Step 1: Effect of excess sodamide.
Heating a vicinal or geminal dibromide with excess sodamide leads to double dehydrohalogenation, forming an alkyne.
Hence, compound \(P\) must be such that it can form an alkyne \(Q\).
Step 2: Reaction with \(HgSO_4/H_2SO_4\).
Hydration of an alkyne in the presence of mercuric sulphate and dilute sulphuric acid gives a ketone via enol–keto tautomerism.
Step 3: Analysis of tests on product \(R\).
Positive iodoform test indicates the presence of a \(\ce{CH3CO-}\) group.
Negative Tollens' test confirms the absence of an aldehyde group.
Thus, \(R\) must be a methyl ketone.
Step 4: Identify the correct structure.
Among the given options, only Option (C) on treatment with sodamide forms an alkyne that, upon mercuric sulphate hydration, yields a methyl ketone satisfying both test conditions.
Final Answer:
\[
\boxed{\text{Option (C)}}
\]