CBSE Class 12 Sample Paper 2022 for Maths Term 1

Subject Code - 041

Sample Question Paper
CLASS: XII
Session: 2021-22
Mathematics (Code-041)
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. 𝜋 1 1
sin [ 3 −sin-1 (− 2)] is equal to:
1 1
a) 2 b) 3
c) -1 d) 1
2. The value of k (k < 0) for which the function 𝑓 defined as 1
1−𝑐𝑜𝑠𝑘𝑥
,𝑥 ≠ 0
𝑓 (𝑥 ) = { 1𝑥𝑠𝑖𝑛𝑥
,𝑥 = 0
2
is continuous at 𝑥 = 0 is:

a) ±1 b) −1
1 1
c) ± 2 d) 2
3. 1, 𝑤ℎ𝑒𝑛 𝑖 ≠ 𝑗 1
If A = [aij] is a square matrix of order 2 such that aij = { , then
0, 𝑤ℎ𝑒𝑛 𝑖 = 𝑗
A2 is:

1 0 1 1
a) [ ] b) | |
1 0 0 0

1 1 1 0
c) | | d) [ ]
1 0 0 1

4. 𝑘 8] 1
Value of 𝑘, for which A = [ is a singular matrix is:
4 2𝑘

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 1
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 1
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)}. 1
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. 2𝑎 + 𝑏 𝑎 − 2𝑏 4 −3 1
If [ ]=[ ], then value of a + b – c + 2d is:
5𝑐 − 𝑑 4𝑐 + 3𝑑 11 24

a) 8 b) 10
c) 4 d) –8

9. 1 1
The point at which the normal to the curve y = 𝑥 + 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)
-1
10. sin (tan x), where |x| < 1, is equal to: 1

𝑥 1
a) √1−𝑥 2 b) √1−𝑥 2

1 𝑥
c) √1+𝑥 2 d) √1+𝑥 2

11. Let the relation R in the set A = {x ∈ Z : 0 ≤ x ≤ 12}, given by R = {(a, b) : |a – 1
b| is a multiple of 4}. Then [1], the equivalence class containing 1, is:

a) {1, 5, 9} b) {0, 1, 2, 5}
c) 𝜙 d) A

12. 𝑑𝑦 1
If ex + ey = ex+y , then 𝑑𝑥 is:

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 1
order of matrix C = 5A +3B is:

a) 3×5 b) 5×3
c) 3×3 d) 5×5

14. 𝑑2 𝑦 1
If y = 5 cos x – 3 sin x, then 𝑑𝑥 2 is equal to:

a) - y b) y
c) 25y d) 9y

15. 2 5], 1
For matrix A =[ (𝑎𝑑𝑗𝐴)′ is equal to:
−11 7
−2 −5] 7 5]
a) [ b) [
11 −7 11 2

7 11 7 −5]
c) [ ] d) [
−5 2 11 2

16. 𝑥2 𝑦2 1
The points on the curve 9 + 16 = 1 at which the tangents are parallel to y-
axis are:

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 1
value of ∑3𝑖=1 𝑎𝑖2 𝐴𝑖2 , where 𝐴𝑖𝑗 denotes the cofactor of element 𝑎𝑖𝑗 is:

a) 7 b) -7
c) 0 d) 49
18. 𝑑𝑦 1
If y = log(cos 𝑒 𝑥 ), then 𝑑𝑥 is:
a) cos 𝑒 𝑥−1 b) 𝑒 −𝑥 cos 𝑒 𝑥
c) 𝑒 𝑥 sin 𝑒 𝑥 d) − 𝑒 𝑥 tan 𝑒 𝑥
19. Based on the given shaded region as the feasible region in the graph, at 1
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

𝜋
20. The least value of the function 𝑓(𝑥 ) = 2𝑐𝑜𝑠𝑥 + 𝑥 in the closed interval [0,2 ] 1
is:
𝜋
a) 2 b) 6 + √3
𝜋
c) d) The least value does not
2
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 𝑓: R⟶R defined as 𝑓(𝑥 ) = 𝑥 3 is: 1

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

22. 𝑑2 𝑦 𝜋 1
If x = a sec 𝜃, y = b tan 𝜃, then 𝑑𝑥 2 at 𝜃 = 6 is:

−3√3𝑏 −2√3𝑏
a) b)
𝑎2 𝑎
−3√3𝑏 −𝑏
c) d) 3 3𝑎2
𝑎 √

23. In the given graph, the feasible region for a LPP is 1
shaded.
The objective function Z = 2x – 3y, will be minimum
at:

a) (4, 10) b) (6, 8)
c) (0, 8) d) (6, 5)
24. 2 1 1 1
The derivative of sin (2𝑥√1 − 𝑥 ) w.r.t sin-1x,
-1
− 2<𝑥< , is:
√ √2

