Question

Find the modulus and amplitude of the following complex numbers.

(1 + 2i)² (1 − i)

Answer 1

Let z = (1 + 2i)² (1 − i)

= (1 + 4i + 4i²) (1 − i)

= [1 + 4i + 4(−1)] (1 − i) ...[∵ i² = −1]

= (−3 + 4i) (1 − i)

= −3 + 3i + 4i − 4i²

= −3 + 7i − 4 (−1)

= − 3 + 7i + 4

∴ z = 1 + 7i

∴ a = 1, b = 7, i. e. a > 0, b > 0

∴ |z| = `sqrt("a"^2 + "b"^2)`

= `sqrt(1^2 + 7^2)`

= `sqrt(1 + 49)`

= `5sqrt(2)`

Here, (1, 7) lies in 1^st quadrant

amp (z) = `tan^-1("b"/"a")`

= tan^–1(7)