The 2024 NHT exams have now been posted (a mere two months after the exams took place). We haven’t looked, but we’ll look if someone suggests in the comments that there’s something to see.
General 1, General 2, Methods 1, Methods 2, Specialist 1, Specialist 2.
UPDATE (25/09/24)
The exam reports (Word, idiots) have now been posted. Thanks to Lancelot for the alert.
General 1, General 2, Methods 1, Methods 2, Specialist 1, Specialist 2.
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From a three-minute skim of the MM papers: Exam 2 MCQ17 doesn’t have a correct answer, and I’m not sure how you do Exam 1 Q3b. and c. without a specified domain! I’ll look at all the papers more closely when I have time.
Thanks, Javert. I’ll look.
I’m sure option D is the intended answer and I think a good lawyer could argue the case. But I think the writer is trying to be too cute and I wonder how many students will simply guess and move on (we won’t know because the response data for MCQ is not provided in the Exam Report).
I’m not a good lawyer, but here’s what I think the argument could be:
Pr(C | B) = Pr(C) means that C is independent of B and so B and C are independent.
Pr(A | B) = 1 implies
. This includes the possibility that A = B, which is also included in 
. (This is the bit I don’t like).
So it looks like Option D is the one.
Options A, B and E are clearly wrong because of the “must”. Option C gave me pause for thought, but I think Pr(A) = 1 is possible therefore “must” not equal 1 is too strong. There might be a tricky subtlety (loophole) that eliminates option D and maybe Javert has found it, but I can’t be bothered looking for it.
As for Exam 1 Q3b and Q3c, I agree. The writer snookered themselves with suboptimal writing. FWIW I think the intent of Q3 is better written as:
Let f(x) = ….
(a) Sketch a graph of y = f(x).
(b) State the maximal domain of f(x).
(c) Find the values ….
A related question: if P(X) = 0, must X be empty? I don’t think it’s a subtlety/loophole at all.
Thank you all. I just glanced at the questions. I think I looked at the probability question previously, thought it smelled wrong and then decided it was ok. But no current thoughts.
The answer is no of course. I think in the context of the question this is somewhat ‘disguised’ and ‘easy’ to overlook (which is what I meant by a ‘loophole’) if you’re trying to find or justify a correct option. Nevertheless, it means that option D “must” not be correct and there’s no correct option. I’m certain D is what the writer had in mind, but it follows a frustratingly familiar pattern:
Writer tries to be too clever –> writer screws up because they’re not smart enough to be too clever –> students pay the price.
What does the writer think they’re testing (apart from our patience)? Surely you can test in a straight forward way and without (trying to) using ‘clever tricks’ that typically shoot the writer in the foot.
I think a lot of students might choose D purely because of the independence bit and they have to choose something – a semi-guess. And we must remember that this is a 1 mark question = 90 seconds. I hate to think if a student ended up spending 5 or more minutes on this question (easy enough to do).
It would be fascinating to know what solution the writer had in mind to justify whatever they think is the correct option. I’ll bet it will be similar to the argument I suggested.
I tell my students that if they see dopey questions like this, just guess and move on (and go back if there’s time later).
Moral of the story: If it’s a VCAA question and smells wrong, it probably is.
Obviously D is the intended answer, but given the textbooks explicitly mention (in dressed down language) the difference between an empty event and one with measure zero, I don’t think it’s a particularly acceptable mistake.
For completeness, here’s an example where all five options are false: take X ~ U[0, 1], let A be the event 0.5 < X < 1, let B be the event 0.5 <= X <= 1, and let C be the event X = 0.
Nice work, Javert.
Thanks, Javert. Of course you are correct: D is the intended answer and is ok in the discrete context but not in the continuous context. Your example is perfect.
The error is very similar to one VCAA made in 2014 (MCQ14), and which was a precursor niggle to me to start this blog. I think the seriousness of the error is debatable, although it’s definitely an error and obviously not good. It’ll definitely get the red in the error list. As you note, option D is true “up to a set of measure zero”, to use the lingo.
In terms of VCE, I think one might reasonably call this a “subtlety” or “loophole” or something. The zero probability of dots is basically not covered at all, to my knowledge. Unlike John, I don’t think the writer was trying to be too clever. I think the question is otherwise too cute, and not very good. But, in regard to the error, I assume the writer simply overlooked the issue in a standard VCE autopilot manner: they simply assumed that
implies that B is a subset of A.
I think in practice the error is unlikely to have misled students, or even distracted more than very few, but definitely the error shouldn’t have occurred. (If it were not NHT I’d be hammering it harder.)
Javert, In your example, is it true that option (C) is false?
Does the attached correctly show, at the VCE level, why Option D must NOT be correct?
Option A is correct for this example. But can a counter-example, consistent with the given data, also be drawn for Option A (and Option C), to demonstrate why Option A (and Option C) must NOT be correct?
Is the “must” (implying always) the problem with this question?
John, others may be able to make sense of what you wrote, but I can’t make heads or tails of it.
The issue with the question seems pretty simple to me, as outlined by Javert.
Hi, Javert. Regarding 3(b) and 3(c) on MM1, yes it’s a bit of a mess. Maybe not quite up to the red pen (I’m pondering), but definitely amounts to an error and, as John notes, it is remarkably poor writing. This is a case where Bennett’s “experts” would presumably be saying “the intent was clear”, and we are all saying “not clear enough, and without excuse”. I really don’t understand how people can write so poorly, and how it can be scrutinised by N pairs of eyes, for some non-trivial value of N, and no one pick it up. The shoddiness of the expression is obvious, and painful.
For those keeping score, the error is akin to 2022 MM1 6(c) (ii) and (iii).
Had a bit of a flick and work through, wish I had a more time to go through more throughly, but alas.
Did pick up Q39 in General Exam 1 (page 24) which felt a little too ambiguous about what the maximum number of students is meant to be: either so that the group of students could walk any possible route from entrance to exit (then it would be 9 – 31 to 9 to 56) or that there exists a route from entrance to exit that the group could walk that fits as many as possible (then it would be 20 – 31 to 20 to 32 to 21).
Thanks, Alex. I’ll look.
Hi Alex. I don’t see the ambiguity (and I don’t understand your notation). I don’t see any reason why the students aren’t permitted to walk together whatever route will accommodate them all. So why isn’t the answer clearly D (20)?
The two possible options I’m thinking of for what they mean by the maximum are:
– a group that is as big as possible, preemptively finds the route they can actually take (yellow path), then follows that
– a group that is as big as possible, can take whatever route they want essentially choosing as they go without any preplanning (green is most restrictive route that they would still need to be able to take)
Since the question don’t mention the maximum until after the question has already said the students have walked through the historical site, I interpret it as they took whatever route they pleased without needing to decide before they set out what that route would be. Hence, a maximum group of 9.
Maybe I’m being too pendantic? 🤷🏻♂️
No, I don’t think you’re being too pedantic. I think you’re being too wrong.
I understand the two readings one might look for in the question, but I think your 9 reading is simply not there. The group was able to walk through together. Clearly there could have been 20 in the group.
For MM Exam 2 MC12, is it really necessary to include “Newton’s method is used to estimate the x-intercept of the function” in the preamble?
For MC19 of the same exam, is it necessary to say the function is ‘differentiable’ when it is already stated that it is ‘continuous’?
Also, how likely is it that the marking guides for these exams will be released, given the Bennett Report?
MCQ 12:
I’ve got no beef with this wording. Do you prefer that they ask for an estimate of the root of f(x)?
MCQ 19:
A function can be continuous at a point but not differentiable at the point. Eg f(x) = |x| is not differentiable at x = 0 but is continuous at x = 0. However, they could/should have simply said that the function is differentiable (because a function that is differentiable at a point is also continuous at the point).
