Secret 2021 Specialist Business: Exam 1 Discussion

Go for it.

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UPDATE (09/11/21)

As for Methods 1, there is not too much to say about this exam except for the final Q9, which is a mess. Commenters, who are better able to judge, have suggested the exam was too long; the exam did feel pretty fiddly in parts. Here are our question by question comments.

Q1 A simple kinematics question.

Q2 A trivial definite integral. As SRK and others have noted, 3 marks – equating to 4.5 minutes – to evaluate \boldsymbol{\int\limits_0^1\frac{2x + 1}{x^2 +1}\, {\bf d}x} is absurd.

Q3 An OK statistical hypothesis question, to the extent that’s not a contradiction in terms. As commenter John Friend noted, the modelling scenario is ridiculous.

Q4 A straight-forward volume of revolution question, which is screwed up. A minor but telling example of the examiners not knowing, or teaching, how maths works. And more than a few students will probably be cheated a mark.

Part (a) has students calculate the volume Vs of a solid resulting from rotating a region around the x-axis. Part (b) then squishes this solid by a factor k in the x direction, meaning the volume gets scaled by 1/k. That’s fine, although squishing in the y direction, resulting in the circle radii being scaled, would have been more interesting and a better test. The problem is that Part (b) asks for the new volume in terms of Vs, which is ridiculous. The point of writing Quantity B in terms of Quantity A is exactly when you don’t know Quantity A, or when Quantity A will vary. But students just calculated the fixed Quantity A, like two seconds ago. Part (b) is just a dumb question, and it seems likely that a number of students, who are smarter than the examiners, will lose their one mark.

Q5 A straight-forward but pretty grungy implicit differentiation question. Compare to Q2, worth the same marks.

Q6  A really, really stupid question. Well, yeah, these linear independence questions are always stupid, and the whole sub-topic is stupid. But including a vector \boldsymbol{\underset{\sim}{c} = 3\underset{\sim}{i} + 2\underset{\sim}{i} + \left|1-p^2\right|\underset{\sim}{k}}, and asking for which \boldsymbol{p} we get independence is a good mile stupider. Plus, as commenter Worm has noted, these exam questions have almost always been couched in terms of dependence in the past. So, yes, VCAA might have put “independent” in bold; but why pass up an opportunity to screw students of a mark and then sermonise on how students should read the question?

Q7 An ok if rather odd initial value problem. As noted by commenter SRK, there is no reason to exclude negative t in Part (b), although it was probably intended that students assume t ≥ 0. Both interpretations should be permitted, and it would be surprising (and reprehensible) if VCAA did otherwise.

Q8 An ok complex polynomial question. Part (b) asks students to solve \boldsymbol{z^2 +2\bar{z} + 2}, which is easily done by splitting \boldsymbol{z} into real and imaginary parts, but will probably be missed by many students.

Q9 Just a mess. The question involves two particles, A and B, travelling respectively according to the equations \boldsymbol{\underset{\sim}{r}(t) = (-1+4\cos t)\underset{\sim}{i} + \left(\frac2{\sqrt3}\sin t\right)\underset{\sim}{j}} and \boldsymbol{\underset{\sim}{s}(t) = (3\sec t - 1)\underset{\sim}{i} + (\tan t) \underset{\sim}{j}}, for \boldsymbol{0\leqslant t\leqslant c}. Which is fine, except we’re not told the value of \boldsymbol{c}. Why aren’t we told? Because the writers can’t write.

Part (b) asks students to show that the particles collide, and then where they collide. Of course, the first thing to do is to find out when the particles collide, which turns out to be when \boldsymbol{t=\pi/6}. Unless, of course, \boldsymbol{c<\pi/6}, in which case the particles don’t collide. Which is stupid. And wrong. So, why not have some specific choice of \boldsymbol{c\geqslant \pi/6}? Because, of course, the particles won’t continue along their paths after they collide, and setting \boldsymbol{c = \pi/6} would give the game away. Apparently no Specialist exam writer could figure a way out of this dilemma, could figure out a clear and correct way to write what they intended. So they left it stupid and wrong. Which is stupid and wrong.

The rest of the question is better, but not by much. Part (a) has students show that the path of particle A is (part of) an ellipse, and then show that the path “in the first quadrant” – i.e. all of it – can be written as \boldsymbol{y =\frac{\sqrt3}{6}\sqrt{-x^2-2x+15}}. Why would anyone care? Well, they wouldn’t, but it means we can now pretend to be interested in the function \boldsymbol{y =\frac{\sqrt3}{6}\sqrt{-x^2-2x+15}}. Which brings us to Part (c).

