Secret 2023 Methods Business: Exam 2 Discussion

When you’re ready.

UPDATE (16/04/24)

The exam report is now posted, here (Word, idiots).

UPDATE (31/12/23)

The exam is now posted, here.

UPDATE (10/11/23)

The Marty is slow but the readers are patient.

Here are my thoughts on the multiple choice questions. In general, the MCQ seem OK, given the limitations of the subject, and are generally well written. As usual, commenters have covered the ground pretty well but there seem to be a couple things to add.

MCQ 3. The specification that the functions are continuous is there for no reason, and thus has no reason being there.

MCQ 6. The construction “The value of BLAH is …” is too clumsy. Better is, “Then BLAH equals …”.

MCQ 8. The condition n ≠ m is there for no reason, and thus has no reason being there.

MCQ 9. A reasonable question utterly perverted by idiotic wording, and probably made worse by an idiotic, uncorrected 2021 exam report. See here. (17/04/24) The solution in the exam report is bad but at least makes clear what was intended. See the updated discussion here.

MCQ 11. Similar to MCQ 9, but not quite as bad. It is still very bad, though. VCAA must stop it with the redundant “continuous and differentiable” bullshit:

A function that is differentiable is automatically continuous.

It is teaching Not Maths to be using the current phrasing.

Furthermore, the question didn’t have to indicate anything whatsoever about differentiability “for all \boldsymbol{x \in R}“: the derivatives (and values) of the functions are specified at x = -2, and that is all that is required.

Finally, the “gradient of  the graph …” framing, rather than simply “the derivative of …”, is needless, clumsy, wordy and muddying. (Afterthought: Is the graph thing why, for jumping at shadows reasons, the writers ventured into general differentiability?)

MCQ 13. It’s pseudocode crap. I don’t do pseudocode crap. But commenters indicate that it is slightly wonky.

MCQ 14. In practice just CAS crap, but the question can be easily figured out with zero working. CAS poisons everything.

MCQ 18. (17/01/24) There is no correct answer. As commenter LH has noted, one of the endpoints will be a local minimum (and the other a local maximum). So, there will always be \boldsymbol{a^2 + 1} local minima. (17/04/24) The exam report is silent on the screw up.

MCQ 19. In practice just CAS crap, but the question can be easily figured out without computing the discriminant. CAS poisons everything.

MCQ 20. The question is overegged, and screwed. By VCAA’s rules (Word, idiots), a composition of functions is either defined or it is not: \boldsymbol{f \circ g} and \boldsymbol{g \circ f} are not. (17/04/24) The exam report is silent on the fundamental error, but their solution to this question is hilarious.

 

UPDATE (11/11/23)

Here are my thoughts on Part B. Not a whole lot to say beyond what commenters have noted. Perhaps it’s a little less weird than in previous years. As always, CAS poisons everything.

Q1. The wording in (d) is appalling, an absurd confusion of what is being defined, and how.

Q3. A repeatedly bad question. Limits as x goes to -∞ are not, or at least were not, kosher; there is no proper indication of the concept in the study design, it is new and it should have been flagged. Then, what is the point of having a CAS-fueled question about the derivatives of nx, which 99% of Methods students will have no clue how to calculate? Part (c) is very poorly worded. Part (h) is a weird, 2-mark Magrittism. (17/04/24) The exam report notes that students “appeared to be confused by the limit notation”. Jesus wept. (17/04/24) After pondering the exam report, I’ve added a WitCH, here.

Q4. As Bryn has noted, (f) should be asking for the maximum permissible standard deviation. This is arguably covered by the question asking for the “required” standard deviation, but the wording, at best, is vague. Part (j) has the “m” for “meters” glitch. It’s hardly a hanging offence, but the sentence does scan awkwardly. Also, this is a recycled question from 2009. It’s not clear that that matters, but it seems very strange. Why would they do this?

Q5. As commenters (and reporters) have noted, (b) is a bad screw-up, with “domain” written when “maximal domain” was intended. We won’t know how big a screw-up, however, because we don’t know yet, and may never know, if VCAA will take responsibility for the error and give full marks for all correct answers. Like the assholes refused to do in 2022. The rest of the question is CAS garbage. An ugly and painful way to end. (17/04/24) The exam report notes the error, although they could have done so with a little more humility.

