Secret 2023 Further-General Business: Exam 2 Discussion

Same as usual. Have fun.

UPDATE (31/12/23)

The exam is now posted, here.


56 Replies to “Secret 2023 Further-General Business: Exam 2 Discussion”

  1. What is going on with the proof reading of these exams? So a minor, insignificant error is noticed and announced to students before the exam (an extra “of” in a sentence, but the intended meaning is perfectly clear anyway). But a more serious error is completely overlooked – the transition matrix in 10b, which students were asked to complete, had a column with sum = 2.

    1. Thanks, SRK. I think that’s the error that was flagged to me.

      Without excusing the proofreading, how significant an error was it? Is the question undoable, or doable in different ways, or what?

      1. I think it’s still doable, because it’s possible to ignore the error, and use the other information in the question to correctly fill out the remainder of the matrix. But the other way of looking at it, is that the error means the matrix can not represent the situation described in the accompanying text, which might cause some confusion. I’d hope that the question just gets thrown out / the mark awarded to all students.

        1. Right. So if students are autopiloting then it doesn’t matter. But an alert student may lose time or composure.

        2. Agree but doubt that will happen.

          VCAA has form here when it comes to matrix questions on Further/General exam papers…

            1. True.

              The matrix was never referred to as a transition matrix though, so I’m not sure how many students (if any) would question the column sum.

              GM exams are button-pushing exercises done under strict time restraints. There is no time to think. This is not making excuses, it is more a statement of hope.

              1. True, the phrase “transition matrix” is not used. But part c. asks about the order in which the circus “will visit the five towns”, which implies that the model is intended to be deterministic, so having two 1s in a single column (and each of the other columns only has one 1) is nonsensical.

  2. My daughter submitted herself to that exam this afternoon. Among other things, I asked her if she thought the exam was fair. Her answer was equivocal. I then talked about my friend MR and his identification of difficulty in the halls of exam paper construction, amongst other things

    1. Thanks, Toby. I know very little about Further-General, and so tend to not comment unless there’s a blatant error to consider. It’s always tricky with a new curriculum. I think it’s difficult for VCAA to try to make the first exam fair, and I think it’s difficult for teachers and students to judge the fairness. Certainly, I can’t judge it.

      1. In this instance, the major change from the previous study design is that now ALL students are required to answer questions on Matrices and Networks, whereas previously students needed to choose to answer two modules from Matrices, Networks, Trig & Measurement, Graphs and Linear Relations. (Although, the overwhelming majority chose Matrices and Networks anyway.) So I can understand VCAA wanting to be cautious about the difficulty of the Matrices and Networks questions – although, interestingly, the final Networks question on Exam 1 replicated the final question on the 2022 Exam 2, which was answered extremely poorly (98% received 0 marks).

        But the Data + Finance parts of the course are pretty much identical to previous years, so I found it a bit strange that this year’s Exam 1 was easy compared to previous. I think Exam 2 had a better difficulty curve for these topics, even though no single question will pose a significant challenge for the strongest students.

    1. Thanks, Chris. By VCAA’s standards, yesterday’s Further errors would appear to be small beer. (I haven’t seen the exam yet.) But the reporters, reasonably enough, smell blood.

      I’m not convinced VCAA will continue to deny Methods-Specialist errors in the manner they have in the past. There was a lot of fallout from last year’s screw-ups (and not nearly all the fallout has yet fallen out). One would hope that the writers-vetters have been more careful this year, but we’ll see. But, if there are significant errors again, I think there may be less of an ostrich strategy to deal with them.

      1. When the Minister says “We unreservedly apologise”, you can be sure there’s some action behind the scenes.

        I am guessing that the previous Minister was assured the problem had been fixed. This one’s new to the portfolio and a Minister on the make – interesting to see how it’s handled, especially after it’s been brought to his attention in this manner!

        1. The previous Minister referred a previous complaint about errors on exams back to the VCAA. You can guess how that turned out. I don’t have a lot of confidence in unreserved apologies, investigations etc. unless they lead to genuine structural and cultural change.

