At 9:30 the two hands of a correct clock divide the whole turn into a smaller angle and a larger (reflex) angle. How big is the LARGER of those two angles?

The minute hand at 30 minutes is at 180° from the 12 mark. The hour hand is at $9\times 30+30\times 0.5=270+15=285$°. The gap between them is 285 − 180 = 105°, so the reflex angle on the other side is 360 − 105 = 255°. The traps: 105° is the smaller angle, which is what this kind of question usually asks for; 285° is simply where the hour hand is standing; 270° comes from leaving the hour hand parked on the 9 and forgetting that it has already crept half way towards the 10.