57 Replies to “Secret 2023 Methods Business: Exam 2 Discussion”

  1. Haven’t seen it in full yet but apparently a conceptually difficult exam with a much better final question than last year’s easy nonsense

    Also students instructed to correct “m” to “metres” hopefully the increased media attention means they are finally vigilantly proofreading exams lol

    1. There might be some debate as to whether it’s an error or an edit. And if it’s an error, whether it’s noteworthy or not. Let’s call it a ‘correction’. What cannot be debated is that the VCAA thought it was noteworthy enough to send instructions to all schools for making the correction. The instructions didn’t refer to it as an error, they simply said to make the following announcement to students (which consisted of instructions on how to make the correction).

      The need for a correction was obviously detected only after the exam was printed, so I don’t think it’s a testament to attentive proof reading (otherwise it would have been fixed earlier).

      I agree that media attention was probably a contributing factor for last-minute vigilance. I suspect there was probably some poor schmuck whose job it was to check and re-check the paper prior to the exam and to detect and report any necessary last minute corrections. Maybe the correction was an over-reaction to the media attention. It’s a shame the poor sap didn’t detect the need for the word ‘maximal’ in Question 5 part (b).

      1. The focus on “errors” is understandable, but it distracts people from the main point: the systemic awfulness.

        1. Yes, the errors are a symptom of the illness.

          “Systemic awfulness” is the disease, structural and cultural change is the cure.

  2. Two issues: The cylinder MCQ requires the assumption that no overlap is formed. A typo with ‘m’ instead of metres.

    Much better than e1 anyhow.

  3. The MCQ seem to be slightly more difficult than normal.
    Q1 CAS Bashing
    Q2: Easy show that + more cas bashing
    Q3: a) State the value of lim x-> -inf (2^x + 5). Never seen a limit question on an exam before.
    Rest of the question: Some cas bashing but a bit of thinking required + the new newton’s method question by filling out a table
    Q4: Standard probability question, learning from their mistake, they didn’t make the mistake of the probability density function in terms of sin not having an area of 1.
    Q5: Pretty good question, requires more skill than CAS.

    1. I didn’t think that limits were part of the study design, but they are there. Don’t know if that’s a thing introduced in the new study design.

      It’s debatable whether the study design actually covers infinite limits. It says “informal concepts of limit, continuity and differentiability.” If they test infinite limits, they should explicitly mention them in the study design.

      For a more pedantic point, why did they write “state the value of”? For the purposes of VCE infinity is not a value, they could have just as easily said “evaluate” or “find”, or “determinate.” I could imagine that tripping up a student who was taught that infinity is not a number, something I remember my teachers saying multiple times during VCE.

      Edit: Oops, misread 2^x as x^2. So the third paragraph is largely irrelevant.

  4. Q5b. I don’t think this has a unique answer. \cosh(x-2) is strictly increasing on any interval that’s a subset of [2, \infty), so the question stem doesn’t uniquely define g_1.

    This has consequences for part c, which seems to assume that the domain of g_1 is the largest interval on which \cosh(x-2) is one-to-one and strictly increasing. But without this unstated assumption, there’s no guarantee that g_1^{-1}(x) = x has a solution, so perhaps the intersection point Q doesn’t exist; similarly, without this assumption there may be no finite region bounded by the graphs of y=g(x), y=g_1^{-1}(x) and g_2^{-1}(x).

    1. I was halfway through writing about this when you commented, and I picked up on this while doing the exam. I doubt anyone was seriously impacted, but the manner in which it completely breaks the question, as you’ve described, is shocking for something that could’ve easily been specified.

    2. Thanks, SRK, and you other guys. Is it reasonable to summarise this as an error that VCAA can get away with? That is, it is something that is definitely wrong, that will really piss off Marty and co, but that will not screw up many/any students?

      1. I think it would bug a mathematician … (using the point P from Exam 1 as a baseline). But I simply cook sausages so …

        Actually I think most students will instinctively find the maximal domain. The better ones will realise that any subset of [1, oo) is also an answer for the domain etc. but I don’t think that will fuss them. The very best will probably say any subset of [1, oo) and be guided by common sense for part (c). But all this is pure speculation.

        I still haven’t finished it (just finished Section A) but I know that MCQ 13 was the only appearance of pseudocode. It was very pleasing to see that common sense prevailed (if such a thing exists for pseudocode in Methods and Specialist). The question made the very best of a bad situation, in my opinion, and I’m grateful to whoever wrote it. It was the question I hoped it would be.

      2. Charlie Pickering mentioned it this morning on ABC radio and it’s come up in the Vic Maths teachers’ FB group as an error – will be interesting to see how broadly it goes. A key question will be if VCAA recognised the error and briefed the Minister. Ministers do NOT like surprises.