VCAA has a long history of writing questions where the wording is over-egged.
Re: “Also, how likely is it that the marking guides for these exams will be released, given the Bennett Report?”
That’s a question I have asked myself. One of six recommendations made in the Bennett Report says:
“6. The VCAA should publish examination material in a more timely manner and update its policies and processes relating to the management of errors and alleged errors as follows:
a. Tighten publication timelines for examination documentation to more closely align with other comparable Australian jurisdictions as follows:
i. Publish the VCE Mathematics examination paper on the VCAA website within 5 days of a paper being sat.
ii. Publish the final version of the Marking Guide used to assess students’ performance on the VCAA website within 5 days of the end of the marking of each Mathematics examination.
iii. Publish the report prepared by the Chief Assessor on the VCAA website before the end of January the following year.”
Given that (a)(i) never happened, it’s possible that some at the VCAA might think that the recommendations of the Bennett Report only apply to the November exams. There are several good people who joined the VCAA after the 2022 debacle who are working hard to implement all of the Bennett Report recommendations, but there are also people instrumental in creating the 2022 debacle who remain at the VCAA.
John, stop with the whiny bullshit or you are banned.
My apologies John, I did indeed mean the other way round: if a function is differentiable it should be continuous, so continuous is unneeded.
For Q12, I just thought that the sentence in the preamble is unnecessary is all. It doesn’t contribute anything to the question, and students should already know that the Newton’s Method returns an estimated x-intercept.
Regarding the release dates for the NHT examinations, all subjects seem to be released at the same time, so this is perhaps not (only) the fault of the Mathematics department.
I agree that marking guides will most likely not be released until the November examinations. However, has the VCAA explicitly said that they will release marking guides at all? They said that they will implement “as many measures as possible”. Could they possibly argue for ‘confidentiality’ reasons that the marking guide will not be ‘possible’ to release?
Thanks, AY. I will look at the questions you raise in a minute, although I think John has it pretty much covered, and this verbiage is standard (and annoying as hell).
Regarding VCAA and Bennett, I would suggest, perhaps surprisingly, some patience and some qualified trust. VCAA is a very big and a very stupid boat to be trying to turn around. Bennett has called for some very large specific changes and, more generally, very large cultural changes. Even with the goodest of good faith, such changes will take time and effort. Even November 2024 I would regard as too soon to be a proper test, although of course everyone will be watching and VCAA knows everyone will be watching.
As it stands, I have a *lot* of faith in the acting CEO of VCAA. I liked her when I heard her on radio and I liked her when I met her. She seemed very smart and absolutely sincere, neither of which was or is remotely true of Gniel. That doesn’t mean she’ll succeed, even if she stays. But I believe she is genuinely trying. I don’t think there is or will be any intention from the acting CEO to try to shortchange the Bennett recommendations.
Thanks, AY. I’ll look at those questions, and all the questions, tomorrow.
I don’t think Bennett will properly kick in until November 2024, at the earliest. In the first place, VCAA has obviously been scrambling to do what they can, and NHT would presumably have been well in the can when the report came out. Secondly, it’s NHT. Who gives a shit?
Hi, AY.
Re MCQ 12, I agree with John on the wording. It seems fine to me (although the question is loathsome). What is your concern?
Re MCQ 19, of course as you intended to say, there is no need for the question to refer to continuity, since it is implied by the differentiability. It is also doubly redundant, since continuity plays no active role in the question. As John notes, this specific redundancy is standard wording in VCE (which doesn’t make it any less stupid or annoying).
Thank you all for your comments. I think I have looked at and responded about everything that has been raised (which, as it stands, is remarkably little, especially for NHT, which tends to be a basket case.)
I was under the impression that, although they entered the exam writing process late in the game, real mathematicians had at least seen these exams. Maybe that explains “remarkably little, especially for NHT, which tends to be a basket case.”
I honestly don’t know.
