BANSALCLASSES TARGET IIT JEE 2007
M A T H E M A T I C S STERLING
QUESTION BANK ON FUNCTION, FUNCTION, LIMIT, LIMIT, CONTINUITY & DERIV DERIVABI ABILIT LITY Y METHOD OF DIFFERENTIATION DIFFERENTIATION INVERSE TRIGONOMETRIC FUNCTION
Time Limit : 5 Sitting Each of 100 Minutes duration approx.
Question bank on Definite, Indefinite Indefin ite Integration, MOD & ITF Select the correct alternative : (Only one is correct)
Q.1
Minimum Minimum period period of the functio function, n, f (x) = | sin32x | + | cos 32x | is (A)
Q. 2
(B)
(C)
2
4
(B) a = 3 & b = 9/2 (D) a = 3 & b = 9/2
If g is the the inve invers rsee of f & f (x) = (B)
1 1 x 5
then g (x) =
1
(C)
1 [g (x )]
5
x m sin x1 A functi function on f(x) f(x) is define defined d as f(x) f(x) = 0 continuous at x = 0 is (A) 1 (B) 2
Q.5
4
3
x0
(A) 1 + [g(x)]5
Q.4
(D)
If Lim (x3 sin 3x + ax2 + b) exists and is equal to zero then : (A) a = 3 & b = 9/2 (C) a = 3 & b = 9/2
Q.3
x
0, if x
1
(D) none
1 [g (x )]5
m N
0
. The least value of m for which f (x) is
(C) 3
(D) none
The The numb number er k is such such that that tanarc tan(2) arc tan(20k ) = k. The sum of all possible values of k is (A) –
19
21
(B) –
40
(C) 0
40
(D)
1 5
x for 0 x 1
Q. 6
Let f 1(x) =
1 0
x 1
for
for otherwise
and and
f 2 (x) = f = f 1 (– x) for all x f 3 (x) = – f – f 2(x) for all x f 4 (x) = f = f 3(– x) for all x Which of the following is necessarily true? (A) f (A) f 4 (x) = f = f 1 (x) for all x (C) f (C) f 2 (–x) = f = f 4 (x) for all x
Q. 7
3x 4 & f (x) = tan x2 then 5x 6
If y = f
(B) f 1 (x) = – f 3 (–x) for all x (D) f 1 (x) + f + f 3 (x) = 0 for for all all x
dy dx
2
3x 4 1 (B) 2 tan . 5x 6 (5x 6) 2
3
(A) tanx
3 tan x 2 4 tanx2 (C) f 2 5 tan x 6
=
(D) none 1
1
x x n e x n e 2 3 Limit x
Q.8 Q.8
The The valu valuee of
(A) l n
2 3
Bansal C lasses
xn
x
(B) 0
( where n N ) is
(C) n l n
2 3
(D) not defined
Q. B. on FLCD, Method of Differentiation & ITF
[2]
Q.9
Q.10 Q.10
Which one of the following following depicts the graph of an odd function? function?
(A)
(B)
(C)
(D)
If sin sin =
12 13
, cos = –
5 13
, 0 < < 2. Consider the following statements.
5 = cos –1 13 12 = – sin –1 13 12 –1 = – tan 5
I. III.
V.
12 = sin –1 13 12 = tan –1 5
II. IV.
then which of the following statements are true? (A) I, II and IV only (B) III and V only (C) I and III only (D) I, III and V only Q.11 Q.11
Let g is the inverse inverse function function of f & f (x) =
(A) Q.12
5 210
(B)
a 10
1 x2
(C)
. If g(2) = a then g (2) is equal to
a
10
1 a
2
(D)
1 a 10 a2
For a certain certain value value of c, Lim [(x5 + 7x4 + 2)C - x] is finite & non zero. The value of c and the value of x
the limit is (A) 1/5, 7/5 Q.13
1 a 2
x10
(B) 0, 1
(C) 1, 7/5
(D) none
Which one one of the following following best represents represents the graph graph of of the function f(x) = Lim n
(A)
Bansal C lasses
(B)
(C )
2
tan 1nx
(D)
Q. B. on FLCD, Method of Differentiation & ITF
[3]
d 3 d2 y Q.14 .14 If y = P(x), is a polynomial of degree 3, then 2 y . 2 equals : dx dx 2
(A) P (x) + P (x)
Q.15 Q.15
The The sum sum
(B) P (x) . P (x) (x) 3
tan 1
n2 n 1
n 1
(A) Q.16
3 4
cot 1 2
(B)
2
(C) P (x) . P (x) (x)
(D) a constan stantt
is equal to
cot 1 3
(C)
(D)
2
tan 1 2
If f (x) is a diffrentiable diffrentiable function and f (2) = 6 , f (1) = 4, f (x) represents the diffrentiation diffrentiat ion of f (x)
f (2 2h h ) f (2) 2
Limit
w.r.t. w.r.t. x then
f (1 h 2 h ) f (1)
h 0
(A) 3 Q.17
Lim x 1
(B) 4
= (C) 6
(D) 14
(C) 2 sin 2
(D)
cos 2 cos 2 x =
x2 | x |
(A) 2 cos 2
(B)
g (x) . cos cos 1 x Q.18 Q.18 Let f(x f(x)) = 0
2 cos 2
sin 2 2 sin
0 where g(x) is an even function differentiable at x = 0, passing x0
if x if
through the origin . Then f (0) : (A) is equal to 1 (B) is equal to 0
(C) is equ equal to 2
(D) does not exist
3x 2 7x 8 Q.19 The domain domain of definition definition of the function function,, f (x) = arc cos greatest where [ *] denotes the greatest 2 1 x integer function, is : (A) (1, 6) Q.20
(B) [0, 6)
(C) [0, 1]
(D) ( 2, 5]
The sum of the infinite infinite terms of the series
2 3 2 3 2 3 + cot 1 2 + cot 1 3 + ..... is equal to : 4 4 4
cot 1 1 (A) tan –1 (1) Q.21
(B) tan –1 (2)
(C) tan –1 (3)
(D) tan –1 (4)
