Secret 2022 Specialist Business: Exam 2 Discussion

And, you’re done (with VCE maths). We hope it went well.

UPDATE (10/11/22)

Well, that was a surprising surprise. After the first three exams, we were starting to mull over a “VCAA has improved” post. Yes, the first three exams contained a few minor but still annoying errors, generally absurd modelling and plenty of CAS crap and very bad wording, but overall the exams were significantly more coherent and less marred by error than previous years’. So, although Specialist Exam 2 was always a greater danger than Exam 1, we were expecting more of the improved same. But, no.

The Specialist Exam 2 is a mess. Apart from the serious errors noted by commenters, many other questions are very bad. A number of questions are just way, way too cute, and some of the wording is appalling. More so than the other exams, Specialist Exam 2 feels as it were a rushed, last minute submission, that even VCAA’s traditional semidemi-vetting had not been applied.

Thanks to all the commenters. Here are our question by question thoughts. As with the previous exams, we won’t comment on poor wording unless it amounts to error, or something close.

MCQ 1. A good absolute value question. Very easy by hand, but presumably a number of students will waste time using CAS.

MCQ 2. A nice trig question. It should be routine by hand, but for current students is probably not.

MCQ 3. An OK asymptotes question. The possibility of cancelling is clear enough that the question seems fair (and is a repeat of such a question, from a previous year).

MCQ 4. As discussed in the comments and here, purely and simply and, after last year, astonishingly wrong.

MCQ 5. A very easy complex argument question, although students seem to find these difficult.

MCQ 6. A nice complex graph question, although it somewhat repeats MCQ 5. A fair amount of work unless you’re quick on complex geometry (which students are not).

MCQ 7. An OK integration by substitution question, although very easy just by looking at the limits and noting the integrand is positive.

MCQ 8. A typically annoying slope field (not “direction field”) question. Forcing students to decide whether a crappy little segment has slope a bit greater or a bit less than 1 is stupid and nasty.

MCQ 9. A very weird inverse Euler’s method question. Too cute, and if you’re going to do cute, don’t do it with not-at-all-cute decimals. A bad question.

MCQ 10. A very bad implicit differentiation question. Not well structured and over-egged. It’s absurd to require work or a graph to determine the effect of the ± in the expression -1 ± √11.

MCQ 11. A routine linear dependence question.

MCQ 12. A nice dot product and trig question.

MCQ 13. A trivial vector kinematics question.

MCQ 14. A very nice but too subtle SUVAT question. Would be much, much better asked purely algebraically as a short answer question (and probably better in Specialist Year 11).

MCQ 15. A too clever inclined plane question. Yes, the answer is immediate using F = ma, but a knee-jerk assumption that the acceleration is along the plane makes one of the distractors unfairly distracting. We’re betting plenty of students will answer C, and we think such students will be right to feel tricked.

MCQ 16. An easy triangle of forces question.

MCQ 17. Another SUVAT question? What’s the point? Much easier than MCQ 14, and so at least the two questions should have been interchanged.

MCQ 18. Stats crap. I don’t do stats crap. Especially to four meaningless decimal places.

MCQ 19. As commenters have noted, the question is stuffed, and pointless. Obtaining a confidence interval requires a sample mean, not the population mean that has been provided. (10/11/22 We decided this question is worth its own WitCH, here.)

MCQ 20. In principle a nice question, but it doesn’t work. Students should not be required to read an essay before answering a multiple choice question.

Q1. An aimless but OK graphing question. Part (d), with the absolute value and three intersection points, is too cute.

Q2. A very nice complex numbers question.

Q3. A very bad differential equations question. Mostly it is absurd but routine, but the wording for (d) and (e) is so clumsy and opaque as to be seriously misleading. Why has the particle “passed through” O at t = 0? Just have the damn thing begin its thoroughly implausible motion at t = 0. And why then have the second particle begin two seconds after the first? You’ve given the second particle’s position at any time t, so why say anything else? Why refer in part (e) to “when” the particles are at the same distance from O, given you’ve already told students in part (d) to confirm the explicit time at which this occurs?

In principle, part (e) is a nice chain rule application, but, as with MCQ 14, such a question would be much better set in a purely algebraic setting. As it is, any niceness has been swamped by the setting and the appalling wording. Also, and amusingly, the very specific required form of the answer to part (e) suggests the writer reads this blog. They’ve missed the point of the criticism, however: the point is not to (correctly) ask for answers in a unique form, the point is to (re)establish a culture where a reasonable form of the final answer can simply be presumed to be part of the question. But in any case, there’s way, way more to worry about in Q3 than the final form of any answer.

