BANSALCLASSES TARGET IIT JEE 2007
M A T H E M A T I C S NUCLEUS
QUESTION BANK ON
FUNCTION, LIMIT, LIMIT, CONTINUITY CONTINUITY & DERIVABILITY
Time Limit : 4 Sitting Each of 75 Minutes duration approx.
Question bank on function limit continuity & derivability There are 105 questions in this question bank. Select the correct alternative : (Only one is correct) Q. 1 If both f(x) & g(x) are differentiable functions at x = x 0, then the function defined as, h(x) = Maximum {f(x), g(x)} (A) is always differentiable at x = x0 (B) is never differentiable at x = x0 (C) is differentiable at x = x0 provided f(x0) g(x0) (D) cannot be differentiable at x = x0 if f(x0) = g(x0) . Q. 2 If Lim (x3 sin 3x + ax2 + b) exists and is equal to zero then : x0 (A) a = 3 & b = 9/2 (B) a = 3 & b = 9/2 (C) a = 3 & b = 9/2 (D) a = 3 & b = 9/2
Q.3
x m sin x1 A function f(x) is defined as f(x) = 0 continuous at x = 0 is (A) 1 (B) 2
Q.4
For x > 0, let h(x) =
1q 0
x
0,
m N
if x
0
. The least value of m for which f (x) is
(C) 3 p
if x q
(D) none
where p & q 0 are relatively prime integers
if x is irrational
then which one does not hold good? (A) h(x) is discontinuous for all x in (0, ) (B) h(x) is continuous for each irrational in (0, ) (C) h(x) is discontinuous for each rational in (0, ) (D) h(x) is not derivable for all x in (0, ) . 1
Q.5
The value of
(A) l n Q.6
x x e x e 2 3 Limit
x
n
x
2 3
n
(B) 0
(where n N ) is
(C) n l n
2 3
(D) not defined
For a certain value of c, Lim [(x5 + 7x4 + 2)C - x] is finite & non zero. The value of c and the value x
of the limit is (A) 1/5, 7/5 Q.7
1
x
n
(B) 0, 1
(C) 1, 7/5
(D) none
Consider the piecewise defined function
f (x) =
x 0 x4
if
x0
if 0 x 4 if
x4
choose the answer which best describes the continuity of this function (A) The function is unbounded and therefore cannot be continuous. (B) The function is right continuous at x = 0 (C) The function has a removable discontinuity at 0 and 4, but is continuous on the rest of the real line. (D) The function is continuous on the entire real line
Bansal C lasses
Q. B. on FLCD
[2]
Q.8
If
,
are the roots of the quadratic equation ax2 + bx + c = 0 then Lim x
(A) 0 Q.9
(B)
()
a2
(C)
2
( )
(D)
2
a2 2
n
(B)
Lim x 1
(A) Q.11
2
2
bx c equals (x )2
Which one of the following best represents the graph of the function f(x) = Lim
(A)
Q.10
1
1 cos ax 2
(C)
1 4 1 3x x 2 2 1 x x 1 x3
1
2
tan nx 1
(D)
1 = x 3 x 1
3 . x4
(B) 3
3
( )2
(C)
1
(D) none
2
ABC is an isosceles triangle inscribed in a circle of radius r . If AB = AC & h is the altitude from A to BC
and P be the perimeter of ABC then Lim equals (where is the area of the triangle) h 0 P 3 (A) Q.12
1
(B)
32r
1
1
(C)
64r
(D) none
128 r
Let the function f , g and h be defined as follows :
f (x) =
g (x) =
1 x
x sin 0
for 1 x 1 and x 0 for
1 x
x 2 sin 0
x0
for 1 x 1 and x 0 for
x0
h (x) = | x |3 for – 1 x 1 Which of these functions are differentiable at x = 0? (A) f and g only (B) f and h only (C) g and h only Q.13
1 If [x] denotes the greatest integer x, then Limit 4 n n
(A) x/2
Bansal C lasses
(B) x/3
(D) none
1 x 2 x ...... n x equals 3
(C) x/6
Q. B. on FLCD
3
3
(D) x/4
[3]
g ( x ) Q.14
Let f (x) =
h (x)
, where g and h are cotinuous functions on the open interval (a, b). Which of the
following statements is true for a < x < b? (A) f is continuous at all x for which x is not zero. (B) f is continuous at all x for which g (x) = 0 (C) f is continuous at all x for which g (x) is not equal to zero. (D) f is continuous at all x for whichh (x) is not equal to zero.
