08 - DIFFERENTIAL CALCULUS
Page 1
( Answers at the end of all questions ) 1 2 1 2 sec n→∞ n n2 (a)
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 ]
θ cos θ ) at any point
The normal to the curve x = a ( cos θ + θ sin θ ), y = a ( sin θ ‘ θ ’ is such that
ce .c
(2)
1 sec 1 2
2
+
lim
om
(1)
[ AIEEE 2005 ]
A function is matched below agains an interval where it is supposed to be increasing. Which of the following pairs is correctly matched ? Interval
Fu ction
Interval
xa
(3)
m
ra
( 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 o igin
x
3
x
3x
3
2
( b ) [ 2, ∞ )
+ 3x + 3
- 2x 2 + 1
(d)
2x
(- ∞, -4)
3
x
- 3x 2 + 12x + 6 3
- 6x 2 + 6
w w
.e
(a) (- ∞, ∞) 1 (c) (- ∞, ) 3
Function
(4)
Let α and β be the distinct roots of the equation ax 1 - cos ( ax 2 + bx + c )
im
w
x→α
(5)
[ AIEEE 2005 ]
(a)
( x - α )2
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 ]
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 ) Let f be differentiable for al x. If f ( 1 ) = - 2 and f ’ ( x ) ≥ 2 for x ∈ [ 1, 6 ], then
(c) 2
(d) 1
2
y) ,
[ AIEEE 2005 ]
ce .c
(b) 0
1 36 π
1 18 π
(c)
1 54 π
d)
5 6π
m
(b)
[ AIEEE 2005 ]
Let f : R → R be a differenti ble function having f ( 2 ) = 6, f ’ ( 2 ) = f( x )
lim
x→2
∫
6
4t 3 dt x - 2
( b ) 36
w w
( 10 ) If the e u tion
anx
( c ) 12
n
+ an-1x
( d ) 18
n-1
pos tive root x = α, then the equation has positive root which is ( a ) greater than α ( c ) greater than or equal to α
( 11 ) If
1 . Then 48
equa s
.e
( a ) 24
w
[ AIEEE 2005 ]
A spherical iron ball 10 cm in radius is coated with laye of ice of uniform 3 thickness that melts at a rate of 50 cm /min. W en the thickness of ice is 5 cm, then the rate at which thickness of ice decreases in cm/min is (a)
(9)
(d) f(6) = 5
If f is a real valued differentiable function satisfying l f ( x ) - f ( y ) l ≤ ( x x, y ∈ R and f ( 0 ) = 0, then f ( 1 ) equals (a) -1
(8)
(c) f(6) < 5
xa
(7)
(b) f(6) < 8
om
(a) f(6) ≥ 8
ra
(6)
a b 1+ + x x→∞ x2 lim
( a ) a ∈ R, b ∈ R ( c ) a ∈ R, b = 2
[ 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 ]
2x 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
(b)
1 2
(c) -
y + ..... ∞ ( 13 ) If x = e y + e ,
(a)
x 1+ x
(b)
1 2
x > 0,
1 x
(c)
2
(d) -1
then 1 - x x
[ AIEEE 2004 ]
dy dx
is
(d)
9 9 (c) - , 8 2
( b ) ( 2, - 4 )
[ AIEEE 2004 ]
9 9 (d) , 8 2
[ AIEEE 2004 ]
xa
( a ) ( 2, 4 )
1+ x x
= 18x at which the ordinate increases at twice the rate of
m
( 14 ) A point on the parabola y the abscissa is
om
is
ce .c
(a) 1
π π . If f ( x ) is continuous in 0, , x ∈ 0, 2 2
ra
then
π , 4
x ≠
.e
( 15 ) A function y = f ( x ) has a second order derivative f ” ( x ) = 6 ( x - 1 ). If its graph passes through the poin ( 2, 1 ) and at that point the tangent to the graph is y = 3x - 5, then the unction is
w w
(a) (x - 1
w
( 16 )
2
(b) (x - 1)
he 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 )
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 ( c ) H. P.
x 1 - tan [ 1 - sin x ] 2 lim π x x→ 1 + tan [ π - 2x ]3 2 2
∞
1 32
(c)
log ( 3 + x ) - log ( 3 - x ) x x→0 lim
(b) -
(a) 0
x x2 + 1! 2!
