In the Australian state of New South Wales, final year mathematics exams were held a few days ago and the Sydney Morning Herald reports the advanced maths exam was “cruel and difficult.” Students on some message board are posting sad messages saying they might as well not have bothered because it was so hard, and some teacher says:
I am appalled that an examination committee could set such a difficult paper which gives the competent student little chance to show what they know
A game is played by throwing darts at a target. A player can choose to throw two or three darts.
Darcy plays two games. In Game 1, he chooses to throw two darts, and wins if he hits the target at least once. In Game 2, he chooses to throw three darts, and wins if he hits the target at least twice.
The probability that Darcy hits the target on any throw is p, where 0 < p < 1.
(i) Show that the probability that Darcy wins Game 1 is 2p – p[squared].
(ii) Show that the probaility that Darcy wins Game 2 is 3p[squared] – 2p[cubed].
(iii) Prove that Darcy is more likely to win Game 1 than Game 2.
(iv) Find the value of p for which Darcy is twice as likely to wine Game 1 as he is to win Game 2.
So I’m interested to know … do my readers think this is challenging? I did it on a single sheet of paper in 10 minutes yesterday, and it really didn’t seem tough. Admittedly I should be able to do this stuff quickly, but when I compare it to the work I did in 1990 it doesn’t seem very hard at all. Questions i and ii are basic applications of probability theory, without even any conditional or joint probability questions; part ii requires use of basic combinatorics but I remember this stuff was not too hard in year 12 when I did. Questions iii and iv are trivial exercises in problem solving with quadratics: you need to do a sign diagram for iv and complete the square of a quadratic but if you can’t identify and solve such a problem in year 12 surely you have stuffed up somewhere? Also, you don’t need to get i and ii right to do iii and iv, which in my opinion is very far from cruel. I would have been very happy to see that option in an exam when I was doing year 12! Basically, the first two questions are year 11 level probability (at most!) and the last two are year 10 functions.
So I’m wondering, have standards slipped in Australia in the last 20 years, or am I turning into one of those teachers I hated when I was at university, who say “this is trivial high school maths” as they introduce a path integral that can only be solved numerically? I’m pretty sure it’s the former (or the question the Herald gave is not representative) and 38% of people who answered the poll on the Herald website agree with me. Dissenting opinions (and reminiscences about the horrors of your own school days) are welcome in comments…
Update: I found on reddit some photos of two other questions: question 5 and question 7. I think these both look tough though I think I could do question 7 (I think you use differentiation and a change of variables in part i, then ii and iii are just straight nasty old manipulation; though maybe part i is induction). I’ve always been terrible at trigonometry, and I remember fluffing a question very similar to (possibly the same as!) number 5 in my exam in 1990. I don’t think I’d do better this time round. But I’m not sure that this material is excessive for a year 12 maths exam; maybe question 7 is more a first year university question …? But I don’t think so. Kids should be doing series and induction in year 12 for sure …