WitCH 37: A Foolproof Argument

We’re amazed we didn’t know about this one, which was brought to our attention by commenter P.N.. It comes from the 2013 Specialist Mathematics Exam 2: The sole comment on this question in the Examination Report is:

“All students were awarded [the] mark for this question.”

Yep, the question is plain stuffed. We think, however, there is more here than the simple wrongness, which is why we’ve made it a WitCH rather than a PoSWW. Happy hunting.

UPDATE (11/05) Steve C’s comment below has inspired an addition:

Update (20/05/20)

The third greatest issue with the exam question is that it is wrong: none of the available answers is correct. The second greatest issue is that the wrongness is obvious: if z^3 lies in a sector then the natural guess is that z will lie in one of three equally spaced sectors of a third the width, so God knows why the alarm bells weren’t ringing. The greatest issue is that VCAA didn’t have the guts or the basic integrity to fess up: not a single word of responsibility or remorse. Assholes.

Those are the elephants stomping through the room but, as commenters as have noted, there is plenty more awfulness in this question:

  • “Letting” z = a + bi is sloppy, confusing and pointless;
  • The term “quadrant” is undefined;
  • The use of “principal” is unnecessary;
  • “argument” is better thought as the measure of an angle not the angle itself;
  • Given z is a single complex number, “the complete set of values for Arg(z)” will consist of a single number.
  • The grammar isn’t.

42 Replies to “WitCH 37: A Foolproof Argument”

  1. So even the 1% that gave no answer (according to the Examination Report: https://www.vcaa.vic.edu.au/Documents/exams/mathematics/2013/specialist2_examrep13.pdf) were awarded the mark? Yep, that decision makes sense. A lot more sense than admitting that the question was defective and omitting it.

    And does the question really need the “Let z = a + ib” bit (and are we meant to assume that a and b are both real?). Why not just

    “Let z be a complex number such that the principal argument of z^3 lies in the interval \displaystyle \left(\frac{\pi}{2}, \pi \right)

    1. BUT… JF the question didn’t say that. It said “Second Quadrant”. The question to me which jumps out when I read “Second Quadrant” is, do they include Pi/2 and Pi as part of the quadrant or not?

      Which would mean square brackets were in order, depending on the answer of course.

      The “Let z=” is just stupid when the question is about polar coordinates, but incredibly common.

      1. So if nothing else, this ambiguity means that an explicit interval should be stated. I really dislike the whole “second quadrant” terminology in this sort of context.

        1. Yep. At the very least, the use of “quadrant” is needlessly confusing, and it is likely worse. Is “quadrant” ever formally defined?

    2. Hi, JF. I’m not sure of the practical difference between everyone receiving 0 and everyone receiving 1. (A place for Terry Mills’ partial credit MCQ?) But, yes, the unwillingness of VCAA to state explicitly that they fucked up after implicit but undeniable concession of the fuck-up is predictable and disgraceful.

      On the “z = a + ib” bit, I agree entirely. It is clumsy, confusing and completely gratuitous. For the life of me, I don’t understand how VCAA considers such sloppy writing, which is endemic, to be acceptable.

      1. I pity the inexperienced teachers who tried to use the Examination Report to get insight into this question. It could have been a really good teaching opportunity for VCAA (and/or the MAV) – admit the error, apologise for the error, and then explain the error.

        There was probably a deluge of questions like this one on trial exams in the following year – all making the same mistake. Note: The typical late release of the Examination Report for Exam 2 means that trial exams are generally finalised without the benefit of this report.

        I wonder what the commercial solutions had to say about it – the itute solutions gave a correct solution and then say that the best option is D. It would be interesting to know what the MAV solutions said – We all know what the MAV said (or, rather, didn’t say!) for the 2019 Exam 2 Q12 ….

        (And the MAV solution to the Maths Methods 2016 Exam 2 Section B Q3 part (h) is infamously terrible for what it doesn’t say about that VCAA error).

        1. Re: “I pity the inexperienced teachers who tried to use the Examination Report to get insight into this question. It could have been a really good teaching opportunity …”

          I would like to propose a solution to this question for the benefit of those inexperienced teachers, and at the same time suggest where the VCAA writers and vetters have probably made their mistake:

          Let \displaystyle z = r \text{cis}(\theta).

