Class 12 Mathematics: CBSE Sample Question Paper (2023-24) | Sample Papers for Class 12 Medical and Non-Medical PDF Download

 Page 1


Page 1 of 8 
 
                              SAMPLE QUESTION PAPER 
Class:-XII 
Session 2023-24 
Mathematics (Code-041) 
Time: 3 hours                                                                                                                            Maximum marks: 80 
General Instructions:  
1. This Question paper contains - five sections A, B, C, D and E. Each section is compulsory. However, there are 
internal choices in some questions.  
2.  Section A has 18 MCQ’s and 02 Assertion-Reason based questions of 1 mark each.  
3.  Section B has 5 Very Short Answer (VSA)-type questions of 2 marks each.  
4.  Section C has 6 Short Answer (SA)-type questions of 3 marks each.  
5.  Section D has 4 Long Answer (LA)-type questions of 5 marks each.  
6.  Section E has 3 source based/case based/passage based/integrated units of assessment of 4 marks each with  
sub-parts. 
___________________________________________________________________________________________ 
 
Section –A 
  (Multiple Choice Questions) 
  Each question carries 1 mark 
Q1.  If 
i j
A a ? ? ?
? ?
is a square matrix of order 2 such that
1, when 
0, when 
i j
i j
a
i j
? ?
?
?
?
?
, then  
2
A is 
(a)  
2 2
1 0
1 0
?
? ?
? ?
? ?
  (b)  
2 2
1 1
0 0
?
? ?
? ?
? ?
  (c)  
2 2
1 1
1 0
?
? ?
? ?
? ?
   (d) 
2 2
1 0
0 1
?
? ?
? ?
? ?
 
Q2.  If A and B are invertible square matrices of the same order, then which of the following is not correct? 
(a) 
-1
| A |
AB
| B |
?
    (b) ? ?
1 1
| | |B|
A B
A
?
? 
(c) ? ?
1
1 1
A B B A
?
? ?
?      (d) ? ?
1
1 1
A B B A
?
? ?
? ? ? 
Q3.  If the area of the triangle with vertices 
? ? ? ? 3 ,0 , 3,0 ? and 
? ? 0, k is 9 squnits, then the value/s of k will 
be  
(a) 9   (b) 3 ?   (c) -9    (d) 6 
Q4.  If 
? ?
, if 0
3, if 0
kx
x
x f x
x
?
?
?
?
?
?
?
?
  is continuous at 0 x ? , then the value of k is 
(a) -3                 (b) 0                (c) 3    (d) any real number              
Page 2


Page 1 of 8 
 
                              SAMPLE QUESTION PAPER 
Class:-XII 
Session 2023-24 
Mathematics (Code-041) 
Time: 3 hours                                                                                                                            Maximum marks: 80 
General Instructions:  
1. This Question paper contains - five sections A, B, C, D and E. Each section is compulsory. However, there are 
internal choices in some questions.  
2.  Section A has 18 MCQ’s and 02 Assertion-Reason based questions of 1 mark each.  
3.  Section B has 5 Very Short Answer (VSA)-type questions of 2 marks each.  
4.  Section C has 6 Short Answer (SA)-type questions of 3 marks each.  
5.  Section D has 4 Long Answer (LA)-type questions of 5 marks each.  
6.  Section E has 3 source based/case based/passage based/integrated units of assessment of 4 marks each with  
sub-parts. 
___________________________________________________________________________________________ 
 
Section –A 
  (Multiple Choice Questions) 
  Each question carries 1 mark 
Q1.  If 
i j
A a ? ? ?
? ?
is a square matrix of order 2 such that
1, when 
0, when 
i j
i j
a
i j
? ?
?
?
?
?
, then  
2
A is 
(a)  
2 2
1 0
1 0
?
? ?
? ?
? ?
  (b)  
2 2
1 1
0 0
?
? ?
? ?
? ?
  (c)  
2 2
1 1
1 0
?
? ?
? ?
? ?
   (d) 
2 2
1 0
0 1
?
? ?
? ?
? ?
 
