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Question

If 1,z1,z2,z3,zn1 are n roots of unity then the value of 13z1+13z2++13zn1 is equal to :

A
n3n13n112
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B
n3n13n1+12
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C
n3n13n11
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D
n3n13n1+1
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Solution

The correct option is A n3n13n112
xn=1
xn1=0
xn1=(x1)(xz1)(xz2)(xzn1) where 1,z2,,zn1 are the n roots
Now taking log both sides
log(xn1)=log(x1)+log(xz1)+log(xz2)++log(xzn1)
differentiating both side w.r.t x, we get
nxn1xn1=1x1+1xz1+1xz2++1xzn1
Putting x=3, we get
13z1+13z2++13zn1=n3n13n112

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