Question

The number of distinct real roots of the equation $\log_{\sqrt{3}} \tan x \sqrt{\log_{\sqrt{3}} 3 \sqrt{3} + l o g_{\tan x} 3}=-1$ in interval $\left[0 , 2 \pi \right]$ is

• A. $0$
• B. $2$
• C. $4$
• D. $6$

Answer 1

Answer: (B)

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