Q. Minimize $Z=-3x+4y$, subject to $x+2y\le 8$, $3x+2y\le 12$, $x\ge 0$, $y\ge 0$.

Q. Maximise $Z=3x+2y$
subject to $x+2y\le 10,3x+y\le 15,x,y\ge 0$

Q. Minimise $Z=x+2y$
subject to $2x+y\ge 3,x+2y\ge 6,x,y\ge 0$

Q. Minimise and Maximise $Z=x+2y$
subject to $x+2y\ge 100,2x-y\le 0,2x+y\le 200;x,y\ge 0$

Q. Maximize $Z=5x+3y$
subject to $3x+5y\le 15,5x+2y\le 10,x\ge 0,y\ge 0$

Q. Maximize $Z=3x+4y$
subject to the constraints: $x+y\le 4,x\ge 0,y\ge 0$

Q. Maximise $Z=-x+2y$, subject to the constraints:
$x\ge 3,x+y\ge 5,x+2y\ge 6,y\ge 0$

Q. Minimise $Z=3x+5y$
such that $x+3y\ge 3,x+y\ge 2,x,y\ge 0$

Q. Minimise and Maximise of $Z=5x+10y$ subject to $x+2y\le 120,x+y\ge 60,x-2y\ge 0,x,y\ge 0$

Q. Maximise $Z=x+y$, subject to $x-y\le -1,x-y\ge 0,x,y\ge 0$

Q. A manufacturer produces nuts and bolts. It takes $1$ hour of work on machine $A$ and $3$ hours on machine $B$ to produce a package of nuts. It takes $3$ hours on machine $A$ and $1$ hour on machine $B$ to produce a package of bolts. He earns a profit, of Rs.$17.50$ per package on nuts and Rs.$7.00$ per package on bolts. How many packages of each should be produced each day so as to maximise his profit, if he operates his machines for at the most $12$ hours a day?

Q. One kind of cake required $200$ g flour and $25$ g of fat, and another kind of cake required $100$ g of flour and $50$ g of fat. Find the maximum number of cakes which can be made from $5$ kg of flour and $1$ kg of fat assuming that there is no shortage of the other ingredients used in making the cakes?

Q. Reshma wishes to mix two types of food $P$ and $Q$ in such a way that the vitamin contents of the mixture contain at least $8$ units of vitamin $A$ and $11$ units of vitamin $B$. Food $P$ costs $Rs.60/kg$ and Food $Q$ costs $Rs.80/kg$. Food $P$ contains $3units/kg$ of vitamin $A$ and $5units/kg$ of vitamin $B$ while food $Q$ contains $4units/kg$ of vitamin $A$ and $2units/kg$ of vitamin $B$. Determine the minimum cost of the mixture.

Q. A factory makes tennis rackets and cricket bats. A tennis racket takes $1.5$ hours of machine time and $3$ hours of craftsman's time in its making while a cricket bat takes $3$ hour of machine time and $1$ hour of craftsman's time. In a day, the factory has the availability of not more than $42$ hours of machine time and $24$ hours of craftsman's time.
(i) What number of rackets and bats must be made if the factory is to work at full capacity?

Q. A company manufactures two types of novelty souvenirs made of plywood. Souvenirs of type $A$ required $5$ minutes each for cutting and $10$ minutes each for assembling. Souvenirs of type $B$ required $8$ minutes each for cutting and $8$ minutes each for assembling. There are $3$ hours $20$ minutes available for cutting and $4$ hours of assembling. The profit is Rs.$5$ each for type $A$ and $Rs.$6$eachfortype$B$ souvenirs. How many souvenirs of each type should the company manufacture in order to maximise the profit?

Q. A diet is to contain at least $80$ units of vitamin $A$ and $100$ units of minerals. Two foods $F_1$ and $F_2$ are available. Food $F_1$ costs Rs.$4$ per unit food and $F_2$s costs Rs.$6$ per unit. One unit of food $F_1$ contains $3$ units of vitamin $A$ and $4$ units of minerals. One unit of food $F_2$ contains $6$ units of vitamin $A$ and $3$ units of minerals. Formulate this as a linear programming problem. Find the minimum cost for diet that consists of mixture of these two foods and also meets the mineral nutritional requirements?

Q. A factory manufactures two types of screws, $A$ and $B$. Each type screw requires the use of two machines, an automatic and a hand operated. It takes $4$ minutes on the automatic and $6$ minutes on hand operated machines to manufacture a package of screws $A$, while takes $6$ minutes on automatic and $3$ minutes on the hand operated machines to manufacture a package of screws $B$. Each machine is available for at the most $4$ hours on any day. The manufacturer can sell a package of screws $A$ at a profit of Rs.$7$ and screws $B$ at a profit of Rs.$10$. Assuming that he can sell all the screws he manufactures, how many packages of each type should the factory owner produce in a day in order to maximise profit? Determine the maximum profit.

Q. If the profit on a racket and on a bat is Rs.$20$ and Rs.$10$ respectively, find the maximum profit of the factory when it works at full capacity

Q. A cottage industry manufactures pedestal lamps and wooden shades, each requiring the use of a grinding/cutting machine and a sprayer. It takes $2$ hours on grinding/cutting machine and $3$ hours on the sprayer to manufacture a pedestal lamp. It takes $1$ hour on the grinding/cutting machine and $2$ hours on the sprayer to manufacture a shade. On any day, the sprayer is available for at least for at the most $20$ hours and the grinding/cutting machine for at the most $12$ 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 the produces , how should he schedule his daily production in order to maximise his profit?

Q. A merchant plans to sell two types of personal computers, a desktop model and a portable model that will cost Rs 25000 and Rs 40000 respectively. He estimates that the total monthly demand of computers will not exceed 250 units. Determine the number of units of each type of computers which the merchant would stock to get maximum profit if he does not want to invest more than Rs 70 lakhs and if his profit on the desktop model is Rs 4500 and on portable model is Rs 5000.