For a linear programming problem,the objective function is $Z = 8000x + 12000y$. If the corner points of the feasible region are $(0,0)$,$(20,0)$,$(12,6)$,and $(0,10)$,then the maximum value of $Z$ occurs at which corner point?

The feasible region for a $LPP$ is shown in the figure. Find the maximum value of $Z=11x+7y$.

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 objective function of a Linear Programming Problem $(LPP)$ defined over a convex set attains its optimum value at:

The minimum value of $Z = 2x + 3y$ for the system of linear constraints: $2x + 4y \leq 12$,$x + y \leq 3$,$x \geq 0$,and $y \geq 0$ is . . . . . . .

The feasible solution for a Linear Programming Problem $(LPP)$ is shown in the figure. Let $z = 3x - 4y$ be the objective function. The value of (Maximum value of $z$ + Minimum value of $z$) is equal to $....$