Q.15
The period of the function f (x) = (A) /2
Q.16
If f(x) =
| sin x | | cos x | | sin x cos x |
(B) /4
x ex
cos 2x x2
(A) f (0) =
(C)
is
(D) 2
, x 0 is continuous at x = 0, then
5
(B) [f(0)] = – 2 (C) {f(0)} = –0.5 (D) [f(0)] . {f(0)} = –1.5 2 where [x] and {x} denotes greatest integer and fractional part function
Q.17
The value of the limit
n 2
(A) 1
Q.18
(B)
The function g (x) =
1 1 is n 2 1
(C)
4
3
1 2
cos x, x 0
can be made differentiable at x = 0. (B) if b is not equal to zero (D) for no value of b
Let f be differentiable at x = 0 and f ' (0) = 1. Then Lim h 0
(A) 3 Q.20
(D)
x b, x 0
(A) if b is equal to zero (C) if b takes any real value Q.19
1
f (h ) f ( 2h ) h
=
(B) 2
(C) 1
(D) – 1
If f (x) = sin –1(sinx) ; x R then f is (A) continuous and differentiable for all x (B) continuous for all x but not differentiable for all x = (2k + 1) (C) neither continuous nor differentiable for x = (2k – 1)
2 (D) neither continuous nor differentiable for x R [ 1,1]
Bansal C lasses
Q. B. on FLCD
2
, k I
; k I
[4]
Q.21
sin x Limit where [ ] denotes greatest integer function , is 1 1 x 2 cos (3 sin x sin 3x ) 4 (A)
Q.22
(B) 1
If Lim
(C)
l n (3 x ) l n (3 x )
x 0
(A)
Q.23
2
1
(B) –
3
(D) does not exist
= k , the value of k is
x
2
4
3
x 2n 1
The function f (x) = Lim 2n n x (A) g (x) = sgn(x – 1) (C) u (x) = sgn( | x | – 1)
1
2
(C) –
(D) 0
3
is identical with the function (B) h (x) = sgn (tan –1x) (D) v (x) = sgn (cot –1x)
Q.24
The functions defined by f(x) = max {x 2, (x 1)2, 2x (1 x)}, 0 x 1 (A) is differentiable for all x (B) is differentiable for all x excetp at one point (C) is differentiable for all x except at two points (D) is not differentiable at more than two points.
