BANSALCLASSES TARGET IIT JEE 2007
M A T H E M A T I C S NUCLEUS
QUESTION BANK ON
METHOD OF DIFFERENTIATION
Time Limit : 3 Sitting Each of 70 Minutes duration approx.
Question bank on Method of differentiation There are 72 questions in this question bank. Select the correct alternative : (Only one is correct)
Q.1
If g is the the inve invers rsee of f & f (x) = (A) 1 + [g(x)]5
Q.2
If y = tan1
(B)
1 x 5
then g (x) =
1 1 [g (x )]
(B) 1
1
2
3x 4 1 (B) 2 tan . 5x 6 (5x 6) 2
3
If y = sin1 x 1 x
(D) none 1 x 2 &
x
dy dx
(B) sin1 x
=
1 2 x (1 x )
+ p, then p =
(C) sin1 x
(D) none of these
dy 2x 1 & f (x) = sin s in x then = dx x 2 1
If y = f
(A)
(C)
1 x
1 x 2
1 x
2x 1 sin 2 x 1
x2 2
x2
1 x 2
(B)
2x 1 x 2 1
sin
2
5
210
(B)
y x
Bansal C lasses
(B)
2
2
2
(D) none
1 a 2 a 10
If sin sin (xy (xy) + cos cos (xy) (xy) = 0 then then (A)
x sin 2x 1 x 1 1 x
2 1 x
2
Let Let g is the invers inversee funct function ion of f & f (x) =
(A)
Q.7
(D)
dy 3x 4 & f (x) = tan x2 then = dx 5x 6
(A) 0
Q.6
(D) none
1 [g (x )]5
(C ) 0
3 tan x 2 4 tanx2 (C) f 2 5 tan x 6
Q.5
1
If y = f
(A) tanx
Q.4
(C)
5
n xe2 d 2y 1 3 2 n x n ex 2 + tan 1 6 n x then dx2 =
(A) 2
Q.3
1
dy dx y x
x10
1 x 2
. If g(2) = a then g (2) is equal to
a
10
(C)
1 a
(C)
2
(D)
(D)
1 a 10 a2
=
x y
Q. B. on Method of differentiation
x y
[2]
Q.8 Q.8
If y = sin sin1
2 then
1 x
2
(A)
Q.9
2x
5
dy
dx x 2
is :
2
(B)
(C)
5
2
(D) none
5
1 1 w.r.t. 1 x 2 at x = is : 2 2 2x 1
The The deri deriva vati tive ve of sec1 (A) 4
(B) 1/4
(C ) 1
(D) none
d 3 d2 y Q.10 Q.10 If y = P(x), is a polynomial of degree 3, then 2 y . 2 equals : dx dx 2
(A) P (x) + P (x)
(B) P (x) . P (x)
(C) P (x) (x) . P (x) (x)
(D) a const onstan antt
Q.11
Let f(x) be be a quadratic expression which is positive for all real real x . If g(x) = f(x) + f (x) + f (x), then for any real x, which one is correct . (A) g(x) < 0 (B) g(x) > 0 (C) g(x) = 0 (D) g(x) 0
Q.12 Q.12
If x p . yq = (x + y) p + q then
dy dx
is :
(A) independe independent nt of p but depend dependent ent on q (C) dependent ent on both p & q
Q.13 Q.13
(B) dependen dependentt on p but indepe independe ndent nt of q (D) independent of p & q both .
g (x) . cos cos 1x if x 0 where g(x) is an even function differentiable at x = 0, passing 0 if x 0
Let Let f(x) f(x) =
through the origin . Then f (0) : (A) is equal to 1 (B) is equal to 0
Q.14 Q.14
If y =
1 1 x n m
xp m
(A) emnp Q.15
log
sin
cos 2x 2
xp n
1 1 x m p
x
n p then
n p
dy
at e m is equal to:
dx
(C) enp/m
(D) none
(C ) 4
(D) none of these
x has the value equal to 2
(B) 2
If f is differenti differentiable able in (0, 6) & f (4) = 5 then Limit x 2 (A) 5
Q.17 Q.17
1 x m n
+
(B) emn/p
(A) 1 Q.16
1
(D) does not exist
log sin 2 x cos x
Lim x 0
+
(C) is equal to 2
(B) 5/4
f cx 2 h = 2x
f (4 )
(C) 10
(D) 20
Let l = = xLim xm (l (l n x)n where m, n N then : 0 (A) l i l is independent of m and n (B) l is is independent of m and depends on m (C) l is is independen independentt of n and and depend dependent ent on on m (D) l is is dependent on both m and n
Bansal C lasses
Q. B. on Method of differentiation
[3]
cos x
Q.18 Q.18
x
Let Let f(x f(x)) = 2 sin x x 2 tan x
x
(A) 2
sin x
2
(C )
cos x
sin 3x
(A) 0
3 cos 3x
(C ) 4
f 2 (x h) f 2 (x) h
where f (x) means [f(x)]2. If f(x) = x l nx nx then
(B)
f (x )
3
g (x )
x 2
(C ) 0
2
is equal to : (D) none
h
(B) 5f (x) x
If y = x + e then x
(A) e
d 2x dy 2
ex
1 e x
3
If x y + y = 2 then the value of
3 4
(C ) 0
(D) none
is :
(B)
2
(A) Q.25 Q.25
(D) none
f (x 3h) f (x 2 h) If f(x) is a different differentiable iable function function of x then Limit = h 0
(A) f (x)
Q.24 Q.24
(C) 4e
If f(4) = g(4) g(4) = 2 ; f (4) = 9 ; g (4) = 6 then Limit x 4 (A) 3 2
Q.23 Q.23
(D) 12
People living living at Mars, Mars, instead of the usual usual definition definition of derivative derivative D f(x), define define a new kind of derivative, D*f(x) by the formula
D * f (x ) has the value x e (A) e (B) 2e
Q.22
(D) 1
2
(B) – 12
D*f(x) = Limit h 0
Q.21
1
Let Let f(x) f(x) = cos 2x sin 2x 2 cos 2x then f = cos 3x
Q.20 Q.2 0
f (x) = 2x . Then Limit x 0 x 1
(B) cos x
Q.19 Q.19
1
(B)
3 8
d 2y dx 2
(C)
3
ex
1 e x
2
5
Bansal C lasses
(B) 1/5
1
1 e
x
3
at the point point (1, 1) is :
(C)
5 12
If f(a) f(a) = 2, f (a) = 1, g(a) = 1, g (a) = 2 then the value of Limit xa (A)
(D)
(C ) 5
Q. B. on Method of differentiation
(D) none
g (a ) . f (x) is: xa
g (x) . f (a )
(D) none
[4]
Q.26
If f is twice twice differen differentiabl tiablee such that f (x) f (x) , f (x) g(x) h (x) f (x)
2
2
g(x) and h (0) 2 , h (1) 4
then the equation y = h(x) represents : (A) a curve of degree 2 (C) (C) a stra straig ight ht line line with ith slo slope 2
Q.27
(B) a curve passing through the origin (D) (D) a stra straig ight ht line line with with y inte interc rcep eptt equ equal to 2.
R T
The derivati derivative ve of the functio function, n, f(x)=co f(x)=coss-1 S w.r.t. 1 x 2 at x = (A)
3
3 4
2
1 13
U W
R T
3 sin x) V + sin1 S
(2 cos x
1 13
U W
3 sin x) V
is : (B)
5
2
10
(C)
(D) 0
3
Q.28
Let f(x) be a polynomial in x . Then the second derivative of f(e x), is : (A) f (ex) . ex + f (ex) (B) f (e x) . e2x + f (ex) . e2x (C) f (ex) e 2x (D) f (ex) . e2x + f (ex) . ex
Q.29
The solutio solution n set of f (x) > g (x), where f(x) = (A) x > 1
(2 cos x
(B) 0 < x < 1
1
(52x + 1 ) & g(x) = 5x + 4x (l (l n 5) is :
2
(C ) x
0
(D) x > 0
dy 1 x2 1 1 + sec , x > 1 then is equal equal to : dx x 1 x2 1 2
Q.30 Q.30
x If y = sin sin1 2
(A)
Q.31 Q.31
Q.33
x4
If y = (A)
Q.32
x
1 x
x
x
(B)
x
x
x2
x
a b a b a b a
ab
2 ay
(B)
(C ) 0
1
x4
...... then
b ab
2 by
dy dx
(C)
(D) 1
= a ab
2 by
(D)
b ab
2 ay
Let f (x) be a polynomial polynomial function function of second second degree. degree. If f (1) = f (–1) and a, b, c are in A.P., A.P., then f '(a), f '(b) and f '(c) are in (A) G.P. (B) H.P. (C) A.G.P. (D) A.P. y
y1
y2
If y = sin mx then the value of y 3
y4
y6
y7
y 5 (where subscripts subscripts of y shows s hows the order of of derivatiive) is: y8
(A) independent independent of x but dependent dependent on m (C) dependent on both m & x
Bansal C lasses
(B) dependent dependent of x but independent independent of m (D) independ endent of m & x .
