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General Instructions : Same as given in PTS-01.

SECTION A
(Question numbers 01 to 20 carry 1 mark each.)
Followings are multiple choice questions. Select the correct option in each one of them.
01. Given that A is a square matrix of order 3 and adj.A  49 , then A 1 is equal to
1 1
(a) 7 (b)  (c)  (d) –7 only
49 7
 cos   sin  
02. For A    , A  A T equals
 sin  cos  
cos  0   cos 2 0   cos  0   0 cos  
(a) 2   (b)   (c)   (d) 2 
 0 cos    0 cos 2  0 cos   cos  0 
 
03. If a and b are parallel vectors, then which of the following is true?
        
(a) a .b  0 (b) a   b (c) a .b  0 (d) a  b  0
  3 
04. Let tan 1 : R   , 1
 . Then tan (1) 
2 2 
 3 5 7
(a)  (b) (c) (d)
4 4 4 4
05. Let A  {i, s, h, a} . If R : A  A is given by R  {(i, i), (s,s), (a, a), (a, h)} then, which of the
following ordered pair must be added to make R a reflexive relation?
(a) (h, a) (b) (s, h) (c) (h, h) (d) (h,i)
06. If m and n respectively, are the order and degree of the differential equation
2 2
 dy  d y
x    2  0 , then (m n) 
 dx  dx
(a) 1 (b) 2 (c) 3 (d) 4
07. The feasible region, for the inequalities x  0, x  y  1 and, y  0 , lies in
(a) IV Quadrant (b) III Quadrant (c) II Quadrant (d) I Quadrant

08. What is the number of vectors of unit length perpendicular to both the vectors a  2iˆ  ˆj  2kˆ

and b  ˆj  kˆ ?
(a) 0 (b) 1
(c) 2 (d) infinitely many unit vectors are possible
a
09. If f (x) is an odd function then, the value of  f (x)dx 
a
a a a/2
(a) 2 f (x)dx (b)  f (x)dx (c) 2  f (x)dx (d) 0
0 0 0

MATHEMATICIA By O.P. GUPTA : A New Approach in Mathematics 1

CBSE XII Sample Papers (2024-25) By O.P. GUPTA (INDIRA Award Winner)

2 4 9
10. Minor of the element 9 in   3 6 9 is
 2 3 1
(a) 0 (b) –3 (c) 3 (d) 1
11. The feasible region in a LPP is as shown in the graph below.
The corner points are denoted by A, B, C and O.

Let Z  2x  5y then, the value of Zmax  Zmin equals

(a) 0 (b) 4
(c) 10 (d) 1

 2sin x 1 3 0
12. If 0  x  , and  , then the values of x is
2 1 sin x 4 sin x
    
(a) (b) , (c) (d)
3 6 2 6 4
13. Given that the matrices A and B of order 3  m and 3  n respectively, are such that AB and
BA both exist, then order of A is
(a) 3  4 (b) 4  3 (c) 3  3 (d) cannot be determined
14. The probability distribution of a random variable X, where k is a constant, is given below :
0.1, if x  0
 2
kx , if x  1
P(X  x)   .
kx, if x  2 or 3
0, otherwise
Then, P(X  2) equals
(a) 0.15 (b) 0.55 (c) 0.25 (d) 0.45
 dy  y
15. Integration factor of the differential equation     x 2 is denoted by f (x) . Then f (x) 
 dx  x
1 1 1 1
(a) (b)  (c) 2 (d)  2
x x x x
dy
16. If y  x e , then 
dx
(a) x e (b) e. x e1 (c) e.x e (d) x e  log x
x 1 y 1 z  2
17. The direction angle made by the line   with positive direction of x-axis, is
1 2 1
    5
(a) (b) (c) (d) ,
3 6 4 6 6
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     
18. If a  2, b  2 3 and a  b , then the value of a  b is
(a) 16 (b) 4 (c) 4 (d) 16
Followings are 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 and 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) : There is no value of ‘b’ for which the function f (x)  x  cos x  b is strictly
decreasing over  (set of real numbers).
Reason (R) : If f (x)  0 in x  [a, b] then, f (x) is an increasing function in x  [a, b].

20. Assertion (A) : i  j  2k is a vector parallel to the line r  i  j  k   (i  j  2k)
 .

Reason (R) : In the vector form of line r  x i  y j  z k  (a i  b j  c k)
1 1 1 1 1 1
 , a vector parallel to

the line is a1i  b1 j  c1 k .

SECTION B
(Question numbers 21 to 25 carry 2 marks each.)
 2   2 
21. Using principal values, evaluate sin 1  sin   cos1  cos  .
 3   3 
OR
Let A  {1, 2,3} . Write all the possible equivalence relations defined on set A.
1 9
22. For the function f (x)  4x  x 2 ,  2  x  , find the absolute maximum value and absolute
2 2
minimum value.
23. Find the vector equation of the line joining the points (1, 2, 3) and (–3, 4, 3). Also write its
Cartesian equation.
OR
If a line makes the angles α, β and γ with the coordinate axes, then evaluate :
cos 2  cos 2  cos 2  .
d2 y 
24. If x  a sec , y  b tan  , then find 2
at   .
dx 6
OR
 log(1  4x)  log(1  x)
 , if x  0
If f (x)   x is continuous at x  0 , then find the value of k.
 k, if x  0
        
25. If p  q  r  0 and p  3, q  5, r  7 then find the angle between p and q .
SECTION C
(Question numbers 26 to 31 carry 3 marks each.)
1
26. Find :  sin(x  a) cos(x  b) dx .
27. Three cards are drawn successively with replacement from a well shuffled pack of 52 cards.
Find the mean of the number of red cards.
OR

MATHEMATICIA By O.P. GUPTA : A New Approach in Mathematics 3

CBSE XII Sample Papers (2024-25) By O.P. GUPTA (INDIRA Award Winner)

A girl throws a die. If she gets a 5 or 6, she tosses a coin three times and notes the number of
heads. If she gets 1, 2, 3 or 4, she tosses a coin once and notes whether a head or tail is
obtained. If she obtained exactly one head, what is the probability that she threw 1, 2, 3 or 4
with the die?
2
28. Evaluate :  cos x
0
dx .

