08 - Differential Calculus

08 - DIFFERENTIAL CALCULUS

Page 1

( Answers at the end of all questions )

(1)

 1 2 1  2 sec n→∞  n n2 (a)

(2)

1 sec 1 2

2

+

lim

n2

sec 2

4 n2

+ .......... +

1 cosec 1 2

(b)

 1 sec 2 1  n 

( c ) tan 1

is 1 tan 1 2

(d)

[ AIEEE 2005 ]

The normal to the curve x = a ( cos θ + θ sin θ ), y = a ( sin θ - θ cos θ ) at any point ‘ θ ’ is such that ( a ) it passes through the origin π ( b ) it makes angle + θ with the X-axis 2 π ( c ) it passes through ( a , - a ) 2 ( d ) it is at a constant distance from the origin [ AIEEE 2005 ]

(3)

A function is matched below against an interval where it is supposed to be increasing. Which of the following pairs is incorrectly matched ? Interval (a) (- ∞, ∞) 1 (c) (- ∞, ) 3

Function x

3

Interval

- 3x 2 + 3x + 3

3x

3

Function

( b ) [ 2, ∞ )

- 2x 2 + 1

(d)

2x

(- ∞, -4)

3

x

- 3x 2 + 12x + 6 3

- 6x 2 + 6 [ AIEEE 2005 ]

(4)

Let α and β be the distinct roots of the equation ax 1 - cos ( ax 2 + bx + c )

lim

( x - α )2

x→α

(a)

a2 2 (α - β) 2

(b) 0

2

+ bx + c = 0. Then

is equal to

(c) -

a2 2 (α - β) 2

(d)

1 2 (α - β) 2 [ AIEEE 2005 ]

(5)

Suppose f ( x ) is differentiable at x = 1 and (a) 3

(b) 4

(c) 5

(d) 6

lim

1

h→0 h

f ( 1 + h ) = 5, then f ’ ( 1 ) equals [ AIEEE 2005 ]

08 - DIFFERENTIAL CALCULUS

Page 2

( Answers at the end of all questions )

(6)

Let f be differentiable for al x. If f ( 1 ) = - 2 and f ’ ( x ) ≥ 2 for x ∈ [ 1, 6 ], then (a) f(6) ≥ 8

(7)

(d) f(6) = 5

[ AIEEE 2005 ]

2

(b) 0

(c) 2

(d) 1

[ AIEEE 2005 ]

A spherical iron ball 10 cm in radius is coated with a layer of ice of uniform 3 thickness that melts at a rate of 50 cm /min. When the thickness of ice is 5 cm, then the rate at which thickness of ice decreases in cm/min is (a)

(9)

(c) f(6) < 5

If f is a real valued differentiable function satisfying l f ( x ) - f ( y ) l ≤ ( x - y ) , x, y ∈ R and f ( 0 ) = 0, then f ( 1 ) equals (a) -1

(8)

(b) f(6) < 8

1 36 π

1 18 π

(b)

(c)

1 54 π

(d)

5 6π

[ AIEEE 2005 ]

Let f : R → R be a differentiable function having f ( 2 ) = 6, f ’ ( 2 ) = f( x )

lim

x→2

∫

6

( a ) 24

4t 3 dt x - 2

equals

( b ) 36

( 10 ) If the equation

anx

( c ) 12

n

+ an-1x

( d ) 18

n-1

positive root x = α, then the equation has a positive root which is ( a ) greater than α ( c ) greater than or equal to α

( 11 ) If

1 . Then 48

 a b 1+ +  x x→∞  x2 lim

( a ) a ∈ R, b ∈ R ( c ) a ∈ R, b = 2

   

2x

[ AIEEE 2005 ]

+ ….. + a 1 x = 0, a 1 ≠ 0, n ≥ 2 has a nan x

n-1

+ (n - 1)an-1x

( b ) smaller than α ( d ) equal to α

n-2

+ …. + a 1 = 0

[ AIEEE 2005 ]

2

= e , then the values of a and b are ( b ) a = 1, b ∈ R ( d ) a = 1, b = 2

[ AIEEE 2004 ]

08 - DIFFERENTIAL CALCULUS

Page 3

( Answers at the end of all questions ) 1 - tan x , 4x - π

( 12 ) Let f ( x ) =  π f    4

then

(a) 1

x ≠

π  π    . If f ( x ) is continuous in  0, , x ∈  0,  2  2   

is

(b)

