The number of states in the minimal deterministic finite automaton corresponding to the regular expression $${\left( {0 + 1} \right)^{\,\, * }}\left( {10} \right)$$ is ________________.

Which of the following languages is/are regular?

$${L_1}:\left\{ {wx{w^R}|w,x\, \in \left\{ {a,b} \right\}{}^ * } \right.$$ and $$\left. {\left| w \right|,\left| x \right| > 0} \right\},\,{w^R}$$ is the reverse of string $$w$$
$${L_2}:\left\{ {{a^n}{b^m}\left| {m \ne n} \right.} \right.$$ and $$m,n \ge \left. 0 \right\}$$
$${L_3}:\left\{ {{a^p}{b^q}{c^r}\left| {p,q,r \ge 0} \right.} \right\}$$

Consider the alphabet $$\sum { = \left\{ {0,1} \right\},} $$ the null/empty string $$\lambda $$ and the sets of strings $${X_0},\,{X_1},$$ and $${X_2}$$ generated by the corresponding non-terminals of a regular grammar. $${X_0},\,\,{X_1},\,$$ and $${X_2}$$ are related as follows. $$$\eqalign{ & {X_0} = 1\,X{}_1 \cr & {X_1} = 0{X_1} + 1\,{X_2} \cr & {X_2} = 0\,{X_1} + \left\{ \lambda \right\} \cr} $$$
Which one of the following choices precisely represents the strings in $${X_0}$$?

Consider the NPDA $$\left\langle {Q = \left\{ {{q_0},{q_1},{q_2}} \right\}} \right.,$$ $$\Sigma = \left \{ 0, 1 \right \},$$ $$\Gamma = \left \{ 0, 1, \perp \right \},$$ $$\delta, q_{0}, \perp,$$ $$\left. {F = \left\{ {{q_2}} \right\}} \right\rangle $$ , where (as per usual convention) $$Q$$ is the set of states, $$\Sigma$$ is the input alphabet, $$\Gamma$$ is the stack alphabet, $$\delta $$ is the state transition function q₀ is the initial state, $$\perp$$ is the initial stack symbol, and F is the set of accepting states. The state transition is as follows:

Which one of the following sequences must follow the string 101100 so that the overall string is accepted by the automaton?