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

In order to supplement daily diet,a person wishes to take some $X$ and some $Y$ tablets. The contents of iron,calcium and vitamins in $X$ and $Y$ (in milligrams per tablet) are given as below:

Tablets	Iron	Calcium	Vitamin
$X$	$6$	$3$	$2$
$Y$	$2$	$3$	$4$

The person needs at least $18$ milligrams of iron,$21$ milligrams of calcium and $16$ milligrams of vitamins. The price of each tablet of $X$ and $Y$ is $Rs. 2$ and $Rs. 1$ respectively. How many tablets of each should the person take in order to satisfy the above requirement at the minimum cost?

For the $LPP$,minimize $z = x_{1} + x_{2}$ subject to the constraints $5x_{1} + 10x_{2} \geq 0$,$x_{1} + x_{2} \leq 1$,$x_{2} \leq 4$ and $x_{1}, x_{2} \geq 0$.

For the following shaded region,the linear constraints are:

The maximum value of $z=4x+2y$,subject to the constraints $3x+4y \geqslant 12$,$x+y \leqslant 5$,$x, y \geqslant 0$ is

The $L$.$P$.$P$. to maximize $z=x+y$,subject to $x+y \leq 30, x \leq 15, y \leq 20, x+y \geq 15$,and $x, y \geq 0$ has