          1. You’re right BiB – getting genuine change is very very hard. These things are a war rather than a battle. However progress has been made – already the strategy has changed with VCAA emailing schools.

            The campaign is having an effect with multiple media articles. A new Minister having to apologise will be very angry and he will want to be seen as fixing this (he won’t care about cultural change). Initially they will reassure him that this is a one-off in spite of new procedures. Another one and he won’t be happy at all!

            Over time such incidents tend to build up and put a spotlight on agencies. Whether positive change results over the longer term is another matter, but this is a necessary first step for change.

            1. Anyone who doesn’t think that cultural change is just as important as structural change is blind to the obvious.

              And you’re right – every Minister only has a finite number of apologies in them. They will not be wanting to waste them on repeated failures by an agency in their portfolio.

              1. The main thing is that if there are more errors – there will be another apology but my guess is that there will be serious action – eg at least someone will lose their job. This Minister is an aspiring Premier and will be very keen to demonstrate who’s in charge.

                1. Maybe. It depends where in the food chain the buck stops. Higher up the food chain you have career bureaucrats. Unless they’ve committed a criminal act (and crimes against mathematics don’t count), they just get promoted to some other department where they can possibly cause less harm.

  3. This is the email that came to schools;

    Dear Principal and VCE Coordinator

    Following the identification of a typographical error in the matrix stimuli in question 9d of the 2023 General Mathematic Examination 2 paper, the Victorian Curriculum and Assessment Authority (VCAA) has taken the decision to award all students who attempted the exam a correct score for this question. This question was worth 1 mark out of a possible 60 for the exam. This decision has been made on the principle that this is the most effective and appropriate way to ensure no student will be disadvantaged and that the assessment process is fair, valid and reliable.

    The VCAA reiterates our apology to students for this error. Please use the text below to notify students who sat the exam of this outcome.

    No further action will be taken following the pre-emptive correction of the typographical error in question 14d where students were instructed before the examination began to remove the word ‘of’ from the question.

    A comprehensive review of the vetting and proofing process for VCE examinations will be undertaken with any recommended changes to be implemented for 2024 examinations.

    We apologise for the undue stress this has caused. We are disappointed that these errors occurred and are committed to reviewing our processes and strengthening our vetting and proofreading of examination papers prior to the 2024 VCE examinations.

    Student Communication

    “Students are advised that, following the identification of a typographical error in the matrix stimuli in question 9d of the 2023 General Mathematic Examination 2 paper, the Victorian Curriculum and Assessment Authority (VCAA) has taken the decision to award all students who attempted the exam a correct score for this question.

    This question was worth 1 mark out of a possible 60 for the exam. This decision has been made on the principle that this is the most effective and appropriate way to ensure no student will be disadvantaged and that the assessment process is fair, valid and reliable.”

    If you have any questions, please contact the VCAA Examinations Unit on (03) 7022 5550 or

    Yours sincerely

    Stephen Gniel
    Chief Executive Officer

    I’m actually shocked they didn’t say it wouldn’t have impacted students as they usually do….

  4. In General Mathematics, Paper 2, Q1, I would have written “Data were collected …” rather than “Data was collected …”.

    1. Data is a collection. The collection is a singular thing. I think “Data was collected” is OK.

      (“Data were collected” sounds off).

  5. I think there are 2 acceptable answers for Q11. B can have either column matrix [9 0] or [9 -9] depending on whether the state matrix is keeping track of workers who have left. Question does not distinguish.

    1. Yes, I agree the question was unclear. But if they’re going to accept \begin{bmatrix}9 \\ 0 \end{bmatrix}, then shouldn’t they also accept any answer of the form \begin{bmatrix}9 \\ x \end{bmatrix} where -9 \leq x \leq 0? For instance, \begin{bmatrix}9 \\ -2 \end{bmatrix} would represent there being 2 workers who were going to leave but then change their mind and continue to work for the next show, and so the circus needs to hire 7 new workers to bring the total up to 180.

      1. I actually think [9 0] is the better answer. It really depends on whether the people who leave are still considered ‘workers’ as the people who went [9 -9] are only arguing that the state matrix should retain a sum of 180, really the question only requires category W to stay at 180. If L is supposed to represent workers who have left [9 0] is right. I think the description that workers “do not return” prevents any of the other answers being reasonable.