          Then \displaystyle z^3 = r^3 \text{cis}(3\theta).

          Case 1: \displaystyle \frac{\pi}{2} < 3 \theta < \pi therefore \displaystyle \frac{\pi}{6} < \theta < \frac{\pi}{3}.

          This is where I think the VCAA writers and vetters stopped, giving Option B as their intended answer.

          Case 2: \displaystyle \frac{\pi}{2} + 2 \pi < 3 \theta < \pi + 2\pi therefore \displaystyle \frac{5\pi}{6} < \theta < \pi.

          I don’t think the VCAA writers and vetters considered this case. Option E is the closest they get to it, and E was clearly intended to be an incorrect option.

          Case 3: \displaystyle \frac{\pi}{2} - 2 \pi < 3 \theta < \pi - 2\pi therefore \displaystyle -\frac{\pi}{2} < \theta < -\frac{\pi}{3}.

          It is hard to imagine that the VCAA writers and vetters considered this case (even though it is Option D) when they did not consider Case 2.

          The three cases give the complete answer to the question.

          I think the VCAA writers and vetters did not think beyond Case 1 and so their intended answer was Option B. But we’ll never know.

          Given that the set of values of principal arguments for z consists of the union of three sets, I doubt very much that there are any ‘typesetting errors’ that VCAA can hide behind.

          1. Hi, JF. Your conclusion, which feels correct, is what puzzles me. A basic understanding of De Moivre suggests the likely form of the answer: if z^3 is within an interval of length π/2, then the natural guess is that z will be within one of three intervals of length π/6. So, even without considering the cases as you have, none of the available options even smells possible, creating what one would hope is a red flag for checking.

            To me this suggests there are only two possibilities. The first possibility is that the third interval was somehow dropped from Option D during printing. The second possibility is that none of the writers or vetters have an utter clue about De Moivre.

            1. I think the natural guess leading to a red flag can be made even blunter:

              Any complex number has three distinct cube roots lying on a circle and spaced by 2pi/3, so you would expect three distinct intervals (each containing one of the cube roots).

              But VCAA vetters are obviously not capable of performing simple tests of reasonableness to ‘smell’ when something’s not right.

        2. Update:

          I’ve just noticed that the 2013 Examination Report has been updated for this question. The following comment now appears:

          The complete solution set [insert correct answer] was not included in the alternatives, so all students who attempted Question 6 were awarded the mark for this question.

          Much better than the original comment “All students awarded mark”. So credit to VCAA for finally acknowledging the error and making the change. BUT …. I’m not even going to try and find out what was so bloody hard about having this comment in the first place!

          By my reckoning the scoreline for amendments to Examination Reports raised by Marty’s posts this year is now 4 from 4. I heard that someone decent and capable in VCAA (yes, proof of existence!) chandled things and took the time and effort to facilitate all these changes (DuLL people were too busy with more important things to do). I tip my hat to her.

    3. There is more than wrongness in the question. There is a complete lack of understanding of principal argument in the brains of the writers and vetters.

      We will never know what option the writer had in mind – B or D – but the writer (and vetters!) clearly thought that simply multiplying the endpoints of an interval by 3 and then converting to principal arguments demonstrates a complete interval for the principal argument of the cube of z.

      Which is why the writer and vetters probably thought Option E was wrong – not realising that the cube of, for example, \displaystyle z = cis\left( \frac{11\pi}{12} \right) has a principal argument in the second quadrant viz \displaystyle \frac{3 \pi}{4}. The writers and vetters clearly did not understand that the second interval in Option E can include principal arguments of z with the required property for z^3.

      So again, we have VCAA exam writers and vetters who are writing and vetting beyond their intellectual capacity. A wise man once said: “A man’s got to know his limitations”. Clearly VCAA writers and vetters don’t.

      1. Hi, JF. I’ve been pondering along those lines: how did they up with no correct answer? Was it a misunderstanding of principal argument, or a misunderstanding of De Moivre, or an overlooked error in typesetting? Of course we can never know for sure, but usually an error will give clues to its origin. Here, I’m really not sure, but if it wasn’t a typesetting error, it is damning.

      2. Somehow, the inclusion of options D and E make me believe the examiners did not intend B to be the answer but a (clever in their mind) distractor.

