## 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 β 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 β A**

In this section, attempt any 16 questions out of Questions 1 β 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 π₯ = 0 is:

a) Β±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) Β±4 d) 0

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

a) (β β, 2) (2, β) b) (2, β) c) (ββ, 2) d) (β β, 2] (2, β)

## 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 = π₯ + 1/x, x > 0 is perpendicular to the line 3x β 4y β 7 = 0 is:

a) (2, 5/2) b) (Β±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 β€ x β€ 12}, given by R = valence class containing 1 {(a, b) : |a β , 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) β e y – x d) 2 e x – y

## 13. Given that matrices A and B are of order 3Γn and mΓ5 respectively, then the order of matrix C = 5A +3B is:

a) 3Γ5 b) 5Γ3

c) 3Γ3 d) 5Γ5

## 14. If y = 5 cos x β 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,Β±4) b) (Β±4,0)

c) (Β±3,0) d) (0, Β±3)

## 17. Given that A = [πππ] is a square matrix of order 3Γ3 and |A| = 7, then the value of , where π΄ππ denotes the cofactor of element πππ is:

a) 7 b) -7

c) 0 d) 49

## 18. If y = log(cosππ₯), then ππ¦/ππ₯ is:

a) cosππ₯β1 b) πβπ₯ cosππ₯

c) ππ₯sin ππ₯ d) β ππ₯ tan ππ₯

## 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 π(π₯) = 2πππ π₯ + π₯ in the closed interval [ is:

a) 2 b) π β

c) d) The least value does not exist.

**SECTION β 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 π(π₯) = π₯3 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 β 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 β 3×2 β 36x + 7 is:

a) Strictly increasing in (ββ, β2) and strictly decreasing in ( β2, β)

b) Strictly decreasing in ( β2, 3)

c) Strictly decreasing in (ββ, 3) and strictly increasing in (3, β)

d) Strictly decreasing in (ββ, β2) βͺ (3, β)

## 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 π (π₯) = π₯ + πππ π₯ + π is strictly decreasing over R is:

a) π < 1 b) No value of b exists c) π β€ 1 d) π β₯ 1

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

a) (2,4) β R b) (3,8) β R

c) (6,8) β R d) (8,7) β R

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

a) π₯πR b) π₯ = 0

c) π₯π R {0} d) = β1and

## 32. If A = , then the values of π, π and respectively are:

a) β6, β12, β18 b) β6, β4, β9

c) β6, 4, 9 d) β6, 12, 18

33. A linear programming problem is as follows:

πππππππ§π π = 30π₯ + 50π¦ subject to the constraints,

3π₯ + 5π¦ β₯ 15

2π₯ + 3π¦ β€ 18

π₯ β₯ 0, π¦ β₯ 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 π and given by π(π₯) = (10 +

, then the area when it is maximised is:

a) 75ππ2 b) 7β3ππ2

c) 75β3ππ2 d) 5ππ2

## 35. If A is square matrix such that A2 = A, then (I + A)Β³ β 7 A is equal to:

a) A b) I + A

c) I A d) I

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

a) β1 < 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 β 11x + 5 at which the tangent is y = x β 11 is/are:

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

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

a) 1 + πΌ2 + π½πΎ = 0 b) 1 β πΌ2 β π½πΎ = 0 c) 3 β πΌ2 β π½πΎ = 0 d) 3 + πΌ2 + π½πΎ = 0

SECTION β 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 π = ππ₯ + ππ¦, where π, π > 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) π β 3π = 0 b) π = 3π

c) π + 2π = 0 d) 2π β π = 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 [π₯( π₯ β 1 ) + 1]3, 0β€ π₯ β€ 1 is:

## 44. In a linear programming problem, the constraints on the decision variables x and y are β 3π¦ β₯ 0, π¦ β₯ 0, 0 β€ π₯ β€ 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 β€ Ξ± β€ 2Ο, then:

a) |A|=0 b) |A| π(2, β)

c) |A| π(2,4) d) |A| π[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 βΉ 48 per hour at speed 16 km per hour and the fixed charges to run the train amount to βΉ 1200 per hour.

Assume the speed of the train as π£ 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) βΉ 3750 b) βΉ 750

c) βΉ 7500 d) βΉ 75000

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

a) βΉ 3750 b) βΉ 75000

c) βΉ 7500 d) βΉ 15000