Balbharati Maharashtra State Board 12th Commerce Maths Solution Book Pdf Chapter 2 Matrices Ex 2.6 Questions and Answers.

## Maharashtra State Board 12th Commerce Maths Solutions Chapter 2 Matrices Ex 2.6

Question 1.

Solve the following equations by the method of inversion:

(i) x + 2y = 2, 2x + 3y = 3

Solution:

The given equations can be written in the matrix form as:

By equality of matrices,

x = 0, y = 1 is the required solution.

(ii) 2x + y = 5, 3x + 5y = -3

Solution:

(iii) 2x – y + z = 1, x + 2y + 3z = 8 and 3x + y – 4z = 1

Solution:

The given equations can be written in the matrix form as:

By equality of matrices,

x = 1, y = 2, z = 1 is the required solution.

(iv) x + y + z = 1, x – y + z = 2 and x + y – z = 3

Solution:

Question 2.

Express the following equations in matrix form and solve them by method of reduction:

(i) x + 3y = 2, 3x + 5y = 4.

Solution:

The given equations can be written in the matrix form as:

Hence, x = \(\frac{1}{2}\), y = \(\frac{1}{2}\) is the required solution.

(ii) 3x – y = 1, 4x + y = 6.

Solution:

The given equations can be written in the matrix form as:

By equality of matrices,

12x – 4y = 4 …..(1)

7y = 14 …..(2)

From (2), y = 2

Substituting y = 2 in (1), we get

12x – 8 = 4

∴ 12x = 12

∴ x = 1

Hence, x = 1, y = 2 is the required solution.

(iii) x + 2y + z = 8, 2x + 3y – z = 11 and 3x – y – 2z = 5.

Solution:

The given equations can be written in the matrix form as:

By equality of matrices,

x + 2y + z = 8 ……(1)

-y – 3z = -5 …….(2)

16z = 16 ……….(3)

From (3), z = 1

Substituting z = 1 in (2), we get

-y – 3= -5

∴ y = 2

Substituting y = 2, z = 1 in (1), we get

x + 4 + 1 = 8

∴ x = 3

Hence, x = 3, y = 2, z = 1 is the required solution.

(iv) x + y + z = 1, 2x + 3y + 2z = 2 and x + y + 2z = 4.

Solution:

The given equations can be written in the matrix form as:

By equality of matrices,

x + y + z = 1 ……(1)

y = 0

z = 3

Substituting y = 0, z = 3 in (1), we get

x + 0 + 3 = 1

∴ x = -2

Hence, x = -2, y = 0, z = 3 is the required solution.

Question 3.

The total cost of 3 T.V. and 2 V.C.R. is ₹ 35000. The shopkeeper wants a profit of ₹ 1000 per T.V. and ₹ 500 per V.C.R. He sells 2 T.V. and 1 V.C.R. and he gets total revenue of ₹ 21500. Find the cost and selling price of T.V. and V.C.R.

Solution:

Let the cost of each T.V. be ₹ x and each V.C.R. be ₹ y.

Then the total cost of 3 T.V. and 2 V.C.R. is ₹ (3x + 2y) which is given to be ₹ 35000.

∴ 3x + 2y = 35000

The shopkeeper wants a profit of ₹ 1000 per T.V. and ₹ 500 per V.C.R.

The selling price of each T.V. is ₹ (x + 1000) and of each V.C.R. is ₹ (y + 500).

∴ selling price of 2 T.V. and 1 V.C.R is

₹ [2(x + 1000) + (y + 500)] which is given to be ₹ 21500.

∴ 2(x + 1000) + (y + 500) = 21500

∴ 2x + 2000 + y + 500 = 21500

∴ 2x + y = 19000

Hence, the system of linear equations is

3x + 2y = 35000

2x + y = 19000

The equations can be written in matrix form as:

By equality of matrices,

-x = -3000 …….(1)

2x + y = 19000 ……….(2)

From (1), x = 3000

Substituting x = 3000 in (2), we get

2(3000) + y = 19000

∴ y = 19000 – 6000 = 13000

Hence, the cost price of one T.V. is ₹ 3000 and of one V.C.R. is ₹ 13000 and the selling price of one T.V. is ₹ 4000 and of one V.C.R. is ₹ 13500.

Question 4.

The sum of the cost of one Economics book, one Cooperation book, and one Account book is ₹ 420. The total cost of an Economic book, 2 Cooperation books, and an Account book is ₹ 480. Also, the total cost of an Economic book, 3 Cooperation books, and 2 Account books is ₹ 600. Find the cost of each book.

Solution:

Let the cost of 1 Economic book, 1 Cooperation book and 1 Account book be ₹ x, ₹ y and ₹ z respectively.

Then, from the given information

x + y + z = 420

x + 2y + z = 480

x + 3y + 2z = 600

These equations can be written in matrix form as:

By equality of matrices,

x + y + z = 420 …….(1)

y = 60

2y + z = 180 ………(2)

Substituting y = 60 in (2), we get

2(60) + z = 180

∴ z = 180 – 120 = 60

Substituting y = 60, z = 60 in (1), we get

x + 60 + 60 = 420

∴ x = 420 – 120 = 300

Hence, the cost of each Economic book is ₹ 300, each Cooperation book is ₹ 60 and each Account book is ₹ 60.