I : The number of solutions of the equation
$z^{2} + {|z|}^{2} = 0$ is $2$
II : The number of solutions of $z^{2} + |z| = 0$ is $3$
Which of the statements are true

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neither I nor II

Both I & II

only II

only I

answer is B.

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Detailed Solution

$z^{2} + | z |^{2} = 0$

$\begin{array}{llrl} ∵ z^{2} + | z |^{2} = 0 or z^{2} + z \bar{z} = 0 \\ \Rightarrow z (z + \bar{z}) = 0 \\ ∴ z = 0 \\ and z + \bar{z} = 0 \Rightarrow 2 Re (z) = 0 \\ ∴ Re (z) = 0 \\ If z = x + i y \\ = 0 + i y, y \in R \\ and i = \sqrt{- 1} \\ given equation has infinite solutions. \end{array}$

$z^{2} + | z | = 0 \Rightarrow {(x + iy)}^{2} + \sqrt{x^{2} + y^{2}} = 0 \Rightarrow (x^{2} - y^{2} + \sqrt{x^{2} + y^{2}}) + i (2 x y) = 0 \Rightarrow x = 0 \Rightarrow - y^{2} + |y| = 0 \begin{array}{l} ∴ | y | = 0, 1 \\ \Rightarrow y = 0, \pm 1 \\ \begin{aligned} y = 0 \Rightarrow x^{2} + \sqrt{x^{2}} = 0 \\ \Rightarrow x^{2} + | x | = 0 \\ y = 0 \Rightarrow | x |^{2} + | x | = 0 \Rightarrow x = 0 \end{aligned} \\ ∴ x + i y are 0 + 0 \cdot i, 0 + i, 0 - i \\ i.e. z = 0, i, - i are the solutions of the given equation. \end{array}$