Question 1 :
If $f(9)=0, f'(9) = 4$, then $\displaystyle \lim_{ x \rightarrow 9} \dfrac{\sqrt{f(x)} - 3}{\sqrt x - 3}$ $=$
Question 2 : Assertion (A): lf $f(x)=\cos^{2}x+\cos^{2}\left(x+\dfrac{\pi}3\right)- \cos x \cos \left(x+\dfrac{\pi}3\right)$ then $f'(x)=0$<div><br/>Reason(R): Derivative of constant function is zero<br/></div>
Question 3 :
$\lim _{ x\rightarrow { 0 }^{ + } }{ \left( { \left( x\cos { x } \right) }^{ x }+{ \left( \cos { x } \right) }^{ \frac { 1 }{ \ln { x } } }+{ \left( x\sin { x } \right) }^{ x } \right) } $ is equal to<br/>
Question 4 :
$ \lim_{ x \rightarrow 0 } \left\{ \tan \left( \frac { \pi } { 4 } + x \right) \right\} ^ { \frac { 1 } { x } } =$
Question 6 : <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 7 :
If $y=\left | \cos x \right |+\left | \sin x \right |$ then $\frac{dy}{dx}$ at $x=\frac{2\pi }{3}$ is
Question 8 :
Let $f\left( xy \right) =f\left( x \right) \cdot f\left( y \right) $ for all $x,y\in R$. If $f^{ ' }\left( 1 \right) =2$ and $f\left( 4 \right) =4$, then $f^{ ' }\left( 4 \right) $ equal to
Question 9 :
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 10 :
Evaluate $\displaystyle \lim_{x\rightarrow 2}\frac{f\left ( x \right )-f\left ( 2 \right )}{x-2}$ where f(x)=$\displaystyle x^{2}-4$
Question 11 :
$ \displaystyle f:R\rightarrow R $ be such that $ \displaystyle \begin{vmatrix} f\left ( x \right )-f\left ( y \right )\end{vmatrix}^2\leq \begin{vmatrix}x-y\end{vmatrix}^{3} $ for all $ \displaystyle x,y\in R $ then the value of $ \displaystyle {f}'\left ( x \right ) $ is
Question 12 :
$\mathop {\lim }\limits_{x \to {x_1}} \dfrac{x}{{x - {x_1}}}\int_{{x_1}}^x {f\left( t \right)dt} $ is equal to
Question 15 :
$\displaystyle \lim_{x\rightarrow 0}\frac{\left ( x+2 \right )^{10}-2^{10}}{\left ( x+2 \right )^{5}-2^{5}}$ is ______
Question 16 :
<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 18 :
$\displaystyle\frac{dy}{dx}$ for $y=\tan^{-1}\left\{\sqrt{\displaystyle\frac{1+\cos x}{1-\cos x}}\right\}$, where $0 < x < \pi$, is?
Question 19 :
The graph of the function$f(x) = x^{3} + 1$ after translation $4$ units to the right and $2$ units up, resulted in a new graph $l(x)$. What is the value of $l(3.7)$?
Question 20 :
Let $\displaystyle f\left ( \frac{x+y}{2} \right )=\frac{1}{2}\left [ f\left ( x \right )+f\left ( y \right ) \right ]$ for real x and y. If ${f}'\left ( 0 \right )$ exists and equals $-1$ and $f(0)=1$ then the value of $f(2)$ is<br/>
Question 21 :
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 23 :
The value of $\displaystyle\lim _{ x\rightarrow \frac { \pi }{ 2 } }{ \tan ^{ 2 }x{ \left( \sqrt { 2\sin ^{ 2 }{ x } +3\sin { x } +4 } -\sqrt { \sin ^{ 2 }{ x } +6\sin { x } +2 } \right) } }$ is equal to:
Question 24 :
If $\displaystyle \lim_{x\rightarrow m }\frac{x^{3}-m^{3}}{x-m}=3$ then find the number of possible values of $m$.
