For a young person, particularly if the doctor had strong anti-drug feelings, it wouldn't be unthinkable that he might do a routine drug screen as a part of the urinalysis. But he probably won't do that routinely. Unless it's specially requested. And even if your mom is going with you and makes the request, which I hope she won't, you're old enough that he's not supposed to give her the results. The doctor is only supposed to discuss them with you.

The best way to make everyone suspicious and to make them more likely to screen your urine for recent drug use is to put up a fuss about giving a urine sample. Urinalysis is a very basic and routine part of a physical. Fortunately, unless you've been a heavy smoker, you may test clean if you are tested.