I don't think there's really any specific method to find that, I try to imagine different words being run on the TM while isolating what causes those words to be either accepted, rejected or loop. After that its finding the pattern that defines the words that fit into each category i.e. all words that start with a double b end up looping.
Each year you get the previous year's exam paper. So, why aren't we getting 2008's exam paper?
I wrote last year, but the only thing I remember is that the format was the same as the previous year's.