The equation whose solutions are the non-zero solutions of the equation $\bar{z}=i z^2$, is

The equation whose solutions are the non-zero solutions of the equation $\bar{z}=i z^2$, is
  1. $z^3+i=0$
  2. $z^3+z+1=0$
  3. $z^3-i=0$
  4. $z^3+i z+1=0$

Solution

$\begin{array}{lccl}\text { } & \bar{z}=i z^2 & & \\ \Rightarrow & z=-i \bar{z}^2 & & \Rightarrow z=-i\left[i z^2\right]^2 \\ \Rightarrow & z=-i i^2 z^4 & & \Rightarrow z=i z^4 \\ \Rightarrow & z^4=\frac{1}{i} z & & z^4=-i z \\ \Rightarrow & z^4+i z=0 & & z z\left(z^3+i\right)=0 \\ \Rightarrow & z^3+i=0 & & (\because z \neq 0)\end{array}$

Asked in: AP EAMCET 2019 (21 Apr Shift 1)

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