CBSE Class 12 Maths Half Yearly Sample Paper 2025 with Solution, Download PDF

No.

Units

Marks

1.

Relations & Functions

08

2.

Algebra

10

3.

Calculus

35

4.

Vectors and Three - Dimensional Geometry

14

5.

Linear Programming 

05

6.

Probability

08

Total

80

Internal Assessment

• Periodic Tests (Best 2 out of 3 tests conducted) 

• Mathematics Activities

20

10

10

Grand Total

100

SECTION-B

21

Find the principal value of cos 13π6.                    OR

Find the value of(3) --2.

2

22

If x=a tan3θ and y=a sec3, then find dydx  .

2

23

Evaluate :

sec2xtan2x+4dx                                         OR                        1-sin 2x dx, 4<x<2.

2

24

If a =2, b =7 and a b =-3i+j+2k, find the angle between a and b .

2

25

Let X be a random variable which assumes values x1, x2, x3, x4 such that 2PX=x1=3PX=x2=PX=x3=5PX=x4. Find the probability distribution of X.

OR

A die, whose faces are marked 1, 2, 3 in red and 4, 5, 6 in green is tossed. Let, A be the event “number obtained is even” and B be the event “number is marked red”. Find whether the events A and B are independent or not.

2

SECTION C

Sl. No.

Question

Marks

26.

Let X denote the number of hours you study during a randomly selected school day. The probability 

         that X can take the values x, has the following form, where k is some unknown constant.

1. (a)  Find the value of k. 

2. (b)  What is the probability that you study at least two hours ?

3. (c) What is the probability that you study at most two hours?

3

27.

Evaluate 5x+3x2+4x+10dx

3

28.

Solve the differential equation (y  –x)dy=(1+y2)dx.

OR

Show that the family of curves for which the slope of the tangent at any point (x,y)

    on it is x2+y22xy, is given by x2-y2=cx

3

29.

Evaluate 04sin x+cos x   9+16sin 2x  dx

OR

    Evaluate 04x+x-2+x-4dx

3

30.

If y=ex , -1≤x≤1, show that1-x2d2ydx2-x dydx-m2y=0

3

31.

Find the maximum value of Z = 3x + 4y subjected to constraints 

x + y ≤ 40, x+ 2y ≤ 60, x ≥ 0 and y ≥ 0.

OR 

Find the point where the minimum value of Z occurs : Z=2x+y, subject to constraints 

3x + y ≥ 9,

x + y ≥ 7,

x + 2y ≥ 8,

x ≥ 0, y ≥ 0.

3

SECTION D

Sl.No.

Question

Marks

32

If A = 2 3 1 -3 2 1 5 -4 -2     , find A-1   and use it to solve the system of equations:

2x – 3y + 5z = 11,     3x + 2y – 4z = - 5,        x + y – 2z = - 3

5

33

5

34

If A(1,2,-3) , B(4,0,3) and C(6,4,9) be the vertices of triangle ABC . If AD be the median then Find (i)Equation of side BC

(ii) Equation of Median AD 

(iii)Angle between AD and BC.(2+3 marks)

                                        OR

Find equation of line which passes through (1, 1, 1) and perpendicular to both the following lines

     x-12=y-23=z-34   and    x+21=3-y-2=z+14.

5

35

Using integration find the area of region bounded by 

 2x-y+2=0,  x=-2,  x=3 and x axis.

5

SECTION E

Sl.No.

Question

Marks

36

The reliability of a COVID PCR test is specified as follows:

Of people having COVID, 90% of the test detects the disease but 10% goes undetected. Of people free of COVID, 99% of the test is judged COVID negative but 1% are diagnosed as showing COVID positive. From a large population of which only 0.1% have COVID, one person is selected at random, given the COVID PCR test, and the pathologist reports him/her as COVID positive

    Based on the above information, answer the following

 (i) What is the probability of the ‘person to be tested as COVID positive’ given that ‘he is actually having COVID?

(ii) What is the probability of the ‘person to be tested as COVID positive’ given that ‘he is actually not having COVID’? 

(iii) What is the probability that the ‘person' is actually having COVID given that ‘he is tested as COVID positive’?

