Subject: Mathematics
Class XII
Time Allowed: 3 hours
Max. Marks: 80
General Instructions:
1. This Question paper contains – five sections A, B, C, D and E. Each section is compulsory. However, there are internal choices in some questions.
2. Section A has 18 MCQ’s and 02 Assertion Reasoning based questions of 1 mark each.
3. Section B has 5 Very Short Answer (VSA)-type questions of 2 marks each.
4. Section C has 6 Short Answer (SA)-type questions of 3 marks each.
5. Section D has 4 Long Answer (LA)-type questions of 5 marks each.
6. Section E has 3 source based/case based/passage based/integrated units of assessment (4 marks each) with sub parts.
SECTION – A (MCQ) 1 Mark Questions
1. Let A = {1, 2, 3} and consider the relation R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (1, 3)}, Then R is
(a) reflexive but not symmetric
(b) reflexive but not transitive
(c) symmetric and transitive
(d) neither symmetric, nor transitive
2. The modulus function f : R → R, given by f(x) = |x| is
(a) one-one but not onto
(b) onto but not one-one
(c) both one-one and onto
(d) neither one-one nor onto
3. If A and B are symmetric matrices of same order, AB − BA is a :
(a) Skew-symmetric matrix
(b) Symmetric matrix
(c) Zero matrix
(d) Identity matrix.
4.
(a) (2, −1) (b) (−2,1) (c) (2,1) (d) (−2, −1)
5. Value of
(a) x + y (b) x − y (c) xy (d) none of these
6. If A B and are square matrices of the same order 3, such that |A| = 2 and AB = 4I, then the value of |B| is:
(a) 1 (b) 2 (c) 4 (d) 32
7. If x =a (cos q + q sin q), y = a (sin q – q cos q), then dy/dx is:
(a) cot q (b) tan q (c) a cotq (d) a tanq
8. If √x + √y = √a, then dy/dx is :
a) -√x/√y b) -1/2. √y/√x c) -√y/√x d) None of these
9.
a) 1/3 b) 2/3 c) 4/3 d) 8/3
10. The volume of a sphere is increasing at 3 cubic cm/s. The rate at which the radius increases when radius is 2 cm is
a) 3/32π cm/s b) 3/16π cm/s c) 3/48π cm/s d) 1/24π cm/s
11. The function f (x) = tan x – x
(a) always increases
(b) always decreases
(c) never increases
(d) sometimes increases and sometimes decreases
12.
13.
(a) tan x + cot x + C
(b) tan x + cosec x + C
(c) –tan x + cot x + C
(d) tan x + sec x + C
14.
15.
16. The value of cos-1(cos 13π/6)
a) 13π/6 b) 7π/6 c) 5π/6 d) π/6
17. If A is a square matrix of order 3 ´ 3, such that |A| =12, then the value of |A adjA| is:
(a) 12 (b) 144 (c) 1728 (d) 36
18. If y = a sin x + b cos x, then value of y2 + (dy/dx)2 is
a) a2 + b2 b) ab c) a2 – b2 d) 1/(a2 + b2)
Assertion Reasoning Based Questions
19. Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R
Assertion A: The domain of the function sin-1 (3x – 1) is [0,3].
Reason R: The domain of the function sin-1 is [−1,1].
In the light of the above statements, choose the most appropriate answer from the options given below:
a. Both A and R are correct and R is the correct explanation of A
b. Both A and R are correct but R is NOT the correct explanation of A
c. A is correct but R is not correct
d. A is not correct but R is correct
20. Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R
Assertion A: f(x) =log (cos x) is decreasing on the interval (0, π/2)
Reason R: If f I(x) ³ 0 then, f(x) is an increasing function.
In the light of the above statements, choose the most appropriate answer from the options given below
a. Both A and R are correct and R is the correct explanation of A
b. Both A and R are correct but R is NOT the correct explanation of A
c. A is correct but R is not correct
d. A is not correct but R is correct
SECTION – B (Very Short Answer (VSA)-type questions) 2 Marks Each
21. Find the value of
22.
23.
OR
Show that the function f(x) = |x − 3|, x Î R , is not differentiable at x = 3.
24. Find dy/dx at x = 1, y = π/4, if sin2 y + cos xy = K
OR
25. The sides of an equilateral triangle are increasing at the rate of 2 cm/s. Find the rate at which the area increases, when the side is 10 cm.
OR
A particle moves along the curve 6y = x3 + 2. Find the points on the curve at which the y-coordinate is changing 8 times as fast as the x-coordinate.
SECTION – C (Short Answer (SA)-type questions) 3 Marks Each
26. Let Z be the set of all integers and R be the relation on Z defined as R = {(a, b) : a, b Î Z and (a −b) is divisible by 5}. Prove that R is an equivalence relation.
OR
Let A = R −{3} and B = R −{1} . Consider the function f: A → B defined by
f(x) = (x – 2)/ (x – 3). Is f one-one and onto? Justify your answer.
27.
28. Find dy/dx, if y = (cos x)x + (sin x)1/x
29. Find the intervals in which the function f(x) = x4/4 – x3 – 5x2 + 24x + 12
(a) increasing (b) decreasing
30. Evaluate:
31.
OR
SECTION – D (Long Answer (LA)-type questions) 5 Marks Each
32.
33.
OR
34. Find the local maxima or minima if any, of the function f(x) = sin4 x + cos4 x, 0 < x < p/2 .
OR
Find the difference between the greatest and the least values of the function
f(x) = sin 2x – x in [-p/2, p/2]
35. Find the following integral:
OR
Evaluate:
SECTION – E (Case Study questions) 4 Marks Each
36. Sanjeev visited the exhibition along with his family. The exhibition had a huge swing, which attracted many children. Sanjeev found that the swing traced the path of a parabola as given by y = x2.
Using the information given above, answer the following:
(a) Let f : R → R be defined as f(x) = x2. Check if f is one-one and onto or not? Justify your answer.
(b) (b) Let f : {1,2,3,…} →{1,4,9,… } be defined as f(x) = x2. Check if f is one-one and onto or not? Justify your answer.
37. Two schools A and B decided to award prizes to their students for three games hockey (x), cricket (y) and tennis (z). School A decided to award a total of ₹ 11,000 for the three games to 5,4 and 3 students respectively, while school B decided to award ₹10,700 for the three games to 4,3 and 5 students respectively. Also, the three prizes together amount to ₹ 2,700.
Using the information given above, answer the following:
(a) Represent the above situation using matrix equation.
(b) Find out the prize amount for hockey, cricket & tennis.
38. A factory makes an open cardboard box for a jewelry shop from a square sheet of side 18 cm by cutting off squares from each corner and folding up the flaps.
(a) What should be the side of the square to be cut off so that the volume is maximum?
(b) What is the maximum volume of the box?