Let $\Sigma=\{1,2,3,4\}$ For $x \in \Sigma^*$, let prod $(x)$ be the product of symbols in $x$ modulo 7 . We take $\operatorname{prod}(\varepsilon)=1$, where $\varepsilon$ is the null string.

For example, $\operatorname{prod}(124)=(1 \times 2 \times 4) \bmod 7=1$.

Define $L=\left\{x \in \Sigma^{\star} \mid \operatorname{prod}(x)=2\right\}$.

The number of states in a minimum state DFA for $L$ is _________ (Answer in integer)

Despite his initial hesitation, Rehman's __________ to contribute to the success of the project never wavered.

Select the most appropriate option to complete the above sentence.

Bird : Nest :: Bee : ________

Select the correct option to complete the analogy.

If $P e^x=Q e^{-x}$ for all real values of $x$, which one of the following statements is true?