KENDRIYA VIDYALAYA SANGTHAN

MONTHLY TEST SEPTEMBER-2023

CLASS: XII MATHEMATICS
CHAPTERS : INTEGRAL, AOI & DIFF Eq.
Time Allowed: 1 Hour 30 Minutes Maximum Marks: 40

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 8 MCQ’s and 01 Assertion -Reason based questions of 1 mark each.
3. Section B has 2 Very short answer type questions of 2 marks each.
4. Section C has 3 Short Answer (SA)-type questions of 3 mark each.
5. Section D has 2 Long Answer (LA)-type questions of 5 mark each.
6. Section E has 2 sources based /case based/passage based/integrated units of assessment (4 marks each) with
sub parts.

SECTION- A
(Multiple Choice Questions)
Each question carries 1 mark
1.  sec x (sec x  tan x ) dx 
(A) tan x  sec x  C (B) sec x  tan x  C (C) tan x  sec x  C (D) tan x  2 sec x  C

2.  sin ax  b cosax  b dx =

cosax  b  C cosax  b  C (C) cos 2ax  b  C cos 2ax  b  C

1 1 1 1
(A) (B)  (D) 
4a 4a 4a 4a

e2x  1
3.  e2x  1 dx 
(A) log(ex  e x )  C (B) log(ex  e x )  C (C) log( e2 x  e2 x )  C (D) log( e2x  e2x )  C

/ 2

 x sin 
x  tan 5 x  1 dx 
4
4.
 / 2
(A) π (B) 0 (C) 1 (D) 2π

e dx 
x
5.
1
(A) 0 (B) e  2 (C) 2e  2 (D) 2
3
 d2y 
 
4
 dy 
6. The order and degree of the differential equation :  2   4   sin e x  y  0 are
 dx   dx 
(A) Order : 2 , Degree : not defined (B) Order : 3 , Degree : 3
(C) Order : 3 , Degree : 2 (D) Order : 2 , Degree : 3

1
 dy
7. Solution of  ex y is
dx
(A) e y  e x  C (B) e y  ex  C (C)  e y  ex  C (D) ey  2ex  C  0

 1  x  y  xy, y1  0 is
dy
8. Solution of
dx
x2 3 3
(A) log 1  y  x   (B) log 1  y  x 
2 2 2
x2 3 x2 3
(C) log 1  y   (D) log 1  y  x  
2 2 2 2

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.

9. Assertion (A) : y = a sin x + b cos x is a general solution of y′′ + y = 0.

Reason(R) : y = a sin x + b cos x is a trigonometric function.

Section –B
[This section comprises of very short answer type questions (VSA) of 2 marks each]

1
10.  5x  3 dx 
0

13 13 10 11
(A) (B) (C) (D)
10 5 13 10

11. Solution of x 1  y 2 dx  y 1  x 2 dy  0 is

(A) 1  x 2  1  y 2  C (B) 1  x 2  1  y 2  C

(C) 1  x 2  2 1  y 2  C (D) 2 1  x 2  1  y 2  C

Section – C
[This section comprises of short answer type questions (SA) of 3 marks each]

xe x
12. Evaluate :  dx .
x  12
π
xsinx
13. Evaluate :  1  cos2x dx .
0

2
 dy
14. Solve the differenti al equation : 2x 2  2xy  y2  0.
dx
Section –D
[This section comprises of long answer type questions (LA) of 5 marks each]

x 2  2x  8
15. Evaluate :  dx
(x  1)(x  2)

16. Find the area lying in the first quadrant and bounded by the circle x2 + y2 = 4 and the lines x = 0 and x = 2.

Section –E
[This section comprises of 2 case- study/passage based questions of 4 marks each with sub parts. The
case study questions have three sub parts (i), (ii), (iii) of marks 1, 1, 2 respectively.

17. A portion of land is as per the given diagram :

(i) Find the equation of the path BD,
(ii) What will be the formulation to find the area bounded by BD,
the x-axis and the ordinates x = –1 and x = 1.
(iii) Find the area of the shaded region using integration.

18. Arun wants to make a garden in his plot as per the diagram with dimensions given below:

(i) What will be the Cartesian equation of the garden?

(ii) What will be the formulation to find the area of the garden ?
(iii) what will be the area of the garden?( Using integration)

3
 ANSWERS

cos 2ax  b  C
1
1. (C) tan x  sec x  C 2. (D)  3. (A) log(ex  e x )  C
4a
4. (B) 0 5. (C) 2e  2 6. (D) Order : 2 , Degree : 3
x2 3
7. (C)  e y  ex  C 8. log 1  y  x   9. (b)
2 2
13 1 x
10. 11. 1  x 2  1  y2  C 12. e C
10 x 1
2 2x
13. 14.  log x  C 15. x  11 log x  1  16 log x  2  C
4 y
2

3 1
13
16. π 17. (i) y = 3x + 2 (ii)  (3x  2) dx  2(3x  2) dx (iii)
2
Sq. units
1

3
a
x 2 y2 b 2
18. (i)  1 (ii) 4 a  x 2 dx (iii) πab Sq. Units
a 2 b2 0
a