Consider the language $L = \{a^n \mid n \ge 0\} \cup \{a^n b^n \mid n \ge 0\}$ and the following statements. I. $L$ is deterministic context-free. II. $L$ is context-free but not deterministic context-free. III. $L$ is not LL($k$) for any $k$. Which of the above statements is/are TRUE?

1. Deterministic Context-Free Language (DCFL): A DPDA can accept $L$. While reading $a$'s, the DPDA pushes each $a$ onto the stack and remains in an accepting state (so any prefix $a^n$ is accepted if the input ends). If a symbol $b$ is encountered, the automaton moves to a non-accepting state and begins popping $a$'s for each $b$. When the stack becomes empty upon consuming the last $b$ at the end of input, it enters an accepting state. Since no non-deterministic choices are needed, $L$ is a DCFL. Thus, Statement I is TRUE and Statement II is FALSE.

2. LL($k$) property: For any fixed lookahead $k$, a predictive top-down parser cannot decide whether to apply the production for $a^n$ or $a^n b^n$ when the number of $a$'s exceeds $k$, because it cannot see whether $b$'s will appear after arbitrarily many $a$'s. Hence, $L$ is not LL($k$) for any $k$. Thus, Statement III is TRUE.

Therefore, statements I and III only are TRUE, which corresponds to option (C).