Question

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

• A. $2$
• B. $3$
• C. $6$
• D. $5$

Answer 1

Answer: (D)

Let, $z=r{e}^{i\theta }$
$\Rightarrow r^3{e}^{i3\theta }+3r{e}^{-i2\theta }=0$
$\Rightarrow r^2{e}^{i5\theta }=-3$
$\Rightarrow r^2=3$ and $e^{i5\theta }=-1$
$\Rightarrow r=\sqrt{3}$ and $\theta =\frac{\pi }{5}+\frac{2k\pi }{5}$ where, $k=0,1,2,3,4$
$\Rightarrow 5$ solutions