The feasible region represented by the given constraints $2x + 3y \geqslant 12$,$-x + y \leqslant 3$,$x \leqslant 4$,$y \geqslant 3$ is denoted by

The difference between the maximum and minimum values of the objective function $Z = 3x + 5y$, subject to the constraints $x + 3y \leq 60$, $x + y \geq 10$, $x - y \leq 0$, and $x, y \geq 0$ is

The Linear Programming Problem ($L$.$P$.$P$.) to minimize $z = 30x + 20y$ subject to the constraints $x + y \leqslant 8$,$x + 2y \geqslant 4$,$6x + 4y \geqslant 12$,$x \geqslant 0$,and $y \geqslant 0$ has:

The shaded area in the given figure is a solution set for some system of inequations. The maximum value of the function $z=10x+25y$ subject to the linear constraints given by the system is

$A$ firm has to transport $1200$ packages using large vans which can carry $200$ packages each and small vans which can take $80$ packages each. The cost for engaging each large van is $Rs. 400$ and each small van is $Rs. 200$. Not more than $Rs. 3000$ is to be spent on the job and the number of large vans cannot exceed the number of small vans. Formulate this problem as an $LPP$ given that the objective is to minimize cost.

If $Z = 7x + y$ subject to $5x + y \geq 5$,$x + y \geq 3$,$x \geq 0$,$y \geq 0$,then the minimum value of $Z$ is