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Let $f: R \rightarrow R$ and $g: R \rightarrow R$ be functions defined by$f(x)=\left\{\begin{array}{cl}x|x| \sin \left(\frac{1}{x}\right), & x \neq 0, \\ 0, & x=0,\end{array}\right.$ and $g(x)=\left\{\begin{array}{cc}1-2 x, & 0 \leq x \leq \frac{1}{2}, \\ 0, & \text { otherwise }\end{array}\right.$
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

• A
  $(P) \rightarrow(4)( Q ) \rightarrow(3)( R ) \rightarrow(1)( S ) \rightarrow(2)$
• B
  $(P) \rightarrow (5) (Q) \rightarrow (2) (R) \rightarrow (4) (S) \rightarrow (3)$
• C
  $(P) \rightarrow (5) (Q) \rightarrow (3) (R) \rightarrow (2) (S) \rightarrow (4)$
• D
  $(P) \rightarrow (4) (Q) \rightarrow (2) (R) \rightarrow (1) (S) \rightarrow (3)$