Part (c)(i) presents the students with the pretty gross antiderivative of \boldsymbol{\frac{\sqrt3}{6}\sqrt{-x^2-2x+15}}, and in effect asks students to verify by differentiation that it is indeed an antiderivative. Pretty painful for 3 marks; compare to Q2. And, what do we do with this antiderivative? Part (c)(ii) asks the students to calculate a chunk of area under \boldsymbol{y =\frac{\sqrt3}{6}\sqrt{-x^2-2x+15}} between a couple who-cares limits.* The area under the path of a particle that would have already collided. How impressive. How it-really-makes-you-want-to-study-maths-for-the-deep-insightish.

*) (11/02/21) As John Friend has noted below, there are infinitely many forms of the answer to (c)(ii). We are aware that one cares about this, but we’ll keep hammering it.

28 Replies to “Secret 2021 Specialist Business: Exam 1 Discussion”

  1. Did not detect any major issues – the amount of time allocated to some questions seemed too short, however (Q9). Although some questions could be done extremely quickly, eg. Q2 worth 3 marks, but could be done in 30 seconds, maybe it balances out…

    1. Q9 was a slog. The rest seemed reasonable (at first glance)

      Also, Q9 the 0 <= t <= c was strange.
      The constant c has to be less than pi/2 for it to make sense, but it’s not clear if you were meant to find or use that restriction in part b.

  2. Don’t know if it was an issue with efficiency, but I ran out of lines for the very last question. (Why were they so stingy with working space? Apologies in advance to whoever marks my paper.) Overall, wasn’t very difficult but felt incredibly tedious.

    1. Yes I agree – nice exam with good questions.
      However, the effort/working and mark allocation ratio was a bit off – some questions worth only 1 mark but required about 2 or 3 marks’ worth of working.
      Good exam, but very tedious (especially that final derivative – damn that took long).

    2. On the subject of working space (and btw I totally agree with all commentators who said there was not enough writing space) …. Why were pages 2 and 3 blank? Was that in homage to the integrity of past Managers.

      Some superficial observations:

      Q2: 3 marks. Seriously? 3 apting marks for 1 minute of work. You have got to be apting kidding me.

      Q3: The context is poorly conceived. For example, the bulb makers are apting idiots if they claim (by implication of the wording of the question) that bulbs last for more than the mean amount of time on a normal curve!! 50% of of the apting bulbs are going to last for less than the mean time, you imbeciles!! And how is the lifetime defined? Continuous use, standard normal use …? Hours is the usual unit that lifetime of light bulbs is measured in. If VCAA are going to use these dumb-ass ‘real life’ contexts, it has to accept criticism when it’s a bogus-life context. VCAA’s light bulb moment is long overdue.

      And why not 4 dp in (b)(i)??

      And suddenly Pr(-1.96 < Z < 1.96) = 0.95 (or more precisely, Pr(-2 < Z < 2) = 0.95 for Exam 1) doesn't have to be memorised!? That's actually sensible and should have been the case from the get-go. But why not alert teachers that common sense has finally prevailed after 4 years …?? Why not put the apting critical values on the Formula Sheet??

      Q4 (b): What's the point!??

      Q6: Seriously? \displaystyle |1 - p^2| … Totally gratuitous (but then again, the inclusion in the Stupid Design of in/dependence of a set of three vectors is totally gratuitous, particularly given the limited way in which it can be tested).

      Q9: (a) The path is only an ellipse if \displaystyle c \geq 2 \pi. Otherwise the path is only part of an ellipse.

      (b)(i) They don't necessarily collide. They only collide if \displaystyle c \geq \frac{\pi}{6}.

      (c)(i) – 2 marks. Q2 – 3 marks. (c)(i) – 6 minutes. Q2 – 1 minute and done. What the apt …!?

      (c)(ii) There's an infinite number of correct answers unless a and b are co-prime.

      1. John – the whole 0 \leq t \leq c thing was strange – another case of the exam writers not plotting the graphs?

        So c \le \frac{\pi}{2} for particle B not to “reach the end” of its asymptote (after some fairly impressive acceleration). And c \leq 2\pi for particle A to actually trace out a full ellipse. And c \geq \frac{\pi}{6} for the collision to occur.

        Why not just set c = \frac{\pi}{4}, which is all that is needed for all parts of the question and be done with it?

        1. Good questions, Simon. We’ll probably never know the answers. I wonder if these sorts of things ever get discussed at the Assessors Training Day – I would have thought they had relevance to the marking scheme.

          They’re the sorts of questions that those who attend Meet the Con-Artists sessions should demand answers to, with the full support of all teachers present.

      2. John, I think 9(a) is ok. I don’t think there’s it’s natural to interpret the cartesian equation as implying the particle travels along the whole of the ellipse. But, 9(b)(i) is of course ridiculous.

