Title Jee 2014 Booklet6 Hwt Integral Calculus 2 Integral Trigonometric Functions Sine Pi Logarithm 205.6 KB 15
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Page 1

Vidyamandir Classes

VMC/Integral Calculus-2 101 HWT-6/Mathematics

DATE : TIME : 40 Minutes MARKS : [ ___ /10] TEST CODE : IC-2 [1]

START TIME : END TIME : TIME TAKEN: PARENT’S SIGNATURE :

 This test contains a total of 10 Objective Type Questions. Each question carries 1 mark. There is NO NEGATIVE marking.

Choose the correct alternative. Only one choice is correct.

1.

0
5 4

dx

cos x

 is :
(A) /2 (B) /4
(C) /3 (D) /6

2. Let f : R R and g : R R be continuous functions.
Then the value of the integral

       
2

2

f x f x g x g x dx



          
(A) 1
(B) 0

(C)
2 2

f f
    
    

   

(D)
2 2

g g
    
    

   
3. Let g (x) be a function satisfying    g x g x and

g (0) = 1 and f (x) be a function that

satisfies     2f x g x x  . Then the value of the

integral    
1

0

f x g x dx is :

(A)
7

4

e 
(B)

3

2

e 

(C)
2 3

2 2

e
e   (D)

2 3

2 2

e
e  

4. If

2

1

e

e

dx
I

log x
  and

2

2

1

xe
I dx

x
  , then :

(A) 1 22I I (B) 2 12I I

(C) 1 2 0I I  (D) 1 2 0I I 

5. Area of the region     21 1x y x y x   , : | | is :
(A) 1/3 (B) 2/3
(C) 4/3 (D) 5/3

6. 3 2 2 2 2 42 ( 1) (1 2 )sin x x sin x x x sin x x x dx



     
 

is:
(A) 1 (B) 1
(C) /4 (D) 0

7.

1

1

1x dx

   , where [ ] represents the greatest integer
function equals :
(A) 0 (B) 1
(C) 1 (D) 2

8.

30

30

cosx dx



 | |
(A) 0 (B) 120
(C) 120 (D) 120

9.

2

1

e
e

e

log x
dx

x is :

(A) 5/2 (B) 3
(C) 0 (D) 5

10. If        3 3f a x g x , g a x h x   
and    3h a x f x  .

Then
 

     

3

0

a
f x

dx
f x g x h x  is :

(A) a (B) 2a
(C) 3a (D) 3a/2

Page 2

Vidyamandir Classes

VMC/Integral Calculus-2 102 HWT-6/Mathematics

DATE : TIME : 40 Minutes MARKS : [ ___ /10] TEST CODE : IC-2 [2]

START TIME : END TIME : TIME TAKEN: PARENT’S SIGNATURE :

 This test contains a total of 10 Objective Type Questions. Each question carries 1 mark. There is NO NEGATIVE marking.

Choose the correct alternative. Only one choice is correct.

1.
1

2
1

1

e
tan x log x

dx
x x

 
 
  
 is :

(A) 1tan e (B)  log tan e (C) 1 1tan
e

  
 
 

(D) 1tan e

2.
 211

1
1 sin x

sin x dx

 is :

(A)
2 8

4

 
(B)

2 8

4

 
(C)

2 8

2

 
(D)

2 8

2

 

3.     
6

2
1

0

sin x d x , where {x} is fractional part function is :

(A)  23 8  (B)  23 8  (C)  
23 8

2

 
(D)

 23 8
2

 

4. The area bounded by  2 22 1y y x  and its vertical asymptotes is :
(A)

2

(B)  (C) 2 (D) 4

5. Area bounded by    52 3 4 4y x x   , the ordinate x = 3, x = 4 and above the x-axis is :

(A)
5

6

(B)

5

8

(C)

3

8

(D)

2

6. Area bounded by x-axis and the curve      xx xf x e e e | |. . between the lines 1x   and x = 2, where [ ] represents greatest
integer function and { } represent fractional part function, is :

(A)
1

2

e 
(B)

2 1

2

e 
(C)

3 1

2

e 
(D)

4 1

2

e 

7. If     
3

2

2 0

1

x x

x

f t dt f z dz t dt

 
       
 

   , [.] represents greatest integer function and  0 1f  , then 2f
 
 
 

is :

(A) 1 (B) 1 (C) 2 (D) 0

8. If    f a b x f x   , then  
b

a

xf x dx is :

(A)  
2

b

a

a b
f x dx

 (B)  2

b

a

a b
f x dx

 (C)    

b

a

a b f x dx  (D)    
b

a

a b f x dx 

Page 7

Vidyamandir Classes

VMC/Integral Calculus-2 107 HWT-6/Mathematics

9. Let

 
1

3 3
1

1
3 3

0
1

n
n

r
nn

n r
dx

, P lim
x n



 
 
 
  
 
 
 
  

/

then log P is :

(A) 2 1log   (B) 2 3 3log   (C) 2 2log  (D) 4 3 2log  

10.