Q.25
f (x) =
(A)
x
and g (x) =
l nx 1
g(x)
(D)
1
and g (x) are identical functions
f ( x )
1 f ( x ) . g ( x )
1 x 0
3 f (x ) is given by : x3
(B) 4
(C) 0
(D) none of these
Which one of the following functions is continuous everywhere in its domain but has atleast one point where it is not differentiable? (A) f (x) = x 1/3
Q.28
x 0
(B)
x f ( 3) If f(3) = 6 & f (3) = 2, then Limit x 3
(A) 6 Q.27
. Then identify the CORRECT statement
x
and f (x) are identical functions
(C) f (x) . g (x) = 1
Q.26
l nx
(B) f (x) =
|x| x
The limiting value of the function f(x) = (A)
2
Bansal C lasses
(B)
1 2
(C) f (x) = e –x
2 2 (cos x sin x ) 1 sin 2 x (C) 3 2
Q. B. on FLCD
(D) f (x) = tan x
3
when x
(D)
4
is
3 2
[5]
Q.29
Let
f (x) =
23 x 6 2 x 21 x
2x
if x 2 then
4 if x 2 x 3x 2 (A) f (2) = 8 f is continuous at x = 2 (C) f (2 – ) f (2+) f is discontinuous x
2
(B) f (2) = 16 f is continuous at x = 2 (D) f has a removable discontinuity at x = 2
(x 1) e |x1| 1x Q.30 On the interval I = [ 2, 2], the function f(x) = 0
0) (x 0)
(x
then which one of the following does not hold good? (A) is continuous for all values of x I (B) is continuous for x I (0) (C) assumes all intermediate values from f( 2) & f(2) (D) has a maximum value equal to 3/e . Q.31
Which of the following function is surjective but not injective (A) f : R R f (x) = x4 + 2x3 – x2 + 1 (B) f : R R f (x) = x3 + x + 1 (C) f : R R + f (x) = 1 x 2
Q.32
Consider the function f (x) =
(D) f : R R f (x) = x3 + 2x2 – x + 1 x [x]
if 1 x 2 if x 2
1 6x
if 2 x 3
where [x] denotes step up function then at x = 2 function (A) has missing point removable discontinuity (B) has isolated point removable discontinuity (C) has non removable discontinuity finite type (D) is continuous Q.33
Suppose that f is continuous on [a, b] and that f (x) is an integer for each x in [a, b]. Then in [a, b] (A) f is injective (B) Range of f may have many elements (C) {x} is zero for all x [a, b] where { } denotes fractional part function (D) f (x) is constant
Q.34
The graph of function f contains the point P (1, 2) and Q(s, r). The equation of the secant line through P and Q is y = (A) 2
Bansal C lasses
s 2 2s 3 s 1 x – 1 – s. The value of f ' (1), is (B) 3
(C) 4
Q. B. on FLCD
(D) non existent
[6]
Q.35
The range of the function f(x) = (A) ( , )
2 e x l n x 5( x 2 ) ( x 2
7 x 10) is 2 x 2 11x 12 3 (C) , 2
(B) [0 , )
2 sin x sin3 x Q.36 Consider f(x) = 2 sin x sin3 x
(D)
3 , 4 2
sin3 x , x for x (0, ) 2 sin x sin3 x sin x
f(/2) = 3 where [ ] denotes the greatest integer function then, (A) f is continuous & differentiable at x = /2 (B) f is continuous but not differentiable at x = /2 (C) f is neither continuous nor differentiable at x = /2 (D) none of these Q.37
The number of points at which the function, f(x) = x – 0.5 + x – 1 + tan x does not have a derivative in the interval (0, 2) is : (A) 1 (B) 2 (C) 3 (D) 4
Q.38
Let [x] denote the integral part of x R. g(x) = x [x]. Let f(x) be any continuous function with f(0) = f(1) then the function h(x) = f(g(x)) : (A) has finitely many discontinuities (B) is discontinuous at some x = c (C) is continuous on R (D) is a constant function .
Q.39
Given the function f(x) = 2x x3 1 + 5 x 1 x 4 + 7x2 x 1 + 3x + 2 then : (A) the function is continuous but not differentiable at x = 1 (B) the function is discontinuous at x = 1 (C) the function is both cont. & differentiable at x = 1 (D) the range of f(x) is R +.