Question 25 :
$f_n(x)=e^{\displaystyle f_{n-1}(x)}$ for all $n\epsilon N$ and $f_0(x)=x$, then $\displaystyle\frac{d}{dx}\left\{f_n(x)\right\}$ is
Question 26 :
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 27 :
If $\displaystyle y=\sum _{ r=1 }^{ x }{ \tan ^{ -1 }{ \frac { 1 }{ 1+r+{ r }^{ 2 } } } } $ then $\displaystyle \frac { dy }{ dx } $ is equal to
Question 28 :
Suppose, $A=\displaystyle \frac {dy}{dx}$ of $x^2+y^2=4$ at $(\sqrt 2, \sqrt 2), B=\displaystyle \frac {dy}{dx}$ of $sin y+sin x=sin x\cdot sin y$ at $(\pi, \pi)$ and $C=\displaystyle \frac {dy}{dx}$ of $2e^{xy}+e^xe^y-e^x=e^{xy+1}$ at $(1, 1)$, then $(A-B-C)$ has the value equal to .....
Question 31 :
If $3x^{2} - 4xy = 1$, then when $x = 1, \dfrac {dy}{dx} =$
Question 32 :
<span class="wysiwyg-font-size-small">Suppose that $ \displaystyle f $ is a differentiable function with the property that $ \displaystyle f\left ( x+y \right )=f\left ( x \right )+f\left ( y \right )+xy $ and $ \displaystyle \lim_{h\rightarrow 0}\frac{1}{h}f\left ( h \right )=3 $ then</span>
Question 33 : <br/>lf $\displaystyle \mathrm{f}(\mathrm{x})=\ {\mathrm{x}^{2}}+\frac{\mathrm{x}^{2}}{(1+\mathrm{x}^{2})}+\frac{\mathrm{x}^{2}}{(1+\mathrm{x}^{2})^{2}}+\ldots+\frac{\mathrm{x}^{2}}{(1+\mathrm{x}^{2})^{\mathrm{n}}}+\ldots$ <div>then at $x=0$ which of the following is correct?<br/></div>
Question 35 :
$\displaystyle \lim _{ x\rightarrow 2 }{ \frac { { x }^{ 2 } }{ \sqrt { x+2 } -\sqrt { 3x-2 } } } =$
Question 37 :
If for all $x, y$ the function f is defined by; $f(x)+f(y)+f(x)\cdot f(y)=1$ and $f(x) > 0$.When $f(x)$ is differentiable $f'(x)= $,<br>
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 44 :
Assertion: Let f be a differentiable function satisfying $f(x+y)=f(x)+f(y)+2xy-1$ for all $x, y\epsilon R$ and ${f}'\left ( 0 \right )=\sin \phi $. Then f(x)> 0 for all x.
Reason: f in statement is of the form $x^{2}+x\sin \phi +1$.
Question 45 :
Evaluate $\displaystyle \lim_{x\rightarrow 1}\frac{f\left ( x \right )-f\left ( 1 \right )}{x-1}$ where f(x)= $\displaystyle x^{2}-2x$
Question 46 : 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 47 : <div><div>Let $f(x)$ be a real valued function not identically zero, such that</div><div>$f\left ( x+y^{n} \right )=f\left ( x \right )+\left ( f\left ( y \right ) \right )^{n}\quad\forall x, y\in R$</div><div>where $n\in N\left ( n\neq 1 \right )$ and ${f}'\left ( 0 \right )\geq 0$. We may get an explicit form of the function $f(x)$.</div></div><br/>$\displaystyle \int_{0}^{1}f\left ( x \right )dx$ is equal to<br/>
Question 48 :
For every integer $n$, let $a_{n}$ and $b_{n}$ be real numbers. Let function $f: IR \rightarrow IR$ be given by<br>$f(x)=\left\{\begin{array}{l}<br>a_{n}+\sin\pi x,\ for\ x\in[2n,\ 2n+1]\\<br>b_{n}+\cos\pi x,\ for\ x\ \in(2n-1,2n) <br>\end{array}\right.$ <br> $, for\ all\ integers\ n.$<br>lf $f$ is continuous, then which of the following hold(s) for all $n$?<br>
Question 49 :
If $x_{1} = 3$ and $x_{n+1} = \sqrt{2 + x_{n}}, n\ge 1$, then $\displaystyle \lim_{n\to\infty}{x_{n}}$ is
Question 50 :
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
