If the vertices of a feasible region are $O(0,0), A(10,0), B(0,20), C(15,15)$,then the minimum value of the objective function $Z = 10x - 20y + 30$ 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 vertices of the feasible region determined by some linear constraints are $(0,2), (1,1), (3,3), (1,5)$. 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 $(3,3)$ and $(1,5)$ is . . . . . . .

Corner points of the bounded feasible region for an $LP$ problem are $(0,4), (6,0), (12,0), (12,16)$ and $(0,10)$. Let $z = 8x + 12y$ be the objective function. Match the following:
$(i)$ Minimum value of $z$ occurs at $\ldots$
$(ii)$ Maximum value of $z$ occurs at $\ldots$
$(iii)$ Maximum of $z$ is $\ldots$
$(iv)$ Minimum of $z$ is $\ldots$

In a Linear Programming Problem ($L$.$P$.$P$.), the corner points of the feasible region defined by the constraints $3x - y \geq 6$, $x \leq 3$, $y \leq 2$, $y \geq 0$, and $x \geq 0$ are:

Maximize $Z=3x+4y$,subject to the constraints: $x+y \leq 1, x \geq 0, y \geq 0$.