Question

Let $f\left(x\right)=\sqrt{\frac{{\log }_{\left\{a\right\}}|x-2|}{|x|}}$, where $a\in (2,3)$ and $\{.\}$ represents the fractional part function. If $x\in Z$, then possible value(s) of $x$ is/are

• A. $1$
• B. $2$
• C. $3$
• D. $-3$

Answer 1

Answer: (A), (C)

The correct options are
A $1$
C $3$

$f\left(x\right)=\sqrt{\frac{{\log }_{\left\{a\right\}}|x-2|}{|x|}}$
$\left\{a\right\}\in (0,1)$

For $f\left(x\right)$ to be defined,
${\log }_{\left\{a\right\}}|x-2|\ge 0,|x-2|\ne 0,x\ne 0\Rightarrow 0<|x-2|\le 1,x\ne 0,2[\because \{a\}\in (0,1\left)\right]$
$\Rightarrow |x-2|\le 1,x\ne 0,2\Rightarrow -1\le x-2\le 1,x\ne 0,2\Rightarrow 1\le x\le 3,x\ne 0,2\Rightarrow x=1,3$