Secret 2022 Methods Business: Exam 2 Discussion

Fire away. (Exam 1 discussion is here.)

UPDATE (06/11/22)

Thank you all for your comments. We don’t have much in the way of general thoughts on the exam. In the main, by subterranean Methods standards, the exam seemed pretty good. Perhaps on the easy side, but there were some good questions, and not, as in past years, question after question that made us scream. As with Exam 1, the writing was very poor, and we’ll be devoting a post (probably two) to this aspect once the dust has settled. We noticed no major errors, although some of the phrasing was sufficiently unclear or imprecise to to earn at least a yellow card. Mostly the exam was bad simply because Methods is an execrable subject. The writers did reasonably well given the constraints and conventions.

Here are our question by question thoughts. We won’t comment on the (frequent) poor wording unless it amounts to error, or close to it.

MCQ 1. Routine and easy.

MCQ 2. Routine and easy.

MCQ 3. Routine and easy.

MCQ 4. Not a difficult question, but not good, and a bit weird. Since the lack of continuity arises simply from not being defined on the whole interval, this is a little fringe for VCE (and, beyond VCE, not quite mathematically kosher).

MCQ 5. Routine and easy.

MCQ 6. An annoying, busy question, made needlessly busier by the function notation. Students have to hunt too long, too carefully to find the correct answer. A very bad question.

MCQ 7. Routine and easy (although students always seem to muck these ones up).

MCQ 8. Routine and pretty easy. (In truth, plain easy but students seem to muck these ones up.) The phrasing is probably sufficiently cumbersome to throw some students.

MCQ 9. A good question warped by idiotic wording. Writing “shortest distance” to mean “distance” falsely suggests some max-min is involved.

MCQ 10. It’s hypothesis crap. I don’t do hypothesis crap.

MCQ 11. As with MCQ 9, a reasonable question made significantly worse by the framing. The factor k is probably not worth the inclusion, and leaving the sine function unintegrated on all but one answer goes against the underlying purpose and thus the naturalness of the question.

MCQ 12. A nice question.

MCQ 13. A nice question. There is some objection to the complement notation, but it did not concern us.

MCQ 14. Mindless button pushing, and involving an improper integral. An utterly pointless question.

MCQ 15. An OK question, but pretty much a repeat of MCQ 13. No point.

MCQ 16. In principle an ok question, made pretty meaningless by CAS. It can be done easily by hand, but presumably few will. It can also be done very easily by hand, by first noting f'(x) = (x + 3)(x-1) (thanks, Laura).

MCQ 17. In principle a very nice question, but it probably doesn’t work. The need to include (and think about) continuity, and the precise meaning of “many-to-one”, makes it too heavy and fringe for an MCQ.

MCQ 18. A nice question made gratuitously worse by CAS. The same question could have been asked with n = 4 and p = 1/3, or whatever, and would have been much, much better.

MCQ 19. A nice question.

MCQ 20. A very nice question, although not easy and a fair amount of work for an MCQ. The parameters might have been better chosen. As suggested in the comments, something like this would have been better placed in Section B.

Q1. A nice enough but repetitive question, which reduces to the usual CAS stuff. Notably, the integral in (e) needn’t be computed, since the integral is then differentiated, but this application of FTC is too subtle for VCE.

Q2. A really, really stupid and depressing question. There’s a famous and lovely model of rabbits and foxes, and it ain’t trig. The questions themselves are trivial and pointless and as boring as dirt, and/or it’s just CAS/Magritte idiocy. To have the analysis of Acos(t) + Bsin(t) replaced with mindless button pushing demonstrates as well as anything how low Victorian senior mathematics has descended. Simply criminal. The fox transformation in (d) and (e) is a pointless add-on, absurdly expressed with matrices, and the notation Q:R^2\to R^2 is probably VCAA-illegal, and definitely should be VCAA-illegal.

Q3. Routine and routinely aimless questions, framed by a boring and absurd mish-mosh of pseudo-modelling crap. Notably, the PDF in part (b) is not continuous, which it needn’t be, but for a decent coin-height model probably should be.

Q4. An OK question, with some serious issues. Commenters indicate that finding the inverse function in (c) can lead to a hyperbolic tangent answer, on which VCAA will have to adjudicate. Have fun with that. Part (e) is treading into the dangerous Magritte waters of h = h^{-1}, and then goes completely nuts. Part (e)(i) can (arguably) be reasonably considered by noting the value of k for which h'(0) =1, but Part (e)(ii) is utterly pointless and makes absolutely no sense whatsoever. They have already asked in (e)(i) for which k the area of the resulting regions is positive, which by its wording implies also contemplating when “the regions” have zero area (i.e. either consisting of a single point or is the empty set, depending upon one’s definition of “region”.) So, it is perfectly reasonable to claim the (possibly empty) regions exist for all permitted k. It’s just one mark, but this is the kind of gratuitous, irritating idiocy that would be eliminated by proper vetting.

Q5. Mostly harmless.

45 Replies to “Secret 2022 Methods Business: Exam 2 Discussion”

  1. Interesting paper.

    A lot less CAS use than in some years which I liked.

    Some decent distractors in MCQs which I also liked.

    And… perhaps for the first time in a few years, there seem to be marks for working in Section B.

    Q1e was particularly nice.

    Q3c… do they really need to say p is between 0 and 1 (also, inclusive or exclusive?)

    Q4 – nice function, boring questions until part (e) which I think is bordering on crap but I’ll need to see the examiners’ report to know what they were looking for in (ii) to know for sure how crap it is.

    Q5 – yeah… nope.

    1. What makes q1e nice? I just casbashed the whole thing and it wasn’t particularly rewarding – did you have a nicer solution?

      1. Using CAS to do the grunt work of the derivative and then defined a new function a(b), set a'(b)=0 and the whole thing just fell out nicely.

  2. Open question… 4d – will examiners accept answers in terms of tanh (hyperbolic) or will they expect answers in terms of e?

    One calculator model returns a tanh function when solving for the inverse.

        1. Unless i screwed something up during the exam, the CASIO just gave in terms of e. Honestly you didn’t even need to do any calculations- you could just use 1 for k using the equation given on the next page, which makes the real question why its a 3 marker

            1. I don’t think it will be! I think the unit of sq cm will be required. That’s what happens when an exam is bland, the the search for discrimination becomes pedantic. Having said this, I think students will still find enough ways to make mistakes that there will still be a natural spread. I suspect Q2(b), Q3, Q5(b) and Q8 will perform the task more than adequately.

  3. As a student I feel pretty shorted by VCAA. Like yesterday felt much easier than previous years, question 5 is a joke with the final question of the paper being essentially to solve cos(2x) and sin(2x)=0. With calculator. Maybe 3 differentiating marks (last part of q4 and stating trans values)

    1. Hey I think is was sin(2x) = sqrt(2)/2 and cos(2x)=0
      The mid probability question was hard where it introduced another random variable D for the minimum distance from the ceiling f was transformed to g

      I could not solve that one
      I guessed r= -1 and s=3 or vice versa
      Something like that

      1. I guessed the same, r =-1 and s=3

        In the moment I had written h+d=3, and thought that g(d) = f(3-h)
        But d and h were the same

  4. Question 5 was… I guess fun? Certainly not challenging as one might expect on a normal exam. As always there seems to have been one question that the TI-Inspire could solve far, far quicker than the CAS. Overall I found the probability question (I think question 3?) easy enough to begin with but confusing as it went on. Q1 and 2 seemed pretty standard, Q4 was quite nice.

  5. found the question 4 ‘find the inverse function’ (3 marks) pretty useless, since they essentially gave us the inverse in the following part but in terms of k. It was noticeable that the new function, h(x), = f(x) (and thus both inverse functions being equal) when k = 1 for all x, thus giving the answer to the inverse in the previous part.

