Let $| z_{i} | = i, i = 1, 2, 3, 4$ and $| 16 z_{1} z_{2} z_{3} + 9 z_{1} z_{2} z_{4} + 4 z_{1} z_{3} z_{4} + z_{2} z_{3} z_{4} | = 48,$ then the value of $∣ ∣ ∣ \frac{1}{{¯ ¯ ¯ z}_{1}} + \frac{4}{{¯ ¯ ¯ z}_{2}} + \frac{9}{{¯ ¯ ¯ z}_{3}} + \frac{16}{{¯ ¯ ¯ z}_{4}} ∣ ∣ ∣ .$

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Solution

The correct option is B $2$
$\begin{matrix} | z_{i} | = i i = 1, 2, 3, 4 | z_{1} | = 1 | z_{2} | = 2 | z_{3} | = 3 | z_{4} | = 4 | 16 z_{1} z_{2} z_{3} + 9 z_{1} z_{2} z_{3} + 4 z_{1} z_{2} z_{3} + z_{2} z_{3} z_{4} | = 48 | z_{1} z_{2} z_{3} z_{4} | ∣ ∣ ¯ ¯¯¯ ¯ z_{4} + ¯ ¯¯¯ ¯ z_{3} + ¯ ¯¯¯ ¯ z_{2} + ¯ ¯¯¯ ¯ z_{1} ∣ ∣ = 48 1 \times 2 \times 3 \times 4 \times | z_{1} + z_{2} + z_{3} + z_{4} | = 48 \Rightarrow | z_{1} + z_{2} + z_{3} + z_{4} | = 2 = ∣ ∣ \frac{1}{z_{1}} + \frac{4}{z_{2}} + \frac{9}{z_{3}} + \frac{16}{z_{4}} ∣ ∣ . ∣ ∣ ∣ \frac{z_{1} {¯ z}_{1}}{z_{1}} + \frac{z_{2} {¯ z}_{2}}{z_{2}} + \frac{z_{3} {¯ z}_{3}}{z_{3}} + \frac{z_{4} {¯ z}_{4}}{z_{4}} ∣ ∣ ∣ = | z_{1} + z_{2} + z_{3} + z_{4} | = 2 O p t i o n B i s c o r r e c t a n s w e r . \end{matrix}$

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Let |zi|=i,i=1,2,3,4 and |16z1z2z3+9z1z2z4+4z1z3z4+z2z3z4|=48, then the value of ∣∣∣1¯¯¯z1+4¯¯¯z2+9¯¯¯z3+16¯¯¯z4∣∣∣.

Let $| z_{i} | = i, i = 1, 2, 3, 4$ and $| 16 z_{1} z_{2} z_{3} + 9 z_{1} z_{2} z_{4} + 4 z_{1} z_{3} z_{4} + z_{2} z_{3} z_{4} | = 48,$ then the value of $∣ ∣ ∣ \frac{1}{{¯ ¯ ¯ z}_{1}} + \frac{4}{{¯ ¯ ¯ z}_{2}} + \frac{9}{{¯ ¯ ¯ z}_{3}} + \frac{16}{{¯ ¯ ¯ z}_{4}} ∣ ∣ ∣ .$