9 - INTEGRAL CALCULUS

Page 1

( Answers at the end of all questions ) 1

1

2



2 x dx ,

I2 =

0

(c) I3 = I4

(c) 3

(d) I3 > I4

(d) 4

2

2

(b) 1:3:1

[ AIEEE 2005 ]

[ AIEEE 2005 ]

(c) 2:

( log x ) 2 + 1

xe x

+ c

(b)

+ c

2

(d)

.e

1+ x

m

log x

:2

(d) 1:1:1

[ AIEEE 2005 ]

is equal to

xa



 ( log x - 1 )    dx  1 + ( log x ) 2 

(c)

x

x2 + 1

+ c

x ( log x ) 2 + 1

+ c

[ AIEEE 2005 ]

Let f ( x ) be a non-negative continuous function such that the area bounded by the π π curv y = f ( x ), X-axis and the ordinates x = and x = β > is 4 4 π π ( β sin β + cos β + 2 β ). Then f ( ) is 4 2

w w w

1

The parabolas y = 4x and x = 4y divide the square re ion bounded by the lines x = 4, y = 4 and the coordinate axes. If S1, S2 S3 a e espectively the area of these parts numbered from top to bottom, then S1: S2 S3 is

(a)

(a)

π + 4

(c) 1 -

2 -1

π 4

The value of





(a) aπ

(b)

(b)

π 4

cos 2 x 1 + ax

π 2

dx ,

(c)

2 + 1 π + 4

(d) 1 -

2 π

(6)



3

2 x dx , then

ce .c

(b) 2

2

(5)

2

I4 =

1

(b) I1 > I2

(a) 1:2:1

(4)



2

2 x dx ,

The area enclosed between the curve y = log e ( x + e ) and the coordinate axes is (a) 1

(3)

I3 =

0

(a) I2 > I1

(2)



2

3

2 x dx ,

om

If I 1 =

ra

(1)

2

[ AIEEE 2005 ]

a > 0 is

π a

(d) 2π

[ AIEEE 2005 ]

9 - INTEGRAL CALCULUS

Page 2

( Answers at the end of all questions )



(b) e - 1

(a) e

(8)

If

is

sin x dx - α)

∫ sin ( x

(c) 1 - e

∫ cos x

dx - sin x

( b ) ( cos α, sin α ) ( d ) ( - cos α, sin α )

is equal to

π  x -  + C log tan  2 8   2

1

(a)

 x 3π  log tan   + C 8   2 2

 x  log cot   + C  2  2

3π   x + log tan   + C 8   2 2

1

(d)

[ AIEEE 2004 ]

xa 3

( 10 ) The value of

[ AIEEE 2004 ]

1

(b

m

1

(c)

[ AIEEE 2004 ]

= Ax + B log sin ( x - α ) + C, then the value of ( A, B ) is

( a ) ( sin α, cos α ) ( c ) ( - sin α, cos α )

(9)

(d) e + 1

om

lim

n→∞ r =1

ce .c

(7)

r n e

ra

n



1 - x 2 dx

is

.e

-2

28 3

(b)

w w

(a)

w

( 11 ) The value of I =

(a) 0

(b) 1

1 3

(c)



(d)

1 3

[ AIEEE 2004 ]

π 2



( sin x + cos x ) 2 1 + sin 2x

0

(c) 2

dx

is

(d) 3

[ AIEEE 2004 ]

π

π ( 12 ) If

7 3

2

x f ( sin x ) dx = A

0

(a) 0



f ( sin x ) dx , then A is equal to

0

(b) π

(c)

π 4

(d) 2π

[ AIEEE 2004 ]

9 - INTEGRAL CALCULUS

Page 3

( Answers at the end of all questions ) f(a)

ex

( 13 ) If f ( x ) =

1 + ex

,



I1 =

f(a)

x g { x ( 1 - x ) } dx

f(- a)

(c) -1

(d) 1

[ AIEEE 2004 ]

(c) 3

x2



(a) 3

sec 2 t dt

0

x→0

(b) 2

and

[ AIEEE 2004 ]

is

x sin x (c) 1

d) 0

m

lim

( 15 ) The value of

(d) 4

x = 3

ra

(b) 2

x = 1,

ce .c

( 14 ) The area of the region bounded by the curves y = l x - 2 l, X-axis is (a) 1

g { x ( 1 - x ) } dx ,

om

(b) -3



f(- a)

I then the value of 2 is I1 (a) 2

I2 =

and

1



xa

( 16 ) The value of the integral I =

x ( 1 - x )n dx

[ AIEEE 2003 ]

is

0

1 n + 1

(b)

w w

.e

(a)

( 17 ) If f ( y ) = e ,

1 n + 2

g ( y ) = y,

(c)

1 1 n + 1 n + 2

t

y > 0

and

F(t) =

1 1 + n + 1 n + 2 [ AIEEE 2003 ]

(d)



f ( t - y ) g ( y ) dy ,

then

0

(a) F(t) = te t

t

(d) F(t) = 1 - e- (1 + t) t

- (1 + t)

w

(c) F(t) = e

(b) F(t) = te-

t

[ AIEEE 2003 ]

b

( 18 ) If f ( a + b - x ) = f ( x ),

then the value of



x f ( x ) dx

is

a

(a)

(c)

b -a 2

b



f ( x ) dx

(b)

a

a + b 2

b



a

f ( b - x ) dx

a + b 2 b

(d)



a

b



f ( x ) dx

a

f ( a + b - x ) dx

[ AIEEE 2003 ]

9 - INTEGRAL CALCULUS

Page 4

( Answers at the end of all questions ) 4

d e sin x F(x ) = , dx x

( 19 ) Let

x > 0.

3 sin x3 e dx = F ( K ) - F ( 1 ), then one of the x



If

1

possible values of K is ( b ) 16

( c ) 63

( d ) 64

[ AIEEE 2003 ]

om

( a ) 15

( 20 ) Let f ( x ) be a function satisfying f ’ ( x ) = f ( x ) with f ( 0 )

e2 5 2 2

(b) e +

e2 2

-

3 2

(c) e -

e2 3 2 2

(d) e +

e2 2

+

5 2

∫ x sin x dx

xa

( b ) cos x + c ( d ) cos x - sin x + c



.e

1 - os 2x dx os 2x 1

w w

( a ) tan x - x + c ( c ) x - tan x + c

w

( 23 ) The



is

dx 2

2

a cos x + b 2 sin 2 x 0

( b ) π ab 2

∫e

( 24 ) The value of

( c ) 3 log ( x

[ AIEEE 2002 ]

( b ) x + tan x + c ( d ) - x - cot x + c

2

(a ) πab

4

[ AIEEE 2003 ]

[ AIEEE 2002 ]

π

alue of

( a ) log ( x

f ( x ) g ( x ) dx

alue of α is

m

= - x cos x + α, then the

( a ) sin x + c ( c ) x cos x + c

( 22 ) The value of



0

ra

(a) e -

ce .c

f ( x ) + g ( x ) = x . The value of the integral

is

( 21 ) If

1

2

function that satisfies

1 and g ( x ) be a

3 log x

+ 1) +c 4

+ 1) +c

( c ) π / ab

( x 4 + 1 ) -1 dx

(b) (d)

is

(d)

π / 2ab

[ AIEEE 2002 ]

is

1 4 log ( x + 1 ) + c 4 - log ( x4 + 1 ) + c

[ AIEEE 2002 ]

9 - INTEGRAL CALCULUS

Page 5

( Answers at the end of all questions ) dx is

( a ) log ( x + 1 ) + c

(b)

( c ) log ( x - 1 ) + c

(d)

sin2 x



( 26 ) The value of

sin

-1

-

1 log ( x + 1 ) + c x 1 log ( x + 1 ) + c 2

π 2



( t ) dt +

cos - 1 ( t ) dt

0

(b) 1

[ AIEEE 2002 ]

cos2 x

0

(a)

om

x2

(c)

π 4

(d)

is

ce .c

log x



( 25 ) The value of

π

[ AIEEE 2002 ]

b2 + 1 -

equal to x - 1

∫ [x

(b)

0

for all b > 1

x + 1

(c)

then f ( x ) is

2 + 1

(d)

x

[ AIEEE 2002 ]

1 + x2

]

3 + 3x 2 + 3x + 3 + ( x + 1 ) cos ( x + 1 ) dx =

xa

( 28 )

2

m

(a)

ra

( 27 ) If the area bounded by the X-axis, the curve y = f ( x ) and the lines x = 1, x = b is

-2

c) -1

(b) 0

(d) 1

[ IIT 2005 ]

.e

(a) 4

2

2

w w

( 29 ) Find the area between the curves y = ( x - 1 ) , y = ( x + 1 ) and y = 1 3

(a)

(b)

2 3

1

w

30

∫t

If

2 f ( t ) dt = 1

(c)

4 3

(d)

1 6

1 4 [ IIT 2005 ]

- sin x, x ∈ [ 0, π / 2 ], then f ( 1 /

3 ) is

sin x

(b) 1/3

(a) 3 t2

( 31 ) If

∫ x f ( x ) dx 0

(a) -

2 5

=

2 5 t 5

(b) 0

(c) 1

(d)

 4  for t > 0, then f    25 

(c)

2 5

(d) 1

[ IIT 2005 ]

3

is

[ IIT 2004 ]

9 - INTEGRAL CALCULUS

Page 6

( Answers at the end of all questions )



0

(a)

1-x dx is equal to 1+ x

π + 1 2

π - 1 2

(b)

(c) 1

(d)

( 33 ) If the area bounded by the curves x = ay 1 3

(b)

3 x2 + 1



( 34 ) If f ( x ) =

1 2

(c)

2

(b) (- ∞, 0)

ra

(c) [-2 2

0

[ IIT 2003 ]

n I [ ( m + 1) , n - ) ] 1+ m

(b)

2n m I [ ( m + 1) , ( n - 1 ) ] 1+ m 1+ n

(c)

2n m 1+ m 1 m

(d)

m I [ ( m + 1) , ( n - 1 ) ] n+1

xa

(a)

+ 1) , ( n - 1 ) ]

w w

.e

[(m

( 36 ) Area bounded by the curves y = (a) 2

w

( d ) nowhere

m n ∈ R, then I ( m, n ) is

m ( 1 + t )n dt ,

m

∫t

[ IIT 2004 ]

e - t dt , then the interval in which f ( x ) is increasing is

1

( 35 ) If I ( m, n ) =

2

3

2004 ]

and y = ax is 1, then a s equal to

(d) 3

x2

( a ) ( 0, ∞ )

[ II

ce .c

1

(a)

2

π

om

1

( 32 )

( b ) 18

[ IIT 2003 ]

x , x = 2y + 3 in the first quadrant and X-axis is

(c) 9

34 3

(d)

[ IIT 2003 ]

( 37 ) The area bounded by the curves y = l x l - 1 and y = - l x l + 1 is (a) 1

(b) 2

(c) 2

2

(d) 4

[ IIT 2002 ]

x

( 38 ) If f ( x ) =



2 2 - t 2 dt , then the real roots of the equation x

- f ’ ( x ) = 0 are

1

(a) ± 1

(b)

±

1 2

(c)

±

1 2

( d ) 0 and 1

[ IIT 2002 ]

9 - INTEGRAL CALCULUS

Page 7

( Answers at the end of all questions )

( 39 ) Let T > 0 be a fixed real number. Suppose f is a continuous function such that for 3 + 3T

T

all x ∈ R, f ( x + T ) = f ( x ). If I =



f ( x ) dx , then the value of

( c ) 3I

I

1 2

-

1 2

(a) -

  [x 1 



( 40 ) The integral equals

]+

( d ) 6I

 1+ x  ln    dx equals  1- x 

2

(b) 0

(c) 1

( d ) 2 ln

x

∫ f(t

1 2

dt

and

2

2

F ( x ) = x ( 1 + x ),

m

( 41 ) If f : ( 0, ∞ ) → R, F ( x ) =

[ IIT 2002 ]

ce .c

(b)

3

ra

3 I 2

(a)

is

om

0

∫ f ( 2x ) dx

[ IIT 2002 ]

then f ( 4 ) equals

0

5 4

(b) 7

π



cos 2 x

.e

( 42 ) The value of

(c) 4

w w

(a) π

w

( 43 ) If

(x) =

(a ) 0

1 + ax -π

(b) aπ

dx,

(c)

π 2

(c) 2

( 44 ) The value of the integral



e- 1

3 2

( d ) 2π

 e cos x sin x for l x l ≤ 2,   2 otherwise,

(b) 1

(b)

5 2

[ IIT 2001 ]

a > 0, is

e2

(a)

(d) 2

xa

(a)

(c) 3

[ IIT 2001 ]

3

then

(d) 5

=

-2

(d) 3

log e x dx x

∫ f ( x ) dx

[ IIT 2000 ]

is

[ IIT 2000 ]

9 - INTEGRAL CALCULUS

Page 8

( Answers at the end of all questions ) ( 45 ) If f ( x ) = ∫ e x ( x - 1 ) ( x - 2 ) dx , then f decreases in the interval

g(x) =

0 ≤ f(t) ≤

(a)

-

(c)

( 47 )

1 2

x ∫ f ( t ) dt , 0

where f is such that

( d ) ( 2, + ∞ )

1 ≤ f(t) ≤ 1 2

for

[ IIT 2000 ]

t ∈ [0

1]

and

for t ∈ [ 1, 2 ]. Then g ( 2 ) satisfies the inequality

ce .c

Let

( c ) ( 1, 2 )

3 1 ≤ g(2) ≤ 2 2 3 5 ≤ g(2) ≤ 2 2

( b ) 0 ≤ g(2) ≤ 2

(d) 2 < g(2) < 4

[ IIT 2000 ]

If for a real number y, [ y ] is the greatest integer less than or equal to y, then the 3π

ra

( 46 )

( b ) ( - 2, - 1 )

om

( a ) ( - ∞, - 2 )

2



[ 2 sin x ] dx is

m

value of the integral

π 2

(b) 0





π 4

dx 1 + cos x

w w

(a) 2

w

( 49 )

( 50 )

π 2

(d)

π 2

[ IIT 1999 ]

(d) -

1 2

[ IIT 1999 ]

=

.e

4 ( 48 )

(c) -

xa

(a) -π

(b) -2

(c)

1 2

For which of the following values of

m, is the area of the region bounded by the 9 curve y = x - x and the line y = mx equals ? 2 (a) -4

2

(b) -2

(c) 2

(d) 4

[ IIT 1999 ]

If f ( x ) = x - [ x ], for every real number x, where [ x ] is the integral part of x, then 1

∫ f ( x ) dx

is

(a) 1

(b) 2

-1

(c) 0

(d)

1 2

[ IIT 1998 ]

9 - INTEGRAL CALCULUS

Page 9

( Answers at the end of all questions ) x

∫ cos

( 51 ) If g ( x ) =

4 t dt , then g ( x +

π ) equals

0

(b) g(x) - g(π)

(c) g(x)g(π) k

( 52 )



Let f be a positive function. If I1 = 2k - 1 > 0, then

(a) 2

(b) k

(c)

is

1 2

and I2 =

[ II

1997 ]

∫ f [ x ( 1 - x ) ] dx ,

1- k

ce .c

where

g( x ) g( π)

k

x f [ x ( 1 - x ) ] dx

1- k

I1 I2

(d)

om

(a) g(x) + g(π)

(d) 1

[ IIT 1997 ]

6 5

(b)

