf8f0\n\uf8fb\n\u222b"
    },
    {
        "Chapter": "1",
        "sentence_range": "4327-4330",
        "Text": "Now, the area of the region OAQBO bounded by curves y2 = 4x and x2 = 4y =\n2\n4\n0\n2\nx4\nx\ndx\n\uf8eb\n\uf8f6\n\u2212\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\u222b\n = \n4\n3\n3\n2\n0\n2\n2\n3\nx12\nx\n\uf8ee\n\uf8f9\n\u00d7\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n= 32\n16\n16\n3\n3\n3\n\u2212\n= (1)\nAgain, the area of the region OPQAO bounded by the curves x2 = 4y, x = 0, x = 4\nand x-axis\n=\n2\n4\n4\n3\n0\n0\n1\n16\n4\n12\n3\nx\ndx\n\uf8eex\n\uf8f9\n=\n=\n\uf8f0\n\uf8fb\n\u222b (2)\nSimilarly, the area of the region OBQRO bounded by the curve y2 = 4x, y-axis,\ny = 0 and y = 4\n=\n2\n4\n4\n4\n3\n0\n0\n0\n1\n16\n4\n12\n3\ny\nxdy\ndy\n\uf8eey\n\uf8f9\n=\n=\n=\n\uf8f0\n\uf8fb\n\u222b\n\u222b"
    },
    {
        "Chapter": "1",
        "sentence_range": "4328-4331",
        "Text": "=\n2\n4\n0\n2\nx4\nx\ndx\n\uf8eb\n\uf8f6\n\u2212\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\u222b\n = \n4\n3\n3\n2\n0\n2\n2\n3\nx12\nx\n\uf8ee\n\uf8f9\n\u00d7\n\u2212\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8f0\n\uf8fb\n= 32\n16\n16\n3\n3\n3\n\u2212\n= (1)\nAgain, the area of the region OPQAO bounded by the curves x2 = 4y, x = 0, x = 4\nand x-axis\n=\n2\n4\n4\n3\n0\n0\n1\n16\n4\n12\n3\nx\ndx\n\uf8eex\n\uf8f9\n=\n=\n\uf8f0\n\uf8fb\n\u222b (2)\nSimilarly, the area of the region OBQRO bounded by the curve y2 = 4x, y-axis,\ny = 0 and y = 4\n=\n2\n4\n4\n4\n3\n0\n0\n0\n1\n16\n4\n12\n3\ny\nxdy\ndy\n\uf8eey\n\uf8f9\n=\n=\n=\n\uf8f0\n\uf8fb\n\u222b\n\u222b (3)\nFrom (1), (2) and (3), it is concluded that the area of the region OAQBO = area of\nthe region OPQAO = area of the region OBQRO, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "4329-4332",
        "Text": "(1)\nAgain, the area of the region OPQAO bounded by the curves x2 = 4y, x = 0, x = 4\nand x-axis\n=\n2\n4\n4\n3\n0\n0\n1\n16\n4\n12\n3\nx\ndx\n\uf8eex\n\uf8f9\n=\n=\n\uf8f0\n\uf8fb\n\u222b (2)\nSimilarly, the area of the region OBQRO bounded by the curve y2 = 4x, y-axis,\ny = 0 and y = 4\n=\n2\n4\n4\n4\n3\n0\n0\n0\n1\n16\n4\n12\n3\ny\nxdy\ndy\n\uf8eey\n\uf8f9\n=\n=\n=\n\uf8f0\n\uf8fb\n\u222b\n\u222b (3)\nFrom (1), (2) and (3), it is concluded that the area of the region OAQBO = area of\nthe region OPQAO = area of the region OBQRO, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "4330-4333",
        "Text": "(2)\nSimilarly, the area of the region OBQRO bounded by the curve y2 = 4x, y-axis,\ny = 0 and y = 4\n=\n2\n4\n4\n4\n3\n0\n0\n0\n1\n16\n4\n12\n3\ny\nxdy\ndy\n\uf8eey\n\uf8f9\n=\n=\n=\n\uf8f0\n\uf8fb\n\u222b\n\u222b (3)\nFrom (1), (2) and (3), it is concluded that the area of the region OAQBO = area of\nthe region OPQAO = area of the region OBQRO, i e , area bounded by parabolas\ny2 = 4x and x2 = 4y divides the area of the square in three equal parts"
    },
    {
        "Chapter": "1",
        "sentence_range": "4331-4334",
        "Text": "(3)\nFrom (1), (2) and (3), it is concluded that the area of the region OAQBO = area of\nthe region OPQAO = area of the region OBQRO, i e , area bounded by parabolas\ny2 = 4x and x2 = 4y divides the area of the square in three equal parts Example 14 Find the area of the region\n{(x, y) : 0 \u2264 y \u2264 x2 + 1, 0 \u2264 y \u2264 x + 1, 0 \u2264 x \u2264 2}\nSolution Let us first sketch the region whose area is to\nbe found out"
    },
    {
        "Chapter": "1",
        "sentence_range": "4332-4335",
        "Text": "e , area bounded by parabolas\ny2 = 4x and x2 = 4y divides the area of the square in three equal parts Example 14 Find the area of the region\n{(x, y) : 0 \u2264 y \u2264 x2 + 1, 0 \u2264 y \u2264 x + 1, 0 \u2264 x \u2264 2}\nSolution Let us first sketch the region whose area is to\nbe found out This region is the intersection of the\nfollowing regions"
    },
    {
        "Chapter": "1",
        "sentence_range": "4333-4336",
        "Text": ", area bounded by parabolas\ny2 = 4x and x2 = 4y divides the area of the square in three equal parts Example 14 Find the area of the region\n{(x, y) : 0 \u2264 y \u2264 x2 + 1, 0 \u2264 y \u2264 x + 1, 0 \u2264 x \u2264 2}\nSolution Let us first sketch the region whose area is to\nbe found out This region is the intersection of the\nfollowing regions A1 = {(x, y) : 0 \u2264 y \u2264 x2 + 1},\nA2 = {(x, y) : 0 \u2264 y \u2264 x + 1}\nand\nA3 = {(x, y) : 0 \u2264 x \u2264 2}\nThe points of intersection of y = x2 + 1 and y = x + 1 are points P(0, 1) and Q(1, 2)"
    },
    {
        "Chapter": "1",
        "sentence_range": "4334-4337",
        "Text": "Example 14 Find the area of the region\n{(x, y) : 0 \u2264 y \u2264 x2 + 1, 0 \u2264 y \u2264 x + 1, 0 \u2264 x \u2264 2}\nSolution Let us first sketch the region whose area is to\nbe found out This region is the intersection of the\nfollowing regions A1 = {(x, y) : 0 \u2264 y \u2264 x2 + 1},\nA2 = {(x, y) : 0 \u2264 y \u2264 x + 1}\nand\nA3 = {(x, y) : 0 \u2264 x \u2264 2}\nThe points of intersection of y = x2 + 1 and y = x + 1 are points P(0, 1) and Q(1, 2) From the Fig 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "4335-4338",
        "Text": "This region is the intersection of the\nfollowing regions A1 = {(x, y) : 0 \u2264 y \u2264 x2 + 1},\nA2 = {(x, y) : 0 \u2264 y \u2264 x + 1}\nand\nA3 = {(x, y) : 0 \u2264 x \u2264 2}\nThe points of intersection of y = x2 + 1 and y = x + 1 are points P(0, 1) and Q(1, 2) From the Fig 8 24, the required region is the shaded region OPQRSTO whose area\n= area of the region OTQPO + area of the region TSRQT\n=\n1\n2\n2\n0\n1\n(\n1)\n(\n1)\nx\ndx\nx\ndx\n+\n+\n+\n\u222b\n\u222b\n(Why"
    },
    {
        "Chapter": "1",
        "sentence_range": "4336-4339",
        "Text": "A1 = {(x, y) : 0 \u2264 y \u2264 x2 + 1},\nA2 = {(x, y) : 0 \u2264 y \u2264 x + 1}\nand\nA3 = {(x, y) : 0 \u2264 x \u2264 2}\nThe points of intersection of y = x2 + 1 and y = x + 1 are points P(0, 1) and Q(1, 2) From the Fig 8 24, the required region is the shaded region OPQRSTO whose area\n= area of the region OTQPO + area of the region TSRQT\n=\n1\n2\n2\n0\n1\n(\n1)\n(\n1)\nx\ndx\nx\ndx\n+\n+\n+\n\u222b\n\u222b\n(Why )\nFig 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "4337-4340",
        "Text": "From the Fig 8 24, the required region is the shaded region OPQRSTO whose area\n= area of the region OTQPO + area of the region TSRQT\n=\n1\n2\n2\n0\n1\n(\n1)\n(\n1)\nx\ndx\nx\ndx\n+\n+\n+\n\u222b\n\u222b\n(Why )\nFig 8 24\nAPPLICATION OF INTEGRALS         375\n=\n1\n2\n3\n2\n1\n0\n3\n2\nx\nx\nx\nx\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n+\n+\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n=\n(\n)\n1\n1\n1\n0\n2\n2\n1\n3\n2\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n+\n\u2212\n+\n+\n\u2212\n+\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n = 23\n6\nMiscellaneous Exercise on Chapter 8\n1"
    },
    {
        "Chapter": "1",
        "sentence_range": "4338-4341",
        "Text": "24, the required region is the shaded region OPQRSTO whose area\n= area of the region OTQPO + area of the region TSRQT\n=\n1\n2\n2\n0\n1\n(\n1)\n(\n1)\nx\ndx\nx\ndx\n+\n+\n+\n\u222b\n\u222b\n(Why )\nFig 8 24\nAPPLICATION OF INTEGRALS         375\n=\n1\n2\n3\n2\n1\n0\n3\n2\nx\nx\nx\nx\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n+\n+\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n=\n(\n)\n1\n1\n1\n0\n2\n2\n1\n3\n2\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n+\n\u2212\n+\n+\n\u2212\n+\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n = 23\n6\nMiscellaneous Exercise on Chapter 8\n1 Find the area under the given curves and given lines:\n(i) y = x2, x = 1, x = 2 and x-axis\n(ii) y = x4, x = 1, x = 5 and x-axis\n2"
    },
    {
        "Chapter": "1",
        "sentence_range": "4339-4342",
        "Text": ")\nFig 8 24\nAPPLICATION OF INTEGRALS         375\n=\n1\n2\n3\n2\n1\n0\n3\n2\nx\nx\nx\nx\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n+\n+\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n=\n(\n)\n1\n1\n1\n0\n2\n2\n1\n3\n2\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n+\n\u2212\n+\n+\n\u2212\n+\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n = 23\n6\nMiscellaneous Exercise on Chapter 8\n1 Find the area under the given curves and given lines:\n(i) y = x2, x = 1, x = 2 and x-axis\n(ii) y = x4, x = 1, x = 5 and x-axis\n2 Find the area between the curves y = x and y = x2"
    },
    {
        "Chapter": "1",
        "sentence_range": "4340-4343",
        "Text": "24\nAPPLICATION OF INTEGRALS         375\n=\n1\n2\n3\n2\n1\n0\n3\n2\nx\nx\nx\nx\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n+\n+\n+\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n=\n(\n)\n1\n1\n1\n0\n2\n2\n1\n3\n2\n\uf8ee\n\uf8f9\n\uf8ee\n\uf8f9\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n+\n\u2212\n+\n+\n\u2212\n+\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8ef\n\uf8fa\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n\uf8f0\n\uf8fb\n\uf8f0\n\uf8fb\n = 23\n6\nMiscellaneous Exercise on Chapter 8\n1 Find the area under the given curves and given lines:\n(i) y = x2, x = 1, x = 2 and x-axis\n(ii) y = x4, x = 1, x = 5 and x-axis\n2 Find the area between the curves y = x and y = x2 3"
    },
    {
        "Chapter": "1",
        "sentence_range": "4341-4344",
        "Text": "Find the area under the given curves and given lines:\n(i) y = x2, x = 1, x = 2 and x-axis\n(ii) y = x4, x = 1, x = 5 and x-axis\n2 Find the area between the curves y = x and y = x2 3 Find the area of the region lying in the first quadrant and bounded by y = 4x2,\nx = 0, y = 1 and y = 4"
    },
    {
        "Chapter": "1",
        "sentence_range": "4342-4345",
        "Text": "Find the area between the curves y = x and y = x2 3 Find the area of the region lying in the first quadrant and bounded by y = 4x2,\nx = 0, y = 1 and y = 4 4"
    },
    {
        "Chapter": "1",
        "sentence_range": "4343-4346",
        "Text": "3 Find the area of the region lying in the first quadrant and bounded by y = 4x2,\nx = 0, y = 1 and y = 4 4 Sketch the graph of y = \nx +3\n and evaluate \n0\n6\n3\n\u2212\n+\n\u222b\nx\ndx"
    },
    {
        "Chapter": "1",
        "sentence_range": "4344-4347",
        "Text": "Find the area of the region lying in the first quadrant and bounded by y = 4x2,\nx = 0, y = 1 and y = 4 4 Sketch the graph of y = \nx +3\n and evaluate \n0\n6\n3\n\u2212\n+\n\u222b\nx\ndx 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "4345-4348",
        "Text": "4 Sketch the graph of y = \nx +3\n and evaluate \n0\n6\n3\n\u2212\n+\n\u222b\nx\ndx 5 Find the area bounded by the curve y = sin x between x = 0 and x = 2\u03c0"
    },
    {
        "Chapter": "1",
        "sentence_range": "4346-4349",
        "Text": "Sketch the graph of y = \nx +3\n and evaluate \n0\n6\n3\n\u2212\n+\n\u222b\nx\ndx 5 Find the area bounded by the curve y = sin x between x = 0 and x = 2\u03c0 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "4347-4350",
        "Text": "5 Find the area bounded by the curve y = sin x between x = 0 and x = 2\u03c0 6 Find the area enclosed between the parabola y2 = 4ax and the line y = mx"
    },
    {
        "Chapter": "1",
        "sentence_range": "4348-4351",
        "Text": "Find the area bounded by the curve y = sin x between x = 0 and x = 2\u03c0 6 Find the area enclosed between the parabola y2 = 4ax and the line y = mx 7"
    },
    {
        "Chapter": "1",
        "sentence_range": "4349-4352",
        "Text": "6 Find the area enclosed between the parabola y2 = 4ax and the line y = mx 7 Find the area enclosed by the parabola 4y = 3x2 and the line 2y = 3x + 12"
    },
    {
        "Chapter": "1",
        "sentence_range": "4350-4353",
        "Text": "Find the area enclosed between the parabola y2 = 4ax and the line y = mx 7 Find the area enclosed by the parabola 4y = 3x2 and the line 2y = 3x + 12 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "4351-4354",
        "Text": "7 Find the area enclosed by the parabola 4y = 3x2 and the line 2y = 3x + 12 8 Find the area of the smaller region bounded by the ellipse \n2\n2\n1\n9\n4\nx\n+y\n=  and the\nline \n1\n3\n2\nx\n+y\n="
    },
    {
        "Chapter": "1",
        "sentence_range": "4352-4355",
        "Text": "Find the area enclosed by the parabola 4y = 3x2 and the line 2y = 3x + 12 8 Find the area of the smaller region bounded by the ellipse \n2\n2\n1\n9\n4\nx\n+y\n=  and the\nline \n1\n3\n2\nx\n+y\n= 9"
    },
    {
        "Chapter": "1",
        "sentence_range": "4353-4356",
        "Text": "8 Find the area of the smaller region bounded by the ellipse \n2\n2\n1\n9\n4\nx\n+y\n=  and the\nline \n1\n3\n2\nx\n+y\n= 9 Find the area of the smaller region bounded by the ellipse \n2\n2\n2\n2\n1\nx\ny\na\nb\n+\n=\n and the\nline \n1\nx\ny\na\nb\n+\n="
    },
    {
        "Chapter": "1",
        "sentence_range": "4354-4357",
        "Text": "Find the area of the smaller region bounded by the ellipse \n2\n2\n1\n9\n4\nx\n+y\n=  and the\nline \n1\n3\n2\nx\n+y\n= 9 Find the area of the smaller region bounded by the ellipse \n2\n2\n2\n2\n1\nx\ny\na\nb\n+\n=\n and the\nline \n1\nx\ny\na\nb\n+\n= 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "4355-4358",
        "Text": "9 Find the area of the smaller region bounded by the ellipse \n2\n2\n2\n2\n1\nx\ny\na\nb\n+\n=\n and the\nline \n1\nx\ny\na\nb\n+\n= 10 Find the area of the region enclosed by the parabola x2 = y, the line y = x + 2 and\nthe x-axis"
    },
    {
        "Chapter": "1",
        "sentence_range": "4356-4359",
        "Text": "Find the area of the smaller region bounded by the ellipse \n2\n2\n2\n2\n1\nx\ny\na\nb\n+\n=\n and the\nline \n1\nx\ny\na\nb\n+\n= 10 Find the area of the region enclosed by the parabola x2 = y, the line y = x + 2 and\nthe x-axis 11"
    },
    {
        "Chapter": "1",
        "sentence_range": "4357-4360",
        "Text": "10 Find the area of the region enclosed by the parabola x2 = y, the line y = x + 2 and\nthe x-axis 11 Using the method of integration find the area bounded by the curve \n1\nx\n+y\n="
    },
    {
        "Chapter": "1",
        "sentence_range": "4358-4361",
        "Text": "Find the area of the region enclosed by the parabola x2 = y, the line y = x + 2 and\nthe x-axis 11 Using the method of integration find the area bounded by the curve \n1\nx\n+y\n= [Hint: The required region is bounded by lines x + y = 1, x\u2013 y = 1, \u2013 x + y = 1 and\n \u2013 x \u2013 y = 1]"
    },
    {
        "Chapter": "1",
        "sentence_range": "4359-4362",
        "Text": "11 Using the method of integration find the area bounded by the curve \n1\nx\n+y\n= [Hint: The required region is bounded by lines x + y = 1, x\u2013 y = 1, \u2013 x + y = 1 and\n \u2013 x \u2013 y = 1] 376\nMATHEMATICS\n12"
    },
    {
        "Chapter": "1",
        "sentence_range": "4360-4363",
        "Text": "Using the method of integration find the area bounded by the curve \n1\nx\n+y\n= [Hint: The required region is bounded by lines x + y = 1, x\u2013 y = 1, \u2013 x + y = 1 and\n \u2013 x \u2013 y = 1] 376\nMATHEMATICS\n12 Find the area bounded by curves {(x, y) : y \u2265 x2 and y = | x|}"
    },
    {
        "Chapter": "1",
        "sentence_range": "4361-4364",
        "Text": "[Hint: The required region is bounded by lines x + y = 1, x\u2013 y = 1, \u2013 x + y = 1 and\n \u2013 x \u2013 y = 1] 376\nMATHEMATICS\n12 Find the area bounded by curves {(x, y) : y \u2265 x2 and y = | x|} 13"
    },
    {
        "Chapter": "1",
        "sentence_range": "4362-4365",
        "Text": "376\nMATHEMATICS\n12 Find the area bounded by curves {(x, y) : y \u2265 x2 and y = | x|} 13 Using the method of integration find the area of the triangle ABC, coordinates of\nwhose vertices are A(2, 0), B (4, 5) and C (6, 3)"
    },
    {
        "Chapter": "1",
        "sentence_range": "4363-4366",
        "Text": "Find the area bounded by curves {(x, y) : y \u2265 x2 and y = | x|} 13 Using the method of integration find the area of the triangle ABC, coordinates of\nwhose vertices are A(2, 0), B (4, 5) and C (6, 3) 14"
    },
    {
        "Chapter": "1",
        "sentence_range": "4364-4367",
        "Text": "13 Using the method of integration find the area of the triangle ABC, coordinates of\nwhose vertices are A(2, 0), B (4, 5) and C (6, 3) 14 Using the method of integration find the area of the region bounded by lines:\n2x + y = 4, 3x \u2013 2y = 6 and x \u2013 3y + 5 = 0\n15"
    },
    {
        "Chapter": "1",
        "sentence_range": "4365-4368",
        "Text": "Using the method of integration find the area of the triangle ABC, coordinates of\nwhose vertices are A(2, 0), B (4, 5) and C (6, 3) 14 Using the method of integration find the area of the region bounded by lines:\n2x + y = 4, 3x \u2013 2y = 6 and x \u2013 3y + 5 = 0\n15 Find the area of the region {(x, y) : y2 \u2264 4x, 4x2 + 4y2 \u2264 9}\nChoose the correct answer in the following Exercises from 16 to 20"
    },
    {
        "Chapter": "1",
        "sentence_range": "4366-4369",
        "Text": "14 Using the method of integration find the area of the region bounded by lines:\n2x + y = 4, 3x \u2013 2y = 6 and x \u2013 3y + 5 = 0\n15 Find the area of the region {(x, y) : y2 \u2264 4x, 4x2 + 4y2 \u2264 9}\nChoose the correct answer in the following Exercises from 16 to 20 16"
    },
    {
        "Chapter": "1",
        "sentence_range": "4367-4370",
        "Text": "Using the method of integration find the area of the region bounded by lines:\n2x + y = 4, 3x \u2013 2y = 6 and x \u2013 3y + 5 = 0\n15 Find the area of the region {(x, y) : y2 \u2264 4x, 4x2 + 4y2 \u2264 9}\nChoose the correct answer in the following Exercises from 16 to 20 16 Area bounded by the curve y = x3, the x-axis and the ordinates x = \u2013 2 and x = 1 is\n(A) \u2013 9\n(B)\n415\n\u2212\n(C)\n415\n(D)\n17\n4\n17"
    },
    {
        "Chapter": "1",
        "sentence_range": "4368-4371",
        "Text": "Find the area of the region {(x, y) : y2 \u2264 4x, 4x2 + 4y2 \u2264 9}\nChoose the correct answer in the following Exercises from 16 to 20 16 Area bounded by the curve y = x3, the x-axis and the ordinates x = \u2013 2 and x = 1 is\n(A) \u2013 9\n(B)\n415\n\u2212\n(C)\n415\n(D)\n17\n4\n17 The area bounded by the curve y = x | x |, x-axis and the ordinates x = \u2013 1 and\nx = 1 is given by\n(A) 0\n(B) 1\n3\n(C)\n32\n(D)\n4\n3\n[Hint : y = x2 if x > 0 and y = \u2013 x2 if x < 0]"
    },
    {
        "Chapter": "1",
        "sentence_range": "4369-4372",
        "Text": "16 Area bounded by the curve y = x3, the x-axis and the ordinates x = \u2013 2 and x = 1 is\n(A) \u2013 9\n(B)\n415\n\u2212\n(C)\n415\n(D)\n17\n4\n17 The area bounded by the curve y = x | x |, x-axis and the ordinates x = \u2013 1 and\nx = 1 is given by\n(A) 0\n(B) 1\n3\n(C)\n32\n(D)\n4\n3\n[Hint : y = x2 if x > 0 and y = \u2013 x2 if x < 0] 18"
    },
    {
        "Chapter": "1",
        "sentence_range": "4370-4373",
        "Text": "Area bounded by the curve y = x3, the x-axis and the ordinates x = \u2013 2 and x = 1 is\n(A) \u2013 9\n(B)\n415\n\u2212\n(C)\n415\n(D)\n17\n4\n17 The area bounded by the curve y = x | x |, x-axis and the ordinates x = \u2013 1 and\nx = 1 is given by\n(A) 0\n(B) 1\n3\n(C)\n32\n(D)\n4\n3\n[Hint : y = x2 if x > 0 and y = \u2013 x2 if x < 0] 18 The area of the circle x2 + y2 = 16 exterior to the parabola y2 = 6x is\n(A)\n4 (4\n3)\n3\n\u03c0 \u2212\n(B)\n4 (4\n3)\n3\n\u03c0 +\n(C)\n4 (8\n3)\n3\n\u03c0 \u2212\n(D)\n4 (8\n3)\n3\n\u03c0 +\n19"
    },
    {
        "Chapter": "1",
        "sentence_range": "4371-4374",
        "Text": "The area bounded by the curve y = x | x |, x-axis and the ordinates x = \u2013 1 and\nx = 1 is given by\n(A) 0\n(B) 1\n3\n(C)\n32\n(D)\n4\n3\n[Hint : y = x2 if x > 0 and y = \u2013 x2 if x < 0] 18 The area of the circle x2 + y2 = 16 exterior to the parabola y2 = 6x is\n(A)\n4 (4\n3)\n3\n\u03c0 \u2212\n(B)\n4 (4\n3)\n3\n\u03c0 +\n(C)\n4 (8\n3)\n3\n\u03c0 \u2212\n(D)\n4 (8\n3)\n3\n\u03c0 +\n19 The area bounded by the y-axis, y = cos x and y = sin x when 0\n2\nx\n\u03c0\n\u2264\n\u2264\n is\n(A) 2 ( 2 1)\n\u2212\n(B)\n2\n\u22121\n(C)\n2\n+1\n(D)\n2\nSummary\n\ufffd The area of the region bounded by the curve y = f (x), x-axis and the lines\nx = a and x = b (b > a) is given by the formula: Area\n( )\nb\nb\na\na\nydx\nf x dx\n=\n=\n\u222b\n\u222b"
    },
    {
        "Chapter": "1",
        "sentence_range": "4372-4375",
        "Text": "18 The area of the circle x2 + y2 = 16 exterior to the parabola y2 = 6x is\n(A)\n4 (4\n3)\n3\n\u03c0 \u2212\n(B)\n4 (4\n3)\n3\n\u03c0 +\n(C)\n4 (8\n3)\n3\n\u03c0 \u2212\n(D)\n4 (8\n3)\n3\n\u03c0 +\n19 The area bounded by the y-axis, y = cos x and y = sin x when 0\n2\nx\n\u03c0\n\u2264\n\u2264\n is\n(A) 2 ( 2 1)\n\u2212\n(B)\n2\n\u22121\n(C)\n2\n+1\n(D)\n2\nSummary\n\ufffd The area of the region bounded by the curve y = f (x), x-axis and the lines\nx = a and x = b (b > a) is given by the formula: Area\n( )\nb\nb\na\na\nydx\nf x dx\n=\n=\n\u222b\n\u222b \ufffd The area of the region bounded by the curve x = \u03c6 (y), y-axis and the lines\ny = c, y = d is given by the formula: Area\n( )\nd\nd\nc\nc\nxdy\ny dy\n=\n=\n\u03c6\n\u222b\n\u222b"
    },
    {
        "Chapter": "1",
        "sentence_range": "4373-4376",
        "Text": "The area of the circle x2 + y2 = 16 exterior to the parabola y2 = 6x is\n(A)\n4 (4\n3)\n3\n\u03c0 \u2212\n(B)\n4 (4\n3)\n3\n\u03c0 +\n(C)\n4 (8\n3)\n3\n\u03c0 \u2212\n(D)\n4 (8\n3)\n3\n\u03c0 +\n19 The area bounded by the y-axis, y = cos x and y = sin x when 0\n2\nx\n\u03c0\n\u2264\n\u2264\n is\n(A) 2 ( 2 1)\n\u2212\n(B)\n2\n\u22121\n(C)\n2\n+1\n(D)\n2\nSummary\n\ufffd The area of the region bounded by the curve y = f (x), x-axis and the lines\nx = a and x = b (b > a) is given by the formula: Area\n( )\nb\nb\na\na\nydx\nf x dx\n=\n=\n\u222b\n\u222b \ufffd The area of the region bounded by the curve x = \u03c6 (y), y-axis and the lines\ny = c, y = d is given by the formula: Area\n( )\nd\nd\nc\nc\nxdy\ny dy\n=\n=\n\u03c6\n\u222b\n\u222b APPLICATION OF INTEGRALS         377\n\ufffd The area of the region enclosed between two curves y = f (x), y = g (x) and\nthe lines x = a, x = b is given by the formula,\n[\n]\nArea\n( )\n( )\nb\na f x\ng x dx\n=\n\u2212\n\u222b\n, where, f (x) \u2265 g (x) in [a, b]\n\ufffd If f (x) \u2265 g (x) in [a, c] and f (x) \u2264  g (x) in [c, b], a < c < b, then\n[\n]\n[\n]\nArea\n( )\n( )\n( )\n( )\nc\nb\na\nc\nf x\ng x dx\ng x\nf x dx\n=\n\u2212\n+\n\u2212\n\u222b\n\u222b"
    },
    {
        "Chapter": "1",
        "sentence_range": "4374-4377",
        "Text": "The area bounded by the y-axis, y = cos x and y = sin x when 0\n2\nx\n\u03c0\n\u2264\n\u2264\n is\n(A) 2 ( 2 1)\n\u2212\n(B)\n2\n\u22121\n(C)\n2\n+1\n(D)\n2\nSummary\n\ufffd The area of the region bounded by the curve y = f (x), x-axis and the lines\nx = a and x = b (b > a) is given by the formula: Area\n( )\nb\nb\na\na\nydx\nf x dx\n=\n=\n\u222b\n\u222b \ufffd The area of the region bounded by the curve x = \u03c6 (y), y-axis and the lines\ny = c, y = d is given by the formula: Area\n( )\nd\nd\nc\nc\nxdy\ny dy\n=\n=\n\u03c6\n\u222b\n\u222b APPLICATION OF INTEGRALS         377\n\ufffd The area of the region enclosed between two curves y = f (x), y = g (x) and\nthe lines x = a, x = b is given by the formula,\n[\n]\nArea\n( )\n( )\nb\na f x\ng x dx\n=\n\u2212\n\u222b\n, where, f (x) \u2265 g (x) in [a, b]\n\ufffd If f (x) \u2265 g (x) in [a, c] and f (x) \u2264  g (x) in [c, b], a < c < b, then\n[\n]\n[\n]\nArea\n( )\n( )\n( )\n( )\nc\nb\na\nc\nf x\ng x dx\ng x\nf x dx\n=\n\u2212\n+\n\u2212\n\u222b\n\u222b Historical Note\nThe origin of the Integral Calculus goes back to the early period of development\nof Mathematics and it is related to the method of exhaustion developed by the\nmathematicians of ancient Greece"
    },
    {
        "Chapter": "1",
        "sentence_range": "4375-4378",
        "Text": "\ufffd The area of the region bounded by the curve x = \u03c6 (y), y-axis and the lines\ny = c, y = d is given by the formula: Area\n( )\nd\nd\nc\nc\nxdy\ny dy\n=\n=\n\u03c6\n\u222b\n\u222b APPLICATION OF INTEGRALS         377\n\ufffd The area of the region enclosed between two curves y = f (x), y = g (x) and\nthe lines x = a, x = b is given by the formula,\n[\n]\nArea\n( )\n( )\nb\na f x\ng x dx\n=\n\u2212\n\u222b\n, where, f (x) \u2265 g (x) in [a, b]\n\ufffd If f (x) \u2265 g (x) in [a, c] and f (x) \u2264  g (x) in [c, b], a < c < b, then\n[\n]\n[\n]\nArea\n( )\n( )\n( )\n( )\nc\nb\na\nc\nf x\ng x dx\ng x\nf x dx\n=\n\u2212\n+\n\u2212\n\u222b\n\u222b Historical Note\nThe origin of the Integral Calculus goes back to the early period of development\nof Mathematics and it is related to the method of exhaustion developed by the\nmathematicians of ancient Greece This method arose in the solution of problems\non calculating areas of plane figures, surface areas and volumes of solid bodies\netc"
    },
    {
        "Chapter": "1",
        "sentence_range": "4376-4379",
        "Text": "APPLICATION OF INTEGRALS         377\n\ufffd The area of the region enclosed between two curves y = f (x), y = g (x) and\nthe lines x = a, x = b is given by the formula,\n[\n]\nArea\n( )\n( )\nb\na f x\ng x dx\n=\n\u2212\n\u222b\n, where, f (x) \u2265 g (x) in [a, b]\n\ufffd If f (x) \u2265 g (x) in [a, c] and f (x) \u2264  g (x) in [c, b], a < c < b, then\n[\n]\n[\n]\nArea\n( )\n( )\n( )\n( )\nc\nb\na\nc\nf x\ng x dx\ng x\nf x dx\n=\n\u2212\n+\n\u2212\n\u222b\n\u222b Historical Note\nThe origin of the Integral Calculus goes back to the early period of development\nof Mathematics and it is related to the method of exhaustion developed by the\nmathematicians of ancient Greece This method arose in the solution of problems\non calculating areas of plane figures, surface areas and volumes of solid bodies\netc In this sense, the method of exhaustion can be regarded as an early method\nof integration"
    },
    {
        "Chapter": "1",
        "sentence_range": "4377-4380",
        "Text": "Historical Note\nThe origin of the Integral Calculus goes back to the early period of development\nof Mathematics and it is related to the method of exhaustion developed by the\nmathematicians of ancient Greece This method arose in the solution of problems\non calculating areas of plane figures, surface areas and volumes of solid bodies\netc In this sense, the method of exhaustion can be regarded as an early method\nof integration The greatest development of method of exhaustion in the early\nperiod was obtained in the works of Eudoxus (440 B"
    },
    {
        "Chapter": "1",
        "sentence_range": "4378-4381",
        "Text": "This method arose in the solution of problems\non calculating areas of plane figures, surface areas and volumes of solid bodies\netc In this sense, the method of exhaustion can be regarded as an early method\nof integration The greatest development of method of exhaustion in the early\nperiod was obtained in the works of Eudoxus (440 B C"
    },
    {
        "Chapter": "1",
        "sentence_range": "4379-4382",
        "Text": "In this sense, the method of exhaustion can be regarded as an early method\nof integration The greatest development of method of exhaustion in the early\nperiod was obtained in the works of Eudoxus (440 B C ) and Archimedes\n(300 B"
    },
    {
        "Chapter": "1",
        "sentence_range": "4380-4383",
        "Text": "The greatest development of method of exhaustion in the early\nperiod was obtained in the works of Eudoxus (440 B C ) and Archimedes\n(300 B C"
    },
    {
        "Chapter": "1",
        "sentence_range": "4381-4384",
        "Text": "C ) and Archimedes\n(300 B C )\nSystematic approach to the theory of Calculus began in the 17th century"
    },
    {
        "Chapter": "1",
        "sentence_range": "4382-4385",
        "Text": ") and Archimedes\n(300 B C )\nSystematic approach to the theory of Calculus began in the 17th century In 1665, Newton began his work on the Calculus described by him as the theory\nof fluxions and used his theory in finding the tangent and radius of curvature at\nany point on a curve"
    },
    {
        "Chapter": "1",
        "sentence_range": "4383-4386",
        "Text": "C )\nSystematic approach to the theory of Calculus began in the 17th century In 1665, Newton began his work on the Calculus described by him as the theory\nof fluxions and used his theory in finding the tangent and radius of curvature at\nany point on a curve Newton introduced the basic notion of inverse function\ncalled the anti derivative (indefinite integral) or the inverse method of tangents"
    },
    {
        "Chapter": "1",
        "sentence_range": "4384-4387",
        "Text": ")\nSystematic approach to the theory of Calculus began in the 17th century In 1665, Newton began his work on the Calculus described by him as the theory\nof fluxions and used his theory in finding the tangent and radius of curvature at\nany point on a curve Newton introduced the basic notion of inverse function\ncalled the anti derivative (indefinite integral) or the inverse method of tangents During 1684-86, Leibnitz published an article in the Acta Eruditorum\nwhich he called Calculas summatorius, since it was connected with the summation\nof a number of infinitely small areas, whose sum, he indicated by the symbol \u2018\u222b\u2019"
    },
    {
        "Chapter": "1",
        "sentence_range": "4385-4388",
        "Text": "In 1665, Newton began his work on the Calculus described by him as the theory\nof fluxions and used his theory in finding the tangent and radius of curvature at\nany point on a curve Newton introduced the basic notion of inverse function\ncalled the anti derivative (indefinite integral) or the inverse method of tangents During 1684-86, Leibnitz published an article in the Acta Eruditorum\nwhich he called Calculas summatorius, since it was connected with the summation\nof a number of infinitely small areas, whose sum, he indicated by the symbol \u2018\u222b\u2019 In 1696, he followed a suggestion made by J"
    },
    {
        "Chapter": "1",
        "sentence_range": "4386-4389",
        "Text": "Newton introduced the basic notion of inverse function\ncalled the anti derivative (indefinite integral) or the inverse method of tangents During 1684-86, Leibnitz published an article in the Acta Eruditorum\nwhich he called Calculas summatorius, since it was connected with the summation\nof a number of infinitely small areas, whose sum, he indicated by the symbol \u2018\u222b\u2019 In 1696, he followed a suggestion made by J Bernoulli and changed this article to\nCalculus integrali"
    },
    {
        "Chapter": "1",
        "sentence_range": "4387-4390",
        "Text": "During 1684-86, Leibnitz published an article in the Acta Eruditorum\nwhich he called Calculas summatorius, since it was connected with the summation\nof a number of infinitely small areas, whose sum, he indicated by the symbol \u2018\u222b\u2019 In 1696, he followed a suggestion made by J Bernoulli and changed this article to\nCalculus integrali This corresponded to Newton\u2019s inverse method of tangents"
    },
    {
        "Chapter": "1",
        "sentence_range": "4388-4391",
        "Text": "In 1696, he followed a suggestion made by J Bernoulli and changed this article to\nCalculus integrali This corresponded to Newton\u2019s inverse method of tangents Both Newton and Leibnitz adopted quite independent lines of approach which\nwas radically different"
    },
    {
        "Chapter": "1",
        "sentence_range": "4389-4392",
        "Text": "Bernoulli and changed this article to\nCalculus integrali This corresponded to Newton\u2019s inverse method of tangents Both Newton and Leibnitz adopted quite independent lines of approach which\nwas radically different However, respective theories accomplished results that\nwere practically identical"
    },
    {
        "Chapter": "1",
        "sentence_range": "4390-4393",
        "Text": "This corresponded to Newton\u2019s inverse method of tangents Both Newton and Leibnitz adopted quite independent lines of approach which\nwas radically different However, respective theories accomplished results that\nwere practically identical Leibnitz used the notion of definite integral and what is\nquite certain is that he first clearly appreciated tie up between the antiderivative\nand the definite integral"
    },
    {
        "Chapter": "1",
        "sentence_range": "4391-4394",
        "Text": "Both Newton and Leibnitz adopted quite independent lines of approach which\nwas radically different However, respective theories accomplished results that\nwere practically identical Leibnitz used the notion of definite integral and what is\nquite certain is that he first clearly appreciated tie up between the antiderivative\nand the definite integral Conclusively, the fundamental concepts and theory of Integral Calculus\nand primarily its relationships with Differential Calculus were developed in the\nwork of P"
    },
    {
        "Chapter": "1",
        "sentence_range": "4392-4395",
        "Text": "However, respective theories accomplished results that\nwere practically identical Leibnitz used the notion of definite integral and what is\nquite certain is that he first clearly appreciated tie up between the antiderivative\nand the definite integral Conclusively, the fundamental concepts and theory of Integral Calculus\nand primarily its relationships with Differential Calculus were developed in the\nwork of P de Fermat, I"
    },
    {
        "Chapter": "1",
        "sentence_range": "4393-4396",
        "Text": "Leibnitz used the notion of definite integral and what is\nquite certain is that he first clearly appreciated tie up between the antiderivative\nand the definite integral Conclusively, the fundamental concepts and theory of Integral Calculus\nand primarily its relationships with Differential Calculus were developed in the\nwork of P de Fermat, I Newton and G"
    },
    {
        "Chapter": "1",
        "sentence_range": "4394-4397",
        "Text": "Conclusively, the fundamental concepts and theory of Integral Calculus\nand primarily its relationships with Differential Calculus were developed in the\nwork of P de Fermat, I Newton and G Leibnitz at the end of 17th century"
    },
    {
        "Chapter": "1",
        "sentence_range": "4395-4398",
        "Text": "de Fermat, I Newton and G Leibnitz at the end of 17th century 378\nMATHEMATICS\n\u2014\ufffd\n\ufffd\n\ufffd\n\ufffd\n\ufffd\u2014\nHowever, this justification by the concept of limit was only developed in the\nworks of A"
    },
    {
        "Chapter": "1",
        "sentence_range": "4396-4399",
        "Text": "Newton and G Leibnitz at the end of 17th century 378\nMATHEMATICS\n\u2014\ufffd\n\ufffd\n\ufffd\n\ufffd\n\ufffd\u2014\nHowever, this justification by the concept of limit was only developed in the\nworks of A L"
    },
    {
        "Chapter": "1",
        "sentence_range": "4397-4400",
        "Text": "Leibnitz at the end of 17th century 378\nMATHEMATICS\n\u2014\ufffd\n\ufffd\n\ufffd\n\ufffd\n\ufffd\u2014\nHowever, this justification by the concept of limit was only developed in the\nworks of A L Cauchy in the early 19th century"
    },
    {
        "Chapter": "1",
        "sentence_range": "4398-4401",
        "Text": "378\nMATHEMATICS\n\u2014\ufffd\n\ufffd\n\ufffd\n\ufffd\n\ufffd\u2014\nHowever, this justification by the concept of limit was only developed in the\nworks of A L Cauchy in the early 19th century Lastly, it is worth mentioning the\nfollowing quotation by Lie Sophie\u2019s:\n\u201cIt may be said that the conceptions of differential quotient and integral which\nin their origin certainly go back to Archimedes were introduced in Science by the\ninvestigations of Kepler, Descartes, Cavalieri, Fermat and Wallis"
    },
    {
        "Chapter": "1",
        "sentence_range": "4399-4402",
        "Text": "L Cauchy in the early 19th century Lastly, it is worth mentioning the\nfollowing quotation by Lie Sophie\u2019s:\n\u201cIt may be said that the conceptions of differential quotient and integral which\nin their origin certainly go back to Archimedes were introduced in Science by the\ninvestigations of Kepler, Descartes, Cavalieri, Fermat and Wallis The discovery\nthat differentiation and integration are inverse operations belongs to Newton\nand Leibnitz\u201d"
    },
    {
        "Chapter": "1",
        "sentence_range": "4400-4403",
        "Text": "Cauchy in the early 19th century Lastly, it is worth mentioning the\nfollowing quotation by Lie Sophie\u2019s:\n\u201cIt may be said that the conceptions of differential quotient and integral which\nin their origin certainly go back to Archimedes were introduced in Science by the\ninvestigations of Kepler, Descartes, Cavalieri, Fermat and Wallis The discovery\nthat differentiation and integration are inverse operations belongs to Newton\nand Leibnitz\u201d DIFFERENTIAL EQUATIONS\n379\n\ufffdHe who seeks for methods without having a definite problem in mind\nseeks for the most part in vain"
    },
    {
        "Chapter": "1",
        "sentence_range": "4401-4404",
        "Text": "Lastly, it is worth mentioning the\nfollowing quotation by Lie Sophie\u2019s:\n\u201cIt may be said that the conceptions of differential quotient and integral which\nin their origin certainly go back to Archimedes were introduced in Science by the\ninvestigations of Kepler, Descartes, Cavalieri, Fermat and Wallis The discovery\nthat differentiation and integration are inverse operations belongs to Newton\nand Leibnitz\u201d DIFFERENTIAL EQUATIONS\n379\n\ufffdHe who seeks for methods without having a definite problem in mind\nseeks for the most part in vain \u2013 D"
    },
    {
        "Chapter": "1",
        "sentence_range": "4402-4405",
        "Text": "The discovery\nthat differentiation and integration are inverse operations belongs to Newton\nand Leibnitz\u201d DIFFERENTIAL EQUATIONS\n379\n\ufffdHe who seeks for methods without having a definite problem in mind\nseeks for the most part in vain \u2013 D HILBERT \ufffd\n9"
    },
    {
        "Chapter": "1",
        "sentence_range": "4403-4406",
        "Text": "DIFFERENTIAL EQUATIONS\n379\n\ufffdHe who seeks for methods without having a definite problem in mind\nseeks for the most part in vain \u2013 D HILBERT \ufffd\n9 1  Introduction\nIn Class XI and in Chapter 5 of the present book, we\ndiscussed how to differentiate a given function f with respect\nto an independent variable, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "4404-4407",
        "Text": "\u2013 D HILBERT \ufffd\n9 1  Introduction\nIn Class XI and in Chapter 5 of the present book, we\ndiscussed how to differentiate a given function f with respect\nto an independent variable, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "4405-4408",
        "Text": "HILBERT \ufffd\n9 1  Introduction\nIn Class XI and in Chapter 5 of the present book, we\ndiscussed how to differentiate a given function f with respect\nto an independent variable, i e , how to find f \u2032(x) for a given\nfunction f at each x in its domain of definition"
    },
    {
        "Chapter": "1",
        "sentence_range": "4406-4409",
        "Text": "1  Introduction\nIn Class XI and in Chapter 5 of the present book, we\ndiscussed how to differentiate a given function f with respect\nto an independent variable, i e , how to find f \u2032(x) for a given\nfunction f at each x in its domain of definition Further, in\nthe chapter on Integral Calculus, we discussed  how to find\na function f whose derivative is the function g, which may\nalso be formulated as follows:\nFor a given function g, find a function f such that\ndy\ndx  = g(x), where y = f(x)"
    },
    {
        "Chapter": "1",
        "sentence_range": "4407-4410",
        "Text": "e , how to find f \u2032(x) for a given\nfunction f at each x in its domain of definition Further, in\nthe chapter on Integral Calculus, we discussed  how to find\na function f whose derivative is the function g, which may\nalso be formulated as follows:\nFor a given function g, find a function f such that\ndy\ndx  = g(x), where y = f(x) (1)\nAn equation of the form (1) is known as a differential\nequation"
    },
    {
        "Chapter": "1",
        "sentence_range": "4408-4411",
        "Text": ", how to find f \u2032(x) for a given\nfunction f at each x in its domain of definition Further, in\nthe chapter on Integral Calculus, we discussed  how to find\na function f whose derivative is the function g, which may\nalso be formulated as follows:\nFor a given function g, find a function f such that\ndy\ndx  = g(x), where y = f(x) (1)\nAn equation of the form (1) is known as a differential\nequation A formal definition will be given later"
    },
    {
        "Chapter": "1",
        "sentence_range": "4409-4412",
        "Text": "Further, in\nthe chapter on Integral Calculus, we discussed  how to find\na function f whose derivative is the function g, which may\nalso be formulated as follows:\nFor a given function g, find a function f such that\ndy\ndx  = g(x), where y = f(x) (1)\nAn equation of the form (1) is known as a differential\nequation A formal definition will be given later These equations arise in a variety of applications, may it be in Physics, Chemistry,\nBiology, Anthropology, Geology,  Economics etc"
    },
    {
        "Chapter": "1",
        "sentence_range": "4410-4413",
        "Text": "(1)\nAn equation of the form (1) is known as a differential\nequation A formal definition will be given later These equations arise in a variety of applications, may it be in Physics, Chemistry,\nBiology, Anthropology, Geology,  Economics etc Hence, an indepth study of differential\nequations has assumed prime importance in all modern scientific investigations"
    },
    {
        "Chapter": "1",
        "sentence_range": "4411-4414",
        "Text": "A formal definition will be given later These equations arise in a variety of applications, may it be in Physics, Chemistry,\nBiology, Anthropology, Geology,  Economics etc Hence, an indepth study of differential\nequations has assumed prime importance in all modern scientific investigations In this chapter, we will study some basic concepts related to differential equation,\ngeneral and particular solutions of a differential equation, formation of differential\nequations, some methods to solve a first order - first degree differential equation and\nsome applications of differential equations in different areas"
    },
    {
        "Chapter": "1",
        "sentence_range": "4412-4415",
        "Text": "These equations arise in a variety of applications, may it be in Physics, Chemistry,\nBiology, Anthropology, Geology,  Economics etc Hence, an indepth study of differential\nequations has assumed prime importance in all modern scientific investigations In this chapter, we will study some basic concepts related to differential equation,\ngeneral and particular solutions of a differential equation, formation of differential\nequations, some methods to solve a first order - first degree differential equation and\nsome applications of differential equations in different areas 9"
    },
    {
        "Chapter": "1",
        "sentence_range": "4413-4416",
        "Text": "Hence, an indepth study of differential\nequations has assumed prime importance in all modern scientific investigations In this chapter, we will study some basic concepts related to differential equation,\ngeneral and particular solutions of a differential equation, formation of differential\nequations, some methods to solve a first order - first degree differential equation and\nsome applications of differential equations in different areas 9 2  Basic Concepts\nWe are already familiar with the equations of the type:\nx2 \u2013 3x + 3 = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "4414-4417",
        "Text": "In this chapter, we will study some basic concepts related to differential equation,\ngeneral and particular solutions of a differential equation, formation of differential\nequations, some methods to solve a first order - first degree differential equation and\nsome applications of differential equations in different areas 9 2  Basic Concepts\nWe are already familiar with the equations of the type:\nx2 \u2013 3x + 3 = 0 (1)\nsin x + cos x = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "4415-4418",
        "Text": "9 2  Basic Concepts\nWe are already familiar with the equations of the type:\nx2 \u2013 3x + 3 = 0 (1)\nsin x + cos x = 0 (2)\nx + y = 7"
    },
    {
        "Chapter": "1",
        "sentence_range": "4416-4419",
        "Text": "2  Basic Concepts\nWe are already familiar with the equations of the type:\nx2 \u2013 3x + 3 = 0 (1)\nsin x + cos x = 0 (2)\nx + y = 7 (3)\nChapter 9\nDIFFERENTIAL  EQUATIONS\nHenri Poincare\n(1854-1912 )\n\u00a9 NCERT\nnot to be republished\nMATHEMATICS\n380\nLet us consider the equation:\nxdy\ndx +y\n = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "4417-4420",
        "Text": "(1)\nsin x + cos x = 0 (2)\nx + y = 7 (3)\nChapter 9\nDIFFERENTIAL  EQUATIONS\nHenri Poincare\n(1854-1912 )\n\u00a9 NCERT\nnot to be republished\nMATHEMATICS\n380\nLet us consider the equation:\nxdy\ndx +y\n = 0 (4)\nWe see that equations (1), (2) and (3) involve independent and/or dependent variable\n(variables) only but equation (4) involves variables as well as derivative of the dependent\nvariable y with respect to the independent variable x"
    },
    {
        "Chapter": "1",
        "sentence_range": "4418-4421",
        "Text": "(2)\nx + y = 7 (3)\nChapter 9\nDIFFERENTIAL  EQUATIONS\nHenri Poincare\n(1854-1912 )\n\u00a9 NCERT\nnot to be republished\nMATHEMATICS\n380\nLet us consider the equation:\nxdy\ndx +y\n = 0 (4)\nWe see that equations (1), (2) and (3) involve independent and/or dependent variable\n(variables) only but equation (4) involves variables as well as derivative of the dependent\nvariable y with respect to the independent variable x Such an equation is called a\ndifferential equation"
    },
    {
        "Chapter": "1",
        "sentence_range": "4419-4422",
        "Text": "(3)\nChapter 9\nDIFFERENTIAL  EQUATIONS\nHenri Poincare\n(1854-1912 )\n\u00a9 NCERT\nnot to be republished\nMATHEMATICS\n380\nLet us consider the equation:\nxdy\ndx +y\n = 0 (4)\nWe see that equations (1), (2) and (3) involve independent and/or dependent variable\n(variables) only but equation (4) involves variables as well as derivative of the dependent\nvariable y with respect to the independent variable x Such an equation is called a\ndifferential equation In general, an equation involving  derivative (derivatives) of the dependent variable\nwith respect to independent variable (variables) is called a differential equation"
    },
    {
        "Chapter": "1",
        "sentence_range": "4420-4423",
        "Text": "(4)\nWe see that equations (1), (2) and (3) involve independent and/or dependent variable\n(variables) only but equation (4) involves variables as well as derivative of the dependent\nvariable y with respect to the independent variable x Such an equation is called a\ndifferential equation In general, an equation involving  derivative (derivatives) of the dependent variable\nwith respect to independent variable (variables) is called a differential equation A differential equation involving derivatives of the dependent variable with respect\nto only one independent variable is called an ordinary differential equation, e"
    },
    {
        "Chapter": "1",
        "sentence_range": "4421-4424",
        "Text": "Such an equation is called a\ndifferential equation In general, an equation involving  derivative (derivatives) of the dependent variable\nwith respect to independent variable (variables) is called a differential equation A differential equation involving derivatives of the dependent variable with respect\nto only one independent variable is called an ordinary differential equation, e g"
    },
    {
        "Chapter": "1",
        "sentence_range": "4422-4425",
        "Text": "In general, an equation involving  derivative (derivatives) of the dependent variable\nwith respect to independent variable (variables) is called a differential equation A differential equation involving derivatives of the dependent variable with respect\nto only one independent variable is called an ordinary differential equation, e g ,\n3\n2\n2 d y2\ndy\ndx\ndx\n\u239b\n\u239e\n+ \u239c\n\u239f\n\u239d\n\u23a0\n = 0  is an ordinary differential equation"
    },
    {
        "Chapter": "1",
        "sentence_range": "4423-4426",
        "Text": "A differential equation involving derivatives of the dependent variable with respect\nto only one independent variable is called an ordinary differential equation, e g ,\n3\n2\n2 d y2\ndy\ndx\ndx\n\u239b\n\u239e\n+ \u239c\n\u239f\n\u239d\n\u23a0\n = 0  is an ordinary differential equation (5)\nOf course, there are differential equations involving derivatives with respect to\nmore than one independent variables, called partial differential equations but at this\nstage we shall confine ourselves to the study of ordinary differential equations only"
    },
    {
        "Chapter": "1",
        "sentence_range": "4424-4427",
        "Text": "g ,\n3\n2\n2 d y2\ndy\ndx\ndx\n\u239b\n\u239e\n+ \u239c\n\u239f\n\u239d\n\u23a0\n = 0  is an ordinary differential equation (5)\nOf course, there are differential equations involving derivatives with respect to\nmore than one independent variables, called partial differential equations but at this\nstage we shall confine ourselves to the study of ordinary differential equations only Now onward, we will use the term \u2018differential equation\u2019 for \u2018ordinary differential\nequation\u2019"
    },
    {
        "Chapter": "1",
        "sentence_range": "4425-4428",
        "Text": ",\n3\n2\n2 d y2\ndy\ndx\ndx\n\u239b\n\u239e\n+ \u239c\n\u239f\n\u239d\n\u23a0\n = 0  is an ordinary differential equation (5)\nOf course, there are differential equations involving derivatives with respect to\nmore than one independent variables, called partial differential equations but at this\nstage we shall confine ourselves to the study of ordinary differential equations only Now onward, we will use the term \u2018differential equation\u2019 for \u2018ordinary differential\nequation\u2019 \ufffdNote\n1"
    },
    {
        "Chapter": "1",
        "sentence_range": "4426-4429",
        "Text": "(5)\nOf course, there are differential equations involving derivatives with respect to\nmore than one independent variables, called partial differential equations but at this\nstage we shall confine ourselves to the study of ordinary differential equations only Now onward, we will use the term \u2018differential equation\u2019 for \u2018ordinary differential\nequation\u2019 \ufffdNote\n1 We shall prefer to use the following notations for derivatives:\n2\n3\n2\n3\n,\n,\ndy\nd y\nd y\ny\ny\ny\ndx\ndx\ndx\n\u2032\n\u2032\u2032\n\u2032\u2032\u2032\n=\n=\n=\n2"
    },
    {
        "Chapter": "1",
        "sentence_range": "4427-4430",
        "Text": "Now onward, we will use the term \u2018differential equation\u2019 for \u2018ordinary differential\nequation\u2019 \ufffdNote\n1 We shall prefer to use the following notations for derivatives:\n2\n3\n2\n3\n,\n,\ndy\nd y\nd y\ny\ny\ny\ndx\ndx\ndx\n\u2032\n\u2032\u2032\n\u2032\u2032\u2032\n=\n=\n=\n2 For derivatives of higher order, it will be inconvenient  to use so many dashes\nas supersuffix therefore, we use the notation yn for nth order derivative \nn\nd yn\ndx"
    },
    {
        "Chapter": "1",
        "sentence_range": "4428-4431",
        "Text": "\ufffdNote\n1 We shall prefer to use the following notations for derivatives:\n2\n3\n2\n3\n,\n,\ndy\nd y\nd y\ny\ny\ny\ndx\ndx\ndx\n\u2032\n\u2032\u2032\n\u2032\u2032\u2032\n=\n=\n=\n2 For derivatives of higher order, it will be inconvenient  to use so many dashes\nas supersuffix therefore, we use the notation yn for nth order derivative \nn\nd yn\ndx 9"
    },
    {
        "Chapter": "1",
        "sentence_range": "4429-4432",
        "Text": "We shall prefer to use the following notations for derivatives:\n2\n3\n2\n3\n,\n,\ndy\nd y\nd y\ny\ny\ny\ndx\ndx\ndx\n\u2032\n\u2032\u2032\n\u2032\u2032\u2032\n=\n=\n=\n2 For derivatives of higher order, it will be inconvenient  to use so many dashes\nas supersuffix therefore, we use the notation yn for nth order derivative \nn\nd yn\ndx 9 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "4430-4433",
        "Text": "For derivatives of higher order, it will be inconvenient  to use so many dashes\nas supersuffix therefore, we use the notation yn for nth order derivative \nn\nd yn\ndx 9 2 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "4431-4434",
        "Text": "9 2 1 Order of a differential equation\nOrder of a differential equation is defined as the order of the highest order derivative of\nthe dependent variable with respect to the independent variable involved in the given\ndifferential equation"
    },
    {
        "Chapter": "1",
        "sentence_range": "4432-4435",
        "Text": "2 1 Order of a differential equation\nOrder of a differential equation is defined as the order of the highest order derivative of\nthe dependent variable with respect to the independent variable involved in the given\ndifferential equation Consider the following differential equations:\ndy\ndx  = ex"
    },
    {
        "Chapter": "1",
        "sentence_range": "4433-4436",
        "Text": "1 Order of a differential equation\nOrder of a differential equation is defined as the order of the highest order derivative of\nthe dependent variable with respect to the independent variable involved in the given\ndifferential equation Consider the following differential equations:\ndy\ndx  = ex (6)\n\u00a9 NCERT\nnot to be republished\nDIFFERENTIAL EQUATIONS\n381\n2\nd y2\ny\ndx\n+\n = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "4434-4437",
        "Text": "Order of a differential equation\nOrder of a differential equation is defined as the order of the highest order derivative of\nthe dependent variable with respect to the independent variable involved in the given\ndifferential equation Consider the following differential equations:\ndy\ndx  = ex (6)\n\u00a9 NCERT\nnot to be republished\nDIFFERENTIAL EQUATIONS\n381\n2\nd y2\ny\ndx\n+\n = 0 (7)\n3\n3\n2\n2\n3\n2\nd y\nd y\nx\ndx\ndx\n\u239b\n\u239e\n\u239b\n\u239e\n+\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\n= 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "4435-4438",
        "Text": "Consider the following differential equations:\ndy\ndx  = ex (6)\n\u00a9 NCERT\nnot to be republished\nDIFFERENTIAL EQUATIONS\n381\n2\nd y2\ny\ndx\n+\n = 0 (7)\n3\n3\n2\n2\n3\n2\nd y\nd y\nx\ndx\ndx\n\u239b\n\u239e\n\u239b\n\u239e\n+\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\n= 0 (8)\nThe equations (6), (7) and (8) involve the highest derivative of first, second and\nthird order respectively"
    },
    {
        "Chapter": "1",
        "sentence_range": "4436-4439",
        "Text": "(6)\n\u00a9 NCERT\nnot to be republished\nDIFFERENTIAL EQUATIONS\n381\n2\nd y2\ny\ndx\n+\n = 0 (7)\n3\n3\n2\n2\n3\n2\nd y\nd y\nx\ndx\ndx\n\u239b\n\u239e\n\u239b\n\u239e\n+\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\n= 0 (8)\nThe equations (6), (7) and (8) involve the highest derivative of first, second and\nthird order respectively Therefore, the order of these equations are 1, 2 and 3 respectively"
    },
    {
        "Chapter": "1",
        "sentence_range": "4437-4440",
        "Text": "(7)\n3\n3\n2\n2\n3\n2\nd y\nd y\nx\ndx\ndx\n\u239b\n\u239e\n\u239b\n\u239e\n+\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\n= 0 (8)\nThe equations (6), (7) and (8) involve the highest derivative of first, second and\nthird order respectively Therefore, the order of these equations are 1, 2 and 3 respectively 9"
    },
    {
        "Chapter": "1",
        "sentence_range": "4438-4441",
        "Text": "(8)\nThe equations (6), (7) and (8) involve the highest derivative of first, second and\nthird order respectively Therefore, the order of these equations are 1, 2 and 3 respectively 9 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "4439-4442",
        "Text": "Therefore, the order of these equations are 1, 2 and 3 respectively 9 2 2  Degree of a differential equation\nTo study the degree of a differential equation, the key point is that the differential\nequation must be a polynomial equation in derivatives, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "4440-4443",
        "Text": "9 2 2  Degree of a differential equation\nTo study the degree of a differential equation, the key point is that the differential\nequation must be a polynomial equation in derivatives, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "4441-4444",
        "Text": "2 2  Degree of a differential equation\nTo study the degree of a differential equation, the key point is that the differential\nequation must be a polynomial equation in derivatives, i e , y\u2032, y\u2033, y\u2033\u2032 etc"
    },
    {
        "Chapter": "1",
        "sentence_range": "4442-4445",
        "Text": "2  Degree of a differential equation\nTo study the degree of a differential equation, the key point is that the differential\nequation must be a polynomial equation in derivatives, i e , y\u2032, y\u2033, y\u2033\u2032 etc Consider the\nfollowing differential equations:\n2\n3\n2\n3\n2\n2\nd y\nd y\ndy\ny\ndx\ndx\ndx\n\u239b\n\u239e\n+\n\u2212\n+\n\u239c\n\u239f\n\u239d\n\u23a0\n = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "4443-4446",
        "Text": "e , y\u2032, y\u2033, y\u2033\u2032 etc Consider the\nfollowing differential equations:\n2\n3\n2\n3\n2\n2\nd y\nd y\ndy\ny\ndx\ndx\ndx\n\u239b\n\u239e\n+\n\u2212\n+\n\u239c\n\u239f\n\u239d\n\u23a0\n = 0 (9)\n2\nsin2\ndy\ndy\ny\ndx\ndx\n\u239b\n\u239e\n\u239b\n\u239e\n+\n\u2212\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\n = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "4444-4447",
        "Text": ", y\u2032, y\u2033, y\u2033\u2032 etc Consider the\nfollowing differential equations:\n2\n3\n2\n3\n2\n2\nd y\nd y\ndy\ny\ndx\ndx\ndx\n\u239b\n\u239e\n+\n\u2212\n+\n\u239c\n\u239f\n\u239d\n\u23a0\n = 0 (9)\n2\nsin2\ndy\ndy\ny\ndx\ndx\n\u239b\n\u239e\n\u239b\n\u239e\n+\n\u2212\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\n = 0 (10)\nsin\ndy\ndy\ndx\n\u239bdx\n\u239e\n+\n\u239c\n\u239f\n\u239d\n\u23a0  = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "4445-4448",
        "Text": "Consider the\nfollowing differential equations:\n2\n3\n2\n3\n2\n2\nd y\nd y\ndy\ny\ndx\ndx\ndx\n\u239b\n\u239e\n+\n\u2212\n+\n\u239c\n\u239f\n\u239d\n\u23a0\n = 0 (9)\n2\nsin2\ndy\ndy\ny\ndx\ndx\n\u239b\n\u239e\n\u239b\n\u239e\n+\n\u2212\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\n = 0 (10)\nsin\ndy\ndy\ndx\n\u239bdx\n\u239e\n+\n\u239c\n\u239f\n\u239d\n\u23a0  = 0 (11)\nWe observe that equation (9) is a polynomial equation in y\u2033\u2032,  y\u2033 and y\u2032, equation (10)\nis a polynomial equation in y\u2032 (not a polynomial in y though)"
    },
    {
        "Chapter": "1",
        "sentence_range": "4446-4449",
        "Text": "(9)\n2\nsin2\ndy\ndy\ny\ndx\ndx\n\u239b\n\u239e\n\u239b\n\u239e\n+\n\u2212\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\n = 0 (10)\nsin\ndy\ndy\ndx\n\u239bdx\n\u239e\n+\n\u239c\n\u239f\n\u239d\n\u23a0  = 0 (11)\nWe observe that equation (9) is a polynomial equation in y\u2033\u2032,  y\u2033 and y\u2032, equation (10)\nis a polynomial equation in y\u2032 (not a polynomial in y though) Degree of such differential\nequations can be defined"
    },
    {
        "Chapter": "1",
        "sentence_range": "4447-4450",
        "Text": "(10)\nsin\ndy\ndy\ndx\n\u239bdx\n\u239e\n+\n\u239c\n\u239f\n\u239d\n\u23a0  = 0 (11)\nWe observe that equation (9) is a polynomial equation in y\u2033\u2032,  y\u2033 and y\u2032, equation (10)\nis a polynomial equation in y\u2032 (not a polynomial in y though) Degree of such differential\nequations can be defined But equation (11) is not a polynomial equation in y\u2032 and\ndegree of such a differential equation can not be defined"
    },
    {
        "Chapter": "1",
        "sentence_range": "4448-4451",
        "Text": "(11)\nWe observe that equation (9) is a polynomial equation in y\u2033\u2032,  y\u2033 and y\u2032, equation (10)\nis a polynomial equation in y\u2032 (not a polynomial in y though) Degree of such differential\nequations can be defined But equation (11) is not a polynomial equation in y\u2032 and\ndegree of such a differential equation can not be defined By the degree of a differential equation, when it is a polynomial equation in\nderivatives, we mean the highest power (positive integral index) of the highest order\nderivative involved in the given differential equation"
    },
    {
        "Chapter": "1",
        "sentence_range": "4449-4452",
        "Text": "Degree of such differential\nequations can be defined But equation (11) is not a polynomial equation in y\u2032 and\ndegree of such a differential equation can not be defined By the degree of a differential equation, when it is a polynomial equation in\nderivatives, we mean the highest power (positive integral index) of the highest order\nderivative involved in the given differential equation In view of the above definition, one may observe that differential equations (6), (7),\n(8) and (9) each are of degree one, equation (10) is of degree two while the degree of\ndifferential equation (11) is not defined"
    },
    {
        "Chapter": "1",
        "sentence_range": "4450-4453",
        "Text": "But equation (11) is not a polynomial equation in y\u2032 and\ndegree of such a differential equation can not be defined By the degree of a differential equation, when it is a polynomial equation in\nderivatives, we mean the highest power (positive integral index) of the highest order\nderivative involved in the given differential equation In view of the above definition, one may observe that differential equations (6), (7),\n(8) and (9) each are of degree one, equation (10) is of degree two while the degree of\ndifferential equation (11) is not defined \ufffdNote   Order and degree (if defined) of a differential equation are always\npositive integers"
    },
    {
        "Chapter": "1",
        "sentence_range": "4451-4454",
        "Text": "By the degree of a differential equation, when it is a polynomial equation in\nderivatives, we mean the highest power (positive integral index) of the highest order\nderivative involved in the given differential equation In view of the above definition, one may observe that differential equations (6), (7),\n(8) and (9) each are of degree one, equation (10) is of degree two while the degree of\ndifferential equation (11) is not defined \ufffdNote   Order and degree (if defined) of a differential equation are always\npositive integers \u00a9 NCERT\nnot to be republished\nMATHEMATICS\n382\nExample 1 Find the order and degree, if defined, of each of the following differential\nequations:\n(i)\ncos\n0\ndy\nx\ndx \u2212\n=\n(ii)  \n2\n2\n2\n0\nd y\ndy\ndy\nxy\nx\ny\ndx\ndx\ndx\n\u239b\n\u239e\n+\n\u2212\n=\n\u239c\n\u239f\n\u239d\n\u23a0\n(iii)\n2\n0\ny\ny\ny\ne \u2032\n\u2032\u2032\u2032 +\n+\n=\nSolution\n(i)\nThe highest order derivative present in the differential equation is dy\ndx , so its\norder is one"
    },
    {
        "Chapter": "1",
        "sentence_range": "4452-4455",
        "Text": "In view of the above definition, one may observe that differential equations (6), (7),\n(8) and (9) each are of degree one, equation (10) is of degree two while the degree of\ndifferential equation (11) is not defined \ufffdNote   Order and degree (if defined) of a differential equation are always\npositive integers \u00a9 NCERT\nnot to be republished\nMATHEMATICS\n382\nExample 1 Find the order and degree, if defined, of each of the following differential\nequations:\n(i)\ncos\n0\ndy\nx\ndx \u2212\n=\n(ii)  \n2\n2\n2\n0\nd y\ndy\ndy\nxy\nx\ny\ndx\ndx\ndx\n\u239b\n\u239e\n+\n\u2212\n=\n\u239c\n\u239f\n\u239d\n\u23a0\n(iii)\n2\n0\ny\ny\ny\ne \u2032\n\u2032\u2032\u2032 +\n+\n=\nSolution\n(i)\nThe highest order derivative present in the differential equation is dy\ndx , so its\norder is one It is a polynomial equation in y\u2032 and the highest power raised to dy\ndx\nis one, so its degree is one"
    },
    {
        "Chapter": "1",
        "sentence_range": "4453-4456",
        "Text": "\ufffdNote   Order and degree (if defined) of a differential equation are always\npositive integers \u00a9 NCERT\nnot to be republished\nMATHEMATICS\n382\nExample 1 Find the order and degree, if defined, of each of the following differential\nequations:\n(i)\ncos\n0\ndy\nx\ndx \u2212\n=\n(ii)  \n2\n2\n2\n0\nd y\ndy\ndy\nxy\nx\ny\ndx\ndx\ndx\n\u239b\n\u239e\n+\n\u2212\n=\n\u239c\n\u239f\n\u239d\n\u23a0\n(iii)\n2\n0\ny\ny\ny\ne \u2032\n\u2032\u2032\u2032 +\n+\n=\nSolution\n(i)\nThe highest order derivative present in the differential equation is dy\ndx , so its\norder is one It is a polynomial equation in y\u2032 and the highest power raised to dy\ndx\nis one, so its degree is one (ii)\nThe highest order derivative present in the given differential equation is \n2\nd y2\ndx\n, so\nits order is two"
    },
    {
        "Chapter": "1",
        "sentence_range": "4454-4457",
        "Text": "\u00a9 NCERT\nnot to be republished\nMATHEMATICS\n382\nExample 1 Find the order and degree, if defined, of each of the following differential\nequations:\n(i)\ncos\n0\ndy\nx\ndx \u2212\n=\n(ii)  \n2\n2\n2\n0\nd y\ndy\ndy\nxy\nx\ny\ndx\ndx\ndx\n\u239b\n\u239e\n+\n\u2212\n=\n\u239c\n\u239f\n\u239d\n\u23a0\n(iii)\n2\n0\ny\ny\ny\ne \u2032\n\u2032\u2032\u2032 +\n+\n=\nSolution\n(i)\nThe highest order derivative present in the differential equation is dy\ndx , so its\norder is one It is a polynomial equation in y\u2032 and the highest power raised to dy\ndx\nis one, so its degree is one (ii)\nThe highest order derivative present in the given differential equation is \n2\nd y2\ndx\n, so\nits order is two It is a polynomial equation in \n2\nd y2\ndx\n and dy\ndx  and the highest\npower raised to \n2\nd y2\ndx\n is one, so its degree is one"
    },
    {
        "Chapter": "1",
        "sentence_range": "4455-4458",
        "Text": "It is a polynomial equation in y\u2032 and the highest power raised to dy\ndx\nis one, so its degree is one (ii)\nThe highest order derivative present in the given differential equation is \n2\nd y2\ndx\n, so\nits order is two It is a polynomial equation in \n2\nd y2\ndx\n and dy\ndx  and the highest\npower raised to \n2\nd y2\ndx\n is one, so its degree is one (iii)\nThe highest order derivative present in the differential equation is y\u2032\u2032\u2032, so its\norder is three"
    },
    {
        "Chapter": "1",
        "sentence_range": "4456-4459",
        "Text": "(ii)\nThe highest order derivative present in the given differential equation is \n2\nd y2\ndx\n, so\nits order is two It is a polynomial equation in \n2\nd y2\ndx\n and dy\ndx  and the highest\npower raised to \n2\nd y2\ndx\n is one, so its degree is one (iii)\nThe highest order derivative present in the differential equation is y\u2032\u2032\u2032, so its\norder is three The given differential equation is not a polynomial equation in its\nderivatives and so its degree is not defined"
    },
    {
        "Chapter": "1",
        "sentence_range": "4457-4460",
        "Text": "It is a polynomial equation in \n2\nd y2\ndx\n and dy\ndx  and the highest\npower raised to \n2\nd y2\ndx\n is one, so its degree is one (iii)\nThe highest order derivative present in the differential equation is y\u2032\u2032\u2032, so its\norder is three The given differential equation is not a polynomial equation in its\nderivatives and so its degree is not defined EXERCISE 9"
    },
    {
        "Chapter": "1",
        "sentence_range": "4458-4461",
        "Text": "(iii)\nThe highest order derivative present in the differential equation is y\u2032\u2032\u2032, so its\norder is three The given differential equation is not a polynomial equation in its\nderivatives and so its degree is not defined EXERCISE 9 1\nDetermine order and degree (if defined) of differential equations given in Exercises\n1 to 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "4459-4462",
        "Text": "The given differential equation is not a polynomial equation in its\nderivatives and so its degree is not defined EXERCISE 9 1\nDetermine order and degree (if defined) of differential equations given in Exercises\n1 to 10 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "4460-4463",
        "Text": "EXERCISE 9 1\nDetermine order and degree (if defined) of differential equations given in Exercises\n1 to 10 1 4\n4\nsin(\n)\n0\nd y\ny\ndx\n\u2032\u2032\u2032\n+\n=\n2"
    },
    {
        "Chapter": "1",
        "sentence_range": "4461-4464",
        "Text": "1\nDetermine order and degree (if defined) of differential equations given in Exercises\n1 to 10 1 4\n4\nsin(\n)\n0\nd y\ny\ndx\n\u2032\u2032\u2032\n+\n=\n2 y\u2032 + 5y = 0\n3"
    },
    {
        "Chapter": "1",
        "sentence_range": "4462-4465",
        "Text": "1 4\n4\nsin(\n)\n0\nd y\ny\ndx\n\u2032\u2032\u2032\n+\n=\n2 y\u2032 + 5y = 0\n3 4\n2\n2\n3\n0\nds\nsd s\ndt\ndt\n\u239b\n\u239e +\n=\n\u239c\n\u239f\n\u239d\n\u23a0\n4"
    },
    {
        "Chapter": "1",
        "sentence_range": "4463-4466",
        "Text": "4\n4\nsin(\n)\n0\nd y\ny\ndx\n\u2032\u2032\u2032\n+\n=\n2 y\u2032 + 5y = 0\n3 4\n2\n2\n3\n0\nds\nsd s\ndt\ndt\n\u239b\n\u239e +\n=\n\u239c\n\u239f\n\u239d\n\u23a0\n4 2\n2\n2\ncos\n0\nd y\ndy\ndx\ndx\n\u239b\n\u239e\n\u239b\n\u239e\n+\n=\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\n5"
    },
    {
        "Chapter": "1",
        "sentence_range": "4464-4467",
        "Text": "y\u2032 + 5y = 0\n3 4\n2\n2\n3\n0\nds\nsd s\ndt\ndt\n\u239b\n\u239e +\n=\n\u239c\n\u239f\n\u239d\n\u23a0\n4 2\n2\n2\ncos\n0\nd y\ndy\ndx\ndx\n\u239b\n\u239e\n\u239b\n\u239e\n+\n=\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\n5 2\n2\ncos3\nsin3\nd y\nx\nx\ndx\n=\n+\n6"
    },
    {
        "Chapter": "1",
        "sentence_range": "4465-4468",
        "Text": "4\n2\n2\n3\n0\nds\nsd s\ndt\ndt\n\u239b\n\u239e +\n=\n\u239c\n\u239f\n\u239d\n\u23a0\n4 2\n2\n2\ncos\n0\nd y\ndy\ndx\ndx\n\u239b\n\u239e\n\u239b\n\u239e\n+\n=\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\n5 2\n2\ncos3\nsin3\nd y\nx\nx\ndx\n=\n+\n6 2\n(\ny\u2032\u2032\u2032)\n + (y\u2033)3 + (y\u2032)4 + y5 = 0\n7"
    },
    {
        "Chapter": "1",
        "sentence_range": "4466-4469",
        "Text": "2\n2\n2\ncos\n0\nd y\ndy\ndx\ndx\n\u239b\n\u239e\n\u239b\n\u239e\n+\n=\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\n5 2\n2\ncos3\nsin3\nd y\nx\nx\ndx\n=\n+\n6 2\n(\ny\u2032\u2032\u2032)\n + (y\u2033)3 + (y\u2032)4 + y5 = 0\n7 y\u2032\u2032\u2032 + 2y\u2033 + y\u2032 = 0\n\u00a9 NCERT\nnot to be republished\nDIFFERENTIAL EQUATIONS\n383\n8"
    },
    {
        "Chapter": "1",
        "sentence_range": "4467-4470",
        "Text": "2\n2\ncos3\nsin3\nd y\nx\nx\ndx\n=\n+\n6 2\n(\ny\u2032\u2032\u2032)\n + (y\u2033)3 + (y\u2032)4 + y5 = 0\n7 y\u2032\u2032\u2032 + 2y\u2033 + y\u2032 = 0\n\u00a9 NCERT\nnot to be republished\nDIFFERENTIAL EQUATIONS\n383\n8 y\u2032 + y = ex\n9"
    },
    {
        "Chapter": "1",
        "sentence_range": "4468-4471",
        "Text": "2\n(\ny\u2032\u2032\u2032)\n + (y\u2033)3 + (y\u2032)4 + y5 = 0\n7 y\u2032\u2032\u2032 + 2y\u2033 + y\u2032 = 0\n\u00a9 NCERT\nnot to be republished\nDIFFERENTIAL EQUATIONS\n383\n8 y\u2032 + y = ex\n9 y\u2033 + (y\u2032)2 + 2y = 0 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "4469-4472",
        "Text": "y\u2032\u2032\u2032 + 2y\u2033 + y\u2032 = 0\n\u00a9 NCERT\nnot to be republished\nDIFFERENTIAL EQUATIONS\n383\n8 y\u2032 + y = ex\n9 y\u2033 + (y\u2032)2 + 2y = 0 10 y\u2033 + 2y\u2032 + sin y = 0\n11"
    },
    {
        "Chapter": "1",
        "sentence_range": "4470-4473",
        "Text": "y\u2032 + y = ex\n9 y\u2033 + (y\u2032)2 + 2y = 0 10 y\u2033 + 2y\u2032 + sin y = 0\n11 The degree of the differential equation\n3\n2\n2\n2\nsin\n1\n0\nd y\ndy\ndy\ndx\ndx\ndx\n\u239b\n\u239e\n\u239b\n\u239e\n\u239b\n\u239e\n+\n+\n+ =\n\u239c\n\u239f\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\n\u239d\n\u23a0\n is\n(A) 3\n(B) 2\n(C) 1\n(D) not defined\n12"
    },
    {
        "Chapter": "1",
        "sentence_range": "4471-4474",
        "Text": "y\u2033 + (y\u2032)2 + 2y = 0 10 y\u2033 + 2y\u2032 + sin y = 0\n11 The degree of the differential equation\n3\n2\n2\n2\nsin\n1\n0\nd y\ndy\ndy\ndx\ndx\ndx\n\u239b\n\u239e\n\u239b\n\u239e\n\u239b\n\u239e\n+\n+\n+ =\n\u239c\n\u239f\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\n\u239d\n\u23a0\n is\n(A) 3\n(B) 2\n(C) 1\n(D) not defined\n12 The order of the differential equation\n2\n2\n2\n2\n3\n0\nd y\ndy\nx\ny\ndx\ndx\n\u2212\n+\n=\n is\n(A) 2\n(B) 1\n(C) 0\n(D) not defined\n9"
    },
    {
        "Chapter": "1",
        "sentence_range": "4472-4475",
        "Text": "y\u2033 + 2y\u2032 + sin y = 0\n11 The degree of the differential equation\n3\n2\n2\n2\nsin\n1\n0\nd y\ndy\ndy\ndx\ndx\ndx\n\u239b\n\u239e\n\u239b\n\u239e\n\u239b\n\u239e\n+\n+\n+ =\n\u239c\n\u239f\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\n\u239d\n\u23a0\n is\n(A) 3\n(B) 2\n(C) 1\n(D) not defined\n12 The order of the differential equation\n2\n2\n2\n2\n3\n0\nd y\ndy\nx\ny\ndx\ndx\n\u2212\n+\n=\n is\n(A) 2\n(B) 1\n(C) 0\n(D) not defined\n9 3"
    },
    {
        "Chapter": "1",
        "sentence_range": "4473-4476",
        "Text": "The degree of the differential equation\n3\n2\n2\n2\nsin\n1\n0\nd y\ndy\ndy\ndx\ndx\ndx\n\u239b\n\u239e\n\u239b\n\u239e\n\u239b\n\u239e\n+\n+\n+ =\n\u239c\n\u239f\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\n\u239d\n\u23a0\n is\n(A) 3\n(B) 2\n(C) 1\n(D) not defined\n12 The order of the differential equation\n2\n2\n2\n2\n3\n0\nd y\ndy\nx\ny\ndx\ndx\n\u2212\n+\n=\n is\n(A) 2\n(B) 1\n(C) 0\n(D) not defined\n9 3 General and Particular Solutions of a Differential Equation\nIn earlier Classes, we have solved the equations of the type:\nx2 + 1 = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "4474-4477",
        "Text": "The order of the differential equation\n2\n2\n2\n2\n3\n0\nd y\ndy\nx\ny\ndx\ndx\n\u2212\n+\n=\n is\n(A) 2\n(B) 1\n(C) 0\n(D) not defined\n9 3 General and Particular Solutions of a Differential Equation\nIn earlier Classes, we have solved the equations of the type:\nx2 + 1 = 0 (1)\nsin2 x \u2013 cos x = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "4475-4478",
        "Text": "3 General and Particular Solutions of a Differential Equation\nIn earlier Classes, we have solved the equations of the type:\nx2 + 1 = 0 (1)\nsin2 x \u2013 cos x = 0 (2)\nSolution of equations (1) and (2) are numbers, real or complex, that will satisfy the\ngiven equation i"
    },
    {
        "Chapter": "1",
        "sentence_range": "4476-4479",
        "Text": "General and Particular Solutions of a Differential Equation\nIn earlier Classes, we have solved the equations of the type:\nx2 + 1 = 0 (1)\nsin2 x \u2013 cos x = 0 (2)\nSolution of equations (1) and (2) are numbers, real or complex, that will satisfy the\ngiven equation i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "4477-4480",
        "Text": "(1)\nsin2 x \u2013 cos x = 0 (2)\nSolution of equations (1) and (2) are numbers, real or complex, that will satisfy the\ngiven equation i e , when that number is substituted for the unknown x in the given\nequation, L"
    },
    {
        "Chapter": "1",
        "sentence_range": "4478-4481",
        "Text": "(2)\nSolution of equations (1) and (2) are numbers, real or complex, that will satisfy the\ngiven equation i e , when that number is substituted for the unknown x in the given\nequation, L H"
    },
    {
        "Chapter": "1",
        "sentence_range": "4479-4482",
        "Text": "e , when that number is substituted for the unknown x in the given\nequation, L H S"
    },
    {
        "Chapter": "1",
        "sentence_range": "4480-4483",
        "Text": ", when that number is substituted for the unknown x in the given\nequation, L H S becomes equal to the R"
    },
    {
        "Chapter": "1",
        "sentence_range": "4481-4484",
        "Text": "H S becomes equal to the R H"
    },
    {
        "Chapter": "1",
        "sentence_range": "4482-4485",
        "Text": "S becomes equal to the R H S"
    },
    {
        "Chapter": "1",
        "sentence_range": "4483-4486",
        "Text": "becomes equal to the R H S Now consider the differential equation \n2\n2\n0\nd y\ny\ndx\n+\n="
    },
    {
        "Chapter": "1",
        "sentence_range": "4484-4487",
        "Text": "H S Now consider the differential equation \n2\n2\n0\nd y\ny\ndx\n+\n= (3)\nIn contrast to the first two equations, the solution of this differential equation is a\nfunction \u03c6 that will satisfy it i"
    },
    {
        "Chapter": "1",
        "sentence_range": "4485-4488",
        "Text": "S Now consider the differential equation \n2\n2\n0\nd y\ny\ndx\n+\n= (3)\nIn contrast to the first two equations, the solution of this differential equation is a\nfunction \u03c6 that will satisfy it i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "4486-4489",
        "Text": "Now consider the differential equation \n2\n2\n0\nd y\ny\ndx\n+\n= (3)\nIn contrast to the first two equations, the solution of this differential equation is a\nfunction \u03c6 that will satisfy it i e , when the function \u03c6 is substituted for the unknown y\n(dependent variable) in the given differential equation, L"
    },
    {
        "Chapter": "1",
        "sentence_range": "4487-4490",
        "Text": "(3)\nIn contrast to the first two equations, the solution of this differential equation is a\nfunction \u03c6 that will satisfy it i e , when the function \u03c6 is substituted for the unknown y\n(dependent variable) in the given differential equation, L H"
    },
    {
        "Chapter": "1",
        "sentence_range": "4488-4491",
        "Text": "e , when the function \u03c6 is substituted for the unknown y\n(dependent variable) in the given differential equation, L H S"
    },
    {
        "Chapter": "1",
        "sentence_range": "4489-4492",
        "Text": ", when the function \u03c6 is substituted for the unknown y\n(dependent variable) in the given differential equation, L H S becomes equal to R"
    },
    {
        "Chapter": "1",
        "sentence_range": "4490-4493",
        "Text": "H S becomes equal to R H"
    },
    {
        "Chapter": "1",
        "sentence_range": "4491-4494",
        "Text": "S becomes equal to R H S"
    },
    {
        "Chapter": "1",
        "sentence_range": "4492-4495",
        "Text": "becomes equal to R H S The curve y = \u03c6 (x) is called the solution curve (integral curve) of the given\ndifferential equation"
    },
    {
        "Chapter": "1",
        "sentence_range": "4493-4496",
        "Text": "H S The curve y = \u03c6 (x) is called the solution curve (integral curve) of the given\ndifferential equation Consider the function given by\ny = \u03c6 (x) = a sin (x + b),"
    },
    {
        "Chapter": "1",
        "sentence_range": "4494-4497",
        "Text": "S The curve y = \u03c6 (x) is called the solution curve (integral curve) of the given\ndifferential equation Consider the function given by\ny = \u03c6 (x) = a sin (x + b), (4)\nwhere a, b \u2208 R"
    },
    {
        "Chapter": "1",
        "sentence_range": "4495-4498",
        "Text": "The curve y = \u03c6 (x) is called the solution curve (integral curve) of the given\ndifferential equation Consider the function given by\ny = \u03c6 (x) = a sin (x + b), (4)\nwhere a, b \u2208 R When this function and its derivative are substituted in equation (3),\nL"
    },
    {
        "Chapter": "1",
        "sentence_range": "4496-4499",
        "Text": "Consider the function given by\ny = \u03c6 (x) = a sin (x + b), (4)\nwhere a, b \u2208 R When this function and its derivative are substituted in equation (3),\nL H"
    },
    {
        "Chapter": "1",
        "sentence_range": "4497-4500",
        "Text": "(4)\nwhere a, b \u2208 R When this function and its derivative are substituted in equation (3),\nL H S"
    },
    {
        "Chapter": "1",
        "sentence_range": "4498-4501",
        "Text": "When this function and its derivative are substituted in equation (3),\nL H S = R"
    },
    {
        "Chapter": "1",
        "sentence_range": "4499-4502",
        "Text": "H S = R H"
    },
    {
        "Chapter": "1",
        "sentence_range": "4500-4503",
        "Text": "S = R H S"
    },
    {
        "Chapter": "1",
        "sentence_range": "4501-4504",
        "Text": "= R H S So it is a solution of the differential equation (3)"
    },
    {
        "Chapter": "1",
        "sentence_range": "4502-4505",
        "Text": "H S So it is a solution of the differential equation (3) Let a and b be given some particular values say a = 2 and \n4\nb\n=\u03c0\n, then we get a\nfunction\ny = \u03c61(x) = 2sin\n4\nx\n\u03c0\n\u239b\n\u239e\n+\n\u239c\n\u239f\n\u239d\n\u23a0"
    },
    {
        "Chapter": "1",
        "sentence_range": "4503-4506",
        "Text": "S So it is a solution of the differential equation (3) Let a and b be given some particular values say a = 2 and \n4\nb\n=\u03c0\n, then we get a\nfunction\ny = \u03c61(x) = 2sin\n4\nx\n\u03c0\n\u239b\n\u239e\n+\n\u239c\n\u239f\n\u239d\n\u23a0 (5)\nWhen this function and its derivative are substituted in equation (3) again\nL"
    },
    {
        "Chapter": "1",
        "sentence_range": "4504-4507",
        "Text": "So it is a solution of the differential equation (3) Let a and b be given some particular values say a = 2 and \n4\nb\n=\u03c0\n, then we get a\nfunction\ny = \u03c61(x) = 2sin\n4\nx\n\u03c0\n\u239b\n\u239e\n+\n\u239c\n\u239f\n\u239d\n\u23a0 (5)\nWhen this function and its derivative are substituted in equation (3) again\nL H"
    },
    {
        "Chapter": "1",
        "sentence_range": "4505-4508",
        "Text": "Let a and b be given some particular values say a = 2 and \n4\nb\n=\u03c0\n, then we get a\nfunction\ny = \u03c61(x) = 2sin\n4\nx\n\u03c0\n\u239b\n\u239e\n+\n\u239c\n\u239f\n\u239d\n\u23a0 (5)\nWhen this function and its derivative are substituted in equation (3) again\nL H S"
    },
    {
        "Chapter": "1",
        "sentence_range": "4506-4509",
        "Text": "(5)\nWhen this function and its derivative are substituted in equation (3) again\nL H S = R"
    },
    {
        "Chapter": "1",
        "sentence_range": "4507-4510",
        "Text": "H S = R H"
    },
    {
        "Chapter": "1",
        "sentence_range": "4508-4511",
        "Text": "S = R H S"
    },
    {
        "Chapter": "1",
        "sentence_range": "4509-4512",
        "Text": "= R H S Therefore \u03c61 is also a solution of equation (3)"
    },
    {
        "Chapter": "1",
        "sentence_range": "4510-4513",
        "Text": "H S Therefore \u03c61 is also a solution of equation (3) \u00a9 NCERT\nnot to be republished\nMATHEMATICS\n384\nFunction \u03c6  consists of two arbitrary constants (parameters) a, b and it is called\ngeneral solution of the given differential equation"
    },
    {
        "Chapter": "1",
        "sentence_range": "4511-4514",
        "Text": "S Therefore \u03c61 is also a solution of equation (3) \u00a9 NCERT\nnot to be republished\nMATHEMATICS\n384\nFunction \u03c6  consists of two arbitrary constants (parameters) a, b and it is called\ngeneral solution of the given differential equation Whereas function \u03c61 contains no\narbitrary constants but only the particular values of the parameters a and b and hence\nis called a particular solution of the given differential equation"
    },
    {
        "Chapter": "1",
        "sentence_range": "4512-4515",
        "Text": "Therefore \u03c61 is also a solution of equation (3) \u00a9 NCERT\nnot to be republished\nMATHEMATICS\n384\nFunction \u03c6  consists of two arbitrary constants (parameters) a, b and it is called\ngeneral solution of the given differential equation Whereas function \u03c61 contains no\narbitrary constants but only the particular values of the parameters a and b and hence\nis called a particular solution of the given differential equation The solution which contains  arbitrary constants is called the general solution\n(primitive) of the differential equation"
    },
    {
        "Chapter": "1",
        "sentence_range": "4513-4516",
        "Text": "\u00a9 NCERT\nnot to be republished\nMATHEMATICS\n384\nFunction \u03c6  consists of two arbitrary constants (parameters) a, b and it is called\ngeneral solution of the given differential equation Whereas function \u03c61 contains no\narbitrary constants but only the particular values of the parameters a and b and hence\nis called a particular solution of the given differential equation The solution which contains  arbitrary constants is called the general solution\n(primitive) of the differential equation The solution free from arbitrary constants i"
    },
    {
        "Chapter": "1",
        "sentence_range": "4514-4517",
        "Text": "Whereas function \u03c61 contains no\narbitrary constants but only the particular values of the parameters a and b and hence\nis called a particular solution of the given differential equation The solution which contains  arbitrary constants is called the general solution\n(primitive) of the differential equation The solution free from arbitrary constants i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "4515-4518",
        "Text": "The solution which contains  arbitrary constants is called the general solution\n(primitive) of the differential equation The solution free from arbitrary constants i e , the solution obtained from the general\nsolution by giving particular values to the arbitrary constants is called a particular\nsolution of the differential equation"
    },
    {
        "Chapter": "1",
        "sentence_range": "4516-4519",
        "Text": "The solution free from arbitrary constants i e , the solution obtained from the general\nsolution by giving particular values to the arbitrary constants is called a particular\nsolution of the differential equation Example 2 Verify that the function y = e\u2013 3x is a solution of the differential equation\n2\n2\n6\n0\nd y\ndy\ny\ndx\ndx\n+\n\u2212\n=\nSolution Given function is  y = e\u2013 3x"
    },
    {
        "Chapter": "1",
        "sentence_range": "4517-4520",
        "Text": "e , the solution obtained from the general\nsolution by giving particular values to the arbitrary constants is called a particular\nsolution of the differential equation Example 2 Verify that the function y = e\u2013 3x is a solution of the differential equation\n2\n2\n6\n0\nd y\ndy\ny\ndx\ndx\n+\n\u2212\n=\nSolution Given function is  y = e\u2013 3x Differentiating both sides of equation with respect\nto x , we get\n3\n3\nx\ndy\ne\ndx\n\u2212\n= \u2212"
    },
    {
        "Chapter": "1",
        "sentence_range": "4518-4521",
        "Text": ", the solution obtained from the general\nsolution by giving particular values to the arbitrary constants is called a particular\nsolution of the differential equation Example 2 Verify that the function y = e\u2013 3x is a solution of the differential equation\n2\n2\n6\n0\nd y\ndy\ny\ndx\ndx\n+\n\u2212\n=\nSolution Given function is  y = e\u2013 3x Differentiating both sides of equation with respect\nto x , we get\n3\n3\nx\ndy\ne\ndx\n\u2212\n= \u2212 (1)\nNow, differentiating (1) with respect to x, we have\n2\nd y2\ndx\n = 9 e \u2013 3x\nSubstituting the values of  \n2\n2 ,\nd y dy\ndx\ndx\nand y in the given differential equation, we get\nL"
    },
    {
        "Chapter": "1",
        "sentence_range": "4519-4522",
        "Text": "Example 2 Verify that the function y = e\u2013 3x is a solution of the differential equation\n2\n2\n6\n0\nd y\ndy\ny\ndx\ndx\n+\n\u2212\n=\nSolution Given function is  y = e\u2013 3x Differentiating both sides of equation with respect\nto x , we get\n3\n3\nx\ndy\ne\ndx\n\u2212\n= \u2212 (1)\nNow, differentiating (1) with respect to x, we have\n2\nd y2\ndx\n = 9 e \u2013 3x\nSubstituting the values of  \n2\n2 ,\nd y dy\ndx\ndx\nand y in the given differential equation, we get\nL H"
    },
    {
        "Chapter": "1",
        "sentence_range": "4520-4523",
        "Text": "Differentiating both sides of equation with respect\nto x , we get\n3\n3\nx\ndy\ne\ndx\n\u2212\n= \u2212 (1)\nNow, differentiating (1) with respect to x, we have\n2\nd y2\ndx\n = 9 e \u2013 3x\nSubstituting the values of  \n2\n2 ,\nd y dy\ndx\ndx\nand y in the given differential equation, we get\nL H S"
    },
    {
        "Chapter": "1",
        "sentence_range": "4521-4524",
        "Text": "(1)\nNow, differentiating (1) with respect to x, we have\n2\nd y2\ndx\n = 9 e \u2013 3x\nSubstituting the values of  \n2\n2 ,\nd y dy\ndx\ndx\nand y in the given differential equation, we get\nL H S = 9 e\u2013 3x + (\u20133e\u2013 3x) \u2013 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "4522-4525",
        "Text": "H S = 9 e\u2013 3x + (\u20133e\u2013 3x) \u2013 6 e\u2013 3x = 9 e\u2013 3x \u2013 9 e\u2013 3x = 0 = R"
    },
    {
        "Chapter": "1",
        "sentence_range": "4523-4526",
        "Text": "S = 9 e\u2013 3x + (\u20133e\u2013 3x) \u2013 6 e\u2013 3x = 9 e\u2013 3x \u2013 9 e\u2013 3x = 0 = R H"
    },
    {
        "Chapter": "1",
        "sentence_range": "4524-4527",
        "Text": "= 9 e\u2013 3x + (\u20133e\u2013 3x) \u2013 6 e\u2013 3x = 9 e\u2013 3x \u2013 9 e\u2013 3x = 0 = R H S"
    },
    {
        "Chapter": "1",
        "sentence_range": "4525-4528",
        "Text": "e\u2013 3x = 9 e\u2013 3x \u2013 9 e\u2013 3x = 0 = R H S Therefore, the given function is a solution of the given differential equation"
    },
    {
        "Chapter": "1",
        "sentence_range": "4526-4529",
        "Text": "H S Therefore, the given function is a solution of the given differential equation Example 3 Verify that the function y = a cos x + b sin x, where, a, b \u2208 R is a solution\nof the differential equation \n2\n2\n0\nd y\ny\ndx\n+\n=\nSolution The given function is\ny = a cos x + b sin x"
    },
    {
        "Chapter": "1",
        "sentence_range": "4527-4530",
        "Text": "S Therefore, the given function is a solution of the given differential equation Example 3 Verify that the function y = a cos x + b sin x, where, a, b \u2208 R is a solution\nof the differential equation \n2\n2\n0\nd y\ny\ndx\n+\n=\nSolution The given function is\ny = a cos x + b sin x (1)\nDifferentiating both sides of equation (1) with respect to x, successively, we get\ndy\ndx  = \u2013 a sinx + b cosx\n2\nd y2\ndx\n = \u2013 a cos x \u2013 b sinx\n\u00a9 NCERT\nnot to be republished\nDIFFERENTIAL EQUATIONS\n385\nSubstituting the values of \n2\nd y2\ndx\n and y in the given differential equation, we get\nL"
    },
    {
        "Chapter": "1",
        "sentence_range": "4528-4531",
        "Text": "Therefore, the given function is a solution of the given differential equation Example 3 Verify that the function y = a cos x + b sin x, where, a, b \u2208 R is a solution\nof the differential equation \n2\n2\n0\nd y\ny\ndx\n+\n=\nSolution The given function is\ny = a cos x + b sin x (1)\nDifferentiating both sides of equation (1) with respect to x, successively, we get\ndy\ndx  = \u2013 a sinx + b cosx\n2\nd y2\ndx\n = \u2013 a cos x \u2013 b sinx\n\u00a9 NCERT\nnot to be republished\nDIFFERENTIAL EQUATIONS\n385\nSubstituting the values of \n2\nd y2\ndx\n and y in the given differential equation, we get\nL H"
    },
    {
        "Chapter": "1",
        "sentence_range": "4529-4532",
        "Text": "Example 3 Verify that the function y = a cos x + b sin x, where, a, b \u2208 R is a solution\nof the differential equation \n2\n2\n0\nd y\ny\ndx\n+\n=\nSolution The given function is\ny = a cos x + b sin x (1)\nDifferentiating both sides of equation (1) with respect to x, successively, we get\ndy\ndx  = \u2013 a sinx + b cosx\n2\nd y2\ndx\n = \u2013 a cos x \u2013 b sinx\n\u00a9 NCERT\nnot to be republished\nDIFFERENTIAL EQUATIONS\n385\nSubstituting the values of \n2\nd y2\ndx\n and y in the given differential equation, we get\nL H S"
    },
    {
        "Chapter": "1",
        "sentence_range": "4530-4533",
        "Text": "(1)\nDifferentiating both sides of equation (1) with respect to x, successively, we get\ndy\ndx  = \u2013 a sinx + b cosx\n2\nd y2\ndx\n = \u2013 a cos x \u2013 b sinx\n\u00a9 NCERT\nnot to be republished\nDIFFERENTIAL EQUATIONS\n385\nSubstituting the values of \n2\nd y2\ndx\n and y in the given differential equation, we get\nL H S = (\u2013 a cos x \u2013 b sin x) + (a cos x + b sin x) = 0 = R"
    },
    {
        "Chapter": "1",
        "sentence_range": "4531-4534",
        "Text": "H S = (\u2013 a cos x \u2013 b sin x) + (a cos x + b sin x) = 0 = R H"
    },
    {
        "Chapter": "1",
        "sentence_range": "4532-4535",
        "Text": "S = (\u2013 a cos x \u2013 b sin x) + (a cos x + b sin x) = 0 = R H S"
    },
    {
        "Chapter": "1",
        "sentence_range": "4533-4536",
        "Text": "= (\u2013 a cos x \u2013 b sin x) + (a cos x + b sin x) = 0 = R H S Therefore, the given function is a solution of the given differential equation"
    },
    {
        "Chapter": "1",
        "sentence_range": "4534-4537",
        "Text": "H S Therefore, the given function is a solution of the given differential equation EXERCISE 9"
    },
    {
        "Chapter": "1",
        "sentence_range": "4535-4538",
        "Text": "S Therefore, the given function is a solution of the given differential equation EXERCISE 9 2\nIn each of the Exercises 1 to 10 verify that the given functions (explicit or implicit) is a\nsolution of the corresponding differential equation:\n1"
    },
    {
        "Chapter": "1",
        "sentence_range": "4536-4539",
        "Text": "Therefore, the given function is a solution of the given differential equation EXERCISE 9 2\nIn each of the Exercises 1 to 10 verify that the given functions (explicit or implicit) is a\nsolution of the corresponding differential equation:\n1 y = ex + 1\n:\ny\u2033 \u2013 y\u2032 = 0\n2"
    },
    {
        "Chapter": "1",
        "sentence_range": "4537-4540",
        "Text": "EXERCISE 9 2\nIn each of the Exercises 1 to 10 verify that the given functions (explicit or implicit) is a\nsolution of the corresponding differential equation:\n1 y = ex + 1\n:\ny\u2033 \u2013 y\u2032 = 0\n2 y = x2 + 2x + C\n:\ny\u2032 \u2013 2x \u2013 2 = 0\n3"
    },
    {
        "Chapter": "1",
        "sentence_range": "4538-4541",
        "Text": "2\nIn each of the Exercises 1 to 10 verify that the given functions (explicit or implicit) is a\nsolution of the corresponding differential equation:\n1 y = ex + 1\n:\ny\u2033 \u2013 y\u2032 = 0\n2 y = x2 + 2x + C\n:\ny\u2032 \u2013 2x \u2013 2 = 0\n3 y = cos x + C\n:\ny\u2032 + sin x = 0\n4"
    },
    {
        "Chapter": "1",
        "sentence_range": "4539-4542",
        "Text": "y = ex + 1\n:\ny\u2033 \u2013 y\u2032 = 0\n2 y = x2 + 2x + C\n:\ny\u2032 \u2013 2x \u2013 2 = 0\n3 y = cos x + C\n:\ny\u2032 + sin x = 0\n4 y = \n2\n1\n+x\n:\ny\u2032 = \n2\n1\nxy\nx\n+\n5"
    },
    {
        "Chapter": "1",
        "sentence_range": "4540-4543",
        "Text": "y = x2 + 2x + C\n:\ny\u2032 \u2013 2x \u2013 2 = 0\n3 y = cos x + C\n:\ny\u2032 + sin x = 0\n4 y = \n2\n1\n+x\n:\ny\u2032 = \n2\n1\nxy\nx\n+\n5 y = Ax\n:\nxy\u2032 = y (x \u2260 0)\n6"
    },
    {
        "Chapter": "1",
        "sentence_range": "4541-4544",
        "Text": "y = cos x + C\n:\ny\u2032 + sin x = 0\n4 y = \n2\n1\n+x\n:\ny\u2032 = \n2\n1\nxy\nx\n+\n5 y = Ax\n:\nxy\u2032 = y (x \u2260 0)\n6 y = x sin x\n:\nxy\u2032 = y + x \n2\n2\nx\n\u2212y\n (x \u2260 0 and x > y or x < \u2013 y)\n7"
    },
    {
        "Chapter": "1",
        "sentence_range": "4542-4545",
        "Text": "y = \n2\n1\n+x\n:\ny\u2032 = \n2\n1\nxy\nx\n+\n5 y = Ax\n:\nxy\u2032 = y (x \u2260 0)\n6 y = x sin x\n:\nxy\u2032 = y + x \n2\n2\nx\n\u2212y\n (x \u2260 0 and x > y or x < \u2013 y)\n7 xy = log y + C\n:\ny\u2032 = \n2\n1\ny\n\u2212xy\n (xy \u2260 1)\n8"
    },
    {
        "Chapter": "1",
        "sentence_range": "4543-4546",
        "Text": "y = Ax\n:\nxy\u2032 = y (x \u2260 0)\n6 y = x sin x\n:\nxy\u2032 = y + x \n2\n2\nx\n\u2212y\n (x \u2260 0 and x > y or x < \u2013 y)\n7 xy = log y + C\n:\ny\u2032 = \n2\n1\ny\n\u2212xy\n (xy \u2260 1)\n8 y \u2013 cos y = x\n:\n(y sin y + cos y + x) y\u2032 = y\n9"
    },
    {
        "Chapter": "1",
        "sentence_range": "4544-4547",
        "Text": "y = x sin x\n:\nxy\u2032 = y + x \n2\n2\nx\n\u2212y\n (x \u2260 0 and x > y or x < \u2013 y)\n7 xy = log y + C\n:\ny\u2032 = \n2\n1\ny\n\u2212xy\n (xy \u2260 1)\n8 y \u2013 cos y = x\n:\n(y sin y + cos y + x) y\u2032 = y\n9 x + y = tan\u20131y\n:\ny2 y\u2032 + y2 + 1 = 0\n10"
    },
    {
        "Chapter": "1",
        "sentence_range": "4545-4548",
        "Text": "xy = log y + C\n:\ny\u2032 = \n2\n1\ny\n\u2212xy\n (xy \u2260 1)\n8 y \u2013 cos y = x\n:\n(y sin y + cos y + x) y\u2032 = y\n9 x + y = tan\u20131y\n:\ny2 y\u2032 + y2 + 1 = 0\n10 y = \n2\n2\na\n\u2212x\nx \u2208 (\u2013a, a) :\nx + y dy\ndx  = 0 (y \u2260 0)\n11"
    },
    {
        "Chapter": "1",
        "sentence_range": "4546-4549",
        "Text": "y \u2013 cos y = x\n:\n(y sin y + cos y + x) y\u2032 = y\n9 x + y = tan\u20131y\n:\ny2 y\u2032 + y2 + 1 = 0\n10 y = \n2\n2\na\n\u2212x\nx \u2208 (\u2013a, a) :\nx + y dy\ndx  = 0 (y \u2260 0)\n11 The number of arbitrary constants in the general solution of a differential equation\nof fourth order are:\n(A) 0\n(B) 2\n(C) 3\n(D) 4\n12"
    },
    {
        "Chapter": "1",
        "sentence_range": "4547-4550",
        "Text": "x + y = tan\u20131y\n:\ny2 y\u2032 + y2 + 1 = 0\n10 y = \n2\n2\na\n\u2212x\nx \u2208 (\u2013a, a) :\nx + y dy\ndx  = 0 (y \u2260 0)\n11 The number of arbitrary constants in the general solution of a differential equation\nof fourth order are:\n(A) 0\n(B) 2\n(C) 3\n(D) 4\n12 The number of arbitrary constants in the particular solution of a differential equation\nof third order are:\n(A) 3\n(B) 2\n(C) 1\n(D) 0\n9"
    },
    {
        "Chapter": "1",
        "sentence_range": "4548-4551",
        "Text": "y = \n2\n2\na\n\u2212x\nx \u2208 (\u2013a, a) :\nx + y dy\ndx  = 0 (y \u2260 0)\n11 The number of arbitrary constants in the general solution of a differential equation\nof fourth order are:\n(A) 0\n(B) 2\n(C) 3\n(D) 4\n12 The number of arbitrary constants in the particular solution of a differential equation\nof third order are:\n(A) 3\n(B) 2\n(C) 1\n(D) 0\n9 4  Formation of a Differential Equation whose General Solution is given\nWe know that the equation\nx2 + y2 + 2x \u2013 4y + 4 = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "4549-4552",
        "Text": "The number of arbitrary constants in the general solution of a differential equation\nof fourth order are:\n(A) 0\n(B) 2\n(C) 3\n(D) 4\n12 The number of arbitrary constants in the particular solution of a differential equation\nof third order are:\n(A) 3\n(B) 2\n(C) 1\n(D) 0\n9 4  Formation of a Differential Equation whose General Solution is given\nWe know that the equation\nx2 + y2 + 2x \u2013 4y + 4 = 0 (1)\nrepresents a circle having centre at (\u20131, 2) and radius 1 unit"
    },
    {
        "Chapter": "1",
        "sentence_range": "4550-4553",
        "Text": "The number of arbitrary constants in the particular solution of a differential equation\nof third order are:\n(A) 3\n(B) 2\n(C) 1\n(D) 0\n9 4  Formation of a Differential Equation whose General Solution is given\nWe know that the equation\nx2 + y2 + 2x \u2013 4y + 4 = 0 (1)\nrepresents a circle having centre at (\u20131, 2) and radius 1 unit \u00a9 NCERT\nnot to be republished\nMATHEMATICS\n386\nDifferentiating equation (1) with respect to x, we get\ndy\ndx  =\n1\n2\nx\ny\n\u2212+\n (y \u2260 2)"
    },
    {
        "Chapter": "1",
        "sentence_range": "4551-4554",
        "Text": "4  Formation of a Differential Equation whose General Solution is given\nWe know that the equation\nx2 + y2 + 2x \u2013 4y + 4 = 0 (1)\nrepresents a circle having centre at (\u20131, 2) and radius 1 unit \u00a9 NCERT\nnot to be republished\nMATHEMATICS\n386\nDifferentiating equation (1) with respect to x, we get\ndy\ndx  =\n1\n2\nx\ny\n\u2212+\n (y \u2260 2) (2)\nwhich is a differential equation"
    },
    {
        "Chapter": "1",
        "sentence_range": "4552-4555",
        "Text": "(1)\nrepresents a circle having centre at (\u20131, 2) and radius 1 unit \u00a9 NCERT\nnot to be republished\nMATHEMATICS\n386\nDifferentiating equation (1) with respect to x, we get\ndy\ndx  =\n1\n2\nx\ny\n\u2212+\n (y \u2260 2) (2)\nwhich is a differential equation You will find later on [See (example 9 section 9"
    },
    {
        "Chapter": "1",
        "sentence_range": "4553-4556",
        "Text": "\u00a9 NCERT\nnot to be republished\nMATHEMATICS\n386\nDifferentiating equation (1) with respect to x, we get\ndy\ndx  =\n1\n2\nx\ny\n\u2212+\n (y \u2260 2) (2)\nwhich is a differential equation You will find later on [See (example 9 section 9 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "4554-4557",
        "Text": "(2)\nwhich is a differential equation You will find later on [See (example 9 section 9 5 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "4555-4558",
        "Text": "You will find later on [See (example 9 section 9 5 1 )]\nthat this equation represents the family of circles and one member of the family is the\ncircle given in equation (1)"
    },
    {
        "Chapter": "1",
        "sentence_range": "4556-4559",
        "Text": "5 1 )]\nthat this equation represents the family of circles and one member of the family is the\ncircle given in equation (1) Let us consider the equation\nx2 + y2 = r2"
    },
    {
        "Chapter": "1",
        "sentence_range": "4557-4560",
        "Text": "1 )]\nthat this equation represents the family of circles and one member of the family is the\ncircle given in equation (1) Let us consider the equation\nx2 + y2 = r2 (3)\nBy giving different values to r, we get different members of the family e"
    },
    {
        "Chapter": "1",
        "sentence_range": "4558-4561",
        "Text": ")]\nthat this equation represents the family of circles and one member of the family is the\ncircle given in equation (1) Let us consider the equation\nx2 + y2 = r2 (3)\nBy giving different values to r, we get different members of the family e g"
    },
    {
        "Chapter": "1",
        "sentence_range": "4559-4562",
        "Text": "Let us consider the equation\nx2 + y2 = r2 (3)\nBy giving different values to r, we get different members of the family e g x2 + y2 = 1, x2 + y2 = 4, x2 + y2 = 9 etc"
    },
    {
        "Chapter": "1",
        "sentence_range": "4560-4563",
        "Text": "(3)\nBy giving different values to r, we get different members of the family e g x2 + y2 = 1, x2 + y2 = 4, x2 + y2 = 9 etc (see Fig 9"
    },
    {
        "Chapter": "1",
        "sentence_range": "4561-4564",
        "Text": "g x2 + y2 = 1, x2 + y2 = 4, x2 + y2 = 9 etc (see Fig 9 1)"
    },
    {
        "Chapter": "1",
        "sentence_range": "4562-4565",
        "Text": "x2 + y2 = 1, x2 + y2 = 4, x2 + y2 = 9 etc (see Fig 9 1) Thus, equation (3) represents a family of concentric\ncircles centered at the origin and having different radii"
    },
    {
        "Chapter": "1",
        "sentence_range": "4563-4566",
        "Text": "(see Fig 9 1) Thus, equation (3) represents a family of concentric\ncircles centered at the origin and having different radii We are interested in finding a differential equation\nthat is satisfied by each member of the family"
    },
    {
        "Chapter": "1",
        "sentence_range": "4564-4567",
        "Text": "1) Thus, equation (3) represents a family of concentric\ncircles centered at the origin and having different radii We are interested in finding a differential equation\nthat is satisfied by each member of the family The\ndifferential equation must be free from r because r is\ndifferent for different members of the family"
    },
    {
        "Chapter": "1",
        "sentence_range": "4565-4568",
        "Text": "Thus, equation (3) represents a family of concentric\ncircles centered at the origin and having different radii We are interested in finding a differential equation\nthat is satisfied by each member of the family The\ndifferential equation must be free from r because r is\ndifferent for different members of the family This\nequation is obtained by differentiating equation (3) with\nrespect to x, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "4566-4569",
        "Text": "We are interested in finding a differential equation\nthat is satisfied by each member of the family The\ndifferential equation must be free from r because r is\ndifferent for different members of the family This\nequation is obtained by differentiating equation (3) with\nrespect to x, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "4567-4570",
        "Text": "The\ndifferential equation must be free from r because r is\ndifferent for different members of the family This\nequation is obtained by differentiating equation (3) with\nrespect to x, i e ,\n2x + 2y dy\ndx  = 0    or    x + y dy\ndx  = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "4568-4571",
        "Text": "This\nequation is obtained by differentiating equation (3) with\nrespect to x, i e ,\n2x + 2y dy\ndx  = 0    or    x + y dy\ndx  = 0 (4)\nwhich represents the family of concentric circles given by equation (3)"
    },
    {
        "Chapter": "1",
        "sentence_range": "4569-4572",
        "Text": "e ,\n2x + 2y dy\ndx  = 0    or    x + y dy\ndx  = 0 (4)\nwhich represents the family of concentric circles given by equation (3) Again, let us consider the equation\ny = mx + c"
    },
    {
        "Chapter": "1",
        "sentence_range": "4570-4573",
        "Text": ",\n2x + 2y dy\ndx  = 0    or    x + y dy\ndx  = 0 (4)\nwhich represents the family of concentric circles given by equation (3) Again, let us consider the equation\ny = mx + c (5)\nBy giving different values to the parameters m and c, we get different members of\nthe family, e"
    },
    {
        "Chapter": "1",
        "sentence_range": "4571-4574",
        "Text": "(4)\nwhich represents the family of concentric circles given by equation (3) Again, let us consider the equation\ny = mx + c (5)\nBy giving different values to the parameters m and c, we get different members of\nthe family, e g"
    },
    {
        "Chapter": "1",
        "sentence_range": "4572-4575",
        "Text": "Again, let us consider the equation\ny = mx + c (5)\nBy giving different values to the parameters m and c, we get different members of\nthe family, e g ,\ny = x\n(m = 1,   c = 0)\ny = \n3 x\n(m = \n3 ,  c = 0)\ny = x + 1\n(m = 1,  c = 1)\ny = \u2013 x\n(m = \u2013 1,  c = 0)\ny = \u2013 x \u2013 1\n(m = \u2013 1,  c = \u2013 1) etc"
    },
    {
        "Chapter": "1",
        "sentence_range": "4573-4576",
        "Text": "(5)\nBy giving different values to the parameters m and c, we get different members of\nthe family, e g ,\ny = x\n(m = 1,   c = 0)\ny = \n3 x\n(m = \n3 ,  c = 0)\ny = x + 1\n(m = 1,  c = 1)\ny = \u2013 x\n(m = \u2013 1,  c = 0)\ny = \u2013 x \u2013 1\n(m = \u2013 1,  c = \u2013 1) etc ( see Fig 9"
    },
    {
        "Chapter": "1",
        "sentence_range": "4574-4577",
        "Text": "g ,\ny = x\n(m = 1,   c = 0)\ny = \n3 x\n(m = \n3 ,  c = 0)\ny = x + 1\n(m = 1,  c = 1)\ny = \u2013 x\n(m = \u2013 1,  c = 0)\ny = \u2013 x \u2013 1\n(m = \u2013 1,  c = \u2013 1) etc ( see Fig 9 2)"
    },
    {
        "Chapter": "1",
        "sentence_range": "4575-4578",
        "Text": ",\ny = x\n(m = 1,   c = 0)\ny = \n3 x\n(m = \n3 ,  c = 0)\ny = x + 1\n(m = 1,  c = 1)\ny = \u2013 x\n(m = \u2013 1,  c = 0)\ny = \u2013 x \u2013 1\n(m = \u2013 1,  c = \u2013 1) etc ( see Fig 9 2) Thus, equation (5) represents the family of straight lines, where m, c are parameters"
    },
    {
        "Chapter": "1",
        "sentence_range": "4576-4579",
        "Text": "( see Fig 9 2) Thus, equation (5) represents the family of straight lines, where m, c are parameters We are now interested in finding a differential equation that is satisfied by each\nmember of the family"
    },
    {
        "Chapter": "1",
        "sentence_range": "4577-4580",
        "Text": "2) Thus, equation (5) represents the family of straight lines, where m, c are parameters We are now interested in finding a differential equation that is satisfied by each\nmember of the family Further,  the equation must be free from m and c because m and\nFig 9"
    },
    {
        "Chapter": "1",
        "sentence_range": "4578-4581",
        "Text": "Thus, equation (5) represents the family of straight lines, where m, c are parameters We are now interested in finding a differential equation that is satisfied by each\nmember of the family Further,  the equation must be free from m and c because m and\nFig 9 1\n\u00a9 NCERT\nnot to be republished\nDIFFERENTIAL EQUATIONS\n387\nX\nX\u2019\nY\nY\u2019\ny = x+1\ny = x\ny = \u2013x\ny = \u2013x\u20131\ny =\nx\n3\nO\nc are different for different members of the family"
    },
    {
        "Chapter": "1",
        "sentence_range": "4579-4582",
        "Text": "We are now interested in finding a differential equation that is satisfied by each\nmember of the family Further,  the equation must be free from m and c because m and\nFig 9 1\n\u00a9 NCERT\nnot to be republished\nDIFFERENTIAL EQUATIONS\n387\nX\nX\u2019\nY\nY\u2019\ny = x+1\ny = x\ny = \u2013x\ny = \u2013x\u20131\ny =\nx\n3\nO\nc are different for different members of the family This is obtained by differentiating equation (5) with\nrespect to x, successively we get\ndy\ndx =m\n, and \n2\n2\n0\nd y\ndx\n="
    },
    {
        "Chapter": "1",
        "sentence_range": "4580-4583",
        "Text": "Further,  the equation must be free from m and c because m and\nFig 9 1\n\u00a9 NCERT\nnot to be republished\nDIFFERENTIAL EQUATIONS\n387\nX\nX\u2019\nY\nY\u2019\ny = x+1\ny = x\ny = \u2013x\ny = \u2013x\u20131\ny =\nx\n3\nO\nc are different for different members of the family This is obtained by differentiating equation (5) with\nrespect to x, successively we get\ndy\ndx =m\n, and \n2\n2\n0\nd y\ndx\n= (6)\nThe equation (6) represents the family of straight\nlines given by equation (5)"
    },
    {
        "Chapter": "1",
        "sentence_range": "4581-4584",
        "Text": "1\n\u00a9 NCERT\nnot to be republished\nDIFFERENTIAL EQUATIONS\n387\nX\nX\u2019\nY\nY\u2019\ny = x+1\ny = x\ny = \u2013x\ny = \u2013x\u20131\ny =\nx\n3\nO\nc are different for different members of the family This is obtained by differentiating equation (5) with\nrespect to x, successively we get\ndy\ndx =m\n, and \n2\n2\n0\nd y\ndx\n= (6)\nThe equation (6) represents the family of straight\nlines given by equation (5) Note that equations (3) and (5) are the general\nsolutions of equations (4) and (6) respectively"
    },
    {
        "Chapter": "1",
        "sentence_range": "4582-4585",
        "Text": "This is obtained by differentiating equation (5) with\nrespect to x, successively we get\ndy\ndx =m\n, and \n2\n2\n0\nd y\ndx\n= (6)\nThe equation (6) represents the family of straight\nlines given by equation (5) Note that equations (3) and (5) are the general\nsolutions of equations (4) and (6) respectively 9"
    },
    {
        "Chapter": "1",
        "sentence_range": "4583-4586",
        "Text": "(6)\nThe equation (6) represents the family of straight\nlines given by equation (5) Note that equations (3) and (5) are the general\nsolutions of equations (4) and (6) respectively 9 4"
    },
    {
        "Chapter": "1",
        "sentence_range": "4584-4587",
        "Text": "Note that equations (3) and (5) are the general\nsolutions of equations (4) and (6) respectively 9 4 1  Procedure to form a differential equation that will represent a given\nfamily of curves\n(a)\nIf the given family F1 of curves depends on only one parameter then it is\nrepresented by an equation of the form\nF1 (x, y, a) = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "4585-4588",
        "Text": "9 4 1  Procedure to form a differential equation that will represent a given\nfamily of curves\n(a)\nIf the given family F1 of curves depends on only one parameter then it is\nrepresented by an equation of the form\nF1 (x, y, a) = 0 (1)\nFor example, the family of parabolas y2 = ax can be represented by an equation\nof the form f (x, y, a) : y2 = ax"
    },
    {
        "Chapter": "1",
        "sentence_range": "4586-4589",
        "Text": "4 1  Procedure to form a differential equation that will represent a given\nfamily of curves\n(a)\nIf the given family F1 of curves depends on only one parameter then it is\nrepresented by an equation of the form\nF1 (x, y, a) = 0 (1)\nFor example, the family of parabolas y2 = ax can be represented by an equation\nof the form f (x, y, a) : y2 = ax Differentiating equation (1) with respect to x, we get an equation involving\ny\u2032, y, x, and a, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "4587-4590",
        "Text": "1  Procedure to form a differential equation that will represent a given\nfamily of curves\n(a)\nIf the given family F1 of curves depends on only one parameter then it is\nrepresented by an equation of the form\nF1 (x, y, a) = 0 (1)\nFor example, the family of parabolas y2 = ax can be represented by an equation\nof the form f (x, y, a) : y2 = ax Differentiating equation (1) with respect to x, we get an equation involving\ny\u2032, y, x, and a, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "4588-4591",
        "Text": "(1)\nFor example, the family of parabolas y2 = ax can be represented by an equation\nof the form f (x, y, a) : y2 = ax Differentiating equation (1) with respect to x, we get an equation involving\ny\u2032, y, x, and a, i e ,\ng (x, y, y\u2032, a) = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "4589-4592",
        "Text": "Differentiating equation (1) with respect to x, we get an equation involving\ny\u2032, y, x, and a, i e ,\ng (x, y, y\u2032, a) = 0 (2)\nThe required differential equation is then obtained by eliminating a from equations\n(1) and (2) as\nF(x, y, y\u2032) = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "4590-4593",
        "Text": "e ,\ng (x, y, y\u2032, a) = 0 (2)\nThe required differential equation is then obtained by eliminating a from equations\n(1) and (2) as\nF(x, y, y\u2032) = 0 (3)\n(b)\nIf the given family F2 of curves depends on the parameters a, b (say) then it is\nrepresented by an equation of the from\nF2 (x, y, a, b) = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "4591-4594",
        "Text": ",\ng (x, y, y\u2032, a) = 0 (2)\nThe required differential equation is then obtained by eliminating a from equations\n(1) and (2) as\nF(x, y, y\u2032) = 0 (3)\n(b)\nIf the given family F2 of curves depends on the parameters a, b (say) then it is\nrepresented by an equation of the from\nF2 (x, y, a, b) = 0 (4)\nDifferentiating equation (4) with respect to x, we get an equation involving\ny\u2032, x, y,  a, b, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "4592-4595",
        "Text": "(2)\nThe required differential equation is then obtained by eliminating a from equations\n(1) and (2) as\nF(x, y, y\u2032) = 0 (3)\n(b)\nIf the given family F2 of curves depends on the parameters a, b (say) then it is\nrepresented by an equation of the from\nF2 (x, y, a, b) = 0 (4)\nDifferentiating equation (4) with respect to x, we get an equation involving\ny\u2032, x, y,  a, b, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "4593-4596",
        "Text": "(3)\n(b)\nIf the given family F2 of curves depends on the parameters a, b (say) then it is\nrepresented by an equation of the from\nF2 (x, y, a, b) = 0 (4)\nDifferentiating equation (4) with respect to x, we get an equation involving\ny\u2032, x, y,  a, b, i e ,\ng (x, y, y\u2032, a, b) = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "4594-4597",
        "Text": "(4)\nDifferentiating equation (4) with respect to x, we get an equation involving\ny\u2032, x, y,  a, b, i e ,\ng (x, y, y\u2032, a, b) = 0 (5)\nBut it is not possible to eliminate two parameters a and b from the two equations\nand so, we need a third equation"
    },
    {
        "Chapter": "1",
        "sentence_range": "4595-4598",
        "Text": "e ,\ng (x, y, y\u2032, a, b) = 0 (5)\nBut it is not possible to eliminate two parameters a and b from the two equations\nand so, we need a third equation This equation is obtained by differentiating\nequation (5), with respect to x, to obtain a relation of the form\nh (x, y, y\u2032, y\u2033, a, b) = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "4596-4599",
        "Text": ",\ng (x, y, y\u2032, a, b) = 0 (5)\nBut it is not possible to eliminate two parameters a and b from the two equations\nand so, we need a third equation This equation is obtained by differentiating\nequation (5), with respect to x, to obtain a relation of the form\nh (x, y, y\u2032, y\u2033, a, b) = 0 (6)\nFig 9"
    },
    {
        "Chapter": "1",
        "sentence_range": "4597-4600",
        "Text": "(5)\nBut it is not possible to eliminate two parameters a and b from the two equations\nand so, we need a third equation This equation is obtained by differentiating\nequation (5), with respect to x, to obtain a relation of the form\nh (x, y, y\u2032, y\u2033, a, b) = 0 (6)\nFig 9 2\n\u00a9 NCERT\nnot to be republished\nMATHEMATICS\n388\nThe required differential equation is then obtained by eliminating a and b from\nequations (4), (5) and (6) as\nF (x, y, y\u2032, y\u2033) = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "4598-4601",
        "Text": "This equation is obtained by differentiating\nequation (5), with respect to x, to obtain a relation of the form\nh (x, y, y\u2032, y\u2033, a, b) = 0 (6)\nFig 9 2\n\u00a9 NCERT\nnot to be republished\nMATHEMATICS\n388\nThe required differential equation is then obtained by eliminating a and b from\nequations (4), (5) and (6) as\nF (x, y, y\u2032, y\u2033) = 0 (7)\n\ufffdNote  The order of a differential equation representing a family of curves is\nsame as the number of arbitrary constants present in the equation corresponding to\nthe family of curves"
    },
    {
        "Chapter": "1",
        "sentence_range": "4599-4602",
        "Text": "(6)\nFig 9 2\n\u00a9 NCERT\nnot to be republished\nMATHEMATICS\n388\nThe required differential equation is then obtained by eliminating a and b from\nequations (4), (5) and (6) as\nF (x, y, y\u2032, y\u2033) = 0 (7)\n\ufffdNote  The order of a differential equation representing a family of curves is\nsame as the number of arbitrary constants present in the equation corresponding to\nthe family of curves Example 4 Form the differential equation representing the family of curves y = mx,\nwhere, m is arbitrary constant"
    },
    {
        "Chapter": "1",
        "sentence_range": "4600-4603",
        "Text": "2\n\u00a9 NCERT\nnot to be republished\nMATHEMATICS\n388\nThe required differential equation is then obtained by eliminating a and b from\nequations (4), (5) and (6) as\nF (x, y, y\u2032, y\u2033) = 0 (7)\n\ufffdNote  The order of a differential equation representing a family of curves is\nsame as the number of arbitrary constants present in the equation corresponding to\nthe family of curves Example 4 Form the differential equation representing the family of curves y = mx,\nwhere, m is arbitrary constant Solution We have\ny = mx"
    },
    {
        "Chapter": "1",
        "sentence_range": "4601-4604",
        "Text": "(7)\n\ufffdNote  The order of a differential equation representing a family of curves is\nsame as the number of arbitrary constants present in the equation corresponding to\nthe family of curves Example 4 Form the differential equation representing the family of curves y = mx,\nwhere, m is arbitrary constant Solution We have\ny = mx (1)\nDifferentiating both sides of equation (1) with respect to x, we get\ndy\ndx  = m\nSubstituting the value of m in equation (1) we get \ndy\ny\nx\ndx  \nor\ndy\nx dx  \u2013 y = 0\nwhich is free from the parameter m and hence this is the required differential equation"
    },
    {
        "Chapter": "1",
        "sentence_range": "4602-4605",
        "Text": "Example 4 Form the differential equation representing the family of curves y = mx,\nwhere, m is arbitrary constant Solution We have\ny = mx (1)\nDifferentiating both sides of equation (1) with respect to x, we get\ndy\ndx  = m\nSubstituting the value of m in equation (1) we get \ndy\ny\nx\ndx  \nor\ndy\nx dx  \u2013 y = 0\nwhich is free from the parameter m and hence this is the required differential equation Example 5 Form the differential equation representing the family of curves\ny = a sin (x + b), where a, b are arbitrary constants"
    },
    {
        "Chapter": "1",
        "sentence_range": "4603-4606",
        "Text": "Solution We have\ny = mx (1)\nDifferentiating both sides of equation (1) with respect to x, we get\ndy\ndx  = m\nSubstituting the value of m in equation (1) we get \ndy\ny\nx\ndx  \nor\ndy\nx dx  \u2013 y = 0\nwhich is free from the parameter m and hence this is the required differential equation Example 5 Form the differential equation representing the family of curves\ny = a sin (x + b), where a, b are arbitrary constants Solution We have\ny = a sin(x + b)"
    },
    {
        "Chapter": "1",
        "sentence_range": "4604-4607",
        "Text": "(1)\nDifferentiating both sides of equation (1) with respect to x, we get\ndy\ndx  = m\nSubstituting the value of m in equation (1) we get \ndy\ny\nx\ndx  \nor\ndy\nx dx  \u2013 y = 0\nwhich is free from the parameter m and hence this is the required differential equation Example 5 Form the differential equation representing the family of curves\ny = a sin (x + b), where a, b are arbitrary constants Solution We have\ny = a sin(x + b) (1)\nDifferentiating both sides of equation (1) with respect to x, successively we get\ndy\ndx  = a cos (x + b)"
    },
    {
        "Chapter": "1",
        "sentence_range": "4605-4608",
        "Text": "Example 5 Form the differential equation representing the family of curves\ny = a sin (x + b), where a, b are arbitrary constants Solution We have\ny = a sin(x + b) (1)\nDifferentiating both sides of equation (1) with respect to x, successively we get\ndy\ndx  = a cos (x + b) (2)\n2\nd y2\ndx\n = \u2013 a sin(x + b)"
    },
    {
        "Chapter": "1",
        "sentence_range": "4606-4609",
        "Text": "Solution We have\ny = a sin(x + b) (1)\nDifferentiating both sides of equation (1) with respect to x, successively we get\ndy\ndx  = a cos (x + b) (2)\n2\nd y2\ndx\n = \u2013 a sin(x + b) (3)\nEliminating a and b from equations (1), (2) and (3), we get\n2\nd y2\ny\ndx\n+\n = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "4607-4610",
        "Text": "(1)\nDifferentiating both sides of equation (1) with respect to x, successively we get\ndy\ndx  = a cos (x + b) (2)\n2\nd y2\ndx\n = \u2013 a sin(x + b) (3)\nEliminating a and b from equations (1), (2) and (3), we get\n2\nd y2\ny\ndx\n+\n = 0 (4)\nwhich is free from the arbitrary constants a and b and hence this the required differential\nequation"
    },
    {
        "Chapter": "1",
        "sentence_range": "4608-4611",
        "Text": "(2)\n2\nd y2\ndx\n = \u2013 a sin(x + b) (3)\nEliminating a and b from equations (1), (2) and (3), we get\n2\nd y2\ny\ndx\n+\n = 0 (4)\nwhich is free from the arbitrary constants a and b and hence this the required differential\nequation \u00a9 NCERT\nnot to be republished\nDIFFERENTIAL EQUATIONS\n389\nExample 6 Form the differential equation\nrepresenting the family of ellipses having foci on\nx-axis and centre at the origin"
    },
    {
        "Chapter": "1",
        "sentence_range": "4609-4612",
        "Text": "(3)\nEliminating a and b from equations (1), (2) and (3), we get\n2\nd y2\ny\ndx\n+\n = 0 (4)\nwhich is free from the arbitrary constants a and b and hence this the required differential\nequation \u00a9 NCERT\nnot to be republished\nDIFFERENTIAL EQUATIONS\n389\nExample 6 Form the differential equation\nrepresenting the family of ellipses having foci on\nx-axis and centre at the origin Solution We know that the equation of said family\nof ellipses (see Fig 9"
    },
    {
        "Chapter": "1",
        "sentence_range": "4610-4613",
        "Text": "(4)\nwhich is free from the arbitrary constants a and b and hence this the required differential\nequation \u00a9 NCERT\nnot to be republished\nDIFFERENTIAL EQUATIONS\n389\nExample 6 Form the differential equation\nrepresenting the family of ellipses having foci on\nx-axis and centre at the origin Solution We know that the equation of said family\nof ellipses (see Fig 9 3) is\n2\n2\n2\n2\nx\ny\na\n+b\n = 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "4611-4614",
        "Text": "\u00a9 NCERT\nnot to be republished\nDIFFERENTIAL EQUATIONS\n389\nExample 6 Form the differential equation\nrepresenting the family of ellipses having foci on\nx-axis and centre at the origin Solution We know that the equation of said family\nof ellipses (see Fig 9 3) is\n2\n2\n2\n2\nx\ny\na\n+b\n = 1 (1)\nDifferentiating equation (1) with respect to x, we get \n2\n2\n2\n2\n0\nx\ny dy\ndx\na\n+b\n=\nor\ny\ndy\nx\n\u239bdx\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0 =\n2\n\u2212ab2"
    },
    {
        "Chapter": "1",
        "sentence_range": "4612-4615",
        "Text": "Solution We know that the equation of said family\nof ellipses (see Fig 9 3) is\n2\n2\n2\n2\nx\ny\na\n+b\n = 1 (1)\nDifferentiating equation (1) with respect to x, we get \n2\n2\n2\n2\n0\nx\ny dy\ndx\na\n+b\n=\nor\ny\ndy\nx\n\u239bdx\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0 =\n2\n\u2212ab2 (2)\nDifferentiating both sides of equation (2) with respect to x, we get\n2\n2\n2\nxdy\ny\ny\nd y\ndy\ndx\nx\ndx\ndx\nx\n \n \n \n \n \n \n \n \n \n \n \n  \n \n \n \n \n  \n \n \n \n = 0\nor\n2\n2\n2\n\u2013\nd y\ndy\ndy\nxy\nx\ny\ndx\ndx\ndx\n \n \n \n \n \n \n \n = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "4613-4616",
        "Text": "3) is\n2\n2\n2\n2\nx\ny\na\n+b\n = 1 (1)\nDifferentiating equation (1) with respect to x, we get \n2\n2\n2\n2\n0\nx\ny dy\ndx\na\n+b\n=\nor\ny\ndy\nx\n\u239bdx\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0 =\n2\n\u2212ab2 (2)\nDifferentiating both sides of equation (2) with respect to x, we get\n2\n2\n2\nxdy\ny\ny\nd y\ndy\ndx\nx\ndx\ndx\nx\n \n \n \n \n \n \n \n \n \n \n \n  \n \n \n \n \n  \n \n \n \n = 0\nor\n2\n2\n2\n\u2013\nd y\ndy\ndy\nxy\nx\ny\ndx\ndx\ndx\n \n \n \n \n \n \n \n = 0 (3)\nwhich is the required differential equation"
    },
    {
        "Chapter": "1",
        "sentence_range": "4614-4617",
        "Text": "(1)\nDifferentiating equation (1) with respect to x, we get \n2\n2\n2\n2\n0\nx\ny dy\ndx\na\n+b\n=\nor\ny\ndy\nx\n\u239bdx\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0 =\n2\n\u2212ab2 (2)\nDifferentiating both sides of equation (2) with respect to x, we get\n2\n2\n2\nxdy\ny\ny\nd y\ndy\ndx\nx\ndx\ndx\nx\n \n \n \n \n \n \n \n \n \n \n \n  \n \n \n \n \n  \n \n \n \n = 0\nor\n2\n2\n2\n\u2013\nd y\ndy\ndy\nxy\nx\ny\ndx\ndx\ndx\n \n \n \n \n \n \n \n = 0 (3)\nwhich is the required differential equation Example 7 Form the differential equation of the family\nof circles touching the x-axis at origin"
    },
    {
        "Chapter": "1",
        "sentence_range": "4615-4618",
        "Text": "(2)\nDifferentiating both sides of equation (2) with respect to x, we get\n2\n2\n2\nxdy\ny\ny\nd y\ndy\ndx\nx\ndx\ndx\nx\n \n \n \n \n \n \n \n \n \n \n \n  \n \n \n \n \n  \n \n \n \n = 0\nor\n2\n2\n2\n\u2013\nd y\ndy\ndy\nxy\nx\ny\ndx\ndx\ndx\n \n \n \n \n \n \n \n = 0 (3)\nwhich is the required differential equation Example 7 Form the differential equation of the family\nof circles touching the x-axis at origin Solution Let C denote the family of circles touching\nx-axis at origin"
    },
    {
        "Chapter": "1",
        "sentence_range": "4616-4619",
        "Text": "(3)\nwhich is the required differential equation Example 7 Form the differential equation of the family\nof circles touching the x-axis at origin Solution Let C denote the family of circles touching\nx-axis at origin Let (0, a) be  the coordinates of the\ncentre of any member of the family (see Fig 9"
    },
    {
        "Chapter": "1",
        "sentence_range": "4617-4620",
        "Text": "Example 7 Form the differential equation of the family\nof circles touching the x-axis at origin Solution Let C denote the family of circles touching\nx-axis at origin Let (0, a) be  the coordinates of the\ncentre of any member of the family (see Fig 9 4)"
    },
    {
        "Chapter": "1",
        "sentence_range": "4618-4621",
        "Text": "Solution Let C denote the family of circles touching\nx-axis at origin Let (0, a) be  the coordinates of the\ncentre of any member of the family (see Fig 9 4) Therefore, equation of family C is\nx2 + (y \u2013 a)2 = a2  or x2 + y2 = 2ay"
    },
    {
        "Chapter": "1",
        "sentence_range": "4619-4622",
        "Text": "Let (0, a) be  the coordinates of the\ncentre of any member of the family (see Fig 9 4) Therefore, equation of family C is\nx2 + (y \u2013 a)2 = a2  or x2 + y2 = 2ay (1)\nwhere, a is an arbitrary constant"
    },
    {
        "Chapter": "1",
        "sentence_range": "4620-4623",
        "Text": "4) Therefore, equation of family C is\nx2 + (y \u2013 a)2 = a2  or x2 + y2 = 2ay (1)\nwhere, a is an arbitrary constant Differentiating both\nsides of equation (1) with respect to x,we get\n2\n2\ndy\nx\ny dx\n \n = 2\ndy\na dx\n   Fig 9"
    },
    {
        "Chapter": "1",
        "sentence_range": "4621-4624",
        "Text": "Therefore, equation of family C is\nx2 + (y \u2013 a)2 = a2  or x2 + y2 = 2ay (1)\nwhere, a is an arbitrary constant Differentiating both\nsides of equation (1) with respect to x,we get\n2\n2\ndy\nx\ny dx\n \n = 2\ndy\na dx\n   Fig 9 3\nFig 9"
    },
    {
        "Chapter": "1",
        "sentence_range": "4622-4625",
        "Text": "(1)\nwhere, a is an arbitrary constant Differentiating both\nsides of equation (1) with respect to x,we get\n2\n2\ndy\nx\ny dx\n \n = 2\ndy\na dx\n   Fig 9 3\nFig 9 4\nX\nX\u2019\nY\u2019\nY\nO\n\u00a9 NCERT\nnot to be republished\nMATHEMATICS\n390\nor\ndy\nx\n y dx\n = \ndy\na dx  or a = \ndy\nx\ndyy dx\ndx"
    },
    {
        "Chapter": "1",
        "sentence_range": "4623-4626",
        "Text": "Differentiating both\nsides of equation (1) with respect to x,we get\n2\n2\ndy\nx\ny dx\n \n = 2\ndy\na dx\n   Fig 9 3\nFig 9 4\nX\nX\u2019\nY\u2019\nY\nO\n\u00a9 NCERT\nnot to be republished\nMATHEMATICS\n390\nor\ndy\nx\n y dx\n = \ndy\na dx  or a = \ndy\nx\ndyy dx\ndx (2)\nSubstituting the value of a from equation (2) in equation (1), we get\nx2 + y2 = 2\ndy\nx\ny dx\ny\ndy\ndx\n \n \n  \n \n \n \nor\n2\n2\n(\n)\ndy x\ny\ndx\n \n =\n2\n2\n2\ndy\nxy\ny dx\n \nor\ndy\ndx  =\n2\n2\n2\n\u2013\nxy\nx\ny\nThis is the required differential equation of the given family of circles"
    },
    {
        "Chapter": "1",
        "sentence_range": "4624-4627",
        "Text": "3\nFig 9 4\nX\nX\u2019\nY\u2019\nY\nO\n\u00a9 NCERT\nnot to be republished\nMATHEMATICS\n390\nor\ndy\nx\n y dx\n = \ndy\na dx  or a = \ndy\nx\ndyy dx\ndx (2)\nSubstituting the value of a from equation (2) in equation (1), we get\nx2 + y2 = 2\ndy\nx\ny dx\ny\ndy\ndx\n \n \n  \n \n \n \nor\n2\n2\n(\n)\ndy x\ny\ndx\n \n =\n2\n2\n2\ndy\nxy\ny dx\n \nor\ndy\ndx  =\n2\n2\n2\n\u2013\nxy\nx\ny\nThis is the required differential equation of the given family of circles Example 8 Form the differential equation representing the family of parabolas having\nvertex at origin and axis along positive direction of x-axis"
    },
    {
        "Chapter": "1",
        "sentence_range": "4625-4628",
        "Text": "4\nX\nX\u2019\nY\u2019\nY\nO\n\u00a9 NCERT\nnot to be republished\nMATHEMATICS\n390\nor\ndy\nx\n y dx\n = \ndy\na dx  or a = \ndy\nx\ndyy dx\ndx (2)\nSubstituting the value of a from equation (2) in equation (1), we get\nx2 + y2 = 2\ndy\nx\ny dx\ny\ndy\ndx\n \n \n  \n \n \n \nor\n2\n2\n(\n)\ndy x\ny\ndx\n \n =\n2\n2\n2\ndy\nxy\ny dx\n \nor\ndy\ndx  =\n2\n2\n2\n\u2013\nxy\nx\ny\nThis is the required differential equation of the given family of circles Example 8 Form the differential equation representing the family of parabolas having\nvertex at origin and axis along positive direction of x-axis Solution Let P denote the family of above said parabolas (see Fig 9"
    },
    {
        "Chapter": "1",
        "sentence_range": "4626-4629",
        "Text": "(2)\nSubstituting the value of a from equation (2) in equation (1), we get\nx2 + y2 = 2\ndy\nx\ny dx\ny\ndy\ndx\n \n \n  \n \n \n \nor\n2\n2\n(\n)\ndy x\ny\ndx\n \n =\n2\n2\n2\ndy\nxy\ny dx\n \nor\ndy\ndx  =\n2\n2\n2\n\u2013\nxy\nx\ny\nThis is the required differential equation of the given family of circles Example 8 Form the differential equation representing the family of parabolas having\nvertex at origin and axis along positive direction of x-axis Solution Let P denote the family of above said parabolas (see Fig 9 5) and let (a, 0) be the\nfocus of a member of the given family, where a is an arbitrary constant"
    },
    {
        "Chapter": "1",
        "sentence_range": "4627-4630",
        "Text": "Example 8 Form the differential equation representing the family of parabolas having\nvertex at origin and axis along positive direction of x-axis Solution Let P denote the family of above said parabolas (see Fig 9 5) and let (a, 0) be the\nfocus of a member of the given family, where a is an arbitrary constant Therefore, equation\nof family P is\ny2 = 4ax"
    },
    {
        "Chapter": "1",
        "sentence_range": "4628-4631",
        "Text": "Solution Let P denote the family of above said parabolas (see Fig 9 5) and let (a, 0) be the\nfocus of a member of the given family, where a is an arbitrary constant Therefore, equation\nof family P is\ny2 = 4ax (1)\nDifferentiating both sides of equation (1) with respect to x, we get\n2\ndy\ny dx  = 4a"
    },
    {
        "Chapter": "1",
        "sentence_range": "4629-4632",
        "Text": "5) and let (a, 0) be the\nfocus of a member of the given family, where a is an arbitrary constant Therefore, equation\nof family P is\ny2 = 4ax (1)\nDifferentiating both sides of equation (1) with respect to x, we get\n2\ndy\ny dx  = 4a (2)\nSubstituting the value of 4a from equation (2)\nin equation (1), we get\ny2 =\n2\n( )\nydy\nx\ndx\n\u239b\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0\nor\n2\n2\ndy\ny\n\u2212xy dx\n = 0\nwhich is the differential equation of the given family\nof parabolas"
    },
    {
        "Chapter": "1",
        "sentence_range": "4630-4633",
        "Text": "Therefore, equation\nof family P is\ny2 = 4ax (1)\nDifferentiating both sides of equation (1) with respect to x, we get\n2\ndy\ny dx  = 4a (2)\nSubstituting the value of 4a from equation (2)\nin equation (1), we get\ny2 =\n2\n( )\nydy\nx\ndx\n\u239b\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0\nor\n2\n2\ndy\ny\n\u2212xy dx\n = 0\nwhich is the differential equation of the given family\nof parabolas Fig 9"
    },
    {
        "Chapter": "1",
        "sentence_range": "4631-4634",
        "Text": "(1)\nDifferentiating both sides of equation (1) with respect to x, we get\n2\ndy\ny dx  = 4a (2)\nSubstituting the value of 4a from equation (2)\nin equation (1), we get\ny2 =\n2\n( )\nydy\nx\ndx\n\u239b\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0\nor\n2\n2\ndy\ny\n\u2212xy dx\n = 0\nwhich is the differential equation of the given family\nof parabolas Fig 9 5\n\u00a9 NCERT\nnot to be republished\nDIFFERENTIAL EQUATIONS\n391\nEXERCISE 9"
    },
    {
        "Chapter": "1",
        "sentence_range": "4632-4635",
        "Text": "(2)\nSubstituting the value of 4a from equation (2)\nin equation (1), we get\ny2 =\n2\n( )\nydy\nx\ndx\n\u239b\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0\nor\n2\n2\ndy\ny\n\u2212xy dx\n = 0\nwhich is the differential equation of the given family\nof parabolas Fig 9 5\n\u00a9 NCERT\nnot to be republished\nDIFFERENTIAL EQUATIONS\n391\nEXERCISE 9 3\nIn each of the Exercises 1 to 5, form a differential equation representing the given\nfamily of curves by eliminating arbitrary constants a and b"
    },
    {
        "Chapter": "1",
        "sentence_range": "4633-4636",
        "Text": "Fig 9 5\n\u00a9 NCERT\nnot to be republished\nDIFFERENTIAL EQUATIONS\n391\nEXERCISE 9 3\nIn each of the Exercises 1 to 5, form a differential equation representing the given\nfamily of curves by eliminating arbitrary constants a and b 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "4634-4637",
        "Text": "5\n\u00a9 NCERT\nnot to be republished\nDIFFERENTIAL EQUATIONS\n391\nEXERCISE 9 3\nIn each of the Exercises 1 to 5, form a differential equation representing the given\nfamily of curves by eliminating arbitrary constants a and b 1 1\nx\ny\na\n+b\n=\n2"
    },
    {
        "Chapter": "1",
        "sentence_range": "4635-4638",
        "Text": "3\nIn each of the Exercises 1 to 5, form a differential equation representing the given\nfamily of curves by eliminating arbitrary constants a and b 1 1\nx\ny\na\n+b\n=\n2 y2 = a (b2 \u2013 x2)\n3"
    },
    {
        "Chapter": "1",
        "sentence_range": "4636-4639",
        "Text": "1 1\nx\ny\na\n+b\n=\n2 y2 = a (b2 \u2013 x2)\n3 y = a e3x + b e\u2013 2x\n4"
    },
    {
        "Chapter": "1",
        "sentence_range": "4637-4640",
        "Text": "1\nx\ny\na\n+b\n=\n2 y2 = a (b2 \u2013 x2)\n3 y = a e3x + b e\u2013 2x\n4 y = e2x (a + bx)\n5"
    },
    {
        "Chapter": "1",
        "sentence_range": "4638-4641",
        "Text": "y2 = a (b2 \u2013 x2)\n3 y = a e3x + b e\u2013 2x\n4 y = e2x (a + bx)\n5 y = ex (a cos x + b sin x)\n6"
    },
    {
        "Chapter": "1",
        "sentence_range": "4639-4642",
        "Text": "y = a e3x + b e\u2013 2x\n4 y = e2x (a + bx)\n5 y = ex (a cos x + b sin x)\n6 Form the differential equation of the family of circles touching the y-axis at\norigin"
    },
    {
        "Chapter": "1",
        "sentence_range": "4640-4643",
        "Text": "y = e2x (a + bx)\n5 y = ex (a cos x + b sin x)\n6 Form the differential equation of the family of circles touching the y-axis at\norigin 7"
    },
    {
        "Chapter": "1",
        "sentence_range": "4641-4644",
        "Text": "y = ex (a cos x + b sin x)\n6 Form the differential equation of the family of circles touching the y-axis at\norigin 7 Form the differential equation of the family of parabolas having vertex at origin\nand axis along positive y-axis"
    },
    {
        "Chapter": "1",
        "sentence_range": "4642-4645",
        "Text": "Form the differential equation of the family of circles touching the y-axis at\norigin 7 Form the differential equation of the family of parabolas having vertex at origin\nand axis along positive y-axis 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "4643-4646",
        "Text": "7 Form the differential equation of the family of parabolas having vertex at origin\nand axis along positive y-axis 8 Form the differential equation of the family of ellipses having foci on y-axis and\ncentre at origin"
    },
    {
        "Chapter": "1",
        "sentence_range": "4644-4647",
        "Text": "Form the differential equation of the family of parabolas having vertex at origin\nand axis along positive y-axis 8 Form the differential equation of the family of ellipses having foci on y-axis and\ncentre at origin 9"
    },
    {
        "Chapter": "1",
        "sentence_range": "4645-4648",
        "Text": "8 Form the differential equation of the family of ellipses having foci on y-axis and\ncentre at origin 9 Form the differential equation of the family of  hyperbolas having foci on x-axis\nand centre at origin"
    },
    {
        "Chapter": "1",
        "sentence_range": "4646-4649",
        "Text": "Form the differential equation of the family of ellipses having foci on y-axis and\ncentre at origin 9 Form the differential equation of the family of  hyperbolas having foci on x-axis\nand centre at origin 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "4647-4650",
        "Text": "9 Form the differential equation of the family of  hyperbolas having foci on x-axis\nand centre at origin 10 Form the differential equation of the family of circles having centre on y-axis\nand radius 3 units"
    },
    {
        "Chapter": "1",
        "sentence_range": "4648-4651",
        "Text": "Form the differential equation of the family of  hyperbolas having foci on x-axis\nand centre at origin 10 Form the differential equation of the family of circles having centre on y-axis\nand radius 3 units 11"
    },
    {
        "Chapter": "1",
        "sentence_range": "4649-4652",
        "Text": "10 Form the differential equation of the family of circles having centre on y-axis\nand radius 3 units 11 Which of the following differential equations has y = c1 ex + c2 e\u2013x as the general\nsolution"
    },
    {
        "Chapter": "1",
        "sentence_range": "4650-4653",
        "Text": "Form the differential equation of the family of circles having centre on y-axis\nand radius 3 units 11 Which of the following differential equations has y = c1 ex + c2 e\u2013x as the general\nsolution (A)\n2\n2\n0\nd y\ny\ndx\n+\n=\n(B)\n2\n2\n0\nd y\ny\ndx\n\u2212\n=\n(C)\n2\n2\n1\n0\nd y\ndx\n+ =\n(D)\n2\n2\n1\n0\nd y\ndx\n\u2212 =\n12"
    },
    {
        "Chapter": "1",
        "sentence_range": "4651-4654",
        "Text": "11 Which of the following differential equations has y = c1 ex + c2 e\u2013x as the general\nsolution (A)\n2\n2\n0\nd y\ny\ndx\n+\n=\n(B)\n2\n2\n0\nd y\ny\ndx\n\u2212\n=\n(C)\n2\n2\n1\n0\nd y\ndx\n+ =\n(D)\n2\n2\n1\n0\nd y\ndx\n\u2212 =\n12 Which of the following differential equations has y = x as one of its particular\nsolution"
    },
    {
        "Chapter": "1",
        "sentence_range": "4652-4655",
        "Text": "Which of the following differential equations has y = c1 ex + c2 e\u2013x as the general\nsolution (A)\n2\n2\n0\nd y\ny\ndx\n+\n=\n(B)\n2\n2\n0\nd y\ny\ndx\n\u2212\n=\n(C)\n2\n2\n1\n0\nd y\ndx\n+ =\n(D)\n2\n2\n1\n0\nd y\ndx\n\u2212 =\n12 Which of the following differential equations has y = x as one of its particular\nsolution (A)\n2\n2\nd y2\nxdy\nxy\nx\ndx\ndx\n\u2212\n+\n=\n(B)\n2\nd y2\nxdy\nxy\nx\ndx\ndx\n+\n+\n=\n(C)\n2\n2\n2\n0\nd y\nxdy\nxy\ndx\ndx\n \n \n \n(D)\n2\n2\n0\nd y\nxdy\nxy\ndx\ndx\n+\n+\n=\n9"
    },
    {
        "Chapter": "1",
        "sentence_range": "4653-4656",
        "Text": "(A)\n2\n2\n0\nd y\ny\ndx\n+\n=\n(B)\n2\n2\n0\nd y\ny\ndx\n\u2212\n=\n(C)\n2\n2\n1\n0\nd y\ndx\n+ =\n(D)\n2\n2\n1\n0\nd y\ndx\n\u2212 =\n12 Which of the following differential equations has y = x as one of its particular\nsolution (A)\n2\n2\nd y2\nxdy\nxy\nx\ndx\ndx\n\u2212\n+\n=\n(B)\n2\nd y2\nxdy\nxy\nx\ndx\ndx\n+\n+\n=\n(C)\n2\n2\n2\n0\nd y\nxdy\nxy\ndx\ndx\n \n \n \n(D)\n2\n2\n0\nd y\nxdy\nxy\ndx\ndx\n+\n+\n=\n9 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "4654-4657",
        "Text": "Which of the following differential equations has y = x as one of its particular\nsolution (A)\n2\n2\nd y2\nxdy\nxy\nx\ndx\ndx\n\u2212\n+\n=\n(B)\n2\nd y2\nxdy\nxy\nx\ndx\ndx\n+\n+\n=\n(C)\n2\n2\n2\n0\nd y\nxdy\nxy\ndx\ndx\n \n \n \n(D)\n2\n2\n0\nd y\nxdy\nxy\ndx\ndx\n+\n+\n=\n9 5 Methods of Solving First Order, First Degree Differential Equations\nIn this section we shall discuss three methods of solving first order first degree differential\nequations"
    },
    {
        "Chapter": "1",
        "sentence_range": "4655-4658",
        "Text": "(A)\n2\n2\nd y2\nxdy\nxy\nx\ndx\ndx\n\u2212\n+\n=\n(B)\n2\nd y2\nxdy\nxy\nx\ndx\ndx\n+\n+\n=\n(C)\n2\n2\n2\n0\nd y\nxdy\nxy\ndx\ndx\n \n \n \n(D)\n2\n2\n0\nd y\nxdy\nxy\ndx\ndx\n+\n+\n=\n9 5 Methods of Solving First Order, First Degree Differential Equations\nIn this section we shall discuss three methods of solving first order first degree differential\nequations 9"
    },
    {
        "Chapter": "1",
        "sentence_range": "4656-4659",
        "Text": "5 Methods of Solving First Order, First Degree Differential Equations\nIn this section we shall discuss three methods of solving first order first degree differential\nequations 9 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "4657-4660",
        "Text": "Methods of Solving First Order, First Degree Differential Equations\nIn this section we shall discuss three methods of solving first order first degree differential\nequations 9 5 1  Differential equations with variables separable\nA first order-first degree differential equation is of the form\ndy\ndx  = F(x, y)"
    },
    {
        "Chapter": "1",
        "sentence_range": "4658-4661",
        "Text": "9 5 1  Differential equations with variables separable\nA first order-first degree differential equation is of the form\ndy\ndx  = F(x, y) (1)\n\u00a9 NCERT\nnot to be republished\nMATHEMATICS\n392\nIf F (x, y) can be expressed as a product g (x) h(y), where, g(x) is a function of x\nand h(y) is a function of y, then the differential equation (1) is said to be of variable\nseparable type"
    },
    {
        "Chapter": "1",
        "sentence_range": "4659-4662",
        "Text": "5 1  Differential equations with variables separable\nA first order-first degree differential equation is of the form\ndy\ndx  = F(x, y) (1)\n\u00a9 NCERT\nnot to be republished\nMATHEMATICS\n392\nIf F (x, y) can be expressed as a product g (x) h(y), where, g(x) is a function of x\nand h(y) is a function of y, then the differential equation (1) is said to be of variable\nseparable type The differential equation (1) then has the form\ndy\ndx  = h (y)"
    },
    {
        "Chapter": "1",
        "sentence_range": "4660-4663",
        "Text": "1  Differential equations with variables separable\nA first order-first degree differential equation is of the form\ndy\ndx  = F(x, y) (1)\n\u00a9 NCERT\nnot to be republished\nMATHEMATICS\n392\nIf F (x, y) can be expressed as a product g (x) h(y), where, g(x) is a function of x\nand h(y) is a function of y, then the differential equation (1) is said to be of variable\nseparable type The differential equation (1) then has the form\ndy\ndx  = h (y) g(x)"
    },
    {
        "Chapter": "1",
        "sentence_range": "4661-4664",
        "Text": "(1)\n\u00a9 NCERT\nnot to be republished\nMATHEMATICS\n392\nIf F (x, y) can be expressed as a product g (x) h(y), where, g(x) is a function of x\nand h(y) is a function of y, then the differential equation (1) is said to be of variable\nseparable type The differential equation (1) then has the form\ndy\ndx  = h (y) g(x) (2)\nIf h(y) \u2260 0, separating the variables, (2) can be rewritten as\n1\n( )\nh y  dy = g(x) dx"
    },
    {
        "Chapter": "1",
        "sentence_range": "4662-4665",
        "Text": "The differential equation (1) then has the form\ndy\ndx  = h (y) g(x) (2)\nIf h(y) \u2260 0, separating the variables, (2) can be rewritten as\n1\n( )\nh y  dy = g(x) dx (3)\nIntegrating both sides of (3), we get\n1\n( ) dy\n\u222bh y\n=\n( )\n\u222bg x dx"
    },
    {
        "Chapter": "1",
        "sentence_range": "4663-4666",
        "Text": "g(x) (2)\nIf h(y) \u2260 0, separating the variables, (2) can be rewritten as\n1\n( )\nh y  dy = g(x) dx (3)\nIntegrating both sides of (3), we get\n1\n( ) dy\n\u222bh y\n=\n( )\n\u222bg x dx (4)\nThus, (4) provides the solutions of given differential equation in the form\nH(y) = G(x) + C\nHere, H (y) and G (x) are the anti derivatives of \n1\n( )\nh y  and g(x) respectively and\nC is the arbitrary constant"
    },
    {
        "Chapter": "1",
        "sentence_range": "4664-4667",
        "Text": "(2)\nIf h(y) \u2260 0, separating the variables, (2) can be rewritten as\n1\n( )\nh y  dy = g(x) dx (3)\nIntegrating both sides of (3), we get\n1\n( ) dy\n\u222bh y\n=\n( )\n\u222bg x dx (4)\nThus, (4) provides the solutions of given differential equation in the form\nH(y) = G(x) + C\nHere, H (y) and G (x) are the anti derivatives of \n1\n( )\nh y  and g(x) respectively and\nC is the arbitrary constant Example 9 Find the general solution of the differential equation \n1\n2\ndy\nx\ndx\n+y\n=\n\u2212\n, (y \u2260 2)\nSolution We have\ndy\ndx  =\n1\nx2\n\u2212+y"
    },
    {
        "Chapter": "1",
        "sentence_range": "4665-4668",
        "Text": "(3)\nIntegrating both sides of (3), we get\n1\n( ) dy\n\u222bh y\n=\n( )\n\u222bg x dx (4)\nThus, (4) provides the solutions of given differential equation in the form\nH(y) = G(x) + C\nHere, H (y) and G (x) are the anti derivatives of \n1\n( )\nh y  and g(x) respectively and\nC is the arbitrary constant Example 9 Find the general solution of the differential equation \n1\n2\ndy\nx\ndx\n+y\n=\n\u2212\n, (y \u2260 2)\nSolution We have\ndy\ndx  =\n1\nx2\n\u2212+y (1)\nSeparating the variables in equation (1), we get\n(2 \u2013 y) dy = (x + 1) dx"
    },
    {
        "Chapter": "1",
        "sentence_range": "4666-4669",
        "Text": "(4)\nThus, (4) provides the solutions of given differential equation in the form\nH(y) = G(x) + C\nHere, H (y) and G (x) are the anti derivatives of \n1\n( )\nh y  and g(x) respectively and\nC is the arbitrary constant Example 9 Find the general solution of the differential equation \n1\n2\ndy\nx\ndx\n+y\n=\n\u2212\n, (y \u2260 2)\nSolution We have\ndy\ndx  =\n1\nx2\n\u2212+y (1)\nSeparating the variables in equation (1), we get\n(2 \u2013 y) dy = (x + 1) dx (2)\nIntegrating both sides of equation (2), we get\n(2\n)\n\u222b\u2212y dy\n=\n(\n1)\nx\ndx\n+\n\u222b\nor\n2\n2\ny \u2212y2\n =\n2\nC1\nx2\n+x\n+\nor\nx2 + y2 + 2x \u2013 4y + 2 C1 = 0\nor\nx2 + y2 + 2x \u2013 4y + C = 0, where C = 2C1\nwhich is the general solution of equation (1)"
    },
    {
        "Chapter": "1",
        "sentence_range": "4667-4670",
        "Text": "Example 9 Find the general solution of the differential equation \n1\n2\ndy\nx\ndx\n+y\n=\n\u2212\n, (y \u2260 2)\nSolution We have\ndy\ndx  =\n1\nx2\n\u2212+y (1)\nSeparating the variables in equation (1), we get\n(2 \u2013 y) dy = (x + 1) dx (2)\nIntegrating both sides of equation (2), we get\n(2\n)\n\u222b\u2212y dy\n=\n(\n1)\nx\ndx\n+\n\u222b\nor\n2\n2\ny \u2212y2\n =\n2\nC1\nx2\n+x\n+\nor\nx2 + y2 + 2x \u2013 4y + 2 C1 = 0\nor\nx2 + y2 + 2x \u2013 4y + C = 0, where C = 2C1\nwhich is the general solution of equation (1) \u00a9 NCERT\nnot to be republished\nDIFFERENTIAL EQUATIONS\n393\nExample 10 Find the general solution of the differential equation \n2\n2\n11\ndy\ny\ndx\nx\n=+\n+"
    },
    {
        "Chapter": "1",
        "sentence_range": "4668-4671",
        "Text": "(1)\nSeparating the variables in equation (1), we get\n(2 \u2013 y) dy = (x + 1) dx (2)\nIntegrating both sides of equation (2), we get\n(2\n)\n\u222b\u2212y dy\n=\n(\n1)\nx\ndx\n+\n\u222b\nor\n2\n2\ny \u2212y2\n =\n2\nC1\nx2\n+x\n+\nor\nx2 + y2 + 2x \u2013 4y + 2 C1 = 0\nor\nx2 + y2 + 2x \u2013 4y + C = 0, where C = 2C1\nwhich is the general solution of equation (1) \u00a9 NCERT\nnot to be republished\nDIFFERENTIAL EQUATIONS\n393\nExample 10 Find the general solution of the differential equation \n2\n2\n11\ndy\ny\ndx\nx\n=+\n+ Solution Since 1 + y2 \u2260 0, therefore separating the variables, the given differential\nequation can be written as\n2\n1\ndy\n+y\n =\n2\n1\ndx\n+x"
    },
    {
        "Chapter": "1",
        "sentence_range": "4669-4672",
        "Text": "(2)\nIntegrating both sides of equation (2), we get\n(2\n)\n\u222b\u2212y dy\n=\n(\n1)\nx\ndx\n+\n\u222b\nor\n2\n2\ny \u2212y2\n =\n2\nC1\nx2\n+x\n+\nor\nx2 + y2 + 2x \u2013 4y + 2 C1 = 0\nor\nx2 + y2 + 2x \u2013 4y + C = 0, where C = 2C1\nwhich is the general solution of equation (1) \u00a9 NCERT\nnot to be republished\nDIFFERENTIAL EQUATIONS\n393\nExample 10 Find the general solution of the differential equation \n2\n2\n11\ndy\ny\ndx\nx\n=+\n+ Solution Since 1 + y2 \u2260 0, therefore separating the variables, the given differential\nequation can be written as\n2\n1\ndy\n+y\n =\n2\n1\ndx\n+x (1)\nIntegrating both sides of equation (1), we get\n2\n1\ndy\n\u222b+y\n =\n2\n1\ndx\n+x\n\u222b\nor\ntan\u20131 y = tan\u20131x + C\nwhich is the general solution of equation (1)"
    },
    {
        "Chapter": "1",
        "sentence_range": "4670-4673",
        "Text": "\u00a9 NCERT\nnot to be republished\nDIFFERENTIAL EQUATIONS\n393\nExample 10 Find the general solution of the differential equation \n2\n2\n11\ndy\ny\ndx\nx\n=+\n+ Solution Since 1 + y2 \u2260 0, therefore separating the variables, the given differential\nequation can be written as\n2\n1\ndy\n+y\n =\n2\n1\ndx\n+x (1)\nIntegrating both sides of equation (1), we get\n2\n1\ndy\n\u222b+y\n =\n2\n1\ndx\n+x\n\u222b\nor\ntan\u20131 y = tan\u20131x + C\nwhich is the general solution of equation (1) Example 11 Find the particular solution of the differential equation \n2\n4\ndy\ndx = \u2212xy\n given\nthat y = 1, when x = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "4671-4674",
        "Text": "Solution Since 1 + y2 \u2260 0, therefore separating the variables, the given differential\nequation can be written as\n2\n1\ndy\n+y\n =\n2\n1\ndx\n+x (1)\nIntegrating both sides of equation (1), we get\n2\n1\ndy\n\u222b+y\n =\n2\n1\ndx\n+x\n\u222b\nor\ntan\u20131 y = tan\u20131x + C\nwhich is the general solution of equation (1) Example 11 Find the particular solution of the differential equation \n2\n4\ndy\ndx = \u2212xy\n given\nthat y = 1, when x = 0 Solution If y \u2260 0, the given differential equation can be written as\ndy2\ny  = \u2013 4x dx"
    },
    {
        "Chapter": "1",
        "sentence_range": "4672-4675",
        "Text": "(1)\nIntegrating both sides of equation (1), we get\n2\n1\ndy\n\u222b+y\n =\n2\n1\ndx\n+x\n\u222b\nor\ntan\u20131 y = tan\u20131x + C\nwhich is the general solution of equation (1) Example 11 Find the particular solution of the differential equation \n2\n4\ndy\ndx = \u2212xy\n given\nthat y = 1, when x = 0 Solution If y \u2260 0, the given differential equation can be written as\ndy2\ny  = \u2013 4x dx (1)\nIntegrating both sides of equation (1), we get\ny\u222bdy2\n =\n4 x dx\n\u2212 \u222b\nor\n\u2212y1\n = \u2013 2x2 + C\nor\ny =\n212\nx \u2212C"
    },
    {
        "Chapter": "1",
        "sentence_range": "4673-4676",
        "Text": "Example 11 Find the particular solution of the differential equation \n2\n4\ndy\ndx = \u2212xy\n given\nthat y = 1, when x = 0 Solution If y \u2260 0, the given differential equation can be written as\ndy2\ny  = \u2013 4x dx (1)\nIntegrating both sides of equation (1), we get\ny\u222bdy2\n =\n4 x dx\n\u2212 \u222b\nor\n\u2212y1\n = \u2013 2x2 + C\nor\ny =\n212\nx \u2212C (2)\nSubstituting y = 1 and x = 0 in equation (2), we get,  C = \u2013 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "4674-4677",
        "Text": "Solution If y \u2260 0, the given differential equation can be written as\ndy2\ny  = \u2013 4x dx (1)\nIntegrating both sides of equation (1), we get\ny\u222bdy2\n =\n4 x dx\n\u2212 \u222b\nor\n\u2212y1\n = \u2013 2x2 + C\nor\ny =\n212\nx \u2212C (2)\nSubstituting y = 1 and x = 0 in equation (2), we get,  C = \u2013 1 Now substituting the value of C in equation (2), we get the particular solution of the\ngiven differential equation as \n212\n1\ny\nx\n=\n+"
    },
    {
        "Chapter": "1",
        "sentence_range": "4675-4678",
        "Text": "(1)\nIntegrating both sides of equation (1), we get\ny\u222bdy2\n =\n4 x dx\n\u2212 \u222b\nor\n\u2212y1\n = \u2013 2x2 + C\nor\ny =\n212\nx \u2212C (2)\nSubstituting y = 1 and x = 0 in equation (2), we get,  C = \u2013 1 Now substituting the value of C in equation (2), we get the particular solution of the\ngiven differential equation as \n212\n1\ny\nx\n=\n+ Example 12 Find the equation of the curve passing through the point (1, 1) whose\ndifferential equation is x dy = (2x2 + 1) dx (x \u2260 0)"
    },
    {
        "Chapter": "1",
        "sentence_range": "4676-4679",
        "Text": "(2)\nSubstituting y = 1 and x = 0 in equation (2), we get,  C = \u2013 1 Now substituting the value of C in equation (2), we get the particular solution of the\ngiven differential equation as \n212\n1\ny\nx\n=\n+ Example 12 Find the equation of the curve passing through the point (1, 1) whose\ndifferential equation is x dy = (2x2 + 1) dx (x \u2260 0) \u00a9 NCERT\nnot to be republished\nMATHEMATICS\n394\nSolution The given differential equation can be expressed as\ndy* =\n22\n1\n*\nx\ndx\nx\n \n \n \n \n \n \n \nor\ndy =\n1\n2x\nxdx\n\u239b\n\u239e\n+\n\u239c\n\u239f\n\u239d\n\u23a0"
    },
    {
        "Chapter": "1",
        "sentence_range": "4677-4680",
        "Text": "Now substituting the value of C in equation (2), we get the particular solution of the\ngiven differential equation as \n212\n1\ny\nx\n=\n+ Example 12 Find the equation of the curve passing through the point (1, 1) whose\ndifferential equation is x dy = (2x2 + 1) dx (x \u2260 0) \u00a9 NCERT\nnot to be republished\nMATHEMATICS\n394\nSolution The given differential equation can be expressed as\ndy* =\n22\n1\n*\nx\ndx\nx\n \n \n \n \n \n \n \nor\ndy =\n1\n2x\nxdx\n\u239b\n\u239e\n+\n\u239c\n\u239f\n\u239d\n\u23a0 (1)\nIntegrating both sides of equation (1), we get\n\u222bdy\n =\n1\n2x\nxdx\n\u239b\n\u239e\n+\n\u239c\n\u239f\n\u239d\n\u23a0\n\u222b\nor\ny = x2 + log |x| + C"
    },
    {
        "Chapter": "1",
        "sentence_range": "4678-4681",
        "Text": "Example 12 Find the equation of the curve passing through the point (1, 1) whose\ndifferential equation is x dy = (2x2 + 1) dx (x \u2260 0) \u00a9 NCERT\nnot to be republished\nMATHEMATICS\n394\nSolution The given differential equation can be expressed as\ndy* =\n22\n1\n*\nx\ndx\nx\n \n \n \n \n \n \n \nor\ndy =\n1\n2x\nxdx\n\u239b\n\u239e\n+\n\u239c\n\u239f\n\u239d\n\u23a0 (1)\nIntegrating both sides of equation (1), we get\n\u222bdy\n =\n1\n2x\nxdx\n\u239b\n\u239e\n+\n\u239c\n\u239f\n\u239d\n\u23a0\n\u222b\nor\ny = x2 + log |x| + C (2)\nEquation (2) represents the family of solution curves of the given differential equation\nbut we are interested in finding the equation of a particular member of the family which\npasses through the point (1, 1)"
    },
    {
        "Chapter": "1",
        "sentence_range": "4679-4682",
        "Text": "\u00a9 NCERT\nnot to be republished\nMATHEMATICS\n394\nSolution The given differential equation can be expressed as\ndy* =\n22\n1\n*\nx\ndx\nx\n \n \n \n \n \n \n \nor\ndy =\n1\n2x\nxdx\n\u239b\n\u239e\n+\n\u239c\n\u239f\n\u239d\n\u23a0 (1)\nIntegrating both sides of equation (1), we get\n\u222bdy\n =\n1\n2x\nxdx\n\u239b\n\u239e\n+\n\u239c\n\u239f\n\u239d\n\u23a0\n\u222b\nor\ny = x2 + log |x| + C (2)\nEquation (2) represents the family of solution curves of the given differential equation\nbut we are interested in finding the equation of a particular member of the family which\npasses through the point (1, 1) Therefore substituting x = 1, y = 1 in equation (2), we\nget C = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "4680-4683",
        "Text": "(1)\nIntegrating both sides of equation (1), we get\n\u222bdy\n =\n1\n2x\nxdx\n\u239b\n\u239e\n+\n\u239c\n\u239f\n\u239d\n\u23a0\n\u222b\nor\ny = x2 + log |x| + C (2)\nEquation (2) represents the family of solution curves of the given differential equation\nbut we are interested in finding the equation of a particular member of the family which\npasses through the point (1, 1) Therefore substituting x = 1, y = 1 in equation (2), we\nget C = 0 Now substituting the value of C in equation (2) we get the equation of the required\ncurve as y = x2 + log |x|"
    },
    {
        "Chapter": "1",
        "sentence_range": "4681-4684",
        "Text": "(2)\nEquation (2) represents the family of solution curves of the given differential equation\nbut we are interested in finding the equation of a particular member of the family which\npasses through the point (1, 1) Therefore substituting x = 1, y = 1 in equation (2), we\nget C = 0 Now substituting the value of C in equation (2) we get the equation of the required\ncurve as y = x2 + log |x| Example 13 Find the equation of a curve passing through the point (\u20132, 3), given that\nthe slope of the tangent to the curve at any point (x, y) is \n2x2\ny"
    },
    {
        "Chapter": "1",
        "sentence_range": "4682-4685",
        "Text": "Therefore substituting x = 1, y = 1 in equation (2), we\nget C = 0 Now substituting the value of C in equation (2) we get the equation of the required\ncurve as y = x2 + log |x| Example 13 Find the equation of a curve passing through the point (\u20132, 3), given that\nthe slope of the tangent to the curve at any point (x, y) is \n2x2\ny Solution We know that the slope of the tangent to a curve is given by dy\ndx"
    },
    {
        "Chapter": "1",
        "sentence_range": "4683-4686",
        "Text": "Now substituting the value of C in equation (2) we get the equation of the required\ncurve as y = x2 + log |x| Example 13 Find the equation of a curve passing through the point (\u20132, 3), given that\nthe slope of the tangent to the curve at any point (x, y) is \n2x2\ny Solution We know that the slope of the tangent to a curve is given by dy\ndx so,\ndy\ndx  =\ny2x2"
    },
    {
        "Chapter": "1",
        "sentence_range": "4684-4687",
        "Text": "Example 13 Find the equation of a curve passing through the point (\u20132, 3), given that\nthe slope of the tangent to the curve at any point (x, y) is \n2x2\ny Solution We know that the slope of the tangent to a curve is given by dy\ndx so,\ndy\ndx  =\ny2x2 (1)\nSeparating the variables, equation (1) can be written as\ny2 dy = 2x dx"
    },
    {
        "Chapter": "1",
        "sentence_range": "4685-4688",
        "Text": "Solution We know that the slope of the tangent to a curve is given by dy\ndx so,\ndy\ndx  =\ny2x2 (1)\nSeparating the variables, equation (1) can be written as\ny2 dy = 2x dx (2)\nIntegrating both sides of equation (2), we get\n2\n\u222by dy\n =\n2x dx\n\u222b\nor\n3\n3\ny  = x2 + C"
    },
    {
        "Chapter": "1",
        "sentence_range": "4686-4689",
        "Text": "so,\ndy\ndx  =\ny2x2 (1)\nSeparating the variables, equation (1) can be written as\ny2 dy = 2x dx (2)\nIntegrating both sides of equation (2), we get\n2\n\u222by dy\n =\n2x dx\n\u222b\nor\n3\n3\ny  = x2 + C (3)\n*\nThe notation\ndy\ndx due to Leibnitz is extremely flexible and useful in many calculation and formal\ntransformations, where, we can deal with symbols dy and dx exactly as if they were ordinary numbers"
    },
    {
        "Chapter": "1",
        "sentence_range": "4687-4690",
        "Text": "(1)\nSeparating the variables, equation (1) can be written as\ny2 dy = 2x dx (2)\nIntegrating both sides of equation (2), we get\n2\n\u222by dy\n =\n2x dx\n\u222b\nor\n3\n3\ny  = x2 + C (3)\n*\nThe notation\ndy\ndx due to Leibnitz is extremely flexible and useful in many calculation and formal\ntransformations, where, we can deal with symbols dy and dx exactly as if they were ordinary numbers By\ntreating dx and dy like separate entities, we can give neater expressions to many calculations"
    },
    {
        "Chapter": "1",
        "sentence_range": "4688-4691",
        "Text": "(2)\nIntegrating both sides of equation (2), we get\n2\n\u222by dy\n =\n2x dx\n\u222b\nor\n3\n3\ny  = x2 + C (3)\n*\nThe notation\ndy\ndx due to Leibnitz is extremely flexible and useful in many calculation and formal\ntransformations, where, we can deal with symbols dy and dx exactly as if they were ordinary numbers By\ntreating dx and dy like separate entities, we can give neater expressions to many calculations Refer: Introduction to Calculus and Analysis, volume-I page 172, By Richard Courant,\nFritz John Spinger \u2013 Verlog New York"
    },
    {
        "Chapter": "1",
        "sentence_range": "4689-4692",
        "Text": "(3)\n*\nThe notation\ndy\ndx due to Leibnitz is extremely flexible and useful in many calculation and formal\ntransformations, where, we can deal with symbols dy and dx exactly as if they were ordinary numbers By\ntreating dx and dy like separate entities, we can give neater expressions to many calculations Refer: Introduction to Calculus and Analysis, volume-I page 172, By Richard Courant,\nFritz John Spinger \u2013 Verlog New York \u00a9 NCERT\nnot to be republished\nDIFFERENTIAL EQUATIONS\n395\nSubstituting x = \u20132, y = 3 in equation (3), we get C = 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "4690-4693",
        "Text": "By\ntreating dx and dy like separate entities, we can give neater expressions to many calculations Refer: Introduction to Calculus and Analysis, volume-I page 172, By Richard Courant,\nFritz John Spinger \u2013 Verlog New York \u00a9 NCERT\nnot to be republished\nDIFFERENTIAL EQUATIONS\n395\nSubstituting x = \u20132, y = 3 in equation (3), we get C = 5 Substituting the value of C in equation (3), we get the equation of the required curve as\n3\n2\n5\ny3\n=x\n+\n   or   \n1\n2\n3\n(3\n15)\ny\nx\n=\n+\nExample 14 In a bank, principal increases continuously at the rate of 5% per year"
    },
    {
        "Chapter": "1",
        "sentence_range": "4691-4694",
        "Text": "Refer: Introduction to Calculus and Analysis, volume-I page 172, By Richard Courant,\nFritz John Spinger \u2013 Verlog New York \u00a9 NCERT\nnot to be republished\nDIFFERENTIAL EQUATIONS\n395\nSubstituting x = \u20132, y = 3 in equation (3), we get C = 5 Substituting the value of C in equation (3), we get the equation of the required curve as\n3\n2\n5\ny3\n=x\n+\n   or   \n1\n2\n3\n(3\n15)\ny\nx\n=\n+\nExample 14 In a bank, principal increases continuously at the rate of 5% per year In\nhow many years Rs 1000 double itself"
    },
    {
        "Chapter": "1",
        "sentence_range": "4692-4695",
        "Text": "\u00a9 NCERT\nnot to be republished\nDIFFERENTIAL EQUATIONS\n395\nSubstituting x = \u20132, y = 3 in equation (3), we get C = 5 Substituting the value of C in equation (3), we get the equation of the required curve as\n3\n2\n5\ny3\n=x\n+\n   or   \n1\n2\n3\n(3\n15)\ny\nx\n=\n+\nExample 14 In a bank, principal increases continuously at the rate of 5% per year In\nhow many years Rs 1000 double itself Solution Let P be the principal at any time t"
    },
    {
        "Chapter": "1",
        "sentence_range": "4693-4696",
        "Text": "Substituting the value of C in equation (3), we get the equation of the required curve as\n3\n2\n5\ny3\n=x\n+\n   or   \n1\n2\n3\n(3\n15)\ny\nx\n=\n+\nExample 14 In a bank, principal increases continuously at the rate of 5% per year In\nhow many years Rs 1000 double itself Solution Let P be the principal at any time t According to the given problem,\ndp\ndt  =\n5\nP\n\u239b100\n\u239e\u00d7\n\u239c\n\u239f\n\u239d\n\u23a0\nor\ndp\ndt  = P\n20"
    },
    {
        "Chapter": "1",
        "sentence_range": "4694-4697",
        "Text": "In\nhow many years Rs 1000 double itself Solution Let P be the principal at any time t According to the given problem,\ndp\ndt  =\n5\nP\n\u239b100\n\u239e\u00d7\n\u239c\n\u239f\n\u239d\n\u23a0\nor\ndp\ndt  = P\n20 (1)\nseparating the variables in equation (1), we get\nP\ndp  = 20\ndt"
    },
    {
        "Chapter": "1",
        "sentence_range": "4695-4698",
        "Text": "Solution Let P be the principal at any time t According to the given problem,\ndp\ndt  =\n5\nP\n\u239b100\n\u239e\u00d7\n\u239c\n\u239f\n\u239d\n\u23a0\nor\ndp\ndt  = P\n20 (1)\nseparating the variables in equation (1), we get\nP\ndp  = 20\ndt (2)\nIntegrating both sides of equation (2), we get\nlog P =\nC1\n20\nt +\nor\nP =\nC1\net20\ne\u22c5\nor\nP =\n20\nC\net\n (where \nC1\nC\ne\n=\n)"
    },
    {
        "Chapter": "1",
        "sentence_range": "4696-4699",
        "Text": "According to the given problem,\ndp\ndt  =\n5\nP\n\u239b100\n\u239e\u00d7\n\u239c\n\u239f\n\u239d\n\u23a0\nor\ndp\ndt  = P\n20 (1)\nseparating the variables in equation (1), we get\nP\ndp  = 20\ndt (2)\nIntegrating both sides of equation (2), we get\nlog P =\nC1\n20\nt +\nor\nP =\nC1\net20\ne\u22c5\nor\nP =\n20\nC\net\n (where \nC1\nC\ne\n=\n) (3)\nNow\nP = 1000,   when t = 0\nSubstituting the values of P and t in (3), we get C = 1000"
    },
    {
        "Chapter": "1",
        "sentence_range": "4697-4700",
        "Text": "(1)\nseparating the variables in equation (1), we get\nP\ndp  = 20\ndt (2)\nIntegrating both sides of equation (2), we get\nlog P =\nC1\n20\nt +\nor\nP =\nC1\net20\ne\u22c5\nor\nP =\n20\nC\net\n (where \nC1\nC\ne\n=\n) (3)\nNow\nP = 1000,   when t = 0\nSubstituting the values of P and t in (3), we get C = 1000 Therefore, equation (3),\ngives\nP = 1000 \nt20\ne\nLet t years be the time required to double the principal"
    },
    {
        "Chapter": "1",
        "sentence_range": "4698-4701",
        "Text": "(2)\nIntegrating both sides of equation (2), we get\nlog P =\nC1\n20\nt +\nor\nP =\nC1\net20\ne\u22c5\nor\nP =\n20\nC\net\n (where \nC1\nC\ne\n=\n) (3)\nNow\nP = 1000,   when t = 0\nSubstituting the values of P and t in (3), we get C = 1000 Therefore, equation (3),\ngives\nP = 1000 \nt20\ne\nLet t years be the time required to double the principal Then\n2000 = 1000\net20\n    \u21d2  t = 20 loge2\nEXERCISE 9"
    },
    {
        "Chapter": "1",
        "sentence_range": "4699-4702",
        "Text": "(3)\nNow\nP = 1000,   when t = 0\nSubstituting the values of P and t in (3), we get C = 1000 Therefore, equation (3),\ngives\nP = 1000 \nt20\ne\nLet t years be the time required to double the principal Then\n2000 = 1000\net20\n    \u21d2  t = 20 loge2\nEXERCISE 9 4\nFor each of the differential equations in Exercises 1 to 10, find the general solution:\n1"
    },
    {
        "Chapter": "1",
        "sentence_range": "4700-4703",
        "Text": "Therefore, equation (3),\ngives\nP = 1000 \nt20\ne\nLet t years be the time required to double the principal Then\n2000 = 1000\net20\n    \u21d2  t = 20 loge2\nEXERCISE 9 4\nFor each of the differential equations in Exercises 1 to 10, find the general solution:\n1 1\ncos\n1\ncos\ndy\nx\ndx\nx\n= +\u2212\n2"
    },
    {
        "Chapter": "1",
        "sentence_range": "4701-4704",
        "Text": "Then\n2000 = 1000\net20\n    \u21d2  t = 20 loge2\nEXERCISE 9 4\nFor each of the differential equations in Exercises 1 to 10, find the general solution:\n1 1\ncos\n1\ncos\ndy\nx\ndx\nx\n= +\u2212\n2 2\n4\n( 2\n2)\ndy\ny\ny\ndx =\n\u2212\n\u2212 <\n<\n\u00a9 NCERT\nnot to be republished\nMATHEMATICS\n396\n3"
    },
    {
        "Chapter": "1",
        "sentence_range": "4702-4705",
        "Text": "4\nFor each of the differential equations in Exercises 1 to 10, find the general solution:\n1 1\ncos\n1\ncos\ndy\nx\ndx\nx\n= +\u2212\n2 2\n4\n( 2\n2)\ndy\ny\ny\ndx =\n\u2212\n\u2212 <\n<\n\u00a9 NCERT\nnot to be republished\nMATHEMATICS\n396\n3 1(\n1)\ndy\ny\ny\ndx +\n=\n\u2260\n4"
    },
    {
        "Chapter": "1",
        "sentence_range": "4703-4706",
        "Text": "1\ncos\n1\ncos\ndy\nx\ndx\nx\n= +\u2212\n2 2\n4\n( 2\n2)\ndy\ny\ny\ndx =\n\u2212\n\u2212 <\n<\n\u00a9 NCERT\nnot to be republished\nMATHEMATICS\n396\n3 1(\n1)\ndy\ny\ny\ndx +\n=\n\u2260\n4 sec2 x tan y dx + sec2 y tan x dy = 0\n5"
    },
    {
        "Chapter": "1",
        "sentence_range": "4704-4707",
        "Text": "2\n4\n( 2\n2)\ndy\ny\ny\ndx =\n\u2212\n\u2212 <\n<\n\u00a9 NCERT\nnot to be republished\nMATHEMATICS\n396\n3 1(\n1)\ndy\ny\ny\ndx +\n=\n\u2260\n4 sec2 x tan y dx + sec2 y tan x dy = 0\n5 (ex + e\u2013x) dy \u2013 (ex \u2013 e\u2013x) dx = 0\n6"
    },
    {
        "Chapter": "1",
        "sentence_range": "4705-4708",
        "Text": "1(\n1)\ndy\ny\ny\ndx +\n=\n\u2260\n4 sec2 x tan y dx + sec2 y tan x dy = 0\n5 (ex + e\u2013x) dy \u2013 (ex \u2013 e\u2013x) dx = 0\n6 2\n2\n(1\n) (1\n)\ndy\nx\ny\ndx =\n+\n+\n7"
    },
    {
        "Chapter": "1",
        "sentence_range": "4706-4709",
        "Text": "sec2 x tan y dx + sec2 y tan x dy = 0\n5 (ex + e\u2013x) dy \u2013 (ex \u2013 e\u2013x) dx = 0\n6 2\n2\n(1\n) (1\n)\ndy\nx\ny\ndx =\n+\n+\n7 y log y dx \u2013 x dy = 0\n8"
    },
    {
        "Chapter": "1",
        "sentence_range": "4707-4710",
        "Text": "(ex + e\u2013x) dy \u2013 (ex \u2013 e\u2013x) dx = 0\n6 2\n2\n(1\n) (1\n)\ndy\nx\ny\ndx =\n+\n+\n7 y log y dx \u2013 x dy = 0\n8 5\n5\nxdy\ny\ndx = \u2212\n9"
    },
    {
        "Chapter": "1",
        "sentence_range": "4708-4711",
        "Text": "2\n2\n(1\n) (1\n)\ndy\nx\ny\ndx =\n+\n+\n7 y log y dx \u2013 x dy = 0\n8 5\n5\nxdy\ny\ndx = \u2212\n9 sin1\ndy\nx\ndx\n\u2212\n=\n10"
    },
    {
        "Chapter": "1",
        "sentence_range": "4709-4712",
        "Text": "y log y dx \u2013 x dy = 0\n8 5\n5\nxdy\ny\ndx = \u2212\n9 sin1\ndy\nx\ndx\n\u2212\n=\n10 ex tan y dx + (1 \u2013 ex) sec2 y dy = 0\nFor each of the differential equations in Exercises 11 to 14, find a particular solution\nsatisfying the given condition:\n11"
    },
    {
        "Chapter": "1",
        "sentence_range": "4710-4713",
        "Text": "5\n5\nxdy\ny\ndx = \u2212\n9 sin1\ndy\nx\ndx\n\u2212\n=\n10 ex tan y dx + (1 \u2013 ex) sec2 y dy = 0\nFor each of the differential equations in Exercises 11 to 14, find a particular solution\nsatisfying the given condition:\n11 3\n2\n(\n1) dy\nx\nx\nx\ndx\n+\n+\n+\n = 2x2 + x; y = 1 when x = 0\n12"
    },
    {
        "Chapter": "1",
        "sentence_range": "4711-4714",
        "Text": "sin1\ndy\nx\ndx\n\u2212\n=\n10 ex tan y dx + (1 \u2013 ex) sec2 y dy = 0\nFor each of the differential equations in Exercises 11 to 14, find a particular solution\nsatisfying the given condition:\n11 3\n2\n(\n1) dy\nx\nx\nx\ndx\n+\n+\n+\n = 2x2 + x; y = 1 when x = 0\n12 (2\n1)\n1\ndy\nx x\ndx\n\u2212\n= ; y = 0 when x = 2\n13"
    },
    {
        "Chapter": "1",
        "sentence_range": "4712-4715",
        "Text": "ex tan y dx + (1 \u2013 ex) sec2 y dy = 0\nFor each of the differential equations in Exercises 11 to 14, find a particular solution\nsatisfying the given condition:\n11 3\n2\n(\n1) dy\nx\nx\nx\ndx\n+\n+\n+\n = 2x2 + x; y = 1 when x = 0\n12 (2\n1)\n1\ndy\nx x\ndx\n\u2212\n= ; y = 0 when x = 2\n13 cos\ndy\na\n\u239bdx\n\u239e =\n\u239c\n\u239f\n\u239d\n\u23a0\n (a \u2208 R); y = 2 when x = 0\n14"
    },
    {
        "Chapter": "1",
        "sentence_range": "4713-4716",
        "Text": "3\n2\n(\n1) dy\nx\nx\nx\ndx\n+\n+\n+\n = 2x2 + x; y = 1 when x = 0\n12 (2\n1)\n1\ndy\nx x\ndx\n\u2212\n= ; y = 0 when x = 2\n13 cos\ndy\na\n\u239bdx\n\u239e =\n\u239c\n\u239f\n\u239d\n\u23a0\n (a \u2208 R); y = 2 when x = 0\n14 tan\ndy\ny\nx\ndx =\n; y = 1 when x = 0\n15"
    },
    {
        "Chapter": "1",
        "sentence_range": "4714-4717",
        "Text": "(2\n1)\n1\ndy\nx x\ndx\n\u2212\n= ; y = 0 when x = 2\n13 cos\ndy\na\n\u239bdx\n\u239e =\n\u239c\n\u239f\n\u239d\n\u23a0\n (a \u2208 R); y = 2 when x = 0\n14 tan\ndy\ny\nx\ndx =\n; y = 1 when x = 0\n15 Find the equation of a curve passing through the point (0, 0) and whose differential\nequation is y\u2032 = ex sin x"
    },
    {
        "Chapter": "1",
        "sentence_range": "4715-4718",
        "Text": "cos\ndy\na\n\u239bdx\n\u239e =\n\u239c\n\u239f\n\u239d\n\u23a0\n (a \u2208 R); y = 2 when x = 0\n14 tan\ndy\ny\nx\ndx =\n; y = 1 when x = 0\n15 Find the equation of a curve passing through the point (0, 0) and whose differential\nequation is y\u2032 = ex sin x 16"
    },
    {
        "Chapter": "1",
        "sentence_range": "4716-4719",
        "Text": "tan\ndy\ny\nx\ndx =\n; y = 1 when x = 0\n15 Find the equation of a curve passing through the point (0, 0) and whose differential\nequation is y\u2032 = ex sin x 16 For the differential equation \n(\n2) (\n2)\nxydy\nx\ny\ndx =\n+\n+\n, find the solution curve\npassing through the point (1, \u20131)"
    },
    {
        "Chapter": "1",
        "sentence_range": "4717-4720",
        "Text": "Find the equation of a curve passing through the point (0, 0) and whose differential\nequation is y\u2032 = ex sin x 16 For the differential equation \n(\n2) (\n2)\nxydy\nx\ny\ndx =\n+\n+\n, find the solution curve\npassing through the point (1, \u20131) 17"
    },
    {
        "Chapter": "1",
        "sentence_range": "4718-4721",
        "Text": "16 For the differential equation \n(\n2) (\n2)\nxydy\nx\ny\ndx =\n+\n+\n, find the solution curve\npassing through the point (1, \u20131) 17 Find the equation of a curve passing through the point (0, \u20132) given that at any\npoint (x, y) on the curve, the product of the slope of its tangent and y coordinate\nof the point is equal to the x coordinate of the point"
    },
    {
        "Chapter": "1",
        "sentence_range": "4719-4722",
        "Text": "For the differential equation \n(\n2) (\n2)\nxydy\nx\ny\ndx =\n+\n+\n, find the solution curve\npassing through the point (1, \u20131) 17 Find the equation of a curve passing through the point (0, \u20132) given that at any\npoint (x, y) on the curve, the product of the slope of its tangent and y coordinate\nof the point is equal to the x coordinate of the point 18"
    },
    {
        "Chapter": "1",
        "sentence_range": "4720-4723",
        "Text": "17 Find the equation of a curve passing through the point (0, \u20132) given that at any\npoint (x, y) on the curve, the product of the slope of its tangent and y coordinate\nof the point is equal to the x coordinate of the point 18 At any point (x, y) of a curve, the slope of the tangent is twice the slope of the\nline segment joining the point of contact to the point (\u2013 4, \u20133)"
    },
    {
        "Chapter": "1",
        "sentence_range": "4721-4724",
        "Text": "Find the equation of a curve passing through the point (0, \u20132) given that at any\npoint (x, y) on the curve, the product of the slope of its tangent and y coordinate\nof the point is equal to the x coordinate of the point 18 At any point (x, y) of a curve, the slope of the tangent is twice the slope of the\nline segment joining the point of contact to the point (\u2013 4, \u20133) Find the equation\nof the curve given that it passes through (\u20132, 1)"
    },
    {
        "Chapter": "1",
        "sentence_range": "4722-4725",
        "Text": "18 At any point (x, y) of a curve, the slope of the tangent is twice the slope of the\nline segment joining the point of contact to the point (\u2013 4, \u20133) Find the equation\nof the curve given that it passes through (\u20132, 1) 19"
    },
    {
        "Chapter": "1",
        "sentence_range": "4723-4726",
        "Text": "At any point (x, y) of a curve, the slope of the tangent is twice the slope of the\nline segment joining the point of contact to the point (\u2013 4, \u20133) Find the equation\nof the curve given that it passes through (\u20132, 1) 19 The volume of spherical balloon being inflated changes at a constant rate"
    },
    {
        "Chapter": "1",
        "sentence_range": "4724-4727",
        "Text": "Find the equation\nof the curve given that it passes through (\u20132, 1) 19 The volume of spherical balloon being inflated changes at a constant rate If\ninitially its radius is 3 units and after 3 seconds it is 6 units"
    },
    {
        "Chapter": "1",
        "sentence_range": "4725-4728",
        "Text": "19 The volume of spherical balloon being inflated changes at a constant rate If\ninitially its radius is 3 units and after 3 seconds it is 6 units Find the radius of\nballoon after t seconds"
    },
    {
        "Chapter": "1",
        "sentence_range": "4726-4729",
        "Text": "The volume of spherical balloon being inflated changes at a constant rate If\ninitially its radius is 3 units and after 3 seconds it is 6 units Find the radius of\nballoon after t seconds \u00a9 NCERT\nnot to be republished\nDIFFERENTIAL EQUATIONS\n397\n20"
    },
    {
        "Chapter": "1",
        "sentence_range": "4727-4730",
        "Text": "If\ninitially its radius is 3 units and after 3 seconds it is 6 units Find the radius of\nballoon after t seconds \u00a9 NCERT\nnot to be republished\nDIFFERENTIAL EQUATIONS\n397\n20 In a bank, principal increases continuously at the rate of r% per year"
    },
    {
        "Chapter": "1",
        "sentence_range": "4728-4731",
        "Text": "Find the radius of\nballoon after t seconds \u00a9 NCERT\nnot to be republished\nDIFFERENTIAL EQUATIONS\n397\n20 In a bank, principal increases continuously at the rate of r% per year Find the\nvalue of r if Rs 100 double itself in 10 years (loge2 = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "4729-4732",
        "Text": "\u00a9 NCERT\nnot to be republished\nDIFFERENTIAL EQUATIONS\n397\n20 In a bank, principal increases continuously at the rate of r% per year Find the\nvalue of r if Rs 100 double itself in 10 years (loge2 = 0 6931)"
    },
    {
        "Chapter": "1",
        "sentence_range": "4730-4733",
        "Text": "In a bank, principal increases continuously at the rate of r% per year Find the\nvalue of r if Rs 100 double itself in 10 years (loge2 = 0 6931) 21"
    },
    {
        "Chapter": "1",
        "sentence_range": "4731-4734",
        "Text": "Find the\nvalue of r if Rs 100 double itself in 10 years (loge2 = 0 6931) 21 In a bank, principal increases continuously at the rate of 5% per year"
    },
    {
        "Chapter": "1",
        "sentence_range": "4732-4735",
        "Text": "6931) 21 In a bank, principal increases continuously at the rate of 5% per year An amount\nof Rs 1000 is deposited with this bank, how much will it worth after 10 years\n(e0"
    },
    {
        "Chapter": "1",
        "sentence_range": "4733-4736",
        "Text": "21 In a bank, principal increases continuously at the rate of 5% per year An amount\nof Rs 1000 is deposited with this bank, how much will it worth after 10 years\n(e0 5 = 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "4734-4737",
        "Text": "In a bank, principal increases continuously at the rate of 5% per year An amount\nof Rs 1000 is deposited with this bank, how much will it worth after 10 years\n(e0 5 = 1 648)"
    },
    {
        "Chapter": "1",
        "sentence_range": "4735-4738",
        "Text": "An amount\nof Rs 1000 is deposited with this bank, how much will it worth after 10 years\n(e0 5 = 1 648) 22"
    },
    {
        "Chapter": "1",
        "sentence_range": "4736-4739",
        "Text": "5 = 1 648) 22 In a culture, the bacteria count is 1,00,000"
    },
    {
        "Chapter": "1",
        "sentence_range": "4737-4740",
        "Text": "648) 22 In a culture, the bacteria count is 1,00,000 The number is increased by 10% in 2\nhours"
    },
    {
        "Chapter": "1",
        "sentence_range": "4738-4741",
        "Text": "22 In a culture, the bacteria count is 1,00,000 The number is increased by 10% in 2\nhours In how many hours will the count reach 2,00,000, if the rate of growth of\nbacteria is proportional to the number present"
    },
    {
        "Chapter": "1",
        "sentence_range": "4739-4742",
        "Text": "In a culture, the bacteria count is 1,00,000 The number is increased by 10% in 2\nhours In how many hours will the count reach 2,00,000, if the rate of growth of\nbacteria is proportional to the number present 23"
    },
    {
        "Chapter": "1",
        "sentence_range": "4740-4743",
        "Text": "The number is increased by 10% in 2\nhours In how many hours will the count reach 2,00,000, if the rate of growth of\nbacteria is proportional to the number present 23 The general solution of the differential equation \nx\ny\ndy\ne\ndx\n+\n=\n is\n(A) ex + e\u2013y = C\n(B) ex + ey = C\n(C) e\u2013x + ey = C\n(D) e\u2013x + e\u2013y = C\n9"
    },
    {
        "Chapter": "1",
        "sentence_range": "4741-4744",
        "Text": "In how many hours will the count reach 2,00,000, if the rate of growth of\nbacteria is proportional to the number present 23 The general solution of the differential equation \nx\ny\ndy\ne\ndx\n+\n=\n is\n(A) ex + e\u2013y = C\n(B) ex + ey = C\n(C) e\u2013x + ey = C\n(D) e\u2013x + e\u2013y = C\n9 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "4742-4745",
        "Text": "23 The general solution of the differential equation \nx\ny\ndy\ne\ndx\n+\n=\n is\n(A) ex + e\u2013y = C\n(B) ex + ey = C\n(C) e\u2013x + ey = C\n(D) e\u2013x + e\u2013y = C\n9 5 2  Homogeneous differential equations\nConsider the following functions in x and y\nF1 (x, y) = y2 + 2xy,\nF2 (x, y) = 2x \u2013 3y,\nF3 (x, y) = cos\ny\nx\n\u239b\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0 ,\nF4 (x, y) = sin x + cos y\nIf we replace x and y by \u03bbx and \u03bby respectively in the above functions, for any nonzero\nconstant \u03bb, we get\nF1 (\u03bbx, \u03bby) = \u03bb2 (y2 + 2xy) = \u03bb2 F1 (x, y)\nF2 (\u03bbx, \u03bby) = \u03bb (2x \u2013 3y) = \u03bb F2 (x, y)\nF3 (\u03bbx, \u03bby) = cos\ncos\ny\ny\nx\nx\n\u239b\u03bb\n\u239e\n\u239b\n\u239e\n=\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\u03bb\n\u23a0\n\u239d\n\u23a0  = \u03bb0  F3 (x, y)\nF4 (\u03bbx, \u03bby) = sin \u03bbx + cos \u03bby \u2260 \u03bbn F4 (x, y), for any n \u2208 N\nHere, we observe that the functions F1, F2, F3 can be written in the form\nF(\u03bbx, \u03bby) = \u03bbn F (x, y) but F4 can not be written in this form"
    },
    {
        "Chapter": "1",
        "sentence_range": "4743-4746",
        "Text": "The general solution of the differential equation \nx\ny\ndy\ne\ndx\n+\n=\n is\n(A) ex + e\u2013y = C\n(B) ex + ey = C\n(C) e\u2013x + ey = C\n(D) e\u2013x + e\u2013y = C\n9 5 2  Homogeneous differential equations\nConsider the following functions in x and y\nF1 (x, y) = y2 + 2xy,\nF2 (x, y) = 2x \u2013 3y,\nF3 (x, y) = cos\ny\nx\n\u239b\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0 ,\nF4 (x, y) = sin x + cos y\nIf we replace x and y by \u03bbx and \u03bby respectively in the above functions, for any nonzero\nconstant \u03bb, we get\nF1 (\u03bbx, \u03bby) = \u03bb2 (y2 + 2xy) = \u03bb2 F1 (x, y)\nF2 (\u03bbx, \u03bby) = \u03bb (2x \u2013 3y) = \u03bb F2 (x, y)\nF3 (\u03bbx, \u03bby) = cos\ncos\ny\ny\nx\nx\n\u239b\u03bb\n\u239e\n\u239b\n\u239e\n=\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\u03bb\n\u23a0\n\u239d\n\u23a0  = \u03bb0  F3 (x, y)\nF4 (\u03bbx, \u03bby) = sin \u03bbx + cos \u03bby \u2260 \u03bbn F4 (x, y), for any n \u2208 N\nHere, we observe that the functions F1, F2, F3 can be written in the form\nF(\u03bbx, \u03bby) = \u03bbn F (x, y) but F4 can not be written in this form This leads to the following\ndefinition:\nA function F(x, y) is said to be homogeneous function of degree n if\nF(\u03bbx, \u03bby) = \u03bbn F(x, y) for any nonzero constant \u03bb"
    },
    {
        "Chapter": "1",
        "sentence_range": "4744-4747",
        "Text": "5 2  Homogeneous differential equations\nConsider the following functions in x and y\nF1 (x, y) = y2 + 2xy,\nF2 (x, y) = 2x \u2013 3y,\nF3 (x, y) = cos\ny\nx\n\u239b\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0 ,\nF4 (x, y) = sin x + cos y\nIf we replace x and y by \u03bbx and \u03bby respectively in the above functions, for any nonzero\nconstant \u03bb, we get\nF1 (\u03bbx, \u03bby) = \u03bb2 (y2 + 2xy) = \u03bb2 F1 (x, y)\nF2 (\u03bbx, \u03bby) = \u03bb (2x \u2013 3y) = \u03bb F2 (x, y)\nF3 (\u03bbx, \u03bby) = cos\ncos\ny\ny\nx\nx\n\u239b\u03bb\n\u239e\n\u239b\n\u239e\n=\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\u03bb\n\u23a0\n\u239d\n\u23a0  = \u03bb0  F3 (x, y)\nF4 (\u03bbx, \u03bby) = sin \u03bbx + cos \u03bby \u2260 \u03bbn F4 (x, y), for any n \u2208 N\nHere, we observe that the functions F1, F2, F3 can be written in the form\nF(\u03bbx, \u03bby) = \u03bbn F (x, y) but F4 can not be written in this form This leads to the following\ndefinition:\nA function F(x, y) is said to be homogeneous function of degree n if\nF(\u03bbx, \u03bby) = \u03bbn F(x, y) for any nonzero constant \u03bb We note that in the above examples, F1, F2, F3 are homogeneous functions of\ndegree 2, 1, 0 respectively but F4 is not a homogeneous function"
    },
    {
        "Chapter": "1",
        "sentence_range": "4745-4748",
        "Text": "2  Homogeneous differential equations\nConsider the following functions in x and y\nF1 (x, y) = y2 + 2xy,\nF2 (x, y) = 2x \u2013 3y,\nF3 (x, y) = cos\ny\nx\n\u239b\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0 ,\nF4 (x, y) = sin x + cos y\nIf we replace x and y by \u03bbx and \u03bby respectively in the above functions, for any nonzero\nconstant \u03bb, we get\nF1 (\u03bbx, \u03bby) = \u03bb2 (y2 + 2xy) = \u03bb2 F1 (x, y)\nF2 (\u03bbx, \u03bby) = \u03bb (2x \u2013 3y) = \u03bb F2 (x, y)\nF3 (\u03bbx, \u03bby) = cos\ncos\ny\ny\nx\nx\n\u239b\u03bb\n\u239e\n\u239b\n\u239e\n=\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\u03bb\n\u23a0\n\u239d\n\u23a0  = \u03bb0  F3 (x, y)\nF4 (\u03bbx, \u03bby) = sin \u03bbx + cos \u03bby \u2260 \u03bbn F4 (x, y), for any n \u2208 N\nHere, we observe that the functions F1, F2, F3 can be written in the form\nF(\u03bbx, \u03bby) = \u03bbn F (x, y) but F4 can not be written in this form This leads to the following\ndefinition:\nA function F(x, y) is said to be homogeneous function of degree n if\nF(\u03bbx, \u03bby) = \u03bbn F(x, y) for any nonzero constant \u03bb We note that in the above examples, F1, F2, F3 are homogeneous functions of\ndegree 2, 1, 0 respectively but F4 is not a homogeneous function \u00a9 NCERT\nnot to be republished\nMATHEMATICS\n398\nWe also observe that\nF1(x, y) =\n2\n2\n2\n1\n2\n2\ny\ny\ny\nx\nx h\nx\nx\nx\n\u239b\n\u239e\n\u239b\n\u239e\n+\n=\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\nor\nF1(x, y) =\n2\n2\n2\n2\n1\nx\nx\ny\ny h\ny\ny\n\u239b\n\u239e\n\u239b\n\u239e\n+\n=\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\nF2(x, y) =\n1\n1\n3\n3\n2\ny\ny\nx\nx h\nx\nx\n\u239b\n\u239e\n\u239b\n\u239e\n\u2212\n=\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\nor\n F2(x, y) =\n1\n1\n4\n2\n3\nx\nx\ny\ny h\ny\ny\n\u239b\n\u239e\n\u239b\n\u239e\n\u2212\n=\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\nF3(x, y) =\n0\n0\n5\ncos y\ny\nx\nx h\nx\nx\n\u239b\n\u239e\n\u239b\n\u239e\n=\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\nF4(x, y) \u2260\n6\nn\ny\nx h\nx\n\u239b\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0 , for any n \u2208 N\nor\nF4 (x, y) \u2260\n7\nn\nx\ny h\n\u239by\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0 , for any n \u2208 N\nTherefore, a function F (x, y) is a homogeneous function of degree n if\nF(x, y) =\nor\nn\nn\ny\nx\nx g\ny h\nx\ny\n\u239b\n\u239e\n\u239b\n\u239e\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\nA differential equation of the form dy\ndx = F (x, y)  is said to be homogenous if\nF(x, y) is a homogenous function of degree zero"
    },
    {
        "Chapter": "1",
        "sentence_range": "4746-4749",
        "Text": "This leads to the following\ndefinition:\nA function F(x, y) is said to be homogeneous function of degree n if\nF(\u03bbx, \u03bby) = \u03bbn F(x, y) for any nonzero constant \u03bb We note that in the above examples, F1, F2, F3 are homogeneous functions of\ndegree 2, 1, 0 respectively but F4 is not a homogeneous function \u00a9 NCERT\nnot to be republished\nMATHEMATICS\n398\nWe also observe that\nF1(x, y) =\n2\n2\n2\n1\n2\n2\ny\ny\ny\nx\nx h\nx\nx\nx\n\u239b\n\u239e\n\u239b\n\u239e\n+\n=\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\nor\nF1(x, y) =\n2\n2\n2\n2\n1\nx\nx\ny\ny h\ny\ny\n\u239b\n\u239e\n\u239b\n\u239e\n+\n=\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\nF2(x, y) =\n1\n1\n3\n3\n2\ny\ny\nx\nx h\nx\nx\n\u239b\n\u239e\n\u239b\n\u239e\n\u2212\n=\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\nor\n F2(x, y) =\n1\n1\n4\n2\n3\nx\nx\ny\ny h\ny\ny\n\u239b\n\u239e\n\u239b\n\u239e\n\u2212\n=\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\nF3(x, y) =\n0\n0\n5\ncos y\ny\nx\nx h\nx\nx\n\u239b\n\u239e\n\u239b\n\u239e\n=\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\nF4(x, y) \u2260\n6\nn\ny\nx h\nx\n\u239b\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0 , for any n \u2208 N\nor\nF4 (x, y) \u2260\n7\nn\nx\ny h\n\u239by\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0 , for any n \u2208 N\nTherefore, a function F (x, y) is a homogeneous function of degree n if\nF(x, y) =\nor\nn\nn\ny\nx\nx g\ny h\nx\ny\n\u239b\n\u239e\n\u239b\n\u239e\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\nA differential equation of the form dy\ndx = F (x, y)  is said to be homogenous if\nF(x, y) is a homogenous function of degree zero To solve a homogeneous differential equation of the type\n(\n)\nF\n,\ndy\nx y\ndx =\n =\ny\ng\nx\n\u239b\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0"
    },
    {
        "Chapter": "1",
        "sentence_range": "4747-4750",
        "Text": "We note that in the above examples, F1, F2, F3 are homogeneous functions of\ndegree 2, 1, 0 respectively but F4 is not a homogeneous function \u00a9 NCERT\nnot to be republished\nMATHEMATICS\n398\nWe also observe that\nF1(x, y) =\n2\n2\n2\n1\n2\n2\ny\ny\ny\nx\nx h\nx\nx\nx\n\u239b\n\u239e\n\u239b\n\u239e\n+\n=\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\nor\nF1(x, y) =\n2\n2\n2\n2\n1\nx\nx\ny\ny h\ny\ny\n\u239b\n\u239e\n\u239b\n\u239e\n+\n=\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\nF2(x, y) =\n1\n1\n3\n3\n2\ny\ny\nx\nx h\nx\nx\n\u239b\n\u239e\n\u239b\n\u239e\n\u2212\n=\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\nor\n F2(x, y) =\n1\n1\n4\n2\n3\nx\nx\ny\ny h\ny\ny\n\u239b\n\u239e\n\u239b\n\u239e\n\u2212\n=\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\nF3(x, y) =\n0\n0\n5\ncos y\ny\nx\nx h\nx\nx\n\u239b\n\u239e\n\u239b\n\u239e\n=\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\nF4(x, y) \u2260\n6\nn\ny\nx h\nx\n\u239b\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0 , for any n \u2208 N\nor\nF4 (x, y) \u2260\n7\nn\nx\ny h\n\u239by\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0 , for any n \u2208 N\nTherefore, a function F (x, y) is a homogeneous function of degree n if\nF(x, y) =\nor\nn\nn\ny\nx\nx g\ny h\nx\ny\n\u239b\n\u239e\n\u239b\n\u239e\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\nA differential equation of the form dy\ndx = F (x, y)  is said to be homogenous if\nF(x, y) is a homogenous function of degree zero To solve a homogeneous differential equation of the type\n(\n)\nF\n,\ndy\nx y\ndx =\n =\ny\ng\nx\n\u239b\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0 (1)\nWe make the substitution\n    y = v"
    },
    {
        "Chapter": "1",
        "sentence_range": "4748-4751",
        "Text": "\u00a9 NCERT\nnot to be republished\nMATHEMATICS\n398\nWe also observe that\nF1(x, y) =\n2\n2\n2\n1\n2\n2\ny\ny\ny\nx\nx h\nx\nx\nx\n\u239b\n\u239e\n\u239b\n\u239e\n+\n=\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\nor\nF1(x, y) =\n2\n2\n2\n2\n1\nx\nx\ny\ny h\ny\ny\n\u239b\n\u239e\n\u239b\n\u239e\n+\n=\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\nF2(x, y) =\n1\n1\n3\n3\n2\ny\ny\nx\nx h\nx\nx\n\u239b\n\u239e\n\u239b\n\u239e\n\u2212\n=\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\nor\n F2(x, y) =\n1\n1\n4\n2\n3\nx\nx\ny\ny h\ny\ny\n\u239b\n\u239e\n\u239b\n\u239e\n\u2212\n=\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\nF3(x, y) =\n0\n0\n5\ncos y\ny\nx\nx h\nx\nx\n\u239b\n\u239e\n\u239b\n\u239e\n=\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\nF4(x, y) \u2260\n6\nn\ny\nx h\nx\n\u239b\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0 , for any n \u2208 N\nor\nF4 (x, y) \u2260\n7\nn\nx\ny h\n\u239by\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0 , for any n \u2208 N\nTherefore, a function F (x, y) is a homogeneous function of degree n if\nF(x, y) =\nor\nn\nn\ny\nx\nx g\ny h\nx\ny\n\u239b\n\u239e\n\u239b\n\u239e\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\nA differential equation of the form dy\ndx = F (x, y)  is said to be homogenous if\nF(x, y) is a homogenous function of degree zero To solve a homogeneous differential equation of the type\n(\n)\nF\n,\ndy\nx y\ndx =\n =\ny\ng\nx\n\u239b\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0 (1)\nWe make the substitution\n    y = v x"
    },
    {
        "Chapter": "1",
        "sentence_range": "4749-4752",
        "Text": "To solve a homogeneous differential equation of the type\n(\n)\nF\n,\ndy\nx y\ndx =\n =\ny\ng\nx\n\u239b\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0 (1)\nWe make the substitution\n    y = v x (2)\nDifferentiating equation (2) with respect to x, we get\ndy\ndx  =\ndv\nv\n+x dx"
    },
    {
        "Chapter": "1",
        "sentence_range": "4750-4753",
        "Text": "(1)\nWe make the substitution\n    y = v x (2)\nDifferentiating equation (2) with respect to x, we get\ndy\ndx  =\ndv\nv\n+x dx (3)\nSubstituting the value of dy\ndx  from equation (3) in equation (1), we get\n\u00a9 NCERT\nnot to be republished\nDIFFERENTIAL EQUATIONS\n399\ndv\nv\n+x dx\n = g (v)\nor\ndv\nx dx  = g (v) \u2013 v"
    },
    {
        "Chapter": "1",
        "sentence_range": "4751-4754",
        "Text": "x (2)\nDifferentiating equation (2) with respect to x, we get\ndy\ndx  =\ndv\nv\n+x dx (3)\nSubstituting the value of dy\ndx  from equation (3) in equation (1), we get\n\u00a9 NCERT\nnot to be republished\nDIFFERENTIAL EQUATIONS\n399\ndv\nv\n+x dx\n = g (v)\nor\ndv\nx dx  = g (v) \u2013 v (4)\nSeparating the variables in equation (4), we get\n( )\ndv\ng v\n\u2212v\n = dx\nx"
    },
    {
        "Chapter": "1",
        "sentence_range": "4752-4755",
        "Text": "(2)\nDifferentiating equation (2) with respect to x, we get\ndy\ndx  =\ndv\nv\n+x dx (3)\nSubstituting the value of dy\ndx  from equation (3) in equation (1), we get\n\u00a9 NCERT\nnot to be republished\nDIFFERENTIAL EQUATIONS\n399\ndv\nv\n+x dx\n = g (v)\nor\ndv\nx dx  = g (v) \u2013 v (4)\nSeparating the variables in equation (4), we get\n( )\ndv\ng v\n\u2212v\n = dx\nx (5)\nIntegrating both sides of equation (5), we get\n( )\ndv\ng v\n\u2212v\n\u222b\n =\n1\nC\nxdx\n+\n\u222b"
    },
    {
        "Chapter": "1",
        "sentence_range": "4753-4756",
        "Text": "(3)\nSubstituting the value of dy\ndx  from equation (3) in equation (1), we get\n\u00a9 NCERT\nnot to be republished\nDIFFERENTIAL EQUATIONS\n399\ndv\nv\n+x dx\n = g (v)\nor\ndv\nx dx  = g (v) \u2013 v (4)\nSeparating the variables in equation (4), we get\n( )\ndv\ng v\n\u2212v\n = dx\nx (5)\nIntegrating both sides of equation (5), we get\n( )\ndv\ng v\n\u2212v\n\u222b\n =\n1\nC\nxdx\n+\n\u222b (6)\nEquation (6) gives general solution (primitive) of the differential equation (1) when\nwe replace v by y\nx"
    },
    {
        "Chapter": "1",
        "sentence_range": "4754-4757",
        "Text": "(4)\nSeparating the variables in equation (4), we get\n( )\ndv\ng v\n\u2212v\n = dx\nx (5)\nIntegrating both sides of equation (5), we get\n( )\ndv\ng v\n\u2212v\n\u222b\n =\n1\nC\nxdx\n+\n\u222b (6)\nEquation (6) gives general solution (primitive) of the differential equation (1) when\nwe replace v by y\nx \ufffdNote  If the homogeneous differential equation is in the form \nF( , )\ndx\nx y\ndy =\nwhere, F (x, y) is homogenous function of  degree zero, then we make substitution\nx\ny =v\n i"
    },
    {
        "Chapter": "1",
        "sentence_range": "4755-4758",
        "Text": "(5)\nIntegrating both sides of equation (5), we get\n( )\ndv\ng v\n\u2212v\n\u222b\n =\n1\nC\nxdx\n+\n\u222b (6)\nEquation (6) gives general solution (primitive) of the differential equation (1) when\nwe replace v by y\nx \ufffdNote  If the homogeneous differential equation is in the form \nF( , )\ndx\nx y\ndy =\nwhere, F (x, y) is homogenous function of  degree zero, then we make substitution\nx\ny =v\n i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "4756-4759",
        "Text": "(6)\nEquation (6) gives general solution (primitive) of the differential equation (1) when\nwe replace v by y\nx \ufffdNote  If the homogeneous differential equation is in the form \nF( , )\ndx\nx y\ndy =\nwhere, F (x, y) is homogenous function of  degree zero, then we make substitution\nx\ny =v\n i e , x = vy and we proceed further to find the general solution as discussed\nabove by writing \nF( , )"
    },
    {
        "Chapter": "1",
        "sentence_range": "4757-4760",
        "Text": "\ufffdNote  If the homogeneous differential equation is in the form \nF( , )\ndx\nx y\ndy =\nwhere, F (x, y) is homogenous function of  degree zero, then we make substitution\nx\ny =v\n i e , x = vy and we proceed further to find the general solution as discussed\nabove by writing \nF( , ) dx\nx\nx y\nh\ndy\n\u239by\n\u239e\n=\n=\n\u239c\n\u239f\n\u239d\n\u23a0\nExample 15  Show that the differential equation (x \u2013 y) dy\ndx  = x + 2y is homogeneous\nand solve it"
    },
    {
        "Chapter": "1",
        "sentence_range": "4758-4761",
        "Text": "e , x = vy and we proceed further to find the general solution as discussed\nabove by writing \nF( , ) dx\nx\nx y\nh\ndy\n\u239by\n\u239e\n=\n=\n\u239c\n\u239f\n\u239d\n\u23a0\nExample 15  Show that the differential equation (x \u2013 y) dy\ndx  = x + 2y is homogeneous\nand solve it Solution The given differential equation can be expressed as\ndy\ndx  =\n2\nx\ny\nx\n+y\n\u2212"
    },
    {
        "Chapter": "1",
        "sentence_range": "4759-4762",
        "Text": ", x = vy and we proceed further to find the general solution as discussed\nabove by writing \nF( , ) dx\nx\nx y\nh\ndy\n\u239by\n\u239e\n=\n=\n\u239c\n\u239f\n\u239d\n\u23a0\nExample 15  Show that the differential equation (x \u2013 y) dy\ndx  = x + 2y is homogeneous\nand solve it Solution The given differential equation can be expressed as\ndy\ndx  =\n2\nx\ny\nx\n+y\n\u2212 (1)\nLet\nF(x, y) =\n2\nx\ny\nx\n y\n \nNow\nF(\u03bbx, \u03bby) =\n0\n(\n2 )\n( , )\n(\n)\nx\ny\nF x y\nx\ny\n \n  \n \n \n \n\u00a9 NCERT\nnot to be republished\nMATHEMATICS\n400\nTherefore, F(x, y) is a homogenous function of degree zero"
    },
    {
        "Chapter": "1",
        "sentence_range": "4760-4763",
        "Text": "dx\nx\nx y\nh\ndy\n\u239by\n\u239e\n=\n=\n\u239c\n\u239f\n\u239d\n\u23a0\nExample 15  Show that the differential equation (x \u2013 y) dy\ndx  = x + 2y is homogeneous\nand solve it Solution The given differential equation can be expressed as\ndy\ndx  =\n2\nx\ny\nx\n+y\n\u2212 (1)\nLet\nF(x, y) =\n2\nx\ny\nx\n y\n \nNow\nF(\u03bbx, \u03bby) =\n0\n(\n2 )\n( , )\n(\n)\nx\ny\nF x y\nx\ny\n \n  \n \n \n \n\u00a9 NCERT\nnot to be republished\nMATHEMATICS\n400\nTherefore, F(x, y) is a homogenous function of degree zero So, the given differential\nequation is a homogenous differential equation"
    },
    {
        "Chapter": "1",
        "sentence_range": "4761-4764",
        "Text": "Solution The given differential equation can be expressed as\ndy\ndx  =\n2\nx\ny\nx\n+y\n\u2212 (1)\nLet\nF(x, y) =\n2\nx\ny\nx\n y\n \nNow\nF(\u03bbx, \u03bby) =\n0\n(\n2 )\n( , )\n(\n)\nx\ny\nF x y\nx\ny\n \n  \n \n \n \n\u00a9 NCERT\nnot to be republished\nMATHEMATICS\n400\nTherefore, F(x, y) is a homogenous function of degree zero So, the given differential\nequation is a homogenous differential equation Alternatively,\n2\n1\n1\ny\ndy\nyx\ndx\nx\n\u239b\n\u239e\n\u239c+\n\u239f\n= \u239c\n\u239f\n\u239c\n\u239f\n\u2212\n\u239d\n\u23a0\n =\ny\ng\nx\n\u239b\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0"
    },
    {
        "Chapter": "1",
        "sentence_range": "4762-4765",
        "Text": "(1)\nLet\nF(x, y) =\n2\nx\ny\nx\n y\n \nNow\nF(\u03bbx, \u03bby) =\n0\n(\n2 )\n( , )\n(\n)\nx\ny\nF x y\nx\ny\n \n  \n \n \n \n\u00a9 NCERT\nnot to be republished\nMATHEMATICS\n400\nTherefore, F(x, y) is a homogenous function of degree zero So, the given differential\nequation is a homogenous differential equation Alternatively,\n2\n1\n1\ny\ndy\nyx\ndx\nx\n\u239b\n\u239e\n\u239c+\n\u239f\n= \u239c\n\u239f\n\u239c\n\u239f\n\u2212\n\u239d\n\u23a0\n =\ny\ng\nx\n\u239b\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0 (2)\nR"
    },
    {
        "Chapter": "1",
        "sentence_range": "4763-4766",
        "Text": "So, the given differential\nequation is a homogenous differential equation Alternatively,\n2\n1\n1\ny\ndy\nyx\ndx\nx\n\u239b\n\u239e\n\u239c+\n\u239f\n= \u239c\n\u239f\n\u239c\n\u239f\n\u2212\n\u239d\n\u23a0\n =\ny\ng\nx\n\u239b\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0 (2)\nR H"
    },
    {
        "Chapter": "1",
        "sentence_range": "4764-4767",
        "Text": "Alternatively,\n2\n1\n1\ny\ndy\nyx\ndx\nx\n\u239b\n\u239e\n\u239c+\n\u239f\n= \u239c\n\u239f\n\u239c\n\u239f\n\u2212\n\u239d\n\u23a0\n =\ny\ng\nx\n\u239b\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0 (2)\nR H S"
    },
    {
        "Chapter": "1",
        "sentence_range": "4765-4768",
        "Text": "(2)\nR H S of differential equation (2) is of the form \ny\ng\nx\n \n \n \n \n \n  and so it is a homogeneous\nfunction of degree zero"
    },
    {
        "Chapter": "1",
        "sentence_range": "4766-4769",
        "Text": "H S of differential equation (2) is of the form \ny\ng\nx\n \n \n \n \n \n  and so it is a homogeneous\nfunction of degree zero Therefore, equation (1) is a homogeneous differential equation"
    },
    {
        "Chapter": "1",
        "sentence_range": "4767-4770",
        "Text": "S of differential equation (2) is of the form \ny\ng\nx\n \n \n \n \n \n  and so it is a homogeneous\nfunction of degree zero Therefore, equation (1) is a homogeneous differential equation To solve it we make the substitution\ny = vx"
    },
    {
        "Chapter": "1",
        "sentence_range": "4768-4771",
        "Text": "of differential equation (2) is of the form \ny\ng\nx\n \n \n \n \n \n  and so it is a homogeneous\nfunction of degree zero Therefore, equation (1) is a homogeneous differential equation To solve it we make the substitution\ny = vx (3)\nDifferentiating equation (3) with respect to, x we get\ndy\ndx  =\ndv\nv\n+x dx"
    },
    {
        "Chapter": "1",
        "sentence_range": "4769-4772",
        "Text": "Therefore, equation (1) is a homogeneous differential equation To solve it we make the substitution\ny = vx (3)\nDifferentiating equation (3) with respect to, x we get\ndy\ndx  =\ndv\nv\n+x dx (4)\nSubstituting the value of y  and dy\ndx in equation (1) we get\ndv\nv\n+x dx\n = 1\n12\n+vv\n\u2212\nor\ndv\nx dx  = 1\n12\nv\nv\n+v\n\u2212\n\u2212\nor\ndv\nx dx  =\n2\n1\nv1\nvv\n \nor\n2\n1\n1\nv\ndv\nv\nv  \n =\ndx\nx\n \nIntegrating both sides of equation (5), we get\n2\n1\n1\nv\ndv\nv\n v\n \n \n =\nxdx\n \nor\n2\n1\n2\n1\n3\n2\n1\nv\ndv\nv\n v\n \n \n \n \n = \u2013 log |x| + C1\n\u00a9 NCERT\nnot to be republished\nDIFFERENTIAL EQUATIONS\n401\nor\n1\n2\n2\n1\n2\n1\n3\n1\nlog\nC\n2\n2\n1\n1\n \n  \n \n \n \n \n \n \n \nv\ndv\ndv\nx\nv\nv\nv\nv\nor\n2\n1\n2\n2\n1\n3\n1\nlog\n1\nlog\nC\n2\n2\n1\n3\n2\n2\n \n \n \n  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nv\nv\ndv\nx\nv\nor\n2\n1\n1\n1\n3\n2\n2\n1\nlog\n1"
    },
    {
        "Chapter": "1",
        "sentence_range": "4770-4773",
        "Text": "To solve it we make the substitution\ny = vx (3)\nDifferentiating equation (3) with respect to, x we get\ndy\ndx  =\ndv\nv\n+x dx (4)\nSubstituting the value of y  and dy\ndx in equation (1) we get\ndv\nv\n+x dx\n = 1\n12\n+vv\n\u2212\nor\ndv\nx dx  = 1\n12\nv\nv\n+v\n\u2212\n\u2212\nor\ndv\nx dx  =\n2\n1\nv1\nvv\n \nor\n2\n1\n1\nv\ndv\nv\nv  \n =\ndx\nx\n \nIntegrating both sides of equation (5), we get\n2\n1\n1\nv\ndv\nv\n v\n \n \n =\nxdx\n \nor\n2\n1\n2\n1\n3\n2\n1\nv\ndv\nv\n v\n \n \n \n \n = \u2013 log |x| + C1\n\u00a9 NCERT\nnot to be republished\nDIFFERENTIAL EQUATIONS\n401\nor\n1\n2\n2\n1\n2\n1\n3\n1\nlog\nC\n2\n2\n1\n1\n \n  \n \n \n \n \n \n \n \nv\ndv\ndv\nx\nv\nv\nv\nv\nor\n2\n1\n2\n2\n1\n3\n1\nlog\n1\nlog\nC\n2\n2\n1\n3\n2\n2\n \n \n \n  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nv\nv\ndv\nx\nv\nor\n2\n1\n1\n1\n3\n2\n2\n1\nlog\n1 tan\nlog\nC\n2\n2\n3\n3\n \n \n \n \n \n \n  \n \n \n \n \n \nv\nv\nv\nx\nor\n2\n2\n1\n1\n1\n1\n2\n1\nlog\n1\nlog\n3 tan\nC\n2\n2\n3\n \n \n \n \n \n \n \n \n \n \n \n \nv\nv\nv\nx\n(Why"
    },
    {
        "Chapter": "1",
        "sentence_range": "4771-4774",
        "Text": "(3)\nDifferentiating equation (3) with respect to, x we get\ndy\ndx  =\ndv\nv\n+x dx (4)\nSubstituting the value of y  and dy\ndx in equation (1) we get\ndv\nv\n+x dx\n = 1\n12\n+vv\n\u2212\nor\ndv\nx dx  = 1\n12\nv\nv\n+v\n\u2212\n\u2212\nor\ndv\nx dx  =\n2\n1\nv1\nvv\n \nor\n2\n1\n1\nv\ndv\nv\nv  \n =\ndx\nx\n \nIntegrating both sides of equation (5), we get\n2\n1\n1\nv\ndv\nv\n v\n \n \n =\nxdx\n \nor\n2\n1\n2\n1\n3\n2\n1\nv\ndv\nv\n v\n \n \n \n \n = \u2013 log |x| + C1\n\u00a9 NCERT\nnot to be republished\nDIFFERENTIAL EQUATIONS\n401\nor\n1\n2\n2\n1\n2\n1\n3\n1\nlog\nC\n2\n2\n1\n1\n \n  \n \n \n \n \n \n \n \nv\ndv\ndv\nx\nv\nv\nv\nv\nor\n2\n1\n2\n2\n1\n3\n1\nlog\n1\nlog\nC\n2\n2\n1\n3\n2\n2\n \n \n \n  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nv\nv\ndv\nx\nv\nor\n2\n1\n1\n1\n3\n2\n2\n1\nlog\n1 tan\nlog\nC\n2\n2\n3\n3\n \n \n \n \n \n \n  \n \n \n \n \n \nv\nv\nv\nx\nor\n2\n2\n1\n1\n1\n1\n2\n1\nlog\n1\nlog\n3 tan\nC\n2\n2\n3\n \n \n \n \n \n \n \n \n \n \n \n \nv\nv\nv\nx\n(Why )\nReplacing v by y\nx , we get\nor\n2\n2\n1\n1\n2\n1\n1\n2\nlog\n1\nlog\n3 tan\nC\n2\n2\n3\n \n \n \n \n \n \n \n \n \n \n \n \n \ny\ny\ny\nx\nx\nx\nx\nx\nor\n2\n2\n1\n1\n2\n1\n2\nlog\n1\n3 tan\nC\n2\n3\ny\ny\ny\nx\nx\nx\nx\nx\n\u2212\n\u239b\n\u239e\n+\n\u239b\n\u239e\n+\n+\n=\n+\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\nor\n2\n2\n1\n1\n2\nlog (\n)\n2 3 tan\n2C\ny3\nx\ny\nxy\nx\nx\n\u2212\n+\n\u239b\n\u239e\n+\n+\n=\n+\n\u239c\n\u239f\n\u239d\n\u23a0\nor\n2\n2\n1\n2\nlog (\n)\n2 3 tan\nC\n3\n\u2212\n+\n\u239b\n\u239e\n+\n+\n=\n+\n\u239c\n\u239f\n\u239d\n\u23a0\nx\ny\nx\nxy\ny\nx\nwhich is the general solution of the differential equation (1)\nExample 16 Show that the differential equation \ncos\ncos\ny\ndy\ny\nx\ny\nx\nx\ndx\nx\n\u239b\n\u239e\n\u239b\n\u239e\n=\n+\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\n is\nhomogeneous and solve it"
    },
    {
        "Chapter": "1",
        "sentence_range": "4772-4775",
        "Text": "(4)\nSubstituting the value of y  and dy\ndx in equation (1) we get\ndv\nv\n+x dx\n = 1\n12\n+vv\n\u2212\nor\ndv\nx dx  = 1\n12\nv\nv\n+v\n\u2212\n\u2212\nor\ndv\nx dx  =\n2\n1\nv1\nvv\n \nor\n2\n1\n1\nv\ndv\nv\nv  \n =\ndx\nx\n \nIntegrating both sides of equation (5), we get\n2\n1\n1\nv\ndv\nv\n v\n \n \n =\nxdx\n \nor\n2\n1\n2\n1\n3\n2\n1\nv\ndv\nv\n v\n \n \n \n \n = \u2013 log |x| + C1\n\u00a9 NCERT\nnot to be republished\nDIFFERENTIAL EQUATIONS\n401\nor\n1\n2\n2\n1\n2\n1\n3\n1\nlog\nC\n2\n2\n1\n1\n \n  \n \n \n \n \n \n \n \nv\ndv\ndv\nx\nv\nv\nv\nv\nor\n2\n1\n2\n2\n1\n3\n1\nlog\n1\nlog\nC\n2\n2\n1\n3\n2\n2\n \n \n \n  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nv\nv\ndv\nx\nv\nor\n2\n1\n1\n1\n3\n2\n2\n1\nlog\n1 tan\nlog\nC\n2\n2\n3\n3\n \n \n \n \n \n \n  \n \n \n \n \n \nv\nv\nv\nx\nor\n2\n2\n1\n1\n1\n1\n2\n1\nlog\n1\nlog\n3 tan\nC\n2\n2\n3\n \n \n \n \n \n \n \n \n \n \n \n \nv\nv\nv\nx\n(Why )\nReplacing v by y\nx , we get\nor\n2\n2\n1\n1\n2\n1\n1\n2\nlog\n1\nlog\n3 tan\nC\n2\n2\n3\n \n \n \n \n \n \n \n \n \n \n \n \n \ny\ny\ny\nx\nx\nx\nx\nx\nor\n2\n2\n1\n1\n2\n1\n2\nlog\n1\n3 tan\nC\n2\n3\ny\ny\ny\nx\nx\nx\nx\nx\n\u2212\n\u239b\n\u239e\n+\n\u239b\n\u239e\n+\n+\n=\n+\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\nor\n2\n2\n1\n1\n2\nlog (\n)\n2 3 tan\n2C\ny3\nx\ny\nxy\nx\nx\n\u2212\n+\n\u239b\n\u239e\n+\n+\n=\n+\n\u239c\n\u239f\n\u239d\n\u23a0\nor\n2\n2\n1\n2\nlog (\n)\n2 3 tan\nC\n3\n\u2212\n+\n\u239b\n\u239e\n+\n+\n=\n+\n\u239c\n\u239f\n\u239d\n\u23a0\nx\ny\nx\nxy\ny\nx\nwhich is the general solution of the differential equation (1)\nExample 16 Show that the differential equation \ncos\ncos\ny\ndy\ny\nx\ny\nx\nx\ndx\nx\n\u239b\n\u239e\n\u239b\n\u239e\n=\n+\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\n is\nhomogeneous and solve it Solution The given differential equation can be written as\ndy\ndx  =\ncos\ncos\ny\ny\nx\nx\ny\nx\nx\n\u239b\n\u239e +\n\u239c\n\u239f\n\u239d\n\u239b\u23a0\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0"
    },
    {
        "Chapter": "1",
        "sentence_range": "4773-4776",
        "Text": "tan\nlog\nC\n2\n2\n3\n3\n \n \n \n \n \n \n  \n \n \n \n \n \nv\nv\nv\nx\nor\n2\n2\n1\n1\n1\n1\n2\n1\nlog\n1\nlog\n3 tan\nC\n2\n2\n3\n \n \n \n \n \n \n \n \n \n \n \n \nv\nv\nv\nx\n(Why )\nReplacing v by y\nx , we get\nor\n2\n2\n1\n1\n2\n1\n1\n2\nlog\n1\nlog\n3 tan\nC\n2\n2\n3\n \n \n \n \n \n \n \n \n \n \n \n \n \ny\ny\ny\nx\nx\nx\nx\nx\nor\n2\n2\n1\n1\n2\n1\n2\nlog\n1\n3 tan\nC\n2\n3\ny\ny\ny\nx\nx\nx\nx\nx\n\u2212\n\u239b\n\u239e\n+\n\u239b\n\u239e\n+\n+\n=\n+\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\nor\n2\n2\n1\n1\n2\nlog (\n)\n2 3 tan\n2C\ny3\nx\ny\nxy\nx\nx\n\u2212\n+\n\u239b\n\u239e\n+\n+\n=\n+\n\u239c\n\u239f\n\u239d\n\u23a0\nor\n2\n2\n1\n2\nlog (\n)\n2 3 tan\nC\n3\n\u2212\n+\n\u239b\n\u239e\n+\n+\n=\n+\n\u239c\n\u239f\n\u239d\n\u23a0\nx\ny\nx\nxy\ny\nx\nwhich is the general solution of the differential equation (1)\nExample 16 Show that the differential equation \ncos\ncos\ny\ndy\ny\nx\ny\nx\nx\ndx\nx\n\u239b\n\u239e\n\u239b\n\u239e\n=\n+\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\n is\nhomogeneous and solve it Solution The given differential equation can be written as\ndy\ndx  =\ncos\ncos\ny\ny\nx\nx\ny\nx\nx\n\u239b\n\u239e +\n\u239c\n\u239f\n\u239d\n\u239b\u23a0\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0 (1)\n\u00a9 NCERT\nnot to be republished\nMATHEMATICS\n402\nIt is a differential equation of the form \nF( ,\n)\ndy\nx y\ndx ="
    },
    {
        "Chapter": "1",
        "sentence_range": "4774-4777",
        "Text": ")\nReplacing v by y\nx , we get\nor\n2\n2\n1\n1\n2\n1\n1\n2\nlog\n1\nlog\n3 tan\nC\n2\n2\n3\n \n \n \n \n \n \n \n \n \n \n \n \n \ny\ny\ny\nx\nx\nx\nx\nx\nor\n2\n2\n1\n1\n2\n1\n2\nlog\n1\n3 tan\nC\n2\n3\ny\ny\ny\nx\nx\nx\nx\nx\n\u2212\n\u239b\n\u239e\n+\n\u239b\n\u239e\n+\n+\n=\n+\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\nor\n2\n2\n1\n1\n2\nlog (\n)\n2 3 tan\n2C\ny3\nx\ny\nxy\nx\nx\n\u2212\n+\n\u239b\n\u239e\n+\n+\n=\n+\n\u239c\n\u239f\n\u239d\n\u23a0\nor\n2\n2\n1\n2\nlog (\n)\n2 3 tan\nC\n3\n\u2212\n+\n\u239b\n\u239e\n+\n+\n=\n+\n\u239c\n\u239f\n\u239d\n\u23a0\nx\ny\nx\nxy\ny\nx\nwhich is the general solution of the differential equation (1)\nExample 16 Show that the differential equation \ncos\ncos\ny\ndy\ny\nx\ny\nx\nx\ndx\nx\n\u239b\n\u239e\n\u239b\n\u239e\n=\n+\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\n is\nhomogeneous and solve it Solution The given differential equation can be written as\ndy\ndx  =\ncos\ncos\ny\ny\nx\nx\ny\nx\nx\n\u239b\n\u239e +\n\u239c\n\u239f\n\u239d\n\u239b\u23a0\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0 (1)\n\u00a9 NCERT\nnot to be republished\nMATHEMATICS\n402\nIt is a differential equation of the form \nF( ,\n)\ndy\nx y\ndx = Here\nF(x, y) =\ncos\ncos\ny\ny\nx\nx\ny\nx\nx\n\u239b\n\u239e +\n\u239c\n\u239f\n\u239d\n\u239b\u23a0\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0\nReplacing x by \u03bbx and y by \u03bby, we get\nF(\u03bbx, \u03bby) =\n0\n[ cos\n]\n[F( , )]\ncos\ny\ny\nx\nx\nx y\ny\nx\nx\n\u239b\n\u239e\n\u03bb\n+\n\u239c\n\u239f\n\u239d\n\u23a0\n= \u03bb\n\u239b\n\u239e\n\u03bb\u239c\n\u239f\n\u239d\n\u23a0\nThus, F(x, y) is a homogeneous function of degree zero"
    },
    {
        "Chapter": "1",
        "sentence_range": "4775-4778",
        "Text": "Solution The given differential equation can be written as\ndy\ndx  =\ncos\ncos\ny\ny\nx\nx\ny\nx\nx\n\u239b\n\u239e +\n\u239c\n\u239f\n\u239d\n\u239b\u23a0\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0 (1)\n\u00a9 NCERT\nnot to be republished\nMATHEMATICS\n402\nIt is a differential equation of the form \nF( ,\n)\ndy\nx y\ndx = Here\nF(x, y) =\ncos\ncos\ny\ny\nx\nx\ny\nx\nx\n\u239b\n\u239e +\n\u239c\n\u239f\n\u239d\n\u239b\u23a0\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0\nReplacing x by \u03bbx and y by \u03bby, we get\nF(\u03bbx, \u03bby) =\n0\n[ cos\n]\n[F( , )]\ncos\ny\ny\nx\nx\nx y\ny\nx\nx\n\u239b\n\u239e\n\u03bb\n+\n\u239c\n\u239f\n\u239d\n\u23a0\n= \u03bb\n\u239b\n\u239e\n\u03bb\u239c\n\u239f\n\u239d\n\u23a0\nThus, F(x, y) is a homogeneous function of degree zero Therefore, the given differential equation is a homogeneous differential equation"
    },
    {
        "Chapter": "1",
        "sentence_range": "4776-4779",
        "Text": "(1)\n\u00a9 NCERT\nnot to be republished\nMATHEMATICS\n402\nIt is a differential equation of the form \nF( ,\n)\ndy\nx y\ndx = Here\nF(x, y) =\ncos\ncos\ny\ny\nx\nx\ny\nx\nx\n\u239b\n\u239e +\n\u239c\n\u239f\n\u239d\n\u239b\u23a0\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0\nReplacing x by \u03bbx and y by \u03bby, we get\nF(\u03bbx, \u03bby) =\n0\n[ cos\n]\n[F( , )]\ncos\ny\ny\nx\nx\nx y\ny\nx\nx\n\u239b\n\u239e\n\u03bb\n+\n\u239c\n\u239f\n\u239d\n\u23a0\n= \u03bb\n\u239b\n\u239e\n\u03bb\u239c\n\u239f\n\u239d\n\u23a0\nThus, F(x, y) is a homogeneous function of degree zero Therefore, the given differential equation is a homogeneous differential equation To solve it we make the substitution\ny = vx"
    },
    {
        "Chapter": "1",
        "sentence_range": "4777-4780",
        "Text": "Here\nF(x, y) =\ncos\ncos\ny\ny\nx\nx\ny\nx\nx\n\u239b\n\u239e +\n\u239c\n\u239f\n\u239d\n\u239b\u23a0\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0\nReplacing x by \u03bbx and y by \u03bby, we get\nF(\u03bbx, \u03bby) =\n0\n[ cos\n]\n[F( , )]\ncos\ny\ny\nx\nx\nx y\ny\nx\nx\n\u239b\n\u239e\n\u03bb\n+\n\u239c\n\u239f\n\u239d\n\u23a0\n= \u03bb\n\u239b\n\u239e\n\u03bb\u239c\n\u239f\n\u239d\n\u23a0\nThus, F(x, y) is a homogeneous function of degree zero Therefore, the given differential equation is a homogeneous differential equation To solve it we make the substitution\ny = vx (2)\nDifferentiating equation (2) with respect to x, we get\ndy\ndx  =\ndv\nv\n+x dx"
    },
    {
        "Chapter": "1",
        "sentence_range": "4778-4781",
        "Text": "Therefore, the given differential equation is a homogeneous differential equation To solve it we make the substitution\ny = vx (2)\nDifferentiating equation (2) with respect to x, we get\ndy\ndx  =\ndv\nv\n+x dx (3)\nSubstituting the value of y and dy\ndx  in equation (1), we get\ndv\nv\n+x dx\n =\ncos\n1\nvcos\nvv\n+\nor\ndv\nx dx  =\ncos\n1\nvcos\nv\nv\nv\n+ \u2212\nor\ndv\nx dx  =\n1\ncosv\nor\ncosv dv = dx\nx\nTherefore\n\u222bcosv dv\n =\n1 dx\nx\u222b\n\u00a9 NCERT\nnot to be republished\nDIFFERENTIAL EQUATIONS\n403\nor\nsin v = log |x| + log |C|\nor\nsin v = log |Cx|\nReplacing v by y\nx , we get\nsin\ny\nx\n\u239b\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0  = log |Cx|\nwhich is the general solution of the differential equation (1)"
    },
    {
        "Chapter": "1",
        "sentence_range": "4779-4782",
        "Text": "To solve it we make the substitution\ny = vx (2)\nDifferentiating equation (2) with respect to x, we get\ndy\ndx  =\ndv\nv\n+x dx (3)\nSubstituting the value of y and dy\ndx  in equation (1), we get\ndv\nv\n+x dx\n =\ncos\n1\nvcos\nvv\n+\nor\ndv\nx dx  =\ncos\n1\nvcos\nv\nv\nv\n+ \u2212\nor\ndv\nx dx  =\n1\ncosv\nor\ncosv dv = dx\nx\nTherefore\n\u222bcosv dv\n =\n1 dx\nx\u222b\n\u00a9 NCERT\nnot to be republished\nDIFFERENTIAL EQUATIONS\n403\nor\nsin v = log |x| + log |C|\nor\nsin v = log |Cx|\nReplacing v by y\nx , we get\nsin\ny\nx\n\u239b\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0  = log |Cx|\nwhich is the general solution of the differential equation (1) Example 17 Show that the differential equation 2\n2\n0\nx\nx\ny\ny\ny e dx\ny\nx e\ndy\n\u239b\n\u239e\n\u239c\n\u239f\n+\n\u2212\n=\n\u239d\n\u23a0\nis\nhomogeneous and find its particular solution, given that, x = 0 when y = 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "4780-4783",
        "Text": "(2)\nDifferentiating equation (2) with respect to x, we get\ndy\ndx  =\ndv\nv\n+x dx (3)\nSubstituting the value of y and dy\ndx  in equation (1), we get\ndv\nv\n+x dx\n =\ncos\n1\nvcos\nvv\n+\nor\ndv\nx dx  =\ncos\n1\nvcos\nv\nv\nv\n+ \u2212\nor\ndv\nx dx  =\n1\ncosv\nor\ncosv dv = dx\nx\nTherefore\n\u222bcosv dv\n =\n1 dx\nx\u222b\n\u00a9 NCERT\nnot to be republished\nDIFFERENTIAL EQUATIONS\n403\nor\nsin v = log |x| + log |C|\nor\nsin v = log |Cx|\nReplacing v by y\nx , we get\nsin\ny\nx\n\u239b\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0  = log |Cx|\nwhich is the general solution of the differential equation (1) Example 17 Show that the differential equation 2\n2\n0\nx\nx\ny\ny\ny e dx\ny\nx e\ndy\n\u239b\n\u239e\n\u239c\n\u239f\n+\n\u2212\n=\n\u239d\n\u23a0\nis\nhomogeneous and find its particular solution, given that, x = 0 when y = 1 Solution The given differential equation can be written as\ndx\ndy  = 2\n2\nx\ny\nx\ny\nx e\ny\ny e\n\u2212"
    },
    {
        "Chapter": "1",
        "sentence_range": "4781-4784",
        "Text": "(3)\nSubstituting the value of y and dy\ndx  in equation (1), we get\ndv\nv\n+x dx\n =\ncos\n1\nvcos\nvv\n+\nor\ndv\nx dx  =\ncos\n1\nvcos\nv\nv\nv\n+ \u2212\nor\ndv\nx dx  =\n1\ncosv\nor\ncosv dv = dx\nx\nTherefore\n\u222bcosv dv\n =\n1 dx\nx\u222b\n\u00a9 NCERT\nnot to be republished\nDIFFERENTIAL EQUATIONS\n403\nor\nsin v = log |x| + log |C|\nor\nsin v = log |Cx|\nReplacing v by y\nx , we get\nsin\ny\nx\n\u239b\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0  = log |Cx|\nwhich is the general solution of the differential equation (1) Example 17 Show that the differential equation 2\n2\n0\nx\nx\ny\ny\ny e dx\ny\nx e\ndy\n\u239b\n\u239e\n\u239c\n\u239f\n+\n\u2212\n=\n\u239d\n\u23a0\nis\nhomogeneous and find its particular solution, given that, x = 0 when y = 1 Solution The given differential equation can be written as\ndx\ndy  = 2\n2\nx\ny\nx\ny\nx e\ny\ny e\n\u2212 (1)\nLet\nF(x, y) = 2\n2\nx\ny\nx\ny\nxe\ny\nye\n\u2212\nThen\nF(\u03bbx, \u03bby) =\n0\n2\n[F( , )]\n2\nx\ny\nx\ny\nxe\ny\nx y\nye\n\u239b\n\u239e\n\u239c\n\u239f\n\u03bb\n\u2212\n\u239c\n\u239f\n\u239d\n\u23a0 =\u03bb\n\u239b\n\u239e\n\u239c\n\u239f\n\u03bb\u239c\n\u239f\n\u239d\n\u23a0\nThus, F(x, y) is a homogeneous function of degree zero"
    },
    {
        "Chapter": "1",
        "sentence_range": "4782-4785",
        "Text": "Example 17 Show that the differential equation 2\n2\n0\nx\nx\ny\ny\ny e dx\ny\nx e\ndy\n\u239b\n\u239e\n\u239c\n\u239f\n+\n\u2212\n=\n\u239d\n\u23a0\nis\nhomogeneous and find its particular solution, given that, x = 0 when y = 1 Solution The given differential equation can be written as\ndx\ndy  = 2\n2\nx\ny\nx\ny\nx e\ny\ny e\n\u2212 (1)\nLet\nF(x, y) = 2\n2\nx\ny\nx\ny\nxe\ny\nye\n\u2212\nThen\nF(\u03bbx, \u03bby) =\n0\n2\n[F( , )]\n2\nx\ny\nx\ny\nxe\ny\nx y\nye\n\u239b\n\u239e\n\u239c\n\u239f\n\u03bb\n\u2212\n\u239c\n\u239f\n\u239d\n\u23a0 =\u03bb\n\u239b\n\u239e\n\u239c\n\u239f\n\u03bb\u239c\n\u239f\n\u239d\n\u23a0\nThus, F(x, y) is a homogeneous function of degree zero Therefore, the given\ndifferential equation is a homogeneous differential equation"