Q.40
If f (x + y) = f (x) + f (y) + | x | y + xy2, x, y R and f ' (0) = 0, then (A) f need not be differentiable at every non zero x (B) f is differentiable for all x R (C) f is twice differentiable at x = 0 (D) none
Q.41
For Lim x 8
sin{x 10} {10 x}
(where { } denotes fractional part function)
(A) LHL exist but RHL does not exist (C) neither LHL nor RHL does not exist
Q.42
Lim n
(A)
12 n 2 2 (n1)32 (n2)..... n 2.1 3
1
1 3
Bansal C lasses
23 33 ......n 3
(B)
2 3
(B) RHL exist but LHL does not exist. (D) both RHL and LHL exist and equals to 1
is equal to :
(C)
1 2
Q. B. on FLCD
(D)
1 6
[7]
Q.43
The domain of definition of the function f (x) = log
(A) {2}
(B)
3 , {2, 3} 4
x 1 x
|x
2
x 6 | + 16–xC2x–1 + 20–3xP2x–5 is
(D)
(C) {2, 3}
1 4
,
Where [x] denotes greatest integer function.
Q.44
If f (x) =
x 2 bx 25 x 2 7 x 10
(A) 0 Q.45
Q.46
(B) 5
(C) 10
(D) 25
Let f be a differentiable function on the open interval (a, b). Which of the following statements must be true? I. f is continuous on the closed interval [a, b] II. f is bounded on the open interval (a, b) III. If a
The value of
Limit x
(A) 1 Q.47
for x 5 and f is continuous at x = 5, then f (5) has the value equal to
cot 1 x a log a x
sec 1 a x log x a
(a > 1) is equal to (C) /2
(B) 0
(D) does not exist
Let f : (1, 2 ) R satisfies the inequality cos( 2 x 4) 33 2
f (x )
x 2 | 4x 8 | x2
, x (1,2) . Then Lim f ( x ) is equal to x 2
(A) 16 (B) –16 (C) cannot be determined from the given information (D) does not exists
Q.48
n sin x cos x Lim Let a = min [x + 2x + 3, x R] and b = . Then the value of a r b n r is x x x 0 e e r 0
2
(A)
2 n 1 1 3 ·2
n
(B)
2 n 1 1 3 ·2
n
(C)
2n 1 3 ·2
n
(D)
4 n 1 1 3 ·2
n
Q.49
Period of f(x) = nx + n [nx + n], (n N where [ ] denotes the greatest integer function is : (A) 1 (B) 1/n (C) n (D) none of these
Q.50
1 x x 3 + cos1 Let f be a real valued function defined by f(x) = sin1 . Then domain of f (x) 3 5 is given by : (A) [ 4, 4]
Bansal C lasses
(B) [0, 4]
(C) [ 3, 3]
Q. B. on FLCD
(D) [ 5, 5]
[8]
Q.51
1 Lim For the function f (x) = n , which of the following holds? 2 1 n sin (x ) (A) The range of f is a singleton set (C) f is discontinuous for all x I
Q.52
Domain of the function f(x) = (A) (cot1 , )
Q.53
(B) f is continuous on R (D) f is discontinuous for some x R
1 l n cot 1 x
is (C) (– ,0) (0,cot1) (D) (– , cot1)
(B) R – {cot1} 2x 1 , x Q f ( x ) x 2 2 x 5 , x Q
The function
is
(A) continuous no where (B) differentiable no where (C) continuous but not differentiable exactly at one point (D) differentiable and continuous only at one point and discontinuous elsewhere
Q.54
1
For the function f (x) =
1
, x 2 which of the following holds?
x 2 ( x 2) (A) f (2) = 1/2 and f is continuous at x =2 (C) f can not be continuous at x = 2
Q.55
x cos(sin 1 x ) Lim is 1 x 1 2 1 tan(sin x ) (A)
Q.56
(B) f (2) 0, 1/2 and f is continuous at x = 2 (D) f (2) = 0 and f is continuous at x = 2.