Q4. Similar to Q3, this question is routine, just a little silly. Part (a) explicitly uses degrees and implicitly uses radians; it probably won’t confuse anyone too much, but it’s not great. The big problem, as commenters have noted, is that Part (d) is very badly ambiguous. In asking “how far does the ball travel”, it seems likely the question is referring to arc length, but it is also reasonable to interpret it as referring to the straight-line distance from starting point to finish point. We’re betting more than a few students will have done this, and they should be eligible for full marks (and thereby penalising those students who spent more time computing the more difficult arc length).

Q5. An OK and easy forces question. A lot of SUVAT, given we’ve already had two multiple choice SUVAT questions.

Q6. Stats crap. I don’t do stats crap. But as commenters have noted, and see here, part (e) (f) is stuffed stats crap.

77 Replies to “Secret 2022 Specialist Business: Exam 2 Discussion”

  1. student here:
    Am i going insane or does q4 of the multis have no correct answers?
    For those who do not have the paper, the question gave a polynomial p(z) = (z-a)(z-b)(z-c)
    where a,b,c e C and Re(a), Re(b), Re(c) =/= 0 and Im(b) = 0 and when expanded all of the coefficients are real
    and asked which was necessarily true out of a few options and I am 99% sure the answer they wanted was |a|=|c| which is not true if a and c are purely real which has not been excluded.
    Maybe I am misremembering or this was intentional and one of the other answers was correct but I spent a good 5+ minutes checking them and none of them seemed correct and they were all fairly simple expressions so I don’t think anything particularly quirky was happening.

    Other than that though it was a very easy paper and I think all of the other questions were fairly reasonable and clearly (for vcaa) worded.

      1. A a+c=0
        B |a|=|c|
        C a-c=0
        D |a|=|b|
        E a+b+c=0

        Also, the question said necessarily correct. Same with MCQ15 (on which I have an opinion but need to finish the paper first…)

        1. The question stem is:

          The polynomial \displaystyle p(z) = (z - a)(z - b)(z - c) has complex roots a, b and c, where \displaystyle \text{Re}(a) \neq 0, \displaystyle \text{Re}(b) \neq 0, \displaystyle \text{Re}(c) \neq 0 and \displaystyle \text{Im}(b) = 0. When expanded, the polynomial is a cubic with real coefficient.
          Which one of the following statements is necessarily true?
          See RF post above.

          Some random observations:

          1) The logical meaning of “necessarily true” is that the statement is always true. So I don’t know why the writers couldn’t simply ask which of the statements was always true.

          2) Those familiar with Vieta’s formula for a cubic equation (which I think should be taught in all Specialist Maths classes) will know that Option E cannot be correct.

          3) \displaystyle \text{Re}(b) \neq 0 and \displaystyle \text{Im}(b) = 0 tells us that b is a non-zero real number. And since the coefficients of the cubic are real, we now know that a and c must be a complex conjugate pair (*) So \displaystyle a = \alpha + i \beta and \displaystyle c = \alpha - i \beta (or are they … *)
          So options A and C might be true sometimes but not \displaystyle always. And D might be true sometimes but not \displaystyle always.

          4) \displaystyle a = \alpha + i \beta and \displaystyle c = \alpha - i \beta. Therefore option B appears to always be true (*).
          The correct answer (apparently) is Option B … (*)

          * Spoiler alert: OR a and c are both real! In which case Option B is not \displaystyle always true. So there is no correct answer.
          By the questions own admission, real numbers are complex numbers: b is given to be a complex number and Im(b) = 0 is given therefore b is real. Therefore VCAA has admitted that real numbers are complex.

          1. Thank you for the response and providing the stem John but I still don’t understand why this is *always* correct since there is no condition specifying that Im(a) or Im(c) are non-zero.
            Unless I am missing something a=1, b=2, c=3 fulfills all of the criteria in the stem of the question but |a|=\=|c| so B cannot be necessarily true.

              1. all vcaa has to do is state that A and C were elements of C and B was an element of R, would make a lot more sense, i understand why vcaa tried to make it confusing with unique conditions like re(A)=0, but they quite literally unraveled themselves, although, i do think that some initiative can be taken for questions like these, this question of clearly focused on the properties of conjeguate numbers, so diving the boundary value of zero for the imaginary part of A and C would be kind of pointless, idk

            1. The question may have wanted to assess a student’s understanding of the conjugate root theorem. That should lead to a =\bar{c}, but their conditions weren’t specific, leading to the scenario you described. WiTCH 74 covers a similar issue (if not near identical) where a quadratic could also have the conditions you described, that is Im(a)=Im(c)=0. Hence, the error.

