Question

Consider $f(x)={\tan }^{-1}\left(\frac{2x}{\sqrt{9-4x^2}}\right)-{\cos }^{-1}\left(\frac{x}{3}\right)$. 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)

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