Question 94 of 157MediumMCQ
If \[\alpha \] and \[\beta \] are the roots of the equation x2−x+1=0, then \[{\alpha ^{2009}} + {\beta ^{2009}} = \] Solutions · 1
Official solution
x2−x+1=0
\[ \Rightarrow x = {{1 \pm \sqrt {1 - 4} } \over 2}\]
\[x = {{1 \pm \sqrt 3 i} \over 2}\]
\[\alpha = {1 \over 2} + i{{\sqrt 3 } \over 2} = - {\omega ^2}\]
\[\beta = {1 \over 2} - {{i\sqrt 3 } \over 2} = - \omega \]
\[{\alpha ^{2009}} + {\beta ^{2009}} = {\left( { - {\omega ^2}} \right)^{2009}} + {\left( { - \omega } \right)^{2009}}\]
\[ = - {\omega ^2} - \omega = 1\] Answer the question to unlock