Class 12 Sample Paper 2025 Mathematics

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Practice Paper
(2024-25)
Class – XII
Mathematics (Code: 041)

Time: 3 hours Maximum Marks: 80

General Instructions :
Read the following instructions very carefully and strictly follow them :

1. This Question paper contains 38 questions divided into five sections A,B,C,D,E. Each
section is compulsory. However, there are internal choices in some questions.
2. Section A has question number (1-18) as MCQ’s and Question number (19-20 )
Assertion-Reason based questions of 1 mark each.
3. Section B has Question number (21-25) of Very Short Answer (VSA)-type questions of
2 marks each.
4. Section C has Question number (26-31) of Short Answer (SA)-type questions of 3 marks
each.
5. Section D has Question number (32-35 ) of Long Answer (LA)-type questions of 5 marks
each.
6. Section E has Question number (36-38) of Source based/Case based/passage
based/integrated units of assessment questions (4 marks each) with sub parts.
7. There is no overall choice however an internal choice have been provided in 2 questions
in Section -B , 3 questions in Section- C and 2 questions in Section- D

Section – A
Question Number 1-18 are of MCQ type question of one mark each.
1. The domain of the function sin −1 ( 4 x )is : 1

(a) [-4,4] (b ) [-2, 2]

(c) [-1,1] (d) [-0.25, 0.25]

2. 1
If a matrix A= [3 k10− 3 2kk+5+5] is symmetric then , the value of k is :
(a) 8 (b ) 5

(c) -0.4 1+ √ 1561
(d)
12

3.
If A= [cos
sin α cos α ]
α − sin α
and A + A T =I , then the value of tan α
1

is:
(a) 1 1
(b)
√3

(c) √ 3 (d) 0

4. 1
For what values of k , the function given below is continuous at x=0?

{
√ 4+ x − 2 , x ≠ 0
f ( x )= x
k, , x=0

(a) 0 (b) 1/4

(c) 1 (d) 4

5. If P ,Q, and PQ are matrices of order 3 × 2 , a × b and 3 × 4 respectively then the number of 1
elements of the matrix Q is?

(a) 6 (b ) 8

(c) 4 (d) 12

6. The function , f(x)=x|x|, at x=0 is : 1

(a) Continuous and differentiable (b ) Continuous but not differentiable

(c) Differentiable but not Continuous (d) Neither differentiable nor Continuous

7. 1
3 2 3

∫ x 2 dx=k ∫ x 2 dx +∫ x 2 dx , then value of k is :
−2 0 2

(a) 2 (b ) 1

(c) 0 1
(d)
2

2
8. Derivative of e sin x with respect to cos x is : 1

2 2
(a) sin x . e sin x (b ) cos x . e sin x

2 2
(c) − 2 cos x . e sin x (d) − 2 sin 2 x cos x . e sin x

9. The value of 1
π
2

∫ ( sin2025 x − cos2025 x ) dx is equal to :
0

(a) 0 π
(b)
2

π (d) π
(c)
4

10. dy 1
The integrating factor of the differential Equation ( 1− y )
2
+ yx=ay ( −1< y <1 ) is :
dx

1 1
(a) (b )
2
y −1 √ y2 − 1
−1 1
(c) (d)
√ 1− y 2 √ 1− y 2

11. dy 1 1
The solution of differential equation = is:
dx logy
(a) logy =x+c (b ) y logy -y=x+c

(c) logy-y=x+c (d) y logy+=x+c

12. If the diagonal of parallelogram are ⃗ ^
d 1=3 iand ⃗
d 2=4 ^jthen its area is given by : 1

(a) 2 sq unit (b )3 sq unit

(c) 6 sq unit (d) 12 sq unit

13. If a^ and b^ be two unit vectors and ' θ ' is the angle between them , then |a^ − b^|: 1

θ θ
(a) sin (b )2 sin
2 2

θ θ
(c) cos (d) 2 cos
2 2

15. The maximum value of the object function Z=5x+10y subject to the constraints 1
x +2 y ≤ 120 , x + y ≥ 60 , x − 2 y ≥ 0 , x ≥ 0 , y ≥ 0 is:
(a) 300 (b ) 600

(c) 400 (d) 800

16. Two events A and B will be independent , if : 1
(a) A and B are mutually exclusive (b)P(A)=P(B)

(c) P ( Ā B̄ )=[ 1 − P ( A ) ][ 1 − P ( B ) ] (d) P(A) +P(B)=1

17. Corner points of the feasible region determined by the system of linear constraints are (0, 10) , 1
(5, 5) ,(15, 15) ,(0, 20) let Z=px+qy where p , q > 0 . Conditions on p and q so that maximum of Z
occurs at both the points (15, 15) and (0 ,20) is :

(a) q=3p (b ) p=2q

(c) q=2p (d) p=q

18. 1

If x + y ≤ 2 ,x , y ≥ 0 ,the point at which maximum value of 3x+2y attained , will be :
(a) (0, 2) (b ) (0, 0)

(c) (2, 0)
(d) ( 12 , 12 )

ASSERTION-REASON BASED QUESTIONS

Question number 19 and 20 each carry one mark

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):Principal value of cos−1 ( −21 ) is 23π 1

Reason (R) : Domain of cos−1 x is R

20. Assertion(A) :Vector equation of a line passing through through the points A(1, 2, 3) ,and B(4, 5, 1
^ ^j+6 k^ ) + λ ( i+
6)is r⃗ =( 4 i+5 ^ ^j+ k^ )

Reason (R) : Equation of a line passing through a point with position vector ⃗a and parallel to a
vector⃗b is ,r⃗ =⃗a + λ ⃗b

Section B
This section contains 5 Very Short Answer (VSA)-type questions of 2
marks each.
21.
Find the value of sin
−1
( )
−
1
2 ( )
+cos−1 −
√ 3 +cot −1 tan 4 π
2 3 ( )
2

OR
Find the domain of the function f ( x )=sin −1 ( x 2 − 4 ). Also find its range.