(b) 1
( 23 ) The value of
( a ) zero
= k
lim
+
x3 3!
+ ... ,
(c) x
(b)
(d) -
(c)
2 3
[ AIEEE 2003 ]
( d ) log a - log b
then the value of
(d) y
n5 1 4
[ AIEEE 2003 ]
is continuous at x = 0, then the value of f ( 0 ) is
1 + 2 4 + 3 4 + ... + n 4
n→∞
1 8
then the value f k is
(c) a - b
(b) a + b
w w w
(a) 0
(d)
log ( 1 + ax ) - log ( 1 - bx ) x
( a ) ab
( 22 ) If y = 1
2 3
(c)
.e
( 21 ) If f( x ) =
1 3
=
xa
( 20 ) If
(b)
[ AIEEE 2003 ]
ce .c
(a) 0
( d ) A. G. P.
ra
( 19 )
( b ) G. P.
om
( a ) A. P.
m
( 18 )
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
(a)
7n 2
(b) 7n(n + 1)
(c)
7 ( n + 1) 2
(d)
om
r =1
7 n ( n + 1) 2
[ AIEEE 2003 ]
(b) -2
(a) 2
(d) -1
(c) 1
3
2
[ AIEEE 2003 ]
2
(a) 3
ra
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 (b) 1
(c) 2
n
m
( 26 )
ce .c
( 25 ) The real number x when added to its inverse gives the min mum value of the sum at x equal to
(d) 4
f ' ( 1) f " ( 1) f ' ' ' ( 1) ( - 1 )n f n ( 1 ) + + + ... + 1! 2! 3! n!
xa
( 27 ) If f ( x ) = x , then the val e of f ( 1 ) is n
(b) 2
n
1
(c) 1
.e
(a) 2
w w
2
( 28 ) If x = t + t π 2
w
(a)
1
(d) 0
[ AIEEE 2003 ]
π π y = sin t + cos t , then at t = 1, the value of 2 2
and
(b) -
π 6
(c)
π 3
(d) -
3
π 4
( b ) cos θ
3
( c ) tan θ
dy is dx
[ AIEEE 2002 ]
( 29 ) If x = 3 cos θ - 2 cos θ and y = 3 sin θ - 2 sin θ, then the value of ( a ) sin θ
[ AIEEE 2003 ]
( d ) cot θ
dy dx
is
[ AIEEE 2002 ]
08 - DIFFERENTIAL CALCULUS
Page 6
( Answers at the end of all questions ) n
n
(b) 2
( 31 ) The value of
10 3
( 1 - cos 2x ) sin 5x
3 10
(c)
6 5
sin 2 α - sin 2 β lim α → β α2 - β2
(a) 0
(c)
lm
1 - cos 2x x
.e
x→0
w w
(b) 1
f ( x ) = 2x
-2
5 6
[ AIEEE 2002 ]
3
is
(c)
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
(d) 4
[ AIEEE 2002 ]
w
(a
sin β β
xa
(b) 1
( 33 ) The value of
( 34 )
(d)
m
( 32 ) The value of
(a) 0
[ AIEEE 2002 ]
is
x 2 sin 3x
x →0
(b)
(d) 0
ra
(a)
lim
(c) 1
ce .c
(a) 4
om
( 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
1 1 - + x x ( 35 ) If f ( x ) = x e , x ≠0, 0, x =0 (a) (b) (c) (d)
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 ) = (b) -π
(c) 0
(d) 1
[ IIT 2005 ]
om
(a) π
( 37 ) f ( x ) = l l x l - 1 l is not differentiable at x = (b) ±1
(c) 0
(d) 1
( 38 ) If f is a differentiable function such that f : R → R then
( 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
m
( 39 )
1 f = 0 ∀ n ∈ I, n ≥ 1, n
ra
( a ) f ( x ) = 0 ∀ x ∈ [ 0, 1 ] (c) f(0) = 0 = f’(0)
[ IIT 2005 ]
ce .c
( a ) 0, ± 1
( b ) f ” ( x ) = f ’ ( x ) = 5, x ∈ [ 1, 3 ] ( d ) f ” ( x ) = 3, x ∈ ( 1, 3 ) [ IIT 2005 ]
xa
( a ) f ” ( x ) = 2, ∀ x ∈ R ( c ) f ” ( x ) = 2 for only x ∈ [ 1 3 ]
S is a set of polynomial of degree less than or equal to 2, f’(x) 0, ∀ x ∈ [ 0, 1 ], then set S =
w w
( 40 )
.e
[ Note: This question should have been better put as ‘polynomial function of degree two rather than wice differentiable function’. ]
2
w
( 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
( 42 )
(a) 0
(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
3
2
0 < b < c, then f ( x )
( b ) has local maxima ( d ) is a bounded curve
