Question 1 :
The product of a matrix and its transpose is an identity matrix. The value of determinant of this matrix is
Question 2 :
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 3 :
If $A$ is any skew-symmetric matrix of odd order then $\left| A \right| $ equals
Question 4 :
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 5 :
If $\begin{vmatrix} 6i & -3i & 1\\ 4 & 3i & -1 \\ 20 & 3 & i\end{vmatrix}=x+iy$, then<br>
Question 6 :
If A $=\begin{bmatrix}<br>0 & c &-b \\ <br> -c& 0& a\\ <br>b & -a & 0<br>\end{bmatrix}$then$\left ( a^{2}+b^{2}-c^{2} \right )\left | A \right |=$
Question 7 :
$\begin{vmatrix} 2^3 & 3^3 & 3.2^2+3.2+1\\ 3^3 & 4^3 & 3.3^2+3.3+1\\ 4^3 & 5^3 & 3.4^2+3.4+1\end{vmatrix}$ is equal to?
Question 8 :
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 9 :
Find the value of the following determinant:<br/>$\begin{vmatrix}3\sqrt{6} & -4\sqrt{2}\\ 5\sqrt{3} & 2\end{vmatrix}$
Question 10 :
If $\begin{bmatrix} x & 1 & 1\\ 2 & 3 & 4\\ 1 & 1 & 1\end{bmatrix}$ has no inverse, then $x=$
Question 11 :
Simplify $\triangle = $ <br> $<br/>\left |<br/>\begin{array}{111}<br/>1 & sin3\theta & sin^3\theta \\<br/> 2cos\theta& sin6\theta & sin^32\theta \\<br/>4cos^2\theta & sin9\theta & sin^33\theta \\<br/>\end {array}<br/>\right |<br/>$ equals
Question 12 :
If a$\neq$ b $\neq$ c are all positive, then the value of the determinants $\begin{vmatrix} a & b & c \\ b & c & a\\ c & a & b\end{vmatrix}$ is.<br>
Question 13 :
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 14 :
If $A$ and $B$ are squares matrices such that $A^{2006} = O$ and $AB = A + B$, then $det (B)$ equals
Question 15 :
If $m = \begin{bmatrix}1 & 0\\ 0 & 1\end{bmatrix}$ and $n = \begin{bmatrix}0 & 1 \\ -1 & 0\end{bmatrix}$, then what is the value of the determinant of $m \cos \theta - n \sin \theta$?
Question 16 :
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 17 :
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 18 :
$f(x)=\left\{\begin{matrix} 2x-1& if &x>2 \\ k & if &x=2 \\  x^{2}-1& if & x<2\end{matrix}\right.$is continuous at $x= 2$ then $k =$
Question 19 :
If $f(x)=\displaystyle\frac{\log(1+ax)-log(1-bx)}{x}$ for $x\neq 0$ and $f(0)=k$ and $f(x)$ is continuous at $x=0$, then $k$ is equal to:<br>
Question 20 :
<br/>The function $f(x)=\begin{cases}(1+x)^{\frac{5}{x}}& for \ {x}\neq 0,\\ e^{5} & for \  {x}=0\end{cases}$ is ........ at ${x}=0$<br/>
Question 21 :
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 22 :
If $ f\left( x \right) =\begin{cases} 1 & x<0 \\ { x }^{ 2 } & x\ge 0 \end{cases}$ then at $x=0$
Question 23 :
If $f(x) = (1 + 2x)^{\frac{1}{x}}$, for $x \neq 0$ is continuous at $x = 0$, then $f(0) = $_______.
Question 24 :
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 25 :
If $f\left( x \right) =\dfrac { 3sinx-sin\left( 3x \right) }{ { 2x }^{ 3 } } ,x\neq 0,f\left( 0 \right) =2,$ at $x=0,f$ is
Question 27 :
Let $f$ be a non-zero real valued continuous function satisfying $f(x + y) = f(x). f(y)$ for all $x, y$ $\epsilon$ $R$. If $f(2) = 9$ then $f(6) =$
Question 28 :
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 29 :
If $f(x) = \left\{\begin{matrix}\dfrac {A + 3\cos x}{x^{2}},& if\ x < 0\\ B\tan \left (\dfrac {\pi}{[x + 3]}\right ),& if\ x \geq 0\end{matrix}\right.$ Where $[.]$ represents the greatest integer function, is continuous at $x = 0$ Then.<br>
Question 30 :
If f is defined by $f(x) = \left\{\begin{matrix} x, for \ 0 \le x < 1  \\ 2 - x, for \ x \ge 1\end{matrix}\right.$ , then at $X = 1$, is Discuss the nature of the function
Question 32 :
$\sin ^ { - 1 } 5 + \cos ^ { - 1 } 5 = \ldots \ldots$
Question 33 :
The value of $\cos^{-1} (\cos 12) - \sin^{-1} (\sin 12)$ is 
Question 34 :
Simplify ${\cot ^{ - 1}}\dfrac{1}{{\sqrt {{x^2} - 1} }}$ for $x <  - 1$
Question 35 :
The value of $\sin^{-1} \left( \dfrac{2 \sqrt 2}{3} \right ) + \sin^{-1} \left( \dfrac{1}{3}\right )$ is equal to
Question 36 :
${ tan }^{ -1 }x+{ tan }^{ -1 }y={ tan }^{ -1 }\dfrac { x+y }{ 1-xy } $,      $xy<1$<br/>                                    $=\pi +{ tan }^{ -1 }\dfrac { x+y }{ 1-xy } $,      $xy>1$.<br/> Evaluate:  ${ tan }^{ -1 }\dfrac { 3sin2\alpha  }{ 5+3cos2\alpha  } +{ tan }^{ -1 }\left( \dfrac { tan\alpha  }{ 4 }  \right) $<br/>                                  where $-\dfrac { \pi  }{ 2 } <\alpha <\dfrac { \pi  }{ 2 } $
Question 37 :
Solve ${\cos ^{ - 1}}\left( {\frac{4}{5}} \right) + {\cos ^{ - 1}}\left( {\frac{{63}}{{65}}} \right) = $
Question 38 :
$\quad \sin ^{ -1 }{ x } +\sin ^{ -1 }{ \cfrac { 1 }{ x } } +\cos ^{ -1 }{ x } +\cos ^{ -1 }{ \cfrac { 1 }{ x } = } $
Question 43 :
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 44 :
Solve:$\displaystyle \sin { \left( { \tan }^{ -1 }x \right) } ,\left| x \right| <1$ is equal to
Question 46 :
What is $\sin { \left[ \sin ^{ -1 }{ \left( \cfrac { 3 }{ 5 } \right) } +\sin ^{ -1 }{ \left( \cfrac { 4 }{ 5 } \right) } \right] } $ equal to?
