First slide

Algebra of complex numbers

Question

If $ω (\neq 1)$ is a root of unity and ${(1 + ω^{2})}^{11} = a + b ω + c ω^{2},$ then ( $a, b, c)$ equals

Moderate

A

$(1, 1, 0)$
B

$(0, 1, 1)$
C

$(1, 0, 1)$
D

$(1, 1, 1)$

Solution

$a + b ω + c ω^{2} = {(1 + ω^{2})}^{11} = (- ω)^{11}$
$\begin{array}{l} = - {(ω^{3})}^{3} ω^{2} = 1 + ω [∵ 1 + ω + ω^{2} = 0] \\ a = 1, b = 1, c = 0 \end{array}$

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