Question

If $\left|z_1\right|=\left|z_2\right|=\left|z_3\right|=1$ and $z_1+z_2+z_3+=\sqrt{2}+i$, then the complex number $z_2\overline{z_3}+z_3+\overline{z_1}+z_1+\overline{z_2}$ is

• A. Purely imaginary
• B. Purely real
• C. Positive real number
• D. None of these

Answer 1

Answer: (A)

Purely imaginary

$\left|z_1\right|=\left|z_2\right|=\left|z_3\right|=1$

$z_1+z_2+z_3=\sqrt{3}+i$

To find $z_2\overline{z_3}+z_3+\overline{z_1}+z_1+\overline{z_2}$

${\left|\right(z_1+z_2+z_3\left)\right|}^2=(\sqrt{3}+i{)}^2$

$(z_1+z_2+z_3)\left(\overline{z_1+z_2+z_3}\right)=(\sqrt{3}+i{)}^2$

$(z_1+z_2+z_3)(\overline{z_1}+\overline{z_2}+\overline{z_3})=(\sqrt{3}+i{)}^2$

$z_1\overline{z_1}+z_1\overline{z_2}+z_1\overline{z_3}+z_2\overline{z_1}+z_2\overline{z_2}+z_2\overline{z_3}+\overline{z_1}{z}_3+z_3\overline{z_2}+z_3\overline{z_3}$

$=3+(i{)}^2+i2\sqrt{3}$

$[\because z_1\overline{z_1}={\left|z_1\right|}^2]={\left|z_1\right|}^2+{\left|z_2\right|}^2+{\left|z_3\right|}^2+z_1\overline{z_2}+z_1\overline{z_3}+z_2\overline{z_1}=2+2i\sqrt{3}$

$+z_2\overline{z_3}+\overline{z_1}{z}_3+z_3\overline{z_2}$

$3+z_3(\overline{z_2}+\overline{z_2})+z_1\overline{z_2}+z_1\overline{z_3}+z_2\overline{z_1}+z_2\overline{z_3}=2+3i\sqrt{3}$

$z_3\overline{z_2}+z_3\overline{z_1}+z_1\overline{z_2}+z_1\overline{z_3}+z_2\overline{z_1}+z_2\overline{z_3}=-1+2i\sqrt{3}$

Error !