Question 2 :
Simplify ${\cot ^{ - 1}}\dfrac{1}{{\sqrt {{x^2} - 1} }}$ for $x <  - 1$
Question 4 :
$\quad \sin ^{ -1 }{ x } +\sin ^{ -1 }{ \cfrac { 1 }{ x } } +\cos ^{ -1 }{ x } +\cos ^{ -1 }{ \cfrac { 1 }{ x } = } $
Question 5 :
Solve ${\cos ^{ - 1}}\left( {\frac{4}{5}} \right) + {\cos ^{ - 1}}\left( {\frac{{63}}{{65}}} \right) = $
Question 7 :
The value of $\cos^{-1} (\cos 12) - \sin^{-1} (\sin 12)$ is 
Question 9 :
Solve $\cos { \left[ \tan ^{ -1 }{ \left[ \sin { \left( \cot ^{ -1 }{ x }  \right)  }  \right]  }  \right]  } $
Question 10 :
The number of real values of x satisfying the equation $\tan^{-1}\left(\dfrac{x}{1-x^2}\right)+\tan^{-1}\left(\dfrac{1}{x^3}\right)=\dfrac{3\pi}{4}$, is?
Question 11 :
If $\dfrac {(x + 1)^{2}}{x^{3} + x} = \dfrac {A}{x} + \dfrac {Bx + C}{x^{2} + 1}$, then $\csc^{-1}\left (\dfrac {1}{A}\right ) + \cot^{-1}\left (\dfrac {1}{B}\right ) + \sec^{-1}C =$ ____
Question 12 :
The value of $\sin ^{ -1 }{ \left( \cos { \cfrac { 53\pi }{ 5 } } \right) } $ is
Question 13 :
$ \sin \left( 2 \sin^{-1} \sqrt{\dfrac{63}{65}} \right) $<br/>is equal to :
Question 14 :
The value of $ \cos \left( \sin^{-1} \left( \dfrac {2}{3} \right) \right) $ is equal to :
Question 16 : A word is represented by only one set of numbers as given in any one of the alternatives. The sets of numbers given in the alternatives are represented by two classes of alphabets as in the two matrices given below. The columns and rows of Matrix I are numbered from $0$ to $4$ and that of Matrix II are numbered from $5$ to $9$. A letter from these matrices can be represented first by its row and next by its column, e.g., $'R'$ can be represented by $04, 42$ etc., and $'D'$ can be represented by $57, 76$ etc. Similarly, you have to identify the set for the word 'ROAD'.<table class="wysiwyg-table"><tbody><tr><td></td><td></td><td>Matrix I</td><td></td><td></td><td></td></tr><tr><td></td><td>$0$</td><td>$1$</td><td>$2$</td><td>$3$</td><td>$4$</td></tr><tr><td>$0$</td><td>$F$</td><td>$O$</td><td>$M$</td><td>$S$</td><td>$R$</td></tr><tr><td>$1$</td><td>$S$</td><td>$R$</td><td>$F$</td><td>$O$</td><td>$M$</td></tr><tr><td>$2$</td><td>$O$</td><td>$M$</td><td>$S$</td><td>$R$</td><td>$F$</td></tr><tr><td>$3$</td><td>$R$</td><td>$F$</td><td>$O$</td><td>$M$</td><td>$S$</td></tr><tr><td>$4$</td><td>$M$</td><td>$S$</td><td>$R$</td><td>$F$</td><td>$O$</td></tr></tbody></table><table class="wysiwyg-table"><tbody><tr><td></td><td></td><td>Matrix II</td><td></td><td></td><td></td></tr><tr><td></td><td>$5$</td><td>$6$</td><td>$7$</td><td>$8$</td><td>$9$</td></tr><tr><td>$5$</td><td>$A$</td><td>$T$</td><td>$D$</td><td>$I$</td><td>$P$</td></tr><tr><td>$6$</td><td>$I$</td><td>$P$</td><td>$A$</td><td>$T$</td><td>$D$</td></tr><tr><td>$7$</td><td>$T$</td><td>$D$</td><td>$I$</td><td>$P$</td><td>$A$</td></tr><tr><td>$8$</td><td>$P$</td><td>$A$</td><td>$T$</td><td>$D$</td><td>$I$</td></tr><tr><td>$9$</td><td>$D$</td><td>$I$</td><td>$P$</td><td>$A$</td><td>$T$</td></tr></tbody></table>
Question 17 :
The Inverse of a square matrix, if it exist is unique.
