Question

If $i{z}^3+z^2-z+i=0$ then the value of $7|z|$ is_______

• A. $7$
• B. $14$
• C. $21$
• D. $28$

Answer 1

The correct option is A $7$

$i{z}^3+z^2-z+i=0$

$\Rightarrow i{z}^3+i^{2}z+z^2+i=0$

$\Rightarrow iz(z^2+i)+(z^2+i)=0$

$\Rightarrow (z^2+i)(1+iz)=0$

$\Rightarrow z^2=-i$ or $z=i$

$\Rightarrow z=\pm e^{\frac{i3\pi }{4}}$ or $z=\pm e^{\frac{i\pi }{2}}$

$\Rightarrow z=i$ or $z=\frac{1}{\sqrt{2}}+\frac{i}{\sqrt{2}}$ or $z=\frac{-1}{\sqrt{2}}-\frac{-i}{\sqrt{2}}$

$\Rightarrow \left|z\right|=\left|i\right|=1$ or $\left|z\right|=|\frac{1}{\sqrt{2}}+\frac{i}{\sqrt{2}}|=1$ or $\left|z\right|=|\frac{-1}{\sqrt{2}}-\frac{-i}{\sqrt{2}}|=1$

$\left|z\right|=1$ for all cases.

$\therefore 7\left|z\right|=7$