An optically active alcohol of formula $C_9H_{12}O_2$ produced the following compound when refluxed with $KMnO_4$. What is the structure of $(A)$? The original compound showed these properties also:
$\mathop {C_9H_{12}O_2}\limits_{(A)} \xrightarrow{Na} H_2 \text{ liberated}$
$\mathop {C_9H_{12}O_2}\limits_{(A)} \xrightarrow{Br_2} \text{ no rapid reaction}$
$\mathop {C_9H_{12}O_2}\limits_{(A)} \xrightarrow[{cool}]{CrO_3 / H^{+}} C_9H_8O_3$

• A
  $1-$($4$-(hydroxymethyl)phenyl)ethanol
• B
  $1-$(p-tolyl)ethane$-1,2-$diol
• C
  $2-$(p-tolyl)propan$-2-$ol
• D
  both $(a)$ and $(b)$