Question 1 :
$f\left( x \right)\, = \mathop {\lim }\limits_{n \to \infty } \sum\limits_{k = 1}^n {\dfrac{n}{{{n^2} + {k^2}{x^2}}}} $, $x > 0$ is equal to
Question 2 :
$\lim _ { x \rightarrow 0 } \dfrac { 2 ^ { x } - 1 } { \sqrt { 1 + x } - 1 }$ equals
Question 5 :
$\displaystyle \frac{d}{dx}[f(x)\cdot g(x)] =f(x) \frac{d}{dx}g(x)+g(x) \frac{d}{dx}f(x)$ is known as _____ rule.
Question 6 :
The value of $\displaystyle \lim_{x\rightarrow 2}\int_{2}^{x} \dfrac {3t^{2}}{x - 2}dt$ is
Question 8 :
What is $\displaystyle \lim _{ x\rightarrow 0 }{ \cfrac { \sqrt { 1+x } -1 }{ x } } $ equal to?
Question 10 :
Consider the differential equation $\frac { d y } { d x } = \cos x$ Then we observe that <br/>
Question 12 :
$\displaystyle \frac{d}{dx}\left ( \tan ^{-1}\left ( \frac{a-x}{1+ax} \right ) \right )$ equals if ax > -1
Question 14 :
What is $\displaystyle \lim_{x \rightarrow -2 } \begin{pmatrix} \dfrac{x+2}{x^3 + 8} \end{pmatrix} $ equal to ?
Question 16 :
The velocity of a particle is given by $v = 12 + 3(t + 7t^{2})$. What is the acceleration of the particle?
Question 17 : Say true or false.<div>The derivative of a constant function is always non-zero.</div>
Question 22 :
If $f(x)=\left\{\begin{matrix}<br>4x, & x < 0\\ <br>1, & x=0\\<br>3x^2, & x > 0<br>\end{matrix}\right.$ then $\displaystyle \lim_{x\rightarrow 0}f(x)$ equals<br>
Question 23 :
If $\displaystyle f\left ( x \right )=\left ( \frac{\sin ^{m}x}{\sin ^{n}x} \right )^{m+n}.\left ( \frac{\sin ^{n}x}{\sin ^{p}x} \right )^{n+p}.\left ( \frac{\sin ^{p}x}{\sin ^{m}x} \right )^{p+m}$ then $\displaystyle f'\left ( x \right )$ is equal to-
Question 24 : Obtain the differential equation whose solution is<div>$\displaystyle y=x\sin \left ( x+A \right ),$ A being constant.</div>
Question 25 :
$\underset{x \rightarrow 0}{Lt}\dfrac{\sqrt{3 + x^5} - \sqrt{3 - x^5}}{\sin x} =$
Question 26 :
$\displaystyle \lim_{x\rightarrow \infty}\frac {\sqrt {x^2+1}-\sqrt [3]{x^2+1}}{\sqrt [4]{x^4+1}-\sqrt [5]{x^4-1}}$ is equal to<br>
Question 27 :
$If\space f(x+y) = f(x) + f(y) +2xy - 6 for\space all\space x,y\space in\space R\space and\space f '(0)=2\space then\space y = f(x)\space will\space be$	<br/>
Question 28 :
If $\displaystyle y=\frac { x }{ a+\displaystyle\frac { x }{ b+\displaystyle\frac { x }{ a+\displaystyle\frac { x }{ b+.....\infty } } } } $, then $\cfrac{dy}{dx} =$<br><br>
Question 30 :
Find the value of $\lim_{x \rightarrow 0} \dfrac{2x^2 + 3x + 4}{2}$
Question 32 :
If $\displaystyle f(x)=|\cos x|$ then $f'\left ( \frac{3\pi }{4} \right )$ is equal to-
Question 33 :
