Question 1 :
If an iso-profit line yielding the optimal solution coincides with a constaint line, then
Question 2 :
<span>In linear programming, lack of points for a solution set is said to</span>
Question 4 :
What is the solution of $x\le 4,y\ge 0$ and $x\le -4,y\le 0$ ?
Question 9 :
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 10 :
The solution of the set of constraints of a linear programming problem is a convex (open or closed) is called ______ region.
Question 12 :
If x + y = 3 and xy = 2, then the value of $\displaystyle x^{3}-y^{3}$ is equal to
Question 13 :
<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 15 : <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 16 :
The order and degree of the differential equation $\displaystyle\frac{d^2y}{dx^2}=\sqrt[3]{1-\displaystyle\left(\frac{dy}{dx}\right)^4}$ are respectively.
Question 18 :
The solution of the differential equation ${e^{ - x}}(y + 1)dy + (co{s^2}x - sin2x)ydx = 0$ subject to the conditions $y(0) = 1$
Question 19 :
Find the value of $k$ for the function: $2x^2y+3xyz+z^k$ to be homogenous.
Question 20 :
Solve $\displaystyle \left ( 4x+6y+3 \right )dx= \left ( 6x+9y+2 \right )dy$
Question 21 :
Check whether the function is homogenous or not. If yes then find the degree of the function<br/>$g(x)=8x^4$.
Question 24 :
Degree and order of the differential equation $\dfrac { { d }^{ 2 }y }{ d{ x }^{ 2 } } ={ \left( \dfrac { dy }{ dx } \right) }^{ 2 }$ are respectively
Question 25 :
Check whether the function is homogenous or not. If yes then find the degree of the function<br/>$g(x)=4-x^2$.
Question 26 :
Solve the differential equation $\displaystyle e^{dy/dx}=x+1$ given thet when $\displaystyle x=0,y=3.$
Question 29 :
Solve the differential equation$\displaystyle x\frac{dy}{dx}= y\left ( \log y-\log x+1 \right )$
Question 31 :
Evaluate  $\int _{ 0 }^{ 1 }{ \sqrt { \cfrac { x }{ 1-{ x }^{ 3 } }  }  } dx=$
Question 36 :
$\displaystyle \int {\frac{{xdx}}{{\sqrt {1 + {x^2} + \sqrt {{{(1 + {x^2})}^3}} } }}} $ is equal to :
Question 43 :
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 \} $, is
Question 45 :
The area included between the parabolas<br>$y=\dfrac { { x }^{ 2 } }{ 4a }$ and $y=\dfrac { 8{ a }^{ 3 } }{ { x }^{ 2 }+4{ a }^{ 2 } }$ is<br>
Question 46 :
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 47 :
What is the area of the region enclosed between the curve $y^2=2x$ and the straight line $y=x$ ?
Question 48 :
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 49 :
The value of $a$ for which the area between the curves ${y^2} = 4ax$ and ${x^2} = 4ay$ is $1\,sq.\,unit$, is-
Question 50 :
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 51 :
The area bounded by $x^2+y^2-2x=0$ & $y=\sin\displaystyle\frac{\pi x}{2}$ in the upper half of the circle is?
Question 52 :
Find the value of 'a' $\displaystyle \left ( 0< a< \frac{\pi }{2} \right )$ for which the area bounded by the curve $\displaystyle f(x)=\sin ^{3}x+\sin x,y=f(a)$ between $x = 0$ & $\displaystyle x=\pi $ is minimum
Question 53 :
For what value of 'a' is the area of the figure bounded by $\displaystyle y=\frac{1}{x}, y=\frac{1}{2x-1}$ $x = 2$ & $x = a$ equal to $\displaystyle ln\frac{4}{\sqrt{5}}$?
Question 55 :
The area bounded by $y=2-\left| 2-x \right|$ and $y=\frac { 3 }{ \left| x \right|  }$ is :
Question 56 :
Area of the region bounded by curves y=x log x and $y={ 2x-x }^{ 2 }$ is
Question 57 :
The area bounded by the curves $\sqrt{x}+\sqrt{y}=1$ and ${x}+{y}=1$ is ?
Question 58 :
The area bounded by $y=\sin ^{ -1 }{ x } , y=\cos ^{ -1 }{ x }$ and the $x-axis$, is given by
Question 59 :
The area bounded by the curve $ y = \dfrac { \sin { x }  }{ { x } } , x-$ axis and the ordinates $ x=0,x=\dfrac { \pi }{ { 4 } }$ is:
Question 60 :
Area enclosed by the graph of the function $y=\ln ^{2}x-1$ lying in the $4th$ quadrant is
Question 61 :
The area bounded by the region by the curves $ \displaystyle \begin{vmatrix}x\end{vmatrix} =1-y^{2} $ and $ \displaystyle \begin{vmatrix}x\end{vmatrix}+\begin{vmatrix}y\end{vmatrix}=1 $ is<br>
Question 62 :
The triangle formed by the tangent to the parabola $y^2=4x$ at the point whose abscissa lies in the interval $\left[a^2, 4a^2\right]$, the ordinate and the x-axis, has the greatest area equal to?