        If only single intervals were given, then I would have assumed they intended B to be the “correct” answer until a teacher/student/parent or anyone with some level of knowledge of complex numbers wrote to VCAA to point out the error (that is how I will consider the story to have unfolded in my little dream world of online learning)

  2. One thing is sure – their mathematics is flawed, undoubtedly.
    And so is their grammar – “All students were awarded mark for this question”. If I wrote a sentence like that, I’d be crucified. Where’s the “a” or “the” between “awarded” and “mark”?

    1. Indeed, Steve. I guess in VCAA’s rush to release the Examiners Report some errors slipped by. Perhaps they should have spent just a little more time on it and then released it early in Term 3 rather than rush it out at the end of Term 2 ….

      Of course with the current DEET focus on literacy in mathematics, such errors are unlikely to occur these days …. We can be content with the knowledge that only mathematical errors will occur.

    1. Ha ha. I was being sarcastic (with both comments), but it’s quite possible that it really is VCAA’s idea of a rush!

      1. It is May and the paper 2 examiners reports are “not available yet”. No-one is in a rush at VCAA.

        Although I have seen worse – I have worked for a university education department… they make schools look efficient!

        1. I don’t want to sound arrogant, but is it possible the “We’re watching you” aspect of this blog is slowing things down? I don’t really think so, but it’s possible.

        2. Well, I imagine it does take some time to think about MC Q12 and then write

          “All students were awarded mark for this question.”

          (Yep, I’m calling it – I expect this to be the only acknowledgement by VCAA that the question was defective. But I also expect an improvement in their grammar).

          After all, with the current DEET decree on literacy, VCAA would be feeling pressure to get their grammar right (not so much the maths).

          So Marty, if there is a “We’re watching you” aspect of this blog that is slowing things down, we might expect my low expectations to be exceeded (or should that be unexceeded …?) when the Report is published later this year.

  3. Thanks, everyone. I’ll try to update this one very soon, so the backlog of WitCHes doesn’t grow. There is one more aspect of the question, which no one has mentioned, and which irritates the hell out of me.

      1. Thanks, Banacek. That also sounded pretty strange to me, although I think it may be acceptable in the school setting. Angles lying in a particular quadrant is standard.

  4. My reactions have been largely covered by others:
    (i) z = a+bi is not needed,
    (ii) a and b are not specified as real anyway,
    (iii) is a quadrant a closed set?
    (iv) there should be 3 intervals in the answer.
    I have one more worry. To me z as specified here is a number, not a set of numbers. So its principal argument will be one of three numbers, not a union of intervals. The question should say “the complete set of {\bf possible} values …”. Or is the wording used some standard that has escaped me?

    By the way. When I was at school we were taught to use upper cases for principal values (Arg, Artan, etc.) and lower cases with the general values (arg, artan etc.). Then at Melb. Uni. the convention was reversed. Strange because M.U. then strictly controlled the Year 12 examinations. Or maybe it was the other way around:-) What is the convention in year 12 now?

    1. Thanks, Tom. Your “one more worry” is the extra thing that really irritates me. There is no question that z in the questions refers to an unknown but specific complex number. As such, “the complete set of values for Arg z” must be a single (also unknown) value. (As such, your rewording also doesn’t save the question.) The wording is “standard” only in the sense that VCAA’s mangling of the English language is standard.

      On your other points:

      (i) and (ii) Yes, as JF mentioned, it’s pointless and confused.

      (iii) God only knows. My guess is that “quadrant” is never defined in the curricula or the textbooks, which is fine for the intuitive usage at lower levels, and which really really sucks on a Year 12 exam.

      (iv) Yes (although one can imagine questions where the intervals end up overlapping).

      As for principal values, I think “Arg” is standard in VCE, and I thought everywhere. If Melbourne uni once used arg etc. for principal values, I think that was pretty eccentric. But back to the question, although “Arg” is standard at VCE, I’m not sure the expression “principal value” is as commonly used.

    2. Hi Tom.

      The ‘conventions’ change every so often without explanation (but it’s probably political). Once upon a time the convention was to use capital letters for stuff like Arctan, \displaystyle \text{Tan}^{-1}, Tan (the restricted tan function). The unrestricted tan function was tan. But notation starting with a capital letter is explicitly not used these days, I don’t know why.

