SR-IIT-NSC-MATHS-INDEFINITE INTEGRATION-ASSIGN-02:2020-06-12

(One or More options Correct Type)

 1 1

01. Let f  x   lim n 2  x n  x n 1  ; x  0 ,
n 
then  x. f  x dx equals to (where c is constant of
 
integration).
x2 x2 x2 x2 x2 x2 x2
A) ln x  c B)  ln x  c C) ln x   c D) ln x   c
2 4 2 2 4 2 4

02. If f  x    b 2   a  1 b  2  x    sin 2 x  cos 4 x dx be an increasing function of x and b  R ,

then
possible value(s) of “a” is/are , (where c is constant of integration)
A) 0 B) 1 C) 2 D) 4
  1
03. If fn  x    n sin 2 x  sin 2 n2 x  cos2 n2 x  dx , f     n1
4 2
 
then which of the following option(s) is/are correct?
 3
A) The derivative of f n  x  when n  2 and x  , is
3 2
n
B) if g  x   lim
n 
 f n  x  then minimum value of g  x  , is 3
n 1

f  x
(C)If n  lim cot 1 t  then  2 n 2 dx = tan x  cot x  3x  c
t  sin  x  .cos  x 

Note: [k] denotes greatest integer less than or equal to k.

D) All of these
3sin x  4 1
04. If   3  4sin x  2
dx  f  x   C where f  0    (C is constant of integration),
3

then which of the following option(s) is/are correct?

1
(A) the minimum value of f  x  is (B) the minimum value of f  x  is 0
3
1
(C) the value of f   is (D) the value of f   is 0
3
 dx  L P
05. If   3  N C then, where  P, C  R  and L and M are co-prime integers
x

4
1 x  M x4
then which of the following option(s) is/are correct?
A) L 1 B) M 2
C) LM  N P5 D) LM  N P3
06. Let f  x and g  x  are differentiable functions satisfying the conditions:
(i) f  0   2, g  0   1 (ii) f  x  g  x and f  x  g x

then which of the following option(s) is/are correct?

A) Range of f  x  is  3, 
B)  f  2   g  2    22 , where [.] represents greatest integer
f  x
C) h  x   is differentiable on R
g  x

g  x
D) h  x   is differentiable on R
f  x

07. A function f  x  continuous on R and periodic with period 2 satisfies

D  cos x
f  x   sin x. f  x     sin 2 x , then  f  x  dx  x  cos x  A tan 1  B tan x   C log E (where
D  cos x
A, B, C , D, E  R )

then which of the following option(s) is/are correct?

A) 2 A2  B 2  0
B)  B  D   0 (where [.] represents greatest Integer function)
1
C)  D 1
2C
D) A  2C  B  D

08. If y  x  y 2  x , then  dx  Aloge f  x, y   B (where A, B  R ) then which of the following

x  3y
option(s) is/are correct?
A) A is a natural number
B) Number of divisor of f  3,5  is same as no.of divisor of f  7,2 

C) Number of divisor of f  2,2 2  is same as no.of divisor of f  3, 3 

D) A is a negative integer

Page. No. 2
09. Which of the following is incorrect ? (where K1 , K 2 , K 3 , K 4 are real constants)
cos sin 2 cos3 sin 4
(A)  e sin cos  cos  d  K1        ...
1 2  2 3  3 2 4  4  6
cos  sin 2 cos3 sin 4
(B)  e sin cos  cos  d  K3        ...
1 22 32 42
cos 2 cos3 cos 4
(C)  ecos .sin  sin   d  K 2  cos     ..........
2  2! 3  3! 4  4!
cos 2 cos 3 cos 4
(D)  ecos .sin  sin   d  K 4  cos     ..........
2  2! 3  3! 4  4!

 
1  x  x1/ 4  qx p / q
10. If  cot  2 tan 1 dx   C ; (where p and q are relatively prime and C is
 1  x  x1/ 4  p

constant of integration), then which of the following option(s) is/are correct?
(A) p  5 (B) p  2q  13 (C) p  q  1 (D) q  5
(Integer Value Correct Type):
dx 2  x P
11. If  3  loge tan     tan 1  sin x  cos x   c then value of L  M  N  P is
sin  cos3 x L M N 3
(where L, M , N , P , c  R )

 3x2  2 x  1  xM  xN  P 
12. If   6 1
dx  tan    K then, L  M  N  P  Q is, (where
x  2 x5  x 4  2 x3  2 x 2  2
 L  Q 
L, M , N , P , Q , K  R )

f  x
13. Suppose f  x  is a quadratic function such that If  2
f  0  1 2
and
dx is a f  1  4 .
x  x  1
rational function then, sum of digit of value of  f 10   is (where . represents greatest
Integer function).
m
x 2009 1  x2 
14. If the primitive of the function f  x  1006
, w.r.to x is equal to   C
1  x 
2 n  1  x2 

(where m, n  N ) then  m  n  is a four digit number abcd then value of

  a  d    b  c   equals,
1  cos x  sin x  x  sin x  cos x 
15. If  dx = f  x  where f  0   0 then Lt f  x  equals
 x  sin x  x  cos x  x 

p  1  sin 2 x cos x cos 2 x  sin x  cos x

16. For x   0,  , If  1  sin 2 x
 ln  ln  dx = A.sin 2 x.ln  B ln cos 2 x  c
 4  1  sin 2 x  cos x  sin x

then the value of A + B equals ( where A, B, c  R )

Page. No. 3
 A 2 A3 1  f  x  g  x  x x
17. If ‘A’ is square matrix such that I + A + + + .... =   where A= 
2! 3! 2 g  x  f  x   x x 
g  x A B
and 0<x<1, I is an identify matrix and  dx  A log  e x  e  x   Bx  c then
f  x A
equals
(where c is the constant of integration)
 
   B 
e x  tan1 2 x 
Ax  dx  e x  tan1 2 x   c
18. If  2   2 
where c is arbitrary constant and

  2

4 x 1   4 x  1

A
A,B are integers then the value of 
B

a 8
f  x  7 and 32g 0  21and  3 x 1  x 4 dx  g  x  c , where a,b are relatively
7 3
19. If g  x  
b
f 0b  a
prime and c is arbitrary constant then .
11

20. Let f  x    x 2  2 x  6 tan x  2 x tan 2 x  cos 2 xdx and f  x  passes through  , 0  then the
number of solutions of the equation f  x   x 3 in x  0, 2  is

****************************************************************************
KEY
41 D 42 ABC 43 AC 44 BC 45 ABCD

46 BD 47 ABD 48 C 49 BD 50 ABC

51 7 52 8 53 8 54 7 55 0

56 1 57 1 58 8 59 1 60 3

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