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

The $LPP$ to maximize $z = 2x + 5y$ subject to the constraints $x + 3y \leq 6$, $2x + 6y \leq 18$, $x \geq 0$, $y \geq 0$ has:

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:

If for a linear programming problem the feasible region is bounded,then the objective function has . . . . . . .

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 maximum value of $Z=x+3y$ subject to the constraints $2x+y \leq 20$,$x+2y \leq 20$,$x \geq 0$,$y \geq 0$ is: