The maximum value of $z=6x+8y$ subject to the constraints $x-y \geq 0$,$x+3y \leq 12$,$x \geq 0$,$y \geq 0$ is:

The minimum value of $Z = 5x + 8y$ subject to the constraints $x + y \geq 5$,$0 \leq x \leq 4$,$y \geq 2$ is:

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

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?

The solution of the linear programming problem,maximize $Z = 3x_{1} + 5x_{2}$ subject to $3x_{1} + 2x_{2} \leq 18$,$x_{1} \leq 4$,$x_{2} \leq 6$,$x_{1} \geq 0$,$x_{2} \geq 0$ is:

$A$ cooperative society of farmers has $50$ hectares of land to grow two crops $X$ and $Y$. The profit from crops $X$ and $Y$ per hectare are estimated as Rs $10,500$ and Rs $9,000$ respectively. To control weeds,a liquid herbicide has to be used for crops $X$ and $Y$ at rates of $20$ litres and $10$ litres per hectare. Further,no more than $800$ litres of herbicide should be used in order to protect fish and wildlife using a pond which collects drainage from this land. How much land should be allocated to each crop so as to maximise the total profit of the society?