Q. B. on Method of differentiation
[5]
Q.34 Q.34
If x2 + y2 = R 2 (R > 0) then k =
y
1 y 2
1
(A) – Q.35
Q.36 Q.36
1
(B) –
2
R
3
where k in terms of R alone is equal to
(C)
R
2
(D) –
R
2 2
R
If f & g are differentia differentiable ble functions functions such that g (a) = 2 & g(a) = b and if fog is an identity function then f (b) has the value equal to : (A) 2/3 (B) 1 (C ) 0 (D) 1/2 Give Given n f(x) f(x) =
x3 3
+ x2 sin 1.5 a x sin a . sin 2a 5 arc sin (a2 8a + 17) then : (B) f (sin8) (sin 8) > 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.37 Q.3 7
A function f, defined for all positive real numbers, satisfies the equation f(x2) = x3 for every x > 0 . Then the value value of f (4) = (A) 12 (B) 3 (C) 3/2 (D) cannot be determined
Q.38
Given Given : f(x) = 4x3 6x2 cos 2a + 3x sin 2a . sin 6a +
2
If y = (A + Bx) e
d y + (m 1)2 ex then
mx
x
dx
mx
(A) e
(B) e
2 a a 2
then :
(B) f (1/2) < 0 (D) f (1/2) > 0
(A) f(x) is not defined at x = 1/2 (C) f (x) is not defined at x = 1/2
Q.39
n
2
2m
dy dx
+ m2y is equal to :
(C) emx
(D) e(1 m) x
Q.40 Q.40
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
Q.41
Let h (x) be differentiabl differentiablee for all x and let f let f (x) = (kx + ex) h(x) where k is some constant. If h (0) = 5, h ' (0) = – 2 and f ' (0) = 18 then the value of k is equal to (A) 5 (B) 4 (C ) 3 (D) 2.2
Q.42 Q.42
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.43
(B) ex + x
(C ) e ( x
ex )
(D) e(x + l n x)
The equatio equation n y2exy = 9e –3·x2 defines y as a differentiable function of x. The value of
dy dx
for
x = – 1 and y = 3 is (A) –
15 2
Bansal C lasses
(B) –
9 5
(C ) 3
Q. B. on Method of differentiation
(D) 15
[6]
Q.44 Q.44
x Let Let f(x f(x)) = x
x
and g(x) = x
xx then :
(A) f (1) = 1 and g (1) = 2 (C) f (1) = 1 and g (1) = 0 Q.45
The functi function on f(x) f(x) = ex + x, being differentiable and one to one, has a differentiable inverse f –1(x). The value of (A)
Q.46 Q.46
(B) f (1) = 2 and g (1) = 1 (D) f (1) = 1 and g (1) = 1
d dx
–1 (f ) at the point f( l n2) is
1
n 2
(B)
1 3
log sin|x| cos3 x
If f (x) f (x) =
log sin|3x| cos
3
x 2
=4
for |x| <
(C)
3
for x = 0
(A) 0
If y =
(B)
If y is a function function of x then
(A)
d y2
+x
(C ) 2
dx dy
(a b ) 2 (a x) (x b ) 2x
d2 y d x2
+y
dy dx
(D) 4
(C)
(a b ) 2 (a x) (x b )
(D)
(a b ) 2 (a x) (x b ) 2x
= 0 . If x is a function of y then the the equation equation becomes : 3
d x (B) = 0 2 +y d y dy d2 x
=0 2
d x (C) =0 2 y d y dy d2 x
is
dy (b x) x b then wherever it is defined is equal to : dx a x x b
(a b) (a x) (x b)
d2 x
, 3 3
a x
x
(A)
Q.48
(B) 3 (a x)
(D) none
4
x 0
then, the number of points of discontinuity of f in
Q.47 Q.47
1
2
d x (D) = 0 2 x d y dy d2 x
Q.49
