The feasible region for an $LPP$ is shown in the figure. Let $z = 3x - 4y$ be the objective function. The minimum value of $Z$ is:

If an $LPP$ admits an optimal solution at two consecutive vertices of a feasible region,then:

The corner points of the feasible region determined by a system of linear constraints are $(0, 3), (1, 1)$ and $(3, 0)$. If the objective function is $z = px + qy$ where $p, q > 0$, then the condition on $p$ and $q$ such that the minimum of $z$ occurs at both $(3, 0)$ and $(1, 1)$ is . . . . . . .

Maximum value of $z = 3x + 4y$ subject to the constraints $x - y \leqslant -1$,$-x + y \leqslant 0$,and $x, y \geqslant 0$ is:

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 . . . . . . .

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: