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Question

If α0,α1,α2,αn1 be the n, nth roots of the unity, then the value of n1i=0αi(3αi) is equal to


A

n3n1

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B

n13n1

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C

n+13n1

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D

n+23n1

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Solution

The correct option is A

n3n1


Let P=n1i=0αi(3αi)=n1i=0(3αi)33αi=3n1i=013αin1i=01(i) zn1=n1i=0(zαi) ln(zn1)=n1i=0ln(zαi)
Differentiating both sides w.r.t. z, then
nzn1(zn1)=n1i=01(zαi)
Using this relation to simplify Eq. (i), we get
3(n.3n13n1)n=n3n3n1n=n3n1


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