Language where pumping does not work -- Pumping lemma

Question

I understand that the length of y must be greater than 0 but I do not understand how to show B_2 is a regular language or why pumping up does not work in part ii.

Answer 1

For (i), the pumping lemma says there is some such division, not that any division you happen to choose has to work. For (ii), because any string starting and ending in $0$ is in the language ($x$ is anything whatsoever in it's definition).