Question

If $i{z}^3+z^2-z+i=0$, then $|z|$ equals

• A. $2$
• B. $1$
• C. $0$
• D. None of these

Answer 1

Answer: (B)

Given $i{z}^3+z^2-z+i=0$
$\Rightarrow i{z}^2\left(z-i\right)-1\left(z-i\right)=0$
$\Rightarrow \left(z-i\right)\left(i{z}^2-1\right)=0$
$\Rightarrow z=i$ or $i{z}^2=1$
$\Rightarrow \left|z\right|=1$ or $i{z}^2=1...\left(i\right)$
$\Rightarrow -i{\bar{z}}^2=1$ (By taking conjugate on both sides) $...\left(ii\right)$
$\therefore$ Multiplying $\left(i\right)$ and $\left(ii\right)$, we get
${\left(z\bar{z}\right)}^2=1$
$\Rightarrow {\left|z\right|}^2=1$
$\Rightarrow \left|z\right|=1$