Question 63 :
The area of the triangle formed by the tan-gent and the normal at $(a,a)$ on the curve ${y^2} = \dfrac{{{x^3}}}{{2a - x}}$ and the line $x=2a$ is (in sq.units)
Question 64 :
Area bounded by curve $y = (x - 1)(x - 2)(x - 3)$ and x-axis between lines $x = 0, x = 3$
Question 65 :
The smaller area enclosed by $y=f(x)$, where $f(x)$ is polynomial of least degree satisfying $\displaystyle{ \left[ \lim _{ x\rightarrow 0 }{ 1+\frac { f\left( x \right)  }{ { x }^{ 3 } }  }  \right]  }^{ \tfrac { 1 }{ x }  }=e$ and the circle $x^2+y^2=2$ above the $x-$axis is
Question 66 :
The area of the smaller region in which the curve $y=\left [ \frac{x^{3}}{100}+\frac{x}{50} \right ],$ where[.] denotes the greatest integer function, divides the circle $\left ( x-2 \right )^{2}+\left ( y+1 \right )^{2}=4,$ is equal to<br><br><br><br><br><br><br><br>
Question 67 :
The area between the curves y=tan x, cot x and axis in the interval $\left[0,\pi   \right/2$]is ?
Question 68 :
Area bounded by Curve ${ y }^{ 2 }=4x,y$ axis and line y=3 is :
Question 69 :
Area bounded by the curves $y=\log _{ e }{ x } \quad$ and  $y={ \left( \log _{ e }{ x }  \right)  }^{ 2 }$ is ?<br/>
Question 70 :
Area of the region bounded by the curve $y = x^{2}$ and $y = sec^{-1} [sin^{2}x]$ (where [ . ] denotes the greatest integer function) is<br>
Question 71 :
The area of the region, bounded by the curves $y = \sin^{-1} x + x (1 - x)$ and $y = \sin^{-1} x - x (1 - x)$ in the first quadrant, is
Question 72 :
The area enclosed by the curves $y=sinx+$cosx and $y=|cosx-$sinx $|$over the interval $[0, \frac{\pi}{2}]$is<br><br>
Question 73 :
Consider two curves $C_1 : (y - \sqrt 3)^2 = 4 ( x - \sqrt2) $ and $ C_2 : x^2 + y^2 = ( 6 + 2 \sqrt2 ) x + 2 \sqrt{3y} - 6 ( 1 + \sqrt2)$ then
Question 74 : Match the following:<br/><table class="wysiwyg-table"><tbody><tr><td>List-I</td><td>List-II</td></tr><tr><td>1. Area of region bounded by $y=2x-x^{2}$ and $x-$axis</td><td>a. $\dfrac13$</td></tr><tr><td>2. Area of the region $\{(x, y):x^{2}\leq y\leq|x|\}$</td><td>b. $\dfrac12$</td></tr><tr><td>3. Area bounded by $y=x$ and $y=x^{3}$</td><td>c. $\dfrac23$</td></tr><tr><td>4. Area bounded by $y=x|x|$, ${x}$-axis and ${x}=-1,\ {x}=1$</td><td>d. $\dfrac43$</td></tr></tbody></table>The correct match for $1\ 2\ 3\ 4$ is
Question 75 :
Find the area bounded by $\displaystyle y = \cos ^{-1}x,y=\sin ^{-1}x$ and $y-$axis
Question 76 :
Let $g(x) = \cos x^{2}, f(x) = \sqrt {x}$, and $\alpha, \beta (\alpha < \beta)$ be the roots of the quadratic equation $18x^{2} - 9\pi x + \pi^{2} = 0$. Then the area (in $sq.\ units$) bounded by the curve $y = (gof)(x)$ and the lines $x = \alpha, x = \beta$ and $y = 0$, is
Question 77 :
The area of the region above the x-axis bounded by the curve $y=tan x, 0 \leq x\leq \frac {\pi}{2}$ and the tangent to the curve at $x=\frac {\pi}{4}$ is :
Question 78 :
The parabola $y=\dfrac{x^2}{2}$ divides the circle $x^2+y^2=8$ into two parts. Find the area of both parts.
Question 79 : Find the area enclosed between the curves $y^2- 2ye^{sin^{- 1}x} + x^2- 1 +[x] +e^{2sin^{- 1}x} = 0$ <div>and line x = 0 and $x=\frac{1}{2}$ is (where [.] denotes greatest integer function)<br/></div>
Question 80 :
Find the equation of the line passing through the origin and dividing the curvilinear triangle with vertex at the origin, bounded by the curves $y=2x-x^2, y=0$ and $x=1$ into two parts of equal area.
