How to train to run 2.4 km in 10 minutes
That's 1:40 per lap, correct? And your current pace is 2:40–2:50 per lap? That's a big difference.
You don't say how much time you have to train before your fitness exam- a week? A year? Also, how much training have you done so far? If you've been training hard for years and are therefore close to your maximum potential, you aren't going to improve by almost seven minutes.
But if your test is sufficiently far into the future and if you still have sufficient potential to tap into, I'd do three running workouts per week:
- Speed day: 200m x 12 with a 1 minute break between reps. Keep track of each rep time and gradually try to bring them down. You should be doing them at a consistent speed (e.g. if you run 200m in 45 seconds your first rep, then 60 seconds for 11 reps, you're doing it wrong) and faster than your target pace of 50 seconds/200m. I'd start with a target of 1:00/rep.
- Simulated test day: 400m x 6 with no break. Try to be consistent with your times across all reps. I'd start with a target of 2:30/lap. If you have energy after this workout (note the ‘if'!), you could add in some speed work. I'm a fan of 100m x 5 with a 3 minute break between reps, each done as fast as possible. Grit your teeth and take off!
- Long run day: 5 km with no break. Make this a quick but relaxing jog. You can throw in a few short sprints if you feel like it, e.g. "I'm going to sprint to that tree 50m away". I wouldn't do this on a track, because 12 laps at a moderate pace is boring, and a change of scenery will do you good.
Also, are you doing this on a proper rubber track? If so, do all your track workouts with spiked running shoes- they're lighter than most normal running shoes, and the spikes will help you run a little faster. Just make sure to wear them in gradually (don't do every rep of the first workout in them), warm-up thoroughly, and self-massage your Achilles heels.