An organic compound 'A' ( C_3H_6O ) gives a positive iodoform test...

The given organic compound 'A' has the molecular formula $\text{C}_3\text{H}_6\text{O}$.

Since it does not reduce Tollens' reagent, it is not an aldehyde. Therefore, it must be a ketone.

Since it gives a positive iodoform test, it must contain a methyl ketone group ($\text{CH}_3\text{CO}-$).

The only ketone with the molecular formula $\text{C}_3\text{H}_6\text{O}$ is propanone (acetone), which is $\text{CH}_3\text{COCH}_3$.

Note that while propan-2-ol ($\text{CH}_3\text{CHOHCH}_3$) also gives a positive iodoform test and does not reduce Tollens' reagent, its molecular formula is $\text{C}_3\text{H}_8\text{O}$, which does not match the given formula.

Answer: $\text{CH}_3\text{COCH}_3$