# Question-1

**(i) (5i) (ii) i**

^{9}+ i^{19 }(iii) i^{-39}

**Solution:**

(i) (5i)

= - 3i^{2}

= -3(-1) = 3

= 3 + i0

**(ii) i ^{9} + i^{19}**

= i^{8}.i + i^{18}.i

= (i^{2})^{4}i + (i^{2})^{9}i

=(-1)^{4}i + (-1)^{9}i

= i - i = 0

= 0 + i0

**(iii) i ^{-39}**

= (i

^{39})

^{-1 }

= [i^{38}.l]^{-1}

= [(l^{2})^{19 }i]^{-1}

= [(-1)^{19 }i]^{-1}

= (-i)^{-1} =

= - = = = i = 0 + i(1)

# Question-2

**(**

**i) 3(7 + i7) +i(7 + i7)**

**(**

**ii) (1 â€“ i) â€“ (-1 + i6)**

**(**

**iii)**

**Solution:**

(i) 3(7 + i7) +i(7 + i7)

= 21 + 21i + 7i + 7i

^{2 }

= 21 + 28i +7(-1)

= 21 + 28i â€“ 7

= 14 + 28i

(ii) (1 â€“ i) â€“ (-1 + i6)

= 1 â€“ i + 1 â€“ i6

= 2 â€“ i7

= 2 +(-7)i

(iii)

=

=

=

= ** **

# Question-3

**(i)**

**(ii)**

**(1 â€“ i)**

^{4}**(iii)**

**(iv)**

**Solution:**

**(i)**

=

=

=

=

**(ii) (1 â€“ i) ^{4} **

=

=

=

= 4i^{2}

= -4

= -4 + i0

**(iii) **

= + 3

=

=

=

=

**(iv) **

=

= -

= -

= - = -

= -

=

# Question-4

**(i) 4 â€“ 3i (ii) (iii) â€“i**

**Solution:**

**(i) 4 â€“ 3i**

=

=

= =

=

**(ii) **

Multiplicative inverse of

=

=

= =

=

**(iii) â€“i**

Multiplicative inverse of â€“i

=

=

= = = I

= 0 + (1) i

# Question-5

**Express the following expression in the form of a + ib:**

**Solution:**

=

= =

= 0 +

# Question-6

**Solve the following equations**

(i) x

(ii) 2x

(iii) x(i) x

^{2}+ 3 = 0(ii) 2x

^{2}+ x + 1 = 0(iii) x

^{2}+ 3x + 9 = 0**Solution:**

**i) x**

^{2}+ 3 = 0x^{2} = -3

x = Â±

= Â±

= Â±

**ii. 2x ^{2} + x + 1 = 0 **

x =

=

=

=

**iii. x ^{2} + 3x + 9 = 0**

x =

=

=

=

# Question-7

**Solve the following equations**

(i) â€“x

(ii) x

(iii) x(i) â€“x

^{2}+ x â€“ 2 = 0(ii) x

^{2}+ 3x + 5 = 0(iii) x

^{2}- x + 2 = 0**Solution:**

**i. â€“x**

^{2}+ x â€“ 2 = 0x =

=

**ii. x ^{2} + 3x + 5 = 0 **

x =

=

=

=

**iii. x ^{2} - x + 2 = 0 **

x =

=

=

# Question-8

**i. = 0 ii. x + 3 = 0**

iii. x

iii. x

^{2}+ x + = 0 iv. x^{2}+ = 0**Solution:**

**i. = 0**

x =

=

=

**ii. x + 3 = 0**

x =

=

**iii. x ^{2} + x + = 0**

= 0

x =

=

=

**iv. x ^{2} + = 0 **

x^{2} + x + = 0

x =

=

=

# Question-9

**Find the modulus and the arguments of each of the complex numbers**

(i) z = -1 - i (ii) z = -+i

(i) z = -1 - i (ii) z = -+i

**Solution:**

**(i) z = -1 - i**

z = -1 - i

x = -1, y = -

r = === 2

q =

|z| = 2, arg z = + 2p k, where k is an integer.

**(ii) z = -+i**

z = - + i

x = -, y = 1

r = === 2

q =

|z| = 2, arg z = + 2p k, where k is an integer.

# Question-10

**Find the modulus and the arguments of each of the complex numbers**

(i) 1 â€“ i (ii) â€“1 + i

(i) 1 â€“ i (ii) â€“1 + i

**Solution:**

**(i) 1 â€“ i**

x= rcosÎ¸=1;y = rsinÎ¸=âˆ’1

x + iy = 1 â€“ i

x = 1, y = -1

r =

and tan Î¸ = y/x = -1, Î¸ = .

Thus, the polar coordinates of 1 â€“ i are (, ) and its polar form is (cos + i sin).

**(ii) â€“1 + i**

x= rcosÎ¸=âˆ’1;y = rsinÎ¸=1

x + iy = -1 + i

x = -1, y = 1

r =

and tan Î¸ = y/x = -1, Î¸ = .

Thus, the polar coordinates of â€“1+ i are (, ) and its polar form is (cos + i sin).

# Question-11

**Find the modulus and the arguments of each of the complex numbers**

(i) â€“3 (ii) (iii) i

(i) â€“3 (ii) (iii) i

**Solution:**

**(i) â€“3**

x + iy = -3

x = -3, y = 0

r =

and tan Î¸ = y/x = 0/3, Î¸ = Ï€ .

Thus, the polar coordinates of â€“1+ i are (3, Ï€ ) and its polar form is 3(cos Ï€ + i sinÏ€ ).

**(ii) **

x + iy =+ i

x = , y = 1

r =

and tan Î¸ = y/x =, Î¸ = Ï€ /6.

Thus, the polar coordinates of + i are (2, Ï€ /6) and its polar form is 2(cos + i sin).

**(iii) i**

x + iy = i

x = 0, y = 1

r =

and tan Î¸ = y/x = âˆž , Î¸ = .

Thus, the polar coordinates of â€“1+ i are (1, ) and its polar form is (cos+ i sin).