Question

If $\alpha$ is a real number for which $f(x)=\ln \left(3{\cos }^{-1}\left(\frac{3x}{7}\right)-\pi \right)$ is defined, then the possible values of $[\alpha ]$ can be
[Note: [k] denotes greatest integer function less than or equal to $k$.]

• A. -3
• B. -2
• C. -1
• D. 0

Answer 1

Answer: (A), (B), (C), (D)

$f(x)=\ln \left(3{\cos }^{-1}\left(\frac{3x}{7}\right)-\pi \right)$
$3{\cos }^{-1}\left(\frac{3x}{7}\right)-\pi >0\Rightarrow {\cos }^{-1}\left(\frac{3x}{7}\right)>\frac{\pi }{3}\Rightarrow \frac{3x}{7}<\frac{1}{2}\Rightarrow x<\frac{7}{6}$
and $-1\le \frac{3x}{7}\le 1\Rightarrow \frac{-7}{3}\le x\le \frac{7}{3}\therefore x\in \left[\frac{-7}{3},\frac{7}{6}\right)$.