The corner points of the feasible region of the objective function $Z = 3x + 9y$ are $(0, 10)$,$(5, 5)$,$(15, 15)$,and $(0, 20)$. Then,the minimum value of $Z$ is:

The corner points of the feasible region determined by the system of linear constraints are $(0,10), (5,5), (15,15), (5,25)$. 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 $(15,15)$ and $(5,25)$ is . . . . . . .

The corner points of the feasible region of an $LPP$ are $(0,2), (3,0), (6,0), (6,8)$ and $(0,5)$. Then the minimum value of $z = 4x + 6y$ occurs at:

The feasible solution for a $LPP$ is shown in the figure. Let $z=3x-4y$ be the objective function. The minimum value of $z$ occurs at:

Show that the minimum of $Z$ occurs at more than two points.
Minimise and Maximise $Z = 5x + 10y$
subject to $x + 2y \leq 120, x + y \geq 60, x - 2y \geq 0, x, y \geq 0$.

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: