Step 1: Find product \([X]\) from propyne.
Propyne undergoes hydration in the presence of \(\mathrm{HgSO_4/H_2SO_4}\).
This is Markovnikov addition of water to the alkyne, giving first an enol, which then tautomerises to a ketone.
\[
\mathrm{CH_3-C \equiv CH \xrightarrow[\mathrm{H_2SO_4}]{\mathrm{HgSO_4}} CH_3COCH_3}
\]
So, \([X]\) is propanone (acetone).
Step 2: Find product \([Y]\) from ethanenitrile.
Ethanenitrile \(\mathrm{CH_3CN}\) reacts with methyl magnesium bromide \(\mathrm{MeMgBr}\).
Grignard reagent adds to nitrile carbon and on acidic hydrolysis gives a ketone.
\[
\mathrm{CH_3CN \xrightarrow[(B)\ H_3O^+]{(A)\ CH_3MgBr} CH_3COCH_3}
\]
So, \([Y]\) is also propanone (acetone).
Step 3: Reaction between \([X]\) and \([Y]\) in presence of \(\mathrm{Ba(OH)_2}\).
Since both \([X]\) and \([Y]\) are acetone, the reaction is aldol condensation of acetone.
Under basic conditions, two molecules of acetone first give diacetone alcohol, which on heating undergoes dehydration.
First aldol addition:
\[
2\mathrm{CH_3COCH_3} \xrightarrow{\mathrm{Ba(OH)_2}}
\mathrm{CH_3COCH_2C(OH)(CH_3)_2}
\]
On heating, dehydration occurs:
\[
\mathrm{CH_3COCH_2C(OH)(CH_3)_2 \xrightarrow{\Delta} CH_3COCH=C(CH_3)_2}
\]
Thus, final product \([Z]\) is:
\[
\mathrm{CH_3COCH=C(CH_3)_2}
\]
Step 4: Write the IUPAC name of \([Z]\).
Now name the compound \(\mathrm{CH_3COCH=C(CH_3)_2}\).
The longest chain containing the ketone group has 5 carbon atoms.
Numbering is done so that the ketone gets the lowest possible locant.
So the structure is named as:
\[
\text{4-Methylpent-3-en-2-one}
\]
Step 5: Match with the given options.
The correct option is:
\[
\boxed{(B)\ 4\text{-Methylpent-3-en-2-one}}
\]
Final Answer:
\[
\boxed{4\text{-Methylpent-3-en-2-one}}
\]