(a) Which of the following are regular languages? Justify your answers.

(i) $\left\{w^{n} \mid w \in\{a, b\}^{*}, n \geqslant 2\right\}$.

(ii) $\left\{w \in\{a, b, c\}^{*} \mid w\right.$ contains an odd number of $b$ 's and an even number of $c$ 's $\}$.

(iii) $\left\{w \in\{0,1\}^{*} \mid w\right.$ contains no more than 7 consecutive 0 's $\}$.

(b) Consider the language $L$ over alphabet $\{a, b\}$ defined via

$$L:=\left\{w a b^{n} \mid w \in\{a, b\}^{*}, n \in \mathbb{K}\right\} \cup\{b\}^{*}$$

Show that $L$ satisfies the pumping lemma for regular languages but is not a regular language itself.