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The sum of all roots of sin(pi(log)3(1/x...

The sum of all roots of `sin(pi(log)_3(1/x))=0` in `(0,2pi)` is `3/2` (b) 4 (c) `9/2` (d) `(13)/3`

A

`3//2`

B

4

C

`9//2`

D

`13//3`

Text Solution

Verified by Experts

The correct Answer is:
C
`pi log_(3) (1/x)=k pi, k in I`
`log_(3) (1/x)=k or x=3^(-k)`
The possible values of k are `-1, 0, 1, 2, 3,...`
`S=3+1+1/3+1/3^(2)+1/3^(3)+...oo=3/(1-1/3)=9/2`
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