The number of solution of the equation $z^{2} = ¯ ¯ ¯ z$ is

1

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2

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3

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4

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Solution

The correct option is D $4$
Given, $z^{2} = ¯ ¯ ¯ z$
$\Rightarrow z^{2} = \frac{1}{z}$
$\Rightarrow z^{3} = 1$
Hence, $z$ is cube root of unity.
Therefore, $z = 1, w, w^{2}$
Also, $z^{2} = ¯ ¯ ¯ z$
$¯ ¯ ¯ z z - ¯ ¯ ¯ z = 0$
Now $| Z | = | ¯ ¯¯ ¯ Z |$
Therefore, $| z^{2} | - | z | = 0$
$| z | (| z | - 1) = 0$
$| z | = 0$
$z = 0$
Hence, there are four solutions.

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