Secret 2021 Specialist Business: Exam 2 Discussion

Last one. We’re keen to hear.


UPDATE (09/05/22)

We’ve now had a chance to look at the exam report. Of course blatantly lying about the screw up in 2(a) is not good, even if it’s par for VCAA’s gutless course. Our comments on this, and everything, are interlaced below, in green.

UPDATE (06/05/22)

The exam report is now available, here. (Thanks, Sai.) Next week, Godot.

We haven’t looked at it yet, but will update with out thoughts asap.

UPDATE (24/12/21)

The exam is now available, here. (Thanks, John.)

UPDATE (11/11/21)

Well, our kissy-kissy with VCAA didn’t last long. Section B has got serious problems. Here are our thoughts:

Q1 A standard and OK partial fractions and graphing question, trivialised to meaninglessness by CAS. Part (d)(i) should have specified exactly two asymptotes.

Q2 As flagged by SRK, Part (a) is just wrong. And, (b) and (c) aren’t great. See here. (09/05/22) The examination report simply lies about the screw up in (a)(i). Belatedly flagging the existence of “an alternative solution” doesn’t cut it. Be a mensch. We make a few more comments here.)

Q3 An ok volumes of revolution and rates question, made worse, of course, by being more numerical than algebraic. Part (a) is poorly worded, a consequence of the micro-cutting of the question; it makes no sense to write “a definite integral in terms of y and H”, since once we integrate, y becomes a dummy variable. It would have been preferable, and preferable anyway, to ask directly for the volume of revolution. Part (b)(ii) would have be an excellent calculus problem if it hadn’t been destroyed by CAS.

Q4 Yes, a car crash. See here. (09/05/22) The question is still a car crash, but the examination report doesn’t provide much to add.

Q5 An ok and pretty straight-forward inclined plane problem, but the scenario is clunky and the wording throughout is extraordinarily clumsy.

Q6 Stats crap. We just cannot get ourselves to care. (11/11/21) But, writing “main daily sales” instead of “mean daily sales” is pretty damn sloppy.) (12/11/21) As SRK has noted, Part (d) to Part (e) switches from a 1% to a 5% confidence level, a repetition that wastes valuable testing time, and a change that is likely to be overlooked by many students.)(18/04/22) As John Friend notes, this error has been corrected without comment in the posted version of the exam. God, these people are sleazy.(09/05/22) The report notes the main-mean error, mostly to claim that students were not disadvantaged. This cannnot excuse the error, and it certainly cannnot excuse the airbrushing of the error from the published exam.


UPDATE (10/11/21)

Section B is clearly going to be a slog, so we’ll first get the multiple choice questions out of the way.

In brief, and with respect to the stressed students newly commenting here, we liked the MCQ on this year’s exam. There were a couple clunks, but in general the questions seemed clear and good, readily solvable after a moment’s thought. Undoubtedly there was plenty of CAS gaming we didn’t see (or look for), but it was notable that we didn’t even think about Stupid CAS Tricks.

Whatever its intrinsic merits, that doesn’t mean there isn’t room to debate the length/difficulty or the unexpectedness of the MCQ as a whole. But as a decent test of mathematical knowledge and understanding, this year’s MCQ seems clearly better than any for years.

(09/05/22) Students did poorly on the MCQ, although maybe not as poorly as might be expected. Our quick calculation indicates an average score of 10.4/20, down from 11.4 in 2020 and 12.0 in 2019. That’s a notable, if hardly cataclysmic decline. Again, it doesn’t bother us. We liked the MCQ part of the exam; it was testing.)

Here are our question by question thoughts, including some quick suggestions of a (CAS-free) way to think about the questions (with no guarantees of being error-free).

MCQ1 A scary but good asymptotes question. It comes down to how many solutions cos(3x) = -3/2 cos(3x) = -2/3 has on [-π/6,π], so how many solutions cos(t) = -3/2 cos(t) = -2/3 has on [-π/2,3π]. (Corrected 13/11/21. Thanks, SRK and John.)

MCQ2 A nice domain question, coming down to bx = e and bx = 1/e.

MCQ3 A very good max-min question, coming down to determining when the square thing in the denominator is smallest, so when cos(ax) = -1.

MCQ4 A good complex algebra question, most easily done by trying z = i, but can also readily be done in general: the numerator is |z|2 and the denominator is 2*Im(z). (09/05/22) The solution in the exam report is bad. Not wrong, but bad.

MCQ5 A good complex geometry problem, which is easy if one thinks about it, um, geometrically. See Damo’s and John Friend’s comments, below.

MCQ6 A good question damaged by poor wording; writing \boldsymbol{z^2 \in R} is needlessly confusing. The question itself is easy: the hypothesis implies that z is either purely imaginary or purely real. (09/05/22) Just 42% of students answered the question correctly. This could be due to the muddy wording. (The wording of the solution in the report is equally muddy. Do you guys never read what you’ve written?) It seems more likely, however, that students are simply not sufficiently familiar with the relationship between complex algebra and complex geometry.

MCQ7 A bad question. Very easy, but way too cute and poorly worded. The answer amounts to the length of quarter way around a circle.

MCQ8 A standard Euler’s method question. Presumably pointless, since any smart student will have Euler programmed into their machine.

MCQ9 A scary but nice, and easy, inflection point question; it’s simply a question of whether f” changes sign at some point.

MCQ10 A standard, and standardly poorly worded, direction slope field question. One cannot possibly tell from a collection of unspecified dashes what the differential equation “is”.

MCQ11 A standard vector question screwed up by an idiotic inclusion of bearings.

MCQ12 A nice, and easy, dot product question. Nike method.

MCQ13 A nice, and easy, vector resolute question. Vector resolute = (scalar resolute) x (unit vector).

MCQ14 A standard and easy dynamics question.

MCQ15 A standard but nice statics question. A little calculation required, but straight-forward.

MCQ16 A very nice but difficult inclined plane problem. As John Friend notes, it’s probably better as an Exam 1 question.

MCQ17 A standard normal distribution question. It may have been preferable to have given the probabilities to ten decimal places rather than four.

MCQ18 An ok confidence interval question. In can be done quickly without CAS by doing rough calculations: the mean is about (and exactly) 73, and the population standard deviation is about 15.

MCQ19 A nice scaling of a distribution question. It can be quickly done approximately (and exactly) without CAS: the standard deviation has been scaled by 7/6, and we’re looking to go slightly above one standard deviation from the mean.

MCQ20 A standard difference of distributions question.

199 Replies to “Secret 2021 Specialist Business: Exam 2 Discussion”

  1. Seriously. wtf. Honestly that was by far the worst VCAA specialist maths exam for this study design. There was hardly enough time to do half the questions, let alone the whole exam. The standard for MC was ridicuolous. How do they expect students to complete those MC within 30 mins and have enough time for the rest of the questions. It seems that if there were just less questions, or less questions that required less working for the marks allocated (some 2and 3 mark questions took like 10 minutes), people could have actually completed some questions. The exam was like all the hard questions from all trial exams and past papers combined, so every single question was death. Individually, they were potentionally doable, just with a lot of thought, but absolutely not together. Not a great way to end my vce maths journey at all. I am extremely dissapointed that all my hard work amounted to almost nothing, and I am angry- this was completely unfair to a cohort that has already had to deal with so much. I am not expecting to get more than 20 marks on the exam. Nothing but another dissapointment from VCAA, thanks for nothing.

  2. Wish I were on that car accelerating off a cliff. VCAA can differentiate these fists when I catch whoever wrote that exam because they should not be walking the streets with a free conscience. Probability was ok, but it was like handing a bandaid to a third-degree oil burn victim.

    1. Regarding \displaystyle that car … what a farce. The writers have been watching too many Blues Brothers re-runs. These idiots at VCAA prattle on about ‘meaningful’, ‘real-life’ contexts and then we get \displaystyle that car … Seriously? Was a motorcycle \displaystyle too much real-life to handle … These idiots wouldn’t know ‘real life’ if it bit them on the arse.

      1. And clearly centre-of-mass is a foreign concept – no place for that sort of foul language in a Specialist Maths exam where the motion of an object sailing through the air is being modelled.

        And “due to a tail wind tail, the effect of air resistance is negligible”. \displaystyle Spare \displaystyle me. What a load of bollocks. If the car was moving in a straight line at a constant speed, that could be the case.

        If a little knowledge is a dangerous thing, then the exam writers are lethal weapons. Weapons of Maths Destruction.

        1. Still with the car that should be a motorcycle that should have a centre of mass that experiences no air resistance due to a miraculous tail wind …

          I’m (un)pleased to announce that VCAA’s new love affair in Specialist Maths is now official: I give you the \displaystyle smooth \displaystyle join. They joined (smoothly) last year in Exam 1 (Question 7) and I’m happy to report that the joining is still going strong – as seen in Question 4 (c). Although the context of the love affair is strained to the point of stupid in Question 4 (c). I hear various inverse functions are furious as VCAA basks in the glory of its latest little fling.

          1. Still with the car that should be a motorcycle that should have a centre of mass that experiences no air resistance due to a miraculous tail wind whose path is having a love affair with a smooth join …

            The preamble to part (d) tells us that this car accelerates from rest with an initial acceleration that is \displaystyle infinite. Wotta car! No wonder the ramp is having a passionate joining with its path.

