Identify the likely transformations in the sequence

The options themselves strongly indicate the nature of the compounds:

• Option C says $\mathbf{P}$ and $\mathbf{S}$ react with hydroxylamine to give oximes. Only aldehydes and ketones form oximes with hydroxylamine.
• Therefore, $\mathbf{P}$ and $\mathbf{S}$ must be carbonyl compounds (aldehyde/ketone), not carboxylic acids.
• Hence option A (that both $\mathbf{P}$ and $\mathbf{Q}$ are carboxylic acids) is inconsistent if $\mathbf{P}$ forms an oxime.

Test each option using standard organic reactions

Option A: $\mathbf{P}$ and $\mathbf{Q}$ are carboxylic acids.

• Carboxylic acids do not react with hydroxylamine to give oximes.
• Since option C explicitly says $\mathbf{P}$ gives an oxime, $\mathbf{P}$ cannot be a carboxylic acid.
• Therefore A is false.

Option B: Compound $\mathbf{S}$ decolorizes bromine water.

• Bromine water is decolorized by compounds containing $\mathrm{C=C}$ unsaturation or some strongly activated systems.
• But if $\mathbf{S}$ reacts with hydroxylamine to form an oxime, then $\mathbf{S}$ is most likely an aldehyde/ketone.
• Ordinary aldehydes/ketones do not decolorize bromine water by addition like alkenes.
• Therefore B is false.

Option C: Compounds $\mathbf{P}$ and $\mathbf{S}$ react with hydroxylamine to give the corresponding oximes.

• Aldehydes and ketones react with hydroxylamine: $R_2C=O + NH_2OH \rightarrow R_2C=NOH + H_2O$
• This is the standard oxime formation reaction.
• Thus C is true.

Option D: Compound $\mathbf{R}$ reacts with dialkylcadmium to give the corresponding tertiary alcohol.

• Dialkylcadmium reagents are less reactive organometallics and are known to react with acid chlorides to give ketones, not tertiary alcohols.
• Tertiary alcohols are usually obtained by reaction of Grignard reagents/organolithium with ketones or esters (with excess reagent), not by dialkylcadmium directly.
• Therefore D is false.

Final selection

The only statement that must be correct is: $\boxed{\text{C only}}$

Comparison with stored answer

• Stored correct answer: A, C
• Derived answer: C only

These do not match. Since $\mathbf{P}$ forms an oxime (from option C), it cannot be a carboxylic acid, so option A cannot be correct.