For a Linear Programming Problem $(LPP)$,if the objective function is $Z = 4x + 3y$ and the corner points of the bounded feasible region are $(0,0), (25,5), (16,16),$ and $(5,24)$,then the maximum value of $Z$ occurs at the point . . . . . . .

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 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 corner points of the feasible region determined by the system of linear constraints are $(0, 10)$, $(5, 5)$, $(15, 15)$, and $(0, 20)$. Let $z = px + qy$ where $p, q > 0$. The condition on $p$ and $q$ such that the maximum value of $z$ occurs at both points $(15, 15)$ and $(0, 20)$ is

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 . . . . . . .

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