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If alpha is an imaginary fifth...

# If alpha is an imaginary fifth root of unity, then log_(2)|1+alpha+alpha^(2)+alpha^(3)-1/alpha|=

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Text Solution

102-1

A
Solution :
We have, <br> alpha=e^(i2pir//5), where r=0,1,2,3,4 <br> Clearly, alpha^(5)=1 and 1+alpha^(2)+alpha^(3)+alpha^(4)=0 <br> Now, alpha^(5)=1 rArr alpha^(4)=1/alpha. <br> therefore |1+alpha+alpha^(2)+alpha^(3)-1/alpha|=|1+alpha+alpha^(2)+alpha^(3)+alpha^(4)|= |-alpha^(4)-alpha^(4)|=2|alpha^(4)|=2|alpha|^(4)=2 [therefore |alpha|=1] <br> therefore log_(2)|1+alpha+alpha^(2)+alpha^(3)-1/alpha|=log_(2)2=1.

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