2

0

1 3

1 2

sin x
dx

sin x

 

   is :

(A)
2

(B)

8

(C) 1 (D)

1

2

Page 8

Vidyamandir Classes

VMC/Integral Calculus-2 108 HWT-6/Mathematics

DATE : TIME : 40 Minutes MARKS : [ ___ /10] TEST CODE : IC-2 [5]

START TIME : END TIME : TIME TAKEN: PARENT’S SIGNATURE :

 This test contains a total of 10 Objective Type Questions. Each question carries 1 mark. There is NO NEGATIVE marking.

Choose the correct alternative. Only one choice is correct.

1. Let f (x) be differentiable in R and      
2 2

2

0 0

2f x x f t dt tf t dt    . Then  
1

0

f x dx is :

(A)
6

19
(B)

3

19
(C)

14

19
(D)

6

19

2. Area enclosed by the curve  
2

2

8

4 4

x
f x

x

 
  
  

, the x-axis and the ordinates 3x   , equals :

(A) 1
4 3

8 2
3 2

tan   (B) 1
4 3

8 2
3 2

tan  

(C) 1
4 3

8
3 2

tan   (D) None of these

*3. If the function   2x xf x Ae Be Cx   satisfies the condition    0 1 2 31f , f log   and   
4

0

39

2

log

f x cx dx  , then :
(A) A = 5 (B) 6B   (C) C = 3 (D) B = 6

*4. If
3 4 3

1 1 2

1 2 3

0 0 1

2 2 2x x xI dx, I dx, I dx     , and
4

2

4

1

2xI   dx, then :
(A) 2 1I I (B) 3 4I I (C) 4 3I I (D) 1 2I I

*5. If

2

0

sin x dx
A

sin x cos x

 and

2

0

cos x dx
B

sin x cos x

 , then :

(A) A + B = 0 (B)
2

A B

  (C) A B   (D)
4

A B

 

6. Let   2
2 1 3 1

3 2 1

x , x
f x

x , x

   
 

 
and   2

4 7 5 0

5 7 0

x , x
g x

x x , x

   
 

  
. Then value of    

2

2

g f x dx

  equals :

(A) 0 (B) 101/12 (C) 1991/6 (D) 1991/12

7.
 

1

20 1

xxe
dx

x


(A)

2

e
(B)  1 1

2
e

 
 

 
(C)  1 2

2
e

 
 

 
(D) None of these

8.    
2

2 4

2
sin x sin x sin x cos x dx


 

/

/
is equal to :

(A)
4

15
(B) 0 (C)

4

15
 (D)

2

15

Page 14

Vidyamandir Classes

VMC/Integral Calculus-2 114 HWT-6/Mathematics

DATE : TIME : 40 Minutes MARKS : [ ___ /10] TEST CODE : IC-2 [10]

START TIME : END TIME : TIME TAKEN: PARENT’S SIGNATURE :

 This test contains a total of 10 Objective Type Questions. Each question carries 1 mark. There is NO NEGATIVE marking.

Choose the correct alternative. Only one choice is correct.

1. The value of the integral

4

0
3 2



/
sin x cos x

dx
sin x

is :

(A) log(2) (B) log(3) (C)
1

4
log(3) (D)

1

8
log(3)

2. The area of the plane figure bounded by the interval 5 6/     of the x-axis, the graph of the function y cos x and the

segment so the straight lines 5 6x /  and x  is :
(A) 3/2 (B) 5/2 (C) 3/4 (D) 7/2

3. If  
2 3 2 3

3 4 3 4

1 1

 
 

sin x sin x sin x sin x sin x

f x sin x sin x

sin x sin x

then the value of  
2

0

/

f x dx

 is :
(A) 3 (B) 0 (C) 2/3 (D) 1/3

4. Let  0f R    and    
0

x

F x f t dt  . If    2 2 1F x x x  then  4f equals :
(A) 5/4 (B) 7 (C) 4 (D) 2

5. The area bounded by the curve   4 3 22 3y f x x x x     , x-axis and the ordinates corresponding to minimum of the
function  f x is :

(A) 1 (B)
91

30
(C)

30

9
(D) 4

6. Suppose that the graph of  y f x contains the points (0 , 4) and (2, 7). If f  is continuous then  
2

0

f x dx is equal to:
(A) 2 (B) –2 (C) 3 (D) None of these

7. The value of

2

0
20

x

x

sint dt

lim
sin x

is :

(A) 1 (B) 0 (C) 2 (D) None of these

8. The area between the curves 2 1 3and y x y x taking 1 1x     is :

(A) 1/2 (B) 2 (C) 3/4 (D) 3/2

Page 15

Vidyamandir Classes

VMC/Integral Calculus-2 115 HWT-6/Mathematics

9. The value of
2 2

3 3 3 3 3 3
1 2

1 2

 
   
    n

n
lim ..........

n n n n
is :

(A)
1

3
(B)  1 2

3
log (C)  1 3

2
log (D)  1 3

3
log

10. If  2 1 0x f x f
x

 
  
 

for all    0 then
sec

cos

x R ~ f x dx

  =
(A)  2sin  (B) 1 (C) sec cos  (D) 0