\( \text{M} \xrightarrow{\text{CH}_3\text{MgBr}} \text{N} + \text{CH}_4 \uparrow \xrightarrow{\text{H}^+} \text{CH}_3\text{COCH}_2\text{COCH}_3 \)
The given reaction sequence involves the following steps:
1. Reaction with Grignard Reagent: Compound \(M\) reacts with \(\text{CH}_3\text{MgBr}\) (a Grignard reagent) to form an intermediate compound \(N\). In this step, the Grignard reagent adds to the carbonyl group of \(M\), creating a new \(\text{C–C}\) bond and releasing \(\text{CH}_4\) gas as a by-product. The reaction mechanism involves nucleophilic attack by \(\text{CH}_3^-\) on the carbonyl carbon.
2. Hydrolysis: The intermediate \(N\) undergoes acid hydrolysis. During this process, the intermediate is converted into \(\text{CH}_3\text{COCH}_2\text{COCH}_3\), which is a \(\beta\)-diketone. Hydrolysis ensures the protonation and stabilization of the newly formed compound.
Key Structural Feature: \(\text{CH}_3\text{COCH}_2\text{COCH}_3\) has a \(\beta\)-diketone structure, where two keto groups are separated by a \(\text{CH}_2\) group. This confirms that compound \(M\) must also be a \(\beta\)-diketone precursor.
Among the options provided, only \(\text{CH}_3\text{COCH}_2\text{COCH}_3\) matches the final product structure.\\
Thus, the correct answer is Option 1.