            To solve the DE, the technique of separating the variables must be used. This requires that \displaystyle v \neq 0. And yet, the initial condition v = 0 when t = 0 and s = 0 must be used to calculate the value of the arbitrary constant of anti-differentiation. Does anyone see a way of resolving this technical infelicity? Perhaps

            \displaystyle c = \lim_{t \rightarrow 0} \left( \frac{v^2}{3} - 60 s \right) …?

            Will VCAA require the use of limits or does it conveniently ignore the v = 0 issue?

  3. I haven’t done the exam myself, but I know my tutoring students who were very well prepared (excellent marks on recent VCAA, Heffernan, NEAP, Kilbaha and MAV papers) were all distraught. I am very angry on their behalf- these students were 40+ calibre (and some 45+). Of course, everyone appears to have struggled, but that is of little comfort to a cohort of students who have been through so much and deserve better.

      1. I understand it was hard but taking into account the 2019 exam was quite difficult and the A+ there was 80%, do we really think the A+ will be that low?

  4. Hmm. This is not sounding good. I have been handballed the exam but, honestly, I have not looked at it yet. There are some very strong Specialist teachers who comment here, so I’ll be very interested to hear what they think.

      1. Yes, and I have thoughts. But, unlike VCAA and ACARA, I like to be sure of what I’m doing before I write anything. I’ll start posting on the exams tomorrow.

    1. Personally, I thought two particular questions of interest were Q2 and Q4e. Q2 looked like a cheap, trivialised cas-ified copy of a Viete expansion problem (given that they hinted P(z) = (z-z1)(z-z2)(z-z3)), while I thought 4e would be nice by itself if it wasn’t nested at the end of a horribly long question + another cas-ified blunder. The idea was to combine the previous formula given in the question, write some questions, and chug out solutions for the final simultaneous question. It could have been simple as “A car attempts to reach 20m/s at B (given this acceleration-displacement formula). The car brakes at point W with a rate of 9m/s^2, reaching zero velocity at B. What is the distance WB?”

      1. Q2 is also interesting for the reason that z1 is explicitly stated to be real whereas z2 and z3 are stated to be complex. I think there’s a clear implication by VCAA that, at least in this question, z2 and z3 are not real and hence the symbol C is reserved for non-real numbers … The only other interpretation is that z2 and/or z3 may or may not be real and VCAA is hedging its bets. This is directly related to understanding what is intended in MCQ 6, although in Q6 I can see why they want to make it clear that z^2 is real.

        1. Thanks, John. This sub-thread can but (needn’t) move over to the WitCH I just posted.

          Not counting this exam (i.e. MCQ6 and Q2), do you have any evidence of VCAA ever using C to mean non-real (i.e. non-zero imaginary part)?

          1. Marty, I’ve had a brief look – I couldn’t see anything in Exam 1 and off-hand I can’t see anything in Exam 2. So I suppose I’m using 2021 as the probably precedent.

            I’m hoping someone will send VCAA a list of ‘demands’ of major issues that the Examination Report needs to provide explicit clarity on:

            1) WHAT was expected and WHY in Question 2 (a)(i) and (a)(ii).
            (I’ve remarked previously that I anticipate a weasel-worded “Other answers were possible”. I’ve also remarked that VCAA might say that the answer to (a)(i) is Im(z2) = – Im(z3) and that they will accept both answers for (a) (ii) based on this).

            Alternatively, admission of and remorse for the error would be welcomed.

            2) An explicit statement that VCAA considers R to be a subset of C. Because Question 2 (a)(i) and (ii) sure don’t suggest this.

            3) Question 4 part (d) – How can a car have infinite initial acceleration? Admission of and remorse for this error would again be welcomed.

            I’m sure these issues have been raised via the formal VCAA exam feedback process, but I’ve found from experience that this feedback is generally ignored or claimed not to exist (and you don’t get an automated receipt and copy of your submission). But maybe a new broom sweeps clean the parting mess from the old broom.

            I wonder how many diligent students were disadvantaged (time-wise, answer-wise) by these (and other) errors?

      2. Hi Tsui M. What do you mean by “cas-ified blunder”?

        I’ve heard that a number of teachers were mewling about how the Casio failed for Question 4 part (c). Including some self-styled ‘gurus’ who perhaps should have known better than to blame the machine but maybe don’t have that capacity of understanding.

        From what I’ve seen, ALL CAS machines were capable of handling that question – IF the user understood what they needed their machine to do. It was necessary to include restrictions on theta \displaystyle AND u to get an answer. I wouldn’t have thought this was difficult to grasp. And I would have thought it was something that students would get routinely taught. But it seems that some teachers were too obsessed with a narrow minded rote-learnt button pushing recipe to think about the mathematics and try doing this.

        I would have thought doing the 2018 NHT Specialist Exam 2 Question 4 would have ironed out any wrinkles with this ‘type’ of question and allowed teachers to ‘refine’ their recipes.

        There was no CAS machine blunder, the blunder was with the user of the machine. As is nearly always the case. I would suggest that some of the teacher angst with Exam 2 was fuelled from guilt.

        So Marty, you’re off the hook. The lynch mob will come after me now.

        1. I’ve heard that some people were complaining about the deckchairs, but that was absurd. The deckchairs were perfectly fine. Sure, you had no know how to open them, but they worked fine.

  5. As someone who majorly derailed on exam 1, I was honestly really glad to get a hard exam. I felt here that I could ‘prove myself’ more easily.

    1. While I agree that it is good if the exam is hard to get an accurate picture, this was hard in the wrong way – a maths exam should not be half English comprehension test.

    2. tgh, at my early stages of going through the exam, I agree. At the risk of getting lynched on my own blog, in sum I liked the MCQ. I thought the questions were in general testing, but good and fair. That seems to me what you want.

  6. My 30 to 33 (raw) student is distraught. Her sentiments have been echoed by others. She too stated that MC was extreme; do 2 w/ confidence, do 10 by guess, wtf for the rest. How many god complexes residing at VCAA do these students have to deal with? Methods 2 was nuts enough; is there a competition for who is going to produce the craziest exam? I’m awaiting to see the exam.

  7. One of the biggest problems with this exam is that there were no easy questions, and very few medium questions. This is just going to be terrible for anyone aiming for a score that is below 35, there were no questions in that exam to separate the middle range students from the lower-aiming students. It’s honestly just going to be lucky multiple choice guessing that will separate the middle-low students from the lower ones. A terrible and luck-based exam.

    1. Your comment on the multiple choice is probably correct. I should probably be more bothered by that, but it is difficult. I liked seeing questions that required a little thought rather than CAS autopilot.

  8. Nice to see an exam that wasn’t completely trivialised by the CAS calculator. (That isn’t to say that the exam wasn’t cas heavy, but I’d say only 70% of the marks were for calculator knowledge and application as opposed to the usual 90-100%.) Only real complaints so far are the exam length, and the amount of reading required, which would have been horrible for most EAL students I presume. There probably are some major mathematical issues with the exam, but of course I was not able to scrutinise it closely during the exam itself.

  9. I think there is a typo (!) in question 6 – it says “main” where I am sure it is intended to say “mean”. Can anyone confirm if I am just reading this funnily or VCAA actually managed to include a typo in a November exam.

    1. “Main daily sales” makes no sense, so I read it to be “mean”.

      Personally (I will wait to see what others have to say before saying any more) I really felt this paper was less-nice to EAL students than past exams, and it has been a pretty low bar some years in Paper 2, Section B questions.

      1. I can’t fathom how they could use the wrong word on such a high stakes exam. Do they even vet these things at all? I’m sure most students would be able to figure out what they intended to say, but still, that’s pretty unacceptable.

        There were definitely some very wordy questions – a maths exam shouldn’t be about testing reading comprehension.

          1. Or maybe ‘Special’ Mathematics …

            The over-whelming feedback I’ve had from students is that the exam was long. Very long. Having briefly looked at it, that’s my feeling too. And I thought the multiple choice questions got a bit easier once you got through the first few, which was strange.

            So to all the commentators who sat the exam, my advice is to have some hope and feel some optimism. You might – just might – be pleasantly surprised with how things turn out. Remember, it’s all relative. Cross it off the list, walk away, and focus on the next objective (which for many of you will be Chem). Keep a cool head and nail what you know.

            And to all the trial exam vetters who continually mewl about how long the trial exam is, how hard it is, where are the ‘easy’ marks yada yada … Trial exams are meant to prepare students for the VCAA exam. Like the current one. So in future, direct your mewling to VCAA.

            I might have more to say once I work through it more carefully and find all the mistakes. And yeah, main versus mean was sloppy checking (which is what we expect from the muppet vettors).

            1. The subject “Special Mathematics” exists in Tasmania – their “equivalent” of Specialist.

              Except with much nicer exam papers.

              1. I think it’s actually called \displaystyle Mathematics \displaystyle Specialised. And they have another subject called \displaystyle Mathematics \displaystyle Methods.
                You’re right about the exams – far less bullshit and far more mathematics. VCE mathematics really has become a joke.

            2. Well, results came out a couple of days ago. I’m hoping many students (and teachers) got a pleasant surprise. My feeling is that a lot of students who struggled during the year might have done a bit better that they expected. Particularly after scaling.

  10. As a student who was probably on track for a 35 SS I genuinely feel as though I will get less than 35% on that exam. The length of the exam infused with the absolute difficulty of not only a few but every single question made the whole exam very stressful, meaning for questions I probably would’ve gotten I was too distracted by my MANY previous failures, making the whole thing a mess. Annoyed as hell but at least everyone except for a few very top students seem to feel the same way.