A functi function on f (x) satisfie satisfiess the condit condition ion,, f (x) = f (x) + f (x) + f (x) + ...... where f (x) is a differentiable differentiable function function indefinitely indefinitely and dash dash denotes denotes the 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.50 Q.50
If y =
cos 6x 6 cos 4 x 15 cos 2 x 10
cos 5x 5 cos 3x 10 cos x (A) 2 sinx + cosx (B) –2sinx
Bansal C lasses
, then
dy
dx (C) cos2x
=
Q. B. on Method of differentiation
(D) sin2x
[7]
Q.51 Q.51
If
d 2 x dy
3
+
dy 2 dx
d 2y dx 2
(A) 1
Q.52 Q.52
= K then the value of K is equal to (B) –1
(C ) 2
(D) 0
1
1 1 x sin 1 2 x (1 x) where x 0 , If f(x) f(x) = 2 sin
2
then f ' (x) has the value equal to 2
(A)
(B) zero
x (1 x)
Q .5 3
e y = f(x) = 0
Let
2
(C)
x (1 x)
(D)
1 x2
x
0
if x
0
if
Then which of of the following can best best represent the graph of of y = f(x) ?
(A)
Q.54
(B)
Diffrenti Diffrential al coefficien coefficientt of (A) 1
Q.55
1
m n m n
. x n
m n
(B) 0
(D)
1
b x hg
(B) (B) 2 f(h) f(h) + hf '(h) '(h)
x
m n
m
2
(B) (B) 24 a (ax (ax + b)2
w.r.t. w.r.t. x is
h
d 3y dx 3
x mn
2h f (h )
(C) (C) hf( hf(h) h) + 2f '(h '(h)
If y = at + 2bt + c and t = ax + 2bx + c, then (A) 24 a2 (at (at + b)
1
(D)
f (x )
2
. x
m
n
(C ) – 1
Let f (x) (x) be diffrentiable diffrentiable at x = h then Lim x h (A) (A) f(h f(h)) + 2hf 2hf '(h) '(h)
Q.56 Q.56
x
(C)
is equal to (D) (D) hf( hf(h) h) – 2f '(h '(h)
equals
(C) 24 a (at + b) 2
(D) 24 a 2 (ax + b)
1 x x has the value equal a a rc t a n b arc t a n Q.57 xLimit equal to 0 x x a b
(A)
ab 3
Bansal C lasses
(B) 0
(C)
(a 2
b2 )
6a 2 b 2
Q. B. on Method of differentiation
(D)
a2
b2
3a 2 b2
[8]
Q.58
x f ( x ) f ( y) for all x, y & y
Let f (x) be defined for all x > 0 & be continuous. continuous. Let f(x) satisfy f f(e) = 1. Then :
Q.59
Q.60 Q.60
(A) f(x) is bounded
1 (B) f 0 as x 0 x
(C) x.f(x)1 as x 0
(D) f(x) = l n x
Suppose Suppose the function 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
If y =
x4 x2 1 x
(A) cot
Q.61
2
3x 1
5
and
dy dx
= ax + b then the value of a + b is equal to
(B) cot
8
5
(C) tan
12
5
(D) tan
12
5 8
Suppos Supposee that that h (x) = f (x)· g g (x) (x) and F(x) = f g ( x) , where f (2) = 3 ; g (2) (2) = 5 ; g '(2) '(2) = 4 ; f '(2) = –2 and f '(5) = 11, then (A) (A) F'(2 F'(2)) = 11 h'(2 h'(2)) (B) (B) F'(2 F'(2)) = 22h' 22h'(2 (2))
(C) (C) F'(2 F'(2)) = 44 h'(2 h'(2))
(D) (D) none one
Q.62 Q.62
Let f (x) = x3 + 8x + 3 which one of the properties of the derivative enables you to conclude that f that f (x) has an inverse? (A) f (A) f ' (x) is a poly polyno nomi mial al of even even degr degree ee.. (B) (B) f ' (x) is self inverse. (C) domain of f ' (x) is the range of f ' (x). (D) f ' (x) is always positive.
