Most of the heat originating in the core comes from
- The formation of the earth (called primordial heat) - Latent heat (released when iron freezes onto the solid core) - Differentiation (e.g. settling of heavy stuff to the bottom)
The mantle is chock full of radiogenic elements though.
I'm open to being told my instructions are wrong in this case though.
Though I haven't run the numbers, I would strongly bet that the core would be _hotter_ than it is right now if the mantle did not have internal heating.
The reason being that, the rate at which you remove heat from the core (e.g. the rate at which the core cools) is entirely determined by the rate at which mantle convection removes heat. Radiogenic heat is a _strong_ driver of mantle convection in the earth, without radiogenic isotopes you would get much more sluggish mantle convection and a much lower rate of heat removal from the core.
The reason for this is twofold:
- Internal heating leads to higher temperatures which leads to lower viscosity and more vigorous mantle convection - Internal heating will locally heat cold "blobs" and make the buoyant
I would also (much less strongly) bet that plate tectonics would not occur without internal heating. We only see plate tectonics in a narrow slice or parameter space in mantle convection models and once you get plate tectonics you get _much_ more heat removal and _much_ faster cooling.