1 2

(c) -

y + ..... ∞ ( 13 ) If x = e y + e ,

(a)

π , 4

x 1+ x

(b)

x > 0,

1 x

(c)

( 14 ) A point on the parabola y the abscissa is

2

(d) -1

then

dy dx

[ AIEEE 2004 ]

is

1 - x x

(d)

1+ x x

[ AIEEE 2004 ]

= 18x at which the ordinate increases at twice the rate of

 9 9  (c)  - ,   8 2 

( b ) ( 2, - 4 )

( a ) ( 2, 4 )

1 2

 9 9  (d)  ,   8 2 

[ AIEEE 2004 ]

( 15 ) A function y = f ( x ) has a second order derivative f ” ( x ) = 6 ( x - 1 ). If its graph passes through the point ( 2, 1 ) and at that point the tangent to the graph is y = 3x - 5, then the function is (a) (x - 1)

2

(b) (x - 1)

( 16 ) The normal to the curve through the fixed point ( a ) ( a, 0 )

3

(c) (x + 1)

3

(d) (x + 1)

[ AIEEE 2004 ]

x = a ( 1 + cos θ ), y = a sin θ at ‘ θ ’ always passes

( b ) ( 0, a )

( c ) ( 0, 0 )

( d ) ( a, a )

( 17 ) If 2a + 3b + 6c = 0, then at least one root of the equation ax the interval ( a ) ( 0, 1 )

2

( b ) ( 1, 2 )

( c ) ( 2, 3 )

( d ) ( 1, 3 )

[ AIEEE 2004 ]

2

+ bx + c = 0 lies in [ AIEEE 2004 ]

08 - DIFFERENTIAL CALCULUS

Page 4

( Answers at the end of all questions )

( 18 )

Let f ( x ) be a polynomial function of second degree. If f ( 1 ) = f ( - 1 ) and a, b, c are in A. P., then f ’ ( a ), f ’ ( b ) and f ’ ( c ) are in ( a ) A. P.

( 19 )

( b ) G. P.

( c ) H. P.

 x   1 - tan    [ 1 - sin x ] 2   lim π  x  x→ 1 + tan    [ π - 2x ]3 2  2  (a) 0

( 20 ) If

(b)

∞

(d)

log ( 3 + x ) - log ( 3 - x ) x x→0

(b) -

( 21 ) If f( x ) =

( 22 ) If y = 1 +

(d) -

x x2 + 1! 2! (b) 1

lim

+

x3 3!

+ ... ,

(c) x

(b)

(c)

[ AIEEE 2003 ]

then the value of

(d) y

n5 1 4

2 3

( d ) log a - log b

1 + 2 4 + 3 4 + ... + n 4

n→∞

[ AIEEE 2003 ]

is continuous at x = 0, then the value of f ( 0 ) is

(c) a - b

(b) a + b

( 23 ) The value of

( a ) zero

2 3

(c)

log ( 1 + ax ) - log ( 1 - bx ) x

( a ) ab

(a) 0

1 3

1 8

= k, then the value f k is

lim

(a) 0

[ AIEEE 2003 ]

=

1 32

(c)

( d ) A. G. P.

1 5

(d)

dy dx

[ AIEEE 2003 ]

is [ AIEEE 2003 ]

is 1 30

[ AIEEE 2003 ]

08 - DIFFERENTIAL CALCULUS

Page 5

( Answers at the end of all questions ) ( 24 ) If f : R → R satisfies f ( x + y ) = f ( x ) + f ( y ), for all x, y ∈ R and f ( 1 ) = 7, n

∑ f '(r )

then the value of

is

r =1

(a)

7n 2

(b) 7n(n + 1)

(c)

7 ( n + 1) 2

(d)

7 n ( n + 1) 2

[ AIEEE 2003 ]

( 25 ) The real number x when added to its inverse gives the minimum value of the sum at x equal to (b) -2

(a) 2

( 26 )

(d) -1

(c) 1

3

2

[ AIEEE 2003 ]

2

If the function f ( x ) = 2x - 9ax + 12a x + 1, where a > 0, attains its maximum and 2 minimum at p and q respectively such that p = q, then a equals (a) 3

(b) 1

(c) 2

(d) 4

[ AIEEE 2003 ]

f ' ' ' ( 1) ( - 1 )n f n ( 1 ) f ' ( 1) f " ( 1) + + + ... + 1! 2! 3! n!

