Question

If $\left|z_1\right|=2,\left|z_2\right|=3,\left|z_3\right|=4and|2z_1+3z_2+4z_3|=4,$ then absolute value of $8z_2{z}_3+27z_3{z}_1+64z_1{z}_2$ equals

• A. $24$`
• B. $48$
• C. $72$
• D. $96$

Answer 1

The correct option is D $96$

$\left|z_1\right|=2,$ $\left|z_2\right|=3,$ $\left|z_3\right|=4$

$\therefore |2z_1+3z_2+4z_3|=4$

$\therefore 8z_2{z}_3+27z_3{z}_4+64z_1{z}_3=\left|z_1\right|\left|z_2\right|\left|z_3\right||\frac{8}{\left|z_1\right|}+\frac{27}{\left|z_2\right|}+\frac{64}{\left|z_3\right|}|$

$=2\times 3\times 4|\frac{8\overline{z_1}}{{\left|z_1\right|}^2}+\frac{27\overline{z_2}}{{\left|z_2\right|}^2}+\frac{64\overline{z_3}}{{\left|z_3\right|}^2}|$

$=24\left|\underset{–}{2\overline{z_1}+3\overline{z_2}+4\overline{z_3}}\right|$

$=24\left|\overline{2\overline{z_1}+3\overline{z_2}+4\overline{z_3}}\right|$

$=24|2\overline{z_1}+3\overline{z_2}+4\overline{z_3}|$

$=24\times 4$

$=96.$

Hence, the answer is 96.