Let z 1 and z 2 be two complex numbers such that z 1 + z 2 = 5 and z 1 3 + z 2 3 = 20 + 15 i . Then | z 1 4 + z 2 4 | equals [2024]

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Solution

Q.
Let $z_{1}$ and $z_{2}$ be two complex numbers such that $z_{1} + z_{2} = 5$ and $z_{1}^{3} + z_{2}^{3} = 20 + 15 i .$ Then $| z_{1}^{4} + z_{2}^{4} |$ equals [2024]

1 $30 \sqrt{3}$

2 $15 \sqrt{15}$

3 $75$

4 $25 \sqrt{3}$

Ans.
(3)

Given, $z_{1} + z_{2} = 5 \Rightarrow {(z_{1} + z_{2})}^{3} = 5^{3}$

$\Rightarrow 20 + 15 i + 3 z_{1} z_{2} (5) = 125$

$\Rightarrow z_{1} z_{2} = 7 - i$

Now $z_{1}^{4} + z_{2}^{4} = {[{(z_{1} + z_{2})}^{2} - 2 z_{1} z_{2}]}^{2} - 2 z_{1}^{2} z_{2}^{2}$

$= {[25 - 2 (7 - i)]}^{2} - 2 {(7 - i)}^{2}$

$= 625 + 4 {(7 - i)}^{2} - 100 (7 - i) - 2 {(7 - i)}^{2}$

$= 625 + (7 - i) [2 (7 - i) - 100]$

$= 625 + (7 - i) [14 - 2 i - 100]$

$= 625 + (7 - i) [- 86 - 2 i]$

$= 625 - 602 - 14 i + 86 i - 2 = 21 + 72 i$

$∴$ $| z_{1}^{4} + z_{2}^{4} | = \sqrt{441 + 5184} = \sqrt{5625} = 75$

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