Corner points of the feasible region for an $\operatorname{LPP}$ are $(0,2), (3,0), (6,0), (6,8)$ and $(0,5)$. Let $F = 4x + 6y$ be the objective function. The minimum value of $F$ occurs at $....$

The point,at which the maximum value of $10x + 6y$ subject to the constraints $x + y \leq 12$,$2x + y \leq 20$,$x \geq 0$,$y \geq 0$ occurs,is

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

$A$ dietician has to develop a special diet using two foods $P$ and $Q$. Each packet (containing $30 \, g$) of food $P$ contains $12$ units of calcium,$4$ units of iron,$6$ units of cholesterol and $6$ units of vitamin $A$. Each packet of the same quantity of food $Q$ contains $3$ units of calcium,$20$ units of iron,$4$ units of cholesterol and $3$ units of vitamin $A$. The diet requires at least $240$ units of calcium,at least $460$ units of iron and at most $300$ units of cholesterol. How many packets of each food should be used to minimise the amount of vitamin $A$ in the diet? What is the minimum amount of vitamin $A$?

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...

$A$ manufacturer produces two models of bikes: Model $X$ and Model $Y$. Model $X$ takes $6$ man-hours to make per unit,while Model $Y$ takes $10$ man-hours per unit. There is a total of $450$ man-hours available per week. Handling and marketing costs are $Rs. 2000$ and $Rs. 1000$ per unit for Models $X$ and $Y$ respectively. The total funds available for these purposes are $Rs. 80,000$ per week. Profits per unit for Models $X$ and $Y$ are $Rs. 1000$ and $Rs. 500$ respectively. How many bikes of each model should the manufacturer produce so as to yield a maximum profit? Find the maximum profit.