Q2.  If A and B are invertible square matrices of the same order, then which of the following is not correct? 
(a) 
-1
| A |
AB
| B |
?
    (b) ? ?
1 1
| | |B|
A B
A
?
? 
(c) ? ?
1
1 1
A B B A
?
? ?
?      (d) ? ?
1
1 1
A B B A
?
? ?
? ? ? 
Q3.  If the area of the triangle with vertices 
? ? ? ? 3 ,0 , 3,0 ? and 
? ? 0, k is 9 squnits, then the value/s of k will 
be  
(a) 9   (b) 3 ?   (c) -9    (d) 6 
Q4.  If 
? ?
, if 0
3, if 0
kx
x
x f x
x
?
?
?
?
?
?
?
?
  is continuous at 0 x ? , then the value of k is 
(a) -3                 (b) 0                (c) 3    (d) any real number              
Page 2 of 8 
 
Q5.  The lines 
? ?
? ?
2 3 6 r i j k i j k ? ? ? ? ? ? ?
?
? ? ? ?
and 
? ?
? ?
2 6 9 18 r i j k i j k ? ? ? ? ? ? ?
?
? ? ? ?
; (where & ? ? are 
scalars) are  
(a) coincident  (b) skew  (c) intersecting   (d) parallel 
Q6.  The degree of the differential equation 
3
2
2
2
2
1
dy d y
i s
d x d x
? ?
? ?
? ?
? ?
? ?
? ? ? ?
? ?
? ? ? ?
? ?
 
(a) 4     (b) 
3
2
    (c) 2       (d) Not defined 
Q7.  The corner points of the bounded feasible region determined by a system of linear constraints are 
? ? ? ? 0, 3 , 1,1 and 
? ? 3,0 . Let      , Z p x q y ? ? where  ,  0 . p q ? The condition on p and q so that the 
minimum of Z occurs at 
? ? 3,0 and 
? ? 1,1 is  
(a) 2 p q ?    (b) 
2
q
p ?  (c) 3 p q ?    (d)  p q ? 
Q8. A B C D is a rhombus whose diagonals intersect at E . Then E A E B E C E D ? ? ?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
equals to 
 (a) 0
?
   (b) A D
? ? ? ?
    (c) 2 B D
? ? ? ?
   (d) 2 A D
? ? ? ?
 
Q9.  For any integer , n the value of 
2
cos x 3
  Sin (2n + 1) x dx
?
?
e
?
?
 is  
 (a) -1   (b) 0    (c) 1     (d) 2 
Q10. The value of , if A
0 2 1
1 2 0 2 ,where ,
2 0
x x
A x x x
x x
?
? ?
?
? ?
? ? ? ? ?
? ?
? ?
? ?
? ?
?  is    
        (a) ? ?
2
2 1 x ?                (b) 0                                (c) ? ?
3
2 1 x ?                           (d) ? ?
2
2 1 x ? 
Q11. The feasible region corresponding to the linear constraints of a Linear Programming Problem is given    
below. 
 
 
 
 
 
 
 
 
 
 
 Which of the following is not a constraint to the given Linear Programming Problem? 
(a) 2 x y ? ?  (b) 2 10 x y ? ?     (c) 1 x y ? ?   (d) 1 x y ? ? 
Page 3


Page 1 of 8 
 
                              SAMPLE QUESTION PAPER 
Class:-XII 
Session 2023-24 
Mathematics (Code-041) 
Time: 3 hours                                                                                                                            Maximum marks: 80 
General Instructions:  
1. This Question paper contains - five sections A, B, C, D and E. Each section is compulsory. However, there are 
internal choices in some questions.  
2.  Section A has 18 MCQ’s and 02 Assertion-Reason based questions of 1 mark each.  
3.  Section B has 5 Very Short Answer (VSA)-type questions of 2 marks each.  
4.  Section C has 6 Short Answer (SA)-type questions of 3 marks each.  
5.  Section D has 4 Long Answer (LA)-type questions of 5 marks each.  
6.  Section E has 3 source based/case based/passage based/integrated units of assessment of 4 marks each with  
sub-parts. 
___________________________________________________________________________________________ 
 
Section –A 
  (Multiple Choice Questions) 
  Each question carries 1 mark 
Q1.  If 
i j
A a ? ? ?
? ?
is a square matrix of order 2 such that
1, when 
0, when 
i j
i j
a
i j
? ?
?
?
?
?
, then  
2
A is 
(a)  
2 2
1 0
1 0
?
? ?
? ?
? ?
  (b)  
2 2
1 1
0 0
?
? ?
? ?
? ?
  (c)  
2 2
1 1
1 0
?
? ?
? ?
? ?
   (d) 
2 2
1 0
0 1
?
? ?
? ?
? ?
 