          2. As a teacher the fact they said b was complex, but then stated it was purely real threw me off, and is misleading. If they stated a and c were complex and b was real it would’ve avoided much of the ambiguity in the question.

        2. But no “necessarily correct” for Q5. Just a plain old simple ” … which one of the following is correct?”

      2. I will do my best but take this with a grain of salt (and the order I am putting them in might be wrong):
        A) a+c=0
        B) |a|=|c|
        C) a+b=0
        D) |a|=|b|
        E) a+b+c=0
        A and C I don’t really remember but I think they were both just sums of two of the roots equals 0 which I am pretty sure means nothing.
        Hopefully a copy of the paper is floating around soon to check.

    1. Yep, I believe it was explicity stated that the other roots were complex in the question stem as similar issues have arisen from purely real solutions in previous exams

  2. Hey wouldn’t a and c be conjugates due to real coefficients and we know that b is the real solution, so and c could be conjugate complex solutions. Also you yourself wrote that a,b,c e C, so a and c cannot be purely real and are conjugates of each other
    So their magnitude must be the same
    a + bi and a-bi have the same magnitude

    1. I don’t see why a real number isn’t also a complex number.
      e.g. Re(0) = 0 but we still consider 0 a real number just as Im(2) = 0 but it is still a complex number (specifically 2+0i).
      I get they wanted you to use conjugate-root theorem but I think that is not “necessarily true” like the question wanted.
      It really isn’t a big issue but I spent far too long trying to figure out the trick to that question to not complain on the internet about it.

      1. herrapose, you are correct. There is no correct option. See my initial comment above.
        The given information allows us to conclude that either a and c are a complex conjugate pair OR that a and c are both real. The latter possibility means that Option B is not always true. Therefore there are no options that are always true.

        I hope you didn’t waste too much time on it. But even if you did, don’t worry because in due course VCAA and it’s ‘psychometric’ analysis will undoubtedly be able to re-assure you that neither you (or anyone else) were disadvantaged.

      1. All the writer(s) needed to say was
        ” .. has complex roots a, b and c where a and c are non-real and b is real.”

        It’s unbelievable that the (alleged) vettor(s) has allowed the same error to be made two years in row!

        1. Do you think they meant to say Im(a) and Im(c) rather than Re(a) and Re(c)?

          If so, the question almost makes sense…

          1. RF, I think you’ve nailed it.

            I’m willing to go out on a limb and say that the question \displaystyle does make sense with that change. But it’s still a lot more complicated than the change I earlier suggested.

          2. I agree with you both. A ridiculously convoluted way of trying to say it, but i think that must be what was intended.

  3. In BQ4, it is really stupid and confusing to mix degrees and radians.
    In BQ4d, is it asking for arc length or Euclidean distance on the plane?

    1. Distance travelled/covered was the phrase used, so arc length.

      If they wanted shortest path, they would say displacement.

      1. No, the exact phrase used was “How far does the ball travel during the first four seconds after passing through \displaystyle O? I agree with Tungsten that there is ambiguity.

  4. I think VCAA “implies” that a and c are not real by stating only Im(b)=0. However, as OP pointed out, stating Im(b)=0 does not mean Im(a) and Im(c) =/=0 there for B is not true if both a and c are real. Ironically VCAA wants to emphasise the “Logic and proof” in their new SD.

    As long as it is not a hot mess as last year exam, I am happy. Will need to see the paper tomorrow for more thoughts.

  5. I have seen some discussion on the final question,

    As to whether the variance in the volume of the liquid is greater than or less than 25.

    Some students have used,
    Var(liquid)= Var(Total)-Var(can)
    And others,
    Var(Liquid)= Var(total) +Var(can)

    I don’t think the second formula is valid, because the total mass is in fact dependent on the volume of the liquid.

    1. I don’t have the paper, but assuming \text{Total} and \text{can} are independent random variables with liquid=total-can, then \mathbb{V}[total-can]=\mathbb{V}[total]+\mathbb{V}[can]. This stems from the fact that \mathbb{V}[aX]=a^2 \mathbb{V}[X].

      1. If T =C+L
        Applying variance gives
        Var(T)=Var(C)+Var(L)

        Rearranging first for
        L=T-C
        Gives
        Var(L)= Var(T) +Var(C)

        Both cannot be true

        1. Okay, but what are you doing when you’re rearranging the terms? They’re not numbers, they’re random variables. Most notably if I’m understanding right, you start by defining the liquid to be the random variable that is the difference of the total and the can.

          1. I don’t think it makes any sense to define the liquid as the difference.

            if indeed that is how it is defined, you are correct. But it seems illogical to me.

            Does it say in the paper that the mass of the liquid is an random independent variable?

            1. The paper does not claim the independence of any of the three random variables, so no. Would it make sense at all for the total mass of a can+liquid to be independent?

              1. I think both interpretations may be valid.

                In the framework of the question, there isn’t an explicit statement that
                T=L+C or L=T-C

              2. The lack of a statement about independence is a VCAA error.

                If independence is not assumed, then it’s impossible to answer the question. The magic word “covariance” makes sure of that.

                My earlier comment says all that needs to be said. In particular, you MUST assume that the random variables

                M (“Mass (g) of can after being filled) and C (“Mass (g) of empty can)

                are independent to be able to answer the question. The assumption is reasonable but should not have to be made, it should be a given fact in the question.

                1. I don’t know. During the exam, I got stuck because I figured that the mass of the can after being filled would be related to the mass of the empty can, and hence would not be independent. I realised that they wanted you to assume independence after a while but it still didn’t quite make sense to me.

        2. The first step – ALWAYS – before being full of vim and vigour and wanting to calculate anything is to *always* define all the relevant random variables:

          1. Let M be the rv “Mass (g) of can after being filled.”

          M ~ Normal(\displaystyle \mu_M = 406, \displaystyle \sigma_M = 5)

          2. Let C be the rv “Mass (g) of empty can.”

          C ~ Normal(\displaystyle \mu_C = 15, \displaystyle \sigma_C = 0.25)

          3. Let L be the random variable “Mass of liquid (g) in a filled can.”
          Then L = M – C.
          \displaystyle Assuming M and C are independent random variables (*):

          L ~ Normal(\displaystyle \mu_L, \displaystyle \sigma_L)

          where \displaystyle \mu_L = 406 - 15 = 391

          \displaystyle \sigma^2_L = 1^2 \sigma^2_M + (-1)^2 \sigma^2_L = \sigma^2_M + \sigma^2_C = 5^2 + 0.25^2 = 25.0625

          using \displaystyle\sigma^2_{aX+bY} = a^2 \sigma^2_X + b^2 \sigma^2_Y when X and Y are independent random variables.

          1 ml = 1.04 g therefore 375 ml = 390 g therefore you need to calculate Pr(L < 390).

          * The question never says that M and C are independent, but if they're not then we can't answer the question (unless (i) we're also told the covariance and (ii) covariance is suddenly and magically part of the course). So we have to assume it. But we shouldn't have to assume, we should be TOLD! Not telling us this is a VCAA error.

            1. What probability are you trying to calculate? Set your work out (like mine) so we can clearly see what you’re trying to calculate and how you’re trying to calculate it. In particular:
              Start by properly defining your random variables.
              State their distributions.
              State the probability you’re trying to find.

              1. Let T be the total mass
                T~N(406,5^2)
                Let C be the initial mass of the can
                C~N(15,0.25^2)
                Let L be the mass of liquid
                T= L+ C And L~N(u,s^2)
                => u =391
                Var(T)= Var(L+C)
                =>Var(L) = Var(T) -Var(C)
                The rest was plug and chug to find Pr(L<390)

    2. Hi Names. I haven’t looked at the question, will not do so tonight, and wish I never had to. I’ll look tomorrow, and if need be ask some smart people.

      I gather we have three variables A, B and C, with C = A + B. Then as far as I can discern from the battling, the key question is, can we reasonably assume two of these variables are independent, and if so, which two? Then, the variance of the third variable can be determined.

      As John Friend wrote, it doesn’t seem kosher for VCAA to leave that reasonable assumptioning to the student, rather than making the independence explicit. But that’s off the point (for now).

      Back to you tomorrow.

  6. Thanks, everyone. I’m not looking at the exam tonight, but it sounds, um, interesting.

    As Sai notes, this WitCH may be worthwhile background to the complex MCQ being discussed.

  7. (This has pretty much all been said, but here are my thoughts):

    Writing Im(b) is non-zero looks like they were trying to imply that a and c are complex conjugates.

    This would make option B the intended answer.

    However, as JF has correctly said, if a and b are both real then option B is not necessarily true.

    What a load of CRAP.

    1. Hmm, yes the question seems conceptually flawed, assuming they want us to make use of the population parameter. Ideally, you’d want the sample mean instead, since that is how confidence intervals are typically constructed, and knowing the actual population mean makes the whole thing…pointless.

