Jee 2014 Booklet6 Hwt Integral Calculus 2

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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. 05 4 dx cos x  



is : (A) /2 (B) /4 (C) /3 (D) /6

2. Let f : R and g : RR  be continuous functions.R

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 f x

   

g x x2. 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 x e I dx x 



, then : (A) I₁2I₂ (B) I₂2I₁ (C) I₁I₂0 (D) I₁I₂0

5. Area of the region





x y,



:



x1



2 y |x1| is :



(A) 1/3 (B) 2/3

(C) 4/3 (D) 5/3

6. sin x3 2x sin x x2 ( 2 1) x sin x2 (1 2x2 x4) dx

    _ _{ } _ _   



is: (A) 1 (B) 1 (C) /4 (D) 0 7. 1 1 1 x 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 f



3ax



g x , g

  

3ax

  

h x and h



3a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

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1. 1 2 1 1 e tan x log x dx x _x        _   



is :

(A) tan1e (B) log tan e





(C) tan 1 1

e        (D) 1 tan e 2.





2 1 1 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) 3



 2 8



(B) 3



 2 8



(C)





2 3 8 2   (D)





2 3 8 2   4. The area bounded by 2y2 

 

1 y2 xand its vertical asymptotes is :

(A) 2



(B)  (C) 2 (D) 4

5. Area bounded by y2



x3

 

4 4x



5, 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 f x

 

e x .e| |x .e x between the lines

x

 

1

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 f

 

0  , then1 2 f _{ }   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

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VMC/Integral Calculus-2 103 HWT-6/Mathematics 9.

 

2 3 2 3 2 0 x cos x dx 



is : (A) 3  (B) 4  (C) 6  (D) 12  10. If

 

4 1 4 f x dx  



and

 

4 2 7 f x dx 7      



, then the value of

 

2 1 f x dx 



is : (A) 5 (B) 4 (C) 3 (D) 3

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1. 1 5 2 1 . x dx      



, where [ ] denotes the greatest integer function, equals :

(A) 2 2 (B) 2 2 (C) 2.25 (D) 1.25 2.





2 0 n log tan x dx 



is : (A) 2  _(B) 3  _(C) ₀ _(D) 2  

3. The number of points at which

2 2 0 5 4 2 x t t t dt e   



has extremum is : (A) 1 (B) 2 (C) 5 (D) 0

4. If p, q, r are constants, then the value of





3 3 2 3 p sin x q cos x r dx     



depends on the value of :

(A) q and r only (B) p and q only (C) p only (D) all of p, q, and r

5. The area of the region in the first quadrant enclosed by the x-axis, the line x 3y and the circle x2y2 is :4 (A) 4  (B) 3  (C) 2  (D) 2 3  6.









5 2 1 1 2

sin cos x cos sin x dx

     _     



is : (A) 2 2  (B) 2 4  (C) 2 8  (D) 2 1 4  _{ }_ 7. If

 

1 0 x x f t dt x tf t dt



then the quadratic equation whose roots are 1 2 f_ _   and f (2) is : (A) 3x27x 2 0 (B) 3x27x 2 0 (C) 3x27x 2 0 (D) 3x27x 2 0 8.





10 1 2 1 2 1 x x  x x dx



is : (A) 110 (B) 110 3 (C) 18 (D) 98 3 

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VMC/Integral Calculus-2 105 HWT-6/Mathematics 9. P(x) is a non-zero polynomial such that P(0) = 0 and P x

 

3 x P x , P4

   

1  and7

 

1 0 1 5 P x dx .



then

   

1 0 P x P x dx



is :

(A) 6 (B) 8 (C) 7.5 (D) None of these

10. If for a non-zero x, af x

 

bf 1 1 5a b x x    _{ }     ; and

 

2 2 2 1 k f x dx a b  



then k is : (A) 2 7 5 2 b a log   a (B) 2 7 5 2 b a log   a (C) 2 7 5 2 b a log   a (D) 2 7 5 2 b a log   a

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1. If [x] is the greatest integer less than or equal to x, then the area bounded by y   x _{  and y}x     x _{ , and the x-axis between}x

2 x  to x = 2 is : (A) 1 (B) 2 (C) 3/2 (D) 1/2 2. 2 4 4 0 x sin x cos x dx sin x cos x   



(A) 2 32  (B) 2 8  (C) 2 16  (D) 16  3. 2 01 x sin x dx cos x   



(A) 4  (B) 2 4  (C) 2 2  (D) 2  4. If

 

2 1 3 2 x g x t g t dt x    



_  _ , then g 

 

2 is : (A) 1 2 (B) 3 4 (C) 1 (D) 3 2 5. Let f x

 

2x315x224xand

 

5 0 0 H f t dt f t dt    









where, 0  the interval in which 5 H

 

 is increasing is :

