Class 12 Maths Sample Question Paper | CBSE Exam 2021-22 Important Questions

CBSE 2021-22, Maths board exam sample question paper for Class 12. Check the important questions which need to be focused on while preparing for the class 12 Maths board exam.

• Subject Code – 041
• CLASS: XII
• Session: 2021-22
• Subject- Mathematics
• Term – 1
• Time Allowed: 90 minutes
• Maximum Marks: 40

General Instructions:
1. This question paper contains three sections – A, B and C. Each part is compulsory.
2. Section – A has 20 MCQs, attempt any 16 out of 20. 3. Section – B has 20 MCQs, attempt any 16 out of 20.
4. Section – C has 10 MCQs, attempt any 8 out of 10.
5. All questions carry equal marks.
6. There is no negative marking.

SECTION – A
In this section, attempt any 16 questions out of Questions 1 – 20.
Each Question is of 1 mark weightage.

1. sin [ )] is equal to:
a) b)
c) -1 d) 1

2. The value of k (k < 0) for which the function defined as is continuous at 𝑥 = 0 is:

a) ±1 b) 1 c) d) 1 2

3. If A = [aij] is a square matrix of order 2 such that aij , then A2 is:

4. Value of k, for which A = is a singular matrix is:

a) 4 b) -4 c) ±4 d) 0

5. Find the intervals in which the function f given by f (x) = x 2 – 4x + 6 is strictly increasing:

a) (– ∞, 2) (2, ∞) b) (2, ∞) c) (−∞, 2) d) (– ∞, 2] (2, ∞)

6. Given that A is a square matrix of order 3 and | A | = – 4, then | adj A | is equal to:

a) -4 b) 4 c) -16 d) 16

7. A relation R in set A = {1,2,3} is defined as R = {(1, 1), (1, 2), (2, 2), (3, 3)}. Which of the following ordered pair in R shall be removed to make it an equivalence relation in A?

a) (1, 1) b) (1, 2) c) (2, 2) d) (3, 3)

8. If , then value of a + b c + 2d is:

a) 8 b) 10 c) 4 d) 8

9. The point at which the normal to the curve y = 𝑥 + 1/x, x > 0 is perpendicular to the line 3x – 4y – 7 = 0 is:

a) (2, 5/2) b) (±2, 5/2)
c) (- 1/2, 5/2) d) (1/2, 5/2)

10. sin (tan-1x), where |x| < 1, is equal to:

11. Let the relation R in the set A =
b| is a multiple of 4}. Then [1], th
x Z e equi : 0 ≤ x ≤ 12}, given by R = valence class containing 1 {(a, b) : |a – , is:
a) {1, 5, 9} b) {0, 1, 2, 5}
c) d) A

12. If ex + ey = ex+y , then :

a) e y – x b) e x + y
c) – e y – x d) 2 e x – y

13. Given that matrices A and B are of order 3×n and m×5 respectively, then the order of matrix C = 5A +3B is:

a) 3×5 b) 5×3
c) 3×3 d) 5×5

14. If y = 5 cos x – 3 sin x, then is equal to:

a) – y b) y
c) 25y d) 9y

15. For matrix A = is equal to:

a) b)

c) d)

16. The points on the curve axis are:
at which the tangents are parallel to y- 1

a) (0,±4) b) (±4,0)
c) (±3,0) d) (0, ±3)

17. Given that A = [𝑎𝑖𝑗] is a square matrix of order 3×3 and |A| = 7, then the value of , where 𝐴𝑖𝑗 denotes the cofactor of element 𝑎𝑖𝑗 is:

a) 7 b) -7
c) 0 d) 49

18. If y = log(cos𝑒𝑥), then 𝑑𝑦/𝑑𝑥 is:

a) cos𝑒𝑥−1 b) 𝑒−𝑥 cos𝑒𝑥
c) 𝑒𝑥sin 𝑒𝑥 d) − 𝑒𝑥 tan 𝑒𝑥

19. Based on the given shaded region as the feasible region in the graph, at which point(s) is the objective function Z = 3x + 9y maximum?

a) Point B b) Point C
c) Point D d) every point on the line segment CD

20. The least value of the function 𝑓(𝑥) = 2𝑐𝑜𝑠𝑥 + 𝑥 in the closed interval [ is:

a) 2 b) 𝜋 √
c) d) The least value does not exist.

SECTION – B
In this section, attempt any 16 questions out of the Questions 21 – 40.
Each question is of 1 mark weightage.

21. The function: R R defined as 𝑓(𝑥) = 𝑥3 is:

a) One-on but not onto b) Not one-one but onto
c) Neither one-one nor onto d) One-one and onto

23. In the given graph, the feasible region for a LPP is shaded. The objective function Z = 2x – 3y, will be minimum at:

a) (4, 10) b) (6, 8)
c) (0, 8) d) (6, 5)

26. The real function f(x) = 2×3 – 3×2 – 36x + 7 is:

a) Strictly increasing in (−∞, −2) and strictly decreasing in ( −2, ∞)
b) Strictly decreasing in ( −2, 3)
c) Strictly decreasing in (−∞, 3) and strictly increasing in (3, ∞)
d) Strictly decreasing in (−∞, −2) ∪ (3, ∞)

