Question 1 :
Given $\vec p= (2,-4,1), \vec q = (3,-1,2), \vec r = (5,5, 4)$. Then $\vec{PQ}$ and $\vec{QR}$ are
Question 2 :
$\vec{a},\vec{b},\vec{c}$ are three non-collinear vectors such that $\vec{a}+\vec{b}$ is parallel to $\vec{c}$ and $\vec{a}+\vec{c}$ is parallel to $\vec{b}$ then:
Question 5 :
Find a vector in which two of the three direction angles are $\alpha=75^{o}$ and $\beta=55^{o}$.
Question 7 :
<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 8 :
Given $f(x) = x^2 + a$, if $x \le 0$<br> $= 2\sqrt{1 + x^2} + b$, if $x > 0$<br>and $f(-1) = 2$, if $f$ is continuous at $x = 0$, then $(a, b) \equiv$____
Question 9 :
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 10 :
If $f(x)=\left\{\begin{matrix}<br/>2, & x > 4\\ <br/>0, & x\leq 4<br/>\end{matrix}\right.$ then $\displaystyle \lim_{x\rightarrow 4}f(x)$ equals to<br/>
Question 11 :
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 12 :
If $f(x) = (1 + 2x)^{\frac{1}{x}}$, for $x \neq 0$ is continuous at $x = 0$, then $f(0) = $_______.
Question 13 :
For the function $\displaystyle f\left( x \right) ={ \left( x+1 \right)  }^{ 1/x }$ to be continuous at $x=0$, $f(0)$ must be defined as:
Question 14 :
If $f(x)$ is continuous and $g(x)$ is discontinuous function, then $f(x)+g(x)$ is-
Question 15 :
Is f defined by $f(x)=\left\{\begin{matrix} \dfrac{\sin 2x}{x} & if & x\neq 0\\ 1 & if & x=0\end{matrix}\right.$ continuous at $0$.
Question 16 :
If the function<br/>$f(x) = \begin{cases} k + x, \,\,\,\text{for}\, x < 1 \\ 4x + 3, \,\,\,\text{for}\, x\geq 1\end{cases}$<br/>is continuous at $x = 1$, then $k =$
Question 17 :
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 19 :
Let $f(x) = \left\{\begin{matrix}ax + 1,& x < 1\\ 3,& x = 1\\ bx^{2} + 1, & x > 1\end{matrix}\right.$. The value of $'a'$ and $'b'$ for which $f(x)$ is continuous at $x = 1$, is given by<br>
Question 20 :
The function $f\left( x \right)=\left[ x \right] ,$  at ${ x }=5$ is:<br/>
Question 21 :
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 23 :
Is the function f, defined by $(x)=\left\{\begin{matrix} x^2 & if & x\leq 1\\ x & if & x > 1\end{matrix}\right.$, continuous on R?
Question 25 :
The value of the determinant $\begin{vmatrix} 5 & 1 \\ 3 & 2 \end{vmatrix}$
Question 26 :
If $P=\begin{bmatrix} 1 & c & 3\\ 1 & 3 & 3\\ 2 & 4 & 4\end{bmatrix}$ is the adjoint of a $3\times 3$ matrix Q and det.(Q)$=4$, then c is equal to.
