Question

Number of solutions of the equation, $z^3+\frac{3{|z|}^2}{z}=0$, where z is a complex number and $\left|z\right|=\sqrt{3}$ is

• A. 2
• B. 3
• C. 6
• D. 4

Answer 1

Answer: (D)

$z^3+\frac{3{|z|}^2}{z}=0\Rightarrow z^3+\frac{3z.\bar{z}}{z}=0\Rightarrow z^3+3\bar{z}=0$
Let $z=r{e}^{i\theta }$
$\Rightarrow r^3{e}^{i3\theta }+3r{e}^{-i\theta }=0$
$\Rightarrow e^{i4\theta }=-1\left[\because r=\sqrt{3}\right]$
$\Rightarrow cos4\theta +isin4\theta =-1$
$\Rightarrow cos4\theta =-1\dots \left(i\right)$
Now $0\le \theta <2\pi$
$\Rightarrow 0\le 4\theta <8\pi$
$\therefore \theta =\pi ,3\pi ,5\pi ,7\pi$