    1. If you noticed it was worth 3 marks
      1 for the function
      1 for the domain
      1 for working
      To get the third mark you needed working
      But I do agree with you on the function being easily found
      However if you solve on the cas you get tanh(x/2) or something like that

      1. The tanh (hyperbolic tan) question is very much a relevant one as I have spoken to teachers of Specialist Mathematics who have written SACs on the hyperbolic functions and if those students were sitting this exam, they would know of the tanh function.

        Will VCAA mark this as correct? I have no idea.

        Is it correct? Yes, I believe it is.

  6. Some random comments and responses:

    Plenty of button pressing to be found in most questions.

    Starting with Section A Q11, the writers are dot happy (but they missed a dot in the integrand of the preamble to Q11). Dots for implied multiplication everywhere. It looked really weird.

    There seems to be a greater emphasis in properties of integrals (Exam 1 Q2b and Exam 2 Section A Q8).

    I hate the notation used in Q13 (asking for a maximal domain) option C and option D: \displaystyle R\[-a, a] and \displaystyle R\(-a, a) respectively. I much prefer a union of intervals.

    There’s always a needlessly verbose and complicated question – Q17 for this exam.

    Q20 – I dislike multiple choice question that require several different skills. This would have been much better as a short answer question worth 2 marks, say.

    Re: “Q3c… do they really need to say p is between 0 and 1 (also, inclusive or exclusive?)”
    No, they do not. But VCAA really loves its redundant wording (it was probably a coin toss between that and saying that p was a real number). Additional question – Do they really need to say that the coin is unbiased in the preamble to part (a)?

    Re: “Q4 – nice function, boring questions until part (e) which I think is bordering on crap”.
    Part (ii) is stupid. The answer is clearly “Because I figured out the values in part (i) and all values of k isn’t the answer, you muppet.”

    At least you can’t guess the answer to get 1 mark. I don’t know why:
    1) Part (ii) is not omitted and make it part (e) – 2 marks.
    2) Why part (i) couldn’t just be written as “Find the values of k such that …”
    And in the part 9e) stem, why say \displaystyle k \in R AND \displaystyle k > 0. Surely \displaystyle k > 0 is sufficient.

    Re: “Open question… 4d – will examiners accept answers in terms of tanh (hyperbolic) or will they expect answers in terms of e?” How can they reasonably justify NOT accepting an answer in terms of tanh!? (And it’s a lot simpler!) An cluey student might note that part of the answer is given in part (e) (substitute k =1).

    The use of the line y = x (in this case, h'(x) (button pressing) is true.

    1. * Comment got garbled at the end:

      The use of the line y = x in intersection of function and its inverse (in this case, h'(x) < 1) is the gift that keeps on giving for VCAA.

    2. I’d take something like Q17 over most of the mcqs any day of the week.

      I take minor issue with the wording of MCQ9… I think it’s safe to say we’re in Euclidean geometry, so ‘shortest distance’ is pretty unmeaningful.

  7. For question 2b, (show that…), I feel like there should have been a restriction placed on a and/or b (something like a>0, b>0), because a = -900, b = -pi/80 also gives the same function f(t) (but is not required to show).

  8. MCQ14 apparently needing an improper integral is interesting. I do not recall, is that common beyond the normal distribution?

    1. Irrelevant since it’s just button pushing.

      But improper integrals have appeared in the past in CAS-free – obviously a careful treatment is required. I always do an example in class to show exactly how to set them out properly. I guess you could stretch a point and claim that improper integrals are fair game because:
      limits and integrals are in the Study Design and students should know that infinity is not a number that can simply be substituted into a function.

      But for me, that’s stretching beyond the elastic limit even of the elastically vague Study Design. Then again, VCAA can be pedantic on the small stuff, but big stuff like this is usually hand-waved through (in other words, they probably don’t care about a formal, careful treatment).

  9. hello i am a student. can someone explain question 2f and 2g. am i the only one whose cas was having a spasm?

