Question 1 :
Consider $x = 4\tan^{-1}\left (\dfrac {1}{5}\right ), y = \tan^{-1} \left (\dfrac {1}{70}\right )$ and $z = \tan^{-1}\left (\dfrac {1}{99}\right )$.What is $x$ equal to?

Question 2 :
Calculate the value of $\displaystyle \sin^{-1} \cos \left ( \sin^{-1} x\right ) + \cos^{-1} \sin \left ( \cos^{-1} x \right ) $. where $\displaystyle\left | x \right | \leq 1$

Question 3 :
What is $\tan ^{ -1 }{ \left( \dfrac { 1 }{ 2 } \right) } +\tan ^{ -1 }{ \left( \dfrac { 1 }{ 3 } \right) } $ equal to?

Question 4 :
Consider the following :<br>1. ${\sin}^{-1}\dfrac{4}{5}+{\sin}^{-1}\dfrac{3}{5}=\dfrac{\pi}{2}$<br>2. ${\tan}^{-1}\sqrt{3}+{\tan}^{-1}1=-{\tan}^{-1}(2+\sqrt{3})$<br>Which of the above is/are correct?

Question 5 :
Let $A=\left\{ 2,3,4,5,....,17,18 \right\} $. Let $\simeq $ be the equivalence relation on $A\times A$, cartesian product of $A$ with itself, defined by $(a,b)\simeq (c,d)$, iff $ad=bc$. The the number of ordered pairs of the equivalence class of $(3,2)$ is

Question 6 :
If the relation is defined on $R-\left\{ 0 \right\} $ by $\left( x,y \right) \in S\Leftrightarrow xy&gt;0$, then $S$ is ________

Question 7 :
Let $E=\{1, 2, 3, 4\}$ and $F=\{1, 2\}$ then the number of onto functions from E to F is

Question 8 :
a R b if "a and b are animals in different zoological parks" then R is

Question 10 :
If $A=\begin{bmatrix} { a }^{ 2 } &amp; ab &amp; ac \\ ab &amp; { b }^{ 2 } &amp; bc \\ ac &amp; bc &amp; { c }^{ 2 } \end{bmatrix}$ and ${a}^{2}+{b}^{2}+{c}^{2}=1$ then ${A}^{2}=$

Question 11 :
If $\bigl(\begin{smallmatrix} 3x+ 7&amp; 5 \\ y + 1 &amp; 2 - 3x\end{smallmatrix}\bigr) = \bigl(\begin{smallmatrix}1 &amp; y - 2 \\ 8 &amp; 8\end{smallmatrix}\bigr)$ then the values of x and y respectively are<br>

Question 12 :
If AB = 0, then for the matrices $A=\begin{bmatrix} { \cos }^{ 2 }\theta&#160; &amp; \cos\theta \sin\theta&#160; \\ \cos\theta \sin\theta&#160; &amp; { \sin }^{ 2 }\theta&#160; \end{bmatrix}\ and \ B=\begin{bmatrix} { \cos }^{ 2 }\phi&#160; &amp; \cos\phi \sin\phi&#160; \\ \cos\phi \sin\phi&#160; &amp; { \sin }^{ 2 }\phi&#160; \end{bmatrix}, \theta - \phi$ is

Question 13 :
If$\displaystyle a_{ij}=0\left ( i\neq j \right )$ and$\displaystyle a_{ij}=1\left ( i= j \right )$ then the matrix A=$\displaystyle \left [ a_{ij} \right ]_{n\times n}$ is a _____ matrix

Question 14 :
<br/>The function $f(x)=\begin{cases}(1+x)^{\frac{5}{x}}&amp; for \ {x}\neq 0,\\ e^{5} &amp;&#160;for \&#160; {x}=0\end{cases}$ is ........&#160;at ${x}=0$<br/>

Question 15 :
If $f(x) = (1 + 2x)^{\frac{1}{x}}$, for $x \neq 0$ is continuous at $x = 0$, then $f(0) = $_______.

Question 17 :
If function $f\left( x \right) =\begin{cases} x\sin { \left( \dfrac { 1 }{ x } \right) } ;\quad x\neq 0 \\ \quad \quad \quad \quad a;\quad x=0 \end{cases}$ is continuous at $x=0$, then the value of $a$ is

Question 19 :
Let $\omega\neq{1}$ be a cube root of unity and $S$ be the set of all non-singular matrices of the form $ \begin{bmatrix} 1 &amp; a &amp; b \\ \omega &amp; 1 &amp; c \\ { \omega }^{ 2 } &amp; \omega &amp; 1 \end{bmatrix}$Where each of $a,\ b$ and $c$ is either $\omega$ or ${\omega}^{2}$. Then the number of distinct matrices in the set $S$ is

Question 20 :
If A $=\begin{bmatrix}<br>0 &amp; c &amp;-b \\ <br> -c&amp; 0&amp; a\\ <br>b &amp; -a &amp; 0<br>\end{bmatrix}$then$\left ( a^{2}+b^{2}-c^{2} \right )\left | A \right |=$

Question 21 :
$\left|\begin{array}{lll}<br>1 &amp; \omega &amp; \omega^{2}\\<br>\omega &amp; \omega^{2} &amp; 1\\<br>\omega^{2} &amp; 1 &amp; \omega<br>\end{array}\right|=\ldots$..<br>

Question 22 :
Let $\begin{vmatrix} x &amp; 2 &amp; x \\ { x }^{ 2 } &amp; x &amp; 6 \\ x &amp; x &amp; 6 \end{vmatrix}=A{ x }^{ 4 }+B{ x }^{ 3 }+C{ x }^{ 2 }+Dx+E$. Then the value of $5A+4B+3C+2D+E$ is equal to<br>

Question 23 :
$\begin{pmatrix} 6i &amp; -3i &amp; 1 \\ 4 &amp; 3i &amp; -1 \\ 20 &amp; 3 &amp; i \end{pmatrix}=x+iy$

Question 24 :
A stone is dropped into a quiet lake and waves move in circles at the speed of $5$ cm/s At the instant when the radius of the circular wave is $8$ cm how fast is the enclosed area increasing?<br/>

Question 25 :
If the displacement of a particle moving in straight line is given by $x=3t^2+2t+1$ at time $t$ then&#160; the acceleration of the particle at time $t=3$ is

Question 26 :
What is the rate of change $\sqrt{x^2 + 16 }$ with respect to $x^2$ at x = 3 ?

Question 27 :
The radius of a circular plate is increased at $ 0.01 \text {cm/sec}.$ If the area is increased at the rate of $\frac{\pi }{{10}}$. Then its radius is&#160;