why does in this case a protic solvent and a weak base go through an E2 mechanism?

 I understand that a) would go to an E1 mechanism since its in a protic solvent and heat
and then there would be a carbon rearrangement which would allow for a major product being a trisubstituted alkene and a minor product being a disubstituted alkene
but why would B) only have one product wouldn't it also proceed through an E1 mechanism?