Let Let the funct function ion f f , g and h be defined as follows :
f (x) =
g (x) (x) =
1 x
x sin 0
for
1 x
x 2 sin 0
for 1 x 1 and x 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 (A) f and g and g o only (B) f and h only (C) g and and h only
Bansal C lasses
(D) none
Q. B. on FLCD, Method of Differentiation & ITF
[4]
3x 2 2 x 1 Q.22 Q.22 Let f (x) = 6 x 2 5 x 1 4 (A) is equal to 9 Q.23
n 1 Lim sin n n 1 n Let f (x) =
1 3
for x
1 3
1 : 3
then f
(B) is equal to 27
(A) e – Q.24 Q.24
for x
(C) (C) is equal qual to 27
n
when Q is equal to (C) e1 –
(B) – g ( x ) h (x)
(D) does not exis existt
(D) e1 +
, where g where g and and h are cotinuous functions on the open interval (a, b). Which of the
following statements is true for a < x < b? (A) f (A) f is is continuous at all x fo forr which x is not zero. (B) f (B) f is is continuous at all x for which g which g (x) (x) = 0 (C) f is f is continuous at all x for which g which g (x) (x) is not equal to zero. (D) f is f is continuous at all x for whichh which h (x) is not equal to zero. Q.25
1 tan 2A +tan 1(cotA)+tan 1(cot3A) for 0 < A < (/4) is 2
The value value of tan tan1 (A) 4 tan1 (1)
Q.26 Q.26
If y =
(B) 2 tan1 (2)
1 1 x n
m
x p m
(A) emnp Q.27 Q.27
+
(C) 0
1 1 x m
n
xp n
+
(B) emn/p
8
Give Given n f (x) =
1 x
8 1 x
and
g (x) =
Q.29 Q.29
(A) 0
Q.31 Q.31
sin 3x
x
f (sin x )
| sin x cos x |
cos x
Let f(x) f(x) = cos 2x sin 2x 2 cos 2x then f cos 3x
Q.30
sin x
p
4
| sin x | | cos x |
(B) /4 cos x
1 x m
n p then
dy dx
n p
at e m is equal to: (D) none
4 f (cos x )
then g(x) is
(B) periodic with period (D) aperiodic
The period period of the function function f (x) = (A) /2
1
(C) enp/m
(A) periodic with period /2 (C) periodic with period 2 Q.28
(D) none
3 cos 3x
(B) – 12
(C)
is
(D) 2
= 2 (C) 4
cos sin sin 1 x and = cos 1 sin sin cos cos 1 x , then : = sin 1 cos (A) tan = cot (B) tan = cot (C) tan = tan x e x cos 2x If f(x) f(x) = , x 0 is continuous at x = 0, then 2
(D) 12
(D) tan = tan
x
(A) f (0) =
5
(B) (B) [f( [f(0)] 0)] = – 2 (C) (C) {f(0) f(0)} } = –0.5 0.5 (D) (D) [f( [f(0)] . {f(0)} (0)} = –1.5 2 where [x] and {x} denotes greatest integer and fractional part function
Bansal C lasses
Q. B. on FLCD, Method of Differentiation & ITF
[5]
Q.32
The functio function n g (x) =
x b, x 0
cos x, x 0
can be made differentiable at x = 0.
(A) if b is equal to zero (C) if b takes any real value Q.33 Q.3 3
(B) if b is not equal to zero (D) for no value of b
People living living at Mars, instead of the usual usual definition of derivative derivative D f(x), f(x), define a new kind of derivative, D*f(x) by the formula D* f(x) = Limit
f 2 (x h) f 2 ( x)
h 0
h
where f (x) means [f(x)]2. If f(x) = x l nx nx then
D * f (x ) x e has the value (A) e (B) 2e
Q.34 Q.3 4
(C) 4e
Which one of the following statement is meaningless? meaningless?
2e 4 3
(B) cosec1
2
(D) sec1 ()
3
(A) cos1 l n (C) cot1
Q.35
sin x Limit where [ ] denotes greatest integer function , is 1 1 x 2 cos (3 sin x sin 3x ) 4 (A)
Q.36
(D) none
2
(B) 1
(C)
4
(D) does not exist
Which one of the following following statement statement is true? true? (A) If Lim f (x ) ·g( x ) and Lim f ( x ) exist, then Lim g ( x ) exists. x c x c x c (B) If Lim f (x ) ·g( x ) exists, then Lim f (x ) and Lim g ( x ) exist. x c x c x c (C) If Lim f ( x ) g( x ) and Lim f (x ) exist, then Lim g ( x ) exist. x c x c x c (D) If Lim f ( x ) g ( x ) exists, then Lim f ( x ) and Lim g ( x ) exist. x c x c x c
Q.37
If f(4) = g(4) g(4) = 2 ; f (4) = 9 ; g (4) = 6 then Limit x 4 (A) 3 2
Q.38 Q.38
f (x) (x) =
(A)
x
and g (x) =
l nx 1
g( x)
3
(B) l nx x
Bansal C lasses
g (x )
x 2
(C) 0
2
is equal to : (D) none
. Then identify the CORRECT statement
and and f (x) are are identi identical cal functi function onss
(C) f (x) . g (x) = 1
f (x )
x 0
(B) (B) (D)
1 f ( x )
and g (x) are identical functions 1
f ( x ) . g ( x )
1 x 0
Q. B. on FLCD, Method of Differentiation & ITF
[6]
Q.39
Which one one of the the following following functions functions is continuou continuouss everywhere everywhere in its domain but has atleast one point where it is not differentiable? (A) f (x) = x 1/3
Q.40 Q.40
(B) y = x
(D) f (x) = tan x
(C) tan x = (4/3 (4/3)) y