Q.63
Which one of the 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 defined on the entire number number line and and are aperiodic aperiodic then the function F(x) = f (x) . g (x) can not be periodic. 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
Select the correct correct alternatives : (More than one are correct) correct)
Q.64
If y = tanx tan x tan tan 2x tan3x then
dy dx
has the value equal to :
(A) 3 sec2 3x tan x tan 2x + sec2 x tan 2x tan 3x + 2 sec2 2x tan3x tanx (B) 2y 2y (cosec (cosec 2x + 2 cosec cosec 4x + 3 cosec cosec 6x) 2 2 2 (C) 3 sec 3x 2 sec 2x sec x (D) sec2 x + 2 sec 2 2x + 3 sec 2 3x Q.65 Q.65
x
If y = e
(A)
e
x
e
e 2 x
Bansal C lasses
x
then
dy dx
x
(B)
equals e x
e 2x
x
(C)
1 2 x
y2 4
Q. B. on Method of differentiation
(D)
1 2 x
y2 4
[9]
Q.66 Q.66
dy
2
If y = xx then (A) 2 l n x . xx
2
(B) (2 l n x + 1). 1). xx
(C) (2 l n x + 1).x Q.67 Q.67
Let y = (A)
Q.68 Q.68
1
2 y 1
2y
(B)
2x
dy dx
(B)
dy dx
=
x x 2 y
1 4x
(D)
y 2x
y
has the value equal to : 1 1 2
2
1
2x 1 2 y
(C) 1 2y
x
dv
du
u
dx
d 2v
dx
2
2
= u + v
(B)
= 2u
dx 2
(D)
2y
x
x
Let f (x) (x) =
d2 u dx2
=2v
(D) none of these
2 x 1 . x then then : x 1 1
(A) f (10) = 1 (C) domain of f (x) is x 1
(B) f (3/2) = 1 (D) none
Two functions functions f & g have 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)
Q.72 Q.72
1
(C)
The The funct function ionss u = e x sinx ; v = ex cos x satisfy the equation equation :
(C)
Q.71
2
2 (D) x x 1 . l n ex2
x x ...... then
1
(A) v
Q.70 Q.70
x2
If 2x + 2y = 2x + y then
(A) Q.69
x
=
dx
If y = x ( (A) (C)
y x
n x )
n ( n x )
n x y
x n x
, then n x
1
(D) none
2
dy dx
is equal to :
2 n x n n x
((l n x)2 + 2 l n (l (l n x))
Bansal C lasses
(B)
(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. B. on Method of differentiation
[10]
ANSWER KEY D , B 2 7 . Q
C , B , A 1 7 . Q
B , A 0 7 . Q
C , B , A 9 6 . Q
, C , B , A 8 6 . Q D
D , C , A 7 6 . Q
D , C 6 6 . Q
C , A 5 6 . Q
C , B , A 4 6 . Q
B 3 6 . Q
D 2 6 . Q
B 1 6 . Q
B 0 6 . Q
A 9 5 . Q
D 8 5 . Q
D 7 5 . Q
D 6 5 . Q
A 5 5 . Q
B 4 5 . Q
C 3 5 . Q
B 2 5 . Q
D 1 5 . Q
5 . Q B 0
4 . Q A 9
4 . Q C 8
B 7 4 . Q
4 . Q C 6
4 . Q B 5
4 . Q D 4
4 . Q D 3
4 . Q C 2
4 . Q C 1
C 0 4 . Q
A 9 3 . Q
D 8 3 . Q
B 7 3 . Q
D 6 3 . Q
D 5 3 . Q
B 4 3 . Q
D 3 3 . Q
D 2 3 . Q
D 1 3 . Q
C 0 3 . Q
D 9 2 . Q
D 8 2 . Q
C 7 2 . Q
C 6 2 . Q
2 . Q C 5
2 . Q B 4
2 . Q B 3
2 . Q B 2
2 . Q A 1
. Q C 0 2
C 9 1 . Q
1 . Q B 8
. Q A 7 1
. Q D 6 1
C 5 1 . Q
D 4 1 . Q
B 3 1 . Q
D 2 1 . Q
B 1 1 . Q
C 0 1 . Q
A 9 . Q
C 8 . Q
B 7 . Q
B 6 . Q
B 5 . Q
. Q D 4
. Q B 3
C 2 . Q
A 1 . Q
Bansal C lasses
Q. B. on Method of differentiation
[11]
NOTES
