Question

For any two complex numbers $z_{1}$ and $z_{2}$ with $\left|z_{1}\right| \neq\left|z_{2}\right|$ $\left|\sqrt{2} z_{1}+i \sqrt{3} \bar{z}_{2}\right|^{2}+\left|\sqrt{3} \bar{z}_{1}+i \sqrt{2} z_{2}\right|^{2}$ is

• A. less than $5\left|z_{1}\right|^{2}+\left|z_{2}\right|^{2}$
• B. greater than $10\left| z _{1} z_{2}\right|$
• C. equal to $2\left|z_{1}\right|^{2}+3\left|z_{2}\right|^{2}$
• D. zero

Answer 1

Answer: (B)

$\left|\sqrt{2} z_{1}+i \sqrt{3} \bar{z}_{2}\right|^{2}+\left|\sqrt{3} \bar{z}_{1}+i \sqrt{2} z_{2}\right|^{2}$
$=\left(\sqrt{2} z_{1}+i \sqrt{3} \bar{z}_{2}\right)\left(\sqrt{2} \bar{z}_{1}-i \sqrt{3} z_{2}\right)+\left(\sqrt{3} \bar{z}_{1}+i \sqrt{2} z_{2}\right)\left(\sqrt{3} z_{1}-i \sqrt{2} \bar{z}_{2}\right)$
$=5\left(\left|z_{1}\right|^{2}+\left|z_{2}\right|^{2}\right)>5 \cdot 2 \sqrt{\left|z_{1}\right|^{2}\left|z_{2}\right|^{2}}$
$=10\left|z_{1} z_{2}\right|$
Since $A \cdot M >G \cdot M$ for $\left|z_{1}\right| \neq\left|z_{2}\right|$