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NZ maths falls another notch (calculus: and some moments of stupidity)

I was worried this would happen.  After the public cries of “stop making our kids feel stupid” after last year’s exam, this year’s calculus scholarship examination would be a large step down.  The questions would no longer demand the highest level of insight; rather, they would have a reasonably apparent method that any able candidate would be able to see.

The furore last year came from the pass mark of 20, that is, 17 per cent.  It was demotivating to top students, maths teachers cried.  I couldn’t do anything, candidates wailed.  I did last year’s exam, I had hell in it, and I walked out having done less than one-fifth of the paper, but I wasn’t complaining.  The highest mathematics award in secondary school has to demand excellence.  So what if the pass mark’s 17 per cent?  Fewer questions that require insightful thinking are better than more that test practice rather than intelligence.  Besides, the standard of mathematics in New Zealand’s low enough as it is.

This year’s exam, even if above level three, required little thinking—it was mostly doing.  There was no need, in this exam, to sit for a while while you’re trying to figure out how to approach a question.  It was just, how much can you do before three hours expires.  The approaches were mostly fairly obvious (or given).  In fact, having met certain people, I figure there’ll be at least one person (if not more) who’ll get full marks this year.  However will they pick a number one?

Now personally, I didn’t have a problem with this—it’s quite fun to be racing through an exam you expected to be stumped by.  Of course, I’m not saying I finished the exam—there was one sub-question I couldn’t solve, and another I didn’t have enough time to get very far in.  But I well surpassed my goal of completing half the paper.

Before I begin boring you with an account of how I fared in this exam, I’ll mention my predictions for the pass and outstanding cut-offs for this year.  The cut-off for scholarship, I think, will be 20 (out of 40).  The cut-off for outstanding will be 32.  Give or take one or two.  (Last year, my prediction was off by ten below the lower margin, but we’ll see about this year: I think I’m being generous with those predictions.)  Now, on with the account.

Calculus: and some moments of stupidity

I had my moments of stupidity in this exam, and while they might not have been as spectular in some previous exams, I will recount them anyway.

I had an interesting time with question one (b).  I thought I had the perfect approach, and came up with an approximation of ln(1+x)≈(3x²+2x)/(2x²+4x+2), which doesn’t at all match the x−½x² they wanted, so I graphed all three on my calculator.  I was slightly amused: my approximation almost seemed to work better than theirs, so I tried what was perhaps a simpler approach, and I got the answer they wanted.  Meh.  The next part of that question would have been impossible if I had tried to use my approximation, anyway.

By the time I had finished with all three parts to question one, one hour had passed; it was at that point where I decided that I could aim to complete three of the five questions.  So I tried question five: a lesson I had learnt from last year, where the last question looked too messy (it was co-ordinate geometry) so I avoided it, to only learn afterwards that it was the easiest question.  I was somewhat surprised: I completed question five in about twenty-five minutes.

Somehow, in question two (a), even though I successfully evaluated (1+7i)/(−3+4i) to 1−i, I somehow managed to get from there to 1 cis π/4, and therefore proceeded to solve z²=1 cis π/4 rather than z²=1 cis −π/4.  One of two things will happen.  The first is that it will be regarded as a “minor error” and I’ll drop a mark, though I find this unlikely: the error was near the beginning of my answer, and was rather stupid, to say the least, and would have affected my answer significantly, so it’s more likely that I’ll only be given credit for my work up to that point, that is, one mark out of four.  Strangely, I’m not kicking myself for this like I normally would.

What I am kicking myself for, though, (I only just realised this) are my brilliant algebraic skills in question four (a).  This was one of the most interesting questions in the entire paper, so I quite wanted to get it right, but from R²=2r²(1+cos 6π/7) I found that R=√(2).r(1+cos 6π/7), and consequently got a final ratio of 1.344 rather than 0.975.  Urgh.  That’s too stupid mistakes (mind the pun).  (I’ll find more, I bet you… unfortunately, I probably won’t be bothered to detail them here.)

I had no idea what question four (b) meant (I got up to dy/dx=−1/(z+2) but I didn’t know what to do with that, having completely forgotten what “implicit equations” are).  By the time two hours and fifty-five minutes had passed, I had finished questions one, two (with that idiotic error), five, and part (a) of questions three and four (with error), and had written on twenty-three of the twenty-five pages on the answer booklet.  I figured I may as well start on question three (b), the only one remaining, and I was quite literally writing at the bottom of the twenty-fifth page of the answer booklet (though I had left the first page blank because I don’t like writing on endpapers) when the supervisor gave the “pens down” signal.  So close to getting to see what those extra bits of paper look like!  Oh well, maybe next time.

And with that, I had finished my first decent exam of this period.  I must say, I’m quite pleased with myself that I’ve finally broken the long-and-hard streak I seem to have had this exam round.  With two chemistry exams in the next three days, though, and being very, very behind preparation for both of them, my efforts switch to them.  If all goes well, this newfound luck will continue with me.  I guess only time will tell.

Permalinks to my blogs on the 2005 calculus scholarship exam
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