Question

Number of solutions of the equation, $z^{3} + \frac{3\left|z\right|^{2}}{z} = 0$, 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)

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