      As for arguments, arg denotes any old argument and Arg denotes the ‘principal’ argument (that is, the argument lying in an interval defined by convention. In VCE that interval is \displaystyle (-\pi, \pi] but \displaystyle [0, 2\pi) is also in common usage outside of VCE). The use of the term ‘principal argument’ is standard and appears in many exams.

      Once upon a time (back in the VCAB days), the ‘Study Design’ included a list of notation used, but not anymore. The curriculum documentation back then was long on mathematical content and short on administrative bullshit – the opposite of what it is today.

      1. Dear John friend:

        Re:“ Once upon a time (back in the VCAB days), the ‘Study Design’ included a list of notation used, but not anymore. The curriculum documentation back then was long on mathematical content and short ……. – the opposite of what it is today.”

        Do you know anywhere we could access those old papers and study designs?

        I made all possible attempts but I could only found up to exams in 1996. No earlier than that unfortunately.

        P.N.

        1. Hi P.N.

          I plan to scan my archive of old (60’s, 70’s, 80’s) exams at some stage, but not any time soon. I’ll let you know when it’s done.

          Re: VCAB. Marty might be able to forward some emails with attachments of that stuff.

        2. Hi, P.N. As JF suggests, I have a VCAB syllabus for Maths A and B from the late 80s. (Courtesy of a follower of this blog.) If you want to contact me (see the “Contact” link above, or send an email via qedcat.com), I can email you a copy.

          As for the old exams, I’d be interested in hosting an archive of them, but I doubt VCAA would give their approval, which I assume would be necessary.

          1. That would be a great resource and do all maths teachers a great service. I wonder if VCAA does hold the copyright (maybe it’s expired)? A shame if it does, because I doubt VCAA would want any of those old exams so readily available – far too embarrassing once teachers start comparing the standards of yesterday to the standards of today.

            It’s a very worthy project the MAV should consider pursuing – but its cowardly ‘maintenance of good relations’ with VCAA (https://mathematicalcrap.com/2020/05/13/mavs-sense-and-censor-ability/#comment-2871) would probably get in the way. These days the MAV is little more than a commercial arm of its master.

            1. Interesting question. This is one case where I might imagine DLL not objecting, but almost certainly VCAA management would, in a knee-jerk autocratic manner. It is also unclear how far back their copyright would extend, but I wouldn’t be guessing and it would be expensive to find out.

              A couple years back I worked really hard trying to get permission to make Fitzpatrick and Galbraith freely available. It failed, for reasons I still don’t really understand, and it was hugely frustrating. So, I think the old exams would be a great resource, but it’s not a battle I have any intention of undertaking.

  5. Hi,

    The much maligned glossary has an entry for Quadrant under Q

    https://www.australiancurriculum.edu.au/f-10-curriculum/mathematics/glossary/?letter=Q

    The wiki entry for Quadrant is better defined with presumably more reviewers

    The wiki entry uses the interval (–π, π] to define the ‘Principal’ argument but as JF mentions there are other conventions

    https://en.wikipedia.org/wiki/Quadrant_(plane_geometry)

    https://en.wikipedia.org/wiki/Argument_(complex_analysis)

    Steve R

    1. Thanks, Steve. Neither reference indicates whether or not the quadrants include the axes. But it also doesn’t matter. It is clear that there is no generally accepted definition of “quadrant”, which makes it ludicrous to assume such a definition in a Year 12 exam.

  6. Thanks John and Steve.
    The principal value range for the argument of complex numbers is of course associated with where you put the branch cut. The range (-\pi, \pi] follows from a cut along the negative real axis for the argument, and so for the complex logarithm. This is the usual cut. But for some contour integration I suppose a different cut is more convenient.
    Your “once upon a time” convention agrees with my memory of Year 12, 1962. We had artan, arsin and arcos for inverse values. At the time I thought this was a Victorian convention as other texts were still using \sin^{-1} etc. Then the rest of the world started to catch up with “arctan” etc. That made some sense with “arc” meaning an angle, but less sense when you get to “arctanh” and the like. So our “ar” prefix, presumably meaning “argument”, was superior 🙂

    When running the syllabuses at Swinburne and at VU I stuck to these notations (the superior ones). But erosion started with immigrant academics who seemed ignorant of our superior notation. Worse followed when Year 12 control was taken by teachers who had been indoctrinated to believe that functions can only have one value for each argument. But that is a rant for another time.

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