    1. Since the question asked for “nearest…” I just used ctrl-enter and it worked fine. Don’t know what would have happened had I been in “exact” mode.

  10. The question asking to show a=900 and b =Pi/80 is partly redundant.

    We are told in the stem that the two functions have the same period, which implies b =Pi/80 (or-Pi/80), by comparison. I certainly hope they don’t expect working out to arrive at b=Pi/8.

    My solution for the b value consisted of:
    We know period is same for rabbits and foxes.
    This implies b =Pi/80

    Would be annoyed if they don’t accept it.

  11. I still haven’t had the chance to sit down and do it in one shot, but my students thought it was ok, especially the specialist students. My battlers obviously struggled but they felt like they had sections they could attempt. Some students did get caught up with a bit of the wording and some of my stronger students found a couple of the MC ‘evil’ (in their words)… I will report back when I complete it myself. At least there wasn’t a 20 x 20 question I guess…

  12. OK, I’ve stewed for long enough. I’m going to state my biggest irk again so that it’s more obvious. It is Question 13:

    The function \displaystyle f(x) = \log_e\left(\frac{x+a}{x-a}\right), where \displaystyle a is a positive real constant, has the maximal domain
    A. \dispaystyle [-a, a]
    B. \dispaystyle (-a, a)
    C. \dispaystyle R \symbol{92} [-a, a]
    D. \dispaystyle R \symbol{92} (-a, a)
    E. \dispaystyle R

    I \dispaystyle loathe the notation used in Options C and D. I tell my students to \displaystyle never use this notation, that they should always use the interval notation \dispaystyle (-\infty , -a) \cup (a, \infty) (for option C) or \dispaystyle (-\infty , -a] \cup [a, \infty) (for option D) in order to avoid stupid mistakes (like incorrectly excluding or including endpoints) and to be clear to an assessor. I \dispaystyle hate it. I tell assessment writers to \displaystyle never use it.

    Why would VCAA do this? Particularly since they use ‘normal’ interval notation in Question 15.
    The answer to Question 13 is option C but students could very easily be ‘tricked’ in the moment and choose option D. So the student who chose Option D is put in the same basket as the student who chose Option E.

    I can only assume that VCAA did this to ‘trick’ students because the output of a CAS is in interval notation. What a ridiculous and pathetic way of VCAA finally deciding, against all historical evidence, to see whether students can correctly interpret the output of their CAS.

    Advice to VCAA: If you want to test whether students can handle algebraic abstraction like what Question 13 contains, put it on Exam 1.

    Unfortunately I predict Question 13 will create a pathetic culture of teachers giving students this dumb notation in assessments. And I expect it will spread like wildfire in trial exams. Typical VCAA – never does things in a simple clear way when it can do things in a confusing, tricky way.

    @Marty: You talk about “drunk monkeys” and trickery in Exam 1. Options C and D in Question 13 were written by a malicious monkey with trickery on its nasty little mind.

    1. Thanks, John. I have no great problem with the notation (although I prefer A ~ B for set subtraction, a notation that seemingly only I use.) Probably the union of intervals would be more natural for this question, but I’m not bothered by the subtraction, and it’s not obvious to me it was intended as a trick. But fair enough it’s primarily a question of VCAA conventions and expectations, and you and the other teachers are better able to judge that.

      1. Thanks, Marty. It may only be me that has this particular beef.
        I’d be interested to know what others think and whether they would encourage/discourage students to use R\{exclude an interval} notation rather than using a union of intervals.

    1. Thanks, Marty. VCAA seem to have made \displaystyle some effort with Exam 2 to minimise gratuitous button pushing – at least compared to previous years. This is a good thing and hopefully the start of a trend.
      BUT … there shouldn’t be any need to do this in Exam 2! I know you don’t want any CAS at all, but if VCAA can implicitly acknowledge that button pushing at the expense of mathematical thinking should be reduced in Exam 2, then surely the logical thing for VCAA to do is to reduce Exam 2 to 90 minutes (by getting rid of the multiple choice questions *) and increase Exam 1 to 90 minutes. Then there will be room for questions like MCQ Q2, Q12, Q13, Q16, Q19 (*) to be asked as short answer questions in Exam 1.

