Step 1: Identify the nature of intermediate and product:
The reaction sequence is given as:
\[
\text{C}_6\text{H}_5\text{CN} \xrightarrow{\text{B}} \text{Y} \xrightarrow{\text{A}} \text{X}
\]
Compound $\text{Y}$ gives a positive test with $2,4$-DNP, which confirms the presence of a carbonyl group (aldehyde or ketone).
Thus, $\text{Y}$ must be benzaldehyde, i.e.,
\[
\text{Y} = \text{C}_6\text{H}_5\text{CHO}.
\]
Further, compound $\text{X}$ is soluble in dilute $\text{HCl}$, indicating that it is basic in nature, most likely a primary amine.
Hence,
\[
\text{X} = \text{C}_6\text{H}_5\text{CH}_2\text{NH}_2.
\]
Step 2: Determine the type of transformations:
Conversion of nitrile to aldehyde represents a partial reduction, while conversion of nitrile to a primary amine represents a complete reduction.
\[
\text{C}_6\text{H}_5\text{CN} \rightarrow \text{C}_6\text{H}_5\text{CHO} (\text{partial reduction})
\]
\[
\text{C}_6\text{H}_5\text{CN} \rightarrow \text{C}_6\text{H}_5\text{CH}_2\text{NH}_2 (\text{complete reduction})
\]
Step 3: Identify the reagents involved:
Partial reduction of nitriles to aldehydes is achieved using
\[
\text{DIBAL-H followed by hydrolysis}.
\]
Hence,
\[
\text{B} = \text{DIBAL-H}/\text{H}_2\text{O}.
\]
Complete reduction of nitriles to primary amines can be carried out using
\[
\text{Na}/\text{Hg}, \text{C}_2\text{H}_5\text{OH}.
\]
Thus,
\[
\text{A} = \text{Na}/\text{Hg}, \text{C}_2\text{H}_5\text{OH}.
\]
Step 4: Conclusion:
\[
\text{A} = \text{Na}/\text{Hg}, \text{C}_2\text{H}_5\text{OH}, \text{B} = \text{DIBAL-H}/\text{H}_2\text{O}
\]
Therefore, the correct option is (B).