The corner points of the feasible region are $(0, 6)$,$(3, 3)$,$(9, 9)$,and $(0, 12)$. What is the maximum value of the objective function $z = 6x + 12y$?

In a linear programming problem $(LPP)$, if the objective function $Z = ax + by$ has the same maximum value at two distinct corner points, then the number of points at which $Z_{max}$ occurs is

The feasible region for a $LPP$ is shown in the following figure. Evaluate $Z = 4x + y$ at each of the corner points of this region. Find the minimum value of $Z$,if it exists.

The corner points of the feasible region determined by the system of linear constraints are $(0,10), (10,15), (15,25), (0,30)$. Let $z = px + qy$,where $p, q > 0$. The condition on $p$ and $q$ so that the maximum of $z$ occurs at both the points $(15,25)$ and $(0,30)$ is . . . . . . .

The corner points of the feasible region determined by the system of linear constraints are $(2, 72)$,$(15, 20)$,and $(40, 15)$. Let $Z = 6x + 3y$ be the objective function. The minimum value of $Z$ occurs at:

The solution for minimizing the function $z = x + y$ under an $L$.$P$.$P$. with constraints $x + y \geqslant 2$,$x + 2y \leqslant 8$,$y \leqslant 3$,$x, y \geqslant 0$ is