(A) (0, 5) (B) 5 5 2,       (C) 5 0 2 ,       (D) (1. 5) 6. 1 0 1 1 n n k k lim log n n    _  _     



equals : (A) e (B) 1 (C) 1 e (D) 0 7. If 2 2 1 t e dt



, then, 4 e e log t dt



equals : (A) e2  2 (B) e4e2 1  (C) 2e4 e  (D)



2e4 e 



8. The area bounded by the curves x2| |y 1 and x = 0 is :

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

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VMC/Integral Calculus-2 107 HWT-6/Mathematics 9. Let



3 3



1 1 1 3 3 01 n n r n n n r dx , P lim x n                         





/ then log P is :

(A) log2 1  (B) log2 3 3 (C) 2log2 (D) log4 3 2

10. 2 0 1 3 1 2 sin x dx sin x      _   



is : (A) 2  (B) 8  (C) 1 (D) 1 2

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

 

2 2 2 0 0 2 f 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 ordinatesx  , equals :3

(A) 4 8 13 2 3 tan 2     (B) 4 8 13 2 3 tan 2     (C) 4 8 13 3 tan 2     (D) None of these

*3. If the function f x

 

Ae2xBexCx satisfies the condition f

 

0  1, f



log2



31 and



 



4 0 39 2 log f x cx dx



, then : (A) A = 5 (B) B 6 (C) C = 3 (D) B = 6 *4. If 3 4 3 1 1 2 1 2 3 0 0 1 2x 2x 2x I 



dx, I 



dx, I 



dx, and 4 2 4 1 2x I 



dx, then : (A) I₂I₁ (B) I₃I₄ (C) I₄I₃ (D) I₁I₂ *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 2 0 ₁ x xe 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

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VMC/Integral Calculus-2 109 HWT-6/Mathematics 9. The values of a and b which satisfy

 

3 0 1 2 7 2x f  ,



f x dx , f x a b, are : (A) 1, 2 (B) 1 0 2 e , log (C)









2 2 3 2 1 1 2 ₇ ₂ e e log , log _log  _      _(D)









2 2 7 2 1 1 2 ₃ ₂ e log , log _log  _      10. If x f x2

 

f 1 2 x    _{ }

  for all x except at x = 0, then

 

3 1 3

f x dx



/

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1.





4 2 4 1 dx sec x sinx     



// (A) /4 (B) /2 (C)  (D) 2

2. Area common to the circle x2y264 and the parabola y212x is equal to : (A) 16



4 3



3    _     (B)





16 8 3 3    _     (C) 16



4 3



3    _     (D) None of these

3. The area bounded by the curves yx2and

 

2 2 1 y x   is : (A)



3 3



3   (B)



3 2



3   (C)



3 2



6   (D) None of these

4. The area enclosed between the curves y = tan x, tangent

drawn to it at 4 x and y is :0 (A)



4 1



4 log  (B)



4 1



2 log  (C)



4 1



4 log  (D) None of these

5. The area bounded by the hyperbola x2y24 between the lines x = 2 and x = 4 is :

(A) 4 32log



2 3



(B) 8 34log



2 3



(C) 8 34log



2 3



(D) 4 32log



2 3



6. 1 1 0 sin x dx



(A) 1 2   (B) 1 2   (C)  1 (D) None 7. The value of

 

4 0 x dx



, where {} denotes the fractional part of x is : (A) 16/3 (B) 25/3 (C) 7/3 (D) None 8. 2 0 x e sin x dx 



/ is equal to : (A) 2 1 2 e/  (B) 2 1 2 e/  (C) 2 1 2 e/  (D) None

9. If f x

 

min,



2sin x,1cos x,1



, then

 

0 f x dx  



(A) 3 1 5 6    (B) 3 1 2 3    (C) 1 3 2 3    (D) 1 3 5 6    10.





b c a c f x c dx    



(A)





b a f xc dx



(B)

 

b a f x dx



(C)





b a f a  b c x dx



(D) None of these

(11)

1. 2 2 1 2 1 2 1 1 2 1 1 x x dx x x      _  _ _  _   _     



// (A) 3 (B) 3



1 1



2 cos (C) 3 1



cos1



(D) None of these 2. Area included between the parabolas y24ax and

2 4 x  ay is equal to : (A) 2 8 3 a (B) 2 16 3 a (C) 2 4 3 a (D) None of these

3. Area bounded by| |y  x 1 8, yx2& x-axis is equal to:

(A)





2 5 2 6 2  (B)



 



2 3 5 2 6 4 2 6 2 24    (C)



 



2 3 5 2 6 4 2 6 4 24    (D) None of these

4. The area of the smaller region bounded by the circlex2y2 and the lines | y | = x + 1 is :1

(A)



2



4   (B)



2



2   (C)



2



2   (D) None of these 5. 2 0 cosx sinx dx   



| | (A) 4 2 (B) 2 2 (C) 2 (D) None 6.





