Question

Which pair of characteristics would you use to place a fish in one of the three fish classes: agnatha, cartilaginous fishes, or bony fishes?

Answer 1

If I understand your question, it boils down to asking:
A) Does it have jaws? If not, then it's an agnatha.
B) Does it have bones? If so, then it's a bony fish.
Everything that doesn't fall into either of these two classifications would be considered a cartilaginous fish.
But I may not fully grok your question.

Answer 2

The answer is "skeleton type and skin covering"