(D) (D) tan tan x = (4/3 (4/3)) y
f (x 3h) f (x 2 h) If f(x) is a different differentiable iable function function of x then Limit = h 0 h
(B) 5f (x)
(C) 0
Let f et f be a function satisfying f (xy) = value of f of f (40) is (A) 15
Q.43 Q.43
x
1 1 1 1 + sin1 ; y = cos cos 1 then : 2 2 8 2
(A) f (x) Q.42 Q.42
(C) f (x) = e –x
If x = tan tan1 1 cos1 (A) x = y
Q.41
|x|
(B) f (x) =
y
for all positive real numbers x and y. If f (30) = 20, then the
(B) 20
|x |
{e sgn x}
Let f (x) (x) = e
f ( x )
(D) none
(C) 40 [ e| x | sgn x ]
and g (x) = e
(D) 60
, x R where { x } and [ ] denotes the fractional part and
integral part functions respectively. Also h (x) = l n f ( x ) + l n g( x ) then for all real x, h (x) is (A) an odd function (B) an even function (C) neith neither er an odd odd nor an even even functi function on (D) (D) both both odd as well well as even even functi function on Q.44 Q.44
d 2x
If y = x + e x then
dy 2
x
(A) e
Q.45
(B)
ex
1 e x
3
1 e x
2
(D)
1 3 1 ex
tan1 tan 8 has the value equal to 7 7
Let
(C) cos
(B) –1
f (x) (x) =
2
23 x 6 2 x 21 x
x
7
(D) 0
if x 2 then
x2 4 x 3x 2
if x 2
(A) f (2) = 8 f is cont contin inuo uous us at x = 2 (C) f (2 – ) f (2+) f is disco disconti ntinu nuou ouss Q.47 Q.4 7
(C)
ex
cos cos1 cos
(A) 1
Q.46 Q.46
is :
(B) (B) f (2) (2) = 16 f is continuous at x = 2 (D) (D) f has a remo remova vabl blee disco discont ntin inui uity ty at x = 2
Which of the the following following function is surjective surjective but not injective 4 3 (A) f : R R f (x) (x) = x + 2x – x2 + 1 (B) f : R R f (x) = x3 + x + 1 (C) f : R R + f (x) = 1 x 2
Bansal C lasses
(D) f : R R f (x) = x3 + 2x2 – x + 1
Q. B. on FLCD, Method of Differentiation & ITF
[7]
Q.48
If f is twice twice differen differentiabl tiablee such such that f (x) h (x)
–1 If f (x) = 2x3 + 7x – 5 then f (4) is (A) equal to 1 (B) equal to 2
The range range of the function function f(x) =
n
(D) non existent is
3 , 4 (D) 2
if x < 1 has the value value equal to :
3
1
1
23 33 ......n 3
3
(B)
2 3
(D) (1 x) 1
(C) 1 x
(B) 1
is equal equal to :
(C)
1 2
(D)
1 6
If x = cos cos –1 (cos 4) ; y = sin –1 (sin 3) then which of the following holds ? (A) x – y = 1 (B) x + y + 1 = 0 (C) x + 2y = 2 (D) tan (x + y) = – tan7 Let f (x) =
tan 6 x 9 tan 4 x 9 tan 2 x 1 3
3 tan x (B) 9
If f (x) =
x 2 bx 25 x 2 7 x 10
(A) 0 Q.58 Q.58
(C) equal to 1/3
3 , (C) 2
1x 1x 2 1x 4 ...... 1x 2
(A) 4 Q.57 .57
(D) non existent
2x 2 11x 12
12 n 2 2 ( n1)32 ( n2)..... n 2.1
Lim n
(A)
Q.56 Q.56
(C) 4
2 e x l n x 5( x 2) (x 2 7 x 10)
(B) [0 , )
(A) 0
Q.55 Q.55
2 .
If f(x) is a twice differentiable function, function, then between two consecutive roots of the equation f (x) = 0, there exists : (A) (A) atle atleas astt one roo root of f(x) f(x) = 0 (B) (B) atm atmost one one roo root of f(x) f(x) = 0 (C) exa exactly tly one roo root of f(x) f(x) = 0 (D) atmost one root of f (x) = 0
Q.53 Limit n
Q.54
and
(B) a curve passing through the origin (D) (D) a stra straig ight ht line line with with y inte interc rcep eptt equa equall to
(B) 3
(A) ( , ) Q.52
2
f (x) g(x) h (0) 2 , h (1) 4
s 2 2s 3 x – 1 – s. The value of f ' (1), is s 1
(A) 2
Q.51
g(x)
The The graph graph of functi function on f contains The equation of the secant line through f contains the point P (1, 2) and Q(s, r). The P and Q is y =
Q.50 .50
, f ( x)
2
then the equation y = h(x) represents : (A) a curve of degree 2 (C) (C) a stra straig ight ht line line with with slop slopee 2 Q.49
f (x
, if f ' (x) = cosec4(2x) then the value of (C) 16
equals
(D) 64
for x 5 and f is f is continuous at x = 5, then f then f (5) has the value equal to (B) 5
(C) 10
(D) 25
Let f be f be a differentiable function on the open interval (a, b). Which Which of the following statements must be be true? I. f is f is continuous on the the closed interval [a, b] II. f is f is bounded on the open interval (a, b) III. If a
Bansal C lasses
Q. B. on FLCD, Method of Differentiation & ITF
[8]
Q.59 Q.59
If y = (sinx (sinx))l n x cosec (ex (a + bx)) and a + b = (A ) (sin1) (sin1) l n sin1
Q.60
(B) 0
The number number of solutions solutions of the equation equation tan
Let
(B) 2
2 sin x sin 2x f (x) = 2 cos x sin 2x
.