  11. Here’s my review of the whole exam:

    Q5: I don’t believe I’ve seen anything like it around. Although one could just ask their CAS of choice to determine \max_{t \in [0,2\pi]} (|2+\sqrt{3}i+\text{cis}(t)|) (or e^{it}).

    Q7: Shortest distance along the graph from A to B…. is that not redundant? This is awkward phrasing, I feel as though the arclength would be a simpler way for expressing it.

    Q16: Cool double angle application.

    Q9: Graph of f…why not just say a function?

    Section B:
    Q1d: This whole section seems like an apology to the NHT mishap with the function degenerating if the factors cancel.

    Q2 aii: A strange question. One can reason that z_2 = \bar{z_3}, then z_2 = -z_3 which means both numbers are real or both have real components equal to zero, and that z_2 = -3 and z_3 =3 (or also z_2 = -3i and z_3 = 3i).

    Death by related rates.

    One part show, three parts CAS smash. 4e had wording that confused me initially, but as Tsui had pointed out, was something of an interesting question. You can think of it as two experiments, one where the vehicle hits 20 by the time it covers the distance AB, and another where it slows to a stand-still. The prior run gives you the total distance, and the latter run wants you to find the point where the acceleration becomes -9, and assumes the car comes to a halt by the end.

    A somewhat involved mechanics question with slack rope and coefficient of friction. I have not looked at it in depth.

    Funny, I had a stats exam today. Would it not be better to state “failure to reject the null hypothesis”? Understandably, the double negative may throw some people off, and yes it’s been on the exams for a while but nonetheless…

    I can see why this exam can be considered long, seeing as some of the MCQ is involved, and there are some tedious section b questions. More thoughts to come.

    1. Hi Sai,

      Good comments. I think that if students were *really* fluent in using the CAS then this exam was doable – if not [or you insisted on doing things by hand and not playing the CAS game], then there were some tough questions.

      MCQ5 – this was definitely a trickier MCQ. I’ve got a few solutions to it now, but it’s not fun. Using the CAS to smash it would be a good idea, but I didn’t teach my students fMax on the TI CAS… definitely next year.

      MCQ7 – re the wording: I think they were testing if students recognise a circle in parametric form. Students who didn’t would pay the price by having to do the integral.

      ERQ1d – lol

      ERQ4d&e – I don’t think you need part d for part e, but then why is d a “show that” question? If you know the deceleration and the initial speed, you can calculate the required distance to stop = WB. We don’t need the distance from AB in order to get WB…

      ERQ5 – it was ok. But parts a, b and c were basically independent

      ERQ6 – Haven’t looked properly yet!

      1. For MCQ5, if you draw a line segment from the origin, through the centre of the circle, C, to a point P on the other side of the circle, then OP will be the maximum value of the modulus. OC=sqrt(7) and CP=1 so OP=sqrt(7)+1. If you take any other point on the circle Q, the angle OQP will be bigger than 90, so OP will be longer than OQ.

        1. Thanks, Damo. I was about to suggest the same ‘solution’. Considered geometrically, the question is trivial. But students who try to impose a ‘CAS solution’ onto it rather than treating it as a mathematics problem will waste a lot of time, probably get nowhere and end up guessing an answer. And it certainly helps to have seen one during the year …

          These sorts of questions (including “Find the value of z that has the largest Argument, smallest Argument, minimum modulus) were common when \displaystyle regions of the complex plane were part of the subsets topic. You will find them some trial exams, old exams and in ‘Checkpoints’. Nowadays, most textbooks devote only a couple of pages to subsets of the complex plane (Cambridge devotes 3 pages) and there are not many questions in the exercises (to its credit, Cambridge has one page of excellent questions plus an extended response question that contains a part (Q16 b) in the same spirit as MCQ5).

          Students \displaystyle should have seen questions like this during the year. But I’ll bet most teachers spent no more than one lesson on it (as opposed to treating it as a separate topic and spending perhaps 4 or more lessons on it).

          I’ll defend VCAA including this question. The blame for its ‘unfamiliarity’ lies with teachers who spend too little time on ‘subsets of the complex plane’. I anticipate more time will be spent on it next year …

          @Sai: If you think back, surely you might recall seeing something very similar to MCQ5 … Certainly I’d be surprised if there was not a similar question set for homework.

          To me, MCQ 5 is indicative of two broader issues:
          1) The experience of the Specialist Maths teacher, and
          2) The quality of the textbook used.

          1. Ah…you’re right! Q29 from the technology free questions in Ch 4 of Cambridge as well as Q16b have a similar premise. And yes, both of them were questions set for homework. How time flies.

          2. In fact, I’ll give a free plug to the MAV: Its 2021 MAV Specialist Maths Trial Exam 2 had a question (Section B Question 5 (b)) that was the same as the VCAA MCQ 5 … And I believe Glen was asking about the proof of the geometric solution – a screenshot is attached from the solutions (the proof is trivial).

            1. Hi John,

              That proof is deceptive. It is using the theorem that

                  \[|x + y| \le |x| + |y|\,,\quad\text{where}\quad x,y\in\mathbb{R}^2,\]

              something known as the *triangle inequality*. This is an important and fundamental piece of mathematics.

              Anyway, my point in making these comments was not that the question is difficult, or that the proof is hard, but that it relies on mathematics that I didn’t know was actually taught in high school. It is more of a question than an accusation.


              1. Glen, the triangle inequality (which I can guarantee you John knows) may be deep, but the inequality in triangles is trivial. This is a multiple choice question, and Damo’s/John’s geometric argument is fine and good for getting this problem straight in the school context.

                1. The “inequality in triangles” is the same thing as the triangle inequality.

                  With that out of the way…

                  I don’t doubt that John knows it. I doubt that it is covered in school, but I don’t know for sure, thus the question. I don’t mind what argument is used “in the school context”, my point was simply to ask if this kind of content is covered and if it is therefore a fair question to ask on the exam.

                    1. Oh, because “triangles” don’t need to have vertices in the plane? They can exist in an abstract sense? That’s what you’re going for? Is Euclid’s geometry even within a ten mile radius of the current curriculum? Is this the same Marty that says it is OK to say R \subset C?

                      The setting for this question *is* points in a plane…

                      So if they are *not* the same thing, then you *can’t* apply the “inequality in triangles”. If they *are* the same thing, then you are talking about the triangle inequality.

                      You could say that the triangle inequality is a corollary of an abstract “inequality in triangles”. But to do that you’d have to actually show that the plane satisfies all of the hypotheses of your abstract setting. Then, you will have the triangle inequality. The triangle inequality is applied to the actual question at hand.

                      All of which means that, no matter how they get there, they still need to know the triangle inequality to use the proof idea that John posted.

                      This is all completely irrelevant to the merit of the question….

                    2. Hi Glen. The geometric approach can be justified as discussed above (there are usually vector and complex questions in textbooks that ask students to prove the triangle inequality and related inequalities). But a similar geometric argument for finding the minimum modulus is less obvious … So I’ve attached a simple proof that came my way for why z with minimum AND maximum moduli are located at points P such that P, the origin and the centre of the circle are collinear.

                      This is also giving a free plug for a mavellous 2022 trial Specialist exam that comes with solutions with a significant point of difference to other trial exam solutions – background theory and teaching points are included and discussed. Demand no less from all trial exams you consider buying!

                      PS – I can recommend a nice little book called ‘An Introduction to Inequalities’ by Edwin Beckenbach and Richard Bellman:

                  1. No, because The Triangle Inequality is a formal theorem (or axiom) in pure mathematics. It is unnecessary and distracting from Damo’s/John’s clear argument for solving the given MCQ.

                    1. What are we talking about? I’m talking about what I wrote above, which is that for points x, y in the plane we have |x + y| \le |x| + |y|. Their argument uses this. Call it something else if you want, to me that’s the triangle inequality.

                  2. I think the knowledge could be covered in topics Vectors and Complex numbers. Actually, Exercise 4B Question 6 in Cambridge is about it. Similar questions appeared in 2021 Specialist NHT MCQ11.
                    Some trial exams had similar questions.

                    Does anyone think there is a problem in 2021 Exam 2 Section B Q3Biii? I was wondering how the water was added to the vessel, at a constant rate, or the same rate as leaking, any varying rate?

                    1. Thanks, Anonymous. I think B(3)(b)(iii) is just supposed to amount calculating the leak rate at its largest, when the vessel is full. But maybe I misunderstood you, or the question, or both.

      2. Re: MCQ 7. Yes, points A and B both lie on a circle and so there are two distances “along the graph from A to B”: the length of the major arc and the length of the minor arc. So the phrase “shortest distance along the graph from A to B” is necessary for a unique answer.

    2. Actually, now that I think about it, MCQ5 was not too bad as there is a simple geometric approach – I think I’ve seen something similar in an old VCAA or trial exam somewhere.

      And I just noticed that Damo posted the same solution (even used the same C and P as me!) Thanks Damo – your explanation is nice and clear.

    3. On 2a ii: even 2a i is off – since both z_2 and z_3 could have zero imaginary part (for all that’s been said so far), what sense is there in asking for *the* relationship between those two roots?

      I guess one could write down the relationship: IF they have non-zero imaginary part … but otherwise …

      1. Re: Q2(a)(i). Indeed. If z2 and z3 are real then there’s no relationship between them. That’s why I think VCAA attach a different meaning to C depending on the context. In this context VCAA clearly intend C to mean non-real numbers. Otherwise the question makes no sense. But do they …?

