Let z=32+i25+32−i25 .If R(z) and I(z) respectively denote the real and imaginary parts of z, then

A
$R (z) > 0$ and $I (z) > 0$
B
$I (z) = 0$
C
$R (z) < 0$ and $I (z) = 0$
D
$R (z) = - 3$

Solution:

We have $z = {(\frac{\sqrt{3}}{2} + \frac{i}{2})}^{5} + {(\frac{\sqrt{3}}{2} - \frac{i}{2})}^{3}$

$\begin{array}{l} = {(\cos \frac{π}{6} + i \sin \frac{π}{6})}^{5} + {(\cos \frac{π}{6} - i \sin \frac{π}{6})}^{5} \\ = (\cos \frac{5 π}{6} + i \sin \frac{5 π}{6}) + (\cos \frac{5 π}{6} - i \sin \frac{5 π}{6}) = 2 \cos \frac{5 π}{6} < 0 \end{array}$

Here, $I (z) > 0$ and $R (z) > 0$ .

Let $z = {(\frac{\sqrt{3}}{2} + \frac{i}{2})}^{5} + {(\frac{\sqrt{3}}{2} - \frac{i}{2})}^{5}$ .If $R (z)$ and $I (z)$ respectively denote the real and imaginary parts of z, then

Solution:

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