1 cos x
2
, n I
Which of the following is correct? (A) (I) and (II) (C) (III) and (IV) Limit x
(A) 1 Q.63
x = tan –1 x 2
+ tan –1
cot 1 x a log a x
sec 1 a x log x a
is
(D) 0
; x R
(II)
Range of f is R
(IV) (IV)
Domain Domain of f is R – (4n – 1)
2
, n I
(B) (II) and (III) (D) (II) , (III) and (IV) (a > 1) is equal to
(C) /2
(B) 0
(D) does not exist
The derivative derivative of the function function,,
R T
f(x)=cos-1 S
(A) Q.64
at x = 1 is dx (D) indeterminate
2/3
1 sin x
(III) (III) Domain of f is R – (4n +1)
The The valu valuee of
x 3
dy
(C) 1
Consider the following statements (I) Domain of f is R
Q.62 Q.62
then the value of
2e (C) l n sin1
–1
(A) 3 Q.61 Q.61
3
1 13
2
(2 cos x
3 sin x) UV + sin1 RS W T
(B)
5
2
1 13 (C)
(2 cos x
3 sin x) UV w.r.t. W
10
1 x 2 at x =
3 4
is
(D) 0
3
Let Let f : (1, (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 t he given information (D) does not exists
Q.65
Which of the the followin following g is the the solution solution set of of the the equation equation (A) (0, 1)
Q.66 Q.66
(B) (–1, 1) – {0}
(C) (–1, 0)
2x 2 1 2 cos cos –1 x = cot –1 2 ? 2x 1 x (D) [–1, 1]
n sin x cos x Let a = min min [x2 + 2x + 3, x R] and b = Lim x . Then the value of a r b n r is x 0 e e x r 0
(A)
2 n 1 1 3 ·2 n
Bansal C lasses
(B)
2 n 1 1 3 ·2 n
(C)
2n 1 3 ·2 n
(D)
4 n 1 1
Q. B. on FLCD, Method of Differentiation & ITF
3 ·2 n
[9]
Q.67
The solutio solution n set of f (x) > g (x), where f(x) = (A) x > 1
(B) 0 < x < 1
1
(52x + 1) & g(x) = 5x + 4x (l (l n 5) is :
2
(C) x
0
(D) x > 0
Q.68
Let Let f(x) f(x) = sin [a ] x (where [ ] denotes the greatest integer function) . If f is periodic with fundamental period , then a belongs to : (A) [2, 3) (B) {4, 5} (C) [4, 5] (D) [4, 5)
Q.69 Q.69
If f(x f(x)) = esin (x [x]) cos x , then f(x) is ([x] denotes denotes the greatest integer function) (A) non periodic periodic (B) period periodic ic with no fundam fundament ental al perio period d (C) periodic with period 2 (D) periodic with period .
Q.70 Q.70
If y =
x
x
x
ab
(B)
2 ay
Q.71 The The valu valuee of tan 1
Q.72
x
x
a b a b a b a
(A)
(A)
x
1 2
b ab
tan 1 (B)
6
...... then
2 by
52 6 1
6
(C)
4
ab
2 by
(D)
b ab
2ay
3
(D) none
Lim x 0
(B) (B) f is con contin tinuous uous on R (D) f is discontinuous for some x R
6x 2 (cot x )(csc 2x )
has the value equal to
1 sec cos x tan 4 sec x (B) – 6
(C) 0 y
If x 2 + y2 = R 2 (R > 0) then k =
2
(A) –
1
(B) –
R 2
1
3
R
2 R
arc cot x x
2
(D) –
2 R 2
1 x
tan1 x is : 1 x (B) { (/4) /4) , 3/4} (C) {/4 , (3/4)}
The domain domain of the functio function n f(x) =
(D) – 3
where k in terms of R alone a lone is equal to
(C)
The range of the function, function, f(x) = tan1 (A) {/4}
Q.76
a
is equal :
1 y
Q.75
=
1 Lim For the functio function n f (x) = n , which of the following holds? 2 1 n sin (x )
(A) 6 Q.74 .74
dx
(C)
(A) (A) The The rang rangee of f is a sing single leto ton n set set (C) f is discontinuous for all x I Q.73
dy
x
2
(D) {3/4}
, where [x] denotes the greatest integer not greater than x, is (B) R {0}
(A) R
(C) R
Bansal C lasses
n : n I
{0}
(D) R {n : n I}
Q. B. on FLCD, Method of Differentiation & ITF
[10]
Q.77 .77
cos –1
(A) Q.78 Q.78
cos 7 sin 2 is equal to 5 5 2
1
23
(B)
20
Give Given n f(x f(x)) =
x3 3
13 20
(D)
20
17 20
+ x2 sin 1.5 a x sin a . sin 2a 5 arc sin (a2 8a + 17) then : (B) f (sin8) > 0 (D) f (sin (sin 8) < 0
(A) f(x) is not defined at x = sin 8 (C) f (x) is not defined at x = sin 8
Q.79
33
(C)
Range Range of the function function f (x) =
1 2 l n ( x e )
1
is , where [*] denotes the greatest integer
1 x2
1 / function and e = Limit (1 )
0
(A) 0, Q.80
e 1 e
{2}
(C) (0, 1]
The range range of of the the func function, tion, f(x) = cot cot –1 log 0 .5 x 4
(B) 0,
(A) (0, ) Q.81
(B) (0, 1)
3 4
(D) (0, 1) {2}
2x 2 3 is: (C)
3 , 4
Given: Given: f(x) f(x) = 4x3 6x2 cos 2a + 3x sin 2a . sin s in 6a + (A) f(x) is not defined at x = 1/2 (C) f (x) is not defined at x = 1/2
{2}
n
2 a a 2
(D)
, 3 2 4
then
(B) f (1/2) < 0 (D) f (1/2) > 0
Q.82
The period period of the function function f (x) = sin (x + 3 – [x + 3 ] ), where [ ] denotes denotes the greatest greatest integer integer function function is (A) 2 + 3 (B) 2 (C) 1 (D) 3
Q.83
Sum of the roots of the equation, equation, arc cot x – arc cot (x + 2) = (A)
Q.84 Q.8 4
(B) 2
3
1
(C) – 2
is (D) –
(B) f(x) =
3
If y = (A + Bx) e
d2y 2 x + (m 1) e then 2 dx
mx
(A) ex
Limit x 0 1 (A) e 1/12
Bansal C lasses
(B) emx
4 x
3
1
1 |x | (D) f(x) = a|x| (a > 1)
1 x 2 (C) f(x) = e –|x|
Q.86
12
Which one of the following following functions best represent the graph graph as shown shown adjacent? (A) f(x) =
Q.85
2m
dy
+ m2y is equal to :
dx (C) emx
(D) e(1 m) x
cos ecx
has the value equal to : (B) e 1/6
(C) e 1/4
(D) e 1/3
Q. B. on FLCD, Method of Differentiation & ITF
[11]
Q.87
Limit x
x x 1 2x 1 sec x 1
cot 1
x 1
(A) 1 Q.88 Q.88
Q.89
(B)
10
Q.92
1 m
1 x
5
4 5
(D) m
1 m
/ m N {1}
2
(D) 0
