Discussion of the 2023 NHT Methods Exams

This is our post for discussion of the 2023 NHT Mathematical Methods exams, which have now been posted, here and here. We haven’t looked, and don’t particularly intend to, unless something is flagged.

UPDATE (11/10/23)

The reports are out, here and here (Word, idiots). They don’t say much, of course, and are silent on the various absurdities.


56 Replies to “Discussion of the 2023 NHT Methods Exams”

  1. Exam 1:

    3.a) There is no specification that the bags chosen on each day are independent of each other. It’s somewhat implied, but it’s not definite.

    8.a) This is silly.

    8.d) There are two possible interpretations, though I suppose it’s clear enough. They should be more precise with language, though.

    Those are the only mistakes I’ve noticed from a precursory glance of the first exam.

      1. Seems similar to most VCAA methods exams in previous years. 8.c) is the only slightly interesting question, and only slightly so.

        1. Thanks. Usually NHT exams are weird, which is why I don’t take much interest. Maybe they’re becoming more standard.

          1. To be fair this is the first time I’ve looked at a VCAA exam in over six months now, so I might be a bit off. From memory it does look a lot more similar to standard VCAA exams than previous NHT exams.

    1. Thanks, Bob.

      Re 3(a): The wording is very clumsy, but doesn’t the use of “at random” suffice for your point?

      Re 8(a): It is certainly something, but why do you think it’s silly? (Not necessarily disagreeing, just curious about your thoughts.)

      Re 8(b): The wording is atrocious, real Fenimore Cooper stuff, but why do you regard it as ambiguous? (If I squint, I think I can see what you’re suggesting.)

  2. Some quick ones on E2:

    MCQ Q16 seems to be similar to that of VCAA 2019 Exam 1 Question 8b, where a composite function is considered to have a ‘maximal’ or ‘largest’ domain to be defined on.
    MCQ Q17 asks for values of k for which a function has more than one ‘solution’, instead of ‘root’ or something like ‘f(x) = … = 0 has more than one solution when…”.

    1. Thanks, anon.

      Re MCQ 16: Yep, that’s a mistake.

      If they had asked for the largest set or interval, they may have escaped on a technicality. But the word “domain” indicates they are regarding g(f(x)) as a new (and composite) function, and thus they lose by their own stupid rules. I’ll add it to the list.

      Re MCQ17: Ugh! I’ll think about whether to list it as an error, since such sloppiness is pretty common and sort of accepted. But it is an error, and it is gauche.

      1. MCQ 17 – It’s not gauche. It’s diabolical (and wrong).

        Given that a common MCQ asks “Which of the following are the roots of …” and at least one of the wrong options always lists factors (and the vice versa question), it’s not unreasonable to expect a much better standard of examination wording when it comes to the difference between a root and a solution. (Doubly so since the DET has mandated literacy in all subjects as a teaching focus and countless hours have been invested/mandated in staff PD). As RF has already said, functions have roots, equations have solutions.

        It’s an extreme example that nevertheless captures the essence of what is a generally poorly worded exam.

        1. How dare you claim, correctly, that I’m wrong!

          Yeah, you’re right, BiB. I think of the root-factor thing as akin to a grammar error, and am not overly fussed by it. BUT, VCAA are of course fussed by it when a student screws up, and will happily score a kid 0/1 or 1/2 for such an error.

          I haven’t hunted, but can you give examples of: (a) a question on this exam (or this year’s NHT exams) where such a question asks for a root/factor, and they would clearly penalise a student for giving a factor/root; (b) an example of an MCQ where root and factor answers are both offered; (c) a question on last year’s (or the most recent year’s) reports for which the examiners whined about this issue?

    2. Adding on to those, MC Q4 forgets that the positive whole numbers are already larger than zero: “n is a positive whole number larger than zero.” Could have just said “n is a positive integer” or n∈ℤ.

      Not a huge fan of MC Q18, feels dodgy. They didn’t need to mention either the fee or the intial investment and the question still works.

      Will get to the Section B later

      1. Thanks again, Alex.

        Re MCQ4: Yeah, that is remarkable. It’s not like it’s the worst sin one could commit, but how does wording like that survive proper vetting?

        Re MCQ18: Jesus H. Christ. What a stupid, nasty little question.

      2. I agree that Q18 is dodgy, but I’m guessing the examiners are just wanting a simultaneous-equation-2-variables type situation.

        Q17 – the function doesn’t have solutions it has roots or x-intercepts.

        Section B Q5e. GIVE? To whom? Does give now mean the same as state?

        1. I think you and Alex are being way too lenient. I think MCQ18 is actively designed to trick students. Twice.

          Yes, MCQ17 was pointed out above.

          Re B5(e): An appallingly worded question. the double “give” is the least of its issues, but is definitely a really stupid cherry on top.

