The corner points of the feasible region are $(0,10), (5,5), (15,15), (0,20)$. The maximum value of $Z = 3x + 9y$ is . . . . . . .

Corner points of the feasible region determined by the system of linear constraints are $(0,3)$,$(1,1)$ and $(3,0)$. Let $Z = px + qy$,where $p, q > 0$. The condition on $p$ and $q$ so that the minimum value of $Z$ occurs at $(3,0)$ and $(1,1)$ is . . . . . . .

The maximum value of $Z = 60x + 10y$ whose corner points are $(10, 0)$,$(2, 4)$,$(1, 5)$,and $(0, 8)$ is . . . . . . .

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

An aeroplane can carry a maximum of $200$ passengers. $A$ profit of $Rs. 1000$ is made on each executive class ticket and a profit of $Rs. 600$ is made on each economy class ticket. The airline reserves at least $20$ seats for executive class. However,at least $4$ times as many passengers prefer to travel by economy class than by the executive class. Determine how many tickets of each type must be sold in order to maximize the profit for the airline. What is the maximum profit?

Show that the minimum of $Z$ occurs at more than two points.
Maximize $Z = x + y$,subject to $x - y \leq -1$,$-x + y \leq 0$,$x, y \geq 0$.