ACHIEVER’S ACADEMY

SESSION: 2023-24
MATHEMATICS (041)
Test Series Paper-4
Class: XII Maximum Marks: 80
Date: 31.12.2023 Time Allowed: 3 Hours
General Instructions
• This Question Paper has 5 Sections A,B,C,D and E.
• Section A has 20 MCQs carrying 1 mark each
• Section B has 5 questions (VSA) carrying 02 marks each.
• Section C has 6 questions (SA) carrying 03 marks each.
• Section D has 4 questions(LA) carrying 05 marks each.
• 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 
1. What is the principal value of sin −1  − ?
 2
 −    −  
(a) [-π, π] (b)  ,  (c) (-∞, ∞) (d)  , 
 2 2  4 4
2. Let A and B be the events associated with the sample space s, then the value of P(A/B) is:
(a) P(A/B) = 1 (b) P(A/B) = P(A) (c) 0 ≤ P(A/B) (d) 0 ≤ P(A/B) ≤ 1
3. The vector equation of the line joining the points (3, -2, -5) and (3, -2, 6) is:
(a) (4iˆ − 4 ˆj + 5kˆ) +  (12kˆ) (b) (4iˆ − 4 ˆj + 5kˆ) +  (12kˆ)
(c) (6iˆ − 2 ˆj + 2kˆ) +  (5kˆ) (d) (9iˆ − 9 ˆj − 2kˆ) +  (2kˆ)
4. What is the angle between vectors a and b if | a |= 1 , | b |= 2 and a  b = i + j + k ?
(a) π/2 (b) π/3 (c) 2π/2 (d) π/6
5. The area of a parallelogram whose one diagonal and one side are represented by 2i and −3 j is:
(a) 6 sq. units (b) 36 sq. units (c) 3 sq. units (d) 3/2 sq. units
6. If 3x + 2y = sin y, then dy/dx is:
3 sin y − 1 2 − sin y 2 − cos y
(a) (b) (c) (d)
cos y − 2 2 3 3
dy
7. If y = a 2 − x 2 , then y is:
dx
(a) 0 (b) x (c) -x (d) 1
2
dx
8. The value of x
1 x2 −1
is:

(a) π/3 (b) π/2 (c) π/4 (d) π/6

9. The area (in sq. units) enclosed by the curve shown in the given figure is:
 (a) 8/3 (b) 24/7 (c) 32/3 (d) 16/3
3 2
log x
10. The value of 
2
x
dx is:

(a) log 6 log(3/2) (b) log(3/2)

1
(c) 2 log 3 (d) log 6
3
dy 1+ y
11. The integrating factor of +y= is:
dx x
ex e− x
(a) (b) (c) xex (d) x2ex
x x
 2 x + y 4 x   7 7 y − 13
12. If  = x + 6 
then the value of x + y is:
 5x − 7 4 x   y
(a) 7 (b) 4 (c) 5 (d) 2
1 1 1
13. The maximum value of ∆ = 1 1 + sin  1 , where θ is a real number is:
1 + cos  1 1
1
(a) 1 (b) (c) 3 (d) -1
2
14. If A is a symmetric matrix, then A3 is:
(a) symmetric matrix (b) skew-symmetric matrix
(c) Identity matrix (d) row matrix
 2 a 5
15. If the matrix B =  −1 4 b  is a symmetric matrix, then a + b + c is:
 c −4 9 
(a) 0 (b) 5 (c) 4 (d) -1
16. If set A contains 5 elements and the set B contains 6 elements, then the number of one-one and
onto mapping from A to B is:
(a) 600 (b) 56 (c) 65 (d) 0
-1
17. The principal value branch of cosec x is:
(a) {−π/2, π/2} (b) {−π/2, π/2} – {0}
(c) {-∞, ∞} (d) {-π, π} – {0}
a b c a d g
18. If d e f = P, then what is the value of b e h , given P = 17?
g h i c f i
(a) 17 (b) -17 (c) 1/17 (d) −1/17
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.
19. Assertion (A): The value of iˆ  ( ˆj  kˆ) + ˆj  (iˆ + kˆ) + kˆ  (iˆ  ˆj ) is 1.
Reason (R): Since, iˆ  iˆ = ˆj  ˆj = kˆ  kˆ = 0
1
20. Assertion (A): If (3x 2 + 2 x + k )  dx = 0 , then the value if k is -1.
0
 x n +1
Reason (R): x n  dx = .
n +1
SECTION – B
Questions 21 to 25 carry 2 marks each.

21. Let S be the set of points in a plane and R be a relation in S defined as R = {(A, B) : d(A, B) < 2}
where d(A, B) represents the distance between the points A and B. Is R an equivalence relation?
OR
 
Find the value of cot  − 2 cot −1 3 
4 

22. In a linear programming problem, objective function, z = x + 2y. The subjective the constraints
x + 2y ≥ 100, 2x – y ≤ 0, 2x + y ≤ 200, x ≥ 0, y ≥ 0
The graph of the following equations is shown below.