Q2.  If A and B are invertible square matrices of the same order, then which of the following is not correct? 
(a) 
-1
| A |
AB
| B |
?
    (b) ? ?
1 1
| | |B|
A B
A
?
? 
(c) ? ?
1
1 1
A B B A
?
? ?
?      (d) ? ?
1
1 1
A B B A
?
? ?
? ? ? 
Q3.  If the area of the triangle with vertices 
? ? ? ? 3 ,0 , 3,0 ? and 
? ? 0, k is 9 squnits, then the value/s of k will 
be  
(a) 9   (b) 3 ?   (c) -9    (d) 6 
Q4.  If 
? ?
, if 0
3, if 0
kx
x
x f x
x
?
?
?
?
?
?
?
?
  is continuous at 0 x ? , then the value of k is 
(a) -3                 (b) 0                (c) 3    (d) any real number              
Page 2 of 8 
 
Q5.  The lines 
? ?
? ?
2 3 6 r i j k i j k ? ? ? ? ? ? ?
?
? ? ? ?
and 
? ?
? ?
2 6 9 18 r i j k i j k ? ? ? ? ? ? ?
?
? ? ? ?
; (where & ? ? are 
scalars) are  
(a) coincident  (b) skew  (c) intersecting   (d) parallel 
Q6.  The degree of the differential equation 
3
2
2
2
2
1
dy d y
i s
d x d x
? ?
? ?
? ?
? ?
? ?
? ? ? ?
? ?
? ? ? ?
? ?
 
(a) 4     (b) 
3
2
    (c) 2       (d) Not defined 
Q7.  The corner points of the bounded feasible region determined by a system of linear constraints are 
? ? ? ? 0, 3 , 1,1 and 
? ? 3,0 . Let      , Z p x q y ? ? where  ,  0 . p q ? The condition on p and q so that the 
minimum of Z occurs at 
? ? 3,0 and 
? ? 1,1 is  
(a) 2 p q ?    (b) 
2
q
p ?  (c) 3 p q ?    (d)  p q ? 
Q8. A B C D is a rhombus whose diagonals intersect at E . Then E A E B E C E D ? ? ?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
equals to 
 (a) 0
?
   (b) A D
? ? ? ?
    (c) 2 B D
? ? ? ?
   (d) 2 A D
? ? ? ?
 
Q9.  For any integer , n the value of 
2
cos x 3
  Sin (2n + 1) x dx
?
?
e
?
?
 is  
 (a) -1   (b) 0    (c) 1     (d) 2 
Q10. The value of , if A
0 2 1
1 2 0 2 ,where ,
2 0
x x
A x x x
x x
?
? ?
?
? ?
? ? ? ? ?
? ?
? ?
? ?
? ?
?  is    
        (a) ? ?
2
2 1 x ?                (b) 0                                (c) ? ?
3
2 1 x ?                           (d) ? ?
2
2 1 x ? 
Q11. The feasible region corresponding to the linear constraints of a Linear Programming Problem is given    
below. 
 
 
 
 
 
 
 
 
 
 
 Which of the following is not a constraint to the given Linear Programming Problem? 
(a) 2 x y ? ?  (b) 2 10 x y ? ?     (c) 1 x y ? ?   (d) 1 x y ? ? 
Page 3 of 8 
 
Q12. If 4 6 a i j ? ?
?
? ?
 and 
?
3 4 , b j k
?
?
? ? then the vector form of the component of a
?
along b
?
 is 
  (a) 
?
? ?
18
3 4
5
i k ?
?
 (b) 
?
? ?
18
3 4
25
j k ?
?
 (c) 
?
? ?
18
3 4
5
i k ?
?
  (d) 
? ?
18
4 6
25
?
? ?
i j  
Q13. Given that A is a square matrix of order 3 and 2, A ? ? then ? ? 2 ad j A is equal to 
  (a) 
6
2 ?                      (b) 4 ?                      (c) 
8
2 ?                         (d) 
8
2  
Q14. A problem in Mathematics is given to three students whose chances of solving it are 
1 1 1
, ,
2 3 4
 
respectively. If the events of their solving the problem are independent
 
then the probability that the 
problem will be solved, is      
 (a) 
1
4
                      (b)
 
1
3
                             (c) 
1
2
                              (d) 
3
4
 
Q15. The general solution of the differential equation 
? ? –   0; Given , 0 , y d x x d y x y ? ? is of the form  
 (a)   x y c ?  (b)  
2
  x c y ? 
 
