# Exercise 13.2

QUESTION 1

Express the following complex numbers in the standard form a+ib :

(i)

Sol :

(ii)

Sol :

(iii)

Sol :

(iv)

Sol :

(v)

Sol :

(vi)

Sol :

(vii)

Sol :

(viii)

Sol :

(ix)

Sol :

(x)

Sol :

(xi)

Sol :

(xii)

Sool :

QUESTION 2

Find the real values of x and y, if

(i)

Sol :

Comparing both the sides :

Multiplying equation (1) by 3 and equation (2) by 2 :

Adding equations (3) and (4) :

Substituting the value of y in equation (1) :

(ii)

Sol :

Comparing both sides:

Multiplying equation (1) with 3 and equation (2) with 2 :

Adding equations (3) and (4) :

Substituting the value of y in equation (1):

(ii)

Sol :

Comparing both sides :

Multiplying equation (1) by 3 and equation (2) by 8 ,

Adding equations (3) and (4):

Substituting the value of x in equation (1) :

(iii)

Sol :

Comparing both the sides :

Multiplying equation (2) by 2 :

Subtracting equation (3) from 1 :

Substituting the value of y in equation (1) :

(iv)

Sol :

Comparing both the sides

Adding equations (1) and (2)

Substituting the value of x in equation (1)

QUESTION 3

Find the conjugates of the following complex numbers:

(i)

Sol :

(ii)

Sol :

(iii)

Sol :

(iv)

Sol :

(v)

Sol :

(vi)

Sol :

QUESTION 4

Find the multiplicative inverse of the following complex numbers:

(i)

Sol :

Let then,

(ii)

Sol :

(iii)

Sol :

(iv)

Sol :

QUESTION 5

If ,, find

Sol :

Also ,

QUESTION 6

If , , find

(i)

(ii)

Sol :

(i)Given ,

(ii) Given ,

[\because~(i^2=-1)]

Since no term containing *i *is present

QUESTION 7

Find the modulus of

Sol :

QUESTION 8

If , prove that

Sol :

Taking mod on the both sides:

squaring both the sides

hence proved

QUESTION 9

Find the least positive integral value of *n *for which

Sol :

For to be real , the least positive value of *n *will be 2 .

As

QUESTION 10

Find the real values of for which the complex number is purely real .

Sol :

For it to be purely real , the imaginary part must be zero .

This is true for odd multiples of

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QUESTION 11

Find the smallest positive integer value of *m *for which

Sol :

For this to be real , the smallest positive value of *m *will be 1

Thus, , which is real

QUESTION 12

If , find

Sol :

Also,

It is given that ,

[from (1) and (2)]

Thus ,

QUESTION 13

If find

Sol :

It is given that

[from (1)]

QUESTION 14

If find (a+b)

Sol :

It is given that

[from (i)]

Thus ,

QUESTION 15

If , find the value of

Sol :

on putting value of

QUESTION 16

Evaluate the following:

(i) when

Sol :

[squaring both the sides]

[putting value of ]

= 4

(ii) when

Sol :

putting the values of

= 5

(iii) when

Sol :

[putting the values of ]

[putting the value of ]

= 12

(iv) when

Sol :

[squaring both the sides]

[putting value of ]

[putting the value of ]

= 0

(v) when

Sol :

putting the value of

on putting the value of

= 6

QUESTION 17

For a positive integer *n * , find the value of

Sol :

Thus , the value of is