**UPDATE (31/12/20) **The exam is now online.

This is our post for teachers and students to discuss Specialist Exam 2 (not online). There are also posts for Methods Exam 1, Methods Exam 2 and Specialist Exam 1.

**UPDATE (03/12/2020) **

We’ve now gone through the multiple choice component of the exam, and we’ve read the comments below. In brief, and ignoring the screw-ups, most of the questions seemed good, and a number of questions were hard (which is good). We haven’t thought much about the extent to which the questions are trivialised by CAS/Mathematica, although this is of course extremely important; the comments below on this aspect are well worth a careful read.

Here are our question-by-question thoughts:

**MCQ1. **A decent and non-trivial stationary point question. A pretty mean way to begin.

**MCQ2. **A contrived and tricky range of function question. A very mean way to continue.

**MCQ3.** A rather weird piecewise constant acceleration question.

**MCQ4. **A good and not so easy composition of functions question.

**MCQ5**. Intrinsically a routine and good complex algebra question, but the presentation is a mess. The notation is introduced, but then plays no role; indeed, the question would have been vastly improved by having the offered answers expressed in terms of and . Requiring some extra algebraic manipulation to obtain the correct answer is needless, and a little contrived.

**MCQ6.** A very easy complex factorisation question.

**MCQ7.** Ugh! See here.

**MCQ8.** A nice complex algebra question.

**MCQ9.** Complete nonsense, as flagged by commenter Red Five, below. See here.

**MCQ10.** A routine tank mixture problem.

**MCQ11.** A screw-up, and perhaps a semi-deliberate one, as flagged by commenter John Friend, below. See here.

**MCQ12.** A straight-forward but nice Euler’s method problem.

**MCQ13.** A standard linear dependence problem. As noted by commenter John Friend, the problem is trivial with 3 x 3 determinants, which is not on the syllabus but which is commonly taught for this very purpose.

**MCQ14.** A straight-forward force component question.

**MCQ15.** A nice parametrised curve question.

**MCQ16.** A nice dot product and double angle formula question.

**MCQ17.** A straight-forward acceleration as a function of distance question.

**MCQ18.** A straight-forward but nice string tension question.

**MCQ19.** A cricket ball with a mass of 0.02 kg? Otherwise, a nice change of momentum question.

**MCQ20.** A straight-forward but nice force and acceleration question.

**UPDATE (04/12/2020) **

We’ve now gone through Section B (extended question) of the exam, and we’ve read the comments below. There do not appear to be any significant screw-ups, but most of it is pretty poor. In the main, the questions are aimless and badly written, with CAS washing away the potentially good effect of any decent content. Nothing is quite a WitCH or PoSWW, but almost everything is close.

Here are our question-by-question thoughts:

**Q1. **A strikingly aimless parametrised motion question. Seriously, who gives a shit about any of it? Part (b)(i) asks for dy/dx as a function of t, to “hence” obtain the equation of the tangent at t = π, when it is more natural and simpler to first evaluate dy/dt and dx/dt at π. Then, (b)(ii) asks for the velocity at π, for which you need … This is stupid with a capital stupid.

**Q2. **An OK complex geometry question, which begins thusly:

*Two complex numbers, u and v are defined as and .*

Jesus. What’s wrong with “Let and “? The symbols and are pretty crappy choices for fixed complex numbers, and the later choice of for the centre of a circle is *really* crappy. Part (d), finding the centre and radius of this circle, would be a nice question in a CAS-free world.

**Q3.** The best question, graphing and then finding the number of inflection points of for . Much of the goodness is killed by CAS. It is not entirely clear what is meant by “asymptotes” in part (b). (See the discussion here.)

**Q4.** Another parametrised motion question, this one involving a pilot seemingly unaware of the third dimension. Pointless and boring CAS nonsense.

