CBSE Class 12 Question Paper 2026 Mathematics

FOR CBSE CLASS 12 EXAM PREPARATION

CBSE Class 12 2026
Question Paper ·
Mathematics
EXAM YEAR TYPE SUBJECT

CBSE Class 12 2026 Question Paper Mathematics

Notes · Sample Papers · Previous Year Papers · Mock Tests

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a SET-1
Series : 1PQRS
- 
Q.P. Code 65/1/1
 .
 -   - 
Roll No.
-    
Candidates must write the Q.P. Code
m
.co
on the title page of the answer-book.
m
m .co s e m
s e  l a
g l a ag
a MATHEMATICS
  : 3    : 80
Time allowed : 3 hours Maximum Marks : 80

       -    23  
-        -    -  -   

       -  38    .co
m
        ,e- m         

las

 
 -     15 
g
a       -     10.15 
   10.15   10.30     -      
  -      
 Please check that this question paper contains 23 printed pages.
 Q.P. Code given on the right hand side of the question paper should be
written on the title page of the answer-book by the candidate.

o m
Please check that this question paper contains 38 questions.
 m Please write down the serial number of the question in the c
. answer-
c. o book at the given place before attempting it. m
se The question
e m  15 minute time has been allotted to read this question lpaper.
a
l as g to 10.30 a.m., the
paper will be distributed at 10.15 a.m. From 10.15 aa.m.
ag candidates will read the question paper only and will not write any answer
on the answer-book during this period. {}

2413-1
65/1/1 * 1 P.T.O.

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a For more Question Papers, Sample Papers, Notes & Syllabus visit Page 1 of 24

  :

             :
(i)  -   38        
(ii)  -      – -, , ,    
(iii)  –     1  18      19  20   
 1     
(iv)  –     21  25   - (VSA)   2     
(v)  –     26  31  - (SA)   3     
(vi)  –     32  35  - (LA)   5     
(vii)  –     36  38    4     
(viii) -         ,  –       
       
(ix)      

 – 
 1  20        1    

1.  2 cos–1x = y ,  1

(A) 0  y   (B) –  y  
(C) 0  y  2 (D) –  y  0

2.       -      ? 1

(A) 2  1 (B) 1  2
(C) 1  1 (D) 1  n

65/1/1 * 2 {}

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General Instructions :
a
Read the following instructions very carefully and strictly follow them :
(i) This Question paper contains 38 questions. All questions are compulsory.
(ii) Question paper is divided into FIVE Sections – Section A, B, C, D and E.
(iii) In Section A – Question Number 1 to 18 are Multiple Choice Questions
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(MCQs) and Question Number 19 & 20 are Assertion-Reason based
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questions of 1 mark each.

m B – Question Number 21 to 25 are Very Short Answer (VSA)las e
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(iv) In Section
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type questions, carrying 2 marks each. a
(v) a In Section C – Question Number 26 to 31 are Short Answer (SA) type
questions, carrying 3 marks each.
(vi) In Section D – Question Number 32 to 35 are Long Answer (LA) type
questions, carrying 5 marks each.
(vii) In Section E – Question Number 36 to 38 are case study based questions,
carrying 4 marks each.
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(viii) There is no overall choice. However, an internal choice has been provided
em
in few questions in all the Sections except Section – A.
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(ix) Use of calculator is NOT allowed.
a
SECTION – A
Question 1 to 20 are multiple choice questions of 1 mark each.

1. If 2 cos–1x = y, then 1
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(A) 0  y   (B) –  y  
m
c. o (C) 0  y  2 (D) –  y  0
s e m
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las 2.
g
a ?
Which of the following cannot be the order of a row-matrix
ag 1
(A) 2  1 (B) 1  2
(C) 1  1 (D) 1  n

65/1/1 * 3 {}

m .
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a For more Question Papers, Sample Papers, Notes & Syllabus visit Page 3 of 24

3.            /   / ? 1

(i) (A + B) = A + B (ii) (A – B) = B – A

(iii) (AB) = AB (iv) (kAB) = kBA (k   )

(A)  (i) (B) (i), (ii)  (iii)

(C) (i)  (ii) (D) (i)  (iv)

