A LPP $max\,z = ax+by$ having bounded feasible region with vertices on $(2,1), (3,2), (5,6)$ and $(1,7)$. Then which of following are TRUE?

A. $max\,z$ occurs at $(1,7)$ and $(5,6)$ if $b = 4a$

B. $max\,z$ occurs at $(1,7)$ and $(5,6)$ if $b = -4a$

C. $max\,z$ does not exist

D. if $max\,z$ occurs at $(1,7)$ and $(5,6)$, then $max\,z$ occurs for infinite number of value of $x$ and $y$

Choose the correct answer from the options given below:

The corner points of feasible region are:

A. (0, 240)

B. (0, 0)

C. (300, 0)

D. (120, 180)

E. (180, 120)

Choose the correct answer from the options given below:

The maximum value of $Z = 3x+2y$ subject to the constraints $x+2y \geq 8$, $x+y \leq 10, x \geq 0, y \geq 0$ is

The corner points of the feasible region for an L.P.P. are $(0, 10)$, $(5, 5)$, $(5, 15)$ and $(0, 30)$. If the objective function is $Z = \alpha x + \beta y$, where $\alpha, \beta > 0$, find the condition on $\alpha$ and $\beta$ so that the maximum of $Z$ occurs at both the corner points $(5, 15)$ and $(0, 30)$.

A feasible solution is :

For a two variables linear programming problem (LPP), which of the following statement is correct ?

The corner points of the bounded feasible region determined by the system of linear constraints of an LPP are (0, 0), (0, 8), (5, 0) and (4, 10). Let $z = 3x - 4y$ be the objective function, then the minimum value of $z$ is: