The maximum value of $z=50x+15y$ subject to the constraints $x+y \leq 60$,$5x+y \leq 100$,$x \geq 0$,$y \geq 0$ is at the point:

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

The maximum value of $z=3x+4y$,subject to the constraints $x+y \leq 40$,$x+2y \leq 60$ and $x, y \geq 0$ is

Corner points of the feasible region for an $\operatorname{LPP}$ are $(0,2), (3,0), (6,0), (6,8)$ and $(0,5)$. Let $F = 4x + 6y$ be the objective function. The minimum value of $F$ occurs at $....$

$A$ toy company manufactures two types of dolls,$A$ and $B$. Market research and available resources have indicated that the combined production level should not exceed $1200$ dolls per week and the demand for dolls of type $B$ is at most half of that for dolls of type $A$. Further,the production level of dolls of type $A$ can exceed three times the production of dolls of type $B$ by at most $600$ units. If the company makes a profit of $Rs. 12$ and $Rs. 16$ per doll respectively on dolls $A$ and $B$,how many of each should be produced weekly in order to maximize the profit?

The graphical solution set for the system of inequations $x-2y \leq 2$,$5x+2y \geq 10$,$4x+5y \leq 20$,$x \geq 0$,$y \geq 0$ is given by