If $z_1,z_2,z_3$ be three complex number such than $z_1+z_2+z_3=0$and $\left|z_1\right|=\left|z_2\right|=\left|z_3\right|=1$ than $\left|z_1^2+2z_2^2+z_3^2\right|$ equals

$\begin{array}{l}{z}_1+z_2+z_3=0\Rightarrow {\overline{z}}_1+{\overline{z}}_2+{\overline{z}}_3=0\\ \Rightarrow \frac{1}{z_1}+\frac{1}{z_2}+\frac{1}{z_3}=0\end{array}\left[\because z_1{\overline{z}}_1=1\text{ etc. }\right]$

Now,  $0={\left(z_1+z_2+z_3\right)}^2=z_1^2+z_2^2+z_3^2+2\left(z_2{z}_3+z_3{z}_1+z_1{z}_2\right)$

$\begin{array}{l}\Rightarrow 0=z_1^2+z_2^2+z_3^2+2z_1{z}_2{z}_3\left(\frac{1}{z_1}+\frac{1}{z_2}+\frac{1}{z_3}\right)\\ =z_1^2+z_2^2+z_3^2\end{array}$

Thus, $\left|z_1^2+2z_2^2+z_3^2\right|=\left|z_1^2+z_2^2+z_3^2+z_2^2\right|$

$=\left|0+z_2^2\right|={\left|z_2\right|}^2=1$