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The prices of three commodities P ,\ Q ,\ a n d\ R are Rs x ,\ y\ a n d\ z per unit respectively. A purchases 4 units of R and sells 3 unit of P and 5 units of Qdot B purchases 3 units of Q and sells 2 units of P and 1 unit of Rdot C purchases 1 unit of P and sells 4 units of Q and 6 units of Rdot In the process A ,\ B\ a n d\ C earn Rs 6000, 5000 and 13000 respectively. If selling the units is positive earning and buying the units is negative earnings, find the price per unit of three commodities by using matrix method.

The area of a rectangle gets reduced by 9 square units if its length is reduced by 5 units and the breadth is increased by 3 units. If we increase the length by 3 units and breadth by 2 units, the area is increased by 67 square units. Find the length and breadth of the rectangle.

The area of a rectangle gets reduced by 9 square units if its length is reduced by 5 units and the breadth is increased by 3 units. If we increase the length by 3 units and breadth by 2 units, the area is increased by 67 square units. Find the length and breadth of the rectangle.

If in a rectangle, the length is increased by 2 units and breadth is reduced is reduced by 2 units, the area is reduced by 28 square units. If however the length is reduced by 1 unit and breadth is increased by 2 units, the area increases by 33 square units. Find the area of the rectangle.

A diet is to contain at least 4000 units of carbohydrates, 500 units of fat and 300 units of protein. Two foods F_1 and F_2 are available. Food F_1 costs 2 Rs. per unit and food F_2 costs 4 Rs. per unit. A unit of food F_1 contains 10 units of carbohydrates, 20 units of fat and 15 units of protein. A unit of food F_2 contains 25 units of carbohydrates, 10 units of fat and 20 units of protein. Find the minimum cost for a diet that consists of a mixture of these two foods and also meets the minimum requirements.

A diet for a sick person must contain at least 4000 units of vitamins, 50 units of minerals and 1400 calories. Two foods A and B, are available at a cost of Rs. 4 and Rs. 3 per unit respectively. If one unit of A contains 200 units of vitamin, 1 unit of mineral and 40 calories and one unit of food B contains 100 units of vitamin, 2 units of minerals and 40 calories, find what combination of foods should be used to have the least cost?

A dietician has to develop a social 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 atleast 240 units of calcium, atleast 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?