IISER Mathematics - Differential Coefficient
Exam Duration: 45 Mins Total Questions : 30
If f(x) = |x|, then f' (x), where x\(\neq \)0 is equal to
- (a)
-1
- (b)
0
- (c)
1
- (d)
\(\frac { \left| x \right| }{ x } \)
If x=2 cos t - cos 2t, y=2 sin t - sin 2t, then \(\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } \) at t = \(\frac { \pi }{ 2 } \) is
- (a)
-5/2
- (b)
-3/2
- (c)
3/2
- (d)
5/2
Let f(x) =\(\left| \begin{matrix} sin\quad 3x & 1 & 2\left( cos\left( \frac { 3x }{ 2 } \right) +sin\left( \frac { 3x }{ 2 } \right) \right) ^{ 2 } \\ cos\quad 3x & -1 & 2\left( cos^{ 2 }\left( \frac { 3x }{ 2 } \right) -sin^{ 2 }\left( \frac { 3x }{ 2 } \right) \right) \\ tan\quad 3x & 4 & 1+2\quad tan\quad 3x \end{matrix} \right| \)
then the value of f' (x) at x = (2n + 1) \(\pi \), n \(\in \) I (the set of integers) is equal to
- (a)
(-1)n
- (b)
3
- (c)
(-1)n+1
- (d)
9
If y2=ax2+bx+c, then y3.\(\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } \) is
- (a)
a constant
- (b)
a function of x only
- (c)
a function of y only
- (d)
a function of x and y
If \(x=sec\theta -cos\theta ,y={ sec }^{ 10 }\theta -{ cos }^{ 10 }\theta \quad and\quad \left( { x }^{ 2 }+4 \right) \left( \frac { dy }{ dx } \right) ^{ 2 }=k({ y }^{ 2 }+4),\)then k is equal to
- (a)
\(\frac { 1 }{ 100 } \)
- (b)
1
- (c)
10
- (d)
100
If xy =ex-y, then \(\frac { dy }{ dx } \) is equal to
- (a)
(1+1n x)-1
- (b)
(1+1n x)-2
- (c)
1n x (1+1n x)-2
- (d)
none of these
The derivative of sin-1 \(\left( \frac { 2x }{ 1+{ x }^{ 2 } } \right) \) with respect to tan-1 \(\left( \frac { 2x }{ 1-{ x }^{ 2 } } \right) \) is
- (a)
0
- (b)
1
- (c)
\(\frac { 1 }{ 1-{ x }^{ 2 } } \)
- (d)
\(\frac { 1 }{ 1+{ x }^{ 2 } } \)
If f(x) = |x - 2| and g(x) = fof(x), then for x > 20, g' (x) is equal to
- (a)
2
- (b)
1
- (c)
3
- (d)
none of these
If \(\Phi \)(x) be a polynomial function of the second degree. If \(\Phi \)(1)=\(\Phi \)(-1) and a1, a2, a3 are in AP, then \(\Phi \)(a1), \(\Phi \)(a2), \(\Phi \)(a3) are in
- (a)
AP
- (b)
GP
- (c)
HP
- (d)
none of these
If y = \(\left( \frac { ax+b }{ cx+d } \right) \), then 2 \(\frac { dy }{ dx } .\frac { { d }^{ 3 }y }{ { dx }^{ 3 } } \) is equal to
- (a)
\(\left( \frac { { d }^{ 2 }y }{ { dx }^{ 2 } } \right) ^{ 2 }\)
- (b)
\(3\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } \)
- (c)
\(3\left( \frac { { d }^{ 2 }y }{ { dx }^{ 2 } } \right) ^{ 2 }\)
- (d)
\(3\frac { { d }^{ 2 }x }{ { dy }^{ 2 } } \)
If xy . yx = 16,then \(\frac { dy }{ dx } \)at (2, 2) is
- (a)
-1
- (b)
0
- (c)
1
- (d)
none of these
If 5 f(x) + 3f\(\left( \frac { 1 }{ x } \right) \)x + 2 and y = x f(x), then\(\left( \frac { dy }{ dx } \right) _{ x=1 }\) is equal to
- (a)
14
- (b)
7/8
- (c)
1
- (d)
none of these
If f(x)=(logcotx tan x) (logtanxcot x)-1 + tan \(\left( \frac { x }{ \sqrt { \left( 4-{ x }^{ 2 } \right) } } \right) \) then f' (0) is equal to
- (a)
-2
- (b)
2
- (c)
\(\frac { 1 }{ 2 } \)
- (d)
0
If y1/n = {x + \(\sqrt { \left( 1+{ x }^{ 2 } \right) } \)}, then (1 + x2) y2 +xy1 is equal to
- (a)
n2y
- (b)
ny2
- (c)
n2y2
- (d)
none of these
If Y = sin xoand u = cos x, then\(\frac { dy }{ du } \)is equal to
- (a)
- cosec X· cos x
- (b)
\(\frac { \pi }{ 180 } \)cosec xocos x
- (c)
\(-\frac { \pi }{ 180 } \)cosec x· cos xo
- (d)
none of these
If sin y = x sin (a + y) and \(\frac{dy}{dx}=\frac {A}{ 1+{ x }^{ 2 }-2xcosa } \), then the value of A is
