You do an exam. But there are no easy rules to follow. We always use do to describe indefinite activities, often with what, thing, anything, nothing, etc and generally speaking we also use do to talk about duties, jobs or (leisure) activities. Look at the following examples:
He didn't do anything. He just sat there. You expect me to do everything around the house. Well, I'm fed up!
I did all my homework last night so tonight I'm going to do the housework.
I did a lot of research and I think I did a good job on that essay. I did my best anyway.
I intend to do lots of walking on holiday this year, and perhaps some bird-watching too.
We tend to use make when we are talking about constructing, creating or performing something. Study the following examples:
'I made three suggestions and left it to him to make the final decision.'
'I've made all the arrangements for the trip and I've made a great effort to get it all right.'
'I'm afraid I'm going to have to make my excuses and leave.'
'I have to make three phone calls.'
make a lasting impression (on someone)
do the shopping and the washing-up do some serious work
do a lot of damage (to something)
make an announcement
make an application (e.g. for a driving test)
make a sound or a noise
do one's hair or one's teeth
do a lot of harm rather than gooddo business (with somebody)
do (somebody) a favour
make love, not war
make a mess, a profit or a fortune
make fun of someone or a fool of someone
make amends for one's behaviour