List of top Theory of Computation Questions on Regular and contex-free languages, pumping lemma

Consider the following languages: \[ L_1 = \{a^n w a^n | w \in \{a, b\}^*\} \] \[ L_2 = \{ w x w^R | w, x \in \{a, b\}^*, |w|, |x| > 0 \} \] Note that \( w^R \) is the reversal of the string \( w \). Which of the following is/are TRUE?

Consider the following languages:

\( L_1 = \{ ww \mid w \in \{a,b\}^* \} \)
\( L_2 = \{ a^n b^n c^m \mid m, n \geq 0 \} \)
\( L_3 = \{ a^m b^n c^n \mid m, n \geq 0 \} \)

Which of the following statements is/are FALSE?

Let \(L \subseteq \{0,1\}^*\) be an arbitrary regular language accepted by a minimal DFA with \(k\) states. Which one of the following languages must necessarily be accepted by a minimal DFA with \(k\) states?

Let \(L_1\) be a regular language and \(L_2\) be a context-free language. Which of the following languages is/are context-free?

Suppose that \(L_1\) is a regular language and \(L_2\) is a context-free language. Which one of the following languages is NOT necessarily context-free?