First slide

Algebra of complex numbers

Question

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

Moderate

A

2
B

3
C

6
D

5

Solution

Given,
$z^{3} + \frac{3 (\bar{z})^{2}}{| z |} = 0$
Let $z = {re}^{i} θ$
$\Rightarrow r^{3} e^{i 3 θ} + 3 {re}^{- i 2 θ} = 0$
Since r cannot be zero, so
$r^{2} e^{i 5 θ} = - 3$
which will hold for $r^{2} = 3$ and five distinct values of $θ$ Thus, there are five solutions

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