Question 18 : <p>The possibility for the formation of rectangular matrices in the matrix algebra are</p>
Question 19 :
If $A=\begin{bmatrix} { a }^{ 2 } & ab & ac \\ ab & { b }^{ 2 } & bc \\ ac & bc & { c }^{ 2 } \end{bmatrix}$ and ${a}^{2}+{b}^{2}+{c}^{2}=1$ then ${A}^{2}=$
Question 20 :
Construct the matrix of order $3 \times 2$ whose elements are given by $a_{ij} = 2i - j$
Question 21 :
If $\displaystyle  \begin{vmatrix} x & y   \\ 1 & 6   \end{vmatrix} $ = $\displaystyle  \begin{vmatrix} 1 & 8   \\ 1 & 6   \end{vmatrix} $ then x+2y=
Question 22 :
If $A$ and $B$ are square matrices such that $AB = I$ and $BA = I$, then $B$ is<br/>
Question 24 :
If every row of a matrix $A$ contains $p$ elements and its column contains $q$ elements, then the order of $A$ is
Question 26 :
If $A$ and $B$ are two matrices of same order, then $A + B$ is equal to
Question 27 :
If $A$ and $B$ are square matrices such that $B = -A^{-1} BA, \,$ then $\, (A + B)^2$ is equal to 
Question 28 :
If $A+B=\begin{bmatrix} 2 & 3 \\ 4 & 5 \end{bmatrix}$ and $A=\begin{bmatrix} 1 & 2 \\ 0 & 3 \end{bmatrix}$, then matrix $B$ is
Question 29 :
If $3\begin{bmatrix} 4 & 2 \\ 1 & 3 \end{bmatrix}-2\begin{bmatrix} -2 & 1 \\ 3 & 2 \end{bmatrix}+\begin{bmatrix} x & -4 \\ 3 & y \end{bmatrix}=0$ then $\left(x,y\right)=$
Question 30 :
A matrix has $18$ elements. Find the number of possible orders of the matrix
Question 32 :
Given the relation R= {(1,2), (2,3) } on the set {1, 2, 3}, the minimum number of ordered pairs which when added to R make it an equivalence relation is
Question 33 :
In the set $Z$ of all integers, which of the following relation $R$ is an equivalence relation?
Question 34 :
The number of reflexive relation in set A = {a, b, c} is equal to
Question 35 :
If $A=\left\{ 1,2,3 \right\} $, then a relation $R=\left\{ \left( 2,3 \right) \right\} $ on $A$ is
Question 37 :
If $R$ is the largest equivalence relation on a set $A$ and $S$ is any relation on $A$, then
Question 38 :
Find number of all such functions $y = f(x)$ which are one-one?
Question 39 :
Let $R$ be the relation over the set of all straight lines in a plane such that ${l}_{1}$ $R$ ${l}_{2}\Leftrightarrow {l}_{1}\bot {l}_{2}$. Then, $R$ is
Question 40 :
Let N denote the set of all natural numbers and R a relation on $N\times N$. Which of the following is an equivalence relation?
Question 41 :
Let N denote the set of all natural numbers and R a relation on $N\times N$. Which of the following is an equivalence relation?
Question 42 :
For any two real numbers $\theta$ $\phi $, we define $\theta R\phi $ if and only if $\sec ^{ 2 }{ \theta } -\tan ^{ 2 }{ \phi } =1$. The relation $R$ is
Question 43 :
Let $S$ be a relation on $\mathbb{R}^{+}$ defined by $xSy\Leftrightarrow { x }^{ 2 }-{ y }^{ 2 }=2\left( y-x \right)$, then $S$ is
Question 44 :
Let $A = [-1,1]= B $then which of the following functions from $A$ to $B$ is bijective function?
