Consider the following problems. $$L(G)$$ denotes the language generated by a grammar $$G.$$ $$L(M)$$ denotes the language accepted by a machine $$M.$$

$$\,\,\,\,\,\,\,\,{\rm I}.\,\,\,\,\,\,\,$$ For an unrestricted grammar $$G$$ and a string $$W,$$ whether $$w \in L\left( G \right)$$
$$\,\,\,\,\,\,{\rm II}.\,\,\,\,\,\,\,$$ Given a Turing machine $$M,$$ whether $$L(M)$$ is regular
$$\,\,\,\,{\rm III}.\,\,\,\,\,\,\,$$ Given two grammars $${G_1}$$ and $${G_2}$$, whether $$L\left( {{G_1}} \right) = L\left( {{G_2}} \right)$$
$$\,\,\,\,{\rm IV}.\,\,\,\,\,\,\,$$ Given an $$NFA$$ $$N,$$ whether there is a deterministic $$PDA$$ $$P$$ such that $$N$$
$$\,\,\,\,\,\,\,\,\,\,\,\,\,$$ $$\,\,\,\,\,\,\,\\,\,\,$$and $$P$$ accept the same language.

Which one of the following statements is correct?

Given a language $$𝐿,$$ define $${L^i}$$ as follows: $${L^0} = \left\{ \varepsilon \right\}$$
$${L^i} = {L^{i - 1}}.\,\,L$$ for all $$i > 0$$

The order of a language $$L$$ is defined as the smallest k such that $${L^k} = {L^{k + 1}}.$$ Consider the language $${L_1}$$ (over alphabet $$0$$) accepted by the following automaton.

The order of $${L_1}$$ is _____.

“From where are they bringing their books? ________ bringing _______ books from _____.”

The words that best fill the blanks in the above sentence are

The area of a square is $$𝑑.$$ What is the area of the circle which has the diagonal of the square as its diameter?