(c)    c y x ?    (d) 
2
  y c x ? ; 
(Where ' ' c is an arbitrary positive constant of integration) 
Q16. The value of ? for which two vectors 
?
2 2 i j k ? ?
? ?
 and 
?
3 i j k ? ? ?
? ?
 are perpendicular is  
  (a) 2   (b) 4   (c) 6    (d) 8 
Q17. The set of all points where the function 
? ? f x x x ? ? is differentiable, is  
  (a) 
? ? 0, ?  (b) 
? ? ,0 ? ?  (c) 
? ? ? ? ,0 0, ? ? ? ?  (d) 
? ? , ? ? ? 
Q18. If the direction cosines of a line are 
1 1 1
, ,
c c c
? ? , then  
  (a) 0 1 c ? ?  (b) 2 c ?  (c) 2 c ? ?    (d) 3 c ? ? 
 
                                                         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 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. 
Q19. Let ? ? f x be a polynomial function of degree 6 such that ? ? ? ? ? ? ? ?
3 2
1 3
d
f x x x
dx
? ? ? , then  
 ASSERTION (A): 
? ? f x has a minimum at 1. x ? 
         REASON (R): When ? ? ? ? ? ? 0, ,a
d
f x x a h
d x
? ? ? ? and ? ? ? ? ? ? 0, , ;
d
f x x a a h
d x
? ? ? ?
 
where 
 
 
' ' h is an infinitesimally small positive quantity, then 
? ? f x has a minimum at , x a ? 
 provided ? ? f x is continuous at . x a ? 
Page 4


Page 1 of 8 
 
                              SAMPLE QUESTION PAPER 
Class:-XII 
Session 2023-24 
Mathematics (Code-041) 
Time: 3 hours                                                                                                                            Maximum marks: 80 
General Instructions:  
1. This Question paper contains - five sections A, B, C, D and E. Each section is compulsory. However, there are 
internal choices in some questions.  
2.  Section A has 18 MCQ’s and 02 Assertion-Reason based questions of 1 mark each.  
3.  Section B has 5 Very Short Answer (VSA)-type questions of 2 marks each.  
4.  Section C has 6 Short Answer (SA)-type questions of 3 marks each.  
5.  Section D has 4 Long Answer (LA)-type questions of 5 marks each.  
6.  Section E has 3 source based/case based/passage based/integrated units of assessment of 4 marks each with  
sub-parts. 
___________________________________________________________________________________________ 
 
Section –A 
  (Multiple Choice Questions) 
  Each question carries 1 mark 
Q1.  If 
i j
A a ? ? ?
? ?
is a square matrix of order 2 such that
1, when 
0, when 
i j
i j
a
i j
? ?
?
?
?
?
, then  
2
A is 
(a)  
2 2
1 0
1 0
?
? ?
? ?
? ?
  (b)  
2 2
1 1
0 0
?
? ?
? ?
? ?
  (c)  
2 2
1 1
1 0
?
? ?
? ?
? ?
   (d) 
2 2
1 0
0 1
?
? ?
? ?
? ?
 
Q2.  If A and B are invertible square matrices of the same order, then which of the following is not correct? 
(a) 
-1
| A |
AB
| B |
?
    (b) ? ?
1 1
| | |B|
A B
A
?
? 
(c) ? ?
1
1 1
A B B A
?
? ?
?      (d) ? ?
1
1 1
A B B A
?
? ?
? ? ? 
Q3.  If the area of the triangle with vertices 
? ? ? ? 3 ,0 , 3,0 ? and 
? ? 0, k is 9 squnits, then the value/s of k will 
be  
(a) 9   (b) 3 ?   (c) -9    (d) 6 
Q4.  If 
? ?
, if 0
3, if 0
kx
x
x f x
x
?
?
?
?
?
?
?
?
  is continuous at 0 x ? , then the value of k is 
(a) -3                 (b) 0                (c) 3    (d) any real number              
Page 2 of 8 
 
Q5.  The lines 
? ?
? ?
2 3 6 r i j k i j k ? ? ? ? ? ? ?
?
? ? ? ?
and 
? ?
? ?
2 6 9 18 r i j k i j k ? ? ? ? ? ? ?
?
? ? ? ?
; (where & ? ? are 
scalars) are  
(a) coincident  (b) skew  (c) intersecting   (d) parallel 
Q6.  The degree of the differential equation 
3
2
2
2
2
1
dy d y
i s
d x d x
? ?
? ?
? ?
? ?
? ?
? ? ? ?
? ?
? ? ? ?
? ?
 
(a) 4     (b) 
3
2
    (c) 2       (d) Not defined 
Q7.  The corner points of the bounded feasible region determined by a system of linear constraints are 
? ? ? ? 0, 3 , 1,1 and 
? ? 3,0 . Let      , Z p x q y ? ? where  ,  0 . p q ? The condition on p and q so that the 
minimum of Z occurs at 
? ? 3,0 and 
? ? 1,1 is  
(a) 2 p q ?    (b) 
2
q
p ?  (c) 3 p q ?    (d)  p q ? 
Q8. A B C D is a rhombus whose diagonals intersect at E . Then E A E B E C E D ? ? ?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
equals to 
 (a) 0
?
   (b) A D
? ? ? ?
    (c) 2 B D
? ? ? ?
   (d) 2 A D
? ? ? ?
 