    2. I agree that this question is a mess. It seems VCAA wants to test as much as they can in a single MC question. The first part is testing the linear combination of a variable where its distribution is known, so we assume they are population parameter. Then 100 samples came out from nowhere and we were asking to construct a CI for a mysterious sample mean to estimate the population mean which we need to use to produce that interval, what a mess!

    3. There is no doubt that the question is defective and so cannot be answered.

      A confidence interval is based on a given set of sample data. It gives an estimated range of values that is likely to include an unknown population parameter (such as population mean) with a certain level of ‘confidence’ (*).

      The question does not give a sample mean. The question actually seems to be suggesting that a confidence interval based on the population mean should be calculated for a sample mean. That’s a load of bollocks.

      Furthermore, the given information enables the population mean to be exactly calculated, which makes finding a confidence interval for the population mean pointless (assuming the sample mean was given in the first place so that such a calculation can actually be done).

      * More can be said about the meaning of the level of ‘confidence’, but it’s not relevant.

  8. Thank you all for your comments, and sorry to be slow with an update. I’ve now swallowed the exam, but digesting it will take a little longer.

    In brief, the exam is an absolute mess: in a number of parts way too cute, in many parts appallingly worded, and in a few parts plain wrong. I’ve post a WitCH and a PoSWW, on the two questions most discussed above. A couple other questions, also discussed above, are borderline for separate posting, but I decided that solid whacks in the update would suffice.

    I’ll try to get the update done either tonight or early tomorrow. There is plenty to say.

  9. Marty, thanks for the update. The amount of SUVAT in the exam really angers me. It angers me because SUVAT was deleted from the Study Design. Why delete something from the course and then cram an exam full of it? This is effectively saying
    “We’ve deleted this stuff from the syllabus but you better spend time teaching it anyway because we’re going to put it on the exam. Lots of it.”

    I’m happy to be swatted here, but the Section B questions should have explicitly required appropriate differential equations to be set up and solved. As for section A, SUVAT shouldn’t be there either.

    Of course, none of this would be relevant if it had been kept in the course. And under the new Study Design, it and mechanics in general won’t even rate a mention. Then again, VCAA is notorious for finding backdoor ways of putting deleted material onto exams. So who knows …

    Since Marty doesn’t do statscrap, I’ll offer a brief commentary on Q18:

    A routine statistics question that nevertheless offers plenty of opportunity for students to make a mistake. The wrong options are all good distractors (arising from common mistakes), making the question a useful diagnostic tool. Options A and C may prove to be the most popular distractors. A correct option (B) exists.

    1. SUVAT in Specialist papers is (in my opinion) the equal of normals in Methods papers.

      The next one will be “medians are not required, but we can ask about the 50th percentile” or words to that effect…

      If you say something is NOT required, DO NOT TEST IT.

      Even better, DON’T SAY SOMETHING IS NOT REQUIRED.

      1. RF, it’s funny you say that because I was thinking of normals in Methods as I was typing that post! I was also thinking of how the median will undoubtedly be reincarnated in future exams! (And to a lesser extent I was also thinking of the Hypergeometric distribution …)

        I think testing SUVAT by stealth is more despicable. But I agree that the backdoor used by VCAA to continue examining normals by stealth in Methods (“Find the equation of the line that is perpendicular to the curve at …”) requires memorising a formulae (\displaystyle m_n \times m_t = -1) which is no longer on the course (*). Which is pretty dishonest and despicable. And if you don’t know the formula, you can’t fall back on a differential equation … Yeah … “SUVAT in Specialist papers is … the equal of normals in Methods papers.” is probably about right.

        I like your advice in the last line. Just plain common sense.

        * Does this still get taught in Yr 10? VCAA would probably say that everything going back to kindergarten is still examinable, even when it’s explicitly deleted from the Study Design.

        1. I explicitly teach perpendicular gradients in Year 10, and finding equations of normals to curves in 11 Methods. I think our 12 Methods teachers teach it.

          Following along this tangent (no pun intended), equations of circles are no longer in the 11 Methods study design, but I don’t feel comfortable omitting it in future years, since it’s part of the Year 10 curriculum.

  10. A very minor point, but what is with the brackets in Section B 5 a.?

    (The question states “show that the acceleration of the object is given by (8 – k) m/s^2)

  11. Q6. Stats crap. I don’t do stats crap. But as commenters have noted, and see here, part (e) is stuffed stats crap.

    Shouldn’t it be part f, not part e?

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