- (a)
2
- (b)
cos a
- (c)
sin a
- (d)
none of these
The third derivative of a function f(x) vanishes for all x. If f(0) = 1, f' (1) = 2 and f" (1) = - 1, then f(x) is equal to
- (a)
(- 3/2) x2 + 3x + 9
- (b)
(- 1/2) x2 - 3x + 1
- (c)
(-1/2)x2+3x+l
- (d)
(-3/2)x2-7x+2
If y = logex (x - 2)2 for x ≠ 0, 2, then y' (3) is equal to
- (a)
1/3
- (b)
2/3
- (c)
4/3
- (d)
none of these
Let xcos y + y cos x = 5. Then
- (a)
at x = 0, y = 0, y' = 0
- (b)
at x = 0, y = 1, y' = 0
- (c)
at x = y = 1, y' = - 1
- (d)
at x = 1, Y = 0, y' = 1
If \(\sqrt { \left( 1-{ x }^{ 6 } \right) } +\sqrt { \left( 1-{ y }^{ 6 } \right) } =a({ x }^{ 3 }-{ y }^{ 3 })\quad and\quad \frac { dy }{ dx } =f(x,y)\sqrt { \left( \frac { 1-{ y }^{ 6 } }{ 1-{ x }^{ 6 } } \right) } \) , then
- (a)
f(x,y)=y/x
- (b)
f(x,y)=y2/x2
- (c)
f(x, y) = 2y2/x2
- (d)
f(x, y) = x2/y2
If y= tan-1\(\left( \frac { { log }_{ e }\left( e/{ x }^{ 2 } \right) }{ { log }_{ e }\left( e{ x }^{ 2 } \right) } \right) +{ tan }^{ -1 }\left( \frac { 3+2{ log }_{ e }x }{ 1-6{ log }_{ e }x } \right) \)then \(\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } \)is
- (a)
2
- (b)
1
- (c)
0
- (d)
-1
If Y = f(x) be a differentiable function of x such that whose second, third, ... , nth derivatives exist.
ie, nth derivative of y is denoted by \(y_n{d^ny\over dx^n},D^n_y, f^n(x)\)
⇒ \({d^ny\over dx^n}=\underset{h\rightarrow0}{lim}{f^{n-1}(x+h)-f^{n-1}(x)\over h}\)
If y = e3x + 7, then the value of yn(O) is
- (a)
1
- (b)
3n
- (c)
3n.e7
- (d)
3n.e7.7!
If x = a cos \(\theta \), y = b sin \(\theta \), then\(\frac { { d }^{ 3 }y }{ { dx }^{ 2 } } \)is equal to
- (a)
\(\left( \frac { -3b }{ { a }^{ 3 } } \right) { cosec }^{ 4 }\theta { cot }^{ 4 }\theta \)
- (b)
\(\left( \frac { 3b }{ { a }^{ 3 } } \right) { cosec }^{ 4 }\theta { cot }^{ 4 }\theta \)
- (c)
\(\left( \frac { -3b }{ { a }^{ 3 } } \right) { cosec }^{ 4 }\theta { cot }\theta \)
- (d)
none of the above
If F(x) =\(\frac { 1 }{ { x }^{ 2 } } \int _{ 4 }^{ x }{ \{ { 4t }^{ 2 }-{ 2F }^{ ' }(t)\} } dt,\)then F'(4) equals
- (a)
32/9
- (b)
64/3
- (c)
64/9
- (d)
none of these
Let f(x)=(ax+b)cosx+(cx+d)sinx and f' (x) = X cos X be an identity in x, then
- (a)
a=0
- (b)
b=1
- (c)
c=1
- (d)
d=0
Let f(t) = Int. Then \({d\over dx}\left\{ \int_{c^2}^{x^3}f(t)dt\right\}\)
- (a)
has a value 0 when X = 0
- (b)
has a value 0 when X = 1, X =\(4\over 9\)
- (c)
has a value ge2 - 4e when x =e
- (d)
has a differential coefficient 27e - 8 when X= e
If f(x) = sin-1(sin x), then
- (a)
\(f'\left(3\pi\over 4\right)=1\)
- (b)
\(f'\left(5\pi\over 4\right)=-1\)
- (c)
\(f'\left(\pi\over2 \right)\)does not exist
- (d)
f' (π) does not exist.
If F(x) = f(x)g(x) and f'(x) g' (x) = c, then
- (a)
\(F'=c\left[{f\over f'}+{g\over g'}\right]\)
- (b)
\({F''\over F}={f''\over f}+{g''\over g}+{2\over fg}\)
- (c)
\({F'''\over F}={f'''\over f}+{g'''\over g}\)
- (d)
\({F'''\over F''}={f'''\over f''}+{g'''\over ''}\)
If f(x) = sin-1\(\left( \frac { 2x }{ 1+{ x }^{ 2 } } \right) \), then
- (a)
f is derivable for all x, with |x| < 1
- (b)
f is not derivable at x= 1
- (c)
f is not derivable at x=-1
- (d)
f is derivable for all x, with |x| > 1
Let f(x) = x2 + xg' (1) + g" (2) and g(x) = x2 + x f' (2)+ f" (3), then
- (a)
f' (1) = 4 + f' (2)
- (b)
g' (2) = 8 + g' (1)
- (c)
g" (2) + f" (3) = 4
- (d)
none of these