# Secret 2022 Specialist Business: Exam 1 Discussion

We hope it went well for youse all.

### UPDATE (13/12/22)

The exam is now up, here.

### UPDATE (06/11/22)

Thank you all for your comments, and thanks in particular to Simon for his very nice solutions.

There’s not too much to say generally about the exam. As with the Methods exams, it was probably too easy. There’s a couple clunks, but no major issues, and the wording was in general reasonable. Here’s our question by question thoughts.

Q1. Routine.

Q2. Routine. The instruction to “Give your answer in the form y = f(x)” is poor wording.

Q3. Stats crap. I don’t do stats crap. Commenters have noted that the careless use of “mean time” to refer to the sample mean, which will probably have led some students into error. (11/11/22 We’ve decided this is bad enough to have earned its own PoSWW.)

Q4. Not a good question. The partial fractions are simple enough to do by inspection, or by the wrong set-up.

Q5. A routine (and depressingly simple) inclined plane problem. Part (b) is boring and pointless, with very poor wording. In particular, it is unclear whether the direction of the force is required, or just the magnitude: if “finding the value of R” is any different from finding R, God knows why.

Q6. Calvin Trillin used to joke about a generic restaurant being named La Maison de la Casa House. He would be delighted with the expression “vector scalar (dot) product”.

Using the term a as a vector in part (a) and as an unrelated scalar in part (b) is, um, stupid. Part (b)(i) is poorly worded, since the vectors can also be expressed without y. Part (b)(ii) would have been better worded as a straight proof question.

Q7. The computations were pretty heavy, and didn’t test much for the effort. The final required form takes work, is unnatural, and there are infinitely many answers (07/11/22) is more than one answer of the required form.

Q8. A poor question, wordy simply for the sake of being wordy.

Q9. A too-simple initial value problem, weirdly and needlessly phrased with function notation.

Q10. Part (a) seems a pointlessly easy trig graph. Part (b) is then an ok volume of revolution, although requiring tan(π/12) without warning is surprising. Asking for the answer in the form (a-√b)π/c, with a, b and c real, is obviously absurd.

## 96 Replies to “Secret 2022 Specialist Business: Exam 1 Discussion”

1. Red Five says:

Q4… doesn’t say “find an antiderivative”, so will VCAA penalise for omission of +C?

Q6 – using as a vector and then as a parameter in the next part… surely they could name the vector instead???

Not a bad paper. Q7 took some thinking about whether or not to use the compound angle formula, but only briefly.

1. James says:

I assume they will accept multiple answers for 6b)

Ie. OP can be expressed in terms of x and y or x and a.
Given they asked for it in terms of x, y,and a, I assumed they wanted me to write xi +yj and leave it as y until the next part.

1. Red Five says:

That was my assumption.

Also, I expect answers in terms of g will be accepted in Q5.

1. John Friend says:

On the topic of Question 5 (a pathetic send-off to mechanics):

Part (a): Yes, one would hope that was accepted! But VCAA’s infamous inconsistency makes this no sure thing.

Part (b): The given definition of is that it’s a vector. So … how do you find the value of a vector?
As has been the case too many times in the past, one must assume that is the of the constant braking force. Or will VCAA only accept (assuming is accepted)?

Again, it will be very interesting to see what the Report says (or doesn’t say).

@Marty: Do you think “After the body has been in motion for two seconds, …” is as tricky and nasty as, say, “Given that the first card drawn is blue …” ?

1. marty says:

No. Should I?

1. John Friend says:

I don’t know. Should you? (I’m doing a Special Agent Gibbs impersonation).

Well, I think it’s tricky. I think it’s more tricky than the “Given the first card drawn is blue …” Since you think the “Given ..” is tricky, I was just wondering. I guess it’s non-transitive.

(I think giving 4 marks for the integration in Question 4 is also tricky, in fact I think the integral itself is ‘tricky’).

1. Red Five says:

I liked Q4. If you took the partial fractions approach and were reasonably thoughtful in your approach, the relevant fractions dropped out rather quickly.

Yes there was a lot of working still to do after this, but overall not the worst integral I’ve seen on a SM paper.