1 0 0 0 2 0
4.  1 = 0 2 0  2 = 1 0 0 ,  1
0 0 3 0 0 6

(A) 1 =22 (B) 2 =–21

(C) 1 =2 (D) 2 =–1

cos x sin x
5. x      = 1  1
 cos x sin x


(A) 0 (B) 4

 
(C) 3 (D) 2

6.   A  B       ,        
  ? 1

(A) AB (B) AB + BA
(C) (A + B)2 (D) A – B

7.  [0, 3]  f(x) = x3 – 12x     1

(A) –16 (B) –9
(C) 0 (D) 16
65/1/1 * 4 {}

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m .co s em
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3.
a
Which of the following properties is/are true for two matrices of suitable
orders ? 1
(i) (A + B) = A + B (ii) (A – B) = B – A

(iii) (AB) = AB (iv) (kAB) = kBA (k is a scalar)
(A) (i) only (B) (i), (ii) and (iii)
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(C) (i) and (ii) (D) (i) and (iv)
m
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s em l a
ag 1
1 0 0 0 2 0
4. g la
If 1 = 0 2 0 and 2 = 1 0 0 , then
a 0 0 3 0 0 6

(A) 1 =22 (B) 2 =–21

(C) 1 =2 (D) 2 =–1

cos x sin x
m = 1 is
.co
5. One of the values of x for which 1
 cos x sin x

s em 
(A) 0
g la (B) 4

a 
(C) 3 (D) 2

6. If A and B are skew symmetric matrices of same order, then which of the
following matrices is also skew symmetric ? 1
(A) AB (B) AB + BA m
m
c. o (C) (A + B) 2 (D) A – B m.co
m s e
s e g l a
g la 7. The least value of f(x) = x3 – 12x, x  [0, 3] is a 1
a (A) –16 (B) –9
(C) 0 (D) 16

65/1/1 * 5 {}

m .
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s em l a
g la ag
a For more Question Papers, Sample Papers, Notes & Syllabus visit Page 5 of 24

. 3ax
 b2 + c2x2 dx = A log | b + c x | + K ,  A    :
 
8. 2 2 2 1
.

3a
(A) 3a (B)
2b2

3a 3a
(C) (D)
b2c2 2c2

1
. x3
9. . 2 dx    : 1
 x + 2| x | + 1
–1

(A) 0 (B) log 2

1
(C) 2 log 2 (D) 2 log 2

10.  y = x| x |, x-   x = –1  x = 1       1

1
(A) 0 (B)
3

2 4
(C) (D)
3 3

dx
11.   R + Px = Q,  P, Q  R, y   ,     1
dy

P
 Q dy Pdy
(A) e (B) e

P P
 R dy  R dx
(C) e (D) e

65/1/1 * 6 {}

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m
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m .co s e m
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. 3ax
a
.b2 + c2x2 dx = A log | b + c x | + K, then the value of A is
If 
8. 2 2 2 1


3a
(A) 3a (B)
2b2

3a 3a
(C) (D)
b2c2 2c2 m
m .co
m. x .co s e m
s e 1
l a
ag 1
3
9.
g l a
The value of 
.
 x + 2| x | + 1
2
dx is
a –1

(A) 0 (B) log 2

1
(C) 2 log 2 (D) 2 log 2

m
10. The area bounded by the curve y = x| x |, x-axis and the ordinates x = –1
.co
and x = 1 is given by
s em 1

g la (B) 1
(A) 0 a 3

2 4
(C) (D)
3 3

dx
11. The integrating factor of differential equation R + Px = Q where P, Q, R
dy m
m
c. o are functions of y is .co
m s em 1

s e P
g la
la
 Q dy
g (A) e (B) e
Pdy
a
a P P
 R dy  R dx
(C) e (D) e

65/1/1 * 7 {}

m .
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s em l a
g la ag
a For more Question Papers, Sample Papers, Notes & Syllabus visit Page 7 of 24