1
(B) –
2
1
(C)
2
2
Which one of the following is not bounded on the intervals as indicated 1
(B) g(x) = x cos
(C) h(x) = xe –x on (0, )
The domain of the function f(x) =
arc cot x x
x, is : (A) R
1
on (– ) x (D) l (x) = arc tan2x on (– , )
(A) f(x) = 2 x 1 on (0, 1)
Q.57
(D) –
2
2
x 2
, where [x] denotes the greatest integer not greater than
(B) R {0}
(C) R
Bansal C lasses
n : n I
{0}
(D) R {n : n I}
Q. B. on FLCD
[9]
Q.58
If f(x) = cos x, x = n , n = 0, 1, 2, 3, ..... = 3, otherwise and
x 2 1 (x) = 3 5
when x 3, x 0 when x 3
(A) 1 Q.59
Q.60
then Limit f((x)) = x 0
when x 0
(B) 3
(C) 5
(D) none
x x = m, then Let Lim sec –1 = l and Lim sec –1 sin x tan x x 0 x 0 (A) l exists but m does not (B) m exists but l does not (C) l and m both exist (D) neither l nor m exists 1 2 l n ( x e )
Range of the function f (x) =
1 1 x2
is , where [*] denotes the greatest integer
1/ function and e = Limit (1 ) 0
0, e 1 {2} (A) e Q.61
(B) (0, 1)
(C) (0, 1]
Lim sin 1[tan x ] = l then { l } is equal to x 0 (B) 1
(A) 0
(C)
{2}
1
2 2 where [ ] and { } denotes greatest integer and fractional part function.
Q.62
Q.63
(D) 2
2
Number of points where the function f (x) = (x 2 – 1) | x2 – x – 2 | + sin( | x | ) is not differentiable, is (A) 0 (B) 1 (C) 2 (D) 3
Limit x
x x 1 2 x 1 sec x 1
cot 1
x 1
(A) 1 If
x f (x) = ax b
(A) 2x0 , – x 02
(C) /2
if x x 0 if x x 0
R {4}
Bansal C lasses
(B) p
(D) non existent
derivable x R then the values of a and b are respectively
(B) – x0 , 2 x 02
1 cos 2 x 1 sin x Q.65 Let f (x) = p 2x 1 4 2 x 1 (A) p
is equal to
(B) 0 2
Q.64
(D) (0, 1) {2}
2
(C) – 2x0 , – x 02
(D) 2 x 02 , – x0
, x 1 2 1 , x 1 . If f (x) is discontinuous at x = , then 2 2 1 , x 2
1
R 4
(C) p
Q. B. on FLCD
R 0
(D) p
R
[10]
Q.66
Let f(x) be a differentiable function which satisfies the equation f(xy) = f(x) + f(y) for all x > 0, y > 0 then f (x) is equal to (A)
Q.67
Q.68
f ' (1)
(B)
x
ex ) . l n ( x 4 e 2 x ) l n ( x
Let f(x) =
2
Lim cos n2 n
Limit x 0
n
x
x
(sin x tan x )
(1 cos 2x ) 4 x 5 7.(tan 1 x )7 (sin 1 x ) 6 3 sin 5 x (B)
(C) is equal to zero
(D) does not exist
1
is equal to
(C)
7
2 Range of the function , f (x) = cot 1 log 4 /5 (5 x
(B)
, 4
[ x ]2 Limit Let x 0 = l & Limit x 0 2 x
The value of Limit x 0
[ x2 ] x2
If f (x) = (A) 0
Bansal C lasses
2 e x 2 x
tan
x
(B) 1
(D) 1
3
8x 4 )
is :
0 , 4
(D)
0 , 2
= m , where [ ] denotes greatest integer , then:
sin {x }
{x } {x } 1
(B) is tan 1
n
(C)
1
(B) m exists but l does not (D) neither l nor m exists .
