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Question

The sum of roots of the equation x8=1 whose real part is positive is

A
1+2
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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 A 1+2
Given x8=1
The roots can be ei2kπ/8, k=0,1,2,,7
Hence x=e0i,eiπ/4,eiπ/2,ei3π/4,eiπ,ei5π/4,ei3π/2,ei7π/4
So, the roots with +ve real part are
e0i,eiπ/4,ei7π/4
Hence, required sum is
e0i+eiπ/4+ei7π/4
=cos0+cosπ/4+isinπ/4+cos(7π/4)+isin(7π/4)=1+12+i12+12i12=1+2

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