Would the SM E2 Q5b be better reworded as “Find *a* Cartesian equation of the plane P”? Any scalar of the equation would give the same plane.
Yes. Not a big deal, but yes.
If the examiners read this blog, how will they mark answers to MM Exam 2 MCQ17? Give every candidate 1 mark?
Of course not. The exam has already been graded and they would have graded D as correct which, all things considered, is the correct choice.
Spec exam 2 Section A multiple choice 2 is not* compliant with https://www.vcaa.vic.edu.au/curriculum/vce/vce-study-designs/Pages/PseudoCode.aspx
“There are some basic principles to be followed in using pseudocode which include…Use indentation to show the hierarchy of processes within an algorithm such as repeating sections and conditional decisions.”
*Technically the examples in that page don’t always show the word “return” indented. Regardless, the lines below it definitely shouldn’t have been indented and they were
Thanks, Jay. I couldn’t give a stuff about the pseudocode garbage, but if VCAA forces this garbage onto others then of course they should respect their own rules. Do you regard this non-compliance as serious, or are you nitpicking?
I think it is somewhat confusing, but I’m also kinda nitpicking
Ok, fair enough.
This pseudocode is inexcusably garbage because “return” means to “return” to the procedure that called it, terminating the current procedure. Nothing after the return statement would be run. This is how it works in every programming language, and how any person in CS would interpret the code.
Add on top of that the senseless indentation from last year.
If this pseudocode were submitted at university it would get a 0.
Thanks, Joe. I still don’t care, but are you saying the pseudocode would not actually run as written?
The pseudocode defines a function that takes an input x and returns x^2. Nothing is printed as everything after this return statement is ignored.
Christ. I guess I gotta look. God, DLL was a disaster.
Thanks, Joe. I’ll look tomorrow.
Joe (and Jay or anyone, if you wish), I’m trying to make sense of this pseudo crap (while my brain is shrieking for me to leave it alone).
On the VCAA page Jay links to above, the first two “define” examples are to compute a factorial and to compute a trapezoid approximation. So, my questions:
1) In your/CS view is “define … return” in these two examples laid out correctly or incorrectly?
2) Is the use of “define … return” in the SM2 question consistent or inconsistent with the VCAA examples?
3) Should the “return” be indented or not, and why or why not? (note the discrepancy in the two VCAA examples?
4) In the SM2 question, which lines/procedures should be further indented, and why?
1. For the factorial one, it is fine. Indenting the “return” statement would be preferred but it’s not a big deal if there’s only one function. If there’s multiple functions the “return” statement should definitely be indented to make it clear when a function starts and ends. (Or, write something like “end function” or something.)
The trapezium question is set out fine, but spacing between the function and the rest of the algorithm would be better to make it clearer. Ideally the algorithm should be defined as its own function, but this is nitpicking.
2. It is technically consistent with the first example, but it’s not acceptable as there is more code after it, and it’s not clear whether this code is part of the function or not.
3. “return” should definitely be indented as it’s part of the body of the function, a statement to be processed. Return statements do not always appear at the end of a function, they often appear earlier, within if statements, etc.
4. This is how it should be set out: https://pastebin.com/HD6BJdfN
The purpose of indentation in pseudocode/code is to group statements together, so it’s clear what statements fall under a definition, or a while loop, or something else. VCAA’s indentation seems to be completely meaningless, with certain types of statements arbitrarily indented and others not.
God, I hate this.
Thanks very much, Joe (and Jay). I’m beginning to get my head around it. So, the major problem with theSM2 pseudocode is the indentation of the lines between the “return” and the “while”? And the problem with that is that the indentation leaves the a,b,h stuff in No Man’s Land?
Then, minor criticisms are:
*) the lack of indentation of “return”?
*) the lack of spacing between groups?
*) the lack of a separate “define” for the algorithm?
