The minimum value for the $LPP$ $Z = 6x + 2y$,subject to $2x + y \geq 16$,$x \geq 6$,$y \geq 1$ is

$A$ dietician wishes to mix two types of foods in such a way that vitamin contents of the mixture contain at least $8$ $units$ of vitamin $A$ and $10$ $units$ of vitamin $C$. Food $'I'$ contains $2$ $units/kg$ of vitamin $A$ and $1$ $unit/kg$ of vitamin $C$. Food $'II'$ contains $1$ $unit/kg$ of vitamin $A$ and $2$ $units/kg$ of vitamin $C$. It costs Rs $50$ per $kg$ to purchase Food $'I'$ and Rs $70$ per $kg$ to purchase Food $'II'$. Formulate this problem as a linear programming problem to minimise the cost of such a mixture.

The minimum value of $Z = 3x + y$, subject to the constraints $2x + 3y \leq 6$, $x + y \geq 1$, $x \geq 0$, and $y \geq 0$ 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?

If $Z = 7x + y$ subject to $5x + y \geq 5$,$x + y \geq 3$,$x \geq 0$,$y \geq 0$,then the minimum value of $Z$ is