{
22. sin 3 x 2
, if x ≠ 0
Find the value of k , If the function f ( x )= x
k, if x=0
is continuous at x=0

√
23. dy 1− y 2 2
If y √ 1 − x 2 + x √ 1− y 2=1 then prove that =−
dx 1− x 2
OR

Find the differential of sin 2 x w.r.t e cosx
24. A point moves along the curve y=x2 , if its abscissa increases at the rate 2 units/sec. At what rate 2
is distance from origin is increasing when point is at (2,4).
2
25. |x| 2
Find ∫ dx
−1 x
OR

Find

∫ x ( 1x−+12 x ) dx

Section C
This section contains 6 Short Answer (SA)-type questions of 3marks each.
2
26. dy cos ( a+ y ) dy 3
If siny=x cos (a+y), Then show that = , also show that =cos a , when x=0
dx cos a dx
27. Consider experiment of tossing a coin . If the coin shows head toss again , but if it shows tail , then throw 3
a die. Find the conditional probability of the event 'the die shows a number greater than 4' given that '
there is atleast one tail'.
OR
A discrete random variable X has the probability distribution as given below:
X 0 1 2 3

2 2
P(X=x) q 4p p 0.7 − 4 p

Find the values of p and q for which the mean of X , (E(x)) is largest .
π
28. 3
3
dx
Solve ∫
π 1+ √ cotx
6

OR
5

Solve ∫ |x +2|dx
−5
29.
Solve the differential equation x cos ()y dy
x dx
= y cos
y
x
+x () 3

OR

Find the particular solution of the differential equation
dy π
+ y cot x=2 x + x 2 cot x given that y=0 when x=
dx 2
30. Find the distance between the lines ^ ^j − 5 k^ ) + λ ( 2 i+3
r⃗ =( 3 i+3 ^ ^j+6 k^ )and 3
^ ^j − 4 k^ ) + μ ( 2 i+3
r⃗ =( i+2 ^ ^j+6 k^ )
31. 3
sinx x dy
If y=x + ( sinx ) , then find
dx
SECTION D

This section contains four Long Answer (LA)-type questions of 5marks each.
32. Using integration find the area enclosed by the curve 4 x 2 − 9 y 2=36 5

33. Let R be a relation defined on the set of natural numbers N as follows: 5
{( x , y ) : x ∈ N , y ∈ N , 2 x + y=41} . Find the domain and range of the relation R . Also verify whether
R is reflexive , symmetric and transitive .

OR

Check whether a function f : R → [ −1 1
]
, defind as f ( x )=
2 2
x
1+ x 2
is one one and onto or or not.

34. Find the image of the point (1, 2, 3) in the line 5
^ ^j+7 k^ + λ ( 3 i+2
r⃗ =6 i+7 ^ ^j − 2 k^ )
OR

^ ( t − 2 ) ^j+ ( 3 − 2 t ) k^ and
Find the shortest distance between the lines given by r⃗ =( 1− t ) i+

^ ( 2 s − 1 ) ^j − ( 2 s+1 ) k^
r⃗ =( s+1 ) i+

35. 5
Solve the following Linear Programming Problem graphically :
Maximize Z=100x +300y subject to constraints
x +2 y ≤ 12
2 x + y ≤ 12

4 x +5 y ≥ 20
x≥0, y≥0
Section E
Source based/Case based/passage based/integrated units of assessment
Questions
36. Utkarsh , Kavyansh and Myiesha appeared for an interview for three vacancies in the same post . The 1+1+2
probability of Utkarsh selection is 1/5. Kavyansh selection is 1/3 and Myiesha's selection is 1/4. The
event of selection is independent of each other .

Based on the above information answer the following questions :
(i)What is the probability that atleast one of them is selected?

(ii)Find P ( H̄G ) where G is the event of Kavyansh selection and H̄ denotes the event that Utkarsh is
not selected.
(iii)Find the probability that atleast one of them is selected.
OR
(III)Find the probability that exactly two of them are selected.

37. 1+1+2

The relation between fuel consumption F(l/100km) and speed V(km/h)
V2 V
under some constraints is given as F= − +14 .
500 4
On the basis of the above information answer the following questions:
(i) Find F, when V=40km/h.
dF
(ii) Find .
dV
(iii) Find the speed V for which fuel consumption F is minimum.
OR
dF
Find the quantity of fuel required to travel 600 km at the speed V at which =− 0.01
dV

38. Three shopkeepers A, B and C go to a store to buy stationary. A purchase 12 dozen 2+2
notebooks, 5 dozen pens and 6 dozen pencils. B purchase 10 dozen notebooks, 6 dozen
pens and 7 dozen pencils. C purchase 11 dozen notebooks, 13 dozen pens and 8 dozen
pencils. A notebook costs ₹ 0, a pen costs ₹ 12 and a pencil costs ₹ 3.

i) Represent the number of items purchased by shopkeepers A, B and C in matrix
form.
ii) If Y represents the matrix formed by the cost of each item, then find XY.

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