Question

If $z_1,z_2$ be two complex numbers with different modulii, then ${|\sqrt{3z_1}+i\sqrt{2z_2}|}^2+{|\sqrt{2z_1}-i\sqrt{3}{z}_2|}^2$ is

• A. $2{\left|z_1\right|}^2+3{\left|z_2\right|}^2$
• B. $<5({\left|z_1\right|}^2+{\left|z_2\right|}^2)$
• C. $>10\left|z_1{z}_2\right|$
• D. $0$

Answer 1

Answer: (C)

$>10\left|z_1{z}_2\right|$

Expression $=(\sqrt{3}{z}_1+$i$\sqrt{2}{z}_2\left)\right(\sqrt{3}{\overline{z}}_1-i\sqrt{2}{\overline{z}}_2)+(\sqrt{2}{z}_1-i\sqrt{3}{z}_2)+(\sqrt{2}{\overline{z}}_1+i\sqrt{3}{z}_2$)
$=3|z_1{|}^2+2|z_2{|}^2+2|z_1{|}^2+3|z_2{|}^2$
$=5[|z_1{|}^2+|z_2{|}^2]$
But $\frac{|z_1{|}^2+|z_2{|}^2}{2}>\{|z_1{|}^2|z_2{|}^2{\}}^{\frac{1}{2}}$
$5(|z_1{|}^2+|z_2{|}^2)>5.2|z_1||z_2|$i.e., $5(|z_1{|}^2+|z_2{|}^2)>10|z_1{z}_2|$