In a linear programming problem $(LPP)$, if the objective function $Z = ax + by$ has the same maximum value at two distinct corner points, then the number of points at which $Z_{max}$ occurs is

If for a linear programming problem the feasible region is bounded,then the objective function has . . . . . . .

For a linear programming problem,the objective function is $Z = 10500x + 9000y$. If the corner points of the bounded feasible region are $(0,0)$,$(40,0)$,$(30,20)$,and $(0,50)$,then the maximum value of $Z$ is . . . . . . .

The point at which the maximum value of $Z = 3x + 2y$ subject to the constraints $x + 2y \leq 2$,$x \geq 0$,$y \geq 0$ occurs is $.....$

Solve the following linear programming problem graphically:
Minimise $Z = 200x + 500y$.......$(1)$
subject to the constraints:
$x + 2y \geqslant 10$.......$(2)$
$3x + 4y \leqslant 24$.....$(3)$
$x \geqslant 0, y \geqslant 0$......$(4)$

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