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

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Correct option is D

x2−x+1=0⇒x=−ω,−ω2⇒x=eiπ/3=Cisπ3⇒xα+1xα=2cos⁡απ3. Similarly xβ+1xβ, xy+1xγ.Req⇒cos⁡απ3+cos⁡βπ3+cos⁡γπ3=0⇒12−1+12=0⇒α=1,β=3,γ=5

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Let x+1x=1 and α,β,γ are distinct positive integers such that xa+1xα+xβ+1xβ+xγ+1xγ=0. Then minimum value of α+β+γ=

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