Question 1 : <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?
Question 2 :
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 6 :
If $x+y \leq 2, x\leq 0, y\leq 0$ the point at which maximum value of $3x+2y$ attained will be.<br/>
Question 7 :
Find the output of the program given below if$ x = 48$<br/>and $y = 60$<br/>10  $ READ x, y$<br/>20  $Let x = x/3$<br/>30  $ Let y = x + y + 8$<br/>40  $ z = \dfrac y4$<br/>50  $PRINT z$<br/>60  $End$
Question 8 :
$\cos ^{-1}\left ( \cos \left ( \frac{5\pi}{4} \right ) \right )$ is given by
Question 10 : <div><span>${ tan }^{ -1 }x+{ tan }^{ -1 }y={ tan }^{ -1 }\dfrac { x+y }{ 1-xy } $,      $xy<1$</span><br/></div><div>                                    $=\pi +{ tan }^{ -1 }\dfrac { x+y }{ 1-xy } $,      $xy>1$.</div><div><br/></div><div><span> Evaluate:  ${ tan }^{ -1 }\dfrac { 3sin2\alpha  }{ 5+3cos2\alpha  } +{ tan }^{ -1 }\left( \dfrac { tan\alpha  }{ 4 }  \right) $</span><br/></div><div>                                  where $-\dfrac { \pi  }{ 2 } <\alpha <\dfrac { \pi  }{ 2 } $</div>
Question 12 :
The value of $\sin \left( \cos ^{ -1 } \left( -\cfrac { 1 }{ 7 }  \right) +\sin ^{ -1 }\left( -\cfrac { 1 }{ 7 }  \right) \right)=$ ____
Question 13 :
The value of $\cos^{-1} (\cos 12) - \sin^{-1} (\sin 12)$ is 
Question 16 :
If $n - 1\sum\limits_{}^\infty {{{\cot }^{ - 1}}\left( {{{{n^2}} \over 8}} \right) = \pi .} $ where ${a \over b}$ is rational number in its lowest, then correct option is/are
Question 17 :
If $x,y,z \in [-1,1]$ such that $\cos^{-1}x +\cos^{-1}y +\cos^{-1}z=0$, find $x+y+z$.
Question 19 :
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 21 :
If $\sin^{-1}\dfrac{1}{3} + \sin^{-1}\dfrac{2}{3} = \sin^{-1}x$, then $x$ is equal to-
Question 22 :
${\tan ^{ - 1}}2 + {\tan ^{ - 1}}3$ is equal to
Question 25 :
The value of $\sin^{-1} \left( \dfrac{2 \sqrt 2}{3} \right ) + \sin^{-1} \left( \dfrac{1}{3}\right )$ is equal to
Question 27 :
Simplify ${\cot ^{ - 1}}\dfrac{1}{{\sqrt {{x^2} - 1} }}$ for $x <  - 1$
Question 28 :
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 31 :
The solution of the differential equation $\dfrac { dy }{ dx } =\dfrac { y\left( \log { y } -\log { x } +1 \right) }{ x } $ is
Question 32 :
The general solution of the differential equation $ydx - xdy + x^2 .\sin ydy + (1 + x^2) dx = 0$, is equal to:
Question 33 : <div>Solve:</div><div>$\left [x\, \cos\, \displaystyle \frac {y}{x}\, +\, y\, \sin\,\displaystyle \frac {y}{x} \right ]\, y\, =x\, \left [y\, \sin\, \displaystyle \frac {y}{x}\, -\, x\, \cos\,\displaystyle \frac {y}{x} \right ]\, \times\, \displaystyle \frac {dy}{dx}$</div>
Question 34 : The solution of differential equation $(x^2+y^2)dy=xy  dx$ is $y=y(x)$. If<div>$y(1)=1$ and $y(x_0)=e$, then $x_0$ is:</div>
Question 36 :
The solution of $x^{2} \dfrac {dy}{dx} - xy = 1 + \cos \dfrac {y}{x}$ is
Question 37 :
Solve: $(3xy + y^{2})dx + (x^{2} + xy) dy = 0$, $y(1) = 1$
Question 38 :
The normal at each point of a curve passes through $(3, 1)$. If the point $(1, 1)$ lies on the curve the equation of the curve is:<br/>
Question 39 :
$(x^{2}+y^{2})dy=xydx$. If $y(x_{0})=1$, then the value of $x_{0}$ is equal to :
Question 40 : <div><span>Solve the following differential equations:</span><br/></div>$xdx + ydy = xdy - ydx$
Question 44 :
The solution of differential equation $\left ( x tan \left(\displaystyle \frac{y}{x}\right) - y sec^2 \left(\frac{y}{x}\right) \right) dx  +  x sec^2 \left(\frac{y}{x}\right) dy = 0$ satisfying the initial condition $ y(1) = \displaystyle \frac{\pi}{4}$ is:
