Question

If $x,4$ and $\left[y\right]$ are the three terms of a G.P., where $x$ is an integer in the domain of $f\left(x\right)={\sin }^{-1}\frac{x}{2}$ and $\left[y\right]$ denotes the greatest integer less than or equal to $y$, then identify the correct statement(s).

• A. Number of such G.P. is $4$.
• B. Infinite number of G.P. is possible.
• C. The set of values of $y$ is $[-16,-15)\cup [-8,-7)\cup [8,9)\cup [16,17)$
• D. The set of values of $y$ is $[8,9)\cup [16,17)$

Answer 1

Answer: (B), (C)

The correct options are
B Infinite number of G.P. is possible.
C The set of values of $y$ is $[-16,-15)\cup [-8,-7)\cup [8,9)\cup [16,17)$

$f\left(x\right)={\sin }^{-1}\frac{x}{2}$
$\Rightarrow -2\le x\le 2\Rightarrow 16=\left[y\right]x$

Now,
$x=\{-2,-1,0,1,2\}x=-2\Rightarrow \left[y\right]=-8\Rightarrow -8\le y<-7x=-1\Rightarrow \left[y\right]=-16\Rightarrow -16\le y<-15x=1\Rightarrow \left[y\right]=16\Rightarrow 16\le y<17x=2\Rightarrow \left[y\right]=8\Rightarrow 8\le y<9\therefore y\in [-16,-15)\cup [-8,-7)\cup [8,9)\cup [16,17)$