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If alpha Is the fifth root of unity, th...

If `alpha` Is the fifth root of unity, then, prove that :
`Log_2|1+alpha+alpha^(2)+alpha^(3)-(1/alpha)|=1`

Text Solution

Verified by Experts

The correct Answer is:
A, B, C
`(a,b,c)` We have `alpha=cos"(2pi)/(5)+isin"(2pi)/(5)`
and `1+alpha+alpha^(2)+alpha^(3)+alpha^(4)=0`
`:. |1+alpha+alpha^(2)+alpha^(3)|=|-alpha^(4)|=|alpha^(4)|=1`
Also, `|1+alpha+alpha^(2)|=|-alpha^(3)(1+alpha)|=|1+alpha|` ..........`(i)`
`=|1+cos'(2pi)/(5)+isin"(2pi)/(5)|`
`=|2cos'(pi)/(5)(cos'(pi)/(5)+isin"(pi)/(5))|`
`=2cos"(pi)/(5)`
Again from `(i)`, `|1+alpha|=2cos"(pi)/(5)`
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