tan {x} 1
(A) is 1
Q.74
(D) l + m = 0
2
(A) l exists but m does not (C) l & m both exist Q.73
(C) 2 l = m
when n is an integer : (B) is equal to 1
(A) (0 , )
Q.72
If Limit f(x) = l and Limit f(x) = m then :
(B) l = 2m
(A) 0 Q.71
(D) f (1).(lnx)
f(x) = b ([x]2 + [x]) + 1 for x 1 = Sin ( (x+a) ) for x < 1 where [x] denotes the integral part of x, then for what values of a, b the function is continuous at x = 1? (A) a = 2n + (3/2) ; b R ; n I (B) a = 4n + 2 ; b R ; n I (C) a = 4n + (3/2) ; b R + ; n I (D) a = 4n + 1 ; b R + ; n I
(A) is equal to 1
Q.70
(C) f (1)
x
Given
(A) l = m
Q.69
1
where { x } denotes the fractional part function: (C) is sin 1
(D) is non existent
is continuous at x = 0 , then f (0) must be equal to : (C) e2
Q. B. on FLCD
(D) 2
[11]
Q.75
2 2x sin 2x Lim x ( 2 x sin 2 x )e sin x is : (A) equal to zero lim x 0
Q.76 The value of
(A)
e
(B) equal to 1 2
cos axcos ec bx
8 b 2 a 2
(B)
e
(C) equal to 1
(D) non existent
is
8a 2 b 2
(C)
e
a 2 2 b 2
(D)
e
b 2 2a 2
Select the correct alternative : (More than one are correct)
Q.77
Lim f(x) does not exist when : x c
(A) f(x) = [[x]] [2x 1], c = 3
(B) f(x) = [x] x, c = 1
(C) f(x) = {x}2 {x}2, c = 0
(D) f(x) =
tan (sgn x) sgn x
, c = 0.
where [x] denotes step up function & {x} fractional part function.
tan2 {x} x2 [ x ]2 Q.78 Let f (x) = 1 {x} cot {x}
0 x0 x0
for x for for
where [ x ] is the step up function and { x } is the fractional
part function of x , then : (A) xLimit 0 f (x) = 1 (C) cot -1
Limit x 0
(B) xLimit 0 f (x) = 1 2
f ( x) = 1
x . n (cos2x ) Q.79 If f(x) = n 1 x 0
x
0
x
0
then :
(A) f is continuous at x = 0 (C) f is differentiable at x = 0 Q.80
(B) f is continuous at x = 0 but not differentiable at x=0 (D) f is not continuous at x = 0.
Which of the following function (s) is/are Transcidental? (A) f (x) = 5 sin (C) f (x) =
Q.81
(D) f is continuous at x = 1 .
x2
x
2x 1
(B) f (x) =
2 sin 3x x
2
2x 1
(D) f (x) = (x2 + 3).2x
Which of the following function(s) is/are periodic? (A) f(x) = x [x] (B) g(x) = sin (1/x) , x 0 & g(0) = 0 (C) h(x) = x cos x (D) w(x) = sin1 (sinx)
Bansal C lasses
Q. B. on FLCD
[12]
Q.82
Which of following pairs of functions are identical : n sec1 x
(A) f(x) = e & g(x) = sec1 x (B) f(x) = tan (tan1 x) & g(x) = cot (cot1 x) (C) f(x) = sgn(x) & g(x) = sgn(sgn(x)) (D) f(x) = cot2 x.cos2 x & g(x)= cot2 x cos2 x Q.83
Q.84
Which of the following functions are homogeneous ? (A) x sin y + y sin x (B) x e y/x + y ex/y (C) x2 xy
If is small & positive number then which of the following is/are correct ? (A)
Q.85
Q.86
sin
(B) < sin < tan
=1
Let f(x) =
x . 2x
x
(C) sin < < tan (D)
1 cos x
(B) Limit g(x) = l n 4 x
(C) Limit f(x) = l n 4 x 0
(D) Limit g(x) = l n 2 x
x 2x2
>
sin
n 2 then : 2x
(A) Limit f(x) = l n 2 x 0
Let f(x) =
tan
& g(x) = 2x sin
1
7x 5
1 (A) Limit f(x) = x 1
. Then :
(B) Limit f(x) = x 0
3
Q.87
(D) arc sin xy
1 5
(C)
Limit x
f(x) = 0
(D) Limit does not exist x 5 / 2
Which of the following limits vanish? 1 1 4 sin (A) Limit x x
(B) Limit (1 sinx) . tan x x /2
