Question

The corner points of the feasible region determined by the following system of linear inequalities $2x+y\le 10,x+3y\le 15,x,y\ge 0$ are $(0,0)(5,0)$ $(3,4)$ and $(0,5)$. Let $Z=px+qy$, where $p,q>0$. Condition on $p$ and $q$, so that the maximum of $Z$ occurs at both $(3,4)$ and $(0,5)$, is

• A. $p=q$
• B. $p=2q$
• C. $p=3q$
• D. $q=3p$

Answer 1

Answer: (D)

The maximum value of $Z$ is unique.
It is given that the maximum value of $Z$ occurs at two points $(3,4)$ and $(0,5)$. Value of $Z$ at $(3,4)=$ Value of $Z$ at $(0,5)$
$\Rightarrow p(3)+q(4)=p(0)+q(5),3p+4q=5q\Rightarrow 3p=q$