Secret Specialist Business: Exam 1 Discussion

This is our post for teachers and students to discuss Specialist Exam 1 (not online). There are also posts for Methods Exam 1, Methods Exam 2 and Specialist Exam 2.

UPDATE (29/11/2020)

We’ve finally gone through the exam, we’ve read the discussion below, and here are our thoughts.

In brief, the exam is OK but no better, and there are issues. There is some decent testing of skills, but the emphasis (as in the Methods 1 exam) appears to be on fiddly computation rather than deeper concepts. That isn’t great for a 1-hour sprint exam, and commenters have suggested the exam was overly long, but of course a 1-hour sprint exam is intrinsically insane. At a deeper level, some of the questions are contrived and aimless, which is standard, but it feels a little worse this year. And, there are screw-ups.

Here are our question-by-question thoughts:

Q1. The kind of pointless and boring mechanics question whose sole purpose is to make mechanics look bad. Part (a) asks students to compute the normal force, but to no end; the normal force is not required for the rest of the question.

Q2. An intrinsically nice question on integration by substitution, which shoots itself in the foot.

Q3. A routine and nice complex roots question.

Q4. A good inequality inequality question involving absolute values. The question is not difficult but, as commenters have suggested, it seems likely that students will do the question poorly.

Q5. A pretty nice vector resolute (projection) question, sort of a coherent version of last year’s debacle. Part (a) is contrived and flawed by having to choose the integer solution from the two roots of the quadratic; it’s not a hanging offence, but it’s the kind of oddity that would make a thoughtful writer think again.

Q6. A mess. See the comments below, and here.

Q7. An OK if (for a Specialist exam) unusual integration question involving continuity and differentiability of a “hybrid function”. The wording is clumsy, since all that is required is to demand that the function be differentiable; continuity of the function is then automatic, and the demanded continuity of the derivative is irrelevant. Sure, spelling out the continuity may simply be being nice, but including the continuity of the derivative suggests the examiners don’t really get it, or are planning a sleight of hand. We’ll see. Given the most authoritative (Methods) textbook makes a complete hash of this topic, it will be interesting to see if the examination report can get it right. We wouldn’t be betting the house on it.

Q8. An ok but ridiculously contrived volume of revolution question. Asking for the volume to be given in the form \boldsymbol{2\pi(\log_e(a) + b)} where \boldsymbol{a, b \in \mathbb R}  is needless, ill-defined and dumb.

Q9. An OK but ridiculously contrived arclength question. The introduction of the symbol \boldsymbol{s} for the arclength is gratuitous and confusing. And (reviews notes), asking for the arclength to be given in the form \boldsymbol{\log_e(m) + n\log_e(p)} where \boldsymbol{m,n, p \in \mathbb Q}  is needless, ill-defined and dumb.

30 Replies to “Secret Specialist Business: Exam 1 Discussion”

  1. OK, I’ll start…

    I liked the exam in many ways, although I did feel it was a bit easier than previous years. Some mildly challenging partial fractions and an arc-length problem that allowed students to miss a negative in the final integration, but all in all a nice test of skill.

    The appearance of arctan(\sqrt{3}) twice was unusual in a small way, to me at least.

    I do want to see if students were expected to label the y-intercept on the “sketch the graph” question. But not holding my breath for the report.

    1. I agree, RF. I liked it too. I thought it was *mostly harmless*. I liked the necessity of partial fraction decomposition (unprompted) in Q8 (furthermore it’s been a while since an irreducible quadratic appeared in that context).

      Re: The y-intercept. It shouldn’t be required on the “sketch the graph” question since it’s not asked for in the question statement:

      Question 6(c): Sketch the graph of [\displaystyle f(x) = \arctan{(3x - 6)} + \pi] on the axis provided below. Label any asymptotes with their equations and the point of inflection with its coordinates. (2 marks)

      However, I think the y-intercept will need to be shown in the ballpark of the correct location relative to the given scale (so a bit above \displaystyle -\frac{\pi}{2} + \pi = \frac{\pi}{2}) or the ‘shape mark’ might not get awarded.

      My main gripes:

      Q1(a): What (secret) form of answer will be required? Will the substitution g = 9.8 be required?