Q9.  For any integer , n the value of 
2
cos x 3
  Sin (2n + 1) x dx
?
?
e
?
?
 is  
 (a) -1   (b) 0    (c) 1     (d) 2 
Q10. The value of , if A
0 2 1
1 2 0 2 ,where ,
2 0
x x
A x x x
x x
?
? ?
?
? ?
? ? ? ? ?
? ?
? ?
? ?
? ?
?  is    
        (a) ? ?
2
2 1 x ?                (b) 0                                (c) ? ?
3
2 1 x ?                           (d) ? ?
2
2 1 x ? 
Q11. The feasible region corresponding to the linear constraints of a Linear Programming Problem is given    
below. 
 
 
 
 
 
 
 
 
 
 
 Which of the following is not a constraint to the given Linear Programming Problem? 
(a) 2 x y ? ?  (b) 2 10 x y ? ?     (c) 1 x y ? ?   (d) 1 x y ? ? 
Page 3 of 8 
 
Q12. If 4 6 a i j ? ?
?
? ?
 and 
?
3 4 , b j k
?
?
? ? then the vector form of the component of a
?
along b
?
 is 
  (a) 
?
? ?
18
3 4
5
i k ?
?
 (b) 
?
? ?
18
3 4
25
j k ?
?
 (c) 
?
? ?
18
3 4
5
i k ?
?
  (d) 
? ?
18
4 6
25
?
? ?
i j  
Q13. Given that A is a square matrix of order 3 and 2, A ? ? then ? ? 2 ad j A is equal to 
  (a) 
6
2 ?                      (b) 4 ?                      (c) 
8
2 ?                         (d) 
8
2  
Q14. A problem in Mathematics is given to three students whose chances of solving it are 
1 1 1
, ,
2 3 4
 
respectively. If the events of their solving the problem are independent
 
then the probability that the 
problem will be solved, is      
 (a) 
1
4
                      (b)
 
1
3
                             (c) 
1
2
                              (d) 
3
4
 
Q15. The general solution of the differential equation 
? ? –   0; Given , 0 , y d x x d y x y ? ? is of the form  
 (a)   x y c ?  (b)  
2
  x c y ? 
 
(c)    c y x ?    (d) 
2
  y c x ? ; 
(Where ' ' c is an arbitrary positive constant of integration) 
Q16. The value of ? for which two vectors 
?
2 2 i j k ? ?
? ?
 and 
?
3 i j k ? ? ?
? ?
 are perpendicular is  
  (a) 2   (b) 4   (c) 6    (d) 8 
Q17. The set of all points where the function 
? ? f x x x ? ? is differentiable, is  
  (a) 
? ? 0, ?  (b) 
? ? ,0 ? ?  (c) 
? ? ? ? ,0 0, ? ? ? ?  (d) 
? ? , ? ? ? 
Q18. If the direction cosines of a line are 
1 1 1
, ,
c c c
? ? , then  
  (a) 0 1 c ? ?  (b) 2 c ?  (c) 2 c ? ?    (d) 3 c ? ? 
 
                                                         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 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. 
Q19. Let ? ? f x be a polynomial function of degree 6 such that ? ? ? ? ? ? ? ?
3 2
1 3
d
f x x x
dx
? ? ? , then  
 ASSERTION (A): 
? ? f x has a minimum at 1. x ? 
         REASON (R): When ? ? ? ? ? ? 0, ,a
d
f x x a h
d x
? ? ? ? and ? ? ? ? ? ? 0, , ;
d
f x x a a h
d x
? ? ? ?
 
where 
 
 
' ' h is an infinitesimally small positive quantity, then 
? ? f x has a minimum at , x a ? 
 provided ? ? f x is continuous at . x a ? 
Page 4 of 8 
 
Q20. ASSERTION (A): The relation ? ? ? ? : 1,2,3,4 , , , f x y z p ? defined by ? ? ? ? ? ? ? ?
1, , 2, , 3, f x y z ? is a            
                                    bijective function. 
         REASON (R): The function ? ? ? ? : 1,2,3 , , , f x y z p ? such that ? ? ? ? ? ? ? ?
1, , 2, , 3, f x y z ? is one-one. 
 
       Section –B 
[This section comprises of very short answer type questions (VSA) of 2 marks each] 
 
Q21. Find the value of 
1
33
sin cos .
5
?
?
? ? ? ?
? ? ? ?
? ? ? ?
  
   OR 
   Find the domain of 
? ?
1 2
sin 4 . x
?
? 
Q22. Find the interval/s in which the function : f ? ? ? defined by
 
? ? ,
x
f x x e ? is increasing. 
Q23. If ? ?
2
1
;
4 2 1
f x x
x x
? ? ?
? ?
, then find the maximum value of ? ?. f x   
      OR 
    Find the maximum profit that a company can make, if the profit function is given by      
    
? ?
? ? ?
2
72 42 , P x x x
 
where
 
x
 
is the number of units and
 
P is the profit in rupees. 
Q24. Evaluate : 
?
? ? ?
? ?
?
? ?
?
1
1
2
log .
2
x
d x
x
 
Q25. Check whether the function : f ? ? ? defined by ? ?
3
, f x x x ? ? has any critical point/s or not ?     
    If yes, then find the point/s. 
 
                                                                      Section – C 
  [This section comprises of short answer type questions (SA) of 3 marks each] 
 
Q26.  Find : 
? ?
2
2 2
2 3
; 0.
9
x
d x x
x x
?
?
?
?
    
Q27. The random variable X has a probability distribution ? ? P X of the following form, where ' ' k is some   
     real number: 
? ?
, if 0
2 , if 1
3 , if 2
0, otherwise
k x
k x
P X
k x
? ?
?
?
?
?
?
?
?
?
?
 
            (i) Determine the value of . k 
            (ii) Find  ? ? 2 . P X ?
 
Page 5


Page 1 of 8 
 
                              SAMPLE QUESTION PAPER 
Class:-XII 
Session 2023-24 
Mathematics (Code-041) 
Time: 3 hours                                                                                                                            Maximum marks: 80 
General Instructions:  
1. This Question paper contains - five sections A, B, C, D and E. Each section is compulsory. However, there are 
internal choices in some questions.  
2.  Section A has 18 MCQ’s and 02 Assertion-Reason based questions of 1 mark each.  
3.  Section B has 5 Very Short Answer (VSA)-type questions of 2 marks each.  
4.  Section C has 6 Short Answer (SA)-type questions of 3 marks each.  
5.  Section D has 4 Long Answer (LA)-type questions of 5 marks each.  
6.  Section E has 3 source based/case based/passage based/integrated units of assessment of 4 marks each with  
sub-parts. 
___________________________________________________________________________________________ 
 
Section –A 
  (Multiple Choice Questions) 
  Each question carries 1 mark 
Q1.  If 
i j
A a ? ? ?
? ?
is a square matrix of order 2 such that
1, when 
0, when 
i j
i j
a
i j
? ?
?
?
?
?
, then  
2
A is 
(a)  
2 2
1 0
1 0
?
? ?
? ?
? ?
  (b)  
2 2
1 1
0 0
?
? ?
? ?
? ?
  (c)  
2 2
1 1
1 0
?
? ?
? ?
? ?
   (d) 
2 2
1 0
0 1
?
? ?
? ?
? ?
 
Q2.  If A and B are invertible square matrices of the same order, then which of the following is not correct? 
(a) 
-1
| A |
AB
| B |
?
    (b) ? ?
1 1
| | |B|
A B
A
?
? 
(c) ? ?
1
1 1
A B B A
?
? ?
?      (d) ? ?
1
1 1
A B B A
?
? ?
? ? ? 
Q3.  If the area of the triangle with vertices 
? ? ? ? 3 ,0 , 3,0 ? and 
? ? 0, k is 9 squnits, then the value/s of k will 
be  
(a) 9   (b) 3 ?   (c) -9    (d) 6 
Q4.  If 
? ?
, if 0
3, if 0
kx
x
x f x
x
?
?
?
?
?
?
?
?
  is continuous at 0 x ? , then the value of k is 
(a) -3                 (b) 0                (c) 3    (d) any real number              
Page 2 of 8 
 
Q5.  The lines 
? ?
? ?
2 3 6 r i j k i j k ? ? ? ? ? ? ?
?
? ? ? ?
and 
? ?
? ?
2 6 9 18 r i j k i j k ? ? ? ? ? ? ?
?
? ? ? ?
; (where & ? ? are 
scalars) are  
(a) coincident  (b) skew  (c) intersecting   (d) parallel 
Q6.  The degree of the differential equation 
3
2
2
2
2
1
dy d y
i s
d x d x
? ?
? ?
? ?
? ?
? ?
? ? ? ?
? ?
? ? ? ?
? ?
 
