The maximum value of $z = 4x + y$ subject to the constraints $x + y \leq 5$, $2x + y \leq 7$, $3x + 2y \leq 11$, $x \geq 0$, $y \geq 0$ is

Two godowns $A$ and $B$ have grain capacity of $100$ quintals and $50$ quintals respectively. They supply to $3$ ration shops,$D$,$E$ and $F$ whose requirements are $60, 50$ and $40$ quintals respectively. The cost of transportation per quintal from the godowns to the shops are given in the following table:

Transportation cost per quintal (in $Rs$)
From/To $A$ $B$
$D$ $6$ $4$
$E$ $3$ $2$
$F$ $2.50$ $3$

How should the supplies be transported in order that the transportation cost is minimum? What is the minimum cost?

The feasible region of an $LPP$ is shown in the figure. If $z=11x+7y$,then the maximum value of $z$ occurs at

The $L$.$P$.$P$. to maximize $Z = x + y$,subject to $x + y \leq 1$,$2x + 2y \geq 6$,$x \geq 0$,$y \geq 0$ has

The constraints $-x_{1} + x_{2} \leq 1$,$-x_{1} + 3x_{2} \leq 9$,$x_{1}, x_{2} \geq 0$ define:

The optimal solution of the $L$.$P$.$P$. Maximize $Z = 8x + 3y$ subject to the constraints $x + y \leq 3, 4x + y \leq 6, x \geq 0, y \geq 0$ is