Step 1: Identify the reaction with \(Br_2/\text{aq. NaOH}\).
The reagent \(Br_2/\text{aq. NaOH}\) gives Hofmann bromamide degradation reaction.
In this reaction, a primary amide converts into a primary amine with one carbon atom less.
\[
RCONH_2 \xrightarrow{Br_2/NaOH} RNH_2
\]
Step 2: Use molecular formula of [X].
The compound [X] has molecular formula:
\[
C_5H_{11}NO.
\]
A primary amide with this formula can be:
\[
CH_3-(CH_2)_3-CONH_2.
\]
This is pentanamide.
Step 3: Find product [Y] from Hofmann bromamide reaction.
If [X] is pentanamide:
\[
CH_3-(CH_2)_3-CONH_2 \xrightarrow{Br_2/NaOH} CH_3-(CH_2)_3-NH_2.
\]
So, [Y] is butan-1-amine.
\[
Y = CH_3CH_2CH_2CH_2NH_2.
\]
Step 4: Confirm with carbylamine test.
[Y] reacts with \(CHCl_3\) and ethanolic KOH to produce a foul smelling compound.
This is the carbylamine reaction, given only by primary amines.
\[
RNH_2 + CHCl_3 + 3KOH \rightarrow RNC + 3KCl + 3H_2O.
\]
The foul smelling compound is an isocyanide.
Thus, [Y] must be a primary amine.
Step 5: Confirm with HONO reaction.
Primary aliphatic amines react with nitrous acid to form alcohol with liberation of nitrogen gas.
\[
RNH_2 + HONO \rightarrow ROH + N_2 + H_2O.
\]
Since [Y] gives Butan-1-ol, [Y] must be butan-1-amine.
\[
CH_3CH_2CH_2CH_2NH_2 \xrightarrow{HONO} CH_3CH_2CH_2CH_2OH + N_2 + H_2O.
\]
Step 6: Trace back [X].
Since Hofmann bromamide degradation decreases carbon chain by one, [X] must be the amide containing one more carbon than butan-1-amine.
Therefore, [X] is pentanamide:
\[
CH_3-(CH_2)_3-CONH_2.
\]
Step 7: Final conclusion.
The compound [X] is:
\[
\boxed{CH_3-(CH_2)_3-CONH_2}
\]