In $L.P.P.$,the maximum value of the objective function $Z = 6x + 3y$ subject to the constraints $x + y \leq 5$,$x + 2y \geq 4$,$4x + y \leq 12$,$x, y \geq 0$ is:

The solution set of the constraints $2x + 3y \leq 6$,$x + 4y \leq 4$,$x \geq 0$,and $y \geq 0$ includes the point $\ldots$ as a corner point.

$A$ diet of a sick person must contain at least $4000$ units of vitamins,$50$ units of proteins,and $1400$ calories. Two foods $A$ and $B$ are available at a cost of ₹ $4$ and ₹ $3$ per unit respectively. If one unit of $A$ contains $200$ units of vitamins,$1$ unit of protein,and $40$ calories,while one unit of food $B$ contains $100$ units of vitamins,$2$ units of protein,and $40$ calories,formulate the problem so that the diet is the cheapest.

The maximum value of $Z = 5x + 2y$,subject to the constraints $2x - y \geq 2$,$x + 2y \leq 8$,and $x, y \geq 0$,is:

For the following shaded area,the linear constraints except $x, y \geq 0$ are

The function to be maximized is given by $Z=2x+y$. The feasible region for this function $Z$ is the shaded region shown in the figure. The maximum value of $Z$ is . . . . . . and occurs at the point . . . . . . .