Mathematics - Differential Coefficient
Exam Duration: 45 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
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 \(x=sec\theta -cos\theta ,y={ sec }^{ 10 }\theta -{ cos }^{ 10 }\theta \quad and\quad \left( { x }^{ 2 }+4 \right) \left( \frac { dy }{ dx } \right) ^{ 2 }=k({ y }^{ 2 }+4),\)then k is equal to
- (a)
\(\frac { 1 }{ 100 } \)
- (b)
1
- (c)
10
- (d)
100
If 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
The solution set of f'(x) > g' (x) where f(x) = (1/2) 52x+1 and g(x) = 5x+ 4x loge 5 is
- (a)
\((1,\infty )\)
- (b)
(0,1)
- (c)
\([0,\infty )\)
- (d)
\((0,\infty )\)
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 = \(\left( \frac { ax+b }{ cx+d } \right) \), 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 \(\frac { { d }^{ 2 }y }{ { dy }^{ 2 } } \left( \frac { dy }{ dx } \right) ^{ 3 }+\frac { { d }^{ 2 }y }{ { dy }^{ 2 } } =k\) then k is equal to
- (a)
0
- (b)
1
- (c)
2
- (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
The differential coefficient of f(loge x) W.r.t. x, where f(x) = loge x is
- (a)
x/loge x
- (b)
loge x/x
- (c)
(x loge x)-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 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 = 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
Let f be a function such that f(x + y) = f(x) + f(y)for all x and y and f(x) = (2x2 + 3x)g(x)for all x where g(x) is continuous and g(0) = 3. Then f' (x) is equal to
- (a)
9
- (b)
3
- (c)
6
- (d)
none of these
If sin (x + y) = loge (x + y), then \(\frac { dy }{ dx } \)is equal to
- (a)
2
- (b)
-2
- (c)
1
- (d)
-1
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 f(x) = (1 + x)n, then the value of f(0) + f' (0) +\(\frac { { f }^{ '' }(0) }{ 2! } +....+\frac { { f }^{ n }(0) }{ n! } \) is
- (a)
n
- (b)
2n
- (c)
2n-1
- (d)
none of these
If \(D*f(x)=\underset{h\rightarrow0}{lim}{{f^2(x+h)}-f^2(x)\over h}\)
Where \(f^2(x)=\{f(x)\}^2\)
If u = f(x), v = g(x) then the value of D*\(\{{u\over v}\}\)is
- (a)
\(u^2D*v-v^2D*u\over v^4\)
- (b)
\(uD*v-vD*u\over v^2\)
- (c)
\(u^2D*u-u^2D*v\over v^4\)
- (d)
\(vD*u-v^2D*v\over v^4\)
If \(D*f(x)=\underset{h\rightarrow0}{lim}{{f^2(x+h)}-f^2(x)\over h}\)
Where \(f^2(x)=\{f(x)\}^2\)
The value of D*f(x) at the point on the curve y = f(x) such that tangent to it are parallel to x-axis, then
- (a)
f(x)
- (b)
zero
- (c)
2f(x)
- (d)
xf(x)
If \(D*f(x)=\underset{h\rightarrow0}{lim}{{f^2(x+h)}-f^2(x)\over h}\)
Where \(f^2(x)=\{f(x)\}^2\)
The value of D* c, where c is constant is
- (a)
non-zero constant
- (b)
2 constant
- (c)
does not exist
- (d)
zero
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 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 \(y={In\ x\over 2-3x}\)then the value of yn(1) is
- (a)
0
- (b)
(-1)n- 3n
- (c)
(-1)n- 3n.n!
- (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(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
Differential coefficient of sin-1 x W.r.t. sin-1(3x - 4 x3)i
- (a)
\({1\over 3}\ if\ -{\pi\over 8}<x<{\pi\over 8}\)
- (b)
\(3\ if\ {-\pi\over 8}<x<{\pi\over 8}\)
- (c)
\({1\over 3}\ if\ -{\pi\over 9}<x<{\pi\over 9}\)
- (d)
\(3\ if\ -{\pi\over 9}<x<{\pi\over 9}\)
If y=\(\frac { \sqrt { (1+{ t }^{ 2 }) } -\sqrt { (1-{ t }^{ 2 }) } }{ \sqrt { (1+{ t }^{ 2 }) } +\sqrt { (1-{ t }^{ 2 }) } } and\quad x=\sqrt { (1-{ t }^{ 4 }) } ,\quad then\quad \frac { dy }{ dx } \) is equal to
- (a)
\(\frac { -1 }{ { t }^{ 2 }\{ 1+\sqrt { 1-{ t }^{ 4 } } \} } \)
- (b)
\(\frac { \{ \sqrt { (1-{ t }^{ 4 }) } -1\} }{ { t }^{ 6 } } \)
- (c)
\(\frac { 1 }{ { t }^{ 2 }\{ 1+\sqrt { 1-{ t }^{ 4 } } \} } \)
- (d)
\(\frac { 1-\sqrt { (1-{ t }^{ 4 }) } }{ { t }^{ 6 } } \)