First, Lets check if we can draw a Deterministic PDA for the given language or not.
Final state C will accept all strings of the form $a^n$ $b^n$ and final state D will accept all the strings of the form $a^n$ and ϵ.
Hence, the given language is deterministic CFL.
I is true and II is false.
Check for III can be done using a simple argument. In LL(k), k represents the number of lookaheads for LL parser to decide which production should be selected for derivation. Looking at the language, suppose strings are "aaaaa" and "aaaaabbbbb". The parser will not be able to differentiate between these until k > 5. Since, n can be a very large number (possibly infinte), it is simply impossible to use such large value for k.
Hence, the given language cannot be parsed by a LL(k) parser.
III is True.
Hence, option (C) is correct.