  3. I felt like there was not much working out room at all! Not sure if they were expecting us to complete the questions in a short and efficent way, but it seems that others felt the same as well.
    BTW, as a side note, this website/blog is pretty interesting space! I have been an observer here for around 18 months, and this is my first comment. Nice work marty and other frequent commentors 🙂

  4. Question 9 seems far too tedious for the number of marks allocated for it, where each part was worth either 1 or 2 marks. Otherwise, nothing to write home about.

  5. This isn’t an issue with the mathematics of the exam itself (I’m sure other commentators have much to say about that), but rather with the excessively constrained amount of time and working space that we are given to complete it.

    From various sources a recommended duration of an exam is two to three times the length of the time it takes for a teacher to complete it. However, my specialist and methods teachers have admitted to not being able to complete recent methods and specialist exams in the amount of time allocated nor within the lines we are given to write our answers. I think this issue would be resolved if we did it as it is done in NSW—3 hour long exams, with a script booklet. And get rid of the CAS too, for good measure.

  6. Thanks, everyone. I’m very much enjoying the rest while people hammer in my absence. A bit exhausted with Life Other than Blog, but I will try to give the exams a proper look over the week-end.

    1. I’m with you there Marty…

      My brain is struggling. I’m ready for holidays. This has been the longest year ever… I know it’s not longer than any other year… But it has been…

      I genuinely feel sorry for the kids. They have had 2 years of online learning, then they are expected to come out and perform under pressure. Some can’t even sit their exams cause of COVID cases…

      In regards to the exam… I thought it was definitely do-able..

      Finding values for linear independence is definitely different… Maybe they should have bolded it? The mark allocation was ridiculous, where I think the average will be pushed higher because the “easy” questions were worth so many marks… And the spacing? what a nightmare… I kept thinking… am I missing something? Is there something I can’t see that simplifies in a much easier way than what I am thinking?

      Anyway. Here are my solutions. Don’t be too harsh. My brain has been slightly retarded by the long year.

      P.S. what the f*ck is apting?

      1. Thanks Worm. Just a few observations, FWIW.

        Q4b. I wonder if \frac{\pi^2}{2k} will be accepted? (I don’t see why not…)

        Q5. I preferred to first divide through by e^{2y} thus avoiding using product rule. Although this makes little difference, it’s an easy computation either way.

        Q6. My students weren’t tricked / bothered by the Q asking for values which gave linear *independence*. They also mentioned that they did this using a 3×3 determinant.

        Q7b. It’s not clear to me that only positive values of t are allowed. Could the displacement function not describe the particle’s motion for all past and future times?

        Q8b. I initially wondered if this could be more nicely done by observing that the roots are conjugates, and so must also be the roots of \overline{z}^2 + 2z+2=0. But solving these eqns simultaneously just lead to the same eqns as with writing out z=x+yi.

        Q9ci. I think it was helpful to use \sqrt{-x^2-2x+15}=\sqrt{16-(x+1)^2}. There was still a good amount of algebra to slog through, but this made it easier to see in advance how everything would simplify.

        Although, the whole question (c i and ii) is rather contrived, just a consequence of substitutions like x=4\sin t - 1 not being on the course (officially speaking, since it’s widely taught anyway)

        1. SRK, I don’t see any reason to assume that \boldsymbol{\pi^2/2k} will be accepted. It is clear what the question is asking for, and what the question is asking for is stupid.

      2. Worm and SRK – thanks for the solutions and observations.

        Just a quick note that Q9c is easiest if you substitute u = \frac{x+1}{4} everywhere – then it basically reduces to the calculation required when integrating \sqrt(1-u^2) which is a standard textbook exercise [there is a good question in the Cambridge text that gets students to integrate all sorts of things of the form (1\pm x^2)^q x^r]. The first time through Q9 (squished in a busy Friday) I didn’t use the substitution but did wonder if it would work about halfway though the calculation!

        \displaystyle \begin{aligned} LHS & =\frac{d}{dx}\left( 8\arcsin\left(\frac{x+1}{4}\right) +\frac{( x+1)\sqrt{-x^{2} -2x+15}}{2}\right)\\  & =\frac{1}{4}\frac{d}{du}\left( 8\arcsin( u) +\frac{4u\times 4\sqrt{1-u^{2}}}{2}\right)\\  & =2\frac{d}{du}\left(\arcsin( u) +u\sqrt{1-u^{2}}\right)\\  & =2\left(\frac{1}{\sqrt{1-u^{2}}} +1\times \sqrt{1-u^{2}} +u\times \frac{1}{2}\frac{-2u}{\sqrt{1-u^{2}}}\right)\\  & =\frac{2}{\sqrt{1-u^{2}}}\left( 1+\left( 1-u^{2}\right) -u^{2}\right)\\  & =\frac{2}{\sqrt{1-u^{2}}}\left( 2-2u^{2}\right) =4\sqrt{1-u^{2}} =\sqrt{-x^{2} -2x+15} =RHS\quad \quad \quad \quad \quad \qed  \end{aligned}

        SRK: You’re right that Q9c parts i and ii are a bit contrived and just awkward. Having integration by parts on the course would help avoid some of the two-part integration by recognition questions.