d
12.   dx(ey) = 0    , :  1

(A) 0, 1 (B) 1, 1

(C) 2, 1 (D) 1,    

     
13. p       i + 2 j + 3k  2 i – p j + k   ,  1

(A) 0 (B) 1

5 5
(C) 2 (D) – 2

     
14. m          – i – j + 2k, 2 i + m j + 5k 
  
3 i + 11 j + 6k ,  ,  1

(A) 8 (B) –8

5
(C) 2 (D) 2

15.  | a | = 8, | b | = 3  | a  b | = 12 ,  | a  b |    1

(A) 6 3 (B) 8 3

(C) 12 3 (D) 3 12

x y z
16.  (2, 5, 7)   1 = 0 = 0        1

(A) 2 (B) 5

(C) 74 (D) 78

65/1/1 * 8 {}

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m
m .co

m .co s e m
se g l a
a d
12. The order and degree of the differential equation (ey) = 0 respectively
dx
are 1

(A) 0, 1 (B) 1, 1

(C) 2, 1 (D) 1, not defined

m
c o m m .co
13. The value of p .for which vectors i + 2 j + 3k and 2 i – p j + k are perpendicular e
     

emis l as
as
to each other
l a g 1

(A)ag0 (B) 1

5 5
(C) 2 (D) – 2

  
14. The value of m for which the points with position vectors – i – j + 2k,
     
m
.co
2 i + m j + 5k and 3 i + 11 j + 6k are collinear, is 1

(A) 8
s em(B) –8
g la (D) 5
(C) 2 a 2

15. If | a | = 8, | b | = 3 and | a  b | = 12, then the value of | a  b | 1

(A) 6 3 (B) 8 3

(C) 12 3 (D) 3 12
m
m .co
.co s e m
x y z
s em 16. The length of perpendicular drawn from point (2, 5, 7) onglline
a 1 = 0 = 0 is 1

g la a
a (A) 2 (B) 5

(C) 74 (D) 78

65/1/1 * 9 {}

m .
.co s e m
s em l a
g la ag
a For more Question Papers, Sample Papers, Notes & Syllabus visit Page 9 of 24

17.  LPP    Z = 5x + 7y ,        : 1

Z    – Z     :
(A) 8 (B) 29
(C) 35 (D) 43

18.  LPP       1
(A) 0 (B) 1
(C) 2 (D)   

 –   
 :   19  20  (A)   (R)          
    (A)     (R)           
    (A), (B), (C)  (D)     
(A)  (A)   (R)      (R),  (A)   
  
(B)  (A)   (R)   ,   (R),  (A)   
   
(C)  (A)  ,   (R)   
(D)  (A)  ,   (R)   

19.  (A) :        ,      
2
,           , 3   1

P(A  B)
 (R) :    A  B   P7(A|B) = P(B)
65/1/1 * 10 {}

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m
m .co

m .co s em
se g la
a
17. The feasible region of a linear programming problem with objective
function Z = 5x + 7y is shown below : 1

m
m .co
.co s e m
s em l a
g la ag
Thea maximum value of Z – minimum value of Z is
(A) 8 (B) 29
(C) 35 (D) 43

18. The degree of an objective function of a linear programming problem is 1
(A) 0 (B) 1
(C) 2 (D) Any natural number
m
m .co
e
Assertion – Reason Based Questions
s
la
Direction : Questions number 19 and 20 are Assertion and Reason
g
a
based questions carrying 1 mark each. Two statements are given, one
labelled Assertion (A) and other labelled Reason (R).
Select the correct answer from the codes (A), (B), (C) and (D) as given below.
(A) Both Assertion (A) and Reason (R) are true and the Reason (R) is the
correct explanation of the Assertion (A).
(B) Both Assertion (A) and Reason (R) are true, but Reason (R) is not the
correct explanation of the Assertion (A).
(C) Assertion (A) is true and Reason (R) is false.
m
m (D) Assertion (A) is false and Reason (R) is true.
c.19.o Assertion (A) : In an experiment of throwing an unbiased m die, the .co
s e
s em g l a given that
probability of getting a prime number
la a 2
ag number appearing on the die being odd is 3 . 1

P(A  B)
Reason (R) : For any two events A and B, P(A|B) = P(B)

65/1/1 * 11 {}

m .
.co s e m
s em l a
g la ag
a For more Question Papers, Sample Papers, Notes & Syllabus visit Page 11 of 24

20.  (A) : x = py + q, z = ry + s  x = p'y + q', z = r'y + s'    
    pp' + rr' = 1   1

