NCERT Solutions Class 12 Maths Chapter-12 (Linear Programming)Exercise 12.2

NCERT Solutions Class 12 Maths from class 12th Students will get the answers of Chapter-12 (Linear Programming)Exercise 12.2 This chapter will help you to learn the basics and you should expect at least one question in your exam from this chapter.
We have given the answers of all the questions of NCERT Board Mathematics Textbook in very easy language, which will be very easy for the students to understand and remember so that you can pass with good marks in your examination.

Exercise 12.2

Q1. 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 ot vitamin A and 11 units of vitamin 3. Food P costs RS 60/kg and Food Q costs RS 80/kg. Food P contains 3 units "kg Of vitamin A and 5 units /kg of vitamin 3 while food Q contains 4 units /kg of vitamin A and 2 units Jkg of vitamin 3. Determine the minimum cost of the mixture?

Answer. Let the mixture contain x kg of food p and y kg of food Q. Therefore, $x\ge 0\text{and}y\ge 0$ The given information can be compiled in a table as follows.  The mixture must contain at least g units of vitamin A and 11 units of vitamin Therefore, the constraints are $\begin{array}{c}3x+4y\ge 8\\ 5x+2y\ge 11\\ \text{Total cost,}Z\text{, of purchasing food is,}z=60x+80y\\ \text{The mathematical formulation of the given problem is}\end{array}$$$\begin{array}{c}\text{Minimise}Z=60x+80y\dots \left(1\right)\\ \text{subject to the constraints,}\\ 3x+4y\ge 8\dots \left(2\right)\\ 5x+2y\ge 11\dots \left(3\right)\\ x,y\ge 0\dots \left(4\right)\\ \text{The feasible region determined by the system of constraints is as follows.}\end{array}$$ It can be seen that the feasible region is unbounded. $A(\frac{8}{3},0),B(2,\frac{1}{2}),\text{and}C(0,\frac{11}{2})$ The values of z at these corner points are as follows.  $\begin{array}{c}\text{As the feasible region is unbounded, therefore, 160}may\text{or may not be the minimum}\\ \text{value of}z\text{.}\end{array}$ $\begin{array}{c}\text{For this, we graph the inequality,}60x+80y<160\text{or}3x+4y<8,\text{and check whether}\\ \text{the resulting half plane has points in common with the feasible region or not.}\\ \text{It can be seen that the feasible region has no common point with}3x+4y<8\\ \text{Therefore, the minimum cost of the mixture will be Rs}160\text{at the line segment joining}\end{array}$ $(\frac{8}{3},0)\text{and}(2,\frac{1}{2})$

Q2. One kind of cake requires 200g of flour and 25g of fat, and another kind of cake requires 100g of flour and 50g 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.

Answer. $\begin{array}{c}\text{Let there be}x\text{cakes of first kind and}y\text{cakes of second kind. Therefore,}\\ x\ge 0\text{and}y\ge 0\\ \text{The given information can be complied in a table as follows.}\end{array}$  $\begin{array}{c}\therefore 200x+100y\le 5000\\ \Rightarrow 2x+y\le 50\\ 25x+50y\le 1000\\ \Rightarrow x+2y\le 40\end{array}$ $\begin{array}{c}\text{Total numbers of cakes,}z\text{, that can be made are,}z=x+y\\ \text{The mathematical formulation of the given problem is}\\ \text{Maximize}Z=x+y\dots \left(1\right)\\ \text{subject to the constraints,}\\ 2x+y\le 50\\ x+2y\le 40\\ x,y\ge 0\end{array}$ 

Q3. 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? (ii) 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.

Answer. $\begin{array}{c}\text{(i) Let the number of rackets and the number of bats to be made be}x\text{and}y\\ \text{respectively.}\\ \text{The machine time is not available for more than}42\text{hours.}\\ \therefore 1.5x+3y\le 42\\ \text{The craftsman's time is not available for more than}24\text{hours.}\\ \therefore 3x+y\le 24\\ \text{The factory is to work at full capacity. Therefore,}\\ 1.5x+y\le 24\\ 3x+y=24\end{array}$ $\begin{array}{c}\text{On solving these equations, we obtain}\\ x=4\text{and}y=12\\ \text{Thus,}4\text{rackets and}12\text{bats must be made.}\\ \text{(1) The given information can be complied in a table as follows.}\end{array}$  $\begin{array}{c}\therefore 1.5x+3y\le 42\\ 3x+y\le 24\\ x,y\ge 0\end{array}$ $\begin{array}{c}\text{The profit on a racket is Rs}20\text{and on a bat is Rs}10\text{.}\\ \therefore Z=20x+10y\\ \text{The mathematical formulation of the given problem is}\\ \text{Maximize}Z=20x+10y\text{(1)}\\ \text{subject to the constraints,}\\ 1.5x+3y\le 42\dots \text{(2)}\end{array}$ $\begin{array}{c}3x+y\le 24\dots \left(3\right)\\ x,y\ge 0\dots \left(4\right)\\ \text{The feasible region determined by the system of constraints is as follows.}\end{array}$ 

Q4. 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 Rs17.50 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?

Answer. $\begin{array}{c}\text{Let the manufacturer produce}x\text{packages of nuts and}y\text{packages of bolts. Therefore,}\\ x\ge 0\text{and}y\ge 0\\ \text{The given information can be compiled in a table as follows.}\end{array}$  $\begin{array}{c}\text{The profit on a package of nuts is Rs}17.50\text{and on a package of bolts is Rs}7.\text{Therefore,}\\ \text{the constraints are}\\ x+3y\le 12\\ 3x+y\le 12\\ \text{Total profit,}z=17.5x+7y\\ \text{The mathematical formulation of the given problem is}\end{array}$ $\begin{array}{c}\text{subject to the constraints,}\\ x+3y\le 12\dots \left(2\right)\\ 3x+y\le 12\dots \left(3\right)\\ x,y\ge 0\dots \left(4\right)\\ \text{The feasible region determined by the system of constraints is as follows.}\end{array}$ 

Q5. A factory manufactures two types of screws, A and B. Each type of 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 it 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 his profit? Determine the maximum profit.

Answer. $\begin{array}{c}\text{Let the factory manufacture}x\text{screws of type}A\text{and}y\text{screws of type}B\text{on each day.}\\ \text{Therefore,}\\ x\ge 0\text{and}y\ge 0\\ \text{The given information can be compiled in a table as follows.}\end{array}$  $\begin{array}{c}\text{The profit on a package of screws}A\text{is Rs}7\text{and on the package of screws}B\text{is Rs}10\text{.}\\ \text{Therefore, the constraints are}\\ 4x+6y\le 240\\ 6x+3y\le 240\\ \text{Total profit,}Z=7x+10y\end{array}$ $\begin{array}{c}\text{Maximize}z=7x+10y\dots \left(1\right)\\ \text{subject to the constraints,}\\ 4x+6y\le 240\\ 6x+3y\le 240\\ 6x+3y\le 240\dots \left(3\right)\\ x,y\ge 0\dots \left(4\right)\\ \text{The feasible region determined by the system of constraints is}\end{array}$  The maximum value of z is 410 at (30, 20). Thus, the factory should produce 30 packages of screws A and 20 packages of screws B to get the maximum profit of Rs 410.

Q6. 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 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 he produces, how should he schedule his daily production in order to maximise his profit?

Answer. $\begin{array}{c}\text{Let the cottage industry manufacture}x\text{pedestal lamps and}y\text{wooden shades. Therefore,}\\ x\ge 0\text{and}y\ge 0\\ \text{The given information can be compelled in a table as follows.}\end{array}$  $\begin{array}{c}\text{The profit on a lamp is Rs}5\text{and on the shades is Rs}3.\text{Therefore, the constraints are}\\ 2x+y\le 12\\ 3x+2y\le 20\\ \text{Total profit,}z=5x+3y\\ \text{The mathematical formulation of the given problem is}\\ \text{Maximize}z=5x+3y\dots \left(1\right)\end{array}$ $\begin{array}{c}\text{Maximize}z=5x+3y\dots \left(1\right)\\ \text{subject to the constraints,}\\ 2x+y\le 12\dots \text{(2)}\\ 3x+2y\le 20\\ x,y\ge 0\dots \left(4\right)\end{array}$ 

Q7. A company manufactures two types of novelty souvenirs made of plywood. Souvenirs of type A require 5 minutes each for cutting and 10 minutes each for assembling. Souvenirs of type B require 8 minutes each for cutting and 8 minutes each for assembling. There are 3 hours 20 minutes available for cutting and 4 hours for assembling. The profit is Rs 5 each for type A and Rs 6 each for type B souvenirs. How many souvenirs of each type should the company manufacture in order to maximise the profit?