      3. @Worm – Yes, this lot of year 12s have had a really tough 2 years. At my school they were absolutely exhausted by the beginning of term 4 & quite a few SACs had got squished up to the last moment. This meant that the prep for trial exams suffered and their performance on the trials couldn’t really be used as an indicator for the finals.

        1. It would have been nice if VCAA had grown a brain and said no Unit 4 SACs. As it was, they said nothing until it was too late – in the meantime schools had to anticipate what would be said (assuming VCAA would make the stupidest decision, which it did). VCAA finally decides to do its job and say something, far too late to be helpful.

          As an aside, if VCAA had have held its nerve and kept the original GAT date instead of blinking and postponing the GAT, we wouldn’t have had the ridiculous farce of the GAT continually be postponed. Leave it to VCAA to be sanctimonious in the good times and silent in the bad times. And to make stupid decisions in all times. If only the IB was more commercially viable and VCAA had true competition instead of, essentially, of a monopoly …

          1. My daughter is sitting the selective school test next weekend…

            It has been postponed 4 times since June…

            Imagine being a selective school trying to cater for kids and get ready for 2022… and the kids not knowing what they’ll be doing next year…

            1. I agree, the continuous postponement was absolutely awful. Felt like some kind of sick psychological mind game. More of us were stressed about the date of the GAT than the actual thing.

              But honestly, I think having a U4 SAC is actually pretty helpful for exam prep because VCAA is still testing that content. Most students I know would just half-arse learning it and never really understand those application questions.

              It’s a sort of a damned if you do and damned if you don’t situation. Term 4 was incredibly tough, had 4 SACs in the 3 days leading up to the GAT, then we were supposed to be doing practice exams and all that. We were also stressed about possibly not being able to sit exams and getting derived scores (never really cared about SAC marks throughout the year so that was worrying). Trial exams were also completed at home so they weren’t very reliable anyway. My teacher is *actually* retiring after this year (he came back in 2020 to teach because he was bored). I think he’s finally had enough.

              @Worm – Good luck to your daughter!

            2. Yes, it’s tough for the students (and schools), who have had to carry the consequences of an incompetent education system (and Federal Government) that has resembled a bunny in the headlights.

              But I don’t have a lot of sympathy for the select entry schools. The selective school test should have been run on-line. I know of some schools that did run some select entry tests on-line. Personally, I think select entry schools should have/take more autonomy in how they select their students. As times change, schools must become more flexible. Schools cannot depend on DET, VCAA, etc. to show leadership. These organisations – our fearless leaders – have FAILED all of us – over the last 2 years in particular (speak to any Principal Class Officer at any school). Loud and obnoxious in good times, weak and silent in bad times. The bad times have shone a bright spotlight on their incompetence.

          2. I should add that when VCAA finally did give it’s long over-due edict on Unit 4 SACs (towards the end of Term 3), it gave schools the option of on-line SACs or waiting until Term 4 started and to do SACs at school (there was no end in sight to the lock-down at this stage). And was in full sanctimonious-and-arrogant mode, reminding us all of its strict authentication rules etc. Most schools had already anticipated VCAA to be bastards and had run their maths SACs on-line (and what fun, that was. Particularly to mark).

            And apparently the new Mathematics Stupid Design has been approved by the VRQA (https://www.vrqa.vic.gov.au/Pages/default.aspx) – I wonder how long it will be before VCAA deigns that teachers get to see it. And I wonder if VCAA will provide timely and practical advice on how to implement all four units at the same time, particularly when VCAA has previously proclaimed that all Units 1&2 content is examinable in Units 3&4 exams (NB: Unit 3&4 students in 2023 won’t have been taught the new Units 1&2 content). (The proclamation was probably complete bullshit from a dickhead – said purely to deny that the NHT exam writers apted up by including the binomial distribution and discrete random variables on the NHT Specialist Exams – nevertheless it’s been said).

            I could say more but it would simply be a rant about an incompetent, putrid anti-teacher organisation that I wouldn’t piss on if it was on fire.

  7. Thanks to everyone who has commented. I’ve updated the post with my thoughts, pretty much just an angry summary of what others have noted.

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