Step 1: Identify the starting material.
We begin with 1,2-dimethylbenzene, also called o-xylene, which is a benzene ring carrying two neighbouring methyl groups. We will watch what happens to each methyl group through the two steps.
Step 2: Do the controlled oxidation.
Treating a methylbenzene with chromic oxide in acetic anhydride at low temperature (273 to 283 K) is a mild, controlled oxidation. It protects the carbon as a gem-diacetate so the methyl is not pushed all the way to an acid. After hydrolysis, the methyl group becomes an aldehyde ($-\text{CHO}$).
Step 3: Find product P.
Under these mild conditions only one methyl group is converted, while the other stays a methyl. So P is a ring with one $-\text{CHO}$ and one $-\text{CH}_3$ sitting next to each other (2-methylbenzaldehyde).
Step 4: Heat P with concentrated NaOH.
An aldehyde with no alpha hydrogen on the ring side, when heated with concentrated NaOH, undergoes the Cannizzaro reaction. In Cannizzaro, one aldehyde is reduced to an alcohol and another is oxidised to an acid. But here we must also think about the leftover methyl group.
Step 5: Track the groups through to Q.
The aldehyde, under the strong base and on hydrolysis, is taken to the carboxylic acid level, and the adjacent methyl is also affected by the harsh oxidising or hydrolysing conditions so that it too ends up as a carboxylic acid. The result is a ring carrying two acid groups side by side, which is phthalic acid.
Step 6: Name the functional groups in Q.
So the final product Q carries two carboxylic acid groups.
\[ \boxed{\text{Q contains two carboxylic acid groups}} \]