Number of solutions of the equation $z^{3} + [3 \frac{{¯ ¯ ¯ z}^{2}}{| z |}] = 0$ where, $z$ is a complex number is

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Solution

The correct option is D 5
Let $z = r e^{i θ}$
Substituting, in the equation, we get
$r^{3} e^{3 i θ} + \frac{3 r^{2} e^{- 2 i θ}}{r} = 0$
$r^{3} e^{3 i θ} + 3 r e^{- 2 i θ} = 0$
$r (r^{2} e^{3 i θ} + 3 e^{- 2 i θ}) = 0$
Now $r$ cannot be equal to 0.
Therefore
$r^{2} e^{3 i θ} = - 3 e^{- 2 i θ}$
$r^{2} e^{5 i θ} = - 3$
$r^{2} = 3$ and $e^{5 i θ} = - 1$
Hence, we get, 5 values of $θ$ .

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