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

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

1. A

$R\left(z\right)>0$ and $I\left(z\right)>0$

2. B

$I\left(z\right)=0$

3. C

$R\left(z\right)<0$ and $I\left(z\right)=0$

4. D

$R\left(z\right)=-3$

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Solution:

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

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

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

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