Step 1: Understanding the Question:
We need to determine the identity of starting material 'A' that reacts with dimethyl cadmium to produce propanone (acetone) and a byproduct of cadmium chloride ($\mathrm{CdCl}_2$).
Step 2: Key Formula or Approach:
Dialkylcadmium reagents ($\mathrm{R}_2\mathrm{Cd}$) react cleanly with acyl chlorides (acid chlorides, $\mathrm{R'COCl}$) to synthesize ketones. The reaction follows this general stoichiometric path:
$$2\mathrm{R'COCl} + \mathrm{R}_2\mathrm{Cd} \rightarrow 2\mathrm{R'COR} + \mathrm{CdCl}_2$$
Step 3: Detailed Explanation:
Let's analyze the structural fragments of the given products:
The desired ketone product is propanone: $\mathrm{CH}_3-\mathrm{CO}-\mathrm{CH}_3$.
The organometallic reagent is dimethyl cadmium: $(\mathrm{CH}_3)_2\mathrm{Cd}$, which supplies the methyl group ($-\mathrm{CH}_3$).
Matching this to our general reaction equation:
$$2\mathrm{R'COCl} + (\mathrm{CH}_3)_2\mathrm{Cd} \rightarrow 2\mathrm{CH}_3-\mathrm{CO}-\mathrm{CH}_3 + \mathrm{CdCl}_2$$
By isolating the fragments, the $\mathrm{R'}$ acyl segment from starting material 'A' must be a two-carbon chain containing the carbonyl unit ($\mathrm{CH}_3-\mathrm{CO}-$).
Therefore, the starting material 'A' must be acetyl chloride, which is systematically named ethanoyl chloride ($\mathrm{CH}_3\mathrm{COCl}$).
$$2\mathrm{CH}_3\mathrm{COCl} + (\mathrm{CH}_3)_2\mathrm{Cd} \rightarrow 2\mathrm{CH}_3\mathrm{COCH}_3 + \mathrm{CdCl}_2$$
Step 4: Final Answer:
The compound 'A' is ethanoyl chloride, which corresponds to option (C).