    },
    {
        "Chapter": "1",
        "sentence_range": "4783-4786",
        "Text": "Solution The given differential equation can be written as\ndx\ndy  = 2\n2\nx\ny\nx\ny\nx e\ny\ny e\n\u2212 (1)\nLet\nF(x, y) = 2\n2\nx\ny\nx\ny\nxe\ny\nye\n\u2212\nThen\nF(\u03bbx, \u03bby) =\n0\n2\n[F( , )]\n2\nx\ny\nx\ny\nxe\ny\nx y\nye\n\u239b\n\u239e\n\u239c\n\u239f\n\u03bb\n\u2212\n\u239c\n\u239f\n\u239d\n\u23a0 =\u03bb\n\u239b\n\u239e\n\u239c\n\u239f\n\u03bb\u239c\n\u239f\n\u239d\n\u23a0\nThus, F(x, y) is a homogeneous function of degree zero Therefore, the given\ndifferential equation is a homogeneous differential equation To solve it, we make the substitution\nx = vy"
    },
    {
        "Chapter": "1",
        "sentence_range": "4784-4787",
        "Text": "(1)\nLet\nF(x, y) = 2\n2\nx\ny\nx\ny\nxe\ny\nye\n\u2212\nThen\nF(\u03bbx, \u03bby) =\n0\n2\n[F( , )]\n2\nx\ny\nx\ny\nxe\ny\nx y\nye\n\u239b\n\u239e\n\u239c\n\u239f\n\u03bb\n\u2212\n\u239c\n\u239f\n\u239d\n\u23a0 =\u03bb\n\u239b\n\u239e\n\u239c\n\u239f\n\u03bb\u239c\n\u239f\n\u239d\n\u23a0\nThus, F(x, y) is a homogeneous function of degree zero Therefore, the given\ndifferential equation is a homogeneous differential equation To solve it, we make the substitution\nx = vy (2)\nDifferentiating equation (2) with respect to y, we get\ndx\ndy  =\n+\ndv\nv\ny dy\n\u00a9 NCERT\nnot to be republished\nMATHEMATICS\n404\nSubstituting the value of \nxand dx\ndy  in equation (1), we get\ndv\nv\n+y dy\n = 2\n1\n2\nv\nv ev\ne\n\u2212\nor\ndv\ny dy  = 2\n1\n2\nv\nv ev\nv\ne\n\u2212 \u2212\nor\ndv\ny dy  =\n2 ve1\n\u2212\nor\n2ev dv =\ndy\ny\n\u2212\nor\n2 ve\n\u22c5dv\n\u222b\n =\nydy\n\u2212\u222b\nor\n2 ev = \u2013 log |y| + C\nand replacing v by \nx\ny  , we get\n2\nyx\ne  + log |y| = C"
    },
    {
        "Chapter": "1",
        "sentence_range": "4785-4788",
        "Text": "Therefore, the given\ndifferential equation is a homogeneous differential equation To solve it, we make the substitution\nx = vy (2)\nDifferentiating equation (2) with respect to y, we get\ndx\ndy  =\n+\ndv\nv\ny dy\n\u00a9 NCERT\nnot to be republished\nMATHEMATICS\n404\nSubstituting the value of \nxand dx\ndy  in equation (1), we get\ndv\nv\n+y dy\n = 2\n1\n2\nv\nv ev\ne\n\u2212\nor\ndv\ny dy  = 2\n1\n2\nv\nv ev\nv\ne\n\u2212 \u2212\nor\ndv\ny dy  =\n2 ve1\n\u2212\nor\n2ev dv =\ndy\ny\n\u2212\nor\n2 ve\n\u22c5dv\n\u222b\n =\nydy\n\u2212\u222b\nor\n2 ev = \u2013 log |y| + C\nand replacing v by \nx\ny  , we get\n2\nyx\ne  + log |y| = C (3)\nSubstituting x = 0 and y = 1 in equation (3), we get\n2 e0 + log |1| = C \u21d2 C = 2\nSubstituting the value of C in equation (3), we get\n2\nyx\ne  + log |y | = 2\nwhich is the particular solution of the given differential equation"
    },
    {
        "Chapter": "1",
        "sentence_range": "4786-4789",
        "Text": "To solve it, we make the substitution\nx = vy (2)\nDifferentiating equation (2) with respect to y, we get\ndx\ndy  =\n+\ndv\nv\ny dy\n\u00a9 NCERT\nnot to be republished\nMATHEMATICS\n404\nSubstituting the value of \nxand dx\ndy  in equation (1), we get\ndv\nv\n+y dy\n = 2\n1\n2\nv\nv ev\ne\n\u2212\nor\ndv\ny dy  = 2\n1\n2\nv\nv ev\nv\ne\n\u2212 \u2212\nor\ndv\ny dy  =\n2 ve1\n\u2212\nor\n2ev dv =\ndy\ny\n\u2212\nor\n2 ve\n\u22c5dv\n\u222b\n =\nydy\n\u2212\u222b\nor\n2 ev = \u2013 log |y| + C\nand replacing v by \nx\ny  , we get\n2\nyx\ne  + log |y| = C (3)\nSubstituting x = 0 and y = 1 in equation (3), we get\n2 e0 + log |1| = C \u21d2 C = 2\nSubstituting the value of C in equation (3), we get\n2\nyx\ne  + log |y | = 2\nwhich is the particular solution of the given differential equation Example 18 Show that the family of curves for which the slope of the tangent at any\npoint (x, y) on it is  \n2\n2\n2\nx\ny\n+xy\n, is given by x2 \u2013 y2 = cx"
    },
    {
        "Chapter": "1",
        "sentence_range": "4787-4790",
        "Text": "(2)\nDifferentiating equation (2) with respect to y, we get\ndx\ndy  =\n+\ndv\nv\ny dy\n\u00a9 NCERT\nnot to be republished\nMATHEMATICS\n404\nSubstituting the value of \nxand dx\ndy  in equation (1), we get\ndv\nv\n+y dy\n = 2\n1\n2\nv\nv ev\ne\n\u2212\nor\ndv\ny dy  = 2\n1\n2\nv\nv ev\nv\ne\n\u2212 \u2212\nor\ndv\ny dy  =\n2 ve1\n\u2212\nor\n2ev dv =\ndy\ny\n\u2212\nor\n2 ve\n\u22c5dv\n\u222b\n =\nydy\n\u2212\u222b\nor\n2 ev = \u2013 log |y| + C\nand replacing v by \nx\ny  , we get\n2\nyx\ne  + log |y| = C (3)\nSubstituting x = 0 and y = 1 in equation (3), we get\n2 e0 + log |1| = C \u21d2 C = 2\nSubstituting the value of C in equation (3), we get\n2\nyx\ne  + log |y | = 2\nwhich is the particular solution of the given differential equation Example 18 Show that the family of curves for which the slope of the tangent at any\npoint (x, y) on it is  \n2\n2\n2\nx\ny\n+xy\n, is given by x2 \u2013 y2 = cx Solution We know that the slope of the tangent at any point on a curve is dy\ndx"
    },
    {
        "Chapter": "1",
        "sentence_range": "4788-4791",
        "Text": "(3)\nSubstituting x = 0 and y = 1 in equation (3), we get\n2 e0 + log |1| = C \u21d2 C = 2\nSubstituting the value of C in equation (3), we get\n2\nyx\ne  + log |y | = 2\nwhich is the particular solution of the given differential equation Example 18 Show that the family of curves for which the slope of the tangent at any\npoint (x, y) on it is  \n2\n2\n2\nx\ny\n+xy\n, is given by x2 \u2013 y2 = cx Solution We know that the slope of the tangent at any point on a curve is dy\ndx Therefore,\ndy\ndx  =\n2\n2\n2\nx\ny\nxy\n+\n\u00a9 NCERT\nnot to be republished\nDIFFERENTIAL EQUATIONS\n405\nor\ndy\ndx  =\n2\n2\n1\n2\nyxy\nx\n+"
    },
    {
        "Chapter": "1",
        "sentence_range": "4789-4792",
        "Text": "Example 18 Show that the family of curves for which the slope of the tangent at any\npoint (x, y) on it is  \n2\n2\n2\nx\ny\n+xy\n, is given by x2 \u2013 y2 = cx Solution We know that the slope of the tangent at any point on a curve is dy\ndx Therefore,\ndy\ndx  =\n2\n2\n2\nx\ny\nxy\n+\n\u00a9 NCERT\nnot to be republished\nDIFFERENTIAL EQUATIONS\n405\nor\ndy\ndx  =\n2\n2\n1\n2\nyxy\nx\n+ (1)\nClearly, (1) is a homogenous differential equation"
    },
    {
        "Chapter": "1",
        "sentence_range": "4790-4793",
        "Text": "Solution We know that the slope of the tangent at any point on a curve is dy\ndx Therefore,\ndy\ndx  =\n2\n2\n2\nx\ny\nxy\n+\n\u00a9 NCERT\nnot to be republished\nDIFFERENTIAL EQUATIONS\n405\nor\ndy\ndx  =\n2\n2\n1\n2\nyxy\nx\n+ (1)\nClearly, (1) is a homogenous differential equation To solve it we make substitution\ny = vx\nDifferentiating y = vx with respect to x, we get\ndy\ndx  =\ndv\nv\nx dx\n+\nor\ndv\nv\n+x dx\n =\n2\n1\n2\nvv\n+\nor\ndv\nx dx =\n2\n1\n2\nvv\n\u2212\n122\nv\n\u2212vdv\n = dx\nx\nor\n22\nv1\nv \u2212dv\n =\nxdx\n\u2212\nTherefore\n22\nv1\n\u222bv \u2212dv\n =\nx1 dx\n\u2212\u222b\nor\nlog |v2 \u2013 1 | = \u2013 log |x| + log |C1|\nor\nlog |(v2 \u2013 1) (x)| = log |C1|\nor\n(v2 \u2013 1) x = \u00b1 C1\nReplacing v by y\nx  , we get\n2\n2\n1\ny\nx\nx\n\u239b\n\u2212\u239e\n\u239c\n\u239f\n\u239d\n\u23a0\n = \u00b1 C1\nor\n(y2 \u2013 x2) = \u00b1 C1 x or x2 \u2013 y2 = Cx\n\u00a9 NCERT\nnot to be republished\nMATHEMATICS\n406\nEXERCISE 9"
    },
    {
        "Chapter": "1",
        "sentence_range": "4791-4794",
        "Text": "Therefore,\ndy\ndx  =\n2\n2\n2\nx\ny\nxy\n+\n\u00a9 NCERT\nnot to be republished\nDIFFERENTIAL EQUATIONS\n405\nor\ndy\ndx  =\n2\n2\n1\n2\nyxy\nx\n+ (1)\nClearly, (1) is a homogenous differential equation To solve it we make substitution\ny = vx\nDifferentiating y = vx with respect to x, we get\ndy\ndx  =\ndv\nv\nx dx\n+\nor\ndv\nv\n+x dx\n =\n2\n1\n2\nvv\n+\nor\ndv\nx dx =\n2\n1\n2\nvv\n\u2212\n122\nv\n\u2212vdv\n = dx\nx\nor\n22\nv1\nv \u2212dv\n =\nxdx\n\u2212\nTherefore\n22\nv1\n\u222bv \u2212dv\n =\nx1 dx\n\u2212\u222b\nor\nlog |v2 \u2013 1 | = \u2013 log |x| + log |C1|\nor\nlog |(v2 \u2013 1) (x)| = log |C1|\nor\n(v2 \u2013 1) x = \u00b1 C1\nReplacing v by y\nx  , we get\n2\n2\n1\ny\nx\nx\n\u239b\n\u2212\u239e\n\u239c\n\u239f\n\u239d\n\u23a0\n = \u00b1 C1\nor\n(y2 \u2013 x2) = \u00b1 C1 x or x2 \u2013 y2 = Cx\n\u00a9 NCERT\nnot to be republished\nMATHEMATICS\n406\nEXERCISE 9 5\nIn each of the Exercises 1 to 10, show that the given differential equation is homogeneous\nand solve each of them"
    },
    {
        "Chapter": "1",
        "sentence_range": "4792-4795",
        "Text": "(1)\nClearly, (1) is a homogenous differential equation To solve it we make substitution\ny = vx\nDifferentiating y = vx with respect to x, we get\ndy\ndx  =\ndv\nv\nx dx\n+\nor\ndv\nv\n+x dx\n =\n2\n1\n2\nvv\n+\nor\ndv\nx dx =\n2\n1\n2\nvv\n\u2212\n122\nv\n\u2212vdv\n = dx\nx\nor\n22\nv1\nv \u2212dv\n =\nxdx\n\u2212\nTherefore\n22\nv1\n\u222bv \u2212dv\n =\nx1 dx\n\u2212\u222b\nor\nlog |v2 \u2013 1 | = \u2013 log |x| + log |C1|\nor\nlog |(v2 \u2013 1) (x)| = log |C1|\nor\n(v2 \u2013 1) x = \u00b1 C1\nReplacing v by y\nx  , we get\n2\n2\n1\ny\nx\nx\n\u239b\n\u2212\u239e\n\u239c\n\u239f\n\u239d\n\u23a0\n = \u00b1 C1\nor\n(y2 \u2013 x2) = \u00b1 C1 x or x2 \u2013 y2 = Cx\n\u00a9 NCERT\nnot to be republished\nMATHEMATICS\n406\nEXERCISE 9 5\nIn each of the Exercises 1 to 10, show that the given differential equation is homogeneous\nand solve each of them 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "4793-4796",
        "Text": "To solve it we make substitution\ny = vx\nDifferentiating y = vx with respect to x, we get\ndy\ndx  =\ndv\nv\nx dx\n+\nor\ndv\nv\n+x dx\n =\n2\n1\n2\nvv\n+\nor\ndv\nx dx =\n2\n1\n2\nvv\n\u2212\n122\nv\n\u2212vdv\n = dx\nx\nor\n22\nv1\nv \u2212dv\n =\nxdx\n\u2212\nTherefore\n22\nv1\n\u222bv \u2212dv\n =\nx1 dx\n\u2212\u222b\nor\nlog |v2 \u2013 1 | = \u2013 log |x| + log |C1|\nor\nlog |(v2 \u2013 1) (x)| = log |C1|\nor\n(v2 \u2013 1) x = \u00b1 C1\nReplacing v by y\nx  , we get\n2\n2\n1\ny\nx\nx\n\u239b\n\u2212\u239e\n\u239c\n\u239f\n\u239d\n\u23a0\n = \u00b1 C1\nor\n(y2 \u2013 x2) = \u00b1 C1 x or x2 \u2013 y2 = Cx\n\u00a9 NCERT\nnot to be republished\nMATHEMATICS\n406\nEXERCISE 9 5\nIn each of the Exercises 1 to 10, show that the given differential equation is homogeneous\nand solve each of them 1 (x2 + xy) dy = (x2 + y2) dx\n2"
    },
    {
        "Chapter": "1",
        "sentence_range": "4794-4797",
        "Text": "5\nIn each of the Exercises 1 to 10, show that the given differential equation is homogeneous\nand solve each of them 1 (x2 + xy) dy = (x2 + y2) dx\n2 x\ny\ny\n+x\n\u2032 =\n3"
    },
    {
        "Chapter": "1",
        "sentence_range": "4795-4798",
        "Text": "1 (x2 + xy) dy = (x2 + y2) dx\n2 x\ny\ny\n+x\n\u2032 =\n3 (x \u2013 y) dy \u2013 (x + y) dx = 0\n4"
    },
    {
        "Chapter": "1",
        "sentence_range": "4796-4799",
        "Text": "(x2 + xy) dy = (x2 + y2) dx\n2 x\ny\ny\n+x\n\u2032 =\n3 (x \u2013 y) dy \u2013 (x + y) dx = 0\n4 (x2 \u2013 y2) dx + 2xy dy = 0\n5"
    },
    {
        "Chapter": "1",
        "sentence_range": "4797-4800",
        "Text": "x\ny\ny\n+x\n\u2032 =\n3 (x \u2013 y) dy \u2013 (x + y) dx = 0\n4 (x2 \u2013 y2) dx + 2xy dy = 0\n5 2\n2\n22\nxdy\nx\ny\nxy\ndx =\n\u2212\n+\n6"
    },
    {
        "Chapter": "1",
        "sentence_range": "4798-4801",
        "Text": "(x \u2013 y) dy \u2013 (x + y) dx = 0\n4 (x2 \u2013 y2) dx + 2xy dy = 0\n5 2\n2\n22\nxdy\nx\ny\nxy\ndx =\n\u2212\n+\n6 x dy \u2013 y dx = \n2\n2\nx\ny dx\n+\n7"
    },
    {
        "Chapter": "1",
        "sentence_range": "4799-4802",
        "Text": "(x2 \u2013 y2) dx + 2xy dy = 0\n5 2\n2\n22\nxdy\nx\ny\nxy\ndx =\n\u2212\n+\n6 x dy \u2013 y dx = \n2\n2\nx\ny dx\n+\n7 cos\nsin\nsin\ncos\ny\ny\ny\ny\nx\ny\ny dx\ny\nx\nx dy\nx\nx\nx\nx\n\u23a7\n\u23ab\n\u23a7\n\u23ab\n\u239b\n\u239e\n\u239b\n\u239e\n\u239b\n\u239e\n\u239b\n\u239e\n+\n=\n\u2212\n\u23a8\n\u23ac\n\u23a8\n\u23ac\n\u239c\n\u239f\n\u239c\n\u239f\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\n\u239d\n\u23a0\n\u239d\n\u23a0\n\u23a9\n\u23ad\n\u23a9\n\u23ad\n8"
    },
    {
        "Chapter": "1",
        "sentence_range": "4800-4803",
        "Text": "2\n2\n22\nxdy\nx\ny\nxy\ndx =\n\u2212\n+\n6 x dy \u2013 y dx = \n2\n2\nx\ny dx\n+\n7 cos\nsin\nsin\ncos\ny\ny\ny\ny\nx\ny\ny dx\ny\nx\nx dy\nx\nx\nx\nx\n\u23a7\n\u23ab\n\u23a7\n\u23ab\n\u239b\n\u239e\n\u239b\n\u239e\n\u239b\n\u239e\n\u239b\n\u239e\n+\n=\n\u2212\n\u23a8\n\u23ac\n\u23a8\n\u23ac\n\u239c\n\u239f\n\u239c\n\u239f\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\n\u239d\n\u23a0\n\u239d\n\u23a0\n\u23a9\n\u23ad\n\u23a9\n\u23ad\n8 sin\n0\ndy\ny\nx\ny\nx\ndx\n\u239bx\n\u239e\n\u2212\n+\n=\n\u239c\n\u239f\n\u239d\n\u23a0\n9"
    },
    {
        "Chapter": "1",
        "sentence_range": "4801-4804",
        "Text": "x dy \u2013 y dx = \n2\n2\nx\ny dx\n+\n7 cos\nsin\nsin\ncos\ny\ny\ny\ny\nx\ny\ny dx\ny\nx\nx dy\nx\nx\nx\nx\n\u23a7\n\u23ab\n\u23a7\n\u23ab\n\u239b\n\u239e\n\u239b\n\u239e\n\u239b\n\u239e\n\u239b\n\u239e\n+\n=\n\u2212\n\u23a8\n\u23ac\n\u23a8\n\u23ac\n\u239c\n\u239f\n\u239c\n\u239f\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\n\u239d\n\u23a0\n\u239d\n\u23a0\n\u23a9\n\u23ad\n\u23a9\n\u23ad\n8 sin\n0\ndy\ny\nx\ny\nx\ndx\n\u239bx\n\u239e\n\u2212\n+\n=\n\u239c\n\u239f\n\u239d\n\u23a0\n9 log\n2\n0\ny\ny dx\nx\ndy\nx dy\n\u239bx\n\u239e\n+\n\u2212\n=\n\u239c\n\u239f\n\u239d\n\u23a0\n10"
    },
    {
        "Chapter": "1",
        "sentence_range": "4802-4805",
        "Text": "cos\nsin\nsin\ncos\ny\ny\ny\ny\nx\ny\ny dx\ny\nx\nx dy\nx\nx\nx\nx\n\u23a7\n\u23ab\n\u23a7\n\u23ab\n\u239b\n\u239e\n\u239b\n\u239e\n\u239b\n\u239e\n\u239b\n\u239e\n+\n=\n\u2212\n\u23a8\n\u23ac\n\u23a8\n\u23ac\n\u239c\n\u239f\n\u239c\n\u239f\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\n\u239d\n\u23a0\n\u239d\n\u23a0\n\u23a9\n\u23ad\n\u23a9\n\u23ad\n8 sin\n0\ndy\ny\nx\ny\nx\ndx\n\u239bx\n\u239e\n\u2212\n+\n=\n\u239c\n\u239f\n\u239d\n\u23a0\n9 log\n2\n0\ny\ny dx\nx\ndy\nx dy\n\u239bx\n\u239e\n+\n\u2212\n=\n\u239c\n\u239f\n\u239d\n\u23a0\n10 1\n1\n0\nx\nx\ny\ny\nx\ne\ndx\ne\ndy\ny\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFor each of the differential equations in Exercises from 11 to 15, find the particular\nsolution satisfying the given condition:\n11"
    },
    {
        "Chapter": "1",
        "sentence_range": "4803-4806",
        "Text": "sin\n0\ndy\ny\nx\ny\nx\ndx\n\u239bx\n\u239e\n\u2212\n+\n=\n\u239c\n\u239f\n\u239d\n\u23a0\n9 log\n2\n0\ny\ny dx\nx\ndy\nx dy\n\u239bx\n\u239e\n+\n\u2212\n=\n\u239c\n\u239f\n\u239d\n\u23a0\n10 1\n1\n0\nx\nx\ny\ny\nx\ne\ndx\ne\ndy\ny\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFor each of the differential equations in Exercises from 11 to 15, find the particular\nsolution satisfying the given condition:\n11 (x + y) dy + (x \u2013 y) dx = 0; y = 1 when x = 1\n12"
    },
    {
        "Chapter": "1",
        "sentence_range": "4804-4807",
        "Text": "log\n2\n0\ny\ny dx\nx\ndy\nx dy\n\u239bx\n\u239e\n+\n\u2212\n=\n\u239c\n\u239f\n\u239d\n\u23a0\n10 1\n1\n0\nx\nx\ny\ny\nx\ne\ndx\ne\ndy\ny\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFor each of the differential equations in Exercises from 11 to 15, find the particular\nsolution satisfying the given condition:\n11 (x + y) dy + (x \u2013 y) dx = 0; y = 1 when x = 1\n12 x2 dy + (xy + y2) dx = 0; y = 1 when x = 1\n13"
    },
    {
        "Chapter": "1",
        "sentence_range": "4805-4808",
        "Text": "1\n1\n0\nx\nx\ny\ny\nx\ne\ndx\ne\ndy\ny\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFor each of the differential equations in Exercises from 11 to 15, find the particular\nsolution satisfying the given condition:\n11 (x + y) dy + (x \u2013 y) dx = 0; y = 1 when x = 1\n12 x2 dy + (xy + y2) dx = 0; y = 1 when x = 1\n13 sin2\n0;\n4\n \n \n \n  \n \n \n \n \n \n \n \n \n \n \n \ny\nx\ny dx\nx dy\ny\nx\n when x = 1\n14"
    },
    {
        "Chapter": "1",
        "sentence_range": "4806-4809",
        "Text": "(x + y) dy + (x \u2013 y) dx = 0; y = 1 when x = 1\n12 x2 dy + (xy + y2) dx = 0; y = 1 when x = 1\n13 sin2\n0;\n4\n \n \n \n  \n \n \n \n \n \n \n \n \n \n \n \ny\nx\ny dx\nx dy\ny\nx\n when x = 1\n14 cosec\n0\ndy\ny\ny\ndx\nx\n\u239bx\n\u239e\n\u2212\n+\n=\n\u239c\n\u239f\n\u239d\n\u23a0\n;  y = 0 when x = 1\n15"
    },
    {
        "Chapter": "1",
        "sentence_range": "4807-4810",
        "Text": "x2 dy + (xy + y2) dx = 0; y = 1 when x = 1\n13 sin2\n0;\n4\n \n \n \n  \n \n \n \n \n \n \n \n \n \n \n \ny\nx\ny dx\nx dy\ny\nx\n when x = 1\n14 cosec\n0\ndy\ny\ny\ndx\nx\n\u239bx\n\u239e\n\u2212\n+\n=\n\u239c\n\u239f\n\u239d\n\u23a0\n;  y = 0 when x = 1\n15 2\n2\n2\n2\n0\ndy\nxy\ny\nx dx\n+\n\u2212\n=\n;  y = 2 when x = 1\n16"
    },
    {
        "Chapter": "1",
        "sentence_range": "4808-4811",
        "Text": "sin2\n0;\n4\n \n \n \n  \n \n \n \n \n \n \n \n \n \n \n \ny\nx\ny dx\nx dy\ny\nx\n when x = 1\n14 cosec\n0\ndy\ny\ny\ndx\nx\n\u239bx\n\u239e\n\u2212\n+\n=\n\u239c\n\u239f\n\u239d\n\u23a0\n;  y = 0 when x = 1\n15 2\n2\n2\n2\n0\ndy\nxy\ny\nx dx\n+\n\u2212\n=\n;  y = 2 when x = 1\n16 A homogeneous differential equation of the from \ndx\nhx\ndy\n\u239by\n\u239e\n=\n\u239c\n\u239f\n\u239d\n\u23a0  can be solved by\nmaking the substitution"
    },
    {
        "Chapter": "1",
        "sentence_range": "4809-4812",
        "Text": "cosec\n0\ndy\ny\ny\ndx\nx\n\u239bx\n\u239e\n\u2212\n+\n=\n\u239c\n\u239f\n\u239d\n\u23a0\n;  y = 0 when x = 1\n15 2\n2\n2\n2\n0\ndy\nxy\ny\nx dx\n+\n\u2212\n=\n;  y = 2 when x = 1\n16 A homogeneous differential equation of the from \ndx\nhx\ndy\n\u239by\n\u239e\n=\n\u239c\n\u239f\n\u239d\n\u23a0  can be solved by\nmaking the substitution (A) y = vx\n(B) v = yx\n(C) x = vy\n(D) x = v\n\u00a9 NCERT\nnot to be republished\nDIFFERENTIAL EQUATIONS\n407\n17"
    },
    {
        "Chapter": "1",
        "sentence_range": "4810-4813",
        "Text": "2\n2\n2\n2\n0\ndy\nxy\ny\nx dx\n+\n\u2212\n=\n;  y = 2 when x = 1\n16 A homogeneous differential equation of the from \ndx\nhx\ndy\n\u239by\n\u239e\n=\n\u239c\n\u239f\n\u239d\n\u23a0  can be solved by\nmaking the substitution (A) y = vx\n(B) v = yx\n(C) x = vy\n(D) x = v\n\u00a9 NCERT\nnot to be republished\nDIFFERENTIAL EQUATIONS\n407\n17 Which of the following is a homogeneous differential equation"
    },
    {
        "Chapter": "1",
        "sentence_range": "4811-4814",
        "Text": "A homogeneous differential equation of the from \ndx\nhx\ndy\n\u239by\n\u239e\n=\n\u239c\n\u239f\n\u239d\n\u23a0  can be solved by\nmaking the substitution (A) y = vx\n(B) v = yx\n(C) x = vy\n(D) x = v\n\u00a9 NCERT\nnot to be republished\nDIFFERENTIAL EQUATIONS\n407\n17 Which of the following is a homogeneous differential equation (A) (4x + 6y + 5) dy \u2013 (3y + 2x + 4) dx = 0\n(B) (xy) dx \u2013 (x3 + y3) dy = 0\n(C) (x3 + 2y2) dx + 2xy dy = 0\n(D) y2 dx + (x2 \u2013 xy \u2013 y2) dy = 0\n9"
    },
    {
        "Chapter": "1",
        "sentence_range": "4812-4815",
        "Text": "(A) y = vx\n(B) v = yx\n(C) x = vy\n(D) x = v\n\u00a9 NCERT\nnot to be republished\nDIFFERENTIAL EQUATIONS\n407\n17 Which of the following is a homogeneous differential equation (A) (4x + 6y + 5) dy \u2013 (3y + 2x + 4) dx = 0\n(B) (xy) dx \u2013 (x3 + y3) dy = 0\n(C) (x3 + 2y2) dx + 2xy dy = 0\n(D) y2 dx + (x2 \u2013 xy \u2013 y2) dy = 0\n9 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "4813-4816",
        "Text": "Which of the following is a homogeneous differential equation (A) (4x + 6y + 5) dy \u2013 (3y + 2x + 4) dx = 0\n(B) (xy) dx \u2013 (x3 + y3) dy = 0\n(C) (x3 + 2y2) dx + 2xy dy = 0\n(D) y2 dx + (x2 \u2013 xy \u2013 y2) dy = 0\n9 5 3  Linear differential equations\nA differential equation of the from\nP\ndy\ndx +y\n = Q\nwhere, P and Q are constants or functions of x only, is known as a first order linear\ndifferential equation"
    },
    {
        "Chapter": "1",
        "sentence_range": "4814-4817",
        "Text": "(A) (4x + 6y + 5) dy \u2013 (3y + 2x + 4) dx = 0\n(B) (xy) dx \u2013 (x3 + y3) dy = 0\n(C) (x3 + 2y2) dx + 2xy dy = 0\n(D) y2 dx + (x2 \u2013 xy \u2013 y2) dy = 0\n9 5 3  Linear differential equations\nA differential equation of the from\nP\ndy\ndx +y\n = Q\nwhere, P and Q are constants or functions of x only, is known as a first order linear\ndifferential equation Some examples of the first order linear differential equation are\ndy\ndx +y\n = sin x\n1\ndy\ny\ndx\n\u239bx\n\u239e\n+ \u239c\n\u239f\n\u239d\n\u23a0\n = ex\nlog\ndy\ny\ndx\nx\nx\n\u239b\n\u239e\n+ \u239c\n\u239f\n\u239d\n\u23a0\n = 1\nx\nAnother form of first order linear differential equation is\n1P\ndx\nx\ndy +\n = Q1\nwhere, P1 and Q1 are constants or functions of y only"
    },
    {
        "Chapter": "1",
        "sentence_range": "4815-4818",
        "Text": "5 3  Linear differential equations\nA differential equation of the from\nP\ndy\ndx +y\n = Q\nwhere, P and Q are constants or functions of x only, is known as a first order linear\ndifferential equation Some examples of the first order linear differential equation are\ndy\ndx +y\n = sin x\n1\ndy\ny\ndx\n\u239bx\n\u239e\n+ \u239c\n\u239f\n\u239d\n\u23a0\n = ex\nlog\ndy\ny\ndx\nx\nx\n\u239b\n\u239e\n+ \u239c\n\u239f\n\u239d\n\u23a0\n = 1\nx\nAnother form of first order linear differential equation is\n1P\ndx\nx\ndy +\n = Q1\nwhere, P1 and Q1 are constants or functions of y only Some examples of this type of\ndifferential equation are\ndx\ndy +x\n= cos y\n2\ndx\nx\ndy\n+\u2212y\n = y2e \u2013 y\nTo solve the first order linear differential equation of the type\nP\ndy\ny\ndx  \n = Q"
    },
    {
        "Chapter": "1",
        "sentence_range": "4816-4819",
        "Text": "3  Linear differential equations\nA differential equation of the from\nP\ndy\ndx +y\n = Q\nwhere, P and Q are constants or functions of x only, is known as a first order linear\ndifferential equation Some examples of the first order linear differential equation are\ndy\ndx +y\n = sin x\n1\ndy\ny\ndx\n\u239bx\n\u239e\n+ \u239c\n\u239f\n\u239d\n\u23a0\n = ex\nlog\ndy\ny\ndx\nx\nx\n\u239b\n\u239e\n+ \u239c\n\u239f\n\u239d\n\u23a0\n = 1\nx\nAnother form of first order linear differential equation is\n1P\ndx\nx\ndy +\n = Q1\nwhere, P1 and Q1 are constants or functions of y only Some examples of this type of\ndifferential equation are\ndx\ndy +x\n= cos y\n2\ndx\nx\ndy\n+\u2212y\n = y2e \u2013 y\nTo solve the first order linear differential equation of the type\nP\ndy\ny\ndx  \n = Q (1)\nMultiply both sides of the equation by a function of x say g (x) to get\ng(x) dy\ndx  + P"
    },
    {
        "Chapter": "1",
        "sentence_range": "4817-4820",
        "Text": "Some examples of the first order linear differential equation are\ndy\ndx +y\n = sin x\n1\ndy\ny\ndx\n\u239bx\n\u239e\n+ \u239c\n\u239f\n\u239d\n\u23a0\n = ex\nlog\ndy\ny\ndx\nx\nx\n\u239b\n\u239e\n+ \u239c\n\u239f\n\u239d\n\u23a0\n = 1\nx\nAnother form of first order linear differential equation is\n1P\ndx\nx\ndy +\n = Q1\nwhere, P1 and Q1 are constants or functions of y only Some examples of this type of\ndifferential equation are\ndx\ndy +x\n= cos y\n2\ndx\nx\ndy\n+\u2212y\n = y2e \u2013 y\nTo solve the first order linear differential equation of the type\nP\ndy\ny\ndx  \n = Q (1)\nMultiply both sides of the equation by a function of x say g (x) to get\ng(x) dy\ndx  + P (g(x)) y = Q"
    },
    {
        "Chapter": "1",
        "sentence_range": "4818-4821",
        "Text": "Some examples of this type of\ndifferential equation are\ndx\ndy +x\n= cos y\n2\ndx\nx\ndy\n+\u2212y\n = y2e \u2013 y\nTo solve the first order linear differential equation of the type\nP\ndy\ny\ndx  \n = Q (1)\nMultiply both sides of the equation by a function of x say g (x) to get\ng(x) dy\ndx  + P (g(x)) y = Q g (x)"
    },
    {
        "Chapter": "1",
        "sentence_range": "4819-4822",
        "Text": "(1)\nMultiply both sides of the equation by a function of x say g (x) to get\ng(x) dy\ndx  + P (g(x)) y = Q g (x) (2)\n\u00a9 NCERT\nnot to be republished\nMATHEMATICS\n408\nChoose g (x) in such a way that R"
    },
    {
        "Chapter": "1",
        "sentence_range": "4820-4823",
        "Text": "(g(x)) y = Q g (x) (2)\n\u00a9 NCERT\nnot to be republished\nMATHEMATICS\n408\nChoose g (x) in such a way that R H"
    },
    {
        "Chapter": "1",
        "sentence_range": "4821-4824",
        "Text": "g (x) (2)\n\u00a9 NCERT\nnot to be republished\nMATHEMATICS\n408\nChoose g (x) in such a way that R H S"
    },
    {
        "Chapter": "1",
        "sentence_range": "4822-4825",
        "Text": "(2)\n\u00a9 NCERT\nnot to be republished\nMATHEMATICS\n408\nChoose g (x) in such a way that R H S becomes a derivative of y"
    },
    {
        "Chapter": "1",
        "sentence_range": "4823-4826",
        "Text": "H S becomes a derivative of y g (x)"
    },
    {
        "Chapter": "1",
        "sentence_range": "4824-4827",
        "Text": "S becomes a derivative of y g (x) i"
    },
    {
        "Chapter": "1",
        "sentence_range": "4825-4828",
        "Text": "becomes a derivative of y g (x) i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "4826-4829",
        "Text": "g (x) i e g(x) dy\ndx  + P"
    },
    {
        "Chapter": "1",
        "sentence_range": "4827-4830",
        "Text": "i e g(x) dy\ndx  + P g(x) y = d\ndx  [y"
    },
    {
        "Chapter": "1",
        "sentence_range": "4828-4831",
        "Text": "e g(x) dy\ndx  + P g(x) y = d\ndx  [y g (x)]\nor\ng(x) dy\ndx  + P"
    },
    {
        "Chapter": "1",
        "sentence_range": "4829-4832",
        "Text": "g(x) dy\ndx  + P g(x) y = d\ndx  [y g (x)]\nor\ng(x) dy\ndx  + P g(x) y = g(x) dy\ndx  + y g\u2032 (x)\n\u21d2\nP"
    },
    {
        "Chapter": "1",
        "sentence_range": "4830-4833",
        "Text": "g(x) y = d\ndx  [y g (x)]\nor\ng(x) dy\ndx  + P g(x) y = g(x) dy\ndx  + y g\u2032 (x)\n\u21d2\nP g(x) = g\u2032 (x)\nor\nP =\n( )\n( )\ng\nx\ng x\n\u2032\nIntegrating both sides with respect to x, we get\n\u222bPdx\n =\n( )\n( )\ng x dx\n\u2032g x\n\u222b\nor\nP dx\n\u222b\u22c5\n = log(g (x))\nor\ng(x) =\nP dx\ne\u222b\nOn multiplying the equation (1) by g(x) = \ne\u222bP dx\n, the L"
    },
    {
        "Chapter": "1",
        "sentence_range": "4831-4834",
        "Text": "g (x)]\nor\ng(x) dy\ndx  + P g(x) y = g(x) dy\ndx  + y g\u2032 (x)\n\u21d2\nP g(x) = g\u2032 (x)\nor\nP =\n( )\n( )\ng\nx\ng x\n\u2032\nIntegrating both sides with respect to x, we get\n\u222bPdx\n =\n( )\n( )\ng x dx\n\u2032g x\n\u222b\nor\nP dx\n\u222b\u22c5\n = log(g (x))\nor\ng(x) =\nP dx\ne\u222b\nOn multiplying the equation (1) by g(x) = \ne\u222bP dx\n, the L H"
    },
    {
        "Chapter": "1",
        "sentence_range": "4832-4835",
        "Text": "g(x) y = g(x) dy\ndx  + y g\u2032 (x)\n\u21d2\nP g(x) = g\u2032 (x)\nor\nP =\n( )\n( )\ng\nx\ng x\n\u2032\nIntegrating both sides with respect to x, we get\n\u222bPdx\n =\n( )\n( )\ng x dx\n\u2032g x\n\u222b\nor\nP dx\n\u222b\u22c5\n = log(g (x))\nor\ng(x) =\nP dx\ne\u222b\nOn multiplying the equation (1) by g(x) = \ne\u222bP dx\n, the L H S"
    },
    {
        "Chapter": "1",
        "sentence_range": "4833-4836",
        "Text": "g(x) = g\u2032 (x)\nor\nP =\n( )\n( )\ng\nx\ng x\n\u2032\nIntegrating both sides with respect to x, we get\n\u222bPdx\n =\n( )\n( )\ng x dx\n\u2032g x\n\u222b\nor\nP dx\n\u222b\u22c5\n = log(g (x))\nor\ng(x) =\nP dx\ne\u222b\nOn multiplying the equation (1) by g(x) = \ne\u222bP dx\n, the L H S becomes the derivative\nof some function of x and y"
    },
    {
        "Chapter": "1",
        "sentence_range": "4834-4837",
        "Text": "H S becomes the derivative\nof some function of x and y This function g(x) = \ne\u222bP dx\n is called Integrating Factor\n(I"
    },
    {
        "Chapter": "1",
        "sentence_range": "4835-4838",
        "Text": "S becomes the derivative\nof some function of x and y This function g(x) = \ne\u222bP dx\n is called Integrating Factor\n(I F"
    },
    {
        "Chapter": "1",
        "sentence_range": "4836-4839",
        "Text": "becomes the derivative\nof some function of x and y This function g(x) = \ne\u222bP dx\n is called Integrating Factor\n(I F )  of the given differential equation"
    },
    {
        "Chapter": "1",
        "sentence_range": "4837-4840",
        "Text": "This function g(x) = \ne\u222bP dx\n is called Integrating Factor\n(I F )  of the given differential equation Substituting the value of g (x) in equation (2), we get\nP\nP\nP\ndx\ndx\ndy\ne\ne\ny\ndx\n \n \n \n =\nP\nQ\ne  dx\n \nor\nPdx \nd\ndxy e\n \n =\nP\nQ\ndx\ne  \nIntegrating both sides with respect to x, we get\ny e  Pdx\n \n =  \n \nP\nQ\ndx\ne\ndx\n  \n \nor\ny =\n \nP\nP\nQ\nC\ndx\ndx\ne\ne\ndx\n  \n \n \n \n \nwhich is the general solution of the differential equation"
    },
    {
        "Chapter": "1",
        "sentence_range": "4838-4841",
        "Text": "F )  of the given differential equation Substituting the value of g (x) in equation (2), we get\nP\nP\nP\ndx\ndx\ndy\ne\ne\ny\ndx\n \n \n \n =\nP\nQ\ne  dx\n \nor\nPdx \nd\ndxy e\n \n =\nP\nQ\ndx\ne  \nIntegrating both sides with respect to x, we get\ny e  Pdx\n \n =  \n \nP\nQ\ndx\ne\ndx\n  \n \nor\ny =\n \nP\nP\nQ\nC\ndx\ndx\ne\ne\ndx\n  \n \n \n \n \nwhich is the general solution of the differential equation \u00a9 NCERT\nnot to be republished\nDIFFERENTIAL EQUATIONS\n409\nSteps involved to solve first order linear differential equation:\n(i)\nWrite the given differential equation in the form \nP\nQ\ndy\ndx +y\n=\n where P, Q are\nconstants or functions of x only"
    },
    {
        "Chapter": "1",
        "sentence_range": "4839-4842",
        "Text": ")  of the given differential equation Substituting the value of g (x) in equation (2), we get\nP\nP\nP\ndx\ndx\ndy\ne\ne\ny\ndx\n \n \n \n =\nP\nQ\ne  dx\n \nor\nPdx \nd\ndxy e\n \n =\nP\nQ\ndx\ne  \nIntegrating both sides with respect to x, we get\ny e  Pdx\n \n =  \n \nP\nQ\ndx\ne\ndx\n  \n \nor\ny =\n \nP\nP\nQ\nC\ndx\ndx\ne\ne\ndx\n  \n \n \n \n \nwhich is the general solution of the differential equation \u00a9 NCERT\nnot to be republished\nDIFFERENTIAL EQUATIONS\n409\nSteps involved to solve first order linear differential equation:\n(i)\nWrite the given differential equation in the form \nP\nQ\ndy\ndx +y\n=\n where P, Q are\nconstants or functions of x only (ii)\nFind the Integrating Factor (I"
    },
    {
        "Chapter": "1",
        "sentence_range": "4840-4843",
        "Text": "Substituting the value of g (x) in equation (2), we get\nP\nP\nP\ndx\ndx\ndy\ne\ne\ny\ndx\n \n \n \n =\nP\nQ\ne  dx\n \nor\nPdx \nd\ndxy e\n \n =\nP\nQ\ndx\ne  \nIntegrating both sides with respect to x, we get\ny e  Pdx\n \n =  \n \nP\nQ\ndx\ne\ndx\n  \n \nor\ny =\n \nP\nP\nQ\nC\ndx\ndx\ne\ne\ndx\n  \n \n \n \n \nwhich is the general solution of the differential equation \u00a9 NCERT\nnot to be republished\nDIFFERENTIAL EQUATIONS\n409\nSteps involved to solve first order linear differential equation:\n(i)\nWrite the given differential equation in the form \nP\nQ\ndy\ndx +y\n=\n where P, Q are\nconstants or functions of x only (ii)\nFind the Integrating Factor (I F) = \nePdx"
    },
    {
        "Chapter": "1",
        "sentence_range": "4841-4844",
        "Text": "\u00a9 NCERT\nnot to be republished\nDIFFERENTIAL EQUATIONS\n409\nSteps involved to solve first order linear differential equation:\n(i)\nWrite the given differential equation in the form \nP\nQ\ndy\ndx +y\n=\n where P, Q are\nconstants or functions of x only (ii)\nFind the Integrating Factor (I F) = \nePdx (iii)\nWrite the solution of the given differential equation as\ny (I"
    },
    {
        "Chapter": "1",
        "sentence_range": "4842-4845",
        "Text": "(ii)\nFind the Integrating Factor (I F) = \nePdx (iii)\nWrite the solution of the given differential equation as\ny (I F) =\nQ \u00d7 I"
    },
    {
        "Chapter": "1",
        "sentence_range": "4843-4846",
        "Text": "F) = \nePdx (iii)\nWrite the solution of the given differential equation as\ny (I F) =\nQ \u00d7 I F\n C\n \ndx\nIn case, the first order linear differential equation is in the form \n1\n1\nP\nQ\ndx\nx\ndy +\n=\n,\nwhere, P1 and Q1 are constants or functions of y only"
    },
    {
        "Chapter": "1",
        "sentence_range": "4844-4847",
        "Text": "(iii)\nWrite the solution of the given differential equation as\ny (I F) =\nQ \u00d7 I F\n C\n \ndx\nIn case, the first order linear differential equation is in the form \n1\n1\nP\nQ\ndx\nx\ndy +\n=\n,\nwhere, P1 and Q1 are constants or functions of y only Then I"
    },
    {
        "Chapter": "1",
        "sentence_range": "4845-4848",
        "Text": "F) =\nQ \u00d7 I F\n C\n \ndx\nIn case, the first order linear differential equation is in the form \n1\n1\nP\nQ\ndx\nx\ndy +\n=\n,\nwhere, P1 and Q1 are constants or functions of y only Then I F = \ne  1P dy\n and the\nsolution of the differential equation is given by\nx"
    },
    {
        "Chapter": "1",
        "sentence_range": "4846-4849",
        "Text": "F\n C\n \ndx\nIn case, the first order linear differential equation is in the form \n1\n1\nP\nQ\ndx\nx\ndy +\n=\n,\nwhere, P1 and Q1 are constants or functions of y only Then I F = \ne  1P dy\n and the\nsolution of the differential equation is given by\nx (I"
    },
    {
        "Chapter": "1",
        "sentence_range": "4847-4850",
        "Text": "Then I F = \ne  1P dy\n and the\nsolution of the differential equation is given by\nx (I F) =\n(\n)\n1\nQ \u00d7 I"
    },
    {
        "Chapter": "1",
        "sentence_range": "4848-4851",
        "Text": "F = \ne  1P dy\n and the\nsolution of the differential equation is given by\nx (I F) =\n(\n)\n1\nQ \u00d7 I F\ndy +C\n\u222b\nExample 19 Find the general solution of the differential equation \ncos\ndy\ny\nx\ndx \u2212\n="
    },
    {
        "Chapter": "1",
        "sentence_range": "4849-4852",
        "Text": "(I F) =\n(\n)\n1\nQ \u00d7 I F\ndy +C\n\u222b\nExample 19 Find the general solution of the differential equation \ncos\ndy\ny\nx\ndx \u2212\n= Solution Given differential equation is of the form\nP\nQ\ndy\ny\ndx +\n=\n, where P = \u20131 and Q  =  cos x\nTherefore\nI"
    },
    {
        "Chapter": "1",
        "sentence_range": "4850-4853",
        "Text": "F) =\n(\n)\n1\nQ \u00d7 I F\ndy +C\n\u222b\nExample 19 Find the general solution of the differential equation \ncos\ndy\ny\nx\ndx \u2212\n= Solution Given differential equation is of the form\nP\nQ\ndy\ny\ndx +\n=\n, where P = \u20131 and Q  =  cos x\nTherefore\nI F =\n1 dx\nx\ne\ne\n  \n \nMultiplying both sides of equation by I"
    },
    {
        "Chapter": "1",
        "sentence_range": "4851-4854",
        "Text": "F\ndy +C\n\u222b\nExample 19 Find the general solution of the differential equation \ncos\ndy\ny\nx\ndx \u2212\n= Solution Given differential equation is of the form\nP\nQ\ndy\ny\ndx +\n=\n, where P = \u20131 and Q  =  cos x\nTherefore\nI F =\n1 dx\nx\ne\ne\n  \n \nMultiplying both sides of equation by I F, we get\nx\nx\ndy\ne\ne\ny\ndx\n\u2212\n\u2212\u2212\n = e\u2013x cos x\nor\n(\ndy yex)\ndx\n\u2212\n= e\u2013x cos x\nOn integrating both sides with respect to x, we get\nye\u2013 x =\ncos\nC\nex\nx dx\n\u2212\n+\n\u222b"
    },
    {
        "Chapter": "1",
        "sentence_range": "4852-4855",
        "Text": "Solution Given differential equation is of the form\nP\nQ\ndy\ny\ndx +\n=\n, where P = \u20131 and Q  =  cos x\nTherefore\nI F =\n1 dx\nx\ne\ne\n  \n \nMultiplying both sides of equation by I F, we get\nx\nx\ndy\ne\ne\ny\ndx\n\u2212\n\u2212\u2212\n = e\u2013x cos x\nor\n(\ndy yex)\ndx\n\u2212\n= e\u2013x cos x\nOn integrating both sides with respect to x, we get\nye\u2013 x =\ncos\nC\nex\nx dx\n\u2212\n+\n\u222b (1)\nLet\nI =\nexcos\nx dx\n\u2212\n\u222b\n= cos\n( sin ) (\n)\n1\nx\nx\nxe\nx\ne\ndx\n\u2212\n\u2212\n\u239b\n\u239e \u2212\n\u2212\n\u2212\n\u239c\n\u239d\u2212\u239f\n\u23a0 \u222b\n\u00a9 NCERT\nnot to be republished\nMATHEMATICS\n410\n=\ncos\nsin\nx\nx\nx e\nx e\ndx\n\u2212\n\u2212\n\u2212\n\u2212\u222b\n=\ncos\nsin (\u2013\n)\ncos\n(\n)\nx\nx\nx\nx e\nx\ne\nx\ne\ndx\n\u2212\n\u2212\n\u2212\n\u23a1\n\u23a4\n\u2212\n\u2212\n\u2212\n\u2212\n\u23a3\n\u23a6\n\u222b\n=\ncos\nsin\ncos\nx\nx\nx\nx e\nx e\nx e\ndx\n\u2212\n\u2212\n\u2212\n\u2212\n+\n\u2212\u222b\nor\nI = \u2013 e\u2013x cos x + sin x e\u2013x \u2013 I\nor\n2I = (sin x \u2013 cos x) e\u2013x\nor\nI = (sin\ncos )\n2\nx\nx\nx e\u2212\n\u2212\nSubstituting the value of I in equation (1), we get\nye\u2013 x =\nsin\ncos\nC\n2\nx\nx\nx e\u2212\n\u2212\n\u239b\n\u239e\n+\n\u239c\n\u239f\n\u239d\n\u23a0\nor\ny =\nsin\ncos\nC\n2\nx\nx\nx\ne\n\u2212\n\u239b\n\u239e +\n\u239c\n\u239f\n\u239d\n\u23a0\nwhich is the general solution of the given differential equation"
    },
    {
        "Chapter": "1",
        "sentence_range": "4853-4856",
        "Text": "F =\n1 dx\nx\ne\ne\n  \n \nMultiplying both sides of equation by I F, we get\nx\nx\ndy\ne\ne\ny\ndx\n\u2212\n\u2212\u2212\n = e\u2013x cos x\nor\n(\ndy yex)\ndx\n\u2212\n= e\u2013x cos x\nOn integrating both sides with respect to x, we get\nye\u2013 x =\ncos\nC\nex\nx dx\n\u2212\n+\n\u222b (1)\nLet\nI =\nexcos\nx dx\n\u2212\n\u222b\n= cos\n( sin ) (\n)\n1\nx\nx\nxe\nx\ne\ndx\n\u2212\n\u2212\n\u239b\n\u239e \u2212\n\u2212\n\u2212\n\u239c\n\u239d\u2212\u239f\n\u23a0 \u222b\n\u00a9 NCERT\nnot to be republished\nMATHEMATICS\n410\n=\ncos\nsin\nx\nx\nx e\nx e\ndx\n\u2212\n\u2212\n\u2212\n\u2212\u222b\n=\ncos\nsin (\u2013\n)\ncos\n(\n)\nx\nx\nx\nx e\nx\ne\nx\ne\ndx\n\u2212\n\u2212\n\u2212\n\u23a1\n\u23a4\n\u2212\n\u2212\n\u2212\n\u2212\n\u23a3\n\u23a6\n\u222b\n=\ncos\nsin\ncos\nx\nx\nx\nx e\nx e\nx e\ndx\n\u2212\n\u2212\n\u2212\n\u2212\n+\n\u2212\u222b\nor\nI = \u2013 e\u2013x cos x + sin x e\u2013x \u2013 I\nor\n2I = (sin x \u2013 cos x) e\u2013x\nor\nI = (sin\ncos )\n2\nx\nx\nx e\u2212\n\u2212\nSubstituting the value of I in equation (1), we get\nye\u2013 x =\nsin\ncos\nC\n2\nx\nx\nx e\u2212\n\u2212\n\u239b\n\u239e\n+\n\u239c\n\u239f\n\u239d\n\u23a0\nor\ny =\nsin\ncos\nC\n2\nx\nx\nx\ne\n\u2212\n\u239b\n\u239e +\n\u239c\n\u239f\n\u239d\n\u23a0\nwhich is the general solution of the given differential equation Example 20  Find the general solution of the differential equation \n2\n2\n(\n0)\nxdy\ny\nx\nx\ndx +\n=\n\u2260"
    },
    {
        "Chapter": "1",
        "sentence_range": "4854-4857",
        "Text": "F, we get\nx\nx\ndy\ne\ne\ny\ndx\n\u2212\n\u2212\u2212\n = e\u2013x cos x\nor\n(\ndy yex)\ndx\n\u2212\n= e\u2013x cos x\nOn integrating both sides with respect to x, we get\nye\u2013 x =\ncos\nC\nex\nx dx\n\u2212\n+\n\u222b (1)\nLet\nI =\nexcos\nx dx\n\u2212\n\u222b\n= cos\n( sin ) (\n)\n1\nx\nx\nxe\nx\ne\ndx\n\u2212\n\u2212\n\u239b\n\u239e \u2212\n\u2212\n\u2212\n\u239c\n\u239d\u2212\u239f\n\u23a0 \u222b\n\u00a9 NCERT\nnot to be republished\nMATHEMATICS\n410\n=\ncos\nsin\nx\nx\nx e\nx e\ndx\n\u2212\n\u2212\n\u2212\n\u2212\u222b\n=\ncos\nsin (\u2013\n)\ncos\n(\n)\nx\nx\nx\nx e\nx\ne\nx\ne\ndx\n\u2212\n\u2212\n\u2212\n\u23a1\n\u23a4\n\u2212\n\u2212\n\u2212\n\u2212\n\u23a3\n\u23a6\n\u222b\n=\ncos\nsin\ncos\nx\nx\nx\nx e\nx e\nx e\ndx\n\u2212\n\u2212\n\u2212\n\u2212\n+\n\u2212\u222b\nor\nI = \u2013 e\u2013x cos x + sin x e\u2013x \u2013 I\nor\n2I = (sin x \u2013 cos x) e\u2013x\nor\nI = (sin\ncos )\n2\nx\nx\nx e\u2212\n\u2212\nSubstituting the value of I in equation (1), we get\nye\u2013 x =\nsin\ncos\nC\n2\nx\nx\nx e\u2212\n\u2212\n\u239b\n\u239e\n+\n\u239c\n\u239f\n\u239d\n\u23a0\nor\ny =\nsin\ncos\nC\n2\nx\nx\nx\ne\n\u2212\n\u239b\n\u239e +\n\u239c\n\u239f\n\u239d\n\u23a0\nwhich is the general solution of the given differential equation Example 20  Find the general solution of the differential equation \n2\n2\n(\n0)\nxdy\ny\nx\nx\ndx +\n=\n\u2260 Solution The given differential equation is\n2\nxdy\ndx +y\n = x2"
    },
    {
        "Chapter": "1",
        "sentence_range": "4855-4858",
        "Text": "(1)\nLet\nI =\nexcos\nx dx\n\u2212\n\u222b\n= cos\n( sin ) (\n)\n1\nx\nx\nxe\nx\ne\ndx\n\u2212\n\u2212\n\u239b\n\u239e \u2212\n\u2212\n\u2212\n\u239c\n\u239d\u2212\u239f\n\u23a0 \u222b\n\u00a9 NCERT\nnot to be republished\nMATHEMATICS\n410\n=\ncos\nsin\nx\nx\nx e\nx e\ndx\n\u2212\n\u2212\n\u2212\n\u2212\u222b\n=\ncos\nsin (\u2013\n)\ncos\n(\n)\nx\nx\nx\nx e\nx\ne\nx\ne\ndx\n\u2212\n\u2212\n\u2212\n\u23a1\n\u23a4\n\u2212\n\u2212\n\u2212\n\u2212\n\u23a3\n\u23a6\n\u222b\n=\ncos\nsin\ncos\nx\nx\nx\nx e\nx e\nx e\ndx\n\u2212\n\u2212\n\u2212\n\u2212\n+\n\u2212\u222b\nor\nI = \u2013 e\u2013x cos x + sin x e\u2013x \u2013 I\nor\n2I = (sin x \u2013 cos x) e\u2013x\nor\nI = (sin\ncos )\n2\nx\nx\nx e\u2212\n\u2212\nSubstituting the value of I in equation (1), we get\nye\u2013 x =\nsin\ncos\nC\n2\nx\nx\nx e\u2212\n\u2212\n\u239b\n\u239e\n+\n\u239c\n\u239f\n\u239d\n\u23a0\nor\ny =\nsin\ncos\nC\n2\nx\nx\nx\ne\n\u2212\n\u239b\n\u239e +\n\u239c\n\u239f\n\u239d\n\u23a0\nwhich is the general solution of the given differential equation Example 20  Find the general solution of the differential equation \n2\n2\n(\n0)\nxdy\ny\nx\nx\ndx +\n=\n\u2260 Solution The given differential equation is\n2\nxdy\ndx +y\n = x2 (1)\nDividing both sides of equation (1) by x, we get\n2\ndy\ny\ndx\n+x\n = x\nwhich is a linear differential equation of the type \nP\nQ\ndy\ndx +y\n=\n, where \n2\nP\n=x\n and Q = x"
    },
    {
        "Chapter": "1",
        "sentence_range": "4856-4859",
        "Text": "Example 20  Find the general solution of the differential equation \n2\n2\n(\n0)\nxdy\ny\nx\nx\ndx +\n=\n\u2260 Solution The given differential equation is\n2\nxdy\ndx +y\n = x2 (1)\nDividing both sides of equation (1) by x, we get\n2\ndy\ny\ndx\n+x\n = x\nwhich is a linear differential equation of the type \nP\nQ\ndy\ndx +y\n=\n, where \n2\nP\n=x\n and Q = x So\nI"
    },
    {
        "Chapter": "1",
        "sentence_range": "4857-4860",
        "Text": "Solution The given differential equation is\n2\nxdy\ndx +y\n = x2 (1)\nDividing both sides of equation (1) by x, we get\n2\ndy\ny\ndx\n+x\n = x\nwhich is a linear differential equation of the type \nP\nQ\ndy\ndx +y\n=\n, where \n2\nP\n=x\n and Q = x So\nI F =\ne\u222bx2 dx\n= e2 log x = \nlog2\n2\nx\ne\n=x\nlog\n( )\n[\n( )]\nf x\nas e\nf x\n=\nTherefore, solution of the given equation is given by\ny"
    },
    {
        "Chapter": "1",
        "sentence_range": "4858-4861",
        "Text": "(1)\nDividing both sides of equation (1) by x, we get\n2\ndy\ny\ndx\n+x\n = x\nwhich is a linear differential equation of the type \nP\nQ\ndy\ndx +y\n=\n, where \n2\nP\n=x\n and Q = x So\nI F =\ne\u222bx2 dx\n= e2 log x = \nlog2\n2\nx\ne\n=x\nlog\n( )\n[\n( )]\nf x\nas e\nf x\n=\nTherefore, solution of the given equation is given by\ny x2 =\n( ) (2\n)\nC\nx\nx\ndx +\n\u222b\n = \n3\nx dx +C\n\u222b\nor\ny =\n2\n2\nC\nx4\nx\u2212\n+\nwhich is the general solution of the given differential equation"
    },
    {
        "Chapter": "1",
        "sentence_range": "4859-4862",
        "Text": "So\nI F =\ne\u222bx2 dx\n= e2 log x = \nlog2\n2\nx\ne\n=x\nlog\n( )\n[\n( )]\nf x\nas e\nf x\n=\nTherefore, solution of the given equation is given by\ny x2 =\n( ) (2\n)\nC\nx\nx\ndx +\n\u222b\n = \n3\nx dx +C\n\u222b\nor\ny =\n2\n2\nC\nx4\nx\u2212\n+\nwhich is the general solution of the given differential equation \u00a9 NCERT\nnot to be republished\nDIFFERENTIAL EQUATIONS\n411\nExample 21 Find the general solution of the differential equation y dx \u2013 (x + 2y2) dy = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "4860-4863",
        "Text": "F =\ne\u222bx2 dx\n= e2 log x = \nlog2\n2\nx\ne\n=x\nlog\n( )\n[\n( )]\nf x\nas e\nf x\n=\nTherefore, solution of the given equation is given by\ny x2 =\n( ) (2\n)\nC\nx\nx\ndx +\n\u222b\n = \n3\nx dx +C\n\u222b\nor\ny =\n2\n2\nC\nx4\nx\u2212\n+\nwhich is the general solution of the given differential equation \u00a9 NCERT\nnot to be republished\nDIFFERENTIAL EQUATIONS\n411\nExample 21 Find the general solution of the differential equation y dx \u2013 (x + 2y2) dy = 0 Solution The given differential equation can be written as\ndx\nx\ndy\n\u2212y\n = 2y\nThis is a linear differential equation of the type \n1\n1\nP\nQ\ndx\nx\ndy +\n=\n, where 1\n1\nP\n= \u2212y\n and\nQ1 = 2y"
    },
    {
        "Chapter": "1",
        "sentence_range": "4861-4864",
        "Text": "x2 =\n( ) (2\n)\nC\nx\nx\ndx +\n\u222b\n = \n3\nx dx +C\n\u222b\nor\ny =\n2\n2\nC\nx4\nx\u2212\n+\nwhich is the general solution of the given differential equation \u00a9 NCERT\nnot to be republished\nDIFFERENTIAL EQUATIONS\n411\nExample 21 Find the general solution of the differential equation y dx \u2013 (x + 2y2) dy = 0 Solution The given differential equation can be written as\ndx\nx\ndy\n\u2212y\n = 2y\nThis is a linear differential equation of the type \n1\n1\nP\nQ\ndx\nx\ndy +\n=\n, where 1\n1\nP\n= \u2212y\n and\nQ1 = 2y Therefore \n1\n1\nlog\nlog( )\n1\nI"
    },
    {
        "Chapter": "1",
        "sentence_range": "4862-4865",
        "Text": "\u00a9 NCERT\nnot to be republished\nDIFFERENTIAL EQUATIONS\n411\nExample 21 Find the general solution of the differential equation y dx \u2013 (x + 2y2) dy = 0 Solution The given differential equation can be written as\ndx\nx\ndy\n\u2212y\n = 2y\nThis is a linear differential equation of the type \n1\n1\nP\nQ\ndx\nx\ndy +\n=\n, where 1\n1\nP\n= \u2212y\n and\nQ1 = 2y Therefore \n1\n1\nlog\nlog( )\n1\nI F\ndy\ny\ny\ny\ne\ne\ne\ny\n\u2212\n\u2212\n\u2212\n=\u222b\n=\n=\n=\nHence, the solution of the given differential equation is\n1\nx y   =\n1\n(2 )\nC\ny\n\u239bydy\n\u239e\n+\n\u239c\n\u239f\n\u239d\n\u23a0\n\u222b\nor\nx\ny  =\n(2\n)\ndy +C\n\u222b\nor\nx\ny  = 2y + C\nor\nx = 2y2 + Cy\nwhich is a general solution of the given differential equation"
    },
    {
        "Chapter": "1",
        "sentence_range": "4863-4866",
        "Text": "Solution The given differential equation can be written as\ndx\nx\ndy\n\u2212y\n = 2y\nThis is a linear differential equation of the type \n1\n1\nP\nQ\ndx\nx\ndy +\n=\n, where 1\n1\nP\n= \u2212y\n and\nQ1 = 2y Therefore \n1\n1\nlog\nlog( )\n1\nI F\ndy\ny\ny\ny\ne\ne\ne\ny\n\u2212\n\u2212\n\u2212\n=\u222b\n=\n=\n=\nHence, the solution of the given differential equation is\n1\nx y   =\n1\n(2 )\nC\ny\n\u239bydy\n\u239e\n+\n\u239c\n\u239f\n\u239d\n\u23a0\n\u222b\nor\nx\ny  =\n(2\n)\ndy +C\n\u222b\nor\nx\ny  = 2y + C\nor\nx = 2y2 + Cy\nwhich is a general solution of the given differential equation Example 22 Find the particular solution of the differential equation\ncot \ndy+\ny\nx\ndx\n = 2x + x2 cot x (x \u2260 0)\ngiven that y = 0 when \n2\nx\n=\u03c0"
    },
    {
        "Chapter": "1",
        "sentence_range": "4864-4867",
        "Text": "Therefore \n1\n1\nlog\nlog( )\n1\nI F\ndy\ny\ny\ny\ne\ne\ne\ny\n\u2212\n\u2212\n\u2212\n=\u222b\n=\n=\n=\nHence, the solution of the given differential equation is\n1\nx y   =\n1\n(2 )\nC\ny\n\u239bydy\n\u239e\n+\n\u239c\n\u239f\n\u239d\n\u23a0\n\u222b\nor\nx\ny  =\n(2\n)\ndy +C\n\u222b\nor\nx\ny  = 2y + C\nor\nx = 2y2 + Cy\nwhich is a general solution of the given differential equation Example 22 Find the particular solution of the differential equation\ncot \ndy+\ny\nx\ndx\n = 2x + x2 cot x (x \u2260 0)\ngiven that y = 0 when \n2\nx\n=\u03c0 Solution The given equation is a linear differential equation of the type \nP\nQ\ndy\ndx +y\n=\n,\nwhere P = cot x and Q = 2x + x2 cot x"
    },
    {
        "Chapter": "1",
        "sentence_range": "4865-4868",
        "Text": "F\ndy\ny\ny\ny\ne\ne\ne\ny\n\u2212\n\u2212\n\u2212\n=\u222b\n=\n=\n=\nHence, the solution of the given differential equation is\n1\nx y   =\n1\n(2 )\nC\ny\n\u239bydy\n\u239e\n+\n\u239c\n\u239f\n\u239d\n\u23a0\n\u222b\nor\nx\ny  =\n(2\n)\ndy +C\n\u222b\nor\nx\ny  = 2y + C\nor\nx = 2y2 + Cy\nwhich is a general solution of the given differential equation Example 22 Find the particular solution of the differential equation\ncot \ndy+\ny\nx\ndx\n = 2x + x2 cot x (x \u2260 0)\ngiven that y = 0 when \n2\nx\n=\u03c0 Solution The given equation is a linear differential equation of the type \nP\nQ\ndy\ndx +y\n=\n,\nwhere P = cot x and Q = 2x + x2 cot x Therefore\nI"
    },
    {
        "Chapter": "1",
        "sentence_range": "4866-4869",
        "Text": "Example 22 Find the particular solution of the differential equation\ncot \ndy+\ny\nx\ndx\n = 2x + x2 cot x (x \u2260 0)\ngiven that y = 0 when \n2\nx\n=\u03c0 Solution The given equation is a linear differential equation of the type \nP\nQ\ndy\ndx +y\n=\n,\nwhere P = cot x and Q = 2x + x2 cot x Therefore\nI F = \ncot\nlog sin\nsin\nx dx\nx\ne\ne\nx\n \n \n \nHence, the solution of the differential equation is given by\ny"
    },
    {
        "Chapter": "1",
        "sentence_range": "4867-4870",
        "Text": "Solution The given equation is a linear differential equation of the type \nP\nQ\ndy\ndx +y\n=\n,\nwhere P = cot x and Q = 2x + x2 cot x Therefore\nI F = \ncot\nlog sin\nsin\nx dx\nx\ne\ne\nx\n \n \n \nHence, the solution of the differential equation is given by\ny sin x = \u222b(2x + x2 cot x) sin x dx + C\n\u00a9 NCERT\nnot to be republished\nMATHEMATICS\n412\nor\ny sin x = \u222b 2x sin x dx + \u222bx2 cos x dx + C\nor\ny sin x =\n2\n2\n2\n2\n2\nsin\ncos\ncos\nC\n2\n2\nx\nx\nx\nx\ndx\nx\nx dx\n\u239b\n\u239e\n\u239b\n\u239e\n\u2212\n+\n+\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\n\u222b\n\u222b\nor\ny sin x =\n2\n2\n2\nsin\ncos\ncos\nC\nx\nx\nx\nx dx\nx\nx dx\n\u2212\n+\n+\n\u222b\n\u222b\nor\ny sin x = x2 sin x + C"
    },
    {
        "Chapter": "1",
        "sentence_range": "4868-4871",
        "Text": "Therefore\nI F = \ncot\nlog sin\nsin\nx dx\nx\ne\ne\nx\n \n \n \nHence, the solution of the differential equation is given by\ny sin x = \u222b(2x + x2 cot x) sin x dx + C\n\u00a9 NCERT\nnot to be republished\nMATHEMATICS\n412\nor\ny sin x = \u222b 2x sin x dx + \u222bx2 cos x dx + C\nor\ny sin x =\n2\n2\n2\n2\n2\nsin\ncos\ncos\nC\n2\n2\nx\nx\nx\nx\ndx\nx\nx dx\n\u239b\n\u239e\n\u239b\n\u239e\n\u2212\n+\n+\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\n\u222b\n\u222b\nor\ny sin x =\n2\n2\n2\nsin\ncos\ncos\nC\nx\nx\nx\nx dx\nx\nx dx\n\u2212\n+\n+\n\u222b\n\u222b\nor\ny sin x = x2 sin x + C (1)\nSubstituting y = 0 and \n2\nx\n=\u03c0\n in equation (1), we get\n0 =\n2\nsin\nC\n2\n2\n\u03c0\n\u03c0\n\u239b\n\u239e\n\u239b\n\u239e +\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\nor\nC =\n2\n4\n\u2212\u03c0\nSubstituting the value of C in equation (1), we get\ny sin x =\n2\n2 sin\n4\nx\nx\n\u03c0\n\u2212\nor\ny =\n2\n2\n(sin\n0)\n4 sin\nx\nx\n\u03c0x\n\u2212\n\u2260\nwhich is the particular solution of the given differential equation"
    },
    {
        "Chapter": "1",
        "sentence_range": "4869-4872",
        "Text": "F = \ncot\nlog sin\nsin\nx dx\nx\ne\ne\nx\n \n \n \nHence, the solution of the differential equation is given by\ny sin x = \u222b(2x + x2 cot x) sin x dx + C\n\u00a9 NCERT\nnot to be republished\nMATHEMATICS\n412\nor\ny sin x = \u222b 2x sin x dx + \u222bx2 cos x dx + C\nor\ny sin x =\n2\n2\n2\n2\n2\nsin\ncos\ncos\nC\n2\n2\nx\nx\nx\nx\ndx\nx\nx dx\n\u239b\n\u239e\n\u239b\n\u239e\n\u2212\n+\n+\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\n\u222b\n\u222b\nor\ny sin x =\n2\n2\n2\nsin\ncos\ncos\nC\nx\nx\nx\nx dx\nx\nx dx\n\u2212\n+\n+\n\u222b\n\u222b\nor\ny sin x = x2 sin x + C (1)\nSubstituting y = 0 and \n2\nx\n=\u03c0\n in equation (1), we get\n0 =\n2\nsin\nC\n2\n2\n\u03c0\n\u03c0\n\u239b\n\u239e\n\u239b\n\u239e +\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\nor\nC =\n2\n4\n\u2212\u03c0\nSubstituting the value of C in equation (1), we get\ny sin x =\n2\n2 sin\n4\nx\nx\n\u03c0\n\u2212\nor\ny =\n2\n2\n(sin\n0)\n4 sin\nx\nx\n\u03c0x\n\u2212\n\u2260\nwhich is the particular solution of the given differential equation Example 23 Find the equation of a curve passing through the point (0, 1)"
    },
    {
        "Chapter": "1",
        "sentence_range": "4870-4873",
        "Text": "sin x = \u222b(2x + x2 cot x) sin x dx + C\n\u00a9 NCERT\nnot to be republished\nMATHEMATICS\n412\nor\ny sin x = \u222b 2x sin x dx + \u222bx2 cos x dx + C\nor\ny sin x =\n2\n2\n2\n2\n2\nsin\ncos\ncos\nC\n2\n2\nx\nx\nx\nx\ndx\nx\nx dx\n\u239b\n\u239e\n\u239b\n\u239e\n\u2212\n+\n+\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\n\u222b\n\u222b\nor\ny sin x =\n2\n2\n2\nsin\ncos\ncos\nC\nx\nx\nx\nx dx\nx\nx dx\n\u2212\n+\n+\n\u222b\n\u222b\nor\ny sin x = x2 sin x + C (1)\nSubstituting y = 0 and \n2\nx\n=\u03c0\n in equation (1), we get\n0 =\n2\nsin\nC\n2\n2\n\u03c0\n\u03c0\n\u239b\n\u239e\n\u239b\n\u239e +\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\nor\nC =\n2\n4\n\u2212\u03c0\nSubstituting the value of C in equation (1), we get\ny sin x =\n2\n2 sin\n4\nx\nx\n\u03c0\n\u2212\nor\ny =\n2\n2\n(sin\n0)\n4 sin\nx\nx\n\u03c0x\n\u2212\n\u2260\nwhich is the particular solution of the given differential equation Example 23 Find the equation of a curve passing through the point (0, 1) If the slope\nof the tangent to the curve at any point (x, y) is equal to the sum of the x coordinate\n(abscissa) and the product of the x coordinate and y coordinate (ordinate) of that point"
    },
    {
        "Chapter": "1",
        "sentence_range": "4871-4874",
        "Text": "(1)\nSubstituting y = 0 and \n2\nx\n=\u03c0\n in equation (1), we get\n0 =\n2\nsin\nC\n2\n2\n\u03c0\n\u03c0\n\u239b\n\u239e\n\u239b\n\u239e +\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\nor\nC =\n2\n4\n\u2212\u03c0\nSubstituting the value of C in equation (1), we get\ny sin x =\n2\n2 sin\n4\nx\nx\n\u03c0\n\u2212\nor\ny =\n2\n2\n(sin\n0)\n4 sin\nx\nx\n\u03c0x\n\u2212\n\u2260\nwhich is the particular solution of the given differential equation Example 23 Find the equation of a curve passing through the point (0, 1) If the slope\nof the tangent to the curve at any point (x, y) is equal to the sum of the x coordinate\n(abscissa) and the product of the x coordinate and y coordinate (ordinate) of that point Solution We know that the slope of the tangent to the curve is dy\ndx"
    },
    {
        "Chapter": "1",
        "sentence_range": "4872-4875",
        "Text": "Example 23 Find the equation of a curve passing through the point (0, 1) If the slope\nof the tangent to the curve at any point (x, y) is equal to the sum of the x coordinate\n(abscissa) and the product of the x coordinate and y coordinate (ordinate) of that point Solution We know that the slope of the tangent to the curve is dy\ndx Therefore,\ndy\ndx  = x + xy\nor\ndy\ndx \u2212xy\n = x"
    },
    {
        "Chapter": "1",
        "sentence_range": "4873-4876",
        "Text": "If the slope\nof the tangent to the curve at any point (x, y) is equal to the sum of the x coordinate\n(abscissa) and the product of the x coordinate and y coordinate (ordinate) of that point Solution We know that the slope of the tangent to the curve is dy\ndx Therefore,\ndy\ndx  = x + xy\nor\ndy\ndx \u2212xy\n = x (1)\nThis is a linear differential equation of the type \nP\nQ\ndy\ndx +y\n=\n, where P = \u2013 x and Q = x"
    },
    {
        "Chapter": "1",
        "sentence_range": "4874-4877",
        "Text": "Solution We know that the slope of the tangent to the curve is dy\ndx Therefore,\ndy\ndx  = x + xy\nor\ndy\ndx \u2212xy\n = x (1)\nThis is a linear differential equation of the type \nP\nQ\ndy\ndx +y\n=\n, where P = \u2013 x and Q = x Therefore,\nI"
    },
    {
        "Chapter": "1",
        "sentence_range": "4875-4878",
        "Text": "Therefore,\ndy\ndx  = x + xy\nor\ndy\ndx \u2212xy\n = x (1)\nThis is a linear differential equation of the type \nP\nQ\ndy\ndx +y\n=\n, where P = \u2013 x and Q = x Therefore,\nI F =\n2\n2\nx\nx dx\ne\ne\n\u2212\n\u2212\u222b\n=\n\u00a9 NCERT\nnot to be republished\nDIFFERENTIAL EQUATIONS\n413\nHence, the solution of equation is given by\n2\n2\nx\ny e\n\u22c5\u2212\n =\n(\n)\n2\n2\n( )\nC\nx\nx\ndx\ne\n\u2212\n+\n\u222b"
    },
    {
        "Chapter": "1",
        "sentence_range": "4876-4879",
        "Text": "(1)\nThis is a linear differential equation of the type \nP\nQ\ndy\ndx +y\n=\n, where P = \u2013 x and Q = x Therefore,\nI F =\n2\n2\nx\nx dx\ne\ne\n\u2212\n\u2212\u222b\n=\n\u00a9 NCERT\nnot to be republished\nDIFFERENTIAL EQUATIONS\n413\nHence, the solution of equation is given by\n2\n2\nx\ny e\n\u22c5\u2212\n =\n(\n)\n2\n2\n( )\nC\nx\nx\ndx\ne\n\u2212\n+\n\u222b (2)\nLet\nI =\n2\n2\n( )\nx\nx\ndx\ne\n\u2212\n\u222b\nLet \n2\nx2\nt\n\u2212\n= , then \u2013 x dx = dt or x dx = \u2013 dt"
    },
    {
        "Chapter": "1",
        "sentence_range": "4877-4880",
        "Text": "Therefore,\nI F =\n2\n2\nx\nx dx\ne\ne\n\u2212\n\u2212\u222b\n=\n\u00a9 NCERT\nnot to be republished\nDIFFERENTIAL EQUATIONS\n413\nHence, the solution of equation is given by\n2\n2\nx\ny e\n\u22c5\u2212\n =\n(\n)\n2\n2\n( )\nC\nx\nx\ndx\ne\n\u2212\n+\n\u222b (2)\nLet\nI =\n2\n2\n( )\nx\nx\ndx\ne\n\u2212\n\u222b\nLet \n2\nx2\nt\n\u2212\n= , then \u2013 x dx = dt or x dx = \u2013 dt Therefore,       I =\n2\n2\n\u2013\nx\nt\nt\ne dt\ne\ne\n\u2212\n\u2212\n= \u2212\n=\n\u222b\nSubstituting the value of I in equation (2), we get\n2\n2\nx\ny e\n\u2212\n =\n2\n2 + C\n \n \nx\ne\nor\ny =\n2\n2\n1\nC\nx\ne\n\u2212 +"
    },
    {
        "Chapter": "1",
        "sentence_range": "4878-4881",
        "Text": "F =\n2\n2\nx\nx dx\ne\ne\n\u2212\n\u2212\u222b\n=\n\u00a9 NCERT\nnot to be republished\nDIFFERENTIAL EQUATIONS\n413\nHence, the solution of equation is given by\n2\n2\nx\ny e\n\u22c5\u2212\n =\n(\n)\n2\n2\n( )\nC\nx\nx\ndx\ne\n\u2212\n+\n\u222b (2)\nLet\nI =\n2\n2\n( )\nx\nx\ndx\ne\n\u2212\n\u222b\nLet \n2\nx2\nt\n\u2212\n= , then \u2013 x dx = dt or x dx = \u2013 dt Therefore,       I =\n2\n2\n\u2013\nx\nt\nt\ne dt\ne\ne\n\u2212\n\u2212\n= \u2212\n=\n\u222b\nSubstituting the value of I in equation (2), we get\n2\n2\nx\ny e\n\u2212\n =\n2\n2 + C\n \n \nx\ne\nor\ny =\n2\n2\n1\nC\nx\ne\n\u2212 + (3)\nNow (3) represents the equation of family of curves"
    },
    {
        "Chapter": "1",
        "sentence_range": "4879-4882",
        "Text": "(2)\nLet\nI =\n2\n2\n( )\nx\nx\ndx\ne\n\u2212\n\u222b\nLet \n2\nx2\nt\n\u2212\n= , then \u2013 x dx = dt or x dx = \u2013 dt Therefore,       I =\n2\n2\n\u2013\nx\nt\nt\ne dt\ne\ne\n\u2212\n\u2212\n= \u2212\n=\n\u222b\nSubstituting the value of I in equation (2), we get\n2\n2\nx\ny e\n\u2212\n =\n2\n2 + C\n \n \nx\ne\nor\ny =\n2\n2\n1\nC\nx\ne\n\u2212 + (3)\nNow (3) represents the equation of family of curves But we are interested in\nfinding a particular member of the family passing through (0, 1)"
    },
    {
        "Chapter": "1",
        "sentence_range": "4880-4883",
        "Text": "Therefore,       I =\n2\n2\n\u2013\nx\nt\nt\ne dt\ne\ne\n\u2212\n\u2212\n= \u2212\n=\n\u222b\nSubstituting the value of I in equation (2), we get\n2\n2\nx\ny e\n\u2212\n =\n2\n2 + C\n \n \nx\ne\nor\ny =\n2\n2\n1\nC\nx\ne\n\u2212 + (3)\nNow (3) represents the equation of family of curves But we are interested in\nfinding a particular member of the family passing through (0, 1) Substituting x = 0 and\ny = 1 in equation (3) we get\n1 = \u2013 1 + C"
    },
    {
        "Chapter": "1",
        "sentence_range": "4881-4884",
        "Text": "(3)\nNow (3) represents the equation of family of curves But we are interested in\nfinding a particular member of the family passing through (0, 1) Substituting x = 0 and\ny = 1 in equation (3) we get\n1 = \u2013 1 + C e0   or   C = 2\nSubstituting the value of C in equation (3), we get\ny =\n2\n2\n1\n2\nx\ne\n\u2212 +\nwhich  is the equation of the required curve"
    },
    {
        "Chapter": "1",
        "sentence_range": "4882-4885",
        "Text": "But we are interested in\nfinding a particular member of the family passing through (0, 1) Substituting x = 0 and\ny = 1 in equation (3) we get\n1 = \u2013 1 + C e0   or   C = 2\nSubstituting the value of C in equation (3), we get\ny =\n2\n2\n1\n2\nx\ne\n\u2212 +\nwhich  is the equation of the required curve EXERCISE 9"
    },
    {
        "Chapter": "1",
        "sentence_range": "4883-4886",
        "Text": "Substituting x = 0 and\ny = 1 in equation (3) we get\n1 = \u2013 1 + C e0   or   C = 2\nSubstituting the value of C in equation (3), we get\ny =\n2\n2\n1\n2\nx\ne\n\u2212 +\nwhich  is the equation of the required curve EXERCISE 9 6\nFor each of the differential equations given in Exercises 1 to 12, find the general solution:\n1"
    },
    {
        "Chapter": "1",
        "sentence_range": "4884-4887",
        "Text": "e0   or   C = 2\nSubstituting the value of C in equation (3), we get\ny =\n2\n2\n1\n2\nx\ne\n\u2212 +\nwhich  is the equation of the required curve EXERCISE 9 6\nFor each of the differential equations given in Exercises 1 to 12, find the general solution:\n1 2\nsin\ndy\ny\nx\ndx +\n=\n2"
    },
    {
        "Chapter": "1",
        "sentence_range": "4885-4888",
        "Text": "EXERCISE 9 6\nFor each of the differential equations given in Exercises 1 to 12, find the general solution:\n1 2\nsin\ndy\ny\nx\ndx +\n=\n2 2\n3\nx\ndy\ny\ne\ndx\n\u2212\n+\n=\n3"
    },
    {
        "Chapter": "1",
        "sentence_range": "4886-4889",
        "Text": "6\nFor each of the differential equations given in Exercises 1 to 12, find the general solution:\n1 2\nsin\ndy\ny\nx\ndx +\n=\n2 2\n3\nx\ndy\ny\ne\ndx\n\u2212\n+\n=\n3 2\ndy\ny\nx\ndx\n+x\n=\n4"
    },
    {
        "Chapter": "1",
        "sentence_range": "4887-4890",
        "Text": "2\nsin\ndy\ny\nx\ndx +\n=\n2 2\n3\nx\ndy\ny\ne\ndx\n\u2212\n+\n=\n3 2\ndy\ny\nx\ndx\n+x\n=\n4 (sec )\ntan\n0\n2\ndy\nx y\nx\nx\ndx\n\u03c0\n\u239b\n\u239e\n+\n=\n\u2264\n<\n\u239c\n\u239f\n\u239d\n\u23a0\n5"
    },
    {
        "Chapter": "1",
        "sentence_range": "4888-4891",
        "Text": "2\n3\nx\ndy\ny\ne\ndx\n\u2212\n+\n=\n3 2\ndy\ny\nx\ndx\n+x\n=\n4 (sec )\ntan\n0\n2\ndy\nx y\nx\nx\ndx\n\u03c0\n\u239b\n\u239e\n+\n=\n\u2264\n<\n\u239c\n\u239f\n\u239d\n\u23a0\n5 cos2\ntan\nxdy\ny\nx\ndx +\n=\n 0\n2\nx\n\u03c0\n\u239b\n\u239e\n\u2264\n<\n\u239c\n\u239f\n\u239d\n\u23a0\n6"
    },
    {
        "Chapter": "1",
        "sentence_range": "4889-4892",
        "Text": "2\ndy\ny\nx\ndx\n+x\n=\n4 (sec )\ntan\n0\n2\ndy\nx y\nx\nx\ndx\n\u03c0\n\u239b\n\u239e\n+\n=\n\u2264\n<\n\u239c\n\u239f\n\u239d\n\u23a0\n5 cos2\ntan\nxdy\ny\nx\ndx +\n=\n 0\n2\nx\n\u03c0\n\u239b\n\u239e\n\u2264\n<\n\u239c\n\u239f\n\u239d\n\u23a0\n6 2\n2\nlog\nxdy\ny\nx\nx\ndx +\n=\n7"
    },
    {
        "Chapter": "1",
        "sentence_range": "4890-4893",
        "Text": "(sec )\ntan\n0\n2\ndy\nx y\nx\nx\ndx\n\u03c0\n\u239b\n\u239e\n+\n=\n\u2264\n<\n\u239c\n\u239f\n\u239d\n\u23a0\n5 cos2\ntan\nxdy\ny\nx\ndx +\n=\n 0\n2\nx\n\u03c0\n\u239b\n\u239e\n\u2264\n<\n\u239c\n\u239f\n\u239d\n\u23a0\n6 2\n2\nlog\nxdy\ny\nx\nx\ndx +\n=\n7 2\nlog\nlog\ndy\nx\nx\ny\nx\ndx\nx\n+\n=\n8"
    },
    {
        "Chapter": "1",
        "sentence_range": "4891-4894",
        "Text": "cos2\ntan\nxdy\ny\nx\ndx +\n=\n 0\n2\nx\n\u03c0\n\u239b\n\u239e\n\u2264\n<\n\u239c\n\u239f\n\u239d\n\u23a0\n6 2\n2\nlog\nxdy\ny\nx\nx\ndx +\n=\n7 2\nlog\nlog\ndy\nx\nx\ny\nx\ndx\nx\n+\n=\n8 (1 + x2) dy + 2xy dx = cot x dx (x \u2260 0)\n\u00a9 NCERT\nnot to be republished\nMATHEMATICS\n414\n9"
    },
    {
        "Chapter": "1",
        "sentence_range": "4892-4895",
        "Text": "2\n2\nlog\nxdy\ny\nx\nx\ndx +\n=\n7 2\nlog\nlog\ndy\nx\nx\ny\nx\ndx\nx\n+\n=\n8 (1 + x2) dy + 2xy dx = cot x dx (x \u2260 0)\n\u00a9 NCERT\nnot to be republished\nMATHEMATICS\n414\n9 cot\n0 (\n0)\nxdy\ny\nx\nxy\nx\nx\ndx +\n\u2212\n+\n=\n\u2260\n10"
    },
    {
        "Chapter": "1",
        "sentence_range": "4893-4896",
        "Text": "2\nlog\nlog\ndy\nx\nx\ny\nx\ndx\nx\n+\n=\n8 (1 + x2) dy + 2xy dx = cot x dx (x \u2260 0)\n\u00a9 NCERT\nnot to be republished\nMATHEMATICS\n414\n9 cot\n0 (\n0)\nxdy\ny\nx\nxy\nx\nx\ndx +\n\u2212\n+\n=\n\u2260\n10 (\n)\n1\ndy\nx\n+y dx\n=\n11"
    },
    {
        "Chapter": "1",
        "sentence_range": "4894-4897",
        "Text": "(1 + x2) dy + 2xy dx = cot x dx (x \u2260 0)\n\u00a9 NCERT\nnot to be republished\nMATHEMATICS\n414\n9 cot\n0 (\n0)\nxdy\ny\nx\nxy\nx\nx\ndx +\n\u2212\n+\n=\n\u2260\n10 (\n)\n1\ndy\nx\n+y dx\n=\n11 y dx + (x \u2013 y2) dy = 0\n12"
    },
    {
        "Chapter": "1",
        "sentence_range": "4895-4898",
        "Text": "cot\n0 (\n0)\nxdy\ny\nx\nxy\nx\nx\ndx +\n\u2212\n+\n=\n\u2260\n10 (\n)\n1\ndy\nx\n+y dx\n=\n11 y dx + (x \u2013 y2) dy = 0\n12 2\n(\n3\n)\n(\n0)\ndy\nx\ny\ny\ny\ndx\n+\n=\n>"
    },
    {
        "Chapter": "1",
        "sentence_range": "4896-4899",
        "Text": "(\n)\n1\ndy\nx\n+y dx\n=\n11 y dx + (x \u2013 y2) dy = 0\n12 2\n(\n3\n)\n(\n0)\ndy\nx\ny\ny\ny\ndx\n+\n=\n> For each of the differential equations given in Exercises 13 to 15, find a particular\nsolution satisfying the given condition:\n13"
    },
    {
        "Chapter": "1",
        "sentence_range": "4897-4900",
        "Text": "y dx + (x \u2013 y2) dy = 0\n12 2\n(\n3\n)\n(\n0)\ndy\nx\ny\ny\ny\ndx\n+\n=\n> For each of the differential equations given in Exercises 13 to 15, find a particular\nsolution satisfying the given condition:\n13 2 tan\nsin ;\n0 when\n3\ndy\ny\nx\nx y\nx\ndx\n\u03c0\n+\n=\n=\n=\n14"
    },
    {
        "Chapter": "1",
        "sentence_range": "4898-4901",
        "Text": "2\n(\n3\n)\n(\n0)\ndy\nx\ny\ny\ny\ndx\n+\n=\n> For each of the differential equations given in Exercises 13 to 15, find a particular\nsolution satisfying the given condition:\n13 2 tan\nsin ;\n0 when\n3\ndy\ny\nx\nx y\nx\ndx\n\u03c0\n+\n=\n=\n=\n14 2\n12\n(1\n)\n2\n;\n0 when\n1\n1\ndy\nx\nxy\ny\nx\ndx\nx\n+\n+\n=\n=\n=\n+\n15"
    },
    {
        "Chapter": "1",
        "sentence_range": "4899-4902",
        "Text": "For each of the differential equations given in Exercises 13 to 15, find a particular\nsolution satisfying the given condition:\n13 2 tan\nsin ;\n0 when\n3\ndy\ny\nx\nx y\nx\ndx\n\u03c0\n+\n=\n=\n=\n14 2\n12\n(1\n)\n2\n;\n0 when\n1\n1\ndy\nx\nxy\ny\nx\ndx\nx\n+\n+\n=\n=\n=\n+\n15 3 cot\nsin2 ;\n2 when\n2\ndy\ny\nx\nx y\nx\ndx\n\u03c0\n\u2212\n=\n=\n=\n16"
    },
    {
        "Chapter": "1",
        "sentence_range": "4900-4903",
        "Text": "2 tan\nsin ;\n0 when\n3\ndy\ny\nx\nx y\nx\ndx\n\u03c0\n+\n=\n=\n=\n14 2\n12\n(1\n)\n2\n;\n0 when\n1\n1\ndy\nx\nxy\ny\nx\ndx\nx\n+\n+\n=\n=\n=\n+\n15 3 cot\nsin2 ;\n2 when\n2\ndy\ny\nx\nx y\nx\ndx\n\u03c0\n\u2212\n=\n=\n=\n16 Find the equation of a curve passing through the origin given that the slope of the\ntangent to the curve at any point (x, y) is equal to the sum of the coordinates of\nthe point"
    },
    {
        "Chapter": "1",
        "sentence_range": "4901-4904",
        "Text": "2\n12\n(1\n)\n2\n;\n0 when\n1\n1\ndy\nx\nxy\ny\nx\ndx\nx\n+\n+\n=\n=\n=\n+\n15 3 cot\nsin2 ;\n2 when\n2\ndy\ny\nx\nx y\nx\ndx\n\u03c0\n\u2212\n=\n=\n=\n16 Find the equation of a curve passing through the origin given that the slope of the\ntangent to the curve at any point (x, y) is equal to the sum of the coordinates of\nthe point 17"
    },
    {
        "Chapter": "1",
        "sentence_range": "4902-4905",
        "Text": "3 cot\nsin2 ;\n2 when\n2\ndy\ny\nx\nx y\nx\ndx\n\u03c0\n\u2212\n=\n=\n=\n16 Find the equation of a curve passing through the origin given that the slope of the\ntangent to the curve at any point (x, y) is equal to the sum of the coordinates of\nthe point 17 Find the equation of a curve passing through the point (0, 2) given that the sum of\nthe coordinates of any point on the curve exceeds the magnitude of the slope of\nthe tangent to the curve at that point by 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "4903-4906",
        "Text": "Find the equation of a curve passing through the origin given that the slope of the\ntangent to the curve at any point (x, y) is equal to the sum of the coordinates of\nthe point 17 Find the equation of a curve passing through the point (0, 2) given that the sum of\nthe coordinates of any point on the curve exceeds the magnitude of the slope of\nthe tangent to the curve at that point by 5 18"
    },
    {
        "Chapter": "1",
        "sentence_range": "4904-4907",
        "Text": "17 Find the equation of a curve passing through the point (0, 2) given that the sum of\nthe coordinates of any point on the curve exceeds the magnitude of the slope of\nthe tangent to the curve at that point by 5 18 The Integrating Factor of the differential equation \n22\nxdy\ny\nx\ndx \u2212\n=\n is\n(A) e\u2013x\n(B) e\u2013y\n(C) 1\nx\n(D) x\n19"
    },
    {
        "Chapter": "1",
        "sentence_range": "4905-4908",
        "Text": "Find the equation of a curve passing through the point (0, 2) given that the sum of\nthe coordinates of any point on the curve exceeds the magnitude of the slope of\nthe tangent to the curve at that point by 5 18 The Integrating Factor of the differential equation \n22\nxdy\ny\nx\ndx \u2212\n=\n is\n(A) e\u2013x\n(B) e\u2013y\n(C) 1\nx\n(D) x\n19 The Integrating Factor of the differential equation\n2\n(1\ny) dx\nyx\ndy\n\u2212\n+\n = \n( 1\n1)\n \n \nay\ny\n is\n(A)\n12\ny  1\n(B)\n2\n1\ny \u22121\n(C)\n112\n\u2212y\n(D)\n2\n1\n1\ny\n\u2212\nMiscellaneous Examples\nExample 24 Verify that the function y = c1 eax cos bx + c2 eax sin bx, where c1, c2 are\narbitrary constants is a solution of the differential equation\n(\n)\n2\n2\n2\n2\n2\n0\nd y\nady\na\nb\ny\ndx\ndx\n\u2212\n+\n+\n=\n\u00a9 NCERT\nnot to be republished\nDIFFERENTIAL EQUATIONS\n415\nSolution The given function is\ny = eax [c1 cosbx + c2 sinbx]"
    },
    {
        "Chapter": "1",
        "sentence_range": "4906-4909",
        "Text": "18 The Integrating Factor of the differential equation \n22\nxdy\ny\nx\ndx \u2212\n=\n is\n(A) e\u2013x\n(B) e\u2013y\n(C) 1\nx\n(D) x\n19 The Integrating Factor of the differential equation\n2\n(1\ny) dx\nyx\ndy\n\u2212\n+\n = \n( 1\n1)\n \n \nay\ny\n is\n(A)\n12\ny  1\n(B)\n2\n1\ny \u22121\n(C)\n112\n\u2212y\n(D)\n2\n1\n1\ny\n\u2212\nMiscellaneous Examples\nExample 24 Verify that the function y = c1 eax cos bx + c2 eax sin bx, where c1, c2 are\narbitrary constants is a solution of the differential equation\n(\n)\n2\n2\n2\n2\n2\n0\nd y\nady\na\nb\ny\ndx\ndx\n\u2212\n+\n+\n=\n\u00a9 NCERT\nnot to be republished\nDIFFERENTIAL EQUATIONS\n415\nSolution The given function is\ny = eax [c1 cosbx + c2 sinbx] (1)\nDifferentiating both sides of equation (1) with respect to x, we get\ndy\ndx  =\n  \n \n1\n2\n1\n2\n\u2013\nsin\ncos\ncos\nsin\nax\nax\ne\nbc\nbx\nbc\nbx\nc\nbx\nc\nbx e\na\n \n \n \nor\ndy\ndx  =\n2\n1\n2\n1\n[(\n)cos\n(\n)sin\n]\neax\nbc\nac\nbx\nac\nbc\nbx\n+\n+\n\u2212"
    },
    {
        "Chapter": "1",
        "sentence_range": "4907-4910",
        "Text": "The Integrating Factor of the differential equation \n22\nxdy\ny\nx\ndx \u2212\n=\n is\n(A) e\u2013x\n(B) e\u2013y\n(C) 1\nx\n(D) x\n19 The Integrating Factor of the differential equation\n2\n(1\ny) dx\nyx\ndy\n\u2212\n+\n = \n( 1\n1)\n \n \nay\ny\n is\n(A)\n12\ny  1\n(B)\n2\n1\ny \u22121\n(C)\n112\n\u2212y\n(D)\n2\n1\n1\ny\n\u2212\nMiscellaneous Examples\nExample 24 Verify that the function y = c1 eax cos bx + c2 eax sin bx, where c1, c2 are\narbitrary constants is a solution of the differential equation\n(\n)\n2\n2\n2\n2\n2\n0\nd y\nady\na\nb\ny\ndx\ndx\n\u2212\n+\n+\n=\n\u00a9 NCERT\nnot to be republished\nDIFFERENTIAL EQUATIONS\n415\nSolution The given function is\ny = eax [c1 cosbx + c2 sinbx] (1)\nDifferentiating both sides of equation (1) with respect to x, we get\ndy\ndx  =\n  \n \n1\n2\n1\n2\n\u2013\nsin\ncos\ncos\nsin\nax\nax\ne\nbc\nbx\nbc\nbx\nc\nbx\nc\nbx e\na\n \n \n \nor\ndy\ndx  =\n2\n1\n2\n1\n[(\n)cos\n(\n)sin\n]\neax\nbc\nac\nbx\nac\nbc\nbx\n+\n+\n\u2212 (2)\nDifferentiating both sides of equation (2) with respect to x, we get\n2\nd y2\ndx\n =\n2\n1\n2\n1\n[(\n) (\nsin\n)\n(\n) ( cos\n)]\neax\nbc\nac\nb\nbx\nac\nbc\nb\nbx\n \n \n \n \n+ \n2\n1\n2\n1\n[(\n) cos\n(\n) sin\n]\nax"
    },
    {
        "Chapter": "1",
        "sentence_range": "4908-4911",
        "Text": "The Integrating Factor of the differential equation\n2\n(1\ny) dx\nyx\ndy\n\u2212\n+\n = \n( 1\n1)\n \n \nay\ny\n is\n(A)\n12\ny  1\n(B)\n2\n1\ny \u22121\n(C)\n112\n\u2212y\n(D)\n2\n1\n1\ny\n\u2212\nMiscellaneous Examples\nExample 24 Verify that the function y = c1 eax cos bx + c2 eax sin bx, where c1, c2 are\narbitrary constants is a solution of the differential equation\n(\n)\n2\n2\n2\n2\n2\n0\nd y\nady\na\nb\ny\ndx\ndx\n\u2212\n+\n+\n=\n\u00a9 NCERT\nnot to be republished\nDIFFERENTIAL EQUATIONS\n415\nSolution The given function is\ny = eax [c1 cosbx + c2 sinbx] (1)\nDifferentiating both sides of equation (1) with respect to x, we get\ndy\ndx  =\n  \n \n1\n2\n1\n2\n\u2013\nsin\ncos\ncos\nsin\nax\nax\ne\nbc\nbx\nbc\nbx\nc\nbx\nc\nbx e\na\n \n \n \nor\ndy\ndx  =\n2\n1\n2\n1\n[(\n)cos\n(\n)sin\n]\neax\nbc\nac\nbx\nac\nbc\nbx\n+\n+\n\u2212 (2)\nDifferentiating both sides of equation (2) with respect to x, we get\n2\nd y2\ndx\n =\n2\n1\n2\n1\n[(\n) (\nsin\n)\n(\n) ( cos\n)]\neax\nbc\nac\nb\nbx\nac\nbc\nb\nbx\n \n \n \n \n+ \n2\n1\n2\n1\n[(\n) cos\n(\n) sin\n]\nax bc\nac\nbx\nac\nbc\nbx e\na\n+\n+\n\u2212\n=\n2\n2\n2\n2\n2\n1\n2\n1\n2\n1\n[(\n2\n) sin\n(\n2\n) cos\n]\neax\na c\nabc\nb c\nbx\na c\nabc\nb c\nbx\n\u2212\n\u2212\n+\n+\n\u2212\nSubstituting the values of \n2\n2 ,\nd y dy\ndx\ndx\n and y in the given differential equation, we get\nL"
    },
    {
        "Chapter": "1",
        "sentence_range": "4909-4912",
        "Text": "(1)\nDifferentiating both sides of equation (1) with respect to x, we get\ndy\ndx  =\n  \n \n1\n2\n1\n2\n\u2013\nsin\ncos\ncos\nsin\nax\nax\ne\nbc\nbx\nbc\nbx\nc\nbx\nc\nbx e\na\n \n \n \nor\ndy\ndx  =\n2\n1\n2\n1\n[(\n)cos\n(\n)sin\n]\neax\nbc\nac\nbx\nac\nbc\nbx\n+\n+\n\u2212 (2)\nDifferentiating both sides of equation (2) with respect to x, we get\n2\nd y2\ndx\n =\n2\n1\n2\n1\n[(\n) (\nsin\n)\n(\n) ( cos\n)]\neax\nbc\nac\nb\nbx\nac\nbc\nb\nbx\n \n \n \n \n+ \n2\n1\n2\n1\n[(\n) cos\n(\n) sin\n]\nax bc\nac\nbx\nac\nbc\nbx e\na\n+\n+\n\u2212\n=\n2\n2\n2\n2\n2\n1\n2\n1\n2\n1\n[(\n2\n) sin\n(\n2\n) cos\n]\neax\na c\nabc\nb c\nbx\na c\nabc\nb c\nbx\n\u2212\n\u2212\n+\n+\n\u2212\nSubstituting the values of \n2\n2 ,\nd y dy\ndx\ndx\n and y in the given differential equation, we get\nL H"
    },
    {
        "Chapter": "1",
        "sentence_range": "4910-4913",
        "Text": "(2)\nDifferentiating both sides of equation (2) with respect to x, we get\n2\nd y2\ndx\n =\n2\n1\n2\n1\n[(\n) (\nsin\n)\n(\n) ( cos\n)]\neax\nbc\nac\nb\nbx\nac\nbc\nb\nbx\n \n \n \n \n+ \n2\n1\n2\n1\n[(\n) cos\n(\n) sin\n]\nax bc\nac\nbx\nac\nbc\nbx e\na\n+\n+\n\u2212\n=\n2\n2\n2\n2\n2\n1\n2\n1\n2\n1\n[(\n2\n) sin\n(\n2\n) cos\n]\neax\na c\nabc\nb c\nbx\na c\nabc\nb c\nbx\n\u2212\n\u2212\n+\n+\n\u2212\nSubstituting the values of \n2\n2 ,\nd y dy\ndx\ndx\n and y in the given differential equation, we get\nL H S"
    },
    {
        "Chapter": "1",
        "sentence_range": "4911-4914",
        "Text": "bc\nac\nbx\nac\nbc\nbx e\na\n+\n+\n\u2212\n=\n2\n2\n2\n2\n2\n1\n2\n1\n2\n1\n[(\n2\n) sin\n(\n2\n) cos\n]\neax\na c\nabc\nb c\nbx\na c\nabc\nb c\nbx\n\u2212\n\u2212\n+\n+\n\u2212\nSubstituting the values of \n2\n2 ,\nd y dy\ndx\ndx\n and y in the given differential equation, we get\nL H S =\n2\n2\n2\n2\n2\n1\n2\n1\n2\n1\n[\n2\n)sin\n(\n2\n)cos\n]\neax\na c\nabc\nb c\nbx\na c\nabc\nb c\nbx\n\u2212\n\u2212\n+\n+\n\u2212\n2\n1\n2\n1\n2\n[(\n)cos\n(\n)sin\n]\naeax\nbc\nac\nbx\nac\nbc\nbx\n \n \n \n \n2\n2\n1\n2\n(\n)\n[\ncos\nsin\n]\nax\na\nb\ne\nc\nbx\nc\nbx\n \n \n \n=\n(\n)\n2\n2\n2\n2\n2\n2\n1\n2\n2\n1\n2\n2\n2\n2\n2\n2\n2\n1\n2\n1\n2\n1\n1\n1\n2\n2\n2\nsin\n(\n2\n2\n2\n)cos\nax\na c\nabc\nb c\na c\nabc\na c\nb c\nbx\ne\na c\nabc\nb c\nabc\na c\na c\nb c\nbx\n\u23a1\n\u23a4\n\u2212\n\u2212\n\u2212\n+\n+\n+\n\u23a2\n\u23a5\n\u23a2\n\u23a5\n+\n+\n\u2212\n\u2212\n\u2212\n+\n+\n\u23a3\n\u23a6\n=\n[0 sin\n0cos\n]\neax\nbx\nbx\n\u00d7\n+\n= eax \u00d7 0 = 0  = R"
    },
    {
        "Chapter": "1",
        "sentence_range": "4912-4915",
        "Text": "H S =\n2\n2\n2\n2\n2\n1\n2\n1\n2\n1\n[\n2\n)sin\n(\n2\n)cos\n]\neax\na c\nabc\nb c\nbx\na c\nabc\nb c\nbx\n\u2212\n\u2212\n+\n+\n\u2212\n2\n1\n2\n1\n2\n[(\n)cos\n(\n)sin\n]\naeax\nbc\nac\nbx\nac\nbc\nbx\n \n \n \n \n2\n2\n1\n2\n(\n)\n[\ncos\nsin\n]\nax\na\nb\ne\nc\nbx\nc\nbx\n \n \n \n=\n(\n)\n2\n2\n2\n2\n2\n2\n1\n2\n2\n1\n2\n2\n2\n2\n2\n2\n2\n1\n2\n1\n2\n1\n1\n1\n2\n2\n2\nsin\n(\n2\n2\n2\n)cos\nax\na c\nabc\nb c\na c\nabc\na c\nb c\nbx\ne\na c\nabc\nb c\nabc\na c\na c\nb c\nbx\n\u23a1\n\u23a4\n\u2212\n\u2212\n\u2212\n+\n+\n+\n\u23a2\n\u23a5\n\u23a2\n\u23a5\n+\n+\n\u2212\n\u2212\n\u2212\n+\n+\n\u23a3\n\u23a6\n=\n[0 sin\n0cos\n]\neax\nbx\nbx\n\u00d7\n+\n= eax \u00d7 0 = 0  = R H"
    },
    {
        "Chapter": "1",
        "sentence_range": "4913-4916",
        "Text": "S =\n2\n2\n2\n2\n2\n1\n2\n1\n2\n1\n[\n2\n)sin\n(\n2\n)cos\n]\neax\na c\nabc\nb c\nbx\na c\nabc\nb c\nbx\n\u2212\n\u2212\n+\n+\n\u2212\n2\n1\n2\n1\n2\n[(\n)cos\n(\n)sin\n]\naeax\nbc\nac\nbx\nac\nbc\nbx\n \n \n \n \n2\n2\n1\n2\n(\n)\n[\ncos\nsin\n]\nax\na\nb\ne\nc\nbx\nc\nbx\n \n \n \n=\n(\n)\n2\n2\n2\n2\n2\n2\n1\n2\n2\n1\n2\n2\n2\n2\n2\n2\n2\n1\n2\n1\n2\n1\n1\n1\n2\n2\n2\nsin\n(\n2\n2\n2\n)cos\nax\na c\nabc\nb c\na c\nabc\na c\nb c\nbx\ne\na c\nabc\nb c\nabc\na c\na c\nb c\nbx\n\u23a1\n\u23a4\n\u2212\n\u2212\n\u2212\n+\n+\n+\n\u23a2\n\u23a5\n\u23a2\n\u23a5\n+\n+\n\u2212\n\u2212\n\u2212\n+\n+\n\u23a3\n\u23a6\n=\n[0 sin\n0cos\n]\neax\nbx\nbx\n\u00d7\n+\n= eax \u00d7 0 = 0  = R H S"
    },
    {
        "Chapter": "1",
        "sentence_range": "4914-4917",
        "Text": "=\n2\n2\n2\n2\n2\n1\n2\n1\n2\n1\n[\n2\n)sin\n(\n2\n)cos\n]\neax\na c\nabc\nb c\nbx\na c\nabc\nb c\nbx\n\u2212\n\u2212\n+\n+\n\u2212\n2\n1\n2\n1\n2\n[(\n)cos\n(\n)sin\n]\naeax\nbc\nac\nbx\nac\nbc\nbx\n \n \n \n \n2\n2\n1\n2\n(\n)\n[\ncos\nsin\n]\nax\na\nb\ne\nc\nbx\nc\nbx\n \n \n \n=\n(\n)\n2\n2\n2\n2\n2\n2\n1\n2\n2\n1\n2\n2\n2\n2\n2\n2\n2\n1\n2\n1\n2\n1\n1\n1\n2\n2\n2\nsin\n(\n2\n2\n2\n)cos\nax\na c\nabc\nb c\na c\nabc\na c\nb c\nbx\ne\na c\nabc\nb c\nabc\na c\na c\nb c\nbx\n\u23a1\n\u23a4\n\u2212\n\u2212\n\u2212\n+\n+\n+\n\u23a2\n\u23a5\n\u23a2\n\u23a5\n+\n+\n\u2212\n\u2212\n\u2212\n+\n+\n\u23a3\n\u23a6\n=\n[0 sin\n0cos\n]\neax\nbx\nbx\n\u00d7\n+\n= eax \u00d7 0 = 0  = R H S Hence, the given function is a solution of the given differential equation"
    },
    {
        "Chapter": "1",
        "sentence_range": "4915-4918",
        "Text": "H S Hence, the given function is a solution of the given differential equation Example 25 Form the differential equation of the family of circles in the second\nquadrant and touching the coordinate axes"
    },
    {
        "Chapter": "1",
        "sentence_range": "4916-4919",
        "Text": "S Hence, the given function is a solution of the given differential equation Example 25 Form the differential equation of the family of circles in the second\nquadrant and touching the coordinate axes Solution Let C denote the family of circles in the second quadrant and touching the\ncoordinate axes"
    },
    {
        "Chapter": "1",
        "sentence_range": "4917-4920",
        "Text": "Hence, the given function is a solution of the given differential equation Example 25 Form the differential equation of the family of circles in the second\nquadrant and touching the coordinate axes Solution Let C denote the family of circles in the second quadrant and touching the\ncoordinate axes Let (\u2013a, a) be the coordinate of the centre of any member of\nthis family (see Fig 9"
    },
    {
        "Chapter": "1",
        "sentence_range": "4918-4921",
        "Text": "Example 25 Form the differential equation of the family of circles in the second\nquadrant and touching the coordinate axes Solution Let C denote the family of circles in the second quadrant and touching the\ncoordinate axes Let (\u2013a, a) be the coordinate of the centre of any member of\nthis family (see Fig 9 6)"
    },
    {
        "Chapter": "1",
        "sentence_range": "4919-4922",
        "Text": "Solution Let C denote the family of circles in the second quadrant and touching the\ncoordinate axes Let (\u2013a, a) be the coordinate of the centre of any member of\nthis family (see Fig 9 6) \u00a9 NCERT\nnot to be republished\nMATHEMATICS\n416\nX\nX\u2019\nY\nY\u2019\n(\u2013 , )\na a\nO\nEquation representing the family C is\n(x + a)2 + (y \u2013 a)2 = a2"
    },
    {
        "Chapter": "1",
        "sentence_range": "4920-4923",
        "Text": "Let (\u2013a, a) be the coordinate of the centre of any member of\nthis family (see Fig 9 6) \u00a9 NCERT\nnot to be republished\nMATHEMATICS\n416\nX\nX\u2019\nY\nY\u2019\n(\u2013 , )\na a\nO\nEquation representing the family C is\n(x + a)2 + (y \u2013 a)2 = a2 (1)\nor\nx2 + y2 + 2ax \u2013 2ay + a2 = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "4921-4924",
        "Text": "6) \u00a9 NCERT\nnot to be republished\nMATHEMATICS\n416\nX\nX\u2019\nY\nY\u2019\n(\u2013 , )\na a\nO\nEquation representing the family C is\n(x + a)2 + (y \u2013 a)2 = a2 (1)\nor\nx2 + y2 + 2ax \u2013 2ay + a2 = 0 (2)\nDifferentiating equation (2) with respect to x, we get\n2\n2\n2\n2\ndy\ndy\nx\ny\na\na\ndx\ndx\n+\n+\n\u2212\n = 0\nor\ndy\nx\n+y dx\n =\n1\ndy\na dx\n\u239b\n\u2212\u239e\n\u239c\n\u239f\n\u239d\n\u23a0\nor\na =\n1\nx\ny y\ny\n\u2032\n+\n\u2032 \u2212\nSubstituting the value of a in equation (1), we get\n2\n2\n1\n1\nx\ny y\nx\ny y\nx\ny\ny\ny\n\u2032\n\u2032\n+\n+\n\u23a1\n\u23a4\n\u23a1\n\u23a4\n+\n+\n\u2212\n\u23a2\n\u23a5\n\u23a2\n\u23a5\n\u2032\n\u2032\n\u2212\n\u2212\n\u23a3\n\u23a6\n\u23a3\n\u23a6\n = \n2\n1\nx\nyy y\n\u2032\n+\n\u23a1\n\u23a4\n\u23a2\n\u23a5\n\u2032 \u2212\n\u23a3\n\u23a6\nor\n[xy\u2032 \u2013 x + x + y y\u2032]2 + [y y\u2032 \u2013 y \u2013 x \u2013 y y\u2032]2 = [x + y y\u2032]2\nor\n(x + y)2 y\u20322 + [x + y]2 = [x + y y\u2032]2\nor\n(x + y)2 [(y\u2032)2 + 1] = [x + y y\u2032]2\nwhich is the differential equation representing the given family of circles"
    },
    {
        "Chapter": "1",
        "sentence_range": "4922-4925",
        "Text": "\u00a9 NCERT\nnot to be republished\nMATHEMATICS\n416\nX\nX\u2019\nY\nY\u2019\n(\u2013 , )\na a\nO\nEquation representing the family C is\n(x + a)2 + (y \u2013 a)2 = a2 (1)\nor\nx2 + y2 + 2ax \u2013 2ay + a2 = 0 (2)\nDifferentiating equation (2) with respect to x, we get\n2\n2\n2\n2\ndy\ndy\nx\ny\na\na\ndx\ndx\n+\n+\n\u2212\n = 0\nor\ndy\nx\n+y dx\n =\n1\ndy\na dx\n\u239b\n\u2212\u239e\n\u239c\n\u239f\n\u239d\n\u23a0\nor\na =\n1\nx\ny y\ny\n\u2032\n+\n\u2032 \u2212\nSubstituting the value of a in equation (1), we get\n2\n2\n1\n1\nx\ny y\nx\ny y\nx\ny\ny\ny\n\u2032\n\u2032\n+\n+\n\u23a1\n\u23a4\n\u23a1\n\u23a4\n+\n+\n\u2212\n\u23a2\n\u23a5\n\u23a2\n\u23a5\n\u2032\n\u2032\n\u2212\n\u2212\n\u23a3\n\u23a6\n\u23a3\n\u23a6\n = \n2\n1\nx\nyy y\n\u2032\n+\n\u23a1\n\u23a4\n\u23a2\n\u23a5\n\u2032 \u2212\n\u23a3\n\u23a6\nor\n[xy\u2032 \u2013 x + x + y y\u2032]2 + [y y\u2032 \u2013 y \u2013 x \u2013 y y\u2032]2 = [x + y y\u2032]2\nor\n(x + y)2 y\u20322 + [x + y]2 = [x + y y\u2032]2\nor\n(x + y)2 [(y\u2032)2 + 1] = [x + y y\u2032]2\nwhich is the differential equation representing the given family of circles Example 26 Find the particular solution of the differential equation log\n3\n4\ndy\nx\ny\n\u239bdx\n\u239e =\n+\n\u239c\n\u239f\n\u239d\n\u23a0\ngiven that y = 0 when x = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "4923-4926",
        "Text": "(1)\nor\nx2 + y2 + 2ax \u2013 2ay + a2 = 0 (2)\nDifferentiating equation (2) with respect to x, we get\n2\n2\n2\n2\ndy\ndy\nx\ny\na\na\ndx\ndx\n+\n+\n\u2212\n = 0\nor\ndy\nx\n+y dx\n =\n1\ndy\na dx\n\u239b\n\u2212\u239e\n\u239c\n\u239f\n\u239d\n\u23a0\nor\na =\n1\nx\ny y\ny\n\u2032\n+\n\u2032 \u2212\nSubstituting the value of a in equation (1), we get\n2\n2\n1\n1\nx\ny y\nx\ny y\nx\ny\ny\ny\n\u2032\n\u2032\n+\n+\n\u23a1\n\u23a4\n\u23a1\n\u23a4\n+\n+\n\u2212\n\u23a2\n\u23a5\n\u23a2\n\u23a5\n\u2032\n\u2032\n\u2212\n\u2212\n\u23a3\n\u23a6\n\u23a3\n\u23a6\n = \n2\n1\nx\nyy y\n\u2032\n+\n\u23a1\n\u23a4\n\u23a2\n\u23a5\n\u2032 \u2212\n\u23a3\n\u23a6\nor\n[xy\u2032 \u2013 x + x + y y\u2032]2 + [y y\u2032 \u2013 y \u2013 x \u2013 y y\u2032]2 = [x + y y\u2032]2\nor\n(x + y)2 y\u20322 + [x + y]2 = [x + y y\u2032]2\nor\n(x + y)2 [(y\u2032)2 + 1] = [x + y y\u2032]2\nwhich is the differential equation representing the given family of circles Example 26 Find the particular solution of the differential equation log\n3\n4\ndy\nx\ny\n\u239bdx\n\u239e =\n+\n\u239c\n\u239f\n\u239d\n\u23a0\ngiven that y = 0 when x = 0 Solution The given differential equation can be written as\ndy\ndx  = e(3x + 4y)\nor\ndy\ndx  = e3x"
    },
    {
        "Chapter": "1",
        "sentence_range": "4924-4927",
        "Text": "(2)\nDifferentiating equation (2) with respect to x, we get\n2\n2\n2\n2\ndy\ndy\nx\ny\na\na\ndx\ndx\n+\n+\n\u2212\n = 0\nor\ndy\nx\n+y dx\n =\n1\ndy\na dx\n\u239b\n\u2212\u239e\n\u239c\n\u239f\n\u239d\n\u23a0\nor\na =\n1\nx\ny y\ny\n\u2032\n+\n\u2032 \u2212\nSubstituting the value of a in equation (1), we get\n2\n2\n1\n1\nx\ny y\nx\ny y\nx\ny\ny\ny\n\u2032\n\u2032\n+\n+\n\u23a1\n\u23a4\n\u23a1\n\u23a4\n+\n+\n\u2212\n\u23a2\n\u23a5\n\u23a2\n\u23a5\n\u2032\n\u2032\n\u2212\n\u2212\n\u23a3\n\u23a6\n\u23a3\n\u23a6\n = \n2\n1\nx\nyy y\n\u2032\n+\n\u23a1\n\u23a4\n\u23a2\n\u23a5\n\u2032 \u2212\n\u23a3\n\u23a6\nor\n[xy\u2032 \u2013 x + x + y y\u2032]2 + [y y\u2032 \u2013 y \u2013 x \u2013 y y\u2032]2 = [x + y y\u2032]2\nor\n(x + y)2 y\u20322 + [x + y]2 = [x + y y\u2032]2\nor\n(x + y)2 [(y\u2032)2 + 1] = [x + y y\u2032]2\nwhich is the differential equation representing the given family of circles Example 26 Find the particular solution of the differential equation log\n3\n4\ndy\nx\ny\n\u239bdx\n\u239e =\n+\n\u239c\n\u239f\n\u239d\n\u23a0\ngiven that y = 0 when x = 0 Solution The given differential equation can be written as\ndy\ndx  = e(3x + 4y)\nor\ndy\ndx  = e3x e4y"
    },
    {
        "Chapter": "1",
        "sentence_range": "4925-4928",
        "Text": "Example 26 Find the particular solution of the differential equation log\n3\n4\ndy\nx\ny\n\u239bdx\n\u239e =\n+\n\u239c\n\u239f\n\u239d\n\u23a0\ngiven that y = 0 when x = 0 Solution The given differential equation can be written as\ndy\ndx  = e(3x + 4y)\nor\ndy\ndx  = e3x e4y (1)\nSeparating the variables, we get\nedy4 y\n= e3x dx\nTherefore\ne4 y\ndy\n\u222b\u2212\n=\n3x\ne dx\n\u222b\nFig 9"
    },
    {
        "Chapter": "1",
        "sentence_range": "4926-4929",
        "Text": "Solution The given differential equation can be written as\ndy\ndx  = e(3x + 4y)\nor\ndy\ndx  = e3x e4y (1)\nSeparating the variables, we get\nedy4 y\n= e3x dx\nTherefore\ne4 y\ndy\n\u222b\u2212\n=\n3x\ne dx\n\u222b\nFig 9 6\n\u00a9 NCERT\nnot to be republished\nDIFFERENTIAL EQUATIONS\n417\nor\n4\n4\ny\ne\u2212\n\u2212\n =\n3\nC\n3\nex\n+\nor\n4 e3x + 3 e\u2013 4y + 12 C = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "4927-4930",
        "Text": "e4y (1)\nSeparating the variables, we get\nedy4 y\n= e3x dx\nTherefore\ne4 y\ndy\n\u222b\u2212\n=\n3x\ne dx\n\u222b\nFig 9 6\n\u00a9 NCERT\nnot to be republished\nDIFFERENTIAL EQUATIONS\n417\nor\n4\n4\ny\ne\u2212\n\u2212\n =\n3\nC\n3\nex\n+\nor\n4 e3x + 3 e\u2013 4y + 12 C = 0 (2)\nSubstituting x = 0 and y = 0 in (2), we get\n4 + 3 + 12 C = 0 or C = \n127\n\u2212\nSubstituting the value of C in equation (2), we get\n4e3x + 3e\u2013 4y \u2013 7 = 0,\nwhich is a particular solution of the given differential equation"
    },
    {
        "Chapter": "1",
        "sentence_range": "4928-4931",
        "Text": "(1)\nSeparating the variables, we get\nedy4 y\n= e3x dx\nTherefore\ne4 y\ndy\n\u222b\u2212\n=\n3x\ne dx\n\u222b\nFig 9 6\n\u00a9 NCERT\nnot to be republished\nDIFFERENTIAL EQUATIONS\n417\nor\n4\n4\ny\ne\u2212\n\u2212\n =\n3\nC\n3\nex\n+\nor\n4 e3x + 3 e\u2013 4y + 12 C = 0 (2)\nSubstituting x = 0 and y = 0 in (2), we get\n4 + 3 + 12 C = 0 or C = \n127\n\u2212\nSubstituting the value of C in equation (2), we get\n4e3x + 3e\u2013 4y \u2013 7 = 0,\nwhich is a particular solution of the given differential equation Example 27 Solve the differential equation\n(x dy \u2013 y dx) y sin \ny\nx\n\u239b\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0  = (y dx + x dy) x cos \ny\nx\n\u239b\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0"
    },
    {
        "Chapter": "1",
        "sentence_range": "4929-4932",
        "Text": "6\n\u00a9 NCERT\nnot to be republished\nDIFFERENTIAL EQUATIONS\n417\nor\n4\n4\ny\ne\u2212\n\u2212\n =\n3\nC\n3\nex\n+\nor\n4 e3x + 3 e\u2013 4y + 12 C = 0 (2)\nSubstituting x = 0 and y = 0 in (2), we get\n4 + 3 + 12 C = 0 or C = \n127\n\u2212\nSubstituting the value of C in equation (2), we get\n4e3x + 3e\u2013 4y \u2013 7 = 0,\nwhich is a particular solution of the given differential equation Example 27 Solve the differential equation\n(x dy \u2013 y dx) y sin \ny\nx\n\u239b\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0  = (y dx + x dy) x cos \ny\nx\n\u239b\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0 Solution The given differential equation can be written as\n2\n2\nsin\ncos\ncos\nsin\ny\ny\ny\ny\nx y\nx\ndy\nxy\ny\ndx\nx\nx\nx\nx\n\u23a1\n\u23a4\n\u23a1\n\u23a4\n\u239b\n\u239e\n\u239b\n\u239e\n\u239b\n\u239e\n\u239b\n\u239e\n\u2212\n=\n+\n\u239c\n\u239f\n\u239c\n\u239f\n\u239c\n\u239f\n\u239c\n\u239f\n\u23a2\n\u23a5\n\u23a2\n\u23a5\n\u239d\n\u23a0\n\u239d\n\u23a0\n\u239d\n\u23a0\n\u239d\n\u23a0\n\u23a3\n\u23a6\n\u23a3\n\u23a6\nor\ndy\ndx  =\n2\n2\ncos\nsin\nsin\ncos\ny\ny\nxy\ny\nx\nx\ny\ny\nxy\nx\nx\nx\n\u239b\n\u239e\n\u239b\n\u239e\n+\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\n\u239b\n\u239e\n\u239b\n\u239e\n\u2212\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\nDividing numerator and denominator on RHS by x2, we get\ndy\ndx  =\n2\n2\ncos\nsin\nsin\ncos\ny\ny\ny\ny\nx\nx\nx\nx\ny\ny\ny\nx\nx\nx\n\u239b\n\u239e\n\u239b\n\u239e\n\u239b\n\u239e\n+\n\u239c\n\u239f\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\n\u239d\n\u23a0\n\u239b\n\u239e\n\u239b\n\u239e\n\u2212\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0"
    },
    {
        "Chapter": "1",
        "sentence_range": "4930-4933",
        "Text": "(2)\nSubstituting x = 0 and y = 0 in (2), we get\n4 + 3 + 12 C = 0 or C = \n127\n\u2212\nSubstituting the value of C in equation (2), we get\n4e3x + 3e\u2013 4y \u2013 7 = 0,\nwhich is a particular solution of the given differential equation Example 27 Solve the differential equation\n(x dy \u2013 y dx) y sin \ny\nx\n\u239b\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0  = (y dx + x dy) x cos \ny\nx\n\u239b\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0 Solution The given differential equation can be written as\n2\n2\nsin\ncos\ncos\nsin\ny\ny\ny\ny\nx y\nx\ndy\nxy\ny\ndx\nx\nx\nx\nx\n\u23a1\n\u23a4\n\u23a1\n\u23a4\n\u239b\n\u239e\n\u239b\n\u239e\n\u239b\n\u239e\n\u239b\n\u239e\n\u2212\n=\n+\n\u239c\n\u239f\n\u239c\n\u239f\n\u239c\n\u239f\n\u239c\n\u239f\n\u23a2\n\u23a5\n\u23a2\n\u23a5\n\u239d\n\u23a0\n\u239d\n\u23a0\n\u239d\n\u23a0\n\u239d\n\u23a0\n\u23a3\n\u23a6\n\u23a3\n\u23a6\nor\ndy\ndx  =\n2\n2\ncos\nsin\nsin\ncos\ny\ny\nxy\ny\nx\nx\ny\ny\nxy\nx\nx\nx\n\u239b\n\u239e\n\u239b\n\u239e\n+\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\n\u239b\n\u239e\n\u239b\n\u239e\n\u2212\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\nDividing numerator and denominator on RHS by x2, we get\ndy\ndx  =\n2\n2\ncos\nsin\nsin\ncos\ny\ny\ny\ny\nx\nx\nx\nx\ny\ny\ny\nx\nx\nx\n\u239b\n\u239e\n\u239b\n\u239e\n\u239b\n\u239e\n+\n\u239c\n\u239f\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\n\u239d\n\u23a0\n\u239b\n\u239e\n\u239b\n\u239e\n\u2212\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0 (1)\nClearly, equation (1) is a homogeneous differential equation of the form dy\ny\ng\ndx\n\u239bx\n\u239e\n=\n\u239c\n\u239f\n\u239d\n\u23a0"
    },
    {
        "Chapter": "1",
        "sentence_range": "4931-4934",
        "Text": "Example 27 Solve the differential equation\n(x dy \u2013 y dx) y sin \ny\nx\n\u239b\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0  = (y dx + x dy) x cos \ny\nx\n\u239b\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0 Solution The given differential equation can be written as\n2\n2\nsin\ncos\ncos\nsin\ny\ny\ny\ny\nx y\nx\ndy\nxy\ny\ndx\nx\nx\nx\nx\n\u23a1\n\u23a4\n\u23a1\n\u23a4\n\u239b\n\u239e\n\u239b\n\u239e\n\u239b\n\u239e\n\u239b\n\u239e\n\u2212\n=\n+\n\u239c\n\u239f\n\u239c\n\u239f\n\u239c\n\u239f\n\u239c\n\u239f\n\u23a2\n\u23a5\n\u23a2\n\u23a5\n\u239d\n\u23a0\n\u239d\n\u23a0\n\u239d\n\u23a0\n\u239d\n\u23a0\n\u23a3\n\u23a6\n\u23a3\n\u23a6\nor\ndy\ndx  =\n2\n2\ncos\nsin\nsin\ncos\ny\ny\nxy\ny\nx\nx\ny\ny\nxy\nx\nx\nx\n\u239b\n\u239e\n\u239b\n\u239e\n+\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\n\u239b\n\u239e\n\u239b\n\u239e\n\u2212\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\nDividing numerator and denominator on RHS by x2, we get\ndy\ndx  =\n2\n2\ncos\nsin\nsin\ncos\ny\ny\ny\ny\nx\nx\nx\nx\ny\ny\ny\nx\nx\nx\n\u239b\n\u239e\n\u239b\n\u239e\n\u239b\n\u239e\n+\n\u239c\n\u239f\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\n\u239d\n\u23a0\n\u239b\n\u239e\n\u239b\n\u239e\n\u2212\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0 (1)\nClearly, equation (1) is a homogeneous differential equation of the form dy\ny\ng\ndx\n\u239bx\n\u239e\n=\n\u239c\n\u239f\n\u239d\n\u23a0 To solve it, we make the substitution\ny = vx"
    },
    {
        "Chapter": "1",
        "sentence_range": "4932-4935",
        "Text": "Solution The given differential equation can be written as\n2\n2\nsin\ncos\ncos\nsin\ny\ny\ny\ny\nx y\nx\ndy\nxy\ny\ndx\nx\nx\nx\nx\n\u23a1\n\u23a4\n\u23a1\n\u23a4\n\u239b\n\u239e\n\u239b\n\u239e\n\u239b\n\u239e\n\u239b\n\u239e\n\u2212\n=\n+\n\u239c\n\u239f\n\u239c\n\u239f\n\u239c\n\u239f\n\u239c\n\u239f\n\u23a2\n\u23a5\n\u23a2\n\u23a5\n\u239d\n\u23a0\n\u239d\n\u23a0\n\u239d\n\u23a0\n\u239d\n\u23a0\n\u23a3\n\u23a6\n\u23a3\n\u23a6\nor\ndy\ndx  =\n2\n2\ncos\nsin\nsin\ncos\ny\ny\nxy\ny\nx\nx\ny\ny\nxy\nx\nx\nx\n\u239b\n\u239e\n\u239b\n\u239e\n+\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\n\u239b\n\u239e\n\u239b\n\u239e\n\u2212\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\nDividing numerator and denominator on RHS by x2, we get\ndy\ndx  =\n2\n2\ncos\nsin\nsin\ncos\ny\ny\ny\ny\nx\nx\nx\nx\ny\ny\ny\nx\nx\nx\n\u239b\n\u239e\n\u239b\n\u239e\n\u239b\n\u239e\n+\n\u239c\n\u239f\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\n\u239d\n\u23a0\n\u239b\n\u239e\n\u239b\n\u239e\n\u2212\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0 (1)\nClearly, equation (1) is a homogeneous differential equation of the form dy\ny\ng\ndx\n\u239bx\n\u239e\n=\n\u239c\n\u239f\n\u239d\n\u23a0 To solve it, we make the substitution\ny = vx (2)\nor\ndy\ndx  =\ndv\nv\nx dx\n+\n\u00a9 NCERT\nnot to be republished\nMATHEMATICS\n418\nor\ndv\nv\n+x dx\n =\n2\ncos\nsin\nsin\ncos\nv\nv\nv\nv\nv\nv\nv\n+\n\u2212\n(using (1) and (2))\nor\ndv\nx dx  =\nsin2 cos\ncos\nv\nv\nv\nv\nv\n\u2212\nor\nsin\ncoscos\nv\nv\nv\ndv\nv\n\u2212v\n\u239b\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0\n = 2 dx\nx\nTherefore\nsin\ncoscos\nv\nv\nv dv\nv\n\u2212v\n\u239b\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0\n\u222b\n =\n1\n2\nxdx\n\u222b\nor\n1\ntanv dv\nvdv\n\u2212\n\u222b\n\u222b\n =\n1\n2\nxdx\n\u222b\nor\nlog sec\nlog | |\nv\nv\n\u2212\n =\n1\n2log |\n|\nlog | C |\nx +\nor\nlogsec2\nv xv\n = log |C1|\nor\nsecv2\nv x  = \u00b1 C1"
    },
    {
        "Chapter": "1",
        "sentence_range": "4933-4936",
        "Text": "(1)\nClearly, equation (1) is a homogeneous differential equation of the form dy\ny\ng\ndx\n\u239bx\n\u239e\n=\n\u239c\n\u239f\n\u239d\n\u23a0 To solve it, we make the substitution\ny = vx (2)\nor\ndy\ndx  =\ndv\nv\nx dx\n+\n\u00a9 NCERT\nnot to be republished\nMATHEMATICS\n418\nor\ndv\nv\n+x dx\n =\n2\ncos\nsin\nsin\ncos\nv\nv\nv\nv\nv\nv\nv\n+\n\u2212\n(using (1) and (2))\nor\ndv\nx dx  =\nsin2 cos\ncos\nv\nv\nv\nv\nv\n\u2212\nor\nsin\ncoscos\nv\nv\nv\ndv\nv\n\u2212v\n\u239b\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0\n = 2 dx\nx\nTherefore\nsin\ncoscos\nv\nv\nv dv\nv\n\u2212v\n\u239b\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0\n\u222b\n =\n1\n2\nxdx\n\u222b\nor\n1\ntanv dv\nvdv\n\u2212\n\u222b\n\u222b\n =\n1\n2\nxdx\n\u222b\nor\nlog sec\nlog | |\nv\nv\n\u2212\n =\n1\n2log |\n|\nlog | C |\nx +\nor\nlogsec2\nv xv\n = log |C1|\nor\nsecv2\nv x  = \u00b1 C1 (3)\nReplacing v by y\nx  in equation (3), we get\n2\nsec\n(\n)\ny\nx\ny\nx\nx\n\u239b\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239b\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0\n = C where, C = \u00b1 C1\nor\nsec y\nx\n\u239b\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0  = C xy\nwhich is the general solution of the given differential equation"
    },
    {
        "Chapter": "1",
        "sentence_range": "4934-4937",
        "Text": "To solve it, we make the substitution\ny = vx (2)\nor\ndy\ndx  =\ndv\nv\nx dx\n+\n\u00a9 NCERT\nnot to be republished\nMATHEMATICS\n418\nor\ndv\nv\n+x dx\n =\n2\ncos\nsin\nsin\ncos\nv\nv\nv\nv\nv\nv\nv\n+\n\u2212\n(using (1) and (2))\nor\ndv\nx dx  =\nsin2 cos\ncos\nv\nv\nv\nv\nv\n\u2212\nor\nsin\ncoscos\nv\nv\nv\ndv\nv\n\u2212v\n\u239b\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0\n = 2 dx\nx\nTherefore\nsin\ncoscos\nv\nv\nv dv\nv\n\u2212v\n\u239b\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0\n\u222b\n =\n1\n2\nxdx\n\u222b\nor\n1\ntanv dv\nvdv\n\u2212\n\u222b\n\u222b\n =\n1\n2\nxdx\n\u222b\nor\nlog sec\nlog | |\nv\nv\n\u2212\n =\n1\n2log |\n|\nlog | C |\nx +\nor\nlogsec2\nv xv\n = log |C1|\nor\nsecv2\nv x  = \u00b1 C1 (3)\nReplacing v by y\nx  in equation (3), we get\n2\nsec\n(\n)\ny\nx\ny\nx\nx\n\u239b\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239b\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0\n = C where, C = \u00b1 C1\nor\nsec y\nx\n\u239b\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0  = C xy\nwhich is the general solution of the given differential equation Example 28 Solve the differential equation\n(tan\u20131y  \u2013 x) dy = (1 + y2) dx"
    },
    {
        "Chapter": "1",
        "sentence_range": "4935-4938",
        "Text": "(2)\nor\ndy\ndx  =\ndv\nv\nx dx\n+\n\u00a9 NCERT\nnot to be republished\nMATHEMATICS\n418\nor\ndv\nv\n+x dx\n =\n2\ncos\nsin\nsin\ncos\nv\nv\nv\nv\nv\nv\nv\n+\n\u2212\n(using (1) and (2))\nor\ndv\nx dx  =\nsin2 cos\ncos\nv\nv\nv\nv\nv\n\u2212\nor\nsin\ncoscos\nv\nv\nv\ndv\nv\n\u2212v\n\u239b\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0\n = 2 dx\nx\nTherefore\nsin\ncoscos\nv\nv\nv dv\nv\n\u2212v\n\u239b\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0\n\u222b\n =\n1\n2\nxdx\n\u222b\nor\n1\ntanv dv\nvdv\n\u2212\n\u222b\n\u222b\n =\n1\n2\nxdx\n\u222b\nor\nlog sec\nlog | |\nv\nv\n\u2212\n =\n1\n2log |\n|\nlog | C |\nx +\nor\nlogsec2\nv xv\n = log |C1|\nor\nsecv2\nv x  = \u00b1 C1 (3)\nReplacing v by y\nx  in equation (3), we get\n2\nsec\n(\n)\ny\nx\ny\nx\nx\n\u239b\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239b\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0\n = C where, C = \u00b1 C1\nor\nsec y\nx\n\u239b\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0  = C xy\nwhich is the general solution of the given differential equation Example 28 Solve the differential equation\n(tan\u20131y  \u2013 x) dy = (1 + y2) dx Solution The given differential equation can be written as\n2\n1\ndx\nx\ndy\ny\n+\n+\n =\n1\n2\n1tan\ny\ny\n\u2212\n+"
    },
    {
        "Chapter": "1",
        "sentence_range": "4936-4939",
        "Text": "(3)\nReplacing v by y\nx  in equation (3), we get\n2\nsec\n(\n)\ny\nx\ny\nx\nx\n\u239b\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239b\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0\n = C where, C = \u00b1 C1\nor\nsec y\nx\n\u239b\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0  = C xy\nwhich is the general solution of the given differential equation Example 28 Solve the differential equation\n(tan\u20131y  \u2013 x) dy = (1 + y2) dx Solution The given differential equation can be written as\n2\n1\ndx\nx\ndy\ny\n+\n+\n =\n1\n2\n1tan\ny\ny\n\u2212\n+ (1)\n\u00a9 NCERT\nnot to be republished\nDIFFERENTIAL EQUATIONS\n419\nNow (1) is a linear differential equation of the form \n1P\ndx\ndy +\n x = Q1,\nwhere,\nP1 =\n112\n+y\n and \n1\n1\n2\ntan\nQ\n1\ny\ny\n\u2212\n=\n+"
    },
    {
        "Chapter": "1",
        "sentence_range": "4937-4940",
        "Text": "Example 28 Solve the differential equation\n(tan\u20131y  \u2013 x) dy = (1 + y2) dx Solution The given differential equation can be written as\n2\n1\ndx\nx\ndy\ny\n+\n+\n =\n1\n2\n1tan\ny\ny\n\u2212\n+ (1)\n\u00a9 NCERT\nnot to be republished\nDIFFERENTIAL EQUATIONS\n419\nNow (1) is a linear differential equation of the form \n1P\ndx\ndy +\n x = Q1,\nwhere,\nP1 =\n112\n+y\n and \n1\n1\n2\ntan\nQ\n1\ny\ny\n\u2212\n=\n+ Therefore,\nI"
    },
    {
        "Chapter": "1",
        "sentence_range": "4938-4941",
        "Text": "Solution The given differential equation can be written as\n2\n1\ndx\nx\ndy\ny\n+\n+\n =\n1\n2\n1tan\ny\ny\n\u2212\n+ (1)\n\u00a9 NCERT\nnot to be republished\nDIFFERENTIAL EQUATIONS\n419\nNow (1) is a linear differential equation of the form \n1P\ndx\ndy +\n x = Q1,\nwhere,\nP1 =\n112\n+y\n and \n1\n1\n2\ntan\nQ\n1\ny\ny\n\u2212\n=\n+ Therefore,\nI F = \n1\n12\ntan\n1\ndy\ny\ny\ne\ne\n\u2212\n\u222b+\n=\nThus, the solution of the given differential equation is\ntan1\ny\nxe\n\u2212\n =\n1\n1\ntan\n2\ntan\nC\n1\ny\ny\ne\ndy\ny\n\u2212\n\u2212\n\u239b\n\u239e\n+\n\u239c\n\u239f\n+\n\u239d\n\u23a0\n\u222b"
    },
    {
        "Chapter": "1",
        "sentence_range": "4939-4942",
        "Text": "(1)\n\u00a9 NCERT\nnot to be republished\nDIFFERENTIAL EQUATIONS\n419\nNow (1) is a linear differential equation of the form \n1P\ndx\ndy +\n x = Q1,\nwhere,\nP1 =\n112\n+y\n and \n1\n1\n2\ntan\nQ\n1\ny\ny\n\u2212\n=\n+ Therefore,\nI F = \n1\n12\ntan\n1\ndy\ny\ny\ne\ne\n\u2212\n\u222b+\n=\nThus, the solution of the given differential equation is\ntan1\ny\nxe\n\u2212\n =\n1\n1\ntan\n2\ntan\nC\n1\ny\ny\ne\ndy\ny\n\u2212\n\u2212\n\u239b\n\u239e\n+\n\u239c\n\u239f\n+\n\u239d\n\u23a0\n\u222b (2)\nLet\nI =\n1\n1\ntan\n2\ntan\n1\ny\ny\ne\ndy\ny\n\u2212\n\u2212\n\u239b\n\u239e\n\u239c\n\u239f\n+\n\u239d\n\u23a0\n\u222b\nSubstituting tan\u20131 y = t so that \n112\ndy\ndt\ny\n\u239b\n\u239e\n=\n\u239c\n\u239f\n\u239d+\n\u23a0\n, we get\nI =\nt\n\u222bt e dt\n= t et \u2013 \u222b1"
    },
    {
        "Chapter": "1",
        "sentence_range": "4940-4943",
        "Text": "Therefore,\nI F = \n1\n12\ntan\n1\ndy\ny\ny\ne\ne\n\u2212\n\u222b+\n=\nThus, the solution of the given differential equation is\ntan1\ny\nxe\n\u2212\n =\n1\n1\ntan\n2\ntan\nC\n1\ny\ny\ne\ndy\ny\n\u2212\n\u2212\n\u239b\n\u239e\n+\n\u239c\n\u239f\n+\n\u239d\n\u23a0\n\u222b (2)\nLet\nI =\n1\n1\ntan\n2\ntan\n1\ny\ny\ne\ndy\ny\n\u2212\n\u2212\n\u239b\n\u239e\n\u239c\n\u239f\n+\n\u239d\n\u23a0\n\u222b\nSubstituting tan\u20131 y = t so that \n112\ndy\ndt\ny\n\u239b\n\u239e\n=\n\u239c\n\u239f\n\u239d+\n\u23a0\n, we get\nI =\nt\n\u222bt e dt\n= t et \u2013 \u222b1 et dt = t et \u2013 et = et (t \u2013 1)\nor\nI =\ntan1\ny\ne\n\u2212\n(tan\u20131y \u20131)\nSubstituting the value of I in equation (2), we get\n1\n1\ntan\ntan\n1"
    },
    {
        "Chapter": "1",
        "sentence_range": "4941-4944",
        "Text": "F = \n1\n12\ntan\n1\ndy\ny\ny\ne\ne\n\u2212\n\u222b+\n=\nThus, the solution of the given differential equation is\ntan1\ny\nxe\n\u2212\n =\n1\n1\ntan\n2\ntan\nC\n1\ny\ny\ne\ndy\ny\n\u2212\n\u2212\n\u239b\n\u239e\n+\n\u239c\n\u239f\n+\n\u239d\n\u23a0\n\u222b (2)\nLet\nI =\n1\n1\ntan\n2\ntan\n1\ny\ny\ne\ndy\ny\n\u2212\n\u2212\n\u239b\n\u239e\n\u239c\n\u239f\n+\n\u239d\n\u23a0\n\u222b\nSubstituting tan\u20131 y = t so that \n112\ndy\ndt\ny\n\u239b\n\u239e\n=\n\u239c\n\u239f\n\u239d+\n\u23a0\n, we get\nI =\nt\n\u222bt e dt\n= t et \u2013 \u222b1 et dt = t et \u2013 et = et (t \u2013 1)\nor\nI =\ntan1\ny\ne\n\u2212\n(tan\u20131y \u20131)\nSubstituting the value of I in equation (2), we get\n1\n1\ntan\ntan\n1 (tan\n1)\nC\ny\ny\nx e\ne\ny\n\u2212\n\u2212\n\u2212\n=\n\u2212\n+\nor\nx =\n1\n1\ntan\n(tan\n1)\nC\ny\ny\ne\n\u2212\n\u2212\n\u2212\n\u2212\n+\nwhich is the general solution of the given differential equation"
    },
    {
        "Chapter": "1",
        "sentence_range": "4942-4945",
        "Text": "(2)\nLet\nI =\n1\n1\ntan\n2\ntan\n1\ny\ny\ne\ndy\ny\n\u2212\n\u2212\n\u239b\n\u239e\n\u239c\n\u239f\n+\n\u239d\n\u23a0\n\u222b\nSubstituting tan\u20131 y = t so that \n112\ndy\ndt\ny\n\u239b\n\u239e\n=\n\u239c\n\u239f\n\u239d+\n\u23a0\n, we get\nI =\nt\n\u222bt e dt\n= t et \u2013 \u222b1 et dt = t et \u2013 et = et (t \u2013 1)\nor\nI =\ntan1\ny\ne\n\u2212\n(tan\u20131y \u20131)\nSubstituting the value of I in equation (2), we get\n1\n1\ntan\ntan\n1 (tan\n1)\nC\ny\ny\nx e\ne\ny\n\u2212\n\u2212\n\u2212\n=\n\u2212\n+\nor\nx =\n1\n1\ntan\n(tan\n1)\nC\ny\ny\ne\n\u2212\n\u2212\n\u2212\n\u2212\n+\nwhich is the general solution of the given differential equation Miscellaneous Exercise on Chapter 9\n1"
    },
    {
        "Chapter": "1",
        "sentence_range": "4943-4946",
        "Text": "et dt = t et \u2013 et = et (t \u2013 1)\nor\nI =\ntan1\ny\ne\n\u2212\n(tan\u20131y \u20131)\nSubstituting the value of I in equation (2), we get\n1\n1\ntan\ntan\n1 (tan\n1)\nC\ny\ny\nx e\ne\ny\n\u2212\n\u2212\n\u2212\n=\n\u2212\n+\nor\nx =\n1\n1\ntan\n(tan\n1)\nC\ny\ny\ne\n\u2212\n\u2212\n\u2212\n\u2212\n+\nwhich is the general solution of the given differential equation Miscellaneous Exercise on Chapter 9\n1 For each of the differential equations given below, indicate its order and degree\n(if defined)"
    },
    {
        "Chapter": "1",
        "sentence_range": "4944-4947",
        "Text": "(tan\n1)\nC\ny\ny\nx e\ne\ny\n\u2212\n\u2212\n\u2212\n=\n\u2212\n+\nor\nx =\n1\n1\ntan\n(tan\n1)\nC\ny\ny\ne\n\u2212\n\u2212\n\u2212\n\u2212\n+\nwhich is the general solution of the given differential equation Miscellaneous Exercise on Chapter 9\n1 For each of the differential equations given below, indicate its order and degree\n(if defined) (i)\n2\n2\n2\n5\n6\nlog\nd y\nxdy\ny\nx\ndx\ndx\n\u239b\n\u239e\n+\n\u2212\n=\n\u239c\n\u239f\n\u239d\n\u23a0\n         (ii)  \n3\n2\n4\n7\nsin\ndy\ndy\ny\nx\ndx\ndx\n\u239b\n\u239e\n\u239b\n\u239e\n\u2212\n+\n=\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\n(iii)\n4\n3\n4\n3\nsin\n0\nd y\nd y\ndx\ndx\n\u239b\n\u239e\n\u2212\n=\n\u239c\n\u239f\n\u239d\n\u23a0\n\u00a9 NCERT\nnot to be republished\nMATHEMATICS\n420\n2"
    },
    {
        "Chapter": "1",
        "sentence_range": "4945-4948",
        "Text": "Miscellaneous Exercise on Chapter 9\n1 For each of the differential equations given below, indicate its order and degree\n(if defined) (i)\n2\n2\n2\n5\n6\nlog\nd y\nxdy\ny\nx\ndx\ndx\n\u239b\n\u239e\n+\n\u2212\n=\n\u239c\n\u239f\n\u239d\n\u23a0\n         (ii)  \n3\n2\n4\n7\nsin\ndy\ndy\ny\nx\ndx\ndx\n\u239b\n\u239e\n\u239b\n\u239e\n\u2212\n+\n=\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\n(iii)\n4\n3\n4\n3\nsin\n0\nd y\nd y\ndx\ndx\n\u239b\n\u239e\n\u2212\n=\n\u239c\n\u239f\n\u239d\n\u23a0\n\u00a9 NCERT\nnot to be republished\nMATHEMATICS\n420\n2 For each of the exercises given below, verify that the given function (implicit or\nexplicit) is a solution of the corresponding differential equation"
    },
    {
        "Chapter": "1",
        "sentence_range": "4946-4949",
        "Text": "For each of the differential equations given below, indicate its order and degree\n(if defined) (i)\n2\n2\n2\n5\n6\nlog\nd y\nxdy\ny\nx\ndx\ndx\n\u239b\n\u239e\n+\n\u2212\n=\n\u239c\n\u239f\n\u239d\n\u23a0\n         (ii)  \n3\n2\n4\n7\nsin\ndy\ndy\ny\nx\ndx\ndx\n\u239b\n\u239e\n\u239b\n\u239e\n\u2212\n+\n=\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\n(iii)\n4\n3\n4\n3\nsin\n0\nd y\nd y\ndx\ndx\n\u239b\n\u239e\n\u2212\n=\n\u239c\n\u239f\n\u239d\n\u23a0\n\u00a9 NCERT\nnot to be republished\nMATHEMATICS\n420\n2 For each of the exercises given below, verify that the given function (implicit or\nexplicit) is a solution of the corresponding differential equation (i) y = a ex + b e\u2013x + x2\n:\n2\n2\n2\n2\n2\n0\nd y\ndy\nx\nxy\nx\ndx\ndx\n+\n\u2212\n+\n\u2212\n=\n(ii) y = ex (a cos x + b sin x)\n:\n2\n2\n2\n2\n0\nd y\ndy\ny\ndx\ndx\n\u2212\n+\n=\n(iii) y = x sin 3x\n:\n2\n2\n9\n6cos3\n0\nd y\ny\nx\ndx\n+\n\u2212\n=\n(iv) x2 = 2y2 log y\n:\n2\n2\n(\n)\n0\ndy\nx\ny\nxy\ndx\n+\n\u2212\n=\n3"
    },
    {
        "Chapter": "1",
        "sentence_range": "4947-4950",
        "Text": "(i)\n2\n2\n2\n5\n6\nlog\nd y\nxdy\ny\nx\ndx\ndx\n\u239b\n\u239e\n+\n\u2212\n=\n\u239c\n\u239f\n\u239d\n\u23a0\n         (ii)  \n3\n2\n4\n7\nsin\ndy\ndy\ny\nx\ndx\ndx\n\u239b\n\u239e\n\u239b\n\u239e\n\u2212\n+\n=\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\n(iii)\n4\n3\n4\n3\nsin\n0\nd y\nd y\ndx\ndx\n\u239b\n\u239e\n\u2212\n=\n\u239c\n\u239f\n\u239d\n\u23a0\n\u00a9 NCERT\nnot to be republished\nMATHEMATICS\n420\n2 For each of the exercises given below, verify that the given function (implicit or\nexplicit) is a solution of the corresponding differential equation (i) y = a ex + b e\u2013x + x2\n:\n2\n2\n2\n2\n2\n0\nd y\ndy\nx\nxy\nx\ndx\ndx\n+\n\u2212\n+\n\u2212\n=\n(ii) y = ex (a cos x + b sin x)\n:\n2\n2\n2\n2\n0\nd y\ndy\ny\ndx\ndx\n\u2212\n+\n=\n(iii) y = x sin 3x\n:\n2\n2\n9\n6cos3\n0\nd y\ny\nx\ndx\n+\n\u2212\n=\n(iv) x2 = 2y2 log y\n:\n2\n2\n(\n)\n0\ndy\nx\ny\nxy\ndx\n+\n\u2212\n=\n3 Form the differential equation representing the family of curves given by\n(x \u2013 a)2 + 2y2 = a2, where a is an arbitrary constant"
    },
    {
        "Chapter": "1",
        "sentence_range": "4948-4951",
        "Text": "For each of the exercises given below, verify that the given function (implicit or\nexplicit) is a solution of the corresponding differential equation (i) y = a ex + b e\u2013x + x2\n:\n2\n2\n2\n2\n2\n0\nd y\ndy\nx\nxy\nx\ndx\ndx\n+\n\u2212\n+\n\u2212\n=\n(ii) y = ex (a cos x + b sin x)\n:\n2\n2\n2\n2\n0\nd y\ndy\ny\ndx\ndx\n\u2212\n+\n=\n(iii) y = x sin 3x\n:\n2\n2\n9\n6cos3\n0\nd y\ny\nx\ndx\n+\n\u2212\n=\n(iv) x2 = 2y2 log y\n:\n2\n2\n(\n)\n0\ndy\nx\ny\nxy\ndx\n+\n\u2212\n=\n3 Form the differential equation representing the family of curves given by\n(x \u2013 a)2 + 2y2 = a2, where a is an arbitrary constant 4"
    },
    {
        "Chapter": "1",
        "sentence_range": "4949-4952",
        "Text": "(i) y = a ex + b e\u2013x + x2\n:\n2\n2\n2\n2\n2\n0\nd y\ndy\nx\nxy\nx\ndx\ndx\n+\n\u2212\n+\n\u2212\n=\n(ii) y = ex (a cos x + b sin x)\n:\n2\n2\n2\n2\n0\nd y\ndy\ny\ndx\ndx\n\u2212\n+\n=\n(iii) y = x sin 3x\n:\n2\n2\n9\n6cos3\n0\nd y\ny\nx\ndx\n+\n\u2212\n=\n(iv) x2 = 2y2 log y\n:\n2\n2\n(\n)\n0\ndy\nx\ny\nxy\ndx\n+\n\u2212\n=\n3 Form the differential equation representing the family of curves given by\n(x \u2013 a)2 + 2y2 = a2, where a is an arbitrary constant 4 Prove that x2 \u2013 y2 = c (x2 + y2)2 is the general solution of differential equation\n(x3 \u2013 3x y2) dx = (y3 \u2013 3x2y) dy, where c is a parameter"
    },
    {
        "Chapter": "1",
        "sentence_range": "4950-4953",
        "Text": "Form the differential equation representing the family of curves given by\n(x \u2013 a)2 + 2y2 = a2, where a is an arbitrary constant 4 Prove that x2 \u2013 y2 = c (x2 + y2)2 is the general solution of differential equation\n(x3 \u2013 3x y2) dx = (y3 \u2013 3x2y) dy, where c is a parameter 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "4951-4954",
        "Text": "4 Prove that x2 \u2013 y2 = c (x2 + y2)2 is the general solution of differential equation\n(x3 \u2013 3x y2) dx = (y3 \u2013 3x2y) dy, where c is a parameter 5 Form the differential equation of the family of circles in the first quadrant which\ntouch the coordinate axes"
    },
    {
        "Chapter": "1",
        "sentence_range": "4952-4955",
        "Text": "Prove that x2 \u2013 y2 = c (x2 + y2)2 is the general solution of differential equation\n(x3 \u2013 3x y2) dx = (y3 \u2013 3x2y) dy, where c is a parameter 5 Form the differential equation of the family of circles in the first quadrant which\ntouch the coordinate axes 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "4953-4956",
        "Text": "5 Form the differential equation of the family of circles in the first quadrant which\ntouch the coordinate axes 6 Find the general solution of the differential equation \n2\n2\n1\n0\n1\ndy\ny\ndx\n\u2212x\n+\n=\n\u2212"
    },
    {
        "Chapter": "1",
        "sentence_range": "4954-4957",
        "Text": "Form the differential equation of the family of circles in the first quadrant which\ntouch the coordinate axes 6 Find the general solution of the differential equation \n2\n2\n1\n0\n1\ndy\ny\ndx\n\u2212x\n+\n=\n\u2212 7"
    },
    {
        "Chapter": "1",
        "sentence_range": "4955-4958",
        "Text": "6 Find the general solution of the differential equation \n2\n2\n1\n0\n1\ndy\ny\ndx\n\u2212x\n+\n=\n\u2212 7 Show that the general solution of the differential equation \n2\n2\n1\n0\n1\ndy\ny\ny\ndx\nx\n+x\n+\n+\n=\n+\n+\n is\ngiven by (x + y + 1) = A (1 \u2013 x \u2013 y \u2013 2xy), where A is parameter"
    },
    {
        "Chapter": "1",
        "sentence_range": "4956-4959",
        "Text": "Find the general solution of the differential equation \n2\n2\n1\n0\n1\ndy\ny\ndx\n\u2212x\n+\n=\n\u2212 7 Show that the general solution of the differential equation \n2\n2\n1\n0\n1\ndy\ny\ny\ndx\nx\n+x\n+\n+\n=\n+\n+\n is\ngiven by (x + y + 1) = A (1 \u2013 x \u2013 y \u2013 2xy), where A is parameter 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "4957-4960",
        "Text": "7 Show that the general solution of the differential equation \n2\n2\n1\n0\n1\ndy\ny\ny\ndx\nx\n+x\n+\n+\n=\n+\n+\n is\ngiven by (x + y + 1) = A (1 \u2013 x \u2013 y \u2013 2xy), where A is parameter 8 Find the equation of the curve passing through the point 0, 4\n\u03c0\n\u239b\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0  whose differential\nequation is sin x cos y dx + cos x sin y dy = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "4958-4961",
        "Text": "Show that the general solution of the differential equation \n2\n2\n1\n0\n1\ndy\ny\ny\ndx\nx\n+x\n+\n+\n=\n+\n+\n is\ngiven by (x + y + 1) = A (1 \u2013 x \u2013 y \u2013 2xy), where A is parameter 8 Find the equation of the curve passing through the point 0, 4\n\u03c0\n\u239b\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0  whose differential\nequation is sin x cos y dx + cos x sin y dy = 0 9"
    },
    {
        "Chapter": "1",
        "sentence_range": "4959-4962",
        "Text": "8 Find the equation of the curve passing through the point 0, 4\n\u03c0\n\u239b\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0  whose differential\nequation is sin x cos y dx + cos x sin y dy = 0 9 Find the particular solution of the differential equation\n(1 + e2x) dy + (1 + y2) ex dx = 0, given that y = 1 when x = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "4960-4963",
        "Text": "Find the equation of the curve passing through the point 0, 4\n\u03c0\n\u239b\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0  whose differential\nequation is sin x cos y dx + cos x sin y dy = 0 9 Find the particular solution of the differential equation\n(1 + e2x) dy + (1 + y2) ex dx = 0, given that y = 1 when x = 0 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "4961-4964",
        "Text": "9 Find the particular solution of the differential equation\n(1 + e2x) dy + (1 + y2) ex dx = 0, given that y = 1 when x = 0 10 Solve the differential equation \n2\n(\n0)\nx\nx\ny\ny\ny e dx\nx e\ny\ndy y\n\u239b\n\u239e\n\u239c\n\u239f\n=\n+\n\u2260\n\u239d\n\u23a0"
    },
    {
        "Chapter": "1",
        "sentence_range": "4962-4965",
        "Text": "Find the particular solution of the differential equation\n(1 + e2x) dy + (1 + y2) ex dx = 0, given that y = 1 when x = 0 10 Solve the differential equation \n2\n(\n0)\nx\nx\ny\ny\ny e dx\nx e\ny\ndy y\n\u239b\n\u239e\n\u239c\n\u239f\n=\n+\n\u2260\n\u239d\n\u23a0 11"
    },
    {
        "Chapter": "1",
        "sentence_range": "4963-4966",
        "Text": "10 Solve the differential equation \n2\n(\n0)\nx\nx\ny\ny\ny e dx\nx e\ny\ndy y\n\u239b\n\u239e\n\u239c\n\u239f\n=\n+\n\u2260\n\u239d\n\u23a0 11 Find a particular solution of the differential equation  (x \u2013 y) (dx + dy) = dx \u2013 dy,\ngiven that y = \u20131, when x = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "4964-4967",
        "Text": "Solve the differential equation \n2\n(\n0)\nx\nx\ny\ny\ny e dx\nx e\ny\ndy y\n\u239b\n\u239e\n\u239c\n\u239f\n=\n+\n\u2260\n\u239d\n\u23a0 11 Find a particular solution of the differential equation  (x \u2013 y) (dx + dy) = dx \u2013 dy,\ngiven that y = \u20131, when x = 0 (Hint: put x \u2013 y = t)\n\u00a9 NCERT\nnot to be republished\nDIFFERENTIAL EQUATIONS\n421\n12"
    },
    {
        "Chapter": "1",
        "sentence_range": "4965-4968",
        "Text": "11 Find a particular solution of the differential equation  (x \u2013 y) (dx + dy) = dx \u2013 dy,\ngiven that y = \u20131, when x = 0 (Hint: put x \u2013 y = t)\n\u00a9 NCERT\nnot to be republished\nDIFFERENTIAL EQUATIONS\n421\n12 Solve the differential equation \n2\n1(\n0)\nx\ne\ny\ndx\nx\ndy\nx\nx\n\u23a1\u2212\n\u23a4\n\u2212\n=\n\u2260\n\u23a2\n\u23a5\n\u23a3\n\u23a6"
    },
    {
        "Chapter": "1",
        "sentence_range": "4966-4969",
        "Text": "Find a particular solution of the differential equation  (x \u2013 y) (dx + dy) = dx \u2013 dy,\ngiven that y = \u20131, when x = 0 (Hint: put x \u2013 y = t)\n\u00a9 NCERT\nnot to be republished\nDIFFERENTIAL EQUATIONS\n421\n12 Solve the differential equation \n2\n1(\n0)\nx\ne\ny\ndx\nx\ndy\nx\nx\n\u23a1\u2212\n\u23a4\n\u2212\n=\n\u2260\n\u23a2\n\u23a5\n\u23a3\n\u23a6 13"
    },
    {
        "Chapter": "1",
        "sentence_range": "4967-4970",
        "Text": "(Hint: put x \u2013 y = t)\n\u00a9 NCERT\nnot to be republished\nDIFFERENTIAL EQUATIONS\n421\n12 Solve the differential equation \n2\n1(\n0)\nx\ne\ny\ndx\nx\ndy\nx\nx\n\u23a1\u2212\n\u23a4\n\u2212\n=\n\u2260\n\u23a2\n\u23a5\n\u23a3\n\u23a6 13 Find a particular solution of the differential equation \ncot\ndy\ny\nx\ndx +\n = 4x cosec x\n(x \u2260 0), given that y = 0 when \n2\nx\n=\u03c0"
    },
    {
        "Chapter": "1",
        "sentence_range": "4968-4971",
        "Text": "Solve the differential equation \n2\n1(\n0)\nx\ne\ny\ndx\nx\ndy\nx\nx\n\u23a1\u2212\n\u23a4\n\u2212\n=\n\u2260\n\u23a2\n\u23a5\n\u23a3\n\u23a6 13 Find a particular solution of the differential equation \ncot\ndy\ny\nx\ndx +\n = 4x cosec x\n(x \u2260 0), given that y = 0 when \n2\nx\n=\u03c0 14"
    },
    {
        "Chapter": "1",
        "sentence_range": "4969-4972",
        "Text": "13 Find a particular solution of the differential equation \ncot\ndy\ny\nx\ndx +\n = 4x cosec x\n(x \u2260 0), given that y = 0 when \n2\nx\n=\u03c0 14 Find a particular solution of the differential equation (x + 1) dy\ndx  = 2 e\u2013y \u2013 1,  given\nthat y = 0 when x = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "4970-4973",
        "Text": "Find a particular solution of the differential equation \ncot\ndy\ny\nx\ndx +\n = 4x cosec x\n(x \u2260 0), given that y = 0 when \n2\nx\n=\u03c0 14 Find a particular solution of the differential equation (x + 1) dy\ndx  = 2 e\u2013y \u2013 1,  given\nthat y = 0 when x = 0 15"
    },
    {
        "Chapter": "1",
        "sentence_range": "4971-4974",
        "Text": "14 Find a particular solution of the differential equation (x + 1) dy\ndx  = 2 e\u2013y \u2013 1,  given\nthat y = 0 when x = 0 15 The population of a village increases continuously at the rate proportional to the\nnumber of its inhabitants present at any time"
    },
    {
        "Chapter": "1",
        "sentence_range": "4972-4975",
        "Text": "Find a particular solution of the differential equation (x + 1) dy\ndx  = 2 e\u2013y \u2013 1,  given\nthat y = 0 when x = 0 15 The population of a village increases continuously at the rate proportional to the\nnumber of its inhabitants present at any time If the population of the village was\n20, 000 in 1999 and 25000 in the year 2004, what will be the population of the\nvillage in 2009"
    },
    {
        "Chapter": "1",
        "sentence_range": "4973-4976",
        "Text": "15 The population of a village increases continuously at the rate proportional to the\nnumber of its inhabitants present at any time If the population of the village was\n20, 000 in 1999 and 25000 in the year 2004, what will be the population of the\nvillage in 2009 16"
    },
    {
        "Chapter": "1",
        "sentence_range": "4974-4977",
        "Text": "The population of a village increases continuously at the rate proportional to the\nnumber of its inhabitants present at any time If the population of the village was\n20, 000 in 1999 and 25000 in the year 2004, what will be the population of the\nvillage in 2009 16 The general solution of the differential equation \n0\ny dx\n\u2212yx dy\n=\n is\n(A) xy = C\n(B) x = Cy2\n(C) y = Cx\n(D) y = Cx2\n17"
    },
    {
        "Chapter": "1",
        "sentence_range": "4975-4978",
        "Text": "If the population of the village was\n20, 000 in 1999 and 25000 in the year 2004, what will be the population of the\nvillage in 2009 16 The general solution of the differential equation \n0\ny dx\n\u2212yx dy\n=\n is\n(A) xy = C\n(B) x = Cy2\n(C) y = Cx\n(D) y = Cx2\n17 The general solution of a differential equation of the type \n1\n1\nP\nQ\ndx\nx\ndy +\n=\n is\n(A)\n(\n)\n1\n1\nP\nP\nQ1\nC\ndy\ndy\ny e\ne\ndy\n\u222b\n\u222b\n=\n+\n\u222b\n(B)\n(\n)\n1\n1\nP\nP\n1"
    },
    {
        "Chapter": "1",
        "sentence_range": "4976-4979",
        "Text": "16 The general solution of the differential equation \n0\ny dx\n\u2212yx dy\n=\n is\n(A) xy = C\n(B) x = Cy2\n(C) y = Cx\n(D) y = Cx2\n17 The general solution of a differential equation of the type \n1\n1\nP\nQ\ndx\nx\ndy +\n=\n is\n(A)\n(\n)\n1\n1\nP\nP\nQ1\nC\ndy\ndy\ny e\ne\ndy\n\u222b\n\u222b\n=\n+\n\u222b\n(B)\n(\n)\n1\n1\nP\nP\n1 Q\nC\ndx\ndx\ny e\ne\ndx\n\u222b\n\u222b\n=\n+\n\u222b\n(C)\n(\n)\n1\n1\nP\nP\nQ1\nC\ndy\ndy\nx e\ne\ndy\n\u222b\n\u222b\n=\n+\n\u222b\n(D)\n(\n)\n1\n1\nP\nP\nQ1\nC\ndx\ndx\nx e\ne\ndx\n\u222b\n\u222b\n=\n+\n\u222b\n18"
    },
    {
        "Chapter": "1",
        "sentence_range": "4977-4980",
        "Text": "The general solution of the differential equation \n0\ny dx\n\u2212yx dy\n=\n is\n(A) xy = C\n(B) x = Cy2\n(C) y = Cx\n(D) y = Cx2\n17 The general solution of a differential equation of the type \n1\n1\nP\nQ\ndx\nx\ndy +\n=\n is\n(A)\n(\n)\n1\n1\nP\nP\nQ1\nC\ndy\ndy\ny e\ne\ndy\n\u222b\n\u222b\n=\n+\n\u222b\n(B)\n(\n)\n1\n1\nP\nP\n1 Q\nC\ndx\ndx\ny e\ne\ndx\n\u222b\n\u222b\n=\n+\n\u222b\n(C)\n(\n)\n1\n1\nP\nP\nQ1\nC\ndy\ndy\nx e\ne\ndy\n\u222b\n\u222b\n=\n+\n\u222b\n(D)\n(\n)\n1\n1\nP\nP\nQ1\nC\ndx\ndx\nx e\ne\ndx\n\u222b\n\u222b\n=\n+\n\u222b\n18 The general solution of the differential equation ex dy + (y ex + 2x) dx = 0 is\n(A) x ey + x2 = C\n(B) x ey + y2 = C\n(C) y ex + x2 = C\n(D) y ey + x2 = C\n\u00a9 NCERT\nnot to be republished\nMATHEMATICS\n422\nSummary\n\ufffd An equation involving derivatives of the dependent variable with respect to\nindependent variable (variables) is known as a differential equation"
    },
    {
        "Chapter": "1",
        "sentence_range": "4978-4981",
        "Text": "The general solution of a differential equation of the type \n1\n1\nP\nQ\ndx\nx\ndy +\n=\n is\n(A)\n(\n)\n1\n1\nP\nP\nQ1\nC\ndy\ndy\ny e\ne\ndy\n\u222b\n\u222b\n=\n+\n\u222b\n(B)\n(\n)\n1\n1\nP\nP\n1 Q\nC\ndx\ndx\ny e\ne\ndx\n\u222b\n\u222b\n=\n+\n\u222b\n(C)\n(\n)\n1\n1\nP\nP\nQ1\nC\ndy\ndy\nx e\ne\ndy\n\u222b\n\u222b\n=\n+\n\u222b\n(D)\n(\n)\n1\n1\nP\nP\nQ1\nC\ndx\ndx\nx e\ne\ndx\n\u222b\n\u222b\n=\n+\n\u222b\n18 The general solution of the differential equation ex dy + (y ex + 2x) dx = 0 is\n(A) x ey + x2 = C\n(B) x ey + y2 = C\n(C) y ex + x2 = C\n(D) y ey + x2 = C\n\u00a9 NCERT\nnot to be republished\nMATHEMATICS\n422\nSummary\n\ufffd An equation involving derivatives of the dependent variable with respect to\nindependent variable (variables) is known as a differential equation \ufffd Order of a differential equation is the order of the highest order derivative\noccurring in the differential equation"
    },
    {
        "Chapter": "1",
        "sentence_range": "4979-4982",
        "Text": "Q\nC\ndx\ndx\ny e\ne\ndx\n\u222b\n\u222b\n=\n+\n\u222b\n(C)\n(\n)\n1\n1\nP\nP\nQ1\nC\ndy\ndy\nx e\ne\ndy\n\u222b\n\u222b\n=\n+\n\u222b\n(D)\n(\n)\n1\n1\nP\nP\nQ1\nC\ndx\ndx\nx e\ne\ndx\n\u222b\n\u222b\n=\n+\n\u222b\n18 The general solution of the differential equation ex dy + (y ex + 2x) dx = 0 is\n(A) x ey + x2 = C\n(B) x ey + y2 = C\n(C) y ex + x2 = C\n(D) y ey + x2 = C\n\u00a9 NCERT\nnot to be republished\nMATHEMATICS\n422\nSummary\n\ufffd An equation involving derivatives of the dependent variable with respect to\nindependent variable (variables) is known as a differential equation \ufffd Order of a differential equation is the order of the highest order derivative\noccurring in the differential equation \ufffd Degree of a differential equation is defined if it is a polynomial equation in its\nderivatives"
    },
    {
        "Chapter": "1",
        "sentence_range": "4980-4983",
        "Text": "The general solution of the differential equation ex dy + (y ex + 2x) dx = 0 is\n(A) x ey + x2 = C\n(B) x ey + y2 = C\n(C) y ex + x2 = C\n(D) y ey + x2 = C\n\u00a9 NCERT\nnot to be republished\nMATHEMATICS\n422\nSummary\n\ufffd An equation involving derivatives of the dependent variable with respect to\nindependent variable (variables) is known as a differential equation \ufffd Order of a differential equation is the order of the highest order derivative\noccurring in the differential equation \ufffd Degree of a differential equation is defined if it is a polynomial equation in its\nderivatives \ufffd Degree (when defined) of a differential equation is the highest power (positive\ninteger only) of the highest order derivative in it"
    },
    {
        "Chapter": "1",
        "sentence_range": "4981-4984",
        "Text": "\ufffd Order of a differential equation is the order of the highest order derivative\noccurring in the differential equation \ufffd Degree of a differential equation is defined if it is a polynomial equation in its\nderivatives \ufffd Degree (when defined) of a differential equation is the highest power (positive\ninteger only) of the highest order derivative in it \ufffd A function which satisfies the given differential equation is called its solution"
    },
    {
        "Chapter": "1",
        "sentence_range": "4982-4985",
        "Text": "\ufffd Degree of a differential equation is defined if it is a polynomial equation in its\nderivatives \ufffd Degree (when defined) of a differential equation is the highest power (positive\ninteger only) of the highest order derivative in it \ufffd A function which satisfies the given differential equation is called its solution The solution which contains as many arbitrary constants as the order of the\ndifferential equation is called a general solution and the solution free from\narbitrary constants is called particular solution"
    },
    {
        "Chapter": "1",
        "sentence_range": "4983-4986",
        "Text": "\ufffd Degree (when defined) of a differential equation is the highest power (positive\ninteger only) of the highest order derivative in it \ufffd A function which satisfies the given differential equation is called its solution The solution which contains as many arbitrary constants as the order of the\ndifferential equation is called a general solution and the solution free from\narbitrary constants is called particular solution \ufffd To form a differential equation from a given function we differentiate the\nfunction successively as many times as the number of arbitrary constants in\nthe given function and then eliminate the arbitrary constants"
    },
    {
        "Chapter": "1",
        "sentence_range": "4984-4987",
        "Text": "\ufffd A function which satisfies the given differential equation is called its solution The solution which contains as many arbitrary constants as the order of the\ndifferential equation is called a general solution and the solution free from\narbitrary constants is called particular solution \ufffd To form a differential equation from a given function we differentiate the\nfunction successively as many times as the number of arbitrary constants in\nthe given function and then eliminate the arbitrary constants \ufffd Variable separable method is used to solve such an equation in which variables\ncan be separated completely i"
    },
    {
        "Chapter": "1",
        "sentence_range": "4985-4988",
        "Text": "The solution which contains as many arbitrary constants as the order of the\ndifferential equation is called a general solution and the solution free from\narbitrary constants is called particular solution \ufffd To form a differential equation from a given function we differentiate the\nfunction successively as many times as the number of arbitrary constants in\nthe given function and then eliminate the arbitrary constants \ufffd Variable separable method is used to solve such an equation in which variables\ncan be separated completely i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "4986-4989",
        "Text": "\ufffd To form a differential equation from a given function we differentiate the\nfunction successively as many times as the number of arbitrary constants in\nthe given function and then eliminate the arbitrary constants \ufffd Variable separable method is used to solve such an equation in which variables\ncan be separated completely i e terms containing y should remain with dy\nand terms containing x should remain with dx"
    },
    {
        "Chapter": "1",
        "sentence_range": "4987-4990",
        "Text": "\ufffd Variable separable method is used to solve such an equation in which variables\ncan be separated completely i e terms containing y should remain with dy\nand terms containing x should remain with dx \ufffd A differential equation which can be expressed in the form\n( , ) or\n( , )\ndy\ndx\nf x y\ng x y\ndx\ndy\n \n \nwhere, f (x, y) and g(x, y) are homogenous\nfunctions of degree zero is called a homogeneous differential equation"
    },
    {
        "Chapter": "1",
        "sentence_range": "4988-4991",
        "Text": "e terms containing y should remain with dy\nand terms containing x should remain with dx \ufffd A differential equation which can be expressed in the form\n( , ) or\n( , )\ndy\ndx\nf x y\ng x y\ndx\ndy\n \n \nwhere, f (x, y) and g(x, y) are homogenous\nfunctions of degree zero is called a homogeneous differential equation \ufffd A differential equation of the form \n+P\nQ\ndy\ny\ndx\n, where P and Q are  constants\nor functions of x only  is called a first order linear differential equation"
    },
    {
        "Chapter": "1",
        "sentence_range": "4989-4992",
        "Text": "terms containing y should remain with dy\nand terms containing x should remain with dx \ufffd A differential equation which can be expressed in the form\n( , ) or\n( , )\ndy\ndx\nf x y\ng x y\ndx\ndy\n \n \nwhere, f (x, y) and g(x, y) are homogenous\nfunctions of degree zero is called a homogeneous differential equation \ufffd A differential equation of the form \n+P\nQ\ndy\ny\ndx\n, where P and Q are  constants\nor functions of x only  is called a first order linear differential equation Historical Note\nOne of the principal languages of Science is that of differential equations"
    },
    {
        "Chapter": "1",
        "sentence_range": "4990-4993",
        "Text": "\ufffd A differential equation which can be expressed in the form\n( , ) or\n( , )\ndy\ndx\nf x y\ng x y\ndx\ndy\n \n \nwhere, f (x, y) and g(x, y) are homogenous\nfunctions of degree zero is called a homogeneous differential equation \ufffd A differential equation of the form \n+P\nQ\ndy\ny\ndx\n, where P and Q are  constants\nor functions of x only  is called a first order linear differential equation Historical Note\nOne of the principal languages of Science is that of differential equations Interestingly, the date of birth of differential equations is taken to be November,\n11,1675, when Gottfried Wilthelm Freiherr Leibnitz (1646 - 1716) first put in black\nand white the identity \n212\ny dy\ny\n=\n\u222b\n, thereby introducing both the symbols \u222b and dy"
    },
    {
        "Chapter": "1",
        "sentence_range": "4991-4994",
        "Text": "\ufffd A differential equation of the form \n+P\nQ\ndy\ny\ndx\n, where P and Q are  constants\nor functions of x only  is called a first order linear differential equation Historical Note\nOne of the principal languages of Science is that of differential equations Interestingly, the date of birth of differential equations is taken to be November,\n11,1675, when Gottfried Wilthelm Freiherr Leibnitz (1646 - 1716) first put in black\nand white the identity \n212\ny dy\ny\n=\n\u222b\n, thereby introducing both the symbols \u222b and dy \u00a9 NCERT\nnot to be republished\nDIFFERENTIAL EQUATIONS\n423\nLeibnitz was actually interested in the problem of finding a curve whose tangents\nwere prescribed"
    },
    {
        "Chapter": "1",
        "sentence_range": "4992-4995",
        "Text": "Historical Note\nOne of the principal languages of Science is that of differential equations Interestingly, the date of birth of differential equations is taken to be November,\n11,1675, when Gottfried Wilthelm Freiherr Leibnitz (1646 - 1716) first put in black\nand white the identity \n212\ny dy\ny\n=\n\u222b\n, thereby introducing both the symbols \u222b and dy \u00a9 NCERT\nnot to be republished\nDIFFERENTIAL EQUATIONS\n423\nLeibnitz was actually interested in the problem of finding a curve whose tangents\nwere prescribed This led him to discover the \u2018method of separation of variables\u2019\n1691"
    },
    {
        "Chapter": "1",
        "sentence_range": "4993-4996",
        "Text": "Interestingly, the date of birth of differential equations is taken to be November,\n11,1675, when Gottfried Wilthelm Freiherr Leibnitz (1646 - 1716) first put in black\nand white the identity \n212\ny dy\ny\n=\n\u222b\n, thereby introducing both the symbols \u222b and dy \u00a9 NCERT\nnot to be republished\nDIFFERENTIAL EQUATIONS\n423\nLeibnitz was actually interested in the problem of finding a curve whose tangents\nwere prescribed This led him to discover the \u2018method of separation of variables\u2019\n1691 A year later he formulated the \u2018method of solving the homogeneous\ndifferential equations of the first order\u2019"
    },
    {
        "Chapter": "1",
        "sentence_range": "4994-4997",
        "Text": "\u00a9 NCERT\nnot to be republished\nDIFFERENTIAL EQUATIONS\n423\nLeibnitz was actually interested in the problem of finding a curve whose tangents\nwere prescribed This led him to discover the \u2018method of separation of variables\u2019\n1691 A year later he formulated the \u2018method of solving the homogeneous\ndifferential equations of the first order\u2019 He went further in a very short time\nto the discovery of the \u2018method of solving a linear differential equation of the\nfirst-order\u2019"
    },
    {
        "Chapter": "1",
        "sentence_range": "4995-4998",
        "Text": "This led him to discover the \u2018method of separation of variables\u2019\n1691 A year later he formulated the \u2018method of solving the homogeneous\ndifferential equations of the first order\u2019 He went further in a very short time\nto the discovery of the \u2018method of solving a linear differential equation of the\nfirst-order\u2019 How surprising is it that all these methods came from a single man\nand that too within 25 years of the birth of differential equations"
    },
    {
        "Chapter": "1",
        "sentence_range": "4996-4999",
        "Text": "A year later he formulated the \u2018method of solving the homogeneous\ndifferential equations of the first order\u2019 He went further in a very short time\nto the discovery of the \u2018method of solving a linear differential equation of the\nfirst-order\u2019 How surprising is it that all these methods came from a single man\nand that too within 25 years of the birth of differential equations In the old days, what we now call the \u2018solution\u2019 of a differential equation,\nwas used to be referred to as \u2018integral\u2019 of the differential equation, the word\nbeing coined by James Bernoulli (1654 - 1705) in 1690"
    },
    {
        "Chapter": "1",
        "sentence_range": "4997-5000",
        "Text": "He went further in a very short time\nto the discovery of the \u2018method of solving a linear differential equation of the\nfirst-order\u2019 How surprising is it that all these methods came from a single man\nand that too within 25 years of the birth of differential equations In the old days, what we now call the \u2018solution\u2019 of a differential equation,\nwas used to be referred to as \u2018integral\u2019 of the differential equation, the word\nbeing coined by James Bernoulli (1654 - 1705) in 1690 The word \u2018solution was\nfirst used by Joseph Louis Lagrange (1736 - 1813) in 1774, which was almost\nhundred years since the birth of differential equations"
    },
    {
        "Chapter": "1",
        "sentence_range": "4998-5001",
        "Text": "How surprising is it that all these methods came from a single man\nand that too within 25 years of the birth of differential equations In the old days, what we now call the \u2018solution\u2019 of a differential equation,\nwas used to be referred to as \u2018integral\u2019 of the differential equation, the word\nbeing coined by James Bernoulli (1654 - 1705) in 1690 The word \u2018solution was\nfirst used by Joseph Louis Lagrange (1736 - 1813) in 1774, which was almost\nhundred years since the birth of differential equations It was Jules Henri Poincare\n(1854 - 1912) who strongly advocated the use of the word \u2018solution\u2019 and thus the\nword \u2018solution\u2019 has found its deserved place in modern terminology"
    },
    {
        "Chapter": "1",
        "sentence_range": "4999-5002",
        "Text": "In the old days, what we now call the \u2018solution\u2019 of a differential equation,\nwas used to be referred to as \u2018integral\u2019 of the differential equation, the word\nbeing coined by James Bernoulli (1654 - 1705) in 1690 The word \u2018solution was\nfirst used by Joseph Louis Lagrange (1736 - 1813) in 1774, which was almost\nhundred years since the birth of differential equations It was Jules Henri Poincare\n(1854 - 1912) who strongly advocated the use of the word \u2018solution\u2019 and thus the\nword \u2018solution\u2019 has found its deserved place in modern terminology The name of\nthe \u2018method of separation of variables\u2019 is due to John Bernoulli (1667 - 1748),\na younger brother of James Bernoulli"
    },
    {
        "Chapter": "1",
        "sentence_range": "5000-5003",
        "Text": "The word \u2018solution was\nfirst used by Joseph Louis Lagrange (1736 - 1813) in 1774, which was almost\nhundred years since the birth of differential equations It was Jules Henri Poincare\n(1854 - 1912) who strongly advocated the use of the word \u2018solution\u2019 and thus the\nword \u2018solution\u2019 has found its deserved place in modern terminology The name of\nthe \u2018method of separation of variables\u2019 is due to John Bernoulli (1667 - 1748),\na younger brother of James Bernoulli Application to geometric problems were also considered"
    },
    {
        "Chapter": "1",
        "sentence_range": "5001-5004",
        "Text": "It was Jules Henri Poincare\n(1854 - 1912) who strongly advocated the use of the word \u2018solution\u2019 and thus the\nword \u2018solution\u2019 has found its deserved place in modern terminology The name of\nthe \u2018method of separation of variables\u2019 is due to John Bernoulli (1667 - 1748),\na younger brother of James Bernoulli Application to geometric problems were also considered It was again John\nBernoulli who first brought into light the intricate nature of differential equations"
    },
    {
        "Chapter": "1",
        "sentence_range": "5002-5005",
        "Text": "The name of\nthe \u2018method of separation of variables\u2019 is due to John Bernoulli (1667 - 1748),\na younger brother of James Bernoulli Application to geometric problems were also considered It was again John\nBernoulli who first brought into light the intricate nature of differential equations In a letter to Leibnitz, dated May 20, 1715, he revealed the solutions of the\ndifferential equation\nx2 y\u2033 = 2y,\nwhich led to three types of curves, viz"
    },
    {
        "Chapter": "1",
        "sentence_range": "5003-5006",
        "Text": "Application to geometric problems were also considered It was again John\nBernoulli who first brought into light the intricate nature of differential equations In a letter to Leibnitz, dated May 20, 1715, he revealed the solutions of the\ndifferential equation\nx2 y\u2033 = 2y,\nwhich led to three types of curves, viz , parabolas, hyperbolas and a class of\ncubic curves"
    },
    {
        "Chapter": "1",
        "sentence_range": "5004-5007",
        "Text": "It was again John\nBernoulli who first brought into light the intricate nature of differential equations In a letter to Leibnitz, dated May 20, 1715, he revealed the solutions of the\ndifferential equation\nx2 y\u2033 = 2y,\nwhich led to three types of curves, viz , parabolas, hyperbolas and a class of\ncubic curves This shows how varied the solutions of such innocent looking\ndifferential equation can be"
    },
    {
        "Chapter": "1",
        "sentence_range": "5005-5008",
        "Text": "In a letter to Leibnitz, dated May 20, 1715, he revealed the solutions of the\ndifferential equation\nx2 y\u2033 = 2y,\nwhich led to three types of curves, viz , parabolas, hyperbolas and a class of\ncubic curves This shows how varied the solutions of such innocent looking\ndifferential equation can be From the second half of the twentieth century attention\nhas been drawn to the investigation of this complicated nature of the solutions of\ndifferential equations, under the heading \u2018qualitative analysis of differential\nequations\u2019"
    },
    {
        "Chapter": "1",
        "sentence_range": "5006-5009",
        "Text": ", parabolas, hyperbolas and a class of\ncubic curves This shows how varied the solutions of such innocent looking\ndifferential equation can be From the second half of the twentieth century attention\nhas been drawn to the investigation of this complicated nature of the solutions of\ndifferential equations, under the heading \u2018qualitative analysis of differential\nequations\u2019 Now-a-days, this has acquired prime importance being absolutely\nnecessary in almost all investigations"
    },
    {
        "Chapter": "1",
        "sentence_range": "5007-5010",
        "Text": "This shows how varied the solutions of such innocent looking\ndifferential equation can be From the second half of the twentieth century attention\nhas been drawn to the investigation of this complicated nature of the solutions of\ndifferential equations, under the heading \u2018qualitative analysis of differential\nequations\u2019 Now-a-days, this has acquired prime importance being absolutely\nnecessary in almost all investigations \u2014\ufffd\n\ufffd\n\ufffd\n\ufffd\n\ufffd\u2014\n\u00a9 NCERT\nnot to be republished\n MATHEMATICS\n424\n\ufffdIn most sciences one generation tears down what another has built and what\none has established another undoes"
    },
    {
        "Chapter": "1",
        "sentence_range": "5008-5011",
        "Text": "From the second half of the twentieth century attention\nhas been drawn to the investigation of this complicated nature of the solutions of\ndifferential equations, under the heading \u2018qualitative analysis of differential\nequations\u2019 Now-a-days, this has acquired prime importance being absolutely\nnecessary in almost all investigations \u2014\ufffd\n\ufffd\n\ufffd\n\ufffd\n\ufffd\u2014\n\u00a9 NCERT\nnot to be republished\n MATHEMATICS\n424\n\ufffdIn most sciences one generation tears down what another has built and what\none has established another undoes In Mathematics alone each generation\nbuilds a new story to the old structure"
    },
    {
        "Chapter": "1",
        "sentence_range": "5009-5012",
        "Text": "Now-a-days, this has acquired prime importance being absolutely\nnecessary in almost all investigations \u2014\ufffd\n\ufffd\n\ufffd\n\ufffd\n\ufffd\u2014\n\u00a9 NCERT\nnot to be republished\n MATHEMATICS\n424\n\ufffdIn most sciences one generation tears down what another has built and what\none has established another undoes In Mathematics alone each generation\nbuilds a new story to the old structure \u2013 HERMAN HANKEL \ufffd\n10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5010-5013",
        "Text": "\u2014\ufffd\n\ufffd\n\ufffd\n\ufffd\n\ufffd\u2014\n\u00a9 NCERT\nnot to be republished\n MATHEMATICS\n424\n\ufffdIn most sciences one generation tears down what another has built and what\none has established another undoes In Mathematics alone each generation\nbuilds a new story to the old structure \u2013 HERMAN HANKEL \ufffd\n10 1  Introduction\nIn our day to day life, we come across many queries such\nas \u2013 What is your height"
    },
    {
        "Chapter": "1",
        "sentence_range": "5011-5014",
        "Text": "In Mathematics alone each generation\nbuilds a new story to the old structure \u2013 HERMAN HANKEL \ufffd\n10 1  Introduction\nIn our day to day life, we come across many queries such\nas \u2013 What is your height How should a football player hit\nthe ball to give a pass to another player of his team"
    },
    {
        "Chapter": "1",
        "sentence_range": "5012-5015",
        "Text": "\u2013 HERMAN HANKEL \ufffd\n10 1  Introduction\nIn our day to day life, we come across many queries such\nas \u2013 What is your height How should a football player hit\nthe ball to give a pass to another player of his team Observe\nthat a possible answer to the first query may be 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "5013-5016",
        "Text": "1  Introduction\nIn our day to day life, we come across many queries such\nas \u2013 What is your height How should a football player hit\nthe ball to give a pass to another player of his team Observe\nthat a possible answer to the first query may be 1 6 meters,\na quantity that involves only one value (magnitude) which\nis a real number"
    },
    {
        "Chapter": "1",
        "sentence_range": "5014-5017",
        "Text": "How should a football player hit\nthe ball to give a pass to another player of his team Observe\nthat a possible answer to the first query may be 1 6 meters,\na quantity that involves only one value (magnitude) which\nis a real number Such quantities are called scalars"
    },
    {
        "Chapter": "1",
        "sentence_range": "5015-5018",
        "Text": "Observe\nthat a possible answer to the first query may be 1 6 meters,\na quantity that involves only one value (magnitude) which\nis a real number Such quantities are called scalars However, an answer to the second query is a quantity (called\nforce) which involves muscular strength (magnitude) and\ndirection (in which another player is positioned)"
    },
    {
        "Chapter": "1",
        "sentence_range": "5016-5019",
        "Text": "6 meters,\na quantity that involves only one value (magnitude) which\nis a real number Such quantities are called scalars However, an answer to the second query is a quantity (called\nforce) which involves muscular strength (magnitude) and\ndirection (in which another player is positioned) Such\nquantities are called vectors"
    },
    {
        "Chapter": "1",
        "sentence_range": "5017-5020",
        "Text": "Such quantities are called scalars However, an answer to the second query is a quantity (called\nforce) which involves muscular strength (magnitude) and\ndirection (in which another player is positioned) Such\nquantities are called vectors In mathematics, physics and\nengineering, we frequently come across with both types of\nquantities, namely, scalar quantities such as length, mass,\ntime, distance, speed, area, volume, temperature, work,\nmoney, voltage, density, resistance etc"
    },
    {
        "Chapter": "1",
        "sentence_range": "5018-5021",
        "Text": "However, an answer to the second query is a quantity (called\nforce) which involves muscular strength (magnitude) and\ndirection (in which another player is positioned) Such\nquantities are called vectors In mathematics, physics and\nengineering, we frequently come across with both types of\nquantities, namely, scalar quantities such as length, mass,\ntime, distance, speed, area, volume, temperature, work,\nmoney, voltage, density, resistance etc and vector quantities like displacement, velocity,\nacceleration,  force, weight, momentum, electric field intensity etc"
    },
    {
        "Chapter": "1",
        "sentence_range": "5019-5022",
        "Text": "Such\nquantities are called vectors In mathematics, physics and\nengineering, we frequently come across with both types of\nquantities, namely, scalar quantities such as length, mass,\ntime, distance, speed, area, volume, temperature, work,\nmoney, voltage, density, resistance etc and vector quantities like displacement, velocity,\nacceleration,  force, weight, momentum, electric field intensity etc In this chapter, we will study some of the basic concepts about vectors, various\noperations on vectors, and their algebraic and geometric properties"
    },
    {
        "Chapter": "1",
        "sentence_range": "5020-5023",
        "Text": "In mathematics, physics and\nengineering, we frequently come across with both types of\nquantities, namely, scalar quantities such as length, mass,\ntime, distance, speed, area, volume, temperature, work,\nmoney, voltage, density, resistance etc and vector quantities like displacement, velocity,\nacceleration,  force, weight, momentum, electric field intensity etc In this chapter, we will study some of the basic concepts about vectors, various\noperations on vectors, and their algebraic and geometric properties These two type of\nproperties, when considered together give a full realisation to the concept of vectors,\nand lead to their vital applicability in various areas as mentioned above"
    },
    {
        "Chapter": "1",
        "sentence_range": "5021-5024",
        "Text": "and vector quantities like displacement, velocity,\nacceleration,  force, weight, momentum, electric field intensity etc In this chapter, we will study some of the basic concepts about vectors, various\noperations on vectors, and their algebraic and geometric properties These two type of\nproperties, when considered together give a full realisation to the concept of vectors,\nand lead to their vital applicability in various areas as mentioned above 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5022-5025",
        "Text": "In this chapter, we will study some of the basic concepts about vectors, various\noperations on vectors, and their algebraic and geometric properties These two type of\nproperties, when considered together give a full realisation to the concept of vectors,\nand lead to their vital applicability in various areas as mentioned above 10 2  Some Basic Concepts\nLet \u2018l\u2019 be any straight line in plane or three dimensional space"
    },
    {
        "Chapter": "1",
        "sentence_range": "5023-5026",
        "Text": "These two type of\nproperties, when considered together give a full realisation to the concept of vectors,\nand lead to their vital applicability in various areas as mentioned above 10 2  Some Basic Concepts\nLet \u2018l\u2019 be any straight line in plane or three dimensional space This line can be given\ntwo directions by means of arrowheads"
    },
    {
        "Chapter": "1",
        "sentence_range": "5024-5027",
        "Text": "10 2  Some Basic Concepts\nLet \u2018l\u2019 be any straight line in plane or three dimensional space This line can be given\ntwo directions by means of arrowheads A line with one of these directions prescribed\nis called a directed line (Fig 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5025-5028",
        "Text": "2  Some Basic Concepts\nLet \u2018l\u2019 be any straight line in plane or three dimensional space This line can be given\ntwo directions by means of arrowheads A line with one of these directions prescribed\nis called a directed line (Fig 10 1 (i), (ii))"
    },
    {
        "Chapter": "1",
        "sentence_range": "5026-5029",
        "Text": "This line can be given\ntwo directions by means of arrowheads A line with one of these directions prescribed\nis called a directed line (Fig 10 1 (i), (ii)) Chapter 10\nVECTOR ALGEBRA\nW"
    },
    {
        "Chapter": "1",
        "sentence_range": "5027-5030",
        "Text": "A line with one of these directions prescribed\nis called a directed line (Fig 10 1 (i), (ii)) Chapter 10\nVECTOR ALGEBRA\nW R"
    },
    {
        "Chapter": "1",
        "sentence_range": "5028-5031",
        "Text": "1 (i), (ii)) Chapter 10\nVECTOR ALGEBRA\nW R Hamilton\n(1805-1865)\n\u00a9 NCERT\nnot to be republished\nVECTOR ALGEBRA\n425\nNow observe that if we restrict the line l to the line segment AB, then a magnitude\nis prescribed on the line l with one of the two directions, so that we obtain a directed\nline segment (Fig 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5029-5032",
        "Text": "Chapter 10\nVECTOR ALGEBRA\nW R Hamilton\n(1805-1865)\n\u00a9 NCERT\nnot to be republished\nVECTOR ALGEBRA\n425\nNow observe that if we restrict the line l to the line segment AB, then a magnitude\nis prescribed on the line l with one of the two directions, so that we obtain a directed\nline segment (Fig 10 1(iii))"
    },
    {
        "Chapter": "1",
        "sentence_range": "5030-5033",
        "Text": "R Hamilton\n(1805-1865)\n\u00a9 NCERT\nnot to be republished\nVECTOR ALGEBRA\n425\nNow observe that if we restrict the line l to the line segment AB, then a magnitude\nis prescribed on the line l with one of the two directions, so that we obtain a directed\nline segment (Fig 10 1(iii)) Thus, a directed line segment has magnitude as well as\ndirection"
    },
    {
        "Chapter": "1",
        "sentence_range": "5031-5034",
        "Text": "Hamilton\n(1805-1865)\n\u00a9 NCERT\nnot to be republished\nVECTOR ALGEBRA\n425\nNow observe that if we restrict the line l to the line segment AB, then a magnitude\nis prescribed on the line l with one of the two directions, so that we obtain a directed\nline segment (Fig 10 1(iii)) Thus, a directed line segment has magnitude as well as\ndirection Definition 1 A quantity that has magnitude as well as direction is called a vector"
    },
    {
        "Chapter": "1",
        "sentence_range": "5032-5035",
        "Text": "1(iii)) Thus, a directed line segment has magnitude as well as\ndirection Definition 1 A quantity that has magnitude as well as direction is called a vector Notice that a directed line segment is a vector (Fig 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5033-5036",
        "Text": "Thus, a directed line segment has magnitude as well as\ndirection Definition 1 A quantity that has magnitude as well as direction is called a vector Notice that a directed line segment is a vector (Fig 10 1(iii)), denoted as AB\nuuur\n or\nsimply as ar , and read as \u2018vector AB\nuuur\n\u2019 or \u2018vector ar \u2019"
    },
    {
        "Chapter": "1",
        "sentence_range": "5034-5037",
        "Text": "Definition 1 A quantity that has magnitude as well as direction is called a vector Notice that a directed line segment is a vector (Fig 10 1(iii)), denoted as AB\nuuur\n or\nsimply as ar , and read as \u2018vector AB\nuuur\n\u2019 or \u2018vector ar \u2019 The point A from where the vector AB\nuuur\n starts is called its initial point, and the\npoint B where it ends is called its terminal point"
    },
    {
        "Chapter": "1",
        "sentence_range": "5035-5038",
        "Text": "Notice that a directed line segment is a vector (Fig 10 1(iii)), denoted as AB\nuuur\n or\nsimply as ar , and read as \u2018vector AB\nuuur\n\u2019 or \u2018vector ar \u2019 The point A from where the vector AB\nuuur\n starts is called its initial point, and the\npoint B where it ends is called its terminal point The distance between initial and\nterminal points of a vector is called the magnitude (or length) of the vector, denoted as\n| AB\nuuur\n|, or | ar |, or a"
    },
    {
        "Chapter": "1",
        "sentence_range": "5036-5039",
        "Text": "1(iii)), denoted as AB\nuuur\n or\nsimply as ar , and read as \u2018vector AB\nuuur\n\u2019 or \u2018vector ar \u2019 The point A from where the vector AB\nuuur\n starts is called its initial point, and the\npoint B where it ends is called its terminal point The distance between initial and\nterminal points of a vector is called the magnitude (or length) of the vector, denoted as\n| AB\nuuur\n|, or | ar |, or a The arrow indicates the direction of the vector"
    },
    {
        "Chapter": "1",
        "sentence_range": "5037-5040",
        "Text": "The point A from where the vector AB\nuuur\n starts is called its initial point, and the\npoint B where it ends is called its terminal point The distance between initial and\nterminal points of a vector is called the magnitude (or length) of the vector, denoted as\n| AB\nuuur\n|, or | ar |, or a The arrow indicates the direction of the vector \ufffdNote   Since the length is never negative, the notation | ar | < 0 has no meaning"
    },
    {
        "Chapter": "1",
        "sentence_range": "5038-5041",
        "Text": "The distance between initial and\nterminal points of a vector is called the magnitude (or length) of the vector, denoted as\n| AB\nuuur\n|, or | ar |, or a The arrow indicates the direction of the vector \ufffdNote   Since the length is never negative, the notation | ar | < 0 has no meaning Position Vector\nFrom Class XI, recall the three dimensional right handed rectangular coordinate\nsystem (Fig 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5039-5042",
        "Text": "The arrow indicates the direction of the vector \ufffdNote   Since the length is never negative, the notation | ar | < 0 has no meaning Position Vector\nFrom Class XI, recall the three dimensional right handed rectangular coordinate\nsystem (Fig 10 2(i))"
    },
    {
        "Chapter": "1",
        "sentence_range": "5040-5043",
        "Text": "\ufffdNote   Since the length is never negative, the notation | ar | < 0 has no meaning Position Vector\nFrom Class XI, recall the three dimensional right handed rectangular coordinate\nsystem (Fig 10 2(i)) Consider a point P in space, having coordinates (x, y, z) with\nrespect to the origin O(0, 0, 0)"
    },
    {
        "Chapter": "1",
        "sentence_range": "5041-5044",
        "Text": "Position Vector\nFrom Class XI, recall the three dimensional right handed rectangular coordinate\nsystem (Fig 10 2(i)) Consider a point P in space, having coordinates (x, y, z) with\nrespect to the origin O(0, 0, 0) Then, the vector OP\nuuur  having O and P as its initial and\nterminal points, respectively, is called the position vector of the point P with respect\nto O"
    },
    {
        "Chapter": "1",
        "sentence_range": "5042-5045",
        "Text": "2(i)) Consider a point P in space, having coordinates (x, y, z) with\nrespect to the origin O(0, 0, 0) Then, the vector OP\nuuur  having O and P as its initial and\nterminal points, respectively, is called the position vector of the point P with respect\nto O Using distance formula (from Class XI), the magnitude of OP\nuuur  (or rr ) is given by\n| OP |\nuuur  =\n2\n2\n2\nx\ny\nz\n+\n+\nIn practice, the position vectors of points A, B, C, etc"
    },
    {
        "Chapter": "1",
        "sentence_range": "5043-5046",
        "Text": "Consider a point P in space, having coordinates (x, y, z) with\nrespect to the origin O(0, 0, 0) Then, the vector OP\nuuur  having O and P as its initial and\nterminal points, respectively, is called the position vector of the point P with respect\nto O Using distance formula (from Class XI), the magnitude of OP\nuuur  (or rr ) is given by\n| OP |\nuuur  =\n2\n2\n2\nx\ny\nz\n+\n+\nIn practice, the position vectors of points A, B, C, etc , with respect to the origin O\nare denoted by ar , ,\nb c\nr r , etc"
    },
    {
        "Chapter": "1",
        "sentence_range": "5044-5047",
        "Text": "Then, the vector OP\nuuur  having O and P as its initial and\nterminal points, respectively, is called the position vector of the point P with respect\nto O Using distance formula (from Class XI), the magnitude of OP\nuuur  (or rr ) is given by\n| OP |\nuuur  =\n2\n2\n2\nx\ny\nz\n+\n+\nIn practice, the position vectors of points A, B, C, etc , with respect to the origin O\nare denoted by ar , ,\nb c\nr r , etc , respectively (Fig 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5045-5048",
        "Text": "Using distance formula (from Class XI), the magnitude of OP\nuuur  (or rr ) is given by\n| OP |\nuuur  =\n2\n2\n2\nx\ny\nz\n+\n+\nIn practice, the position vectors of points A, B, C, etc , with respect to the origin O\nare denoted by ar , ,\nb c\nr r , etc , respectively (Fig 10 2 (ii))"
    },
    {
        "Chapter": "1",
        "sentence_range": "5046-5049",
        "Text": ", with respect to the origin O\nare denoted by ar , ,\nb c\nr r , etc , respectively (Fig 10 2 (ii)) Fig 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5047-5050",
        "Text": ", respectively (Fig 10 2 (ii)) Fig 10 1\n\u00a9 NCERT\nnot to be republished\n MATHEMATICS\n426\nA\nO\nP\n90\u00b0\nX\nY\nZ\nX\nA\nO\nB\nP(\n)\nx,y,z\nC\nP(\nx,y,z)\nr\nx\ny\nz\nDirection Cosines\nConsider the position vector \nOP or \nuuur\nrr  of a point P(x, y, z) as in Fig 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5048-5051",
        "Text": "2 (ii)) Fig 10 1\n\u00a9 NCERT\nnot to be republished\n MATHEMATICS\n426\nA\nO\nP\n90\u00b0\nX\nY\nZ\nX\nA\nO\nB\nP(\n)\nx,y,z\nC\nP(\nx,y,z)\nr\nx\ny\nz\nDirection Cosines\nConsider the position vector \nOP or \nuuur\nrr  of a point P(x, y, z) as in Fig 10 3"
    },
    {
        "Chapter": "1",
        "sentence_range": "5049-5052",
        "Text": "Fig 10 1\n\u00a9 NCERT\nnot to be republished\n MATHEMATICS\n426\nA\nO\nP\n90\u00b0\nX\nY\nZ\nX\nA\nO\nB\nP(\n)\nx,y,z\nC\nP(\nx,y,z)\nr\nx\ny\nz\nDirection Cosines\nConsider the position vector \nOP or \nuuur\nrr  of a point P(x, y, z) as in Fig 10 3 The angles \u03b1,\n\u03b2, \u03b3 made by the vector rr  with the positive directions of x, y and z-axes respectively,\nare called its direction angles"
    },
    {
        "Chapter": "1",
        "sentence_range": "5050-5053",
        "Text": "1\n\u00a9 NCERT\nnot to be republished\n MATHEMATICS\n426\nA\nO\nP\n90\u00b0\nX\nY\nZ\nX\nA\nO\nB\nP(\n)\nx,y,z\nC\nP(\nx,y,z)\nr\nx\ny\nz\nDirection Cosines\nConsider the position vector \nOP or \nuuur\nrr  of a point P(x, y, z) as in Fig 10 3 The angles \u03b1,\n\u03b2, \u03b3 made by the vector rr  with the positive directions of x, y and z-axes respectively,\nare called its direction angles The cosine values of these angles, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "5051-5054",
        "Text": "3 The angles \u03b1,\n\u03b2, \u03b3 made by the vector rr  with the positive directions of x, y and z-axes respectively,\nare called its direction angles The cosine values of these angles, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "5052-5055",
        "Text": "The angles \u03b1,\n\u03b2, \u03b3 made by the vector rr  with the positive directions of x, y and z-axes respectively,\nare called its direction angles The cosine values of these angles, i e , cos\u03b1, cos\u03b2 and\ncos \u03b3 are called direction cosines of the vector rr , and usually denoted by l, m and n,\nrespectively"
    },
    {
        "Chapter": "1",
        "sentence_range": "5053-5056",
        "Text": "The cosine values of these angles, i e , cos\u03b1, cos\u03b2 and\ncos \u03b3 are called direction cosines of the vector rr , and usually denoted by l, m and n,\nrespectively Fig 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5054-5057",
        "Text": "e , cos\u03b1, cos\u03b2 and\ncos \u03b3 are called direction cosines of the vector rr , and usually denoted by l, m and n,\nrespectively Fig 10 3\nFrom  Fig 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5055-5058",
        "Text": ", cos\u03b1, cos\u03b2 and\ncos \u03b3 are called direction cosines of the vector rr , and usually denoted by l, m and n,\nrespectively Fig 10 3\nFrom  Fig 10 3, one may note that the triangle OAP is right angled, and in it, we\nhave \n(\n)\ncos\n \n stands for | |\nx\nr\nr\n\u03b1 =r\nr"
    },
    {
        "Chapter": "1",
        "sentence_range": "5056-5059",
        "Text": "Fig 10 3\nFrom  Fig 10 3, one may note that the triangle OAP is right angled, and in it, we\nhave \n(\n)\ncos\n \n stands for | |\nx\nr\nr\n\u03b1 =r\nr Similarly, from the right angled triangles OBP and\nOCP, we may write cos\ny and cos\nz\nr\nr\n\u03b2 =\n\u03b3 ="
    },
    {
        "Chapter": "1",
        "sentence_range": "5057-5060",
        "Text": "3\nFrom  Fig 10 3, one may note that the triangle OAP is right angled, and in it, we\nhave \n(\n)\ncos\n \n stands for | |\nx\nr\nr\n\u03b1 =r\nr Similarly, from the right angled triangles OBP and\nOCP, we may write cos\ny and cos\nz\nr\nr\n\u03b2 =\n\u03b3 = Thus, the coordinates of the point P may\nalso be expressed as (lr, mr,nr)"
    },
    {
        "Chapter": "1",
        "sentence_range": "5058-5061",
        "Text": "3, one may note that the triangle OAP is right angled, and in it, we\nhave \n(\n)\ncos\n \n stands for | |\nx\nr\nr\n\u03b1 =r\nr Similarly, from the right angled triangles OBP and\nOCP, we may write cos\ny and cos\nz\nr\nr\n\u03b2 =\n\u03b3 = Thus, the coordinates of the point P may\nalso be expressed as (lr, mr,nr) The numbers lr, mr and nr, proportional to the direction\ncosines are called as direction ratios of vector rr , and denoted as a, b and c, respectively"
    },
    {
        "Chapter": "1",
        "sentence_range": "5059-5062",
        "Text": "Similarly, from the right angled triangles OBP and\nOCP, we may write cos\ny and cos\nz\nr\nr\n\u03b2 =\n\u03b3 = Thus, the coordinates of the point P may\nalso be expressed as (lr, mr,nr) The numbers lr, mr and nr, proportional to the direction\ncosines are called as direction ratios of vector rr , and denoted as a, b and c, respectively Fig 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5060-5063",
        "Text": "Thus, the coordinates of the point P may\nalso be expressed as (lr, mr,nr) The numbers lr, mr and nr, proportional to the direction\ncosines are called as direction ratios of vector rr , and denoted as a, b and c, respectively Fig 10 2\n\u00a9 NCERT\nnot to be republished\nVECTOR ALGEBRA\n427\n\ufffdNote   One may note that l2 + m2 + n2 = 1 but a2 + b2 + c2 \u2260 1, in general"
    },
    {
        "Chapter": "1",
        "sentence_range": "5061-5064",
        "Text": "The numbers lr, mr and nr, proportional to the direction\ncosines are called as direction ratios of vector rr , and denoted as a, b and c, respectively Fig 10 2\n\u00a9 NCERT\nnot to be republished\nVECTOR ALGEBRA\n427\n\ufffdNote   One may note that l2 + m2 + n2 = 1 but a2 + b2 + c2 \u2260 1, in general 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5062-5065",
        "Text": "Fig 10 2\n\u00a9 NCERT\nnot to be republished\nVECTOR ALGEBRA\n427\n\ufffdNote   One may note that l2 + m2 + n2 = 1 but a2 + b2 + c2 \u2260 1, in general 10 3  Types of Vectors\nZero Vector A vector whose initial and terminal points coincide, is called a zero\nvector (or null vector), and denoted as 0\nr"
    },
    {
        "Chapter": "1",
        "sentence_range": "5063-5066",
        "Text": "2\n\u00a9 NCERT\nnot to be republished\nVECTOR ALGEBRA\n427\n\ufffdNote   One may note that l2 + m2 + n2 = 1 but a2 + b2 + c2 \u2260 1, in general 10 3  Types of Vectors\nZero Vector A vector whose initial and terminal points coincide, is called a zero\nvector (or null vector), and denoted as 0\nr Zero vector can not be assigned a definite\ndirection as it has zero magnitude"
    },
    {
        "Chapter": "1",
        "sentence_range": "5064-5067",
        "Text": "10 3  Types of Vectors\nZero Vector A vector whose initial and terminal points coincide, is called a zero\nvector (or null vector), and denoted as 0\nr Zero vector can not be assigned a definite\ndirection as it has zero magnitude Or, alternatively otherwise, it may be regarded as\nhaving any direction"
    },
    {
        "Chapter": "1",
        "sentence_range": "5065-5068",
        "Text": "3  Types of Vectors\nZero Vector A vector whose initial and terminal points coincide, is called a zero\nvector (or null vector), and denoted as 0\nr Zero vector can not be assigned a definite\ndirection as it has zero magnitude Or, alternatively otherwise, it may be regarded as\nhaving any direction The vectors AA, BB\nuuur uuur\n represent the zero vector,\nUnit Vector A vector whose magnitude is unity (i"
    },
    {
        "Chapter": "1",
        "sentence_range": "5066-5069",
        "Text": "Zero vector can not be assigned a definite\ndirection as it has zero magnitude Or, alternatively otherwise, it may be regarded as\nhaving any direction The vectors AA, BB\nuuur uuur\n represent the zero vector,\nUnit Vector A vector whose magnitude is unity (i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "5067-5070",
        "Text": "Or, alternatively otherwise, it may be regarded as\nhaving any direction The vectors AA, BB\nuuur uuur\n represent the zero vector,\nUnit Vector A vector whose magnitude is unity (i e , 1 unit) is called a unit vector"
    },
    {
        "Chapter": "1",
        "sentence_range": "5068-5071",
        "Text": "The vectors AA, BB\nuuur uuur\n represent the zero vector,\nUnit Vector A vector whose magnitude is unity (i e , 1 unit) is called a unit vector The\nunit vector in the direction of a given vector ar  is denoted by \u02c6a"
    },
    {
        "Chapter": "1",
        "sentence_range": "5069-5072",
        "Text": "e , 1 unit) is called a unit vector The\nunit vector in the direction of a given vector ar  is denoted by \u02c6a Coinitial Vectors Two or more vectors having the same initial point are called coinitial\nvectors"
    },
    {
        "Chapter": "1",
        "sentence_range": "5070-5073",
        "Text": ", 1 unit) is called a unit vector The\nunit vector in the direction of a given vector ar  is denoted by \u02c6a Coinitial Vectors Two or more vectors having the same initial point are called coinitial\nvectors Collinear Vectors Two or more vectors are said to be collinear if they are parallel to\nthe same line, irrespective of their magnitudes and directions"
    },
    {
        "Chapter": "1",
        "sentence_range": "5071-5074",
        "Text": "The\nunit vector in the direction of a given vector ar  is denoted by \u02c6a Coinitial Vectors Two or more vectors having the same initial point are called coinitial\nvectors Collinear Vectors Two or more vectors are said to be collinear if they are parallel to\nthe same line, irrespective of their magnitudes and directions Equal Vectors Two vectors \naand \nrb\nr\n are said to be equal, if they have the same\nmagnitude and direction regardless of the positions of their initial points, and written\nas \na =\nrb\nr"
    },
    {
        "Chapter": "1",
        "sentence_range": "5072-5075",
        "Text": "Coinitial Vectors Two or more vectors having the same initial point are called coinitial\nvectors Collinear Vectors Two or more vectors are said to be collinear if they are parallel to\nthe same line, irrespective of their magnitudes and directions Equal Vectors Two vectors \naand \nrb\nr\n are said to be equal, if they have the same\nmagnitude and direction regardless of the positions of their initial points, and written\nas \na =\nrb\nr Negative of a Vector A vector whose magnitude is the same as that of a given vector\n(say, AB\nuuur ), but direction is opposite to that of it, is called negative of the given vector"
    },
    {
        "Chapter": "1",
        "sentence_range": "5073-5076",
        "Text": "Collinear Vectors Two or more vectors are said to be collinear if they are parallel to\nthe same line, irrespective of their magnitudes and directions Equal Vectors Two vectors \naand \nrb\nr\n are said to be equal, if they have the same\nmagnitude and direction regardless of the positions of their initial points, and written\nas \na =\nrb\nr Negative of a Vector A vector whose magnitude is the same as that of a given vector\n(say, AB\nuuur ), but direction is opposite to that of it, is called negative of the given vector For example, vector BA\nuuur  is negative of the vector AB\nuuur , and written as BA\nuuur= \u2212AB\nuuur"
    },
    {
        "Chapter": "1",
        "sentence_range": "5074-5077",
        "Text": "Equal Vectors Two vectors \naand \nrb\nr\n are said to be equal, if they have the same\nmagnitude and direction regardless of the positions of their initial points, and written\nas \na =\nrb\nr Negative of a Vector A vector whose magnitude is the same as that of a given vector\n(say, AB\nuuur ), but direction is opposite to that of it, is called negative of the given vector For example, vector BA\nuuur  is negative of the vector AB\nuuur , and written as BA\nuuur= \u2212AB\nuuur Remark The vectors defined above are such that any of them may be subject to its\nparallel displacement without changing its magnitude and direction"
    },
    {
        "Chapter": "1",
        "sentence_range": "5075-5078",
        "Text": "Negative of a Vector A vector whose magnitude is the same as that of a given vector\n(say, AB\nuuur ), but direction is opposite to that of it, is called negative of the given vector For example, vector BA\nuuur  is negative of the vector AB\nuuur , and written as BA\nuuur= \u2212AB\nuuur Remark The vectors defined above are such that any of them may be subject to its\nparallel displacement without changing its magnitude and direction Such vectors are\ncalled free vectors"
    },
    {
        "Chapter": "1",
        "sentence_range": "5076-5079",
        "Text": "For example, vector BA\nuuur  is negative of the vector AB\nuuur , and written as BA\nuuur= \u2212AB\nuuur Remark The vectors defined above are such that any of them may be subject to its\nparallel displacement without changing its magnitude and direction Such vectors are\ncalled free vectors Throughout this chapter, we will be dealing with free vectors only"
    },
    {
        "Chapter": "1",
        "sentence_range": "5077-5080",
        "Text": "Remark The vectors defined above are such that any of them may be subject to its\nparallel displacement without changing its magnitude and direction Such vectors are\ncalled free vectors Throughout this chapter, we will be dealing with free vectors only Example 1 Represent graphically a displacement\nof 40 km, 30\u00b0 west of south"
    },
    {
        "Chapter": "1",
        "sentence_range": "5078-5081",
        "Text": "Such vectors are\ncalled free vectors Throughout this chapter, we will be dealing with free vectors only Example 1 Represent graphically a displacement\nof 40 km, 30\u00b0 west of south Solution The vector OP\nuuur  represents the required\ndisplacement (Fig 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5079-5082",
        "Text": "Throughout this chapter, we will be dealing with free vectors only Example 1 Represent graphically a displacement\nof 40 km, 30\u00b0 west of south Solution The vector OP\nuuur  represents the required\ndisplacement (Fig 10 4)"
    },
    {
        "Chapter": "1",
        "sentence_range": "5080-5083",
        "Text": "Example 1 Represent graphically a displacement\nof 40 km, 30\u00b0 west of south Solution The vector OP\nuuur  represents the required\ndisplacement (Fig 10 4) Example 2 Classify the following measures as\nscalars and vectors"
    },
    {
        "Chapter": "1",
        "sentence_range": "5081-5084",
        "Text": "Solution The vector OP\nuuur  represents the required\ndisplacement (Fig 10 4) Example 2 Classify the following measures as\nscalars and vectors (i) 5 seconds\n(ii) 1000 cm3\nFig 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5082-5085",
        "Text": "4) Example 2 Classify the following measures as\nscalars and vectors (i) 5 seconds\n(ii) 1000 cm3\nFig 10 4\n\u00a9 NCERT\nnot to be republished\n MATHEMATICS\n428\nFig 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5083-5086",
        "Text": "Example 2 Classify the following measures as\nscalars and vectors (i) 5 seconds\n(ii) 1000 cm3\nFig 10 4\n\u00a9 NCERT\nnot to be republished\n MATHEMATICS\n428\nFig 10 5\n(iii) 10 Newton\n(iv) 30 km/hr\n(v) 10 g/cm3\n(vi) 20 m/s towards north\nSolution\n(i) Time-scalar\n(ii) Volume-scalar\n(iii) Force-vector\n(iv) Speed-scalar\n(v) Density-scalar\n(vi) Velocity-vector\nExample 3 In Fig 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5084-5087",
        "Text": "(i) 5 seconds\n(ii) 1000 cm3\nFig 10 4\n\u00a9 NCERT\nnot to be republished\n MATHEMATICS\n428\nFig 10 5\n(iii) 10 Newton\n(iv) 30 km/hr\n(v) 10 g/cm3\n(vi) 20 m/s towards north\nSolution\n(i) Time-scalar\n(ii) Volume-scalar\n(iii) Force-vector\n(iv) Speed-scalar\n(v) Density-scalar\n(vi) Velocity-vector\nExample 3 In Fig 10 5, which of the vectors are:\n(i) Collinear\n(ii) Equal\n(iii) Coinitial\nSolution\n(i) Collinear vectors : ,\na cand\nrd\nr\nr"
    },
    {
        "Chapter": "1",
        "sentence_range": "5085-5088",
        "Text": "4\n\u00a9 NCERT\nnot to be republished\n MATHEMATICS\n428\nFig 10 5\n(iii) 10 Newton\n(iv) 30 km/hr\n(v) 10 g/cm3\n(vi) 20 m/s towards north\nSolution\n(i) Time-scalar\n(ii) Volume-scalar\n(iii) Force-vector\n(iv) Speed-scalar\n(v) Density-scalar\n(vi) Velocity-vector\nExample 3 In Fig 10 5, which of the vectors are:\n(i) Collinear\n(ii) Equal\n(iii) Coinitial\nSolution\n(i) Collinear vectors : ,\na cand\nrd\nr\nr (ii) Equal vectors : \nand"
    },
    {
        "Chapter": "1",
        "sentence_range": "5086-5089",
        "Text": "5\n(iii) 10 Newton\n(iv) 30 km/hr\n(v) 10 g/cm3\n(vi) 20 m/s towards north\nSolution\n(i) Time-scalar\n(ii) Volume-scalar\n(iii) Force-vector\n(iv) Speed-scalar\n(v) Density-scalar\n(vi) Velocity-vector\nExample 3 In Fig 10 5, which of the vectors are:\n(i) Collinear\n(ii) Equal\n(iii) Coinitial\nSolution\n(i) Collinear vectors : ,\na cand\nrd\nr\nr (ii) Equal vectors : \nand a\nc\nr\nr\n(iii) Coinitial vectors : ,\nand"
    },
    {
        "Chapter": "1",
        "sentence_range": "5087-5090",
        "Text": "5, which of the vectors are:\n(i) Collinear\n(ii) Equal\n(iii) Coinitial\nSolution\n(i) Collinear vectors : ,\na cand\nrd\nr\nr (ii) Equal vectors : \nand a\nc\nr\nr\n(iii) Coinitial vectors : ,\nand b c\nd\nr\nr\nr\nEXERCISE 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5088-5091",
        "Text": "(ii) Equal vectors : \nand a\nc\nr\nr\n(iii) Coinitial vectors : ,\nand b c\nd\nr\nr\nr\nEXERCISE 10 1\n1"
    },
    {
        "Chapter": "1",
        "sentence_range": "5089-5092",
        "Text": "a\nc\nr\nr\n(iii) Coinitial vectors : ,\nand b c\nd\nr\nr\nr\nEXERCISE 10 1\n1 Represent graphically a displacement of 40 km, 30\u00b0 east of north"
    },
    {
        "Chapter": "1",
        "sentence_range": "5090-5093",
        "Text": "b c\nd\nr\nr\nr\nEXERCISE 10 1\n1 Represent graphically a displacement of 40 km, 30\u00b0 east of north 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "5091-5094",
        "Text": "1\n1 Represent graphically a displacement of 40 km, 30\u00b0 east of north 2 Classify the following measures as scalars and vectors"
    },
    {
        "Chapter": "1",
        "sentence_range": "5092-5095",
        "Text": "Represent graphically a displacement of 40 km, 30\u00b0 east of north 2 Classify the following measures as scalars and vectors (i) 10 kg\n(ii) 2 meters north-west\n(iii) 40\u00b0\n(iv) 40 watt\n(v) 10\u201319 coulomb\n(vi) 20 m/s2\n3"
    },
    {
        "Chapter": "1",
        "sentence_range": "5093-5096",
        "Text": "2 Classify the following measures as scalars and vectors (i) 10 kg\n(ii) 2 meters north-west\n(iii) 40\u00b0\n(iv) 40 watt\n(v) 10\u201319 coulomb\n(vi) 20 m/s2\n3 Classify the following as scalar and vector quantities"
    },
    {
        "Chapter": "1",
        "sentence_range": "5094-5097",
        "Text": "Classify the following measures as scalars and vectors (i) 10 kg\n(ii) 2 meters north-west\n(iii) 40\u00b0\n(iv) 40 watt\n(v) 10\u201319 coulomb\n(vi) 20 m/s2\n3 Classify the following as scalar and vector quantities (i) time period\n(ii) distance\n(iii) force\n(iv) velocity\n(v) work done\n4"
    },
    {
        "Chapter": "1",
        "sentence_range": "5095-5098",
        "Text": "(i) 10 kg\n(ii) 2 meters north-west\n(iii) 40\u00b0\n(iv) 40 watt\n(v) 10\u201319 coulomb\n(vi) 20 m/s2\n3 Classify the following as scalar and vector quantities (i) time period\n(ii) distance\n(iii) force\n(iv) velocity\n(v) work done\n4 In Fig 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5096-5099",
        "Text": "Classify the following as scalar and vector quantities (i) time period\n(ii) distance\n(iii) force\n(iv) velocity\n(v) work done\n4 In Fig 10 6 (a square),  identify the following vectors"
    },
    {
        "Chapter": "1",
        "sentence_range": "5097-5100",
        "Text": "(i) time period\n(ii) distance\n(iii) force\n(iv) velocity\n(v) work done\n4 In Fig 10 6 (a square),  identify the following vectors (i) Coinitial\n(ii) Equal\n(iii) Collinear but not equal\n5"
    },
    {
        "Chapter": "1",
        "sentence_range": "5098-5101",
        "Text": "In Fig 10 6 (a square),  identify the following vectors (i) Coinitial\n(ii) Equal\n(iii) Collinear but not equal\n5 Answer the following as true or false"
    },
    {
        "Chapter": "1",
        "sentence_range": "5099-5102",
        "Text": "6 (a square),  identify the following vectors (i) Coinitial\n(ii) Equal\n(iii) Collinear but not equal\n5 Answer the following as true or false (i) ar  and \na\n\u2212 r  are collinear"
    },
    {
        "Chapter": "1",
        "sentence_range": "5100-5103",
        "Text": "(i) Coinitial\n(ii) Equal\n(iii) Collinear but not equal\n5 Answer the following as true or false (i) ar  and \na\n\u2212 r  are collinear (ii) Two collinear vectors are always equal in\nmagnitude"
    },
    {
        "Chapter": "1",
        "sentence_range": "5101-5104",
        "Text": "Answer the following as true or false (i) ar  and \na\n\u2212 r  are collinear (ii) Two collinear vectors are always equal in\nmagnitude (iii) Two vectors having same magnitude are collinear"
    },
    {
        "Chapter": "1",
        "sentence_range": "5102-5105",
        "Text": "(i) ar  and \na\n\u2212 r  are collinear (ii) Two collinear vectors are always equal in\nmagnitude (iii) Two vectors having same magnitude are collinear (iv) Two collinear vectors having the same magnitude are equal"
    },
    {
        "Chapter": "1",
        "sentence_range": "5103-5106",
        "Text": "(ii) Two collinear vectors are always equal in\nmagnitude (iii) Two vectors having same magnitude are collinear (iv) Two collinear vectors having the same magnitude are equal Fig 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5104-5107",
        "Text": "(iii) Two vectors having same magnitude are collinear (iv) Two collinear vectors having the same magnitude are equal Fig 10 6\n\u00a9 NCERT\nnot to be republished\nVECTOR ALGEBRA\n429\n10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5105-5108",
        "Text": "(iv) Two collinear vectors having the same magnitude are equal Fig 10 6\n\u00a9 NCERT\nnot to be republished\nVECTOR ALGEBRA\n429\n10 4  Addition of Vectors\nA vector AB\nuuur\n simply means the displacement from a\npoint A to the point B"
    },
    {
        "Chapter": "1",
        "sentence_range": "5106-5109",
        "Text": "Fig 10 6\n\u00a9 NCERT\nnot to be republished\nVECTOR ALGEBRA\n429\n10 4  Addition of Vectors\nA vector AB\nuuur\n simply means the displacement from a\npoint A to the point B Now consider a situation that a\ngirl moves from A to B and then from B to C\n(Fig 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5107-5110",
        "Text": "6\n\u00a9 NCERT\nnot to be republished\nVECTOR ALGEBRA\n429\n10 4  Addition of Vectors\nA vector AB\nuuur\n simply means the displacement from a\npoint A to the point B Now consider a situation that a\ngirl moves from A to B and then from B to C\n(Fig 10 7)"
    },
    {
        "Chapter": "1",
        "sentence_range": "5108-5111",
        "Text": "4  Addition of Vectors\nA vector AB\nuuur\n simply means the displacement from a\npoint A to the point B Now consider a situation that a\ngirl moves from A to B and then from B to C\n(Fig 10 7) The net displacement made by the girl from\npoint A to the point C, is given by the vector AC\nuuur  and\nexpressed as\nAC\nuuur  = AB\nBC\n+\nuuur\nuuur\nThis is known as the triangle law of vector addition"
    },
    {
        "Chapter": "1",
        "sentence_range": "5109-5112",
        "Text": "Now consider a situation that a\ngirl moves from A to B and then from B to C\n(Fig 10 7) The net displacement made by the girl from\npoint A to the point C, is given by the vector AC\nuuur  and\nexpressed as\nAC\nuuur  = AB\nBC\n+\nuuur\nuuur\nThis is known as the triangle law of vector addition In general, if we have two vectors ar  and b\nr\n (Fig 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5110-5113",
        "Text": "7) The net displacement made by the girl from\npoint A to the point C, is given by the vector AC\nuuur  and\nexpressed as\nAC\nuuur  = AB\nBC\n+\nuuur\nuuur\nThis is known as the triangle law of vector addition In general, if we have two vectors ar  and b\nr\n (Fig 10 8 (i)), then to add them, they\nare positioned so that the initial point of one coincides with the terminal point of the\nother (Fig 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5111-5114",
        "Text": "The net displacement made by the girl from\npoint A to the point C, is given by the vector AC\nuuur  and\nexpressed as\nAC\nuuur  = AB\nBC\n+\nuuur\nuuur\nThis is known as the triangle law of vector addition In general, if we have two vectors ar  and b\nr\n (Fig 10 8 (i)), then to add them, they\nare positioned so that the initial point of one coincides with the terminal point of the\nother (Fig 10 8(ii))"
    },
    {
        "Chapter": "1",
        "sentence_range": "5112-5115",
        "Text": "In general, if we have two vectors ar  and b\nr\n (Fig 10 8 (i)), then to add them, they\nare positioned so that the initial point of one coincides with the terminal point of the\nother (Fig 10 8(ii)) Fig 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5113-5116",
        "Text": "8 (i)), then to add them, they\nare positioned so that the initial point of one coincides with the terminal point of the\nother (Fig 10 8(ii)) Fig 10 8\nFor example, in Fig 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5114-5117",
        "Text": "8(ii)) Fig 10 8\nFor example, in Fig 10 8 (ii), we have shifted vector b\nr  without changing its magnitude\nand direction, so that it\u2019s initial point coincides with the terminal point of ar"
    },
    {
        "Chapter": "1",
        "sentence_range": "5115-5118",
        "Text": "Fig 10 8\nFor example, in Fig 10 8 (ii), we have shifted vector b\nr  without changing its magnitude\nand direction, so that it\u2019s initial point coincides with the terminal point of ar Then, the\nvector a\n+b\nr\nr\n, represented by the third side AC of the triangle ABC, gives us the sum\n(or resultant) of the vectors ar  and b\nr\ni"
    },
    {
        "Chapter": "1",
        "sentence_range": "5116-5119",
        "Text": "8\nFor example, in Fig 10 8 (ii), we have shifted vector b\nr  without changing its magnitude\nand direction, so that it\u2019s initial point coincides with the terminal point of ar Then, the\nvector a\n+b\nr\nr\n, represented by the third side AC of the triangle ABC, gives us the sum\n(or resultant) of the vectors ar  and b\nr\ni e"
    },
    {
        "Chapter": "1",
        "sentence_range": "5117-5120",
        "Text": "8 (ii), we have shifted vector b\nr  without changing its magnitude\nand direction, so that it\u2019s initial point coincides with the terminal point of ar Then, the\nvector a\n+b\nr\nr\n, represented by the third side AC of the triangle ABC, gives us the sum\n(or resultant) of the vectors ar  and b\nr\ni e , in triangle ABC (Fig 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5118-5121",
        "Text": "Then, the\nvector a\n+b\nr\nr\n, represented by the third side AC of the triangle ABC, gives us the sum\n(or resultant) of the vectors ar  and b\nr\ni e , in triangle ABC (Fig 10 8 (ii)), we have\nAB\nBC\n+\nuuur\nuuur  = AC\nuuur\nNow again, since AC\nCA\n= \u2212\nuuur\nuuur\n, from the above equation, we have\nAB\nBC\nCA\n+\n+\nuuur\nuuur\nuuur\n = AA\n0\n=\nuuur\nr\nThis means that when the sides of a triangle are taken in order, it leads to zero\nresultant as the initial and terminal points get coincided (Fig 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5119-5122",
        "Text": "e , in triangle ABC (Fig 10 8 (ii)), we have\nAB\nBC\n+\nuuur\nuuur  = AC\nuuur\nNow again, since AC\nCA\n= \u2212\nuuur\nuuur\n, from the above equation, we have\nAB\nBC\nCA\n+\n+\nuuur\nuuur\nuuur\n = AA\n0\n=\nuuur\nr\nThis means that when the sides of a triangle are taken in order, it leads to zero\nresultant as the initial and terminal points get coincided (Fig 10 8(iii))"
    },
    {
        "Chapter": "1",
        "sentence_range": "5120-5123",
        "Text": ", in triangle ABC (Fig 10 8 (ii)), we have\nAB\nBC\n+\nuuur\nuuur  = AC\nuuur\nNow again, since AC\nCA\n= \u2212\nuuur\nuuur\n, from the above equation, we have\nAB\nBC\nCA\n+\n+\nuuur\nuuur\nuuur\n = AA\n0\n=\nuuur\nr\nThis means that when the sides of a triangle are taken in order, it leads to zero\nresultant as the initial and terminal points get coincided (Fig 10 8(iii)) Fig 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5121-5124",
        "Text": "8 (ii)), we have\nAB\nBC\n+\nuuur\nuuur  = AC\nuuur\nNow again, since AC\nCA\n= \u2212\nuuur\nuuur\n, from the above equation, we have\nAB\nBC\nCA\n+\n+\nuuur\nuuur\nuuur\n = AA\n0\n=\nuuur\nr\nThis means that when the sides of a triangle are taken in order, it leads to zero\nresultant as the initial and terminal points get coincided (Fig 10 8(iii)) Fig 10 7\na\nb\na\nb\n(i)\n(iii)\nA\nC\na\nb\n(ii)\na\nb\n+\nA\nC\nB\nB\na\nb\n\u2013\n\u2013b\nC\u2019\n\u00a9 NCERT\nnot to be republished\n MATHEMATICS\n430\nNow, construct a vector BC\u2032\nuuuur\n so that its magnitude is same as the vector BC\nuuur\n, but\nthe direction opposite to that of it (Fig 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5122-5125",
        "Text": "8(iii)) Fig 10 7\na\nb\na\nb\n(i)\n(iii)\nA\nC\na\nb\n(ii)\na\nb\n+\nA\nC\nB\nB\na\nb\n\u2013\n\u2013b\nC\u2019\n\u00a9 NCERT\nnot to be republished\n MATHEMATICS\n430\nNow, construct a vector BC\u2032\nuuuur\n so that its magnitude is same as the vector BC\nuuur\n, but\nthe direction opposite to that of it (Fig 10 8 (iii)), i"
    },
    {
        "Chapter": "1",
        "sentence_range": "5123-5126",
        "Text": "Fig 10 7\na\nb\na\nb\n(i)\n(iii)\nA\nC\na\nb\n(ii)\na\nb\n+\nA\nC\nB\nB\na\nb\n\u2013\n\u2013b\nC\u2019\n\u00a9 NCERT\nnot to be republished\n MATHEMATICS\n430\nNow, construct a vector BC\u2032\nuuuur\n so that its magnitude is same as the vector BC\nuuur\n, but\nthe direction opposite to that of it (Fig 10 8 (iii)), i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "5124-5127",
        "Text": "7\na\nb\na\nb\n(i)\n(iii)\nA\nC\na\nb\n(ii)\na\nb\n+\nA\nC\nB\nB\na\nb\n\u2013\n\u2013b\nC\u2019\n\u00a9 NCERT\nnot to be republished\n MATHEMATICS\n430\nNow, construct a vector BC\u2032\nuuuur\n so that its magnitude is same as the vector BC\nuuur\n, but\nthe direction opposite to that of it (Fig 10 8 (iii)), i e ,\nBC\u2032\nuuuur\n =\nBC\n\u2212\nuuur\nThen, on applying triangle law from the Fig 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5125-5128",
        "Text": "8 (iii)), i e ,\nBC\u2032\nuuuur\n =\nBC\n\u2212\nuuur\nThen, on applying triangle law from the Fig 10 8 (iii), we have\nAC\nAB\nBC\n\u2032\n\u2032\n=\n+\nuuuur\nuuur\nuuuur  = AB\n( BC)\n+ \u2212\nuuur\nuuur\na\nb\n=\n\u2212\nr\nr\nThe vector AC\u2032\nuuuur  is said to represent the difference of \naand\nrb\nr"
    },
    {
        "Chapter": "1",
        "sentence_range": "5126-5129",
        "Text": "e ,\nBC\u2032\nuuuur\n =\nBC\n\u2212\nuuur\nThen, on applying triangle law from the Fig 10 8 (iii), we have\nAC\nAB\nBC\n\u2032\n\u2032\n=\n+\nuuuur\nuuur\nuuuur  = AB\n( BC)\n+ \u2212\nuuur\nuuur\na\nb\n=\n\u2212\nr\nr\nThe vector AC\u2032\nuuuur  is said to represent the difference of \naand\nrb\nr Now, consider a boat in a river going from one bank of the river to the other in a\ndirection perpendicular to the flow of the river"
    },
    {
        "Chapter": "1",
        "sentence_range": "5127-5130",
        "Text": ",\nBC\u2032\nuuuur\n =\nBC\n\u2212\nuuur\nThen, on applying triangle law from the Fig 10 8 (iii), we have\nAC\nAB\nBC\n\u2032\n\u2032\n=\n+\nuuuur\nuuur\nuuuur  = AB\n( BC)\n+ \u2212\nuuur\nuuur\na\nb\n=\n\u2212\nr\nr\nThe vector AC\u2032\nuuuur  is said to represent the difference of \naand\nrb\nr Now, consider a boat in a river going from one bank of the river to the other in a\ndirection perpendicular to the flow of the river Then, it is acted upon by two velocity\nvectors\u2013one is the velocity imparted to the boat by its engine and other one is the\nvelocity of the flow of river water"
    },
    {
        "Chapter": "1",
        "sentence_range": "5128-5131",
        "Text": "8 (iii), we have\nAC\nAB\nBC\n\u2032\n\u2032\n=\n+\nuuuur\nuuur\nuuuur  = AB\n( BC)\n+ \u2212\nuuur\nuuur\na\nb\n=\n\u2212\nr\nr\nThe vector AC\u2032\nuuuur  is said to represent the difference of \naand\nrb\nr Now, consider a boat in a river going from one bank of the river to the other in a\ndirection perpendicular to the flow of the river Then, it is acted upon by two velocity\nvectors\u2013one is the velocity imparted to the boat by its engine and other one is the\nvelocity of the flow of river water Under the simultaneous influence of these two\nvelocities, the boat in actual starts travelling with a different velocity"
    },
    {
        "Chapter": "1",
        "sentence_range": "5129-5132",
        "Text": "Now, consider a boat in a river going from one bank of the river to the other in a\ndirection perpendicular to the flow of the river Then, it is acted upon by two velocity\nvectors\u2013one is the velocity imparted to the boat by its engine and other one is the\nvelocity of the flow of river water Under the simultaneous influence of these two\nvelocities, the boat in actual starts travelling with a different velocity To have a precise\nidea about the effective speed and direction\n(i"
    },
    {
        "Chapter": "1",
        "sentence_range": "5130-5133",
        "Text": "Then, it is acted upon by two velocity\nvectors\u2013one is the velocity imparted to the boat by its engine and other one is the\nvelocity of the flow of river water Under the simultaneous influence of these two\nvelocities, the boat in actual starts travelling with a different velocity To have a precise\nidea about the effective speed and direction\n(i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "5131-5134",
        "Text": "Under the simultaneous influence of these two\nvelocities, the boat in actual starts travelling with a different velocity To have a precise\nidea about the effective speed and direction\n(i e , the resultant velocity) of the boat, we have\nthe following law of vector addition"
    },
    {
        "Chapter": "1",
        "sentence_range": "5132-5135",
        "Text": "To have a precise\nidea about the effective speed and direction\n(i e , the resultant velocity) of the boat, we have\nthe following law of vector addition If we have two vectors \naand\nrb\nr\nrepresented\nby the two adjacent sides of a parallelogram in\nmagnitude and direction (Fig 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5133-5136",
        "Text": "e , the resultant velocity) of the boat, we have\nthe following law of vector addition If we have two vectors \naand\nrb\nr\nrepresented\nby the two adjacent sides of a parallelogram in\nmagnitude and direction (Fig 10 9), then their\nsum \na+\nrb\nr\n is represented in magnitude and\ndirection by the diagonal of the parallelogram\nthrough their common point"
    },
    {
        "Chapter": "1",
        "sentence_range": "5134-5137",
        "Text": ", the resultant velocity) of the boat, we have\nthe following law of vector addition If we have two vectors \naand\nrb\nr\nrepresented\nby the two adjacent sides of a parallelogram in\nmagnitude and direction (Fig 10 9), then their\nsum \na+\nrb\nr\n is represented in magnitude and\ndirection by the diagonal of the parallelogram\nthrough their common point This is known as\nthe parallelogram law of vector addition"
    },
    {
        "Chapter": "1",
        "sentence_range": "5135-5138",
        "Text": "If we have two vectors \naand\nrb\nr\nrepresented\nby the two adjacent sides of a parallelogram in\nmagnitude and direction (Fig 10 9), then their\nsum \na+\nrb\nr\n is represented in magnitude and\ndirection by the diagonal of the parallelogram\nthrough their common point This is known as\nthe parallelogram law of vector addition \ufffdNote    From Fig 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5136-5139",
        "Text": "9), then their\nsum \na+\nrb\nr\n is represented in magnitude and\ndirection by the diagonal of the parallelogram\nthrough their common point This is known as\nthe parallelogram law of vector addition \ufffdNote    From Fig 10 9, using the triangle law, one may note that\nOA\nAC\n+\nuuur\nuuur  = OC\nuuur\nor\nOA\nOB\n+\nuuur\nuuur  = OC\nuuur                                 (since AC\nuuur=OB\nuuur\n)\nwhich is parallelogram law"
    },
    {
        "Chapter": "1",
        "sentence_range": "5137-5140",
        "Text": "This is known as\nthe parallelogram law of vector addition \ufffdNote    From Fig 10 9, using the triangle law, one may note that\nOA\nAC\n+\nuuur\nuuur  = OC\nuuur\nor\nOA\nOB\n+\nuuur\nuuur  = OC\nuuur                                 (since AC\nuuur=OB\nuuur\n)\nwhich is parallelogram law Thus, we may say that the two laws of vector\naddition are equivalent to each other"
    },
    {
        "Chapter": "1",
        "sentence_range": "5138-5141",
        "Text": "\ufffdNote    From Fig 10 9, using the triangle law, one may note that\nOA\nAC\n+\nuuur\nuuur  = OC\nuuur\nor\nOA\nOB\n+\nuuur\nuuur  = OC\nuuur                                 (since AC\nuuur=OB\nuuur\n)\nwhich is parallelogram law Thus, we may say that the two laws of vector\naddition are equivalent to each other Properties of vector addition\nProperty 1 For any two vectors \naand\nrb\nr\n,\na\nb\n+\nr\nr\n = b\na\n+\nr\nr\n(Commutative property)\nFig 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5139-5142",
        "Text": "9, using the triangle law, one may note that\nOA\nAC\n+\nuuur\nuuur  = OC\nuuur\nor\nOA\nOB\n+\nuuur\nuuur  = OC\nuuur                                 (since AC\nuuur=OB\nuuur\n)\nwhich is parallelogram law Thus, we may say that the two laws of vector\naddition are equivalent to each other Properties of vector addition\nProperty 1 For any two vectors \naand\nrb\nr\n,\na\nb\n+\nr\nr\n = b\na\n+\nr\nr\n(Commutative property)\nFig 10 9\n\u00a9 NCERT\nnot to be republished\nVECTOR ALGEBRA\n431\nProof Consider the parallelogram ABCD\n(Fig 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5140-5143",
        "Text": "Thus, we may say that the two laws of vector\naddition are equivalent to each other Properties of vector addition\nProperty 1 For any two vectors \naand\nrb\nr\n,\na\nb\n+\nr\nr\n = b\na\n+\nr\nr\n(Commutative property)\nFig 10 9\n\u00a9 NCERT\nnot to be republished\nVECTOR ALGEBRA\n431\nProof Consider the parallelogram ABCD\n(Fig 10 10)"
    },
    {
        "Chapter": "1",
        "sentence_range": "5141-5144",
        "Text": "Properties of vector addition\nProperty 1 For any two vectors \naand\nrb\nr\n,\na\nb\n+\nr\nr\n = b\na\n+\nr\nr\n(Commutative property)\nFig 10 9\n\u00a9 NCERT\nnot to be republished\nVECTOR ALGEBRA\n431\nProof Consider the parallelogram ABCD\n(Fig 10 10) Let AB\nand BC\n,\na\nb\n \n \nuuur\nuuur\nr\nr\n then using\nthe triangle law, from triangle ABC, we have\nAC\nuuur  =\na+\nrb\nr\nNow, since the opposite sides of a\nparallelogram are equal and parallel, from\nFig 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5142-5145",
        "Text": "9\n\u00a9 NCERT\nnot to be republished\nVECTOR ALGEBRA\n431\nProof Consider the parallelogram ABCD\n(Fig 10 10) Let AB\nand BC\n,\na\nb\n \n \nuuur\nuuur\nr\nr\n then using\nthe triangle law, from triangle ABC, we have\nAC\nuuur  =\na+\nrb\nr\nNow, since the opposite sides of a\nparallelogram are equal and parallel, from\nFig 10 10, we have, AD = BC = b\nuuur\nuuur\nr\n and\nDC = AB = a\nuuur\nuuur\nr"
    },
    {
        "Chapter": "1",
        "sentence_range": "5143-5146",
        "Text": "10) Let AB\nand BC\n,\na\nb\n \n \nuuur\nuuur\nr\nr\n then using\nthe triangle law, from triangle ABC, we have\nAC\nuuur  =\na+\nrb\nr\nNow, since the opposite sides of a\nparallelogram are equal and parallel, from\nFig 10 10, we have, AD = BC = b\nuuur\nuuur\nr\n and\nDC = AB = a\nuuur\nuuur\nr Again using triangle law, from\ntriangle ADC, we have\nAC\nuuuur  = AD + DC =\nb+\na\nuuur\nuuur\nr\nr\nHence\na\nb\n+\nr\nr\n = b\na\n+\nr\nr\nProperty 2 For any three vectors ,\na band\nc\nrr\nr\n(\n)\na\nb\nc\n+\n+\nr\nr\nr  =\n(\n)\na\nb\nc\n+\n+\nr\nr\nr\n(Associative property)\nProof Let the vectors ,\na band\nc\nrr\nr be represented by PQ,  QR  and  RS\nuuur\nuuur\nuuur , respectively,\nas shown in Fig 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5144-5147",
        "Text": "Let AB\nand BC\n,\na\nb\n \n \nuuur\nuuur\nr\nr\n then using\nthe triangle law, from triangle ABC, we have\nAC\nuuur  =\na+\nrb\nr\nNow, since the opposite sides of a\nparallelogram are equal and parallel, from\nFig 10 10, we have, AD = BC = b\nuuur\nuuur\nr\n and\nDC = AB = a\nuuur\nuuur\nr Again using triangle law, from\ntriangle ADC, we have\nAC\nuuuur  = AD + DC =\nb+\na\nuuur\nuuur\nr\nr\nHence\na\nb\n+\nr\nr\n = b\na\n+\nr\nr\nProperty 2 For any three vectors ,\na band\nc\nrr\nr\n(\n)\na\nb\nc\n+\n+\nr\nr\nr  =\n(\n)\na\nb\nc\n+\n+\nr\nr\nr\n(Associative property)\nProof Let the vectors ,\na band\nc\nrr\nr be represented by PQ,  QR  and  RS\nuuur\nuuur\nuuur , respectively,\nas shown in Fig 10 11(i) and (ii)"
    },
    {
        "Chapter": "1",
        "sentence_range": "5145-5148",
        "Text": "10, we have, AD = BC = b\nuuur\nuuur\nr\n and\nDC = AB = a\nuuur\nuuur\nr Again using triangle law, from\ntriangle ADC, we have\nAC\nuuuur  = AD + DC =\nb+\na\nuuur\nuuur\nr\nr\nHence\na\nb\n+\nr\nr\n = b\na\n+\nr\nr\nProperty 2 For any three vectors ,\na band\nc\nrr\nr\n(\n)\na\nb\nc\n+\n+\nr\nr\nr  =\n(\n)\na\nb\nc\n+\n+\nr\nr\nr\n(Associative property)\nProof Let the vectors ,\na band\nc\nrr\nr be represented by PQ,  QR  and  RS\nuuur\nuuur\nuuur , respectively,\nas shown in Fig 10 11(i) and (ii) Fig 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5146-5149",
        "Text": "Again using triangle law, from\ntriangle ADC, we have\nAC\nuuuur  = AD + DC =\nb+\na\nuuur\nuuur\nr\nr\nHence\na\nb\n+\nr\nr\n = b\na\n+\nr\nr\nProperty 2 For any three vectors ,\na band\nc\nrr\nr\n(\n)\na\nb\nc\n+\n+\nr\nr\nr  =\n(\n)\na\nb\nc\n+\n+\nr\nr\nr\n(Associative property)\nProof Let the vectors ,\na band\nc\nrr\nr be represented by PQ,  QR  and  RS\nuuur\nuuur\nuuur , respectively,\nas shown in Fig 10 11(i) and (ii) Fig 10 11\nThen\na\nb\n+\nr\nr\n = PQ + QR = PR\nuuur\nuuur\nuuur\nand\nb\nc\n+\nr\nr  = QR + RS = QS\nuuur\nuuur\nuuur\nSo\n(\n)\n \na\nb\nc\n+\nr+\nr\nr  = PR + RS = PS\nuuur\nuuur\nuur\nFig 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5147-5150",
        "Text": "11(i) and (ii) Fig 10 11\nThen\na\nb\n+\nr\nr\n = PQ + QR = PR\nuuur\nuuur\nuuur\nand\nb\nc\n+\nr\nr  = QR + RS = QS\nuuur\nuuur\nuuur\nSo\n(\n)\n \na\nb\nc\n+\nr+\nr\nr  = PR + RS = PS\nuuur\nuuur\nuur\nFig 10 10\n\u00a9 NCERT\nnot to be republished\n MATHEMATICS\n432\na\na\n1\n2\n1\n2\na\n\u20132\na\na\n2\nand\n(\n)\na\nb\nc\n+\n+\nr\nr\nr  = PQ + QS = PS\nuuur\nuuur\nuur\nHence\n(\n)\na\nb\nc\n+\n+\nr\nr\nr  =\n(\n)\na\nb\nc\n+\n+\nr\nr\nr\nRemark The associative property of vector addition enables us to write the sum of\nthree vectors \n,\n,\n  as \na\nb\nc\na\nb\nc\n+\n+\nr\nr\nr\nr\nr\nr  without using brackets"
    },
    {
        "Chapter": "1",
        "sentence_range": "5148-5151",
        "Text": "Fig 10 11\nThen\na\nb\n+\nr\nr\n = PQ + QR = PR\nuuur\nuuur\nuuur\nand\nb\nc\n+\nr\nr  = QR + RS = QS\nuuur\nuuur\nuuur\nSo\n(\n)\n \na\nb\nc\n+\nr+\nr\nr  = PR + RS = PS\nuuur\nuuur\nuur\nFig 10 10\n\u00a9 NCERT\nnot to be republished\n MATHEMATICS\n432\na\na\n1\n2\n1\n2\na\n\u20132\na\na\n2\nand\n(\n)\na\nb\nc\n+\n+\nr\nr\nr  = PQ + QS = PS\nuuur\nuuur\nuur\nHence\n(\n)\na\nb\nc\n+\n+\nr\nr\nr  =\n(\n)\na\nb\nc\n+\n+\nr\nr\nr\nRemark The associative property of vector addition enables us to write the sum of\nthree vectors \n,\n,\n  as \na\nb\nc\na\nb\nc\n+\n+\nr\nr\nr\nr\nr\nr  without using brackets Note that for any vector ar, we have\na +0\nr\nr\n = 0\na\na\n+\n=\nr\nr\nr\nHere, the zero vector 0\nr  is called the additive identity for the vector addition"
    },
    {
        "Chapter": "1",
        "sentence_range": "5149-5152",
        "Text": "11\nThen\na\nb\n+\nr\nr\n = PQ + QR = PR\nuuur\nuuur\nuuur\nand\nb\nc\n+\nr\nr  = QR + RS = QS\nuuur\nuuur\nuuur\nSo\n(\n)\n \na\nb\nc\n+\nr+\nr\nr  = PR + RS = PS\nuuur\nuuur\nuur\nFig 10 10\n\u00a9 NCERT\nnot to be republished\n MATHEMATICS\n432\na\na\n1\n2\n1\n2\na\n\u20132\na\na\n2\nand\n(\n)\na\nb\nc\n+\n+\nr\nr\nr  = PQ + QS = PS\nuuur\nuuur\nuur\nHence\n(\n)\na\nb\nc\n+\n+\nr\nr\nr  =\n(\n)\na\nb\nc\n+\n+\nr\nr\nr\nRemark The associative property of vector addition enables us to write the sum of\nthree vectors \n,\n,\n  as \na\nb\nc\na\nb\nc\n+\n+\nr\nr\nr\nr\nr\nr  without using brackets Note that for any vector ar, we have\na +0\nr\nr\n = 0\na\na\n+\n=\nr\nr\nr\nHere, the zero vector 0\nr  is called the additive identity for the vector addition 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5150-5153",
        "Text": "10\n\u00a9 NCERT\nnot to be republished\n MATHEMATICS\n432\na\na\n1\n2\n1\n2\na\n\u20132\na\na\n2\nand\n(\n)\na\nb\nc\n+\n+\nr\nr\nr  = PQ + QS = PS\nuuur\nuuur\nuur\nHence\n(\n)\na\nb\nc\n+\n+\nr\nr\nr  =\n(\n)\na\nb\nc\n+\n+\nr\nr\nr\nRemark The associative property of vector addition enables us to write the sum of\nthree vectors \n,\n,\n  as \na\nb\nc\na\nb\nc\n+\n+\nr\nr\nr\nr\nr\nr  without using brackets Note that for any vector ar, we have\na +0\nr\nr\n = 0\na\na\n+\n=\nr\nr\nr\nHere, the zero vector 0\nr  is called the additive identity for the vector addition 10 5  Multiplication of a Vector by a Scalar\nLet ar  be a given vector and \u03bb a scalar"
    },
    {
        "Chapter": "1",
        "sentence_range": "5151-5154",
        "Text": "Note that for any vector ar, we have\na +0\nr\nr\n = 0\na\na\n+\n=\nr\nr\nr\nHere, the zero vector 0\nr  is called the additive identity for the vector addition 10 5  Multiplication of a Vector by a Scalar\nLet ar  be a given vector and \u03bb a scalar Then the product of the vector ar  by the scalar\n\u03bb, denoted as \u03bb ar , is called the multiplication of vector ar  by the scalar \u03bb"
    },
    {
        "Chapter": "1",
        "sentence_range": "5152-5155",
        "Text": "10 5  Multiplication of a Vector by a Scalar\nLet ar  be a given vector and \u03bb a scalar Then the product of the vector ar  by the scalar\n\u03bb, denoted as \u03bb ar , is called the multiplication of vector ar  by the scalar \u03bb Note that,\n\u03bb ar  is also a vector, collinear to the vector ar"
    },
    {
        "Chapter": "1",
        "sentence_range": "5153-5156",
        "Text": "5  Multiplication of a Vector by a Scalar\nLet ar  be a given vector and \u03bb a scalar Then the product of the vector ar  by the scalar\n\u03bb, denoted as \u03bb ar , is called the multiplication of vector ar  by the scalar \u03bb Note that,\n\u03bb ar  is also a vector, collinear to the vector ar The vector \u03bb ar  has the direction same\n(or opposite) to that of vector ar  according as the value of \u03bb is positive (or negative)"
    },
    {
        "Chapter": "1",
        "sentence_range": "5154-5157",
        "Text": "Then the product of the vector ar  by the scalar\n\u03bb, denoted as \u03bb ar , is called the multiplication of vector ar  by the scalar \u03bb Note that,\n\u03bb ar  is also a vector, collinear to the vector ar The vector \u03bb ar  has the direction same\n(or opposite) to that of vector ar  according as the value of \u03bb is positive (or negative) Also, the magnitude of vector \u03bb ar  is |\u03bb| times the magnitude of the vector ar , i"
    },
    {
        "Chapter": "1",
        "sentence_range": "5155-5158",
        "Text": "Note that,\n\u03bb ar  is also a vector, collinear to the vector ar The vector \u03bb ar  has the direction same\n(or opposite) to that of vector ar  according as the value of \u03bb is positive (or negative) Also, the magnitude of vector \u03bb ar  is |\u03bb| times the magnitude of the vector ar , i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "5156-5159",
        "Text": "The vector \u03bb ar  has the direction same\n(or opposite) to that of vector ar  according as the value of \u03bb is positive (or negative) Also, the magnitude of vector \u03bb ar  is |\u03bb| times the magnitude of the vector ar , i e ,\n|\na|\n\u03bbr  = |\n||\na|\n\u03bb\nr\nA geometric visualisation of multiplication of a vector by a scalar is given\nin Fig 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5157-5160",
        "Text": "Also, the magnitude of vector \u03bb ar  is |\u03bb| times the magnitude of the vector ar , i e ,\n|\na|\n\u03bbr  = |\n||\na|\n\u03bb\nr\nA geometric visualisation of multiplication of a vector by a scalar is given\nin Fig 10 12"
    },
    {
        "Chapter": "1",
        "sentence_range": "5158-5161",
        "Text": "e ,\n|\na|\n\u03bbr  = |\n||\na|\n\u03bb\nr\nA geometric visualisation of multiplication of a vector by a scalar is given\nin Fig 10 12 Fig 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5159-5162",
        "Text": ",\n|\na|\n\u03bbr  = |\n||\na|\n\u03bb\nr\nA geometric visualisation of multiplication of a vector by a scalar is given\nin Fig 10 12 Fig 10 12\nWhen \u03bb = \u20131, then a\n\u03bb = \u2212a\nr\nr, which is a vector having magnitude equal to the\nmagnitude of ar  and direction opposite to that of the direction of ar"
    },
    {
        "Chapter": "1",
        "sentence_range": "5160-5163",
        "Text": "12 Fig 10 12\nWhen \u03bb = \u20131, then a\n\u03bb = \u2212a\nr\nr, which is a vector having magnitude equal to the\nmagnitude of ar  and direction opposite to that of the direction of ar The vector \u2013 ar  is\ncalled the negative (or additive inverse) of vector ar  and we always have\n(\u2013 )\na\na\nr+\nr  = (\u2013 )\n0\na\n+a\n=\nr\nr\nr\nAlso, if \n= |1\na|\n\u03bb\nr , provided \n0,  i"
    },
    {
        "Chapter": "1",
        "sentence_range": "5161-5164",
        "Text": "Fig 10 12\nWhen \u03bb = \u20131, then a\n\u03bb = \u2212a\nr\nr, which is a vector having magnitude equal to the\nmagnitude of ar  and direction opposite to that of the direction of ar The vector \u2013 ar  is\ncalled the negative (or additive inverse) of vector ar  and we always have\n(\u2013 )\na\na\nr+\nr  = (\u2013 )\n0\na\n+a\n=\nr\nr\nr\nAlso, if \n= |1\na|\n\u03bb\nr , provided \n0,  i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "5162-5165",
        "Text": "12\nWhen \u03bb = \u20131, then a\n\u03bb = \u2212a\nr\nr, which is a vector having magnitude equal to the\nmagnitude of ar  and direction opposite to that of the direction of ar The vector \u2013 ar  is\ncalled the negative (or additive inverse) of vector ar  and we always have\n(\u2013 )\na\na\nr+\nr  = (\u2013 )\n0\na\n+a\n=\nr\nr\nr\nAlso, if \n= |1\na|\n\u03bb\nr , provided \n0,  i e r\nr\na\na  is not a null vector, then\n|\n| |\n||\n|\na\na\n\u03bb\nr= \u03bb\nr  =\n1 |\n| 1\n|\na| a\nr\nr\n\u00a9 NCERT\nnot to be republished\nVECTOR ALGEBRA\n433\nSo, \u03bb ar  represents the unit vector in the direction of ar"
    },
    {
        "Chapter": "1",
        "sentence_range": "5163-5166",
        "Text": "The vector \u2013 ar  is\ncalled the negative (or additive inverse) of vector ar  and we always have\n(\u2013 )\na\na\nr+\nr  = (\u2013 )\n0\na\n+a\n=\nr\nr\nr\nAlso, if \n= |1\na|\n\u03bb\nr , provided \n0,  i e r\nr\na\na  is not a null vector, then\n|\n| |\n||\n|\na\na\n\u03bb\nr= \u03bb\nr  =\n1 |\n| 1\n|\na| a\nr\nr\n\u00a9 NCERT\nnot to be republished\nVECTOR ALGEBRA\n433\nSo, \u03bb ar  represents the unit vector in the direction of ar We write it as\n\u02c6a  =\n|1\n| a\na\nr\nr\n\ufffdNote  For any scalar k, 0 = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "5164-5167",
        "Text": "e r\nr\na\na  is not a null vector, then\n|\n| |\n||\n|\na\na\n\u03bb\nr= \u03bb\nr  =\n1 |\n| 1\n|\na| a\nr\nr\n\u00a9 NCERT\nnot to be republished\nVECTOR ALGEBRA\n433\nSo, \u03bb ar  represents the unit vector in the direction of ar We write it as\n\u02c6a  =\n|1\n| a\na\nr\nr\n\ufffdNote  For any scalar k, 0 = 0 k\nr\nr\n10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5165-5168",
        "Text": "r\nr\na\na  is not a null vector, then\n|\n| |\n||\n|\na\na\n\u03bb\nr= \u03bb\nr  =\n1 |\n| 1\n|\na| a\nr\nr\n\u00a9 NCERT\nnot to be republished\nVECTOR ALGEBRA\n433\nSo, \u03bb ar  represents the unit vector in the direction of ar We write it as\n\u02c6a  =\n|1\n| a\na\nr\nr\n\ufffdNote  For any scalar k, 0 = 0 k\nr\nr\n10 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "5166-5169",
        "Text": "We write it as\n\u02c6a  =\n|1\n| a\na\nr\nr\n\ufffdNote  For any scalar k, 0 = 0 k\nr\nr\n10 5 1  Components of a vector\nLet us take the points A(1, 0, 0), B(0, 1, 0) and C(0, 0, 1) on the x-axis, y-axis and\nz-axis, respectively"
    },
    {
        "Chapter": "1",
        "sentence_range": "5167-5170",
        "Text": "k\nr\nr\n10 5 1  Components of a vector\nLet us take the points A(1, 0, 0), B(0, 1, 0) and C(0, 0, 1) on the x-axis, y-axis and\nz-axis, respectively Then, clearly\n| OA | 1,| OB|\nuuur=\nuuur  = 1 and | OC |\n=1\nuuur\nThe vectors OA,  OB and OC\nuuur\nuuur\nuuur , each having magnitude 1,\nare called unit vectors along the axes OX, OY and OZ,\nrespectively, and denoted by \n\u02c6\n\u02c6 \u02c6\n,   and  \ni j\nk , respectively\n(Fig 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5168-5171",
        "Text": "5 1  Components of a vector\nLet us take the points A(1, 0, 0), B(0, 1, 0) and C(0, 0, 1) on the x-axis, y-axis and\nz-axis, respectively Then, clearly\n| OA | 1,| OB|\nuuur=\nuuur  = 1 and | OC |\n=1\nuuur\nThe vectors OA,  OB and OC\nuuur\nuuur\nuuur , each having magnitude 1,\nare called unit vectors along the axes OX, OY and OZ,\nrespectively, and denoted by \n\u02c6\n\u02c6 \u02c6\n,   and  \ni j\nk , respectively\n(Fig 10 13)"
    },
    {
        "Chapter": "1",
        "sentence_range": "5169-5172",
        "Text": "1  Components of a vector\nLet us take the points A(1, 0, 0), B(0, 1, 0) and C(0, 0, 1) on the x-axis, y-axis and\nz-axis, respectively Then, clearly\n| OA | 1,| OB|\nuuur=\nuuur  = 1 and | OC |\n=1\nuuur\nThe vectors OA,  OB and OC\nuuur\nuuur\nuuur , each having magnitude 1,\nare called unit vectors along the axes OX, OY and OZ,\nrespectively, and denoted by \n\u02c6\n\u02c6 \u02c6\n,   and  \ni j\nk , respectively\n(Fig 10 13) Now, consider the position vector OP\nuuur\n of a point P(x, y, z) as in Fig 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5170-5173",
        "Text": "Then, clearly\n| OA | 1,| OB|\nuuur=\nuuur  = 1 and | OC |\n=1\nuuur\nThe vectors OA,  OB and OC\nuuur\nuuur\nuuur , each having magnitude 1,\nare called unit vectors along the axes OX, OY and OZ,\nrespectively, and denoted by \n\u02c6\n\u02c6 \u02c6\n,   and  \ni j\nk , respectively\n(Fig 10 13) Now, consider the position vector OP\nuuur\n of a point P(x, y, z) as in Fig 10 14"
    },
    {
        "Chapter": "1",
        "sentence_range": "5171-5174",
        "Text": "13) Now, consider the position vector OP\nuuur\n of a point P(x, y, z) as in Fig 10 14 Let P1\nbe the foot of the perpendicular from P on the plane XOY"
    },
    {
        "Chapter": "1",
        "sentence_range": "5172-5175",
        "Text": "Now, consider the position vector OP\nuuur\n of a point P(x, y, z) as in Fig 10 14 Let P1\nbe the foot of the perpendicular from P on the plane XOY We, thus, see that P1 P is\nparallel to z-axis"
    },
    {
        "Chapter": "1",
        "sentence_range": "5173-5176",
        "Text": "14 Let P1\nbe the foot of the perpendicular from P on the plane XOY We, thus, see that P1 P is\nparallel to z-axis As \n\u02c6\n\u02c6 \u02c6\n,  and \ni j\nk  are the unit vectors along the x, y and z-axes,\nrespectively, and by the definition of the coordinates of P, we have \n1\n\u02c6\nP P\nOR\nzk\n=\n=\nuuur\nuuur"
    },
    {
        "Chapter": "1",
        "sentence_range": "5174-5177",
        "Text": "Let P1\nbe the foot of the perpendicular from P on the plane XOY We, thus, see that P1 P is\nparallel to z-axis As \n\u02c6\n\u02c6 \u02c6\n,  and \ni j\nk  are the unit vectors along the x, y and z-axes,\nrespectively, and by the definition of the coordinates of P, we have \n1\n\u02c6\nP P\nOR\nzk\n=\n=\nuuur\nuuur Similarly, \n1\n\u02c6\nQP\nOS\nyj\n=\n=\nuuur\nuuur\n and \n\u02c6\nOQ\nxi\n=\nuuur"
    },
    {
        "Chapter": "1",
        "sentence_range": "5175-5178",
        "Text": "We, thus, see that P1 P is\nparallel to z-axis As \n\u02c6\n\u02c6 \u02c6\n,  and \ni j\nk  are the unit vectors along the x, y and z-axes,\nrespectively, and by the definition of the coordinates of P, we have \n1\n\u02c6\nP P\nOR\nzk\n=\n=\nuuur\nuuur Similarly, \n1\n\u02c6\nQP\nOS\nyj\n=\n=\nuuur\nuuur\n and \n\u02c6\nOQ\nxi\n=\nuuur Fig 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5176-5179",
        "Text": "As \n\u02c6\n\u02c6 \u02c6\n,  and \ni j\nk  are the unit vectors along the x, y and z-axes,\nrespectively, and by the definition of the coordinates of P, we have \n1\n\u02c6\nP P\nOR\nzk\n=\n=\nuuur\nuuur Similarly, \n1\n\u02c6\nQP\nOS\nyj\n=\n=\nuuur\nuuur\n and \n\u02c6\nOQ\nxi\n=\nuuur Fig 10 13\nFig 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5177-5180",
        "Text": "Similarly, \n1\n\u02c6\nQP\nOS\nyj\n=\n=\nuuur\nuuur\n and \n\u02c6\nOQ\nxi\n=\nuuur Fig 10 13\nFig 10 14\n\u00a9 NCERT\nnot to be republished\n MATHEMATICS\n434\nTherefore, it follows that\nOP1\nuuur  =\n1\n\u02c6\n\u02c6\nOQ + QP\nxi\nyj\n=\n+\nuuur\nuuur\nand\nOP\nuuur  =\n1\n1\n\u02c6\n\u02c6\n\u02c6\nOP + P P\nxi\nyj\nzk\n=\n+\n+\nuuur\nuuuur\nHence, the position vector of P with reference to O is given by\nOP (or\n)\nr\nuuur\nr  =\n\u02c6\n\u02c6\n\u02c6\nxi\nyj\nzk\n+\n+\nThis form of any vector is called its component form"
    },
    {
        "Chapter": "1",
        "sentence_range": "5178-5181",
        "Text": "Fig 10 13\nFig 10 14\n\u00a9 NCERT\nnot to be republished\n MATHEMATICS\n434\nTherefore, it follows that\nOP1\nuuur  =\n1\n\u02c6\n\u02c6\nOQ + QP\nxi\nyj\n=\n+\nuuur\nuuur\nand\nOP\nuuur  =\n1\n1\n\u02c6\n\u02c6\n\u02c6\nOP + P P\nxi\nyj\nzk\n=\n+\n+\nuuur\nuuuur\nHence, the position vector of P with reference to O is given by\nOP (or\n)\nr\nuuur\nr  =\n\u02c6\n\u02c6\n\u02c6\nxi\nyj\nzk\n+\n+\nThis form of any vector is called its component form Here, x, y and z are called\nas the scalar components of rr , and \n\u02c6\n\u02c6\n\u02c6\n,\n  and\nxi\nyj\nzk  are called the vector components\nof rr along the respective axes"
    },
    {
        "Chapter": "1",
        "sentence_range": "5179-5182",
        "Text": "13\nFig 10 14\n\u00a9 NCERT\nnot to be republished\n MATHEMATICS\n434\nTherefore, it follows that\nOP1\nuuur  =\n1\n\u02c6\n\u02c6\nOQ + QP\nxi\nyj\n=\n+\nuuur\nuuur\nand\nOP\nuuur  =\n1\n1\n\u02c6\n\u02c6\n\u02c6\nOP + P P\nxi\nyj\nzk\n=\n+\n+\nuuur\nuuuur\nHence, the position vector of P with reference to O is given by\nOP (or\n)\nr\nuuur\nr  =\n\u02c6\n\u02c6\n\u02c6\nxi\nyj\nzk\n+\n+\nThis form of any vector is called its component form Here, x, y and z are called\nas the scalar components of rr , and \n\u02c6\n\u02c6\n\u02c6\n,\n  and\nxi\nyj\nzk  are called the vector components\nof rr along the respective axes Sometimes x, y and z are also termed as rectangular\ncomponents"
    },
    {
        "Chapter": "1",
        "sentence_range": "5180-5183",
        "Text": "14\n\u00a9 NCERT\nnot to be republished\n MATHEMATICS\n434\nTherefore, it follows that\nOP1\nuuur  =\n1\n\u02c6\n\u02c6\nOQ + QP\nxi\nyj\n=\n+\nuuur\nuuur\nand\nOP\nuuur  =\n1\n1\n\u02c6\n\u02c6\n\u02c6\nOP + P P\nxi\nyj\nzk\n=\n+\n+\nuuur\nuuuur\nHence, the position vector of P with reference to O is given by\nOP (or\n)\nr\nuuur\nr  =\n\u02c6\n\u02c6\n\u02c6\nxi\nyj\nzk\n+\n+\nThis form of any vector is called its component form Here, x, y and z are called\nas the scalar components of rr , and \n\u02c6\n\u02c6\n\u02c6\n,\n  and\nxi\nyj\nzk  are called the vector components\nof rr along the respective axes Sometimes x, y and z are also termed as rectangular\ncomponents The length of any vector \n\u02c6\n\u02c6\n\u02c6\nr\nxi\nyj\nzk\n=\n+\n+\nr\n, is readily determined by applying the\nPythagoras theorem twice"
    },
    {
        "Chapter": "1",
        "sentence_range": "5181-5184",
        "Text": "Here, x, y and z are called\nas the scalar components of rr , and \n\u02c6\n\u02c6\n\u02c6\n,\n  and\nxi\nyj\nzk  are called the vector components\nof rr along the respective axes Sometimes x, y and z are also termed as rectangular\ncomponents The length of any vector \n\u02c6\n\u02c6\n\u02c6\nr\nxi\nyj\nzk\n=\n+\n+\nr\n, is readily determined by applying the\nPythagoras theorem twice We note that in the right angle triangle OQP1 (Fig 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5182-5185",
