Question

In the interval $\left(0,2\pi \right)$ , sum of all the roots of the equation $\sin \left(\pi {\log }_3\left(\frac{1}{x}\right)\right)=0$ is

• A. $\frac{3}{2}$
• B. $4$
• C. $\frac{9}{2}$
• D. $\frac{13}{3}$

Answer 1

Answer: (C)

$\pi {\left(\text{ log}\right)}_3\left(\frac{1}{\text{x}}\right)=\text{k}\pi \text{, }\text{k}\in \text{I}$
${\left(\text{log}\right)}_3\left(\frac{1}{\text{x}}\right)=\text{k}\Rightarrow \text{ x}=3^{-\text{k}}$
Possible values of $k$ are $-1,0,1,2,3,.......$
$\text{S}=\left(3+1\right)+\left(\frac{1}{3}+\frac{1}{3^2}+\frac{1}{3^3}+\text{.....}\infty \right)$
$=4+\frac{\left(1/3\right)}{1-\left(1/3\right)}=4+\frac{1}{2}=\frac{9}{2}$