The constraints $-x+y \leq 1, -x+3y \leq 9, x \geq 0, y \geq 0$ define a:

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 determined by the system of linear constraints are $(0,10), (5,5), (15,15), (0,20)$. 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 $(0,20)$ is $\ldots \ldots$

Minimise $Z = 3x + 2y$ subject to the constraints:
$x + y \geqslant 8$ ... $(1)$
$3x + 5y \leqslant 15$ ... $(2)$
$x \geqslant 0, y \geqslant 0$ ... $(3)$

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 $\ldots$

If $x+y \leq 2, x \geq 0, y \geq 0$,the point at which the maximum value of $3x+2y$ is attained will be: