Question 1 :
The value of $a$ for which the area between the curves ${y^2} = 4ax$ and ${x^2} = 4ay$ is $1\,sq.\,unit$, is-
Question 3 :
The area bounded by the $x-$axis, the curve $y=f\left(x\right)$ and the lines $x=1$ and $x=b$ is equal to $\left(\sqrt{{b}^{2}+1}-\sqrt{2}\right)$ for all $b>1$, then $f\left(x\right)$ is
Question 4 :
The area (in sq. units) of the region $\{ x \in R:x \ge ,y \ge 0,y \ge x - 2\ $ and $y \le \sqrt x \} $, is
Question 5 :
What is the area of the region enclosed between the curve $y^2=2x$ and the straight line $y=x$ ?
Question 6 :
If the curves $y=x^3+ax$ and $y=bx^2+c$ pass through the point $(-1, 0)$ and have common tangent line at this point, then the value of $a+b$ is?
Question 7 :
Area bounded by curve $x\left( { x }^{ 2 }+p \right) =y-1$ with $y=1$ $p<0$is -
Question 8 :
The area of the region bounded by the curve $x={ y }^{ 2 }-2$ and $x=y$ is
Question 9 :
Find the area of the closed figure bounded by the following curves $y = 2$ $\cos^2 x (1 \, + \,  \sin^2 x)$ on the interval $[0, 2\pi]$ and the abscissa axis.
Question 10 :
$\displaystyle\int_0^{\pi /2} {\dfrac{{\sin x}}{{\sqrt {1 + {\mathop{\rm cosx}\nolimits} } }}} dx = $
Question 11 :
$\int _{ 0 }^{ 1 }{ \dfrac { { e }^{ x }.x }{ { \left( x+1 \right) }^{ 2 } } dx= }$
Question 13 :
$\displaystyle \int _{ 0 }^{ 1 }{ \tan ^{ -1 }{ \left( \dfrac { x }{ \sqrt { 1-{ x }^{ 2 } } } \right) } dx }$
Question 14 :
The quadratic polynomial $p(x)$ has the following properties: $p(x)\ge 0$ for all real number, $p(1)=$ and $p(2)=2$ value of $p(0)+p(3)$ is equal to
Question 17 :
$ \int_0^{ \pi /2 } \sin^5 x \cos ^6 x dx = $
Question 20 :
The vector which can give unit along the x-axis is <span> $\overrightarrow { A } =2\hat { i } -4\hat { j } +7\hat { k } , \overrightarrow { B } =7\hat { i } +2\hat { j } -5\hat { k } $. Find C?</span>
Question 21 :
For $O$ being the origin and $3$ points $P,Q$ and $R$ lie on a plane. If $\displaystyle \vec{PO}+\vec{OQ}=\vec{QO}+\vec{OR}$, then $P, Q, R$ are <br/>
Question 22 :
If $\vec a=\hat i+2\hat j$ and $\vec b = 3\hat j$, then $\vec a\cdot\vec b=$<br/>
Question 23 :
If $\vec a$ is parallel to $\vec b \times \vec c$, then $(\vec a \times \vec b) \cdot (\vec a \times \vec c)$ is equal to
Question 24 :
Let $\vec{A}=\hat{i}+2\hat{j}{+}3\hat{k},\ \vec{B}=4\hat{i}+2\hat{j},\ \vec{C}=2\hat{i}+2\hat{j}{+}2\hat{k}$. Then the ratio in which $C$ divides $AB$ is<br/>
Question 25 :
Consider two vectors $\vec{F_{1}}=2\hat{i}+5\hat{k}$ and $\vec{F_{2}}=3\hat{j}+4\hat{k}$. The magnitude of the scalar product of these vectors is
Question 26 :
If $\vec {a} . \hat {i} = \vec {a} . (\hat {i} + \hat {j}) = \vec {a} (\hat {i} + \hat {j} + \hat {k})$, thus $\vec {a}=$
Question 29 :
Let $\vec{a}=\hat{i}+\hat{j}+\hat{k}$ and $\vec{b}$ is a vector such that $\vec{a}.\vec{b}=0$ and $\vec{a}\times \vec{b}=0$. Then which of following is correct?
