Question

The sum of all roots of the equation

$\sin \left(\pi {\log }_3\left(\frac{1}{x}\right)\right)=0$

in the interval $(0,2\pi )$ is

• A. $3/2$
• B. $4$
• C. $9/2$
• D. $13/3$

Answer 1

Answer: (C)

$9/2$

General solution of this equation is,
$\pi {\log }_3\left(\frac{1}{x}\right)=n\pi$, where n is any integer
$\Rightarrow x=3^{-n}$
In $(0,2\pi )$ n can take values,$-1,0,1,2,3$,.........$\infty$
So sum of all roots is,
$=3+1+1/3+1/3^2+1/3^3+...................\infty$
$=4+\frac{1/3}{1-1/3}$ $=4+\frac{1}{2}$ $=\frac{9}{2}$
Hence, option 'C' is correct.