        1. Marty, I think your wires are crossed. You’re talking about the error (yes, I believe its fair to call it that) in Question 7 part (c) of the General Maths Exam 2.

          The error raised by SRK that is being discussed here is an error in the Maths Methods Exam 2 Question 5 part (b). The error is that the word “maximal” is missing. Without declaring this, there is no unique answer and the point Q in the preamble to part (c) suffers an existential crisis.

        2. Oh, geez. Thanks BiB. Too much going on. Thanks, BiB. I’ll delete my comment now (which will also delete your reply).

          1. The following comment does not excuse the omission that SRK notes.
            The fact that the question includes the graphs of the inverses of g_1 and g_2, would likely result in most student who were able to tackle part c), tacking it knowing what was intended.

            1. Understood. But let’s be clear:

              One of several on-going issues is students having to guess or assume what the question intends, as opposed to the wording of the question being precise enough that no guess-work is necessary.

              The first thing any attentive vetter \displaystyle should have noticed about part (b) is that the maximal domain and range are intended. The attentive vetter then provides feedback that insertion of the word “maximal” is necessary.

              VCAA cannot argue that the intent of the question was clear and therefore there’s no problem – VCAA pays obvious attention to detail when statements such as \displaystyle a, b, c \in R are included in their questions. One could easily argue that such statements can be assumed and are superfluous in a Mathematical Methods exam. They are relatively pedantic and redundant compared to other details (such as “maximal” or the definition of the point P in Exam 1 Question 9) that are often omitted.

              Yes, most students may have assumed (without even realising they were) that the maximal domain and range were intended. Just as many students may have assumed in the 2022 Specialist Mathematics Exam 2 that \displaystyle a and \displaystyle c are not real in MCQ 4, or that you just shut up and do a nonsense calculation in MCQ 19, or that the necessary random variables in Section B Question 6 parts (e) and (f) are independent etc.

              But what of the students who are smart enough to realise that a question cannot be uniquely answered (or answered at all because it makes no sense or there is no answer) without making assumptions? How much disadvantage do they suffer whilst wrestling with this dilemma?

                1. Zero is too much. The question is wrong, and glaringly so, in a manner no mathematician (or solid teacher) would miss.

  5. MCQ12 is interesting. Haven’t seen a question like this before in a VCAA exam (but I could be wrong).

    Not sure if I like it or not yet.

      1. I liked that you had to think a bit. Just looking for the maximum value of the quadratic would not work and a few students may have been left wondering why.

        Yes, probably better on a non-CAS exam, but what question isn’t?

  6. I feel like MCQ 20 missed the mark because it didn’t assess what I assume it was meant to.

    I assume it wanted students to find a domain restriction on f such that the new range of f (while f was under that restriction) was a subset of the domain of g, and vice versa. However, f isn’t defined for four of the MCQ answers, so there’s no need to think about anything.

    Better interval options would’ve made the question slightly better, but I really feel like this type of question should always be short answer so that you need to think about the domain restrictions that give the biggest possible ranges that work, rather than just checking the validity of the 5 ranges produced by the 5 intervals.

  7. Does anyone attempt Q3h. I couldn’t find the exact value as indicated by the question. I like it as it can’r be just solve by using CAS, but how on earth an exact answer can be produced?
    In general, I think this paper is more difficult than recents papers, especially the MC part. I just feel for students who completed MC and felt deflated coming to part B. Even bashing CAS questions require some advanced skills with the CAS (Q1d). It will be interesting to see the examiner report on this paper.

    1. The solution was n=e. If you play around with a slider, you can see the solution is roughly 2.71 which hints (especially since log base e was involved) that e may be the solution.

      Given that the minimum is an xintercept, we set f(x)=0 to get x^n=n^x. By inspection x=n is a solution to this equation (there are others which aren’t of this form!). So we will try to find a solution when n and x are equal.

      Solving f'(x)=0 we get
      ln(n)n^{x}-nx^{n-1}=0
      ln(n)n^{x}=nx^{n-1}
      ln(n)n^{x-1}=x^{n-1}

      Since we are only looking for solutions of the form n=x, we know n^{x-1}=x^{n-1}. Dividing then gives

      ln(n)=1
      n=e

      I must admit I’m not entirely happy with this solution but every solution I think of requires either some simplifying assumption or solving by inspection. I’m not sure if there is a nice way to solve this. CAS certainly doesn’t work here.

      1. Thanks, I got the decimal and guessing it is e but the 2 marks allocation bothers me. What do they expect students to approaches. If by inspection, should be 1 mark, if algebraic/calculus route, should deserve more mark than that.

