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### Instruction: help me answer this question Find the value of sin (1005°)., think step by step ### Assistant: We know that the values of sin x repeat after an interval of 2π or 360°. Therefore, sin (1005°) = sin (1005° - 900°) = sin 105° = sin (90° + 15°) = -cos 15°. Hence, sin (1005°) = -1/√2. , The answer is The value of sin (1005°) is -1/√2.

### Instruction: help me answer this question Find the sum to 20 terms in the geometric progression 0.15, 0.015, 0.0015 ..., think step by step ### Assistant: , The answer is The sum to 20 terms in the geometric progression 0.15, 0.015, 0.0015 ...

### Instruction: help me answer this question (iii) If the length of a pendulum is 75 cm and the tip describes an arc of length 21 cm, what angle does the pendulum swing in radians?, think step by step ### Assistant: , The answer is θ = 21/75 radian = 7/25 radian

### Instruction: help me answer this question `f(x) = sqrt(x - 1)`, think step by step ### Assistant: , The answer is The domain of the function is `x ≥ 1`.

### Instruction: help me answer this question `f(x) = (x + 3)/(x - 4)`, think step by step ### Assistant: , The answer is The domain of the function is all real numbers except `x = 4`.

### Instruction: help me answer this question Show that the statement p: “If x is a real number such that x^3 + 4x = 0, then x is 0” is true by (i) direct method, think step by step ### Assistant: In this method, we assume the given condition is true and using this assumption if we can prove the result then the statement is true. We assume q is true, which gives us x^3 + 4x = 0. Further, this results in x = 0 or x^2 + 4 = 0. Since x is real, x = 0, making statement r true. Thus, statement p is true. , The answer is The statement p is true by the direct method.

### Instruction: help me answer this question (5.1) A solution of 8% boric acid is to be diluted by adding a 2% boric acid solution to it. The resulting mixture is to be more than 4% but less than 6% boric acid. If we have 640 litres of the 8% solution, how many litres of the 2% solution will have to be added?, think step by step ### Assistant: , The answer is The number of litres of 2% of boric acid solution that has to be added will be more than 320 litres but less than 1280 litres.

### Instruction: help me answer this question `f(x) = sqrt(x - 1)`, think step by step ### Assistant: , The answer is The domain of the function is `x ≥ 1`.

### Instruction: help me answer this question , think step by step ### Assistant: N/A , The answer is N/A

### Instruction: help me answer this question Since, the vertices are (± 7, 0), so, a = 7, think step by step ### Assistant: , The answer is Comparing the given vertices (±7, 0) with the general form of the hyperbola, we find that a = 7.

### Instruction: help me answer this question Checking for the origin (0, 0) 0 ≤ 6 this is true , think step by step ### Assistant: 0 ≤ 6 this is true , The answer is The origin lies in the solution area, and the area would be on the left of the line graph.

### Instruction: help me answer this question `f(x) = 1/(x^2 + 1)`, think step by step ### Assistant: , The answer is The domain of the function is all real numbers.

### Instruction: help me answer this question What is the number of subsets of the set A × B?, think step by step ### Assistant: , The answer is The set A × B has 16 subsets

### Instruction: help me answer this question A coin is tossed, and then a die is rolled only in case a head is shown on the coin. Describe the sample space., think step by step ### Assistant: , The answer is When the head is encountered, the number of possible outcomes equals (1 × 6) = 6. So, the sample space when a head is encountered, SH, is {H1, H2, H3, H4, H5, H6}. If the tail is encountered, the sample space ST = {T}. Hence, the total sample space S = {H1, H2, H3, H4, H5, H6, T}.

### Instruction: help me answer this question Let us make the table of the given data and append other columns after calculations., think step by step ### Assistant: This question is not a typical question with a definitive answer. Instead, it is a instruction for the reader to create a table by calculating mid-points, fixi, (xi – x̅), (xi – x̅)2, and fi(xi – x̅)2 for the given classes and their frequencies. , The answer is The answer is a set of instructions to be performed on the given data.

### Instruction: help me answer this question Given, 2x – 3y > 6, think step by step ### Assistant: , The answer is The solution region of the given inequality is below the line 2x – 3y > 6. (Away from the origin.)