x
(C) Limit x
3 . sgn (x) x2 x 5 2 x2
2 [ x] (D) Limit x 3 2
x
9 9
where [ ] denotes greatest integer function Q.88
Limit [1 + cos2m (n! x)] is given by If x is a real number in [0, 1] then the value of Limit m n
(A) 1 or 2 according as x is rational or irrational (B) 2 or 1 according as x is rational or irrational (C) 1 for all x (D) 2 for all x . Q.89
If f(x) is a polynomial function satisfying the condition f(x) . f(1/x) = f(x) + f(1/x) and f(2) = 9 then : (A) 2 f(4) = 3 f(6) (B) 14 f(1) = f(3) (C) 9 f(3) = 2 f(5) (D) f(10) = f(11)
Q.90
Which of the following function(s) not defined at x = 0 has/have removable discontinuity at x = 0 ? (A) f(x) =
1 1 2 cotx
Bansal C lasses
| sin x | (C) f(x) = x sin (B) f(x)=cos x x
Q. B. on FLCD
(D) f(x) =
1 n
x
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Q.91
The function f(x) =
x3 x 3x 13 4 2 4 2
(A) continuous at x = 1 (C) continuous at x = 3
Q.92
If f(x) = cos
, x 1
is : (B) diff. at x = 1 (D) differentiable at x = 3
x 1 ; where [x] is the greatest integerr function of x, then f(x) is cos x 2
continuous at : (A) x = 0 Q.93
, x 1
(B) x = 1
(C) x = 2
(D) none of these
Identify the pair(s) of functions which are identical . (A) y = tan (cos 1 x); y = (C) y = sin (arc tan x); y =
1 x 2 x
x 1 x2
(B) y = tan (cot 1 x); y =
1 x
(D) y = cos (arc tan x); y = sin (arc cot x)
Q.94
The function, f (x) = [x] [x] where [ x ] denotes greatest integer function (A) is continuous for all positive integers (B) is discontinuous for all non positive integers (C) has finite number of elements in its range (D) is such that its graph does not lie above the x axis.
Q.95
Let f (x + y) = f (x) + f (y) for all x , y R. Then : (A) f (x) must be continuous x R (B) f (x) may be continuous x R (C) f (x) must be discontinuous x R (D) f (x) may be discontinuous x R
Q.96
The function f(x) =
Q.97
Let f(x) be defined in [–2, 2] by f(x) = max (4 – x 2, 1 + x 2), –2 < x < 0 = min (4 – x2, 1 + x 2), 0 < x < 2 The f(x) (A) is continuous at all points (B) has a point of discontinuity (C) is not differentiable only at one point. (D) is not differentiable at more than one point
1 1 x 2 (A) has its domain –1 < x < 1. (B) has finite one sided derivates at the point x = 0. (C) is continuous and differentiable at x = 0. (D) is continuous but not differentiable at x = 0.
Bansal C lasses
Q. B. on FLCD
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Q.98
The function f(x) = sgnx.sinx is (A) discontinuous no where. (C) aperiodic
(B) an even function (D) differentiable for all x
1
Q.99
l n x
The function f(x) = x (A) is a constant function (C) is such that limit f(x) exist x1
(B) has a domain (0, 1) U (e, ) (D) is aperiodic
Q.100 Which pair(s) of function(s) is/are equal? –1
(A) f(x) = cos(2tan x) ; g(x) =
1 x2 1 x
(B) f(x) =
2
2x 1 x2
; g(x) = sin(2cot –1x) 1
1 x ) n 1 x n (sgn c ot (C) f(x) = e ; g(x) = e (D) f(x) = X a , a > 0; g(x) = a x , a > 0 where {x} and [x] denotes the fractional part & integral part functions.