Answer. $\begin{array}{c}\text{Let the company manufacture}x\text{souvenirs of type}A\text{and}y\text{souvenirs of type}B\\ \text{Therefore,}\\ x\ge 0\text{and}y\ge 0\\ \text{The given information can be complied in a table as follows.}\end{array}$  $\begin{array}{c}\text{The profit on type A souvenirs is Rs}5\text{and on type}B\text{souvenirs is Rs}6.\text{Therefore, the}\\ \text{constraints are}\\ 5x+8y\le 200\\ 10x+8y\le 240\\ \text{Total profit,}Z=5x+6y\end{array}$ $\begin{array}{c}\text{The mathematical formulation of the given problem is}\\ \text{Maximize}Z=5x+6y\dots \left(1\right)\\ \text{subject to the constraints,}\\ 5x+8y\le 200\\ 5x+4y\le 120\\ x,y\ge 0\dots \left(4\right)\end{array}$  Thus, e souvenirs Of type A and 20 souvenirs Of type B should be produced each day to get the maximum profit of Rs 160.

Q8. 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 should 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.

Answer. $\begin{array}{c}\text{Let the merchant stock}x\text{desktop models and}y\text{portable models. Therefore,}\\ x\ge 0\text{and}y\ge 0\\ \text{The cost of a desktop model is Rs}25000\text{and of a portable model is Rs 4000. However,}\\ \text{the merchant can invest a maximum of Rs}70\text{lakhs.}\\ \therefore 25000x+40000y\le 7000000\\ 5x+8y\le 1400\\ \therefore x+y\le 250\end{array}$ $\begin{array}{c}\text{The monthly demand of computers will not exceed}250\text{units.}\\ \therefore x+y\le 250\\ \text{The profit on a desktop model is Rs}4500\text{and the profit on a portable model is Rs 5000.}\\ \text{Total profit,}z=4500x+5000y\\ \text{Thus, the mathematical formulation of the given problem is}\\ \text{Maximum}Z=4500x+5000y\end{array}$ $\begin{array}{c}\text{Thus, the mathematical formulation of the given problem is}\\ \text{Maximum}Z=4500x+5000y\\ \text{subject to the constraints,}\\ 5x+8y\le 1400\\ x+y\le 250\\ x+y\le 250\\ x,y\ge 0\\ \text{The feasible region determined by the system of constraints is as follows.}\end{array}$  The maximum value of Z Is 1150000 at (200, SO). Thus, the merchant should stock 200 desktop models and 50 portable models to get the maximum profit of Rs 1150000.

Q9. A diet is to contain at least 80 units of vitamin A and 100 units of minerals. Two foods F1 and F2 are available. Food F1 costs Rs 4 per unit food and F2 costs Rs 6 per unit. One unit of food F1 contains 3 units of vitamin A and 4 units of minerals. One unit of food F2 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 minimal nutritional requirements.

Answer. $\begin{array}{c}\text{Let the diet contain}x\text{units of food}{F}_1\text{and}y\text{units of food}{F}_2\text{. Therefore,}\\ x\ge 0\text{and}y\ge 0\\ \text{The given information can be complied in a table as follows.}\end{array}$  $\begin{array}{c}\text{The cost of food}{F}_1\text{is Rs}4\text{per unit and of Food}{F}_2\text{is Rs}6\text{per unit. Therefore, the}\\ \text{constraints are}\\ 3x+6y\ge 80\\ 4x+3y\ge 100\\ \text{x,}y\ge 0\\ \text{Total cost of the diet,}z=4x+6y\end{array}$ $\begin{array}{c}\text{The mathematical formulation of the given problem is}\\ \text{Minimise}z=4x+6y\dots \left(1\right)\\ \text{subject to the constraints,}\\ 3x+6y\ge 80\dots \left(2\right)\\ 4x+3y\ge 100\dots \left(3\right)\\ x,y\ge 0\dots \left(4\right)\\ \text{The feasible region determined by the constraints is as follows.}\end{array}$  ${}^A(\frac{8}{3},0),B(2,\frac{1}{2}),\text{and}C(0,\frac{11}{2})$  $\begin{array}{c}\text{For this, we draw a graph of the inequality,}4x+6y<104\text{or}2x+3y<52,\text{and check}\\ \text{whether the resulting half plane has points in common with the feasible region or not.}\\ \text{It can be seen that the feasible region has no common point with}2x+3y<52\\ \text{Therefore, the minimum cost of the mixture will be Rs}104\text{.}\end{array}$