### Instruction: help me answer this question Given x + y ≤ 9, what are the required points?, think step by step ### Assistant: These points are obtained by putting values of x = 0 and y = 0 in the equation one by one. , The answer is The required points are (0, 9) and (9, 0).

### Instruction: help me answer this question `f(x) = sqrt(x - 1)`, think step by step ### Assistant: , The answer is The domain of the function is `x ≥ 1`.

### Instruction: help me answer this question Find the 13th term in the expansion of , think step by step ### Assistant: , The answer is Not provided in the data

### Instruction: help me answer this question Show the ratio of the sum of the first n terms of a G.P. to the sum of terms from (n+1)th to (2n)th term is given by, think step by step ### Assistant: We first find the sum of the first n terms and the sum of terms from (n+1)th to (2n)th term using the formula for the sum of a G.P. Then, expressing the latter sum as a G.P., we show that the given ratio simplifies to r^(n^2). , The answer is The ratio of the sum of the first n terms of a G.P. to the sum of terms from (n+1)th to (2n)th term is given by

### Instruction: help me answer this question We know that n(S) = 21, n(T) = 32, and n(S ∩ T) = 11. Find n(S ∪ T)., think step by step ### Assistant: This is based on the formula n(S ∪ T) = n(S) + n(T) - n(S ∩ T) , The answer is n(S ∪ T) = 21 + 32 - 11 = 42

### Instruction: help me answer this question `f(x) = 1/(x^2 + 1)`, think step by step ### Assistant: , The answer is The domain of the function is all real numbers.

### Instruction: help me answer this question Write the first five terms of the sequence whose nth term is an = n (n + 2), think step by step ### Assistant: , The answer is 3, 8, 15, 24, 35

### Instruction: help me answer this question A: a number less than 7, think step by step ### Assistant: All the numbers in the die are less than 7. , The answer is A = (1, 2, 3, 4, 5, 6)

### Instruction: help me answer this question Are {a, e, i, o, u } and {a, b, c, d} disjoint?, think step by step ### Assistant: {a, e, i, o, u} ∩ {a, b, c, d} = {a} , The answer is Not disjoint

### Instruction: help me answer this question `f(x) = 1/(x^2 + 1)`, think step by step ### Assistant: , The answer is The domain of the function is all real numbers.

### Instruction: help me answer this question `f(x) = 1/(x^2 + 1)`, think step by step ### Assistant: , The answer is The domain of the function is all real numbers.

### Instruction: help me answer this question Find X – Y, think step by step ### Assistant: X – Y represents the set of elements that are present in X but not in Y. , The answer is {a, c}

### Instruction: help me answer this question Let A = {1, 2, 3, ... , 14}. Define a relation R from A to A by R = {(x, y): 3x – y = 0, where x, y ∈ A}. Write down its domain, codomain and range., think step by step ### Assistant: The range of R is {3, 6, 9, 12} , The answer is The relation R from A to A is given as R = {(x, y): 3x – y = 0, where x, y ∈ A} = {(x, y): 3x = y, where x, y ∈ A} So, R = {(1, 3), (2, 6), (3, 9), (4, 12)} Now, The domain of R is the set of all first elements of the ordered pairs in the relation. Hence, Domain of R = {1, 2, 3, 4} The whole set A is the codomain of the relation R. Hence, Codomain of R = A = {1, 2, 3, ..., 14} The range of R is the set of all second elements of the ordered pairs in the relation.

### Instruction: help me answer this question `f(x) = sqrt(x - 1)`, think step by step ### Assistant: , The answer is The domain of the function is `x ≥ 1`.

### Instruction: help me answer this question Find the value of: (i) sin 75o, (ii) tan 15o, think step by step ### Assistant: N/A , The answer is {'part1': {'subpart': '(i)', 'content': 'sin 75o = sin (45o + 30o) = sin 45o cos 30o + cos 45o sin 30o = √3 + 1 / 2√2'}, 'part2': {'subpart': '(ii)', 'content': 'tan 15o can be written as tan (45o – 30o) = (tan 45o – tan 30o) / (1 + (tan 45o * tan 30o)) = √3 / (√3 + 1)'}}

### Instruction: help me answer this question What can you say about the given triangle?, think step by step ### Assistant: , The answer is The given triangle is right-angled at A (4, 4).