f ( 2 + 2h + h 2 ) - f ( 2 )
h →0
f ( 1 + h - h2 ) - f ( 1 )
(b) -3
sin nx [ ( a - n ) nx - tan x ]
x →0
a is equal to
.e
n+1 n
(c
3
’(2)
(d)
[ IIT 2004 ]
6, where f ’ ( c ) means the
3 2
[ IIT 2003 ]
= 0, where n is a non-zero positive integer, then
x
(b) n
2
(c)
w w
(a)
xa
lim
m
( a ) does not exist
If
ra
If f ( x ) is a differentiable function, f ’ ( 1 ) = 1, derivative of the function at x = c, then lim
( 46 )
[ IIT 2004 ]
2
+ bx + cx + d,
( a ) is strictly increasing ( c ) has local minima
( 45 )
(d) 2
ce .c
( 44 ) Let f ( x ) = x
(c) 0
is
om
(b) -1
(a) 1
lim
1 n
(d) n +
1 n
[ IIT 2003 ]
w
( 47 ) Which functi n 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 )
( 50 )
If
(b) 2
f: R → R
xn
x→0
(c) 3
(d) 4
be such that
(a) 1
(b)
(c) e
2
3
(d) e
2
11 , 0 (b) ± 3
f ’ ( 1 ) = 6,
Let f : R → R be a functio f ( x ) is not differentiable is (
.e
( a ) { - 1, 1 }
then
3
1
f (1 + x ) x lim f ( 1) x→0
[ IIT 2002 ]
= 12y wh re the tangent is vertical, is / ( are ) ( c ) ( 0, 0 )
m
4 ( d ) ± , 2 3
[ IIT 2002 ]
3
defined by f ( x ) = { x, x }. The set of all points where
xa
( 52 )
and
ra
( 51 ) The point ( s ) on the curve y + 3x 4 ( a ) ± , -2 3
[ I T 2002 ]
f(1) = 3
equals 1 2 e
is a finite non-zero number is
om
(a) 1
( cos x - 1 ) ( cos x - e x )
lim
ce .c
( 49 ) The integer n for which
) { - 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 - 1)π k
w w
(a) (-
(c) (-1
w
54
( 55 )
k
kπ
k - 1
(b) (-1) (d) (-1)
k - 1
(k - 1)π kπ
[ IIT 2001 ]
The left hand derivative of f ( x ) = [ x ] sin ( πx ) at x = k, where k is an integer, is (a) (-1) (k - 1)π k
(c) (-1) kπ k
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 ) 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
( 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
lim
(b) e-
x
[ IIT 2000 ]
m
x - 3 x→∞ x + 2
ra
( b ) yy” + ( y’ ) + 1 = 0 2 ( d ) yy” + 2( y’ ) + 1 = 0
1
=
(c) e-
xa
(a) e
[ IIT 2001 ]
2
( a ) yy” - 2 ( y’ ) + 1 = 0 2 - 1 = 0 ( c ) yy” + ( y’ )
( 59 ) For x ∈ R,
[ IIT 2001 ]
ce .c
( 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
om
( 56 ) If f ( x ) = x e
5
(d) e
5
[ IIT 2000 ]
.e
( 60 ) Consider the fo owing 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 ? Both sin x and cos x are decreasing functions in the interval
w
w w
S:
( 61 )
(a (b) (c) (d)
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
( a ) a local maximum ( c ) a local minimum
then at x = 0, f has
( b ) no local maximum ( d ) no extremum
( 63 ) For all x ∈ ( 0, 1 ), which of the following is true ? ( b ) loge ( 1 + x ) < x ( d ) loge x > x
4
ce .c
x
(a) e < 1 + x ( c ) sin x > x
[ I T 2000 ]
om
( 62 ) If f ( x ) =
4
[ IIT 2000 ]
( 64 ) The function f ( x ) = sin x + cos x increases if
3π 5π < x < 8 8
(d)
The function f ( x ) = [ x ] to y, is discontinuous at
2
( 66 ) The function f ( x ) = ( x
w w
(a) -1
w
lim
2
(b) 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
(c) 1
x tan 2x - 2x tan x
x→0
( b ) all integers except 0 and 1 ( d ) all integers except 1
p 0
.e
( a ) all integers ( c ) all integers ex
( 67 )
[ IIT 1999 ]
