A) \[\frac{3}{2}\]
B) \[4\]
C) \[\frac{9}{2}\]
D) \[\frac{13}{3}\]
Correct Answer: C
Solution :
| Given, \[\sin \left[ \pi {{\log }_{3}}\left( \frac{1}{x} \right) \right]=0\] |
| \[\Rightarrow \] \[\pi {{\log }_{3}}\left( \frac{1}{x} \right)=n\pi \] |
| \[\Rightarrow \] \[{{\log }_{3}}\left( \frac{1}{x} \right)=n\] |
| \[\Rightarrow \] \[x={{3}^{-n}}\] |
| Sum of roots \[=3+1+\frac{1}{3}+\frac{1}{{{3}^{2}}}+\frac{1}{{{3}^{3}}}+...+\infty \] |
| \[[\because x\in (0,2\pi )]\] |
| \[=4+\frac{1}{3}\left( \frac{1}{1-\frac{1}{3}} \right)=4+\frac{1}{2}=\frac{9}{2}\] |
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