Translation:The explanation had been written on the test.
I think the words are almost exactly synonyms for "test" and "exam" in English. So a "test" can be the same thing as an "exam" (in a school or a university, for example). But you can also "test" a product to see if it's fit for purpose (and "test/i" works too in that sense). And you can "examine" something closely to discover more about it (and "ekzamen/i" works for that too).
Well they are not exact synonyms. An exam/ekzameno is the complete set of tests/testoj, that you are graded by. e.g. if your complete maths exam consists of four individual tests: addition, substraction, multiplication, division. But fairly often test is used as a synonym for exam.