There are two types of fertilisers $F_{1}$ and $F_{2}$. $F_{1}$ consists of $10\%$ nitrogen and $6\%$ phosphoric acid,and $F_{2}$ consists of $5\%$ nitrogen and $10\%$ phosphoric acid. After testing the soil conditions,a farmer finds that she needs at least $14\,kg$ of nitrogen and $14\,kg$ of phosphoric acid for her crop. If $F_{1}$ costs $Rs\,6/kg$ and $F_{2}$ costs $Rs\,5/kg$,determine how much of each type of fertiliser should be used so that nutrient requirements are met at a minimum cost. What is the minimum cost?

$A$ dietician wishes to mix together two kinds of food $X$ and $Y$ in such a way that the mixture contains at least $10$ units of vitamin $A$,$12$ units of vitamin $B$,and $8$ units of vitamin $C$. The vitamin contents of one $kg$ of food are given below:

Food	Vitamin $A$	Vitamin $B$	Vitamin $C$
$X$	$1$	$2$	$3$
$Y$	$2$	$2$	$1$

One $kg$ of food $X$ costs $Rs. 16$ and one $kg$ of food $Y$ costs $Rs. 20$. Find the least cost of the mixture which will produce the required diet?

For the inequalities $x+y \leq 3$,$2x+5y \geq 10$,$x \geq 0$,$y \geq 0$,which of the following points lies in the feasible region?

The objective function $z=x_1+x_2$,subject to $x_1+x_2 \leq 10, -2x_1+3x_2 \leq 15, x_1 \leq 6, x_1, x_2 \geq 0$,has a maximum value at:

The shaded region in the following figure is the solution set of the inequations:

$A$ cottage industry manufactures pedestal lamps and wooden shades,each requiring the use of a grinding/cutting machine and a sprayer. It takes $2 \text{ hours}$ on the grinding/cutting machine and $3 \text{ hours}$ on the sprayer to manufacture a pedestal lamp. It takes $1 \text{ hour}$ on the grinding/cutting machine and $2 \text{ hours}$ on the sprayer to manufacture a shade. On any day,the sprayer is available for at most $20 \text{ hours}$ and the grinding/cutting machine for at most $12 \text{ hours}$. The profit from the sale of a lamp is $Rs. 5$ and that from a shade is $Rs. 3$. Assuming that the manufacturer can sell all the lamps and shades that he produces,how should he schedule his daily production in order to maximise his profit?