Consider the following context-free grammars:
$$\eqalign{ & {G_1}:\,\,\,\,\,S \to aS|B,\,\,B \to b|bB \cr & {G_2}:\,\,\,\,\,S \to aA|bB,\,\,A \to aA|B|\varepsilon ,\,\,B \to bB|\varepsilon \cr} $$

Which one of the following pairs of languages is generated by $${G_1}$$ and $${G_2}$$, respectively?

Which of the following languages are context-free? $$$\eqalign{ & {L_1} = \left\{ {{a^m}{b^n}{a^n}{b^m}|m,n \ge 1} \right\} \cr & {L_2} = \left\{ {{a^m}{b^n}{a^m}{b^n}|m,n \ge 1} \right\} \cr & {L_3} = \left\{ {{a^m}{b^n}|m = 2n + 1} \right\} \cr} $$$

Consider the following languages over the alphabet $$\sum { = \left\{ {0,\,1,\,c} \right\}:} $$
$$\eqalign{ & {L_1} = \left\{ {{0^n}\,{1^n}\,\left| {n \ge } \right.0} \right\} \cr & {L_2} = \left\{ {wc{w^r}\,\left| {w \in \left\{ {0,\,1} \right\}{}^ * } \right.} \right\} \cr & {L_3} = \left\{ {w{w^r}\,\left| {w \in \left\{ {0,\,1} \right\}{}^ * } \right.} \right\} \cr} $$

Here, $${w^r}$$ is the reverse of the string $$w.$$ Which of these languages are deterministic Context- free languages?

Consider the $$DFA$$ $$A$$ given below.

Which of the following are FALSE?
$$1.$$ Complement of $$L(A)$$ is context - free.
$$2.$$ $$L(A)$$ $$ = \left( {{{11}^ * }0 + 0} \right)\left( {0 + 1} \right){}^ * {0^ * }\left. {{1^ * }} \right)$$
$$3.$$ For the language accepted by $$A, A$$ is the minimal $$DFA.$$
$$4.$$ $$A$$ accepts all strings over $$\left\{ {0,1} \right\}$$ of length at least $$2.$$