Question 45 :
If function $f$ has an inverse, then which of the following conditions is necessary and sufficient
Question 46 :
If $\begin{vmatrix} x & y\\ 4 & 2 \end{vmatrix}=7$ and $\begin{vmatrix} 2 & 3\\ y & x \end{vmatrix}=4$ then<br>
Question 47 :
The value of the determinant$\begin{vmatrix} 5 & 1 \\ 3 & 2 \end{vmatrix}$
Question 48 :
The determinant $\begin{vmatrix}a & b & a\alpha +b\\ b & c & b\alpha +c\\ a\alpha +b & b\alpha +c & 0\end{vmatrix}$ is equal to zero, if.
Question 49 :
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 & a & b \\ \omega & 1 & c \\ { \omega }^{ 2 } & \omega & 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 50 :
If $A$ is a skew symmetric matrix, then $\left| A \right| $ is
Question 51 :
$A=\begin{bmatrix} 5 & 5a & a \\ 0 & a & 5a \\ 0 & 0 & 5 \end{bmatrix}$ If $\left| A^{ 2 } \right| =25$ then $|a|=$
Question 52 :
If $\omega$ is a non-real cube root of unity and n is not a multiple of 3, then $\displaystyle \Delta =\left | \begin{matrix}<br>1 & \omega^{n} &\omega^{2n} \\ <br>\omega^{2n}&1 &\omega^{n} \\ <br>\omega^{n}&\omega^{2n} &1 <br>\end{matrix} \right |$ is equal to<br>
Question 53 :
$\displaystyle \Delta = \begin{vmatrix}1 & \log_{x}y  & \log_{x}z \\ \log_{y}x & 1 & \log_{y}z \\ \log _{z}x  &\log _{z}y  & 1\end{vmatrix}$ is equal to <br/> <br/>
Question 54 :
$x = \left| \begin{gathered}   - 1\,\,\,\,\,\, - 2\,\,\,\,\,\,\, - 2 \hfill \\  \,\,\,2\,\,\,\,\,\,\,\,\,\,1\,\,\,\,\,\,\, - 2 \hfill \\  \,\,\,2\,\,\,\,\,\, - 2\,\,\,\,\,\,\,\,\,\,1 \hfill \\ \end{gathered}  \right|$, then $x=$
Question 55 :
Find the value of the following determinant:<br/>$\begin{vmatrix}\displaystyle \frac{-4}{7} & \displaystyle \frac{-6}{35}\\ 5 & \displaystyle \frac{-2}{5}\end{vmatrix}$
Question 56 :
If$p{\lambda ^4} + p{\lambda ^3} + p{\lambda ^2} + s\lambda + t = $ $\left| {\begin{array}{*{20}{c}}{{\lambda ^2} + 3\lambda } & {\lambda + 1} & {\lambda + 3}\\{\lambda + 1} & {2 - \lambda } & {\lambda - 4}\\{\lambda - 3} & {\lambda + 4} & {3\lambda }\end{array}} \right|$, then value of t is
Question 57 :
The value of the determinant<br/>$\displaystyle \Delta =\left| \begin{matrix} \log { x }  \\ \log { 2x }  \\ \log { 3x }  \end{matrix}\,\,\,\begin{matrix} \log { y }  \\ \log { 2y }  \\ \log { 3y }  \end{matrix}\,\,\,\begin{matrix} \log { z }  \\ \log { 2z }  \\ \log { 3z }  \end{matrix} \right| $<br/>
Question 58 :
If $f(x)=\begin{vmatrix} 1 & x & x+1\\ 2x & x(x-1) & (x+1)x\\ 3x(x-1) & x(x-1)(x-2) & (x+1)x(x-1)\end{vmatrix}$ then $f(100)$ is equal to?
