Question

Consider $f(x)=\tan ^{-1}\left(\frac{2 x}{\sqrt{9-4 x^2}}\right)-\cos ^{-1}\left(\frac{x}{3}\right)$. Identify which of the following statement(s) is(are) correct?

• A. Number of solutions of the equation $f ( x )=\ln (- x )$ is 2 .
• B. Number of solutions of the equation $f ( x )=\ln (- x )$ is 1 .
• C. If $f(x)-k=0$ has a solution then number of integral values of $k$ is 4 .
• D. If $f ( x )- k =0$ has a solution then number of integral values of $k$ is 3 .

Answer 1

Answer: (B), (C)

$ f(x)=\tan -1\left(\frac{2 x}{\sqrt{9-4 x^2}}\right)-\cos ^{-1}\left(\frac{x}{3}\right)$
Domain of $f(x)$ is $x \in\left(\frac{-3}{2}, \frac{3}{2}\right)$
$f(x)=\sin ^{-1}\left(\frac{2 x}{3}\right)-\cos ^{-1}\left(\frac{x}{3}\right)$
When $x \in\left(\frac{-3}{2}, \frac{3}{2}\right)$
$\left(\frac{-\pi}{2}, \frac{\pi}{2}\right)-\left(\frac{2 \pi}{3}, \frac{\pi}{3}\right)$
Range of $f ( x )$
$\left(\frac{-7 \pi}{6}+\frac{\pi}{6}\right)$

Number of solution of $f ( x )=\ln (- x )$
Has only one solution (B) $f ( x )- k =0$ has number of integral solution $k =\{-3,-2,-1,0\} \Rightarrow 4$ value (C)