**Q5.** An absolute mess of a dynamics question. The diagram is shoddy. The appropriate range of the frictional parameter should be given or determined *before* asking students to compute a Fantasyland acceleration. Part (e), which feels like an afterthought, involves a jarring and needless switch from the algebraic to numeric, with a specific velocity and implausible force plucked from thin air.

It was good to see a few *new* multiple choice questions that didn’t reward blatant button-pushing. Of particular note, I thought Q4, 5 and 8 were pretty nice (come on – tell me why I’m wrong…)

Q9 though… I think I know what the intended answer was (B) but with the absence of any reference to a differential equation, I am a bit lost as to exactly what “curve” the question is actually referring to.

It seemed a bit… odd. But I do not know enough about differential equations to know if this is an error or not (and I get the feeling some regular readers will soon be able to assist)

I agree, RF. And I also liked Section A Q2. But Q8 was trivial with button pushing: Just let x = 2 and y = 1, say, calculate the powers and then choose the consistent option.

Re: Q9. A simple diagram facilitates getting the DE . Then you can either button push to get the slope field and choose the compatible option (trivial with Mathematica using StreamPlot), or you can use your brain and eliminate all the wrong options in the following way:

therefore only options B, C and E are viable.

and therefore only option B remains.

A couple of superficial observations (I haven’t had a chance to sit down and do the exam in detail):

I disliked Q13 (“The vectors … will be linearly dependent when the value of is”) because well-taught students can just set the determinant of an appropriate 3×3 matrix equal to zero – absolutely no understanding is required.

I thought Section B was *mostly harmless*. It’s been a long time between drinks when students last had to state the geometrical interpretation (Q2 (c)) of as ‘the line that is the perpendicular bisector of the line segment joining z = u and z = v.’

I thought Q3 was very interesting – parts (d) and (e) are clearly intended to ‘force’ teachers to include a generalisation in their SACs …

I found Q8 to be pretty simple without choosing specific values, but yes, in this case it would be an easy way to get a correct answer.

The geometry of the Argand diagram has been so much about circles and rays lately that seeing a perpendicular bisector was nice, although all my students will know this ad nauseum from Year 11.

Overall, I felt Part B was a lot easier than previous years. Part of me wonders why (given what happened in Methods)

Still not convinced MCQ9 actually works as a question, mainly because I’m not convinced that “a curve” and a solution to a DE have to be the same thing just because a slope field is given. Don’t have the mathematical knowledge to know where to begin to understand if the question is valid or not – hence I come to Marty’s blog!

I should add that Q11 is an improper integral (I wonder if the exam setting panel even understand what these are since they keep appearing) but no calculation is required so I suppose it can go through to the wicket keeper.

And a lucky thing no calculation was required. The integral doesn’t converge! Maybe that was deliberate so that students couldn’t compute the value with a CAS and then compare with the value from the options … (Still, it’s a cheap trick). Here is the question:

Question 11

With a suitable substitution can be expressed as

Thanks, John. It’s worse than a cheap trick. Posting on it now.

I should add that Q11 is an improper integral (I wonder if the exam setting panel even understand what these are since they keep appearing) but no calculation is required so I suppose it can go through to the wicket keeper …

Thanks, JF. Q2(c) puzzled me, because I thought of the geometric interpretation as the natural way to get the equation of the line in 2(a). Isn’t that typically taught?

Definitely B3 is the nicest question on the exam, although, as always, it’d be infinitely better without CAS. I’m also assuming the final table of inflection points requires only answers; if some explicit justification is required, that’s a hell of a lot of work for 2 marks.

Hi Marty.

Re: Q2(c). Yes, it *is* typically taught AFAIK. And many students will have chosen to answer (a) using this method. So I agree that part (c) is puzzling – it will be interesting to see the performance of the cohort in the Examination Report.

Re: Q3. Well, 7/10 of the marks are Maths Methods marks so I don’t think it’s much chop as a *Specialist* Maths question. And most of those 7 marks are trivialised by the CAS. I can’t see any requirement for showing change in concavity so even the 3/10 Specialist-specific marks are weak as piss.

