$ A $ is a set having $ 6 $ distinct elements. The number of distinct functions from $ A $ to $ A $ which are not bijections is

Let $S=(0,1) \cup(1,2) \cup(3,4)$ and $T=\{0,1,2,3\}$. Then which of the following statements is(are) true?
$(A)$ There are infinitely many functions from $S$ to $T$.
$(B)$ There are infinitely many strictly increasing functions from $S$ to $T$.
$(C)$ The number of continuous functions from $S$ to $T$ is at most $120$.
$(D)$ Every continuous function from $S$ to $T$ is differentiable.

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

If $f: R \rightarrow [-1, 1]$ and $g: R \rightarrow A$ are two surjective mappings and $\sin \left(g(x) - \frac{\pi}{3}\right) = \frac{f(x)}{2} \sqrt{4 - f^2(x)}$,then $A =$

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 $P(x)$ be a polynomial with real coefficients such that $P(\sin^2 x) = P(\cos^2 x)$ for all $x \in [0, \pi/2)$. Consider the following statements:
$I.$ $P(x)$ is an even function.
$II.$ $P(x)$ can be expressed as a polynomial in $(2x - 1)^2$.
$III.$ $P(x)$ is a polynomial of even degree.
Then,