1. John Friend says:

So why didn’t VCAA the partial fraction approach? Changing the coefficient of from 3 to 4 in the numerator doesn’t make things any more onerous.

2. Anonymous says:

I hadn’t noticed the specificity of ‘value’… I had taken that to just mean the magnitude. I assumed that if they wanted a direction they would have asked for the breaking force where it is implied to have a direction.

2. Anonymous says:

Another concern is for question 3i), the stats question.

When we arrive at Pr(z>-2), using a 0.95 interval the answer is 0.975, and the question is to round up or down… given that 1.96 is expected knowledge for 0.95 interval, would it be expected to round up to 0.98 and not 0.97?

1. John Friend says:

There was a similar issue on the 2019 NHT Specialist Maths Exam 1. I quote from here:

Hard SEL: The Specialist Error List

“The question asks for an answer to 2 decimal places, but the precise answer (using the standard approximation) is 0.025. The examination report states, without explanation, that both 0.02 and 0.03 were accepted as correct.”

So there is a very clear precedent by VCAA for both answers being accepted. It will be interesting to what is said (or not said) in the Examination Report.

Having said this, it be argued that since -2 -2) will be a bit bigger than Pr(Z > -1.96) (*) and so rounding UP to 0.98 is required. Furthermore, there is a convention (?) that if the trailing number in the decimal is 5, then you round up.

But as I already noted, VCAA have set a very clear precedent. I wonder if it remembers its own precedents?

At least the rounding is more straightforward in part (b) (and it doesn’t even matter whether you use z = 1.96 or z = 2 to get your answer. Of course, you’re not to know that until after the fact).

* In fact, using a CAS yields 0.977, correct to three dp.

1. John Friend says:

* Funny formatting things happening again:

“… since negative 2 is less than negative 1.96, Pr(Z > – 2) will be a bit bigger than Pr(Z > -1.96)”

1. James says:

Yes that was my reasoning to provide 0.98

2. Amber says:

For Q4 One of my students split the numerator into 3x^2+12 and 4x, and then cancelled factors, getting to the necessary expression to integrate without using partial fractions. It means you can get to the answer in about 3 lines rather that the whole pages provided. Do you think they’ll get the full 4 marks?

1. marty says:

Hi, Amber. I have thoughts on this question, which I’ll save until others have had the opportunity to comment. But briefly, I don’t see why your student shouldn’t be eligible for full marks. Nothing stated in the question suggests a requirement for any specific partial fraction working.

1. John Friend says:

Agreed. I wonder if VCAA intended the cleverer students to note this … (4 marks fr not much work suggests VCAA either didn’t realise or wanted to send students down the wrong fork in the road).

2. Red Five says:

I think your student’s choice of approach is excellent and should get full marks.

Personally I went straight for partial fractions and then realised pretty quickly that by comparing coefficients.

I *really* wonder though about the +C business since past VCAA exams have said “find AN antiderivative” to avoid the issue, but here they have chosen not to…

3. W says:

off course it should be full mark!
I’d say it is even better.

3. John Friend says:

Re: “Q4… doesn’t say “find an antiderivative”, so will VCAA penalise for omission of +C?”

My understanding is that something like “Find .” requires a “+C” in the answer.

Re: “Q6 – using a as a vector and then as a parameter in the next part… surely they could name the vector v instead???”

In fairness, the a in part (a) is not italicised and it is in part (b). But I agree, calling the vectors u and v in part (a) would have been much better. Or using instead of n part (b).

Q10 – Nice sting in the tail having to get in surd form before being able to give the answer in the required form. But 3 marks is a bit short. (I’ve seen VCAA questions worth 3 marks simply to find a surd value for , let alone putting it in the context of finding the volume of a solid of revolution).

1. Red Five says:

is reasonably commonly calculated in Specialist Unit 2, so I wouldn’t be surprised if some students had memorised the value.

I didn’t mind this question, except for the business. That was (and was in the past, too) needless and weird.

1. John Friend says:

I wonder whether the marking scheme will require some working for the value, or whether it can just be plucked from the air and used without justification (I’m very unhappy about this *). The latter would potentially mean:
Correct substitutions into the formula for V – 1 mark.
Correct evaluation of integral with tan(pi/12) – 1 mark.

