JEE Main · 2023 · 06 Apr 2023 · Shift 2 · Sets Relations and Functions

The ceiling function $\lceil x\rceil$ equals $x$ when $x$ is an integer and equals $[x]+1$ (the next integer) otherwise. For $f(x)$ to be defined, we need $\lceil x\rceil -x>0$. If $x$ is an integer, $\lceil x\rceil =x$, so $\lceil x\rceil -x=0$, which is not positive. Thus integers are excluded from the domain. For non-integer $x$, $\lceil x\rceil -x=1-\{x\}$ where $\{x\}$ is the fractional part, and $0<\{x\}<1$, so $0<1-\{x\}<1$. Hence $f(x)=\frac{1}{\sqrt{1-\{x\}}}>1$. The domain is $A=\mathbb{R}-\mathbb{Z}$ and the range is $B=(1,\infty )$. Then $A\cap B=(\mathbb{R}-\mathbb{Z})\cap (1,\infty )=(1,\infty )-\mathbb{N}$, so (S1) is true. But $A\cup B=(\mathbb{R}-\mathbb{Z})\cup (1,\infty )\ne (1,\infty )$ (it includes all non-integers), so (S2) is false. Only (S1) is true.