1 Lim x sin x x is equal to x 1 | x |3 2
(B) – 1
(C) 0
The range of values of p for which the equation equation
1 2
,
2
1
(B) [0, 1)
(D) non existent
1 sin cos –1 cos(tan x) = p has a solution is:
(C)
1 2
, 1
(D) (– 1, 1)
Let f(x) be a differen differentiable tiable function function which which satisfies satisfies the equation equation f(xy) = f(x) + f(y) for all x > 0, y > 0 then f (x) is equal to
f ' (1) x
(B)
1 x
(C) f (1)
(D) f (1).(lnx)
Let ef(x) = l n x . If If g(x) is the inverse function of f(x) then g (x) equals to : (A) ex
Q.97
(D)
Let Let f(x) f(x) = xn , n being a non-negative non-negative integer . The number of values of n for which f (p + q) = f (p) + f (q) is valid for all p, q > 0 is : (A) 0 (B) 1 (C) 2 (D) none of these
(A) Q.96 Q.96
2
has the value equal to Limit 1 (arc cos x ) x 1 (A) 4 (B) 2 (C) 1
(A) Q.95
(C)
5
(A) 1 Q.94
/ m I {0}
4
Q.93
(D) non existent
The solution solution set for [x] [x] {x} = 1 where {x} and and [x] are fraction fractional al part part & integral integral part part of x, is is (A) R + – (0, 1) (B) R+ – {1} (C) m
Q.91
(C) /2
(B) 0
Supp Suppos osee f (x) = eax + e bx, where a b, and that f '' (x) – 2 f 2 f ' (x) – 15 f 15 f (x) = 0 for all x. Then the product ab is equal equal to (A) 25 (B) 9 (C) – 15 (D) – 9 x 2 –1 –1 is There There exists exists a positive positive real number number x satisfying satisfying cos(tan cos(tan x) = x. The value of cos 2 (A)
Q.90
is equal to
(B) ex + x
(C) e ( x
ex )
(D) e(x + l n x)
The domain domain of definition definition of the function function : f (x) (x) = l n ( x2 5 x 24 – x – 2) is (A) (– , –3]
Bansal C lasses
(B) (– , –3 ] U [8, ) (C)
, 28 9
(D) none
Q. B. on FLCD, Method of Differentiation & ITF
[12]
tan 3x 5 1 cos 6x ; g(x) is a function having the same time period as that of f(x), then which Q.98 Q.98 If f (x) 2 ta
Q.99 Q.99
of the following can be g(x). (A) (sec23x + cosec23x)tan23x
(B) 2 sin3x + 3cos3x
(C) 2 1 cos2 3x + cosec3x
(D) 3 cosec3x + 2 tan3x
2 cot cot cot 1 (3) cot 1 (7) cot 1 (13) cot 1(21) has the value equal to (A) 1 (B) 2 (C) 3 (D) 4
dy Q.100 The equation equation y2exy = 9e –3·x2 defines y as a differentiable function of x. The value of for dx x = – 1 and y = 3 is (A) – 15/2 (B) – 9/5 (C) 3 (D) 15 Q.101 Let f(x) =
l n ( x 2 e x ) l n ( x
4
e2x )
(A) l = l = m
. If Limit f(x) = l and Limit f(x) = m then : x
x
(B) l = l = 2m
(C) 2 l = m
(D) l + + m = 0
Q.102 Which one one of the following following statements is NOT CORRECT ? (A) The derivative of a diffrentiable periodic function is a periodic function with the same period. (B) If f (x) and g (x) both are defined on the entire number line and are aperiodic then the function F(x) = f (x) . g (x) can not be periodic. (C) Derivative of an even differentiable function is an odd function and derivative of an odd differentiable function is an even function. (D) Every function f (x) can be represented as the t he sum of an even and an odd function Q.103 Lim cos n 2 n when n is an integer : n (A) (A) is is equa equall to 1 Q.104 The value of tan 1 (where a, b, c > 0) (A) /4
(B) (B) is is equa equall to 1 a ( a b c) b c
(B)
+ tan 1
/2
(C) (C) is is equa equall to zero zero b ( a b c) ca
(C)
+ tan 1
(D) (D) does does not not exis existt c (a
b ab
c)
is :
(D) 0
Q.105 The function function f(x) f(x) = e x + x, being differentiable and one to one, has a differentiable inverse f –1(x). The value of (A)
d dx
–1 (f ) at the point f( l n2) is
1 n 2
(B)
1 3
(C)
1 4
(D) none
Q.106 Given the the graphs graphs of the two functions, y = f(x) & y = g(x). In the adjacent figure from point A on the graph of the function y = f(x) corresponding to the given value of the independent variable (say x0), a straight line is drawn parallel to the X-axis to intersect the bisector of the first and the third quadrants at point B . From the point B a straight line parallel to the Y-axis is drawn to intersect the graph of the function y = g(x) at C. Again a straight line is drawn from the point C parallel to the X-axis, to intersect the line NN at D . If the straight stra ight line NN is parallel parallel to Y-axis, then the co-ord co-ordinates inates of the the point point D are (A) f(x0), g(f(x0)) (B) x0, g(x0) (C) x0, g(f(x0)) (D) f(x0), f(g f(g (x0))
Bansal C lasses
Q. B. on FLCD, Method of Differentiation & ITF
[13]
Q.107 A functio function n f :R
R,
f(x) =
2x 1 x2
is
(A) inj inject ective ive by not sur surjec jective ive (C) (C) inje inject ctiv ivee as well well as surj surjec ecti tive ve [ x ]2 Limit Q.108 Q.108 Let Let x 0 = l & Limit x 0 2 x
(B) surjec jectiv tive but not inj injectiv tive (D) (D) neit neithe herr inje inject ctiv ivee nor nor surj surjec ecti tive ve
[ x2 ] x2
= m , where [ ] denotes denotes greatest integer , then:
(A) l exists but m does not (C) l & m both exist
(B) m exists but l does does not (D) neither l nor m exists .