Hi Marty. I’m not sure if I’m too late to post a comment on the NHT 2024 Gen Maths exam Paper 1 or not. Anyway, it’s regarding MCQ 38. Is the correct answer B or C ? I have repeatedly tried to get a flow of 77 noting the obvious cut of 77 formed by the two edges near the exit. However, after numerous attempts I can only come up with a flow of 76 and I am unable to identify any cut of 76.
It’s never too late. Thanks, Rob. I’ll take a look later today.
Hi, Rob. I think the maximum is 76. The key here seems to be that the 34-edge is pretty useless: only 13 people can get to use this edge. So, the maximum number of people who can get to the start of the 56-edge is 13 + 16 + 9 + 17 = 55.
I cannot remember the efficient or algorithmic methods of doing these problems, so am not suggesting how a student should approach the question. I can ponder if you wish.
Thanks for that Marty. I agree with you and think the answer is 76. However, I thought the Maximum flow has to equal the Minimum cut; and since I was unable to find a cut of 76 after looking for at least 30 minutes I was not sure. Another one of those MCQ’s impossible to do in a reasonable amount of time
Thanks, Rob. Maybe it comes down to the definition of a “cut”. If you cut at the 13, 16, 9, 17, and 21 edges, I think that may constitute a “cut”. That is, the remaining edges do not allow any flow from the source to the sink.
As Marty has noted, the minimum cut is through 13+0+16+9+17+21=76 where the zero is fr the reverse flow of the edge marked 12, despite it appearing to be heading “forward”. It actually flows from the sink side toward the source for this cut. I used the process whereby you reduce every path so that it has a zero. Done by subtracting the smallest value in a path from all values on that path. Eventually you find a clear corridor of zeroes tha will indicate the minimum cut.
I’m not keen on teaching kids to follow a process unless they understand it but Marty’s explanation is a step toward it.
I’m intrigued by this method, I haven’t seen it before. I don’t believe the 21 edge is the smallest on any path. I know this post is old, but if you see this, I’d appreciate some info on how your method works.
Hi Samantha, I’ll give RobV and others a chance to respond. But if not much comes of it, or it seems worth it anyway, I can do a post on it. Just comment (or email) when you wish to prod me in that regard.
Hi Samantha, apologies for the late reply to your question about this technique. Hopefully you’ve already solved it, however if not; I have always referred to it as the ‘tally method’. Shitloads easier than cuts, more methodical too, imo. My stronger students have always preferred this method.
If you need to find the cut, you look for a ‘cut’ with all zeros. I doubt you would ever be asked to do this as I don’t see the point but VCAA do some weird crap.
Happy to help if you need more help with it.
Messy pic attached of referenced question.
MM1 Q2c… I’ve been thinking about this a bit…
The lines are perpendicular if the product of their gradients is
. Fine. I get one answer for 
by this method.
But… will the examiners be expecting/allowing
as well? If 
then one line is horizontal and the other vertical, so are they not perpendicular?
I am anticipating both
and 
will be allowed since the question says “a value” (suggesting there may be more than one answer, any of which will be correct), but I am never sure on these matters.
Thanks, RF. I’ll look.
Hi, RF. Well-spotted, and it is puzzling. Here are my thoughts.
First, I am as sure as I can be that both answers were accepted. There is zero argument for not accepting a = 0. And, true, VCAA has relied upon zero argument many times in the past. But now, in this new era of scrutiny, I think they will be much more reluctant to do so. NHT makes it dodgier, but I’d still bet on sanity here.
The more interesting question is whether VCAA expected both/either answers. On the basis of zero evidence, I suspect that VCAA did, but only belatedly.
Note that (a) is worded,
“Find the value of a …”,
and (b) is worded,
“Find the values of a …”.
Then (c) is worded
“Find a value of a …”.
So, in distinction to (a), the wording in (c) implicitly acknowledges at least the possibility of more than one value. But, why then, contrary to (b), ask for just one of these values?