1

m

5 6

(c)

xa

(a)

ra

( 53 ) The slope of the tangent to a curve y = f ( x ) at [ x, f ( x ) ] is 2x + 1. If the curve passes through the point ( 1, 2 ), then the area of the region bounded by the curve, the X-axis and the line x = 1 is (d) 6

[ IIT 1995 ]



( 54 ) The value of



[ 2 sin x ] dx

where [ . ] represents the greatest integer function, is

-

5π 3

(b) -π

w w

(a)

.e

π

π 2

w

( 55 ) The value of

(a) 0

dx



1 + tan 3 x 0

(b) 1

(c)

(c)

5π 3

(d) -2π

[ IIT 1995 ]

is

π 2

(d)

π 4

[ IIT 1993 ]

( 56 ) If f : R → R be a differentiable function and f ( 1 ) = 4, then the value of f(x) 2t lim dt is ∫ x - 1 x →1 4 (a) 8f’(1)

(b) 4f’(1)

(c) 2f’(1)

(d) f’(1)

[ IIT 1990 ]

9 - INTEGRAL CALCULUS

Page 10

( Answers at the end of all questions ) If f : R → R and g : R → R are continuous functions, then the value of the integral π 2



is

π 2

(a) π

(b) 1

(c) -1

(d) 0 π



( 58 ) For any integer n, the integral

(a) π

(b) 1

(c) 0

ra

2



π 4

(b)

π 2

is

( d ) none of these

[ IIT 1983 ]

.e

If the area bounded by the curves y = f ( x ), the X-axis and the ordinates x = 1 and x = b is ( b - 1 ) sin ( 3b + 4 ), then f ( x ) is

w w w

dx

t nx

(c) π

( a ) ( x - 1 ) co ( 3x + 4 ) ( c ) sin ( 3x + 4 )

( 61 )

cot x

[ IIT 1985 ]

xa

(a)

cot x

m

0

has the value

( d ) none of hese

π ( 59 ) The value of the integral

[ IIT 1990 ]

2 e cos x cos 3 ( 2n + 1 ) x dx

0

( 60 )

om

-

[ f ( x ) + f ( - x ) ] [ g ( x ) + g ( - x ) ] dx

ce .c

( 57 )

( b ) sin ( 3x + 4 ) + 3 ( x - 1 ) cos ( 3x + 4 ) ( d ) none of these

1

he value of the definite integral

(a) -1



[ IIT 1982 ]

2

( 1 + e - x ) dx is

0

(b) 2

(c) 1 + e–

1

( d ) none of these

[ IIT 1981 ]

9 - INTEGRAL CALCULUS

Page 11

( Answers at the end of all questions )

2 a

3 d

4 d

5 d

6 b

7 b

8 b

9 a,d

10 a

11 2

12 b

13 a

14 a

21 a

22 c

23 d

24 b

25 b

26 c

27 d

28 a

29 a

30 a

31 c

32 b

33 a

34 b

41 c

42 c

43 c

44 b

45 c

46 b

47 c

48 a

49 b,d

50 a

51 a

52 c

53 a

61 d

62

63

64

65

66

67

68

69

70

71

72

73

ra m xa .e w w w

15 c

16 c

17

18 b

19 d

20 b

35 c

36 b

37 b

38 a

39 c

40 a

54 a

55 d

56 a

57 d

58 c

59 a

60 b

74

75

76

77

78

79

80

ce .c

1 b

om

Answers

Mathematics-Integral-MCQ.pdf

x 1. xe. 2. x. +. +. ( d ) c. 1 ) x log (. x. 2 +. +. [ AIEEE 2005 ]. ( 5 ) Let f ( x ) be a non-negative continuous function such that the area bounded by the. curv y = f ( x ), X-axis and the ordinates x = 4. π and x = β >. 4. π is. ( β sin β +. 4. π. cos β + 2 β ). Then f ( 2. π ) is. ( a ) 4. π. + 2 -1 ( b ) 4. π - 2 + 1. ( c ) 1 - 4. π - 2 ( d ) 1 - 4. π.

74KB Sizes 1 Downloads 44 Views

Recommend Documents

No documents