Question 1 :
Let f be a twice differentiable such that $f''(x)=-f(x)$ and $f'(x)=g(x)$. If $h(x)=\left \{f(x)\right \}^2+\left \{g(x)\right \}^2$, where $h(5)=11$. Find $h(10)$
Question 2 :
Let $f(x) = \sqrt{x - 1} + \sqrt{x + 24 - 10\sqrt{x - 1}}, 1 \le x \le 26$ be a real valued function, then $f'(x)$ for $1 < x < 26$ is
Question 4 :
Suppose $A=\displaystyle \frac{dy}{dx}$ when $x^2+y^2=4$ at $(\sqrt{2},\sqrt{2})$,$ B=\displaystyle \frac{dy}{dx}$ when $\sin y+ \sin x=\sin x-\sin y$ at $(\pi,\pi)$ and $C=\displaystyle \frac{dy}{dx}$ when $2e^{xy}+e^x e^y-e^x-e^y=e^{xy+1}$ at $(1,1)$, then $(A+B+C)$ has the value equal to
Question 5 :
If $y=\left | \cos x \right |+\left | \sin x \right |$ then $\frac{dy}{dx}$ at $x=\frac{2\pi }{3}$ is
Question 6 :
If the prime sign (') represents differentiation w.r.t. $x$ and $f^{'}=\sin x+\sin 4x.\cos x$, then $f^{'}\left ( 2x^{2}+\cfrac{\pi }{2} \right )$ at $x=\sqrt{\dfrac{\pi }{2}}$ is equal to
Question 7 :
Let f and g be differentiable function such that ${f}'\left ( x \right )=2g\left ( x \right )$ and ${g}'\left ( x \right )=-f\left ( x \right )$, and let $T\left ( x \right )=\left ( f\left ( x \right ) \right )^{2}-\left ( g\left ( x \right ) \right )^{2}$. Then ${T}'\left ( x \right )$ is equal to<br/>
Question 8 :
Assertion: $\displaystyle f\left ( x \right )=\sin ^{2}x+\sin ^{2}\left ( x+\frac{\pi }{3} \right )+\cos x\cos \left ( x+\frac{\pi }{3} \right )$ then ${f}'\left ( x \right )=0$
Reason: Derivative of a constant function is always zero
Question 9 :
If $\displaystyle y=\sum _{ r=1 }^{ x }{ \tan ^{ -1 }{ \frac { 1 }{ 1+r+{ r }^{ 2 } } } } $ then $\displaystyle \frac { dy }{ dx } $ is equal to
Question 11 :
$f:R\rightarrow R$ and $f(x)=2ax+sin 2x$, then the set of values of a for which $f(x)$ is one-one and onto is
Question 12 :
The equation ${ \left( x-n \right) }^{ m }+{ \left( x-{ n }^{ 2 } \right) }^{ m }+{ \left( x-{ n }^{ 3 } \right) }^{ m }+...+{ \left( x-{ n }^{ m } \right) }^{ m }=0$ $(m$ is odd positive integer$),$ has
Question 13 :
Let $f$ be a differential function such that $f(x)=f(4-x)$ and $g(x)=f(2+x)$ for all $x\in R,$ then
Question 14 :
If $y\left ( n \right )=e^{x}e^{x^{2}}...e^{x^{n}}, 0< x< 1$. Then $\displaystyle \lim_{n\rightarrow \infty }\frac{dy\left ( n \right )}{dx}$ at $x=\dfrac12$ is<br/>
Question 15 :
If $y=\cos 2x\cos 3x$, then $y_n$ is equal to $\\$<br/>Where, $y_n$ denotes the $n^{th} $ derivative of $y$.<br/>
Question 16 :
$f_n(x)=e^{\displaystyle f_{n-1}(x)}$ for all $n\epsilon N$ and $f_0(x)=x$, then $\displaystyle\frac{d}{dx}\left\{f_n(x)\right\}$ is
Question 18 :
If for a continuous function $f,f(0)=f(1)=0,f'(1)=2$ and $g\left( x \right)=f\left( { e }^{ x } \right) { e }^{ f\left( x \right) }$, then $g'(0)$ is equal to
Question 20 :
If ${ e }^{ x }\left( 1+x \right) \sec ^{ 2 }{ x{ e }^{ x } } dx =f\left( x \right) +$ constant, then $f\left( x \right) $ is equal to
Question 21 :
If $\displaystyle \int f(x) dx = \frac {3}{55} \sqrt[3]{\tan^5 x} (5 \tan^2 x + 11) + C$ then $f(x)$ is equal to
Question 22 : <p>The slope(s) of common tangent(s) to the curves $ \displaystyle y={ e }^{ -x }$ and $ \displaystyle y={ e }^{ -x }\sin { x } $ can be -</p>
Question 23 :
The derivative of $\sin ^{ 2 }{ x } $ with respect to $\cos ^{ 2 }{ x } $ is