          1. I think I was being lenient to make sure I wasn’t missing something and that it was as bad as I thought it was.

            I’d feel stinged if I invested $10k and had to pay 4% in fees before getting any interest out of it, let alone only ~2% p.a. Takes two years just to get back to your initial investment 🤦🏻‍♂️

            1. I’d feel stinged if I misinterpreted a question because I took it to be what it appeared to be, a bog standard compound question, or if I forgot to consider the idiotic fifth decimal place.

              What the hell do they think such nastiness is teaching kids other than to hate “mathematics”?

              1. There can be little doubt that MCQ18 was intended to trick students. For exactly those two reasons but particularly for the fifth decimal place business. Its another (one of the many over the years) MCQ thats doing too much. But at least there’s a correct answer 🙂

      3. Section B Q1 they need to decide if they are going to state the sets each parameter is in or not. They do it for part a and b, not c or d, then do for f. Also, “using algebra” in part d is completely unnecessary with the answer to part b.

        Q5 needed a better through line. This one is all over the shop. The f(x)-1 seems silly just to remove the +1 on 2sin(3x)+1 when they could just write the integral of 2sin(3x). Some context of why we’re equating an average value to g(a)/a would be nice (might be an application I haven’t seen before, but seems arbitrary). And part e is…something.

        1. Thanks, Alex. Q1 is pretty funny. Of course it demonstrates that declarations such as “where x is an element of the entirely obvious world” are nothing other than ritual.

          Q5 is appalling throughout. Part (d)(iii) smells of some deeper meaning, but I haven’t had the stomach to look closely.

          1. Q5 (d) (iii) is equating the average value of \displaystyle 2 \sin(3x) over two different intervals, viz \displaystyle \left[a, \frac{\pi}{2} \right] and \displaystyle [0, a]. But I can’t immediately see it’s raison d’etre . Is there an obscure connection to part (e)?

        2. I’m just now looking at Q5 d(iii) and e and I honestly can’t figure out what the hell is going on with either question.
          Anyone care to enlighten me? There’s no solutions published yet.

          1. Hi, Ron. It’s been a while, but isn’t d(iii) just calculator monkey work? (What it means is indicated by BiB in a comment above, but I don’t see that knowing what it means helps do the question.)

            For (e), it’s a completely insane question, and I wouldn’t worry too much. (It seems now standard to have these 1-mark idiocies to end the Methods Exam 2.) Perhaps there’s a stupid calculator way to do it, but here’s how I thought about it (and haven’t checked it all out).

            The f – 1 function in the integral is just a fancy name for 2sin(3x – kπ). The integral of 2sin(3x) from 0 to pi is 4/3 (stretch of 2 and a shrink of 3). So, the most the integral of 2sin(3x – kπ) from 0 to a can be is 4/3. And, that can only occur if 2sin(3x – kπ) starts at 0 and goes up. So, we need k = 2nπ.

  3. Re: MCQ18.

    Just for some fun, I asked Mathematica to solve v(10)=12000 and v(20)=15000 simultaneously for a,b.

    Amongst a large number of complex solutions (expected) I got a=0.0225652, b=9600

    This ignores the second and fourth sentences, but if I was in a rush… choose answer A and move on.

    1. You should include a domain:

      Clear[V, a, b, f]
      V[t_] := (b – f)*(1 + a)^t
      NSolve[{V[10] == 12000, V[20] == 15000, b == 10000}, {a, b, f}, Reals]

      and get rid of all those pesky non-real solutions:

      {{b -> 10000., a -> 0.0225652, f -> 400.}}

      (Since Mathematica treats everything as complex, sometimes NSolve won’t even give an answer unless the domain Reals is included).

      1. Thought about using the Reals qualifier (there is even a package called RealOnly which I’ve seen VCE students use) but decided that this was an exercise in speed as much as anything else.

        Did use NSolve though. Seemed to work fine.

        Whether it gave the correct/intended answer of course is another matter!

        1. I think the problem with some of those packages is that they require internet connectivity, which students won’t have in the exams. As for the correct/intended answer, time will tell.

          1. You install them once, then you just need to “activate” similar to the TI NSpire library/UDFs.

            Finding the packages in the first place is the challenge as the software is not exactly designed for school exams.

            That aside, defining the v(t) function on any CAS and then solving simultaneously for a, b will give an answer (hopefully).

            I still think this may have been what VCAA was intending when the question was written. NHT papers don’t give any commentary on solutions, especially with MCQs so we may never know.

