Question

If $\left|z_1\right|=2,\left|z_2\right|=3,\left|z_3\right|=4$ and $|2z_1+3z_2+4z_3|=10$, then absolute value $8z_2{z}_3+27z_3{z}_1+64z_1{z}_2$ must be equal to-

Answer 1

$|z_1|=2,z_1\overline{z_1}=4$

$|z_2|=3,z_2\overline{z_2}=9$

$|z_3|=4,z_3\overline{z_3}=16$

$|8z_2{z}_3+27z_3{z}_1+64z_1{z}_2|$

$⟹|2z_1\overline{z_1}{z}_2{z}_3+3z_2\overline{z_2}{z}_3{z}_1+4z_3\overline{z_3}{z}_1{z}_2|$

$⟹|z_1{z}_2{z}_3||2\overline{z_1}+3\overline{z_2}+4\overline{z_3}|$

$⟹2\times 3\times 4|2z_1+3z_2+4z_3|$

$⟹24\times 10=240$