$\underset { x\rightarrow 0 }{ \lim } { \left( 1+\dfrac { 2 }{ { x }^{ 2 } } \right) }^{ { x }^{ 2 } }=$
Question 34 :
The value of $\displaystyle \lim_{x \rightarrow 2} \frac{\sqrt{1 + \sqrt{2 + x}} - \sqrt 3}{x- 2}$ is
Question 36 :
The value of the limit $\displaystyle\lim _{ x\rightarrow 1 }{ \dfrac { \sin { \left( { e }^{ x-1 }-1 \right) } }{ \log { x } } } $ is
Question 39 :
Find $\dfrac{{dy}}{{dx}}$ of the following $y = 1 + 2x + 3{x^2} + \left( {n - 1} \right){x^{n - 2}}$<br/><br/>
Question 40 :
Let $f(x + y) = f(x) \times f(y)$ for all $x$ and $y$ and $f(7) = 5, f'(0) = 2$ then $f'(7)$ will be
Question 41 : <div>State whether the given statement is True or False.</div>Derivative of $y=2x^5$ with respect to $x$ is $10x^4$.<br/>
Question 43 :
If function $f(x)=\cfrac{{x}^{3}-{a}^{3}}{x-a}$, is continuous at $x=a$ then the value of $f(a)$ is-
Question 48 : Say true or false.<div>If $y=2 \sec x$, then $\dfrac{dy}{dx} $ is $2 \sec x \tan x$.</div>
Question 49 :
If $y = \left( 1 + x ^ { 1 / 4 } \right) \left( 1 + x ^ { 1 / 2 } \right) \left( 1 - x ^ { 1 / 4 } \right) ,$ then $\frac { d y } { d x } =$
Question 51 :
The value of $\lim _{ x\rightarrow 1 }{ \dfrac { x+{ x }^{ 2 }+{ x }^{ 3 }+......{ x }^{ n } }{ x-1 } }$ is
Question 52 :
Function f(x)=$\left| {x - 2} \right| - 2\left| {x - 4} \right|\,is$ discontinous at:
Question 53 :
$\lim\limits_{x \to 1-} \frac{\sqrt{\pi} - \sqrt{2\sin^{-1}x}}{\sqrt{1 - x}}$ is equal to
Question 54 :
For the function $f(x) = \displaystyle \frac{x^{100}}{100} + \frac{x^{99}}{99} + ........... + \frac{x^2}{2} + x+1$, $f'(1) =$<br/>
Question 55 :
If the function $f(x)$ defined by $f(x) = \dfrac {x^{100}}{100} + \dfrac {x^{99}}{99} + ..... + \dfrac {x^{2}}{2} + x + 1$, then $f'(0) =$
Question 56 :
If $\underset { x\rightarrow 0 }{ lim } \dfrac { \sin 2x+\cos 2x }{ { x }^{ 3 }+5 }=k $ then $k=$
Question 57 :
$\lim_{x\rightarrow \infty} \left( x^3 \displaystyle \int_{-1/x}^{1/x} \dfrac{ln(1+t^2)}{1+e^t}dt \right )$ is equal to
Question 60 :
If $y=|\cos x|+|\sin x|$, then $\displaystyle \dfrac {dy}{dx}$ at $x=\dfrac {2\pi}{3}$ is
Question 61 :
If $f'(x)=\sin x+\sin 4x\cdot \cos x$, then $f'\left (2x^2+\displaystyle \frac {\pi}{2}\right )$ is
Question 62 : The value of $\displaystyle\lim_{x\to0}\frac{(\tan(\left \{ x \right \}-1))\sin\left \{ x \right \}}{\left \{ x \right \}(\left \{ x \right \}-1)}$ is given by :<div> where $\left \{ x \right \}$ denotes the fractional part function</div>
Question 63 :
The value of $\displaystyle \lim_{x\to2}{\displaystyle \frac{\sqrt{1 + \sqrt{2 + x}} - \sqrt{3}}{x - 2}}$ is
