Question 1 :
If $f(x),$ $\phi \left ( x \right ),\varphi \left ( x \right )$&#160;are continuous on&#160;[a, b] and&#160;differentiable on&#160;$\left ( a, b \right )\exists \ &#160;c\epsilon \left ( a, b \right )$, then<div>$\begin{vmatrix} f(a) &amp; \phi(a) &amp; \varphi(a) \\ f(b) &amp; \phi(b) &amp; \varphi(b) \\ f'(c) &amp; \phi'(c) &amp; \varphi'(c)\end{vmatrix}=$</div>

Question 2 :
The value of $c$ in the lagranges mean value&#160;theorem for&#160;$f\left ( x \right )= x^{3},a=1, h=\dfrac{1}{2}$ is

Question 3 :
Consider the function $f(x) =max\left[ x^2,\; (1-x)^2,\; 2x (1 -x)\right]$ where $0 \le x \le 1$. Let Rolle's Theorem is applicable for $f(x)$ on greatest interval $[a,b]$ then $b - a$ is equal to

Question 4 :
Let $f:\left[ 2,7 \right] \rightarrow [0,\infty )$ be a continuous and differentiable function. Then,<div>$\displaystyle&#160;\left( f\left( 7 \right) -f\left( 2 \right) &#160;\right) \frac { \left( \left( { f\left( 7 \right) &#160;} \right) ^{ 2 }+\left( { f\left( 2 \right) &#160;} \right) ^{ 2 }+f\left( 2 \right) f\left( 7 \right) &#160;\right) &#160;}{ 3 } $ is equal to ( where $c\in \left( 2,7 \right) $)</div>

Question 5 :
If &nbsp;$2a+3b+6c=0$, &nbsp;then at least one root of&nbsp;the equation &nbsp;$ax^{2}+bx+c=0$ &nbsp;lies in the interval

Question 6 :
There are $4$ balls of different colours &amp; $4$ boxes of colours same as those of the balls. The number of ways in which the balls, one in each box, could be placed such that exactly no ball go to the box of its own colour is:

Question 7 :
A number is selected at random from the first $1,000$ natural numbers. What is the probability that the number so selected would be a multiple of $7$ or $11$?

Question 8 :
A box contains $6$ red marbles numbers from $1$ through $6$ and $4$ white marbles $12$ through $15$. Find the probability that a marble drawn 'at random' is white and odd numbered

Question 9 :
A bag contains 5 red and 3 blue balls. If 3 balls are drawn at random without replacement then the probability that two of the three balls were red, the first ball being red, is

Question 10 :
The are included between the curves $y^2 = 4ax \,$ and $\, x^2 = 4 ay$ is ____&#160; sq units.

Question 11 :
The area inside the parabola $5x^2-y=0$ but outside the parabola $2x^2-y+9=0$, is

Question 12 :
Let $A_n$ be the constant number such that $c&gt;1$. If the least area of the figure given by the line passing through the point (1,c) with gradient 'm' and the parabola $y=x^2$ is 36 sq.units find the value of $(c^2+m^2)$.

Question 13 :
The area enclosed between the curves $y=a{ x }^{ 2 }$ and $x=a{ y }^{ 2 }$ $\\ (a&gt;0)$ is $1sq.unit$. then $a=$

Question 14 :
If $|\vec{a}|=5, |\vec{a}-\vec{b}|=8$ and $|\vec{a}+\vec{b}|=10$, then $|\vec{b}|$ is equal to :

Question 15 :
If a, b, c are position vectors of the vertices of a $\displaystyle \Delta ABC,$ then $ \displaystyle \overrightarrow{AB}+\overrightarrow{BC}+\overrightarrow{CA}=$

Question 16 :
The perimeter of the triangle whose vertices are the points&nbsp;$2\bar{i}-\bar{j}+\bar{k}, \bar{i}-3\bar{j}-5\bar{K},3\bar{i}-4\bar{j}-4\bar{j}-4\bar{k}$ is&nbsp;

Question 18 :
Let $f(x) = |x - 2|$, where $x$ is a real number. Which one of the following is true?

Question 19 :
Let X be the set of all citizens of India. Elements x, y in X are said to be related if the difference of their age is 5 years. Which one of the following is correct ?

Question 20 :
Which of the following is not a binary operation on $R$?

Question 21 :
The value of $\left| \begin{matrix} 1+w &amp; { w }^{ 2 } &amp; -w \\ 1+{ w }^{ 2 } &amp; w &amp; -{ w }^{ 2 } \\ { w }^{ 2 }+w &amp; w &amp; -{ w }^{ 2 } \end{matrix} \right| $ is equal to&nbsp;

Question 22 :
If a. b, c are negative and different real numbers then $\displaystyle \Delta=\begin{vmatrix}a &amp;b&nbsp; &amp;c \\ b &amp;c&nbsp; &amp;a \\c&nbsp; &amp;a&nbsp; &amp;b \end{vmatrix}$ is<br>

Question 23 :
If $\begin{vmatrix}1 &amp; sin \theta &amp;1 \\ -sin \theta &amp; 1 &amp; sin \theta\\&nbsp;-1 &amp; -sin \theta &amp; 1\end{vmatrix}$ then,

Question 24 :
If $A=\begin{bmatrix} x &amp; x-1 \\ 2x &amp; 1 \end{bmatrix}$ and if $\text{det}A=-9$, then the values of $x$ are

Question 27 :
The value of $\tan ^{ -1 }{ \left( 2\sin { \left( \sec ^{ -1 }{ \left( 2 \right)&nbsp; }&nbsp; \right)&nbsp; }&nbsp; \right)&nbsp; } $ is

Question 28 :
Values of $\displaystyle \cos^{-1}\cos \frac{13\pi }{2}+\tan^{-1}\tan \frac{13\pi }{5}+\sec^{-1}\sec \frac{13\pi }{3}+\cot^{-1}\cot \frac{17\pi }{2}+\sin^{-1}\sin \frac{33\pi }{5}$ is<br>

Question 29 :
Which of the following is not true, if $\mathrm{A}$and $\mathrm{B}$ are two matrices each of order $n\times n$, then&nbsp;<br><br>

Question 30 :
The number of all possible matrices of order $3 \times 3$ with each entry $0$ or $1$ is:

Question 31 :
$\displaystyle (x^{2}-y^{2})dx+2xy dy=0$,&#160; the solution to this differential equation represents which curve:

Question 32 :
<span>Find the order and degree of&#160;</span>$\left [&#160;\displaystyle \frac {d^2x}{dt^2} \right ]^3\, +\,&#160;\left [&#160;\displaystyle \frac {dx}{dt} \right ]^4\, -\, xt\, =\, 0$.

Question 33 :
<div>The order and degree of :</div><div><span>A) $y=1+\left ( \dfrac{dy}{dx} \right )^{1}+\dfrac{1}{2!}\left ( \dfrac{dy}{dx} \right )^{2}+\dfrac{1}{3!}\left ( \dfrac{dy}{dx} \right )^{3}+.....$</span><br/></div><div><span>B) $\dfrac{d^{2}y}{dx^{2}}=xlog\left ( \dfrac{dy}{dx} \right )$</span><br/></div>

Question 34 :
Order and degree of the D.E&#160;$\left [ \left ( \dfrac{dy}{dx} \right )^{2}+\dfrac{d^{2}y}{dx^{2}}\right ]^{3/4}=K\dfrac{d^{3}y}{dx^{3}}$ is:

Question 35 :
Match vector operations between two vectors A and B in column I with&nbsp;angles between the two vectors in column II :&nbsp;<table class="wysiwyg-table"><tbody><tr><td>Column-I</td><td>Column-II</td></tr><tr><td>$\mathrm{a})|\vec{\mathrm{A}}+\vec{\mathrm{B}}|=|\vec{\mathrm{A}}-\vec{\mathrm{B}}|$</td><td>e) $45^{0}$</td></tr><tr><td>$\mathrm{b}) |\vec{\mathrm{A}}\times\vec{\mathrm{B}}|=\vec{\mathrm{A}}.\vec{\mathrm{B}}$ </td><td>f) $30^{0}$</td></tr><tr><td>$\mathrm{c}) \displaystyle \vec{\mathrm{A}}.\vec{\mathrm{B}}=\frac{\mathrm{A}\mathrm{B}}{2}$</td><td>g) $90^{0}$</td></tr><tr><td>$\mathrm{d}) |\displaystyle \vec{\mathrm{A}}\times\vec{\mathrm{B}}|=\frac{\mathrm{A}\mathrm{B}}{2}$ </td><td>h) $60^{0}$</td></tr></tbody></table>

Question 36 :
If $\vec{\mathrm{A}}\times\vec{\mathrm{B}}=\vec{\mathrm{B}}\times\vec{\mathrm{A}}$ , then the angle between $A$ and $\mathrm{B}$ is&nbsp;<br><br>

Question 37 :
The lines &nbsp;$\displaystyle \frac{x+3}{-2}=\frac{y}{1}=\frac{z-4}{3}$ &nbsp;and $\displaystyle \frac{x}{\lambda }=\frac{y-1}{\lambda +1}=\frac{z}{\lambda +2}$ are perpendicular to each other. Then $\lambda$ is equal to&nbsp;&nbsp;

Question 38 :
<span>Apply linear programming to this problem. A firm wants to determine how many units of each of two products (products D and E) they should produce to make the most money. The profit in the manufacture of a unit of product D is $100 and the profit in the manufacture of a unit of product E is $87. The firm is limited by its total available labor hours and total available machine hours. The total labor hours per week are 4,000. Product D takes 5 hours per unit of labor and product E takes 7 hours per unit. The total machine hours are 5,000 per week. Product D takes 9 hours per unit of machine time and product E takes 3 hours per unit. Which of the following is one of the constraints for this linear program?</span>

Question 41 :
Given a system of inequation:<div>$\displaystyle 2y-x\le 4$<br/>$\displaystyle -2x+y\ge -4$</div><div>Find the value of $s$, which is the greatest possible sum of the $x$ and $y$ co-ordinates of the point which satisfies the following inequalities as graphed in the $xy$ plane.<br/></div>

Question 42 :
Corner points of the bounded feasible region for an LP problem are $A(0,5) B(0,3) C(1,0) D(6,0)$. Let $z = -50x + 20y$ be the objective function. Minimum value of z occurs at ______ center point.

Question 47 :
A spherical iron ball $10 cm$ in radius is coated with&#160;a layer of ice of uniform thickness that melts at a&#160;rate of $50 cm^3/min$. When the thickness of ice is $5 cm$, then the rate of which the thickness of ice&#160;decreases, is.

Question 48 :
The distance moved by a particle travelling in a straight line in $t$ seconds is given by $s=45t+11{ t }^{ 2 }-{ t }^{ 3 }$. The time taken by the particle to come to rest is

Question 49 :
The rate of change of volume of sphere with respect to its radius is ___________ when its radius is $r=7$ cm.

Question 50 :
If the rate of increase of the population of the city is $5\%$ per year. In t time population of city is P, then expression of P in terms of t is _________.