        There’s actually a contradiction of meanings within the question itself! The first usage of \displaystyle z \in C clearly allows \displaystyle z_1 \in C. But the second usage clearly does not.

        1. John, that is just muddying things. First things first: ignoring the current exam, is there any evidence of VCAA having ever used C to mean non-real complex number?

          1. Actually Question 2 (a) (i) has a ‘perfectly good answer’ – the relationship between them is that Im(z2) = – Im(z3) …

    4. Thanks, Sai. Brief replies on the MCQ.

      MCQ5, see Damo and John’s approach below. A very good example of how CAS poisons everything.

      MCQ7 is overly cute and is a bad question, but see John’s comment below.

      MCQ9, yeah the “the graph of …” stuff is unnecessary and annoying, but also pretty standard for VCE. The properties are considered attached to the pictures rather than the functions. Long live Magritte!

      MCQ16, yes a very nice question, but very difficult for a 90-second MCQ.

      1. In fact, it’s a nice question that gets trivialised by Mathematica (if you understand what your doing). See attached screenshot. Gone in 60 Seconds. It would be a nice question for Exam 1.

    5. Question 5 includes coefficient of friction by stealth.

      Coefficient of friction should never have been deleted from the Specialist sillibus (but that’s another story) – including it by stealth is the consequence of this moronic decision.

      Just like the hypergeometric distribution was deleted from the Methods sillibus some years ago but continues to get used (particularly in sampling) by stealth. It’s interesting to note what has been deleted from the sillibus over the years yet still appears by stealth on exam questions.

  12. Is the notion of “implied domain” something students are trained to understand? What is the definition of “implied domain”? Is it the same as “maximal domain”?

    I’m wondering this because of Q2.

    1. The two expressions are often used interchangeably.

      There is also a domain “feature” an the TI CAS machines which would render this question pretty useless at testing much more than button-pushing.

        1. Yep. Picture attached as proof. Longest part was typing the word “domain” with the keypad because there isn’t a shortcut…

  13. Q5 is strange. Is this kind of question in the syllabus? Finding the maximum of an associated function evaluated along an implicitly defined subset of the complex plane?

    1. No, I don’t think in general it is part of the syllabus (as that would require things like Lagrange multipliers). However, for the circle it can be reasoned using high school geometry the way Glen described it…

      Questions involving geometry with rays, circles and points on the complex plane are part of the syllabus – it used to include hyperbolas and ellipses too.

    2. Glen, see earlier comments. I find myself in the strange and unfamiliar position of defending VCAA – the question is fair. The question is ‘trivial’ and tests some nice mathematics. Blame for its ‘unfamiliarity’ lies with teachers who spend too little time on ‘subsets of the complex plane’.

      1. Interesting position, I can see where you’re coming from, but do you not worry about proving the intuitively obvious geometric solution, or “solution by picture”? Such a thing still needs some mathematics and still needs to be taught…. unless students just take things on faith (should not be the case in a mathematics class).

        1. There is no time for idealism in VCE Mathematics…

          JF is correct, in that there are some typical “types” of questions about regions of the complex plane that come up again and again in VCAA, although there is some level of effort made to make questions look different, there is a surprising level of familiarity.

          The really good students will ask why (when we study complex numbers in Term 1, not when they are preparing for the exams) and a lot of other nearly-as-good students will learn the tricks, practice them and do well on these questions without too much thought.

        2. Glen, I don’t ask my students to take a leap of faith (for something like this, anyway). The proof is elegant and simple to explain to students. The same sort of question appeared on the 2021 MAV Trial Exam 2. So, to save some effort, I’ve simply attached a screenshot from the solutions to support my statement.

  14. A genuine question. If z is an element of the complex numbers and the imaginary part of z is equal to 0, can we then state that z is an element of the real numbers (MC6 and ER2)? I would have thought there was a difference between the imaginary part being equal to 0 and there being no imaginary part: {0} is not the null set. I’m not sure if this is an important distinction or if I’m being precious…

    1. Hi Damo — you are correct. There is a “real axis”, let’s call it A, and it is defined to be

          \[A = \{ z \in C\,:\, \Im z = 0\}\,.\]

      This is a set of complex numbers whose imaginary part vanishes, and it is correct to say that A \subset C.

      The *real numbers* R on the other hand are different. Later in your mathematics education (apologies for the assumption) you may learn about embeddings, and there is a particularly nice embedding (sometimes called an identification) \iota : R \rightarrow A which takes a real number from R and maps it to its corresponding point on the real axis A. Once we understand how this works, then we can sometimes drop the \iota and pretend as though R and A are the same thing. However, we should only do so with the understanding that there is an embedding working in the background that makes all of this rigorous.

      No, you’re not being precious. Your brain is working, and has noticed a logical inconsistency! You should be congratulated for it.

      1. Thanks Glen, that’s what I thought. I actually completed my Maths degree over 20 years ago and have been submerged in High School Maths ever since. While I can still spot things that don’t feel right, I don’t necessarily have confidence in my conclusions, so it’s good to tap in to the more learned folk on here. One of the benefits of this site, for me, is that it will often scratch something that I haven’t thought about for years – when I get really motivated, I’ll pull out old texts and delve into it a bit more detail. For teachers who read this site, I believe that this prompt into thinking about maths more deeply is about the best PD you can get.

      2. Ah, so we actually mean to say there’s a subset of \mathbb{C} that’s isomorphic to \mathbb{R}? Or more generally, a homomorphism from the bigger set to the smaller set?

        1. Hi Sai — it depends on what setting you are working in. For example, if you are viewing Euclidean space \mathbb{R}^n and the complex numbers \mathbb{C}^n as groups, then homomorphism is the right idea. If you want to do calculus, then I would suggest viewing them as differentiable manifolds, in which case the correct notion is diffeomorphism. But that all comes much later.

  15. I’m concerned about these “arg” questions in the exam. How is the “arg” covered in the curriculum? Are the students taught that

        \[r e^{it} \ne r e^{i (t + 2\pi)}\]

    ? Are they taught that if

        \[z = w\]


        \[Arg(z) = Arg(w)\]

    ? While arg(z) = arg(w) may not follow?

    Question 4 has “Arg” and question 6 has “arg”, are they taught that Arg is the principal argument? It looks like it would be in (-\pi,\pi].

    What does “z^2” in Q6 mean? Is it normal abuse of notation to define “w\in R” to mean “\Im w = 0“? Because C and R are distinct and it is not correct to write R \subset C.

    What are polar coordinates for these students? Because if we have principal arguments in our treatment of complex arithmetic, then polar coordinates and algebra on C occurs with respect to this choice. For instance, the polar form of z is r e^{it} where t \in (-\pi,\pi]. Then, in Q6, we have z^2 = r^2 e^{i s} where s is “2t \mod (-\pi,\pi]” (hopefully this notation is clear, the way we normally do this is by taking a quotient, but that notation probably won’t be helpful here). If the imaginary part vanishes, that means s =0 or s=\pi, which then mean t = 0 or t = \pm\pi/2. In other words, if we do arithmetic where each complex number has one polar form, then none of the responses are correct.

    I know what they want students to do here. My point is that I don’t think it is good for the student’s mathematical education to mix and match different approaches to complex numbers within the syllabus, let alone a few questions apart in an exam. Similarly with Q5…. anyway.

    1. Hi Glen.

      The distinction between arg and Arg, that is, argument and principal argument, is not explicitly made in the Mathematics Stupid Design. Nor is a domain convention for the principal argument explicitly stated. However, it would be a \displaystyle very negligent Specialist Maths teacher that did not teach the distinction and did not explicitly state the domain convention for Arg that VCAA assumes (as well as saying that other conventions are in common usage).

      I’ll add that past, present and future Stupid Designs have no emphasis on the \displaystyle construction of a number system in which there is a number \displaystyle i with the property that \displaystyle i^2 = −1 – this is a damning indictment on VCAA.

      But now I’ll put a target on myself and say that I consider \displaystyle R \subset C. We consider natural numbers to be a subset of integers, we consider integers to be a subset of rational numbers, we consider rational numbers to be a subset of real numbers. I see no value in NOT considering real numbers to be a subset of complex numbers.

      1. That’s concerning. So when you talk about the equation

            \[x^4 = 2,\]

        and you ask students to solve it, you’re not thinking of 2 (and x) as a real number? Or is 2 going to be complex here? Or is it somehow is “everything all at once”? Or maybe students should just guess that because we used an “x” instead of a “z” or “n” that it will be real? That sounds so much harder than just saying that here, we are working in the real numbers, so we can do things like take positive and negative roots of non-negative reals (which 2 here denotes). Or saying that here we are working in the natural numbers, etc etc.

        The main problem is that setting up the space in which we are working is a fundamental task and something that often needs to be tailored to the problem at hand. This step becomes more and more important as the student’s mathematical education develops. While it isn’t of huge value to give lessons about this early on in primary or secondary, it is definitely of value to be consistent and set a good example…

        The secondary problem is that it just isn’t correct. The same symbol is used for many things, which is definitely confusing for students, but it doesn’t have to be. Confusing, that is… it definitely should be the same symbol.

        I suspect you are just trolling me!

        1. I’m not trolling. (But I thought you might be!)

          I agree that there’s a lot that gets assumed, usually within a reasonable context. For examples:

          1) If I asked a student to solve \displaystyle x^4 = 1 (let’s keep it simple) in Methods, I probably wouldn’t have the preamble “Find all real solutions to”. The context assures me that the student understands this (complex numbers are not taught in Methods, x is conventionally used to represent real numbers etc.) There’s no confusion.