Question 24 : A function f: $R\rightarrow R$ satisfies $\sin x \cos y [f(2x+2y)- f(2x-2y)]= \cos x \sin y[f(2x+2y)+f(2x-2y)]$. <div>If $f'(0)=\dfrac {1}{2}$, then<br/></div>
Question 26 :
If $y= \displaystyle \frac{x^4-x^2+1}{x^2+\sqrt{3}x+1}$ and $\displaystyle \frac{dy}{dx}=ax+b$ then the value of $a+b$ is equal to
Question 27 :
If $\displaystyle (1-y)'''\cdot (1+y)''=1+a_{1}y+a_{2}y^{2}+...+a_{m+n}y^{m+n}$ where $m\:\in\:N$ and $a_{1}=a_{2}=10$, then $(m,n)$ is
Question 29 :
If $f(x) = \log_e\left\{\displaystyle\frac{u(x)}{v(x)}\right\}, u(1) = v(1)$ and $u'(x) = v'(x) = 2$, then $f'(1)$ is equal to
Question 30 :
Assertion: If $y=\left( 1+x \right) \left( 1+{ x }^{ 2 } \right) \left( 1+{ x }^{ 4 } \right) ...\left( 1+{ x }^{ { 2 }^{ n } } \right) $, then $\displaystyle \frac { dy }{ dx } $ at $x=0$ is $1$
Reason: $\displaystyle y=\dfrac { 1-{ x }^{ { 2 }^{ n+1 } } }{ 1-x } $
Question 31 :
If $y=\displaystyle \frac {(a-x)\sqrt {a-x}-(b-x)\sqrt {x-b}}{\sqrt {a-x}+\sqrt {x-b}}$, then find $\displaystyle \frac {dy}{dx}$ wherever defined.
Question 32 :
If $y=\left ( 1+x \right )\left ( 1+x^{2} \right )\left ( 1+x^{4} \right )...\left ( 1+x^{2^{n}} \right )$, then $\cfrac{dy}{dx}$ at $x=0$ is
Question 33 :
Two functions $f$ & $g$ have first & second derivatives at $x = 0$ & satisfy the relations, $f\left( 0 \right) = \displaystyle\frac { 2 }{ g\left( 0 \right) } , { f }^{ \prime }\left( 0 \right) = 2 { g }^{ \prime }\left( 0 \right) = 4 g\left( 0 \right) , { g }^{ \prime \prime }\left( 0 \right) = 5 { f }^{ \prime \prime }\left( 0 \right) = 6 f\left( 0 \right) = 3$ then
Question 34 :
If $f(x)=\cos x\cdot \cos 2x\cdot \cos 4x\cdot \cos 8x\cdot \cos 16x$, then $f'\left (\dfrac {\pi}{4}\right )$ is
Question 35 :
The Function $f(x)=\displaystyle \frac{1-\cos x(\cos2x)^{\displaystyle \frac{1}{2}}(\cos 3x)^{\displaystyle \frac{1}{3}}}{x^2}$ is not defined at $x=0$. If $f(x)$ is continuous at $x=0$ then $f(0)$ equals
Question 36 :
If $f(x)=\cos x\cdot \cos2x \cdot \cos4x \cdot \cos8x \cdot \cos16x$, then $f'(\pi/4)$ equals
Question 39 :
If $y= \tan x .\ tan 2x .\ tan 3x$, then $\frac {dy}{dx}$ has the value equal to
Question 40 :
<br/> Let $\mathrm{f}(\mathrm{x})=\left\{\begin{array}{l}\mathrm{x}^{\mathrm{n}}\sin\frac{1}{\mathrm{x}},\quad \mathrm{x}\neq 0\\0, \quad \mathrm{x}=0\end{array}\right.$ , then f(x) is continuous but not differentiable at x=0 if
Question 41 :
lf $ \displaystyle \mathrm{f}(\mathrm{x})=\mathrm{x}.\sin \frac{1}{x}$ for $x \neq 0,\ \mathrm{f}(\mathrm{0})=0$ then?<br/>
Question 42 :
For $n\epsilon N$, let $f\left ( x \right )=min\:\left \{ 1-\tan ^{n}x, 1-\sin ^{n}x, 1-x^{n} \right \}$, $x\epsilon \left ( -\cfrac {\pi}{2}, \cfrac {\pi}{2} \right )$. The left hand derivative of $f$ at $x=\cfrac {\pi}{4}$ is<br/>
Question 43 :
The derivative of $f\left( \tan { x } \right) $ with respect to $g(\sec x)$ at $\quad x=\cfrac { \pi }{ 4 } $, where $f'(1)=2;\quad g'(\sqrt { 2 } )=4$ is
Question 45 :
$\displaystyle\frac{dy}{dx}$ for $y=\tan^{-1}\left\{\sqrt{\displaystyle\frac{1+\cos x}{1-\cos x}}\right\}$, where $0 < x < \pi$, is?
Question 47 :
$\cfrac { d }{ dx } \left( 3\cos { \left( \cfrac { \pi }{ 6 } +{ x }^{ 0 } \right) } -4\cos ^{ 3 }{ \left( \cfrac { \pi }{ 6 } +{ x }^{ 0 } \right) } \right) =....\quad $
