Step 1: We need two groups of amino acids.
Set X must have an alcohol type -OH group on the side chain. Set Y must have an amide type -CONH$_2$ group on the side chain. Let us scan the list.
Step 2: Find the -OH side chains.
Serine (III) has a -CH$_2$OH side chain. Tyrosine (V) has a phenol -OH on its ring. Both carry the -OH group, so X = III, V.
Step 3: Note what cysteine has.
Cysteine (IV) carries an -SH (thiol) group, not -OH, so it does not belong in X. This rules out the options that put IV in X.
Step 4: Find the -CONH$_2$ side chains.
Glutamine (II) and asparagine (VI) both have an amide -CONH$_2$ group in their side chains. So Y = II, VI.
Step 5: Note what lysine has.
Lysine (I) has a plain amine -NH$_2$ group, not an amide, so it is not in Y.
Step 6: Conclusion.
So the answer is X = III, V and Y = II, VI. \[ \boxed{\text{III, V; II, VI}} \]