3. If A and , then 4. If A and , then 5. If A and then 6. If A, then B is called as super set of A. It is denoted as 7. If A and Then A is called as proper subset of B. 8. If then total number of subsets of A = and number of proper subsets is -1 Finite Set: Set A is said to be finite if it contains countable number of elements Finite sets are sets that have a finite number of members. If the elements of a finite set are listed one after another, the process will eventually “run out” of elements to list. Example: A = {0, 2, 4, 6, 8, …, 100} C = {x : x is an integer, 1 < x < 10} Infinite Set: Set A is said to be infinite if it contains uncountable number of elements. Eg: and N={1, 2, 3, …..} An infinite set is a set which is not finite. It is not possible to explicitly list out all the elements of an infinite set. Example: T = {x : x is a triangle} N is the set of natural numbers A is the set of fractions Equal sets: Two sets A and B are said to be equal if A and B have the same element . A = , B = Equivalent sets: Two sets A and B are said to be equivalent if n(A) = n(B) A={1,2,3} B={a,b,c} clearly n(A) = n(B)=3 Universal set: A set which is superset of all sets under consideration is called universal set .It is denoted by U. A={1,2,3} B={2,3,5} C={1,4,6}
Module 3 : Principles of Mathematical Induction Introduction: The principle of mathematical induction is a technique to verify the validity of a pre-derived formula or statement in terms of natural number n. Principles of Mathematical Induction: Suppose is a given statement P(n), where n is a natural number, such that P(n) is true for n=1, i.e. P(1) is true, and If P(n) is true for n = k (where k is some positive integer), i.e. if P(k) is true, then P(n) is also true for n= k+1, i.e. the truth of P(k) implies the truth of P(k+1), then the statement P(n) is true for all nN. Exercises for Practice Prove the following by using mathematical induction, for all + = Level-II Prove the following by using mathematical induction, for all
Module 5 (Linear Inequalities) LINEAR INEQUALITIES for solving Linear Inequalities The following rules can be applied to any inequality Add or subtract the same n umber or expressions to both the sides. Multiply or divide both the sides by same positive number The inequality reverses if it is multiplied or divide by a negative number a < b b > a a < b 1/a > 1/b x2 ≤ a2 x ≤ a and x ≥ -a Points to remember x = 0 is y-axis y = 0 is x-axis x = k is a line parallel to y-axis passing through (k, 0) of x-axis. y = k is a line parallel to x-axis passing through (0, k) of y-axis. Method to find Graphical Solution Draw lines corresponding to each equation treating it as equality. Find the feasible region: intersection of all the inequalities. LEVEL – I Solve 24x < 100 when x is a natural number. Solve the inequalities and draw the graph of the solution on number line: 3x – 2 < 2x + 1. Solve the following system of inequalities: 2x + y ≥ 8, x + 2y ≥ 10. the following system of inequalities graphically: x ≥ 3, y ≥ 2. the following system of inequalities graphically: x + y ≤ 9, y>x, x≥0. Find all of consecutive odd positive integers both of which are smaller than 10 such that there sum is more than 11. LEVEL – II the inequalities and draw the graph of the solution on number line: 5(2x – 7) – 3(2x + 3) ≤ 0. the following system of inequalities graphically: x + y ≤ 400, 2x + y ≤ 600.
Solve the following system of inequalities graphically: 2x + 19 ≤ 6x + 47. The longest side of a triangle is 3 times the shortest side and the third side is 2 cm shorter than the longest side. If the perimeter of the triangle is at least 61 cm, find the minimum length of the shortest side. LEVEL – III Solve the following system of inequalities: x ≥ 3, y ≥ 2. the following system of inequalities: 2x + y ≥ 3, 3x + 5y ≥ 2. Solve the following system of inequalities: 3x – 7 < 5 + x, 11 -5x ≤ 1. How many litres of water will have to be added to 1125 litres of 45% solution of acid so that the resulting mixture will contain more than 25% but less than 30% of acid content? LEVEL – IV Solve the following system of inequalities: 3( 1 – x) < 2(x + 4) Solve the following system of inequalities: x + y ≥ 1, x ≤ 5, y≤ 4 the following system of inequalities: > A manufacturer has 600 litres of a 12% solution of acid. How many litres of a 30% acid solution must be added to it so that acid content in the resulting mixture will be more than 15% but less than 18%?
Time: 3 hours Max Marks: 100 Sl No. Topics Very Answer (1 Short Answer (4 marks) Long Answer (6 marks) Total Marks Unit I 1. Sets 1(1) 4(1) 6(1)+V 11(3) 2. Relations and Functions _ 4(2)+V _ 8(2) 3. Trigonometry 1(2) 4(1) 6(2) 18(5) Unit II 4. Principle of Mathematical Induction _ _ 6(1) 6(1) 5. Complex Numbers and Quadratic Equations 1(1) 4(2) _ 9(3) 6. Linear Inequalities _ 4(1)+V 6(1) 10(2) 7. Permutations and Combinations _ 4(3)+V _ 12(3) 8. Binomial Theorem _ 4(1) 6(1) 10(2) 9. Sequences and Series 1(2) 4(2) 6(1) 16(5) Questions 1(6) 4(13) 6(7) 100(26) Very easy-30% Easy-40% Difficult-30% Unit I (37 Unit II (63marks) KENDRIYA VIDYALAYA SANGATHAN, RAIPUR REGION BLUE PRINT Sub:- Mathematics CLASS-XI Session Ending Examination
SL. NO NAME OF THE CHAPTER KNOWLEDGE APPLICATION TOTAL VSA SA LA VSA SA LA VSA SA LA 1 SETS 1(4) 1(6) 10 2 RELATION AND FUNCTIONS 1(4) 4 3 TRIGONOMETRY 1(1) 1(4) 1(4) 1(6) 15 4 PMI 1(6) 6 5 COMPLEX NUMBERS AND QUARDRATIC EQUATIONS 1(1) 1(4) 5 6 LINEAR INEQUALITIES 1(6) 6 7 PERMUTATIONS AND COMBINATIONS 1(4) 1(6) 10 8 BINOMIAL THEOREM 1(1) 1(4) 5 9 SEQUENCE AND SERIES 1(6) 6 10 STRAIGHT LINES 1(1) 1(4) 5 11 CONIC SECTIONS 1(4) 4 12 INTRODUCTION TO 3-D 1(4) 4 13 AND DERIVATIVE 1(4) 1(1) 5 14 MATHEMATICAL REASONING 1(4) 4 15 STATISTICS 1(6) 6 16 PROBABILITY 1(1) 1(4) 5 TOTAL 5(5) 5(20) 1(1)_ 7(28) 1(4) 7(42) 26(100)