Question

If $\alpha$ is a real number for which $f(x)=\ln \left(3 \cos ^{-1}\left(\frac{3 x}{7}\right)-\pi\right)$ 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{3 x }{7}\right)-\pi\right)$
$3 \cos ^{-1}\left(\frac{3 x}{7}\right)-\pi>0 \Rightarrow \cos ^{-1}\left(\frac{3 x}{7}\right)>\frac{\pi}{3} \Rightarrow \frac{3 x}{7}<\frac{1}{2} \Rightarrow x<\frac{7}{6}$
and $-1 \leq \frac{3 x}{7} \leq 1 \Rightarrow \frac{-7}{3} \leq x \leq \frac{7}{3} \therefore x \in\left[\frac{-7}{3}, \frac{7}{6}\right)$.