Q.109 Which of the following following is the solution set set of the equation equation sin –1x = cos –1x + sin –1(3x – 2)? (A)
1 , 1 2
1 , 1 2
(B)
d2 y
Q.110 If y is a function of x then
(A)
(C)
d2 x d y2 d2 x d y2
+x
dx dy
(A) cot
(B)
cos x
5 2
3x 1
(D)
dy
Q.113 The value of Limit x 0
d2 x
1 , 1 3
d y2 d2 x d y2
3
d x + y = 0 d y 2
d x x = 0 d y
(C) (C) is equa equall to 289 289
5
(C) tan
12
tan {x } 1
sin {x }
{x } {x } 1
(A) is 1
(D)
(D) is non exist xisteent
= ax + b then the value of a + b is equal to
dx
(B) cot
8
2
(B) (B) is equa equall to 9 and
1 , 1 3
= 0 . If x is a function of y then the equation equation becomes becomes
2
x 4 x2 1 x
dx
d x y = 0 d y
(A) (A) is equa equall to 4 Q.112 Q.112 If y =
dy
(C)
=0
1 log2cos x 2
Q.111 Limit x 0
d x2
+y
(B) is tan 1
5 12
(D) tan
5 8
where { x } denotes denotes the fractional fractional part function: function: (C) is sin 1
(D) is non existent
Q.114 If f(x) = cosec –1(cosecx) and cosec(cosec –1x) are equal functions then maximum range of values of x is
,1 1, 2 2 (C) , 1 1 , (A)
,0 0 , 2 2 (D) 1, 0 0 ,1 f (x) = f (x) + f (x) + f (x) + ......
(B)
Q.115 Q.115 A function function f (x) satisfies the condition, condition, where f (x) is a differentiable differentiable function function indefinitely indefinitely and dash dash denotes denotes the order order of of derivative derivative . If If f (0) = 1, then f (x) is : x/2 x 2x (A) e (B) e (C) e (D) e4x Q.116 Q.116 Let f : R R
f (x) (x) =
x 1 |x |
. Then f (x) is :
(A) (A) inje inject ctiv ivee but not not surj surjec ecti tive ve (C) (C) inje inject ctiv ivee as well well as surj surjec ecti tive ve
Bansal C lasses
(B) (B) surj surjec ecti tiv ve but but not inje inject ctiv ivee (D) (D) neit neithe herr inje inject ctiv ivee nor nor surj surjec ecti tive ve .
Q. B. on FLCD, Method of Differentiation & ITF
[14]
1 x x 2
–1
–1
1 x 2 + cos x = cot
Q.117 The solution solution set set of the equation equation sin
(B) (B) (0, (0, 1] U {–1}
(A) (A) [–1 [–1, 1] – {0} {0}
–1
(C) [–1, 0) U {1}
– sin –1x
(D) [–1, 1]
Q.118 Suppose the function f (x) – f – f (2x) has the derivative 5 at x = 1 and derivative 7 at x = 2. The derivative of the function f function f (x) – f – f (4x) at x = 1, has the value equal to (A) 19 (B) 9 (C) 17 (D) 14 2 2x sin 2x Lim Q.119 x is : ( 2x sin 2x )e sin x (A) equal to zero
y = f(x) f(x) =
Q.12 Q.120 0 Le Let
(C) equal to 1
(B) equal to 1 e
(D) non existent
1 x2
x
0
if x
0
if
0
Then which of of the following can best represent the graph of y = f(x) ?
(A)
(B)
lim x 0
Q.121 The value value of
(A)
e
(C) 2
cos axcos ec bx
8 b 2 a 2
(B)
e
(D)
is
8a 2 b 2
(C)
e
a 2 2 b 2
(D)
x 1 2 2
Q.122 The set of values values of x for for which the the equation equation cos –1x + cos –1
(A) [0, 1]
Q.123 xLimit 0
(A)
1 x x
(B)
a arc tan
ab
x a
0, 1 2
1 , 1 (C) 2
b arc tan
e
b 2 2a 2
3 3x 2 =
3
holds good is
(D) {–1, 0, 1}
x
has the value equal to
b
Q.125 Diffrential coefficient of (A) 1
Bansal C lasses
x
(B) 0
(C)
a2
b2
(D) 3a 2 b2 6a 2 b 2 Q.124 Q.124 If f If f (x) is a function from R R, we say that f (x) has property I if f (f (x) ) = x for all real number x, and we we say that f (x) has property II if f (–f(x)) = – x for all real number x. How many linear functions, have both property I and II? (A) exactly one (B) exactly two (C) exac xactly tly three (D) infinite
3
(B) 0
b2 )
(a 2
1
m n m n
n x m n
.