Fill in the blanks:
Q.101 A function f is defined as follows, f(x) =
sin x ax b
c where c is a known quantity. If f is derivable xc
if x if
at x = c, then the values of 'a' & 'b' are _____ &______ respectively . Q.102 A weight hangs by a spring & is caused to vibrate by a sinusoidal force . Its displacement s(t) at time t is given by an equation of the form, s(t) =
A 2
c
k 2
(sin kt sin ct) where A, c & k are positive constants
with c k, then the limiting value of the displacement as c k is ______.
(cos )x (sin )x cos 2 Limit Q.103 x 4 where 0 < < is ______ . 2 x4 2
3/ x cos 2 x Q.104 Limit has the value equal to ______ . x 0 Q.105 If f(x) = sin x, x n , n = 0, ±1, ±2, ±3,.... = 2, otherwise and g(x) = x² + 1, x 0, 2 = 4, x=0 = 5, x=2 then Limit g [f(x)] is ______ x 0
Bansal C lasses
Q. B. on FLCD
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ANSWER KEY 1 5 0 1 . Q e 4 . Q 0 1
6 -
2 0 1 . Q
n n s o c n l l i s n i s 4 4
s
o
c
. Q 3 0 1
c s o i s & c s o c 1 0 1 . Q c c - c n
C , B , A 0 0 1 . Q
C , A 9 9 . Q
C , B , A 8 9 . Q
D , B 7 9 . Q
D , B , A 6 9 . Q
D , B 5 9 . Q
, B , A 4 9 . Q D , C
, B , A 3 9 . Q D , C
, B 2 9 . Q C
, B , A 1 9 . Q C
, C , B 0 9 . Q D
C , B 9 8 . Q
D , B 8 8 . Q
D , B , A 7 8 . Q
, B , A 6 8 . Q D , C
D , C 5 8 . Q
D , C 4 8 . Q
C , B 3 8 . Q
D , C , B 2 8 . Q
D , A 1 8 . Q
D , B , A 0 8 . Q
C , A 9 7 . Q
C , A 8 7 . Q
C , B 7 7 . Q C 6 7 . Q
7 . Q D 5
7 . Q D 4
7 . Q D 3
7 . Q B 2
7 . Q B 1
C 0 7 . Q
C 9 6 . Q
A 8 6 . Q
A 7 6 . Q
A 6 6 . Q
A 5 6 . Q
A 4 6 . Q
A 3 6 . Q
C 2 6 . Q
D 1 6 . Q
D 0 6 . Q
5 . Q A 9
. Q B 8 5
. Q C 7 5
B 6 5 . Q
B 5 5 . Q
C 4 5 . Q
D 3 5 . Q
D 2 5 . Q
C 1 5 . Q
5 . Q A 0
4 . Q B 9
4 . Q D 8
4 . Q B 7
4 . Q A 6
D 5 4 . Q
A 4 4 . Q
A 3 4 . Q
A 2 4 . Q
B 1 4 . Q
B 0 4 . Q
B 9 3 . Q
C 8 3 . Q
C 7 3 . Q
A 6 3 . Q
A 5 3 . Q
C 4 3 . Q
D 3 3 . Q
B 2 3 . Q
D 1 3 . Q
A 0 3 . Q
C 9 2 . Q
D 8 2 . Q
A 7 2 . Q
C 6 2 . Q
A 5 2 . Q
C 4 2 . Q
C 3 2 . Q
A 2 2 . Q
A 1 2 . Q
. Q B 0 2
. Q A 9 1
D 8 1 . Q
1 . Q D 7
. Q D 6 1
C 5 1 . Q
D 4 1 . Q
D 3 1 . Q
C 2 1 . Q
C 1 1 . Q
1 . Q B 0
. Q A 9
. Q C 8
. Q D 7
. Q A 6
B 5 . Q
A 4 . Q
C 3 . Q
A 2 . Q
C 1 . Q
Bansal C lasses
Q. B. on FLCD
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