For the $LPP$,maximize $z=x+4y$ subject to the constraints $x+2y \leq 2$,$x+2y \geq 8$,$x, y \geq 0$.

The difference between the maximum value and minimum value of the objective function $z = 3x + 5y$ of a linear programming problem subject to constraints $5x + 10y \leq 50$, $x + y \geq 1$, $y \leq 4$ and $x \geq 0, y \geq 0$ is $3\lambda$. Then the value of $\lambda$ is

For the following shaded region,the linear constraints are:

Solution of the following $LP$ problem
Minimize $z = -3x + 2y$
subject to $0 \leq x \leq 4, 1 \leq y \leq 6, x + y \leq 5$ is $.....$

The difference between the maximum and minimum values of the objective function $Z = 3x + 5y$, subject to the constraints $x + 3y \leq 60$, $x + y \geq 10$, $x - y \leq 0$, and $x, y \geq 0$, is

For the linear programming problem, $x + 2y \leq 10$, $3x + y \leq 12$, $x, y \geq 0$, the maximum value of $z = 5x + 10y$ occurs at every point on the line segment joining the points..