1
I have to make an exam tomorrow, so I need to study tonight.
I have to take an exam tomorrow, so I need to study tonight.
Use take, not make, when you are the student doing an exam. A teacher prepares an exam; a student takes it.
2
I passed my maths exam this morning, but I won’t know my result until next week.
I took my maths exam this morning, but I won’t know my result until next week.
Take an exam means do the test. Pass an exam means get a good enough result. Here, the student does not know the result yet, so use took, not passed.