If $z \in C$ and $i z^3+4 z^2-z+4 i=0$, then a complex root of this equation having minimum magnitude is

If $z \in C$ and $i z^3+4 z^2-z+4 i=0$, then a complex root of this equation having minimum magnitude is
  1. $4 i$
  2. $\frac{1-i}{\sqrt{2}}$
  3. $\frac{\sqrt{3}+i}{2}$
  4. $\frac{1+i}{\sqrt{2}}$

Solution

Given complex equation, $ \begin{array}{cc} & i z^3+4 z^2-z+4 i=0 \\ \Rightarrow & z^2(i z+4)+i(i z+4)=0 \\ \Rightarrow & (i z+4)\left(z^2+i\right)=0 \\ \Rightarrow & z=4 i \text { or } z^2=i \\ \Rightarrow & z=4 i \\ \text { or } & \pm\left(\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{2}} i\right) \end{array} $ So, the complex root having minimum magnitude is $ \frac{1-i}{\sqrt{2}} \text { or } \frac{-1+i}{\sqrt{2}} $

Asked in: AP EAMCET 2018 (22 Apr Shift 2)

Practice more Complex Number questions on Aicharya