Question 30 : <p class="MsoNormal">If a fair die is rolled $4$ times, then what is the probability that there are at least $2$ sixes ?</p>
Question 31 :
The probability that a leap year will have $53$ sundays is
Question 32 :
If $P\left( A \right) = \displaystyle\frac { 6 }{ 11 }, P\left( B \right) = \displaystyle\frac { 5 }{ 11 }$ and $ P\left( A \cup B \right) = \displaystyle\frac { 7 }{ 11 }$, find <br>(i) $ P\left( A \cap B \right)$<br>(ii) $ P\left( A | B \right)$<br>(iii) $ P\left( B | A \right) $
Question 33 :
A coin is tossed three times, where<br>(i) $E$ : head on third toss, $F$ : heads on first two tosses<br>(ii) $E$ : at least tow heads, $F$ : at most two heads<br>(iii) $E$ : at most two tails, $F$ : at least one tail<br>Determine $ P\left( E|F \right) $
Question 34 :
A random variable X has its range $X = \{3, 2, 1\}$ with the probabilities,<br/>$\dfrac{1}{2},\dfrac{1}{3}$ and $\dfrac{1}{6}$ respectively. The mean value of X is<br/>
Question 35 :
Box I contains $2$ white and $3$ red balls and box II contains $4$ white and $5$ red balls. One ball is drawn at random from one of the boxes and is found to be red. Then, the probability that it was from box II, is?
Question 36 :
In $5$ throws of a die, getting $1$ or $2$ is a success. The mean number of successes is
Question 37 : <p class="MsoNormal">A box contains $10$ items, $3$ of which are defective. If $4$ are selected at random without replacement, what is the probability that at least $2$ of the $4$ are defective?</p>
Question 39 : The probability distribution of a discrete random variable $X$ is:<table class="wysiwyg-table"><tbody><tr><td>$X = x$</td><td>$1$</td><td>$2$</td><td>$3$</td><td>$4$</td><td>$5$</td></tr><tr><td>$P(X = x)$</td><td>$k$</td><td>$2k$</td><td>$3k$</td><td>$4k$</td><td>$5k$</td></tr></tbody></table>Find $P (X\leq 4)$
Question 40 :
$f : R\rightarrow R, f(x) = 3x + 2$<br>$g : R \rightarrow R, g(x) = 6x + 5$<br>for the given functions $(gof^{-1})(10) =$ _______.
Question 41 :
If $f(x) = \sqrt {x^{2} - 3x + 6}$ and $g(x) = \dfrac {156}{x +17}$, find the value of the composite function $g(f(4))$.
Question 42 :
If $f(x) = 3x + 2, g(x) = x^2 + 1$, then the value of $(fog) (x^2 +1)$ is
Question 43 :
Find the correct <span>expression for $\displaystyle f\left( g\left( x \right)  \right) $ given that </span>$\displaystyle f\left( x \right) =4x+1$ and $\displaystyle g\left( x \right) ={ x }^{ 2 }-2$
Question 44 :
If $f: R \rightarrow R$ and $g: R \rightarrow R$ are defined by $f(x) =2x +3, g(x)=x^2 + 7$, what are the values of $x$ such that $g(f(x))=8$?<br/>
Question 45 :
If $ f (x) =  \dfrac {1}{1-x} , x \ne 1 $ then $(fofof)(x) = $
Question 46 :
Find the value of $\displaystyle \left( g\circ f \right) \left( 6 \right) $ if $\displaystyle g\left( x \right) ={ x }^{ 2 }+\frac { 5 }{ 2 } $ and $\displaystyle f\left( x \right) =\frac { x }{ 4 } -1$. <br/>
Question 47 :
Let $f(x)=\cfrac { 1 }{ 1-x } $. Then $\left\{ f\circ \left( f\circ f \right) \right\} (x)$
Question 48 :
Read the following information and answer the three items that follow :<br>Let $f(x) = x^2 + 2x - 5 $ and $g(x) = 5x + 30$<br>If $h(x) = 5f(x) - xg (x)$, then what is the derivative of $h(x)$ ?
Question 49 :
Read the following information and answer the three items that follow :<br>Let $f(x) = x^2 + 2x - 5 $ and $g(x) = 5x + 30$<br>Consider the following statements:<br>1. $f[g(x)]$ is a polynomial of degree 3.<br>2. $g[g(x)]$ is a polynomial of degree 2.<br>Which of the above statements is/are correct ?