I think (e)(ii) is intended to push the not-so-secret agenda of getting teachers to include generalisations in their SACs. It wouldn’t be the first time VCAA has tried to ‘nudge’ teachers through exam questions. I’d guess 1/2 a mark for each answer – justification is clearly NOT required because it simply says “… stating the value(s) of n”. Plus there are no writing lines, only working space. Now, if it had said “… *finding* the values of n” …

So sorry, Marty. I think the question is a shit Specialist question. If it was on a Methods Exam 1 and reference to points of inflection deleted, it would be a nice question.

Yes, that’s a very good point. I still think it’s a nice question (in a CAS-free world), but it’s a nice Methods question, not a Specialist question.

Re: Q2(c). The state mean was 0.55.

I have more to say about Q3 below …

Thanks, John. I removed your double-comment. Does the 0.55 suggest anything?

Re: Q4

If we’re going by the maximal capability of what any of the allowed CAS calculators can do (i.e Mathematica) then unfortunately, Mathematica can completely steamroll the question. The code below does just that:

FullSimplify[f[g[x]], 0 < x < \[Pi]/2] // TrigReduce

FunctionRange[{1/2 Sin[2 x], 0 < x < \[Pi]/2}, x, y]

If we're talking about the TI-nspire, it will also simplify the function (albeit with a mod sign) and all that's left is to determine the range of the function, which is more or less trivial. Nothing really tested if any student just threw the expression into their CAS.

Re: Q5

I don't really have any issues here, it's a decent way of testing properties of complex numbers.

Re: Q8

Just a question here, would it be frowned upon if instead of manipulating the expression into one that relates to the original expression, a student chose to plug in arbitrary values of x and y?

Re: Q9

What JF said, although personally using StreamPlot requires some good eyesight… In case the derivation was not so obvious, you would do it like so:

Substitute then rearrange for x as follows:

Then finally equate that for the value at that point:

Finally, rearrange for and you're ready to squint your eyes at 5 different options. I'm not really certain on the validity of the question myself however….

Hey Sai,

Q4 for MCQ would’ve been a nice composite functions question if it wasn’t trivialised by Mathematica/CAS. I solved the question in reading time (it caught my interest!) whereas my friends just brute forced it on the TI Inspire, LOL!

I was surprised that the geometry of circles didn’t come up this time for Q5. I felt that this exam was very physics themed and would be nicer with more creative geometry.

Indeed. It would have made a nice Exam 1 question.

Please allow me to reiterate my objection to multiple choice questions. The purpose of an examination is to assess where students are up to in learning about the subject. Suppose that my answer to a multiple choice question is (C). What does that tell you about my learning?

If (C) is the correct answer, can you infer that I understand what is being tested? Not really – I might have guessed, or I might have got to (C) because I eliminated the alternatives but have no idea why (C) was correct, or I might have meant (B) but accidentally wrote (C). In any case, my score is 1.

If (C) is not the correct answer, can you infer that I don’t understand what is being tested? Not really – perhaps I made a mistake in a calculation, or I knew the correct answer and accidentally wrote (C). In any case, my score is zero.

Now the response of experts in assessment is that you don’t assess a student’s learning about a topic by a single MC question. You should ask several MC questions on the same topic to get an understanding of the student’s learning on the topic. This process is called triangulation. But in an examination, one does not ask several questions on the same topic.

What does (C) tell us? Thus endeth the lesson.

I should add that multiple choice questions have one, and only one, advantage.

TM, as far as the VCAA exams are concerned, I am in complete agreement with you. I hate them on the VCAA exam – they are purely a stupid cost saving measure. I’d like to see them wiped from the face of the Earth and get either a shorter Exam 2 (90 minutes) or two extra extended-response questions.

Thanks, RF. I’m just going through the exam now. I pretty much agree, although MCQ5 annoyed me in the way it was presented. MCQ9 is, as you suggest, ridiculous. I’ll post on it soon.

Would u lose a mark if it asks to state the function and instead u used y=… then said domain of x is…

Which question are you talking about?

The complex numbers question where the argument of u is pie on 4 and it asked for the equation of the line. I had y=x+1 and stated the correct domain but it wasn’t in the full like function format

Your answer is fine. Your 1 mark is in the mail.

I’m curious how Section B Question 4d (asking if the drone makes contact with the aeroplane, giving reasons) will be marked – in particular, what will count as sufficient evidence.

I imagine many students will just use CAS to solve when the horizontal components are equal and then to solve when the vertical components are equal; CAS gives approximate solutions only, but there is no common solution, so no collision. But since CAS only gives approximate solutions, will students be required to give evidence that there aren’t OTHER solutions that perhaps the CAS is not finding? For instance, perhaps by sketching appropriate graphs and showing the locations of zeroes.

Good question. Who knows what pinheaded reasons will be required. However if common sense prevails (a huge assumption), then

“use CAS to solve when the horizontal components are equal and then to solve when the vertical components are equal; CAS gives approximate solutions only, but there is no common solution, so no collision”

will be sufficient to get the 3 marks:

1 mark for both equations,

1 mark for the approximate solutions to each equation,

1 mark for a conclusion based on those solutions.

And I suspect explicit inclusion of the domain 0 leq t leq 40 somewhere will be required.

A CAS calculator does not warn that other solutions might be possible. Mathematica gives all of them (this is certainly expected from Mathematica and assesors won’t know who’s using Mathematica …)

The plot of y = 450 – 150 Sin[pi*t/6] and y = 30*t for 0 leq t leq 40 is interesting – there is *nearly* a second solution which is very close to one of the solutions to 400 – 200 Cos[pi*t/6] = -t^2 + 40*t ….

SRK, what is the alternative to what you are suggesting? There’s no way to get exact solutions to the two equations.

I generally get a bit nervous when my TI just gives me approximate solutions (ie. perhaps it is missing some, although I know that it is supposed to give an error / warning message in these cases), so for this question I used the CAS to display the graphs of and and roughly sketched those on the page, along with approximate location of zeroes, just to confirm that there’s none in common.

So, what do u guys think the A+ cut-offs are for both of the exams????

I’ll answer this question, only because the cut-offs generally reflect how difficult the cohort found the exam. (Otherwise who gives a flying truck??? This isn’t ATAR notes. Study Scores depend on rank not artificial lines drawn in the sand).

Answer: I expect the cut-offs will be similar to what they are every year (https://www.vcaa.vic.edu.au/Documents/statistics/2019/section3/vce_specialist_mathematics_ga19.pdf). Ta da!

JF what about parts of question 5 where they ask for the time and distance in terms of k?? Are answers which are in terms of g acceptable or do we have to further simplify it??

Who knows. VCAA will make it up on the day like it always does. Consistency and transparency is not VCAA’s strong suit. See the brief discussion here: https://mathematicalcrap.com/2020/11/20/secret-specialist-business-exam-1-discussion/#comment-5301

Roronoazoro, it’s an excellent question. (I means yours is an excellent question. The exam question is woeful.)

Given (b)(i) leaves as , I think the natural interpretation of (c) and (d) is to do the same. I don’t see how substituting could be regarded as incorrect, but who knows with these clowns?) It is all muddied, however, by the idiotic part (e), which of course requires using .

Question 2 (11 marks)

Two complex numbers, and , are defined as and .

b. Plot the points that represent and … on the Argand diagram below. (2 marks)

d. i. Sketch the ray given by on the Argand diagram in part b. (1 mark)

Hands up those who see the conundrum here. (If was a cat it would surely be called Schrodinger).

Incidentally, surely the preamble would have been much better as “Let two complex numbers and .” (But VCAA loves its bloated sentences).

