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Question

If α=cos(8π11)+isin(8π11), then Re(α+α2+α3+α4+α5) is equal to

A
12
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B
12
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C
0
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D
1
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Solution

The correct option is B 12
We have,
α=cos(8π11)+isin(8π11)=e8πi/11
Hence α is one of the 11th root of the unity
and α11=1,(¯¯¯¯α)11r=(α)r
Re(α+α2+α3+α4+α5)=α+α2+α3+α4+α5+(¯¯¯¯α)+(¯¯¯¯α)2+(¯¯¯¯α)3+(¯¯¯¯α)4+(¯¯¯¯α)52
=1+(1+α+α2+α3+α4+α5+α10+α9+α8+α7+α6)2
=1+02
[ Sum of all roots of unity is zero]
=12

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