The minimum value of the objective function $z = x + 2y$ of an L.P.P. subject to constraints $2x + y \geq 3, \frac {x} {2} + 2y \geq 6, x \geq 0, y \geq 0$ is:

If the corner points of bounded feasible region for an LPP are (0,2) (3,0) (6,0) (6,8) and (0, 5) then the minimum value of the objective function f=4x+6y occur at

The corner points of the bounded feasible region determined by the system of linear constraints are (0, 10), (5, 5), (15, 15), (0, 20). Let z = px + qy where p, q > 0. Then the condition on p and q so that the maximum value of z occurs at (15, 15) and (0, 20) is

The linear constraints for which the shaded region in the figure are:

For the linear programming problem (LPP),

$Maximize Z = 7x + 9y$, subject to constraints,

$x - y \le -1, -x + y \le 0, x, y \ge 0.$

Which of the following is correct?

For the objective function Z=-4x + 6y subject to the constraints 3x + 2y ≥ 5, 7x + 2y ≤ 9, x ≥ 0, y ≥ 0, the maximum value of Z occurs at $(a, b)$ and the minimum value of Z occurs at $(p, q)$ then the value of $\frac{a}{p} + \frac{b}{q}$ is:

For LPP: Maximize $z = 2x + 3y$ subject to the constraints

$x + y \geq 2$, $x + 2y \geq 3$, $x \geq 0$, $y \geq 0$,

which of the following graph represents the feasible region of the above LPP as shaded portion?