Let $R = \{(1, 3), (2, 2), (3, 2)\}$ and $S = \{(2, 1), (3, 2), (2, 3)\}$ be two relations on set $A = \{1, 2, 3\}$. Then $RoS =$

Let $g(x) = 1 + x - [x]$ and $f(x) = \begin{cases} -1, & x < 0 \\ 0, & x = 0 \\ 1, & x > 0 \end{cases}$,where $[x]$ denotes the greatest integer less than or equal to $x$. Then for all $x$,$f(g(x)) = $

Given $f'(x) > 0$ and $g'(x) < 0$ for all $x \in R$,then which of the following is true?

The range of the function $f(x) = \left[ \frac{1}{\ln(x^2 + e)} \right] + \frac{1}{\sqrt{1 + x^2}}$ is,where $[*]$ denotes the greatest integer function and $e = \lim_{\alpha \to 0} (1 + \alpha)^{1/\alpha}$.

If $f(x) = x \left( \frac{1}{x-1} + \frac{1}{x} + \frac{1}{x+1} \right)$ for $x > 1$, then:

Answer the following by appropriately matching the lists based on the information given in the paragraph.
Let $f(x) = \sin(\pi \cos x)$ and $g(x) = \cos(2\pi \sin x)$ be two functions defined for $x > 0$. Define the following sets whose elements are written in increasing order:
$X = \{x : f(x) = 0\}, Y = \{x : f'(x) = 0\}$
$Z = \{x : g(x) = 0\}, W = \{x : g'(x) = 0\}$
$List-I$ contains the sets $X, Y, Z$ and $W$. $List-II$ contains some information regarding these sets.

$List-I$	$List-II$
$(I) X$	$(P) \supseteq \{\frac{\pi}{2}, \frac{3\pi}{2}, 4\pi, 7\pi\}$
$(II) Y$	$(Q) \text{ an arithmetic progression}$
$(III) Z$	$(R) \text{ NOT an arithmetic progression}$
$(IV) W$	$(S) \supseteq \{\frac{\pi}{6}, \frac{7\pi}{6}, \frac{13\pi}{6}\}$
	$(T) \supseteq \{\frac{\pi}{3}, \frac{2\pi}{3}, \pi\}$
	$(U) \supseteq \{\frac{\pi}{6}, \frac{3\pi}{4}\}$

$(1)$ Which of the following is the only $CORRECT$ combination?
$(1) (II), (R), (S)$ $(2) (I), (P), (R)$ $(3) (II), (Q), (T)$ $(4) (I), (Q), (U)$
$(2)$ Which of the following is the only $CORRECT$ combination?
$(1) (IV), (Q), (T)$ $(2) (IV), (P), (R), (S)$ $(3) (III), (R), (U)$ $(4) (III), (P), (Q), (U)$