**NCERT Solutions for Maths Class 12 Exercise 12.2**

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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.

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**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.

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 F_{1}
and F_{2} are available. Food F_{1} costs Rs. 4 per unit food
and F_{2} 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 minimal nutritional requirements.

**Solution:**

Let the number of
units of food F_{1} be x units and the number of units of food F_{2}
be y units.

There are two
types of fertilizers F_{1} and F_{2}. F_{1} consists of
10% nitrogen and 6% phosphoric acid and F_{2} 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 F_{1} costs Rs. 6/kg and F_{2} 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
F_{1} and y kg of fertilizer F_{2} 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).