(a) 4     (b) 
3
2
    (c) 2       (d) Not defined 
Q7.  The corner points of the bounded feasible region determined by a system of linear constraints are 
? ? ? ? 0, 3 , 1,1 and 
? ? 3,0 . Let      , Z p x q y ? ? where  ,  0 . p q ? The condition on p and q so that the 
minimum of Z occurs at 
? ? 3,0 and 
? ? 1,1 is  
(a) 2 p q ?    (b) 
2
q
p ?  (c) 3 p q ?    (d)  p q ? 
Q8. A B C D is a rhombus whose diagonals intersect at E . Then E A E B E C E D ? ? ?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
equals to 
 (a) 0
?
   (b) A D
? ? ? ?
    (c) 2 B D
? ? ? ?
   (d) 2 A D
? ? ? ?
 
Q9.  For any integer , n the value of 
2
cos x 3
  Sin (2n + 1) x dx
?
?
e
?
?
 is  
 (a) -1   (b) 0    (c) 1     (d) 2 
Q10. The value of , if A
0 2 1
1 2 0 2 ,where ,
2 0
x x
A x x x
x x
?
? ?
?
? ?
? ? ? ? ?
? ?
? ?
? ?
? ?
?  is    
        (a) ? ?
2
2 1 x ?                (b) 0                                (c) ? ?
3
2 1 x ?                           (d) ? ?
2
2 1 x ? 
Q11. The feasible region corresponding to the linear constraints of a Linear Programming Problem is given    
below. 
 
 
 
 
 
 
 
 
 
 
 Which of the following is not a constraint to the given Linear Programming Problem? 
(a) 2 x y ? ?  (b) 2 10 x y ? ?     (c) 1 x y ? ?   (d) 1 x y ? ? 
Page 3 of 8 
 
Q12. If 4 6 a i j ? ?
?
? ?
 and 
?
3 4 , b j k
?
?
? ? then the vector form of the component of a
?
along b
?
 is 
  (a) 
?
? ?
18
3 4
5
i k ?
?
 (b) 
?
? ?
18
3 4
25
j k ?
?
 (c) 
?
? ?
18
3 4
5
i k ?
?
  (d) 
? ?
18
4 6
25
?
? ?
i j  
Q13. Given that A is a square matrix of order 3 and 2, A ? ? then ? ? 2 ad j A is equal to 
  (a) 
6
2 ?                      (b) 4 ?                      (c) 
8
2 ?                         (d) 
8
2  
Q14. A problem in Mathematics is given to three students whose chances of solving it are 
1 1 1
, ,
2 3 4
 
respectively. If the events of their solving the problem are independent
 
then the probability that the 
problem will be solved, is      
 (a) 
1
4
                      (b)
 
1
3
                             (c) 
1
2
                              (d) 
3
4
 
Q15. The general solution of the differential equation 
? ? –   0; Given , 0 , y d x x d y x y ? ? is of the form  
 (a)   x y c ?  (b)  
2
  x c y ? 
 
(c)    c y x ?    (d) 
2
  y c x ? ; 
(Where ' ' c is an arbitrary positive constant of integration) 
Q16. The value of ? for which two vectors 
?
2 2 i j k ? ?
? ?
 and 
?
3 i j k ? ? ?
? ?
 are perpendicular is  
  (a) 2   (b) 4   (c) 6    (d) 8 
Q17. The set of all points where the function 
? ? f x x x ? ? is differentiable, is  
  (a) 
? ? 0, ?  (b) 
? ? ,0 ? ?  (c) 
? ? ? ? ,0 0, ? ? ? ?  (d) 
? ? , ? ? ? 
Q18. If the direction cosines of a line are 
1 1 1
, ,
c c c
? ? , then  
  (a) 0 1 c ? ?  (b) 2 c ?  (c) 2 c ? ?    (d) 3 c ? ? 
 