Not a good marking scheme in my opinion.

* So now students have to memorise more obscure ‘special’ angles …?

2. John Friend says:

Re: re: “Re: “Q4… doesn’t say “find an antiderivative”, so will VCAA penalise for omission of +C?”

One other thing – the VCAA formula sheet includes the “+C” on its antiderivatives.

2. John Friend says:

Two things stand out to me at the moment:

1) Question 3 (b). I’ve attached a screenshot.

The wording of the second sentence is bad. The intent is probably to give the sample mean as 9 seconds. But the key word “sample” is missing. I think some (many?) students could be misled into thinking that the mean time for one cup of coffee is 9/25 and be put off by this … The Examination Report will be interesting for what it says (or doesn’t say!).

2) Q10 (b) asks for an answer in the form where . As has commonly been the case in the past, there is no unique answer (there are an infinite number of possible values of a,b,c).

And is it just me, or is the phrase “vector scalar (dot) product” (in Q6(b)(ii) at least two words too many, or even necessary?

1. Red Five says:

It has just occurred to me… the is redundant in the expression given since if then is also real… ditto .

Yes, I’m being pedantic. VCAA has taught me that this is necessary.

1. John Friend says:

I’ll suggest that VCAA kept the to be helpful. What I don’t know is why they didn’t say , unless they wanted a greater variety of infinite possible answers to choose from.

1. John Friend says:

Can anyone suggest why VCAA wouldn’t ask for the answer in the form

where

It still serves VCAA’s assumed purpose(s) and the tighter constraint means that:

1) It’s unique.

2) It will give students re-assurance that they got the right answer (and perhaps help those who didn’t go back and find what might be a simple mistake).

2. marty says:

It ain’t just you.

3. anon student says:

I am one of the students who interpreted this as the mean being 9/25 seconds… majority of my friends interpreted it as intended of 9 seconds. disappointed with the wording on that one

3. Terry Mills says:

>JF: Inserting the word “sample” as you suggest would be an improvement. But I think that the meaning is clear enough as it stands.

1. marty says:

If “sample” would have improved it then it is not clear enough as it stands.

2. Red Five says:

Is it though? A population mean and a sample mean are quite different things. Are they not?

1. John Friend says:

Yes, they are. But I don’t see your point, RF.

There’s three different means in Question 3:

The mean time in the normal distribution followed by the population of cups of coffee, pre-serviced machine.

The mean time in the normal distribution followed by the population of cups of coffee, post-serviced machine.

The mean time of the sample of cups of coffee in the second sentence of the stem to part (b). This is what requires clarification. It should have been stated as the sample mean time.

And on this topic, I do not understand why VCAA refuses to include critical values on the Formula Sheet. And do they want students to memorise z = 1.96 or z = 2? I’ve raised this before – in Methods it seems students must memorise z = 2, but in Specialist it seems they must memorise z = 1.96. None of this has ever been explicitly stated by VCAA, you have to join the dots from exam questions.

1. Red Five says:

I have no point. Just being a pedant.

2. Terry Mills says:

A more serious error is, in what sense can the sample be regarded as a random sample?

1. John Friend says:

Indeed.
Answer: In the sense of having to assume that every VCAA sample is a random sample!

(But yes, these are the subtleties that get ignored in the puerile statscrap that has been imposed/inflicted upon Specialist Maths).

1. Terry Mills says:

This generates bad habits when people apply statistics; they ignore these important points.

In mathematics, one would not ignore issues such as whether a function is continuous or differentiable.

Even the definition of a random sample is wrong in many books that I have checked.

1. Red Five says:

Randomness as a concept is beyond the SM study design.

It shouldn’t be, but it is.

2. John Friend says:

The whole Study Design generates bad habits, it ignores many important points.
Issues such as whether a function is continuous or differentiable are barely addressed in the Study Design and even then only superficially (“Informal treatment of …”).

3. John Friend says:

anon student above would disagree with you, TM. I think it’s a bad omission. I wonder what VCAA’s ‘psychometric’ testing will reveal and what will be said (or not said) in the Report. anon student above provides evidence that this error affected at least one student, I am aware of others.