      Q6(a): How much trivial working (for 1 mark) will VCAA require to

      “Show that [the derivative of \displaystyle f(x) = \arctan{(3x - 6)} + \pi] is \displaystyle f'(x) = \frac{3}{9x^2 - 36x + 37}“. (1 mark)

      Q6(b): Not a lot of writing space to get f”(x), show f”(2) = 0 AND show change in concavity across x = 2 ….

      I also thought the exam was a bit long – it won’t surprise me if quite a few students didn’t finish it.

      I predict Q4, although simple, will be poorly done (and I’m OK with that):

      Solve the inequality \displaystyle 3 - x > \frac{1}{|x-4|}. (4 marks)

      1. Re: 6(b)

        Hi John friend!

        I wonder if VCAA do not intend for the change in concavity to be shown/tested for? The “Hence” that prefaces the question definitely threw me.

        Does the following check? If so, could this be the intended method for solution?

        Completing the square on the denominator of part (a) allows you to show that the derivative is positive for all values of x, so f is an increasing function.
        Setting f”(x) = 0 allows us to find that x = 2 is the point of maximum gradient.
        These facts together imply that there is a point of inflection at x = 2, as an increasing function with a unique maximum derivative must change concavity.
        So perhaps you were able to get away without testing/showing concavity changes about x = 2?

        1. It will depend on the intelligence of VCAA … Hopefully some of the assessors have a brain might have raised this.

        2. MyCool, your understanding of Q6 is much clearer than that of the examiners. God knows what the examiners intended for 6(b), but note MCQ 4 of the 2014 Methods 2 Exam (and similarly on the 2007 exam), which is noted here, and which is discussed in detail here.

      2. Thanks, JF. I mostly agree. It was pretty computationally heavy, which isn’t great for a sprint exam. As with Methods 1, I wonder if this was a conscious strategy, an attempt to be “nice” to students who hadn’t had as much explanation this year; if so, I think the attempt probably failed. I agree on Q4 and the Three-card Monte of Q1. Q6 is a complete mess, which I’ll address soon.

    2. I’m surprised to hear you say that you thought it was easier than previous years. I thought it was definitely harder than last year’s exam 1 (which I thought was the easiest for some time), and no easier than the other Exam 1s from the current study design (2016-2018).

    3. Thanks, RF. I wonder if the not asking for the y-intercept was an oversight. If so, I can’t imagine the VCAA would still require it.

      1. Marty, I don’t think it was an oversight. Maybe they thought that students might (wrongly) think that there’s no exact coordinate and therefore screw up 1 of the 2 marks (a small bone tossed to them in a pandemic year). After all surely \arctan(-6) + \pi is not exact. Where are the surds …?

        Or maybe they thought that requiring it would make the question worth an extra mark – a mark that couldn’t be spared.

        I think it’s disappointing that there’s little emphasis in Specialist Maths on exact values written in symbolic form (like \arctan(-6) + \pi). What I’d love to see on Exam 1 is something simple like

        Solve cot(x) = -3 for 0 < x < 2pi

        or the same but no restricted domain.

        Save the tan(x) = Sqrt[3] stuff for the rote learners in Methods.

      2. No , I don’t think it was. A few years ago (maybe pre-2016 even) there was a question involving a trig graph and the question didn’t specifically ask for x-intercepts to be labelled. The examiners report (smugly, in my opinion) commented that a lot of students wasted time finding and labelling x-intercepts that were not specifically required by the question.

    4. Thanks, RF. Having finally gotten to go through the exam, I’m less positive on the exam, although, except for one question, I’m not strongly negative. I’ll post my question-by-question thoughts today or tomorrow.

  2. Yes, I really liked Q4 but admit that I think I accidentally helped myself by quickly sketching both graphs to get a sense of (1) how many solutions and (b) whether they were positive or negative or crossed from one to the other.

    Q1a is a permanent bug-bear. To me, g is exact, but then they say g=9.8, so I’m always puzzled as to whether I’m meant to substitute (and the examiner reports don’t help)

    Q6 – I’m not sure how harshly they will require showing the change in concavity. It is kind of obvious for a tan graph (or its reflection, in this case) – maybe?