      2. For what it’s worth:

        Stationary points:
        \displaystyle f'(x) = \ln(n) n^x - n x^{n-1} = 0 \implies  \ln(n) n^{x-1} - x^{n-1} = 0. …. (1)

        x-intercept:
        \displaystyle f(x) = 0 \implies n^x - x^n = 0 \implies n^x = x^n. …. (2)

        Equation (2) is interesting and more can be read here: https://en.wikipedia.org/wiki/Equation_xy_%3D_yx

        There is an obvious set of solutions to equation (2) of the form \displaystyle x = n. There are other solutions. But for a 2 mark question we take a leap of faith and hope these other solutions don’t need to be considered.

        Substitute \displaystyle x = n into equation (1):

        \displaystyle \ln(n) n^{n-1} - n^{n-1} = 0 \implies n^{n-1} (\ln(n) - 1) = 0.

        Case 1: \displaystyle n^{n-1} = 0. No real solution (\displaystyle n \rightarrow  0 is indeterminate).

        Case 2: \displaystyle \ln(n) - 1 = 0 \implies n = e.

        Therefore \displaystyle f(x) = e^x - x^e.

        It should now be checked in the usual way that the stationary point is a minimum turning point (it can also be seen by inspection of the graph). (If it’s not, we’re in trouble because those “other solutions” to equation (2) will need to be investigated). Luckily it’s a minimum turning point.

        We only had to find a value and we have so the question is answered. The question suggests that the answer \displaystyle n = e is unique, which requires either proof or faith.

        1. Did Douglas Adams not once say “…proof denies faith…”?

          OK, totally out of context, but in a rather odd way, I think it works here.

    1. I don’t know if the question is stuffed in the way you suggest, but it is garbage in at least one manner.

      This is madness.

      1. Continuity of the derivative at x = a is not required for the function to be differentiable at x = a. So I think I take back what I’ve said about the derivative not existing. But VCAA has made the whole ‘smooth’ thing very confusing – if they use ‘smooth’ as a word to denote continuity of the derivative, I’m not sure that the derivative is continuous at 2pi …
        I’m too tired to contemplate the “at least one manner”, it’s on my ‘To Do’ list.

        Edit: So I don’t look like a complete fool, I think continuity of f(x) at x = a plus existence of limit of f'(x) as x –> a is sufficient for continuity of f'(x) at x = a. So I might be barking up the wrong tree for what I thought was an error.

        1. You raise a good point which I hadn’t considered: what actually does ‘smooth’ mean here? I’ve always taken it to mean infinitely differentiable without a second thought, but it seems VCAA uses it to mean continuously differentiable (e.g. the function here is only C^2, and Google led me to the MM21 exam report’s discussion MCQ19). I checked both the Cambridge and Jacaranda textbooks for clarification; the only thing I could find was Jacaranda’s statement that “The gradient of a function only exists where the graph is smooth and continuous.”

          Your edit is correct: this follows from the mean value theorem.

          1. Thanks, Javert. “Smooth” means what you think it means, and VCAA thinks it means what you think they think it means. And, VCAA don’t know how to prove a function is (what they think is) smooth.

      2. Ok, I take it back (sort of). I misread the question, and thought it was worse than the usual very bad. It is simply the usual very bad.

        I clearly have to write something about this stuff (again), but am too tired now. I’ll post something tomorrow.

  8. Apologies if this has been raised already but I couldn’t see it. Question 4f seems to be missing something about finding the “maximum” standard deviation. If I wanted to ensure that more than 99% of balls are in the required range and I could choose any standard deviation I would choose 0 cm (whereupon 100% of balls will be in the range), or 0.01, 0.02, etc. up to 0.06 cm. Seems odd to ask for “the” required standard deviation. I might be missing something though?

    1. Hi Bryn. I think you are correct.
      This error was also made in the 2009 Methods Exam 2 Question 3 part (d), upon which the 2023 question is so obviously heavily based upon. (The answer given in the 2009 Examination Report confirms that VCAA intended the maximum standard deviation).

      Another error. That’s four errors by my count. So far …

      1. Thank you both, and everyone. I take it these “errors” are significantly larger than “m for meters” but smaller than last year’s Big Five?

  9. Thanks again to everyone for their comments. I’ve updated the post with a few thoughts on Part B, but I didn’t have much to add.

  10. I’ve updated the post (and to the MM Error List) with brief comments on MCQ9, MCQ18, MCQ20, Q3 and Q5.

    The exam report has also raised another question part to address, which I’ll get back to and which I may make a separate post. I may also write a post on a separate aspect of the exam report.

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