### Instruction: help me answer this question Find the derivative of (vi) (2 / x + 1) – x2 / 3x – 1, think step by step ### Assistant: , The answer is The derivative of (2 / x + 1) – x2 / 3x – 1 is not provided.

### Instruction: help me answer this question What are the coordinates of the vertices?, think step by step ### Assistant: , The answer is The coordinates of the vertices are (0, 5) and (0, -5)

### Instruction: help me answer this question (ii) (a, b) ∈ R, implies (b, a) ∈ R, think step by step ### Assistant: , The answer is No, it is not true that (a, b) ∈ R, implies (b, a) ∈ R.

### Instruction: help me answer this question `f(x) = (x + 3)/(x - 4)`, think step by step ### Assistant: , The answer is The domain of the function is all real numbers except `x = 4`.

### Instruction: help me answer this question If the sum of the first p terms of an A.P. is equal to the sum of the first q terms, then find the sum of the first (p + q) terms., think step by step ### Assistant: Explanation provided in the markdown text outlines the steps to derive the answer. It includes the sum of p and q terms of the A.P. and their simplification leading to the relationship between p and q, and finally the sum of (p + q) terms. , The answer is The sum of (p + q) terms of the A.P. is 0.

### Instruction: help me answer this question Which of the following cannot be a valid assignment of probabilities for outcomes of sample Space S = {ω1, ω2, ω3, ω4, ω5, ω6, ω7}?, think step by step ### Assistant: The axiomatic approach to probability requires that each probability assigned to the outcomes be non-negative and the sum of all the probabilities of all outcomes should be equal to 1. , The answer is

### Instruction: help me answer this question `f(x) = 1/(x^2 + 1)`, think step by step ### Assistant: , The answer is The domain of the function is all real numbers.

### Instruction: help me answer this question `f(x) = (x + 3)/(x - 4)`, think step by step ### Assistant: , The answer is The domain of the function is all real numbers except `x = 4`.

### Instruction: help me answer this question Find the multiplicative inverse of the given number: -21, think step by step ### Assistant: , The answer is The multiplicative inverse of -21 is -1/21.

### Instruction: help me answer this question The sample space if we are interested in the number of girls in the family when there are two children in the family, then Sample space S = {2, 1, 0}., think step by step ### Assistant: N/A , The answer is S = {2, 1, 0}

### Instruction: help me answer this question Given, y + 8 ≥ 2x, think step by step ### Assistant: , The answer is The solution region of the given inequality is above the line y + 8 = 2x. (That is, the origin is included in the region.)

### Instruction: help me answer this question Is it an order?, think step by step ### Assistant: None. , The answer is No, it is not an order.

### Instruction: help me answer this question `f(x) = 1/(x^2 + 1)`, think step by step ### Assistant: , The answer is The domain of the function is all real numbers.

### Instruction: help me answer this question Write the first three terms of the arithmetic sequence for which an = n + 10., think step by step ### Assistant: The first three terms of the arithmetic sequence are obtained by substituting n = 1, 2, and 3 into the formula an = n + 10. , The answer is The first three terms are 11, 12, and 13.

### Instruction: help me answer this question What is the length of the shortest piece to be?, think step by step ### Assistant: , The answer is Let us assume the length of the shortest piece to be x cm.

### Instruction: help me answer this question A committee of 7 has to be formed from 9 boys and 4 girls. In how many ways can this be done when the committee consists of at most 3 girls?, think step by step ### Assistant: , The answer is The total number of ways in which a committee can have at most 3 girls are 1632.

### Instruction: help me answer this question Are the events BI and C mutually exclusive?, think step by step ### Assistant: BI ∩ C = {(1, 2), (3, 2), (4, 2), (6, 2)} , The answer is No, they are not mutually exclusive.

### Instruction: help me answer this question b = 3, c = 4, centre at the origin; foci on the x axis. What is the equation of the ellipse?, think step by step ### Assistant: The foci are on the x-axis, so the major axis is along the x-axis. The equation of the ellipse will be of the form x2/a2 + y2/b2 = 1. Since b = 3 and c = 4, a = 5. Using the formula a2 = b2 + c2, we get a2 = 25. Therefore, the equation of the ellipse is x2/25 + y2/9 = 1. , The answer is The equation of the ellipse is x2/25 + y2/9 = 1.

