NCERT Solutions for Maths Class 12 Exercise 12.2

# NCERT Solutions for Maths Class 12 Exercise 12.2

## NCERT Solutions for Maths Class 12 Exercise 12.2

Hello Students. Welcome to maths-formula.com. In this post, you will find the complete NCERT Solutions for Maths Class 12 Exercise 12.2.

You can download the PDF of NCERT Books Maths Chapter 10 for your easy reference while studying NCERT Solutions for Maths Class 12 Exercise 12.2.

Class 12th is a very crucial stage of your student’s life, since you take all important decisions about your career on this stage. Mathematics plays a vital role to take decision for your career because if you are good in mathematics, you can choose engineering and technology field as your career.

NCERT Solutions for Maths Class 12 Exercise 12.2 helps you to solve each and every problem with step by step explanation which makes you strong in mathematics.

All the schools affiliated with CBSE, follow the NCERT books for all subjects. You can check your syllabus from NCERT Syllabus for Mathematics Class 12.

NCERT Solutions for Maths Class 12 Exercise 12.2 are prepared by the experienced teachers of CBSE board. If you are preparing for JEE Mains and NEET level exams, then it will definitely make your foundation strong.

If you want to recall All Maths Formulas for Class 12, you can find it by clicking this link.

If you want to recall All Maths Formulas for Class 11, you can find it by clicking this link.

NCERT Solutions for Maths Class 12 Exercise 12.1

NCERT Solutions for Maths Class 12 Exercise 12.2

Maths Class 12 Ex 12.2 Question 1.

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 3 units/kg of vitamin A and 5 units/kg of vitamin B while food Q contains 4 units/kg of vitamin A and 2 units/kg of vitamin B. Determine the minimum cost of the mixture.

Solution:

Let Reshma mixes x kg of food P and y kg of food Q.

Maths Class 12 Ex 12.2 Question 2.

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

Solution:

Let the number of cakes made of first kind are x and that of second kind is y.
∴ Let to maximize Z = x + y Maths Class 12 Ex 12.2 Question 3.

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 hours 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.

Solution:

Let x tennis rackets and y cricket bats are produced in one day in the factory.

Maths Class 12 Ex 12.2 Question 4.

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 maximize his profit, if he operates his machines for at the most 12 hours a day?

Solution:

Let x packages of nuts and y packages of bolts are produced.

Maths Class 12 Ex 12.2 Question 5.

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 a day in order to maximize his profit? Determine the maximum profit.

Solution:

Let the manufacturer produces x packages of screws A and y packages of screw B, then the time taken by x packages of screw A and y packages of screw B on automatic machine = (4x + 6y) minutes.
And on hand operated machine = (6x + 3y) minutes

Maths Class 12 Ex 12.2 Question 6.

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, while 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 maximize his profit?

Solution:

Let the manufacturer produces x pedestal lamps and y wooden shades. Then, the time taken by x pedestal lamps and y wooden shades on grinding/cutting machines = (2x + y) hours and time taken by x pedestal lamps and y shades on the sprayer = (3x + 2y) hours.
Since grinding/cutting machine is available for at the most 12 hours, we have: 2x + y ≤ 12.

Since sprayer is available for at the most 20 hours, we have: 3x + 2y ≤ 20.
Profit from the sale of x lamps and y shades.
Z = 5x + 3y
So, our problem is to maximize Z = 5x + 3y subject to constraints 3x + 2y ≤ 20, 2x + y ≤ 12, x, y ≥ 0.
Consider 3x + 2y ≤ 20
Let 3x + 2y = 20

Maths Class 12 Ex 12.2 Question 7.

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 maximize the profit?

Solution:

Let the company manufactures x souvenirs of type A and y souvenirs of type B.

Then, the time taken for cutting x souvenirs of type A and y souvenirs of type B = (5x + 8y) minutes.
Since 3 hours 20 minutes, i.e., 200 minutes are available for cutting, so we should have 5x + 8y ≤ 200.
Also, as 4 hours, i.e., 240 minutes are available for assembling, so we have 10x + 8y ≤ 240, i.e., 5x + 4y ≤ 120.

Maths Class 12 Ex 12.2 Question 8.

A merchant plans to sell two types of personal computers – a desktop model and a portable model that will cost Rs. 25,000 and Rs. 40,000 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.

Solution:

Let the number of desktop model be x and the number of portable model be y.

Total monthly demand of computer does not exceed 250.

It means x + y ≤ 250,

The cost of 1 desktop computer if Rs. 25,000 and 1 portable computer is Rs. 40,000
∴ the cost of x desktop and y portable computer = Rs. (25,000 x + 40,000 y)
Maximum investment = Rs. 70 lakhs = Rs. 70,00,000
25,000 x + 40,000 y ≤ 70,00,000 or 5x + 8y ≤ 1400
Profit on 1 desktop computer is Rs. 4500 and on 1 portable computer is Rs. 5000.

Total profit,
Z = 4500 x + 5000 y

Maths Class 12 Ex 12.2 Question 9.

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.

Solution:

Let the number of units of food F1 be x units and the number of units of food F2 be y units.

Maths Class 12 Ex 12.2 Question 10.

There are two types of fertilizers 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 fertilizer should be used so that nutrient requirements are met at a minimum cost. What is the minimum cost?

Solution:

Let x kg of fertilizer F1 and y kg of fertilizer F2 be required.

Maths Class 12 Ex 12.2 Question 11.

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

Solution:

Maximum value of Z = px + qy occurs at (3, 4) and (0, 5).
At (3, 4), Z = px + qy = 3p + 4q
At (0, 5), Z = 0 + q.5 = 5q
Both are the maximum values
Therefore, 3p + 4q = 5q or q = 3p
Hence, the correct answer is option (D).