            1. And the packages are entirely unnecessary. Only basic code is needed but most students think that they need a nuclear bomb to kill a fly. Worse:
              1) They don’t understand the nuclear bomb.
              2) They can’t even see the fly.
              It is very disheartening. Students don’t want to understand how to answer a question using mathematics. They impatiently say “Yes, but how would I do it on [Mathematica, TI-nspire, ClassPad].”
              To state the bleeding obvious, the education system is producing generations of students that are subservient to coding and button pushing without any understanding of what’s going on. Pseudo-understanding. All under the guise of ’21st century thinking’.

              1. “Pseudo-understanding” will be my new catchphrase when I give a quiz on the new pseudocode… nice one.

            1. I wrote “A” as the answer, but with VCAA papers I am never confident that I have interpreted the question and answers in the same way the paper setter has.

              So, the answer is clear. Provided it is the answer.

  4. Exam 2 page 15

    e) What is the point of specifying “for t>0” ? A function can be continuous, or continuous on an interval: the wording here suggests that the continuity of a function is some pointwise condition, when of course it isn’t.

    g) The derivative clearly doesn’t exist everywhere, and I don’t like glossing over this fact (“find the derivative C_2′(t)”) even if there is some talk of “relevant domains”.

    1. Thanks, A.

      I also don’t like the wording of (e), but why do you consider continuity to not be a pointwise condition?

      On (g), yes I don’t like this kind of framing either, but I think it’s pretty standard in VCE.

  5. Exam 2, Section B, Question 2.

    Just some gripes about the wording.
    The question states “students suggest” a model and the rest of the question is written as if this model is definitely correct. The intention is clear, but there’s no need for such clumsy wording. Just make the model definite from the beginning.

    On page 14 the model is for “…the amount of caffeine…”, on page 15 a new model is suggested for “… the absorption of caffeine…”, and parts f and h of the question refer to “…the amount of active caffeine…”. Why?

    1. Thanks, tej.

      I’ll leave it to others to comment in detail. The questions has been discussed a lot above, because it’s a bad, nasty question. But I don’t think anybody raised tech concerns.

      These “not solvable by Machine X” issues come up periodically. Usually it seems that it’s not exactly that the question is not solvable, but that Machine X requires more fiddling than Machine Y, and maybe significantly more.

    2. I was able to solve it using the ClassPad.
      To do it with very little thought at all, just pop it into the simultaneous equation template in the Main application and hey presto!

    1. Thanks, Guardian. I’ve added links to the reports, and also to the Specialist Reports on the other discussion page.

  6. Hi marty,

    Exam 2 MCQ 3, would it be correct to say that the function is strictly decreasing over both the domains (-1,1) and [-1,1]? The exam report gives [-1,1] as the correct answer, presumably as it’s the ‘largest’ interval, but the question just asks for which domain the function is strictly decreasing for.

    The wording seems a bit dodgy to me, although I admit that I only picked up on it because I carelessly got it wrong during my exam revision.


    1. Hi Bugle, and thanks very much. You are absolutely correct: both (-1,1) and [-1,1], i.e. both B and D, are correct for the question as asked. The question is screwed.

      Moreover, the examiners obviously know the question is screwed. Which is why they pretended in the exam report that the “maximal” question had been asked.

      I’ll add the question (and report) to the error list post.

  7. Exam 1 Question 8 part (d): Exam Report.

    The Report provides two methods for answering the question. The second (alternative) method makes little sense to me after the third line and gives two apparently different values for c. It’s not an error but I think it’s absurd and misleading that two answers are given when in fact they are equivalent.

    We note that the equivalency is somewhat obscured (certainly for students) by the use of the same parameter k in both answers.

    1. I think the second method can be made valid, but it is fiddly to do so and there is a serious leap in the argument as written.

      The basic argument they’re making is that if sin(A) = – sin(B) then A and B must differ by an odd multiple of π. That is generally false (e.g A = π/6, B = -π/6). However, with A and B depending upon x, and the equation needing to be true for all x, I think the “odd multiple” thing can be concluded.

      1. Surely after line 3 (which sets the stage) it would have been more logical and clearer to consider the two cases (that exploit the symmetry of the unit circle):

        Case 1: \displaystyle \pi x - c = -\pi x - c + 2 n \pi where \displaystyle n \in Z.
        No good because the c’s cancel.

        Case 2: \displaystyle \pi x - c = \pi - (-\pi x - c) + 2 n \pi where \displaystyle n \in Z
        and we’re off and running.

        1. Back in the VUSEB days of Pure Maths, questions that boiled down to solving sin(A) = sin(B) or cos(A) = cos(B) were a dime and dozen. It’s a shame that the techniques for answering such questions have been ‘lost’ over the last 40 years or so. Technically, such questions are still examinable (as this question shows) but I don’t see examples in any textbook or trial exams and I doubt they are mentioned in most classrooms.

          It would be interesting to see what happened if a question such as

          Solve \displaystyle \sin (2x) = \cos (5x)

          appeared on a Specialist Maths Exam 1 paper.

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