QUOTE(Cyrilla @ Apr 29 2007, 11:48 PM)

The last sharp is 'te' and the last flat is 'fa'...

And whether you learn it via solfa or tonic, supertonic, mediant, subdominant, dominant etc that rule always applies to major keys.
Look at the key signature (if given) this will tell you whether (for the sake of arguement a piece is in E major or c# minor) Then look for accidentals, particularly surrounding the cadences. If there are none bar an obvious modulation, its major, if there are, and they fit the pattern for melodic and harmonic minor variants then the chance is its minor. If harmonised look at the final cadence. If it is not harmonised try to write the final cadence, that should establish the tonality.
If there is not a keys signature list every sharp or flat in the order in which you would build the key signature, if there are accidentals consistent with the relative minor then then chances are its minor, if it fits a pattern for the major key its major.
There is no shortcut to learning all keys major or minor using the circle of fifths. The pupils who have really grasped that have all got merits and disttinctions in their theory exams.
Whilst working out examples at home why not double check your answer by playing it. You may not be able to do this in the exam, but it should reassure you are doing things along the right lines.