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 $-1,0,1,2,3,.......$
$\text{S} = \left(3 + 1\right) + \left(\frac{1}{3} + \frac{1}{3^{2}} + \frac{1}{3^{3}} + \text{.....} \infty \right)$
$= 4 + \frac{\left(1 / 3\right)}{1 - \left(1 / 3\right)} = 4 + \frac{1}{2} = \frac{9}{2}$