Let $f : \mathbb { R } \rightarrow \mathbb { R }$ and $g : \mathbb { R } \rightarrow \mathbb { R }$ be functions defined by

$$f ( x ) = \left\{ \begin{array} { l l } x | x | \sin \left( \frac { 1 } { x } \right) , & x \neq 0 , \\ 0 , & x = 0 , \end{array} \quad \text { and } \quad g ( x ) = \begin{cases} 1 - 2 x , & 0 \leq x \leq \frac { 1 } { 2 } \\ 0 , & \text { otherwise } \end{cases} \right.$$

Let $a , b , c , d \in \mathbb { R }$. Define the function $h : \mathbb { R } \rightarrow \mathbb { R }$ by

$$h ( x ) = a f ( x ) + b \left( g ( x ) + g \left( \frac { 1 } { 2 } - x \right) \right) + c ( x - g ( x ) ) + d g ( x ) , x \in \mathbb { R }$$

Match each entry in List-I to the correct entry in List-II.

\textbf{List-I}

(P) If $a = 0 , b = 1 , c = 0$, and $d = 0$, then

(Q) If $a = 1 , b = 0 , c = 0$, and $d = 0$, then

(R) If $a = 0 , b = 0 , c = 1$, and $d = 0$, then

(S) If $a = 0 , b = 0 , c = 0$, and $d = 1$, then

\textbf{List-II}

(1) $h$ is one-one.

(2) $h$ is onto.

(3) $h$ is differentiable on $\mathbb { R }$.

(4) the range of $h$ is $[ 0,1 ]$.

(5) the range of $h$ is $\{ 0,1 \}$.

The correct option is

(A) $(\mathrm{P}) \rightarrow (4)$, $(\mathrm{Q}) \rightarrow (3)$, $(\mathrm{R}) \rightarrow (1)$, $(\mathrm{S}) \rightarrow (2)$

(B) $(\mathrm{P}) \rightarrow (5)$, $(\mathrm{Q}) \rightarrow (2)$, $(\mathrm{R}) \rightarrow (4)$, $(\mathrm{S}) \rightarrow (3)$

(C) $(\mathrm{P}) \rightarrow (5)$, $(\mathrm{Q}) \rightarrow (3)$, $(\mathrm{R}) \rightarrow (2)$, $(\mathrm{S}) \rightarrow (4)$

(D) $(\mathrm{P}) \rightarrow (4)$, $(\mathrm{Q}) \rightarrow (2)$, $(\mathrm{R}) \rightarrow (1)$, $(\mathrm{S}) \rightarrow (3)$