Question 4 :
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 $(0, 30)$is _______.

Question 7 :
What is the solution of $x\le 4,y\ge 0$ and $x\le -4,y\le 0$ ?

Question 8 :
The number of points in $\\ \left( -\infty ,\infty \right) $ for which ${ x }^{ 2 }-x\sin { x } -\cos { x } =0$, is

Question 9 :
The given table shows the number of cars manufactured in four different colours on a particular day. Study it carefully and answer the question.<br/><table class="table table-bordered"><tbody><tr><td rowspan="2">&#160;<b>Colour</b></td><td colspan="3"><b>&#160; &#160;Number of cars manufactured</b></td></tr><tr><td><b>&#160;Vento</b></td><td><b>&#160;Creta</b></td><td><b>WagonR&#160;</b></td></tr><tr><td>&#160;Red</td><td>&#160;65</td><td>&#160;88</td><td>&#160;93</td></tr><tr><td>&#160;White</td><td>&#160;54</td><td>&#160;42</td><td>&#160;80</td></tr><tr><td>&#160;Black</td><td>&#160;66</td><td>&#160;52</td><td>&#160;88</td></tr><tr><td>&#160;Sliver</td><td>37</td><td>&#160;49</td><td>&#160;74</td></tr></tbody></table>What was the total number of black cars manufactured?

Question 10 :
Solving an integer programming problem by rounding off answers obtained by solving it as a linear programming problem (using simplex), we find that

Question 12 :
In order for a linear programming problem to have a unique solution, the solution must exist

Question 16 :
In Graphical solution the feasible solution is any solution to a LPP which satisfies

Question 18 :
The corner points of the feasible region are $A(0,0),B(16,0),C(8,16)$ and $D(0,24)$. The minimum value of the objective function $z=300x+190y$ is _______

Question 19 :
Which of the following is an essential condition in a situation for linear programming to be useful?