Question

The number of solutions of the equation $z^{3}+\frac{3 \left(\bar{z}\right)^{2}}{\left|z\right|}=0$ (where, $z$ is a complex number) are

• A. $2$
• B. $3$
• C. $6$
• D. $5$

Answer 1

Answer: (D)

Let, $z=re^{i \theta }$
$\Rightarrow r^{3}e^{i 3 \theta }+3re^{- i 2 \theta }=0$
$\Rightarrow r^{2}e^{i 5 \theta }=-3$
$\Rightarrow r^{2}=3$ and $e^{i 5 \theta }=-1$
$\Rightarrow r=\sqrt{3}$ and $\theta =\frac{\pi }{5}+\frac{2 k \pi }{5}$ where, $k=0,1,2,3,4$
$\Rightarrow 5$ solutions