### Instruction: help me answer this question Verify the following: 3.1 (i) (0, 7, –10), (1, 6, –6), and (4, 9, –6) are the vertices of an isosceles triangle. 3.2 (ii) (0, 7, 10), (–1, 6, 6), and (–4, 9, 6) are the vertices of a right-angled triangle. 3.3 (iii) (–1, 2, 1), (1, –2, 5), (4, –7, 8), and (2, –3, 4) are the vertices of a parallelogram., think step by step ### Assistant: , The answer is 3.1 (i) The given points are the vertices of an isosceles triangle. 3.2 (ii) The given points are the vertices of a right-angled triangle. 3.3 (iii) The given points are not the vertices of a parallelogram.

### Instruction: help me answer this question `f(x) = 1/(x^2 + 1)`, think step by step ### Assistant: , The answer is The domain of the function is all real numbers.

### Instruction: help me answer this question Find the coordinates of the points which trisect the line segment joining the points P (4, 2, – 6) and Q (10, –16, 6)., think step by step ### Assistant: , The answer is The coordinates of the points which trisect the line segment joining the points P (4, 2, – 6) and Q (10, –16, 6) are R(6.67, -5.33, -4) and S(7.33, -10.67, 4).

### Instruction: help me answer this question (iv) [–23, 5), think step by step ### Assistant: , The answer is {x: x ∈ R, –23 ≤ x < 5}

### Instruction: help me answer this question `f(x) = sqrt(x - 1)`, think step by step ### Assistant: , The answer is The domain of the function is `x ≥ 1`.

### Instruction: help me answer this question (iv) (2, –1, 3) and (–2, 1, 3), think step by step ### Assistant: , The answer is 2√11 units

### Instruction: help me answer this question Find the equation of the circle with given end points of its diameter as (2, 3) and (6, -5)., think step by step ### Assistant: , The answer is (x - 4)2 + (y + 1)2 = 25

### Instruction: help me answer this question - 15 ≤ 3(x – 2)/5 ≤ 0, think step by step ### Assistant: , The answer is x ∈ [-20, -14]

### Instruction: help me answer this question `f(x) = (x + 3)/(x - 4)`, think step by step ### Assistant: , The answer is The domain of the function is all real numbers except `x = 4`.

### Instruction: help me answer this question Checking for the origin (0, 0) 0 ≤ 3, this is true , think step by step ### Assistant: 0 ≤ 3, this is true , The answer is The solution area would include the origin and hence would be on the left side of the line graph.

### Instruction: help me answer this question `f(x) = (x + 3)/(x - 4)`, think step by step ### Assistant: , The answer is The domain of the function is all real numbers except `x = 4`.

### Instruction: help me answer this question `f(x) = 1/(x^2 + 1)`, think step by step ### Assistant: , The answer is The domain of the function is all real numbers.

### Instruction: help me answer this question cosec x = – 2, think step by step ### Assistant: , The answer is The solution to this equation is not provided in the data.

### Instruction: help me answer this question Find the equation of the circle with the given condition: Centre (1, 1) and radius √2, think step by step ### Assistant: , The answer is The equation of the circle is x2 + y2 – 2x -2y = 0

### Instruction: help me answer this question Is 102n – 1 + 1 divisible by 11 for all natural numbers, i.e., n?, think step by step ### Assistant: This can be proved using the principle of mathematical induction. For n=1, 102n – 1 + 1 = 11, which is divisible by 11. Assuming P (k) to be true for some positive integer k, it can be shown that P (k + 1) is also true, hence P (n) is true for all natural numbers, i.e., n. , The answer is Yes, 102n – 1 + 1 is divisible by 11 for all natural numbers.

### Instruction: help me answer this question The base of an equilateral triangle with side 2a lies along the y-axis, such that the mid-point of the base is at the origin. Find the vertices of the triangle., think step by step ### Assistant: , The answer is Let the triangle be ABC with base BC lying along the y-axis. O (0, 0) is the mid-point of the base BC. Let the third vertex be A (x, y). Side of the triangle is 2a units. Applying distance formula, we get x^2 + (y-a)^2 = a^2. Therefore, the coordinates of A are (a, 0) and the coordinates of B and C are (0, a) and (0, -a) respectively.