- [ x2 ] where [ y ] is the greatest integer less than or equal
xa
( 65 )
3π π < x < 4 8 5π 3π < x < 8 4
(b)
m
(c)
π 8
ra
(a) 0 < x <
(d) 2
[ IIT 1999 ]
= (c)
1 2
(d) -
1 2
[ IIT 1999 ]
x
( 68 ) The function f ( x ) =
∫ t(e
t
- 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
( 70 ) If
x
∫ f ( t ) dt
= x +
0
then the value of f ( 1 ) is
1
1 2
(a)
∫ t f ( t ) dt ,
(b) 0
[ IIT 1998 ]
ce .c
x
2
om
( 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
(d) -
(c) 1
2
1 2
[ IIT 1998 ]
ra
( 71 ) Let h ( x ) = min [ x, x ], for every real number x, then
( b ) h is differentiable for all x ( d ) h is not differentiable at two values of x
[ IIT 1998 ]
m
( a ) h is continuous for all x ( c ) h’ ( x ) = 1 for all x > 1
h is increasing whenev r f is increasing h is increasing wh ever f is decreasing h is decreasing whenever f is decreasing nothing ca be said in general
w w w
( 73 ) If f ( x ) =
( 74 )
for every real number x, then
.e
(a) (b) (c) (d)
( ) (b) (c) (d)
2
xa
( 72 ) If h ( x ) = f ( x ) - [ f ( 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
[ IIT 1997 ]
equals
n2 + r 2 5
(b) -1 +
5
(c) -1 +
2
(d) 1 +
2
[ IIT 1997 ]
08 - DIFFERENTIAL CALCULUS
Page 13
( Answers at the end of all questions )
( 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
ce .c
2x - 1 ( 76 ) The function f ( x ) = [ x ] cos π, where [ . ] denote 2 function, is discontinuous at ( a ) all x ( c ) no x
[ f ( x ) ] at x = 0 is
om
x3 6
( b ) all integer points ( d ) x which is not an integer
[ IIT 1997 ]
the greatest integer
[ IIT 1995 ]
1 → 0 as x → 0 x ( d ) f ( x ) = log x (b)
m
( a ) f ( x ) is bounded
ra
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
xa
( c ) x f ( x ) → 1 as x → 0
( 78 ) On the interval [ 0, (b)
1 4
w w
.e
(a) 0
], he function x (c)
1 2
25
(1 - x)
(d)
75
[ IIT 1995 ]
attains maximum value at the point
1 3
[ IIT 1995 ]
( 79 ) Th func ion 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
w
(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
om
[ IIT 1995 ]
2
( 82 ) Let [ . ] denote the greatest integer function and f ( x ) = [ tan x ] Then, lim f ( x ) does not exist
( b ) f ( x ) is contin ous
x→0
( c ) f ( x ) is not differentiable at x = 0 3x 2 + 12x - 1, 37 - x,
t x = 0
(d) f’(0) = 1
-1 ≤ x ≤ 2 , 2 < x ≤ 3
lim
x→0
(b) -1
(
) 0
[ IIT 1993 ]
is
( d ) none of these
.e
(a) 1
1 ( 1 - cos 2x) 2 x
xa
( 84 ) The value of
( b ) f ( x ) is continuous on [ - 1, 3 ] ( d f ’ ( 2 ) does not exist
m
( a ) f ( x ) is increasing on [ - 1, 2 ] ( c ) f ( x ) is maximum at x = 2
[ IIT 1993 ]
hen
ra
( 83 ) If f ( x ) =
ce .c
(a)
[ IIT 1991 ]
w w
( 85 ) The follow ng functions are continuous on ( 0, π ). ( a ) tan x (
) 1,
w
2 sin
2x , 9
(b) 0 < x ≤
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 (a) are both continuous (b) are both discontinuous 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
( b ) P ’’ ( x ) P ’’’ ( x ) ( d ) a constant
( 88 ) The function f ( x ) =
( a ) continuous at x = 1 ( c ) continuous at x = 3
x ≥ 1 x < 1
ra
is differentiable is
∞ , 0 ) ∪ ( 0, ∞ ) [ IIT 1987 ]