Question 27 :
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 28 :
If $ A = \begin{bmatrix} \log x & -1 \\ - \log x & 2 \end{bmatrix} $ and if $ \text{det} (A) = 2$ , then the value of $x$ is equal to :
Question 29 :
 If $\left| {\begin{array}{*{20}{c}}  a&b&c \\   b&c&a \\   c&a&b \end{array}} \right| = k\left( {a + b + c} \right)\left( {{a^2} + {b^2} + {c^2} - bc - ca - ab} \right)$, then $k=$
Question 30 :
$x = \left| \begin{gathered}   - 1\,\,\,\,\,\, - 2\,\,\,\,\,\,\, - 2 \hfill \\  \,\,\,2\,\,\,\,\,\,\,\,\,\,1\,\,\,\,\,\,\, - 2 \hfill \\  \,\,\,2\,\,\,\,\,\, - 2\,\,\,\,\,\,\,\,\,\,1 \hfill \\ \end{gathered}  \right|$, then $x=$
Question 31 : <div><span>Find the value of the following determinant:</span><br/></div>$\begin{vmatrix}\displaystyle \frac{-4}{7} & \displaystyle \frac{-6}{35}\\ 5 & \displaystyle \frac{-2}{5}\end{vmatrix}$
Question 33 :
If $\Delta_1=\begin{vmatrix} 1 & 0\\ a & b\end{vmatrix}$ and $\Delta_2=\begin{vmatrix} 1 & 0\\ c & d\end{vmatrix}$ then $\Delta_2 \Delta_1$ is equal to<br>
Question 34 :
<i></i>What is the determinant of the matrix $\left [\begin{matrix} 3& 6\\ -1 & 2\end {matrix} \right]$?<br/>
Question 35 : <div><span>Find the value of the following determinant:</span><br/></div>$\begin{vmatrix}-3 & 8\\ 6 & 0\end{vmatrix}$
Question 36 :
The value of $\dfrac {1}{x-y}\begin{vmatrix} 1 & 0 & 0\\ 3 & x^3 & 1 \\ 5 & y^3 & 1\end{vmatrix}$ is<br>
Question 37 :
For positive numbers $x, y$ and $z$ the numerical value of the determinant $\begin{vmatrix} 1 & \log_x y & \log_x z \\ \log_y x & 1 & \log_y z \\ \log_z x & \log_z y & 1 \end{vmatrix}$ is<br>
Question 38 :
The product of a matrix and its transpose is an identity matrix. The value of determinant of this matrix is
Question 39 :
If$\displaystyle \left | \begin{matrix}-12 &0   &\lambda  \\  0&  2& -1\\  2& 1 &15 \end{matrix} \right |=-360$, then the value of $\lambda$,is
Question 40 :
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 41 :
If ${ \Delta }_{ 1 }=\begin{vmatrix} x & b & b \\ a & x & b \\ a & a & x \end{vmatrix}$ and $ { \Delta }_{ 2 }=\begin{vmatrix} x & b \\ a & x \end{vmatrix}$ are the given determinants, then<br>
Question 42 :
$D\mathrm{e}\mathrm{t} \left\{\begin{array}{lll}<br>2 & 45 & 55\\<br>1 & 29 & 32\\<br>3 & 68 & 87<br>\end{array}\right\}=\ldots.$ .<br>
Question 43 :
Find the value of $x$ in $\begin{vmatrix} 2 & 4 \\ 5 & 1 \end{vmatrix}=\begin{vmatrix} 2x & 4 \\ 6 & x \end{vmatrix}$.