### Instruction: help me answer this question `f(x) = 1/(x^2 + 1)`, think step by step ### Assistant: , The answer is The domain of the function is all real numbers.

### Instruction: help me answer this question Simplify the expression ((n+1)!) * (n!) / ((n+2)!)., think step by step ### Assistant: The expression can be simplified by using the definition of n!, which is n! = n*(n-1)!. The expression reduces to ((n+1)! * n!) / ((n+2)!) = ((n+1)*n! * n!) / ((n+2)*(n+1)!) = (n+1) because all the other terms cancel out. , The answer is The simplified expression is (n+1).

### Instruction: help me answer this question i-39, think step by step ### Assistant: i-39 = 1/ i39 = 1/ i4 x 9 + 3 = 1/ (19 x i3) = 1/ i3 = 1/ (-i) [i4 = 1, i3 = -I and i2 = -1] Now, multiplying the numerator and denominator by i, we get i-39 = 1 x i / (-i x i) = i/ 1 = i , The answer is 0 + i

### Instruction: help me answer this question Simplify the expression: (2x + 3)(x - 1) + 2., think step by step ### Assistant: , The answer is The simplified expression is 2x^2 + x + 1.

### Instruction: help me answer this question Given 5x + 4y ≤ 20, what are the required points?, think step by step ### Assistant: These points are obtained by putting values of x = 0 and y = 0 in the equation one by one. , The answer is The required points are (0, 5) and (4, 0).

### Instruction: help me answer this question Solve -5/4(13/b²) = -5/4., think step by step ### Assistant: , The answer is b² = 13

### Instruction: help me answer this question Checking for x ≥ 1, The value of x would be unaffected by any value of y, The origin would not lie on the plane, think step by step ### Assistant: The required area to be included would be on the left of the graph x ≥1 , The answer is ⇒ 0 ≥ 1, which is not true

### Instruction: help me answer this question Find the slope of the line AB., think step by step ### Assistant: , The answer is The slope of the line AB (m1) = (5-4)/(3-4) = 1/-1 = -1

### Instruction: help me answer this question `f(x) = (x + 3)/(x - 4)`, think step by step ### Assistant: , The answer is The domain of the function is all real numbers except `x = 4`.

### Instruction: help me answer this question Simplify the given expression: (2x)^3 * (3x)^2, think step by step ### Assistant: , The answer is The simplified form of the given expression is 216x^5.

### Instruction: help me answer this question (1.1) Which line should be drawn in the graph?, think step by step ### Assistant: N/A , The answer is The line x = – 3 should be drawn in the graph

### Instruction: help me answer this question (iv) {a} ⊂ {a. b, c}, think step by step ### Assistant: {a} is a subset of {a, b, c}. , The answer is True

### Instruction: help me answer this question Is n (n + 1) (n + 5) a multiple of 3 for all natural numbers, i.e., n?, think step by step ### Assistant: This can be proved using the principle of mathematical induction. For n=1, n (n + 1) (n + 5) = 12, which is divisible by 3. Assuming P (k) to be true for some positive integer k, it can be shown that P (k + 1) is also true, hence P (n) is true for all natural numbers, i.e., n. , The answer is Yes, n (n + 1) (n + 5) is a multiple of 3 for all natural numbers.

### Instruction: help me answer this question Find the coordinates of the foci and the vertices, the eccentricity and the length of the latus rectum of the hyperbola y2/9 – x2/27 = 1, think step by step ### Assistant: Given the equation y2/9 – x2/27 = 1, we can compare it with the standard equation of a hyperbola y2/a2 – x2/b2 = 1. By doing so, we find that a = 3 and b = √27. We then calculate c using a2 + b2 = c2, which gives us c = 6. The coordinates of the foci are (0, ±c), so they are (0, 6) and (0, -6). The coordinates of the vertices are (0, ±a), so they are (0, 3) and (0, -3). The eccentricity is e = c/a = 6/3 = 2. The length of the latus rectum is given by 2b2/a = (2 × 27)/3 = 18. , The answer is The coordinates of the foci are (0, 6) and (0, -6), the coordinates of the vertices are (0, 3) and (0, -3), the eccentricity is e = c/a = 2, and the length of the latus rectum is 18.

