First slide

Algebra of complex numbers

Question

$Let x + \frac{1}{x} = 1 and α, β, γ are distinct positive integers such that$ $(x^{a} + \frac{1}{x^{α}}) + (x^{β} + \frac{1}{x^{β}}) + (x^{γ} + \frac{1}{x^{γ}}) = 0 . Then minimum value of α + β + γ =$

Moderate

A

15
B

12
C

10
D

9

Solution

$x^{2} - x + 1 = 0 \Rightarrow x = - ω, - ω^{2} \Rightarrow x = e^{i π / 3} = C i s \frac{π}{3} \Rightarrow x^{α} + \frac{1}{x^{α}} = 2 \cos \frac{α π}{3} . Similarly x^{β} + \frac{1}{x^{β}}, x^{y} + \frac{1}{x^{γ}}$ .
$Req \Rightarrow \cos \frac{α π}{3} + \cos \frac{β π}{3} + \cos \frac{γ π}{3} = 0 \Rightarrow \frac{1}{2} - 1 + \frac{1}{2} = 0 \Rightarrow α = 1, β = 3, γ = 5$

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