        "Text": "Sometimes x, y and z are also termed as rectangular\ncomponents The length of any vector \n\u02c6\n\u02c6\n\u02c6\nr\nxi\nyj\nzk\n=\n+\n+\nr\n, is readily determined by applying the\nPythagoras theorem twice We note that in the right angle triangle OQP1 (Fig 10 14)\n| OP |1\nuuuur  =\n2\n2\n2\n2\n|OQ| +|QP |1\nx\ny\n=\n+\nuuuur\nuuur\n,\nand in the right angle triangle OP1P, we have\nOP\nuuur  =\n2\n2\n2\n2\n2\n1\n1\n| OP |\n| P P |\n(\n)\nx\ny\nz\n \n \n \n \nuuur\nuuur\nHence, the length of any vector \n\u02c6\n\u02c6\n\u02c6+\nr\nxi\nyj\nzk\n=\n+\nr\n is given by\n|\n|\nrr  =\n2\n2\n2\n\u02c6\n\u02c6\n\u02c6\n|\n| =\nxi\nyj\nzk\nx\ny\nz\n+\n+\n+\n+\nIf  and \na\nrb\nr\n are any two vectors given in the component form \n1\n2\n3 \u02c6\n\u02c6\n\u02c6+\na i\na j\na k\n+\n and\n1\n2\n3 \u02c6\n\u02c6\n\u02c6\nb i\nb j\nb k\n+\n+\n, respectively, then\n(i)\nthe sum (or resultant) of the vectors \naand \nrb\nr\n is given by\na\nb\n+\nr\nr\n =\n1\n1\n2\n2\n3\n3 \u02c6\n\u02c6\n\u02c6\n(\n)\n(\n)\n(\n)\na\nb i\na\nb\nj\na\nb k\n+\n+\n+\n+\n+\n(ii)\nthe difference of the vector \naand \nrb\nr\n is given by\na\n\u2212b\nr\nr\n=\n1\n1\n2\n2\n3\n3 \u02c6\n\u02c6\n\u02c6\n(\n)\n(\n)\n(\n)\na\nb i\na\nb\nj\na\nb k\n\u2212\n+\n\u2212\n+\n\u2212\n(iii)\nthe vectors \naand \nrb\nr\n are equal if and only if\na1 = b1, a2 = b2   and   a3 = b3\n(iv)\nthe multiplication of vector ar  by any scalar \u03bb is given by\na\n\u03bbr  =\n1\n2\n3 \u02c6\n\u02c6\n\u02c6\n(\n)\n(\n)\n(\n)\na i\na\nj\na k\n \n \n \n \n \n\u00a9 NCERT\nnot to be republished\nVECTOR ALGEBRA\n435\nThe addition of vectors and the multiplication of a vector by a scalar together give\nthe following distributive laws:\nLet \na and \nrb\nr\n be any two vectors, and k and m be any scalars"
    },
    {
        "Chapter": "1",
        "sentence_range": "5183-5186",
        "Text": "The length of any vector \n\u02c6\n\u02c6\n\u02c6\nr\nxi\nyj\nzk\n=\n+\n+\nr\n, is readily determined by applying the\nPythagoras theorem twice We note that in the right angle triangle OQP1 (Fig 10 14)\n| OP |1\nuuuur  =\n2\n2\n2\n2\n|OQ| +|QP |1\nx\ny\n=\n+\nuuuur\nuuur\n,\nand in the right angle triangle OP1P, we have\nOP\nuuur  =\n2\n2\n2\n2\n2\n1\n1\n| OP |\n| P P |\n(\n)\nx\ny\nz\n \n \n \n \nuuur\nuuur\nHence, the length of any vector \n\u02c6\n\u02c6\n\u02c6+\nr\nxi\nyj\nzk\n=\n+\nr\n is given by\n|\n|\nrr  =\n2\n2\n2\n\u02c6\n\u02c6\n\u02c6\n|\n| =\nxi\nyj\nzk\nx\ny\nz\n+\n+\n+\n+\nIf  and \na\nrb\nr\n are any two vectors given in the component form \n1\n2\n3 \u02c6\n\u02c6\n\u02c6+\na i\na j\na k\n+\n and\n1\n2\n3 \u02c6\n\u02c6\n\u02c6\nb i\nb j\nb k\n+\n+\n, respectively, then\n(i)\nthe sum (or resultant) of the vectors \naand \nrb\nr\n is given by\na\nb\n+\nr\nr\n =\n1\n1\n2\n2\n3\n3 \u02c6\n\u02c6\n\u02c6\n(\n)\n(\n)\n(\n)\na\nb i\na\nb\nj\na\nb k\n+\n+\n+\n+\n+\n(ii)\nthe difference of the vector \naand \nrb\nr\n is given by\na\n\u2212b\nr\nr\n=\n1\n1\n2\n2\n3\n3 \u02c6\n\u02c6\n\u02c6\n(\n)\n(\n)\n(\n)\na\nb i\na\nb\nj\na\nb k\n\u2212\n+\n\u2212\n+\n\u2212\n(iii)\nthe vectors \naand \nrb\nr\n are equal if and only if\na1 = b1, a2 = b2   and   a3 = b3\n(iv)\nthe multiplication of vector ar  by any scalar \u03bb is given by\na\n\u03bbr  =\n1\n2\n3 \u02c6\n\u02c6\n\u02c6\n(\n)\n(\n)\n(\n)\na i\na\nj\na k\n \n \n \n \n \n\u00a9 NCERT\nnot to be republished\nVECTOR ALGEBRA\n435\nThe addition of vectors and the multiplication of a vector by a scalar together give\nthe following distributive laws:\nLet \na and \nrb\nr\n be any two vectors, and k and m be any scalars Then\n(i)\n(\n)\nka\nma\nk\nm a\n+\n=\n+\nr\nr\nr\n(ii)\n(\n)\n(\n)\nk ma\nr=km a\nr\n(iii)\n(\n)\nk a\nb\nka\nkb\n \n \n \nr\nr\nr\nr\nRemarks\n(i)\nOne may observe that whatever be the value of \u03bb, the vector a\n\u03bbr  is always\ncollinear to the vector ar"
    },
    {
        "Chapter": "1",
        "sentence_range": "5184-5187",
        "Text": "We note that in the right angle triangle OQP1 (Fig 10 14)\n| OP |1\nuuuur  =\n2\n2\n2\n2\n|OQ| +|QP |1\nx\ny\n=\n+\nuuuur\nuuur\n,\nand in the right angle triangle OP1P, we have\nOP\nuuur  =\n2\n2\n2\n2\n2\n1\n1\n| OP |\n| P P |\n(\n)\nx\ny\nz\n \n \n \n \nuuur\nuuur\nHence, the length of any vector \n\u02c6\n\u02c6\n\u02c6+\nr\nxi\nyj\nzk\n=\n+\nr\n is given by\n|\n|\nrr  =\n2\n2\n2\n\u02c6\n\u02c6\n\u02c6\n|\n| =\nxi\nyj\nzk\nx\ny\nz\n+\n+\n+\n+\nIf  and \na\nrb\nr\n are any two vectors given in the component form \n1\n2\n3 \u02c6\n\u02c6\n\u02c6+\na i\na j\na k\n+\n and\n1\n2\n3 \u02c6\n\u02c6\n\u02c6\nb i\nb j\nb k\n+\n+\n, respectively, then\n(i)\nthe sum (or resultant) of the vectors \naand \nrb\nr\n is given by\na\nb\n+\nr\nr\n =\n1\n1\n2\n2\n3\n3 \u02c6\n\u02c6\n\u02c6\n(\n)\n(\n)\n(\n)\na\nb i\na\nb\nj\na\nb k\n+\n+\n+\n+\n+\n(ii)\nthe difference of the vector \naand \nrb\nr\n is given by\na\n\u2212b\nr\nr\n=\n1\n1\n2\n2\n3\n3 \u02c6\n\u02c6\n\u02c6\n(\n)\n(\n)\n(\n)\na\nb i\na\nb\nj\na\nb k\n\u2212\n+\n\u2212\n+\n\u2212\n(iii)\nthe vectors \naand \nrb\nr\n are equal if and only if\na1 = b1, a2 = b2   and   a3 = b3\n(iv)\nthe multiplication of vector ar  by any scalar \u03bb is given by\na\n\u03bbr  =\n1\n2\n3 \u02c6\n\u02c6\n\u02c6\n(\n)\n(\n)\n(\n)\na i\na\nj\na k\n \n \n \n \n \n\u00a9 NCERT\nnot to be republished\nVECTOR ALGEBRA\n435\nThe addition of vectors and the multiplication of a vector by a scalar together give\nthe following distributive laws:\nLet \na and \nrb\nr\n be any two vectors, and k and m be any scalars Then\n(i)\n(\n)\nka\nma\nk\nm a\n+\n=\n+\nr\nr\nr\n(ii)\n(\n)\n(\n)\nk ma\nr=km a\nr\n(iii)\n(\n)\nk a\nb\nka\nkb\n \n \n \nr\nr\nr\nr\nRemarks\n(i)\nOne may observe that whatever be the value of \u03bb, the vector a\n\u03bbr  is always\ncollinear to the vector ar In fact, two vectors  and \na\nrb\nr\n are collinear if and only\nif there exists a nonzero scalar \u03bb such that b\nr= \u03bba\nr"
    },
    {
        "Chapter": "1",
        "sentence_range": "5185-5188",
        "Text": "14)\n| OP |1\nuuuur  =\n2\n2\n2\n2\n|OQ| +|QP |1\nx\ny\n=\n+\nuuuur\nuuur\n,\nand in the right angle triangle OP1P, we have\nOP\nuuur  =\n2\n2\n2\n2\n2\n1\n1\n| OP |\n| P P |\n(\n)\nx\ny\nz\n \n \n \n \nuuur\nuuur\nHence, the length of any vector \n\u02c6\n\u02c6\n\u02c6+\nr\nxi\nyj\nzk\n=\n+\nr\n is given by\n|\n|\nrr  =\n2\n2\n2\n\u02c6\n\u02c6\n\u02c6\n|\n| =\nxi\nyj\nzk\nx\ny\nz\n+\n+\n+\n+\nIf  and \na\nrb\nr\n are any two vectors given in the component form \n1\n2\n3 \u02c6\n\u02c6\n\u02c6+\na i\na j\na k\n+\n and\n1\n2\n3 \u02c6\n\u02c6\n\u02c6\nb i\nb j\nb k\n+\n+\n, respectively, then\n(i)\nthe sum (or resultant) of the vectors \naand \nrb\nr\n is given by\na\nb\n+\nr\nr\n =\n1\n1\n2\n2\n3\n3 \u02c6\n\u02c6\n\u02c6\n(\n)\n(\n)\n(\n)\na\nb i\na\nb\nj\na\nb k\n+\n+\n+\n+\n+\n(ii)\nthe difference of the vector \naand \nrb\nr\n is given by\na\n\u2212b\nr\nr\n=\n1\n1\n2\n2\n3\n3 \u02c6\n\u02c6\n\u02c6\n(\n)\n(\n)\n(\n)\na\nb i\na\nb\nj\na\nb k\n\u2212\n+\n\u2212\n+\n\u2212\n(iii)\nthe vectors \naand \nrb\nr\n are equal if and only if\na1 = b1, a2 = b2   and   a3 = b3\n(iv)\nthe multiplication of vector ar  by any scalar \u03bb is given by\na\n\u03bbr  =\n1\n2\n3 \u02c6\n\u02c6\n\u02c6\n(\n)\n(\n)\n(\n)\na i\na\nj\na k\n \n \n \n \n \n\u00a9 NCERT\nnot to be republished\nVECTOR ALGEBRA\n435\nThe addition of vectors and the multiplication of a vector by a scalar together give\nthe following distributive laws:\nLet \na and \nrb\nr\n be any two vectors, and k and m be any scalars Then\n(i)\n(\n)\nka\nma\nk\nm a\n+\n=\n+\nr\nr\nr\n(ii)\n(\n)\n(\n)\nk ma\nr=km a\nr\n(iii)\n(\n)\nk a\nb\nka\nkb\n \n \n \nr\nr\nr\nr\nRemarks\n(i)\nOne may observe that whatever be the value of \u03bb, the vector a\n\u03bbr  is always\ncollinear to the vector ar In fact, two vectors  and \na\nrb\nr\n are collinear if and only\nif there exists a nonzero scalar \u03bb such that b\nr= \u03bba\nr If the vectors \na and \nrb\nr\n are\ngiven in the component form, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "5186-5189",
        "Text": "Then\n(i)\n(\n)\nka\nma\nk\nm a\n+\n=\n+\nr\nr\nr\n(ii)\n(\n)\n(\n)\nk ma\nr=km a\nr\n(iii)\n(\n)\nk a\nb\nka\nkb\n \n \n \nr\nr\nr\nr\nRemarks\n(i)\nOne may observe that whatever be the value of \u03bb, the vector a\n\u03bbr  is always\ncollinear to the vector ar In fact, two vectors  and \na\nrb\nr\n are collinear if and only\nif there exists a nonzero scalar \u03bb such that b\nr= \u03bba\nr If the vectors \na and \nrb\nr\n are\ngiven in the component form, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "5187-5190",
        "Text": "In fact, two vectors  and \na\nrb\nr\n are collinear if and only\nif there exists a nonzero scalar \u03bb such that b\nr= \u03bba\nr If the vectors \na and \nrb\nr\n are\ngiven in the component form, i e 1\n2\n3 \u02c6\n\u02c6\n\u02c6\na\na i\na j\na k\n=\n+\n+\nr\n and \n1\n2\n3 \u02c6\n\u02c6\n\u02c6\nb\nb i\nb j\nb k\n=\n+\n+\nr\n,\nthen the two vectors are collinear if and only if\n1\n2\n3 \u02c6\n\u02c6\n\u02c6\nb i\nb j\nb k\n+\n+\n =\n1\n2\n3 \u02c6\n\u02c6\n\u02c6\n(\n)\na i\na j\na k\n\u03bb\n+\n+\n\u21d4\n1\n2\n3 \u02c6\n\u02c6\n\u02c6\nb i\nb j\nb k\n+\n+\n =\n1\n2\n3 \u02c6\n\u02c6\n\u02c6\n(\n)\n(\n)\n(\n)\na i\na\nj\na k\n\u03bb\n+ \u03bb\n+ \u03bb\n\u21d4\n1\n1\nb\n= \u03bba\n,\n2\n2\n3\n3\n,\nb\na\nb\na\n= \u03bb\n= \u03bb\n\u21d4\n1\n1\nb\na  =\n3\n2\n2\n3\nb\nab\n=a\n= \u03bb\n(ii)\nIf \n1\n2\n3 \u02c6\n\u02c6\n\u02c6\na\na i\na j\na k\n \n \n \nr\n, then a1, a2, a3 are also called direction ratios of ar"
    },
    {
        "Chapter": "1",
        "sentence_range": "5188-5191",
        "Text": "If the vectors \na and \nrb\nr\n are\ngiven in the component form, i e 1\n2\n3 \u02c6\n\u02c6\n\u02c6\na\na i\na j\na k\n=\n+\n+\nr\n and \n1\n2\n3 \u02c6\n\u02c6\n\u02c6\nb\nb i\nb j\nb k\n=\n+\n+\nr\n,\nthen the two vectors are collinear if and only if\n1\n2\n3 \u02c6\n\u02c6\n\u02c6\nb i\nb j\nb k\n+\n+\n =\n1\n2\n3 \u02c6\n\u02c6\n\u02c6\n(\n)\na i\na j\na k\n\u03bb\n+\n+\n\u21d4\n1\n2\n3 \u02c6\n\u02c6\n\u02c6\nb i\nb j\nb k\n+\n+\n =\n1\n2\n3 \u02c6\n\u02c6\n\u02c6\n(\n)\n(\n)\n(\n)\na i\na\nj\na k\n\u03bb\n+ \u03bb\n+ \u03bb\n\u21d4\n1\n1\nb\n= \u03bba\n,\n2\n2\n3\n3\n,\nb\na\nb\na\n= \u03bb\n= \u03bb\n\u21d4\n1\n1\nb\na  =\n3\n2\n2\n3\nb\nab\n=a\n= \u03bb\n(ii)\nIf \n1\n2\n3 \u02c6\n\u02c6\n\u02c6\na\na i\na j\na k\n \n \n \nr\n, then a1, a2, a3 are also called direction ratios of ar (iii)\nIn case if it is given that  l, m, n are direction cosines of a vector, then \n\u02c6\n\u02c6\n\u02c6\nli\nmj\nnk\n+\n+\n= \n\u02c6\n\u02c6\n\u02c6\n(cos )\n(cos )\n(cos )\ni\nj\nk\n\u03b1\n+\n\u03b2\n+\n\u03b3\n is the unit vector in the direction of that vector,\nwhere \u03b1, \u03b2 and \u03b3 are the angles which the vector makes with x, y and z axes\nrespectively"
    },
    {
        "Chapter": "1",
        "sentence_range": "5189-5192",
        "Text": "e 1\n2\n3 \u02c6\n\u02c6\n\u02c6\na\na i\na j\na k\n=\n+\n+\nr\n and \n1\n2\n3 \u02c6\n\u02c6\n\u02c6\nb\nb i\nb j\nb k\n=\n+\n+\nr\n,\nthen the two vectors are collinear if and only if\n1\n2\n3 \u02c6\n\u02c6\n\u02c6\nb i\nb j\nb k\n+\n+\n =\n1\n2\n3 \u02c6\n\u02c6\n\u02c6\n(\n)\na i\na j\na k\n\u03bb\n+\n+\n\u21d4\n1\n2\n3 \u02c6\n\u02c6\n\u02c6\nb i\nb j\nb k\n+\n+\n =\n1\n2\n3 \u02c6\n\u02c6\n\u02c6\n(\n)\n(\n)\n(\n)\na i\na\nj\na k\n\u03bb\n+ \u03bb\n+ \u03bb\n\u21d4\n1\n1\nb\n= \u03bba\n,\n2\n2\n3\n3\n,\nb\na\nb\na\n= \u03bb\n= \u03bb\n\u21d4\n1\n1\nb\na  =\n3\n2\n2\n3\nb\nab\n=a\n= \u03bb\n(ii)\nIf \n1\n2\n3 \u02c6\n\u02c6\n\u02c6\na\na i\na j\na k\n \n \n \nr\n, then a1, a2, a3 are also called direction ratios of ar (iii)\nIn case if it is given that  l, m, n are direction cosines of a vector, then \n\u02c6\n\u02c6\n\u02c6\nli\nmj\nnk\n+\n+\n= \n\u02c6\n\u02c6\n\u02c6\n(cos )\n(cos )\n(cos )\ni\nj\nk\n\u03b1\n+\n\u03b2\n+\n\u03b3\n is the unit vector in the direction of that vector,\nwhere \u03b1, \u03b2 and \u03b3 are the angles which the vector makes with x, y and z axes\nrespectively Example 4 Find the values of x, y and z so that the vectors \n\u02c6\n\u02c6\n2\u02c6\na\nxi\nj\nzk\n=\n+\n+\nr\n and\n\u02c6\n\u02c6\n\u02c6\n2\nb\ni\nyj\nk\n=\n+\n+\nr\n are equal"
    },
    {
        "Chapter": "1",
        "sentence_range": "5190-5193",
        "Text": "1\n2\n3 \u02c6\n\u02c6\n\u02c6\na\na i\na j\na k\n=\n+\n+\nr\n and \n1\n2\n3 \u02c6\n\u02c6\n\u02c6\nb\nb i\nb j\nb k\n=\n+\n+\nr\n,\nthen the two vectors are collinear if and only if\n1\n2\n3 \u02c6\n\u02c6\n\u02c6\nb i\nb j\nb k\n+\n+\n =\n1\n2\n3 \u02c6\n\u02c6\n\u02c6\n(\n)\na i\na j\na k\n\u03bb\n+\n+\n\u21d4\n1\n2\n3 \u02c6\n\u02c6\n\u02c6\nb i\nb j\nb k\n+\n+\n =\n1\n2\n3 \u02c6\n\u02c6\n\u02c6\n(\n)\n(\n)\n(\n)\na i\na\nj\na k\n\u03bb\n+ \u03bb\n+ \u03bb\n\u21d4\n1\n1\nb\n= \u03bba\n,\n2\n2\n3\n3\n,\nb\na\nb\na\n= \u03bb\n= \u03bb\n\u21d4\n1\n1\nb\na  =\n3\n2\n2\n3\nb\nab\n=a\n= \u03bb\n(ii)\nIf \n1\n2\n3 \u02c6\n\u02c6\n\u02c6\na\na i\na j\na k\n \n \n \nr\n, then a1, a2, a3 are also called direction ratios of ar (iii)\nIn case if it is given that  l, m, n are direction cosines of a vector, then \n\u02c6\n\u02c6\n\u02c6\nli\nmj\nnk\n+\n+\n= \n\u02c6\n\u02c6\n\u02c6\n(cos )\n(cos )\n(cos )\ni\nj\nk\n\u03b1\n+\n\u03b2\n+\n\u03b3\n is the unit vector in the direction of that vector,\nwhere \u03b1, \u03b2 and \u03b3 are the angles which the vector makes with x, y and z axes\nrespectively Example 4 Find the values of x, y and z so that the vectors \n\u02c6\n\u02c6\n2\u02c6\na\nxi\nj\nzk\n=\n+\n+\nr\n and\n\u02c6\n\u02c6\n\u02c6\n2\nb\ni\nyj\nk\n=\n+\n+\nr\n are equal Solution Note that two vectors are equal if and only if their corresponding components\nare equal"
    },
    {
        "Chapter": "1",
        "sentence_range": "5191-5194",
        "Text": "(iii)\nIn case if it is given that  l, m, n are direction cosines of a vector, then \n\u02c6\n\u02c6\n\u02c6\nli\nmj\nnk\n+\n+\n= \n\u02c6\n\u02c6\n\u02c6\n(cos )\n(cos )\n(cos )\ni\nj\nk\n\u03b1\n+\n\u03b2\n+\n\u03b3\n is the unit vector in the direction of that vector,\nwhere \u03b1, \u03b2 and \u03b3 are the angles which the vector makes with x, y and z axes\nrespectively Example 4 Find the values of x, y and z so that the vectors \n\u02c6\n\u02c6\n2\u02c6\na\nxi\nj\nzk\n=\n+\n+\nr\n and\n\u02c6\n\u02c6\n\u02c6\n2\nb\ni\nyj\nk\n=\n+\n+\nr\n are equal Solution Note that two vectors are equal if and only if their corresponding components\nare equal Thus, the given vectors \naand \nrb\nr\n will be equal if and only if\nx = 2, y = 2, z = 1\n\u00a9 NCERT\nnot to be republished\n MATHEMATICS\n436\nExample 5 Let \n\u02c6\n2\u02c6\na\ni\nj\nr= +\n and \n\u02c6\n\u02c6\n2\nb\ni\nj\n=\n+\nr"
    },
    {
        "Chapter": "1",
        "sentence_range": "5192-5195",
        "Text": "Example 4 Find the values of x, y and z so that the vectors \n\u02c6\n\u02c6\n2\u02c6\na\nxi\nj\nzk\n=\n+\n+\nr\n and\n\u02c6\n\u02c6\n\u02c6\n2\nb\ni\nyj\nk\n=\n+\n+\nr\n are equal Solution Note that two vectors are equal if and only if their corresponding components\nare equal Thus, the given vectors \naand \nrb\nr\n will be equal if and only if\nx = 2, y = 2, z = 1\n\u00a9 NCERT\nnot to be republished\n MATHEMATICS\n436\nExample 5 Let \n\u02c6\n2\u02c6\na\ni\nj\nr= +\n and \n\u02c6\n\u02c6\n2\nb\ni\nj\n=\n+\nr Is |\n|\n|\n|\na\nb\n=\nr\nr"
    },
    {
        "Chapter": "1",
        "sentence_range": "5193-5196",
        "Text": "Solution Note that two vectors are equal if and only if their corresponding components\nare equal Thus, the given vectors \naand \nrb\nr\n will be equal if and only if\nx = 2, y = 2, z = 1\n\u00a9 NCERT\nnot to be republished\n MATHEMATICS\n436\nExample 5 Let \n\u02c6\n2\u02c6\na\ni\nj\nr= +\n and \n\u02c6\n\u02c6\n2\nb\ni\nj\n=\n+\nr Is |\n|\n|\n|\na\nb\n=\nr\nr Are the vectors \na and \nrb\nr\nequal"
    },
    {
        "Chapter": "1",
        "sentence_range": "5194-5197",
        "Text": "Thus, the given vectors \naand \nrb\nr\n will be equal if and only if\nx = 2, y = 2, z = 1\n\u00a9 NCERT\nnot to be republished\n MATHEMATICS\n436\nExample 5 Let \n\u02c6\n2\u02c6\na\ni\nj\nr= +\n and \n\u02c6\n\u02c6\n2\nb\ni\nj\n=\n+\nr Is |\n|\n|\n|\na\nb\n=\nr\nr Are the vectors \na and \nrb\nr\nequal Solution We have \n2\n2\n|\n|\n1\n2\n5\na =\n+\n=\nr\n and \n2\n2\n|\n|\n2\n1\n5\nb  \n \n \nr\nSo, |\n| |\n|\na\n=b\nr\nr"
    },
    {
        "Chapter": "1",
        "sentence_range": "5195-5198",
        "Text": "Is |\n|\n|\n|\na\nb\n=\nr\nr Are the vectors \na and \nrb\nr\nequal Solution We have \n2\n2\n|\n|\n1\n2\n5\na =\n+\n=\nr\n and \n2\n2\n|\n|\n2\n1\n5\nb  \n \n \nr\nSo, |\n| |\n|\na\n=b\nr\nr But, the two vectors are not equal since their corresponding components\nare distinct"
    },
    {
        "Chapter": "1",
        "sentence_range": "5196-5199",
        "Text": "Are the vectors \na and \nrb\nr\nequal Solution We have \n2\n2\n|\n|\n1\n2\n5\na =\n+\n=\nr\n and \n2\n2\n|\n|\n2\n1\n5\nb  \n \n \nr\nSo, |\n| |\n|\na\n=b\nr\nr But, the two vectors are not equal since their corresponding components\nare distinct Example 6 Find unit vector in the direction of vector \n\u02c6\n\u02c6\n\u02c6\n2\n3\na\ni\nj\nk\n=\n+\n+\nr\nSolution The unit vector in the direction of a vector ar  is given by \n1\n\u02c6\n|\n|\na\n=aa\nrr"
    },
    {
        "Chapter": "1",
        "sentence_range": "5197-5200",
        "Text": "Solution We have \n2\n2\n|\n|\n1\n2\n5\na =\n+\n=\nr\n and \n2\n2\n|\n|\n2\n1\n5\nb  \n \n \nr\nSo, |\n| |\n|\na\n=b\nr\nr But, the two vectors are not equal since their corresponding components\nare distinct Example 6 Find unit vector in the direction of vector \n\u02c6\n\u02c6\n\u02c6\n2\n3\na\ni\nj\nk\n=\n+\n+\nr\nSolution The unit vector in the direction of a vector ar  is given by \n1\n\u02c6\n|\n|\na\n=aa\nrr Now\n|\n|\nar  =\n2\n2\n2\n2\n3\n1\n14\n+\n+\n=\nTherefore\n1\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(2\n3\n)\n14\na\ni\nj\nk\n=\n+\n+\n =\n2\n3\n1\n\u02c6\n\u02c6\n\u02c6\n14\n14\n14\ni\nj\nk\n+\n+\nExample 7 Find a vector in the direction of vector \n\u02c6\n2\u02c6\na\ni\nj\nr= \u2212\n that has magnitude\n7 units"
    },
    {
        "Chapter": "1",
        "sentence_range": "5198-5201",
        "Text": "But, the two vectors are not equal since their corresponding components\nare distinct Example 6 Find unit vector in the direction of vector \n\u02c6\n\u02c6\n\u02c6\n2\n3\na\ni\nj\nk\n=\n+\n+\nr\nSolution The unit vector in the direction of a vector ar  is given by \n1\n\u02c6\n|\n|\na\n=aa\nrr Now\n|\n|\nar  =\n2\n2\n2\n2\n3\n1\n14\n+\n+\n=\nTherefore\n1\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(2\n3\n)\n14\na\ni\nj\nk\n=\n+\n+\n =\n2\n3\n1\n\u02c6\n\u02c6\n\u02c6\n14\n14\n14\ni\nj\nk\n+\n+\nExample 7 Find a vector in the direction of vector \n\u02c6\n2\u02c6\na\ni\nj\nr= \u2212\n that has magnitude\n7 units Solution The unit vector in the direction of the given vector ar  is\n1\n\u02c6\n|\n|\na\n=aa\nrr\n =\n1\n1\n2\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(\n2 )\n5\n5\n5\ni\nj\ni\nj\n\u2212\n=\n\u2212\nTherefore, the vector having magnitude equal to 7 and in the direction of ar is\n7a\n\u2227  =\n1\n2\n7\n5\n5\ni\nj\n\u2227\n\u2227\n\u239b\n\u239e\n\u2212\n\u239c\n\u239f\n\u239d\n\u23a0\n = \n7\n14\n\u02c6\n\u02c6\n5\n5\ni\nj\n\u2212\nExample 8 Find the unit vector in the direction of the sum of the vectors,\n\u02c6\n\u02c6\n\u02c6\n2\n2 \u2013 5\na\ni\nj\nk\n=\n+\nr\n and \n\u02c6\n\u02c6\n\u02c6\n2\n3\nb\ni\nj\nk\n=\n+\n+\nr"
    },
    {
        "Chapter": "1",
        "sentence_range": "5199-5202",
        "Text": "Example 6 Find unit vector in the direction of vector \n\u02c6\n\u02c6\n\u02c6\n2\n3\na\ni\nj\nk\n=\n+\n+\nr\nSolution The unit vector in the direction of a vector ar  is given by \n1\n\u02c6\n|\n|\na\n=aa\nrr Now\n|\n|\nar  =\n2\n2\n2\n2\n3\n1\n14\n+\n+\n=\nTherefore\n1\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(2\n3\n)\n14\na\ni\nj\nk\n=\n+\n+\n =\n2\n3\n1\n\u02c6\n\u02c6\n\u02c6\n14\n14\n14\ni\nj\nk\n+\n+\nExample 7 Find a vector in the direction of vector \n\u02c6\n2\u02c6\na\ni\nj\nr= \u2212\n that has magnitude\n7 units Solution The unit vector in the direction of the given vector ar  is\n1\n\u02c6\n|\n|\na\n=aa\nrr\n =\n1\n1\n2\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(\n2 )\n5\n5\n5\ni\nj\ni\nj\n\u2212\n=\n\u2212\nTherefore, the vector having magnitude equal to 7 and in the direction of ar is\n7a\n\u2227  =\n1\n2\n7\n5\n5\ni\nj\n\u2227\n\u2227\n\u239b\n\u239e\n\u2212\n\u239c\n\u239f\n\u239d\n\u23a0\n = \n7\n14\n\u02c6\n\u02c6\n5\n5\ni\nj\n\u2212\nExample 8 Find the unit vector in the direction of the sum of the vectors,\n\u02c6\n\u02c6\n\u02c6\n2\n2 \u2013 5\na\ni\nj\nk\n=\n+\nr\n and \n\u02c6\n\u02c6\n\u02c6\n2\n3\nb\ni\nj\nk\n=\n+\n+\nr Solution The sum of the given vectors is\n\u02c6\n\u02c6\n\u02c6\n(\n, say) = 4\n3\n2\n \n \n \n \nr\nr\nr\na\nb\nc\ni\nj\nk\nand\n|\n|\ncr  =\n2\n2\n2\n4\n3\n( 2)\n29\n+\n+ \u2212\n=\n\u00a9 NCERT\nnot to be republished\nVECTOR ALGEBRA\n437\nThus, the required unit vector is\n1\n1\n4\n3\n2\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(4\n3\n2 )\n|\n|\n29\n29\n29\n29\nc\nc\ni\nj\nk\ni\nj\nk\n=c\n=\n+\n\u2212\n=\n+\n\u2212\nr\nr\nExample 9  Write the direction ratio\u2019s of the vector \n\u02c6\n\u02c6\n\u02c6\n2\na\ni\nj\nk\n= +\n\u2212\nr\n and hence calculate\nits direction cosines"
    },
    {
        "Chapter": "1",
        "sentence_range": "5200-5203",
        "Text": "Now\n|\n|\nar  =\n2\n2\n2\n2\n3\n1\n14\n+\n+\n=\nTherefore\n1\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(2\n3\n)\n14\na\ni\nj\nk\n=\n+\n+\n =\n2\n3\n1\n\u02c6\n\u02c6\n\u02c6\n14\n14\n14\ni\nj\nk\n+\n+\nExample 7 Find a vector in the direction of vector \n\u02c6\n2\u02c6\na\ni\nj\nr= \u2212\n that has magnitude\n7 units Solution The unit vector in the direction of the given vector ar  is\n1\n\u02c6\n|\n|\na\n=aa\nrr\n =\n1\n1\n2\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(\n2 )\n5\n5\n5\ni\nj\ni\nj\n\u2212\n=\n\u2212\nTherefore, the vector having magnitude equal to 7 and in the direction of ar is\n7a\n\u2227  =\n1\n2\n7\n5\n5\ni\nj\n\u2227\n\u2227\n\u239b\n\u239e\n\u2212\n\u239c\n\u239f\n\u239d\n\u23a0\n = \n7\n14\n\u02c6\n\u02c6\n5\n5\ni\nj\n\u2212\nExample 8 Find the unit vector in the direction of the sum of the vectors,\n\u02c6\n\u02c6\n\u02c6\n2\n2 \u2013 5\na\ni\nj\nk\n=\n+\nr\n and \n\u02c6\n\u02c6\n\u02c6\n2\n3\nb\ni\nj\nk\n=\n+\n+\nr Solution The sum of the given vectors is\n\u02c6\n\u02c6\n\u02c6\n(\n, say) = 4\n3\n2\n \n \n \n \nr\nr\nr\na\nb\nc\ni\nj\nk\nand\n|\n|\ncr  =\n2\n2\n2\n4\n3\n( 2)\n29\n+\n+ \u2212\n=\n\u00a9 NCERT\nnot to be republished\nVECTOR ALGEBRA\n437\nThus, the required unit vector is\n1\n1\n4\n3\n2\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(4\n3\n2 )\n|\n|\n29\n29\n29\n29\nc\nc\ni\nj\nk\ni\nj\nk\n=c\n=\n+\n\u2212\n=\n+\n\u2212\nr\nr\nExample 9  Write the direction ratio\u2019s of the vector \n\u02c6\n\u02c6\n\u02c6\n2\na\ni\nj\nk\n= +\n\u2212\nr\n and hence calculate\nits direction cosines Solution Note that the direction ratio\u2019s a, b, c of a vector \n\u02c6\n\u02c6\n\u02c6\nr\nxi\nyj\nzk\n=\n+\n+\nr\n are just\nthe respective components x, y and z of the vector"
    },
    {
        "Chapter": "1",
        "sentence_range": "5201-5204",
        "Text": "Solution The unit vector in the direction of the given vector ar  is\n1\n\u02c6\n|\n|\na\n=aa\nrr\n =\n1\n1\n2\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(\n2 )\n5\n5\n5\ni\nj\ni\nj\n\u2212\n=\n\u2212\nTherefore, the vector having magnitude equal to 7 and in the direction of ar is\n7a\n\u2227  =\n1\n2\n7\n5\n5\ni\nj\n\u2227\n\u2227\n\u239b\n\u239e\n\u2212\n\u239c\n\u239f\n\u239d\n\u23a0\n = \n7\n14\n\u02c6\n\u02c6\n5\n5\ni\nj\n\u2212\nExample 8 Find the unit vector in the direction of the sum of the vectors,\n\u02c6\n\u02c6\n\u02c6\n2\n2 \u2013 5\na\ni\nj\nk\n=\n+\nr\n and \n\u02c6\n\u02c6\n\u02c6\n2\n3\nb\ni\nj\nk\n=\n+\n+\nr Solution The sum of the given vectors is\n\u02c6\n\u02c6\n\u02c6\n(\n, say) = 4\n3\n2\n \n \n \n \nr\nr\nr\na\nb\nc\ni\nj\nk\nand\n|\n|\ncr  =\n2\n2\n2\n4\n3\n( 2)\n29\n+\n+ \u2212\n=\n\u00a9 NCERT\nnot to be republished\nVECTOR ALGEBRA\n437\nThus, the required unit vector is\n1\n1\n4\n3\n2\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(4\n3\n2 )\n|\n|\n29\n29\n29\n29\nc\nc\ni\nj\nk\ni\nj\nk\n=c\n=\n+\n\u2212\n=\n+\n\u2212\nr\nr\nExample 9  Write the direction ratio\u2019s of the vector \n\u02c6\n\u02c6\n\u02c6\n2\na\ni\nj\nk\n= +\n\u2212\nr\n and hence calculate\nits direction cosines Solution Note that the direction ratio\u2019s a, b, c of a vector \n\u02c6\n\u02c6\n\u02c6\nr\nxi\nyj\nzk\n=\n+\n+\nr\n are just\nthe respective components x, y and z of the vector So, for the given vector, we have\na = 1, b = 1 and c = \u20132"
    },
    {
        "Chapter": "1",
        "sentence_range": "5202-5205",
        "Text": "Solution The sum of the given vectors is\n\u02c6\n\u02c6\n\u02c6\n(\n, say) = 4\n3\n2\n \n \n \n \nr\nr\nr\na\nb\nc\ni\nj\nk\nand\n|\n|\ncr  =\n2\n2\n2\n4\n3\n( 2)\n29\n+\n+ \u2212\n=\n\u00a9 NCERT\nnot to be republished\nVECTOR ALGEBRA\n437\nThus, the required unit vector is\n1\n1\n4\n3\n2\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(4\n3\n2 )\n|\n|\n29\n29\n29\n29\nc\nc\ni\nj\nk\ni\nj\nk\n=c\n=\n+\n\u2212\n=\n+\n\u2212\nr\nr\nExample 9  Write the direction ratio\u2019s of the vector \n\u02c6\n\u02c6\n\u02c6\n2\na\ni\nj\nk\n= +\n\u2212\nr\n and hence calculate\nits direction cosines Solution Note that the direction ratio\u2019s a, b, c of a vector \n\u02c6\n\u02c6\n\u02c6\nr\nxi\nyj\nzk\n=\n+\n+\nr\n are just\nthe respective components x, y and z of the vector So, for the given vector, we have\na = 1, b = 1 and c = \u20132 Further, if l, m and n are the direction cosines of the given\nvector, then\n1\n1\n2\n,\n,\n  as  | |\n6\n|\n|\n|\n|\n|\n|\n6\n6\n6\na\nb\nc\nl\nm\nn\nr\nr\nr\nr\n\u2212\n=\n=\n=\n=\n=\n=\nr=\nr\nr\nr\nThus, the direction cosines are \n1\n1\n2\n,\n,\u2013\n6\n6\n6\n\u239b\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0"
    },
    {
        "Chapter": "1",
        "sentence_range": "5203-5206",
        "Text": "Solution Note that the direction ratio\u2019s a, b, c of a vector \n\u02c6\n\u02c6\n\u02c6\nr\nxi\nyj\nzk\n=\n+\n+\nr\n are just\nthe respective components x, y and z of the vector So, for the given vector, we have\na = 1, b = 1 and c = \u20132 Further, if l, m and n are the direction cosines of the given\nvector, then\n1\n1\n2\n,\n,\n  as  | |\n6\n|\n|\n|\n|\n|\n|\n6\n6\n6\na\nb\nc\nl\nm\nn\nr\nr\nr\nr\n\u2212\n=\n=\n=\n=\n=\n=\nr=\nr\nr\nr\nThus, the direction cosines are \n1\n1\n2\n,\n,\u2013\n6\n6\n6\n\u239b\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5204-5207",
        "Text": "So, for the given vector, we have\na = 1, b = 1 and c = \u20132 Further, if l, m and n are the direction cosines of the given\nvector, then\n1\n1\n2\n,\n,\n  as  | |\n6\n|\n|\n|\n|\n|\n|\n6\n6\n6\na\nb\nc\nl\nm\nn\nr\nr\nr\nr\n\u2212\n=\n=\n=\n=\n=\n=\nr=\nr\nr\nr\nThus, the direction cosines are \n1\n1\n2\n,\n,\u2013\n6\n6\n6\n\u239b\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0 10 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "5205-5208",
        "Text": "Further, if l, m and n are the direction cosines of the given\nvector, then\n1\n1\n2\n,\n,\n  as  | |\n6\n|\n|\n|\n|\n|\n|\n6\n6\n6\na\nb\nc\nl\nm\nn\nr\nr\nr\nr\n\u2212\n=\n=\n=\n=\n=\n=\nr=\nr\nr\nr\nThus, the direction cosines are \n1\n1\n2\n,\n,\u2013\n6\n6\n6\n\u239b\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0 10 5 2  Vector joining two points\nIf P1(x1, y1, z1) and P2(x2, y2, z2) are any two points, then the vector joining P1 and P2\nis the vector \n1 2\nP P\nuuuur  (Fig 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5206-5209",
        "Text": "10 5 2  Vector joining two points\nIf P1(x1, y1, z1) and P2(x2, y2, z2) are any two points, then the vector joining P1 and P2\nis the vector \n1 2\nP P\nuuuur  (Fig 10 15)"
    },
    {
        "Chapter": "1",
        "sentence_range": "5207-5210",
        "Text": "5 2  Vector joining two points\nIf P1(x1, y1, z1) and P2(x2, y2, z2) are any two points, then the vector joining P1 and P2\nis the vector \n1 2\nP P\nuuuur  (Fig 10 15) Joining the points P1 and P2 with the origin\nO, and applying triangle law, from the triangle\nOP1P2, we have\n1\n1 2\nOP\nuuur+P P\nuuuur  =\nOP"
    },
    {
        "Chapter": "1",
        "sentence_range": "5208-5211",
        "Text": "2  Vector joining two points\nIf P1(x1, y1, z1) and P2(x2, y2, z2) are any two points, then the vector joining P1 and P2\nis the vector \n1 2\nP P\nuuuur  (Fig 10 15) Joining the points P1 and P2 with the origin\nO, and applying triangle law, from the triangle\nOP1P2, we have\n1\n1 2\nOP\nuuur+P P\nuuuur  =\nOP 2\nuuuur\nUsing the properties of vector addition, the\nabove equation becomes\n1 2\nP P\nuuuur  =\n2\n1\nOP\nOP\n\u2212\nuuuur\nuuur\ni"
    },
    {
        "Chapter": "1",
        "sentence_range": "5209-5212",
        "Text": "15) Joining the points P1 and P2 with the origin\nO, and applying triangle law, from the triangle\nOP1P2, we have\n1\n1 2\nOP\nuuur+P P\nuuuur  =\nOP 2\nuuuur\nUsing the properties of vector addition, the\nabove equation becomes\n1 2\nP P\nuuuur  =\n2\n1\nOP\nOP\n\u2212\nuuuur\nuuur\ni e"
    },
    {
        "Chapter": "1",
        "sentence_range": "5210-5213",
        "Text": "Joining the points P1 and P2 with the origin\nO, and applying triangle law, from the triangle\nOP1P2, we have\n1\n1 2\nOP\nuuur+P P\nuuuur  =\nOP 2\nuuuur\nUsing the properties of vector addition, the\nabove equation becomes\n1 2\nP P\nuuuur  =\n2\n1\nOP\nOP\n\u2212\nuuuur\nuuur\ni e 1 2\nP P\nuuuur  =\n2\n2\n2\n1\n1\n1\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(\n)\n(\n)\nx i\ny j\nz k\nx i\ny j\nz k\n+\n+\n\u2212\n+\n+\n=\n2\n1\n2\n1\n2\n1 \u02c6\n\u02c6\n\u02c6\n(\n)\n(\n)\n(\n)\nx\nx i\ny\ny\nj\nz\nz k\n\u2212\n+\n\u2212\n+\n\u2212\nThe magnitude of vector \n1 2\nP P\nuuuur\n is given by\n1 2\nP P\nuuuur  =\n2\n2\n2\n2\n1\n2\n1\n2\n1\n(\n)\n(\n)\n(\n)\nx\nx\ny\ny\nz\nz\n\u2212\n+\n\u2212\n+\n\u2212\nFig 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5211-5214",
        "Text": "2\nuuuur\nUsing the properties of vector addition, the\nabove equation becomes\n1 2\nP P\nuuuur  =\n2\n1\nOP\nOP\n\u2212\nuuuur\nuuur\ni e 1 2\nP P\nuuuur  =\n2\n2\n2\n1\n1\n1\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(\n)\n(\n)\nx i\ny j\nz k\nx i\ny j\nz k\n+\n+\n\u2212\n+\n+\n=\n2\n1\n2\n1\n2\n1 \u02c6\n\u02c6\n\u02c6\n(\n)\n(\n)\n(\n)\nx\nx i\ny\ny\nj\nz\nz k\n\u2212\n+\n\u2212\n+\n\u2212\nThe magnitude of vector \n1 2\nP P\nuuuur\n is given by\n1 2\nP P\nuuuur  =\n2\n2\n2\n2\n1\n2\n1\n2\n1\n(\n)\n(\n)\n(\n)\nx\nx\ny\ny\nz\nz\n\u2212\n+\n\u2212\n+\n\u2212\nFig 10 15\n\u00a9 NCERT\nnot to be republished\n MATHEMATICS\n438\nExample 10 Find the vector joining the points P(2, 3, 0) and Q(\u2013 1, \u2013 2, \u2013 4) directed\nfrom P to Q"
    },
    {
        "Chapter": "1",
        "sentence_range": "5212-5215",
        "Text": "e 1 2\nP P\nuuuur  =\n2\n2\n2\n1\n1\n1\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(\n)\n(\n)\nx i\ny j\nz k\nx i\ny j\nz k\n+\n+\n\u2212\n+\n+\n=\n2\n1\n2\n1\n2\n1 \u02c6\n\u02c6\n\u02c6\n(\n)\n(\n)\n(\n)\nx\nx i\ny\ny\nj\nz\nz k\n\u2212\n+\n\u2212\n+\n\u2212\nThe magnitude of vector \n1 2\nP P\nuuuur\n is given by\n1 2\nP P\nuuuur  =\n2\n2\n2\n2\n1\n2\n1\n2\n1\n(\n)\n(\n)\n(\n)\nx\nx\ny\ny\nz\nz\n\u2212\n+\n\u2212\n+\n\u2212\nFig 10 15\n\u00a9 NCERT\nnot to be republished\n MATHEMATICS\n438\nExample 10 Find the vector joining the points P(2, 3, 0) and Q(\u2013 1, \u2013 2, \u2013 4) directed\nfrom P to Q Solution Since the vector is to be directed from P to Q, clearly P is the initial point\nand Q is the terminal point"
    },
    {
        "Chapter": "1",
        "sentence_range": "5213-5216",
        "Text": "1 2\nP P\nuuuur  =\n2\n2\n2\n1\n1\n1\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(\n)\n(\n)\nx i\ny j\nz k\nx i\ny j\nz k\n+\n+\n\u2212\n+\n+\n=\n2\n1\n2\n1\n2\n1 \u02c6\n\u02c6\n\u02c6\n(\n)\n(\n)\n(\n)\nx\nx i\ny\ny\nj\nz\nz k\n\u2212\n+\n\u2212\n+\n\u2212\nThe magnitude of vector \n1 2\nP P\nuuuur\n is given by\n1 2\nP P\nuuuur  =\n2\n2\n2\n2\n1\n2\n1\n2\n1\n(\n)\n(\n)\n(\n)\nx\nx\ny\ny\nz\nz\n\u2212\n+\n\u2212\n+\n\u2212\nFig 10 15\n\u00a9 NCERT\nnot to be republished\n MATHEMATICS\n438\nExample 10 Find the vector joining the points P(2, 3, 0) and Q(\u2013 1, \u2013 2, \u2013 4) directed\nfrom P to Q Solution Since the vector is to be directed from P to Q, clearly P is the initial point\nand Q is the terminal point So, the required vector joining P and Q is the vector PQ\nuuur ,\ngiven by\nPQ\nuuur  =\n\u02c6\n\u02c6\n\u02c6\n( 1\n2)\n( 2\n3)\n( 4\n0)\ni\nj\nk\n\u2212 \u2212\n+ \u2212 \u2212\n+ \u2212 \u2212\ni"
    },
    {
        "Chapter": "1",
        "sentence_range": "5214-5217",
        "Text": "15\n\u00a9 NCERT\nnot to be republished\n MATHEMATICS\n438\nExample 10 Find the vector joining the points P(2, 3, 0) and Q(\u2013 1, \u2013 2, \u2013 4) directed\nfrom P to Q Solution Since the vector is to be directed from P to Q, clearly P is the initial point\nand Q is the terminal point So, the required vector joining P and Q is the vector PQ\nuuur ,\ngiven by\nPQ\nuuur  =\n\u02c6\n\u02c6\n\u02c6\n( 1\n2)\n( 2\n3)\n( 4\n0)\ni\nj\nk\n\u2212 \u2212\n+ \u2212 \u2212\n+ \u2212 \u2212\ni e"
    },
    {
        "Chapter": "1",
        "sentence_range": "5215-5218",
        "Text": "Solution Since the vector is to be directed from P to Q, clearly P is the initial point\nand Q is the terminal point So, the required vector joining P and Q is the vector PQ\nuuur ,\ngiven by\nPQ\nuuur  =\n\u02c6\n\u02c6\n\u02c6\n( 1\n2)\n( 2\n3)\n( 4\n0)\ni\nj\nk\n\u2212 \u2212\n+ \u2212 \u2212\n+ \u2212 \u2212\ni e PQ\nuuur  =\n\u02c6\n\u02c6\n\u02c6\n3\n5\n4"
    },
    {
        "Chapter": "1",
        "sentence_range": "5216-5219",
        "Text": "So, the required vector joining P and Q is the vector PQ\nuuur ,\ngiven by\nPQ\nuuur  =\n\u02c6\n\u02c6\n\u02c6\n( 1\n2)\n( 2\n3)\n( 4\n0)\ni\nj\nk\n\u2212 \u2212\n+ \u2212 \u2212\n+ \u2212 \u2212\ni e PQ\nuuur  =\n\u02c6\n\u02c6\n\u02c6\n3\n5\n4 i\nj\nk\n\u2212\n\u2212\n\u2212\n10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5217-5220",
        "Text": "e PQ\nuuur  =\n\u02c6\n\u02c6\n\u02c6\n3\n5\n4 i\nj\nk\n\u2212\n\u2212\n\u2212\n10 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "5218-5221",
        "Text": "PQ\nuuur  =\n\u02c6\n\u02c6\n\u02c6\n3\n5\n4 i\nj\nk\n\u2212\n\u2212\n\u2212\n10 5 3  Section formula\nLet P and Q be two points represented by the position vectorsOP and OQ\nuuur\nuuur , respectively,\nwith respect to the origin O"
    },
    {
        "Chapter": "1",
        "sentence_range": "5219-5222",
        "Text": "i\nj\nk\n\u2212\n\u2212\n\u2212\n10 5 3  Section formula\nLet P and Q be two points represented by the position vectorsOP and OQ\nuuur\nuuur , respectively,\nwith respect to the origin O Then the line segment\njoining the points P and Q may be divided by a third\npoint, say R, in two ways \u2013 internally (Fig 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5220-5223",
        "Text": "5 3  Section formula\nLet P and Q be two points represented by the position vectorsOP and OQ\nuuur\nuuur , respectively,\nwith respect to the origin O Then the line segment\njoining the points P and Q may be divided by a third\npoint, say R, in two ways \u2013 internally (Fig 10 16)\nand externally (Fig 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5221-5224",
        "Text": "3  Section formula\nLet P and Q be two points represented by the position vectorsOP and OQ\nuuur\nuuur , respectively,\nwith respect to the origin O Then the line segment\njoining the points P and Q may be divided by a third\npoint, say R, in two ways \u2013 internally (Fig 10 16)\nand externally (Fig 10 17)"
    },
    {
        "Chapter": "1",
        "sentence_range": "5222-5225",
        "Text": "Then the line segment\njoining the points P and Q may be divided by a third\npoint, say R, in two ways \u2013 internally (Fig 10 16)\nand externally (Fig 10 17) Here, we intend to find\nthe position vector OR\nuuur for the point R with respect\nto the origin O"
    },
    {
        "Chapter": "1",
        "sentence_range": "5223-5226",
        "Text": "16)\nand externally (Fig 10 17) Here, we intend to find\nthe position vector OR\nuuur for the point R with respect\nto the origin O We take the two cases one by one"
    },
    {
        "Chapter": "1",
        "sentence_range": "5224-5227",
        "Text": "17) Here, we intend to find\nthe position vector OR\nuuur for the point R with respect\nto the origin O We take the two cases one by one Case I When R divides PQ internally (Fig 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5225-5228",
        "Text": "Here, we intend to find\nthe position vector OR\nuuur for the point R with respect\nto the origin O We take the two cases one by one Case I When R divides PQ internally (Fig 10 16)"
    },
    {
        "Chapter": "1",
        "sentence_range": "5226-5229",
        "Text": "We take the two cases one by one Case I When R divides PQ internally (Fig 10 16) If R divides PQ\nuuur\n such that \nRQ\nm\nuuur  = \nnPR\nuuur ,\nwhere m and n are positive scalars, we say that the point R divides PQ\nuuur  internally in the\nratio of m : n"
    },
    {
        "Chapter": "1",
        "sentence_range": "5227-5230",
        "Text": "Case I When R divides PQ internally (Fig 10 16) If R divides PQ\nuuur\n such that \nRQ\nm\nuuur  = \nnPR\nuuur ,\nwhere m and n are positive scalars, we say that the point R divides PQ\nuuur  internally in the\nratio of m : n Now  from triangles ORQ and OPR, we have\nRQ\nuuur  = OQ\nOR\nb\nr\n\u2212\n=\n\u2212\nuuur\nuuur\nr\nr\nand\nPR\nuuur  = OR\nOP\nr\na\n\u2212\n=\n\u2212\nuuur\nuuur\nr\nr ,\nTherefore, we have\n(\n)\nm b\nr\n\u2212\nr\nr  =\n(\n)\nn r\nr\u2212a\nr     (Why"
    },
    {
        "Chapter": "1",
        "sentence_range": "5228-5231",
        "Text": "16) If R divides PQ\nuuur\n such that \nRQ\nm\nuuur  = \nnPR\nuuur ,\nwhere m and n are positive scalars, we say that the point R divides PQ\nuuur  internally in the\nratio of m : n Now  from triangles ORQ and OPR, we have\nRQ\nuuur  = OQ\nOR\nb\nr\n\u2212\n=\n\u2212\nuuur\nuuur\nr\nr\nand\nPR\nuuur  = OR\nOP\nr\na\n\u2212\n=\n\u2212\nuuur\nuuur\nr\nr ,\nTherefore, we have\n(\n)\nm b\nr\n\u2212\nr\nr  =\n(\n)\nn r\nr\u2212a\nr     (Why )\nor\nrr  = mb\nna\nm\n+n\n+\nr\nr\n(on simplification)\nHence, the position vector of the point R which divides P and Q internally in the\nratio of m : n is given by\nOR\nuuur  = mb\nna\nm\n+n\n+\nr\nr\nFig 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5229-5232",
        "Text": "If R divides PQ\nuuur\n such that \nRQ\nm\nuuur  = \nnPR\nuuur ,\nwhere m and n are positive scalars, we say that the point R divides PQ\nuuur  internally in the\nratio of m : n Now  from triangles ORQ and OPR, we have\nRQ\nuuur  = OQ\nOR\nb\nr\n\u2212\n=\n\u2212\nuuur\nuuur\nr\nr\nand\nPR\nuuur  = OR\nOP\nr\na\n\u2212\n=\n\u2212\nuuur\nuuur\nr\nr ,\nTherefore, we have\n(\n)\nm b\nr\n\u2212\nr\nr  =\n(\n)\nn r\nr\u2212a\nr     (Why )\nor\nrr  = mb\nna\nm\n+n\n+\nr\nr\n(on simplification)\nHence, the position vector of the point R which divides P and Q internally in the\nratio of m : n is given by\nOR\nuuur  = mb\nna\nm\n+n\n+\nr\nr\nFig 10 16\n\u00a9 NCERT\nnot to be republished\nVECTOR ALGEBRA\n439\nCase II When R divides PQ externally (Fig 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5230-5233",
        "Text": "Now  from triangles ORQ and OPR, we have\nRQ\nuuur  = OQ\nOR\nb\nr\n\u2212\n=\n\u2212\nuuur\nuuur\nr\nr\nand\nPR\nuuur  = OR\nOP\nr\na\n\u2212\n=\n\u2212\nuuur\nuuur\nr\nr ,\nTherefore, we have\n(\n)\nm b\nr\n\u2212\nr\nr  =\n(\n)\nn r\nr\u2212a\nr     (Why )\nor\nrr  = mb\nna\nm\n+n\n+\nr\nr\n(on simplification)\nHence, the position vector of the point R which divides P and Q internally in the\nratio of m : n is given by\nOR\nuuur  = mb\nna\nm\n+n\n+\nr\nr\nFig 10 16\n\u00a9 NCERT\nnot to be republished\nVECTOR ALGEBRA\n439\nCase II When R divides PQ externally (Fig 10 17)"
    },
    {
        "Chapter": "1",
        "sentence_range": "5231-5234",
        "Text": ")\nor\nrr  = mb\nna\nm\n+n\n+\nr\nr\n(on simplification)\nHence, the position vector of the point R which divides P and Q internally in the\nratio of m : n is given by\nOR\nuuur  = mb\nna\nm\n+n\n+\nr\nr\nFig 10 16\n\u00a9 NCERT\nnot to be republished\nVECTOR ALGEBRA\n439\nCase II When R divides PQ externally (Fig 10 17) We leave it to the reader as an exercise to verify\nthat the position vector of the point R which divides\nthe line segment PQ externally in the ratio\nm : n \nPR\n i"
    },
    {
        "Chapter": "1",
        "sentence_range": "5232-5235",
        "Text": "16\n\u00a9 NCERT\nnot to be republished\nVECTOR ALGEBRA\n439\nCase II When R divides PQ externally (Fig 10 17) We leave it to the reader as an exercise to verify\nthat the position vector of the point R which divides\nthe line segment PQ externally in the ratio\nm : n \nPR\n i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "5233-5236",
        "Text": "17) We leave it to the reader as an exercise to verify\nthat the position vector of the point R which divides\nthe line segment PQ externally in the ratio\nm : n \nPR\n i e QR\n \n \n \n \n \n \nnm\n is given by\nOR\nuuur  =\nmb\nna\nm\n\u2212n\n\u2212\nr\nr\nRemark If R is the midpoint of PQ , then m = n"
    },
    {
        "Chapter": "1",
        "sentence_range": "5234-5237",
        "Text": "We leave it to the reader as an exercise to verify\nthat the position vector of the point R which divides\nthe line segment PQ externally in the ratio\nm : n \nPR\n i e QR\n \n \n \n \n \n \nnm\n is given by\nOR\nuuur  =\nmb\nna\nm\n\u2212n\n\u2212\nr\nr\nRemark If R is the midpoint of PQ , then m = n And therefore, from Case I, the\nmidpoint R of  PQ\nuuur\n, will have its position vector as\nOR\nuuur  =\na2\nb\n+\nr\nr\nExample 11 Consider two points P and Q with position vectors OP\n3\n2\na\nb\n=\n\u2212\nuuur\nr\nr\n and\nOQ\na\nb\n \n \nuuur\nr\nr"
    },
    {
        "Chapter": "1",
        "sentence_range": "5235-5238",
        "Text": "e QR\n \n \n \n \n \n \nnm\n is given by\nOR\nuuur  =\nmb\nna\nm\n\u2212n\n\u2212\nr\nr\nRemark If R is the midpoint of PQ , then m = n And therefore, from Case I, the\nmidpoint R of  PQ\nuuur\n, will have its position vector as\nOR\nuuur  =\na2\nb\n+\nr\nr\nExample 11 Consider two points P and Q with position vectors OP\n3\n2\na\nb\n=\n\u2212\nuuur\nr\nr\n and\nOQ\na\nb\n \n \nuuur\nr\nr Find the position vector of a point R which divides the line joining P and Q\nin the ratio 2:1, (i) internally, and (ii) externally"
    },
    {
        "Chapter": "1",
        "sentence_range": "5236-5239",
        "Text": "QR\n \n \n \n \n \n \nnm\n is given by\nOR\nuuur  =\nmb\nna\nm\n\u2212n\n\u2212\nr\nr\nRemark If R is the midpoint of PQ , then m = n And therefore, from Case I, the\nmidpoint R of  PQ\nuuur\n, will have its position vector as\nOR\nuuur  =\na2\nb\n+\nr\nr\nExample 11 Consider two points P and Q with position vectors OP\n3\n2\na\nb\n=\n\u2212\nuuur\nr\nr\n and\nOQ\na\nb\n \n \nuuur\nr\nr Find the position vector of a point R which divides the line joining P and Q\nin the ratio 2:1, (i) internally, and (ii) externally Solution\n(i)\nThe position vector of the point R dividing the join of P and Q internally in the\nratio 2:1 is\nOR\nuuur  = 2(\n)\n(3\n2 )\n5\n2\n1\n3\na\nb\na\nb\na\n+\n+\n\u2212\n=\n+\nr\nr\nr\nr\nr\n(ii)\nThe position vector of the point R dividing the join of P and Q externally in the\nratio 2:1 is\nOR\nuuur  = 2(\n)\n(3\n2 )\n4\n2\n1\na\nb\na\nb\nb\na\n+\n\u2212\n\u2212\n=\n\u2212\n\u2212\nr\nr\nr\nr\nr\nr\nExample 12 Show that the points \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\nA(2\n), B(\n3\n5 ), C(3\n4\n4 )\ni\nj\nk\ni\nj\nk\ni\nj\nk\n \n \n \n \n \n \n are\nthe vertices of a right angled triangle"
    },
    {
        "Chapter": "1",
        "sentence_range": "5237-5240",
        "Text": "And therefore, from Case I, the\nmidpoint R of  PQ\nuuur\n, will have its position vector as\nOR\nuuur  =\na2\nb\n+\nr\nr\nExample 11 Consider two points P and Q with position vectors OP\n3\n2\na\nb\n=\n\u2212\nuuur\nr\nr\n and\nOQ\na\nb\n \n \nuuur\nr\nr Find the position vector of a point R which divides the line joining P and Q\nin the ratio 2:1, (i) internally, and (ii) externally Solution\n(i)\nThe position vector of the point R dividing the join of P and Q internally in the\nratio 2:1 is\nOR\nuuur  = 2(\n)\n(3\n2 )\n5\n2\n1\n3\na\nb\na\nb\na\n+\n+\n\u2212\n=\n+\nr\nr\nr\nr\nr\n(ii)\nThe position vector of the point R dividing the join of P and Q externally in the\nratio 2:1 is\nOR\nuuur  = 2(\n)\n(3\n2 )\n4\n2\n1\na\nb\na\nb\nb\na\n+\n\u2212\n\u2212\n=\n\u2212\n\u2212\nr\nr\nr\nr\nr\nr\nExample 12 Show that the points \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\nA(2\n), B(\n3\n5 ), C(3\n4\n4 )\ni\nj\nk\ni\nj\nk\ni\nj\nk\n \n \n \n \n \n \n are\nthe vertices of a right angled triangle Solution We have\n AB\nuuur  =\n\u02c6\n\u02c6\n\u02c6\n(1\n2)\n( 3 1)\n( 5 1)\ni\nj\nk\n\u2212\n+ \u2212 +\n+ \u2212 \u2212\n\u02c6\n\u02c6\n2\u02c6\n6\ni\nj\nk\n  \n \n \n  BC\nuuur  =\n\u02c6\n\u02c6\n\u02c6\n(3 1)\n( 4\n3)\n( 4\n5)\ni\nj\nk\n\u2212\n+ \u2212 +\n+ \u2212 +\n\u02c6\n\u02c6\n\u02c6\n2i\nj\nk\n=\n\u2212\n+\nand\nCA\nuuur\n =\n\u02c6\n\u02c6\n\u02c6\n(2\n3)\n( 1\n4)\n(1\n4)\ni\nj\nk\n\u2212\n+ \u2212 +\n+\n+\n \n\u02c6\n\u02c6\n3\u02c6\n5\ni\nj\nk\n= \u2212 +\n+\nFig 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5238-5241",
        "Text": "Find the position vector of a point R which divides the line joining P and Q\nin the ratio 2:1, (i) internally, and (ii) externally Solution\n(i)\nThe position vector of the point R dividing the join of P and Q internally in the\nratio 2:1 is\nOR\nuuur  = 2(\n)\n(3\n2 )\n5\n2\n1\n3\na\nb\na\nb\na\n+\n+\n\u2212\n=\n+\nr\nr\nr\nr\nr\n(ii)\nThe position vector of the point R dividing the join of P and Q externally in the\nratio 2:1 is\nOR\nuuur  = 2(\n)\n(3\n2 )\n4\n2\n1\na\nb\na\nb\nb\na\n+\n\u2212\n\u2212\n=\n\u2212\n\u2212\nr\nr\nr\nr\nr\nr\nExample 12 Show that the points \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\nA(2\n), B(\n3\n5 ), C(3\n4\n4 )\ni\nj\nk\ni\nj\nk\ni\nj\nk\n \n \n \n \n \n \n are\nthe vertices of a right angled triangle Solution We have\n AB\nuuur  =\n\u02c6\n\u02c6\n\u02c6\n(1\n2)\n( 3 1)\n( 5 1)\ni\nj\nk\n\u2212\n+ \u2212 +\n+ \u2212 \u2212\n\u02c6\n\u02c6\n2\u02c6\n6\ni\nj\nk\n  \n \n \n  BC\nuuur  =\n\u02c6\n\u02c6\n\u02c6\n(3 1)\n( 4\n3)\n( 4\n5)\ni\nj\nk\n\u2212\n+ \u2212 +\n+ \u2212 +\n\u02c6\n\u02c6\n\u02c6\n2i\nj\nk\n=\n\u2212\n+\nand\nCA\nuuur\n =\n\u02c6\n\u02c6\n\u02c6\n(2\n3)\n( 1\n4)\n(1\n4)\ni\nj\nk\n\u2212\n+ \u2212 +\n+\n+\n \n\u02c6\n\u02c6\n3\u02c6\n5\ni\nj\nk\n= \u2212 +\n+\nFig 10 17\n\u00a9 NCERT\nnot to be republished\n MATHEMATICS\n440\nFurther, note that\n| AB|2\nuuur\n =\n2\n2\n41\n6\n35 | BC |\n| CA |\n=\n+\n=\n+\nuuur\nuuur\nHence, the triangle is a right angled triangle"
    },
    {
        "Chapter": "1",
        "sentence_range": "5239-5242",
        "Text": "Solution\n(i)\nThe position vector of the point R dividing the join of P and Q internally in the\nratio 2:1 is\nOR\nuuur  = 2(\n)\n(3\n2 )\n5\n2\n1\n3\na\nb\na\nb\na\n+\n+\n\u2212\n=\n+\nr\nr\nr\nr\nr\n(ii)\nThe position vector of the point R dividing the join of P and Q externally in the\nratio 2:1 is\nOR\nuuur  = 2(\n)\n(3\n2 )\n4\n2\n1\na\nb\na\nb\nb\na\n+\n\u2212\n\u2212\n=\n\u2212\n\u2212\nr\nr\nr\nr\nr\nr\nExample 12 Show that the points \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\nA(2\n), B(\n3\n5 ), C(3\n4\n4 )\ni\nj\nk\ni\nj\nk\ni\nj\nk\n \n \n \n \n \n \n are\nthe vertices of a right angled triangle Solution We have\n AB\nuuur  =\n\u02c6\n\u02c6\n\u02c6\n(1\n2)\n( 3 1)\n( 5 1)\ni\nj\nk\n\u2212\n+ \u2212 +\n+ \u2212 \u2212\n\u02c6\n\u02c6\n2\u02c6\n6\ni\nj\nk\n  \n \n \n  BC\nuuur  =\n\u02c6\n\u02c6\n\u02c6\n(3 1)\n( 4\n3)\n( 4\n5)\ni\nj\nk\n\u2212\n+ \u2212 +\n+ \u2212 +\n\u02c6\n\u02c6\n\u02c6\n2i\nj\nk\n=\n\u2212\n+\nand\nCA\nuuur\n =\n\u02c6\n\u02c6\n\u02c6\n(2\n3)\n( 1\n4)\n(1\n4)\ni\nj\nk\n\u2212\n+ \u2212 +\n+\n+\n \n\u02c6\n\u02c6\n3\u02c6\n5\ni\nj\nk\n= \u2212 +\n+\nFig 10 17\n\u00a9 NCERT\nnot to be republished\n MATHEMATICS\n440\nFurther, note that\n| AB|2\nuuur\n =\n2\n2\n41\n6\n35 | BC |\n| CA |\n=\n+\n=\n+\nuuur\nuuur\nHence, the triangle is a right angled triangle EXERCISE 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5240-5243",
        "Text": "Solution We have\n AB\nuuur  =\n\u02c6\n\u02c6\n\u02c6\n(1\n2)\n( 3 1)\n( 5 1)\ni\nj\nk\n\u2212\n+ \u2212 +\n+ \u2212 \u2212\n\u02c6\n\u02c6\n2\u02c6\n6\ni\nj\nk\n  \n \n \n  BC\nuuur  =\n\u02c6\n\u02c6\n\u02c6\n(3 1)\n( 4\n3)\n( 4\n5)\ni\nj\nk\n\u2212\n+ \u2212 +\n+ \u2212 +\n\u02c6\n\u02c6\n\u02c6\n2i\nj\nk\n=\n\u2212\n+\nand\nCA\nuuur\n =\n\u02c6\n\u02c6\n\u02c6\n(2\n3)\n( 1\n4)\n(1\n4)\ni\nj\nk\n\u2212\n+ \u2212 +\n+\n+\n \n\u02c6\n\u02c6\n3\u02c6\n5\ni\nj\nk\n= \u2212 +\n+\nFig 10 17\n\u00a9 NCERT\nnot to be republished\n MATHEMATICS\n440\nFurther, note that\n| AB|2\nuuur\n =\n2\n2\n41\n6\n35 | BC |\n| CA |\n=\n+\n=\n+\nuuur\nuuur\nHence, the triangle is a right angled triangle EXERCISE 10 2\n1"
    },
    {
        "Chapter": "1",
        "sentence_range": "5241-5244",
        "Text": "17\n\u00a9 NCERT\nnot to be republished\n MATHEMATICS\n440\nFurther, note that\n| AB|2\nuuur\n =\n2\n2\n41\n6\n35 | BC |\n| CA |\n=\n+\n=\n+\nuuur\nuuur\nHence, the triangle is a right angled triangle EXERCISE 10 2\n1 Compute the magnitude of the following vectors:\n1\n1\n1\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n;\n2\n7\n3 ;   \n3\n3\n3\na\ni\nj\nk\nb\ni\nj\nk\nc\ni\nj\nk\n= +\n+\n=\n\u2212\n\u2212\n=\n+\n\u2212\nr\nr\nr\n2"
    },
    {
        "Chapter": "1",
        "sentence_range": "5242-5245",
        "Text": "EXERCISE 10 2\n1 Compute the magnitude of the following vectors:\n1\n1\n1\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n;\n2\n7\n3 ;   \n3\n3\n3\na\ni\nj\nk\nb\ni\nj\nk\nc\ni\nj\nk\n= +\n+\n=\n\u2212\n\u2212\n=\n+\n\u2212\nr\nr\nr\n2 Write two different vectors having same magnitude"
    },
    {
        "Chapter": "1",
        "sentence_range": "5243-5246",
        "Text": "2\n1 Compute the magnitude of the following vectors:\n1\n1\n1\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n;\n2\n7\n3 ;   \n3\n3\n3\na\ni\nj\nk\nb\ni\nj\nk\nc\ni\nj\nk\n= +\n+\n=\n\u2212\n\u2212\n=\n+\n\u2212\nr\nr\nr\n2 Write two different vectors having same magnitude 3"
    },
    {
        "Chapter": "1",
        "sentence_range": "5244-5247",
        "Text": "Compute the magnitude of the following vectors:\n1\n1\n1\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n;\n2\n7\n3 ;   \n3\n3\n3\na\ni\nj\nk\nb\ni\nj\nk\nc\ni\nj\nk\n= +\n+\n=\n\u2212\n\u2212\n=\n+\n\u2212\nr\nr\nr\n2 Write two different vectors having same magnitude 3 Write two different vectors having same direction"
    },
    {
        "Chapter": "1",
        "sentence_range": "5245-5248",
        "Text": "Write two different vectors having same magnitude 3 Write two different vectors having same direction 4"
    },
    {
        "Chapter": "1",
        "sentence_range": "5246-5249",
        "Text": "3 Write two different vectors having same direction 4 Find the values of x and y so that the vectors \u02c6\n\u02c6\n\u02c6\n\u02c6\n2\n3   and  \ni\nj\nxi\nyj\n+\n+\n are equal"
    },
    {
        "Chapter": "1",
        "sentence_range": "5247-5250",
        "Text": "Write two different vectors having same direction 4 Find the values of x and y so that the vectors \u02c6\n\u02c6\n\u02c6\n\u02c6\n2\n3   and  \ni\nj\nxi\nyj\n+\n+\n are equal 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "5248-5251",
        "Text": "4 Find the values of x and y so that the vectors \u02c6\n\u02c6\n\u02c6\n\u02c6\n2\n3   and  \ni\nj\nxi\nyj\n+\n+\n are equal 5 Find the scalar and vector components of the vector with initial point (2, 1) and\nterminal point (\u2013 5, 7)"
    },
    {
        "Chapter": "1",
        "sentence_range": "5249-5252",
        "Text": "Find the values of x and y so that the vectors \u02c6\n\u02c6\n\u02c6\n\u02c6\n2\n3   and  \ni\nj\nxi\nyj\n+\n+\n are equal 5 Find the scalar and vector components of the vector with initial point (2, 1) and\nterminal point (\u2013 5, 7) 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "5250-5253",
        "Text": "5 Find the scalar and vector components of the vector with initial point (2, 1) and\nterminal point (\u2013 5, 7) 6 Find the sum of the vectors \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n2\n,\n2\n4\n5\na\ni\nj\nk b\ni\nj\nk\n= \u2212\n+\n= \u2212\n+\n+\nr\nr\nand \n\u02c6\n\u02c6\n\u02c6\n6  \u2013 7\nc\ni\nj\nk\n=\n\u2212\nr"
    },
    {
        "Chapter": "1",
        "sentence_range": "5251-5254",
        "Text": "Find the scalar and vector components of the vector with initial point (2, 1) and\nterminal point (\u2013 5, 7) 6 Find the sum of the vectors \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n2\n,\n2\n4\n5\na\ni\nj\nk b\ni\nj\nk\n= \u2212\n+\n= \u2212\n+\n+\nr\nr\nand \n\u02c6\n\u02c6\n\u02c6\n6  \u2013 7\nc\ni\nj\nk\n=\n\u2212\nr 7"
    },
    {
        "Chapter": "1",
        "sentence_range": "5252-5255",
        "Text": "6 Find the sum of the vectors \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n2\n,\n2\n4\n5\na\ni\nj\nk b\ni\nj\nk\n= \u2212\n+\n= \u2212\n+\n+\nr\nr\nand \n\u02c6\n\u02c6\n\u02c6\n6  \u2013 7\nc\ni\nj\nk\n=\n\u2212\nr 7 Find the unit vector in the direction of the vector \n\u02c6\n\u02c6\n\u02c6\n2\na\ni\nj\nk\n= +\n+\nr"
    },
    {
        "Chapter": "1",
        "sentence_range": "5253-5256",
        "Text": "Find the sum of the vectors \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n2\n,\n2\n4\n5\na\ni\nj\nk b\ni\nj\nk\n= \u2212\n+\n= \u2212\n+\n+\nr\nr\nand \n\u02c6\n\u02c6\n\u02c6\n6  \u2013 7\nc\ni\nj\nk\n=\n\u2212\nr 7 Find the unit vector in the direction of the vector \n\u02c6\n\u02c6\n\u02c6\n2\na\ni\nj\nk\n= +\n+\nr 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "5254-5257",
        "Text": "7 Find the unit vector in the direction of the vector \n\u02c6\n\u02c6\n\u02c6\n2\na\ni\nj\nk\n= +\n+\nr 8 Find the unit vector in the direction of vector PQ,\nuuur\n where P and Q are the points\n(1, 2, 3) and (4, 5, 6), respectively"
    },
    {
        "Chapter": "1",
        "sentence_range": "5255-5258",
        "Text": "Find the unit vector in the direction of the vector \n\u02c6\n\u02c6\n\u02c6\n2\na\ni\nj\nk\n= +\n+\nr 8 Find the unit vector in the direction of vector PQ,\nuuur\n where P and Q are the points\n(1, 2, 3) and (4, 5, 6), respectively 9"
    },
    {
        "Chapter": "1",
        "sentence_range": "5256-5259",
        "Text": "8 Find the unit vector in the direction of vector PQ,\nuuur\n where P and Q are the points\n(1, 2, 3) and (4, 5, 6), respectively 9 For given vectors, \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n2\n2   and\na\ni\nj\nk\nb\ni\nj\nk\n=\n\u2212\n+\n= \u2212 +\n\u2212\nr\nr\n, find the unit vector in the\ndirection of the vector a\nb\n+\nr\nr"
    },
    {
        "Chapter": "1",
        "sentence_range": "5257-5260",
        "Text": "Find the unit vector in the direction of vector PQ,\nuuur\n where P and Q are the points\n(1, 2, 3) and (4, 5, 6), respectively 9 For given vectors, \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n2\n2   and\na\ni\nj\nk\nb\ni\nj\nk\n=\n\u2212\n+\n= \u2212 +\n\u2212\nr\nr\n, find the unit vector in the\ndirection of the vector a\nb\n+\nr\nr 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5258-5261",
        "Text": "9 For given vectors, \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n2\n2   and\na\ni\nj\nk\nb\ni\nj\nk\n=\n\u2212\n+\n= \u2212 +\n\u2212\nr\nr\n, find the unit vector in the\ndirection of the vector a\nb\n+\nr\nr 10 Find a vector  in the direction of vector \n\u02c6\n\u02c6\n\u02c6\n5\n2\ni\nj\nk\n\u2212\n+\n which has magnitude 8 units"
    },
    {
        "Chapter": "1",
        "sentence_range": "5259-5262",
        "Text": "For given vectors, \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n2\n2   and\na\ni\nj\nk\nb\ni\nj\nk\n=\n\u2212\n+\n= \u2212 +\n\u2212\nr\nr\n, find the unit vector in the\ndirection of the vector a\nb\n+\nr\nr 10 Find a vector  in the direction of vector \n\u02c6\n\u02c6\n\u02c6\n5\n2\ni\nj\nk\n\u2212\n+\n which has magnitude 8 units 11"
    },
    {
        "Chapter": "1",
        "sentence_range": "5260-5263",
        "Text": "10 Find a vector  in the direction of vector \n\u02c6\n\u02c6\n\u02c6\n5\n2\ni\nj\nk\n\u2212\n+\n which has magnitude 8 units 11 Show that the vectors \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n2\n3\n4   and\n4\n6\n8\ni\nj\nk\ni\nj\nk\n\u2212\n+\n\u2212\n+\n\u2212\n are collinear"
    },
    {
        "Chapter": "1",
        "sentence_range": "5261-5264",
        "Text": "Find a vector  in the direction of vector \n\u02c6\n\u02c6\n\u02c6\n5\n2\ni\nj\nk\n\u2212\n+\n which has magnitude 8 units 11 Show that the vectors \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n2\n3\n4   and\n4\n6\n8\ni\nj\nk\ni\nj\nk\n\u2212\n+\n\u2212\n+\n\u2212\n are collinear 12"
    },
    {
        "Chapter": "1",
        "sentence_range": "5262-5265",
        "Text": "11 Show that the vectors \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n2\n3\n4   and\n4\n6\n8\ni\nj\nk\ni\nj\nk\n\u2212\n+\n\u2212\n+\n\u2212\n are collinear 12 Find the direction cosines of the vector \n\u02c6\n\u02c6\n2\u02c6\n3\ni\nj\nk\n+\n+"
    },
    {
        "Chapter": "1",
        "sentence_range": "5263-5266",
        "Text": "Show that the vectors \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n2\n3\n4   and\n4\n6\n8\ni\nj\nk\ni\nj\nk\n\u2212\n+\n\u2212\n+\n\u2212\n are collinear 12 Find the direction cosines of the vector \n\u02c6\n\u02c6\n2\u02c6\n3\ni\nj\nk\n+\n+ 13"
    },
    {
        "Chapter": "1",
        "sentence_range": "5264-5267",
        "Text": "12 Find the direction cosines of the vector \n\u02c6\n\u02c6\n2\u02c6\n3\ni\nj\nk\n+\n+ 13 Find the direction cosines of the vector joining the points A (1, 2, \u20133) and\nB(\u20131, \u20132, 1), directed from A to B"
    },
    {
        "Chapter": "1",
        "sentence_range": "5265-5268",
        "Text": "Find the direction cosines of the vector \n\u02c6\n\u02c6\n2\u02c6\n3\ni\nj\nk\n+\n+ 13 Find the direction cosines of the vector joining the points A (1, 2, \u20133) and\nB(\u20131, \u20132, 1), directed from A to B 14"
    },
    {
        "Chapter": "1",
        "sentence_range": "5266-5269",
        "Text": "13 Find the direction cosines of the vector joining the points A (1, 2, \u20133) and\nB(\u20131, \u20132, 1), directed from A to B 14 Show that the vector \n\u02c6\n\u02c6\n\u02c6\ni\nj\nk\n+\n+\n is equally inclined to the axes OX, OY and OZ"
    },
    {
        "Chapter": "1",
        "sentence_range": "5267-5270",
        "Text": "Find the direction cosines of the vector joining the points A (1, 2, \u20133) and\nB(\u20131, \u20132, 1), directed from A to B 14 Show that the vector \n\u02c6\n\u02c6\n\u02c6\ni\nj\nk\n+\n+\n is equally inclined to the axes OX, OY and OZ 15"
    },
    {
        "Chapter": "1",
        "sentence_range": "5268-5271",
        "Text": "14 Show that the vector \n\u02c6\n\u02c6\n\u02c6\ni\nj\nk\n+\n+\n is equally inclined to the axes OX, OY and OZ 15 Find the position vector of a point R which divides the line joining two points  P\nand Q whose position vectors are \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n2\n  and \u2013\ni\nj\nk\ni\nj\nk\n+\n\u2212\n+\n+\n respectively, in the\nratio 2 : 1\n(i) internally\n(ii) externally\n\u00a9 NCERT\nnot to be republished\nVECTOR ALGEBRA\n441\n16"
    },
    {
        "Chapter": "1",
        "sentence_range": "5269-5272",
        "Text": "Show that the vector \n\u02c6\n\u02c6\n\u02c6\ni\nj\nk\n+\n+\n is equally inclined to the axes OX, OY and OZ 15 Find the position vector of a point R which divides the line joining two points  P\nand Q whose position vectors are \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n2\n  and \u2013\ni\nj\nk\ni\nj\nk\n+\n\u2212\n+\n+\n respectively, in the\nratio 2 : 1\n(i) internally\n(ii) externally\n\u00a9 NCERT\nnot to be republished\nVECTOR ALGEBRA\n441\n16 Find the position vector of the mid point of the vector joining the points  P(2, 3, 4)\nand Q(4, 1, \u20132)"
    },
    {
        "Chapter": "1",
        "sentence_range": "5270-5273",
        "Text": "15 Find the position vector of a point R which divides the line joining two points  P\nand Q whose position vectors are \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n2\n  and \u2013\ni\nj\nk\ni\nj\nk\n+\n\u2212\n+\n+\n respectively, in the\nratio 2 : 1\n(i) internally\n(ii) externally\n\u00a9 NCERT\nnot to be republished\nVECTOR ALGEBRA\n441\n16 Find the position vector of the mid point of the vector joining the points  P(2, 3, 4)\nand Q(4, 1, \u20132) 17"
    },
    {
        "Chapter": "1",
        "sentence_range": "5271-5274",
        "Text": "Find the position vector of a point R which divides the line joining two points  P\nand Q whose position vectors are \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n2\n  and \u2013\ni\nj\nk\ni\nj\nk\n+\n\u2212\n+\n+\n respectively, in the\nratio 2 : 1\n(i) internally\n(ii) externally\n\u00a9 NCERT\nnot to be republished\nVECTOR ALGEBRA\n441\n16 Find the position vector of the mid point of the vector joining the points  P(2, 3, 4)\nand Q(4, 1, \u20132) 17 Show that the points A, B and C with position vectors, \n\u02c6\n\u02c6\n\u02c6\n3\n4\n4 ,\na\ni\nj\nk\n=\n\u2212\n\u2212\nr\n\u02c6\n\u02c6\n\u02c6\n2\nb\ni\nj\nk\n=\n\u2212\n+\nr\n and \n\u02c6\n\u02c6\n3\u02c6\n5\nc\ni\nj\nk\n= \u2212\n\u2212\nr\n, respectively form the vertices of a right angled\ntriangle"
    },
    {
        "Chapter": "1",
        "sentence_range": "5272-5275",
        "Text": "Find the position vector of the mid point of the vector joining the points  P(2, 3, 4)\nand Q(4, 1, \u20132) 17 Show that the points A, B and C with position vectors, \n\u02c6\n\u02c6\n\u02c6\n3\n4\n4 ,\na\ni\nj\nk\n=\n\u2212\n\u2212\nr\n\u02c6\n\u02c6\n\u02c6\n2\nb\ni\nj\nk\n=\n\u2212\n+\nr\n and \n\u02c6\n\u02c6\n3\u02c6\n5\nc\ni\nj\nk\n= \u2212\n\u2212\nr\n, respectively form the vertices of a right angled\ntriangle 18"
    },
    {
        "Chapter": "1",
        "sentence_range": "5273-5276",
        "Text": "17 Show that the points A, B and C with position vectors, \n\u02c6\n\u02c6\n\u02c6\n3\n4\n4 ,\na\ni\nj\nk\n=\n\u2212\n\u2212\nr\n\u02c6\n\u02c6\n\u02c6\n2\nb\ni\nj\nk\n=\n\u2212\n+\nr\n and \n\u02c6\n\u02c6\n3\u02c6\n5\nc\ni\nj\nk\n= \u2212\n\u2212\nr\n, respectively form the vertices of a right angled\ntriangle 18 In triangle ABC (Fig 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5274-5277",
        "Text": "Show that the points A, B and C with position vectors, \n\u02c6\n\u02c6\n\u02c6\n3\n4\n4 ,\na\ni\nj\nk\n=\n\u2212\n\u2212\nr\n\u02c6\n\u02c6\n\u02c6\n2\nb\ni\nj\nk\n=\n\u2212\n+\nr\n and \n\u02c6\n\u02c6\n3\u02c6\n5\nc\ni\nj\nk\n= \u2212\n\u2212\nr\n, respectively form the vertices of a right angled\ntriangle 18 In triangle ABC (Fig 10 18), which of the following is not true:\n(A) AB + BC + CA = 0\nuuur\nuuuur\nuuur\nr\n(B) AB\nBC\nAC\n0\n+\n\u2212\n=\nuuur\nuuur\nuuur\nr\n(C) AB\nBC\nCA\n0\n+\n\u2212\n=\nuuur\nuuur\nuuur\nr\n(D) AB\nCB\nCA\n0\n\u2212\n+\n=\nuuur\nuuur\nuuur\nr\n19"
    },
    {
        "Chapter": "1",
        "sentence_range": "5275-5278",
        "Text": "18 In triangle ABC (Fig 10 18), which of the following is not true:\n(A) AB + BC + CA = 0\nuuur\nuuuur\nuuur\nr\n(B) AB\nBC\nAC\n0\n+\n\u2212\n=\nuuur\nuuur\nuuur\nr\n(C) AB\nBC\nCA\n0\n+\n\u2212\n=\nuuur\nuuur\nuuur\nr\n(D) AB\nCB\nCA\n0\n\u2212\n+\n=\nuuur\nuuur\nuuur\nr\n19 If \naand\nrb\nr\n are two collinear vectors, then which of the following are incorrect:\n(A)\n,  for  some scalar \nb\n= \u03bba\n\u03bb\nr\nr\n(B) a\n= \u00b1b\nr\nr\n(C) the respective components of \naand\nrb\nr\n are not proportional\n(D) both the vectors \naand\nrb\nr\n have same direction, but different magnitudes"
    },
    {
        "Chapter": "1",
        "sentence_range": "5276-5279",
        "Text": "In triangle ABC (Fig 10 18), which of the following is not true:\n(A) AB + BC + CA = 0\nuuur\nuuuur\nuuur\nr\n(B) AB\nBC\nAC\n0\n+\n\u2212\n=\nuuur\nuuur\nuuur\nr\n(C) AB\nBC\nCA\n0\n+\n\u2212\n=\nuuur\nuuur\nuuur\nr\n(D) AB\nCB\nCA\n0\n\u2212\n+\n=\nuuur\nuuur\nuuur\nr\n19 If \naand\nrb\nr\n are two collinear vectors, then which of the following are incorrect:\n(A)\n,  for  some scalar \nb\n= \u03bba\n\u03bb\nr\nr\n(B) a\n= \u00b1b\nr\nr\n(C) the respective components of \naand\nrb\nr\n are not proportional\n(D) both the vectors \naand\nrb\nr\n have same direction, but different magnitudes 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5277-5280",
        "Text": "18), which of the following is not true:\n(A) AB + BC + CA = 0\nuuur\nuuuur\nuuur\nr\n(B) AB\nBC\nAC\n0\n+\n\u2212\n=\nuuur\nuuur\nuuur\nr\n(C) AB\nBC\nCA\n0\n+\n\u2212\n=\nuuur\nuuur\nuuur\nr\n(D) AB\nCB\nCA\n0\n\u2212\n+\n=\nuuur\nuuur\nuuur\nr\n19 If \naand\nrb\nr\n are two collinear vectors, then which of the following are incorrect:\n(A)\n,  for  some scalar \nb\n= \u03bba\n\u03bb\nr\nr\n(B) a\n= \u00b1b\nr\nr\n(C) the respective components of \naand\nrb\nr\n are not proportional\n(D) both the vectors \naand\nrb\nr\n have same direction, but different magnitudes 10 6  Product of Two Vectors\nSo far we have studied about addition and subtraction of vectors"
    },
    {
        "Chapter": "1",
        "sentence_range": "5278-5281",
        "Text": "If \naand\nrb\nr\n are two collinear vectors, then which of the following are incorrect:\n(A)\n,  for  some scalar \nb\n= \u03bba\n\u03bb\nr\nr\n(B) a\n= \u00b1b\nr\nr\n(C) the respective components of \naand\nrb\nr\n are not proportional\n(D) both the vectors \naand\nrb\nr\n have same direction, but different magnitudes 10 6  Product of Two Vectors\nSo far we have studied about addition and subtraction of vectors An other algebraic\noperation which we intend to discuss regarding vectors is their product"
    },
    {
        "Chapter": "1",
        "sentence_range": "5279-5282",
        "Text": "10 6  Product of Two Vectors\nSo far we have studied about addition and subtraction of vectors An other algebraic\noperation which we intend to discuss regarding vectors is their product We may\nrecall that product of two numbers is a number, product of two matrices is again a\nmatrix"
    },
    {
        "Chapter": "1",
        "sentence_range": "5280-5283",
        "Text": "6  Product of Two Vectors\nSo far we have studied about addition and subtraction of vectors An other algebraic\noperation which we intend to discuss regarding vectors is their product We may\nrecall that product of two numbers is a number, product of two matrices is again a\nmatrix But in case of functions, we may multiply them in two ways, namely,\nmultiplication of two functions pointwise and composition of two functions"
    },
    {
        "Chapter": "1",
        "sentence_range": "5281-5284",
        "Text": "An other algebraic\noperation which we intend to discuss regarding vectors is their product We may\nrecall that product of two numbers is a number, product of two matrices is again a\nmatrix But in case of functions, we may multiply them in two ways, namely,\nmultiplication of two functions pointwise and composition of two functions Similarly,\nmultiplication of two vectors is also defined in two ways, namely, scalar (or dot)\nproduct where the result is a scalar, and vector (or cross) product where the\nresult is a vector"
    },
    {
        "Chapter": "1",
        "sentence_range": "5282-5285",
        "Text": "We may\nrecall that product of two numbers is a number, product of two matrices is again a\nmatrix But in case of functions, we may multiply them in two ways, namely,\nmultiplication of two functions pointwise and composition of two functions Similarly,\nmultiplication of two vectors is also defined in two ways, namely, scalar (or dot)\nproduct where the result is a scalar, and vector (or cross) product where the\nresult is a vector Based upon these two types of products for vectors, they have\nfound various applications in geometry, mechanics and engineering"
    },
    {
        "Chapter": "1",
        "sentence_range": "5283-5286",
        "Text": "But in case of functions, we may multiply them in two ways, namely,\nmultiplication of two functions pointwise and composition of two functions Similarly,\nmultiplication of two vectors is also defined in two ways, namely, scalar (or dot)\nproduct where the result is a scalar, and vector (or cross) product where the\nresult is a vector Based upon these two types of products for vectors, they have\nfound various applications in geometry, mechanics and engineering In this section,\nwe will discuss these two types of products"
    },
    {
        "Chapter": "1",
        "sentence_range": "5284-5287",
        "Text": "Similarly,\nmultiplication of two vectors is also defined in two ways, namely, scalar (or dot)\nproduct where the result is a scalar, and vector (or cross) product where the\nresult is a vector Based upon these two types of products for vectors, they have\nfound various applications in geometry, mechanics and engineering In this section,\nwe will discuss these two types of products 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5285-5288",
        "Text": "Based upon these two types of products for vectors, they have\nfound various applications in geometry, mechanics and engineering In this section,\nwe will discuss these two types of products 10 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "5286-5289",
        "Text": "In this section,\nwe will discuss these two types of products 10 6 1  Scalar (or dot) product of two vectors\nDefinition 2 The scalar product of two nonzero vectors \naand\nrb\nr\n, denoted by a b\n\u22c5\nr\nr\n, is\nFig 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5287-5290",
        "Text": "10 6 1  Scalar (or dot) product of two vectors\nDefinition 2 The scalar product of two nonzero vectors \naand\nrb\nr\n, denoted by a b\n\u22c5\nr\nr\n, is\nFig 10 18\n\u00a9 NCERT\nnot to be republished\n MATHEMATICS\n442\ndefined as\na b\n\u22c5\nr\nr\n = |\n| |\n| cos ,\na\nb\n\u03b8\nr\nr\nwhere, \u03b8 is the angle between \nand\n, 0\na\nb\n   \nr\nr\n (Fig 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5288-5291",
        "Text": "6 1  Scalar (or dot) product of two vectors\nDefinition 2 The scalar product of two nonzero vectors \naand\nrb\nr\n, denoted by a b\n\u22c5\nr\nr\n, is\nFig 10 18\n\u00a9 NCERT\nnot to be republished\n MATHEMATICS\n442\ndefined as\na b\n\u22c5\nr\nr\n = |\n| |\n| cos ,\na\nb\n\u03b8\nr\nr\nwhere, \u03b8 is the angle between \nand\n, 0\na\nb\n   \nr\nr\n (Fig 10 19)"
    },
    {
        "Chapter": "1",
        "sentence_range": "5289-5292",
        "Text": "1  Scalar (or dot) product of two vectors\nDefinition 2 The scalar product of two nonzero vectors \naand\nrb\nr\n, denoted by a b\n\u22c5\nr\nr\n, is\nFig 10 18\n\u00a9 NCERT\nnot to be republished\n MATHEMATICS\n442\ndefined as\na b\n\u22c5\nr\nr\n = |\n| |\n| cos ,\na\nb\n\u03b8\nr\nr\nwhere, \u03b8 is the angle between \nand\n, 0\na\nb\n   \nr\nr\n (Fig 10 19) If either \n0 or\n0,\na\nb\n=\n=\nr\nr\nr\nr\n then \u03b8 is not defined, and in this case,\nwe define \n0\na b\n \nr \nr\nObservations\n1"
    },
    {
        "Chapter": "1",
        "sentence_range": "5290-5293",
        "Text": "18\n\u00a9 NCERT\nnot to be republished\n MATHEMATICS\n442\ndefined as\na b\n\u22c5\nr\nr\n = |\n| |\n| cos ,\na\nb\n\u03b8\nr\nr\nwhere, \u03b8 is the angle between \nand\n, 0\na\nb\n   \nr\nr\n (Fig 10 19) If either \n0 or\n0,\na\nb\n=\n=\nr\nr\nr\nr\n then \u03b8 is not defined, and in this case,\nwe define \n0\na b\n \nr \nr\nObservations\n1 a b\n\u22c5\nr\nr\n is a real number"
    },
    {
        "Chapter": "1",
        "sentence_range": "5291-5294",
        "Text": "19) If either \n0 or\n0,\na\nb\n=\n=\nr\nr\nr\nr\n then \u03b8 is not defined, and in this case,\nwe define \n0\na b\n \nr \nr\nObservations\n1 a b\n\u22c5\nr\nr\n is a real number 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "5292-5295",
        "Text": "If either \n0 or\n0,\na\nb\n=\n=\nr\nr\nr\nr\n then \u03b8 is not defined, and in this case,\nwe define \n0\na b\n \nr \nr\nObservations\n1 a b\n\u22c5\nr\nr\n is a real number 2 Let \naand\nrb\nr\nbe two nonzero vectors, then \n0\na b\n\u22c5\nr=\nr\n if and only if \naand\nrb\nr\n are\nperpendicular to each other"
    },
    {
        "Chapter": "1",
        "sentence_range": "5293-5296",
        "Text": "a b\n\u22c5\nr\nr\n is a real number 2 Let \naand\nrb\nr\nbe two nonzero vectors, then \n0\na b\n\u22c5\nr=\nr\n if and only if \naand\nrb\nr\n are\nperpendicular to each other i"
    },
    {
        "Chapter": "1",
        "sentence_range": "5294-5297",
        "Text": "2 Let \naand\nrb\nr\nbe two nonzero vectors, then \n0\na b\n\u22c5\nr=\nr\n if and only if \naand\nrb\nr\n are\nperpendicular to each other i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "5295-5298",
        "Text": "Let \naand\nrb\nr\nbe two nonzero vectors, then \n0\na b\n\u22c5\nr=\nr\n if and only if \naand\nrb\nr\n are\nperpendicular to each other i e 0\n \na b\na\nb\n\u22c5\n=\n\u21d4\n\u22a5\nr\nr\nr\nr\n3"
    },
    {
        "Chapter": "1",
        "sentence_range": "5296-5299",
        "Text": "i e 0\n \na b\na\nb\n\u22c5\n=\n\u21d4\n\u22a5\nr\nr\nr\nr\n3 If \u03b8 = 0, then \n|\n| |\n|\na b\na\nb\n\u22c5\nr=\nr\nr\nr\nIn particular, \n2\n|\n| ,\na a\na\n\u22c5\nr r=\nr\n as \u03b8 in this case is 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "5297-5300",
        "Text": "e 0\n \na b\na\nb\n\u22c5\n=\n\u21d4\n\u22a5\nr\nr\nr\nr\n3 If \u03b8 = 0, then \n|\n| |\n|\na b\na\nb\n\u22c5\nr=\nr\nr\nr\nIn particular, \n2\n|\n| ,\na a\na\n\u22c5\nr r=\nr\n as \u03b8 in this case is 0 4"
    },
    {
        "Chapter": "1",
        "sentence_range": "5298-5301",
        "Text": "0\n \na b\na\nb\n\u22c5\n=\n\u21d4\n\u22a5\nr\nr\nr\nr\n3 If \u03b8 = 0, then \n|\n| |\n|\na b\na\nb\n\u22c5\nr=\nr\nr\nr\nIn particular, \n2\n|\n| ,\na a\na\n\u22c5\nr r=\nr\n as \u03b8 in this case is 0 4 If \u03b8 = \u03c0, then \n|\n| |\n|\na b\na\nb\n\u22c5\nr= \u2212\nr\nr\nr\nIn particular, \n2\n(\n)\n|\n|\na\na\na\n  \n  \nr\nr\nr\n, as \u03b8 in this case is \u03c0"
    },
    {
        "Chapter": "1",
        "sentence_range": "5299-5302",
        "Text": "If \u03b8 = 0, then \n|\n| |\n|\na b\na\nb\n\u22c5\nr=\nr\nr\nr\nIn particular, \n2\n|\n| ,\na a\na\n\u22c5\nr r=\nr\n as \u03b8 in this case is 0 4 If \u03b8 = \u03c0, then \n|\n| |\n|\na b\na\nb\n\u22c5\nr= \u2212\nr\nr\nr\nIn particular, \n2\n(\n)\n|\n|\na\na\na\n  \n  \nr\nr\nr\n, as \u03b8 in this case is \u03c0 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "5300-5303",
        "Text": "4 If \u03b8 = \u03c0, then \n|\n| |\n|\na b\na\nb\n\u22c5\nr= \u2212\nr\nr\nr\nIn particular, \n2\n(\n)\n|\n|\na\na\na\n  \n  \nr\nr\nr\n, as \u03b8 in this case is \u03c0 5 In view of the Observations 2 and 3, for mutually perpendicular unit vectors\n\u02c6\n,\u02c6 \u02c6\nand\n,\ni\nj\nk  we have\n\u02c6 \u02c6\n\u02c6 \u02c6\ni i\n\u22c5 =j j\n\u22c5\n = \u02c6 \u02c6\n1,\nk k\n\u22c5\n=\n\u02c6\n\u02c6 \u02c6\n\u02c6\ni\nj\nj k\n\u22c5\n=\n\u22c5\n = \u02c6 \u02c6\n0\nk i  \n6"
    },
    {
        "Chapter": "1",
        "sentence_range": "5301-5304",
        "Text": "If \u03b8 = \u03c0, then \n|\n| |\n|\na b\na\nb\n\u22c5\nr= \u2212\nr\nr\nr\nIn particular, \n2\n(\n)\n|\n|\na\na\na\n  \n  \nr\nr\nr\n, as \u03b8 in this case is \u03c0 5 In view of the Observations 2 and 3, for mutually perpendicular unit vectors\n\u02c6\n,\u02c6 \u02c6\nand\n,\ni\nj\nk  we have\n\u02c6 \u02c6\n\u02c6 \u02c6\ni i\n\u22c5 =j j\n\u22c5\n = \u02c6 \u02c6\n1,\nk k\n\u22c5\n=\n\u02c6\n\u02c6 \u02c6\n\u02c6\ni\nj\nj k\n\u22c5\n=\n\u22c5\n = \u02c6 \u02c6\n0\nk i  \n6 The angle between two nonzero vectors ra  and \nr\nb is given by\ncos\n,\n|\n||\na b|\na b\n  \nr\nr\nr\nr\n or\n\u20131"
    },
    {
        "Chapter": "1",
        "sentence_range": "5302-5305",
        "Text": "5 In view of the Observations 2 and 3, for mutually perpendicular unit vectors\n\u02c6\n,\u02c6 \u02c6\nand\n,\ni\nj\nk  we have\n\u02c6 \u02c6\n\u02c6 \u02c6\ni i\n\u22c5 =j j\n\u22c5\n = \u02c6 \u02c6\n1,\nk k\n\u22c5\n=\n\u02c6\n\u02c6 \u02c6\n\u02c6\ni\nj\nj k\n\u22c5\n=\n\u22c5\n = \u02c6 \u02c6\n0\nk i  \n6 The angle between two nonzero vectors ra  and \nr\nb is given by\ncos\n,\n|\n||\na b|\na b\n  \nr\nr\nr\nr\n or\n\u20131 cos\n|\n||\na b|\na b\n\u239b\n\u239e\n\u03b8 =\n\u239c\n\u239f\n\u239d\n\u23a0\nrr\nr\nr\n7"
    },
    {
        "Chapter": "1",
        "sentence_range": "5303-5306",
        "Text": "In view of the Observations 2 and 3, for mutually perpendicular unit vectors\n\u02c6\n,\u02c6 \u02c6\nand\n,\ni\nj\nk  we have\n\u02c6 \u02c6\n\u02c6 \u02c6\ni i\n\u22c5 =j j\n\u22c5\n = \u02c6 \u02c6\n1,\nk k\n\u22c5\n=\n\u02c6\n\u02c6 \u02c6\n\u02c6\ni\nj\nj k\n\u22c5\n=\n\u22c5\n = \u02c6 \u02c6\n0\nk i  \n6 The angle between two nonzero vectors ra  and \nr\nb is given by\ncos\n,\n|\n||\na b|\na b\n  \nr\nr\nr\nr\n or\n\u20131 cos\n|\n||\na b|\na b\n\u239b\n\u239e\n\u03b8 =\n\u239c\n\u239f\n\u239d\n\u23a0\nrr\nr\nr\n7 The scalar product is commutative"
    },
    {
        "Chapter": "1",
        "sentence_range": "5304-5307",
        "Text": "The angle between two nonzero vectors ra  and \nr\nb is given by\ncos\n,\n|\n||\na b|\na b\n  \nr\nr\nr\nr\n or\n\u20131 cos\n|\n||\na b|\na b\n\u239b\n\u239e\n\u03b8 =\n\u239c\n\u239f\n\u239d\n\u23a0\nrr\nr\nr\n7 The scalar product is commutative i"
    },
    {
        "Chapter": "1",
        "sentence_range": "5305-5308",
        "Text": "cos\n|\n||\na b|\na b\n\u239b\n\u239e\n\u03b8 =\n\u239c\n\u239f\n\u239d\n\u23a0\nrr\nr\nr\n7 The scalar product is commutative i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "5306-5309",
        "Text": "The scalar product is commutative i e a b\n\u22c5\nrr\n= b a\n\u22c5\nr r           (Why"
    },
    {
        "Chapter": "1",
        "sentence_range": "5307-5310",
        "Text": "i e a b\n\u22c5\nrr\n= b a\n\u22c5\nr r           (Why )\nTwo important properties of scalar product\nProperty 1 (Distributivity of scalar product over addition) Let ,   and  \na b\nc\nrr\nr  be\nany three vectors, then\n(\n)\na\nb\nc\n\u22c5\nr+\nr\nr  =\n \n \na b\na c\n\u22c5\n+\n\u22c5\nrr\nr r\nFig 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5308-5311",
        "Text": "e a b\n\u22c5\nrr\n= b a\n\u22c5\nr r           (Why )\nTwo important properties of scalar product\nProperty 1 (Distributivity of scalar product over addition) Let ,   and  \na b\nc\nrr\nr  be\nany three vectors, then\n(\n)\na\nb\nc\n\u22c5\nr+\nr\nr  =\n \n \na b\na c\n\u22c5\n+\n\u22c5\nrr\nr r\nFig 10 19\n\u00a9 NCERT\nnot to be republished\nVECTOR ALGEBRA\n443\n(i)\nB\nC\nA\nl\nB\nl\nA\nC\n(ii)\nA\nB\nC\nl\n(iv)\nl\nC\nB\nA\n(iii)\n\u03b8\n\u03b8\n\u03b8\n\u03b8\np\np\np\np\na\na\na\na\n(90 <\n< 180 )\n0\n0\n\u03b8\n(0 <\n< 90 )\n0\n0\n\u03b8\n(270 <\n< 360 )\n0\n0\n\u03b8\n(180 <\n< 270 )\n0\n0\n\u03b8\nProperty 2 Let  \naand \nrb\nr\n be any two vectors, and \u03bb be any scalar"
    },
    {
        "Chapter": "1",
        "sentence_range": "5309-5312",
        "Text": "a b\n\u22c5\nrr\n= b a\n\u22c5\nr r           (Why )\nTwo important properties of scalar product\nProperty 1 (Distributivity of scalar product over addition) Let ,   and  \na b\nc\nrr\nr  be\nany three vectors, then\n(\n)\na\nb\nc\n\u22c5\nr+\nr\nr  =\n \n \na b\na c\n\u22c5\n+\n\u22c5\nrr\nr r\nFig 10 19\n\u00a9 NCERT\nnot to be republished\nVECTOR ALGEBRA\n443\n(i)\nB\nC\nA\nl\nB\nl\nA\nC\n(ii)\nA\nB\nC\nl\n(iv)\nl\nC\nB\nA\n(iii)\n\u03b8\n\u03b8\n\u03b8\n\u03b8\np\np\np\np\na\na\na\na\n(90 <\n< 180 )\n0\n0\n\u03b8\n(0 <\n< 90 )\n0\n0\n\u03b8\n(270 <\n< 360 )\n0\n0\n\u03b8\n(180 <\n< 270 )\n0\n0\n\u03b8\nProperty 2 Let  \naand \nrb\nr\n be any two vectors, and \u03bb be any scalar Then\n(\na)\nb\n\u03bb\n\u22c5\nr\nr\n = (\n)\n(\n)\n(\n)\na\nb\na b\na\nb\n \n  \n \n \n  \nr\nr\nr\nr\nr\nr\nIf two vectors \naand \nrb\nr\n are given in component form as \n1\n2\n3 \u02c6\n\u02c6\n\u02c6\na i\na j\na k\n+\n+\n and\n1\n2\n3 \u02c6\n\u02c6\n\u02c6\nb i\nb j\nb k\n+\n+\n, then their scalar product is given as\na b\n\u22c5\nrr\n =\n1\n2\n3\n1\n2\n3\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(\n) (\n)\na i\na j\na k\nb i\nb j\nb k\n+\n+\n\u22c5\n+\n+\n=\n1\n1\n2\n3\n2\n1\n2\n3\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(\n)\n(\n)\na i\nb i\nb j\nb k\na j\nb i\nb j\nb k\n\u22c5\n+\n+\n+\n\u22c5\n+\n+\n +\n3\n1\n2\n3\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(\n)\na k\nb i\nb j\nb k\n\u22c5\n+\n+\n=\n1 1\n1 2\n1 3\n2 1\n2 2\n2 3\n\u02c6\n\u02c6\n\u02c6 \u02c6\n\u02c6 \u02c6\n\u02c6\n\u02c6 \u02c6\n\u02c6 \u02c6\n\u02c6\n(\n)\n(\n)\n(\n)\n(\n)\n(\n)\n(\n)\na b i i\na b i\nj\na b i k\na b\nj i\na b\nj j\na b\nj k\n\u22c5\n+\n\u22c5\n+\n\u22c5\n+\n\u22c5\n+\n\u22c5\n+\n\u22c5\n + \n3 1\n3 2\n3 3\n\u02c6\n\u02c6\n\u02c6 \u02c6\n\u02c6\n\u02c6\n(\n)\n(\n)\n(\n)\na b k i\na b k j\na b k k\n\u22c5\n+\n\u22c5\n+\n\u22c5\n(Using the above Properties 1 and 2)\n= a1b1 + a2b2 + a3b3\n(Using Observation 5)\nThus\na b\n\u22c5\nrr\n =  \n1 1\n2 2\n3 3\na b\na b\na b\n+\n+\n10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5310-5313",
        "Text": ")\nTwo important properties of scalar product\nProperty 1 (Distributivity of scalar product over addition) Let ,   and  \na b\nc\nrr\nr  be\nany three vectors, then\n(\n)\na\nb\nc\n\u22c5\nr+\nr\nr  =\n \n \na b\na c\n\u22c5\n+\n\u22c5\nrr\nr r\nFig 10 19\n\u00a9 NCERT\nnot to be republished\nVECTOR ALGEBRA\n443\n(i)\nB\nC\nA\nl\nB\nl\nA\nC\n(ii)\nA\nB\nC\nl\n(iv)\nl\nC\nB\nA\n(iii)\n\u03b8\n\u03b8\n\u03b8\n\u03b8\np\np\np\np\na\na\na\na\n(90 <\n< 180 )\n0\n0\n\u03b8\n(0 <\n< 90 )\n0\n0\n\u03b8\n(270 <\n< 360 )\n0\n0\n\u03b8\n(180 <\n< 270 )\n0\n0\n\u03b8\nProperty 2 Let  \naand \nrb\nr\n be any two vectors, and \u03bb be any scalar Then\n(\na)\nb\n\u03bb\n\u22c5\nr\nr\n = (\n)\n(\n)\n(\n)\na\nb\na b\na\nb\n \n  \n \n \n  \nr\nr\nr\nr\nr\nr\nIf two vectors \naand \nrb\nr\n are given in component form as \n1\n2\n3 \u02c6\n\u02c6\n\u02c6\na i\na j\na k\n+\n+\n and\n1\n2\n3 \u02c6\n\u02c6\n\u02c6\nb i\nb j\nb k\n+\n+\n, then their scalar product is given as\na b\n\u22c5\nrr\n =\n1\n2\n3\n1\n2\n3\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(\n) (\n)\na i\na j\na k\nb i\nb j\nb k\n+\n+\n\u22c5\n+\n+\n=\n1\n1\n2\n3\n2\n1\n2\n3\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(\n)\n(\n)\na i\nb i\nb j\nb k\na j\nb i\nb j\nb k\n\u22c5\n+\n+\n+\n\u22c5\n+\n+\n +\n3\n1\n2\n3\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(\n)\na k\nb i\nb j\nb k\n\u22c5\n+\n+\n=\n1 1\n1 2\n1 3\n2 1\n2 2\n2 3\n\u02c6\n\u02c6\n\u02c6 \u02c6\n\u02c6 \u02c6\n\u02c6\n\u02c6 \u02c6\n\u02c6 \u02c6\n\u02c6\n(\n)\n(\n)\n(\n)\n(\n)\n(\n)\n(\n)\na b i i\na b i\nj\na b i k\na b\nj i\na b\nj j\na b\nj k\n\u22c5\n+\n\u22c5\n+\n\u22c5\n+\n\u22c5\n+\n\u22c5\n+\n\u22c5\n + \n3 1\n3 2\n3 3\n\u02c6\n\u02c6\n\u02c6 \u02c6\n\u02c6\n\u02c6\n(\n)\n(\n)\n(\n)\na b k i\na b k j\na b k k\n\u22c5\n+\n\u22c5\n+\n\u22c5\n(Using the above Properties 1 and 2)\n= a1b1 + a2b2 + a3b3\n(Using Observation 5)\nThus\na b\n\u22c5\nrr\n =  \n1 1\n2 2\n3 3\na b\na b\na b\n+\n+\n10 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "5311-5314",
        "Text": "19\n\u00a9 NCERT\nnot to be republished\nVECTOR ALGEBRA\n443\n(i)\nB\nC\nA\nl\nB\nl\nA\nC\n(ii)\nA\nB\nC\nl\n(iv)\nl\nC\nB\nA\n(iii)\n\u03b8\n\u03b8\n\u03b8\n\u03b8\np\np\np\np\na\na\na\na\n(90 <\n< 180 )\n0\n0\n\u03b8\n(0 <\n< 90 )\n0\n0\n\u03b8\n(270 <\n< 360 )\n0\n0\n\u03b8\n(180 <\n< 270 )\n0\n0\n\u03b8\nProperty 2 Let  \naand \nrb\nr\n be any two vectors, and \u03bb be any scalar Then\n(\na)\nb\n\u03bb\n\u22c5\nr\nr\n = (\n)\n(\n)\n(\n)\na\nb\na b\na\nb\n \n  \n \n \n  \nr\nr\nr\nr\nr\nr\nIf two vectors \naand \nrb\nr\n are given in component form as \n1\n2\n3 \u02c6\n\u02c6\n\u02c6\na i\na j\na k\n+\n+\n and\n1\n2\n3 \u02c6\n\u02c6\n\u02c6\nb i\nb j\nb k\n+\n+\n, then their scalar product is given as\na b\n\u22c5\nrr\n =\n1\n2\n3\n1\n2\n3\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(\n) (\n)\na i\na j\na k\nb i\nb j\nb k\n+\n+\n\u22c5\n+\n+\n=\n1\n1\n2\n3\n2\n1\n2\n3\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(\n)\n(\n)\na i\nb i\nb j\nb k\na j\nb i\nb j\nb k\n\u22c5\n+\n+\n+\n\u22c5\n+\n+\n +\n3\n1\n2\n3\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(\n)\na k\nb i\nb j\nb k\n\u22c5\n+\n+\n=\n1 1\n1 2\n1 3\n2 1\n2 2\n2 3\n\u02c6\n\u02c6\n\u02c6 \u02c6\n\u02c6 \u02c6\n\u02c6\n\u02c6 \u02c6\n\u02c6 \u02c6\n\u02c6\n(\n)\n(\n)\n(\n)\n(\n)\n(\n)\n(\n)\na b i i\na b i\nj\na b i k\na b\nj i\na b\nj j\na b\nj k\n\u22c5\n+\n\u22c5\n+\n\u22c5\n+\n\u22c5\n+\n\u22c5\n+\n\u22c5\n + \n3 1\n3 2\n3 3\n\u02c6\n\u02c6\n\u02c6 \u02c6\n\u02c6\n\u02c6\n(\n)\n(\n)\n(\n)\na b k i\na b k j\na b k k\n\u22c5\n+\n\u22c5\n+\n\u22c5\n(Using the above Properties 1 and 2)\n= a1b1 + a2b2 + a3b3\n(Using Observation 5)\nThus\na b\n\u22c5\nrr\n =  \n1 1\n2 2\n3 3\na b\na b\na b\n+\n+\n10 6 2  Projection of a vector on a line\nSuppose a vector AB\nuuur\n makes an angle \u03b8 with a given directed line l (say), in the\nanticlockwise direction  (Fig 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5312-5315",
        "Text": "Then\n(\na)\nb\n\u03bb\n\u22c5\nr\nr\n = (\n)\n(\n)\n(\n)\na\nb\na b\na\nb\n \n  \n \n \n  \nr\nr\nr\nr\nr\nr\nIf two vectors \naand \nrb\nr\n are given in component form as \n1\n2\n3 \u02c6\n\u02c6\n\u02c6\na i\na j\na k\n+\n+\n and\n1\n2\n3 \u02c6\n\u02c6\n\u02c6\nb i\nb j\nb k\n+\n+\n, then their scalar product is given as\na b\n\u22c5\nrr\n =\n1\n2\n3\n1\n2\n3\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(\n) (\n)\na i\na j\na k\nb i\nb j\nb k\n+\n+\n\u22c5\n+\n+\n=\n1\n1\n2\n3\n2\n1\n2\n3\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(\n)\n(\n)\na i\nb i\nb j\nb k\na j\nb i\nb j\nb k\n\u22c5\n+\n+\n+\n\u22c5\n+\n+\n +\n3\n1\n2\n3\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(\n)\na k\nb i\nb j\nb k\n\u22c5\n+\n+\n=\n1 1\n1 2\n1 3\n2 1\n2 2\n2 3\n\u02c6\n\u02c6\n\u02c6 \u02c6\n\u02c6 \u02c6\n\u02c6\n\u02c6 \u02c6\n\u02c6 \u02c6\n\u02c6\n(\n)\n(\n)\n(\n)\n(\n)\n(\n)\n(\n)\na b i i\na b i\nj\na b i k\na b\nj i\na b\nj j\na b\nj k\n\u22c5\n+\n\u22c5\n+\n\u22c5\n+\n\u22c5\n+\n\u22c5\n+\n\u22c5\n + \n3 1\n3 2\n3 3\n\u02c6\n\u02c6\n\u02c6 \u02c6\n\u02c6\n\u02c6\n(\n)\n(\n)\n(\n)\na b k i\na b k j\na b k k\n\u22c5\n+\n\u22c5\n+\n\u22c5\n(Using the above Properties 1 and 2)\n= a1b1 + a2b2 + a3b3\n(Using Observation 5)\nThus\na b\n\u22c5\nrr\n =  \n1 1\n2 2\n3 3\na b\na b\na b\n+\n+\n10 6 2  Projection of a vector on a line\nSuppose a vector AB\nuuur\n makes an angle \u03b8 with a given directed line l (say), in the\nanticlockwise direction  (Fig 10 20)"
    },
    {
        "Chapter": "1",
        "sentence_range": "5313-5316",
        "Text": "6 2  Projection of a vector on a line\nSuppose a vector AB\nuuur\n makes an angle \u03b8 with a given directed line l (say), in the\nanticlockwise direction  (Fig 10 20) Then the projection of AB\nuuur\n on l is a vector pr\n(say) with magnitude | AB | cos\u03b8\nuuur\n, and the direction of pr  being the same (or opposite)\nto that of the line l, depending upon whether cos\u03b8 is positive or negative"
    },
    {
        "Chapter": "1",
        "sentence_range": "5314-5317",
        "Text": "2  Projection of a vector on a line\nSuppose a vector AB\nuuur\n makes an angle \u03b8 with a given directed line l (say), in the\nanticlockwise direction  (Fig 10 20) Then the projection of AB\nuuur\n on l is a vector pr\n(say) with magnitude | AB | cos\u03b8\nuuur\n, and the direction of pr  being the same (or opposite)\nto that of the line l, depending upon whether cos\u03b8 is positive or negative The vector pr\nFig 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5315-5318",
        "Text": "20) Then the projection of AB\nuuur\n on l is a vector pr\n(say) with magnitude | AB | cos\u03b8\nuuur\n, and the direction of pr  being the same (or opposite)\nto that of the line l, depending upon whether cos\u03b8 is positive or negative The vector pr\nFig 10 20\n\u00a9 NCERT\nnot to be republished\n MATHEMATICS\n444\nis called the projection vector, and its magnitude | pr | is simply called as the projection\nof the vector AB\nuuur\n on the directed line l"
    },
    {
        "Chapter": "1",
        "sentence_range": "5316-5319",
        "Text": "Then the projection of AB\nuuur\n on l is a vector pr\n(say) with magnitude | AB | cos\u03b8\nuuur\n, and the direction of pr  being the same (or opposite)\nto that of the line l, depending upon whether cos\u03b8 is positive or negative The vector pr\nFig 10 20\n\u00a9 NCERT\nnot to be republished\n MATHEMATICS\n444\nis called the projection vector, and its magnitude | pr | is simply called as the projection\nof the vector AB\nuuur\n on the directed line l For example, in each of the following figures (Fig 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5317-5320",
        "Text": "The vector pr\nFig 10 20\n\u00a9 NCERT\nnot to be republished\n MATHEMATICS\n444\nis called the projection vector, and its magnitude | pr | is simply called as the projection\nof the vector AB\nuuur\n on the directed line l For example, in each of the following figures (Fig 10 20(i) to (iv)), projection vector\nof AB\nuuur\n along the line l is vector AC\nuuur"
    },
    {
        "Chapter": "1",
        "sentence_range": "5318-5321",
        "Text": "20\n\u00a9 NCERT\nnot to be republished\n MATHEMATICS\n444\nis called the projection vector, and its magnitude | pr | is simply called as the projection\nof the vector AB\nuuur\n on the directed line l For example, in each of the following figures (Fig 10 20(i) to (iv)), projection vector\nof AB\nuuur\n along the line l is vector AC\nuuur Observations\n1"
    },
    {
        "Chapter": "1",
        "sentence_range": "5319-5322",
        "Text": "For example, in each of the following figures (Fig 10 20(i) to (iv)), projection vector\nof AB\nuuur\n along the line l is vector AC\nuuur Observations\n1 If \u02c6p  is the unit vector along a line l, then the projection of a vector ar  on the line\nl is given by \na p\u02c6\nr"
    },
    {
        "Chapter": "1",
        "sentence_range": "5320-5323",
        "Text": "20(i) to (iv)), projection vector\nof AB\nuuur\n along the line l is vector AC\nuuur Observations\n1 If \u02c6p  is the unit vector along a line l, then the projection of a vector ar  on the line\nl is given by \na p\u02c6\nr 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "5321-5324",
        "Text": "Observations\n1 If \u02c6p  is the unit vector along a line l, then the projection of a vector ar  on the line\nl is given by \na p\u02c6\nr 2 Projection of a vector ar  on other vector b\nr , is given by\na b\u02c6,\nr\u22c5\n   or   \n,  or  1\n(\n)\n|\n|\n|\n|\nb\na\na b\nb\nb\n\u239b\n\u239e\n\u22c5\n\u22c5\n\u239c\n\u239f\n\u239d\n\u23a0\nr\nr\nr\nr\nr\nr\n3"
    },
    {
        "Chapter": "1",
        "sentence_range": "5322-5325",
        "Text": "If \u02c6p  is the unit vector along a line l, then the projection of a vector ar  on the line\nl is given by \na p\u02c6\nr 2 Projection of a vector ar  on other vector b\nr , is given by\na b\u02c6,\nr\u22c5\n   or   \n,  or  1\n(\n)\n|\n|\n|\n|\nb\na\na b\nb\nb\n\u239b\n\u239e\n\u22c5\n\u22c5\n\u239c\n\u239f\n\u239d\n\u23a0\nr\nr\nr\nr\nr\nr\n3 If  \u03b8 = 0, then the projection vector of AB\nuuur\n will be AB\nuuur\n itself and if \u03b8 = \u03c0, then the\nprojection vector of AB\nuuur\n will be BA\nuuur"
    },
    {
        "Chapter": "1",
        "sentence_range": "5323-5326",
        "Text": "2 Projection of a vector ar  on other vector b\nr , is given by\na b\u02c6,\nr\u22c5\n   or   \n,  or  1\n(\n)\n|\n|\n|\n|\nb\na\na b\nb\nb\n\u239b\n\u239e\n\u22c5\n\u22c5\n\u239c\n\u239f\n\u239d\n\u23a0\nr\nr\nr\nr\nr\nr\n3 If  \u03b8 = 0, then the projection vector of AB\nuuur\n will be AB\nuuur\n itself and if \u03b8 = \u03c0, then the\nprojection vector of AB\nuuur\n will be BA\nuuur 4"
    },
    {
        "Chapter": "1",
        "sentence_range": "5324-5327",
        "Text": "Projection of a vector ar  on other vector b\nr , is given by\na b\u02c6,\nr\u22c5\n   or   \n,  or  1\n(\n)\n|\n|\n|\n|\nb\na\na b\nb\nb\n\u239b\n\u239e\n\u22c5\n\u22c5\n\u239c\n\u239f\n\u239d\n\u23a0\nr\nr\nr\nr\nr\nr\n3 If  \u03b8 = 0, then the projection vector of AB\nuuur\n will be AB\nuuur\n itself and if \u03b8 = \u03c0, then the\nprojection vector of AB\nuuur\n will be BA\nuuur 4 If \n= 2\n\u03c0\n\u03b8\n or \n3\n= 2\n\u03c0\n\u03b8\n, then the projection vector of AB\nuuur\n will be zero vector"
    },
    {
        "Chapter": "1",
        "sentence_range": "5325-5328",
        "Text": "If  \u03b8 = 0, then the projection vector of AB\nuuur\n will be AB\nuuur\n itself and if \u03b8 = \u03c0, then the\nprojection vector of AB\nuuur\n will be BA\nuuur 4 If \n= 2\n\u03c0\n\u03b8\n or \n3\n= 2\n\u03c0\n\u03b8\n, then the projection vector of AB\nuuur\n will be zero vector Remark If \u03b1, \u03b2 and \u03b3 are the direction angles of vector \n1\n2\n3 \u02c6\n\u02c6\n\u02c6\na\na i\na j\na k\n=\n+\n+\nr\n, then its\ndirection cosines may be given as\n3\n1\n2\n\u02c6\ncos\n, cos\n,  and  cos\n\u02c6\n|\n|\n|\n|\n|\n|\n|\n|| |\na\na\na\na i\na\na\na\na i\n \n  \n \n  \n  \nr\nr\nr\nr\nr\nAlso, note that |\na| cos ,  | |cos  and | |cos\na\na\n\u03b1\n\u03b2\n\u03b3\nr\nr\nr\n are respectively the projections of\nar  along OX, OY and OZ"
    },
    {
        "Chapter": "1",
        "sentence_range": "5326-5329",
        "Text": "4 If \n= 2\n\u03c0\n\u03b8\n or \n3\n= 2\n\u03c0\n\u03b8\n, then the projection vector of AB\nuuur\n will be zero vector Remark If \u03b1, \u03b2 and \u03b3 are the direction angles of vector \n1\n2\n3 \u02c6\n\u02c6\n\u02c6\na\na i\na j\na k\n=\n+\n+\nr\n, then its\ndirection cosines may be given as\n3\n1\n2\n\u02c6\ncos\n, cos\n,  and  cos\n\u02c6\n|\n|\n|\n|\n|\n|\n|\n|| |\na\na\na\na i\na\na\na\na i\n \n  \n \n  \n  \nr\nr\nr\nr\nr\nAlso, note that |\na| cos ,  | |cos  and | |cos\na\na\n\u03b1\n\u03b2\n\u03b3\nr\nr\nr\n are respectively the projections of\nar  along OX, OY and OZ i"
    },
    {
        "Chapter": "1",
        "sentence_range": "5327-5330",
        "Text": "If \n= 2\n\u03c0\n\u03b8\n or \n3\n= 2\n\u03c0\n\u03b8\n, then the projection vector of AB\nuuur\n will be zero vector Remark If \u03b1, \u03b2 and \u03b3 are the direction angles of vector \n1\n2\n3 \u02c6\n\u02c6\n\u02c6\na\na i\na j\na k\n=\n+\n+\nr\n, then its\ndirection cosines may be given as\n3\n1\n2\n\u02c6\ncos\n, cos\n,  and  cos\n\u02c6\n|\n|\n|\n|\n|\n|\n|\n|| |\na\na\na\na i\na\na\na\na i\n \n  \n \n  \n  \nr\nr\nr\nr\nr\nAlso, note that |\na| cos ,  | |cos  and | |cos\na\na\n\u03b1\n\u03b2\n\u03b3\nr\nr\nr\n are respectively the projections of\nar  along OX, OY and OZ i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "5328-5331",
        "Text": "Remark If \u03b1, \u03b2 and \u03b3 are the direction angles of vector \n1\n2\n3 \u02c6\n\u02c6\n\u02c6\na\na i\na j\na k\n=\n+\n+\nr\n, then its\ndirection cosines may be given as\n3\n1\n2\n\u02c6\ncos\n, cos\n,  and  cos\n\u02c6\n|\n|\n|\n|\n|\n|\n|\n|| |\na\na\na\na i\na\na\na\na i\n \n  \n \n  \n  \nr\nr\nr\nr\nr\nAlso, note that |\na| cos ,  | |cos  and | |cos\na\na\n\u03b1\n\u03b2\n\u03b3\nr\nr\nr\n are respectively the projections of\nar  along OX, OY and OZ i e , the scalar components a1, a2 and a3 of the vector ar ,\nare precisely the projections of ar  along x-axis, y-axis and z-axis, respectively"
    },
    {
        "Chapter": "1",
        "sentence_range": "5329-5332",
        "Text": "i e , the scalar components a1, a2 and a3 of the vector ar ,\nare precisely the projections of ar  along x-axis, y-axis and z-axis, respectively Further,\nif ar  is a unit vector, then it may be expressed in terms of its direction cosines as\n\u02c6\n\u02c6\n\u02c6\ncos\ncos\ncos\na\ni\nj\nk\n=\n\u03b1 +\n\u03b2 +\n\u03b3\nr\nExample 13 Find the angle between two vectors  and \na\nrb\nr\n with magnitudes 1 and 2\nrespectively and  when  \na b1\n\u22c5\n=\nrr"
    },
    {
        "Chapter": "1",
        "sentence_range": "5330-5333",
        "Text": "e , the scalar components a1, a2 and a3 of the vector ar ,\nare precisely the projections of ar  along x-axis, y-axis and z-axis, respectively Further,\nif ar  is a unit vector, then it may be expressed in terms of its direction cosines as\n\u02c6\n\u02c6\n\u02c6\ncos\ncos\ncos\na\ni\nj\nk\n=\n\u03b1 +\n\u03b2 +\n\u03b3\nr\nExample 13 Find the angle between two vectors  and \na\nrb\nr\n with magnitudes 1 and 2\nrespectively and  when  \na b1\n\u22c5\n=\nrr Solution Given \n1,|\n|\n1and | |\n2\na b\na\nb\n \n \n \nr\nr\nr\nr"
    },
    {
        "Chapter": "1",
        "sentence_range": "5331-5334",
        "Text": ", the scalar components a1, a2 and a3 of the vector ar ,\nare precisely the projections of ar  along x-axis, y-axis and z-axis, respectively Further,\nif ar  is a unit vector, then it may be expressed in terms of its direction cosines as\n\u02c6\n\u02c6\n\u02c6\ncos\ncos\ncos\na\ni\nj\nk\n=\n\u03b1 +\n\u03b2 +\n\u03b3\nr\nExample 13 Find the angle between two vectors  and \na\nrb\nr\n with magnitudes 1 and 2\nrespectively and  when  \na b1\n\u22c5\n=\nrr Solution Given \n1,|\n|\n1and | |\n2\na b\na\nb\n \n \n \nr\nr\nr\nr We have\n1\n1 1\ncos\ncos\n2\n3\n|\n||\na b|\na b\n \n \n \n \n \n \n \n  \n \n \n \n \n \n \n \n \n \n \nrr\nr\nr\n\u00a9 NCERT\nnot to be republished\nVECTOR ALGEBRA\n445\nExample 14 Find angle \u2018\u03b8\u2019 between the vectors \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n   and  \na\ni\nj\nk\nb\ni\nj\nk\n= +\n\u2212\n= \u2212\n+\nr\nr"
    },
    {
        "Chapter": "1",
        "sentence_range": "5332-5335",
        "Text": "Further,\nif ar  is a unit vector, then it may be expressed in terms of its direction cosines as\n\u02c6\n\u02c6\n\u02c6\ncos\ncos\ncos\na\ni\nj\nk\n=\n\u03b1 +\n\u03b2 +\n\u03b3\nr\nExample 13 Find the angle between two vectors  and \na\nrb\nr\n with magnitudes 1 and 2\nrespectively and  when  \na b1\n\u22c5\n=\nrr Solution Given \n1,|\n|\n1and | |\n2\na b\na\nb\n \n \n \nr\nr\nr\nr We have\n1\n1 1\ncos\ncos\n2\n3\n|\n||\na b|\na b\n \n \n \n \n \n \n \n  \n \n \n \n \n \n \n \n \n \n \nrr\nr\nr\n\u00a9 NCERT\nnot to be republished\nVECTOR ALGEBRA\n445\nExample 14 Find angle \u2018\u03b8\u2019 between the vectors \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n   and  \na\ni\nj\nk\nb\ni\nj\nk\n= +\n\u2212\n= \u2212\n+\nr\nr Solution The angle \u03b8 between two vectors \naand \nrb\nr\n  is given by\ncos\u03b8 = |\n||\na b|\na b\n\u22c5 r\nr\nr\nr\nNow\na b\n\u22c5\nrr\n =\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(\n) (\n)\n1 1 1\n1\ni\nj\nk\ni\nj\nk\n+\n\u2212\n\u22c5\n\u2212\n+\n= \u2212 \u2212 = \u2212"
    },
    {
        "Chapter": "1",
        "sentence_range": "5333-5336",
        "Text": "Solution Given \n1,|\n|\n1and | |\n2\na b\na\nb\n \n \n \nr\nr\nr\nr We have\n1\n1 1\ncos\ncos\n2\n3\n|\n||\na b|\na b\n \n \n \n \n \n \n \n  \n \n \n \n \n \n \n \n \n \n \nrr\nr\nr\n\u00a9 NCERT\nnot to be republished\nVECTOR ALGEBRA\n445\nExample 14 Find angle \u2018\u03b8\u2019 between the vectors \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n   and  \na\ni\nj\nk\nb\ni\nj\nk\n= +\n\u2212\n= \u2212\n+\nr\nr Solution The angle \u03b8 between two vectors \naand \nrb\nr\n  is given by\ncos\u03b8 = |\n||\na b|\na b\n\u22c5 r\nr\nr\nr\nNow\na b\n\u22c5\nrr\n =\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(\n) (\n)\n1 1 1\n1\ni\nj\nk\ni\nj\nk\n+\n\u2212\n\u22c5\n\u2212\n+\n= \u2212 \u2212 = \u2212 Therefore, we have\n  cos\u03b8 =\n31\n\u2212\nhence the required angle is\n\u03b8 =\n1\n1\ncos\n3\n \n \n \n  \nExample 15 If \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n5\n3   and  \n3\n5\na\ni\nj\nk\nb\ni\nj\nk\n=\n\u2212\n\u2212\n=\n+\n\u2212\nr\nr\n, then show that the vectors\n and  \na\nb\na\nb\n+\n\u2212\nr\nr\nr\nr\n are perpendicular"
    },
    {
        "Chapter": "1",
        "sentence_range": "5334-5337",
        "Text": "We have\n1\n1 1\ncos\ncos\n2\n3\n|\n||\na b|\na b\n \n \n \n \n \n \n \n  \n \n \n \n \n \n \n \n \n \n \nrr\nr\nr\n\u00a9 NCERT\nnot to be republished\nVECTOR ALGEBRA\n445\nExample 14 Find angle \u2018\u03b8\u2019 between the vectors \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n   and  \na\ni\nj\nk\nb\ni\nj\nk\n= +\n\u2212\n= \u2212\n+\nr\nr Solution The angle \u03b8 between two vectors \naand \nrb\nr\n  is given by\ncos\u03b8 = |\n||\na b|\na b\n\u22c5 r\nr\nr\nr\nNow\na b\n\u22c5\nrr\n =\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(\n) (\n)\n1 1 1\n1\ni\nj\nk\ni\nj\nk\n+\n\u2212\n\u22c5\n\u2212\n+\n= \u2212 \u2212 = \u2212 Therefore, we have\n  cos\u03b8 =\n31\n\u2212\nhence the required angle is\n\u03b8 =\n1\n1\ncos\n3\n \n \n \n  \nExample 15 If \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n5\n3   and  \n3\n5\na\ni\nj\nk\nb\ni\nj\nk\n=\n\u2212\n\u2212\n=\n+\n\u2212\nr\nr\n, then show that the vectors\n and  \na\nb\na\nb\n+\n\u2212\nr\nr\nr\nr\n are perpendicular Solution We know that two nonzero vectors are perpendicular if their scalar product\nis zero"
    },
    {
        "Chapter": "1",
        "sentence_range": "5335-5338",
        "Text": "Solution The angle \u03b8 between two vectors \naand \nrb\nr\n  is given by\ncos\u03b8 = |\n||\na b|\na b\n\u22c5 r\nr\nr\nr\nNow\na b\n\u22c5\nrr\n =\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(\n) (\n)\n1 1 1\n1\ni\nj\nk\ni\nj\nk\n+\n\u2212\n\u22c5\n\u2212\n+\n= \u2212 \u2212 = \u2212 Therefore, we have\n  cos\u03b8 =\n31\n\u2212\nhence the required angle is\n\u03b8 =\n1\n1\ncos\n3\n \n \n \n  \nExample 15 If \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n5\n3   and  \n3\n5\na\ni\nj\nk\nb\ni\nj\nk\n=\n\u2212\n\u2212\n=\n+\n\u2212\nr\nr\n, then show that the vectors\n and  \na\nb\na\nb\n+\n\u2212\nr\nr\nr\nr\n are perpendicular Solution We know that two nonzero vectors are perpendicular if their scalar product\nis zero Here\na\nb\n+\nr\nr\n =\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(5\n3 )\n(\n3\n5 )\n6\n2\n8\ni\nj\nk\ni\nj\nk\ni\nj\nk\n\u2212\n\u2212\n+\n+\n\u2212\n=\n+\n\u2212\nand\na\n\u2212b\nr\nr\n =\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(5\n3 )\n(\n3\n5 )\n4\n4\n2\ni\nj\nk\ni\nj\nk\ni\nj\nk\n\u2212\n\u2212\n\u2212\n+\n\u2212\n=\n\u2212\n+\nSo\n      \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(\n) (\n)\n(6\n2\n8 ) (4\n4\n2 )\n24\n8 16\n0"
    },
    {
        "Chapter": "1",
        "sentence_range": "5336-5339",
        "Text": "Therefore, we have\n  cos\u03b8 =\n31\n\u2212\nhence the required angle is\n\u03b8 =\n1\n1\ncos\n3\n \n \n \n  \nExample 15 If \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n5\n3   and  \n3\n5\na\ni\nj\nk\nb\ni\nj\nk\n=\n\u2212\n\u2212\n=\n+\n\u2212\nr\nr\n, then show that the vectors\n and  \na\nb\na\nb\n+\n\u2212\nr\nr\nr\nr\n are perpendicular Solution We know that two nonzero vectors are perpendicular if their scalar product\nis zero Here\na\nb\n+\nr\nr\n =\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(5\n3 )\n(\n3\n5 )\n6\n2\n8\ni\nj\nk\ni\nj\nk\ni\nj\nk\n\u2212\n\u2212\n+\n+\n\u2212\n=\n+\n\u2212\nand\na\n\u2212b\nr\nr\n =\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(5\n3 )\n(\n3\n5 )\n4\n4\n2\ni\nj\nk\ni\nj\nk\ni\nj\nk\n\u2212\n\u2212\n\u2212\n+\n\u2212\n=\n\u2212\n+\nSo\n      \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(\n) (\n)\n(6\n2\n8 ) (4\n4\n2 )\n24\n8 16\n0 a\nb\na\nb\ni\nj\nk\ni\nj\nk\n+\n\u22c5\n\u2212\n=\n+\n\u2212\n\u22c5\n\u2212\n+\n=\n\u2212\n\u2212\n=\nr\nr\nr\nr\nHence\n  and  \na\nb\na\nb\n+\n\u2212\nr\nr\nr\nr\n are perpendicular vectors"
    },
    {
        "Chapter": "1",
        "sentence_range": "5337-5340",
        "Text": "Solution We know that two nonzero vectors are perpendicular if their scalar product\nis zero Here\na\nb\n+\nr\nr\n =\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(5\n3 )\n(\n3\n5 )\n6\n2\n8\ni\nj\nk\ni\nj\nk\ni\nj\nk\n\u2212\n\u2212\n+\n+\n\u2212\n=\n+\n\u2212\nand\na\n\u2212b\nr\nr\n =\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(5\n3 )\n(\n3\n5 )\n4\n4\n2\ni\nj\nk\ni\nj\nk\ni\nj\nk\n\u2212\n\u2212\n\u2212\n+\n\u2212\n=\n\u2212\n+\nSo\n      \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(\n) (\n)\n(6\n2\n8 ) (4\n4\n2 )\n24\n8 16\n0 a\nb\na\nb\ni\nj\nk\ni\nj\nk\n+\n\u22c5\n\u2212\n=\n+\n\u2212\n\u22c5\n\u2212\n+\n=\n\u2212\n\u2212\n=\nr\nr\nr\nr\nHence\n  and  \na\nb\na\nb\n+\n\u2212\nr\nr\nr\nr\n are perpendicular vectors Example 16 Find the projection of the vector \n\u02c6\n\u02c6\n\u02c6\n2\n3\n2\na\ni\nj\nk\n=\n+\n+\nr\n on the vector\n\u02c6\n\u02c6\n2\u02c6\nb\ni\nj\nk\n=\n+\n+\nr"
    },
    {
        "Chapter": "1",
        "sentence_range": "5338-5341",
        "Text": "Here\na\nb\n+\nr\nr\n =\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(5\n3 )\n(\n3\n5 )\n6\n2\n8\ni\nj\nk\ni\nj\nk\ni\nj\nk\n\u2212\n\u2212\n+\n+\n\u2212\n=\n+\n\u2212\nand\na\n\u2212b\nr\nr\n =\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(5\n3 )\n(\n3\n5 )\n4\n4\n2\ni\nj\nk\ni\nj\nk\ni\nj\nk\n\u2212\n\u2212\n\u2212\n+\n\u2212\n=\n\u2212\n+\nSo\n      \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(\n) (\n)\n(6\n2\n8 ) (4\n4\n2 )\n24\n8 16\n0 a\nb\na\nb\ni\nj\nk\ni\nj\nk\n+\n\u22c5\n\u2212\n=\n+\n\u2212\n\u22c5\n\u2212\n+\n=\n\u2212\n\u2212\n=\nr\nr\nr\nr\nHence\n  and  \na\nb\na\nb\n+\n\u2212\nr\nr\nr\nr\n are perpendicular vectors Example 16 Find the projection of the vector \n\u02c6\n\u02c6\n\u02c6\n2\n3\n2\na\ni\nj\nk\n=\n+\n+\nr\n on the vector\n\u02c6\n\u02c6\n2\u02c6\nb\ni\nj\nk\n=\n+\n+\nr Solution The projection of vector ar  on  the vector b\nr\n is given by\n1 (\n)\n|\n|\na b\nb\n\u22c5\nrr\nr\n = \n2\n2\n2\n(2 1\n3 2\n2 1)\n10\n5\n6\n3\n6\n(1)\n(2)\n(1)\n\u00d7 + \u00d7\n+\n\u00d7\n=\n=\n+\n+\nExample 17 Find |\n|\na\nb\n\u2212\nr\nr\n, if two vectors \naand\nrb\nr\n are such that |\n|\n2, |\n|\n3\na\nb\n \nr \nr\nand \n4\na b\n\u22c5\nrr="
    },
    {
        "Chapter": "1",
        "sentence_range": "5339-5342",
        "Text": "a\nb\na\nb\ni\nj\nk\ni\nj\nk\n+\n\u22c5\n\u2212\n=\n+\n\u2212\n\u22c5\n\u2212\n+\n=\n\u2212\n\u2212\n=\nr\nr\nr\nr\nHence\n  and  \na\nb\na\nb\n+\n\u2212\nr\nr\nr\nr\n are perpendicular vectors Example 16 Find the projection of the vector \n\u02c6\n\u02c6\n\u02c6\n2\n3\n2\na\ni\nj\nk\n=\n+\n+\nr\n on the vector\n\u02c6\n\u02c6\n2\u02c6\nb\ni\nj\nk\n=\n+\n+\nr Solution The projection of vector ar  on  the vector b\nr\n is given by\n1 (\n)\n|\n|\na b\nb\n\u22c5\nrr\nr\n = \n2\n2\n2\n(2 1\n3 2\n2 1)\n10\n5\n6\n3\n6\n(1)\n(2)\n(1)\n\u00d7 + \u00d7\n+\n\u00d7\n=\n=\n+\n+\nExample 17 Find |\n|\na\nb\n\u2212\nr\nr\n, if two vectors \naand\nrb\nr\n are such that |\n|\n2, |\n|\n3\na\nb\n \nr \nr\nand \n4\na b\n\u22c5\nrr= Solution We have\n2\n|\n|\na\nb\n \nr\nr\n = (\n) (\n)\na\nb\na\nb\n\u2212\n\u22c5\n\u2212\nr\nr\nr\nr\n="
    },
    {
        "Chapter": "1",
        "sentence_range": "5340-5343",
        "Text": "Example 16 Find the projection of the vector \n\u02c6\n\u02c6\n\u02c6\n2\n3\n2\na\ni\nj\nk\n=\n+\n+\nr\n on the vector\n\u02c6\n\u02c6\n2\u02c6\nb\ni\nj\nk\n=\n+\n+\nr Solution The projection of vector ar  on  the vector b\nr\n is given by\n1 (\n)\n|\n|\na b\nb\n\u22c5\nrr\nr\n = \n2\n2\n2\n(2 1\n3 2\n2 1)\n10\n5\n6\n3\n6\n(1)\n(2)\n(1)\n\u00d7 + \u00d7\n+\n\u00d7\n=\n=\n+\n+\nExample 17 Find |\n|\na\nb\n\u2212\nr\nr\n, if two vectors \naand\nrb\nr\n are such that |\n|\n2, |\n|\n3\na\nb\n \nr \nr\nand \n4\na b\n\u22c5\nrr= Solution We have\n2\n|\n|\na\nb\n \nr\nr\n = (\n) (\n)\na\nb\na\nb\n\u2212\n\u22c5\n\u2212\nr\nr\nr\nr\n= a a\na b\nb a\nb b\n\u2212\n\u22c5\n\u2212\n\u22c5\n+\n\u22c5\nr\nr\nr r\nr r\nr\nr\n\u00a9 NCERT\nnot to be republished\n MATHEMATICS\n446\nB\nC\nA\na\nb\n+\na\nb\n=\n2\n2\n|\n|\n2(\n) |\n|\na\na b\nb\n\u2212\n\u22c5\n+\nr\nr\nr\nr\n=\n2\n2\n(2)\n2(4)\n(3)\n\u2212\n+\nTherefore\n|\n|\na\n\u2212b\nr\nr\n =\n5\nExample 18 If ar  is a unit vector and (\n) (\n)\n8\nx\na\nx\na\n\u2212\n\u22c5\n+\n=\nr\nr\nr\nr\n, then find |\n|\nxr"
    },
    {
        "Chapter": "1",
        "sentence_range": "5341-5344",
        "Text": "Solution The projection of vector ar  on  the vector b\nr\n is given by\n1 (\n)\n|\n|\na b\nb\n\u22c5\nrr\nr\n = \n2\n2\n2\n(2 1\n3 2\n2 1)\n10\n5\n6\n3\n6\n(1)\n(2)\n(1)\n\u00d7 + \u00d7\n+\n\u00d7\n=\n=\n+\n+\nExample 17 Find |\n|\na\nb\n\u2212\nr\nr\n, if two vectors \naand\nrb\nr\n are such that |\n|\n2, |\n|\n3\na\nb\n \nr \nr\nand \n4\na b\n\u22c5\nrr= Solution We have\n2\n|\n|\na\nb\n \nr\nr\n = (\n) (\n)\na\nb\na\nb\n\u2212\n\u22c5\n\u2212\nr\nr\nr\nr\n= a a\na b\nb a\nb b\n\u2212\n\u22c5\n\u2212\n\u22c5\n+\n\u22c5\nr\nr\nr r\nr r\nr\nr\n\u00a9 NCERT\nnot to be republished\n MATHEMATICS\n446\nB\nC\nA\na\nb\n+\na\nb\n=\n2\n2\n|\n|\n2(\n) |\n|\na\na b\nb\n\u2212\n\u22c5\n+\nr\nr\nr\nr\n=\n2\n2\n(2)\n2(4)\n(3)\n\u2212\n+\nTherefore\n|\n|\na\n\u2212b\nr\nr\n =\n5\nExample 18 If ar  is a unit vector and (\n) (\n)\n8\nx\na\nx\na\n\u2212\n\u22c5\n+\n=\nr\nr\nr\nr\n, then find |\n|\nxr Solution Since ar  is a unit vector, |\nra =| 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "5342-5345",
        "Text": "Solution We have\n2\n|\n|\na\nb\n \nr\nr\n = (\n) (\n)\na\nb\na\nb\n\u2212\n\u22c5\n\u2212\nr\nr\nr\nr\n= a a\na b\nb a\nb b\n\u2212\n\u22c5\n\u2212\n\u22c5\n+\n\u22c5\nr\nr\nr r\nr r\nr\nr\n\u00a9 NCERT\nnot to be republished\n MATHEMATICS\n446\nB\nC\nA\na\nb\n+\na\nb\n=\n2\n2\n|\n|\n2(\n) |\n|\na\na b\nb\n\u2212\n\u22c5\n+\nr\nr\nr\nr\n=\n2\n2\n(2)\n2(4)\n(3)\n\u2212\n+\nTherefore\n|\n|\na\n\u2212b\nr\nr\n =\n5\nExample 18 If ar  is a unit vector and (\n) (\n)\n8\nx\na\nx\na\n\u2212\n\u22c5\n+\n=\nr\nr\nr\nr\n, then find |\n|\nxr Solution Since ar  is a unit vector, |\nra =| 1 Also,\n(\n) (\n)\nx\na\nx\na\n\u2212\n\u22c5\n+\nr\nr\nr\nr  = 8\nor\nx x\nx a\na x\na a\n\u22c5\n+\n\u22c5\n\u2212\n\u22c5\n\u2212\n\u22c5\nr r\nr r\nr r\nr r  = 8\nor\n2\n|\n|\n1\nrx\n = 8  i"
    },
    {
        "Chapter": "1",
        "sentence_range": "5343-5346",
        "Text": "a a\na b\nb a\nb b\n\u2212\n\u22c5\n\u2212\n\u22c5\n+\n\u22c5\nr\nr\nr r\nr r\nr\nr\n\u00a9 NCERT\nnot to be republished\n MATHEMATICS\n446\nB\nC\nA\na\nb\n+\na\nb\n=\n2\n2\n|\n|\n2(\n) |\n|\na\na b\nb\n\u2212\n\u22c5\n+\nr\nr\nr\nr\n=\n2\n2\n(2)\n2(4)\n(3)\n\u2212\n+\nTherefore\n|\n|\na\n\u2212b\nr\nr\n =\n5\nExample 18 If ar  is a unit vector and (\n) (\n)\n8\nx\na\nx\na\n\u2212\n\u22c5\n+\n=\nr\nr\nr\nr\n, then find |\n|\nxr Solution Since ar  is a unit vector, |\nra =| 1 Also,\n(\n) (\n)\nx\na\nx\na\n\u2212\n\u22c5\n+\nr\nr\nr\nr  = 8\nor\nx x\nx a\na x\na a\n\u22c5\n+\n\u22c5\n\u2212\n\u22c5\n\u2212\n\u22c5\nr r\nr r\nr r\nr r  = 8\nor\n2\n|\n|\n1\nrx\n = 8  i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "5344-5347",
        "Text": "Solution Since ar  is a unit vector, |\nra =| 1 Also,\n(\n) (\n)\nx\na\nx\na\n\u2212\n\u22c5\n+\nr\nr\nr\nr  = 8\nor\nx x\nx a\na x\na a\n\u22c5\n+\n\u22c5\n\u2212\n\u22c5\n\u2212\n\u22c5\nr r\nr r\nr r\nr r  = 8\nor\n2\n|\n|\n1\nrx\n = 8  i e | rx |2 = 9\nTherefore\n|\n|\nxr  = 3 (as magnitude of a vector is non negative)"
    },
    {
        "Chapter": "1",
        "sentence_range": "5345-5348",
        "Text": "Also,\n(\n) (\n)\nx\na\nx\na\n\u2212\n\u22c5\n+\nr\nr\nr\nr  = 8\nor\nx x\nx a\na x\na a\n\u22c5\n+\n\u22c5\n\u2212\n\u22c5\n\u2212\n\u22c5\nr r\nr r\nr r\nr r  = 8\nor\n2\n|\n|\n1\nrx\n = 8  i e | rx |2 = 9\nTherefore\n|\n|\nxr  = 3 (as magnitude of a vector is non negative) Example 19 For any two vectors \naand\nrb\nr\n, we always have |\n|\n|\n||\n|\na b\na\nb\n\u22c5\n\u2264\nr\nr\nr\nr\n  (Cauchy-\nSchwartz inequality)"
    },
    {
        "Chapter": "1",
        "sentence_range": "5346-5349",
        "Text": "e | rx |2 = 9\nTherefore\n|\n|\nxr  = 3 (as magnitude of a vector is non negative) Example 19 For any two vectors \naand\nrb\nr\n, we always have |\n|\n|\n||\n|\na b\na\nb\n\u22c5\n\u2264\nr\nr\nr\nr\n  (Cauchy-\nSchwartz inequality) Solution The inequality holds trivially when either \n0  or  \n0\na\nb\n=\nr=\nr\nr\nr"
    },
    {
        "Chapter": "1",
        "sentence_range": "5347-5350",
        "Text": "| rx |2 = 9\nTherefore\n|\n|\nxr  = 3 (as magnitude of a vector is non negative) Example 19 For any two vectors \naand\nrb\nr\n, we always have |\n|\n|\n||\n|\na b\na\nb\n\u22c5\n\u2264\nr\nr\nr\nr\n  (Cauchy-\nSchwartz inequality) Solution The inequality holds trivially when either \n0  or  \n0\na\nb\n=\nr=\nr\nr\nr Actually, in such a\nsituation we have |\n|\n0\n|\n||\n|\na b\na\nb\n\u22c5\n=\n=\nr\nr\nr\nr"
    },
    {
        "Chapter": "1",
        "sentence_range": "5348-5351",
        "Text": "Example 19 For any two vectors \naand\nrb\nr\n, we always have |\n|\n|\n||\n|\na b\na\nb\n\u22c5\n\u2264\nr\nr\nr\nr\n  (Cauchy-\nSchwartz inequality) Solution The inequality holds trivially when either \n0  or  \n0\na\nb\n=\nr=\nr\nr\nr Actually, in such a\nsituation we have |\n|\n0\n|\n||\n|\na b\na\nb\n\u22c5\n=\n=\nr\nr\nr\nr So, let us assume that |\n|\n0\n|\n|\na\nb\n\u2260\n\u2260\nr\nr"
    },
    {
        "Chapter": "1",
        "sentence_range": "5349-5352",
        "Text": "Solution The inequality holds trivially when either \n0  or  \n0\na\nb\n=\nr=\nr\nr\nr Actually, in such a\nsituation we have |\n|\n0\n|\n||\n|\na b\na\nb\n\u22c5\n=\n=\nr\nr\nr\nr So, let us assume that |\n|\n0\n|\n|\na\nb\n\u2260\n\u2260\nr\nr Then, we have\n|\n|\n|\n||\na b|\na b\n\u22c5\nrr\nr\nr\n = | cos |\n1\n\u03b8 \u2264\nTherefore\n|\na b|\n\u22c5\nrr\n\u2264 |\n||\n|\na\nrb\nr\nExample 20 For any two vectors \naand\nrb\nr\n, we always\nhave |\n|\n|\n|\n|\n|\na\nb\na\nb\n+\n\u2264\n+\nr\nr\nr\nr\n(triangle inequality)"
    },
    {
        "Chapter": "1",
        "sentence_range": "5350-5353",
        "Text": "Actually, in such a\nsituation we have |\n|\n0\n|\n||\n|\na b\na\nb\n\u22c5\n=\n=\nr\nr\nr\nr So, let us assume that |\n|\n0\n|\n|\na\nb\n\u2260\n\u2260\nr\nr Then, we have\n|\n|\n|\n||\na b|\na b\n\u22c5\nrr\nr\nr\n = | cos |\n1\n\u03b8 \u2264\nTherefore\n|\na b|\n\u22c5\nrr\n\u2264 |\n||\n|\na\nrb\nr\nExample 20 For any two vectors \naand\nrb\nr\n, we always\nhave |\n|\n|\n|\n|\n|\na\nb\na\nb\n+\n\u2264\n+\nr\nr\nr\nr\n(triangle inequality) Solution The inequality holds trivially in case either\n0 or\n0\na\nb\n=\nr=\nr\nr\nr\n (How"
    },
    {
        "Chapter": "1",
        "sentence_range": "5351-5354",
        "Text": "So, let us assume that |\n|\n0\n|\n|\na\nb\n\u2260\n\u2260\nr\nr Then, we have\n|\n|\n|\n||\na b|\na b\n\u22c5\nrr\nr\nr\n = | cos |\n1\n\u03b8 \u2264\nTherefore\n|\na b|\n\u22c5\nrr\n\u2264 |\n||\n|\na\nrb\nr\nExample 20 For any two vectors \naand\nrb\nr\n, we always\nhave |\n|\n|\n|\n|\n|\na\nb\na\nb\n+\n\u2264\n+\nr\nr\nr\nr\n(triangle inequality) Solution The inequality holds trivially in case either\n0 or\n0\na\nb\n=\nr=\nr\nr\nr\n (How )"
    },
    {
        "Chapter": "1",
        "sentence_range": "5352-5355",
        "Text": "Then, we have\n|\n|\n|\n||\na b|\na b\n\u22c5\nrr\nr\nr\n = | cos |\n1\n\u03b8 \u2264\nTherefore\n|\na b|\n\u22c5\nrr\n\u2264 |\n||\n|\na\nrb\nr\nExample 20 For any two vectors \naand\nrb\nr\n, we always\nhave |\n|\n|\n|\n|\n|\na\nb\na\nb\n+\n\u2264\n+\nr\nr\nr\nr\n(triangle inequality) Solution The inequality holds trivially in case either\n0 or\n0\na\nb\n=\nr=\nr\nr\nr\n (How ) So, let |\n|\n0\n| |\na\nb\n \n \nr\nr\nr"
    },
    {
        "Chapter": "1",
        "sentence_range": "5353-5356",
        "Text": "Solution The inequality holds trivially in case either\n0 or\n0\na\nb\n=\nr=\nr\nr\nr\n (How ) So, let |\n|\n0\n| |\na\nb\n \n \nr\nr\nr Then,\n2\n|\n|\na\n+b\nr\nr\n =\n2\n(\n)\n(\n) (\n)\na\nb\na\nb\na\nb\n+\n=\n+\n\u22c5\n+\nr\nr\nr\nr\nr\nr\n= a a\na b\nb a\nb b\n\u22c5\n+\n\u22c5\n+\n\u22c5\n+\n\u22c5\nr\nr\nr r\nr r\nr\nr\n=\n2\n2\n|\n|\n2\n|\n|\na\na b\nb\n+\n\u22c5 +\nr\nr\nr\nr\n (scalar product is commutative)\n\u2264\n2\n2\n|\n|\n2|\n|\n|\n|\na\na b\nb\n+\n\u22c5\nr+\nr\nr\nr\n(since \n|\n|\nx\nx\nx\n\u2264\n\u2200 \u2208 R )\n\u2264\n2\n2\n|\n|\n2|\n||\n|\n|\n|\na\na b\nb\n+\n+\nr\nr\nr\nr\n(from Example 19)\n=\n2\n(|\n|\n|\n|)\na\nb\n \nr\nr\nFig 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5354-5357",
        "Text": ") So, let |\n|\n0\n| |\na\nb\n \n \nr\nr\nr Then,\n2\n|\n|\na\n+b\nr\nr\n =\n2\n(\n)\n(\n) (\n)\na\nb\na\nb\na\nb\n+\n=\n+\n\u22c5\n+\nr\nr\nr\nr\nr\nr\n= a a\na b\nb a\nb b\n\u22c5\n+\n\u22c5\n+\n\u22c5\n+\n\u22c5\nr\nr\nr r\nr r\nr\nr\n=\n2\n2\n|\n|\n2\n|\n|\na\na b\nb\n+\n\u22c5 +\nr\nr\nr\nr\n (scalar product is commutative)\n\u2264\n2\n2\n|\n|\n2|\n|\n|\n|\na\na b\nb\n+\n\u22c5\nr+\nr\nr\nr\n(since \n|\n|\nx\nx\nx\n\u2264\n\u2200 \u2208 R )\n\u2264\n2\n2\n|\n|\n2|\n||\n|\n|\n|\na\na b\nb\n+\n+\nr\nr\nr\nr\n(from Example 19)\n=\n2\n(|\n|\n|\n|)\na\nb\n \nr\nr\nFig 10 21\n\u00a9 NCERT\nnot to be republished\nVECTOR ALGEBRA\n447\nHence\n|\n|\na\nb\n \nr\nr\n\u2264 |\n|\n|\n|\na\nb\n \nr\nr\nRemark If the equality holds in triangle inequality (in the above Example 20), i"
    },
    {
        "Chapter": "1",
        "sentence_range": "5355-5358",
        "Text": "So, let |\n|\n0\n| |\na\nb\n \n \nr\nr\nr Then,\n2\n|\n|\na\n+b\nr\nr\n =\n2\n(\n)\n(\n) (\n)\na\nb\na\nb\na\nb\n+\n=\n+\n\u22c5\n+\nr\nr\nr\nr\nr\nr\n= a a\na b\nb a\nb b\n\u22c5\n+\n\u22c5\n+\n\u22c5\n+\n\u22c5\nr\nr\nr r\nr r\nr\nr\n=\n2\n2\n|\n|\n2\n|\n|\na\na b\nb\n+\n\u22c5 +\nr\nr\nr\nr\n (scalar product is commutative)\n\u2264\n2\n2\n|\n|\n2|\n|\n|\n|\na\na b\nb\n+\n\u22c5\nr+\nr\nr\nr\n(since \n|\n|\nx\nx\nx\n\u2264\n\u2200 \u2208 R )\n\u2264\n2\n2\n|\n|\n2|\n||\n|\n|\n|\na\na b\nb\n+\n+\nr\nr\nr\nr\n(from Example 19)\n=\n2\n(|\n|\n|\n|)\na\nb\n \nr\nr\nFig 10 21\n\u00a9 NCERT\nnot to be republished\nVECTOR ALGEBRA\n447\nHence\n|\n|\na\nb\n \nr\nr\n\u2264 |\n|\n|\n|\na\nb\n \nr\nr\nRemark If the equality holds in triangle inequality (in the above Example 20), i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "5356-5359",
        "Text": "Then,\n2\n|\n|\na\n+b\nr\nr\n =\n2\n(\n)\n(\n) (\n)\na\nb\na\nb\na\nb\n+\n=\n+\n\u22c5\n+\nr\nr\nr\nr\nr\nr\n= a a\na b\nb a\nb b\n\u22c5\n+\n\u22c5\n+\n\u22c5\n+\n\u22c5\nr\nr\nr r\nr r\nr\nr\n=\n2\n2\n|\n|\n2\n|\n|\na\na b\nb\n+\n\u22c5 +\nr\nr\nr\nr\n (scalar product is commutative)\n\u2264\n2\n2\n|\n|\n2|\n|\n|\n|\na\na b\nb\n+\n\u22c5\nr+\nr\nr\nr\n(since \n|\n|\nx\nx\nx\n\u2264\n\u2200 \u2208 R )\n\u2264\n2\n2\n|\n|\n2|\n||\n|\n|\n|\na\na b\nb\n+\n+\nr\nr\nr\nr\n(from Example 19)\n=\n2\n(|\n|\n|\n|)\na\nb\n \nr\nr\nFig 10 21\n\u00a9 NCERT\nnot to be republished\nVECTOR ALGEBRA\n447\nHence\n|\n|\na\nb\n \nr\nr\n\u2264 |\n|\n|\n|\na\nb\n \nr\nr\nRemark If the equality holds in triangle inequality (in the above Example 20), i e |\n|\na\nb\n+\nr\nr\n = |\n|\n|\n|\na\n+b\nr\nr\n,\nthen\n| AC|\nuuur\n = | AB|\nuuur+| BC |\nuuur\nshowing that the points A, B and C are collinear"
    },
    {
        "Chapter": "1",
        "sentence_range": "5357-5360",
        "Text": "21\n\u00a9 NCERT\nnot to be republished\nVECTOR ALGEBRA\n447\nHence\n|\n|\na\nb\n \nr\nr\n\u2264 |\n|\n|\n|\na\nb\n \nr\nr\nRemark If the equality holds in triangle inequality (in the above Example 20), i e |\n|\na\nb\n+\nr\nr\n = |\n|\n|\n|\na\n+b\nr\nr\n,\nthen\n| AC|\nuuur\n = | AB|\nuuur+| BC |\nuuur\nshowing that the points A, B and C are collinear Example 21 Show that the points \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\nA( 2\n3\n5 ), B(\n2\n3 )\ni\nj\nk\ni\nj\nk\n\u2212\n+\n+\n+\n+\n and \n\u02c6\nC(7\u02c6\n)\ni\nk\n\u2212\nare collinear"
    },
    {
        "Chapter": "1",
        "sentence_range": "5358-5361",
        "Text": "e |\n|\na\nb\n+\nr\nr\n = |\n|\n|\n|\na\n+b\nr\nr\n,\nthen\n| AC|\nuuur\n = | AB|\nuuur+| BC |\nuuur\nshowing that the points A, B and C are collinear Example 21 Show that the points \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\nA( 2\n3\n5 ), B(\n2\n3 )\ni\nj\nk\ni\nj\nk\n\u2212\n+\n+\n+\n+\n and \n\u02c6\nC(7\u02c6\n)\ni\nk\n\u2212\nare collinear Solution We have\n AB\nuuur\n =\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(1\n2)\n(2\n3)\n(3\n5)\n3\n2\ni\nj\nk\ni\nj\nk\n \n \n \n \n \n \n \n,\nBC\nuuur\n =\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(7\n1)\n(0\n2)\n( 1\n3)\n6\n2\n4\ni\nj\nk\ni\nj\nk\n \n \n \n \n \n \n \n \n \n,\nAC\nuuur\n =\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(7\n2)\n(0\n3)\n( 1\n5)\n9\n3\n6\ni\nj\nk\ni\nj\nk\n \n \n \n \n \n \n \n \n \n| AB|\nuuur\n =\n14, | BC|\n 2 14  and  | AC| 3 14\n \nuuur\nuuur\nTherefore\nAC\nuuur\n = | AB|\nuuur+| BC |\nuuur\nHence the points A, B and C are collinear"
    },
    {
        "Chapter": "1",
        "sentence_range": "5359-5362",
        "Text": "|\n|\na\nb\n+\nr\nr\n = |\n|\n|\n|\na\n+b\nr\nr\n,\nthen\n| AC|\nuuur\n = | AB|\nuuur+| BC |\nuuur\nshowing that the points A, B and C are collinear Example 21 Show that the points \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\nA( 2\n3\n5 ), B(\n2\n3 )\ni\nj\nk\ni\nj\nk\n\u2212\n+\n+\n+\n+\n and \n\u02c6\nC(7\u02c6\n)\ni\nk\n\u2212\nare collinear Solution We have\n AB\nuuur\n =\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(1\n2)\n(2\n3)\n(3\n5)\n3\n2\ni\nj\nk\ni\nj\nk\n \n \n \n \n \n \n \n,\nBC\nuuur\n =\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(7\n1)\n(0\n2)\n( 1\n3)\n6\n2\n4\ni\nj\nk\ni\nj\nk\n \n \n \n \n \n \n \n \n \n,\nAC\nuuur\n =\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(7\n2)\n(0\n3)\n( 1\n5)\n9\n3\n6\ni\nj\nk\ni\nj\nk\n \n \n \n \n \n \n \n \n \n| AB|\nuuur\n =\n14, | BC|\n 2 14  and  | AC| 3 14\n \nuuur\nuuur\nTherefore\nAC\nuuur\n = | AB|\nuuur+| BC |\nuuur\nHence the points A, B and C are collinear \ufffdNote   In Example 21, one may note that although AB\nBC\nCA\n0\n+\n+\n=\nuuur\nuuur\nuuur\nr  but the\npoints A, B and C do not form the vertices of a triangle"
    },
    {
        "Chapter": "1",
        "sentence_range": "5360-5363",
        "Text": "Example 21 Show that the points \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\nA( 2\n3\n5 ), B(\n2\n3 )\ni\nj\nk\ni\nj\nk\n\u2212\n+\n+\n+\n+\n and \n\u02c6\nC(7\u02c6\n)\ni\nk\n\u2212\nare collinear Solution We have\n AB\nuuur\n =\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(1\n2)\n(2\n3)\n(3\n5)\n3\n2\ni\nj\nk\ni\nj\nk\n \n \n \n \n \n \n \n,\nBC\nuuur\n =\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(7\n1)\n(0\n2)\n( 1\n3)\n6\n2\n4\ni\nj\nk\ni\nj\nk\n \n \n \n \n \n \n \n \n \n,\nAC\nuuur\n =\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(7\n2)\n(0\n3)\n( 1\n5)\n9\n3\n6\ni\nj\nk\ni\nj\nk\n \n \n \n \n \n \n \n \n \n| AB|\nuuur\n =\n14, | BC|\n 2 14  and  | AC| 3 14\n \nuuur\nuuur\nTherefore\nAC\nuuur\n = | AB|\nuuur+| BC |\nuuur\nHence the points A, B and C are collinear \ufffdNote   In Example 21, one may note that although AB\nBC\nCA\n0\n+\n+\n=\nuuur\nuuur\nuuur\nr  but the\npoints A, B and C do not form the vertices of a triangle EXERCISE 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5361-5364",
        "Text": "Solution We have\n AB\nuuur\n =\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(1\n2)\n(2\n3)\n(3\n5)\n3\n2\ni\nj\nk\ni\nj\nk\n \n \n \n \n \n \n \n,\nBC\nuuur\n =\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(7\n1)\n(0\n2)\n( 1\n3)\n6\n2\n4\ni\nj\nk\ni\nj\nk\n \n \n \n \n \n \n \n \n \n,\nAC\nuuur\n =\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(7\n2)\n(0\n3)\n( 1\n5)\n9\n3\n6\ni\nj\nk\ni\nj\nk\n \n \n \n \n \n \n \n \n \n| AB|\nuuur\n =\n14, | BC|\n 2 14  and  | AC| 3 14\n \nuuur\nuuur\nTherefore\nAC\nuuur\n = | AB|\nuuur+| BC |\nuuur\nHence the points A, B and C are collinear \ufffdNote   In Example 21, one may note that although AB\nBC\nCA\n0\n+\n+\n=\nuuur\nuuur\nuuur\nr  but the\npoints A, B and C do not form the vertices of a triangle EXERCISE 10 3\n1"
    },
    {
        "Chapter": "1",
        "sentence_range": "5362-5365",
        "Text": "\ufffdNote   In Example 21, one may note that although AB\nBC\nCA\n0\n+\n+\n=\nuuur\nuuur\nuuur\nr  but the\npoints A, B and C do not form the vertices of a triangle EXERCISE 10 3\n1 Find the angle between two vectors \naand \nrb\nr\nwith magnitudes \n3  and 2 ,\nrespectively having \n6\na b\n\u22c5\n=\nrr"
    },
    {
        "Chapter": "1",
        "sentence_range": "5363-5366",
        "Text": "EXERCISE 10 3\n1 Find the angle between two vectors \naand \nrb\nr\nwith magnitudes \n3  and 2 ,\nrespectively having \n6\na b\n\u22c5\n=\nrr 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "5364-5367",
        "Text": "3\n1 Find the angle between two vectors \naand \nrb\nr\nwith magnitudes \n3  and 2 ,\nrespectively having \n6\na b\n\u22c5\n=\nrr 2 Find the angle between the vectors \n\u02c6\n\u02c6\n2\u02c6\n3  \ni\nj\nk\n\u2212\n+\n and \n\u02c6\n\u02c6\n\u02c6\n3\n2\ni\nj\nk\n\u2212\n+\n3"
    },
    {
        "Chapter": "1",
        "sentence_range": "5365-5368",
        "Text": "Find the angle between two vectors \naand \nrb\nr\nwith magnitudes \n3  and 2 ,\nrespectively having \n6\na b\n\u22c5\n=\nrr 2 Find the angle between the vectors \n\u02c6\n\u02c6\n2\u02c6\n3  \ni\nj\nk\n\u2212\n+\n and \n\u02c6\n\u02c6\n\u02c6\n3\n2\ni\nj\nk\n\u2212\n+\n3 Find the projection of the vector \u02c6\n\u02c6\ni\n\u2212j\n on the vector \u02c6\n\u02c6\ni\n+j"
    },
    {
        "Chapter": "1",
        "sentence_range": "5366-5369",
        "Text": "2 Find the angle between the vectors \n\u02c6\n\u02c6\n2\u02c6\n3  \ni\nj\nk\n\u2212\n+\n and \n\u02c6\n\u02c6\n\u02c6\n3\n2\ni\nj\nk\n\u2212\n+\n3 Find the projection of the vector \u02c6\n\u02c6\ni\n\u2212j\n on the vector \u02c6\n\u02c6\ni\n+j 4"
    },
    {
        "Chapter": "1",
        "sentence_range": "5367-5370",
        "Text": "Find the angle between the vectors \n\u02c6\n\u02c6\n2\u02c6\n3  \ni\nj\nk\n\u2212\n+\n and \n\u02c6\n\u02c6\n\u02c6\n3\n2\ni\nj\nk\n\u2212\n+\n3 Find the projection of the vector \u02c6\n\u02c6\ni\n\u2212j\n on the vector \u02c6\n\u02c6\ni\n+j 4 Find the projection of the vector \n\u02c6\n\u02c6\n3\u02c6\n7\ni\nj\nk\n+\n+\n on the vector \n\u02c6\n\u02c6\n\u02c6\n7\n8\ni\nj\nk\n\u2212\n+"
    },
    {
        "Chapter": "1",
        "sentence_range": "5368-5371",
        "Text": "Find the projection of the vector \u02c6\n\u02c6\ni\n\u2212j\n on the vector \u02c6\n\u02c6\ni\n+j 4 Find the projection of the vector \n\u02c6\n\u02c6\n3\u02c6\n7\ni\nj\nk\n+\n+\n on the vector \n\u02c6\n\u02c6\n\u02c6\n7\n8\ni\nj\nk\n\u2212\n+ 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "5369-5372",
        "Text": "4 Find the projection of the vector \n\u02c6\n\u02c6\n3\u02c6\n7\ni\nj\nk\n+\n+\n on the vector \n\u02c6\n\u02c6\n\u02c6\n7\n8\ni\nj\nk\n\u2212\n+ 5 Show that each of the given three vectors is a unit vector:\n1\n1\n1\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(2\n3\n6 ), \n(3\n6\n2 ),  \n(6\n2\n3 )\n7\n7\n7\ni\nj\nk\ni\nj\nk\ni\nj\nk\n+\n+\n\u2212\n+\n+\n\u2212\nAlso, show that they are mutually perpendicular to each other"
    },
    {
        "Chapter": "1",
        "sentence_range": "5370-5373",
        "Text": "Find the projection of the vector \n\u02c6\n\u02c6\n3\u02c6\n7\ni\nj\nk\n+\n+\n on the vector \n\u02c6\n\u02c6\n\u02c6\n7\n8\ni\nj\nk\n\u2212\n+ 5 Show that each of the given three vectors is a unit vector:\n1\n1\n1\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(2\n3\n6 ), \n(3\n6\n2 ),  \n(6\n2\n3 )\n7\n7\n7\ni\nj\nk\ni\nj\nk\ni\nj\nk\n+\n+\n\u2212\n+\n+\n\u2212\nAlso, show that they are mutually perpendicular to each other \u00a9 NCERT\nnot to be republished\n MATHEMATICS\n448\n6"
    },
    {
        "Chapter": "1",
        "sentence_range": "5371-5374",
        "Text": "5 Show that each of the given three vectors is a unit vector:\n1\n1\n1\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(2\n3\n6 ), \n(3\n6\n2 ),  \n(6\n2\n3 )\n7\n7\n7\ni\nj\nk\ni\nj\nk\ni\nj\nk\n+\n+\n\u2212\n+\n+\n\u2212\nAlso, show that they are mutually perpendicular to each other \u00a9 NCERT\nnot to be republished\n MATHEMATICS\n448\n6 Find |\n|  and |\n|\na\nrb\nr\n, if (\n) (\n)\n8  and | | 8|\n|\na\nb\na\nb\na\nb\n+\n\u22c5\n\u2212\n=\n=\nr\nr\nr\nr\nr\nr"
    },
    {
        "Chapter": "1",
        "sentence_range": "5372-5375",
        "Text": "Show that each of the given three vectors is a unit vector:\n1\n1\n1\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(2\n3\n6 ), \n(3\n6\n2 ),  \n(6\n2\n3 )\n7\n7\n7\ni\nj\nk\ni\nj\nk\ni\nj\nk\n+\n+\n\u2212\n+\n+\n\u2212\nAlso, show that they are mutually perpendicular to each other \u00a9 NCERT\nnot to be republished\n MATHEMATICS\n448\n6 Find |\n|  and |\n|\na\nrb\nr\n, if (\n) (\n)\n8  and | | 8|\n|\na\nb\na\nb\na\nb\n+\n\u22c5\n\u2212\n=\n=\nr\nr\nr\nr\nr\nr 7"
    },
    {
        "Chapter": "1",
        "sentence_range": "5373-5376",
        "Text": "\u00a9 NCERT\nnot to be republished\n MATHEMATICS\n448\n6 Find |\n|  and |\n|\na\nrb\nr\n, if (\n) (\n)\n8  and | | 8|\n|\na\nb\na\nb\na\nb\n+\n\u22c5\n\u2212\n=\n=\nr\nr\nr\nr\nr\nr 7 Evaluate the product (3\n5 ) (2\n7 )\na\nb\na\nb\n\u2212\n\u22c5\n+\nr\nr\nr\nr"
    },
    {
        "Chapter": "1",
        "sentence_range": "5374-5377",
        "Text": "Find |\n|  and |\n|\na\nrb\nr\n, if (\n) (\n)\n8  and | | 8|\n|\na\nb\na\nb\na\nb\n+\n\u22c5\n\u2212\n=\n=\nr\nr\nr\nr\nr\nr 7 Evaluate the product (3\n5 ) (2\n7 )\na\nb\na\nb\n\u2212\n\u22c5\n+\nr\nr\nr\nr 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "5375-5378",
        "Text": "7 Evaluate the product (3\n5 ) (2\n7 )\na\nb\na\nb\n\u2212\n\u22c5\n+\nr\nr\nr\nr 8 Find the magnitude of two vectors \naand \nrb\nr\n, having the same magnitude and\nsuch that the angle between them is 60o and their scalar product is 1\n2"
    },
    {
        "Chapter": "1",
        "sentence_range": "5376-5379",
        "Text": "Evaluate the product (3\n5 ) (2\n7 )\na\nb\na\nb\n\u2212\n\u22c5\n+\nr\nr\nr\nr 8 Find the magnitude of two vectors \naand \nrb\nr\n, having the same magnitude and\nsuch that the angle between them is 60o and their scalar product is 1\n2 9"
    },
    {
        "Chapter": "1",
        "sentence_range": "5377-5380",
        "Text": "8 Find the magnitude of two vectors \naand \nrb\nr\n, having the same magnitude and\nsuch that the angle between them is 60o and their scalar product is 1\n2 9 Find |\n|\nxr , if for a unit vector ar , (\n) (\n)\n12\nx\na\nx\na\n\u2212\n\u22c5\n+\n=\nr\nr\nr\nr"
    },
    {
        "Chapter": "1",
        "sentence_range": "5378-5381",
        "Text": "Find the magnitude of two vectors \naand \nrb\nr\n, having the same magnitude and\nsuch that the angle between them is 60o and their scalar product is 1\n2 9 Find |\n|\nxr , if for a unit vector ar , (\n) (\n)\n12\nx\na\nx\na\n\u2212\n\u22c5\n+\n=\nr\nr\nr\nr 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5379-5382",
        "Text": "9 Find |\n|\nxr , if for a unit vector ar , (\n) (\n)\n12\nx\na\nx\na\n\u2212\n\u22c5\n+\n=\nr\nr\nr\nr 10 If \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n2\n2\n3 ,\n2\nand\n3\na\ni\nj\nk\nb\ni\nj\nk\nc\ni\nj\n=\n+\n+\n= \u2212 +\n+\n=\n+\nr\nr\nr\nare such that a\n+ \u03bbb\nr\nr\n is\nperpendicular to cr , then find the value of \u03bb"
    },
    {
        "Chapter": "1",
        "sentence_range": "5380-5383",
        "Text": "Find |\n|\nxr , if for a unit vector ar , (\n) (\n)\n12\nx\na\nx\na\n\u2212\n\u22c5\n+\n=\nr\nr\nr\nr 10 If \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n2\n2\n3 ,\n2\nand\n3\na\ni\nj\nk\nb\ni\nj\nk\nc\ni\nj\n=\n+\n+\n= \u2212 +\n+\n=\n+\nr\nr\nr\nare such that a\n+ \u03bbb\nr\nr\n is\nperpendicular to cr , then find the value of \u03bb 11"
    },
    {
        "Chapter": "1",
        "sentence_range": "5381-5384",
        "Text": "10 If \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n2\n2\n3 ,\n2\nand\n3\na\ni\nj\nk\nb\ni\nj\nk\nc\ni\nj\n=\n+\n+\n= \u2212 +\n+\n=\n+\nr\nr\nr\nare such that a\n+ \u03bbb\nr\nr\n is\nperpendicular to cr , then find the value of \u03bb 11 Show that |\n|\n|\n|\na b\nr+b a\nr\nr\nr  is perpendicular to |\n|\n|\n|\na b\nr\u2212b a\nr\nr\nr , for any two nonzero\nvectors \naand \nrb\nr"
    },
    {
        "Chapter": "1",
        "sentence_range": "5382-5385",
        "Text": "If \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n2\n2\n3 ,\n2\nand\n3\na\ni\nj\nk\nb\ni\nj\nk\nc\ni\nj\n=\n+\n+\n= \u2212 +\n+\n=\n+\nr\nr\nr\nare such that a\n+ \u03bbb\nr\nr\n is\nperpendicular to cr , then find the value of \u03bb 11 Show that |\n|\n|\n|\na b\nr+b a\nr\nr\nr  is perpendicular to |\n|\n|\n|\na b\nr\u2212b a\nr\nr\nr , for any two nonzero\nvectors \naand \nrb\nr 12"
    },
    {
        "Chapter": "1",
        "sentence_range": "5383-5386",
        "Text": "11 Show that |\n|\n|\n|\na b\nr+b a\nr\nr\nr  is perpendicular to |\n|\n|\n|\na b\nr\u2212b a\nr\nr\nr , for any two nonzero\nvectors \naand \nrb\nr 12 If \n0  and  \n0\na a\na b\n\u22c5\n=\n\u22c5\n=\nr\nr r\nr\n, then what can be concluded about the vector b\nr"
    },
    {
        "Chapter": "1",
        "sentence_range": "5384-5387",
        "Text": "Show that |\n|\n|\n|\na b\nr+b a\nr\nr\nr  is perpendicular to |\n|\n|\n|\na b\nr\u2212b a\nr\nr\nr , for any two nonzero\nvectors \naand \nrb\nr 12 If \n0  and  \n0\na a\na b\n\u22c5\n=\n\u22c5\n=\nr\nr r\nr\n, then what can be concluded about the vector b\nr 13"
    },
    {
        "Chapter": "1",
        "sentence_range": "5385-5388",
        "Text": "12 If \n0  and  \n0\na a\na b\n\u22c5\n=\n\u22c5\n=\nr\nr r\nr\n, then what can be concluded about the vector b\nr 13 If \n, ,\na b c\nrr\nr  are unit vectors such that \n0\na\nb\nc\n+\n+\n=\nr\nr\nr\nr\n, find the value of\na b\nb c\nc a\n\u22c5\n+\n\u22c5\n+\n\u22c5\nr\nr\nr\nr\nr r"
    },
    {
        "Chapter": "1",
        "sentence_range": "5386-5389",
        "Text": "If \n0  and  \n0\na a\na b\n\u22c5\n=\n\u22c5\n=\nr\nr r\nr\n, then what can be concluded about the vector b\nr 13 If \n, ,\na b c\nrr\nr  are unit vectors such that \n0\na\nb\nc\n+\n+\n=\nr\nr\nr\nr\n, find the value of\na b\nb c\nc a\n\u22c5\n+\n\u22c5\n+\n\u22c5\nr\nr\nr\nr\nr r 14"
    },
    {
        "Chapter": "1",
        "sentence_range": "5387-5390",
        "Text": "13 If \n, ,\na b c\nrr\nr  are unit vectors such that \n0\na\nb\nc\n+\n+\n=\nr\nr\nr\nr\n, find the value of\na b\nb c\nc a\n\u22c5\n+\n\u22c5\n+\n\u22c5\nr\nr\nr\nr\nr r 14 If either vector \n0  or \n0,  then  \n0\na\nb\na b\n=\n=\n\u22c5\n=\nr\nr\nr\nr\nr\nr"
    },
    {
        "Chapter": "1",
        "sentence_range": "5388-5391",
        "Text": "If \n, ,\na b c\nrr\nr  are unit vectors such that \n0\na\nb\nc\n+\n+\n=\nr\nr\nr\nr\n, find the value of\na b\nb c\nc a\n\u22c5\n+\n\u22c5\n+\n\u22c5\nr\nr\nr\nr\nr r 14 If either vector \n0  or \n0,  then  \n0\na\nb\na b\n=\n=\n\u22c5\n=\nr\nr\nr\nr\nr\nr But the converse need not be\ntrue"
    },
    {
        "Chapter": "1",
        "sentence_range": "5389-5392",
        "Text": "14 If either vector \n0  or \n0,  then  \n0\na\nb\na b\n=\n=\n\u22c5\n=\nr\nr\nr\nr\nr\nr But the converse need not be\ntrue Justify your answer with an example"
    },
    {
        "Chapter": "1",
        "sentence_range": "5390-5393",
        "Text": "If either vector \n0  or \n0,  then  \n0\na\nb\na b\n=\n=\n\u22c5\n=\nr\nr\nr\nr\nr\nr But the converse need not be\ntrue Justify your answer with an example 15"
    },
    {
        "Chapter": "1",
        "sentence_range": "5391-5394",
        "Text": "But the converse need not be\ntrue Justify your answer with an example 15 If the vertices A, B, C of a triangle ABC are (1, 2, 3), (\u20131, 0, 0), (0, 1, 2),\nrespectively, then find \u2220ABC"
    },
    {
        "Chapter": "1",
        "sentence_range": "5392-5395",
        "Text": "Justify your answer with an example 15 If the vertices A, B, C of a triangle ABC are (1, 2, 3), (\u20131, 0, 0), (0, 1, 2),\nrespectively, then find \u2220ABC [\u2220ABC is the angle between the vectors BA\nuuur\nand BC\nuuur ]"
    },
    {
        "Chapter": "1",
        "sentence_range": "5393-5396",
        "Text": "15 If the vertices A, B, C of a triangle ABC are (1, 2, 3), (\u20131, 0, 0), (0, 1, 2),\nrespectively, then find \u2220ABC [\u2220ABC is the angle between the vectors BA\nuuur\nand BC\nuuur ] 16"
    },
    {
        "Chapter": "1",
        "sentence_range": "5394-5397",
        "Text": "If the vertices A, B, C of a triangle ABC are (1, 2, 3), (\u20131, 0, 0), (0, 1, 2),\nrespectively, then find \u2220ABC [\u2220ABC is the angle between the vectors BA\nuuur\nand BC\nuuur ] 16 Show that the points A(1, 2, 7), B(2, 6, 3) and C(3, 10, \u20131) are collinear"
    },
    {
        "Chapter": "1",
        "sentence_range": "5395-5398",
        "Text": "[\u2220ABC is the angle between the vectors BA\nuuur\nand BC\nuuur ] 16 Show that the points A(1, 2, 7), B(2, 6, 3) and C(3, 10, \u20131) are collinear 17"
    },
    {
        "Chapter": "1",
        "sentence_range": "5396-5399",
        "Text": "16 Show that the points A(1, 2, 7), B(2, 6, 3) and C(3, 10, \u20131) are collinear 17 Show that the vectors \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n2\n,\n3\n5\nand 3\n4\n4\ni\nj\nk i\nj\nk\ni\nj\nk\n\u2212\n+\n\u2212\n\u2212\n\u2212\n\u2212\n form the vertices\nof a right angled triangle"
    },
    {
        "Chapter": "1",
        "sentence_range": "5397-5400",
        "Text": "Show that the points A(1, 2, 7), B(2, 6, 3) and C(3, 10, \u20131) are collinear 17 Show that the vectors \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n2\n,\n3\n5\nand 3\n4\n4\ni\nj\nk i\nj\nk\ni\nj\nk\n\u2212\n+\n\u2212\n\u2212\n\u2212\n\u2212\n form the vertices\nof a right angled triangle 18"
    },
    {
        "Chapter": "1",
        "sentence_range": "5398-5401",
        "Text": "17 Show that the vectors \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n2\n,\n3\n5\nand 3\n4\n4\ni\nj\nk i\nj\nk\ni\nj\nk\n\u2212\n+\n\u2212\n\u2212\n\u2212\n\u2212\n form the vertices\nof a right angled triangle 18 If ar  is a nonzero vector of magnitude \u2018a\u2019 and \u03bb a nonzero scalar, then \u03bb ar  is unit\nvector if\n(A) \u03bb = 1\n(B) \u03bb = \u2013 1\n(C) a = |\u03bb|\n(D) a = 1/|\u03bb|\n10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5399-5402",
        "Text": "Show that the vectors \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n2\n,\n3\n5\nand 3\n4\n4\ni\nj\nk i\nj\nk\ni\nj\nk\n\u2212\n+\n\u2212\n\u2212\n\u2212\n\u2212\n form the vertices\nof a right angled triangle 18 If ar  is a nonzero vector of magnitude \u2018a\u2019 and \u03bb a nonzero scalar, then \u03bb ar  is unit\nvector if\n(A) \u03bb = 1\n(B) \u03bb = \u2013 1\n(C) a = |\u03bb|\n(D) a = 1/|\u03bb|\n10 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "5400-5403",
        "Text": "18 If ar  is a nonzero vector of magnitude \u2018a\u2019 and \u03bb a nonzero scalar, then \u03bb ar  is unit\nvector if\n(A) \u03bb = 1\n(B) \u03bb = \u2013 1\n(C) a = |\u03bb|\n(D) a = 1/|\u03bb|\n10 6 3  Vector (or cross) product of two vectors\nIn Section 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5401-5404",
        "Text": "If ar  is a nonzero vector of magnitude \u2018a\u2019 and \u03bb a nonzero scalar, then \u03bb ar  is unit\nvector if\n(A) \u03bb = 1\n(B) \u03bb = \u2013 1\n(C) a = |\u03bb|\n(D) a = 1/|\u03bb|\n10 6 3  Vector (or cross) product of two vectors\nIn Section 10 2, we have discussed on the three dimensional right handed rectangular\ncoordinate system"
    },
    {
        "Chapter": "1",
        "sentence_range": "5402-5405",
        "Text": "6 3  Vector (or cross) product of two vectors\nIn Section 10 2, we have discussed on the three dimensional right handed rectangular\ncoordinate system In this system, when the positive x-axis is rotated counterclockwise\n\u00a9 NCERT\nnot to be republished\nVECTOR ALGEBRA\n449\ninto the positive y-axis, a right handed (standard) screw would advance in the direction\nof the positive z-axis (Fig 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5403-5406",
        "Text": "3  Vector (or cross) product of two vectors\nIn Section 10 2, we have discussed on the three dimensional right handed rectangular\ncoordinate system In this system, when the positive x-axis is rotated counterclockwise\n\u00a9 NCERT\nnot to be republished\nVECTOR ALGEBRA\n449\ninto the positive y-axis, a right handed (standard) screw would advance in the direction\nof the positive z-axis (Fig 10 22(i))"
    },
    {
        "Chapter": "1",
        "sentence_range": "5404-5407",
        "Text": "2, we have discussed on the three dimensional right handed rectangular\ncoordinate system In this system, when the positive x-axis is rotated counterclockwise\n\u00a9 NCERT\nnot to be republished\nVECTOR ALGEBRA\n449\ninto the positive y-axis, a right handed (standard) screw would advance in the direction\nof the positive z-axis (Fig 10 22(i)) In a right handed coordinate system, the thumb of the right hand points in the\ndirection of the positive z-axis when the fingers are curled in the direction away from\nthe positive x-axis toward the positive y-axis (Fig 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5405-5408",
        "Text": "In this system, when the positive x-axis is rotated counterclockwise\n\u00a9 NCERT\nnot to be republished\nVECTOR ALGEBRA\n449\ninto the positive y-axis, a right handed (standard) screw would advance in the direction\nof the positive z-axis (Fig 10 22(i)) In a right handed coordinate system, the thumb of the right hand points in the\ndirection of the positive z-axis when the fingers are curled in the direction away from\nthe positive x-axis toward the positive y-axis (Fig 10 22(ii))"
    },
    {
        "Chapter": "1",
        "sentence_range": "5406-5409",
        "Text": "22(i)) In a right handed coordinate system, the thumb of the right hand points in the\ndirection of the positive z-axis when the fingers are curled in the direction away from\nthe positive x-axis toward the positive y-axis (Fig 10 22(ii)) Fig 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5407-5410",
        "Text": "In a right handed coordinate system, the thumb of the right hand points in the\ndirection of the positive z-axis when the fingers are curled in the direction away from\nthe positive x-axis toward the positive y-axis (Fig 10 22(ii)) Fig 10 22 (i), (ii)\nDefinition 3 The vector product of two nonzero vectors \naand\nrb\nr\n, is denoted by a\n b\nr\nr\nand defined as\na\nb\n\u00d7\nr\nr\n =\n\u02c6\n|\n||\na b|sin\nn\n\u03b8\nr\nr\n,\nwhere, \u03b8 is the angle between  \naand\nrb\nr\n, 0 \u2264 \u03b8 \u2264 \u03c0  and \u02c6n  is\na unit vector perpendicular to both \na and  \nrb\nr\n, such that\n\u02c6\n,\na b and  \nn\nrr\n form a right handed system (Fig 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5408-5411",
        "Text": "22(ii)) Fig 10 22 (i), (ii)\nDefinition 3 The vector product of two nonzero vectors \naand\nrb\nr\n, is denoted by a\n b\nr\nr\nand defined as\na\nb\n\u00d7\nr\nr\n =\n\u02c6\n|\n||\na b|sin\nn\n\u03b8\nr\nr\n,\nwhere, \u03b8 is the angle between  \naand\nrb\nr\n, 0 \u2264 \u03b8 \u2264 \u03c0  and \u02c6n  is\na unit vector perpendicular to both \na and  \nrb\nr\n, such that\n\u02c6\n,\na b and  \nn\nrr\n form a right handed system (Fig 10 23)"
    },
    {
        "Chapter": "1",
        "sentence_range": "5409-5412",
        "Text": "Fig 10 22 (i), (ii)\nDefinition 3 The vector product of two nonzero vectors \naand\nrb\nr\n, is denoted by a\n b\nr\nr\nand defined as\na\nb\n\u00d7\nr\nr\n =\n\u02c6\n|\n||\na b|sin\nn\n\u03b8\nr\nr\n,\nwhere, \u03b8 is the angle between  \naand\nrb\nr\n, 0 \u2264 \u03b8 \u2264 \u03c0  and \u02c6n  is\na unit vector perpendicular to both \na and  \nrb\nr\n, such that\n\u02c6\n,\na b and  \nn\nrr\n form a right handed system (Fig 10 23) i"
    },
    {
        "Chapter": "1",
        "sentence_range": "5410-5413",
        "Text": "22 (i), (ii)\nDefinition 3 The vector product of two nonzero vectors \naand\nrb\nr\n, is denoted by a\n b\nr\nr\nand defined as\na\nb\n\u00d7\nr\nr\n =\n\u02c6\n|\n||\na b|sin\nn\n\u03b8\nr\nr\n,\nwhere, \u03b8 is the angle between  \naand\nrb\nr\n, 0 \u2264 \u03b8 \u2264 \u03c0  and \u02c6n  is\na unit vector perpendicular to both \na and  \nrb\nr\n, such that\n\u02c6\n,\na b and  \nn\nrr\n form a right handed system (Fig 10 23) i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "5411-5414",
        "Text": "23) i e , the\nright handed system rotated from \nato\nrb\nr\n moves in the\ndirection of \u02c6n"
    },
    {
        "Chapter": "1",
        "sentence_range": "5412-5415",
        "Text": "i e , the\nright handed system rotated from \nato\nrb\nr\n moves in the\ndirection of \u02c6n If either \n0 or\n0\na\nb\n=\nr=\nr\nr\nr\n, then \u03b8 is not defined and in this case, we define \n0\na\n\u00d7b\nr=\nr\nr"
    },
    {
        "Chapter": "1",
        "sentence_range": "5413-5416",
        "Text": "e , the\nright handed system rotated from \nato\nrb\nr\n moves in the\ndirection of \u02c6n If either \n0 or\n0\na\nb\n=\nr=\nr\nr\nr\n, then \u03b8 is not defined and in this case, we define \n0\na\n\u00d7b\nr=\nr\nr Observations\n1"
    },
    {
        "Chapter": "1",
        "sentence_range": "5414-5417",
        "Text": ", the\nright handed system rotated from \nato\nrb\nr\n moves in the\ndirection of \u02c6n If either \n0 or\n0\na\nb\n=\nr=\nr\nr\nr\n, then \u03b8 is not defined and in this case, we define \n0\na\n\u00d7b\nr=\nr\nr Observations\n1 a\n\u00d7b\nr\nr\n is a vector"
    },
    {
        "Chapter": "1",
        "sentence_range": "5415-5418",
        "Text": "If either \n0 or\n0\na\nb\n=\nr=\nr\nr\nr\n, then \u03b8 is not defined and in this case, we define \n0\na\n\u00d7b\nr=\nr\nr Observations\n1 a\n\u00d7b\nr\nr\n is a vector 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "5416-5419",
        "Text": "Observations\n1 a\n\u00d7b\nr\nr\n is a vector 2 Let \naand\nrb\nr\n be two nonzero vectors"
    },
    {
        "Chapter": "1",
        "sentence_range": "5417-5420",
        "Text": "a\n\u00d7b\nr\nr\n is a vector 2 Let \naand\nrb\nr\n be two nonzero vectors Then \n0\na\n\u00d7b\nr=\nr\nr\n if and only if \na  and  \nrb\nr\nare parallel (or collinear) to each other, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "5418-5421",
        "Text": "2 Let \naand\nrb\nr\n be two nonzero vectors Then \n0\na\n\u00d7b\nr=\nr\nr\n if and only if \na  and  \nrb\nr\nare parallel (or collinear) to each other, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "5419-5422",
        "Text": "Let \naand\nrb\nr\n be two nonzero vectors Then \n0\na\n\u00d7b\nr=\nr\nr\n if and only if \na  and  \nrb\nr\nare parallel (or collinear) to each other, i e ,\na\nb\n\u00d7\nr\nr\n = 0\n\u21d4a b\nr\nr\nr \ufffd\nFig 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5420-5423",
        "Text": "Then \n0\na\n\u00d7b\nr=\nr\nr\n if and only if \na  and  \nrb\nr\nare parallel (or collinear) to each other, i e ,\na\nb\n\u00d7\nr\nr\n = 0\n\u21d4a b\nr\nr\nr \ufffd\nFig 10 23\n\u00a9 NCERT\nnot to be republished\n MATHEMATICS\n450\nIn particular, \n0\na\n\u00d7a\n=\nr\nr\nr\n  and \n(\n)\n0\na\n\u00d7 \u2212a\n=\nr\nr\nr\n, since in the first situation, \u03b8 = 0\nand in the second one, \u03b8 = \u03c0, making the value of sin \u03b8 to be 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "5421-5424",
        "Text": "e ,\na\nb\n\u00d7\nr\nr\n = 0\n\u21d4a b\nr\nr\nr \ufffd\nFig 10 23\n\u00a9 NCERT\nnot to be republished\n MATHEMATICS\n450\nIn particular, \n0\na\n\u00d7a\n=\nr\nr\nr\n  and \n(\n)\n0\na\n\u00d7 \u2212a\n=\nr\nr\nr\n, since in the first situation, \u03b8 = 0\nand in the second one, \u03b8 = \u03c0, making the value of sin \u03b8 to be 0 3"
    },
    {
        "Chapter": "1",
        "sentence_range": "5422-5425",
        "Text": ",\na\nb\n\u00d7\nr\nr\n = 0\n\u21d4a b\nr\nr\nr \ufffd\nFig 10 23\n\u00a9 NCERT\nnot to be republished\n MATHEMATICS\n450\nIn particular, \n0\na\n\u00d7a\n=\nr\nr\nr\n  and \n(\n)\n0\na\n\u00d7 \u2212a\n=\nr\nr\nr\n, since in the first situation, \u03b8 = 0\nand in the second one, \u03b8 = \u03c0, making the value of sin \u03b8 to be 0 3 If \n2\n   \n then \n|\n||\n|\na\nb\na b\n \nr \nr\nr\nr"
    },
    {
        "Chapter": "1",
        "sentence_range": "5423-5426",
        "Text": "23\n\u00a9 NCERT\nnot to be republished\n MATHEMATICS\n450\nIn particular, \n0\na\n\u00d7a\n=\nr\nr\nr\n  and \n(\n)\n0\na\n\u00d7 \u2212a\n=\nr\nr\nr\n, since in the first situation, \u03b8 = 0\nand in the second one, \u03b8 = \u03c0, making the value of sin \u03b8 to be 0 3 If \n2\n   \n then \n|\n||\n|\na\nb\na b\n \nr \nr\nr\nr 4"
    },
    {
        "Chapter": "1",
        "sentence_range": "5424-5427",
        "Text": "3 If \n2\n   \n then \n|\n||\n|\na\nb\na b\n \nr \nr\nr\nr 4 In view of the Observations 2 and 3, for mutually perpendicular\nunit vectors \n\u02c6\n\u02c6\n,\u02c6\nand\ni\nj\nk  (Fig 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5425-5428",
        "Text": "If \n2\n   \n then \n|\n||\n|\na\nb\na b\n \nr \nr\nr\nr 4 In view of the Observations 2 and 3, for mutually perpendicular\nunit vectors \n\u02c6\n\u02c6\n,\u02c6\nand\ni\nj\nk  (Fig 10 24), we have\n\u02c6\n\u02c6\ni\ni\n\u00d7  = \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n0\nj\nj\nk\nk\n\u00d7\n=\n\u00d7\n=\nr\n\u02c6\n\u02c6\ni\n\u00d7j\n= \u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n,\n,  \nk\nj\nk\ni\nk\ni\nj\n\u00d7\n=\n\u00d7 =\n5"
    },
    {
        "Chapter": "1",
        "sentence_range": "5426-5429",
        "Text": "4 In view of the Observations 2 and 3, for mutually perpendicular\nunit vectors \n\u02c6\n\u02c6\n,\u02c6\nand\ni\nj\nk  (Fig 10 24), we have\n\u02c6\n\u02c6\ni\ni\n\u00d7  = \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n0\nj\nj\nk\nk\n\u00d7\n=\n\u00d7\n=\nr\n\u02c6\n\u02c6\ni\n\u00d7j\n= \u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n,\n,  \nk\nj\nk\ni\nk\ni\nj\n\u00d7\n=\n\u00d7 =\n5 In terms of vector product, the angle between two vectors   and  \na\nrb\nr\n  may be\ngiven as\nsin \u03b8 = |\n|\n|\n||\n|\na\nb\na b\n\u00d7\nr\nr\nr\nr\n6"
    },
    {
        "Chapter": "1",
        "sentence_range": "5427-5430",
        "Text": "In view of the Observations 2 and 3, for mutually perpendicular\nunit vectors \n\u02c6\n\u02c6\n,\u02c6\nand\ni\nj\nk  (Fig 10 24), we have\n\u02c6\n\u02c6\ni\ni\n\u00d7  = \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n0\nj\nj\nk\nk\n\u00d7\n=\n\u00d7\n=\nr\n\u02c6\n\u02c6\ni\n\u00d7j\n= \u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n,\n,  \nk\nj\nk\ni\nk\ni\nj\n\u00d7\n=\n\u00d7 =\n5 In terms of vector product, the angle between two vectors   and  \na\nrb\nr\n  may be\ngiven as\nsin \u03b8 = |\n|\n|\n||\n|\na\nb\na b\n\u00d7\nr\nr\nr\nr\n6 It is always true that the vector product is not commutative, as a\n\u00d7b\nr\nr\n = \nb\na\n\u2212\nr\u00d7\nr"
    },
    {
        "Chapter": "1",
        "sentence_range": "5428-5431",
        "Text": "24), we have\n\u02c6\n\u02c6\ni\ni\n\u00d7  = \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n0\nj\nj\nk\nk\n\u00d7\n=\n\u00d7\n=\nr\n\u02c6\n\u02c6\ni\n\u00d7j\n= \u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n,\n,  \nk\nj\nk\ni\nk\ni\nj\n\u00d7\n=\n\u00d7 =\n5 In terms of vector product, the angle between two vectors   and  \na\nrb\nr\n  may be\ngiven as\nsin \u03b8 = |\n|\n|\n||\n|\na\nb\na b\n\u00d7\nr\nr\nr\nr\n6 It is always true that the vector product is not commutative, as a\n\u00d7b\nr\nr\n = \nb\na\n\u2212\nr\u00d7\nr Indeed, \n\u02c6\n|\n||\n| sin\na\nb\na b\nn\n\u00d7\n=\n\u03b8\nr\nr\nr\nr\n, where \n\u02c6\n,\na b and  \nn\nrr\n form a right handed system,\ni"
    },
    {
        "Chapter": "1",
        "sentence_range": "5429-5432",
        "Text": "In terms of vector product, the angle between two vectors   and  \na\nrb\nr\n  may be\ngiven as\nsin \u03b8 = |\n|\n|\n||\n|\na\nb\na b\n\u00d7\nr\nr\nr\nr\n6 It is always true that the vector product is not commutative, as a\n\u00d7b\nr\nr\n = \nb\na\n\u2212\nr\u00d7\nr Indeed, \n\u02c6\n|\n||\n| sin\na\nb\na b\nn\n\u00d7\n=\n\u03b8\nr\nr\nr\nr\n, where \n\u02c6\n,\na b and  \nn\nrr\n form a right handed system,\ni e"
    },
    {
        "Chapter": "1",
        "sentence_range": "5430-5433",
        "Text": "It is always true that the vector product is not commutative, as a\n\u00d7b\nr\nr\n = \nb\na\n\u2212\nr\u00d7\nr Indeed, \n\u02c6\n|\n||\n| sin\na\nb\na b\nn\n\u00d7\n=\n\u03b8\nr\nr\nr\nr\n, where \n\u02c6\n,\na b and  \nn\nrr\n form a right handed system,\ni e , \u03b8 is traversed from \nato\nrb\nr\n, Fig 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5431-5434",
        "Text": "Indeed, \n\u02c6\n|\n||\n| sin\na\nb\na b\nn\n\u00d7\n=\n\u03b8\nr\nr\nr\nr\n, where \n\u02c6\n,\na b and  \nn\nrr\n form a right handed system,\ni e , \u03b8 is traversed from \nato\nrb\nr\n, Fig 10 25 (i)"
    },
    {
        "Chapter": "1",
        "sentence_range": "5432-5435",
        "Text": "e , \u03b8 is traversed from \nato\nrb\nr\n, Fig 10 25 (i) While, \n1\u02c6\n|\n||\n| sin\nb\na\na b\nn\n\u00d7\n=\n\u03b8\nr\nr\nr\nr\n, where\n1\u02c6\n,\nb aand\nn\nr r\n form a right handed system i"
    },
    {
        "Chapter": "1",
        "sentence_range": "5433-5436",
        "Text": ", \u03b8 is traversed from \nato\nrb\nr\n, Fig 10 25 (i) While, \n1\u02c6\n|\n||\n| sin\nb\na\na b\nn\n\u00d7\n=\n\u03b8\nr\nr\nr\nr\n, where\n1\u02c6\n,\nb aand\nn\nr r\n form a right handed system i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "5434-5437",
        "Text": "25 (i) While, \n1\u02c6\n|\n||\n| sin\nb\na\na b\nn\n\u00d7\n=\n\u03b8\nr\nr\nr\nr\n, where\n1\u02c6\n,\nb aand\nn\nr r\n form a right handed system i e \u03b8 is traversed from \nbto\na\nr\nr ,\nFig 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5435-5438",
        "Text": "While, \n1\u02c6\n|\n||\n| sin\nb\na\na b\nn\n\u00d7\n=\n\u03b8\nr\nr\nr\nr\n, where\n1\u02c6\n,\nb aand\nn\nr r\n form a right handed system i e \u03b8 is traversed from \nbto\na\nr\nr ,\nFig 10 25(ii)"
    },
    {
        "Chapter": "1",
        "sentence_range": "5436-5439",
        "Text": "e \u03b8 is traversed from \nbto\na\nr\nr ,\nFig 10 25(ii) Fig 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5437-5440",
        "Text": "\u03b8 is traversed from \nbto\na\nr\nr ,\nFig 10 25(ii) Fig 10 25 (i), (ii)\nThus, if we assume \naand\nrb\nr\n to lie in the plane of the paper, then \n1\n\u02c6\nn and \u02c6\nn  both\nwill be perpendicular to the plane of the paper"
    },
    {
        "Chapter": "1",
        "sentence_range": "5438-5441",
        "Text": "25(ii) Fig 10 25 (i), (ii)\nThus, if we assume \naand\nrb\nr\n to lie in the plane of the paper, then \n1\n\u02c6\nn and \u02c6\nn  both\nwill be perpendicular to the plane of the paper But, \u02c6n being directed above the\npaper while \n1\u02c6n  directed below the paper"
    },
    {
        "Chapter": "1",
        "sentence_range": "5439-5442",
        "Text": "Fig 10 25 (i), (ii)\nThus, if we assume \naand\nrb\nr\n to lie in the plane of the paper, then \n1\n\u02c6\nn and \u02c6\nn  both\nwill be perpendicular to the plane of the paper But, \u02c6n being directed above the\npaper while \n1\u02c6n  directed below the paper i"
    },
    {
        "Chapter": "1",
        "sentence_range": "5440-5443",
        "Text": "25 (i), (ii)\nThus, if we assume \naand\nrb\nr\n to lie in the plane of the paper, then \n1\n\u02c6\nn and \u02c6\nn  both\nwill be perpendicular to the plane of the paper But, \u02c6n being directed above the\npaper while \n1\u02c6n  directed below the paper i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "5441-5444",
        "Text": "But, \u02c6n being directed above the\npaper while \n1\u02c6n  directed below the paper i e 1\u02c6\n\u02c6\nn\n= \u2212n"
    },
    {
        "Chapter": "1",
        "sentence_range": "5442-5445",
        "Text": "i e 1\u02c6\n\u02c6\nn\n= \u2212n Fig 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5443-5446",
        "Text": "e 1\u02c6\n\u02c6\nn\n= \u2212n Fig 10 24\n\u00a9 NCERT\nnot to be republished\nVECTOR ALGEBRA\n451\nHence\na b\n\u00d7\nr\nr\n =\n\u02c6\n|\n||\na b|sin\nn\n \nr\nr\n=\n1\u02c6\n|\n||\na b|sin\nn\n\u2212\n\u03b8\nr\nr\nb\na\n= \u2212\nr\u00d7\nr\n7"
    },
    {
        "Chapter": "1",
        "sentence_range": "5444-5447",
        "Text": "1\u02c6\n\u02c6\nn\n= \u2212n Fig 10 24\n\u00a9 NCERT\nnot to be republished\nVECTOR ALGEBRA\n451\nHence\na b\n\u00d7\nr\nr\n =\n\u02c6\n|\n||\na b|sin\nn\n \nr\nr\n=\n1\u02c6\n|\n||\na b|sin\nn\n\u2212\n\u03b8\nr\nr\nb\na\n= \u2212\nr\u00d7\nr\n7 In view of the Observations 4 and 6, we have\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n,\n and"
    },
    {
        "Chapter": "1",
        "sentence_range": "5445-5448",
        "Text": "Fig 10 24\n\u00a9 NCERT\nnot to be republished\nVECTOR ALGEBRA\n451\nHence\na b\n\u00d7\nr\nr\n =\n\u02c6\n|\n||\na b|sin\nn\n \nr\nr\n=\n1\u02c6\n|\n||\na b|sin\nn\n\u2212\n\u03b8\nr\nr\nb\na\n= \u2212\nr\u00d7\nr\n7 In view of the Observations 4 and 6, we have\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n,\n and j\ni\nk\nk\nj\ni\ni\nk\nj\n\u00d7 = \u2212\n\u00d7\n= \u2212\n\u00d7\n= \u2212\n8"
    },
    {
        "Chapter": "1",
        "sentence_range": "5446-5449",
        "Text": "24\n\u00a9 NCERT\nnot to be republished\nVECTOR ALGEBRA\n451\nHence\na b\n\u00d7\nr\nr\n =\n\u02c6\n|\n||\na b|sin\nn\n \nr\nr\n=\n1\u02c6\n|\n||\na b|sin\nn\n\u2212\n\u03b8\nr\nr\nb\na\n= \u2212\nr\u00d7\nr\n7 In view of the Observations 4 and 6, we have\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n,\n and j\ni\nk\nk\nj\ni\ni\nk\nj\n\u00d7 = \u2212\n\u00d7\n= \u2212\n\u00d7\n= \u2212\n8 If \naand\nrb\nr\nrepresent the adjacent sides of a triangle then its area is given as\n1 |\n|\n2 a\nb\n \nr\nr"
    },
    {
        "Chapter": "1",
        "sentence_range": "5447-5450",
        "Text": "In view of the Observations 4 and 6, we have\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n,\n and j\ni\nk\nk\nj\ni\ni\nk\nj\n\u00d7 = \u2212\n\u00d7\n= \u2212\n\u00d7\n= \u2212\n8 If \naand\nrb\nr\nrepresent the adjacent sides of a triangle then its area is given as\n1 |\n|\n2 a\nb\n \nr\nr By definition of the area of a triangle, we have from\nFig 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5448-5451",
        "Text": "j\ni\nk\nk\nj\ni\ni\nk\nj\n\u00d7 = \u2212\n\u00d7\n= \u2212\n\u00d7\n= \u2212\n8 If \naand\nrb\nr\nrepresent the adjacent sides of a triangle then its area is given as\n1 |\n|\n2 a\nb\n \nr\nr By definition of the area of a triangle, we have from\nFig 10 26,\nArea of triangle ABC = 1 AB CD"
    },
    {
        "Chapter": "1",
        "sentence_range": "5449-5452",
        "Text": "If \naand\nrb\nr\nrepresent the adjacent sides of a triangle then its area is given as\n1 |\n|\n2 a\nb\n \nr\nr By definition of the area of a triangle, we have from\nFig 10 26,\nArea of triangle ABC = 1 AB CD 2\n\u22c5\nBut AB\n|\nb|\n=\nr  (as given), and CD = |\n|\nar sin\u03b8"
    },
    {
        "Chapter": "1",
        "sentence_range": "5450-5453",
        "Text": "By definition of the area of a triangle, we have from\nFig 10 26,\nArea of triangle ABC = 1 AB CD 2\n\u22c5\nBut AB\n|\nb|\n=\nr  (as given), and CD = |\n|\nar sin\u03b8 Thus,  Area of triangle ABC = 1 |\n||\n| sin\n2 b\na\n\u03b8\nr\nr\n \n1 |\n|"
    },
    {
        "Chapter": "1",
        "sentence_range": "5451-5454",
        "Text": "26,\nArea of triangle ABC = 1 AB CD 2\n\u22c5\nBut AB\n|\nb|\n=\nr  (as given), and CD = |\n|\nar sin\u03b8 Thus,  Area of triangle ABC = 1 |\n||\n| sin\n2 b\na\n\u03b8\nr\nr\n \n1 |\n| 2 a\nb\n=\n\u00d7\nr\nr\n9"
    },
    {
        "Chapter": "1",
        "sentence_range": "5452-5455",
        "Text": "2\n\u22c5\nBut AB\n|\nb|\n=\nr  (as given), and CD = |\n|\nar sin\u03b8 Thus,  Area of triangle ABC = 1 |\n||\n| sin\n2 b\na\n\u03b8\nr\nr\n \n1 |\n| 2 a\nb\n=\n\u00d7\nr\nr\n9 If \na  and  \nrb\nr\n represent the adjacent sides of a  parallelogram, then its area is\ngiven by |\n|\na\nb\n\u00d7\nr\nr"
    },
    {
        "Chapter": "1",
        "sentence_range": "5453-5456",
        "Text": "Thus,  Area of triangle ABC = 1 |\n||\n| sin\n2 b\na\n\u03b8\nr\nr\n \n1 |\n| 2 a\nb\n=\n\u00d7\nr\nr\n9 If \na  and  \nrb\nr\n represent the adjacent sides of a  parallelogram, then its area is\ngiven by |\n|\na\nb\n\u00d7\nr\nr From Fig 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5454-5457",
        "Text": "2 a\nb\n=\n\u00d7\nr\nr\n9 If \na  and  \nrb\nr\n represent the adjacent sides of a  parallelogram, then its area is\ngiven by |\n|\na\nb\n\u00d7\nr\nr From Fig 10 27, we have\nArea of parallelogram ABCD = AB"
    },
    {
        "Chapter": "1",
        "sentence_range": "5455-5458",
        "Text": "If \na  and  \nrb\nr\n represent the adjacent sides of a  parallelogram, then its area is\ngiven by |\n|\na\nb\n\u00d7\nr\nr From Fig 10 27, we have\nArea of parallelogram ABCD = AB DE"
    },
    {
        "Chapter": "1",
        "sentence_range": "5456-5459",
        "Text": "From Fig 10 27, we have\nArea of parallelogram ABCD = AB DE But AB\n|\n=b|\nr\n (as given), and\nDE\n|\n=a|sin\n\u03b8\nr"
    },
    {
        "Chapter": "1",
        "sentence_range": "5457-5460",
        "Text": "27, we have\nArea of parallelogram ABCD = AB DE But AB\n|\n=b|\nr\n (as given), and\nDE\n|\n=a|sin\n\u03b8\nr Thus,\nArea of parallelogram ABCD =  |\n||\n| sin\nb\na\n\u03b8\nr\nr\n \n|\n|"
    },
    {
        "Chapter": "1",
        "sentence_range": "5458-5461",
        "Text": "DE But AB\n|\n=b|\nr\n (as given), and\nDE\n|\n=a|sin\n\u03b8\nr Thus,\nArea of parallelogram ABCD =  |\n||\n| sin\nb\na\n\u03b8\nr\nr\n \n|\n| a\nb\n=\n\u00d7\nr\nr\nWe now state two important properties of vector product"
    },
    {
        "Chapter": "1",
        "sentence_range": "5459-5462",
        "Text": "But AB\n|\n=b|\nr\n (as given), and\nDE\n|\n=a|sin\n\u03b8\nr Thus,\nArea of parallelogram ABCD =  |\n||\n| sin\nb\na\n\u03b8\nr\nr\n \n|\n| a\nb\n=\n\u00d7\nr\nr\nWe now state two important properties of vector product Property 3 (Distributivity of vector product over addition): If ,\na band\nc\nrr\nr\nare any three vectors and \u03bb be a scalar, then\n(i)\n(\n)\na\nb\nc\n\u00d7\nr+\nr\nr  = a\nb\na\nc\n \n \nr\nr\nr\nr\n(ii)\n(\n\u03bba b)\n\u00d7\nr\nr\n = (\n)\n(\n)\na\nb\na\nb\n\u03bb\n\u00d7\n=\n\u00d7 \u03bb\nr\nr\nr\nr\nFig 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5460-5463",
        "Text": "Thus,\nArea of parallelogram ABCD =  |\n||\n| sin\nb\na\n\u03b8\nr\nr\n \n|\n| a\nb\n=\n\u00d7\nr\nr\nWe now state two important properties of vector product Property 3 (Distributivity of vector product over addition): If ,\na band\nc\nrr\nr\nare any three vectors and \u03bb be a scalar, then\n(i)\n(\n)\na\nb\nc\n\u00d7\nr+\nr\nr  = a\nb\na\nc\n \n \nr\nr\nr\nr\n(ii)\n(\n\u03bba b)\n\u00d7\nr\nr\n = (\n)\n(\n)\na\nb\na\nb\n\u03bb\n\u00d7\n=\n\u00d7 \u03bb\nr\nr\nr\nr\nFig 10 26\n Fig 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5461-5464",
        "Text": "a\nb\n=\n\u00d7\nr\nr\nWe now state two important properties of vector product Property 3 (Distributivity of vector product over addition): If ,\na band\nc\nrr\nr\nare any three vectors and \u03bb be a scalar, then\n(i)\n(\n)\na\nb\nc\n\u00d7\nr+\nr\nr  = a\nb\na\nc\n \n \nr\nr\nr\nr\n(ii)\n(\n\u03bba b)\n\u00d7\nr\nr\n = (\n)\n(\n)\na\nb\na\nb\n\u03bb\n\u00d7\n=\n\u00d7 \u03bb\nr\nr\nr\nr\nFig 10 26\n Fig 10 27\n\u00a9 NCERT\nnot to be republished\n MATHEMATICS\n452\nLet \naand\nrb\nr\n be two vectors given in component form as \n1\n2\n3 \u02c6\n\u02c6\n\u02c6\na i\na j\na k\n+\n+\nand\n1\n2\n3 \u02c6\n\u02c6\n\u02c6\nb i\nb j\nb k\n+\n+\n, respectively"
    },
    {
        "Chapter": "1",
        "sentence_range": "5462-5465",
        "Text": "Property 3 (Distributivity of vector product over addition): If ,\na band\nc\nrr\nr\nare any three vectors and \u03bb be a scalar, then\n(i)\n(\n)\na\nb\nc\n\u00d7\nr+\nr\nr  = a\nb\na\nc\n \n \nr\nr\nr\nr\n(ii)\n(\n\u03bba b)\n\u00d7\nr\nr\n = (\n)\n(\n)\na\nb\na\nb\n\u03bb\n\u00d7\n=\n\u00d7 \u03bb\nr\nr\nr\nr\nFig 10 26\n Fig 10 27\n\u00a9 NCERT\nnot to be republished\n MATHEMATICS\n452\nLet \naand\nrb\nr\n be two vectors given in component form as \n1\n2\n3 \u02c6\n\u02c6\n\u02c6\na i\na j\na k\n+\n+\nand\n1\n2\n3 \u02c6\n\u02c6\n\u02c6\nb i\nb j\nb k\n+\n+\n, respectively Then their cross product may be given by\na\nb\n\u00d7\nr\nr\n =\n1\n2\n3\n1\n2\n3\n\u02c6\n\u02c6\n\u02c6\ni\nj\nk\na\na\na\nb\nb\nb\nExplanation We have\na b\n\u00d7\nr\nr\n =\n1\n2\n3\n1\n2\n3\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(\n)\n(\n)\na i\na j\na k\nb i\nb j\nb k\n+\n+\n\u00d7\n+\n+\n=\n1 1\n1 2\n1 3\n2 1\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(\n)\n(\n)\n(\n)\n(\n)\na b i\ni\na b i\nj\na b i\nk\na b\nj\ni\n\u00d7\n+\n\u00d7\n+\n\u00d7\n+\n\u00d7\n+ \n2 2\n2 3\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(\n)\n(\n)\na b\nj\nj\na b\nj\nk\n\u00d7\n+\n\u00d7\n+  \n3 1\n3 2\n3 3\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(\n)\n(\n)\n(\n)\na b k\ni\na b k\nj\na b k\nk\n\u00d7\n+\n\u00d7\n+\n\u00d7\n(by Property 1)\n=\n1 2\n1 3\n2 1\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(\n)\n(\n)\n(\n)\na b i\nj\na b k\ni\na b i\nj\n\u00d7\n\u2212\n\u00d7\n\u2212\n\u00d7\n+  \n2 3\n3 1\n3 2\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(\n)\n(\n)\n(\n)\na b\nj\nk\na b k\ni\na b\nj\nk\n\u00d7\n+\n\u00d7\n\u2212\n\u00d7\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6 \u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(as\n0  and  \n,\n  and  \n)\ni\ni\nj\nj\nk\nk\ni\nk\nk\ni\nj\ni\ni\nj\nk\nj\nj\nk\n\u00d7 =\n\u00d7\n=\n\u00d7\n=\n\u00d7\n= \u2212 \u00d7\n\u00d7 = \u2212 \u00d7\n\u00d7\n= \u2212 \u00d7\n=\n1 2\n1 3\n2 1\n2 3\n3 1\n3 2\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\na b k\na b j\na b k\na b i\na b j\na b i\n\u2212\n\u2212\n+\n+\n\u2212\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(as\n,\n  and  \n)\ni\nj\nk\nj\nk\ni\nk\ni\nj\n\u00d7\n=\n\u00d7\n=\n\u00d7 =\n=\n2 3\n3 2\n1 3\n3 1\n1 2\n2 1 \u02c6\n\u02c6\n\u02c6\n(\n)\n(\n)\n(\n)\na b\na b i\na b\na b\nj\na b\na b k\n\u2212\n\u2212\n\u2212\n+\n\u2212\n=\n1\n2\n3\n1\n2\n3\n\u02c6\n\u02c6\n\u02c6\ni\nj\nk\na\na\na\nb\nb\nb\nExample 22 Find \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n|\n|,  if  \n2\n3   and  \n3\n5\n2\na b\na\ni\nj\nk\nb\ni\nj\nk\n\u00d7\n=\n+\n+\n=\n+\n\u2212\nr\nr\nr\nr\nSolution We have\na\nb\n\u00d7\nr\nr\n =\n\u02c6\n\u02c6\n\u02c6\n2\n1\n3\n3\n5\n2\ni\nj\nk\n\u2212\n=\n\u02c6\n\u02c6\n\u02c6\n( 2\n15)\n( 4\n9)\n(10 \u2013 3)\ni\nj\nk\n\u2212 \u2212\n\u2212 \u2212 \u2212\n+\n\u02c6\n\u02c6\n\u02c6\n17\n13\n7\ni\nj\nk\n= \u2212\n+\n+\nHence\n|\n|\n \nr\nra b  =\n2\n2\n2\n( 17)\n(13)\n(7)\n507\n\u2212\n+\n+\n=\n\u00a9 NCERT\nnot to be republished\nVECTOR ALGEBRA\n453\nExample 23 Find a unit vector perpendicular to each of the vectors (\n)\na\nb\n+\nr\nr\n and\n(\n),\na\n\u2212b\nr\nr\nwhere \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n,\n2\n3\na\ni\nj\nk\nb\ni\nj\nk\n= +\n+\n= +\n+\nr\nr"
    },
    {
        "Chapter": "1",
        "sentence_range": "5463-5466",
        "Text": "26\n Fig 10 27\n\u00a9 NCERT\nnot to be republished\n MATHEMATICS\n452\nLet \naand\nrb\nr\n be two vectors given in component form as \n1\n2\n3 \u02c6\n\u02c6\n\u02c6\na i\na j\na k\n+\n+\nand\n1\n2\n3 \u02c6\n\u02c6\n\u02c6\nb i\nb j\nb k\n+\n+\n, respectively Then their cross product may be given by\na\nb\n\u00d7\nr\nr\n =\n1\n2\n3\n1\n2\n3\n\u02c6\n\u02c6\n\u02c6\ni\nj\nk\na\na\na\nb\nb\nb\nExplanation We have\na b\n\u00d7\nr\nr\n =\n1\n2\n3\n1\n2\n3\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(\n)\n(\n)\na i\na j\na k\nb i\nb j\nb k\n+\n+\n\u00d7\n+\n+\n=\n1 1\n1 2\n1 3\n2 1\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(\n)\n(\n)\n(\n)\n(\n)\na b i\ni\na b i\nj\na b i\nk\na b\nj\ni\n\u00d7\n+\n\u00d7\n+\n\u00d7\n+\n\u00d7\n+ \n2 2\n2 3\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(\n)\n(\n)\na b\nj\nj\na b\nj\nk\n\u00d7\n+\n\u00d7\n+  \n3 1\n3 2\n3 3\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(\n)\n(\n)\n(\n)\na b k\ni\na b k\nj\na b k\nk\n\u00d7\n+\n\u00d7\n+\n\u00d7\n(by Property 1)\n=\n1 2\n1 3\n2 1\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(\n)\n(\n)\n(\n)\na b i\nj\na b k\ni\na b i\nj\n\u00d7\n\u2212\n\u00d7\n\u2212\n\u00d7\n+  \n2 3\n3 1\n3 2\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(\n)\n(\n)\n(\n)\na b\nj\nk\na b k\ni\na b\nj\nk\n\u00d7\n+\n\u00d7\n\u2212\n\u00d7\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6 \u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(as\n0  and  \n,\n  and  \n)\ni\ni\nj\nj\nk\nk\ni\nk\nk\ni\nj\ni\ni\nj\nk\nj\nj\nk\n\u00d7 =\n\u00d7\n=\n\u00d7\n=\n\u00d7\n= \u2212 \u00d7\n\u00d7 = \u2212 \u00d7\n\u00d7\n= \u2212 \u00d7\n=\n1 2\n1 3\n2 1\n2 3\n3 1\n3 2\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\na b k\na b j\na b k\na b i\na b j\na b i\n\u2212\n\u2212\n+\n+\n\u2212\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(as\n,\n  and  \n)\ni\nj\nk\nj\nk\ni\nk\ni\nj\n\u00d7\n=\n\u00d7\n=\n\u00d7 =\n=\n2 3\n3 2\n1 3\n3 1\n1 2\n2 1 \u02c6\n\u02c6\n\u02c6\n(\n)\n(\n)\n(\n)\na b\na b i\na b\na b\nj\na b\na b k\n\u2212\n\u2212\n\u2212\n+\n\u2212\n=\n1\n2\n3\n1\n2\n3\n\u02c6\n\u02c6\n\u02c6\ni\nj\nk\na\na\na\nb\nb\nb\nExample 22 Find \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n|\n|,  if  \n2\n3   and  \n3\n5\n2\na b\na\ni\nj\nk\nb\ni\nj\nk\n\u00d7\n=\n+\n+\n=\n+\n\u2212\nr\nr\nr\nr\nSolution We have\na\nb\n\u00d7\nr\nr\n =\n\u02c6\n\u02c6\n\u02c6\n2\n1\n3\n3\n5\n2\ni\nj\nk\n\u2212\n=\n\u02c6\n\u02c6\n\u02c6\n( 2\n15)\n( 4\n9)\n(10 \u2013 3)\ni\nj\nk\n\u2212 \u2212\n\u2212 \u2212 \u2212\n+\n\u02c6\n\u02c6\n\u02c6\n17\n13\n7\ni\nj\nk\n= \u2212\n+\n+\nHence\n|\n|\n \nr\nra b  =\n2\n2\n2\n( 17)\n(13)\n(7)\n507\n\u2212\n+\n+\n=\n\u00a9 NCERT\nnot to be republished\nVECTOR ALGEBRA\n453\nExample 23 Find a unit vector perpendicular to each of the vectors (\n)\na\nb\n+\nr\nr\n and\n(\n),\na\n\u2212b\nr\nr\nwhere \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n,\n2\n3\na\ni\nj\nk\nb\ni\nj\nk\n= +\n+\n= +\n+\nr\nr Solution We have \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n2\n3\n4\nand\n2\na\nb\ni\nj\nk\na\nb\nj\nk\n+\n=\n+\n+\n\u2212\n= \u2212 \u2212\nr\nr\nr\nr\nA vector which is perpendicular to both \nand\na\nb\na\nb\n+\n\u2212\nr\nr\nr\nr\n is given by\n(\n)\n(\n)\na\nb\na\nb\n+\n\u00d7\n\u2212\nr\nr\nr\nr\n =\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n2\n3\n4\n2\n4\n2\n(\n, say)\n0\n1\n2\ni\nj\nk\ni\nj\nk\nc\n= \u2212\n+\n\u2212\n=\n\u2212\n\u2212\nr\nNow\n|\n|\ncr  =\n4\n16\n4\n24\n2 6\n+\n+\n=\n=\nTherefore, the required unit vector is\n|\nc|\nc\nr\nr  =\n1\n2\n1 \u02c6\n\u02c6\n\u02c6\n6\n6\n6\ni\nj\nk\n\u2212\n+\n\u2212\n\ufffdNote   There are two perpendicular directions to any plane"
    },
    {
        "Chapter": "1",
        "sentence_range": "5464-5467",
        "Text": "27\n\u00a9 NCERT\nnot to be republished\n MATHEMATICS\n452\nLet \naand\nrb\nr\n be two vectors given in component form as \n1\n2\n3 \u02c6\n\u02c6\n\u02c6\na i\na j\na k\n+\n+\nand\n1\n2\n3 \u02c6\n\u02c6\n\u02c6\nb i\nb j\nb k\n+\n+\n, respectively Then their cross product may be given by\na\nb\n\u00d7\nr\nr\n =\n1\n2\n3\n1\n2\n3\n\u02c6\n\u02c6\n\u02c6\ni\nj\nk\na\na\na\nb\nb\nb\nExplanation We have\na b\n\u00d7\nr\nr\n =\n1\n2\n3\n1\n2\n3\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(\n)\n(\n)\na i\na j\na k\nb i\nb j\nb k\n+\n+\n\u00d7\n+\n+\n=\n1 1\n1 2\n1 3\n2 1\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(\n)\n(\n)\n(\n)\n(\n)\na b i\ni\na b i\nj\na b i\nk\na b\nj\ni\n\u00d7\n+\n\u00d7\n+\n\u00d7\n+\n\u00d7\n+ \n2 2\n2 3\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(\n)\n(\n)\na b\nj\nj\na b\nj\nk\n\u00d7\n+\n\u00d7\n+  \n3 1\n3 2\n3 3\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(\n)\n(\n)\n(\n)\na b k\ni\na b k\nj\na b k\nk\n\u00d7\n+\n\u00d7\n+\n\u00d7\n(by Property 1)\n=\n1 2\n1 3\n2 1\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(\n)\n(\n)\n(\n)\na b i\nj\na b k\ni\na b i\nj\n\u00d7\n\u2212\n\u00d7\n\u2212\n\u00d7\n+  \n2 3\n3 1\n3 2\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(\n)\n(\n)\n(\n)\na b\nj\nk\na b k\ni\na b\nj\nk\n\u00d7\n+\n\u00d7\n\u2212\n\u00d7\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6 \u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(as\n0  and  \n,\n  and  \n)\ni\ni\nj\nj\nk\nk\ni\nk\nk\ni\nj\ni\ni\nj\nk\nj\nj\nk\n\u00d7 =\n\u00d7\n=\n\u00d7\n=\n\u00d7\n= \u2212 \u00d7\n\u00d7 = \u2212 \u00d7\n\u00d7\n= \u2212 \u00d7\n=\n1 2\n1 3\n2 1\n2 3\n3 1\n3 2\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\na b k\na b j\na b k\na b i\na b j\na b i\n\u2212\n\u2212\n+\n+\n\u2212\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(as\n,\n  and  \n)\ni\nj\nk\nj\nk\ni\nk\ni\nj\n\u00d7\n=\n\u00d7\n=\n\u00d7 =\n=\n2 3\n3 2\n1 3\n3 1\n1 2\n2 1 \u02c6\n\u02c6\n\u02c6\n(\n)\n(\n)\n(\n)\na b\na b i\na b\na b\nj\na b\na b k\n\u2212\n\u2212\n\u2212\n+\n\u2212\n=\n1\n2\n3\n1\n2\n3\n\u02c6\n\u02c6\n\u02c6\ni\nj\nk\na\na\na\nb\nb\nb\nExample 22 Find \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n|\n|,  if  \n2\n3   and  \n3\n5\n2\na b\na\ni\nj\nk\nb\ni\nj\nk\n\u00d7\n=\n+\n+\n=\n+\n\u2212\nr\nr\nr\nr\nSolution We have\na\nb\n\u00d7\nr\nr\n =\n\u02c6\n\u02c6\n\u02c6\n2\n1\n3\n3\n5\n2\ni\nj\nk\n\u2212\n=\n\u02c6\n\u02c6\n\u02c6\n( 2\n15)\n( 4\n9)\n(10 \u2013 3)\ni\nj\nk\n\u2212 \u2212\n\u2212 \u2212 \u2212\n+\n\u02c6\n\u02c6\n\u02c6\n17\n13\n7\ni\nj\nk\n= \u2212\n+\n+\nHence\n|\n|\n \nr\nra b  =\n2\n2\n2\n( 17)\n(13)\n(7)\n507\n\u2212\n+\n+\n=\n\u00a9 NCERT\nnot to be republished\nVECTOR ALGEBRA\n453\nExample 23 Find a unit vector perpendicular to each of the vectors (\n)\na\nb\n+\nr\nr\n and\n(\n),\na\n\u2212b\nr\nr\nwhere \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n,\n2\n3\na\ni\nj\nk\nb\ni\nj\nk\n= +\n+\n= +\n+\nr\nr Solution We have \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n2\n3\n4\nand\n2\na\nb\ni\nj\nk\na\nb\nj\nk\n+\n=\n+\n+\n\u2212\n= \u2212 \u2212\nr\nr\nr\nr\nA vector which is perpendicular to both \nand\na\nb\na\nb\n+\n\u2212\nr\nr\nr\nr\n is given by\n(\n)\n(\n)\na\nb\na\nb\n+\n\u00d7\n\u2212\nr\nr\nr\nr\n =\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n2\n3\n4\n2\n4\n2\n(\n, say)\n0\n1\n2\ni\nj\nk\ni\nj\nk\nc\n= \u2212\n+\n\u2212\n=\n\u2212\n\u2212\nr\nNow\n|\n|\ncr  =\n4\n16\n4\n24\n2 6\n+\n+\n=\n=\nTherefore, the required unit vector is\n|\nc|\nc\nr\nr  =\n1\n2\n1 \u02c6\n\u02c6\n\u02c6\n6\n6\n6\ni\nj\nk\n\u2212\n+\n\u2212\n\ufffdNote   There are two perpendicular directions to any plane Thus, another unit\nvector perpendicular to \nand\na\nb\na\nb\n+\n\u2212\nr\nr\nr\nr\n will be 1\n2\n1 \u02c6\n\u02c6\n\u02c6"
    },
    {
        "Chapter": "1",
        "sentence_range": "5465-5468",
        "Text": "Then their cross product may be given by\na\nb\n\u00d7\nr\nr\n =\n1\n2\n3\n1\n2\n3\n\u02c6\n\u02c6\n\u02c6\ni\nj\nk\na\na\na\nb\nb\nb\nExplanation We have\na b\n\u00d7\nr\nr\n =\n1\n2\n3\n1\n2\n3\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(\n)\n(\n)\na i\na j\na k\nb i\nb j\nb k\n+\n+\n\u00d7\n+\n+\n=\n1 1\n1 2\n1 3\n2 1\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(\n)\n(\n)\n(\n)\n(\n)\na b i\ni\na b i\nj\na b i\nk\na b\nj\ni\n\u00d7\n+\n\u00d7\n+\n\u00d7\n+\n\u00d7\n+ \n2 2\n2 3\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(\n)\n(\n)\na b\nj\nj\na b\nj\nk\n\u00d7\n+\n\u00d7\n+  \n3 1\n3 2\n3 3\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(\n)\n(\n)\n(\n)\na b k\ni\na b k\nj\na b k\nk\n\u00d7\n+\n\u00d7\n+\n\u00d7\n(by Property 1)\n=\n1 2\n1 3\n2 1\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(\n)\n(\n)\n(\n)\na b i\nj\na b k\ni\na b i\nj\n\u00d7\n\u2212\n\u00d7\n\u2212\n\u00d7\n+  \n2 3\n3 1\n3 2\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(\n)\n(\n)\n(\n)\na b\nj\nk\na b k\ni\na b\nj\nk\n\u00d7\n+\n\u00d7\n\u2212\n\u00d7\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6 \u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(as\n0  and  \n,\n  and  \n)\ni\ni\nj\nj\nk\nk\ni\nk\nk\ni\nj\ni\ni\nj\nk\nj\nj\nk\n\u00d7 =\n\u00d7\n=\n\u00d7\n=\n\u00d7\n= \u2212 \u00d7\n\u00d7 = \u2212 \u00d7\n\u00d7\n= \u2212 \u00d7\n=\n1 2\n1 3\n2 1\n2 3\n3 1\n3 2\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\na b k\na b j\na b k\na b i\na b j\na b i\n\u2212\n\u2212\n+\n+\n\u2212\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(as\n,\n  and  \n)\ni\nj\nk\nj\nk\ni\nk\ni\nj\n\u00d7\n=\n\u00d7\n=\n\u00d7 =\n=\n2 3\n3 2\n1 3\n3 1\n1 2\n2 1 \u02c6\n\u02c6\n\u02c6\n(\n)\n(\n)\n(\n)\na b\na b i\na b\na b\nj\na b\na b k\n\u2212\n\u2212\n\u2212\n+\n\u2212\n=\n1\n2\n3\n1\n2\n3\n\u02c6\n\u02c6\n\u02c6\ni\nj\nk\na\na\na\nb\nb\nb\nExample 22 Find \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n|\n|,  if  \n2\n3   and  \n3\n5\n2\na b\na\ni\nj\nk\nb\ni\nj\nk\n\u00d7\n=\n+\n+\n=\n+\n\u2212\nr\nr\nr\nr\nSolution We have\na\nb\n\u00d7\nr\nr\n =\n\u02c6\n\u02c6\n\u02c6\n2\n1\n3\n3\n5\n2\ni\nj\nk\n\u2212\n=\n\u02c6\n\u02c6\n\u02c6\n( 2\n15)\n( 4\n9)\n(10 \u2013 3)\ni\nj\nk\n\u2212 \u2212\n\u2212 \u2212 \u2212\n+\n\u02c6\n\u02c6\n\u02c6\n17\n13\n7\ni\nj\nk\n= \u2212\n+\n+\nHence\n|\n|\n \nr\nra b  =\n2\n2\n2\n( 17)\n(13)\n(7)\n507\n\u2212\n+\n+\n=\n\u00a9 NCERT\nnot to be republished\nVECTOR ALGEBRA\n453\nExample 23 Find a unit vector perpendicular to each of the vectors (\n)\na\nb\n+\nr\nr\n and\n(\n),\na\n\u2212b\nr\nr\nwhere \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n,\n2\n3\na\ni\nj\nk\nb\ni\nj\nk\n= +\n+\n= +\n+\nr\nr Solution We have \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n2\n3\n4\nand\n2\na\nb\ni\nj\nk\na\nb\nj\nk\n+\n=\n+\n+\n\u2212\n= \u2212 \u2212\nr\nr\nr\nr\nA vector which is perpendicular to both \nand\na\nb\na\nb\n+\n\u2212\nr\nr\nr\nr\n is given by\n(\n)\n(\n)\na\nb\na\nb\n+\n\u00d7\n\u2212\nr\nr\nr\nr\n =\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n2\n3\n4\n2\n4\n2\n(\n, say)\n0\n1\n2\ni\nj\nk\ni\nj\nk\nc\n= \u2212\n+\n\u2212\n=\n\u2212\n\u2212\nr\nNow\n|\n|\ncr  =\n4\n16\n4\n24\n2 6\n+\n+\n=\n=\nTherefore, the required unit vector is\n|\nc|\nc\nr\nr  =\n1\n2\n1 \u02c6\n\u02c6\n\u02c6\n6\n6\n6\ni\nj\nk\n\u2212\n+\n\u2212\n\ufffdNote   There are two perpendicular directions to any plane Thus, another unit\nvector perpendicular to \nand\na\nb\na\nb\n+\n\u2212\nr\nr\nr\nr\n will be 1\n2\n1 \u02c6\n\u02c6\n\u02c6 6\n6\n6\ni\nj\nk\n\u2212\n+\n But that will\nbe a consequence of (\n)\n(\n)\na\nb\na\nb\n\u2212\n\u00d7\n+\nr\nr\nr\nr"
    },
    {
        "Chapter": "1",
        "sentence_range": "5466-5469",
        "Text": "Solution We have \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n2\n3\n4\nand\n2\na\nb\ni\nj\nk\na\nb\nj\nk\n+\n=\n+\n+\n\u2212\n= \u2212 \u2212\nr\nr\nr\nr\nA vector which is perpendicular to both \nand\na\nb\na\nb\n+\n\u2212\nr\nr\nr\nr\n is given by\n(\n)\n(\n)\na\nb\na\nb\n+\n\u00d7\n\u2212\nr\nr\nr\nr\n =\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n2\n3\n4\n2\n4\n2\n(\n, say)\n0\n1\n2\ni\nj\nk\ni\nj\nk\nc\n= \u2212\n+\n\u2212\n=\n\u2212\n\u2212\nr\nNow\n|\n|\ncr  =\n4\n16\n4\n24\n2 6\n+\n+\n=\n=\nTherefore, the required unit vector is\n|\nc|\nc\nr\nr  =\n1\n2\n1 \u02c6\n\u02c6\n\u02c6\n6\n6\n6\ni\nj\nk\n\u2212\n+\n\u2212\n\ufffdNote   There are two perpendicular directions to any plane Thus, another unit\nvector perpendicular to \nand\na\nb\na\nb\n+\n\u2212\nr\nr\nr\nr\n will be 1\n2\n1 \u02c6\n\u02c6\n\u02c6 6\n6\n6\ni\nj\nk\n\u2212\n+\n But that will\nbe a consequence of (\n)\n(\n)\na\nb\na\nb\n\u2212\n\u00d7\n+\nr\nr\nr\nr Example 24 Find the area of a triangle having the points A(1, 1, 1), B(1, 2, 3)\nand C(2, 3, 1) as its vertices"
    },
    {
        "Chapter": "1",
        "sentence_range": "5467-5470",
        "Text": "Thus, another unit\nvector perpendicular to \nand\na\nb\na\nb\n+\n\u2212\nr\nr\nr\nr\n will be 1\n2\n1 \u02c6\n\u02c6\n\u02c6 6\n6\n6\ni\nj\nk\n\u2212\n+\n But that will\nbe a consequence of (\n)\n(\n)\na\nb\na\nb\n\u2212\n\u00d7\n+\nr\nr\nr\nr Example 24 Find the area of a triangle having the points A(1, 1, 1), B(1, 2, 3)\nand C(2, 3, 1) as its vertices Solution We have \n\u02c6\n\u02c6\n\u02c6\n\u02c6\nAB\n2\nand AC\n2\nj\nk\ni\nj\n=\n+\n= +\nuuur\nuuur"
    },
    {
        "Chapter": "1",
        "sentence_range": "5468-5471",
        "Text": "6\n6\n6\ni\nj\nk\n\u2212\n+\n But that will\nbe a consequence of (\n)\n(\n)\na\nb\na\nb\n\u2212\n\u00d7\n+\nr\nr\nr\nr Example 24 Find the area of a triangle having the points A(1, 1, 1), B(1, 2, 3)\nand C(2, 3, 1) as its vertices Solution We have \n\u02c6\n\u02c6\n\u02c6\n\u02c6\nAB\n2\nand AC\n2\nj\nk\ni\nj\n=\n+\n= +\nuuur\nuuur The area of the given triangle\nis 1 | AB AC |\n2\nuuur\u00d7\nuuur"
    },
    {
        "Chapter": "1",
        "sentence_range": "5469-5472",
        "Text": "Example 24 Find the area of a triangle having the points A(1, 1, 1), B(1, 2, 3)\nand C(2, 3, 1) as its vertices Solution We have \n\u02c6\n\u02c6\n\u02c6\n\u02c6\nAB\n2\nand AC\n2\nj\nk\ni\nj\n=\n+\n= +\nuuur\nuuur The area of the given triangle\nis 1 | AB AC |\n2\nuuur\u00d7\nuuur Now,\nAB AC\n\u00d7\nuuur\nuuur  =\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n0\n1\n2\n4\n2\n1\n2\n0\ni\nj\nk\ni\nj\nk\n= \u2212\n+\n\u2212\nTherefore\n| AB AC|\nuuur\u00d7\nuuur  =\n16\n4\n1\n21\n+\n+ =\nThus, the required area is \n1\n21\n2\n\u00a9 NCERT\nnot to be republished\n MATHEMATICS\n454\nExample 25 Find the area of a parallelogram whose adjacent sides are given\nby the vectors \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n3\n4\nand\na\ni\nj\nk\nb\ni\nj\nk\n=\n+\n+\n= \u2212\n+\nr\nr\nSolution The area of a parallelogram with \naand\nrb\nr\n as its adjacent sides is given\nby |\n|\na\n\u00d7b\nr\nr"
    },
    {
        "Chapter": "1",
        "sentence_range": "5470-5473",
        "Text": "Solution We have \n\u02c6\n\u02c6\n\u02c6\n\u02c6\nAB\n2\nand AC\n2\nj\nk\ni\nj\n=\n+\n= +\nuuur\nuuur The area of the given triangle\nis 1 | AB AC |\n2\nuuur\u00d7\nuuur Now,\nAB AC\n\u00d7\nuuur\nuuur  =\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n0\n1\n2\n4\n2\n1\n2\n0\ni\nj\nk\ni\nj\nk\n= \u2212\n+\n\u2212\nTherefore\n| AB AC|\nuuur\u00d7\nuuur  =\n16\n4\n1\n21\n+\n+ =\nThus, the required area is \n1\n21\n2\n\u00a9 NCERT\nnot to be republished\n MATHEMATICS\n454\nExample 25 Find the area of a parallelogram whose adjacent sides are given\nby the vectors \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n3\n4\nand\na\ni\nj\nk\nb\ni\nj\nk\n=\n+\n+\n= \u2212\n+\nr\nr\nSolution The area of a parallelogram with \naand\nrb\nr\n as its adjacent sides is given\nby |\n|\na\n\u00d7b\nr\nr Now\na\nb\n\u00d7\nr\nr\n =\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n3\n1\n4\n5\n4\n1\n1\n1\ni\nj\nk\ni\nj\nk\n=\n+\n\u2212\n\u2212\nTherefore\n|\n|\na\nb\n\u00d7\nr\nr\n =\n25 1 16\n42\n+ +\n=\nand hence, the required area is \n42"
    },
    {
        "Chapter": "1",
        "sentence_range": "5471-5474",
        "Text": "The area of the given triangle\nis 1 | AB AC |\n2\nuuur\u00d7\nuuur Now,\nAB AC\n\u00d7\nuuur\nuuur  =\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n0\n1\n2\n4\n2\n1\n2\n0\ni\nj\nk\ni\nj\nk\n= \u2212\n+\n\u2212\nTherefore\n| AB AC|\nuuur\u00d7\nuuur  =\n16\n4\n1\n21\n+\n+ =\nThus, the required area is \n1\n21\n2\n\u00a9 NCERT\nnot to be republished\n MATHEMATICS\n454\nExample 25 Find the area of a parallelogram whose adjacent sides are given\nby the vectors \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n3\n4\nand\na\ni\nj\nk\nb\ni\nj\nk\n=\n+\n+\n= \u2212\n+\nr\nr\nSolution The area of a parallelogram with \naand\nrb\nr\n as its adjacent sides is given\nby |\n|\na\n\u00d7b\nr\nr Now\na\nb\n\u00d7\nr\nr\n =\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n3\n1\n4\n5\n4\n1\n1\n1\ni\nj\nk\ni\nj\nk\n=\n+\n\u2212\n\u2212\nTherefore\n|\n|\na\nb\n\u00d7\nr\nr\n =\n25 1 16\n42\n+ +\n=\nand hence, the required area is \n42 EXERCISE 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5472-5475",
        "Text": "Now,\nAB AC\n\u00d7\nuuur\nuuur  =\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n0\n1\n2\n4\n2\n1\n2\n0\ni\nj\nk\ni\nj\nk\n= \u2212\n+\n\u2212\nTherefore\n| AB AC|\nuuur\u00d7\nuuur  =\n16\n4\n1\n21\n+\n+ =\nThus, the required area is \n1\n21\n2\n\u00a9 NCERT\nnot to be republished\n MATHEMATICS\n454\nExample 25 Find the area of a parallelogram whose adjacent sides are given\nby the vectors \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n3\n4\nand\na\ni\nj\nk\nb\ni\nj\nk\n=\n+\n+\n= \u2212\n+\nr\nr\nSolution The area of a parallelogram with \naand\nrb\nr\n as its adjacent sides is given\nby |\n|\na\n\u00d7b\nr\nr Now\na\nb\n\u00d7\nr\nr\n =\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n3\n1\n4\n5\n4\n1\n1\n1\ni\nj\nk\ni\nj\nk\n=\n+\n\u2212\n\u2212\nTherefore\n|\n|\na\nb\n\u00d7\nr\nr\n =\n25 1 16\n42\n+ +\n=\nand hence, the required area is \n42 EXERCISE 10 4\n1"
    },
    {
        "Chapter": "1",
        "sentence_range": "5473-5476",
        "Text": "Now\na\nb\n\u00d7\nr\nr\n =\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n3\n1\n4\n5\n4\n1\n1\n1\ni\nj\nk\ni\nj\nk\n=\n+\n\u2212\n\u2212\nTherefore\n|\n|\na\nb\n\u00d7\nr\nr\n =\n25 1 16\n42\n+ +\n=\nand hence, the required area is \n42 EXERCISE 10 4\n1 Find \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n|\n|,  if  \n7\n7\nand\n3\n2\n2\na b\na\ni\nj\nk\nb\ni\nj\nk\n\u00d7\n= \u2212\n+\n=\n\u2212\n+\nr\nr\nr\nr"
    },
    {
        "Chapter": "1",
        "sentence_range": "5474-5477",
        "Text": "EXERCISE 10 4\n1 Find \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n|\n|,  if  \n7\n7\nand\n3\n2\n2\na b\na\ni\nj\nk\nb\ni\nj\nk\n\u00d7\n= \u2212\n+\n=\n\u2212\n+\nr\nr\nr\nr 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "5475-5478",
        "Text": "4\n1 Find \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n|\n|,  if  \n7\n7\nand\n3\n2\n2\na b\na\ni\nj\nk\nb\ni\nj\nk\n\u00d7\n= \u2212\n+\n=\n\u2212\n+\nr\nr\nr\nr 2 Find a unit vector perpendicular to each of the vector \nand\na\nb\na\nb\n+\n\u2212\nr\nr\nr\nr\n, where\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n3\n2\n2\nand\n2\n2\na\ni\nj\nk\nb\ni\nj\nk\n=\n+\n+\n= +\n\u2212\nr\nr"
    },
    {
        "Chapter": "1",
        "sentence_range": "5476-5479",
        "Text": "Find \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n|\n|,  if  \n7\n7\nand\n3\n2\n2\na b\na\ni\nj\nk\nb\ni\nj\nk\n\u00d7\n= \u2212\n+\n=\n\u2212\n+\nr\nr\nr\nr 2 Find a unit vector perpendicular to each of the vector \nand\na\nb\na\nb\n+\n\u2212\nr\nr\nr\nr\n, where\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n3\n2\n2\nand\n2\n2\na\ni\nj\nk\nb\ni\nj\nk\n=\n+\n+\n= +\n\u2212\nr\nr 3"
    },
    {
        "Chapter": "1",
        "sentence_range": "5477-5480",
        "Text": "2 Find a unit vector perpendicular to each of the vector \nand\na\nb\na\nb\n+\n\u2212\nr\nr\nr\nr\n, where\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n3\n2\n2\nand\n2\n2\na\ni\nj\nk\nb\ni\nj\nk\n=\n+\n+\n= +\n\u2212\nr\nr 3 If a unit vector ar  makes angles \n\u02c6\n\u02c6\nwith ,\nwith\n3\ni4\nj\n\u03c0\n\u03c0\n and an acute angle \u03b8 with\n\u02c6k , then find \u03b8 and hence, the components of ar"
    },
    {
        "Chapter": "1",
        "sentence_range": "5478-5481",
        "Text": "Find a unit vector perpendicular to each of the vector \nand\na\nb\na\nb\n+\n\u2212\nr\nr\nr\nr\n, where\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n3\n2\n2\nand\n2\n2\na\ni\nj\nk\nb\ni\nj\nk\n=\n+\n+\n= +\n\u2212\nr\nr 3 If a unit vector ar  makes angles \n\u02c6\n\u02c6\nwith ,\nwith\n3\ni4\nj\n\u03c0\n\u03c0\n and an acute angle \u03b8 with\n\u02c6k , then find \u03b8 and hence, the components of ar 4"
    },
    {
        "Chapter": "1",
        "sentence_range": "5479-5482",
        "Text": "3 If a unit vector ar  makes angles \n\u02c6\n\u02c6\nwith ,\nwith\n3\ni4\nj\n\u03c0\n\u03c0\n and an acute angle \u03b8 with\n\u02c6k , then find \u03b8 and hence, the components of ar 4 Show that\n(\n)\n(\n)\na\nb\na\nb\n\u2212\n\u00d7\n+\nr\nr\nr\nr\n = 2(\na b)\n\u00d7\nr\nr\n5"
    },
    {
        "Chapter": "1",
        "sentence_range": "5480-5483",
        "Text": "If a unit vector ar  makes angles \n\u02c6\n\u02c6\nwith ,\nwith\n3\ni4\nj\n\u03c0\n\u03c0\n and an acute angle \u03b8 with\n\u02c6k , then find \u03b8 and hence, the components of ar 4 Show that\n(\n)\n(\n)\na\nb\na\nb\n\u2212\n\u00d7\n+\nr\nr\nr\nr\n = 2(\na b)\n\u00d7\nr\nr\n5 Find \u03bb and \u03bc if \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(2\n6\n27 )\n(\n)\n0\ni\nj\nk\ni\nj\nk\n+\n+\n\u00d7\n+ \u03bb + \u03bc\n=\nr"
    },
    {
        "Chapter": "1",
        "sentence_range": "5481-5484",
        "Text": "4 Show that\n(\n)\n(\n)\na\nb\na\nb\n\u2212\n\u00d7\n+\nr\nr\nr\nr\n = 2(\na b)\n\u00d7\nr\nr\n5 Find \u03bb and \u03bc if \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(2\n6\n27 )\n(\n)\n0\ni\nj\nk\ni\nj\nk\n+\n+\n\u00d7\n+ \u03bb + \u03bc\n=\nr 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "5482-5485",
        "Text": "Show that\n(\n)\n(\n)\na\nb\na\nb\n\u2212\n\u00d7\n+\nr\nr\nr\nr\n = 2(\na b)\n\u00d7\nr\nr\n5 Find \u03bb and \u03bc if \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(2\n6\n27 )\n(\n)\n0\ni\nj\nk\ni\nj\nk\n+\n+\n\u00d7\n+ \u03bb + \u03bc\n=\nr 6 Given that \n0\na b\n \nrr \n and \n0\na\n\u00d7b\nr=\nr\nr"
    },
    {
        "Chapter": "1",
        "sentence_range": "5483-5486",
        "Text": "Find \u03bb and \u03bc if \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(2\n6\n27 )\n(\n)\n0\ni\nj\nk\ni\nj\nk\n+\n+\n\u00d7\n+ \u03bb + \u03bc\n=\nr 6 Given that \n0\na b\n \nrr \n and \n0\na\n\u00d7b\nr=\nr\nr What can you conclude about the vectors\naand\nrb\nr"
    },
    {
        "Chapter": "1",
        "sentence_range": "5484-5487",
        "Text": "6 Given that \n0\na b\n \nrr \n and \n0\na\n\u00d7b\nr=\nr\nr What can you conclude about the vectors\naand\nrb\nr 7"
    },
    {
        "Chapter": "1",
        "sentence_range": "5485-5488",
        "Text": "Given that \n0\na b\n \nrr \n and \n0\na\n\u00d7b\nr=\nr\nr What can you conclude about the vectors\naand\nrb\nr 7 Let the vectors \n, ,\na b c\nrr\nr  be given as \n1\n2\n3\n1\n2\n3\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n,\n,\na i\na j\na k b i\nb j\nb k\n+\n+\n+\n+\n1\n2\n3 \u02c6\n\u02c6\n\u02c6\nc i\nc j\nc k\n+\n+"
    },
    {
        "Chapter": "1",
        "sentence_range": "5486-5489",
        "Text": "What can you conclude about the vectors\naand\nrb\nr 7 Let the vectors \n, ,\na b c\nrr\nr  be given as \n1\n2\n3\n1\n2\n3\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n,\n,\na i\na j\na k b i\nb j\nb k\n+\n+\n+\n+\n1\n2\n3 \u02c6\n\u02c6\n\u02c6\nc i\nc j\nc k\n+\n+ Then show that \n(\n)\na\nb\nc\na b\na\nc\n\u00d7\n+\n= \u00d7\n+\n\u00d7\nr\nr\nr\nr\nr\nr\nr"
    },
    {
        "Chapter": "1",
        "sentence_range": "5487-5490",
        "Text": "7 Let the vectors \n, ,\na b c\nrr\nr  be given as \n1\n2\n3\n1\n2\n3\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n,\n,\na i\na j\na k b i\nb j\nb k\n+\n+\n+\n+\n1\n2\n3 \u02c6\n\u02c6\n\u02c6\nc i\nc j\nc k\n+\n+ Then show that \n(\n)\na\nb\nc\na b\na\nc\n\u00d7\n+\n= \u00d7\n+\n\u00d7\nr\nr\nr\nr\nr\nr\nr 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "5488-5491",
        "Text": "Let the vectors \n, ,\na b c\nrr\nr  be given as \n1\n2\n3\n1\n2\n3\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n,\n,\na i\na j\na k b i\nb j\nb k\n+\n+\n+\n+\n1\n2\n3 \u02c6\n\u02c6\n\u02c6\nc i\nc j\nc k\n+\n+ Then show that \n(\n)\na\nb\nc\na b\na\nc\n\u00d7\n+\n= \u00d7\n+\n\u00d7\nr\nr\nr\nr\nr\nr\nr 8 If either \n0  or  \n0,\na\nb\n=\nr=\nr\nr\nr\n then \n0\na\n\u00d7b\n=\nr\nr\nr"
    },
    {
        "Chapter": "1",
        "sentence_range": "5489-5492",
        "Text": "Then show that \n(\n)\na\nb\nc\na b\na\nc\n\u00d7\n+\n= \u00d7\n+\n\u00d7\nr\nr\nr\nr\nr\nr\nr 8 If either \n0  or  \n0,\na\nb\n=\nr=\nr\nr\nr\n then \n0\na\n\u00d7b\n=\nr\nr\nr Is the converse true"
    },
    {
        "Chapter": "1",
        "sentence_range": "5490-5493",
        "Text": "8 If either \n0  or  \n0,\na\nb\n=\nr=\nr\nr\nr\n then \n0\na\n\u00d7b\n=\nr\nr\nr Is the converse true Justify your\nanswer with an example"
    },
    {
        "Chapter": "1",
        "sentence_range": "5491-5494",
        "Text": "If either \n0  or  \n0,\na\nb\n=\nr=\nr\nr\nr\n then \n0\na\n\u00d7b\n=\nr\nr\nr Is the converse true Justify your\nanswer with an example 9"
    },
    {
        "Chapter": "1",
        "sentence_range": "5492-5495",
        "Text": "Is the converse true Justify your\nanswer with an example 9 Find the area of the triangle with vertices A(1, 1, 2), B(2, 3, 5) and C(1, 5, 5)"
    },
    {
        "Chapter": "1",
        "sentence_range": "5493-5496",
        "Text": "Justify your\nanswer with an example 9 Find the area of the triangle with vertices A(1, 1, 2), B(2, 3, 5) and C(1, 5, 5) \u00a9 NCERT\nnot to be republished\nVECTOR ALGEBRA\n455\n10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5494-5497",
        "Text": "9 Find the area of the triangle with vertices A(1, 1, 2), B(2, 3, 5) and C(1, 5, 5) \u00a9 NCERT\nnot to be republished\nVECTOR ALGEBRA\n455\n10 Find the area of the parallelogram whose adjacent sides are determined by the\nvectors \n\u02c6\n\u02c6\n\u02c6\n3\na\ni\nj\nk\n=\n\u2212\n+\nr\nand \n\u02c6\n\u02c6\n\u02c6\n2\n7\nb\ni\nj\nk\n=\n\u2212\n+\nr"
    },
    {
        "Chapter": "1",
        "sentence_range": "5495-5498",
        "Text": "Find the area of the triangle with vertices A(1, 1, 2), B(2, 3, 5) and C(1, 5, 5) \u00a9 NCERT\nnot to be republished\nVECTOR ALGEBRA\n455\n10 Find the area of the parallelogram whose adjacent sides are determined by the\nvectors \n\u02c6\n\u02c6\n\u02c6\n3\na\ni\nj\nk\n=\n\u2212\n+\nr\nand \n\u02c6\n\u02c6\n\u02c6\n2\n7\nb\ni\nj\nk\n=\n\u2212\n+\nr 11"
    },
    {
        "Chapter": "1",
        "sentence_range": "5496-5499",
        "Text": "\u00a9 NCERT\nnot to be republished\nVECTOR ALGEBRA\n455\n10 Find the area of the parallelogram whose adjacent sides are determined by the\nvectors \n\u02c6\n\u02c6\n\u02c6\n3\na\ni\nj\nk\n=\n\u2212\n+\nr\nand \n\u02c6\n\u02c6\n\u02c6\n2\n7\nb\ni\nj\nk\n=\n\u2212\n+\nr 11 Let the vectors \na  and  \nrb\nr\n be such that \n2\n|\n| 3  and |\n|\n3\na\nb\n=\nr=\nr\n, then a\n\u00d7b\nr\nr\n is a\nunit vector, if the angle between   and  \na\nrb\nr\n is\n(A) \u03c0/6\n(B) \u03c0/4\n(C) \u03c0/3\n(D) \u03c0/2\n12"
    },
    {
        "Chapter": "1",
        "sentence_range": "5497-5500",
        "Text": "Find the area of the parallelogram whose adjacent sides are determined by the\nvectors \n\u02c6\n\u02c6\n\u02c6\n3\na\ni\nj\nk\n=\n\u2212\n+\nr\nand \n\u02c6\n\u02c6\n\u02c6\n2\n7\nb\ni\nj\nk\n=\n\u2212\n+\nr 11 Let the vectors \na  and  \nrb\nr\n be such that \n2\n|\n| 3  and |\n|\n3\na\nb\n=\nr=\nr\n, then a\n\u00d7b\nr\nr\n is a\nunit vector, if the angle between   and  \na\nrb\nr\n is\n(A) \u03c0/6\n(B) \u03c0/4\n(C) \u03c0/3\n(D) \u03c0/2\n12 Area of a rectangle having vertices A, B, C and D with position vectors\n1\n1\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u2013\n4 ,\n4\n2\n2\ni\nj\nk i\nj\nk\n+\n+\n+\n+\n, \n1\n\u02c6\n\u02c6\n\u02c6\n4\n2\ni\nj\nk\n\u2212\n+\n and \n1\n\u02c6\n\u02c6\n\u02c6\n\u2013\n4\n2\ni\nj\nk\n\u2212\n+\n, respectively is\n(A) 1\n2\n(B) 1\n(C) 2\n(D) 4\nMiscellaneous Examples\nExample 26 Write all the unit vectors in XY-plane"
    },
    {
        "Chapter": "1",
        "sentence_range": "5498-5501",
        "Text": "11 Let the vectors \na  and  \nrb\nr\n be such that \n2\n|\n| 3  and |\n|\n3\na\nb\n=\nr=\nr\n, then a\n\u00d7b\nr\nr\n is a\nunit vector, if the angle between   and  \na\nrb\nr\n is\n(A) \u03c0/6\n(B) \u03c0/4\n(C) \u03c0/3\n(D) \u03c0/2\n12 Area of a rectangle having vertices A, B, C and D with position vectors\n1\n1\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u2013\n4 ,\n4\n2\n2\ni\nj\nk i\nj\nk\n+\n+\n+\n+\n, \n1\n\u02c6\n\u02c6\n\u02c6\n4\n2\ni\nj\nk\n\u2212\n+\n and \n1\n\u02c6\n\u02c6\n\u02c6\n\u2013\n4\n2\ni\nj\nk\n\u2212\n+\n, respectively is\n(A) 1\n2\n(B) 1\n(C) 2\n(D) 4\nMiscellaneous Examples\nExample 26 Write all the unit vectors in XY-plane Solution Let r\nx i\ny j\n\u2227\n\u2227\n=\n+\nr\n be a unit vector in XY-plane (Fig 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5499-5502",
        "Text": "Let the vectors \na  and  \nrb\nr\n be such that \n2\n|\n| 3  and |\n|\n3\na\nb\n=\nr=\nr\n, then a\n\u00d7b\nr\nr\n is a\nunit vector, if the angle between   and  \na\nrb\nr\n is\n(A) \u03c0/6\n(B) \u03c0/4\n(C) \u03c0/3\n(D) \u03c0/2\n12 Area of a rectangle having vertices A, B, C and D with position vectors\n1\n1\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u2013\n4 ,\n4\n2\n2\ni\nj\nk i\nj\nk\n+\n+\n+\n+\n, \n1\n\u02c6\n\u02c6\n\u02c6\n4\n2\ni\nj\nk\n\u2212\n+\n and \n1\n\u02c6\n\u02c6\n\u02c6\n\u2013\n4\n2\ni\nj\nk\n\u2212\n+\n, respectively is\n(A) 1\n2\n(B) 1\n(C) 2\n(D) 4\nMiscellaneous Examples\nExample 26 Write all the unit vectors in XY-plane Solution Let r\nx i\ny j\n\u2227\n\u2227\n=\n+\nr\n be a unit vector in XY-plane (Fig 10 28)"
    },
    {
        "Chapter": "1",
        "sentence_range": "5500-5503",
        "Text": "Area of a rectangle having vertices A, B, C and D with position vectors\n1\n1\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u2013\n4 ,\n4\n2\n2\ni\nj\nk i\nj\nk\n+\n+\n+\n+\n, \n1\n\u02c6\n\u02c6\n\u02c6\n4\n2\ni\nj\nk\n\u2212\n+\n and \n1\n\u02c6\n\u02c6\n\u02c6\n\u2013\n4\n2\ni\nj\nk\n\u2212\n+\n, respectively is\n(A) 1\n2\n(B) 1\n(C) 2\n(D) 4\nMiscellaneous Examples\nExample 26 Write all the unit vectors in XY-plane Solution Let r\nx i\ny j\n\u2227\n\u2227\n=\n+\nr\n be a unit vector in XY-plane (Fig 10 28) Then, from the\nfigure, we have x = cos \u03b8 and y = sin \u03b8 (since | rr | = 1)"
    },
    {
        "Chapter": "1",
        "sentence_range": "5501-5504",
        "Text": "Solution Let r\nx i\ny j\n\u2227\n\u2227\n=\n+\nr\n be a unit vector in XY-plane (Fig 10 28) Then, from the\nfigure, we have x = cos \u03b8 and y = sin \u03b8 (since | rr | = 1) So, we may write the vector rr as\n(\nOP)\nr =\nuuur\nr\n=\n\u02c6\n\u02c6\ncos\nsin\ni\nj"
    },
    {
        "Chapter": "1",
        "sentence_range": "5502-5505",
        "Text": "28) Then, from the\nfigure, we have x = cos \u03b8 and y = sin \u03b8 (since | rr | = 1) So, we may write the vector rr as\n(\nOP)\nr =\nuuur\nr\n=\n\u02c6\n\u02c6\ncos\nsin\ni\nj (1)\nClearly,\n|\n|\nrr  =\n2\n2\ncos\nsin\n1\n\u03b8 +\n\u03b8 =\nFig 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5503-5506",
        "Text": "Then, from the\nfigure, we have x = cos \u03b8 and y = sin \u03b8 (since | rr | = 1) So, we may write the vector rr as\n(\nOP)\nr =\nuuur\nr\n=\n\u02c6\n\u02c6\ncos\nsin\ni\nj (1)\nClearly,\n|\n|\nrr  =\n2\n2\ncos\nsin\n1\n\u03b8 +\n\u03b8 =\nFig 10 28\nAlso, as \u03b8 varies from 0 to 2\u03c0, the point P (Fig 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5504-5507",
        "Text": "So, we may write the vector rr as\n(\nOP)\nr =\nuuur\nr\n=\n\u02c6\n\u02c6\ncos\nsin\ni\nj (1)\nClearly,\n|\n|\nrr  =\n2\n2\ncos\nsin\n1\n\u03b8 +\n\u03b8 =\nFig 10 28\nAlso, as \u03b8 varies from 0 to 2\u03c0, the point P (Fig 10 28) traces the circle x2 + y2 = 1\ncounterclockwise, and this covers all possible directions"
    },
    {
        "Chapter": "1",
        "sentence_range": "5505-5508",
        "Text": "(1)\nClearly,\n|\n|\nrr  =\n2\n2\ncos\nsin\n1\n\u03b8 +\n\u03b8 =\nFig 10 28\nAlso, as \u03b8 varies from 0 to 2\u03c0, the point P (Fig 10 28) traces the circle x2 + y2 = 1\ncounterclockwise, and this covers all possible directions So, (1) gives every unit vector\nin the XY-plane"
    },
    {
        "Chapter": "1",
        "sentence_range": "5506-5509",
        "Text": "28\nAlso, as \u03b8 varies from 0 to 2\u03c0, the point P (Fig 10 28) traces the circle x2 + y2 = 1\ncounterclockwise, and this covers all possible directions So, (1) gives every unit vector\nin the XY-plane \u00a9 NCERT\nnot to be republished\n MATHEMATICS\n456\nExample 27 If \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n, 2\n5 , 3\n2\n3   and  \n6\n \n \n \n \n \n \ni\nj\nk\ni\nj\ni\nj\nk\ni\nj\nk  are the position\nvectors of points A, B, C and D respectively, then find the angle between AB\nuuur\n and\nCD\nuuur"
    },
    {
        "Chapter": "1",
        "sentence_range": "5507-5510",
        "Text": "28) traces the circle x2 + y2 = 1\ncounterclockwise, and this covers all possible directions So, (1) gives every unit vector\nin the XY-plane \u00a9 NCERT\nnot to be republished\n MATHEMATICS\n456\nExample 27 If \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n, 2\n5 , 3\n2\n3   and  \n6\n \n \n \n \n \n \ni\nj\nk\ni\nj\ni\nj\nk\ni\nj\nk  are the position\nvectors of points A, B, C and D respectively, then find the angle between AB\nuuur\n and\nCD\nuuur Deduce that AB\nuuur\n and CD\nuuur\n are collinear"
    },
    {
        "Chapter": "1",
        "sentence_range": "5508-5511",
        "Text": "So, (1) gives every unit vector\nin the XY-plane \u00a9 NCERT\nnot to be republished\n MATHEMATICS\n456\nExample 27 If \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n, 2\n5 , 3\n2\n3   and  \n6\n \n \n \n \n \n \ni\nj\nk\ni\nj\ni\nj\nk\ni\nj\nk  are the position\nvectors of points A, B, C and D respectively, then find the angle between AB\nuuur\n and\nCD\nuuur Deduce that AB\nuuur\n and CD\nuuur\n are collinear Solution Note that if \u03b8 is the angle between AB and CD, then \u03b8 is also the angle\nbetween AB  and  CD\nuuur\nuuur"
    },
    {
        "Chapter": "1",
        "sentence_range": "5509-5512",
        "Text": "\u00a9 NCERT\nnot to be republished\n MATHEMATICS\n456\nExample 27 If \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n, 2\n5 , 3\n2\n3   and  \n6\n \n \n \n \n \n \ni\nj\nk\ni\nj\ni\nj\nk\ni\nj\nk  are the position\nvectors of points A, B, C and D respectively, then find the angle between AB\nuuur\n and\nCD\nuuur Deduce that AB\nuuur\n and CD\nuuur\n are collinear Solution Note that if \u03b8 is the angle between AB and CD, then \u03b8 is also the angle\nbetween AB  and  CD\nuuur\nuuur Now\nAB\nuuur\n = Position vector of B \u2013 Position vector of A\n=\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(2\n5 )\n(\n)\n4\ni\nj\ni\nj\nk\ni\nj\nk\n+\n\u2212\n+\n+\n= +\n\u2212\nTherefore\n| AB|\nuuur  =\n2\n2\n2\n(1)\n(4)\n( 1)\n3 2\n+\n+ \u2212\n=\nSimilarly\nCD\nuuur  =\n\u02c6\n\u02c6\n\u02c6\n2\n8\n2   and  |CD | 6 2\ni\nj\nk\n\u2212\n\u2212\n+\n=\nuuur\nThus\ncos \u03b8 =\nAB CD\n|AB||CD|\n \nuuur uuur\nuuur uuur\n=\n1( 2)\n4( 8)\n( 1)(2)\n36\n1\n36\n(3 2)(6 2)\n\u2212\n+\n\u2212\n+ \u2212\n=\u2212\n= \u2212\nSince 0 \u2264 \u03b8 \u2264 \u03c0, it follows that \u03b8 = \u03c0"
    },
    {
        "Chapter": "1",
        "sentence_range": "5510-5513",
        "Text": "Deduce that AB\nuuur\n and CD\nuuur\n are collinear Solution Note that if \u03b8 is the angle between AB and CD, then \u03b8 is also the angle\nbetween AB  and  CD\nuuur\nuuur Now\nAB\nuuur\n = Position vector of B \u2013 Position vector of A\n=\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(2\n5 )\n(\n)\n4\ni\nj\ni\nj\nk\ni\nj\nk\n+\n\u2212\n+\n+\n= +\n\u2212\nTherefore\n| AB|\nuuur  =\n2\n2\n2\n(1)\n(4)\n( 1)\n3 2\n+\n+ \u2212\n=\nSimilarly\nCD\nuuur  =\n\u02c6\n\u02c6\n\u02c6\n2\n8\n2   and  |CD | 6 2\ni\nj\nk\n\u2212\n\u2212\n+\n=\nuuur\nThus\ncos \u03b8 =\nAB CD\n|AB||CD|\n \nuuur uuur\nuuur uuur\n=\n1( 2)\n4( 8)\n( 1)(2)\n36\n1\n36\n(3 2)(6 2)\n\u2212\n+\n\u2212\n+ \u2212\n=\u2212\n= \u2212\nSince 0 \u2264 \u03b8 \u2264 \u03c0, it follows that \u03b8 = \u03c0 This shows that AB\nuuur\n and CD\nuuur\n are collinear"
    },
    {
        "Chapter": "1",
        "sentence_range": "5511-5514",
        "Text": "Solution Note that if \u03b8 is the angle between AB and CD, then \u03b8 is also the angle\nbetween AB  and  CD\nuuur\nuuur Now\nAB\nuuur\n = Position vector of B \u2013 Position vector of A\n=\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(2\n5 )\n(\n)\n4\ni\nj\ni\nj\nk\ni\nj\nk\n+\n\u2212\n+\n+\n= +\n\u2212\nTherefore\n| AB|\nuuur  =\n2\n2\n2\n(1)\n(4)\n( 1)\n3 2\n+\n+ \u2212\n=\nSimilarly\nCD\nuuur  =\n\u02c6\n\u02c6\n\u02c6\n2\n8\n2   and  |CD | 6 2\ni\nj\nk\n\u2212\n\u2212\n+\n=\nuuur\nThus\ncos \u03b8 =\nAB CD\n|AB||CD|\n \nuuur uuur\nuuur uuur\n=\n1( 2)\n4( 8)\n( 1)(2)\n36\n1\n36\n(3 2)(6 2)\n\u2212\n+\n\u2212\n+ \u2212\n=\u2212\n= \u2212\nSince 0 \u2264 \u03b8 \u2264 \u03c0, it follows that \u03b8 = \u03c0 This shows that AB\nuuur\n and CD\nuuur\n are collinear Alternatively, \n1\nAB\n2CD\n  \nuuur\nuuur\n which implies that AB and CD\nuuur\nuuur  are collinear vectors"
    },
    {
        "Chapter": "1",
        "sentence_range": "5512-5515",
        "Text": "Now\nAB\nuuur\n = Position vector of B \u2013 Position vector of A\n=\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(2\n5 )\n(\n)\n4\ni\nj\ni\nj\nk\ni\nj\nk\n+\n\u2212\n+\n+\n= +\n\u2212\nTherefore\n| AB|\nuuur  =\n2\n2\n2\n(1)\n(4)\n( 1)\n3 2\n+\n+ \u2212\n=\nSimilarly\nCD\nuuur  =\n\u02c6\n\u02c6\n\u02c6\n2\n8\n2   and  |CD | 6 2\ni\nj\nk\n\u2212\n\u2212\n+\n=\nuuur\nThus\ncos \u03b8 =\nAB CD\n|AB||CD|\n \nuuur uuur\nuuur uuur\n=\n1( 2)\n4( 8)\n( 1)(2)\n36\n1\n36\n(3 2)(6 2)\n\u2212\n+\n\u2212\n+ \u2212\n=\u2212\n= \u2212\nSince 0 \u2264 \u03b8 \u2264 \u03c0, it follows that \u03b8 = \u03c0 This shows that AB\nuuur\n and CD\nuuur\n are collinear Alternatively, \n1\nAB\n2CD\n  \nuuur\nuuur\n which implies that AB and CD\nuuur\nuuur  are collinear vectors Example 28 Let \n,\na band\nc\nrr\nr  be three vectors such that |\n| 3, |\n| 4, | | 5\na\nb\nc\n=\n=\n=\nr\nr\nr\n and\neach one of them being perpendicular to the sum of the other two, find |\na b c|\n+ +\nr\nr\nr"
    },
    {
        "Chapter": "1",
        "sentence_range": "5513-5516",
        "Text": "This shows that AB\nuuur\n and CD\nuuur\n are collinear Alternatively, \n1\nAB\n2CD\n  \nuuur\nuuur\n which implies that AB and CD\nuuur\nuuur  are collinear vectors Example 28 Let \n,\na band\nc\nrr\nr  be three vectors such that |\n| 3, |\n| 4, | | 5\na\nb\nc\n=\n=\n=\nr\nr\nr\n and\neach one of them being perpendicular to the sum of the other two, find |\na b c|\n+ +\nr\nr\nr Solution Given \n(\n)\na\nb\nc\n\u22c5\nr+\nr\nr  = 0,\n(\n)\n0,\n(\n)\n0"
    },
    {
        "Chapter": "1",
        "sentence_range": "5514-5517",
        "Text": "Alternatively, \n1\nAB\n2CD\n  \nuuur\nuuur\n which implies that AB and CD\nuuur\nuuur  are collinear vectors Example 28 Let \n,\na band\nc\nrr\nr  be three vectors such that |\n| 3, |\n| 4, | | 5\na\nb\nc\n=\n=\n=\nr\nr\nr\n and\neach one of them being perpendicular to the sum of the other two, find |\na b c|\n+ +\nr\nr\nr Solution Given \n(\n)\na\nb\nc\n\u22c5\nr+\nr\nr  = 0,\n(\n)\n0,\n(\n)\n0 b\nc\na\nc\na\nb\n\u22c5\n+\n=\n\u22c5\n+\n=\nr\nr\nr\nr\nr\nr\nNow\n2\n|\n|\na\nb\nc\n+\n+\nr\nr\nr\n =\n2\n(\n)\n(\n) (\n)\na\nb\nc\na\nb\nc\na\nb\nc\n+\n+\n=\n+\n+\n\u22c5\n+\n+\nr\nr\nr\nr\nr\nr\nr\nr\nr\n=\n(\n)\n(\n)\na a\na\nb\nc\nb b\nb\na\nc\n\u22c5\n+\n\u22c5\n+\n+\n\u22c5\n+\n\u22c5\n+\nr\nr r\nr\nr r\nr\nr\nr\nr\n+"
    },
    {
        "Chapter": "1",
        "sentence_range": "5515-5518",
        "Text": "Example 28 Let \n,\na band\nc\nrr\nr  be three vectors such that |\n| 3, |\n| 4, | | 5\na\nb\nc\n=\n=\n=\nr\nr\nr\n and\neach one of them being perpendicular to the sum of the other two, find |\na b c|\n+ +\nr\nr\nr Solution Given \n(\n)\na\nb\nc\n\u22c5\nr+\nr\nr  = 0,\n(\n)\n0,\n(\n)\n0 b\nc\na\nc\na\nb\n\u22c5\n+\n=\n\u22c5\n+\n=\nr\nr\nr\nr\nr\nr\nNow\n2\n|\n|\na\nb\nc\n+\n+\nr\nr\nr\n =\n2\n(\n)\n(\n) (\n)\na\nb\nc\na\nb\nc\na\nb\nc\n+\n+\n=\n+\n+\n\u22c5\n+\n+\nr\nr\nr\nr\nr\nr\nr\nr\nr\n=\n(\n)\n(\n)\na a\na\nb\nc\nb b\nb\na\nc\n\u22c5\n+\n\u22c5\n+\n+\n\u22c5\n+\n\u22c5\n+\nr\nr r\nr\nr r\nr\nr\nr\nr\n+ (\n)"
    },
    {
        "Chapter": "1",
        "sentence_range": "5516-5519",
        "Text": "Solution Given \n(\n)\na\nb\nc\n\u22c5\nr+\nr\nr  = 0,\n(\n)\n0,\n(\n)\n0 b\nc\na\nc\na\nb\n\u22c5\n+\n=\n\u22c5\n+\n=\nr\nr\nr\nr\nr\nr\nNow\n2\n|\n|\na\nb\nc\n+\n+\nr\nr\nr\n =\n2\n(\n)\n(\n) (\n)\na\nb\nc\na\nb\nc\na\nb\nc\n+\n+\n=\n+\n+\n\u22c5\n+\n+\nr\nr\nr\nr\nr\nr\nr\nr\nr\n=\n(\n)\n(\n)\na a\na\nb\nc\nb b\nb\na\nc\n\u22c5\n+\n\u22c5\n+\n+\n\u22c5\n+\n\u22c5\n+\nr\nr r\nr\nr r\nr\nr\nr\nr\n+ (\n) c a\nb\nc c\n+\nr+\nr r\nr r\n=\n2\n2\n2\n|\n|\n|\n|\n|\n|\na\nb\nc\n+\n+\nr\nr\nr\n= 9 + 16 + 25 = 50\nTherefore\n|\n|\na\nb\nc\n+\nr+\nr\nr  =\n50\n5 2\n=\n\u00a9 NCERT\nnot to be republished\nVECTOR ALGEBRA\n457\nExample 29 Three vectors ,    and  \na\nb\nc\nr\nr\nr  satisfy the condition \n0\na\nb\nc\n+\n+\n=\nr\nr\nr\nr"
    },
    {
        "Chapter": "1",
        "sentence_range": "5517-5520",
        "Text": "b\nc\na\nc\na\nb\n\u22c5\n+\n=\n\u22c5\n+\n=\nr\nr\nr\nr\nr\nr\nNow\n2\n|\n|\na\nb\nc\n+\n+\nr\nr\nr\n =\n2\n(\n)\n(\n) (\n)\na\nb\nc\na\nb\nc\na\nb\nc\n+\n+\n=\n+\n+\n\u22c5\n+\n+\nr\nr\nr\nr\nr\nr\nr\nr\nr\n=\n(\n)\n(\n)\na a\na\nb\nc\nb b\nb\na\nc\n\u22c5\n+\n\u22c5\n+\n+\n\u22c5\n+\n\u22c5\n+\nr\nr r\nr\nr r\nr\nr\nr\nr\n+ (\n) c a\nb\nc c\n+\nr+\nr r\nr r\n=\n2\n2\n2\n|\n|\n|\n|\n|\n|\na\nb\nc\n+\n+\nr\nr\nr\n= 9 + 16 + 25 = 50\nTherefore\n|\n|\na\nb\nc\n+\nr+\nr\nr  =\n50\n5 2\n=\n\u00a9 NCERT\nnot to be republished\nVECTOR ALGEBRA\n457\nExample 29 Three vectors ,    and  \na\nb\nc\nr\nr\nr  satisfy the condition \n0\na\nb\nc\n+\n+\n=\nr\nr\nr\nr Evaluate\nthe quantity \n,  if  |\n| 1, |\n| 4  and |\n| 2\na b\nb c\nc a\na\nb\nc\n\u03bc =\n\u22c5\n+\n\u22c5\n+ \u22c5\n=\n=\n=\nr\nr\nr\nr\nr\nr r\nr\nr"
    },
    {
        "Chapter": "1",
        "sentence_range": "5518-5521",
        "Text": "(\n) c a\nb\nc c\n+\nr+\nr r\nr r\n=\n2\n2\n2\n|\n|\n|\n|\n|\n|\na\nb\nc\n+\n+\nr\nr\nr\n= 9 + 16 + 25 = 50\nTherefore\n|\n|\na\nb\nc\n+\nr+\nr\nr  =\n50\n5 2\n=\n\u00a9 NCERT\nnot to be republished\nVECTOR ALGEBRA\n457\nExample 29 Three vectors ,    and  \na\nb\nc\nr\nr\nr  satisfy the condition \n0\na\nb\nc\n+\n+\n=\nr\nr\nr\nr Evaluate\nthe quantity \n,  if  |\n| 1, |\n| 4  and |\n| 2\na b\nb c\nc a\na\nb\nc\n\u03bc =\n\u22c5\n+\n\u22c5\n+ \u22c5\n=\n=\n=\nr\nr\nr\nr\nr\nr r\nr\nr Solution Since \n0\na\nb\nc\n+\n+\n=\nr\nr\nr\nr\n, we have\n(\n)\na\na\nb\nc\n \n \nr\nr\nr\nr  = 0\nor\na a\na b\na c\n\u22c5\n+\n\u22c5\n+\n\u22c5\nr\nr\nr r\nr\nr\n = 0\nTherefore\na b\na c\n\u22c5\n+\n\u22c5\nr\nr\nr\nr\n =\n2\n1\n\u2212a\n= \u2212\nr"
    },
    {
        "Chapter": "1",
        "sentence_range": "5519-5522",
        "Text": "c a\nb\nc c\n+\nr+\nr r\nr r\n=\n2\n2\n2\n|\n|\n|\n|\n|\n|\na\nb\nc\n+\n+\nr\nr\nr\n= 9 + 16 + 25 = 50\nTherefore\n|\n|\na\nb\nc\n+\nr+\nr\nr  =\n50\n5 2\n=\n\u00a9 NCERT\nnot to be republished\nVECTOR ALGEBRA\n457\nExample 29 Three vectors ,    and  \na\nb\nc\nr\nr\nr  satisfy the condition \n0\na\nb\nc\n+\n+\n=\nr\nr\nr\nr Evaluate\nthe quantity \n,  if  |\n| 1, |\n| 4  and |\n| 2\na b\nb c\nc a\na\nb\nc\n\u03bc =\n\u22c5\n+\n\u22c5\n+ \u22c5\n=\n=\n=\nr\nr\nr\nr\nr\nr r\nr\nr Solution Since \n0\na\nb\nc\n+\n+\n=\nr\nr\nr\nr\n, we have\n(\n)\na\na\nb\nc\n \n \nr\nr\nr\nr  = 0\nor\na a\na b\na c\n\u22c5\n+\n\u22c5\n+\n\u22c5\nr\nr\nr r\nr\nr\n = 0\nTherefore\na b\na c\n\u22c5\n+\n\u22c5\nr\nr\nr\nr\n =\n2\n1\n\u2212a\n= \u2212\nr (1)\nAgain,\n(\n)\nb\na\nb\nc\n\u22c5\n+\n+\nr\nr\nr\nr  = 0\nor\na b\nb c\n\u22c5\n+\n\u22c5\nr\nr r\nr\n =\n2\n16\n\u2212b\n= \u2212\nr"
    },
    {
        "Chapter": "1",
        "sentence_range": "5520-5523",
        "Text": "Evaluate\nthe quantity \n,  if  |\n| 1, |\n| 4  and |\n| 2\na b\nb c\nc a\na\nb\nc\n\u03bc =\n\u22c5\n+\n\u22c5\n+ \u22c5\n=\n=\n=\nr\nr\nr\nr\nr\nr r\nr\nr Solution Since \n0\na\nb\nc\n+\n+\n=\nr\nr\nr\nr\n, we have\n(\n)\na\na\nb\nc\n \n \nr\nr\nr\nr  = 0\nor\na a\na b\na c\n\u22c5\n+\n\u22c5\n+\n\u22c5\nr\nr\nr r\nr\nr\n = 0\nTherefore\na b\na c\n\u22c5\n+\n\u22c5\nr\nr\nr\nr\n =\n2\n1\n\u2212a\n= \u2212\nr (1)\nAgain,\n(\n)\nb\na\nb\nc\n\u22c5\n+\n+\nr\nr\nr\nr  = 0\nor\na b\nb c\n\u22c5\n+\n\u22c5\nr\nr r\nr\n =\n2\n16\n\u2212b\n= \u2212\nr (2)\nSimilarly\na c\nb c\n\u22c5\n+\nr\u22c5\nr r\nr  = \u2013 4"
    },
    {
        "Chapter": "1",
        "sentence_range": "5521-5524",
        "Text": "Solution Since \n0\na\nb\nc\n+\n+\n=\nr\nr\nr\nr\n, we have\n(\n)\na\na\nb\nc\n \n \nr\nr\nr\nr  = 0\nor\na a\na b\na c\n\u22c5\n+\n\u22c5\n+\n\u22c5\nr\nr\nr r\nr\nr\n = 0\nTherefore\na b\na c\n\u22c5\n+\n\u22c5\nr\nr\nr\nr\n =\n2\n1\n\u2212a\n= \u2212\nr (1)\nAgain,\n(\n)\nb\na\nb\nc\n\u22c5\n+\n+\nr\nr\nr\nr  = 0\nor\na b\nb c\n\u22c5\n+\n\u22c5\nr\nr r\nr\n =\n2\n16\n\u2212b\n= \u2212\nr (2)\nSimilarly\na c\nb c\n\u22c5\n+\nr\u22c5\nr r\nr  = \u2013 4 (3)\nAdding (1), (2) and (3), we have\n2 (\n)\na b\nb\na\nc\nc\n\u22c5\n+\n\u22c5 +\n\u22c5\nr\nr\nr\nr\nr\nr\n = \u2013 21\nor\n2\u03bc = \u2013 21, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "5522-5525",
        "Text": "(1)\nAgain,\n(\n)\nb\na\nb\nc\n\u22c5\n+\n+\nr\nr\nr\nr  = 0\nor\na b\nb c\n\u22c5\n+\n\u22c5\nr\nr r\nr\n =\n2\n16\n\u2212b\n= \u2212\nr (2)\nSimilarly\na c\nb c\n\u22c5\n+\nr\u22c5\nr r\nr  = \u2013 4 (3)\nAdding (1), (2) and (3), we have\n2 (\n)\na b\nb\na\nc\nc\n\u22c5\n+\n\u22c5 +\n\u22c5\nr\nr\nr\nr\nr\nr\n = \u2013 21\nor\n2\u03bc = \u2013 21, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "5523-5526",
        "Text": "(2)\nSimilarly\na c\nb c\n\u22c5\n+\nr\u22c5\nr r\nr  = \u2013 4 (3)\nAdding (1), (2) and (3), we have\n2 (\n)\na b\nb\na\nc\nc\n\u22c5\n+\n\u22c5 +\n\u22c5\nr\nr\nr\nr\nr\nr\n = \u2013 21\nor\n2\u03bc = \u2013 21, i e , \u03bc =\n221\n\u2212\nExample 30 If with reference to the right handed system of mutually perpendicular\nunit vectors \n\u02c6\n\u02c6\n\u02c6 \u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n,\n and ,\n3\n,  \n2\n\u2013 3\ni\nj\nk\ni\nj\ni\nj\nk\n\u03b1 =\n\u2212\n\u03b2 =\n+\nr\nr\n, then express \u03b2\nr  in the form\n1\n2\n, where  1\n   \n  \n \nr\nr\nr\nr\nis parallel to \n  and  2\nr\nr\n is perpendicular to \u03b1r"
    },
    {
        "Chapter": "1",
        "sentence_range": "5524-5527",
        "Text": "(3)\nAdding (1), (2) and (3), we have\n2 (\n)\na b\nb\na\nc\nc\n\u22c5\n+\n\u22c5 +\n\u22c5\nr\nr\nr\nr\nr\nr\n = \u2013 21\nor\n2\u03bc = \u2013 21, i e , \u03bc =\n221\n\u2212\nExample 30 If with reference to the right handed system of mutually perpendicular\nunit vectors \n\u02c6\n\u02c6\n\u02c6 \u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n,\n and ,\n3\n,  \n2\n\u2013 3\ni\nj\nk\ni\nj\ni\nj\nk\n\u03b1 =\n\u2212\n\u03b2 =\n+\nr\nr\n, then express \u03b2\nr  in the form\n1\n2\n, where  1\n   \n  \n \nr\nr\nr\nr\nis parallel to \n  and  2\nr\nr\n is perpendicular to \u03b1r Solution Let \n1\n   ,  \n \nr\nr\n is a scalar, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "5525-5528",
        "Text": "e , \u03bc =\n221\n\u2212\nExample 30 If with reference to the right handed system of mutually perpendicular\nunit vectors \n\u02c6\n\u02c6\n\u02c6 \u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n,\n and ,\n3\n,  \n2\n\u2013 3\ni\nj\nk\ni\nj\ni\nj\nk\n\u03b1 =\n\u2212\n\u03b2 =\n+\nr\nr\n, then express \u03b2\nr  in the form\n1\n2\n, where  1\n   \n  \n \nr\nr\nr\nr\nis parallel to \n  and  2\nr\nr\n is perpendicular to \u03b1r Solution Let \n1\n   ,  \n \nr\nr\n is a scalar, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "5526-5529",
        "Text": ", \u03bc =\n221\n\u2212\nExample 30 If with reference to the right handed system of mutually perpendicular\nunit vectors \n\u02c6\n\u02c6\n\u02c6 \u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n,\n and ,\n3\n,  \n2\n\u2013 3\ni\nj\nk\ni\nj\ni\nj\nk\n\u03b1 =\n\u2212\n\u03b2 =\n+\nr\nr\n, then express \u03b2\nr  in the form\n1\n2\n, where  1\n   \n  \n \nr\nr\nr\nr\nis parallel to \n  and  2\nr\nr\n is perpendicular to \u03b1r Solution Let \n1\n   ,  \n \nr\nr\n is a scalar, i e , \n1\n\u02c6\n\u02c6\n3 i\nr\u03b2 = \u03bb \u2212 \u03bbj"
    },
    {
        "Chapter": "1",
        "sentence_range": "5527-5530",
        "Text": "Solution Let \n1\n   ,  \n \nr\nr\n is a scalar, i e , \n1\n\u02c6\n\u02c6\n3 i\nr\u03b2 = \u03bb \u2212 \u03bbj Now\n2\nr\u03b2 = \u03b2 \u2212\u03b21\nr\nr  =\n\u02c6\n\u02c6\n\u02c6\n(2\n3 )\n(1\n)\n3\ni\nj\nk\n\u2212 \u03bb\n+\n+ \u03bb\n\u2212"
    },
    {
        "Chapter": "1",
        "sentence_range": "5528-5531",
        "Text": "e , \n1\n\u02c6\n\u02c6\n3 i\nr\u03b2 = \u03bb \u2212 \u03bbj Now\n2\nr\u03b2 = \u03b2 \u2212\u03b21\nr\nr  =\n\u02c6\n\u02c6\n\u02c6\n(2\n3 )\n(1\n)\n3\ni\nj\nk\n\u2212 \u03bb\n+\n+ \u03bb\n\u2212 Now, since \n\u03b22\nr\n is to be perpendicular to \u03b1r , we should have \n2\n\u03b1\u22c5\u03b2 =0\nrr"
    },
    {
        "Chapter": "1",
        "sentence_range": "5529-5532",
        "Text": ", \n1\n\u02c6\n\u02c6\n3 i\nr\u03b2 = \u03bb \u2212 \u03bbj Now\n2\nr\u03b2 = \u03b2 \u2212\u03b21\nr\nr  =\n\u02c6\n\u02c6\n\u02c6\n(2\n3 )\n(1\n)\n3\ni\nj\nk\n\u2212 \u03bb\n+\n+ \u03bb\n\u2212 Now, since \n\u03b22\nr\n is to be perpendicular to \u03b1r , we should have \n2\n\u03b1\u22c5\u03b2 =0\nrr i"
    },
    {
        "Chapter": "1",
        "sentence_range": "5530-5533",
        "Text": "Now\n2\nr\u03b2 = \u03b2 \u2212\u03b21\nr\nr  =\n\u02c6\n\u02c6\n\u02c6\n(2\n3 )\n(1\n)\n3\ni\nj\nk\n\u2212 \u03bb\n+\n+ \u03bb\n\u2212 Now, since \n\u03b22\nr\n is to be perpendicular to \u03b1r , we should have \n2\n\u03b1\u22c5\u03b2 =0\nrr i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "5531-5534",
        "Text": "Now, since \n\u03b22\nr\n is to be perpendicular to \u03b1r , we should have \n2\n\u03b1\u22c5\u03b2 =0\nrr i e ,\n3(2\n3 )\n(1\n)\n\u2212 \u03bb \u2212\n+ \u03bb  = 0\nor\n\u03bb = 1\n2\nTherefore\n\u03b21\nr  = 3\n1\n\u02c6\n\u02c6\n2\n2\ni\nj\n\u2212\n  and \n2\n1\n3\n\u02c6\n\u02c6\n\u02c6 \u2013 3\n2\n2\ni\nj\nk\n\u03b2 =\n+\nr\n\u00a9 NCERT\nnot to be republished\n MATHEMATICS\n458\nMiscellaneous Exercise on Chapter 10\n1"
    },
    {
        "Chapter": "1",
        "sentence_range": "5532-5535",
        "Text": "i e ,\n3(2\n3 )\n(1\n)\n\u2212 \u03bb \u2212\n+ \u03bb  = 0\nor\n\u03bb = 1\n2\nTherefore\n\u03b21\nr  = 3\n1\n\u02c6\n\u02c6\n2\n2\ni\nj\n\u2212\n  and \n2\n1\n3\n\u02c6\n\u02c6\n\u02c6 \u2013 3\n2\n2\ni\nj\nk\n\u03b2 =\n+\nr\n\u00a9 NCERT\nnot to be republished\n MATHEMATICS\n458\nMiscellaneous Exercise on Chapter 10\n1 Write down a unit vector in XY-plane, making an angle of 30\u00b0 with the positive\ndirection of x-axis"
    },
    {
        "Chapter": "1",
        "sentence_range": "5533-5536",
        "Text": "e ,\n3(2\n3 )\n(1\n)\n\u2212 \u03bb \u2212\n+ \u03bb  = 0\nor\n\u03bb = 1\n2\nTherefore\n\u03b21\nr  = 3\n1\n\u02c6\n\u02c6\n2\n2\ni\nj\n\u2212\n  and \n2\n1\n3\n\u02c6\n\u02c6\n\u02c6 \u2013 3\n2\n2\ni\nj\nk\n\u03b2 =\n+\nr\n\u00a9 NCERT\nnot to be republished\n MATHEMATICS\n458\nMiscellaneous Exercise on Chapter 10\n1 Write down a unit vector in XY-plane, making an angle of 30\u00b0 with the positive\ndirection of x-axis 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "5534-5537",
        "Text": ",\n3(2\n3 )\n(1\n)\n\u2212 \u03bb \u2212\n+ \u03bb  = 0\nor\n\u03bb = 1\n2\nTherefore\n\u03b21\nr  = 3\n1\n\u02c6\n\u02c6\n2\n2\ni\nj\n\u2212\n  and \n2\n1\n3\n\u02c6\n\u02c6\n\u02c6 \u2013 3\n2\n2\ni\nj\nk\n\u03b2 =\n+\nr\n\u00a9 NCERT\nnot to be republished\n MATHEMATICS\n458\nMiscellaneous Exercise on Chapter 10\n1 Write down a unit vector in XY-plane, making an angle of 30\u00b0 with the positive\ndirection of x-axis 2 Find the scalar components and magnitude of the vector joining the points\nP(x1, y1, z1) and Q(x2, y2, z2)"
    },
    {
        "Chapter": "1",
        "sentence_range": "5535-5538",
        "Text": "Write down a unit vector in XY-plane, making an angle of 30\u00b0 with the positive\ndirection of x-axis 2 Find the scalar components and magnitude of the vector joining the points\nP(x1, y1, z1) and Q(x2, y2, z2) 3"
    },
    {
        "Chapter": "1",
        "sentence_range": "5536-5539",
        "Text": "2 Find the scalar components and magnitude of the vector joining the points\nP(x1, y1, z1) and Q(x2, y2, z2) 3 A girl walks 4 km towards west, then she walks 3 km in a direction 30\u00b0 east of\nnorth and stops"
    },
    {
        "Chapter": "1",
        "sentence_range": "5537-5540",
        "Text": "Find the scalar components and magnitude of the vector joining the points\nP(x1, y1, z1) and Q(x2, y2, z2) 3 A girl walks 4 km towards west, then she walks 3 km in a direction 30\u00b0 east of\nnorth and stops Determine the girl\u2019s displacement from her initial point of\ndeparture"
    },
    {
        "Chapter": "1",
        "sentence_range": "5538-5541",
        "Text": "3 A girl walks 4 km towards west, then she walks 3 km in a direction 30\u00b0 east of\nnorth and stops Determine the girl\u2019s displacement from her initial point of\ndeparture 4"
    },
    {
        "Chapter": "1",
        "sentence_range": "5539-5542",
        "Text": "A girl walks 4 km towards west, then she walks 3 km in a direction 30\u00b0 east of\nnorth and stops Determine the girl\u2019s displacement from her initial point of\ndeparture 4 If a\nb\nc\n=\nr+\nr\nr , then is it true that |\n| |\n|\n|\n|\na\nb\nc\n=\n+\nr\nr\nr"
    },
    {
        "Chapter": "1",
        "sentence_range": "5540-5543",
        "Text": "Determine the girl\u2019s displacement from her initial point of\ndeparture 4 If a\nb\nc\n=\nr+\nr\nr , then is it true that |\n| |\n|\n|\n|\na\nb\nc\n=\n+\nr\nr\nr Justify your answer"
    },
    {
        "Chapter": "1",
        "sentence_range": "5541-5544",
        "Text": "4 If a\nb\nc\n=\nr+\nr\nr , then is it true that |\n| |\n|\n|\n|\na\nb\nc\n=\n+\nr\nr\nr Justify your answer 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "5542-5545",
        "Text": "If a\nb\nc\n=\nr+\nr\nr , then is it true that |\n| |\n|\n|\n|\na\nb\nc\n=\n+\nr\nr\nr Justify your answer 5 Find the value of x for which \n\u02c6\n\u02c6\n\u02c6\n(\n)\nx i\nj\nk\n+\n+\n is a unit vector"
    },
    {
        "Chapter": "1",
        "sentence_range": "5543-5546",
        "Text": "Justify your answer 5 Find the value of x for which \n\u02c6\n\u02c6\n\u02c6\n(\n)\nx i\nj\nk\n+\n+\n is a unit vector 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "5544-5547",
        "Text": "5 Find the value of x for which \n\u02c6\n\u02c6\n\u02c6\n(\n)\nx i\nj\nk\n+\n+\n is a unit vector 6 Find a vector of magnitude 5 units, and parallel to the resultant of the vectors\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n2\n3\n  and  \n2\na\ni\nj\nk\nb\ni\nj\nk\n=\n+\n\u2212\n= \u2212\n+\nr\nr"
    },
    {
        "Chapter": "1",
        "sentence_range": "5545-5548",
        "Text": "Find the value of x for which \n\u02c6\n\u02c6\n\u02c6\n(\n)\nx i\nj\nk\n+\n+\n is a unit vector 6 Find a vector of magnitude 5 units, and parallel to the resultant of the vectors\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n2\n3\n  and  \n2\na\ni\nj\nk\nb\ni\nj\nk\n=\n+\n\u2212\n= \u2212\n+\nr\nr 7"
    },
    {
        "Chapter": "1",
        "sentence_range": "5546-5549",