          2) If I asked a student to solve \displaystyle z^4 = 1 (let’s keep it simple) in Specialist, I probably would add that z was complex. But if I didn’t, the context would still assure me that the student (should!) understands this (complex numbers are taught in Specialist, z is conventionally used to represent complex numbers etc.)

          3) When I solve a quadratic equation in methods, I always talk about the discriminant being less than zero meaning that there are no REAL solutions. I never say there’s no solutions.

          There’s a fine line between using precise language that’s helpful and precise language that’s not helpful. If I taught Methods or Specialist with the rigour that a third year real (or complex analysis) class was taught, I’d never cover the sillibus, and what I did cover would be confusing for 90% of the students. (Don’t get me started on the mixture of ability in classes).

          I understand your points and in a utopian world you’d get no argument from me. But teachers live in a real, dystopian world and are forced to make judgement calls on what to compromise on a daily basis. And if you assume that the sillibus from earlier years has been taught and taught well, you’re in for a very nasty shock when teaching VCE mathematics. And the last 2 years has made this problem much worse.

          I think we’re gonna agree to disagree on this. And that’s OK.

    2. Someone correct me if I’m wrong.
      Re: MCQ 6 (for those without an exam) …

      If \displaystyle z \in C, \displaystyle z \neq 0 and \displaystyle z^2 \in R, then possible values of \displaystyle \arg{(z)} are

      A. \displaystyle \frac{k \pi}{2}, \displaystyle k \in Z

      B. \displaystyle k \pi, \displaystyle k \in Z

      C. \displaystyle \frac{(2k+1) \pi}{2}, \displaystyle k \in Z

      D. \displaystyle \frac{(4k+1) \pi}{2}, \displaystyle k \in Z

      E. \displaystyle \frac{4k-1) \pi}{2}, \displaystyle k \in Z

      If \displaystyle z^2 \in R and \displaystyle z is complex (and so implied to be non-real) then \displaystyle z^2 < 0 and so
      \displaystyle z=bi , where \displaystyle b is real. That is, \displaystyle z is purely imaginary.

      So options C, D and E all give \displaystyle possible values of \displaystyle \arg{(z)} … (my emphasis). Can anyone see why, on the basis of the wording of the question, options D and E would be rejected? Three correct answers?

      (A is rejected because if \displaystyle k is even then \displaystyle z is real).

      1. I ran into this issue as well, though I disregarded it since I was rather tired when I looked at it. I believe D and E are subsets of C, but it seems as though the meaning was bungled when they said *possible*, because they are all valid answers.

      2. JF: this relates to points that Damo & Glen discuss above, but I’ve always interpreted VCAA’s use of “z \in \mathbb{C}” to mean that z could be of the form a + 0i, where a \in \mathbb{R}.

        1. Yes, of course it includes that possibility. The issue is muddied by VCAA’s needless introduction of the “element of R” confusion.

      3. I think that the only way this question makes sense is if we accept that the Real numbers are a subset of the Complex numbers (which I think is wrong). If we make that assumption, you can’t then conclude that, because z is complex, it must be non-real. It can be both real and complex.

        However, this then means that all of the options give possible values of arg(z). So everyone gets the mark?

        1. Then the correct answer is option F: ALL of the above.

          In fact, VCAA will have a weasel-worded statement in the Examination Report:
          There will be an implication that z is non-real and that C is the correct option since options D and E are subsets of C.

          I’d ask for clarification on all this, but the way I hear it, some fellow contacted VCAA about the error in 2019 Specialist Maths Question 12 and was subsequently blamed for delaying the publication of the 2019 Exam 2 Report whilst the question was investigated.

        2. No, Damo, this is getting too cute. The question is poorly worded, in two distinct ways, but in the context of VCE mathematics the Reals are a subset of the Complexes.

          1. Then they aren’t really talking about the reals, but rather the real axis in the complex plane. So much for being correct.

            Anyway, that’s not the main problem with this question….

            1. Glen, this is high school. In high school the reals are a subset of the complexes, just as the rationals are a subset of the reals. Yes, they worded the question stupidly, but the question is clear enough, and I think fine.

              1. OK let’s keep disagreeing.

                I am not asking for much, just an acknowledgement in the appropriate class (where complex numbers are taught) that the real axis in the complex plane is not the exact same thing as the real numbers that they learnt before. This can then proceed with a statement like “ yes, it is not be correct to say that R \subset C. That said, we can often manipulate members of the real axis as if they were real numbers, so we can often think of the real axis as being the real numbers.”

                It doesn’t take much time and sets out an important mindset for later on.

      4. Prediction: option A is listed as the correct answer.

        The examiners report may then say something along the lines of “if z is real then, by definition, z is a complex number with Re(z) equal to zero…”

        Of course, if A is marked as correct then every option gives a possible set of values.

        1. I see we have a contest, RF. It adds just that little bit more excitement and anticipation. But I’m not sure I can wait until 2024 when the Report is likely to be published (I thought 2020 Reports would be published sooner as a result of a new broom, but was wrong. Hopefully there’s a 12 month lag in the sweeping)

          1. Ah yes, you are correct, with or without arithmetic on the complex plane with the principal argument setting, we don’t end up with zero correct solutions. We actually end up with multiple correct solutions! Argh, I thought I had read this one carefully. I have not answered a question correctly that I actually thought about. It tricked me.


            1. Glen, you’ve raised an interesting and important issue: What happens to the student that’s smarter than the exam writer? The answer is that the student gets disadvantaged because the student sees what the writer doesn’t and gets penalised for this.

          2. Just in case you read this… the report is out (6/May/2021) and the report does list A as the correct answer with a small sentence, just as I predicted.

            So… I win.

            Not that winning in this case fills me with anything but more questions!

            1. As I wrote above, the question is poorly worded but intrinsically fine. The correct answer is A.

              1. All good – I was having a friendly (pun intended) poke at JF since he said “we have a contest”.

                1. I can’t even remember what the contest was! All I know is that with this Report, every teacher loses. But I’m hoping that VCAA will be dragged kicking and screaming to admit that the current Report attempts to cover up multiple defects in Exam 2 questions. The maths never lies (unlike VCAA).

                  The standard we walk past is the standard we accept. None of us should walk past what VCAA has done in this Report.

      5. Very good point, John. They should have worded it “the possible values …”. I think the intended answer, and the answer to the extractible question, is A. But, as written, all answers are correct.

        1. The question *is* written as “the possible values” – I suspect JF just omitted that in his haste to post the comment.

            1. No the question is not fine. But the writer should be fined.

              It should have been worded “…, then \displaystyle all \displaystyle of the possible values of arg(z) are” (my emphasis).

              1. This is nitpicking. The “all” is clearly implied by the “the”. Worse, is the “element of R” stuff, but the meaning of the question is clear.

                1. This one now seems like a grey area. Is it clear that the “all” is implied by the “the”? I can see that. But I can also see a student saying to themselves:

                  “But if they meant ‘all of the possible values’ then why did they only say ‘the possible values’?”

                  Not a good question either way.

                  1. The question clearly could have been worded more carefully. The element of reals thing is ugly and dumb. But I think the question is basically fine. And I think you guys are wielding a double-edged sword.

                    I am the last person to defend VCAA. They are consistently arrogant and inept, particularly in the structure of and the wording of exam questions. But one of the issues with the exam wording is that the writers are so paranoid about misinterpretation, they tie themselves in knots and refuse to write plain, clear English. They don’t believe that plain English is permitted. They’ve never heard of Gowers.

                    Now some of this is due to the writers being inept and/or lazy. Thee floating c in the final question of Specialist 1, for example, is simply due to incompetence. But writing clear and unambiguous exam questions is difficult. It’s not helped if people bash them with contrived readings.

  16. Q10…. again, this is not a direction field, it is a slope field. Not only this, but the magnitude of the vector field at each point is not indicated, making it completely impossible to answer the question as asked. (The issue is with the word “the” that indicates uniqueness.)

    1. You’re right, it is a slope (gradient) field. However, assuming that one of the options actually generated the field, then the word “the” is justified as you can (I think) eliminate all but one.

      1. Glen, I agree that it’s a slope field not a direction field. I think VCAA has made this error several times. But I disagree that the question is impossible to answer: It’s easy to eliminate all but one option.

        The slopes are negative when y = 0 and x > 0 therefore options A, B and E are rejected.

        So it’s a choice between options C and D.
        Substitute (1, 0.5) into C: the slope is zero. This is inconsistent with the given slope field so option C is rejected.

        The correct answer is option D. Does anyone see anything inconsistent between option D and the given slope field?

        1. No no no, my point is not that we can’t work out which option the exam setters want us to choose. Yes, we can work out what option they want us to pick.

          My point is that we can’t rule out that the picture given is only the result of option D. There could be (and I would say, certainly are) many other vector fields that give the same (or at least to us indistinguishable) picture.

          Little Jimmy picks up some marbles. The marbles are either blue or red.

          “Miss, can you guess what colour the marble is in my pocket?”





          “Oh then what colour is the marble in your pocket?”

          “Gotcha miss, I picked up TWO marbles!”

        2. John, this seems to be deliberately missing Glen’s point, and we’ve had this debate before. Being able to decipher what VCAA means in order to answer the question does not imply the question is well-formed. Glen’s objection to the question is valid.

          1. Fair enough.
            Questions like this should be worded “\displaystyle A \displaystyle possible differential equation that has the diagram above as its \displaystyle slope field is” (my emphasis).

