The key reagents involved are:

• $\mathrm{CrO_3}$: a strong oxidizing agent, used for oxidation of side chains/allied benzylic oxidation under suitable conditions.
• $\mathrm{CrO_2Cl_2}$: chromyl chloride, specifically known for the Etard reaction, which oxidizes the benzylic methyl group of an aromatic ring to an aldehyde.

To identify $A$ and $B$, we use the characteristic reaction of chromyl chloride:

$\mathrm{Ar-CH_3} \xrightarrow[ ]{CrO_2Cl_2} \mathrm{Ar-CHO}$

So whenever the conversion is from a methyl-substituted aromatic compound to the corresponding aldehyde, the reagent must be $\mathrm{CrO_2Cl_2}$.

A stronger oxidation with $\mathrm{CrO_3}$ generally takes benzylic side chains further to the carboxylic acid:

$\mathrm{Ar-CH_3} \xrightarrow[ ]{CrO_3} \mathrm{Ar-COOH}$

Therefore:

• The reagent giving aldehyde formation must be $B = \mathrm{CrO_2Cl_2}$.
• The other oxidation corresponds to $A = \mathrm{CrO_3}$.

Hence the correct combination is:

$A = \mathrm{CrO_3}, \qquad B = \mathrm{CrO_2Cl_2}$

This matches Option B.