Name the feasible region, and find the corner point at which the objective function is minimum.
 
23. Evaluate:  1 − sin 2 xdx, x
4 2
OR
dx
Evaluate:  2
sin xcos 2 x
24. Find the vector in the direction of the vector i − 2 j + 2k that has magnitude 9.
25. For the differential equation, find a particular solution satisfying the given condition (1 + sin 2x)
dy + (1 + y2) cos x dx = 0, given that when x = π/2, y = 0

SECTION – C
Questions 26 to 31 carry 3 marks each.
26. 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.
x +1
27. Evaluate:  dx
( x + 2)( x + 3)
OR

x sin x
Evaluate:  dx
0
1 + cos 2 x
dy  y
28. Find the particular solution of the differential equation: x = y − x tan   , x ≠ 0. Given that
dx x

y= , when x = 1.
4
OR
 dy
Find the particular solution of the differential equation: + 2 y tan x = sin x , given that y = 0
dx

when x = .
3
29. A class has 15 students whose ages are 14, 17, 15, 14, 21, 17, 19, 20, 16, 18, 20, 17, 16, 19, 20
years one student is selected in such a manner that each has the same chance of being chosen and
the age X of the selected student is recorded. What is the probability distribution of random
variable X
30. The feasible region of a ∠PR is given as follows:

(i) Write the constraints with respect to the above in terms of x and y.
(ii) Find the coordinate of B and C and maximize, z = x + y.
31. Find the shortest distance between the lines
r = (i + 2 j + k ) +  (i − j + k ) and r = (2i − j − k ) +  (2i + j + 2k )
SECTION – D
Questions 32 to 35 carry 5 marks each.
x2 y2
32. Find the area of the smaller region bounded by the ellipse + = 1 and the straight line
16 9
3x + 4y = 12.
OR
Using integration, find the area bounded by the tangent to the curve 4y = x2 at the point (2, 1) and
the lines whose equations are x = 2y and x = 3y – 3.
1 −1 2   −2 0 1 
33. Use product 0 2 −3  9 2 −3 to solve the system of equations
 3 −2 4   6 1 −2 
x – y + 2z = 1 , 2y – 3z = 1 , 3x – 2y + 4z = 2
OR
2 3
Show that the matrix A =   satisfies the equation A – 4A + I = O, where I is 2 × 2 identity
2

 1 2 
matrix and O is 2 × 2 zero matrix. Using this equation, find A–1.
34. Prove that the height of the cylinder of maximum volume that can be inscribed in a sphere of
radius R is 2R/√3. Also find the maximum volume. L
35. Find the equation of a line passing through the point P(2, -1, 3) and perpendicular to the lines:
r = (iˆ + ˆj − kˆ) +  (2iˆ − 2 ˆj + kˆ) and r = (2iˆ − ˆj − 3kˆ) +  (iˆ + 2 ˆj + 2kˆ)
SECTION – E(Case Study Based Questions)
Questions 36 to 38 carry 4 marks each.
36. Case-Study 1: Read the following passage and answer the questions given below.
An electric circuit includes a device that gives energy to the charged particles constituting the
current, such as a battery or a generator; devices that use current, such as lamps, electric motors,
or computers; and the connecting wires or transmission lines.
An electric circuit consists of two subsystems say A and B as shown below:
 For previous testing procedures, the following probabilities are assumed to be known.
P(A fails) = 0.2, P(B fails alone) = 0.15, P(A and B fail) = 0.15
Based on the above information answer the following questions:
(a) What is the probability that B fails? [1]
(b) What is the probability that A fails alone? [1]
(c) Find the probability that the whole of the electric system fails? [2]
OR
Find the conditional probability that B fails when A has already failed. [2]
37. Case-Study 2:
Mohini purchased a rectangular parallelopiped shaped box and a spherical ball inside it as a
showpiece. The sides of the box are x, 2x and x/3, and the radius of the sphere is y.

The sum of the surface area of the parallelopiped and sphere is given to be constant.
Based on the above information, answer the following questions:
(a) Let the constant surface area given to be S, then what is the relation between x and y? [1]
(b) If the combined volume is denoted by V, then what is the value of V? [1]
(c) If volume V is minimum, then how are x and y is related to each other? [2]
OR
If the shape has minimum volume when x = 2y, then what is the difference in the volume and
surface area of the shape? [2]
38. Case-Study3: Read the following passage and answer the questions given below.
A general election of Lok Sabha is a gigantic exercise. About 911 million people were eligible to
vote and voter turnout was about 67%, the highest ever.
Let f(x) be the set of all citizens of India who were eligible to exercise their voting right in the
general election held in 2019. A relation ‘R’ is defined on I as follows:
b b
If f(x) is a continuous function defined on [a, b]  f ( x)dx =  f (a + b − x)dx on the basis of the
a a

above information answer the following equations:


2
cos x
(a) Evaluate:  1 + e
−
x
dx [2]
2

cos 2 x
(b) Find the value of  1 + a x dx , a > 0.
−
[2]