I suspect it is because VCAA only realised at the last minute that there was more than one value. Obviously (c) is a poorly framed question, but almost seems consciously so. It feels to me as if the question was fixed as well as could be at the last minute, that it originally asked for “the value”. The alternative fix would have been to have asked for “the values”, including a = 0, but that would have made the question too fiddly for 1 mark.
Just a guess, but that’s my guess.
Thanks. When I first read the question, I actually thought
was the obvious answer, but then went and set the product of the gradients to 
to see if there was another (I expected there would be).
Of course, a student in exam-mode will behave differently.
Interesting. If VCAA initially expected one answer, I doubt that it was a = 0.
I pondered a little further, and am thinking maybe VCAA always knew there were two answers around. If so, they presumably decided, at some stage, that requiring both answers was too much for 1 mark. Then, for whatever bad reason, they watered down, and muddied, the question rather than bump it to 2 marks.
Methods Paper 2 Q3c. What will happen if students include 0 in their domain for the second part of the piecewise function? It is only 1 mark, so it seems a bit harsh if they mark a student as wrong for including 0 but then again, if 0 is part of both domains then
is no longer a function…
Will wait for the official answers from VCAA before knowing if this is an error or not I guess.
You mean 4(c)? The entire question is so revolting, I’ve been trying hard to pretend it doesn’t exist.
As to your specific point, given the function to be defined is continuous, it is perfectly legitimate (if not good style) to include 0 in the domain of the second bit. There is no question that including 0 (or excluding 0) *should* have been permitted in the grading. But, given this is NHT, we’ll likely never know how the question was graded.
Yes, 4c.
I’m only just this week finding time to actually work through these NHT papers, so I may add a few more queries next week when I get to the SM papers…
Hey Team,
GM exam 2, question 13C. Is the question asking that the students find 28% of east each year and then divide the total, thus 2% for the answer?
Are we a Team?
Anyway, I think you ask a very good question. I’ll ponder this a bit, but I think it’s probably worth a WitCH (so I don’t have to think).
Hi Paul. I’ve thought about this, and that’s not how I read the question. Here’s my thoughts (which may be wrong).
First, the question is asking about the chickens in East field after three days. So, cube the transition matrix and add the “now in East” entries. multiplying by the starting 10000 in each field, that comes to 11293 chickens?
Then, of those 11293 chickens, you want to know how many were in east field after day 1 and day 2. Is it easiest to get that number by going back to the transition matrix: find the number of chickens in east field after day 1: is that 11100? Then 0.28 of those chickens will stay there for day 2, and then 0.28 of *those* chickens will stay put for another day, giving (0.28)^2 * 11100 = 870 chickens.
So 870 out of 1129 chickens will have just sat in the east field for the previous two days? That comes to about 8%?
Exam report is out now.
https://www.vcaa.vic.edu.au/assessment/vce-assessment/past-examinations/NHT-past-exams/Pages/NHT-past-exams.aspx
MM1 report is the most detailed. Very good.
SM1 report is the 2nd detailed amongst all maths reports but it is precise and succinct. Very good.
The SM2 and MM2 reports are on the brief side of spectrum. The SM2 report gave no explanation to any MCQ.
Report to MM2 gave a lot of explanations to the MCQs…particular the second half. The ERQs just had brief answers without much comments or insights.
GM1+2 reports are not awfully different compared to the previous years…
Thanks very much, Lancelot. I’ll add the links and check a few things.
I never expect much detail in the NHT reports, and don’t really blame VCAA for paying it little attention, even if some past reports were pretty farcical. The entire NHT program is (was) only to keep happy a few rich-school twats.
Ok, I’ve updated this pot and the error posts with links to the exam reports. Thanks to Lancelot for the flagging.
Unless I missed something, there are only the two Methods errors that were flagged by Javert: Q3 on Exam 1 and MCQ17 on Exam 2. (Please remind me if I’ve forgotten something.)
