I live in Australia and I will be doing my final high school mathematics test in 5 months from today. The test is about the following topics:

  • Differentiation and Integration and their applications (Maxima/minima problems, volumes, motion [simple harmonic, projectile, circular and resisted motion], conical pendulum and banked tracks)
  • Complex numbers

  • Polynomials

  • Conic sections

  • Series and sequences, mathematical induction and analysis

  • Inequalities (similar to ones in IMO)

  • Circle geometry

  • Probability, combinatorics and binomial probability.

Here are some examples of past exam papers:

The test starts of very easy then gradually gets harder. I'm confident that I can get most of the questions correct but timing is a problem as it is difficult to get the whole test done in the time allowed.

Many people, including my teacher, told me that the best way to prepare is to do as many of these papers as possible in timed conditions. Do you agree with that? The problem is that doing the same thing with slightly different questions over and over is very boring :(

  • $\begingroup$ Hi. I've taken this exam several years ago. Doing as many of the past papers as possible is a good way to guarantee a decent mark. If you're aiming for the top, you'll need to understand the material very well. Knowing the proofs of several key theorems may give you additional insight as well. $\endgroup$ – fixedp May 13 '13 at 8:42
  • $\begingroup$ Could you please list some of those theorems? Do you mean proofs like Gregory/Leibniz Series, Sine Integral and the Riemann-Lebesgue Lemma, Wallis product, $\zeta(2n)$, Taylor series, fundamental theorem of algebra, Cauchy-Schwarz inequality, etc. Thank you for the reply :) $\endgroup$ – please delete me May 13 '13 at 8:46
  • $\begingroup$ Perhaps theorems was not the best thing to say here. Formulas is what I meant. You use a lot of formulas in this exam. It is good to be able to derive most of them from basic definitions. For example, if you are able to derive everything you've learnt about conic sections from the definitions, you'll have a good idea how to proceed with most conic sections proofs. And by deriving I don't mean merely understanding. I mean being able to walk up to the board and explain it insightfully to everyone in your class. $\endgroup$ – fixedp May 13 '13 at 9:03
  • $\begingroup$ @fixedp I think understanding the concepts is relatively easy but the hardest thing is that for the last questions in the paper I don't only need speed and accuracy but also good problem solving skills. Do you think doing mathematical olympiad questions is a good idea? As they can help me improve problem solving skills. $\endgroup$ – please delete me May 13 '13 at 9:11
  • $\begingroup$ I think Olympiad type questions may be overkill. By all means go for it, but make sure it does not interfere with your preparations for any of your exams. $\endgroup$ – fixedp May 13 '13 at 9:16

Yes, it is the best way to get the highest mark on the test and yes it is very boring. At the very high percentiles, the ranks are decided by who did the most past papers, who saw the most familiar questions and who has practiced their speed - not necessarily by who has the most solid understanding of all the mathematical principles involved. With enough practice you should aim to get to the point where you don't even need to think about how to do the problems until Q8, everything before that is just writing down what comes to your mind as fast as you can.

Not saying is this the best way to learn or enjoy mathematics, but it is the best way to rank the highest on the tests. Unfortunately sometimes we all have to jump through some hoops.

  • $\begingroup$ I agree with this comment. To get into the highest percentiles you need pretty good understanding of the material anyways. After that, it's up to practice, and a bit of luck. $\endgroup$ – fixedp May 13 '13 at 8:54
  • $\begingroup$ Thank you a lot for the reply! I will keep doing past papers then and asking questions here to understand how to approach more difficult questions. Do you know Jonathan Le and Sean Gomes? They both go/went to USYD maths. $\endgroup$ – please delete me May 13 '13 at 8:54
  • $\begingroup$ @AlexanderJones I've met Sean once and we had a small chat, but I don't know either beyond that. I think they were at least two years above me. $\endgroup$ – Ragib Zaman May 13 '13 at 8:58
  • $\begingroup$ @RagibZaman I know them because they are active on boredofstudies.org maths section. Thank you for the help, I have been spending a lot of time on trying to self learn undergrad maths but I will spend my time concentrating on preparing for the test even though it is boring. $\endgroup$ – please delete me May 13 '13 at 9:02
  • $\begingroup$ If we're thinking of the same Sean Gomes, he's at ANU now. $\endgroup$ – fixedp May 13 '13 at 9:03

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