Not sure what the exam answers are, but in real life ( I would like to think that it should be the same, but in flying who knows? ), if the departure plate says "after take-off contact XXX" then do it ( this is common in Germany ). A "goodbye" at the end of reading back the take-off clearance whilst still on the runway should remove any ambiguity here.
As 30W says at Birmingham and perhaps others there is a conditional clearance to contact departure when passing 2000ft.
Unless there is an automatic frequency change as part of your departure clearance, then for any other scenario you wait until told what frequency to contact. The departure plate may or may not have the next frequency. It may have more than one frequency, but you most definitely wait until told.