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
Re(z)=0
B
$\left.\operatorname{Re}( z )>0, I _{ m } z \right)>$
C
$\operatorname{Re}( z )>0, I _{ rs }( z )<0$
D
$I _{ m }( z )=0$

✓

Answer

(d) $I _{ m }( z )=0$
Explanation: From given,
$z=2^5 C _0{\frac{\sqrt{3}^2}{2}}^5+{ }^5 C _2 \frac{\sqrt{3}}{2}^3 \frac{i^2}{2}+{ }^5 C _4 \frac{\sqrt{3}}{2} \frac{i}{2}$
Since $t^2=-1$ and $t^4=1$, will not contain any $i$ and hence $I_m(z)=0$

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