JEE Main Mathematics - Differential Coefficient
Exam Duration: 60 Mins Total Questions : 30
If y=tan-1\(\sqrt { \left( \frac { 1+sin\quad x }{ 1-sin\quad x } \right) } ,\frac { \pi }{ 2 } <x<\pi ,then\frac { dy }{ dx } \)equals
- (a)
-1/2
- (b)
-1
- (c)
1/2
- (d)
1
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
\(If\quad x={ e }^{ y+e...\infty },\quad then\quad \frac { dy }{ dx } is\)
- (a)
\(\frac { 1 }{ x } \)
- (b)
\(\frac { 1-x }{ x } \)
- (c)
\(\frac { x }{ 1+x } \)
- (d)
none of these
If f(x) = sin-1 (sin x) + cos-1 (sin x) and \(\Phi \) (x) = f (f(f(x))), then \(\Phi \)' (x) is equal to
- (a)
1
- (b)
sin x
- (c)
0
- (d)
none of these
If variables x and y are related by the equation \(x=\int _{ 0 }^{ y }{ \frac { 1 }{ \sqrt { (1+{ 9u }^{ 2 }) } } } \) du, then \(\frac { dy }{ dx } \) is equal to
- (a)
\(\frac { 1 }{ \sqrt { (1+{ 9y }^{ 2 }) } } \)
- (b)
\(\sqrt { (1+{ 9y }^{ 2 }) } \)
- (c)
\((1+{ 9y }^{ 2 })\)
- (d)
\(\frac { 1 }{ (1+{ 9y }^{ 2 }) } \)
If P(x) is a polynomial such that P(x2 + 1) = {P(x)}2 + 1 and P(0) = 0, then P' (0) is equal to
- (a)
-1
- (b)
0
- (c)
1
- (d)
none of these
If y = (1 + x) (1 + x2)(1 + x4) .... (1 + x2n),then \(\frac { dy }{ dx } \) at x=0 is
- (a)
0
- (b)
-1
- (c)
1
- (d)
none of these
If xy . yx = 16,then \(\frac { dy }{ dx } \)at (2, 2) is
- (a)
-1
- (b)
0
- (c)
1
- (d)
none of these
If y2 = P(x) is a polynomial of degree 3, then \(2\frac { d }{ dx } \left[ { y }^{ 3 }\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } \right] \)equals
- (a)
P'" (x) + P' x
- (b)
P" (x)· P''' (x)
- (c)
P (x) . P'" (x)
- (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 f(x) = cot-1\(\left( \frac { { x }^{ x }-{ x }^{ -x } }{ 2 } \right) \)then r (1) is
- (a)
-1
- (b)
1
- (c)
log 2
- (d)
-log 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
If y=tan-1 \(\left( \frac { 1 }{ \left( 1+x+{ x }^{ 2 } \right) } \right) +{ tan }^{ -1 }\left( \frac { 1 }{ { x }^{ 2 }+3x+3 } \right) +{ tan }^{ -1 }\left( \frac { 1 }{ { x }^{ 2 }+5x+7 } \right) +...+\) up to n terms. then y' (0) is equal to
- (a)
\(-\frac { 1 }{ 1+{ n }^{ 2 } } \)
- (b)
\(-\frac { { n }^{ 2 } }{ 1+{ n }^{ 2 } } \)
- (c)
\(\frac { n }{ 1+{ n }^{ 2 } } \)
- (d)
none of these
If \(\sqrt { (x+y) } +\sqrt { (y-x) } =1\quad then\quad \frac { { d }^{ 2 }y }{ { dx }^{ 2 } } \) equals
- (a)
2/a
- (b)
-2/a2
- (c)
2/a2
- (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 P(x) be a polynomial of degree 4, with P(2) = - 1, P' (2) = 0, P" (2) = 2, p'" (2) = - 12 and piv (2) = 24, then P" (1) is equal to
- (a)
22
- (b)
24
- (c)
26
- (d)
28
If f(x) = sin \(\left\{ \frac { \pi }{ 3 } [x]-{ x }^{ 2 } \right\} \) for 2 < x < 3 and [x] denotes the greatest integer less than or equal to x, then \(\left( { f }^{ ' }\sqrt { \pi /3 } \right) \) is equal to
- (a)
\(\sqrt { \pi /3 } \)
- (b)
\(-\sqrt { \pi /3 } \)
- (c)
\(-\sqrt { \pi } \)
- (d)
none of these
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 \(y={In\ x\over x}\)then the value of y" (e) is
- (a)
1
- (b)
-1/e
- (c)
-1/e2
- (d)
-1/e3
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 n = 4p+3, p ∈ I and y = tan-1 x, then yn (0) is
- (a)
0
- (b)
n!
- (c)
-(n-1)!
- (d)
(n-1)!
If y=\(\sqrt { x+\sqrt { y+\sqrt { x+\sqrt { y+...\infty , } } } } then\quad \frac { dy }{ dx } \) is equal to
- (a)
\(\frac { 1 }{ 2y-1 } \)
- (b)
\(\frac { { y }^{ 2 }-x }{ { 2y }^{ 3 }-2xy-1 } \)
- (c)
(2y - 1)
- (d)
none of these
If \(\int _{ \pi /2 }^{ x }{ \sqrt { (3-2{ sin }^{ 2 }t) } +\int _{ 0 }^{ y }{ cos\quad t\quad dt=0, } } then\left( \frac { dy }{ dx } \right) _{ \pi ,\pi }\)is
- (a)
-3
- (b)
0
- (c)
\(\sqrt { 3 } \)
- (d)
none of these
If fn(x)=efn-l(X) for all n ∈ N and f0(x)= x, then \({d\over dx}\{f_n(x)\}\) is equal to
- (a)
\(f_n(x).{d\over dx}\{f_{n-1}(x)\}\)
- (b)
fn(x).f(n-1)(x)
- (c)
fn(x).f(n-1)(x)....f2(x).f1(x)
- (d)
\(\prod_{i=1}^nf_i(x)\)
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