n

( 27 ) If f ( x ) = x , then the value of f ( 1 ) is (a) 2

n

(b) 2

2

( 28 ) If x = t + t + 1

(a)

π 2

n -1

(c) 1

[ AIEEE 2003 ]

 π   π  y = sin  t  + cos  t  , then at t = 1, the value of  2   2 

and

(b) -

(d) 0

π 6

(c)

π 3

(d) -

3

π 4

[ AIEEE 2002 ]

3

( 29 ) If x = 3 cos θ - 2 cos θ and y = 3 sin θ - 2 sin θ, then the value of ( a ) sin θ

( b ) cos θ

( c ) tan θ

dy is dx

( d ) cot θ

dy dx

is

[ AIEEE 2002 ]

08 - DIFFERENTIAL CALCULUS

Page 6

( Answers at the end of all questions ) n

n

( 30 ) Let f ( a ) = g ( a ) = k and their nth derivatives f ( a ), g ( a ) exist and are not equal f ( a )g( x ) - f (a ) - g( a )f ( x ) + g(a ) for some n. Further if = 4, then the value lim g( x ) - f ( x ) x →a of k is (a) 4

(b) 2

( 31 ) The value of

(a)

10 3

lim

(c) 1

( 1 - cos 2x ) sin 5x

x →0

(b)

(d) 0

(c)

6 5

( 32 ) The value of

 sin 2 α - sin 2 β   lim  α → β   α2 - β2

(a) 0

(c)

(b) 1

lim

( 33 ) The value of

(a) 0

(b) 1

( 34 ) If f ( x ) = 2x (a) -2

x→0

3

sin β β

1 - cos 2x x (c)

(d)

5 6

[ AIEEE 2002 ]

is

sin 2β 2β

[ AIEEE 2002 ]

( d ) does not exist

[ AIEEE 2002 ]

(d)

is

2

- 3x2 - 12x + 5 on [ - 2, 4 ], then relative maximum occurs at x =

(b) -1

(c) 2

  1 1 - +    x x  ( 35 ) If f ( x ) =  x e  , x ≠0,   0, x =0 (a) (b) (c) (d)

is

x 2 sin 3x

3 10

[ AIEEE 2002 ]

(d) 4

[ AIEEE 2002 ]

then f ( x ) is

discontinuous everywhere continuous as well as differentiable for all x neither differentiable nor continuous at x = 0 continuous at all x but not differentiable at x = 0

[ AIEEE 2002 ]

08 - DIFFERENTIAL CALCULUS

Page 7

( Answers at the end of all questions ) ( 36 ) If y is a twice differentiable function and x cos y + y cos x = π, then y” ( 0 ) = (a) π

(b) -π

(c) 0

(d) 1

[ IIT 2005 ]

( 37 ) f ( x ) = l l x l - 1 l is not differentiable at x = ( a ) 0, ± 1

(b) ±1

(c) 0

(d) 1

[ IIT 2005 ]

 1  ( 38 ) If f is a differentiable function such that f : R → R, f   = 0 ∀ n ∈ I, n ≥ 1,  n  then ( a ) f ( x ) = 0 ∀ x ∈ [ 0, 1 ] (c) f(0) = 0 = f’(0)

( 39 )

( b ) f ( 0 ) = 0, but f ’ ( 0 ) may or may not be 0 ( d ) l f ( x ) l ≤ 1 ∀ x ∈ [ 0, 1 ] [ IIT 2005 ]

f is a twice differentiable polynomial function of x such that f ( 1 ) = 1, f ( 2 ) = 4 and f ( 3 ) = 9, then ( a ) f ” ( x ) = 2, ∀ x ∈ R ( c ) f ” ( x ) = 2 for only x ∈ [ 1, 3 ]

( b ) f ” ( x ) = f ’ ( x ) = 5, x ∈ [ 1, 3 ] ( d ) f ” ( x ) = 3, x ∈ ( 1, 3 ) [ IIT 2005 ]

[ Note: This question should have been better put as ‘polynomial function of degree two rather than twice differentiable function’. ]

( 40 )

S is a set of polynomial of degree less than or equal to 2, f ’ ( x ) > 0, ∀ x ∈ [ 0, 1 ], then set S = 2

( a ) ax + ( 1 - a ) x , a ∈ R 2 ( c ) ax + ( 1 - a ) x , 0 < a < ∞

f ( 0 ) = 0,

f ( 1 ) = 1,

2

( b ) ax + ( 1 - a ) x , 0 < a < 2 (d) φ

[ IIT 2005 ]

