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?

A single tape Turing Machine $$M$$ has two states $${q_0}$$ and $${q_1}$$, of which $${q_0}$$ is the starting state. The tape alphabet of $$M$$ is $$\left\{ {0,\,\,1,\,\,B} \right\}$$ and its input alphabet is $$\left\{ {0,\,\,1} \right\}$$. The symbol $$B$$ is the blank symbol used to indicate end of an input string. The transition function of $$M$$ is described in the following table.

The table is interpreted as illustrated below. The entry $$\left( {{q_1},1,\,R} \right)$$ in row $${{q_0}}$$ and column $$1$$ signifies that if $$M$$ is in state $${{q_0}}$$ and reads $$1$$ on the current tape square, then it writes $$1$$ on the same tape square, moves its tape head one position to the right and transitions to state $${{q_1}}$$.

Which of the following statements is true about $$M?$$