If $z = ax + by$ where $a, b > 0$ subject to the constraints $x \leq 2, y \leq 2, x + y \geq 3, x \geq 0, y \geq 0$ has a minimum value at $(2, 1)$ only,then...

$A$ firm has to transport $1200$ packages using large vans which can carry $200$ packages each and small vans which can take $80$ packages each. The cost for engaging each large van is $Rs. 400$ and each small van is $Rs. 200$. Not more than $Rs. 3000$ is to be spent on the job and the number of large vans cannot exceed the number of small vans. Formulate this problem as an $LPP$ given that the objective is to minimize cost.

The maximum value of $z = 4x + y$ subject to the constraints $x + y \leq 5$, $2x + y \leq 7$, $3x + 2y \leq 11$, $x \geq 0$, $y \geq 0$ is

The maximum value of $z=3x+4y$,subject to the constraints $x+y \leq 40$,$x+2y \leq 60$ and $x, y \geq 0$ is

The difference between the maximum value and the minimum value of the objective function $z = 3x + y$ subject to the constraints $2x + 3y \leq 6$, $x + y \geq 1$, $x \geq 0$, $y \geq 0$ is....

$A$ wholesale merchant wants to start a cereal business with $Rs \ 24000$. Wheat costs $Rs \ 400$ per quintal and rice costs $Rs \ 600$ per quintal. He has a storage capacity of $200$ quintals of cereal. He earns a profit of $Rs \ 25$ per quintal on wheat and $Rs \ 40$ per quintal on rice. If he stores $x$ quintals of rice and $y$ quintals of wheat,then for maximum profit,the objective function is: