Consider the following context-free grammar $G$, where $S, A$, and $B$ are the variables (nonterminals), $a$ and $b$ are the terminal symbols, $S$ is the start variable, and the rules of $G$ are described as:

$$\begin{aligned} & S \rightarrow a a B \mid A b b \\ & A \rightarrow a \mid a A \\ & B \rightarrow b \mid b B \end{aligned}$$

Which ONE of the languages $L(G)$ is accepted by $G$ ?

A regular language $L$ is accepted by a non-deterministic finite automaton (NFA) with $n$ states. Which of the following statement(s) is/are FALSE?

Consider the following two languages over the alphabet $\{a, b\}$ :

$$\begin{aligned} & L_1=\left\{\alpha \beta \alpha \mid \alpha \in\{a, b\}^{+} \text {AND } \beta \in\{a, b\}^{+}\right\} \\ & L_2=\left\{\alpha \beta \alpha \mid \alpha \in\{a\}^{+} \text {AND } \beta \in\{a, b\}^{+}\right\} \end{aligned}$$

Which ONE of the following statements is CORRECT?

Consider the following two languages over the alphabet $\{a, b, c\}$, where $m$ and $n$ are natural numbers.

$$\begin{aligned} & L_1=\left\{a^m b^m c^{m+n} \mid m, n \geq 1\right\} \\ & L_2=\left\{a^m b^n c^{m+n} \mid m, n \geq 1\right\} \end{aligned}$$

Which ONE of the following statements is CORRECT?