      Use the CAS in Exam 2 for genuine calculations that would not be reasonable to do ‘by hand’ (such as MCQ 20 – but as a short answer question).

      Re: MCQ 11. What makes you think you know the purpose? What makes you think VCAA had any “naturalness” in mind?
      The factor k is included purely to trap careless students.
      Leaving the sine function unintegrated is (I assume) to stop students from simply using a CAS to get the antiderivative. I think the intent here was to see if students could set up one of those (moronic) “integration by recognition”. (***)

      * Apart from reducing costs in marking, can anyone tell me what good purpose most of VCAA’s multiple choice questions serve?

      ** Ones that are trivialised by the CAS (such as Q5, Q18 [but use ‘nicer’ numbers]) can also be included (at an average of 3 marks per question).

      *** Much better tested in Exam 1. And we can be glad (live in hope) that this will/should no longer be part of the Specialist Maths syllabus now that integration by parts has (finally!) been added to the new Study Design.

      1. Thanks, John. Obviously the optimal amount of CAS is zero. Failing that, Exam 1 should be two hours, and include the multiple choice. Exam 2 should consist solely of a shortened B section.

        On MCQ 11, the underlying point is that a known derivative enables us to evaluate a non-obvious integral. The question kind of does that, but leaves off halfway, for no reason. Or, if as you, the reason was to avoid the use of CAS, it seemly becomes argument #374 that CAS poisons everything.

        I have no problem in principle with MCQ. I don’t care if they go. What has to go is CAS, together with the arrogant clowns that let loose this hell.

        1. Thanks, Marty.

          Re: “Q1 … the integral in (e) needn’t be computed, since the integral is then differentiated, but this application of FTC is too subtle for VCE.”

          It looks to me like the application of the FTC would require using the chain rule (since the terminals are functions of a and b), which would make using the FTC too subtle for Yr 12. In fact, I think the question is beyond the scope of Maths Methods for a number of reasons, including:

          1) There is no minimum area, only a limiting minimum area (of zero).

          2) Since a is also a variable, how VCAA intends the question to be done is wrong – I think VCAA is pretending that the area is a function only of b, but it’s a function of two variables, b \displaystyle and a. So I think this question is making students equate the \displaystyle partial derivative of area with respect to b to zero:

          1. Get the equation of the normal.
          2. Get the intersection points of the normal with y = g(x).
          3. Get the enclosed area A. A is a function of two variables, a and b.
          4. Solve \displaystyle \frac{dA}{db} = 0 for b. (But actually it’s \displaystyle \frac{\partial A}{\partial b} = 0).

          It is step 4 (and a bit of step 3) that criticism 2) is aimed at.
          Criticism 1) is aimed at the question implying the existence of a minimum area in the first place.

          I think the question is idiotic and wrong.

          1. John, please be way less cute and more direct with your criticism. I’m quite happy to bash VCAA, and possibly you have a point, which I’ll consider. But please make your point clearly.

            You seem to be suggesting that the question is not asking what was intended. Please state clearly why.

              1. OK, I think I understand what you’re saying, but I don’t buy it.

                If someone asks me to differentiate y=cx^2, I will naturally assume c is constant unless there is something to indicate otherwise. I don’t see how the exam question treating a as constant is any different.

                1. Why should a be treated as the constant rather than b? Why can’t there be an unknown fixed point where x = b?
                  a or b constant, there’s an assumption being made. I think both should be treated as variables. We will probably continue to disagree on this.

                  But be that as it may, do you think the minimum area exists. If so, what do you think it’s value is?

                  1. It’s not a matter of a difference of opinion.

                    I didn’t try to calculate the minimum since it got too lengthy to do by hand (or I couldn’t see the trick). I didn’t smell any issues, so currently have no reason to doubt that the problem is OK (modulo being CAS junk).

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