    1. Re 1a: My understanding – and hopefully someone with more credibility can confirm this – is that (1) if no form for the answer is prescribed then leaving answers in terms of g is accepted but (2) if the question asks for the answer to be written correct to one decimal place or the nearest integer, or something like that, then you must substitute g=9.8.

      1. Sure. On paper 2 I have no issue as the questions often ask for a set number of decimal places. Paper 1 I have an issue in that:

        Either they want you to substitute 9.8 for g

        Or they shouldn’t write g = 9.8 on the exam paper.

      2. Hi SRK. Your understanding is what any *sane* person would think. But you must remember that we are dealing with idiots. Take the Specialist Maths 2015 Exam 1 Question 2(a):

        The question does not prescribe a form for the answer. But the *snort* Confidential Marking Guide (a battered copy of which blew right into my face while I was exercising on a VERY windy day and then blew out of my hands after I’d read and memorised it) *EXPLICITLY* said that an answer in terms of g is NOT to be accepted. g = 9.8 had to be substituted and the answer simplified to 220 Newtons. The deliberately deceitful and untrustworthy Examination Report fails to mention this important piece of information.

        So my understanding (based on information that VCAA refuses to disclose in its Examination Reports) is that g = 9.8 must be substituted if the resulting arithmetic calculation is not ‘difficult’. To put ‘difficult’ into perspective, the answer g/13 WAS accepted for the Specialist Maths 2018 Exam 1 Question 1(b) (and this is also stated in the Examination Report). So simplifying 9.8/13 = 98/130 = 49/65 is presumably considered ‘difficult’.

        Personally, I think VCAA just make it up each year. There is no consistency and you’re potentially damned if you do and damned if you don’t each year.

        1. Hi, John. The 2015 examination report does indicate the intended answer of 220, so I’m not sure how the report is being deceitful, although the extra half-line of calculation wouldn’t have killed them.

          Of course your main point is that VCAA should be clear and consistent in whether they want g or 9.8, and the whole 2015 question is weird and a little silly. To begin, 2 marks for either 20 x 9.8 or 20 x g is pretty ludicrous. Secondly, the examination report whines about students writing the force as negative, but no positive direction has been indicated in the question. Moreover, in part (b) they explicitly ask for the “acceleration of the lift downwards”; with hindsight it is clear from the examination report that they wanted the magnitude of the acceleration, but that interpretation of the question isn’t totally obvious to me.

          There’s always conventions and tricky wording with up-down acceleration problems, but this all feels very clunky.

          1. OK. They give the answer of 220 but never explicitly say that an answer in terms of g was not accepted and lost 1 mark. How hard to make this clear. Of course, VCAA engages in this sort of deceit so that it doesn’t have to answer questions such as when it is and isn’t OK to give an answer in terms of g. Because, as I noted with the 2018 exam, suddenly it IS acceptable. This is a non-trivial issue because you don’t want a student to things they don’t have to do and hence expose themselves to increased risk of making a careless error and hence needlessly losing a mark. And yet this is exactly what VCAA seems to want.

    1. It is the subject with the highest number of enrolments amongst the subjects classified as “Mathematics” by VCAA.

      Truth is, so much of what the FM exams actually test is not really Mathematics (opinion) although it could be if:

      1. The questions weren’t so repetitive from year to year (again, opinion)
      2. Calculators and notes were not allowed for every single assessment (again, opinion)
      3. The statistics section actually asked meaningful questions requiring a minimum level of thought, unlike the “memorize (or write in your bound references) these sentences and write them at the appropriate times to get full marks for the explanation” questions which are currently seen (again, opinion)

      In short: see what Marty said in response.

      1. I like the idea of the bound reference. Used well, it can be a great benefit to the student. I usually keep notes in a bound reference when I read a non-fiction book. This certainly slows me down in reading but the process helps me to understand the work.

        Some students create good bound references. Such resources don’t need to be polished or works of art to be useful.

        Some students have no idea how to do this. I have noticed that students are not particularly good at taking notes of any sort. One might blame the ubiquitous laptop for this.

        Some students buy a bound reference from students who completed the subject in the previous year.

        I have seen professionally designed bound references for Specialist Mathematics. These are in the form of semi-notes where students can fill in the blanks.

        The real value of the bound reference lies in the preparation rather than as an aid in the examination.