        "Text": "6 Find a vector of magnitude 5 units, and parallel to the resultant of the vectors\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n2\n3\n  and  \n2\na\ni\nj\nk\nb\ni\nj\nk\n=\n+\n\u2212\n= \u2212\n+\nr\nr 7 If \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n,  \n2\n3\nand   \n2\na\ni\nj\nk\nb\ni\nj\nk\nc\ni\nj\nk\n= +\n+\n=\n\u2212\n+\n= \u2212\n+\nr\nr\nr\n, find a unit vector parallel\nto the vector 2\n\u2013  \n3\na\nb\nr+c\nr\nr"
    },
    {
        "Chapter": "1",
        "sentence_range": "5547-5550",
        "Text": "Find a vector of magnitude 5 units, and parallel to the resultant of the vectors\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n2\n3\n  and  \n2\na\ni\nj\nk\nb\ni\nj\nk\n=\n+\n\u2212\n= \u2212\n+\nr\nr 7 If \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n,  \n2\n3\nand   \n2\na\ni\nj\nk\nb\ni\nj\nk\nc\ni\nj\nk\n= +\n+\n=\n\u2212\n+\n= \u2212\n+\nr\nr\nr\n, find a unit vector parallel\nto the vector 2\n\u2013  \n3\na\nb\nr+c\nr\nr 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "5548-5551",
        "Text": "7 If \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n,  \n2\n3\nand   \n2\na\ni\nj\nk\nb\ni\nj\nk\nc\ni\nj\nk\n= +\n+\n=\n\u2212\n+\n= \u2212\n+\nr\nr\nr\n, find a unit vector parallel\nto the vector 2\n\u2013  \n3\na\nb\nr+c\nr\nr 8 Show that the points A(1, \u2013 2, \u2013 8), B(5, 0, \u20132) and C(11, 3, 7) are collinear, and\nfind the ratio in which B divides AC"
    },
    {
        "Chapter": "1",
        "sentence_range": "5549-5552",
        "Text": "If \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n,  \n2\n3\nand   \n2\na\ni\nj\nk\nb\ni\nj\nk\nc\ni\nj\nk\n= +\n+\n=\n\u2212\n+\n= \u2212\n+\nr\nr\nr\n, find a unit vector parallel\nto the vector 2\n\u2013  \n3\na\nb\nr+c\nr\nr 8 Show that the points A(1, \u2013 2, \u2013 8), B(5, 0, \u20132) and C(11, 3, 7) are collinear, and\nfind the ratio in which B divides AC 9"
    },
    {
        "Chapter": "1",
        "sentence_range": "5550-5553",
        "Text": "8 Show that the points A(1, \u2013 2, \u2013 8), B(5, 0, \u20132) and C(11, 3, 7) are collinear, and\nfind the ratio in which B divides AC 9 Find the position vector of a point R which divides the line joining two points\nP and Q whose position vectors are (2\n) and ( \u2013 3 )\na\nb\na\nb\n+\nr\nr\nr\nr\n externally in the ratio\n1 : 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "5551-5554",
        "Text": "Show that the points A(1, \u2013 2, \u2013 8), B(5, 0, \u20132) and C(11, 3, 7) are collinear, and\nfind the ratio in which B divides AC 9 Find the position vector of a point R which divides the line joining two points\nP and Q whose position vectors are (2\n) and ( \u2013 3 )\na\nb\na\nb\n+\nr\nr\nr\nr\n externally in the ratio\n1 : 2 Also, show that P is the mid point of the line segment RQ"
    },
    {
        "Chapter": "1",
        "sentence_range": "5552-5555",
        "Text": "9 Find the position vector of a point R which divides the line joining two points\nP and Q whose position vectors are (2\n) and ( \u2013 3 )\na\nb\na\nb\n+\nr\nr\nr\nr\n externally in the ratio\n1 : 2 Also, show that P is the mid point of the line segment RQ 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5553-5556",
        "Text": "Find the position vector of a point R which divides the line joining two points\nP and Q whose position vectors are (2\n) and ( \u2013 3 )\na\nb\na\nb\n+\nr\nr\nr\nr\n externally in the ratio\n1 : 2 Also, show that P is the mid point of the line segment RQ 10 The two adjacent sides of a parallelogram are \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n2\n4\n5  and  \n2\n3\ni\nj\nk\ni\nj\nk\n\u2212\n+\n\u2212\n\u2212"
    },
    {
        "Chapter": "1",
        "sentence_range": "5554-5557",
        "Text": "Also, show that P is the mid point of the line segment RQ 10 The two adjacent sides of a parallelogram are \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n2\n4\n5  and  \n2\n3\ni\nj\nk\ni\nj\nk\n\u2212\n+\n\u2212\n\u2212 Find the unit vector parallel to its diagonal"
    },
    {
        "Chapter": "1",
        "sentence_range": "5555-5558",
        "Text": "10 The two adjacent sides of a parallelogram are \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n2\n4\n5  and  \n2\n3\ni\nj\nk\ni\nj\nk\n\u2212\n+\n\u2212\n\u2212 Find the unit vector parallel to its diagonal Also, find its area"
    },
    {
        "Chapter": "1",
        "sentence_range": "5556-5559",
        "Text": "The two adjacent sides of a parallelogram are \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n2\n4\n5  and  \n2\n3\ni\nj\nk\ni\nj\nk\n\u2212\n+\n\u2212\n\u2212 Find the unit vector parallel to its diagonal Also, find its area 11"
    },
    {
        "Chapter": "1",
        "sentence_range": "5557-5560",
        "Text": "Find the unit vector parallel to its diagonal Also, find its area 11 Show that the direction cosines of a vector equally inclined to the axes OX, OY\nand OZ are 1\n1\n1\n,\n,"
    },
    {
        "Chapter": "1",
        "sentence_range": "5558-5561",
        "Text": "Also, find its area 11 Show that the direction cosines of a vector equally inclined to the axes OX, OY\nand OZ are 1\n1\n1\n,\n, 3\n3\n3\n12"
    },
    {
        "Chapter": "1",
        "sentence_range": "5559-5562",
        "Text": "11 Show that the direction cosines of a vector equally inclined to the axes OX, OY\nand OZ are 1\n1\n1\n,\n, 3\n3\n3\n12 Let \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n4\n2 ,\n3\n2\n7   and  \n2\n4\na\ni\nj\nk b\ni\nj\nk\nc\ni\nj\nk\n= +\n+\n=\n\u2212\n+\n=\n\u2212\n+\nr\nr\nr"
    },
    {
        "Chapter": "1",
        "sentence_range": "5560-5563",
        "Text": "Show that the direction cosines of a vector equally inclined to the axes OX, OY\nand OZ are 1\n1\n1\n,\n, 3\n3\n3\n12 Let \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n4\n2 ,\n3\n2\n7   and  \n2\n4\na\ni\nj\nk b\ni\nj\nk\nc\ni\nj\nk\n= +\n+\n=\n\u2212\n+\n=\n\u2212\n+\nr\nr\nr Find a vector d\nr\nwhich is perpendicular to both \na and \nrb\nr\n, and \n15\nc d\n\u22c5\nrr="
    },
    {
        "Chapter": "1",
        "sentence_range": "5561-5564",
        "Text": "3\n3\n3\n12 Let \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n4\n2 ,\n3\n2\n7   and  \n2\n4\na\ni\nj\nk b\ni\nj\nk\nc\ni\nj\nk\n= +\n+\n=\n\u2212\n+\n=\n\u2212\n+\nr\nr\nr Find a vector d\nr\nwhich is perpendicular to both \na and \nrb\nr\n, and \n15\nc d\n\u22c5\nrr= 13"
    },
    {
        "Chapter": "1",
        "sentence_range": "5562-5565",
        "Text": "Let \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n4\n2 ,\n3\n2\n7   and  \n2\n4\na\ni\nj\nk b\ni\nj\nk\nc\ni\nj\nk\n= +\n+\n=\n\u2212\n+\n=\n\u2212\n+\nr\nr\nr Find a vector d\nr\nwhich is perpendicular to both \na and \nrb\nr\n, and \n15\nc d\n\u22c5\nrr= 13 The scalar product of the vector \n\u02c6\n\u02c6\n\u02c6\ni\nj\nk\n+\n+\n with a unit vector along the sum of\nvectors \n\u02c6\n\u02c6\n\u02c6\n2\n4\n5\ni\nj\nk\n+\n\u2212\n and \n\u02c6\n\u02c6\n2\u02c6\n3\ni\nj\nk\n\u03bb +\n+\n is equal to one"
    },
    {
        "Chapter": "1",
        "sentence_range": "5563-5566",
        "Text": "Find a vector d\nr\nwhich is perpendicular to both \na and \nrb\nr\n, and \n15\nc d\n\u22c5\nrr= 13 The scalar product of the vector \n\u02c6\n\u02c6\n\u02c6\ni\nj\nk\n+\n+\n with a unit vector along the sum of\nvectors \n\u02c6\n\u02c6\n\u02c6\n2\n4\n5\ni\nj\nk\n+\n\u2212\n and \n\u02c6\n\u02c6\n2\u02c6\n3\ni\nj\nk\n\u03bb +\n+\n is equal to one Find the value of \u03bb"
    },
    {
        "Chapter": "1",
        "sentence_range": "5564-5567",
        "Text": "13 The scalar product of the vector \n\u02c6\n\u02c6\n\u02c6\ni\nj\nk\n+\n+\n with a unit vector along the sum of\nvectors \n\u02c6\n\u02c6\n\u02c6\n2\n4\n5\ni\nj\nk\n+\n\u2212\n and \n\u02c6\n\u02c6\n2\u02c6\n3\ni\nj\nk\n\u03bb +\n+\n is equal to one Find the value of \u03bb 14"
    },
    {
        "Chapter": "1",
        "sentence_range": "5565-5568",
        "Text": "The scalar product of the vector \n\u02c6\n\u02c6\n\u02c6\ni\nj\nk\n+\n+\n with a unit vector along the sum of\nvectors \n\u02c6\n\u02c6\n\u02c6\n2\n4\n5\ni\nj\nk\n+\n\u2212\n and \n\u02c6\n\u02c6\n2\u02c6\n3\ni\nj\nk\n\u03bb +\n+\n is equal to one Find the value of \u03bb 14 If ,\na b, c\nr r\nr\n are mutually perpendicular vectors of equal magnitudes, show that\nthe vector a\nb\nc\n+\nr+\nr\nr  is equally inclined to ,\nand\na\nb\nc\nr\nr\nr"
    },
    {
        "Chapter": "1",
        "sentence_range": "5566-5569",
        "Text": "Find the value of \u03bb 14 If ,\na b, c\nr r\nr\n are mutually perpendicular vectors of equal magnitudes, show that\nthe vector a\nb\nc\n+\nr+\nr\nr  is equally inclined to ,\nand\na\nb\nc\nr\nr\nr \u00a9 NCERT\nnot to be republished\nVECTOR ALGEBRA\n459\n15"
    },
    {
        "Chapter": "1",
        "sentence_range": "5567-5570",
        "Text": "14 If ,\na b, c\nr r\nr\n are mutually perpendicular vectors of equal magnitudes, show that\nthe vector a\nb\nc\n+\nr+\nr\nr  is equally inclined to ,\nand\na\nb\nc\nr\nr\nr \u00a9 NCERT\nnot to be republished\nVECTOR ALGEBRA\n459\n15 Prove that  \n2\n2\n(\n) (\n) |\n|\n| |\na\nb\na\nb\na\nb\n+\n\u22c5\n+\n=\n+\nr\nr\nr\nr\nr\nr\n, if and only if ,a b\nrr\n are perpendicular,\ngiven \n0,\n0\na\nb\n\u2260\nrr\u2260\nr\nr"
    },
    {
        "Chapter": "1",
        "sentence_range": "5568-5571",
        "Text": "If ,\na b, c\nr r\nr\n are mutually perpendicular vectors of equal magnitudes, show that\nthe vector a\nb\nc\n+\nr+\nr\nr  is equally inclined to ,\nand\na\nb\nc\nr\nr\nr \u00a9 NCERT\nnot to be republished\nVECTOR ALGEBRA\n459\n15 Prove that  \n2\n2\n(\n) (\n) |\n|\n| |\na\nb\na\nb\na\nb\n+\n\u22c5\n+\n=\n+\nr\nr\nr\nr\nr\nr\n, if and only if ,a b\nrr\n are perpendicular,\ngiven \n0,\n0\na\nb\n\u2260\nrr\u2260\nr\nr Choose the correct answer in Exercises 16 to 19"
    },
    {
        "Chapter": "1",
        "sentence_range": "5569-5572",
        "Text": "\u00a9 NCERT\nnot to be republished\nVECTOR ALGEBRA\n459\n15 Prove that  \n2\n2\n(\n) (\n) |\n|\n| |\na\nb\na\nb\na\nb\n+\n\u22c5\n+\n=\n+\nr\nr\nr\nr\nr\nr\n, if and only if ,a b\nrr\n are perpendicular,\ngiven \n0,\n0\na\nb\n\u2260\nrr\u2260\nr\nr Choose the correct answer in Exercises 16 to 19 16"
    },
    {
        "Chapter": "1",
        "sentence_range": "5570-5573",
        "Text": "Prove that  \n2\n2\n(\n) (\n) |\n|\n| |\na\nb\na\nb\na\nb\n+\n\u22c5\n+\n=\n+\nr\nr\nr\nr\nr\nr\n, if and only if ,a b\nrr\n are perpendicular,\ngiven \n0,\n0\na\nb\n\u2260\nrr\u2260\nr\nr Choose the correct answer in Exercises 16 to 19 16 If \u03b8 is the angle between two vectors \na and \nrb\nr\n, then \n0\na b\n\u22c5\nrr\u2265\n only when\n(A) 0\n< \u03b8 <\u03c02\n(B) 0\n\u03c02\n\u2264 \u03b8 \u2264\n(C) 0 < \u03b8 < \u03c0\n(D) 0 \u2264 \u03b8 \u2264 \u03c0\n17"
    },
    {
        "Chapter": "1",
        "sentence_range": "5571-5574",
        "Text": "Choose the correct answer in Exercises 16 to 19 16 If \u03b8 is the angle between two vectors \na and \nrb\nr\n, then \n0\na b\n\u22c5\nrr\u2265\n only when\n(A) 0\n< \u03b8 <\u03c02\n(B) 0\n\u03c02\n\u2264 \u03b8 \u2264\n(C) 0 < \u03b8 < \u03c0\n(D) 0 \u2264 \u03b8 \u2264 \u03c0\n17 Let   and \na\nrb\nr\n be two unit vectors and \u03b8 is the angle between them"
    },
    {
        "Chapter": "1",
        "sentence_range": "5572-5575",
        "Text": "16 If \u03b8 is the angle between two vectors \na and \nrb\nr\n, then \n0\na b\n\u22c5\nrr\u2265\n only when\n(A) 0\n< \u03b8 <\u03c02\n(B) 0\n\u03c02\n\u2264 \u03b8 \u2264\n(C) 0 < \u03b8 < \u03c0\n(D) 0 \u2264 \u03b8 \u2264 \u03c0\n17 Let   and \na\nrb\nr\n be two unit vectors and \u03b8 is the angle between them Then a\n+b\nr\nr\nis a unit vector if\n(A)\n\u03b8 =\u03c04\n(B)\n\u03b8 =\u03c03\n(C)\n\u03b8 =\u03c02\n(D)\n\u03c032\n\u03b8 =\n18"
    },
    {
        "Chapter": "1",
        "sentence_range": "5573-5576",
        "Text": "If \u03b8 is the angle between two vectors \na and \nrb\nr\n, then \n0\na b\n\u22c5\nrr\u2265\n only when\n(A) 0\n< \u03b8 <\u03c02\n(B) 0\n\u03c02\n\u2264 \u03b8 \u2264\n(C) 0 < \u03b8 < \u03c0\n(D) 0 \u2264 \u03b8 \u2264 \u03c0\n17 Let   and \na\nrb\nr\n be two unit vectors and \u03b8 is the angle between them Then a\n+b\nr\nr\nis a unit vector if\n(A)\n\u03b8 =\u03c04\n(B)\n\u03b8 =\u03c03\n(C)\n\u03b8 =\u03c02\n(D)\n\u03c032\n\u03b8 =\n18 The value of \n\u02c6\n\u02c6\n\u02c6\n\u02c6 \u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6"
    },
    {
        "Chapter": "1",
        "sentence_range": "5574-5577",
        "Text": "Let   and \na\nrb\nr\n be two unit vectors and \u03b8 is the angle between them Then a\n+b\nr\nr\nis a unit vector if\n(A)\n\u03b8 =\u03c04\n(B)\n\u03b8 =\u03c03\n(C)\n\u03b8 =\u03c02\n(D)\n\u03c032\n\u03b8 =\n18 The value of \n\u02c6\n\u02c6\n\u02c6\n\u02c6 \u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6 (\n)\n(\n)\n(\n)\ni\nj\nk\nj\ni\nk\nk\ni\nj\n \n \n \n \n \n \n \n is\n(A) 0\n(B) \u20131\n(C) 1\n(D) 3\n19"
    },
    {
        "Chapter": "1",
        "sentence_range": "5575-5578",
        "Text": "Then a\n+b\nr\nr\nis a unit vector if\n(A)\n\u03b8 =\u03c04\n(B)\n\u03b8 =\u03c03\n(C)\n\u03b8 =\u03c02\n(D)\n\u03c032\n\u03b8 =\n18 The value of \n\u02c6\n\u02c6\n\u02c6\n\u02c6 \u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6 (\n)\n(\n)\n(\n)\ni\nj\nk\nj\ni\nk\nk\ni\nj\n \n \n \n \n \n \n \n is\n(A) 0\n(B) \u20131\n(C) 1\n(D) 3\n19 If \u03b8 is the angle between any two vectors   and \na\nrb\nr\n, then |\n| |\n|\na b\na b\n\u22c5\n=\n\u00d7\nr\nr\nr\nr\n when\n\u03b8 is equal to\n(A) 0\n(B)\n\u03c04\n(C)\n\u03c02\n(D) \u03c0\nSummary\n\ufffd Position vector of a point P(x, y, z) is given as \n\u02c6\n\u02c6\n\u02c6\nOP(\nr)\nxi\nyj\nzk\n=\n=\n+\n+\nuuur\nr\n, and its\nmagnitude by \n2\n2\n2\nx\ny\nz\n+\n+"
    },
    {
        "Chapter": "1",
        "sentence_range": "5576-5579",
        "Text": "The value of \n\u02c6\n\u02c6\n\u02c6\n\u02c6 \u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6 (\n)\n(\n)\n(\n)\ni\nj\nk\nj\ni\nk\nk\ni\nj\n \n \n \n \n \n \n \n is\n(A) 0\n(B) \u20131\n(C) 1\n(D) 3\n19 If \u03b8 is the angle between any two vectors   and \na\nrb\nr\n, then |\n| |\n|\na b\na b\n\u22c5\n=\n\u00d7\nr\nr\nr\nr\n when\n\u03b8 is equal to\n(A) 0\n(B)\n\u03c04\n(C)\n\u03c02\n(D) \u03c0\nSummary\n\ufffd Position vector of a point P(x, y, z) is given as \n\u02c6\n\u02c6\n\u02c6\nOP(\nr)\nxi\nyj\nzk\n=\n=\n+\n+\nuuur\nr\n, and its\nmagnitude by \n2\n2\n2\nx\ny\nz\n+\n+ \ufffd The scalar components of a vector are its direction ratios, and represent its\nprojections along the respective axes"
    },
    {
        "Chapter": "1",
        "sentence_range": "5577-5580",
        "Text": "(\n)\n(\n)\n(\n)\ni\nj\nk\nj\ni\nk\nk\ni\nj\n \n \n \n \n \n \n \n is\n(A) 0\n(B) \u20131\n(C) 1\n(D) 3\n19 If \u03b8 is the angle between any two vectors   and \na\nrb\nr\n, then |\n| |\n|\na b\na b\n\u22c5\n=\n\u00d7\nr\nr\nr\nr\n when\n\u03b8 is equal to\n(A) 0\n(B)\n\u03c04\n(C)\n\u03c02\n(D) \u03c0\nSummary\n\ufffd Position vector of a point P(x, y, z) is given as \n\u02c6\n\u02c6\n\u02c6\nOP(\nr)\nxi\nyj\nzk\n=\n=\n+\n+\nuuur\nr\n, and its\nmagnitude by \n2\n2\n2\nx\ny\nz\n+\n+ \ufffd The scalar components of a vector are its direction ratios, and represent its\nprojections along the respective axes \ufffd The magnitude (r), direction ratios (a, b, c) and direction cosines (l, m, n) of\nany vector are related as:\n,\n,\na\nb\nc\nl\nm\nn\nr\nr\nr\n=\n=\n=\n\ufffd The vector sum of the three sides of a triangle taken in order is 0\nr"
    },
    {
        "Chapter": "1",
        "sentence_range": "5578-5581",
        "Text": "If \u03b8 is the angle between any two vectors   and \na\nrb\nr\n, then |\n| |\n|\na b\na b\n\u22c5\n=\n\u00d7\nr\nr\nr\nr\n when\n\u03b8 is equal to\n(A) 0\n(B)\n\u03c04\n(C)\n\u03c02\n(D) \u03c0\nSummary\n\ufffd Position vector of a point P(x, y, z) is given as \n\u02c6\n\u02c6\n\u02c6\nOP(\nr)\nxi\nyj\nzk\n=\n=\n+\n+\nuuur\nr\n, and its\nmagnitude by \n2\n2\n2\nx\ny\nz\n+\n+ \ufffd The scalar components of a vector are its direction ratios, and represent its\nprojections along the respective axes \ufffd The magnitude (r), direction ratios (a, b, c) and direction cosines (l, m, n) of\nany vector are related as:\n,\n,\na\nb\nc\nl\nm\nn\nr\nr\nr\n=\n=\n=\n\ufffd The vector sum of the three sides of a triangle taken in order is 0\nr \u00a9 NCERT\nnot to be republished\n MATHEMATICS\n460\n\ufffd The vector sum of two coinitial vectors is given by the diagonal of the\nparallelogram whose adjacent sides are the given vectors"
    },
    {
        "Chapter": "1",
        "sentence_range": "5579-5582",
        "Text": "\ufffd The scalar components of a vector are its direction ratios, and represent its\nprojections along the respective axes \ufffd The magnitude (r), direction ratios (a, b, c) and direction cosines (l, m, n) of\nany vector are related as:\n,\n,\na\nb\nc\nl\nm\nn\nr\nr\nr\n=\n=\n=\n\ufffd The vector sum of the three sides of a triangle taken in order is 0\nr \u00a9 NCERT\nnot to be republished\n MATHEMATICS\n460\n\ufffd The vector sum of two coinitial vectors is given by the diagonal of the\nparallelogram whose adjacent sides are the given vectors \ufffd The multiplication of a given vector by a scalar \u03bb, changes the magnitude of\nthe vector by the multiple |\u03bb|, and keeps the direction same (or makes it\nopposite) according as the value of \u03bb is positive (or negative)"
    },
    {
        "Chapter": "1",
        "sentence_range": "5580-5583",
        "Text": "\ufffd The magnitude (r), direction ratios (a, b, c) and direction cosines (l, m, n) of\nany vector are related as:\n,\n,\na\nb\nc\nl\nm\nn\nr\nr\nr\n=\n=\n=\n\ufffd The vector sum of the three sides of a triangle taken in order is 0\nr \u00a9 NCERT\nnot to be republished\n MATHEMATICS\n460\n\ufffd The vector sum of two coinitial vectors is given by the diagonal of the\nparallelogram whose adjacent sides are the given vectors \ufffd The multiplication of a given vector by a scalar \u03bb, changes the magnitude of\nthe vector by the multiple |\u03bb|, and keeps the direction same (or makes it\nopposite) according as the value of \u03bb is positive (or negative) \ufffd For a given vector ar , the vector \u02c6\n|\na|\na\n=a\nr\nr  gives the unit vector in the direction\nof ar"
    },
    {
        "Chapter": "1",
        "sentence_range": "5581-5584",
        "Text": "\u00a9 NCERT\nnot to be republished\n MATHEMATICS\n460\n\ufffd The vector sum of two coinitial vectors is given by the diagonal of the\nparallelogram whose adjacent sides are the given vectors \ufffd The multiplication of a given vector by a scalar \u03bb, changes the magnitude of\nthe vector by the multiple |\u03bb|, and keeps the direction same (or makes it\nopposite) according as the value of \u03bb is positive (or negative) \ufffd For a given vector ar , the vector \u02c6\n|\na|\na\n=a\nr\nr  gives the unit vector in the direction\nof ar \ufffd The position vector of a point R dividing a line segment joining the points\nP and Q whose position vectors are \naand\nrb\nr\n respectively, in the ratio m : n\n(i)\ninternally, is given by na\nmmb\n+n\n+\nr\nr"
    },
    {
        "Chapter": "1",
        "sentence_range": "5582-5585",
        "Text": "\ufffd The multiplication of a given vector by a scalar \u03bb, changes the magnitude of\nthe vector by the multiple |\u03bb|, and keeps the direction same (or makes it\nopposite) according as the value of \u03bb is positive (or negative) \ufffd For a given vector ar , the vector \u02c6\n|\na|\na\n=a\nr\nr  gives the unit vector in the direction\nof ar \ufffd The position vector of a point R dividing a line segment joining the points\nP and Q whose position vectors are \naand\nrb\nr\n respectively, in the ratio m : n\n(i)\ninternally, is given by na\nmmb\n+n\n+\nr\nr (ii)\nexternally, is given by mb\nna\nm\n\u2212n\n\u2212\nr\nr"
    },
    {
        "Chapter": "1",
        "sentence_range": "5583-5586",
        "Text": "\ufffd For a given vector ar , the vector \u02c6\n|\na|\na\n=a\nr\nr  gives the unit vector in the direction\nof ar \ufffd The position vector of a point R dividing a line segment joining the points\nP and Q whose position vectors are \naand\nrb\nr\n respectively, in the ratio m : n\n(i)\ninternally, is given by na\nmmb\n+n\n+\nr\nr (ii)\nexternally, is given by mb\nna\nm\n\u2212n\n\u2212\nr\nr \ufffd The scalar product of two given vectors \naand\nrb\nr\n having angle \u03b8 between\nthem is defined as\n|\n||\n| cos\na b\na b\n\u22c5\n=\n\u03b8\nr\nr\nr\nr"
    },
    {
        "Chapter": "1",
        "sentence_range": "5584-5587",
        "Text": "\ufffd The position vector of a point R dividing a line segment joining the points\nP and Q whose position vectors are \naand\nrb\nr\n respectively, in the ratio m : n\n(i)\ninternally, is given by na\nmmb\n+n\n+\nr\nr (ii)\nexternally, is given by mb\nna\nm\n\u2212n\n\u2212\nr\nr \ufffd The scalar product of two given vectors \naand\nrb\nr\n having angle \u03b8 between\nthem is defined as\n|\n||\n| cos\na b\na b\n\u22c5\n=\n\u03b8\nr\nr\nr\nr Also, when a b\n\u22c5\nrr\n is given, the angle \u2018\u03b8\u2019 between the vectors \naand\nrb\nr\n may be\ndetermined by\ncos\u03b8 = |\n||\na b|\na b\n\u22c5\nrr\nr\nr\n\ufffd If \u03b8 is the angle between two vectors \naand\nrb\nr\n, then their cross product is\ngiven as\na\nb\n\u00d7\nr\nr\n=\n\u02c6\n|\n||\na b|sin\nn\n\u03b8\nr\nr\nwhere \u02c6n is a unit vector perpendicular to the plane containing \naand\nrb\nr"
    },
    {
        "Chapter": "1",
        "sentence_range": "5585-5588",
        "Text": "(ii)\nexternally, is given by mb\nna\nm\n\u2212n\n\u2212\nr\nr \ufffd The scalar product of two given vectors \naand\nrb\nr\n having angle \u03b8 between\nthem is defined as\n|\n||\n| cos\na b\na b\n\u22c5\n=\n\u03b8\nr\nr\nr\nr Also, when a b\n\u22c5\nrr\n is given, the angle \u2018\u03b8\u2019 between the vectors \naand\nrb\nr\n may be\ndetermined by\ncos\u03b8 = |\n||\na b|\na b\n\u22c5\nrr\nr\nr\n\ufffd If \u03b8 is the angle between two vectors \naand\nrb\nr\n, then their cross product is\ngiven as\na\nb\n\u00d7\nr\nr\n=\n\u02c6\n|\n||\na b|sin\nn\n\u03b8\nr\nr\nwhere \u02c6n is a unit vector perpendicular to the plane containing \naand\nrb\nr Such\nthat \n\u02c6\n, ,\nr\nra b n form right handed system of coordinate axes"
    },
    {
        "Chapter": "1",
        "sentence_range": "5586-5589",
        "Text": "\ufffd The scalar product of two given vectors \naand\nrb\nr\n having angle \u03b8 between\nthem is defined as\n|\n||\n| cos\na b\na b\n\u22c5\n=\n\u03b8\nr\nr\nr\nr Also, when a b\n\u22c5\nrr\n is given, the angle \u2018\u03b8\u2019 between the vectors \naand\nrb\nr\n may be\ndetermined by\ncos\u03b8 = |\n||\na b|\na b\n\u22c5\nrr\nr\nr\n\ufffd If \u03b8 is the angle between two vectors \naand\nrb\nr\n, then their cross product is\ngiven as\na\nb\n\u00d7\nr\nr\n=\n\u02c6\n|\n||\na b|sin\nn\n\u03b8\nr\nr\nwhere \u02c6n is a unit vector perpendicular to the plane containing \naand\nrb\nr Such\nthat \n\u02c6\n, ,\nr\nra b n form right handed system of coordinate axes \ufffd If we have two vectors \naand\nrb\nr\n, given in component form as\n1\n2\n3 \u02c6\n\u02c6\n\u02c6\na\na i\na j\na k\n=\n+\n+\nr\n and \n1\n2\n3 \u02c6\n\u02c6\n\u02c6\nb\nb i\nb j\nb k\n=\n+\n+\nr\n and \u03bb any scalar,\n\u00a9 NCERT\nnot to be republished\nVECTOR ALGEBRA\n461\nthen\na\nb\n+\nr\nr\n =\n1\n1\n2\n2\n3\n3 \u02c6\n\u02c6\n\u02c6\n(\n)\n(\n)\n(\n)\na\nb i\na\nb\nj\na\nb\nk\n+\n+\n+\n+\n+\n;\na\n\u03bbr  =\n1\n2\n3 \u02c6\n\u02c6\n\u02c6\n(\n)\n(\n)\n(\n)\na i\na\nj\na k\n\u03bb\n+ \u03bb\n+ \u03bb\n;\nrr"
    },
    {
        "Chapter": "1",
        "sentence_range": "5587-5590",
        "Text": "Also, when a b\n\u22c5\nrr\n is given, the angle \u2018\u03b8\u2019 between the vectors \naand\nrb\nr\n may be\ndetermined by\ncos\u03b8 = |\n||\na b|\na b\n\u22c5\nrr\nr\nr\n\ufffd If \u03b8 is the angle between two vectors \naand\nrb\nr\n, then their cross product is\ngiven as\na\nb\n\u00d7\nr\nr\n=\n\u02c6\n|\n||\na b|sin\nn\n\u03b8\nr\nr\nwhere \u02c6n is a unit vector perpendicular to the plane containing \naand\nrb\nr Such\nthat \n\u02c6\n, ,\nr\nra b n form right handed system of coordinate axes \ufffd If we have two vectors \naand\nrb\nr\n, given in component form as\n1\n2\n3 \u02c6\n\u02c6\n\u02c6\na\na i\na j\na k\n=\n+\n+\nr\n and \n1\n2\n3 \u02c6\n\u02c6\n\u02c6\nb\nb i\nb j\nb k\n=\n+\n+\nr\n and \u03bb any scalar,\n\u00a9 NCERT\nnot to be republished\nVECTOR ALGEBRA\n461\nthen\na\nb\n+\nr\nr\n =\n1\n1\n2\n2\n3\n3 \u02c6\n\u02c6\n\u02c6\n(\n)\n(\n)\n(\n)\na\nb i\na\nb\nj\na\nb\nk\n+\n+\n+\n+\n+\n;\na\n\u03bbr  =\n1\n2\n3 \u02c6\n\u02c6\n\u02c6\n(\n)\n(\n)\n(\n)\na i\na\nj\na k\n\u03bb\n+ \u03bb\n+ \u03bb\n;\nrr a b\n =\n1 1\n2 2\n3 3\na b\na b\na b\n+\n+\n;\nand\na\nb\n\u00d7\nr\nr\n =\n1\n1\n1\n2\n2\n2\n\u02c6\n\u02c6\n\u02c6"
    },
    {
        "Chapter": "1",
        "sentence_range": "5588-5591",
        "Text": "Such\nthat \n\u02c6\n, ,\nr\nra b n form right handed system of coordinate axes \ufffd If we have two vectors \naand\nrb\nr\n, given in component form as\n1\n2\n3 \u02c6\n\u02c6\n\u02c6\na\na i\na j\na k\n=\n+\n+\nr\n and \n1\n2\n3 \u02c6\n\u02c6\n\u02c6\nb\nb i\nb j\nb k\n=\n+\n+\nr\n and \u03bb any scalar,\n\u00a9 NCERT\nnot to be republished\nVECTOR ALGEBRA\n461\nthen\na\nb\n+\nr\nr\n =\n1\n1\n2\n2\n3\n3 \u02c6\n\u02c6\n\u02c6\n(\n)\n(\n)\n(\n)\na\nb i\na\nb\nj\na\nb\nk\n+\n+\n+\n+\n+\n;\na\n\u03bbr  =\n1\n2\n3 \u02c6\n\u02c6\n\u02c6\n(\n)\n(\n)\n(\n)\na i\na\nj\na k\n\u03bb\n+ \u03bb\n+ \u03bb\n;\nrr a b\n =\n1 1\n2 2\n3 3\na b\na b\na b\n+\n+\n;\nand\na\nb\n\u00d7\nr\nr\n =\n1\n1\n1\n2\n2\n2\n\u02c6\n\u02c6\n\u02c6 i\nj\nk\na\nb\nc\na\nb\nc\nHistorical Note\nThe word vector has been derived from a Latin word vectus, which means\n\u201cto carry\u201d"
    },
    {
        "Chapter": "1",
        "sentence_range": "5589-5592",
        "Text": "\ufffd If we have two vectors \naand\nrb\nr\n, given in component form as\n1\n2\n3 \u02c6\n\u02c6\n\u02c6\na\na i\na j\na k\n=\n+\n+\nr\n and \n1\n2\n3 \u02c6\n\u02c6\n\u02c6\nb\nb i\nb j\nb k\n=\n+\n+\nr\n and \u03bb any scalar,\n\u00a9 NCERT\nnot to be republished\nVECTOR ALGEBRA\n461\nthen\na\nb\n+\nr\nr\n =\n1\n1\n2\n2\n3\n3 \u02c6\n\u02c6\n\u02c6\n(\n)\n(\n)\n(\n)\na\nb i\na\nb\nj\na\nb\nk\n+\n+\n+\n+\n+\n;\na\n\u03bbr  =\n1\n2\n3 \u02c6\n\u02c6\n\u02c6\n(\n)\n(\n)\n(\n)\na i\na\nj\na k\n\u03bb\n+ \u03bb\n+ \u03bb\n;\nrr a b\n =\n1 1\n2 2\n3 3\na b\na b\na b\n+\n+\n;\nand\na\nb\n\u00d7\nr\nr\n =\n1\n1\n1\n2\n2\n2\n\u02c6\n\u02c6\n\u02c6 i\nj\nk\na\nb\nc\na\nb\nc\nHistorical Note\nThe word vector has been derived from a Latin word vectus, which means\n\u201cto carry\u201d The germinal ideas of modern vector theory date from around 1800\nwhen Caspar Wessel (1745-1818) and Jean Robert Argand (1768-1822) described\nthat how a complex number a + ib could be given a geometric interpretation with\nthe help of a directed line segment in a coordinate plane"
    },
    {
        "Chapter": "1",
        "sentence_range": "5590-5593",
        "Text": "a b\n =\n1 1\n2 2\n3 3\na b\na b\na b\n+\n+\n;\nand\na\nb\n\u00d7\nr\nr\n =\n1\n1\n1\n2\n2\n2\n\u02c6\n\u02c6\n\u02c6 i\nj\nk\na\nb\nc\na\nb\nc\nHistorical Note\nThe word vector has been derived from a Latin word vectus, which means\n\u201cto carry\u201d The germinal ideas of modern vector theory date from around 1800\nwhen Caspar Wessel (1745-1818) and Jean Robert Argand (1768-1822) described\nthat how a complex number a + ib could be given a geometric interpretation with\nthe help of a directed line segment in a coordinate plane William Rowen Hamilton\n(1805-1865) an Irish mathematician was the first to use the term vector for a\ndirected line segment in his book Lectures on Quaternions (1853)"
    },
    {
        "Chapter": "1",
        "sentence_range": "5591-5594",
        "Text": "i\nj\nk\na\nb\nc\na\nb\nc\nHistorical Note\nThe word vector has been derived from a Latin word vectus, which means\n\u201cto carry\u201d The germinal ideas of modern vector theory date from around 1800\nwhen Caspar Wessel (1745-1818) and Jean Robert Argand (1768-1822) described\nthat how a complex number a + ib could be given a geometric interpretation with\nthe help of a directed line segment in a coordinate plane William Rowen Hamilton\n(1805-1865) an Irish mathematician was the first to use the term vector for a\ndirected line segment in his book Lectures on Quaternions (1853) Hamilton\u2019s\nmethod of quaternions (an ordered set of four real numbers given as:\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6 \u02c6\n, , ,\na\nbi\ncj\ndk i\nj k\n+\n+\n+\n following certain algebraic rules) was a solution to the\nproblem of multiplying vectors in three dimensional space"
    },
    {
        "Chapter": "1",
        "sentence_range": "5592-5595",
        "Text": "The germinal ideas of modern vector theory date from around 1800\nwhen Caspar Wessel (1745-1818) and Jean Robert Argand (1768-1822) described\nthat how a complex number a + ib could be given a geometric interpretation with\nthe help of a directed line segment in a coordinate plane William Rowen Hamilton\n(1805-1865) an Irish mathematician was the first to use the term vector for a\ndirected line segment in his book Lectures on Quaternions (1853) Hamilton\u2019s\nmethod of quaternions (an ordered set of four real numbers given as:\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6 \u02c6\n, , ,\na\nbi\ncj\ndk i\nj k\n+\n+\n+\n following certain algebraic rules) was a solution to the\nproblem of multiplying vectors in three dimensional space Though, we must\nmention here that in practice, the idea of vector concept and their addition was\nknown much earlier ever since the time of Aristotle (384-322 B"
    },
    {
        "Chapter": "1",
        "sentence_range": "5593-5596",
        "Text": "William Rowen Hamilton\n(1805-1865) an Irish mathematician was the first to use the term vector for a\ndirected line segment in his book Lectures on Quaternions (1853) Hamilton\u2019s\nmethod of quaternions (an ordered set of four real numbers given as:\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6 \u02c6\n, , ,\na\nbi\ncj\ndk i\nj k\n+\n+\n+\n following certain algebraic rules) was a solution to the\nproblem of multiplying vectors in three dimensional space Though, we must\nmention here that in practice, the idea of vector concept and their addition was\nknown much earlier ever since the time of Aristotle (384-322 B C"
    },
    {
        "Chapter": "1",
        "sentence_range": "5594-5597",
        "Text": "Hamilton\u2019s\nmethod of quaternions (an ordered set of four real numbers given as:\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6 \u02c6\n, , ,\na\nbi\ncj\ndk i\nj k\n+\n+\n+\n following certain algebraic rules) was a solution to the\nproblem of multiplying vectors in three dimensional space Though, we must\nmention here that in practice, the idea of vector concept and their addition was\nknown much earlier ever since the time of Aristotle (384-322 B C ), a Greek\nphilosopher, and pupil of Plato (427-348 B"
    },
    {
        "Chapter": "1",
        "sentence_range": "5595-5598",
        "Text": "Though, we must\nmention here that in practice, the idea of vector concept and their addition was\nknown much earlier ever since the time of Aristotle (384-322 B C ), a Greek\nphilosopher, and pupil of Plato (427-348 B C"
    },
    {
        "Chapter": "1",
        "sentence_range": "5596-5599",
        "Text": "C ), a Greek\nphilosopher, and pupil of Plato (427-348 B C )"
    },
    {
        "Chapter": "1",
        "sentence_range": "5597-5600",
        "Text": "), a Greek\nphilosopher, and pupil of Plato (427-348 B C ) That time it was supposed to be\nknown that the combined action of two or more forces could be seen by adding\nthem according to parallelogram law"
    },
    {
        "Chapter": "1",
        "sentence_range": "5598-5601",
        "Text": "C ) That time it was supposed to be\nknown that the combined action of two or more forces could be seen by adding\nthem according to parallelogram law The correct law for the composition of\nforces, that forces add vectorially, had been discovered in the case of perpendicular\nforces by Stevin-Simon (1548-1620)"
    },
    {
        "Chapter": "1",
        "sentence_range": "5599-5602",
        "Text": ") That time it was supposed to be\nknown that the combined action of two or more forces could be seen by adding\nthem according to parallelogram law The correct law for the composition of\nforces, that forces add vectorially, had been discovered in the case of perpendicular\nforces by Stevin-Simon (1548-1620) In 1586 A"
    },
    {
        "Chapter": "1",
        "sentence_range": "5600-5603",
        "Text": "That time it was supposed to be\nknown that the combined action of two or more forces could be seen by adding\nthem according to parallelogram law The correct law for the composition of\nforces, that forces add vectorially, had been discovered in the case of perpendicular\nforces by Stevin-Simon (1548-1620) In 1586 A D"
    },
    {
        "Chapter": "1",
        "sentence_range": "5601-5604",
        "Text": "The correct law for the composition of\nforces, that forces add vectorially, had been discovered in the case of perpendicular\nforces by Stevin-Simon (1548-1620) In 1586 A D , he analysed the principle of\ngeometric addition of forces in his treatise DeBeghinselen der Weeghconst\n(\u201cPrinciples of the Art of Weighing\u201d), which caused a major breakthrough in the\ndevelopment of mechanics"
    },
    {
        "Chapter": "1",
        "sentence_range": "5602-5605",
        "Text": "In 1586 A D , he analysed the principle of\ngeometric addition of forces in his treatise DeBeghinselen der Weeghconst\n(\u201cPrinciples of the Art of Weighing\u201d), which caused a major breakthrough in the\ndevelopment of mechanics But it took another 200 years for the general concept\nof vectors to form"
    },
    {
        "Chapter": "1",
        "sentence_range": "5603-5606",
        "Text": "D , he analysed the principle of\ngeometric addition of forces in his treatise DeBeghinselen der Weeghconst\n(\u201cPrinciples of the Art of Weighing\u201d), which caused a major breakthrough in the\ndevelopment of mechanics But it took another 200 years for the general concept\nof vectors to form In the 1880, Josaih Willard Gibbs (1839-1903), an American physicist\nand mathematician, and Oliver Heaviside (1850-1925), an English engineer, created\nwhat we now know as vector analysis, essentially by separating the real (scalar)\n\u00a9 NCERT\nnot to be republished\n MATHEMATICS\n462\npart of quaternion from its imaginary (vector) part"
    },
    {
        "Chapter": "1",
        "sentence_range": "5604-5607",
        "Text": ", he analysed the principle of\ngeometric addition of forces in his treatise DeBeghinselen der Weeghconst\n(\u201cPrinciples of the Art of Weighing\u201d), which caused a major breakthrough in the\ndevelopment of mechanics But it took another 200 years for the general concept\nof vectors to form In the 1880, Josaih Willard Gibbs (1839-1903), an American physicist\nand mathematician, and Oliver Heaviside (1850-1925), an English engineer, created\nwhat we now know as vector analysis, essentially by separating the real (scalar)\n\u00a9 NCERT\nnot to be republished\n MATHEMATICS\n462\npart of quaternion from its imaginary (vector) part In 1881 and 1884, Gibbs\nprinted a treatise entitled Element of Vector Analysis"
    },
    {
        "Chapter": "1",
        "sentence_range": "5605-5608",
        "Text": "But it took another 200 years for the general concept\nof vectors to form In the 1880, Josaih Willard Gibbs (1839-1903), an American physicist\nand mathematician, and Oliver Heaviside (1850-1925), an English engineer, created\nwhat we now know as vector analysis, essentially by separating the real (scalar)\n\u00a9 NCERT\nnot to be republished\n MATHEMATICS\n462\npart of quaternion from its imaginary (vector) part In 1881 and 1884, Gibbs\nprinted a treatise entitled Element of Vector Analysis This book gave a systematic\nand concise account of vectors"
    },
    {
        "Chapter": "1",
        "sentence_range": "5606-5609",
        "Text": "In the 1880, Josaih Willard Gibbs (1839-1903), an American physicist\nand mathematician, and Oliver Heaviside (1850-1925), an English engineer, created\nwhat we now know as vector analysis, essentially by separating the real (scalar)\n\u00a9 NCERT\nnot to be republished\n MATHEMATICS\n462\npart of quaternion from its imaginary (vector) part In 1881 and 1884, Gibbs\nprinted a treatise entitled Element of Vector Analysis This book gave a systematic\nand concise account of vectors However, much of the credit for demonstrating\nthe applications of vectors is due to the D"
    },
    {
        "Chapter": "1",
        "sentence_range": "5607-5610",
        "Text": "In 1881 and 1884, Gibbs\nprinted a treatise entitled Element of Vector Analysis This book gave a systematic\nand concise account of vectors However, much of the credit for demonstrating\nthe applications of vectors is due to the D Heaviside and P"
    },
    {
        "Chapter": "1",
        "sentence_range": "5608-5611",
        "Text": "This book gave a systematic\nand concise account of vectors However, much of the credit for demonstrating\nthe applications of vectors is due to the D Heaviside and P G"
    },
    {
        "Chapter": "1",
        "sentence_range": "5609-5612",
        "Text": "However, much of the credit for demonstrating\nthe applications of vectors is due to the D Heaviside and P G Tait (1831-1901)\nwho contributed significantly to this subject"
    },
    {
        "Chapter": "1",
        "sentence_range": "5610-5613",
        "Text": "Heaviside and P G Tait (1831-1901)\nwho contributed significantly to this subject \u2014\ufffd\n\ufffd\n\ufffd\n\ufffd\n\ufffd\u2014\n\u00a9 NCERT\nnot to be republished\nTHREE D IMENSIONAL G EOMETRY\n463\n\ufffd The moving power of mathematical invention is not\nreasoning but imagination"
    },
    {
        "Chapter": "1",
        "sentence_range": "5611-5614",
        "Text": "G Tait (1831-1901)\nwho contributed significantly to this subject \u2014\ufffd\n\ufffd\n\ufffd\n\ufffd\n\ufffd\u2014\n\u00a9 NCERT\nnot to be republished\nTHREE D IMENSIONAL G EOMETRY\n463\n\ufffd The moving power of mathematical invention is not\nreasoning but imagination \u2013 A"
    },
    {
        "Chapter": "1",
        "sentence_range": "5612-5615",
        "Text": "Tait (1831-1901)\nwho contributed significantly to this subject \u2014\ufffd\n\ufffd\n\ufffd\n\ufffd\n\ufffd\u2014\n\u00a9 NCERT\nnot to be republished\nTHREE D IMENSIONAL G EOMETRY\n463\n\ufffd The moving power of mathematical invention is not\nreasoning but imagination \u2013 A DEMORGAN \ufffd\n11"
    },
    {
        "Chapter": "1",
        "sentence_range": "5613-5616",
        "Text": "\u2014\ufffd\n\ufffd\n\ufffd\n\ufffd\n\ufffd\u2014\n\u00a9 NCERT\nnot to be republished\nTHREE D IMENSIONAL G EOMETRY\n463\n\ufffd The moving power of mathematical invention is not\nreasoning but imagination \u2013 A DEMORGAN \ufffd\n11 1  Introduction\nIn Class XI, while studying Analytical Geometry in two\ndimensions, and the introduction to three dimensional\ngeometry, we confined to the Cartesian methods only"
    },
    {
        "Chapter": "1",
        "sentence_range": "5614-5617",
        "Text": "\u2013 A DEMORGAN \ufffd\n11 1  Introduction\nIn Class XI, while studying Analytical Geometry in two\ndimensions, and the introduction to three dimensional\ngeometry, we confined to the Cartesian methods only In\nthe previous chapter of this book, we have studied some\nbasic concepts of vectors"
    },
    {
        "Chapter": "1",
        "sentence_range": "5615-5618",
        "Text": "DEMORGAN \ufffd\n11 1  Introduction\nIn Class XI, while studying Analytical Geometry in two\ndimensions, and the introduction to three dimensional\ngeometry, we confined to the Cartesian methods only In\nthe previous chapter of this book, we have studied some\nbasic concepts of vectors We will now use vector algebra\nto three dimensional geometry"
    },
    {
        "Chapter": "1",
        "sentence_range": "5616-5619",
        "Text": "1  Introduction\nIn Class XI, while studying Analytical Geometry in two\ndimensions, and the introduction to three dimensional\ngeometry, we confined to the Cartesian methods only In\nthe previous chapter of this book, we have studied some\nbasic concepts of vectors We will now use vector algebra\nto three dimensional geometry The purpose of this\napproach to 3-dimensional geometry is that it makes the\nstudy simple and elegant*"
    },
    {
        "Chapter": "1",
        "sentence_range": "5617-5620",
        "Text": "In\nthe previous chapter of this book, we have studied some\nbasic concepts of vectors We will now use vector algebra\nto three dimensional geometry The purpose of this\napproach to 3-dimensional geometry is that it makes the\nstudy simple and elegant* In this chapter, we shall study the direction cosines\nand direction ratios of a line joining two points and also\ndiscuss about the equations of lines and planes in space\nunder different conditions, angle between two lines, two\nplanes, a line and a plane, shortest distance between two\nskew lines and distance of a point from a plane"
    },
    {
        "Chapter": "1",
        "sentence_range": "5618-5621",
        "Text": "We will now use vector algebra\nto three dimensional geometry The purpose of this\napproach to 3-dimensional geometry is that it makes the\nstudy simple and elegant* In this chapter, we shall study the direction cosines\nand direction ratios of a line joining two points and also\ndiscuss about the equations of lines and planes in space\nunder different conditions, angle between two lines, two\nplanes, a line and a plane, shortest distance between two\nskew lines and distance of a point from a plane Most of\nthe above results are obtained in vector form"
    },
    {
        "Chapter": "1",
        "sentence_range": "5619-5622",
        "Text": "The purpose of this\napproach to 3-dimensional geometry is that it makes the\nstudy simple and elegant* In this chapter, we shall study the direction cosines\nand direction ratios of a line joining two points and also\ndiscuss about the equations of lines and planes in space\nunder different conditions, angle between two lines, two\nplanes, a line and a plane, shortest distance between two\nskew lines and distance of a point from a plane Most of\nthe above results are obtained in vector form Nevertheless, we shall also translate\nthese results in the Cartesian form which, at times, presents a more clear geometric\nand analytic picture of the situation"
    },
    {
        "Chapter": "1",
        "sentence_range": "5620-5623",
        "Text": "In this chapter, we shall study the direction cosines\nand direction ratios of a line joining two points and also\ndiscuss about the equations of lines and planes in space\nunder different conditions, angle between two lines, two\nplanes, a line and a plane, shortest distance between two\nskew lines and distance of a point from a plane Most of\nthe above results are obtained in vector form Nevertheless, we shall also translate\nthese results in the Cartesian form which, at times, presents a more clear geometric\nand analytic picture of the situation 11"
    },
    {
        "Chapter": "1",
        "sentence_range": "5621-5624",
        "Text": "Most of\nthe above results are obtained in vector form Nevertheless, we shall also translate\nthese results in the Cartesian form which, at times, presents a more clear geometric\nand analytic picture of the situation 11 2  Direction Cosines and Direction Ratios of a Line\nFrom Chapter 10, recall that if a directed line L passing through the origin makes\nangles \u03b1, \u03b2 and \u03b3 with x, y and z-axes, respectively, called direction angles, then cosine\nof these angles, namely, cos \u03b1, cos \u03b2 and cos \u03b3 are called direction cosines of the\ndirected line L"
    },
    {
        "Chapter": "1",
        "sentence_range": "5622-5625",
        "Text": "Nevertheless, we shall also translate\nthese results in the Cartesian form which, at times, presents a more clear geometric\nand analytic picture of the situation 11 2  Direction Cosines and Direction Ratios of a Line\nFrom Chapter 10, recall that if a directed line L passing through the origin makes\nangles \u03b1, \u03b2 and \u03b3 with x, y and z-axes, respectively, called direction angles, then cosine\nof these angles, namely, cos \u03b1, cos \u03b2 and cos \u03b3 are called direction cosines of the\ndirected line L If we reverse the direction of L, then the direction angles are replaced by their supplements,\ni"
    },
    {
        "Chapter": "1",
        "sentence_range": "5623-5626",
        "Text": "11 2  Direction Cosines and Direction Ratios of a Line\nFrom Chapter 10, recall that if a directed line L passing through the origin makes\nangles \u03b1, \u03b2 and \u03b3 with x, y and z-axes, respectively, called direction angles, then cosine\nof these angles, namely, cos \u03b1, cos \u03b2 and cos \u03b3 are called direction cosines of the\ndirected line L If we reverse the direction of L, then the direction angles are replaced by their supplements,\ni e"
    },
    {
        "Chapter": "1",
        "sentence_range": "5624-5627",
        "Text": "2  Direction Cosines and Direction Ratios of a Line\nFrom Chapter 10, recall that if a directed line L passing through the origin makes\nangles \u03b1, \u03b2 and \u03b3 with x, y and z-axes, respectively, called direction angles, then cosine\nof these angles, namely, cos \u03b1, cos \u03b2 and cos \u03b3 are called direction cosines of the\ndirected line L If we reverse the direction of L, then the direction angles are replaced by their supplements,\ni e , \u03c0 \u03b1\n\u2212 , \u03c0 \u03b2\n\u2212  and \u03c0 \u03b3\n\u2212"
    },
    {
        "Chapter": "1",
        "sentence_range": "5625-5628",
        "Text": "If we reverse the direction of L, then the direction angles are replaced by their supplements,\ni e , \u03c0 \u03b1\n\u2212 , \u03c0 \u03b2\n\u2212  and \u03c0 \u03b3\n\u2212 Thus, the signs of the direction cosines are reversed"
    },
    {
        "Chapter": "1",
        "sentence_range": "5626-5629",
        "Text": "e , \u03c0 \u03b1\n\u2212 , \u03c0 \u03b2\n\u2212  and \u03c0 \u03b3\n\u2212 Thus, the signs of the direction cosines are reversed Chapter 11\nTHREE DIMENSIONAL GEOMETRY\n* Fo r vari ou s acti vi ti es  i n th ree di mens io nal geomet ry, on e may refer to  t he Boo k\n\u201cA Hand Book for designing Mathematics Laboratory in Schools\u201d, NCERT, 2005\nLeonhard Euler\n(1707-1783)\n\u00a9 NCERT\nnot to be republished\n MATHEMATI CS\n464\nNote that a given line in space can be extended in two opposite directions and so it\nhas two sets of direction cosines"
    },
    {
        "Chapter": "1",
        "sentence_range": "5627-5630",
        "Text": ", \u03c0 \u03b1\n\u2212 , \u03c0 \u03b2\n\u2212  and \u03c0 \u03b3\n\u2212 Thus, the signs of the direction cosines are reversed Chapter 11\nTHREE DIMENSIONAL GEOMETRY\n* Fo r vari ou s acti vi ti es  i n th ree di mens io nal geomet ry, on e may refer to  t he Boo k\n\u201cA Hand Book for designing Mathematics Laboratory in Schools\u201d, NCERT, 2005\nLeonhard Euler\n(1707-1783)\n\u00a9 NCERT\nnot to be republished\n MATHEMATI CS\n464\nNote that a given line in space can be extended in two opposite directions and so it\nhas two sets of direction cosines In order to have a unique set of direction cosines for\na given line in space, we must take the given line as a directed line"
    },
    {
        "Chapter": "1",
        "sentence_range": "5628-5631",
        "Text": "Thus, the signs of the direction cosines are reversed Chapter 11\nTHREE DIMENSIONAL GEOMETRY\n* Fo r vari ou s acti vi ti es  i n th ree di mens io nal geomet ry, on e may refer to  t he Boo k\n\u201cA Hand Book for designing Mathematics Laboratory in Schools\u201d, NCERT, 2005\nLeonhard Euler\n(1707-1783)\n\u00a9 NCERT\nnot to be republished\n MATHEMATI CS\n464\nNote that a given line in space can be extended in two opposite directions and so it\nhas two sets of direction cosines In order to have a unique set of direction cosines for\na given line in space, we must take the given line as a directed line These unique\ndirection cosines are denoted by l, m and n"
    },
    {
        "Chapter": "1",
        "sentence_range": "5629-5632",
        "Text": "Chapter 11\nTHREE DIMENSIONAL GEOMETRY\n* Fo r vari ou s acti vi ti es  i n th ree di mens io nal geomet ry, on e may refer to  t he Boo k\n\u201cA Hand Book for designing Mathematics Laboratory in Schools\u201d, NCERT, 2005\nLeonhard Euler\n(1707-1783)\n\u00a9 NCERT\nnot to be republished\n MATHEMATI CS\n464\nNote that a given line in space can be extended in two opposite directions and so it\nhas two sets of direction cosines In order to have a unique set of direction cosines for\na given line in space, we must take the given line as a directed line These unique\ndirection cosines are denoted by l, m and n Remark If the given line in space does not pass through the origin, then, in order to find\nits direction cosines, we draw a line through the origin and parallel to the given line"
    },
    {
        "Chapter": "1",
        "sentence_range": "5630-5633",
        "Text": "In order to have a unique set of direction cosines for\na given line in space, we must take the given line as a directed line These unique\ndirection cosines are denoted by l, m and n Remark If the given line in space does not pass through the origin, then, in order to find\nits direction cosines, we draw a line through the origin and parallel to the given line Now take one of the directed lines from the origin and find its direction cosines as two\nparallel line have same set of direction cosines"
    },
    {
        "Chapter": "1",
        "sentence_range": "5631-5634",
        "Text": "These unique\ndirection cosines are denoted by l, m and n Remark If the given line in space does not pass through the origin, then, in order to find\nits direction cosines, we draw a line through the origin and parallel to the given line Now take one of the directed lines from the origin and find its direction cosines as two\nparallel line have same set of direction cosines Any three numbers which are proportional to the direction cosines of a line are\ncalled the direction ratios of the line"
    },
    {
        "Chapter": "1",
        "sentence_range": "5632-5635",
        "Text": "Remark If the given line in space does not pass through the origin, then, in order to find\nits direction cosines, we draw a line through the origin and parallel to the given line Now take one of the directed lines from the origin and find its direction cosines as two\nparallel line have same set of direction cosines Any three numbers which are proportional to the direction cosines of a line are\ncalled the direction ratios of the line If l, m, n are direction cosines and a, b, c are\ndirection ratios of a line, then a = \u03bbl, b=\u03bbm and c = \u03bbn, for any nonzero \u03bb \u2208 R"
    },
    {
        "Chapter": "1",
        "sentence_range": "5633-5636",
        "Text": "Now take one of the directed lines from the origin and find its direction cosines as two\nparallel line have same set of direction cosines Any three numbers which are proportional to the direction cosines of a line are\ncalled the direction ratios of the line If l, m, n are direction cosines and a, b, c are\ndirection ratios of a line, then a = \u03bbl, b=\u03bbm and c = \u03bbn, for any nonzero \u03bb \u2208 R \ufffdNote   Some authors also call direction ratios as direction numbers"
    },
    {
        "Chapter": "1",
        "sentence_range": "5634-5637",
        "Text": "Any three numbers which are proportional to the direction cosines of a line are\ncalled the direction ratios of the line If l, m, n are direction cosines and a, b, c are\ndirection ratios of a line, then a = \u03bbl, b=\u03bbm and c = \u03bbn, for any nonzero \u03bb \u2208 R \ufffdNote   Some authors also call direction ratios as direction numbers Let a, b, c be direction ratios of a line and let l, m and n be the direction cosines\n(d"
    },
    {
        "Chapter": "1",
        "sentence_range": "5635-5638",
        "Text": "If l, m, n are direction cosines and a, b, c are\ndirection ratios of a line, then a = \u03bbl, b=\u03bbm and c = \u03bbn, for any nonzero \u03bb \u2208 R \ufffdNote   Some authors also call direction ratios as direction numbers Let a, b, c be direction ratios of a line and let l, m and n be the direction cosines\n(d c\u2019s) of the line"
    },
    {
        "Chapter": "1",
        "sentence_range": "5636-5639",
        "Text": "\ufffdNote   Some authors also call direction ratios as direction numbers Let a, b, c be direction ratios of a line and let l, m and n be the direction cosines\n(d c\u2019s) of the line Then\nl\na  = m\nb  = n\nk\nc\n=\n (say), k being a constant"
    },
    {
        "Chapter": "1",
        "sentence_range": "5637-5640",
        "Text": "Let a, b, c be direction ratios of a line and let l, m and n be the direction cosines\n(d c\u2019s) of the line Then\nl\na  = m\nb  = n\nk\nc\n=\n (say), k being a constant Therefore\nl = ak, m = bk, n = ck"
    },
    {
        "Chapter": "1",
        "sentence_range": "5638-5641",
        "Text": "c\u2019s) of the line Then\nl\na  = m\nb  = n\nk\nc\n=\n (say), k being a constant Therefore\nl = ak, m = bk, n = ck (1)\nBut\nl2 + m 2 + n2 = 1\nTherefore\nk2 (a2 + b2 + c2) = 1\nor\nk =\n2\n2\n2\n1\na\nb\nc\n\u00b1\n+\n+\nFig 11"
    },
    {
        "Chapter": "1",
        "sentence_range": "5639-5642",
        "Text": "Then\nl\na  = m\nb  = n\nk\nc\n=\n (say), k being a constant Therefore\nl = ak, m = bk, n = ck (1)\nBut\nl2 + m 2 + n2 = 1\nTherefore\nk2 (a2 + b2 + c2) = 1\nor\nk =\n2\n2\n2\n1\na\nb\nc\n\u00b1\n+\n+\nFig 11 1\n\u00a9 NCERT\nnot to be republished\nTHREE D IMENSIONAL G EOMETRY\n465\nHence, from (1), the d"
    },
    {
        "Chapter": "1",
        "sentence_range": "5640-5643",
        "Text": "Therefore\nl = ak, m = bk, n = ck (1)\nBut\nl2 + m 2 + n2 = 1\nTherefore\nk2 (a2 + b2 + c2) = 1\nor\nk =\n2\n2\n2\n1\na\nb\nc\n\u00b1\n+\n+\nFig 11 1\n\u00a9 NCERT\nnot to be republished\nTHREE D IMENSIONAL G EOMETRY\n465\nHence, from (1), the d c"
    },
    {
        "Chapter": "1",
        "sentence_range": "5641-5644",
        "Text": "(1)\nBut\nl2 + m 2 + n2 = 1\nTherefore\nk2 (a2 + b2 + c2) = 1\nor\nk =\n2\n2\n2\n1\na\nb\nc\n\u00b1\n+\n+\nFig 11 1\n\u00a9 NCERT\nnot to be republished\nTHREE D IMENSIONAL G EOMETRY\n465\nHence, from (1), the d c \u2019s of the line are\n2\n2\n2\n2\n2\n2\n2\n2\n2\n,\n,\na\nb\nc\nl\nm\nn\na\nb\nc\na\nb\nc\na\nb\nc\n=\u00b1\n= \u00b1\n= \u00b1\n+\n+\n+\n+\n+\n+\nwhere, depending on the desired sign of k, either a positive or a negative sign is to be\ntaken for l, m and n"
    },
    {
        "Chapter": "1",
        "sentence_range": "5642-5645",
        "Text": "1\n\u00a9 NCERT\nnot to be republished\nTHREE D IMENSIONAL G EOMETRY\n465\nHence, from (1), the d c \u2019s of the line are\n2\n2\n2\n2\n2\n2\n2\n2\n2\n,\n,\na\nb\nc\nl\nm\nn\na\nb\nc\na\nb\nc\na\nb\nc\n=\u00b1\n= \u00b1\n= \u00b1\n+\n+\n+\n+\n+\n+\nwhere, depending on the desired sign of k, either a positive or a negative sign is to be\ntaken for l, m and n For any line, if a, b, c are direction ratios of a line, then ka, kb, kc; k \u2260 0 is also a\nset of direction ratios"
    },
    {
        "Chapter": "1",
        "sentence_range": "5643-5646",
        "Text": "c \u2019s of the line are\n2\n2\n2\n2\n2\n2\n2\n2\n2\n,\n,\na\nb\nc\nl\nm\nn\na\nb\nc\na\nb\nc\na\nb\nc\n=\u00b1\n= \u00b1\n= \u00b1\n+\n+\n+\n+\n+\n+\nwhere, depending on the desired sign of k, either a positive or a negative sign is to be\ntaken for l, m and n For any line, if a, b, c are direction ratios of a line, then ka, kb, kc; k \u2260 0 is also a\nset of direction ratios So, any two sets of direction ratios of a line are also proportional"
    },
    {
        "Chapter": "1",
        "sentence_range": "5644-5647",
        "Text": "\u2019s of the line are\n2\n2\n2\n2\n2\n2\n2\n2\n2\n,\n,\na\nb\nc\nl\nm\nn\na\nb\nc\na\nb\nc\na\nb\nc\n=\u00b1\n= \u00b1\n= \u00b1\n+\n+\n+\n+\n+\n+\nwhere, depending on the desired sign of k, either a positive or a negative sign is to be\ntaken for l, m and n For any line, if a, b, c are direction ratios of a line, then ka, kb, kc; k \u2260 0 is also a\nset of direction ratios So, any two sets of direction ratios of a line are also proportional Also, for any line there are infinitely many sets of direction ratios"
    },
    {
        "Chapter": "1",
        "sentence_range": "5645-5648",
        "Text": "For any line, if a, b, c are direction ratios of a line, then ka, kb, kc; k \u2260 0 is also a\nset of direction ratios So, any two sets of direction ratios of a line are also proportional Also, for any line there are infinitely many sets of direction ratios 11"
    },
    {
        "Chapter": "1",
        "sentence_range": "5646-5649",
        "Text": "So, any two sets of direction ratios of a line are also proportional Also, for any line there are infinitely many sets of direction ratios 11 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "5647-5650",
        "Text": "Also, for any line there are infinitely many sets of direction ratios 11 2 1  Relation between the direction cosines of a line\nConsider a line RS with direction cosines l, m, n"
    },
    {
        "Chapter": "1",
        "sentence_range": "5648-5651",
        "Text": "11 2 1  Relation between the direction cosines of a line\nConsider a line RS with direction cosines l, m, n Through\nthe origin draw a line parallel to the given line and take a\npoint P(x, y, z) on this line"
    },
    {
        "Chapter": "1",
        "sentence_range": "5649-5652",
        "Text": "2 1  Relation between the direction cosines of a line\nConsider a line RS with direction cosines l, m, n Through\nthe origin draw a line parallel to the given line and take a\npoint P(x, y, z) on this line From P draw a perpendicular\nPA on the x-axis (Fig"
    },
    {
        "Chapter": "1",
        "sentence_range": "5650-5653",
        "Text": "1  Relation between the direction cosines of a line\nConsider a line RS with direction cosines l, m, n Through\nthe origin draw a line parallel to the given line and take a\npoint P(x, y, z) on this line From P draw a perpendicular\nPA on the x-axis (Fig 11"
    },
    {
        "Chapter": "1",
        "sentence_range": "5651-5654",
        "Text": "Through\nthe origin draw a line parallel to the given line and take a\npoint P(x, y, z) on this line From P draw a perpendicular\nPA on the x-axis (Fig 11 2)"
    },
    {
        "Chapter": "1",
        "sentence_range": "5652-5655",
        "Text": "From P draw a perpendicular\nPA on the x-axis (Fig 11 2) Let OP = r"
    },
    {
        "Chapter": "1",
        "sentence_range": "5653-5656",
        "Text": "11 2) Let OP = r Then\nOA\ncos\n\u03b1=OP\nx\nr="
    },
    {
        "Chapter": "1",
        "sentence_range": "5654-5657",
        "Text": "2) Let OP = r Then\nOA\ncos\n\u03b1=OP\nx\nr= This gives x = lr"
    },
    {
        "Chapter": "1",
        "sentence_range": "5655-5658",
        "Text": "Let OP = r Then\nOA\ncos\n\u03b1=OP\nx\nr= This gives x = lr Similarly,\ny = mr and z = nr\nThus\nx2 + y2 + z2 = r2 (l2 + m 2 + n2)\nBut\nx2 + y2 + z2 = r2\nHence\nl2 + m2 + n2 = 1\n11"
    },
    {
        "Chapter": "1",
        "sentence_range": "5656-5659",
        "Text": "Then\nOA\ncos\n\u03b1=OP\nx\nr= This gives x = lr Similarly,\ny = mr and z = nr\nThus\nx2 + y2 + z2 = r2 (l2 + m 2 + n2)\nBut\nx2 + y2 + z2 = r2\nHence\nl2 + m2 + n2 = 1\n11 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "5657-5660",
        "Text": "This gives x = lr Similarly,\ny = mr and z = nr\nThus\nx2 + y2 + z2 = r2 (l2 + m 2 + n2)\nBut\nx2 + y2 + z2 = r2\nHence\nl2 + m2 + n2 = 1\n11 2 2  Direction cosines of a line passing through two points\nSince one and only one line passes through two given points, we can determine the\ndirection cosines of a line passing through the given points P(x1, y1, z1) and Q(x2, y2, z2)\nas follows (Fig 11"
    },
    {
        "Chapter": "1",
        "sentence_range": "5658-5661",
        "Text": "Similarly,\ny = mr and z = nr\nThus\nx2 + y2 + z2 = r2 (l2 + m 2 + n2)\nBut\nx2 + y2 + z2 = r2\nHence\nl2 + m2 + n2 = 1\n11 2 2  Direction cosines of a line passing through two points\nSince one and only one line passes through two given points, we can determine the\ndirection cosines of a line passing through the given points P(x1, y1, z1) and Q(x2, y2, z2)\nas follows (Fig 11 3 (a))"
    },
    {
        "Chapter": "1",
        "sentence_range": "5659-5662",
        "Text": "2 2  Direction cosines of a line passing through two points\nSince one and only one line passes through two given points, we can determine the\ndirection cosines of a line passing through the given points P(x1, y1, z1) and Q(x2, y2, z2)\nas follows (Fig 11 3 (a)) Fig 11"
    },
    {
        "Chapter": "1",
        "sentence_range": "5660-5663",
        "Text": "2  Direction cosines of a line passing through two points\nSince one and only one line passes through two given points, we can determine the\ndirection cosines of a line passing through the given points P(x1, y1, z1) and Q(x2, y2, z2)\nas follows (Fig 11 3 (a)) Fig 11 3\nr\nZ\nX\nY\nR\nS\nP ( , , )\nx y z\nA\nO\u03b1\nA\nO\nP\nx\n\u03b1\n\u03b1\nFig 11"
    },
    {
        "Chapter": "1",
        "sentence_range": "5661-5664",
        "Text": "3 (a)) Fig 11 3\nr\nZ\nX\nY\nR\nS\nP ( , , )\nx y z\nA\nO\u03b1\nA\nO\nP\nx\n\u03b1\n\u03b1\nFig 11 2\n\u00a9 NCERT\nnot to be republished\n MATHEMATI CS\n466\nLet l, m, n be the direction cosines of the line PQ and let it makes angles \u03b1, \u03b2 and  \u03b3\nwith the x, y and z-axis, respectively"
    },
    {
        "Chapter": "1",
        "sentence_range": "5662-5665",
        "Text": "Fig 11 3\nr\nZ\nX\nY\nR\nS\nP ( , , )\nx y z\nA\nO\u03b1\nA\nO\nP\nx\n\u03b1\n\u03b1\nFig 11 2\n\u00a9 NCERT\nnot to be republished\n MATHEMATI CS\n466\nLet l, m, n be the direction cosines of the line PQ and let it makes angles \u03b1, \u03b2 and  \u03b3\nwith the x, y and z-axis, respectively Draw perpendiculars from P and Q to XY-plane to meet at R and S"
    },
    {
        "Chapter": "1",
        "sentence_range": "5663-5666",
        "Text": "3\nr\nZ\nX\nY\nR\nS\nP ( , , )\nx y z\nA\nO\u03b1\nA\nO\nP\nx\n\u03b1\n\u03b1\nFig 11 2\n\u00a9 NCERT\nnot to be republished\n MATHEMATI CS\n466\nLet l, m, n be the direction cosines of the line PQ and let it makes angles \u03b1, \u03b2 and  \u03b3\nwith the x, y and z-axis, respectively Draw perpendiculars from P and Q to XY-plane to meet at R and S Draw a\nperpendicular from P to QS to meet at N"
    },
    {
        "Chapter": "1",
        "sentence_range": "5664-5667",
        "Text": "2\n\u00a9 NCERT\nnot to be republished\n MATHEMATI CS\n466\nLet l, m, n be the direction cosines of the line PQ and let it makes angles \u03b1, \u03b2 and  \u03b3\nwith the x, y and z-axis, respectively Draw perpendiculars from P and Q to XY-plane to meet at R and S Draw a\nperpendicular from P to QS to meet at N Now, in right angle triangle PNQ, \u2220PQN=\n\u03b3  (Fig 11"
    },
    {
        "Chapter": "1",
        "sentence_range": "5665-5668",
        "Text": "Draw perpendiculars from P and Q to XY-plane to meet at R and S Draw a\nperpendicular from P to QS to meet at N Now, in right angle triangle PNQ, \u2220PQN=\n\u03b3  (Fig 11 3 (b)"
    },
    {
        "Chapter": "1",
        "sentence_range": "5666-5669",
        "Text": "Draw a\nperpendicular from P to QS to meet at N Now, in right angle triangle PNQ, \u2220PQN=\n\u03b3  (Fig 11 3 (b) Therefore,\ncos\u03b3 =\n2\n1\nNQ\nPQ\nzPQ\n\u2212z\n=\nSimilarly\ncos\u03b1 =\n2\n1\n2\n1\nand cos\nPQ\nPQ\nx\nx\ny\ny\n\u2212\n\u2212\n\u03b2 =\nHence, the direction cosines of the line segment joining the points P(x1, y1, z1) and\nQ(x2, y2, z2) are\n2\n1\nPQ\nx\n\u2212x\n, \n2\n1\nPQ\ny\n\u2212y\n, \n2\n1\nPQ\nz\nz\n\u2212\nwhere\nPQ =\n(\n)\n2\n2\n2\n2\n1\n2\n1\n2\n1\n(\n)\n(\n)\nx\nx\ny\ny\nz\nz\n\u2212\n+ \u2212\n+ \u2212\n\ufffdNote   The direction ratios of the line segment joining P(x1, y1, z1) and Q(x2, y2, z2)\nmay be taken as\nx2 \u2013 x1, y2 \u2013 y1, z2 \u2013 z1 or x1 \u2013 x2, y1 \u2013 y2, z1 \u2013 z2\nExample 1 If a line makes angle 90\u00b0, 60\u00b0 and 30\u00b0 with the positive direction of x, y and\nz-axis respectively, find its direction cosines"
    },
    {
        "Chapter": "1",
        "sentence_range": "5667-5670",
        "Text": "Now, in right angle triangle PNQ, \u2220PQN=\n\u03b3  (Fig 11 3 (b) Therefore,\ncos\u03b3 =\n2\n1\nNQ\nPQ\nzPQ\n\u2212z\n=\nSimilarly\ncos\u03b1 =\n2\n1\n2\n1\nand cos\nPQ\nPQ\nx\nx\ny\ny\n\u2212\n\u2212\n\u03b2 =\nHence, the direction cosines of the line segment joining the points P(x1, y1, z1) and\nQ(x2, y2, z2) are\n2\n1\nPQ\nx\n\u2212x\n, \n2\n1\nPQ\ny\n\u2212y\n, \n2\n1\nPQ\nz\nz\n\u2212\nwhere\nPQ =\n(\n)\n2\n2\n2\n2\n1\n2\n1\n2\n1\n(\n)\n(\n)\nx\nx\ny\ny\nz\nz\n\u2212\n+ \u2212\n+ \u2212\n\ufffdNote   The direction ratios of the line segment joining P(x1, y1, z1) and Q(x2, y2, z2)\nmay be taken as\nx2 \u2013 x1, y2 \u2013 y1, z2 \u2013 z1 or x1 \u2013 x2, y1 \u2013 y2, z1 \u2013 z2\nExample 1 If a line makes angle 90\u00b0, 60\u00b0 and 30\u00b0 with the positive direction of x, y and\nz-axis respectively, find its direction cosines Solution Let the d"
    },
    {
        "Chapter": "1",
        "sentence_range": "5668-5671",
        "Text": "3 (b) Therefore,\ncos\u03b3 =\n2\n1\nNQ\nPQ\nzPQ\n\u2212z\n=\nSimilarly\ncos\u03b1 =\n2\n1\n2\n1\nand cos\nPQ\nPQ\nx\nx\ny\ny\n\u2212\n\u2212\n\u03b2 =\nHence, the direction cosines of the line segment joining the points P(x1, y1, z1) and\nQ(x2, y2, z2) are\n2\n1\nPQ\nx\n\u2212x\n, \n2\n1\nPQ\ny\n\u2212y\n, \n2\n1\nPQ\nz\nz\n\u2212\nwhere\nPQ =\n(\n)\n2\n2\n2\n2\n1\n2\n1\n2\n1\n(\n)\n(\n)\nx\nx\ny\ny\nz\nz\n\u2212\n+ \u2212\n+ \u2212\n\ufffdNote   The direction ratios of the line segment joining P(x1, y1, z1) and Q(x2, y2, z2)\nmay be taken as\nx2 \u2013 x1, y2 \u2013 y1, z2 \u2013 z1 or x1 \u2013 x2, y1 \u2013 y2, z1 \u2013 z2\nExample 1 If a line makes angle 90\u00b0, 60\u00b0 and 30\u00b0 with the positive direction of x, y and\nz-axis respectively, find its direction cosines Solution Let the d c"
    },
    {
        "Chapter": "1",
        "sentence_range": "5669-5672",
        "Text": "Therefore,\ncos\u03b3 =\n2\n1\nNQ\nPQ\nzPQ\n\u2212z\n=\nSimilarly\ncos\u03b1 =\n2\n1\n2\n1\nand cos\nPQ\nPQ\nx\nx\ny\ny\n\u2212\n\u2212\n\u03b2 =\nHence, the direction cosines of the line segment joining the points P(x1, y1, z1) and\nQ(x2, y2, z2) are\n2\n1\nPQ\nx\n\u2212x\n, \n2\n1\nPQ\ny\n\u2212y\n, \n2\n1\nPQ\nz\nz\n\u2212\nwhere\nPQ =\n(\n)\n2\n2\n2\n2\n1\n2\n1\n2\n1\n(\n)\n(\n)\nx\nx\ny\ny\nz\nz\n\u2212\n+ \u2212\n+ \u2212\n\ufffdNote   The direction ratios of the line segment joining P(x1, y1, z1) and Q(x2, y2, z2)\nmay be taken as\nx2 \u2013 x1, y2 \u2013 y1, z2 \u2013 z1 or x1 \u2013 x2, y1 \u2013 y2, z1 \u2013 z2\nExample 1 If a line makes angle 90\u00b0, 60\u00b0 and 30\u00b0 with the positive direction of x, y and\nz-axis respectively, find its direction cosines Solution Let the d c 's of the lines be l , m, n"
    },
    {
        "Chapter": "1",
        "sentence_range": "5670-5673",
        "Text": "Solution Let the d c 's of the lines be l , m, n Then l = cos 900 = 0, m = cos 600 = 1\n2,\nn = cos 300 = 2\n3"
    },
    {
        "Chapter": "1",
        "sentence_range": "5671-5674",
        "Text": "c 's of the lines be l , m, n Then l = cos 900 = 0, m = cos 600 = 1\n2,\nn = cos 300 = 2\n3 Example 2 If a line has direction ratios 2, \u2013 1, \u2013 2, determine its direction cosines"
    },
    {
        "Chapter": "1",
        "sentence_range": "5672-5675",
        "Text": "'s of the lines be l , m, n Then l = cos 900 = 0, m = cos 600 = 1\n2,\nn = cos 300 = 2\n3 Example 2 If a line has direction ratios 2, \u2013 1, \u2013 2, determine its direction cosines Solution Direction cosines are\n2\n2\n2\n)2\n(\n)1\n(\n2\n2\n+\u2212\n+\u2212\n,  \n2\n2\n2\n(2)\n)1\n(\n2\n1\n+\u2212\n\u2212\n+\n\u2212\n,  \n(\n)\n2\n2\n2\n)2\n(\n1\n2\n2\n+\u2212\n\u2212\n+\n\u2212\nor\n 2\n1\n2\n,\n,\n3\n3\n3\n\u2212\n\u2212\nExample 3 Find the direction cosines of the line passing through  the two points\n(\u2013 2, 4, \u2013 5) and (1, 2, 3)"
    },
    {
        "Chapter": "1",
        "sentence_range": "5673-5676",
        "Text": "Then l = cos 900 = 0, m = cos 600 = 1\n2,\nn = cos 300 = 2\n3 Example 2 If a line has direction ratios 2, \u2013 1, \u2013 2, determine its direction cosines Solution Direction cosines are\n2\n2\n2\n)2\n(\n)1\n(\n2\n2\n+\u2212\n+\u2212\n,  \n2\n2\n2\n(2)\n)1\n(\n2\n1\n+\u2212\n\u2212\n+\n\u2212\n,  \n(\n)\n2\n2\n2\n)2\n(\n1\n2\n2\n+\u2212\n\u2212\n+\n\u2212\nor\n 2\n1\n2\n,\n,\n3\n3\n3\n\u2212\n\u2212\nExample 3 Find the direction cosines of the line passing through  the two points\n(\u2013 2, 4, \u2013 5) and (1, 2, 3) \u00a9 NCERT\nnot to be republished\nTHREE D IMENSIONAL G EOMETRY\n467\nSolution We know the direction cosines of the line passing through  two points\nP(x1, y1, z1) and Q(x2, y2, z2) are given by\n2\n1\n2\n1\n2\n1\n,\n,\nPQ\nPQ\nPQ\nx\nx\ny\ny\nz\nz\n\u2212\n\u2212\n\u2212\nwhere\nPQ =\n(\n)\n12\n2\n12\n2\n2\n1\n2\n)\n(\n)\n(\nz\nz\ny\ny\nx\nx\n\u2212\n+\n\u2212\n+\n\u2212\nHere P is (\u2013 2, 4, \u2013 5) and Q is (1, 2, 3)"
    },
    {
        "Chapter": "1",
        "sentence_range": "5674-5677",
        "Text": "Example 2 If a line has direction ratios 2, \u2013 1, \u2013 2, determine its direction cosines Solution Direction cosines are\n2\n2\n2\n)2\n(\n)1\n(\n2\n2\n+\u2212\n+\u2212\n,  \n2\n2\n2\n(2)\n)1\n(\n2\n1\n+\u2212\n\u2212\n+\n\u2212\n,  \n(\n)\n2\n2\n2\n)2\n(\n1\n2\n2\n+\u2212\n\u2212\n+\n\u2212\nor\n 2\n1\n2\n,\n,\n3\n3\n3\n\u2212\n\u2212\nExample 3 Find the direction cosines of the line passing through  the two points\n(\u2013 2, 4, \u2013 5) and (1, 2, 3) \u00a9 NCERT\nnot to be republished\nTHREE D IMENSIONAL G EOMETRY\n467\nSolution We know the direction cosines of the line passing through  two points\nP(x1, y1, z1) and Q(x2, y2, z2) are given by\n2\n1\n2\n1\n2\n1\n,\n,\nPQ\nPQ\nPQ\nx\nx\ny\ny\nz\nz\n\u2212\n\u2212\n\u2212\nwhere\nPQ =\n(\n)\n12\n2\n12\n2\n2\n1\n2\n)\n(\n)\n(\nz\nz\ny\ny\nx\nx\n\u2212\n+\n\u2212\n+\n\u2212\nHere P is (\u2013 2, 4, \u2013 5) and Q is (1, 2, 3) So\nPQ =\n2\n2\n2\n(1\n( 2))\n(2\n4)\n(3\n( 5))\n\u2212 \u2212\n+\n\u2212\n+\n\u2212 \u2212\n = \n77\nThus, the direction cosines of the line joining two points is\n3\n2\n8\n,\n,\n77\n77\n77\n\u2212\nExample 4 Find the direction cosines of x, y and z-axis"
    },
    {
        "Chapter": "1",
        "sentence_range": "5675-5678",
        "Text": "Solution Direction cosines are\n2\n2\n2\n)2\n(\n)1\n(\n2\n2\n+\u2212\n+\u2212\n,  \n2\n2\n2\n(2)\n)1\n(\n2\n1\n+\u2212\n\u2212\n+\n\u2212\n,  \n(\n)\n2\n2\n2\n)2\n(\n1\n2\n2\n+\u2212\n\u2212\n+\n\u2212\nor\n 2\n1\n2\n,\n,\n3\n3\n3\n\u2212\n\u2212\nExample 3 Find the direction cosines of the line passing through  the two points\n(\u2013 2, 4, \u2013 5) and (1, 2, 3) \u00a9 NCERT\nnot to be republished\nTHREE D IMENSIONAL G EOMETRY\n467\nSolution We know the direction cosines of the line passing through  two points\nP(x1, y1, z1) and Q(x2, y2, z2) are given by\n2\n1\n2\n1\n2\n1\n,\n,\nPQ\nPQ\nPQ\nx\nx\ny\ny\nz\nz\n\u2212\n\u2212\n\u2212\nwhere\nPQ =\n(\n)\n12\n2\n12\n2\n2\n1\n2\n)\n(\n)\n(\nz\nz\ny\ny\nx\nx\n\u2212\n+\n\u2212\n+\n\u2212\nHere P is (\u2013 2, 4, \u2013 5) and Q is (1, 2, 3) So\nPQ =\n2\n2\n2\n(1\n( 2))\n(2\n4)\n(3\n( 5))\n\u2212 \u2212\n+\n\u2212\n+\n\u2212 \u2212\n = \n77\nThus, the direction cosines of the line joining two points is\n3\n2\n8\n,\n,\n77\n77\n77\n\u2212\nExample 4 Find the direction cosines of x, y and z-axis Solution The x-axis makes angles 0\u00b0, 90\u00b0 and 90\u00b0 respectively with x, y and z-axis"
    },
    {
        "Chapter": "1",
        "sentence_range": "5676-5679",
        "Text": "\u00a9 NCERT\nnot to be republished\nTHREE D IMENSIONAL G EOMETRY\n467\nSolution We know the direction cosines of the line passing through  two points\nP(x1, y1, z1) and Q(x2, y2, z2) are given by\n2\n1\n2\n1\n2\n1\n,\n,\nPQ\nPQ\nPQ\nx\nx\ny\ny\nz\nz\n\u2212\n\u2212\n\u2212\nwhere\nPQ =\n(\n)\n12\n2\n12\n2\n2\n1\n2\n)\n(\n)\n(\nz\nz\ny\ny\nx\nx\n\u2212\n+\n\u2212\n+\n\u2212\nHere P is (\u2013 2, 4, \u2013 5) and Q is (1, 2, 3) So\nPQ =\n2\n2\n2\n(1\n( 2))\n(2\n4)\n(3\n( 5))\n\u2212 \u2212\n+\n\u2212\n+\n\u2212 \u2212\n = \n77\nThus, the direction cosines of the line joining two points is\n3\n2\n8\n,\n,\n77\n77\n77\n\u2212\nExample 4 Find the direction cosines of x, y and z-axis Solution The x-axis makes angles 0\u00b0, 90\u00b0 and 90\u00b0 respectively with x, y and z-axis Therefore, the direction cosines of x-axis are cos 0\u00b0, cos 90\u00b0, cos 90\u00b0 i"
    },
    {
        "Chapter": "1",
        "sentence_range": "5677-5680",
        "Text": "So\nPQ =\n2\n2\n2\n(1\n( 2))\n(2\n4)\n(3\n( 5))\n\u2212 \u2212\n+\n\u2212\n+\n\u2212 \u2212\n = \n77\nThus, the direction cosines of the line joining two points is\n3\n2\n8\n,\n,\n77\n77\n77\n\u2212\nExample 4 Find the direction cosines of x, y and z-axis Solution The x-axis makes angles 0\u00b0, 90\u00b0 and 90\u00b0 respectively with x, y and z-axis Therefore, the direction cosines of x-axis are cos 0\u00b0, cos 90\u00b0, cos 90\u00b0 i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "5678-5681",
        "Text": "Solution The x-axis makes angles 0\u00b0, 90\u00b0 and 90\u00b0 respectively with x, y and z-axis Therefore, the direction cosines of x-axis are cos 0\u00b0, cos 90\u00b0, cos 90\u00b0 i e , 1,0,0"
    },
    {
        "Chapter": "1",
        "sentence_range": "5679-5682",
        "Text": "Therefore, the direction cosines of x-axis are cos 0\u00b0, cos 90\u00b0, cos 90\u00b0 i e , 1,0,0 Similarly, direction cosines of y-axis and z-axis are 0, 1, 0 and 0, 0, 1 respectively"
    },
    {
        "Chapter": "1",
        "sentence_range": "5680-5683",
        "Text": "e , 1,0,0 Similarly, direction cosines of y-axis and z-axis are 0, 1, 0 and 0, 0, 1 respectively Example 5 Show that the points A (2, 3, \u2013 4), B (1, \u2013 2, 3) and C (3, 8, \u2013 11) are\ncollinear"
    },
    {
        "Chapter": "1",
        "sentence_range": "5681-5684",
        "Text": ", 1,0,0 Similarly, direction cosines of y-axis and z-axis are 0, 1, 0 and 0, 0, 1 respectively Example 5 Show that the points A (2, 3, \u2013 4), B (1, \u2013 2, 3) and C (3, 8, \u2013 11) are\ncollinear Solution Direction ratios of line joining A and B are\n1 \u2013 2, \u2013 2 \u2013 3, 3 + 4 i"
    },
    {
        "Chapter": "1",
        "sentence_range": "5682-5685",
        "Text": "Similarly, direction cosines of y-axis and z-axis are 0, 1, 0 and 0, 0, 1 respectively Example 5 Show that the points A (2, 3, \u2013 4), B (1, \u2013 2, 3) and C (3, 8, \u2013 11) are\ncollinear Solution Direction ratios of line joining A and B are\n1 \u2013 2, \u2013 2 \u2013 3, 3 + 4 i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "5683-5686",
        "Text": "Example 5 Show that the points A (2, 3, \u2013 4), B (1, \u2013 2, 3) and C (3, 8, \u2013 11) are\ncollinear Solution Direction ratios of line joining A and B are\n1 \u2013 2, \u2013 2 \u2013 3, 3 + 4 i e , \u2013 1, \u2013 5, 7"
    },
    {
        "Chapter": "1",
        "sentence_range": "5684-5687",
        "Text": "Solution Direction ratios of line joining A and B are\n1 \u2013 2, \u2013 2 \u2013 3, 3 + 4 i e , \u2013 1, \u2013 5, 7 The direction ratios of line joining B and C are\n3 \u20131, 8 + 2, \u2013 11 \u2013 3, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "5685-5688",
        "Text": "e , \u2013 1, \u2013 5, 7 The direction ratios of line joining B and C are\n3 \u20131, 8 + 2, \u2013 11 \u2013 3, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "5686-5689",
        "Text": ", \u2013 1, \u2013 5, 7 The direction ratios of line joining B and C are\n3 \u20131, 8 + 2, \u2013 11 \u2013 3, i e , 2, 10, \u2013 14"
    },
    {
        "Chapter": "1",
        "sentence_range": "5687-5690",
        "Text": "The direction ratios of line joining B and C are\n3 \u20131, 8 + 2, \u2013 11 \u2013 3, i e , 2, 10, \u2013 14 It is clear that direction ratios of AB and BC are proportional, hence, AB is parallel\nto BC"
    },
    {
        "Chapter": "1",
        "sentence_range": "5688-5691",
        "Text": "e , 2, 10, \u2013 14 It is clear that direction ratios of AB and BC are proportional, hence, AB is parallel\nto BC But point B is common to both AB and BC"
    },
    {
        "Chapter": "1",
        "sentence_range": "5689-5692",
        "Text": ", 2, 10, \u2013 14 It is clear that direction ratios of AB and BC are proportional, hence, AB is parallel\nto BC But point B is common to both AB and BC Therefore, A, B, C are\ncollinear points"
    },
    {
        "Chapter": "1",
        "sentence_range": "5690-5693",
        "Text": "It is clear that direction ratios of AB and BC are proportional, hence, AB is parallel\nto BC But point B is common to both AB and BC Therefore, A, B, C are\ncollinear points EXERCISE 11"
    },
    {
        "Chapter": "1",
        "sentence_range": "5691-5694",
        "Text": "But point B is common to both AB and BC Therefore, A, B, C are\ncollinear points EXERCISE 11 1\n1"
    },
    {
        "Chapter": "1",
        "sentence_range": "5692-5695",
        "Text": "Therefore, A, B, C are\ncollinear points EXERCISE 11 1\n1 If a line makes angles 90\u00b0, 135\u00b0, 45\u00b0 with the x, y and z-axes respectively, find its\ndirection cosines"
    },
    {
        "Chapter": "1",
        "sentence_range": "5693-5696",
        "Text": "EXERCISE 11 1\n1 If a line makes angles 90\u00b0, 135\u00b0, 45\u00b0 with the x, y and z-axes respectively, find its\ndirection cosines 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "5694-5697",
        "Text": "1\n1 If a line makes angles 90\u00b0, 135\u00b0, 45\u00b0 with the x, y and z-axes respectively, find its\ndirection cosines 2 Find the direction cosines of a line which makes equal angles with the coordinate\naxes"
    },
    {
        "Chapter": "1",
        "sentence_range": "5695-5698",
        "Text": "If a line makes angles 90\u00b0, 135\u00b0, 45\u00b0 with the x, y and z-axes respectively, find its\ndirection cosines 2 Find the direction cosines of a line which makes equal angles with the coordinate\naxes 3"
    },
    {
        "Chapter": "1",
        "sentence_range": "5696-5699",
        "Text": "2 Find the direction cosines of a line which makes equal angles with the coordinate\naxes 3 If a line has the direction ratios \u201318, 12, \u2013 4, then what are its direction cosines"
    },
    {
        "Chapter": "1",
        "sentence_range": "5697-5700",
        "Text": "Find the direction cosines of a line which makes equal angles with the coordinate\naxes 3 If a line has the direction ratios \u201318, 12, \u2013 4, then what are its direction cosines 4"
    },
    {
        "Chapter": "1",
        "sentence_range": "5698-5701",
        "Text": "3 If a line has the direction ratios \u201318, 12, \u2013 4, then what are its direction cosines 4 Show that the points (2, 3, 4), (\u2013 1, \u2013 2, 1), (5, 8, 7) are collinear"
    },
    {
        "Chapter": "1",
        "sentence_range": "5699-5702",
        "Text": "If a line has the direction ratios \u201318, 12, \u2013 4, then what are its direction cosines 4 Show that the points (2, 3, 4), (\u2013 1, \u2013 2, 1), (5, 8, 7) are collinear 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "5700-5703",
        "Text": "4 Show that the points (2, 3, 4), (\u2013 1, \u2013 2, 1), (5, 8, 7) are collinear 5 Find the direction cosines of the sides of the triangle whose vertices are\n(3, 5, \u2013 4), (\u2013 1, 1, 2) and (\u2013 5, \u2013 5, \u2013 2)"
    },
    {
        "Chapter": "1",
        "sentence_range": "5701-5704",
        "Text": "Show that the points (2, 3, 4), (\u2013 1, \u2013 2, 1), (5, 8, 7) are collinear 5 Find the direction cosines of the sides of the triangle whose vertices are\n(3, 5, \u2013 4), (\u2013 1, 1, 2) and (\u2013 5, \u2013 5, \u2013 2) \u00a9 NCERT\nnot to be republished\n MATHEMATI CS\n468\n11"
    },
    {
        "Chapter": "1",
        "sentence_range": "5702-5705",
        "Text": "5 Find the direction cosines of the sides of the triangle whose vertices are\n(3, 5, \u2013 4), (\u2013 1, 1, 2) and (\u2013 5, \u2013 5, \u2013 2) \u00a9 NCERT\nnot to be republished\n MATHEMATI CS\n468\n11 3 Equation of a Line in Space\nWe have studied equation of  lines in two dimensions in Class XI, we shall now study\nthe vector and cartesian equations of a line in space"
    },
    {
        "Chapter": "1",
        "sentence_range": "5703-5706",
        "Text": "Find the direction cosines of the sides of the triangle whose vertices are\n(3, 5, \u2013 4), (\u2013 1, 1, 2) and (\u2013 5, \u2013 5, \u2013 2) \u00a9 NCERT\nnot to be republished\n MATHEMATI CS\n468\n11 3 Equation of a Line in Space\nWe have studied equation of  lines in two dimensions in Class XI, we shall now study\nthe vector and cartesian equations of a line in space (i)A line is uniquely determined if\nit passes through a given point and has given direction, or\n(ii)\nit passes through two given points"
    },
    {
        "Chapter": "1",
        "sentence_range": "5704-5707",
        "Text": "\u00a9 NCERT\nnot to be republished\n MATHEMATI CS\n468\n11 3 Equation of a Line in Space\nWe have studied equation of  lines in two dimensions in Class XI, we shall now study\nthe vector and cartesian equations of a line in space (i)A line is uniquely determined if\nit passes through a given point and has given direction, or\n(ii)\nit passes through two given points 11"
    },
    {
        "Chapter": "1",
        "sentence_range": "5705-5708",
        "Text": "3 Equation of a Line in Space\nWe have studied equation of  lines in two dimensions in Class XI, we shall now study\nthe vector and cartesian equations of a line in space (i)A line is uniquely determined if\nit passes through a given point and has given direction, or\n(ii)\nit passes through two given points 11 3"
    },
    {
        "Chapter": "1",
        "sentence_range": "5706-5709",
        "Text": "(i)A line is uniquely determined if\nit passes through a given point and has given direction, or\n(ii)\nit passes through two given points 11 3 1Equation of a line through a given point and parallel to a given vector \nr\nb\nLet ar be the position vector of the given point\nA wit h respect  to the origin O of t he\nrectangular coordinate system"
    },
    {
        "Chapter": "1",
        "sentence_range": "5707-5710",
        "Text": "11 3 1Equation of a line through a given point and parallel to a given vector \nr\nb\nLet ar be the position vector of the given point\nA wit h respect  to the origin O of t he\nrectangular coordinate system Let l be the\nline which passes through the point A and is\nparallel to a given vector b\nr"
    },
    {
        "Chapter": "1",
        "sentence_range": "5708-5711",
        "Text": "3 1Equation of a line through a given point and parallel to a given vector \nr\nb\nLet ar be the position vector of the given point\nA wit h respect  to the origin O of t he\nrectangular coordinate system Let l be the\nline which passes through the point A and is\nparallel to a given vector b\nr Let  rr  be the\nposition vector of an arbitrary point P on the\nline (Fig 11"
    },
    {
        "Chapter": "1",
        "sentence_range": "5709-5712",
        "Text": "1Equation of a line through a given point and parallel to a given vector \nr\nb\nLet ar be the position vector of the given point\nA wit h respect  to the origin O of t he\nrectangular coordinate system Let l be the\nline which passes through the point A and is\nparallel to a given vector b\nr Let  rr  be the\nposition vector of an arbitrary point P on the\nline (Fig 11 4)"
    },
    {
        "Chapter": "1",
        "sentence_range": "5710-5713",
        "Text": "Let l be the\nline which passes through the point A and is\nparallel to a given vector b\nr Let  rr  be the\nposition vector of an arbitrary point P on the\nline (Fig 11 4) Then AP\nuuur\n is parallel to the vector b\nr\n,  i"
    },
    {
        "Chapter": "1",
        "sentence_range": "5711-5714",
        "Text": "Let  rr  be the\nposition vector of an arbitrary point P on the\nline (Fig 11 4) Then AP\nuuur\n is parallel to the vector b\nr\n,  i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "5712-5715",
        "Text": "4) Then AP\nuuur\n is parallel to the vector b\nr\n,  i e ,\nAP\nuuur\n= \u03bbb\nr\n,  where \u03bb is some real number"
    },
    {
        "Chapter": "1",
        "sentence_range": "5713-5716",
        "Text": "Then AP\nuuur\n is parallel to the vector b\nr\n,  i e ,\nAP\nuuur\n= \u03bbb\nr\n,  where \u03bb is some real number But\nAP\nuuur\n = OP \u2013 OA\nuuur\nuuur\ni"
    },
    {
        "Chapter": "1",
        "sentence_range": "5714-5717",
        "Text": "e ,\nAP\nuuur\n= \u03bbb\nr\n,  where \u03bb is some real number But\nAP\nuuur\n = OP \u2013 OA\nuuur\nuuur\ni e"
    },
    {
        "Chapter": "1",
        "sentence_range": "5715-5718",
        "Text": ",\nAP\nuuur\n= \u03bbb\nr\n,  where \u03bb is some real number But\nAP\nuuur\n = OP \u2013 OA\nuuur\nuuur\ni e \u03bbb\nr\n = r\na\u2212\nr\nr\nConversely, for each value of the parameter \u03bb, this equation gives the position\nvector of a point P on the line"
    },
    {
        "Chapter": "1",
        "sentence_range": "5716-5719",
        "Text": "But\nAP\nuuur\n = OP \u2013 OA\nuuur\nuuur\ni e \u03bbb\nr\n = r\na\u2212\nr\nr\nConversely, for each value of the parameter \u03bb, this equation gives the position\nvector of a point P on the line Hence, the vector equation of the line is given by\nrr  =\n\u03bb\nr\nra +\nb"
    },
    {
        "Chapter": "1",
        "sentence_range": "5717-5720",
        "Text": "e \u03bbb\nr\n = r\na\u2212\nr\nr\nConversely, for each value of the parameter \u03bb, this equation gives the position\nvector of a point P on the line Hence, the vector equation of the line is given by\nrr  =\n\u03bb\nr\nra +\nb (1)\nRemark If \n\u02c6\n\u02c6\n\u02c6\nb\nai\nbj\nck\n= + +\nr\n, then a, b, c are direction ratios of the line and conversely,,\nif a, b, c are direction ratios of a line, then \n\u02c6\n\u02c6\n\u02c6\n=\n+\n+\nbr\nai\nbj\nck will be the parallel to\nthe line"
    },
    {
        "Chapter": "1",
        "sentence_range": "5718-5721",
        "Text": "\u03bbb\nr\n = r\na\u2212\nr\nr\nConversely, for each value of the parameter \u03bb, this equation gives the position\nvector of a point P on the line Hence, the vector equation of the line is given by\nrr  =\n\u03bb\nr\nra +\nb (1)\nRemark If \n\u02c6\n\u02c6\n\u02c6\nb\nai\nbj\nck\n= + +\nr\n, then a, b, c are direction ratios of the line and conversely,,\nif a, b, c are direction ratios of a line, then \n\u02c6\n\u02c6\n\u02c6\n=\n+\n+\nbr\nai\nbj\nck will be the parallel to\nthe line Here, b should not be confused with |\nr\nb|"
    },
    {
        "Chapter": "1",
        "sentence_range": "5719-5722",
        "Text": "Hence, the vector equation of the line is given by\nrr  =\n\u03bb\nr\nra +\nb (1)\nRemark If \n\u02c6\n\u02c6\n\u02c6\nb\nai\nbj\nck\n= + +\nr\n, then a, b, c are direction ratios of the line and conversely,,\nif a, b, c are direction ratios of a line, then \n\u02c6\n\u02c6\n\u02c6\n=\n+\n+\nbr\nai\nbj\nck will be the parallel to\nthe line Here, b should not be confused with |\nr\nb| Derivation of cartesian form from vector form\nLet the coordinates of the given point A be (x1, y1, z1) and the direction  ratios of\nthe line be a, b, c"
    },
    {
        "Chapter": "1",
        "sentence_range": "5720-5723",
        "Text": "(1)\nRemark If \n\u02c6\n\u02c6\n\u02c6\nb\nai\nbj\nck\n= + +\nr\n, then a, b, c are direction ratios of the line and conversely,,\nif a, b, c are direction ratios of a line, then \n\u02c6\n\u02c6\n\u02c6\n=\n+\n+\nbr\nai\nbj\nck will be the parallel to\nthe line Here, b should not be confused with |\nr\nb| Derivation of cartesian form from vector form\nLet the coordinates of the given point A be (x1, y1, z1) and the direction  ratios of\nthe line be a, b, c Consider the coordinates of any point P be (x, y, z)"
    },
    {
        "Chapter": "1",
        "sentence_range": "5721-5724",
        "Text": "Here, b should not be confused with |\nr\nb| Derivation of cartesian form from vector form\nLet the coordinates of the given point A be (x1, y1, z1) and the direction  ratios of\nthe line be a, b, c Consider the coordinates of any point P be (x, y, z) Then\nzk\njy\nix\nr\n\u02c6\n\u02c6\n\u02c6\n+\n+\nr=\n; \nk\nz\nj\ny\nxi\na\n\u02c6\n\u02c6\n\u02c6\n1\n1\n1\n+\n+\n=\nr\nand\n\u02c6\n\u02c6\n\u02c6\nb\na i\nb j\nc k\n=\n+\n+\nr\nSubstituting these values in (1) and equating the coefficients of \u02c6\n\u02c6\n,i\nj  and k\u02c6 , we get\nx = x1 + \u03bba;  y = y1 + \u03bb b;  z = z1+ \u03bbc"
    },
    {
        "Chapter": "1",
        "sentence_range": "5722-5725",
        "Text": "Derivation of cartesian form from vector form\nLet the coordinates of the given point A be (x1, y1, z1) and the direction  ratios of\nthe line be a, b, c Consider the coordinates of any point P be (x, y, z) Then\nzk\njy\nix\nr\n\u02c6\n\u02c6\n\u02c6\n+\n+\nr=\n; \nk\nz\nj\ny\nxi\na\n\u02c6\n\u02c6\n\u02c6\n1\n1\n1\n+\n+\n=\nr\nand\n\u02c6\n\u02c6\n\u02c6\nb\na i\nb j\nc k\n=\n+\n+\nr\nSubstituting these values in (1) and equating the coefficients of \u02c6\n\u02c6\n,i\nj  and k\u02c6 , we get\nx = x1 + \u03bba;  y = y1 + \u03bb b;  z = z1+ \u03bbc (2)\nFig 11"
    },
    {
        "Chapter": "1",
        "sentence_range": "5723-5726",
        "Text": "Consider the coordinates of any point P be (x, y, z) Then\nzk\njy\nix\nr\n\u02c6\n\u02c6\n\u02c6\n+\n+\nr=\n; \nk\nz\nj\ny\nxi\na\n\u02c6\n\u02c6\n\u02c6\n1\n1\n1\n+\n+\n=\nr\nand\n\u02c6\n\u02c6\n\u02c6\nb\na i\nb j\nc k\n=\n+\n+\nr\nSubstituting these values in (1) and equating the coefficients of \u02c6\n\u02c6\n,i\nj  and k\u02c6 , we get\nx = x1 + \u03bba;  y = y1 + \u03bb b;  z = z1+ \u03bbc (2)\nFig 11 4\n\u00a9 NCERT\nnot to be republished\nTHREE D IMENSIONAL G EOMETRY\n469\nThese are parametric equations of the line"
    },
    {
        "Chapter": "1",
        "sentence_range": "5724-5727",
        "Text": "Then\nzk\njy\nix\nr\n\u02c6\n\u02c6\n\u02c6\n+\n+\nr=\n; \nk\nz\nj\ny\nxi\na\n\u02c6\n\u02c6\n\u02c6\n1\n1\n1\n+\n+\n=\nr\nand\n\u02c6\n\u02c6\n\u02c6\nb\na i\nb j\nc k\n=\n+\n+\nr\nSubstituting these values in (1) and equating the coefficients of \u02c6\n\u02c6\n,i\nj  and k\u02c6 , we get\nx = x1 + \u03bba;  y = y1 + \u03bb b;  z = z1+ \u03bbc (2)\nFig 11 4\n\u00a9 NCERT\nnot to be republished\nTHREE D IMENSIONAL G EOMETRY\n469\nThese are parametric equations of the line Eliminating the parameter \u03bb from (2),\nwe get\n1\nx \u2013 x\na\n =\n1\n1\ny \u2013 y\n=z \u2013 z\nb\nc"
    },
    {
        "Chapter": "1",
        "sentence_range": "5725-5728",
        "Text": "(2)\nFig 11 4\n\u00a9 NCERT\nnot to be republished\nTHREE D IMENSIONAL G EOMETRY\n469\nThese are parametric equations of the line Eliminating the parameter \u03bb from (2),\nwe get\n1\nx \u2013 x\na\n =\n1\n1\ny \u2013 y\n=z \u2013 z\nb\nc (3)\nThis is the Cartesian equation of the line"
    },
    {
        "Chapter": "1",
        "sentence_range": "5726-5729",
        "Text": "4\n\u00a9 NCERT\nnot to be republished\nTHREE D IMENSIONAL G EOMETRY\n469\nThese are parametric equations of the line Eliminating the parameter \u03bb from (2),\nwe get\n1\nx \u2013 x\na\n =\n1\n1\ny \u2013 y\n=z \u2013 z\nb\nc (3)\nThis is the Cartesian equation of the line \ufffdNote   If l, m, n are the direction cosines of the line, the equation of the line is\n1\nx \u2013 x\nl\n =\n1\n1\ny \u2013 y\n=z \u2013 z\nm\nn\nExample 6 Find the vector and the Cartesian equations of the line through the point\n(5, 2, \u2013 4) and which is parallel to the vector \n\u02c6\n\u02c6\n\u02c6\n3\n2\n8\ni\nj\nk\n+\n\u2212"
    },
    {
        "Chapter": "1",
        "sentence_range": "5727-5730",
        "Text": "Eliminating the parameter \u03bb from (2),\nwe get\n1\nx \u2013 x\na\n =\n1\n1\ny \u2013 y\n=z \u2013 z\nb\nc (3)\nThis is the Cartesian equation of the line \ufffdNote   If l, m, n are the direction cosines of the line, the equation of the line is\n1\nx \u2013 x\nl\n =\n1\n1\ny \u2013 y\n=z \u2013 z\nm\nn\nExample 6 Find the vector and the Cartesian equations of the line through the point\n(5, 2, \u2013 4) and which is parallel to the vector \n\u02c6\n\u02c6\n\u02c6\n3\n2\n8\ni\nj\nk\n+\n\u2212 Solution We have\nar  =\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n5\n2\n4\nand\n3\n2\n8\ni\nj\nk\nb\ni\nj\nk\n+\n\u2212\n=\n+\n\u2212\nr\nTherefore, the vector equation of the line is\nrr =\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n5\n2\n4\n( 3\n2\n8\n)\ni\nj\nk\ni\nj\nk\n+\n\u2212\n+ \u03bb\n+\n\u2212\nNow, rr is the position vector of any point P(x, y, z) on the line"
    },
    {
        "Chapter": "1",
        "sentence_range": "5728-5731",
        "Text": "(3)\nThis is the Cartesian equation of the line \ufffdNote   If l, m, n are the direction cosines of the line, the equation of the line is\n1\nx \u2013 x\nl\n =\n1\n1\ny \u2013 y\n=z \u2013 z\nm\nn\nExample 6 Find the vector and the Cartesian equations of the line through the point\n(5, 2, \u2013 4) and which is parallel to the vector \n\u02c6\n\u02c6\n\u02c6\n3\n2\n8\ni\nj\nk\n+\n\u2212 Solution We have\nar  =\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n5\n2\n4\nand\n3\n2\n8\ni\nj\nk\nb\ni\nj\nk\n+\n\u2212\n=\n+\n\u2212\nr\nTherefore, the vector equation of the line is\nrr =\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n5\n2\n4\n( 3\n2\n8\n)\ni\nj\nk\ni\nj\nk\n+\n\u2212\n+ \u03bb\n+\n\u2212\nNow, rr is the position vector of any point P(x, y, z) on the line Therefore,\n\u02c6\n\u02c6\n\u02c6\nxi\ny j\nz k\n+\n+\n =\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n5\n2\n4\n( 3\n2\n8\n)\ni\nj\nk\ni\nj\nk\n+\n\u2212\n+ \u03bb\n+\n\u2212\n=\n$\n$\n(5\n3 )\n(2\n2 )\n( 4\n8 )\ni\nj\nk\n+ \u03bb\n+\n+ \u03bb\n+ \u2212\n\u2212 \u03bb\n$\nEliminating \u03bb , we get\n5\n3\nx \u2212\n =\n2\n4\n2\n8\ny\nz\n\u2212\n+\n=\n\u2212\nwhich is the equation of the line in Cartesian form"
    },
    {
        "Chapter": "1",
        "sentence_range": "5729-5732",
        "Text": "\ufffdNote   If l, m, n are the direction cosines of the line, the equation of the line is\n1\nx \u2013 x\nl\n =\n1\n1\ny \u2013 y\n=z \u2013 z\nm\nn\nExample 6 Find the vector and the Cartesian equations of the line through the point\n(5, 2, \u2013 4) and which is parallel to the vector \n\u02c6\n\u02c6\n\u02c6\n3\n2\n8\ni\nj\nk\n+\n\u2212 Solution We have\nar  =\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n5\n2\n4\nand\n3\n2\n8\ni\nj\nk\nb\ni\nj\nk\n+\n\u2212\n=\n+\n\u2212\nr\nTherefore, the vector equation of the line is\nrr =\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n5\n2\n4\n( 3\n2\n8\n)\ni\nj\nk\ni\nj\nk\n+\n\u2212\n+ \u03bb\n+\n\u2212\nNow, rr is the position vector of any point P(x, y, z) on the line Therefore,\n\u02c6\n\u02c6\n\u02c6\nxi\ny j\nz k\n+\n+\n =\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n5\n2\n4\n( 3\n2\n8\n)\ni\nj\nk\ni\nj\nk\n+\n\u2212\n+ \u03bb\n+\n\u2212\n=\n$\n$\n(5\n3 )\n(2\n2 )\n( 4\n8 )\ni\nj\nk\n+ \u03bb\n+\n+ \u03bb\n+ \u2212\n\u2212 \u03bb\n$\nEliminating \u03bb , we get\n5\n3\nx \u2212\n =\n2\n4\n2\n8\ny\nz\n\u2212\n+\n=\n\u2212\nwhich is the equation of the line in Cartesian form 11"
    },
    {
        "Chapter": "1",
        "sentence_range": "5730-5733",
        "Text": "Solution We have\nar  =\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n5\n2\n4\nand\n3\n2\n8\ni\nj\nk\nb\ni\nj\nk\n+\n\u2212\n=\n+\n\u2212\nr\nTherefore, the vector equation of the line is\nrr =\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n5\n2\n4\n( 3\n2\n8\n)\ni\nj\nk\ni\nj\nk\n+\n\u2212\n+ \u03bb\n+\n\u2212\nNow, rr is the position vector of any point P(x, y, z) on the line Therefore,\n\u02c6\n\u02c6\n\u02c6\nxi\ny j\nz k\n+\n+\n =\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n5\n2\n4\n( 3\n2\n8\n)\ni\nj\nk\ni\nj\nk\n+\n\u2212\n+ \u03bb\n+\n\u2212\n=\n$\n$\n(5\n3 )\n(2\n2 )\n( 4\n8 )\ni\nj\nk\n+ \u03bb\n+\n+ \u03bb\n+ \u2212\n\u2212 \u03bb\n$\nEliminating \u03bb , we get\n5\n3\nx \u2212\n =\n2\n4\n2\n8\ny\nz\n\u2212\n+\n=\n\u2212\nwhich is the equation of the line in Cartesian form 11 3"
    },
    {
        "Chapter": "1",
        "sentence_range": "5731-5734",
        "Text": "Therefore,\n\u02c6\n\u02c6\n\u02c6\nxi\ny j\nz k\n+\n+\n =\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n5\n2\n4\n( 3\n2\n8\n)\ni\nj\nk\ni\nj\nk\n+\n\u2212\n+ \u03bb\n+\n\u2212\n=\n$\n$\n(5\n3 )\n(2\n2 )\n( 4\n8 )\ni\nj\nk\n+ \u03bb\n+\n+ \u03bb\n+ \u2212\n\u2212 \u03bb\n$\nEliminating \u03bb , we get\n5\n3\nx \u2212\n =\n2\n4\n2\n8\ny\nz\n\u2212\n+\n=\n\u2212\nwhich is the equation of the line in Cartesian form 11 3 2 Equation of a line passing through two given points\nLet ar  and b\nr\n be the position vectors of two\npoints A (x1, y1, z1) and B(x 2, y 2, z2),\nrespectively that are lying on a line (Fig 11"
    },
    {
        "Chapter": "1",
        "sentence_range": "5732-5735",
        "Text": "11 3 2 Equation of a line passing through two given points\nLet ar  and b\nr\n be the position vectors of two\npoints A (x1, y1, z1) and B(x 2, y 2, z2),\nrespectively that are lying on a line (Fig 11 5)"
    },
    {
        "Chapter": "1",
        "sentence_range": "5733-5736",
        "Text": "3 2 Equation of a line passing through two given points\nLet ar  and b\nr\n be the position vectors of two\npoints A (x1, y1, z1) and B(x 2, y 2, z2),\nrespectively that are lying on a line (Fig 11 5) Let rr  be t he position vector of an\narbitrary point P(x, y, z), then P is a point on\nthe line if and only if AP\nr\na\n= \u2212\nuuur\nr\nr  and\nAB\nb\na\n=\n\u2212\nuuur\nr\nr  are collinear vectors"
    },
    {
        "Chapter": "1",
        "sentence_range": "5734-5737",
        "Text": "2 Equation of a line passing through two given points\nLet ar  and b\nr\n be the position vectors of two\npoints A (x1, y1, z1) and B(x 2, y 2, z2),\nrespectively that are lying on a line (Fig 11 5) Let rr  be t he position vector of an\narbitrary point P(x, y, z), then P is a point on\nthe line if and only if AP\nr\na\n= \u2212\nuuur\nr\nr  and\nAB\nb\na\n=\n\u2212\nuuur\nr\nr  are collinear vectors Therefore,\nP is on the line if and only if\n(\n)\nr\na\nb\na\n\u2212=\u03bb \u2212\nr\nr\nr\nr\nFig 11"
    },
    {
        "Chapter": "1",
        "sentence_range": "5735-5738",
        "Text": "5) Let rr  be t he position vector of an\narbitrary point P(x, y, z), then P is a point on\nthe line if and only if AP\nr\na\n= \u2212\nuuur\nr\nr  and\nAB\nb\na\n=\n\u2212\nuuur\nr\nr  are collinear vectors Therefore,\nP is on the line if and only if\n(\n)\nr\na\nb\na\n\u2212=\u03bb \u2212\nr\nr\nr\nr\nFig 11 5\n\u00a9 NCERT\nnot to be republished\n MATHEMATI CS\n470\nor\n(\n)\nr\na\nb\na\n=\n+ \u03bb\n\u2212\nr\nr\nr\nr , \u03bb \u2208 R"
    },
    {
        "Chapter": "1",
        "sentence_range": "5736-5739",
        "Text": "Let rr  be t he position vector of an\narbitrary point P(x, y, z), then P is a point on\nthe line if and only if AP\nr\na\n= \u2212\nuuur\nr\nr  and\nAB\nb\na\n=\n\u2212\nuuur\nr\nr  are collinear vectors Therefore,\nP is on the line if and only if\n(\n)\nr\na\nb\na\n\u2212=\u03bb \u2212\nr\nr\nr\nr\nFig 11 5\n\u00a9 NCERT\nnot to be republished\n MATHEMATI CS\n470\nor\n(\n)\nr\na\nb\na\n=\n+ \u03bb\n\u2212\nr\nr\nr\nr , \u03bb \u2208 R (1)\nThis is the vector equation of the line"
    },
    {
        "Chapter": "1",
        "sentence_range": "5737-5740",
        "Text": "Therefore,\nP is on the line if and only if\n(\n)\nr\na\nb\na\n\u2212=\u03bb \u2212\nr\nr\nr\nr\nFig 11 5\n\u00a9 NCERT\nnot to be republished\n MATHEMATI CS\n470\nor\n(\n)\nr\na\nb\na\n=\n+ \u03bb\n\u2212\nr\nr\nr\nr , \u03bb \u2208 R (1)\nThis is the vector equation of the line Derivation of cartesian form from vector form\nWe have\n1\n1\n1\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n,\nr\nxi\ny j\nz k a\nx i\ny j\nz k\n= + +\n= + +\nr\nr\nand \n2\n2\n2 \u02c6\n\u02c6\n\u02c6\n,\nb\nx i\ny j\nz k\n= + +\nr\nSubstituting these values in (1), we get\n$\n$\n$\n$\n$\n$\n1\n1\n1\n2\n1\n2\n1\n2\n1\n[(\n)\n(\n)\n(\n) ]\nxi\ny j\nz k\nx i\ny j\nz k\nx\nx i\ny\ny\nj\nz\nz k\n+\n+\n=\n+\n+\n+ \u03bb\n\u2212\n+\n\u2212\n+\n\u2212\n$\n$\n$\nEquating the like coefficients of \njk\ni\n\u02c6\n,\u02c6\n,\u02c6\n, we get\nx = x1 + \u03bb (x2 \u2013 x1); y = y1 + \u03bb (y2 \u2013 y1); z = z1 + \u03bb (z2 \u2013 z1)\nOn eliminating  \u03bb, we obtain\n1\n1\n1\n2\n1\n2\n1\n2\n1\nx\nx\ny\ny\nz\nz\nx\nx\ny\ny\nz\nz\n\u2212\n\u2212\n\u2212\n=\n=\n\u2212\n\u2212\n\u2212\nwhich is the equation of the line in Cartesian form"
    },
    {
        "Chapter": "1",
        "sentence_range": "5738-5741",
        "Text": "5\n\u00a9 NCERT\nnot to be republished\n MATHEMATI CS\n470\nor\n(\n)\nr\na\nb\na\n=\n+ \u03bb\n\u2212\nr\nr\nr\nr , \u03bb \u2208 R (1)\nThis is the vector equation of the line Derivation of cartesian form from vector form\nWe have\n1\n1\n1\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n,\nr\nxi\ny j\nz k a\nx i\ny j\nz k\n= + +\n= + +\nr\nr\nand \n2\n2\n2 \u02c6\n\u02c6\n\u02c6\n,\nb\nx i\ny j\nz k\n= + +\nr\nSubstituting these values in (1), we get\n$\n$\n$\n$\n$\n$\n1\n1\n1\n2\n1\n2\n1\n2\n1\n[(\n)\n(\n)\n(\n) ]\nxi\ny j\nz k\nx i\ny j\nz k\nx\nx i\ny\ny\nj\nz\nz k\n+\n+\n=\n+\n+\n+ \u03bb\n\u2212\n+\n\u2212\n+\n\u2212\n$\n$\n$\nEquating the like coefficients of \njk\ni\n\u02c6\n,\u02c6\n,\u02c6\n, we get\nx = x1 + \u03bb (x2 \u2013 x1); y = y1 + \u03bb (y2 \u2013 y1); z = z1 + \u03bb (z2 \u2013 z1)\nOn eliminating  \u03bb, we obtain\n1\n1\n1\n2\n1\n2\n1\n2\n1\nx\nx\ny\ny\nz\nz\nx\nx\ny\ny\nz\nz\n\u2212\n\u2212\n\u2212\n=\n=\n\u2212\n\u2212\n\u2212\nwhich is the equation of the line in Cartesian form Example 7 Find the vector equation for the line passing through the points (\u20131, 0, 2)\nand (3, 4, 6)"
    },
    {
        "Chapter": "1",
        "sentence_range": "5739-5742",
        "Text": "(1)\nThis is the vector equation of the line Derivation of cartesian form from vector form\nWe have\n1\n1\n1\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n,\nr\nxi\ny j\nz k a\nx i\ny j\nz k\n= + +\n= + +\nr\nr\nand \n2\n2\n2 \u02c6\n\u02c6\n\u02c6\n,\nb\nx i\ny j\nz k\n= + +\nr\nSubstituting these values in (1), we get\n$\n$\n$\n$\n$\n$\n1\n1\n1\n2\n1\n2\n1\n2\n1\n[(\n)\n(\n)\n(\n) ]\nxi\ny j\nz k\nx i\ny j\nz k\nx\nx i\ny\ny\nj\nz\nz k\n+\n+\n=\n+\n+\n+ \u03bb\n\u2212\n+\n\u2212\n+\n\u2212\n$\n$\n$\nEquating the like coefficients of \njk\ni\n\u02c6\n,\u02c6\n,\u02c6\n, we get\nx = x1 + \u03bb (x2 \u2013 x1); y = y1 + \u03bb (y2 \u2013 y1); z = z1 + \u03bb (z2 \u2013 z1)\nOn eliminating  \u03bb, we obtain\n1\n1\n1\n2\n1\n2\n1\n2\n1\nx\nx\ny\ny\nz\nz\nx\nx\ny\ny\nz\nz\n\u2212\n\u2212\n\u2212\n=\n=\n\u2212\n\u2212\n\u2212\nwhich is the equation of the line in Cartesian form Example 7 Find the vector equation for the line passing through the points (\u20131, 0, 2)\nand (3, 4, 6) Solution Let  ar  and  b\nr\n be the position vectors of the point A(\u2013 1, 0, 2) and B(3, 4, 6)"
    },
    {
        "Chapter": "1",
        "sentence_range": "5740-5743",
        "Text": "Derivation of cartesian form from vector form\nWe have\n1\n1\n1\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n,\nr\nxi\ny j\nz k a\nx i\ny j\nz k\n= + +\n= + +\nr\nr\nand \n2\n2\n2 \u02c6\n\u02c6\n\u02c6\n,\nb\nx i\ny j\nz k\n= + +\nr\nSubstituting these values in (1), we get\n$\n$\n$\n$\n$\n$\n1\n1\n1\n2\n1\n2\n1\n2\n1\n[(\n)\n(\n)\n(\n) ]\nxi\ny j\nz k\nx i\ny j\nz k\nx\nx i\ny\ny\nj\nz\nz k\n+\n+\n=\n+\n+\n+ \u03bb\n\u2212\n+\n\u2212\n+\n\u2212\n$\n$\n$\nEquating the like coefficients of \njk\ni\n\u02c6\n,\u02c6\n,\u02c6\n, we get\nx = x1 + \u03bb (x2 \u2013 x1); y = y1 + \u03bb (y2 \u2013 y1); z = z1 + \u03bb (z2 \u2013 z1)\nOn eliminating  \u03bb, we obtain\n1\n1\n1\n2\n1\n2\n1\n2\n1\nx\nx\ny\ny\nz\nz\nx\nx\ny\ny\nz\nz\n\u2212\n\u2212\n\u2212\n=\n=\n\u2212\n\u2212\n\u2212\nwhich is the equation of the line in Cartesian form Example 7 Find the vector equation for the line passing through the points (\u20131, 0, 2)\nand (3, 4, 6) Solution Let  ar  and  b\nr\n be the position vectors of the point A(\u2013 1, 0, 2) and B(3, 4, 6) Then\n\u02c6\n\u02c6\n2\na\ni\n=\u2212+k\nr\nand\n\u02c6\n\u02c6\n\u02c6\n3\n4\n6\nb\ni\nj\nk\n= + +\nr\nTherefore\n\u02c6\n\u02c6\n\u02c6\n4\n4\n4\nb\na\ni\nj\nk\n\u2212\n=\n+\n+\nr\nr\nLet rr  be the position vector of any point on the line"
    },
    {
        "Chapter": "1",
        "sentence_range": "5741-5744",
        "Text": "Example 7 Find the vector equation for the line passing through the points (\u20131, 0, 2)\nand (3, 4, 6) Solution Let  ar  and  b\nr\n be the position vectors of the point A(\u2013 1, 0, 2) and B(3, 4, 6) Then\n\u02c6\n\u02c6\n2\na\ni\n=\u2212+k\nr\nand\n\u02c6\n\u02c6\n\u02c6\n3\n4\n6\nb\ni\nj\nk\n= + +\nr\nTherefore\n\u02c6\n\u02c6\n\u02c6\n4\n4\n4\nb\na\ni\nj\nk\n\u2212\n=\n+\n+\nr\nr\nLet rr  be the position vector of any point on the line Then the vector equation of\nthe line is\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n2\n(4\n4\n4 )\nr\ni\nk\ni\nj\n=\u2212+ +\u03bb + +k\nr\nExample 8 The Cartesian equation of a line is\n3\n5\n6\n2\n4\n2\nx\ny\nz\n+ \u2212 +\n=\n=\nFind the vector equation for the line"
    },
    {
        "Chapter": "1",
        "sentence_range": "5742-5745",
        "Text": "Solution Let  ar  and  b\nr\n be the position vectors of the point A(\u2013 1, 0, 2) and B(3, 4, 6) Then\n\u02c6\n\u02c6\n2\na\ni\n=\u2212+k\nr\nand\n\u02c6\n\u02c6\n\u02c6\n3\n4\n6\nb\ni\nj\nk\n= + +\nr\nTherefore\n\u02c6\n\u02c6\n\u02c6\n4\n4\n4\nb\na\ni\nj\nk\n\u2212\n=\n+\n+\nr\nr\nLet rr  be the position vector of any point on the line Then the vector equation of\nthe line is\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n2\n(4\n4\n4 )\nr\ni\nk\ni\nj\n=\u2212+ +\u03bb + +k\nr\nExample 8 The Cartesian equation of a line is\n3\n5\n6\n2\n4\n2\nx\ny\nz\n+ \u2212 +\n=\n=\nFind the vector equation for the line Solution Comparing the given equation with the standard form\n1\n1\n1\nx\nx\ny\ny\nz\nz\na\nb\nc\n\u2212\n\u2212\n\u2212\n=\n=\nWe observe that\nx1 = \u2013 3, y1 = 5, z1 = \u2013 6; a = 2, b = 4, c = 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "5743-5746",
        "Text": "Then\n\u02c6\n\u02c6\n2\na\ni\n=\u2212+k\nr\nand\n\u02c6\n\u02c6\n\u02c6\n3\n4\n6\nb\ni\nj\nk\n= + +\nr\nTherefore\n\u02c6\n\u02c6\n\u02c6\n4\n4\n4\nb\na\ni\nj\nk\n\u2212\n=\n+\n+\nr\nr\nLet rr  be the position vector of any point on the line Then the vector equation of\nthe line is\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n2\n(4\n4\n4 )\nr\ni\nk\ni\nj\n=\u2212+ +\u03bb + +k\nr\nExample 8 The Cartesian equation of a line is\n3\n5\n6\n2\n4\n2\nx\ny\nz\n+ \u2212 +\n=\n=\nFind the vector equation for the line Solution Comparing the given equation with the standard form\n1\n1\n1\nx\nx\ny\ny\nz\nz\na\nb\nc\n\u2212\n\u2212\n\u2212\n=\n=\nWe observe that\nx1 = \u2013 3, y1 = 5, z1 = \u2013 6; a = 2, b = 4, c = 2 \u00a9 NCERT\nnot to be republished\nTHREE D IMENSIONAL G EOMETRY\n471\nThus, the required line passes through the point (\u2013 3, 5, \u2013 6) and is parallel to the\nvector \n\u02c6\n\u02c6\n\u02c6\n2\n4\n2\ni\nj\nk\n+\n+"
    },
    {
        "Chapter": "1",
        "sentence_range": "5744-5747",
        "Text": "Then the vector equation of\nthe line is\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n2\n(4\n4\n4 )\nr\ni\nk\ni\nj\n=\u2212+ +\u03bb + +k\nr\nExample 8 The Cartesian equation of a line is\n3\n5\n6\n2\n4\n2\nx\ny\nz\n+ \u2212 +\n=\n=\nFind the vector equation for the line Solution Comparing the given equation with the standard form\n1\n1\n1\nx\nx\ny\ny\nz\nz\na\nb\nc\n\u2212\n\u2212\n\u2212\n=\n=\nWe observe that\nx1 = \u2013 3, y1 = 5, z1 = \u2013 6; a = 2, b = 4, c = 2 \u00a9 NCERT\nnot to be republished\nTHREE D IMENSIONAL G EOMETRY\n471\nThus, the required line passes through the point (\u2013 3, 5, \u2013 6) and is parallel to the\nvector \n\u02c6\n\u02c6\n\u02c6\n2\n4\n2\ni\nj\nk\n+\n+ Let rr  be the position vector of any point on the line, then the\nvector equation of the line is given by\n\u02c6\n\u02c6\n\u02c6\n(\n3\n5\n6 )\nr\ni\nj\nr=\u2212 + \u2212k\n+ \u03bb \n\u02c6\n\u02c6\n\u02c6\n(2\n4\n2 )\ni\nj\nk\n+\n+\n11"
    },
    {
        "Chapter": "1",
        "sentence_range": "5745-5748",
        "Text": "Solution Comparing the given equation with the standard form\n1\n1\n1\nx\nx\ny\ny\nz\nz\na\nb\nc\n\u2212\n\u2212\n\u2212\n=\n=\nWe observe that\nx1 = \u2013 3, y1 = 5, z1 = \u2013 6; a = 2, b = 4, c = 2 \u00a9 NCERT\nnot to be republished\nTHREE D IMENSIONAL G EOMETRY\n471\nThus, the required line passes through the point (\u2013 3, 5, \u2013 6) and is parallel to the\nvector \n\u02c6\n\u02c6\n\u02c6\n2\n4\n2\ni\nj\nk\n+\n+ Let rr  be the position vector of any point on the line, then the\nvector equation of the line is given by\n\u02c6\n\u02c6\n\u02c6\n(\n3\n5\n6 )\nr\ni\nj\nr=\u2212 + \u2212k\n+ \u03bb \n\u02c6\n\u02c6\n\u02c6\n(2\n4\n2 )\ni\nj\nk\n+\n+\n11 4  Angle between Two Lines\nLet L1 and L2 be two lines passing through the origin\nand with direction ratios a1, b1, c1 and a2, b2, c2,\nrespectively"
    },
    {
        "Chapter": "1",
        "sentence_range": "5746-5749",
        "Text": "\u00a9 NCERT\nnot to be republished\nTHREE D IMENSIONAL G EOMETRY\n471\nThus, the required line passes through the point (\u2013 3, 5, \u2013 6) and is parallel to the\nvector \n\u02c6\n\u02c6\n\u02c6\n2\n4\n2\ni\nj\nk\n+\n+ Let rr  be the position vector of any point on the line, then the\nvector equation of the line is given by\n\u02c6\n\u02c6\n\u02c6\n(\n3\n5\n6 )\nr\ni\nj\nr=\u2212 + \u2212k\n+ \u03bb \n\u02c6\n\u02c6\n\u02c6\n(2\n4\n2 )\ni\nj\nk\n+\n+\n11 4  Angle between Two Lines\nLet L1 and L2 be two lines passing through the origin\nand with direction ratios a1, b1, c1 and a2, b2, c2,\nrespectively Let P be a point on L1 and Q be a point\non L2"
    },
    {
        "Chapter": "1",
        "sentence_range": "5747-5750",
        "Text": "Let rr  be the position vector of any point on the line, then the\nvector equation of the line is given by\n\u02c6\n\u02c6\n\u02c6\n(\n3\n5\n6 )\nr\ni\nj\nr=\u2212 + \u2212k\n+ \u03bb \n\u02c6\n\u02c6\n\u02c6\n(2\n4\n2 )\ni\nj\nk\n+\n+\n11 4  Angle between Two Lines\nLet L1 and L2 be two lines passing through the origin\nand with direction ratios a1, b1, c1 and a2, b2, c2,\nrespectively Let P be a point on L1 and Q be a point\non L2 Consider the directed lines OP and OQ as\ngiven in Fig 11"
    },
    {
        "Chapter": "1",
        "sentence_range": "5748-5751",
        "Text": "4  Angle between Two Lines\nLet L1 and L2 be two lines passing through the origin\nand with direction ratios a1, b1, c1 and a2, b2, c2,\nrespectively Let P be a point on L1 and Q be a point\non L2 Consider the directed lines OP and OQ as\ngiven in Fig 11 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "5749-5752",
        "Text": "Let P be a point on L1 and Q be a point\non L2 Consider the directed lines OP and OQ as\ngiven in Fig 11 6 Let \u03b8 be the acute angle between\nOP and OQ"
    },
    {
        "Chapter": "1",
        "sentence_range": "5750-5753",
        "Text": "Consider the directed lines OP and OQ as\ngiven in Fig 11 6 Let \u03b8 be the acute angle between\nOP and OQ Now recall that the directed line\nsegments OP and OQ are vectors with components\na1, b1, c1 and a2, b2, c2, respectively"
    },
    {
        "Chapter": "1",
        "sentence_range": "5751-5754",
        "Text": "6 Let \u03b8 be the acute angle between\nOP and OQ Now recall that the directed line\nsegments OP and OQ are vectors with components\na1, b1, c1 and a2, b2, c2, respectively Therefore, the\nangle \u03b8 between them is given by\ncos \u03b8 =\n1 2\n1 2\n1 2\n2\n2\n2\n2\n2\n2\n1\n1\n1\n2\n2\n2\na a\nb b\nc c\na\nb\nc\na\nb\nc\n+\n+\n+\n+\n+\n+"
    },
    {
        "Chapter": "1",
        "sentence_range": "5752-5755",
        "Text": "Let \u03b8 be the acute angle between\nOP and OQ Now recall that the directed line\nsegments OP and OQ are vectors with components\na1, b1, c1 and a2, b2, c2, respectively Therefore, the\nangle \u03b8 between them is given by\ncos \u03b8 =\n1 2\n1 2\n1 2\n2\n2\n2\n2\n2\n2\n1\n1\n1\n2\n2\n2\na a\nb b\nc c\na\nb\nc\na\nb\nc\n+\n+\n+\n+\n+\n+ (1)\nThe angle between the lines in terms of sin \u03b8 is given by\nsin \u03b8 =\n2\n1\ncos\n\u2212 \u03b8\n=\n(\n)(\n)\n2\n1 2\n1 2\n1 2\n2\n2\n2\n2\n2\n2\n1\n1\n1\n2\n2\n2\n(\n)\n1\na a\nbb\nc c\na\nb\nc\na\nb\nc\n+\n+\n\u2212\n+\n+\n+\n+\n=\n(\n)(\n) (\n)\n(\n) (\n)\n2\n2\n2\n2\n2\n2\n2\n1\n1\n1\n2\n2\n2\n1 2\n1 2\n1 2\n2\n2\n2\n2\n2\n2\n1\n1\n1\n2\n2\n2\na\nb\nc\na\nb\nc\na a\nbb\nc c\na\nb\nc\na\nb\nc\n+\n+\n+\n+\n\u2212\n+\n+\n+\n+\n+\n+\n=\n2\n2\n2\n1\n2\n2\n1\n1\n2\n2\n1\n1\n2\n2\n1\n2\n2\n2\n2\n2\n2\n1\n1\n1\n2\n2\n2\n(\n)\n(\n)\n(\n)\n\u2212\n+\n\u2212\n+\n\u2212\n+ +\n+ +\na b\na b\nb c\nb c\nc a\nc a\na\nb\nc\na\nb\nc"
    },
    {
        "Chapter": "1",
        "sentence_range": "5753-5756",
        "Text": "Now recall that the directed line\nsegments OP and OQ are vectors with components\na1, b1, c1 and a2, b2, c2, respectively Therefore, the\nangle \u03b8 between them is given by\ncos \u03b8 =\n1 2\n1 2\n1 2\n2\n2\n2\n2\n2\n2\n1\n1\n1\n2\n2\n2\na a\nb b\nc c\na\nb\nc\na\nb\nc\n+\n+\n+\n+\n+\n+ (1)\nThe angle between the lines in terms of sin \u03b8 is given by\nsin \u03b8 =\n2\n1\ncos\n\u2212 \u03b8\n=\n(\n)(\n)\n2\n1 2\n1 2\n1 2\n2\n2\n2\n2\n2\n2\n1\n1\n1\n2\n2\n2\n(\n)\n1\na a\nbb\nc c\na\nb\nc\na\nb\nc\n+\n+\n\u2212\n+\n+\n+\n+\n=\n(\n)(\n) (\n)\n(\n) (\n)\n2\n2\n2\n2\n2\n2\n2\n1\n1\n1\n2\n2\n2\n1 2\n1 2\n1 2\n2\n2\n2\n2\n2\n2\n1\n1\n1\n2\n2\n2\na\nb\nc\na\nb\nc\na a\nbb\nc c\na\nb\nc\na\nb\nc\n+\n+\n+\n+\n\u2212\n+\n+\n+\n+\n+\n+\n=\n2\n2\n2\n1\n2\n2\n1\n1\n2\n2\n1\n1\n2\n2\n1\n2\n2\n2\n2\n2\n2\n1\n1\n1\n2\n2\n2\n(\n)\n(\n)\n(\n)\n\u2212\n+\n\u2212\n+\n\u2212\n+ +\n+ +\na b\na b\nb c\nb c\nc a\nc a\na\nb\nc\na\nb\nc (2)\n\ufffdNote   In case the lines L1 and L2 do not pass through the origin, we may take\nlines \n1\n\u2032L andL2\n\u2032 which are parallel to L1 and L2 respectively and pass through\nthe origin"
    },
    {
        "Chapter": "1",
        "sentence_range": "5754-5757",
        "Text": "Therefore, the\nangle \u03b8 between them is given by\ncos \u03b8 =\n1 2\n1 2\n1 2\n2\n2\n2\n2\n2\n2\n1\n1\n1\n2\n2\n2\na a\nb b\nc c\na\nb\nc\na\nb\nc\n+\n+\n+\n+\n+\n+ (1)\nThe angle between the lines in terms of sin \u03b8 is given by\nsin \u03b8 =\n2\n1\ncos\n\u2212 \u03b8\n=\n(\n)(\n)\n2\n1 2\n1 2\n1 2\n2\n2\n2\n2\n2\n2\n1\n1\n1\n2\n2\n2\n(\n)\n1\na a\nbb\nc c\na\nb\nc\na\nb\nc\n+\n+\n\u2212\n+\n+\n+\n+\n=\n(\n)(\n) (\n)\n(\n) (\n)\n2\n2\n2\n2\n2\n2\n2\n1\n1\n1\n2\n2\n2\n1 2\n1 2\n1 2\n2\n2\n2\n2\n2\n2\n1\n1\n1\n2\n2\n2\na\nb\nc\na\nb\nc\na a\nbb\nc c\na\nb\nc\na\nb\nc\n+\n+\n+\n+\n\u2212\n+\n+\n+\n+\n+\n+\n=\n2\n2\n2\n1\n2\n2\n1\n1\n2\n2\n1\n1\n2\n2\n1\n2\n2\n2\n2\n2\n2\n1\n1\n1\n2\n2\n2\n(\n)\n(\n)\n(\n)\n\u2212\n+\n\u2212\n+\n\u2212\n+ +\n+ +\na b\na b\nb c\nb c\nc a\nc a\na\nb\nc\na\nb\nc (2)\n\ufffdNote   In case the lines L1 and L2 do not pass through the origin, we may take\nlines \n1\n\u2032L andL2\n\u2032 which are parallel to L1 and L2 respectively and pass through\nthe origin Fig 11"
    },
    {
        "Chapter": "1",
        "sentence_range": "5755-5758",
        "Text": "(1)\nThe angle between the lines in terms of sin \u03b8 is given by\nsin \u03b8 =\n2\n1\ncos\n\u2212 \u03b8\n=\n(\n)(\n)\n2\n1 2\n1 2\n1 2\n2\n2\n2\n2\n2\n2\n1\n1\n1\n2\n2\n2\n(\n)\n1\na a\nbb\nc c\na\nb\nc\na\nb\nc\n+\n+\n\u2212\n+\n+\n+\n+\n=\n(\n)(\n) (\n)\n(\n) (\n)\n2\n2\n2\n2\n2\n2\n2\n1\n1\n1\n2\n2\n2\n1 2\n1 2\n1 2\n2\n2\n2\n2\n2\n2\n1\n1\n1\n2\n2\n2\na\nb\nc\na\nb\nc\na a\nbb\nc c\na\nb\nc\na\nb\nc\n+\n+\n+\n+\n\u2212\n+\n+\n+\n+\n+\n+\n=\n2\n2\n2\n1\n2\n2\n1\n1\n2\n2\n1\n1\n2\n2\n1\n2\n2\n2\n2\n2\n2\n1\n1\n1\n2\n2\n2\n(\n)\n(\n)\n(\n)\n\u2212\n+\n\u2212\n+\n\u2212\n+ +\n+ +\na b\na b\nb c\nb c\nc a\nc a\na\nb\nc\na\nb\nc (2)\n\ufffdNote   In case the lines L1 and L2 do not pass through the origin, we may take\nlines \n1\n\u2032L andL2\n\u2032 which are parallel to L1 and L2 respectively and pass through\nthe origin Fig 11 6\n\u00a9 NCERT\nnot to be republished\n MATHEMATI CS\n472\nIf instead of direction ratios for the lines L1 and L2, direction cosines, namely,\nl1, m 1, n1 for L1 and l2, m 2, n2 for L2 are given, then (1) and (2) takes the following form:\ncos \u03b8 = |l1 l2 + m 1m 2 + n1n2|    (as \n2\n2\n2\n1\n1\n1\n1\nl\nm\nn\n+\n+\n=\n2\n2\n2\n2\n2\n2\nl\nm\nn\n=\n+\n+\n)"
    },
    {
        "Chapter": "1",
        "sentence_range": "5756-5759",
        "Text": "(2)\n\ufffdNote   In case the lines L1 and L2 do not pass through the origin, we may take\nlines \n1\n\u2032L andL2\n\u2032 which are parallel to L1 and L2 respectively and pass through\nthe origin Fig 11 6\n\u00a9 NCERT\nnot to be republished\n MATHEMATI CS\n472\nIf instead of direction ratios for the lines L1 and L2, direction cosines, namely,\nl1, m 1, n1 for L1 and l2, m 2, n2 for L2 are given, then (1) and (2) takes the following form:\ncos \u03b8 = |l1 l2 + m 1m 2 + n1n2|    (as \n2\n2\n2\n1\n1\n1\n1\nl\nm\nn\n+\n+\n=\n2\n2\n2\n2\n2\n2\nl\nm\nn\n=\n+\n+\n) (3)\nand\nsin \u03b8 =\n(\n)\n2\n2\n2\n1\n2\n2\n1\n1\n2\n2\n1\n1 2\n2 1\n(\n)\n(\n)\nl m\nl m\nm n\nm n\nn l\nn l\n\u2212\n\u2212\n\u2212\n+\n\u2212"
    },
    {
        "Chapter": "1",
        "sentence_range": "5757-5760",
        "Text": "Fig 11 6\n\u00a9 NCERT\nnot to be republished\n MATHEMATI CS\n472\nIf instead of direction ratios for the lines L1 and L2, direction cosines, namely,\nl1, m 1, n1 for L1 and l2, m 2, n2 for L2 are given, then (1) and (2) takes the following form:\ncos \u03b8 = |l1 l2 + m 1m 2 + n1n2|    (as \n2\n2\n2\n1\n1\n1\n1\nl\nm\nn\n+\n+\n=\n2\n2\n2\n2\n2\n2\nl\nm\nn\n=\n+\n+\n) (3)\nand\nsin \u03b8 =\n(\n)\n2\n2\n2\n1\n2\n2\n1\n1\n2\n2\n1\n1 2\n2 1\n(\n)\n(\n)\nl m\nl m\nm n\nm n\nn l\nn l\n\u2212\n\u2212\n\u2212\n+\n\u2212 (4)\nTwo lines with direction ratios a1, b1, c1 and a2, b2, c2 are\n(i)\nperpendicular i"
    },
    {
        "Chapter": "1",
        "sentence_range": "5758-5761",
        "Text": "6\n\u00a9 NCERT\nnot to be republished\n MATHEMATI CS\n472\nIf instead of direction ratios for the lines L1 and L2, direction cosines, namely,\nl1, m 1, n1 for L1 and l2, m 2, n2 for L2 are given, then (1) and (2) takes the following form:\ncos \u03b8 = |l1 l2 + m 1m 2 + n1n2|    (as \n2\n2\n2\n1\n1\n1\n1\nl\nm\nn\n+\n+\n=\n2\n2\n2\n2\n2\n2\nl\nm\nn\n=\n+\n+\n) (3)\nand\nsin \u03b8 =\n(\n)\n2\n2\n2\n1\n2\n2\n1\n1\n2\n2\n1\n1 2\n2 1\n(\n)\n(\n)\nl m\nl m\nm n\nm n\nn l\nn l\n\u2212\n\u2212\n\u2212\n+\n\u2212 (4)\nTwo lines with direction ratios a1, b1, c1 and a2, b2, c2 are\n(i)\nperpendicular i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "5759-5762",
        "Text": "(3)\nand\nsin \u03b8 =\n(\n)\n2\n2\n2\n1\n2\n2\n1\n1\n2\n2\n1\n1 2\n2 1\n(\n)\n(\n)\nl m\nl m\nm n\nm n\nn l\nn l\n\u2212\n\u2212\n\u2212\n+\n\u2212 (4)\nTwo lines with direction ratios a1, b1, c1 and a2, b2, c2 are\n(i)\nperpendicular i e if \u03b8 = 90\u00b0 by (1)\na1a2 + b1b2 + c1c2 = 0\n(ii)\nparallel i"
    },
    {
        "Chapter": "1",
        "sentence_range": "5760-5763",
        "Text": "(4)\nTwo lines with direction ratios a1, b1, c1 and a2, b2, c2 are\n(i)\nperpendicular i e if \u03b8 = 90\u00b0 by (1)\na1a2 + b1b2 + c1c2 = 0\n(ii)\nparallel i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "5761-5764",
        "Text": "e if \u03b8 = 90\u00b0 by (1)\na1a2 + b1b2 + c1c2 = 0\n(ii)\nparallel i e if \u03b8 = 0 by (2)\n1\n2\na\na  =\n1\n1\n2\n2\nb\nc\nb\nc\n=\nNow, we find the angle between two lines when their equations are given"
    },
    {
        "Chapter": "1",
        "sentence_range": "5762-5765",
        "Text": "if \u03b8 = 90\u00b0 by (1)\na1a2 + b1b2 + c1c2 = 0\n(ii)\nparallel i e if \u03b8 = 0 by (2)\n1\n2\na\na  =\n1\n1\n2\n2\nb\nc\nb\nc\n=\nNow, we find the angle between two lines when their equations are given If \u03b8 is\nacute the angle between the lines\nrr  =\n1\n1\na\nb\n+\u03bb\nur\n and  rr  = \n2\n2\na\nb\n+\u00b5\nr\nr\nthen\ncos\u03b8 =\n1\n2\n1\n2\nb\nb\nb\nb\n\u22c5\nr\nr\nr\nr\nIn Cartesian form, if \u03b8 is the angle between the lines\n1\n1\nx\nx\na\n\u2212\n =\n1\n1\n1\n1\ny\ny\nz\nz\nb\nc\n\u2212\n\u2212\n="
    },
    {
        "Chapter": "1",
        "sentence_range": "5763-5766",
        "Text": "e if \u03b8 = 0 by (2)\n1\n2\na\na  =\n1\n1\n2\n2\nb\nc\nb\nc\n=\nNow, we find the angle between two lines when their equations are given If \u03b8 is\nacute the angle between the lines\nrr  =\n1\n1\na\nb\n+\u03bb\nur\n and  rr  = \n2\n2\na\nb\n+\u00b5\nr\nr\nthen\ncos\u03b8 =\n1\n2\n1\n2\nb\nb\nb\nb\n\u22c5\nr\nr\nr\nr\nIn Cartesian form, if \u03b8 is the angle between the lines\n1\n1\nx\nx\na\n\u2212\n =\n1\n1\n1\n1\ny\ny\nz\nz\nb\nc\n\u2212\n\u2212\n= (1)\nand\n2\n2\nx\nx\na\n\u2212\n =\n2\n2\n2\n2\ny\ny\nz\nz\nb\nc\n\u2212\n\u2212\n="
    },
    {
        "Chapter": "1",
        "sentence_range": "5764-5767",
        "Text": "if \u03b8 = 0 by (2)\n1\n2\na\na  =\n1\n1\n2\n2\nb\nc\nb\nc\n=\nNow, we find the angle between two lines when their equations are given If \u03b8 is\nacute the angle between the lines\nrr  =\n1\n1\na\nb\n+\u03bb\nur\n and  rr  = \n2\n2\na\nb\n+\u00b5\nr\nr\nthen\ncos\u03b8 =\n1\n2\n1\n2\nb\nb\nb\nb\n\u22c5\nr\nr\nr\nr\nIn Cartesian form, if \u03b8 is the angle between the lines\n1\n1\nx\nx\na\n\u2212\n =\n1\n1\n1\n1\ny\ny\nz\nz\nb\nc\n\u2212\n\u2212\n= (1)\nand\n2\n2\nx\nx\na\n\u2212\n =\n2\n2\n2\n2\ny\ny\nz\nz\nb\nc\n\u2212\n\u2212\n= (2)\nwhere, a1, b1, c1 and a2, b2, c2 are the direction ratios of the lines (1) and (2), respectively,\nthen\ncos \u03b8 =\n1 2\n1 2\n1 2\n2\n2\n2\n2\n2\n2\n1\n1\n1\n2\n2\n2\na a\nb b\nc c\na\nb\nc\na\nb\nc\n+\n+\n+\n+\n+\n+\nExample 9 Find the angle between the pair of lines given by\nrr  =\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n3\n2\n4\n(\n2\n2 )\ni\nj\nk\ni\nj\nk\n+\n\u2212\n+ \u03bb\n+\n+\nand\nrr  =\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n5\n2\n(3\n2\n6 )\ni\nj\ni\nj\nk\n\u2212\n+ \u00b5\n+\n+\n\u00a9 NCERT\nnot to be republished\nTHREE D IMENSIONAL G EOMETRY\n473\nSolution Here 1b\nr\n = \n\u02c6\n\u02c6\n2\u02c6\n2\ni\nj\nk\n+\n+\n and \n2b\nr\n = \n\u02c6\n\u02c6\n\u02c6\n3\n2\n6\ni\nj\nk\n+\n+\nThe angle \u03b8 between the two lines is given by\ncos \u03b8 =\n1\n2\n1\n2\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(\n2\n2 ) (3\n2\n6 )\n1\n4\n4\n9\n4\n36\nb b\ni\nj\nk\ni\nj\nk\nb b\n\u22c5\n+\n+\n\u22c5\n+\n+\n=\n+ +\n+\n+\nr\nr\nr\nr\n=\n3\n4 12\n19\n3 7\n21\n+\n+\n=\n\u00d7\nHence\n\u03b8 = cos\u20131 \n19\n21\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\nExample 10 Find the angle between the pair of lines\n3\n3\nx +\n =\n1\n3\n5\n4\ny\nz\n\u2212\n+\n=\nand\n1\n1\nx +\n =\n4\n5\n1\n2\ny\nz\n\u2212\n\u2212\n=\nSolution The direction ratios of the first line are 3, 5, 4 and the direction ratios of the\nsecond line are 1, 1, 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "5765-5768",
        "Text": "If \u03b8 is\nacute the angle between the lines\nrr  =\n1\n1\na\nb\n+\u03bb\nur\n and  rr  = \n2\n2\na\nb\n+\u00b5\nr\nr\nthen\ncos\u03b8 =\n1\n2\n1\n2\nb\nb\nb\nb\n\u22c5\nr\nr\nr\nr\nIn Cartesian form, if \u03b8 is the angle between the lines\n1\n1\nx\nx\na\n\u2212\n =\n1\n1\n1\n1\ny\ny\nz\nz\nb\nc\n\u2212\n\u2212\n= (1)\nand\n2\n2\nx\nx\na\n\u2212\n =\n2\n2\n2\n2\ny\ny\nz\nz\nb\nc\n\u2212\n\u2212\n= (2)\nwhere, a1, b1, c1 and a2, b2, c2 are the direction ratios of the lines (1) and (2), respectively,\nthen\ncos \u03b8 =\n1 2\n1 2\n1 2\n2\n2\n2\n2\n2\n2\n1\n1\n1\n2\n2\n2\na a\nb b\nc c\na\nb\nc\na\nb\nc\n+\n+\n+\n+\n+\n+\nExample 9 Find the angle between the pair of lines given by\nrr  =\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n3\n2\n4\n(\n2\n2 )\ni\nj\nk\ni\nj\nk\n+\n\u2212\n+ \u03bb\n+\n+\nand\nrr  =\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n5\n2\n(3\n2\n6 )\ni\nj\ni\nj\nk\n\u2212\n+ \u00b5\n+\n+\n\u00a9 NCERT\nnot to be republished\nTHREE D IMENSIONAL G EOMETRY\n473\nSolution Here 1b\nr\n = \n\u02c6\n\u02c6\n2\u02c6\n2\ni\nj\nk\n+\n+\n and \n2b\nr\n = \n\u02c6\n\u02c6\n\u02c6\n3\n2\n6\ni\nj\nk\n+\n+\nThe angle \u03b8 between the two lines is given by\ncos \u03b8 =\n1\n2\n1\n2\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(\n2\n2 ) (3\n2\n6 )\n1\n4\n4\n9\n4\n36\nb b\ni\nj\nk\ni\nj\nk\nb b\n\u22c5\n+\n+\n\u22c5\n+\n+\n=\n+ +\n+\n+\nr\nr\nr\nr\n=\n3\n4 12\n19\n3 7\n21\n+\n+\n=\n\u00d7\nHence\n\u03b8 = cos\u20131 \n19\n21\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\nExample 10 Find the angle between the pair of lines\n3\n3\nx +\n =\n1\n3\n5\n4\ny\nz\n\u2212\n+\n=\nand\n1\n1\nx +\n =\n4\n5\n1\n2\ny\nz\n\u2212\n\u2212\n=\nSolution The direction ratios of the first line are 3, 5, 4 and the direction ratios of the\nsecond line are 1, 1, 2 If \u03b8 is the angle between them, then\ncos \u03b8 =\n2\n2\n2\n2\n2\n2\n3"
    },
    {
        "Chapter": "1",
        "sentence_range": "5766-5769",
        "Text": "(1)\nand\n2\n2\nx\nx\na\n\u2212\n =\n2\n2\n2\n2\ny\ny\nz\nz\nb\nc\n\u2212\n\u2212\n= (2)\nwhere, a1, b1, c1 and a2, b2, c2 are the direction ratios of the lines (1) and (2), respectively,\nthen\ncos \u03b8 =\n1 2\n1 2\n1 2\n2\n2\n2\n2\n2\n2\n1\n1\n1\n2\n2\n2\na a\nb b\nc c\na\nb\nc\na\nb\nc\n+\n+\n+\n+\n+\n+\nExample 9 Find the angle between the pair of lines given by\nrr  =\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n3\n2\n4\n(\n2\n2 )\ni\nj\nk\ni\nj\nk\n+\n\u2212\n+ \u03bb\n+\n+\nand\nrr  =\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n5\n2\n(3\n2\n6 )\ni\nj\ni\nj\nk\n\u2212\n+ \u00b5\n+\n+\n\u00a9 NCERT\nnot to be republished\nTHREE D IMENSIONAL G EOMETRY\n473\nSolution Here 1b\nr\n = \n\u02c6\n\u02c6\n2\u02c6\n2\ni\nj\nk\n+\n+\n and \n2b\nr\n = \n\u02c6\n\u02c6\n\u02c6\n3\n2\n6\ni\nj\nk\n+\n+\nThe angle \u03b8 between the two lines is given by\ncos \u03b8 =\n1\n2\n1\n2\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(\n2\n2 ) (3\n2\n6 )\n1\n4\n4\n9\n4\n36\nb b\ni\nj\nk\ni\nj\nk\nb b\n\u22c5\n+\n+\n\u22c5\n+\n+\n=\n+ +\n+\n+\nr\nr\nr\nr\n=\n3\n4 12\n19\n3 7\n21\n+\n+\n=\n\u00d7\nHence\n\u03b8 = cos\u20131 \n19\n21\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\nExample 10 Find the angle between the pair of lines\n3\n3\nx +\n =\n1\n3\n5\n4\ny\nz\n\u2212\n+\n=\nand\n1\n1\nx +\n =\n4\n5\n1\n2\ny\nz\n\u2212\n\u2212\n=\nSolution The direction ratios of the first line are 3, 5, 4 and the direction ratios of the\nsecond line are 1, 1, 2 If \u03b8 is the angle between them, then\ncos \u03b8 =\n2\n2\n2\n2\n2\n2\n3 1\n5"
    },
    {
        "Chapter": "1",
        "sentence_range": "5767-5770",
        "Text": "(2)\nwhere, a1, b1, c1 and a2, b2, c2 are the direction ratios of the lines (1) and (2), respectively,\nthen\ncos \u03b8 =\n1 2\n1 2\n1 2\n2\n2\n2\n2\n2\n2\n1\n1\n1\n2\n2\n2\na a\nb b\nc c\na\nb\nc\na\nb\nc\n+\n+\n+\n+\n+\n+\nExample 9 Find the angle between the pair of lines given by\nrr  =\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n3\n2\n4\n(\n2\n2 )\ni\nj\nk\ni\nj\nk\n+\n\u2212\n+ \u03bb\n+\n+\nand\nrr  =\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n5\n2\n(3\n2\n6 )\ni\nj\ni\nj\nk\n\u2212\n+ \u00b5\n+\n+\n\u00a9 NCERT\nnot to be republished\nTHREE D IMENSIONAL G EOMETRY\n473\nSolution Here 1b\nr\n = \n\u02c6\n\u02c6\n2\u02c6\n2\ni\nj\nk\n+\n+\n and \n2b\nr\n = \n\u02c6\n\u02c6\n\u02c6\n3\n2\n6\ni\nj\nk\n+\n+\nThe angle \u03b8 between the two lines is given by\ncos \u03b8 =\n1\n2\n1\n2\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(\n2\n2 ) (3\n2\n6 )\n1\n4\n4\n9\n4\n36\nb b\ni\nj\nk\ni\nj\nk\nb b\n\u22c5\n+\n+\n\u22c5\n+\n+\n=\n+ +\n+\n+\nr\nr\nr\nr\n=\n3\n4 12\n19\n3 7\n21\n+\n+\n=\n\u00d7\nHence\n\u03b8 = cos\u20131 \n19\n21\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\nExample 10 Find the angle between the pair of lines\n3\n3\nx +\n =\n1\n3\n5\n4\ny\nz\n\u2212\n+\n=\nand\n1\n1\nx +\n =\n4\n5\n1\n2\ny\nz\n\u2212\n\u2212\n=\nSolution The direction ratios of the first line are 3, 5, 4 and the direction ratios of the\nsecond line are 1, 1, 2 If \u03b8 is the angle between them, then\ncos \u03b8 =\n2\n2\n2\n2\n2\n2\n3 1\n5 1\n4"
    },
    {
        "Chapter": "1",
        "sentence_range": "5768-5771",
        "Text": "If \u03b8 is the angle between them, then\ncos \u03b8 =\n2\n2\n2\n2\n2\n2\n3 1\n5 1\n4 2\n16\n16\n8 3\n15\n50\n6\n5 2\n6\n3\n5\n4\n1\n1\n2\n+\n+\n=\n=\n=\n+\n+\n+\n+\nHence, the required angle is cos\u20131 8\n3\n15\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8"
    },
    {
        "Chapter": "1",
        "sentence_range": "5769-5772",
        "Text": "1\n5 1\n4 2\n16\n16\n8 3\n15\n50\n6\n5 2\n6\n3\n5\n4\n1\n1\n2\n+\n+\n=\n=\n=\n+\n+\n+\n+\nHence, the required angle is cos\u20131 8\n3\n15\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8 11"
    },
    {
        "Chapter": "1",
        "sentence_range": "5770-5773",
        "Text": "1\n4 2\n16\n16\n8 3\n15\n50\n6\n5 2\n6\n3\n5\n4\n1\n1\n2\n+\n+\n=\n=\n=\n+\n+\n+\n+\nHence, the required angle is cos\u20131 8\n3\n15\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8 11 5  Shortest Distance between Two Lines\nIf two lines in space intersect at a point, then the shortest distance between them is\nzero"
    },
    {
        "Chapter": "1",
        "sentence_range": "5771-5774",
        "Text": "2\n16\n16\n8 3\n15\n50\n6\n5 2\n6\n3\n5\n4\n1\n1\n2\n+\n+\n=\n=\n=\n+\n+\n+\n+\nHence, the required angle is cos\u20131 8\n3\n15\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8 11 5  Shortest Distance between Two Lines\nIf two lines in space intersect at a point, then the shortest distance between them is\nzero Also, if two lines in space are parallel,\nthen the shortest distance between them\nwill be the perpendicular distance, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "5772-5775",
        "Text": "11 5  Shortest Distance between Two Lines\nIf two lines in space intersect at a point, then the shortest distance between them is\nzero Also, if two lines in space are parallel,\nthen the shortest distance between them\nwill be the perpendicular distance, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "5773-5776",
        "Text": "5  Shortest Distance between Two Lines\nIf two lines in space intersect at a point, then the shortest distance between them is\nzero Also, if two lines in space are parallel,\nthen the shortest distance between them\nwill be the perpendicular distance, i e the\nlength of the perpendicular drawn from a\npoint on one line onto the other line"
    },
    {
        "Chapter": "1",
        "sentence_range": "5774-5777",
        "Text": "Also, if two lines in space are parallel,\nthen the shortest distance between them\nwill be the perpendicular distance, i e the\nlength of the perpendicular drawn from a\npoint on one line onto the other line Further, in a space, there are lines which\nare neither intersecting nor parallel"
    },
    {
        "Chapter": "1",
        "sentence_range": "5775-5778",
        "Text": "e the\nlength of the perpendicular drawn from a\npoint on one line onto the other line Further, in a space, there are lines which\nare neither intersecting nor parallel In fact,\nsuch pair of lines are non coplanar and\nare called skew lines"
    },
    {
        "Chapter": "1",
        "sentence_range": "5776-5779",
        "Text": "the\nlength of the perpendicular drawn from a\npoint on one line onto the other line Further, in a space, there are lines which\nare neither intersecting nor parallel In fact,\nsuch pair of lines are non coplanar and\nare called skew lines For example, let us\nconsider a room of size 1, 3, 2 units along\nx, y and z-axes respectively Fig 11"
    },
    {
        "Chapter": "1",
        "sentence_range": "5777-5780",
        "Text": "Further, in a space, there are lines which\nare neither intersecting nor parallel In fact,\nsuch pair of lines are non coplanar and\nare called skew lines For example, let us\nconsider a room of size 1, 3, 2 units along\nx, y and z-axes respectively Fig 11 7"
    },
    {
        "Chapter": "1",
        "sentence_range": "5778-5781",
        "Text": "In fact,\nsuch pair of lines are non coplanar and\nare called skew lines For example, let us\nconsider a room of size 1, 3, 2 units along\nx, y and z-axes respectively Fig 11 7 Fig 11"
    },
    {
        "Chapter": "1",
        "sentence_range": "5779-5782",
        "Text": "For example, let us\nconsider a room of size 1, 3, 2 units along\nx, y and z-axes respectively Fig 11 7 Fig 11 7\n\u00a9 NCERT\nnot to be republished\n MATHEMATI CS\n474\nl2\nS\nT\nQ\nP\nl1\nThe line GE that goes diagonally across the ceiling and the line DB passes through\none corner of the ceiling directly above A and goes diagonally down the wall"
    },
    {
        "Chapter": "1",
        "sentence_range": "5780-5783",
        "Text": "7 Fig 11 7\n\u00a9 NCERT\nnot to be republished\n MATHEMATI CS\n474\nl2\nS\nT\nQ\nP\nl1\nThe line GE that goes diagonally across the ceiling and the line DB passes through\none corner of the ceiling directly above A and goes diagonally down the wall These\nlines are skew because they are not parallel and also never meet"
    },
    {
        "Chapter": "1",
        "sentence_range": "5781-5784",
        "Text": "Fig 11 7\n\u00a9 NCERT\nnot to be republished\n MATHEMATI CS\n474\nl2\nS\nT\nQ\nP\nl1\nThe line GE that goes diagonally across the ceiling and the line DB passes through\none corner of the ceiling directly above A and goes diagonally down the wall These\nlines are skew because they are not parallel and also never meet By the shortest distance between two lines we mean the join of a point in one line\nwith one point on the other line so that the length of the segment so obtained is the\nsmallest"
    },
    {
        "Chapter": "1",
        "sentence_range": "5782-5785",
        "Text": "7\n\u00a9 NCERT\nnot to be republished\n MATHEMATI CS\n474\nl2\nS\nT\nQ\nP\nl1\nThe line GE that goes diagonally across the ceiling and the line DB passes through\none corner of the ceiling directly above A and goes diagonally down the wall These\nlines are skew because they are not parallel and also never meet By the shortest distance between two lines we mean the join of a point in one line\nwith one point on the other line so that the length of the segment so obtained is the\nsmallest For skew lines, the line of the shortest distance will be perpendicular to both\nthe lines"
    },
    {
        "Chapter": "1",
        "sentence_range": "5783-5786",
        "Text": "These\nlines are skew because they are not parallel and also never meet By the shortest distance between two lines we mean the join of a point in one line\nwith one point on the other line so that the length of the segment so obtained is the\nsmallest For skew lines, the line of the shortest distance will be perpendicular to both\nthe lines 11"
    },
    {
        "Chapter": "1",
        "sentence_range": "5784-5787",
        "Text": "By the shortest distance between two lines we mean the join of a point in one line\nwith one point on the other line so that the length of the segment so obtained is the\nsmallest For skew lines, the line of the shortest distance will be perpendicular to both\nthe lines 11 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "5785-5788",
        "Text": "For skew lines, the line of the shortest distance will be perpendicular to both\nthe lines 11 5 1  Distance between two skew lines\nWe now determine the shortest distance between two skew lines in the following way:\nLet l1 and l2  be two skew lines with equations (Fig"
    },
    {
        "Chapter": "1",
        "sentence_range": "5786-5789",
        "Text": "11 5 1  Distance between two skew lines\nWe now determine the shortest distance between two skew lines in the following way:\nLet l1 and l2  be two skew lines with equations (Fig 11"
    },
    {
        "Chapter": "1",
        "sentence_range": "5787-5790",
        "Text": "5 1  Distance between two skew lines\nWe now determine the shortest distance between two skew lines in the following way:\nLet l1 and l2  be two skew lines with equations (Fig 11 8)\nrr  =\n1\n1\na\nb\n+ \u03bb\nr\nr"
    },
    {
        "Chapter": "1",
        "sentence_range": "5788-5791",
        "Text": "1  Distance between two skew lines\nWe now determine the shortest distance between two skew lines in the following way:\nLet l1 and l2  be two skew lines with equations (Fig 11 8)\nrr  =\n1\n1\na\nb\n+ \u03bb\nr\nr (1)\nand\nrr  =\n2\n2\na\nb\n+ \u00b5\nr\nr"
    },
    {
        "Chapter": "1",
        "sentence_range": "5789-5792",
        "Text": "11 8)\nrr  =\n1\n1\na\nb\n+ \u03bb\nr\nr (1)\nand\nrr  =\n2\n2\na\nb\n+ \u00b5\nr\nr (2)\nTake any point S on l1 with position vector \n1ar\n and T on l2, with position vector  \nar"
    },
    {
        "Chapter": "1",
        "sentence_range": "5790-5793",
        "Text": "8)\nrr  =\n1\n1\na\nb\n+ \u03bb\nr\nr (1)\nand\nrr  =\n2\n2\na\nb\n+ \u00b5\nr\nr (2)\nTake any point S on l1 with position vector \n1ar\n and T on l2, with position vector  \nar 2\nThen the magnitude of the shortest distance vector\nwill be equal to that of the projection of ST along the\ndirection of the line of shortest distance (See 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5791-5794",
        "Text": "(1)\nand\nrr  =\n2\n2\na\nb\n+ \u00b5\nr\nr (2)\nTake any point S on l1 with position vector \n1ar\n and T on l2, with position vector  \nar 2\nThen the magnitude of the shortest distance vector\nwill be equal to that of the projection of ST along the\ndirection of the line of shortest distance (See 10 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "5792-5795",
        "Text": "(2)\nTake any point S on l1 with position vector \n1ar\n and T on l2, with position vector  \nar 2\nThen the magnitude of the shortest distance vector\nwill be equal to that of the projection of ST along the\ndirection of the line of shortest distance (See 10 6 2)"
    },
    {
        "Chapter": "1",
        "sentence_range": "5793-5796",
        "Text": "2\nThen the magnitude of the shortest distance vector\nwill be equal to that of the projection of ST along the\ndirection of the line of shortest distance (See 10 6 2) If PQ\nuuur\n is the shortest distance vector between\nl1 and l2 , then it being perpendicular to both  1b\nr\n and\n2b\nr\n, the unit vector n\u02c6  along PQ\nuuur\n would therefore be\n\u02c6n =\n1\n2\n1\n2\n|\n|\nb\nb\nb\nb\n\u00d7\n\u00d7\nr\nr\nr\nr"
    },
    {
        "Chapter": "1",
        "sentence_range": "5794-5797",
        "Text": "6 2) If PQ\nuuur\n is the shortest distance vector between\nl1 and l2 , then it being perpendicular to both  1b\nr\n and\n2b\nr\n, the unit vector n\u02c6  along PQ\nuuur\n would therefore be\n\u02c6n =\n1\n2\n1\n2\n|\n|\nb\nb\nb\nb\n\u00d7\n\u00d7\nr\nr\nr\nr (3)\nThen\nPQ\nuuur\n = d n\u02c6\nwhere, d is the magnitude of the shortest distance vector"
    },
    {
        "Chapter": "1",
        "sentence_range": "5795-5798",
        "Text": "2) If PQ\nuuur\n is the shortest distance vector between\nl1 and l2 , then it being perpendicular to both  1b\nr\n and\n2b\nr\n, the unit vector n\u02c6  along PQ\nuuur\n would therefore be\n\u02c6n =\n1\n2\n1\n2\n|\n|\nb\nb\nb\nb\n\u00d7\n\u00d7\nr\nr\nr\nr (3)\nThen\nPQ\nuuur\n = d n\u02c6\nwhere, d is the magnitude of the shortest distance vector Let \u03b8 be the angle between\nST\nuur\n and PQ\nuuur"
    },
    {
        "Chapter": "1",
        "sentence_range": "5796-5799",
        "Text": "If PQ\nuuur\n is the shortest distance vector between\nl1 and l2 , then it being perpendicular to both  1b\nr\n and\n2b\nr\n, the unit vector n\u02c6  along PQ\nuuur\n would therefore be\n\u02c6n =\n1\n2\n1\n2\n|\n|\nb\nb\nb\nb\n\u00d7\n\u00d7\nr\nr\nr\nr (3)\nThen\nPQ\nuuur\n = d n\u02c6\nwhere, d is the magnitude of the shortest distance vector Let \u03b8 be the angle between\nST\nuur\n and PQ\nuuur Then\nPQ = ST |cos \u03b8|\nBut\ncos \u03b8 =\nPQ ST\n| PQ | | ST |\n\u22c5\nuuur uur\nuuuur\nuur\n =\n2\n1\n\u02c6 (\n)\nST\nd n\na\na\n\u22c5d\n\u2212\nr\nr\n(since  \n2\n1\nST\n)\na\na\n=\n\u2212\nuur\nr\nr\n=\n1\n2\n2\n1\n1\n2\n(\n) (\n)\nST\nb\nb\na\na\nb\nb\n\u00d7\n\u22c5\n\u2212\n\u00d7\nr\nr\nr\nr\nr\nr\n[From (3)]\nFig 11"
    },
    {
        "Chapter": "1",
        "sentence_range": "5797-5800",
        "Text": "(3)\nThen\nPQ\nuuur\n = d n\u02c6\nwhere, d is the magnitude of the shortest distance vector Let \u03b8 be the angle between\nST\nuur\n and PQ\nuuur Then\nPQ = ST |cos \u03b8|\nBut\ncos \u03b8 =\nPQ ST\n| PQ | | ST |\n\u22c5\nuuur uur\nuuuur\nuur\n =\n2\n1\n\u02c6 (\n)\nST\nd n\na\na\n\u22c5d\n\u2212\nr\nr\n(since  \n2\n1\nST\n)\na\na\n=\n\u2212\nuur\nr\nr\n=\n1\n2\n2\n1\n1\n2\n(\n) (\n)\nST\nb\nb\na\na\nb\nb\n\u00d7\n\u22c5\n\u2212\n\u00d7\nr\nr\nr\nr\nr\nr\n[From (3)]\nFig 11 8\n\u00a9 NCERT\nnot to be republished\nTHREE D IMENSIONAL G EOMETRY\n475\nHence, the required shortest distance is\nd = PQ = ST |cos \u03b8|\nor\nd =\n1\n2\n2\n1\n1\n2\n(\n)"
    },
    {
        "Chapter": "1",
        "sentence_range": "5798-5801",
        "Text": "Let \u03b8 be the angle between\nST\nuur\n and PQ\nuuur Then\nPQ = ST |cos \u03b8|\nBut\ncos \u03b8 =\nPQ ST\n| PQ | | ST |\n\u22c5\nuuur uur\nuuuur\nuur\n =\n2\n1\n\u02c6 (\n)\nST\nd n\na\na\n\u22c5d\n\u2212\nr\nr\n(since  \n2\n1\nST\n)\na\na\n=\n\u2212\nuur\nr\nr\n=\n1\n2\n2\n1\n1\n2\n(\n) (\n)\nST\nb\nb\na\na\nb\nb\n\u00d7\n\u22c5\n\u2212\n\u00d7\nr\nr\nr\nr\nr\nr\n[From (3)]\nFig 11 8\n\u00a9 NCERT\nnot to be republished\nTHREE D IMENSIONAL G EOMETRY\n475\nHence, the required shortest distance is\nd = PQ = ST |cos \u03b8|\nor\nd =\n1\n2\n2\n1\n1\n2\n(\n) (\n)\n|\n|\nb\nb\na\na\nb\nb\n\u00d7\n\u2212\n\u00d7\nr\nr\nr\nr\nr\nr\nCartesian form\nThe shortest distance between the lines\nl1 : \n1\n1\nx\nx\n\u2212a\n =\n1\n1\n1\n1\ny\ny\nz\nz\nb\nc\n\u2212\n\u2212\n=\nand\nl2  : \n2\n2\nx\nx\n\u2212a\n =\n2\n2\n2\n2\ny\ny\nz\nz\nb\nc\n\u2212\n\u2212\n=\nis\n2\n1\n2\n1\n2\n1\n1\n1\n1\n2\n2\n2\n2\n2\n2\n1 2\n2 1\n1\n2\n2 1\n1 2\n2 1\n(\n)\n(\n)\n(\n)\nx\nx\ny\ny\nz\nz\na\nb\nc\na\nb\nc\nb c\nb c\nc a\nc a\na b\na b\n\u2212\n\u2212\n\u2212\n\u2212\n+\n\u2212\n+\n\u2212\n11"
    },
    {
        "Chapter": "1",
        "sentence_range": "5799-5802",
        "Text": "Then\nPQ = ST |cos \u03b8|\nBut\ncos \u03b8 =\nPQ ST\n| PQ | | ST |\n\u22c5\nuuur uur\nuuuur\nuur\n =\n2\n1\n\u02c6 (\n)\nST\nd n\na\na\n\u22c5d\n\u2212\nr\nr\n(since  \n2\n1\nST\n)\na\na\n=\n\u2212\nuur\nr\nr\n=\n1\n2\n2\n1\n1\n2\n(\n) (\n)\nST\nb\nb\na\na\nb\nb\n\u00d7\n\u22c5\n\u2212\n\u00d7\nr\nr\nr\nr\nr\nr\n[From (3)]\nFig 11 8\n\u00a9 NCERT\nnot to be republished\nTHREE D IMENSIONAL G EOMETRY\n475\nHence, the required shortest distance is\nd = PQ = ST |cos \u03b8|\nor\nd =\n1\n2\n2\n1\n1\n2\n(\n) (\n)\n|\n|\nb\nb\na\na\nb\nb\n\u00d7\n\u2212\n\u00d7\nr\nr\nr\nr\nr\nr\nCartesian form\nThe shortest distance between the lines\nl1 : \n1\n1\nx\nx\n\u2212a\n =\n1\n1\n1\n1\ny\ny\nz\nz\nb\nc\n\u2212\n\u2212\n=\nand\nl2  : \n2\n2\nx\nx\n\u2212a\n =\n2\n2\n2\n2\ny\ny\nz\nz\nb\nc\n\u2212\n\u2212\n=\nis\n2\n1\n2\n1\n2\n1\n1\n1\n1\n2\n2\n2\n2\n2\n2\n1 2\n2 1\n1\n2\n2 1\n1 2\n2 1\n(\n)\n(\n)\n(\n)\nx\nx\ny\ny\nz\nz\na\nb\nc\na\nb\nc\nb c\nb c\nc a\nc a\na b\na b\n\u2212\n\u2212\n\u2212\n\u2212\n+\n\u2212\n+\n\u2212\n11 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "5800-5803",
        "Text": "8\n\u00a9 NCERT\nnot to be republished\nTHREE D IMENSIONAL G EOMETRY\n475\nHence, the required shortest distance is\nd = PQ = ST |cos \u03b8|\nor\nd =\n1\n2\n2\n1\n1\n2\n(\n) (\n)\n|\n|\nb\nb\na\na\nb\nb\n\u00d7\n\u2212\n\u00d7\nr\nr\nr\nr\nr\nr\nCartesian form\nThe shortest distance between the lines\nl1 : \n1\n1\nx\nx\n\u2212a\n =\n1\n1\n1\n1\ny\ny\nz\nz\nb\nc\n\u2212\n\u2212\n=\nand\nl2  : \n2\n2\nx\nx\n\u2212a\n =\n2\n2\n2\n2\ny\ny\nz\nz\nb\nc\n\u2212\n\u2212\n=\nis\n2\n1\n2\n1\n2\n1\n1\n1\n1\n2\n2\n2\n2\n2\n2\n1 2\n2 1\n1\n2\n2 1\n1 2\n2 1\n(\n)\n(\n)\n(\n)\nx\nx\ny\ny\nz\nz\na\nb\nc\na\nb\nc\nb c\nb c\nc a\nc a\na b\na b\n\u2212\n\u2212\n\u2212\n\u2212\n+\n\u2212\n+\n\u2212\n11 5 2 Distance between parallel lines\nIf two lines l1 and  l2 are parallel, then they are coplanar"
    },
    {
        "Chapter": "1",
        "sentence_range": "5801-5804",
        "Text": "(\n)\n|\n|\nb\nb\na\na\nb\nb\n\u00d7\n\u2212\n\u00d7\nr\nr\nr\nr\nr\nr\nCartesian form\nThe shortest distance between the lines\nl1 : \n1\n1\nx\nx\n\u2212a\n =\n1\n1\n1\n1\ny\ny\nz\nz\nb\nc\n\u2212\n\u2212\n=\nand\nl2  : \n2\n2\nx\nx\n\u2212a\n =\n2\n2\n2\n2\ny\ny\nz\nz\nb\nc\n\u2212\n\u2212\n=\nis\n2\n1\n2\n1\n2\n1\n1\n1\n1\n2\n2\n2\n2\n2\n2\n1 2\n2 1\n1\n2\n2 1\n1 2\n2 1\n(\n)\n(\n)\n(\n)\nx\nx\ny\ny\nz\nz\na\nb\nc\na\nb\nc\nb c\nb c\nc a\nc a\na b\na b\n\u2212\n\u2212\n\u2212\n\u2212\n+\n\u2212\n+\n\u2212\n11 5 2 Distance between parallel lines\nIf two lines l1 and  l2 are parallel, then they are coplanar Let the lines be given by\nrr  =\n1a\nb\n+ \u03bb\nr\nr"
    },
    {
        "Chapter": "1",
        "sentence_range": "5802-5805",
        "Text": "5 2 Distance between parallel lines\nIf two lines l1 and  l2 are parallel, then they are coplanar Let the lines be given by\nrr  =\n1a\nb\n+ \u03bb\nr\nr (1)\nand\nrr  =\n2a\nb\n+ \u00b5\nr\nr\n\u2026 (2)\nwhere, \n1ar is the position vector of a point S on l1 and\n2\nar\n is the position vector of a point T on l2 Fig 11"
    },
    {
        "Chapter": "1",
        "sentence_range": "5803-5806",
        "Text": "2 Distance between parallel lines\nIf two lines l1 and  l2 are parallel, then they are coplanar Let the lines be given by\nrr  =\n1a\nb\n+ \u03bb\nr\nr (1)\nand\nrr  =\n2a\nb\n+ \u00b5\nr\nr\n\u2026 (2)\nwhere, \n1ar is the position vector of a point S on l1 and\n2\nar\n is the position vector of a point T on l2 Fig 11 9"
    },
    {
        "Chapter": "1",
        "sentence_range": "5804-5807",
        "Text": "Let the lines be given by\nrr  =\n1a\nb\n+ \u03bb\nr\nr (1)\nand\nrr  =\n2a\nb\n+ \u00b5\nr\nr\n\u2026 (2)\nwhere, \n1ar is the position vector of a point S on l1 and\n2\nar\n is the position vector of a point T on l2 Fig 11 9 As l1, l2 are coplanar, if the foot of the perpendicular\nfrom T on the line l1 is P, then the distance between the\nlines l1 and l2 =  |TP |"
    },
    {
        "Chapter": "1",
        "sentence_range": "5805-5808",
        "Text": "(1)\nand\nrr  =\n2a\nb\n+ \u00b5\nr\nr\n\u2026 (2)\nwhere, \n1ar is the position vector of a point S on l1 and\n2\nar\n is the position vector of a point T on l2 Fig 11 9 As l1, l2 are coplanar, if the foot of the perpendicular\nfrom T on the line l1 is P, then the distance between the\nlines l1 and l2 =  |TP | Let \u03b8 be the angle between the vectors ST\nuur\nand b\nr"
    },
    {
        "Chapter": "1",
        "sentence_range": "5806-5809",
        "Text": "9 As l1, l2 are coplanar, if the foot of the perpendicular\nfrom T on the line l1 is P, then the distance between the\nlines l1 and l2 =  |TP | Let \u03b8 be the angle between the vectors ST\nuur\nand b\nr Then\nST\nb \u00d7\nuur\nr\n =\n\u02c6\n(|\nb|| ST| sin )\nn\n\u03b8\nuur\nr"
    },
    {
        "Chapter": "1",
        "sentence_range": "5807-5810",
        "Text": "As l1, l2 are coplanar, if the foot of the perpendicular\nfrom T on the line l1 is P, then the distance between the\nlines l1 and l2 =  |TP | Let \u03b8 be the angle between the vectors ST\nuur\nand b\nr Then\nST\nb \u00d7\nuur\nr\n =\n\u02c6\n(|\nb|| ST| sin )\nn\n\u03b8\nuur\nr (3)\nwhere n\u02c6  is the unit vector perpendicular to the plane of the lines l1 and l2"
    },
    {
        "Chapter": "1",
        "sentence_range": "5808-5811",
        "Text": "Let \u03b8 be the angle between the vectors ST\nuur\nand b\nr Then\nST\nb \u00d7\nuur\nr\n =\n\u02c6\n(|\nb|| ST| sin )\nn\n\u03b8\nuur\nr (3)\nwhere n\u02c6  is the unit vector perpendicular to the plane of the lines l1 and l2 But\nST\nuur\n=\n2\n1\na\na\u2212\nr\nr\nFig 11"
    },
    {
        "Chapter": "1",
        "sentence_range": "5809-5812",
        "Text": "Then\nST\nb \u00d7\nuur\nr\n =\n\u02c6\n(|\nb|| ST| sin )\nn\n\u03b8\nuur\nr (3)\nwhere n\u02c6  is the unit vector perpendicular to the plane of the lines l1 and l2 But\nST\nuur\n=\n2\n1\na\na\u2212\nr\nr\nFig 11 9\n\u00a9 NCERT\nnot to be republished\n MATHEMATI CS\n476\nTherefore, from (3), we get\n2\n1\n(\n)\nb\na\na\n\u00d7\n\u2212\nr\nr\nr  =\n\u02c6\n|\nb| PT\nn\nr\n        (since PT = ST sin \u03b8)\ni"
    },
    {
        "Chapter": "1",
        "sentence_range": "5810-5813",
        "Text": "(3)\nwhere n\u02c6  is the unit vector perpendicular to the plane of the lines l1 and l2 But\nST\nuur\n=\n2\n1\na\na\u2212\nr\nr\nFig 11 9\n\u00a9 NCERT\nnot to be republished\n MATHEMATI CS\n476\nTherefore, from (3), we get\n2\n1\n(\n)\nb\na\na\n\u00d7\n\u2212\nr\nr\nr  =\n\u02c6\n|\nb| PT\nn\nr\n        (since PT = ST sin \u03b8)\ni e"
    },
    {
        "Chapter": "1",
        "sentence_range": "5811-5814",
        "Text": "But\nST\nuur\n=\n2\n1\na\na\u2212\nr\nr\nFig 11 9\n\u00a9 NCERT\nnot to be republished\n MATHEMATI CS\n476\nTherefore, from (3), we get\n2\n1\n(\n)\nb\na\na\n\u00d7\n\u2212\nr\nr\nr  =\n\u02c6\n|\nb| PT\nn\nr\n        (since PT = ST sin \u03b8)\ni e ,\n2\n1\n|\n(\n)|\nb\na\na\n\u00d7\n\u2212\nr\nr\nr\n = |\nb| PT 1\n\u22c5\nr\n       (as |\n|\n\u02c6n  = 1)\nHence, the distance between the given parallel lines is\nd =\n2\n1\n(\n)\n| PT |\n|\n|\nb\na\na\nb\n\u00d7\n\u2212\n=\nr\nr\nr\nuuur\nr\nExample 11 Find the shortest distance between the lines l1 and l2 whose vector\nequations are\nrr  =\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(2\n)\ni\nj\ni\nj\nk\n+\n+ \u03bb\n\u2212\n+"
    },
    {
        "Chapter": "1",
        "sentence_range": "5812-5815",
        "Text": "9\n\u00a9 NCERT\nnot to be republished\n MATHEMATI CS\n476\nTherefore, from (3), we get\n2\n1\n(\n)\nb\na\na\n\u00d7\n\u2212\nr\nr\nr  =\n\u02c6\n|\nb| PT\nn\nr\n        (since PT = ST sin \u03b8)\ni e ,\n2\n1\n|\n(\n)|\nb\na\na\n\u00d7\n\u2212\nr\nr\nr\n = |\nb| PT 1\n\u22c5\nr\n       (as |\n|\n\u02c6n  = 1)\nHence, the distance between the given parallel lines is\nd =\n2\n1\n(\n)\n| PT |\n|\n|\nb\na\na\nb\n\u00d7\n\u2212\n=\nr\nr\nr\nuuur\nr\nExample 11 Find the shortest distance between the lines l1 and l2 whose vector\nequations are\nrr  =\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(2\n)\ni\nj\ni\nj\nk\n+\n+ \u03bb\n\u2212\n+ (1)\nand\nrr  =\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n2\n(3\n5\n2\n)\ni\nj\nk\ni\nj\nk\n+\n\u2212\n+ \u00b5\n\u2212\n+"
    },
    {
        "Chapter": "1",
        "sentence_range": "5813-5816",
        "Text": "e ,\n2\n1\n|\n(\n)|\nb\na\na\n\u00d7\n\u2212\nr\nr\nr\n = |\nb| PT 1\n\u22c5\nr\n       (as |\n|\n\u02c6n  = 1)\nHence, the distance between the given parallel lines is\nd =\n2\n1\n(\n)\n| PT |\n|\n|\nb\na\na\nb\n\u00d7\n\u2212\n=\nr\nr\nr\nuuur\nr\nExample 11 Find the shortest distance between the lines l1 and l2 whose vector\nequations are\nrr  =\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(2\n)\ni\nj\ni\nj\nk\n+\n+ \u03bb\n\u2212\n+ (1)\nand\nrr  =\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n2\n(3\n5\n2\n)\ni\nj\nk\ni\nj\nk\n+\n\u2212\n+ \u00b5\n\u2212\n+ (2)\nSolution Comparing (1) and (2) with rr  = \n1\n1\na\nb\n+ \u03bb\nr\nr\n and  \n2\n2\nb\na\nr\nr\nr\nr\n+\u00b5\n=\n respectively,,\nwe get\n1ar  =\n1\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n,\n2\ni\nj\nb\ni\nj\nk\n+\n=\n\u2212\n+\nr\n2\nar\n = 2 \u02c6i  + \u02c6j \u2013 \u02c6k  and \n2b\nr\n = 3 \u02c6i  \u2013 5 \u02c6j  + 2 \u02c6k\nTherefore\n2\n1\na\nr\u2212a\nr  =\n\u02c6\n\u02c6i\nk\n\u2212\nand\n1\n2\nb\nb\n\u00d7\nr\nr\n =\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n( 2\n)\n( 3\n5\n2\n)\ni\nj\nk\ni\nj\nk\n\u2212\n+\n\u00d7\n\u2212\n+\n=\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n2\n1\n1\n3\n7\n3\n5\n2\ni\nj\nk\ni\nj\nk\n\u2212\n=\n\u2212\n\u2212\n\u2212\nSo\n1\n2\n|\n|\nb\nb\n\u00d7\nr\nr\n =\n9\n1\n49\n59\n+ +\n=\nHence, the shortest distance between the given lines is given by\nd  =\n|\n|\n)\n)"
    },
    {
        "Chapter": "1",
        "sentence_range": "5814-5817",
        "Text": ",\n2\n1\n|\n(\n)|\nb\na\na\n\u00d7\n\u2212\nr\nr\nr\n = |\nb| PT 1\n\u22c5\nr\n       (as |\n|\n\u02c6n  = 1)\nHence, the distance between the given parallel lines is\nd =\n2\n1\n(\n)\n| PT |\n|\n|\nb\na\na\nb\n\u00d7\n\u2212\n=\nr\nr\nr\nuuur\nr\nExample 11 Find the shortest distance between the lines l1 and l2 whose vector\nequations are\nrr  =\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(2\n)\ni\nj\ni\nj\nk\n+\n+ \u03bb\n\u2212\n+ (1)\nand\nrr  =\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n2\n(3\n5\n2\n)\ni\nj\nk\ni\nj\nk\n+\n\u2212\n+ \u00b5\n\u2212\n+ (2)\nSolution Comparing (1) and (2) with rr  = \n1\n1\na\nb\n+ \u03bb\nr\nr\n and  \n2\n2\nb\na\nr\nr\nr\nr\n+\u00b5\n=\n respectively,,\nwe get\n1ar  =\n1\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n,\n2\ni\nj\nb\ni\nj\nk\n+\n=\n\u2212\n+\nr\n2\nar\n = 2 \u02c6i  + \u02c6j \u2013 \u02c6k  and \n2b\nr\n = 3 \u02c6i  \u2013 5 \u02c6j  + 2 \u02c6k\nTherefore\n2\n1\na\nr\u2212a\nr  =\n\u02c6\n\u02c6i\nk\n\u2212\nand\n1\n2\nb\nb\n\u00d7\nr\nr\n =\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n( 2\n)\n( 3\n5\n2\n)\ni\nj\nk\ni\nj\nk\n\u2212\n+\n\u00d7\n\u2212\n+\n=\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n2\n1\n1\n3\n7\n3\n5\n2\ni\nj\nk\ni\nj\nk\n\u2212\n=\n\u2212\n\u2212\n\u2212\nSo\n1\n2\n|\n|\nb\nb\n\u00d7\nr\nr\n =\n9\n1\n49\n59\n+ +\n=\nHence, the shortest distance between the given lines is given by\nd  =\n|\n|\n)\n) (\n(\n2\n1\n1\n2\n2\n1\nb\nb\na\na\nb\nb\nr\nr\nr\nr\nr\nr\n\u00d7\n\u2212\n\u00d7\n  \n59\n10\n59\n7|\n0\n|3\n=\n+\n\u2212\n=\nExample 12 Find the distance between the lines l1 and l2 given by\nrr  =\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n2\n4\n( 2\n3\n6\n)\ni\nj\nk\ni\nj\nk\n+\n\u2212\n+ \u03bb\n+\n+\nand\nrr  =\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n3\n3\n5\n( 2\n3\n6\n)\ni\nj\nk\ni\nj\nk\n+\n\u2212\n+ \u00b5\n+\n+\n\u00a9 NCERT\nnot to be republished\nTHREE D IMENSIONAL G EOMETRY\n477\nSolution The two lines are parallel (Why"
    },
    {
        "Chapter": "1",
        "sentence_range": "5815-5818",
        "Text": "(1)\nand\nrr  =\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n2\n(3\n5\n2\n)\ni\nj\nk\ni\nj\nk\n+\n\u2212\n+ \u00b5\n\u2212\n+ (2)\nSolution Comparing (1) and (2) with rr  = \n1\n1\na\nb\n+ \u03bb\nr\nr\n and  \n2\n2\nb\na\nr\nr\nr\nr\n+\u00b5\n=\n respectively,,\nwe get\n1ar  =\n1\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n,\n2\ni\nj\nb\ni\nj\nk\n+\n=\n\u2212\n+\nr\n2\nar\n = 2 \u02c6i  + \u02c6j \u2013 \u02c6k  and \n2b\nr\n = 3 \u02c6i  \u2013 5 \u02c6j  + 2 \u02c6k\nTherefore\n2\n1\na\nr\u2212a\nr  =\n\u02c6\n\u02c6i\nk\n\u2212\nand\n1\n2\nb\nb\n\u00d7\nr\nr\n =\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n( 2\n)\n( 3\n5\n2\n)\ni\nj\nk\ni\nj\nk\n\u2212\n+\n\u00d7\n\u2212\n+\n=\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n2\n1\n1\n3\n7\n3\n5\n2\ni\nj\nk\ni\nj\nk\n\u2212\n=\n\u2212\n\u2212\n\u2212\nSo\n1\n2\n|\n|\nb\nb\n\u00d7\nr\nr\n =\n9\n1\n49\n59\n+ +\n=\nHence, the shortest distance between the given lines is given by\nd  =\n|\n|\n)\n) (\n(\n2\n1\n1\n2\n2\n1\nb\nb\na\na\nb\nb\nr\nr\nr\nr\nr\nr\n\u00d7\n\u2212\n\u00d7\n  \n59\n10\n59\n7|\n0\n|3\n=\n+\n\u2212\n=\nExample 12 Find the distance between the lines l1 and l2 given by\nrr  =\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n2\n4\n( 2\n3\n6\n)\ni\nj\nk\ni\nj\nk\n+\n\u2212\n+ \u03bb\n+\n+\nand\nrr  =\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n3\n3\n5\n( 2\n3\n6\n)\ni\nj\nk\ni\nj\nk\n+\n\u2212\n+ \u00b5\n+\n+\n\u00a9 NCERT\nnot to be republished\nTHREE D IMENSIONAL G EOMETRY\n477\nSolution The two lines are parallel (Why ) We have\n1ar  =\n\u02c6\n\u02c6\n2\u02c6\n4\ni\nj\nk\n+\n\u2212\n, \n2ar\n = \n\u02c6\n\u02c6\n\u02c6\n3\n3\n5\ni\nj\nk\n+\n\u2212\n and b\nr\n = \n\u02c6\n\u02c6\n\u02c6\n2\n3\n6\ni\nj\nk\n+\n+\nTherefore, the distance between the lines is given by\nd =\n2\n1\n(\n)\n| |\nb\na\na\nb\n\u00d7\n\u2212\nr\nr\nr\nr\n = \n\u02c6\n\u02c6\n\u02c6\n2\n3\n6\n2\n1\n1\n4\n9\n36\ni\nj\nk\n\u2212\n+\n+\nor\n=\n\u02c6\n\u02c6\n\u02c6\n|\n9\n14\n4\n|\n293\n7293\n49\n49\ni\nj\nk\n\u2212\n+\n\u2212\n=\n=\nEXERCISE 11"
    },
    {
        "Chapter": "1",
        "sentence_range": "5816-5819",
        "Text": "(2)\nSolution Comparing (1) and (2) with rr  = \n1\n1\na\nb\n+ \u03bb\nr\nr\n and  \n2\n2\nb\na\nr\nr\nr\nr\n+\u00b5\n=\n respectively,,\nwe get\n1ar  =\n1\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n,\n2\ni\nj\nb\ni\nj\nk\n+\n=\n\u2212\n+\nr\n2\nar\n = 2 \u02c6i  + \u02c6j \u2013 \u02c6k  and \n2b\nr\n = 3 \u02c6i  \u2013 5 \u02c6j  + 2 \u02c6k\nTherefore\n2\n1\na\nr\u2212a\nr  =\n\u02c6\n\u02c6i\nk\n\u2212\nand\n1\n2\nb\nb\n\u00d7\nr\nr\n =\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n( 2\n)\n( 3\n5\n2\n)\ni\nj\nk\ni\nj\nk\n\u2212\n+\n\u00d7\n\u2212\n+\n=\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n2\n1\n1\n3\n7\n3\n5\n2\ni\nj\nk\ni\nj\nk\n\u2212\n=\n\u2212\n\u2212\n\u2212\nSo\n1\n2\n|\n|\nb\nb\n\u00d7\nr\nr\n =\n9\n1\n49\n59\n+ +\n=\nHence, the shortest distance between the given lines is given by\nd  =\n|\n|\n)\n) (\n(\n2\n1\n1\n2\n2\n1\nb\nb\na\na\nb\nb\nr\nr\nr\nr\nr\nr\n\u00d7\n\u2212\n\u00d7\n  \n59\n10\n59\n7|\n0\n|3\n=\n+\n\u2212\n=\nExample 12 Find the distance between the lines l1 and l2 given by\nrr  =\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n2\n4\n( 2\n3\n6\n)\ni\nj\nk\ni\nj\nk\n+\n\u2212\n+ \u03bb\n+\n+\nand\nrr  =\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n3\n3\n5\n( 2\n3\n6\n)\ni\nj\nk\ni\nj\nk\n+\n\u2212\n+ \u00b5\n+\n+\n\u00a9 NCERT\nnot to be republished\nTHREE D IMENSIONAL G EOMETRY\n477\nSolution The two lines are parallel (Why ) We have\n1ar  =\n\u02c6\n\u02c6\n2\u02c6\n4\ni\nj\nk\n+\n\u2212\n, \n2ar\n = \n\u02c6\n\u02c6\n\u02c6\n3\n3\n5\ni\nj\nk\n+\n\u2212\n and b\nr\n = \n\u02c6\n\u02c6\n\u02c6\n2\n3\n6\ni\nj\nk\n+\n+\nTherefore, the distance between the lines is given by\nd =\n2\n1\n(\n)\n| |\nb\na\na\nb\n\u00d7\n\u2212\nr\nr\nr\nr\n = \n\u02c6\n\u02c6\n\u02c6\n2\n3\n6\n2\n1\n1\n4\n9\n36\ni\nj\nk\n\u2212\n+\n+\nor\n=\n\u02c6\n\u02c6\n\u02c6\n|\n9\n14\n4\n|\n293\n7293\n49\n49\ni\nj\nk\n\u2212\n+\n\u2212\n=\n=\nEXERCISE 11 2\n1"
    },
    {
        "Chapter": "1",
        "sentence_range": "5817-5820",
        "Text": "(\n(\n2\n1\n1\n2\n2\n1\nb\nb\na\na\nb\nb\nr\nr\nr\nr\nr\nr\n\u00d7\n\u2212\n\u00d7\n  \n59\n10\n59\n7|\n0\n|3\n=\n+\n\u2212\n=\nExample 12 Find the distance between the lines l1 and l2 given by\nrr  =\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n2\n4\n( 2\n3\n6\n)\ni\nj\nk\ni\nj\nk\n+\n\u2212\n+ \u03bb\n+\n+\nand\nrr  =\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n3\n3\n5\n( 2\n3\n6\n)\ni\nj\nk\ni\nj\nk\n+\n\u2212\n+ \u00b5\n+\n+\n\u00a9 NCERT\nnot to be republished\nTHREE D IMENSIONAL G EOMETRY\n477\nSolution The two lines are parallel (Why ) We have\n1ar  =\n\u02c6\n\u02c6\n2\u02c6\n4\ni\nj\nk\n+\n\u2212\n, \n2ar\n = \n\u02c6\n\u02c6\n\u02c6\n3\n3\n5\ni\nj\nk\n+\n\u2212\n and b\nr\n = \n\u02c6\n\u02c6\n\u02c6\n2\n3\n6\ni\nj\nk\n+\n+\nTherefore, the distance between the lines is given by\nd =\n2\n1\n(\n)\n| |\nb\na\na\nb\n\u00d7\n\u2212\nr\nr\nr\nr\n = \n\u02c6\n\u02c6\n\u02c6\n2\n3\n6\n2\n1\n1\n4\n9\n36\ni\nj\nk\n\u2212\n+\n+\nor\n=\n\u02c6\n\u02c6\n\u02c6\n|\n9\n14\n4\n|\n293\n7293\n49\n49\ni\nj\nk\n\u2212\n+\n\u2212\n=\n=\nEXERCISE 11 2\n1 Show that the three lines with direction cosines\n12\n3\n4\n4\n12\n3\n3\n4\n12\n,\n,\n;\n,\n,\n;\n,\n,\n13\n13\n13\n13\n13\n13\n13\n13\n13\n\u2212\n\u2212\n\u2212\n are mutually perpendicular"
    },
    {
        "Chapter": "1",
        "sentence_range": "5818-5821",
        "Text": ") We have\n1ar  =\n\u02c6\n\u02c6\n2\u02c6\n4\ni\nj\nk\n+\n\u2212\n, \n2ar\n = \n\u02c6\n\u02c6\n\u02c6\n3\n3\n5\ni\nj\nk\n+\n\u2212\n and b\nr\n = \n\u02c6\n\u02c6\n\u02c6\n2\n3\n6\ni\nj\nk\n+\n+\nTherefore, the distance between the lines is given by\nd =\n2\n1\n(\n)\n| |\nb\na\na\nb\n\u00d7\n\u2212\nr\nr\nr\nr\n = \n\u02c6\n\u02c6\n\u02c6\n2\n3\n6\n2\n1\n1\n4\n9\n36\ni\nj\nk\n\u2212\n+\n+\nor\n=\n\u02c6\n\u02c6\n\u02c6\n|\n9\n14\n4\n|\n293\n7293\n49\n49\ni\nj\nk\n\u2212\n+\n\u2212\n=\n=\nEXERCISE 11 2\n1 Show that the three lines with direction cosines\n12\n3\n4\n4\n12\n3\n3\n4\n12\n,\n,\n;\n,\n,\n;\n,\n,\n13\n13\n13\n13\n13\n13\n13\n13\n13\n\u2212\n\u2212\n\u2212\n are mutually perpendicular 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "5819-5822",
        "Text": "2\n1 Show that the three lines with direction cosines\n12\n3\n4\n4\n12\n3\n3\n4\n12\n,\n,\n;\n,\n,\n;\n,\n,\n13\n13\n13\n13\n13\n13\n13\n13\n13\n\u2212\n\u2212\n\u2212\n are mutually perpendicular 2 Show that the line through the points (1, \u2013 1, 2), (3, 4, \u2013 2) is perpendicular to the\nline through the points (0, 3, 2) and (3, 5, 6)"
    },
    {
        "Chapter": "1",
        "sentence_range": "5820-5823",
        "Text": "Show that the three lines with direction cosines\n12\n3\n4\n4\n12\n3\n3\n4\n12\n,\n,\n;\n,\n,\n;\n,\n,\n13\n13\n13\n13\n13\n13\n13\n13\n13\n\u2212\n\u2212\n\u2212\n are mutually perpendicular 2 Show that the line through the points (1, \u2013 1, 2), (3, 4, \u2013 2) is perpendicular to the\nline through the points (0, 3, 2) and (3, 5, 6) 3"
    },
    {
        "Chapter": "1",
        "sentence_range": "5821-5824",
        "Text": "2 Show that the line through the points (1, \u2013 1, 2), (3, 4, \u2013 2) is perpendicular to the\nline through the points (0, 3, 2) and (3, 5, 6) 3 Show that the line through the points (4, 7, 8), (2, 3, 4) is parallel to the line\nthrough the points (\u2013 1, \u2013 2, 1), (1, 2, 5)"
    },
    {
        "Chapter": "1",
        "sentence_range": "5822-5825",
        "Text": "Show that the line through the points (1, \u2013 1, 2), (3, 4, \u2013 2) is perpendicular to the\nline through the points (0, 3, 2) and (3, 5, 6) 3 Show that the line through the points (4, 7, 8), (2, 3, 4) is parallel to the line\nthrough the points (\u2013 1, \u2013 2, 1), (1, 2, 5) 4"
    },
    {
        "Chapter": "1",
        "sentence_range": "5823-5826",
        "Text": "3 Show that the line through the points (4, 7, 8), (2, 3, 4) is parallel to the line\nthrough the points (\u2013 1, \u2013 2, 1), (1, 2, 5) 4 Find the equation of the line which passes through the point (1, 2, 3) and is\nparallel to the vector \n\u02c6\n\u02c6\n\u02c6\n3\n2\n2\ni\nj\nk\n+\n\u2212"
    },
    {
        "Chapter": "1",
        "sentence_range": "5824-5827",
        "Text": "Show that the line through the points (4, 7, 8), (2, 3, 4) is parallel to the line\nthrough the points (\u2013 1, \u2013 2, 1), (1, 2, 5) 4 Find the equation of the line which passes through the point (1, 2, 3) and is\nparallel to the vector \n\u02c6\n\u02c6\n\u02c6\n3\n2\n2\ni\nj\nk\n+\n\u2212 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "5825-5828",
        "Text": "4 Find the equation of the line which passes through the point (1, 2, 3) and is\nparallel to the vector \n\u02c6\n\u02c6\n\u02c6\n3\n2\n2\ni\nj\nk\n+\n\u2212 5 Find the equation of the line in vector and in cartesian form that passes through\nthe point with position vector \n\u02c6\n2\u02c6\n4\ni\nj\nk\n\u2212\n+\nand is in the direction \n\u02c6\n\u02c6\n2\u02c6\ni\nj\nk\n+\n\u2212"
    },
    {
        "Chapter": "1",
        "sentence_range": "5826-5829",
        "Text": "Find the equation of the line which passes through the point (1, 2, 3) and is\nparallel to the vector \n\u02c6\n\u02c6\n\u02c6\n3\n2\n2\ni\nj\nk\n+\n\u2212 5 Find the equation of the line in vector and in cartesian form that passes through\nthe point with position vector \n\u02c6\n2\u02c6\n4\ni\nj\nk\n\u2212\n+\nand is in the direction \n\u02c6\n\u02c6\n2\u02c6\ni\nj\nk\n+\n\u2212 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "5827-5830",
        "Text": "5 Find the equation of the line in vector and in cartesian form that passes through\nthe point with position vector \n\u02c6\n2\u02c6\n4\ni\nj\nk\n\u2212\n+\nand is in the direction \n\u02c6\n\u02c6\n2\u02c6\ni\nj\nk\n+\n\u2212 6 Find the cartesian equation of the line which passes through the point (\u2013 2, 4, \u2013 5)\nand parallel to the line given by \n3\n4\n8\n3\n5\n6\nx\ny\nz\n+\n\u2212\n+\n=\n="
    },
    {
        "Chapter": "1",
        "sentence_range": "5828-5831",
        "Text": "Find the equation of the line in vector and in cartesian form that passes through\nthe point with position vector \n\u02c6\n2\u02c6\n4\ni\nj\nk\n\u2212\n+\nand is in the direction \n\u02c6\n\u02c6\n2\u02c6\ni\nj\nk\n+\n\u2212 6 Find the cartesian equation of the line which passes through the point (\u2013 2, 4, \u2013 5)\nand parallel to the line given by \n3\n4\n8\n3\n5\n6\nx\ny\nz\n+\n\u2212\n+\n=\n= 7"
    },
    {
        "Chapter": "1",
        "sentence_range": "5829-5832",
        "Text": "6 Find the cartesian equation of the line which passes through the point (\u2013 2, 4, \u2013 5)\nand parallel to the line given by \n3\n4\n8\n3\n5\n6\nx\ny\nz\n+\n\u2212\n+\n=\n= 7 The cartesian equation of a line is \n5\n4\n6\n3\n7\n2\nx\ny\nz\n\u2212\n+\n\u2212\n=\n="
    },
    {
        "Chapter": "1",
        "sentence_range": "5830-5833",
        "Text": "Find the cartesian equation of the line which passes through the point (\u2013 2, 4, \u2013 5)\nand parallel to the line given by \n3\n4\n8\n3\n5\n6\nx\ny\nz\n+\n\u2212\n+\n=\n= 7 The cartesian equation of a line is \n5\n4\n6\n3\n7\n2\nx\ny\nz\n\u2212\n+\n\u2212\n=\n= Write its vector form"
    },
    {
        "Chapter": "1",
        "sentence_range": "5831-5834",
        "Text": "7 The cartesian equation of a line is \n5\n4\n6\n3\n7\n2\nx\ny\nz\n\u2212\n+\n\u2212\n=\n= Write its vector form 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "5832-5835",
        "Text": "The cartesian equation of a line is \n5\n4\n6\n3\n7\n2\nx\ny\nz\n\u2212\n+\n\u2212\n=\n= Write its vector form 8 Find the vector and the cartesian equations of the lines that passes through the\norigin and (5, \u2013 2, 3)"
    },
    {
        "Chapter": "1",
        "sentence_range": "5833-5836",
        "Text": "Write its vector form 8 Find the vector and the cartesian equations of the lines that passes through the\norigin and (5, \u2013 2, 3) \u00a9 NCERT\nnot to be republished\n MATHEMATI CS\n478\n9"
    },
    {
        "Chapter": "1",
        "sentence_range": "5834-5837",
        "Text": "8 Find the vector and the cartesian equations of the lines that passes through the\norigin and (5, \u2013 2, 3) \u00a9 NCERT\nnot to be republished\n MATHEMATI CS\n478\n9 Find the vector and the cartesian equations of the line that passes through the\npoints (3, \u2013 2, \u2013 5), (3, \u2013 2, 6)"
    },
    {
        "Chapter": "1",
        "sentence_range": "5835-5838",
        "Text": "Find the vector and the cartesian equations of the lines that passes through the\norigin and (5, \u2013 2, 3) \u00a9 NCERT\nnot to be republished\n MATHEMATI CS\n478\n9 Find the vector and the cartesian equations of the line that passes through the\npoints (3, \u2013 2, \u2013 5), (3, \u2013 2, 6) 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5836-5839",
        "Text": "\u00a9 NCERT\nnot to be republished\n MATHEMATI CS\n478\n9 Find the vector and the cartesian equations of the line that passes through the\npoints (3, \u2013 2, \u2013 5), (3, \u2013 2, 6) 10 Find the angle between the following pairs of lines:\n(i)\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n2\n5\n(3\n2\n6 )\nr\ni\nj\nk\ni\nj\nk\n=\n\u2212\n+\n+ \u03bb\n+\n+\nr\n and\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n7\n6\n(\n2\n2 )\nr\ni\nk\ni\nj\nk\n=\n\u2212\n+ \u00b5\n+\n+\nr\n(ii)\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n3\n2\n(\n2 )\nr\ni\nj\nk\ni\nj\nk\n=\n+\n\u2212\n+ \u03bb\n\u2212\n\u2212\nr\n and\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n2\n56\n(3\n5\n4 )\nr\ni\nj\nk\ni\nj\nk\n=\n\u2212\n\u2212\n+ \u00b5\n\u2212\n\u2212\nr\n11"
    },
    {
        "Chapter": "1",
        "sentence_range": "5837-5840",
        "Text": "Find the vector and the cartesian equations of the line that passes through the\npoints (3, \u2013 2, \u2013 5), (3, \u2013 2, 6) 10 Find the angle between the following pairs of lines:\n(i)\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n2\n5\n(3\n2\n6 )\nr\ni\nj\nk\ni\nj\nk\n=\n\u2212\n+\n+ \u03bb\n+\n+\nr\n and\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n7\n6\n(\n2\n2 )\nr\ni\nk\ni\nj\nk\n=\n\u2212\n+ \u00b5\n+\n+\nr\n(ii)\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n3\n2\n(\n2 )\nr\ni\nj\nk\ni\nj\nk\n=\n+\n\u2212\n+ \u03bb\n\u2212\n\u2212\nr\n and\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n2\n56\n(3\n5\n4 )\nr\ni\nj\nk\ni\nj\nk\n=\n\u2212\n\u2212\n+ \u00b5\n\u2212\n\u2212\nr\n11 Find the angle between the following pair of lines:\n(i)\n2\n1\n3\n2\n4\n5\nand\n2\n5\n3\n1\n8\n4\nx\ny\nz\nx\ny\nz\n\u2212\n\u2212 +\n+\n\u2212\n\u2212\n=\n=\n=\n=\n\u2212\n\u2212\n(ii)\n5\n2\n3\nand\n2\n2\n1\n4\n1\n8\nx\ny\nz\nx\ny\nz\n\u2212\n\u2212\n\u2212\n=\n=\n=\n=\n12"
    },
    {
        "Chapter": "1",
        "sentence_range": "5838-5841",
        "Text": "10 Find the angle between the following pairs of lines:\n(i)\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n2\n5\n(3\n2\n6 )\nr\ni\nj\nk\ni\nj\nk\n=\n\u2212\n+\n+ \u03bb\n+\n+\nr\n and\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n7\n6\n(\n2\n2 )\nr\ni\nk\ni\nj\nk\n=\n\u2212\n+ \u00b5\n+\n+\nr\n(ii)\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n3\n2\n(\n2 )\nr\ni\nj\nk\ni\nj\nk\n=\n+\n\u2212\n+ \u03bb\n\u2212\n\u2212\nr\n and\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n2\n56\n(3\n5\n4 )\nr\ni\nj\nk\ni\nj\nk\n=\n\u2212\n\u2212\n+ \u00b5\n\u2212\n\u2212\nr\n11 Find the angle between the following pair of lines:\n(i)\n2\n1\n3\n2\n4\n5\nand\n2\n5\n3\n1\n8\n4\nx\ny\nz\nx\ny\nz\n\u2212\n\u2212 +\n+\n\u2212\n\u2212\n=\n=\n=\n=\n\u2212\n\u2212\n(ii)\n5\n2\n3\nand\n2\n2\n1\n4\n1\n8\nx\ny\nz\nx\ny\nz\n\u2212\n\u2212\n\u2212\n=\n=\n=\n=\n12 Find the values of p so that the lines 1\n7\n14\n3\n3\n2\n2\nx\ny\nz\np\n\u2212\n\u2212\n\u2212\n=\n=\nand  7\n7\n5\n6\n3\n1\n5\nx\ny\nz\np\n\u2212\n\u2212\n\u2212\n=\n=\n are at right angles"
    },
    {
        "Chapter": "1",
        "sentence_range": "5839-5842",
        "Text": "Find the angle between the following pairs of lines:\n(i)\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n2\n5\n(3\n2\n6 )\nr\ni\nj\nk\ni\nj\nk\n=\n\u2212\n+\n+ \u03bb\n+\n+\nr\n and\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n7\n6\n(\n2\n2 )\nr\ni\nk\ni\nj\nk\n=\n\u2212\n+ \u00b5\n+\n+\nr\n(ii)\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n3\n2\n(\n2 )\nr\ni\nj\nk\ni\nj\nk\n=\n+\n\u2212\n+ \u03bb\n\u2212\n\u2212\nr\n and\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n2\n56\n(3\n5\n4 )\nr\ni\nj\nk\ni\nj\nk\n=\n\u2212\n\u2212\n+ \u00b5\n\u2212\n\u2212\nr\n11 Find the angle between the following pair of lines:\n(i)\n2\n1\n3\n2\n4\n5\nand\n2\n5\n3\n1\n8\n4\nx\ny\nz\nx\ny\nz\n\u2212\n\u2212 +\n+\n\u2212\n\u2212\n=\n=\n=\n=\n\u2212\n\u2212\n(ii)\n5\n2\n3\nand\n2\n2\n1\n4\n1\n8\nx\ny\nz\nx\ny\nz\n\u2212\n\u2212\n\u2212\n=\n=\n=\n=\n12 Find the values of p so that the lines 1\n7\n14\n3\n3\n2\n2\nx\ny\nz\np\n\u2212\n\u2212\n\u2212\n=\n=\nand  7\n7\n5\n6\n3\n1\n5\nx\ny\nz\np\n\u2212\n\u2212\n\u2212\n=\n=\n are at right angles 13"
    },
    {
        "Chapter": "1",
        "sentence_range": "5840-5843",
        "Text": "Find the angle between the following pair of lines:\n(i)\n2\n1\n3\n2\n4\n5\nand\n2\n5\n3\n1\n8\n4\nx\ny\nz\nx\ny\nz\n\u2212\n\u2212 +\n+\n\u2212\n\u2212\n=\n=\n=\n=\n\u2212\n\u2212\n(ii)\n5\n2\n3\nand\n2\n2\n1\n4\n1\n8\nx\ny\nz\nx\ny\nz\n\u2212\n\u2212\n\u2212\n=\n=\n=\n=\n12 Find the values of p so that the lines 1\n7\n14\n3\n3\n2\n2\nx\ny\nz\np\n\u2212\n\u2212\n\u2212\n=\n=\nand  7\n7\n5\n6\n3\n1\n5\nx\ny\nz\np\n\u2212\n\u2212\n\u2212\n=\n=\n are at right angles 13 Show that the lines \n5\n2\n7\n5\n1\nx\ny\nz\n\u2212\n+\n=\n=\n\u2212\n and \n1\n2\n3\nx\ny\nz\n=\n=\n are perpendicular to\neach other"
    },
    {
        "Chapter": "1",
        "sentence_range": "5841-5844",
        "Text": "Find the values of p so that the lines 1\n7\n14\n3\n3\n2\n2\nx\ny\nz\np\n\u2212\n\u2212\n\u2212\n=\n=\nand  7\n7\n5\n6\n3\n1\n5\nx\ny\nz\np\n\u2212\n\u2212\n\u2212\n=\n=\n are at right angles 13 Show that the lines \n5\n2\n7\n5\n1\nx\ny\nz\n\u2212\n+\n=\n=\n\u2212\n and \n1\n2\n3\nx\ny\nz\n=\n=\n are perpendicular to\neach other 14"
    },
    {
        "Chapter": "1",
        "sentence_range": "5842-5845",
        "Text": "13 Show that the lines \n5\n2\n7\n5\n1\nx\ny\nz\n\u2212\n+\n=\n=\n\u2212\n and \n1\n2\n3\nx\ny\nz\n=\n=\n are perpendicular to\neach other 14 Find the shortest distance between the lines\n\u02c6\n\u02c6\n\u02c6\n(\n2\n)\nr\ni\nj\nk\n=\n+\n+\nr\n + \n\u02c6\n\u02c6\n\u02c6\n(\n)\ni\nj\nk\n\u03bb\n\u2212\n+\n and\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n2\n(2\n2 )\nr\ni\nj\nk\ni\nj\nk\n=\n\u2212\n\u2212\n+ \u00b5\n+\n+\nr\n15"
    },
    {
        "Chapter": "1",
        "sentence_range": "5843-5846",
        "Text": "Show that the lines \n5\n2\n7\n5\n1\nx\ny\nz\n\u2212\n+\n=\n=\n\u2212\n and \n1\n2\n3\nx\ny\nz\n=\n=\n are perpendicular to\neach other 14 Find the shortest distance between the lines\n\u02c6\n\u02c6\n\u02c6\n(\n2\n)\nr\ni\nj\nk\n=\n+\n+\nr\n + \n\u02c6\n\u02c6\n\u02c6\n(\n)\ni\nj\nk\n\u03bb\n\u2212\n+\n and\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n2\n(2\n2 )\nr\ni\nj\nk\ni\nj\nk\n=\n\u2212\n\u2212\n+ \u00b5\n+\n+\nr\n15 Find the shortest distance between the lines\n1\n1\n1\n7\n6\n1\nx\ny\nz\n+\n+\n+\n=\n=\n\u2212\n   and  \n3\n5\n7\n1\n2\n1\nx\ny\nz\n\u2212\n\u2212\n\u2212\n=\n=\n\u2212\n16"
    },
    {
        "Chapter": "1",
        "sentence_range": "5844-5847",
        "Text": "14 Find the shortest distance between the lines\n\u02c6\n\u02c6\n\u02c6\n(\n2\n)\nr\ni\nj\nk\n=\n+\n+\nr\n + \n\u02c6\n\u02c6\n\u02c6\n(\n)\ni\nj\nk\n\u03bb\n\u2212\n+\n and\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n2\n(2\n2 )\nr\ni\nj\nk\ni\nj\nk\n=\n\u2212\n\u2212\n+ \u00b5\n+\n+\nr\n15 Find the shortest distance between the lines\n1\n1\n1\n7\n6\n1\nx\ny\nz\n+\n+\n+\n=\n=\n\u2212\n   and  \n3\n5\n7\n1\n2\n1\nx\ny\nz\n\u2212\n\u2212\n\u2212\n=\n=\n\u2212\n16 Find the shortest distance between the lines whose vector equations are\n\u02c6\n\u02c6\n\u02c6\n(\n2\n3 )\nr\ni\nj\nk\n=\n+\n+\nr\n + \n\u02c6\n\u02c6\n\u02c6\n(\n3\n2 )\ni\nj\nk\n\u03bb\n\u2212\n+\nand  \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n4\n5\n6\n(2\n3\n)\nr\ni\nj\nk\ni\nj\nk\n=\n+\n+\n+ \u00b5\n+\n+\nr\n17"
    },
    {
        "Chapter": "1",
        "sentence_range": "5845-5848",
        "Text": "Find the shortest distance between the lines\n\u02c6\n\u02c6\n\u02c6\n(\n2\n)\nr\ni\nj\nk\n=\n+\n+\nr\n + \n\u02c6\n\u02c6\n\u02c6\n(\n)\ni\nj\nk\n\u03bb\n\u2212\n+\n and\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n2\n(2\n2 )\nr\ni\nj\nk\ni\nj\nk\n=\n\u2212\n\u2212\n+ \u00b5\n+\n+\nr\n15 Find the shortest distance between the lines\n1\n1\n1\n7\n6\n1\nx\ny\nz\n+\n+\n+\n=\n=\n\u2212\n   and  \n3\n5\n7\n1\n2\n1\nx\ny\nz\n\u2212\n\u2212\n\u2212\n=\n=\n\u2212\n16 Find the shortest distance between the lines whose vector equations are\n\u02c6\n\u02c6\n\u02c6\n(\n2\n3 )\nr\ni\nj\nk\n=\n+\n+\nr\n + \n\u02c6\n\u02c6\n\u02c6\n(\n3\n2 )\ni\nj\nk\n\u03bb\n\u2212\n+\nand  \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n4\n5\n6\n(2\n3\n)\nr\ni\nj\nk\ni\nj\nk\n=\n+\n+\n+ \u00b5\n+\n+\nr\n17 Find the shortest distance between the lines whose vector equations are\n\u02c6\n\u02c6\n\u02c6\n(1\n)\n(\n2)\n(3\n2 )\nr\nt i\nt\nj\nt k\n=\n\u2212\n+\n\u2212\n+\n\u2212\nr\n and\n\u02c6\n\u02c6\n\u02c6\n(\n1)\n(2\n1)\n(2\n1)\nr\ns\ni\ns\nj\ns\nk\n=\n+\n+\n\u2212\n\u2212\n+\nr\n\u00a9 NCERT\nnot to be republished\nTHREE D IMENSIONAL G EOMETRY\n479\n11"
    },
    {
        "Chapter": "1",
        "sentence_range": "5846-5849",
        "Text": "Find the shortest distance between the lines\n1\n1\n1\n7\n6\n1\nx\ny\nz\n+\n+\n+\n=\n=\n\u2212\n   and  \n3\n5\n7\n1\n2\n1\nx\ny\nz\n\u2212\n\u2212\n\u2212\n=\n=\n\u2212\n16 Find the shortest distance between the lines whose vector equations are\n\u02c6\n\u02c6\n\u02c6\n(\n2\n3 )\nr\ni\nj\nk\n=\n+\n+\nr\n + \n\u02c6\n\u02c6\n\u02c6\n(\n3\n2 )\ni\nj\nk\n\u03bb\n\u2212\n+\nand  \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n4\n5\n6\n(2\n3\n)\nr\ni\nj\nk\ni\nj\nk\n=\n+\n+\n+ \u00b5\n+\n+\nr\n17 Find the shortest distance between the lines whose vector equations are\n\u02c6\n\u02c6\n\u02c6\n(1\n)\n(\n2)\n(3\n2 )\nr\nt i\nt\nj\nt k\n=\n\u2212\n+\n\u2212\n+\n\u2212\nr\n and\n\u02c6\n\u02c6\n\u02c6\n(\n1)\n(2\n1)\n(2\n1)\nr\ns\ni\ns\nj\ns\nk\n=\n+\n+\n\u2212\n\u2212\n+\nr\n\u00a9 NCERT\nnot to be republished\nTHREE D IMENSIONAL G EOMETRY\n479\n11 6  Plane\nA plane is determined uniquely if any one of the following is known:\n(i)\nthe normal to the plane and its distance from the origin is given, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "5847-5850",
        "Text": "Find the shortest distance between the lines whose vector equations are\n\u02c6\n\u02c6\n\u02c6\n(\n2\n3 )\nr\ni\nj\nk\n=\n+\n+\nr\n + \n\u02c6\n\u02c6\n\u02c6\n(\n3\n2 )\ni\nj\nk\n\u03bb\n\u2212\n+\nand  \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n4\n5\n6\n(2\n3\n)\nr\ni\nj\nk\ni\nj\nk\n=\n+\n+\n+ \u00b5\n+\n+\nr\n17 Find the shortest distance between the lines whose vector equations are\n\u02c6\n\u02c6\n\u02c6\n(1\n)\n(\n2)\n(3\n2 )\nr\nt i\nt\nj\nt k\n=\n\u2212\n+\n\u2212\n+\n\u2212\nr\n and\n\u02c6\n\u02c6\n\u02c6\n(\n1)\n(2\n1)\n(2\n1)\nr\ns\ni\ns\nj\ns\nk\n=\n+\n+\n\u2212\n\u2212\n+\nr\n\u00a9 NCERT\nnot to be republished\nTHREE D IMENSIONAL G EOMETRY\n479\n11 6  Plane\nA plane is determined uniquely if any one of the following is known:\n(i)\nthe normal to the plane and its distance from the origin is given, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "5848-5851",
        "Text": "Find the shortest distance between the lines whose vector equations are\n\u02c6\n\u02c6\n\u02c6\n(1\n)\n(\n2)\n(3\n2 )\nr\nt i\nt\nj\nt k\n=\n\u2212\n+\n\u2212\n+\n\u2212\nr\n and\n\u02c6\n\u02c6\n\u02c6\n(\n1)\n(2\n1)\n(2\n1)\nr\ns\ni\ns\nj\ns\nk\n=\n+\n+\n\u2212\n\u2212\n+\nr\n\u00a9 NCERT\nnot to be republished\nTHREE D IMENSIONAL G EOMETRY\n479\n11 6  Plane\nA plane is determined uniquely if any one of the following is known:\n(i)\nthe normal to the plane and its distance from the origin is given, i e , equation of\na plane in normal form"
    },
    {
        "Chapter": "1",
        "sentence_range": "5849-5852",
        "Text": "6  Plane\nA plane is determined uniquely if any one of the following is known:\n(i)\nthe normal to the plane and its distance from the origin is given, i e , equation of\na plane in normal form (ii)\nit passes through a point and is perpendicular to a given direction"
    },
    {
        "Chapter": "1",
        "sentence_range": "5850-5853",
        "Text": "e , equation of\na plane in normal form (ii)\nit passes through a point and is perpendicular to a given direction (iii)\nit passes through three given non collinear points"
    },
    {
        "Chapter": "1",
        "sentence_range": "5851-5854",
        "Text": ", equation of\na plane in normal form (ii)\nit passes through a point and is perpendicular to a given direction (iii)\nit passes through three given non collinear points Now we shall find vector and Cartesian equations of the planes"
    },
    {
        "Chapter": "1",
        "sentence_range": "5852-5855",
        "Text": "(ii)\nit passes through a point and is perpendicular to a given direction (iii)\nit passes through three given non collinear points Now we shall find vector and Cartesian equations of the planes 11"
    },
    {
        "Chapter": "1",
        "sentence_range": "5853-5856",
        "Text": "(iii)\nit passes through three given non collinear points Now we shall find vector and Cartesian equations of the planes 11 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "5854-5857",
        "Text": "Now we shall find vector and Cartesian equations of the planes 11 6 1Equation of a plane in normal form\nConsider a plane whose perpendicular distance from  the origin is d (d \u2260 0)"
    },
    {
        "Chapter": "1",
        "sentence_range": "5855-5858",
        "Text": "11 6 1Equation of a plane in normal form\nConsider a plane whose perpendicular distance from  the origin is d (d \u2260 0) Fig 11"
    },
    {
        "Chapter": "1",
        "sentence_range": "5856-5859",
        "Text": "6 1Equation of a plane in normal form\nConsider a plane whose perpendicular distance from  the origin is d (d \u2260 0) Fig 11 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "5857-5860",
        "Text": "1Equation of a plane in normal form\nConsider a plane whose perpendicular distance from  the origin is d (d \u2260 0) Fig 11 10 If ON\nuuur\n is the normal from the origin to the plane, and n\u02c6  is the unit normal vector\nalong ON\nuuur"
    },
    {
        "Chapter": "1",
        "sentence_range": "5858-5861",
        "Text": "Fig 11 10 If ON\nuuur\n is the normal from the origin to the plane, and n\u02c6  is the unit normal vector\nalong ON\nuuur Then ON\nuuur\n= d n\u02c6"
    },
    {
        "Chapter": "1",
        "sentence_range": "5859-5862",
        "Text": "10 If ON\nuuur\n is the normal from the origin to the plane, and n\u02c6  is the unit normal vector\nalong ON\nuuur Then ON\nuuur\n= d n\u02c6 Let P be any\npoint  on the plane"
    },
    {
        "Chapter": "1",
        "sentence_range": "5860-5863",
        "Text": "If ON\nuuur\n is the normal from the origin to the plane, and n\u02c6  is the unit normal vector\nalong ON\nuuur Then ON\nuuur\n= d n\u02c6 Let P be any\npoint  on the plane T herefore, NP\nuuur\nis\nperpendicular to  ON\nuuur\nTherefore,  NP ON"
    },
    {
        "Chapter": "1",
        "sentence_range": "5861-5864",
        "Text": "Then ON\nuuur\n= d n\u02c6 Let P be any\npoint  on the plane T herefore, NP\nuuur\nis\nperpendicular to  ON\nuuur\nTherefore,  NP ON \u22c5\nuuur uuur\n = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "5862-5865",
        "Text": "Let P be any\npoint  on the plane T herefore, NP\nuuur\nis\nperpendicular to  ON\nuuur\nTherefore,  NP ON \u22c5\nuuur uuur\n = 0 (1)\nLet rr be the position vector of  the point P,,\nthen NP\nuuur\n= \ndn\nr\n\u02c6\nr\u2212\n (as ON\nNP\nOP\n+ =\nuuur\nuuur\nuuur\n)\nTherefore, (1) becomes\n(\n)\nr\nd n\nd n\n\u2227\n\u2227\n\u2212\n\u22c5\nr\n = 0\nor\n(\n)\nr\nd n\nn\n\u2227\n\u2227\n\u2212\n\u22c5\nr\n = 0\n(d \u2260 0)\nor\nr n\nd n n\n\u2227\n\u2227\n\u2227\n\u22c5\n\u2212\n\u22c5\nr\n = 0\ni"
    },
    {
        "Chapter": "1",
        "sentence_range": "5863-5866",
        "Text": "T herefore, NP\nuuur\nis\nperpendicular to  ON\nuuur\nTherefore,  NP ON \u22c5\nuuur uuur\n = 0 (1)\nLet rr be the position vector of  the point P,,\nthen NP\nuuur\n= \ndn\nr\n\u02c6\nr\u2212\n (as ON\nNP\nOP\n+ =\nuuur\nuuur\nuuur\n)\nTherefore, (1) becomes\n(\n)\nr\nd n\nd n\n\u2227\n\u2227\n\u2212\n\u22c5\nr\n = 0\nor\n(\n)\nr\nd n\nn\n\u2227\n\u2227\n\u2212\n\u22c5\nr\n = 0\n(d \u2260 0)\nor\nr n\nd n n\n\u2227\n\u2227\n\u2227\n\u22c5\n\u2212\n\u22c5\nr\n = 0\ni e"
    },
    {
        "Chapter": "1",
        "sentence_range": "5864-5867",
        "Text": "\u22c5\nuuur uuur\n = 0 (1)\nLet rr be the position vector of  the point P,,\nthen NP\nuuur\n= \ndn\nr\n\u02c6\nr\u2212\n (as ON\nNP\nOP\n+ =\nuuur\nuuur\nuuur\n)\nTherefore, (1) becomes\n(\n)\nr\nd n\nd n\n\u2227\n\u2227\n\u2212\n\u22c5\nr\n = 0\nor\n(\n)\nr\nd n\nn\n\u2227\n\u2227\n\u2212\n\u22c5\nr\n = 0\n(d \u2260 0)\nor\nr n\nd n n\n\u2227\n\u2227\n\u2227\n\u22c5\n\u2212\n\u22c5\nr\n = 0\ni e ,\nr n\nr\u22c5\u2227\n = d\n(as\n1)\n\u2227n n\n\u2227\u22c5\n=\n\u2026 (2)\nThis is the vector form of the equation of the plane"
    },
    {
        "Chapter": "1",
        "sentence_range": "5865-5868",
        "Text": "(1)\nLet rr be the position vector of  the point P,,\nthen NP\nuuur\n= \ndn\nr\n\u02c6\nr\u2212\n (as ON\nNP\nOP\n+ =\nuuur\nuuur\nuuur\n)\nTherefore, (1) becomes\n(\n)\nr\nd n\nd n\n\u2227\n\u2227\n\u2212\n\u22c5\nr\n = 0\nor\n(\n)\nr\nd n\nn\n\u2227\n\u2227\n\u2212\n\u22c5\nr\n = 0\n(d \u2260 0)\nor\nr n\nd n n\n\u2227\n\u2227\n\u2227\n\u22c5\n\u2212\n\u22c5\nr\n = 0\ni e ,\nr n\nr\u22c5\u2227\n = d\n(as\n1)\n\u2227n n\n\u2227\u22c5\n=\n\u2026 (2)\nThis is the vector form of the equation of the plane Cartesian form\nEquation (2) gives the vector equation of a plane, where n\u02c6  is the unit vector normal to\nthe plane"
    },
    {
        "Chapter": "1",
        "sentence_range": "5866-5869",
        "Text": "e ,\nr n\nr\u22c5\u2227\n = d\n(as\n1)\n\u2227n n\n\u2227\u22c5\n=\n\u2026 (2)\nThis is the vector form of the equation of the plane Cartesian form\nEquation (2) gives the vector equation of a plane, where n\u02c6  is the unit vector normal to\nthe plane Let P(x, y, z) be any point on the plane"
    },
    {
        "Chapter": "1",
        "sentence_range": "5867-5870",
        "Text": ",\nr n\nr\u22c5\u2227\n = d\n(as\n1)\n\u2227n n\n\u2227\u22c5\n=\n\u2026 (2)\nThis is the vector form of the equation of the plane Cartesian form\nEquation (2) gives the vector equation of a plane, where n\u02c6  is the unit vector normal to\nthe plane Let P(x, y, z) be any point on the plane Then\nOP\nuuur\n = \n\u02c6\n\u02c6\n\u02c6\nr\nx i\ny j\n= + +z k\nr\nLet l, m, n be the direction cosines of n\u02c6"
    },
    {
        "Chapter": "1",
        "sentence_range": "5868-5871",
        "Text": "Cartesian form\nEquation (2) gives the vector equation of a plane, where n\u02c6  is the unit vector normal to\nthe plane Let P(x, y, z) be any point on the plane Then\nOP\nuuur\n = \n\u02c6\n\u02c6\n\u02c6\nr\nx i\ny j\n= + +z k\nr\nLet l, m, n be the direction cosines of n\u02c6 Then\n\u02c6n = \n\u02c6\n\u02c6\n\u02c6\nl i\nm j\nn k\n+ +\nX\nY\nZ\nN\nP(\n)\nx, y,z\n r\nd\nO\nFig 11"
    },
    {
        "Chapter": "1",
        "sentence_range": "5869-5872",
        "Text": "Let P(x, y, z) be any point on the plane Then\nOP\nuuur\n = \n\u02c6\n\u02c6\n\u02c6\nr\nx i\ny j\n= + +z k\nr\nLet l, m, n be the direction cosines of n\u02c6 Then\n\u02c6n = \n\u02c6\n\u02c6\n\u02c6\nl i\nm j\nn k\n+ +\nX\nY\nZ\nN\nP(\n)\nx, y,z\n r\nd\nO\nFig 11 10\n\u00a9 NCERT\nnot to be republished\n MATHEMATI CS\n480\nTherefore, (2) gives\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(\n) (\n)\nx i\ny j\nz k\nl i\nm j\nn k\nd\n+\n+\n\u22c5\n+\n+\n=\ni"
    },
    {
        "Chapter": "1",
        "sentence_range": "5870-5873",
        "Text": "Then\nOP\nuuur\n = \n\u02c6\n\u02c6\n\u02c6\nr\nx i\ny j\n= + +z k\nr\nLet l, m, n be the direction cosines of n\u02c6 Then\n\u02c6n = \n\u02c6\n\u02c6\n\u02c6\nl i\nm j\nn k\n+ +\nX\nY\nZ\nN\nP(\n)\nx, y,z\n r\nd\nO\nFig 11 10\n\u00a9 NCERT\nnot to be republished\n MATHEMATI CS\n480\nTherefore, (2) gives\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(\n) (\n)\nx i\ny j\nz k\nl i\nm j\nn k\nd\n+\n+\n\u22c5\n+\n+\n=\ni e"
    },
    {
        "Chapter": "1",
        "sentence_range": "5871-5874",
        "Text": "Then\n\u02c6n = \n\u02c6\n\u02c6\n\u02c6\nl i\nm j\nn k\n+ +\nX\nY\nZ\nN\nP(\n)\nx, y,z\n r\nd\nO\nFig 11 10\n\u00a9 NCERT\nnot to be republished\n MATHEMATI CS\n480\nTherefore, (2) gives\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(\n) (\n)\nx i\ny j\nz k\nl i\nm j\nn k\nd\n+\n+\n\u22c5\n+\n+\n=\ni e ,\n lx + my + nz = d"
    },
    {
        "Chapter": "1",
        "sentence_range": "5872-5875",
        "Text": "10\n\u00a9 NCERT\nnot to be republished\n MATHEMATI CS\n480\nTherefore, (2) gives\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(\n) (\n)\nx i\ny j\nz k\nl i\nm j\nn k\nd\n+\n+\n\u22c5\n+\n+\n=\ni e ,\n lx + my + nz = d (3)\nThis is the cartesian equation of the plane in the normal form"
    },
    {
        "Chapter": "1",
        "sentence_range": "5873-5876",
        "Text": "e ,\n lx + my + nz = d (3)\nThis is the cartesian equation of the plane in the normal form \ufffdNote   Equation (3) shows that if \n\u02c6\n\u02c6\n\u02c6\n(\n)\nr\na i\nb j\nc k\n\u22c5\n+\n+\nr\n= d is the vector equation\nof a plane, then ax + by + cz = d is the Cartesian equation of the plane, where a, b\nand c are the direction ratios of the normal to the plane"
    },
    {
        "Chapter": "1",
        "sentence_range": "5874-5877",
        "Text": ",\n lx + my + nz = d (3)\nThis is the cartesian equation of the plane in the normal form \ufffdNote   Equation (3) shows that if \n\u02c6\n\u02c6\n\u02c6\n(\n)\nr\na i\nb j\nc k\n\u22c5\n+\n+\nr\n= d is the vector equation\nof a plane, then ax + by + cz = d is the Cartesian equation of the plane, where a, b\nand c are the direction ratios of the normal to the plane Example 13 Find the vector equation of the plane which is at a distance of \n29\n6\nfrom the origin and its normal vector from the origin is \n\u02c6\n\u02c6\n\u02c6\n2\n3\n4\ni\nj\n\u2212 +k"
    },
    {
        "Chapter": "1",
        "sentence_range": "5875-5878",
        "Text": "(3)\nThis is the cartesian equation of the plane in the normal form \ufffdNote   Equation (3) shows that if \n\u02c6\n\u02c6\n\u02c6\n(\n)\nr\na i\nb j\nc k\n\u22c5\n+\n+\nr\n= d is the vector equation\nof a plane, then ax + by + cz = d is the Cartesian equation of the plane, where a, b\nand c are the direction ratios of the normal to the plane Example 13 Find the vector equation of the plane which is at a distance of \n29\n6\nfrom the origin and its normal vector from the origin is \n\u02c6\n\u02c6\n\u02c6\n2\n3\n4\ni\nj\n\u2212 +k Solution Let \nk\nj\ni\nn\n4\u02c6\n3\u02c6\n2\u02c6\n+\n\u2212\nr="
    },
    {
        "Chapter": "1",
        "sentence_range": "5876-5879",
        "Text": "\ufffdNote   Equation (3) shows that if \n\u02c6\n\u02c6\n\u02c6\n(\n)\nr\na i\nb j\nc k\n\u22c5\n+\n+\nr\n= d is the vector equation\nof a plane, then ax + by + cz = d is the Cartesian equation of the plane, where a, b\nand c are the direction ratios of the normal to the plane Example 13 Find the vector equation of the plane which is at a distance of \n29\n6\nfrom the origin and its normal vector from the origin is \n\u02c6\n\u02c6\n\u02c6\n2\n3\n4\ni\nj\n\u2212 +k Solution Let \nk\nj\ni\nn\n4\u02c6\n3\u02c6\n2\u02c6\n+\n\u2212\nr= Then\n|\n|\n\u02c6\nnn\nn\nr\nr\n=\n =\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n2\n3\n4\n2\n3\n4\n4\n9\n16\n29\ni\nj\nk\ni\nj\nk\n\u2212\n+\n\u2212\n+\n=\n+\n+\nHence, the required equation of the plane is\n2\n3\n4\n6\n\u02c6\n\u02c6\n\u02c6\n29\n29\n29\n29\nr\ni\nj\nk\n\u2212\n\uf8eb\n\uf8f6\n\u22c5\n+\n+\n=\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\nr\nExample 14 Find the direction cosines of the unit vector perpendicular to the plane\n\u02c6\n\u02c6\n\u02c6\n(6\n3\n2 )\n1\nr\ni\nj\nk\n\u22c5\n\u2212\n\u2212\n+\nr\n = 0  passing through the origin"
    },
    {
        "Chapter": "1",
        "sentence_range": "5877-5880",
        "Text": "Example 13 Find the vector equation of the plane which is at a distance of \n29\n6\nfrom the origin and its normal vector from the origin is \n\u02c6\n\u02c6\n\u02c6\n2\n3\n4\ni\nj\n\u2212 +k Solution Let \nk\nj\ni\nn\n4\u02c6\n3\u02c6\n2\u02c6\n+\n\u2212\nr= Then\n|\n|\n\u02c6\nnn\nn\nr\nr\n=\n =\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n2\n3\n4\n2\n3\n4\n4\n9\n16\n29\ni\nj\nk\ni\nj\nk\n\u2212\n+\n\u2212\n+\n=\n+\n+\nHence, the required equation of the plane is\n2\n3\n4\n6\n\u02c6\n\u02c6\n\u02c6\n29\n29\n29\n29\nr\ni\nj\nk\n\u2212\n\uf8eb\n\uf8f6\n\u22c5\n+\n+\n=\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\nr\nExample 14 Find the direction cosines of the unit vector perpendicular to the plane\n\u02c6\n\u02c6\n\u02c6\n(6\n3\n2 )\n1\nr\ni\nj\nk\n\u22c5\n\u2212\n\u2212\n+\nr\n = 0  passing through the origin Solution The given equation can be written as\n\u02c6\n\u02c6\n\u02c6\n(\n6\n3\n2\n\u22c5\n\u2212\n+\n+\nrr\ni\nj\nk ) = 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "5878-5881",
        "Text": "Solution Let \nk\nj\ni\nn\n4\u02c6\n3\u02c6\n2\u02c6\n+\n\u2212\nr= Then\n|\n|\n\u02c6\nnn\nn\nr\nr\n=\n =\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n2\n3\n4\n2\n3\n4\n4\n9\n16\n29\ni\nj\nk\ni\nj\nk\n\u2212\n+\n\u2212\n+\n=\n+\n+\nHence, the required equation of the plane is\n2\n3\n4\n6\n\u02c6\n\u02c6\n\u02c6\n29\n29\n29\n29\nr\ni\nj\nk\n\u2212\n\uf8eb\n\uf8f6\n\u22c5\n+\n+\n=\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\nr\nExample 14 Find the direction cosines of the unit vector perpendicular to the plane\n\u02c6\n\u02c6\n\u02c6\n(6\n3\n2 )\n1\nr\ni\nj\nk\n\u22c5\n\u2212\n\u2212\n+\nr\n = 0  passing through the origin Solution The given equation can be written as\n\u02c6\n\u02c6\n\u02c6\n(\n6\n3\n2\n\u22c5\n\u2212\n+\n+\nrr\ni\nj\nk ) = 1 (1)\nNow\n\u02c6\n\u02c6\n\u02c6\n|\n6\n3\n2\n|\ni\nj\nk\n\u2212\n+\n+\n =\n36\n9\n4\n7\n+ + =\nTherefore,  dividing both sides of (1) by 7, we get\n6\n3\n2 \u02c6\n\u02c6\n\u02c6\n7\n7\n7\nr\ni\nj\nk\n\uf8eb\n\uf8f6\n\u22c5 \u2212\n+\n+\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\nr\n = 1\n7\nwhich is the equation of the plane in the form \nr n\u02c6\nr\u22c5 =d"
    },
    {
        "Chapter": "1",
        "sentence_range": "5879-5882",
        "Text": "Then\n|\n|\n\u02c6\nnn\nn\nr\nr\n=\n =\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n2\n3\n4\n2\n3\n4\n4\n9\n16\n29\ni\nj\nk\ni\nj\nk\n\u2212\n+\n\u2212\n+\n=\n+\n+\nHence, the required equation of the plane is\n2\n3\n4\n6\n\u02c6\n\u02c6\n\u02c6\n29\n29\n29\n29\nr\ni\nj\nk\n\u2212\n\uf8eb\n\uf8f6\n\u22c5\n+\n+\n=\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\nr\nExample 14 Find the direction cosines of the unit vector perpendicular to the plane\n\u02c6\n\u02c6\n\u02c6\n(6\n3\n2 )\n1\nr\ni\nj\nk\n\u22c5\n\u2212\n\u2212\n+\nr\n = 0  passing through the origin Solution The given equation can be written as\n\u02c6\n\u02c6\n\u02c6\n(\n6\n3\n2\n\u22c5\n\u2212\n+\n+\nrr\ni\nj\nk ) = 1 (1)\nNow\n\u02c6\n\u02c6\n\u02c6\n|\n6\n3\n2\n|\ni\nj\nk\n\u2212\n+\n+\n =\n36\n9\n4\n7\n+ + =\nTherefore,  dividing both sides of (1) by 7, we get\n6\n3\n2 \u02c6\n\u02c6\n\u02c6\n7\n7\n7\nr\ni\nj\nk\n\uf8eb\n\uf8f6\n\u22c5 \u2212\n+\n+\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\nr\n = 1\n7\nwhich is the equation of the plane in the form \nr n\u02c6\nr\u22c5 =d This shows that \nk\nj\ni\nn\n\u02c6\n27\n37\u02c6\n67\u02c6\n\u02c6\n+\n+\n=\u2212\n is a unit vector perpendicular to the\nplane through the origin"
    },
    {
        "Chapter": "1",
        "sentence_range": "5880-5883",
        "Text": "Solution The given equation can be written as\n\u02c6\n\u02c6\n\u02c6\n(\n6\n3\n2\n\u22c5\n\u2212\n+\n+\nrr\ni\nj\nk ) = 1 (1)\nNow\n\u02c6\n\u02c6\n\u02c6\n|\n6\n3\n2\n|\ni\nj\nk\n\u2212\n+\n+\n =\n36\n9\n4\n7\n+ + =\nTherefore,  dividing both sides of (1) by 7, we get\n6\n3\n2 \u02c6\n\u02c6\n\u02c6\n7\n7\n7\nr\ni\nj\nk\n\uf8eb\n\uf8f6\n\u22c5 \u2212\n+\n+\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\nr\n = 1\n7\nwhich is the equation of the plane in the form \nr n\u02c6\nr\u22c5 =d This shows that \nk\nj\ni\nn\n\u02c6\n27\n37\u02c6\n67\u02c6\n\u02c6\n+\n+\n=\u2212\n is a unit vector perpendicular to the\nplane through the origin Hence, the direction cosines of n\u02c6  are \n27\n,\n37\n,\n7\n\u22126"
    },
    {
        "Chapter": "1",
        "sentence_range": "5881-5884",
        "Text": "(1)\nNow\n\u02c6\n\u02c6\n\u02c6\n|\n6\n3\n2\n|\ni\nj\nk\n\u2212\n+\n+\n =\n36\n9\n4\n7\n+ + =\nTherefore,  dividing both sides of (1) by 7, we get\n6\n3\n2 \u02c6\n\u02c6\n\u02c6\n7\n7\n7\nr\ni\nj\nk\n\uf8eb\n\uf8f6\n\u22c5 \u2212\n+\n+\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\nr\n = 1\n7\nwhich is the equation of the plane in the form \nr n\u02c6\nr\u22c5 =d This shows that \nk\nj\ni\nn\n\u02c6\n27\n37\u02c6\n67\u02c6\n\u02c6\n+\n+\n=\u2212\n is a unit vector perpendicular to the\nplane through the origin Hence, the direction cosines of n\u02c6  are \n27\n,\n37\n,\n7\n\u22126 \u00a9 NCERT\nnot to be republished\nTHREE D IMENSIONAL G EOMETRY\n481\nZ\nY\nX\nO\nP(x1, y1, z1)\nExample 15 Find the distance of the plane 2x \u2013  3y + 4z \u2013 6 = 0 from the origin"
    },
    {
        "Chapter": "1",
        "sentence_range": "5882-5885",
        "Text": "This shows that \nk\nj\ni\nn\n\u02c6\n27\n37\u02c6\n67\u02c6\n\u02c6\n+\n+\n=\u2212\n is a unit vector perpendicular to the\nplane through the origin Hence, the direction cosines of n\u02c6  are \n27\n,\n37\n,\n7\n\u22126 \u00a9 NCERT\nnot to be republished\nTHREE D IMENSIONAL G EOMETRY\n481\nZ\nY\nX\nO\nP(x1, y1, z1)\nExample 15 Find the distance of the plane 2x \u2013  3y + 4z \u2013 6 = 0 from the origin Solution Since the direction ratios of the normal to the plane are 2, \u20133, 4; the direction\ncosines of it are\n2\n2\n2\n2\n2\n2\n2\n2\n2\n2\n3\n4\n,\n,\n2\n( 3)\n4\n2\n( 3)\n4\n2\n( 3)\n4\n\u2212\n+ \u2212\n+\n+ \u2212\n+\n+ \u2212\n+\n , i"
    },
    {
        "Chapter": "1",
        "sentence_range": "5883-5886",
        "Text": "Hence, the direction cosines of n\u02c6  are \n27\n,\n37\n,\n7\n\u22126 \u00a9 NCERT\nnot to be republished\nTHREE D IMENSIONAL G EOMETRY\n481\nZ\nY\nX\nO\nP(x1, y1, z1)\nExample 15 Find the distance of the plane 2x \u2013  3y + 4z \u2013 6 = 0 from the origin Solution Since the direction ratios of the normal to the plane are 2, \u20133, 4; the direction\ncosines of it are\n2\n2\n2\n2\n2\n2\n2\n2\n2\n2\n3\n4\n,\n,\n2\n( 3)\n4\n2\n( 3)\n4\n2\n( 3)\n4\n\u2212\n+ \u2212\n+\n+ \u2212\n+\n+ \u2212\n+\n , i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "5884-5887",
        "Text": "\u00a9 NCERT\nnot to be republished\nTHREE D IMENSIONAL G EOMETRY\n481\nZ\nY\nX\nO\nP(x1, y1, z1)\nExample 15 Find the distance of the plane 2x \u2013  3y + 4z \u2013 6 = 0 from the origin Solution Since the direction ratios of the normal to the plane are 2, \u20133, 4; the direction\ncosines of it are\n2\n2\n2\n2\n2\n2\n2\n2\n2\n2\n3\n4\n,\n,\n2\n( 3)\n4\n2\n( 3)\n4\n2\n( 3)\n4\n\u2212\n+ \u2212\n+\n+ \u2212\n+\n+ \u2212\n+\n , i e , \n2\n3\n4\n,\n,\n29\n29\n29\n\u2212\nHence, dividing the equation 2x \u2013 3y + 4z \u2013 6 = 0 i"
    },
    {
        "Chapter": "1",
        "sentence_range": "5885-5888",
        "Text": "Solution Since the direction ratios of the normal to the plane are 2, \u20133, 4; the direction\ncosines of it are\n2\n2\n2\n2\n2\n2\n2\n2\n2\n2\n3\n4\n,\n,\n2\n( 3)\n4\n2\n( 3)\n4\n2\n( 3)\n4\n\u2212\n+ \u2212\n+\n+ \u2212\n+\n+ \u2212\n+\n , i e , \n2\n3\n4\n,\n,\n29\n29\n29\n\u2212\nHence, dividing the equation 2x \u2013 3y + 4z \u2013 6 = 0 i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "5886-5889",
        "Text": "e , \n2\n3\n4\n,\n,\n29\n29\n29\n\u2212\nHence, dividing the equation 2x \u2013 3y + 4z \u2013 6 = 0 i e ,  2x \u2013 3y + 4z = 6 throughout by\n29 , we get\n2\n3\n4\n6\n29\n29\n29\n29\nx\ny\nz\n\u2212\n+\n+\n=\nThis is of the form lx + my + nz = d, where d is the distance of the plane from the\norigin"
    },
    {
        "Chapter": "1",
        "sentence_range": "5887-5890",
        "Text": ", \n2\n3\n4\n,\n,\n29\n29\n29\n\u2212\nHence, dividing the equation 2x \u2013 3y + 4z \u2013 6 = 0 i e ,  2x \u2013 3y + 4z = 6 throughout by\n29 , we get\n2\n3\n4\n6\n29\n29\n29\n29\nx\ny\nz\n\u2212\n+\n+\n=\nThis is of the form lx + my + nz = d, where d is the distance of the plane from the\norigin So, the distance of the plane from the origin is \n29\n6"
    },
    {
        "Chapter": "1",
        "sentence_range": "5888-5891",
        "Text": "e ,  2x \u2013 3y + 4z = 6 throughout by\n29 , we get\n2\n3\n4\n6\n29\n29\n29\n29\nx\ny\nz\n\u2212\n+\n+\n=\nThis is of the form lx + my + nz = d, where d is the distance of the plane from the\norigin So, the distance of the plane from the origin is \n29\n6 Example 16 Find the coordinates of the foot of the perpendicular drawn from the\norigin to the plane 2x \u2013 3y + 4z \u2013 6 = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "5889-5892",
        "Text": ",  2x \u2013 3y + 4z = 6 throughout by\n29 , we get\n2\n3\n4\n6\n29\n29\n29\n29\nx\ny\nz\n\u2212\n+\n+\n=\nThis is of the form lx + my + nz = d, where d is the distance of the plane from the\norigin So, the distance of the plane from the origin is \n29\n6 Example 16 Find the coordinates of the foot of the perpendicular drawn from the\norigin to the plane 2x \u2013 3y + 4z \u2013 6 = 0 Solution Let the coordinates of the foot of the perpendicular P from the origin to the\nplane is (x1, y1, z1) (Fig 11"
    },
    {
        "Chapter": "1",
        "sentence_range": "5890-5893",
        "Text": "So, the distance of the plane from the origin is \n29\n6 Example 16 Find the coordinates of the foot of the perpendicular drawn from the\norigin to the plane 2x \u2013 3y + 4z \u2013 6 = 0 Solution Let the coordinates of the foot of the perpendicular P from the origin to the\nplane is (x1, y1, z1) (Fig 11 11)"
    },
    {
        "Chapter": "1",
        "sentence_range": "5891-5894",
        "Text": "Example 16 Find the coordinates of the foot of the perpendicular drawn from the\norigin to the plane 2x \u2013 3y + 4z \u2013 6 = 0 Solution Let the coordinates of the foot of the perpendicular P from the origin to the\nplane is (x1, y1, z1) (Fig 11 11) Then, the direction ratios of the line OP are\nx1, y1, z1"
    },
    {
        "Chapter": "1",
        "sentence_range": "5892-5895",
        "Text": "Solution Let the coordinates of the foot of the perpendicular P from the origin to the\nplane is (x1, y1, z1) (Fig 11 11) Then, the direction ratios of the line OP are\nx1, y1, z1 Writing the equation of the plane in the normal\nform, we have\n2\n3\n4\n6\n29\n29\n29\n29\nx\ny\nz\n\u2212\n+\n=\nwhere, \n2\n3\n4\n,\n,\n29\n29\n29\n\u2212\n are the direct ion\ncosines of the OP"
    },
    {
        "Chapter": "1",
        "sentence_range": "5893-5896",
        "Text": "11) Then, the direction ratios of the line OP are\nx1, y1, z1 Writing the equation of the plane in the normal\nform, we have\n2\n3\n4\n6\n29\n29\n29\n29\nx\ny\nz\n\u2212\n+\n=\nwhere, \n2\n3\n4\n,\n,\n29\n29\n29\n\u2212\n are the direct ion\ncosines of the OP Since d"
    },
    {
        "Chapter": "1",
        "sentence_range": "5894-5897",
        "Text": "Then, the direction ratios of the line OP are\nx1, y1, z1 Writing the equation of the plane in the normal\nform, we have\n2\n3\n4\n6\n29\n29\n29\n29\nx\ny\nz\n\u2212\n+\n=\nwhere, \n2\n3\n4\n,\n,\n29\n29\n29\n\u2212\n are the direct ion\ncosines of the OP Since d c"
    },
    {
        "Chapter": "1",
        "sentence_range": "5895-5898",
        "Text": "Writing the equation of the plane in the normal\nform, we have\n2\n3\n4\n6\n29\n29\n29\n29\nx\ny\nz\n\u2212\n+\n=\nwhere, \n2\n3\n4\n,\n,\n29\n29\n29\n\u2212\n are the direct ion\ncosines of the OP Since d c \u2019s and direction ratios of a line are proportional, we have\n21\n29\nx\n =\n1\n1\n3\n4\n29\n29\ny\nz\n=\n\u2212\n= k\ni"
    },
    {
        "Chapter": "1",
        "sentence_range": "5896-5899",
        "Text": "Since d c \u2019s and direction ratios of a line are proportional, we have\n21\n29\nx\n =\n1\n1\n3\n4\n29\n29\ny\nz\n=\n\u2212\n= k\ni e"
    },
    {
        "Chapter": "1",
        "sentence_range": "5897-5900",
        "Text": "c \u2019s and direction ratios of a line are proportional, we have\n21\n29\nx\n =\n1\n1\n3\n4\n29\n29\ny\nz\n=\n\u2212\n= k\ni e ,\nx1 =\n29\n2k , y1 = \n1\n3\n4\n,\n29\n29\nk\nk\nz\n\u2212\n=\nFig 11"
    },
    {
        "Chapter": "1",
        "sentence_range": "5898-5901",
        "Text": "\u2019s and direction ratios of a line are proportional, we have\n21\n29\nx\n =\n1\n1\n3\n4\n29\n29\ny\nz\n=\n\u2212\n= k\ni e ,\nx1 =\n29\n2k , y1 = \n1\n3\n4\n,\n29\n29\nk\nk\nz\n\u2212\n=\nFig 11 11\n\u00a9 NCERT\nnot to be republished\n MATHEMATI CS\n482\nSubstituting these in the equation of the plane, we get k = \n29\n6"
    },
    {
        "Chapter": "1",
        "sentence_range": "5899-5902",
        "Text": "e ,\nx1 =\n29\n2k , y1 = \n1\n3\n4\n,\n29\n29\nk\nk\nz\n\u2212\n=\nFig 11 11\n\u00a9 NCERT\nnot to be republished\n MATHEMATI CS\n482\nSubstituting these in the equation of the plane, we get k = \n29\n6 Hence, the foot of the perpendicular is 12\n18 24\n,\n,\n29\n29\n29\n\u2212\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8"
    },
    {
        "Chapter": "1",
        "sentence_range": "5900-5903",
        "Text": ",\nx1 =\n29\n2k , y1 = \n1\n3\n4\n,\n29\n29\nk\nk\nz\n\u2212\n=\nFig 11 11\n\u00a9 NCERT\nnot to be republished\n MATHEMATI CS\n482\nSubstituting these in the equation of the plane, we get k = \n29\n6 Hence, the foot of the perpendicular is 12\n18 24\n,\n,\n29\n29\n29\n\u2212\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8 \ufffdNote   If d is the distance from the origin and l, m, n are the direction cosines of\nthe normal to the plane through the origin, then the foot of the perpendicular is\n(ld, md, nd)"
    },
    {
        "Chapter": "1",
        "sentence_range": "5901-5904",
        "Text": "11\n\u00a9 NCERT\nnot to be republished\n MATHEMATI CS\n482\nSubstituting these in the equation of the plane, we get k = \n29\n6 Hence, the foot of the perpendicular is 12\n18 24\n,\n,\n29\n29\n29\n\u2212\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8 \ufffdNote   If d is the distance from the origin and l, m, n are the direction cosines of\nthe normal to the plane through the origin, then the foot of the perpendicular is\n(ld, md, nd) 11"
    },
    {
        "Chapter": "1",
        "sentence_range": "5902-5905",
        "Text": "Hence, the foot of the perpendicular is 12\n18 24\n,\n,\n29\n29\n29\n\u2212\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8 \ufffdNote   If d is the distance from the origin and l, m, n are the direction cosines of\nthe normal to the plane through the origin, then the foot of the perpendicular is\n(ld, md, nd) 11 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "5903-5906",
        "Text": "\ufffdNote   If d is the distance from the origin and l, m, n are the direction cosines of\nthe normal to the plane through the origin, then the foot of the perpendicular is\n(ld, md, nd) 11 6 2Equation of a plane perpendicular to a\ngiven vector and passing through a given point\nIn the space, there can be many planes that are\nperpendicular to the given vector, but through a given\npoint P(x1, y1, z1), only one such plane exists (see\nFig 11"
    },
    {
        "Chapter": "1",
        "sentence_range": "5904-5907",
        "Text": "11 6 2Equation of a plane perpendicular to a\ngiven vector and passing through a given point\nIn the space, there can be many planes that are\nperpendicular to the given vector, but through a given\npoint P(x1, y1, z1), only one such plane exists (see\nFig 11 12)"
    },
    {
        "Chapter": "1",
        "sentence_range": "5905-5908",
        "Text": "6 2Equation of a plane perpendicular to a\ngiven vector and passing through a given point\nIn the space, there can be many planes that are\nperpendicular to the given vector, but through a given\npoint P(x1, y1, z1), only one such plane exists (see\nFig 11 12) Let a plane pass  through a point A with position\nvector ar  and perpendicular to the vector N\nur"
    },
    {
        "Chapter": "1",
        "sentence_range": "5906-5909",
        "Text": "2Equation of a plane perpendicular to a\ngiven vector and passing through a given point\nIn the space, there can be many planes that are\nperpendicular to the given vector, but through a given\npoint P(x1, y1, z1), only one such plane exists (see\nFig 11 12) Let a plane pass  through a point A with position\nvector ar  and perpendicular to the vector N\nur Let rr be the position vector of any point P(x, y, z) in the plane"
    },
    {
        "Chapter": "1",
        "sentence_range": "5907-5910",
        "Text": "12) Let a plane pass  through a point A with position\nvector ar  and perpendicular to the vector N\nur Let rr be the position vector of any point P(x, y, z) in the plane (Fig 11"
    },
    {
        "Chapter": "1",
        "sentence_range": "5908-5911",
        "Text": "Let a plane pass  through a point A with position\nvector ar  and perpendicular to the vector N\nur Let rr be the position vector of any point P(x, y, z) in the plane (Fig 11 13)"
    },
    {
        "Chapter": "1",
        "sentence_range": "5909-5912",
        "Text": "Let rr be the position vector of any point P(x, y, z) in the plane (Fig 11 13) Then the point P lies in the plane if and only if\nAP\nuuur\n is perpendicular to N\nur"
    },
    {
        "Chapter": "1",
        "sentence_range": "5910-5913",
        "Text": "(Fig 11 13) Then the point P lies in the plane if and only if\nAP\nuuur\n is perpendicular to N\nur i"
    },
    {
        "Chapter": "1",
        "sentence_range": "5911-5914",
        "Text": "13) Then the point P lies in the plane if and only if\nAP\nuuur\n is perpendicular to N\nur i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "5912-5915",
        "Text": "Then the point P lies in the plane if and only if\nAP\nuuur\n is perpendicular to N\nur i e , AP\nuuur"
    },
    {
        "Chapter": "1",
        "sentence_range": "5913-5916",
        "Text": "i e , AP\nuuur N\nur\n= 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "5914-5917",
        "Text": "e , AP\nuuur N\nur\n= 0 But\nAP\nr\na\n=\n\u2212\nuuur\nr\nr"
    },
    {
        "Chapter": "1",
        "sentence_range": "5915-5918",
        "Text": ", AP\nuuur N\nur\n= 0 But\nAP\nr\na\n=\n\u2212\nuuur\nr\nr Therefore, (\n) N\n0\nr\n\u2212a\n\u22c5\n=\nr\nr\nr\n       \u2026 (1)\nThis is the vector equation of the plane"
    },
    {
        "Chapter": "1",
        "sentence_range": "5916-5919",
        "Text": "N\nur\n= 0 But\nAP\nr\na\n=\n\u2212\nuuur\nr\nr Therefore, (\n) N\n0\nr\n\u2212a\n\u22c5\n=\nr\nr\nr\n       \u2026 (1)\nThis is the vector equation of the plane Cartesian form\nLet the given point A be (x1, y1, z1), P be (x, y, z)\nand direction ratios of  N\nur\n are A, B and C"
    },
    {
        "Chapter": "1",
        "sentence_range": "5917-5920",
        "Text": "But\nAP\nr\na\n=\n\u2212\nuuur\nr\nr Therefore, (\n) N\n0\nr\n\u2212a\n\u22c5\n=\nr\nr\nr\n       \u2026 (1)\nThis is the vector equation of the plane Cartesian form\nLet the given point A be (x1, y1, z1), P be (x, y, z)\nand direction ratios of  N\nur\n are A, B and C Then,\n1\n1\n1 \u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n,\na\nx i\ny j\nz k\nr\nxi\ny j\nz k\n=\n+\n+\n=\n+\n+\nr\nr\n and \n\u02c6\n\u02c6\n\u02c6\nN\nA\nB\nC\ni\nj\nk\n= + +\nr\nNow\nr( \u2013 ) N= 0\na \u22c5\nr\nr\nr\nSo\n(\n)\n(\n)\n(\n)\n1\n1\n1 \u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(A\nB\nC ) 0\nx\nx i\ny\ny\nj\nz\nz\nk\ni\nj\nk\n\uf8ee\n\uf8f9\n\u2212\n+\n\u2212\n+\n\u2212\n\u22c5\n+\n+\n=\n\uf8f0\n\uf8fb\ni"
    },
    {
        "Chapter": "1",
        "sentence_range": "5918-5921",
        "Text": "Therefore, (\n) N\n0\nr\n\u2212a\n\u22c5\n=\nr\nr\nr\n       \u2026 (1)\nThis is the vector equation of the plane Cartesian form\nLet the given point A be (x1, y1, z1), P be (x, y, z)\nand direction ratios of  N\nur\n are A, B and C Then,\n1\n1\n1 \u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n,\na\nx i\ny j\nz k\nr\nxi\ny j\nz k\n=\n+\n+\n=\n+\n+\nr\nr\n and \n\u02c6\n\u02c6\n\u02c6\nN\nA\nB\nC\ni\nj\nk\n= + +\nr\nNow\nr( \u2013 ) N= 0\na \u22c5\nr\nr\nr\nSo\n(\n)\n(\n)\n(\n)\n1\n1\n1 \u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(A\nB\nC ) 0\nx\nx i\ny\ny\nj\nz\nz\nk\ni\nj\nk\n\uf8ee\n\uf8f9\n\u2212\n+\n\u2212\n+\n\u2212\n\u22c5\n+\n+\n=\n\uf8f0\n\uf8fb\ni e"
    },
    {
        "Chapter": "1",
        "sentence_range": "5919-5922",
        "Text": "Cartesian form\nLet the given point A be (x1, y1, z1), P be (x, y, z)\nand direction ratios of  N\nur\n are A, B and C Then,\n1\n1\n1 \u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n,\na\nx i\ny j\nz k\nr\nxi\ny j\nz k\n=\n+\n+\n=\n+\n+\nr\nr\n and \n\u02c6\n\u02c6\n\u02c6\nN\nA\nB\nC\ni\nj\nk\n= + +\nr\nNow\nr( \u2013 ) N= 0\na \u22c5\nr\nr\nr\nSo\n(\n)\n(\n)\n(\n)\n1\n1\n1 \u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(A\nB\nC ) 0\nx\nx i\ny\ny\nj\nz\nz\nk\ni\nj\nk\n\uf8ee\n\uf8f9\n\u2212\n+\n\u2212\n+\n\u2212\n\u22c5\n+\n+\n=\n\uf8f0\n\uf8fb\ni e A (x \u2013 x1) + B (y \u2013 y1) + C (z \u2013 z1) = 0\nExample 17 Find the vector and cartesian equations of the plane which passes through\nthe point (5, 2, \u2013 4) and perpendicular to the line with direction ratios 2, 3, \u2013 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "5920-5923",
        "Text": "Then,\n1\n1\n1 \u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n,\na\nx i\ny j\nz k\nr\nxi\ny j\nz k\n=\n+\n+\n=\n+\n+\nr\nr\n and \n\u02c6\n\u02c6\n\u02c6\nN\nA\nB\nC\ni\nj\nk\n= + +\nr\nNow\nr( \u2013 ) N= 0\na \u22c5\nr\nr\nr\nSo\n(\n)\n(\n)\n(\n)\n1\n1\n1 \u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(A\nB\nC ) 0\nx\nx i\ny\ny\nj\nz\nz\nk\ni\nj\nk\n\uf8ee\n\uf8f9\n\u2212\n+\n\u2212\n+\n\u2212\n\u22c5\n+\n+\n=\n\uf8f0\n\uf8fb\ni e A (x \u2013 x1) + B (y \u2013 y1) + C (z \u2013 z1) = 0\nExample 17 Find the vector and cartesian equations of the plane which passes through\nthe point (5, 2, \u2013 4) and perpendicular to the line with direction ratios 2, 3, \u2013 1 Fig 11"
    },
    {
        "Chapter": "1",
        "sentence_range": "5921-5924",
        "Text": "e A (x \u2013 x1) + B (y \u2013 y1) + C (z \u2013 z1) = 0\nExample 17 Find the vector and cartesian equations of the plane which passes through\nthe point (5, 2, \u2013 4) and perpendicular to the line with direction ratios 2, 3, \u2013 1 Fig 11 12\nFig 11"
    },
    {
        "Chapter": "1",
        "sentence_range": "5922-5925",
        "Text": "A (x \u2013 x1) + B (y \u2013 y1) + C (z \u2013 z1) = 0\nExample 17 Find the vector and cartesian equations of the plane which passes through\nthe point (5, 2, \u2013 4) and perpendicular to the line with direction ratios 2, 3, \u2013 1 Fig 11 12\nFig 11 13\n\u00a9 NCERT\nnot to be republished\nTHREE D IMENSIONAL G EOMETRY\n483\nY\nZ\nO\na\nr\nR\nP\nbS\nc\n(RS  RT)\nX\nX\nT\nSolution We have the position vector of point (5, 2, \u2013 4) as \n\u02c6\n\u02c6\n\u02c6\n5\n2\n4\na\ni\nj\nk\n=\n+\n\u2212\nr\n and the\nnormal vector N\nr\n perpendicular to the plane as \n\u02c6\n\u02c6\nN =2 + 3\u02c6\ni\nj\nk\u2212\nr\nTherefore, the vector equation of the plane is given by  (\n)"
    },
    {
        "Chapter": "1",
        "sentence_range": "5923-5926",
        "Text": "Fig 11 12\nFig 11 13\n\u00a9 NCERT\nnot to be republished\nTHREE D IMENSIONAL G EOMETRY\n483\nY\nZ\nO\na\nr\nR\nP\nbS\nc\n(RS  RT)\nX\nX\nT\nSolution We have the position vector of point (5, 2, \u2013 4) as \n\u02c6\n\u02c6\n\u02c6\n5\n2\n4\na\ni\nj\nk\n=\n+\n\u2212\nr\n and the\nnormal vector N\nr\n perpendicular to the plane as \n\u02c6\n\u02c6\nN =2 + 3\u02c6\ni\nj\nk\u2212\nr\nTherefore, the vector equation of the plane is given by  (\n) N\n0\nr\na\u2212\n=\nr\nr\nr\nor\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n[\n(5\n2\n4 )] (2\n3\n)\n0\nr\ni\nj\nk\ni\nj\nk\n\u2212\n+\n\u2212\n\u22c5\n+\n\u2212\n=\nr"
    },
    {
        "Chapter": "1",
        "sentence_range": "5924-5927",
        "Text": "12\nFig 11 13\n\u00a9 NCERT\nnot to be republished\nTHREE D IMENSIONAL G EOMETRY\n483\nY\nZ\nO\na\nr\nR\nP\nbS\nc\n(RS  RT)\nX\nX\nT\nSolution We have the position vector of point (5, 2, \u2013 4) as \n\u02c6\n\u02c6\n\u02c6\n5\n2\n4\na\ni\nj\nk\n=\n+\n\u2212\nr\n and the\nnormal vector N\nr\n perpendicular to the plane as \n\u02c6\n\u02c6\nN =2 + 3\u02c6\ni\nj\nk\u2212\nr\nTherefore, the vector equation of the plane is given by  (\n) N\n0\nr\na\u2212\n=\nr\nr\nr\nor\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n[\n(5\n2\n4 )] (2\n3\n)\n0\nr\ni\nj\nk\ni\nj\nk\n\u2212\n+\n\u2212\n\u22c5\n+\n\u2212\n=\nr (1)\nTransforming (1) into Cartesian form, we have\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n[( \u20135)\n(\n2)\n(\n4) ] (2\n3\n)\n0\nx\ni\ny\nj\nz\nk\ni\nj\nk\n+\n\u2212\n+\n+\n\u22c5\n+\n\u2212\n=\nor\n2(\n5) 3(\n2) 1(\n4) 0\nx\ny\nz\n\u2212 + \u2212 \u2212 + =\ni"
    },
    {
        "Chapter": "1",
        "sentence_range": "5925-5928",
        "Text": "13\n\u00a9 NCERT\nnot to be republished\nTHREE D IMENSIONAL G EOMETRY\n483\nY\nZ\nO\na\nr\nR\nP\nbS\nc\n(RS  RT)\nX\nX\nT\nSolution We have the position vector of point (5, 2, \u2013 4) as \n\u02c6\n\u02c6\n\u02c6\n5\n2\n4\na\ni\nj\nk\n=\n+\n\u2212\nr\n and the\nnormal vector N\nr\n perpendicular to the plane as \n\u02c6\n\u02c6\nN =2 + 3\u02c6\ni\nj\nk\u2212\nr\nTherefore, the vector equation of the plane is given by  (\n) N\n0\nr\na\u2212\n=\nr\nr\nr\nor\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n[\n(5\n2\n4 )] (2\n3\n)\n0\nr\ni\nj\nk\ni\nj\nk\n\u2212\n+\n\u2212\n\u22c5\n+\n\u2212\n=\nr (1)\nTransforming (1) into Cartesian form, we have\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n[( \u20135)\n(\n2)\n(\n4) ] (2\n3\n)\n0\nx\ni\ny\nj\nz\nk\ni\nj\nk\n+\n\u2212\n+\n+\n\u22c5\n+\n\u2212\n=\nor\n2(\n5) 3(\n2) 1(\n4) 0\nx\ny\nz\n\u2212 + \u2212 \u2212 + =\ni e"
    },
    {
        "Chapter": "1",
        "sentence_range": "5926-5929",
        "Text": "N\n0\nr\na\u2212\n=\nr\nr\nr\nor\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n[\n(5\n2\n4 )] (2\n3\n)\n0\nr\ni\nj\nk\ni\nj\nk\n\u2212\n+\n\u2212\n\u22c5\n+\n\u2212\n=\nr (1)\nTransforming (1) into Cartesian form, we have\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n[( \u20135)\n(\n2)\n(\n4) ] (2\n3\n)\n0\nx\ni\ny\nj\nz\nk\ni\nj\nk\n+\n\u2212\n+\n+\n\u22c5\n+\n\u2212\n=\nor\n2(\n5) 3(\n2) 1(\n4) 0\nx\ny\nz\n\u2212 + \u2212 \u2212 + =\ni e 2x + 3y \u2013 z = 20\nwhich is the cartesian equation of the plane"
    },
    {
        "Chapter": "1",
        "sentence_range": "5927-5930",
        "Text": "(1)\nTransforming (1) into Cartesian form, we have\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n[( \u20135)\n(\n2)\n(\n4) ] (2\n3\n)\n0\nx\ni\ny\nj\nz\nk\ni\nj\nk\n+\n\u2212\n+\n+\n\u22c5\n+\n\u2212\n=\nor\n2(\n5) 3(\n2) 1(\n4) 0\nx\ny\nz\n\u2212 + \u2212 \u2212 + =\ni e 2x + 3y \u2013 z = 20\nwhich is the cartesian equation of the plane 11"
    },
    {
        "Chapter": "1",
        "sentence_range": "5928-5931",
        "Text": "e 2x + 3y \u2013 z = 20\nwhich is the cartesian equation of the plane 11 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "5929-5932",
        "Text": "2x + 3y \u2013 z = 20\nwhich is the cartesian equation of the plane 11 6 3  Equation of a plane passing through three non collinear points\nLet R, S and T be three non collinear points on the plane with position vectors ar ,b\nr\nand\ncrrespectively (Fig 11"
    },
    {
        "Chapter": "1",
        "sentence_range": "5930-5933",
        "Text": "11 6 3  Equation of a plane passing through three non collinear points\nLet R, S and T be three non collinear points on the plane with position vectors ar ,b\nr\nand\ncrrespectively (Fig 11 14)"
    },
    {
        "Chapter": "1",
        "sentence_range": "5931-5934",
        "Text": "6 3  Equation of a plane passing through three non collinear points\nLet R, S and T be three non collinear points on the plane with position vectors ar ,b\nr\nand\ncrrespectively (Fig 11 14) Fig 11"
    },
    {
        "Chapter": "1",
        "sentence_range": "5932-5935",
        "Text": "3  Equation of a plane passing through three non collinear points\nLet R, S and T be three non collinear points on the plane with position vectors ar ,b\nr\nand\ncrrespectively (Fig 11 14) Fig 11 14\nThe vectors RS\nuuur\n and RT\nuuur\n are in the given plane"
    },
    {
        "Chapter": "1",
        "sentence_range": "5933-5936",
        "Text": "14) Fig 11 14\nThe vectors RS\nuuur\n and RT\nuuur\n are in the given plane Therefore, the vector RS\nRT\n\u00d7\nuuur\nuuur\nis perpendicular to the plane containing points R, S and T"
    },
    {
        "Chapter": "1",
        "sentence_range": "5934-5937",
        "Text": "Fig 11 14\nThe vectors RS\nuuur\n and RT\nuuur\n are in the given plane Therefore, the vector RS\nRT\n\u00d7\nuuur\nuuur\nis perpendicular to the plane containing points R, S and T Let rr be the position vector\nof any point P in the plane"
    },
    {
        "Chapter": "1",
        "sentence_range": "5935-5938",
        "Text": "14\nThe vectors RS\nuuur\n and RT\nuuur\n are in the given plane Therefore, the vector RS\nRT\n\u00d7\nuuur\nuuur\nis perpendicular to the plane containing points R, S and T Let rr be the position vector\nof any point P in the plane Therefore, the equation of the plane passing through R and\nperpendicular to the vector RS\nRT\n\u00d7\nuuur\nuuur\n is\n(\n) (RS\nRT)\nr\n\u2212a\n\u22c5\n\u00d7\nuuur\nuuur\nr\nr\n = 0\nor\nr\nr\nr\nr\nr\nr\n(\n)"
    },
    {
        "Chapter": "1",
        "sentence_range": "5936-5939",
        "Text": "Therefore, the vector RS\nRT\n\u00d7\nuuur\nuuur\nis perpendicular to the plane containing points R, S and T Let rr be the position vector\nof any point P in the plane Therefore, the equation of the plane passing through R and\nperpendicular to the vector RS\nRT\n\u00d7\nuuur\nuuur\n is\n(\n) (RS\nRT)\nr\n\u2212a\n\u22c5\n\u00d7\nuuur\nuuur\nr\nr\n = 0\nor\nr\nr\nr\nr\nr\nr\n(\n) [(\n)\u00d7(\n)]\nr \u2013a\nb \u2013a\nc \u2013a\n = 0                                                       \u2026 (1)\n\u00a9 NCERT\nnot to be republished\n MATHEMATI CS\n484\nFig 11"
    },
    {
        "Chapter": "1",
        "sentence_range": "5937-5940",
        "Text": "Let rr be the position vector\nof any point P in the plane Therefore, the equation of the plane passing through R and\nperpendicular to the vector RS\nRT\n\u00d7\nuuur\nuuur\n is\n(\n) (RS\nRT)\nr\n\u2212a\n\u22c5\n\u00d7\nuuur\nuuur\nr\nr\n = 0\nor\nr\nr\nr\nr\nr\nr\n(\n) [(\n)\u00d7(\n)]\nr \u2013a\nb \u2013a\nc \u2013a\n = 0                                                       \u2026 (1)\n\u00a9 NCERT\nnot to be republished\n MATHEMATI CS\n484\nFig 11 15\nThis is the equation of the plane in vector form passing through three noncollinear\npoints"
    },
    {
        "Chapter": "1",
        "sentence_range": "5938-5941",
