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Let $[x]$ denote the greatest integer $\leq x$, where $x \in R$. If the domain of the real valued function $\mathrm{f}(\mathrm{x})=\sqrt{\frac{[\mathrm{x}] \mid-2}{\sqrt{[\mathrm{x}] \mid-3}}}$ is $(-\infty, \mathrm{a}) \cup[\mathrm{b}, \mathrm{c}) \cup[4, \infty), \mathrm{a}\,<\,\mathrm{b}\,<\,\mathrm{c}$, then the value of $\mathrm{a}+\mathrm{b}+\mathrm{c}$ is:
$-3$
$1$
$-2$
$8$
Solution
For domain,
$\frac{|[x]|-2}{|[x]|-3} \geq 0$
Case $I:$ When $|[x]|-2 \geq 0$
and $|[x]|-3\,>\,0$
$\therefore x \in(-\infty,-3) \cup[4, \infty] \ldots . .(1)$
Case $II:$ When $|[x]|-2 \leq 0$
and $|[x]|-3\,<\,0$
$\therefore \mathrm{x} \in[-2,3) \quad \ldots(2)$
So, from $(1)$ and $(2)$
We get
Domain of function
$=(-\infty,-3) \cup[-2,3) \cup[4, \infty)$
$\therefore(a+b+c)=-3+(-2)+3=-2(a\,<\,b\,<\,c)$