                                                         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 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. 
Q19. Let ? ? f x be a polynomial function of degree 6 such that ? ? ? ? ? ? ? ?
3 2
1 3
d
f x x x
dx
? ? ? , then  
 ASSERTION (A): 
? ? f x has a minimum at 1. x ? 
         REASON (R): When ? ? ? ? ? ? 0, ,a
d
f x x a h
d x
? ? ? ? and ? ? ? ? ? ? 0, , ;
d
f x x a a h
d x
? ? ? ?
 
where 
 
 
' ' h is an infinitesimally small positive quantity, then 
? ? f x has a minimum at , x a ? 
 provided ? ? f x is continuous at . x a ? 
Page 4 of 8 
 
Q20. ASSERTION (A): The relation ? ? ? ? : 1,2,3,4 , , , f x y z p ? defined by ? ? ? ? ? ? ? ?
1, , 2, , 3, f x y z ? is a            
                                    bijective function. 
         REASON (R): The function ? ? ? ? : 1,2,3 , , , f x y z p ? such that ? ? ? ? ? ? ? ?
1, , 2, , 3, f x y z ? is one-one. 
 
       Section –B 
[This section comprises of very short answer type questions (VSA) of 2 marks each] 
 
Q21. Find the value of 
1
33
sin cos .
5
?
?
? ? ? ?
? ? ? ?
? ? ? ?
  
   OR 
   Find the domain of 
? ?
1 2
sin 4 . x
?
? 
Q22. Find the interval/s in which the function : f ? ? ? defined by
 
? ? ,
x
f x x e ? is increasing. 
Q23. If ? ?
2
1
;
4 2 1
f x x
x x
? ? ?
? ?
, then find the maximum value of ? ?. f x   
      OR 
    Find the maximum profit that a company can make, if the profit function is given by      
    
? ?
? ? ?
2
72 42 , P x x x
 
where
 
x
 
is the number of units and
 
P is the profit in rupees. 
Q24. Evaluate : 
?
? ? ?
? ?
?
? ?
?
1
1
2
log .
2
x
d x
x
 
Q25. Check whether the function : f ? ? ? defined by ? ?
3
, f x x x ? ? has any critical point/s or not ?     
    If yes, then find the point/s. 
 
                                                                      Section – C 
  [This section comprises of short answer type questions (SA) of 3 marks each] 
 
Q26.  Find : 
? ?
2
2 2
2 3
; 0.
9
x
d x x
x x
?
?
?
?
    
Q27. The random variable X has a probability distribution ? ? P X of the following form, where ' ' k is some   
     real number: 
? ?
, if 0
2 , if 1
3 , if 2
0, otherwise
k x
k x
P X
k x
? ?
?
?
?
?
?
?
?
?
?
 
            (i) Determine the value of . k 
            (ii) Find  ? ? 2 . P X ?
 
Page 5 of 8 
 
            (iii)  Find  ? ? 2 . P X ? 
Q28.  Find : 
? ?
3
; 0,1 .
1
x
d x x
x
?
?
?
 
OR 
   Evaluate: ? ?
4
0
log 1 tan . x d x
?
?
?
 
Q29. Solve the differential equation: ? ?
2
, 0 .
x x
y y
y e dx x e y dy y
? ?
? ? ? ? ?
? ?
? ?
 
OR 
    Solve the differential equation: 
? ?
2
cos tan ; 0 .
2
d y
x y x x
d x
? ? ?
? ? ? ?
? ?
? ?
 
Q30. Solve the following Linear Programming Problem graphically: 
   Minimize: 2 z x y ? ? ,  
    subject to the constraints: 2 100, 2 0, 2 200, , 0. x y x y x y x y ? ? ? ? ? ? ?       
OR 
 Solve the following Linear Programming Problem graphically: 
   Maximize: 2 z x y ? ? ? ,  
    subject to the constraints: 3, 5, 2 6, 0. x x y x y y ? ? ? ? ? ?       
Q31. If ? ? ? ?
y
x
a b x e x then  prove that 
2
2
2
.
d y a
x
d x a bx
? ?
?
? ?
?
? ?
 
 
                           Section –D 
           [This section comprises of long answer type questions (LA) of 5 marks each] 
 
Q32.  Make a rough sketch of the region ? ? ? ?
2
, : 0 1, 0 1, 0 2 x y y x y x x ? ? ? ? ? ? ? ? and find the    
     area of the region, using the method of integration. 
Q33. Let ? be the set of all natural numbers and R be a relation on ? ? ? defined by    
   ? ? ? ? , , a b R c d a d b c ? ? for all ? ? ? ? , , , a b c d ? ? ? ? . Show that R is an equivalence relation on   
    ? ? ? . Also, find the equivalence class of ? ? 2,6 , i.e.,  ? ? 2,6 ? ?
? ?
. 
   OR 
   Show that the function
? ? : : 1 1 f x x ? ? ? ? ? ? ? defined by 
? ? ,
1
x
f x x
x
? ?
?
? is one-one and    
    onto function. 
Q34.  Using the matrix method, solve the following system of linear equations :