( 41 ) Let y be a function of x, such that log ( x + y ) = 2xy, then y ’ ( 0 ) is (a) 0

( 42 )

(b) 1

Let f ( x ) = x then α is (a) -2

α

(c)

1 2

log x for x >

(b) -1

(c) 0

(d)

3 2

[ IIT 2004 ]

and f ( 0 ) = 0 follows Rolle’s theorem for x ∈ [ 0, 1],

(d)

1 2

[ IIT 2004 ]

08 - DIFFERENTIAL CALCULUS

Page 8

( Answers at the end of all questions ) f ( x2 ) - f ( x ) x →0 f (x ) - f (0)

( 43 ) If f ( x ) is strictly increasing and differentiable, then (b) -1

(a) 1

( 44 ) Let f ( x ) = x

3

(c) 0

2

+ bx + cx + d,

( a ) is strictly increasing ( c ) has local minima

( 45 )

(d) 2

is [ IIT 2004 ]

2

0 < b < c, then f ( x )

( b ) has local maxima ( d ) is a bounded curve

[ IIT 2004 ]

If f ( x ) is a differentiable function, f ’ ( 1 ) = 1, f ’ ( 2 ) = 6, where f ’ ( c ) means the derivative of the function at x = c, then lim

f ( 2 + 2h + h 2 ) - f ( 2 )

h →0

f ( 1 + h - h2 ) - f ( 1 )

(b) -3

( a ) does not exist

( 46 )

lim

If

lim

(c) 3

sin nx [ ( a - n ) nx - tan x ]

x →0

x2

(d)

3 2

[ IIT 2003 ]

= 0, where n is a non-zero positive integer, then

a is equal to (a)

n+1 n

(b) n

2

(c)

1 n

(d) n +

1 n

[ IIT 2003 ]

( 47 ) Which function does not obey Mean Value Theorem in [ 0, 1 ] ? 1  1 x <  2 - x, 2  (a) f(x) =  2   1 - x  , x ≥ 1   2 2  (c) f(x) = x lxl

 sin x ,  (b) f(x) =  x  1,

(b) R - {1}

x = 0

(d) f(x) = lxl

( 48 ) The domain of the derivative of the function f ( x ) =

(a) R - {0}

x ≠ 0

(c) R - {-1}

[ IIT 2003 ]

 tan - 1 x, if l x l ≤ 1   1  ( l x l - 1 ), if l x l > 1 2

( d ) R - { - 1, 1 }

is

[ IIT 2002 ]

08 - DIFFERENTIAL CALCULUS

Page 9

( Answers at the end of all questions )

( 49 ) The integer n for which (a) 1

( 50 )

If

(b) 2

f: R → R

( cos x - 1 ) ( cos x - e x )

lim

xn

x→0

(c) 3

is a finite non-zero number is

(d) 4

be such that

[ IIT 2002 ]

f(1) = 3

and

f ’ ( 1 ) = 6,

then

 f (1 + x ) lim  f ( 1) x→0

1

x  

equals (a) 1

1 2 e

(b)

(c) e

2

3

( 51 ) The point ( s ) on the curve y + 3x  4 , -2 ( a )  ± 3 

( 52 )

   

(d) e

2

  11 , 0 (b)  ±   3  

3

[ IIT 2002 ]

= 12y where the tangent is vertical, is / ( are ) ( c ) ( 0, 0 )

  4 ( d )  ± , 2  3  

[ IIT 2002 ]

3

Let f : R → R be a function defined by f ( x ) = { x, x }. The set of all points where f ( x ) is not differentiable is ( a ) { - 1, 1 }

( b ) { - 1, 0 }

( c ) { 0, 1 }

( d ) { - 1, 0, 1 }

[ IIT 2001 ]

( 53 ) The left hand derivative of f ( x ) = [ x ] sin ( πx ) at x = k, where k is an integer, is k

(a) (-1) (k - 1)π (c) (-1)

k

kπ

(b) (-1)

k - 1

(d) (-1)

k - 1

(k - 1)π kπ

[ IIT 2001 ]

( 54 ) The left hand derivative of f ( x ) = [ x ] sin ( πx ) at x = k, where k is an integer, is k

(a) (-1) (k - 1)π k

(c) (-1) kπ

( 55 )

lim

sin ( π cos 2 x )

x→0

(a) - π

x2 (b) π

(b) (-1)

k - 1

(d) (-1)

k - 1

(k - 1)π kπ

[ IIT 2001 ]

equals (c) π/2

(d) 1

[ IIT 2001 ]

08 - DIFFERENTIAL CALCULUS

Page 10

( Answers at the end of all questions )

( 56 ) If f ( x ) = x e

x(1 - x)

, then f ( x ) is

 1   - 2 , 1  

( a ) increasing on

( b ) decreasing on R  1  ( d ) decreasing on  - , 1   2 

( c ) increasing on R

[ IIT 2001 ]

( 57 ) Which of the following functions is differentiable at x = 0 ? ( a ) cos ( l x l ) + l x l ( c ) sin ( l x l ) + l x l

( 58 ) If x

2

2

+ y

( b ) cos ( l x l ) - l x l ( d ) sin ( l x l ) - l x l

= 1, then 2

2

( a ) yy” - 2 ( y’ ) + 1 = 0 2 - 1 = 0 ( c ) yy” + ( y’ )

( 59 ) For x ∈ R,

(a) e

[ IIT 2001 ]

( b ) yy” + ( y’ ) + 1 = 0 2 ( d ) yy” + 2( y’ ) + 1 = 0

 x - 3    x→∞  x + 2 

x

lim

(b) e-

1

[ IIT 2000 ]

=

(c) e-

5

(d) e

5

[ IIT 2000 ]

( 60 ) Consider the following statements in S and R:  π  , π  2 R: If a differentiable function decreases in an interval ( a, b ), then its decreases in ( a, b ). Which of the following is true ? S:

(a) (b) (c) (d)

( 61 )

Both sin x and cos x are decreasing functions in the interval

Both S and Both S and S is correct S is correct

   derivative also

R are wrong. R are correct, but R is not the correct explanation of S. and R is correct explanation of S. and R is wrong.

If the normal to the curve y = f ( x ) at the point ( 3, 4 ) makes an angle

[ IIT 2000 ]

3π with the 4

positive X-axis, then f ’ ( 3 ) = (a) -1

(b) -3/4

(c) 4/3

(d) 1

[ IIT 2000 ]

08 - DIFFERENTIAL CALCULUS

Page 11

( Answers at the end of all questions )  l x l for 0 < l x l ≤ 2 ,  x = 0  1 for

( 62 ) If f ( x ) =

( a ) a local maximum ( c ) a local minimum

then at x = 0, f has

( b ) no local maximum ( d ) no extremum

[ IIT 2000 ]

( 63 ) For all x ∈ ( 0, 1 ), which of the following is true ? x

(a) e < 1 + x ( c ) sin x > x

( b ) loge ( 1 + x ) < x ( d ) loge x > x

4

[ IIT 2000 ]

4

( 64 ) The function f ( x ) = sin x + cos x increases if (a) 0 < x < (c)

( 65 )

π 8

3π π < x < 4 8 5π 3π < x < 8 4

(b)

3π 5π < x < 8 8

(d)

The function f ( x ) = [ x ] to y, is discontinuous at

2

- [ x2 ] where [ y ] is the greatest integer less than or equal

( a ) all integers ( c ) all integers except 0

( 66 ) The function f ( x ) = ( x (a) -1

( 67 )

lim

2

(b) 0

x→0

(a) 2

(1 -

cos 2x ) 2

(b) -2

[ IIT 1999 ]

- 1 ) l x2 - 3x + 2 l + cos ( l x l ) is NOT differentiable at (d) 2

[ IIT 1999 ]

= (c)

x

( 68 ) The function f ( x ) =

( b ) all integers except 0 and 1 ( d ) all integers except 1

(c) 1

x tan 2x - 2x tan x

[ IIT 1999 ]

∫ t(e

t

1 2

(d) -

1 2

[ IIT 1999 ]

- 1 ) ( t - 1 ) ( t - 2 ) 3 ( t - 3 ) 5 dt has a local minimum at x =

-1

(a) 0

(b) 1

(c) 2

(d) 3

[ IIT 1999 ]

08 - DIFFERENTIAL CALCULUS

Page 12

( Answers at the end of all questions ) ( 69 )

1 - cos 2 ( x - 1 )

lim

x - 1

x →1

( a ) exists and is equal to ( b ) exists and is equal to 2 ( c ) does not exist because x - 1 → 0 ( d ) does not exist because left hand limit ≠ right hand limit x