1
m
. x
(C) – 1
n
1
m n
m
w.r.t. w.r.t. x is (D)
x mn
Q. B. on FLCD, Method of Differentiation & ITF
[15]
Q.126 Let f Let f (x) =
x
and let g let g (x)= (x)=
r x
. Let S be the set of all real numbers r such that f that f (g(x)) = g ( f (x)) 1 x 1 x for infinitely many real number x. The number of elements in set s et S is (A) 1 (B) 2 (C) 3 (D) 5
Q.127 Q.127 Let Let f (x) be a linear function with the properties that f that f (1) f (2), f (2), f (3) f (4), and f and f (5) = 5. Which of the following statements is true? (A) f (0) < 0 (B) f (0) = 0 (C) f (1) < f (0) < f (–1) (D) f (0) = 5 Q.128 Let f (x) (x) be diffrentiable diffrentiable at x = h then then Lim x h (A) (A) f(h f(h)) + 2hf '(h '(h)
(B) (B) 2 f(h) f(h) + hf '(h) '(h)
bx hg
f (x ) x
2h f (h )
h
(C) (C) hf( hf(h h) + 2f '(h '(h)
is equal to (D) (D) hf( hf(h) h) – 2f '(h) (h)
Limit [1 + cos2m (n! x)] is given by Q.129 If x is a real 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 . 2
2
Q.130 If y = at + 2bt + c and t = ax a x + 2bx + c, then (A) 24 a2 (at + b)
(B) 24 a (ax + b)2
d 3y dx 3
equals
(C) 24 a (at + b) 2
(D) 24 a2 (ax + b)
Direction for Q.131 and Q.132 The graph of a relation is (i) Symmetric Symmetric with respect to the x-axis provided provided that whenever whenever (a, b) is a point on the graph, graph, so is (a, – b) (ii) ii) Symmetric Symmetric with respect to the the y-axis y-axis provided provided that that whenever whenever (a, (a, b) b) is a point point on the the graph, graph, so is (– a, b) (iii) ii) Symmetric Symmetric with with respect respect to the origin provided provided that that whenever whenever (a, b) is a point on the the graph, graph, so is (– a, – b) (iv) (iv) Symmetric with respect to the line y = x, provided that whenever whenever (a, b) b) is a point point on the graph, graph, so is (b, a)
Q.131 The graph graph of the the relation relation x4 + y3 = 1 is symmetric with respect to (A) the x-axis (B) the y-axis (C) the origin (E) both the x-axis and y-axis
(D) the line y = x
Q.132 Suppose R is a relation whose whose graph is symmetric symmetric to both the x-axis and y-axis, y-axis, and that the point (1, 2) is on the graph of R. Which one of the following foll owing points is NOT necessarily on the graph of R? (A) (–1, 2) (B) (1, – 2) (C) (–1, –2) (D) (2, 1) (E) all of these points are on the graph of R. Select the correct correct alternatives : (More than one one are correct) correct)
Q.133 If y = tan x tan 2x tan3x then
dy dx
has the value equal to :
(A) 3 sec2 3x tan x tan2x tan 2x + sec2 x tan2x tan 2x tan 3x + 2 sec2 2x tan3x tanx (B) 2y 2y (cosec (cosec 2x + 2 cosec cosec 4x + 3 cosec cosec 6x) (C) 3 sec2 3x 2 sec2 2x sec2 x (D) sec2 x + 2 sec2 2x + 3 sec2 3x
Bansal C lasses
Q. B. on FLCD, Method of Differentiation & ITF
[16]
Q.134 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 tan (sgn (sgn x) sgn x
, c = 0.
where [x] denotes step up function & {x} fractional part function.
tan2 {x} x2 [ x ]2 Q.135 Let Let f (x) = 1 {x} cot {x}
0 x0 x0
for x for for
where [ x ] is the step up function function and { x } is the the fractional fractional
part functi function on of of x , then : (A)
Limit
(C) cot -1
Limit
f (x) = 1
x 0
Limit Lim x 0
(B) x 0 f (x) (x) = 1 2
f ( x) = 1
(D) f is continuous at x = 1 .
Q.136 Q.1 36 Which of the following following function function (s) is/are Transcidental? Transcidental? (A) f (x) = 5 sin (C) f (x) =
x2
(B) f (x) =
x
(A)
2y
2x
x2
2x 1
(D) f (x) = (x2 + 3).2x
2x 1
Q.137 Q.137 If 2x + 2y = 2x + y then
2 sin 3x
dy
has the the value equal to :
dx
(B)
1
(C) 1 2y
1 2x
3 cot 1 2x 3 3 x2 Q.138 Given f(x) = 2 1 /x x cos e
for for x
0
for x
0
2
1
2x 1 2y
(D)
2y
x
where { } & [ ] denotes denotes the fractional fractional part part and and the the
integral part functions respectively res pectively,, then which of the following statement does not hold good. (A) f(0 ) = 0 (B) f(0+) = 3 (C) f(0) = 0 continuity continuity of f at x = 0 (D) (D) irrem irremov ovab able le disco disconti ntinu nuity ity of f at x = 0 Q.139 Q.1 39 The graphs graphs of which of the following pairs differ . (A) y =
sin x 1 tan 2 x
+
cos x 1 cot 2 x
; y = sin 2x
(B) y = tan x cot x ; y = sin x cosec x (C) y = cos x + sin x ; y =
sec x
cos ec x
sec x cos ec x
(D) none of these
Bansal C lasses
Q. B. on FLCD, Method of Differentiation & ITF
[17]
1 14 Q.140 The value of cos c os 1 cos is : 5 2 (A) cos
7 5
(B) sin
2 5
10
(C) cos
Q.141 Q.1 41 Which of the following following functions functions are homogene homogeneous ous ? y/x x/y (A) (A) x siny siny + y sinx sinx (B) (B) x e + y e (C) x2 xy
3
cos 5
(D) arc arc sin sin xy
2 x 1 . x then : x 1 1
x
Q.142 Q.142 Let Let f (x) =
(D)