            1. Yeah. A lot can be saved with correct wording. It’s funny, the curriculum continually hammers the point that words are important in mathematics, but then they get it wrong themselves when it really matters.

  17. I honestly thought the entire point of exams was to distinguish which kids have worked hard and studied throughout the year from those who rely on their base intelligence, so that universities can see which students are committed to their studies. The wankers who created the spec paper have completely lost the plot because that exam was impossible to prepare for as no other vcaa papers from the study design matched its level of difficulty. The paper did not benefit the students who have laboured all year, rather those who are mathematic prodigies. I’m just extremely sad that an entire two years worth of work has come down to how many mc answers I was able to guess correctly.

    1. It sucks, and I’m sorry. One thing that you might take heart in is that plenty of “math prodigies” would be confused by it as well.

    2. You have a good point. Whatever the intrinsic merits of the paper, and at this stage I’m still sorting out my thoughts, if the style of paper was not reasonably expected, that is a big issue. But then there is a very important question, probably worth its own post, or book.

      The current VCE mathematics culture is unmitigated garbage. How, then, can one transform the current culture to a sane culture? Of course there’s no chance of that happening, since the number of teachers and students who are aware of the awfulness is small and smallering. But it is interesting to ponder what a benevolent dictator might do.

  18. This one might be my fault, but MCQ 11 annoys me: why “south 30° west”? Does anyone really bother with bearings any more? I certainly don’t.

      1. Not so sure about that. It kind of appears around Year 10 but, thankfully, disappears after that. SRK is correct: it probably won’t put off that many students, but it was stupid wording.

      2. Yeah, it made me think about what “south 30° west” means – does it mean 30° west from the direction south, or 30° south from the direction west? Honestly, I had to google just to confirm. This is mean for students who know how to resolve vectors but aren’t familiar with (or have forgotten) bearings.

        As for it being in the curriculum, it’s explicitly mentioned in the Year 11 syllabus. For Year 12, there’s a dot point on “standard contexts for the application of vectors to the motion of a particle and geometric problems” which perhaps could be stretched to include bearings.

          1. If only VCAA could find its ball-bearings, that is, its marbles. Because they got lost a long time ago.

  19. Thanks to everyone who has commented so far. I’ve vaguely been listening while getting the Exam 1s out of the way. Given the anger, I’ll look at this exam before Methods 2. I plan to post on the multiple choice questions in the morning and then on the extended questions later in the day.

  20. Hi,

    A third party has had a go at the MCQ on paper 2

    If it is legit it may have some value for those who have yet to see the questions

    Steve R

    1. OMG I think my brain just imploded. Is this what students are supposed to do to complete this exam?!? Translate questions into computer code??? What on earth is this garbage masquerading for mathematics!

      1. Glen, that is what we Victorians refer to as “VCE”.

        The vast majority of VCE maths teachers’ policy is: shoot CAS at it first and think about it second or, usually, never. There are exceptions, a number of whom comment on this blog, but we here live in a State of lunacy.

        The CAS-first policy is pretty much demanded by the appalling exams and the mandated appallingness of the SACs. It is all garbage, from beginning to end, and it poisons almost everyone. The brightest, could-become-a-mathematician students, from the strongest schools, typically present me this mindless CAS idiocy as if it were mathematics.

        I’m still in the early stages of going though Exam 2, only quickly through the MCQ so far. What is notable to me, however, is that the majority of MCQ can be handled quickly and easily with a little bit of thought. This is definitely not the norm, and I wonder if some (the majority?) of the discomfort/anger with the exam is that it was a mathematics exam, and that the teachers and the students were, understandably, not expecting that.

        1. Marty, I think that’s right. Previous MCQ sections have been mostly about knowing how to use your CAS efficiently to answer a question in fewer than 90 seconds. As for the ways in which the questions on this exam can be “handled quickly and easily with a little bit of thought” – I suspect teachers have taught their students these methods, but students will only resort to those after wrangling with the CAS for a minute or so, and then time pressure really kicks in. Good examples of this (to my mind) are: Q1, Q3, Q7, Q9, Q12, Q13.

          Notwithstanding other issues with specific questions, the big problem with this one was the length: too much reading, stingy mark allocations.

        2. I’m so sorry to hear that. I thought that there were special “CAS” exams, I didn’t know that it permeated so deeply to completely distort and disfigure what mathematics is.

          1. Well, of the two exams, one of them is “CAS active” and the other is not.

            But the CAS active exam is worth DOUBLE the amount of marks in the non-CAS exam. Furthermore, ALL internal assessment (one third of the overall subject mark) is (meant to be) CAS active. Overall, 78% of the subject’s mark is from CAS active assessment.

  21. SRK, I suspect you are generally wrong about teachers having “taught their students these methods” (i.e. to think), in any reasonable sense of “taught”.

    1. “It may have been preferable to have given the probabilities to ten decimal places rather than four.” I enjoyed this one. Thanks for the update.

  22. Re: Section B.
    Thanks for your thoughts, Marty. Yeah, one-night stands often turn out like that … (so I hear).

    Re: Question 1 (a).
    Such an easy mark to lose if VCAA only accepts \displaystyle 2 + \frac{-5x+11}{(x-1)(x+2)} and NOT \displaystyle 2 - \frac{5x-11}{(x-1)(x+2)}
    And it’s a black day for Specialist Maths if this question is intended to scaffold part (b) (which it was) …
    In fact, it’s a black day for Specialist Maths if parts (a) and (b) are intended to scaffold part (c) (which they are) …

  23. Another complaint about Section B Question 6:

    Part d. asks for the smallest sample mean such that the null hypothesis \mu = 60000 is rejected at 1% significance.

    Part e. asks for the probability that a null hypothesis \mu = 60000 is accepted, given that the actual mean is 63000. BUT now we are testing at 5% significance level.

    1) Mean and tricky, since many students will overlook the change in significance level, and use their answer for d. to calculate their answer for e. That’s 2 marks gone, presumably.

    2) Even if students catch the change, what’s the point in requiring students to recalculate the smallest sample mean such that the null hypothesis is rejected? They’ve already shown they can do that in part d.

    1. God, I hate this crap.

      Thanks, SRK. I’ll add a note, but I want to make sure I understand what (e) is asking. In the preamble to (e) we are told that “mean daily sales is now 63,000”. What, then, is being tested in (e)?

      1. Sorry Marty if I’m being dense (I assume you know how VCAA expects the question to be answered), but I’m not quite sure what you are asking? I would say that what is *actually* being tested in (e) is whether a student realises that they are supposed to continue assuming a null hypothesis of \mu = 60000 even though the question says “the mean daily sales is now \mu = 630000“.

        1. No, SRK, it’s not you being dense. Just think of me as one of those students who really, really doesn’t want to listen. I put as little time as possible into thinking about the VCE stats, and I honestly don’t know how to interpret (e).

          Is it the idea that the mean has *in fact* been raised to 63,000 but that the company/advertisers don’t know that? So, is (e) then asking what is the probability of (incorrectly) accepting the same null hypothesis, even though the mean has increased to the stated level?

              1. 1) Yes, this question has been asked before – although always with students being able to use their answer to the previous part (ie. the significance level has not changed). That in particular is what bugged me about this one.

                2) I think the wording of the question is sufficiently clear, for what they are trying to ask. But in my experience, students find it very difficult to wrap their heads around the situation being described, even if they get how to answer the question. Perhaps I am doing a poor job explaining it, but I also think students think “how come I know that the mean is now 63000 but the person doing the statistical test doesn’t?”

                While I’ve no education / experience in probability / statistics beyond what I learned in high school, I think I can appreciate the importance of understanding the chance that a statistical test might return a certain kind of error. But I’m not convinced this is the right way to go about assessing that understanding, given what is in the study design. Seems very convoluted.

                1. Thanks very much, SRK. I get your point about the change in confidence level.

                  As for the wording, if it is reasonably standard then that’s fine. Or fine enough. In reality, the wording sucks: it is obvious that one should make clear and to distinguish between what is true and known by the testers, and what is true but unknown by the testers. It’s not that hard to write clearly. It’s just no one cares.

                  As for the topic, it is appalling, and the ABS should be ashamed for unleashing this corrupting horror on us.

                  1. I agree. There’s a few things happening in (e) – none of them good.

                    1) The switch in the level of significance is unnecessary and and is dirty trick. The question is actually asking for the probability of making a Type 2 error (more on this below), so the point of the question is not be trivialised by staying with the 1% level of significance. It’s a pointless bait-and-switch.

                    2) The question is asking for the probability of making a Type 2 error, although it can’t use that succinct language because Type 2 error is not on the Stupid Design (I wonder if it will be included in the new Stupid Design). It’s a defect of the question that it does not make it clear that the new mean (63,000) is UNknown by whoever is doing the new test.

                    3) There’s a fair bit to do in this question, certainly more than is suggested by 2 marks:
                    (i) The critical value \displaystyle \overline{x}^{*} at the 5% level of significance has to be calculated using the mean (under H0) of 60,000.
                    (ii) The probability that the sample mean is less than \displaystyle \overline{x}^{*} using the ‘true’ mean of 63,000 has to be calculated.

                    And of course, it has to be realised that this is what the question is asking.

                    The question would be a lot clearer IF the Stupid Design explicitly included Type 2 errors. Then there would be an appropriate language for asking this sort of question (which has appeared on Exam 2 most years). It’s worth noting that this sort of question, when asked in previous years, first prompted students to calculate the critical value …

                    Having said this, there have been enough questions of this type on past exams that a prepared student should be able to do it. And a competent teacher will have done an example in class (because it is such a common exam question). The key words that students should have been taught to recognise are “probability null hypothesis incorrectly accepted”.