( 70 ) If

= x +

0

1 2

[ IIT 1998 ]

x

∫ f ( t ) dt

(a)

2

∫ t f ( t ) dt ,

then the value of f ( 1 ) is

1

(b) 0

(d) -

(c) 1

1 2

[ IIT 1998 ]

2

( 71 ) Let h ( x ) = min [ x, x ], for every real number x, then ( a ) h is continuous for all x ( c ) h’ ( x ) = 1 for all x > 1

( 72 ) If h ( x ) = f ( x ) - [ f ( x ) ] (a) (b) (c) (d)

( 74 )

[ IIT 1998 ]

for every real number x, then

h is increasing whenever f is increasing h is increasing whenever f is decreasing h is decreasing whenever f is decreasing nothing can be said in general

( 73 ) If f ( x ) = (a) (b) (c) (d)

2

( b ) h is differentiable for all x ( d ) h is not differentiable at two values of x

x sin x

and g ( x ) =

[ IIT 1998 ]

x , where 0 < x ≤ 1, then in this interval tan x

both f ( x ) and g ( x ) are increasing functions both f ( x ) and g ( x ) are decreasing functions f ( x ) is an increasing function g ( x ) is an increasing function

1 n→p n lim

2n

∑ 1

(a) 1 +

r n2 + r 2 5

[ IIT 1997 ]

equals

(b) -1 +

5

(c) -1 +

2

(d) 1 +

2

[ IIT 1997 ]

08 - DIFFERENTIAL CALCULUS

Page 13

( Answers at the end of all questions ) x3 6

( 75 ) If f ( x ) =

sin x -1 p2

p (a) p

cos x 0 , ( p is a constant ), then

(b) p + p

p3 2

(c) p + p

3

d3 dx 3

( d ) independent of p

[ f ( x ) ] at x = 0 is

[ IIT 1997 ]

 2x - 1  ( 76 ) The function f ( x ) = [ x ] cos   π, where [ . ] denotes the greatest integer 2   function, is discontinuous at ( a ) all x ( c ) no x

( b ) all integer points ( d ) x which is not an integer

[ IIT 1995 ]

 x   = f ( x ) - f ( y ) for ( 77 ) If f ( x ) is defined and continuous for all x > 0 and satisfy f   y  all x, y and f ( e ) = 1, then  1   → 0 as x → 0   x  ( d ) f ( x ) = log x

( a ) f ( x ) is bounded

(b)

( c ) x f ( x ) → 1 as x → 0

( 78 ) On the interval [ 0, 1 ], the function x (a) 0

(b)

1 4

(c)

1 2

25

(1 - x)

(d)

75

[ IIT 1995 ]

attains maximum value at the point

1 3

[ IIT 1995 ]

( 79 ) The function f ( x ) = l px - q l + r l x l, x ∈ ( - ∞ , ∞ ) where p > 0, q > 0, r > 0, assumes its minimum value only at one point if (a) p ≠ q

(b) r ≠ q

( 80 ) The function f ( x ) =

ln ( π + x ) ln ( e + x )

(c) r ≠ p

( d ) p =q = r

[ IIT 1995 ]

is

( a ) increasing on [ 0, ∞ ) π   ( c ) increasing on  0, e  

( b ) decreasing on [ 0, ∞ )  π  and decreasing on  , ∞   e 

π   ( d ) decreasing on  0,  e  

and increasing on

 π  e, 

∞  

[ IIT 1995 ]

08 - DIFFERENTIAL CALCULUS

Page 14

( Answers at the end of all questions ) ( 81 ) The function f ( x ) = max { ( 1 - x ), ( 1 + x ), 2 }, x ∈ ( - ∞ , ∞ ), is ( a ) continuous at all points ( b ) differentiable at all points ( c ) differentiable at all points except at x = 1 and x = - 1 ( d ) continuous at all points except at x = 1 and x = - 1

[ IIT 1995 ]

2

( 82 ) Let [ . ] denote the greatest integer function and f ( x ) = [ tan x ]. Then, (a)

lim f ( x ) does not exist

( b ) f ( x ) is continuous at x = 0

x→0

( c ) f ( x ) is not differentiable at x = 0

( 83 ) If f ( x ) =

 3x 2 + 12x - 1,   37 - x,

-1 ≤ x ≤ 2 , 2 < x ≤ 3

( a ) f ( x ) is increasing on [ - 1, 2 ] ( c ) f ( x ) is maximum at x = 2

( 84 ) The value of

lim

x→0

(b) -1

(a) 1

[ IIT 1993 ]

then

( b ) f ( x ) is continuous on [ - 1, 3 ] ( d ) f ’ ( 2 ) does not exist

1 ( 1 - cos 2x) 2 x (c) 0

(d) f’(0) = 1

[ IIT 1993 ]

is ( d ) none of these

[ IIT 1991 ]