Question 59 :
If $A + B + C = \pi$, then $\begin{vmatrix}\sin (A + B + C)& \sin B& \cos C\\ -\sin B& 0 & \tan A\\ \cos (A + B)& -\tan A& 0\end{vmatrix}$ is equal to<br>
Question 60 :
If a. b, c are negative and different real numbers then $\displaystyle \Delta=\begin{vmatrix}a &b &c \\ b &c &a \\c &a &b \end{vmatrix}$ is<br>
Question 61 :
If as continuous function 'f' satisfies the realation <br/>$\int_{0}^{t}(f(x) \, - \, \sqrt{f^1(x)})dx \, = \, 0 \, and \, f(0) \, = \, {-1}$<br/>the f(x) is equal to
Question 62 :
If $f(x) = \begin{cases}\dfrac{x^2-(a+2)x+a}{x-2} & x\ne 2\\ 2 & x = 2 \end{cases}$ is continuous at $x = 2$, then the value of $a$ is
Question 63 :
Function $f(x) = \left\{\begin{matrix}(\log_{2}2x)^{\log_{x} 8};& x\neq 1\\(k - 1)^{3};& x = 1\end{matrix}\right.$ is continuous at $x = 1$, then $k =$ _______.<br>
Question 64 :
If the function $\displaystyle f(x)=\begin{cases} 2x^{2}-3 & \text{ if } 0<x\leq 1 \\ x^{2}+bx-1 & \text{ if } 1<x<2 \end{cases}$ is continuous at every point of its domain then the value of $b$ is 
Question 65 :
$f(x)=\left\{\begin{matrix} 2x-1& if &x>2 \\ k & if &x=2 \\  x^{2}-1& if & x<2\end{matrix}\right.$is continuous at $x= 2$ then $k =$
Question 66 :
Following function is continous at the point $x=2$ <br/>          $f\left( x \right) = \left\{ \begin{array}{l}1 + x,\,\,\,\,when\,\,x < 2\\5 - x,\,\,\,\,when\,\,x \ge 2\end{array} \right.$
Question 67 :
If $f(x)=\displaystyle \frac{\sin x}{x},x\neq 0$ is to be continuous at ${x}=0$ then $\mathrm{f}({0})=$<br/>
Question 68 :
The function $f(x)=\dfrac{1-\sin x+\cos x}{1+\sin x+\cos x}$ is not defined at $x=\pi$. The value of $f(\pi)$ so that $f(x)$ is continuous at $x=\pi$ is
Question 70 :
The integer $'n'$ for which $\mathop {\lim }\limits_{x \to 0} \dfrac{{\cos 2x - 1}}{{{x^n}}}$ is a finite non-zero number is
Question 71 :
Let $f(x) = \left\{\begin{matrix} -2,& -3 \leq x \leq 0\\ x - 2,  & x < x \leq 3\end{matrix}\right.$ and $g(x) = f(|x|) + |f(x)|$<br/>Which of the following statements are correct?<br/>1. $g(x)$ is continuous at $x = 0$.<br/>2. $g(x)$ is continuous at $x = 2$.<br/>3. $g(x)$ is continuous at $x = -1$.<br/>Select the correct answer using the code given below
Question 72 :
Given a function $f(x) = \left\{\begin{matrix}-1 & if & x \leq 0\\ ax + b & if & 0 < x < 1\\ 1 & if & x \geq 1\end{matrix}\right.$ where $a, b$ are constants. The function is continuous everywhere.What is the value of $b$?
Question 73 :
If $\displaystyle f\left( x \right)=\begin{cases} \begin{matrix} x+\lambda , & -1<x<3 \end{matrix} \\ \begin{matrix} 4, & x=3 \end{matrix} \\ \begin{matrix} 3x-5, & x>3 \end{matrix} \end{cases}$ is continuous at $x=3$ then tha value of $\lambda$ is
Question 74 :
If $f(x)=\begin{cases} x+2,\quad \quad when\quad x<1 \\ 4x-1,\quad when\quad 1\le x\le 3 \\ { x }^{ 2 }+5,\quad when\quad x>3 \end{cases}$, then correct statement is-
Question 75 :
$f(x)=\dfrac{p+q^{\frac{1}{x}}}{r+s^{\frac{1}{x}}},<br>s<1, q<1,r\neq 0, \mathrm{f}(\mathrm{0})=1$, is left continuous at $x =0$ then<br>
Question 77 :
If x + y = 3 and xy = 2, then the value of$\displaystyle x^{3}-y^{3}$ is equal to
Question 78 :
The corner points of the feasible region determined by the system of linear constraints are $(0, 10),(5, 5), (25, 20)$ and $(0, 30)$. Let $Z = px + qy$, where $p, q > 0$. Condition on $p$ and $q$ so that the maximum of $Z$ occurs at both the points $(25, 20)$ and $(0, 30)$is _______.