        Of course the scenario of using a matrix to track this type of thing in real life is absurd so there are no clues from what would be realistic.

  6. Thank you all for your comments. I’ve had no time to look at the exam(s), even the matrix error that has received all of the attention. I will try to put up a brief post tonight on the various responses to this error, but I’m dead tired. So, if the post begins “The General examfghtrefcwetiofe …”, you know I’ve fallen asleep at the keyboard.

    I’ll carefully read all your comments on the week-end.

  7. I was alerted to the following video. It suggests there’s another error on Exam 2: more subtle, but worse. I haven’t checked it out, and cannot vouch for it, but Bryn is usually careful and correct:

    1. I guess my only question is what does it mean to fully repay a loan? Does it mean that the last payment puts you exactly at zero, or does it mean that the last payment can put you a bit below zero?

      Either way, you’ve repaid the loan. If the former, yes the question is broken – there’s no correct answer (no rounding puts you exactly zero). But if the latter, then there’s an answer (there’s a way of rounding that puts you a bit below zero).

      But I agree that the model doesn’t work if only two decimal place accuracy is used. In real life our repayments are probably made to more than two decimal places because it’s all digital. You can make payments to more than two decimal places. And I suppose it’s all figured so that the amount owing after the last payment is somewhere between 0.004 and -0.004 dollars … Which rounds to 0.00.

      it seems to me that the question requires a little bit of maths, a lot of button pushing and an understanding of financial matters beyond mathematics.

      PS – It’s a very nice clip. Concise and clear.

      1. I haven’t watched this carefully or tried to really get it. But I think Bryn’s main point is, there’s been an accepted interpretation in VCE of what “repay the loan” means. However, that interpretation doesn’t work, or at least not automatically or cleanly, for this variation of the problem.

  8. I see this issue as similar to that in Specialist Exam 2 2018, Question 6e. That question asks for the smallest sample mean that would result in the null hypothesis not being rejected, rounded to two decimal places. The answer is something like 146.513…, so 146.51 to two dp. However, a sample mean of 146.51 would result in the null hypothesis being rejected, hence a lot of students gave their answer as 146.52. Both answers were accepted.

    In both exam questions, it seems to me that the issue is whether the *rounded* answer should also meet the specified condition (resulting in null hypothesis not being rejected; bringing a debt down to zero), or whether we regard the rounding instruction as something which is done primarily for convenience (ie. to avoid writing down heaps of digits or some otherwise too complicated expression).

    I agree with everything Bryn says – he’s correct that the 2016 – 2022 exams were more pedantic about this issue, in either the wording of questions, or the numbers used; I agree that both 350.01 and 350.02 should be accepted. However, I actually prefer the 2023 way (and 2006 – 2015 way) of wording the question compared to the tedious “155 equal fortnightly payments with one final payment of a different amount” – it should just be a standing assumption that the final payment may be different, and there’s no need to consider the final payment adjustment unless specifically instructed. In the context of working out a weekly payment to pay off a nearly $50000 loan in 3 years, who cares about coughing up another 52 cents at the end?

    1. Instead of an “accepted \displaystyle interpretation in VCE of what “repay the loan” means”” (my emphasis), there should be a clear \displaystyle definition stated somewhere. And maybe there is, does anyone know? I can’t find anything in the VCAA Mathematics Glossary or Study Design.

    2. I think there is a distinction between the cases where there is a genuinely continuous quantity being rounded and when the underlying quantity is actually discrete (such as where there must be an integer number of cents).

      Here is an example of the issue occurring in a continuous question

      Q: Find the smallest number, k, such that k × k is at least 10. Round your answer to the nearest integer.
      A: 3 (even though this particular value of k doesn’t satisfy the description, it is the answer to the question of “round your answer, sqrt(10), to the nearest integer”)

      But in the case of loans, I do not believe it is possible to pay the bank a fraction of a cent in a monthly payment, or in fact in any transaction between distinct parties using standard currencies.

