Question

Number of integral solutions of the equation $\operatorname{sgn}\left(\sin ^{-1}\left[\frac{\pi x}{6}\right]\right)=1$, where $[ x ]$ denotes the greatest integer less than or equal to $x$ and $\operatorname{sgn} x$ denotes signum function of $x$.

• A. 2
• B. 3
• C. 5
• D. 7

Answer 1

Answer: (A)

We have $\sin ^{-1}\left[\frac{\pi x}{6}\right]>0 \Rightarrow\left[\frac{\pi x}{6}\right]=1 $
$\Rightarrow 1 \leq \frac{\pi x}{6}<2 \Rightarrow \frac{6}{\pi} \leq x<\frac{12}{\pi}$
$\therefore x =2,3$ only.
Hence two integral solution will satisfy above equation