Question 50 :
The value of $\cos^{-1} (\cos 12) - \sin^{-1} (\sin 12)$ is 
Question 52 :
${\tan ^{ - 1}}2 + {\tan ^{ - 1}}3$ is equal to
Question 53 :
Simplify ${\cot ^{ - 1}}\dfrac{1}{{\sqrt {{x^2} - 1} }}$ for $x <  - 1$
Question 54 :
The value of $\sin \left( \cos ^{ -1 } \left( -\cfrac { 1 }{ 7 }  \right) +\sin ^{ -1 }\left( -\cfrac { 1 }{ 7 }  \right) \right)=$ ____
Question 55 :
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 58 :
$\cos ^{-1}\left ( \cos \left ( \frac{5\pi}{4} \right ) \right )$ is given by
Question 59 :
Consider the following statements:<br/>1. $\tan^{-1} 1+ \tan^{-1} (0.5) = \dfrac {\pi}2$<br/>2. $\sin^{-1}{\cfrac{1}{3} }+ \cos^{-1}{\cfrac{1}{3}} =\cfrac{\pi}{2}$<br/>Which of the above statements is/are correct ? 
Question 60 :
What is the output for the following matrix multiplication $A_{3 \times 2}\times B_{2\times 3}$?<br>
Question 61 :
If $A$ is any matrix, then the product $AA$ is defined only when A is a matrix of order $m \times n$ where : <span><br/></span>
Question 62 :
If $\mathrm{A}=\left[\begin{array}{lll}<br/>1 & -3 & -4\\<br/>-1 & 3 & 4\\<br/>1 & -3 & -4<br/>\end{array}\right]$, then $\mathrm{A}^{2}=$<br/>
Question 63 :
Given $A$ is a matrix of order $3\times 2$. If order of $AB$ is $3\times 3$, then order of $B$ will be 
Question 64 :
If $A$ is of order $3\times 4$ and $B$ is of order $4\times 3$ , then the order of $BA$ is :
Question 67 :
What is the output order for the following matrix multiplication $A_{2 \times 1}\times B_{1\times 2}$?<br/>
Question 68 :
If $A=\begin{bmatrix} 2 & 3 \\ 1 & 2 \end{bmatrix}$ and $B=\begin{bmatrix} 1 & 3 & 2 \\ 2 & 3 & 4 \end{bmatrix}$, then $AB$ equal to 
Question 70 :
If the area of the triangle formed by $ (0,0), (a,0) $ and $ \left( \dfrac{1}{2} , a \right) $ is equal to $ \dfrac {1}{2} $ sq unit, then the values of $a$ are :
Question 71 :
Which of the given values of $x$ and $y$ make the following pair of matrices equal.<br>$\displaystyle \begin{bmatrix} 3x+7 & 5 \\ y+1 & 2-3x \end{bmatrix}=\begin{bmatrix} 0 & y-2 \\ 8 & 4 \end{bmatrix}$
Question 72 :
If $-9$ is a root of the equation $\begin{vmatrix} x & 3 & 7 \\ 2 & x & 2 \\ 7 & 6 & x \end{vmatrix}=0$, then the other two roots are
Question 73 :
The system of equations which can be solved by matrix inversion method have_______.
Question 74 :
The points (2, -3), (4,3) and (5, k/2) are on the same straight line. The value(s) of k is (are):
Question 75 :
The number of line segments possible with three collinear points is ________.
Question 77 :
Two points $(a, 0)$ and $(0, b)$ are joined by a straight line. Another point on this line is
Question 78 :
The value of k for which $kx+3y-k+3=0$ and $12x+ky=k$, have infinite solutions, is?
Question 82 :
Let $y = x^{x^{x .......}},$ then $\displaystyle \frac{dy}{dx}$ is equal to
Question 83 :
Derivative of $\log { \left( \sec { \theta } +\tan { \theta } \right) } $ with respect to $\sec { \theta } $ at $\theta =\cfrac { \pi }{ 4 } $ is ...........
Question 84 :
Find the derivative with respect to x of the function$\displaystyle (\log_{\cos x}\sin x)(\log_{\sin x}\cos x)^{-1}+\sin ^{-1}\frac{2x}{1+x^{2}} \mbox{at} x=\frac{\pi }{4}$
Question 85 :
If $y = \sqrt {x{{\log }_e}x} $, then $\dfrac{{dy}}{{dx}}$ at $x = e$ is
Question 86 :
If the tangent to the curve $x=a \, (\theta + \sin \, \theta), y=a (1+ \cos \,\theta)$ at $ \theta=\dfrac{\pi}{3}$ makes an angle $\alpha (0 \leq\alpha < \pi)$ with x-axis, then $\alpha$ =
Question 87 :
If $y=\ln { \left( { e }^{ mx }+{ e }^{ -mx } \right) } $, then what is $\cfrac { dy }{ dx } $ at $x=0$ equal to?