Q10. There are two types of fertilisers F1 and F2. F1 consists of 10% nitrogen and 6% phosphoric acid and F2 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 F1 costs Rs 6/kg and F2 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?

Answer. $\begin{array}{c}\text{Let the farmer buy}x\text{kg of fertilizer}{F}_1\text{and}y\text{kg of fertilizer}{F}_2\text{. Therefore,}\\ x\ge 0\text{and}y\ge 0\\ \text{The given information can be complied in a table as follows.}\end{array}$  $\begin{array}{c}{F}_1\text{consists of}10\%\text{nitrogen and}{F}_2\text{consists of}5\%\text{nitrogen. However, the farmer}\\ \text{requires at least}14\text{kg of nitrogen.}\\ \therefore 10\%\text{of}x+5\%\text{of}y\ge 14\end{array}$ $\begin{array}{c}\frac{x}{10}+\frac{y}{20}\ge 14\\ 2x+y\ge 280\\ F_1\text{consists of}6\%\text{phosphoric acid and}{F}_2\text{consists of}10\%\text{phosphoric acid. However, the}\\ \text{farmer requires at least}14\text{kg of phosphoric acid.}\end{array}$ $\begin{array}{c}\frac{6x}{100}+\frac{10y}{100}\ge 14\\ 3x+56y\ge 700\\ \text{Total cost of fertilizers,}Z=6x+5y\\ \text{The mathematical formulation of the given problem is}\\ \text{Minimize}z=6x+5y\dots \left(1\right)\\ \text{subject to the constraints,}\end{array}$ $\begin{array}{c}\text{Total cost of fertilizers,}z=6x+5y\\ \text{The mathematical formulation of the given problem is}\\ \text{Minimize}Z=6x+5y\dots \left(1\right)\\ \text{subject to the constraints,}\\ 2x+y\ge 280\dots \left(2\right)\\ 3x+5y\ge 700\dots \left(3\right)\\ x,y\ge 0\dots \left(4\right)\end{array}$  $\begin{array}{c}\text{It can be seen that the feasible region is unbounded.}\\ \text{The corner points are}(\frac{700}{3},0),B(100,80),\text{and}C(0,280)\\ \text{The values of}z\text{at these points are as follows.}\end{array}$  $\begin{array}{c}\text{For this, we draw a graph of the inequality,}6x+5y<1000,\text{and check whether the}\\ \text{resulting half plane has points in common with the feasible region or not.}\\ \text{It can be seen that the feasible region has no common point with}\\ 6x+5y<1000\end{array}$

Q11. The corner points of the feasible region determined by the following system of linear inequalities: 2x + y ≤ 10, x + 3y ≤ 15, x, y ≥ 0 are (0, 0), (5, 0), (3, 4) and (0, 5). Let Z = px + qy, where p, q > 0. Condition on p and q so that the maximum of Z occurs at both (3, 4) and (0, 5) is (A) p = q (B) p = 2q (C) p = 3q (D) q = 3p

Answer. $\begin{array}{c}\text{The maximum value of}Z\text{is unique.}\\ \text{It is given that the maximum value of}z\text{occurs at two points,}(3,4)\text{and}(0,5)\text{.}\end{array}$ $\begin{array}{c}\therefore \text{value of}z\text{at}(3,4)=\text{value of}z\text{at}(0,5)\\ \Rightarrow p\left(3\right)+q\left(4\right)=p\left(0\right)+q\left(5\right)\\ \Rightarrow 3p+4q=5q\\ \Rightarrow q=3p\\ \text{Hence, the correct answer is D.}\end{array}$

Chapter-12 (Linear Programming)

• Exercise 12.1