Question

Numbers of solutions of the equation $z^3+\frac{3(\overline{z}{)}^2}{|z|}=0$ where $z$ is a complex number is

• A. $2$
• B. $3$
• C. $6$
• D. $5$

Answer 1

The correct option is D $5$

$z^3+\frac{3{\overline{z}}^2}{\left|z\right|}=0$ ...(1)
$\left|z\right|\ne 0$
$\Rightarrow {\left|z\right|}^3=|-\frac{3{\overline{z}}^2}{\left|z\right|}|\Rightarrow {\left|z\right|}^3=3\left|z\right|\because \left|z\right|\ne 0\Rightarrow {\left|z\right|}^2=3\Rightarrow z\overline{z}=3$
$\Rightarrow z^3+\frac{3}{\sqrt{3}}{\left(\frac{3}{z}\right)}^2=0\Rightarrow z^5+9\sqrt{3}=0$
Therefore equation has five roots.
Ans: D