Yes you are correct. "in a month" implies that the exam will take place at the end of the stated time period whereas "within" a month implies that it can be taken anytime within the stated time period. but the correct sentence would be number 1 number 2 would be better if you said "I have to take my exam within a month" because "I've got an exam within a month" just doesnt sound right youre implying you can take it anytime within a month, but you arent stating a date using "got" implies confirmation, "have to" implies more flexibility

An example: Today is the 04.08.2017 I have an exam in a month. (It means that you have an exam on 04.09.2017) I have an exam within a month (It means that your exam will be at any day until the 04.09.2017. For example on 11.08.2017 or 20.08.2017 or 03.09.2017) Could I help you? Is your question answered? Thomas :-)

Some good answers so far. Personally, I would not say either of those two things. Instead, I would say: "I've got an exam in a month's time" This one means that I have an exam in 30 days or 31 days, as that is the duration of "a month's [worth] of time". or "I've got an exam next month" This one means that I have an exam that falls at some point within the next calendar month. For example, if it is currently September and I have an exam in October, I can say "I've got an exam next month". It just means that my exam is going to be some time in October. I can say "I've got an exam next month" even if it is currently the end of September and the exam is right at the beginning of October, and there is therefore only one week left before the exam. The exam is still "next month" because it is currently September but the exam is going to be in October.