If z = ( 3 2 + i 2 )^5 + ( 3 2 - i 2 )^5 , then - Vidyadip

MCQ

If $z = {\left( {\frac{{\sqrt 3 }}{2} + \frac{i}{2}} \right)^5} + {\left( {\frac{{\sqrt 3 }}{2} - \frac{i}{2}} \right)^5}$, then

A
${\mathop{\rm Re}\nolimits} (z) = 0$
✓
${\mathop{\rm Im}\nolimits} (z) = 0$
C
${\mathop{\rm Re}\nolimits} (z) > 0,{\mathop{\rm Im}\nolimits} (z) > 0$
D
${\mathop{\rm Re}\nolimits} (z) > 0,{\mathop{\rm Im}\nolimits} (z) < 0$

✓

Answer

Correct option: B.

${\mathop{\rm Im}\nolimits} (z) = 0$

b
(b) Given that $z = {\left( {\frac{{\sqrt 3 }}{2} + i\frac{1}{2}} \right)^5} + {\left( {\frac{{\sqrt 3 }}{2} - i\frac{1}{2}} \right)^5}$
$ = {\left[ {\cos \left( {\frac{\pi }{6}} \right) + i\sin \left( {\frac{\pi }{6}} \right)} \right]^5} + {\left[ {\cos \left( {\frac{\pi }{6}} \right) - i\sin \left( {\frac{\pi }{6}} \right)} \right]^5}$
$ = \cos \frac{{5\pi }}{6} + i\sin \frac{{5\pi }}{6} + \cos \frac{{5\pi }}{6} - i\sin \frac{{5\pi }}{6}$.
Heance $Im (z) = 0.$

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