The number of all possible solutions of the equation $\mathrm{z}^3+\overline{\mathrm{z}}=0$ is

The number of all possible solutions of the equation $\mathrm{z}^3+\overline{\mathrm{z}}=0$ is
  1. 4
  2. 5
  3. 3
  4. 6

Solution

$ \begin{aligned} & \text { Given that } z^3+\bar{z}=0 \Rightarrow z^3=-\bar{z} \\ & \Rightarrow|z|^3=|-\bar{z}|=|z| \Rightarrow|z|\left(|z|^2-1\right)=0 \\ & |z|=0 \text { or }|z|^2=1 \Rightarrow z \bar{z}=1 \Rightarrow \bar{z}=\frac{1}{z} \\ & \therefore z^3+\bar{z}=0 \Rightarrow z^3+\frac{1}{z}=0 \\ & \Rightarrow z^4+1=0 \rightarrow 4 \text { solution } \\ & |z|=0 \rightarrow \text { one solution } \end{aligned} $ Total 5 solution

Asked in: AP EAMCET 2023 (15 May Shift 1)

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