The feasible region for a $LPP$ is shown in the figure. Find the minimum value of $Z=11x+7y$.

The following five inequalities form the feasible region: $2x - y \leq 8$,$x + y \leq 20$,$-x + y \geq -10$,$x \geq 0$,$y \geq 0$. Which of the following is a redundant constraint?

The corner points of the feasible region determined by the system of linear constraints are $(0,10), (5,5), (15,15), (5,25)$. Let $z = px + qy$ where $p, q > 0$. The condition on $p$ and $q$ so that the maximum of $z$ occurs at both the points $(15,15)$ and $(5,25)$ is . . . . . . .

$A$ fruit grower can use two types of fertilizer in his garden,brand $P$ and brand $Q$. The amounts (in $kg$) of nitrogen,phosphoric acid,potash,and chlorine in a bag of each brand are given in the table. Tests indicate that the garden needs at least $240\,kg$ of phosphoric acid,at least $270\,kg$ of potash and at most $310\,kg$ of chlorine. If the grower wants to minimize the amount of nitrogen added to the garden,how many bags of each brand should be used? What is the minimum amount of nitrogen added in the garden (in $,kg$)?

	Brand $P$ ($kg$ per bag)	Brand $Q$ ($kg$ per bag)
Nitrogen	$3$	$3.5$
Phosphoric acid	$1$	$2$
Potash	$3$	$1.5$
Chlorine	$1.5$	$2$

The corner points of the feasible region determined by the system of linear constraints are $(2, 72)$,$(15, 20)$,and $(40, 15)$. Let $Z = 6x + 3y$ be the objective function. The minimum value of $Z$ occurs at:

Let $x$ and $y$ be optimal solutions of a Linear Programming $(LP)$ problem. Then,which of the following is true?