Question 1 :
$\displaystyle z=10x+25y$ subject to&nbsp;$\displaystyle 0\le x\le 3$ and&nbsp;$\displaystyle 0\le y\le 3,x+y\le 5$ then the maximum value of z is<br>

Question 2 :
<span>LP theory states that the optimal solution to any problem will lie at</span>

Question 3 :
<span>Which of the following is an essential condition in a situation for linear programming to be useful?</span>

Question 4 :
<span>Which of the following is a property of all linear programming problems?</span>

Question 5 :
A firm manufactures three products $A,B$ and $C$. Time to manufacture product $A$ is twice that for $B$ and thrice that for $C$ and if the entire labour is engaged in making product $A,1600$ units of this product can be produced.These products are to be produced in the ratio $3:4:5.$ There is demand for at least $300,250$ and $200$ units of products $A,B$ and $C$ and the profit earned per unit is Rs.$90,$&nbsp;Rs$40$ and Rs.$30$ respectively.<br><table class="wysiwyg-table"><tbody><tr><td>Raw<br>material</td><td>Requirement per unit product(Kg)<br>A</td><td>Requirement per unit product(Kg)<br>B</td><td>Requirement per unit product(Kg)<br>C</td><td>Total availability (kg)</td></tr><tr><td>$P$</td><td>$6$</td><td>$5$</td><td>$2$</td><td>$5,000$</td></tr><tr><td>$Q$</td><td>$4$</td><td>$7$</td><td>$3$</td><td>$6,000$</td></tr></tbody></table>Formulate the problem as a linear programming problem and find all the constraints for the above product mix problem.

Question 6 :
Equation of normal drawn to the graph of the function defined as $f(x)=\dfrac{\sin x^2}{x}$, $x\neq 0$ and $f(0)=0$ at the origin is?

Question 7 :
One disadvantage of using North-West Corner rule to find initial solution to the transportation&#160;problem is that

Question 9 :
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 10 :
To write the dual; it should be ensured that&#160;&#160;<br/>I. All the primal variables are non-negative.<br/>II. All the bi values are non-negative.<br/><span>III.&#160;All the constraints are $&#8804;$ type if it is maximization problem and $&#8805;$ type if it is a minimization problem.</span>

Question 13 :
The solution of the differential equation&nbsp;${e^{ - x}}(y + 1)dy + (co{s^2}x - sin2x)ydx = 0$ subject to the conditions $y(0) = 1$

Question 15 :
Check whether the function is homogenous or not. If yes then find the degree of the function<br/>$g(x)=8x^4$.

Question 18 :
Which of the following is true regarding the function $f(x, y)= x^4 \sin \dfrac{x}{y}$ ?

Question 19 :
The degree of the differential equation<br>$\displaystyle { \left[ 1+{ \left( \frac { dy }{ dx } &nbsp;\right) &nbsp;}^{ 2 } \right] &nbsp;}^{ { 5 }/{ 3 } }=\frac { { d }^{ 2 }y }{ { dx }^{ 2 } }&nbsp;$<br>

Question 21 :
Find the value of $k$ for the function: $2x^2y+3xyz+z^k$ to be homogenous.

Question 22 :
Solve $\displaystyle \left ( 4x+6y+3 \right )dx= \left ( 6x+9y+2 \right )dy$

Question 24 :
If $\displaystyle{\int \frac{\displaystyle dx}{\displaystyle&#160;\sqrt{x}+\displaystyle \sqrt[3]{x}}}=a\sqrt{x}+b(\sqrt[3]{x})+c(\sqrt[6]{x})+d\: \ln(\sqrt[6]{x}+1)+e$, $e$ being arbitrary constant then. Find the value of $20a + b + c + d.$<br/>

Question 25 :
The integral $\displaystyle\int {\dfrac{{2{x^{12}} + 5{x^9}}}{{{{\left( {{x^5} + {x^3} + 1} \right)}^3}}}dx} $ is equal to

Question 26 :
If $f\left( \cfrac { 3x-4 }{ 3x+4 }&nbsp; \right) =x+2$, then $\int { f(x) } dx$ is

Question 27 :
If $ \displaystyle f(x)=\lim_{n\rightarrow \infty }(2x+4x^{3}+......+2^{n}x^{2n-1})\left ( 0&lt;x&lt;\frac{1}{\sqrt{2}} \right )$, then the value of $\displaystyle\int f(x) dx$ is equal to<br/><div>$\textbf{Note}$: $c$ is the constant of integration.</div>

Question 28 :
$\displaystyle \int \sqrt {1+x \sqrt {1+(x+1) \sqrt {1+(x+2) (x+4)}}}$ $dx$ is equal to

Question 31 :
Solution of the differential equation<br>$\left \{\dfrac {1}{x} - \dfrac {y^{2}}{(x - y)^{2}}\right \} dx + \left \{\dfrac {x^{2}}{(x - y)^{2}} - \dfrac {1}{y}\right \} dy = 0$ is<br>(where $c$ is arbitrary constant).

Question 34 :
The area (in sq. units) of the region&nbsp;$\{ x \in R:x \ge ,y \ge 0,y \ge x - 2\ $&nbsp; and&nbsp; $y \le \sqrt x \} $, is

Question 35 :
The value of $a$ for which the area between the curves ${y^2} = 4ax$ and ${x^2} = 4ay$ is $1\,sq.\,unit$, is-

Question 37 :
Find the area of the closed figure bounded by the following curves $y = 2$ $\cos^2 x (1 \, + \,&#160; \sin^2 x)$ on the interval $[0, 2\pi]$ and the abscissa axis.

Question 38 :
What is the area of the region enclosed between the curve $y^2=2x$ and the straight line $y=x$ ?

Question 39 :
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 40 :
The area of the figure bounded by $f\left(x\right)=\sin{x}, g\left(x\right)=\cos{x}$ in the first quadrant is:

Question 41 :
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 $b&gt;1$, then $f\left(x\right)$ is

Question 42 :
If the area bounded by the x-axis, curve $y=f(x)$ and the lines $x=1$, $x=b$ is equal to $\sqrt{b^2+1}-\sqrt{2}$ for all $b &gt; 1$, then $f(x)$ is

Question 43 :
The area bounded by the curve $y = f\left( x \right)$, above the $x$-axis, between $x = a$ and $x = b$ is:

Question 44 :
Points of inflexion of the curve<br>$y = x^4 - 6x^3 + 12x^2 + 5x + 7$ are

Question 45 :
The area of the region bounded by the curve $x={ y }^{ 2 }-2$ and $x=y$ is

Question 46 :
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 47 :
If area bounded by the curves $x=at^2$ and $y=ax^2$ is $1$, then a$=$ __________.

Question 48 :
Area bounded by curve $x\left( { x }^{ 2 }+p \right) =y-1$ with $y=1$ $p&lt;0$is -

Question 49 :
Find the area of the closed figure bounded by the following curves y = cos x (0 $\leqslant&#160;&#160;x \leqslant&#160;&#160;\pi/2)$, y = 0, x = 0, and a straight. line which is a tangent to the curve y = cos x at the point x = $\pi/4$.