For $$s \in {\left( {0 + 1} \right)^ * },$$ let $$d(s)$$ denote the decimal value of $$s(e. g.d(101)=5)$$
Let $$L = \left\{ {s \in {{\left( {0 + 1} \right)}^ * }\left| {\,d\left( s \right)\,} \right.} \right.$$ mod $$5=2$$ and $$d(s)$$ mod $$\left. {7 \ne 4} \right\}$$

Which of the following statement is true?

Let $${L_1}$$ be a regular language, $${L_2}$$ be a deterministic context-free language and $${L_3}$$ a recursively enumerable, but not recursive, language. Which one of the following statement is false?

Let $${L_1}$$ be a recursive language, and Let $${L_2}$$ be a recursively enumerable but not a recursive language. Which one of the following is TRUE?

Define Languages $${L_0}$$ and $${L_1}$$ as follows
$${L_0} = \left\{ { < M,\,w,\,0 > \left| {M\,\,} \right.} \right.$$ halts on $$\left. w \right\}$$
$${L_1} = \left\{ { < M,w,1 > \left| M \right.} \right.$$ does not halts on $$\left. w \right\}$$

Here $$ < M,\,w,\,i > $$ is a triplet, whose first component, $$M$$ is an encoding of a Turing Machine, second component, $$w$$, is a string, and third component, $$t,$$ is a bit.
Let $$L = {L_0} \cup {L_1}.$$ Which of the following is true?