It's still an idea I'm working on, but it's plausible that any AI which is trying to accomplish something complicated in the material world will pursue knowledge of math, physics, and engineering. Even an AI which doesn't have an explicitly physical goal (maybe it's a chess program) still might get into physics and engineering in order to improve its own functioning.

This does seem very likely. See Steve Omohundro's "The Basic AI Drives" for one discussion.

What I'm wondering is whether Friendliness might shake out from more general goals.

This seems very unlikely; see Value is Fragile.

(One exception: It seems conceivable that game theory, plus the possibility of being in a simulation, might give rise to a general rule like "treat your inferiors as you would be treated by your superiors" that would restrain arbitrary AIs.)