The following is our second WitCH from the 2021 NHT Methods Exam 2.

# Tag: polynomials

## WitCH 64: Decreasing Intelligence

Just in case anybody got the wrong impression and hoped or feared we’d turned over a new leaf, we’ll be posting a number of WitCHes in the next few days. We’ve finally had a chance to look at the 2021 NHT exams (although the exam reports have still not appeared). As usual, the exams are clunky and eccentric, and we’ll be posting a brief question-by-question overview of the exams. But, first, some highlights. Continue reading “WitCH 64: Decreasing Intelligence”

## PoSWW 17: Blessed are the Cheesemakers

This one is old, which is not in keeping with the spirit of our PoSWWs and WitCHes. And, we’ve already written on it and talked about it. But, as the GOAT PoSWW, it really deserves its own post. It is an exercise from the textbook *Heinemann Maths Zone 9 *(2011), which does not appear to still exist. (And yes, the accompanying photo appeared alongside the question in the text book.)

## WitCH 57: Tunnel Vision

The following is just a dumb exercise, and so is probably more of a PoSWW. It seems so lemmingly stupid, however, that it comes around full cycle to be a WitCH. It is an exercise from *Maths Quest Mathematical Methods 11*. The exercise appears in a pre-calculus, CAS-permitted chapter, *Cubic Polynomials*. The suggested answers are (a) , and (b) 81/32 km.

## WitCH 5: What a West

This one’s shooting a smelly fish in a barrel, almost a POSWW. Sometimes, however, it’s easier for a tired blogger to let the readers do the shooting. (For those interested in more substantial fish, WitCH 2, WitCH 3 and Tweel’s Mathematical Puzzle still require attention.)

Our latest WitCH comes courtesy of two nameless (but maybe not unknown) Western troublemakers. Earlier this year we got stuck into Western Australia’s 2017 Mathematics Applications exam. This year, it’s the SCSA‘s Mathematical Methods exam (not online. **Update: **now online here and here.) that wins the idiocy prize. The whole exam is predictably awful, but Question 15 is the real winner:

**The population of mosquitos, P (in thousands), in an artificial lake in a housing estate is measured at the beginning of the year. The population after t months is given by the function,** **.**

**The rate of growth of the population is initially increasing. It then slows to be momentarily stationary in mid-winter (at t = 6), then continues to increase again in the last half of the year. **

**Determine the values of a and b.**

Go to it.

## Update

As Number 8 and Steve R hinted at and as Damo nailed, the central idiocy concerns the expression “the rate of population growth”, which means P'(t) and which then makes the problem unsolvable as written. Specifically:

- In the second paragraph, “it” has a stationary point of inflection when t = 6, which is impossible if “it” refers to the
**quadratic**P'(t). - On the other hand, if “it” refers to P(t) then solving gives a < 0. That implies P”(0) = 2a < 0, which means “the rate of population growth” (i.e. P’) is initially decreasing, contradicting the first claim of the second paragraph.

The most generous interpretation is that the examiners intended for the population P, not the rate P’, to be initially increasing. Other interpretations are less generous.

No matter the intent, the question is inexcusable. It is also worth noting that even if corrected the question is awful, a trivial inflection problem dressed up with idiotic modelling:

- Modelling population growth with a cubic is hilarious.
- Months is a pretty stupid unit of time.
- The
~~rate of~~population~~growth~~initially increasing is irrelevant. - Why is the lake artificial? Who gives a shit?
- Why is the lake in a housing estate? Who gives a shit?

Finally, it’s “latter half” or “second half”, not “last half”. Yes, with all else awful here, it hardly matters. But it’s wrong.

## Further Update

The marking schemes for the exam are now up, here and here. As was predicted, “the rate of growth of the population” was intended to mean “population”. As is predictable, the grading scheme gives no indication that the question is garbled garbage.

The gutless contempt with which certain educational authorities repeatedly treat students and teachers is a wonder to behold.

## An Untrustworthy T-Shirt

Among the many Australian mathematics organisations that are making matters worse rather than better, the Australian Mathematics Trust must not be included. AMT is great, a rare beacon of hope. A beacon somewhat dimmed, it is true, by the fact that the AMT guys have an average age of about 95. Still, any beacon in a storm, or whatever.

Apart from their wonderful work on the Australian Mathematics Competition and their associated endeavours, AMT sells excellent books on problem-solving, as well as some very cool (and some very uncool) mathematical t-shirts. One shirt, however, is particularly eye-catching:

This shirt celebrates Norwegian mathematician Niels Henrik Abel and his 1824 proof of the non-existence of a solution in radicals of the general quintic equation. That is, in contrast to the quadratic formula, and to the cubic and the quartic analogues, there does not exist a quintic formula. It’s a pretty shirt.

It’s also a pretty weird shirt. AMT’s blurb reads

*This T Shirt features Abel’s proof that **polynomials of order five or higher cannot be solved algebraically*.

Stylewise, it is probably a good thing that Abel’s “proof” doesn’t actually appear on the shirt. What is not so good is the sloppy statement of what Abel supposedly proved.

Abel didn’t prove that “[polynomial equations] of order five or higher cannot be solved algebraically”. What he proved was that such equations could not *generally* be solved, that there’s no *general *quintic formula. In particular, Abel’s theorem does not automatically rule out any particular equation from being solved in terms of radicals. As a very simple example, the quintic equation

is easily shown to have the solutions .

Which brings us back to AMT’s t-shirt. Why on Earth would one choose to illustrate the general unsolvability of the quintic with a specific equation that is solvable, and very obviously so?

Even good guys can screw up, of course. It’s preferable, however, not to emblazon one’s screw-up on a t-shirt.