JF what made that part d. i. so annoying was that the ray plays absolutely no role in the rest of the question – it could just as well have been Arg(z – i) = pi/4 without any loss.

I read part d. ii. – which strongly suggests that students need to write down the domain of the line – as a tacit admission that it’ll be difficult to tell from responses to d. i. if students have drawn the graph correctly, so hopefully this means students won’t be penalised for an ambiguous graph at the endpoint.

You’re right about the ray. And good point (pardon the pun) about part d. ii.

SRK, rereading the question, I think you’re probably correct. The wording “Write down the function” is reasonably read to require a domain as well. The casual “write down”, however, subliminally suggests otherwise. I’m guessing that the domain is required and that many students will not include the domain, costing them, presumably, 100% of the marks on this question.

Yes, the non sequiturishness of the question is also notable, although pretty much the whole exam is one big non sequitur. I wonder if it is intended as a semi-hint to part (e), which can nicely be done by hand. But probably not: why think when you can CAS?

JF, apart from part (d), I really liked this question because it was refreshingly different to the way loci of the complex plane have been examined in recent years (what with the areas of sectors and over-reliance on rays and circles).

Sure, it isn’t anything a middling Specialist Unit 2 student couldn’t handle (but that could be said for any of the non-calculus parts to a certain extent). I just felt this was actually a decent question…

…until part (d).

JF, the preamble is indeed awful.

My hand is nervously raised for (d)(i). Is your concern that is sometimes used as the real part of ? No question that is a crappy choice for a fixed complex number, and the wording is pretty sloppy.

Marty, think about how you would plot the point u in part (b). Now think about how you would sketch the ray in part (d)(i) – in particular, what has to happen at the terminus …

Oh, the problem being that has already been marked on the graph, but is not part of the ray that is now being drawn?

Bingo! How dumb is that.

And none of the vettors spotted this. Including the blind reviewer who is meant to do the exam exactly as a student would. Keystone cops stuff.

Yeah, I see. Pretty dumb. Not that hard to make clear in a diagram, and I imagine the examiners will have no choice but to ignore the endpoint of the ray. But gratuitous confusion in a gratuitous question.

Not pretty dumb. Ugly dumb. Dumbness that makes the question a mess. The assessors meeting would have been interesting. It gets back to SRK’s astute insight:

“I read part d. ii. – which strongly suggests that students need to write down the domain of the line – as a tacit admission that it’ll be difficult to tell from responses to d. i. if students have drawn the graph correctly, so hopefully this means students won’t be penalised for an ambiguous graph at the endpoint.”

So what do you reckon I’d get if I score 37/40 in exam 1 and 74-76/80 in exam 2.

This is based on me being ranked 1 in a cohort with a sac average of 93%.

Hypothetical questions like this are discussed at ATAR notes.

Re: Q3.

And the rest of the goodness was killed by having to answer part (e)(ii) in 3 minutes (2 marks at 1.5 minutes per mark). Even forgoing a proper analysis (not that the question is asking for a proper analysis, just some lucky guesses) and simply relying on some pictures and some solutions to a general equation, there’s no chance of getting correct answers in 3 minutes. VCAA stupidity at its finest. I’d love to see the dipshit who thought this was a good idea answer one of *my* 2 mark questions in 3 minutes (or even 6 minutes).

Except for part (e)(ii), NONE of Q3 is Specialist – it’s Methods (even part (b) is Methods in my view). So get rid of all the Methods bullshit prior to (e)(ii), just have one or two key parts that lead up to (e)(ii), and make (e)(ii) worth 5 marks (= 8 minutes. Still not ideal but at least students can attempt a proper analysis).

No surprise that the average mark for (e)(ii) was 0.05 marks out of a possible 2 marks. Yes, 0.05.

The only reason this question was on the exam was to ‘nudge’ teachers to include this sort of stuff on their SACs (if it’s on the exam, it will get put on every teachers SAC). Pure VCAA manipulation of teachers at the expense of students.