The corner points of the feasible region determined by the system of linear constraints are $(0,10), (10,15), (15,25), (0,30)$. Let $z = px + qy$,where $p, q > 0$. The condition on $p$ and $q$ so that the maximum of $z$ occurs at both the points $(15,25)$ and $(0,30)$ is . . . . . . .

The feasible region represented by the constraints $y - 2x \leq 4$, $x + y \geq 5$, $x \leq 4$, $y \geq 2$, and $x, y \geq 0$ 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?

For a linear programming problem, the objective function is $z = px + qy$, where $p, q > 0$. If at the corner points $(0, 10)$ and $(5, 5)$ the values of $z$ are $90$ and $60$ respectively, then the relation between $p$ and $q$ is . . . . . . .

Consider the following statements:
Statement $(I)$: In a $LPP$,the objective function is always linear.
Statement $(II)$: In a $LPP$,the linear inequalities on variables are called constraints.
Which of the following is correct?

Minimize $Z = 13x - 15y$ subject to the constraints: $x + y \leq 7$,$2x - 3y + 6 \geq 0$,$x \geq 0$,$y \geq 0$.