Define L.P.P and its advantages.

A manufacturing company makes two types of teaching aids $M$ and $N$ of mathematics for class XII. Each type of $M$ requires $9$ labour hours of fabricating and $1$ labour hour for finishing. Each type of $N$ requires $12$ labour hour for fabricating and $3$ labour hour for finishing. For fabricating and finishing, the maximum labour hours available per week are $180$ and $30$ respectively. The company makes a profit of $₹80$ on each piece of type $M$ and $₹120$ on each piece of type $N$. How many pieces of type $M$ and type $N$ should be manufactured per week to get a maximum profit? Make it as an LPP and solve graphically. What is the maximum profit per week?

A manufacturer of a line of patent medicines is preparing a production plan on medicines $A$ and $B$. There are sufficient ingredients available to make $20000$ bottles of $A$ and $40000$ bottles of $B$ but there are only $45000$ bottles into which either of the medicines can be put. Furthermore, it takes $3$ hours to prepare enough material to fill $1000$ bottles of $A$ and it takes $1$ hour to prepare enough material to fill $1000$ bottles of $B$, and there are $66$ hours available for this operation. The profit is $₹8$ per bottle for $A$ and $₹7$ per bottle for $B$. How should the manufacture schedule the production in order to maximize his profit? Also, find the maximum profit.

Solve the systems of simultaneous inequations : $2x+y>1$and $2x-y\ge -3$.

Anil wants to invest at the most $₹12000$ in bonds $M$ and $N$. According to rules, he has to invest at least $₹2000$ in bond $M$ and at least $₹4000$ in bond $N$. if the rate of interest of bond $M$ is $8\%$ per annum and on bond $N$, it is $10\%$per annum, how should he invest his money for maximum interest? Solve the problem graphically.

A firm manufactures two types of products, $A$ and $B$, and sells them at a profit of $₹2$ on type $A$ and $₹2$on type $B$. Each product is processed on two machines, $M_1$ and $M_2$. Type $A$ requires one minute of processing time on $M_1$ and two minutes on $M_2$. Type $B$ requires one minute on $M_1$ and one minute on $M_2$. The machine $M_1$ is available for not more than $6$ hours $40$ minutes while $M_2$ is available for at most $10$ hours a day. Find how many products of each type the firm should produce each day in order to get maximum profit.

Find the linear constraints for which the shaded area in the figure given is the solution set.

A medicine company has factories at two places, $X$ and $Y$. From these places, supply is made to each of its three agencies situated at $P, Q$ and $R$. the monthly requirement of the agencies are respectively $40$ packets, $40$ packets and $50$ packets of medicine, while the production capacity of the factories at $X$ and $Y$ are $60$ packets and $70$packets respectively. The transportation costs per packet from the factories to the agencies are given as follows.

How many packets from each factory should be transported to each agency so that the cost of transportation is minimum? Also, find the minimum cost.

Graph the solution set of the inequality $x+y\ge 4$.

A dealer wishes to purchase a number of fans and sewing machines. He has only $₹5760$ to invest and space for at most $20$ items. A fan costs him $₹360$ and a sewing machine,$₹240$. He expects to gain $₹22$ on a fan and $₹18$ on a sewing machine. Assuming that he can sell all the items he can buy, how should he invest the money in order to maximise the profit?

Graph the solution set of the inequality $x+2y>1$.

A company manufactures two types of toys, $a$ and $b$. Type $a$ requires $5$ minutes each for cutting and $10$ minutes each for assembling. Type $b$ required $8$ minutes each for cutting and $8$ minutes each for assembling. There are $3$ hours available for cutting and $4$ hours available for assembling in a day. The profit is $₹50$ each on type $a$ and $₹60$ each on type $b$. how many toys of each types should the company manufactures in a day to maximise the profit?

Find the maximum and minimum values of $Z=2x+y$, subject to the constraints

$x+3y\ge 6, x-3y\le 3, 3x+4y\le 24,$

$-3x+2y\le 6, 5x+y\ge 5, x\ge 0 \mathrm{and} y\ge 0.$

Show that the solution set of the following linear constraints is empty: $x-2y\ge 0, 2x-y\le -2, x\ge 0$ and $y\ge 0$.

Maximise $Z=60x+15y$, subject to the constraints $x+y\le 50,3x+y\le 90,x\ge 0,y\ge 0.$

Solve the system of simultaneous inequations : $x-2y\ge 0, 2x-y\le -2$.

Mr. Dass wants to invest $₹12000$ in public provident fund ($\mathrm{PPF}$) and in national bonds. He has to invest at least $₹1000$ in $\mathrm{PPF}$ and at least $₹2000$ in bonds. If the rate of interest on $\mathrm{PPF}$ is $12\%$ per annum and that on bonds is $15\%$ per annum, how should he invest the money to earn maximum annual income? Also find the maximum annual income.

Graph the solution set of the inequality $x - y\le 3$.

Solve the system of simultaneous inequations : $3x+4y\ge 12, x\ge 0, y\ge 1$ and $4x+7y\le 28$.

A dealer wishes to purchase a number of fans and sewing machines. He has only $₹5760$ to invest and has space for at most $20$ items. A fan costs him $₹360$ and a sewing machine $₹240$. He expects to sell a fan at a profit of $₹22$ and a sewing machine at a profit of $₹18$. Assuming that he can sell all the items that he buys, how should he invest his money to maximize the profit? Solve the graphically and find the maximum profit.