        "Text": "Therefore, the equation of the plane passing through R and\nperpendicular to the vector RS\nRT\n\u00d7\nuuur\nuuur\n is\n(\n) (RS\nRT)\nr\n\u2212a\n\u22c5\n\u00d7\nuuur\nuuur\nr\nr\n = 0\nor\nr\nr\nr\nr\nr\nr\n(\n) [(\n)\u00d7(\n)]\nr \u2013a\nb \u2013a\nc \u2013a\n = 0                                                       \u2026 (1)\n\u00a9 NCERT\nnot to be republished\n MATHEMATI CS\n484\nFig 11 15\nThis is the equation of the plane in vector form passing through three noncollinear\npoints \ufffdNote   Why was it necessary to say that the three points\nhad to be non collinear"
    },
    {
        "Chapter": "1",
        "sentence_range": "5939-5942",
        "Text": "[(\n)\u00d7(\n)]\nr \u2013a\nb \u2013a\nc \u2013a\n = 0                                                       \u2026 (1)\n\u00a9 NCERT\nnot to be republished\n MATHEMATI CS\n484\nFig 11 15\nThis is the equation of the plane in vector form passing through three noncollinear\npoints \ufffdNote   Why was it necessary to say that the three points\nhad to be non collinear If the three points were on the same\nline, then there will be many planes that will contain them\n(Fig 11"
    },
    {
        "Chapter": "1",
        "sentence_range": "5940-5943",
        "Text": "15\nThis is the equation of the plane in vector form passing through three noncollinear\npoints \ufffdNote   Why was it necessary to say that the three points\nhad to be non collinear If the three points were on the same\nline, then there will be many planes that will contain them\n(Fig 11 15)"
    },
    {
        "Chapter": "1",
        "sentence_range": "5941-5944",
        "Text": "\ufffdNote   Why was it necessary to say that the three points\nhad to be non collinear If the three points were on the same\nline, then there will be many planes that will contain them\n(Fig 11 15) These planes will resemble the pages of a book where the\nline containing the points R, S and T are members in the binding\nof the book"
    },
    {
        "Chapter": "1",
        "sentence_range": "5942-5945",
        "Text": "If the three points were on the same\nline, then there will be many planes that will contain them\n(Fig 11 15) These planes will resemble the pages of a book where the\nline containing the points R, S and T are members in the binding\nof the book Cartesian form\nLet (x1, y1, z1), (x2, y2, z2) and (x3,  y3,  z3) be the coordinates of the points R, S and T\nrespectively"
    },
    {
        "Chapter": "1",
        "sentence_range": "5943-5946",
        "Text": "15) These planes will resemble the pages of a book where the\nline containing the points R, S and T are members in the binding\nof the book Cartesian form\nLet (x1, y1, z1), (x2, y2, z2) and (x3,  y3,  z3) be the coordinates of the points R, S and T\nrespectively Let (x, y, z) be the coordinates of any point P on the plane with position\nvector rr"
    },
    {
        "Chapter": "1",
        "sentence_range": "5944-5947",
        "Text": "These planes will resemble the pages of a book where the\nline containing the points R, S and T are members in the binding\nof the book Cartesian form\nLet (x1, y1, z1), (x2, y2, z2) and (x3,  y3,  z3) be the coordinates of the points R, S and T\nrespectively Let (x, y, z) be the coordinates of any point P on the plane with position\nvector rr Then\nRP\nuuur  = (x \u2013 x1) \u02c6i  + (y \u2013 y1) \u02c6j  + (z \u2013 z1) \u02c6k\nRS\nuuur  = (x2 \u2013 x1) \u02c6i  + (y2 \u2013 y1) \u02c6j  + (z2 \u2013 z1) \u02c6k\nRT\nuuur  = (x3 \u2013 x1) \u02c6i + (y3 \u2013 y1) \u02c6j  + (z3 \u2013 z1) \u02c6k\nSubstituting these values in equation (1) of the vector form and expressing it in the\nform of a determinant, we have\n1\n1\n1\n2\n1\n2\n1\n2\n1\n3\n1\n3\n1\n3\n1\n0\nx\nx\ny\ny\nz\nz\nx\nx\ny\ny\nz\nz\nx\nx\ny\ny\nz\nz\n\u2212\n\u2212\n\u2212\n\u2212\n\u2212\n\u2212\n=\n\u2212\n\u2212\n\u2212\nwhich is the equation of the plane in Cartesian form passing through three non collinear\npoints (x1, y1, z1), (x2, y2, z2) and (x3, y3, z3)"
    },
    {
        "Chapter": "1",
        "sentence_range": "5945-5948",
        "Text": "Cartesian form\nLet (x1, y1, z1), (x2, y2, z2) and (x3,  y3,  z3) be the coordinates of the points R, S and T\nrespectively Let (x, y, z) be the coordinates of any point P on the plane with position\nvector rr Then\nRP\nuuur  = (x \u2013 x1) \u02c6i  + (y \u2013 y1) \u02c6j  + (z \u2013 z1) \u02c6k\nRS\nuuur  = (x2 \u2013 x1) \u02c6i  + (y2 \u2013 y1) \u02c6j  + (z2 \u2013 z1) \u02c6k\nRT\nuuur  = (x3 \u2013 x1) \u02c6i + (y3 \u2013 y1) \u02c6j  + (z3 \u2013 z1) \u02c6k\nSubstituting these values in equation (1) of the vector form and expressing it in the\nform of a determinant, we have\n1\n1\n1\n2\n1\n2\n1\n2\n1\n3\n1\n3\n1\n3\n1\n0\nx\nx\ny\ny\nz\nz\nx\nx\ny\ny\nz\nz\nx\nx\ny\ny\nz\nz\n\u2212\n\u2212\n\u2212\n\u2212\n\u2212\n\u2212\n=\n\u2212\n\u2212\n\u2212\nwhich is the equation of the plane in Cartesian form passing through three non collinear\npoints (x1, y1, z1), (x2, y2, z2) and (x3, y3, z3) Example 18 Find the vector equations of the plane passing through the points\nR(2, 5, \u2013 3), S(\u2013 2, \u2013 3, 5) and T(5, 3,\u2013 3)"
    },
    {
        "Chapter": "1",
        "sentence_range": "5946-5949",
        "Text": "Let (x, y, z) be the coordinates of any point P on the plane with position\nvector rr Then\nRP\nuuur  = (x \u2013 x1) \u02c6i  + (y \u2013 y1) \u02c6j  + (z \u2013 z1) \u02c6k\nRS\nuuur  = (x2 \u2013 x1) \u02c6i  + (y2 \u2013 y1) \u02c6j  + (z2 \u2013 z1) \u02c6k\nRT\nuuur  = (x3 \u2013 x1) \u02c6i + (y3 \u2013 y1) \u02c6j  + (z3 \u2013 z1) \u02c6k\nSubstituting these values in equation (1) of the vector form and expressing it in the\nform of a determinant, we have\n1\n1\n1\n2\n1\n2\n1\n2\n1\n3\n1\n3\n1\n3\n1\n0\nx\nx\ny\ny\nz\nz\nx\nx\ny\ny\nz\nz\nx\nx\ny\ny\nz\nz\n\u2212\n\u2212\n\u2212\n\u2212\n\u2212\n\u2212\n=\n\u2212\n\u2212\n\u2212\nwhich is the equation of the plane in Cartesian form passing through three non collinear\npoints (x1, y1, z1), (x2, y2, z2) and (x3, y3, z3) Example 18 Find the vector equations of the plane passing through the points\nR(2, 5, \u2013 3), S(\u2013 2, \u2013 3, 5) and T(5, 3,\u2013 3) Solution Let \n\u02c6\n\u02c6\n\u02c6\n2\n5\n3\na\ni\nj\nr= + \u2212k\n, \n\u02c6\n\u02c6\n\u02c6\n2\n3\n5\nb\ni\nj\nk\n= \u2212\n\u2212\n+\nr\n, \n\u02c6\n\u02c6\n\u02c6\n5\n3\n3\nc\ni\nj\n= + \u2212k\nr\nThen the vector equation of the plane passing through ar , b\nr\n and crand  is\ngiven by\n(\n) (RS RT)\nr\n\u2212a\n\u22c5\n\u00d7\nuuur uuur\nr\nr\n = 0     (Why"
    },
    {
        "Chapter": "1",
        "sentence_range": "5947-5950",
        "Text": "Then\nRP\nuuur  = (x \u2013 x1) \u02c6i  + (y \u2013 y1) \u02c6j  + (z \u2013 z1) \u02c6k\nRS\nuuur  = (x2 \u2013 x1) \u02c6i  + (y2 \u2013 y1) \u02c6j  + (z2 \u2013 z1) \u02c6k\nRT\nuuur  = (x3 \u2013 x1) \u02c6i + (y3 \u2013 y1) \u02c6j  + (z3 \u2013 z1) \u02c6k\nSubstituting these values in equation (1) of the vector form and expressing it in the\nform of a determinant, we have\n1\n1\n1\n2\n1\n2\n1\n2\n1\n3\n1\n3\n1\n3\n1\n0\nx\nx\ny\ny\nz\nz\nx\nx\ny\ny\nz\nz\nx\nx\ny\ny\nz\nz\n\u2212\n\u2212\n\u2212\n\u2212\n\u2212\n\u2212\n=\n\u2212\n\u2212\n\u2212\nwhich is the equation of the plane in Cartesian form passing through three non collinear\npoints (x1, y1, z1), (x2, y2, z2) and (x3, y3, z3) Example 18 Find the vector equations of the plane passing through the points\nR(2, 5, \u2013 3), S(\u2013 2, \u2013 3, 5) and T(5, 3,\u2013 3) Solution Let \n\u02c6\n\u02c6\n\u02c6\n2\n5\n3\na\ni\nj\nr= + \u2212k\n, \n\u02c6\n\u02c6\n\u02c6\n2\n3\n5\nb\ni\nj\nk\n= \u2212\n\u2212\n+\nr\n, \n\u02c6\n\u02c6\n\u02c6\n5\n3\n3\nc\ni\nj\n= + \u2212k\nr\nThen the vector equation of the plane passing through ar , b\nr\n and crand  is\ngiven by\n(\n) (RS RT)\nr\n\u2212a\n\u22c5\n\u00d7\nuuur uuur\nr\nr\n = 0     (Why )\nor\n(\n) [(\n) (\n)]\nr\na\nb\na\nc\na\n\u2212\n\u22c5\n\u2212\n\u00d7\n\u2212\nr\nr\nr\nr\nr\nr\n = 0\ni"
    },
    {
        "Chapter": "1",
        "sentence_range": "5948-5951",
        "Text": "Example 18 Find the vector equations of the plane passing through the points\nR(2, 5, \u2013 3), S(\u2013 2, \u2013 3, 5) and T(5, 3,\u2013 3) Solution Let \n\u02c6\n\u02c6\n\u02c6\n2\n5\n3\na\ni\nj\nr= + \u2212k\n, \n\u02c6\n\u02c6\n\u02c6\n2\n3\n5\nb\ni\nj\nk\n= \u2212\n\u2212\n+\nr\n, \n\u02c6\n\u02c6\n\u02c6\n5\n3\n3\nc\ni\nj\n= + \u2212k\nr\nThen the vector equation of the plane passing through ar , b\nr\n and crand  is\ngiven by\n(\n) (RS RT)\nr\n\u2212a\n\u22c5\n\u00d7\nuuur uuur\nr\nr\n = 0     (Why )\nor\n(\n) [(\n) (\n)]\nr\na\nb\na\nc\na\n\u2212\n\u22c5\n\u2212\n\u00d7\n\u2212\nr\nr\nr\nr\nr\nr\n = 0\ni e"
    },
    {
        "Chapter": "1",
        "sentence_range": "5949-5952",
        "Text": "Solution Let \n\u02c6\n\u02c6\n\u02c6\n2\n5\n3\na\ni\nj\nr= + \u2212k\n, \n\u02c6\n\u02c6\n\u02c6\n2\n3\n5\nb\ni\nj\nk\n= \u2212\n\u2212\n+\nr\n, \n\u02c6\n\u02c6\n\u02c6\n5\n3\n3\nc\ni\nj\n= + \u2212k\nr\nThen the vector equation of the plane passing through ar , b\nr\n and crand  is\ngiven by\n(\n) (RS RT)\nr\n\u2212a\n\u22c5\n\u00d7\nuuur uuur\nr\nr\n = 0     (Why )\nor\n(\n) [(\n) (\n)]\nr\na\nb\na\nc\na\n\u2212\n\u22c5\n\u2212\n\u00d7\n\u2212\nr\nr\nr\nr\nr\nr\n = 0\ni e \u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n[\n(2\n5\n3 )] [( 4\n8\n8 )\n(3\n2 )]\n0\nr\ni\nj\nk\ni\nj\nk\ni\nj\n\u2212 + \u2212 \u22c5 \u2212 \u2212 + \u00d7 \u2212 =\nr\nR \nS  \nT\n\u00a9 NCERT\nnot to be republished\nTHREE D IMENSIONAL G EOMETRY\n485\n11"
    },
    {
        "Chapter": "1",
        "sentence_range": "5950-5953",
        "Text": ")\nor\n(\n) [(\n) (\n)]\nr\na\nb\na\nc\na\n\u2212\n\u22c5\n\u2212\n\u00d7\n\u2212\nr\nr\nr\nr\nr\nr\n = 0\ni e \u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n[\n(2\n5\n3 )] [( 4\n8\n8 )\n(3\n2 )]\n0\nr\ni\nj\nk\ni\nj\nk\ni\nj\n\u2212 + \u2212 \u22c5 \u2212 \u2212 + \u00d7 \u2212 =\nr\nR \nS  \nT\n\u00a9 NCERT\nnot to be republished\nTHREE D IMENSIONAL G EOMETRY\n485\n11 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "5951-5954",
        "Text": "e \u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n[\n(2\n5\n3 )] [( 4\n8\n8 )\n(3\n2 )]\n0\nr\ni\nj\nk\ni\nj\nk\ni\nj\n\u2212 + \u2212 \u22c5 \u2212 \u2212 + \u00d7 \u2212 =\nr\nR \nS  \nT\n\u00a9 NCERT\nnot to be republished\nTHREE D IMENSIONAL G EOMETRY\n485\n11 6 4  Intercept form of the equation of a plane\nIn this section, we shall deduce the equation of a plane in terms of the intercepts made\nby the plane on the coordinate axes"
    },
    {
        "Chapter": "1",
        "sentence_range": "5952-5955",
        "Text": "\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n[\n(2\n5\n3 )] [( 4\n8\n8 )\n(3\n2 )]\n0\nr\ni\nj\nk\ni\nj\nk\ni\nj\n\u2212 + \u2212 \u22c5 \u2212 \u2212 + \u00d7 \u2212 =\nr\nR \nS  \nT\n\u00a9 NCERT\nnot to be republished\nTHREE D IMENSIONAL G EOMETRY\n485\n11 6 4  Intercept form of the equation of a plane\nIn this section, we shall deduce the equation of a plane in terms of the intercepts made\nby the plane on the coordinate axes Let the equation of the plane be\nAx + By + Cz + D = 0  (D \u2260 0)"
    },
    {
        "Chapter": "1",
        "sentence_range": "5953-5956",
        "Text": "6 4  Intercept form of the equation of a plane\nIn this section, we shall deduce the equation of a plane in terms of the intercepts made\nby the plane on the coordinate axes Let the equation of the plane be\nAx + By + Cz + D = 0  (D \u2260 0) (1)\nLet the plane make intercepts a, b, c on x, y and z axes, respectively  (Fig 11"
    },
    {
        "Chapter": "1",
        "sentence_range": "5954-5957",
        "Text": "4  Intercept form of the equation of a plane\nIn this section, we shall deduce the equation of a plane in terms of the intercepts made\nby the plane on the coordinate axes Let the equation of the plane be\nAx + By + Cz + D = 0  (D \u2260 0) (1)\nLet the plane make intercepts a, b, c on x, y and z axes, respectively  (Fig 11 16)"
    },
    {
        "Chapter": "1",
        "sentence_range": "5955-5958",
        "Text": "Let the equation of the plane be\nAx + By + Cz + D = 0  (D \u2260 0) (1)\nLet the plane make intercepts a, b, c on x, y and z axes, respectively  (Fig 11 16) Hence, the plane meets x, y and z-axes at (a, 0, 0),\n(0, b, 0), (0, 0, c), respectively"
    },
    {
        "Chapter": "1",
        "sentence_range": "5956-5959",
        "Text": "(1)\nLet the plane make intercepts a, b, c on x, y and z axes, respectively  (Fig 11 16) Hence, the plane meets x, y and z-axes at (a, 0, 0),\n(0, b, 0), (0, 0, c), respectively Therefore\nAa + D = 0 or A =\nD\na\u2212\nBb + D = 0 or B =\nD\nb\u2212\nCc + D = 0 or C =\nD\nc\u2212\nSubstituting these values in the equation (1) of the\nplane and simplifying, we get\nx\ny\nz\na\nb\nc\n+\n+\n = 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "5957-5960",
        "Text": "16) Hence, the plane meets x, y and z-axes at (a, 0, 0),\n(0, b, 0), (0, 0, c), respectively Therefore\nAa + D = 0 or A =\nD\na\u2212\nBb + D = 0 or B =\nD\nb\u2212\nCc + D = 0 or C =\nD\nc\u2212\nSubstituting these values in the equation (1) of the\nplane and simplifying, we get\nx\ny\nz\na\nb\nc\n+\n+\n = 1 (1)\nwhich is the required equation of the plane in the intercept form"
    },
    {
        "Chapter": "1",
        "sentence_range": "5958-5961",
        "Text": "Hence, the plane meets x, y and z-axes at (a, 0, 0),\n(0, b, 0), (0, 0, c), respectively Therefore\nAa + D = 0 or A =\nD\na\u2212\nBb + D = 0 or B =\nD\nb\u2212\nCc + D = 0 or C =\nD\nc\u2212\nSubstituting these values in the equation (1) of the\nplane and simplifying, we get\nx\ny\nz\na\nb\nc\n+\n+\n = 1 (1)\nwhich is the required equation of the plane in the intercept form Example 19 Find the equation of the plane with intercepts 2, 3 and 4 on the x, y and\nz-axis respectively"
    },
    {
        "Chapter": "1",
        "sentence_range": "5959-5962",
        "Text": "Therefore\nAa + D = 0 or A =\nD\na\u2212\nBb + D = 0 or B =\nD\nb\u2212\nCc + D = 0 or C =\nD\nc\u2212\nSubstituting these values in the equation (1) of the\nplane and simplifying, we get\nx\ny\nz\na\nb\nc\n+\n+\n = 1 (1)\nwhich is the required equation of the plane in the intercept form Example 19 Find the equation of the plane with intercepts 2, 3 and 4 on the x, y and\nz-axis respectively Solution Let the equation of the plane be\nx\ny\nz\na\nb\nc\n+\n+\n = 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "5960-5963",
        "Text": "(1)\nwhich is the required equation of the plane in the intercept form Example 19 Find the equation of the plane with intercepts 2, 3 and 4 on the x, y and\nz-axis respectively Solution Let the equation of the plane be\nx\ny\nz\na\nb\nc\n+\n+\n = 1 (1)\nHere\na = 2, b = 3, c = 4"
    },
    {
        "Chapter": "1",
        "sentence_range": "5961-5964",
        "Text": "Example 19 Find the equation of the plane with intercepts 2, 3 and 4 on the x, y and\nz-axis respectively Solution Let the equation of the plane be\nx\ny\nz\na\nb\nc\n+\n+\n = 1 (1)\nHere\na = 2, b = 3, c = 4 Substituting the values of a, b and c in (1), we get the required equation of the\nplane as \n1\n2\n3\n4\nx\ny\nz\n+\n+\n=  or 6x + 4y + 3z = 12"
    },
    {
        "Chapter": "1",
        "sentence_range": "5962-5965",
        "Text": "Solution Let the equation of the plane be\nx\ny\nz\na\nb\nc\n+\n+\n = 1 (1)\nHere\na = 2, b = 3, c = 4 Substituting the values of a, b and c in (1), we get the required equation of the\nplane as \n1\n2\n3\n4\nx\ny\nz\n+\n+\n=  or 6x + 4y + 3z = 12 11"
    },
    {
        "Chapter": "1",
        "sentence_range": "5963-5966",
        "Text": "(1)\nHere\na = 2, b = 3, c = 4 Substituting the values of a, b and c in (1), we get the required equation of the\nplane as \n1\n2\n3\n4\nx\ny\nz\n+\n+\n=  or 6x + 4y + 3z = 12 11 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "5964-5967",
        "Text": "Substituting the values of a, b and c in (1), we get the required equation of the\nplane as \n1\n2\n3\n4\nx\ny\nz\n+\n+\n=  or 6x + 4y + 3z = 12 11 6 5  Plane passing through the intersection\nof two given planes\nLet  \u03c01 and \u03c0 2 be two planes with equat ions\nrr n\u22c51\u02c6\n = d1 and \nrr n\u22c52\u02c6\n = d2 respectively"
    },
    {
        "Chapter": "1",
        "sentence_range": "5965-5968",
        "Text": "11 6 5  Plane passing through the intersection\nof two given planes\nLet  \u03c01 and \u03c0 2 be two planes with equat ions\nrr n\u22c51\u02c6\n = d1 and \nrr n\u22c52\u02c6\n = d2 respectively The position\nvector of any point on the line of intersection must\nsatisfy both the equations (Fig 11"
    },
    {
        "Chapter": "1",
        "sentence_range": "5966-5969",
        "Text": "6 5  Plane passing through the intersection\nof two given planes\nLet  \u03c01 and \u03c0 2 be two planes with equat ions\nrr n\u22c51\u02c6\n = d1 and \nrr n\u22c52\u02c6\n = d2 respectively The position\nvector of any point on the line of intersection must\nsatisfy both the equations (Fig 11 17)"
    },
    {
        "Chapter": "1",
        "sentence_range": "5967-5970",
        "Text": "5  Plane passing through the intersection\nof two given planes\nLet  \u03c01 and \u03c0 2 be two planes with equat ions\nrr n\u22c51\u02c6\n = d1 and \nrr n\u22c52\u02c6\n = d2 respectively The position\nvector of any point on the line of intersection must\nsatisfy both the equations (Fig 11 17) Fig 11"
    },
    {
        "Chapter": "1",
        "sentence_range": "5968-5971",
        "Text": "The position\nvector of any point on the line of intersection must\nsatisfy both the equations (Fig 11 17) Fig 11 16\nFig 11"
    },
    {
        "Chapter": "1",
        "sentence_range": "5969-5972",
        "Text": "17) Fig 11 16\nFig 11 17\n\u00a9 NCERT\nnot to be republished\n MATHEMATI CS\n486\nIf tr  is the position vector of a point on the line, then\n\u22c5rt n1\u02c6\n = d1 and  \n\u22c5rt n2\u02c6\n = d2\nTherefore, for all real values of \u03bb, we have\n1\n2\n\u02c6\n\u02c6\n(\n)\nt\nn\nn\n\u22c5\n+\u03bb\nr\n = \n1\n2\nd\nd\n+\u03bb\nSince tr is arbitrary, it satisfies for any point on the line"
    },
    {
        "Chapter": "1",
        "sentence_range": "5970-5973",
        "Text": "Fig 11 16\nFig 11 17\n\u00a9 NCERT\nnot to be republished\n MATHEMATI CS\n486\nIf tr  is the position vector of a point on the line, then\n\u22c5rt n1\u02c6\n = d1 and  \n\u22c5rt n2\u02c6\n = d2\nTherefore, for all real values of \u03bb, we have\n1\n2\n\u02c6\n\u02c6\n(\n)\nt\nn\nn\n\u22c5\n+\u03bb\nr\n = \n1\n2\nd\nd\n+\u03bb\nSince tr is arbitrary, it satisfies for any point on the line Hence, the equation \n1\n2\n1\n2\n(\n)\nr\nn\nn\nd\nd\n\u22c5\n+ \u03bb\n=\n+\u03bb\nr\nr\nr\nrepresents a plane \u03c03 which is such\nthat if any vector rr satisfies both the equations \u03c01 and \u03c02, it also satisfies the equation\n\u03c03 i"
    },
    {
        "Chapter": "1",
        "sentence_range": "5971-5974",
        "Text": "16\nFig 11 17\n\u00a9 NCERT\nnot to be republished\n MATHEMATI CS\n486\nIf tr  is the position vector of a point on the line, then\n\u22c5rt n1\u02c6\n = d1 and  \n\u22c5rt n2\u02c6\n = d2\nTherefore, for all real values of \u03bb, we have\n1\n2\n\u02c6\n\u02c6\n(\n)\nt\nn\nn\n\u22c5\n+\u03bb\nr\n = \n1\n2\nd\nd\n+\u03bb\nSince tr is arbitrary, it satisfies for any point on the line Hence, the equation \n1\n2\n1\n2\n(\n)\nr\nn\nn\nd\nd\n\u22c5\n+ \u03bb\n=\n+\u03bb\nr\nr\nr\nrepresents a plane \u03c03 which is such\nthat if any vector rr satisfies both the equations \u03c01 and \u03c02, it also satisfies the equation\n\u03c03 i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "5972-5975",
        "Text": "17\n\u00a9 NCERT\nnot to be republished\n MATHEMATI CS\n486\nIf tr  is the position vector of a point on the line, then\n\u22c5rt n1\u02c6\n = d1 and  \n\u22c5rt n2\u02c6\n = d2\nTherefore, for all real values of \u03bb, we have\n1\n2\n\u02c6\n\u02c6\n(\n)\nt\nn\nn\n\u22c5\n+\u03bb\nr\n = \n1\n2\nd\nd\n+\u03bb\nSince tr is arbitrary, it satisfies for any point on the line Hence, the equation \n1\n2\n1\n2\n(\n)\nr\nn\nn\nd\nd\n\u22c5\n+ \u03bb\n=\n+\u03bb\nr\nr\nr\nrepresents a plane \u03c03 which is such\nthat if any vector rr satisfies both the equations \u03c01 and \u03c02, it also satisfies the equation\n\u03c03 i e , any plane passing through the intersection of the planes\nr n1\n\u22c5r r  =\n1\n2\n2\ndand\nr n\nd\n\u22c5\n=\nr r\nhas the equation\n1\n2\n(\n)\nr\nn\nn\n\u22c5\n+ \u03bb\nr\nr\nr\n= d1 + \u03bbd2"
    },
    {
        "Chapter": "1",
        "sentence_range": "5973-5976",
        "Text": "Hence, the equation \n1\n2\n1\n2\n(\n)\nr\nn\nn\nd\nd\n\u22c5\n+ \u03bb\n=\n+\u03bb\nr\nr\nr\nrepresents a plane \u03c03 which is such\nthat if any vector rr satisfies both the equations \u03c01 and \u03c02, it also satisfies the equation\n\u03c03 i e , any plane passing through the intersection of the planes\nr n1\n\u22c5r r  =\n1\n2\n2\ndand\nr n\nd\n\u22c5\n=\nr r\nhas the equation\n1\n2\n(\n)\nr\nn\nn\n\u22c5\n+ \u03bb\nr\nr\nr\n= d1 + \u03bbd2 (1)\nCartesian form\nIn Cartesian system, let\n1nr  =\n1\n2\n1 \u02c6\n\u02c6\n\u02c6\nA\nB\nC\ni\nj\nk\n+\n+\n2nr  =\n2\n2\n2 \u02c6\n\u02c6\n\u02c6\nA\nB\nC\ni\nj\nk\n+\n+\nand\nrr  =\n\u02c6\n\u02c6\n\u02c6\nxi\ny j\nz k\n+\n+\nThen (1) becomes\nx (A1 +  \u03bbA2) + y (B1 +  \u03bbB2) + z (C1 +  \u03bbC2) = d1 +  \u03bbd2\nor\n(A1x + B1y + C 1z \u2013 d1) +  \u03bb(A2x + B2 y + C 2z \u2013 d2) = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "5974-5977",
        "Text": "e , any plane passing through the intersection of the planes\nr n1\n\u22c5r r  =\n1\n2\n2\ndand\nr n\nd\n\u22c5\n=\nr r\nhas the equation\n1\n2\n(\n)\nr\nn\nn\n\u22c5\n+ \u03bb\nr\nr\nr\n= d1 + \u03bbd2 (1)\nCartesian form\nIn Cartesian system, let\n1nr  =\n1\n2\n1 \u02c6\n\u02c6\n\u02c6\nA\nB\nC\ni\nj\nk\n+\n+\n2nr  =\n2\n2\n2 \u02c6\n\u02c6\n\u02c6\nA\nB\nC\ni\nj\nk\n+\n+\nand\nrr  =\n\u02c6\n\u02c6\n\u02c6\nxi\ny j\nz k\n+\n+\nThen (1) becomes\nx (A1 +  \u03bbA2) + y (B1 +  \u03bbB2) + z (C1 +  \u03bbC2) = d1 +  \u03bbd2\nor\n(A1x + B1y + C 1z \u2013 d1) +  \u03bb(A2x + B2 y + C 2z \u2013 d2) = 0 (2)\nwhich is the required Cartesian form of the equation of the plane passing through the\nintersection of the given planes for each value of \u03bb"
    },
    {
        "Chapter": "1",
        "sentence_range": "5975-5978",
        "Text": ", any plane passing through the intersection of the planes\nr n1\n\u22c5r r  =\n1\n2\n2\ndand\nr n\nd\n\u22c5\n=\nr r\nhas the equation\n1\n2\n(\n)\nr\nn\nn\n\u22c5\n+ \u03bb\nr\nr\nr\n= d1 + \u03bbd2 (1)\nCartesian form\nIn Cartesian system, let\n1nr  =\n1\n2\n1 \u02c6\n\u02c6\n\u02c6\nA\nB\nC\ni\nj\nk\n+\n+\n2nr  =\n2\n2\n2 \u02c6\n\u02c6\n\u02c6\nA\nB\nC\ni\nj\nk\n+\n+\nand\nrr  =\n\u02c6\n\u02c6\n\u02c6\nxi\ny j\nz k\n+\n+\nThen (1) becomes\nx (A1 +  \u03bbA2) + y (B1 +  \u03bbB2) + z (C1 +  \u03bbC2) = d1 +  \u03bbd2\nor\n(A1x + B1y + C 1z \u2013 d1) +  \u03bb(A2x + B2 y + C 2z \u2013 d2) = 0 (2)\nwhich is the required Cartesian form of the equation of the plane passing through the\nintersection of the given planes for each value of \u03bb Example 20 Find the vector equation of the plane passing through the intersection of\nthe planes \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(\n) 6 and\n(2\n3\n4 )\n5,\nr\ni\nj\nk\nr\ni\nj\nk\n\u22c5\n+ +\n=\n\u22c5\n+\n+\n= \u2212\nr\nr\nand the point (1, 1, 1)"
    },
    {
        "Chapter": "1",
        "sentence_range": "5976-5979",
        "Text": "(1)\nCartesian form\nIn Cartesian system, let\n1nr  =\n1\n2\n1 \u02c6\n\u02c6\n\u02c6\nA\nB\nC\ni\nj\nk\n+\n+\n2nr  =\n2\n2\n2 \u02c6\n\u02c6\n\u02c6\nA\nB\nC\ni\nj\nk\n+\n+\nand\nrr  =\n\u02c6\n\u02c6\n\u02c6\nxi\ny j\nz k\n+\n+\nThen (1) becomes\nx (A1 +  \u03bbA2) + y (B1 +  \u03bbB2) + z (C1 +  \u03bbC2) = d1 +  \u03bbd2\nor\n(A1x + B1y + C 1z \u2013 d1) +  \u03bb(A2x + B2 y + C 2z \u2013 d2) = 0 (2)\nwhich is the required Cartesian form of the equation of the plane passing through the\nintersection of the given planes for each value of \u03bb Example 20 Find the vector equation of the plane passing through the intersection of\nthe planes \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(\n) 6 and\n(2\n3\n4 )\n5,\nr\ni\nj\nk\nr\ni\nj\nk\n\u22c5\n+ +\n=\n\u22c5\n+\n+\n= \u2212\nr\nr\nand the point (1, 1, 1) Solution Here, \n1\n\u02c6\n\u02c6\n\u02c6\nn\ni\nj\nk\n=\n+ +\nr\n and \n2\nnr  = \n\u02c6\n\u02c6\n\u02c6\n2\n3\n4 ;\ni\nj\nk\n+\n+\nand\nd1 = 6 and d2 = \u20135\nHence, using the relation \n1\n2\n1\n2\n(\n)\nr\nn\nn\nd\nd\n\u22c5\n+ \u03bb\n=\n+\u03bb\nr\nr\nr\n, we get\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n[\n(2\n3\n4 )]\nr\ni\nj\nk\ni\nj\nk\n\u22c5\n+ +\n+\u03bb\n+\n+\nr\n = 6 5\n\u2212 \u03bb\nor\n\u02c6\n\u02c6\n\u02c6\n[(1 2 )\n(1 3 )\n(1 4 ) ]\nr\ni\nj\nk\n\u22c5\n+ \u03bb\n+\n+ \u03bb\n+\n+ \u03bb\nr\n = 6 5\n\u2212 \u03bb                                \u2026 (1)\nwhere, \u03bb is some real number"
    },
    {
        "Chapter": "1",
        "sentence_range": "5977-5980",
        "Text": "(2)\nwhich is the required Cartesian form of the equation of the plane passing through the\nintersection of the given planes for each value of \u03bb Example 20 Find the vector equation of the plane passing through the intersection of\nthe planes \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(\n) 6 and\n(2\n3\n4 )\n5,\nr\ni\nj\nk\nr\ni\nj\nk\n\u22c5\n+ +\n=\n\u22c5\n+\n+\n= \u2212\nr\nr\nand the point (1, 1, 1) Solution Here, \n1\n\u02c6\n\u02c6\n\u02c6\nn\ni\nj\nk\n=\n+ +\nr\n and \n2\nnr  = \n\u02c6\n\u02c6\n\u02c6\n2\n3\n4 ;\ni\nj\nk\n+\n+\nand\nd1 = 6 and d2 = \u20135\nHence, using the relation \n1\n2\n1\n2\n(\n)\nr\nn\nn\nd\nd\n\u22c5\n+ \u03bb\n=\n+\u03bb\nr\nr\nr\n, we get\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n[\n(2\n3\n4 )]\nr\ni\nj\nk\ni\nj\nk\n\u22c5\n+ +\n+\u03bb\n+\n+\nr\n = 6 5\n\u2212 \u03bb\nor\n\u02c6\n\u02c6\n\u02c6\n[(1 2 )\n(1 3 )\n(1 4 ) ]\nr\ni\nj\nk\n\u22c5\n+ \u03bb\n+\n+ \u03bb\n+\n+ \u03bb\nr\n = 6 5\n\u2212 \u03bb                                \u2026 (1)\nwhere, \u03bb is some real number \u00a9 NCERT\nnot to be republished\nTHREE D IMENSIONAL G EOMETRY\n487\nTaking\n\u02c6\n\u02c6\n\u02c6\nr\nxi\ny j\nr= + +z k\n, we get\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(\n) [(1 2 )\n(1 3 )\n(1 4 ) ] 6 5\nxi\ny j\nz k\ni\nj\nk\n+\n+\n\u22c5\n+ \u03bb\n+\n+ \u03bb\n+\n+ \u03bb\n= \u2212 \u03bb\nor\n(1 + 2\u03bb ) x + (1 + 3\u03bb) y + (1 + 4\u03bb) z = 6 \u2013 5\u03bb\nor\n(x + y + z \u2013 6 ) + \u03bb  (2x + 3y + 4 z + 5) = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "5978-5981",
        "Text": "Example 20 Find the vector equation of the plane passing through the intersection of\nthe planes \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(\n) 6 and\n(2\n3\n4 )\n5,\nr\ni\nj\nk\nr\ni\nj\nk\n\u22c5\n+ +\n=\n\u22c5\n+\n+\n= \u2212\nr\nr\nand the point (1, 1, 1) Solution Here, \n1\n\u02c6\n\u02c6\n\u02c6\nn\ni\nj\nk\n=\n+ +\nr\n and \n2\nnr  = \n\u02c6\n\u02c6\n\u02c6\n2\n3\n4 ;\ni\nj\nk\n+\n+\nand\nd1 = 6 and d2 = \u20135\nHence, using the relation \n1\n2\n1\n2\n(\n)\nr\nn\nn\nd\nd\n\u22c5\n+ \u03bb\n=\n+\u03bb\nr\nr\nr\n, we get\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n[\n(2\n3\n4 )]\nr\ni\nj\nk\ni\nj\nk\n\u22c5\n+ +\n+\u03bb\n+\n+\nr\n = 6 5\n\u2212 \u03bb\nor\n\u02c6\n\u02c6\n\u02c6\n[(1 2 )\n(1 3 )\n(1 4 ) ]\nr\ni\nj\nk\n\u22c5\n+ \u03bb\n+\n+ \u03bb\n+\n+ \u03bb\nr\n = 6 5\n\u2212 \u03bb                                \u2026 (1)\nwhere, \u03bb is some real number \u00a9 NCERT\nnot to be republished\nTHREE D IMENSIONAL G EOMETRY\n487\nTaking\n\u02c6\n\u02c6\n\u02c6\nr\nxi\ny j\nr= + +z k\n, we get\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(\n) [(1 2 )\n(1 3 )\n(1 4 ) ] 6 5\nxi\ny j\nz k\ni\nj\nk\n+\n+\n\u22c5\n+ \u03bb\n+\n+ \u03bb\n+\n+ \u03bb\n= \u2212 \u03bb\nor\n(1 + 2\u03bb ) x + (1 + 3\u03bb) y + (1 + 4\u03bb) z = 6 \u2013 5\u03bb\nor\n(x + y + z \u2013 6 ) + \u03bb  (2x + 3y + 4 z + 5) = 0 (2)\nGiven that the plane passes through the point (1,1,1), it must satisfy (2), i"
    },
    {
        "Chapter": "1",
        "sentence_range": "5979-5982",
        "Text": "Solution Here, \n1\n\u02c6\n\u02c6\n\u02c6\nn\ni\nj\nk\n=\n+ +\nr\n and \n2\nnr  = \n\u02c6\n\u02c6\n\u02c6\n2\n3\n4 ;\ni\nj\nk\n+\n+\nand\nd1 = 6 and d2 = \u20135\nHence, using the relation \n1\n2\n1\n2\n(\n)\nr\nn\nn\nd\nd\n\u22c5\n+ \u03bb\n=\n+\u03bb\nr\nr\nr\n, we get\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n[\n(2\n3\n4 )]\nr\ni\nj\nk\ni\nj\nk\n\u22c5\n+ +\n+\u03bb\n+\n+\nr\n = 6 5\n\u2212 \u03bb\nor\n\u02c6\n\u02c6\n\u02c6\n[(1 2 )\n(1 3 )\n(1 4 ) ]\nr\ni\nj\nk\n\u22c5\n+ \u03bb\n+\n+ \u03bb\n+\n+ \u03bb\nr\n = 6 5\n\u2212 \u03bb                                \u2026 (1)\nwhere, \u03bb is some real number \u00a9 NCERT\nnot to be republished\nTHREE D IMENSIONAL G EOMETRY\n487\nTaking\n\u02c6\n\u02c6\n\u02c6\nr\nxi\ny j\nr= + +z k\n, we get\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(\n) [(1 2 )\n(1 3 )\n(1 4 ) ] 6 5\nxi\ny j\nz k\ni\nj\nk\n+\n+\n\u22c5\n+ \u03bb\n+\n+ \u03bb\n+\n+ \u03bb\n= \u2212 \u03bb\nor\n(1 + 2\u03bb ) x + (1 + 3\u03bb) y + (1 + 4\u03bb) z = 6 \u2013 5\u03bb\nor\n(x + y + z \u2013 6 ) + \u03bb  (2x + 3y + 4 z + 5) = 0 (2)\nGiven that the plane passes through the point (1,1,1), it must satisfy (2), i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "5980-5983",
        "Text": "\u00a9 NCERT\nnot to be republished\nTHREE D IMENSIONAL G EOMETRY\n487\nTaking\n\u02c6\n\u02c6\n\u02c6\nr\nxi\ny j\nr= + +z k\n, we get\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(\n) [(1 2 )\n(1 3 )\n(1 4 ) ] 6 5\nxi\ny j\nz k\ni\nj\nk\n+\n+\n\u22c5\n+ \u03bb\n+\n+ \u03bb\n+\n+ \u03bb\n= \u2212 \u03bb\nor\n(1 + 2\u03bb ) x + (1 + 3\u03bb) y + (1 + 4\u03bb) z = 6 \u2013 5\u03bb\nor\n(x + y + z \u2013 6 ) + \u03bb  (2x + 3y + 4 z + 5) = 0 (2)\nGiven that the plane passes through the point (1,1,1), it must satisfy (2), i e (1 + 1 + 1 \u2013 6) + \u03bb (2 + 3 + 4 + 5) = 0\nor\n\u03bb = 3\n14\nPutting the values of  \u03bb in (1), we get\n3\n9\n6 \u02c6\n\u02c6\n\u02c6\n1\n1\n1\n7\n14\n7\nr\ni\nj\nk\n\uf8ee\n\uf8f9\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n+\n+\n+\n+\n+\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n\uf8f0\n\uf8fb\nr\n = \n15\n6 14\n\u2212\nor\n10\n23\n13 \u02c6\n\u02c6\n\u02c6\n7\n14\n7\nr\ni\nj\nk\n\uf8eb\n\uf8f6\n+\n+\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\nr\n = 69\n14\nor\n\u02c6\n\u02c6\n\u02c6\n(20\n23\n26 )\nr\ni\nj\nk\n\u22c5\n+\n+\nr\n = 69\nwhich is the required vector equation of the plane"
    },
    {
        "Chapter": "1",
        "sentence_range": "5981-5984",
        "Text": "(2)\nGiven that the plane passes through the point (1,1,1), it must satisfy (2), i e (1 + 1 + 1 \u2013 6) + \u03bb (2 + 3 + 4 + 5) = 0\nor\n\u03bb = 3\n14\nPutting the values of  \u03bb in (1), we get\n3\n9\n6 \u02c6\n\u02c6\n\u02c6\n1\n1\n1\n7\n14\n7\nr\ni\nj\nk\n\uf8ee\n\uf8f9\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n+\n+\n+\n+\n+\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n\uf8f0\n\uf8fb\nr\n = \n15\n6 14\n\u2212\nor\n10\n23\n13 \u02c6\n\u02c6\n\u02c6\n7\n14\n7\nr\ni\nj\nk\n\uf8eb\n\uf8f6\n+\n+\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\nr\n = 69\n14\nor\n\u02c6\n\u02c6\n\u02c6\n(20\n23\n26 )\nr\ni\nj\nk\n\u22c5\n+\n+\nr\n = 69\nwhich is the required vector equation of the plane 11"
    },
    {
        "Chapter": "1",
        "sentence_range": "5982-5985",
        "Text": "e (1 + 1 + 1 \u2013 6) + \u03bb (2 + 3 + 4 + 5) = 0\nor\n\u03bb = 3\n14\nPutting the values of  \u03bb in (1), we get\n3\n9\n6 \u02c6\n\u02c6\n\u02c6\n1\n1\n1\n7\n14\n7\nr\ni\nj\nk\n\uf8ee\n\uf8f9\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n+\n+\n+\n+\n+\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n\uf8f0\n\uf8fb\nr\n = \n15\n6 14\n\u2212\nor\n10\n23\n13 \u02c6\n\u02c6\n\u02c6\n7\n14\n7\nr\ni\nj\nk\n\uf8eb\n\uf8f6\n+\n+\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\nr\n = 69\n14\nor\n\u02c6\n\u02c6\n\u02c6\n(20\n23\n26 )\nr\ni\nj\nk\n\u22c5\n+\n+\nr\n = 69\nwhich is the required vector equation of the plane 11 7  Coplanarity of Two Lines\nLet the given lines be\nrr  = \n1\n1\na\nb\n+\u03bb\nr\nr"
    },
    {
        "Chapter": "1",
        "sentence_range": "5983-5986",
        "Text": "(1 + 1 + 1 \u2013 6) + \u03bb (2 + 3 + 4 + 5) = 0\nor\n\u03bb = 3\n14\nPutting the values of  \u03bb in (1), we get\n3\n9\n6 \u02c6\n\u02c6\n\u02c6\n1\n1\n1\n7\n14\n7\nr\ni\nj\nk\n\uf8ee\n\uf8f9\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n\uf8eb\n\uf8f6\n+\n+\n+\n+\n+\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ef\n\uf8fa\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n\uf8ed\n\uf8f8\n\uf8f0\n\uf8fb\nr\n = \n15\n6 14\n\u2212\nor\n10\n23\n13 \u02c6\n\u02c6\n\u02c6\n7\n14\n7\nr\ni\nj\nk\n\uf8eb\n\uf8f6\n+\n+\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\nr\n = 69\n14\nor\n\u02c6\n\u02c6\n\u02c6\n(20\n23\n26 )\nr\ni\nj\nk\n\u22c5\n+\n+\nr\n = 69\nwhich is the required vector equation of the plane 11 7  Coplanarity of Two Lines\nLet the given lines be\nrr  = \n1\n1\na\nb\n+\u03bb\nr\nr (1)\nand\nrr  = \n2\n2\na\nb\n+ \u00b5\nr\nr"
    },
    {
        "Chapter": "1",
        "sentence_range": "5984-5987",
        "Text": "11 7  Coplanarity of Two Lines\nLet the given lines be\nrr  = \n1\n1\na\nb\n+\u03bb\nr\nr (1)\nand\nrr  = \n2\n2\na\nb\n+ \u00b5\nr\nr (2)\nThe line (1) passes through the point, say A, with position vector \n1\nar and is parallel\nto \n1b\nr"
    },
    {
        "Chapter": "1",
        "sentence_range": "5985-5988",
        "Text": "7  Coplanarity of Two Lines\nLet the given lines be\nrr  = \n1\n1\na\nb\n+\u03bb\nr\nr (1)\nand\nrr  = \n2\n2\na\nb\n+ \u00b5\nr\nr (2)\nThe line (1) passes through the point, say A, with position vector \n1\nar and is parallel\nto \n1b\nr The line (2) passes through the point, say B with position vector \n2\nar and is parallel\nto \n2b\nr"
    },
    {
        "Chapter": "1",
        "sentence_range": "5986-5989",
        "Text": "(1)\nand\nrr  = \n2\n2\na\nb\n+ \u00b5\nr\nr (2)\nThe line (1) passes through the point, say A, with position vector \n1\nar and is parallel\nto \n1b\nr The line (2) passes through the point, say B with position vector \n2\nar and is parallel\nto \n2b\nr Thus,\nAB\nuuur\n = \n2\n1\na\na\u2212\nr\nr\nThe given lines are coplanar if and only if  AB\nuuur\n is perpendicular to 1\n2\nb\nb\n\u00d7\nr\nr"
    },
    {
        "Chapter": "1",
        "sentence_range": "5987-5990",
        "Text": "(2)\nThe line (1) passes through the point, say A, with position vector \n1\nar and is parallel\nto \n1b\nr The line (2) passes through the point, say B with position vector \n2\nar and is parallel\nto \n2b\nr Thus,\nAB\nuuur\n = \n2\n1\na\na\u2212\nr\nr\nThe given lines are coplanar if and only if  AB\nuuur\n is perpendicular to 1\n2\nb\nb\n\u00d7\nr\nr i"
    },
    {
        "Chapter": "1",
        "sentence_range": "5988-5991",
        "Text": "The line (2) passes through the point, say B with position vector \n2\nar and is parallel\nto \n2b\nr Thus,\nAB\nuuur\n = \n2\n1\na\na\u2212\nr\nr\nThe given lines are coplanar if and only if  AB\nuuur\n is perpendicular to 1\n2\nb\nb\n\u00d7\nr\nr i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "5989-5992",
        "Text": "Thus,\nAB\nuuur\n = \n2\n1\na\na\u2212\nr\nr\nThe given lines are coplanar if and only if  AB\nuuur\n is perpendicular to 1\n2\nb\nb\n\u00d7\nr\nr i e 1\n2\nAB"
    },
    {
        "Chapter": "1",
        "sentence_range": "5990-5993",
        "Text": "i e 1\n2\nAB (\n)\nb\nb\n\u00d7\nuuur r\nr\n = 0 or \n2\n1\n1\n2\n(\n) (\n)\na\na\nb\nb\n\u2212\n\u22c5\n\u00d7\nr\nr\nr\nr\n = 0\nCartesian form\nLet (x1, y1, z1) and (x2, y2, z2) be the coordinates of the points A and B respectively"
    },
    {
        "Chapter": "1",
        "sentence_range": "5991-5994",
        "Text": "e 1\n2\nAB (\n)\nb\nb\n\u00d7\nuuur r\nr\n = 0 or \n2\n1\n1\n2\n(\n) (\n)\na\na\nb\nb\n\u2212\n\u22c5\n\u00d7\nr\nr\nr\nr\n = 0\nCartesian form\nLet (x1, y1, z1) and (x2, y2, z2) be the coordinates of the points A and B respectively \u00a9 NCERT\nnot to be republished\n MATHEMATI CS\n488\nLet a1, b1, c1 and a2, b2, c2 be the direction ratios of 1b\nr\nand \n2b\nr\n, respectively"
    },
    {
        "Chapter": "1",
        "sentence_range": "5992-5995",
        "Text": "1\n2\nAB (\n)\nb\nb\n\u00d7\nuuur r\nr\n = 0 or \n2\n1\n1\n2\n(\n) (\n)\na\na\nb\nb\n\u2212\n\u22c5\n\u00d7\nr\nr\nr\nr\n = 0\nCartesian form\nLet (x1, y1, z1) and (x2, y2, z2) be the coordinates of the points A and B respectively \u00a9 NCERT\nnot to be republished\n MATHEMATI CS\n488\nLet a1, b1, c1 and a2, b2, c2 be the direction ratios of 1b\nr\nand \n2b\nr\n, respectively Then\n2\n1\n2\n1\n2\n1 \u02c6\n\u02c6\n\u02c6\nAB (\n)\n(\n)\n(\n)\nx\nx i\ny\ny\nj\nz\nz k\n= \u2212 + \u2212 + \u2212\nuuur\n1\n1\n1\n1\n2\n2\n2\n2\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\nand\nb\na i\nb j\nc k\nb\na i\nb j\nc k\n= + +\n= + +\nr\nr\nThe given lines are coplanar if and only if \n(\n)\n1\n2\nAB\n0\n\u22c5b b\n\u00d7\n=\nuuur\nr\nr"
    },
    {
        "Chapter": "1",
        "sentence_range": "5993-5996",
        "Text": "(\n)\nb\nb\n\u00d7\nuuur r\nr\n = 0 or \n2\n1\n1\n2\n(\n) (\n)\na\na\nb\nb\n\u2212\n\u22c5\n\u00d7\nr\nr\nr\nr\n = 0\nCartesian form\nLet (x1, y1, z1) and (x2, y2, z2) be the coordinates of the points A and B respectively \u00a9 NCERT\nnot to be republished\n MATHEMATI CS\n488\nLet a1, b1, c1 and a2, b2, c2 be the direction ratios of 1b\nr\nand \n2b\nr\n, respectively Then\n2\n1\n2\n1\n2\n1 \u02c6\n\u02c6\n\u02c6\nAB (\n)\n(\n)\n(\n)\nx\nx i\ny\ny\nj\nz\nz k\n= \u2212 + \u2212 + \u2212\nuuur\n1\n1\n1\n1\n2\n2\n2\n2\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\nand\nb\na i\nb j\nc k\nb\na i\nb j\nc k\n= + +\n= + +\nr\nr\nThe given lines are coplanar if and only if \n(\n)\n1\n2\nAB\n0\n\u22c5b b\n\u00d7\n=\nuuur\nr\nr In the cartesian form,\nit can be expressed as\n2\n1\n2\n1\n2\n1\n1\n1\n1\n2\n2\n2\n0\nx\nx\ny\ny\nz\nz\na\nb\nc\na\nb\nc\n\u2212\n\u2212\n\u2212 ="
    },
    {
        "Chapter": "1",
        "sentence_range": "5994-5997",
        "Text": "\u00a9 NCERT\nnot to be republished\n MATHEMATI CS\n488\nLet a1, b1, c1 and a2, b2, c2 be the direction ratios of 1b\nr\nand \n2b\nr\n, respectively Then\n2\n1\n2\n1\n2\n1 \u02c6\n\u02c6\n\u02c6\nAB (\n)\n(\n)\n(\n)\nx\nx i\ny\ny\nj\nz\nz k\n= \u2212 + \u2212 + \u2212\nuuur\n1\n1\n1\n1\n2\n2\n2\n2\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\nand\nb\na i\nb j\nc k\nb\na i\nb j\nc k\n= + +\n= + +\nr\nr\nThe given lines are coplanar if and only if \n(\n)\n1\n2\nAB\n0\n\u22c5b b\n\u00d7\n=\nuuur\nr\nr In the cartesian form,\nit can be expressed as\n2\n1\n2\n1\n2\n1\n1\n1\n1\n2\n2\n2\n0\nx\nx\ny\ny\nz\nz\na\nb\nc\na\nb\nc\n\u2212\n\u2212\n\u2212 = (4)\nExample 21 Show that the lines\n+3\n1\n5\n\u20133\n1\n5\nx\ny\nz\n\u2212\n\u2212\n=\n=\n and  \n+1\n2\n5\n\u20131\n2\n5\nx\ny\nz\n\u2212\n\u2212\n=\n=\n are coplanar"
    },
    {
        "Chapter": "1",
        "sentence_range": "5995-5998",
        "Text": "Then\n2\n1\n2\n1\n2\n1 \u02c6\n\u02c6\n\u02c6\nAB (\n)\n(\n)\n(\n)\nx\nx i\ny\ny\nj\nz\nz k\n= \u2212 + \u2212 + \u2212\nuuur\n1\n1\n1\n1\n2\n2\n2\n2\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\nand\nb\na i\nb j\nc k\nb\na i\nb j\nc k\n= + +\n= + +\nr\nr\nThe given lines are coplanar if and only if \n(\n)\n1\n2\nAB\n0\n\u22c5b b\n\u00d7\n=\nuuur\nr\nr In the cartesian form,\nit can be expressed as\n2\n1\n2\n1\n2\n1\n1\n1\n1\n2\n2\n2\n0\nx\nx\ny\ny\nz\nz\na\nb\nc\na\nb\nc\n\u2212\n\u2212\n\u2212 = (4)\nExample 21 Show that the lines\n+3\n1\n5\n\u20133\n1\n5\nx\ny\nz\n\u2212\n\u2212\n=\n=\n and  \n+1\n2\n5\n\u20131\n2\n5\nx\ny\nz\n\u2212\n\u2212\n=\n=\n are coplanar Solution Here,x1 = \u2013 3, y1 = 1, z1 = 5, a1 = \u2013 3, b1 = 1, c1 = 5\nx2 = \u2013 1, y2 = 2, z2 = 5, a2 = \u20131, b2 = 2, c2 = 5\nNow, consider the determinant\n2\n1\n2\n1\n2\n1\n1\n1\n1\n2\n2\n2\n2 1\n0\n3 1 5\n0\n1\n2\n5\nx\nx\ny\ny\nz\nz\na\nb\nc\na\nb\nc\n\u2212\n\u2212\n\u2212 =\u2212\n=\n\u2212\nTherefore, lines are coplanar"
    },
    {
        "Chapter": "1",
        "sentence_range": "5996-5999",
        "Text": "In the cartesian form,\nit can be expressed as\n2\n1\n2\n1\n2\n1\n1\n1\n1\n2\n2\n2\n0\nx\nx\ny\ny\nz\nz\na\nb\nc\na\nb\nc\n\u2212\n\u2212\n\u2212 = (4)\nExample 21 Show that the lines\n+3\n1\n5\n\u20133\n1\n5\nx\ny\nz\n\u2212\n\u2212\n=\n=\n and  \n+1\n2\n5\n\u20131\n2\n5\nx\ny\nz\n\u2212\n\u2212\n=\n=\n are coplanar Solution Here,x1 = \u2013 3, y1 = 1, z1 = 5, a1 = \u2013 3, b1 = 1, c1 = 5\nx2 = \u2013 1, y2 = 2, z2 = 5, a2 = \u20131, b2 = 2, c2 = 5\nNow, consider the determinant\n2\n1\n2\n1\n2\n1\n1\n1\n1\n2\n2\n2\n2 1\n0\n3 1 5\n0\n1\n2\n5\nx\nx\ny\ny\nz\nz\na\nb\nc\na\nb\nc\n\u2212\n\u2212\n\u2212 =\u2212\n=\n\u2212\nTherefore, lines are coplanar 11"
    },
    {
        "Chapter": "1",
        "sentence_range": "5997-6000",
        "Text": "(4)\nExample 21 Show that the lines\n+3\n1\n5\n\u20133\n1\n5\nx\ny\nz\n\u2212\n\u2212\n=\n=\n and  \n+1\n2\n5\n\u20131\n2\n5\nx\ny\nz\n\u2212\n\u2212\n=\n=\n are coplanar Solution Here,x1 = \u2013 3, y1 = 1, z1 = 5, a1 = \u2013 3, b1 = 1, c1 = 5\nx2 = \u2013 1, y2 = 2, z2 = 5, a2 = \u20131, b2 = 2, c2 = 5\nNow, consider the determinant\n2\n1\n2\n1\n2\n1\n1\n1\n1\n2\n2\n2\n2 1\n0\n3 1 5\n0\n1\n2\n5\nx\nx\ny\ny\nz\nz\na\nb\nc\na\nb\nc\n\u2212\n\u2212\n\u2212 =\u2212\n=\n\u2212\nTherefore, lines are coplanar 11 8  Angle between Two Planes\nDefinition 2 The angle between two planes is defined as the angle between their\nnormals (Fig 11"
    },
    {
        "Chapter": "1",
        "sentence_range": "5998-6001",
        "Text": "Solution Here,x1 = \u2013 3, y1 = 1, z1 = 5, a1 = \u2013 3, b1 = 1, c1 = 5\nx2 = \u2013 1, y2 = 2, z2 = 5, a2 = \u20131, b2 = 2, c2 = 5\nNow, consider the determinant\n2\n1\n2\n1\n2\n1\n1\n1\n1\n2\n2\n2\n2 1\n0\n3 1 5\n0\n1\n2\n5\nx\nx\ny\ny\nz\nz\na\nb\nc\na\nb\nc\n\u2212\n\u2212\n\u2212 =\u2212\n=\n\u2212\nTherefore, lines are coplanar 11 8  Angle between Two Planes\nDefinition 2 The angle between two planes is defined as the angle between their\nnormals (Fig 11 18 (a))"
    },
    {
        "Chapter": "1",
        "sentence_range": "5999-6002",
        "Text": "11 8  Angle between Two Planes\nDefinition 2 The angle between two planes is defined as the angle between their\nnormals (Fig 11 18 (a)) Observe that if \u03b8 is an angle between the two planes, then so\nis 180 \u2013 \u03b8 (Fig 11"
    },
    {
        "Chapter": "1",
        "sentence_range": "6000-6003",
        "Text": "8  Angle between Two Planes\nDefinition 2 The angle between two planes is defined as the angle between their\nnormals (Fig 11 18 (a)) Observe that if \u03b8 is an angle between the two planes, then so\nis 180 \u2013 \u03b8 (Fig 11 18 (b))"
    },
    {
        "Chapter": "1",
        "sentence_range": "6001-6004",
        "Text": "18 (a)) Observe that if \u03b8 is an angle between the two planes, then so\nis 180 \u2013 \u03b8 (Fig 11 18 (b)) We shall take the acute angle as the angles between\ntwo planes"
    },
    {
        "Chapter": "1",
        "sentence_range": "6002-6005",
        "Text": "Observe that if \u03b8 is an angle between the two planes, then so\nis 180 \u2013 \u03b8 (Fig 11 18 (b)) We shall take the acute angle as the angles between\ntwo planes Fig 11"
    },
    {
        "Chapter": "1",
        "sentence_range": "6003-6006",
        "Text": "18 (b)) We shall take the acute angle as the angles between\ntwo planes Fig 11 18\n\u00a9 NCERT\nnot to be republished\nTHREE D IMENSIONAL G EOMETRY\n489\nIf \n1nr  and \n2nr  are normals to the planes and \u03b8 be the angle between the planes\nr n\u22c51\nr r  = d1 and \n2\n2"
    },
    {
        "Chapter": "1",
        "sentence_range": "6004-6007",
        "Text": "We shall take the acute angle as the angles between\ntwo planes Fig 11 18\n\u00a9 NCERT\nnot to be republished\nTHREE D IMENSIONAL G EOMETRY\n489\nIf \n1nr  and \n2nr  are normals to the planes and \u03b8 be the angle between the planes\nr n\u22c51\nr r  = d1 and \n2\n2 d\nn\nr\nr=\nr"
    },
    {
        "Chapter": "1",
        "sentence_range": "6005-6008",
        "Text": "Fig 11 18\n\u00a9 NCERT\nnot to be republished\nTHREE D IMENSIONAL G EOMETRY\n489\nIf \n1nr  and \n2nr  are normals to the planes and \u03b8 be the angle between the planes\nr n\u22c51\nr r  = d1 and \n2\n2 d\nn\nr\nr=\nr Then \u03b8 is the angle between the normals to the planes drawn from some common\npoint"
    },
    {
        "Chapter": "1",
        "sentence_range": "6006-6009",
        "Text": "18\n\u00a9 NCERT\nnot to be republished\nTHREE D IMENSIONAL G EOMETRY\n489\nIf \n1nr  and \n2nr  are normals to the planes and \u03b8 be the angle between the planes\nr n\u22c51\nr r  = d1 and \n2\n2 d\nn\nr\nr=\nr Then \u03b8 is the angle between the normals to the planes drawn from some common\npoint We have,\ncos \u03b8 =\n1\n2\n1\n2\n|\n| |\n|\nn\nn\nn\nn\nr\u22c5\nr\nr\nr\n\ufffdNote  The planes are perpendicular to each other if \n1nr"
    },
    {
        "Chapter": "1",
        "sentence_range": "6007-6010",
        "Text": "d\nn\nr\nr=\nr Then \u03b8 is the angle between the normals to the planes drawn from some common\npoint We have,\ncos \u03b8 =\n1\n2\n1\n2\n|\n| |\n|\nn\nn\nn\nn\nr\u22c5\nr\nr\nr\n\ufffdNote  The planes are perpendicular to each other if \n1nr 2nr  = 0 and parallel if\n1nr  is parallel to  \n2nr"
    },
    {
        "Chapter": "1",
        "sentence_range": "6008-6011",
        "Text": "Then \u03b8 is the angle between the normals to the planes drawn from some common\npoint We have,\ncos \u03b8 =\n1\n2\n1\n2\n|\n| |\n|\nn\nn\nn\nn\nr\u22c5\nr\nr\nr\n\ufffdNote  The planes are perpendicular to each other if \n1nr 2nr  = 0 and parallel if\n1nr  is parallel to  \n2nr Cartesian form Let \u03b8 be the angle between the planes,\nA1 x + B1 y + C1z + D1 = 0 and  A2x + B2 y + C2 z + D2 = 0\nThe direction ratios of the normal to the planes are A1, B1, C1 and A2, B2, C2\nrespectively"
    },
    {
        "Chapter": "1",
        "sentence_range": "6009-6012",
        "Text": "We have,\ncos \u03b8 =\n1\n2\n1\n2\n|\n| |\n|\nn\nn\nn\nn\nr\u22c5\nr\nr\nr\n\ufffdNote  The planes are perpendicular to each other if \n1nr 2nr  = 0 and parallel if\n1nr  is parallel to  \n2nr Cartesian form Let \u03b8 be the angle between the planes,\nA1 x + B1 y + C1z + D1 = 0 and  A2x + B2 y + C2 z + D2 = 0\nThe direction ratios of the normal to the planes are A1, B1, C1 and A2, B2, C2\nrespectively Therefore,   cos \u03b8 = \n1\n2\n1\n2\n1\n2\n2\n2\n2\n2\n2\n2\n1\n1\n1\n2\n2\n2\nA A\nB B\nC C\nA\nB\nC\nA\nB\nC\n+\n+\n+\n+\n+\n+\n\ufffdNote\n1"
    },
    {
        "Chapter": "1",
        "sentence_range": "6010-6013",
        "Text": "2nr  = 0 and parallel if\n1nr  is parallel to  \n2nr Cartesian form Let \u03b8 be the angle between the planes,\nA1 x + B1 y + C1z + D1 = 0 and  A2x + B2 y + C2 z + D2 = 0\nThe direction ratios of the normal to the planes are A1, B1, C1 and A2, B2, C2\nrespectively Therefore,   cos \u03b8 = \n1\n2\n1\n2\n1\n2\n2\n2\n2\n2\n2\n2\n1\n1\n1\n2\n2\n2\nA A\nB B\nC C\nA\nB\nC\nA\nB\nC\n+\n+\n+\n+\n+\n+\n\ufffdNote\n1 If the planes are at right angles, t hen \u03b8 = 90o and so cos \u03b8 = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "6011-6014",
        "Text": "Cartesian form Let \u03b8 be the angle between the planes,\nA1 x + B1 y + C1z + D1 = 0 and  A2x + B2 y + C2 z + D2 = 0\nThe direction ratios of the normal to the planes are A1, B1, C1 and A2, B2, C2\nrespectively Therefore,   cos \u03b8 = \n1\n2\n1\n2\n1\n2\n2\n2\n2\n2\n2\n2\n1\n1\n1\n2\n2\n2\nA A\nB B\nC C\nA\nB\nC\nA\nB\nC\n+\n+\n+\n+\n+\n+\n\ufffdNote\n1 If the planes are at right angles, t hen \u03b8 = 90o and so cos \u03b8 = 0 Hence, cos \u03b8 = A1A2 + B1B2 + C1C2 = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "6012-6015",
        "Text": "Therefore,   cos \u03b8 = \n1\n2\n1\n2\n1\n2\n2\n2\n2\n2\n2\n2\n1\n1\n1\n2\n2\n2\nA A\nB B\nC C\nA\nB\nC\nA\nB\nC\n+\n+\n+\n+\n+\n+\n\ufffdNote\n1 If the planes are at right angles, t hen \u03b8 = 90o and so cos \u03b8 = 0 Hence, cos \u03b8 = A1A2 + B1B2 + C1C2 = 0 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "6013-6016",
        "Text": "If the planes are at right angles, t hen \u03b8 = 90o and so cos \u03b8 = 0 Hence, cos \u03b8 = A1A2 + B1B2 + C1C2 = 0 2 If the planes are parallel, then \n1\n1\n1\n2\n2\n2\nA\nB\nC\nA\nB\n= =C"
    },
    {
        "Chapter": "1",
        "sentence_range": "6014-6017",
        "Text": "Hence, cos \u03b8 = A1A2 + B1B2 + C1C2 = 0 2 If the planes are parallel, then \n1\n1\n1\n2\n2\n2\nA\nB\nC\nA\nB\n= =C Example 22 Find the angle between the two planes 2x + y \u2013 2z = 5 and 3x \u2013 6y \u2013 2z = 7\nusing vector method"
    },
    {
        "Chapter": "1",
        "sentence_range": "6015-6018",
        "Text": "2 If the planes are parallel, then \n1\n1\n1\n2\n2\n2\nA\nB\nC\nA\nB\n= =C Example 22 Find the angle between the two planes 2x + y \u2013 2z = 5 and 3x \u2013 6y \u2013 2z = 7\nusing vector method Solution The angle between two planes is the angle between their normals"
    },
    {
        "Chapter": "1",
        "sentence_range": "6016-6019",
        "Text": "If the planes are parallel, then \n1\n1\n1\n2\n2\n2\nA\nB\nC\nA\nB\n= =C Example 22 Find the angle between the two planes 2x + y \u2013 2z = 5 and 3x \u2013 6y \u2013 2z = 7\nusing vector method Solution The angle between two planes is the angle between their normals From the\nequation of the planes, the normal vectors are\nN1\nur\n =\n\u02c6\n\u02c6\n\u02c6\n2\n2\ni\nj\nk\n+\n\u2212\n and  \n2\n\u02c6\n\u02c6\n\u02c6\nN\n3\n6\n2\ni\nj\nk\n=\n\u2212\n\u2212\nur\nTherefore\ncos \u03b8 =\n1\n2\n1\n2\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\nN\nN\n(2\n2\n) (3\n6\n2\n)\n| N | |N |\n4\n1\n4\n9\n36\n4\ni\nj\nk\ni\nj\nk\n\u22c5\n+\n\u2212\n\u22c5\n\u2212\n\u2212\n=\n+\n+\n+\n+\nur\nur\nur\nur\n = \n4\n21\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\nHence\n\u03b8 = cos \u2013 1\n\uf8f6\uf8f8\uf8f7\n\uf8eb\uf8ed\uf8ec\n21\n4\n\u00a9 NCERT\nnot to be republished\n MATHEMATI CS\n490\nExample 23 Find the angle between the two planes 3x \u2013 6y + 2z = 7 and 2x + 2y \u2013 2z =5"
    },
    {
        "Chapter": "1",
        "sentence_range": "6017-6020",
        "Text": "Example 22 Find the angle between the two planes 2x + y \u2013 2z = 5 and 3x \u2013 6y \u2013 2z = 7\nusing vector method Solution The angle between two planes is the angle between their normals From the\nequation of the planes, the normal vectors are\nN1\nur\n =\n\u02c6\n\u02c6\n\u02c6\n2\n2\ni\nj\nk\n+\n\u2212\n and  \n2\n\u02c6\n\u02c6\n\u02c6\nN\n3\n6\n2\ni\nj\nk\n=\n\u2212\n\u2212\nur\nTherefore\ncos \u03b8 =\n1\n2\n1\n2\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\nN\nN\n(2\n2\n) (3\n6\n2\n)\n| N | |N |\n4\n1\n4\n9\n36\n4\ni\nj\nk\ni\nj\nk\n\u22c5\n+\n\u2212\n\u22c5\n\u2212\n\u2212\n=\n+\n+\n+\n+\nur\nur\nur\nur\n = \n4\n21\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\nHence\n\u03b8 = cos \u2013 1\n\uf8f6\uf8f8\uf8f7\n\uf8eb\uf8ed\uf8ec\n21\n4\n\u00a9 NCERT\nnot to be republished\n MATHEMATI CS\n490\nExample 23 Find the angle between the two planes 3x \u2013 6y + 2z = 7 and 2x + 2y \u2013 2z =5 Solution Comparing the given equations of the planes with the equations\nA1 x + B1 y + C1 z + D1 = 0  and  A2 x + B2 y + C2 z + D2 = 0\nWe get\nA1 = 3, B1 = \u2013 6, C1 = 2\n A2 = 2, B2 = 2, C2 = \u2013 2\ncos \u03b8 =\n(\n)\n(\n)\n2\n2\n2\n2\n2\n2\n3\n2\n( 6) (2)\n(2) ( 2)\n3\n( 6)\n( 2)\n2\n2\n( 2)\n\u00d7\n+ \u2212\n+\n\u2212\n+ \u2212\n+ \u2212\n+\n+ \u2212\n=\n10\n5\n5 3\n21\n7\n2 3\n7 3\n\u2212\n=\n=\n\u00d7\nTherefore,\n\u03b8 = cos-1  5 3\n21\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n11"
    },
    {
        "Chapter": "1",
        "sentence_range": "6018-6021",
        "Text": "Solution The angle between two planes is the angle between their normals From the\nequation of the planes, the normal vectors are\nN1\nur\n =\n\u02c6\n\u02c6\n\u02c6\n2\n2\ni\nj\nk\n+\n\u2212\n and  \n2\n\u02c6\n\u02c6\n\u02c6\nN\n3\n6\n2\ni\nj\nk\n=\n\u2212\n\u2212\nur\nTherefore\ncos \u03b8 =\n1\n2\n1\n2\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\nN\nN\n(2\n2\n) (3\n6\n2\n)\n| N | |N |\n4\n1\n4\n9\n36\n4\ni\nj\nk\ni\nj\nk\n\u22c5\n+\n\u2212\n\u22c5\n\u2212\n\u2212\n=\n+\n+\n+\n+\nur\nur\nur\nur\n = \n4\n21\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\nHence\n\u03b8 = cos \u2013 1\n\uf8f6\uf8f8\uf8f7\n\uf8eb\uf8ed\uf8ec\n21\n4\n\u00a9 NCERT\nnot to be republished\n MATHEMATI CS\n490\nExample 23 Find the angle between the two planes 3x \u2013 6y + 2z = 7 and 2x + 2y \u2013 2z =5 Solution Comparing the given equations of the planes with the equations\nA1 x + B1 y + C1 z + D1 = 0  and  A2 x + B2 y + C2 z + D2 = 0\nWe get\nA1 = 3, B1 = \u2013 6, C1 = 2\n A2 = 2, B2 = 2, C2 = \u2013 2\ncos \u03b8 =\n(\n)\n(\n)\n2\n2\n2\n2\n2\n2\n3\n2\n( 6) (2)\n(2) ( 2)\n3\n( 6)\n( 2)\n2\n2\n( 2)\n\u00d7\n+ \u2212\n+\n\u2212\n+ \u2212\n+ \u2212\n+\n+ \u2212\n=\n10\n5\n5 3\n21\n7\n2 3\n7 3\n\u2212\n=\n=\n\u00d7\nTherefore,\n\u03b8 = cos-1  5 3\n21\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n11 9  Distance of a Point from a Plane\nVector form\nConsider a point P with position vector ar  and a plane \u03c0 1 whose equation is\n\u22c5rr n\u02c6\n = d (Fig 11"
    },
    {
        "Chapter": "1",
        "sentence_range": "6019-6022",
        "Text": "From the\nequation of the planes, the normal vectors are\nN1\nur\n =\n\u02c6\n\u02c6\n\u02c6\n2\n2\ni\nj\nk\n+\n\u2212\n and  \n2\n\u02c6\n\u02c6\n\u02c6\nN\n3\n6\n2\ni\nj\nk\n=\n\u2212\n\u2212\nur\nTherefore\ncos \u03b8 =\n1\n2\n1\n2\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\nN\nN\n(2\n2\n) (3\n6\n2\n)\n| N | |N |\n4\n1\n4\n9\n36\n4\ni\nj\nk\ni\nj\nk\n\u22c5\n+\n\u2212\n\u22c5\n\u2212\n\u2212\n=\n+\n+\n+\n+\nur\nur\nur\nur\n = \n4\n21\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\nHence\n\u03b8 = cos \u2013 1\n\uf8f6\uf8f8\uf8f7\n\uf8eb\uf8ed\uf8ec\n21\n4\n\u00a9 NCERT\nnot to be republished\n MATHEMATI CS\n490\nExample 23 Find the angle between the two planes 3x \u2013 6y + 2z = 7 and 2x + 2y \u2013 2z =5 Solution Comparing the given equations of the planes with the equations\nA1 x + B1 y + C1 z + D1 = 0  and  A2 x + B2 y + C2 z + D2 = 0\nWe get\nA1 = 3, B1 = \u2013 6, C1 = 2\n A2 = 2, B2 = 2, C2 = \u2013 2\ncos \u03b8 =\n(\n)\n(\n)\n2\n2\n2\n2\n2\n2\n3\n2\n( 6) (2)\n(2) ( 2)\n3\n( 6)\n( 2)\n2\n2\n( 2)\n\u00d7\n+ \u2212\n+\n\u2212\n+ \u2212\n+ \u2212\n+\n+ \u2212\n=\n10\n5\n5 3\n21\n7\n2 3\n7 3\n\u2212\n=\n=\n\u00d7\nTherefore,\n\u03b8 = cos-1  5 3\n21\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n11 9  Distance of a Point from a Plane\nVector form\nConsider a point P with position vector ar  and a plane \u03c0 1 whose equation is\n\u22c5rr n\u02c6\n = d (Fig 11 19)"
    },
    {
        "Chapter": "1",
        "sentence_range": "6020-6023",
        "Text": "Solution Comparing the given equations of the planes with the equations\nA1 x + B1 y + C1 z + D1 = 0  and  A2 x + B2 y + C2 z + D2 = 0\nWe get\nA1 = 3, B1 = \u2013 6, C1 = 2\n A2 = 2, B2 = 2, C2 = \u2013 2\ncos \u03b8 =\n(\n)\n(\n)\n2\n2\n2\n2\n2\n2\n3\n2\n( 6) (2)\n(2) ( 2)\n3\n( 6)\n( 2)\n2\n2\n( 2)\n\u00d7\n+ \u2212\n+\n\u2212\n+ \u2212\n+ \u2212\n+\n+ \u2212\n=\n10\n5\n5 3\n21\n7\n2 3\n7 3\n\u2212\n=\n=\n\u00d7\nTherefore,\n\u03b8 = cos-1  5 3\n21\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\n11 9  Distance of a Point from a Plane\nVector form\nConsider a point P with position vector ar  and a plane \u03c0 1 whose equation is\n\u22c5rr n\u02c6\n = d (Fig 11 19) Consider a plane \u03c02 through P parallel to the plane  \u03c01"
    },
    {
        "Chapter": "1",
        "sentence_range": "6021-6024",
        "Text": "9  Distance of a Point from a Plane\nVector form\nConsider a point P with position vector ar  and a plane \u03c0 1 whose equation is\n\u22c5rr n\u02c6\n = d (Fig 11 19) Consider a plane \u03c02 through P parallel to the plane  \u03c01 The unit vector normal to\n\u03c02 is n\u02c6"
    },
    {
        "Chapter": "1",
        "sentence_range": "6022-6025",
        "Text": "19) Consider a plane \u03c02 through P parallel to the plane  \u03c01 The unit vector normal to\n\u03c02 is n\u02c6 Hence, its equation is \n\u02c6\n(\n)\n0\nr\na\nn\n\u2212\n\u22c5\n=\nr\nr\ni"
    },
    {
        "Chapter": "1",
        "sentence_range": "6023-6026",
        "Text": "Consider a plane \u03c02 through P parallel to the plane  \u03c01 The unit vector normal to\n\u03c02 is n\u02c6 Hence, its equation is \n\u02c6\n(\n)\n0\nr\na\nn\n\u2212\n\u22c5\n=\nr\nr\ni e"
    },
    {
        "Chapter": "1",
        "sentence_range": "6024-6027",
        "Text": "The unit vector normal to\n\u03c02 is n\u02c6 Hence, its equation is \n\u02c6\n(\n)\n0\nr\na\nn\n\u2212\n\u22c5\n=\nr\nr\ni e ,\n\u22c5rr n\u02c6\n =\na n\u02c6\n\u22c5r\nThus, the distance ON\u2032 of this plane from the origin is \n\u02c6\n|\na n|\nr\u22c5"
    },
    {
        "Chapter": "1",
        "sentence_range": "6025-6028",
        "Text": "Hence, its equation is \n\u02c6\n(\n)\n0\nr\na\nn\n\u2212\n\u22c5\n=\nr\nr\ni e ,\n\u22c5rr n\u02c6\n =\na n\u02c6\n\u22c5r\nThus, the distance ON\u2032 of this plane from the origin is \n\u02c6\n|\na n|\nr\u22c5 Therefore, the distance\nPQ from the plane \u03c01 is (Fig"
    },
    {
        "Chapter": "1",
        "sentence_range": "6026-6029",
        "Text": "e ,\n\u22c5rr n\u02c6\n =\na n\u02c6\n\u22c5r\nThus, the distance ON\u2032 of this plane from the origin is \n\u02c6\n|\na n|\nr\u22c5 Therefore, the distance\nPQ from the plane \u03c01 is (Fig 11"
    },
    {
        "Chapter": "1",
        "sentence_range": "6027-6030",
        "Text": ",\n\u22c5rr n\u02c6\n =\na n\u02c6\n\u22c5r\nThus, the distance ON\u2032 of this plane from the origin is \n\u02c6\n|\na n|\nr\u22c5 Therefore, the distance\nPQ from the plane \u03c01 is (Fig 11 21 (a))\ni"
    },
    {
        "Chapter": "1",
        "sentence_range": "6028-6031",
        "Text": "Therefore, the distance\nPQ from the plane \u03c01 is (Fig 11 21 (a))\ni e"
    },
    {
        "Chapter": "1",
        "sentence_range": "6029-6032",
        "Text": "11 21 (a))\ni e ,\nON \u2013 ON\u2032 = |d \u2013 \na n\u22c5\u02c6 |\nr\nFig 11"
    },
    {
        "Chapter": "1",
        "sentence_range": "6030-6033",
        "Text": "21 (a))\ni e ,\nON \u2013 ON\u2032 = |d \u2013 \na n\u22c5\u02c6 |\nr\nFig 11 19\n(a)\na\nZ\nX\nY\nd\nN\u2019\nP\nO\n\u03c01\n\u03c02\nN\nQ\n\u03c01\n(b)\nP\nX\nZ\nY\nO\nd\nN\u2019\nN\n\u03c02\na\n\u00a9 NCERT\nnot to be republished\nTHREE D IMENSIONAL G EOMETRY\n491\nwhich is the length of the perpendicular from a point to the given plane"
    },
    {
        "Chapter": "1",
        "sentence_range": "6031-6034",
        "Text": "e ,\nON \u2013 ON\u2032 = |d \u2013 \na n\u22c5\u02c6 |\nr\nFig 11 19\n(a)\na\nZ\nX\nY\nd\nN\u2019\nP\nO\n\u03c01\n\u03c02\nN\nQ\n\u03c01\n(b)\nP\nX\nZ\nY\nO\nd\nN\u2019\nN\n\u03c02\na\n\u00a9 NCERT\nnot to be republished\nTHREE D IMENSIONAL G EOMETRY\n491\nwhich is the length of the perpendicular from a point to the given plane We may establish the similar results for (Fig 11"
    },
    {
        "Chapter": "1",
        "sentence_range": "6032-6035",
        "Text": ",\nON \u2013 ON\u2032 = |d \u2013 \na n\u22c5\u02c6 |\nr\nFig 11 19\n(a)\na\nZ\nX\nY\nd\nN\u2019\nP\nO\n\u03c01\n\u03c02\nN\nQ\n\u03c01\n(b)\nP\nX\nZ\nY\nO\nd\nN\u2019\nN\n\u03c02\na\n\u00a9 NCERT\nnot to be republished\nTHREE D IMENSIONAL G EOMETRY\n491\nwhich is the length of the perpendicular from a point to the given plane We may establish the similar results for (Fig 11 19 (b))"
    },
    {
        "Chapter": "1",
        "sentence_range": "6033-6036",
        "Text": "19\n(a)\na\nZ\nX\nY\nd\nN\u2019\nP\nO\n\u03c01\n\u03c02\nN\nQ\n\u03c01\n(b)\nP\nX\nZ\nY\nO\nd\nN\u2019\nN\n\u03c02\na\n\u00a9 NCERT\nnot to be republished\nTHREE D IMENSIONAL G EOMETRY\n491\nwhich is the length of the perpendicular from a point to the given plane We may establish the similar results for (Fig 11 19 (b)) \ufffdNote\n1"
    },
    {
        "Chapter": "1",
        "sentence_range": "6034-6037",
        "Text": "We may establish the similar results for (Fig 11 19 (b)) \ufffdNote\n1 If the equation of the plane \u03c02 is in the form \nrN\n\u22c5 =d\nrur\n, where N\nur\n is normal\nto the plane, then the perpendicular distance is |\nN\n|\n| N |\na\n\u22c5 \u2212d\nur\nr\nur"
    },
    {
        "Chapter": "1",
        "sentence_range": "6035-6038",
        "Text": "19 (b)) \ufffdNote\n1 If the equation of the plane \u03c02 is in the form \nrN\n\u22c5 =d\nrur\n, where N\nur\n is normal\nto the plane, then the perpendicular distance is |\nN\n|\n| N |\na\n\u22c5 \u2212d\nur\nr\nur 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "6036-6039",
        "Text": "\ufffdNote\n1 If the equation of the plane \u03c02 is in the form \nrN\n\u22c5 =d\nrur\n, where N\nur\n is normal\nto the plane, then the perpendicular distance is |\nN\n|\n| N |\na\n\u22c5 \u2212d\nur\nr\nur 2 The length of the perpendicular from origin O to the plane \nrrurN\u22c5 =\nd  is |\n|\n| N |\ndur\n(since ar  = 0)"
    },
    {
        "Chapter": "1",
        "sentence_range": "6037-6040",
        "Text": "If the equation of the plane \u03c02 is in the form \nrN\n\u22c5 =d\nrur\n, where N\nur\n is normal\nto the plane, then the perpendicular distance is |\nN\n|\n| N |\na\n\u22c5 \u2212d\nur\nr\nur 2 The length of the perpendicular from origin O to the plane \nrrurN\u22c5 =\nd  is |\n|\n| N |\ndur\n(since ar  = 0) Cartesian form\nLet P(x1, y1, z1) be the given point with position vector ar and\nAx + By + Cz = D\nbe the Cartesian equation of the given plane"
    },
    {
        "Chapter": "1",
        "sentence_range": "6038-6041",
        "Text": "2 The length of the perpendicular from origin O to the plane \nrrurN\u22c5 =\nd  is |\n|\n| N |\ndur\n(since ar  = 0) Cartesian form\nLet P(x1, y1, z1) be the given point with position vector ar and\nAx + By + Cz = D\nbe the Cartesian equation of the given plane Then\nar  =\n1\n1\n1 \u02c6\n\u02c6\n\u02c6\nx i\ny\nj\nz k\n+\n+\nN\nur\n =\n\u02c6\n\u02c6\n\u02c6\nA\nB\nC\ni\nj\nk\n+\n+\nHence, from Note 1, the perpendicular from P to the plane is\n1\n1\n1\n2\n2\n2\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(\n) ( A\nB\nC\n)\nD\nA\nB\nC\nx i\ny\nj\nz k\ni\nj\nk\n+\n+\n\u22c5\n+\n+\n\u2212\n+\n+\n=\n1\n1\n1\n2\n2\n2\nA\nB\nC\nD\nA\nB\nC\nx\ny\nz\n+\n+\n\u2212\n+ +\nExample 24 Find the distance of a point (2, 5, \u2013 3) from the plane\n\u02c6\n\u02c6\n\u02c6\n( 6\n3\n2\n)\nr\ni\nj\nk\n\u22c5\n\u2212\n+\nr\n = 4\nSolution Here,  \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n2\n5\n3\n, N\n6\n3\n2\n= + \u2212\n= \u2212 +\nur\nra\ni\nj\nk\ni\nj\nk and d = 4"
    },
    {
        "Chapter": "1",
        "sentence_range": "6039-6042",
        "Text": "The length of the perpendicular from origin O to the plane \nrrurN\u22c5 =\nd  is |\n|\n| N |\ndur\n(since ar  = 0) Cartesian form\nLet P(x1, y1, z1) be the given point with position vector ar and\nAx + By + Cz = D\nbe the Cartesian equation of the given plane Then\nar  =\n1\n1\n1 \u02c6\n\u02c6\n\u02c6\nx i\ny\nj\nz k\n+\n+\nN\nur\n =\n\u02c6\n\u02c6\n\u02c6\nA\nB\nC\ni\nj\nk\n+\n+\nHence, from Note 1, the perpendicular from P to the plane is\n1\n1\n1\n2\n2\n2\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(\n) ( A\nB\nC\n)\nD\nA\nB\nC\nx i\ny\nj\nz k\ni\nj\nk\n+\n+\n\u22c5\n+\n+\n\u2212\n+\n+\n=\n1\n1\n1\n2\n2\n2\nA\nB\nC\nD\nA\nB\nC\nx\ny\nz\n+\n+\n\u2212\n+ +\nExample 24 Find the distance of a point (2, 5, \u2013 3) from the plane\n\u02c6\n\u02c6\n\u02c6\n( 6\n3\n2\n)\nr\ni\nj\nk\n\u22c5\n\u2212\n+\nr\n = 4\nSolution Here,  \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n2\n5\n3\n, N\n6\n3\n2\n= + \u2212\n= \u2212 +\nur\nra\ni\nj\nk\ni\nj\nk and d = 4 Therefore, the distance of the point (2, 5, \u2013 3) from the given plane is\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n| (2\n5\n3 ) (6\n3\n2\n)\n4|\n\u02c6\n\u02c6\n\u02c6\n| 6\n3\n2\n|\ni\nj\nk\ni\nj\nk\ni\nj\nk\n+\n\u2212\n\u22c5\n\u2212\n+\n\u2212\n\u2212\n+\n = \n| 12\n15\n6\n4 |\n713\n36\n9\n4\n\u2212\n\u2212\n\u2212\n=\n+\n+\n\u00a9 NCERT\nnot to be republished\n MATHEMATI CS\n492\n11"
    },
    {
        "Chapter": "1",
        "sentence_range": "6040-6043",
        "Text": "Cartesian form\nLet P(x1, y1, z1) be the given point with position vector ar and\nAx + By + Cz = D\nbe the Cartesian equation of the given plane Then\nar  =\n1\n1\n1 \u02c6\n\u02c6\n\u02c6\nx i\ny\nj\nz k\n+\n+\nN\nur\n =\n\u02c6\n\u02c6\n\u02c6\nA\nB\nC\ni\nj\nk\n+\n+\nHence, from Note 1, the perpendicular from P to the plane is\n1\n1\n1\n2\n2\n2\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(\n) ( A\nB\nC\n)\nD\nA\nB\nC\nx i\ny\nj\nz k\ni\nj\nk\n+\n+\n\u22c5\n+\n+\n\u2212\n+\n+\n=\n1\n1\n1\n2\n2\n2\nA\nB\nC\nD\nA\nB\nC\nx\ny\nz\n+\n+\n\u2212\n+ +\nExample 24 Find the distance of a point (2, 5, \u2013 3) from the plane\n\u02c6\n\u02c6\n\u02c6\n( 6\n3\n2\n)\nr\ni\nj\nk\n\u22c5\n\u2212\n+\nr\n = 4\nSolution Here,  \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n2\n5\n3\n, N\n6\n3\n2\n= + \u2212\n= \u2212 +\nur\nra\ni\nj\nk\ni\nj\nk and d = 4 Therefore, the distance of the point (2, 5, \u2013 3) from the given plane is\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n| (2\n5\n3 ) (6\n3\n2\n)\n4|\n\u02c6\n\u02c6\n\u02c6\n| 6\n3\n2\n|\ni\nj\nk\ni\nj\nk\ni\nj\nk\n+\n\u2212\n\u22c5\n\u2212\n+\n\u2212\n\u2212\n+\n = \n| 12\n15\n6\n4 |\n713\n36\n9\n4\n\u2212\n\u2212\n\u2212\n=\n+\n+\n\u00a9 NCERT\nnot to be republished\n MATHEMATI CS\n492\n11 10  Angle between a Line and a Plane\nDefinition 3 The angle between a line and a plane is\nthe complement of the angle between the line and\nnormal to the plane (Fig 11"
    },
    {
        "Chapter": "1",
        "sentence_range": "6041-6044",
        "Text": "Then\nar  =\n1\n1\n1 \u02c6\n\u02c6\n\u02c6\nx i\ny\nj\nz k\n+\n+\nN\nur\n =\n\u02c6\n\u02c6\n\u02c6\nA\nB\nC\ni\nj\nk\n+\n+\nHence, from Note 1, the perpendicular from P to the plane is\n1\n1\n1\n2\n2\n2\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(\n) ( A\nB\nC\n)\nD\nA\nB\nC\nx i\ny\nj\nz k\ni\nj\nk\n+\n+\n\u22c5\n+\n+\n\u2212\n+\n+\n=\n1\n1\n1\n2\n2\n2\nA\nB\nC\nD\nA\nB\nC\nx\ny\nz\n+\n+\n\u2212\n+ +\nExample 24 Find the distance of a point (2, 5, \u2013 3) from the plane\n\u02c6\n\u02c6\n\u02c6\n( 6\n3\n2\n)\nr\ni\nj\nk\n\u22c5\n\u2212\n+\nr\n = 4\nSolution Here,  \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n2\n5\n3\n, N\n6\n3\n2\n= + \u2212\n= \u2212 +\nur\nra\ni\nj\nk\ni\nj\nk and d = 4 Therefore, the distance of the point (2, 5, \u2013 3) from the given plane is\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n| (2\n5\n3 ) (6\n3\n2\n)\n4|\n\u02c6\n\u02c6\n\u02c6\n| 6\n3\n2\n|\ni\nj\nk\ni\nj\nk\ni\nj\nk\n+\n\u2212\n\u22c5\n\u2212\n+\n\u2212\n\u2212\n+\n = \n| 12\n15\n6\n4 |\n713\n36\n9\n4\n\u2212\n\u2212\n\u2212\n=\n+\n+\n\u00a9 NCERT\nnot to be republished\n MATHEMATI CS\n492\n11 10  Angle between a Line and a Plane\nDefinition 3 The angle between a line and a plane is\nthe complement of the angle between the line and\nnormal to the plane (Fig 11 20)"
    },
    {
        "Chapter": "1",
        "sentence_range": "6042-6045",
        "Text": "Therefore, the distance of the point (2, 5, \u2013 3) from the given plane is\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n| (2\n5\n3 ) (6\n3\n2\n)\n4|\n\u02c6\n\u02c6\n\u02c6\n| 6\n3\n2\n|\ni\nj\nk\ni\nj\nk\ni\nj\nk\n+\n\u2212\n\u22c5\n\u2212\n+\n\u2212\n\u2212\n+\n = \n| 12\n15\n6\n4 |\n713\n36\n9\n4\n\u2212\n\u2212\n\u2212\n=\n+\n+\n\u00a9 NCERT\nnot to be republished\n MATHEMATI CS\n492\n11 10  Angle between a Line and a Plane\nDefinition 3 The angle between a line and a plane is\nthe complement of the angle between the line and\nnormal to the plane (Fig 11 20) Vector form If the equation of the line is\nb\na\nr\nr\nr\nr\n+\u03bb\n=\n and the equation of the plane is\nr n\nd\n\u22c5\nr r="
    },
    {
        "Chapter": "1",
        "sentence_range": "6043-6046",
        "Text": "10  Angle between a Line and a Plane\nDefinition 3 The angle between a line and a plane is\nthe complement of the angle between the line and\nnormal to the plane (Fig 11 20) Vector form If the equation of the line is\nb\na\nr\nr\nr\nr\n+\u03bb\n=\n and the equation of the plane is\nr n\nd\n\u22c5\nr r= Then the angle \u03b8 between the line and the\nnormal to the plane is\ncos \u03b8 =\n|\n| |\n|\nbb n\nn\n\u22c5\n\u22c5\nrr r\nr\nand so the angle \u03c6 between the line and the plane is given by 90 \u2013 \u03b8, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "6044-6047",
        "Text": "20) Vector form If the equation of the line is\nb\na\nr\nr\nr\nr\n+\u03bb\n=\n and the equation of the plane is\nr n\nd\n\u22c5\nr r= Then the angle \u03b8 between the line and the\nnormal to the plane is\ncos \u03b8 =\n|\n| |\n|\nbb n\nn\n\u22c5\n\u22c5\nrr r\nr\nand so the angle \u03c6 between the line and the plane is given by 90 \u2013 \u03b8, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "6045-6048",
        "Text": "Vector form If the equation of the line is\nb\na\nr\nr\nr\nr\n+\u03bb\n=\n and the equation of the plane is\nr n\nd\n\u22c5\nr r= Then the angle \u03b8 between the line and the\nnormal to the plane is\ncos \u03b8 =\n|\n| |\n|\nbb n\nn\n\u22c5\n\u22c5\nrr r\nr\nand so the angle \u03c6 between the line and the plane is given by 90 \u2013 \u03b8, i e ,\nsin (90 \u2013 \u03b8) = cos \u03b8\ni"
    },
    {
        "Chapter": "1",
        "sentence_range": "6046-6049",
        "Text": "Then the angle \u03b8 between the line and the\nnormal to the plane is\ncos \u03b8 =\n|\n| |\n|\nbb n\nn\n\u22c5\n\u22c5\nrr r\nr\nand so the angle \u03c6 between the line and the plane is given by 90 \u2013 \u03b8, i e ,\nsin (90 \u2013 \u03b8) = cos \u03b8\ni e"
    },
    {
        "Chapter": "1",
        "sentence_range": "6047-6050",
        "Text": "e ,\nsin (90 \u2013 \u03b8) = cos \u03b8\ni e sin \u03c6 =\n|\n| |\n|\nbb n\nn\n\u22c5\nrr r\nr\n or \u03c6 = \nsin\u20131\nb n\nb n\n\u22c5\nExample 25 Find the angle between the line\n1\n2\nx +\n =\n3\n3\n6\ny\nz \u2212\n=\nand the plane 10 x + 2y \u2013 11 z = 3"
    },
    {
        "Chapter": "1",
        "sentence_range": "6048-6051",
        "Text": ",\nsin (90 \u2013 \u03b8) = cos \u03b8\ni e sin \u03c6 =\n|\n| |\n|\nbb n\nn\n\u22c5\nrr r\nr\n or \u03c6 = \nsin\u20131\nb n\nb n\n\u22c5\nExample 25 Find the angle between the line\n1\n2\nx +\n =\n3\n3\n6\ny\nz \u2212\n=\nand the plane 10 x + 2y \u2013 11 z = 3 Solution Let \u03b8 be the angle between the line and the normal to the plane"
    },
    {
        "Chapter": "1",
        "sentence_range": "6049-6052",
        "Text": "e sin \u03c6 =\n|\n| |\n|\nbb n\nn\n\u22c5\nrr r\nr\n or \u03c6 = \nsin\u20131\nb n\nb n\n\u22c5\nExample 25 Find the angle between the line\n1\n2\nx +\n =\n3\n3\n6\ny\nz \u2212\n=\nand the plane 10 x + 2y \u2013 11 z = 3 Solution Let \u03b8 be the angle between the line and the normal to the plane Converting the\ngiven equations into vector form, we have\nrr  =\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n( \u2013\n3 )\n( 2\n3\n6\n)\ni\nk\ni\nj\nk\n+\n+ \u03bb\n+\n+\nand\n\u02c6\n\u02c6\n\u02c6\n( 10\n2\n11\n)\nr\ni\nj\nk\n\u22c5\n+\n\u2212\nr\n = 3\nHere\nb\nr\n =\n\u02c6\n\u02c6\n\u02c6\n2\n3\n6\ni\nj\nk\n+\n+\n   and  \nk\nj\ni\nn\n11\u02c6\n2\u02c6\n10\u02c6\n\u2212\n+\n=\nr\nsin \u03c6 =\n2\n2\n2\n2\n2\n2\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(2\n3\n6\n) (10\n2\n11 )\n2\n3\n6\n10\n2\n11\ni\nj\nk\ni\nj\nk\n+\n+\n\u22c5\n+\n\u2212\n+\n+\n+\n+\n=\n40\n7\n15\n\u2212\n\u00d7\n = \n\u2212218\n = 8\n21   or \u03c6 = \n1\n8\nsin\n21\n\u2212 \uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\nFig 11"
    },
    {
        "Chapter": "1",
        "sentence_range": "6050-6053",
        "Text": "sin \u03c6 =\n|\n| |\n|\nbb n\nn\n\u22c5\nrr r\nr\n or \u03c6 = \nsin\u20131\nb n\nb n\n\u22c5\nExample 25 Find the angle between the line\n1\n2\nx +\n =\n3\n3\n6\ny\nz \u2212\n=\nand the plane 10 x + 2y \u2013 11 z = 3 Solution Let \u03b8 be the angle between the line and the normal to the plane Converting the\ngiven equations into vector form, we have\nrr  =\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n( \u2013\n3 )\n( 2\n3\n6\n)\ni\nk\ni\nj\nk\n+\n+ \u03bb\n+\n+\nand\n\u02c6\n\u02c6\n\u02c6\n( 10\n2\n11\n)\nr\ni\nj\nk\n\u22c5\n+\n\u2212\nr\n = 3\nHere\nb\nr\n =\n\u02c6\n\u02c6\n\u02c6\n2\n3\n6\ni\nj\nk\n+\n+\n   and  \nk\nj\ni\nn\n11\u02c6\n2\u02c6\n10\u02c6\n\u2212\n+\n=\nr\nsin \u03c6 =\n2\n2\n2\n2\n2\n2\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(2\n3\n6\n) (10\n2\n11 )\n2\n3\n6\n10\n2\n11\ni\nj\nk\ni\nj\nk\n+\n+\n\u22c5\n+\n\u2212\n+\n+\n+\n+\n=\n40\n7\n15\n\u2212\n\u00d7\n = \n\u2212218\n = 8\n21   or \u03c6 = \n1\n8\nsin\n21\n\u2212 \uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\nFig 11 20\n\u00a9 NCERT\nnot to be republished\nTHREE D IMENSIONAL G EOMETRY\n493\nEXERCISE 11"
    },
    {
        "Chapter": "1",
        "sentence_range": "6051-6054",
        "Text": "Solution Let \u03b8 be the angle between the line and the normal to the plane Converting the\ngiven equations into vector form, we have\nrr  =\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n( \u2013\n3 )\n( 2\n3\n6\n)\ni\nk\ni\nj\nk\n+\n+ \u03bb\n+\n+\nand\n\u02c6\n\u02c6\n\u02c6\n( 10\n2\n11\n)\nr\ni\nj\nk\n\u22c5\n+\n\u2212\nr\n = 3\nHere\nb\nr\n =\n\u02c6\n\u02c6\n\u02c6\n2\n3\n6\ni\nj\nk\n+\n+\n   and  \nk\nj\ni\nn\n11\u02c6\n2\u02c6\n10\u02c6\n\u2212\n+\n=\nr\nsin \u03c6 =\n2\n2\n2\n2\n2\n2\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(2\n3\n6\n) (10\n2\n11 )\n2\n3\n6\n10\n2\n11\ni\nj\nk\ni\nj\nk\n+\n+\n\u22c5\n+\n\u2212\n+\n+\n+\n+\n=\n40\n7\n15\n\u2212\n\u00d7\n = \n\u2212218\n = 8\n21   or \u03c6 = \n1\n8\nsin\n21\n\u2212 \uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\nFig 11 20\n\u00a9 NCERT\nnot to be republished\nTHREE D IMENSIONAL G EOMETRY\n493\nEXERCISE 11 3\n1"
    },
    {
        "Chapter": "1",
        "sentence_range": "6052-6055",
        "Text": "Converting the\ngiven equations into vector form, we have\nrr  =\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n( \u2013\n3 )\n( 2\n3\n6\n)\ni\nk\ni\nj\nk\n+\n+ \u03bb\n+\n+\nand\n\u02c6\n\u02c6\n\u02c6\n( 10\n2\n11\n)\nr\ni\nj\nk\n\u22c5\n+\n\u2212\nr\n = 3\nHere\nb\nr\n =\n\u02c6\n\u02c6\n\u02c6\n2\n3\n6\ni\nj\nk\n+\n+\n   and  \nk\nj\ni\nn\n11\u02c6\n2\u02c6\n10\u02c6\n\u2212\n+\n=\nr\nsin \u03c6 =\n2\n2\n2\n2\n2\n2\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n(2\n3\n6\n) (10\n2\n11 )\n2\n3\n6\n10\n2\n11\ni\nj\nk\ni\nj\nk\n+\n+\n\u22c5\n+\n\u2212\n+\n+\n+\n+\n=\n40\n7\n15\n\u2212\n\u00d7\n = \n\u2212218\n = 8\n21   or \u03c6 = \n1\n8\nsin\n21\n\u2212 \uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\nFig 11 20\n\u00a9 NCERT\nnot to be republished\nTHREE D IMENSIONAL G EOMETRY\n493\nEXERCISE 11 3\n1 In each of the following cases, determine the direction cosines of the normal to\nthe plane and the distance from the origin"
    },
    {
        "Chapter": "1",
        "sentence_range": "6053-6056",
        "Text": "20\n\u00a9 NCERT\nnot to be republished\nTHREE D IMENSIONAL G EOMETRY\n493\nEXERCISE 11 3\n1 In each of the following cases, determine the direction cosines of the normal to\nthe plane and the distance from the origin (a)\nz = 2\n(b)\nx + y + z = 1\n(c)\n2x + 3y \u2013 z = 5\n(d)\n5y + 8 = 0\n2"
    },
    {
        "Chapter": "1",
        "sentence_range": "6054-6057",
        "Text": "3\n1 In each of the following cases, determine the direction cosines of the normal to\nthe plane and the distance from the origin (a)\nz = 2\n(b)\nx + y + z = 1\n(c)\n2x + 3y \u2013 z = 5\n(d)\n5y + 8 = 0\n2 Find the vector equation of a plane which is at a distance of 7 units from the\norigin and normal to the vector  \nk\nj\ni\n6\u02c6\n5\u02c6\n3\u02c6\n\u2212\n+"
    },
    {
        "Chapter": "1",
        "sentence_range": "6055-6058",
        "Text": "In each of the following cases, determine the direction cosines of the normal to\nthe plane and the distance from the origin (a)\nz = 2\n(b)\nx + y + z = 1\n(c)\n2x + 3y \u2013 z = 5\n(d)\n5y + 8 = 0\n2 Find the vector equation of a plane which is at a distance of 7 units from the\norigin and normal to the vector  \nk\nj\ni\n6\u02c6\n5\u02c6\n3\u02c6\n\u2212\n+ 3"
    },
    {
        "Chapter": "1",
        "sentence_range": "6056-6059",
        "Text": "(a)\nz = 2\n(b)\nx + y + z = 1\n(c)\n2x + 3y \u2013 z = 5\n(d)\n5y + 8 = 0\n2 Find the vector equation of a plane which is at a distance of 7 units from the\norigin and normal to the vector  \nk\nj\ni\n6\u02c6\n5\u02c6\n3\u02c6\n\u2212\n+ 3 Find the Cartesian equation of the following planes:\n(a)\n\u02c6\n\u02c6\n\u02c6\n(\n)\n2\nr\ni\nj\nk\n\u22c5\n+\n\u2212\n=\nr\n(b)\n\u02c6\n\u02c6\n\u02c6\n(2\n3\n4 )\n1\nr\ni\nj\nk\n\u22c5\n+\n\u2212\n=\nr\n(c)\n\u02c6\n\u02c6\n\u02c6\n[(\n2 )\n(3\n)\n(2\n)\n]\n15\nr\ns\nt i\nt\nj\ns\nt k\n\u22c5\n\u2212\n+\n\u2212\n+\n+\n=\nr\n4"
    },
    {
        "Chapter": "1",
        "sentence_range": "6057-6060",
        "Text": "Find the vector equation of a plane which is at a distance of 7 units from the\norigin and normal to the vector  \nk\nj\ni\n6\u02c6\n5\u02c6\n3\u02c6\n\u2212\n+ 3 Find the Cartesian equation of the following planes:\n(a)\n\u02c6\n\u02c6\n\u02c6\n(\n)\n2\nr\ni\nj\nk\n\u22c5\n+\n\u2212\n=\nr\n(b)\n\u02c6\n\u02c6\n\u02c6\n(2\n3\n4 )\n1\nr\ni\nj\nk\n\u22c5\n+\n\u2212\n=\nr\n(c)\n\u02c6\n\u02c6\n\u02c6\n[(\n2 )\n(3\n)\n(2\n)\n]\n15\nr\ns\nt i\nt\nj\ns\nt k\n\u22c5\n\u2212\n+\n\u2212\n+\n+\n=\nr\n4 In the following cases, find the coordinates of the foot of the perpendicular\ndrawn from the origin"
    },
    {
        "Chapter": "1",
        "sentence_range": "6058-6061",
        "Text": "3 Find the Cartesian equation of the following planes:\n(a)\n\u02c6\n\u02c6\n\u02c6\n(\n)\n2\nr\ni\nj\nk\n\u22c5\n+\n\u2212\n=\nr\n(b)\n\u02c6\n\u02c6\n\u02c6\n(2\n3\n4 )\n1\nr\ni\nj\nk\n\u22c5\n+\n\u2212\n=\nr\n(c)\n\u02c6\n\u02c6\n\u02c6\n[(\n2 )\n(3\n)\n(2\n)\n]\n15\nr\ns\nt i\nt\nj\ns\nt k\n\u22c5\n\u2212\n+\n\u2212\n+\n+\n=\nr\n4 In the following cases, find the coordinates of the foot of the perpendicular\ndrawn from the origin (a)\n2x + 3y + 4z \u2013 12 = 0\n(b)\n3y + 4z \u2013 6 = 0\n(c)\nx + y + z = 1\n(d)\n5y + 8 = 0\n5"
    },
    {
        "Chapter": "1",
        "sentence_range": "6059-6062",
        "Text": "Find the Cartesian equation of the following planes:\n(a)\n\u02c6\n\u02c6\n\u02c6\n(\n)\n2\nr\ni\nj\nk\n\u22c5\n+\n\u2212\n=\nr\n(b)\n\u02c6\n\u02c6\n\u02c6\n(2\n3\n4 )\n1\nr\ni\nj\nk\n\u22c5\n+\n\u2212\n=\nr\n(c)\n\u02c6\n\u02c6\n\u02c6\n[(\n2 )\n(3\n)\n(2\n)\n]\n15\nr\ns\nt i\nt\nj\ns\nt k\n\u22c5\n\u2212\n+\n\u2212\n+\n+\n=\nr\n4 In the following cases, find the coordinates of the foot of the perpendicular\ndrawn from the origin (a)\n2x + 3y + 4z \u2013 12 = 0\n(b)\n3y + 4z \u2013 6 = 0\n(c)\nx + y + z = 1\n(d)\n5y + 8 = 0\n5 Find the vector and cartesian equations of the planes\n(a)\nthat passes through the point (1, 0, \u2013 2) and the normal to the plane is\n\u02c6\n\u02c6\n\u02c6"
    },
    {
        "Chapter": "1",
        "sentence_range": "6060-6063",
        "Text": "In the following cases, find the coordinates of the foot of the perpendicular\ndrawn from the origin (a)\n2x + 3y + 4z \u2013 12 = 0\n(b)\n3y + 4z \u2013 6 = 0\n(c)\nx + y + z = 1\n(d)\n5y + 8 = 0\n5 Find the vector and cartesian equations of the planes\n(a)\nthat passes through the point (1, 0, \u2013 2) and the normal to the plane is\n\u02c6\n\u02c6\n\u02c6 i\nj\nk\n+ \u2212\n(b)\nthat passes through the point (1,4, 6) and the normal vector to the plane is\n\u02c6\n\u02c6\n2\u02c6"
    },
    {
        "Chapter": "1",
        "sentence_range": "6061-6064",
        "Text": "(a)\n2x + 3y + 4z \u2013 12 = 0\n(b)\n3y + 4z \u2013 6 = 0\n(c)\nx + y + z = 1\n(d)\n5y + 8 = 0\n5 Find the vector and cartesian equations of the planes\n(a)\nthat passes through the point (1, 0, \u2013 2) and the normal to the plane is\n\u02c6\n\u02c6\n\u02c6 i\nj\nk\n+ \u2212\n(b)\nthat passes through the point (1,4, 6) and the normal vector to the plane is\n\u02c6\n\u02c6\n2\u02c6 i\nj\nk\n\u2212 +\n6"
    },
    {
        "Chapter": "1",
        "sentence_range": "6062-6065",
        "Text": "Find the vector and cartesian equations of the planes\n(a)\nthat passes through the point (1, 0, \u2013 2) and the normal to the plane is\n\u02c6\n\u02c6\n\u02c6 i\nj\nk\n+ \u2212\n(b)\nthat passes through the point (1,4, 6) and the normal vector to the plane is\n\u02c6\n\u02c6\n2\u02c6 i\nj\nk\n\u2212 +\n6 Find the equations of the planes that passes through three points"
    },
    {
        "Chapter": "1",
        "sentence_range": "6063-6066",
        "Text": "i\nj\nk\n+ \u2212\n(b)\nthat passes through the point (1,4, 6) and the normal vector to the plane is\n\u02c6\n\u02c6\n2\u02c6 i\nj\nk\n\u2212 +\n6 Find the equations of the planes that passes through three points (a)\n(1, 1, \u2013 1),  (6, 4, \u2013 5), (\u2013 4, \u2013 2, 3)\n(b)\n(1, 1, 0), (1, 2, 1), (\u2013 2, 2, \u2013 1)\n7"
    },
    {
        "Chapter": "1",
        "sentence_range": "6064-6067",
        "Text": "i\nj\nk\n\u2212 +\n6 Find the equations of the planes that passes through three points (a)\n(1, 1, \u2013 1),  (6, 4, \u2013 5), (\u2013 4, \u2013 2, 3)\n(b)\n(1, 1, 0), (1, 2, 1), (\u2013 2, 2, \u2013 1)\n7 Find the intercepts cut off by the plane 2x + y \u2013 z = 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "6065-6068",
        "Text": "Find the equations of the planes that passes through three points (a)\n(1, 1, \u2013 1),  (6, 4, \u2013 5), (\u2013 4, \u2013 2, 3)\n(b)\n(1, 1, 0), (1, 2, 1), (\u2013 2, 2, \u2013 1)\n7 Find the intercepts cut off by the plane 2x + y \u2013 z = 5 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "6066-6069",
        "Text": "(a)\n(1, 1, \u2013 1),  (6, 4, \u2013 5), (\u2013 4, \u2013 2, 3)\n(b)\n(1, 1, 0), (1, 2, 1), (\u2013 2, 2, \u2013 1)\n7 Find the intercepts cut off by the plane 2x + y \u2013 z = 5 8 Find the equation of the plane with intercept 3 on the y-axis and parallel to ZOX\nplane"
    },
    {
        "Chapter": "1",
        "sentence_range": "6067-6070",
        "Text": "Find the intercepts cut off by the plane 2x + y \u2013 z = 5 8 Find the equation of the plane with intercept 3 on the y-axis and parallel to ZOX\nplane 9"
    },
    {
        "Chapter": "1",
        "sentence_range": "6068-6071",
        "Text": "8 Find the equation of the plane with intercept 3 on the y-axis and parallel to ZOX\nplane 9 Find the equation of t he plane through the intersection of the planes\n3x \u2013 y + 2z \u2013 4 = 0 and x + y + z \u2013 2 = 0 and the point (2, 2, 1)"
    },
    {
        "Chapter": "1",
        "sentence_range": "6069-6072",
        "Text": "Find the equation of the plane with intercept 3 on the y-axis and parallel to ZOX\nplane 9 Find the equation of t he plane through the intersection of the planes\n3x \u2013 y + 2z \u2013 4 = 0 and x + y + z \u2013 2 = 0 and the point (2, 2, 1) 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "6070-6073",
        "Text": "9 Find the equation of t he plane through the intersection of the planes\n3x \u2013 y + 2z \u2013 4 = 0 and x + y + z \u2013 2 = 0 and the point (2, 2, 1) 10 Find the vector equation of the plane passing through the intersection of the\nplanes \n\u02c6\n\u02c6\n\u02c6"
    },
    {
        "Chapter": "1",
        "sentence_range": "6071-6074",
        "Text": "Find the equation of t he plane through the intersection of the planes\n3x \u2013 y + 2z \u2013 4 = 0 and x + y + z \u2013 2 = 0 and the point (2, 2, 1) 10 Find the vector equation of the plane passing through the intersection of the\nplanes \n\u02c6\n\u02c6\n\u02c6 (2\n2\n3\n)\n7\nr\ni\nj\nk\n+\n\u2212\n=\nr\n, \n\u02c6\n\u02c6\n\u02c6"
    },
    {
        "Chapter": "1",
        "sentence_range": "6072-6075",
        "Text": "10 Find the vector equation of the plane passing through the intersection of the\nplanes \n\u02c6\n\u02c6\n\u02c6 (2\n2\n3\n)\n7\nr\ni\nj\nk\n+\n\u2212\n=\nr\n, \n\u02c6\n\u02c6\n\u02c6 (2\n5\n3\n)\n9\nr\ni\nj\nk\n+\n+\n=\nr\nand through the point\n(2, 1, 3)"
    },
    {
        "Chapter": "1",
        "sentence_range": "6073-6076",
        "Text": "Find the vector equation of the plane passing through the intersection of the\nplanes \n\u02c6\n\u02c6\n\u02c6 (2\n2\n3\n)\n7\nr\ni\nj\nk\n+\n\u2212\n=\nr\n, \n\u02c6\n\u02c6\n\u02c6 (2\n5\n3\n)\n9\nr\ni\nj\nk\n+\n+\n=\nr\nand through the point\n(2, 1, 3) 11"
    },
    {
        "Chapter": "1",
        "sentence_range": "6074-6077",
        "Text": "(2\n2\n3\n)\n7\nr\ni\nj\nk\n+\n\u2212\n=\nr\n, \n\u02c6\n\u02c6\n\u02c6 (2\n5\n3\n)\n9\nr\ni\nj\nk\n+\n+\n=\nr\nand through the point\n(2, 1, 3) 11 Find the equation of the plane through the line of intersection of the\nplanes x + y + z = 1 and 2x + 3y + 4z = 5 which is perpendicular to the plane\nx \u2013 y + z = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "6075-6078",
        "Text": "(2\n5\n3\n)\n9\nr\ni\nj\nk\n+\n+\n=\nr\nand through the point\n(2, 1, 3) 11 Find the equation of the plane through the line of intersection of the\nplanes x + y + z = 1 and 2x + 3y + 4z = 5 which is perpendicular to the plane\nx \u2013 y + z = 0 \u00a9 NCERT\nnot to be republished\n MATHEMATI CS\n494\n12"
    },
    {
        "Chapter": "1",
        "sentence_range": "6076-6079",
        "Text": "11 Find the equation of the plane through the line of intersection of the\nplanes x + y + z = 1 and 2x + 3y + 4z = 5 which is perpendicular to the plane\nx \u2013 y + z = 0 \u00a9 NCERT\nnot to be republished\n MATHEMATI CS\n494\n12 Find the angle between the planes whose vector equations are\n\u02c6\n\u02c6\n\u02c6\n(2\n2\n3 )\n5\nr\ni\nj\nk\n\u22c5\n+\n\u2212\n=\nr\nand  \n\u02c6\n\u02c6\n\u02c6\n(3\n3\n5 )\n3\nr\ni\nj\nk\n\u22c5\n\u2212\n+\n=\nr"
    },
    {
        "Chapter": "1",
        "sentence_range": "6077-6080",
        "Text": "Find the equation of the plane through the line of intersection of the\nplanes x + y + z = 1 and 2x + 3y + 4z = 5 which is perpendicular to the plane\nx \u2013 y + z = 0 \u00a9 NCERT\nnot to be republished\n MATHEMATI CS\n494\n12 Find the angle between the planes whose vector equations are\n\u02c6\n\u02c6\n\u02c6\n(2\n2\n3 )\n5\nr\ni\nj\nk\n\u22c5\n+\n\u2212\n=\nr\nand  \n\u02c6\n\u02c6\n\u02c6\n(3\n3\n5 )\n3\nr\ni\nj\nk\n\u22c5\n\u2212\n+\n=\nr 13"
    },
    {
        "Chapter": "1",
        "sentence_range": "6078-6081",
        "Text": "\u00a9 NCERT\nnot to be republished\n MATHEMATI CS\n494\n12 Find the angle between the planes whose vector equations are\n\u02c6\n\u02c6\n\u02c6\n(2\n2\n3 )\n5\nr\ni\nj\nk\n\u22c5\n+\n\u2212\n=\nr\nand  \n\u02c6\n\u02c6\n\u02c6\n(3\n3\n5 )\n3\nr\ni\nj\nk\n\u22c5\n\u2212\n+\n=\nr 13 In the following cases, determine whether the given planes are parallel or\nperpendicular, and in case they are neither, find the angles between them"
    },
    {
        "Chapter": "1",
        "sentence_range": "6079-6082",
        "Text": "Find the angle between the planes whose vector equations are\n\u02c6\n\u02c6\n\u02c6\n(2\n2\n3 )\n5\nr\ni\nj\nk\n\u22c5\n+\n\u2212\n=\nr\nand  \n\u02c6\n\u02c6\n\u02c6\n(3\n3\n5 )\n3\nr\ni\nj\nk\n\u22c5\n\u2212\n+\n=\nr 13 In the following cases, determine whether the given planes are parallel or\nperpendicular, and in case they are neither, find the angles between them (a)\n7x + 5y + 6z + 30 = 0 and 3x \u2013 y \u2013 10z + 4 = 0\n(b)\n2x + y + 3z \u2013 2 = 0\nand x \u2013 2y + 5 = 0\n(c)\n2x \u2013 2y + 4z + 5 = 0\nand 3x \u2013 3y + 6z \u2013 1 = 0\n(d)\n2x \u2013 y + 3z \u2013 1 = 0\nand 2x \u2013 y + 3z + 3 = 0\n(e)\n4x + 8y + z \u2013 8 = 0\nand y + z \u2013 4 = 0\n14"
    },
    {
        "Chapter": "1",
        "sentence_range": "6080-6083",
        "Text": "13 In the following cases, determine whether the given planes are parallel or\nperpendicular, and in case they are neither, find the angles between them (a)\n7x + 5y + 6z + 30 = 0 and 3x \u2013 y \u2013 10z + 4 = 0\n(b)\n2x + y + 3z \u2013 2 = 0\nand x \u2013 2y + 5 = 0\n(c)\n2x \u2013 2y + 4z + 5 = 0\nand 3x \u2013 3y + 6z \u2013 1 = 0\n(d)\n2x \u2013 y + 3z \u2013 1 = 0\nand 2x \u2013 y + 3z + 3 = 0\n(e)\n4x + 8y + z \u2013 8 = 0\nand y + z \u2013 4 = 0\n14 In the following cases, find the distance of each of the given points from the\ncorresponding given plane"
    },
    {
        "Chapter": "1",
        "sentence_range": "6081-6084",
        "Text": "In the following cases, determine whether the given planes are parallel or\nperpendicular, and in case they are neither, find the angles between them (a)\n7x + 5y + 6z + 30 = 0 and 3x \u2013 y \u2013 10z + 4 = 0\n(b)\n2x + y + 3z \u2013 2 = 0\nand x \u2013 2y + 5 = 0\n(c)\n2x \u2013 2y + 4z + 5 = 0\nand 3x \u2013 3y + 6z \u2013 1 = 0\n(d)\n2x \u2013 y + 3z \u2013 1 = 0\nand 2x \u2013 y + 3z + 3 = 0\n(e)\n4x + 8y + z \u2013 8 = 0\nand y + z \u2013 4 = 0\n14 In the following cases, find the distance of each of the given points from the\ncorresponding given plane Point\nPlane\n(a)\n(0, 0, 0)\n3x \u2013 4y + 12 z = 3\n(b)\n(3, \u2013 2, 1)\n2x \u2013 y + 2z + 3 = 0\n(c)\n(2, 3, \u2013 5)\nx + 2y \u2013 2z = 9\n(d)\n(\u2013 6, 0, 0)\n2x \u2013 3y + 6z \u2013 2 = 0\nMiscellaneous Examples\nExample 26 A line makes angles  \u03b1,  \u03b2, \u03b3 and  \u03b4 with the diagonals of a cube, prove that\ncos2 \u03b1 + cos2 \u03b2 + cos2 \u03b3 + cos2 \u03b4 = 4\n3\nSolution A cube is a rectangular parallelopiped having equal length, breadth and height"
    },
    {
        "Chapter": "1",
        "sentence_range": "6082-6085",
        "Text": "(a)\n7x + 5y + 6z + 30 = 0 and 3x \u2013 y \u2013 10z + 4 = 0\n(b)\n2x + y + 3z \u2013 2 = 0\nand x \u2013 2y + 5 = 0\n(c)\n2x \u2013 2y + 4z + 5 = 0\nand 3x \u2013 3y + 6z \u2013 1 = 0\n(d)\n2x \u2013 y + 3z \u2013 1 = 0\nand 2x \u2013 y + 3z + 3 = 0\n(e)\n4x + 8y + z \u2013 8 = 0\nand y + z \u2013 4 = 0\n14 In the following cases, find the distance of each of the given points from the\ncorresponding given plane Point\nPlane\n(a)\n(0, 0, 0)\n3x \u2013 4y + 12 z = 3\n(b)\n(3, \u2013 2, 1)\n2x \u2013 y + 2z + 3 = 0\n(c)\n(2, 3, \u2013 5)\nx + 2y \u2013 2z = 9\n(d)\n(\u2013 6, 0, 0)\n2x \u2013 3y + 6z \u2013 2 = 0\nMiscellaneous Examples\nExample 26 A line makes angles  \u03b1,  \u03b2, \u03b3 and  \u03b4 with the diagonals of a cube, prove that\ncos2 \u03b1 + cos2 \u03b2 + cos2 \u03b3 + cos2 \u03b4 = 4\n3\nSolution A cube is a rectangular parallelopiped having equal length, breadth and height Let OADBFEGC be the cube with each side of length a units"
    },
    {
        "Chapter": "1",
        "sentence_range": "6083-6086",
        "Text": "In the following cases, find the distance of each of the given points from the\ncorresponding given plane Point\nPlane\n(a)\n(0, 0, 0)\n3x \u2013 4y + 12 z = 3\n(b)\n(3, \u2013 2, 1)\n2x \u2013 y + 2z + 3 = 0\n(c)\n(2, 3, \u2013 5)\nx + 2y \u2013 2z = 9\n(d)\n(\u2013 6, 0, 0)\n2x \u2013 3y + 6z \u2013 2 = 0\nMiscellaneous Examples\nExample 26 A line makes angles  \u03b1,  \u03b2, \u03b3 and  \u03b4 with the diagonals of a cube, prove that\ncos2 \u03b1 + cos2 \u03b2 + cos2 \u03b3 + cos2 \u03b4 = 4\n3\nSolution A cube is a rectangular parallelopiped having equal length, breadth and height Let OADBFEGC be the cube with each side of length a units (Fig 11"
    },
    {
        "Chapter": "1",
        "sentence_range": "6084-6087",
        "Text": "Point\nPlane\n(a)\n(0, 0, 0)\n3x \u2013 4y + 12 z = 3\n(b)\n(3, \u2013 2, 1)\n2x \u2013 y + 2z + 3 = 0\n(c)\n(2, 3, \u2013 5)\nx + 2y \u2013 2z = 9\n(d)\n(\u2013 6, 0, 0)\n2x \u2013 3y + 6z \u2013 2 = 0\nMiscellaneous Examples\nExample 26 A line makes angles  \u03b1,  \u03b2, \u03b3 and  \u03b4 with the diagonals of a cube, prove that\ncos2 \u03b1 + cos2 \u03b2 + cos2 \u03b3 + cos2 \u03b4 = 4\n3\nSolution A cube is a rectangular parallelopiped having equal length, breadth and height Let OADBFEGC be the cube with each side of length a units (Fig 11 21)\nThe four diagonals are OE, AF, BG and CD"
    },
    {
        "Chapter": "1",
        "sentence_range": "6085-6088",
        "Text": "Let OADBFEGC be the cube with each side of length a units (Fig 11 21)\nThe four diagonals are OE, AF, BG and CD The direction cosines of the diagonal OE which\nis the line joining two points O and E are\n2\n2\n2\n2\n2\n2\n2\n2\n2\n0\n0\n0\n,\n,\na\na\na\na\na\na\na\na\na\na\na\na\n\u2212\n\u2212\n\u2212\n+ +\n+ +\n+ +\ni"
    },
    {
        "Chapter": "1",
        "sentence_range": "6086-6089",
        "Text": "(Fig 11 21)\nThe four diagonals are OE, AF, BG and CD The direction cosines of the diagonal OE which\nis the line joining two points O and E are\n2\n2\n2\n2\n2\n2\n2\n2\n2\n0\n0\n0\n,\n,\na\na\na\na\na\na\na\na\na\na\na\na\n\u2212\n\u2212\n\u2212\n+ +\n+ +\n+ +\ni e"
    },
    {
        "Chapter": "1",
        "sentence_range": "6087-6090",
        "Text": "21)\nThe four diagonals are OE, AF, BG and CD The direction cosines of the diagonal OE which\nis the line joining two points O and E are\n2\n2\n2\n2\n2\n2\n2\n2\n2\n0\n0\n0\n,\n,\na\na\na\na\na\na\na\na\na\na\na\na\n\u2212\n\u2212\n\u2212\n+ +\n+ +\n+ +\ni e ,\n3\n1\n, \n3\n1\n,  \n3\n1\nB(0, , 0)\n a  \nC(0, 0, )\na\na( , 0, )G\na\nF(0, , )\na a\nX\nD( , , 0)\na\na\nY\nZ\nO\nFig 11"
    },
    {
        "Chapter": "1",
        "sentence_range": "6088-6091",
        "Text": "The direction cosines of the diagonal OE which\nis the line joining two points O and E are\n2\n2\n2\n2\n2\n2\n2\n2\n2\n0\n0\n0\n,\n,\na\na\na\na\na\na\na\na\na\na\na\na\n\u2212\n\u2212\n\u2212\n+ +\n+ +\n+ +\ni e ,\n3\n1\n, \n3\n1\n,  \n3\n1\nB(0, , 0)\n a  \nC(0, 0, )\na\na( , 0, )G\na\nF(0, , )\na a\nX\nD( , , 0)\na\na\nY\nZ\nO\nFig 11 21\n\u00a9 NCERT\nnot to be republished\nTHREE D IMENSIONAL G EOMETRY\n495\nSimilarly, the direction cosines of AF, BG and CD are \u20131\n3\n, \n3\n1 , \n3\n1 ; \n3\n1 ,\n\u20131\n3\n, \n3\n1   and  \n3\n1  , \n3\n1 , \u20131\n3\n, respectively"
    },
    {
        "Chapter": "1",
        "sentence_range": "6089-6092",
        "Text": "e ,\n3\n1\n, \n3\n1\n,  \n3\n1\nB(0, , 0)\n a  \nC(0, 0, )\na\na( , 0, )G\na\nF(0, , )\na a\nX\nD( , , 0)\na\na\nY\nZ\nO\nFig 11 21\n\u00a9 NCERT\nnot to be republished\nTHREE D IMENSIONAL G EOMETRY\n495\nSimilarly, the direction cosines of AF, BG and CD are \u20131\n3\n, \n3\n1 , \n3\n1 ; \n3\n1 ,\n\u20131\n3\n, \n3\n1   and  \n3\n1  , \n3\n1 , \u20131\n3\n, respectively Let l, m, n be the direction cosines of the given line which makes angles  \u03b1, \u03b2, \u03b3, \u03b4\nwith OE, AF, BG, CD, respectively"
    },
    {
        "Chapter": "1",
        "sentence_range": "6090-6093",
        "Text": ",\n3\n1\n, \n3\n1\n,  \n3\n1\nB(0, , 0)\n a  \nC(0, 0, )\na\na( , 0, )G\na\nF(0, , )\na a\nX\nD( , , 0)\na\na\nY\nZ\nO\nFig 11 21\n\u00a9 NCERT\nnot to be republished\nTHREE D IMENSIONAL G EOMETRY\n495\nSimilarly, the direction cosines of AF, BG and CD are \u20131\n3\n, \n3\n1 , \n3\n1 ; \n3\n1 ,\n\u20131\n3\n, \n3\n1   and  \n3\n1  , \n3\n1 , \u20131\n3\n, respectively Let l, m, n be the direction cosines of the given line which makes angles  \u03b1, \u03b2, \u03b3, \u03b4\nwith OE, AF, BG, CD, respectively Then\ncos\u03b1  = 1\n3\n (l + m+ n); cos \u03b2 = \n1\n3 (\u2013 l + m + n);\ncos\u03b3 =\n1\n3\n(l \u2013 m + n); cos \u03b4 = \n1\n3 (l + m \u2013 n)     (Why"
    },
    {
        "Chapter": "1",
        "sentence_range": "6091-6094",
        "Text": "21\n\u00a9 NCERT\nnot to be republished\nTHREE D IMENSIONAL G EOMETRY\n495\nSimilarly, the direction cosines of AF, BG and CD are \u20131\n3\n, \n3\n1 , \n3\n1 ; \n3\n1 ,\n\u20131\n3\n, \n3\n1   and  \n3\n1  , \n3\n1 , \u20131\n3\n, respectively Let l, m, n be the direction cosines of the given line which makes angles  \u03b1, \u03b2, \u03b3, \u03b4\nwith OE, AF, BG, CD, respectively Then\ncos\u03b1  = 1\n3\n (l + m+ n); cos \u03b2 = \n1\n3 (\u2013 l + m + n);\ncos\u03b3 =\n1\n3\n(l \u2013 m + n); cos \u03b4 = \n1\n3 (l + m \u2013 n)     (Why )\nSquaring and adding, we get\ncos2\u03b1 + cos2 \u03b2 + cos2 \u03b3 + cos2 \u03b4\n = 1\n3   [ (l + m + n )2 + (\u2013l + m + n)2 ]  + (l \u2013 m + n)2 + (l + m \u2013n)2]\n = 1\n3  [ 4 (l2 + m 2 + n2 ) ]  = 3\n4\n          (as l2 + m 2 + n2 = 1)\nExample 27 Find the equation of the plane that contains the point (1, \u2013 1, 2) and is\nperpendicular to each of the planes 2x + 3y \u2013 2z = 5 and x + 2y \u2013 3z = 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "6092-6095",
        "Text": "Let l, m, n be the direction cosines of the given line which makes angles  \u03b1, \u03b2, \u03b3, \u03b4\nwith OE, AF, BG, CD, respectively Then\ncos\u03b1  = 1\n3\n (l + m+ n); cos \u03b2 = \n1\n3 (\u2013 l + m + n);\ncos\u03b3 =\n1\n3\n(l \u2013 m + n); cos \u03b4 = \n1\n3 (l + m \u2013 n)     (Why )\nSquaring and adding, we get\ncos2\u03b1 + cos2 \u03b2 + cos2 \u03b3 + cos2 \u03b4\n = 1\n3   [ (l + m + n )2 + (\u2013l + m + n)2 ]  + (l \u2013 m + n)2 + (l + m \u2013n)2]\n = 1\n3  [ 4 (l2 + m 2 + n2 ) ]  = 3\n4\n          (as l2 + m 2 + n2 = 1)\nExample 27 Find the equation of the plane that contains the point (1, \u2013 1, 2) and is\nperpendicular to each of the planes 2x + 3y \u2013 2z = 5 and x + 2y \u2013 3z = 8 Solution The equation of the plane containing the given point is\nA (x \u2013 1) + B(y + 1) + C (z \u2013 2) = 0\nApplying the condition of perpendicularly to the plane given in (1) with the planes"
    },
    {
        "Chapter": "1",
        "sentence_range": "6093-6096",
        "Text": "Then\ncos\u03b1  = 1\n3\n (l + m+ n); cos \u03b2 = \n1\n3 (\u2013 l + m + n);\ncos\u03b3 =\n1\n3\n(l \u2013 m + n); cos \u03b4 = \n1\n3 (l + m \u2013 n)     (Why )\nSquaring and adding, we get\ncos2\u03b1 + cos2 \u03b2 + cos2 \u03b3 + cos2 \u03b4\n = 1\n3   [ (l + m + n )2 + (\u2013l + m + n)2 ]  + (l \u2013 m + n)2 + (l + m \u2013n)2]\n = 1\n3  [ 4 (l2 + m 2 + n2 ) ]  = 3\n4\n          (as l2 + m 2 + n2 = 1)\nExample 27 Find the equation of the plane that contains the point (1, \u2013 1, 2) and is\nperpendicular to each of the planes 2x + 3y \u2013 2z = 5 and x + 2y \u2013 3z = 8 Solution The equation of the plane containing the given point is\nA (x \u2013 1) + B(y + 1) + C (z \u2013 2) = 0\nApplying the condition of perpendicularly to the plane given in (1) with the planes (1)\n2x + 3y \u2013 2z = 5 and x + 2y \u2013 3z = 8, we have\n2A + 3B \u2013 2C = 0 and A + 2B \u2013 3C = 0\nSolving these equations, we find A = \u2013 5C and B = 4C"
    },
    {
        "Chapter": "1",
        "sentence_range": "6094-6097",
        "Text": ")\nSquaring and adding, we get\ncos2\u03b1 + cos2 \u03b2 + cos2 \u03b3 + cos2 \u03b4\n = 1\n3   [ (l + m + n )2 + (\u2013l + m + n)2 ]  + (l \u2013 m + n)2 + (l + m \u2013n)2]\n = 1\n3  [ 4 (l2 + m 2 + n2 ) ]  = 3\n4\n          (as l2 + m 2 + n2 = 1)\nExample 27 Find the equation of the plane that contains the point (1, \u2013 1, 2) and is\nperpendicular to each of the planes 2x + 3y \u2013 2z = 5 and x + 2y \u2013 3z = 8 Solution The equation of the plane containing the given point is\nA (x \u2013 1) + B(y + 1) + C (z \u2013 2) = 0\nApplying the condition of perpendicularly to the plane given in (1) with the planes (1)\n2x + 3y \u2013 2z = 5 and x + 2y \u2013 3z = 8, we have\n2A + 3B \u2013 2C = 0 and A + 2B \u2013 3C = 0\nSolving these equations, we find A = \u2013 5C and B = 4C Hence, the required\nequation is\n\u2013 5C (x \u2013 1) + 4 C (y + 1) + C(z \u2013 2) = 0\ni"
    },
    {
        "Chapter": "1",
        "sentence_range": "6095-6098",
        "Text": "Solution The equation of the plane containing the given point is\nA (x \u2013 1) + B(y + 1) + C (z \u2013 2) = 0\nApplying the condition of perpendicularly to the plane given in (1) with the planes (1)\n2x + 3y \u2013 2z = 5 and x + 2y \u2013 3z = 8, we have\n2A + 3B \u2013 2C = 0 and A + 2B \u2013 3C = 0\nSolving these equations, we find A = \u2013 5C and B = 4C Hence, the required\nequation is\n\u2013 5C (x \u2013 1) + 4 C (y + 1) + C(z \u2013 2) = 0\ni e"
    },
    {
        "Chapter": "1",
        "sentence_range": "6096-6099",
        "Text": "(1)\n2x + 3y \u2013 2z = 5 and x + 2y \u2013 3z = 8, we have\n2A + 3B \u2013 2C = 0 and A + 2B \u2013 3C = 0\nSolving these equations, we find A = \u2013 5C and B = 4C Hence, the required\nequation is\n\u2013 5C (x \u2013 1) + 4 C (y + 1) + C(z \u2013 2) = 0\ni e 5x \u2013 4y \u2013 z = 7\nExample 28 Find the distance between the point P(6, 5, 9) and the plane determined\nby the points A (3, \u2013 1, 2), B (5, 2, 4) and C(\u2013 1, \u2013 1, 6)"
    },
    {
        "Chapter": "1",
        "sentence_range": "6097-6100",
        "Text": "Hence, the required\nequation is\n\u2013 5C (x \u2013 1) + 4 C (y + 1) + C(z \u2013 2) = 0\ni e 5x \u2013 4y \u2013 z = 7\nExample 28 Find the distance between the point P(6, 5, 9) and the plane determined\nby the points A (3, \u2013 1, 2), B (5, 2, 4) and C(\u2013 1, \u2013 1, 6) Solution Let A, B, C be the three points in the plane"
    },
    {
        "Chapter": "1",
        "sentence_range": "6098-6101",
        "Text": "e 5x \u2013 4y \u2013 z = 7\nExample 28 Find the distance between the point P(6, 5, 9) and the plane determined\nby the points A (3, \u2013 1, 2), B (5, 2, 4) and C(\u2013 1, \u2013 1, 6) Solution Let A, B, C be the three points in the plane D is the foot of the perpendicular\ndrawn from a point P to the plane"
    },
    {
        "Chapter": "1",
        "sentence_range": "6099-6102",
        "Text": "5x \u2013 4y \u2013 z = 7\nExample 28 Find the distance between the point P(6, 5, 9) and the plane determined\nby the points A (3, \u2013 1, 2), B (5, 2, 4) and C(\u2013 1, \u2013 1, 6) Solution Let A, B, C be the three points in the plane D is the foot of the perpendicular\ndrawn from a point P to the plane PD is the required distance to be determined, which\nis the projection of AP\nuuur\n on AB\nAC\n\u00d7\nuuur\nuuur"
    },
    {
        "Chapter": "1",
        "sentence_range": "6100-6103",
        "Text": "Solution Let A, B, C be the three points in the plane D is the foot of the perpendicular\ndrawn from a point P to the plane PD is the required distance to be determined, which\nis the projection of AP\nuuur\n on AB\nAC\n\u00d7\nuuur\nuuur \u00a9 NCERT\nnot to be republished\n MATHEMATI CS\n496\nHence, PD = the dot product of AP\nuuur\n  with the unit vector along AB\nAC\n\u00d7\nuuur\nuuur"
    },
    {
        "Chapter": "1",
        "sentence_range": "6101-6104",
        "Text": "D is the foot of the perpendicular\ndrawn from a point P to the plane PD is the required distance to be determined, which\nis the projection of AP\nuuur\n on AB\nAC\n\u00d7\nuuur\nuuur \u00a9 NCERT\nnot to be republished\n MATHEMATI CS\n496\nHence, PD = the dot product of AP\nuuur\n  with the unit vector along AB\nAC\n\u00d7\nuuur\nuuur So\nAP\nuuur\n= 3 \nk\nj\ni\n7\u02c6\n6\u02c6\n\u02c6\n+\n+\nand\nAB\nAC\n\u00d7\nuuur\nuuur\n=\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n2\n3\n2\n12\n16\n12\n4\n0\n4\ni\nj\nk\ni\nj\nk\n=\n\u2212\n+\n\u2212\nUnit vector along AB\nAC\n\u00d7\nuuur\nuuur\n=\n\u02c6\n\u02c6\n\u02c6\n3\n4\n3\n34\ni\nj\nk\n\u2212\n+\nHence\nPD = (\n)\u02c6\n7\n6\u02c6\n3\u02c6\nk\nj\ni\n+\n+"
    },
    {
        "Chapter": "1",
        "sentence_range": "6102-6105",
        "Text": "PD is the required distance to be determined, which\nis the projection of AP\nuuur\n on AB\nAC\n\u00d7\nuuur\nuuur \u00a9 NCERT\nnot to be republished\n MATHEMATI CS\n496\nHence, PD = the dot product of AP\nuuur\n  with the unit vector along AB\nAC\n\u00d7\nuuur\nuuur So\nAP\nuuur\n= 3 \nk\nj\ni\n7\u02c6\n6\u02c6\n\u02c6\n+\n+\nand\nAB\nAC\n\u00d7\nuuur\nuuur\n=\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n2\n3\n2\n12\n16\n12\n4\n0\n4\ni\nj\nk\ni\nj\nk\n=\n\u2212\n+\n\u2212\nUnit vector along AB\nAC\n\u00d7\nuuur\nuuur\n=\n\u02c6\n\u02c6\n\u02c6\n3\n4\n3\n34\ni\nj\nk\n\u2212\n+\nHence\nPD = (\n)\u02c6\n7\n6\u02c6\n3\u02c6\nk\nj\ni\n+\n+ \u02c6\n\u02c6\n\u02c6\n3\n4\n3\n34\ni\nj\nk\n\u2212\n+\n=\n17\n34\n3\nAlternatively, find the equation of the plane passing through A, B and C and then\ncompute the distance of the point P from the plane"
    },
    {
        "Chapter": "1",
        "sentence_range": "6103-6106",
        "Text": "\u00a9 NCERT\nnot to be republished\n MATHEMATI CS\n496\nHence, PD = the dot product of AP\nuuur\n  with the unit vector along AB\nAC\n\u00d7\nuuur\nuuur So\nAP\nuuur\n= 3 \nk\nj\ni\n7\u02c6\n6\u02c6\n\u02c6\n+\n+\nand\nAB\nAC\n\u00d7\nuuur\nuuur\n=\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n2\n3\n2\n12\n16\n12\n4\n0\n4\ni\nj\nk\ni\nj\nk\n=\n\u2212\n+\n\u2212\nUnit vector along AB\nAC\n\u00d7\nuuur\nuuur\n=\n\u02c6\n\u02c6\n\u02c6\n3\n4\n3\n34\ni\nj\nk\n\u2212\n+\nHence\nPD = (\n)\u02c6\n7\n6\u02c6\n3\u02c6\nk\nj\ni\n+\n+ \u02c6\n\u02c6\n\u02c6\n3\n4\n3\n34\ni\nj\nk\n\u2212\n+\n=\n17\n34\n3\nAlternatively, find the equation of the plane passing through A, B and C and then\ncompute the distance of the point P from the plane Example 29 Show that the lines\nx\na\nd\n\u2212\n+\n\u03b1 \u2212 \u03b4\n = y\na\nz\na\nd\n\u2212\n\u2212\n\u2212\n=\n\u03b1\n\u03b1 + \u03b4\nand\nx\nb\nc\n\u2212\n+\n\u03b2 \u2212 \u03b3\n = y\nb\nz\nb\nc\n\u2212\n\u2212\n\u2212\n=\n\u03b2\n\u03b2 + \u03b3\n are coplanar"
    },
    {
        "Chapter": "1",
        "sentence_range": "6104-6107",
        "Text": "So\nAP\nuuur\n= 3 \nk\nj\ni\n7\u02c6\n6\u02c6\n\u02c6\n+\n+\nand\nAB\nAC\n\u00d7\nuuur\nuuur\n=\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n2\n3\n2\n12\n16\n12\n4\n0\n4\ni\nj\nk\ni\nj\nk\n=\n\u2212\n+\n\u2212\nUnit vector along AB\nAC\n\u00d7\nuuur\nuuur\n=\n\u02c6\n\u02c6\n\u02c6\n3\n4\n3\n34\ni\nj\nk\n\u2212\n+\nHence\nPD = (\n)\u02c6\n7\n6\u02c6\n3\u02c6\nk\nj\ni\n+\n+ \u02c6\n\u02c6\n\u02c6\n3\n4\n3\n34\ni\nj\nk\n\u2212\n+\n=\n17\n34\n3\nAlternatively, find the equation of the plane passing through A, B and C and then\ncompute the distance of the point P from the plane Example 29 Show that the lines\nx\na\nd\n\u2212\n+\n\u03b1 \u2212 \u03b4\n = y\na\nz\na\nd\n\u2212\n\u2212\n\u2212\n=\n\u03b1\n\u03b1 + \u03b4\nand\nx\nb\nc\n\u2212\n+\n\u03b2 \u2212 \u03b3\n = y\nb\nz\nb\nc\n\u2212\n\u2212\n\u2212\n=\n\u03b2\n\u03b2 + \u03b3\n are coplanar Solution\nHere\nx1 = a \u2013 d\nx2 = b \u2013 c\ny1 = a\ny2 = b\nz1 = a + d\nz2 = b + c\na1 = \u03b1 \u2013 \u03b4\na2 = \u03b2 \u2013 \u03b3\nb1 = \u03b1\nb2 = \u03b2\nc1 = \u03b1 + \u03b4\nc2 = \u03b2 + \u03b3\nNow consider the determinant\n2\n1\n2\n1\n2\n1\n1\n1\n1\n2\n2\n2\nx\nx\ny\ny\nz\nz\na\nb\nc\na\nb\nc\n\u2212\n\u2212\n\u2212\n = \nb\nc\na\nd\nb\na\nb\nc\na\nd\n\u2212 \u2212\n+\n\u2212\n+\n\u2212\n\u2212\n\u03b1 \u2212 \u03b4\n\u03b1\n\u03b1 + \u03b4\n\u03b2 \u2212 \u03b3\n\u03b2\n\u03b2 + \u03b3\n\u00a9 NCERT\nnot to be republished\nTHREE D IMENSIONAL G EOMETRY\n497\nAdding third column to the first column, we get\n2 \nb\na\nb\na\nb\nc\na\nd\n\u2212\n\u2212\n+\n\u2212\n\u2212\n\u03b1\n\u03b1\n\u03b1 + \u03b4\n\u03b2\n\u03b2\n\u03b2 + \u03b3\n = 0\nSince the first and second columns are identical"
    },
    {
        "Chapter": "1",
        "sentence_range": "6105-6108",
        "Text": "\u02c6\n\u02c6\n\u02c6\n3\n4\n3\n34\ni\nj\nk\n\u2212\n+\n=\n17\n34\n3\nAlternatively, find the equation of the plane passing through A, B and C and then\ncompute the distance of the point P from the plane Example 29 Show that the lines\nx\na\nd\n\u2212\n+\n\u03b1 \u2212 \u03b4\n = y\na\nz\na\nd\n\u2212\n\u2212\n\u2212\n=\n\u03b1\n\u03b1 + \u03b4\nand\nx\nb\nc\n\u2212\n+\n\u03b2 \u2212 \u03b3\n = y\nb\nz\nb\nc\n\u2212\n\u2212\n\u2212\n=\n\u03b2\n\u03b2 + \u03b3\n are coplanar Solution\nHere\nx1 = a \u2013 d\nx2 = b \u2013 c\ny1 = a\ny2 = b\nz1 = a + d\nz2 = b + c\na1 = \u03b1 \u2013 \u03b4\na2 = \u03b2 \u2013 \u03b3\nb1 = \u03b1\nb2 = \u03b2\nc1 = \u03b1 + \u03b4\nc2 = \u03b2 + \u03b3\nNow consider the determinant\n2\n1\n2\n1\n2\n1\n1\n1\n1\n2\n2\n2\nx\nx\ny\ny\nz\nz\na\nb\nc\na\nb\nc\n\u2212\n\u2212\n\u2212\n = \nb\nc\na\nd\nb\na\nb\nc\na\nd\n\u2212 \u2212\n+\n\u2212\n+\n\u2212\n\u2212\n\u03b1 \u2212 \u03b4\n\u03b1\n\u03b1 + \u03b4\n\u03b2 \u2212 \u03b3\n\u03b2\n\u03b2 + \u03b3\n\u00a9 NCERT\nnot to be republished\nTHREE D IMENSIONAL G EOMETRY\n497\nAdding third column to the first column, we get\n2 \nb\na\nb\na\nb\nc\na\nd\n\u2212\n\u2212\n+\n\u2212\n\u2212\n\u03b1\n\u03b1\n\u03b1 + \u03b4\n\u03b2\n\u03b2\n\u03b2 + \u03b3\n = 0\nSince the first and second columns are identical Hence, the given two lines are\ncoplanar"
    },
    {
        "Chapter": "1",
        "sentence_range": "6106-6109",
        "Text": "Example 29 Show that the lines\nx\na\nd\n\u2212\n+\n\u03b1 \u2212 \u03b4\n = y\na\nz\na\nd\n\u2212\n\u2212\n\u2212\n=\n\u03b1\n\u03b1 + \u03b4\nand\nx\nb\nc\n\u2212\n+\n\u03b2 \u2212 \u03b3\n = y\nb\nz\nb\nc\n\u2212\n\u2212\n\u2212\n=\n\u03b2\n\u03b2 + \u03b3\n are coplanar Solution\nHere\nx1 = a \u2013 d\nx2 = b \u2013 c\ny1 = a\ny2 = b\nz1 = a + d\nz2 = b + c\na1 = \u03b1 \u2013 \u03b4\na2 = \u03b2 \u2013 \u03b3\nb1 = \u03b1\nb2 = \u03b2\nc1 = \u03b1 + \u03b4\nc2 = \u03b2 + \u03b3\nNow consider the determinant\n2\n1\n2\n1\n2\n1\n1\n1\n1\n2\n2\n2\nx\nx\ny\ny\nz\nz\na\nb\nc\na\nb\nc\n\u2212\n\u2212\n\u2212\n = \nb\nc\na\nd\nb\na\nb\nc\na\nd\n\u2212 \u2212\n+\n\u2212\n+\n\u2212\n\u2212\n\u03b1 \u2212 \u03b4\n\u03b1\n\u03b1 + \u03b4\n\u03b2 \u2212 \u03b3\n\u03b2\n\u03b2 + \u03b3\n\u00a9 NCERT\nnot to be republished\nTHREE D IMENSIONAL G EOMETRY\n497\nAdding third column to the first column, we get\n2 \nb\na\nb\na\nb\nc\na\nd\n\u2212\n\u2212\n+\n\u2212\n\u2212\n\u03b1\n\u03b1\n\u03b1 + \u03b4\n\u03b2\n\u03b2\n\u03b2 + \u03b3\n = 0\nSince the first and second columns are identical Hence, the given two lines are\ncoplanar Example 30 Find the coordinates of the point where the line through the points\nA (3, 4, 1) and B(5, 1, 6) crosses the XY-plane"
    },
    {
        "Chapter": "1",
        "sentence_range": "6107-6110",
        "Text": "Solution\nHere\nx1 = a \u2013 d\nx2 = b \u2013 c\ny1 = a\ny2 = b\nz1 = a + d\nz2 = b + c\na1 = \u03b1 \u2013 \u03b4\na2 = \u03b2 \u2013 \u03b3\nb1 = \u03b1\nb2 = \u03b2\nc1 = \u03b1 + \u03b4\nc2 = \u03b2 + \u03b3\nNow consider the determinant\n2\n1\n2\n1\n2\n1\n1\n1\n1\n2\n2\n2\nx\nx\ny\ny\nz\nz\na\nb\nc\na\nb\nc\n\u2212\n\u2212\n\u2212\n = \nb\nc\na\nd\nb\na\nb\nc\na\nd\n\u2212 \u2212\n+\n\u2212\n+\n\u2212\n\u2212\n\u03b1 \u2212 \u03b4\n\u03b1\n\u03b1 + \u03b4\n\u03b2 \u2212 \u03b3\n\u03b2\n\u03b2 + \u03b3\n\u00a9 NCERT\nnot to be republished\nTHREE D IMENSIONAL G EOMETRY\n497\nAdding third column to the first column, we get\n2 \nb\na\nb\na\nb\nc\na\nd\n\u2212\n\u2212\n+\n\u2212\n\u2212\n\u03b1\n\u03b1\n\u03b1 + \u03b4\n\u03b2\n\u03b2\n\u03b2 + \u03b3\n = 0\nSince the first and second columns are identical Hence, the given two lines are\ncoplanar Example 30 Find the coordinates of the point where the line through the points\nA (3, 4, 1) and B(5, 1, 6) crosses the XY-plane Solution The vector equation of the line through the points A and B is\n rr  =\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n3\n4\n[ (5\n3)\n(1\n4)\n(6\n1)\n]\ni\nj\nk\ni\nj\nk\n+\n+\n+ \u03bb\n\u2212\n+\n\u2212\n+\n\u2212\ni"
    },
    {
        "Chapter": "1",
        "sentence_range": "6108-6111",
        "Text": "Hence, the given two lines are\ncoplanar Example 30 Find the coordinates of the point where the line through the points\nA (3, 4, 1) and B(5, 1, 6) crosses the XY-plane Solution The vector equation of the line through the points A and B is\n rr  =\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n3\n4\n[ (5\n3)\n(1\n4)\n(6\n1)\n]\ni\nj\nk\ni\nj\nk\n+\n+\n+ \u03bb\n\u2212\n+\n\u2212\n+\n\u2212\ni e"
    },
    {
        "Chapter": "1",
        "sentence_range": "6109-6112",
        "Text": "Example 30 Find the coordinates of the point where the line through the points\nA (3, 4, 1) and B(5, 1, 6) crosses the XY-plane Solution The vector equation of the line through the points A and B is\n rr  =\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n3\n4\n[ (5\n3)\n(1\n4)\n(6\n1)\n]\ni\nj\nk\ni\nj\nk\n+\n+\n+ \u03bb\n\u2212\n+\n\u2212\n+\n\u2212\ni e rr  =\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n3\n4\n( 2\n3\n5\n)\ni\nj\nk\ni\nj\nk\n+\n+\n+ \u03bb\n\u2212\n+"
    },
    {
        "Chapter": "1",
        "sentence_range": "6110-6113",
        "Text": "Solution The vector equation of the line through the points A and B is\n rr  =\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n3\n4\n[ (5\n3)\n(1\n4)\n(6\n1)\n]\ni\nj\nk\ni\nj\nk\n+\n+\n+ \u03bb\n\u2212\n+\n\u2212\n+\n\u2212\ni e rr  =\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n3\n4\n( 2\n3\n5\n)\ni\nj\nk\ni\nj\nk\n+\n+\n+ \u03bb\n\u2212\n+ (1)\nLet P be the point where the line AB crosses the XY-plane"
    },
    {
        "Chapter": "1",
        "sentence_range": "6111-6114",
        "Text": "e rr  =\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n3\n4\n( 2\n3\n5\n)\ni\nj\nk\ni\nj\nk\n+\n+\n+ \u03bb\n\u2212\n+ (1)\nLet P be the point where the line AB crosses the XY-plane Then the position\nvector of the point P is of the form \nyj\nxi\n\u02c6\n\u02c6 +"
    },
    {
        "Chapter": "1",
        "sentence_range": "6112-6115",
        "Text": "rr  =\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n3\n4\n( 2\n3\n5\n)\ni\nj\nk\ni\nj\nk\n+\n+\n+ \u03bb\n\u2212\n+ (1)\nLet P be the point where the line AB crosses the XY-plane Then the position\nvector of the point P is of the form \nyj\nxi\n\u02c6\n\u02c6 + This point must satisfy the equation (1)"
    },
    {
        "Chapter": "1",
        "sentence_range": "6113-6116",
        "Text": "(1)\nLet P be the point where the line AB crosses the XY-plane Then the position\nvector of the point P is of the form \nyj\nxi\n\u02c6\n\u02c6 + This point must satisfy the equation (1) (Why"
    },
    {
        "Chapter": "1",
        "sentence_range": "6114-6117",
        "Text": "Then the position\nvector of the point P is of the form \nyj\nxi\n\u02c6\n\u02c6 + This point must satisfy the equation (1) (Why )\ni"
    },
    {
        "Chapter": "1",
        "sentence_range": "6115-6118",
        "Text": "This point must satisfy the equation (1) (Why )\ni e"
    },
    {
        "Chapter": "1",
        "sentence_range": "6116-6119",
        "Text": "(Why )\ni e \u02c6\n\u02c6\nx i\n+y j\n =\n\u02c6\n\u02c6\n\u02c6\n(3\n2\n)\n( 4\n3 )\n(1\n5\n)\ni\nj\nk\n+\n\u03bb\n+\n\u2212\n\u03bb\n+\n+\n\u03bb\nEquating the like coefficients of \n\u02c6\n\u02c6\n,\u02c6\nand\ni\nj\nk , we have\nx = 3 + 2 \u03bb\ny = 4 \u2013 3 \u03bb\n0 = 1 + 5 \u03bb\nSolving the above equations, we get\nx = 13\n23\nand\n5\n5\ny =\nHence, the coordinates of the required point are \n\uf8f6\uf8f8\uf8f7\n\uf8eb\uf8ed\uf8ec\n5,0\n135,23"
    },
    {
        "Chapter": "1",
        "sentence_range": "6117-6120",
        "Text": ")\ni e \u02c6\n\u02c6\nx i\n+y j\n =\n\u02c6\n\u02c6\n\u02c6\n(3\n2\n)\n( 4\n3 )\n(1\n5\n)\ni\nj\nk\n+\n\u03bb\n+\n\u2212\n\u03bb\n+\n+\n\u03bb\nEquating the like coefficients of \n\u02c6\n\u02c6\n,\u02c6\nand\ni\nj\nk , we have\nx = 3 + 2 \u03bb\ny = 4 \u2013 3 \u03bb\n0 = 1 + 5 \u03bb\nSolving the above equations, we get\nx = 13\n23\nand\n5\n5\ny =\nHence, the coordinates of the required point are \n\uf8f6\uf8f8\uf8f7\n\uf8eb\uf8ed\uf8ec\n5,0\n135,23 Miscellaneous Exercise on Chapter 11\n1"
    },
    {
        "Chapter": "1",
        "sentence_range": "6118-6121",
        "Text": "e \u02c6\n\u02c6\nx i\n+y j\n =\n\u02c6\n\u02c6\n\u02c6\n(3\n2\n)\n( 4\n3 )\n(1\n5\n)\ni\nj\nk\n+\n\u03bb\n+\n\u2212\n\u03bb\n+\n+\n\u03bb\nEquating the like coefficients of \n\u02c6\n\u02c6\n,\u02c6\nand\ni\nj\nk , we have\nx = 3 + 2 \u03bb\ny = 4 \u2013 3 \u03bb\n0 = 1 + 5 \u03bb\nSolving the above equations, we get\nx = 13\n23\nand\n5\n5\ny =\nHence, the coordinates of the required point are \n\uf8f6\uf8f8\uf8f7\n\uf8eb\uf8ed\uf8ec\n5,0\n135,23 Miscellaneous Exercise on Chapter 11\n1 Show that the line joining the origin to the point (2, 1, 1) is perpendicular to the\nline determined by the points (3, 5, \u2013 1), (4, 3, \u2013 1)"
    },
    {
        "Chapter": "1",
        "sentence_range": "6119-6122",
        "Text": "\u02c6\n\u02c6\nx i\n+y j\n =\n\u02c6\n\u02c6\n\u02c6\n(3\n2\n)\n( 4\n3 )\n(1\n5\n)\ni\nj\nk\n+\n\u03bb\n+\n\u2212\n\u03bb\n+\n+\n\u03bb\nEquating the like coefficients of \n\u02c6\n\u02c6\n,\u02c6\nand\ni\nj\nk , we have\nx = 3 + 2 \u03bb\ny = 4 \u2013 3 \u03bb\n0 = 1 + 5 \u03bb\nSolving the above equations, we get\nx = 13\n23\nand\n5\n5\ny =\nHence, the coordinates of the required point are \n\uf8f6\uf8f8\uf8f7\n\uf8eb\uf8ed\uf8ec\n5,0\n135,23 Miscellaneous Exercise on Chapter 11\n1 Show that the line joining the origin to the point (2, 1, 1) is perpendicular to the\nline determined by the points (3, 5, \u2013 1), (4, 3, \u2013 1) 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "6120-6123",
        "Text": "Miscellaneous Exercise on Chapter 11\n1 Show that the line joining the origin to the point (2, 1, 1) is perpendicular to the\nline determined by the points (3, 5, \u2013 1), (4, 3, \u2013 1) 2 If l1, m 1, n1 and l2, m 2, n2 are the direction cosines of two mutually perpendicular\nlines, show that the direction cosines of the line perpendicular to both of these\nare  \n1\n2\n2\n1\n1\n2\n2\n1\n1\n2\n2\n1\n,\n,\nm n\nm n\nn l\nn l\nl m\nl m\n\u2212\n\u2212\n\u2212\n\u00a9 NCERT\nnot to be republished\n MATHEMATI CS\n498\n3"
    },
    {
        "Chapter": "1",
        "sentence_range": "6121-6124",
        "Text": "Show that the line joining the origin to the point (2, 1, 1) is perpendicular to the\nline determined by the points (3, 5, \u2013 1), (4, 3, \u2013 1) 2 If l1, m 1, n1 and l2, m 2, n2 are the direction cosines of two mutually perpendicular\nlines, show that the direction cosines of the line perpendicular to both of these\nare  \n1\n2\n2\n1\n1\n2\n2\n1\n1\n2\n2\n1\n,\n,\nm n\nm n\nn l\nn l\nl m\nl m\n\u2212\n\u2212\n\u2212\n\u00a9 NCERT\nnot to be republished\n MATHEMATI CS\n498\n3 Find the angle between the lines whose direction ratios are a, b, c and\nb \u2013 c, c \u2013 a, a \u2013 b"
    },
    {
        "Chapter": "1",
        "sentence_range": "6122-6125",
        "Text": "2 If l1, m 1, n1 and l2, m 2, n2 are the direction cosines of two mutually perpendicular\nlines, show that the direction cosines of the line perpendicular to both of these\nare  \n1\n2\n2\n1\n1\n2\n2\n1\n1\n2\n2\n1\n,\n,\nm n\nm n\nn l\nn l\nl m\nl m\n\u2212\n\u2212\n\u2212\n\u00a9 NCERT\nnot to be republished\n MATHEMATI CS\n498\n3 Find the angle between the lines whose direction ratios are a, b, c and\nb \u2013 c, c \u2013 a, a \u2013 b 4"
    },
    {
        "Chapter": "1",
        "sentence_range": "6123-6126",
        "Text": "If l1, m 1, n1 and l2, m 2, n2 are the direction cosines of two mutually perpendicular\nlines, show that the direction cosines of the line perpendicular to both of these\nare  \n1\n2\n2\n1\n1\n2\n2\n1\n1\n2\n2\n1\n,\n,\nm n\nm n\nn l\nn l\nl m\nl m\n\u2212\n\u2212\n\u2212\n\u00a9 NCERT\nnot to be republished\n MATHEMATI CS\n498\n3 Find the angle between the lines whose direction ratios are a, b, c and\nb \u2013 c, c \u2013 a, a \u2013 b 4 Find the equation of a line parallel to x-axis and passing through the origin"
    },
    {
        "Chapter": "1",
        "sentence_range": "6124-6127",
        "Text": "Find the angle between the lines whose direction ratios are a, b, c and\nb \u2013 c, c \u2013 a, a \u2013 b 4 Find the equation of a line parallel to x-axis and passing through the origin 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "6125-6128",
        "Text": "4 Find the equation of a line parallel to x-axis and passing through the origin 5 If the coordinates of the points A, B, C, D be (1, 2, 3), (4, 5, 7), (\u2013 4, 3, \u2013 6) and\n(2, 9, 2) respectively, then find the angle between the lines AB and CD"
    },
    {
        "Chapter": "1",
        "sentence_range": "6126-6129",
        "Text": "Find the equation of a line parallel to x-axis and passing through the origin 5 If the coordinates of the points A, B, C, D be (1, 2, 3), (4, 5, 7), (\u2013 4, 3, \u2013 6) and\n(2, 9, 2) respectively, then find the angle between the lines AB and CD 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "6127-6130",
        "Text": "5 If the coordinates of the points A, B, C, D be (1, 2, 3), (4, 5, 7), (\u2013 4, 3, \u2013 6) and\n(2, 9, 2) respectively, then find the angle between the lines AB and CD 6 If the lines \n1\n2\n3\n1\n1\n6\nand\n3\n2\n2\n3\n1\n5\nx\ny\nz\nx\ny\nz\nk\nk\n\u2212\n\u2212\n\u2212\n\u2212\n\u2212\n\u2212\n=\n=\n=\n=\n\u2212\n\u2212\n are perpendicular,,\nfind the value of k"
    },
    {
        "Chapter": "1",
        "sentence_range": "6128-6131",
        "Text": "If the coordinates of the points A, B, C, D be (1, 2, 3), (4, 5, 7), (\u2013 4, 3, \u2013 6) and\n(2, 9, 2) respectively, then find the angle between the lines AB and CD 6 If the lines \n1\n2\n3\n1\n1\n6\nand\n3\n2\n2\n3\n1\n5\nx\ny\nz\nx\ny\nz\nk\nk\n\u2212\n\u2212\n\u2212\n\u2212\n\u2212\n\u2212\n=\n=\n=\n=\n\u2212\n\u2212\n are perpendicular,,\nfind the value of k 7"
    },
    {
        "Chapter": "1",
        "sentence_range": "6129-6132",
        "Text": "6 If the lines \n1\n2\n3\n1\n1\n6\nand\n3\n2\n2\n3\n1\n5\nx\ny\nz\nx\ny\nz\nk\nk\n\u2212\n\u2212\n\u2212\n\u2212\n\u2212\n\u2212\n=\n=\n=\n=\n\u2212\n\u2212\n are perpendicular,,\nfind the value of k 7 Find the vector equation of the line passing through (1, 2, 3) and perpendicular to\nthe plane \n0\n9\n)\u02c6\n5\n2\u02c6\n\u02c6\n("
    },
    {
        "Chapter": "1",
        "sentence_range": "6130-6133",
        "Text": "If the lines \n1\n2\n3\n1\n1\n6\nand\n3\n2\n2\n3\n1\n5\nx\ny\nz\nx\ny\nz\nk\nk\n\u2212\n\u2212\n\u2212\n\u2212\n\u2212\n\u2212\n=\n=\n=\n=\n\u2212\n\u2212\n are perpendicular,,\nfind the value of k 7 Find the vector equation of the line passing through (1, 2, 3) and perpendicular to\nthe plane \n0\n9\n)\u02c6\n5\n2\u02c6\n\u02c6\n( =\n+\n\u2212\n+\nk\nj\ni\nrr"
    },
    {
        "Chapter": "1",
        "sentence_range": "6131-6134",
        "Text": "7 Find the vector equation of the line passing through (1, 2, 3) and perpendicular to\nthe plane \n0\n9\n)\u02c6\n5\n2\u02c6\n\u02c6\n( =\n+\n\u2212\n+\nk\nj\ni\nrr 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "6132-6135",
        "Text": "Find the vector equation of the line passing through (1, 2, 3) and perpendicular to\nthe plane \n0\n9\n)\u02c6\n5\n2\u02c6\n\u02c6\n( =\n+\n\u2212\n+\nk\nj\ni\nrr 8 Find the equation of the plane passing through (a, b, c) and parallel to the plane\n\u02c6\n\u02c6\n\u02c6\n(\n)\n2"
    },
    {
        "Chapter": "1",
        "sentence_range": "6133-6136",
        "Text": "=\n+\n\u2212\n+\nk\nj\ni\nrr 8 Find the equation of the plane passing through (a, b, c) and parallel to the plane\n\u02c6\n\u02c6\n\u02c6\n(\n)\n2 r\ni\nj\nk\n\u22c5 + + =\nr\n9"
    },
    {
        "Chapter": "1",
        "sentence_range": "6134-6137",
        "Text": "8 Find the equation of the plane passing through (a, b, c) and parallel to the plane\n\u02c6\n\u02c6\n\u02c6\n(\n)\n2 r\ni\nj\nk\n\u22c5 + + =\nr\n9 Find the shortest distance between lines \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n6\n2\n2\n(\n2\n2\n)\nr\ni\nj\nk\ni\nj\nk\n=\n+\n+\n+ \u03bb\n\u2212\n+\nr\nand \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n4\n(3\n2\n2\n)\nr\ni\nk\ni\nj\nk\n= \u2212\n\u2212\n+ \u00b5\n\u2212\n\u2212\nr"
    },
    {
        "Chapter": "1",
        "sentence_range": "6135-6138",
        "Text": "Find the equation of the plane passing through (a, b, c) and parallel to the plane\n\u02c6\n\u02c6\n\u02c6\n(\n)\n2 r\ni\nj\nk\n\u22c5 + + =\nr\n9 Find the shortest distance between lines \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n6\n2\n2\n(\n2\n2\n)\nr\ni\nj\nk\ni\nj\nk\n=\n+\n+\n+ \u03bb\n\u2212\n+\nr\nand \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n4\n(3\n2\n2\n)\nr\ni\nk\ni\nj\nk\n= \u2212\n\u2212\n+ \u00b5\n\u2212\n\u2212\nr 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "6136-6139",
        "Text": "r\ni\nj\nk\n\u22c5 + + =\nr\n9 Find the shortest distance between lines \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n6\n2\n2\n(\n2\n2\n)\nr\ni\nj\nk\ni\nj\nk\n=\n+\n+\n+ \u03bb\n\u2212\n+\nr\nand \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n4\n(3\n2\n2\n)\nr\ni\nk\ni\nj\nk\n= \u2212\n\u2212\n+ \u00b5\n\u2212\n\u2212\nr 10 Find the coordinates of the point where the line through (5, 1, 6) and (3, 4,1)\ncrosses the YZ-plane"
    },
    {
        "Chapter": "1",
        "sentence_range": "6137-6140",
        "Text": "Find the shortest distance between lines \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n6\n2\n2\n(\n2\n2\n)\nr\ni\nj\nk\ni\nj\nk\n=\n+\n+\n+ \u03bb\n\u2212\n+\nr\nand \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n4\n(3\n2\n2\n)\nr\ni\nk\ni\nj\nk\n= \u2212\n\u2212\n+ \u00b5\n\u2212\n\u2212\nr 10 Find the coordinates of the point where the line through (5, 1, 6) and (3, 4,1)\ncrosses the YZ-plane 11"
    },
    {
        "Chapter": "1",
        "sentence_range": "6138-6141",
        "Text": "10 Find the coordinates of the point where the line through (5, 1, 6) and (3, 4,1)\ncrosses the YZ-plane 11 Find the coordinates of the point where the line through (5, 1, 6) and (3, 4, 1)\ncrosses the ZX-plane"
    },
    {
        "Chapter": "1",
        "sentence_range": "6139-6142",
        "Text": "Find the coordinates of the point where the line through (5, 1, 6) and (3, 4,1)\ncrosses the YZ-plane 11 Find the coordinates of the point where the line through (5, 1, 6) and (3, 4, 1)\ncrosses the ZX-plane 12"
    },
    {
        "Chapter": "1",
        "sentence_range": "6140-6143",
        "Text": "11 Find the coordinates of the point where the line through (5, 1, 6) and (3, 4, 1)\ncrosses the ZX-plane 12 Find the coordinates of the point where the line through (3, \u2013 4, \u2013 5) and\n(2, \u2013 3, 1) crosses the plane 2x + y + z = 7"
    },
    {
        "Chapter": "1",
        "sentence_range": "6141-6144",
        "Text": "Find the coordinates of the point where the line through (5, 1, 6) and (3, 4, 1)\ncrosses the ZX-plane 12 Find the coordinates of the point where the line through (3, \u2013 4, \u2013 5) and\n(2, \u2013 3, 1) crosses the plane 2x + y + z = 7 13"
    },
    {
        "Chapter": "1",
        "sentence_range": "6142-6145",
        "Text": "12 Find the coordinates of the point where the line through (3, \u2013 4, \u2013 5) and\n(2, \u2013 3, 1) crosses the plane 2x + y + z = 7 13 Find the equation of the plane passing through the point (\u2013 1, 3, 2) and perpendicular\nto each of the planes x + 2y + 3z = 5 and 3x + 3y + z = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "6143-6146",
        "Text": "Find the coordinates of the point where the line through (3, \u2013 4, \u2013 5) and\n(2, \u2013 3, 1) crosses the plane 2x + y + z = 7 13 Find the equation of the plane passing through the point (\u2013 1, 3, 2) and perpendicular\nto each of the planes x + 2y + 3z = 5 and 3x + 3y + z = 0 14"
    },
    {
        "Chapter": "1",
        "sentence_range": "6144-6147",
        "Text": "13 Find the equation of the plane passing through the point (\u2013 1, 3, 2) and perpendicular\nto each of the planes x + 2y + 3z = 5 and 3x + 3y + z = 0 14 If the points (1, 1, p) and (\u2013 3, 0, 1) be equidist ant from the plane\n\u02c6\n\u02c6\n\u02c6\n(3\n4\n12 )\n13\n0,\n\u22c5 + \u2212\n+ =\nrr\ni\nj\nk\nthen find the value of p"
    },
    {
        "Chapter": "1",
        "sentence_range": "6145-6148",
        "Text": "Find the equation of the plane passing through the point (\u2013 1, 3, 2) and perpendicular\nto each of the planes x + 2y + 3z = 5 and 3x + 3y + z = 0 14 If the points (1, 1, p) and (\u2013 3, 0, 1) be equidist ant from the plane\n\u02c6\n\u02c6\n\u02c6\n(3\n4\n12 )\n13\n0,\n\u22c5 + \u2212\n+ =\nrr\ni\nj\nk\nthen find the value of p 15"
    },
    {
        "Chapter": "1",
        "sentence_range": "6146-6149",
        "Text": "14 If the points (1, 1, p) and (\u2013 3, 0, 1) be equidist ant from the plane\n\u02c6\n\u02c6\n\u02c6\n(3\n4\n12 )\n13\n0,\n\u22c5 + \u2212\n+ =\nrr\ni\nj\nk\nthen find the value of p 15 Find the equation of the plane passing through the line of intersection of the\nplanes \n\u02c6\n\u02c6\n\u02c6\n(\n)\n1\nr\ni\nj\nk\n\u22c5\n+\n+\n=\nr\nand \n\u02c6\n\u02c6\n\u02c6\n(2\n3\n)\n4\n0\nr\ni\nj\nk\n\u22c5\n+\n\u2212\n+\n=\nr\n and parallel to x-axis"
    },
    {
        "Chapter": "1",
        "sentence_range": "6147-6150",
        "Text": "If the points (1, 1, p) and (\u2013 3, 0, 1) be equidist ant from the plane\n\u02c6\n\u02c6\n\u02c6\n(3\n4\n12 )\n13\n0,\n\u22c5 + \u2212\n+ =\nrr\ni\nj\nk\nthen find the value of p 15 Find the equation of the plane passing through the line of intersection of the\nplanes \n\u02c6\n\u02c6\n\u02c6\n(\n)\n1\nr\ni\nj\nk\n\u22c5\n+\n+\n=\nr\nand \n\u02c6\n\u02c6\n\u02c6\n(2\n3\n)\n4\n0\nr\ni\nj\nk\n\u22c5\n+\n\u2212\n+\n=\nr\n and parallel to x-axis 16"
    },
    {
        "Chapter": "1",
        "sentence_range": "6148-6151",
        "Text": "15 Find the equation of the plane passing through the line of intersection of the\nplanes \n\u02c6\n\u02c6\n\u02c6\n(\n)\n1\nr\ni\nj\nk\n\u22c5\n+\n+\n=\nr\nand \n\u02c6\n\u02c6\n\u02c6\n(2\n3\n)\n4\n0\nr\ni\nj\nk\n\u22c5\n+\n\u2212\n+\n=\nr\n and parallel to x-axis 16 If O be the origin and the coordinates of P be (1, 2, \u2013 3), then find the equation of\nthe plane passing through P and perpendicular to OP"
    },
    {
        "Chapter": "1",
        "sentence_range": "6149-6152",
        "Text": "Find the equation of the plane passing through the line of intersection of the\nplanes \n\u02c6\n\u02c6\n\u02c6\n(\n)\n1\nr\ni\nj\nk\n\u22c5\n+\n+\n=\nr\nand \n\u02c6\n\u02c6\n\u02c6\n(2\n3\n)\n4\n0\nr\ni\nj\nk\n\u22c5\n+\n\u2212\n+\n=\nr\n and parallel to x-axis 16 If O be the origin and the coordinates of P be (1, 2, \u2013 3), then find the equation of\nthe plane passing through P and perpendicular to OP 17"
    },
    {
        "Chapter": "1",
        "sentence_range": "6150-6153",
        "Text": "16 If O be the origin and the coordinates of P be (1, 2, \u2013 3), then find the equation of\nthe plane passing through P and perpendicular to OP 17 Find the equation of the plane which contains the line of intersection of the planes\n\u02c6\n\u02c6\n\u02c6\n(\n2\n3 )\n4\n0\nr\ni\nj\nk\n\u22c5\n+\n+\n\u2212\n=\nr\n,\n\u02c6\n\u02c6\n\u02c6\n(2\n)\n5\n0\nr\ni\nj\nk\n\u22c5\n+\n\u2212\n+\n=\nr\nand which is perpendicular to the\nplane \n\u02c6\n\u02c6\n\u02c6\n(5\n3\n6 )\n8\n0\nr\ni\nj\nk\n\u22c5\n+\n\u2212\n+\n=\nr"
    },
    {
        "Chapter": "1",
        "sentence_range": "6151-6154",
        "Text": "If O be the origin and the coordinates of P be (1, 2, \u2013 3), then find the equation of\nthe plane passing through P and perpendicular to OP 17 Find the equation of the plane which contains the line of intersection of the planes\n\u02c6\n\u02c6\n\u02c6\n(\n2\n3 )\n4\n0\nr\ni\nj\nk\n\u22c5\n+\n+\n\u2212\n=\nr\n,\n\u02c6\n\u02c6\n\u02c6\n(2\n)\n5\n0\nr\ni\nj\nk\n\u22c5\n+\n\u2212\n+\n=\nr\nand which is perpendicular to the\nplane \n\u02c6\n\u02c6\n\u02c6\n(5\n3\n6 )\n8\n0\nr\ni\nj\nk\n\u22c5\n+\n\u2212\n+\n=\nr \u00a9 NCERT\nnot to be republished\nTHREE D IMENSIONAL G EOMETRY\n499\n18"
    },
    {
        "Chapter": "1",
        "sentence_range": "6152-6155",
        "Text": "17 Find the equation of the plane which contains the line of intersection of the planes\n\u02c6\n\u02c6\n\u02c6\n(\n2\n3 )\n4\n0\nr\ni\nj\nk\n\u22c5\n+\n+\n\u2212\n=\nr\n,\n\u02c6\n\u02c6\n\u02c6\n(2\n)\n5\n0\nr\ni\nj\nk\n\u22c5\n+\n\u2212\n+\n=\nr\nand which is perpendicular to the\nplane \n\u02c6\n\u02c6\n\u02c6\n(5\n3\n6 )\n8\n0\nr\ni\nj\nk\n\u22c5\n+\n\u2212\n+\n=\nr \u00a9 NCERT\nnot to be republished\nTHREE D IMENSIONAL G EOMETRY\n499\n18 Find the distance of the point (\u2013 1, \u2013 5, \u2013 10) from the point of intersection of the\nline \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n2\n2\n(3\n4\n2\n)\nr\ni\nj\nk\ni\nj\nk\n=\n\u2212\n+\n+ \u03bb\n+\n+\nr\nand the plane \n\u02c6\n\u02c6\n\u02c6\n(\n)\n5\nr\ni\nj\nk\n\u22c5\n\u2212\n+\n=\nr"
    },
    {
        "Chapter": "1",
        "sentence_range": "6153-6156",
        "Text": "Find the equation of the plane which contains the line of intersection of the planes\n\u02c6\n\u02c6\n\u02c6\n(\n2\n3 )\n4\n0\nr\ni\nj\nk\n\u22c5\n+\n+\n\u2212\n=\nr\n,\n\u02c6\n\u02c6\n\u02c6\n(2\n)\n5\n0\nr\ni\nj\nk\n\u22c5\n+\n\u2212\n+\n=\nr\nand which is perpendicular to the\nplane \n\u02c6\n\u02c6\n\u02c6\n(5\n3\n6 )\n8\n0\nr\ni\nj\nk\n\u22c5\n+\n\u2212\n+\n=\nr \u00a9 NCERT\nnot to be republished\nTHREE D IMENSIONAL G EOMETRY\n499\n18 Find the distance of the point (\u2013 1, \u2013 5, \u2013 10) from the point of intersection of the\nline \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n2\n2\n(3\n4\n2\n)\nr\ni\nj\nk\ni\nj\nk\n=\n\u2212\n+\n+ \u03bb\n+\n+\nr\nand the plane \n\u02c6\n\u02c6\n\u02c6\n(\n)\n5\nr\ni\nj\nk\n\u22c5\n\u2212\n+\n=\nr 19"
    },
    {
        "Chapter": "1",
        "sentence_range": "6154-6157",
        "Text": "\u00a9 NCERT\nnot to be republished\nTHREE D IMENSIONAL G EOMETRY\n499\n18 Find the distance of the point (\u2013 1, \u2013 5, \u2013 10) from the point of intersection of the\nline \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n2\n2\n(3\n4\n2\n)\nr\ni\nj\nk\ni\nj\nk\n=\n\u2212\n+\n+ \u03bb\n+\n+\nr\nand the plane \n\u02c6\n\u02c6\n\u02c6\n(\n)\n5\nr\ni\nj\nk\n\u22c5\n\u2212\n+\n=\nr 19 Find the vector equation of the line passing through (1, 2, 3) and parallel to  the\nplanes \n\u02c6\n\u02c6\n\u02c6\n(\n2 )\n5\nr\ni\nj\nk\n\u22c5\n\u2212\n+\n=\nr\n and \n\u02c6\n\u02c6\n\u02c6\n(3\n)\n6\nr\ni\nj\nk\n\u22c5\n+\n+\n=\nr"
    },
    {
        "Chapter": "1",
        "sentence_range": "6155-6158",
        "Text": "Find the distance of the point (\u2013 1, \u2013 5, \u2013 10) from the point of intersection of the\nline \n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n\u02c6\n2\n2\n(3\n4\n2\n)\nr\ni\nj\nk\ni\nj\nk\n=\n\u2212\n+\n+ \u03bb\n+\n+\nr\nand the plane \n\u02c6\n\u02c6\n\u02c6\n(\n)\n5\nr\ni\nj\nk\n\u22c5\n\u2212\n+\n=\nr 19 Find the vector equation of the line passing through (1, 2, 3) and parallel to  the\nplanes \n\u02c6\n\u02c6\n\u02c6\n(\n2 )\n5\nr\ni\nj\nk\n\u22c5\n\u2212\n+\n=\nr\n and \n\u02c6\n\u02c6\n\u02c6\n(3\n)\n6\nr\ni\nj\nk\n\u22c5\n+\n+\n=\nr 20"
    },
    {
        "Chapter": "1",
        "sentence_range": "6156-6159",
        "Text": "19 Find the vector equation of the line passing through (1, 2, 3) and parallel to  the\nplanes \n\u02c6\n\u02c6\n\u02c6\n(\n2 )\n5\nr\ni\nj\nk\n\u22c5\n\u2212\n+\n=\nr\n and \n\u02c6\n\u02c6\n\u02c6\n(3\n)\n6\nr\ni\nj\nk\n\u22c5\n+\n+\n=\nr 20 Find the vector equation of the line passing through the point (1, 2, \u2013 4) and\nperpendicular to the two lines:\n7\n10\n16\n19\n3\n8\n\u2212\n=\n\u2212\n+\n=\n\u2212\nz\ny\nx\nand \n15\n3\nx \u2212\n = \n29\n5\n8\n5\ny\nz\n\u2212\n\u2212\n=\n\u2212"
    },
    {
        "Chapter": "1",
        "sentence_range": "6157-6160",
        "Text": "Find the vector equation of the line passing through (1, 2, 3) and parallel to  the\nplanes \n\u02c6\n\u02c6\n\u02c6\n(\n2 )\n5\nr\ni\nj\nk\n\u22c5\n\u2212\n+\n=\nr\n and \n\u02c6\n\u02c6\n\u02c6\n(3\n)\n6\nr\ni\nj\nk\n\u22c5\n+\n+\n=\nr 20 Find the vector equation of the line passing through the point (1, 2, \u2013 4) and\nperpendicular to the two lines:\n7\n10\n16\n19\n3\n8\n\u2212\n=\n\u2212\n+\n=\n\u2212\nz\ny\nx\nand \n15\n3\nx \u2212\n = \n29\n5\n8\n5\ny\nz\n\u2212\n\u2212\n=\n\u2212 21"
    },
    {
        "Chapter": "1",
        "sentence_range": "6158-6161",
        "Text": "20 Find the vector equation of the line passing through the point (1, 2, \u2013 4) and\nperpendicular to the two lines:\n7\n10\n16\n19\n3\n8\n\u2212\n=\n\u2212\n+\n=\n\u2212\nz\ny\nx\nand \n15\n3\nx \u2212\n = \n29\n5\n8\n5\ny\nz\n\u2212\n\u2212\n=\n\u2212 21 Prove that if a plane has the intercepts a, b, c and is at a distance of p units from\nthe origin, then \n2\n2\n2\n2\n1\n1\n1\n1\np\nc\nb\na\n=\n+\n+"
    },
    {
        "Chapter": "1",
        "sentence_range": "6159-6162",
        "Text": "Find the vector equation of the line passing through the point (1, 2, \u2013 4) and\nperpendicular to the two lines:\n7\n10\n16\n19\n3\n8\n\u2212\n=\n\u2212\n+\n=\n\u2212\nz\ny\nx\nand \n15\n3\nx \u2212\n = \n29\n5\n8\n5\ny\nz\n\u2212\n\u2212\n=\n\u2212 21 Prove that if a plane has the intercepts a, b, c and is at a distance of p units from\nthe origin, then \n2\n2\n2\n2\n1\n1\n1\n1\np\nc\nb\na\n=\n+\n+ Choose the correct answer in Exercises 22 and 23"
    },
    {
        "Chapter": "1",
        "sentence_range": "6160-6163",
        "Text": "21 Prove that if a plane has the intercepts a, b, c and is at a distance of p units from\nthe origin, then \n2\n2\n2\n2\n1\n1\n1\n1\np\nc\nb\na\n=\n+\n+ Choose the correct answer in Exercises 22 and 23 22"
    },
    {
        "Chapter": "1",
        "sentence_range": "6161-6164",
        "Text": "Prove that if a plane has the intercepts a, b, c and is at a distance of p units from\nthe origin, then \n2\n2\n2\n2\n1\n1\n1\n1\np\nc\nb\na\n=\n+\n+ Choose the correct answer in Exercises 22 and 23 22 Distance between the two planes: 2x + 3y + 4z = 4 and 4x + 6y + 8z = 12 is\n(A) 2 units\n(B) 4 units\n(C) 8 units\n(D)\n2\n29\n units\n23"
    },
    {
        "Chapter": "1",
        "sentence_range": "6162-6165",
        "Text": "Choose the correct answer in Exercises 22 and 23 22 Distance between the two planes: 2x + 3y + 4z = 4 and 4x + 6y + 8z = 12 is\n(A) 2 units\n(B) 4 units\n(C) 8 units\n(D)\n2\n29\n units\n23 The planes: 2x \u2013 y + 4z = 5 and 5x \u2013 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "6163-6166",
        "Text": "22 Distance between the two planes: 2x + 3y + 4z = 4 and 4x + 6y + 8z = 12 is\n(A) 2 units\n(B) 4 units\n(C) 8 units\n(D)\n2\n29\n units\n23 The planes: 2x \u2013 y + 4z = 5 and 5x \u2013 2 5y + 10z = 6 are\n(A) Perpendicular\n(B) Parallel\n(C) intersect y-axis\n(D) passes through \n5\n0,0,\n4\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\nSummary\n\ufffd Direction cosines of a line are the cosines of the angles made by the line\nwith the positive directions of the coordinate axes"
    },
    {
        "Chapter": "1",
        "sentence_range": "6164-6167",
        "Text": "Distance between the two planes: 2x + 3y + 4z = 4 and 4x + 6y + 8z = 12 is\n(A) 2 units\n(B) 4 units\n(C) 8 units\n(D)\n2\n29\n units\n23 The planes: 2x \u2013 y + 4z = 5 and 5x \u2013 2 5y + 10z = 6 are\n(A) Perpendicular\n(B) Parallel\n(C) intersect y-axis\n(D) passes through \n5\n0,0,\n4\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\nSummary\n\ufffd Direction cosines of a line are the cosines of the angles made by the line\nwith the positive directions of the coordinate axes \ufffd If l, m, n are the direction cosines of a line, then l2 + m 2 + n2 = 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "6165-6168",
        "Text": "The planes: 2x \u2013 y + 4z = 5 and 5x \u2013 2 5y + 10z = 6 are\n(A) Perpendicular\n(B) Parallel\n(C) intersect y-axis\n(D) passes through \n5\n0,0,\n4\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\nSummary\n\ufffd Direction cosines of a line are the cosines of the angles made by the line\nwith the positive directions of the coordinate axes \ufffd If l, m, n are the direction cosines of a line, then l2 + m 2 + n2 = 1 \ufffd Direction cosines of a line joining two points P(x1, y1, z1) and Q(x2, y2, z2) are\n2\n1\n2\n1\n2\n1\n,\n,\nPQ\nPQ\nPQ\nx\nx\ny\ny\nz\nz\n\u2212\n\u2212\n\u2212\nwhere PQ =   \n(\n)\n12\n2\n2\n1\n2\n2\n1\n2\n)\n(\n)\n(\nz\nz\ny\ny\nx\nx\n\u2212\n+\n\u2212\n+\n\u2212\n\ufffd Direction ratios of a line are the numbers which are proportional to the\ndirection cosines of a line"
    },
    {
        "Chapter": "1",
        "sentence_range": "6166-6169",
        "Text": "5y + 10z = 6 are\n(A) Perpendicular\n(B) Parallel\n(C) intersect y-axis\n(D) passes through \n5\n0,0,\n4\n\uf8eb\n\uf8f6\n\uf8ec\n\uf8f7\n\uf8ed\n\uf8f8\nSummary\n\ufffd Direction cosines of a line are the cosines of the angles made by the line\nwith the positive directions of the coordinate axes \ufffd If l, m, n are the direction cosines of a line, then l2 + m 2 + n2 = 1 \ufffd Direction cosines of a line joining two points P(x1, y1, z1) and Q(x2, y2, z2) are\n2\n1\n2\n1\n2\n1\n,\n,\nPQ\nPQ\nPQ\nx\nx\ny\ny\nz\nz\n\u2212\n\u2212\n\u2212\nwhere PQ =   \n(\n)\n12\n2\n2\n1\n2\n2\n1\n2\n)\n(\n)\n(\nz\nz\ny\ny\nx\nx\n\u2212\n+\n\u2212\n+\n\u2212\n\ufffd Direction ratios of a line are the numbers which are proportional to the\ndirection cosines of a line \ufffd If l, m, n are the direction cosines and a, b, c are the direction ratios of a line\n\u00a9 NCERT\nnot to be republished\n MATHEMATI CS\n500\nthen\nl = \n2\n2\n2\nc\nb\na\na\n+\n+\n; m = \n2\n2\n2\nc\nb\na\nb\n+\n+\n; n = \n2\n2\n2\nc\nb\na\nc\n+\n+\n\ufffd Skew lines are lines in space which are neither parallel nor intersecting"
    },
    {
        "Chapter": "1",
        "sentence_range": "6167-6170",
        "Text": "\ufffd If l, m, n are the direction cosines of a line, then l2 + m 2 + n2 = 1 \ufffd Direction cosines of a line joining two points P(x1, y1, z1) and Q(x2, y2, z2) are\n2\n1\n2\n1\n2\n1\n,\n,\nPQ\nPQ\nPQ\nx\nx\ny\ny\nz\nz\n\u2212\n\u2212\n\u2212\nwhere PQ =   \n(\n)\n12\n2\n2\n1\n2\n2\n1\n2\n)\n(\n)\n(\nz\nz\ny\ny\nx\nx\n\u2212\n+\n\u2212\n+\n\u2212\n\ufffd Direction ratios of a line are the numbers which are proportional to the\ndirection cosines of a line \ufffd If l, m, n are the direction cosines and a, b, c are the direction ratios of a line\n\u00a9 NCERT\nnot to be republished\n MATHEMATI CS\n500\nthen\nl = \n2\n2\n2\nc\nb\na\na\n+\n+\n; m = \n2\n2\n2\nc\nb\na\nb\n+\n+\n; n = \n2\n2\n2\nc\nb\na\nc\n+\n+\n\ufffd Skew lines are lines in space which are neither parallel nor intersecting They lie in different planes"
    },
    {
        "Chapter": "1",
        "sentence_range": "6168-6171",
        "Text": "\ufffd Direction cosines of a line joining two points P(x1, y1, z1) and Q(x2, y2, z2) are\n2\n1\n2\n1\n2\n1\n,\n,\nPQ\nPQ\nPQ\nx\nx\ny\ny\nz\nz\n\u2212\n\u2212\n\u2212\nwhere PQ =   \n(\n)\n12\n2\n2\n1\n2\n2\n1\n2\n)\n(\n)\n(\nz\nz\ny\ny\nx\nx\n\u2212\n+\n\u2212\n+\n\u2212\n\ufffd Direction ratios of a line are the numbers which are proportional to the\ndirection cosines of a line \ufffd If l, m, n are the direction cosines and a, b, c are the direction ratios of a line\n\u00a9 NCERT\nnot to be republished\n MATHEMATI CS\n500\nthen\nl = \n2\n2\n2\nc\nb\na\na\n+\n+\n; m = \n2\n2\n2\nc\nb\na\nb\n+\n+\n; n = \n2\n2\n2\nc\nb\na\nc\n+\n+\n\ufffd Skew lines are lines in space which are neither parallel nor intersecting They lie in different planes \ufffd Angle between skew lines is the angle between two intersecting lines\ndrawn from any point (preferably through the origin) parallel to each of the\nskew lines"
    },
    {
        "Chapter": "1",
        "sentence_range": "6169-6172",
        "Text": "\ufffd If l, m, n are the direction cosines and a, b, c are the direction ratios of a line\n\u00a9 NCERT\nnot to be republished\n MATHEMATI CS\n500\nthen\nl = \n2\n2\n2\nc\nb\na\na\n+\n+\n; m = \n2\n2\n2\nc\nb\na\nb\n+\n+\n; n = \n2\n2\n2\nc\nb\na\nc\n+\n+\n\ufffd Skew lines are lines in space which are neither parallel nor intersecting They lie in different planes \ufffd Angle between skew lines is the angle between two intersecting lines\ndrawn from any point (preferably through the origin) parallel to each of the\nskew lines \ufffd If  l1, m 1, n1 and l2, m 2, n2 are the direction cosines of two lines; and  \u03b8  is the\nacute angle between the two lines; then\ncos\u03b8  =  |l1l2 + m 1m 2 + n1n2|\n\ufffd If a1, b1, c1 and a2, b2, c2 are the direction ratios of two lines and \u03b8 is the\nacute angle between the two lines; then\ncos\u03b8 = \n1\n2\n1\n2\n1\n2\n2\n2\n2\n2\n2\n2\n1\n1\n1\n2\n2\n2\na a\nb b\nc c\na\nb\nc\na\nb\nc\n+\n+\n+\n+\n+\n+\n\ufffd Vector equation of a line that passes through the given point whose position\nvector is ar  and parallel to a given vector b\nr\n is r\na\nb\n=\n+ \u03bb\nr\nr\nr"
    },
    {
        "Chapter": "1",
        "sentence_range": "6170-6173",
        "Text": "They lie in different planes \ufffd Angle between skew lines is the angle between two intersecting lines\ndrawn from any point (preferably through the origin) parallel to each of the\nskew lines \ufffd If  l1, m 1, n1 and l2, m 2, n2 are the direction cosines of two lines; and  \u03b8  is the\nacute angle between the two lines; then\ncos\u03b8  =  |l1l2 + m 1m 2 + n1n2|\n\ufffd If a1, b1, c1 and a2, b2, c2 are the direction ratios of two lines and \u03b8 is the\nacute angle between the two lines; then\ncos\u03b8 = \n1\n2\n1\n2\n1\n2\n2\n2\n2\n2\n2\n2\n1\n1\n1\n2\n2\n2\na a\nb b\nc c\na\nb\nc\na\nb\nc\n+\n+\n+\n+\n+\n+\n\ufffd Vector equation of a line that passes through the given point whose position\nvector is ar  and parallel to a given vector b\nr\n is r\na\nb\n=\n+ \u03bb\nr\nr\nr \ufffd Equation of a line through a point (x1, y1, z1) and having direction cosines l, m, n is\n1\n1\n1\nx\nx\ny\ny\nz\nz\nl\nm\nn\n\u2212\n\u2212\n\u2212\n=\n=\n\ufffd The vector equation of a line which passes through two points whose position\nvectors are ar  and b\nr\n is \n(\n)\nr\na\nb\na\n=\n+ \u03bb\n\u2212\nr\nr\nr\nr"
    },
    {
        "Chapter": "1",
        "sentence_range": "6171-6174",
        "Text": "\ufffd Angle between skew lines is the angle between two intersecting lines\ndrawn from any point (preferably through the origin) parallel to each of the\nskew lines \ufffd If  l1, m 1, n1 and l2, m 2, n2 are the direction cosines of two lines; and  \u03b8  is the\nacute angle between the two lines; then\ncos\u03b8  =  |l1l2 + m 1m 2 + n1n2|\n\ufffd If a1, b1, c1 and a2, b2, c2 are the direction ratios of two lines and \u03b8 is the\nacute angle between the two lines; then\ncos\u03b8 = \n1\n2\n1\n2\n1\n2\n2\n2\n2\n2\n2\n2\n1\n1\n1\n2\n2\n2\na a\nb b\nc c\na\nb\nc\na\nb\nc\n+\n+\n+\n+\n+\n+\n\ufffd Vector equation of a line that passes through the given point whose position\nvector is ar  and parallel to a given vector b\nr\n is r\na\nb\n=\n+ \u03bb\nr\nr\nr \ufffd Equation of a line through a point (x1, y1, z1) and having direction cosines l, m, n is\n1\n1\n1\nx\nx\ny\ny\nz\nz\nl\nm\nn\n\u2212\n\u2212\n\u2212\n=\n=\n\ufffd The vector equation of a line which passes through two points whose position\nvectors are ar  and b\nr\n is \n(\n)\nr\na\nb\na\n=\n+ \u03bb\n\u2212\nr\nr\nr\nr \ufffd Cartesian equation of a line that passes through two points (x1, y1, z1) and\n (x2, y2, z2) is \n1\n1\n1\n2\n1\n2\n1\n2\n1\nx\nx\ny\ny\nz\nz\nx\nx\ny\ny\nz\nz\n\u2212\n\u2212\n\u2212\n=\n=\n\u2212\n\u2212\n\u2212"
    },
    {
        "Chapter": "1",
        "sentence_range": "6172-6175",
        "Text": "\ufffd If  l1, m 1, n1 and l2, m 2, n2 are the direction cosines of two lines; and  \u03b8  is the\nacute angle between the two lines; then\ncos\u03b8  =  |l1l2 + m 1m 2 + n1n2|\n\ufffd If a1, b1, c1 and a2, b2, c2 are the direction ratios of two lines and \u03b8 is the\nacute angle between the two lines; then\ncos\u03b8 = \n1\n2\n1\n2\n1\n2\n2\n2\n2\n2\n2\n2\n1\n1\n1\n2\n2\n2\na a\nb b\nc c\na\nb\nc\na\nb\nc\n+\n+\n+\n+\n+\n+\n\ufffd Vector equation of a line that passes through the given point whose position\nvector is ar  and parallel to a given vector b\nr\n is r\na\nb\n=\n+ \u03bb\nr\nr\nr \ufffd Equation of a line through a point (x1, y1, z1) and having direction cosines l, m, n is\n1\n1\n1\nx\nx\ny\ny\nz\nz\nl\nm\nn\n\u2212\n\u2212\n\u2212\n=\n=\n\ufffd The vector equation of a line which passes through two points whose position\nvectors are ar  and b\nr\n is \n(\n)\nr\na\nb\na\n=\n+ \u03bb\n\u2212\nr\nr\nr\nr \ufffd Cartesian equation of a line that passes through two points (x1, y1, z1) and\n (x2, y2, z2) is \n1\n1\n1\n2\n1\n2\n1\n2\n1\nx\nx\ny\ny\nz\nz\nx\nx\ny\ny\nz\nz\n\u2212\n\u2212\n\u2212\n=\n=\n\u2212\n\u2212\n\u2212 \ufffd If \u03b8 is the acute angle between \n1\n1\nr\na\nb\n=\n+ \u03bb\nr\nr\nr\n and \n2\n2\nr\na\nb\n=\n+ \u03bb\nr\nr\nr\n, then\n1\n2\n1\n2\ncos\n|\n| |\n|\nbb b\nb\n\u22c5\n\u03b8 =\nr\nr\nr\nr\n\ufffd If  \n1\n1\n1\n1\n1\n1\nn\nz\nz\nm\ny\ny\nl\nx\nx\n\u2212\n=\n\u2212\n=\n\u2212\n and  \n2\n2\n2\n2\n2\n2\nn\nz\nz\nm\ny\ny\nl\nx\nx\n\u2212\n=\n\u2212\n=\n\u2212\nare the equations of two lines,  then the  acute angle between the two lines is\ngiven by cos \u03b8  = |l1l2 + m 1m 2 + n1n2|"
    },
    {
        "Chapter": "1",
        "sentence_range": "6173-6176",
        "Text": "\ufffd Equation of a line through a point (x1, y1, z1) and having direction cosines l, m, n is\n1\n1\n1\nx\nx\ny\ny\nz\nz\nl\nm\nn\n\u2212\n\u2212\n\u2212\n=\n=\n\ufffd The vector equation of a line which passes through two points whose position\nvectors are ar  and b\nr\n is \n(\n)\nr\na\nb\na\n=\n+ \u03bb\n\u2212\nr\nr\nr\nr \ufffd Cartesian equation of a line that passes through two points (x1, y1, z1) and\n (x2, y2, z2) is \n1\n1\n1\n2\n1\n2\n1\n2\n1\nx\nx\ny\ny\nz\nz\nx\nx\ny\ny\nz\nz\n\u2212\n\u2212\n\u2212\n=\n=\n\u2212\n\u2212\n\u2212 \ufffd If \u03b8 is the acute angle between \n1\n1\nr\na\nb\n=\n+ \u03bb\nr\nr\nr\n and \n2\n2\nr\na\nb\n=\n+ \u03bb\nr\nr\nr\n, then\n1\n2\n1\n2\ncos\n|\n| |\n|\nbb b\nb\n\u22c5\n\u03b8 =\nr\nr\nr\nr\n\ufffd If  \n1\n1\n1\n1\n1\n1\nn\nz\nz\nm\ny\ny\nl\nx\nx\n\u2212\n=\n\u2212\n=\n\u2212\n and  \n2\n2\n2\n2\n2\n2\nn\nz\nz\nm\ny\ny\nl\nx\nx\n\u2212\n=\n\u2212\n=\n\u2212\nare the equations of two lines,  then the  acute angle between the two lines is\ngiven by cos \u03b8  = |l1l2 + m 1m 2 + n1n2| \u00a9 NCERT\nnot to be republished\nTHREE D IMENSIONAL G EOMETRY\n501\n\ufffd Shortest distance between two skew lines is the line segment perpendicular\nto both the lines"
    },
    {
        "Chapter": "1",
        "sentence_range": "6174-6177",
        "Text": "\ufffd Cartesian equation of a line that passes through two points (x1, y1, z1) and\n (x2, y2, z2) is \n1\n1\n1\n2\n1\n2\n1\n2\n1\nx\nx\ny\ny\nz\nz\nx\nx\ny\ny\nz\nz\n\u2212\n\u2212\n\u2212\n=\n=\n\u2212\n\u2212\n\u2212 \ufffd If \u03b8 is the acute angle between \n1\n1\nr\na\nb\n=\n+ \u03bb\nr\nr\nr\n and \n2\n2\nr\na\nb\n=\n+ \u03bb\nr\nr\nr\n, then\n1\n2\n1\n2\ncos\n|\n| |\n|\nbb b\nb\n\u22c5\n\u03b8 =\nr\nr\nr\nr\n\ufffd If  \n1\n1\n1\n1\n1\n1\nn\nz\nz\nm\ny\ny\nl\nx\nx\n\u2212\n=\n\u2212\n=\n\u2212\n and  \n2\n2\n2\n2\n2\n2\nn\nz\nz\nm\ny\ny\nl\nx\nx\n\u2212\n=\n\u2212\n=\n\u2212\nare the equations of two lines,  then the  acute angle between the two lines is\ngiven by cos \u03b8  = |l1l2 + m 1m 2 + n1n2| \u00a9 NCERT\nnot to be republished\nTHREE D IMENSIONAL G EOMETRY\n501\n\ufffd Shortest distance between two skew lines is the line segment perpendicular\nto both the lines \ufffd Shortest distance between \n1\n1\nr\na\nb\n= +\u03bb\nr\nr\nr\n and \n2\n2\nr\na\nb\n= +\u00b5\nr\nr\nr\n is\n1\n2\n2\n1\n1\n2\n(\n) (\n\u2013\n)\n|\n|\nb\nb\na\na\nb\nb\n\u00d7\n\u22c5\n\u00d7\nr\nr\nr\nr\nr\nr\n\ufffd Shortest distance between the lines: \n1\n1\n1\n1\n1\n1\nx\nx\ny\ny\nz\nz\na\nb\nc\n\u2212\n\u2212\n\u2212\n=\n=\n and\n2\n2\n2\n2\nx\nx\ny\ny\na\nb\n\u2212\n\u2212\n=\n = \n2\n2\nz\nz\n\u2212c\n is\n2\n1\n2\n1\n2\n1\n1\n1\n1\n2\n2\n2\n2\n2\n2\n1 2\n2 1\n1 2\n2 1\n1 2\n2 1\n(\n)\n(\n)\n(\n)\nx\nx\ny\ny\nz\nz\na\nb\nc\na\nb\nc\nbc\nb c\nc a\nc a\na b\na b\n\u2212\n\u2212\n\u2212\n\u2212\n+\n\u2212\n+\n\u2212\n\ufffd Distance between parallel lines \n1\nr\na\nb\n= +\u03bb\nr\nr\nr\nand \n2\nr\na\nb\n= +\u00b5\nr\nr\nr\nis\n2\n1\n(\n)\n|\n|\nb\na\na\nb\n\u00d7\n\u2212\nr\nr\nr\nr\n\ufffd In the vector form, equation of a plane which is at a distance d from the\norigin, and n\u02c6  is the unit vector normal to the plane through the origin is\nr n\u02c6\nd\n\u22c5\n=\nr"
    },
    {
        "Chapter": "1",
        "sentence_range": "6175-6178",
        "Text": "\ufffd If \u03b8 is the acute angle between \n1\n1\nr\na\nb\n=\n+ \u03bb\nr\nr\nr\n and \n2\n2\nr\na\nb\n=\n+ \u03bb\nr\nr\nr\n, then\n1\n2\n1\n2\ncos\n|\n| |\n|\nbb b\nb\n\u22c5\n\u03b8 =\nr\nr\nr\nr\n\ufffd If  \n1\n1\n1\n1\n1\n1\nn\nz\nz\nm\ny\ny\nl\nx\nx\n\u2212\n=\n\u2212\n=\n\u2212\n and  \n2\n2\n2\n2\n2\n2\nn\nz\nz\nm\ny\ny\nl\nx\nx\n\u2212\n=\n\u2212\n=\n\u2212\nare the equations of two lines,  then the  acute angle between the two lines is\ngiven by cos \u03b8  = |l1l2 + m 1m 2 + n1n2| \u00a9 NCERT\nnot to be republished\nTHREE D IMENSIONAL G EOMETRY\n501\n\ufffd Shortest distance between two skew lines is the line segment perpendicular\nto both the lines \ufffd Shortest distance between \n1\n1\nr\na\nb\n= +\u03bb\nr\nr\nr\n and \n2\n2\nr\na\nb\n= +\u00b5\nr\nr\nr\n is\n1\n2\n2\n1\n1\n2\n(\n) (\n\u2013\n)\n|\n|\nb\nb\na\na\nb\nb\n\u00d7\n\u22c5\n\u00d7\nr\nr\nr\nr\nr\nr\n\ufffd Shortest distance between the lines: \n1\n1\n1\n1\n1\n1\nx\nx\ny\ny\nz\nz\na\nb\nc\n\u2212\n\u2212\n\u2212\n=\n=\n and\n2\n2\n2\n2\nx\nx\ny\ny\na\nb\n\u2212\n\u2212\n=\n = \n2\n2\nz\nz\n\u2212c\n is\n2\n1\n2\n1\n2\n1\n1\n1\n1\n2\n2\n2\n2\n2\n2\n1 2\n2 1\n1 2\n2 1\n1 2\n2 1\n(\n)\n(\n)\n(\n)\nx\nx\ny\ny\nz\nz\na\nb\nc\na\nb\nc\nbc\nb c\nc a\nc a\na b\na b\n\u2212\n\u2212\n\u2212\n\u2212\n+\n\u2212\n+\n\u2212\n\ufffd Distance between parallel lines \n1\nr\na\nb\n= +\u03bb\nr\nr\nr\nand \n2\nr\na\nb\n= +\u00b5\nr\nr\nr\nis\n2\n1\n(\n)\n|\n|\nb\na\na\nb\n\u00d7\n\u2212\nr\nr\nr\nr\n\ufffd In the vector form, equation of a plane which is at a distance d from the\norigin, and n\u02c6  is the unit vector normal to the plane through the origin is\nr n\u02c6\nd\n\u22c5\n=\nr \ufffd Equation of a plane which is at a distance of d from the origin and the direction\ncosines of the normal to the plane as l, m, n is lx + my + nz = d"
    },
    {
        "Chapter": "1",
        "sentence_range": "6176-6179",
        "Text": "\u00a9 NCERT\nnot to be republished\nTHREE D IMENSIONAL G EOMETRY\n501\n\ufffd Shortest distance between two skew lines is the line segment perpendicular\nto both the lines \ufffd Shortest distance between \n1\n1\nr\na\nb\n= +\u03bb\nr\nr\nr\n and \n2\n2\nr\na\nb\n= +\u00b5\nr\nr\nr\n is\n1\n2\n2\n1\n1\n2\n(\n) (\n\u2013\n)\n|\n|\nb\nb\na\na\nb\nb\n\u00d7\n\u22c5\n\u00d7\nr\nr\nr\nr\nr\nr\n\ufffd Shortest distance between the lines: \n1\n1\n1\n1\n1\n1\nx\nx\ny\ny\nz\nz\na\nb\nc\n\u2212\n\u2212\n\u2212\n=\n=\n and\n2\n2\n2\n2\nx\nx\ny\ny\na\nb\n\u2212\n\u2212\n=\n = \n2\n2\nz\nz\n\u2212c\n is\n2\n1\n2\n1\n2\n1\n1\n1\n1\n2\n2\n2\n2\n2\n2\n1 2\n2 1\n1 2\n2 1\n1 2\n2 1\n(\n)\n(\n)\n(\n)\nx\nx\ny\ny\nz\nz\na\nb\nc\na\nb\nc\nbc\nb c\nc a\nc a\na b\na b\n\u2212\n\u2212\n\u2212\n\u2212\n+\n\u2212\n+\n\u2212\n\ufffd Distance between parallel lines \n1\nr\na\nb\n= +\u03bb\nr\nr\nr\nand \n2\nr\na\nb\n= +\u00b5\nr\nr\nr\nis\n2\n1\n(\n)\n|\n|\nb\na\na\nb\n\u00d7\n\u2212\nr\nr\nr\nr\n\ufffd In the vector form, equation of a plane which is at a distance d from the\norigin, and n\u02c6  is the unit vector normal to the plane through the origin is\nr n\u02c6\nd\n\u22c5\n=\nr \ufffd Equation of a plane which is at a distance of d from the origin and the direction\ncosines of the normal to the plane as l, m, n is lx + my + nz = d \ufffd The equation of a plane through a point whose position vector is ar  and\nperpendicular to the vector N\nur\n is (\n)"
    },
    {
        "Chapter": "1",
        "sentence_range": "6177-6180",
        "Text": "\ufffd Shortest distance between \n1\n1\nr\na\nb\n= +\u03bb\nr\nr\nr\n and \n2\n2\nr\na\nb\n= +\u00b5\nr\nr\nr\n is\n1\n2\n2\n1\n1\n2\n(\n) (\n\u2013\n)\n|\n|\nb\nb\na\na\nb\nb\n\u00d7\n\u22c5\n\u00d7\nr\nr\nr\nr\nr\nr\n\ufffd Shortest distance between the lines: \n1\n1\n1\n1\n1\n1\nx\nx\ny\ny\nz\nz\na\nb\nc\n\u2212\n\u2212\n\u2212\n=\n=\n and\n2\n2\n2\n2\nx\nx\ny\ny\na\nb\n\u2212\n\u2212\n=\n = \n2\n2\nz\nz\n\u2212c\n is\n2\n1\n2\n1\n2\n1\n1\n1\n1\n2\n2\n2\n2\n2\n2\n1 2\n2 1\n1 2\n2 1\n1 2\n2 1\n(\n)\n(\n)\n(\n)\nx\nx\ny\ny\nz\nz\na\nb\nc\na\nb\nc\nbc\nb c\nc a\nc a\na b\na b\n\u2212\n\u2212\n\u2212\n\u2212\n+\n\u2212\n+\n\u2212\n\ufffd Distance between parallel lines \n1\nr\na\nb\n= +\u03bb\nr\nr\nr\nand \n2\nr\na\nb\n= +\u00b5\nr\nr\nr\nis\n2\n1\n(\n)\n|\n|\nb\na\na\nb\n\u00d7\n\u2212\nr\nr\nr\nr\n\ufffd In the vector form, equation of a plane which is at a distance d from the\norigin, and n\u02c6  is the unit vector normal to the plane through the origin is\nr n\u02c6\nd\n\u22c5\n=\nr \ufffd Equation of a plane which is at a distance of d from the origin and the direction\ncosines of the normal to the plane as l, m, n is lx + my + nz = d \ufffd The equation of a plane through a point whose position vector is ar  and\nperpendicular to the vector N\nur\n is (\n) N\n0\nr\n\u2212a\n=\nur\nr\nr"
    },
    {
        "Chapter": "1",
        "sentence_range": "6178-6181",
        "Text": "\ufffd Equation of a plane which is at a distance of d from the origin and the direction\ncosines of the normal to the plane as l, m, n is lx + my + nz = d \ufffd The equation of a plane through a point whose position vector is ar  and\nperpendicular to the vector N\nur\n is (\n) N\n0\nr\n\u2212a\n=\nur\nr\nr \ufffd Equation of a plane perpendicular to a given line with direction ratios A, B, C\nand passing through a given point (x1, y1, z1) is\nA (x \u2013 x1) + B (y \u2013 y1) + C (z \u2013 z1 ) = 0\n\ufffd Equation of a plane passing through three non collinear points (x1, y1, z1),\n\u00a9 NCERT\nnot to be republished\n MATHEMATI CS\n502\n(x2, y2, z2) and (x3, y3, z3) is\n1\n3\n1\n3\n1\n3\n1\n2\n1\n2\n1\n2\n1\n1\n1\nz\nz\ny\ny\nx\nx\nz\nz\ny\ny\nx\nx\nz\nz\ny\ny\nx\nx\n\u2212\n\u2212\n\u2212\n\u2212\n\u2212\n\u2212\n\u2212\n\u2212\n\u2212\n = 0\n\ufffd Vector equation of a plane that contains three non collinear points having\nposition vectors \nb\na\nr,r\n and cr is  (\n)"
    },
    {
        "Chapter": "1",
        "sentence_range": "6179-6182",
        "Text": "\ufffd The equation of a plane through a point whose position vector is ar  and\nperpendicular to the vector N\nur\n is (\n) N\n0\nr\n\u2212a\n=\nur\nr\nr \ufffd Equation of a plane perpendicular to a given line with direction ratios A, B, C\nand passing through a given point (x1, y1, z1) is\nA (x \u2013 x1) + B (y \u2013 y1) + C (z \u2013 z1 ) = 0\n\ufffd Equation of a plane passing through three non collinear points (x1, y1, z1),\n\u00a9 NCERT\nnot to be republished\n MATHEMATI CS\n502\n(x2, y2, z2) and (x3, y3, z3) is\n1\n3\n1\n3\n1\n3\n1\n2\n1\n2\n1\n2\n1\n1\n1\nz\nz\ny\ny\nx\nx\nz\nz\ny\ny\nx\nx\nz\nz\ny\ny\nx\nx\n\u2212\n\u2212\n\u2212\n\u2212\n\u2212\n\u2212\n\u2212\n\u2212\n\u2212\n = 0\n\ufffd Vector equation of a plane that contains three non collinear points having\nposition vectors \nb\na\nr,r\n and cr is  (\n) [(\n)\n(\n) ]\n0\nr\na\nb\na\nc\na\n\u2212\n\u2212\n\u00d7\n\u2212\n=\nr\nr\nr\nr\nr\nr\n\ufffd Equation of a plane that cuts the coordinates axes at (a, 0, 0), (0, b, 0) and\n(0, 0, c) is\n=1\n+\n+\nc\nz\nb\ny\na\nx\n\ufffd Vector equation of a plane t hat passes through t he intersection of\nplanes\n1\n1\n2\n2\nand\nr n\nd\nr n\nd\n\u22c5\n=\n\u22c5\n=\nr r\nr r\n  is \n1\n2\n1\n2\n(\n)\nr\nn\nn\nd\nd\n\u22c5\n+ \u03bb\n=\n+ \u03bb\nr\nr\nr\n, where \u03bb is any\nnonzero constant"
    },
    {
        "Chapter": "1",
        "sentence_range": "6180-6183",
        "Text": "N\n0\nr\n\u2212a\n=\nur\nr\nr \ufffd Equation of a plane perpendicular to a given line with direction ratios A, B, C\nand passing through a given point (x1, y1, z1) is\nA (x \u2013 x1) + B (y \u2013 y1) + C (z \u2013 z1 ) = 0\n\ufffd Equation of a plane passing through three non collinear points (x1, y1, z1),\n\u00a9 NCERT\nnot to be republished\n MATHEMATI CS\n502\n(x2, y2, z2) and (x3, y3, z3) is\n1\n3\n1\n3\n1\n3\n1\n2\n1\n2\n1\n2\n1\n1\n1\nz\nz\ny\ny\nx\nx\nz\nz\ny\ny\nx\nx\nz\nz\ny\ny\nx\nx\n\u2212\n\u2212\n\u2212\n\u2212\n\u2212\n\u2212\n\u2212\n\u2212\n\u2212\n = 0\n\ufffd Vector equation of a plane that contains three non collinear points having\nposition vectors \nb\na\nr,r\n and cr is  (\n) [(\n)\n(\n) ]\n0\nr\na\nb\na\nc\na\n\u2212\n\u2212\n\u00d7\n\u2212\n=\nr\nr\nr\nr\nr\nr\n\ufffd Equation of a plane that cuts the coordinates axes at (a, 0, 0), (0, b, 0) and\n(0, 0, c) is\n=1\n+\n+\nc\nz\nb\ny\na\nx\n\ufffd Vector equation of a plane t hat passes through t he intersection of\nplanes\n1\n1\n2\n2\nand\nr n\nd\nr n\nd\n\u22c5\n=\n\u22c5\n=\nr r\nr r\n  is \n1\n2\n1\n2\n(\n)\nr\nn\nn\nd\nd\n\u22c5\n+ \u03bb\n=\n+ \u03bb\nr\nr\nr\n, where \u03bb is any\nnonzero constant \ufffd Cartesian equation of a plane that passes through the intersection of two\ngiven planes A1 x + B1 y + C1 z + D1 = 0 and   A2 x + B2 y + C2 z + D2 = 0\nis (A1 x + B1 y + C1 z + D1) + \u03bb(A2 x + B2 y + C2 z + D2) = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "6181-6184",
        "Text": "\ufffd Equation of a plane perpendicular to a given line with direction ratios A, B, C\nand passing through a given point (x1, y1, z1) is\nA (x \u2013 x1) + B (y \u2013 y1) + C (z \u2013 z1 ) = 0\n\ufffd Equation of a plane passing through three non collinear points (x1, y1, z1),\n\u00a9 NCERT\nnot to be republished\n MATHEMATI CS\n502\n(x2, y2, z2) and (x3, y3, z3) is\n1\n3\n1\n3\n1\n3\n1\n2\n1\n2\n1\n2\n1\n1\n1\nz\nz\ny\ny\nx\nx\nz\nz\ny\ny\nx\nx\nz\nz\ny\ny\nx\nx\n\u2212\n\u2212\n\u2212\n\u2212\n\u2212\n\u2212\n\u2212\n\u2212\n\u2212\n = 0\n\ufffd Vector equation of a plane that contains three non collinear points having\nposition vectors \nb\na\nr,r\n and cr is  (\n) [(\n)\n(\n) ]\n0\nr\na\nb\na\nc\na\n\u2212\n\u2212\n\u00d7\n\u2212\n=\nr\nr\nr\nr\nr\nr\n\ufffd Equation of a plane that cuts the coordinates axes at (a, 0, 0), (0, b, 0) and\n(0, 0, c) is\n=1\n+\n+\nc\nz\nb\ny\na\nx\n\ufffd Vector equation of a plane t hat passes through t he intersection of\nplanes\n1\n1\n2\n2\nand\nr n\nd\nr n\nd\n\u22c5\n=\n\u22c5\n=\nr r\nr r\n  is \n1\n2\n1\n2\n(\n)\nr\nn\nn\nd\nd\n\u22c5\n+ \u03bb\n=\n+ \u03bb\nr\nr\nr\n, where \u03bb is any\nnonzero constant \ufffd Cartesian equation of a plane that passes through the intersection of two\ngiven planes A1 x + B1 y + C1 z + D1 = 0 and   A2 x + B2 y + C2 z + D2 = 0\nis (A1 x + B1 y + C1 z + D1) + \u03bb(A2 x + B2 y + C2 z + D2) = 0 \ufffd Two lines \n1\n1\nr\na\nb\n=\n+ \u03bb\nr\nr\nr\n and \n2\n2\nr\na\nb\n=\n+ \u00b5\nr\nr\nr\n are coplanar if\n2\n1\n1\n2\n(\n) (\n)\na\na\nb\nb\n\u2212\n\u22c5\n\u00d7\nr\nr\nr\nr\n= 0\n\ufffd In the cartesian form above lines passing through the points A (x1, y1, z1) and\nB \n2\n2\n2\n(\n,\n,\n)\nx\ny\nz\n2\n2\n2\n2\n\u2013\n\u2013\ny\ny\nz\nz\nb\nC\n=\n=\nare coplanar if \n2\n2\n2\n1\n1\n1\n1\n2\n1\n2\n1\n2\nc\nb\na\nc\nb\na\nz\nz\ny\ny\nx\nx\n\u2212\n\u2212\n\u2212\n = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "6182-6185",
        "Text": "[(\n)\n(\n) ]\n0\nr\na\nb\na\nc\na\n\u2212\n\u2212\n\u00d7\n\u2212\n=\nr\nr\nr\nr\nr\nr\n\ufffd Equation of a plane that cuts the coordinates axes at (a, 0, 0), (0, b, 0) and\n(0, 0, c) is\n=1\n+\n+\nc\nz\nb\ny\na\nx\n\ufffd Vector equation of a plane t hat passes through t he intersection of\nplanes\n1\n1\n2\n2\nand\nr n\nd\nr n\nd\n\u22c5\n=\n\u22c5\n=\nr r\nr r\n  is \n1\n2\n1\n2\n(\n)\nr\nn\nn\nd\nd\n\u22c5\n+ \u03bb\n=\n+ \u03bb\nr\nr\nr\n, where \u03bb is any\nnonzero constant \ufffd Cartesian equation of a plane that passes through the intersection of two\ngiven planes A1 x + B1 y + C1 z + D1 = 0 and   A2 x + B2 y + C2 z + D2 = 0\nis (A1 x + B1 y + C1 z + D1) + \u03bb(A2 x + B2 y + C2 z + D2) = 0 \ufffd Two lines \n1\n1\nr\na\nb\n=\n+ \u03bb\nr\nr\nr\n and \n2\n2\nr\na\nb\n=\n+ \u00b5\nr\nr\nr\n are coplanar if\n2\n1\n1\n2\n(\n) (\n)\na\na\nb\nb\n\u2212\n\u22c5\n\u00d7\nr\nr\nr\nr\n= 0\n\ufffd In the cartesian form above lines passing through the points A (x1, y1, z1) and\nB \n2\n2\n2\n(\n,\n,\n)\nx\ny\nz\n2\n2\n2\n2\n\u2013\n\u2013\ny\ny\nz\nz\nb\nC\n=\n=\nare coplanar if \n2\n2\n2\n1\n1\n1\n1\n2\n1\n2\n1\n2\nc\nb\na\nc\nb\na\nz\nz\ny\ny\nx\nx\n\u2212\n\u2212\n\u2212\n = 0 \ufffd In the vector form, if  \u03b8   is the angle between the two planes, \n1\n1\nr n\nd\n\u22c5\nr r=\n  and\n2\n2\nr n\nd\n\u22c5\nr r=\n, then \u03b8 = cos\u20131 \n1\n2\n1\n2\n|\n|\n|\n||\n|\nn\nn\nn\nr\u22c5n\nr\nr\nr"
    },
    {
        "Chapter": "1",
        "sentence_range": "6183-6186",
        "Text": "\ufffd Cartesian equation of a plane that passes through the intersection of two\ngiven planes A1 x + B1 y + C1 z + D1 = 0 and   A2 x + B2 y + C2 z + D2 = 0\nis (A1 x + B1 y + C1 z + D1) + \u03bb(A2 x + B2 y + C2 z + D2) = 0 \ufffd Two lines \n1\n1\nr\na\nb\n=\n+ \u03bb\nr\nr\nr\n and \n2\n2\nr\na\nb\n=\n+ \u00b5\nr\nr\nr\n are coplanar if\n2\n1\n1\n2\n(\n) (\n)\na\na\nb\nb\n\u2212\n\u22c5\n\u00d7\nr\nr\nr\nr\n= 0\n\ufffd In the cartesian form above lines passing through the points A (x1, y1, z1) and\nB \n2\n2\n2\n(\n,\n,\n)\nx\ny\nz\n2\n2\n2\n2\n\u2013\n\u2013\ny\ny\nz\nz\nb\nC\n=\n=\nare coplanar if \n2\n2\n2\n1\n1\n1\n1\n2\n1\n2\n1\n2\nc\nb\na\nc\nb\na\nz\nz\ny\ny\nx\nx\n\u2212\n\u2212\n\u2212\n = 0 \ufffd In the vector form, if  \u03b8   is the angle between the two planes, \n1\n1\nr n\nd\n\u22c5\nr r=\n  and\n2\n2\nr n\nd\n\u22c5\nr r=\n, then \u03b8 = cos\u20131 \n1\n2\n1\n2\n|\n|\n|\n||\n|\nn\nn\nn\nr\u22c5n\nr\nr\nr \ufffd The angle \u03c6 between the line r\na\nb\n=\n+ \u03bb\nr\nr\nr\nand the plane \nr n\u02c6\nd\n\u22c5\n=\nr\nis\n\u00a9 NCERT\nnot to be republished\nTHREE D IMENSIONAL G EOMETRY\n503\n\u02c6\nsin\n| | | |\u02c6\nb n\nb n\n\u22c5\n\u03c6 =\nr\nr\n\ufffd The angle \u03b8 between the planes A1x + B1y + C1z + D1 = 0 and\nA2 x + B2 y + C2 z + D2 = 0 is given by\ncos \u03b8 =\n1\n2\n1\n2\n1\n2\n2\n2\n2\n2\n2\n2\n1\n1\n1\n2\n2\n2\nA A\nB B\nC C\nA\nB\nC\nA\nB\nC\n+\n+\n+\n+\n+\n+\n\ufffd The distance of a point whose position vector is ar from the plane \nr n\u02c6\nd\n\u22c5\n=\nr\n is\n\u02c6\n|\n|\nd\n\u2212a n\n\u22c5\nr\n\ufffd The distance from a point (x1, y1, z1) to the plane Ax + By + Cz + D = 0 is\n1\n1\n1\n2\n2\n2\nA\nB\nC\nD\nA\nB\nC\nx\ny\nz\n+\n+\n+\n+\n+"
    },
    {
        "Chapter": "1",
        "sentence_range": "6184-6187",
        "Text": "\ufffd Two lines \n1\n1\nr\na\nb\n=\n+ \u03bb\nr\nr\nr\n and \n2\n2\nr\na\nb\n=\n+ \u00b5\nr\nr\nr\n are coplanar if\n2\n1\n1\n2\n(\n) (\n)\na\na\nb\nb\n\u2212\n\u22c5\n\u00d7\nr\nr\nr\nr\n= 0\n\ufffd In the cartesian form above lines passing through the points A (x1, y1, z1) and\nB \n2\n2\n2\n(\n,\n,\n)\nx\ny\nz\n2\n2\n2\n2\n\u2013\n\u2013\ny\ny\nz\nz\nb\nC\n=\n=\nare coplanar if \n2\n2\n2\n1\n1\n1\n1\n2\n1\n2\n1\n2\nc\nb\na\nc\nb\na\nz\nz\ny\ny\nx\nx\n\u2212\n\u2212\n\u2212\n = 0 \ufffd In the vector form, if  \u03b8   is the angle between the two planes, \n1\n1\nr n\nd\n\u22c5\nr r=\n  and\n2\n2\nr n\nd\n\u22c5\nr r=\n, then \u03b8 = cos\u20131 \n1\n2\n1\n2\n|\n|\n|\n||\n|\nn\nn\nn\nr\u22c5n\nr\nr\nr \ufffd The angle \u03c6 between the line r\na\nb\n=\n+ \u03bb\nr\nr\nr\nand the plane \nr n\u02c6\nd\n\u22c5\n=\nr\nis\n\u00a9 NCERT\nnot to be republished\nTHREE D IMENSIONAL G EOMETRY\n503\n\u02c6\nsin\n| | | |\u02c6\nb n\nb n\n\u22c5\n\u03c6 =\nr\nr\n\ufffd The angle \u03b8 between the planes A1x + B1y + C1z + D1 = 0 and\nA2 x + B2 y + C2 z + D2 = 0 is given by\ncos \u03b8 =\n1\n2\n1\n2\n1\n2\n2\n2\n2\n2\n2\n2\n1\n1\n1\n2\n2\n2\nA A\nB B\nC C\nA\nB\nC\nA\nB\nC\n+\n+\n+\n+\n+\n+\n\ufffd The distance of a point whose position vector is ar from the plane \nr n\u02c6\nd\n\u22c5\n=\nr\n is\n\u02c6\n|\n|\nd\n\u2212a n\n\u22c5\nr\n\ufffd The distance from a point (x1, y1, z1) to the plane Ax + By + Cz + D = 0 is\n1\n1\n1\n2\n2\n2\nA\nB\nC\nD\nA\nB\nC\nx\ny\nz\n+\n+\n+\n+\n+ \u2014\ufffd\u2014\n\u00a9 NCERT\nnot to be republished\n504\nMATHEMATICS\n\ufffdThe mathematical experience of the student is incomplete if he never had\nthe opportunity to solve a problem invented by himself"
    },
    {
        "Chapter": "1",
        "sentence_range": "6185-6188",
        "Text": "\ufffd In the vector form, if  \u03b8   is the angle between the two planes, \n1\n1\nr n\nd\n\u22c5\nr r=\n  and\n2\n2\nr n\nd\n\u22c5\nr r=\n, then \u03b8 = cos\u20131 \n1\n2\n1\n2\n|\n|\n|\n||\n|\nn\nn\nn\nr\u22c5n\nr\nr\nr \ufffd The angle \u03c6 between the line r\na\nb\n=\n+ \u03bb\nr\nr\nr\nand the plane \nr n\u02c6\nd\n\u22c5\n=\nr\nis\n\u00a9 NCERT\nnot to be republished\nTHREE D IMENSIONAL G EOMETRY\n503\n\u02c6\nsin\n| | | |\u02c6\nb n\nb n\n\u22c5\n\u03c6 =\nr\nr\n\ufffd The angle \u03b8 between the planes A1x + B1y + C1z + D1 = 0 and\nA2 x + B2 y + C2 z + D2 = 0 is given by\ncos \u03b8 =\n1\n2\n1\n2\n1\n2\n2\n2\n2\n2\n2\n2\n1\n1\n1\n2\n2\n2\nA A\nB B\nC C\nA\nB\nC\nA\nB\nC\n+\n+\n+\n+\n+\n+\n\ufffd The distance of a point whose position vector is ar from the plane \nr n\u02c6\nd\n\u22c5\n=\nr\n is\n\u02c6\n|\n|\nd\n\u2212a n\n\u22c5\nr\n\ufffd The distance from a point (x1, y1, z1) to the plane Ax + By + Cz + D = 0 is\n1\n1\n1\n2\n2\n2\nA\nB\nC\nD\nA\nB\nC\nx\ny\nz\n+\n+\n+\n+\n+ \u2014\ufffd\u2014\n\u00a9 NCERT\nnot to be republished\n504\nMATHEMATICS\n\ufffdThe mathematical experience of the student is incomplete if he never had\nthe opportunity to solve a problem invented by himself \u2013 G"
    },
    {
        "Chapter": "1",
        "sentence_range": "6186-6189",
        "Text": "\ufffd The angle \u03c6 between the line r\na\nb\n=\n+ \u03bb\nr\nr\nr\nand the plane \nr n\u02c6\nd\n\u22c5\n=\nr\nis\n\u00a9 NCERT\nnot to be republished\nTHREE D IMENSIONAL G EOMETRY\n503\n\u02c6\nsin\n| | | |\u02c6\nb n\nb n\n\u22c5\n\u03c6 =\nr\nr\n\ufffd The angle \u03b8 between the planes A1x + B1y + C1z + D1 = 0 and\nA2 x + B2 y + C2 z + D2 = 0 is given by\ncos \u03b8 =\n1\n2\n1\n2\n1\n2\n2\n2\n2\n2\n2\n2\n1\n1\n1\n2\n2\n2\nA A\nB B\nC C\nA\nB\nC\nA\nB\nC\n+\n+\n+\n+\n+\n+\n\ufffd The distance of a point whose position vector is ar from the plane \nr n\u02c6\nd\n\u22c5\n=\nr\n is\n\u02c6\n|\n|\nd\n\u2212a n\n\u22c5\nr\n\ufffd The distance from a point (x1, y1, z1) to the plane Ax + By + Cz + D = 0 is\n1\n1\n1\n2\n2\n2\nA\nB\nC\nD\nA\nB\nC\nx\ny\nz\n+\n+\n+\n+\n+ \u2014\ufffd\u2014\n\u00a9 NCERT\nnot to be republished\n504\nMATHEMATICS\n\ufffdThe mathematical experience of the student is incomplete if he never had\nthe opportunity to solve a problem invented by himself \u2013 G POLYA \ufffd\n12"
    },
    {
        "Chapter": "1",
        "sentence_range": "6187-6190",
        "Text": "\u2014\ufffd\u2014\n\u00a9 NCERT\nnot to be republished\n504\nMATHEMATICS\n\ufffdThe mathematical experience of the student is incomplete if he never had\nthe opportunity to solve a problem invented by himself \u2013 G POLYA \ufffd\n12 1  Introduction\nIn earlier classes, we have discussed systems of linear\nequations and their applications in day to day problems"
    },
    {
        "Chapter": "1",
        "sentence_range": "6188-6191",
        "Text": "\u2013 G POLYA \ufffd\n12 1  Introduction\nIn earlier classes, we have discussed systems of linear\nequations and their applications in day to day problems In\nClass XI, we have studied linear inequalities and systems\nof linear inequalities in two variables and their solutions by\ngraphical method"
    },
    {
        "Chapter": "1",
        "sentence_range": "6189-6192",
        "Text": "POLYA \ufffd\n12 1  Introduction\nIn earlier classes, we have discussed systems of linear\nequations and their applications in day to day problems In\nClass XI, we have studied linear inequalities and systems\nof linear inequalities in two variables and their solutions by\ngraphical method Many applications in mathematics\ninvolve systems of inequalities/equations"
    },
    {
        "Chapter": "1",
        "sentence_range": "6190-6193",
        "Text": "1  Introduction\nIn earlier classes, we have discussed systems of linear\nequations and their applications in day to day problems In\nClass XI, we have studied linear inequalities and systems\nof linear inequalities in two variables and their solutions by\ngraphical method Many applications in mathematics\ninvolve systems of inequalities/equations In this chapter,\nwe shall apply the systems of linear inequalities/equations\nto solve some real life problems of the type as given below:\nA furniture dealer deals in only two items\u2013tables and\nchairs"
    },
    {
        "Chapter": "1",
        "sentence_range": "6191-6194",
        "Text": "In\nClass XI, we have studied linear inequalities and systems\nof linear inequalities in two variables and their solutions by\ngraphical method Many applications in mathematics\ninvolve systems of inequalities/equations In this chapter,\nwe shall apply the systems of linear inequalities/equations\nto solve some real life problems of the type as given below:\nA furniture dealer deals in only two items\u2013tables and\nchairs He has Rs 50,000 to invest and has storage space\nof at most 60 pieces"
    },
    {
        "Chapter": "1",
        "sentence_range": "6192-6195",
        "Text": "Many applications in mathematics\ninvolve systems of inequalities/equations In this chapter,\nwe shall apply the systems of linear inequalities/equations\nto solve some real life problems of the type as given below:\nA furniture dealer deals in only two items\u2013tables and\nchairs He has Rs 50,000 to invest and has storage space\nof at most 60 pieces A table costs Rs 2500 and a chair\nRs 500"
    },
    {
        "Chapter": "1",
        "sentence_range": "6193-6196",
        "Text": "In this chapter,\nwe shall apply the systems of linear inequalities/equations\nto solve some real life problems of the type as given below:\nA furniture dealer deals in only two items\u2013tables and\nchairs He has Rs 50,000 to invest and has storage space\nof at most 60 pieces A table costs Rs 2500 and a chair\nRs 500 He estimates that from the sale of one table, he\ncan make a profit of Rs 250 and that from the sale of one\nchair a profit of Rs 75"
    },
    {
        "Chapter": "1",
        "sentence_range": "6194-6197",
        "Text": "He has Rs 50,000 to invest and has storage space\nof at most 60 pieces A table costs Rs 2500 and a chair\nRs 500 He estimates that from the sale of one table, he\ncan make a profit of Rs 250 and that from the sale of one\nchair a profit of Rs 75 He wants to know how many tables and chairs he should buy\nfrom the available money so as to maximise his total profit, assuming that he can sell all\nthe items which he buys"
    },
    {
        "Chapter": "1",
        "sentence_range": "6195-6198",
        "Text": "A table costs Rs 2500 and a chair\nRs 500 He estimates that from the sale of one table, he\ncan make a profit of Rs 250 and that from the sale of one\nchair a profit of Rs 75 He wants to know how many tables and chairs he should buy\nfrom the available money so as to maximise his total profit, assuming that he can sell all\nthe items which he buys Such type of problems which seek to maximise (or, minimise) profit (or, cost) form\na general class of problems called optimisation problems"
    },
    {
        "Chapter": "1",
        "sentence_range": "6196-6199",
        "Text": "He estimates that from the sale of one table, he\ncan make a profit of Rs 250 and that from the sale of one\nchair a profit of Rs 75 He wants to know how many tables and chairs he should buy\nfrom the available money so as to maximise his total profit, assuming that he can sell all\nthe items which he buys Such type of problems which seek to maximise (or, minimise) profit (or, cost) form\na general class of problems called optimisation problems Thus, an optimisation\nproblem may involve finding maximum profit, minimum cost, or minimum use of\nresources etc"
    },
    {
        "Chapter": "1",
        "sentence_range": "6197-6200",
        "Text": "He wants to know how many tables and chairs he should buy\nfrom the available money so as to maximise his total profit, assuming that he can sell all\nthe items which he buys Such type of problems which seek to maximise (or, minimise) profit (or, cost) form\na general class of problems called optimisation problems Thus, an optimisation\nproblem may involve finding maximum profit, minimum cost, or minimum use of\nresources etc A special but a very important class of optimisation problems is linear programming\nproblem"
    },
    {
        "Chapter": "1",
        "sentence_range": "6198-6201",
        "Text": "Such type of problems which seek to maximise (or, minimise) profit (or, cost) form\na general class of problems called optimisation problems Thus, an optimisation\nproblem may involve finding maximum profit, minimum cost, or minimum use of\nresources etc A special but a very important class of optimisation problems is linear programming\nproblem The above stated optimisation problem is an example of linear programming\nproblem"
    },
    {
        "Chapter": "1",
        "sentence_range": "6199-6202",
        "Text": "Thus, an optimisation\nproblem may involve finding maximum profit, minimum cost, or minimum use of\nresources etc A special but a very important class of optimisation problems is linear programming\nproblem The above stated optimisation problem is an example of linear programming\nproblem Linear programming problems are of much interest because of their wide\napplicability in industry, commerce, management science etc"
    },
    {
        "Chapter": "1",
        "sentence_range": "6200-6203",
        "Text": "A special but a very important class of optimisation problems is linear programming\nproblem The above stated optimisation problem is an example of linear programming\nproblem Linear programming problems are of much interest because of their wide\napplicability in industry, commerce, management science etc In this chapter, we shall study some linear programming problems and their solutions\nby graphical method only, though there are many other methods also to solve such\nproblems"
    },
    {
        "Chapter": "1",
        "sentence_range": "6201-6204",
        "Text": "The above stated optimisation problem is an example of linear programming\nproblem Linear programming problems are of much interest because of their wide\napplicability in industry, commerce, management science etc In this chapter, we shall study some linear programming problems and their solutions\nby graphical method only, though there are many other methods also to solve such\nproblems Chapter 12\nLINEAR PROGRAMMING\nL"
    },
    {
        "Chapter": "1",
        "sentence_range": "6202-6205",
        "Text": "Linear programming problems are of much interest because of their wide\napplicability in industry, commerce, management science etc In this chapter, we shall study some linear programming problems and their solutions\nby graphical method only, though there are many other methods also to solve such\nproblems Chapter 12\nLINEAR PROGRAMMING\nL Kantorovich\n\u00a9 NCERT\nnot to be republished\nLINEAR PROGRAMMING         505\n12"
    },
    {
        "Chapter": "1",
        "sentence_range": "6203-6206",
        "Text": "In this chapter, we shall study some linear programming problems and their solutions\nby graphical method only, though there are many other methods also to solve such\nproblems Chapter 12\nLINEAR PROGRAMMING\nL Kantorovich\n\u00a9 NCERT\nnot to be republished\nLINEAR PROGRAMMING         505\n12 2  Linear Programming Problem and its Mathematical Formulation\nWe begin our discussion with the above example of furniture dealer which will further\nlead to a mathematical formulation of the problem in two variables"
    },
    {
        "Chapter": "1",
        "sentence_range": "6204-6207",
        "Text": "Chapter 12\nLINEAR PROGRAMMING\nL Kantorovich\n\u00a9 NCERT\nnot to be republished\nLINEAR PROGRAMMING         505\n12 2  Linear Programming Problem and its Mathematical Formulation\nWe begin our discussion with the above example of furniture dealer which will further\nlead to a mathematical formulation of the problem in two variables In this example, we\nobserve\n(i)\nThe dealer can invest his money in buying tables or chairs or combination thereof"
    },
    {
        "Chapter": "1",
        "sentence_range": "6205-6208",
        "Text": "Kantorovich\n\u00a9 NCERT\nnot to be republished\nLINEAR PROGRAMMING         505\n12 2  Linear Programming Problem and its Mathematical Formulation\nWe begin our discussion with the above example of furniture dealer which will further\nlead to a mathematical formulation of the problem in two variables In this example, we\nobserve\n(i)\nThe dealer can invest his money in buying tables or chairs or combination thereof Further he would earn different profits by following different investment\nstrategies"
    },
    {
        "Chapter": "1",
        "sentence_range": "6206-6209",
        "Text": "2  Linear Programming Problem and its Mathematical Formulation\nWe begin our discussion with the above example of furniture dealer which will further\nlead to a mathematical formulation of the problem in two variables In this example, we\nobserve\n(i)\nThe dealer can invest his money in buying tables or chairs or combination thereof Further he would earn different profits by following different investment\nstrategies (ii)\nThere are certain overriding conditions or constraints viz"
    },
    {
        "Chapter": "1",
        "sentence_range": "6207-6210",
        "Text": "In this example, we\nobserve\n(i)\nThe dealer can invest his money in buying tables or chairs or combination thereof Further he would earn different profits by following different investment\nstrategies (ii)\nThere are certain overriding conditions or constraints viz , his investment is\nlimited to a maximum of Rs 50,000 and so is his storage space which is for a\nmaximum of 60 pieces"
    },
    {
        "Chapter": "1",
        "sentence_range": "6208-6211",
        "Text": "Further he would earn different profits by following different investment\nstrategies (ii)\nThere are certain overriding conditions or constraints viz , his investment is\nlimited to a maximum of Rs 50,000 and so is his storage space which is for a\nmaximum of 60 pieces Suppose he decides to buy tables only and no chairs, so he can buy 50000 \u00f7 2500,\ni"
    },
    {
        "Chapter": "1",
        "sentence_range": "6209-6212",
        "Text": "(ii)\nThere are certain overriding conditions or constraints viz , his investment is\nlimited to a maximum of Rs 50,000 and so is his storage space which is for a\nmaximum of 60 pieces Suppose he decides to buy tables only and no chairs, so he can buy 50000 \u00f7 2500,\ni e"
    },
    {
        "Chapter": "1",
        "sentence_range": "6210-6213",
        "Text": ", his investment is\nlimited to a maximum of Rs 50,000 and so is his storage space which is for a\nmaximum of 60 pieces Suppose he decides to buy tables only and no chairs, so he can buy 50000 \u00f7 2500,\ni e , 20 tables"
    },
    {
        "Chapter": "1",
        "sentence_range": "6211-6214",
        "Text": "Suppose he decides to buy tables only and no chairs, so he can buy 50000 \u00f7 2500,\ni e , 20 tables His profit in this case will be Rs (250 \u00d7 20), i"
    },
    {
        "Chapter": "1",
        "sentence_range": "6212-6215",
        "Text": "e , 20 tables His profit in this case will be Rs (250 \u00d7 20), i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "6213-6216",
        "Text": ", 20 tables His profit in this case will be Rs (250 \u00d7 20), i e , Rs 5000"
    },
    {
        "Chapter": "1",
        "sentence_range": "6214-6217",
        "Text": "His profit in this case will be Rs (250 \u00d7 20), i e , Rs 5000 Suppose he chooses to buy chairs only and no tables"
    },
    {
        "Chapter": "1",
        "sentence_range": "6215-6218",
        "Text": "e , Rs 5000 Suppose he chooses to buy chairs only and no tables With his capital of Rs 50,000,\nhe can buy 50000 \u00f7 500, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "6216-6219",
        "Text": ", Rs 5000 Suppose he chooses to buy chairs only and no tables With his capital of Rs 50,000,\nhe can buy 50000 \u00f7 500, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "6217-6220",
        "Text": "Suppose he chooses to buy chairs only and no tables With his capital of Rs 50,000,\nhe can buy 50000 \u00f7 500, i e 100 chairs"
    },
    {
        "Chapter": "1",
        "sentence_range": "6218-6221",
        "Text": "With his capital of Rs 50,000,\nhe can buy 50000 \u00f7 500, i e 100 chairs But he can store only 60 pieces"
    },
    {
        "Chapter": "1",
        "sentence_range": "6219-6222",
        "Text": "e 100 chairs But he can store only 60 pieces Therefore, he\nis forced to buy only 60 chairs which will give him a total profit of Rs (60 \u00d7 75), i"
    },
    {
        "Chapter": "1",
        "sentence_range": "6220-6223",
        "Text": "100 chairs But he can store only 60 pieces Therefore, he\nis forced to buy only 60 chairs which will give him a total profit of Rs (60 \u00d7 75), i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "6221-6224",
        "Text": "But he can store only 60 pieces Therefore, he\nis forced to buy only 60 chairs which will give him a total profit of Rs (60 \u00d7 75), i e ,\nRs 4500"
    },
    {
        "Chapter": "1",
        "sentence_range": "6222-6225",
        "Text": "Therefore, he\nis forced to buy only 60 chairs which will give him a total profit of Rs (60 \u00d7 75), i e ,\nRs 4500 There are many other possibilities, for instance, he may choose to buy 10 tables\nand 50 chairs, as he can store only 60 pieces"
    },
    {
        "Chapter": "1",
        "sentence_range": "6223-6226",
        "Text": "e ,\nRs 4500 There are many other possibilities, for instance, he may choose to buy 10 tables\nand 50 chairs, as he can store only 60 pieces Total profit in this case would be\nRs (10 \u00d7 250 + 50 \u00d7 75), i"
    },
    {
        "Chapter": "1",
        "sentence_range": "6224-6227",
        "Text": ",\nRs 4500 There are many other possibilities, for instance, he may choose to buy 10 tables\nand 50 chairs, as he can store only 60 pieces Total profit in this case would be\nRs (10 \u00d7 250 + 50 \u00d7 75), i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "6225-6228",
        "Text": "There are many other possibilities, for instance, he may choose to buy 10 tables\nand 50 chairs, as he can store only 60 pieces Total profit in this case would be\nRs (10 \u00d7 250 + 50 \u00d7 75), i e , Rs 6250 and so on"
    },
    {
        "Chapter": "1",
        "sentence_range": "6226-6229",
        "Text": "Total profit in this case would be\nRs (10 \u00d7 250 + 50 \u00d7 75), i e , Rs 6250 and so on We, thus, find that the dealer can invest his money in different ways and he would\nearn different profits by following different investment strategies"
    },
    {
        "Chapter": "1",
        "sentence_range": "6227-6230",
        "Text": "e , Rs 6250 and so on We, thus, find that the dealer can invest his money in different ways and he would\nearn different profits by following different investment strategies Now the problem is : How should he invest his money in order to get maximum\nprofit"
    },
    {
        "Chapter": "1",
        "sentence_range": "6228-6231",
        "Text": ", Rs 6250 and so on We, thus, find that the dealer can invest his money in different ways and he would\nearn different profits by following different investment strategies Now the problem is : How should he invest his money in order to get maximum\nprofit To answer this question, let us try to formulate the problem mathematically"
    },
    {
        "Chapter": "1",
        "sentence_range": "6229-6232",
        "Text": "We, thus, find that the dealer can invest his money in different ways and he would\nearn different profits by following different investment strategies Now the problem is : How should he invest his money in order to get maximum\nprofit To answer this question, let us try to formulate the problem mathematically 12"
    },
    {
        "Chapter": "1",
        "sentence_range": "6230-6233",
        "Text": "Now the problem is : How should he invest his money in order to get maximum\nprofit To answer this question, let us try to formulate the problem mathematically 12 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "6231-6234",
        "Text": "To answer this question, let us try to formulate the problem mathematically 12 2 1 Mathematical formulation of the problem\nLet x be the number of tables and y be the number of chairs that the dealer buys"
    },
    {
        "Chapter": "1",
        "sentence_range": "6232-6235",
        "Text": "12 2 1 Mathematical formulation of the problem\nLet x be the number of tables and y be the number of chairs that the dealer buys Obviously, x and y must be non-negative, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "6233-6236",
        "Text": "2 1 Mathematical formulation of the problem\nLet x be the number of tables and y be the number of chairs that the dealer buys Obviously, x and y must be non-negative, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "6234-6237",
        "Text": "1 Mathematical formulation of the problem\nLet x be the number of tables and y be the number of chairs that the dealer buys Obviously, x and y must be non-negative, i e ,\n0"
    },
    {
        "Chapter": "1",
        "sentence_range": "6235-6238",
        "Text": "Obviously, x and y must be non-negative, i e ,\n0 (1)\n(Non-negative constraints)"
    },
    {
        "Chapter": "1",
        "sentence_range": "6236-6239",
        "Text": "e ,\n0 (1)\n(Non-negative constraints) (2)\n0\nyx\n \n  \n \n \nThe dealer is constrained by the maximum amount he can invest (Here it is\nRs 50,000) and by the maximum number of items he can store (Here it is 60)"
    },
    {
        "Chapter": "1",
        "sentence_range": "6237-6240",
        "Text": ",\n0 (1)\n(Non-negative constraints) (2)\n0\nyx\n \n  \n \n \nThe dealer is constrained by the maximum amount he can invest (Here it is\nRs 50,000) and by the maximum number of items he can store (Here it is 60) Stated mathematically,\n2500x + 500y \u2264 50000 (investment constraint)\nor\n5x + y \u2264 100"
    },
    {
        "Chapter": "1",
        "sentence_range": "6238-6241",
        "Text": "(1)\n(Non-negative constraints) (2)\n0\nyx\n \n  \n \n \nThe dealer is constrained by the maximum amount he can invest (Here it is\nRs 50,000) and by the maximum number of items he can store (Here it is 60) Stated mathematically,\n2500x + 500y \u2264 50000 (investment constraint)\nor\n5x + y \u2264 100 (3)\nand\nx + y \u2264 60  (storage constraint)"
    },
    {
        "Chapter": "1",
        "sentence_range": "6239-6242",
        "Text": "(2)\n0\nyx\n \n  \n \n \nThe dealer is constrained by the maximum amount he can invest (Here it is\nRs 50,000) and by the maximum number of items he can store (Here it is 60) Stated mathematically,\n2500x + 500y \u2264 50000 (investment constraint)\nor\n5x + y \u2264 100 (3)\nand\nx + y \u2264 60  (storage constraint) (4)\n\u00a9 NCERT\nnot to be republished\n506\nMATHEMATICS\nThe dealer wants to invest in such a way so as to maximise his profit, say, Z which\nstated as a function of x and y is given by\nZ = 250x + 75y (called objective function)"
    },
    {
        "Chapter": "1",
        "sentence_range": "6240-6243",
        "Text": "Stated mathematically,\n2500x + 500y \u2264 50000 (investment constraint)\nor\n5x + y \u2264 100 (3)\nand\nx + y \u2264 60  (storage constraint) (4)\n\u00a9 NCERT\nnot to be republished\n506\nMATHEMATICS\nThe dealer wants to invest in such a way so as to maximise his profit, say, Z which\nstated as a function of x and y is given by\nZ = 250x + 75y (called objective function) (5)\nMathematically, the given problems now reduces to:\nMaximise Z = 250x + 75y\nsubject to the constraints:\n5x + y \u2264 100\nx + y \u2264 60\nx \u2265 0,  y \u2265 0\nSo, we have to maximise the linear function Z subject to certain conditions determined\nby a set of linear inequalities with variables as non-negative"
    },
    {
        "Chapter": "1",
        "sentence_range": "6241-6244",
        "Text": "(3)\nand\nx + y \u2264 60  (storage constraint) (4)\n\u00a9 NCERT\nnot to be republished\n506\nMATHEMATICS\nThe dealer wants to invest in such a way so as to maximise his profit, say, Z which\nstated as a function of x and y is given by\nZ = 250x + 75y (called objective function) (5)\nMathematically, the given problems now reduces to:\nMaximise Z = 250x + 75y\nsubject to the constraints:\n5x + y \u2264 100\nx + y \u2264 60\nx \u2265 0,  y \u2265 0\nSo, we have to maximise the linear function Z subject to certain conditions determined\nby a set of linear inequalities with variables as non-negative There are also some other\nproblems where we have to minimise a linear function subject to certain conditions\ndetermined by a set of linear inequalities with variables as non-negative"
    },
    {
        "Chapter": "1",
        "sentence_range": "6242-6245",
        "Text": "(4)\n\u00a9 NCERT\nnot to be republished\n506\nMATHEMATICS\nThe dealer wants to invest in such a way so as to maximise his profit, say, Z which\nstated as a function of x and y is given by\nZ = 250x + 75y (called objective function) (5)\nMathematically, the given problems now reduces to:\nMaximise Z = 250x + 75y\nsubject to the constraints:\n5x + y \u2264 100\nx + y \u2264 60\nx \u2265 0,  y \u2265 0\nSo, we have to maximise the linear function Z subject to certain conditions determined\nby a set of linear inequalities with variables as non-negative There are also some other\nproblems where we have to minimise a linear function subject to certain conditions\ndetermined by a set of linear inequalities with variables as non-negative Such problems\nare called Linear Programming Problems"
    },
    {
        "Chapter": "1",
        "sentence_range": "6243-6246",
        "Text": "(5)\nMathematically, the given problems now reduces to:\nMaximise Z = 250x + 75y\nsubject to the constraints:\n5x + y \u2264 100\nx + y \u2264 60\nx \u2265 0,  y \u2265 0\nSo, we have to maximise the linear function Z subject to certain conditions determined\nby a set of linear inequalities with variables as non-negative There are also some other\nproblems where we have to minimise a linear function subject to certain conditions\ndetermined by a set of linear inequalities with variables as non-negative Such problems\nare called Linear Programming Problems Thus, a Linear Programming Problem is one that is concerned with finding the\noptimal value (maximum or minimum value) of  a linear function (called objective\nfunction) of several variables (say x and y), subject to the conditions that the variables\nare non-negative and satisfy a set of linear inequalities (called linear constraints)"
    },
    {
        "Chapter": "1",
        "sentence_range": "6244-6247",
        "Text": "There are also some other\nproblems where we have to minimise a linear function subject to certain conditions\ndetermined by a set of linear inequalities with variables as non-negative Such problems\nare called Linear Programming Problems Thus, a Linear Programming Problem is one that is concerned with finding the\noptimal value (maximum or minimum value) of  a linear function (called objective\nfunction) of several variables (say x and y), subject to the conditions that the variables\nare non-negative and satisfy a set of linear inequalities (called linear constraints) The term linear implies that all the mathematical relations used in the problem are\nlinear relations while the term programming refers to the method of determining a\nparticular programme or plan of action"
    },
    {
        "Chapter": "1",
        "sentence_range": "6245-6248",
        "Text": "Such problems\nare called Linear Programming Problems Thus, a Linear Programming Problem is one that is concerned with finding the\noptimal value (maximum or minimum value) of  a linear function (called objective\nfunction) of several variables (say x and y), subject to the conditions that the variables\nare non-negative and satisfy a set of linear inequalities (called linear constraints) The term linear implies that all the mathematical relations used in the problem are\nlinear relations while the term programming refers to the method of determining a\nparticular programme or plan of action Before we proceed further, we now formally define some terms (which have been\nused above) which we shall be using in the linear programming problems:\nObjective function Linear function Z = ax + by, where a, b are constants, which has\nto be maximised or minimized is called a linear objective function"
    },
    {
        "Chapter": "1",
        "sentence_range": "6246-6249",
        "Text": "Thus, a Linear Programming Problem is one that is concerned with finding the\noptimal value (maximum or minimum value) of  a linear function (called objective\nfunction) of several variables (say x and y), subject to the conditions that the variables\nare non-negative and satisfy a set of linear inequalities (called linear constraints) The term linear implies that all the mathematical relations used in the problem are\nlinear relations while the term programming refers to the method of determining a\nparticular programme or plan of action Before we proceed further, we now formally define some terms (which have been\nused above) which we shall be using in the linear programming problems:\nObjective function Linear function Z = ax + by, where a, b are constants, which has\nto be maximised or minimized is called a linear objective function In the above example, Z = 250x + 75y is a linear objective function"
    },
    {
        "Chapter": "1",
        "sentence_range": "6247-6250",
        "Text": "The term linear implies that all the mathematical relations used in the problem are\nlinear relations while the term programming refers to the method of determining a\nparticular programme or plan of action Before we proceed further, we now formally define some terms (which have been\nused above) which we shall be using in the linear programming problems:\nObjective function Linear function Z = ax + by, where a, b are constants, which has\nto be maximised or minimized is called a linear objective function In the above example, Z = 250x + 75y is a linear objective function Variables x and\ny are called decision variables"
    },
    {
        "Chapter": "1",
        "sentence_range": "6248-6251",
        "Text": "Before we proceed further, we now formally define some terms (which have been\nused above) which we shall be using in the linear programming problems:\nObjective function Linear function Z = ax + by, where a, b are constants, which has\nto be maximised or minimized is called a linear objective function In the above example, Z = 250x + 75y is a linear objective function Variables x and\ny are called decision variables Constraints The linear inequalities or equations or restrictions on the variables of a\nlinear programming problem are called constraints"
    },
    {
        "Chapter": "1",
        "sentence_range": "6249-6252",
        "Text": "In the above example, Z = 250x + 75y is a linear objective function Variables x and\ny are called decision variables Constraints The linear inequalities or equations or restrictions on the variables of a\nlinear programming problem are called constraints The conditions x \u2265 0, y \u2265 0 are\ncalled non-negative restrictions"
    },
    {
        "Chapter": "1",
        "sentence_range": "6250-6253",
        "Text": "Variables x and\ny are called decision variables Constraints The linear inequalities or equations or restrictions on the variables of a\nlinear programming problem are called constraints The conditions x \u2265 0, y \u2265 0 are\ncalled non-negative restrictions In the above example, the set of inequalities (1) to (4)\nare constraints"
    },
    {
        "Chapter": "1",
        "sentence_range": "6251-6254",
        "Text": "Constraints The linear inequalities or equations or restrictions on the variables of a\nlinear programming problem are called constraints The conditions x \u2265 0, y \u2265 0 are\ncalled non-negative restrictions In the above example, the set of inequalities (1) to (4)\nare constraints Optimisation problem A problem which seeks to maximise or minimise a linear\nfunction (say of two variables x and y) subject to certain constraints as determined by\na set of linear inequalities is called an optimisation problem"
    },
    {
        "Chapter": "1",
        "sentence_range": "6252-6255",
        "Text": "The conditions x \u2265 0, y \u2265 0 are\ncalled non-negative restrictions In the above example, the set of inequalities (1) to (4)\nare constraints Optimisation problem A problem which seeks to maximise or minimise a linear\nfunction (say of two variables x and y) subject to certain constraints as determined by\na set of linear inequalities is called an optimisation problem Linear programming\nproblems are special type of optimisation problems"
    },
    {
        "Chapter": "1",
        "sentence_range": "6253-6256",
        "Text": "In the above example, the set of inequalities (1) to (4)\nare constraints Optimisation problem A problem which seeks to maximise or minimise a linear\nfunction (say of two variables x and y) subject to certain constraints as determined by\na set of linear inequalities is called an optimisation problem Linear programming\nproblems are special type of optimisation problems The above problem of investing a\n\u00a9 NCERT\nnot to be republished\nLINEAR PROGRAMMING         507\ngiven sum by the dealer in purchasing chairs and tables is an example of an optimisation\nproblem as well as of a linear programming problem"
    },
    {
        "Chapter": "1",
        "sentence_range": "6254-6257",
        "Text": "Optimisation problem A problem which seeks to maximise or minimise a linear\nfunction (say of two variables x and y) subject to certain constraints as determined by\na set of linear inequalities is called an optimisation problem Linear programming\nproblems are special type of optimisation problems The above problem of investing a\n\u00a9 NCERT\nnot to be republished\nLINEAR PROGRAMMING         507\ngiven sum by the dealer in purchasing chairs and tables is an example of an optimisation\nproblem as well as of a linear programming problem We will now discuss how to find solutions to a linear programming problem"
    },
    {
        "Chapter": "1",
        "sentence_range": "6255-6258",
        "Text": "Linear programming\nproblems are special type of optimisation problems The above problem of investing a\n\u00a9 NCERT\nnot to be republished\nLINEAR PROGRAMMING         507\ngiven sum by the dealer in purchasing chairs and tables is an example of an optimisation\nproblem as well as of a linear programming problem We will now discuss how to find solutions to a linear programming problem In this\nchapter, we will be concerned only with the graphical method"
    },
    {
        "Chapter": "1",
        "sentence_range": "6256-6259",
        "Text": "The above problem of investing a\n\u00a9 NCERT\nnot to be republished\nLINEAR PROGRAMMING         507\ngiven sum by the dealer in purchasing chairs and tables is an example of an optimisation\nproblem as well as of a linear programming problem We will now discuss how to find solutions to a linear programming problem In this\nchapter, we will be concerned only with the graphical method 12"
    },
    {
        "Chapter": "1",
        "sentence_range": "6257-6260",
        "Text": "We will now discuss how to find solutions to a linear programming problem In this\nchapter, we will be concerned only with the graphical method 12 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "6258-6261",
        "Text": "In this\nchapter, we will be concerned only with the graphical method 12 2 2 Graphical method of solving linear programming problems\nIn Class XI, we have learnt how to graph a system of linear inequalities involving two\nvariables x and y and to find its solutions graphically"
    },
    {
        "Chapter": "1",
        "sentence_range": "6259-6262",
        "Text": "12 2 2 Graphical method of solving linear programming problems\nIn Class XI, we have learnt how to graph a system of linear inequalities involving two\nvariables x and y and to find its solutions graphically Let us refer to the problem of\ninvestment in tables and chairs discussed in Section 12"
    },
    {
        "Chapter": "1",
        "sentence_range": "6260-6263",
        "Text": "2 2 Graphical method of solving linear programming problems\nIn Class XI, we have learnt how to graph a system of linear inequalities involving two\nvariables x and y and to find its solutions graphically Let us refer to the problem of\ninvestment in tables and chairs discussed in Section 12 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "6261-6264",
        "Text": "2 Graphical method of solving linear programming problems\nIn Class XI, we have learnt how to graph a system of linear inequalities involving two\nvariables x and y and to find its solutions graphically Let us refer to the problem of\ninvestment in tables and chairs discussed in Section 12 2 We will now solve this problem\ngraphically"
    },
    {
        "Chapter": "1",
        "sentence_range": "6262-6265",
        "Text": "Let us refer to the problem of\ninvestment in tables and chairs discussed in Section 12 2 We will now solve this problem\ngraphically Let us graph the constraints stated as linear inequalities:\n5x + y \u2264 100"
    },
    {
        "Chapter": "1",
        "sentence_range": "6263-6266",
        "Text": "2 We will now solve this problem\ngraphically Let us graph the constraints stated as linear inequalities:\n5x + y \u2264 100 (1)\nx + y \u2264 60"
    },
    {
        "Chapter": "1",
        "sentence_range": "6264-6267",
        "Text": "We will now solve this problem\ngraphically Let us graph the constraints stated as linear inequalities:\n5x + y \u2264 100 (1)\nx + y \u2264 60 (2)\nx \u2265 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "6265-6268",
        "Text": "Let us graph the constraints stated as linear inequalities:\n5x + y \u2264 100 (1)\nx + y \u2264 60 (2)\nx \u2265 0 (3)\ny \u2265 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "6266-6269",
        "Text": "(1)\nx + y \u2264 60 (2)\nx \u2265 0 (3)\ny \u2265 0 (4)\nThe graph of this system (shaded region) consists of the points common to all half\nplanes determined by the inequalities (1) to (4) (Fig 12"
    },
    {
        "Chapter": "1",
        "sentence_range": "6267-6270",
        "Text": "(2)\nx \u2265 0 (3)\ny \u2265 0 (4)\nThe graph of this system (shaded region) consists of the points common to all half\nplanes determined by the inequalities (1) to (4) (Fig 12 1)"
    },
    {
        "Chapter": "1",
        "sentence_range": "6268-6271",
        "Text": "(3)\ny \u2265 0 (4)\nThe graph of this system (shaded region) consists of the points common to all half\nplanes determined by the inequalities (1) to (4) (Fig 12 1) Each point in this region\nrepresents a feasible choice open to the dealer for investing in tables and chairs"
    },
    {
        "Chapter": "1",
        "sentence_range": "6269-6272",
        "Text": "(4)\nThe graph of this system (shaded region) consists of the points common to all half\nplanes determined by the inequalities (1) to (4) (Fig 12 1) Each point in this region\nrepresents a feasible choice open to the dealer for investing in tables and chairs The\nregion, therefore, is called the feasible region for the problem"
    },
    {
        "Chapter": "1",
        "sentence_range": "6270-6273",
        "Text": "1) Each point in this region\nrepresents a feasible choice open to the dealer for investing in tables and chairs The\nregion, therefore, is called the feasible region for the problem Every point of this\nregion is called a feasible solution to the problem"
    },
    {
        "Chapter": "1",
        "sentence_range": "6271-6274",
        "Text": "Each point in this region\nrepresents a feasible choice open to the dealer for investing in tables and chairs The\nregion, therefore, is called the feasible region for the problem Every point of this\nregion is called a feasible solution to the problem Thus, we have,\nFeasible region The common region determined by all the constraints including\nnon-negative constraints x, y \u2265 0 of a linear programming problem is called the feasible\nregion (or solution region) for the problem"
    },
    {
        "Chapter": "1",
        "sentence_range": "6272-6275",
        "Text": "The\nregion, therefore, is called the feasible region for the problem Every point of this\nregion is called a feasible solution to the problem Thus, we have,\nFeasible region The common region determined by all the constraints including\nnon-negative constraints x, y \u2265 0 of a linear programming problem is called the feasible\nregion (or solution region) for the problem In Fig 12"
    },
    {
        "Chapter": "1",
        "sentence_range": "6273-6276",
        "Text": "Every point of this\nregion is called a feasible solution to the problem Thus, we have,\nFeasible region The common region determined by all the constraints including\nnon-negative constraints x, y \u2265 0 of a linear programming problem is called the feasible\nregion (or solution region) for the problem In Fig 12 1, the region OABC (shaded) is\nthe feasible region for the problem"
    },
    {
        "Chapter": "1",
        "sentence_range": "6274-6277",
        "Text": "Thus, we have,\nFeasible region The common region determined by all the constraints including\nnon-negative constraints x, y \u2265 0 of a linear programming problem is called the feasible\nregion (or solution region) for the problem In Fig 12 1, the region OABC (shaded) is\nthe feasible region for the problem The region other than feasible region is called an\ninfeasible region"
    },
    {
        "Chapter": "1",
        "sentence_range": "6275-6278",
        "Text": "In Fig 12 1, the region OABC (shaded) is\nthe feasible region for the problem The region other than feasible region is called an\ninfeasible region Feasible solutions Points within and on the\nboundary of the feasible region represent\nfeasible solutions of the constraints"
    },
    {
        "Chapter": "1",
        "sentence_range": "6276-6279",
        "Text": "1, the region OABC (shaded) is\nthe feasible region for the problem The region other than feasible region is called an\ninfeasible region Feasible solutions Points within and on the\nboundary of the feasible region represent\nfeasible solutions of the constraints In\nFig 12"
    },
    {
        "Chapter": "1",
        "sentence_range": "6277-6280",
        "Text": "The region other than feasible region is called an\ninfeasible region Feasible solutions Points within and on the\nboundary of the feasible region represent\nfeasible solutions of the constraints In\nFig 12 1, every point within and on the\nboundary of the feasible region OABC\nrepresents feasible solution to the problem"
    },
    {
        "Chapter": "1",
        "sentence_range": "6278-6281",
        "Text": "Feasible solutions Points within and on the\nboundary of the feasible region represent\nfeasible solutions of the constraints In\nFig 12 1, every point within and on the\nboundary of the feasible region OABC\nrepresents feasible solution to the problem For example, the point (10, 50) is a feasible\nsolution of the problem and so are the points\n(0, 60), (20, 0) etc"
    },
    {
        "Chapter": "1",
        "sentence_range": "6279-6282",
        "Text": "In\nFig 12 1, every point within and on the\nboundary of the feasible region OABC\nrepresents feasible solution to the problem For example, the point (10, 50) is a feasible\nsolution of the problem and so are the points\n(0, 60), (20, 0) etc Any point outside the feasible region is\ncalled an  infeasible solution"
    },
    {
        "Chapter": "1",
        "sentence_range": "6280-6283",
        "Text": "1, every point within and on the\nboundary of the feasible region OABC\nrepresents feasible solution to the problem For example, the point (10, 50) is a feasible\nsolution of the problem and so are the points\n(0, 60), (20, 0) etc Any point outside the feasible region is\ncalled an  infeasible solution For example,\nthe point (25, 40) is an infeasible solution of\nthe problem"
    },
    {
        "Chapter": "1",
        "sentence_range": "6281-6284",
        "Text": "For example, the point (10, 50) is a feasible\nsolution of the problem and so are the points\n(0, 60), (20, 0) etc Any point outside the feasible region is\ncalled an  infeasible solution For example,\nthe point (25, 40) is an infeasible solution of\nthe problem Fig 12"
    },
    {
        "Chapter": "1",
        "sentence_range": "6282-6285",
        "Text": "Any point outside the feasible region is\ncalled an  infeasible solution For example,\nthe point (25, 40) is an infeasible solution of\nthe problem Fig 12 1\n\u00a9 NCERT\nnot to be republished\n508\nMATHEMATICS\nOptimal (feasible) solution: Any point in the feasible region that gives the optimal\nvalue (maximum or minimum) of the objective function is called an optimal solution"
    },
    {
        "Chapter": "1",
        "sentence_range": "6283-6286",
        "Text": "For example,\nthe point (25, 40) is an infeasible solution of\nthe problem Fig 12 1\n\u00a9 NCERT\nnot to be republished\n508\nMATHEMATICS\nOptimal (feasible) solution: Any point in the feasible region that gives the optimal\nvalue (maximum or minimum) of the objective function is called an optimal solution Now, we see that every point in the feasible region OABC satisfies all the constraints\nas given in (1) to (4), and since there are infinitely many points, it is not evident how\nwe should go about finding a point that gives a maximum value of the objective function\nZ = 250x + 75y"
    },
    {
        "Chapter": "1",
        "sentence_range": "6284-6287",
        "Text": "Fig 12 1\n\u00a9 NCERT\nnot to be republished\n508\nMATHEMATICS\nOptimal (feasible) solution: Any point in the feasible region that gives the optimal\nvalue (maximum or minimum) of the objective function is called an optimal solution Now, we see that every point in the feasible region OABC satisfies all the constraints\nas given in (1) to (4), and since there are infinitely many points, it is not evident how\nwe should go about finding a point that gives a maximum value of the objective function\nZ = 250x + 75y To handle this situation, we use the following theorems which are\nfundamental in solving linear programming problems"
    },
    {
        "Chapter": "1",
        "sentence_range": "6285-6288",
        "Text": "1\n\u00a9 NCERT\nnot to be republished\n508\nMATHEMATICS\nOptimal (feasible) solution: Any point in the feasible region that gives the optimal\nvalue (maximum or minimum) of the objective function is called an optimal solution Now, we see that every point in the feasible region OABC satisfies all the constraints\nas given in (1) to (4), and since there are infinitely many points, it is not evident how\nwe should go about finding a point that gives a maximum value of the objective function\nZ = 250x + 75y To handle this situation, we use the following theorems which are\nfundamental in solving linear programming problems The proofs of these theorems\nare beyond the scope of the book"
    },
    {
        "Chapter": "1",
        "sentence_range": "6286-6289",
        "Text": "Now, we see that every point in the feasible region OABC satisfies all the constraints\nas given in (1) to (4), and since there are infinitely many points, it is not evident how\nwe should go about finding a point that gives a maximum value of the objective function\nZ = 250x + 75y To handle this situation, we use the following theorems which are\nfundamental in solving linear programming problems The proofs of these theorems\nare beyond the scope of the book Theorem 1 Let R be the feasible region (convex polygon) for a linear programming\nproblem and let Z = ax + by be the objective function"
    },
    {
        "Chapter": "1",
        "sentence_range": "6287-6290",
        "Text": "To handle this situation, we use the following theorems which are\nfundamental in solving linear programming problems The proofs of these theorems\nare beyond the scope of the book Theorem 1 Let R be the feasible region (convex polygon) for a linear programming\nproblem and let Z = ax + by be the objective function When Z has an optimal value\n(maximum or minimum), where the variables x and y are subject to constraints described\nby linear inequalities, this optimal value must occur at a corner point* (vertex) of the\nfeasible region"
    },
    {
        "Chapter": "1",
        "sentence_range": "6288-6291",
        "Text": "The proofs of these theorems\nare beyond the scope of the book Theorem 1 Let R be the feasible region (convex polygon) for a linear programming\nproblem and let Z = ax + by be the objective function When Z has an optimal value\n(maximum or minimum), where the variables x and y are subject to constraints described\nby linear inequalities, this optimal value must occur at a corner point* (vertex) of the\nfeasible region Theorem 2 Let R be the feasible region for a linear programming problem, and let\nZ = ax + by be the objective function"
    },
    {
        "Chapter": "1",
        "sentence_range": "6289-6292",
        "Text": "Theorem 1 Let R be the feasible region (convex polygon) for a linear programming\nproblem and let Z = ax + by be the objective function When Z has an optimal value\n(maximum or minimum), where the variables x and y are subject to constraints described\nby linear inequalities, this optimal value must occur at a corner point* (vertex) of the\nfeasible region Theorem 2 Let R be the feasible region for a linear programming problem, and let\nZ = ax + by be the objective function If R is bounded**, then the objective function\nZ has both a maximum and a minimum value on R and each of these occurs at a\ncorner point (vertex) of R"
    },
    {
        "Chapter": "1",
        "sentence_range": "6290-6293",
        "Text": "When Z has an optimal value\n(maximum or minimum), where the variables x and y are subject to constraints described\nby linear inequalities, this optimal value must occur at a corner point* (vertex) of the\nfeasible region Theorem 2 Let R be the feasible region for a linear programming problem, and let\nZ = ax + by be the objective function If R is bounded**, then the objective function\nZ has both a maximum and a minimum value on R and each of these occurs at a\ncorner point (vertex) of R Remark If R is unbounded, then a maximum or a minimum value of the objective\nfunction may not exist"
    },
    {
        "Chapter": "1",
        "sentence_range": "6291-6294",
        "Text": "Theorem 2 Let R be the feasible region for a linear programming problem, and let\nZ = ax + by be the objective function If R is bounded**, then the objective function\nZ has both a maximum and a minimum value on R and each of these occurs at a\ncorner point (vertex) of R Remark If R is unbounded, then a maximum or a minimum value of the objective\nfunction may not exist However, if it exists, it must occur at a corner point of R"
    },
    {
        "Chapter": "1",
        "sentence_range": "6292-6295",
        "Text": "If R is bounded**, then the objective function\nZ has both a maximum and a minimum value on R and each of these occurs at a\ncorner point (vertex) of R Remark If R is unbounded, then a maximum or a minimum value of the objective\nfunction may not exist However, if it exists, it must occur at a corner point of R (By Theorem 1)"
    },
    {
        "Chapter": "1",
        "sentence_range": "6293-6296",
        "Text": "Remark If R is unbounded, then a maximum or a minimum value of the objective\nfunction may not exist However, if it exists, it must occur at a corner point of R (By Theorem 1) In the above example, the corner points (vertices) of the bounded (feasible) region\nare: O, A, B and C and it is easy to find their coordinates as (0, 0), (20, 0), (10, 50) and\n(0, 60) respectively"
    },
    {
        "Chapter": "1",
        "sentence_range": "6294-6297",
        "Text": "However, if it exists, it must occur at a corner point of R (By Theorem 1) In the above example, the corner points (vertices) of the bounded (feasible) region\nare: O, A, B and C and it is easy to find their coordinates as (0, 0), (20, 0), (10, 50) and\n(0, 60) respectively Let us now compute the values of Z at these points"
    },
    {
        "Chapter": "1",
        "sentence_range": "6295-6298",
        "Text": "(By Theorem 1) In the above example, the corner points (vertices) of the bounded (feasible) region\nare: O, A, B and C and it is easy to find their coordinates as (0, 0), (20, 0), (10, 50) and\n(0, 60) respectively Let us now compute the values of Z at these points We have\n*\nA corner point of a feasible region is a point in the region which is the intersection of two boundary lines"
    },
    {
        "Chapter": "1",
        "sentence_range": "6296-6299",
        "Text": "In the above example, the corner points (vertices) of the bounded (feasible) region\nare: O, A, B and C and it is easy to find their coordinates as (0, 0), (20, 0), (10, 50) and\n(0, 60) respectively Let us now compute the values of Z at these points We have\n*\nA corner point of a feasible region is a point in the region which is the intersection of two boundary lines **\nA feasible region of a system of linear inequalities is said to be bounded if it can be enclosed within a\ncircle"
    },
    {
        "Chapter": "1",
        "sentence_range": "6297-6300",
        "Text": "Let us now compute the values of Z at these points We have\n*\nA corner point of a feasible region is a point in the region which is the intersection of two boundary lines **\nA feasible region of a system of linear inequalities is said to be bounded if it can be enclosed within a\ncircle Otherwise, it is called unbounded"
    },
    {
        "Chapter": "1",
        "sentence_range": "6298-6301",
        "Text": "We have\n*\nA corner point of a feasible region is a point in the region which is the intersection of two boundary lines **\nA feasible region of a system of linear inequalities is said to be bounded if it can be enclosed within a\ncircle Otherwise, it is called unbounded Unbounded means that the feasible region does extend\nindefinitely in any direction"
    },
    {
        "Chapter": "1",
        "sentence_range": "6299-6302",
        "Text": "**\nA feasible region of a system of linear inequalities is said to be bounded if it can be enclosed within a\ncircle Otherwise, it is called unbounded Unbounded means that the feasible region does extend\nindefinitely in any direction Vertex of the\nCorresponding value\nFeasible Region\nof Z (in Rs)\nO (0,0)\n0\nC (0,60)\n4500\nB (10,50)\n6250\nA (20,0)\n 5000\nMaximum\n\u2190\n\u00a9 NCERT\nnot to be republished\nLINEAR PROGRAMMING         509\nWe observe that the maximum profit to the dealer results from the investment\nstrategy (10, 50), i"
    },
    {
        "Chapter": "1",
        "sentence_range": "6300-6303",
        "Text": "Otherwise, it is called unbounded Unbounded means that the feasible region does extend\nindefinitely in any direction Vertex of the\nCorresponding value\nFeasible Region\nof Z (in Rs)\nO (0,0)\n0\nC (0,60)\n4500\nB (10,50)\n6250\nA (20,0)\n 5000\nMaximum\n\u2190\n\u00a9 NCERT\nnot to be republished\nLINEAR PROGRAMMING         509\nWe observe that the maximum profit to the dealer results from the investment\nstrategy (10, 50), i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "6301-6304",
        "Text": "Unbounded means that the feasible region does extend\nindefinitely in any direction Vertex of the\nCorresponding value\nFeasible Region\nof Z (in Rs)\nO (0,0)\n0\nC (0,60)\n4500\nB (10,50)\n6250\nA (20,0)\n 5000\nMaximum\n\u2190\n\u00a9 NCERT\nnot to be republished\nLINEAR PROGRAMMING         509\nWe observe that the maximum profit to the dealer results from the investment\nstrategy (10, 50), i e buying 10 tables and 50 chairs"
    },
    {
        "Chapter": "1",
        "sentence_range": "6302-6305",
        "Text": "Vertex of the\nCorresponding value\nFeasible Region\nof Z (in Rs)\nO (0,0)\n0\nC (0,60)\n4500\nB (10,50)\n6250\nA (20,0)\n 5000\nMaximum\n\u2190\n\u00a9 NCERT\nnot to be republished\nLINEAR PROGRAMMING         509\nWe observe that the maximum profit to the dealer results from the investment\nstrategy (10, 50), i e buying 10 tables and 50 chairs This method of solving linear programming problem is referred as Corner Point\nMethod"
    },
    {
        "Chapter": "1",
        "sentence_range": "6303-6306",
        "Text": "e buying 10 tables and 50 chairs This method of solving linear programming problem is referred as Corner Point\nMethod The method comprises of the following steps:\n1"
    },
    {
        "Chapter": "1",
        "sentence_range": "6304-6307",
        "Text": "buying 10 tables and 50 chairs This method of solving linear programming problem is referred as Corner Point\nMethod The method comprises of the following steps:\n1 Find the feasible region of the linear programming problem and determine its\ncorner points (vertices) either by inspection or by solving the two equations of\nthe lines intersecting at that point"
    },
    {
        "Chapter": "1",
        "sentence_range": "6305-6308",
        "Text": "This method of solving linear programming problem is referred as Corner Point\nMethod The method comprises of the following steps:\n1 Find the feasible region of the linear programming problem and determine its\ncorner points (vertices) either by inspection or by solving the two equations of\nthe lines intersecting at that point 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "6306-6309",
        "Text": "The method comprises of the following steps:\n1 Find the feasible region of the linear programming problem and determine its\ncorner points (vertices) either by inspection or by solving the two equations of\nthe lines intersecting at that point 2 Evaluate the objective function Z = ax + by at each corner point"
    },
    {
        "Chapter": "1",
        "sentence_range": "6307-6310",
        "Text": "Find the feasible region of the linear programming problem and determine its\ncorner points (vertices) either by inspection or by solving the two equations of\nthe lines intersecting at that point 2 Evaluate the objective function Z = ax + by at each corner point Let M and m,\nrespectively denote the largest and smallest values of these points"
    },
    {
        "Chapter": "1",
        "sentence_range": "6308-6311",
        "Text": "2 Evaluate the objective function Z = ax + by at each corner point Let M and m,\nrespectively denote the largest and smallest values of these points 3"
    },
    {
        "Chapter": "1",
        "sentence_range": "6309-6312",
        "Text": "Evaluate the objective function Z = ax + by at each corner point Let M and m,\nrespectively denote the largest and smallest values of these points 3 (i)\nWhen the feasible region is bounded, M and m are the maximum and\nminimum values of Z"
    },
    {
        "Chapter": "1",
        "sentence_range": "6310-6313",
        "Text": "Let M and m,\nrespectively denote the largest and smallest values of these points 3 (i)\nWhen the feasible region is bounded, M and m are the maximum and\nminimum values of Z (ii) In case, the feasible region is unbounded, we have:\n4"
    },
    {
        "Chapter": "1",
        "sentence_range": "6311-6314",
        "Text": "3 (i)\nWhen the feasible region is bounded, M and m are the maximum and\nminimum values of Z (ii) In case, the feasible region is unbounded, we have:\n4 (a) M is the maximum value of Z, if the open half plane determined by\nax + by > M has no point in common with the feasible region"
    },
    {
        "Chapter": "1",
        "sentence_range": "6312-6315",
        "Text": "(i)\nWhen the feasible region is bounded, M and m are the maximum and\nminimum values of Z (ii) In case, the feasible region is unbounded, we have:\n4 (a) M is the maximum value of Z, if the open half plane determined by\nax + by > M has no point in common with the feasible region Otherwise, Z\nhas no maximum value"
    },
    {
        "Chapter": "1",
        "sentence_range": "6313-6316",
        "Text": "(ii) In case, the feasible region is unbounded, we have:\n4 (a) M is the maximum value of Z, if the open half plane determined by\nax + by > M has no point in common with the feasible region Otherwise, Z\nhas no maximum value (b) Similarly, m is the minimum value of Z, if the open half plane determined by\nax + by < m has no point in common with the feasible region"
    },
    {
        "Chapter": "1",
        "sentence_range": "6314-6317",
        "Text": "(a) M is the maximum value of Z, if the open half plane determined by\nax + by > M has no point in common with the feasible region Otherwise, Z\nhas no maximum value (b) Similarly, m is the minimum value of Z, if the open half plane determined by\nax + by < m has no point in common with the feasible region Otherwise, Z\nhas no minimum value"
    },
    {
        "Chapter": "1",
        "sentence_range": "6315-6318",
        "Text": "Otherwise, Z\nhas no maximum value (b) Similarly, m is the minimum value of Z, if the open half plane determined by\nax + by < m has no point in common with the feasible region Otherwise, Z\nhas no minimum value We will now illustrate these steps of Corner Point Method by considering some\nexamples:\nExample 1 Solve the following linear programming problem graphically:\nMaximise Z = 4x + y"
    },
    {
        "Chapter": "1",
        "sentence_range": "6316-6319",
        "Text": "(b) Similarly, m is the minimum value of Z, if the open half plane determined by\nax + by < m has no point in common with the feasible region Otherwise, Z\nhas no minimum value We will now illustrate these steps of Corner Point Method by considering some\nexamples:\nExample 1 Solve the following linear programming problem graphically:\nMaximise Z = 4x + y (1)\nsubject to the constraints:\nx + y \u2264 50"
    },
    {
        "Chapter": "1",
        "sentence_range": "6317-6320",
        "Text": "Otherwise, Z\nhas no minimum value We will now illustrate these steps of Corner Point Method by considering some\nexamples:\nExample 1 Solve the following linear programming problem graphically:\nMaximise Z = 4x + y (1)\nsubject to the constraints:\nx + y \u2264 50 (2)\n3x + y \u2264 90"
    },
    {
        "Chapter": "1",
        "sentence_range": "6318-6321",
        "Text": "We will now illustrate these steps of Corner Point Method by considering some\nexamples:\nExample 1 Solve the following linear programming problem graphically:\nMaximise Z = 4x + y (1)\nsubject to the constraints:\nx + y \u2264 50 (2)\n3x + y \u2264 90 (3)\nx \u2265 0, y \u2265 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "6319-6322",
        "Text": "(1)\nsubject to the constraints:\nx + y \u2264 50 (2)\n3x + y \u2264 90 (3)\nx \u2265 0, y \u2265 0 (4)\nSolution The shaded region in Fig 12"
    },
    {
        "Chapter": "1",
        "sentence_range": "6320-6323",
        "Text": "(2)\n3x + y \u2264 90 (3)\nx \u2265 0, y \u2265 0 (4)\nSolution The shaded region in Fig 12 2 is the feasible region determined by the system\nof constraints (2) to (4)"
    },
    {
        "Chapter": "1",
        "sentence_range": "6321-6324",
        "Text": "(3)\nx \u2265 0, y \u2265 0 (4)\nSolution The shaded region in Fig 12 2 is the feasible region determined by the system\nof constraints (2) to (4) We observe that the feasible region OABC is bounded"
    },
    {
        "Chapter": "1",
        "sentence_range": "6322-6325",
        "Text": "(4)\nSolution The shaded region in Fig 12 2 is the feasible region determined by the system\nof constraints (2) to (4) We observe that the feasible region OABC is bounded So,\nwe now use Corner Point Method to determine the maximum value of Z"
    },
    {
        "Chapter": "1",
        "sentence_range": "6323-6326",
        "Text": "2 is the feasible region determined by the system\nof constraints (2) to (4) We observe that the feasible region OABC is bounded So,\nwe now use Corner Point Method to determine the maximum value of Z The coordinates of the corner points O, A, B and C are (0, 0), (30, 0), (20, 30) and\n(0, 50) respectively"
    },
    {
        "Chapter": "1",
        "sentence_range": "6324-6327",
        "Text": "We observe that the feasible region OABC is bounded So,\nwe now use Corner Point Method to determine the maximum value of Z The coordinates of the corner points O, A, B and C are (0, 0), (30, 0), (20, 30) and\n(0, 50) respectively Now we evaluate Z at each corner point"
    },
    {
        "Chapter": "1",
        "sentence_range": "6325-6328",
        "Text": "So,\nwe now use Corner Point Method to determine the maximum value of Z The coordinates of the corner points O, A, B and C are (0, 0), (30, 0), (20, 30) and\n(0, 50) respectively Now we evaluate Z at each corner point \u00a9 NCERT\nnot to be republished\n510\nMATHEMATICS\nFig 12"
    },
    {
        "Chapter": "1",
        "sentence_range": "6326-6329",
        "Text": "The coordinates of the corner points O, A, B and C are (0, 0), (30, 0), (20, 30) and\n(0, 50) respectively Now we evaluate Z at each corner point \u00a9 NCERT\nnot to be republished\n510\nMATHEMATICS\nFig 12 2\nHence, maximum value of Z is 120 at the point (30, 0)"
    },
    {
        "Chapter": "1",
        "sentence_range": "6327-6330",
        "Text": "Now we evaluate Z at each corner point \u00a9 NCERT\nnot to be republished\n510\nMATHEMATICS\nFig 12 2\nHence, maximum value of Z is 120 at the point (30, 0) Example 2 Solve the following linear programming problem graphically:\nMinimise Z = 200 x + 500 y"
    },
    {
        "Chapter": "1",
        "sentence_range": "6328-6331",
        "Text": "\u00a9 NCERT\nnot to be republished\n510\nMATHEMATICS\nFig 12 2\nHence, maximum value of Z is 120 at the point (30, 0) Example 2 Solve the following linear programming problem graphically:\nMinimise Z = 200 x + 500 y (1)\nsubject to the constraints:\nx + 2y \u2265 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "6329-6332",
        "Text": "2\nHence, maximum value of Z is 120 at the point (30, 0) Example 2 Solve the following linear programming problem graphically:\nMinimise Z = 200 x + 500 y (1)\nsubject to the constraints:\nx + 2y \u2265 10 (2)\n3x + 4y \u2264 24"
    },
    {
        "Chapter": "1",
        "sentence_range": "6330-6333",
        "Text": "Example 2 Solve the following linear programming problem graphically:\nMinimise Z = 200 x + 500 y (1)\nsubject to the constraints:\nx + 2y \u2265 10 (2)\n3x + 4y \u2264 24 (3)\nx \u2265 0, y \u2265 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "6331-6334",
        "Text": "(1)\nsubject to the constraints:\nx + 2y \u2265 10 (2)\n3x + 4y \u2264 24 (3)\nx \u2265 0, y \u2265 0 (4)\nSolution The shaded region in Fig 12"
    },
    {
        "Chapter": "1",
        "sentence_range": "6332-6335",
        "Text": "(2)\n3x + 4y \u2264 24 (3)\nx \u2265 0, y \u2265 0 (4)\nSolution The shaded region in Fig 12 3 is the feasible region ABC determined by the\nsystem of constraints (2) to (4), which is bounded"
    },
    {
        "Chapter": "1",
        "sentence_range": "6333-6336",
        "Text": "(3)\nx \u2265 0, y \u2265 0 (4)\nSolution The shaded region in Fig 12 3 is the feasible region ABC determined by the\nsystem of constraints (2) to (4), which is bounded The coordinates of corner points\nCorner Point Corresponding value\nof Z\n(0, 0)\n0\n(30, 0)\n120 \u2190\nMaximum\n(20, 30)\n110\n(0, 50)\n50\nCorner Point Corresponding value\nof Z\n(0, 5)\n2500\n(4, 3)\n2300\n(0, 6)\n3000\nMinimum\n\u2190\nFig 12"
    },
    {
        "Chapter": "1",
        "sentence_range": "6334-6337",
        "Text": "(4)\nSolution The shaded region in Fig 12 3 is the feasible region ABC determined by the\nsystem of constraints (2) to (4), which is bounded The coordinates of corner points\nCorner Point Corresponding value\nof Z\n(0, 0)\n0\n(30, 0)\n120 \u2190\nMaximum\n(20, 30)\n110\n(0, 50)\n50\nCorner Point Corresponding value\nof Z\n(0, 5)\n2500\n(4, 3)\n2300\n(0, 6)\n3000\nMinimum\n\u2190\nFig 12 3\n\u00a9 NCERT\nnot to be republished\nLINEAR PROGRAMMING         511\nA, B and C are (0,5), (4,3) and (0,6) respectively"
    },
    {
        "Chapter": "1",
        "sentence_range": "6335-6338",
        "Text": "3 is the feasible region ABC determined by the\nsystem of constraints (2) to (4), which is bounded The coordinates of corner points\nCorner Point Corresponding value\nof Z\n(0, 0)\n0\n(30, 0)\n120 \u2190\nMaximum\n(20, 30)\n110\n(0, 50)\n50\nCorner Point Corresponding value\nof Z\n(0, 5)\n2500\n(4, 3)\n2300\n(0, 6)\n3000\nMinimum\n\u2190\nFig 12 3\n\u00a9 NCERT\nnot to be republished\nLINEAR PROGRAMMING         511\nA, B and C are (0,5), (4,3) and (0,6) respectively Now we evaluate Z  = 200x + 500y\nat these points"
    },
    {
        "Chapter": "1",
        "sentence_range": "6336-6339",
        "Text": "The coordinates of corner points\nCorner Point Corresponding value\nof Z\n(0, 0)\n0\n(30, 0)\n120 \u2190\nMaximum\n(20, 30)\n110\n(0, 50)\n50\nCorner Point Corresponding value\nof Z\n(0, 5)\n2500\n(4, 3)\n2300\n(0, 6)\n3000\nMinimum\n\u2190\nFig 12 3\n\u00a9 NCERT\nnot to be republished\nLINEAR PROGRAMMING         511\nA, B and C are (0,5), (4,3) and (0,6) respectively Now we evaluate Z  = 200x + 500y\nat these points Hence, minimum value of Z is 2300 attained at the point (4, 3)\nExample 3 Solve the following problem graphically:\nMinimise and Maximise Z = 3x + 9y"
    },
    {
        "Chapter": "1",
        "sentence_range": "6337-6340",
        "Text": "3\n\u00a9 NCERT\nnot to be republished\nLINEAR PROGRAMMING         511\nA, B and C are (0,5), (4,3) and (0,6) respectively Now we evaluate Z  = 200x + 500y\nat these points Hence, minimum value of Z is 2300 attained at the point (4, 3)\nExample 3 Solve the following problem graphically:\nMinimise and Maximise Z = 3x + 9y (1)\nsubject to the constraints:\nx + 3y \u2264 60"
    },
    {
        "Chapter": "1",
        "sentence_range": "6338-6341",
        "Text": "Now we evaluate Z  = 200x + 500y\nat these points Hence, minimum value of Z is 2300 attained at the point (4, 3)\nExample 3 Solve the following problem graphically:\nMinimise and Maximise Z = 3x + 9y (1)\nsubject to the constraints:\nx + 3y \u2264 60 (2)\nx + y \u2265 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "6339-6342",
        "Text": "Hence, minimum value of Z is 2300 attained at the point (4, 3)\nExample 3 Solve the following problem graphically:\nMinimise and Maximise Z = 3x + 9y (1)\nsubject to the constraints:\nx + 3y \u2264 60 (2)\nx + y \u2265 10 (3)\nx \u2264 y"
    },
    {
        "Chapter": "1",
        "sentence_range": "6340-6343",
        "Text": "(1)\nsubject to the constraints:\nx + 3y \u2264 60 (2)\nx + y \u2265 10 (3)\nx \u2264 y (4)\nx \u2265 0, y \u2265 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "6341-6344",
        "Text": "(2)\nx + y \u2265 10 (3)\nx \u2264 y (4)\nx \u2265 0, y \u2265 0 (5)\nSolution First of all, let us graph the feasible region of the system of linear inequalities\n(2) to (5)"
    },
    {
        "Chapter": "1",
        "sentence_range": "6342-6345",
        "Text": "(3)\nx \u2264 y (4)\nx \u2265 0, y \u2265 0 (5)\nSolution First of all, let us graph the feasible region of the system of linear inequalities\n(2) to (5) The feasible region ABCD is shown in the Fig 12"
    },
    {
        "Chapter": "1",
        "sentence_range": "6343-6346",
        "Text": "(4)\nx \u2265 0, y \u2265 0 (5)\nSolution First of all, let us graph the feasible region of the system of linear inequalities\n(2) to (5) The feasible region ABCD is shown in the Fig 12 4"
    },
    {
        "Chapter": "1",
        "sentence_range": "6344-6347",
        "Text": "(5)\nSolution First of all, let us graph the feasible region of the system of linear inequalities\n(2) to (5) The feasible region ABCD is shown in the Fig 12 4 Note that the region is\nbounded"
    },
    {
        "Chapter": "1",
        "sentence_range": "6345-6348",
        "Text": "The feasible region ABCD is shown in the Fig 12 4 Note that the region is\nbounded The coordinates of the corner points A, B, C and D are (0, 10), (5, 5), (15,15)\nand (0, 20) respectively"
    },
    {
        "Chapter": "1",
        "sentence_range": "6346-6349",
        "Text": "4 Note that the region is\nbounded The coordinates of the corner points A, B, C and D are (0, 10), (5, 5), (15,15)\nand (0, 20) respectively Fig 12"
    },
    {
        "Chapter": "1",
        "sentence_range": "6347-6350",
        "Text": "Note that the region is\nbounded The coordinates of the corner points A, B, C and D are (0, 10), (5, 5), (15,15)\nand (0, 20) respectively Fig 12 4\nCorner\nCorresponding value of\nPoint\n Z = 3x + 9y\nA (0, 10)\n90\nB (5, 5)\n60\nC (15, 15)\n180\nD (0, 20)\n180\nMinimum\nMaximum\n(Multiple\noptimal\nsolutions)\n\u2190\n}\u2190\nWe now find the minimum and maximum value of Z"
    },
    {
        "Chapter": "1",
        "sentence_range": "6348-6351",
        "Text": "The coordinates of the corner points A, B, C and D are (0, 10), (5, 5), (15,15)\nand (0, 20) respectively Fig 12 4\nCorner\nCorresponding value of\nPoint\n Z = 3x + 9y\nA (0, 10)\n90\nB (5, 5)\n60\nC (15, 15)\n180\nD (0, 20)\n180\nMinimum\nMaximum\n(Multiple\noptimal\nsolutions)\n\u2190\n}\u2190\nWe now find the minimum and maximum value of Z From the table, we find that\nthe minimum value of Z is 60 at the point B (5, 5) of the feasible region"
    },
    {
        "Chapter": "1",
        "sentence_range": "6349-6352",
        "Text": "Fig 12 4\nCorner\nCorresponding value of\nPoint\n Z = 3x + 9y\nA (0, 10)\n90\nB (5, 5)\n60\nC (15, 15)\n180\nD (0, 20)\n180\nMinimum\nMaximum\n(Multiple\noptimal\nsolutions)\n\u2190\n}\u2190\nWe now find the minimum and maximum value of Z From the table, we find that\nthe minimum value of Z is 60 at the point B (5, 5) of the feasible region The maximum value of Z on the feasible region occurs at the two corner points\nC (15, 15) and D (0, 20) and it is 180 in each case"
    },
    {
        "Chapter": "1",
        "sentence_range": "6350-6353",
        "Text": "4\nCorner\nCorresponding value of\nPoint\n Z = 3x + 9y\nA (0, 10)\n90\nB (5, 5)\n60\nC (15, 15)\n180\nD (0, 20)\n180\nMinimum\nMaximum\n(Multiple\noptimal\nsolutions)\n\u2190\n}\u2190\nWe now find the minimum and maximum value of Z From the table, we find that\nthe minimum value of Z is 60 at the point B (5, 5) of the feasible region The maximum value of Z on the feasible region occurs at the two corner points\nC (15, 15) and D (0, 20) and it is 180 in each case Remark Observe that in the above example, the  problem  has multiple optimal solutions\nat the corner points C and D, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "6351-6354",
        "Text": "From the table, we find that\nthe minimum value of Z is 60 at the point B (5, 5) of the feasible region The maximum value of Z on the feasible region occurs at the two corner points\nC (15, 15) and D (0, 20) and it is 180 in each case Remark Observe that in the above example, the  problem  has multiple optimal solutions\nat the corner points C and D, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "6352-6355",
        "Text": "The maximum value of Z on the feasible region occurs at the two corner points\nC (15, 15) and D (0, 20) and it is 180 in each case Remark Observe that in the above example, the  problem  has multiple optimal solutions\nat the corner points C and D, i e the both points produce same maximum value 180"
    },
    {
        "Chapter": "1",
        "sentence_range": "6353-6356",
        "Text": "Remark Observe that in the above example, the  problem  has multiple optimal solutions\nat the corner points C and D, i e the both points produce same maximum value 180 In\nsuch cases, you can see that every point on the line segment CD joining the two corner\npoints C and D also give the same maximum value"
    },
    {
        "Chapter": "1",
        "sentence_range": "6354-6357",
        "Text": "e the both points produce same maximum value 180 In\nsuch cases, you can see that every point on the line segment CD joining the two corner\npoints C and D also give the same maximum value Same is also true in the case if the\ntwo points produce same minimum value"
    },
    {
        "Chapter": "1",
        "sentence_range": "6355-6358",
        "Text": "the both points produce same maximum value 180 In\nsuch cases, you can see that every point on the line segment CD joining the two corner\npoints C and D also give the same maximum value Same is also true in the case if the\ntwo points produce same minimum value \u00a9 NCERT\nnot to be republished\n512\nMATHEMATICS\nExample 4 Determine graphically the minimum value of the objective function\nZ = \u2013 50x + 20y"
    },
    {
        "Chapter": "1",
        "sentence_range": "6356-6359",
        "Text": "In\nsuch cases, you can see that every point on the line segment CD joining the two corner\npoints C and D also give the same maximum value Same is also true in the case if the\ntwo points produce same minimum value \u00a9 NCERT\nnot to be republished\n512\nMATHEMATICS\nExample 4 Determine graphically the minimum value of the objective function\nZ = \u2013 50x + 20y (1)\nsubject to the constraints:\n2x \u2013 y \u2265 \u2013 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "6357-6360",
        "Text": "Same is also true in the case if the\ntwo points produce same minimum value \u00a9 NCERT\nnot to be republished\n512\nMATHEMATICS\nExample 4 Determine graphically the minimum value of the objective function\nZ = \u2013 50x + 20y (1)\nsubject to the constraints:\n2x \u2013 y \u2265 \u2013 5 (2)\n3x + y \u2265 3"
    },
    {
        "Chapter": "1",
        "sentence_range": "6358-6361",
        "Text": "\u00a9 NCERT\nnot to be republished\n512\nMATHEMATICS\nExample 4 Determine graphically the minimum value of the objective function\nZ = \u2013 50x + 20y (1)\nsubject to the constraints:\n2x \u2013 y \u2265 \u2013 5 (2)\n3x + y \u2265 3 (3)\n2x \u2013 3y \u2264 12"
    },
    {
        "Chapter": "1",
        "sentence_range": "6359-6362",
        "Text": "(1)\nsubject to the constraints:\n2x \u2013 y \u2265 \u2013 5 (2)\n3x + y \u2265 3 (3)\n2x \u2013 3y \u2264 12 (4)\nx \u2265 0, y \u2265 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "6360-6363",
        "Text": "(2)\n3x + y \u2265 3 (3)\n2x \u2013 3y \u2264 12 (4)\nx \u2265 0, y \u2265 0 (5)\nSolution First of all, let us graph the feasible region of the system of inequalities (2) to\n(5)"
    },
    {
        "Chapter": "1",
        "sentence_range": "6361-6364",
        "Text": "(3)\n2x \u2013 3y \u2264 12 (4)\nx \u2265 0, y \u2265 0 (5)\nSolution First of all, let us graph the feasible region of the system of inequalities (2) to\n(5) The feasible region (shaded) is shown in the Fig 12"
    },
    {
        "Chapter": "1",
        "sentence_range": "6362-6365",
        "Text": "(4)\nx \u2265 0, y \u2265 0 (5)\nSolution First of all, let us graph the feasible region of the system of inequalities (2) to\n(5) The feasible region (shaded) is shown in the Fig 12 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "6363-6366",
        "Text": "(5)\nSolution First of all, let us graph the feasible region of the system of inequalities (2) to\n(5) The feasible region (shaded) is shown in the Fig 12 5 Observe that the feasible\nregion is unbounded"
    },
    {
        "Chapter": "1",
        "sentence_range": "6364-6367",
        "Text": "The feasible region (shaded) is shown in the Fig 12 5 Observe that the feasible\nregion is unbounded We now evaluate Z at the corner points"
    },
    {
        "Chapter": "1",
        "sentence_range": "6365-6368",
        "Text": "5 Observe that the feasible\nregion is unbounded We now evaluate Z at the corner points From this table, we find that  \u2013 300 is the smallest value of Z at the corner point\n(6, 0)"
    },
    {
        "Chapter": "1",
        "sentence_range": "6366-6369",
        "Text": "Observe that the feasible\nregion is unbounded We now evaluate Z at the corner points From this table, we find that  \u2013 300 is the smallest value of Z at the corner point\n(6, 0) Can we say that minimum value of Z is \u2013 300"
    },
    {
        "Chapter": "1",
        "sentence_range": "6367-6370",
        "Text": "We now evaluate Z at the corner points From this table, we find that  \u2013 300 is the smallest value of Z at the corner point\n(6, 0) Can we say that minimum value of Z is \u2013 300 Note that if the region would\nhave been bounded, this smallest value of Z is the minimum value of Z (Theorem 2)"
    },
    {
        "Chapter": "1",
        "sentence_range": "6368-6371",
        "Text": "From this table, we find that  \u2013 300 is the smallest value of Z at the corner point\n(6, 0) Can we say that minimum value of Z is \u2013 300 Note that if the region would\nhave been bounded, this smallest value of Z is the minimum value of Z (Theorem 2) But here we see that the feasible region is unbounded"
    },
    {
        "Chapter": "1",
        "sentence_range": "6369-6372",
        "Text": "Can we say that minimum value of Z is \u2013 300 Note that if the region would\nhave been bounded, this smallest value of Z is the minimum value of Z (Theorem 2) But here we see that the feasible region is unbounded Therefore, \u2013 300 may or may\nnot be the minimum value of Z"
    },
    {
        "Chapter": "1",
        "sentence_range": "6370-6373",
        "Text": "Note that if the region would\nhave been bounded, this smallest value of Z is the minimum value of Z (Theorem 2) But here we see that the feasible region is unbounded Therefore, \u2013 300 may or may\nnot be the minimum value of Z To decide this issue, we graph the inequality\n\u2013 50x + 20y < \u2013 300 (see Step 3(ii) of corner Point Method"
    },
    {
        "Chapter": "1",
        "sentence_range": "6371-6374",
        "Text": "But here we see that the feasible region is unbounded Therefore, \u2013 300 may or may\nnot be the minimum value of Z To decide this issue, we graph the inequality\n\u2013 50x + 20y < \u2013 300 (see Step 3(ii) of corner Point Method )\ni"
    },
    {
        "Chapter": "1",
        "sentence_range": "6372-6375",
        "Text": "Therefore, \u2013 300 may or may\nnot be the minimum value of Z To decide this issue, we graph the inequality\n\u2013 50x + 20y < \u2013 300 (see Step 3(ii) of corner Point Method )\ni e"
    },
    {
        "Chapter": "1",
        "sentence_range": "6373-6376",
        "Text": "To decide this issue, we graph the inequality\n\u2013 50x + 20y < \u2013 300 (see Step 3(ii) of corner Point Method )\ni e ,\n\u2013 5x + 2y < \u2013 30\nand check whether the resulting open half plane has points in common with feasible\nregion or not"
    },
    {
        "Chapter": "1",
        "sentence_range": "6374-6377",
        "Text": ")\ni e ,\n\u2013 5x + 2y < \u2013 30\nand check whether the resulting open half plane has points in common with feasible\nregion or not If it has common points, then \u2013300 will not be the minimum value of Z"
    },
    {
        "Chapter": "1",
        "sentence_range": "6375-6378",
        "Text": "e ,\n\u2013 5x + 2y < \u2013 30\nand check whether the resulting open half plane has points in common with feasible\nregion or not If it has common points, then \u2013300 will not be the minimum value of Z Otherwise, \u2013300 will be the minimum value of Z"
    },
    {
        "Chapter": "1",
        "sentence_range": "6376-6379",
        "Text": ",\n\u2013 5x + 2y < \u2013 30\nand check whether the resulting open half plane has points in common with feasible\nregion or not If it has common points, then \u2013300 will not be the minimum value of Z Otherwise, \u2013300 will be the minimum value of Z Fig 12"
    },
    {
        "Chapter": "1",
        "sentence_range": "6377-6380",
        "Text": "If it has common points, then \u2013300 will not be the minimum value of Z Otherwise, \u2013300 will be the minimum value of Z Fig 12 5\nCorner Point\nZ = \u2013 50x + 20y\n(0, 5)\n100\n(0, 3)\n60\n(1, 0)\n\u201350\n(6, 0)\n\u2013 300\nsmallest\n\u2190\n\u00a9 NCERT\nnot to be republished\nLINEAR PROGRAMMING         513\nAs shown in the Fig 12"
    },
    {
        "Chapter": "1",
        "sentence_range": "6378-6381",
        "Text": "Otherwise, \u2013300 will be the minimum value of Z Fig 12 5\nCorner Point\nZ = \u2013 50x + 20y\n(0, 5)\n100\n(0, 3)\n60\n(1, 0)\n\u201350\n(6, 0)\n\u2013 300\nsmallest\n\u2190\n\u00a9 NCERT\nnot to be republished\nLINEAR PROGRAMMING         513\nAs shown in the Fig 12 5, it has common points"
    },
    {
        "Chapter": "1",
        "sentence_range": "6379-6382",
        "Text": "Fig 12 5\nCorner Point\nZ = \u2013 50x + 20y\n(0, 5)\n100\n(0, 3)\n60\n(1, 0)\n\u201350\n(6, 0)\n\u2013 300\nsmallest\n\u2190\n\u00a9 NCERT\nnot to be republished\nLINEAR PROGRAMMING         513\nAs shown in the Fig 12 5, it has common points Therefore, Z = \u201350 x + 20 y\nhas no minimum value subject to the given constraints"
    },
    {
        "Chapter": "1",
        "sentence_range": "6380-6383",
        "Text": "5\nCorner Point\nZ = \u2013 50x + 20y\n(0, 5)\n100\n(0, 3)\n60\n(1, 0)\n\u201350\n(6, 0)\n\u2013 300\nsmallest\n\u2190\n\u00a9 NCERT\nnot to be republished\nLINEAR PROGRAMMING         513\nAs shown in the Fig 12 5, it has common points Therefore, Z = \u201350 x + 20 y\nhas no minimum value subject to the given constraints In the above example, can you say whether z = \u2013 50 x + 20 y has the maximum\nvalue 100 at (0,5)"
    },
    {
        "Chapter": "1",
        "sentence_range": "6381-6384",
        "Text": "5, it has common points Therefore, Z = \u201350 x + 20 y\nhas no minimum value subject to the given constraints In the above example, can you say whether z = \u2013 50 x + 20 y has the maximum\nvalue 100 at (0,5) For this, check whether the graph of \u2013 50 x + 20 y > 100 has points\nin common with the feasible region"
    },
    {
        "Chapter": "1",
        "sentence_range": "6382-6385",
        "Text": "Therefore, Z = \u201350 x + 20 y\nhas no minimum value subject to the given constraints In the above example, can you say whether z = \u2013 50 x + 20 y has the maximum\nvalue 100 at (0,5) For this, check whether the graph of \u2013 50 x + 20 y > 100 has points\nin common with the feasible region (Why"
    },
    {
        "Chapter": "1",
        "sentence_range": "6383-6386",
        "Text": "In the above example, can you say whether z = \u2013 50 x + 20 y has the maximum\nvalue 100 at (0,5) For this, check whether the graph of \u2013 50 x + 20 y > 100 has points\nin common with the feasible region (Why )\nExample 5 Minimise Z = 3x + 2y\nsubject to the constraints:\nx + y \u2265 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "6384-6387",
        "Text": "For this, check whether the graph of \u2013 50 x + 20 y > 100 has points\nin common with the feasible region (Why )\nExample 5 Minimise Z = 3x + 2y\nsubject to the constraints:\nx + y \u2265 8 (1)\n3x + 5y \u2264 15"
    },
    {
        "Chapter": "1",
        "sentence_range": "6385-6388",
        "Text": "(Why )\nExample 5 Minimise Z = 3x + 2y\nsubject to the constraints:\nx + y \u2265 8 (1)\n3x + 5y \u2264 15 (2)\nx \u2265 0, y \u2265 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "6386-6389",
        "Text": ")\nExample 5 Minimise Z = 3x + 2y\nsubject to the constraints:\nx + y \u2265 8 (1)\n3x + 5y \u2264 15 (2)\nx \u2265 0, y \u2265 0 (3)\nSolution Let us graph the inequalities (1) to (3) (Fig 12"
    },
    {
        "Chapter": "1",
        "sentence_range": "6387-6390",
        "Text": "(1)\n3x + 5y \u2264 15 (2)\nx \u2265 0, y \u2265 0 (3)\nSolution Let us graph the inequalities (1) to (3) (Fig 12 6)"
    },
    {
        "Chapter": "1",
        "sentence_range": "6388-6391",
        "Text": "(2)\nx \u2265 0, y \u2265 0 (3)\nSolution Let us graph the inequalities (1) to (3) (Fig 12 6) Is there any feasible region"
    },
    {
        "Chapter": "1",
        "sentence_range": "6389-6392",
        "Text": "(3)\nSolution Let us graph the inequalities (1) to (3) (Fig 12 6) Is there any feasible region Why is so"
    },
    {
        "Chapter": "1",
        "sentence_range": "6390-6393",
        "Text": "6) Is there any feasible region Why is so From Fig 12"
    },
    {
        "Chapter": "1",
        "sentence_range": "6391-6394",
        "Text": "Is there any feasible region Why is so From Fig 12 6, you can see that\nthere is no point satisfying all the\nconstraints simultaneously"
    },
    {
        "Chapter": "1",
        "sentence_range": "6392-6395",
        "Text": "Why is so From Fig 12 6, you can see that\nthere is no point satisfying all the\nconstraints simultaneously Thus, the\nproblem is having no feasible region and\nhence no feasible solution"
    },
    {
        "Chapter": "1",
        "sentence_range": "6393-6396",
        "Text": "From Fig 12 6, you can see that\nthere is no point satisfying all the\nconstraints simultaneously Thus, the\nproblem is having no feasible region and\nhence no feasible solution Remarks From the examples which we\nhave discussed so far, we notice some\ngeneral features of linear programming\nproblems:\n(i)\nThe feasible region is always a\nconvex region"
    },
    {
        "Chapter": "1",
        "sentence_range": "6394-6397",
        "Text": "6, you can see that\nthere is no point satisfying all the\nconstraints simultaneously Thus, the\nproblem is having no feasible region and\nhence no feasible solution Remarks From the examples which we\nhave discussed so far, we notice some\ngeneral features of linear programming\nproblems:\n(i)\nThe feasible region is always a\nconvex region (ii)\nThe maximum (or minimum)\nsolution of the objective function occurs at the vertex (corner) of the feasible\nregion"
    },
    {
        "Chapter": "1",
        "sentence_range": "6395-6398",
        "Text": "Thus, the\nproblem is having no feasible region and\nhence no feasible solution Remarks From the examples which we\nhave discussed so far, we notice some\ngeneral features of linear programming\nproblems:\n(i)\nThe feasible region is always a\nconvex region (ii)\nThe maximum (or minimum)\nsolution of the objective function occurs at the vertex (corner) of the feasible\nregion If two corner points produce the same maximum (or minimum) value\nof the objective function, then every point on the line segment joining these\npoints will also give the same maximum (or minimum) value"
    },
    {
        "Chapter": "1",
        "sentence_range": "6396-6399",
        "Text": "Remarks From the examples which we\nhave discussed so far, we notice some\ngeneral features of linear programming\nproblems:\n(i)\nThe feasible region is always a\nconvex region (ii)\nThe maximum (or minimum)\nsolution of the objective function occurs at the vertex (corner) of the feasible\nregion If two corner points produce the same maximum (or minimum) value\nof the objective function, then every point on the line segment joining these\npoints will also give the same maximum (or minimum) value EXERCISE 12"
    },
    {
        "Chapter": "1",
        "sentence_range": "6397-6400",
        "Text": "(ii)\nThe maximum (or minimum)\nsolution of the objective function occurs at the vertex (corner) of the feasible\nregion If two corner points produce the same maximum (or minimum) value\nof the objective function, then every point on the line segment joining these\npoints will also give the same maximum (or minimum) value EXERCISE 12 1\nSolve the following Linear Programming Problems graphically:\n1"
    },
    {
        "Chapter": "1",
        "sentence_range": "6398-6401",
        "Text": "If two corner points produce the same maximum (or minimum) value\nof the objective function, then every point on the line segment joining these\npoints will also give the same maximum (or minimum) value EXERCISE 12 1\nSolve the following Linear Programming Problems graphically:\n1 Maximise Z =  3x + 4y\nsubject to the constraints : x + y \u2264 4, x  \u2265  0, y \u2265 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "6399-6402",
        "Text": "EXERCISE 12 1\nSolve the following Linear Programming Problems graphically:\n1 Maximise Z =  3x + 4y\nsubject to the constraints : x + y \u2264 4, x  \u2265  0, y \u2265 0 Fig 12"
    },
    {
        "Chapter": "1",
        "sentence_range": "6400-6403",
        "Text": "1\nSolve the following Linear Programming Problems graphically:\n1 Maximise Z =  3x + 4y\nsubject to the constraints : x + y \u2264 4, x  \u2265  0, y \u2265 0 Fig 12 6\n\u00a9 NCERT\nnot to be republished\n514\nMATHEMATICS\n2"
    },
    {
        "Chapter": "1",
        "sentence_range": "6401-6404",
        "Text": "Maximise Z =  3x + 4y\nsubject to the constraints : x + y \u2264 4, x  \u2265  0, y \u2265 0 Fig 12 6\n\u00a9 NCERT\nnot to be republished\n514\nMATHEMATICS\n2 Minimise Z = \u2013 3x + 4 y\nsubject to x + 2y \u2264 8, 3x + 2y \u2264 12,  x  \u2265  0, y \u2265 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "6402-6405",
        "Text": "Fig 12 6\n\u00a9 NCERT\nnot to be republished\n514\nMATHEMATICS\n2 Minimise Z = \u2013 3x + 4 y\nsubject to x + 2y \u2264 8, 3x + 2y \u2264 12,  x  \u2265  0, y \u2265 0 3"
    },
    {
        "Chapter": "1",
        "sentence_range": "6403-6406",
        "Text": "6\n\u00a9 NCERT\nnot to be republished\n514\nMATHEMATICS\n2 Minimise Z = \u2013 3x + 4 y\nsubject to x + 2y \u2264 8, 3x + 2y \u2264 12,  x  \u2265  0, y \u2265 0 3 Maximise Z = 5x + 3y\nsubject to 3x + 5y  \u2264 15, 5x + 2y \u2264 10, x  \u2265 0, y \u2265 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "6404-6407",
        "Text": "Minimise Z = \u2013 3x + 4 y\nsubject to x + 2y \u2264 8, 3x + 2y \u2264 12,  x  \u2265  0, y \u2265 0 3 Maximise Z = 5x + 3y\nsubject to 3x + 5y  \u2264 15, 5x + 2y \u2264 10, x  \u2265 0, y \u2265 0 4"
    },
    {
        "Chapter": "1",
        "sentence_range": "6405-6408",
        "Text": "3 Maximise Z = 5x + 3y\nsubject to 3x + 5y  \u2264 15, 5x + 2y \u2264 10, x  \u2265 0, y \u2265 0 4 Minimise Z = 3x + 5y\nsuch that x + 3y  \u2265 3, x + y  \u2265 2, x, y \u2265 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "6406-6409",
        "Text": "Maximise Z = 5x + 3y\nsubject to 3x + 5y  \u2264 15, 5x + 2y \u2264 10, x  \u2265 0, y \u2265 0 4 Minimise Z = 3x + 5y\nsuch that x + 3y  \u2265 3, x + y  \u2265 2, x, y \u2265 0 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "6407-6410",
        "Text": "4 Minimise Z = 3x + 5y\nsuch that x + 3y  \u2265 3, x + y  \u2265 2, x, y \u2265 0 5 Maximise Z = 3x + 2y\nsubject to x + 2y \u2264 10, 3x + y \u2264 15, x, y \u2265 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "6408-6411",
        "Text": "Minimise Z = 3x + 5y\nsuch that x + 3y  \u2265 3, x + y  \u2265 2, x, y \u2265 0 5 Maximise Z = 3x + 2y\nsubject to x + 2y \u2264 10, 3x + y \u2264 15, x, y \u2265 0 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "6409-6412",
        "Text": "5 Maximise Z = 3x + 2y\nsubject to x + 2y \u2264 10, 3x + y \u2264 15, x, y \u2265 0 6 Minimise Z = x + 2y\nsubject to 2x + y \u2265 3, x + 2y \u2265 6, x, y \u2265 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "6410-6413",
        "Text": "Maximise Z = 3x + 2y\nsubject to x + 2y \u2264 10, 3x + y \u2264 15, x, y \u2265 0 6 Minimise Z = x + 2y\nsubject to 2x + y \u2265 3, x + 2y \u2265 6, x, y \u2265 0 Show that the minimum of Z occurs at more than two points"
    },
    {
        "Chapter": "1",
        "sentence_range": "6411-6414",
        "Text": "6 Minimise Z = x + 2y\nsubject to 2x + y \u2265 3, x + 2y \u2265 6, x, y \u2265 0 Show that the minimum of Z occurs at more than two points 7"
    },
    {
        "Chapter": "1",
        "sentence_range": "6412-6415",
        "Text": "Minimise Z = x + 2y\nsubject to 2x + y \u2265 3, x + 2y \u2265 6, x, y \u2265 0 Show that the minimum of Z occurs at more than two points 7 Minimise and Maximise Z = 5x + 10 y\nsubject to x + 2y  \u2264 120, x + y \u2265 60, x \u2013 2y \u2265 0, x, y \u2265 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "6413-6416",
        "Text": "Show that the minimum of Z occurs at more than two points 7 Minimise and Maximise Z = 5x + 10 y\nsubject to x + 2y  \u2264 120, x + y \u2265 60, x \u2013 2y \u2265 0, x, y \u2265 0 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "6414-6417",
        "Text": "7 Minimise and Maximise Z = 5x + 10 y\nsubject to x + 2y  \u2264 120, x + y \u2265 60, x \u2013 2y \u2265 0, x, y \u2265 0 8 Minimise and Maximise Z = x + 2y\nsubject to x + 2y \u2265 100, 2x \u2013 y \u2264 0, 2x + y \u2264 200; x, y \u2265 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "6415-6418",
        "Text": "Minimise and Maximise Z = 5x + 10 y\nsubject to x + 2y  \u2264 120, x + y \u2265 60, x \u2013 2y \u2265 0, x, y \u2265 0 8 Minimise and Maximise Z = x + 2y\nsubject to x + 2y \u2265 100, 2x \u2013 y \u2264 0, 2x + y \u2264 200; x, y \u2265 0 9"
    },
    {
        "Chapter": "1",
        "sentence_range": "6416-6419",
        "Text": "8 Minimise and Maximise Z = x + 2y\nsubject to x + 2y \u2265 100, 2x \u2013 y \u2264 0, 2x + y \u2264 200; x, y \u2265 0 9 Maximise Z = \u2013 x + 2y, subject to the constraints:\nx \u2265 3, x + y \u2265 5, x + 2y \u2265 6, y \u2265 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "6417-6420",
        "Text": "Minimise and Maximise Z = x + 2y\nsubject to x + 2y \u2265 100, 2x \u2013 y \u2264 0, 2x + y \u2264 200; x, y \u2265 0 9 Maximise Z = \u2013 x + 2y, subject to the constraints:\nx \u2265 3, x + y \u2265 5, x + 2y \u2265 6, y \u2265 0 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "6418-6421",
        "Text": "9 Maximise Z = \u2013 x + 2y, subject to the constraints:\nx \u2265 3, x + y \u2265 5, x + 2y \u2265 6, y \u2265 0 10 Maximise Z = x + y, subject to x \u2013 y \u2264 \u20131, \u2013x + y \u2264  0,  x, y  \u2265 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "6419-6422",
        "Text": "Maximise Z = \u2013 x + 2y, subject to the constraints:\nx \u2265 3, x + y \u2265 5, x + 2y \u2265 6, y \u2265 0 10 Maximise Z = x + y, subject to x \u2013 y \u2264 \u20131, \u2013x + y \u2264  0,  x, y  \u2265 0 12"
    },
    {
        "Chapter": "1",
        "sentence_range": "6420-6423",
        "Text": "10 Maximise Z = x + y, subject to x \u2013 y \u2264 \u20131, \u2013x + y \u2264  0,  x, y  \u2265 0 12 3  Different Types of Linear Programming Problems\nA few important linear programming problems are listed below:\n1"
    },
    {
        "Chapter": "1",
        "sentence_range": "6421-6424",
        "Text": "Maximise Z = x + y, subject to x \u2013 y \u2264 \u20131, \u2013x + y \u2264  0,  x, y  \u2265 0 12 3  Different Types of Linear Programming Problems\nA few important linear programming problems are listed below:\n1 Manufacturing problems In these problems, we determine the number of units\nof different products which should be produced and sold by a firm\nwhen each product requires a fixed manpower, machine hours, labour hour per\nunit of product, warehouse space per unit of the output etc"
    },
    {
        "Chapter": "1",
        "sentence_range": "6422-6425",
        "Text": "12 3  Different Types of Linear Programming Problems\nA few important linear programming problems are listed below:\n1 Manufacturing problems In these problems, we determine the number of units\nof different products which should be produced and sold by a firm\nwhen each product requires a fixed manpower, machine hours, labour hour per\nunit of product, warehouse space per unit of the output etc , in order to make\nmaximum profit"
    },
    {
        "Chapter": "1",
        "sentence_range": "6423-6426",
        "Text": "3  Different Types of Linear Programming Problems\nA few important linear programming problems are listed below:\n1 Manufacturing problems In these problems, we determine the number of units\nof different products which should be produced and sold by a firm\nwhen each product requires a fixed manpower, machine hours, labour hour per\nunit of product, warehouse space per unit of the output etc , in order to make\nmaximum profit 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "6424-6427",
        "Text": "Manufacturing problems In these problems, we determine the number of units\nof different products which should be produced and sold by a firm\nwhen each product requires a fixed manpower, machine hours, labour hour per\nunit of product, warehouse space per unit of the output etc , in order to make\nmaximum profit 2 Diet problems In these problems, we determine the amount of different kinds\nof constituents/nutrients which should be included in a diet so as to minimise the\ncost of the desired diet such that it contains a certain minimum amount of each\nconstituent/nutrients"
    },
    {
        "Chapter": "1",
        "sentence_range": "6425-6428",
        "Text": ", in order to make\nmaximum profit 2 Diet problems In these problems, we determine the amount of different kinds\nof constituents/nutrients which should be included in a diet so as to minimise the\ncost of the desired diet such that it contains a certain minimum amount of each\nconstituent/nutrients 3"
    },
    {
        "Chapter": "1",
        "sentence_range": "6426-6429",
        "Text": "2 Diet problems In these problems, we determine the amount of different kinds\nof constituents/nutrients which should be included in a diet so as to minimise the\ncost of the desired diet such that it contains a certain minimum amount of each\nconstituent/nutrients 3 Transportation problems In these problems, we determine a transportation\nschedule in order to find the cheapest way of transporting a product from\nplants/factories situated at different locations to different markets"
    },
    {
        "Chapter": "1",
        "sentence_range": "6427-6430",
        "Text": "Diet problems In these problems, we determine the amount of different kinds\nof constituents/nutrients which should be included in a diet so as to minimise the\ncost of the desired diet such that it contains a certain minimum amount of each\nconstituent/nutrients 3 Transportation problems In these problems, we determine a transportation\nschedule in order to find the cheapest way of transporting a product from\nplants/factories situated at different locations to different markets \u00a9 NCERT\nnot to be republished\nLINEAR PROGRAMMING         515\nLet us now solve some of these types of linear programming problems:\nExample 6 (Diet problem): A dietician wishes to mix two types of foods in such a\nway that vitamin contents of the mixture contain atleast 8 units of vitamin A and 10\nunits of vitamin C"
    },
    {
        "Chapter": "1",
        "sentence_range": "6428-6431",
        "Text": "3 Transportation problems In these problems, we determine a transportation\nschedule in order to find the cheapest way of transporting a product from\nplants/factories situated at different locations to different markets \u00a9 NCERT\nnot to be republished\nLINEAR PROGRAMMING         515\nLet us now solve some of these types of linear programming problems:\nExample 6 (Diet problem): A dietician wishes to mix two types of foods in such a\nway that vitamin contents of the mixture contain atleast 8 units of vitamin A and 10\nunits of vitamin C Food \u2018I\u2019 contains 2 units/kg of vitamin A and 1 unit/kg of vitamin C"
    },
    {
        "Chapter": "1",
        "sentence_range": "6429-6432",
        "Text": "Transportation problems In these problems, we determine a transportation\nschedule in order to find the cheapest way of transporting a product from\nplants/factories situated at different locations to different markets \u00a9 NCERT\nnot to be republished\nLINEAR PROGRAMMING         515\nLet us now solve some of these types of linear programming problems:\nExample 6 (Diet problem): A dietician wishes to mix two types of foods in such a\nway that vitamin contents of the mixture contain atleast 8 units of vitamin A and 10\nunits of vitamin C Food \u2018I\u2019 contains 2 units/kg of vitamin A and 1 unit/kg of vitamin C Food \u2018II\u2019 contains 1 unit/kg of vitamin A and 2 units/kg of vitamin C"
    },
    {
        "Chapter": "1",
        "sentence_range": "6430-6433",
        "Text": "\u00a9 NCERT\nnot to be republished\nLINEAR PROGRAMMING         515\nLet us now solve some of these types of linear programming problems:\nExample 6 (Diet problem): A dietician wishes to mix two types of foods in such a\nway that vitamin contents of the mixture contain atleast 8 units of vitamin A and 10\nunits of vitamin C Food \u2018I\u2019 contains 2 units/kg of vitamin A and 1 unit/kg of vitamin C Food \u2018II\u2019 contains 1 unit/kg of vitamin A and 2 units/kg of vitamin C It costs\nRs 50 per kg to purchase Food \u2018I\u2019 and Rs 70 per kg to purchase Food \u2018II\u2019"
    },
    {
        "Chapter": "1",
        "sentence_range": "6431-6434",
        "Text": "Food \u2018I\u2019 contains 2 units/kg of vitamin A and 1 unit/kg of vitamin C Food \u2018II\u2019 contains 1 unit/kg of vitamin A and 2 units/kg of vitamin C It costs\nRs 50 per kg to purchase Food \u2018I\u2019 and Rs 70 per kg to purchase Food \u2018II\u2019 Formulate\nthis problem as a linear programming problem to minimise the cost of such a mixture"
    },
    {
        "Chapter": "1",
        "sentence_range": "6432-6435",
        "Text": "Food \u2018II\u2019 contains 1 unit/kg of vitamin A and 2 units/kg of vitamin C It costs\nRs 50 per kg to purchase Food \u2018I\u2019 and Rs 70 per kg to purchase Food \u2018II\u2019 Formulate\nthis problem as a linear programming problem to minimise the cost of such a mixture Solution Let the mixture contain x kg of Food \u2018I\u2019 and y kg of Food \u2018II\u2019"
    },
    {
        "Chapter": "1",
        "sentence_range": "6433-6436",
        "Text": "It costs\nRs 50 per kg to purchase Food \u2018I\u2019 and Rs 70 per kg to purchase Food \u2018II\u2019 Formulate\nthis problem as a linear programming problem to minimise the cost of such a mixture Solution Let the mixture contain x kg of Food \u2018I\u2019 and y kg of Food \u2018II\u2019 Clearly, x \u2265 0,\ny \u2265 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "6434-6437",
        "Text": "Formulate\nthis problem as a linear programming problem to minimise the cost of such a mixture Solution Let the mixture contain x kg of Food \u2018I\u2019 and y kg of Food \u2018II\u2019 Clearly, x \u2265 0,\ny \u2265 0 We make the following table from the given data:\nResources\nFood\nRequirement\nI\nII\n(x)\n(y)\nVitamin A\n2\n1\n8\n(units/kg)\nVitamin C\n1\n2\n10\n(units/kg)\nCost (Rs/kg)\n50\n70\nSince the mixture must contain at least 8 units of vitamin A and 10 units of\nvitamin C, we have the constraints:\n2x + y \u2265 8\nx + 2y \u2265 10\nTotal cost Z of purchasing x kg of food \u2018I\u2019 and y kg of Food \u2018II\u2019 is\nZ = 50x + 70y\nHence, the mathematical formulation of the problem is:\nMinimise\nZ = 50x + 70y"
    },
    {
        "Chapter": "1",
        "sentence_range": "6435-6438",
        "Text": "Solution Let the mixture contain x kg of Food \u2018I\u2019 and y kg of Food \u2018II\u2019 Clearly, x \u2265 0,\ny \u2265 0 We make the following table from the given data:\nResources\nFood\nRequirement\nI\nII\n(x)\n(y)\nVitamin A\n2\n1\n8\n(units/kg)\nVitamin C\n1\n2\n10\n(units/kg)\nCost (Rs/kg)\n50\n70\nSince the mixture must contain at least 8 units of vitamin A and 10 units of\nvitamin C, we have the constraints:\n2x + y \u2265 8\nx + 2y \u2265 10\nTotal cost Z of purchasing x kg of food \u2018I\u2019 and y kg of Food \u2018II\u2019 is\nZ = 50x + 70y\nHence, the mathematical formulation of the problem is:\nMinimise\nZ = 50x + 70y (1)\nsubject to the constraints:\n2x + y \u2265 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "6436-6439",
        "Text": "Clearly, x \u2265 0,\ny \u2265 0 We make the following table from the given data:\nResources\nFood\nRequirement\nI\nII\n(x)\n(y)\nVitamin A\n2\n1\n8\n(units/kg)\nVitamin C\n1\n2\n10\n(units/kg)\nCost (Rs/kg)\n50\n70\nSince the mixture must contain at least 8 units of vitamin A and 10 units of\nvitamin C, we have the constraints:\n2x + y \u2265 8\nx + 2y \u2265 10\nTotal cost Z of purchasing x kg of food \u2018I\u2019 and y kg of Food \u2018II\u2019 is\nZ = 50x + 70y\nHence, the mathematical formulation of the problem is:\nMinimise\nZ = 50x + 70y (1)\nsubject to the constraints:\n2x + y \u2265 8 (2)\nx + 2y \u2265 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "6437-6440",
        "Text": "We make the following table from the given data:\nResources\nFood\nRequirement\nI\nII\n(x)\n(y)\nVitamin A\n2\n1\n8\n(units/kg)\nVitamin C\n1\n2\n10\n(units/kg)\nCost (Rs/kg)\n50\n70\nSince the mixture must contain at least 8 units of vitamin A and 10 units of\nvitamin C, we have the constraints:\n2x + y \u2265 8\nx + 2y \u2265 10\nTotal cost Z of purchasing x kg of food \u2018I\u2019 and y kg of Food \u2018II\u2019 is\nZ = 50x + 70y\nHence, the mathematical formulation of the problem is:\nMinimise\nZ = 50x + 70y (1)\nsubject to the constraints:\n2x + y \u2265 8 (2)\nx + 2y \u2265 10 (3)\nx, y \u2265 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "6438-6441",
        "Text": "(1)\nsubject to the constraints:\n2x + y \u2265 8 (2)\nx + 2y \u2265 10 (3)\nx, y \u2265 0 (4)\nLet us graph the inequalities (2) to (4)"
    },
    {
        "Chapter": "1",
        "sentence_range": "6439-6442",
        "Text": "(2)\nx + 2y \u2265 10 (3)\nx, y \u2265 0 (4)\nLet us graph the inequalities (2) to (4) The feasible region determined by the\nsystem is shown in the Fig 12"
    },
    {
        "Chapter": "1",
        "sentence_range": "6440-6443",
        "Text": "(3)\nx, y \u2265 0 (4)\nLet us graph the inequalities (2) to (4) The feasible region determined by the\nsystem is shown in the Fig 12 7"
    },
    {
        "Chapter": "1",
        "sentence_range": "6441-6444",
        "Text": "(4)\nLet us graph the inequalities (2) to (4) The feasible region determined by the\nsystem is shown in the Fig 12 7 Here again, observe that the feasible region is\nunbounded"
    },
    {
        "Chapter": "1",
        "sentence_range": "6442-6445",
        "Text": "The feasible region determined by the\nsystem is shown in the Fig 12 7 Here again, observe that the feasible region is\nunbounded \u00a9 NCERT\nnot to be republished\n516\nMATHEMATICS\nLet us evaluate Z at the corner points A(0,8), B(2,4) and C(10,0)"
    },
    {
        "Chapter": "1",
        "sentence_range": "6443-6446",
        "Text": "7 Here again, observe that the feasible region is\nunbounded \u00a9 NCERT\nnot to be republished\n516\nMATHEMATICS\nLet us evaluate Z at the corner points A(0,8), B(2,4) and C(10,0) Fig 12"
    },
    {
        "Chapter": "1",
        "sentence_range": "6444-6447",
        "Text": "Here again, observe that the feasible region is\nunbounded \u00a9 NCERT\nnot to be republished\n516\nMATHEMATICS\nLet us evaluate Z at the corner points A(0,8), B(2,4) and C(10,0) Fig 12 7\nIn the table, we find that smallest value of Z is 380 at the point (2,4)"
    },
    {
        "Chapter": "1",
        "sentence_range": "6445-6448",
        "Text": "\u00a9 NCERT\nnot to be republished\n516\nMATHEMATICS\nLet us evaluate Z at the corner points A(0,8), B(2,4) and C(10,0) Fig 12 7\nIn the table, we find that smallest value of Z is 380 at the point (2,4) Can we say\nthat the minimum value of Z is 380"
    },
    {
        "Chapter": "1",
        "sentence_range": "6446-6449",
        "Text": "Fig 12 7\nIn the table, we find that smallest value of Z is 380 at the point (2,4) Can we say\nthat the minimum value of Z is 380 Remember that the feasible region is unbounded"
    },
    {
        "Chapter": "1",
        "sentence_range": "6447-6450",
        "Text": "7\nIn the table, we find that smallest value of Z is 380 at the point (2,4) Can we say\nthat the minimum value of Z is 380 Remember that the feasible region is unbounded Therefore, we have to draw the graph of the inequality\n50x + 70y < 380  i"
    },
    {
        "Chapter": "1",
        "sentence_range": "6448-6451",
        "Text": "Can we say\nthat the minimum value of Z is 380 Remember that the feasible region is unbounded Therefore, we have to draw the graph of the inequality\n50x + 70y < 380  i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "6449-6452",
        "Text": "Remember that the feasible region is unbounded Therefore, we have to draw the graph of the inequality\n50x + 70y < 380  i e ,\n5x + 7y < 38\nto check whether the resulting open half plane has any point common with the feasible\nregion"
    },
    {
        "Chapter": "1",
        "sentence_range": "6450-6453",
        "Text": "Therefore, we have to draw the graph of the inequality\n50x + 70y < 380  i e ,\n5x + 7y < 38\nto check whether the resulting open half plane has any point common with the feasible\nregion From the Fig 12"
    },
    {
        "Chapter": "1",
        "sentence_range": "6451-6454",
        "Text": "e ,\n5x + 7y < 38\nto check whether the resulting open half plane has any point common with the feasible\nregion From the Fig 12 7, we see that it has no points in common"
    },
    {
        "Chapter": "1",
        "sentence_range": "6452-6455",
        "Text": ",\n5x + 7y < 38\nto check whether the resulting open half plane has any point common with the feasible\nregion From the Fig 12 7, we see that it has no points in common Thus, the minimum value of Z is 380 attained at the point (2, 4)"
    },
    {
        "Chapter": "1",
        "sentence_range": "6453-6456",
        "Text": "From the Fig 12 7, we see that it has no points in common Thus, the minimum value of Z is 380 attained at the point (2, 4) Hence, the optimal\nmixing strategy for the dietician would be to mix 2 kg of Food \u2018I\u2019 and 4 kg of Food \u2018II\u2019,\nand with this strategy, the minimum cost of the mixture will be Rs 380"
    },
    {
        "Chapter": "1",
        "sentence_range": "6454-6457",
        "Text": "7, we see that it has no points in common Thus, the minimum value of Z is 380 attained at the point (2, 4) Hence, the optimal\nmixing strategy for the dietician would be to mix 2 kg of Food \u2018I\u2019 and 4 kg of Food \u2018II\u2019,\nand with this strategy, the minimum cost of the mixture will be Rs 380 Example 7 (Allocation problem) A cooperative society of farmers has 50 hectare\nof land to grow two crops X and Y"
    },
    {
        "Chapter": "1",
        "sentence_range": "6455-6458",
        "Text": "Thus, the minimum value of Z is 380 attained at the point (2, 4) Hence, the optimal\nmixing strategy for the dietician would be to mix 2 kg of Food \u2018I\u2019 and 4 kg of Food \u2018II\u2019,\nand with this strategy, the minimum cost of the mixture will be Rs 380 Example 7 (Allocation problem) A cooperative society of farmers has 50 hectare\nof land to grow two crops X and Y The profit from crops X and Y per hectare are\nestimated as Rs 10,500 and Rs 9,000 respectively"
    },
    {
        "Chapter": "1",
        "sentence_range": "6456-6459",
        "Text": "Hence, the optimal\nmixing strategy for the dietician would be to mix 2 kg of Food \u2018I\u2019 and 4 kg of Food \u2018II\u2019,\nand with this strategy, the minimum cost of the mixture will be Rs 380 Example 7 (Allocation problem) A cooperative society of farmers has 50 hectare\nof land to grow two crops X and Y The profit from crops X and Y per hectare are\nestimated as Rs 10,500 and Rs 9,000 respectively To control weeds, a liquid herbicide\nhas to be used for crops X and Y at rates of 20 litres and 10 litres per hectare"
    },
    {
        "Chapter": "1",
        "sentence_range": "6457-6460",
        "Text": "Example 7 (Allocation problem) A cooperative society of farmers has 50 hectare\nof land to grow two crops X and Y The profit from crops X and Y per hectare are\nestimated as Rs 10,500 and Rs 9,000 respectively To control weeds, a liquid herbicide\nhas to be used for crops X and Y at rates of 20 litres and 10 litres per hectare Further,\nno more than 800 litres of herbicide should be used in order to protect fish and wild life\nusing a pond which collects drainage from this land"
    },
    {
        "Chapter": "1",
        "sentence_range": "6458-6461",
        "Text": "The profit from crops X and Y per hectare are\nestimated as Rs 10,500 and Rs 9,000 respectively To control weeds, a liquid herbicide\nhas to be used for crops X and Y at rates of 20 litres and 10 litres per hectare Further,\nno more than 800 litres of herbicide should be used in order to protect fish and wild life\nusing a pond which collects drainage from this land How much land should be allocated\nto each crop so as to maximise the total profit of the society"
    },
    {
        "Chapter": "1",
        "sentence_range": "6459-6462",
        "Text": "To control weeds, a liquid herbicide\nhas to be used for crops X and Y at rates of 20 litres and 10 litres per hectare Further,\nno more than 800 litres of herbicide should be used in order to protect fish and wild life\nusing a pond which collects drainage from this land How much land should be allocated\nto each crop so as to maximise the total profit of the society Solution Let x hectare of land be allocated to crop X and y hectare to crop Y"
    },
    {
        "Chapter": "1",
        "sentence_range": "6460-6463",
        "Text": "Further,\nno more than 800 litres of herbicide should be used in order to protect fish and wild life\nusing a pond which collects drainage from this land How much land should be allocated\nto each crop so as to maximise the total profit of the society Solution Let x hectare of land be allocated to crop X and y hectare to crop Y Obviously,\nx \u2265 0, y \u2265 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "6461-6464",
        "Text": "How much land should be allocated\nto each crop so as to maximise the total profit of the society Solution Let x hectare of land be allocated to crop X and y hectare to crop Y Obviously,\nx \u2265 0, y \u2265 0 Profit per hectare on crop X = Rs 10500\nProfit per hectare on crop Y = Rs 9000\nTherefore, total profit\n= Rs (10500x + 9000y)\nCorner Point\nZ = 50x + 70y\n(0,8)\n560\n(2,4)\n380\n(10,0)\n500\nMinimum\n\u2190\n\u00a9 NCERT\nnot to be republished\nLINEAR PROGRAMMING         517\nThe mathematical formulation of the problem is as follows:\nMaximise\nZ = 10500 x + 9000 y\nsubject  to the constraints:\nx + y \u2264 50   (constraint related to land)"
    },
    {
        "Chapter": "1",
        "sentence_range": "6462-6465",
        "Text": "Solution Let x hectare of land be allocated to crop X and y hectare to crop Y Obviously,\nx \u2265 0, y \u2265 0 Profit per hectare on crop X = Rs 10500\nProfit per hectare on crop Y = Rs 9000\nTherefore, total profit\n= Rs (10500x + 9000y)\nCorner Point\nZ = 50x + 70y\n(0,8)\n560\n(2,4)\n380\n(10,0)\n500\nMinimum\n\u2190\n\u00a9 NCERT\nnot to be republished\nLINEAR PROGRAMMING         517\nThe mathematical formulation of the problem is as follows:\nMaximise\nZ = 10500 x + 9000 y\nsubject  to the constraints:\nx + y \u2264 50   (constraint related to land) (1)\n20x + 10y \u2264 800 (constraint related to use of herbicide)\ni"
    },
    {
        "Chapter": "1",
        "sentence_range": "6463-6466",
        "Text": "Obviously,\nx \u2265 0, y \u2265 0 Profit per hectare on crop X = Rs 10500\nProfit per hectare on crop Y = Rs 9000\nTherefore, total profit\n= Rs (10500x + 9000y)\nCorner Point\nZ = 50x + 70y\n(0,8)\n560\n(2,4)\n380\n(10,0)\n500\nMinimum\n\u2190\n\u00a9 NCERT\nnot to be republished\nLINEAR PROGRAMMING         517\nThe mathematical formulation of the problem is as follows:\nMaximise\nZ = 10500 x + 9000 y\nsubject  to the constraints:\nx + y \u2264 50   (constraint related to land) (1)\n20x + 10y \u2264 800 (constraint related to use of herbicide)\ni e"
    },
    {
        "Chapter": "1",
        "sentence_range": "6464-6467",
        "Text": "Profit per hectare on crop X = Rs 10500\nProfit per hectare on crop Y = Rs 9000\nTherefore, total profit\n= Rs (10500x + 9000y)\nCorner Point\nZ = 50x + 70y\n(0,8)\n560\n(2,4)\n380\n(10,0)\n500\nMinimum\n\u2190\n\u00a9 NCERT\nnot to be republished\nLINEAR PROGRAMMING         517\nThe mathematical formulation of the problem is as follows:\nMaximise\nZ = 10500 x + 9000 y\nsubject  to the constraints:\nx + y \u2264 50   (constraint related to land) (1)\n20x + 10y \u2264 800 (constraint related to use of herbicide)\ni e 2x + y \u2264 80"
    },
    {
        "Chapter": "1",
        "sentence_range": "6465-6468",
        "Text": "(1)\n20x + 10y \u2264 800 (constraint related to use of herbicide)\ni e 2x + y \u2264 80 (2)\nx \u2265 0, y \u2265 0\n    (non negative constraint)"
    },
    {
        "Chapter": "1",
        "sentence_range": "6466-6469",
        "Text": "e 2x + y \u2264 80 (2)\nx \u2265 0, y \u2265 0\n    (non negative constraint) (3)\nLet us draw the graph of the system of inequalities (1) to (3)"
    },
    {
        "Chapter": "1",
        "sentence_range": "6467-6470",
        "Text": "2x + y \u2264 80 (2)\nx \u2265 0, y \u2265 0\n    (non negative constraint) (3)\nLet us draw the graph of the system of inequalities (1) to (3) The feasible region\nOABC is shown (shaded) in the Fig 12"
    },
    {
        "Chapter": "1",
        "sentence_range": "6468-6471",
        "Text": "(2)\nx \u2265 0, y \u2265 0\n    (non negative constraint) (3)\nLet us draw the graph of the system of inequalities (1) to (3) The feasible region\nOABC is shown (shaded) in the Fig 12 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "6469-6472",
        "Text": "(3)\nLet us draw the graph of the system of inequalities (1) to (3) The feasible region\nOABC is shown (shaded) in the Fig 12 8 Observe that the feasible region is bounded"
    },
    {
        "Chapter": "1",
        "sentence_range": "6470-6473",
        "Text": "The feasible region\nOABC is shown (shaded) in the Fig 12 8 Observe that the feasible region is bounded The coordinates of the corner points O, A, B and C are (0, 0), (40, 0), (30, 20) and\n(0, 50) respectively"
    },
    {
        "Chapter": "1",
        "sentence_range": "6471-6474",
        "Text": "8 Observe that the feasible region is bounded The coordinates of the corner points O, A, B and C are (0, 0), (40, 0), (30, 20) and\n(0, 50) respectively Let us evaluate the objective function Z = 10500 x  + 9000y at\nthese vertices to find which one gives the maximum profit"
    },
    {
        "Chapter": "1",
        "sentence_range": "6472-6475",
        "Text": "Observe that the feasible region is bounded The coordinates of the corner points O, A, B and C are (0, 0), (40, 0), (30, 20) and\n(0, 50) respectively Let us evaluate the objective function Z = 10500 x  + 9000y at\nthese vertices to find which one gives the maximum profit Fig 12"
    },
    {
        "Chapter": "1",
        "sentence_range": "6473-6476",
        "Text": "The coordinates of the corner points O, A, B and C are (0, 0), (40, 0), (30, 20) and\n(0, 50) respectively Let us evaluate the objective function Z = 10500 x  + 9000y at\nthese vertices to find which one gives the maximum profit Fig 12 8\nHence, the society will get the maximum profit of Rs 4,95,000 by allocating 30\nhectares for crop X and 20 hectares for crop Y"
    },
    {
        "Chapter": "1",
        "sentence_range": "6474-6477",
        "Text": "Let us evaluate the objective function Z = 10500 x  + 9000y at\nthese vertices to find which one gives the maximum profit Fig 12 8\nHence, the society will get the maximum profit of Rs 4,95,000 by allocating 30\nhectares for crop X and 20 hectares for crop Y Example 8 (Manufacturing problem) A manufacturing company makes two models\nA and B of a product"
    },
    {
        "Chapter": "1",
        "sentence_range": "6475-6478",
        "Text": "Fig 12 8\nHence, the society will get the maximum profit of Rs 4,95,000 by allocating 30\nhectares for crop X and 20 hectares for crop Y Example 8 (Manufacturing problem) A manufacturing company makes two models\nA and B of a product Each piece of Model A requires 9 labour hours for fabricating\nand 1 labour hour for finishing"
    },
    {
        "Chapter": "1",
        "sentence_range": "6476-6479",
        "Text": "8\nHence, the society will get the maximum profit of Rs 4,95,000 by allocating 30\nhectares for crop X and 20 hectares for crop Y Example 8 (Manufacturing problem) A manufacturing company makes two models\nA and B of a product Each piece of Model A requires 9 labour hours for fabricating\nand 1 labour hour for finishing Each piece of Model B requires 12 labour hours for\nfabricating and 3 labour hours for finishing"
    },
    {
        "Chapter": "1",
        "sentence_range": "6477-6480",
        "Text": "Example 8 (Manufacturing problem) A manufacturing company makes two models\nA and B of a product Each piece of Model A requires 9 labour hours for fabricating\nand 1 labour hour for finishing Each piece of Model B requires 12 labour hours for\nfabricating and 3 labour hours for finishing For fabricating and finishing, the maximum\nlabour hours available are 180 and 30 respectively"
    },
    {
        "Chapter": "1",
        "sentence_range": "6478-6481",
        "Text": "Each piece of Model A requires 9 labour hours for fabricating\nand 1 labour hour for finishing Each piece of Model B requires 12 labour hours for\nfabricating and 3 labour hours for finishing For fabricating and finishing, the maximum\nlabour hours available are 180 and 30 respectively The company makes a profit of\nRs 8000 on each piece of model A and Rs 12000 on each piece of Model B"
    },
    {
        "Chapter": "1",
        "sentence_range": "6479-6482",
        "Text": "Each piece of Model B requires 12 labour hours for\nfabricating and 3 labour hours for finishing For fabricating and finishing, the maximum\nlabour hours available are 180 and 30 respectively The company makes a profit of\nRs 8000 on each piece of model A and Rs 12000 on each piece of Model B How many\npieces of Model A and Model B should be manufactured per week to realise a maximum\nprofit"
    },
    {
        "Chapter": "1",
        "sentence_range": "6480-6483",
        "Text": "For fabricating and finishing, the maximum\nlabour hours available are 180 and 30 respectively The company makes a profit of\nRs 8000 on each piece of model A and Rs 12000 on each piece of Model B How many\npieces of Model A and Model B should be manufactured per week to realise a maximum\nprofit What is the maximum profit per week"
    },
    {
        "Chapter": "1",
        "sentence_range": "6481-6484",
        "Text": "The company makes a profit of\nRs 8000 on each piece of model A and Rs 12000 on each piece of Model B How many\npieces of Model A and Model B should be manufactured per week to realise a maximum\nprofit What is the maximum profit per week Corner Point Z = 10500x + 9000y\nO(0, 0)\n0\nA( 40, 0)\n420000\nB(30, 20)\n495000\nC(0,50)\n450000\n\u2190     Maximum\n\u00a9 NCERT\nnot to be republished\n518\nMATHEMATICS\n\u2190\nSolution Suppose x is the number of pieces of Model A and y is the number of pieces\nof Model B"
    },
    {
        "Chapter": "1",
        "sentence_range": "6482-6485",
        "Text": "How many\npieces of Model A and Model B should be manufactured per week to realise a maximum\nprofit What is the maximum profit per week Corner Point Z = 10500x + 9000y\nO(0, 0)\n0\nA( 40, 0)\n420000\nB(30, 20)\n495000\nC(0,50)\n450000\n\u2190     Maximum\n\u00a9 NCERT\nnot to be republished\n518\nMATHEMATICS\n\u2190\nSolution Suppose x is the number of pieces of Model A and y is the number of pieces\nof Model B Then\nTotal profit (in Rs) = 8000 x + 12000 y\nLet\nZ = 8000 x + 12000 y\nWe now have the following mathematical model for the given problem"
    },
    {
        "Chapter": "1",
        "sentence_range": "6483-6486",
        "Text": "What is the maximum profit per week Corner Point Z = 10500x + 9000y\nO(0, 0)\n0\nA( 40, 0)\n420000\nB(30, 20)\n495000\nC(0,50)\n450000\n\u2190     Maximum\n\u00a9 NCERT\nnot to be republished\n518\nMATHEMATICS\n\u2190\nSolution Suppose x is the number of pieces of Model A and y is the number of pieces\nof Model B Then\nTotal profit (in Rs) = 8000 x + 12000 y\nLet\nZ = 8000 x + 12000 y\nWe now have the following mathematical model for the given problem Maximise Z = 8000 x + 12000 y"
    },
    {
        "Chapter": "1",
        "sentence_range": "6484-6487",
        "Text": "Corner Point Z = 10500x + 9000y\nO(0, 0)\n0\nA( 40, 0)\n420000\nB(30, 20)\n495000\nC(0,50)\n450000\n\u2190     Maximum\n\u00a9 NCERT\nnot to be republished\n518\nMATHEMATICS\n\u2190\nSolution Suppose x is the number of pieces of Model A and y is the number of pieces\nof Model B Then\nTotal profit (in Rs) = 8000 x + 12000 y\nLet\nZ = 8000 x + 12000 y\nWe now have the following mathematical model for the given problem Maximise Z = 8000 x + 12000 y (1)\nsubject to the constraints:\n9x + 12y \u2264 180     (Fabricating constraint)\ni"
    },
    {
        "Chapter": "1",
        "sentence_range": "6485-6488",
        "Text": "Then\nTotal profit (in Rs) = 8000 x + 12000 y\nLet\nZ = 8000 x + 12000 y\nWe now have the following mathematical model for the given problem Maximise Z = 8000 x + 12000 y (1)\nsubject to the constraints:\n9x + 12y \u2264 180     (Fabricating constraint)\ni e"
    },
    {
        "Chapter": "1",
        "sentence_range": "6486-6489",
        "Text": "Maximise Z = 8000 x + 12000 y (1)\nsubject to the constraints:\n9x + 12y \u2264 180     (Fabricating constraint)\ni e 3x + 4y \u2264 60"
    },
    {
        "Chapter": "1",
        "sentence_range": "6487-6490",
        "Text": "(1)\nsubject to the constraints:\n9x + 12y \u2264 180     (Fabricating constraint)\ni e 3x + 4y \u2264 60 (2)\nx + 3y \u2264 30\n(Finishing constraint)"
    },
    {
        "Chapter": "1",
        "sentence_range": "6488-6491",
        "Text": "e 3x + 4y \u2264 60 (2)\nx + 3y \u2264 30\n(Finishing constraint) (3)\nx \u2265 0, y \u2265 0\n(non-negative constraint)"
    },
    {
        "Chapter": "1",
        "sentence_range": "6489-6492",
        "Text": "3x + 4y \u2264 60 (2)\nx + 3y \u2264 30\n(Finishing constraint) (3)\nx \u2265 0, y \u2265 0\n(non-negative constraint) (4)\nThe feasible region (shaded) OABC determined by the linear inequalities (2) to (4)\nis shown in the Fig 12"
    },
    {
        "Chapter": "1",
        "sentence_range": "6490-6493",
        "Text": "(2)\nx + 3y \u2264 30\n(Finishing constraint) (3)\nx \u2265 0, y \u2265 0\n(non-negative constraint) (4)\nThe feasible region (shaded) OABC determined by the linear inequalities (2) to (4)\nis shown in the Fig 12 9"
    },
    {
        "Chapter": "1",
        "sentence_range": "6491-6494",
        "Text": "(3)\nx \u2265 0, y \u2265 0\n(non-negative constraint) (4)\nThe feasible region (shaded) OABC determined by the linear inequalities (2) to (4)\nis shown in the Fig 12 9 Note that the feasible region is bounded"
    },
    {
        "Chapter": "1",
        "sentence_range": "6492-6495",
        "Text": "(4)\nThe feasible region (shaded) OABC determined by the linear inequalities (2) to (4)\nis shown in the Fig 12 9 Note that the feasible region is bounded Fig 12"
    },
    {
        "Chapter": "1",
        "sentence_range": "6493-6496",
        "Text": "9 Note that the feasible region is bounded Fig 12 9\nLet us evaluate the objective function Z at each corner point as shown below:\nCorner Point\nZ = 8000 x + 12000 y\n0 (0, 0)\n0\nA (20, 0)\n160000\nB (12, 6)\n168000\nMaximum\nC (0, 10)\n120000\nWe find that maximum value of Z is 1,68,000 at B (12, 6)"
    },
    {
        "Chapter": "1",
        "sentence_range": "6494-6497",
        "Text": "Note that the feasible region is bounded Fig 12 9\nLet us evaluate the objective function Z at each corner point as shown below:\nCorner Point\nZ = 8000 x + 12000 y\n0 (0, 0)\n0\nA (20, 0)\n160000\nB (12, 6)\n168000\nMaximum\nC (0, 10)\n120000\nWe find that maximum value of Z is 1,68,000 at B (12, 6) Hence, the company\nshould produce 12 pieces of Model A and 6 pieces of Model B to realise maximum\nprofit and maximum profit then will be Rs 1,68,000"
    },
    {
        "Chapter": "1",
        "sentence_range": "6495-6498",
        "Text": "Fig 12 9\nLet us evaluate the objective function Z at each corner point as shown below:\nCorner Point\nZ = 8000 x + 12000 y\n0 (0, 0)\n0\nA (20, 0)\n160000\nB (12, 6)\n168000\nMaximum\nC (0, 10)\n120000\nWe find that maximum value of Z is 1,68,000 at B (12, 6) Hence, the company\nshould produce 12 pieces of Model A and 6 pieces of Model B to realise maximum\nprofit and maximum profit then will be Rs 1,68,000 \u00a9 NCERT\nnot to be republished\nLINEAR PROGRAMMING         519\nEXERCISE 12"
    },
    {
        "Chapter": "1",
        "sentence_range": "6496-6499",
        "Text": "9\nLet us evaluate the objective function Z at each corner point as shown below:\nCorner Point\nZ = 8000 x + 12000 y\n0 (0, 0)\n0\nA (20, 0)\n160000\nB (12, 6)\n168000\nMaximum\nC (0, 10)\n120000\nWe find that maximum value of Z is 1,68,000 at B (12, 6) Hence, the company\nshould produce 12 pieces of Model A and 6 pieces of Model B to realise maximum\nprofit and maximum profit then will be Rs 1,68,000 \u00a9 NCERT\nnot to be republished\nLINEAR PROGRAMMING         519\nEXERCISE 12 2\n1"
    },
    {
        "Chapter": "1",
        "sentence_range": "6497-6500",
        "Text": "Hence, the company\nshould produce 12 pieces of Model A and 6 pieces of Model B to realise maximum\nprofit and maximum profit then will be Rs 1,68,000 \u00a9 NCERT\nnot to be republished\nLINEAR PROGRAMMING         519\nEXERCISE 12 2\n1 Reshma wishes to mix two types of food P and Q in such a way that the vitamin\ncontents of the mixture contain at least 8 units of vitamin A and 11 units of\nvitamin B"
    },
    {
        "Chapter": "1",
        "sentence_range": "6498-6501",
        "Text": "\u00a9 NCERT\nnot to be republished\nLINEAR PROGRAMMING         519\nEXERCISE 12 2\n1 Reshma wishes to mix two types of food P and Q in such a way that the vitamin\ncontents of the mixture contain at least 8 units of vitamin A and 11 units of\nvitamin B Food P costs Rs 60/kg and Food Q costs Rs 80/kg"
    },
    {
        "Chapter": "1",
        "sentence_range": "6499-6502",
        "Text": "2\n1 Reshma wishes to mix two types of food P and Q in such a way that the vitamin\ncontents of the mixture contain at least 8 units of vitamin A and 11 units of\nvitamin B Food P costs Rs 60/kg and Food Q costs Rs 80/kg Food P contains\n3 units/kg of Vitamin A and 5 units / kg of Vitamin B while food Q contains\n4 units/kg of Vitamin A and 2 units/kg of vitamin B"
    },
    {
        "Chapter": "1",
        "sentence_range": "6500-6503",
        "Text": "Reshma wishes to mix two types of food P and Q in such a way that the vitamin\ncontents of the mixture contain at least 8 units of vitamin A and 11 units of\nvitamin B Food P costs Rs 60/kg and Food Q costs Rs 80/kg Food P contains\n3 units/kg of Vitamin A and 5 units / kg of Vitamin B while food Q contains\n4 units/kg of Vitamin A and 2 units/kg of vitamin B Determine the minimum cost\nof the mixture"
    },
    {
        "Chapter": "1",
        "sentence_range": "6501-6504",
        "Text": "Food P costs Rs 60/kg and Food Q costs Rs 80/kg Food P contains\n3 units/kg of Vitamin A and 5 units / kg of Vitamin B while food Q contains\n4 units/kg of Vitamin A and 2 units/kg of vitamin B Determine the minimum cost\nof the mixture 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "6502-6505",
        "Text": "Food P contains\n3 units/kg of Vitamin A and 5 units / kg of Vitamin B while food Q contains\n4 units/kg of Vitamin A and 2 units/kg of vitamin B Determine the minimum cost\nof the mixture 2 One kind of cake requires 200g of flour and 25g of fat, and another kind of cake\nrequires 100g of flour and 50g of fat"
    },
    {
        "Chapter": "1",
        "sentence_range": "6503-6506",
        "Text": "Determine the minimum cost\nof the mixture 2 One kind of cake requires 200g of flour and 25g of fat, and another kind of cake\nrequires 100g of flour and 50g of fat Find the maximum number of cakes which\ncan be made from 5kg of flour and 1 kg of fat assuming that there is no shortage\nof the other ingredients used in making the cakes"
    },
    {
        "Chapter": "1",
        "sentence_range": "6504-6507",
        "Text": "2 One kind of cake requires 200g of flour and 25g of fat, and another kind of cake\nrequires 100g of flour and 50g of fat Find the maximum number of cakes which\ncan be made from 5kg of flour and 1 kg of fat assuming that there is no shortage\nof the other ingredients used in making the cakes 3"
    },
    {
        "Chapter": "1",
        "sentence_range": "6505-6508",
        "Text": "One kind of cake requires 200g of flour and 25g of fat, and another kind of cake\nrequires 100g of flour and 50g of fat Find the maximum number of cakes which\ncan be made from 5kg of flour and 1 kg of fat assuming that there is no shortage\nof the other ingredients used in making the cakes 3 A factory makes tennis rackets and cricket bats"
    },
    {
        "Chapter": "1",
        "sentence_range": "6506-6509",
        "Text": "Find the maximum number of cakes which\ncan be made from 5kg of flour and 1 kg of fat assuming that there is no shortage\nof the other ingredients used in making the cakes 3 A factory makes tennis rackets and cricket bats A tennis racket takes 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "6507-6510",
        "Text": "3 A factory makes tennis rackets and cricket bats A tennis racket takes 1 5 hours\nof machine time and 3 hours of craftman\u2019s time in its making while a cricket bat\ntakes 3 hour of machine time and 1 hour of craftman\u2019s time"
    },
    {
        "Chapter": "1",
        "sentence_range": "6508-6511",
        "Text": "A factory makes tennis rackets and cricket bats A tennis racket takes 1 5 hours\nof machine time and 3 hours of craftman\u2019s time in its making while a cricket bat\ntakes 3 hour of machine time and 1 hour of craftman\u2019s time In a day, the factory\nhas the availability of not more than 42 hours of machine time and 24 hours of\ncraftsman\u2019s time"
    },
    {
        "Chapter": "1",
        "sentence_range": "6509-6512",
        "Text": "A tennis racket takes 1 5 hours\nof machine time and 3 hours of craftman\u2019s time in its making while a cricket bat\ntakes 3 hour of machine time and 1 hour of craftman\u2019s time In a day, the factory\nhas the availability of not more than 42 hours of machine time and 24 hours of\ncraftsman\u2019s time (i) What number of rackets and bats must be made if  the factory is to work\nat full capacity"
    },
    {
        "Chapter": "1",
        "sentence_range": "6510-6513",
        "Text": "5 hours\nof machine time and 3 hours of craftman\u2019s time in its making while a cricket bat\ntakes 3 hour of machine time and 1 hour of craftman\u2019s time In a day, the factory\nhas the availability of not more than 42 hours of machine time and 24 hours of\ncraftsman\u2019s time (i) What number of rackets and bats must be made if  the factory is to work\nat full capacity (ii) If the profit on a racket and on a bat is Rs 20 and Rs 10 respectively, find\nthe maximum profit of the factory when it works at full capacity"
    },
    {
        "Chapter": "1",
        "sentence_range": "6511-6514",
        "Text": "In a day, the factory\nhas the availability of not more than 42 hours of machine time and 24 hours of\ncraftsman\u2019s time (i) What number of rackets and bats must be made if  the factory is to work\nat full capacity (ii) If the profit on a racket and on a bat is Rs 20 and Rs 10 respectively, find\nthe maximum profit of the factory when it works at full capacity 4"
    },
    {
        "Chapter": "1",
        "sentence_range": "6512-6515",
        "Text": "(i) What number of rackets and bats must be made if  the factory is to work\nat full capacity (ii) If the profit on a racket and on a bat is Rs 20 and Rs 10 respectively, find\nthe maximum profit of the factory when it works at full capacity 4 A manufacturer produces nuts and bolts"
    },
    {
        "Chapter": "1",
        "sentence_range": "6513-6516",
        "Text": "(ii) If the profit on a racket and on a bat is Rs 20 and Rs 10 respectively, find\nthe maximum profit of the factory when it works at full capacity 4 A manufacturer produces nuts and bolts It takes 1 hour of work on machine A\nand 3 hours on machine B to produce a package of nuts"
    },
    {
        "Chapter": "1",
        "sentence_range": "6514-6517",
        "Text": "4 A manufacturer produces nuts and bolts It takes 1 hour of work on machine A\nand 3 hours on machine B to produce a package of nuts It takes 3 hours on\nmachine A and 1 hour on machine B to produce a package of bolts"
    },
    {
        "Chapter": "1",
        "sentence_range": "6515-6518",
        "Text": "A manufacturer produces nuts and bolts It takes 1 hour of work on machine A\nand 3 hours on machine B to produce a package of nuts It takes 3 hours on\nmachine A and 1 hour on machine B to produce a package of bolts He earns a\nprofit of Rs17"
    },
    {
        "Chapter": "1",
        "sentence_range": "6516-6519",
        "Text": "It takes 1 hour of work on machine A\nand 3 hours on machine B to produce a package of nuts It takes 3 hours on\nmachine A and 1 hour on machine B to produce a package of bolts He earns a\nprofit of Rs17 50 per package on nuts and Rs 7"
    },
    {
        "Chapter": "1",
        "sentence_range": "6517-6520",
        "Text": "It takes 3 hours on\nmachine A and 1 hour on machine B to produce a package of bolts He earns a\nprofit of Rs17 50 per package on nuts and Rs 7 00 per package on bolts"
    },
    {
        "Chapter": "1",
        "sentence_range": "6518-6521",
        "Text": "He earns a\nprofit of Rs17 50 per package on nuts and Rs 7 00 per package on bolts How\nmany packages of each should be produced each day so as to maximise his\nprofit, if he operates his machines for at the most 12 hours a day"
    },
    {
        "Chapter": "1",
        "sentence_range": "6519-6522",
        "Text": "50 per package on nuts and Rs 7 00 per package on bolts How\nmany packages of each should be produced each day so as to maximise his\nprofit, if he operates his machines for at the most 12 hours a day 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "6520-6523",
        "Text": "00 per package on bolts How\nmany packages of each should be produced each day so as to maximise his\nprofit, if he operates his machines for at the most 12 hours a day 5 A factory manufactures two types of screws, A and B"
    },
    {
        "Chapter": "1",
        "sentence_range": "6521-6524",
        "Text": "How\nmany packages of each should be produced each day so as to maximise his\nprofit, if he operates his machines for at the most 12 hours a day 5 A factory manufactures two types of screws, A and B Each type of screw\nrequires the use of two machines, an automatic and a hand operated"
    },
    {
        "Chapter": "1",
        "sentence_range": "6522-6525",
        "Text": "5 A factory manufactures two types of screws, A and B Each type of screw\nrequires the use of two machines, an automatic and a hand operated It takes\n4 minutes on the automatic and 6 minutes on hand operated machines to\nmanufacture a package of screws A, while it takes 6 minutes on automatic and\n3 minutes on the hand operated machines to manufacture a package of screws\nB"
    },
    {
        "Chapter": "1",
        "sentence_range": "6523-6526",
        "Text": "A factory manufactures two types of screws, A and B Each type of screw\nrequires the use of two machines, an automatic and a hand operated It takes\n4 minutes on the automatic and 6 minutes on hand operated machines to\nmanufacture a package of screws A, while it takes 6 minutes on automatic and\n3 minutes on the hand operated machines to manufacture a package of screws\nB Each machine is available for at the most 4 hours on any day"
    },
    {
        "Chapter": "1",
        "sentence_range": "6524-6527",
        "Text": "Each type of screw\nrequires the use of two machines, an automatic and a hand operated It takes\n4 minutes on the automatic and 6 minutes on hand operated machines to\nmanufacture a package of screws A, while it takes 6 minutes on automatic and\n3 minutes on the hand operated machines to manufacture a package of screws\nB Each machine is available for at the most 4 hours on any day The manufacturer\ncan sell a package of screws A at a profit of Rs 7 and screws B at a profit of\nRs 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "6525-6528",
        "Text": "It takes\n4 minutes on the automatic and 6 minutes on hand operated machines to\nmanufacture a package of screws A, while it takes 6 minutes on automatic and\n3 minutes on the hand operated machines to manufacture a package of screws\nB Each machine is available for at the most 4 hours on any day The manufacturer\ncan sell a package of screws A at a profit of Rs 7 and screws B at a profit of\nRs 10 Assuming that he can sell all the screws he manufactures, how many\npackages of each type should the factory owner produce in a day in order to\nmaximise his profit"
    },
    {
        "Chapter": "1",
        "sentence_range": "6526-6529",
        "Text": "Each machine is available for at the most 4 hours on any day The manufacturer\ncan sell a package of screws A at a profit of Rs 7 and screws B at a profit of\nRs 10 Assuming that he can sell all the screws he manufactures, how many\npackages of each type should the factory owner produce in a day in order to\nmaximise his profit Determine the maximum profit"
    },
    {
        "Chapter": "1",
        "sentence_range": "6527-6530",
        "Text": "The manufacturer\ncan sell a package of screws A at a profit of Rs 7 and screws B at a profit of\nRs 10 Assuming that he can sell all the screws he manufactures, how many\npackages of each type should the factory owner produce in a day in order to\nmaximise his profit Determine the maximum profit \u00a9 NCERT\nnot to be republished\n520\nMATHEMATICS\n6"
    },
    {
        "Chapter": "1",
        "sentence_range": "6528-6531",
        "Text": "Assuming that he can sell all the screws he manufactures, how many\npackages of each type should the factory owner produce in a day in order to\nmaximise his profit Determine the maximum profit \u00a9 NCERT\nnot to be republished\n520\nMATHEMATICS\n6 A cottage industry manufactures pedestal lamps and wooden shades, each\nrequiring the use of a grinding/cutting machine and a sprayer"
    },
    {
        "Chapter": "1",
        "sentence_range": "6529-6532",
        "Text": "Determine the maximum profit \u00a9 NCERT\nnot to be republished\n520\nMATHEMATICS\n6 A cottage industry manufactures pedestal lamps and wooden shades, each\nrequiring the use of a grinding/cutting machine and a sprayer It takes 2 hours on\ngrinding/cutting machine and 3 hours on the sprayer to manufacture a pedestal\nlamp"
    },
    {
        "Chapter": "1",
        "sentence_range": "6530-6533",
        "Text": "\u00a9 NCERT\nnot to be republished\n520\nMATHEMATICS\n6 A cottage industry manufactures pedestal lamps and wooden shades, each\nrequiring the use of a grinding/cutting machine and a sprayer It takes 2 hours on\ngrinding/cutting machine and 3 hours on the sprayer to manufacture a pedestal\nlamp It takes 1 hour on the grinding/cutting machine and 2 hours on the sprayer\nto manufacture a shade"
    },
    {
        "Chapter": "1",
        "sentence_range": "6531-6534",
        "Text": "A cottage industry manufactures pedestal lamps and wooden shades, each\nrequiring the use of a grinding/cutting machine and a sprayer It takes 2 hours on\ngrinding/cutting machine and 3 hours on the sprayer to manufacture a pedestal\nlamp It takes 1 hour on the grinding/cutting machine and 2 hours on the sprayer\nto manufacture a shade On any day, the sprayer is available for at the most 20\nhours and the grinding/cutting machine for at the most 12 hours"
    },
    {
        "Chapter": "1",
        "sentence_range": "6532-6535",
        "Text": "It takes 2 hours on\ngrinding/cutting machine and 3 hours on the sprayer to manufacture a pedestal\nlamp It takes 1 hour on the grinding/cutting machine and 2 hours on the sprayer\nto manufacture a shade On any day, the sprayer is available for at the most 20\nhours and the grinding/cutting machine for at the most 12 hours The profit from\nthe sale of a lamp is Rs 5 and that from a shade is Rs 3"
    },
    {
        "Chapter": "1",
        "sentence_range": "6533-6536",
        "Text": "It takes 1 hour on the grinding/cutting machine and 2 hours on the sprayer\nto manufacture a shade On any day, the sprayer is available for at the most 20\nhours and the grinding/cutting machine for at the most 12 hours The profit from\nthe sale of a lamp is Rs 5 and that from a shade is Rs 3 Assuming that the\nmanufacturer can sell all the lamps and shades that he produces, how should he\nschedule his daily production in order to maximise his profit"
    },
    {
        "Chapter": "1",
        "sentence_range": "6534-6537",
        "Text": "On any day, the sprayer is available for at the most 20\nhours and the grinding/cutting machine for at the most 12 hours The profit from\nthe sale of a lamp is Rs 5 and that from a shade is Rs 3 Assuming that the\nmanufacturer can sell all the lamps and shades that he produces, how should he\nschedule his daily production in order to maximise his profit 7"
    },
    {
        "Chapter": "1",
        "sentence_range": "6535-6538",
        "Text": "The profit from\nthe sale of a lamp is Rs 5 and that from a shade is Rs 3 Assuming that the\nmanufacturer can sell all the lamps and shades that he produces, how should he\nschedule his daily production in order to maximise his profit 7 A company manufactures two types of novelty souvenirs made of plywood"
    },
    {
        "Chapter": "1",
        "sentence_range": "6536-6539",
        "Text": "Assuming that the\nmanufacturer can sell all the lamps and shades that he produces, how should he\nschedule his daily production in order to maximise his profit 7 A company manufactures two types of novelty souvenirs made of plywood Souvenirs of type A require 5 minutes each for cutting and 10 minutes each for\nassembling"
    },
    {
        "Chapter": "1",
        "sentence_range": "6537-6540",
        "Text": "7 A company manufactures two types of novelty souvenirs made of plywood Souvenirs of type A require 5 minutes each for cutting and 10 minutes each for\nassembling Souvenirs of type B require 8 minutes each for cutting and 8 minutes\neach for assembling"
    },
    {
        "Chapter": "1",
        "sentence_range": "6538-6541",
        "Text": "A company manufactures two types of novelty souvenirs made of plywood Souvenirs of type A require 5 minutes each for cutting and 10 minutes each for\nassembling Souvenirs of type B require 8 minutes each for cutting and 8 minutes\neach for assembling There are 3 hours 20 minutes available for cutting and 4\nhours for assembling"
    },
    {
        "Chapter": "1",
        "sentence_range": "6539-6542",
        "Text": "Souvenirs of type A require 5 minutes each for cutting and 10 minutes each for\nassembling Souvenirs of type B require 8 minutes each for cutting and 8 minutes\neach for assembling There are 3 hours 20 minutes available for cutting and 4\nhours for assembling The profit is Rs 5 each for type A and Rs 6 each for type\nB souvenirs"
    },
    {
        "Chapter": "1",
        "sentence_range": "6540-6543",
        "Text": "Souvenirs of type B require 8 minutes each for cutting and 8 minutes\neach for assembling There are 3 hours 20 minutes available for cutting and 4\nhours for assembling The profit is Rs 5 each for type A and Rs 6 each for type\nB souvenirs How many souvenirs of each type should the company manufacture\nin order to maximise the profit"
    },
    {
        "Chapter": "1",
        "sentence_range": "6541-6544",
        "Text": "There are 3 hours 20 minutes available for cutting and 4\nhours for assembling The profit is Rs 5 each for type A and Rs 6 each for type\nB souvenirs How many souvenirs of each type should the company manufacture\nin order to maximise the profit 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "6542-6545",
        "Text": "The profit is Rs 5 each for type A and Rs 6 each for type\nB souvenirs How many souvenirs of each type should the company manufacture\nin order to maximise the profit 8 A merchant plans to sell two types of personal computers \u2013 a desktop model and\na portable model that will cost Rs 25000 and Rs 40000 respectively"
    },
    {
        "Chapter": "1",
        "sentence_range": "6543-6546",
        "Text": "How many souvenirs of each type should the company manufacture\nin order to maximise the profit 8 A merchant plans to sell two types of personal computers \u2013 a desktop model and\na portable model that will cost Rs 25000 and Rs 40000 respectively He estimates\nthat the total monthly demand of computers will not exceed 250 units"
    },
    {
        "Chapter": "1",
        "sentence_range": "6544-6547",
        "Text": "8 A merchant plans to sell two types of personal computers \u2013 a desktop model and\na portable model that will cost Rs 25000 and Rs 40000 respectively He estimates\nthat the total monthly demand of computers will not exceed 250 units Determine\nthe number of units of each type of computers which the merchant should stock\nto get maximum profit if he does not want to invest more than Rs 70 lakhs and if\nhis profit on the desktop model is Rs 4500 and on portable model is Rs 5000"
    },
    {
        "Chapter": "1",
        "sentence_range": "6545-6548",
        "Text": "A merchant plans to sell two types of personal computers \u2013 a desktop model and\na portable model that will cost Rs 25000 and Rs 40000 respectively He estimates\nthat the total monthly demand of computers will not exceed 250 units Determine\nthe number of units of each type of computers which the merchant should stock\nto get maximum profit if he does not want to invest more than Rs 70 lakhs and if\nhis profit on the desktop model is Rs 4500 and on portable model is Rs 5000 9"
    },
    {
        "Chapter": "1",
        "sentence_range": "6546-6549",
        "Text": "He estimates\nthat the total monthly demand of computers will not exceed 250 units Determine\nthe number of units of each type of computers which the merchant should stock\nto get maximum profit if he does not want to invest more than Rs 70 lakhs and if\nhis profit on the desktop model is Rs 4500 and on portable model is Rs 5000 9 A diet is to contain at least 80 units of vitamin A and 100 units of minerals"
    },
    {
        "Chapter": "1",
        "sentence_range": "6547-6550",
        "Text": "Determine\nthe number of units of each type of computers which the merchant should stock\nto get maximum profit if he does not want to invest more than Rs 70 lakhs and if\nhis profit on the desktop model is Rs 4500 and on portable model is Rs 5000 9 A diet is to contain at least 80 units of vitamin A and 100 units of minerals Two\nfoods F1 and F2 are available"
    },
    {
        "Chapter": "1",
        "sentence_range": "6548-6551",
        "Text": "9 A diet is to contain at least 80 units of vitamin A and 100 units of minerals Two\nfoods F1 and F2 are available Food F1 costs Rs 4 per unit food and F2 costs\nRs 6 per unit"
    },
    {
        "Chapter": "1",
        "sentence_range": "6549-6552",
        "Text": "A diet is to contain at least 80 units of vitamin A and 100 units of minerals Two\nfoods F1 and F2 are available Food F1 costs Rs 4 per unit food and F2 costs\nRs 6 per unit One unit of food F1 contains 3 units of vitamin A and 4 units of\nminerals"
    },
    {
        "Chapter": "1",
        "sentence_range": "6550-6553",
        "Text": "Two\nfoods F1 and F2 are available Food F1 costs Rs 4 per unit food and F2 costs\nRs 6 per unit One unit of food F1 contains 3 units of vitamin A and 4 units of\nminerals One unit of food F2 contains 6 units of vitamin A and 3 units of minerals"
    },
    {
        "Chapter": "1",
        "sentence_range": "6551-6554",
        "Text": "Food F1 costs Rs 4 per unit food and F2 costs\nRs 6 per unit One unit of food F1 contains 3 units of vitamin A and 4 units of\nminerals One unit of food F2 contains 6 units of vitamin A and 3 units of minerals Formulate this as a linear programming problem"
    },
    {
        "Chapter": "1",
        "sentence_range": "6552-6555",
        "Text": "One unit of food F1 contains 3 units of vitamin A and 4 units of\nminerals One unit of food F2 contains 6 units of vitamin A and 3 units of minerals Formulate this as a linear programming problem Find the minimum cost for diet\nthat consists of mixture of these two foods and also meets the minimal nutritional\nrequirements"
    },
    {
        "Chapter": "1",
        "sentence_range": "6553-6556",
        "Text": "One unit of food F2 contains 6 units of vitamin A and 3 units of minerals Formulate this as a linear programming problem Find the minimum cost for diet\nthat consists of mixture of these two foods and also meets the minimal nutritional\nrequirements 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "6554-6557",
        "Text": "Formulate this as a linear programming problem Find the minimum cost for diet\nthat consists of mixture of these two foods and also meets the minimal nutritional\nrequirements 10 There are two types of fertilisers F1 and F2"
    },
    {
        "Chapter": "1",
        "sentence_range": "6555-6558",
        "Text": "Find the minimum cost for diet\nthat consists of mixture of these two foods and also meets the minimal nutritional\nrequirements 10 There are two types of fertilisers F1 and F2 F1 consists of 10% nitrogen and 6%\nphosphoric acid and F2 consists of 5% nitrogen and 10% phosphoric acid"
    },
    {
        "Chapter": "1",
        "sentence_range": "6556-6559",
        "Text": "10 There are two types of fertilisers F1 and F2 F1 consists of 10% nitrogen and 6%\nphosphoric acid and F2 consists of 5% nitrogen and 10% phosphoric acid After\ntesting the soil conditions, a farmer finds that she needs atleast 14 kg of nitrogen\nand 14 kg of phosphoric acid for her crop"
    },
    {
        "Chapter": "1",
        "sentence_range": "6557-6560",
        "Text": "There are two types of fertilisers F1 and F2 F1 consists of 10% nitrogen and 6%\nphosphoric acid and F2 consists of 5% nitrogen and 10% phosphoric acid After\ntesting the soil conditions, a farmer finds that she needs atleast 14 kg of nitrogen\nand 14 kg of phosphoric acid for her crop If F1 costs Rs 6/kg and F2 costs\nRs 5/kg, determine how much of each type of fertiliser should be used so that\nnutrient requirements are met at a minimum cost"
    },
    {
        "Chapter": "1",
        "sentence_range": "6558-6561",
        "Text": "F1 consists of 10% nitrogen and 6%\nphosphoric acid and F2 consists of 5% nitrogen and 10% phosphoric acid After\ntesting the soil conditions, a farmer finds that she needs atleast 14 kg of nitrogen\nand 14 kg of phosphoric acid for her crop If F1 costs Rs 6/kg and F2 costs\nRs 5/kg, determine how much of each type of fertiliser should be used so that\nnutrient requirements are met at a minimum cost What is the minimum cost"
    },
    {
        "Chapter": "1",
        "sentence_range": "6559-6562",
        "Text": "After\ntesting the soil conditions, a farmer finds that she needs atleast 14 kg of nitrogen\nand 14 kg of phosphoric acid for her crop If F1 costs Rs 6/kg and F2 costs\nRs 5/kg, determine how much of each type of fertiliser should be used so that\nnutrient requirements are met at a minimum cost What is the minimum cost 11"
    },
    {
        "Chapter": "1",
        "sentence_range": "6560-6563",
        "Text": "If F1 costs Rs 6/kg and F2 costs\nRs 5/kg, determine how much of each type of fertiliser should be used so that\nnutrient requirements are met at a minimum cost What is the minimum cost 11 The corner points of  the feasible region determined by the following system of\nlinear inequalities:\n2x + y \u2264 10, x + 3y \u2264 15, x, y \u2265 0 are (0, 0), (5, 0), (3, 4) and (0, 5)"
    },
    {
        "Chapter": "1",
        "sentence_range": "6561-6564",
        "Text": "What is the minimum cost 11 The corner points of  the feasible region determined by the following system of\nlinear inequalities:\n2x + y \u2264 10, x + 3y \u2264 15, x, y \u2265 0 are (0, 0), (5, 0), (3, 4) and (0, 5) Let\nZ = px + qy, where p, q > 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "6562-6565",
        "Text": "11 The corner points of  the feasible region determined by the following system of\nlinear inequalities:\n2x + y \u2264 10, x + 3y \u2264 15, x, y \u2265 0 are (0, 0), (5, 0), (3, 4) and (0, 5) Let\nZ = px + qy, where p, q > 0 Condition on p and q so that the maximum of Z\noccurs at both (3, 4) and (0, 5) is\n(A) p = q\n(B) p = 2q\n(C) p = 3q\n(D) q = 3p\n\u00a9 NCERT\nnot to be republished\nLINEAR PROGRAMMING         521\nMiscellaneous Examples\nExample 9 (Diet problem) A dietician has to develop a special diet using two foods\nP and Q"
    },
    {
        "Chapter": "1",
        "sentence_range": "6563-6566",
        "Text": "The corner points of  the feasible region determined by the following system of\nlinear inequalities:\n2x + y \u2264 10, x + 3y \u2264 15, x, y \u2265 0 are (0, 0), (5, 0), (3, 4) and (0, 5) Let\nZ = px + qy, where p, q > 0 Condition on p and q so that the maximum of Z\noccurs at both (3, 4) and (0, 5) is\n(A) p = q\n(B) p = 2q\n(C) p = 3q\n(D) q = 3p\n\u00a9 NCERT\nnot to be republished\nLINEAR PROGRAMMING         521\nMiscellaneous Examples\nExample 9 (Diet problem) A dietician has to develop a special diet using two foods\nP and Q Each packet (containing 30 g) of food P contains 12 units of calcium, 4 units\nof iron, 6 units of cholesterol and 6 units of vitamin A"
    },
    {
        "Chapter": "1",
        "sentence_range": "6564-6567",
        "Text": "Let\nZ = px + qy, where p, q > 0 Condition on p and q so that the maximum of Z\noccurs at both (3, 4) and (0, 5) is\n(A) p = q\n(B) p = 2q\n(C) p = 3q\n(D) q = 3p\n\u00a9 NCERT\nnot to be republished\nLINEAR PROGRAMMING         521\nMiscellaneous Examples\nExample 9 (Diet problem) A dietician has to develop a special diet using two foods\nP and Q Each packet (containing 30 g) of food P contains 12 units of calcium, 4 units\nof iron, 6 units of cholesterol and 6 units of vitamin A Each packet of the same quantity\nof food Q contains 3 units of calcium, 20 units of iron, 4 units of cholesterol and 3 units\nof vitamin A"
    },
    {
        "Chapter": "1",
        "sentence_range": "6565-6568",
        "Text": "Condition on p and q so that the maximum of Z\noccurs at both (3, 4) and (0, 5) is\n(A) p = q\n(B) p = 2q\n(C) p = 3q\n(D) q = 3p\n\u00a9 NCERT\nnot to be republished\nLINEAR PROGRAMMING         521\nMiscellaneous Examples\nExample 9 (Diet problem) A dietician has to develop a special diet using two foods\nP and Q Each packet (containing 30 g) of food P contains 12 units of calcium, 4 units\nof iron, 6 units of cholesterol and 6 units of vitamin A Each packet of the same quantity\nof food Q contains 3 units of calcium, 20 units of iron, 4 units of cholesterol and 3 units\nof vitamin A The diet requires atleast 240 units of calcium, atleast 460 units of iron and\nat most 300 units of cholesterol"
    },
    {
        "Chapter": "1",
        "sentence_range": "6566-6569",
        "Text": "Each packet (containing 30 g) of food P contains 12 units of calcium, 4 units\nof iron, 6 units of cholesterol and 6 units of vitamin A Each packet of the same quantity\nof food Q contains 3 units of calcium, 20 units of iron, 4 units of cholesterol and 3 units\nof vitamin A The diet requires atleast 240 units of calcium, atleast 460 units of iron and\nat most 300 units of cholesterol How many packets of each food should be used to\nminimise the amount of vitamin A in the diet"
    },
    {
        "Chapter": "1",
        "sentence_range": "6567-6570",
        "Text": "Each packet of the same quantity\nof food Q contains 3 units of calcium, 20 units of iron, 4 units of cholesterol and 3 units\nof vitamin A The diet requires atleast 240 units of calcium, atleast 460 units of iron and\nat most 300 units of cholesterol How many packets of each food should be used to\nminimise the amount of vitamin A in the diet What is the minimum amount of vitamin A"
    },
    {
        "Chapter": "1",
        "sentence_range": "6568-6571",
        "Text": "The diet requires atleast 240 units of calcium, atleast 460 units of iron and\nat most 300 units of cholesterol How many packets of each food should be used to\nminimise the amount of vitamin A in the diet What is the minimum amount of vitamin A Solution Let x and y be the number of packets of food P and Q respectively"
    },
    {
        "Chapter": "1",
        "sentence_range": "6569-6572",
        "Text": "How many packets of each food should be used to\nminimise the amount of vitamin A in the diet What is the minimum amount of vitamin A Solution Let x and y be the number of packets of food P and Q respectively Obviously\nx \u2265 0, y \u2265 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "6570-6573",
        "Text": "What is the minimum amount of vitamin A Solution Let x and y be the number of packets of food P and Q respectively Obviously\nx \u2265 0, y \u2265 0 Mathematical formulation of the given problem is as follows:\nMinimise  Z = 6x + 3y (vitamin A)\nsubject to the constraints\n12x + 3y \u2265 240 (constraint on calcium), i"
    },
    {
        "Chapter": "1",
        "sentence_range": "6571-6574",
        "Text": "Solution Let x and y be the number of packets of food P and Q respectively Obviously\nx \u2265 0, y \u2265 0 Mathematical formulation of the given problem is as follows:\nMinimise  Z = 6x + 3y (vitamin A)\nsubject to the constraints\n12x + 3y \u2265 240 (constraint on calcium), i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "6572-6575",
        "Text": "Obviously\nx \u2265 0, y \u2265 0 Mathematical formulation of the given problem is as follows:\nMinimise  Z = 6x + 3y (vitamin A)\nsubject to the constraints\n12x + 3y \u2265 240 (constraint on calcium), i e 4x + y \u2265 80"
    },
    {
        "Chapter": "1",
        "sentence_range": "6573-6576",
        "Text": "Mathematical formulation of the given problem is as follows:\nMinimise  Z = 6x + 3y (vitamin A)\nsubject to the constraints\n12x + 3y \u2265 240 (constraint on calcium), i e 4x + y \u2265 80 (1)\n4x + 20y \u2265 460 (constraint on iron), i"
    },
    {
        "Chapter": "1",
        "sentence_range": "6574-6577",
        "Text": "e 4x + y \u2265 80 (1)\n4x + 20y \u2265 460 (constraint on iron), i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "6575-6578",
        "Text": "4x + y \u2265 80 (1)\n4x + 20y \u2265 460 (constraint on iron), i e x + 5y \u2265 115"
    },
    {
        "Chapter": "1",
        "sentence_range": "6576-6579",
        "Text": "(1)\n4x + 20y \u2265 460 (constraint on iron), i e x + 5y \u2265 115 (2)\n6x + 4y \u2264 300 (constraint on cholesterol), i"
    },
    {
        "Chapter": "1",
        "sentence_range": "6577-6580",
        "Text": "e x + 5y \u2265 115 (2)\n6x + 4y \u2264 300 (constraint on cholesterol), i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "6578-6581",
        "Text": "x + 5y \u2265 115 (2)\n6x + 4y \u2264 300 (constraint on cholesterol), i e 3x + 2y \u2264 150"
    },
    {
        "Chapter": "1",
        "sentence_range": "6579-6582",
        "Text": "(2)\n6x + 4y \u2264 300 (constraint on cholesterol), i e 3x + 2y \u2264 150 (3)\nx \u2265 0, y \u2265 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "6580-6583",
        "Text": "e 3x + 2y \u2264 150 (3)\nx \u2265 0, y \u2265 0 (4)\nLet us graph the inequalities (1) to (4)"
    },
    {
        "Chapter": "1",
        "sentence_range": "6581-6584",
        "Text": "3x + 2y \u2264 150 (3)\nx \u2265 0, y \u2265 0 (4)\nLet us graph the inequalities (1) to (4) The feasible region (shaded) determined by the constraints (1) to (4) is shown in\nFig 12"
    },
    {
        "Chapter": "1",
        "sentence_range": "6582-6585",
        "Text": "(3)\nx \u2265 0, y \u2265 0 (4)\nLet us graph the inequalities (1) to (4) The feasible region (shaded) determined by the constraints (1) to (4) is shown in\nFig 12 10 and note that it is bounded"
    },
    {
        "Chapter": "1",
        "sentence_range": "6583-6586",
        "Text": "(4)\nLet us graph the inequalities (1) to (4) The feasible region (shaded) determined by the constraints (1) to (4) is shown in\nFig 12 10 and note that it is bounded Fig 12"
    },
    {
        "Chapter": "1",
        "sentence_range": "6584-6587",
        "Text": "The feasible region (shaded) determined by the constraints (1) to (4) is shown in\nFig 12 10 and note that it is bounded Fig 12 10\n\u00a9 NCERT\nnot to be republished\n522\nMATHEMATICS\nThe coordinates of the corner points L, M and N are (2, 72), (15, 20) and (40, 15)\nrespectively"
    },
    {
        "Chapter": "1",
        "sentence_range": "6585-6588",
        "Text": "10 and note that it is bounded Fig 12 10\n\u00a9 NCERT\nnot to be republished\n522\nMATHEMATICS\nThe coordinates of the corner points L, M and N are (2, 72), (15, 20) and (40, 15)\nrespectively Let us evaluate Z at these points:\nCorner Point\nZ = 6 x + 3 y\n(2, 72)\n228\n(15, 20)\n150 \u2190\nMinimum\n(40, 15)\n285\nFrom the table, we find that Z is minimum at the point (15, 20)"
    },
    {
        "Chapter": "1",
        "sentence_range": "6586-6589",
        "Text": "Fig 12 10\n\u00a9 NCERT\nnot to be republished\n522\nMATHEMATICS\nThe coordinates of the corner points L, M and N are (2, 72), (15, 20) and (40, 15)\nrespectively Let us evaluate Z at these points:\nCorner Point\nZ = 6 x + 3 y\n(2, 72)\n228\n(15, 20)\n150 \u2190\nMinimum\n(40, 15)\n285\nFrom the table, we find that Z is minimum at the point (15, 20) Hence, the amount\nof vitamin A under the constraints given in the problem will be minimum, if 15 packets\nof food P and 20 packets of food Q are used in the special diet"
    },
    {
        "Chapter": "1",
        "sentence_range": "6587-6590",
        "Text": "10\n\u00a9 NCERT\nnot to be republished\n522\nMATHEMATICS\nThe coordinates of the corner points L, M and N are (2, 72), (15, 20) and (40, 15)\nrespectively Let us evaluate Z at these points:\nCorner Point\nZ = 6 x + 3 y\n(2, 72)\n228\n(15, 20)\n150 \u2190\nMinimum\n(40, 15)\n285\nFrom the table, we find that Z is minimum at the point (15, 20) Hence, the amount\nof vitamin A under the constraints given in the problem will be minimum, if 15 packets\nof food P and 20 packets of food Q are used in the special diet The minimum amount\nof vitamin A will be 150 units"
    },
    {
        "Chapter": "1",
        "sentence_range": "6588-6591",
        "Text": "Let us evaluate Z at these points:\nCorner Point\nZ = 6 x + 3 y\n(2, 72)\n228\n(15, 20)\n150 \u2190\nMinimum\n(40, 15)\n285\nFrom the table, we find that Z is minimum at the point (15, 20) Hence, the amount\nof vitamin A under the constraints given in the problem will be minimum, if 15 packets\nof food P and 20 packets of food Q are used in the special diet The minimum amount\nof vitamin A will be 150 units Example 10 (Manufacturing problem) A manufacturer has three machines I, II\nand III installed in his factory"
    },
    {
        "Chapter": "1",
        "sentence_range": "6589-6592",
        "Text": "Hence, the amount\nof vitamin A under the constraints given in the problem will be minimum, if 15 packets\nof food P and 20 packets of food Q are used in the special diet The minimum amount\nof vitamin A will be 150 units Example 10 (Manufacturing problem) A manufacturer has three machines I, II\nand III installed in his factory Machines I and II are capable of being operated for\nat most 12 hours whereas machine III must be operated for atleast 5 hours a day"
    },
    {
        "Chapter": "1",
        "sentence_range": "6590-6593",
        "Text": "The minimum amount\nof vitamin A will be 150 units Example 10 (Manufacturing problem) A manufacturer has three machines I, II\nand III installed in his factory Machines I and II are capable of being operated for\nat most 12 hours whereas machine III must be operated for atleast 5 hours a day She\nproduces only two items M and N each requiring the use of all the three machines"
    },
    {
        "Chapter": "1",
        "sentence_range": "6591-6594",
        "Text": "Example 10 (Manufacturing problem) A manufacturer has three machines I, II\nand III installed in his factory Machines I and II are capable of being operated for\nat most 12 hours whereas machine III must be operated for atleast 5 hours a day She\nproduces only two items M and N each requiring the use of all the three machines The number of hours required for producing 1 unit of each of M and N on the three\nmachines are given in the following table:\nItems\nNumber of hours required on machines\nI\nII\nIII\nM\n1\n2\n1\nN\n2\n1\n1"
    },
    {
        "Chapter": "1",
        "sentence_range": "6592-6595",
        "Text": "Machines I and II are capable of being operated for\nat most 12 hours whereas machine III must be operated for atleast 5 hours a day She\nproduces only two items M and N each requiring the use of all the three machines The number of hours required for producing 1 unit of each of M and N on the three\nmachines are given in the following table:\nItems\nNumber of hours required on machines\nI\nII\nIII\nM\n1\n2\n1\nN\n2\n1\n1 25\nShe makes a profit of Rs 600 and Rs 400 on items M and N respectively"
    },
    {
        "Chapter": "1",
        "sentence_range": "6593-6596",
        "Text": "She\nproduces only two items M and N each requiring the use of all the three machines The number of hours required for producing 1 unit of each of M and N on the three\nmachines are given in the following table:\nItems\nNumber of hours required on machines\nI\nII\nIII\nM\n1\n2\n1\nN\n2\n1\n1 25\nShe makes a profit of Rs 600 and Rs 400 on items M and N respectively How many\nof each item should she produce so as to maximise her profit assuming that she can sell\nall the items that she produced"
    },
    {
        "Chapter": "1",
        "sentence_range": "6594-6597",
        "Text": "The number of hours required for producing 1 unit of each of M and N on the three\nmachines are given in the following table:\nItems\nNumber of hours required on machines\nI\nII\nIII\nM\n1\n2\n1\nN\n2\n1\n1 25\nShe makes a profit of Rs 600 and Rs 400 on items M and N respectively How many\nof each item should she produce so as to maximise her profit assuming that she can sell\nall the items that she produced What will be the maximum profit"
    },
    {
        "Chapter": "1",
        "sentence_range": "6595-6598",
        "Text": "25\nShe makes a profit of Rs 600 and Rs 400 on items M and N respectively How many\nof each item should she produce so as to maximise her profit assuming that she can sell\nall the items that she produced What will be the maximum profit Solution Let x and y be the number of items M and N respectively"
    },
    {
        "Chapter": "1",
        "sentence_range": "6596-6599",
        "Text": "How many\nof each item should she produce so as to maximise her profit assuming that she can sell\nall the items that she produced What will be the maximum profit Solution Let x and y be the number of items M and N respectively Total profit on the production = Rs (600 x + 400 y)\nMathematical formulation of the given problem is as follows:\nMaximise Z = 600 x + 400 y\nsubject to the constraints:\nx + 2y \u2264 12 (constraint on Machine I)"
    },
    {
        "Chapter": "1",
        "sentence_range": "6597-6600",
        "Text": "What will be the maximum profit Solution Let x and y be the number of items M and N respectively Total profit on the production = Rs (600 x + 400 y)\nMathematical formulation of the given problem is as follows:\nMaximise Z = 600 x + 400 y\nsubject to the constraints:\nx + 2y \u2264 12 (constraint on Machine I) (1)\n2x + y \u2264 12 (constraint on Machine II)"
    },
    {
        "Chapter": "1",
        "sentence_range": "6598-6601",
        "Text": "Solution Let x and y be the number of items M and N respectively Total profit on the production = Rs (600 x + 400 y)\nMathematical formulation of the given problem is as follows:\nMaximise Z = 600 x + 400 y\nsubject to the constraints:\nx + 2y \u2264 12 (constraint on Machine I) (1)\n2x + y \u2264 12 (constraint on Machine II) (2)\nx + 5\n4  y \u2265 5 (constraint on Machine III)"
    },
    {
        "Chapter": "1",
        "sentence_range": "6599-6602",
        "Text": "Total profit on the production = Rs (600 x + 400 y)\nMathematical formulation of the given problem is as follows:\nMaximise Z = 600 x + 400 y\nsubject to the constraints:\nx + 2y \u2264 12 (constraint on Machine I) (1)\n2x + y \u2264 12 (constraint on Machine II) (2)\nx + 5\n4  y \u2265 5 (constraint on Machine III) (3)\nx  \u2265 0,  y \u2265 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "6600-6603",
        "Text": "(1)\n2x + y \u2264 12 (constraint on Machine II) (2)\nx + 5\n4  y \u2265 5 (constraint on Machine III) (3)\nx  \u2265 0,  y \u2265 0 (4)\n\u00a9 NCERT\nnot to be republished\nLINEAR PROGRAMMING         523\nLet us draw the graph of constraints (1) to (4)"
    },
    {
        "Chapter": "1",
        "sentence_range": "6601-6604",
        "Text": "(2)\nx + 5\n4  y \u2265 5 (constraint on Machine III) (3)\nx  \u2265 0,  y \u2265 0 (4)\n\u00a9 NCERT\nnot to be republished\nLINEAR PROGRAMMING         523\nLet us draw the graph of constraints (1) to (4) ABCDE is the feasible region\n(shaded) as shown in Fig 12"
    },
    {
        "Chapter": "1",
        "sentence_range": "6602-6605",
        "Text": "(3)\nx  \u2265 0,  y \u2265 0 (4)\n\u00a9 NCERT\nnot to be republished\nLINEAR PROGRAMMING         523\nLet us draw the graph of constraints (1) to (4) ABCDE is the feasible region\n(shaded) as shown in Fig 12 11 determined by the constraints (1) to (4)"
    },
    {
        "Chapter": "1",
        "sentence_range": "6603-6606",
        "Text": "(4)\n\u00a9 NCERT\nnot to be republished\nLINEAR PROGRAMMING         523\nLet us draw the graph of constraints (1) to (4) ABCDE is the feasible region\n(shaded) as shown in Fig 12 11 determined by the constraints (1) to (4) Observe that\nthe feasible region is bounded, coordinates of the corner points A, B, C, D and E are\n(5, 0) (6, 0), (4, 4), (0, 6) and (0, 4) respectively"
    },
    {
        "Chapter": "1",
        "sentence_range": "6604-6607",
        "Text": "ABCDE is the feasible region\n(shaded) as shown in Fig 12 11 determined by the constraints (1) to (4) Observe that\nthe feasible region is bounded, coordinates of the corner points A, B, C, D and E are\n(5, 0) (6, 0), (4, 4), (0, 6) and (0, 4) respectively Fig 12"
    },
    {
        "Chapter": "1",
        "sentence_range": "6605-6608",
        "Text": "11 determined by the constraints (1) to (4) Observe that\nthe feasible region is bounded, coordinates of the corner points A, B, C, D and E are\n(5, 0) (6, 0), (4, 4), (0, 6) and (0, 4) respectively Fig 12 11\nLet us evaluate Z = 600 x + 400 y at these corner points"
    },
    {
        "Chapter": "1",
        "sentence_range": "6606-6609",
        "Text": "Observe that\nthe feasible region is bounded, coordinates of the corner points A, B, C, D and E are\n(5, 0) (6, 0), (4, 4), (0, 6) and (0, 4) respectively Fig 12 11\nLet us evaluate Z = 600 x + 400 y at these corner points Corner point\nZ = 600 x + 400 y\n(5, 0)\n3000\n(6, 0)\n3600\n(4, 4)\n4000 \u2190\nMaximum\n(0, 6)\n2400\n(0, 4)\n1600\nWe see that the point (4, 4) is giving the maximum value of Z"
    },
    {
        "Chapter": "1",
        "sentence_range": "6607-6610",
        "Text": "Fig 12 11\nLet us evaluate Z = 600 x + 400 y at these corner points Corner point\nZ = 600 x + 400 y\n(5, 0)\n3000\n(6, 0)\n3600\n(4, 4)\n4000 \u2190\nMaximum\n(0, 6)\n2400\n(0, 4)\n1600\nWe see that the point (4, 4) is giving the maximum value of Z Hence, the\nmanufacturer has to produce 4 units of each item to get the maximum profit of Rs 4000"
    },
    {
        "Chapter": "1",
        "sentence_range": "6608-6611",
        "Text": "11\nLet us evaluate Z = 600 x + 400 y at these corner points Corner point\nZ = 600 x + 400 y\n(5, 0)\n3000\n(6, 0)\n3600\n(4, 4)\n4000 \u2190\nMaximum\n(0, 6)\n2400\n(0, 4)\n1600\nWe see that the point (4, 4) is giving the maximum value of Z Hence, the\nmanufacturer has to produce 4 units of each item to get the maximum profit of Rs 4000 Example 11 (Transportation problem) There are two factories  located one at\nplace P and the other at place Q"
    },
    {
        "Chapter": "1",
        "sentence_range": "6609-6612",
        "Text": "Corner point\nZ = 600 x + 400 y\n(5, 0)\n3000\n(6, 0)\n3600\n(4, 4)\n4000 \u2190\nMaximum\n(0, 6)\n2400\n(0, 4)\n1600\nWe see that the point (4, 4) is giving the maximum value of Z Hence, the\nmanufacturer has to produce 4 units of each item to get the maximum profit of Rs 4000 Example 11 (Transportation problem) There are two factories  located one at\nplace P and the other at place Q From these locations, a certain commodity is to be\ndelivered to each of the three depots situated at A, B and C"
    },
    {
        "Chapter": "1",
        "sentence_range": "6610-6613",
        "Text": "Hence, the\nmanufacturer has to produce 4 units of each item to get the maximum profit of Rs 4000 Example 11 (Transportation problem) There are two factories  located one at\nplace P and the other at place Q From these locations, a certain commodity is to be\ndelivered to each of the three depots situated at A, B and C The weekly requirements\nof the depots are respectively 5, 5 and 4 units of the commodity while the production\ncapacity of the factories at P and Q are respectively 8 and 6 units"
    },
    {
        "Chapter": "1",
        "sentence_range": "6611-6614",
        "Text": "Example 11 (Transportation problem) There are two factories  located one at\nplace P and the other at place Q From these locations, a certain commodity is to be\ndelivered to each of the three depots situated at A, B and C The weekly requirements\nof the depots are respectively 5, 5 and 4 units of the commodity while the production\ncapacity of the factories at P and Q are respectively 8 and 6 units The cost of\n\u00a9 NCERT\nnot to be republished\n524\nMATHEMATICS\nP\n8 units\nA\n5 units\nC\n4 units\nQ\n6 units\nFactory\nFactory\nDepot\nDepot\nB\n5 units\nRs 1005 \u2013 y\nRs 120\nRs 100\nRs 150\nRs 160\n6\n[(5\n) + (5\n)]\nx\ny\n\u2013\n\u2013\n\u2013\nDepot\ny\nRs 100\n8 \u2013\nx\u2013\ny\n5\nx\n\u2013\nx\ntransportation per unit is given below:\nFrom/To\nCost (in Rs)\nA\nB\nC\nP\n160\n100\n150\nQ\n100\n120\n100\nHow many units should be transported from each factory to each depot in order that\nthe transportation cost is minimum"
    },
    {
        "Chapter": "1",
        "sentence_range": "6612-6615",
        "Text": "From these locations, a certain commodity is to be\ndelivered to each of the three depots situated at A, B and C The weekly requirements\nof the depots are respectively 5, 5 and 4 units of the commodity while the production\ncapacity of the factories at P and Q are respectively 8 and 6 units The cost of\n\u00a9 NCERT\nnot to be republished\n524\nMATHEMATICS\nP\n8 units\nA\n5 units\nC\n4 units\nQ\n6 units\nFactory\nFactory\nDepot\nDepot\nB\n5 units\nRs 1005 \u2013 y\nRs 120\nRs 100\nRs 150\nRs 160\n6\n[(5\n) + (5\n)]\nx\ny\n\u2013\n\u2013\n\u2013\nDepot\ny\nRs 100\n8 \u2013\nx\u2013\ny\n5\nx\n\u2013\nx\ntransportation per unit is given below:\nFrom/To\nCost (in Rs)\nA\nB\nC\nP\n160\n100\n150\nQ\n100\n120\n100\nHow many units should be transported from each factory to each depot in order that\nthe transportation cost is minimum What will be the minimum transportation cost"
    },
    {
        "Chapter": "1",
        "sentence_range": "6613-6616",
        "Text": "The weekly requirements\nof the depots are respectively 5, 5 and 4 units of the commodity while the production\ncapacity of the factories at P and Q are respectively 8 and 6 units The cost of\n\u00a9 NCERT\nnot to be republished\n524\nMATHEMATICS\nP\n8 units\nA\n5 units\nC\n4 units\nQ\n6 units\nFactory\nFactory\nDepot\nDepot\nB\n5 units\nRs 1005 \u2013 y\nRs 120\nRs 100\nRs 150\nRs 160\n6\n[(5\n) + (5\n)]\nx\ny\n\u2013\n\u2013\n\u2013\nDepot\ny\nRs 100\n8 \u2013\nx\u2013\ny\n5\nx\n\u2013\nx\ntransportation per unit is given below:\nFrom/To\nCost (in Rs)\nA\nB\nC\nP\n160\n100\n150\nQ\n100\n120\n100\nHow many units should be transported from each factory to each depot in order that\nthe transportation cost is minimum What will be the minimum transportation cost Solution The problem can be explained diagrammatically as follows (Fig 12"
    },
    {
        "Chapter": "1",
        "sentence_range": "6614-6617",
        "Text": "The cost of\n\u00a9 NCERT\nnot to be republished\n524\nMATHEMATICS\nP\n8 units\nA\n5 units\nC\n4 units\nQ\n6 units\nFactory\nFactory\nDepot\nDepot\nB\n5 units\nRs 1005 \u2013 y\nRs 120\nRs 100\nRs 150\nRs 160\n6\n[(5\n) + (5\n)]\nx\ny\n\u2013\n\u2013\n\u2013\nDepot\ny\nRs 100\n8 \u2013\nx\u2013\ny\n5\nx\n\u2013\nx\ntransportation per unit is given below:\nFrom/To\nCost (in Rs)\nA\nB\nC\nP\n160\n100\n150\nQ\n100\n120\n100\nHow many units should be transported from each factory to each depot in order that\nthe transportation cost is minimum What will be the minimum transportation cost Solution The problem can be explained diagrammatically as follows (Fig 12 12):\nLet x units and y units of the commodity be transported from the factory at P to\nthe depots at A and B respectively"
    },
    {
        "Chapter": "1",
        "sentence_range": "6615-6618",
        "Text": "What will be the minimum transportation cost Solution The problem can be explained diagrammatically as follows (Fig 12 12):\nLet x units and y units of the commodity be transported from the factory at P to\nthe depots at A and B respectively Then (8 \u2013 x \u2013 y) units will be transported to depot\nat C (Why"
    },
    {
        "Chapter": "1",
        "sentence_range": "6616-6619",
        "Text": "Solution The problem can be explained diagrammatically as follows (Fig 12 12):\nLet x units and y units of the commodity be transported from the factory at P to\nthe depots at A and B respectively Then (8 \u2013 x \u2013 y) units will be transported to depot\nat C (Why )\nHence, we have\nx \u2265 0, y \u2265 0\nand\n8 \u2013 x \u2013 y \u2265 0\ni"
    },
    {
        "Chapter": "1",
        "sentence_range": "6617-6620",
        "Text": "12):\nLet x units and y units of the commodity be transported from the factory at P to\nthe depots at A and B respectively Then (8 \u2013 x \u2013 y) units will be transported to depot\nat C (Why )\nHence, we have\nx \u2265 0, y \u2265 0\nand\n8 \u2013 x \u2013 y \u2265 0\ni e"
    },
    {
        "Chapter": "1",
        "sentence_range": "6618-6621",
        "Text": "Then (8 \u2013 x \u2013 y) units will be transported to depot\nat C (Why )\nHence, we have\nx \u2265 0, y \u2265 0\nand\n8 \u2013 x \u2013 y \u2265 0\ni e x \u2265 0, y \u2265 0     and     x + y \u2264 8\nNow, the weekly requirement of the depot at A is 5 units of the commodity"
    },
    {
        "Chapter": "1",
        "sentence_range": "6619-6622",
        "Text": ")\nHence, we have\nx \u2265 0, y \u2265 0\nand\n8 \u2013 x \u2013 y \u2265 0\ni e x \u2265 0, y \u2265 0     and     x + y \u2264 8\nNow, the weekly requirement of the depot at A is 5 units of the commodity Since\nx units are transported from the factory at P, the remaining (5 \u2013 x) units need to be\ntransported from the factory at Q"
    },
    {
        "Chapter": "1",
        "sentence_range": "6620-6623",
        "Text": "e x \u2265 0, y \u2265 0     and     x + y \u2264 8\nNow, the weekly requirement of the depot at A is 5 units of the commodity Since\nx units are transported from the factory at P, the remaining (5 \u2013 x) units need to be\ntransported from the factory at Q Obviously, 5 \u2013 x \u2265 0, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "6621-6624",
        "Text": "x \u2265 0, y \u2265 0     and     x + y \u2264 8\nNow, the weekly requirement of the depot at A is 5 units of the commodity Since\nx units are transported from the factory at P, the remaining (5 \u2013 x) units need to be\ntransported from the factory at Q Obviously, 5 \u2013 x \u2265 0, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "6622-6625",
        "Text": "Since\nx units are transported from the factory at P, the remaining (5 \u2013 x) units need to be\ntransported from the factory at Q Obviously, 5 \u2013 x \u2265 0, i e x \u2264 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "6623-6626",
        "Text": "Obviously, 5 \u2013 x \u2265 0, i e x \u2264 5 Similarly, (5 \u2013 y) and 6 \u2013 (5 \u2013 x + 5 \u2013 y) = x + y \u2013 4 units are to be transported from\nthe factory at Q to the depots at B and C respectively"
    },
    {
        "Chapter": "1",
        "sentence_range": "6624-6627",
        "Text": "e x \u2264 5 Similarly, (5 \u2013 y) and 6 \u2013 (5 \u2013 x + 5 \u2013 y) = x + y \u2013 4 units are to be transported from\nthe factory at Q to the depots at B and C respectively Thus,\n5 \u2013 y \u2265 0  ,  x + y \u2013 4 \u22650\ni"
    },
    {
        "Chapter": "1",
        "sentence_range": "6625-6628",
        "Text": "x \u2264 5 Similarly, (5 \u2013 y) and 6 \u2013 (5 \u2013 x + 5 \u2013 y) = x + y \u2013 4 units are to be transported from\nthe factory at Q to the depots at B and C respectively Thus,\n5 \u2013 y \u2265 0  ,  x + y \u2013 4 \u22650\ni e"
    },
    {
        "Chapter": "1",
        "sentence_range": "6626-6629",
        "Text": "Similarly, (5 \u2013 y) and 6 \u2013 (5 \u2013 x + 5 \u2013 y) = x + y \u2013 4 units are to be transported from\nthe factory at Q to the depots at B and C respectively Thus,\n5 \u2013 y \u2265 0  ,  x + y \u2013 4 \u22650\ni e y \u2264 5  ,  x + y \u2265\n4\nFig 12"
    },
    {
        "Chapter": "1",
        "sentence_range": "6627-6630",
        "Text": "Thus,\n5 \u2013 y \u2265 0  ,  x + y \u2013 4 \u22650\ni e y \u2264 5  ,  x + y \u2265\n4\nFig 12 12\n\u00a9 NCERT\nnot to be republished\nLINEAR PROGRAMMING         525\nTotal transportation cost Z is given by\nZ = 160 x + 100 y + 100 ( 5 \u2013 x) + 120 (5 \u2013 y) + 100 (x + y \u2013 4) + 150 (8 \u2013 x \u2013 y)\n= 10 (x \u2013 7 y + 190)\nTherefore, the problem reduces to\nMinimise Z = 10 (x \u2013 7y + 190)\nsubject to the constraints:\nx \u2265 0, y \u2265 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "6628-6631",
        "Text": "e y \u2264 5  ,  x + y \u2265\n4\nFig 12 12\n\u00a9 NCERT\nnot to be republished\nLINEAR PROGRAMMING         525\nTotal transportation cost Z is given by\nZ = 160 x + 100 y + 100 ( 5 \u2013 x) + 120 (5 \u2013 y) + 100 (x + y \u2013 4) + 150 (8 \u2013 x \u2013 y)\n= 10 (x \u2013 7 y + 190)\nTherefore, the problem reduces to\nMinimise Z = 10 (x \u2013 7y + 190)\nsubject to the constraints:\nx \u2265 0, y \u2265 0 (1)\nx + y \u2264 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "6629-6632",
        "Text": "y \u2264 5  ,  x + y \u2265\n4\nFig 12 12\n\u00a9 NCERT\nnot to be republished\nLINEAR PROGRAMMING         525\nTotal transportation cost Z is given by\nZ = 160 x + 100 y + 100 ( 5 \u2013 x) + 120 (5 \u2013 y) + 100 (x + y \u2013 4) + 150 (8 \u2013 x \u2013 y)\n= 10 (x \u2013 7 y + 190)\nTherefore, the problem reduces to\nMinimise Z = 10 (x \u2013 7y + 190)\nsubject to the constraints:\nx \u2265 0, y \u2265 0 (1)\nx + y \u2264 8 (2)\nx \u2264 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "6630-6633",
        "Text": "12\n\u00a9 NCERT\nnot to be republished\nLINEAR PROGRAMMING         525\nTotal transportation cost Z is given by\nZ = 160 x + 100 y + 100 ( 5 \u2013 x) + 120 (5 \u2013 y) + 100 (x + y \u2013 4) + 150 (8 \u2013 x \u2013 y)\n= 10 (x \u2013 7 y + 190)\nTherefore, the problem reduces to\nMinimise Z = 10 (x \u2013 7y + 190)\nsubject to the constraints:\nx \u2265 0, y \u2265 0 (1)\nx + y \u2264 8 (2)\nx \u2264 5 (3)\ny \u2264 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "6631-6634",
        "Text": "(1)\nx + y \u2264 8 (2)\nx \u2264 5 (3)\ny \u2264 5 (4)\nand\nx + y \u2265 4"
    },
    {
        "Chapter": "1",
        "sentence_range": "6632-6635",
        "Text": "(2)\nx \u2264 5 (3)\ny \u2264 5 (4)\nand\nx + y \u2265 4 (5)\nThe shaded region ABCDEF\nrepresented by the constraints (1) to\n(5) is the feasible region (Fig 12"
    },
    {
        "Chapter": "1",
        "sentence_range": "6633-6636",
        "Text": "(3)\ny \u2264 5 (4)\nand\nx + y \u2265 4 (5)\nThe shaded region ABCDEF\nrepresented by the constraints (1) to\n(5) is the feasible region (Fig 12 13)"
    },
    {
        "Chapter": "1",
        "sentence_range": "6634-6637",
        "Text": "(4)\nand\nx + y \u2265 4 (5)\nThe shaded region ABCDEF\nrepresented by the constraints (1) to\n(5) is the feasible region (Fig 12 13) Observe that the feasible region is bounded"
    },
    {
        "Chapter": "1",
        "sentence_range": "6635-6638",
        "Text": "(5)\nThe shaded region ABCDEF\nrepresented by the constraints (1) to\n(5) is the feasible region (Fig 12 13) Observe that the feasible region is bounded The coordinates of the corner points\nof the feasible region are (0, 4), (0, 5), (3, 5), (5, 3), (5, 0) and (4, 0)"
    },
    {
        "Chapter": "1",
        "sentence_range": "6636-6639",
        "Text": "13) Observe that the feasible region is bounded The coordinates of the corner points\nof the feasible region are (0, 4), (0, 5), (3, 5), (5, 3), (5, 0) and (4, 0) Let us evaluate Z at these points"
    },
    {
        "Chapter": "1",
        "sentence_range": "6637-6640",
        "Text": "Observe that the feasible region is bounded The coordinates of the corner points\nof the feasible region are (0, 4), (0, 5), (3, 5), (5, 3), (5, 0) and (4, 0) Let us evaluate Z at these points Corner Point\nZ = 10 (x \u2013 7 y + 190)\n(0, 4)\n1620\n(0, 5)\n1550 \u2190\n\u2190\n\u2190\n\u2190\n\u2190\nMinimum\n(3, 5)\n1580\n(5, 3)\n1740\n(5, 0)\n1950\n(4, 0)\n1940\nFrom the table, we see that the minimum value of Z  is 1550 at the point (0, 5)"
    },
    {
        "Chapter": "1",
        "sentence_range": "6638-6641",
        "Text": "The coordinates of the corner points\nof the feasible region are (0, 4), (0, 5), (3, 5), (5, 3), (5, 0) and (4, 0) Let us evaluate Z at these points Corner Point\nZ = 10 (x \u2013 7 y + 190)\n(0, 4)\n1620\n(0, 5)\n1550 \u2190\n\u2190\n\u2190\n\u2190\n\u2190\nMinimum\n(3, 5)\n1580\n(5, 3)\n1740\n(5, 0)\n1950\n(4, 0)\n1940\nFrom the table, we see that the minimum value of Z  is 1550 at the point (0, 5) Hence, the optimal transportation strategy will be to deliver 0, 5 and 3 units from\nthe factory at P and 5, 0 and 1 units from the factory at Q to the depots at A, B and C\nrespectively"
    },
    {
        "Chapter": "1",
        "sentence_range": "6639-6642",
        "Text": "Let us evaluate Z at these points Corner Point\nZ = 10 (x \u2013 7 y + 190)\n(0, 4)\n1620\n(0, 5)\n1550 \u2190\n\u2190\n\u2190\n\u2190\n\u2190\nMinimum\n(3, 5)\n1580\n(5, 3)\n1740\n(5, 0)\n1950\n(4, 0)\n1940\nFrom the table, we see that the minimum value of Z  is 1550 at the point (0, 5) Hence, the optimal transportation strategy will be to deliver 0, 5 and 3 units from\nthe factory at P and 5, 0 and 1 units from the factory at Q to the depots at A, B and C\nrespectively Corresponding to this strategy, the transportation cost would be minimum,\ni"
    },
    {
        "Chapter": "1",
        "sentence_range": "6640-6643",
        "Text": "Corner Point\nZ = 10 (x \u2013 7 y + 190)\n(0, 4)\n1620\n(0, 5)\n1550 \u2190\n\u2190\n\u2190\n\u2190\n\u2190\nMinimum\n(3, 5)\n1580\n(5, 3)\n1740\n(5, 0)\n1950\n(4, 0)\n1940\nFrom the table, we see that the minimum value of Z  is 1550 at the point (0, 5) Hence, the optimal transportation strategy will be to deliver 0, 5 and 3 units from\nthe factory at P and 5, 0 and 1 units from the factory at Q to the depots at A, B and C\nrespectively Corresponding to this strategy, the transportation cost would be minimum,\ni e"
    },
    {
        "Chapter": "1",
        "sentence_range": "6641-6644",
        "Text": "Hence, the optimal transportation strategy will be to deliver 0, 5 and 3 units from\nthe factory at P and 5, 0 and 1 units from the factory at Q to the depots at A, B and C\nrespectively Corresponding to this strategy, the transportation cost would be minimum,\ni e , Rs 1550"
    },
    {
        "Chapter": "1",
        "sentence_range": "6642-6645",
        "Text": "Corresponding to this strategy, the transportation cost would be minimum,\ni e , Rs 1550 Miscellaneous Exercise on Chapter 12\n1"
    },
    {
        "Chapter": "1",
        "sentence_range": "6643-6646",
        "Text": "e , Rs 1550 Miscellaneous Exercise on Chapter 12\n1 Refer to Example 9"
    },
    {
        "Chapter": "1",
        "sentence_range": "6644-6647",
        "Text": ", Rs 1550 Miscellaneous Exercise on Chapter 12\n1 Refer to Example 9 How many packets of each food should be used to maximise\nthe amount of vitamin A in the diet"
    },
    {
        "Chapter": "1",
        "sentence_range": "6645-6648",
        "Text": "Miscellaneous Exercise on Chapter 12\n1 Refer to Example 9 How many packets of each food should be used to maximise\nthe amount of vitamin A in the diet What is the maximum amount of vitamin A\nin the diet"
    },
    {
        "Chapter": "1",
        "sentence_range": "6646-6649",
        "Text": "Refer to Example 9 How many packets of each food should be used to maximise\nthe amount of vitamin A in the diet What is the maximum amount of vitamin A\nin the diet Fig 12"
    },
    {
        "Chapter": "1",
        "sentence_range": "6647-6650",
        "Text": "How many packets of each food should be used to maximise\nthe amount of vitamin A in the diet What is the maximum amount of vitamin A\nin the diet Fig 12 13\n\u00a9 NCERT\nnot to be republished\n526\nMATHEMATICS\n2"
    },
    {
        "Chapter": "1",
        "sentence_range": "6648-6651",
        "Text": "What is the maximum amount of vitamin A\nin the diet Fig 12 13\n\u00a9 NCERT\nnot to be republished\n526\nMATHEMATICS\n2 A farmer mixes two brands P and Q of cattle feed"
    },
    {
        "Chapter": "1",
        "sentence_range": "6649-6652",
        "Text": "Fig 12 13\n\u00a9 NCERT\nnot to be republished\n526\nMATHEMATICS\n2 A farmer mixes two brands P and Q of cattle feed Brand P, costing Rs 250 per\nbag, contains 3 units of nutritional element A, 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "6650-6653",
        "Text": "13\n\u00a9 NCERT\nnot to be republished\n526\nMATHEMATICS\n2 A farmer mixes two brands P and Q of cattle feed Brand P, costing Rs 250 per\nbag, contains 3 units of nutritional element A, 2 5 units of element B and 2 units\nof element C"
    },
    {
        "Chapter": "1",
        "sentence_range": "6651-6654",
        "Text": "A farmer mixes two brands P and Q of cattle feed Brand P, costing Rs 250 per\nbag, contains 3 units of nutritional element A, 2 5 units of element B and 2 units\nof element C Brand Q costing Rs 200 per bag contains 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "6652-6655",
        "Text": "Brand P, costing Rs 250 per\nbag, contains 3 units of nutritional element A, 2 5 units of element B and 2 units\nof element C Brand Q costing Rs 200 per bag contains 1 5 units of nutritional\nelement A, 11"
    },
    {
        "Chapter": "1",
        "sentence_range": "6653-6656",
        "Text": "5 units of element B and 2 units\nof element C Brand Q costing Rs 200 per bag contains 1 5 units of nutritional\nelement A, 11 25 units of element B, and 3 units of element C"
    },
    {
        "Chapter": "1",
        "sentence_range": "6654-6657",
        "Text": "Brand Q costing Rs 200 per bag contains 1 5 units of nutritional\nelement A, 11 25 units of element B, and 3 units of element C The minimum\nrequirements of nutrients A, B and C are 18 units, 45 units and 24 units respectively"
    },
    {
        "Chapter": "1",
        "sentence_range": "6655-6658",
        "Text": "5 units of nutritional\nelement A, 11 25 units of element B, and 3 units of element C The minimum\nrequirements of nutrients A, B and C are 18 units, 45 units and 24 units respectively Determine the number of bags of each brand which should be mixed in order to\nproduce a mixture having a minimum cost per bag"
    },
    {
        "Chapter": "1",
        "sentence_range": "6656-6659",
        "Text": "25 units of element B, and 3 units of element C The minimum\nrequirements of nutrients A, B and C are 18 units, 45 units and 24 units respectively Determine the number of bags of each brand which should be mixed in order to\nproduce a mixture having a minimum cost per bag What is the minimum cost of\nthe mixture per bag"
    },
    {
        "Chapter": "1",
        "sentence_range": "6657-6660",
        "Text": "The minimum\nrequirements of nutrients A, B and C are 18 units, 45 units and 24 units respectively Determine the number of bags of each brand which should be mixed in order to\nproduce a mixture having a minimum cost per bag What is the minimum cost of\nthe mixture per bag 3"
    },
    {
        "Chapter": "1",
        "sentence_range": "6658-6661",
        "Text": "Determine the number of bags of each brand which should be mixed in order to\nproduce a mixture having a minimum cost per bag What is the minimum cost of\nthe mixture per bag 3 A dietician wishes to mix together two kinds of food X and Y in such a way that\nthe mixture contains at least 10 units of vitamin A, 12 units of vitamin B and\n8 units of vitamin C"
    },
    {
        "Chapter": "1",
        "sentence_range": "6659-6662",
        "Text": "What is the minimum cost of\nthe mixture per bag 3 A dietician wishes to mix together two kinds of food X and Y in such a way that\nthe mixture contains at least 10 units of vitamin A, 12 units of vitamin B and\n8 units of vitamin C The vitamin contents of one kg food is given below:\nFood\nVitamin A Vitamin B\nVitamin C\nX\n1\n2\n3\nY\n2\n2\n1\nOne kg of food X costs Rs 16 and one kg of food Y costs Rs 20"
    },
    {
        "Chapter": "1",
        "sentence_range": "6660-6663",
        "Text": "3 A dietician wishes to mix together two kinds of food X and Y in such a way that\nthe mixture contains at least 10 units of vitamin A, 12 units of vitamin B and\n8 units of vitamin C The vitamin contents of one kg food is given below:\nFood\nVitamin A Vitamin B\nVitamin C\nX\n1\n2\n3\nY\n2\n2\n1\nOne kg of food X costs Rs 16 and one kg of food Y costs Rs 20 Find the least\ncost of the mixture which will produce the required diet"
    },
    {
        "Chapter": "1",
        "sentence_range": "6661-6664",
        "Text": "A dietician wishes to mix together two kinds of food X and Y in such a way that\nthe mixture contains at least 10 units of vitamin A, 12 units of vitamin B and\n8 units of vitamin C The vitamin contents of one kg food is given below:\nFood\nVitamin A Vitamin B\nVitamin C\nX\n1\n2\n3\nY\n2\n2\n1\nOne kg of food X costs Rs 16 and one kg of food Y costs Rs 20 Find the least\ncost of the mixture which will produce the required diet 4"
    },
    {
        "Chapter": "1",
        "sentence_range": "6662-6665",
        "Text": "The vitamin contents of one kg food is given below:\nFood\nVitamin A Vitamin B\nVitamin C\nX\n1\n2\n3\nY\n2\n2\n1\nOne kg of food X costs Rs 16 and one kg of food Y costs Rs 20 Find the least\ncost of the mixture which will produce the required diet 4 A manufacturer makes two types of toys A and B"
    },
    {
        "Chapter": "1",
        "sentence_range": "6663-6666",
        "Text": "Find the least\ncost of the mixture which will produce the required diet 4 A manufacturer makes two types of toys A and B Three machines are needed\nfor this purpose and the time (in minutes) required for each toy on the machines\nis given below:\nTypes of Toys\nMachines\nI\nII\nIII\nA\n12\n18\n6\nB\n6\n0\n9\nEach machine is available for a maximum of 6 hours per day"
    },
    {
        "Chapter": "1",
        "sentence_range": "6664-6667",
        "Text": "4 A manufacturer makes two types of toys A and B Three machines are needed\nfor this purpose and the time (in minutes) required for each toy on the machines\nis given below:\nTypes of Toys\nMachines\nI\nII\nIII\nA\n12\n18\n6\nB\n6\n0\n9\nEach machine is available for a maximum of 6 hours per day If the profit on\neach toy of type A is Rs 7"
    },
    {
        "Chapter": "1",
        "sentence_range": "6665-6668",
        "Text": "A manufacturer makes two types of toys A and B Three machines are needed\nfor this purpose and the time (in minutes) required for each toy on the machines\nis given below:\nTypes of Toys\nMachines\nI\nII\nIII\nA\n12\n18\n6\nB\n6\n0\n9\nEach machine is available for a maximum of 6 hours per day If the profit on\neach toy of type A is Rs 7 50 and that on each toy of type B is Rs 5, show that 15\ntoys of type A and 30 of type B should be manufactured in a day to get maximum\nprofit"
    },
    {
        "Chapter": "1",
        "sentence_range": "6666-6669",
        "Text": "Three machines are needed\nfor this purpose and the time (in minutes) required for each toy on the machines\nis given below:\nTypes of Toys\nMachines\nI\nII\nIII\nA\n12\n18\n6\nB\n6\n0\n9\nEach machine is available for a maximum of 6 hours per day If the profit on\neach toy of type A is Rs 7 50 and that on each toy of type B is Rs 5, show that 15\ntoys of type A and 30 of type B should be manufactured in a day to get maximum\nprofit 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "6667-6670",
        "Text": "If the profit on\neach toy of type A is Rs 7 50 and that on each toy of type B is Rs 5, show that 15\ntoys of type A and 30 of type B should be manufactured in a day to get maximum\nprofit 5 An aeroplane can carry a maximum of 200 passengers"
    },
    {
        "Chapter": "1",
        "sentence_range": "6668-6671",
        "Text": "50 and that on each toy of type B is Rs 5, show that 15\ntoys of type A and 30 of type B should be manufactured in a day to get maximum\nprofit 5 An aeroplane can carry a maximum of 200 passengers A profit of Rs 1000 is\nmade on each executive class ticket and a profit of Rs 600 is made on each\neconomy class ticket"
    },
    {
        "Chapter": "1",
        "sentence_range": "6669-6672",
        "Text": "5 An aeroplane can carry a maximum of 200 passengers A profit of Rs 1000 is\nmade on each executive class ticket and a profit of Rs 600 is made on each\neconomy class ticket The airline reserves at least 20 seats for executive class"
    },
    {
        "Chapter": "1",
        "sentence_range": "6670-6673",
        "Text": "An aeroplane can carry a maximum of 200 passengers A profit of Rs 1000 is\nmade on each executive class ticket and a profit of Rs 600 is made on each\neconomy class ticket The airline reserves at least 20 seats for executive class However, at least 4 times as many passengers prefer to travel by economy class\nthan by the executive class"
    },
    {
        "Chapter": "1",
        "sentence_range": "6671-6674",
        "Text": "A profit of Rs 1000 is\nmade on each executive class ticket and a profit of Rs 600 is made on each\neconomy class ticket The airline reserves at least 20 seats for executive class However, at least 4 times as many passengers prefer to travel by economy class\nthan by the executive class Determine how many tickets of each type must be\nsold in order to maximise the profit for the airline"
    },
    {
        "Chapter": "1",
        "sentence_range": "6672-6675",
        "Text": "The airline reserves at least 20 seats for executive class However, at least 4 times as many passengers prefer to travel by economy class\nthan by the executive class Determine how many tickets of each type must be\nsold in order to maximise the profit for the airline What is the maximum profit"
    },
    {
        "Chapter": "1",
        "sentence_range": "6673-6676",
        "Text": "However, at least 4 times as many passengers prefer to travel by economy class\nthan by the executive class Determine how many tickets of each type must be\nsold in order to maximise the profit for the airline What is the maximum profit \u00a9 NCERT\nnot to be republished\nLINEAR PROGRAMMING         527\n6"
    },
    {
        "Chapter": "1",
        "sentence_range": "6674-6677",
        "Text": "Determine how many tickets of each type must be\nsold in order to maximise the profit for the airline What is the maximum profit \u00a9 NCERT\nnot to be republished\nLINEAR PROGRAMMING         527\n6 Two godowns A and B have grain capacity of 100 quintals and 50 quintals\nrespectively"
    },
    {
        "Chapter": "1",
        "sentence_range": "6675-6678",
        "Text": "What is the maximum profit \u00a9 NCERT\nnot to be republished\nLINEAR PROGRAMMING         527\n6 Two godowns A and B have grain capacity of 100 quintals and 50 quintals\nrespectively They supply to 3 ration shops, D, E and F whose requirements are\n60, 50 and 40 quintals respectively"
    },
    {
        "Chapter": "1",
        "sentence_range": "6676-6679",
        "Text": "\u00a9 NCERT\nnot to be republished\nLINEAR PROGRAMMING         527\n6 Two godowns A and B have grain capacity of 100 quintals and 50 quintals\nrespectively They supply to 3 ration shops, D, E and F whose requirements are\n60, 50 and 40 quintals respectively The cost of transportation per quintal from\nthe godowns to the shops are given in the following table:\nTransportation cost per quintal (in Rs)\nFrom/To\nA\nB\nD\n6\n4\nE\n3\n2\nF\n2"
    },
    {
        "Chapter": "1",
        "sentence_range": "6677-6680",
        "Text": "Two godowns A and B have grain capacity of 100 quintals and 50 quintals\nrespectively They supply to 3 ration shops, D, E and F whose requirements are\n60, 50 and 40 quintals respectively The cost of transportation per quintal from\nthe godowns to the shops are given in the following table:\nTransportation cost per quintal (in Rs)\nFrom/To\nA\nB\nD\n6\n4\nE\n3\n2\nF\n2 50\n3\nHow should the supplies be transported in order that the transportation cost is\nminimum"
    },
    {
        "Chapter": "1",
        "sentence_range": "6678-6681",
        "Text": "They supply to 3 ration shops, D, E and F whose requirements are\n60, 50 and 40 quintals respectively The cost of transportation per quintal from\nthe godowns to the shops are given in the following table:\nTransportation cost per quintal (in Rs)\nFrom/To\nA\nB\nD\n6\n4\nE\n3\n2\nF\n2 50\n3\nHow should the supplies be transported in order that the transportation cost is\nminimum What is the minimum cost"
    },
    {
        "Chapter": "1",
        "sentence_range": "6679-6682",
        "Text": "The cost of transportation per quintal from\nthe godowns to the shops are given in the following table:\nTransportation cost per quintal (in Rs)\nFrom/To\nA\nB\nD\n6\n4\nE\n3\n2\nF\n2 50\n3\nHow should the supplies be transported in order that the transportation cost is\nminimum What is the minimum cost 7"
    },
    {
        "Chapter": "1",
        "sentence_range": "6680-6683",
        "Text": "50\n3\nHow should the supplies be transported in order that the transportation cost is\nminimum What is the minimum cost 7 An oil company has two depots A and B with capacities of 7000 L and 4000 L\nrespectively"
    },
    {
        "Chapter": "1",
        "sentence_range": "6681-6684",
        "Text": "What is the minimum cost 7 An oil company has two depots A and B with capacities of 7000 L and 4000 L\nrespectively The company is to supply oil to three petrol pumps, D, E and F\nwhose requirements are 4500L, 3000L and 3500L respectively"
    },
    {
        "Chapter": "1",
        "sentence_range": "6682-6685",
        "Text": "7 An oil company has two depots A and B with capacities of 7000 L and 4000 L\nrespectively The company is to supply oil to three petrol pumps, D, E and F\nwhose requirements are 4500L, 3000L and 3500L respectively The distances\n(in km) between the depots and the petrol pumps is given in the following table:\nDistance in (km"
    },
    {
        "Chapter": "1",
        "sentence_range": "6683-6686",
        "Text": "An oil company has two depots A and B with capacities of 7000 L and 4000 L\nrespectively The company is to supply oil to three petrol pumps, D, E and F\nwhose requirements are 4500L, 3000L and 3500L respectively The distances\n(in km) between the depots and the petrol pumps is given in the following table:\nDistance in (km )\nFrom / To\nA\nB\nD\n7\n3\nE\n6\n4\nF\n3\n2\nAssuming that the transportation cost of 10 litres of oil is Re 1 per km, how\nshould the delivery be scheduled in order that the transportation cost is minimum"
    },
    {
        "Chapter": "1",
        "sentence_range": "6684-6687",
        "Text": "The company is to supply oil to three petrol pumps, D, E and F\nwhose requirements are 4500L, 3000L and 3500L respectively The distances\n(in km) between the depots and the petrol pumps is given in the following table:\nDistance in (km )\nFrom / To\nA\nB\nD\n7\n3\nE\n6\n4\nF\n3\n2\nAssuming that the transportation cost of 10 litres of oil is Re 1 per km, how\nshould the delivery be scheduled in order that the transportation cost is minimum What is the minimum cost"
    },
    {
        "Chapter": "1",
        "sentence_range": "6685-6688",
        "Text": "The distances\n(in km) between the depots and the petrol pumps is given in the following table:\nDistance in (km )\nFrom / To\nA\nB\nD\n7\n3\nE\n6\n4\nF\n3\n2\nAssuming that the transportation cost of 10 litres of oil is Re 1 per km, how\nshould the delivery be scheduled in order that the transportation cost is minimum What is the minimum cost 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "6686-6689",
        "Text": ")\nFrom / To\nA\nB\nD\n7\n3\nE\n6\n4\nF\n3\n2\nAssuming that the transportation cost of 10 litres of oil is Re 1 per km, how\nshould the delivery be scheduled in order that the transportation cost is minimum What is the minimum cost 8 A fruit grower can use two types of fertilizer in his garden, brand P and brand Q"
    },
    {
        "Chapter": "1",
        "sentence_range": "6687-6690",
        "Text": "What is the minimum cost 8 A fruit grower can use two types of fertilizer in his garden, brand P and brand Q The amounts (in kg) of nitrogen, phosphoric acid, potash, and chlorine in a bag of\neach brand are given in the table"
    },
    {
        "Chapter": "1",
        "sentence_range": "6688-6691",
        "Text": "8 A fruit grower can use two types of fertilizer in his garden, brand P and brand Q The amounts (in kg) of nitrogen, phosphoric acid, potash, and chlorine in a bag of\neach brand are given in the table Tests indicate that the garden needs at least\n240 kg of phosphoric acid, at least 270 kg of potash and at most 310 kg of\nchlorine"
    },
    {
        "Chapter": "1",
        "sentence_range": "6689-6692",
        "Text": "A fruit grower can use two types of fertilizer in his garden, brand P and brand Q The amounts (in kg) of nitrogen, phosphoric acid, potash, and chlorine in a bag of\neach brand are given in the table Tests indicate that the garden needs at least\n240 kg of phosphoric acid, at least 270 kg of potash and at most 310 kg of\nchlorine If the grower wants to minimise the amount of nitrogen added to the garden,\nhow many bags of each brand should be used"
    },
    {
        "Chapter": "1",
        "sentence_range": "6690-6693",
        "Text": "The amounts (in kg) of nitrogen, phosphoric acid, potash, and chlorine in a bag of\neach brand are given in the table Tests indicate that the garden needs at least\n240 kg of phosphoric acid, at least 270 kg of potash and at most 310 kg of\nchlorine If the grower wants to minimise the amount of nitrogen added to the garden,\nhow many bags of each brand should be used What is the minimum amount of\nnitrogen added in the garden"
    },
    {
        "Chapter": "1",
        "sentence_range": "6691-6694",
        "Text": "Tests indicate that the garden needs at least\n240 kg of phosphoric acid, at least 270 kg of potash and at most 310 kg of\nchlorine If the grower wants to minimise the amount of nitrogen added to the garden,\nhow many bags of each brand should be used What is the minimum amount of\nnitrogen added in the garden \u00a9 NCERT\nnot to be republished\n528\nMATHEMATICS\nkg per bag\nBrand P\nBrand Q\nNitrogen\n3\n3"
    },
    {
        "Chapter": "1",
        "sentence_range": "6692-6695",
        "Text": "If the grower wants to minimise the amount of nitrogen added to the garden,\nhow many bags of each brand should be used What is the minimum amount of\nnitrogen added in the garden \u00a9 NCERT\nnot to be republished\n528\nMATHEMATICS\nkg per bag\nBrand P\nBrand Q\nNitrogen\n3\n3 5\nPhosphoric acid\n1\n2\nPotash\n3\n1"
    },
    {
        "Chapter": "1",
        "sentence_range": "6693-6696",
        "Text": "What is the minimum amount of\nnitrogen added in the garden \u00a9 NCERT\nnot to be republished\n528\nMATHEMATICS\nkg per bag\nBrand P\nBrand Q\nNitrogen\n3\n3 5\nPhosphoric acid\n1\n2\nPotash\n3\n1 5\nChlorine\n1"
    },
    {
        "Chapter": "1",
        "sentence_range": "6694-6697",
        "Text": "\u00a9 NCERT\nnot to be republished\n528\nMATHEMATICS\nkg per bag\nBrand P\nBrand Q\nNitrogen\n3\n3 5\nPhosphoric acid\n1\n2\nPotash\n3\n1 5\nChlorine\n1 5\n2\n9"
    },
    {
        "Chapter": "1",
        "sentence_range": "6695-6698",
        "Text": "5\nPhosphoric acid\n1\n2\nPotash\n3\n1 5\nChlorine\n1 5\n2\n9 Refer to Question 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "6696-6699",
        "Text": "5\nChlorine\n1 5\n2\n9 Refer to Question 8 If the grower wants to maximise the amount of nitrogen\nadded to the garden, how many bags of each brand should be added"
    },
    {
        "Chapter": "1",
        "sentence_range": "6697-6700",
        "Text": "5\n2\n9 Refer to Question 8 If the grower wants to maximise the amount of nitrogen\nadded to the garden, how many bags of each brand should be added What is\nthe maximum amount of nitrogen added"
    },
    {
        "Chapter": "1",
        "sentence_range": "6698-6701",
        "Text": "Refer to Question 8 If the grower wants to maximise the amount of nitrogen\nadded to the garden, how many bags of each brand should be added What is\nthe maximum amount of nitrogen added 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "6699-6702",
        "Text": "If the grower wants to maximise the amount of nitrogen\nadded to the garden, how many bags of each brand should be added What is\nthe maximum amount of nitrogen added 10 A toy company manufactures two types of dolls, A and B"
    },
    {
        "Chapter": "1",
        "sentence_range": "6700-6703",
        "Text": "What is\nthe maximum amount of nitrogen added 10 A toy company manufactures two types of dolls, A and B Market tests and available\nresources have indicated that the combined production level should not exceed 1200\ndolls per week and the demand for dolls of type B is at most half of that for dolls of\ntype A"
    },
    {
        "Chapter": "1",
        "sentence_range": "6701-6704",
        "Text": "10 A toy company manufactures two types of dolls, A and B Market tests and available\nresources have indicated that the combined production level should not exceed 1200\ndolls per week and the demand for dolls of type B is at most half of that for dolls of\ntype A Further, the production level of dolls of type A can exceed three times the\nproduction of dolls of other type by at most 600 units"
    },
    {
        "Chapter": "1",
        "sentence_range": "6702-6705",
        "Text": "A toy company manufactures two types of dolls, A and B Market tests and available\nresources have indicated that the combined production level should not exceed 1200\ndolls per week and the demand for dolls of type B is at most half of that for dolls of\ntype A Further, the production level of dolls of type A can exceed three times the\nproduction of dolls of other type by at most 600 units If the company makes profit of\nRs 12 and Rs 16 per doll respectively on dolls A and B, how many of each should be\nproduced weekly in order to maximise the profit"
    },
    {
        "Chapter": "1",
        "sentence_range": "6703-6706",
        "Text": "Market tests and available\nresources have indicated that the combined production level should not exceed 1200\ndolls per week and the demand for dolls of type B is at most half of that for dolls of\ntype A Further, the production level of dolls of type A can exceed three times the\nproduction of dolls of other type by at most 600 units If the company makes profit of\nRs 12 and Rs 16 per doll respectively on dolls A and B, how many of each should be\nproduced weekly in order to maximise the profit Summary\n\ufffd A linear programming problem is one that is concerned with finding the optimal\nvalue (maximum or minimum) of a linear function of several variables (called\nobjective function) subject to the conditions that the variables are\nnon-negative and satisfy a set of linear inequalities (called linear constraints)"
    },
    {
        "Chapter": "1",
        "sentence_range": "6704-6707",
        "Text": "Further, the production level of dolls of type A can exceed three times the\nproduction of dolls of other type by at most 600 units If the company makes profit of\nRs 12 and Rs 16 per doll respectively on dolls A and B, how many of each should be\nproduced weekly in order to maximise the profit Summary\n\ufffd A linear programming problem is one that is concerned with finding the optimal\nvalue (maximum or minimum) of a linear function of several variables (called\nobjective function) subject to the conditions that the variables are\nnon-negative and satisfy a set of linear inequalities (called linear constraints) Variables are sometimes called decision variables and are non-negative"
    },
    {
        "Chapter": "1",
        "sentence_range": "6705-6708",
        "Text": "If the company makes profit of\nRs 12 and Rs 16 per doll respectively on dolls A and B, how many of each should be\nproduced weekly in order to maximise the profit Summary\n\ufffd A linear programming problem is one that is concerned with finding the optimal\nvalue (maximum or minimum) of a linear function of several variables (called\nobjective function) subject to the conditions that the variables are\nnon-negative and satisfy a set of linear inequalities (called linear constraints) Variables are sometimes called decision variables and are non-negative \ufffd A few important linear programming problems are:\n(i) Diet problems\n(ii) Manufacturing problems\n(iii) Transportation problems\n\ufffd The common region determined by all the constraints including the non-negative\nconstraints x \u2265 0, y \u2265 0 of a linear programming problem is called the feasible\nregion (or solution region) for the problem"
    },
    {
        "Chapter": "1",
        "sentence_range": "6706-6709",
        "Text": "Summary\n\ufffd A linear programming problem is one that is concerned with finding the optimal\nvalue (maximum or minimum) of a linear function of several variables (called\nobjective function) subject to the conditions that the variables are\nnon-negative and satisfy a set of linear inequalities (called linear constraints) Variables are sometimes called decision variables and are non-negative \ufffd A few important linear programming problems are:\n(i) Diet problems\n(ii) Manufacturing problems\n(iii) Transportation problems\n\ufffd The common region determined by all the constraints including the non-negative\nconstraints x \u2265 0, y \u2265 0 of a linear programming problem is called the feasible\nregion (or solution region) for the problem \ufffd Points within and on the boundary of the feasible region represent feasible\nsolutions of the constraints"
    },
    {
        "Chapter": "1",
        "sentence_range": "6707-6710",
        "Text": "Variables are sometimes called decision variables and are non-negative \ufffd A few important linear programming problems are:\n(i) Diet problems\n(ii) Manufacturing problems\n(iii) Transportation problems\n\ufffd The common region determined by all the constraints including the non-negative\nconstraints x \u2265 0, y \u2265 0 of a linear programming problem is called the feasible\nregion (or solution region) for the problem \ufffd Points within and on the boundary of the feasible region represent feasible\nsolutions of the constraints Any point outside the feasible region is an infeasible solution"
    },
    {
        "Chapter": "1",
        "sentence_range": "6708-6711",
        "Text": "\ufffd A few important linear programming problems are:\n(i) Diet problems\n(ii) Manufacturing problems\n(iii) Transportation problems\n\ufffd The common region determined by all the constraints including the non-negative\nconstraints x \u2265 0, y \u2265 0 of a linear programming problem is called the feasible\nregion (or solution region) for the problem \ufffd Points within and on the boundary of the feasible region represent feasible\nsolutions of the constraints Any point outside the feasible region is an infeasible solution \u00a9 NCERT\nnot to be republished\nLINEAR PROGRAMMING         529\n\ufffd Any point in the feasible region that gives the optimal value (maximum or\nminimum) of the objective function is called an optimal solution"
    },
    {
        "Chapter": "1",
        "sentence_range": "6709-6712",
        "Text": "\ufffd Points within and on the boundary of the feasible region represent feasible\nsolutions of the constraints Any point outside the feasible region is an infeasible solution \u00a9 NCERT\nnot to be republished\nLINEAR PROGRAMMING         529\n\ufffd Any point in the feasible region that gives the optimal value (maximum or\nminimum) of the objective function is called an optimal solution \ufffd The following Theorems are fundamental in solving linear programming\nproblems:\nTheorem 1 Let R be the feasible region (convex polygon) for a linear\nprogramming problem and let Z = ax + by be the objective function"
    },
    {
        "Chapter": "1",
        "sentence_range": "6710-6713",
        "Text": "Any point outside the feasible region is an infeasible solution \u00a9 NCERT\nnot to be republished\nLINEAR PROGRAMMING         529\n\ufffd Any point in the feasible region that gives the optimal value (maximum or\nminimum) of the objective function is called an optimal solution \ufffd The following Theorems are fundamental in solving linear programming\nproblems:\nTheorem 1 Let R be the feasible region (convex polygon) for a linear\nprogramming problem and let Z = ax + by be the objective function When Z\nhas an optimal value (maximum or minimum), where the variables x and y\nare subject to constraints described by linear inequalities, this optimal value\nmust occur at a corner point (vertex) of the feasible region"
    },
    {
        "Chapter": "1",
        "sentence_range": "6711-6714",
        "Text": "\u00a9 NCERT\nnot to be republished\nLINEAR PROGRAMMING         529\n\ufffd Any point in the feasible region that gives the optimal value (maximum or\nminimum) of the objective function is called an optimal solution \ufffd The following Theorems are fundamental in solving linear programming\nproblems:\nTheorem 1 Let R be the feasible region (convex polygon) for a linear\nprogramming problem and let Z = ax + by be the objective function When Z\nhas an optimal value (maximum or minimum), where the variables x and y\nare subject to constraints described by linear inequalities, this optimal value\nmust occur at a corner point (vertex) of the feasible region Theorem 2 Let R be the feasible region for a linear programming problem,\nand let Z = ax + by be the objective function"
    },
    {
        "Chapter": "1",
        "sentence_range": "6712-6715",
        "Text": "\ufffd The following Theorems are fundamental in solving linear programming\nproblems:\nTheorem 1 Let R be the feasible region (convex polygon) for a linear\nprogramming problem and let Z = ax + by be the objective function When Z\nhas an optimal value (maximum or minimum), where the variables x and y\nare subject to constraints described by linear inequalities, this optimal value\nmust occur at a corner point (vertex) of the feasible region Theorem 2 Let R be the feasible region for a linear programming problem,\nand let Z = ax + by be the objective function If R is bounded, then the\nobjective function Z has both a maximum and a minimum value on R and\neach of these occurs at a corner point (vertex) of R"
    },
    {
        "Chapter": "1",
        "sentence_range": "6713-6716",
        "Text": "When Z\nhas an optimal value (maximum or minimum), where the variables x and y\nare subject to constraints described by linear inequalities, this optimal value\nmust occur at a corner point (vertex) of the feasible region Theorem 2 Let R be the feasible region for a linear programming problem,\nand let Z = ax + by be the objective function If R is bounded, then the\nobjective function Z has both a maximum and a minimum value on R and\neach of these occurs at a corner point (vertex) of R \ufffd If the feasible region is unbounded, then a maximum or a minimum may not\nexist"
    },
    {
        "Chapter": "1",
        "sentence_range": "6714-6717",
        "Text": "Theorem 2 Let R be the feasible region for a linear programming problem,\nand let Z = ax + by be the objective function If R is bounded, then the\nobjective function Z has both a maximum and a minimum value on R and\neach of these occurs at a corner point (vertex) of R \ufffd If the feasible region is unbounded, then a maximum or a minimum may not\nexist However, if it exists, it must occur at a corner point of R"
    },
    {
        "Chapter": "1",
        "sentence_range": "6715-6718",
        "Text": "If R is bounded, then the\nobjective function Z has both a maximum and a minimum value on R and\neach of these occurs at a corner point (vertex) of R \ufffd If the feasible region is unbounded, then a maximum or a minimum may not\nexist However, if it exists, it must occur at a corner point of R \ufffd Corner point method for solving a linear programming problem"
    },
    {
        "Chapter": "1",
        "sentence_range": "6716-6719",
        "Text": "\ufffd If the feasible region is unbounded, then a maximum or a minimum may not\nexist However, if it exists, it must occur at a corner point of R \ufffd Corner point method for solving a linear programming problem The method\ncomprises of the following steps:\n(i) Find the feasible region of the linear programming problem and determine\nits corner points (vertices)"
    },
    {
        "Chapter": "1",
        "sentence_range": "6717-6720",
        "Text": "However, if it exists, it must occur at a corner point of R \ufffd Corner point method for solving a linear programming problem The method\ncomprises of the following steps:\n(i) Find the feasible region of the linear programming problem and determine\nits corner points (vertices) (ii) Evaluate the objective function Z = ax + by at each corner point"
    },
    {
        "Chapter": "1",
        "sentence_range": "6718-6721",
        "Text": "\ufffd Corner point method for solving a linear programming problem The method\ncomprises of the following steps:\n(i) Find the feasible region of the linear programming problem and determine\nits corner points (vertices) (ii) Evaluate the objective function Z = ax + by at each corner point Let M\nand m respectively be the largest and smallest values at these points"
    },
    {
        "Chapter": "1",
        "sentence_range": "6719-6722",
        "Text": "The method\ncomprises of the following steps:\n(i) Find the feasible region of the linear programming problem and determine\nits corner points (vertices) (ii) Evaluate the objective function Z = ax + by at each corner point Let M\nand m respectively be the largest and smallest values at these points (iii) If the feasible region is bounded, M and m respectively are the maximum\nand minimum values of the objective function"
    },
    {
        "Chapter": "1",
        "sentence_range": "6720-6723",
        "Text": "(ii) Evaluate the objective function Z = ax + by at each corner point Let M\nand m respectively be the largest and smallest values at these points (iii) If the feasible region is bounded, M and m respectively are the maximum\nand minimum values of the objective function If the feasible region is unbounded, then\n(i) M is the maximum value of the objective function, if the open half plane\ndetermined by ax + by > M has no point in common with the feasible\nregion"
    },
    {
        "Chapter": "1",
        "sentence_range": "6721-6724",
        "Text": "Let M\nand m respectively be the largest and smallest values at these points (iii) If the feasible region is bounded, M and m respectively are the maximum\nand minimum values of the objective function If the feasible region is unbounded, then\n(i) M is the maximum value of the objective function, if the open half plane\ndetermined by ax + by > M has no point in common with the feasible\nregion Otherwise, the objective function has no maximum value"
    },
    {
        "Chapter": "1",
        "sentence_range": "6722-6725",
        "Text": "(iii) If the feasible region is bounded, M and m respectively are the maximum\nand minimum values of the objective function If the feasible region is unbounded, then\n(i) M is the maximum value of the objective function, if the open half plane\ndetermined by ax + by > M has no point in common with the feasible\nregion Otherwise, the objective function has no maximum value (ii) m is the minimum value of the objective function, if the open half plane\ndetermined by ax + by < m has no point in common with the feasible\nregion"
    },
    {
        "Chapter": "1",
        "sentence_range": "6723-6726",
        "Text": "If the feasible region is unbounded, then\n(i) M is the maximum value of the objective function, if the open half plane\ndetermined by ax + by > M has no point in common with the feasible\nregion Otherwise, the objective function has no maximum value (ii) m is the minimum value of the objective function, if the open half plane\ndetermined by ax + by < m has no point in common with the feasible\nregion Otherwise, the objective function has no minimum value"
    },
    {
        "Chapter": "1",
        "sentence_range": "6724-6727",
        "Text": "Otherwise, the objective function has no maximum value (ii) m is the minimum value of the objective function, if the open half plane\ndetermined by ax + by < m has no point in common with the feasible\nregion Otherwise, the objective function has no minimum value \ufffd If two corner points of the feasible region are both optimal solutions of the\nsame type, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "6725-6728",
        "Text": "(ii) m is the minimum value of the objective function, if the open half plane\ndetermined by ax + by < m has no point in common with the feasible\nregion Otherwise, the objective function has no minimum value \ufffd If two corner points of the feasible region are both optimal solutions of the\nsame type, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "6726-6729",
        "Text": "Otherwise, the objective function has no minimum value \ufffd If two corner points of the feasible region are both optimal solutions of the\nsame type, i e , both produce the same maximum or minimum, then any point\non the line segment joining these two points is also an optimal solution of the\nsame type"
    },
    {
        "Chapter": "1",
        "sentence_range": "6727-6730",
        "Text": "\ufffd If two corner points of the feasible region are both optimal solutions of the\nsame type, i e , both produce the same maximum or minimum, then any point\non the line segment joining these two points is also an optimal solution of the\nsame type \u00a9 NCERT\nnot to be republished\n530\nMATHEMATICS\nHistorical Note\nIn the World War II, when the war operations had to be planned to economise\nexpenditure, maximise damage to the enemy, linear programming problems\ncame to the forefront"
    },
    {
        "Chapter": "1",
        "sentence_range": "6728-6731",
        "Text": "e , both produce the same maximum or minimum, then any point\non the line segment joining these two points is also an optimal solution of the\nsame type \u00a9 NCERT\nnot to be republished\n530\nMATHEMATICS\nHistorical Note\nIn the World War II, when the war operations had to be planned to economise\nexpenditure, maximise damage to the enemy, linear programming problems\ncame to the forefront The first problem in linear programming was formulated in 1941 by the Russian\nmathematician, L"
    },
    {
        "Chapter": "1",
        "sentence_range": "6729-6732",
        "Text": ", both produce the same maximum or minimum, then any point\non the line segment joining these two points is also an optimal solution of the\nsame type \u00a9 NCERT\nnot to be republished\n530\nMATHEMATICS\nHistorical Note\nIn the World War II, when the war operations had to be planned to economise\nexpenditure, maximise damage to the enemy, linear programming problems\ncame to the forefront The first problem in linear programming was formulated in 1941 by the Russian\nmathematician, L Kantorovich and the American economist, F"
    },
    {
        "Chapter": "1",
        "sentence_range": "6730-6733",
        "Text": "\u00a9 NCERT\nnot to be republished\n530\nMATHEMATICS\nHistorical Note\nIn the World War II, when the war operations had to be planned to economise\nexpenditure, maximise damage to the enemy, linear programming problems\ncame to the forefront The first problem in linear programming was formulated in 1941 by the Russian\nmathematician, L Kantorovich and the American economist, F L"
    },
    {
        "Chapter": "1",
        "sentence_range": "6731-6734",
        "Text": "The first problem in linear programming was formulated in 1941 by the Russian\nmathematician, L Kantorovich and the American economist, F L Hitchcock,\nboth of whom worked at it independently of each other"
    },
    {
        "Chapter": "1",
        "sentence_range": "6732-6735",
        "Text": "Kantorovich and the American economist, F L Hitchcock,\nboth of whom worked at it independently of each other This was the well\nknown transportation problem"
    },
    {
        "Chapter": "1",
        "sentence_range": "6733-6736",
        "Text": "L Hitchcock,\nboth of whom worked at it independently of each other This was the well\nknown transportation problem In 1945, an English economist, G"
    },
    {
        "Chapter": "1",
        "sentence_range": "6734-6737",
        "Text": "Hitchcock,\nboth of whom worked at it independently of each other This was the well\nknown transportation problem In 1945, an English economist, G Stigler,\ndescribed yet another linear programming problem \u2013 that of determining an\noptimal diet"
    },
    {
        "Chapter": "1",
        "sentence_range": "6735-6738",
        "Text": "This was the well\nknown transportation problem In 1945, an English economist, G Stigler,\ndescribed yet another linear programming problem \u2013 that of determining an\noptimal diet In 1947, the American economist, G"
    },
    {
        "Chapter": "1",
        "sentence_range": "6736-6739",
        "Text": "In 1945, an English economist, G Stigler,\ndescribed yet another linear programming problem \u2013 that of determining an\noptimal diet In 1947, the American economist, G B"
    },
    {
        "Chapter": "1",
        "sentence_range": "6737-6740",
        "Text": "Stigler,\ndescribed yet another linear programming problem \u2013 that of determining an\noptimal diet In 1947, the American economist, G B Dantzig suggested an efficient method\nknown as the simplex method which is an iterative procedure to solve any\nlinear programming problem in a finite number of steps"
    },
    {
        "Chapter": "1",
        "sentence_range": "6738-6741",
        "Text": "In 1947, the American economist, G B Dantzig suggested an efficient method\nknown as the simplex method which is an iterative procedure to solve any\nlinear programming problem in a finite number of steps L"
    },
    {
        "Chapter": "1",
        "sentence_range": "6739-6742",
        "Text": "B Dantzig suggested an efficient method\nknown as the simplex method which is an iterative procedure to solve any\nlinear programming problem in a finite number of steps L Katorovich and American mathematical economist, T"
    },
    {
        "Chapter": "1",
        "sentence_range": "6740-6743",
        "Text": "Dantzig suggested an efficient method\nknown as the simplex method which is an iterative procedure to solve any\nlinear programming problem in a finite number of steps L Katorovich and American mathematical economist, T C"
    },
    {
        "Chapter": "1",
        "sentence_range": "6741-6744",
        "Text": "L Katorovich and American mathematical economist, T C Koopmans were\nawarded the nobel prize in the year 1975 in economics for their pioneering\nwork in linear programming"
    },
    {
        "Chapter": "1",
        "sentence_range": "6742-6745",
        "Text": "Katorovich and American mathematical economist, T C Koopmans were\nawarded the nobel prize in the year 1975 in economics for their pioneering\nwork in linear programming With the advent of computers and the necessary\nsoftwares, it has become possible to apply linear programming model to\nincreasingly complex problems in many areas"
    },
    {
        "Chapter": "1",
        "sentence_range": "6743-6746",
        "Text": "C Koopmans were\nawarded the nobel prize in the year 1975 in economics for their pioneering\nwork in linear programming With the advent of computers and the necessary\nsoftwares, it has become possible to apply linear programming model to\nincreasingly complex problems in many areas \u2014\ufffd\n\ufffd\n\ufffd\n\ufffd\n\ufffd\u2014\n\u00a9 NCERT\nnot to be republished\nPROBABILITY      531\n\ufffdThe theory of probabilities is simply the Science of logic\nquantitatively treated"
    },
    {
        "Chapter": "1",
        "sentence_range": "6744-6747",
        "Text": "Koopmans were\nawarded the nobel prize in the year 1975 in economics for their pioneering\nwork in linear programming With the advent of computers and the necessary\nsoftwares, it has become possible to apply linear programming model to\nincreasingly complex problems in many areas \u2014\ufffd\n\ufffd\n\ufffd\n\ufffd\n\ufffd\u2014\n\u00a9 NCERT\nnot to be republished\nPROBABILITY      531\n\ufffdThe theory of probabilities is simply the Science of logic\nquantitatively treated \u2013 C"
    },
    {
        "Chapter": "1",
        "sentence_range": "6745-6748",
        "Text": "With the advent of computers and the necessary\nsoftwares, it has become possible to apply linear programming model to\nincreasingly complex problems in many areas \u2014\ufffd\n\ufffd\n\ufffd\n\ufffd\n\ufffd\u2014\n\u00a9 NCERT\nnot to be republished\nPROBABILITY      531\n\ufffdThe theory of probabilities is simply the Science of logic\nquantitatively treated \u2013 C S"
    },
    {
        "Chapter": "1",
        "sentence_range": "6746-6749",
        "Text": "\u2014\ufffd\n\ufffd\n\ufffd\n\ufffd\n\ufffd\u2014\n\u00a9 NCERT\nnot to be republished\nPROBABILITY      531\n\ufffdThe theory of probabilities is simply the Science of logic\nquantitatively treated \u2013 C S PEIRCE \ufffd\n13"
    },
    {
        "Chapter": "1",
        "sentence_range": "6747-6750",
        "Text": "\u2013 C S PEIRCE \ufffd\n13 1  Introduction\nIn earlier Classes, we have studied the probability as a\nmeasure of uncertainty of events in a random experiment"
    },
    {
        "Chapter": "1",
        "sentence_range": "6748-6751",
        "Text": "S PEIRCE \ufffd\n13 1  Introduction\nIn earlier Classes, we have studied the probability as a\nmeasure of uncertainty of events in a random experiment We discussed the axiomatic approach formulated by\nRussian Mathematician, A"
    },
    {
        "Chapter": "1",
        "sentence_range": "6749-6752",
        "Text": "PEIRCE \ufffd\n13 1  Introduction\nIn earlier Classes, we have studied the probability as a\nmeasure of uncertainty of events in a random experiment We discussed the axiomatic approach formulated by\nRussian Mathematician, A N"
    },
    {
        "Chapter": "1",
        "sentence_range": "6750-6753",
        "Text": "1  Introduction\nIn earlier Classes, we have studied the probability as a\nmeasure of uncertainty of events in a random experiment We discussed the axiomatic approach formulated by\nRussian Mathematician, A N Kolmogorov (1903-1987)\nand treated probability as a function of outcomes of the\nexperiment"
    },
    {
        "Chapter": "1",
        "sentence_range": "6751-6754",
        "Text": "We discussed the axiomatic approach formulated by\nRussian Mathematician, A N Kolmogorov (1903-1987)\nand treated probability as a function of outcomes of the\nexperiment We have also established equivalence between\nthe axiomatic theory and the classical theory of probability\nin case of equally likely outcomes"
    },
    {
        "Chapter": "1",
        "sentence_range": "6752-6755",
        "Text": "N Kolmogorov (1903-1987)\nand treated probability as a function of outcomes of the\nexperiment We have also established equivalence between\nthe axiomatic theory and the classical theory of probability\nin case of equally likely outcomes On the basis of this\nrelationship, we obtained probabilities of events associated\nwith discrete sample spaces"
    },
    {
        "Chapter": "1",
        "sentence_range": "6753-6756",
        "Text": "Kolmogorov (1903-1987)\nand treated probability as a function of outcomes of the\nexperiment We have also established equivalence between\nthe axiomatic theory and the classical theory of probability\nin case of equally likely outcomes On the basis of this\nrelationship, we obtained probabilities of events associated\nwith discrete sample spaces We have also studied the\naddition rule of probability"
    },
    {
        "Chapter": "1",
        "sentence_range": "6754-6757",
        "Text": "We have also established equivalence between\nthe axiomatic theory and the classical theory of probability\nin case of equally likely outcomes On the basis of this\nrelationship, we obtained probabilities of events associated\nwith discrete sample spaces We have also studied the\naddition rule of probability In this chapter, we shall discuss\nthe important concept of conditional probability of an event\ngiven that another event has occurred, which will be helpful\nin understanding the Bayes' theorem, multiplication rule of\nprobability and independence of events"
    },
    {
        "Chapter": "1",
        "sentence_range": "6755-6758",
        "Text": "On the basis of this\nrelationship, we obtained probabilities of events associated\nwith discrete sample spaces We have also studied the\naddition rule of probability In this chapter, we shall discuss\nthe important concept of conditional probability of an event\ngiven that another event has occurred, which will be helpful\nin understanding the Bayes' theorem, multiplication rule of\nprobability and independence of events We shall also learn\nan  important concept of  random variable and its probability\ndistribution and also the mean and variance of a probability  distribution"
    },
    {
        "Chapter": "1",
        "sentence_range": "6756-6759",
        "Text": "We have also studied the\naddition rule of probability In this chapter, we shall discuss\nthe important concept of conditional probability of an event\ngiven that another event has occurred, which will be helpful\nin understanding the Bayes' theorem, multiplication rule of\nprobability and independence of events We shall also learn\nan  important concept of  random variable and its probability\ndistribution and also the mean and variance of a probability  distribution In the last\nsection of the chapter, we shall study an important discrete probability distribution\ncalled  Binomial distribution"
    },
    {
        "Chapter": "1",
        "sentence_range": "6757-6760",
        "Text": "In this chapter, we shall discuss\nthe important concept of conditional probability of an event\ngiven that another event has occurred, which will be helpful\nin understanding the Bayes' theorem, multiplication rule of\nprobability and independence of events We shall also learn\nan  important concept of  random variable and its probability\ndistribution and also the mean and variance of a probability  distribution In the last\nsection of the chapter, we shall study an important discrete probability distribution\ncalled  Binomial distribution Throughout this chapter, we shall take up the experiments\nhaving equally likely outcomes, unless stated otherwise"
    },
    {
        "Chapter": "1",
        "sentence_range": "6758-6761",
        "Text": "We shall also learn\nan  important concept of  random variable and its probability\ndistribution and also the mean and variance of a probability  distribution In the last\nsection of the chapter, we shall study an important discrete probability distribution\ncalled  Binomial distribution Throughout this chapter, we shall take up the experiments\nhaving equally likely outcomes, unless stated otherwise 13"
    },
    {
        "Chapter": "1",
        "sentence_range": "6759-6762",
        "Text": "In the last\nsection of the chapter, we shall study an important discrete probability distribution\ncalled  Binomial distribution Throughout this chapter, we shall take up the experiments\nhaving equally likely outcomes, unless stated otherwise 13 2  Conditional Probability\nUptill now in probability, we have discussed the methods of finding the probability of\nevents"
    },
    {
        "Chapter": "1",
        "sentence_range": "6760-6763",
        "Text": "Throughout this chapter, we shall take up the experiments\nhaving equally likely outcomes, unless stated otherwise 13 2  Conditional Probability\nUptill now in probability, we have discussed the methods of finding the probability of\nevents If we have two events from the same sample space, does the information\nabout the occurrence of one of the events affect the probability of the other event"
    },
    {
        "Chapter": "1",
        "sentence_range": "6761-6764",
        "Text": "13 2  Conditional Probability\nUptill now in probability, we have discussed the methods of finding the probability of\nevents If we have two events from the same sample space, does the information\nabout the occurrence of one of the events affect the probability of the other event Let\nus try to answer this question by taking up a random experiment in which the outcomes\nare equally likely to occur"
    },
    {
        "Chapter": "1",
        "sentence_range": "6762-6765",
        "Text": "2  Conditional Probability\nUptill now in probability, we have discussed the methods of finding the probability of\nevents If we have two events from the same sample space, does the information\nabout the occurrence of one of the events affect the probability of the other event Let\nus try to answer this question by taking up a random experiment in which the outcomes\nare equally likely to occur Consider the experiment of tossing three fair coins"
    },
    {
        "Chapter": "1",
        "sentence_range": "6763-6766",
        "Text": "If we have two events from the same sample space, does the information\nabout the occurrence of one of the events affect the probability of the other event Let\nus try to answer this question by taking up a random experiment in which the outcomes\nare equally likely to occur Consider the experiment of tossing three fair coins The sample space of the\nexperiment is\nS = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}\nChapter 13\nPROBABILITY\nPierre de Fermat\n(1601-1665)\n\u00a9 NCERT\nnot to be republished\n 532\nMATHEMATICS\nSince the coins are fair, we can assign the probability  1\n8   to each sample point"
    },
    {
        "Chapter": "1",
        "sentence_range": "6764-6767",
        "Text": "Let\nus try to answer this question by taking up a random experiment in which the outcomes\nare equally likely to occur Consider the experiment of tossing three fair coins The sample space of the\nexperiment is\nS = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}\nChapter 13\nPROBABILITY\nPierre de Fermat\n(1601-1665)\n\u00a9 NCERT\nnot to be republished\n 532\nMATHEMATICS\nSince the coins are fair, we can assign the probability  1\n8   to each sample point Let\nE be the event \u2018at least two heads appear\u2019 and  F be the event \u2018first coin shows tail\u2019"
    },
    {
        "Chapter": "1",
        "sentence_range": "6765-6768",
        "Text": "Consider the experiment of tossing three fair coins The sample space of the\nexperiment is\nS = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}\nChapter 13\nPROBABILITY\nPierre de Fermat\n(1601-1665)\n\u00a9 NCERT\nnot to be republished\n 532\nMATHEMATICS\nSince the coins are fair, we can assign the probability  1\n8   to each sample point Let\nE be the event \u2018at least two heads appear\u2019 and  F be the event \u2018first coin shows tail\u2019 Then\nE = {HHH, HHT, HTH, THH}\nand\nF = {THH, THT, TTH, TTT}\nTherefore\nP(E) = P ({HHH}) + P ({HHT}) + P ({HTH}) + P ({THH})\n= 1\n1\n1\n1\n1\n8\n8\n8\n8\n2\n+\n+\n+\n=\n (Why"
    },
    {
        "Chapter": "1",
        "sentence_range": "6766-6769",
        "Text": "The sample space of the\nexperiment is\nS = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}\nChapter 13\nPROBABILITY\nPierre de Fermat\n(1601-1665)\n\u00a9 NCERT\nnot to be republished\n 532\nMATHEMATICS\nSince the coins are fair, we can assign the probability  1\n8   to each sample point Let\nE be the event \u2018at least two heads appear\u2019 and  F be the event \u2018first coin shows tail\u2019 Then\nE = {HHH, HHT, HTH, THH}\nand\nF = {THH, THT, TTH, TTT}\nTherefore\nP(E) = P ({HHH}) + P ({HHT}) + P ({HTH}) + P ({THH})\n= 1\n1\n1\n1\n1\n8\n8\n8\n8\n2\n+\n+\n+\n=\n (Why )\nand\nP(F) = P ({THH}) + P ({THT}) + P ({TTH}) + P ({TTT})\n= 1\n1\n1\n1\n1\n8\n8\n8\n8\n2\n+\n+\n+\n=\nAlso\nE \u2229 F = {THH}\nwith\nP(E \u2229 F) = P({THH}) = 1\n8\nNow, suppose we are given that the first coin shows tail, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "6767-6770",
        "Text": "Let\nE be the event \u2018at least two heads appear\u2019 and  F be the event \u2018first coin shows tail\u2019 Then\nE = {HHH, HHT, HTH, THH}\nand\nF = {THH, THT, TTH, TTT}\nTherefore\nP(E) = P ({HHH}) + P ({HHT}) + P ({HTH}) + P ({THH})\n= 1\n1\n1\n1\n1\n8\n8\n8\n8\n2\n+\n+\n+\n=\n (Why )\nand\nP(F) = P ({THH}) + P ({THT}) + P ({TTH}) + P ({TTT})\n= 1\n1\n1\n1\n1\n8\n8\n8\n8\n2\n+\n+\n+\n=\nAlso\nE \u2229 F = {THH}\nwith\nP(E \u2229 F) = P({THH}) = 1\n8\nNow, suppose we are given that the first coin shows tail, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "6768-6771",
        "Text": "Then\nE = {HHH, HHT, HTH, THH}\nand\nF = {THH, THT, TTH, TTT}\nTherefore\nP(E) = P ({HHH}) + P ({HHT}) + P ({HTH}) + P ({THH})\n= 1\n1\n1\n1\n1\n8\n8\n8\n8\n2\n+\n+\n+\n=\n (Why )\nand\nP(F) = P ({THH}) + P ({THT}) + P ({TTH}) + P ({TTT})\n= 1\n1\n1\n1\n1\n8\n8\n8\n8\n2\n+\n+\n+\n=\nAlso\nE \u2229 F = {THH}\nwith\nP(E \u2229 F) = P({THH}) = 1\n8\nNow, suppose we are given that the first coin shows tail, i e F occurs, then what is\nthe probability of occurrence of E"
    },
    {
        "Chapter": "1",
        "sentence_range": "6769-6772",
        "Text": ")\nand\nP(F) = P ({THH}) + P ({THT}) + P ({TTH}) + P ({TTT})\n= 1\n1\n1\n1\n1\n8\n8\n8\n8\n2\n+\n+\n+\n=\nAlso\nE \u2229 F = {THH}\nwith\nP(E \u2229 F) = P({THH}) = 1\n8\nNow, suppose we are given that the first coin shows tail, i e F occurs, then what is\nthe probability of occurrence of E With the information of occurrence of F, we are\nsure that the cases in which first coin does not result into a tail should not be considered\nwhile finding the probability of E"
    },
    {
        "Chapter": "1",
        "sentence_range": "6770-6773",
        "Text": "e F occurs, then what is\nthe probability of occurrence of E With the information of occurrence of F, we are\nsure that the cases in which first coin does not result into a tail should not be considered\nwhile finding the probability of E This information reduces our sample space from the\nset S to its subset F for the event E"
    },
    {
        "Chapter": "1",
        "sentence_range": "6771-6774",
        "Text": "F occurs, then what is\nthe probability of occurrence of E With the information of occurrence of F, we are\nsure that the cases in which first coin does not result into a tail should not be considered\nwhile finding the probability of E This information reduces our sample space from the\nset S to its subset F for the event E In other words, the additional information really\namounts to telling us that the situation may be considered as being that of a new\nrandom experiment for which the sample space consists of all those outcomes only\nwhich are favourable to the occurrence of the event F"
    },
    {
        "Chapter": "1",
        "sentence_range": "6772-6775",
        "Text": "With the information of occurrence of F, we are\nsure that the cases in which first coin does not result into a tail should not be considered\nwhile finding the probability of E This information reduces our sample space from the\nset S to its subset F for the event E In other words, the additional information really\namounts to telling us that the situation may be considered as being that of a new\nrandom experiment for which the sample space consists of all those outcomes only\nwhich are favourable to the occurrence of the event F Now, the sample point of F which is favourable to event E is THH"
    },
    {
        "Chapter": "1",
        "sentence_range": "6773-6776",
        "Text": "This information reduces our sample space from the\nset S to its subset F for the event E In other words, the additional information really\namounts to telling us that the situation may be considered as being that of a new\nrandom experiment for which the sample space consists of all those outcomes only\nwhich are favourable to the occurrence of the event F Now, the sample point of F which is favourable to event E is THH Thus, Probability of E considering F as the sample space = 1\n4 ,\nor\nProbability of E given that the event F has occurred = 1\n4\nThis probability of the event E is called the conditional probability of E given\nthat F has already occurred, and is denoted by P (E|F)"
    },
    {
        "Chapter": "1",
        "sentence_range": "6774-6777",
        "Text": "In other words, the additional information really\namounts to telling us that the situation may be considered as being that of a new\nrandom experiment for which the sample space consists of all those outcomes only\nwhich are favourable to the occurrence of the event F Now, the sample point of F which is favourable to event E is THH Thus, Probability of E considering F as the sample space = 1\n4 ,\nor\nProbability of E given that the event F has occurred = 1\n4\nThis probability of the event E is called the conditional probability of E given\nthat F has already occurred, and is denoted by P (E|F) Thus\nP(E|F) = 1\n4\nNote that the elements of F which favour the event E are the common elements of\nE and F, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "6775-6778",
        "Text": "Now, the sample point of F which is favourable to event E is THH Thus, Probability of E considering F as the sample space = 1\n4 ,\nor\nProbability of E given that the event F has occurred = 1\n4\nThis probability of the event E is called the conditional probability of E given\nthat F has already occurred, and is denoted by P (E|F) Thus\nP(E|F) = 1\n4\nNote that the elements of F which favour the event E are the common elements of\nE and F, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "6776-6779",
        "Text": "Thus, Probability of E considering F as the sample space = 1\n4 ,\nor\nProbability of E given that the event F has occurred = 1\n4\nThis probability of the event E is called the conditional probability of E given\nthat F has already occurred, and is denoted by P (E|F) Thus\nP(E|F) = 1\n4\nNote that the elements of F which favour the event E are the common elements of\nE and F, i e the sample points of E \u2229 F"
    },
    {
        "Chapter": "1",
        "sentence_range": "6777-6780",
        "Text": "Thus\nP(E|F) = 1\n4\nNote that the elements of F which favour the event E are the common elements of\nE and F, i e the sample points of E \u2229 F \u00a9 NCERT\nnot to be republished\nPROBABILITY      533\nThus, we can also write the conditional probability of E given that F has occurred as\nP(E|F) =\nNumberof elementaryeventsfavourableto E\nF\nNumberof elementaryeventswhicharefavourableto F\n\u2229\n=\n(E\n(F)F)\nn\nn\n\u2229\nDividing the numerator and the denominator by total number of elementary events\nof the sample space, we see that P(E|F) can also be written as\nP(E|F) =\n(E\nF)\nP(E\nF)\n(S)\n(F)\nP(F)\n(S)\nn\nnn\nn\n\u2229\n\u2229\n="
    },
    {
        "Chapter": "1",
        "sentence_range": "6778-6781",
        "Text": "e the sample points of E \u2229 F \u00a9 NCERT\nnot to be republished\nPROBABILITY      533\nThus, we can also write the conditional probability of E given that F has occurred as\nP(E|F) =\nNumberof elementaryeventsfavourableto E\nF\nNumberof elementaryeventswhicharefavourableto F\n\u2229\n=\n(E\n(F)F)\nn\nn\n\u2229\nDividing the numerator and the denominator by total number of elementary events\nof the sample space, we see that P(E|F) can also be written as\nP(E|F) =\n(E\nF)\nP(E\nF)\n(S)\n(F)\nP(F)\n(S)\nn\nnn\nn\n\u2229\n\u2229\n= (1)\nNote that (1) is valid only when P(F) \u2260 0 i"
    },
    {
        "Chapter": "1",
        "sentence_range": "6779-6782",
        "Text": "the sample points of E \u2229 F \u00a9 NCERT\nnot to be republished\nPROBABILITY      533\nThus, we can also write the conditional probability of E given that F has occurred as\nP(E|F) =\nNumberof elementaryeventsfavourableto E\nF\nNumberof elementaryeventswhicharefavourableto F\n\u2229\n=\n(E\n(F)F)\nn\nn\n\u2229\nDividing the numerator and the denominator by total number of elementary events\nof the sample space, we see that P(E|F) can also be written as\nP(E|F) =\n(E\nF)\nP(E\nF)\n(S)\n(F)\nP(F)\n(S)\nn\nnn\nn\n\u2229\n\u2229\n= (1)\nNote that (1) is valid only when P(F) \u2260 0 i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "6780-6783",
        "Text": "\u00a9 NCERT\nnot to be republished\nPROBABILITY      533\nThus, we can also write the conditional probability of E given that F has occurred as\nP(E|F) =\nNumberof elementaryeventsfavourableto E\nF\nNumberof elementaryeventswhicharefavourableto F\n\u2229\n=\n(E\n(F)F)\nn\nn\n\u2229\nDividing the numerator and the denominator by total number of elementary events\nof the sample space, we see that P(E|F) can also be written as\nP(E|F) =\n(E\nF)\nP(E\nF)\n(S)\n(F)\nP(F)\n(S)\nn\nnn\nn\n\u2229\n\u2229\n= (1)\nNote that (1) is valid only when P(F) \u2260 0 i e , F \u2260 \u03c6 (Why"
    },
    {
        "Chapter": "1",
        "sentence_range": "6781-6784",
        "Text": "(1)\nNote that (1) is valid only when P(F) \u2260 0 i e , F \u2260 \u03c6 (Why )\nThus, we can define the conditional probability as follows :\nDefinition 1 If  E and F are two events associated with the same sample space of a\nrandom experiment, the conditional probability of the event E given that F has occurred,\ni"
    },
    {
        "Chapter": "1",
        "sentence_range": "6782-6785",
        "Text": "e , F \u2260 \u03c6 (Why )\nThus, we can define the conditional probability as follows :\nDefinition 1 If  E and F are two events associated with the same sample space of a\nrandom experiment, the conditional probability of the event E given that F has occurred,\ni e"
    },
    {
        "Chapter": "1",
        "sentence_range": "6783-6786",
        "Text": ", F \u2260 \u03c6 (Why )\nThus, we can define the conditional probability as follows :\nDefinition 1 If  E and F are two events associated with the same sample space of a\nrandom experiment, the conditional probability of the event E given that F has occurred,\ni e P (E|F) is given by\nP(E|F) = P(E\nF)\nP(F)\n\u2229\n provided P(F) \u2260 0\n13"
    },
    {
        "Chapter": "1",
        "sentence_range": "6784-6787",
        "Text": ")\nThus, we can define the conditional probability as follows :\nDefinition 1 If  E and F are two events associated with the same sample space of a\nrandom experiment, the conditional probability of the event E given that F has occurred,\ni e P (E|F) is given by\nP(E|F) = P(E\nF)\nP(F)\n\u2229\n provided P(F) \u2260 0\n13 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "6785-6788",
        "Text": "e P (E|F) is given by\nP(E|F) = P(E\nF)\nP(F)\n\u2229\n provided P(F) \u2260 0\n13 2 1  Properties of conditional probability\nLet E and F be events of a sample space S of an experiment, then we have\nProperty 1 P(S|F) = P(F|F) = 1\nWe know that\nP(S|F) = P(S\nF)\nP(F)\n1\nP(F)\nP(F)\n\u2229\n=\n=\nAlso\nP(F|F) = P(F\nF)\nP(F) 1\nP(F)\nP(F)\n\u2229\n=\n=\nThus\nP(S|F) = P(F|F) = 1\nProperty 2 If A and B are any two events of a sample space S and F is an event\nof S such that P(F) \u2260 0, then\nP((A \u222a B)|F) = P(A|F) + P(B|F) \u2013 P((A \u2229 B)|F)\n\u00a9 NCERT\nnot to be republished\n 534\nMATHEMATICS\nIn particular, if A and B are disjoint events, then\nP((A\u222aB)|F) = P(A|F) + P(B|F)\nWe have\nP((A\u222aB)|F) = P[(A\nB)\nF]\nP(F)\n\u222a\n\u2229\n= P[(A\nF)\n(B\nF)]\n\u2229P(F)\n\u222a\n\u2229\n(by distributive law of union of sets over  intersection)\n=\nP(A\nF)+P(B\nF)\u2013P(A\nB \nF)\nP(F)\n\u2229\n\u2229\n\u2229\n\u2229\n= P(A\nF)\nP(B\nF)\nP[(A\nB) \nF]\nP(F)\nP(F)\nP(F)\n\u2229\n\u2229\n\u2229\n\u2229\n+\n\u2212\n= P(A|F) + P(B|F) \u2013 P((A \u2229B)|F)\nWhen A and B are disjoint events, then\nP((A \u2229 B)|F) = 0\n\u21d2\nP((A \u222a B)|F) = P(A|F) + P(B|F)\nProperty 3 P(E\u2032|F) = 1 \u2212 P(E|F)\nFrom Property 1, we know that P(S|F) = 1\n\u21d2\nP(E \u222a E\u2032|F) = 1\n   since  S = E \u222a E\u2032\n\u21d2\nP(E|F) + P (E\u2032|F) = 1\n    since E and E\u2032 are disjoint events\nThus,\nP(E\u2032|F) = 1 \u2212 P(E|F)\nLet us now take up some examples"
    },
    {
        "Chapter": "1",
        "sentence_range": "6786-6789",
        "Text": "P (E|F) is given by\nP(E|F) = P(E\nF)\nP(F)\n\u2229\n provided P(F) \u2260 0\n13 2 1  Properties of conditional probability\nLet E and F be events of a sample space S of an experiment, then we have\nProperty 1 P(S|F) = P(F|F) = 1\nWe know that\nP(S|F) = P(S\nF)\nP(F)\n1\nP(F)\nP(F)\n\u2229\n=\n=\nAlso\nP(F|F) = P(F\nF)\nP(F) 1\nP(F)\nP(F)\n\u2229\n=\n=\nThus\nP(S|F) = P(F|F) = 1\nProperty 2 If A and B are any two events of a sample space S and F is an event\nof S such that P(F) \u2260 0, then\nP((A \u222a B)|F) = P(A|F) + P(B|F) \u2013 P((A \u2229 B)|F)\n\u00a9 NCERT\nnot to be republished\n 534\nMATHEMATICS\nIn particular, if A and B are disjoint events, then\nP((A\u222aB)|F) = P(A|F) + P(B|F)\nWe have\nP((A\u222aB)|F) = P[(A\nB)\nF]\nP(F)\n\u222a\n\u2229\n= P[(A\nF)\n(B\nF)]\n\u2229P(F)\n\u222a\n\u2229\n(by distributive law of union of sets over  intersection)\n=\nP(A\nF)+P(B\nF)\u2013P(A\nB \nF)\nP(F)\n\u2229\n\u2229\n\u2229\n\u2229\n= P(A\nF)\nP(B\nF)\nP[(A\nB) \nF]\nP(F)\nP(F)\nP(F)\n\u2229\n\u2229\n\u2229\n\u2229\n+\n\u2212\n= P(A|F) + P(B|F) \u2013 P((A \u2229B)|F)\nWhen A and B are disjoint events, then\nP((A \u2229 B)|F) = 0\n\u21d2\nP((A \u222a B)|F) = P(A|F) + P(B|F)\nProperty 3 P(E\u2032|F) = 1 \u2212 P(E|F)\nFrom Property 1, we know that P(S|F) = 1\n\u21d2\nP(E \u222a E\u2032|F) = 1\n   since  S = E \u222a E\u2032\n\u21d2\nP(E|F) + P (E\u2032|F) = 1\n    since E and E\u2032 are disjoint events\nThus,\nP(E\u2032|F) = 1 \u2212 P(E|F)\nLet us now take up some examples Example 1 If P(A) = 7\n13 , P(B) = 9\n13  and P(A \u2229 B) = 4\n13 , evaluate P(A|B)"
    },
    {
        "Chapter": "1",
        "sentence_range": "6787-6790",
        "Text": "2 1  Properties of conditional probability\nLet E and F be events of a sample space S of an experiment, then we have\nProperty 1 P(S|F) = P(F|F) = 1\nWe know that\nP(S|F) = P(S\nF)\nP(F)\n1\nP(F)\nP(F)\n\u2229\n=\n=\nAlso\nP(F|F) = P(F\nF)\nP(F) 1\nP(F)\nP(F)\n\u2229\n=\n=\nThus\nP(S|F) = P(F|F) = 1\nProperty 2 If A and B are any two events of a sample space S and F is an event\nof S such that P(F) \u2260 0, then\nP((A \u222a B)|F) = P(A|F) + P(B|F) \u2013 P((A \u2229 B)|F)\n\u00a9 NCERT\nnot to be republished\n 534\nMATHEMATICS\nIn particular, if A and B are disjoint events, then\nP((A\u222aB)|F) = P(A|F) + P(B|F)\nWe have\nP((A\u222aB)|F) = P[(A\nB)\nF]\nP(F)\n\u222a\n\u2229\n= P[(A\nF)\n(B\nF)]\n\u2229P(F)\n\u222a\n\u2229\n(by distributive law of union of sets over  intersection)\n=\nP(A\nF)+P(B\nF)\u2013P(A\nB \nF)\nP(F)\n\u2229\n\u2229\n\u2229\n\u2229\n= P(A\nF)\nP(B\nF)\nP[(A\nB) \nF]\nP(F)\nP(F)\nP(F)\n\u2229\n\u2229\n\u2229\n\u2229\n+\n\u2212\n= P(A|F) + P(B|F) \u2013 P((A \u2229B)|F)\nWhen A and B are disjoint events, then\nP((A \u2229 B)|F) = 0\n\u21d2\nP((A \u222a B)|F) = P(A|F) + P(B|F)\nProperty 3 P(E\u2032|F) = 1 \u2212 P(E|F)\nFrom Property 1, we know that P(S|F) = 1\n\u21d2\nP(E \u222a E\u2032|F) = 1\n   since  S = E \u222a E\u2032\n\u21d2\nP(E|F) + P (E\u2032|F) = 1\n    since E and E\u2032 are disjoint events\nThus,\nP(E\u2032|F) = 1 \u2212 P(E|F)\nLet us now take up some examples Example 1 If P(A) = 7\n13 , P(B) = 9\n13  and P(A \u2229 B) = 4\n13 , evaluate P(A|B) Solution We have \n4\nP(A\nB)\n4\n13\nP(A|B)=\n9\nP(B)\n9\n13\n\u2229\n=\n=\nExample 2 A family has two children"
    },
    {
        "Chapter": "1",
        "sentence_range": "6788-6791",
        "Text": "1  Properties of conditional probability\nLet E and F be events of a sample space S of an experiment, then we have\nProperty 1 P(S|F) = P(F|F) = 1\nWe know that\nP(S|F) = P(S\nF)\nP(F)\n1\nP(F)\nP(F)\n\u2229\n=\n=\nAlso\nP(F|F) = P(F\nF)\nP(F) 1\nP(F)\nP(F)\n\u2229\n=\n=\nThus\nP(S|F) = P(F|F) = 1\nProperty 2 If A and B are any two events of a sample space S and F is an event\nof S such that P(F) \u2260 0, then\nP((A \u222a B)|F) = P(A|F) + P(B|F) \u2013 P((A \u2229 B)|F)\n\u00a9 NCERT\nnot to be republished\n 534\nMATHEMATICS\nIn particular, if A and B are disjoint events, then\nP((A\u222aB)|F) = P(A|F) + P(B|F)\nWe have\nP((A\u222aB)|F) = P[(A\nB)\nF]\nP(F)\n\u222a\n\u2229\n= P[(A\nF)\n(B\nF)]\n\u2229P(F)\n\u222a\n\u2229\n(by distributive law of union of sets over  intersection)\n=\nP(A\nF)+P(B\nF)\u2013P(A\nB \nF)\nP(F)\n\u2229\n\u2229\n\u2229\n\u2229\n= P(A\nF)\nP(B\nF)\nP[(A\nB) \nF]\nP(F)\nP(F)\nP(F)\n\u2229\n\u2229\n\u2229\n\u2229\n+\n\u2212\n= P(A|F) + P(B|F) \u2013 P((A \u2229B)|F)\nWhen A and B are disjoint events, then\nP((A \u2229 B)|F) = 0\n\u21d2\nP((A \u222a B)|F) = P(A|F) + P(B|F)\nProperty 3 P(E\u2032|F) = 1 \u2212 P(E|F)\nFrom Property 1, we know that P(S|F) = 1\n\u21d2\nP(E \u222a E\u2032|F) = 1\n   since  S = E \u222a E\u2032\n\u21d2\nP(E|F) + P (E\u2032|F) = 1\n    since E and E\u2032 are disjoint events\nThus,\nP(E\u2032|F) = 1 \u2212 P(E|F)\nLet us now take up some examples Example 1 If P(A) = 7\n13 , P(B) = 9\n13  and P(A \u2229 B) = 4\n13 , evaluate P(A|B) Solution We have \n4\nP(A\nB)\n4\n13\nP(A|B)=\n9\nP(B)\n9\n13\n\u2229\n=\n=\nExample 2 A family has two children What is the probability that both the children are\nboys given that at least one of them is a boy"
    },
    {
        "Chapter": "1",
        "sentence_range": "6789-6792",
        "Text": "Example 1 If P(A) = 7\n13 , P(B) = 9\n13  and P(A \u2229 B) = 4\n13 , evaluate P(A|B) Solution We have \n4\nP(A\nB)\n4\n13\nP(A|B)=\n9\nP(B)\n9\n13\n\u2229\n=\n=\nExample 2 A family has two children What is the probability that both the children are\nboys given that at least one of them is a boy \u00a9 NCERT\nnot to be republished\nPROBABILITY      535\nSolution Let b stand for boy and g for girl"
    },
    {
        "Chapter": "1",
        "sentence_range": "6790-6793",
        "Text": "Solution We have \n4\nP(A\nB)\n4\n13\nP(A|B)=\n9\nP(B)\n9\n13\n\u2229\n=\n=\nExample 2 A family has two children What is the probability that both the children are\nboys given that at least one of them is a boy \u00a9 NCERT\nnot to be republished\nPROBABILITY      535\nSolution Let b stand for boy and g for girl The sample space of the experiment is\nS = {(b, b), (g, b), (b, g), (g, g)}\nLet E and F denote the following events :\nE : \u2018both the children are boys\u2019\nF : \u2018at least one of the child is a boy\u2019\nThen\nE = {(b,b)} and F = {(b,b), (g,b), (b,g)}\nNow\nE \u2229 F = {(b,b)}\nThus\nP(F) = 3\n4  and P (E \u2229 F )= 1\n4\nTherefore\nP(E|F) =\n1\nP(E\nF)\n1\n34\nP(F)\n3\n4\n\u2229\n=\n=\nExample 3  Ten cards  numbered 1 to 10 are placed in a box, mixed up thoroughly and\nthen one card is drawn randomly"
    },
    {
        "Chapter": "1",
        "sentence_range": "6791-6794",
        "Text": "What is the probability that both the children are\nboys given that at least one of them is a boy \u00a9 NCERT\nnot to be republished\nPROBABILITY      535\nSolution Let b stand for boy and g for girl The sample space of the experiment is\nS = {(b, b), (g, b), (b, g), (g, g)}\nLet E and F denote the following events :\nE : \u2018both the children are boys\u2019\nF : \u2018at least one of the child is a boy\u2019\nThen\nE = {(b,b)} and F = {(b,b), (g,b), (b,g)}\nNow\nE \u2229 F = {(b,b)}\nThus\nP(F) = 3\n4  and P (E \u2229 F )= 1\n4\nTherefore\nP(E|F) =\n1\nP(E\nF)\n1\n34\nP(F)\n3\n4\n\u2229\n=\n=\nExample 3  Ten cards  numbered 1 to 10 are placed in a box, mixed up thoroughly and\nthen one card is drawn randomly If it is known that the number on the drawn card is\nmore than 3, what is the probability that it is an even number"
    },
    {
        "Chapter": "1",
        "sentence_range": "6792-6795",
        "Text": "\u00a9 NCERT\nnot to be republished\nPROBABILITY      535\nSolution Let b stand for boy and g for girl The sample space of the experiment is\nS = {(b, b), (g, b), (b, g), (g, g)}\nLet E and F denote the following events :\nE : \u2018both the children are boys\u2019\nF : \u2018at least one of the child is a boy\u2019\nThen\nE = {(b,b)} and F = {(b,b), (g,b), (b,g)}\nNow\nE \u2229 F = {(b,b)}\nThus\nP(F) = 3\n4  and P (E \u2229 F )= 1\n4\nTherefore\nP(E|F) =\n1\nP(E\nF)\n1\n34\nP(F)\n3\n4\n\u2229\n=\n=\nExample 3  Ten cards  numbered 1 to 10 are placed in a box, mixed up thoroughly and\nthen one card is drawn randomly If it is known that the number on the drawn card is\nmore than 3, what is the probability that it is an even number Solution Let A be the event \u2018the number on the card drawn is even\u2019 and B be the\nevent \u2018the number on the card drawn is  greater than 3\u2019"
    },
    {
        "Chapter": "1",
        "sentence_range": "6793-6796",
        "Text": "The sample space of the experiment is\nS = {(b, b), (g, b), (b, g), (g, g)}\nLet E and F denote the following events :\nE : \u2018both the children are boys\u2019\nF : \u2018at least one of the child is a boy\u2019\nThen\nE = {(b,b)} and F = {(b,b), (g,b), (b,g)}\nNow\nE \u2229 F = {(b,b)}\nThus\nP(F) = 3\n4  and P (E \u2229 F )= 1\n4\nTherefore\nP(E|F) =\n1\nP(E\nF)\n1\n34\nP(F)\n3\n4\n\u2229\n=\n=\nExample 3  Ten cards  numbered 1 to 10 are placed in a box, mixed up thoroughly and\nthen one card is drawn randomly If it is known that the number on the drawn card is\nmore than 3, what is the probability that it is an even number Solution Let A be the event \u2018the number on the card drawn is even\u2019 and B be the\nevent \u2018the number on the card drawn is  greater than 3\u2019 We have to find P(A|B)"
    },
    {
        "Chapter": "1",
        "sentence_range": "6794-6797",
        "Text": "If it is known that the number on the drawn card is\nmore than 3, what is the probability that it is an even number Solution Let A be the event \u2018the number on the card drawn is even\u2019 and B be the\nevent \u2018the number on the card drawn is  greater than 3\u2019 We have to find P(A|B) Now, the sample space of the experiment is S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}\nThen\nA = {2, 4, 6, 8, 10},  B = {4, 5, 6, 7, 8, 9, 10}\nand\nA \u2229 B = {4, 6, 8, 10}\nAlso\nP(A) = 5\n7\n4\n, P(B) =\nand P(A\nB)\n10\n10\n10\n\u2229\n=\nThen\nP(A|B) =\n4\nP(A\nB)\n4\n710\nP(B)\n7\n10\n\u2229\n=\n=\nExample 4 In a school, there are 1000 students, out of which 430 are girls"
    },
    {
        "Chapter": "1",
        "sentence_range": "6795-6798",
        "Text": "Solution Let A be the event \u2018the number on the card drawn is even\u2019 and B be the\nevent \u2018the number on the card drawn is  greater than 3\u2019 We have to find P(A|B) Now, the sample space of the experiment is S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}\nThen\nA = {2, 4, 6, 8, 10},  B = {4, 5, 6, 7, 8, 9, 10}\nand\nA \u2229 B = {4, 6, 8, 10}\nAlso\nP(A) = 5\n7\n4\n, P(B) =\nand P(A\nB)\n10\n10\n10\n\u2229\n=\nThen\nP(A|B) =\n4\nP(A\nB)\n4\n710\nP(B)\n7\n10\n\u2229\n=\n=\nExample 4 In a school, there are 1000 students, out of which 430 are girls It is known\nthat out of 430, 10% of the girls study in class XII"
    },
    {
        "Chapter": "1",
        "sentence_range": "6796-6799",
        "Text": "We have to find P(A|B) Now, the sample space of the experiment is S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}\nThen\nA = {2, 4, 6, 8, 10},  B = {4, 5, 6, 7, 8, 9, 10}\nand\nA \u2229 B = {4, 6, 8, 10}\nAlso\nP(A) = 5\n7\n4\n, P(B) =\nand P(A\nB)\n10\n10\n10\n\u2229\n=\nThen\nP(A|B) =\n4\nP(A\nB)\n4\n710\nP(B)\n7\n10\n\u2229\n=\n=\nExample 4 In a school, there are 1000 students, out of which 430 are girls It is known\nthat out of 430, 10% of the girls study in class XII What is the probability that a student\nchosen randomly studies in Class XII given that the chosen student is a girl"
    },
    {
        "Chapter": "1",
        "sentence_range": "6797-6800",
        "Text": "Now, the sample space of the experiment is S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}\nThen\nA = {2, 4, 6, 8, 10},  B = {4, 5, 6, 7, 8, 9, 10}\nand\nA \u2229 B = {4, 6, 8, 10}\nAlso\nP(A) = 5\n7\n4\n, P(B) =\nand P(A\nB)\n10\n10\n10\n\u2229\n=\nThen\nP(A|B) =\n4\nP(A\nB)\n4\n710\nP(B)\n7\n10\n\u2229\n=\n=\nExample 4 In a school, there are 1000 students, out of which 430 are girls It is known\nthat out of 430, 10% of the girls study in class XII What is the probability that a student\nchosen randomly studies in Class XII given that the chosen student is a girl Solution Let E denote the event that a student chosen randomly studies in Class XII\nand F be the event that the randomly chosen student is a girl"
    },
    {
        "Chapter": "1",
        "sentence_range": "6798-6801",
        "Text": "It is known\nthat out of 430, 10% of the girls study in class XII What is the probability that a student\nchosen randomly studies in Class XII given that the chosen student is a girl Solution Let E denote the event that a student chosen randomly studies in Class XII\nand F be the event that the randomly chosen student is a girl We have to find P (E|F)"
    },
    {
        "Chapter": "1",
        "sentence_range": "6799-6802",
        "Text": "What is the probability that a student\nchosen randomly studies in Class XII given that the chosen student is a girl Solution Let E denote the event that a student chosen randomly studies in Class XII\nand F be the event that the randomly chosen student is a girl We have to find P (E|F) \u00a9 NCERT\nnot to be republished\n 536\nMATHEMATICS\nNow\n P(F) = 430\n1000 =0"
    },
    {
        "Chapter": "1",
        "sentence_range": "6800-6803",
        "Text": "Solution Let E denote the event that a student chosen randomly studies in Class XII\nand F be the event that the randomly chosen student is a girl We have to find P (E|F) \u00a9 NCERT\nnot to be republished\n 536\nMATHEMATICS\nNow\n P(F) = 430\n1000 =0 43\n and  \n43\nP(E\nF)=\n0"
    },
    {
        "Chapter": "1",
        "sentence_range": "6801-6804",
        "Text": "We have to find P (E|F) \u00a9 NCERT\nnot to be republished\n 536\nMATHEMATICS\nNow\n P(F) = 430\n1000 =0 43\n and  \n43\nP(E\nF)=\n0 043\n1000\n \n \n  (Why"
    },
    {
        "Chapter": "1",
        "sentence_range": "6802-6805",
        "Text": "\u00a9 NCERT\nnot to be republished\n 536\nMATHEMATICS\nNow\n P(F) = 430\n1000 =0 43\n and  \n43\nP(E\nF)=\n0 043\n1000\n \n \n  (Why )\nThen\n P(E|F) = P(E\nF)\n0"
    },
    {
        "Chapter": "1",
        "sentence_range": "6803-6806",
        "Text": "43\n and  \n43\nP(E\nF)=\n0 043\n1000\n \n \n  (Why )\nThen\n P(E|F) = P(E\nF)\n0 043\n0"
    },
    {
        "Chapter": "1",
        "sentence_range": "6804-6807",
        "Text": "043\n1000\n \n \n  (Why )\nThen\n P(E|F) = P(E\nF)\n0 043\n0 1\nP(F)\n0"
    },
    {
        "Chapter": "1",
        "sentence_range": "6805-6808",
        "Text": ")\nThen\n P(E|F) = P(E\nF)\n0 043\n0 1\nP(F)\n0 43\n\u2229\n=\n=\nExample 5 A die is thrown three times"
    },
    {
        "Chapter": "1",
        "sentence_range": "6806-6809",
        "Text": "043\n0 1\nP(F)\n0 43\n\u2229\n=\n=\nExample 5 A die is thrown three times Events A and B are defined as below:\nA : 4 on the third throw\nB : 6 on the first and 5 on the second throw\nFind the probability of A given that B has already occurred"
    },
    {
        "Chapter": "1",
        "sentence_range": "6807-6810",
        "Text": "1\nP(F)\n0 43\n\u2229\n=\n=\nExample 5 A die is thrown three times Events A and B are defined as below:\nA : 4 on the third throw\nB : 6 on the first and 5 on the second throw\nFind the probability of A given that B has already occurred Solution The sample space has 216 outcomes"
    },
    {
        "Chapter": "1",
        "sentence_range": "6808-6811",
        "Text": "43\n\u2229\n=\n=\nExample 5 A die is thrown three times Events A and B are defined as below:\nA : 4 on the third throw\nB : 6 on the first and 5 on the second throw\nFind the probability of A given that B has already occurred Solution The sample space has 216 outcomes Now\nA =\n(1,1,4)   (1,2,4)"
    },
    {
        "Chapter": "1",
        "sentence_range": "6809-6812",
        "Text": "Events A and B are defined as below:\nA : 4 on the third throw\nB : 6 on the first and 5 on the second throw\nFind the probability of A given that B has already occurred Solution The sample space has 216 outcomes Now\nA =\n(1,1,4)   (1,2,4) (1,6,4) (2,1,4) (2,2,4)"
    },
    {
        "Chapter": "1",
        "sentence_range": "6810-6813",
        "Text": "Solution The sample space has 216 outcomes Now\nA =\n(1,1,4)   (1,2,4) (1,6,4) (2,1,4) (2,2,4) (2,6,4)\n(3,1,4) (3,2,4)"
    },
    {
        "Chapter": "1",
        "sentence_range": "6811-6814",
        "Text": "Now\nA =\n(1,1,4)   (1,2,4) (1,6,4) (2,1,4) (2,2,4) (2,6,4)\n(3,1,4) (3,2,4) (3,6,4) (4,1,4) (4,2,4)"
    },
    {
        "Chapter": "1",
        "sentence_range": "6812-6815",
        "Text": "(1,6,4) (2,1,4) (2,2,4) (2,6,4)\n(3,1,4) (3,2,4) (3,6,4) (4,1,4) (4,2,4) (4,6,4)\n(5,1,4) (5,2,4)"
    },
    {
        "Chapter": "1",
        "sentence_range": "6813-6816",
        "Text": "(2,6,4)\n(3,1,4) (3,2,4) (3,6,4) (4,1,4) (4,2,4) (4,6,4)\n(5,1,4) (5,2,4) (5,6,4) (6,1,4) (6,2,4)"
    },
    {
        "Chapter": "1",
        "sentence_range": "6814-6817",
        "Text": "(3,6,4) (4,1,4) (4,2,4) (4,6,4)\n(5,1,4) (5,2,4) (5,6,4) (6,1,4) (6,2,4) (6,6,4)\n\u23a7\n\u23ab\n\u23aa\n\u23aa\n\u23a8\n\u23ac\n\u23aa\n\u23aa\n\u23a9\n\u23ad\nB = {(6,5,1), (6,5,2), (6,5,3), (6,5,4), (6,5,5), (6,5,6)}\nand\nA \u2229 B = {(6,5,4)}"
    },
    {
        "Chapter": "1",
        "sentence_range": "6815-6818",
        "Text": "(4,6,4)\n(5,1,4) (5,2,4) (5,6,4) (6,1,4) (6,2,4) (6,6,4)\n\u23a7\n\u23ab\n\u23aa\n\u23aa\n\u23a8\n\u23ac\n\u23aa\n\u23aa\n\u23a9\n\u23ad\nB = {(6,5,1), (6,5,2), (6,5,3), (6,5,4), (6,5,5), (6,5,6)}\nand\nA \u2229 B = {(6,5,4)} Now\nP(B) =\n6\n216  and  P (A \u2229 B) = 1\n216\nThen\nP(A|B) =\n1\nP(A\nB)\n1\n216\n6\nP(B)\n6\n216\n\u2229\n=\n=\nExample 6  A die is thrown twice and the sum of the numbers appearing is observed\nto be 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "6816-6819",
        "Text": "(5,6,4) (6,1,4) (6,2,4) (6,6,4)\n\u23a7\n\u23ab\n\u23aa\n\u23aa\n\u23a8\n\u23ac\n\u23aa\n\u23aa\n\u23a9\n\u23ad\nB = {(6,5,1), (6,5,2), (6,5,3), (6,5,4), (6,5,5), (6,5,6)}\nand\nA \u2229 B = {(6,5,4)} Now\nP(B) =\n6\n216  and  P (A \u2229 B) = 1\n216\nThen\nP(A|B) =\n1\nP(A\nB)\n1\n216\n6\nP(B)\n6\n216\n\u2229\n=\n=\nExample 6  A die is thrown twice and the sum of the numbers appearing is observed\nto be 6 What is the conditional probability that the number 4 has appeared at least\nonce"
    },
    {
        "Chapter": "1",
        "sentence_range": "6817-6820",
        "Text": "(6,6,4)\n\u23a7\n\u23ab\n\u23aa\n\u23aa\n\u23a8\n\u23ac\n\u23aa\n\u23aa\n\u23a9\n\u23ad\nB = {(6,5,1), (6,5,2), (6,5,3), (6,5,4), (6,5,5), (6,5,6)}\nand\nA \u2229 B = {(6,5,4)} Now\nP(B) =\n6\n216  and  P (A \u2229 B) = 1\n216\nThen\nP(A|B) =\n1\nP(A\nB)\n1\n216\n6\nP(B)\n6\n216\n\u2229\n=\n=\nExample 6  A die is thrown twice and the sum of the numbers appearing is observed\nto be 6 What is the conditional probability that the number 4 has appeared at least\nonce Solution  Let E be the event that \u2018number 4 appears at least once\u2019 and F be the event\nthat \u2018the sum of the numbers appearing is 6\u2019"
    },
    {
        "Chapter": "1",
        "sentence_range": "6818-6821",
        "Text": "Now\nP(B) =\n6\n216  and  P (A \u2229 B) = 1\n216\nThen\nP(A|B) =\n1\nP(A\nB)\n1\n216\n6\nP(B)\n6\n216\n\u2229\n=\n=\nExample 6  A die is thrown twice and the sum of the numbers appearing is observed\nto be 6 What is the conditional probability that the number 4 has appeared at least\nonce Solution  Let E be the event that \u2018number 4 appears at least once\u2019 and F be the event\nthat \u2018the sum of the numbers appearing is 6\u2019 Then,\nE = {(4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (1,4), (2,4), (3,4), (5,4), (6,4)}\nand\nF = {(1,5), (2,4), (3,3), (4,2), (5,1)}\nWe have\nP(E) = 11\n36 and P(F) = 5\n36\nAlso\nE\u2229F = {(2,4), (4,2)}\n\u00a9 NCERT\nnot to be republished\nPROBABILITY      537\nTherefore\nP(E\u2229F) =\n2\n36\nHence, the required probability\nP(E|F) =\n2\nP(E\nF)\n2\n536\nP(F)\n5\n36\n\u2229\n=\n=\nFor the conditional probability discussed above, we have considered the elemen-\ntary events of the experiment to be equally likely and the corresponding definition of\nthe probability of an event was used"
    },
    {
        "Chapter": "1",
        "sentence_range": "6819-6822",
        "Text": "What is the conditional probability that the number 4 has appeared at least\nonce Solution  Let E be the event that \u2018number 4 appears at least once\u2019 and F be the event\nthat \u2018the sum of the numbers appearing is 6\u2019 Then,\nE = {(4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (1,4), (2,4), (3,4), (5,4), (6,4)}\nand\nF = {(1,5), (2,4), (3,3), (4,2), (5,1)}\nWe have\nP(E) = 11\n36 and P(F) = 5\n36\nAlso\nE\u2229F = {(2,4), (4,2)}\n\u00a9 NCERT\nnot to be republished\nPROBABILITY      537\nTherefore\nP(E\u2229F) =\n2\n36\nHence, the required probability\nP(E|F) =\n2\nP(E\nF)\n2\n536\nP(F)\n5\n36\n\u2229\n=\n=\nFor the conditional probability discussed above, we have considered the elemen-\ntary events of the experiment to be equally likely and the corresponding definition of\nthe probability of an event was used However, the same definition can also be used in\nthe general case where the elementary events of the sample space are not equally\nlikely, the probabilities P(E\u2229F) and P(F) being calculated accordingly"
    },
    {
        "Chapter": "1",
        "sentence_range": "6820-6823",
        "Text": "Solution  Let E be the event that \u2018number 4 appears at least once\u2019 and F be the event\nthat \u2018the sum of the numbers appearing is 6\u2019 Then,\nE = {(4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (1,4), (2,4), (3,4), (5,4), (6,4)}\nand\nF = {(1,5), (2,4), (3,3), (4,2), (5,1)}\nWe have\nP(E) = 11\n36 and P(F) = 5\n36\nAlso\nE\u2229F = {(2,4), (4,2)}\n\u00a9 NCERT\nnot to be republished\nPROBABILITY      537\nTherefore\nP(E\u2229F) =\n2\n36\nHence, the required probability\nP(E|F) =\n2\nP(E\nF)\n2\n536\nP(F)\n5\n36\n\u2229\n=\n=\nFor the conditional probability discussed above, we have considered the elemen-\ntary events of the experiment to be equally likely and the corresponding definition of\nthe probability of an event was used However, the same definition can also be used in\nthe general case where the elementary events of the sample space are not equally\nlikely, the probabilities P(E\u2229F) and P(F) being calculated accordingly Let us take up\nthe following example"
    },
    {
        "Chapter": "1",
        "sentence_range": "6821-6824",
        "Text": "Then,\nE = {(4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (1,4), (2,4), (3,4), (5,4), (6,4)}\nand\nF = {(1,5), (2,4), (3,3), (4,2), (5,1)}\nWe have\nP(E) = 11\n36 and P(F) = 5\n36\nAlso\nE\u2229F = {(2,4), (4,2)}\n\u00a9 NCERT\nnot to be republished\nPROBABILITY      537\nTherefore\nP(E\u2229F) =\n2\n36\nHence, the required probability\nP(E|F) =\n2\nP(E\nF)\n2\n536\nP(F)\n5\n36\n\u2229\n=\n=\nFor the conditional probability discussed above, we have considered the elemen-\ntary events of the experiment to be equally likely and the corresponding definition of\nthe probability of an event was used However, the same definition can also be used in\nthe general case where the elementary events of the sample space are not equally\nlikely, the probabilities P(E\u2229F) and P(F) being calculated accordingly Let us take up\nthe following example Example 7 Consider the experiment of tossing a coin"
    },
    {
        "Chapter": "1",
        "sentence_range": "6822-6825",
        "Text": "However, the same definition can also be used in\nthe general case where the elementary events of the sample space are not equally\nlikely, the probabilities P(E\u2229F) and P(F) being calculated accordingly Let us take up\nthe following example Example 7 Consider the experiment of tossing a coin If the coin shows head, toss it\nagain but if it shows tail, then throw a die"
    },
    {
        "Chapter": "1",
        "sentence_range": "6823-6826",
        "Text": "Let us take up\nthe following example Example 7 Consider the experiment of tossing a coin If the coin shows head, toss it\nagain but if it shows tail, then throw a die Find the\nconditional probability of the event that \u2018the die shows\na number greater than 4\u2019 given that \u2018there is at least\none tail\u2019"
    },
    {
        "Chapter": "1",
        "sentence_range": "6824-6827",
        "Text": "Example 7 Consider the experiment of tossing a coin If the coin shows head, toss it\nagain but if it shows tail, then throw a die Find the\nconditional probability of the event that \u2018the die shows\na number greater than 4\u2019 given that \u2018there is at least\none tail\u2019 Solution The outcomes of the experiment can be\nrepresented in following diagrammatic manner called\nthe \u2018tree diagram\u2019"
    },
    {
        "Chapter": "1",
        "sentence_range": "6825-6828",
        "Text": "If the coin shows head, toss it\nagain but if it shows tail, then throw a die Find the\nconditional probability of the event that \u2018the die shows\na number greater than 4\u2019 given that \u2018there is at least\none tail\u2019 Solution The outcomes of the experiment can be\nrepresented in following diagrammatic manner called\nthe \u2018tree diagram\u2019 The sample space of the experiment may be\ndescribed as\nS = {(H,H), (H,T), (T,1), (T,2), (T,3), (T,4), (T,5), (T,6)}\nwhere (H, H) denotes that both the tosses result into\nhead and (T, i) denote the first toss result into a tail and\nthe number i appeared on the die for i = 1,2,3,4,5,6"
    },
    {
        "Chapter": "1",
        "sentence_range": "6826-6829",
        "Text": "Find the\nconditional probability of the event that \u2018the die shows\na number greater than 4\u2019 given that \u2018there is at least\none tail\u2019 Solution The outcomes of the experiment can be\nrepresented in following diagrammatic manner called\nthe \u2018tree diagram\u2019 The sample space of the experiment may be\ndescribed as\nS = {(H,H), (H,T), (T,1), (T,2), (T,3), (T,4), (T,5), (T,6)}\nwhere (H, H) denotes that both the tosses result into\nhead and (T, i) denote the first toss result into a tail and\nthe number i appeared on the die for i = 1,2,3,4,5,6 Thus, the probabilities assigned to the 8 elementary\nevents\n(H, H), (H, T), (T, 1), (T, 2), (T, 3) (T, 4), (T, 5), (T, 6)\nare 1 1 1\n1\n1\n1\n1\n1\n,\n,\n,\n,\n,\n,\n,\n4 4 12 12 12 12 12 12  respectively which  is\nclear from the Fig 13"
    },
    {
        "Chapter": "1",
        "sentence_range": "6827-6830",
        "Text": "Solution The outcomes of the experiment can be\nrepresented in following diagrammatic manner called\nthe \u2018tree diagram\u2019 The sample space of the experiment may be\ndescribed as\nS = {(H,H), (H,T), (T,1), (T,2), (T,3), (T,4), (T,5), (T,6)}\nwhere (H, H) denotes that both the tosses result into\nhead and (T, i) denote the first toss result into a tail and\nthe number i appeared on the die for i = 1,2,3,4,5,6 Thus, the probabilities assigned to the 8 elementary\nevents\n(H, H), (H, T), (T, 1), (T, 2), (T, 3) (T, 4), (T, 5), (T, 6)\nare 1 1 1\n1\n1\n1\n1\n1\n,\n,\n,\n,\n,\n,\n,\n4 4 12 12 12 12 12 12  respectively which  is\nclear from the Fig 13 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "6828-6831",
        "Text": "The sample space of the experiment may be\ndescribed as\nS = {(H,H), (H,T), (T,1), (T,2), (T,3), (T,4), (T,5), (T,6)}\nwhere (H, H) denotes that both the tosses result into\nhead and (T, i) denote the first toss result into a tail and\nthe number i appeared on the die for i = 1,2,3,4,5,6 Thus, the probabilities assigned to the 8 elementary\nevents\n(H, H), (H, T), (T, 1), (T, 2), (T, 3) (T, 4), (T, 5), (T, 6)\nare 1 1 1\n1\n1\n1\n1\n1\n,\n,\n,\n,\n,\n,\n,\n4 4 12 12 12 12 12 12  respectively which  is\nclear from the Fig 13 2 Fig 13"
    },
    {
        "Chapter": "1",
        "sentence_range": "6829-6832",
        "Text": "Thus, the probabilities assigned to the 8 elementary\nevents\n(H, H), (H, T), (T, 1), (T, 2), (T, 3) (T, 4), (T, 5), (T, 6)\nare 1 1 1\n1\n1\n1\n1\n1\n,\n,\n,\n,\n,\n,\n,\n4 4 12 12 12 12 12 12  respectively which  is\nclear from the Fig 13 2 Fig 13 1\nFig 13"
    },
    {
        "Chapter": "1",
        "sentence_range": "6830-6833",
        "Text": "2 Fig 13 1\nFig 13 2\n\u00a9 NCERT\nnot to be republished\n 538\nMATHEMATICS\nLet F be the event that \u2018there is at least one tail\u2019 and E be the event \u2018the die shows\na number greater than 4\u2019"
    },
    {
        "Chapter": "1",
        "sentence_range": "6831-6834",
        "Text": "Fig 13 1\nFig 13 2\n\u00a9 NCERT\nnot to be republished\n 538\nMATHEMATICS\nLet F be the event that \u2018there is at least one tail\u2019 and E be the event \u2018the die shows\na number greater than 4\u2019 Then\nF = {(H,T), (T,1), (T,2), (T,3), (T,4), (T,5), (T,6)}\nE = {(T,5), (T,6)} and E \u2229 F = {(T,5), (T,6)}\nNow\nP(F) = P({(H,T)}) + P ({(T,1)}) + P ({(T,2)}) + P ({(T,3)})\n+ P ({(T,4)}) + P({(T,5)}) + P({(T,6)})\n= 1\n1\n1\n1\n1\n1\n1\n3\n4\n12\n12\n12\n12\n12\n12\n4\n \n \n \n \n \n \nand\nP(E \u2229 F) = P ({(T,5)}) + P ({(T,6)}) = 1\n1\n1\n12\n12\n6\n \n \nHence\nP(E|F) =\n1\nP(E\nF)\n2\n36\nP(F)\n9\n4\n\u2229\n=\n=\nEXERCISE 13"
    },
    {
        "Chapter": "1",
        "sentence_range": "6832-6835",
        "Text": "1\nFig 13 2\n\u00a9 NCERT\nnot to be republished\n 538\nMATHEMATICS\nLet F be the event that \u2018there is at least one tail\u2019 and E be the event \u2018the die shows\na number greater than 4\u2019 Then\nF = {(H,T), (T,1), (T,2), (T,3), (T,4), (T,5), (T,6)}\nE = {(T,5), (T,6)} and E \u2229 F = {(T,5), (T,6)}\nNow\nP(F) = P({(H,T)}) + P ({(T,1)}) + P ({(T,2)}) + P ({(T,3)})\n+ P ({(T,4)}) + P({(T,5)}) + P({(T,6)})\n= 1\n1\n1\n1\n1\n1\n1\n3\n4\n12\n12\n12\n12\n12\n12\n4\n \n \n \n \n \n \nand\nP(E \u2229 F) = P ({(T,5)}) + P ({(T,6)}) = 1\n1\n1\n12\n12\n6\n \n \nHence\nP(E|F) =\n1\nP(E\nF)\n2\n36\nP(F)\n9\n4\n\u2229\n=\n=\nEXERCISE 13 1\n1"
    },
    {
        "Chapter": "1",
        "sentence_range": "6833-6836",
        "Text": "2\n\u00a9 NCERT\nnot to be republished\n 538\nMATHEMATICS\nLet F be the event that \u2018there is at least one tail\u2019 and E be the event \u2018the die shows\na number greater than 4\u2019 Then\nF = {(H,T), (T,1), (T,2), (T,3), (T,4), (T,5), (T,6)}\nE = {(T,5), (T,6)} and E \u2229 F = {(T,5), (T,6)}\nNow\nP(F) = P({(H,T)}) + P ({(T,1)}) + P ({(T,2)}) + P ({(T,3)})\n+ P ({(T,4)}) + P({(T,5)}) + P({(T,6)})\n= 1\n1\n1\n1\n1\n1\n1\n3\n4\n12\n12\n12\n12\n12\n12\n4\n \n \n \n \n \n \nand\nP(E \u2229 F) = P ({(T,5)}) + P ({(T,6)}) = 1\n1\n1\n12\n12\n6\n \n \nHence\nP(E|F) =\n1\nP(E\nF)\n2\n36\nP(F)\n9\n4\n\u2229\n=\n=\nEXERCISE 13 1\n1 Given that E and F are events such that P(E) = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "6834-6837",
        "Text": "Then\nF = {(H,T), (T,1), (T,2), (T,3), (T,4), (T,5), (T,6)}\nE = {(T,5), (T,6)} and E \u2229 F = {(T,5), (T,6)}\nNow\nP(F) = P({(H,T)}) + P ({(T,1)}) + P ({(T,2)}) + P ({(T,3)})\n+ P ({(T,4)}) + P({(T,5)}) + P({(T,6)})\n= 1\n1\n1\n1\n1\n1\n1\n3\n4\n12\n12\n12\n12\n12\n12\n4\n \n \n \n \n \n \nand\nP(E \u2229 F) = P ({(T,5)}) + P ({(T,6)}) = 1\n1\n1\n12\n12\n6\n \n \nHence\nP(E|F) =\n1\nP(E\nF)\n2\n36\nP(F)\n9\n4\n\u2229\n=\n=\nEXERCISE 13 1\n1 Given that E and F are events such that P(E) = 0 6, P(F) = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "6835-6838",
        "Text": "1\n1 Given that E and F are events such that P(E) = 0 6, P(F) = 0 3 and\nP(E \u2229 F) = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "6836-6839",
        "Text": "Given that E and F are events such that P(E) = 0 6, P(F) = 0 3 and\nP(E \u2229 F) = 0 2, find P(E|F) and P(F|E)\n2"
    },
    {
        "Chapter": "1",
        "sentence_range": "6837-6840",
        "Text": "6, P(F) = 0 3 and\nP(E \u2229 F) = 0 2, find P(E|F) and P(F|E)\n2 Compute P(A|B), if P(B) = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "6838-6841",
        "Text": "3 and\nP(E \u2229 F) = 0 2, find P(E|F) and P(F|E)\n2 Compute P(A|B), if P(B) = 0 5 and  P (A \u2229 B) = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "6839-6842",
        "Text": "2, find P(E|F) and P(F|E)\n2 Compute P(A|B), if P(B) = 0 5 and  P (A \u2229 B) = 0 32\n3"
    },
    {
        "Chapter": "1",
        "sentence_range": "6840-6843",
        "Text": "Compute P(A|B), if P(B) = 0 5 and  P (A \u2229 B) = 0 32\n3 If  P(A) = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "6841-6844",
        "Text": "5 and  P (A \u2229 B) = 0 32\n3 If  P(A) = 0 8,  P (B) = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "6842-6845",
        "Text": "32\n3 If  P(A) = 0 8,  P (B) = 0 5 and P(B|A) = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "6843-6846",
        "Text": "If  P(A) = 0 8,  P (B) = 0 5 and P(B|A) = 0 4, find\n(i) P(A \u2229 B)\n(ii) P(A|B)\n(iii) P(A \u222a B)\n4"
    },
    {
        "Chapter": "1",
        "sentence_range": "6844-6847",
        "Text": "8,  P (B) = 0 5 and P(B|A) = 0 4, find\n(i) P(A \u2229 B)\n(ii) P(A|B)\n(iii) P(A \u222a B)\n4 Evaluate P(A \u222a B), if 2P(A) = P(B) = 5\n13  and P(A|B) = 2\n5\n5"
    },
    {
        "Chapter": "1",
        "sentence_range": "6845-6848",
        "Text": "5 and P(B|A) = 0 4, find\n(i) P(A \u2229 B)\n(ii) P(A|B)\n(iii) P(A \u222a B)\n4 Evaluate P(A \u222a B), if 2P(A) = P(B) = 5\n13  and P(A|B) = 2\n5\n5 If P(A) = 6\n11 , P(B) = 5\n11 and P(A \u222a B) \n 117\n, find\n(i) P(A\u2229B)\n(ii) P(A|B)\n(iii) P(B|A)\nDetermine P(E|F) in Exercises 6 to 9"
    },
    {
        "Chapter": "1",
        "sentence_range": "6846-6849",
        "Text": "4, find\n(i) P(A \u2229 B)\n(ii) P(A|B)\n(iii) P(A \u222a B)\n4 Evaluate P(A \u222a B), if 2P(A) = P(B) = 5\n13  and P(A|B) = 2\n5\n5 If P(A) = 6\n11 , P(B) = 5\n11 and P(A \u222a B) \n 117\n, find\n(i) P(A\u2229B)\n(ii) P(A|B)\n(iii) P(B|A)\nDetermine P(E|F) in Exercises 6 to 9 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "6847-6850",
        "Text": "Evaluate P(A \u222a B), if 2P(A) = P(B) = 5\n13  and P(A|B) = 2\n5\n5 If P(A) = 6\n11 , P(B) = 5\n11 and P(A \u222a B) \n 117\n, find\n(i) P(A\u2229B)\n(ii) P(A|B)\n(iii) P(B|A)\nDetermine P(E|F) in Exercises 6 to 9 6 A coin is tossed three times, where\n(i) E : head on third toss  ,      F : heads on first two tosses\n(ii) E : at least two heads  ,     F : at most two heads\n(iii) E : at most two tails    ,      F : at least one tail\n\u00a9 NCERT\nnot to be republished\nPROBABILITY      539\n7"
    },
    {
        "Chapter": "1",
        "sentence_range": "6848-6851",
        "Text": "If P(A) = 6\n11 , P(B) = 5\n11 and P(A \u222a B) \n 117\n, find\n(i) P(A\u2229B)\n(ii) P(A|B)\n(iii) P(B|A)\nDetermine P(E|F) in Exercises 6 to 9 6 A coin is tossed three times, where\n(i) E : head on third toss  ,      F : heads on first two tosses\n(ii) E : at least two heads  ,     F : at most two heads\n(iii) E : at most two tails    ,      F : at least one tail\n\u00a9 NCERT\nnot to be republished\nPROBABILITY      539\n7 Two coins are tossed once, where\n(i)\nE : tail appears on one coin,\nF : one coin shows head\n(ii)\nE :  no tail appears,\nF : no head appears\n8"
    },
    {
        "Chapter": "1",
        "sentence_range": "6849-6852",
        "Text": "6 A coin is tossed three times, where\n(i) E : head on third toss  ,      F : heads on first two tosses\n(ii) E : at least two heads  ,     F : at most two heads\n(iii) E : at most two tails    ,      F : at least one tail\n\u00a9 NCERT\nnot to be republished\nPROBABILITY      539\n7 Two coins are tossed once, where\n(i)\nE : tail appears on one coin,\nF : one coin shows head\n(ii)\nE :  no tail appears,\nF : no head appears\n8 A die is thrown three times,\nE : 4 appears on the third toss,\nF : 6 and 5 appears respectively\non first two tosses\n9"
    },
    {
        "Chapter": "1",
        "sentence_range": "6850-6853",
        "Text": "A coin is tossed three times, where\n(i) E : head on third toss  ,      F : heads on first two tosses\n(ii) E : at least two heads  ,     F : at most two heads\n(iii) E : at most two tails    ,      F : at least one tail\n\u00a9 NCERT\nnot to be republished\nPROBABILITY      539\n7 Two coins are tossed once, where\n(i)\nE : tail appears on one coin,\nF : one coin shows head\n(ii)\nE :  no tail appears,\nF : no head appears\n8 A die is thrown three times,\nE : 4 appears on the third toss,\nF : 6 and 5 appears respectively\non first two tosses\n9 Mother, father and son line up at random for a family picture\nE : son on one end,\nF : father in middle\n10"
    },
    {
        "Chapter": "1",
        "sentence_range": "6851-6854",
        "Text": "Two coins are tossed once, where\n(i)\nE : tail appears on one coin,\nF : one coin shows head\n(ii)\nE :  no tail appears,\nF : no head appears\n8 A die is thrown three times,\nE : 4 appears on the third toss,\nF : 6 and 5 appears respectively\non first two tosses\n9 Mother, father and son line up at random for a family picture\nE : son on one end,\nF : father in middle\n10 A black and a red dice are rolled"
    },
    {
        "Chapter": "1",
        "sentence_range": "6852-6855",
        "Text": "A die is thrown three times,\nE : 4 appears on the third toss,\nF : 6 and 5 appears respectively\non first two tosses\n9 Mother, father and son line up at random for a family picture\nE : son on one end,\nF : father in middle\n10 A black and a red dice are rolled (a) Find the conditional probability of obtaining a sum greater than 9, given\nthat the black die resulted in a 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "6853-6856",
        "Text": "Mother, father and son line up at random for a family picture\nE : son on one end,\nF : father in middle\n10 A black and a red dice are rolled (a) Find the conditional probability of obtaining a sum greater than 9, given\nthat the black die resulted in a 5 (b) Find the conditional probability of obtaining the sum 8, given that the red die\nresulted in a number less than 4"
    },
    {
        "Chapter": "1",
        "sentence_range": "6854-6857",
        "Text": "A black and a red dice are rolled (a) Find the conditional probability of obtaining a sum greater than 9, given\nthat the black die resulted in a 5 (b) Find the conditional probability of obtaining the sum 8, given that the red die\nresulted in a number less than 4 11"
    },
    {
        "Chapter": "1",
        "sentence_range": "6855-6858",
        "Text": "(a) Find the conditional probability of obtaining a sum greater than 9, given\nthat the black die resulted in a 5 (b) Find the conditional probability of obtaining the sum 8, given that the red die\nresulted in a number less than 4 11 A fair die is rolled"
    },
    {
        "Chapter": "1",
        "sentence_range": "6856-6859",
        "Text": "(b) Find the conditional probability of obtaining the sum 8, given that the red die\nresulted in a number less than 4 11 A fair die is rolled Consider events E = {1,3,5},  F = {2,3} and G = {2,3,4,5}\nFind\n(i) P(E|F) and P(F|E)\n(ii) P(E|G) and P(G|E)\n(iii) P((E \u222a F)|G) and P((E \u2229 F)|G)\n12"
    },
    {
        "Chapter": "1",
        "sentence_range": "6857-6860",
        "Text": "11 A fair die is rolled Consider events E = {1,3,5},  F = {2,3} and G = {2,3,4,5}\nFind\n(i) P(E|F) and P(F|E)\n(ii) P(E|G) and P(G|E)\n(iii) P((E \u222a F)|G) and P((E \u2229 F)|G)\n12 Assume that each born child is equally likely to be a boy or a girl"
    },
    {
        "Chapter": "1",
        "sentence_range": "6858-6861",
        "Text": "A fair die is rolled Consider events E = {1,3,5},  F = {2,3} and G = {2,3,4,5}\nFind\n(i) P(E|F) and P(F|E)\n(ii) P(E|G) and P(G|E)\n(iii) P((E \u222a F)|G) and P((E \u2229 F)|G)\n12 Assume that each born child is equally likely to be a boy or a girl If a family has\ntwo children, what is the conditional probability that both are girls given that\n(i) the youngest is a girl, (ii) at least one is a girl"
    },
    {
        "Chapter": "1",
        "sentence_range": "6859-6862",
        "Text": "Consider events E = {1,3,5},  F = {2,3} and G = {2,3,4,5}\nFind\n(i) P(E|F) and P(F|E)\n(ii) P(E|G) and P(G|E)\n(iii) P((E \u222a F)|G) and P((E \u2229 F)|G)\n12 Assume that each born child is equally likely to be a boy or a girl If a family has\ntwo children, what is the conditional probability that both are girls given that\n(i) the youngest is a girl, (ii) at least one is a girl 13"
    },
    {
        "Chapter": "1",
        "sentence_range": "6860-6863",
        "Text": "Assume that each born child is equally likely to be a boy or a girl If a family has\ntwo children, what is the conditional probability that both are girls given that\n(i) the youngest is a girl, (ii) at least one is a girl 13 An instructor has a question bank consisting of 300 easy True / False questions,\n200 difficult True / False questions, 500 easy multiple choice questions and 400\ndifficult multiple choice questions"
    },
    {
        "Chapter": "1",
        "sentence_range": "6861-6864",
        "Text": "If a family has\ntwo children, what is the conditional probability that both are girls given that\n(i) the youngest is a girl, (ii) at least one is a girl 13 An instructor has a question bank consisting of 300 easy True / False questions,\n200 difficult True / False questions, 500 easy multiple choice questions and 400\ndifficult multiple choice questions If a question is selected at random from the\nquestion bank, what is the probability that it will be an easy question given that it\nis a multiple choice question"
    },
    {
        "Chapter": "1",
        "sentence_range": "6862-6865",
        "Text": "13 An instructor has a question bank consisting of 300 easy True / False questions,\n200 difficult True / False questions, 500 easy multiple choice questions and 400\ndifficult multiple choice questions If a question is selected at random from the\nquestion bank, what is the probability that it will be an easy question given that it\nis a multiple choice question 14"
    },
    {
        "Chapter": "1",
        "sentence_range": "6863-6866",
        "Text": "An instructor has a question bank consisting of 300 easy True / False questions,\n200 difficult True / False questions, 500 easy multiple choice questions and 400\ndifficult multiple choice questions If a question is selected at random from the\nquestion bank, what is the probability that it will be an easy question given that it\nis a multiple choice question 14 Given that the two numbers appearing on throwing two dice are different"
    },
    {
        "Chapter": "1",
        "sentence_range": "6864-6867",
        "Text": "If a question is selected at random from the\nquestion bank, what is the probability that it will be an easy question given that it\nis a multiple choice question 14 Given that the two numbers appearing on throwing two dice are different Find\nthe probability of the event \u2018the sum of numbers on the dice is 4\u2019"
    },
    {
        "Chapter": "1",
        "sentence_range": "6865-6868",
        "Text": "14 Given that the two numbers appearing on throwing two dice are different Find\nthe probability of the event \u2018the sum of numbers on the dice is 4\u2019 15"
    },
    {
        "Chapter": "1",
        "sentence_range": "6866-6869",
        "Text": "Given that the two numbers appearing on throwing two dice are different Find\nthe probability of the event \u2018the sum of numbers on the dice is 4\u2019 15 Consider the experiment of throwing a die, if a multiple of 3 comes up, throw the\ndie again and if any other number comes, toss a coin"
    },
    {
        "Chapter": "1",
        "sentence_range": "6867-6870",
        "Text": "Find\nthe probability of the event \u2018the sum of numbers on the dice is 4\u2019 15 Consider the experiment of throwing a die, if a multiple of 3 comes up, throw the\ndie again and if any other number comes, toss a coin Find the conditional probability\nof the event \u2018the coin shows a tail\u2019, given that \u2018at least one die shows a 3\u2019"
    },
    {
        "Chapter": "1",
        "sentence_range": "6868-6871",
        "Text": "15 Consider the experiment of throwing a die, if a multiple of 3 comes up, throw the\ndie again and if any other number comes, toss a coin Find the conditional probability\nof the event \u2018the coin shows a tail\u2019, given that \u2018at least one die shows a 3\u2019 In each of the Exercises 16 and 17 choose the correct answer:\n16"
    },
    {
        "Chapter": "1",
        "sentence_range": "6869-6872",
        "Text": "Consider the experiment of throwing a die, if a multiple of 3 comes up, throw the\ndie again and if any other number comes, toss a coin Find the conditional probability\nof the event \u2018the coin shows a tail\u2019, given that \u2018at least one die shows a 3\u2019 In each of the Exercises 16 and 17 choose the correct answer:\n16 If  P(A) = 1\n2 , P(B) = 0, then P(A|B) is\n(A) 0\n(B) 1\n2\n(C) not defined\n(D) 1\n\u00a9 NCERT\nnot to be republished\n 540\nMATHEMATICS\n17"
    },
    {
        "Chapter": "1",
        "sentence_range": "6870-6873",
        "Text": "Find the conditional probability\nof the event \u2018the coin shows a tail\u2019, given that \u2018at least one die shows a 3\u2019 In each of the Exercises 16 and 17 choose the correct answer:\n16 If  P(A) = 1\n2 , P(B) = 0, then P(A|B) is\n(A) 0\n(B) 1\n2\n(C) not defined\n(D) 1\n\u00a9 NCERT\nnot to be republished\n 540\nMATHEMATICS\n17 If A and B are events such that P(A|B) = P(B|A), then\n(A) A \u2282 B but A \u2260 B\n(B) A = B\n(C) A \u2229 B = \u03c6\n(D) P(A) = P(B)\n13"
    },
    {
        "Chapter": "1",
        "sentence_range": "6871-6874",
        "Text": "In each of the Exercises 16 and 17 choose the correct answer:\n16 If  P(A) = 1\n2 , P(B) = 0, then P(A|B) is\n(A) 0\n(B) 1\n2\n(C) not defined\n(D) 1\n\u00a9 NCERT\nnot to be republished\n 540\nMATHEMATICS\n17 If A and B are events such that P(A|B) = P(B|A), then\n(A) A \u2282 B but A \u2260 B\n(B) A = B\n(C) A \u2229 B = \u03c6\n(D) P(A) = P(B)\n13 3  Multiplication Theorem on Probability\nLet E and F be two events associated with a sample space S"
    },
    {
        "Chapter": "1",
        "sentence_range": "6872-6875",
        "Text": "If  P(A) = 1\n2 , P(B) = 0, then P(A|B) is\n(A) 0\n(B) 1\n2\n(C) not defined\n(D) 1\n\u00a9 NCERT\nnot to be republished\n 540\nMATHEMATICS\n17 If A and B are events such that P(A|B) = P(B|A), then\n(A) A \u2282 B but A \u2260 B\n(B) A = B\n(C) A \u2229 B = \u03c6\n(D) P(A) = P(B)\n13 3  Multiplication Theorem on Probability\nLet E and F be two events associated with a sample space S Clearly, the set E \u2229 F\ndenotes the event that both E and F have occurred"
    },
    {
        "Chapter": "1",
        "sentence_range": "6873-6876",
        "Text": "If A and B are events such that P(A|B) = P(B|A), then\n(A) A \u2282 B but A \u2260 B\n(B) A = B\n(C) A \u2229 B = \u03c6\n(D) P(A) = P(B)\n13 3  Multiplication Theorem on Probability\nLet E and F be two events associated with a sample space S Clearly, the set E \u2229 F\ndenotes the event that both E and F have occurred In other words, E \u2229 F denotes the\nsimultaneous occurrence of the events E and F"
    },
    {
        "Chapter": "1",
        "sentence_range": "6874-6877",
        "Text": "3  Multiplication Theorem on Probability\nLet E and F be two events associated with a sample space S Clearly, the set E \u2229 F\ndenotes the event that both E and F have occurred In other words, E \u2229 F denotes the\nsimultaneous occurrence of the events E and F The event E \u2229 F is also written as EF"
    },
    {
        "Chapter": "1",
        "sentence_range": "6875-6878",
        "Text": "Clearly, the set E \u2229 F\ndenotes the event that both E and F have occurred In other words, E \u2229 F denotes the\nsimultaneous occurrence of the events E and F The event E \u2229 F is also written as EF Very often we need to find the probability of  the event EF"
    },
    {
        "Chapter": "1",
        "sentence_range": "6876-6879",
        "Text": "In other words, E \u2229 F denotes the\nsimultaneous occurrence of the events E and F The event E \u2229 F is also written as EF Very often we need to find the probability of  the event EF For example, in the\nexperiment of drawing two cards one after the other, we may be interested in finding\nthe probability of the event \u2018a king and a queen\u2019"
    },
    {
        "Chapter": "1",
        "sentence_range": "6877-6880",
        "Text": "The event E \u2229 F is also written as EF Very often we need to find the probability of  the event EF For example, in the\nexperiment of drawing two cards one after the other, we may be interested in finding\nthe probability of the event \u2018a king and a queen\u2019 The probability of event EF is obtained\nby using the conditional probability as obtained below :\nWe know that the conditional probability of event E given that F has occurred is\ndenoted by P(E|F) and is given by\nP(E|F) = P(E\nP(F)F) ,P(F) 0\n\u2229\n\u2260\nFrom this result, we can write\nP(E \u2229 F) = P(F)"
    },
    {
        "Chapter": "1",
        "sentence_range": "6878-6881",
        "Text": "Very often we need to find the probability of  the event EF For example, in the\nexperiment of drawing two cards one after the other, we may be interested in finding\nthe probability of the event \u2018a king and a queen\u2019 The probability of event EF is obtained\nby using the conditional probability as obtained below :\nWe know that the conditional probability of event E given that F has occurred is\ndenoted by P(E|F) and is given by\nP(E|F) = P(E\nP(F)F) ,P(F) 0\n\u2229\n\u2260\nFrom this result, we can write\nP(E \u2229 F) = P(F) P(E|F)"
    },
    {
        "Chapter": "1",
        "sentence_range": "6879-6882",
        "Text": "For example, in the\nexperiment of drawing two cards one after the other, we may be interested in finding\nthe probability of the event \u2018a king and a queen\u2019 The probability of event EF is obtained\nby using the conditional probability as obtained below :\nWe know that the conditional probability of event E given that F has occurred is\ndenoted by P(E|F) and is given by\nP(E|F) = P(E\nP(F)F) ,P(F) 0\n\u2229\n\u2260\nFrom this result, we can write\nP(E \u2229 F) = P(F) P(E|F) (1)\nAlso, we know that\nP(F|E) = P(F\nP(E)E) ,P(E) 0\n\u2229\n\u2260\nor\nP(F|E) = P(E\nP(E)F)\n\u2229\n (since E \u2229 F = F \u2229 E)\nThus,\nP(E \u2229 F) = P(E)"
    },
    {
        "Chapter": "1",
        "sentence_range": "6880-6883",
        "Text": "The probability of event EF is obtained\nby using the conditional probability as obtained below :\nWe know that the conditional probability of event E given that F has occurred is\ndenoted by P(E|F) and is given by\nP(E|F) = P(E\nP(F)F) ,P(F) 0\n\u2229\n\u2260\nFrom this result, we can write\nP(E \u2229 F) = P(F) P(E|F) (1)\nAlso, we know that\nP(F|E) = P(F\nP(E)E) ,P(E) 0\n\u2229\n\u2260\nor\nP(F|E) = P(E\nP(E)F)\n\u2229\n (since E \u2229 F = F \u2229 E)\nThus,\nP(E \u2229 F) = P(E) P(F|E)"
    },
    {
        "Chapter": "1",
        "sentence_range": "6881-6884",
        "Text": "P(E|F) (1)\nAlso, we know that\nP(F|E) = P(F\nP(E)E) ,P(E) 0\n\u2229\n\u2260\nor\nP(F|E) = P(E\nP(E)F)\n\u2229\n (since E \u2229 F = F \u2229 E)\nThus,\nP(E \u2229 F) = P(E) P(F|E) (2)\nCombining (1) and (2), we find that\nP(E \u2229 F) = P(E) P(F|E)\n= P(F) P(E|F) provided P(E) \u2260 0 and P(F) \u2260 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "6882-6885",
        "Text": "(1)\nAlso, we know that\nP(F|E) = P(F\nP(E)E) ,P(E) 0\n\u2229\n\u2260\nor\nP(F|E) = P(E\nP(E)F)\n\u2229\n (since E \u2229 F = F \u2229 E)\nThus,\nP(E \u2229 F) = P(E) P(F|E) (2)\nCombining (1) and (2), we find that\nP(E \u2229 F) = P(E) P(F|E)\n= P(F) P(E|F) provided P(E) \u2260 0 and P(F) \u2260 0 The above result is known as the multiplication rule of probability"
    },
    {
        "Chapter": "1",
        "sentence_range": "6883-6886",
        "Text": "P(F|E) (2)\nCombining (1) and (2), we find that\nP(E \u2229 F) = P(E) P(F|E)\n= P(F) P(E|F) provided P(E) \u2260 0 and P(F) \u2260 0 The above result is known as the multiplication rule of probability Let us now take up an example"
    },
    {
        "Chapter": "1",
        "sentence_range": "6884-6887",
        "Text": "(2)\nCombining (1) and (2), we find that\nP(E \u2229 F) = P(E) P(F|E)\n= P(F) P(E|F) provided P(E) \u2260 0 and P(F) \u2260 0 The above result is known as the multiplication rule of probability Let us now take up an example Example 8  An urn contains 10 black and 5 white balls"
    },
    {
        "Chapter": "1",
        "sentence_range": "6885-6888",
        "Text": "The above result is known as the multiplication rule of probability Let us now take up an example Example 8  An urn contains 10 black and 5 white balls Two balls are drawn from the\nurn one after the other without replacement"
    },
    {
        "Chapter": "1",
        "sentence_range": "6886-6889",
        "Text": "Let us now take up an example Example 8  An urn contains 10 black and 5 white balls Two balls are drawn from the\nurn one after the other without replacement What is the probability that both drawn\nballs are black"
    },
    {
        "Chapter": "1",
        "sentence_range": "6887-6890",
        "Text": "Example 8  An urn contains 10 black and 5 white balls Two balls are drawn from the\nurn one after the other without replacement What is the probability that both drawn\nballs are black Solution Let E and F denote respectively the events that first and second ball drawn\nare black"
    },
    {
        "Chapter": "1",
        "sentence_range": "6888-6891",
        "Text": "Two balls are drawn from the\nurn one after the other without replacement What is the probability that both drawn\nballs are black Solution Let E and F denote respectively the events that first and second ball drawn\nare black We have to find P(E \u2229 F) or P (EF)"
    },
    {
        "Chapter": "1",
        "sentence_range": "6889-6892",
        "Text": "What is the probability that both drawn\nballs are black Solution Let E and F denote respectively the events that first and second ball drawn\nare black We have to find P(E \u2229 F) or P (EF) \u00a9 NCERT\nnot to be republished\nPROBABILITY      541\nNow\nP(E) = P (black ball in first draw) = 10\n15\nAlso given that the first ball drawn is black, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "6890-6893",
        "Text": "Solution Let E and F denote respectively the events that first and second ball drawn\nare black We have to find P(E \u2229 F) or P (EF) \u00a9 NCERT\nnot to be republished\nPROBABILITY      541\nNow\nP(E) = P (black ball in first draw) = 10\n15\nAlso given that the first ball drawn is black, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "6891-6894",
        "Text": "We have to find P(E \u2229 F) or P (EF) \u00a9 NCERT\nnot to be republished\nPROBABILITY      541\nNow\nP(E) = P (black ball in first draw) = 10\n15\nAlso given that the first ball drawn is black, i e , event E has occurred, now there\nare 9 black balls and five white balls left in the urn"
    },
    {
        "Chapter": "1",
        "sentence_range": "6892-6895",
        "Text": "\u00a9 NCERT\nnot to be republished\nPROBABILITY      541\nNow\nP(E) = P (black ball in first draw) = 10\n15\nAlso given that the first ball drawn is black, i e , event E has occurred, now there\nare 9 black balls and five white balls left in the urn Therefore, the probability that the\nsecond ball drawn is black, given that the ball in the first draw is black, is nothing but\nthe conditional probability of F given that E has occurred"
    },
    {
        "Chapter": "1",
        "sentence_range": "6893-6896",
        "Text": "e , event E has occurred, now there\nare 9 black balls and five white balls left in the urn Therefore, the probability that the\nsecond ball drawn is black, given that the ball in the first draw is black, is nothing but\nthe conditional probability of F given that E has occurred i"
    },
    {
        "Chapter": "1",
        "sentence_range": "6894-6897",
        "Text": ", event E has occurred, now there\nare 9 black balls and five white balls left in the urn Therefore, the probability that the\nsecond ball drawn is black, given that the ball in the first draw is black, is nothing but\nthe conditional probability of F given that E has occurred i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "6895-6898",
        "Text": "Therefore, the probability that the\nsecond ball drawn is black, given that the ball in the first draw is black, is nothing but\nthe conditional probability of F given that E has occurred i e P(F|E) = 9\n14\nBy multiplication rule of probability, we have\nP (E \u2229 F) = P(E) P(F|E)\n= 10\n9\n3\n15\n14\n7\n \nMultiplication rule of probability for more than two events If E, F and G are\nthree events of sample space, we have\nP(E \u2229 F \u2229 G) = P(E) P(F|E) P(G|(E \u2229 F)) = P(E) P(F|E) P(G|EF)\nSimilarly, the multiplication rule of probability can be extended for four or\nmore events"
    },
    {
        "Chapter": "1",
        "sentence_range": "6896-6899",
        "Text": "i e P(F|E) = 9\n14\nBy multiplication rule of probability, we have\nP (E \u2229 F) = P(E) P(F|E)\n= 10\n9\n3\n15\n14\n7\n \nMultiplication rule of probability for more than two events If E, F and G are\nthree events of sample space, we have\nP(E \u2229 F \u2229 G) = P(E) P(F|E) P(G|(E \u2229 F)) = P(E) P(F|E) P(G|EF)\nSimilarly, the multiplication rule of probability can be extended for four or\nmore events The following example illustrates the extension of multiplication rule of probability\nfor three events"
    },
    {
        "Chapter": "1",
        "sentence_range": "6897-6900",
        "Text": "e P(F|E) = 9\n14\nBy multiplication rule of probability, we have\nP (E \u2229 F) = P(E) P(F|E)\n= 10\n9\n3\n15\n14\n7\n \nMultiplication rule of probability for more than two events If E, F and G are\nthree events of sample space, we have\nP(E \u2229 F \u2229 G) = P(E) P(F|E) P(G|(E \u2229 F)) = P(E) P(F|E) P(G|EF)\nSimilarly, the multiplication rule of probability can be extended for four or\nmore events The following example illustrates the extension of multiplication rule of probability\nfor three events Example 9 Three cards are drawn successively, without replacement from a pack of\n52 well shuffled cards"
    },
    {
        "Chapter": "1",
        "sentence_range": "6898-6901",
        "Text": "P(F|E) = 9\n14\nBy multiplication rule of probability, we have\nP (E \u2229 F) = P(E) P(F|E)\n= 10\n9\n3\n15\n14\n7\n \nMultiplication rule of probability for more than two events If E, F and G are\nthree events of sample space, we have\nP(E \u2229 F \u2229 G) = P(E) P(F|E) P(G|(E \u2229 F)) = P(E) P(F|E) P(G|EF)\nSimilarly, the multiplication rule of probability can be extended for four or\nmore events The following example illustrates the extension of multiplication rule of probability\nfor three events Example 9 Three cards are drawn successively, without replacement from a pack of\n52 well shuffled cards What is the probability that first two cards are kings and the\nthird card drawn is an ace"
    },
    {
        "Chapter": "1",
        "sentence_range": "6899-6902",
        "Text": "The following example illustrates the extension of multiplication rule of probability\nfor three events Example 9 Three cards are drawn successively, without replacement from a pack of\n52 well shuffled cards What is the probability that first two cards are kings and the\nthird card drawn is an ace Solution Let K denote the event that the card drawn is king and A be the event that\nthe card drawn is an ace"
    },
    {
        "Chapter": "1",
        "sentence_range": "6900-6903",
        "Text": "Example 9 Three cards are drawn successively, without replacement from a pack of\n52 well shuffled cards What is the probability that first two cards are kings and the\nthird card drawn is an ace Solution Let K denote the event that the card drawn is king and A be the event that\nthe card drawn is an ace Clearly, we have to find P (KKA)\nNow\nP(K) = 4\n52\nAlso, P (K|K) is the probability of second king with the condition that one king has\nalready been drawn"
    },
    {
        "Chapter": "1",
        "sentence_range": "6901-6904",
        "Text": "What is the probability that first two cards are kings and the\nthird card drawn is an ace Solution Let K denote the event that the card drawn is king and A be the event that\nthe card drawn is an ace Clearly, we have to find P (KKA)\nNow\nP(K) = 4\n52\nAlso, P (K|K) is the probability of second king with the condition that one king has\nalready been drawn Now there are three kings in (52 \u2212 1) = 51 cards"
    },
    {
        "Chapter": "1",
        "sentence_range": "6902-6905",
        "Text": "Solution Let K denote the event that the card drawn is king and A be the event that\nthe card drawn is an ace Clearly, we have to find P (KKA)\nNow\nP(K) = 4\n52\nAlso, P (K|K) is the probability of second king with the condition that one king has\nalready been drawn Now there are three kings in (52 \u2212 1) = 51 cards Therefore\nP(K|K) = 3\n51\nLastly, P(A|KK) is the probability of third drawn card to be an ace, with the condition\nthat two kings have already been drawn"
    },
    {
        "Chapter": "1",
        "sentence_range": "6903-6906",
        "Text": "Clearly, we have to find P (KKA)\nNow\nP(K) = 4\n52\nAlso, P (K|K) is the probability of second king with the condition that one king has\nalready been drawn Now there are three kings in (52 \u2212 1) = 51 cards Therefore\nP(K|K) = 3\n51\nLastly, P(A|KK) is the probability of third drawn card to be an ace, with the condition\nthat two kings have already been drawn Now there are four aces in left 50 cards"
    },
    {
        "Chapter": "1",
        "sentence_range": "6904-6907",
        "Text": "Now there are three kings in (52 \u2212 1) = 51 cards Therefore\nP(K|K) = 3\n51\nLastly, P(A|KK) is the probability of third drawn card to be an ace, with the condition\nthat two kings have already been drawn Now there are four aces in left 50 cards \u00a9 NCERT\nnot to be republished\n 542\nMATHEMATICS\nTherefore\nP(A|KK) = 4\n50\nBy multiplication law of probability, we have\nP(KKA) = P(K)   P(K|K)  P(A|KK)\n= 4\n3\n4\n2\n52\n51\n50\n5525\n \n \n \n13"
    },
    {
        "Chapter": "1",
        "sentence_range": "6905-6908",
        "Text": "Therefore\nP(K|K) = 3\n51\nLastly, P(A|KK) is the probability of third drawn card to be an ace, with the condition\nthat two kings have already been drawn Now there are four aces in left 50 cards \u00a9 NCERT\nnot to be republished\n 542\nMATHEMATICS\nTherefore\nP(A|KK) = 4\n50\nBy multiplication law of probability, we have\nP(KKA) = P(K)   P(K|K)  P(A|KK)\n= 4\n3\n4\n2\n52\n51\n50\n5525\n \n \n \n13 4  Independent Events\nConsider the experiment of drawing a card from a deck of 52 playing cards, in which\nthe elementary events are assumed to be equally likely"
    },
    {
        "Chapter": "1",
        "sentence_range": "6906-6909",
        "Text": "Now there are four aces in left 50 cards \u00a9 NCERT\nnot to be republished\n 542\nMATHEMATICS\nTherefore\nP(A|KK) = 4\n50\nBy multiplication law of probability, we have\nP(KKA) = P(K)   P(K|K)  P(A|KK)\n= 4\n3\n4\n2\n52\n51\n50\n5525\n \n \n \n13 4  Independent Events\nConsider the experiment of drawing a card from a deck of 52 playing cards, in which\nthe elementary events are assumed to be equally likely If E and F denote the events\n'the card drawn is a spade' and 'the card drawn is an ace' respectively, then\nP(E) = 13\n1\n4\n1\nand P(F)\n52\n4\n52\n13\n \n \n \nAlso E and F is the event ' the card drawn is the ace of spades' so that\nP(E \u2229F) = 1\n52\nHence\nP(E|F) =\n1\nP(E\nF)\n1\n152\nP(F)\n4\n13\n \n \n \nSince P(E) = 1\n4 = P (E|F), we can say that the occurrence of event F has not\naffected the probability of occurrence of the event E"
    },
    {
        "Chapter": "1",
        "sentence_range": "6907-6910",
        "Text": "\u00a9 NCERT\nnot to be republished\n 542\nMATHEMATICS\nTherefore\nP(A|KK) = 4\n50\nBy multiplication law of probability, we have\nP(KKA) = P(K)   P(K|K)  P(A|KK)\n= 4\n3\n4\n2\n52\n51\n50\n5525\n \n \n \n13 4  Independent Events\nConsider the experiment of drawing a card from a deck of 52 playing cards, in which\nthe elementary events are assumed to be equally likely If E and F denote the events\n'the card drawn is a spade' and 'the card drawn is an ace' respectively, then\nP(E) = 13\n1\n4\n1\nand P(F)\n52\n4\n52\n13\n \n \n \nAlso E and F is the event ' the card drawn is the ace of spades' so that\nP(E \u2229F) = 1\n52\nHence\nP(E|F) =\n1\nP(E\nF)\n1\n152\nP(F)\n4\n13\n \n \n \nSince P(E) = 1\n4 = P (E|F), we can say that the occurrence of event F has not\naffected the probability of occurrence of the event E We also have\nP(F|E) =\n1\nP(E\nF)\n1\n52\nP(F)\n1\nP(E)\n13\n4\n \n \n \n \nAgain,  P(F) = 1\n13  = P(F|E) shows that occurrence of event E has not affected\nthe probability of occurrence of the event F"
    },
    {
        "Chapter": "1",
        "sentence_range": "6908-6911",
        "Text": "4  Independent Events\nConsider the experiment of drawing a card from a deck of 52 playing cards, in which\nthe elementary events are assumed to be equally likely If E and F denote the events\n'the card drawn is a spade' and 'the card drawn is an ace' respectively, then\nP(E) = 13\n1\n4\n1\nand P(F)\n52\n4\n52\n13\n \n \n \nAlso E and F is the event ' the card drawn is the ace of spades' so that\nP(E \u2229F) = 1\n52\nHence\nP(E|F) =\n1\nP(E\nF)\n1\n152\nP(F)\n4\n13\n \n \n \nSince P(E) = 1\n4 = P (E|F), we can say that the occurrence of event F has not\naffected the probability of occurrence of the event E We also have\nP(F|E) =\n1\nP(E\nF)\n1\n52\nP(F)\n1\nP(E)\n13\n4\n \n \n \n \nAgain,  P(F) = 1\n13  = P(F|E) shows that occurrence of event E has not affected\nthe probability of occurrence of the event F Thus, E and F are two events such that the probability of occurrence of one of\nthem is not affected by occurrence of the other"
    },
    {
        "Chapter": "1",
        "sentence_range": "6909-6912",
        "Text": "If E and F denote the events\n'the card drawn is a spade' and 'the card drawn is an ace' respectively, then\nP(E) = 13\n1\n4\n1\nand P(F)\n52\n4\n52\n13\n \n \n \nAlso E and F is the event ' the card drawn is the ace of spades' so that\nP(E \u2229F) = 1\n52\nHence\nP(E|F) =\n1\nP(E\nF)\n1\n152\nP(F)\n4\n13\n \n \n \nSince P(E) = 1\n4 = P (E|F), we can say that the occurrence of event F has not\naffected the probability of occurrence of the event E We also have\nP(F|E) =\n1\nP(E\nF)\n1\n52\nP(F)\n1\nP(E)\n13\n4\n \n \n \n \nAgain,  P(F) = 1\n13  = P(F|E) shows that occurrence of event E has not affected\nthe probability of occurrence of the event F Thus, E and F are two events such that the probability of occurrence of one of\nthem is not affected by occurrence of the other Such events are called independent events"
    },
    {
        "Chapter": "1",
        "sentence_range": "6910-6913",
        "Text": "We also have\nP(F|E) =\n1\nP(E\nF)\n1\n52\nP(F)\n1\nP(E)\n13\n4\n \n \n \n \nAgain,  P(F) = 1\n13  = P(F|E) shows that occurrence of event E has not affected\nthe probability of occurrence of the event F Thus, E and F are two events such that the probability of occurrence of one of\nthem is not affected by occurrence of the other Such events are called independent events \u00a9 NCERT\nnot to be republished\nPROBABILITY      543\nDefinition 2 Two events E and F are said to be independent, if\nP(F|E) = P (F)  provided P (E)  \u2260 0\nand\nP (E|F) = P (E)  provided P (F)  \u2260 0\nThus, in this definition we need to have P (E) \u2260 0 and  P(F) \u2260 0\nNow, by the multiplication rule of probability, we have\nP(E \u2229 F) = P(E)"
    },
    {
        "Chapter": "1",
        "sentence_range": "6911-6914",
        "Text": "Thus, E and F are two events such that the probability of occurrence of one of\nthem is not affected by occurrence of the other Such events are called independent events \u00a9 NCERT\nnot to be republished\nPROBABILITY      543\nDefinition 2 Two events E and F are said to be independent, if\nP(F|E) = P (F)  provided P (E)  \u2260 0\nand\nP (E|F) = P (E)  provided P (F)  \u2260 0\nThus, in this definition we need to have P (E) \u2260 0 and  P(F) \u2260 0\nNow, by the multiplication rule of probability, we have\nP(E \u2229 F) = P(E) P (F|E)"
    },
    {
        "Chapter": "1",
        "sentence_range": "6912-6915",
        "Text": "Such events are called independent events \u00a9 NCERT\nnot to be republished\nPROBABILITY      543\nDefinition 2 Two events E and F are said to be independent, if\nP(F|E) = P (F)  provided P (E)  \u2260 0\nand\nP (E|F) = P (E)  provided P (F)  \u2260 0\nThus, in this definition we need to have P (E) \u2260 0 and  P(F) \u2260 0\nNow, by the multiplication rule of probability, we have\nP(E \u2229 F) = P(E) P (F|E) (1)\nIf E and F are independent, then (1) becomes\nP(E \u2229 F) = P(E)"
    },
    {
        "Chapter": "1",
        "sentence_range": "6913-6916",
        "Text": "\u00a9 NCERT\nnot to be republished\nPROBABILITY      543\nDefinition 2 Two events E and F are said to be independent, if\nP(F|E) = P (F)  provided P (E)  \u2260 0\nand\nP (E|F) = P (E)  provided P (F)  \u2260 0\nThus, in this definition we need to have P (E) \u2260 0 and  P(F) \u2260 0\nNow, by the multiplication rule of probability, we have\nP(E \u2229 F) = P(E) P (F|E) (1)\nIf E and F are independent, then (1) becomes\nP(E \u2229 F) = P(E) P(F)"
    },
    {
        "Chapter": "1",
        "sentence_range": "6914-6917",
        "Text": "P (F|E) (1)\nIf E and F are independent, then (1) becomes\nP(E \u2229 F) = P(E) P(F) (2)\nThus, using (2), the independence of two events is also defined as follows:\nDefinition 3 Let E and F be two events associated with the same random experiment,\nthen E and F are said to be independent if\nP(E \u2229 F) = P(E)"
    },
    {
        "Chapter": "1",
        "sentence_range": "6915-6918",
        "Text": "(1)\nIf E and F are independent, then (1) becomes\nP(E \u2229 F) = P(E) P(F) (2)\nThus, using (2), the independence of two events is also defined as follows:\nDefinition 3 Let E and F be two events associated with the same random experiment,\nthen E and F are said to be independent if\nP(E \u2229 F) = P(E) P (F)\nRemarks\n(i)\nTwo events E and F are said to be dependent if they are not independent, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "6916-6919",
        "Text": "P(F) (2)\nThus, using (2), the independence of two events is also defined as follows:\nDefinition 3 Let E and F be two events associated with the same random experiment,\nthen E and F are said to be independent if\nP(E \u2229 F) = P(E) P (F)\nRemarks\n(i)\nTwo events E and F are said to be dependent if they are not independent, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "6917-6920",
        "Text": "(2)\nThus, using (2), the independence of two events is also defined as follows:\nDefinition 3 Let E and F be two events associated with the same random experiment,\nthen E and F are said to be independent if\nP(E \u2229 F) = P(E) P (F)\nRemarks\n(i)\nTwo events E and F are said to be dependent if they are not independent, i e if\nP(E \u2229 F ) \u2260 P(E)"
    },
    {
        "Chapter": "1",
        "sentence_range": "6918-6921",
        "Text": "P (F)\nRemarks\n(i)\nTwo events E and F are said to be dependent if they are not independent, i e if\nP(E \u2229 F ) \u2260 P(E) P (F)\n(ii)\nSometimes there is a confusion between independent events and mutually\nexclusive events"
    },
    {
        "Chapter": "1",
        "sentence_range": "6919-6922",
        "Text": "e if\nP(E \u2229 F ) \u2260 P(E) P (F)\n(ii)\nSometimes there is a confusion between independent events and mutually\nexclusive events Term \u2018independent\u2019 is defined in terms of \u2018probability of events\u2019\nwhereas mutually exclusive is defined in term of events (subset of sample space)"
    },
    {
        "Chapter": "1",
        "sentence_range": "6920-6923",
        "Text": "if\nP(E \u2229 F ) \u2260 P(E) P (F)\n(ii)\nSometimes there is a confusion between independent events and mutually\nexclusive events Term \u2018independent\u2019 is defined in terms of \u2018probability of events\u2019\nwhereas mutually exclusive is defined in term of events (subset of sample space) Moreover, mutually exclusive events never have an outcome common, but\nindependent events, may have common outcome"
    },
    {
        "Chapter": "1",
        "sentence_range": "6921-6924",
        "Text": "P (F)\n(ii)\nSometimes there is a confusion between independent events and mutually\nexclusive events Term \u2018independent\u2019 is defined in terms of \u2018probability of events\u2019\nwhereas mutually exclusive is defined in term of events (subset of sample space) Moreover, mutually exclusive events never have an outcome common, but\nindependent events, may have common outcome Clearly, \u2018independent\u2019 and\n\u2018mutually exclusive\u2019 do not have the same meaning"
    },
    {
        "Chapter": "1",
        "sentence_range": "6922-6925",
        "Text": "Term \u2018independent\u2019 is defined in terms of \u2018probability of events\u2019\nwhereas mutually exclusive is defined in term of events (subset of sample space) Moreover, mutually exclusive events never have an outcome common, but\nindependent events, may have common outcome Clearly, \u2018independent\u2019 and\n\u2018mutually exclusive\u2019 do not have the same meaning In other words, two independent events having nonzero probabilities of occurrence\ncan not be mutually exclusive, and conversely, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "6923-6926",
        "Text": "Moreover, mutually exclusive events never have an outcome common, but\nindependent events, may have common outcome Clearly, \u2018independent\u2019 and\n\u2018mutually exclusive\u2019 do not have the same meaning In other words, two independent events having nonzero probabilities of occurrence\ncan not be mutually exclusive, and conversely, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "6924-6927",
        "Text": "Clearly, \u2018independent\u2019 and\n\u2018mutually exclusive\u2019 do not have the same meaning In other words, two independent events having nonzero probabilities of occurrence\ncan not be mutually exclusive, and conversely, i e two mutually exclusive events\nhaving nonzero probabilities of occurrence can not be independent"
    },
    {
        "Chapter": "1",
        "sentence_range": "6925-6928",
        "Text": "In other words, two independent events having nonzero probabilities of occurrence\ncan not be mutually exclusive, and conversely, i e two mutually exclusive events\nhaving nonzero probabilities of occurrence can not be independent (iii)\nTwo experiments are said to be independent if for every pair of events E and F,\nwhere E is associated with the first experiment and F with the second experiment,\nthe probability of the simultaneous occurrence of the events E and F when the\ntwo experiments are performed is the product of P(E) and P(F) calculated\nseparately on the basis of two experiments, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "6926-6929",
        "Text": "e two mutually exclusive events\nhaving nonzero probabilities of occurrence can not be independent (iii)\nTwo experiments are said to be independent if for every pair of events E and F,\nwhere E is associated with the first experiment and F with the second experiment,\nthe probability of the simultaneous occurrence of the events E and F when the\ntwo experiments are performed is the product of P(E) and P(F) calculated\nseparately on the basis of two experiments, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "6927-6930",
        "Text": "two mutually exclusive events\nhaving nonzero probabilities of occurrence can not be independent (iii)\nTwo experiments are said to be independent if for every pair of events E and F,\nwhere E is associated with the first experiment and F with the second experiment,\nthe probability of the simultaneous occurrence of the events E and F when the\ntwo experiments are performed is the product of P(E) and P(F) calculated\nseparately on the basis of two experiments, i e , P (E \u2229 F) = P (E)"
    },
    {
        "Chapter": "1",
        "sentence_range": "6928-6931",
        "Text": "(iii)\nTwo experiments are said to be independent if for every pair of events E and F,\nwhere E is associated with the first experiment and F with the second experiment,\nthe probability of the simultaneous occurrence of the events E and F when the\ntwo experiments are performed is the product of P(E) and P(F) calculated\nseparately on the basis of two experiments, i e , P (E \u2229 F) = P (E) P(F)\n(iv)\nThree events A, B and C are said to be mutually independent, if\nP(A \u2229 B) = P(A) P(B)\nP(A \u2229 C) = P(A) P(C)\nP(B \u2229 C) = P(B) P(C)\nand\nP(A \u2229 B \u2229 C) = P(A) P(B) P(C)\n\u00a9 NCERT\nnot to be republished\n 544\nMATHEMATICS\nIf at least one of the above is not true for three given events, we say that the\nevents are not independent"
    },
    {
        "Chapter": "1",
        "sentence_range": "6929-6932",
        "Text": "e , P (E \u2229 F) = P (E) P(F)\n(iv)\nThree events A, B and C are said to be mutually independent, if\nP(A \u2229 B) = P(A) P(B)\nP(A \u2229 C) = P(A) P(C)\nP(B \u2229 C) = P(B) P(C)\nand\nP(A \u2229 B \u2229 C) = P(A) P(B) P(C)\n\u00a9 NCERT\nnot to be republished\n 544\nMATHEMATICS\nIf at least one of the above is not true for three given events, we say that the\nevents are not independent Example 10 A die is thrown"
    },
    {
        "Chapter": "1",
        "sentence_range": "6930-6933",
        "Text": ", P (E \u2229 F) = P (E) P(F)\n(iv)\nThree events A, B and C are said to be mutually independent, if\nP(A \u2229 B) = P(A) P(B)\nP(A \u2229 C) = P(A) P(C)\nP(B \u2229 C) = P(B) P(C)\nand\nP(A \u2229 B \u2229 C) = P(A) P(B) P(C)\n\u00a9 NCERT\nnot to be republished\n 544\nMATHEMATICS\nIf at least one of the above is not true for three given events, we say that the\nevents are not independent Example 10 A die is thrown If E is the event \u2018the number appearing is a multiple of\n3\u2019 and F be the event \u2018the number appearing is even\u2019 then find whether E and F are\nindependent"
    },
    {
        "Chapter": "1",
        "sentence_range": "6931-6934",
        "Text": "P(F)\n(iv)\nThree events A, B and C are said to be mutually independent, if\nP(A \u2229 B) = P(A) P(B)\nP(A \u2229 C) = P(A) P(C)\nP(B \u2229 C) = P(B) P(C)\nand\nP(A \u2229 B \u2229 C) = P(A) P(B) P(C)\n\u00a9 NCERT\nnot to be republished\n 544\nMATHEMATICS\nIf at least one of the above is not true for three given events, we say that the\nevents are not independent Example 10 A die is thrown If E is the event \u2018the number appearing is a multiple of\n3\u2019 and F be the event \u2018the number appearing is even\u2019 then find whether E and F are\nindependent Solution We know that the sample space is S = {1, 2, 3, 4, 5, 6}\nNow\nE = { 3, 6}, F = { 2, 4, 6} and  E \u2229 F = {6}\nThen\nP(E) = 2\n1\n3\n1\n1\n, P(F)\nand P(E\n F)\n6\n3\n6\n2\n6\n=\n=\n=\n\u2229\n=\nClearly\nP(E \u2229 F) = P(E)"
    },
    {
        "Chapter": "1",
        "sentence_range": "6932-6935",
        "Text": "Example 10 A die is thrown If E is the event \u2018the number appearing is a multiple of\n3\u2019 and F be the event \u2018the number appearing is even\u2019 then find whether E and F are\nindependent Solution We know that the sample space is S = {1, 2, 3, 4, 5, 6}\nNow\nE = { 3, 6}, F = { 2, 4, 6} and  E \u2229 F = {6}\nThen\nP(E) = 2\n1\n3\n1\n1\n, P(F)\nand P(E\n F)\n6\n3\n6\n2\n6\n=\n=\n=\n\u2229\n=\nClearly\nP(E \u2229 F) = P(E) P (F)\nHence\nE and F are independent events"
    },
    {
        "Chapter": "1",
        "sentence_range": "6933-6936",
        "Text": "If E is the event \u2018the number appearing is a multiple of\n3\u2019 and F be the event \u2018the number appearing is even\u2019 then find whether E and F are\nindependent Solution We know that the sample space is S = {1, 2, 3, 4, 5, 6}\nNow\nE = { 3, 6}, F = { 2, 4, 6} and  E \u2229 F = {6}\nThen\nP(E) = 2\n1\n3\n1\n1\n, P(F)\nand P(E\n F)\n6\n3\n6\n2\n6\n=\n=\n=\n\u2229\n=\nClearly\nP(E \u2229 F) = P(E) P (F)\nHence\nE and F are independent events Example 11 An unbiased die is thrown twice"
    },
    {
        "Chapter": "1",
        "sentence_range": "6934-6937",
        "Text": "Solution We know that the sample space is S = {1, 2, 3, 4, 5, 6}\nNow\nE = { 3, 6}, F = { 2, 4, 6} and  E \u2229 F = {6}\nThen\nP(E) = 2\n1\n3\n1\n1\n, P(F)\nand P(E\n F)\n6\n3\n6\n2\n6\n=\n=\n=\n\u2229\n=\nClearly\nP(E \u2229 F) = P(E) P (F)\nHence\nE and F are independent events Example 11 An unbiased die is thrown twice Let the event A be \u2018odd number on the\nfirst throw\u2019 and B the event \u2018odd number on the second throw\u2019"
    },
    {
        "Chapter": "1",
        "sentence_range": "6935-6938",
        "Text": "P (F)\nHence\nE and F are independent events Example 11 An unbiased die is thrown twice Let the event A be \u2018odd number on the\nfirst throw\u2019 and B the event \u2018odd number on the second throw\u2019 Check the independence\nof the events A and B"
    },
    {
        "Chapter": "1",
        "sentence_range": "6936-6939",
        "Text": "Example 11 An unbiased die is thrown twice Let the event A be \u2018odd number on the\nfirst throw\u2019 and B the event \u2018odd number on the second throw\u2019 Check the independence\nof the events A and B Solution If all the 36 elementary events of the experiment are considered to be equally\nlikely, we have\nP(A) = 18\n1\n36\n=2\n and \n18\n1\nP(B)\n36\n2\n \n \nAlso\nP(A \u2229 B) = P (odd number on both throws)\n= 9\n1\n36\n4\n=\nNow\nP(A) P(B) = 1\n1\n1\n2\n2\n4\n\u00d7\n=\nClearly\nP(A \u2229 B) = P(A) \u00d7 P(B)\nThus,\nA and B are independent events\nExample 12 Three coins are tossed simultaneously"
    },
    {
        "Chapter": "1",
        "sentence_range": "6937-6940",
        "Text": "Let the event A be \u2018odd number on the\nfirst throw\u2019 and B the event \u2018odd number on the second throw\u2019 Check the independence\nof the events A and B Solution If all the 36 elementary events of the experiment are considered to be equally\nlikely, we have\nP(A) = 18\n1\n36\n=2\n and \n18\n1\nP(B)\n36\n2\n \n \nAlso\nP(A \u2229 B) = P (odd number on both throws)\n= 9\n1\n36\n4\n=\nNow\nP(A) P(B) = 1\n1\n1\n2\n2\n4\n\u00d7\n=\nClearly\nP(A \u2229 B) = P(A) \u00d7 P(B)\nThus,\nA and B are independent events\nExample 12 Three coins are tossed simultaneously Consider the event E \u2018three heads\nor three tails\u2019, F \u2018at least two heads\u2019 and G \u2018at most two heads\u2019"
    },
    {
        "Chapter": "1",
        "sentence_range": "6938-6941",
        "Text": "Check the independence\nof the events A and B Solution If all the 36 elementary events of the experiment are considered to be equally\nlikely, we have\nP(A) = 18\n1\n36\n=2\n and \n18\n1\nP(B)\n36\n2\n \n \nAlso\nP(A \u2229 B) = P (odd number on both throws)\n= 9\n1\n36\n4\n=\nNow\nP(A) P(B) = 1\n1\n1\n2\n2\n4\n\u00d7\n=\nClearly\nP(A \u2229 B) = P(A) \u00d7 P(B)\nThus,\nA and B are independent events\nExample 12 Three coins are tossed simultaneously Consider the event E \u2018three heads\nor three tails\u2019, F \u2018at least two heads\u2019 and G \u2018at most two heads\u2019 Of the pairs (E,F),\n(E,G) and (F,G), which are independent"
    },
    {
        "Chapter": "1",
        "sentence_range": "6939-6942",
        "Text": "Solution If all the 36 elementary events of the experiment are considered to be equally\nlikely, we have\nP(A) = 18\n1\n36\n=2\n and \n18\n1\nP(B)\n36\n2\n \n \nAlso\nP(A \u2229 B) = P (odd number on both throws)\n= 9\n1\n36\n4\n=\nNow\nP(A) P(B) = 1\n1\n1\n2\n2\n4\n\u00d7\n=\nClearly\nP(A \u2229 B) = P(A) \u00d7 P(B)\nThus,\nA and B are independent events\nExample 12 Three coins are tossed simultaneously Consider the event E \u2018three heads\nor three tails\u2019, F \u2018at least two heads\u2019 and G \u2018at most two heads\u2019 Of the pairs (E,F),\n(E,G) and (F,G), which are independent which are dependent"
    },
    {
        "Chapter": "1",
        "sentence_range": "6940-6943",
        "Text": "Consider the event E \u2018three heads\nor three tails\u2019, F \u2018at least two heads\u2019 and G \u2018at most two heads\u2019 Of the pairs (E,F),\n(E,G) and (F,G), which are independent which are dependent Solution The sample space of the experiment is given by\nS = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}\nClearly\nE = {HHH, TTT}, F= {HHH, HHT, HTH, THH}\n\u00a9 NCERT\nnot to be republished\nPROBABILITY      545\nand\nG = {HHT, HTH, THH, HTT, THT, TTH, TTT}\nAlso\nE \u2229 F = {HHH}, E \u2229 G = {TTT}, F \u2229 G = { HHT, HTH, THH}\nTherefore\nP(E) = 2\n1\n4\n1\n7\n, P(F)\n, P(G)\n8\n4\n8\n2\n8\n=\n=\n=\n=\nand\nP(E\u2229F) = 1\n1\n3\n, P(E\nG)\n, P(F\nG)\n8\n8\n8\n\u2229\n=\n\u2229\n=\nAlso\nP(E)"
    },
    {
        "Chapter": "1",
        "sentence_range": "6941-6944",
        "Text": "Of the pairs (E,F),\n(E,G) and (F,G), which are independent which are dependent Solution The sample space of the experiment is given by\nS = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}\nClearly\nE = {HHH, TTT}, F= {HHH, HHT, HTH, THH}\n\u00a9 NCERT\nnot to be republished\nPROBABILITY      545\nand\nG = {HHT, HTH, THH, HTT, THT, TTH, TTT}\nAlso\nE \u2229 F = {HHH}, E \u2229 G = {TTT}, F \u2229 G = { HHT, HTH, THH}\nTherefore\nP(E) = 2\n1\n4\n1\n7\n, P(F)\n, P(G)\n8\n4\n8\n2\n8\n=\n=\n=\n=\nand\nP(E\u2229F) = 1\n1\n3\n, P(E\nG)\n, P(F\nG)\n8\n8\n8\n\u2229\n=\n\u2229\n=\nAlso\nP(E) P(F) = 1\n1\n1\n1\n7\n7\n, P(E) P(G)\n4\n2\n8\n4\n8\n32\n \n \n \n \n \n \nand\nP(F)"
    },
    {
        "Chapter": "1",
        "sentence_range": "6942-6945",
        "Text": "which are dependent Solution The sample space of the experiment is given by\nS = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}\nClearly\nE = {HHH, TTT}, F= {HHH, HHT, HTH, THH}\n\u00a9 NCERT\nnot to be republished\nPROBABILITY      545\nand\nG = {HHT, HTH, THH, HTT, THT, TTH, TTT}\nAlso\nE \u2229 F = {HHH}, E \u2229 G = {TTT}, F \u2229 G = { HHT, HTH, THH}\nTherefore\nP(E) = 2\n1\n4\n1\n7\n, P(F)\n, P(G)\n8\n4\n8\n2\n8\n=\n=\n=\n=\nand\nP(E\u2229F) = 1\n1\n3\n, P(E\nG)\n, P(F\nG)\n8\n8\n8\n\u2229\n=\n\u2229\n=\nAlso\nP(E) P(F) = 1\n1\n1\n1\n7\n7\n, P(E) P(G)\n4\n2\n8\n4\n8\n32\n \n \n \n \n \n \nand\nP(F) P(G) = 1\n7\n7\n2\n8\n16\n \n \nThus\nP(E \u2229 F) = P(E)"
    },
    {
        "Chapter": "1",
        "sentence_range": "6943-6946",
        "Text": "Solution The sample space of the experiment is given by\nS = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}\nClearly\nE = {HHH, TTT}, F= {HHH, HHT, HTH, THH}\n\u00a9 NCERT\nnot to be republished\nPROBABILITY      545\nand\nG = {HHT, HTH, THH, HTT, THT, TTH, TTT}\nAlso\nE \u2229 F = {HHH}, E \u2229 G = {TTT}, F \u2229 G = { HHT, HTH, THH}\nTherefore\nP(E) = 2\n1\n4\n1\n7\n, P(F)\n, P(G)\n8\n4\n8\n2\n8\n=\n=\n=\n=\nand\nP(E\u2229F) = 1\n1\n3\n, P(E\nG)\n, P(F\nG)\n8\n8\n8\n\u2229\n=\n\u2229\n=\nAlso\nP(E) P(F) = 1\n1\n1\n1\n7\n7\n, P(E) P(G)\n4\n2\n8\n4\n8\n32\n \n \n \n \n \n \nand\nP(F) P(G) = 1\n7\n7\n2\n8\n16\n \n \nThus\nP(E \u2229 F) = P(E) P(F)\nP(E \u2229 G) \u2260 P(E)"
    },
    {
        "Chapter": "1",
        "sentence_range": "6944-6947",
        "Text": "P(F) = 1\n1\n1\n1\n7\n7\n, P(E) P(G)\n4\n2\n8\n4\n8\n32\n \n \n \n \n \n \nand\nP(F) P(G) = 1\n7\n7\n2\n8\n16\n \n \nThus\nP(E \u2229 F) = P(E) P(F)\nP(E \u2229 G) \u2260 P(E) P(G)\nand\nP(F \u2229 G) \u2260 P (F)"
    },
    {
        "Chapter": "1",
        "sentence_range": "6945-6948",
        "Text": "P(G) = 1\n7\n7\n2\n8\n16\n \n \nThus\nP(E \u2229 F) = P(E) P(F)\nP(E \u2229 G) \u2260 P(E) P(G)\nand\nP(F \u2229 G) \u2260 P (F) P(G)\nHence, the events (E and F) are independent, and the events (E and G) and\n(F and G) are dependent"
    },
    {
        "Chapter": "1",
        "sentence_range": "6946-6949",
        "Text": "P(F)\nP(E \u2229 G) \u2260 P(E) P(G)\nand\nP(F \u2229 G) \u2260 P (F) P(G)\nHence, the events (E and F) are independent, and the events (E and G) and\n(F and G) are dependent Example 13 Prove that if E and F are independent events, then so are the events\nE and F\u2032"
    },
    {
        "Chapter": "1",
        "sentence_range": "6947-6950",
        "Text": "P(G)\nand\nP(F \u2229 G) \u2260 P (F) P(G)\nHence, the events (E and F) are independent, and the events (E and G) and\n(F and G) are dependent Example 13 Prove that if E and F are independent events, then so are the events\nE and F\u2032 Solution Since E and F are independent, we have\nP(E \u2229 F) = P(E)"
    },
    {
        "Chapter": "1",
        "sentence_range": "6948-6951",
        "Text": "P(G)\nHence, the events (E and F) are independent, and the events (E and G) and\n(F and G) are dependent Example 13 Prove that if E and F are independent events, then so are the events\nE and F\u2032 Solution Since E and F are independent, we have\nP(E \u2229 F) = P(E) P(F)"
    },
    {
        "Chapter": "1",
        "sentence_range": "6949-6952",
        "Text": "Example 13 Prove that if E and F are independent events, then so are the events\nE and F\u2032 Solution Since E and F are independent, we have\nP(E \u2229 F) = P(E) P(F) (1)\nFrom the venn diagram in Fig 13"
    },
    {
        "Chapter": "1",
        "sentence_range": "6950-6953",
        "Text": "Solution Since E and F are independent, we have\nP(E \u2229 F) = P(E) P(F) (1)\nFrom the venn diagram in Fig 13 3, it is clear\nthat E \u2229 F and E \u2229 F\u2032 are mutually exclusive events\nand also  E =(E \u2229 F) \u222a (E \u2229 F\u2032)"
    },
    {
        "Chapter": "1",
        "sentence_range": "6951-6954",
        "Text": "P(F) (1)\nFrom the venn diagram in Fig 13 3, it is clear\nthat E \u2229 F and E \u2229 F\u2032 are mutually exclusive events\nand also  E =(E \u2229 F) \u222a (E \u2229 F\u2032) Therefore\nP(E) = P(E \u2229 F) + P(E \u2229 F\u2032)\nor\nP(E \u2229 F\u2032) = P(E) \u2212 P(E \u2229 F)\n= P(E) \u2212 P(E)"
    },
    {
        "Chapter": "1",
        "sentence_range": "6952-6955",
        "Text": "(1)\nFrom the venn diagram in Fig 13 3, it is clear\nthat E \u2229 F and E \u2229 F\u2032 are mutually exclusive events\nand also  E =(E \u2229 F) \u222a (E \u2229 F\u2032) Therefore\nP(E) = P(E \u2229 F) + P(E \u2229 F\u2032)\nor\nP(E \u2229 F\u2032) = P(E) \u2212 P(E \u2229 F)\n= P(E) \u2212 P(E) P(F)\n(by  (1))\n= P(E) (1\u2212P(F))\n= P(E)"
    },
    {
        "Chapter": "1",
        "sentence_range": "6953-6956",
        "Text": "3, it is clear\nthat E \u2229 F and E \u2229 F\u2032 are mutually exclusive events\nand also  E =(E \u2229 F) \u222a (E \u2229 F\u2032) Therefore\nP(E) = P(E \u2229 F) + P(E \u2229 F\u2032)\nor\nP(E \u2229 F\u2032) = P(E) \u2212 P(E \u2229 F)\n= P(E) \u2212 P(E) P(F)\n(by  (1))\n= P(E) (1\u2212P(F))\n= P(E) P(F\u2032)\nHence,  E and F\u2032 are independent\n(E\n\u2229F )\u2019\n(E\nF)\n\u2019\u2229\nE\nF\nS\n(E\nF)\n\u2229\n(E\nF )\n\u2019\n\u2019\n\u2229\nFig 13"
    },
    {
        "Chapter": "1",
        "sentence_range": "6954-6957",
        "Text": "Therefore\nP(E) = P(E \u2229 F) + P(E \u2229 F\u2032)\nor\nP(E \u2229 F\u2032) = P(E) \u2212 P(E \u2229 F)\n= P(E) \u2212 P(E) P(F)\n(by  (1))\n= P(E) (1\u2212P(F))\n= P(E) P(F\u2032)\nHence,  E and F\u2032 are independent\n(E\n\u2229F )\u2019\n(E\nF)\n\u2019\u2229\nE\nF\nS\n(E\nF)\n\u2229\n(E\nF )\n\u2019\n\u2019\n\u2229\nFig 13 3\n\u00a9 NCERT\nnot to be republished\n 546\nMATHEMATICS\n\ufffdNote In a similar manner, it can be shown that if the events E and F are\nindependent, then\n(a)\nE\u2032 and F are independent,\n(b)\nE\u2032 and F\u2032 are independent\nExample 14 If A and B are two independent events, then the probability of occurrence\nof at least one of A and B is given by 1\u2013 P(A\u2032) P(B\u2032)\nSolution We have\nP(at least one of A and B) = P(A \u222a B)\n= P(A) + P(B) \u2212 P(A \u2229 B)\n= P(A) + P(B) \u2212 P(A) P(B)\n= P(A) + P(B) [1\u2212P(A)]\n= P(A)  +  P(B)"
    },
    {
        "Chapter": "1",
        "sentence_range": "6955-6958",
        "Text": "P(F)\n(by  (1))\n= P(E) (1\u2212P(F))\n= P(E) P(F\u2032)\nHence,  E and F\u2032 are independent\n(E\n\u2229F )\u2019\n(E\nF)\n\u2019\u2229\nE\nF\nS\n(E\nF)\n\u2229\n(E\nF )\n\u2019\n\u2019\n\u2229\nFig 13 3\n\u00a9 NCERT\nnot to be republished\n 546\nMATHEMATICS\n\ufffdNote In a similar manner, it can be shown that if the events E and F are\nindependent, then\n(a)\nE\u2032 and F are independent,\n(b)\nE\u2032 and F\u2032 are independent\nExample 14 If A and B are two independent events, then the probability of occurrence\nof at least one of A and B is given by 1\u2013 P(A\u2032) P(B\u2032)\nSolution We have\nP(at least one of A and B) = P(A \u222a B)\n= P(A) + P(B) \u2212 P(A \u2229 B)\n= P(A) + P(B) \u2212 P(A) P(B)\n= P(A) + P(B) [1\u2212P(A)]\n= P(A)  +  P(B) P(A\u2032)\n= 1\u2212 P(A\u2032) + P(B) P(A\u2032)\n= 1\u2212 P(A\u2032)  [1\u2212 P(B)]\n= 1\u2212 P(A\u2032) P (B\u2032)\nEXERCISE 13"
    },
    {
        "Chapter": "1",
        "sentence_range": "6956-6959",
        "Text": "P(F\u2032)\nHence,  E and F\u2032 are independent\n(E\n\u2229F )\u2019\n(E\nF)\n\u2019\u2229\nE\nF\nS\n(E\nF)\n\u2229\n(E\nF )\n\u2019\n\u2019\n\u2229\nFig 13 3\n\u00a9 NCERT\nnot to be republished\n 546\nMATHEMATICS\n\ufffdNote In a similar manner, it can be shown that if the events E and F are\nindependent, then\n(a)\nE\u2032 and F are independent,\n(b)\nE\u2032 and F\u2032 are independent\nExample 14 If A and B are two independent events, then the probability of occurrence\nof at least one of A and B is given by 1\u2013 P(A\u2032) P(B\u2032)\nSolution We have\nP(at least one of A and B) = P(A \u222a B)\n= P(A) + P(B) \u2212 P(A \u2229 B)\n= P(A) + P(B) \u2212 P(A) P(B)\n= P(A) + P(B) [1\u2212P(A)]\n= P(A)  +  P(B) P(A\u2032)\n= 1\u2212 P(A\u2032) + P(B) P(A\u2032)\n= 1\u2212 P(A\u2032)  [1\u2212 P(B)]\n= 1\u2212 P(A\u2032) P (B\u2032)\nEXERCISE 13 2\n1"
    },
    {
        "Chapter": "1",
        "sentence_range": "6957-6960",
        "Text": "3\n\u00a9 NCERT\nnot to be republished\n 546\nMATHEMATICS\n\ufffdNote In a similar manner, it can be shown that if the events E and F are\nindependent, then\n(a)\nE\u2032 and F are independent,\n(b)\nE\u2032 and F\u2032 are independent\nExample 14 If A and B are two independent events, then the probability of occurrence\nof at least one of A and B is given by 1\u2013 P(A\u2032) P(B\u2032)\nSolution We have\nP(at least one of A and B) = P(A \u222a B)\n= P(A) + P(B) \u2212 P(A \u2229 B)\n= P(A) + P(B) \u2212 P(A) P(B)\n= P(A) + P(B) [1\u2212P(A)]\n= P(A)  +  P(B) P(A\u2032)\n= 1\u2212 P(A\u2032) + P(B) P(A\u2032)\n= 1\u2212 P(A\u2032)  [1\u2212 P(B)]\n= 1\u2212 P(A\u2032) P (B\u2032)\nEXERCISE 13 2\n1 If  P(A) \n 53\n and P (B) \n 51\n, find P (A  \u2229 B) if A and B are independent events"
    },
    {
        "Chapter": "1",
        "sentence_range": "6958-6961",
        "Text": "P(A\u2032)\n= 1\u2212 P(A\u2032) + P(B) P(A\u2032)\n= 1\u2212 P(A\u2032)  [1\u2212 P(B)]\n= 1\u2212 P(A\u2032) P (B\u2032)\nEXERCISE 13 2\n1 If  P(A) \n 53\n and P (B) \n 51\n, find P (A  \u2229 B) if A and B are independent events 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "6959-6962",
        "Text": "2\n1 If  P(A) \n 53\n and P (B) \n 51\n, find P (A  \u2229 B) if A and B are independent events 2 Two cards are drawn at random and without replacement from a pack of 52\nplaying cards"
    },
    {
        "Chapter": "1",
        "sentence_range": "6960-6963",
        "Text": "If  P(A) \n 53\n and P (B) \n 51\n, find P (A  \u2229 B) if A and B are independent events 2 Two cards are drawn at random and without replacement from a pack of 52\nplaying cards Find the probability that both the cards are black"
    },
    {
        "Chapter": "1",
        "sentence_range": "6961-6964",
        "Text": "2 Two cards are drawn at random and without replacement from a pack of 52\nplaying cards Find the probability that both the cards are black 3"
    },
    {
        "Chapter": "1",
        "sentence_range": "6962-6965",
        "Text": "Two cards are drawn at random and without replacement from a pack of 52\nplaying cards Find the probability that both the cards are black 3 A box of oranges is inspected by examining three randomly selected oranges\ndrawn without replacement"
    },
    {
        "Chapter": "1",
        "sentence_range": "6963-6966",
        "Text": "Find the probability that both the cards are black 3 A box of oranges is inspected by examining three randomly selected oranges\ndrawn without replacement If all the three oranges are good, the box is approved\nfor sale, otherwise, it is rejected"
    },
    {
        "Chapter": "1",
        "sentence_range": "6964-6967",
        "Text": "3 A box of oranges is inspected by examining three randomly selected oranges\ndrawn without replacement If all the three oranges are good, the box is approved\nfor sale, otherwise, it is rejected Find the probability that a box containing 15\noranges out of which 12 are  good and 3 are bad ones will be approved for sale"
    },
    {
        "Chapter": "1",
        "sentence_range": "6965-6968",
        "Text": "A box of oranges is inspected by examining three randomly selected oranges\ndrawn without replacement If all the three oranges are good, the box is approved\nfor sale, otherwise, it is rejected Find the probability that a box containing 15\noranges out of which 12 are  good and 3 are bad ones will be approved for sale 4"
    },
    {
        "Chapter": "1",
        "sentence_range": "6966-6969",
        "Text": "If all the three oranges are good, the box is approved\nfor sale, otherwise, it is rejected Find the probability that a box containing 15\noranges out of which 12 are  good and 3 are bad ones will be approved for sale 4 A fair coin and an unbiased die are tossed"
    },
    {
        "Chapter": "1",
        "sentence_range": "6967-6970",
        "Text": "Find the probability that a box containing 15\noranges out of which 12 are  good and 3 are bad ones will be approved for sale 4 A fair coin and an unbiased die are tossed Let A be the event \u2018head appears on\nthe coin\u2019 and B be the event \u20183 on the die\u2019"
    },
    {
        "Chapter": "1",
        "sentence_range": "6968-6971",
        "Text": "4 A fair coin and an unbiased die are tossed Let A be the event \u2018head appears on\nthe coin\u2019 and B be the event \u20183 on the die\u2019 Check whether A and B are\nindependent events or not"
    },
    {
        "Chapter": "1",
        "sentence_range": "6969-6972",
        "Text": "A fair coin and an unbiased die are tossed Let A be the event \u2018head appears on\nthe coin\u2019 and B be the event \u20183 on the die\u2019 Check whether A and B are\nindependent events or not 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "6970-6973",
        "Text": "Let A be the event \u2018head appears on\nthe coin\u2019 and B be the event \u20183 on the die\u2019 Check whether A and B are\nindependent events or not 5 A die marked 1, 2, 3 in red and 4, 5, 6 in green is tossed"
    },
    {
        "Chapter": "1",
        "sentence_range": "6971-6974",
        "Text": "Check whether A and B are\nindependent events or not 5 A die marked 1, 2, 3 in red and 4, 5, 6 in green is tossed Let A be the event,\n\u2018the number is even,\u2019 and B be the event, \u2018the number is red\u2019"
    },
    {
        "Chapter": "1",
        "sentence_range": "6972-6975",
        "Text": "5 A die marked 1, 2, 3 in red and 4, 5, 6 in green is tossed Let A be the event,\n\u2018the number is even,\u2019 and B be the event, \u2018the number is red\u2019 Are A and B\nindependent"
    },
    {
        "Chapter": "1",
        "sentence_range": "6973-6976",
        "Text": "A die marked 1, 2, 3 in red and 4, 5, 6 in green is tossed Let A be the event,\n\u2018the number is even,\u2019 and B be the event, \u2018the number is red\u2019 Are A and B\nindependent 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "6974-6977",
        "Text": "Let A be the event,\n\u2018the number is even,\u2019 and B be the event, \u2018the number is red\u2019 Are A and B\nindependent 6 Let E and F be events with P(E) \n 53\n, P(F) \n=103\n and P (E \u2229 F) = 1\n5"
    },
    {
        "Chapter": "1",
        "sentence_range": "6975-6978",
        "Text": "Are A and B\nindependent 6 Let E and F be events with P(E) \n 53\n, P(F) \n=103\n and P (E \u2229 F) = 1\n5 Are\nE and F independent"
    },
    {
        "Chapter": "1",
        "sentence_range": "6976-6979",
        "Text": "6 Let E and F be events with P(E) \n 53\n, P(F) \n=103\n and P (E \u2229 F) = 1\n5 Are\nE and F independent \u00a9 NCERT\nnot to be republished\nPROBABILITY      547\n7"
    },
    {
        "Chapter": "1",
        "sentence_range": "6977-6980",
        "Text": "Let E and F be events with P(E) \n 53\n, P(F) \n=103\n and P (E \u2229 F) = 1\n5 Are\nE and F independent \u00a9 NCERT\nnot to be republished\nPROBABILITY      547\n7 Given that the events A and B are such that P(A) = 1\n2 ,  P(A \u222a B) = 3\n5  and\nP(B) = p"
    },
    {
        "Chapter": "1",
        "sentence_range": "6978-6981",
        "Text": "Are\nE and F independent \u00a9 NCERT\nnot to be republished\nPROBABILITY      547\n7 Given that the events A and B are such that P(A) = 1\n2 ,  P(A \u222a B) = 3\n5  and\nP(B) = p Find p if they are (i) mutually exclusive (ii) independent"
    },
    {
        "Chapter": "1",
        "sentence_range": "6979-6982",
        "Text": "\u00a9 NCERT\nnot to be republished\nPROBABILITY      547\n7 Given that the events A and B are such that P(A) = 1\n2 ,  P(A \u222a B) = 3\n5  and\nP(B) = p Find p if they are (i) mutually exclusive (ii) independent 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "6980-6983",
        "Text": "Given that the events A and B are such that P(A) = 1\n2 ,  P(A \u222a B) = 3\n5  and\nP(B) = p Find p if they are (i) mutually exclusive (ii) independent 8 Let A and B be independent events with P(A) = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "6981-6984",
        "Text": "Find p if they are (i) mutually exclusive (ii) independent 8 Let A and B be independent events with P(A) = 0 3 and P(B) = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "6982-6985",
        "Text": "8 Let A and B be independent events with P(A) = 0 3 and P(B) = 0 4"
    },
    {
        "Chapter": "1",
        "sentence_range": "6983-6986",
        "Text": "Let A and B be independent events with P(A) = 0 3 and P(B) = 0 4 Find\n(i) P(A \u2229 B)\n(ii) P(A \u222a B)\n(iii) P(A|B)\n(iv) P(B|A)\n9"
    },
    {
        "Chapter": "1",
        "sentence_range": "6984-6987",
        "Text": "3 and P(B) = 0 4 Find\n(i) P(A \u2229 B)\n(ii) P(A \u222a B)\n(iii) P(A|B)\n(iv) P(B|A)\n9 If A and B are two events such that  P(A) = 1\n4 , P (B) = 1\n2  and  P(A \u2229 B) = 1\n8 ,\nfind P (not A and not B)"
    },
    {
        "Chapter": "1",
        "sentence_range": "6985-6988",
        "Text": "4 Find\n(i) P(A \u2229 B)\n(ii) P(A \u222a B)\n(iii) P(A|B)\n(iv) P(B|A)\n9 If A and B are two events such that  P(A) = 1\n4 , P (B) = 1\n2  and  P(A \u2229 B) = 1\n8 ,\nfind P (not A and not B) 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "6986-6989",
        "Text": "Find\n(i) P(A \u2229 B)\n(ii) P(A \u222a B)\n(iii) P(A|B)\n(iv) P(B|A)\n9 If A and B are two events such that  P(A) = 1\n4 , P (B) = 1\n2  and  P(A \u2229 B) = 1\n8 ,\nfind P (not A and not B) 10 Events A and B are such that P (A) = 1\n2 , P(B) = 7\n12  and P(not A or not B) = 1\n4"
    },
    {
        "Chapter": "1",
        "sentence_range": "6987-6990",
        "Text": "If A and B are two events such that  P(A) = 1\n4 , P (B) = 1\n2  and  P(A \u2229 B) = 1\n8 ,\nfind P (not A and not B) 10 Events A and B are such that P (A) = 1\n2 , P(B) = 7\n12  and P(not A or not B) = 1\n4 State whether A and B are independent"
    },
    {
        "Chapter": "1",
        "sentence_range": "6988-6991",
        "Text": "10 Events A and B are such that P (A) = 1\n2 , P(B) = 7\n12  and P(not A or not B) = 1\n4 State whether A and B are independent 11"
    },
    {
        "Chapter": "1",
        "sentence_range": "6989-6992",
        "Text": "Events A and B are such that P (A) = 1\n2 , P(B) = 7\n12  and P(not A or not B) = 1\n4 State whether A and B are independent 11 Given two independent events A and B such that P(A) = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "6990-6993",
        "Text": "State whether A and B are independent 11 Given two independent events A and B such that P(A) = 0 3,  P(B) = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "6991-6994",
        "Text": "11 Given two independent events A and B such that P(A) = 0 3,  P(B) = 0 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "6992-6995",
        "Text": "Given two independent events A and B such that P(A) = 0 3,  P(B) = 0 6 Find\n(i) P(A and B)\n(ii) P(A and not B)\n(iii) P(A or B)\n(iv) P(neither A nor B)\n12"
    },
    {
        "Chapter": "1",
        "sentence_range": "6993-6996",
        "Text": "3,  P(B) = 0 6 Find\n(i) P(A and B)\n(ii) P(A and not B)\n(iii) P(A or B)\n(iv) P(neither A nor B)\n12 A die is tossed thrice"
    },
    {
        "Chapter": "1",
        "sentence_range": "6994-6997",
        "Text": "6 Find\n(i) P(A and B)\n(ii) P(A and not B)\n(iii) P(A or B)\n(iv) P(neither A nor B)\n12 A die is tossed thrice Find the probability of getting an odd number at least once"
    },
    {
        "Chapter": "1",
        "sentence_range": "6995-6998",
        "Text": "Find\n(i) P(A and B)\n(ii) P(A and not B)\n(iii) P(A or B)\n(iv) P(neither A nor B)\n12 A die is tossed thrice Find the probability of getting an odd number at least once 13"
    },
    {
        "Chapter": "1",
        "sentence_range": "6996-6999",
        "Text": "A die is tossed thrice Find the probability of getting an odd number at least once 13 Two balls are drawn at random with replacement from a box containing 10 black\nand 8 red balls"
    },
    {
        "Chapter": "1",
        "sentence_range": "6997-7000",
        "Text": "Find the probability of getting an odd number at least once 13 Two balls are drawn at random with replacement from a box containing 10 black\nand 8 red balls Find the probability that\n(i) both balls are red"
    },
    {
        "Chapter": "1",
        "sentence_range": "6998-7001",
        "Text": "13 Two balls are drawn at random with replacement from a box containing 10 black\nand 8 red balls Find the probability that\n(i) both balls are red (ii) first ball is black and second is red"
    },
    {
        "Chapter": "1",
        "sentence_range": "6999-7002",
        "Text": "Two balls are drawn at random with replacement from a box containing 10 black\nand 8 red balls Find the probability that\n(i) both balls are red (ii) first ball is black and second is red (iii) one of them is black and other is red"
    },
    {
        "Chapter": "1",
        "sentence_range": "7000-7003",
        "Text": "Find the probability that\n(i) both balls are red (ii) first ball is black and second is red (iii) one of them is black and other is red 14"
    },
    {
        "Chapter": "1",
        "sentence_range": "7001-7004",
        "Text": "(ii) first ball is black and second is red (iii) one of them is black and other is red 14 Probability of solving specific problem independently by A and B are 1\n2  and 1\n3\nrespectively"
    },
    {
        "Chapter": "1",
        "sentence_range": "7002-7005",
        "Text": "(iii) one of them is black and other is red 14 Probability of solving specific problem independently by A and B are 1\n2  and 1\n3\nrespectively If both try to solve the problem independently, find the probability\nthat\n(i) the problem is solved\n(ii) exactly one of them solves the problem"
    },
    {
        "Chapter": "1",
        "sentence_range": "7003-7006",
        "Text": "14 Probability of solving specific problem independently by A and B are 1\n2  and 1\n3\nrespectively If both try to solve the problem independently, find the probability\nthat\n(i) the problem is solved\n(ii) exactly one of them solves the problem 15"
    },
    {
        "Chapter": "1",
        "sentence_range": "7004-7007",
        "Text": "Probability of solving specific problem independently by A and B are 1\n2  and 1\n3\nrespectively If both try to solve the problem independently, find the probability\nthat\n(i) the problem is solved\n(ii) exactly one of them solves the problem 15 One card is drawn at random from a well shuffled deck of 52 cards"
    },
    {
        "Chapter": "1",
        "sentence_range": "7005-7008",
        "Text": "If both try to solve the problem independently, find the probability\nthat\n(i) the problem is solved\n(ii) exactly one of them solves the problem 15 One card is drawn at random from a well shuffled deck of 52 cards In which of\nthe following cases are the events E and F independent"
    },
    {
        "Chapter": "1",
        "sentence_range": "7006-7009",
        "Text": "15 One card is drawn at random from a well shuffled deck of 52 cards In which of\nthe following cases are the events E and F independent (i) E : \u2018the card drawn is a spade\u2019\nF : \u2018the card drawn is an ace\u2019\n(ii) E : \u2018the card drawn is black\u2019\nF : \u2018the card drawn is a king\u2019\n(iii) E : \u2018the card drawn is a king or queen\u2019\nF : \u2018the card drawn is a queen or jack\u2019"
    },
    {
        "Chapter": "1",
        "sentence_range": "7007-7010",
        "Text": "One card is drawn at random from a well shuffled deck of 52 cards In which of\nthe following cases are the events E and F independent (i) E : \u2018the card drawn is a spade\u2019\nF : \u2018the card drawn is an ace\u2019\n(ii) E : \u2018the card drawn is black\u2019\nF : \u2018the card drawn is a king\u2019\n(iii) E : \u2018the card drawn is a king or queen\u2019\nF : \u2018the card drawn is a queen or jack\u2019 \u00a9 NCERT\nnot to be republished\n 548\nMATHEMATICS\n16"
    },
    {
        "Chapter": "1",
        "sentence_range": "7008-7011",
        "Text": "In which of\nthe following cases are the events E and F independent (i) E : \u2018the card drawn is a spade\u2019\nF : \u2018the card drawn is an ace\u2019\n(ii) E : \u2018the card drawn is black\u2019\nF : \u2018the card drawn is a king\u2019\n(iii) E : \u2018the card drawn is a king or queen\u2019\nF : \u2018the card drawn is a queen or jack\u2019 \u00a9 NCERT\nnot to be republished\n 548\nMATHEMATICS\n16 In a hostel, 60% of the students read Hindi news paper, 40% read English news\npaper and 20% read both Hindi and English news papers"
    },
    {
        "Chapter": "1",
        "sentence_range": "7009-7012",
        "Text": "(i) E : \u2018the card drawn is a spade\u2019\nF : \u2018the card drawn is an ace\u2019\n(ii) E : \u2018the card drawn is black\u2019\nF : \u2018the card drawn is a king\u2019\n(iii) E : \u2018the card drawn is a king or queen\u2019\nF : \u2018the card drawn is a queen or jack\u2019 \u00a9 NCERT\nnot to be republished\n 548\nMATHEMATICS\n16 In a hostel, 60% of the students read Hindi news paper, 40% read English news\npaper and 20% read both Hindi and English news papers A student is selected\nat random"
    },
    {
        "Chapter": "1",
        "sentence_range": "7010-7013",
        "Text": "\u00a9 NCERT\nnot to be republished\n 548\nMATHEMATICS\n16 In a hostel, 60% of the students read Hindi news paper, 40% read English news\npaper and 20% read both Hindi and English news papers A student is selected\nat random (a) Find the probability that she reads neither Hindi nor English news papers"
    },
    {
        "Chapter": "1",
        "sentence_range": "7011-7014",
        "Text": "In a hostel, 60% of the students read Hindi news paper, 40% read English news\npaper and 20% read both Hindi and English news papers A student is selected\nat random (a) Find the probability that she reads neither Hindi nor English news papers (b) If she reads Hindi news paper, find the probability that she reads English\nnews paper"
    },
    {
        "Chapter": "1",
        "sentence_range": "7012-7015",
        "Text": "A student is selected\nat random (a) Find the probability that she reads neither Hindi nor English news papers (b) If she reads Hindi news paper, find the probability that she reads English\nnews paper (c) If she reads English news paper, find the probability that she reads Hindi\nnews paper"
    },
    {
        "Chapter": "1",
        "sentence_range": "7013-7016",
        "Text": "(a) Find the probability that she reads neither Hindi nor English news papers (b) If she reads Hindi news paper, find the probability that she reads English\nnews paper (c) If she reads English news paper, find the probability that she reads Hindi\nnews paper Choose the correct answer in Exercises 17 and 18"
    },
    {
        "Chapter": "1",
        "sentence_range": "7014-7017",
        "Text": "(b) If she reads Hindi news paper, find the probability that she reads English\nnews paper (c) If she reads English news paper, find the probability that she reads Hindi\nnews paper Choose the correct answer in Exercises 17 and 18 17"
    },
    {
        "Chapter": "1",
        "sentence_range": "7015-7018",
        "Text": "(c) If she reads English news paper, find the probability that she reads Hindi\nnews paper Choose the correct answer in Exercises 17 and 18 17 The probability of obtaining an even prime number on each die, when a pair of\ndice is rolled is\n(A) 0\n(B) 1\n3\n(C)\n121\n(D)\n1\n36\n18"
    },
    {
        "Chapter": "1",
        "sentence_range": "7016-7019",
        "Text": "Choose the correct answer in Exercises 17 and 18 17 The probability of obtaining an even prime number on each die, when a pair of\ndice is rolled is\n(A) 0\n(B) 1\n3\n(C)\n121\n(D)\n1\n36\n18 Two events A and B will be independent, if\n(A) A and B are mutually exclusive\n(B) P(A\u2032B\u2032) = [1 \u2013 P(A)] [1 \u2013 P(B)]\n(C) P(A) = P(B)\n(D) P(A) + P(B) = 1\n13"
    },
    {
        "Chapter": "1",
        "sentence_range": "7017-7020",
        "Text": "17 The probability of obtaining an even prime number on each die, when a pair of\ndice is rolled is\n(A) 0\n(B) 1\n3\n(C)\n121\n(D)\n1\n36\n18 Two events A and B will be independent, if\n(A) A and B are mutually exclusive\n(B) P(A\u2032B\u2032) = [1 \u2013 P(A)] [1 \u2013 P(B)]\n(C) P(A) = P(B)\n(D) P(A) + P(B) = 1\n13 5  Bayes' Theorem\nConsider that there are two bags I and II"
    },
    {
        "Chapter": "1",
        "sentence_range": "7018-7021",
        "Text": "The probability of obtaining an even prime number on each die, when a pair of\ndice is rolled is\n(A) 0\n(B) 1\n3\n(C)\n121\n(D)\n1\n36\n18 Two events A and B will be independent, if\n(A) A and B are mutually exclusive\n(B) P(A\u2032B\u2032) = [1 \u2013 P(A)] [1 \u2013 P(B)]\n(C) P(A) = P(B)\n(D) P(A) + P(B) = 1\n13 5  Bayes' Theorem\nConsider that there are two bags I and II Bag I contains 2 white and 3 red balls and\nBag II contains 4 white and 5 red balls"
    },
    {
        "Chapter": "1",
        "sentence_range": "7019-7022",
        "Text": "Two events A and B will be independent, if\n(A) A and B are mutually exclusive\n(B) P(A\u2032B\u2032) = [1 \u2013 P(A)] [1 \u2013 P(B)]\n(C) P(A) = P(B)\n(D) P(A) + P(B) = 1\n13 5  Bayes' Theorem\nConsider that there are two bags I and II Bag I contains 2 white and 3 red balls and\nBag II contains 4 white and 5 red balls One ball is drawn at random from one of the\nbags"
    },
    {
        "Chapter": "1",
        "sentence_range": "7020-7023",
        "Text": "5  Bayes' Theorem\nConsider that there are two bags I and II Bag I contains 2 white and 3 red balls and\nBag II contains 4 white and 5 red balls One ball is drawn at random from one of the\nbags We can find the probability of selecting any of the bags (i"
    },
    {
        "Chapter": "1",
        "sentence_range": "7021-7024",
        "Text": "Bag I contains 2 white and 3 red balls and\nBag II contains 4 white and 5 red balls One ball is drawn at random from one of the\nbags We can find the probability of selecting any of the bags (i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "7022-7025",
        "Text": "One ball is drawn at random from one of the\nbags We can find the probability of selecting any of the bags (i e 1\n2 ) or probability of\ndrawing a ball of a particular colour (say white) from a particular bag (say Bag I)"
    },
    {
        "Chapter": "1",
        "sentence_range": "7023-7026",
        "Text": "We can find the probability of selecting any of the bags (i e 1\n2 ) or probability of\ndrawing a ball of a particular colour (say white) from a particular bag (say Bag I) In\nother words, we can find the probability that the ball drawn is of a particular colour, if\nwe are given the bag from which the ball is drawn"
    },
    {
        "Chapter": "1",
        "sentence_range": "7024-7027",
        "Text": "e 1\n2 ) or probability of\ndrawing a ball of a particular colour (say white) from a particular bag (say Bag I) In\nother words, we can find the probability that the ball drawn is of a particular colour, if\nwe are given the bag from which the ball is drawn But, can we find the probability that\nthe ball drawn is from a particular bag (say Bag II), if the colour of the ball drawn is\ngiven"
    },
    {
        "Chapter": "1",
        "sentence_range": "7025-7028",
        "Text": "1\n2 ) or probability of\ndrawing a ball of a particular colour (say white) from a particular bag (say Bag I) In\nother words, we can find the probability that the ball drawn is of a particular colour, if\nwe are given the bag from which the ball is drawn But, can we find the probability that\nthe ball drawn is from a particular bag (say Bag II), if the colour of the ball drawn is\ngiven Here, we have to find the reverse probability of Bag II to be selected when an\nevent occurred after it is known"
    },
    {
        "Chapter": "1",
        "sentence_range": "7026-7029",
        "Text": "In\nother words, we can find the probability that the ball drawn is of a particular colour, if\nwe are given the bag from which the ball is drawn But, can we find the probability that\nthe ball drawn is from a particular bag (say Bag II), if the colour of the ball drawn is\ngiven Here, we have to find the reverse probability of Bag II to be selected when an\nevent occurred after it is known Famous mathematician, John Bayes' solved the problem\nof finding reverse probability by using conditional probability"
    },
    {
        "Chapter": "1",
        "sentence_range": "7027-7030",
        "Text": "But, can we find the probability that\nthe ball drawn is from a particular bag (say Bag II), if the colour of the ball drawn is\ngiven Here, we have to find the reverse probability of Bag II to be selected when an\nevent occurred after it is known Famous mathematician, John Bayes' solved the problem\nof finding reverse probability by using conditional probability The formula developed\nby him is known as \u2018Bayes theorem\u2019 which was published posthumously in 1763"
    },
    {
        "Chapter": "1",
        "sentence_range": "7028-7031",
        "Text": "Here, we have to find the reverse probability of Bag II to be selected when an\nevent occurred after it is known Famous mathematician, John Bayes' solved the problem\nof finding reverse probability by using conditional probability The formula developed\nby him is known as \u2018Bayes theorem\u2019 which was published posthumously in 1763 Before stating and proving the Bayes' theorem, let us first take up a definition and\nsome preliminary results"
    },
    {
        "Chapter": "1",
        "sentence_range": "7029-7032",
        "Text": "Famous mathematician, John Bayes' solved the problem\nof finding reverse probability by using conditional probability The formula developed\nby him is known as \u2018Bayes theorem\u2019 which was published posthumously in 1763 Before stating and proving the Bayes' theorem, let us first take up a definition and\nsome preliminary results 13"
    },
    {
        "Chapter": "1",
        "sentence_range": "7030-7033",
        "Text": "The formula developed\nby him is known as \u2018Bayes theorem\u2019 which was published posthumously in 1763 Before stating and proving the Bayes' theorem, let us first take up a definition and\nsome preliminary results 13 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "7031-7034",
        "Text": "Before stating and proving the Bayes' theorem, let us first take up a definition and\nsome preliminary results 13 5 1 Partition of a sample space\nA set of events E1, E2,"
    },
    {
        "Chapter": "1",
        "sentence_range": "7032-7035",
        "Text": "13 5 1 Partition of a sample space\nA set of events E1, E2, , En is said to represent a partition of the sample space S if\n(a) Ei \u2229 Ej = \u03c6, i \u2260 j, i, j = 1, 2, 3,"
    },
    {
        "Chapter": "1",
        "sentence_range": "7033-7036",
        "Text": "5 1 Partition of a sample space\nA set of events E1, E2, , En is said to represent a partition of the sample space S if\n(a) Ei \u2229 Ej = \u03c6, i \u2260 j, i, j = 1, 2, 3, , n\n\u00a9 NCERT\nnot to be republished\nPROBABILITY      549\nFig 13"
    },
    {
        "Chapter": "1",
        "sentence_range": "7034-7037",
        "Text": "1 Partition of a sample space\nA set of events E1, E2, , En is said to represent a partition of the sample space S if\n(a) Ei \u2229 Ej = \u03c6, i \u2260 j, i, j = 1, 2, 3, , n\n\u00a9 NCERT\nnot to be republished\nPROBABILITY      549\nFig 13 4\n(b) E1 \u222a \u03952 \u222a"
    },
    {
        "Chapter": "1",
        "sentence_range": "7035-7038",
        "Text": ", En is said to represent a partition of the sample space S if\n(a) Ei \u2229 Ej = \u03c6, i \u2260 j, i, j = 1, 2, 3, , n\n\u00a9 NCERT\nnot to be republished\nPROBABILITY      549\nFig 13 4\n(b) E1 \u222a \u03952 \u222a \u222a En= S and\n(c) P(Ei) > 0 for all i = 1, 2,"
    },
    {
        "Chapter": "1",
        "sentence_range": "7036-7039",
        "Text": ", n\n\u00a9 NCERT\nnot to be republished\nPROBABILITY      549\nFig 13 4\n(b) E1 \u222a \u03952 \u222a \u222a En= S and\n(c) P(Ei) > 0 for all i = 1, 2, , n"
    },
    {
        "Chapter": "1",
        "sentence_range": "7037-7040",
        "Text": "4\n(b) E1 \u222a \u03952 \u222a \u222a En= S and\n(c) P(Ei) > 0 for all i = 1, 2, , n In other words, the events E1, E2,"
    },
    {
        "Chapter": "1",
        "sentence_range": "7038-7041",
        "Text": "\u222a En= S and\n(c) P(Ei) > 0 for all i = 1, 2, , n In other words, the events E1, E2, , En represent a partition of the sample space\nS if they are pairwise disjoint, exhaustive and have nonzero probabilities"
    },
    {
        "Chapter": "1",
        "sentence_range": "7039-7042",
        "Text": ", n In other words, the events E1, E2, , En represent a partition of the sample space\nS if they are pairwise disjoint, exhaustive and have nonzero probabilities As an example, we see that any nonempty event E and its complement E\u2032 form a\npartition of the sample space S since they satisfy E \u2229 E\u2032 = \u03c6 and E \u222a E\u2032 = S"
    },
    {
        "Chapter": "1",
        "sentence_range": "7040-7043",
        "Text": "In other words, the events E1, E2, , En represent a partition of the sample space\nS if they are pairwise disjoint, exhaustive and have nonzero probabilities As an example, we see that any nonempty event E and its complement E\u2032 form a\npartition of the sample space S since they satisfy E \u2229 E\u2032 = \u03c6 and E \u222a E\u2032 = S From the Venn diagram in Fig 13"
    },
    {
        "Chapter": "1",
        "sentence_range": "7041-7044",
        "Text": ", En represent a partition of the sample space\nS if they are pairwise disjoint, exhaustive and have nonzero probabilities As an example, we see that any nonempty event E and its complement E\u2032 form a\npartition of the sample space S since they satisfy E \u2229 E\u2032 = \u03c6 and E \u222a E\u2032 = S From the Venn diagram in Fig 13 3, one can easily observe that if E and F are any\ntwo events associated with a sample space S, then the set  {E \u2229 F\u2032, E \u2229 F, E\u2032 \u2229 F, E\u2032 \u2229 F\u2032}\nis a partition of the sample space S"
    },
    {
        "Chapter": "1",
        "sentence_range": "7042-7045",
        "Text": "As an example, we see that any nonempty event E and its complement E\u2032 form a\npartition of the sample space S since they satisfy E \u2229 E\u2032 = \u03c6 and E \u222a E\u2032 = S From the Venn diagram in Fig 13 3, one can easily observe that if E and F are any\ntwo events associated with a sample space S, then the set  {E \u2229 F\u2032, E \u2229 F, E\u2032 \u2229 F, E\u2032 \u2229 F\u2032}\nis a partition of the sample space S It may be mentioned that the partition of a sample\nspace is not unique"
    },
    {
        "Chapter": "1",
        "sentence_range": "7043-7046",
        "Text": "From the Venn diagram in Fig 13 3, one can easily observe that if E and F are any\ntwo events associated with a sample space S, then the set  {E \u2229 F\u2032, E \u2229 F, E\u2032 \u2229 F, E\u2032 \u2229 F\u2032}\nis a partition of the sample space S It may be mentioned that the partition of a sample\nspace is not unique There can be several partitions of the same sample space"
    },
    {
        "Chapter": "1",
        "sentence_range": "7044-7047",
        "Text": "3, one can easily observe that if E and F are any\ntwo events associated with a sample space S, then the set  {E \u2229 F\u2032, E \u2229 F, E\u2032 \u2229 F, E\u2032 \u2229 F\u2032}\nis a partition of the sample space S It may be mentioned that the partition of a sample\nspace is not unique There can be several partitions of the same sample space We shall now prove a theorem known as Theorem of total probability"
    },
    {
        "Chapter": "1",
        "sentence_range": "7045-7048",
        "Text": "It may be mentioned that the partition of a sample\nspace is not unique There can be several partitions of the same sample space We shall now prove a theorem known as Theorem of total probability 13"
    },
    {
        "Chapter": "1",
        "sentence_range": "7046-7049",
        "Text": "There can be several partitions of the same sample space We shall now prove a theorem known as Theorem of total probability 13 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "7047-7050",
        "Text": "We shall now prove a theorem known as Theorem of total probability 13 5 2  Theorem of total probability\nLet   {E1, E2,"
    },
    {
        "Chapter": "1",
        "sentence_range": "7048-7051",
        "Text": "13 5 2  Theorem of total probability\nLet   {E1, E2, ,En}  be a partition of the sample space S,  and suppose that each of the\nevents E1, E2,"
    },
    {
        "Chapter": "1",
        "sentence_range": "7049-7052",
        "Text": "5 2  Theorem of total probability\nLet   {E1, E2, ,En}  be a partition of the sample space S,  and suppose that each of the\nevents E1, E2, , En has nonzero probability of occurrence"
    },
    {
        "Chapter": "1",
        "sentence_range": "7050-7053",
        "Text": "2  Theorem of total probability\nLet   {E1, E2, ,En}  be a partition of the sample space S,  and suppose that each of the\nevents E1, E2, , En has nonzero probability of occurrence Let A be any event associated\nwith S, then\nP(A) = P(E1) P(A|E1) + P(E2) P(A|E2) +"
    },
    {
        "Chapter": "1",
        "sentence_range": "7051-7054",
        "Text": ",En}  be a partition of the sample space S,  and suppose that each of the\nevents E1, E2, , En has nonzero probability of occurrence Let A be any event associated\nwith S, then\nP(A) = P(E1) P(A|E1) + P(E2) P(A|E2) + + P(En) P(A|En)\n=\n1\nP(E )P(A|E )\nn\nj\nj\nj=\u2211\nProof  Given that E1, E2,"
    },
    {
        "Chapter": "1",
        "sentence_range": "7052-7055",
        "Text": ", En has nonzero probability of occurrence Let A be any event associated\nwith S, then\nP(A) = P(E1) P(A|E1) + P(E2) P(A|E2) + + P(En) P(A|En)\n=\n1\nP(E )P(A|E )\nn\nj\nj\nj=\u2211\nProof  Given that E1, E2, , En is a partition of the sample space S (Fig 13"
    },
    {
        "Chapter": "1",
        "sentence_range": "7053-7056",
        "Text": "Let A be any event associated\nwith S, then\nP(A) = P(E1) P(A|E1) + P(E2) P(A|E2) + + P(En) P(A|En)\n=\n1\nP(E )P(A|E )\nn\nj\nj\nj=\u2211\nProof  Given that E1, E2, , En is a partition of the sample space S (Fig 13 4)"
    },
    {
        "Chapter": "1",
        "sentence_range": "7054-7057",
        "Text": "+ P(En) P(A|En)\n=\n1\nP(E )P(A|E )\nn\nj\nj\nj=\u2211\nProof  Given that E1, E2, , En is a partition of the sample space S (Fig 13 4) Therefore,\nS = E1 \u222a E2 \u222a"
    },
    {
        "Chapter": "1",
        "sentence_range": "7055-7058",
        "Text": ", En is a partition of the sample space S (Fig 13 4) Therefore,\nS = E1 \u222a E2 \u222a \u222a En"
    },
    {
        "Chapter": "1",
        "sentence_range": "7056-7059",
        "Text": "4) Therefore,\nS = E1 \u222a E2 \u222a \u222a En (1)\nand\nEi \u2229 Ej = \u03c6, i \u2260 j, i, j = 1, 2,"
    },
    {
        "Chapter": "1",
        "sentence_range": "7057-7060",
        "Text": "Therefore,\nS = E1 \u222a E2 \u222a \u222a En (1)\nand\nEi \u2229 Ej = \u03c6, i \u2260 j, i, j = 1, 2, , n\nNow, we know that for any event A,\nA = A \u2229 S\n= A \u2229 (E1 \u222a E2 \u222a"
    },
    {
        "Chapter": "1",
        "sentence_range": "7058-7061",
        "Text": "\u222a En (1)\nand\nEi \u2229 Ej = \u03c6, i \u2260 j, i, j = 1, 2, , n\nNow, we know that for any event A,\nA = A \u2229 S\n= A \u2229 (E1 \u222a E2 \u222a \u222a En)\n= (A \u2229 E1) \u222a (A \u2229 E2) \u222a"
    },
    {
        "Chapter": "1",
        "sentence_range": "7059-7062",
        "Text": "(1)\nand\nEi \u2229 Ej = \u03c6, i \u2260 j, i, j = 1, 2, , n\nNow, we know that for any event A,\nA = A \u2229 S\n= A \u2229 (E1 \u222a E2 \u222a \u222a En)\n= (A \u2229 E1) \u222a (A \u2229 E2) \u222a \u222a (A \u2229 En)\nAlso A \u2229 Ei and A \u2229 Ej are respectively the subsets of Ei and Ej"
    },
    {
        "Chapter": "1",
        "sentence_range": "7060-7063",
        "Text": ", n\nNow, we know that for any event A,\nA = A \u2229 S\n= A \u2229 (E1 \u222a E2 \u222a \u222a En)\n= (A \u2229 E1) \u222a (A \u2229 E2) \u222a \u222a (A \u2229 En)\nAlso A \u2229 Ei and A \u2229 Ej are respectively the subsets of Ei and Ej We know that\nEi  and Ej are disjoint, for i\n\u2260j\n, therefore, A  \u2229 Ei and A  \u2229 Ej  are also disjoint for all\ni \u2260 j,  i, j = 1, 2,"
    },
    {
        "Chapter": "1",
        "sentence_range": "7061-7064",
        "Text": "\u222a En)\n= (A \u2229 E1) \u222a (A \u2229 E2) \u222a \u222a (A \u2229 En)\nAlso A \u2229 Ei and A \u2229 Ej are respectively the subsets of Ei and Ej We know that\nEi  and Ej are disjoint, for i\n\u2260j\n, therefore, A  \u2229 Ei and A  \u2229 Ej  are also disjoint for all\ni \u2260 j,  i, j = 1, 2, , n"
    },
    {
        "Chapter": "1",
        "sentence_range": "7062-7065",
        "Text": "\u222a (A \u2229 En)\nAlso A \u2229 Ei and A \u2229 Ej are respectively the subsets of Ei and Ej We know that\nEi  and Ej are disjoint, for i\n\u2260j\n, therefore, A  \u2229 Ei and A  \u2229 Ej  are also disjoint for all\ni \u2260 j,  i, j = 1, 2, , n Thus,\nP(A) = P [(A \u2229 E1) \u222a (A \u2229 E2)\u222a"
    },
    {
        "Chapter": "1",
        "sentence_range": "7063-7066",
        "Text": "We know that\nEi  and Ej are disjoint, for i\n\u2260j\n, therefore, A  \u2229 Ei and A  \u2229 Ej  are also disjoint for all\ni \u2260 j,  i, j = 1, 2, , n Thus,\nP(A) = P [(A \u2229 E1) \u222a (A \u2229 E2)\u222a \u222a (A \u2229 En)]\n= P (A \u2229 E1) + P (A \u2229 E2) +"
    },
    {
        "Chapter": "1",
        "sentence_range": "7064-7067",
        "Text": ", n Thus,\nP(A) = P [(A \u2229 E1) \u222a (A \u2229 E2)\u222a \u222a (A \u2229 En)]\n= P (A \u2229 E1) + P (A \u2229 E2) + + P (A \u2229 En)\nNow, by multiplication rule of probability, we have\nP(A \u2229 Ei) = P(Ei) P(A|Ei) as  P (Ei) \u2260 0\u2200i = 1,2,"
    },
    {
        "Chapter": "1",
        "sentence_range": "7065-7068",
        "Text": "Thus,\nP(A) = P [(A \u2229 E1) \u222a (A \u2229 E2)\u222a \u222a (A \u2229 En)]\n= P (A \u2229 E1) + P (A \u2229 E2) + + P (A \u2229 En)\nNow, by multiplication rule of probability, we have\nP(A \u2229 Ei) = P(Ei) P(A|Ei) as  P (Ei) \u2260 0\u2200i = 1,2, , n\n\u00a9 NCERT\nnot to be republished\n 550\nMATHEMATICS\nTherefore,\nP (A) = P (E1) P (A|E1) + P (E2) P (A|E2) +"
    },
    {
        "Chapter": "1",
        "sentence_range": "7066-7069",
        "Text": "\u222a (A \u2229 En)]\n= P (A \u2229 E1) + P (A \u2229 E2) + + P (A \u2229 En)\nNow, by multiplication rule of probability, we have\nP(A \u2229 Ei) = P(Ei) P(A|Ei) as  P (Ei) \u2260 0\u2200i = 1,2, , n\n\u00a9 NCERT\nnot to be republished\n 550\nMATHEMATICS\nTherefore,\nP (A) = P (E1) P (A|E1) + P (E2) P (A|E2) + + P (En)P(A|En)\nor\nP(A) =\n1\nP(E )P(A|E )\nn\nj\nj\nj=\u2211\nExample 15 A person has undertaken a construction job"
    },
    {
        "Chapter": "1",
        "sentence_range": "7067-7070",
        "Text": "+ P (A \u2229 En)\nNow, by multiplication rule of probability, we have\nP(A \u2229 Ei) = P(Ei) P(A|Ei) as  P (Ei) \u2260 0\u2200i = 1,2, , n\n\u00a9 NCERT\nnot to be republished\n 550\nMATHEMATICS\nTherefore,\nP (A) = P (E1) P (A|E1) + P (E2) P (A|E2) + + P (En)P(A|En)\nor\nP(A) =\n1\nP(E )P(A|E )\nn\nj\nj\nj=\u2211\nExample 15 A person has undertaken a construction job The probabilities are 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7068-7071",
        "Text": ", n\n\u00a9 NCERT\nnot to be republished\n 550\nMATHEMATICS\nTherefore,\nP (A) = P (E1) P (A|E1) + P (E2) P (A|E2) + + P (En)P(A|En)\nor\nP(A) =\n1\nP(E )P(A|E )\nn\nj\nj\nj=\u2211\nExample 15 A person has undertaken a construction job The probabilities are 0 65\nthat there will be strike, 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7069-7072",
        "Text": "+ P (En)P(A|En)\nor\nP(A) =\n1\nP(E )P(A|E )\nn\nj\nj\nj=\u2211\nExample 15 A person has undertaken a construction job The probabilities are 0 65\nthat there will be strike, 0 80 that the construction job will be completed on time if there\nis no strike, and 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7070-7073",
        "Text": "The probabilities are 0 65\nthat there will be strike, 0 80 that the construction job will be completed on time if there\nis no strike, and 0 32 that the construction job will be completed on time if there is a\nstrike"
    },
    {
        "Chapter": "1",
        "sentence_range": "7071-7074",
        "Text": "65\nthat there will be strike, 0 80 that the construction job will be completed on time if there\nis no strike, and 0 32 that the construction job will be completed on time if there is a\nstrike Determine the probability that the construction job will be completed on time"
    },
    {
        "Chapter": "1",
        "sentence_range": "7072-7075",
        "Text": "80 that the construction job will be completed on time if there\nis no strike, and 0 32 that the construction job will be completed on time if there is a\nstrike Determine the probability that the construction job will be completed on time Solution Let A be the event that the construction job will be completed on time, and B\nbe the event that there will be a strike"
    },
    {
        "Chapter": "1",
        "sentence_range": "7073-7076",
        "Text": "32 that the construction job will be completed on time if there is a\nstrike Determine the probability that the construction job will be completed on time Solution Let A be the event that the construction job will be completed on time, and B\nbe the event that there will be a strike We have to find P(A)"
    },
    {
        "Chapter": "1",
        "sentence_range": "7074-7077",
        "Text": "Determine the probability that the construction job will be completed on time Solution Let A be the event that the construction job will be completed on time, and B\nbe the event that there will be a strike We have to find P(A) We have\nP(B) = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7075-7078",
        "Text": "Solution Let A be the event that the construction job will be completed on time, and B\nbe the event that there will be a strike We have to find P(A) We have\nP(B) = 0 65, P(no strike) = P(B\u2032) = 1 \u2212 P(B) = 1 \u2212 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7076-7079",
        "Text": "We have to find P(A) We have\nP(B) = 0 65, P(no strike) = P(B\u2032) = 1 \u2212 P(B) = 1 \u2212 0 65 = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7077-7080",
        "Text": "We have\nP(B) = 0 65, P(no strike) = P(B\u2032) = 1 \u2212 P(B) = 1 \u2212 0 65 = 0 35\nP(A|B) = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7078-7081",
        "Text": "65, P(no strike) = P(B\u2032) = 1 \u2212 P(B) = 1 \u2212 0 65 = 0 35\nP(A|B) = 0 32, P(A|B\u2032) = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7079-7082",
        "Text": "65 = 0 35\nP(A|B) = 0 32, P(A|B\u2032) = 0 80\nSince events B and B\u2032 form a partition of the sample space S, therefore, by theorem\non total probability, we have\nP(A) = P(B) P(A|B) + P(B\u2032) P(A|B\u2032)\n         = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7080-7083",
        "Text": "35\nP(A|B) = 0 32, P(A|B\u2032) = 0 80\nSince events B and B\u2032 form a partition of the sample space S, therefore, by theorem\non total probability, we have\nP(A) = P(B) P(A|B) + P(B\u2032) P(A|B\u2032)\n         = 0 65 \u00d7 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7081-7084",
        "Text": "32, P(A|B\u2032) = 0 80\nSince events B and B\u2032 form a partition of the sample space S, therefore, by theorem\non total probability, we have\nP(A) = P(B) P(A|B) + P(B\u2032) P(A|B\u2032)\n         = 0 65 \u00d7 0 32 + 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7082-7085",
        "Text": "80\nSince events B and B\u2032 form a partition of the sample space S, therefore, by theorem\non total probability, we have\nP(A) = P(B) P(A|B) + P(B\u2032) P(A|B\u2032)\n         = 0 65 \u00d7 0 32 + 0 35 \u00d7 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7083-7086",
        "Text": "65 \u00d7 0 32 + 0 35 \u00d7 0 8\n         = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7084-7087",
        "Text": "32 + 0 35 \u00d7 0 8\n         = 0 208 + 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7085-7088",
        "Text": "35 \u00d7 0 8\n         = 0 208 + 0 28 = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7086-7089",
        "Text": "8\n         = 0 208 + 0 28 = 0 488\nThus, the probability that the construction job will be completed in time is 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7087-7090",
        "Text": "208 + 0 28 = 0 488\nThus, the probability that the construction job will be completed in time is 0 488"
    },
    {
        "Chapter": "1",
        "sentence_range": "7088-7091",
        "Text": "28 = 0 488\nThus, the probability that the construction job will be completed in time is 0 488 We shall now state and prove the Bayes' theorem"
    },
    {
        "Chapter": "1",
        "sentence_range": "7089-7092",
        "Text": "488\nThus, the probability that the construction job will be completed in time is 0 488 We shall now state and prove the Bayes' theorem Bayes\u2019 Theorem If E1, E2 ,"
    },
    {
        "Chapter": "1",
        "sentence_range": "7090-7093",
        "Text": "488 We shall now state and prove the Bayes' theorem Bayes\u2019 Theorem If E1, E2 , , En are n non empty events which constitute a partition\nof sample space S, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "7091-7094",
        "Text": "We shall now state and prove the Bayes' theorem Bayes\u2019 Theorem If E1, E2 , , En are n non empty events which constitute a partition\nof sample space S, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "7092-7095",
        "Text": "Bayes\u2019 Theorem If E1, E2 , , En are n non empty events which constitute a partition\nof sample space S, i e E1, E2 ,"
    },
    {
        "Chapter": "1",
        "sentence_range": "7093-7096",
        "Text": ", En are n non empty events which constitute a partition\nof sample space S, i e E1, E2 , , En are pairwise disjoint and E1\u222a E2\u222a"
    },
    {
        "Chapter": "1",
        "sentence_range": "7094-7097",
        "Text": "e E1, E2 , , En are pairwise disjoint and E1\u222a E2\u222a \u222a En = S and\nA is any event of nonzero probability, then\nP(Ei|A) =\n1\nP(E )P(A|E )\nP(E )P(A|E )\ni\ni\nn\nj\nj\nj=\u2211\n   for any i = 1, 2, 3,"
    },
    {
        "Chapter": "1",
        "sentence_range": "7095-7098",
        "Text": "E1, E2 , , En are pairwise disjoint and E1\u222a E2\u222a \u222a En = S and\nA is any event of nonzero probability, then\nP(Ei|A) =\n1\nP(E )P(A|E )\nP(E )P(A|E )\ni\ni\nn\nj\nj\nj=\u2211\n   for any i = 1, 2, 3, , n\nProof By formula of conditional probability, we know that\nP(Ei|A) = P(A\nE )\nP(A)\ni\n\u2229\n= P(E )P(A|E )\nP(A)\ni\ni  (by multiplication rule of probability)\n=\n1\nP(E )P(A|E )\nP(E )P(A|E )\ni\ni\nn\nj\nj\nj=\u2211\n (by the result of theorem of total probability)\n\u00a9 NCERT\nnot to be republished\nPROBABILITY      551\nRemark  The following terminology is generally used when Bayes' theorem is applied"
    },
    {
        "Chapter": "1",
        "sentence_range": "7096-7099",
        "Text": ", En are pairwise disjoint and E1\u222a E2\u222a \u222a En = S and\nA is any event of nonzero probability, then\nP(Ei|A) =\n1\nP(E )P(A|E )\nP(E )P(A|E )\ni\ni\nn\nj\nj\nj=\u2211\n   for any i = 1, 2, 3, , n\nProof By formula of conditional probability, we know that\nP(Ei|A) = P(A\nE )\nP(A)\ni\n\u2229\n= P(E )P(A|E )\nP(A)\ni\ni  (by multiplication rule of probability)\n=\n1\nP(E )P(A|E )\nP(E )P(A|E )\ni\ni\nn\nj\nj\nj=\u2211\n (by the result of theorem of total probability)\n\u00a9 NCERT\nnot to be republished\nPROBABILITY      551\nRemark  The following terminology is generally used when Bayes' theorem is applied The events E1, E2,"
    },
    {
        "Chapter": "1",
        "sentence_range": "7097-7100",
        "Text": "\u222a En = S and\nA is any event of nonzero probability, then\nP(Ei|A) =\n1\nP(E )P(A|E )\nP(E )P(A|E )\ni\ni\nn\nj\nj\nj=\u2211\n   for any i = 1, 2, 3, , n\nProof By formula of conditional probability, we know that\nP(Ei|A) = P(A\nE )\nP(A)\ni\n\u2229\n= P(E )P(A|E )\nP(A)\ni\ni  (by multiplication rule of probability)\n=\n1\nP(E )P(A|E )\nP(E )P(A|E )\ni\ni\nn\nj\nj\nj=\u2211\n (by the result of theorem of total probability)\n\u00a9 NCERT\nnot to be republished\nPROBABILITY      551\nRemark  The following terminology is generally used when Bayes' theorem is applied The events E1, E2, , En are  called hypotheses"
    },
    {
        "Chapter": "1",
        "sentence_range": "7098-7101",
        "Text": ", n\nProof By formula of conditional probability, we know that\nP(Ei|A) = P(A\nE )\nP(A)\ni\n\u2229\n= P(E )P(A|E )\nP(A)\ni\ni  (by multiplication rule of probability)\n=\n1\nP(E )P(A|E )\nP(E )P(A|E )\ni\ni\nn\nj\nj\nj=\u2211\n (by the result of theorem of total probability)\n\u00a9 NCERT\nnot to be republished\nPROBABILITY      551\nRemark  The following terminology is generally used when Bayes' theorem is applied The events E1, E2, , En are  called hypotheses The probability P(Ei) is called the priori probability of the hypothesis Ei\nThe conditional probability P(Ei |A) is called a posteriori probability of the\nhypothesis Ei"
    },
    {
        "Chapter": "1",
        "sentence_range": "7099-7102",
        "Text": "The events E1, E2, , En are  called hypotheses The probability P(Ei) is called the priori probability of the hypothesis Ei\nThe conditional probability P(Ei |A) is called a posteriori probability of the\nhypothesis Ei Bayes' theorem is also called the formula for the probability of \"causes\""
    },
    {
        "Chapter": "1",
        "sentence_range": "7100-7103",
        "Text": ", En are  called hypotheses The probability P(Ei) is called the priori probability of the hypothesis Ei\nThe conditional probability P(Ei |A) is called a posteriori probability of the\nhypothesis Ei Bayes' theorem is also called the formula for the probability of \"causes\" Since the\nEi's are a partition of the sample space S, one and only one of the events Ei occurs (i"
    },
    {
        "Chapter": "1",
        "sentence_range": "7101-7104",
        "Text": "The probability P(Ei) is called the priori probability of the hypothesis Ei\nThe conditional probability P(Ei |A) is called a posteriori probability of the\nhypothesis Ei Bayes' theorem is also called the formula for the probability of \"causes\" Since the\nEi's are a partition of the sample space S, one and only one of the events Ei occurs (i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "7102-7105",
        "Text": "Bayes' theorem is also called the formula for the probability of \"causes\" Since the\nEi's are a partition of the sample space S, one and only one of the events Ei occurs (i e one of the events Ei must occur and only one can occur)"
    },
    {
        "Chapter": "1",
        "sentence_range": "7103-7106",
        "Text": "Since the\nEi's are a partition of the sample space S, one and only one of the events Ei occurs (i e one of the events Ei must occur and only one can occur) Hence, the above formula\ngives us the probability of a particular Ei (i"
    },
    {
        "Chapter": "1",
        "sentence_range": "7104-7107",
        "Text": "e one of the events Ei must occur and only one can occur) Hence, the above formula\ngives us the probability of a particular Ei (i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "7105-7108",
        "Text": "one of the events Ei must occur and only one can occur) Hence, the above formula\ngives us the probability of a particular Ei (i e a \"Cause\"), given that the event A has\noccurred"
    },
    {
        "Chapter": "1",
        "sentence_range": "7106-7109",
        "Text": "Hence, the above formula\ngives us the probability of a particular Ei (i e a \"Cause\"), given that the event A has\noccurred The Bayes'  theorem has its applications in variety of situations, few of which are\nillustrated in following examples"
    },
    {
        "Chapter": "1",
        "sentence_range": "7107-7110",
        "Text": "e a \"Cause\"), given that the event A has\noccurred The Bayes'  theorem has its applications in variety of situations, few of which are\nillustrated in following examples Example 16 Bag I contains 3 red and 4 black balls while another Bag II contains 5 red\nand 6 black balls"
    },
    {
        "Chapter": "1",
        "sentence_range": "7108-7111",
        "Text": "a \"Cause\"), given that the event A has\noccurred The Bayes'  theorem has its applications in variety of situations, few of which are\nillustrated in following examples Example 16 Bag I contains 3 red and 4 black balls while another Bag II contains 5 red\nand 6 black balls One ball is drawn at random from one of the bags and it is found to\nbe red"
    },
    {
        "Chapter": "1",
        "sentence_range": "7109-7112",
        "Text": "The Bayes'  theorem has its applications in variety of situations, few of which are\nillustrated in following examples Example 16 Bag I contains 3 red and 4 black balls while another Bag II contains 5 red\nand 6 black balls One ball is drawn at random from one of the bags and it is found to\nbe red Find the probability that it was drawn from Bag II"
    },
    {
        "Chapter": "1",
        "sentence_range": "7110-7113",
        "Text": "Example 16 Bag I contains 3 red and 4 black balls while another Bag II contains 5 red\nand 6 black balls One ball is drawn at random from one of the bags and it is found to\nbe red Find the probability that it was drawn from Bag II Solution Let E1 be the event of choosing the bag I, E2 the event of choosing the bag II\nand A be the event of drawing a red ball"
    },
    {
        "Chapter": "1",
        "sentence_range": "7111-7114",
        "Text": "One ball is drawn at random from one of the bags and it is found to\nbe red Find the probability that it was drawn from Bag II Solution Let E1 be the event of choosing the bag I, E2 the event of choosing the bag II\nand A be the event of drawing a red ball Then\nP(E1) = P(E2) =  1\n2\nAlso\nP(A|E1) = P(drawing a red ball from Bag I) = 3\n7\nand\nP(A|E2) = P(drawing a red ball from Bag II) = 5\n11\nNow, the probability of drawing a ball from Bag II, being given that it is red,\nis P(E2|A)\nBy using Bayes' theorem, we have\nP(E2|A) =\n2\n2\n1\n1\n2\n2\nP(E )P(A|E )\nP(E )P(A|E )+P(E )P(A|E )  = \n1\n5\n35\n2 11\n1\n3\n1\n5\n68\n2\n7\n2 11\n\u00d7\n=\n\u00d7\n+\n\u00d7\nExample 17 Given three identical boxes I, II and III, each containing two coins"
    },
    {
        "Chapter": "1",
        "sentence_range": "7112-7115",
        "Text": "Find the probability that it was drawn from Bag II Solution Let E1 be the event of choosing the bag I, E2 the event of choosing the bag II\nand A be the event of drawing a red ball Then\nP(E1) = P(E2) =  1\n2\nAlso\nP(A|E1) = P(drawing a red ball from Bag I) = 3\n7\nand\nP(A|E2) = P(drawing a red ball from Bag II) = 5\n11\nNow, the probability of drawing a ball from Bag II, being given that it is red,\nis P(E2|A)\nBy using Bayes' theorem, we have\nP(E2|A) =\n2\n2\n1\n1\n2\n2\nP(E )P(A|E )\nP(E )P(A|E )+P(E )P(A|E )  = \n1\n5\n35\n2 11\n1\n3\n1\n5\n68\n2\n7\n2 11\n\u00d7\n=\n\u00d7\n+\n\u00d7\nExample 17 Given three identical boxes I, II and III, each containing two coins In\nbox I, both coins are gold coins, in box II, both are silver coins and in the box III, there\nis one gold and one silver coin"
    },
    {
        "Chapter": "1",
        "sentence_range": "7113-7116",
        "Text": "Solution Let E1 be the event of choosing the bag I, E2 the event of choosing the bag II\nand A be the event of drawing a red ball Then\nP(E1) = P(E2) =  1\n2\nAlso\nP(A|E1) = P(drawing a red ball from Bag I) = 3\n7\nand\nP(A|E2) = P(drawing a red ball from Bag II) = 5\n11\nNow, the probability of drawing a ball from Bag II, being given that it is red,\nis P(E2|A)\nBy using Bayes' theorem, we have\nP(E2|A) =\n2\n2\n1\n1\n2\n2\nP(E )P(A|E )\nP(E )P(A|E )+P(E )P(A|E )  = \n1\n5\n35\n2 11\n1\n3\n1\n5\n68\n2\n7\n2 11\n\u00d7\n=\n\u00d7\n+\n\u00d7\nExample 17 Given three identical boxes I, II and III, each containing two coins In\nbox I, both coins are gold coins, in box II, both are silver coins and in the box III, there\nis one gold and one silver coin A person chooses a box at random and takes out a coin"
    },
    {
        "Chapter": "1",
        "sentence_range": "7114-7117",
        "Text": "Then\nP(E1) = P(E2) =  1\n2\nAlso\nP(A|E1) = P(drawing a red ball from Bag I) = 3\n7\nand\nP(A|E2) = P(drawing a red ball from Bag II) = 5\n11\nNow, the probability of drawing a ball from Bag II, being given that it is red,\nis P(E2|A)\nBy using Bayes' theorem, we have\nP(E2|A) =\n2\n2\n1\n1\n2\n2\nP(E )P(A|E )\nP(E )P(A|E )+P(E )P(A|E )  = \n1\n5\n35\n2 11\n1\n3\n1\n5\n68\n2\n7\n2 11\n\u00d7\n=\n\u00d7\n+\n\u00d7\nExample 17 Given three identical boxes I, II and III, each containing two coins In\nbox I, both coins are gold coins, in box II, both are silver coins and in the box III, there\nis one gold and one silver coin A person chooses a box at random and takes out a coin If the coin is of gold, what is the probability that the other coin in the box is also of gold"
    },
    {
        "Chapter": "1",
        "sentence_range": "7115-7118",
        "Text": "In\nbox I, both coins are gold coins, in box II, both are silver coins and in the box III, there\nis one gold and one silver coin A person chooses a box at random and takes out a coin If the coin is of gold, what is the probability that the other coin in the box is also of gold \u00a9 NCERT\nnot to be republished\n 552\nMATHEMATICS\nSolution Let E1, E2 and E3 be the events that boxes I, II and III are chosen, respectively"
    },
    {
        "Chapter": "1",
        "sentence_range": "7116-7119",
        "Text": "A person chooses a box at random and takes out a coin If the coin is of gold, what is the probability that the other coin in the box is also of gold \u00a9 NCERT\nnot to be republished\n 552\nMATHEMATICS\nSolution Let E1, E2 and E3 be the events that boxes I, II and III are chosen, respectively Then\nP(E1) = P(E2) = P(E3) = 1\n3\nAlso, let A be the event that \u2018the coin drawn is of gold\u2019\nThen\nP(A|E1) = P(a gold coin from bag I) = 2\n2  = 1\nP(A|E2) = P(a gold coin from bag II) = 0\nP(A|E3) = P(a gold coin from bag III) = 1\n2\nNow, the probability that the other coin in the box is of gold\n= the probability that gold coin is drawn from the box I"
    },
    {
        "Chapter": "1",
        "sentence_range": "7117-7120",
        "Text": "If the coin is of gold, what is the probability that the other coin in the box is also of gold \u00a9 NCERT\nnot to be republished\n 552\nMATHEMATICS\nSolution Let E1, E2 and E3 be the events that boxes I, II and III are chosen, respectively Then\nP(E1) = P(E2) = P(E3) = 1\n3\nAlso, let A be the event that \u2018the coin drawn is of gold\u2019\nThen\nP(A|E1) = P(a gold coin from bag I) = 2\n2  = 1\nP(A|E2) = P(a gold coin from bag II) = 0\nP(A|E3) = P(a gold coin from bag III) = 1\n2\nNow, the probability that the other coin in the box is of gold\n= the probability that gold coin is drawn from the box I = P(E1|A)\nBy Bayes' theorem, we know that\nP(E1|A) =\n1\n1\n1\n1\n2\n2\n3\n3\nP(E )P(A|E )\nP(E )P(A|E )+P(E )P(A|E )+P(E )P(A|E )\n=\n1 1\n2\n3\n1\n1\n1\n1\n3\n1\n0\n3\n3\n3\n2\n\u00d7\n=\n\u00d7 + \u00d7 + \u00d7\nExample 18 Suppose that the reliability of a HIV test is specified as follows:\nOf people having HIV, 90% of the test detect the disease but 10% go undetected"
    },
    {
        "Chapter": "1",
        "sentence_range": "7118-7121",
        "Text": "\u00a9 NCERT\nnot to be republished\n 552\nMATHEMATICS\nSolution Let E1, E2 and E3 be the events that boxes I, II and III are chosen, respectively Then\nP(E1) = P(E2) = P(E3) = 1\n3\nAlso, let A be the event that \u2018the coin drawn is of gold\u2019\nThen\nP(A|E1) = P(a gold coin from bag I) = 2\n2  = 1\nP(A|E2) = P(a gold coin from bag II) = 0\nP(A|E3) = P(a gold coin from bag III) = 1\n2\nNow, the probability that the other coin in the box is of gold\n= the probability that gold coin is drawn from the box I = P(E1|A)\nBy Bayes' theorem, we know that\nP(E1|A) =\n1\n1\n1\n1\n2\n2\n3\n3\nP(E )P(A|E )\nP(E )P(A|E )+P(E )P(A|E )+P(E )P(A|E )\n=\n1 1\n2\n3\n1\n1\n1\n1\n3\n1\n0\n3\n3\n3\n2\n\u00d7\n=\n\u00d7 + \u00d7 + \u00d7\nExample 18 Suppose that the reliability of a HIV test is specified as follows:\nOf people having HIV, 90% of the test detect the disease but 10% go undetected Of\npeople free of HIV, 99% of the test are judged HIV\u2013ive but 1% are diagnosed as\nshowing HIV+ive"
    },
    {
        "Chapter": "1",
        "sentence_range": "7119-7122",
        "Text": "Then\nP(E1) = P(E2) = P(E3) = 1\n3\nAlso, let A be the event that \u2018the coin drawn is of gold\u2019\nThen\nP(A|E1) = P(a gold coin from bag I) = 2\n2  = 1\nP(A|E2) = P(a gold coin from bag II) = 0\nP(A|E3) = P(a gold coin from bag III) = 1\n2\nNow, the probability that the other coin in the box is of gold\n= the probability that gold coin is drawn from the box I = P(E1|A)\nBy Bayes' theorem, we know that\nP(E1|A) =\n1\n1\n1\n1\n2\n2\n3\n3\nP(E )P(A|E )\nP(E )P(A|E )+P(E )P(A|E )+P(E )P(A|E )\n=\n1 1\n2\n3\n1\n1\n1\n1\n3\n1\n0\n3\n3\n3\n2\n\u00d7\n=\n\u00d7 + \u00d7 + \u00d7\nExample 18 Suppose that the reliability of a HIV test is specified as follows:\nOf people having HIV, 90% of the test detect the disease but 10% go undetected Of\npeople free of HIV, 99% of the test are judged HIV\u2013ive but 1% are diagnosed as\nshowing HIV+ive From a large population of which only 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7120-7123",
        "Text": "= P(E1|A)\nBy Bayes' theorem, we know that\nP(E1|A) =\n1\n1\n1\n1\n2\n2\n3\n3\nP(E )P(A|E )\nP(E )P(A|E )+P(E )P(A|E )+P(E )P(A|E )\n=\n1 1\n2\n3\n1\n1\n1\n1\n3\n1\n0\n3\n3\n3\n2\n\u00d7\n=\n\u00d7 + \u00d7 + \u00d7\nExample 18 Suppose that the reliability of a HIV test is specified as follows:\nOf people having HIV, 90% of the test detect the disease but 10% go undetected Of\npeople free of HIV, 99% of the test are judged HIV\u2013ive but 1% are diagnosed as\nshowing HIV+ive From a large population of which only 0 1% have HIV, one person\nis selected at random, given the HIV test, and the pathologist reports him/her as\nHIV+ive"
    },
    {
        "Chapter": "1",
        "sentence_range": "7121-7124",
        "Text": "Of\npeople free of HIV, 99% of the test are judged HIV\u2013ive but 1% are diagnosed as\nshowing HIV+ive From a large population of which only 0 1% have HIV, one person\nis selected at random, given the HIV test, and the pathologist reports him/her as\nHIV+ive What is the probability that the person actually has HIV"
    },
    {
        "Chapter": "1",
        "sentence_range": "7122-7125",
        "Text": "From a large population of which only 0 1% have HIV, one person\nis selected at random, given the HIV test, and the pathologist reports him/her as\nHIV+ive What is the probability that the person actually has HIV Solution Let E denote the event that the person selected is actually having HIV and A\nthe event that the person's HIV test is diagnosed as +ive"
    },
    {
        "Chapter": "1",
        "sentence_range": "7123-7126",
        "Text": "1% have HIV, one person\nis selected at random, given the HIV test, and the pathologist reports him/her as\nHIV+ive What is the probability that the person actually has HIV Solution Let E denote the event that the person selected is actually having HIV and A\nthe event that the person's HIV test is diagnosed as +ive We need to find P(E|A)"
    },
    {
        "Chapter": "1",
        "sentence_range": "7124-7127",
        "Text": "What is the probability that the person actually has HIV Solution Let E denote the event that the person selected is actually having HIV and A\nthe event that the person's HIV test is diagnosed as +ive We need to find P(E|A) Also E\u2032 denotes the event that the person selected is actually not having HIV"
    },
    {
        "Chapter": "1",
        "sentence_range": "7125-7128",
        "Text": "Solution Let E denote the event that the person selected is actually having HIV and A\nthe event that the person's HIV test is diagnosed as +ive We need to find P(E|A) Also E\u2032 denotes the event that the person selected is actually not having HIV Clearly, {E, E\u2032} is a partition of the sample space of all people in the population"
    },
    {
        "Chapter": "1",
        "sentence_range": "7126-7129",
        "Text": "We need to find P(E|A) Also E\u2032 denotes the event that the person selected is actually not having HIV Clearly, {E, E\u2032} is a partition of the sample space of all people in the population We are given that\nP(E) = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7127-7130",
        "Text": "Also E\u2032 denotes the event that the person selected is actually not having HIV Clearly, {E, E\u2032} is a partition of the sample space of all people in the population We are given that\nP(E) = 0 1% \n0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7128-7131",
        "Text": "Clearly, {E, E\u2032} is a partition of the sample space of all people in the population We are given that\nP(E) = 0 1% \n0 1\n0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7129-7132",
        "Text": "We are given that\nP(E) = 0 1% \n0 1\n0 001\n 100\n \n\u00a9 NCERT\nnot to be republished\nPROBABILITY      553\nP(E\u2032) = 1 \u2013 P(E) = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7130-7133",
        "Text": "1% \n0 1\n0 001\n 100\n \n\u00a9 NCERT\nnot to be republished\nPROBABILITY      553\nP(E\u2032) = 1 \u2013 P(E) = 0 999\nP(A|E) = P(Person tested as HIV+ive given that he/she\nis actually having HIV)\n= 90% \n90\n0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7131-7134",
        "Text": "1\n0 001\n 100\n \n\u00a9 NCERT\nnot to be republished\nPROBABILITY      553\nP(E\u2032) = 1 \u2013 P(E) = 0 999\nP(A|E) = P(Person tested as HIV+ive given that he/she\nis actually having HIV)\n= 90% \n90\n0 9\n 100\n \nand\nP(A|E\u2032) = P(Person tested as HIV +ive given that he/she\nis actually not having HIV)\n= 1% = 1\n100 = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7132-7135",
        "Text": "001\n 100\n \n\u00a9 NCERT\nnot to be republished\nPROBABILITY      553\nP(E\u2032) = 1 \u2013 P(E) = 0 999\nP(A|E) = P(Person tested as HIV+ive given that he/she\nis actually having HIV)\n= 90% \n90\n0 9\n 100\n \nand\nP(A|E\u2032) = P(Person tested as HIV +ive given that he/she\nis actually not having HIV)\n= 1% = 1\n100 = 0 01\nNow, by Bayes' theorem\nP(E|A) =\nP(E)P(A|E)\nP(E)P(A|E)+P(E )P(A|E )\n \n \n=\n0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7133-7136",
        "Text": "999\nP(A|E) = P(Person tested as HIV+ive given that he/she\nis actually having HIV)\n= 90% \n90\n0 9\n 100\n \nand\nP(A|E\u2032) = P(Person tested as HIV +ive given that he/she\nis actually not having HIV)\n= 1% = 1\n100 = 0 01\nNow, by Bayes' theorem\nP(E|A) =\nP(E)P(A|E)\nP(E)P(A|E)+P(E )P(A|E )\n \n \n=\n0 001 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7134-7137",
        "Text": "9\n 100\n \nand\nP(A|E\u2032) = P(Person tested as HIV +ive given that he/she\nis actually not having HIV)\n= 1% = 1\n100 = 0 01\nNow, by Bayes' theorem\nP(E|A) =\nP(E)P(A|E)\nP(E)P(A|E)+P(E )P(A|E )\n \n \n=\n0 001 0 9\n90\n0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7135-7138",
        "Text": "01\nNow, by Bayes' theorem\nP(E|A) =\nP(E)P(A|E)\nP(E)P(A|E)+P(E )P(A|E )\n \n \n=\n0 001 0 9\n90\n0 001 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7136-7139",
        "Text": "001 0 9\n90\n0 001 0 9 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7137-7140",
        "Text": "9\n90\n0 001 0 9 0 999 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7138-7141",
        "Text": "001 0 9 0 999 0 01\n1089\n\u00d7\n=\n\u00d7\n+\n\u00d7\n= 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7139-7142",
        "Text": "9 0 999 0 01\n1089\n\u00d7\n=\n\u00d7\n+\n\u00d7\n= 0 083 approx"
    },
    {
        "Chapter": "1",
        "sentence_range": "7140-7143",
        "Text": "999 0 01\n1089\n\u00d7\n=\n\u00d7\n+\n\u00d7\n= 0 083 approx Thus,  the probability that a person selected at random is actually having HIV\ngiven that he/she is tested HIV+ive is 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7141-7144",
        "Text": "01\n1089\n\u00d7\n=\n\u00d7\n+\n\u00d7\n= 0 083 approx Thus,  the probability that a person selected at random is actually having HIV\ngiven that he/she is tested HIV+ive is 0 083"
    },
    {
        "Chapter": "1",
        "sentence_range": "7142-7145",
        "Text": "083 approx Thus,  the probability that a person selected at random is actually having HIV\ngiven that he/she is tested HIV+ive is 0 083 Example 19 In a factory which manufactures bolts, machines A, B and C manufacture\nrespectively 25%, 35% and 40% of the bolts"
    },
    {
        "Chapter": "1",
        "sentence_range": "7143-7146",
        "Text": "Thus,  the probability that a person selected at random is actually having HIV\ngiven that he/she is tested HIV+ive is 0 083 Example 19 In a factory which manufactures bolts, machines A, B and C manufacture\nrespectively 25%, 35% and 40% of the bolts Of their outputs, 5, 4 and 2 percent are\nrespectively defective bolts"
    },
    {
        "Chapter": "1",
        "sentence_range": "7144-7147",
        "Text": "083 Example 19 In a factory which manufactures bolts, machines A, B and C manufacture\nrespectively 25%, 35% and 40% of the bolts Of their outputs, 5, 4 and 2 percent are\nrespectively defective bolts A bolt is drawn at random from the product and is found\nto be defective"
    },
    {
        "Chapter": "1",
        "sentence_range": "7145-7148",
        "Text": "Example 19 In a factory which manufactures bolts, machines A, B and C manufacture\nrespectively 25%, 35% and 40% of the bolts Of their outputs, 5, 4 and 2 percent are\nrespectively defective bolts A bolt is drawn at random from the product and is found\nto be defective What is the probability that it is manufactured by the machine B"
    },
    {
        "Chapter": "1",
        "sentence_range": "7146-7149",
        "Text": "Of their outputs, 5, 4 and 2 percent are\nrespectively defective bolts A bolt is drawn at random from the product and is found\nto be defective What is the probability that it is manufactured by the machine B Solution Let events B1, B2, B3 be the following :\nB1 : the bolt is manufactured by machine A\nB2 : the bolt is manufactured by machine B\nB3 : the bolt is manufactured by machine C\nClearly, B1, B2, B3 are mutually exclusive and exhaustive events and hence, they\nrepresent a partition of the sample space"
    },
    {
        "Chapter": "1",
        "sentence_range": "7147-7150",
        "Text": "A bolt is drawn at random from the product and is found\nto be defective What is the probability that it is manufactured by the machine B Solution Let events B1, B2, B3 be the following :\nB1 : the bolt is manufactured by machine A\nB2 : the bolt is manufactured by machine B\nB3 : the bolt is manufactured by machine C\nClearly, B1, B2, B3 are mutually exclusive and exhaustive events and hence, they\nrepresent a partition of the sample space Let the event E be \u2018the bolt is defective\u2019"
    },
    {
        "Chapter": "1",
        "sentence_range": "7148-7151",
        "Text": "What is the probability that it is manufactured by the machine B Solution Let events B1, B2, B3 be the following :\nB1 : the bolt is manufactured by machine A\nB2 : the bolt is manufactured by machine B\nB3 : the bolt is manufactured by machine C\nClearly, B1, B2, B3 are mutually exclusive and exhaustive events and hence, they\nrepresent a partition of the sample space Let the event E be \u2018the bolt is defective\u2019 The event E occurs with B1 or with B2 or with B3"
    },
    {
        "Chapter": "1",
        "sentence_range": "7149-7152",
        "Text": "Solution Let events B1, B2, B3 be the following :\nB1 : the bolt is manufactured by machine A\nB2 : the bolt is manufactured by machine B\nB3 : the bolt is manufactured by machine C\nClearly, B1, B2, B3 are mutually exclusive and exhaustive events and hence, they\nrepresent a partition of the sample space Let the event E be \u2018the bolt is defective\u2019 The event E occurs with B1 or with B2 or with B3 Given that,\nP(B1) = 25% = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7150-7153",
        "Text": "Let the event E be \u2018the bolt is defective\u2019 The event E occurs with B1 or with B2 or with B3 Given that,\nP(B1) = 25% = 0 25,  P (B2) = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7151-7154",
        "Text": "The event E occurs with B1 or with B2 or with B3 Given that,\nP(B1) = 25% = 0 25,  P (B2) = 0 35 and P(B3) = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7152-7155",
        "Text": "Given that,\nP(B1) = 25% = 0 25,  P (B2) = 0 35 and P(B3) = 0 40\nAgain P(E|B1) = Probability that the bolt drawn is defective given that it is manu-\nfactured by machine A = 5% = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7153-7156",
        "Text": "25,  P (B2) = 0 35 and P(B3) = 0 40\nAgain P(E|B1) = Probability that the bolt drawn is defective given that it is manu-\nfactured by machine A = 5% = 0 05\nSimilarly,\nP(E|B2) = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7154-7157",
        "Text": "35 and P(B3) = 0 40\nAgain P(E|B1) = Probability that the bolt drawn is defective given that it is manu-\nfactured by machine A = 5% = 0 05\nSimilarly,\nP(E|B2) = 0 04,  P(E|B3) = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7155-7158",
        "Text": "40\nAgain P(E|B1) = Probability that the bolt drawn is defective given that it is manu-\nfactured by machine A = 5% = 0 05\nSimilarly,\nP(E|B2) = 0 04,  P(E|B3) = 0 02"
    },
    {
        "Chapter": "1",
        "sentence_range": "7156-7159",
        "Text": "05\nSimilarly,\nP(E|B2) = 0 04,  P(E|B3) = 0 02 \u00a9 NCERT\nnot to be republished\n 554\nMATHEMATICS\nHence, by Bayes' Theorem, we have\nP(B2|E) =\n2\n2\n1\n1\n2\n2\n3\n3\nP(B )P(E|B )\nP(B )P(E|B )+P(B )P(E|B )+P(B )P(E|B )\n=\n0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7157-7160",
        "Text": "04,  P(E|B3) = 0 02 \u00a9 NCERT\nnot to be republished\n 554\nMATHEMATICS\nHence, by Bayes' Theorem, we have\nP(B2|E) =\n2\n2\n1\n1\n2\n2\n3\n3\nP(B )P(E|B )\nP(B )P(E|B )+P(B )P(E|B )+P(B )P(E|B )\n=\n0 35\n0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7158-7161",
        "Text": "02 \u00a9 NCERT\nnot to be republished\n 554\nMATHEMATICS\nHence, by Bayes' Theorem, we have\nP(B2|E) =\n2\n2\n1\n1\n2\n2\n3\n3\nP(B )P(E|B )\nP(B )P(E|B )+P(B )P(E|B )+P(B )P(E|B )\n=\n0 35\n0 04\n0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7159-7162",
        "Text": "\u00a9 NCERT\nnot to be republished\n 554\nMATHEMATICS\nHence, by Bayes' Theorem, we have\nP(B2|E) =\n2\n2\n1\n1\n2\n2\n3\n3\nP(B )P(E|B )\nP(B )P(E|B )+P(B )P(E|B )+P(B )P(E|B )\n=\n0 35\n0 04\n0 25\n0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7160-7163",
        "Text": "35\n0 04\n0 25\n0 05\n0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7161-7164",
        "Text": "04\n0 25\n0 05\n0 35\n0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7162-7165",
        "Text": "25\n0 05\n0 35\n0 04\n0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7163-7166",
        "Text": "05\n0 35\n0 04\n0 40\n0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7164-7167",
        "Text": "35\n0 04\n0 40\n0 02\n\u00d7\n\u00d7\n+\n\u00d7\n+\n\u00d7\n= 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7165-7168",
        "Text": "04\n0 40\n0 02\n\u00d7\n\u00d7\n+\n\u00d7\n+\n\u00d7\n= 0 0140\n28\n0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7166-7169",
        "Text": "40\n0 02\n\u00d7\n\u00d7\n+\n\u00d7\n+\n\u00d7\n= 0 0140\n28\n0 0345\n69\n=\nExample 20 A doctor is to visit a patient"
    },
    {
        "Chapter": "1",
        "sentence_range": "7167-7170",
        "Text": "02\n\u00d7\n\u00d7\n+\n\u00d7\n+\n\u00d7\n= 0 0140\n28\n0 0345\n69\n=\nExample 20 A doctor is to visit a patient From the past experience, it is known that\nthe probabilities that he will come by train, bus, scooter or by other means of transport\nare respectively 3 1 1\n2\n, ,\n10 5 10and\n5"
    },
    {
        "Chapter": "1",
        "sentence_range": "7168-7171",
        "Text": "0140\n28\n0 0345\n69\n=\nExample 20 A doctor is to visit a patient From the past experience, it is known that\nthe probabilities that he will come by train, bus, scooter or by other means of transport\nare respectively 3 1 1\n2\n, ,\n10 5 10and\n5 The probabilities that he will be late are 1 1\n, , and1\n4 3\n12,\nif he comes by train, bus and scooter respectively, but if he comes by other means of\ntransport, then he will not be late"
    },
    {
        "Chapter": "1",
        "sentence_range": "7169-7172",
        "Text": "0345\n69\n=\nExample 20 A doctor is to visit a patient From the past experience, it is known that\nthe probabilities that he will come by train, bus, scooter or by other means of transport\nare respectively 3 1 1\n2\n, ,\n10 5 10and\n5 The probabilities that he will be late are 1 1\n, , and1\n4 3\n12,\nif he comes by train, bus and scooter respectively, but if he comes by other means of\ntransport, then he will not be late When he arrives, he is late"
    },
    {
        "Chapter": "1",
        "sentence_range": "7170-7173",
        "Text": "From the past experience, it is known that\nthe probabilities that he will come by train, bus, scooter or by other means of transport\nare respectively 3 1 1\n2\n, ,\n10 5 10and\n5 The probabilities that he will be late are 1 1\n, , and1\n4 3\n12,\nif he comes by train, bus and scooter respectively, but if he comes by other means of\ntransport, then he will not be late When he arrives, he is late What is the probability\nthat he comes by train"
    },
    {
        "Chapter": "1",
        "sentence_range": "7171-7174",
        "Text": "The probabilities that he will be late are 1 1\n, , and1\n4 3\n12,\nif he comes by train, bus and scooter respectively, but if he comes by other means of\ntransport, then he will not be late When he arrives, he is late What is the probability\nthat he comes by train Solution Let E be the event that the doctor visits the patient late and let T1, T2, T3, T4\nbe the events  that the doctor comes by train, bus, scooter, and other means of transport\nrespectively"
    },
    {
        "Chapter": "1",
        "sentence_range": "7172-7175",
        "Text": "When he arrives, he is late What is the probability\nthat he comes by train Solution Let E be the event that the doctor visits the patient late and let T1, T2, T3, T4\nbe the events  that the doctor comes by train, bus, scooter, and other means of transport\nrespectively Then\nP(T1) =\n2\n3\n4\n3\n1\n1\n2\n, P(T )\n,P(T )\nand P(T )\n10\n5\n10\n5\n=\n=\n=\n(given)\nP(E|T1) = Probability that the doctor arriving late comes by train = 1\n4\nSimilarly, P(E|T2) = 1\n3 , P(E|T3) = 1\n12  and P(E|T4) = 0, since he is not late if  he\ncomes by other means of transport"
    },
    {
        "Chapter": "1",
        "sentence_range": "7173-7176",
        "Text": "What is the probability\nthat he comes by train Solution Let E be the event that the doctor visits the patient late and let T1, T2, T3, T4\nbe the events  that the doctor comes by train, bus, scooter, and other means of transport\nrespectively Then\nP(T1) =\n2\n3\n4\n3\n1\n1\n2\n, P(T )\n,P(T )\nand P(T )\n10\n5\n10\n5\n=\n=\n=\n(given)\nP(E|T1) = Probability that the doctor arriving late comes by train = 1\n4\nSimilarly, P(E|T2) = 1\n3 , P(E|T3) = 1\n12  and P(E|T4) = 0, since he is not late if  he\ncomes by other means of transport Therefore, by Bayes' Theorem, we have\nP(T1|E) = Probability that the doctor arriving late comes by train\n=\n1\n1\n1\n1\n2\n2\n3\n3\n4\n4\nP(T )P(E|T )\nP(T )P(E|T )+P(T )P(E|T )+P(T )P(E|T )+P(T )P(E|T )\n=\n3\n1\n10\n4\n3\n1\n1\n1\n1\n1\n2\n0\n10\n4\n5\n3\n10\n12\n5\n \n \n \n \n \n \n \n \n = 3\n120\n1\n40\n18\n2\n\u00d7\n=\nHence, the required probability is 1\n2"
    },
    {
        "Chapter": "1",
        "sentence_range": "7174-7177",
        "Text": "Solution Let E be the event that the doctor visits the patient late and let T1, T2, T3, T4\nbe the events  that the doctor comes by train, bus, scooter, and other means of transport\nrespectively Then\nP(T1) =\n2\n3\n4\n3\n1\n1\n2\n, P(T )\n,P(T )\nand P(T )\n10\n5\n10\n5\n=\n=\n=\n(given)\nP(E|T1) = Probability that the doctor arriving late comes by train = 1\n4\nSimilarly, P(E|T2) = 1\n3 , P(E|T3) = 1\n12  and P(E|T4) = 0, since he is not late if  he\ncomes by other means of transport Therefore, by Bayes' Theorem, we have\nP(T1|E) = Probability that the doctor arriving late comes by train\n=\n1\n1\n1\n1\n2\n2\n3\n3\n4\n4\nP(T )P(E|T )\nP(T )P(E|T )+P(T )P(E|T )+P(T )P(E|T )+P(T )P(E|T )\n=\n3\n1\n10\n4\n3\n1\n1\n1\n1\n1\n2\n0\n10\n4\n5\n3\n10\n12\n5\n \n \n \n \n \n \n \n \n = 3\n120\n1\n40\n18\n2\n\u00d7\n=\nHence, the required probability is 1\n2 \u00a9 NCERT\nnot to be republished\nPROBABILITY      555\nExample 21 A man is known to speak truth 3 out of 4 times"
    },
    {
        "Chapter": "1",
        "sentence_range": "7175-7178",
        "Text": "Then\nP(T1) =\n2\n3\n4\n3\n1\n1\n2\n, P(T )\n,P(T )\nand P(T )\n10\n5\n10\n5\n=\n=\n=\n(given)\nP(E|T1) = Probability that the doctor arriving late comes by train = 1\n4\nSimilarly, P(E|T2) = 1\n3 , P(E|T3) = 1\n12  and P(E|T4) = 0, since he is not late if  he\ncomes by other means of transport Therefore, by Bayes' Theorem, we have\nP(T1|E) = Probability that the doctor arriving late comes by train\n=\n1\n1\n1\n1\n2\n2\n3\n3\n4\n4\nP(T )P(E|T )\nP(T )P(E|T )+P(T )P(E|T )+P(T )P(E|T )+P(T )P(E|T )\n=\n3\n1\n10\n4\n3\n1\n1\n1\n1\n1\n2\n0\n10\n4\n5\n3\n10\n12\n5\n \n \n \n \n \n \n \n \n = 3\n120\n1\n40\n18\n2\n\u00d7\n=\nHence, the required probability is 1\n2 \u00a9 NCERT\nnot to be republished\nPROBABILITY      555\nExample 21 A man is known to speak truth 3 out of 4 times He throws a die and\nreports that it is a six"
    },
    {
        "Chapter": "1",
        "sentence_range": "7176-7179",
        "Text": "Therefore, by Bayes' Theorem, we have\nP(T1|E) = Probability that the doctor arriving late comes by train\n=\n1\n1\n1\n1\n2\n2\n3\n3\n4\n4\nP(T )P(E|T )\nP(T )P(E|T )+P(T )P(E|T )+P(T )P(E|T )+P(T )P(E|T )\n=\n3\n1\n10\n4\n3\n1\n1\n1\n1\n1\n2\n0\n10\n4\n5\n3\n10\n12\n5\n \n \n \n \n \n \n \n \n = 3\n120\n1\n40\n18\n2\n\u00d7\n=\nHence, the required probability is 1\n2 \u00a9 NCERT\nnot to be republished\nPROBABILITY      555\nExample 21 A man is known to speak truth 3 out of 4 times He throws a die and\nreports that it is a six Find the probability that it is actually a six"
    },
    {
        "Chapter": "1",
        "sentence_range": "7177-7180",
        "Text": "\u00a9 NCERT\nnot to be republished\nPROBABILITY      555\nExample 21 A man is known to speak truth 3 out of 4 times He throws a die and\nreports that it is a six Find the probability that it is actually a six Solution Let E be the event that the man reports that six occurs in the throwing of the\ndie and let S1 be the event that six occurs and S2 be the event that six does not occur"
    },
    {
        "Chapter": "1",
        "sentence_range": "7178-7181",
        "Text": "He throws a die and\nreports that it is a six Find the probability that it is actually a six Solution Let E be the event that the man reports that six occurs in the throwing of the\ndie and let S1 be the event that six occurs and S2 be the event that six does not occur Then\nP(S1) = Probability that six occurs = 1\n6\nP(S2) = Probability that six does not occur = 5\n6\nP(E|S1) = Probability that the man reports that six occurs when six has\nactually occurred on the die\n= Probability that the man speaks the truth = 3\n4\nP(E|S2) = Probability that the man reports that six occurs  when six has\nnot actually occurred on the die\n= Probability that the man does not speak the truth \n3\n1\n1\n4\n4\n \n \n \nThus, by Bayes' theorem, we get\nP(S1|E) = Probability that the report of the man that six has occurred is\nactually a six\n=\n1\n1\n1\n1\n2\n2\nP(S )P(E |S )\nP(S )P(E|S )+P(S )P(E|S )\n=\n1\n3\n1\n24\n3\n6\n4\n1\n3\n5\n1\n8\n8\n8\n6\n4\n6\n4\n \n \n \n \n \n \n \nHence, the required probability is 3"
    },
    {
        "Chapter": "1",
        "sentence_range": "7179-7182",
        "Text": "Find the probability that it is actually a six Solution Let E be the event that the man reports that six occurs in the throwing of the\ndie and let S1 be the event that six occurs and S2 be the event that six does not occur Then\nP(S1) = Probability that six occurs = 1\n6\nP(S2) = Probability that six does not occur = 5\n6\nP(E|S1) = Probability that the man reports that six occurs when six has\nactually occurred on the die\n= Probability that the man speaks the truth = 3\n4\nP(E|S2) = Probability that the man reports that six occurs  when six has\nnot actually occurred on the die\n= Probability that the man does not speak the truth \n3\n1\n1\n4\n4\n \n \n \nThus, by Bayes' theorem, we get\nP(S1|E) = Probability that the report of the man that six has occurred is\nactually a six\n=\n1\n1\n1\n1\n2\n2\nP(S )P(E |S )\nP(S )P(E|S )+P(S )P(E|S )\n=\n1\n3\n1\n24\n3\n6\n4\n1\n3\n5\n1\n8\n8\n8\n6\n4\n6\n4\n \n \n \n \n \n \n \nHence, the required probability is 3 8\nEXERCISE 13"
    },
    {
        "Chapter": "1",
        "sentence_range": "7180-7183",
        "Text": "Solution Let E be the event that the man reports that six occurs in the throwing of the\ndie and let S1 be the event that six occurs and S2 be the event that six does not occur Then\nP(S1) = Probability that six occurs = 1\n6\nP(S2) = Probability that six does not occur = 5\n6\nP(E|S1) = Probability that the man reports that six occurs when six has\nactually occurred on the die\n= Probability that the man speaks the truth = 3\n4\nP(E|S2) = Probability that the man reports that six occurs  when six has\nnot actually occurred on the die\n= Probability that the man does not speak the truth \n3\n1\n1\n4\n4\n \n \n \nThus, by Bayes' theorem, we get\nP(S1|E) = Probability that the report of the man that six has occurred is\nactually a six\n=\n1\n1\n1\n1\n2\n2\nP(S )P(E |S )\nP(S )P(E|S )+P(S )P(E|S )\n=\n1\n3\n1\n24\n3\n6\n4\n1\n3\n5\n1\n8\n8\n8\n6\n4\n6\n4\n \n \n \n \n \n \n \nHence, the required probability is 3 8\nEXERCISE 13 3\n1"
    },
    {
        "Chapter": "1",
        "sentence_range": "7181-7184",
        "Text": "Then\nP(S1) = Probability that six occurs = 1\n6\nP(S2) = Probability that six does not occur = 5\n6\nP(E|S1) = Probability that the man reports that six occurs when six has\nactually occurred on the die\n= Probability that the man speaks the truth = 3\n4\nP(E|S2) = Probability that the man reports that six occurs  when six has\nnot actually occurred on the die\n= Probability that the man does not speak the truth \n3\n1\n1\n4\n4\n \n \n \nThus, by Bayes' theorem, we get\nP(S1|E) = Probability that the report of the man that six has occurred is\nactually a six\n=\n1\n1\n1\n1\n2\n2\nP(S )P(E |S )\nP(S )P(E|S )+P(S )P(E|S )\n=\n1\n3\n1\n24\n3\n6\n4\n1\n3\n5\n1\n8\n8\n8\n6\n4\n6\n4\n \n \n \n \n \n \n \nHence, the required probability is 3 8\nEXERCISE 13 3\n1 An urn contains 5 red and 5 black balls"
    },
    {
        "Chapter": "1",
        "sentence_range": "7182-7185",
        "Text": "8\nEXERCISE 13 3\n1 An urn contains 5 red and 5 black balls A ball is drawn at random, its colour is\nnoted and is returned to the urn"
    },
    {
        "Chapter": "1",
        "sentence_range": "7183-7186",
        "Text": "3\n1 An urn contains 5 red and 5 black balls A ball is drawn at random, its colour is\nnoted and is returned to the urn Moreover, 2 additional balls of the colour drawn\nare put in the urn and then a ball is drawn at random"
    },
    {
        "Chapter": "1",
        "sentence_range": "7184-7187",
        "Text": "An urn contains 5 red and 5 black balls A ball is drawn at random, its colour is\nnoted and is returned to the urn Moreover, 2 additional balls of the colour drawn\nare put in the urn and then a ball is drawn at random What is the probability that\nthe second ball is red"
    },
    {
        "Chapter": "1",
        "sentence_range": "7185-7188",
        "Text": "A ball is drawn at random, its colour is\nnoted and is returned to the urn Moreover, 2 additional balls of the colour drawn\nare put in the urn and then a ball is drawn at random What is the probability that\nthe second ball is red \u00a9 NCERT\nnot to be republished\n 556\nMATHEMATICS\n2"
    },
    {
        "Chapter": "1",
        "sentence_range": "7186-7189",
        "Text": "Moreover, 2 additional balls of the colour drawn\nare put in the urn and then a ball is drawn at random What is the probability that\nthe second ball is red \u00a9 NCERT\nnot to be republished\n 556\nMATHEMATICS\n2 A bag contains 4 red and 4 black balls, another bag contains 2 red and 6 black\nballs"
    },
    {
        "Chapter": "1",
        "sentence_range": "7187-7190",
        "Text": "What is the probability that\nthe second ball is red \u00a9 NCERT\nnot to be republished\n 556\nMATHEMATICS\n2 A bag contains 4 red and 4 black balls, another bag contains 2 red and 6 black\nballs One of the two bags is selected at random and a ball is drawn from the bag\nwhich is found to be red"
    },
    {
        "Chapter": "1",
        "sentence_range": "7188-7191",
        "Text": "\u00a9 NCERT\nnot to be republished\n 556\nMATHEMATICS\n2 A bag contains 4 red and 4 black balls, another bag contains 2 red and 6 black\nballs One of the two bags is selected at random and a ball is drawn from the bag\nwhich is found to be red Find the probability that the ball is drawn from the\nfirst bag"
    },
    {
        "Chapter": "1",
        "sentence_range": "7189-7192",
        "Text": "A bag contains 4 red and 4 black balls, another bag contains 2 red and 6 black\nballs One of the two bags is selected at random and a ball is drawn from the bag\nwhich is found to be red Find the probability that the ball is drawn from the\nfirst bag 3"
    },
    {
        "Chapter": "1",
        "sentence_range": "7190-7193",
        "Text": "One of the two bags is selected at random and a ball is drawn from the bag\nwhich is found to be red Find the probability that the ball is drawn from the\nfirst bag 3 Of the students in a college, it is known that 60% reside in hostel and 40% are\nday scholars (not residing in hostel)"
    },
    {
        "Chapter": "1",
        "sentence_range": "7191-7194",
        "Text": "Find the probability that the ball is drawn from the\nfirst bag 3 Of the students in a college, it is known that 60% reside in hostel and 40% are\nday scholars (not residing in hostel) Previous year results report that 30% of all\nstudents who reside in hostel attain A grade and 20% of day scholars attain A\ngrade in their annual examination"
    },
    {
        "Chapter": "1",
        "sentence_range": "7192-7195",
        "Text": "3 Of the students in a college, it is known that 60% reside in hostel and 40% are\nday scholars (not residing in hostel) Previous year results report that 30% of all\nstudents who reside in hostel attain A grade and 20% of day scholars attain A\ngrade in their annual examination At the end of the year, one student is chosen\nat random from the college and he has an A grade, what is the probability that the\nstudent is a hostlier"
    },
    {
        "Chapter": "1",
        "sentence_range": "7193-7196",
        "Text": "Of the students in a college, it is known that 60% reside in hostel and 40% are\nday scholars (not residing in hostel) Previous year results report that 30% of all\nstudents who reside in hostel attain A grade and 20% of day scholars attain A\ngrade in their annual examination At the end of the year, one student is chosen\nat random from the college and he has an A grade, what is the probability that the\nstudent is a hostlier 4"
    },
    {
        "Chapter": "1",
        "sentence_range": "7194-7197",
        "Text": "Previous year results report that 30% of all\nstudents who reside in hostel attain A grade and 20% of day scholars attain A\ngrade in their annual examination At the end of the year, one student is chosen\nat random from the college and he has an A grade, what is the probability that the\nstudent is a hostlier 4 In answering a question on a multiple choice test, a student either knows the\nanswer or guesses"
    },
    {
        "Chapter": "1",
        "sentence_range": "7195-7198",
        "Text": "At the end of the year, one student is chosen\nat random from the college and he has an A grade, what is the probability that the\nstudent is a hostlier 4 In answering a question on a multiple choice test, a student either knows the\nanswer or guesses Let 3\n4  be the probability that he knows the answer and 1\n4\nbe the probability that he guesses"
    },
    {
        "Chapter": "1",
        "sentence_range": "7196-7199",
        "Text": "4 In answering a question on a multiple choice test, a student either knows the\nanswer or guesses Let 3\n4  be the probability that he knows the answer and 1\n4\nbe the probability that he guesses Assuming that a student who guesses at the\nanswer will be correct with probability 1\n4"
    },
    {
        "Chapter": "1",
        "sentence_range": "7197-7200",
        "Text": "In answering a question on a multiple choice test, a student either knows the\nanswer or guesses Let 3\n4  be the probability that he knows the answer and 1\n4\nbe the probability that he guesses Assuming that a student who guesses at the\nanswer will be correct with probability 1\n4 What is the probability that the stu-\ndent knows the answer given that he answered it correctly"
    },
    {
        "Chapter": "1",
        "sentence_range": "7198-7201",
        "Text": "Let 3\n4  be the probability that he knows the answer and 1\n4\nbe the probability that he guesses Assuming that a student who guesses at the\nanswer will be correct with probability 1\n4 What is the probability that the stu-\ndent knows the answer given that he answered it correctly 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "7199-7202",
        "Text": "Assuming that a student who guesses at the\nanswer will be correct with probability 1\n4 What is the probability that the stu-\ndent knows the answer given that he answered it correctly 5 A laboratory blood test is 99% effective in detecting a certain disease when it is\nin fact, present"
    },
    {
        "Chapter": "1",
        "sentence_range": "7200-7203",
        "Text": "What is the probability that the stu-\ndent knows the answer given that he answered it correctly 5 A laboratory blood test is 99% effective in detecting a certain disease when it is\nin fact, present However, the test also yields a false positive result for 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7201-7204",
        "Text": "5 A laboratory blood test is 99% effective in detecting a certain disease when it is\nin fact, present However, the test also yields a false positive result for 0 5% of\nthe healthy person tested (i"
    },
    {
        "Chapter": "1",
        "sentence_range": "7202-7205",
        "Text": "A laboratory blood test is 99% effective in detecting a certain disease when it is\nin fact, present However, the test also yields a false positive result for 0 5% of\nthe healthy person tested (i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "7203-7206",
        "Text": "However, the test also yields a false positive result for 0 5% of\nthe healthy person tested (i e if a healthy person is tested, then, with probability\n0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7204-7207",
        "Text": "5% of\nthe healthy person tested (i e if a healthy person is tested, then, with probability\n0 005, the test will imply he has the disease)"
    },
    {
        "Chapter": "1",
        "sentence_range": "7205-7208",
        "Text": "e if a healthy person is tested, then, with probability\n0 005, the test will imply he has the disease) If 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7206-7209",
        "Text": "if a healthy person is tested, then, with probability\n0 005, the test will imply he has the disease) If 0 1 percent of the population\nactually has the disease, what is the probability that a person has the disease\ngiven that his test result is positive"
    },
    {
        "Chapter": "1",
        "sentence_range": "7207-7210",
        "Text": "005, the test will imply he has the disease) If 0 1 percent of the population\nactually has the disease, what is the probability that a person has the disease\ngiven that his test result is positive 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "7208-7211",
        "Text": "If 0 1 percent of the population\nactually has the disease, what is the probability that a person has the disease\ngiven that his test result is positive 6 There are three coins"
    },
    {
        "Chapter": "1",
        "sentence_range": "7209-7212",
        "Text": "1 percent of the population\nactually has the disease, what is the probability that a person has the disease\ngiven that his test result is positive 6 There are three coins One is a two headed coin (having head on both faces),\nanother is a biased coin that comes up heads 75% of the time and third is an\nunbiased coin"
    },
    {
        "Chapter": "1",
        "sentence_range": "7210-7213",
        "Text": "6 There are three coins One is a two headed coin (having head on both faces),\nanother is a biased coin that comes up heads 75% of the time and third is an\nunbiased coin One of the three coins is chosen at random and tossed, it shows\nheads, what is the probability that it was the two headed coin"
    },
    {
        "Chapter": "1",
        "sentence_range": "7211-7214",
        "Text": "There are three coins One is a two headed coin (having head on both faces),\nanother is a biased coin that comes up heads 75% of the time and third is an\nunbiased coin One of the three coins is chosen at random and tossed, it shows\nheads, what is the probability that it was the two headed coin 7"
    },
    {
        "Chapter": "1",
        "sentence_range": "7212-7215",
        "Text": "One is a two headed coin (having head on both faces),\nanother is a biased coin that comes up heads 75% of the time and third is an\nunbiased coin One of the three coins is chosen at random and tossed, it shows\nheads, what is the probability that it was the two headed coin 7 An insurance company insured 2000 scooter drivers, 4000 car drivers and 6000\ntruck drivers"
    },
    {
        "Chapter": "1",
        "sentence_range": "7213-7216",
        "Text": "One of the three coins is chosen at random and tossed, it shows\nheads, what is the probability that it was the two headed coin 7 An insurance company insured 2000 scooter drivers, 4000 car drivers and 6000\ntruck drivers The probability of an accidents are 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7214-7217",
        "Text": "7 An insurance company insured 2000 scooter drivers, 4000 car drivers and 6000\ntruck drivers The probability of an accidents are 0 01, 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7215-7218",
        "Text": "An insurance company insured 2000 scooter drivers, 4000 car drivers and 6000\ntruck drivers The probability of an accidents are 0 01, 0 03 and 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7216-7219",
        "Text": "The probability of an accidents are 0 01, 0 03 and 0 15 respectively"
    },
    {
        "Chapter": "1",
        "sentence_range": "7217-7220",
        "Text": "01, 0 03 and 0 15 respectively One of the insured persons meets with an accident"
    },
    {
        "Chapter": "1",
        "sentence_range": "7218-7221",
        "Text": "03 and 0 15 respectively One of the insured persons meets with an accident What is the probability that\nhe is a scooter driver"
    },
    {
        "Chapter": "1",
        "sentence_range": "7219-7222",
        "Text": "15 respectively One of the insured persons meets with an accident What is the probability that\nhe is a scooter driver 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "7220-7223",
        "Text": "One of the insured persons meets with an accident What is the probability that\nhe is a scooter driver 8 A factory has two machines A and B"
    },
    {
        "Chapter": "1",
        "sentence_range": "7221-7224",
        "Text": "What is the probability that\nhe is a scooter driver 8 A factory has two machines A and B Past record shows that machine A produced\n60% of the items of output and machine B produced 40% of the items"
    },
    {
        "Chapter": "1",
        "sentence_range": "7222-7225",
        "Text": "8 A factory has two machines A and B Past record shows that machine A produced\n60% of the items of output and machine B produced 40% of the items Further,\n2% of the items produced by machine A and 1% produced by machine B were\ndefective"
    },
    {
        "Chapter": "1",
        "sentence_range": "7223-7226",
        "Text": "A factory has two machines A and B Past record shows that machine A produced\n60% of the items of output and machine B produced 40% of the items Further,\n2% of the items produced by machine A and 1% produced by machine B were\ndefective All the items are put into one stockpile and then one item is chosen at\nrandom from this and is found to be defective"
    },
    {
        "Chapter": "1",
        "sentence_range": "7224-7227",
        "Text": "Past record shows that machine A produced\n60% of the items of output and machine B produced 40% of the items Further,\n2% of the items produced by machine A and 1% produced by machine B were\ndefective All the items are put into one stockpile and then one item is chosen at\nrandom from this and is found to be defective What is the probability that it was\nproduced by machine B"
    },
    {
        "Chapter": "1",
        "sentence_range": "7225-7228",
        "Text": "Further,\n2% of the items produced by machine A and 1% produced by machine B were\ndefective All the items are put into one stockpile and then one item is chosen at\nrandom from this and is found to be defective What is the probability that it was\nproduced by machine B 9"
    },
    {
        "Chapter": "1",
        "sentence_range": "7226-7229",
        "Text": "All the items are put into one stockpile and then one item is chosen at\nrandom from this and is found to be defective What is the probability that it was\nproduced by machine B 9 Two groups are competing for the position on the Board of directors of a\ncorporation"
    },
    {
        "Chapter": "1",
        "sentence_range": "7227-7230",
        "Text": "What is the probability that it was\nproduced by machine B 9 Two groups are competing for the position on the Board of directors of a\ncorporation The probabilities that the first and the second groups will win are\n\u00a9 NCERT\nnot to be republished\nPROBABILITY      557\n0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7228-7231",
        "Text": "9 Two groups are competing for the position on the Board of directors of a\ncorporation The probabilities that the first and the second groups will win are\n\u00a9 NCERT\nnot to be republished\nPROBABILITY      557\n0 6 and 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7229-7232",
        "Text": "Two groups are competing for the position on the Board of directors of a\ncorporation The probabilities that the first and the second groups will win are\n\u00a9 NCERT\nnot to be republished\nPROBABILITY      557\n0 6 and 0 4 respectively"
    },
    {
        "Chapter": "1",
        "sentence_range": "7230-7233",
        "Text": "The probabilities that the first and the second groups will win are\n\u00a9 NCERT\nnot to be republished\nPROBABILITY      557\n0 6 and 0 4 respectively Further, if the first group wins, the probability of\nintroducing a new product is 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7231-7234",
        "Text": "6 and 0 4 respectively Further, if the first group wins, the probability of\nintroducing a new product is 0 7 and the corresponding probability is 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7232-7235",
        "Text": "4 respectively Further, if the first group wins, the probability of\nintroducing a new product is 0 7 and the corresponding probability is 0 3 if the\nsecond group wins"
    },
    {
        "Chapter": "1",
        "sentence_range": "7233-7236",
        "Text": "Further, if the first group wins, the probability of\nintroducing a new product is 0 7 and the corresponding probability is 0 3 if the\nsecond group wins Find the probability that the new product introduced was by\nthe second group"
    },
    {
        "Chapter": "1",
        "sentence_range": "7234-7237",
        "Text": "7 and the corresponding probability is 0 3 if the\nsecond group wins Find the probability that the new product introduced was by\nthe second group 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "7235-7238",
        "Text": "3 if the\nsecond group wins Find the probability that the new product introduced was by\nthe second group 10 Suppose a girl throws a die"
    },
    {
        "Chapter": "1",
        "sentence_range": "7236-7239",
        "Text": "Find the probability that the new product introduced was by\nthe second group 10 Suppose a girl throws a die If she gets a 5 or 6, she tosses a coin three times and\nnotes the number of heads"
    },
    {
        "Chapter": "1",
        "sentence_range": "7237-7240",
        "Text": "10 Suppose a girl throws a die If she gets a 5 or 6, she tosses a coin three times and\nnotes the number of heads If she gets 1, 2, 3 or 4, she tosses a coin once and\nnotes whether a head or tail is obtained"
    },
    {
        "Chapter": "1",
        "sentence_range": "7238-7241",
        "Text": "Suppose a girl throws a die If she gets a 5 or 6, she tosses a coin three times and\nnotes the number of heads If she gets 1, 2, 3 or 4, she tosses a coin once and\nnotes whether a head or tail is obtained If she obtained exactly one head, what\nis the probability that she threw 1, 2, 3 or 4 with the die"
    },
    {
        "Chapter": "1",
        "sentence_range": "7239-7242",
        "Text": "If she gets a 5 or 6, she tosses a coin three times and\nnotes the number of heads If she gets 1, 2, 3 or 4, she tosses a coin once and\nnotes whether a head or tail is obtained If she obtained exactly one head, what\nis the probability that she threw 1, 2, 3 or 4 with the die 11"
    },
    {
        "Chapter": "1",
        "sentence_range": "7240-7243",
        "Text": "If she gets 1, 2, 3 or 4, she tosses a coin once and\nnotes whether a head or tail is obtained If she obtained exactly one head, what\nis the probability that she threw 1, 2, 3 or 4 with the die 11 A manufacturer has three machine operators A, B  and C"
    },
    {
        "Chapter": "1",
        "sentence_range": "7241-7244",
        "Text": "If she obtained exactly one head, what\nis the probability that she threw 1, 2, 3 or 4 with the die 11 A manufacturer has three machine operators A, B  and C The first operator A\nproduces 1% defective items, where as the other two operators B and C pro-\nduce 5% and 7% defective items respectively"
    },
    {
        "Chapter": "1",
        "sentence_range": "7242-7245",
        "Text": "11 A manufacturer has three machine operators A, B  and C The first operator A\nproduces 1% defective items, where as the other two operators B and C pro-\nduce 5% and 7% defective items respectively A is on the job for 50% of the\ntime, B is on the job for 30% of the time and C is on the job for 20% of the time"
    },
    {
        "Chapter": "1",
        "sentence_range": "7243-7246",
        "Text": "A manufacturer has three machine operators A, B  and C The first operator A\nproduces 1% defective items, where as the other two operators B and C pro-\nduce 5% and 7% defective items respectively A is on the job for 50% of the\ntime, B is on the job for 30% of the time and C is on the job for 20% of the time A defective item is produced, what is the probability that it was produced by A"
    },
    {
        "Chapter": "1",
        "sentence_range": "7244-7247",
        "Text": "The first operator A\nproduces 1% defective items, where as the other two operators B and C pro-\nduce 5% and 7% defective items respectively A is on the job for 50% of the\ntime, B is on the job for 30% of the time and C is on the job for 20% of the time A defective item is produced, what is the probability that it was produced by A 12"
    },
    {
        "Chapter": "1",
        "sentence_range": "7245-7248",
        "Text": "A is on the job for 50% of the\ntime, B is on the job for 30% of the time and C is on the job for 20% of the time A defective item is produced, what is the probability that it was produced by A 12 A card from a pack of 52 cards is lost"
    },
    {
        "Chapter": "1",
        "sentence_range": "7246-7249",
        "Text": "A defective item is produced, what is the probability that it was produced by A 12 A card from a pack of 52 cards is lost From the remaining cards of the pack,\ntwo cards  are drawn and are found to be both diamonds"
    },
    {
        "Chapter": "1",
        "sentence_range": "7247-7250",
        "Text": "12 A card from a pack of 52 cards is lost From the remaining cards of the pack,\ntwo cards  are drawn and are found to be both diamonds Find the probability of\nthe lost card being a diamond"
    },
    {
        "Chapter": "1",
        "sentence_range": "7248-7251",
        "Text": "A card from a pack of 52 cards is lost From the remaining cards of the pack,\ntwo cards  are drawn and are found to be both diamonds Find the probability of\nthe lost card being a diamond 13"
    },
    {
        "Chapter": "1",
        "sentence_range": "7249-7252",
        "Text": "From the remaining cards of the pack,\ntwo cards  are drawn and are found to be both diamonds Find the probability of\nthe lost card being a diamond 13 Probability that A speaks truth is 4\n5"
    },
    {
        "Chapter": "1",
        "sentence_range": "7250-7253",
        "Text": "Find the probability of\nthe lost card being a diamond 13 Probability that A speaks truth is 4\n5 A coin is tossed"
    },
    {
        "Chapter": "1",
        "sentence_range": "7251-7254",
        "Text": "13 Probability that A speaks truth is 4\n5 A coin is tossed A reports that a head\nappears"
    },
    {
        "Chapter": "1",
        "sentence_range": "7252-7255",
        "Text": "Probability that A speaks truth is 4\n5 A coin is tossed A reports that a head\nappears The probability that actually there was head is\n(A) 4\n5\n(B) 1\n2\n(C)\n51\n(D) 2\n5\n14"
    },
    {
        "Chapter": "1",
        "sentence_range": "7253-7256",
        "Text": "A coin is tossed A reports that a head\nappears The probability that actually there was head is\n(A) 4\n5\n(B) 1\n2\n(C)\n51\n(D) 2\n5\n14 If A and B are two events such that A \u2282 B and P(B) \u2260  0, then which of the\nfollowing is correct"
    },
    {
        "Chapter": "1",
        "sentence_range": "7254-7257",
        "Text": "A reports that a head\nappears The probability that actually there was head is\n(A) 4\n5\n(B) 1\n2\n(C)\n51\n(D) 2\n5\n14 If A and B are two events such that A \u2282 B and P(B) \u2260  0, then which of the\nfollowing is correct (A)\nP(B)\nP(A | B)\n P(A)\n(B) P(A|B) < P(A)\n(C) P(A|B) \u2265 P(A)\n(D) None of these\n13"
    },
    {
        "Chapter": "1",
        "sentence_range": "7255-7258",
        "Text": "The probability that actually there was head is\n(A) 4\n5\n(B) 1\n2\n(C)\n51\n(D) 2\n5\n14 If A and B are two events such that A \u2282 B and P(B) \u2260  0, then which of the\nfollowing is correct (A)\nP(B)\nP(A | B)\n P(A)\n(B) P(A|B) < P(A)\n(C) P(A|B) \u2265 P(A)\n(D) None of these\n13 6  Random Variables and its Probability Distributions\nWe have already learnt about random experiments and formation of sample spaces"
    },
    {
        "Chapter": "1",
        "sentence_range": "7256-7259",
        "Text": "If A and B are two events such that A \u2282 B and P(B) \u2260  0, then which of the\nfollowing is correct (A)\nP(B)\nP(A | B)\n P(A)\n(B) P(A|B) < P(A)\n(C) P(A|B) \u2265 P(A)\n(D) None of these\n13 6  Random Variables and its Probability Distributions\nWe have already learnt about random experiments and formation of sample spaces In\nmost of these experiments, we were not only interested in the particular outcome that\noccurs but rather in some number associated with that outcomes  as shown in following\nexamples/experiments"
    },
    {
        "Chapter": "1",
        "sentence_range": "7257-7260",
        "Text": "(A)\nP(B)\nP(A | B)\n P(A)\n(B) P(A|B) < P(A)\n(C) P(A|B) \u2265 P(A)\n(D) None of these\n13 6  Random Variables and its Probability Distributions\nWe have already learnt about random experiments and formation of sample spaces In\nmost of these experiments, we were not only interested in the particular outcome that\noccurs but rather in some number associated with that outcomes  as shown in following\nexamples/experiments (i)\nIn tossing two dice, we may be interested in the sum of the numbers on the\ntwo dice"
    },
    {
        "Chapter": "1",
        "sentence_range": "7258-7261",
        "Text": "6  Random Variables and its Probability Distributions\nWe have already learnt about random experiments and formation of sample spaces In\nmost of these experiments, we were not only interested in the particular outcome that\noccurs but rather in some number associated with that outcomes  as shown in following\nexamples/experiments (i)\nIn tossing two dice, we may be interested in the sum of the numbers on the\ntwo dice (ii)\nIn tossing a coin 50 times, we may want the number of heads obtained"
    },
    {
        "Chapter": "1",
        "sentence_range": "7259-7262",
        "Text": "In\nmost of these experiments, we were not only interested in the particular outcome that\noccurs but rather in some number associated with that outcomes  as shown in following\nexamples/experiments (i)\nIn tossing two dice, we may be interested in the sum of the numbers on the\ntwo dice (ii)\nIn tossing a coin 50 times, we may want the number of heads obtained \u00a9 NCERT\nnot to be republished\n 558\nMATHEMATICS\n(iii)\nIn the experiment of taking out four articles (one after the other) at random\nfrom a lot of 20 articles in which 6 are defective, we want to know the\nnumber of defectives in the sample of four and not in the particular sequence\nof defective and nondefective articles"
    },
    {
        "Chapter": "1",
        "sentence_range": "7260-7263",
        "Text": "(i)\nIn tossing two dice, we may be interested in the sum of the numbers on the\ntwo dice (ii)\nIn tossing a coin 50 times, we may want the number of heads obtained \u00a9 NCERT\nnot to be republished\n 558\nMATHEMATICS\n(iii)\nIn the experiment of taking out four articles (one after the other) at random\nfrom a lot of 20 articles in which 6 are defective, we want to know the\nnumber of defectives in the sample of four and not in the particular sequence\nof defective and nondefective articles In all of the above experiments, we have a rule which assigns to each outcome of\nthe experiment a single real number"
    },
    {
        "Chapter": "1",
        "sentence_range": "7261-7264",
        "Text": "(ii)\nIn tossing a coin 50 times, we may want the number of heads obtained \u00a9 NCERT\nnot to be republished\n 558\nMATHEMATICS\n(iii)\nIn the experiment of taking out four articles (one after the other) at random\nfrom a lot of 20 articles in which 6 are defective, we want to know the\nnumber of defectives in the sample of four and not in the particular sequence\nof defective and nondefective articles In all of the above experiments, we have a rule which assigns to each outcome of\nthe experiment a single real number This single real number may vary with different\noutcomes of the experiment"
    },
    {
        "Chapter": "1",
        "sentence_range": "7262-7265",
        "Text": "\u00a9 NCERT\nnot to be republished\n 558\nMATHEMATICS\n(iii)\nIn the experiment of taking out four articles (one after the other) at random\nfrom a lot of 20 articles in which 6 are defective, we want to know the\nnumber of defectives in the sample of four and not in the particular sequence\nof defective and nondefective articles In all of the above experiments, we have a rule which assigns to each outcome of\nthe experiment a single real number This single real number may vary with different\noutcomes of the experiment Hence, it is a variable"
    },
    {
        "Chapter": "1",
        "sentence_range": "7263-7266",
        "Text": "In all of the above experiments, we have a rule which assigns to each outcome of\nthe experiment a single real number This single real number may vary with different\noutcomes of the experiment Hence, it is a variable Also its value depends upon the\noutcome of a random experiment and, hence, is called random variable"
    },
    {
        "Chapter": "1",
        "sentence_range": "7264-7267",
        "Text": "This single real number may vary with different\noutcomes of the experiment Hence, it is a variable Also its value depends upon the\noutcome of a random experiment and, hence, is called random variable A random\nvariable is usually denoted by X"
    },
    {
        "Chapter": "1",
        "sentence_range": "7265-7268",
        "Text": "Hence, it is a variable Also its value depends upon the\noutcome of a random experiment and, hence, is called random variable A random\nvariable is usually denoted by X If you recall the definition of a function, you will realise that the random variable X\nis really speaking a function whose domain is the set of outcomes (or sample space) of\na random experiment"
    },
    {
        "Chapter": "1",
        "sentence_range": "7266-7269",
        "Text": "Also its value depends upon the\noutcome of a random experiment and, hence, is called random variable A random\nvariable is usually denoted by X If you recall the definition of a function, you will realise that the random variable X\nis really speaking a function whose domain is the set of outcomes (or sample space) of\na random experiment A random variable can take any real value, therefore, its\nco-domain is the set of real numbers"
    },
    {
        "Chapter": "1",
        "sentence_range": "7267-7270",
        "Text": "A random\nvariable is usually denoted by X If you recall the definition of a function, you will realise that the random variable X\nis really speaking a function whose domain is the set of outcomes (or sample space) of\na random experiment A random variable can take any real value, therefore, its\nco-domain is the set of real numbers Hence, a random variable can be defined as\nfollows :\nDefinition 4 A random variable is a real valued function whose domain is the sample\nspace of a random experiment"
    },
    {
        "Chapter": "1",
        "sentence_range": "7268-7271",
        "Text": "If you recall the definition of a function, you will realise that the random variable X\nis really speaking a function whose domain is the set of outcomes (or sample space) of\na random experiment A random variable can take any real value, therefore, its\nco-domain is the set of real numbers Hence, a random variable can be defined as\nfollows :\nDefinition 4 A random variable is a real valued function whose domain is the sample\nspace of a random experiment For example, let us consider the experiment of tossing a coin two times in succession"
    },
    {
        "Chapter": "1",
        "sentence_range": "7269-7272",
        "Text": "A random variable can take any real value, therefore, its\nco-domain is the set of real numbers Hence, a random variable can be defined as\nfollows :\nDefinition 4 A random variable is a real valued function whose domain is the sample\nspace of a random experiment For example, let us consider the experiment of tossing a coin two times in succession The sample space of the experiment is  S = {HH, HT, TH, TT}"
    },
    {
        "Chapter": "1",
        "sentence_range": "7270-7273",
        "Text": "Hence, a random variable can be defined as\nfollows :\nDefinition 4 A random variable is a real valued function whose domain is the sample\nspace of a random experiment For example, let us consider the experiment of tossing a coin two times in succession The sample space of the experiment is  S = {HH, HT, TH, TT} If X denotes the number of heads obtained, then X is a random variable and for\neach outcome, its value is as given below :\nX(HH) = 2, X (HT) = 1, X (TH) = 1, X (TT) = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7271-7274",
        "Text": "For example, let us consider the experiment of tossing a coin two times in succession The sample space of the experiment is  S = {HH, HT, TH, TT} If X denotes the number of heads obtained, then X is a random variable and for\neach outcome, its value is as given below :\nX(HH) = 2, X (HT) = 1, X (TH) = 1, X (TT) = 0 More than  one random variables can be defined on the same sample space"
    },
    {
        "Chapter": "1",
        "sentence_range": "7272-7275",
        "Text": "The sample space of the experiment is  S = {HH, HT, TH, TT} If X denotes the number of heads obtained, then X is a random variable and for\neach outcome, its value is as given below :\nX(HH) = 2, X (HT) = 1, X (TH) = 1, X (TT) = 0 More than  one random variables can be defined on the same sample space For\nexample, let Y denote the number of heads minus the number of tails for each outcome\nof the above sample space S"
    },
    {
        "Chapter": "1",
        "sentence_range": "7273-7276",
        "Text": "If X denotes the number of heads obtained, then X is a random variable and for\neach outcome, its value is as given below :\nX(HH) = 2, X (HT) = 1, X (TH) = 1, X (TT) = 0 More than  one random variables can be defined on the same sample space For\nexample, let Y denote the number of heads minus the number of tails for each outcome\nof the above sample space S Then\nY(HH) = 2, Y (HT) = 0, Y (TH) = 0, Y (TT) = \u2013 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "7274-7277",
        "Text": "More than  one random variables can be defined on the same sample space For\nexample, let Y denote the number of heads minus the number of tails for each outcome\nof the above sample space S Then\nY(HH) = 2, Y (HT) = 0, Y (TH) = 0, Y (TT) = \u2013 2 Thus, X and Y are two different random variables defined on the same sample\nspace S"
    },
    {
        "Chapter": "1",
        "sentence_range": "7275-7278",
        "Text": "For\nexample, let Y denote the number of heads minus the number of tails for each outcome\nof the above sample space S Then\nY(HH) = 2, Y (HT) = 0, Y (TH) = 0, Y (TT) = \u2013 2 Thus, X and Y are two different random variables defined on the same sample\nspace S Example  22 A person plays a game of tossing a coin thrice"
    },
    {
        "Chapter": "1",
        "sentence_range": "7276-7279",
        "Text": "Then\nY(HH) = 2, Y (HT) = 0, Y (TH) = 0, Y (TT) = \u2013 2 Thus, X and Y are two different random variables defined on the same sample\nspace S Example  22 A person plays a game of tossing a coin thrice For each head, he is\ngiven Rs 2 by the organiser of the game and for each tail, he has to give Rs 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "7277-7280",
        "Text": "Thus, X and Y are two different random variables defined on the same sample\nspace S Example  22 A person plays a game of tossing a coin thrice For each head, he is\ngiven Rs 2 by the organiser of the game and for each tail, he has to give Rs 1 50 to the\norganiser"
    },
    {
        "Chapter": "1",
        "sentence_range": "7278-7281",
        "Text": "Example  22 A person plays a game of tossing a coin thrice For each head, he is\ngiven Rs 2 by the organiser of the game and for each tail, he has to give Rs 1 50 to the\norganiser Let X denote the amount  gained or lost by the person"
    },
    {
        "Chapter": "1",
        "sentence_range": "7279-7282",
        "Text": "For each head, he is\ngiven Rs 2 by the organiser of the game and for each tail, he has to give Rs 1 50 to the\norganiser Let X denote the amount  gained or lost by the person Show that X is a\nrandom variable and exhibit it as a function on the sample space of the experiment"
    },
    {
        "Chapter": "1",
        "sentence_range": "7280-7283",
        "Text": "50 to the\norganiser Let X denote the amount  gained or lost by the person Show that X is a\nrandom variable and exhibit it as a function on the sample space of the experiment Solution X is a number whose values are defined on the outcomes of a random\nexperiment"
    },
    {
        "Chapter": "1",
        "sentence_range": "7281-7284",
        "Text": "Let X denote the amount  gained or lost by the person Show that X is a\nrandom variable and exhibit it as a function on the sample space of the experiment Solution X is a number whose values are defined on the outcomes of a random\nexperiment Therefore, X is a random variable"
    },
    {
        "Chapter": "1",
        "sentence_range": "7282-7285",
        "Text": "Show that X is a\nrandom variable and exhibit it as a function on the sample space of the experiment Solution X is a number whose values are defined on the outcomes of a random\nexperiment Therefore, X is a random variable Now, sample space of the experiment is\nS = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}\n\u00a9 NCERT\nnot to be republished\nPROBABILITY      559\nThen\nX (HHH) = Rs (2 \u00d7 3) = Rs 6\nX(HHT) = X (HTH) = X(THH) = Rs (2 \u00d7 2 \u2212 1 \u00d7 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "7283-7286",
        "Text": "Solution X is a number whose values are defined on the outcomes of a random\nexperiment Therefore, X is a random variable Now, sample space of the experiment is\nS = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}\n\u00a9 NCERT\nnot to be republished\nPROBABILITY      559\nThen\nX (HHH) = Rs (2 \u00d7 3) = Rs 6\nX(HHT) = X (HTH) = X(THH) = Rs (2 \u00d7 2 \u2212 1 \u00d7 1 50) = Rs 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "7284-7287",
        "Text": "Therefore, X is a random variable Now, sample space of the experiment is\nS = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}\n\u00a9 NCERT\nnot to be republished\nPROBABILITY      559\nThen\nX (HHH) = Rs (2 \u00d7 3) = Rs 6\nX(HHT) = X (HTH) = X(THH) = Rs (2 \u00d7 2 \u2212 1 \u00d7 1 50) = Rs 2 50\nX(HTT) = X(THT) = (TTH) = Rs (1 \u00d7 2) \u2013 (2 \u00d7 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "7285-7288",
        "Text": "Now, sample space of the experiment is\nS = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}\n\u00a9 NCERT\nnot to be republished\nPROBABILITY      559\nThen\nX (HHH) = Rs (2 \u00d7 3) = Rs 6\nX(HHT) = X (HTH) = X(THH) = Rs (2 \u00d7 2 \u2212 1 \u00d7 1 50) = Rs 2 50\nX(HTT) = X(THT) = (TTH) = Rs (1 \u00d7 2) \u2013 (2 \u00d7 1 50) = \u2013 Re 1\nand\nX(TTT) = \u2212 Rs (3 \u00d7 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "7286-7289",
        "Text": "50) = Rs 2 50\nX(HTT) = X(THT) = (TTH) = Rs (1 \u00d7 2) \u2013 (2 \u00d7 1 50) = \u2013 Re 1\nand\nX(TTT) = \u2212 Rs (3 \u00d7 1 50) =  \u2212 Rs 4"
    },
    {
        "Chapter": "1",
        "sentence_range": "7287-7290",
        "Text": "50\nX(HTT) = X(THT) = (TTH) = Rs (1 \u00d7 2) \u2013 (2 \u00d7 1 50) = \u2013 Re 1\nand\nX(TTT) = \u2212 Rs (3 \u00d7 1 50) =  \u2212 Rs 4 50\nwhere, minus sign shows the loss to the player"
    },
    {
        "Chapter": "1",
        "sentence_range": "7288-7291",
        "Text": "50) = \u2013 Re 1\nand\nX(TTT) = \u2212 Rs (3 \u00d7 1 50) =  \u2212 Rs 4 50\nwhere, minus sign shows the loss to the player Thus, for each element of the sample\nisspace, X takes a unique value, hence, X is a function on the sample space whose range\n{\u20131,  2"
    },
    {
        "Chapter": "1",
        "sentence_range": "7289-7292",
        "Text": "50) =  \u2212 Rs 4 50\nwhere, minus sign shows the loss to the player Thus, for each element of the sample\nisspace, X takes a unique value, hence, X is a function on the sample space whose range\n{\u20131,  2 50,  \u2013 4"
    },
    {
        "Chapter": "1",
        "sentence_range": "7290-7293",
        "Text": "50\nwhere, minus sign shows the loss to the player Thus, for each element of the sample\nisspace, X takes a unique value, hence, X is a function on the sample space whose range\n{\u20131,  2 50,  \u2013 4 50,  6}\nExample 23  A  bag contains 2 white and 1 red balls"
    },
    {
        "Chapter": "1",
        "sentence_range": "7291-7294",
        "Text": "Thus, for each element of the sample\nisspace, X takes a unique value, hence, X is a function on the sample space whose range\n{\u20131,  2 50,  \u2013 4 50,  6}\nExample 23  A  bag contains 2 white and 1 red balls One ball is drawn at random and\nthen put back in the box after noting its colour"
    },
    {
        "Chapter": "1",
        "sentence_range": "7292-7295",
        "Text": "50,  \u2013 4 50,  6}\nExample 23  A  bag contains 2 white and 1 red balls One ball is drawn at random and\nthen put back in the box after noting its colour The process is repeated again"
    },
    {
        "Chapter": "1",
        "sentence_range": "7293-7296",
        "Text": "50,  6}\nExample 23  A  bag contains 2 white and 1 red balls One ball is drawn at random and\nthen put back in the box after noting its colour The process is repeated again If X\ndenotes the number of red balls recorded in the two draws, describe X"
    },
    {
        "Chapter": "1",
        "sentence_range": "7294-7297",
        "Text": "One ball is drawn at random and\nthen put back in the box after noting its colour The process is repeated again If X\ndenotes the number of red balls recorded in the two draws, describe X Solution Let the balls in the bag be denoted by w1, w2, r"
    },
    {
        "Chapter": "1",
        "sentence_range": "7295-7298",
        "Text": "The process is repeated again If X\ndenotes the number of red balls recorded in the two draws, describe X Solution Let the balls in the bag be denoted by w1, w2, r Then the sample space is\nS = {w1 w1, w1 w2, w2 w2, w2 w1, w1 r, w2 r, r w1, r w2, r r}\nNow, for\n\u03c9 \u2208 S\nX (\u03c9) = number of red balls\nTherefore\nX({w1 w1}) = X({w1 w2}) = X({w2 w2}) = X({w2 w1}) = 0\nX({w1 r}) = X({w2 r}) = X({r w1}) = X({r w2}) = 1 and X({r r}) = 2\nThus, X is a random variable which can take values 0,  1 or 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "7296-7299",
        "Text": "If X\ndenotes the number of red balls recorded in the two draws, describe X Solution Let the balls in the bag be denoted by w1, w2, r Then the sample space is\nS = {w1 w1, w1 w2, w2 w2, w2 w1, w1 r, w2 r, r w1, r w2, r r}\nNow, for\n\u03c9 \u2208 S\nX (\u03c9) = number of red balls\nTherefore\nX({w1 w1}) = X({w1 w2}) = X({w2 w2}) = X({w2 w1}) = 0\nX({w1 r}) = X({w2 r}) = X({r w1}) = X({r w2}) = 1 and X({r r}) = 2\nThus, X is a random variable which can take values 0,  1 or 2 13"
    },
    {
        "Chapter": "1",
        "sentence_range": "7297-7300",
        "Text": "Solution Let the balls in the bag be denoted by w1, w2, r Then the sample space is\nS = {w1 w1, w1 w2, w2 w2, w2 w1, w1 r, w2 r, r w1, r w2, r r}\nNow, for\n\u03c9 \u2208 S\nX (\u03c9) = number of red balls\nTherefore\nX({w1 w1}) = X({w1 w2}) = X({w2 w2}) = X({w2 w1}) = 0\nX({w1 r}) = X({w2 r}) = X({r w1}) = X({r w2}) = 1 and X({r r}) = 2\nThus, X is a random variable which can take values 0,  1 or 2 13 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "7298-7301",
        "Text": "Then the sample space is\nS = {w1 w1, w1 w2, w2 w2, w2 w1, w1 r, w2 r, r w1, r w2, r r}\nNow, for\n\u03c9 \u2208 S\nX (\u03c9) = number of red balls\nTherefore\nX({w1 w1}) = X({w1 w2}) = X({w2 w2}) = X({w2 w1}) = 0\nX({w1 r}) = X({w2 r}) = X({r w1}) = X({r w2}) = 1 and X({r r}) = 2\nThus, X is a random variable which can take values 0,  1 or 2 13 6 1 Probability distribution of a random variable\nLet us  look at the experiment of selecting one family out of ten families f1,  f2 ,"
    },
    {
        "Chapter": "1",
        "sentence_range": "7299-7302",
        "Text": "13 6 1 Probability distribution of a random variable\nLet us  look at the experiment of selecting one family out of ten families f1,  f2 , , f10 in\nsuch a manner that each family  is equally likely to be selected"
    },
    {
        "Chapter": "1",
        "sentence_range": "7300-7303",
        "Text": "6 1 Probability distribution of a random variable\nLet us  look at the experiment of selecting one family out of ten families f1,  f2 , , f10 in\nsuch a manner that each family  is equally likely to be selected Let the families f1,  f2,"
    },
    {
        "Chapter": "1",
        "sentence_range": "7301-7304",
        "Text": "1 Probability distribution of a random variable\nLet us  look at the experiment of selecting one family out of ten families f1,  f2 , , f10 in\nsuch a manner that each family  is equally likely to be selected Let the families f1,  f2, , f10  have 3, 4, 3, 2, 5, 4, 3, 6, 4, 5 members, respectively"
    },
    {
        "Chapter": "1",
        "sentence_range": "7302-7305",
        "Text": ", f10 in\nsuch a manner that each family  is equally likely to be selected Let the families f1,  f2, , f10  have 3, 4, 3, 2, 5, 4, 3, 6, 4, 5 members, respectively Let us select a family and note down the number of members in the family denoting\nX"
    },
    {
        "Chapter": "1",
        "sentence_range": "7303-7306",
        "Text": "Let the families f1,  f2, , f10  have 3, 4, 3, 2, 5, 4, 3, 6, 4, 5 members, respectively Let us select a family and note down the number of members in the family denoting\nX Clearly, X is a random variable defined as below :\nX(f1) = 3, X(f2) = 4, X(f3) = 3, X(f4) = 2, X(f5) = 5,\nX(f6) = 4, X(f7) = 3, X(f8) = 6, X(f9) = 4, X(f10) = 5\nThus, X can take any value 2,3,4,5 or 6 depending upon which family is selected"
    },
    {
        "Chapter": "1",
        "sentence_range": "7304-7307",
        "Text": ", f10  have 3, 4, 3, 2, 5, 4, 3, 6, 4, 5 members, respectively Let us select a family and note down the number of members in the family denoting\nX Clearly, X is a random variable defined as below :\nX(f1) = 3, X(f2) = 4, X(f3) = 3, X(f4) = 2, X(f5) = 5,\nX(f6) = 4, X(f7) = 3, X(f8) = 6, X(f9) = 4, X(f10) = 5\nThus, X can take any value 2,3,4,5 or 6 depending upon which family is selected Now,  X will take the value 2 when the family f4 is selected"
    },
    {
        "Chapter": "1",
        "sentence_range": "7305-7308",
        "Text": "Let us select a family and note down the number of members in the family denoting\nX Clearly, X is a random variable defined as below :\nX(f1) = 3, X(f2) = 4, X(f3) = 3, X(f4) = 2, X(f5) = 5,\nX(f6) = 4, X(f7) = 3, X(f8) = 6, X(f9) = 4, X(f10) = 5\nThus, X can take any value 2,3,4,5 or 6 depending upon which family is selected Now,  X will take the value 2 when the family f4 is selected X can take the value\n3 when any one of the families f1, f3, f7 is selected"
    },
    {
        "Chapter": "1",
        "sentence_range": "7306-7309",
        "Text": "Clearly, X is a random variable defined as below :\nX(f1) = 3, X(f2) = 4, X(f3) = 3, X(f4) = 2, X(f5) = 5,\nX(f6) = 4, X(f7) = 3, X(f8) = 6, X(f9) = 4, X(f10) = 5\nThus, X can take any value 2,3,4,5 or 6 depending upon which family is selected Now,  X will take the value 2 when the family f4 is selected X can take the value\n3 when any one of the families f1, f3, f7 is selected Similarly,\nX = 4,  when family f2, f6 or f9 is selected,\nX = 5,  when family f5 or f10 is selected\nand\nX = 6, when family f8  is selected"
    },
    {
        "Chapter": "1",
        "sentence_range": "7307-7310",
        "Text": "Now,  X will take the value 2 when the family f4 is selected X can take the value\n3 when any one of the families f1, f3, f7 is selected Similarly,\nX = 4,  when family f2, f6 or f9 is selected,\nX = 5,  when family f5 or f10 is selected\nand\nX = 6, when family f8  is selected \u00a9 NCERT\nnot to be republished\n 560\nMATHEMATICS\nSince we had assumed that each family is equally likely to be selected, the probability\nthat family f4 is selected is 1\n10"
    },
    {
        "Chapter": "1",
        "sentence_range": "7308-7311",
        "Text": "X can take the value\n3 when any one of the families f1, f3, f7 is selected Similarly,\nX = 4,  when family f2, f6 or f9 is selected,\nX = 5,  when family f5 or f10 is selected\nand\nX = 6, when family f8  is selected \u00a9 NCERT\nnot to be republished\n 560\nMATHEMATICS\nSince we had assumed that each family is equally likely to be selected, the probability\nthat family f4 is selected is 1\n10 Thus, the probability that X can take the value 2 is 1\n10"
    },
    {
        "Chapter": "1",
        "sentence_range": "7309-7312",
        "Text": "Similarly,\nX = 4,  when family f2, f6 or f9 is selected,\nX = 5,  when family f5 or f10 is selected\nand\nX = 6, when family f8  is selected \u00a9 NCERT\nnot to be republished\n 560\nMATHEMATICS\nSince we had assumed that each family is equally likely to be selected, the probability\nthat family f4 is selected is 1\n10 Thus, the probability that X can take the value 2 is 1\n10 We write P(X = 2) = 1\n10\nAlso, the probability that any one of the families f1, f3 or f7 is selected is\nP({f1, f3, f7}) = 3\n10\nThus, the probability that X can take the value 3 = 3\n10\nWe write\nP(X = 3) = 3\n10\nSimilarly, we obtain\nP(X = 4) = P({f2, f6, f9}) = 3\n10\nP(X = 5) = P({f5, f10}) = 2\n10\nand\nP(X = 6) = P({f8}) = 1\n10\nSuch a description giving the values of the random variable along with the\ncorresponding probabilities is called the probability distribution of the random\nvariable X"
    },
    {
        "Chapter": "1",
        "sentence_range": "7310-7313",
