Let $A = \{2, 3, 4, 5, 6\}$. Let $R$ be a relation on the set $A \times A$ defined by $(x, y) R (z, w)$ if and only if $x$ divides $z$ and $y \le w$. Then the number of elements in $R$ is . . . . . . .

If $X$ and $Y$ are two non-empty sets where $f: X \to Y$ is a function defined such that $f(C) = \{f(x) : x \in C\}$ for $C \subseteq X$ and $f^{-1}(D) = \{x : f(x) \in D\}$ for $D \subseteq Y$,then for any $A \subseteq X$ and $B \subseteq Y$,which of the following is true?

If the set of all values of $a$ is $[\alpha, \beta] \cup [\gamma, \delta]$ for which the function $f(x) = \begin{cases} 3x + |a^2 - 4|; & a \leqslant x < 1 \\ 5 - x^2; & x \geqslant 1 \end{cases}$ has its largest value at $x = 1$,then find the value of $(\alpha + \beta + \gamma + \delta)$.

Let $f: R \rightarrow R$ and $g: R \rightarrow R$ be functions defined by
$f(x)=\begin{cases} x|x| \sin \left(\frac{1}{x}\right), & x \neq 0 \\ 0, & x=0 \end{cases}$ and $g(x)=\begin{cases} 1-2x, & 0 \leq x \leq \frac{1}{2} \\ 0, & \text{otherwise} \end{cases}$
Let $a, b, c, d \in R$. Define the function $h: R \rightarrow R$ by
$h(x)=a f(x)+b\left(g(x)+g\left(\frac{1}{2}-x\right)\right)+c(x-g(x))+d g(x), x \in R$
Match each entry in $List-I$ to the correct entry in $List-II$.

$List-I$	$List-II$
$(P)$ If $a=0, b=1, c=0$ and $d=0$,then	$(1)$ $h$ is one-one
$(Q)$ If $a=1, b=0, c=0$ and $d=0$,then	$(2)$ $h$ is onto
$(R)$ If $a=0, b=0, c=1$ and $d=0$,then	$(3)$ $h$ is differentiable on $R$
$(S)$ If $a=0, b=0, c=0$ and $d=1$,then	$(4)$ the range of $h$ is $[0,1]$
	$(5)$ the range of $h$ is $\{0,1\}$

The correct option is

Consider a function $f : N \rightarrow R$,satisfying $f(1)+2 f(2)+3 f(3)+\ldots+x f(x)=x(x+1) f(x)$ for $x \geq 2$,with $f(1)=1$. Then $\frac{1}{f(2022)}+\frac{1}{f(2028)}$ is equal to

Let $S$ be a finite set. Then a non-identity function $f: S \rightarrow S$ can be