Consider the following languages.

$$\,\,\,\,\,\,\,\,\,\,\,\,$$ $${L_1} = \left\{ {\left\langle M \right\rangle |M} \right.$$ takes at least $$2016$$ steps on some input $$\left. \, \right\},$$
$$\,\,\,\,\,\,\,\,\,\,\,\,$$ $${L_2} = \left\{ {\left\langle M \right\rangle |M} \right.$$ takes at least $$2016$$ steps on all inputs $$\left. \, \right\}$$ and
$$\,\,\,\,\,\,\,\,\,\,\,\,$$ $${L_3} = \left\{ {\left\langle M \right\rangle |M} \right.$$ accepts $$\left. \varepsilon \right\},$$

where for each Turing machine $${M,\left\langle M \right\rangle }$$ denotes a specific encoding of $$M.$$ Which one of the following is TRUE?

Language $${L_1}$$ is defined by the grammar: $$S{}_1 \to a{S_1}b|\varepsilon $$
Language $${L_2}$$ is defined by the grammar: $$S{}_2 \to ab{S_2}|\varepsilon $$

Consider the following statements:
$$P:$$ $${L_1}$$ is regular
$$Q:$$ $${L_2}$$ is regular

Which one of the following is TRUE?

The number of states in the minimum sized $$DFA$$ that accepts the language defined by the regular expression $$${\left( {0 + 1} \right)^ * }\left( {0 + 1} \right){\left( {0 + 1} \right)^ * }$$$
is ___________________.

Consider the following types of languages: $${L_1}:$$ Regular, $${L_2}:$$ Context-free, $${L_3}:$$ Recursive, $${L_4}:$$ Recursively enumerable. Which of the following is/are TRUE?

$$\,\,\,\,\,\,\,{\rm I}.\,\,\,\,\,\,\,$$ $$\overline {{L_3}} \cup {L_4}$$ is recursively enumerable
$$\,\,\,\,\,{\rm I}{\rm I}.\,\,\,\,\,\,\,$$ $$\overline {{L_2}} \cup {L_3}$$ is recursive
$$\,\,\,{\rm I}{\rm I}{\rm I}.\,\,\,\,\,\,\,$$ $$L_1^ * \cap {L_2}$$ is context-free
$$\,\,\,{\rm I}V.\,\,\,\,\,\,\,$$ $${L_1} \cup \overline {{L_2}} $$ is context-free