### Instruction: help me answer this question P (k + 1) is true whenever P (k) is true., think step by step ### Assistant: Principle of Mathematical Induction , The answer is Therefore, by the principle of mathematical induction, statement P (n) is true for all natural numbers, i.e., n.

### Instruction: help me answer this question (i) (–3, 0), think step by step ### Assistant: , The answer is {x: x ∈ R, –3 < x < 0}

### Instruction: help me answer this question (vi) {x: x is a perfect square}, think step by step ### Assistant: , The answer is {x: x ∈ N and x is not a perfect square}

### Instruction: help me answer this question Find the value of `^5√((0.99)`., think step by step ### Assistant: The given text provides a detailed explanation for this calculation. , The answer is An approximation of (0.99)⁵ using the first three terms of its expansion is 0.951.

### Instruction: help me answer this question `f(x) = 1/(x^2 + 1)`, think step by step ### Assistant: , The answer is The domain of the function is all real numbers.

### Instruction: help me answer this question `f(x) = (x + 3)/(x - 4)`, think step by step ### Assistant: , The answer is The domain of the function is all real numbers except `x = 4`.

### Instruction: help me answer this question Expand (101)4, think step by step ### Assistant: Solution: Given, (101)4 101 can be expressed as the sum or difference of two numbers, and then the binomial theorem can be applied. , The answer is 1030401

### Instruction: help me answer this question Which of the following is the HCF of 16 and 24?, think step by step ### Assistant: , The answer is The HCF of 16 and 24 is 8.

### Instruction: help me answer this question `f(x) = 1/(x^2 + 1)`, think step by step ### Assistant: , The answer is The domain of the function is all real numbers.

### Instruction: help me answer this question `f(x) = (x + 3)/(x - 4)`, think step by step ### Assistant: , The answer is The domain of the function is all real numbers except `x = 4`.

### Instruction: help me answer this question False., think step by step ### Assistant: , The answer is Elements of {a, b, c} are a, b, c. Hence, {a} ⊂ {a, b, c}

### Instruction: help me answer this question `f(x) = 1/(x^2 + 1)`, think step by step ### Assistant: , The answer is The domain of the function is all real numbers.

### Instruction: help me answer this question `f(x) = 1/(x^2 + 1)`, think step by step ### Assistant: , The answer is The domain of the function is all real numbers.

### Instruction: help me answer this question What is the simplified equation?, think step by step ### Assistant: , The answer is 4x + 3 ≤ 91

### Instruction: help me answer this question What is the probability of getting 3 tails?, think step by step ### Assistant: , The answer is The probability of getting 3 tails is 1/8.

### Instruction: help me answer this question Question: P (k + 1) is true whenever P (k) is true., think step by step ### Assistant: Explaination: N/A , The answer is Answer: Therefore, by the principle of mathematical induction, statement P (n) is true for all natural numbers, i.e., n.

### Instruction: help me answer this question 41n – 14n is a multiple of 27., think step by step ### Assistant: N/A , The answer is The given statement is true for all natural numbers.

### Instruction: help me answer this question (iii) (–1, 3, –4) and (1, –3, 4), think step by step ### Assistant: , The answer is 2√26 units

### Instruction: help me answer this question Find the coordinates of the point which divides the line segment joining the points (– 2, 3, 5) and (1, – 4, 6) in the ratio (i) 2: 3 internally, (ii) 2: 3 externally., think step by step ### Assistant: The formulas used for the coordinates of a point dividing two points internally and externally come directly from the section formula. The section formula gives the coordinates of a point that divides the line joining two points in a ratio, internally or externally. In this case, the two points are P (x1, y1, z1) = (– 2, 3, 5) and Q (x2, y2, z2) = (1, – 4, 6), and the given ratio is m : n = 2 : 3. Replacing these values in the section formula provides the required coordinates of the point dividing the two points in the given ratio, both internally and externally. , The answer is

### Instruction: help me answer this question `f(x) = 1/(x^2 + 1)`, think step by step ### Assistant: , The answer is The domain of the function is all real numbers.