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If ω is a complex cube root of unity, then what is ω10 + ω-10 equal to? 1. 22. -13. -24. 1 |
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Answer» Correct Answer - Option 2 : -1 Concept: Properties of cube root of unity: ω3 = 1 and (ω2 + ω + 1) = 0 Calculations: Consider, ω10 + ω-10 = ω10 + \(\dfrac {1}{ω^{10}}\) = \(\dfrac {ω^{20}+1}{ω^{10}}\) = \(\dfrac {ω^{18}.ω^{2}+1}{ω^{9.}ω}\) = \(\dfrac {{(ω^3)^6}.ω^{2}+1}{(ω^3)^3ω}\) = \(\dfrac {{(1)^6}.ω^{2}+1}{(1)^3ω}\) (∵ ω3 = 1) = \(\dfrac {ω^{2}+1}{ω}\) From equation (1), we have = \(\dfrac {-ω}{ω}\) (∵ 1 + ω2 = -ω) = - 1 Hence, if ω is a complex cube root of unity, then ω10 + ω-10 equal to -1. |
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