The examiner talks about the engine failure case which is the reason why V1 was designed anyway. Fairly obvious A is correct I would say.
The question is clearly with a fixed V2. V2 and V1 have regulatory no direct link as far as I know (and I presume this is a question for an official exam). V2 is limited by rotation speed, minimum control speed (air) and stall speed. If a company uses fixed V1/V2 relationships than that is their choice as long as it makes sure the minimum climb requirements, etc, are fullfilled.
Improved climb uses available excess field to increase Vr, hence V2, and hence creating better climb gradients. Different story that has initially nothing to do with V1 changes (although V1 is likely to change because Vr and V2 change). The note about the field limit might be confusing, but it's only there to show V1 can be increased, field length is not limiting your stopping distance because of a higher V1.