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Let \( \alpha \) be a root of the equation \( 1+x^{2}+x^{4}=0 \). Then the value of \( \alpha^{1011}+\alpha^{2022}-\alpha^{3033} \) is equal to : |
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Answer» Let ω = x2 then α be root of 1 + ω + ω2 = 0, where ω is cubic Hence, α = √ω ⇒ α2 = ω ⇒ α6 = ω3 = 1 Now, α1011 + α2022 - α3033 = α6 x 168 + 3 + α333 x 6 + 4 - α6 x 505 +3 = (α6)168α3 + (α6)333α4 - (α6)505α3 = α3 + α4 - α3 (\(\because \) α6 = 1) = α4 = ω2 (\(\because\) α2 = ω) = \(\frac{-1-\sqrt3i}2\) |
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