1 ₉₉ 0 1 x x dx



(A) 11 10100 (B) 1 10010 (C) 1 10100 (D) None

7. Let f be real valued function such that f

 

2 2 andf 

 

2  , then1

 

3 2 2 4 2 f x x t lim dt x 



  (A) 6 (B) 12 (C) 32 (D) None 8. 4 1 4 1 1 / tan x dx x x  _  _    



(A) 1 (B) 0 (C) 1 (D) None

9. The value of the integral





1 0 1 1 n k f k x dx   



is : (A)

 

1 0 f x dx



(B)

 

2 0 f x dx



(C)

 

0 n f x dx



(D)

 

1 0 n



f x dx 10. If 1 2

   

0 0 a a I 



 _{ }x dx, I 



x dx, x fractional part of x,

 



= G.I.F., aZ, then : (A) ₂





2 1 a I I   (B) I₁aI₁ (C) I₁



a1



I₂ (D) I₂aI₁

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1. Area bounded by the curves y|x1|,y0 and 2

x 

| | is given by :

(A) 5 (B) 4

(C) 9/2 (D) None

2. The area bounded by the curves

2 ₂ ₁ x ₁ ₀ yx  x , e    and ordinatesy

x

 

1

and x = 1 is : (A)





2 3 2 3 3 e e e   (B)

 

2 ₁ e e  (C)





2 3 2 3 3 e  e (D) None of these

3. Let the circle x2y2 divide the area bounded by4 tangent and normal at

 

1, 3 and x-axis in A1 and A2.

Then A1/A2 = (A)



3 3



   (B)



3 3



   (C)



3 3



   (D) None of these 4. The value of 2 0 1



/ dx cot x  is equal to : (A)  (B) /2 (C) /4 (D) /3 5. The value of 2 2 04 9 x dx cos x sin x  



is equal to : (A) 2/12 (B) 2/4 (C) 2/6 (D) 2/3

6. If f     R R g R R are continuous functions then the value of the integral

 







 



2 2 / / f x f x g x g x dx            



(A) 1 (B) 0 (C) 1 (D)  7. If



2



2



0 4 9 dx k x x     



, then the value of k is :

(A) 1/60 (B) 1/80

(C) 1/40 (D) 1/20

8. The area of the figure bounded by the lines

 

0 2 and

x  x /  f x sin x g x cos x is:

(A) 2



21



(B) 31

(C) 2



31



(D) 2



21



9. An inflection point on the graph of the function



 



2 0 1 2 x y



t t dt is : (A) x 1 (B) x3 2/ (C) x4 3/ (D) x1 10. The value of

 

2 ₄ 0 / sin x dx 



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

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1. Let f be a periodic continuous function with period

0

T  . If

 

0

T

I 



f x dx, then the value of

 

4 4 1 4 3 T I f x dx  



is : (A) I (B) 2I (C) 3I (D) 4I 2. The value of 2 2 1 0 x x dx  _     



, where [x] is the greatest integer less than or equal to x is :

(A) 2 (B) 8/3

(C) 4 (D) None of these

3. The area bounded by the curves y 5x2 and 1 y | x  is :| (A) 5 2 4   _      (B) 5 2 4         (C) 5 2 2         (D) 2 5   _      4. If

 

2 2 2 4 x t x f x e dt  





, then the function f x

 

increases in : (A)



 0



(B)



0 



(C)



 1 2



(D)



 2



5.





4 2 0 1 2 



/ log tan tan d

    (A) log

 

2 (B)

 

2 2log  (C)

 

2 4log  (D) log

 

2 6. If f x

 

 e₂ecos x . sin x  for| x |2, then

 

3 2 f x dx  



otherwise : (A) 0 (B) 1 (C) 2 (D) 3 7. 3 4 41 / / dx cos x  



 is equal to : (A) 2 (B) 2 (C) 1/2 (D) 1 2/

8. The area of the plane region bounded by the curves 2 2 0 x y  and x3y2 is equal to :1 (A) 5/3 (B) 1/3 (C) 2/3 (D) 4/3 9. Let

 

2 0 6 x

f x 



x dx, then the real roots of the equation x2 f

 

x  are :0

(A) x  6 (B) x 3

(C) x  2 (D) x 1

10. Let f be an odd function, then



 



1 1 | x| f x cos x dx  



is equal to : (A) 0 (B) 1 (C) 2 (D) None of these

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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 x 5/6 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 f    and



0



R

 

0 x F x 



f t dt. If F x

 

2 x2



1 x



then f

 

4 equals : (A) 5/4 (B) 7 (C) 4 (D) 2

5. The area bounded by the curve y f x

 

x42x3x2 , x-axis and the ordinates corresponding to minimum of the3 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 2 0 x x sin t dt lim sin x 



is :

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

8. The area between the curves yx2andyx1 3 taking x  _ 1 1_{ is :}

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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 3log (C)

 

1 3 2log (D)

 

1 3 3log 10. If x f x2

 

f 1 0 x    _{ }

  for all

 

0 then

 

sec cos x R ~ f x dx   



=