OPERATIONS OF COMPLEX NUMBERS

Choose the correct or the most suitable answer from the given four alternatives :

Problem 1 :

iⁿ + i^{n + 1} + i^{n+ 2} + i^{n + 3} is

(1)  0   (2)  1  (3)  -1  (4)  i

Solution :

Given, iⁿ + i^{n + 1} + i^{n+ 2} + i^{n + 3}

= iⁿ(1 + i + i² + i³)

= iⁿ(1 + i - 1 - i)

= iⁿ(0)

= 0

So, option (1) is correct.

Problem 2 :

(1)  1 + i    (2)  i   (3)  1   (4)   0

Solution :

= (i + i² + i³ + ... + i¹³) + (i⁰ + i + i² + i³ + ... + i¹²)

= i⁰  + 2(i + i² + i³ + ... + i¹²) + i¹³

= 1 + 2(0) + i

= i + 1

So, option (1) is correct.

Problem 3 :

The area of the triangle formed by the complex numbers z, iz, and z + iz in the Argand’s diagram is

(1)  1/2 |z|²   (2)  |z|²    (3)  3/2  |z|²   (4)  2|z|²

Solution :

Area of the triangle = 1/2 × bh

= 1/2 × |z| × |iz|

= 1/2 × |z|²

So, option (1) is correct.

Problem 4 :

The conjugate of a complex numbers is 1/(I - 2). Then, the complex number is

Solution :

So, option (2) is correct.

Problem 5 :

(1)   0   (2)  1  (3)  2  (4)  3

Solution :

So, option (3) is correct.