CLASS XII SESSION 2023-24

SAMPLE QUESTION PAPER 12

SUBJECT: MATHEMATICS (041) MAX. MARKS : 80
CLASS : XII DURATION: 3 HRS
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-Reason 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
Questions 1 to 20 carry 1 mark each.

1. Let A be a square matrix of order 2 × 2, then |KA| is equal to

(a) K|A| (b) K2|A| (c) K3|A| (d) 2K|A|

2 x −1 3 0
2. If = , then find the value of x
4 2 2 1
(a) 3 (b) 2/3 (c) 3/2 (d) -1/4

3. If A is a square matrix such that A2 = A, then (I + A)2 – 3A is

(a) I (b) 2A (c) 3I (d) A

d2y
4. If y = Ae5x + Be–5x then is equal to
dx 2
(a) 25y (b) 5y (c) –25y (d) 10y

5. The value of λ such that the vector a = 2iˆ +  ˆj + k and b = iˆ + 2 ˆj + 3k are orthogonal is:
(a) 3/2 (b) −5/2 (c) −1/2 (d) 1/2

 0 2b −2 
6. The matrix A =  3 1 3  is a symmetric matrix. Then the value of a and b respectively are:
3a 3 −1
−2 3 −1 1 3 1
(a) , (b) , (c) -2, 2 (d) ,
3 2 2 2 2 2
7. The area (in sq. m) of the shaded region as shown in the figure is:
 (a) 32/3 sq. units (b) 16/3 sq. units (c) 4 sq. units (d) 16 sq. units
2
 d2y  dy d 3 y
8. If p and q are the degree and order of the differential equation  2  + 3 + 3 = 4 , then the
 dx  dx dx
value of 2p – 3q is
(a) 7 (b) –7 (c) 3 (d) –3

9. A set of values of decision variables that satisfies the linear constraints and non-negativity
conditions of an L.P.P. is called its:
(a) Unbounded solution (b) Optimum solution
(c) Feasible solution (d) None of these
 /2
1
10. The value of  1 + tan
0
3
x
dx is

(a) 0 (b) 1 (c) π/4 (d) π/2

11. For any vector a , the value of | a  iˆ |2 + | a  ˆj |2 + | a  k |2 is:

(a) a (b) a2 (c) 1 (d) 0

12. Solution of LPP

To maximise Z = 4x + 8y
subject to constraints : 2x + y ≤ 30, x + 2y ≤ 24, x ≥ 3, y ≤ 9, y ≥ 0 is
(a) x = 12, y = 6 (b) x = 6, y = 12 (c) x = 9, y = 6 (d) none of these

13. If events A and B are independent, P(A) = 0.35, P(A ∪ B) = 0.60 then P(B) is
(a) 0.25 (b) 0 (c) 0.95 (d) none of these

dy 2
14. General solution of differential equation = x5 + x3 − is
dx x
x6 x4 x 6
x 4
(a) y = + − 2log | x | (b) y = + − 2log | x | +1
6 4 6 4
2 x6 x4
(c) y = 5 x 4 + 3x 2 + 2 + C (d) y = + − 2log | x | +C
x 6 4

15. The domain, for which tan-1x > cot-1x holds true, is:
(a) x = 1 (b) x > 1 (c) x < 1 (d) Not defined

16. Direction ratios of a line are 2, 3, –6. Then direction cosines of a line making obtuse angle with
the y-axis are
2 −3 −6 −2 3 −6 −2 −3 6 −2 −3 −6
(a) , , (b) , , (c) , , (d) , ,
7 7 7 7 7 7 7 7 7 7 7 7
17. A function f : Z → Z given by f (x) = 5x + 3 is
(a) one-one but not onto. (b) bijective
(c) onto but not one-one (d) None of these

18. The area of a parallelogram whose one diagonal is 2i + j − 2k and one side is 3i + j − k is
(a) i − 4 j − k (b) 3√2 sq units (c) 6√2 sq units (d) 6 sq units
 ASSERTION-REASON BASED QUESTIONS
In the following questions, a statement of assertion (A) is followed by a statement of Reason (R).
Choose the correct answer out of the following choices.
(a) Both A and R are true and R is the correct explanation of A.
(b) Both A and R are true but R is not the correct explanation of A.
(c) A is true but R is false.
(d) A is false but R is true.

x +1 y − 2 z + 3
19. Assertion (A): The angle between the straight lines = = and
2 5 4
x −1 y + 2 z − 3
= = is 90°
1 2 −3
Reason (R): Skew lines are lines in different planes which are parallel and intersecting.

20. Assertion (A): We can write sin–1x = (sin x) –1.

Reason (R): Any value in the range of principal value branch is called principal value of that
inverse trigonometric function.

SECTION – B
Questions 21 to 25 carry 2 marks each.

21. If sin [cot–1 (x + 1)] = cos (tan–1x), then find x.

OR
Prove that the Greatest Integer Function f : R → R, given by f(x) = [x] is neither one-one nor
onto. Where [x] denotes the greatest integer less than or equal to x.

22. Find the rate of change of volume of sphere with respect to its surface area, when radius is 2 cm.

23. Discuss the continuity of the following function at x = 0 :

 x 4 + +2 x 3 + x 2
 , if x  0
f ( x) =  tan −1 x
 0, if x = 0


x +1 y − 2 z + 3 1− x y + 2 3 − z
24. Find the angle between the straight lines = = and = = .
2 5 4 −1 2 3

25. If a = i + j + k , b = 4i − 2 j + 3k and c = i − 2 j + k , find a vector of magnitude 6 units which is

parallel to the vector 2a − b + 3c .
OR
If a = 7i + j − 4k and b = 2i + 6 j + 3k , then find the projection of b on a .