𝜋
a) 2 b) 2 − 2
𝜋
c) d) −2
2
25. 1
1 −1 0 2 2 −4
If A = [2 3 4] and B = [−4 2 −4], then:
0 1 2 2 −1 5

a) A-1 = B b) A-1 = 6B
c) B-1 = B 1
d) B-1 = 6A

26. The real function f(x) = 2x3 – 3x2 – 36x + 7 is: 1

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, ∞)

27. √1+𝑐𝑜𝑠𝑥+√1−𝑐𝑜𝑠𝑥
Simplest form of tan-1 ( 1+𝑐𝑜𝑠𝑥− 1−𝑐𝑜𝑠𝑥) , 𝜋 < 𝑥 < 2 is:
3𝜋 1
√ √

𝜋 𝑥 3𝜋 𝑥
a) 4 − 2 b) −2
2

𝑥 𝑥
c) − 2 d) 𝜋 − 2

28. Given that A is a non-singular matrix of order 3 such that A2 = 2A, then value 1
of |2A| is:

a) 4 b) 8
c) 64 d) 16

29. The value of 𝑏 for which the function 𝑓(𝑥 ) = 𝑥 + 𝑐𝑜𝑠𝑥 + 𝑏 is strictly 1
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: 1
a) (2,4) ∈ R b) (3,8) ∈ R
c) (6,8) ∈ R d) (8,7) ∈ R

𝑥
31. ,𝑥 < 0 1
The point(s), at which the function f given by 𝑓(𝑥) ={ |𝑥|
−1, 𝑥 ≥ 0
is continuous, is/are:

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

32. 0 2 0 3𝑎 1
If A = [ ] and 𝑘A = [ ], then the values of 𝑘, 𝑎 and 𝑏 respectively
3 −4 2𝑏 24
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: 1
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 + 1
𝑥)√100 − 𝑥 2 , 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: 1

a) A b) I + A
c) I − A d) I
36. If tan-1 x = y, then: 1
−𝜋 𝜋
a) −1 < y < 1 b) ≤y ≤ 2
2

−𝜋 𝜋 −𝜋 𝜋
c) <y < 2 d) y 𝜖{ 2 , 2 }
2

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

a) Surjective function b) Injective function
c) Bijective function d) function
38. 3 1 1
For A = [ ], then 14A-1 is given by:
−1 2

2 −1 4 −2
a) 14 [ ] b) [ ]
1 3 2 6

2 −1 −3 −1
c) 2 [ ] d) 2[ ]
1 −3 1 −2

39. The point(s) on the curve y = x 3 – 11x + 5 at which the tangent is y = x – 11 1
is/are:

a) (-2,19) b) (2, - 9)
c) (±2, 19) d) (-2, 19) and (2, -9)
40. 𝛼 𝛽 1
Given that A = [ ]and A2 = 3I, then:
𝛾 −𝛼

a) 1 + 𝛼 2 + 𝛽𝛾 = 0 b) 1 − 𝛼 2 − 𝛽𝛾 = 0
c) 3 − 𝛼 2 − 𝛽𝛾 = 0 d) 3 + 𝛼 2 + 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 1
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? 1

1
a) 2 b) 1

c) 2 d) 3
1
43. The maximum value of [𝑥(𝑥 − 1) + 1]3 , 0≤ 𝑥 ≤ 1 is: 1

1
a) 0 b) 2
c) 1 3 1
d) √3

44. In a linear programming problem, the constraints on the decision variables x 1
and y are 𝑥 − 3𝑦 ≥ 0, 𝑦 ≥ 0, 0 ≤ 𝑥 ≤ 3. The feasible region

a) is not in the first b) is bounded in the first
quadrant quadrant
c) is unbounded in the d) does not exist
first quadrant
45. 1 sinα 1 1
Let A = [−sinα 1 sinα], where 0 ≤ α ≤ 2π, then:
−1 −sinα 1

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 𝑘 times the square of the speed the train 1
generates in km/h, the value of 𝑘 is:

16 1
a) 3 b) 3
c) 3 3
d) 16
47. If the train has travelled a distance of 500km, then the total cost of running 1
the train is given by function:

15 600000 375 600000
a) 𝑣+ b) 𝑣+
16 𝑣 4 𝑣

5 150000 3 6000
c) 16 𝑣 2 + d) 16 𝑣 + 𝑣
𝑣

48. The most economical speed to run the train is: 1

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: 1

a) ₹ 3750 b) ₹ 750
c) ₹ 7500 d) ₹ 75000
50. The total cost of the train to travel 500km at the most economical speed is: 1

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

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