 (R) :   r = a 1 + b 1  r = a 2 + b 2     b 1 b 2 = 0


 – 
    -         2   

21. (a)     f(x)     2

 x 3
 , x3
f(x) =  2( x  3 ) x = 3      
 x 6 , x 3
 6


1 3 dy
(b)  3(x2 + y2) = 4xy ,  2 , 2   dx    2

22.               ,   

       1 mm3/min.          6
            ,       10 mm
  2

65/1/1 * 12 {}

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m
m .co

m .co s em
se g la
a
20. Assertion (A) : Lines given by x = py + q, z = ry + s and x = p'y + q',
z = r'y + s' are perpendicular to each other when
pp' + rr' = 1. 1

Reason (R) : Two lines r = a 1 + b 1 and r = a 2 + b 2 are perpendicular

to each other if b 1 b 2 = 0.

m
m .co
.co SECTION – B
m comprises Very Short Answer (VSA) type questions of las e m
This section
s e g
2 marks
g l a each.
a
21. (a) a Check whether function f(x) defined as 2

 x 3
 , x3
f(x) =  2( x  3 ) is continuous at x = 3 or not ?
 x  6
 6 , x3

OR
m
(b) If 3(x2 + y2) = 4xy, then find
dy 1
.co 3
at 2 , 2 . 2
s em
dx  

g l a
22. A room freshner bottle in athe shape of an inverted cone sprays the
perfume at regular intervals such that volume of the perfume in the bottle
decreases at the steady rate of 1 mm3/min. Find the rate at which level of
perfume is dropping at an instant when level of perfume in the bottle is

10 mm, if the semi-vertical angle of conical bottle is 6 . 2

m
m .co
m .co s e m
s e g l a
g la a
a

65/1/1 * 13 {}

m .
.co s e m
s em l a
g la ag
a For more Question Papers, Sample Papers, Notes & Syllabus visit Page 13 of 24


23.   P(1, 3, 2)  Q(–1, 0, 8)   QP     14    

  2
    
24.  a = 3 i – 2 j + 2k  b = i + 2k         
               :   
     2

 cos 2x  sin 2x  
25. (a)   : tan–1   , 0 < x < 4. 2
 cos 2x  sin 2x 

  1 
(b)    : tan  sin 1 1  cos 1     2
  2 

 – 
   -         3   
1
.
26.    : 
.x tan–1x dx.

3
0

. x+2
27. (a)   : . dx 3
 x–2

. x2
(b)   : .(x2 + 9) (x2 + 16) dx 3


/4 1/2
. dx .
28.  I1 = .  I2 = .| x | dx ,    I1 – 4I2 = 0. 3
 1 + cos 2x 
–/4 –1/2

dy
29. (a)   x2 dx = x2 + xy + y2       3

dy
(b)   xy dx = (x + 2) (y + 2),     ,   
y(1) = –1. 3

65/1/1 * 14 {}

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m
m .co

m .co s e m
se g l a
a 
23. Find the vector of magnitude 14 in the direction of QP, where P and Q are
the points (1, 3, 2) and (–1, 0, 8) respectively. 2

    
24. Vectors a = 3 i – 2 j + 2k and b = i + 2k represent the two adjacent sides of
a parallelogram. Find the vectors representing its diagonals and hence
find their lengths. 2
m
m
 cos 2x  sin 2x  .co
.co

25. (a) Simplify : tan–1  , 0 < x < 4 . 2
e m
em
 cos 2x  sin 2x 
l as
l as OR
ag
(b) agEvaluate : tan  sin 1  cos    
  1 
1 1
2
  2 

SECTION – C
This section comprises Short Answer (SA) type questions of 3 marks each.
1
.
26. Evaluate : .x tan–1x dx. 3

0
m
m .co
27. (a)
.
Find .
x+2
dx s e
la
3
 x–2
OR ag
. x2
(b) Find : . 2 dx 3
 (x + 9) (x2 + 16)