OR
3
Evaluate :   x  x  1 dx .
0

29. Solve the differential equation : x 2 ydx  (x 3  y3 )dy  0 .

OR
Find the particular solution of the following differential equation :

cos y dx  (1  2e  x ) sin y dy  0; y(0)  .
4
30. A linear programming problem is as follows.
To maximize: Z = (x  y)
Subject to constraints: 2x  y  50 , x  2y  40 , x  0, y  0 .
In the feasible region, find the point at which maximum value of Z occurs. Solve graphically.
x dx
31. Find :  2 .
x  3x  2
SECTION D
(Question numbers 32 to 35 carry 5 marks each.)
32. Using integration, find the area of the smaller region bounded between y  36  x 2 and x  4 .
33. Using differentiation, find two positive numbers whose sum is 15 and the sum of whose squares
is minimum.
OR
Two equal sides of an isosceles triangle with fixed base b (in centimeter) are decreasing at the
rate of 3 cm/s. How fast is the area decreasing when two equal sides are equal to the base?
34. Determine the equations of a line passing through the point (1, 2, –4) and perpendicular to the
x  8 y  19 z  10 x  15 y  29 5  z
two lines   and;   .
3 16 7 3 8 5
OR
      
If   3ˆi  ˆj and   2ˆi  ˆj  3kˆ , then express  in the form of   1  2 where 1 is parallel
  
to  and  2 is perpendicular to  .
2 3 10 
 
35. If A   4 6 5  , find A–1. Using A–1 solve the following system of equations :
6 9 20 

2 3 10 4 6 5 6 9 20
   2,    5,    4; x, y, z  0 .
x y z x y z x y z
SECTION E
(Question numbers 36 to 38 carry 4 marks each.)
This section contains three Case-study / Passage based questions.
First two questions have three sub-parts (i), (ii) and (iii) of marks 1, 1 and 2 respectively.
Third question has two sub-parts of 2 marks each.
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36. CASE STUDY I : Read the following passage and the answer the questions given below.
An organization conducted bike race under two different categories – Boys and Girls.
There were 28 participants in all. Among all of them, finally three from category 1 and two from
category 2 were selected for the final race. Ravi forms two sets B and G with these participants
for his college project.
Let B  {b1 , b 2 , b3 } and G  {g1 , g 2 }, where B represents the set of Boys selected and G the set
of Girls selected for the final race.

(i) How many relations are possible from B to G?

(ii) Among all the possible relations which are defined from B to G, how many functions can be
formed from B to G?
(iii) Let R : B  B be defined by R  {(x , y) : x and y are students of the same sex}.
Check if R is an equivalence relation.
OR
(iii) A function f : B  G be defined by f  {(b1 , g1 ), (b 2 , g 2 ), (b3 , g1 )}.
Check if f is bijective (i.e., one-one and onto both). Justify your answer.
37. CASE STUDY II : Read the following passage and answer the questions given below.

Mr Neeraj Jha is a business analyst. He offers his expert-views to the companies.

A ball manufacturing company hires Mr Jha for his services.
Mr Jha observed that P(x)  5x 2  1250x  30 (in `) is the total profit function of this ball
manufacturing company, where x is the production of the company.
(i) Differentiate P (x) with respect to x.
(ii) What will be the production when the profit is maximum?
(iii) What will be the maximum profit?
OR
(iii) Check if the profit function P(x) is strictly increasing in the interval x  (0, 125) ?
38. CASE STUDY III : Read the following passage and answer the questions given below.

MATHEMATICIA By O.P. GUPTA : A New Approach in Mathematics 5

CBSE XII Sample Papers (2024-25) By O.P. GUPTA (INDIRA Award Winner)

An electronic assembly consists of two kinds of sub-systems say, A and B.

From 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 .
(i) Find P(B fails) and, P(A fails alone) .
(ii) Find P(A fails | B has failed) .

6 MATHEMATICIA By O.P. GUPTA : A New Approach in Mathematics

 A BRIEF SYNOPSIS
Of CONTENTS IN

CBSE 21 SAMPLE PAPERS

For CBSE 2024-25 Exams  Class 12 Maths (041)

Pleasure Test Series

By O.P. Gupta

 Multiple Choices Questions

 Assertion-Reason (A-R) Questions
 Subjective type Questions (2 Markers, 3 Markers & 5 Markers)
 CASE STUDY QUESTIONS (As per Latest format for 2024-25)
 More H.O.T.S. Questions (As per official CBSE Sample Paper)
 Detailed Solutions of 16 Sample Papers
 ANSWERS of 5 Unsolved Sample Papers (with PDF / Video Solutions)

Most of the Pleasure Tests (PTS) are based on the Blueprint - same as that of CBSE
Official Sample Question Paper. Though, in some of the PTS we have adopted
different Blueprint : keeping in mind that the Unitwise weightage is not altered.

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