(A) f (10) = 1 (C) domain of f (x) is x 1
(B) f (3/2) = 1 (D) none
Q.143 If is small & positive number then which which of the following is/are correct ? (A)
sin
(B) < sin < tan
=1 x
Q.144 Let f(x) =
2 x2
1
7x 5
1 (A) Limit f(x) = x 1
tan
>
sin
. Then :
(B) Limit f(x) = x 0
3
(C) sin < < tan (D)
1 5
(C) Limit f(x) = 0 x
(D) Limit does not exist x 5 / 2
Q.145 If f(x) is a polynom polynomial ial function function satisfying satisfying the condition f(x) . f(1/x) f(1/x) = f(x) + f(1/x) and and f(2) = 9 then : (A) (A) 2 f(4) f(4) = 3 f(6) f(6) (B) (B) 14 14 f(1) f(1) = f(3) f(3) (C) (C) 9 f(3) f(3) = 2 f(5) f(5) (D) (D) f(1 f(10 0) = f(1 f(11) Q.146 Two functions f & g have first & second derivatives at x = 0 & satisfy the relations, relations, f(0) =
2 g(0)
, f (0) (0) = 2 g (0) = 4g (0) , g (0) = 5 f (0) = 6 f(0) = 3 then :
(A) if h(x) =
f (x) g(x)
then h (0) =
15
(B) if k(x) = f(x) . g(x) sin x then k (0) = 2
4
1 g (x) (C) Limit = x 0 f (x)
(D) none
2
Q.147 Which of of the following following function(s) function(s) not defined defined at x = 0 has/have removable discontinuity discontinuity at at x = 0 ? (A) f(x) =
| sin x | (B) f(x)=cos (C) f(x) = x sin x x
1 1 2 cotx
(D) f(x) =
1 n
x
Q.148 For the equation equation 2x = tan(2tan –1a) + 2tan(tan –1a + tan –1a3), which of the following is invalid? (A) a2x + 2a = x (B) a2 + 2ax 2ax + 1 = 0 (C) (C) a 0 (D) a –1, 1
x3 Q.149 The function function f(x) = x 3x 13 4 2 4 2
(A) continuous at x = 1 (C) continuous at x = 3
Bansal C lasses
, x 1 , x 1
is : (B) diff. at x = 1 (D) differentiable at x = 3
Q. B. on FLCD, Method of Differentiation & ITF
[18]
Q.150 Q.1 50 Identify the pair(s) of functions which are identical . (A) y = tan (cos 1 x); y = (C) y = sin (arc tan x); y =
Q.151 If y = x ( (A) (C)
y x
n x )
n ( n x )
, then
n x y
x n x
n x
1
dy dx
1 x 2
(B) y = tan (cot 1 x) ; y =
x
x 1 x
1 x
(D) y = cos (arc tan x) ; y = sin (arc cot x)
2
is equal to :
2 n x n n x
(B)
((l n x)2 + 2 l n (l (l n x))
(D)
y
(l n x) (l (l n x) l n (l (2 l n (l ( l n x) + 1)
x y n y x n x
(2 l n (l (l n x) + 1)
Q.152 The function, 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.153 Q.1 53 The graph of a function y = f(x) defined in [–1, 3] is as shown. Then Then which of the following statement(s) is(are) True? (A) f is continuous at x = –1. (B) f has an isolated discontinuity at x = 1. (C) f has a missing point discontinuity at x = 2. (D) f has a non removable discontinuity at the origin. Q.154 Q.1 54 Which of the following following function(s) function(s) has/have has/have the same range? range? (A) f(x) =
1
(B) f(x) =
1 x
1 1 x2
Q.155 The function function f(x) f(x) = (sgn x) (sin x) is (A) discontinuous no where. (C) aperiodic
(C) f(x) =
1 1
x
(D) f(x) =
1 3 x
(B) an even function (D) differentiable for all x
Q.156 If cos –1x + cos –1y + cos –1z = , then (A) x2 + y2 + z2 + 2xyz = 1 (B) 2(sin –1x + sin –1y + sin –1z) = cos –1x + cos –1y + cos –1z (C) xy + yz + zx = x + y + z – 1
x 1 y 1 z 1 > 6 (D) x + y + z 1
Q.157 The function function f(x) = x l n x (A) is a constant function (C) is such that Lim f(x) exist x 1
Bansal C lasses
(B) has a domain (0, 1) U (e, ) (D) is aperiodic
Q. B. on FLCD, Method of Differentiation & ITF
[19]
ANSWER KEY KEY Q. 1
C
Q. 2
A
Q.3
A
Q. 4
C
Q.5
A
Q. 6
B
Q.7
B
Q. 8
B
Q. 9
D
Q .1 0
D
Q.11
B
Q .1 2
A
Q.13
A
Q .1 4
C
Q.15
A
Q.16
A
Q .1 7
C
Q.18
B
Q .1 9
A
Q.20
B
Q .2 1
C
Q.22
B
Q.23
C
Q .2 4
D
Q.25
A
Q .2 6
D
Q.27
A
Q .2 8
C
Q.29
C
Q.30
A
Q .3 1
D
Q.32
D
Q .3 3
C
Q.34
A
Q .3 5
A
Q.36
C
Q.37
A
Q .3 8
A
Q.39
A
Q .4 0
C
Q.41
B
Q .4 2
A
Q.43
A
Q.44
B
Q .4 5
B
Q.46
C
Q .4 7
D
Q.48
C
Q .4 9
C
Q.50
A
Q.51
A
Q .5 2
B
Q.53
D
Q .5 4
A
Q.55
D
Q .5 6
C
Q.57
A
Q.58
D
Q .5 9
C
Q.60
A
Q .6 1
C
Q.62
A
Q .6 3
C
Q.64
B
Q.65
A
Q .6 6
D
Q.67
D
Q .6 8
D
Q.69
C
Q .7 0
D
Q.71
A
Q.72
C
Q .7 3
D
Q.74
B
Q .7 5
C
Q.76
C
Q .7 7
D
Q.78
D
Q.79
D
Q .8 0
C
Q.81
D
Q .8 2
C
Q.83
C
Q .8 4
C
Q.85
A
Q.86
A
Q .8 7
A
Q.88
C
Q .8 9
C
Q.90
D
Q .9 1
A
Q.92
C
Q.93
B
Q .9 4
B
Q.95
A
Q .9 6
C
Q.97
A
Q .9 8
A
Q.99
C
Q.100 D
Q.101 A
Q.102 B
Q.103 C
Q.104 C
Q.105 B
Q.106 C
Q.107 D
Q.108 B
Q.109 A
Q.110 C
Q.111 C
Q.112 B
Q.113 D
Q.114 A
Q.115 A
Q.116 A
Q.117 C
Q.118 A
Q.119 D
Q.120 C
Q.121 C
Q.122 C
Q.123 D
Q.124 B
Q.125 B
Q.126 B
Q.127 D
Q.128 A
Q.129 B
Q.130 D
Q.131 B
Q.132 D
Q.133 A,B,C
Q.134 B,C
Q.135 A,C
Q.136 A,B,D
Q.137 A,B,C,D
Q.138 B,D
Q.139 A,B,C
Q.140 B,C,D
Q.141 B,C
Q.142 A,B
Q.143 C,D
Q.144 A,B,C,D
Q.145 B,C
Q.146 A,B,C
Q.147 B,C,D
Q.148 B,C
Q.149 A,B,C
Q.150 A,B,C,D
Q.151 B,D
Q.152 A,B,C,D
Q.153 A,B,C,D
Q.154 B,C
Q.155 A,B,C
Q.156 A,B
Q.157 Q.157 A,C A,C
Bansal C lasses
Q. B. on FLCD, Method of Differentiation & ITF
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