CBSE Class 12th Math 12 – Linear Programmimg MCQs

A dietitian wishes to mix two types of foods, R and S in such a way that the vitamin contents of the mixture contains at least 4 units of vitamin A and 10 units of vitamin B. Food R costs Rs 6/kg and Food S costs Rs 8/kg. Food R contains 3 units/kg of Vitamin A and 4 units/kg of Vitamin B, while food S contains 4 units/kg of Vitamin A and 5 units/kg of vitamin B. If the mixture contains x kg of food ‘R’ and y kg of food ‘S’, then the non-negative constraint of this problem as a linear programming problem to minimum cost of the mixture is _________.

Correct! Wrong!

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The mixture contains x kg of food ‘R’ and y kg of food ‘S’, and the amount of food ‘R’ and food ‘S’ cannot be negative, i.e. x, y ≥ 0. Hence, if the mixture contains x kg of food ‘R’ and y kg of food ‘S’, then the non-negative constraint of this problem as a linear programming problem to minimum cost of the mixture is x, y ≥ 0.

If Z = 250x + 75y is a linear objective function in linear programming problem, then the variables x and y are called ____.

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In the above example, where Z = 250x + 75y is a linear objective function in linear programming problem, the variables x and y are called decision variables.

One type of biscuit requires 10 g of sugar and 15 g of fat, and another type of biscuit requires 20 g of sugar and 45 g of fat. To make the maximum number of biscuits from 5 kg of sugar and 3 kg of fat, the constraints are ________, ___________ and the non-negative constraints.

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To make the maximum number of biscuits from 5 kg of sugar and 3 kg of fat, the constraints are x + 2y ≤ 500, x + 3y ≤ 200 and the non-negative constraints. Let x and y be the number of first kind of biscuits and the second kind of biscuits respectively. Obviously, x ≥ 0, y ≥ 0. Now, 10 x + 20y ≤ 5000, i.e. x + 2y ≤ 500 (Sugar constraints) and 15x + 45 y ≤ 3000 i.e. x + 3y ≤ 200 (Fat constraints) Hence, to make the maximum number of biscuits from 5 kg of sugar and 3 kg of fat, the constraints are x + 2y ≤ 500, x + 3y ≤ 200 and the non-negative constraints.

A dealer deals in only two items fans and refrigerators. He has Rs 1,00,000 to invest. A fan costs Rs 2,000 and a refrigerator Rs 15,000. He estimates that from the sale of one fan, he can make a profit of Rs 250 and that from the sale of one refrigerator, a profit of Rs 1000. He wants to know how many fans and refrigerators he should buy with the available money so as to maximize his total profit, assuming that he can sell all the items which he buys. Then the objective function of profit Z = ____.

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Let x be the number of fans and y be the number of refrigerators that the dealer buys. Obviously, x and y must be non-negative. The dealer wants to invest in such a way so as to maximize his profit, say, Z which is stated as a function of x and y given by Z = 250x + 1000y (called objective function) Mathematically, the given problems now reduces to maximize Z = 250x + 1000y. Hence, the objective function of profit Z =250x + 1000y.

The problems where we have to maximize/minimize a linear function, subject to certain conditions determined by a set of linear inequalities with variables as non-negative, are called ____ problems.

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By definition, linear programming problems are problems where we have to maximize/minimize a linear function, subject to certain conditions determined by a set of linear inequalities with non-negative variables.

John goes to market with Rs 250 to buy sugar which is available in packets of 1 kg. The price of one packet of sugar is Rs 40. If x denotes the number of packets of sugar which he buys, then the constraints are ____ and ____.

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Given, John goes to market with Rs 250 to buy sugar which is available in packets of 1 kg. The price of one packet of sugar is Rs 40. If x denotes the number of packets of sugar then the constraint is x ≥ 0 (Non-negative constraint) and the total amount spent by him is Rs 40x. Since, he has to buy sugar in packets only, he may not be able to spend the entire amount of Rs 250 (Because 250 is not divisible by 40) Therefore, 40x < 250 Hence, the constraints are x ≥ 0 and 40x < 250.

The linear inequalities or equations or restrictions on the variables of a linear programming problem are called ____.

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By definition, the linear inequalities or equations or restrictions on the variables of a linear programming problem are called constraints.

A type of problems which seek to maximize (or, minimize) profit (or, cost) form a general class of problems called ____ problems.

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By definition, optimization problems are a type of problems which seek to maximize (or, minimize) profit (or, cost). Thus, an optimization problem may involve finding maximum profit, minimum cost, or minimum use of resources, etc.

A linear function Z = ax + by, where a, b are constants, which has to be maximized or minimized is called a linear ____.

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By definition, a linear function Z = ax + by, where a, b are constants, which has to be maximized or minimized is called a linear objective function.

A dietitian wishes to mix two types of juices in such a way that the vitamin contents of the mixture contain at least 5 units of vitamin A and 7 units of vitamin C. Juice ‘I’ contains 3 units/liter of vitamin A and 1 unit/liter of vitamin C. Juice ‘II’ contains 5 unit/liter of vitamin A and 2 units/liter of vitamin C. It costs Rs 30 per litre to purchase juice ‘I’ and Rs 40 per litre to purchase juice ‘II’. If the mixture contain x liters of juice ‘I’ and y liters of juice ‘II’, then the objective function of this problem as a linear programming problem to minimize the cost of such a mixture is minimize Z = __________.

Correct! Wrong!

The objective function of this problem as a linear programming problem to minimize the cost of such a mixture is, minimize Z = 30x + 40y. If the mixture contains x litre of juice ‘I’ and y litre of juice ‘II’, then Cost x litre of juice ‘I’ = 30x and Cost x litre of juice ‘I’ = 40y Hence, cost of the mixture containing x litre of juice ‘I’ and y litre of juice ‘II’ = 30x + 40y. Hence, if the mixture contain x liters of juice ‘I’ and y liters of juice ‘II’ , then the objective function of this problem as a linear programming problem to minimize the cost of such a mixture is, minimize Z = 30x + 40y.

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