Which one of the following regular expressions is equivalent to the language accepted by the DFA given below?

Let the initial state be $q_0$ and the final state be $q_1$.

1. Transitions from the DFA:

   • At state $q_0$ (start state): a loop on $0$, and a transition on $1$ to $q_1$.

   • At state $q_1$ (accepting state): a loop on $0$, and a transition on $1$ to $q_0$.

2. Reaching the accepting state $q_1$ for the first time:

   • From $q_0$, the automaton can loop on $0$ zero or more times, followed by a $1$ to transition to $q_1$. This path is represented by the regular expression $0^*1$.

3. Looping back to $q_1$:

   • Once in state $q_1$, any sequence that leaves and returns to $q_1$ (without staying in $q_0$) can be repeated arbitrarily:

     • Reading $0$ loops directly on $q_1$.

     • Reading $1$ goes to $q_0$, followed by any number of $0$s ($0^*$), and another $1$ transitions back to $q_1$, which corresponds to the sequence $10^*1$.

   • Thus, any sequence of loops at $q_1$ is given by $(0 + 10^*1)^*$.

4. Combining the paths:

   $$R = 0^*1(0 + 10^*1)^*$$

Therefore, option A is the correct regular expression.