Question 88 :
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 89 :
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 90 :
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 91 :
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 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 :
What is the rate of change $\sqrt{x^2 + 16 }$ with respect to $x^2$ at x = 3 ?
Question 95 :
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 96 :
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 97 :
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 98 :
The solution of $\left( { x }^{ 2 }{ y }^{ 3 }+{ x }^{ 2 } \right) dx+\left( { y }^{ 2 }{ x }^{ 3 }+{ y }^{ 2 } \right) dy=0$ is
Question 102 :
The equation of the curve passing through $(0,1)$ which is a solution of the differential equation $\left( 1+{ y }^{ 2 } \right) dx+\left( 1+{ x }^{ 2 } \right) dy=0$ is given by
Question 106 :
Solution of differential equation $\displaystyle \frac{dy}{dx} = sin x + 2x$, is
Question 107 :
The solution of the differential equaton $3{e^x}\tan ydx + \left( {1 - {e^x}} \right){\sec ^2}ydy = 0$ is 
Question 108 :
If a line makes the angles $ \alpha , \beta$ and $\gamma$ with the axes, then what is the value of $1+\cos 2\alpha +\cos 2\beta+\cos 2\gamma$<span> equal to ?</span>
Question 109 :
If a line has the direction ratio $18, 12, 4 $, then its direction cosines are:<br/>
Question 110 :
The direction angles of the line $x = 4z + 3, y = 2 - 3z$ are $\alpha, \beta$ and $\gamma$, then $\cos \alpha + \cos \beta + \cos \gamma =$ ________.
Question 111 : <span>From the point $P(3, -1, 11)$, a perpendicular is drawn on the line $L$ given by the equation $\dfrac {x}{2} = \dfrac {y - 2}{3} = \dfrac {z - 3}{4}$. Let $Q$ be the foot of the perpendicular.</span><div>What are the direction ratios of the line segment $PQ$?</div>
Question 112 :
If the dr's the line are $(1+\lambda, 1-\lambda, 2)$ and it makes an angle ${60}^{o}$ with the Y-axis then $\lambda$ is
Question 113 : <table class="table table-bordered"><tbody><tr><td> List I</td><td>List II </td></tr><tr><td><span>1) d.c's of $x -$ axis</span></td><td><span>a) $(1,1,1)$</span> </td></tr><tr><td><span>2) d.c's of $y -$ axis</span></td><td><span>b)$\left(\displaystyle \frac{]}{\sqrt{3}}\frac{]}{\sqrt{3}},\frac{]}{\sqrt{3}}\right)$</span></td></tr><tr><td><span>3) d.c's of $z -$ axis</span></td><td><span>c) $(1,0,0)$</span><br/></td></tr><tr><td><span>4) d.c's of a line makes </span><span>equal angles with axes</span></td><td><span>d) $(0,1,0)$</span></td></tr><tr><td> </td><td><span>e) $(0,0,1)$</span></td></tr></tbody></table>The correct order for 1, 2, 3, 4 is
Question 114 :
A vector is equally inclined to the $x$-axis, $y$-axis and $z$-axis respectively, its direction cosines are
Question 115 :
Direction cosines $l, m, n$ of two lines are connected by the equation $l-5m+3n=0$ and $7l^{2}+5m^{2}-3n^{2}=0$. The direction cosines of one of the lines are
Question 117 :
The direction ratios of the diagonal of the cube joining the origin to the opposite corner are (when the $3$ concurrent edges of the cube are coordinate axes)<br/>
Question 121 :
If x + y = 3 and xy = 2, then the value of $\displaystyle x^{3}-y^{3}$ is equal to
Question 122 :
If an iso-profit line yielding the optimal solution coincides with a constaint line, then
Question 123 :
<span>In linear programming, lack of points for a solution set is said to</span>
Question 124 :
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 $(0, 30)$ <span>is _______.</span>
Question 126 :
<span>Solving an integer programming problem by rounding off answers obtained by solving it as a linear programming problem (using simplex), we find that</span>
Question 127 : <div><span>The given table shows the number of cars manufactured in four different colours on a particular day. Study it carefully and answer the question.</span><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></div>What was the total number of black cars manufactured?
