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?

Consider a $LPP$ given by minimize $Z = 6x + 10y$. Subject to $x \geq 6, y \geq 2, 2x + y \geq 10, x \geq 0, y \geq 0$. Redundant constraints in this $LPP$ are $....$

For the objective function $Z = 4x + y$ subject to the constraints $x + y \leq 50$,$3x + y \leq 90$,$x \geq 0$,$y \geq 0$,whose corner points of the feasible region are $(0,0)$,$(30,0)$,$(20,30)$,and $(0,50)$,the maximum value of $Z$ is . . . . . . .

Which of the following terms is not used in a linear programming problem?

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 . . . . . . .

Solve the following Linear Programming Problem graphically:
Maximise $Z = 3x + 2y$
subject to the constraints:
$x + 2y \leq 10$
$3x + y \leq 15$
$x, y \geq 0$