Question 141 of 157MediumMCQ
Let \[\alpha\] be a root of the equation 1 + x2 + x4 = 0. Then, the value of \[\alpha\]1011 + \[\alpha\]2022 - \[\alpha\]3033 is equal to :
Solutions · 1
Official solution
Given, \[\alpha\] is a root of the equation 1 + x2 + x4 = 0
\[\therefore\] \[\alpha\] will satisfy the equation.
\[\therefore\] 1 + \[\alpha\]2 + \[\alpha\]4 = 0
\[{\alpha ^2} = {{ - 1 \pm \sqrt {1 - 4} } \over 2}\]
\[ = {{ - 1 \pm \sqrt 3 i} \over 2}\]
\[\therefore\] \[{\alpha ^2} = \omega \,ar\,{\omega ^2}\]
Now,
\[{\alpha ^{1011}} + {\alpha ^{2022}} - {\alpha ^{3033}}\]
\[ = \alpha \,.\,{({\alpha ^2})^{505}} + {({\alpha ^2})^{1011}} - \alpha \,.\,{({\alpha ^2})^{1516}}\]
\[ = \alpha {(\omega )^{505}} + {(\omega )^{1011}} - \alpha \,.\,{(\omega )^{1516}}\]
\[ = \alpha \,.\,{({\omega ^3})^{168}}\,.\,\omega + {({\omega ^3})^{337}} - \alpha \,.\,{({\omega ^3})^{505}}\,.\,\omega \]
\[ = \alpha \,\omega + 1 - \alpha \,\omega \]
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