        "Text": "\u00a9 NCERT\nnot to be republished\n 560\nMATHEMATICS\nSince we had assumed that each family is equally likely to be selected, the probability\nthat family f4 is selected is 1\n10 Thus, the probability that X can take the value 2 is 1\n10 We write P(X = 2) = 1\n10\nAlso, the probability that any one of the families f1, f3 or f7 is selected is\nP({f1, f3, f7}) = 3\n10\nThus, the probability that X can take the value 3 = 3\n10\nWe write\nP(X = 3) = 3\n10\nSimilarly, we obtain\nP(X = 4) = P({f2, f6, f9}) = 3\n10\nP(X = 5) = P({f5, f10}) = 2\n10\nand\nP(X = 6) = P({f8}) = 1\n10\nSuch a description giving the values of the random variable along with the\ncorresponding probabilities is called the probability distribution of the random\nvariable X In general, the probability distribution of a random variable X is defined as follows:\nDefinition 5 The probability distribution of a random variable X is the system of numbers\nX\n:\nx1\nx2"
    },
    {
        "Chapter": "1",
        "sentence_range": "7311-7314",
        "Text": "Thus, the probability that X can take the value 2 is 1\n10 We write P(X = 2) = 1\n10\nAlso, the probability that any one of the families f1, f3 or f7 is selected is\nP({f1, f3, f7}) = 3\n10\nThus, the probability that X can take the value 3 = 3\n10\nWe write\nP(X = 3) = 3\n10\nSimilarly, we obtain\nP(X = 4) = P({f2, f6, f9}) = 3\n10\nP(X = 5) = P({f5, f10}) = 2\n10\nand\nP(X = 6) = P({f8}) = 1\n10\nSuch a description giving the values of the random variable along with the\ncorresponding probabilities is called the probability distribution of the random\nvariable X In general, the probability distribution of a random variable X is defined as follows:\nDefinition 5 The probability distribution of a random variable X is the system of numbers\nX\n:\nx1\nx2 xn\nP(X)\n:\np 1\np 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "7312-7315",
        "Text": "We write P(X = 2) = 1\n10\nAlso, the probability that any one of the families f1, f3 or f7 is selected is\nP({f1, f3, f7}) = 3\n10\nThus, the probability that X can take the value 3 = 3\n10\nWe write\nP(X = 3) = 3\n10\nSimilarly, we obtain\nP(X = 4) = P({f2, f6, f9}) = 3\n10\nP(X = 5) = P({f5, f10}) = 2\n10\nand\nP(X = 6) = P({f8}) = 1\n10\nSuch a description giving the values of the random variable along with the\ncorresponding probabilities is called the probability distribution of the random\nvariable X In general, the probability distribution of a random variable X is defined as follows:\nDefinition 5 The probability distribution of a random variable X is the system of numbers\nX\n:\nx1\nx2 xn\nP(X)\n:\np 1\np 2 p n\nwhere,\n1\n0,\nn\ni\ni\ni\np\np\n \n \n = 1, i = 1, 2,"
    },
    {
        "Chapter": "1",
        "sentence_range": "7313-7316",
        "Text": "In general, the probability distribution of a random variable X is defined as follows:\nDefinition 5 The probability distribution of a random variable X is the system of numbers\nX\n:\nx1\nx2 xn\nP(X)\n:\np 1\np 2 p n\nwhere,\n1\n0,\nn\ni\ni\ni\np\np\n \n \n = 1, i = 1, 2, , n\nThe real numbers x1, x2,"
    },
    {
        "Chapter": "1",
        "sentence_range": "7314-7317",
        "Text": "xn\nP(X)\n:\np 1\np 2 p n\nwhere,\n1\n0,\nn\ni\ni\ni\np\np\n \n \n = 1, i = 1, 2, , n\nThe real numbers x1, x2, , xn are the possible values of the random variable X and\npi (i = 1,2,"
    },
    {
        "Chapter": "1",
        "sentence_range": "7315-7318",
        "Text": "p n\nwhere,\n1\n0,\nn\ni\ni\ni\np\np\n \n \n = 1, i = 1, 2, , n\nThe real numbers x1, x2, , xn are the possible values of the random variable X and\npi (i = 1,2, , n) is the probability of the random variable X taking the value xi i"
    },
    {
        "Chapter": "1",
        "sentence_range": "7316-7319",
        "Text": ", n\nThe real numbers x1, x2, , xn are the possible values of the random variable X and\npi (i = 1,2, , n) is the probability of the random variable X taking the value xi i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "7317-7320",
        "Text": ", xn are the possible values of the random variable X and\npi (i = 1,2, , n) is the probability of the random variable X taking the value xi i e ,\nP(X = xi) = pi\n\u00a9 NCERT\nnot to be republished\nPROBABILITY      561\n\ufffdNote  If xi is one of the possible values of a random variable X, the statement\nX = xi is true only at some point (s) of the sample space"
    },
    {
        "Chapter": "1",
        "sentence_range": "7318-7321",
        "Text": ", n) is the probability of the random variable X taking the value xi i e ,\nP(X = xi) = pi\n\u00a9 NCERT\nnot to be republished\nPROBABILITY      561\n\ufffdNote  If xi is one of the possible values of a random variable X, the statement\nX = xi is true only at some point (s) of the sample space Hence, the probability that\nX takes value xi is always nonzero, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "7319-7322",
        "Text": "e ,\nP(X = xi) = pi\n\u00a9 NCERT\nnot to be republished\nPROBABILITY      561\n\ufffdNote  If xi is one of the possible values of a random variable X, the statement\nX = xi is true only at some point (s) of the sample space Hence, the probability that\nX takes value xi is always nonzero, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "7320-7323",
        "Text": ",\nP(X = xi) = pi\n\u00a9 NCERT\nnot to be republished\nPROBABILITY      561\n\ufffdNote  If xi is one of the possible values of a random variable X, the statement\nX = xi is true only at some point (s) of the sample space Hence, the probability that\nX takes value xi is always nonzero, i e P(X = xi) \u2260 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7321-7324",
        "Text": "Hence, the probability that\nX takes value xi is always nonzero, i e P(X = xi) \u2260 0 Also for all possible values of the random variable X, all elements of the sample\nspace are covered"
    },
    {
        "Chapter": "1",
        "sentence_range": "7322-7325",
        "Text": "e P(X = xi) \u2260 0 Also for all possible values of the random variable X, all elements of the sample\nspace are covered Hence, the sum of all the probabilities in a probability distribution\nmust be one"
    },
    {
        "Chapter": "1",
        "sentence_range": "7323-7326",
        "Text": "P(X = xi) \u2260 0 Also for all possible values of the random variable X, all elements of the sample\nspace are covered Hence, the sum of all the probabilities in a probability distribution\nmust be one Example 24 Two cards are drawn successively with replacement from a well-shuffled\ndeck of 52 cards"
    },
    {
        "Chapter": "1",
        "sentence_range": "7324-7327",
        "Text": "Also for all possible values of the random variable X, all elements of the sample\nspace are covered Hence, the sum of all the probabilities in a probability distribution\nmust be one Example 24 Two cards are drawn successively with replacement from a well-shuffled\ndeck of 52 cards Find the probability distribution of the number of aces"
    },
    {
        "Chapter": "1",
        "sentence_range": "7325-7328",
        "Text": "Hence, the sum of all the probabilities in a probability distribution\nmust be one Example 24 Two cards are drawn successively with replacement from a well-shuffled\ndeck of 52 cards Find the probability distribution of the number of aces Solution The number of aces is a random variable"
    },
    {
        "Chapter": "1",
        "sentence_range": "7326-7329",
        "Text": "Example 24 Two cards are drawn successively with replacement from a well-shuffled\ndeck of 52 cards Find the probability distribution of the number of aces Solution The number of aces is a random variable Let it be denoted by X"
    },
    {
        "Chapter": "1",
        "sentence_range": "7327-7330",
        "Text": "Find the probability distribution of the number of aces Solution The number of aces is a random variable Let it be denoted by X Clearly, X\ncan take the values 0, 1, or 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "7328-7331",
        "Text": "Solution The number of aces is a random variable Let it be denoted by X Clearly, X\ncan take the values 0, 1, or 2 Now, since the draws are done with replacement, therefore, the two draws form\nindependent experiments"
    },
    {
        "Chapter": "1",
        "sentence_range": "7329-7332",
        "Text": "Let it be denoted by X Clearly, X\ncan take the values 0, 1, or 2 Now, since the draws are done with replacement, therefore, the two draws form\nindependent experiments Therefore,\nP(X = 0) = P(non-ace and non-ace)\n= P(non-ace) \u00d7 P(non-ace)\n= 48\n48\n144\n52\n52\n169\n\u00d7\n=\nP(X = 1) = P(ace and non-ace or non-ace and ace)\n= P(ace and non-ace) + P(non-ace and ace)\n= P(ace)"
    },
    {
        "Chapter": "1",
        "sentence_range": "7330-7333",
        "Text": "Clearly, X\ncan take the values 0, 1, or 2 Now, since the draws are done with replacement, therefore, the two draws form\nindependent experiments Therefore,\nP(X = 0) = P(non-ace and non-ace)\n= P(non-ace) \u00d7 P(non-ace)\n= 48\n48\n144\n52\n52\n169\n\u00d7\n=\nP(X = 1) = P(ace and non-ace or non-ace and ace)\n= P(ace and non-ace) + P(non-ace and ace)\n= P(ace) P(non-ace) + P (non-ace)"
    },
    {
        "Chapter": "1",
        "sentence_range": "7331-7334",
        "Text": "Now, since the draws are done with replacement, therefore, the two draws form\nindependent experiments Therefore,\nP(X = 0) = P(non-ace and non-ace)\n= P(non-ace) \u00d7 P(non-ace)\n= 48\n48\n144\n52\n52\n169\n\u00d7\n=\nP(X = 1) = P(ace and non-ace or non-ace and ace)\n= P(ace and non-ace) + P(non-ace and ace)\n= P(ace) P(non-ace) + P (non-ace) P(ace)\n= 4\n48\n48\n4\n24\n52\n52\n52\n52\n169\n\u00d7\n+\n\u00d7\n=\nand\nP(X = 2) = P (ace and ace)\n= 4\n4\n1\n52\n52\n169\n \n \nThus, the required probability distribution is\nX\n0\n1\n2\nP(X)\n144\n169\n24\n169\n1\n169\nExample 25 Find the probability distribution of number of doublets in three throws of\na pair of dice"
    },
    {
        "Chapter": "1",
        "sentence_range": "7332-7335",
        "Text": "Therefore,\nP(X = 0) = P(non-ace and non-ace)\n= P(non-ace) \u00d7 P(non-ace)\n= 48\n48\n144\n52\n52\n169\n\u00d7\n=\nP(X = 1) = P(ace and non-ace or non-ace and ace)\n= P(ace and non-ace) + P(non-ace and ace)\n= P(ace) P(non-ace) + P (non-ace) P(ace)\n= 4\n48\n48\n4\n24\n52\n52\n52\n52\n169\n\u00d7\n+\n\u00d7\n=\nand\nP(X = 2) = P (ace and ace)\n= 4\n4\n1\n52\n52\n169\n \n \nThus, the required probability distribution is\nX\n0\n1\n2\nP(X)\n144\n169\n24\n169\n1\n169\nExample 25 Find the probability distribution of number of doublets in three throws of\na pair of dice \u00a9 NCERT\nnot to be republished\n 562\nMATHEMATICS\nSolution Let X denote the number of doublets"
    },
    {
        "Chapter": "1",
        "sentence_range": "7333-7336",
        "Text": "P(non-ace) + P (non-ace) P(ace)\n= 4\n48\n48\n4\n24\n52\n52\n52\n52\n169\n\u00d7\n+\n\u00d7\n=\nand\nP(X = 2) = P (ace and ace)\n= 4\n4\n1\n52\n52\n169\n \n \nThus, the required probability distribution is\nX\n0\n1\n2\nP(X)\n144\n169\n24\n169\n1\n169\nExample 25 Find the probability distribution of number of doublets in three throws of\na pair of dice \u00a9 NCERT\nnot to be republished\n 562\nMATHEMATICS\nSolution Let X denote the number of doublets Possible doublets are\n(1,1) , (2,2),  (3,3),  (4,4),  (5,5), (6,6)\nClearly, X can take the value 0, 1, 2, or 3"
    },
    {
        "Chapter": "1",
        "sentence_range": "7334-7337",
        "Text": "P(ace)\n= 4\n48\n48\n4\n24\n52\n52\n52\n52\n169\n\u00d7\n+\n\u00d7\n=\nand\nP(X = 2) = P (ace and ace)\n= 4\n4\n1\n52\n52\n169\n \n \nThus, the required probability distribution is\nX\n0\n1\n2\nP(X)\n144\n169\n24\n169\n1\n169\nExample 25 Find the probability distribution of number of doublets in three throws of\na pair of dice \u00a9 NCERT\nnot to be republished\n 562\nMATHEMATICS\nSolution Let X denote the number of doublets Possible doublets are\n(1,1) , (2,2),  (3,3),  (4,4),  (5,5), (6,6)\nClearly, X can take the value 0, 1, 2, or 3 Probability of getting a doublet \n6\n1\n36\n6\n \n \nProbability of not getting a doublet \n1\n5\n1\n6\n6\n \n \n \nNow\nP(X = 0) = P (no doublet) = 5\n5\n5\n125\n6\n6\n6\n216\n \n \n \nP(X = 1) = P (one doublet and two non-doublets)\n= 1\n5\n5\n5\n1\n5\n5\n5\n1\n6\n6\n6\n6\n6\n6\n6\n6\n6\n \n \n \n \n \n \n \n \n=\n2\n2\n1\n5\n75\n3 6\n216\n6\n \n \n \n \n \n \n \n \nP(X = 2) = P (two doublets and one non-doublet)\n=\n2\n1\n1\n5\n1\n5\n1\n5\n1\n1\n1\n5\n15\n3\n6\n6\n6\n6\n6\n6\n6\n6\n6\n6\n216\n 6\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nand\nP(X = 3) = P (three doublets)\n= 1\n1\n1\n1\n6\n6\n6\n216\n \n \n \nThus,  the required probability distribution is\nX\n0\n1\n2\n3\nP(X)\n125\n216\n75\n216\n15\n216\n1\n216\nVerification Sum of the probabilities\n1\nn\ni\ni\np\n=\u2211\n = 125\n75\n15\n1\n216\n216\n216\n216\n \n \n \n= 125\n75\n15\n1\n216\n1\n216\n216\n \n \n \n \n\u00a9 NCERT\nnot to be republished\nPROBABILITY      563\nExample 26 Let X denote the number of hours you study during a randomly selected\nschool day"
    },
    {
        "Chapter": "1",
        "sentence_range": "7335-7338",
        "Text": "\u00a9 NCERT\nnot to be republished\n 562\nMATHEMATICS\nSolution Let X denote the number of doublets Possible doublets are\n(1,1) , (2,2),  (3,3),  (4,4),  (5,5), (6,6)\nClearly, X can take the value 0, 1, 2, or 3 Probability of getting a doublet \n6\n1\n36\n6\n \n \nProbability of not getting a doublet \n1\n5\n1\n6\n6\n \n \n \nNow\nP(X = 0) = P (no doublet) = 5\n5\n5\n125\n6\n6\n6\n216\n \n \n \nP(X = 1) = P (one doublet and two non-doublets)\n= 1\n5\n5\n5\n1\n5\n5\n5\n1\n6\n6\n6\n6\n6\n6\n6\n6\n6\n \n \n \n \n \n \n \n \n=\n2\n2\n1\n5\n75\n3 6\n216\n6\n \n \n \n \n \n \n \n \nP(X = 2) = P (two doublets and one non-doublet)\n=\n2\n1\n1\n5\n1\n5\n1\n5\n1\n1\n1\n5\n15\n3\n6\n6\n6\n6\n6\n6\n6\n6\n6\n6\n216\n 6\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nand\nP(X = 3) = P (three doublets)\n= 1\n1\n1\n1\n6\n6\n6\n216\n \n \n \nThus,  the required probability distribution is\nX\n0\n1\n2\n3\nP(X)\n125\n216\n75\n216\n15\n216\n1\n216\nVerification Sum of the probabilities\n1\nn\ni\ni\np\n=\u2211\n = 125\n75\n15\n1\n216\n216\n216\n216\n \n \n \n= 125\n75\n15\n1\n216\n1\n216\n216\n \n \n \n \n\u00a9 NCERT\nnot to be republished\nPROBABILITY      563\nExample 26 Let X denote the number of hours you study during a randomly selected\nschool day The probability that X can take the values x, has the following form, where\nk is some unknown constant"
    },
    {
        "Chapter": "1",
        "sentence_range": "7336-7339",
        "Text": "Possible doublets are\n(1,1) , (2,2),  (3,3),  (4,4),  (5,5), (6,6)\nClearly, X can take the value 0, 1, 2, or 3 Probability of getting a doublet \n6\n1\n36\n6\n \n \nProbability of not getting a doublet \n1\n5\n1\n6\n6\n \n \n \nNow\nP(X = 0) = P (no doublet) = 5\n5\n5\n125\n6\n6\n6\n216\n \n \n \nP(X = 1) = P (one doublet and two non-doublets)\n= 1\n5\n5\n5\n1\n5\n5\n5\n1\n6\n6\n6\n6\n6\n6\n6\n6\n6\n \n \n \n \n \n \n \n \n=\n2\n2\n1\n5\n75\n3 6\n216\n6\n \n \n \n \n \n \n \n \nP(X = 2) = P (two doublets and one non-doublet)\n=\n2\n1\n1\n5\n1\n5\n1\n5\n1\n1\n1\n5\n15\n3\n6\n6\n6\n6\n6\n6\n6\n6\n6\n6\n216\n 6\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nand\nP(X = 3) = P (three doublets)\n= 1\n1\n1\n1\n6\n6\n6\n216\n \n \n \nThus,  the required probability distribution is\nX\n0\n1\n2\n3\nP(X)\n125\n216\n75\n216\n15\n216\n1\n216\nVerification Sum of the probabilities\n1\nn\ni\ni\np\n=\u2211\n = 125\n75\n15\n1\n216\n216\n216\n216\n \n \n \n= 125\n75\n15\n1\n216\n1\n216\n216\n \n \n \n \n\u00a9 NCERT\nnot to be republished\nPROBABILITY      563\nExample 26 Let X denote the number of hours you study during a randomly selected\nschool day The probability that X can take the values x, has the following form, where\nk is some unknown constant P(X = x) =\n0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7337-7340",
        "Text": "Probability of getting a doublet \n6\n1\n36\n6\n \n \nProbability of not getting a doublet \n1\n5\n1\n6\n6\n \n \n \nNow\nP(X = 0) = P (no doublet) = 5\n5\n5\n125\n6\n6\n6\n216\n \n \n \nP(X = 1) = P (one doublet and two non-doublets)\n= 1\n5\n5\n5\n1\n5\n5\n5\n1\n6\n6\n6\n6\n6\n6\n6\n6\n6\n \n \n \n \n \n \n \n \n=\n2\n2\n1\n5\n75\n3 6\n216\n6\n \n \n \n \n \n \n \n \nP(X = 2) = P (two doublets and one non-doublet)\n=\n2\n1\n1\n5\n1\n5\n1\n5\n1\n1\n1\n5\n15\n3\n6\n6\n6\n6\n6\n6\n6\n6\n6\n6\n216\n 6\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nand\nP(X = 3) = P (three doublets)\n= 1\n1\n1\n1\n6\n6\n6\n216\n \n \n \nThus,  the required probability distribution is\nX\n0\n1\n2\n3\nP(X)\n125\n216\n75\n216\n15\n216\n1\n216\nVerification Sum of the probabilities\n1\nn\ni\ni\np\n=\u2211\n = 125\n75\n15\n1\n216\n216\n216\n216\n \n \n \n= 125\n75\n15\n1\n216\n1\n216\n216\n \n \n \n \n\u00a9 NCERT\nnot to be republished\nPROBABILITY      563\nExample 26 Let X denote the number of hours you study during a randomly selected\nschool day The probability that X can take the values x, has the following form, where\nk is some unknown constant P(X = x) =\n0 1, if\n0\n, if\n1or2\n(5\n), if\n3or4\n0, otherwise\n=\n\u23aa\u23a7\n=\n\u23aa\u23a8\n\u2212\n=\n\u23aa\n\u23aa\u23a9\nx\nkx\nx\nk\nx\nx\n(a)\nFind the value of k"
    },
    {
        "Chapter": "1",
        "sentence_range": "7338-7341",
        "Text": "The probability that X can take the values x, has the following form, where\nk is some unknown constant P(X = x) =\n0 1, if\n0\n, if\n1or2\n(5\n), if\n3or4\n0, otherwise\n=\n\u23aa\u23a7\n=\n\u23aa\u23a8\n\u2212\n=\n\u23aa\n\u23aa\u23a9\nx\nkx\nx\nk\nx\nx\n(a)\nFind the value of k (b)\nWhat is the probability that you study at least two hours"
    },
    {
        "Chapter": "1",
        "sentence_range": "7339-7342",
        "Text": "P(X = x) =\n0 1, if\n0\n, if\n1or2\n(5\n), if\n3or4\n0, otherwise\n=\n\u23aa\u23a7\n=\n\u23aa\u23a8\n\u2212\n=\n\u23aa\n\u23aa\u23a9\nx\nkx\nx\nk\nx\nx\n(a)\nFind the value of k (b)\nWhat is the probability that you study at least two hours Exactly two hours"
    },
    {
        "Chapter": "1",
        "sentence_range": "7340-7343",
        "Text": "1, if\n0\n, if\n1or2\n(5\n), if\n3or4\n0, otherwise\n=\n\u23aa\u23a7\n=\n\u23aa\u23a8\n\u2212\n=\n\u23aa\n\u23aa\u23a9\nx\nkx\nx\nk\nx\nx\n(a)\nFind the value of k (b)\nWhat is the probability that you study at least two hours Exactly two hours At\nmost two hours"
    },
    {
        "Chapter": "1",
        "sentence_range": "7341-7344",
        "Text": "(b)\nWhat is the probability that you study at least two hours Exactly two hours At\nmost two hours Solution The probability distribution of X is\nX\n0\n1\n2\n3\n4\nP(X)\n0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7342-7345",
        "Text": "Exactly two hours At\nmost two hours Solution The probability distribution of X is\nX\n0\n1\n2\n3\n4\nP(X)\n0 1\nk\n2k\n2k\nk\n(a)\nWe know that\n1\nn\ni\ni\np\n=\u2211\n = 1\n       Therefore\n0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7343-7346",
        "Text": "At\nmost two hours Solution The probability distribution of X is\nX\n0\n1\n2\n3\n4\nP(X)\n0 1\nk\n2k\n2k\nk\n(a)\nWe know that\n1\nn\ni\ni\np\n=\u2211\n = 1\n       Therefore\n0 1 + k + 2k + 2k + k = 1\n        i"
    },
    {
        "Chapter": "1",
        "sentence_range": "7344-7347",
        "Text": "Solution The probability distribution of X is\nX\n0\n1\n2\n3\n4\nP(X)\n0 1\nk\n2k\n2k\nk\n(a)\nWe know that\n1\nn\ni\ni\np\n=\u2211\n = 1\n       Therefore\n0 1 + k + 2k + 2k + k = 1\n        i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "7345-7348",
        "Text": "1\nk\n2k\n2k\nk\n(a)\nWe know that\n1\nn\ni\ni\np\n=\u2211\n = 1\n       Therefore\n0 1 + k + 2k + 2k + k = 1\n        i e k = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7346-7349",
        "Text": "1 + k + 2k + 2k + k = 1\n        i e k = 0 15\n(b)\nP(you study at least two hours)\n= P(X \u2265 2)\n= P(X = 2) + P (X = 3) + P (X = 4)\n= 2k + 2k + k = 5k = 5 \u00d7 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7347-7350",
        "Text": "e k = 0 15\n(b)\nP(you study at least two hours)\n= P(X \u2265 2)\n= P(X = 2) + P (X = 3) + P (X = 4)\n= 2k + 2k + k = 5k = 5 \u00d7 0 15 = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7348-7351",
        "Text": "k = 0 15\n(b)\nP(you study at least two hours)\n= P(X \u2265 2)\n= P(X = 2) + P (X = 3) + P (X = 4)\n= 2k + 2k + k = 5k = 5 \u00d7 0 15 = 0 75\nP(you study exactly two hours)\n= P(X = 2)\n= 2k = 2 \u00d7 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7349-7352",
        "Text": "15\n(b)\nP(you study at least two hours)\n= P(X \u2265 2)\n= P(X = 2) + P (X = 3) + P (X = 4)\n= 2k + 2k + k = 5k = 5 \u00d7 0 15 = 0 75\nP(you study exactly two hours)\n= P(X = 2)\n= 2k = 2 \u00d7 0 15 = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7350-7353",
        "Text": "15 = 0 75\nP(you study exactly two hours)\n= P(X = 2)\n= 2k = 2 \u00d7 0 15 = 0 3\nP(you study at most two hours)\n= P(X \u2264 2)\n= P (X = 0) + P(X = 1) + P(X = 2)\n= 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7351-7354",
        "Text": "75\nP(you study exactly two hours)\n= P(X = 2)\n= 2k = 2 \u00d7 0 15 = 0 3\nP(you study at most two hours)\n= P(X \u2264 2)\n= P (X = 0) + P(X = 1) + P(X = 2)\n= 0 1 + k + 2k = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7352-7355",
        "Text": "15 = 0 3\nP(you study at most two hours)\n= P(X \u2264 2)\n= P (X = 0) + P(X = 1) + P(X = 2)\n= 0 1 + k + 2k = 0 1 + 3k = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7353-7356",
        "Text": "3\nP(you study at most two hours)\n= P(X \u2264 2)\n= P (X = 0) + P(X = 1) + P(X = 2)\n= 0 1 + k + 2k = 0 1 + 3k = 0 1 + 3 \u00d7 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7354-7357",
        "Text": "1 + k + 2k = 0 1 + 3k = 0 1 + 3 \u00d7 0 15\n= 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7355-7358",
        "Text": "1 + 3k = 0 1 + 3 \u00d7 0 15\n= 0 55\n13"
    },
    {
        "Chapter": "1",
        "sentence_range": "7356-7359",
        "Text": "1 + 3 \u00d7 0 15\n= 0 55\n13 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "7357-7360",
        "Text": "15\n= 0 55\n13 6 2  Mean of a random variable\nIn many problems, it is desirable to describe some feature of the random variable by\nmeans of a single number that can be computed from its probability distribution"
    },
    {
        "Chapter": "1",
        "sentence_range": "7358-7361",
        "Text": "55\n13 6 2  Mean of a random variable\nIn many problems, it is desirable to describe some feature of the random variable by\nmeans of a single number that can be computed from its probability distribution Few\nsuch numbers are mean, median and mode"
    },
    {
        "Chapter": "1",
        "sentence_range": "7359-7362",
        "Text": "6 2  Mean of a random variable\nIn many problems, it is desirable to describe some feature of the random variable by\nmeans of a single number that can be computed from its probability distribution Few\nsuch numbers are mean, median and mode In this section, we shall discuss mean only"
    },
    {
        "Chapter": "1",
        "sentence_range": "7360-7363",
        "Text": "2  Mean of a random variable\nIn many problems, it is desirable to describe some feature of the random variable by\nmeans of a single number that can be computed from its probability distribution Few\nsuch numbers are mean, median and mode In this section, we shall discuss mean only Mean is a measure of location or central tendency in the sense that it roughly locates a\nmiddle or average value of the random variable"
    },
    {
        "Chapter": "1",
        "sentence_range": "7361-7364",
        "Text": "Few\nsuch numbers are mean, median and mode In this section, we shall discuss mean only Mean is a measure of location or central tendency in the sense that it roughly locates a\nmiddle or average value of the random variable \u00a9 NCERT\nnot to be republished\n 564\nMATHEMATICS\nDefinition 6  Let X be a random variable whose possible values x1, x2, x3,"
    },
    {
        "Chapter": "1",
        "sentence_range": "7362-7365",
        "Text": "In this section, we shall discuss mean only Mean is a measure of location or central tendency in the sense that it roughly locates a\nmiddle or average value of the random variable \u00a9 NCERT\nnot to be republished\n 564\nMATHEMATICS\nDefinition 6  Let X be a random variable whose possible values x1, x2, x3, , xn occur\nwith probabilities p1, p2, p3,"
    },
    {
        "Chapter": "1",
        "sentence_range": "7363-7366",
        "Text": "Mean is a measure of location or central tendency in the sense that it roughly locates a\nmiddle or average value of the random variable \u00a9 NCERT\nnot to be republished\n 564\nMATHEMATICS\nDefinition 6  Let X be a random variable whose possible values x1, x2, x3, , xn occur\nwith probabilities p1, p2, p3, , pn, respectively"
    },
    {
        "Chapter": "1",
        "sentence_range": "7364-7367",
        "Text": "\u00a9 NCERT\nnot to be republished\n 564\nMATHEMATICS\nDefinition 6  Let X be a random variable whose possible values x1, x2, x3, , xn occur\nwith probabilities p1, p2, p3, , pn, respectively The mean of X, denoted by \u03bc, is the\nnumber \n1\nn\ni\ni\ni\nx p\n=\u2211\ni"
    },
    {
        "Chapter": "1",
        "sentence_range": "7365-7368",
        "Text": ", xn occur\nwith probabilities p1, p2, p3, , pn, respectively The mean of X, denoted by \u03bc, is the\nnumber \n1\nn\ni\ni\ni\nx p\n=\u2211\ni e"
    },
    {
        "Chapter": "1",
        "sentence_range": "7366-7369",
        "Text": ", pn, respectively The mean of X, denoted by \u03bc, is the\nnumber \n1\nn\ni\ni\ni\nx p\n=\u2211\ni e the mean of X is the weighted average of the possible values of X,\neach value being weighted by its probability with which it occurs"
    },
    {
        "Chapter": "1",
        "sentence_range": "7367-7370",
        "Text": "The mean of X, denoted by \u03bc, is the\nnumber \n1\nn\ni\ni\ni\nx p\n=\u2211\ni e the mean of X is the weighted average of the possible values of X,\neach value being weighted by its probability with which it occurs The mean of a random variable X is also called the expectation of X, denoted by\nE(X)"
    },
    {
        "Chapter": "1",
        "sentence_range": "7368-7371",
        "Text": "e the mean of X is the weighted average of the possible values of X,\neach value being weighted by its probability with which it occurs The mean of a random variable X is also called the expectation of X, denoted by\nE(X) Thus,\nE (X) = \u03bc =\n1\nn\ni\ni\ni\nx p\n  \n= x1 p1+ x2 p2 +"
    },
    {
        "Chapter": "1",
        "sentence_range": "7369-7372",
        "Text": "the mean of X is the weighted average of the possible values of X,\neach value being weighted by its probability with which it occurs The mean of a random variable X is also called the expectation of X, denoted by\nE(X) Thus,\nE (X) = \u03bc =\n1\nn\ni\ni\ni\nx p\n  \n= x1 p1+ x2 p2 + + xn pn"
    },
    {
        "Chapter": "1",
        "sentence_range": "7370-7373",
        "Text": "The mean of a random variable X is also called the expectation of X, denoted by\nE(X) Thus,\nE (X) = \u03bc =\n1\nn\ni\ni\ni\nx p\n  \n= x1 p1+ x2 p2 + + xn pn In other words, the mean or expectation of a random variable X is the sum of the\nproducts of all possible values of X by their respective probabilities"
    },
    {
        "Chapter": "1",
        "sentence_range": "7371-7374",
        "Text": "Thus,\nE (X) = \u03bc =\n1\nn\ni\ni\ni\nx p\n  \n= x1 p1+ x2 p2 + + xn pn In other words, the mean or expectation of a random variable X is the sum of the\nproducts of all possible values of X by their respective probabilities Example 27 Let a pair of dice be thrown and the random variable X be the sum of the\nnumbers that appear on the two dice"
    },
    {
        "Chapter": "1",
        "sentence_range": "7372-7375",
        "Text": "+ xn pn In other words, the mean or expectation of a random variable X is the sum of the\nproducts of all possible values of X by their respective probabilities Example 27 Let a pair of dice be thrown and the random variable X be the sum of the\nnumbers that appear on the two dice Find the mean or expectation of X"
    },
    {
        "Chapter": "1",
        "sentence_range": "7373-7376",
        "Text": "In other words, the mean or expectation of a random variable X is the sum of the\nproducts of all possible values of X by their respective probabilities Example 27 Let a pair of dice be thrown and the random variable X be the sum of the\nnumbers that appear on the two dice Find the mean or expectation of X Solution The sample space of the experiment consists of 36 elementary events in the\nform of ordered pairs (xi, yi), where xi = 1, 2, 3, 4, 5, 6 and yi = 1, 2, 3, 4, 5, 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "7374-7377",
        "Text": "Example 27 Let a pair of dice be thrown and the random variable X be the sum of the\nnumbers that appear on the two dice Find the mean or expectation of X Solution The sample space of the experiment consists of 36 elementary events in the\nform of ordered pairs (xi, yi), where xi = 1, 2, 3, 4, 5, 6 and yi = 1, 2, 3, 4, 5, 6 The random variable X i"
    },
    {
        "Chapter": "1",
        "sentence_range": "7375-7378",
        "Text": "Find the mean or expectation of X Solution The sample space of the experiment consists of 36 elementary events in the\nform of ordered pairs (xi, yi), where xi = 1, 2, 3, 4, 5, 6 and yi = 1, 2, 3, 4, 5, 6 The random variable X i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "7376-7379",
        "Text": "Solution The sample space of the experiment consists of 36 elementary events in the\nform of ordered pairs (xi, yi), where xi = 1, 2, 3, 4, 5, 6 and yi = 1, 2, 3, 4, 5, 6 The random variable X i e the sum of the numbers on the two dice takes the\nvalues 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 or 12"
    },
    {
        "Chapter": "1",
        "sentence_range": "7377-7380",
        "Text": "The random variable X i e the sum of the numbers on the two dice takes the\nvalues 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 or 12 Now\nP(X = 2) = P({(1,1)}) \n361\n \nP(X = 3) = P({(1,2), (2,1)}) \n362\n \nP(X = 4) = P({(1,3), (2,2), (3,1)}) \n363\n \nP(X = 5) = P({(1,4), (2,3), (3,2), (4,1)})\n364\n \nP(X = 6) = P({(1,5), (2,4), (3,3), (4,2), (5,1)})\n365\n \nP(X = 7) = P({(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)})\n366\n \nP(X = 8) = P({(2,6), (3,5), (4,4), (5,3), (6,2)})\n365\n \n\u00a9 NCERT\nnot to be republished\nPROBABILITY      565\nP(X = 9) = P({(3,6), (4,5), (5,4), (6,3)})\n364\n \nP(X = 10) = P({(4,6), (5,5), (6,4)})\n363\n \nP(X = 11) = P({(5,6), (6,5)})\n362\n \nP(X = 12) = P({(6,6)}) \n361\n \nThe probability distribution of X is\nX or xi\n2\n3\n4\n5\n6\n7\n8\n9\n10\n11\n12\nP(X) or pi\n1\n36\n2\n36\n3\n36\n4\n36\n5\n36\n6\n36\n5\n36\n4\n36\n3\n36\n2\n36\n1\n36\nTherefore,\n\u03bc = E(X) =\n1\n1\n2\n3\n4\n2\n3\n4\n5\n36\n36\n36\n36\nn\ni\ni\ni\nx p\n=\n= \u00d7\n+ \u00d7\n+ \u00d7\n+ \u00d7\n\u2211\n5\n6\n5\n6\n7\n8\n36\n36\n36\n \n \n \n \n \n \n4\n3\n2\n1\n9\n10\n11\n12\n36\n36\n36\n36\n \n \n \n \n \n \n \n \n= 2\n6\n12\n20\n30\n42\n40\n36\n30\n22\n12\n36\n \n \n \n \n \n \n \n \n \n \n = 7\nThus, the mean of the sum of the numbers that appear on throwing two fair dice is 7"
    },
    {
        "Chapter": "1",
        "sentence_range": "7378-7381",
        "Text": "e the sum of the numbers on the two dice takes the\nvalues 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 or 12 Now\nP(X = 2) = P({(1,1)}) \n361\n \nP(X = 3) = P({(1,2), (2,1)}) \n362\n \nP(X = 4) = P({(1,3), (2,2), (3,1)}) \n363\n \nP(X = 5) = P({(1,4), (2,3), (3,2), (4,1)})\n364\n \nP(X = 6) = P({(1,5), (2,4), (3,3), (4,2), (5,1)})\n365\n \nP(X = 7) = P({(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)})\n366\n \nP(X = 8) = P({(2,6), (3,5), (4,4), (5,3), (6,2)})\n365\n \n\u00a9 NCERT\nnot to be republished\nPROBABILITY      565\nP(X = 9) = P({(3,6), (4,5), (5,4), (6,3)})\n364\n \nP(X = 10) = P({(4,6), (5,5), (6,4)})\n363\n \nP(X = 11) = P({(5,6), (6,5)})\n362\n \nP(X = 12) = P({(6,6)}) \n361\n \nThe probability distribution of X is\nX or xi\n2\n3\n4\n5\n6\n7\n8\n9\n10\n11\n12\nP(X) or pi\n1\n36\n2\n36\n3\n36\n4\n36\n5\n36\n6\n36\n5\n36\n4\n36\n3\n36\n2\n36\n1\n36\nTherefore,\n\u03bc = E(X) =\n1\n1\n2\n3\n4\n2\n3\n4\n5\n36\n36\n36\n36\nn\ni\ni\ni\nx p\n=\n= \u00d7\n+ \u00d7\n+ \u00d7\n+ \u00d7\n\u2211\n5\n6\n5\n6\n7\n8\n36\n36\n36\n \n \n \n \n \n \n4\n3\n2\n1\n9\n10\n11\n12\n36\n36\n36\n36\n \n \n \n \n \n \n \n \n= 2\n6\n12\n20\n30\n42\n40\n36\n30\n22\n12\n36\n \n \n \n \n \n \n \n \n \n \n = 7\nThus, the mean of the sum of the numbers that appear on throwing two fair dice is 7 13"
    },
    {
        "Chapter": "1",
        "sentence_range": "7379-7382",
        "Text": "the sum of the numbers on the two dice takes the\nvalues 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 or 12 Now\nP(X = 2) = P({(1,1)}) \n361\n \nP(X = 3) = P({(1,2), (2,1)}) \n362\n \nP(X = 4) = P({(1,3), (2,2), (3,1)}) \n363\n \nP(X = 5) = P({(1,4), (2,3), (3,2), (4,1)})\n364\n \nP(X = 6) = P({(1,5), (2,4), (3,3), (4,2), (5,1)})\n365\n \nP(X = 7) = P({(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)})\n366\n \nP(X = 8) = P({(2,6), (3,5), (4,4), (5,3), (6,2)})\n365\n \n\u00a9 NCERT\nnot to be republished\nPROBABILITY      565\nP(X = 9) = P({(3,6), (4,5), (5,4), (6,3)})\n364\n \nP(X = 10) = P({(4,6), (5,5), (6,4)})\n363\n \nP(X = 11) = P({(5,6), (6,5)})\n362\n \nP(X = 12) = P({(6,6)}) \n361\n \nThe probability distribution of X is\nX or xi\n2\n3\n4\n5\n6\n7\n8\n9\n10\n11\n12\nP(X) or pi\n1\n36\n2\n36\n3\n36\n4\n36\n5\n36\n6\n36\n5\n36\n4\n36\n3\n36\n2\n36\n1\n36\nTherefore,\n\u03bc = E(X) =\n1\n1\n2\n3\n4\n2\n3\n4\n5\n36\n36\n36\n36\nn\ni\ni\ni\nx p\n=\n= \u00d7\n+ \u00d7\n+ \u00d7\n+ \u00d7\n\u2211\n5\n6\n5\n6\n7\n8\n36\n36\n36\n \n \n \n \n \n \n4\n3\n2\n1\n9\n10\n11\n12\n36\n36\n36\n36\n \n \n \n \n \n \n \n \n= 2\n6\n12\n20\n30\n42\n40\n36\n30\n22\n12\n36\n \n \n \n \n \n \n \n \n \n \n = 7\nThus, the mean of the sum of the numbers that appear on throwing two fair dice is 7 13 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "7380-7383",
        "Text": "Now\nP(X = 2) = P({(1,1)}) \n361\n \nP(X = 3) = P({(1,2), (2,1)}) \n362\n \nP(X = 4) = P({(1,3), (2,2), (3,1)}) \n363\n \nP(X = 5) = P({(1,4), (2,3), (3,2), (4,1)})\n364\n \nP(X = 6) = P({(1,5), (2,4), (3,3), (4,2), (5,1)})\n365\n \nP(X = 7) = P({(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)})\n366\n \nP(X = 8) = P({(2,6), (3,5), (4,4), (5,3), (6,2)})\n365\n \n\u00a9 NCERT\nnot to be republished\nPROBABILITY      565\nP(X = 9) = P({(3,6), (4,5), (5,4), (6,3)})\n364\n \nP(X = 10) = P({(4,6), (5,5), (6,4)})\n363\n \nP(X = 11) = P({(5,6), (6,5)})\n362\n \nP(X = 12) = P({(6,6)}) \n361\n \nThe probability distribution of X is\nX or xi\n2\n3\n4\n5\n6\n7\n8\n9\n10\n11\n12\nP(X) or pi\n1\n36\n2\n36\n3\n36\n4\n36\n5\n36\n6\n36\n5\n36\n4\n36\n3\n36\n2\n36\n1\n36\nTherefore,\n\u03bc = E(X) =\n1\n1\n2\n3\n4\n2\n3\n4\n5\n36\n36\n36\n36\nn\ni\ni\ni\nx p\n=\n= \u00d7\n+ \u00d7\n+ \u00d7\n+ \u00d7\n\u2211\n5\n6\n5\n6\n7\n8\n36\n36\n36\n \n \n \n \n \n \n4\n3\n2\n1\n9\n10\n11\n12\n36\n36\n36\n36\n \n \n \n \n \n \n \n \n= 2\n6\n12\n20\n30\n42\n40\n36\n30\n22\n12\n36\n \n \n \n \n \n \n \n \n \n \n = 7\nThus, the mean of the sum of the numbers that appear on throwing two fair dice is 7 13 6 3  Variance of a random variable\nThe mean of a random variable does not give us information about the variability in the\nvalues of the random variable"
    },
    {
        "Chapter": "1",
        "sentence_range": "7381-7384",
        "Text": "13 6 3  Variance of a random variable\nThe mean of a random variable does not give us information about the variability in the\nvalues of the random variable In fact, if the variance is small, then the values of the\nrandom variable are close to the mean"
    },
    {
        "Chapter": "1",
        "sentence_range": "7382-7385",
        "Text": "6 3  Variance of a random variable\nThe mean of a random variable does not give us information about the variability in the\nvalues of the random variable In fact, if the variance is small, then the values of the\nrandom variable are close to the mean Also random variables with different probability\ndistributions can have equal means, as shown in the following distributions of X and Y"
    },
    {
        "Chapter": "1",
        "sentence_range": "7383-7386",
        "Text": "3  Variance of a random variable\nThe mean of a random variable does not give us information about the variability in the\nvalues of the random variable In fact, if the variance is small, then the values of the\nrandom variable are close to the mean Also random variables with different probability\ndistributions can have equal means, as shown in the following distributions of X and Y X\n1\n2\n3\n4\nP(X)\n1\n8\n2\n8\n3\n8\n2\n8\n\u00a9 NCERT\nnot to be republished\n 566\nMATHEMATICS\nY\n\u20131\n0\n4\n5\n6\nP(Y)\n1\n8\n2\n8\n3\n8\n1\n8\n1\n8\nClearly\nE(X) =\n1\n2\n3\n2\n22\n1\n2\n3\n4\n2"
    },
    {
        "Chapter": "1",
        "sentence_range": "7384-7387",
        "Text": "In fact, if the variance is small, then the values of the\nrandom variable are close to the mean Also random variables with different probability\ndistributions can have equal means, as shown in the following distributions of X and Y X\n1\n2\n3\n4\nP(X)\n1\n8\n2\n8\n3\n8\n2\n8\n\u00a9 NCERT\nnot to be republished\n 566\nMATHEMATICS\nY\n\u20131\n0\n4\n5\n6\nP(Y)\n1\n8\n2\n8\n3\n8\n1\n8\n1\n8\nClearly\nE(X) =\n1\n2\n3\n2\n22\n1\n2\n3\n4\n2 75\n8\n8\n8\n8\n8\n\u00d7 + \u00d7\n+ \u00d7 + \u00d7\n=\n=\nand\nE(Y) =\n1\n2\n3\n1\n1\n22\n1\n0\n4\n5\n6\n2"
    },
    {
        "Chapter": "1",
        "sentence_range": "7385-7388",
        "Text": "Also random variables with different probability\ndistributions can have equal means, as shown in the following distributions of X and Y X\n1\n2\n3\n4\nP(X)\n1\n8\n2\n8\n3\n8\n2\n8\n\u00a9 NCERT\nnot to be republished\n 566\nMATHEMATICS\nY\n\u20131\n0\n4\n5\n6\nP(Y)\n1\n8\n2\n8\n3\n8\n1\n8\n1\n8\nClearly\nE(X) =\n1\n2\n3\n2\n22\n1\n2\n3\n4\n2 75\n8\n8\n8\n8\n8\n\u00d7 + \u00d7\n+ \u00d7 + \u00d7\n=\n=\nand\nE(Y) =\n1\n2\n3\n1\n1\n22\n1\n0\n4\n5\n6\n2 75\n8\n8\n8\n8\n8\n8\n\u2212 \u00d7 + \u00d7\n+ \u00d7 + \u00d7 =\n\u00d7 =\n=\nThe variables X and Y are different, however their means are same"
    },
    {
        "Chapter": "1",
        "sentence_range": "7386-7389",
        "Text": "X\n1\n2\n3\n4\nP(X)\n1\n8\n2\n8\n3\n8\n2\n8\n\u00a9 NCERT\nnot to be republished\n 566\nMATHEMATICS\nY\n\u20131\n0\n4\n5\n6\nP(Y)\n1\n8\n2\n8\n3\n8\n1\n8\n1\n8\nClearly\nE(X) =\n1\n2\n3\n2\n22\n1\n2\n3\n4\n2 75\n8\n8\n8\n8\n8\n\u00d7 + \u00d7\n+ \u00d7 + \u00d7\n=\n=\nand\nE(Y) =\n1\n2\n3\n1\n1\n22\n1\n0\n4\n5\n6\n2 75\n8\n8\n8\n8\n8\n8\n\u2212 \u00d7 + \u00d7\n+ \u00d7 + \u00d7 =\n\u00d7 =\n=\nThe variables X and Y are different, however their means are same It is also\neasily observable from the diagramatic representation of these distributions (Fig 13"
    },
    {
        "Chapter": "1",
        "sentence_range": "7387-7390",
        "Text": "75\n8\n8\n8\n8\n8\n\u00d7 + \u00d7\n+ \u00d7 + \u00d7\n=\n=\nand\nE(Y) =\n1\n2\n3\n1\n1\n22\n1\n0\n4\n5\n6\n2 75\n8\n8\n8\n8\n8\n8\n\u2212 \u00d7 + \u00d7\n+ \u00d7 + \u00d7 =\n\u00d7 =\n=\nThe variables X and Y are different, however their means are same It is also\neasily observable from the diagramatic representation of these distributions (Fig 13 5)"
    },
    {
        "Chapter": "1",
        "sentence_range": "7388-7391",
        "Text": "75\n8\n8\n8\n8\n8\n8\n\u2212 \u00d7 + \u00d7\n+ \u00d7 + \u00d7 =\n\u00d7 =\n=\nThe variables X and Y are different, however their means are same It is also\neasily observable from the diagramatic representation of these distributions (Fig 13 5) Fig 13"
    },
    {
        "Chapter": "1",
        "sentence_range": "7389-7392",
        "Text": "It is also\neasily observable from the diagramatic representation of these distributions (Fig 13 5) Fig 13 5\nTo distinguish X from Y, we require a measure of the extent to which the values of\nthe random variables spread out"
    },
    {
        "Chapter": "1",
        "sentence_range": "7390-7393",
        "Text": "5) Fig 13 5\nTo distinguish X from Y, we require a measure of the extent to which the values of\nthe random variables spread out In Statistics, we have studied that the variance is a\nmeasure of the spread or scatter in data"
    },
    {
        "Chapter": "1",
        "sentence_range": "7391-7394",
        "Text": "Fig 13 5\nTo distinguish X from Y, we require a measure of the extent to which the values of\nthe random variables spread out In Statistics, we have studied that the variance is a\nmeasure of the spread or scatter in data Likewise, the variability or spread in the\nvalues of a random variable may be measured by variance"
    },
    {
        "Chapter": "1",
        "sentence_range": "7392-7395",
        "Text": "5\nTo distinguish X from Y, we require a measure of the extent to which the values of\nthe random variables spread out In Statistics, we have studied that the variance is a\nmeasure of the spread or scatter in data Likewise, the variability or spread in the\nvalues of a random variable may be measured by variance Definition 7  Let X be a random variable whose possible values x1, x2,"
    },
    {
        "Chapter": "1",
        "sentence_range": "7393-7396",
        "Text": "In Statistics, we have studied that the variance is a\nmeasure of the spread or scatter in data Likewise, the variability or spread in the\nvalues of a random variable may be measured by variance Definition 7  Let X be a random variable whose possible values x1, x2, ,xn occur with\nprobabilities p(x1), p(x2),"
    },
    {
        "Chapter": "1",
        "sentence_range": "7394-7397",
        "Text": "Likewise, the variability or spread in the\nvalues of a random variable may be measured by variance Definition 7  Let X be a random variable whose possible values x1, x2, ,xn occur with\nprobabilities p(x1), p(x2), , p(xn) respectively"
    },
    {
        "Chapter": "1",
        "sentence_range": "7395-7398",
        "Text": "Definition 7  Let X be a random variable whose possible values x1, x2, ,xn occur with\nprobabilities p(x1), p(x2), , p(xn) respectively Let \u03bc = E (X) be the mean of X"
    },
    {
        "Chapter": "1",
        "sentence_range": "7396-7399",
        "Text": ",xn occur with\nprobabilities p(x1), p(x2), , p(xn) respectively Let \u03bc = E (X) be the mean of X The variance of X, denoted by Var (X) or \n2\n x\n is\ndefined as\n2\n\u03c3x\n =\n2\n1\nVar(X)=\n(\n\u03bc)\n(\n)\nn\ni\ni\ni\nx\np x\n=\n\u2212\n\u2211\nor equivalently\n2\n x\n = E(X \u2013 \u03bc)2\nO\n1 8\n2 8\n3 8\nP(Y)\nO\n1 8\n2 8\n3 8\nP(X)\n1\n2\n3\n4\n1\n2\n3\n4\n\u20131\n5\n6\n(i)\n(ii)\n\u00a9 NCERT\nnot to be republished\nPROBABILITY      567\nThe non-negative number\n\u03c3x =\n2\n1\nVar(X) =\n(\n\u03bc)\n(\n)\nn\ni\ni\ni\nx\np x\n=\n\u2212\n\u2211\nis called the standard deviation of the  random variable X"
    },
    {
        "Chapter": "1",
        "sentence_range": "7397-7400",
        "Text": ", p(xn) respectively Let \u03bc = E (X) be the mean of X The variance of X, denoted by Var (X) or \n2\n x\n is\ndefined as\n2\n\u03c3x\n =\n2\n1\nVar(X)=\n(\n\u03bc)\n(\n)\nn\ni\ni\ni\nx\np x\n=\n\u2212\n\u2211\nor equivalently\n2\n x\n = E(X \u2013 \u03bc)2\nO\n1 8\n2 8\n3 8\nP(Y)\nO\n1 8\n2 8\n3 8\nP(X)\n1\n2\n3\n4\n1\n2\n3\n4\n\u20131\n5\n6\n(i)\n(ii)\n\u00a9 NCERT\nnot to be republished\nPROBABILITY      567\nThe non-negative number\n\u03c3x =\n2\n1\nVar(X) =\n(\n\u03bc)\n(\n)\nn\ni\ni\ni\nx\np x\n=\n\u2212\n\u2211\nis called the standard deviation of the  random variable X Another formula to find the variance of a random variable"
    },
    {
        "Chapter": "1",
        "sentence_range": "7398-7401",
        "Text": "Let \u03bc = E (X) be the mean of X The variance of X, denoted by Var (X) or \n2\n x\n is\ndefined as\n2\n\u03c3x\n =\n2\n1\nVar(X)=\n(\n\u03bc)\n(\n)\nn\ni\ni\ni\nx\np x\n=\n\u2212\n\u2211\nor equivalently\n2\n x\n = E(X \u2013 \u03bc)2\nO\n1 8\n2 8\n3 8\nP(Y)\nO\n1 8\n2 8\n3 8\nP(X)\n1\n2\n3\n4\n1\n2\n3\n4\n\u20131\n5\n6\n(i)\n(ii)\n\u00a9 NCERT\nnot to be republished\nPROBABILITY      567\nThe non-negative number\n\u03c3x =\n2\n1\nVar(X) =\n(\n\u03bc)\n(\n)\nn\ni\ni\ni\nx\np x\n=\n\u2212\n\u2211\nis called the standard deviation of the  random variable X Another formula to find the variance of a random variable We know that,\nVar (X) =\n2\n1\n(\n\u03bc)\n(\n)\nn\ni\ni\ni\nx\np x\n=\n\u2212\n\u2211\n=\n2\n2\n1\n(\n\u03bc\n2\u03bc\n) (\n)\nn\ni\ni\ni\ni\nx\nx\np x\n \n \n \n \n=\n2\n2\n1\n1\n1\n(\n)\n\u03bc\n(\n)\n2\u03bc\n(\n)\nn\nn\nn\ni\ni\ni\ni\ni\ni\ni\ni\nx\np x\np x\nx p x\n=\n=\n=\n+\n\u2212\n\u2211\n\u2211\n\u2211\n=\n2\n2\n1\n1\n1\n(\n) \u03bc\n(\n)\n2\u03bc\n(\n)\nn\nn\nn\ni\ni\ni\ni\ni\ni\ni\ni\nx\np x\np x\nx p x\n=\n=\n=\n+\n\u2212\n\u2211\n\u2211\n\u2211\n=\n2\n2\n2\n1\n=1\n1\n(\n) \u03bc\n2\u03bc\nsince\n ( )=1and\u03bc=\n( )\nn\nn\nn\ni\ni\ni\ni\ni\ni\ni\ni\nx\np x\np x\nx p x\n=\n=\n\u23a1\n\u23a4\n+\n\u2212\n\u23a2\n\u23a5\n\u23a3\n\u23a6\n\u2211\n\u2211\n\u2211\n=\n2\n2\n1\n(\n) \u03bc\nn\ni\ni\ni\nx\np x\n=\n\u2212\n\u2211\nor\nVar (X) =\n2\n2\n1\n1\n(\n)\n(\n)\nn\nn\ni\ni\ni\ni\ni\ni\nx\np x\nx p x\n=\n=\n\u239b\n\u239e\n\u2212\u239c\n\u239f\n\u239d\n\u23a0\n\u2211\n\u2211\nor\nVar (X) = E(X2) \u2013 [E(X)]2, where E(X2) =\n2\n1\n(\n)\nn\ni\ni\ni\nx\np x\n=\u2211\nExample 28 Find the variance of the number obtained on a throw of an unbiased die"
    },
    {
        "Chapter": "1",
        "sentence_range": "7399-7402",
        "Text": "The variance of X, denoted by Var (X) or \n2\n x\n is\ndefined as\n2\n\u03c3x\n =\n2\n1\nVar(X)=\n(\n\u03bc)\n(\n)\nn\ni\ni\ni\nx\np x\n=\n\u2212\n\u2211\nor equivalently\n2\n x\n = E(X \u2013 \u03bc)2\nO\n1 8\n2 8\n3 8\nP(Y)\nO\n1 8\n2 8\n3 8\nP(X)\n1\n2\n3\n4\n1\n2\n3\n4\n\u20131\n5\n6\n(i)\n(ii)\n\u00a9 NCERT\nnot to be republished\nPROBABILITY      567\nThe non-negative number\n\u03c3x =\n2\n1\nVar(X) =\n(\n\u03bc)\n(\n)\nn\ni\ni\ni\nx\np x\n=\n\u2212\n\u2211\nis called the standard deviation of the  random variable X Another formula to find the variance of a random variable We know that,\nVar (X) =\n2\n1\n(\n\u03bc)\n(\n)\nn\ni\ni\ni\nx\np x\n=\n\u2212\n\u2211\n=\n2\n2\n1\n(\n\u03bc\n2\u03bc\n) (\n)\nn\ni\ni\ni\ni\nx\nx\np x\n \n \n \n \n=\n2\n2\n1\n1\n1\n(\n)\n\u03bc\n(\n)\n2\u03bc\n(\n)\nn\nn\nn\ni\ni\ni\ni\ni\ni\ni\ni\nx\np x\np x\nx p x\n=\n=\n=\n+\n\u2212\n\u2211\n\u2211\n\u2211\n=\n2\n2\n1\n1\n1\n(\n) \u03bc\n(\n)\n2\u03bc\n(\n)\nn\nn\nn\ni\ni\ni\ni\ni\ni\ni\ni\nx\np x\np x\nx p x\n=\n=\n=\n+\n\u2212\n\u2211\n\u2211\n\u2211\n=\n2\n2\n2\n1\n=1\n1\n(\n) \u03bc\n2\u03bc\nsince\n ( )=1and\u03bc=\n( )\nn\nn\nn\ni\ni\ni\ni\ni\ni\ni\ni\nx\np x\np x\nx p x\n=\n=\n\u23a1\n\u23a4\n+\n\u2212\n\u23a2\n\u23a5\n\u23a3\n\u23a6\n\u2211\n\u2211\n\u2211\n=\n2\n2\n1\n(\n) \u03bc\nn\ni\ni\ni\nx\np x\n=\n\u2212\n\u2211\nor\nVar (X) =\n2\n2\n1\n1\n(\n)\n(\n)\nn\nn\ni\ni\ni\ni\ni\ni\nx\np x\nx p x\n=\n=\n\u239b\n\u239e\n\u2212\u239c\n\u239f\n\u239d\n\u23a0\n\u2211\n\u2211\nor\nVar (X) = E(X2) \u2013 [E(X)]2, where E(X2) =\n2\n1\n(\n)\nn\ni\ni\ni\nx\np x\n=\u2211\nExample 28 Find the variance of the number obtained on a throw of an unbiased die Solution The sample space of the experiment is S = {1, 2, 3, 4, 5, 6}"
    },
    {
        "Chapter": "1",
        "sentence_range": "7400-7403",
        "Text": "Another formula to find the variance of a random variable We know that,\nVar (X) =\n2\n1\n(\n\u03bc)\n(\n)\nn\ni\ni\ni\nx\np x\n=\n\u2212\n\u2211\n=\n2\n2\n1\n(\n\u03bc\n2\u03bc\n) (\n)\nn\ni\ni\ni\ni\nx\nx\np x\n \n \n \n \n=\n2\n2\n1\n1\n1\n(\n)\n\u03bc\n(\n)\n2\u03bc\n(\n)\nn\nn\nn\ni\ni\ni\ni\ni\ni\ni\ni\nx\np x\np x\nx p x\n=\n=\n=\n+\n\u2212\n\u2211\n\u2211\n\u2211\n=\n2\n2\n1\n1\n1\n(\n) \u03bc\n(\n)\n2\u03bc\n(\n)\nn\nn\nn\ni\ni\ni\ni\ni\ni\ni\ni\nx\np x\np x\nx p x\n=\n=\n=\n+\n\u2212\n\u2211\n\u2211\n\u2211\n=\n2\n2\n2\n1\n=1\n1\n(\n) \u03bc\n2\u03bc\nsince\n ( )=1and\u03bc=\n( )\nn\nn\nn\ni\ni\ni\ni\ni\ni\ni\ni\nx\np x\np x\nx p x\n=\n=\n\u23a1\n\u23a4\n+\n\u2212\n\u23a2\n\u23a5\n\u23a3\n\u23a6\n\u2211\n\u2211\n\u2211\n=\n2\n2\n1\n(\n) \u03bc\nn\ni\ni\ni\nx\np x\n=\n\u2212\n\u2211\nor\nVar (X) =\n2\n2\n1\n1\n(\n)\n(\n)\nn\nn\ni\ni\ni\ni\ni\ni\nx\np x\nx p x\n=\n=\n\u239b\n\u239e\n\u2212\u239c\n\u239f\n\u239d\n\u23a0\n\u2211\n\u2211\nor\nVar (X) = E(X2) \u2013 [E(X)]2, where E(X2) =\n2\n1\n(\n)\nn\ni\ni\ni\nx\np x\n=\u2211\nExample 28 Find the variance of the number obtained on a throw of an unbiased die Solution The sample space of the experiment is S = {1, 2, 3, 4, 5, 6} Let X denote the number obtained on the throw"
    },
    {
        "Chapter": "1",
        "sentence_range": "7401-7404",
        "Text": "We know that,\nVar (X) =\n2\n1\n(\n\u03bc)\n(\n)\nn\ni\ni\ni\nx\np x\n=\n\u2212\n\u2211\n=\n2\n2\n1\n(\n\u03bc\n2\u03bc\n) (\n)\nn\ni\ni\ni\ni\nx\nx\np x\n \n \n \n \n=\n2\n2\n1\n1\n1\n(\n)\n\u03bc\n(\n)\n2\u03bc\n(\n)\nn\nn\nn\ni\ni\ni\ni\ni\ni\ni\ni\nx\np x\np x\nx p x\n=\n=\n=\n+\n\u2212\n\u2211\n\u2211\n\u2211\n=\n2\n2\n1\n1\n1\n(\n) \u03bc\n(\n)\n2\u03bc\n(\n)\nn\nn\nn\ni\ni\ni\ni\ni\ni\ni\ni\nx\np x\np x\nx p x\n=\n=\n=\n+\n\u2212\n\u2211\n\u2211\n\u2211\n=\n2\n2\n2\n1\n=1\n1\n(\n) \u03bc\n2\u03bc\nsince\n ( )=1and\u03bc=\n( )\nn\nn\nn\ni\ni\ni\ni\ni\ni\ni\ni\nx\np x\np x\nx p x\n=\n=\n\u23a1\n\u23a4\n+\n\u2212\n\u23a2\n\u23a5\n\u23a3\n\u23a6\n\u2211\n\u2211\n\u2211\n=\n2\n2\n1\n(\n) \u03bc\nn\ni\ni\ni\nx\np x\n=\n\u2212\n\u2211\nor\nVar (X) =\n2\n2\n1\n1\n(\n)\n(\n)\nn\nn\ni\ni\ni\ni\ni\ni\nx\np x\nx p x\n=\n=\n\u239b\n\u239e\n\u2212\u239c\n\u239f\n\u239d\n\u23a0\n\u2211\n\u2211\nor\nVar (X) = E(X2) \u2013 [E(X)]2, where E(X2) =\n2\n1\n(\n)\nn\ni\ni\ni\nx\np x\n=\u2211\nExample 28 Find the variance of the number obtained on a throw of an unbiased die Solution The sample space of the experiment is S = {1, 2, 3, 4, 5, 6} Let X denote the number obtained on the throw Then X is a random variable\nwhich can take values 1, 2, 3, 4, 5, or 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "7402-7405",
        "Text": "Solution The sample space of the experiment is S = {1, 2, 3, 4, 5, 6} Let X denote the number obtained on the throw Then X is a random variable\nwhich can take values 1, 2, 3, 4, 5, or 6 \u00a9 NCERT\nnot to be republished\n 568\nMATHEMATICS\nAlso\nP(1) = P(2) = P(3) = P(4) = P(5) = P(6) = 1\n6\nTherefore, the Probability distribution of  X  is\nX\n1\n2\n3\n4\n5\n6\nP(X)\n1\n6\n1\n6\n1\n6\n1\n6\n1\n6\n1\n6\nNow\nE(X) =\n1\n(\n)\nn\ni\ni\ni\nx p x\n  \n=\n1\n1\n1\n1\n1\n1\n21\n1\n2\n3\n4\n5\n6\n6\n6\n6\n6\n6\n6\n6\n\u00d7\n+ \u00d7\n+ \u00d7\n+ \u00d7\n+ \u00d7\n+ \u00d7\n=\nAlso\nE(X2) =\n2\n2\n2\n2\n2\n2\n1\n1\n1\n1\n1\n1\n91\n1\n2\n3\n4\n5\n6\n6\n6\n6\n6\n6\n6\n6\n\u00d7\n+\n\u00d7\n+\n\u00d7\n+\n\u00d7\n+\n\u00d7\n+\n\u00d7\n=\nThus,\nVar (X) = E (X2) \u2013 (E(X))2\n=\n2\n91\n21\n91\n441\n6\n6\n6\n36\n\u239b\n\u239e\n\u2212\n=\n\u2212\n\u239c\n\u239f\n\u239d\n\u23a0\n  \n35\n12\n \nExample 29 Two cards are drawn simultaneously (or successively without replacement)\nfrom a well shuffled pack of 52 cards"
    },
    {
        "Chapter": "1",
        "sentence_range": "7403-7406",
        "Text": "Let X denote the number obtained on the throw Then X is a random variable\nwhich can take values 1, 2, 3, 4, 5, or 6 \u00a9 NCERT\nnot to be republished\n 568\nMATHEMATICS\nAlso\nP(1) = P(2) = P(3) = P(4) = P(5) = P(6) = 1\n6\nTherefore, the Probability distribution of  X  is\nX\n1\n2\n3\n4\n5\n6\nP(X)\n1\n6\n1\n6\n1\n6\n1\n6\n1\n6\n1\n6\nNow\nE(X) =\n1\n(\n)\nn\ni\ni\ni\nx p x\n  \n=\n1\n1\n1\n1\n1\n1\n21\n1\n2\n3\n4\n5\n6\n6\n6\n6\n6\n6\n6\n6\n\u00d7\n+ \u00d7\n+ \u00d7\n+ \u00d7\n+ \u00d7\n+ \u00d7\n=\nAlso\nE(X2) =\n2\n2\n2\n2\n2\n2\n1\n1\n1\n1\n1\n1\n91\n1\n2\n3\n4\n5\n6\n6\n6\n6\n6\n6\n6\n6\n\u00d7\n+\n\u00d7\n+\n\u00d7\n+\n\u00d7\n+\n\u00d7\n+\n\u00d7\n=\nThus,\nVar (X) = E (X2) \u2013 (E(X))2\n=\n2\n91\n21\n91\n441\n6\n6\n6\n36\n\u239b\n\u239e\n\u2212\n=\n\u2212\n\u239c\n\u239f\n\u239d\n\u23a0\n  \n35\n12\n \nExample 29 Two cards are drawn simultaneously (or successively without replacement)\nfrom a well shuffled pack of 52 cards Find the mean, variance and standard deviation\nof the number of kings"
    },
    {
        "Chapter": "1",
        "sentence_range": "7404-7407",
        "Text": "Then X is a random variable\nwhich can take values 1, 2, 3, 4, 5, or 6 \u00a9 NCERT\nnot to be republished\n 568\nMATHEMATICS\nAlso\nP(1) = P(2) = P(3) = P(4) = P(5) = P(6) = 1\n6\nTherefore, the Probability distribution of  X  is\nX\n1\n2\n3\n4\n5\n6\nP(X)\n1\n6\n1\n6\n1\n6\n1\n6\n1\n6\n1\n6\nNow\nE(X) =\n1\n(\n)\nn\ni\ni\ni\nx p x\n  \n=\n1\n1\n1\n1\n1\n1\n21\n1\n2\n3\n4\n5\n6\n6\n6\n6\n6\n6\n6\n6\n\u00d7\n+ \u00d7\n+ \u00d7\n+ \u00d7\n+ \u00d7\n+ \u00d7\n=\nAlso\nE(X2) =\n2\n2\n2\n2\n2\n2\n1\n1\n1\n1\n1\n1\n91\n1\n2\n3\n4\n5\n6\n6\n6\n6\n6\n6\n6\n6\n\u00d7\n+\n\u00d7\n+\n\u00d7\n+\n\u00d7\n+\n\u00d7\n+\n\u00d7\n=\nThus,\nVar (X) = E (X2) \u2013 (E(X))2\n=\n2\n91\n21\n91\n441\n6\n6\n6\n36\n\u239b\n\u239e\n\u2212\n=\n\u2212\n\u239c\n\u239f\n\u239d\n\u23a0\n  \n35\n12\n \nExample 29 Two cards are drawn simultaneously (or successively without replacement)\nfrom a well shuffled pack of 52 cards Find the mean, variance and standard deviation\nof the number of kings Solution Let X denote the number of kings in a draw of two cards"
    },
    {
        "Chapter": "1",
        "sentence_range": "7405-7408",
        "Text": "\u00a9 NCERT\nnot to be republished\n 568\nMATHEMATICS\nAlso\nP(1) = P(2) = P(3) = P(4) = P(5) = P(6) = 1\n6\nTherefore, the Probability distribution of  X  is\nX\n1\n2\n3\n4\n5\n6\nP(X)\n1\n6\n1\n6\n1\n6\n1\n6\n1\n6\n1\n6\nNow\nE(X) =\n1\n(\n)\nn\ni\ni\ni\nx p x\n  \n=\n1\n1\n1\n1\n1\n1\n21\n1\n2\n3\n4\n5\n6\n6\n6\n6\n6\n6\n6\n6\n\u00d7\n+ \u00d7\n+ \u00d7\n+ \u00d7\n+ \u00d7\n+ \u00d7\n=\nAlso\nE(X2) =\n2\n2\n2\n2\n2\n2\n1\n1\n1\n1\n1\n1\n91\n1\n2\n3\n4\n5\n6\n6\n6\n6\n6\n6\n6\n6\n\u00d7\n+\n\u00d7\n+\n\u00d7\n+\n\u00d7\n+\n\u00d7\n+\n\u00d7\n=\nThus,\nVar (X) = E (X2) \u2013 (E(X))2\n=\n2\n91\n21\n91\n441\n6\n6\n6\n36\n\u239b\n\u239e\n\u2212\n=\n\u2212\n\u239c\n\u239f\n\u239d\n\u23a0\n  \n35\n12\n \nExample 29 Two cards are drawn simultaneously (or successively without replacement)\nfrom a well shuffled pack of 52 cards Find the mean, variance and standard deviation\nof the number of kings Solution Let X denote the number of kings in a draw of two cards X is a random\nvariable which can assume the values 0, 1 or 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "7406-7409",
        "Text": "Find the mean, variance and standard deviation\nof the number of kings Solution Let X denote the number of kings in a draw of two cards X is a random\nvariable which can assume the values 0, 1 or 2 Now\nP(X = 0) = P (no king) \n48\n2\n52\n2\n48"
    },
    {
        "Chapter": "1",
        "sentence_range": "7407-7410",
        "Text": "Solution Let X denote the number of kings in a draw of two cards X is a random\nvariable which can assume the values 0, 1 or 2 Now\nP(X = 0) = P (no king) \n48\n2\n52\n2\n48 C\n48 47\n188\n2"
    },
    {
        "Chapter": "1",
        "sentence_range": "7408-7411",
        "Text": "X is a random\nvariable which can assume the values 0, 1 or 2 Now\nP(X = 0) = P (no king) \n48\n2\n52\n2\n48 C\n48 47\n188\n2 (48\n52"
    },
    {
        "Chapter": "1",
        "sentence_range": "7409-7412",
        "Text": "Now\nP(X = 0) = P (no king) \n48\n2\n52\n2\n48 C\n48 47\n188\n2 (48\n52 2)"
    },
    {
        "Chapter": "1",
        "sentence_range": "7410-7413",
        "Text": "C\n48 47\n188\n2 (48\n52 2) 52 51\n221\nC\n2"
    },
    {
        "Chapter": "1",
        "sentence_range": "7411-7414",
        "Text": "(48\n52 2) 52 51\n221\nC\n2 (52\n2)"
    },
    {
        "Chapter": "1",
        "sentence_range": "7412-7415",
        "Text": "2) 52 51\n221\nC\n2 (52\n2) P(X = 1) = P (one king and one non-king) \n4\n148\n1\n52\n2\nC\nCC\n=\n   = 4 48 2\n32\n52 51\n221\n\u00d7\n\u00d7 =\n\u00d7\n\u00a9 NCERT\nnot to be republished\nPROBABILITY      569\nand\nP(X = 2) = P (two kings) =\n4\n2\n52\n2\nC\n4 3\n1\n52 51\n221\nC\n\u00d7\n=\n=\n\u00d7\nThus,  the probability distribution of  X is\nX\n0\n1\n2\nP(X)\n188\n221\n32\n221\n1\n221\nNow                           Mean of\nX = E(X) = \n1\n(\n)\nn\ni\ni\ni\nx p x\n  \n=\n188\n32\n1\n34\n0\n1\n2\n221\n221\n221\n221\n\u00d7\n+ \u00d7\n+ \u00d7\n=\nAlso\nE(X2) =\n2\n1\n(\n)\nn\ni\ni\ni\nx p x\n=\u2211\n=\n2\n2\n2\n188\n32\n1\n36\n0\n1\n2\n221\n221\n221\n221\n\u00d7\n+\n\u00d7\n+\n\u00d7\n=\nNow\nVar(X) = E(X2) \u2013 [E(X)]2\n=\n2\n2\n36\n34\n6800\n221\u2013\n221\n(221)\n\u239b\n\u239e =\n\u239c\n\u239f\n\u239d\n\u23a0\nTherefore\n\u03c3x =\n6800\nVar(X)\n0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7413-7416",
        "Text": "52 51\n221\nC\n2 (52\n2) P(X = 1) = P (one king and one non-king) \n4\n148\n1\n52\n2\nC\nCC\n=\n   = 4 48 2\n32\n52 51\n221\n\u00d7\n\u00d7 =\n\u00d7\n\u00a9 NCERT\nnot to be republished\nPROBABILITY      569\nand\nP(X = 2) = P (two kings) =\n4\n2\n52\n2\nC\n4 3\n1\n52 51\n221\nC\n\u00d7\n=\n=\n\u00d7\nThus,  the probability distribution of  X is\nX\n0\n1\n2\nP(X)\n188\n221\n32\n221\n1\n221\nNow                           Mean of\nX = E(X) = \n1\n(\n)\nn\ni\ni\ni\nx p x\n  \n=\n188\n32\n1\n34\n0\n1\n2\n221\n221\n221\n221\n\u00d7\n+ \u00d7\n+ \u00d7\n=\nAlso\nE(X2) =\n2\n1\n(\n)\nn\ni\ni\ni\nx p x\n=\u2211\n=\n2\n2\n2\n188\n32\n1\n36\n0\n1\n2\n221\n221\n221\n221\n\u00d7\n+\n\u00d7\n+\n\u00d7\n=\nNow\nVar(X) = E(X2) \u2013 [E(X)]2\n=\n2\n2\n36\n34\n6800\n221\u2013\n221\n(221)\n\u239b\n\u239e =\n\u239c\n\u239f\n\u239d\n\u23a0\nTherefore\n\u03c3x =\n6800\nVar(X)\n0 37\n221\n=\n=\nEXERCISE 13"
    },
    {
        "Chapter": "1",
        "sentence_range": "7414-7417",
        "Text": "(52\n2) P(X = 1) = P (one king and one non-king) \n4\n148\n1\n52\n2\nC\nCC\n=\n   = 4 48 2\n32\n52 51\n221\n\u00d7\n\u00d7 =\n\u00d7\n\u00a9 NCERT\nnot to be republished\nPROBABILITY      569\nand\nP(X = 2) = P (two kings) =\n4\n2\n52\n2\nC\n4 3\n1\n52 51\n221\nC\n\u00d7\n=\n=\n\u00d7\nThus,  the probability distribution of  X is\nX\n0\n1\n2\nP(X)\n188\n221\n32\n221\n1\n221\nNow                           Mean of\nX = E(X) = \n1\n(\n)\nn\ni\ni\ni\nx p x\n  \n=\n188\n32\n1\n34\n0\n1\n2\n221\n221\n221\n221\n\u00d7\n+ \u00d7\n+ \u00d7\n=\nAlso\nE(X2) =\n2\n1\n(\n)\nn\ni\ni\ni\nx p x\n=\u2211\n=\n2\n2\n2\n188\n32\n1\n36\n0\n1\n2\n221\n221\n221\n221\n\u00d7\n+\n\u00d7\n+\n\u00d7\n=\nNow\nVar(X) = E(X2) \u2013 [E(X)]2\n=\n2\n2\n36\n34\n6800\n221\u2013\n221\n(221)\n\u239b\n\u239e =\n\u239c\n\u239f\n\u239d\n\u23a0\nTherefore\n\u03c3x =\n6800\nVar(X)\n0 37\n221\n=\n=\nEXERCISE 13 4\n1"
    },
    {
        "Chapter": "1",
        "sentence_range": "7415-7418",
        "Text": "P(X = 1) = P (one king and one non-king) \n4\n148\n1\n52\n2\nC\nCC\n=\n   = 4 48 2\n32\n52 51\n221\n\u00d7\n\u00d7 =\n\u00d7\n\u00a9 NCERT\nnot to be republished\nPROBABILITY      569\nand\nP(X = 2) = P (two kings) =\n4\n2\n52\n2\nC\n4 3\n1\n52 51\n221\nC\n\u00d7\n=\n=\n\u00d7\nThus,  the probability distribution of  X is\nX\n0\n1\n2\nP(X)\n188\n221\n32\n221\n1\n221\nNow                           Mean of\nX = E(X) = \n1\n(\n)\nn\ni\ni\ni\nx p x\n  \n=\n188\n32\n1\n34\n0\n1\n2\n221\n221\n221\n221\n\u00d7\n+ \u00d7\n+ \u00d7\n=\nAlso\nE(X2) =\n2\n1\n(\n)\nn\ni\ni\ni\nx p x\n=\u2211\n=\n2\n2\n2\n188\n32\n1\n36\n0\n1\n2\n221\n221\n221\n221\n\u00d7\n+\n\u00d7\n+\n\u00d7\n=\nNow\nVar(X) = E(X2) \u2013 [E(X)]2\n=\n2\n2\n36\n34\n6800\n221\u2013\n221\n(221)\n\u239b\n\u239e =\n\u239c\n\u239f\n\u239d\n\u23a0\nTherefore\n\u03c3x =\n6800\nVar(X)\n0 37\n221\n=\n=\nEXERCISE 13 4\n1 State which of the following are not the probability distributions of a random\nvariable"
    },
    {
        "Chapter": "1",
        "sentence_range": "7416-7419",
        "Text": "37\n221\n=\n=\nEXERCISE 13 4\n1 State which of the following are not the probability distributions of a random\nvariable Give reasons for your answer"
    },
    {
        "Chapter": "1",
        "sentence_range": "7417-7420",
        "Text": "4\n1 State which of the following are not the probability distributions of a random\nvariable Give reasons for your answer (i)\nX\n0\n1\n2\nP(X)\n0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7418-7421",
        "Text": "State which of the following are not the probability distributions of a random\nvariable Give reasons for your answer (i)\nX\n0\n1\n2\nP(X)\n0 4\n0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7419-7422",
        "Text": "Give reasons for your answer (i)\nX\n0\n1\n2\nP(X)\n0 4\n0 4\n0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7420-7423",
        "Text": "(i)\nX\n0\n1\n2\nP(X)\n0 4\n0 4\n0 2\n(ii)\nX\n0\n1\n2\n3\n4\nP(X)\n0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7421-7424",
        "Text": "4\n0 4\n0 2\n(ii)\nX\n0\n1\n2\n3\n4\nP(X)\n0 1\n0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7422-7425",
        "Text": "4\n0 2\n(ii)\nX\n0\n1\n2\n3\n4\nP(X)\n0 1\n0 5\n0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7423-7426",
        "Text": "2\n(ii)\nX\n0\n1\n2\n3\n4\nP(X)\n0 1\n0 5\n0 2\n\u2013 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7424-7427",
        "Text": "1\n0 5\n0 2\n\u2013 0 1\n0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7425-7428",
        "Text": "5\n0 2\n\u2013 0 1\n0 3\n\u00a9 NCERT\nnot to be republished\n 570\nMATHEMATICS\n(iii)\nY\n\u2013 1\n0\n1\nP(Y)\n0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7426-7429",
        "Text": "2\n\u2013 0 1\n0 3\n\u00a9 NCERT\nnot to be republished\n 570\nMATHEMATICS\n(iii)\nY\n\u2013 1\n0\n1\nP(Y)\n0 6\n0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7427-7430",
        "Text": "1\n0 3\n\u00a9 NCERT\nnot to be republished\n 570\nMATHEMATICS\n(iii)\nY\n\u2013 1\n0\n1\nP(Y)\n0 6\n0 1\n0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7428-7431",
        "Text": "3\n\u00a9 NCERT\nnot to be republished\n 570\nMATHEMATICS\n(iii)\nY\n\u2013 1\n0\n1\nP(Y)\n0 6\n0 1\n0 2\n(iv)\nZ\n3\n2\n1\n0\n\u20131\nP(Z)\n0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7429-7432",
        "Text": "6\n0 1\n0 2\n(iv)\nZ\n3\n2\n1\n0\n\u20131\nP(Z)\n0 3\n0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7430-7433",
        "Text": "1\n0 2\n(iv)\nZ\n3\n2\n1\n0\n\u20131\nP(Z)\n0 3\n0 2\n0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7431-7434",
        "Text": "2\n(iv)\nZ\n3\n2\n1\n0\n\u20131\nP(Z)\n0 3\n0 2\n0 4\n0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7432-7435",
        "Text": "3\n0 2\n0 4\n0 1\n0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7433-7436",
        "Text": "2\n0 4\n0 1\n0 05\n2"
    },
    {
        "Chapter": "1",
        "sentence_range": "7434-7437",
        "Text": "4\n0 1\n0 05\n2 An urn contains 5 red and 2 black balls"
    },
    {
        "Chapter": "1",
        "sentence_range": "7435-7438",
        "Text": "1\n0 05\n2 An urn contains 5 red and 2 black balls Two balls are randomly drawn"
    },
    {
        "Chapter": "1",
        "sentence_range": "7436-7439",
        "Text": "05\n2 An urn contains 5 red and 2 black balls Two balls are randomly drawn Let X\nrepresent the number of black balls"
    },
    {
        "Chapter": "1",
        "sentence_range": "7437-7440",
        "Text": "An urn contains 5 red and 2 black balls Two balls are randomly drawn Let X\nrepresent the number of black balls What are the possible values of X"
    },
    {
        "Chapter": "1",
        "sentence_range": "7438-7441",
        "Text": "Two balls are randomly drawn Let X\nrepresent the number of black balls What are the possible values of X Is X a\nrandom variable"
    },
    {
        "Chapter": "1",
        "sentence_range": "7439-7442",
        "Text": "Let X\nrepresent the number of black balls What are the possible values of X Is X a\nrandom variable 3"
    },
    {
        "Chapter": "1",
        "sentence_range": "7440-7443",
        "Text": "What are the possible values of X Is X a\nrandom variable 3 Let X represent the difference between the number of heads and the number of\ntails obtained when a coin is tossed 6 times"
    },
    {
        "Chapter": "1",
        "sentence_range": "7441-7444",
        "Text": "Is X a\nrandom variable 3 Let X represent the difference between the number of heads and the number of\ntails obtained when a coin is tossed 6 times What are possible values of X"
    },
    {
        "Chapter": "1",
        "sentence_range": "7442-7445",
        "Text": "3 Let X represent the difference between the number of heads and the number of\ntails obtained when a coin is tossed 6 times What are possible values of X 4"
    },
    {
        "Chapter": "1",
        "sentence_range": "7443-7446",
        "Text": "Let X represent the difference between the number of heads and the number of\ntails obtained when a coin is tossed 6 times What are possible values of X 4 Find the probability distribution of\n(i) number of heads in two tosses of a coin"
    },
    {
        "Chapter": "1",
        "sentence_range": "7444-7447",
        "Text": "What are possible values of X 4 Find the probability distribution of\n(i) number of heads in two tosses of a coin (ii) number of tails in the simultaneous tosses of three coins"
    },
    {
        "Chapter": "1",
        "sentence_range": "7445-7448",
        "Text": "4 Find the probability distribution of\n(i) number of heads in two tosses of a coin (ii) number of tails in the simultaneous tosses of three coins (iii) number of heads in four tosses of a coin"
    },
    {
        "Chapter": "1",
        "sentence_range": "7446-7449",
        "Text": "Find the probability distribution of\n(i) number of heads in two tosses of a coin (ii) number of tails in the simultaneous tosses of three coins (iii) number of heads in four tosses of a coin 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "7447-7450",
        "Text": "(ii) number of tails in the simultaneous tosses of three coins (iii) number of heads in four tosses of a coin 5 Find the probability distribution of the number of successes in two tosses of a die,\nwhere a success is defined as\n(i) number greater than 4\n(ii) six appears on at least one die\n6"
    },
    {
        "Chapter": "1",
        "sentence_range": "7448-7451",
        "Text": "(iii) number of heads in four tosses of a coin 5 Find the probability distribution of the number of successes in two tosses of a die,\nwhere a success is defined as\n(i) number greater than 4\n(ii) six appears on at least one die\n6 From a lot of 30 bulbs which include 6 defectives, a sample of 4 bulbs is drawn\nat random with replacement"
    },
    {
        "Chapter": "1",
        "sentence_range": "7449-7452",
        "Text": "5 Find the probability distribution of the number of successes in two tosses of a die,\nwhere a success is defined as\n(i) number greater than 4\n(ii) six appears on at least one die\n6 From a lot of 30 bulbs which include 6 defectives, a sample of 4 bulbs is drawn\nat random with replacement Find the probability distribution of the number of\ndefective bulbs"
    },
    {
        "Chapter": "1",
        "sentence_range": "7450-7453",
        "Text": "Find the probability distribution of the number of successes in two tosses of a die,\nwhere a success is defined as\n(i) number greater than 4\n(ii) six appears on at least one die\n6 From a lot of 30 bulbs which include 6 defectives, a sample of 4 bulbs is drawn\nat random with replacement Find the probability distribution of the number of\ndefective bulbs 7"
    },
    {
        "Chapter": "1",
        "sentence_range": "7451-7454",
        "Text": "From a lot of 30 bulbs which include 6 defectives, a sample of 4 bulbs is drawn\nat random with replacement Find the probability distribution of the number of\ndefective bulbs 7 A coin is biased so that the head is 3 times as likely to occur as tail"
    },
    {
        "Chapter": "1",
        "sentence_range": "7452-7455",
        "Text": "Find the probability distribution of the number of\ndefective bulbs 7 A coin is biased so that the head is 3 times as likely to occur as tail If the coin is\ntossed twice, find the probability distribution of number of tails"
    },
    {
        "Chapter": "1",
        "sentence_range": "7453-7456",
        "Text": "7 A coin is biased so that the head is 3 times as likely to occur as tail If the coin is\ntossed twice, find the probability distribution of number of tails 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "7454-7457",
        "Text": "A coin is biased so that the head is 3 times as likely to occur as tail If the coin is\ntossed twice, find the probability distribution of number of tails 8 A random variable X has the following probability distribution:\nX\n0\n1\n2\n3\n4\n5\n6\n7\nP(X)\n0\nk\n2k\n2k\n3k\nk 2 2k2 7k2+k\nDetermine\n(i) k\n(ii) P(X < 3)\n(iii)  P(X > 6)\n(iv) P(0 < X < 3)\n\u00a9 NCERT\nnot to be republished\nPROBABILITY      571\n9"
    },
    {
        "Chapter": "1",
        "sentence_range": "7455-7458",
        "Text": "If the coin is\ntossed twice, find the probability distribution of number of tails 8 A random variable X has the following probability distribution:\nX\n0\n1\n2\n3\n4\n5\n6\n7\nP(X)\n0\nk\n2k\n2k\n3k\nk 2 2k2 7k2+k\nDetermine\n(i) k\n(ii) P(X < 3)\n(iii)  P(X > 6)\n(iv) P(0 < X < 3)\n\u00a9 NCERT\nnot to be republished\nPROBABILITY      571\n9 The random variable X has a probability distribution P(X) of the following form,\nwhere k is some number :\nP(X) =\n,\n0\n2 ,\n1\n3 ,\n2\n0, otherwise\nk\nif x\nk if x\nk if x\n=\n\u23aa\u23a7\n=\n\u23aa\u23a8\n=\n\u23aa\n\u23aa\u23a9\n(a)\nDetermine the value of k"
    },
    {
        "Chapter": "1",
        "sentence_range": "7456-7459",
        "Text": "8 A random variable X has the following probability distribution:\nX\n0\n1\n2\n3\n4\n5\n6\n7\nP(X)\n0\nk\n2k\n2k\n3k\nk 2 2k2 7k2+k\nDetermine\n(i) k\n(ii) P(X < 3)\n(iii)  P(X > 6)\n(iv) P(0 < X < 3)\n\u00a9 NCERT\nnot to be republished\nPROBABILITY      571\n9 The random variable X has a probability distribution P(X) of the following form,\nwhere k is some number :\nP(X) =\n,\n0\n2 ,\n1\n3 ,\n2\n0, otherwise\nk\nif x\nk if x\nk if x\n=\n\u23aa\u23a7\n=\n\u23aa\u23a8\n=\n\u23aa\n\u23aa\u23a9\n(a)\nDetermine the value of k (b)\nFind P (X < 2),  P (X \u2264 2), P(X \u2265 2)"
    },
    {
        "Chapter": "1",
        "sentence_range": "7457-7460",
        "Text": "A random variable X has the following probability distribution:\nX\n0\n1\n2\n3\n4\n5\n6\n7\nP(X)\n0\nk\n2k\n2k\n3k\nk 2 2k2 7k2+k\nDetermine\n(i) k\n(ii) P(X < 3)\n(iii)  P(X > 6)\n(iv) P(0 < X < 3)\n\u00a9 NCERT\nnot to be republished\nPROBABILITY      571\n9 The random variable X has a probability distribution P(X) of the following form,\nwhere k is some number :\nP(X) =\n,\n0\n2 ,\n1\n3 ,\n2\n0, otherwise\nk\nif x\nk if x\nk if x\n=\n\u23aa\u23a7\n=\n\u23aa\u23a8\n=\n\u23aa\n\u23aa\u23a9\n(a)\nDetermine the value of k (b)\nFind P (X < 2),  P (X \u2264 2), P(X \u2265 2) 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "7458-7461",
        "Text": "The random variable X has a probability distribution P(X) of the following form,\nwhere k is some number :\nP(X) =\n,\n0\n2 ,\n1\n3 ,\n2\n0, otherwise\nk\nif x\nk if x\nk if x\n=\n\u23aa\u23a7\n=\n\u23aa\u23a8\n=\n\u23aa\n\u23aa\u23a9\n(a)\nDetermine the value of k (b)\nFind P (X < 2),  P (X \u2264 2), P(X \u2265 2) 10 Find the mean number of heads in three tosses of a fair coin"
    },
    {
        "Chapter": "1",
        "sentence_range": "7459-7462",
        "Text": "(b)\nFind P (X < 2),  P (X \u2264 2), P(X \u2265 2) 10 Find the mean number of heads in three tosses of a fair coin 11"
    },
    {
        "Chapter": "1",
        "sentence_range": "7460-7463",
        "Text": "10 Find the mean number of heads in three tosses of a fair coin 11 Two dice are thrown simultaneously"
    },
    {
        "Chapter": "1",
        "sentence_range": "7461-7464",
        "Text": "Find the mean number of heads in three tosses of a fair coin 11 Two dice are thrown simultaneously If X denotes the number of sixes, find  the\nexpectation of X"
    },
    {
        "Chapter": "1",
        "sentence_range": "7462-7465",
        "Text": "11 Two dice are thrown simultaneously If X denotes the number of sixes, find  the\nexpectation of X 12"
    },
    {
        "Chapter": "1",
        "sentence_range": "7463-7466",
        "Text": "Two dice are thrown simultaneously If X denotes the number of sixes, find  the\nexpectation of X 12 Two numbers are selected at random (without replacement) from the first six\npositive integers"
    },
    {
        "Chapter": "1",
        "sentence_range": "7464-7467",
        "Text": "If X denotes the number of sixes, find  the\nexpectation of X 12 Two numbers are selected at random (without replacement) from the first six\npositive integers Let X denote the larger of the two numbers obtained"
    },
    {
        "Chapter": "1",
        "sentence_range": "7465-7468",
        "Text": "12 Two numbers are selected at random (without replacement) from the first six\npositive integers Let X denote the larger of the two numbers obtained Find\nE(X)"
    },
    {
        "Chapter": "1",
        "sentence_range": "7466-7469",
        "Text": "Two numbers are selected at random (without replacement) from the first six\npositive integers Let X denote the larger of the two numbers obtained Find\nE(X) 13"
    },
    {
        "Chapter": "1",
        "sentence_range": "7467-7470",
        "Text": "Let X denote the larger of the two numbers obtained Find\nE(X) 13 Let X denote the sum of the numbers obtained when two fair dice are rolled"
    },
    {
        "Chapter": "1",
        "sentence_range": "7468-7471",
        "Text": "Find\nE(X) 13 Let X denote the sum of the numbers obtained when two fair dice are rolled Find the variance and standard deviation of X"
    },
    {
        "Chapter": "1",
        "sentence_range": "7469-7472",
        "Text": "13 Let X denote the sum of the numbers obtained when two fair dice are rolled Find the variance and standard deviation of X 14"
    },
    {
        "Chapter": "1",
        "sentence_range": "7470-7473",
        "Text": "Let X denote the sum of the numbers obtained when two fair dice are rolled Find the variance and standard deviation of X 14 A class has 15 students whose ages are 14, 17, 15, 14, 21, 17, 19, 20, 16, 18, 20,\n17, 16, 19 and 20 years"
    },
    {
        "Chapter": "1",
        "sentence_range": "7471-7474",
        "Text": "Find the variance and standard deviation of X 14 A class has 15 students whose ages are 14, 17, 15, 14, 21, 17, 19, 20, 16, 18, 20,\n17, 16, 19 and 20 years One student is selected in such a manner that each has\nthe same chance of being chosen and the age X of the selected student is\nrecorded"
    },
    {
        "Chapter": "1",
        "sentence_range": "7472-7475",
        "Text": "14 A class has 15 students whose ages are 14, 17, 15, 14, 21, 17, 19, 20, 16, 18, 20,\n17, 16, 19 and 20 years One student is selected in such a manner that each has\nthe same chance of being chosen and the age X of the selected student is\nrecorded What is the probability  distribution of the random variable X"
    },
    {
        "Chapter": "1",
        "sentence_range": "7473-7476",
        "Text": "A class has 15 students whose ages are 14, 17, 15, 14, 21, 17, 19, 20, 16, 18, 20,\n17, 16, 19 and 20 years One student is selected in such a manner that each has\nthe same chance of being chosen and the age X of the selected student is\nrecorded What is the probability  distribution of the random variable X Find\nmean, variance and standard deviation of X"
    },
    {
        "Chapter": "1",
        "sentence_range": "7474-7477",
        "Text": "One student is selected in such a manner that each has\nthe same chance of being chosen and the age X of the selected student is\nrecorded What is the probability  distribution of the random variable X Find\nmean, variance and standard deviation of X 15"
    },
    {
        "Chapter": "1",
        "sentence_range": "7475-7478",
        "Text": "What is the probability  distribution of the random variable X Find\nmean, variance and standard deviation of X 15 In a meeting, 70% of the members favour and 30% oppose a certain proposal"
    },
    {
        "Chapter": "1",
        "sentence_range": "7476-7479",
        "Text": "Find\nmean, variance and standard deviation of X 15 In a meeting, 70% of the members favour and 30% oppose a certain proposal A member is selected at random and we take X = 0 if he opposed, and X = 1 if\nhe is in favour"
    },
    {
        "Chapter": "1",
        "sentence_range": "7477-7480",
        "Text": "15 In a meeting, 70% of the members favour and 30% oppose a certain proposal A member is selected at random and we take X = 0 if he opposed, and X = 1 if\nhe is in favour Find E(X) and Var (X)"
    },
    {
        "Chapter": "1",
        "sentence_range": "7478-7481",
        "Text": "In a meeting, 70% of the members favour and 30% oppose a certain proposal A member is selected at random and we take X = 0 if he opposed, and X = 1 if\nhe is in favour Find E(X) and Var (X) Choose the correct answer in each of the following:\n16"
    },
    {
        "Chapter": "1",
        "sentence_range": "7479-7482",
        "Text": "A member is selected at random and we take X = 0 if he opposed, and X = 1 if\nhe is in favour Find E(X) and Var (X) Choose the correct answer in each of the following:\n16 The mean of the numbers obtained on throwing a die having written 1 on three\nfaces, 2 on two faces and 5 on one face is\n(A) 1\n(B) 2\n(C) 5\n(D) 8\n3\n17"
    },
    {
        "Chapter": "1",
        "sentence_range": "7480-7483",
        "Text": "Find E(X) and Var (X) Choose the correct answer in each of the following:\n16 The mean of the numbers obtained on throwing a die having written 1 on three\nfaces, 2 on two faces and 5 on one face is\n(A) 1\n(B) 2\n(C) 5\n(D) 8\n3\n17 Suppose that two cards are drawn at random from a deck of cards"
    },
    {
        "Chapter": "1",
        "sentence_range": "7481-7484",
        "Text": "Choose the correct answer in each of the following:\n16 The mean of the numbers obtained on throwing a die having written 1 on three\nfaces, 2 on two faces and 5 on one face is\n(A) 1\n(B) 2\n(C) 5\n(D) 8\n3\n17 Suppose that two cards are drawn at random from a deck of cards Let X be the\nnumber of aces obtained"
    },
    {
        "Chapter": "1",
        "sentence_range": "7482-7485",
        "Text": "The mean of the numbers obtained on throwing a die having written 1 on three\nfaces, 2 on two faces and 5 on one face is\n(A) 1\n(B) 2\n(C) 5\n(D) 8\n3\n17 Suppose that two cards are drawn at random from a deck of cards Let X be the\nnumber of aces obtained Then the value of E(X) is\n(A)\n37\n221\n(B)\n135\n(C)\n131\n(D)\n2\n13\n\u00a9 NCERT\nnot to be republished\n 572\nMATHEMATICS\n13"
    },
    {
        "Chapter": "1",
        "sentence_range": "7483-7486",
        "Text": "Suppose that two cards are drawn at random from a deck of cards Let X be the\nnumber of aces obtained Then the value of E(X) is\n(A)\n37\n221\n(B)\n135\n(C)\n131\n(D)\n2\n13\n\u00a9 NCERT\nnot to be republished\n 572\nMATHEMATICS\n13 7  Bernoulli Trials and Binomial Distribution\n13"
    },
    {
        "Chapter": "1",
        "sentence_range": "7484-7487",
        "Text": "Let X be the\nnumber of aces obtained Then the value of E(X) is\n(A)\n37\n221\n(B)\n135\n(C)\n131\n(D)\n2\n13\n\u00a9 NCERT\nnot to be republished\n 572\nMATHEMATICS\n13 7  Bernoulli Trials and Binomial Distribution\n13 7"
    },
    {
        "Chapter": "1",
        "sentence_range": "7485-7488",
        "Text": "Then the value of E(X) is\n(A)\n37\n221\n(B)\n135\n(C)\n131\n(D)\n2\n13\n\u00a9 NCERT\nnot to be republished\n 572\nMATHEMATICS\n13 7  Bernoulli Trials and Binomial Distribution\n13 7 1  Bernoulli trials\nMany experiments are dichotomous in nature"
    },
    {
        "Chapter": "1",
        "sentence_range": "7486-7489",
        "Text": "7  Bernoulli Trials and Binomial Distribution\n13 7 1  Bernoulli trials\nMany experiments are dichotomous in nature For example, a tossed coin shows a\n\u2018head\u2019 or \u2018tail\u2019, a manufactured item can be \u2018defective\u2019 or \u2018non-defective\u2019, the response\nto a question might be \u2018yes\u2019 or \u2018no\u2019, an egg has \u2018hatched\u2019 or \u2018not hatched\u2019, the decision\nis \u2018yes\u2019 or \u2018no\u2019 etc"
    },
    {
        "Chapter": "1",
        "sentence_range": "7487-7490",
        "Text": "7 1  Bernoulli trials\nMany experiments are dichotomous in nature For example, a tossed coin shows a\n\u2018head\u2019 or \u2018tail\u2019, a manufactured item can be \u2018defective\u2019 or \u2018non-defective\u2019, the response\nto a question might be \u2018yes\u2019 or \u2018no\u2019, an egg has \u2018hatched\u2019 or \u2018not hatched\u2019, the decision\nis \u2018yes\u2019 or \u2018no\u2019 etc In such cases, it is customary to call one of the outcomes a \u2018success\u2019\nand the other \u2018not success\u2019 or \u2018failure\u2019"
    },
    {
        "Chapter": "1",
        "sentence_range": "7488-7491",
        "Text": "1  Bernoulli trials\nMany experiments are dichotomous in nature For example, a tossed coin shows a\n\u2018head\u2019 or \u2018tail\u2019, a manufactured item can be \u2018defective\u2019 or \u2018non-defective\u2019, the response\nto a question might be \u2018yes\u2019 or \u2018no\u2019, an egg has \u2018hatched\u2019 or \u2018not hatched\u2019, the decision\nis \u2018yes\u2019 or \u2018no\u2019 etc In such cases, it is customary to call one of the outcomes a \u2018success\u2019\nand the other \u2018not success\u2019 or \u2018failure\u2019 For example, in tossing a coin, if the occurrence\nof the head is considered a success, then occurrence of tail is a failure"
    },
    {
        "Chapter": "1",
        "sentence_range": "7489-7492",
        "Text": "For example, a tossed coin shows a\n\u2018head\u2019 or \u2018tail\u2019, a manufactured item can be \u2018defective\u2019 or \u2018non-defective\u2019, the response\nto a question might be \u2018yes\u2019 or \u2018no\u2019, an egg has \u2018hatched\u2019 or \u2018not hatched\u2019, the decision\nis \u2018yes\u2019 or \u2018no\u2019 etc In such cases, it is customary to call one of the outcomes a \u2018success\u2019\nand the other \u2018not success\u2019 or \u2018failure\u2019 For example, in tossing a coin, if the occurrence\nof the head is considered a success, then occurrence of tail is a failure Each time we toss a coin or roll a die or perform any other experiment, we call it a\ntrial"
    },
    {
        "Chapter": "1",
        "sentence_range": "7490-7493",
        "Text": "In such cases, it is customary to call one of the outcomes a \u2018success\u2019\nand the other \u2018not success\u2019 or \u2018failure\u2019 For example, in tossing a coin, if the occurrence\nof the head is considered a success, then occurrence of tail is a failure Each time we toss a coin or roll a die or perform any other experiment, we call it a\ntrial If a coin is tossed, say, 4 times, the number of trials is 4, each having exactly two\noutcomes, namely, success or failure"
    },
    {
        "Chapter": "1",
        "sentence_range": "7491-7494",
        "Text": "For example, in tossing a coin, if the occurrence\nof the head is considered a success, then occurrence of tail is a failure Each time we toss a coin or roll a die or perform any other experiment, we call it a\ntrial If a coin is tossed, say, 4 times, the number of trials is 4, each having exactly two\noutcomes, namely, success or failure The outcome of any trial is independent of the\noutcome of any other trial"
    },
    {
        "Chapter": "1",
        "sentence_range": "7492-7495",
        "Text": "Each time we toss a coin or roll a die or perform any other experiment, we call it a\ntrial If a coin is tossed, say, 4 times, the number of trials is 4, each having exactly two\noutcomes, namely, success or failure The outcome of any trial is independent of the\noutcome of any other trial In each of such trials, the probability of success or failure\nremains constant"
    },
    {
        "Chapter": "1",
        "sentence_range": "7493-7496",
        "Text": "If a coin is tossed, say, 4 times, the number of trials is 4, each having exactly two\noutcomes, namely, success or failure The outcome of any trial is independent of the\noutcome of any other trial In each of such trials, the probability of success or failure\nremains constant Such independent trials which have only two outcomes usually\nreferred as \u2018success\u2019 or \u2018failure\u2019 are called Bernoulli trials"
    },
    {
        "Chapter": "1",
        "sentence_range": "7494-7497",
        "Text": "The outcome of any trial is independent of the\noutcome of any other trial In each of such trials, the probability of success or failure\nremains constant Such independent trials which have only two outcomes usually\nreferred as \u2018success\u2019 or \u2018failure\u2019 are called Bernoulli trials Definition 8 Trials of a random experiment are called Bernoulli trials, if they satisfy\nthe following conditions :\n(i)\nThere should be a finite number of trials"
    },
    {
        "Chapter": "1",
        "sentence_range": "7495-7498",
        "Text": "In each of such trials, the probability of success or failure\nremains constant Such independent trials which have only two outcomes usually\nreferred as \u2018success\u2019 or \u2018failure\u2019 are called Bernoulli trials Definition 8 Trials of a random experiment are called Bernoulli trials, if they satisfy\nthe following conditions :\n(i)\nThere should be a finite number of trials (ii)\nThe trials should be independent"
    },
    {
        "Chapter": "1",
        "sentence_range": "7496-7499",
        "Text": "Such independent trials which have only two outcomes usually\nreferred as \u2018success\u2019 or \u2018failure\u2019 are called Bernoulli trials Definition 8 Trials of a random experiment are called Bernoulli trials, if they satisfy\nthe following conditions :\n(i)\nThere should be a finite number of trials (ii)\nThe trials should be independent (iii)\nEach trial has exactly two outcomes : success or failure"
    },
    {
        "Chapter": "1",
        "sentence_range": "7497-7500",
        "Text": "Definition 8 Trials of a random experiment are called Bernoulli trials, if they satisfy\nthe following conditions :\n(i)\nThere should be a finite number of trials (ii)\nThe trials should be independent (iii)\nEach trial has exactly two outcomes : success or failure (iv)\nThe probability of  success remains the same in each trial"
    },
    {
        "Chapter": "1",
        "sentence_range": "7498-7501",
        "Text": "(ii)\nThe trials should be independent (iii)\nEach trial has exactly two outcomes : success or failure (iv)\nThe probability of  success remains the same in each trial For example, throwing a die 50 times is a case of 50 Bernoulli trials, in which each\ntrial results in success (say an even number) or failure (an odd number) and the\nprobability of success (p) is same for all 50 throws"
    },
    {
        "Chapter": "1",
        "sentence_range": "7499-7502",
        "Text": "(iii)\nEach trial has exactly two outcomes : success or failure (iv)\nThe probability of  success remains the same in each trial For example, throwing a die 50 times is a case of 50 Bernoulli trials, in which each\ntrial results in success (say an even number) or failure (an odd number) and the\nprobability of success (p) is same for all 50 throws Obviously, the successive throws\nof the die are independent experiments"
    },
    {
        "Chapter": "1",
        "sentence_range": "7500-7503",
        "Text": "(iv)\nThe probability of  success remains the same in each trial For example, throwing a die 50 times is a case of 50 Bernoulli trials, in which each\ntrial results in success (say an even number) or failure (an odd number) and the\nprobability of success (p) is same for all 50 throws Obviously, the successive throws\nof the die are independent experiments If the die is fair and have six numbers 1 to 6\nwritten on six faces, then  p = 1\n2  and q = 1 \u2013 p = 1\n2  = probability of failure"
    },
    {
        "Chapter": "1",
        "sentence_range": "7501-7504",
        "Text": "For example, throwing a die 50 times is a case of 50 Bernoulli trials, in which each\ntrial results in success (say an even number) or failure (an odd number) and the\nprobability of success (p) is same for all 50 throws Obviously, the successive throws\nof the die are independent experiments If the die is fair and have six numbers 1 to 6\nwritten on six faces, then  p = 1\n2  and q = 1 \u2013 p = 1\n2  = probability of failure Example 30 Six balls are drawn successively from an urn containing 7 red and 9 black\nballs"
    },
    {
        "Chapter": "1",
        "sentence_range": "7502-7505",
        "Text": "Obviously, the successive throws\nof the die are independent experiments If the die is fair and have six numbers 1 to 6\nwritten on six faces, then  p = 1\n2  and q = 1 \u2013 p = 1\n2  = probability of failure Example 30 Six balls are drawn successively from an urn containing 7 red and 9 black\nballs Tell whether or not the trials of drawing balls are Bernoulli trials when after each\ndraw the ball drawn is\n(i)\nreplaced\n(ii)\nnot replaced in the urn"
    },
    {
        "Chapter": "1",
        "sentence_range": "7503-7506",
        "Text": "If the die is fair and have six numbers 1 to 6\nwritten on six faces, then  p = 1\n2  and q = 1 \u2013 p = 1\n2  = probability of failure Example 30 Six balls are drawn successively from an urn containing 7 red and 9 black\nballs Tell whether or not the trials of drawing balls are Bernoulli trials when after each\ndraw the ball drawn is\n(i)\nreplaced\n(ii)\nnot replaced in the urn Solution\n(i)\nThe number of trials is finite"
    },
    {
        "Chapter": "1",
        "sentence_range": "7504-7507",
        "Text": "Example 30 Six balls are drawn successively from an urn containing 7 red and 9 black\nballs Tell whether or not the trials of drawing balls are Bernoulli trials when after each\ndraw the ball drawn is\n(i)\nreplaced\n(ii)\nnot replaced in the urn Solution\n(i)\nThe number of trials is finite When the drawing is done with replacement, the\nprobability of success (say, red ball) is  p = 7\n16  which is same for all six trials\n(draws)"
    },
    {
        "Chapter": "1",
        "sentence_range": "7505-7508",
        "Text": "Tell whether or not the trials of drawing balls are Bernoulli trials when after each\ndraw the ball drawn is\n(i)\nreplaced\n(ii)\nnot replaced in the urn Solution\n(i)\nThe number of trials is finite When the drawing is done with replacement, the\nprobability of success (say, red ball) is  p = 7\n16  which is same for all six trials\n(draws) Hence, the drawing of balls with replacements are Bernoulli trials"
    },
    {
        "Chapter": "1",
        "sentence_range": "7506-7509",
        "Text": "Solution\n(i)\nThe number of trials is finite When the drawing is done with replacement, the\nprobability of success (say, red ball) is  p = 7\n16  which is same for all six trials\n(draws) Hence, the drawing of balls with replacements are Bernoulli trials \u00a9 NCERT\nnot to be republished\nPROBABILITY      573\n(ii)\nWhen the drawing is done without replacement, the probability of success\n(i"
    },
    {
        "Chapter": "1",
        "sentence_range": "7507-7510",
        "Text": "When the drawing is done with replacement, the\nprobability of success (say, red ball) is  p = 7\n16  which is same for all six trials\n(draws) Hence, the drawing of balls with replacements are Bernoulli trials \u00a9 NCERT\nnot to be republished\nPROBABILITY      573\n(ii)\nWhen the drawing is done without replacement, the probability of success\n(i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "7508-7511",
        "Text": "Hence, the drawing of balls with replacements are Bernoulli trials \u00a9 NCERT\nnot to be republished\nPROBABILITY      573\n(ii)\nWhen the drawing is done without replacement, the probability of success\n(i e , red ball) in first trial is 7\n16 , in 2nd trial is 6\n15 if the first ball drawn is red or\n7\n15  if the first ball drawn is black and so on"
    },
    {
        "Chapter": "1",
        "sentence_range": "7509-7512",
        "Text": "\u00a9 NCERT\nnot to be republished\nPROBABILITY      573\n(ii)\nWhen the drawing is done without replacement, the probability of success\n(i e , red ball) in first trial is 7\n16 , in 2nd trial is 6\n15 if the first ball drawn is red or\n7\n15  if the first ball drawn is black and so on Clearly, the probability of success is\nnot same for all trials, hence the trials are not Bernoulli trials"
    },
    {
        "Chapter": "1",
        "sentence_range": "7510-7513",
        "Text": "e , red ball) in first trial is 7\n16 , in 2nd trial is 6\n15 if the first ball drawn is red or\n7\n15  if the first ball drawn is black and so on Clearly, the probability of success is\nnot same for all trials, hence the trials are not Bernoulli trials 13"
    },
    {
        "Chapter": "1",
        "sentence_range": "7511-7514",
        "Text": ", red ball) in first trial is 7\n16 , in 2nd trial is 6\n15 if the first ball drawn is red or\n7\n15  if the first ball drawn is black and so on Clearly, the probability of success is\nnot same for all trials, hence the trials are not Bernoulli trials 13 7"
    },
    {
        "Chapter": "1",
        "sentence_range": "7512-7515",
        "Text": "Clearly, the probability of success is\nnot same for all trials, hence the trials are not Bernoulli trials 13 7 2  Binomial distribution\nConsider the experiment of tossing a coin in which each trial results in success (say,\nheads) or failure (tails)"
    },
    {
        "Chapter": "1",
        "sentence_range": "7513-7516",
        "Text": "13 7 2  Binomial distribution\nConsider the experiment of tossing a coin in which each trial results in success (say,\nheads) or failure (tails) Let S and F denote respectively success and failure in each\ntrial"
    },
    {
        "Chapter": "1",
        "sentence_range": "7514-7517",
        "Text": "7 2  Binomial distribution\nConsider the experiment of tossing a coin in which each trial results in success (say,\nheads) or failure (tails) Let S and F denote respectively success and failure in each\ntrial Suppose we are interested in finding the ways in which we have one success in\nsix trials"
    },
    {
        "Chapter": "1",
        "sentence_range": "7515-7518",
        "Text": "2  Binomial distribution\nConsider the experiment of tossing a coin in which each trial results in success (say,\nheads) or failure (tails) Let S and F denote respectively success and failure in each\ntrial Suppose we are interested in finding the ways in which we have one success in\nsix trials Clearly, six different cases are there as listed below:\nSFFFFF,  FSFFFF, FFSFFF, FFFSFF, FFFFSF,  FFFFFS"
    },
    {
        "Chapter": "1",
        "sentence_range": "7516-7519",
        "Text": "Let S and F denote respectively success and failure in each\ntrial Suppose we are interested in finding the ways in which we have one success in\nsix trials Clearly, six different cases are there as listed below:\nSFFFFF,  FSFFFF, FFSFFF, FFFSFF, FFFFSF,  FFFFFS Similarly, two successes and four failures can have \n6"
    },
    {
        "Chapter": "1",
        "sentence_range": "7517-7520",
        "Text": "Suppose we are interested in finding the ways in which we have one success in\nsix trials Clearly, six different cases are there as listed below:\nSFFFFF,  FSFFFF, FFSFFF, FFFSFF, FFFFSF,  FFFFFS Similarly, two successes and four failures can have \n6 4"
    },
    {
        "Chapter": "1",
        "sentence_range": "7518-7521",
        "Text": "Clearly, six different cases are there as listed below:\nSFFFFF,  FSFFFF, FFSFFF, FFFSFF, FFFFSF,  FFFFFS Similarly, two successes and four failures can have \n6 4 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "7519-7522",
        "Text": "Similarly, two successes and four failures can have \n6 4 2 combinations"
    },
    {
        "Chapter": "1",
        "sentence_range": "7520-7523",
        "Text": "4 2 combinations It will be\nlengthy job to list all of these ways"
    },
    {
        "Chapter": "1",
        "sentence_range": "7521-7524",
        "Text": "2 combinations It will be\nlengthy job to list all of these ways Therefore, calculation of probabilities of 0, 1, 2,"
    },
    {
        "Chapter": "1",
        "sentence_range": "7522-7525",
        "Text": "combinations It will be\nlengthy job to list all of these ways Therefore, calculation of probabilities of 0, 1, 2, ,\nn number of successes may be lengthy and time consuming"
    },
    {
        "Chapter": "1",
        "sentence_range": "7523-7526",
        "Text": "It will be\nlengthy job to list all of these ways Therefore, calculation of probabilities of 0, 1, 2, ,\nn number of successes may be lengthy and time consuming To avoid the lengthy\ncalculations and listing of all the possible cases, for the probabilities  of number of\nsuccesses in n-Bernoulli trials, a formula is derived"
    },
    {
        "Chapter": "1",
        "sentence_range": "7524-7527",
        "Text": "Therefore, calculation of probabilities of 0, 1, 2, ,\nn number of successes may be lengthy and time consuming To avoid the lengthy\ncalculations and listing of all the possible cases, for the probabilities  of number of\nsuccesses in n-Bernoulli trials, a formula is derived For this purpose, let us take the\nexperiment made up of three Bernoulli trials with probabilities p and q = 1 \u2013 p for\nsuccess and failure respectively in each trial"
    },
    {
        "Chapter": "1",
        "sentence_range": "7525-7528",
        "Text": ",\nn number of successes may be lengthy and time consuming To avoid the lengthy\ncalculations and listing of all the possible cases, for the probabilities  of number of\nsuccesses in n-Bernoulli trials, a formula is derived For this purpose, let us take the\nexperiment made up of three Bernoulli trials with probabilities p and q = 1 \u2013 p for\nsuccess and failure respectively in each trial The sample space of the experiment is\nthe set\nS = {SSS, SSF, SFS, FSS, SFF, FSF, FFS, FFF}\nThe number of successes is a random variable X and can take values 0, 1, 2, or 3"
    },
    {
        "Chapter": "1",
        "sentence_range": "7526-7529",
        "Text": "To avoid the lengthy\ncalculations and listing of all the possible cases, for the probabilities  of number of\nsuccesses in n-Bernoulli trials, a formula is derived For this purpose, let us take the\nexperiment made up of three Bernoulli trials with probabilities p and q = 1 \u2013 p for\nsuccess and failure respectively in each trial The sample space of the experiment is\nthe set\nS = {SSS, SSF, SFS, FSS, SFF, FSF, FFS, FFF}\nThe number of successes is a random variable X and can take values 0, 1, 2, or 3 The probability distribution of the number of successes is as below :\nP(X = 0) = P(no success)\n= P({FFF}) = P(F) P(F) P(F)\n= q"
    },
    {
        "Chapter": "1",
        "sentence_range": "7527-7530",
        "Text": "For this purpose, let us take the\nexperiment made up of three Bernoulli trials with probabilities p and q = 1 \u2013 p for\nsuccess and failure respectively in each trial The sample space of the experiment is\nthe set\nS = {SSS, SSF, SFS, FSS, SFF, FSF, FFS, FFF}\nThe number of successes is a random variable X and can take values 0, 1, 2, or 3 The probability distribution of the number of successes is as below :\nP(X = 0) = P(no success)\n= P({FFF}) = P(F) P(F) P(F)\n= q q"
    },
    {
        "Chapter": "1",
        "sentence_range": "7528-7531",
        "Text": "The sample space of the experiment is\nthe set\nS = {SSS, SSF, SFS, FSS, SFF, FSF, FFS, FFF}\nThe number of successes is a random variable X and can take values 0, 1, 2, or 3 The probability distribution of the number of successes is as below :\nP(X = 0) = P(no success)\n= P({FFF}) = P(F) P(F) P(F)\n= q q q = q3 since  the trials are independent\nP(X = 1) = P(one successes)\n= P({SFF, FSF, FFS})\n= P({SFF}) + P({FSF}) + P({FFS})\n= P(S) P(F) P(F) + P(F) P(S) P(F) + P(F) P(F) P(S)\n= p"
    },
    {
        "Chapter": "1",
        "sentence_range": "7529-7532",
        "Text": "The probability distribution of the number of successes is as below :\nP(X = 0) = P(no success)\n= P({FFF}) = P(F) P(F) P(F)\n= q q q = q3 since  the trials are independent\nP(X = 1) = P(one successes)\n= P({SFF, FSF, FFS})\n= P({SFF}) + P({FSF}) + P({FFS})\n= P(S) P(F) P(F) + P(F) P(S) P(F) + P(F) P(F) P(S)\n= p q"
    },
    {
        "Chapter": "1",
        "sentence_range": "7530-7533",
        "Text": "q q = q3 since  the trials are independent\nP(X = 1) = P(one successes)\n= P({SFF, FSF, FFS})\n= P({SFF}) + P({FSF}) + P({FFS})\n= P(S) P(F) P(F) + P(F) P(S) P(F) + P(F) P(F) P(S)\n= p q q + q"
    },
    {
        "Chapter": "1",
        "sentence_range": "7531-7534",
        "Text": "q = q3 since  the trials are independent\nP(X = 1) = P(one successes)\n= P({SFF, FSF, FFS})\n= P({SFF}) + P({FSF}) + P({FFS})\n= P(S) P(F) P(F) + P(F) P(S) P(F) + P(F) P(F) P(S)\n= p q q + q p"
    },
    {
        "Chapter": "1",
        "sentence_range": "7532-7535",
        "Text": "q q + q p q + q"
    },
    {
        "Chapter": "1",
        "sentence_range": "7533-7536",
        "Text": "q + q p q + q q"
    },
    {
        "Chapter": "1",
        "sentence_range": "7534-7537",
        "Text": "p q + q q p = 3pq2\nP(X = 2) = P (two successes)\n= P({SSF, SFS, FSS})\n= P({SSF}) + P ({SFS}) + P({FSS})\n\u00a9 NCERT\nnot to be republished\n 574\nMATHEMATICS\n= P(S) P(S) P(F) + P(S) P(F) P(S) + P(F) P(S) P(S)\n= p"
    },
    {
        "Chapter": "1",
        "sentence_range": "7535-7538",
        "Text": "q + q q p = 3pq2\nP(X = 2) = P (two successes)\n= P({SSF, SFS, FSS})\n= P({SSF}) + P ({SFS}) + P({FSS})\n\u00a9 NCERT\nnot to be republished\n 574\nMATHEMATICS\n= P(S) P(S) P(F) + P(S) P(F) P(S) + P(F) P(S) P(S)\n= p p"
    },
    {
        "Chapter": "1",
        "sentence_range": "7536-7539",
        "Text": "q p = 3pq2\nP(X = 2) = P (two successes)\n= P({SSF, SFS, FSS})\n= P({SSF}) + P ({SFS}) + P({FSS})\n\u00a9 NCERT\nnot to be republished\n 574\nMATHEMATICS\n= P(S) P(S) P(F) + P(S) P(F) P(S) + P(F) P(S) P(S)\n= p p q"
    },
    {
        "Chapter": "1",
        "sentence_range": "7537-7540",
        "Text": "p = 3pq2\nP(X = 2) = P (two successes)\n= P({SSF, SFS, FSS})\n= P({SSF}) + P ({SFS}) + P({FSS})\n\u00a9 NCERT\nnot to be republished\n 574\nMATHEMATICS\n= P(S) P(S) P(F) + P(S) P(F) P(S) + P(F) P(S) P(S)\n= p p q +  p"
    },
    {
        "Chapter": "1",
        "sentence_range": "7538-7541",
        "Text": "p q +  p q"
    },
    {
        "Chapter": "1",
        "sentence_range": "7539-7542",
        "Text": "q +  p q p  +  q"
    },
    {
        "Chapter": "1",
        "sentence_range": "7540-7543",
        "Text": "+  p q p  +  q p"
    },
    {
        "Chapter": "1",
        "sentence_range": "7541-7544",
        "Text": "q p  +  q p p  =  3p2q\nand\nP(X = 3) = P(three success) = P ({SSS})\n= P(S)"
    },
    {
        "Chapter": "1",
        "sentence_range": "7542-7545",
        "Text": "p  +  q p p  =  3p2q\nand\nP(X = 3) = P(three success) = P ({SSS})\n= P(S) P(S)"
    },
    {
        "Chapter": "1",
        "sentence_range": "7543-7546",
        "Text": "p p  =  3p2q\nand\nP(X = 3) = P(three success) = P ({SSS})\n= P(S) P(S) P(S) = p3\nThus, the probability distribution of X is\nX\n0\n1\n2\n3\nP(X)\nq 3\n3q2p\n3qp2\np 3\nAlso, the binominal expansion of (q + p)3 is\nq\nq\np\nqp\np\n3\n3 2\n3\n2\n3\n+\n+\n+\nNote that the probabilities of 0, 1, 2 or 3 successes are respectively the 1st, 2nd,\n3rd and 4th term in the expansion of (q + p)3"
    },
    {
        "Chapter": "1",
        "sentence_range": "7544-7547",
        "Text": "p  =  3p2q\nand\nP(X = 3) = P(three success) = P ({SSS})\n= P(S) P(S) P(S) = p3\nThus, the probability distribution of X is\nX\n0\n1\n2\n3\nP(X)\nq 3\n3q2p\n3qp2\np 3\nAlso, the binominal expansion of (q + p)3 is\nq\nq\np\nqp\np\n3\n3 2\n3\n2\n3\n+\n+\n+\nNote that the probabilities of 0, 1, 2 or 3 successes are respectively the 1st, 2nd,\n3rd and 4th term in the expansion of (q + p)3 Also, since q + p = 1, it follows that the sum of these probabilities, as expected, is 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "7545-7548",
        "Text": "P(S) P(S) = p3\nThus, the probability distribution of X is\nX\n0\n1\n2\n3\nP(X)\nq 3\n3q2p\n3qp2\np 3\nAlso, the binominal expansion of (q + p)3 is\nq\nq\np\nqp\np\n3\n3 2\n3\n2\n3\n+\n+\n+\nNote that the probabilities of 0, 1, 2 or 3 successes are respectively the 1st, 2nd,\n3rd and 4th term in the expansion of (q + p)3 Also, since q + p = 1, it follows that the sum of these probabilities, as expected, is 1 Thus, we may conclude that in an experiment of n-Bernoulli trials, the probabilities\nof 0, 1, 2,"
    },
    {
        "Chapter": "1",
        "sentence_range": "7546-7549",
        "Text": "P(S) = p3\nThus, the probability distribution of X is\nX\n0\n1\n2\n3\nP(X)\nq 3\n3q2p\n3qp2\np 3\nAlso, the binominal expansion of (q + p)3 is\nq\nq\np\nqp\np\n3\n3 2\n3\n2\n3\n+\n+\n+\nNote that the probabilities of 0, 1, 2 or 3 successes are respectively the 1st, 2nd,\n3rd and 4th term in the expansion of (q + p)3 Also, since q + p = 1, it follows that the sum of these probabilities, as expected, is 1 Thus, we may conclude that in an experiment of n-Bernoulli trials, the probabilities\nof 0, 1, 2, , n successes can be obtained as 1st, 2nd,"
    },
    {
        "Chapter": "1",
        "sentence_range": "7547-7550",
        "Text": "Also, since q + p = 1, it follows that the sum of these probabilities, as expected, is 1 Thus, we may conclude that in an experiment of n-Bernoulli trials, the probabilities\nof 0, 1, 2, , n successes can be obtained as 1st, 2nd, ,(n + 1)th terms in the expansion\nof (q + p)n"
    },
    {
        "Chapter": "1",
        "sentence_range": "7548-7551",
        "Text": "Thus, we may conclude that in an experiment of n-Bernoulli trials, the probabilities\nof 0, 1, 2, , n successes can be obtained as 1st, 2nd, ,(n + 1)th terms in the expansion\nof (q + p)n To  prove this assertion (result), let us find the probability of x-successes in\nan experiment of n-Bernoulli trials"
    },
    {
        "Chapter": "1",
        "sentence_range": "7549-7552",
        "Text": ", n successes can be obtained as 1st, 2nd, ,(n + 1)th terms in the expansion\nof (q + p)n To  prove this assertion (result), let us find the probability of x-successes in\nan experiment of n-Bernoulli trials Clearly, in case of x successes (S), there will be (n \u2013 x) failures (F)"
    },
    {
        "Chapter": "1",
        "sentence_range": "7550-7553",
        "Text": ",(n + 1)th terms in the expansion\nof (q + p)n To  prove this assertion (result), let us find the probability of x-successes in\nan experiment of n-Bernoulli trials Clearly, in case of x successes (S), there will be (n \u2013 x) failures (F) Now, x successes (S) and (n \u2013 x) failures (F) can be obtained in"
    },
    {
        "Chapter": "1",
        "sentence_range": "7551-7554",
        "Text": "To  prove this assertion (result), let us find the probability of x-successes in\nan experiment of n-Bernoulli trials Clearly, in case of x successes (S), there will be (n \u2013 x) failures (F) Now, x successes (S) and (n \u2013 x) failures (F) can be obtained in ("
    },
    {
        "Chapter": "1",
        "sentence_range": "7552-7555",
        "Text": "Clearly, in case of x successes (S), there will be (n \u2013 x) failures (F) Now, x successes (S) and (n \u2013 x) failures (F) can be obtained in ( )"
    },
    {
        "Chapter": "1",
        "sentence_range": "7553-7556",
        "Text": "Now, x successes (S) and (n \u2013 x) failures (F) can be obtained in ( ) n\nx n\n x\n ways"
    },
    {
        "Chapter": "1",
        "sentence_range": "7554-7557",
        "Text": "( ) n\nx n\n x\n ways In each of these ways, the probability of x successes and (n \u2212 x) failures is\n= P(x successes)"
    },
    {
        "Chapter": "1",
        "sentence_range": "7555-7558",
        "Text": ") n\nx n\n x\n ways In each of these ways, the probability of x successes and (n \u2212 x) failures is\n= P(x successes) P(n\u2013x) failures is\n=\ntimes\n(\n) times\nP(S)"
    },
    {
        "Chapter": "1",
        "sentence_range": "7556-7559",
        "Text": "n\nx n\n x\n ways In each of these ways, the probability of x successes and (n \u2212 x) failures is\n= P(x successes) P(n\u2013x) failures is\n=\ntimes\n(\n) times\nP(S) P(S)"
    },
    {
        "Chapter": "1",
        "sentence_range": "7557-7560",
        "Text": "In each of these ways, the probability of x successes and (n \u2212 x) failures is\n= P(x successes) P(n\u2013x) failures is\n=\ntimes\n(\n) times\nP(S) P(S) P(S)\nP(F)"
    },
    {
        "Chapter": "1",
        "sentence_range": "7558-7561",
        "Text": "P(n\u2013x) failures is\n=\ntimes\n(\n) times\nP(S) P(S) P(S)\nP(F) P(F)"
    },
    {
        "Chapter": "1",
        "sentence_range": "7559-7562",
        "Text": "P(S) P(S)\nP(F) P(F) P(F)\nx\nn x\n \n1442443\n1442443  = px qn\u2013x\nThus, the probability of x successes in n-Bernoulli trials is"
    },
    {
        "Chapter": "1",
        "sentence_range": "7560-7563",
        "Text": "P(S)\nP(F) P(F) P(F)\nx\nn x\n \n1442443\n1442443  = px qn\u2013x\nThus, the probability of x successes in n-Bernoulli trials is ("
    },
    {
        "Chapter": "1",
        "sentence_range": "7561-7564",
        "Text": "P(F) P(F)\nx\nn x\n \n1442443\n1442443  = px qn\u2013x\nThus, the probability of x successes in n-Bernoulli trials is ( )"
    },
    {
        "Chapter": "1",
        "sentence_range": "7562-7565",
        "Text": "P(F)\nx\nn x\n \n1442443\n1442443  = px qn\u2013x\nThus, the probability of x successes in n-Bernoulli trials is ( ) n\nx n\n\u2212x\npx qn\u2013x\nor nCx\n px  qn\u2013x\nThus\nP(x successes) =\nnC\nx\nn x\nx p q \u2212 ,    x = 0, 1, 2,"
    },
    {
        "Chapter": "1",
        "sentence_range": "7563-7566",
        "Text": "( ) n\nx n\n\u2212x\npx qn\u2013x\nor nCx\n px  qn\u2013x\nThus\nP(x successes) =\nnC\nx\nn x\nx p q \u2212 ,    x = 0, 1, 2, ,n"
    },
    {
        "Chapter": "1",
        "sentence_range": "7564-7567",
        "Text": ") n\nx n\n\u2212x\npx qn\u2013x\nor nCx\n px  qn\u2013x\nThus\nP(x successes) =\nnC\nx\nn x\nx p q \u2212 ,    x = 0, 1, 2, ,n (q = 1 \u2013 p)\nClearly, P(x successes), i"
    },
    {
        "Chapter": "1",
        "sentence_range": "7565-7568",
        "Text": "n\nx n\n\u2212x\npx qn\u2013x\nor nCx\n px  qn\u2013x\nThus\nP(x successes) =\nnC\nx\nn x\nx p q \u2212 ,    x = 0, 1, 2, ,n (q = 1 \u2013 p)\nClearly, P(x successes), i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "7566-7569",
        "Text": ",n (q = 1 \u2013 p)\nClearly, P(x successes), i e C\nn\nx\nn x\nx p q \u2212 is the (x + 1)th term in the binomial\nexpansion of (q + p)n"
    },
    {
        "Chapter": "1",
        "sentence_range": "7567-7570",
        "Text": "(q = 1 \u2013 p)\nClearly, P(x successes), i e C\nn\nx\nn x\nx p q \u2212 is the (x + 1)th term in the binomial\nexpansion of (q + p)n Thus, the probability distribution of number of successes in an experiment consisting\nof n Bernoulli trials may be obtained by the binomial expansion of (q + p)n"
    },
    {
        "Chapter": "1",
        "sentence_range": "7568-7571",
        "Text": "e C\nn\nx\nn x\nx p q \u2212 is the (x + 1)th term in the binomial\nexpansion of (q + p)n Thus, the probability distribution of number of successes in an experiment consisting\nof n Bernoulli trials may be obtained by the binomial expansion of (q + p)n Hence, this\n\u00a9 NCERT\nnot to be republished\nPROBABILITY      575\ndistribution of number of successes X can be written as\nX\n0\n1\n2"
    },
    {
        "Chapter": "1",
        "sentence_range": "7569-7572",
        "Text": "C\nn\nx\nn x\nx p q \u2212 is the (x + 1)th term in the binomial\nexpansion of (q + p)n Thus, the probability distribution of number of successes in an experiment consisting\nof n Bernoulli trials may be obtained by the binomial expansion of (q + p)n Hence, this\n\u00a9 NCERT\nnot to be republished\nPROBABILITY      575\ndistribution of number of successes X can be written as\nX\n0\n1\n2 x"
    },
    {
        "Chapter": "1",
        "sentence_range": "7570-7573",
        "Text": "Thus, the probability distribution of number of successes in an experiment consisting\nof n Bernoulli trials may be obtained by the binomial expansion of (q + p)n Hence, this\n\u00a9 NCERT\nnot to be republished\nPROBABILITY      575\ndistribution of number of successes X can be written as\nX\n0\n1\n2 x n\nP(X)\nnC0 qn\nnC1 qn\u20131p1\nnC2 qn\u20132p2\nnCx qn\u2013xpx\nnCn pn\nThe above probability distribution is known as binomial distribution with parameters\nn and p, because for given values of n and p, we can find the complete probability\ndistribution"
    },
    {
        "Chapter": "1",
        "sentence_range": "7571-7574",
        "Text": "Hence, this\n\u00a9 NCERT\nnot to be republished\nPROBABILITY      575\ndistribution of number of successes X can be written as\nX\n0\n1\n2 x n\nP(X)\nnC0 qn\nnC1 qn\u20131p1\nnC2 qn\u20132p2\nnCx qn\u2013xpx\nnCn pn\nThe above probability distribution is known as binomial distribution with parameters\nn and p, because for given values of n and p, we can find the complete probability\ndistribution The probability of x successes P(X = x) is also denoted by P(x) and is given by\nP(x) = nCx qn\u2013xpx,    x = 0, 1,"
    },
    {
        "Chapter": "1",
        "sentence_range": "7572-7575",
        "Text": "x n\nP(X)\nnC0 qn\nnC1 qn\u20131p1\nnC2 qn\u20132p2\nnCx qn\u2013xpx\nnCn pn\nThe above probability distribution is known as binomial distribution with parameters\nn and p, because for given values of n and p, we can find the complete probability\ndistribution The probability of x successes P(X = x) is also denoted by P(x) and is given by\nP(x) = nCx qn\u2013xpx,    x = 0, 1, , n"
    },
    {
        "Chapter": "1",
        "sentence_range": "7573-7576",
        "Text": "n\nP(X)\nnC0 qn\nnC1 qn\u20131p1\nnC2 qn\u20132p2\nnCx qn\u2013xpx\nnCn pn\nThe above probability distribution is known as binomial distribution with parameters\nn and p, because for given values of n and p, we can find the complete probability\ndistribution The probability of x successes P(X = x) is also denoted by P(x) and is given by\nP(x) = nCx qn\u2013xpx,    x = 0, 1, , n (q = 1 \u2013 p)\nThis P(x) is called the probability function of the binomial distribution"
    },
    {
        "Chapter": "1",
        "sentence_range": "7574-7577",
        "Text": "The probability of x successes P(X = x) is also denoted by P(x) and is given by\nP(x) = nCx qn\u2013xpx,    x = 0, 1, , n (q = 1 \u2013 p)\nThis P(x) is called the probability function of the binomial distribution A binomial distribution with n-Bernoulli trials and probability of success in each\ntrial as p, is denoted by B(n, p)"
    },
    {
        "Chapter": "1",
        "sentence_range": "7575-7578",
        "Text": ", n (q = 1 \u2013 p)\nThis P(x) is called the probability function of the binomial distribution A binomial distribution with n-Bernoulli trials and probability of success in each\ntrial as p, is denoted by B(n, p) Let  us now take up some examples"
    },
    {
        "Chapter": "1",
        "sentence_range": "7576-7579",
        "Text": "(q = 1 \u2013 p)\nThis P(x) is called the probability function of the binomial distribution A binomial distribution with n-Bernoulli trials and probability of success in each\ntrial as p, is denoted by B(n, p) Let  us now take up some examples Example 31 If a fair coin  is tossed 10 times, find the probability of\n(i)\nexactly six heads\n(ii)\nat least six heads\n(iii)\nat most six heads\nSolution The repeated tosses of a coin are Bernoulli trials"
    },
    {
        "Chapter": "1",
        "sentence_range": "7577-7580",
        "Text": "A binomial distribution with n-Bernoulli trials and probability of success in each\ntrial as p, is denoted by B(n, p) Let  us now take up some examples Example 31 If a fair coin  is tossed 10 times, find the probability of\n(i)\nexactly six heads\n(ii)\nat least six heads\n(iii)\nat most six heads\nSolution The repeated tosses of a coin are Bernoulli trials Let X denote the number\nof heads in an experiment of 10 trials"
    },
    {
        "Chapter": "1",
        "sentence_range": "7578-7581",
        "Text": "Let  us now take up some examples Example 31 If a fair coin  is tossed 10 times, find the probability of\n(i)\nexactly six heads\n(ii)\nat least six heads\n(iii)\nat most six heads\nSolution The repeated tosses of a coin are Bernoulli trials Let X denote the number\nof heads in an experiment of 10 trials Clearly, X has the binomial distribution with n = 10 and p = 1\n2\nTherefore\nP(X = x) = nCxqn\u2013xpx, x = 0, 1, 2,"
    },
    {
        "Chapter": "1",
        "sentence_range": "7579-7582",
        "Text": "Example 31 If a fair coin  is tossed 10 times, find the probability of\n(i)\nexactly six heads\n(ii)\nat least six heads\n(iii)\nat most six heads\nSolution The repeated tosses of a coin are Bernoulli trials Let X denote the number\nof heads in an experiment of 10 trials Clearly, X has the binomial distribution with n = 10 and p = 1\n2\nTherefore\nP(X = x) = nCxqn\u2013xpx, x = 0, 1, 2, ,n\nHere\nn = 10, \n21\np\n , q = 1 \u2013 p = 1\n2\nTherefore\nP(X = x) =\n10\n10\n10\n10\n1\n1\n1\nC\nC\n2\n2\n2\nx\nx\nx\nx\n\u2212\n\u239b\n\u239e\n\u239b\n\u239e\n\u239b\n\u239e\n=\n\u239c\n\u239f\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\n\u239d\n\u23a0\nNow\n(i) P(X = 6) =\n10\n10\n6\n10\n1\n10"
    },
    {
        "Chapter": "1",
        "sentence_range": "7580-7583",
        "Text": "Let X denote the number\nof heads in an experiment of 10 trials Clearly, X has the binomial distribution with n = 10 and p = 1\n2\nTherefore\nP(X = x) = nCxqn\u2013xpx, x = 0, 1, 2, ,n\nHere\nn = 10, \n21\np\n , q = 1 \u2013 p = 1\n2\nTherefore\nP(X = x) =\n10\n10\n10\n10\n1\n1\n1\nC\nC\n2\n2\n2\nx\nx\nx\nx\n\u2212\n\u239b\n\u239e\n\u239b\n\u239e\n\u239b\n\u239e\n=\n\u239c\n\u239f\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\n\u239d\n\u23a0\nNow\n(i) P(X = 6) =\n10\n10\n6\n10\n1\n10 1\n105\nC\n2\n6"
    },
    {
        "Chapter": "1",
        "sentence_range": "7581-7584",
        "Text": "Clearly, X has the binomial distribution with n = 10 and p = 1\n2\nTherefore\nP(X = x) = nCxqn\u2013xpx, x = 0, 1, 2, ,n\nHere\nn = 10, \n21\np\n , q = 1 \u2013 p = 1\n2\nTherefore\nP(X = x) =\n10\n10\n10\n10\n1\n1\n1\nC\nC\n2\n2\n2\nx\nx\nx\nx\n\u2212\n\u239b\n\u239e\n\u239b\n\u239e\n\u239b\n\u239e\n=\n\u239c\n\u239f\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\n\u239d\n\u23a0\nNow\n(i) P(X = 6) =\n10\n10\n6\n10\n1\n10 1\n105\nC\n2\n6 4"
    },
    {
        "Chapter": "1",
        "sentence_range": "7582-7585",
        "Text": ",n\nHere\nn = 10, \n21\np\n , q = 1 \u2013 p = 1\n2\nTherefore\nP(X = x) =\n10\n10\n10\n10\n1\n1\n1\nC\nC\n2\n2\n2\nx\nx\nx\nx\n\u2212\n\u239b\n\u239e\n\u239b\n\u239e\n\u239b\n\u239e\n=\n\u239c\n\u239f\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\n\u239d\n\u23a0\nNow\n(i) P(X = 6) =\n10\n10\n6\n10\n1\n10 1\n105\nC\n2\n6 4 512\n2\n\u239b\n\u239e\n=\n=\n\u239c\n\u239f\n\u00d7\n\u239d\n\u23a0\n(ii) P(at least six heads) = P(X \u2265 6)\n= P (X = 6) + P (X = 7) + P (X = 8) + P(X = 9) + P (X = 10)\n\u00a9 NCERT\nnot to be republished\n 576\nMATHEMATICS\n=\n10\n10\n10\n10\n10\n10\n10\n10\n10\n10\n6\n7\n8\n9\n10\n1\n1\n1\n1\n1\nC\nC\nC\nC\nC\n2\n2\n2\n2\n2\n\u239b\n\u239e\n\u239b\n\u239e\n\u239b\n\u239e\n\u239b\n\u239e\n\u239b\n\u239e\n+\n+\n+\n+\n\u239c\n\u239f\n\u239c\n\u239f\n\u239c\n\u239f\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\n\u239d\n\u23a0\n\u239d\n\u23a0\n\u239d\n\u23a0\n=\n10\n10"
    },
    {
        "Chapter": "1",
        "sentence_range": "7583-7586",
        "Text": "1\n105\nC\n2\n6 4 512\n2\n\u239b\n\u239e\n=\n=\n\u239c\n\u239f\n\u00d7\n\u239d\n\u23a0\n(ii) P(at least six heads) = P(X \u2265 6)\n= P (X = 6) + P (X = 7) + P (X = 8) + P(X = 9) + P (X = 10)\n\u00a9 NCERT\nnot to be republished\n 576\nMATHEMATICS\n=\n10\n10\n10\n10\n10\n10\n10\n10\n10\n10\n6\n7\n8\n9\n10\n1\n1\n1\n1\n1\nC\nC\nC\nC\nC\n2\n2\n2\n2\n2\n\u239b\n\u239e\n\u239b\n\u239e\n\u239b\n\u239e\n\u239b\n\u239e\n\u239b\n\u239e\n+\n+\n+\n+\n\u239c\n\u239f\n\u239c\n\u239f\n\u239c\n\u239f\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\n\u239d\n\u23a0\n\u239d\n\u23a0\n\u239d\n\u23a0\n=\n10\n10 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "7584-7587",
        "Text": "4 512\n2\n\u239b\n\u239e\n=\n=\n\u239c\n\u239f\n\u00d7\n\u239d\n\u23a0\n(ii) P(at least six heads) = P(X \u2265 6)\n= P (X = 6) + P (X = 7) + P (X = 8) + P(X = 9) + P (X = 10)\n\u00a9 NCERT\nnot to be republished\n 576\nMATHEMATICS\n=\n10\n10\n10\n10\n10\n10\n10\n10\n10\n10\n6\n7\n8\n9\n10\n1\n1\n1\n1\n1\nC\nC\nC\nC\nC\n2\n2\n2\n2\n2\n\u239b\n\u239e\n\u239b\n\u239e\n\u239b\n\u239e\n\u239b\n\u239e\n\u239b\n\u239e\n+\n+\n+\n+\n\u239c\n\u239f\n\u239c\n\u239f\n\u239c\n\u239f\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\n\u239d\n\u23a0\n\u239d\n\u23a0\n\u239d\n\u23a0\n=\n10\n10 10 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "7585-7588",
        "Text": "512\n2\n\u239b\n\u239e\n=\n=\n\u239c\n\u239f\n\u00d7\n\u239d\n\u23a0\n(ii) P(at least six heads) = P(X \u2265 6)\n= P (X = 6) + P (X = 7) + P (X = 8) + P(X = 9) + P (X = 10)\n\u00a9 NCERT\nnot to be republished\n 576\nMATHEMATICS\n=\n10\n10\n10\n10\n10\n10\n10\n10\n10\n10\n6\n7\n8\n9\n10\n1\n1\n1\n1\n1\nC\nC\nC\nC\nC\n2\n2\n2\n2\n2\n\u239b\n\u239e\n\u239b\n\u239e\n\u239b\n\u239e\n\u239b\n\u239e\n\u239b\n\u239e\n+\n+\n+\n+\n\u239c\n\u239f\n\u239c\n\u239f\n\u239c\n\u239f\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\n\u239d\n\u23a0\n\u239d\n\u23a0\n\u239d\n\u23a0\n=\n10\n10 10 10 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "7586-7589",
        "Text": "10 10 10 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "7587-7590",
        "Text": "10 10 10 1\n6"
    },
    {
        "Chapter": "1",
        "sentence_range": "7588-7591",
        "Text": "10 10 1\n6 4"
    },
    {
        "Chapter": "1",
        "sentence_range": "7589-7592",
        "Text": "10 1\n6 4 7"
    },
    {
        "Chapter": "1",
        "sentence_range": "7590-7593",
        "Text": "1\n6 4 7 3"
    },
    {
        "Chapter": "1",
        "sentence_range": "7591-7594",
        "Text": "4 7 3 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "7592-7595",
        "Text": "7 3 8 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "7593-7596",
        "Text": "3 8 2 9"
    },
    {
        "Chapter": "1",
        "sentence_range": "7594-7597",
        "Text": "8 2 9 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "7595-7598",
        "Text": "2 9 1 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "7596-7599",
        "Text": "9 1 10 2\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n  \n193\n512\n=\n(iii) P(at most six heads) = P(X \u2264 6)\n= P (X = 0) + P (X = 1) + P (X = 2) + P (X = 3)\n+ P (X = 4) + P (X = 5) + P (X = 6)\n=\n10\n10\n10\n10\n10\n10\n10\n1\n2\n3\n1\n1\n1\n1\nC\nC\nC\n2\n2\n2\n2\n\u239b\n\u239e\n\u239b\n\u239e\n\u239b\n\u239e\n\u239b\n\u239e\n+\n+\n+\n\u239c\n\u239f\n\u239c\n\u239f\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\n\u239d\n\u23a0\n\u239d\n\u23a0\n+ \n10\n10\n10\n10\n10\n10\n4\n5\n6\n1\n1\n1\nC\nC\nC\n2\n2\n2\n\u239b\n\u239e\n\u239b\n\u239e\n\u239b\n\u239e\n+\n+\n\u239c\n\u239f\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\n\u239d\n\u23a0\n= 848\n53\n1024\n64\n=\nExample 32 Ten eggs are drawn successively with replacement from a lot containing\n10% defective eggs"
    },
    {
        "Chapter": "1",
        "sentence_range": "7597-7600",
        "Text": "1 10 2\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n  \n193\n512\n=\n(iii) P(at most six heads) = P(X \u2264 6)\n= P (X = 0) + P (X = 1) + P (X = 2) + P (X = 3)\n+ P (X = 4) + P (X = 5) + P (X = 6)\n=\n10\n10\n10\n10\n10\n10\n10\n1\n2\n3\n1\n1\n1\n1\nC\nC\nC\n2\n2\n2\n2\n\u239b\n\u239e\n\u239b\n\u239e\n\u239b\n\u239e\n\u239b\n\u239e\n+\n+\n+\n\u239c\n\u239f\n\u239c\n\u239f\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\n\u239d\n\u23a0\n\u239d\n\u23a0\n+ \n10\n10\n10\n10\n10\n10\n4\n5\n6\n1\n1\n1\nC\nC\nC\n2\n2\n2\n\u239b\n\u239e\n\u239b\n\u239e\n\u239b\n\u239e\n+\n+\n\u239c\n\u239f\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\n\u239d\n\u23a0\n= 848\n53\n1024\n64\n=\nExample 32 Ten eggs are drawn successively with replacement from a lot containing\n10% defective eggs Find the probability that there is at least one defective egg"
    },
    {
        "Chapter": "1",
        "sentence_range": "7598-7601",
        "Text": "10 2\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n  \n193\n512\n=\n(iii) P(at most six heads) = P(X \u2264 6)\n= P (X = 0) + P (X = 1) + P (X = 2) + P (X = 3)\n+ P (X = 4) + P (X = 5) + P (X = 6)\n=\n10\n10\n10\n10\n10\n10\n10\n1\n2\n3\n1\n1\n1\n1\nC\nC\nC\n2\n2\n2\n2\n\u239b\n\u239e\n\u239b\n\u239e\n\u239b\n\u239e\n\u239b\n\u239e\n+\n+\n+\n\u239c\n\u239f\n\u239c\n\u239f\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\n\u239d\n\u23a0\n\u239d\n\u23a0\n+ \n10\n10\n10\n10\n10\n10\n4\n5\n6\n1\n1\n1\nC\nC\nC\n2\n2\n2\n\u239b\n\u239e\n\u239b\n\u239e\n\u239b\n\u239e\n+\n+\n\u239c\n\u239f\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\n\u239d\n\u23a0\n= 848\n53\n1024\n64\n=\nExample 32 Ten eggs are drawn successively with replacement from a lot containing\n10% defective eggs Find the probability that there is at least one defective egg Solution Let X denote the number of defective eggs in the 10 eggs drawn"
    },
    {
        "Chapter": "1",
        "sentence_range": "7599-7602",
        "Text": "2\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n  \n193\n512\n=\n(iii) P(at most six heads) = P(X \u2264 6)\n= P (X = 0) + P (X = 1) + P (X = 2) + P (X = 3)\n+ P (X = 4) + P (X = 5) + P (X = 6)\n=\n10\n10\n10\n10\n10\n10\n10\n1\n2\n3\n1\n1\n1\n1\nC\nC\nC\n2\n2\n2\n2\n\u239b\n\u239e\n\u239b\n\u239e\n\u239b\n\u239e\n\u239b\n\u239e\n+\n+\n+\n\u239c\n\u239f\n\u239c\n\u239f\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\n\u239d\n\u23a0\n\u239d\n\u23a0\n+ \n10\n10\n10\n10\n10\n10\n4\n5\n6\n1\n1\n1\nC\nC\nC\n2\n2\n2\n\u239b\n\u239e\n\u239b\n\u239e\n\u239b\n\u239e\n+\n+\n\u239c\n\u239f\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\n\u239d\n\u23a0\n= 848\n53\n1024\n64\n=\nExample 32 Ten eggs are drawn successively with replacement from a lot containing\n10% defective eggs Find the probability that there is at least one defective egg Solution Let X denote the number of defective eggs in the 10 eggs drawn Since the\ndrawing is done with replacement, the trials are Bernoulli trials"
    },
    {
        "Chapter": "1",
        "sentence_range": "7600-7603",
        "Text": "Find the probability that there is at least one defective egg Solution Let X denote the number of defective eggs in the 10 eggs drawn Since the\ndrawing is done with replacement, the trials are Bernoulli trials Clearly, X has the\nbinomial distribution with n = 10 and \n10\n1\n100\n10\np"
    },
    {
        "Chapter": "1",
        "sentence_range": "7601-7604",
        "Text": "Solution Let X denote the number of defective eggs in the 10 eggs drawn Since the\ndrawing is done with replacement, the trials are Bernoulli trials Clearly, X has the\nbinomial distribution with n = 10 and \n10\n1\n100\n10\np Therefore\nq =\n9\n1\n\u2212p10\n=\nNow\nP(at least one defective egg) = P(X \u2265 1) = 1 \u2013 P (X = 0)\n=\n10\n10\n0\n9\n1\nC\n\u239b10\n\u239e\n\u2212\n\u239c\n\u239f\n\u239d\n\u23a0  = \n10\n910\n1\n10\n\u2212\nEXERCISE 13"
    },
    {
        "Chapter": "1",
        "sentence_range": "7602-7605",
        "Text": "Since the\ndrawing is done with replacement, the trials are Bernoulli trials Clearly, X has the\nbinomial distribution with n = 10 and \n10\n1\n100\n10\np Therefore\nq =\n9\n1\n\u2212p10\n=\nNow\nP(at least one defective egg) = P(X \u2265 1) = 1 \u2013 P (X = 0)\n=\n10\n10\n0\n9\n1\nC\n\u239b10\n\u239e\n\u2212\n\u239c\n\u239f\n\u239d\n\u23a0  = \n10\n910\n1\n10\n\u2212\nEXERCISE 13 5\n1"
    },
    {
        "Chapter": "1",
        "sentence_range": "7603-7606",
        "Text": "Clearly, X has the\nbinomial distribution with n = 10 and \n10\n1\n100\n10\np Therefore\nq =\n9\n1\n\u2212p10\n=\nNow\nP(at least one defective egg) = P(X \u2265 1) = 1 \u2013 P (X = 0)\n=\n10\n10\n0\n9\n1\nC\n\u239b10\n\u239e\n\u2212\n\u239c\n\u239f\n\u239d\n\u23a0  = \n10\n910\n1\n10\n\u2212\nEXERCISE 13 5\n1 A die is thrown 6 times"
    },
    {
        "Chapter": "1",
        "sentence_range": "7604-7607",
        "Text": "Therefore\nq =\n9\n1\n\u2212p10\n=\nNow\nP(at least one defective egg) = P(X \u2265 1) = 1 \u2013 P (X = 0)\n=\n10\n10\n0\n9\n1\nC\n\u239b10\n\u239e\n\u2212\n\u239c\n\u239f\n\u239d\n\u23a0  = \n10\n910\n1\n10\n\u2212\nEXERCISE 13 5\n1 A die is thrown 6 times If  \u2018getting an odd number\u2019 is a success, what is the\nprobability of\n(i) 5 successes"
    },
    {
        "Chapter": "1",
        "sentence_range": "7605-7608",
        "Text": "5\n1 A die is thrown 6 times If  \u2018getting an odd number\u2019 is a success, what is the\nprobability of\n(i) 5 successes (ii) at least 5 successes"
    },
    {
        "Chapter": "1",
        "sentence_range": "7606-7609",
        "Text": "A die is thrown 6 times If  \u2018getting an odd number\u2019 is a success, what is the\nprobability of\n(i) 5 successes (ii) at least 5 successes (iii) at most 5 successes"
    },
    {
        "Chapter": "1",
        "sentence_range": "7607-7610",
        "Text": "If  \u2018getting an odd number\u2019 is a success, what is the\nprobability of\n(i) 5 successes (ii) at least 5 successes (iii) at most 5 successes \u00a9 NCERT\nnot to be republished\nPROBABILITY      577\n2"
    },
    {
        "Chapter": "1",
        "sentence_range": "7608-7611",
        "Text": "(ii) at least 5 successes (iii) at most 5 successes \u00a9 NCERT\nnot to be republished\nPROBABILITY      577\n2 A pair of dice is thrown 4 times"
    },
    {
        "Chapter": "1",
        "sentence_range": "7609-7612",
        "Text": "(iii) at most 5 successes \u00a9 NCERT\nnot to be republished\nPROBABILITY      577\n2 A pair of dice is thrown 4 times If getting a doublet is considered a success, find\nthe probability of two successes"
    },
    {
        "Chapter": "1",
        "sentence_range": "7610-7613",
        "Text": "\u00a9 NCERT\nnot to be republished\nPROBABILITY      577\n2 A pair of dice is thrown 4 times If getting a doublet is considered a success, find\nthe probability of two successes 3"
    },
    {
        "Chapter": "1",
        "sentence_range": "7611-7614",
        "Text": "A pair of dice is thrown 4 times If getting a doublet is considered a success, find\nthe probability of two successes 3 There are 5% defective items in a large bulk of items"
    },
    {
        "Chapter": "1",
        "sentence_range": "7612-7615",
        "Text": "If getting a doublet is considered a success, find\nthe probability of two successes 3 There are 5% defective items in a large bulk of items What is the probability\nthat a sample of 10 items will include not more than one defective item"
    },
    {
        "Chapter": "1",
        "sentence_range": "7613-7616",
        "Text": "3 There are 5% defective items in a large bulk of items What is the probability\nthat a sample of 10 items will include not more than one defective item 4"
    },
    {
        "Chapter": "1",
        "sentence_range": "7614-7617",
        "Text": "There are 5% defective items in a large bulk of items What is the probability\nthat a sample of 10 items will include not more than one defective item 4 Five cards are drawn successively with replacement from a well-shuffled deck\nof 52 cards"
    },
    {
        "Chapter": "1",
        "sentence_range": "7615-7618",
        "Text": "What is the probability\nthat a sample of 10 items will include not more than one defective item 4 Five cards are drawn successively with replacement from a well-shuffled deck\nof 52 cards What is the probability that\n(i) all the five cards are spades"
    },
    {
        "Chapter": "1",
        "sentence_range": "7616-7619",
        "Text": "4 Five cards are drawn successively with replacement from a well-shuffled deck\nof 52 cards What is the probability that\n(i) all the five cards are spades (ii) only 3 cards are spades"
    },
    {
        "Chapter": "1",
        "sentence_range": "7617-7620",
        "Text": "Five cards are drawn successively with replacement from a well-shuffled deck\nof 52 cards What is the probability that\n(i) all the five cards are spades (ii) only 3 cards are spades (iii) none is a spade"
    },
    {
        "Chapter": "1",
        "sentence_range": "7618-7621",
        "Text": "What is the probability that\n(i) all the five cards are spades (ii) only 3 cards are spades (iii) none is a spade 5"
    },
    {
        "Chapter": "1",
        "sentence_range": "7619-7622",
        "Text": "(ii) only 3 cards are spades (iii) none is a spade 5 The probability that a bulb produced by a factory will fuse after 150 days of use\nis 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7620-7623",
        "Text": "(iii) none is a spade 5 The probability that a bulb produced by a factory will fuse after 150 days of use\nis 0 05"
    },
    {
        "Chapter": "1",
        "sentence_range": "7621-7624",
        "Text": "5 The probability that a bulb produced by a factory will fuse after 150 days of use\nis 0 05 Find the probability that out of 5 such bulbs\n(i) none\n(ii) not more than one\n(iii) more than one\n(iv) at least one\nwill fuse after 150 days of use"
    },
    {
        "Chapter": "1",
        "sentence_range": "7622-7625",
        "Text": "The probability that a bulb produced by a factory will fuse after 150 days of use\nis 0 05 Find the probability that out of 5 such bulbs\n(i) none\n(ii) not more than one\n(iii) more than one\n(iv) at least one\nwill fuse after 150 days of use 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "7623-7626",
        "Text": "05 Find the probability that out of 5 such bulbs\n(i) none\n(ii) not more than one\n(iii) more than one\n(iv) at least one\nwill fuse after 150 days of use 6 A bag consists of 10 balls each marked with one of the digits 0 to 9"
    },
    {
        "Chapter": "1",
        "sentence_range": "7624-7627",
        "Text": "Find the probability that out of 5 such bulbs\n(i) none\n(ii) not more than one\n(iii) more than one\n(iv) at least one\nwill fuse after 150 days of use 6 A bag consists of 10 balls each marked with one of the digits 0 to 9 If four balls\nare drawn successively with replacement from the bag, what is the probability\nthat none is marked with the digit 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7625-7628",
        "Text": "6 A bag consists of 10 balls each marked with one of the digits 0 to 9 If four balls\nare drawn successively with replacement from the bag, what is the probability\nthat none is marked with the digit 0 7"
    },
    {
        "Chapter": "1",
        "sentence_range": "7626-7629",
        "Text": "A bag consists of 10 balls each marked with one of the digits 0 to 9 If four balls\nare drawn successively with replacement from the bag, what is the probability\nthat none is marked with the digit 0 7 In an examination, 20 questions of true-false type are asked"
    },
    {
        "Chapter": "1",
        "sentence_range": "7627-7630",
        "Text": "If four balls\nare drawn successively with replacement from the bag, what is the probability\nthat none is marked with the digit 0 7 In an examination, 20 questions of true-false type are asked Suppose a student\ntosses a fair coin to determine his answer to each question"
    },
    {
        "Chapter": "1",
        "sentence_range": "7628-7631",
        "Text": "7 In an examination, 20 questions of true-false type are asked Suppose a student\ntosses a fair coin to determine his answer to each question If the coin falls\nheads, he answers 'true'; if it falls tails, he answers 'false'"
    },
    {
        "Chapter": "1",
        "sentence_range": "7629-7632",
        "Text": "In an examination, 20 questions of true-false type are asked Suppose a student\ntosses a fair coin to determine his answer to each question If the coin falls\nheads, he answers 'true'; if it falls tails, he answers 'false' Find the probability\nthat he answers at least 12 questions correctly"
    },
    {
        "Chapter": "1",
        "sentence_range": "7630-7633",
        "Text": "Suppose a student\ntosses a fair coin to determine his answer to each question If the coin falls\nheads, he answers 'true'; if it falls tails, he answers 'false' Find the probability\nthat he answers at least 12 questions correctly 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "7631-7634",
        "Text": "If the coin falls\nheads, he answers 'true'; if it falls tails, he answers 'false' Find the probability\nthat he answers at least 12 questions correctly 8 Suppose X has a binomial distribution \nB 6, 21"
    },
    {
        "Chapter": "1",
        "sentence_range": "7632-7635",
        "Text": "Find the probability\nthat he answers at least 12 questions correctly 8 Suppose X has a binomial distribution \nB 6, 21 Show that  X = 3 is the most\nlikely outcome"
    },
    {
        "Chapter": "1",
        "sentence_range": "7633-7636",
        "Text": "8 Suppose X has a binomial distribution \nB 6, 21 Show that  X = 3 is the most\nlikely outcome (Hint : P(X = 3) is the maximum among all P(xi), xi = 0,1,2,3,4,5,6)\n9"
    },
    {
        "Chapter": "1",
        "sentence_range": "7634-7637",
        "Text": "Suppose X has a binomial distribution \nB 6, 21 Show that  X = 3 is the most\nlikely outcome (Hint : P(X = 3) is the maximum among all P(xi), xi = 0,1,2,3,4,5,6)\n9 On a multiple choice examination with three possible answers for each of the\nfive questions, what is the probability that a candidate would get four or more\ncorrect answers just by guessing"
    },
    {
        "Chapter": "1",
        "sentence_range": "7635-7638",
        "Text": "Show that  X = 3 is the most\nlikely outcome (Hint : P(X = 3) is the maximum among all P(xi), xi = 0,1,2,3,4,5,6)\n9 On a multiple choice examination with three possible answers for each of the\nfive questions, what is the probability that a candidate would get four or more\ncorrect answers just by guessing 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "7636-7639",
        "Text": "(Hint : P(X = 3) is the maximum among all P(xi), xi = 0,1,2,3,4,5,6)\n9 On a multiple choice examination with three possible answers for each of the\nfive questions, what is the probability that a candidate would get four or more\ncorrect answers just by guessing 10 A person buys a lottery ticket in 50 lotteries, in each of which his chance of\nwinning a prize is \n1\n100"
    },
    {
        "Chapter": "1",
        "sentence_range": "7637-7640",
        "Text": "On a multiple choice examination with three possible answers for each of the\nfive questions, what is the probability that a candidate would get four or more\ncorrect answers just by guessing 10 A person buys a lottery ticket in 50 lotteries, in each of which his chance of\nwinning a prize is \n1\n100 What is the probability that he will win a prize\n(a) at least once (b) exactly once (c) at least twice"
    },
    {
        "Chapter": "1",
        "sentence_range": "7638-7641",
        "Text": "10 A person buys a lottery ticket in 50 lotteries, in each of which his chance of\nwinning a prize is \n1\n100 What is the probability that he will win a prize\n(a) at least once (b) exactly once (c) at least twice \u00a9 NCERT\nnot to be republished\n 578\nMATHEMATICS\n11"
    },
    {
        "Chapter": "1",
        "sentence_range": "7639-7642",
        "Text": "A person buys a lottery ticket in 50 lotteries, in each of which his chance of\nwinning a prize is \n1\n100 What is the probability that he will win a prize\n(a) at least once (b) exactly once (c) at least twice \u00a9 NCERT\nnot to be republished\n 578\nMATHEMATICS\n11 Find the probability of getting 5 exactly twice in 7 throws of a die"
    },
    {
        "Chapter": "1",
        "sentence_range": "7640-7643",
        "Text": "What is the probability that he will win a prize\n(a) at least once (b) exactly once (c) at least twice \u00a9 NCERT\nnot to be republished\n 578\nMATHEMATICS\n11 Find the probability of getting 5 exactly twice in 7 throws of a die 12"
    },
    {
        "Chapter": "1",
        "sentence_range": "7641-7644",
        "Text": "\u00a9 NCERT\nnot to be republished\n 578\nMATHEMATICS\n11 Find the probability of getting 5 exactly twice in 7 throws of a die 12 Find the probability of throwing at most 2 sixes in 6 throws of a single die"
    },
    {
        "Chapter": "1",
        "sentence_range": "7642-7645",
        "Text": "Find the probability of getting 5 exactly twice in 7 throws of a die 12 Find the probability of throwing at most 2 sixes in 6 throws of a single die 13"
    },
    {
        "Chapter": "1",
        "sentence_range": "7643-7646",
        "Text": "12 Find the probability of throwing at most 2 sixes in 6 throws of a single die 13 It is known that 10% of certain articles manufactured are defective"
    },
    {
        "Chapter": "1",
        "sentence_range": "7644-7647",
        "Text": "Find the probability of throwing at most 2 sixes in 6 throws of a single die 13 It is known that 10% of certain articles manufactured are defective What is the\nprobability that in a random sample of 12 such articles, 9 are defective"
    },
    {
        "Chapter": "1",
        "sentence_range": "7645-7648",
        "Text": "13 It is known that 10% of certain articles manufactured are defective What is the\nprobability that in a random sample of 12 such articles, 9 are defective In each of the following, choose the correct answer:\n14"
    },
    {
        "Chapter": "1",
        "sentence_range": "7646-7649",
        "Text": "It is known that 10% of certain articles manufactured are defective What is the\nprobability that in a random sample of 12 such articles, 9 are defective In each of the following, choose the correct answer:\n14 In a box containing 100 bulbs, 10 are defective"
    },
    {
        "Chapter": "1",
        "sentence_range": "7647-7650",
        "Text": "What is the\nprobability that in a random sample of 12 such articles, 9 are defective In each of the following, choose the correct answer:\n14 In a box containing 100 bulbs, 10 are defective The probability that out of a\nsample of 5 bulbs, none is defective is\n(A) 10\u20131\n(B)\n\u239b215\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0\n(C)\n\u239b1095\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0\n(D)\n9\n10\n15"
    },
    {
        "Chapter": "1",
        "sentence_range": "7648-7651",
        "Text": "In each of the following, choose the correct answer:\n14 In a box containing 100 bulbs, 10 are defective The probability that out of a\nsample of 5 bulbs, none is defective is\n(A) 10\u20131\n(B)\n\u239b215\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0\n(C)\n\u239b1095\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0\n(D)\n9\n10\n15 The probability that a student is not a swimmer is 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "7649-7652",
        "Text": "In a box containing 100 bulbs, 10 are defective The probability that out of a\nsample of 5 bulbs, none is defective is\n(A) 10\u20131\n(B)\n\u239b215\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0\n(C)\n\u239b1095\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0\n(D)\n9\n10\n15 The probability that a student is not a swimmer is 1 5  Then the probability that\nout of five students, four are swimmers is\n(A)\n4\n5\n4\n4\n1\nC\n5\n5\n\u239b\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0\n(B)\n44\n1\n5\n5\n\u239b\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0\n(C)\n4\n5\n1\n1\n4\nC 5\n\u239b5\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0\n(D) None of these\nMiscellaneous Examples\nExample 33 Coloured balls are distributed in four boxes as shown in the following\ntable:\nBox\nColour\n    Black      White       Red\n Blue\nI\n3\n4\n5\n6\nII\n2\n2\n2\n2\nIII\n1\n2\n3\n1\nIV\n4\n3\n1\n5\nA box is selected at random and then a ball is randomly drawn from the selected\nbox"
    },
    {
        "Chapter": "1",
        "sentence_range": "7650-7653",
        "Text": "The probability that out of a\nsample of 5 bulbs, none is defective is\n(A) 10\u20131\n(B)\n\u239b215\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0\n(C)\n\u239b1095\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0\n(D)\n9\n10\n15 The probability that a student is not a swimmer is 1 5  Then the probability that\nout of five students, four are swimmers is\n(A)\n4\n5\n4\n4\n1\nC\n5\n5\n\u239b\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0\n(B)\n44\n1\n5\n5\n\u239b\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0\n(C)\n4\n5\n1\n1\n4\nC 5\n\u239b5\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0\n(D) None of these\nMiscellaneous Examples\nExample 33 Coloured balls are distributed in four boxes as shown in the following\ntable:\nBox\nColour\n    Black      White       Red\n Blue\nI\n3\n4\n5\n6\nII\n2\n2\n2\n2\nIII\n1\n2\n3\n1\nIV\n4\n3\n1\n5\nA box is selected at random and then a ball is randomly drawn from the selected\nbox The colour of the ball is black, what is the probability that ball drawn is from the\nbox III"
    },
    {
        "Chapter": "1",
        "sentence_range": "7651-7654",
        "Text": "The probability that a student is not a swimmer is 1 5  Then the probability that\nout of five students, four are swimmers is\n(A)\n4\n5\n4\n4\n1\nC\n5\n5\n\u239b\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0\n(B)\n44\n1\n5\n5\n\u239b\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0\n(C)\n4\n5\n1\n1\n4\nC 5\n\u239b5\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0\n(D) None of these\nMiscellaneous Examples\nExample 33 Coloured balls are distributed in four boxes as shown in the following\ntable:\nBox\nColour\n    Black      White       Red\n Blue\nI\n3\n4\n5\n6\nII\n2\n2\n2\n2\nIII\n1\n2\n3\n1\nIV\n4\n3\n1\n5\nA box is selected at random and then a ball is randomly drawn from the selected\nbox The colour of the ball is black, what is the probability that ball drawn is from the\nbox III \u00a9 NCERT\nnot to be republished\nPROBABILITY      579\nSolution Let A, E1, E2, E3 and E4 be the events as defined below :\nA : a black ball is selected\nE1 : box I is selected\nE2 : box II is selected\nE3 : box III is selected\nE4 : box IV is selected\nSince the boxes are chosen at random,\nTherefore\nP(E1) = P(E2) = P(E3) = P(E4) = 1\n4\nAlso\nP(A|E1) = 3\n18 , P(A|E2) = 2\n8 , P(A|E3) = 1\n7  and  P(A|E4) = 4\n13\nP(box III is selected, given that the drawn ball is black) = P(E3|A)"
    },
    {
        "Chapter": "1",
        "sentence_range": "7652-7655",
        "Text": "5  Then the probability that\nout of five students, four are swimmers is\n(A)\n4\n5\n4\n4\n1\nC\n5\n5\n\u239b\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0\n(B)\n44\n1\n5\n5\n\u239b\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0\n(C)\n4\n5\n1\n1\n4\nC 5\n\u239b5\n\u239e\n\u239c\n\u239f\n\u239d\n\u23a0\n(D) None of these\nMiscellaneous Examples\nExample 33 Coloured balls are distributed in four boxes as shown in the following\ntable:\nBox\nColour\n    Black      White       Red\n Blue\nI\n3\n4\n5\n6\nII\n2\n2\n2\n2\nIII\n1\n2\n3\n1\nIV\n4\n3\n1\n5\nA box is selected at random and then a ball is randomly drawn from the selected\nbox The colour of the ball is black, what is the probability that ball drawn is from the\nbox III \u00a9 NCERT\nnot to be republished\nPROBABILITY      579\nSolution Let A, E1, E2, E3 and E4 be the events as defined below :\nA : a black ball is selected\nE1 : box I is selected\nE2 : box II is selected\nE3 : box III is selected\nE4 : box IV is selected\nSince the boxes are chosen at random,\nTherefore\nP(E1) = P(E2) = P(E3) = P(E4) = 1\n4\nAlso\nP(A|E1) = 3\n18 , P(A|E2) = 2\n8 , P(A|E3) = 1\n7  and  P(A|E4) = 4\n13\nP(box III is selected, given that the drawn ball is black) = P(E3|A) By Bayes'\ntheorem,\nP(E3|A) = \n3\n3\n1\n1\n2\n2\n3\n3\n4\n4\nP(E ) P(A|E )\nP(E )P(A|E )\nP(E )P(A|E )+P(E )P(A|E )\nP(E )P(A|E )\n \n \n \n= \n1\n1\n4\n7\n0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7653-7656",
        "Text": "The colour of the ball is black, what is the probability that ball drawn is from the\nbox III \u00a9 NCERT\nnot to be republished\nPROBABILITY      579\nSolution Let A, E1, E2, E3 and E4 be the events as defined below :\nA : a black ball is selected\nE1 : box I is selected\nE2 : box II is selected\nE3 : box III is selected\nE4 : box IV is selected\nSince the boxes are chosen at random,\nTherefore\nP(E1) = P(E2) = P(E3) = P(E4) = 1\n4\nAlso\nP(A|E1) = 3\n18 , P(A|E2) = 2\n8 , P(A|E3) = 1\n7  and  P(A|E4) = 4\n13\nP(box III is selected, given that the drawn ball is black) = P(E3|A) By Bayes'\ntheorem,\nP(E3|A) = \n3\n3\n1\n1\n2\n2\n3\n3\n4\n4\nP(E ) P(A|E )\nP(E )P(A|E )\nP(E )P(A|E )+P(E )P(A|E )\nP(E )P(A|E )\n \n \n \n= \n1\n1\n4\n7\n0 165\n1\n3\n1\n1\n1\n1\n1\n4\n4\n18\n4\n4\n4\n7\n4\n13\n\u00d7\n=\n\u00d7\n+\n\u00d7\n+\n\u00d7\n+\n\u00d7\nExample 34 Find the mean of the Binomial distribution \nB 4, 31"
    },
    {
        "Chapter": "1",
        "sentence_range": "7654-7657",
        "Text": "\u00a9 NCERT\nnot to be republished\nPROBABILITY      579\nSolution Let A, E1, E2, E3 and E4 be the events as defined below :\nA : a black ball is selected\nE1 : box I is selected\nE2 : box II is selected\nE3 : box III is selected\nE4 : box IV is selected\nSince the boxes are chosen at random,\nTherefore\nP(E1) = P(E2) = P(E3) = P(E4) = 1\n4\nAlso\nP(A|E1) = 3\n18 , P(A|E2) = 2\n8 , P(A|E3) = 1\n7  and  P(A|E4) = 4\n13\nP(box III is selected, given that the drawn ball is black) = P(E3|A) By Bayes'\ntheorem,\nP(E3|A) = \n3\n3\n1\n1\n2\n2\n3\n3\n4\n4\nP(E ) P(A|E )\nP(E )P(A|E )\nP(E )P(A|E )+P(E )P(A|E )\nP(E )P(A|E )\n \n \n \n= \n1\n1\n4\n7\n0 165\n1\n3\n1\n1\n1\n1\n1\n4\n4\n18\n4\n4\n4\n7\n4\n13\n\u00d7\n=\n\u00d7\n+\n\u00d7\n+\n\u00d7\n+\n\u00d7\nExample 34 Find the mean of the Binomial distribution \nB 4, 31 Solution Let X be the random variable whose probability distribution is \nB 4,31"
    },
    {
        "Chapter": "1",
        "sentence_range": "7655-7658",
        "Text": "By Bayes'\ntheorem,\nP(E3|A) = \n3\n3\n1\n1\n2\n2\n3\n3\n4\n4\nP(E ) P(A|E )\nP(E )P(A|E )\nP(E )P(A|E )+P(E )P(A|E )\nP(E )P(A|E )\n \n \n \n= \n1\n1\n4\n7\n0 165\n1\n3\n1\n1\n1\n1\n1\n4\n4\n18\n4\n4\n4\n7\n4\n13\n\u00d7\n=\n\u00d7\n+\n\u00d7\n+\n\u00d7\n+\n\u00d7\nExample 34 Find the mean of the Binomial distribution \nB 4, 31 Solution Let X be the random variable whose probability distribution is \nB 4,31 Here\nn = 4, p = 1\n3  and  q = \n1\n2\n1\n3\n3\n\u2212\n=\nWe know that\nP(X = x) =\n4\n4\n2\n1\nC\n3\n3\nx\nx\nx\n\u2212\n\u239b\n\u239e\n\u239b\n\u239e\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\n, x = 0, 1, 2, 3, 4"
    },
    {
        "Chapter": "1",
        "sentence_range": "7656-7659",
        "Text": "165\n1\n3\n1\n1\n1\n1\n1\n4\n4\n18\n4\n4\n4\n7\n4\n13\n\u00d7\n=\n\u00d7\n+\n\u00d7\n+\n\u00d7\n+\n\u00d7\nExample 34 Find the mean of the Binomial distribution \nB 4, 31 Solution Let X be the random variable whose probability distribution is \nB 4,31 Here\nn = 4, p = 1\n3  and  q = \n1\n2\n1\n3\n3\n\u2212\n=\nWe know that\nP(X = x) =\n4\n4\n2\n1\nC\n3\n3\nx\nx\nx\n\u2212\n\u239b\n\u239e\n\u239b\n\u239e\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\n, x = 0, 1, 2, 3, 4 i"
    },
    {
        "Chapter": "1",
        "sentence_range": "7657-7660",
        "Text": "Solution Let X be the random variable whose probability distribution is \nB 4,31 Here\nn = 4, p = 1\n3  and  q = \n1\n2\n1\n3\n3\n\u2212\n=\nWe know that\nP(X = x) =\n4\n4\n2\n1\nC\n3\n3\nx\nx\nx\n\u2212\n\u239b\n\u239e\n\u239b\n\u239e\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\n, x = 0, 1, 2, 3, 4 i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "7658-7661",
        "Text": "Here\nn = 4, p = 1\n3  and  q = \n1\n2\n1\n3\n3\n\u2212\n=\nWe know that\nP(X = x) =\n4\n4\n2\n1\nC\n3\n3\nx\nx\nx\n\u2212\n\u239b\n\u239e\n\u239b\n\u239e\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0\n, x = 0, 1, 2, 3, 4 i e the distribution of X is\nxi\nP(xi)\nxi P(xi)\n0\n4\n4\n0\n2\nC\n 3\n \n \n \n \n \n0\n1\n3\n4\n1\n2\n1\nC\n3\n  3\n \n  \n \n  \n \n3\n4\n1\n2\n1\nC\n3\n3\n \n  \n \n \n  \n \n \n  \n \n\u00a9 NCERT\nnot to be republished\n 580\nMATHEMATICS\n2\n2\n2\n4\n2\n2\n1\nC\n3\n3\n \n \n \n \n \n \n \n \n \n2\n2\n4\n2\n2\n1\n2\nC\n3\n3\n \n \n \n \n \n \n \n \n  \n \n \n \n  \n3\n3\n4\n3\n2\n1\nC\n3\n3\n \n  \n \n \n  \n \n \n  \n \n3\n4\n3\n2\n1\n3\nC\n3\n3\n \n \n  \n \n  \n \n \n \n  \n \n \n \n4\n4\n4\n4\n1\nC\n 3\n \n \n \n \n \n4\n4\n4\n1\n4\nC\n3\n \n \n \n \n \n \n \n \n \n \n \n \nNow Mean (\u03bc) =\n4\n1\n(\n)\ni\ni\ni\nx p x\n=\u2211\n=\n3\n2\n2\n4\n4\n1\n2\n2\n1\n2\n1\n0\nC\n2\nC\n3\n3\n3\n3\n\u239b\n\u239e \u239b\n\u239e\n\u239b\n\u239e \u239b\n\u239e\n+\n+ \u22c5\n\u239c\n\u239f \u239c\n\u239f\n\u239c\n\u239f \u239c\n\u239f\n\u239d\n\u23a0 \u239d\n\u23a0\n\u239d\n\u23a0 \u239d\n\u23a0\n+\n3\n4\n4\n4\n3\n4\n2\n1\n1\n3\nC\n4\nC\n3\n3\n3\n \n  \n \n \n \n \n \n \n \n  \n \n \n \n \n  \n \n \n \n=\n3\n2\n4\n4\n4\n4\n2\n2\n2\n1\n4\n2\n6\n3\n4\n4 1\n3\n3\n3\n3\n\u00d7\n+\n\u00d7\n\u00d7\n+\n\u00d7\n\u00d7\n+\n\u00d7 \u00d7\n=\n4\n32\n48\n24\n4\n108\n4\n81\n3\n3\n+\n+\n+\n=\n=\nExample 35 The probability of a shooter hitting a target is 3\n4"
    },
    {
        "Chapter": "1",
        "sentence_range": "7659-7662",
        "Text": "i e the distribution of X is\nxi\nP(xi)\nxi P(xi)\n0\n4\n4\n0\n2\nC\n 3\n \n \n \n \n \n0\n1\n3\n4\n1\n2\n1\nC\n3\n  3\n \n  \n \n  \n \n3\n4\n1\n2\n1\nC\n3\n3\n \n  \n \n \n  \n \n \n  \n \n\u00a9 NCERT\nnot to be republished\n 580\nMATHEMATICS\n2\n2\n2\n4\n2\n2\n1\nC\n3\n3\n \n \n \n \n \n \n \n \n \n2\n2\n4\n2\n2\n1\n2\nC\n3\n3\n \n \n \n \n \n \n \n \n  \n \n \n \n  \n3\n3\n4\n3\n2\n1\nC\n3\n3\n \n  \n \n \n  \n \n \n  \n \n3\n4\n3\n2\n1\n3\nC\n3\n3\n \n \n  \n \n  \n \n \n \n  \n \n \n \n4\n4\n4\n4\n1\nC\n 3\n \n \n \n \n \n4\n4\n4\n1\n4\nC\n3\n \n \n \n \n \n \n \n \n \n \n \n \nNow Mean (\u03bc) =\n4\n1\n(\n)\ni\ni\ni\nx p x\n=\u2211\n=\n3\n2\n2\n4\n4\n1\n2\n2\n1\n2\n1\n0\nC\n2\nC\n3\n3\n3\n3\n\u239b\n\u239e \u239b\n\u239e\n\u239b\n\u239e \u239b\n\u239e\n+\n+ \u22c5\n\u239c\n\u239f \u239c\n\u239f\n\u239c\n\u239f \u239c\n\u239f\n\u239d\n\u23a0 \u239d\n\u23a0\n\u239d\n\u23a0 \u239d\n\u23a0\n+\n3\n4\n4\n4\n3\n4\n2\n1\n1\n3\nC\n4\nC\n3\n3\n3\n \n  \n \n \n \n \n \n \n \n  \n \n \n \n \n  \n \n \n \n=\n3\n2\n4\n4\n4\n4\n2\n2\n2\n1\n4\n2\n6\n3\n4\n4 1\n3\n3\n3\n3\n\u00d7\n+\n\u00d7\n\u00d7\n+\n\u00d7\n\u00d7\n+\n\u00d7 \u00d7\n=\n4\n32\n48\n24\n4\n108\n4\n81\n3\n3\n+\n+\n+\n=\n=\nExample 35 The probability of a shooter hitting a target is 3\n4 How many minimum\nnumber of times must he/she fire so that the probability of hitting the target at least\nonce is more than 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7660-7663",
        "Text": "e the distribution of X is\nxi\nP(xi)\nxi P(xi)\n0\n4\n4\n0\n2\nC\n 3\n \n \n \n \n \n0\n1\n3\n4\n1\n2\n1\nC\n3\n  3\n \n  \n \n  \n \n3\n4\n1\n2\n1\nC\n3\n3\n \n  \n \n \n  \n \n \n  \n \n\u00a9 NCERT\nnot to be republished\n 580\nMATHEMATICS\n2\n2\n2\n4\n2\n2\n1\nC\n3\n3\n \n \n \n \n \n \n \n \n \n2\n2\n4\n2\n2\n1\n2\nC\n3\n3\n \n \n \n \n \n \n \n \n  \n \n \n \n  \n3\n3\n4\n3\n2\n1\nC\n3\n3\n \n  \n \n \n  \n \n \n  \n \n3\n4\n3\n2\n1\n3\nC\n3\n3\n \n \n  \n \n  \n \n \n \n  \n \n \n \n4\n4\n4\n4\n1\nC\n 3\n \n \n \n \n \n4\n4\n4\n1\n4\nC\n3\n \n \n \n \n \n \n \n \n \n \n \n \nNow Mean (\u03bc) =\n4\n1\n(\n)\ni\ni\ni\nx p x\n=\u2211\n=\n3\n2\n2\n4\n4\n1\n2\n2\n1\n2\n1\n0\nC\n2\nC\n3\n3\n3\n3\n\u239b\n\u239e \u239b\n\u239e\n\u239b\n\u239e \u239b\n\u239e\n+\n+ \u22c5\n\u239c\n\u239f \u239c\n\u239f\n\u239c\n\u239f \u239c\n\u239f\n\u239d\n\u23a0 \u239d\n\u23a0\n\u239d\n\u23a0 \u239d\n\u23a0\n+\n3\n4\n4\n4\n3\n4\n2\n1\n1\n3\nC\n4\nC\n3\n3\n3\n \n  \n \n \n \n \n \n \n \n  \n \n \n \n \n  \n \n \n \n=\n3\n2\n4\n4\n4\n4\n2\n2\n2\n1\n4\n2\n6\n3\n4\n4 1\n3\n3\n3\n3\n\u00d7\n+\n\u00d7\n\u00d7\n+\n\u00d7\n\u00d7\n+\n\u00d7 \u00d7\n=\n4\n32\n48\n24\n4\n108\n4\n81\n3\n3\n+\n+\n+\n=\n=\nExample 35 The probability of a shooter hitting a target is 3\n4 How many minimum\nnumber of times must he/she fire so that the probability of hitting the target at least\nonce is more than 0 99"
    },
    {
        "Chapter": "1",
        "sentence_range": "7661-7664",
        "Text": "the distribution of X is\nxi\nP(xi)\nxi P(xi)\n0\n4\n4\n0\n2\nC\n 3\n \n \n \n \n \n0\n1\n3\n4\n1\n2\n1\nC\n3\n  3\n \n  \n \n  \n \n3\n4\n1\n2\n1\nC\n3\n3\n \n  \n \n \n  \n \n \n  \n \n\u00a9 NCERT\nnot to be republished\n 580\nMATHEMATICS\n2\n2\n2\n4\n2\n2\n1\nC\n3\n3\n \n \n \n \n \n \n \n \n \n2\n2\n4\n2\n2\n1\n2\nC\n3\n3\n \n \n \n \n \n \n \n \n  \n \n \n \n  \n3\n3\n4\n3\n2\n1\nC\n3\n3\n \n  \n \n \n  \n \n \n  \n \n3\n4\n3\n2\n1\n3\nC\n3\n3\n \n \n  \n \n  \n \n \n \n  \n \n \n \n4\n4\n4\n4\n1\nC\n 3\n \n \n \n \n \n4\n4\n4\n1\n4\nC\n3\n \n \n \n \n \n \n \n \n \n \n \n \nNow Mean (\u03bc) =\n4\n1\n(\n)\ni\ni\ni\nx p x\n=\u2211\n=\n3\n2\n2\n4\n4\n1\n2\n2\n1\n2\n1\n0\nC\n2\nC\n3\n3\n3\n3\n\u239b\n\u239e \u239b\n\u239e\n\u239b\n\u239e \u239b\n\u239e\n+\n+ \u22c5\n\u239c\n\u239f \u239c\n\u239f\n\u239c\n\u239f \u239c\n\u239f\n\u239d\n\u23a0 \u239d\n\u23a0\n\u239d\n\u23a0 \u239d\n\u23a0\n+\n3\n4\n4\n4\n3\n4\n2\n1\n1\n3\nC\n4\nC\n3\n3\n3\n \n  \n \n \n \n \n \n \n \n  \n \n \n \n \n  \n \n \n \n=\n3\n2\n4\n4\n4\n4\n2\n2\n2\n1\n4\n2\n6\n3\n4\n4 1\n3\n3\n3\n3\n\u00d7\n+\n\u00d7\n\u00d7\n+\n\u00d7\n\u00d7\n+\n\u00d7 \u00d7\n=\n4\n32\n48\n24\n4\n108\n4\n81\n3\n3\n+\n+\n+\n=\n=\nExample 35 The probability of a shooter hitting a target is 3\n4 How many minimum\nnumber of times must he/she fire so that the probability of hitting the target at least\nonce is more than 0 99 Solution Let the shooter fire n times"
    },
    {
        "Chapter": "1",
        "sentence_range": "7662-7665",
        "Text": "How many minimum\nnumber of times must he/she fire so that the probability of hitting the target at least\nonce is more than 0 99 Solution Let the shooter fire n times Obviously, n fires are n Bernoulli trials"
    },
    {
        "Chapter": "1",
        "sentence_range": "7663-7666",
        "Text": "99 Solution Let the shooter fire n times Obviously, n fires are n Bernoulli trials In each\ntrial, p = probability of hitting the target = 3\n4  and q  =  probability of not hitting the\ntarget = 1\n4"
    },
    {
        "Chapter": "1",
        "sentence_range": "7664-7667",
        "Text": "Solution Let the shooter fire n times Obviously, n fires are n Bernoulli trials In each\ntrial, p = probability of hitting the target = 3\n4  and q  =  probability of not hitting the\ntarget = 1\n4 Then  P(X = x) = \n1\n3\n3\nC\nC\nC\n4\n4\n4\nn x\nx\nx\nn\nn x\nx\nn\nn\nx\nx\nx\nn\nq\np\n\u2212\n\u2212\n\u239b\n\u239e\n\u239b\n\u239e\n=\n=\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7665-7668",
        "Text": "Obviously, n fires are n Bernoulli trials In each\ntrial, p = probability of hitting the target = 3\n4  and q  =  probability of not hitting the\ntarget = 1\n4 Then  P(X = x) = \n1\n3\n3\nC\nC\nC\n4\n4\n4\nn x\nx\nx\nn\nn x\nx\nn\nn\nx\nx\nx\nn\nq\np\n\u2212\n\u2212\n\u239b\n\u239e\n\u239b\n\u239e\n=\n=\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0 Now, given that,\nP(hitting the target at least once) > 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7666-7669",
        "Text": "In each\ntrial, p = probability of hitting the target = 3\n4  and q  =  probability of not hitting the\ntarget = 1\n4 Then  P(X = x) = \n1\n3\n3\nC\nC\nC\n4\n4\n4\nn x\nx\nx\nn\nn x\nx\nn\nn\nx\nx\nx\nn\nq\np\n\u2212\n\u2212\n\u239b\n\u239e\n\u239b\n\u239e\n=\n=\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0 Now, given that,\nP(hitting the target at least once) > 0 99\ni"
    },
    {
        "Chapter": "1",
        "sentence_range": "7667-7670",
        "Text": "Then  P(X = x) = \n1\n3\n3\nC\nC\nC\n4\n4\n4\nn x\nx\nx\nn\nn x\nx\nn\nn\nx\nx\nx\nn\nq\np\n\u2212\n\u2212\n\u239b\n\u239e\n\u239b\n\u239e\n=\n=\n\u239c\n\u239f\n\u239c\n\u239f\n\u239d\n\u23a0\n\u239d\n\u23a0 Now, given that,\nP(hitting the target at least once) > 0 99\ni e"
    },
    {
        "Chapter": "1",
        "sentence_range": "7668-7671",
        "Text": "Now, given that,\nP(hitting the target at least once) > 0 99\ni e P(x \u2265 1) > 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7669-7672",
        "Text": "99\ni e P(x \u2265 1) > 0 99\n\u00a9 NCERT\nnot to be republished\nPROBABILITY      581\nTherefore,\n1 \u2013 P (x = 0) > 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7670-7673",
        "Text": "e P(x \u2265 1) > 0 99\n\u00a9 NCERT\nnot to be republished\nPROBABILITY      581\nTherefore,\n1 \u2013 P (x = 0) > 0 99\nor\n0\n1\n1\nC\n4\nn\nn\n\u2212\n > 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7671-7674",
        "Text": "P(x \u2265 1) > 0 99\n\u00a9 NCERT\nnot to be republished\nPROBABILITY      581\nTherefore,\n1 \u2013 P (x = 0) > 0 99\nor\n0\n1\n1\nC\n4\nn\nn\n\u2212\n > 0 99\nor\n0\n1\n1\nC\n0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7672-7675",
        "Text": "99\n\u00a9 NCERT\nnot to be republished\nPROBABILITY      581\nTherefore,\n1 \u2013 P (x = 0) > 0 99\nor\n0\n1\n1\nC\n4\nn\nn\n\u2212\n > 0 99\nor\n0\n1\n1\nC\n0 01 i"
    },
    {
        "Chapter": "1",
        "sentence_range": "7673-7676",
        "Text": "99\nor\n0\n1\n1\nC\n4\nn\nn\n\u2212\n > 0 99\nor\n0\n1\n1\nC\n0 01 i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "7674-7677",
        "Text": "99\nor\n0\n1\n1\nC\n0 01 i e 4\n4\n \nn\nn\nn  < 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7675-7678",
        "Text": "01 i e 4\n4\n \nn\nn\nn  < 0 01\nor\n4n > \n1\n0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7676-7679",
        "Text": "e 4\n4\n \nn\nn\nn  < 0 01\nor\n4n > \n1\n0 01  = 100"
    },
    {
        "Chapter": "1",
        "sentence_range": "7677-7680",
        "Text": "4\n4\n \nn\nn\nn  < 0 01\nor\n4n > \n1\n0 01  = 100 (1)\nThe minimum value of n to satisfy the inequality (1) is 4"
    },
    {
        "Chapter": "1",
        "sentence_range": "7678-7681",
        "Text": "01\nor\n4n > \n1\n0 01  = 100 (1)\nThe minimum value of n to satisfy the inequality (1) is 4 Thus,  the shooter must fire 4 times"
    },
    {
        "Chapter": "1",
        "sentence_range": "7679-7682",
        "Text": "01  = 100 (1)\nThe minimum value of n to satisfy the inequality (1) is 4 Thus,  the shooter must fire 4 times Example 36 A and B throw a die alternatively  till one of them gets a \u20186\u2019 and wins the\ngame"
    },
    {
        "Chapter": "1",
        "sentence_range": "7680-7683",
        "Text": "(1)\nThe minimum value of n to satisfy the inequality (1) is 4 Thus,  the shooter must fire 4 times Example 36 A and B throw a die alternatively  till one of them gets a \u20186\u2019 and wins the\ngame Find their respective probabilities of winning, if A starts first"
    },
    {
        "Chapter": "1",
        "sentence_range": "7681-7684",
        "Text": "Thus,  the shooter must fire 4 times Example 36 A and B throw a die alternatively  till one of them gets a \u20186\u2019 and wins the\ngame Find their respective probabilities of winning, if A starts first Solution Let S denote the success (getting a \u20186\u2019) and F denote the failure (not getting\na \u20186\u2019)"
    },
    {
        "Chapter": "1",
        "sentence_range": "7682-7685",
        "Text": "Example 36 A and B throw a die alternatively  till one of them gets a \u20186\u2019 and wins the\ngame Find their respective probabilities of winning, if A starts first Solution Let S denote the success (getting a \u20186\u2019) and F denote the failure (not getting\na \u20186\u2019) Thus,\nP(S) = 1\n5\n6, P(F)\n6\n=\nP(A wins in the first throw) = P(S) = 1\n6\nA gets the third throw, when the first throw by A and second throw by B result into\nfailures"
    },
    {
        "Chapter": "1",
        "sentence_range": "7683-7686",
        "Text": "Find their respective probabilities of winning, if A starts first Solution Let S denote the success (getting a \u20186\u2019) and F denote the failure (not getting\na \u20186\u2019) Thus,\nP(S) = 1\n5\n6, P(F)\n6\n=\nP(A wins in the first throw) = P(S) = 1\n6\nA gets the third throw, when the first throw by A and second throw by B result into\nfailures Therefore,\nP(A wins in the 3rd throw) = P(FFS) = \n5\n5\n1\nP(F)P(F)P(S)= 6\n6\n6\n \n \n=\n52\n1\n6\n6\n\u239b\n\u239e \u00d7\n\u239c\n\u239f\n\u239d\n\u23a0\nP(A wins in the 5th throw) = P (FFFFS) \n54\n1\n6\n6\n \n \n \n \n  \n \n \n \n \n \n \n and so on"
    },
    {
        "Chapter": "1",
        "sentence_range": "7684-7687",
        "Text": "Solution Let S denote the success (getting a \u20186\u2019) and F denote the failure (not getting\na \u20186\u2019) Thus,\nP(S) = 1\n5\n6, P(F)\n6\n=\nP(A wins in the first throw) = P(S) = 1\n6\nA gets the third throw, when the first throw by A and second throw by B result into\nfailures Therefore,\nP(A wins in the 3rd throw) = P(FFS) = \n5\n5\n1\nP(F)P(F)P(S)= 6\n6\n6\n \n \n=\n52\n1\n6\n6\n\u239b\n\u239e \u00d7\n\u239c\n\u239f\n\u239d\n\u23a0\nP(A wins in the 5th throw) = P (FFFFS) \n54\n1\n6\n6\n \n \n \n \n  \n \n \n \n \n \n \n and so on Hence,\nP(A wins) =\n2\n4\n1\n5\n1\n5\n1"
    },
    {
        "Chapter": "1",
        "sentence_range": "7685-7688",
        "Text": "Thus,\nP(S) = 1\n5\n6, P(F)\n6\n=\nP(A wins in the first throw) = P(S) = 1\n6\nA gets the third throw, when the first throw by A and second throw by B result into\nfailures Therefore,\nP(A wins in the 3rd throw) = P(FFS) = \n5\n5\n1\nP(F)P(F)P(S)= 6\n6\n6\n \n \n=\n52\n1\n6\n6\n\u239b\n\u239e \u00d7\n\u239c\n\u239f\n\u239d\n\u23a0\nP(A wins in the 5th throw) = P (FFFFS) \n54\n1\n6\n6\n \n \n \n \n  \n \n \n \n \n \n \n and so on Hence,\nP(A wins) =\n2\n4\n1\n5\n1\n5\n1 6\n6\n6\n6\n6\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n=\n61\n25\n1 36\n\u2212\n = 6\n11\n\u00a9 NCERT\nnot to be republished\n 582\nMATHEMATICS\nP(B wins) = 1 \u2013 P (A wins) = \n6\n 1 11 115\n \nRemark If a + ar + ar2 +"
    },
    {
        "Chapter": "1",
        "sentence_range": "7686-7689",
        "Text": "Therefore,\nP(A wins in the 3rd throw) = P(FFS) = \n5\n5\n1\nP(F)P(F)P(S)= 6\n6\n6\n \n \n=\n52\n1\n6\n6\n\u239b\n\u239e \u00d7\n\u239c\n\u239f\n\u239d\n\u23a0\nP(A wins in the 5th throw) = P (FFFFS) \n54\n1\n6\n6\n \n \n \n \n  \n \n \n \n \n \n \n and so on Hence,\nP(A wins) =\n2\n4\n1\n5\n1\n5\n1 6\n6\n6\n6\n6\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n=\n61\n25\n1 36\n\u2212\n = 6\n11\n\u00a9 NCERT\nnot to be republished\n 582\nMATHEMATICS\nP(B wins) = 1 \u2013 P (A wins) = \n6\n 1 11 115\n \nRemark If a + ar + ar2 + + arn\u20131 +"
    },
    {
        "Chapter": "1",
        "sentence_range": "7687-7690",
        "Text": "Hence,\nP(A wins) =\n2\n4\n1\n5\n1\n5\n1 6\n6\n6\n6\n6\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n=\n61\n25\n1 36\n\u2212\n = 6\n11\n\u00a9 NCERT\nnot to be republished\n 582\nMATHEMATICS\nP(B wins) = 1 \u2013 P (A wins) = \n6\n 1 11 115\n \nRemark If a + ar + ar2 + + arn\u20131 + , where |r| < 1, then sum of this infinite G"
    },
    {
        "Chapter": "1",
        "sentence_range": "7688-7691",
        "Text": "6\n6\n6\n6\n6\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n=\n61\n25\n1 36\n\u2212\n = 6\n11\n\u00a9 NCERT\nnot to be republished\n 582\nMATHEMATICS\nP(B wins) = 1 \u2013 P (A wins) = \n6\n 1 11 115\n \nRemark If a + ar + ar2 + + arn\u20131 + , where |r| < 1, then sum of this infinite G P"
    },
    {
        "Chapter": "1",
        "sentence_range": "7689-7692",
        "Text": "+ arn\u20131 + , where |r| < 1, then sum of this infinite G P is given by"
    },
    {
        "Chapter": "1",
        "sentence_range": "7690-7693",
        "Text": ", where |r| < 1, then sum of this infinite G P is given by 1\na\n\u2212r\n (Refer A"
    },
    {
        "Chapter": "1",
        "sentence_range": "7691-7694",
        "Text": "P is given by 1\na\n\u2212r\n (Refer A 1"
    },
    {
        "Chapter": "1",
        "sentence_range": "7692-7695",
        "Text": "is given by 1\na\n\u2212r\n (Refer A 1 3 of Class XI Text book)"
    },
    {
        "Chapter": "1",
        "sentence_range": "7693-7696",
        "Text": "1\na\n\u2212r\n (Refer A 1 3 of Class XI Text book) Example 37 If a machine is correctly set up, it produces 90% acceptable items"
    },
    {
        "Chapter": "1",
        "sentence_range": "7694-7697",
        "Text": "1 3 of Class XI Text book) Example 37 If a machine is correctly set up, it produces 90% acceptable items If it is\nincorrectly set up, it produces only 40% acceptable items"
    },
    {
        "Chapter": "1",
        "sentence_range": "7695-7698",
        "Text": "3 of Class XI Text book) Example 37 If a machine is correctly set up, it produces 90% acceptable items If it is\nincorrectly set up, it produces only 40% acceptable items Past experience shows that\n80% of the set ups are correctly done"
    },
    {
        "Chapter": "1",
        "sentence_range": "7696-7699",
        "Text": "Example 37 If a machine is correctly set up, it produces 90% acceptable items If it is\nincorrectly set up, it produces only 40% acceptable items Past experience shows that\n80% of the set ups are correctly done If after a certain set up, the machine produces\n2 acceptable items, find the probability that the machine is correctly setup"
    },
    {
        "Chapter": "1",
        "sentence_range": "7697-7700",
        "Text": "If it is\nincorrectly set up, it produces only 40% acceptable items Past experience shows that\n80% of the set ups are correctly done If after a certain set up, the machine produces\n2 acceptable items, find the probability that the machine is correctly setup Solution Let A be the event that the machine produces 2 acceptable items"
    },
    {
        "Chapter": "1",
        "sentence_range": "7698-7701",
        "Text": "Past experience shows that\n80% of the set ups are correctly done If after a certain set up, the machine produces\n2 acceptable items, find the probability that the machine is correctly setup Solution Let A be the event that the machine produces 2 acceptable items Also let B1 represent the event of correct set up and B2 represent the event of\nincorrect setup"
    },
    {
        "Chapter": "1",
        "sentence_range": "7699-7702",
        "Text": "If after a certain set up, the machine produces\n2 acceptable items, find the probability that the machine is correctly setup Solution Let A be the event that the machine produces 2 acceptable items Also let B1 represent the event of correct set up and B2 represent the event of\nincorrect setup Now\nP(B1) = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7700-7703",
        "Text": "Solution Let A be the event that the machine produces 2 acceptable items Also let B1 represent the event of correct set up and B2 represent the event of\nincorrect setup Now\nP(B1) = 0 8, P(B2) = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7701-7704",
        "Text": "Also let B1 represent the event of correct set up and B2 represent the event of\nincorrect setup Now\nP(B1) = 0 8, P(B2) = 0 2\nP(A|B1) = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7702-7705",
        "Text": "Now\nP(B1) = 0 8, P(B2) = 0 2\nP(A|B1) = 0 9 \u00d7 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7703-7706",
        "Text": "8, P(B2) = 0 2\nP(A|B1) = 0 9 \u00d7 0 9  and P(A|B2) =  0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7704-7707",
        "Text": "2\nP(A|B1) = 0 9 \u00d7 0 9  and P(A|B2) =  0 4 \u00d7 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7705-7708",
        "Text": "9 \u00d7 0 9  and P(A|B2) =  0 4 \u00d7 0 4\nTherefore\nP(B1|A) =\n1\n1\n1\n1\n2\n2\nP(B ) P(A|B )\nP(B ) P(A|B ) + P(B ) P(A|B )\n=\n0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7706-7709",
        "Text": "9  and P(A|B2) =  0 4 \u00d7 0 4\nTherefore\nP(B1|A) =\n1\n1\n1\n1\n2\n2\nP(B ) P(A|B )\nP(B ) P(A|B ) + P(B ) P(A|B )\n=\n0 8\u00d7 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7707-7710",
        "Text": "4 \u00d7 0 4\nTherefore\nP(B1|A) =\n1\n1\n1\n1\n2\n2\nP(B ) P(A|B )\nP(B ) P(A|B ) + P(B ) P(A|B )\n=\n0 8\u00d7 0 9\u00d7 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7708-7711",
        "Text": "4\nTherefore\nP(B1|A) =\n1\n1\n1\n1\n2\n2\nP(B ) P(A|B )\nP(B ) P(A|B ) + P(B ) P(A|B )\n=\n0 8\u00d7 0 9\u00d7 0 9\n648\n0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7709-7712",
        "Text": "8\u00d7 0 9\u00d7 0 9\n648\n0 95\n0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7710-7713",
        "Text": "9\u00d7 0 9\n648\n0 95\n0 8\u00d7 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7711-7714",
        "Text": "9\n648\n0 95\n0 8\u00d7 0 9\u00d7 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7712-7715",
        "Text": "95\n0 8\u00d7 0 9\u00d7 0 9 + 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7713-7716",
        "Text": "8\u00d7 0 9\u00d7 0 9 + 0 2\u00d7 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7714-7717",
        "Text": "9\u00d7 0 9 + 0 2\u00d7 0 4\u00d7 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7715-7718",
        "Text": "9 + 0 2\u00d7 0 4\u00d7 0 4\n=680\n=\nMiscellaneous Exercise on Chapter 13\n1"
    },
    {
        "Chapter": "1",
        "sentence_range": "7716-7719",
        "Text": "2\u00d7 0 4\u00d7 0 4\n=680\n=\nMiscellaneous Exercise on Chapter 13\n1 A and B are two events such that P (A) \u2260 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7717-7720",
        "Text": "4\u00d7 0 4\n=680\n=\nMiscellaneous Exercise on Chapter 13\n1 A and B are two events such that P (A) \u2260 0 Find P(B|A), if\n(i) A is a subset of B\n(ii) A \u2229 B = \u03c6\n2"
    },
    {
        "Chapter": "1",
        "sentence_range": "7718-7721",
        "Text": "4\n=680\n=\nMiscellaneous Exercise on Chapter 13\n1 A and B are two events such that P (A) \u2260 0 Find P(B|A), if\n(i) A is a subset of B\n(ii) A \u2229 B = \u03c6\n2 A couple has two children,\n(i) Find the probability that both children are males, if it is known that at least\none of the children is male"
    },
    {
        "Chapter": "1",
        "sentence_range": "7719-7722",
        "Text": "A and B are two events such that P (A) \u2260 0 Find P(B|A), if\n(i) A is a subset of B\n(ii) A \u2229 B = \u03c6\n2 A couple has two children,\n(i) Find the probability that both children are males, if it is known that at least\none of the children is male (ii) Find the probability that both children are females, if it is known that the\nelder child is a female"
    },
    {
        "Chapter": "1",
        "sentence_range": "7720-7723",
        "Text": "Find P(B|A), if\n(i) A is a subset of B\n(ii) A \u2229 B = \u03c6\n2 A couple has two children,\n(i) Find the probability that both children are males, if it is known that at least\none of the children is male (ii) Find the probability that both children are females, if it is known that the\nelder child is a female 3"
    },
    {
        "Chapter": "1",
        "sentence_range": "7721-7724",
        "Text": "A couple has two children,\n(i) Find the probability that both children are males, if it is known that at least\none of the children is male (ii) Find the probability that both children are females, if it is known that the\nelder child is a female 3 Suppose that 5% of men and 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7722-7725",
        "Text": "(ii) Find the probability that both children are females, if it is known that the\nelder child is a female 3 Suppose that 5% of men and 0 25% of women have grey hair"
    },
    {
        "Chapter": "1",
        "sentence_range": "7723-7726",
        "Text": "3 Suppose that 5% of men and 0 25% of women have grey hair A grey haired\nperson is selected at random"
    },
    {
        "Chapter": "1",
        "sentence_range": "7724-7727",
        "Text": "Suppose that 5% of men and 0 25% of women have grey hair A grey haired\nperson is selected at random What is the probability of this person being male"
    },
    {
        "Chapter": "1",
        "sentence_range": "7725-7728",
        "Text": "25% of women have grey hair A grey haired\nperson is selected at random What is the probability of this person being male Assume that there are equal number of males and females"
    },
    {
        "Chapter": "1",
        "sentence_range": "7726-7729",
        "Text": "A grey haired\nperson is selected at random What is the probability of this person being male Assume that there are equal number of males and females 4"
    },
    {
        "Chapter": "1",
        "sentence_range": "7727-7730",
        "Text": "What is the probability of this person being male Assume that there are equal number of males and females 4 Suppose that 90% of people are right-handed"
    },
    {
        "Chapter": "1",
        "sentence_range": "7728-7731",
        "Text": "Assume that there are equal number of males and females 4 Suppose that 90% of people are right-handed What  is the probability that\nat most 6 of a random sample of 10 people are right-handed"
    },
    {
        "Chapter": "1",
        "sentence_range": "7729-7732",
        "Text": "4 Suppose that 90% of people are right-handed What  is the probability that\nat most 6 of a random sample of 10 people are right-handed \u00a9 NCERT\nnot to be republished\nPROBABILITY      583\n5"
    },
    {
        "Chapter": "1",
        "sentence_range": "7730-7733",
        "Text": "Suppose that 90% of people are right-handed What  is the probability that\nat most 6 of a random sample of 10 people are right-handed \u00a9 NCERT\nnot to be republished\nPROBABILITY      583\n5 An urn contains 25 balls of which 10 balls bear a mark 'X' and the remaining 15\nbear a mark 'Y'"
    },
    {
        "Chapter": "1",
        "sentence_range": "7731-7734",
        "Text": "What  is the probability that\nat most 6 of a random sample of 10 people are right-handed \u00a9 NCERT\nnot to be republished\nPROBABILITY      583\n5 An urn contains 25 balls of which 10 balls bear a mark 'X' and the remaining 15\nbear a mark 'Y' A ball is drawn at random from the urn, its mark is noted down\nand it is replaced"
    },
    {
        "Chapter": "1",
        "sentence_range": "7732-7735",
        "Text": "\u00a9 NCERT\nnot to be republished\nPROBABILITY      583\n5 An urn contains 25 balls of which 10 balls bear a mark 'X' and the remaining 15\nbear a mark 'Y' A ball is drawn at random from the urn, its mark is noted down\nand it is replaced If 6 balls are drawn in this way, find the probability that\n(i) all will bear 'X' mark"
    },
    {
        "Chapter": "1",
        "sentence_range": "7733-7736",
        "Text": "An urn contains 25 balls of which 10 balls bear a mark 'X' and the remaining 15\nbear a mark 'Y' A ball is drawn at random from the urn, its mark is noted down\nand it is replaced If 6 balls are drawn in this way, find the probability that\n(i) all will bear 'X' mark (ii) not more than 2 will bear 'Y' mark"
    },
    {
        "Chapter": "1",
        "sentence_range": "7734-7737",
        "Text": "A ball is drawn at random from the urn, its mark is noted down\nand it is replaced If 6 balls are drawn in this way, find the probability that\n(i) all will bear 'X' mark (ii) not more than 2 will bear 'Y' mark (iii) at least one ball will bear 'Y' mark"
    },
    {
        "Chapter": "1",
        "sentence_range": "7735-7738",
        "Text": "If 6 balls are drawn in this way, find the probability that\n(i) all will bear 'X' mark (ii) not more than 2 will bear 'Y' mark (iii) at least one ball will bear 'Y' mark (iv) the number of balls with 'X' mark and 'Y' mark will be equal"
    },
    {
        "Chapter": "1",
        "sentence_range": "7736-7739",
        "Text": "(ii) not more than 2 will bear 'Y' mark (iii) at least one ball will bear 'Y' mark (iv) the number of balls with 'X' mark and 'Y' mark will be equal 6"
    },
    {
        "Chapter": "1",
        "sentence_range": "7737-7740",
        "Text": "(iii) at least one ball will bear 'Y' mark (iv) the number of balls with 'X' mark and 'Y' mark will be equal 6 In a hurdle race, a player has to cross 10 hurdles"
    },
    {
        "Chapter": "1",
        "sentence_range": "7738-7741",
        "Text": "(iv) the number of balls with 'X' mark and 'Y' mark will be equal 6 In a hurdle race, a player has to cross 10 hurdles The probability that he will\nclear each hurdle is 5\n6"
    },
    {
        "Chapter": "1",
        "sentence_range": "7739-7742",
        "Text": "6 In a hurdle race, a player has to cross 10 hurdles The probability that he will\nclear each hurdle is 5\n6 What is the probability that he will knock down fewer\nthan 2 hurdles"
    },
    {
        "Chapter": "1",
        "sentence_range": "7740-7743",
        "Text": "In a hurdle race, a player has to cross 10 hurdles The probability that he will\nclear each hurdle is 5\n6 What is the probability that he will knock down fewer\nthan 2 hurdles 7"
    },
    {
        "Chapter": "1",
        "sentence_range": "7741-7744",
        "Text": "The probability that he will\nclear each hurdle is 5\n6 What is the probability that he will knock down fewer\nthan 2 hurdles 7 A die is thrown again and again until three sixes are obtained"
    },
    {
        "Chapter": "1",
        "sentence_range": "7742-7745",
        "Text": "What is the probability that he will knock down fewer\nthan 2 hurdles 7 A die is thrown again and again until three sixes are obtained Find the probabil-\nity of obtaining the third six in the sixth throw of the die"
    },
    {
        "Chapter": "1",
        "sentence_range": "7743-7746",
        "Text": "7 A die is thrown again and again until three sixes are obtained Find the probabil-\nity of obtaining the third six in the sixth throw of the die 8"
    },
    {
        "Chapter": "1",
        "sentence_range": "7744-7747",
        "Text": "A die is thrown again and again until three sixes are obtained Find the probabil-\nity of obtaining the third six in the sixth throw of the die 8 If a leap year is selected at random, what is the chance that it will contain 53\ntuesdays"
    },
    {
        "Chapter": "1",
        "sentence_range": "7745-7748",
        "Text": "Find the probabil-\nity of obtaining the third six in the sixth throw of the die 8 If a leap year is selected at random, what is the chance that it will contain 53\ntuesdays 9"
    },
    {
        "Chapter": "1",
        "sentence_range": "7746-7749",
        "Text": "8 If a leap year is selected at random, what is the chance that it will contain 53\ntuesdays 9 An experiment succeeds twice as often as it fails"
    },
    {
        "Chapter": "1",
        "sentence_range": "7747-7750",
        "Text": "If a leap year is selected at random, what is the chance that it will contain 53\ntuesdays 9 An experiment succeeds twice as often as it fails Find the probability that in the\nnext six trials, there will be atleast 4 successes"
    },
    {
        "Chapter": "1",
        "sentence_range": "7748-7751",
        "Text": "9 An experiment succeeds twice as often as it fails Find the probability that in the\nnext six trials, there will be atleast 4 successes 10"
    },
    {
        "Chapter": "1",
        "sentence_range": "7749-7752",
        "Text": "An experiment succeeds twice as often as it fails Find the probability that in the\nnext six trials, there will be atleast 4 successes 10 How many times must a man toss a fair coin so that the probability of having\nat least one head is more than 90%"
    },
    {
        "Chapter": "1",
        "sentence_range": "7750-7753",
        "Text": "Find the probability that in the\nnext six trials, there will be atleast 4 successes 10 How many times must a man toss a fair coin so that the probability of having\nat least one head is more than 90% 11"
    },
    {
        "Chapter": "1",
        "sentence_range": "7751-7754",
        "Text": "10 How many times must a man toss a fair coin so that the probability of having\nat least one head is more than 90% 11 In a game, a man wins a rupee for a six and loses a rupee for any other number\nwhen a fair die is thrown"
    },
    {
        "Chapter": "1",
        "sentence_range": "7752-7755",
        "Text": "How many times must a man toss a fair coin so that the probability of having\nat least one head is more than 90% 11 In a game, a man wins a rupee for a six and loses a rupee for any other number\nwhen a fair die is thrown The man decided to throw a die thrice but to quit as\nand when he gets a six"
    },
    {
        "Chapter": "1",
        "sentence_range": "7753-7756",
        "Text": "11 In a game, a man wins a rupee for a six and loses a rupee for any other number\nwhen a fair die is thrown The man decided to throw a die thrice but to quit as\nand when he gets a six Find the expected value of the amount he wins / loses"
    },
    {
        "Chapter": "1",
        "sentence_range": "7754-7757",
        "Text": "In a game, a man wins a rupee for a six and loses a rupee for any other number\nwhen a fair die is thrown The man decided to throw a die thrice but to quit as\nand when he gets a six Find the expected value of the amount he wins / loses 12"
    },
    {
        "Chapter": "1",
        "sentence_range": "7755-7758",
        "Text": "The man decided to throw a die thrice but to quit as\nand when he gets a six Find the expected value of the amount he wins / loses 12 Suppose we have four boxes A,B,C and D containing coloured marbles as given\nbelow:\nBox\nMarble colour\nRed\nWhite\nBlack\nA\n1\n6\n3\nB\n6\n2\n2\nC\n8\n1\n1\nD\n0\n6\n4\n   One of the boxes has been selected at random and a single marble is drawn from\nit"
    },
    {
        "Chapter": "1",
        "sentence_range": "7756-7759",
        "Text": "Find the expected value of the amount he wins / loses 12 Suppose we have four boxes A,B,C and D containing coloured marbles as given\nbelow:\nBox\nMarble colour\nRed\nWhite\nBlack\nA\n1\n6\n3\nB\n6\n2\n2\nC\n8\n1\n1\nD\n0\n6\n4\n   One of the boxes has been selected at random and a single marble is drawn from\nit If the marble is red, what is the probability that it was drawn from box A"
    },
    {
        "Chapter": "1",
        "sentence_range": "7757-7760",
        "Text": "12 Suppose we have four boxes A,B,C and D containing coloured marbles as given\nbelow:\nBox\nMarble colour\nRed\nWhite\nBlack\nA\n1\n6\n3\nB\n6\n2\n2\nC\n8\n1\n1\nD\n0\n6\n4\n   One of the boxes has been selected at random and a single marble is drawn from\nit If the marble is red, what is the probability that it was drawn from box A , box B"
    },
    {
        "Chapter": "1",
        "sentence_range": "7758-7761",
        "Text": "Suppose we have four boxes A,B,C and D containing coloured marbles as given\nbelow:\nBox\nMarble colour\nRed\nWhite\nBlack\nA\n1\n6\n3\nB\n6\n2\n2\nC\n8\n1\n1\nD\n0\n6\n4\n   One of the boxes has been selected at random and a single marble is drawn from\nit If the marble is red, what is the probability that it was drawn from box A , box B ,\nbox C"
    },
    {
        "Chapter": "1",
        "sentence_range": "7759-7762",
        "Text": "If the marble is red, what is the probability that it was drawn from box A , box B ,\nbox C \u00a9 NCERT\nnot to be republished\n 584\nMATHEMATICS\n13"
    },
    {
        "Chapter": "1",
        "sentence_range": "7760-7763",
        "Text": ", box B ,\nbox C \u00a9 NCERT\nnot to be republished\n 584\nMATHEMATICS\n13 Assume that the chances of a patient having a heart attack is 40%"
    },
    {
        "Chapter": "1",
        "sentence_range": "7761-7764",
        "Text": ",\nbox C \u00a9 NCERT\nnot to be republished\n 584\nMATHEMATICS\n13 Assume that the chances of a patient having a heart attack is 40% It is also\nassumed that a meditation and yoga course reduce the risk of heart attack by\n30% and prescription of certain drug reduces its chances by 25%"
    },
    {
        "Chapter": "1",
        "sentence_range": "7762-7765",
        "Text": "\u00a9 NCERT\nnot to be republished\n 584\nMATHEMATICS\n13 Assume that the chances of a patient having a heart attack is 40% It is also\nassumed that a meditation and yoga course reduce the risk of heart attack by\n30% and prescription of certain drug reduces its chances by 25% At a time a\npatient can choose any one of the two options with equal probabilities"
    },
    {
        "Chapter": "1",
        "sentence_range": "7763-7766",
        "Text": "Assume that the chances of a patient having a heart attack is 40% It is also\nassumed that a meditation and yoga course reduce the risk of heart attack by\n30% and prescription of certain drug reduces its chances by 25% At a time a\npatient can choose any one of the two options with equal probabilities It is given\nthat after going through one of the two options the patient selected at random\nsuffers a heart attack"
    },
    {
        "Chapter": "1",
        "sentence_range": "7764-7767",
        "Text": "It is also\nassumed that a meditation and yoga course reduce the risk of heart attack by\n30% and prescription of certain drug reduces its chances by 25% At a time a\npatient can choose any one of the two options with equal probabilities It is given\nthat after going through one of the two options the patient selected at random\nsuffers a heart attack Find the probability that the patient followed a course of\nmeditation and yoga"
    },
    {
        "Chapter": "1",
        "sentence_range": "7765-7768",
        "Text": "At a time a\npatient can choose any one of the two options with equal probabilities It is given\nthat after going through one of the two options the patient selected at random\nsuffers a heart attack Find the probability that the patient followed a course of\nmeditation and yoga 14"
    },
    {
        "Chapter": "1",
        "sentence_range": "7766-7769",
        "Text": "It is given\nthat after going through one of the two options the patient selected at random\nsuffers a heart attack Find the probability that the patient followed a course of\nmeditation and yoga 14 If each element of a second order determinant is either zero or one, what is the\nprobability that the value of the determinant is positive"
    },
    {
        "Chapter": "1",
        "sentence_range": "7767-7770",
        "Text": "Find the probability that the patient followed a course of\nmeditation and yoga 14 If each element of a second order determinant is either zero or one, what is the\nprobability that the value of the determinant is positive (Assume that the indi-\nvidual entries of the determinant are chosen independently, each value being\nassumed with probability 1\n2 )"
    },
    {
        "Chapter": "1",
        "sentence_range": "7768-7771",
        "Text": "14 If each element of a second order determinant is either zero or one, what is the\nprobability that the value of the determinant is positive (Assume that the indi-\nvidual entries of the determinant are chosen independently, each value being\nassumed with probability 1\n2 ) 15"
    },
    {
        "Chapter": "1",
        "sentence_range": "7769-7772",
        "Text": "If each element of a second order determinant is either zero or one, what is the\nprobability that the value of the determinant is positive (Assume that the indi-\nvidual entries of the determinant are chosen independently, each value being\nassumed with probability 1\n2 ) 15 An electronic assembly consists of two subsystems, say, A and B"
    },
    {
        "Chapter": "1",
        "sentence_range": "7770-7773",
        "Text": "(Assume that the indi-\nvidual entries of the determinant are chosen independently, each value being\nassumed with probability 1\n2 ) 15 An electronic assembly consists of two subsystems, say, A and B From previ-\nous testing procedures, the following probabilities are assumed to be known:\nP(A fails) = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7771-7774",
        "Text": "15 An electronic assembly consists of two subsystems, say, A and B From previ-\nous testing procedures, the following probabilities are assumed to be known:\nP(A fails) = 0 2\nP(B fails alone) = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7772-7775",
        "Text": "An electronic assembly consists of two subsystems, say, A and B From previ-\nous testing procedures, the following probabilities are assumed to be known:\nP(A fails) = 0 2\nP(B fails alone) = 0 15\nP(A and B fail) = 0"
    },
    {
        "Chapter": "1",
        "sentence_range": "7773-7776",
        "Text": "From previ-\nous testing procedures, the following probabilities are assumed to be known:\nP(A fails) = 0 2\nP(B fails alone) = 0 15\nP(A and B fail) = 0 15\nEvaluate the following probabilities\n(i) P(A fails|B has failed)\n(ii) P(A fails alone)\n16"
    },
    {
        "Chapter": "1",
        "sentence_range": "7774-7777",
        "Text": "2\nP(B fails alone) = 0 15\nP(A and B fail) = 0 15\nEvaluate the following probabilities\n(i) P(A fails|B has failed)\n(ii) P(A fails alone)\n16 Bag I contains 3 red and 4 black balls and Bag II contains 4 red and 5 black balls"
    },
    {
        "Chapter": "1",
        "sentence_range": "7775-7778",
        "Text": "15\nP(A and B fail) = 0 15\nEvaluate the following probabilities\n(i) P(A fails|B has failed)\n(ii) P(A fails alone)\n16 Bag I contains 3 red and 4 black balls and Bag II contains 4 red and 5 black balls One ball is transferred from Bag I to Bag II and then a ball is drawn from Bag II"
    },
    {
        "Chapter": "1",
        "sentence_range": "7776-7779",
        "Text": "15\nEvaluate the following probabilities\n(i) P(A fails|B has failed)\n(ii) P(A fails alone)\n16 Bag I contains 3 red and 4 black balls and Bag II contains 4 red and 5 black balls One ball is transferred from Bag I to Bag II and then a ball is drawn from Bag II The ball so drawn is found to be red in colour"
    },
    {
        "Chapter": "1",
        "sentence_range": "7777-7780",
        "Text": "Bag I contains 3 red and 4 black balls and Bag II contains 4 red and 5 black balls One ball is transferred from Bag I to Bag II and then a ball is drawn from Bag II The ball so drawn is found to be red in colour Find the probability that the\ntransferred ball is black"
    },
    {
        "Chapter": "1",
        "sentence_range": "7778-7781",
        "Text": "One ball is transferred from Bag I to Bag II and then a ball is drawn from Bag II The ball so drawn is found to be red in colour Find the probability that the\ntransferred ball is black Choose the correct answer in each of the following:\n17"
    },
    {
        "Chapter": "1",
        "sentence_range": "7779-7782",
        "Text": "The ball so drawn is found to be red in colour Find the probability that the\ntransferred ball is black Choose the correct answer in each of the following:\n17 If A and B are two events such that P(A) \u2260 0 and P(B | A) = 1, then\n(A) A \u2282 B\n(B) B \u2282 A\n(C) B = \u03c6\n(D) A = \u03c6\n18"
    },
    {
        "Chapter": "1",
        "sentence_range": "7780-7783",
        "Text": "Find the probability that the\ntransferred ball is black Choose the correct answer in each of the following:\n17 If A and B are two events such that P(A) \u2260 0 and P(B | A) = 1, then\n(A) A \u2282 B\n(B) B \u2282 A\n(C) B = \u03c6\n(D) A = \u03c6\n18 If P(A|B) > P(A), then which of the following is correct :\n(A) P(B|A) < P(B)\n(B) P(A \u2229 B) < P(A)"
    },
    {
        "Chapter": "1",
        "sentence_range": "7781-7784",
        "Text": "Choose the correct answer in each of the following:\n17 If A and B are two events such that P(A) \u2260 0 and P(B | A) = 1, then\n(A) A \u2282 B\n(B) B \u2282 A\n(C) B = \u03c6\n(D) A = \u03c6\n18 If P(A|B) > P(A), then which of the following is correct :\n(A) P(B|A) < P(B)\n(B) P(A \u2229 B) < P(A) P(B)\n(C) P(B|A) > P(B)\n(D) P(B|A) = P(B)\n19"
    },
    {
        "Chapter": "1",
        "sentence_range": "7782-7785",
        "Text": "If A and B are two events such that P(A) \u2260 0 and P(B | A) = 1, then\n(A) A \u2282 B\n(B) B \u2282 A\n(C) B = \u03c6\n(D) A = \u03c6\n18 If P(A|B) > P(A), then which of the following is correct :\n(A) P(B|A) < P(B)\n(B) P(A \u2229 B) < P(A) P(B)\n(C) P(B|A) > P(B)\n(D) P(B|A) = P(B)\n19 If A and B are any two events such that P(A) + P(B) \u2013 P(A and B) = P(A), then\n(A) P(B|A) = 1\n(B) P(A|B) = 1\n(C) P(B|A) = 0\n(D) P(A|B) = 0\n\u00a9 NCERT\nnot to be republished\nPROBABILITY      585\nSummary\nThe salient features of the chapter are \u2013\n\ufffd The conditional probability of an event E, given the occurrence of the event F\nis given by \nP(E\nF)\nP(E | F)\nP(F)\n\u2229\n=\n, P(F) \u2260 0\n\ufffd 0 \u2264 P (E|F) \u2264 1,\nP (E\u2032|F) = 1 \u2013 P (E|F)\nP ((E \u222a F)|G) = P (E|G) + P (F|G) \u2013 P ((E \u2229 F)|G)\n\ufffd P (E \u2229 F) = P (E) P (F|E), P (E) \u2260 0\nP (E \u2229 F) = P (F)  P (E|F), P (F) \u2260 0\n\ufffd If E and F are independent, then\nP (E \u2229 F) = P (E) P (F)\nP (E|F) = P (E), P (F) \u2260 0\nP (F|E) = P (F), P(E) \u2260 0\n\ufffd Theorem of total probability\nLet {E1, E2,"
    },
    {
        "Chapter": "1",
        "sentence_range": "7783-7786",
        "Text": "If P(A|B) > P(A), then which of the following is correct :\n(A) P(B|A) < P(B)\n(B) P(A \u2229 B) < P(A) P(B)\n(C) P(B|A) > P(B)\n(D) P(B|A) = P(B)\n19 If A and B are any two events such that P(A) + P(B) \u2013 P(A and B) = P(A), then\n(A) P(B|A) = 1\n(B) P(A|B) = 1\n(C) P(B|A) = 0\n(D) P(A|B) = 0\n\u00a9 NCERT\nnot to be republished\nPROBABILITY      585\nSummary\nThe salient features of the chapter are \u2013\n\ufffd The conditional probability of an event E, given the occurrence of the event F\nis given by \nP(E\nF)\nP(E | F)\nP(F)\n\u2229\n=\n, P(F) \u2260 0\n\ufffd 0 \u2264 P (E|F) \u2264 1,\nP (E\u2032|F) = 1 \u2013 P (E|F)\nP ((E \u222a F)|G) = P (E|G) + P (F|G) \u2013 P ((E \u2229 F)|G)\n\ufffd P (E \u2229 F) = P (E) P (F|E), P (E) \u2260 0\nP (E \u2229 F) = P (F)  P (E|F), P (F) \u2260 0\n\ufffd If E and F are independent, then\nP (E \u2229 F) = P (E) P (F)\nP (E|F) = P (E), P (F) \u2260 0\nP (F|E) = P (F), P(E) \u2260 0\n\ufffd Theorem of total probability\nLet {E1, E2, ,En) be a partition of a sample space and suppose that each of\nE1, E2,"
    },
    {
        "Chapter": "1",
        "sentence_range": "7784-7787",
        "Text": "P(B)\n(C) P(B|A) > P(B)\n(D) P(B|A) = P(B)\n19 If A and B are any two events such that P(A) + P(B) \u2013 P(A and B) = P(A), then\n(A) P(B|A) = 1\n(B) P(A|B) = 1\n(C) P(B|A) = 0\n(D) P(A|B) = 0\n\u00a9 NCERT\nnot to be republished\nPROBABILITY      585\nSummary\nThe salient features of the chapter are \u2013\n\ufffd The conditional probability of an event E, given the occurrence of the event F\nis given by \nP(E\nF)\nP(E | F)\nP(F)\n\u2229\n=\n, P(F) \u2260 0\n\ufffd 0 \u2264 P (E|F) \u2264 1,\nP (E\u2032|F) = 1 \u2013 P (E|F)\nP ((E \u222a F)|G) = P (E|G) + P (F|G) \u2013 P ((E \u2229 F)|G)\n\ufffd P (E \u2229 F) = P (E) P (F|E), P (E) \u2260 0\nP (E \u2229 F) = P (F)  P (E|F), P (F) \u2260 0\n\ufffd If E and F are independent, then\nP (E \u2229 F) = P (E) P (F)\nP (E|F) = P (E), P (F) \u2260 0\nP (F|E) = P (F), P(E) \u2260 0\n\ufffd Theorem of total probability\nLet {E1, E2, ,En) be a partition of a sample space and suppose that each of\nE1, E2, , En has nonzero probability"
    },
    {
        "Chapter": "1",
        "sentence_range": "7785-7788",
        "Text": "If A and B are any two events such that P(A) + P(B) \u2013 P(A and B) = P(A), then\n(A) P(B|A) = 1\n(B) P(A|B) = 1\n(C) P(B|A) = 0\n(D) P(A|B) = 0\n\u00a9 NCERT\nnot to be republished\nPROBABILITY      585\nSummary\nThe salient features of the chapter are \u2013\n\ufffd The conditional probability of an event E, given the occurrence of the event F\nis given by \nP(E\nF)\nP(E | F)\nP(F)\n\u2229\n=\n, P(F) \u2260 0\n\ufffd 0 \u2264 P (E|F) \u2264 1,\nP (E\u2032|F) = 1 \u2013 P (E|F)\nP ((E \u222a F)|G) = P (E|G) + P (F|G) \u2013 P ((E \u2229 F)|G)\n\ufffd P (E \u2229 F) = P (E) P (F|E), P (E) \u2260 0\nP (E \u2229 F) = P (F)  P (E|F), P (F) \u2260 0\n\ufffd If E and F are independent, then\nP (E \u2229 F) = P (E) P (F)\nP (E|F) = P (E), P (F) \u2260 0\nP (F|E) = P (F), P(E) \u2260 0\n\ufffd Theorem of total probability\nLet {E1, E2, ,En) be a partition of a sample space and suppose that each of\nE1, E2, , En has nonzero probability Let A be any event associated with S,\nthen\nP(A) = P(E1) P (A|E1) + P (E2) P (A|E2) +"
    },
    {
        "Chapter": "1",
        "sentence_range": "7786-7789",
        "Text": ",En) be a partition of a sample space and suppose that each of\nE1, E2, , En has nonzero probability Let A be any event associated with S,\nthen\nP(A) = P(E1) P (A|E1) + P (E2) P (A|E2) + + P (En) P(A|En)\n\ufffd Bayes' theorem If E1, E2,"
    },
    {
        "Chapter": "1",
        "sentence_range": "7787-7790",
        "Text": ", En has nonzero probability Let A be any event associated with S,\nthen\nP(A) = P(E1) P (A|E1) + P (E2) P (A|E2) + + P (En) P(A|En)\n\ufffd Bayes' theorem If E1, E2, , En are events which constitute a partition of\nsample space S, i"
    },
    {
        "Chapter": "1",
        "sentence_range": "7788-7791",
        "Text": "Let A be any event associated with S,\nthen\nP(A) = P(E1) P (A|E1) + P (E2) P (A|E2) + + P (En) P(A|En)\n\ufffd Bayes' theorem If E1, E2, , En are events which constitute a partition of\nsample space S, i e"
    },
    {
        "Chapter": "1",
        "sentence_range": "7789-7792",
        "Text": "+ P (En) P(A|En)\n\ufffd Bayes' theorem If E1, E2, , En are events which constitute a partition of\nsample space S, i e E1, E2,"
    },
    {
        "Chapter": "1",
        "sentence_range": "7790-7793",
        "Text": ", En are events which constitute a partition of\nsample space S, i e E1, E2, , En are pairwise disjoint and E1 4 E2 4"
    },
    {
        "Chapter": "1",
        "sentence_range": "7791-7794",
        "Text": "e E1, E2, , En are pairwise disjoint and E1 4 E2 4 4 En = S\nand A be any event with nonzero probability, then\ni\ni\n1\nP(E )P(A|E )\nP(E | A)\nP(E )P(A|E )\ni\nn\nj\nj\nj \n \n \n\ufffd A random variable is a real valued function whose domain is the sample\nspace of a random experiment"
    },
    {
        "Chapter": "1",
        "sentence_range": "7792-7795",
        "Text": "E1, E2, , En are pairwise disjoint and E1 4 E2 4 4 En = S\nand A be any event with nonzero probability, then\ni\ni\n1\nP(E )P(A|E )\nP(E | A)\nP(E )P(A|E )\ni\nn\nj\nj\nj \n \n \n\ufffd A random variable is a real valued function whose domain is the sample\nspace of a random experiment \ufffd The probability distribution of a random variable X is the system of numbers\nX\n:\nx1\nx2"
    },
    {
        "Chapter": "1",
        "sentence_range": "7793-7796",
        "Text": ", En are pairwise disjoint and E1 4 E2 4 4 En = S\nand A be any event with nonzero probability, then\ni\ni\n1\nP(E )P(A|E )\nP(E | A)\nP(E )P(A|E )\ni\nn\nj\nj\nj \n \n \n\ufffd A random variable is a real valued function whose domain is the sample\nspace of a random experiment \ufffd The probability distribution of a random variable X is the system of numbers\nX\n:\nx1\nx2 xn\nP(X)\n:\np 1\np 2"
    },
    {
        "Chapter": "1",
        "sentence_range": "7794-7797",
        "Text": "4 En = S\nand A be any event with nonzero probability, then\ni\ni\n1\nP(E )P(A|E )\nP(E | A)\nP(E )P(A|E )\ni\nn\nj\nj\nj \n \n \n\ufffd A random variable is a real valued function whose domain is the sample\nspace of a random experiment \ufffd The probability distribution of a random variable X is the system of numbers\nX\n:\nx1\nx2 xn\nP(X)\n:\np 1\np 2 p n\nwhere,\n1\n0,\n1,\n1, 2,"
    },
    {
        "Chapter": "1",
        "sentence_range": "7795-7798",
        "Text": "\ufffd The probability distribution of a random variable X is the system of numbers\nX\n:\nx1\nx2 xn\nP(X)\n:\np 1\np 2 p n\nwhere,\n1\n0,\n1,\n1, 2, ,\nn\ni\ni\ni\np\np\ni\nn\n=\n>\n=\n=\n\u2211\n\u00a9 NCERT\nnot to be republished\n 586\nMATHEMATICS\n\ufffd Let X be a random variable whose possible values x1, x2, x3,"
    },
    {
        "Chapter": "1",
        "sentence_range": "7796-7799",
        "Text": "xn\nP(X)\n:\np 1\np 2 p n\nwhere,\n1\n0,\n1,\n1, 2, ,\nn\ni\ni\ni\np\np\ni\nn\n=\n>\n=\n=\n\u2211\n\u00a9 NCERT\nnot to be republished\n 586\nMATHEMATICS\n\ufffd Let X be a random variable whose possible values x1, x2, x3, , xn occur with\nprobabilities p1,  p2, p3,"
    },
    {
        "Chapter": "1",
        "sentence_range": "7797-7800",
        "Text": "p n\nwhere,\n1\n0,\n1,\n1, 2, ,\nn\ni\ni\ni\np\np\ni\nn\n=\n>\n=\n=\n\u2211\n\u00a9 NCERT\nnot to be republished\n 586\nMATHEMATICS\n\ufffd Let X be a random variable whose possible values x1, x2, x3, , xn occur with\nprobabilities p1,  p2, p3, pn respectively"
    },
    {
        "Chapter": "1",
        "sentence_range": "7798-7801",
        "Text": ",\nn\ni\ni\ni\np\np\ni\nn\n=\n>\n=\n=\n\u2211\n\u00a9 NCERT\nnot to be republished\n 586\nMATHEMATICS\n\ufffd Let X be a random variable whose possible values x1, x2, x3, , xn occur with\nprobabilities p1,  p2, p3, pn respectively The mean of X, denoted by \u03bc, is\nthe number \n1\nn\ni\ni\ni\nx p"
    },
    {
        "Chapter": "1",
        "sentence_range": "7799-7802",
        "Text": ", xn occur with\nprobabilities p1,  p2, p3, pn respectively The mean of X, denoted by \u03bc, is\nthe number \n1\nn\ni\ni\ni\nx p The mean of a random variable X is also called the expectation of X, denoted\nby E (X)"
    },
    {
        "Chapter": "1",
        "sentence_range": "7800-7803",
        "Text": "pn respectively The mean of X, denoted by \u03bc, is\nthe number \n1\nn\ni\ni\ni\nx p The mean of a random variable X is also called the expectation of X, denoted\nby E (X) \ufffd Let X be a random variable whose possible values x1, x2,"
    },
    {
        "Chapter": "1",
        "sentence_range": "7801-7804",
        "Text": "The mean of X, denoted by \u03bc, is\nthe number \n1\nn\ni\ni\ni\nx p The mean of a random variable X is also called the expectation of X, denoted\nby E (X) \ufffd Let X be a random variable whose possible values x1, x2, , xn occur with\nprobabilities p(x1), p(x2),"
    },
    {
        "Chapter": "1",
        "sentence_range": "7802-7805",
        "Text": "The mean of a random variable X is also called the expectation of X, denoted\nby E (X) \ufffd Let X be a random variable whose possible values x1, x2, , xn occur with\nprobabilities p(x1), p(x2), , p(xn) respectively"
    },
    {
        "Chapter": "1",
        "sentence_range": "7803-7806",
        "Text": "\ufffd Let X be a random variable whose possible values x1, x2, , xn occur with\nprobabilities p(x1), p(x2), , p(xn) respectively Let \u03bc = E(X) be the mean of X"
    },
    {
        "Chapter": "1",
        "sentence_range": "7804-7807",
        "Text": ", xn occur with\nprobabilities p(x1), p(x2), , p(xn) respectively Let \u03bc = E(X) be the mean of X The variance of X, denoted by Var (X) or\n\u03c3x\n2, is defined as \n2\n2\n1\nVar(X)=\n(\n\u03bc)\n(\n)\nn\nx\ni\ni\ni\nx\np x\n \n \n \n \n \nor equivalently  \u03c3x\n2 = E (X \u2013 \u03bc)2\nThe non-negative number\n2\n1\nVa r(X) =\n(\n\u03bc)\n(\n)\nn\nx\ni\ni\ni\nx\np x\n \n \n \n \n \nis called the standard deviation of the  random variable X"
    },
    {
        "Chapter": "1",
        "sentence_range": "7805-7808",
        "Text": ", p(xn) respectively Let \u03bc = E(X) be the mean of X The variance of X, denoted by Var (X) or\n\u03c3x\n2, is defined as \n2\n2\n1\nVar(X)=\n(\n\u03bc)\n(\n)\nn\nx\ni\ni\ni\nx\np x\n \n \n \n \n \nor equivalently  \u03c3x\n2 = E (X \u2013 \u03bc)2\nThe non-negative number\n2\n1\nVa r(X) =\n(\n\u03bc)\n(\n)\nn\nx\ni\ni\ni\nx\np x\n \n \n \n \n \nis called the standard deviation of the  random variable X \ufffd Var (X) = E (X2) \u2013 [E(X)]2\n\ufffd Trials of a random experiment are called Bernoulli trials, if they satisfy the\nfollowing conditions :\n(i) There should be a finite number of trials"
    },
    {
        "Chapter": "1",
        "sentence_range": "7806-7809",
        "Text": "Let \u03bc = E(X) be the mean of X The variance of X, denoted by Var (X) or\n\u03c3x\n2, is defined as \n2\n2\n1\nVar(X)=\n(\n\u03bc)\n(\n)\nn\nx\ni\ni\ni\nx\np x\n \n \n \n \n \nor equivalently  \u03c3x\n2 = E (X \u2013 \u03bc)2\nThe non-negative number\n2\n1\nVa r(X) =\n(\n\u03bc)\n(\n)\nn\nx\ni\ni\ni\nx\np x\n \n \n \n \n \nis called the standard deviation of the  random variable X \ufffd Var (X) = E (X2) \u2013 [E(X)]2\n\ufffd Trials of a random experiment are called Bernoulli trials, if they satisfy the\nfollowing conditions :\n(i) There should be a finite number of trials (ii) The trials should be independent"
    },
    {
        "Chapter": "1",
        "sentence_range": "7807-7810",
        "Text": "The variance of X, denoted by Var (X) or\n\u03c3x\n2, is defined as \n2\n2\n1\nVar(X)=\n(\n\u03bc)\n(\n)\nn\nx\ni\ni\ni\nx\np x\n \n \n \n \n \nor equivalently  \u03c3x\n2 = E (X \u2013 \u03bc)2\nThe non-negative number\n2\n1\nVa r(X) =\n(\n\u03bc)\n(\n)\nn\nx\ni\ni\ni\nx\np x\n \n \n \n \n \nis called the standard deviation of the  random variable X \ufffd Var (X) = E (X2) \u2013 [E(X)]2\n\ufffd Trials of a random experiment are called Bernoulli trials, if they satisfy the\nfollowing conditions :\n(i) There should be a finite number of trials (ii) The trials should be independent (iii) Each trial has exactly two outcomes : success or failure"
    },
    {
        "Chapter": "1",
        "sentence_range": "7808-7811",
        "Text": "\ufffd Var (X) = E (X2) \u2013 [E(X)]2\n\ufffd Trials of a random experiment are called Bernoulli trials, if they satisfy the\nfollowing conditions :\n(i) There should be a finite number of trials (ii) The trials should be independent (iii) Each trial has exactly two outcomes : success or failure (iv) The probability of success remains the same in each trial"
    },
    {
        "Chapter": "1",
        "sentence_range": "7809-7812",
        "Text": "(ii) The trials should be independent (iii) Each trial has exactly two outcomes : success or failure (iv) The probability of success remains the same in each trial For Binomial distribution B (n, p), P (X = x) = nCx q n\u2013x px, x = 0,  1,"
    },
    {
        "Chapter": "1",
        "sentence_range": "7810-7813",
        "Text": "(iii) Each trial has exactly two outcomes : success or failure (iv) The probability of success remains the same in each trial For Binomial distribution B (n, p), P (X = x) = nCx q n\u2013x px, x = 0,  1, , n\n(q = 1 \u2013 p)\n Historical Note\nThe earliest indication on measurement of chances in game of dice appeared\nin 1477 in a commentary on Dante's Divine Comedy"
    },
    {
        "Chapter": "1",
        "sentence_range": "7811-7814",
        "Text": "(iv) The probability of success remains the same in each trial For Binomial distribution B (n, p), P (X = x) = nCx q n\u2013x px, x = 0,  1, , n\n(q = 1 \u2013 p)\n Historical Note\nThe earliest indication on measurement of chances in game of dice appeared\nin 1477 in a commentary on Dante's Divine Comedy A treatise on gambling\nnamed liber de Ludo Alcae, by Geronimo Carden (1501-1576) was published\nposthumously in 1663"
    },
    {
        "Chapter": "1",
        "sentence_range": "7812-7815",
        "Text": "For Binomial distribution B (n, p), P (X = x) = nCx q n\u2013x px, x = 0,  1, , n\n(q = 1 \u2013 p)\n Historical Note\nThe earliest indication on measurement of chances in game of dice appeared\nin 1477 in a commentary on Dante's Divine Comedy A treatise on gambling\nnamed liber de Ludo Alcae, by Geronimo Carden (1501-1576) was published\nposthumously in 1663 In this treatise, he gives the number of favourable cases\nfor each event when two dice are thrown"
    },
    {
        "Chapter": "1",
        "sentence_range": "7813-7816",
        "Text": ", n\n(q = 1 \u2013 p)\n Historical Note\nThe earliest indication on measurement of chances in game of dice appeared\nin 1477 in a commentary on Dante's Divine Comedy A treatise on gambling\nnamed liber de Ludo Alcae, by Geronimo Carden (1501-1576) was published\nposthumously in 1663 In this treatise, he gives the number of favourable cases\nfor each event when two dice are thrown \u00a9 NCERT\nnot to be republished\nPROBABILITY      587\nGalileo (1564-1642) gave casual remarks concerning the correct evaluation\nof chance in a game of three dice"
    },
    {
        "Chapter": "1",
        "sentence_range": "7814-7817",
        "Text": "A treatise on gambling\nnamed liber de Ludo Alcae, by Geronimo Carden (1501-1576) was published\nposthumously in 1663 In this treatise, he gives the number of favourable cases\nfor each event when two dice are thrown \u00a9 NCERT\nnot to be republished\nPROBABILITY      587\nGalileo (1564-1642) gave casual remarks concerning the correct evaluation\nof chance in a game of three dice Galileo analysed that when three dice are\nthrown, the sum of the number that appear is more likely to be 10 than the sum 9,\nbecause the number of cases favourable to 10 are more than the number of\ncases for the appearance of number 9"
    },
    {
        "Chapter": "1",
        "sentence_range": "7815-7818",
        "Text": "In this treatise, he gives the number of favourable cases\nfor each event when two dice are thrown \u00a9 NCERT\nnot to be republished\nPROBABILITY      587\nGalileo (1564-1642) gave casual remarks concerning the correct evaluation\nof chance in a game of three dice Galileo analysed that when three dice are\nthrown, the sum of the number that appear is more likely to be 10 than the sum 9,\nbecause the number of cases favourable to 10 are more than the number of\ncases for the appearance of number 9 Apart from these early contributions, it is generally acknowledged that the\ntrue origin of the science of probability lies in the correspondence between two\ngreat men of the seventeenth century, Pascal (1623-1662) and Pierre de Fermat\n(1601-1665)"
    },
    {
        "Chapter": "1",
        "sentence_range": "7816-7819",
        "Text": "\u00a9 NCERT\nnot to be republished\nPROBABILITY      587\nGalileo (1564-1642) gave casual remarks concerning the correct evaluation\nof chance in a game of three dice Galileo analysed that when three dice are\nthrown, the sum of the number that appear is more likely to be 10 than the sum 9,\nbecause the number of cases favourable to 10 are more than the number of\ncases for the appearance of number 9 Apart from these early contributions, it is generally acknowledged that the\ntrue origin of the science of probability lies in the correspondence between two\ngreat men of the seventeenth century, Pascal (1623-1662) and Pierre de Fermat\n(1601-1665) A French gambler, Chevalier de Metre asked Pascal to explain\nsome seeming contradiction between his theoretical reasoning and the\nobservation gathered from gambling"
    },
    {
        "Chapter": "1",
        "sentence_range": "7817-7820",
        "Text": "Galileo analysed that when three dice are\nthrown, the sum of the number that appear is more likely to be 10 than the sum 9,\nbecause the number of cases favourable to 10 are more than the number of\ncases for the appearance of number 9 Apart from these early contributions, it is generally acknowledged that the\ntrue origin of the science of probability lies in the correspondence between two\ngreat men of the seventeenth century, Pascal (1623-1662) and Pierre de Fermat\n(1601-1665) A French gambler, Chevalier de Metre asked Pascal to explain\nsome seeming contradiction between his theoretical reasoning and the\nobservation gathered from gambling In a series of letters written around 1654,\nPascal and Fermat laid the first foundation of science of probability"
    },
    {
        "Chapter": "1",
        "sentence_range": "7818-7821",
        "Text": "Apart from these early contributions, it is generally acknowledged that the\ntrue origin of the science of probability lies in the correspondence between two\ngreat men of the seventeenth century, Pascal (1623-1662) and Pierre de Fermat\n(1601-1665) A French gambler, Chevalier de Metre asked Pascal to explain\nsome seeming contradiction between his theoretical reasoning and the\nobservation gathered from gambling In a series of letters written around 1654,\nPascal and Fermat laid the first foundation of science of probability Pascal solved\nthe problem in algebraic manner while Fermat used the method of combinations"
    },
    {
        "Chapter": "1",
        "sentence_range": "7819-7822",
        "Text": "A French gambler, Chevalier de Metre asked Pascal to explain\nsome seeming contradiction between his theoretical reasoning and the\nobservation gathered from gambling In a series of letters written around 1654,\nPascal and Fermat laid the first foundation of science of probability Pascal solved\nthe problem in algebraic manner while Fermat used the method of combinations Great Dutch Scientist, Huygens (1629-1695), became acquainted with the\ncontent of the correspondence between Pascal and Fermat and published a first\nbook on probability, \"De Ratiociniis in Ludo Aleae\" containing solution of many\ninteresting rather than difficult problems on probability in games of chances"
    },
    {
        "Chapter": "1",
        "sentence_range": "7820-7823",
        "Text": "In a series of letters written around 1654,\nPascal and Fermat laid the first foundation of science of probability Pascal solved\nthe problem in algebraic manner while Fermat used the method of combinations Great Dutch Scientist, Huygens (1629-1695), became acquainted with the\ncontent of the correspondence between Pascal and Fermat and published a first\nbook on probability, \"De Ratiociniis in Ludo Aleae\" containing solution of many\ninteresting rather than difficult problems on probability in games of chances The next great work on probability theory is by Jacob Bernoulli (1654-1705),\nin the form of a great book, \"Ars Conjectendi\" published posthumously in 1713\nby his nephew, Nicholes Bernoulli"
    },
    {
        "Chapter": "1",
        "sentence_range": "7821-7824",
        "Text": "Pascal solved\nthe problem in algebraic manner while Fermat used the method of combinations Great Dutch Scientist, Huygens (1629-1695), became acquainted with the\ncontent of the correspondence between Pascal and Fermat and published a first\nbook on probability, \"De Ratiociniis in Ludo Aleae\" containing solution of many\ninteresting rather than difficult problems on probability in games of chances The next great work on probability theory is by Jacob Bernoulli (1654-1705),\nin the form of a great book, \"Ars Conjectendi\" published posthumously in 1713\nby his nephew, Nicholes Bernoulli To him is due the discovery of one of the most\nimportant probability distribution known as Binomial distribution"
    },
    {
        "Chapter": "1",
        "sentence_range": "7822-7825",
        "Text": "Great Dutch Scientist, Huygens (1629-1695), became acquainted with the\ncontent of the correspondence between Pascal and Fermat and published a first\nbook on probability, \"De Ratiociniis in Ludo Aleae\" containing solution of many\ninteresting rather than difficult problems on probability in games of chances The next great work on probability theory is by Jacob Bernoulli (1654-1705),\nin the form of a great book, \"Ars Conjectendi\" published posthumously in 1713\nby his nephew, Nicholes Bernoulli To him is due the discovery of one of the most\nimportant probability distribution known as Binomial distribution The next\nremarkable work on probability lies in 1993"
    },
    {
        "Chapter": "1",
        "sentence_range": "7823-7826",
        "Text": "The next great work on probability theory is by Jacob Bernoulli (1654-1705),\nin the form of a great book, \"Ars Conjectendi\" published posthumously in 1713\nby his nephew, Nicholes Bernoulli To him is due the discovery of one of the most\nimportant probability distribution known as Binomial distribution The next\nremarkable work on probability lies in 1993 A"
    },
    {
        "Chapter": "1",
        "sentence_range": "7824-7827",
        "Text": "To him is due the discovery of one of the most\nimportant probability distribution known as Binomial distribution The next\nremarkable work on probability lies in 1993 A N"
    },
    {
        "Chapter": "1",
        "sentence_range": "7825-7828",
        "Text": "The next\nremarkable work on probability lies in 1993 A N Kolmogorov (1903-1987) is\ncredited with the axiomatic theory of probability"
    },
    {
        "Chapter": "1",
        "sentence_range": "7826-7829",
        "Text": "A N Kolmogorov (1903-1987) is\ncredited with the axiomatic theory of probability His book, \u2018Foundations of\nprobability\u2019 published in 1933, introduces probability as a set function and is\nconsidered a \u2018classic"
    },
    {
        "Chapter": "1",
        "sentence_range": "7827-7830",
        "Text": "N Kolmogorov (1903-1987) is\ncredited with the axiomatic theory of probability His book, \u2018Foundations of\nprobability\u2019 published in 1933, introduces probability as a set function and is\nconsidered a \u2018classic \u2019"
    },
    {
        "Chapter": "1",
        "sentence_range": "7828-7831",
        "Text": "Kolmogorov (1903-1987) is\ncredited with the axiomatic theory of probability His book, \u2018Foundations of\nprobability\u2019 published in 1933, introduces probability as a set function and is\nconsidered a \u2018classic \u2019 \u2014\ufffd\n\ufffd\n\ufffd\n\ufffd\n\ufffd\u2014\n\u00a9 NCERT\nnot to be republished"
    }
]