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Question-1

(i) (5i)       (ii) i9 + i19        (iii) i-39

 


Solution:
(i) (5i)

= - 3i2

= -3(-1) = 3

= 3 + i0

(ii) i9 + i19

= i8.i + i18.i

= (i2)4i + (i2)9i

=(-1)4i + (-1)9i

= i - i = 0

= 0 + i0

(iii) i-39

= (i39)-1

= [i38.l]-1

= [(l2)19 i]-1

= [(-1)19 i]-1

= (-i)-1 =

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

Question-2

(i) 3(7 + i7) +i(7 + i7)           

(ii) (1 – i) – (-1 + i6)        

(iii)

 


Solution:
(i) 3(7 + i7) +i(7 + i7)

= 21 + 21i + 7i + 7i
2

= 21 + 28i +7(-1)

= 21 + 28i – 7

= 14 + 28i

(ii) (1 – i) – (-1 + i6)

= 1 – i + 1 – i6

= 2 – i7

= 2 +(-7)i

(iii)

=

=

=

=

Question-3

(i)     (ii) (1 – i)4        (iii)      (iv)        

 


Solution:
(i)   

=

=

=

=

(ii) (1 – i)4  

=

=

=

= 4i2

= -4

= -4 + i0

(iii)

= + 3

=

=
=

=

(iv)   

=

= -

= -

= - = -

= -

=

Question-4

(i) 4 – 3i              (ii)                  (iii) –i

 


Solution:
(i) 4 – 3i  

=

=

= =

=

(ii)   

Multiplicative inverse of

=

=

= =

=

(iii) –i

Multiplicative inverse of –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) x2 + 3 = 0              

(ii) 2x2 + x + 1 = 0            

(iii) x2 + 3x + 9 = 0

Solution:
i) x2 + 3 = 0

x2 = -3

x = ±

   = ±

   = ±

ii. 2x2 + x + 1 = 0

x =
  =
  =
  =

iii. x2 + 3x + 9 = 0

x =
  =
  =
  =

Question-7

Solve the following equations
(i) –x2 + x – 2 = 0               

(ii) x2 + 3x + 5 = 0          

(iii) x2 - x + 2 = 0

Solution:
i. –x2 + x – 2 = 0

x =

  =

ii. x2 + 3x + 5 = 0

x =

  =

  =

  =

iii. x2 - x + 2 = 0

x =

  =

  =

Question-8

i. = 0        ii. x + 3 = 0            
 iii. x2 + x + = 0         iv. x2 + = 0

Solution:
i. = 0

x =

  =

  =

ii. x + 3 = 0

x =

  =

iii. x2 + x + = 0

= 0

x =

  =

  =

iv. x2 + = 0

x2 + x + = 0

x =

  =

  =

Question-9

Find the modulus and the arguments of each of the complex numbers
(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 – i   (ii) –1 + i

Solution:
(i) 1 – i   
x= rcosθ=1;y = rsinθ=−1
x + iy = 1 – i

x = 1, y = -1
r =

and tan θ = y/x = -1, θ = .
Thus, the polar coordinates of 1 – i are (, ) and its polar form is (cos + i sin).

(ii) –1 + i
x= rcosθ=−1;y = rsinθ=1
x + iy = -1 + i

x = -1, y = 1

r =

and tan θ = y/x = -1, θ = .
Thus, the polar coordinates of –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) –3     (ii)         (iii) i

Solution:
(i) –3     
x + iy = -3
x = -3, y = 0
r =
and tan θ = y/x = 0/3, θ = π .
Thus, the polar coordinates of –1+ i are (3, π ) and its polar form is 3(cos π + i sinπ ).

(ii)      
x + iy =+ i
x = , y = 1
r =
and tan θ = y/x =, θ = π /6.
Thus, the polar coordinates of + i are (2, π /6) and its polar form is 2(cos + i sin).   

(iii) i
x + iy = i
x = 0, y = 1
r =
and tan θ = y/x = , θ = .
Thus, the polar coordinates of –1+ i are (1, ) and its polar form is (cos+ i sin).





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