Question 79 :
Solving an integer programming problem by rounding off answers obtained by solving it as a linear programming problem (using simplex), we find that
Question 80 : The given table shows the number of cars manufactured in four different colours on a particular day. Study it carefully and answer the question.<br/><table class="table table-bordered"><tbody><tr><td rowspan="2"> <b>Colour</b></td><td colspan="3"><b>   Number of cars manufactured</b></td></tr><tr><td><b> Vento</b></td><td><b> Creta</b></td><td><b>WagonR </b></td></tr><tr><td> Red</td><td> 65</td><td> 88</td><td> 93</td></tr><tr><td> White</td><td> 54</td><td> 42</td><td> 80</td></tr><tr><td> Black</td><td> 66</td><td> 52</td><td> 88</td></tr><tr><td> Sliver</td><td>37</td><td> 49</td><td> 74</td></tr></tbody></table>What was the total number of black cars manufactured?
Question 81 :
The number of points in $\\ \left( -\infty ,\infty \right) $ for which ${ x }^{ 2 }-x\sin { x } -\cos { x } =0$, is
Question 85 :
In linear programming, lack of points for a solution set is said to
Question 86 :
The interval in which the function $x^3$ increases less rapidly than $6x^2+15x+15$ is
Question 87 :
A particle moves along a curve so that its coordinates at time $t$ are $\displaystyle x = t, y = \frac{1}{2} t^{2}, z =\frac{1}{3}t^{3}$ acceleration at $ t=1 $ is<br>
Question 88 :
If the radius of a sphere is measured as $8\ cm$ with a error of $0.03\ cm$, then the approximate error calculate its volume is
Question 89 :
If the distance $s$ travelled by a particle in time $t$ is given by $s=t^2-2t+5$, then its acceleration is
Question 90 :
A particle moves along a straight line according to the law $s=16-2t+3t^3$, where $s$ metres is the distance of the particle from a fixed point at the end of $t$ second. The acceleration of the particle at the end of $2$ s is
Question 91 :
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 
Question 92 :
What is the rate of change of the area of a circle with respect to its radius $r$ at $r = 6$ $cm$.
Question 93 :
The area of an equilateral triangle of side $'a'$ feet is increasing at the rate of $4 sq.ft./sec$. The rate at which the perimeter is increasing is
Question 94 :
The rate of change of surface area of a sphere of radius $r$ when the radius is increasing at the rate of $2 cm/sec$ is proportional to
Question 95 :
What is the rate of change $\sqrt{x^2 + 16 }$ with respect to $x^2$ at x = 3 ?
Question 96 : <p>A particle moves in a straight line with a velocity given by $\displaystyle \frac{{dx}}{{dt}} = x + 1$  <span class="wysiwyg-font-size-medium">(x is the distance described). The time taken by a particle to traverse a distance of 99 metres is:<br/></p>
Question 97 :
Half of a large cylindrical tank open at the top is filled with water and identical heavy spherical balls are to be dropped into the tank without spilling water out. If the radius and the height of the tank are equal and each is four times the radius of a ball, what is the maximum number of balls that can be dropped?
Question 98 :
A cylindrical tank of radius $10m$ is being filled with wheat at the rate of $314$ cubic metre per hour. then the depth of the wheat is increasing at the rate of
Question 99 :
The side of a square is increased by $20 \% $ . Find the $ \% $ change in its area.
Question 100 :
A dynamite blast blows a heavy rock straight up with a launch velocity of $160$  $m/sec$. It reaches a height of $s= 160t\, -\, 16t^{2}$ after $t$ $sec$. The velocity of the rock when it is $256$ $m$ above the ground on the way up is<br/>
