Class 12 - Mathematics

Sample Paper - 01 (2024-25)

Maximum Marks: 80
Time Allowed: : 3 hours

General Instructions:

i. This Question paper contains 38 questions. All questions are compulsory.

ii. This Question paper is divided into five Sections - A, B, C, D and E.
iii. In Section A, Questions no. 1 to 18 are multiple choice questions (MCQs) and Questions no. 19 and 20 are Assertion-
Reason based questions of 1 mark each.
iv. In Section B, Questions no. 21 to 25 are Very Short Answer (VSA)-type questions, carrying 2 marks each.
v. In Section C, Questions no. 26 to 31 are Short Answer (SA)-type questions, carrying 3 marks each.
vi. In Section D, Questions no. 32 to 35 are Long Answer (LA)-type questions, carrying 5 marks each.
vii. In Section E, Questions no. 36 to 38 are Case study-based questions, carrying 4 marks each.
viii. There is no overall choice. However, an internal choice has been provided in 2 questions in Section B, 3 questions in Section
C, 2 questions in Section D and one subpart each in 2 questions of Section E.
ix. Use of calculators is not allowed.

Section A

[ ]
0 1
1. If A = 2023 is equal to
0 0 , then A

a)
[
2023
0 ] 0
2023

[ ]
0 1
b)
0 0

[ ]
0 2023
c)
0 0

[ ]
0 0
d)
0 0

2. If A is a 3 × 3 matrix and |A| = -2, then value of |A (adj A)| is

a) -2
b) 8
c) 2
d) -8

3. The system of linear equations

5x + ky = 5,
 3x + 3y = 5;
will be consistent if:

a) k = 5
b) k = -5
c) k ≠ -3
d) k ≠ 5

[ ]
3 − 2x x+1
4. If the matrix A = 2 4 is singular then x = ?.

a) 1
b) 0
c) -1
d) -2

5. The direction ratios of a line parallel to z-axis are:

a) < 0, 0, 0 >
b) < 1, 1, 0 >
c) < 1, 1, 1 >
d) < 0, 0, 1 >

6. If y = e-x (A cos x + B sin x), then y is a solution of

d 2y dy
a) +2 + 2y = 0
dx 2 dx
d 2y dy
b) −2 + 2y = 0
dx 2 dx
d 2y
c) + 2y = 0
dx 2
d 2y dy
d) 2 + 2 dx = 0
dx

7. In an LPP, if the objective function z = ax + by has the same maximum value on two corner points of the feasible region,
then the number of points at which zmax occurs is:

a) finite
b) 0
c) infinite
d) 2

→ → →
8. In △ABC, AB = î + ĵ + 2k̂ and AC = 3î − ĵ + 4k̂. If D is mid-point of BC, then vector AD is equal to:

a) î − ĵ + k̂
b) 2î − 2ĵ + 2k̂
c) 4î + 6k̂
d) 2î + 3k̂
 π
8
9. ∫ tan2 (2x) is equal to
0

4+π
a)
8
4−π
b)
8
4−π
c) 2
4−π
d) 4

10. If A =
[ ] 1
0
0
0
and B =
[ ]
1 1
0 0
, then B'A' is equal to:

[ ]
1 1
a)
0 0

[ ]
1 1
b)
1 1

[ ]
1 0
c) 1 0

d)
[ 0 0
0 0 ]
11. Objective function of an LPP is

a) a function to be optimized
b) a function between the variables
c) a constraint
d) a relation between the variables

→ π → →
12. If the angle between the vectors →
a and b is and | →
a × b | = 1, then →
a ⋅ b is equal to
4

a) -1
1
b)
√2
c) √2
d) 1

13. Find the area of the triangle with vertices (0,0), (4,2), and (1,1).

a) 1 sq.unit
b) 2 sq.unit
c) 0 sq.unit
d) 5 sq.unit
 ()
¯
1 2 1 A
ˉ = , P(B)
14. For any two events A and B, if P(A) ˉ = and P(A ∩ B) = 4 , then P equals:
2 3 ¯
B

1
a)
4
3
b)
8
8
c)
9
1
d)
8

dz z z
15. Which of the following transformations reduce the differential equation dx
+ x logz = (logz) 2 into the form
x2
du
+ P(x)u = Q(x)
dx

a) u = ex
b) u = log x
c) u = (log z)2
d) u = (log z)-1

→ → → →
16. ABCD is a rhombus whose diagonals intersect at E. Then EA + EB + EC + ED equals

→
a) 0
→
b) AD
→
c) 2AD
→
d) 2BC

dy
17. If y = √sinx + √sinx + √sinx + …∞ then dx = ?
sin x
a)
( 2y − 1 )
cos x
b) (y−1)
cos x
c)
( 2y − 1 )
sin x
d)
( 2y + 1 )

18. The direction ratios of two lines are a, b, c and (b - c), (c - a), (a - b) respectively. The angle between these lines is

π
a)
2
π
b)
4
 π
c) 3
3π
d)
4

19. Assertion (A): If two positive numbers are such that sum is 16 and sum of their cubes is minimum, then numbers are 8,
8.
Reason (R): If f be a function defined on an interval I and c ∈ l and let f be twice differentiable at c, then x = c is a
point of local minima if f'(c) = 0 and f"(c) > 0 and f(c) is local minimum value of f.

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.

{
n
if n is even
2
20. Assertion (A): A function f: N → N be defined by f(n) = (n+1) for all n ∈ N; is one-one.
if n is odd
2

Reason (R): A function f: A → B is said to be injective if a ≠ b then f(a) ≠ f(b).

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.
Section B

1 1
21. Find the value of cos − 1 + 2sin − 1 .
2 2

OR

Find the value of tan − 1 −

( ) 1

√3
+ cot − 1
()
1

√3 [ ( )]
+ tan − 1 sin
−π
2
.

22. Find the absolute maximum and minimum values of the function f given by f(x) = cos2 x + sin x, x ∈ [0, π]

23. Find the rate of change of the area of a circle with respect to its radius r when

a. r = 3 cm
b. r = 4 cm

OR

Find the intervals of function f(x) = (x - 1)(x - 2)2 is

a. increasing
b. decreasing.

24. Evaluate: ∫ 41f(x)dx, where f(x) =|x - 1| + |x - 2| + |x - 3|

25. A ladder 13 m long is leaning against a vertical wall. The bottom of the ladder is dragged away from the wall along the
ground at the rate of 2 cm/sec. How fast is the height on the wall decreasing when the foot of the ladder is 5 m away
from the wall?

Section C

2x + 5
26. Evaluate ∫ dx.
√7 − 6x − x2
27. There are three coins. One is a coin having tails on both faces, another is a biased coin that comes up tails 70% of the
time and the third is an unbiased coin. One of the coins is chosen at random and tossed, it shows tail. Find the probability
that it was a coin with tail on both the faces.

5x 2
28. Evaluate the definite integral ∫ 21
x 2 + 4x + 3

OR

1
Evaluate: ∫
−2
√5 − 4x − x2 dx

[ () ]
29. Find the particular solution of the differential equation xsin 2
y
x
π
− y dx + x dy = 0, given that y = when x = 1
4

OR

dy 2
Find the general solution of xlogx dx + y = x logx

30. Find the Maximum and Minimum value of 2x + y

Subject to the constraints:
x + 3y ≥ 6,
x - 3y ≥ 3,
3x + 4y ≤ 24,
- 3x + 2y ≤ 6,
5x + y ≥ 5,
where non-negative restrictions are x, y ≥ 0.

OR

Solved the linear programming problem graphically:

Maximize Z = 60x + 15y
Subject to constraints
x + y ≤ 50
3x + y ≤ 90
x, y ≥ 0

{
1, if x ≤ 3
31. Find the values of a and b so that the function f given by f(x) = ax + b, if 3 < x < 5 is continuous at x = 3 and x = 5
7, if x ≥ 5
 Section D

32. Using integration, find the area of the triangle formed by positive X-axis and tangent and normal to the circle x2 + y2 = 4
at (1, √3).
OR,
x
33. Show that the function f : R → {x ∈ R : -1 < x < 1} defined by f(x) = 1+ |x|
,x ∈ R is one-one and onto function.

OR

​ et N be the set of all natural numbers and let R be a relation on N × N, defined by

L
{a, b) R (c, d) ⇔ ad = bc for all (a, b), (c, d) ∈ N × N
Show that R is an equivalence relation on N × N. Also, find the equivalence class [(2, 6)]

[ ]
2 −3 5
34. If A = 3 2 − 4 , find A-1. Using A-1 solve the system of equations 2x - 3y + 5z = 11; 3x + 2y - 4z = -5; x + y - 2z
1 1 −2
= -3

35. Show that the straight lines whose direction cosines are given by the

OR

Find the shortest distance between the lines l1 and l2 whose vector equations are

→
r = î + ĵ + λ(2î − ĵ + k̂) ...(1)
→
and r = 2î + ĵ − k̂ + μ(3î − 5ĵ + 2k̂) ...(2)

Section E

36. Read the following text carefully and answer the questions that follow:
Akshat and his friend Aditya were playing the snake and ladder game. They had their own dice to play the game. Akshat
was having red dice whereas Aditya had black dice. In the beginning, they were using their own dice to play the game.
But then they decided to make it faster and started playing with two dice together.

Aditya rolled down both black and red die together.

First die is black and second is red.

i. Find the conditional probability of obtaining a sum greater than 9, given that the black die resulted in a 5. (1)
ii. Find the conditional probability of obtaining the sum 8, given that the red die resulted in a number less than 4. (1)
iii. Find the conditional probability of obtaining the sum 10, given that the black die resulted in even number. (2)
OR
 Find the conditional probability of obtaining the doublet, given that the red die resulted in a number more than 4. (2)

37. Read the following text carefully and answer the questions that follow:
The slogans on chart papers are to be placed on a school bulletin board at the points A, B and C displaying A (follow
Rules), B (Respect your elders) and C (Be a good human). The coordinates of these points are (1, 4, 2), (3, -3, -2) and
(-2, 2, 6), respectively.

→ →
i. If →
a, b and →
c be the position vectors of points A, B, C, respectively, then find | →
a +b +→
c | . (1)
ii. If →
a = 4î + 6ĵ + 12k̂, then find the unit vector in direction of →
a. (1)
iii. Find area of △ABC. (2)
OR
Write the triangle law of addition for △ABC. Suppose, if the given slogans are to be placed on a straight line, then
→ →
the value of | →
a ×b +b ×→
c +→
c ×→
a | . (2)

38. Read the following text carefully and answer the questions that follow:
A gardener wants to construct a rectangular bed of garden in a circular patch of land. He takes the maximum perimeter
of the rectangular region as possible. (Refer to the images given below for calculations)

i. Find the perimeter of rectangle in terms of any one side and radius of circle. (1)
ii. Find critical points to maximize the perimeter of rectangle? (1)
iii. Check for maximum or minimum value of perimeter at critical point. (2)
OR
If a rectangle of the maximum perimeter which can be inscribed in a circle of radius 10 cm is square, then the
perimeter of region. (2)
.