First slide

Algebra of complex numbers

Question

$Let α & β are roots of x^{2} + ω x + ω^{2} = 0, where ω is an imaginary cube root of unity and$
$z = α^{9} + i β^{9}, then value of | z | is$

Moderate

A

$\sqrt{2}$
B

2
C

1
D

$\frac{\sqrt{15}}{2}$

Solution

$\begin{array}{l} Given equation is x^{2} + ω x + ω^{2} = 0 \\ \Rightarrow x^{2} - (1 + ω^{2}) x + ω^{2} = 0 (∴ 1 + ω + ω^{2} = 0 \Rightarrow ω = - (1 + ω^{2})) \end{array}$
$\begin{array}{l} \Rightarrow α = 1, β = ω^{2} are roots of the equation \\ ∴ z = α^{9} + i β^{9} \\ = (1)^{9} + i {(ω^{2})}^{9} \\ = 1 + i \\ Hence, | z | = \sqrt{1 + 1} = \sqrt{2} \end{array}$

Get Instant Solutions

When in doubt download our app. Now available Google Play Store- Doubts App

Recieve an sms with download link

Doubts App

OR

SCAN ME

Doubts App

Doubts App