Step 1: Understanding the Concept:
This problem requires structural identification of organic compounds using diagnostic chemical tests.
- A positive iodoform test indicates the structural presence of a methyl ketone ($\text{CH}_3\text{C=O}$) group or a methyl carbinol ($\text{CH}_3\text{CH(OH)}$) group.
- A positive Tollens' test indicates the presence of an aldehyde ($\text{-CHO}$) functional group, while a negative test indicates its absence (e.g., ketones or protected aldehydes).
- Acid hydrolysis of an acetal or ketal functional group ($\text{-CH(OR)}_2$ or $\text{-C(OR)}_2\text{-}$) converts it back into its corresponding aldehyde or ketone along with alcohols.
Step 2: Key Formula or Approach:
Analyze the structural clues for Compound P and Compound Q systematically:
1. Compound P ($\text{C}_6\text{H}_{12}\text{O}_3$): Has a $\text{CH}_3\text{CO-}$ group (positive iodoform) but lacks a $\text{-CHO}$ group (negative Tollens').
2. Acid Hydrolysis of P: Dilute acid breaks down any acetal linkages present in P.
3. Compound Q: Possesses both an aldehyde group (positive Tollens') and a methyl ketone group (positive iodoform).
Step 3: Detailed Explanation:
Let's test Option (2), $\text{CH}_3\text{COCH}_2\text{CH(OCH}_3)_2$:
- It contains the methyl ketone group ($\text{CH}_3\text{CO-}$), which accounts for Compound P's positive iodoform test.
- The aldehyde group is protected as a dimethyl acetal group ($\text{-CH(OCH}_3)_2$). Because it contains a ketone and a protected acetal rather than a free aldehyde, it will give a negative Tollens' test. This completely matches all given behaviors for Compound P.
Now, let's look at what happens when Option (2) is treated with dilute acid (hydrolysis):
The dimethyl acetal group ($\text{-CH(OCH}_3)_2$) undergoes clean cleavage to restore the parent aldehyde group ($\text{-CHO}$), eliminating two molecules of methanol ($\text{CH}_3\text{OH}$):
\[ \text{CH}_3\text{COCH}_2\text{CH(OCH}_3)_2 + \text{H}_2\text{O} \xrightarrow{\text{dil. H}^+} \text{CH}_3\text{COCH}_2\text{CHO} + 2\text{CH}_3\text{OH} \]
The resulting hydrolyzed product is Compound Q: $\text{CH}_3\text{COCH}_2\text{CHO}$.
Let's check the structural features of Compound Q ($\text{CH}_3\text{COCH}_2\text{CHO}$):
1. It contains a free terminal aldehyde group ($\text{-CHO}$), meaning it will yield a positive Tollens' test.
2. It simultaneously retains its methyl ketone segment ($\text{CH}_3\text{CO-}$), which yields a positive iodoform test.
This perfectly satisfies all condition criteria outlined in the problem. Let's briefly check why other options fail:
- Option (1) and (4) do not possess an acetal group that can hydrolyze into a free aldehyde, so their products will fail Tollens' test.
- Option (3) contains a formyl hydrogen ($\text{HCO-}$), which behaves as an active aldehyde group, meaning P would mistakenly give a positive Tollens' test initially.
Step 4: Final Answer:
The structure of compound P is $\text{CH}_3\text{COCH}_2\text{CH(OCH}_3)_2$.