/4 1/2
. dx .
28. If I1 = . and I2 = .| x | dx, then show that I1 – 4I2 = 0. 3
 1 + cos 2x 
–/4 –1/2

o: m
m . c
. co
29. (a) Find
dy
the general solution of the following differential equation
s e m 3

s em x
dx
2
= x + xy + y 2 2

g l a
g la OR a
a (b) Find the particular solution of the differential equation 3
dy
xy = (x + 2) (y + 2), given that y(1) = –1.
dx

65/1/1 * 15 {}

m .
.co s e m
s em l a
g la ag
a For more Question Papers, Sample Papers, Notes & Syllabus visit Page 15 of 24

30.     (LLP)      : 3

 x + y  7,

2x – 3y + 6  0,

x  0, y  0

  Z = 13x – 15y    

31. (a)      I  3   4   ,   II  8   6  
        3       I      
    II               
              3



(b)   X  Y             a     X,
Y        b ,     P(X) + P(Y) = 2 – 2a + b. 3

 – 
   -         5   
32. (a)   R,     Z       R = {(x, y) : |x – y|
   ‘p’   , x, y  Z}  

   R        5


 3   3  3x + 2
(b)  f : R –  5   R –  5 ,     f(x) = 5x – 3 .  
   

f        5

65/1/1 * 16 {}

For more Question Papers, Sample Papers, Notes & Syllabus visit Page 16 of 24

m
m .co

m .co s em
se g la
a
30. Solve the following linear programming problem graphically : 3

Minimize Z = 13x – 15y

Subject to constraints

x + y  7,

2x – 3y + 6  0,
m
m
c. xo 0, y  0 m .co
s e
s em l a
la of two bags, bag I contains 3 red and 4 white balls and bag IIag
31. (a) gOut
a contains 8 red and 6 white balls. A die is thrown. If it shows a
number less than 3 then a ball is drawn at random from bag I,
otherwise a ball is drawn at random from bag II. Find the probability
that the ball drawn from one of the bags is a red ball. 3

OR
m
(b)
.co
The probability of simultaneous occurrence of atleast one of the two

em
events X and Y is a. If the probability that exactly one of the events
s
g a ) + P(Y) = 2 – 2a + b.
X, Y occurs is b, prove thatlP(X 3
a
SECTION – D

This section comprises Long Answer (LA) type questions of 5 marks each.

32. (a) A relation R is defined on Z, the set of integers, as 5

m
.co
R = {(x, y) : |x – y| is divisible by a prime number ‘p’, x, y  Z}
m
m .co check whether R is an equivalence relation or not.
s e m
s e g l a
la
OR
g a
a (b)
 3   3 
A function f : R –  5   R –  5  is defined as f(x) =
3x + 2
. Show
    5x – 3
that f is one-one and onto. 5

65/1/1 * 17 {}

m .
.co s e m
s em l a
g la ag
a For more Question Papers, Sample Papers, Notes & Syllabus visit Page 17 of 24

 0 2 1
33. (a)  A =   2  1  2 ,  A–1   

 1  1 0
A–1          :
–2y + z = 7, 2x – y – z = 8, x – 2y = 10 5

 3  1 sin 3x 
(b)    7 4 cos 2x    (singular)  ,  x  

  11 7 2 

    x  0,    5
 2

d2y dy
34.  x = cos t, y = cos mt ,     (1 – x2) –x + m2y = 0. 5
dx2 dx

x–1 y–2 z–3 x–4 y–1
35.     2 = 3 = 4  5 = 2 = z,    
    ,       ,     ,  
     5

 – 
   3          4   

36.  -       5000        
    ` 300    

              ` 1   
10            ` x    

65/1/1 * 18 {}

For more Question Papers, Sample Papers, Notes & Syllabus visit Page 18 of 24

m
m .co

m .co s e m
se g l a
 0 2 1
a
33. (a)
 
If A =  2  1  2 , find A–1 and use it to solve the following
 
 1  1 0
system of equations :
–2y + z = 7, 2x – y – z = 8, x – 2y = 10 5
OR
m
 3 m
 1 sin 3x 
.co
(b)

