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Q.

For $Z_{1} = \sqrt[6]{\frac{1 - i}{1 + i \sqrt{3}}}; Z_{2} = \sqrt[6]{\frac{1 - i}{\sqrt{3} + i}}; Z_{3} = \sqrt[6]{\frac{1 + i}{\sqrt{3} - i}}$ which of the following holds good?

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a

$\sum {|Z_{1}|}^{2} = \frac{3}{2}$

b

$\sum {|Z_{1}|}^{3} + {|Z_{2}|}^{3} = {|Z_{3}|}^{- 6}$

c

${|z_{1}|}^{- 4} + {|z_{2}|}^{- 4} = {|z_{3}|}^{- 8}$

d

${|Z_{1}|}^{4} + {|Z_{2}|}^{4} = {|Z_{3}|}^{8}$

answer is B.

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Detailed Solution

$\begin{array}{l} |z_{1}| = {|\frac{1 - i}{1 + i \sqrt{3}}|}^{\frac{1}{6}} = {|\frac{\sqrt{2}}{2}|}^{\frac{1}{6}} = 2^{- \frac{1}{12}} |llly |z_{2}| = 2^{- \frac{1}{12}} \\ |z_{3}| = 2^{- \frac{1}{12}} \Rightarrow {|z_{1}|}^{- 4} + {|z_{2}|}^{- 4} = 2^{2 / 3} = {|z_{3}|}^{- 8} \end{array}$

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For Z1=1−i1+i36;Z2=1−i3+i6;Z3=1+i3−i6which of the following holds good?