                    I think Exam 2 has caught out a lot of teachers who have been allowed to take a minimalist approach to what they explicitly teach and have emphasised button pushing on a CAS.

                    By the way, I’ve heard reports of a lot of teachers lamenting not being able to do Question 4 (b) on the Crapio … (Apparently it’s fine for the Insipid and Mathematicrap).
                    a) I wonder if there actually IS a machine problem.
                    b) I wonder if the very similar Question 4 on the NHT 2018 Exam 2 caused similar consternation.
                    c) I wonder if students are being taught Plan B button pushing.
                    d) I wonder if there’s been an over-reliance on Plan A button pushing (and button pushing in general) that’s caught some people with their pants down.

                    The question to ask VCAA is “Is Exam 2 vetted using all common CAS – Insipid, Crapio, Mathematicrap, Spewlett-Hackard?? If not, WHY NOT?!

                    (By the way, every school had many opportunities to adopt and use Mathematicrap rather than continue using the hand-held s%!t. But most declined. And now they mewl about alleged ‘shortcomings’ of the hand-held s%!t … I have very little sympathy)

  24. I’m a Yr 12 Specialist teacher and I’m not here to comment on the maths of Exam 2 because I can’t face doing it just yet. What I’m so upset about was how this exam affected my students, and anecdotally students across the state. My class were so upset and demoralised coming out – students across a spectrum of abilities but who have worked hard at maths across two awful years, with no concessions of any kind from VCAA. They basically came out feeling crap and like they were crap at maths (including my potential raw 40+ student). VCAA gave them nothing at any point and then slammed the door behind them on the way out. I’m left feeling like I have taken a group of students who were interested and in some cases actually passionate about maths, I’ve nurtured that interest through junior school and built them up, and then I’ve got them into my senior class and basically broken them. I know with the grade distribution it will all actually be ‘fine’ in the wash up but what a grim way to end their high school maths education. Ironically I failed my Unit 4 audit because my SAC was not accessible and equitable (which means not open ended enough apparently) – but how was that exam equitable and accessible? My lower-mid range students were wiped out, so rather than being proud of themselves for sticking with a challenging subject and seeing it through they feel like the whole experience was a waste of time. So much for caring about the mental well being of young people in an unprecedented global pandemic….

    1. Thanks very much, Amber. I find this issue very tricky, for lots of reasons.

      I have read the students’ angry comments above, and I very much appreciate the students, and you, for sharing their point of view. I am also the first one to kick VCAA in the balls when it is warranted, which is pretty much always. But I have problems with some, although definitely not all, of the criticisms of the exam.

      As I’ve indicated in the post updates, my view is:

      a) The MCQ section was generally difficult, but not *that* difficult, and for the right reasons. The questions were mostly good and they tested mathematics.

      b) Section B was generally difficult for all the wrong reasons, containing plenty of inexcusable nonsense, and errors.

      Now, even accepting the quality of Section A, of course that doesn’t mean the exam cannot be criticised. There is Section B, and there is the pandemic, and there is VCAA continually screwing up their response to the pandemic. And there is a natural expectation of continuity: who could have guessed that Section A was going to test mathematics?

      I understand that students are upset, and I understand that the exam may have worked poorly for the mid and lower students. These are important aspects. But the underlying fact is that Specialist Mathematics has been a very bad subject, very badly examined. The syllabus is appalling, the textbooks are appalling, the CAS is appalling, the SACs are appalling, the exams are appalling. The subject is largely ritual, and it is a woefully misleading, inadequate and inept subject for introducing keen and/or intelligent students to the nature of proper mathematical thought.

      This year’s Section A was the first sign for a very very long time of some genuine mathematical culture in VCE mathematics. Any of the much valid criticism of the VCAA and this year’s Specialist Exam 2 that don’t acknowledge this is ignoring a bigger picture.

    2. Hi Amber.

      Here’s my opinion: The one primary goal of an exam is to test the learning outcomes of a given curriculum, or part of a curriculum. In order to do this properly it absolutely must have three things: (1) be clear and precise; (2) be correct; and (3) be aligned to the learning outcomes.

      This exam has failed in many questions on each of (1), (2) and (3) above, and quite rightly should be condemned for it. You might observe us disagreeing about details, and Marty certainly has his own take on the MCQ part (which he has already explained) and that some of us may or may not agree with. Personally, I disagree with Marty because of the concerns I’ve raised about the MCQ here already and also point (3) above.

      Mathematics is beautiful, mathematics is fun, mathematics should be clear and accessible. I’m really sorry that your students have suffered under this big bad wolf that calls itself mathematics.

      1. Thanks, Glen. Which specific MCQ do you object to, and why? Also, how have you gained an understanding of the “learning outcomes” (sheesh) for Specialist Mathematics, and how do you see the MCQ conflicting with those “learning outcomes”?

        1. For the learning outcomes aspect, I am looking at the evidence here on this blog. Specifically posts like Amber’s above.

          I don’t like MCQ 2, 4, 5, 6, 7, 10, 11 for instance. I think 16 could have a word or two on its assumptions (depending on how students are taught). I spent a bit of time looking through the questions but I did not do them in detail. I probably missed some other confusing aspects of these questions as well.

          Disclaimer: I am CAS-ignorant and only know about Victorian High School Math from what I’ve read here and on linked pages, so you might wish to ignore what I’m saying on that basis, in which case, I don’t mind at all.

          1. Student expectation is not the same thing as “learning outcomes”. The former is important but it is to be managed, not to be dictated by. As for the MCQ you have noted, any reasonable objections to them, and I’d have to be convinced on some, is almost entirely irrelevant to this debate. As a whole, the wording of the questions is no worse than in previous years.

          2. Hi Glen.

            With respect, have you read through the Stupid Design? Unless you have, any discussion on ‘learning outcomes’ and whether questions do or don’t conform to them is moot:

            Click to access 2016MathematicsSD.pdf

            Enjoy. You’ll need to look at Maths Methods Units 1 – 4 and Specialist Units 1 – 2 because its all pre-requisite for Specialist 3 – 4. And make sure to read the sections called Outcomes.

            I’d also recommend looking at the exams from 2016 – :


            to join the dots from what you see in the Stupid Design.

            If you can walk away afterwards and still be \displaystyle compos \displaystyle mentis, you’re one step ahead of most of us! This is the world VCE teachers are forced to live in. Be glad you have the NSW syllabus. (There was a time, long ago, when the NSW curriculum was crap compared to Victoria – I worked with students from both states and could always spot the one from NSW).

            I don’t think any of the MCQ you’ve mentioned fail to link to the Stupid Design, which is not to say that I don’t think the wording of some of them is poor or the question is stupid (Question 6 and Question 7, for example). But there’s a distinction between poor and stupid wording and being outside of the scope of the course.

            1. I’m not saying all of the questions I flagged fail to link to learning outcomes. I can see how it reads like that, but I was only talking about (3) in the first two and last sentences of my comment.

              Marty, I am well aware that student expectations are not the same as learning outcomes. But Amber is not a student. Anyway, I’ve been upfront about my evidence and I have my disclaimer there. I really don’t care at all if you dismiss my argument on the basis that I haven’t gone to the primary source material that John has helpfully linked to justify my point.

              Also. Since when does “no worse than usual” work as a defence for anything? Especially a mathematics exam. If the “usual” is “poor”, and the next exam is “poor”, I’m still not going to like it. Yes there are degrees of awfulness, but bad is still bad. At least in my opinion.

              1. Christ, Glen. I’m not defending the awfulness of Specialist but you’re ignoring the points. You introduced “learning outcomes”, which was absurd. You criticised a bunch of MCQ, with plenty of inaccuracy, which was absurd. And you’re ignoring why students hated the MCQ, which has absolutely nothing to do with your (and my) objections to some of the MCQ.

                The students hated the MCQ because it tested mathematics in a way they did not expect and for which they were not prepared. One can argue the reasonableness and fairness of that, but not the nature of the fundamental issue.

    3. Amber, I know the following will sound harsh. It’s not directed at you. It’s my opinion:

      I think a lot of teachers have got by on a minimalist approach to what they explicitly teach. I think a lot of teachers have got by on teaching their students what buttons to press for what type of question. I think this exam is a wake up call for how the subject is (or should be) taught. I think this exam has caught a lot of teachers with their pants down.

      Students have to face life’s brutal reality – hard work doesn’t always get (or seem to get) you the result you wanted or ‘deserve’. That’s life. Sometimes all you can do is console yourself with the simple fact that you did your best. Look back and have no regrets. If you can look back and say “I gave it my best shot, there was nothing more I could do” then that’s a personal victory above and beyond what the scoreboard shows. Despite bleeding heart efforts to the contrary (and I say bleeding hearts because they’re setting students up for failure), not everyone can be above average and not everyone wins a prize.

      I empathise with you and your students. But Specialist Maths is (or should be) a demanding and challenging subject. It’s not (or shouldn’t be) a button pushing subject. Students should be going into this subject with their eyes wide open. Under no misapprehension that hard work will guarantee success. I say to students that I can’t guarantee success if they work hard, but I can guarantee failure if they don’t.