      So now it is more like this question:

      Q: Find the smallest number of people in a room such that, if you were to square that number you get at least 10. Round your answer to the nearest whole number.
      A: 4

      Because the possible numbers of people in a room are {0, 1, 2, 3, 4, …} you are forced in this case to just choose the smallest integer satisfying the description (4), and then in fact there is no need to do anything with the instruction to round.

      1. Bryn, I agree with you but I want to play the Devil’s Advocate (and please tell me where I go wrong):

        We’re assuming that the payments are essentially a cash transaction and hence can only be made to the nearest cent. But can the payments be considered a digital transaction, in which case the payment can carry any number of decimal places and we simply round at the end to get an answer (if so, would it have made sense for the question to have asked for, say, an answer to four decimal places?). I’m not fluent with the content, so I’m asking just to learn more about this.

        1) Is this the first time this sort of ‘issue’ has arisen?
        2) Has ‘repay the loan’ been defined by the VCAA anywhere?

        1. I think VCAA has never come out and said that we are working with cash (or perhaps more precisely, just Australian Dollars between real bank accounts, since cash itself is rounded to the nearest 5 cents in Australia).

          However, they have always had it so the regular payment is given the two decimal places since Recursion and Financial Modelling was introduced in 2016, and combined with the fact that this is how banking usually works in Australia, I guess most teachers have just assumed it (one of the key skills in the study design does involve using technology “to solve practical problems associated with compound investments and loans”).

          The concept of “fully repaying the loan” meaning repay it to zero can be inferred from the way it has been asked in almost every prior exam combined with the corresponding examination reports. For instance, most recently 2022 question 8 talks about a final repayment required to fully pay off the loan, which here means get it down to zero. In fact, the concept is even there in question 5 of this very exam, where they understand that the final payment needs to be different in order to get the result to zero.

          It could make for an interesting defence of the question if VCAA claims that they were talking about some form of cryptocurrency where in fact transactions can be conducted to more decimal places, but they just asked for the final answer to the nearest cent, and it’s just been a coincidence that they never have chosen to have the regular payment given to more than two decimal places. In some ways I would have preferred if that’s how financial maths had been set up (as it’s a lot easier to do the Maths then) but it seems odd to introduce it after seven years of being in the study design.

  9. Regarding question 7 (discussed by Bryn in the video linked by Marty), I think two solutions are possible for d.
    The ‘expected’ solution is d=0, for which the sequence V_n is a geometric sequence with common ratio = 1.0015.
    Setting d=90 results in the sequence V_n being constant/a geometric sequence with common ratio = 1.

    Not sure if the (somewhat trivial) case in which the common ratio is one is within the General Maths definition of “geometric sequence” (and/or the definition generally accepted by the broader maths community)? The question implies only one value of d exists.

  10. A comment which follows on from a few of the previous replies:

    1) On the issue raised by Bryn’s video about the repayment amount required to pay off the loan: apparently both 350.01 and 350.02 will be accepted as correct answers.

    2) On the issue raised by Marcus that d = 90 results in a geometric sequence with common ratio = 1: apparently d = 90 will NOT be accepted.

    3) On Q11b, apparently both \begin{bmatrix}9 \\ -9\end{bmatrix} and \begin{bmatrix}9 \\ 0 \end{bmatrix} will be accepted.

    4) Fresh hell: the very first question in the Matrices section states that the matrix N = \begin{bmatrix}36 \\ 15 \\ 8\end{bmatrix} lists the prices of, respectively, family, adult, and child tickets to a circus. Furthermore, it is stated that “The element in row i and column j of matrix N is n_{ij}“. Part a. is “Which element shows the cost of one child ticket”. Apparently accepted answers include “n_{31}” and “8” but not “N_{31}“.

    5) Old hell, but still diabolical: Question 2, part b ii asks students to explain why the farmer is correct to believe that farm A has greater capacity to grow larger oysters than farm B. The intended answer is something like “21% of oysters grown at farm A are large, which is larger than the 14% of oysters grown at farm B which are large”. However, not accepted is “21% of oysters grown at farm A are large, whereas 14% of oysters grown at farm B are large”.