SECTION – C
Questions 13 to 22 carry 3 marks each.
 /3
1
26. Evaluate the following integral: 
 1+ /6 tan x
dx

OR

x
Evaluate the following integral:  1 + sin x dx
0
 sin x
27. Evaluate:  (1 − cos x)(2 − cos x) dx
28. Solve the following problem graphically: Minimise and Maximise Z = 3x + 9y subject to the
constraints: x + 3y ≤ 60; x + y ≥ 10; x ≤ y; x ≥ 0, y ≥ 0

29. A family has 2 children. Find the probability that both are boys, if it is known that (i) at least one
of the children is a boy. (ii) the elder child is a boy.

OR

Bag I contains 3 red and 4 black balls and bag II contains 4 red and 5 black balls. One ball is
transferred from bag I to bag II and then a ball is drawn from bag II at random. The balls so
drawn is found to be red in colour. Find the probability that the transferred ball is black.
 /2
dx
30. Evaluate:  1+
0 tan x
dy
31. Find the particular solution of the differential equation = 1 + x + y + xy, given that y = 0
dx
when x = 1.
OR

  y 
Solve the following differential equation:  x sin 2   − y  dx + xdy = 0
 x 

SECTION – D
Questions 32 to 35 carry 5 marks each.

32. Using integration, find the smaller area enclosed by the circle x2 + y2 = 4 and the line x + y = 2.

33. Find the vector equation of the line through the point (1, 2, –4) and perpendicular to the two lines
r = (8i − 19 j + 10k ) +  (3i − 16 j + 7k ) and r = (15i + 29 j + 5k ) +  (3i + 8 j − 5k )
OR
Find the shortest distance between the following lines :
l1 : r = (i + 2 j − 4k ) +  (2i + 3 j + 6k )
l2 : r = (3i + 3 j − 5k ) +  (4i + 6 j + 12k )

34. Show that the relation S in the set R of real numbers defined as S = {(a, b): a, b ∈ R and a ≤ b3}
is neither reflexive nor symmetric and nor transitive.
OR
Given a non-empty set X, define the relation R in P(X) as follows:
For A, B ∈ 𝑃(𝑋), (𝐴, 𝐵) ∈ 𝑅 iff 𝐴 ⊂ 𝐵. Prove that R is reflexive, transitive and not symmetric.

1 −1 0   2 2 −4 
35. Given A =  2 3 4  and B =  −4 2 −4  , verify that BA = 6I, use the result to solve the
 
 0 1 2   2 −1 5 
system x – y = 3, 2x + 3y + 4z = 17, y + 2z = 7.
 SECTION – E(Case Study Based Questions)
Questions 36 to 38 carry 4 marks each.

36. Case-Study 3:
Shalini wants to prepare a handmade gift box for her friend's birthday at home. For making lower
part of box, she takes a square piece of cardboard of side 20 cm.

If x cm be the length of each side of the square cardboard which is to be cut off from corners of
the square piece of side 20 cm and Volume of the box is V then, answer the following questions.
dV
(a) Find the value of V for which = 0 [1]
dx
(b) Shalini is interested in maximising the volume of the box. So, what should be the side of the
square to be cut off so that the volume of the box is maximum? [2]
(c) Find the maximum value of the volume. [1]

37. Case-Study 2: Read the following passage and answer the questions given below.

The temperature of a person during an intestinal illness is given by

1 493
𝑓(𝑥) = − 𝑥² + 𝑚𝑥 + , 0 ≤ 𝑥 ≤ 12, m being a constant, where f(x) is the temperature in °F at
10 50
x days.
(i) Is the function differentiable in the interval (0, 12)? Justify your answer.
(ii) If 6 is the critical point of the function, then find the value of the constant

(iii) Find the intervals in which the function is strictly increasing/strictly decreasing.
OR
(iii) Find the points of local maximum/local minimum, if any, in the interval (0, 12) as well as
the points of absolute maximum/absolute minimum in the interval [0, 12]. Also, find the
corresponding local maximum/local minimum and the absolute maximum/absolute minimum
values of the function.
38. Case-Study 1:
Let d1, d2, d3 be three mutually exclusive diseases.
Let S = {S1, S2, S3, S4, S5, S6} be the set of observable symptoms of these diseases. For example,
S1 is the shortness of breath, S2 is the loss of weight, S3 is the fatigue, etc. Suppose a random
sample of 10,000 patients contains 3200 patients with disease d1, 3500 patients with disease d2,
and 3300 patients with disease d3. Also, 3100 patients with disease d1, 3300 patients with disease
d2, and 3000 patients with disease d3 show the symptom S.

Based on the above information answer the following questions:

(a) A person is chosen at random from the sample of 10,000. What is The probability that the
person chosen does not suffer from disease d3? [1]
(b) Find the conditional probability that the patient shows the symptom S given that he suffers
from disease d1 and also calculate the conditional probability that the patient shows the symptom
S given that he suffers from disease d2.
OR
If a person chosen at random shows the symptom S, then what is the probability that he does
suffer from disease d1? [2]
(c) Let Di denote the event that the patient has disease di (i = 1, 2, 3) and S be the event that the
3
d 
patient shows the symptom S. Then find the value of P  i  . [1]
1  s