On both errors, the exam reports are silent. These are both bad (if not huge) errors, although the reasonable treatment of them differs. The intended meaning in Q3 is clear enough and it is not a huge deal (while still being unprofessional and pretty gutless) for the exam report to be silent on the error.
On MCQ17, however, the intended argument is simply invalid, and the report just paints over the issue, presenting the invalid argument as if it were valid. Even if one regards the flaw as a minor nitpick, and that is highly debatable, it is simply inexcusable for VCAA to be writing something in an exam report that is wrong, particularly if they know it is wrong, and it is difficult to believe that they do not.
I’m not so fussed by the occurrence of such an error, particularly on NHT. But VCAA’s repeated deceit on the reports is perverting and exhausting.
Hello,
I’ve just done this Methods exam 2 after using it as a trial exam and have a question if anyone has answers…
Is there anyway to do the last question 5e without messing around with CAS sliders on graphs and trial and error? Students know a=1/e gets one point of intersection and the a = -4 has none so there are some clues. I hate using sliders its so clunky – I don’t see how students are meant to work with the complexity in this question if that’s their only option?
Thanks
Sigh. CAS poisons everything.
Thanks, Amber. I’m not going to even pretend know what idiocy was intended, but hopefully someone can guide you.
May I ask in general what you thought of the exam?
I haven’t taught Methods for a few years I usually teach Spesh so I haven’t done as many Methods exams. I thought it was OK? Kind of easy with nothing over 2 marks and lots of questions were ‘show that’ questions for no reason I could see as the answers were straightforward. There were plenty of very gettable marks but the downside is there’s nothing very meaty in there. That’s why I found 5e annoying because there isn’t much that benefits the stronger thinking student which is where I think the separator questions should be aimed.
Thanks, Amber.
NHT GM exam 1 2024 Q14.
It’s a simple smoothing question, the report says the answer is D, yet everyone I’ve spoken to is sure the answer is B.
Thanks, Floz. I’ll look.
Thanks, again, Floz. I’ve checked, and I also had a General-ish friend check. Yes, the answer clearly seems to be B. I also was not able to obtain D by trying various plausible alternatives. Weird. In any case I’ve added it to the General Error List.
A very late comment, but I was just having a look at the 2024 SM Exam 1 and the “report”. I have a question / comment about Question 3b, which asks for the values of x on which the graph of
is concave up. The report gives the answer as x > 1 or x < -1. (Edit: inequality signs aren't showing up? should be "x less than -1 or x greater than 1")
The issue here is that VCAA does not define "concave up". The report's answer assumes that a function's graph is concave up on an interval if and only if the function's second derivative is positive. But, the definition I am familiar with is that a function's graph is concave up on an interval if and only if the graph is below the line segment joining the points on the graph at the endpoints of that interval. This means that a function's graph is concave up if the second derivative is non-negative, in which case the answer to that exam question is x ≤ -1 or x ≥ 1.
This is frustrating, because I think VCAA is aware of a similar issue with increasing / decreasing functions and the sign of the first derivative. eg. if asked for the values where y = x^2 is increasing, I believe VCAA would require the answer to be x ≥ 0.
Thanks, SRK.
Quickly on the LaTeX glitch, I wonder if you copied your comment from a draft on Word or something. The inequalities are fine when I type them in a comment.
More importantly, you are correct, that VCAA has bollocksed this. As you note, without a clear definition of “concave” there can be no clear answer. I’ll add it to the list.
My own sense of “concave” is the same as yours: concavity in terms of the graph lying above line segments. I don’t think there is a broadly accepted definition, at least not at the undergraduate or school level, and I’m fine with a more vague use of “concave” in terms of (increasing) first derivative or (positive) second derivative. But if it’s gonna be loose then the moral is don’t ask for a tight answer: stick to ensuring graphs are drawn “concave” or “convex”, and so on.
This seems to be an NHT thing, where no one cared enough about Haileybury’s twenty brats to properly proofread the exam. I don’t see than VCAA has specifically asked about concavity for a long time, if ever.