    1. Thanks, SRK. I’ll try to look tomorrow. Two quick questions:

      a) Are 2 and 3 reasonable decisions?

      b) Are 4 and 5 as nasty and as nuts as they would appear to be?

      1. To me, (4) doesn’t seem nasty/nuts. In MM/SM, students are expected to distinguish between lowercase and capital letters, e.g., writing f(3) or F(3) seems different (or t vs. T in temperature/time situations). So it seems reasonable that if you’re told the notation is n then you should use n and not N. And maybe they have to accept 8 as an answer because that is an element of the matrix so the student has literally answered the question that is asked.

        With (5) I would be curious about whether students need to specify it’s larger in order to get the mark, or whether it’s enough to say it’s just a different value. I guess it’s implied by saying “whereas” but perhaps they want students to explicitly highlight that they are different, and one way to do so is to say that one is larger than the other.

          1. I haven’t formed a view on them, sorry.

            My main concern is actually that the rounding issue in financial maths has highlighted that some people are of the view that payments are a continuous variable (rather than a discrete: number of cents being transferred from one party to another).

            I am assuming that everyone knows that N (number of periods) is discrete and therefore even when the calculator spits out N=8.231 you have to think about whether 8 or 9 is the correct answer.

            But having heard people say “normal rounding rules always apply” I am concerned that the distinction between continuous and discrete has been lost and it becomes unclear how one ought to teach financial maths in the light of this confusion.

            1. Thanks, Bryn, and no problem. General-Further is not my world, so prefer to get smart opinions before I offer my stupid opinions. Having said that, I intend to to slap you, and presumably the graders, on (4) and (5).

        1. (5), as SRK as suggested it will be graded, is plain nuts, and plain nasty. There is no stylistic or logical reason to include “greater” in the answer.

          If I ask “Why is Bill older than Ted?” then a perfectly reasonable form of answer is “Because Bill is 5 and Ted is 3.” There is no need to include “5 is greater than 3” because the inequality is already indicated by the “older” in the question.

          Same thing here.

        2. (4) is both nasty and nuts.

          There is nothing remotely ambiguous about writing \boldsymbol{N_{ij}} for a matrix entry, and is in fact very standard notation. The fake confusion here doesn’t remotely compare to t for time and T for temperature, or even to t for variable time and T for final time.

          Sure, the exam specifies the notation \boldsymbol{n_{ij}}, but is \boldsymbol{N_{31}} closer to true or false? Is it closer to \boldsymbol{n_{31}} or to “Bob ate a fish”? But you’re telling me it’s fine that \boldsymbol{N_{31}} scores the same 0 as “Bob ate a fish”?

          One of the problems, of course, is asking trivial questions, which you have to do in General.

          But the real problem is the assessors. The utter lack of empathy, of basic humanness, astonishes me. It really feels as if they are always looking to be nasty, and there are numerous cases on the record when they obviously were.

      2. 2) No. The only thing close to a definition of geometric sequence in the study design is from Unit 1 General Mathematics, where the topic of “Arithmetic and geometric sequences, first-order linear recurrence relations and financial mathematics” is stated to include: “use of a first-order recurrence relation of the form u_o=a,\;u_{n+1}=Ru_n where a and R are constants, to generate the values of a geometric sequence”. No conditions / restrictions are placed on the possible values of R, hence the study design does not rule out the sequence 60000, 60000, 60000, … (which is what we get from setting d = 90 in the 2023 Exam question) from being geometric.

        3) Yes. The question was poorly written. Although – and this applies to point (1) as well – in general it’s a bad sign when this sort of thing is happening often, since it inevitably means that students will waste time in the exam second-guessing what they are supposed to do.

        4) Yes, I think so. I’m happy for “8” to be accepted as an answer, but disallowing N_{31} seems nuts to me. I appreciate Bryn’s point, but I don’t think those considerations apply to this question or to General Maths – it’s perfectly clear and unambiguous what is meant by “N_{31}” in this context.

        5) Yes, the expectation is that students state that 21% is larger than 14% – otherwise, no mark. This is insane, but it is an insanity of which any reasonably experienced or well-informed General Maths teacher (and well trained student) will be aware.

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