Let f and g be increasing nd decreasing functions respectively from ( 0, ∞ ) to ( 0, ∞ ). Let h ( x ) = f [ g ( x ) ] If h ( 0 ) = 0, h ( x ) - h ( 1 ) is
.e
( a ) always zero ( d ) strictly increasing
2
( b ) always negative ( ) none of these
4
( c ) always positive
2n
( a ) neit er 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
w
[ IIT 1987 ]
Let P ( x ) = a0 + a1x + a2x + … + a nx be a polynomial in a real variable x with 0 < a0 < a < a2 < … < a n. The function P ( x ) has
w w
( 91 )
x + lxl
[ IIT 1988 ]
m
( b ) ( 0, ∞ ) (c) ( ( e ) none of these
xa
( 90 )
is
( b ) differentiable at x = 1 ( d ) differentiable at x = 3
( 89 ) The set of all points where the function f ( x ) = (a) (- ∞, ∞) ( d ) ( 0, ∞ )
[ IIT 1988 ]
om
x - 3 2 x 3x 13 + 2 4 4
equals
ce .c
( a ) P ’’’ ( x ) + P ’ ( x ) ( c ) P ( x ) P ’’’ ( x )
d 3 d 2 y y dx dx 2
[ IIT 1986 ]
( 92 ) The function f ( x ) = 1 + l sin x l is
( 93 )
( a ) continuous nowhere ( b ) continuous everywhere ( c ) differentiable ( d ) not differentiable at x = 0 ( e ) not differentiable at infinite number of points [ IIT 1986 ] 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 ) =
(c) -1
(b) 0
x + 1 ), then
x -
( 95 ) If f ( x ) = x (
( d ) none of these
om
(a) 1
lim f ( x ) equals
x→0
ce .c
where
( 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
(a) 0
+
2 1 - n2 1 2
(b) -
1 - n2 n
+ ... +
is
qual to
ra
1 n → ∞ 1 - n2 lim
1 2
(c)
[ IIT 1985 ]
( d ) none of these
[ IIT 1984 ]
m
( 96 )
[ IIT 1985 ]
xa
( 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
.e
( c ) differentiable for al
w w
( 98 ) If G ( x ) = -
(a)
1 24
25
(b)
1 5
x 2 , then
(c) -
G ( x ) - G ( 1) x - 1 x →1 lim
24
[ IIT 1984 ]
has the value
( d ) none of these
[ IIT 1983 ]
w
( 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
2
om
( 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 ]
( 102 ) If y = a ln x + bx + x has its extremum values at x = - 1 and x = 2 then
( c ) a = - 2, b =
1 2
ce .c
( a ) a = 2, b = - 1
( b ) a = 2, b = -
1 2
( d ) none of these
[ IIT 1983 ]
( b - < f ” ( x ) < 0 for all x d ) f ” ( x ) < - 2 for all x
[ IIT 1982 ]
xa
m
( a ) f ” ( x ) > 0 for all x ( c ) - 2 ≤ f ” ( x ) ≤ - 1 for all x
ra
( 103 ) There exists a function f ( x ) satisfying f ( 0 ) = 1, f ’ ( 0 ) = - 1, f ( x ) > 0 for all x and
discontinuous a 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 ]
w w
(a) (b) (c) (d)
.e
( 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
w
( 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 3 c
4 a
5 c
6 a
7 b
8 b
9 d
10 b
11 b
12 c
13 c
14 d
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
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
61 d
62 d
63 b
68 b,d
69 d
70 a
98 d
83 a,b,c,d 99 c
66 d 84 d
100 b
67 c 85 b,c
101 c
86 b
102 b
87 c 103 a
88 a,b,c 104 d
w
w w
.e
xa
m
97 a,b,d
82 b
65 b
71 a,c,d
72 ac
89 a
90 a
ra
81 a,c
64 b
15 b
16 a
17 a
18 a
19 c
20 c
35 d
36 a
37 a
38 c
39 a
40 b
5 b
5 a
57 d
58 b
59 c
60 d
79 c
80 b
om
2 d
ce .c
1 c
105 c
106
73 c
107
74
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
