Question

The number of distinct real roots of the equation ${\log }_{\sqrt{3}}\tan x\sqrt{{\log }_{\sqrt{3}}3\sqrt{3}+lo{g}_{\tan x}3}=-1$ in interval $\left[0,2\pi \right]$ is

• A. $0$
• B. $2$
• C. $4$
• D. $6$

Answer 1

Answer: (B)

Put $lo{g}_{\sqrt{3}}tanx=t,t<0$
$t\sqrt{\frac{2}{t}+3}=-1$
Squaring on both sides
$\Rightarrow t^2\left(3+\frac{2}{t}\right)=1$
$\Rightarrow 3t^2+2t-1=0\Rightarrow \left(t+1\right)\left(3t-1\right)=0$
$\therefore t=-1\left(t=\frac{1}{3}\text{is rejected since t < 0}\right)$
So, $lo{g}_{\sqrt{3}}tanx=-1\Rightarrow tanx={\left(\sqrt{3}\right)}^{-1}$
$\Rightarrow tanx=\frac{1}{\sqrt{3}}\Rightarrow x=\frac{\pi }{6}\text{ or }\frac{7\pi }{6}$