4

37

An architect designs a garden in a residential complex. The garden is in the shape of a rectangle inscribed in a large circle of radius 20 m as shown in the following figure. If the length and breadth of rectangle garden are 2x and 2y meters respectively.

Based on the above information answer the following 

(i) find the area A of the green grass of garden also find dAdx

(ii) find the maximum area of the garden.

4

38

Ishaan left his village on the weekend. First, he travelled up to the temple. After this , he left for the Zoo. After this, he left to go shopping in a mall. The position of Ishaan at different Places is given in the following graph 

Based on the above information, answer the following

(i) find the position vector of B and D in term i & j

(ii) find the length of vector AD .

(iii) find the vector BC . and if M =4 j+3 k then find the unit vector of M .

4

SECTION A

Sl. no.

Answer

Marks

1

D

1

2

A

1

3

C

1

4

A

1

5

B

1

6

D

1

7

A

1

8

C

1

9

D

1

10

B

1

11

A

1

12

D

1

13

D

1

14

C

1

15

C

1

16

B

1

17

B

1

18

B

1

19

D

1

20

A

1

SECTION B

Sl.No.

Answer

Marks

21

We know that, cos x =x, if x0, . 

Now, 

cos 136

=cos 2+6 , ∵ 1360,

=cos 6 , ∵cos 2+x=cos x 

=6, ∵60,

OR

(3) --2

=3--2 , ∵-x=-x 

=3--3

=3-23

=-3

2

2

22

x = a tan 3 θ 

Differentiating w.r.t.  θ

             dxdθ = a 3 tan2 θ sec2 θ 

Again, y = a sec3 θ

Differentiating w.r.t.  θ

            dydθ = a 3 sec2 θ sec θ tan θ  = 3 a tan θ sec3 θ

             dydx = dydθdxdθ=3 a tan θ sec3 θ/ a 3 tan2 θ sec2 θ 

                    =  sec θ  tan θ

                    = cosec θ

2

23

sec2xtan2x+4dx

Let,tan x=z

∴sec2x dx=dz

=dzz2+22

=log z+z2+22+c

=log tan x +tan2x+4+c

OR

1-sin 2x dx, 4<x<2

=sin2x+cos2x-2.sin x .cos x  dx

=sin x -cos x 2dx

=sin x -cos x dx,  in 4<x<2, sin x  is greater than cos x 

=-cos x -sin x +c

2

24

a b =a b sin θ  n , where n is a unit vector.

a b =a b sin θ n

-32+12+22=a b sin   n

9+1+4=2.7.sin   .1

14=14 sin  

1414= sin  

114= sin  

sin   =114

sin =±114

θ=114

2

25

Let, 2PX=x1=3PX=x2=PX=x3=5PX=x4=k

∴PX=x1=k2, PX=x2=k3, PX=x3=k, PX=x4=k5   

Now,

PX=x1+PX=x2+PX=x3+PX=x4=1

k2+k3+k+k5=1

15k+10k+30k+6k30=1

61k30=1

k=3061

OR

S=1, 2, 3, 4, 5, 6

A= 2, 4, 6

B=1, 2, 3

A∩B=2

PA=36=12

PB=36=12

PA∩B=16

PA.PB=12.12=14

∴PA∩B≠PA.PB

 Hence,A and B are not independent

2

2

SECTION C

Sl.No.

Answer

Marks

26

The probability distribution of X is 

3

27

3

28

3

29

OR 

For correct solution = 20

3

30

(1-x2)dydx =my

Again diff. w.r.t. x and solve we get 

3

31

Maximum value = 140

OR 

From the table, we find that 8 is the minimum value of Z at G(1, 6). Since the region is unbounded we have to check that the inequality 2x + y < 8 in the open half plane has any point in common or not. 

Since, it has no point in common. So, Z is minimum at G (1, 6) and the minimum value of Z is 8.

3

SECTION D

Sl.No.

Answer

Marks

32

5

33

5

34

OR

5

35

5

CHECK: CBSE Class12 Maths Half Yearly Sample Paper 2025 with Solution, Download PDF