If  7 .co
4 cos 2x  is a singular matrix, then find all values of x
s e m
em

s   11 7 2  l a
g la   ag 5
a where x  0,  .
 2

d2y dy
34. If x = cos t, y = cos mt, prove that (1 – x2) –x + m2y = 0. 5
dx2 dx

x–1 y–2 z–3 x–4 y–1
35. Check whether the lines given by 2 = 3 = 4 and 5 = 2 = z
m
.co
are parallel or not. If parallel, find the distance between them, otherwise
find their point of intersection, if the lines are intersecting. 5

s em
g l a
SECTION –E
a
This section comprises of 3 case study based questions of 4 marks each.
36. An online delivery company in a city has 5000 subscribers and collects
annual subscription fees of ` 300 per subscriber for unlimited free
deliveries.

m
m .co
m .co s e m
s e g l a
g la a
a The company wishes to increase the annual subscription fee. It is
predicted that, for every increase of ` 1, ten subscribers will discontinue.
Assume that the company increased the annual fee by ` x.
65/1/1 * 19 {}

m .
.co s e m
s em l a
g la ag
a For more Question Papers, Sample Papers, Notes & Syllabus visit Page 19 of 24

          :
(i)    ` x        ? 1

(ii)  R(x)    ` x           , 
R(x)  x         1

(iii) (a) x     ,   R(x)    2


(iii) (b)      (0, 5000)  R(x)      2

37.  -   ` 3,00,000   , ` 2,00,000     
` 50,000           ` 100   1,00,000  
  ` 1,00,00,000   

     
          :
(i)   -       ? 1

(ii) (a)       ` 2,00,000      ? 2


(ii) (b)            ? 2

(iii)        ` 5,00,000 ,         
     1,00,000  1         
         1

65/1/1 * 20 {}

For more Question Papers, Sample Papers, Notes & Syllabus visit Page 20 of 24

m
m .co

m .co s em
se g la
a
Based on the given information, answer the following questions :
(i) How many subscribers will discontinue after an increase of ` x in
annual fee ? 1
(ii) If R(x) denotes the total revenue collected after the increase of ` x in
subscription fee, express R(x) as a function of x. 1
(iii) (a) Find the value of x for which R(x) is maximum. 2 m
m .co
.co m
OR
(iii) (b) e m the sub-intervals of (0, 5000) in which R(x) is increasing las
Find
e
l as and decreasing. ag2
ag
37. In an online jackpot, there is one first prize of ` 3,00,000, two second
prizes of ` 2,00,000 each and three third prizes of ` 50,000 each.

m
.co
s em
g la
a each costing ` 100 were sold there by
A total of 1,00,000 jackpot tickets
raising a fund of ` 1,00,00,000.
Rohan bought one ticket.
Based on given information, answer the following questions :
(i) What are the possible amounts, the person can win ? 1
(ii) (a) What is the probability that the person wins atleast ` 2,00,000 ?
om
2

o m OR . c
. c (ii) (b) What is the probability that the person does not winsany
e m amount ? 2
s em (iii) In another jackpot, Rohan also bought a ticket havingg l a a prize money
la a
ag of ` 5,00,000. The chances of winning the jackpot are 1 in 1,00,000.
Find the probability that on exactly one of tickets he wins the
jackpot. 1

65/1/1 * 21 {}

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a For more Question Papers, Sample Papers, Notes & Syllabus visit Page 21 of 24

38.                   

            C1 : x2 + y2 = 64    
               ,  
C2 : x2 + y2 = 4    

          :
(i)  C1  C2      1

(ii)  C1  C2   y  x        (y = f(x)) 1

(iii) (a)                2

(iii) (b)               2
__________

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38. Roundabouts are often made on busy roads to ease the traffic and avoid
red lights.

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agsuch round-about is made such that equation representing its
One
boundary is given by C1 ; x2 + y2 = 64.
There is a circular pond with a fountain in the middle of the roundabout
whose equation is given by C2 : x2 + y2 = 4.
Based on the given information, answer the following questions :
(i) Represent the given equations C1 and C2 with the help of a diagram. 1
m
(ii)
.co
Express y as a function of x, (y = f(x)), for both C1 an C2.
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1
(iii) (a) e
Using integration find the area of region covered by the
s
roundabout.
g la 2
a OR
(iii) (b) Using integration, find the area of region covered by circular
pond. 2
_____________

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a

65/1/1 * 23 {}

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a For more Question Papers, Sample Papers, Notes & Syllabus visit Page 23 of 24

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