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

## UPDATE (30/08/21)

Just a quick word of advice to VCE students: don’t take any psychedelics before your maths exams. You’ll be in for a very bad trip.

The question is nuts, of course, and is more of a PoSWW than a WitCH. In principle, including extraneous information in a question is fine, but this is not the way to do it. Moreover, if you’re gonna do it then you gotta know your limitations.

Using “repeated root” in this context is absurd, since the expression only makes sense for polynomials (and, after VCE, for suitably polynomial-like functions). **(05/09/21 – “zeroes” not “roots”.) **The expression “continuous and differentiable” is also absurd, since a differentiable function is automatically continuous. Finally, the lurking Mr. Big pointed out how helpful it was for the question to specify that *a* and *b* are real, saving students precious time that would have been wasted considering the alternatives.

Most of the first sentence of the preamble is extraneous. The only property needs is continuity over the interval [1, 6].

“Let be a continuous function over . Given that …”

I’m very late (only saw this linked in a recent post), but for those interested, continuity isn’t even required, though this is beyond the curriculum: *(Riemann) integrability* does not imply continuity, and only the former is needed here.

Yes. But, as you say, this is not in the curriculum. Functions are not referred to as “integrable”: they are simply integrated. So, continuity is a natural enough assumption.

What on earth is with the conditions on ? Cruel and unusual punishment. The only one that makes sense is continuous, as JF points out also.

I’d just additionally note that I can even forget entirely about the and just integrate the constant. Only one choice has .

The question is stupid, the answers are stupid. Blegh.

This is just bottom-of-the-barrel in terms of quality. Like someone else said on the other one — is there no checking or proof reading of this garbage?!?

The garbage does get checked. Either the vettors are clueless or the garbage is a vast improvement from the original offal. Toss the coin.

The question should specify that these are the only roots. This is Glen’s point I think?

Does a “repeated root” allow say, a triple root?. If so there is insufficient information.

Further, the function is not defined to be a polynomial. I guess we could define a repeated root to be where both and are 0, but then we really need extra conditions to answer this question.

Three errors in one question. Is this a record Marty? All fixable by saying that is a polynomial of degree 4.

Thanks, tom. Yes, “repeated root” makes no sense here. But I’m not sure what you mean by “error” and what you regard as the three errors here. I’m also not sure what you mean by “insufficient information”. There’s sufficient information to do the problem.

Of course the question is weird and stupid, including one stupidity that no one has yet mentioned. But the only error I can see is the repeated root thing.

Using extraneous brackets:

versus

.

This is something I was going to mention earlier. It’s something that really irks me. There is no need for extraneous brackets. They make the expression inelegant and difficult to read for no good reason. The integration sign and the act as defacto brackets (and VCAA have actually publicly explicitly stated this in the past!!).

I usually equate using brackets in this context with a try-hard wanker with delusions of grandeur.

JF, in general I agree that the brackets in integrals only clutter things. Here, the f(x) + 1 seems not so obvious an integrand, so I’m fine with the brackets. In any case, that’s not the other annoyance I indicated.

Hmm … Well, I’m less than fine with them. Anyway … would it be the lack of definition of (as opposed to the definition of )?

No, is preferable, but I’m fine with either, at least in secondary school.

It’s a judgment call (and we agree it doesn’t really matter), but I think it’s an error. It is written as if the concept of repeated root is generally applicable, which is false.

Oops my error.

I’m not defending the question (because I can’t).

But these multiple choice questions are so incredibly common, surely the way through this for students is just to drill themselves on these and not over-think them?

I know how counter-intuitive that is considering some of the questions that come up on the ends of these Methods paper 2 exams… but I don’t see a better way.

RF, I can sort of defend the question. It’s worthwhile to learn to recognise extraneous information, to be able to focus upon what is required to answer a question. In some sense all word problems have that element, and of course learning to extract the required information from the words is not easy.

It seems to me the exam question is intended to pretty much only test this skill and, once the focus is there, the question is intentionally trivial. The trouble is, the extraneousness is so contrived and clunky, with error.

As to how to get a student to cope with such aspects of questions, it’s tricky. Probably there’s nothing better than what you suggest. The problem is, so many VCE questions are so poorly composed and so poorly written, it is a hugely difficult task to distinguish irrelevance from incompetence.

I’m not sure there’s an error amid the contrived extraneousness of the first sentence. The repeated root is not an error. A perfectly good rule for the function is .

The idea of a repeated root is meaningless outside of polynomial functions, which f is not stated as being.

What about ?

Sure, JF, which amounts to considering power series, but the point still holds. It’s plain weird to refer to the “repeated root” of an otherwise unspecified “continuous and differentiable function”, and it makes the “continuous and differentiable” redundant.

Yes, we all agree that’s it’s weird and extraneous. But that doesn’t make it erroneous …? And it’s no more weird and extraneous than giving the unrepeated (I assume …) roots.

I’m pretty sure I could construct a function that was not continuous over but nevertheless had the given roots, so I don’t think stating the roots makes the statement of continuity redundant.

I would never defend this question, but I think it is best to give a warning here. An important concept in (later) analysis to talk about the degree of a root of a function. (Not only power series.) It is common to use the word “multiplicity” when that degree is a whole number, and then it is also common to call something with multiplicity greater than 1 a repeated root.

So it is not correct to say that “The idea of a repeated root is meaningless outside of polynomial functions”.

Thanks, Glen, although I think you’re being a bit legalistic, and subtly shifting the question. anonymous is more right than wrong, particularly given the context.

I wasn’t going to reply to Anonymous until I saw JF and then your reply to JF. It felt like it needed to be pointed out that the concept is actually quite important and not restricted to polynomials (or power series).

Definitely tangent to a tangent.

I assume they are trying to test the difference between integrating and finding the bound area as well as integral properties.

Ah, I hadn’t thought of that. I don’t think that’s exactly quite it, Alex, since the information is insufficient to determine the trapped area of either or . But you’re probably correct, that the notion of bounded area was intended to be part of the extra-info trickery.

As I have explained before, I am opposed to multiple choice questions in examinations. I do use such questions in class as a learning tool by asking my pupils to explain why the wrong answers are wrong, and why the correct answer is correct.

As for this one, I have always thought that functions have zeros, and equations have roots. But maybe I am out of date.

You are probably out of date, but you are also correct.