Question 45 :
The differential equation of all conics with centre at origin is of order
Question 46 :
Solve: $(x - y\ln y + y\ln x) dx + x(\ln y - \ln x) dy = 0$
Question 47 :
Let $P(x)$ be a polynomial of least degree whose graph has three points of inflection $(-1, - 1), (1, 1)$ and a point with abscissa $0$ at which the curve is inclined to the axis of abscissa at an angle of $60^{0}$. Then $\int _{ 0 }^{ 1 } P(x)dx$ equals to:<br/>
Question 48 :
If $\displaystyle x\frac{dy}{dx}=y\left ( \log y-\log x+1 \right )$, then the solution of the equation is<br>
Question 51 :
The solution of the differential equation<br>$\dfrac { dy }{ dx } +\dfrac { 2yx }{ 1+{ x }^{ 2 } } =\dfrac { 1 }{ { \left( 1+{ x }^{ 2 } \right) }^{ 2 } } $
Question 52 :
Solution of $y^{2}dx=(xy-x^{2})dy$, given that $y = 1$ when $x =1$, is:
Question 53 :
A curve passes through the point $\displaystyle (1,\ \frac{\pi}{6})$. Let the slope of the curve at each point $\displaystyle (x,\ y)$be $\displaystyle <br/>\frac{y}{x}+\sec(\frac{y}{x})$, $x>0$. Then the equation of the curve is:<br/>
Question 54 :
At present, a firm is manufacturing 2000 items. It is estimated that the rate of change of production $P$ w.r.t. additional number of workers $x$ is given by $\frac{dP}{dx}=100-12\sqrt{x}$. If the firm employs 25 more workers, then the new level of production of items is:<br/><br/>
Question 55 :
If the curve satisfying $(1+e^{x/y})dx+e^{x/y}(1-x/y)dy=0$ passes through $(1, 1)$, then $9 +y(2)e^{2/y(2)}-e$ is equal to<br/>
Question 58 :
The solution of primitive integral equation $\left ( x^{2}+y^{2} \right )dy=xy.dx$ is $y=y(x)$. If $y(1)=1$ and $y(x_{0})=e$ then $x_{0}$ is
Question 59 :
A curve passing through the point $(1,1)$ has the property that the perpendicular distance of the origin from the normal at any point $P$ of the curve is equal to the distance of $P$ from the x-axis. The equation of the curve represents
Question 60 :
Order and degree of the differential equation, $\left(\dfrac{d^3y}{dx^3}\right)^{\dfrac{3}{2}} + \left(\dfrac{d^3y}{dx^3}\right)^{\dfrac{-2}{3}} = 0$ are respectively
Question 62 :
Show that the differential equation $2ye^{x/y} dx + (y 2x e^{x/y}) dy = 0$ is homogeneous.<br/>Find the particular solution of this differential equation, given that $x = 0 $ when $y = 1$.
Question 64 :
Let the population of rabbits surviving at a time $ t$ be governed by the differential equation $ \displaystyle \frac{dp(t)}{dt}=\displaystyle \frac{1}{2}p(t)-200$. If $ p(0)=100$, then $ p(t)$ equals:
Question 67 :
The solution of differential equation $\left( { x }^{ 2 }-xy \right) dy=\left( xy+{ y }^{ 2 } \right) dx$ is :
Question 68 :
$\displaystyle \frac{dy}{dx}=\frac{x^{2}+xy}{x^{2}+y^{2}}$<br/>Solving this gives $\displaystyle c(x-y)^{2/3}(x^{3}+xy+y^{3})^{1/5}=\left [ \frac{1}{\sqrt{k}}tan^{-1}\frac{x+2y}{x\sqrt{k}} \right ]$ where $ exp (x)=e^{x}$, then what is k?
Question 69 :
If $\dfrac { dy }{ dx } =\dfrac { x-y }{ x+y } $ then:
Question 70 :
The solution of $x^{2}y \: dx-\left ( x^{3} +y^{3}\right ) \: dy= 0$ is:
Question 71 :
The solution of the differential equation $3xy'=3y-{ \left( { x }^{ 2 }-{ y }^{ 2 } \right)  }^{ 1/2 }=0$, satisfying the condition $y(1)=1$ is:
Question 72 :
The solution of the D.E. $\displaystyle(x^{3}-3xy^{2}) dx = (y^{3}-3x^{2}y) dy$, is:
Question 73 :
If for the differential equation $y{}'= \displaystyle \frac{y}{x}+\phi \displaystyle \left ( \frac{x}{y} \right )$ the general solution is $y= \displaystyle \frac{x}{\log \left | Cx \right |}$ then $\phi \left ( x/y \right )$ is given by:
