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Inequality

If z 1, z 2 a...

Question

If $z_{1}, z_{2}$ are complex numbers such that $z_{1}^{3} - 3 z_{1} z_{2}^{2}$ = 2 and $3 z_{1}^{2} z_{2} - z_{2}^{3}$ = 11 then $| z_{1}^{2} + z_{2}^{2} |$ =

A

3

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B

4

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C

5

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D

6

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Solution

The correct option is C
5

$z_{1}^{3} - 3 z_{1} z_{2}^{2} + 3 i z_{1}^{2} z_{2} - i z_{2}^{3} = 2 + 11 i$ $\Rightarrow$ $(z_{1} + i z_{2})^{3}$ = 2 + 11i

Similarly $(z_{1} - i z_{2})^{3}$ = 2 - 11i

$| z_{1}^{2} + z_{2}^{2} |$ = $| (z_{1} + i z_{2}) (z_{1} - i z_{2}) |$ = $| (2 + 11 i)^{\frac{1}{3}} (2 - 11 i)^{\frac{1}{3}} |$ = 5

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