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QUESTION: 1

The solution of the following system of equation is

2x + 3y = 5

5x – 2y = 3

Solution:

2x + 3y = 5

Multiply equation with ‘5’, we get

10x + 15y = 25……………………(1)

5x – 2y = 3

Multiply equation with ‘2’, we get

10x - 4y = 6………………………(2)

Subtracting (1) from (2), we get

19y = 19

y = 1

Put the value of y in eq(1)

10x + 15(1) = 25

10x = 10

x = 1

QUESTION: 2

One third of sum of two angles is 60° and one fourth of their difference is 28°. The angles are

Solution:

Solution:-

Let the two angles be 'x' and 'y'.

So, according to the question,

One third of the sum of two angles :- 1/3(x+y) = 60

x/3 + y/3 = 60 ......(1)

Quarter of their difference :- 1/4(x-y) = 28

x/4 - y/4 = 28...........(2)

Multiplying the equation (1) by 3 and equation (2) by 4, we get

x + y =180 .......(3) x - y = 112 ......(4)

Subtracting (4) from (3),

x + y = 180

x - y = 112

- + -

_________

2y = 68

_________

2y = 68

y = 34

Putting the value of y = 34 in the equation (3)

x + y = 180

x + 34 = 180

x = 180 - 34

x = 146

The two angles are 146° and 34°.

QUESTION: 3

For a square matrix A in a matrix equation AX = B, if │A│≠ 0, then

Solution:

QUESTION: 4

The following system of equations has

x + 3y + 3z = 2

x + 4y + 3z = 1

x + 3y + 4z = 2

Solution:

Let A = {(1,3,3) (1,4,3) (1,3,4)}

|A| = 1(16-9) -3(4-3) +3(3-4)

|A| = 1(7) -3(1) +3(-1)

= 7 - 3 - 3

= 1

Therefore, A is not equal to zero, it has unique solution.

QUESTION: 5

Inverse of a matrix A is given by

Solution:

Inverse of matrix (A^{-1}) = (adj A)/|A|

QUESTION: 6

If , then A^{-1} =

Solution:

QUESTION: 7

The system of equations kx + 2y – z = 1,

(k – 1)y – 2z = 2

(k + 2)z = 3 has a unique solution, if k is

Solution:

This system of equations has a unique solution, if

QUESTION: 8

System of equations AX = B is inconsistent if

Solution:

If (adj A) B ≠ 0 (zero matrix), then the solution does not exist. The system of equations is inconsistent. Else, if (adj A) B = 0 then the system will either have infinitely many solutions (consistent system) or no solution (inconsistent system).

QUESTION: 9

Inverse of , is

Solution:

A = {(6,7) (8,9)}

|A| = (6 * 9) - (8 * 7)

= 54 - 56

|A| = -2

A^{-1} = -½{(9,-7) (-8,6)}

A^{-1} = {(-9/2, 7/2) (4,-3)}

QUESTION: 10

A system of linear equations AX = B is said to be inconsistent, if the system of equations has

Solution:

A linear system is said to be consistent if it has at least one solution; and is said to be inconsistent if it has no solution. have no solution, a unique solution, and infinitely many solutions, respectively.

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