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Question

The value of 10k=1(sin2kπ11icos2kπ11) is

A
1
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B
1
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C
i
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D
i
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Solution

The correct option is C i
We have,
10k=1(sin2kπ11icos2kπ11)
=i10k=1(cos2kπ11+isin2kπ11)=i10k=1ei2kπ/11
and we know that sum of 11,11th roots of unity is
10k=0ei2kπ/11=0
i10k=1ei2kπ/11=i[10k=0ei2kπ/11e0]=i

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