      The whole VCAA system is cruel. The fact that some students are guaranteed to fail (raw score less than 25) in EVERY subject simply because every subject is marked on a normal curve with a mean of 30 and a standard deviation of 7 is abominable and diabolical. Even if only the very brightest students did Specialist Maths, half will still get a raw study score less than 30. Maybe VCE results should be absolute rather that relative …?

      If students pigheadedly refuse to believe that “with the grade distribution it will all actually be ‘fine’ in the wash up” then there’s nothing for it except for them to stew in their own juices. Something I constantly say to students is that it’s impossible for them to know what their result will be, regardless of how hard or how easy they think the exam is. And I constantly reinforce this fact after SACs. I try to teach my students to accept failure, learn from it, and move it. We do our students no favours by constantly letting them succeed. We must also let them fail. Otherwise, the first time they ‘fail’ will be in the exams.

      The sad reality is that numbers in Specialist Maths are declining (and numbers will probably be even lower next year because of the word-of-mouth bad press this exam got). Some schools, particularly schools in rural Victoria, struggle to justify having a Specialist Maths class. There are several reasons for this. But a consequence is that some students doing Specialist Maths shouldn’t be doing it. They’ve been convinced to do it so that there’s a critical number of students doing it that justifies the class going ahead.

      If an exam is unfair for everyone, then it’s fair for everyone … No-one knows what their result will be until they get it.

      1. John – I think you’re making a bunch of assumptions in there that might not be correct.

        First, you a right that “success” for students should be more viewed as making the right decisions and carrying out a study plan with effort and discipline – not on the final result (study scores) which depends on factors you can’t control.

        Your characterization of a “raw score less than 25 = fail” is wrong – and you pretty much said as much below that statement. However, as the study score is the only number reported back to students by VCAA, without guidance it’s reasonable for them to focus on it. The Outcome statements in (all?) VCE subjects give no real definition of what is a pass or fail. This is probably on purpose and many schools consequentially almost never actually fail a student = give them an N.

        The study score and scaling system is probably the best you can do to fairly provide a single rank for students’ academic performance. But that leaves aside the issues about whether that is a good thing to want; and the lack of other usable scores to judge the learning of a student in a subject.

        Most teachers I’ve met who teach specialist care about the maths and do not just teach button pressing. They take the same pragmatic approach that you do – try to balance learning mathematics with success in the final exam. John, you have worked hard to obtain your mathematical knowledge and to gain your unique position of teaching almost exclusively VCE methods and specialist maths. Not everyone can have the same focus.

          1. I’m always overly optimistic about people and motivations, so you and John might be correct 🤷🏽‍♂️ but I hope not

    4. @Amber: Nothing you’ve said is wrong – this cohort did get it tough and that exam was a rude ending. The student comments at the beginning were basically right – individually the questions were not too bad, but trying to do them all under exam conditions and timing is not easy.

      I don’t think that many of the non-student commenters here actually sat down and tried to the whole exam in exam conditions. Feel free to tell me otherwise, everyone! How long did it take and what do you think your mark was? My effort in writing up solutions (not the same as doing the exam) was broken up over the evening – but it took me over two hours to get through it all and I made a few arithmetic errors and went the wrong direction on things MCQ5 and Q4e [still kicking myself re MCQ5].

      Unit 4 is tough to be audited in – the tension between what VCAA wants and the need to prepare students for exam-style questions is huge. I got audited in Unit 3 and so based my SAC around the MAV curvature one. Most students did not enjoy it and struggled – but at least it gave them a chance to think about what it means to succeed in this subject. We talked about seeing the maths in a context vs answering questions – and about scaling & study scores etc. I hope they had a realistic expectation of what to expect in the exams and about how their efforts translate into scores. They weren’t happy after the exam, but they also didn’t seem defeated…

      1. Simon, your final paragraph hits the nail on the head. I think there’s too little of that intent happening in the Specialist and Methods classrooms. To the detriment of the students.

        And yes, Unit 4 is a tough audit because there are two SACs that have to ‘pass’. Then again, any VCAA audit is tough – because of the pedantic justifying-my-existence attitude of the sanctimonious know-all auditor. You can fail an audit because you include writing lines for a question. Seriously. I have copies of SACs written by some of these auditors – it would be a lot of fun to publicly dissect those error-ridden, poorly written and formatted SACs line by line and then wonder why the writers have a right to judge and fail others.

        The whole SAC system is a complete farce, as has been blogged and commented on:

        SACs of Shit

        Content Warning: MA 15+. The above blog (and its comments) contains strong language and violence. J(N)F recommends viewing by a mature audience.

        1. John – re the SAC system, yes. As for the auditors – I don’t know them. But I do remember the pedanticness you experienced around the weighting of outcomes a while back – forcing the marks/SAC to unnaturally conform.

          To be honest, I don’t object to the intent of what I think VCAA is trying to do. Having students being able to explore and apply mathematics in a less directed way – moving past basic skill questions and faux applications. The current system and constraints make achieving this not ideal though.

          This year, I wrote a SAC on kinematics (we hadn’t quite done dynamics on time) that I was quite happy with. I shared it with Marty, and am happy to send you a copy for you to dissect 🙂

      2. Once I went to a meeting with the examiners in Methods – I was a pre-service teacher at the time. We were told that to find the inverse of a function, the first step is to interchange x and y. I admitted that I did not start this way, and asked “Would I lose marks?” The examiner said “Probably.” This was a turning point for me.

        1. Terry, did you propose how \displaystyle you would start finding the inverse of a function? If so, what was the reaction of the ‘examiner’ to your proposal?

          This is the whole problem. Garbage like the “interchange x and y” statement gets made in Examination Reports, espoused in pre-service courses, propagates and becomes gospel.

          Education Ministers mewl and bleat about standards, convene ‘expert’ groups to find answers to raising standards etc. The solution is simple: Stop garbage like “interchange x and y” being espoused.

          By the way … Marty, I disagree with the description “poorly functioning robots”. Such a description abrogates the personal responsibility of examiners. Including their \displaystyle choice whether or not to propagate garbage.

          1. I did not propose to the examiner how I did the problem. At the same time, I was helping my grandson with his VCE mathematics. When it came to finding the inverse function, I told him that the first thing to do is to interchange x and y. He asked “Why would you do that?” Excellent question.

            (I am about to steal the ball.)

            It’s nearly the end of the year for my Year 8 students. We are playing games – today, they had Sudoku – they all knew how this works. I watched them knuckle down to 70 minutes of thinking. It was wonderful. The sheer joy of thinking. At the end of the lesson one student said “I have never been so extended.”

            (You can have the ball back again.)

            1. Hi Terry.
              Re: I did not propose to the examiner how I did the problem.

              You should have. Opportunities like this are rare. The ensuing discussion (assuming the Examiner and the lecturer allowed a discussion …) would have been more valuable for your fellow students then a Semester’s worth of cagle from the lecturer.

              The pity is that there are teachers out there who chant “interchange x and y” and have no idea what it means (or doesn’t mean). (“But that’s just what you do …”) And this sort of meaningless ritualistic crap gets reinforced in the Examination Reports (and by Assessors who visit pre-service teaching courses as guest speakers).

              On a related note, and at the definite risk of being off-topic, I’ve been thinking about the utter stupidity of people such as yourself receiving NO credit for their prior teaching experience. The rigidity in the so-called Master of Teaching is utterly insane. I know people who have left the higher education sector to do the Master of Education and pursue a career in secondary school teaching who will NEVER recommend this to their former colleagues. The Education Faculties are incredibly petty and officious in failing to recognise teaching experience. The question of how to attract quality teaching applicants from industry has a simple answer: Education Faculties show more flexibility. I find it ironic (and extremely hypocritical) that these Faculties mewl and bleat about catering for individual differences, differentiated teaching etc etc and yet do the exact opposite in their Master of Education degrees. If I was a cynic I’d think it was all about the money.

            2. Here is something salient:

              The first sentence is vey surprising:
              “The conclusion of today’s Centre for Independent Studies report on initial teacher education (ITE) is that there is need for reform in teacher education courses.”

              Another article of interest:

  25. I’ll say it again: any criticism of Exam 2 and VCAA, valid or otherwise, that doesn’t acknowledge the significant improvement in mathematical sense of the MCQ section is ignoring a bigger picture.

    Oh, and the MAV curvature SAC is a crime against humanity.

    1. I’m not familiar with the MAV Curvature SAC. I don’t look at commercial SACs at all, I find it easier to write my own than to spend time editing someone elses.

      Re: Dissection of kinematics SAC. That’s generous, but I wouldn’t have the time. I barely have the time to dissect my own SACs and solutions (which I always do after students have sat them and have given me feedback).

      Re: Pedantry. I will confidently claim that every single teacher that’s been audited will have a story of genuine pedantry.

    1. re Q2:
      I do wonder if the alternative answer of Im(z2) = -Im(z3) was accepted.
      Scratch that, what I interpret as “Where working was correct and complete across Questions 2ai. and 2aii., these answers were accepted.” seems to be acknowledging (?) that it was possible to determine that z could be real. Huh, but then why not list both potential relationships in the first part?

      1. I wonder how VCAA reconciles its sole official answer to Q2(a)(i) with the possible solutions \displaystyle z_2 = 3 and \displaystyle z_3 = -3 (and vice versa) in part (a)(ii)?? Then again, maybe the new Maths Mangler honestly thinks that \displaystyle \overline{3} = -3.
        VCAA is so wrong it’s not even wrong.

        And I see that VCAA supports dividing by zero, according to the Report comments for Q4(d).

        New Maths Mangler, same old weasel-worded, deceitful and misleading Examination Reports.

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