Question 47 :
The value of $ \cos \left( \sin^{-1} \left( \dfrac {2}{3} \right) \right) $ is equal to :
Question 48 :
What is $\tan ^{ -1 }{ \left( \dfrac { 1 }{ 2 } \right) } +\tan ^{ -1 }{ \left( \dfrac { 1 }{ 3 } \right) } $ equal to?
Question 49 :
If $\sin ^{ -1 }{ \left( \cfrac { x }{ 13 } \right) } +co\sec ^{ -1 }{ \left( \cfrac { 13 }{ 12 } \right) } =\cfrac { \pi }{ 2 } $, then the value if $x$ is
Question 50 :
$ \sin \left( 2 \sin^{-1} \sqrt{\dfrac{63}{65}} \right) $<br/>is equal to :
Question 52 :
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 54 :
The number of points in $\\ \left( -\infty ,\infty \right) $ for which ${ x }^{ 2 }-x\sin { x } -\cos { x } =0$, is
Question 55 :
In linear programming, lack of points for a solution set is said to
Question 56 :
If x + y = 3 and xy = 2, then the value of$\displaystyle x^{3}-y^{3}$ is equal to
Question 57 : 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 60 :
Solving an integer programming problem by rounding off answers obtained by solving it as a linear programming problem (using simplex), we find that
Question 61 :
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 63 :
The interval in which the function $x^3$ increases less rapidly than $6x^2+15x+15$ is
Question 64 :
The position of a particle is given by $s={ t }^{ 3 }-6{ t }^{ 2 }-15t$ where $s$ in metres, $t$ is in seconds. If the particle is at rest, then time $t=.....$
Question 65 :
What is the rate of change $\sqrt{x^2 + 16 }$ with respect to $x^2$ at x = 3 ?
Question 66 :
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 67 :
If $y = 6x -x^3$ and $x$ increases at the rate of $5$ units per second, the rate of change of slope when $x = 3$ is
Question 68 :
The radius of a circle is uniformly increasing at the rate of $3cm/s$. What is the rate of increase in area, when the radius is $10cm$?
Question 69 :
If the distance $s$ travelled by a particle in time $t$ is given by $s=t^2-2t+5$, then its acceleration is
Question 70 :
The sides of two squares are $x$ and $y$ respectively, such that $y = x + x^{2}$. The rate of change of area of second square with respect to area of first square is ________.
Question 71 :
A spherical iron ball$ 10$ cm in radius is with a layer of ice of uniform thickness that melts at a rate of $50$ cm$^3$/min. When the thickness of ice is $5$ cm, then the rate at which the thickness of ice decreases, is
Question 72 :
The equation of motion of a particle moving along a straight line is $s=2t^3-9t^2+12t$, where the units of s and t are centimetre and second. The acceleration of the particle will be zero after.
Question 73 :
The side of a square is increased by $20 \% $ . Find the $ \% $ change in its area.
Question 74 :
The rate of change of the volume of a cone withrespect to the radius of its base is.
Question 75 :
Find the equation of the quadratic function $f$ whose graph increases over the interval $(-\infty, -2)$ and decreases over the interval $(-2,+\infty)$, $f(0)=23$ and $f(1)=8$
Question 76 :
A cube of ice melts without changing its shape at the uniform rate of $4\>cm^3/min$. The rate of change of the surface area of the cube, in $cm^2/min$, when the volume of the cube is $125\>cm^3$, is
Question 77 :
If the rate of change in the circumference of a circle of $0.3 cm/s$, then the rate of change in the area of the circle when the radius is $5$cm, is:
Question 78 :
Let $f"(x) > 0$ and $\phi (x) = f(x) + f(2 - x) , x \in (0,2)$ be a function, then the function $\phi (x)$ is
Question 79 :
The side of an equilateral triangle is '$a$' units and is increasing at the rate of $ \lambda $ units /sec. The rate of increase of its area is
Question 80 :
A man on a wharf 12 mt above the water level pulls in a rope to which a boat is attached at the rate of $1$ mt per second. At what rate is the boat approaching the shore, when there is still $13$ mt rope out ?
