Which one of the following languages over $\Sigma = \{a, b\}$ is NOT context-free?

Let us analyze each language:

• (A) $L_1 = \{w w^R \mid w \in \{a, b\}^*\}$: This is the language of even-length palindromes, which is standard non-deterministic context-free (a PDA pushes $w$ and pops on guessing the midpoint to match $w^R$).

• (B) $L_2 = \{w a^n b^n w^R \mid w \in \{a, b\}^*, n \ge 0\}$: A PDA can push $w$ onto the stack, then push $a^n$ and pop on matching $b^n$, and finally pop the remaining $w$ to match $w^R$. Because the stack operates in LIFO order, $a^n b^n$ is fully processed and cleared before matching $w$ with $w^R$. Hence, this is context-free.

• (C) $L_3 = \{w a^n w^R b^n \mid w \in \{a, b\}^*, n \ge 0\}$: To check $w$ and $w^R$, the stack must be used to match them. However, the count $n$ of $a^n$ must also be compared with $b^n$ appearing *after* $w^R$. Since the stack must clear/compare $w$ with $w^R$ first, the count of $n$ cannot be retained simultaneously using a single pushdown stack. Using the Pumping Lemma for CFLs, it can be proven that $L_3$ is NOT context-free.

• (D) $L_4 = \{a^n b^i \mid i \in \{n, 3n, 5n\}, n \ge 0\} = \{a^n b^n\} \cup \{a^n b^{3n}\} \cup \{a^n b^{5n}\}$: This is a finite union of three context-free languages, and since CFLs are closed under union, $L_4$ is context-free.

Therefore, the language in option (C) is not context-free.