          1. Once upon a time, students could take in an A4 sheet of notes annotated on both sides. Then the CAS calculator was piloted. The CAS and non-CAS (graphing calculator) students did the same Specialist exam and the Methods Exam was more or less the same too. It was pointed out that the CAS calculator could store the equivalent of 100 or so pages of notes. So it was decided that everybody could bring in a bound reference in order to even out this. The CAS calculator ultimately replaced the graphing calculator but the bound reference stayed because … well I don’t know why. Either because an dickheaded moron decided it was a good idea to retain it, or a dickheaded moron was incompetent in not thinking things through. So it’s a vestigial of the Pilot CAS calculator era.

            Apart from all its other detrimental influences, the introduction of the CAS calculator facilitated the establishment of massively over-inflated reputations of many teachers as mathematical gurus that I doubt would ever have happened otherwise.

            And now you have Mathematica … students can bring in a bound set of notes AND a USB stick storing as many nb (notebook) and pac (paclet) files as they want.

  3. On Q7, I wonder if the reason why we are told that the derivative is continuous (and not merely that it exists) is so it is obvious that m, n can be found by solving m = \frac{-8x}{(1+x^2)^2} at x = 1, rather than by considering the difference quotient. In this case if the derivative exists then it is continuous, but I can appreciate that it’s better to not require students to recognise that. (Although, more generally, I did find this an odd question for a SM exam)

    1. Thanks, SRK. Yes, I think this is probably the point of the wording. “Given that” the function has a continuous derivative, then your suggested method of matching derivatives on the left and right is legitimate. But, as I was suggesting, this amounts to sleight of hand: nothing in the Cambridge text nor, I venture, in any other VCE text, clarifies the relationship between this matching of derivatives, the question of differentiability at the problem point, and the question of continuous differentiability. There will be very few students, or teachers, who will understand why the legitimate approach is indeed legitimate.

  4. Re: Update (29/11/2020) question-by-question thoughts.

    1) Well, it’s hard to do anything inspiring with mechanics given the systematic erosion of its content over the years. And the proposed Stupid Design shows why this happened.

    5) I wonder if “having to choose the integer solution from the two roots of the quadratic” was deliberate in order to nudge students towards trying to factorise the quadratic. If so, I’m not sure it was successful – I know many students who attempted to use the quadratic formula rather than attempting a routine factorisation (probably put off by the superficial difficulty of factorising). One could argue it would have been better to cook the question better so that the quadratic factorisation was more obvious.

    6) Yes, 6(b) is pure dumb-ass wording.

    7) Re: “continuity of the function is then automatic” when a function is differentiable.

    I doubt it’s automatic in the minds of many students. So I prefer your hypothesis that “spelling out the continuity [of the function] may simply be [VCAA] being nice”, rather than the hypothesis that VCAA does not understand the statement is irrelevant. I think most students would use the continuity of the function with or without the explicit nudge – but only out of desperation to get a second equation.

    Regarding the oddness. Yes, I can think of better things to ask. Alternatively, I can think of a different function to use (to *ahem* differentiate part (a) from simply being a Methods question). As written, I don’t see the point of part (a) – it belongs on a Methods exam.

    2), 8) and 9) Yep, the whole “Give you answer in the form …” instruction is really starting to wear thin. Three questions … VCAA made the most of an opportunity to shoot itself in both feet and also the hand. I think you’re right – it’s just VCAA trying to make the marking easier. But they missed a trick to be supercilious with Q8 – only the very worst students will forget the 2pi …

    Re: 8). Yes, the function is contrived. But I’m OK with that. I’m OK with using a volume question as a vehicle for testing integration and hence the function being contrived.

    1. Thanks, John. Re (1), yes the mechanics in Specialist has been fifth-columned to death. I was planning to make this comment on the draft post at some point. Re (5), you might be correct, although it feels to me more of a question that naturally gives a two-solution quadratic, and the restriction to integer was just a clunky way to specify one solution. Re (7), yes I think it likely that including continuity was conscious spelling out, and I’m basically willing to give it a pass. Although, the in-fact reasonableness of spelling it out demonstrates the whole topic is mush theory. Re (8) and (9), I’m not usually too nitpicky on contrived functions, especially for arclength and revolution. Still these were pretty damn contrived, and they concluded an exam brimful of contrivance and aimlessness.

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