The vertices of the feasible region for the constraints $x+y \leq 4$,$x \leq 2$,$y \leq 1$,$x+y \geq 1$,$x, y \geq 0$ are

$A$ firm has to transport $1200$ packages using large vans which can carry $200$ packages each and small vans which can take $80$ packages each. The cost for engaging each large van is $Rs. 400$ and each small van is $Rs. 200$. Not more than $Rs. 3000$ is to be spent on the job and the number of large vans cannot exceed the number of small vans. Formulate this problem as an $LPP$ given that the objective is to minimize cost.

The solution set for minimizing the function $z = x + y$ with constraints $x + y \geqslant 2$,$x + 2y \leqslant 8$,$y \leqslant 3$,$x, y \geqslant 0$ contains

The shaded area in the figure given below is a solution set of a system of inequations. The minimum value of the objective function $Z = 3x + 5y$,subject to the linear constraints given by this system of inequations,is:

The maximum value of $Z = 10x + 25y$ subject to $0 \leq x \leq 3$,$0 \leq y \leq 3$,and $x + y \leq 5$ occurs at the point:

$A$ manufacturer makes two types of toys $A$ and $B$. Three machines are needed for this purpose and the time (in $minutes$) required for each toy on the machines is given below:

Types of Toys	Machine-$I$	Machine-$II$	Machine-$III$
$A$	$12$	$18$	$6$
$B$	$6$	$0$	$9$

Each machine is available for a maximum of $6 \, hours$ $(360 \, minutes)$ per day. If the profit on each toy of type $A$ is $Rs. \, 7.50$ and that on each toy of type $B$ is $Rs. \, 5$,find the number of toys of each type that should be manufactured in a day to get maximum profit.