Identify the likely transformations involved

From the reagent sets, we recognize two standard reactions of alkyl benzenes:

• $\mathrm{Cr_2O_3, 770~K, 20~atm}$ : this is the catalytic dehydrogenation condition used to convert an alkyl side chain into an alkenyl side chain, e.g. $\mathrm{C_6H_5CH_2CH_3 \rightarrow C_6H_5CH{=}CH_2}$ (ethylbenzene to styrene)

• $\mathrm{CrO_2Cl_2/H_3O^+}$ : Etard oxidation, which converts a benzylic methyl group into an aldehyde: $\mathrm{C_6H_5CH_3 \rightarrow C_6H_5CHO}$

• $\mathrm{NaOH}$ followed by $\mathrm{H_3O^+}$ : this strongly suggests a Cannizzaro reaction followed by acidification, applicable to benzaldehyde: $\mathrm{2C_6H_5CHO + OH^- \rightarrow C_6H_5CH_2OH + C_6H_5COO^-}$ and on acidification, $\mathrm{C_6H_5COO^- \xrightarrow{H_3O^+} C_6H_5COOH}$

Build the correct sequence logically

Since steps (iii) and (iv) are $\mathrm{NaOH}$ and then $\mathrm{H_3O^+}$, the compound formed in step (ii) must be benzaldehyde (or a similar non-enolizable aldehyde) so that Cannizzaro reaction can occur.

Therefore step (ii) should be: $\mathrm{CrO_2Cl_2/H_3O^+}$

Now, what should step (i) be before that?

To get benzaldehyde by Etard oxidation, the substrate for step (ii) should be toluene.

Among the options, the sequence where step (i) first forms toluene appropriately from the earlier aromatic hydrocarbon setup is the one beginning with: $\mathrm{Cr_2O_3, 770~K, 20~atm}$

This matches the standard conversion pattern used in such hydrocarbon sequences.

Check options one by one

Option A

$\mathrm{(i)Cr_2O_3,770K,20~atm;\ (ii)CrO_2Cl_2,H_3O^+;\ (iii)NaOH;\ (iv)H_3O^+}$

• Step (ii) gives benzaldehyde.
• Steps (iii), (iv) correctly convert benzaldehyde via Cannizzaro reaction to alcohol + acid.
• Sequence is chemically consistent.

Option B

$\mathrm{(i)CrO_2Cl_2,H_3O^+;\ (ii)Cr_2O_3,770K,20~atm;\ (iii)NaOH;\ (iv)H_3O^+}$

• If Etard oxidation is done first, an aldehyde is formed.
• Applying dehydrogenation conditions after aldehyde formation is not the logical sequence here.
• Inconsistent.

Option C

$\mathrm{(i)KMnO_4,OH^-;\ (ii)Mo_2O_3,\Delta;\ (iii)NaOH;\ (iv)H_3O^+}$

• $\mathrm{KMnO_4/OH^-}$ would strongly oxidize benzylic side chains to carboxylic acids, not aldehydes.
• Cannizzaro after that is not possible.
• Incorrect.

Option D

$\mathrm{(i)Mo_2O_3,\Delta;\ (ii)CrO_2Cl_2,H_3O^+;\ (iii)NaOH;\ (iv)H_3O^+}$

• The first reagent does not fit the needed transformation as well as option A.
• Less chemically appropriate.
• Incorrect.

Conclusion

The only chemically correct sequence is: $\boxed{\text{A}}$