Step 1: Understanding the nature of compound [A].
The compound [A] is given as an unsaturated hydrocarbon.
It may be an alkene or an alkyne.
The reaction with \(Na/\text{liquid }NH_3\) gives a trans product, which is a characteristic reaction of alkynes.
Therefore, [A] is most likely an alkyne.
Step 2: Using the \(Na/\text{liquid }NH_3\) reaction.
Alkynes undergo partial reduction with \(Na/\text{liquid }NH_3\) to form trans alkenes.
\[
RC \equiv CR' \xrightarrow{Na/\text{liquid }NH_3} RCH=CHR'
\]
This confirms that [A] contains a carbon-carbon triple bond.
So, option (D) But-2-ene is eliminated because it is already an alkene.
Step 3: Using ammoniacal \(AgNO_3\) test.
Terminal alkynes react with ammoniacal \(AgNO_3\) to form silver acetylide precipitate.
Internal alkynes do not react with ammoniacal \(AgNO_3\).
The question states:
\[
[A] \xrightarrow{\text{ammoniacal }AgNO_3} \text{No reaction}
\]
Therefore, [A] cannot be a terminal alkyne.
So, propyne and but-1-yne are eliminated.
Step 4: Identifying the possible alkyne.
After eliminating terminal alkynes, the remaining alkyne option is:
\[
\mathrm{CH_3-C \equiv C-CH_3}
\]
This compound is But-2-yne.
It is an internal symmetrical alkyne.
Step 5: Checking oxidation with alkaline \(KMnO_4\) at 298 K.
Alkynes on oxidation with cold alkaline \(KMnO_4\) can form diketones.
For But-2-yne:
\[
\mathrm{CH_3-C \equiv C-CH_3 \xrightarrow[\;298K\;]{alkaline \ KMnO_4} CH_3COCOCH_3}
\]
The product is butane-2,3-dione, which is a diketone.
This matches the given product [B].
Step 6: Checking ozonolysis followed by \(Zn/H_2O\).
Ozonolysis of But-2-yne gives the same diketone under the given reductive work-up conditions.
\[
\mathrm{CH_3-C \equiv C-CH_3 \xrightarrow[(ii)\ Zn/H_2O]{(i)\ O_3} CH_3COCOCH_3}
\]
Thus, [B] is again a diketone.
This observation supports the identification of [A] as But-2-yne.
Step 7: Checking hot alkaline \(KMnO_4\) oxidation.
On strong oxidation with alkaline \(KMnO_4\) at \(353K\), alkynes undergo oxidative cleavage.
But-2-yne gives two moles of acetic acid:
\[
\mathrm{CH_3-C \equiv C-CH_3 \xrightarrow[\;353K\;]{alkaline \ KMnO_4} 2CH_3COOH}
\]
The molecular formula of acetic acid is:
\[
\mathrm{C_2H_4O_2}
\]
This exactly matches the given condition:
\[
2 \text{ moles of carboxylic acid } (C_2H_4O_2)
\]
Step 8: Final conclusion.
All the given reactions are satisfied only by But-2-yne.
It is an internal alkyne, so it does not react with ammoniacal \(AgNO_3\).
It gives a trans alkene with \(Na/\text{liquid }NH_3\), a diketone with mild oxidation, and two moles of acetic acid with strong oxidation.
Therefore, compound [A] is:
\[
\boxed{\text{But-2-yne}}
\]
Hence, the correct answer is option (B).