( 85 ) The following functions are continuous on ( 0, π ). ( a ) tan x

(b)

( c ) 1,

0 < x ≤

2 sin

2x , 9

3π 4

3π < x ≤ π 4

π 1 ∫ t sin dt t 0

( d ) x sin x, π sin ( π + x ), 2

0 < x ≤

π 2

π < x < π 2

[ IIT 1991 ]

x - 1, then, on the interval [ 0, π ], tan [ f ( x ) ] and 2 1 1 are both continuous (b) are both discontinuous (a) f(x) f(x)

( 86 ) If f ( x ) =

(c) f

-1

( x ) are both continuous

(d) f

-1

( x ) are both discontinuous

[ IIT 1989 ]

08 - DIFFERENTIAL CALCULUS

Page 15

( Answers at the end of all questions ) 2

( 87 ) If y = P ( x ), a polynomial of degree 3, then 2

( a ) P ’’’ ( x ) + P ’ ( x ) ( c ) P ( x ) P ’’’ ( x )

 x - 3  2  x 3x 13 +  2 4  4

( a ) continuous at x = 1 ( c ) continuous at x = 3

x < 1

is

( b ) differentiable at x = 1 ( d ) differentiable at x = 3 x 1 + lxl

[ IIT 1988 ]

is differentiable is

( b ) ( 0, ∞ ) ( c ) ( - ∞ , 0 ) ∪ ( 0, ∞ ) ( e ) none of these

[ IIT 1987 ]

Let f and g be increasing and decreasing functions respectively from ( 0, ∞ ) to ( 0, ∞ ). Let h ( x ) = f [ g ( x ) ]. If h ( 0 ) = 0, h ( x ) - h ( 1 ) is ( a ) always zero ( d ) strictly increasing

( 91 )

[ IIT 1988 ]

x ≥ 1

( 89 ) The set of all points where the function f ( x ) =

( 90 )

equals

( b ) P ’’ ( x ) P ’’’ ( x ) ( d ) a constant

( 88 ) The function f ( x ) =

(a) (- ∞, ∞) ( d ) ( 0, ∞ )

d  3 d 2 y  y dx  dx 2  

2

( b ) always negative ( e ) none of these

4

( c ) always positive [ IIT 1987 ]

2n

Let P ( x ) = a0 + a1x + a2x + … + a nx be a polynomial in a real variable x with 0 < a0 < a1 < a2 < … < a n. The function P ( x ) has ( a ) neither a maximum nor a minimum ( b ) only one maximum ( c ) only one minimum ( d ) only one maximum and only one minimum ( e ) none of these

[ IIT 1986 ]

( 92 ) The function f ( x ) = 1 + l sin x l is ( a ) continuous nowhere ( b ) continuous everywhere ( c ) differentiable ( d ) not differentiable at x = 0 ( e ) not differentiable at infinite number of points [ IIT 1986 ]

( 93 )

Let [ x ] denote the greatest integer less than or equal to x. If f ( x ) = [ x sin πx ], then f ( x ) is ( a ) continuous at x = 0 ( b ) continuous in ( -1, 0 ) ( c ) differentiable at x = 1 ( d ) differentiable in ( - 1, 1 ) ( e ) none of these [ IIT 1986 ]

08 - DIFFERENTIAL CALCULUS

Page 16

( Answers at the end of all questions ) sin [ x ] , [x] ≠ 0 [x] = 0, [ x ] = 0, x ] denotes the greatest integer less than or equal to x, then

( 94 ) If f ( x ) = where (a) 1

(c) -1

(b) 0

x -

( 95 ) If f ( x ) = x (

lim f ( x ) equals

x→0

( d ) none of these

x + 1 ), then

( a ) f ( x ) is continuous but not differentiable at x = 0 ( b ) f ( x ) is differentiable at x = 0 ( c ) f ( x ) is not differentiable at x = 0 ( d ) none of these

( 96 )

 1  n → ∞  1 - n2 lim

(a) 0

+

(b) -

[ IIT 1985 ]