28. Given that A is a non-singular matrix of order 3 such that A2 = 2A, then value of |2A| is:

a) b)
c) 64 d) 16

29. The value of for which the function 𝑓 (𝑥) = 𝑥 + 𝑐𝑜𝑠𝑥 + 𝑏 is strictly decreasing over R is:

a) 𝑏 < 1 b) No value of b exists c) 𝑏 ≤ 1 d) 𝑏 ≥ 1

30. Let R be the relation in the set N given by R = {(a, b) : a = b – 2, b > 6}, then:

a) (2,4) ∈ R b) (3,8) ∈ R
c) (6,8) ∈ R d) (8,7) ∈ R

31. The point(s), at which the function f gi ven by is continuous, is/are:

a) 𝑥𝜖R b) 𝑥 = 0
c) 𝑥𝜖 R {0} d) = −1and

32. If A = , then the values of 𝑘, 𝑎 and respectively are:

a) −6, −12, −18 b) −6, −4, −9
c) −6, 4, 9 d) −6, 12, 18

33. A linear programming problem is as follows:
𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒 𝑍 = 30𝑥 + 50𝑦 subject to the constraints,
3𝑥 + 5𝑦 ≥ 15
2𝑥 + 3𝑦 ≤ 18
𝑥 ≥ 0, 𝑦 ≥ 0
In the feasible region, the minimum value of Z occurs at
a) a unique point b) no point

c) infinitely many points d) two points only

34. The area of a trapezium is defined by function 𝑓 and given by 𝑓(𝑥) = (10 +
, then the area when it is maximised is:
a) 75𝑐𝑚2 b) 7√3𝑐𝑚2
c) 75√3𝑐𝑚2 d) 5𝑐𝑚2

35. If A is square matrix such that A2 = A, then (I + A)³ – 7 A is equal to:

a) A b) I + A
c) I A d) I

36. If tan-1 x = y, then:

a) −1 < y < 1 b) c) d) y

37. Let A = {1, 2, 3}, B = {4, 5, 6, 7} and let f= {(1, 4), (2, 5), (3, 6)} be a function from A to B. Based on the given infor mation, is best defined as:

a) Surjective function b) Injective function c) Bijective function d) function

38. For A = , then 14A-1 is given b y:

a) b) c) d)

39. The point(s) on the curve y = x 3 – 11x + 5 at which the tangent is y = x – 11 is/are:

a) (-2,19) b) (2, – 9) c) (±2, 19) d) (-2, 19) and (2, -9)

40. Given that A = and A 2 = 3 , then:

a) 1 + 𝛼2 + 𝛽𝛾 = 0 b) 1 − 𝛼2 − 𝛽𝛾 = 0 c) 3 − 𝛼2 − 𝛽𝛾 = 0 d) 3 + 𝛼2 + 𝛽𝛾 = 0

SECTION – C In this section, attempt any 8 questions. Each question is of 1-mark weightage. Questions 46-50 are based on a Case-Study.

41. For an objective function 𝑍 = 𝑎𝑥 + 𝑏𝑦, where 𝑎, 𝑏 > 0; the corner points of the feasible region determined by a set of constraints (linear inequalities) are (0, 20), (10, 10), (30, 30) and (0, 40). The condition on a and b such that the maximum Z occurs at both the points (30, 30) and (0, 40) is:

a) 𝑏 − 3𝑎 = 0 b) 𝑎 = 3𝑏

c) 𝑎 + 2𝑏 = 0 d) 2𝑎 − 𝑏 = 0

42. For which value of m is the line y = mx + 1 a tangent to the curve y 2 = 4x?

a) b)

c) d)

43. The maximum value of [𝑥( 𝑥 − 1 ) + 1]3, 0≤ 𝑥 ≤ 1 is:

44. In a linear programming problem, the constraints on the decision variables x and y are − 3𝑦 ≥ 0, 𝑦 ≥ 0, 0 ≤ 𝑥 ≤ 3. The feasible region

a) is not in the first quadrant b) is bounded in the first quadrant
c) is unbounded in the first quadrant d) does not exist

45. Let A = , where 0 ≤ α ≤ 2π, then:

a) |A|=0 b) |A| 𝜖(2, ∞)
c) |A| 𝜖(2,4) d) |A| 𝜖[2,4]

CASE STUDY
The fuel cost per hour for running a train is proportional to the square of the speed it generates in km per hour. If the fuel costs ₹ 48 per hour at speed 16 km per hour and the fixed charges to run the train amount to ₹ 1200 per hour.
Assume the speed of the train as 𝑣 km/h.

Based on the given information, answer the following questions.

46. Given that the fuel cost per hour is times the square of the speed the train generates in km/h, the value of is:

a) 16/3 b) 1/3
c) 3 d) 3/16

47. If the train has travelled a distance of 500km, then the total cost of running the train is given by function:

48. The most economical speed to run the train is:

a) 18km/h b) 5km/h
c) 80km/h d) 40km/h

49. The fuel cost for the train to travel 500km at the most economical speed is:

a) ₹ 3750 b) ₹ 750
c) ₹ 7500 d) ₹ 75000

50. The total cost of the train to travel 500km at the most economical speed is:

a) ₹ 3750 b) ₹ 75000
c) ₹ 7500 d) ₹ 15000

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