If $α$ is the fifth root of unity then

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$| 1 + α + α^{2} + α^{3} + α^{4} | = 0$

$| 1 + α + α^{2} + α^{3} | = 1$

$| 1 + α + α^{2} | = 2 \cos \frac{π}{5}$

$| 1 + α | = 2 \cos \frac{π}{10}$

answer is Û.

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Detailed Solution

we have $α = \cos \frac{2 π}{5} + i \sin \frac{2 π}{5}$ and $1 + α + α^{2} + α^{3} + α^{4} = 0$

$∴ | 1 + α + α^{2} + α^{3} | = | - α^{4} | = | α^{4} | = 1$

Also, $| 1 + α + α^{2} | = | - α^{3} (1 + α) | = | 1 + α | ................... (i)$

$= | 1 + \cos \frac{2 π}{5} + i \sin \frac{2 π}{5} |$

$= | 2 \cos \frac{π}{5} (\cos \frac{π}{5} + i \sin \frac{π}{5}) | = 2 \cos \frac{π}{5}$

Again from (i), $| 1 + α | = | 1 + α + α^{2} | = 2 \cos \frac{π}{5}$

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