( 3 + i 2 )^6 + ( i - 3 2 )^6 is equal to - Vidyadip

MCQ

${\left( {\frac{{\sqrt 3 + i}}{2}} \right)^6} + {\left( {\frac{{i - \sqrt 3 }}{2}} \right)^6}$is equal to

✓
$- 2$
B
$0$
C
$2$
D
$1$

✓

Answer

Correct option: A.

$- 2$

a
(a) ${\left( {\frac{{\sqrt 3 + i}}{2}} \right)^6} + {\left( {\frac{{i - \sqrt 3 }}{2}} \right)^6} = {\left( {\frac{{ - 1 + \sqrt 3 i}}{{2i}}} \right)^6} + {\left( {\frac{{ - 1 - \sqrt 3 i}}{{2i}}} \right)^6}$
$ = \frac{1}{{{i^6}}}[{(\omega )^6} + {({\omega ^2})^6}] = - [{({\omega ^3})^2} + {({\omega ^3})^4}]$

$\left( {\because \,\,\,\omega = \frac{{ - 1 + \sqrt 3 i}}{2},{\omega ^2} = \frac{{ - 1 - \sqrt 3 i}}{2}} \right)$

$ = - (1 + 1) = - 2$.

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