Question 44 :
What is the value of the determinant<br>$\begin{vmatrix} 1!& 2! & 3!\\ 2! & 3! & 4! \\ 3!& 4!& 5!\end{vmatrix}$ <br> $?$
Question 46 :
Let a be the square matrix of order 2 such that $A^2 - 4A + 4I =0$ where I is an identify matrix of order 2. .If$ B = A ^5 - 4A^4 + 6 A^3 + 4A^2 + A $ then Det (B) is equal to
Question 47 :
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 48 :
If $A$ is a square matrix such that $\displaystyle \left| A \right| =2$, then for any positive integer <span>$\displaystyle n,\left| { A }^{ n } \right| $ is equal to</span><br/>
Question 49 :
If $f(x)$ and $g(x)$ are functions such that $f(x+)=f(x)g(y)+g(x)f(y)$, then the value of $\begin{vmatrix} f\left( \alpha \right) & g\left( \alpha \right) & f\left( \alpha +\theta \right) \\ f\left( \beta \right) & g\left( \beta \right) & f\left( \beta +\theta \right) \\ f\left( \gamma \right) & g\left( \gamma \right) & f\left( \gamma +\theta \right) \end{vmatrix}$ is
Question 50 :
If $a\neq6,b,c$ satisfy $\begin{vmatrix}  a&2b&2c  \\ 3&b&c \\ 4&a&b \end{vmatrix}=0,$ then $abc =$ 
Question 51 :
If $\begin{bmatrix} x & 1 & 1\\ 2 & 3 & 4\\ 1 & 1 & 1\end{bmatrix}$ has no inverse, then $x=$
Question 52 :
Maximum value of a second order determinant whose every element is either 0,1 or 2 only is:
Question 53 :
If $\begin{vmatrix} a & a & x\\ m & m & m\\ b & x & b\end{vmatrix}=0$ then $x=?$
Question 55 :
If $A=[{a}_{ij}]$ is scalar matrix of order $n\times n$ such that ${a}_{ij}=k$ for all $i$, then $\left| A \right| $ equals
Question 56 :
If the coordinates of the vertices of an equilateral triangle with sides of length $a$ are $({x}_{1},{y}_{1}),({x}_{2},{y}_{2})$ and $({x}_{3},{y}_{3})$, then ${ \begin{vmatrix} { x }_{ 1 } & { y }_{ 1 } & 1 \\ { x }_{ 2 } & { y }_{ 2 } & 1 \\ { x }_{ 3 } & y_{ 3 } & 1 \end{vmatrix} }^{ 2 }=$
Question 57 :
For what value of x the matrix A is singular? <br><span>$A= \begin{bmatrix} 1+x & 7 \\ 3-x & 8 \end{bmatrix}$</span>
Question 58 :
If the trivial solution is the only solution of the system of equations $x-ky+z=0,kx+3y-kz=0, 3x+y-z=0$. Then the set of all values of k is:<br>
Question 59 :
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 60 :
If a particle moves such that the displacement is proportional to the square of the velocity acquired, then its acceleration is
Question 61 :
What is the rate of change of the area of a circle with respect to its radius $r$ at $r = 6$ $cm$.
Question 62 :
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 63 :
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 64 :
If the distance $s$ travelled by a particle in time $t$ is given by $s=t^2-2t+5$, then its acceleration is
Question 65 :
A ladder, 5 meter long, standing on a horizontal floor, leans against vertical wall. If the top of the ladder slides downwards at the rate of 10cm/sec, then the rate at which the angle between the floor and the ladder is decreasing when lower end of ladder is 2 metres from the wall is:
Question 66 :
The interval in which the function $x^3$ increases less rapidly than $6x^2+15x+15$ is
Question 67 :
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 68 :
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 69 :
If the displacement of a particle moving in straight line is given by $x=3t^2+2t+1$ at time $t$ then  the acceleration of the particle at time $t=3$ is
Question 70 :
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 71 :
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 72 :
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 73 :
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 74 :
Consider the following statements:<br/>1. $\dfrac {dy}{dx}$ at a point on the curve gives slope of the tangent at that point.<br/>2. If $a(t)$ denotes acceleration of a particle, then $\displaystyle \int a(t) dt + c$ give velocity of the particle.<br/>3. If $s(t)$ gives displacement of a particle at time $t$, then $\dfrac {ds}{dt}$ gives its acceleration at that instant.<br/>Which of the above statements is/ are correct?
Question 75 :
The sides of two squares are $x$ and $y$ respectively, such that $y = x + x^{2}$. The rate of change of area of <span>second square with respect to area of first square is ________.</span>
Question 76 :
What is the rate of change $\sqrt{x^2 + 16 }$ with respect to $x^2$ at x = 3 ?
Question 77 :
The side of a square sheet is increasing at the rate of $4 cm$ per minute. The rate by which the area increasing when the side is $8 cm$ long is.
Question 78 :
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 79 :
The interval in which the function $f(x) = {x^3}$ increases less rapidly than $\,g(x) = 6{x^2} + 15x + 5$ is :