2

  1 - n2  n

+ ... +

1 - n2 1 2

1 2

(c)

[ IIT 1985 ]

is equal to

( d ) none of these

[ IIT 1984 ]

( 97 ) If x + l y l = 2y, then y as a function of x is ( a ) defined for all real x

( b ) continuous at x = 0 dy 1 = ( d ) such that for x < 0 dx 3

( c ) differentiable for all x 25 - x 2 , then

( 98 ) If G ( x ) = -

(a)

1 24

(b)

1 5

(c) -

G ( x ) - G ( 1) x - 1 x →1 lim

24

[ IIT 1984 ]

has the value

( d ) none of these

[ IIT 1983 ]

( 99 ) If f ( a ) = 2, f ’ ( a ) = 1, g ( a ) = - 1, g’ ( a ) = 2, then the value of g( x )f (a ) - g(a ) f ( x ) is lim x - a x→a (a) -5

(b)

1 5

(c) 5

( d ) none of these

[ IIT 1983 ]

ln ( 1 + ax ) - ln ( 1 - bx ) is not defined at x = 0. The value which x should be assigned to f at x = 0, so that it is continuous at x = 0, is

( 100 ) The function f ( x ) =

(a) a - b

(b) a + b

( c ) ln a + ln b

( d ) none of these

[ IIT 1983 ]

08 - DIFFERENTIAL CALCULUS

Page 17

( Answers at the end of all questions ) ( 101 ) The normal to the curve x = a ( cos θ + θ sin θ ), y = a ( sin θ - θ cos θ ) at any point ‘ θ ’ is such that ( a ) it makes a constant angle with the X-axis ( b ) it passes through the origin ( c ) it is at a constant distance from the origin ( d ) none of these [ IIT 1983 ] 2

( 102 ) If y = a ln x + bx + x has its extremum values at x = - 1 and x = 2, then ( a ) a = 2, b = - 1 ( c ) a = - 2, b =

1 2

( b ) a = 2, b = -

1 2

( d ) none of these

[ IIT 1983 ]

( 103 ) There exists a function f ( x ) satisfying f ( 0 ) = 1, f ’ ( 0 ) = - 1, f ( x ) > 0 for all x and ( a ) f ” ( x ) > 0 for all x ( c ) - 2 ≤ f ” ( x ) ≤ - 1 for all x

( b ) - 1 < f ” ( x ) < 0 for all x ( d ) f ” ( x ) < - 2 for all x

[ IIT 1982 ]

( 104 ) For a real number y, let [ y ] denote the greatest integer less than or equal to y. Then tan [ π ( x - π ) ] is the function f ( x ) = 1 + [ x ]2 (a) (b) (c) (d)

discontinuous at some x continuous at all x, but the derivative f ” ( x ) does not exist for some x f ’ ( x ) exists for all x, but the derivative f ” ( x ) does not exist for some x f ” ( x ) exists for all x [ IIT 1981 ]

( 105 ) If f ( x ) = (a) 0

x - sin x x + cos 2 x (b) ∞

, then (c) 1

lim

x→∞

f ( x ) is

( d ) none of these

[ IIT 1979 ]

08 - DIFFERENTIAL CALCULUS

Page 18

( Answers at the end of all questions )

Answers 1 c

2 d

3 c

4 a

5 c

6 a

7 b

8 b

9 d

10 b

11 b

12 c

13 c

14 d

15 b

16 a

17 a

18 a

19 c

20 c

21 b

22 d

23 c

24 d

25 c

26 c

27 d

28 b

29 d

30 a

31 a

32 d

33 d

34 d

35 d

36 a

37 a

38 c

39 a

40 b

41 b

42 d

43 b

44 a

45 c

46 d

47 a

48 d

49 c

50 c

51 d

52 d

53 a

54 a

55 b

56 a

57 d

58 b

59 c

60 d

61 d

62 d

63 b

68 b,d

69 d

70 a

71 a,c,d

72 a,c

79 c

80 b

89 a

90 a

81 a,c 97 a,b,d

82 b 98 d

64 b

65 b

83 a,b,c,d 99 c

100 b

66 d 84 d

67 c 85 b,c

101 c

86 b

102 b

87 c 103 a

88 a,b,c 104 d

105 c

106

73 c

107

74 b

91 c 108

75 d

92 b,d,e 109

76 b

77 d

78 b

93 a,d

94 d

95 a

96 b

110

111

112

113