1. marty says:

Jesus. Does this mean I’m actually gonna have to look at this stats crap?

1. John Friend says:

Only if you want to, Marty. Only if you want to. (But why should we be the only ones that have to swim in the statscrap pool?)

2. anon student says:

anon student here,

Just got given my statement of marks and I received full marks for this question despite my different interpretation of it.

1. John Friend says:

That’s very good news, anon student. I assume you paid the fee to see this?

I’d love to hear from any student that used 9/25 for their confidence interval in Exam 1 Question 3(b), paid the fee to see their Exam 1 statement of marks and can comment on whether or not it appears they got penalised for using 9/25.

PS – I hope your results allow you to continue on your desired post-school pathway.

EDIT – I stupidly thought you were talking about Exam 2 Q6(f). I just realised that you were talking about the Exam 1 Q3(b) 9/25 business. In which case, it’s very excellent news! Common-sense news! VCAA-would-have-an-ugly-fight-on-its-hands-if-it-was-otherwise news. *Ahem* You used 9/25 not 9, right?

1. anon student says:

Statement of marks was free for me – no fee paid. I think inspecting my exam itself required a fee to be paid

2. marty says:

Thanks, anon s. Just to be absolutely clear, you took the mean to 9/25 in 3(b) and received full marks?

1. John Friend says:

Thankyou for this comment, Marty. It’s a very important clarifying question!

2. anon student says:

Yep, I took the mean as 9/25 and received full marks for question 3b

1. marty says:

Thanks again, anon. Very interesting.

4. Red Five says:

As far as the vs business… not so long ago in a galaxy not so far away, examiners would write “use an integer multiple of the standard deviation” in questions.

Not this time.

I do wonder what changed.

1.96 is an approximation anyway (as is 2) when taking 95% confidence intervals (as anyone using the CAS on Thursday’s Methods 3&4 exam will know – using the stat-confidence intervals feature gave a *slightly* different result to using the CI formula with 1.96…)

1. Tungsten says:

I do not necessarily see an issue with not specifying “integer multiple” in this case since the confidence interval is reasonably easy, and asking for the answer to one decimal place should require only as much accuracy in the z-value as needed for an answer in the appropriate form. However I agree with JF that it is stupid do not include such and similar values on the formula sheet. For example, I decided to memorise the probabilities of Z within 1,2,3 standard deviations to 4 decimal places for safety, but should I really need to?

1. John Friend says:

RF is right, examiners did used to write “use an integer multiple of the standard deviation” in questions. This unofficial ‘policy’ changed without explanation. We may well wonder what changed (*). And the fact that z = 2 is used in Methods and z = 1.96 is used in Specialist is just idiocy.

A formal statement by VCAA on what value to memorise, z = 1.96 or z = 2, for both Methods and Specialist would be a good start (ideally it would be the same value). In a CAS-Free exam, is there any point in using 1.96 rather than 2?

* Nothing’s changed, of course. It’s just another example of VCAA’s infamous inconsistency. Enabled by an apathetic teaching cohort. The whole statscrap is a joke, and little things like this make it an even bigger joke.

1. Red Five says:

1.96 was written in the draft Methods study design pre-2016 (for first exams in 2016). I’m not sure if that made the final cut or not.

Also not sure what the current study design says but will check now!

1. steve r says:

RF,

Perhaps I am a pre CAS dinosaur … but is it not possible to provide tables of Z values to the students prior to the exam so they can get a feel for the Standard Normal ?

Steve R

STU Z Table.doc

1. Red Five says:

Once upon a time, such a table was provided.

We are not allowed to have nice things anymore.

We get a useless formula sheet and a cover page that says “acceleration due to gravity is g ms/^2 where g=9.8”

So, it IS possible, but there must be a willingness on the part of the paper-setting authority.

1. Terry Mills says:

I show my Year 9 students log tables. A few of them prefer to use them over the calculator.

1. John Friend says:

I think the log tables (Kaye and Laby Four Figure Mathematical Tables, natch) are a brilliant and fun way of making logarithms and their rules a student’s friend.

2. Terry Mills says:

Students could copy this table into their summary books,

1. Red Five says:

Which are not allowed in Paper 1 for Methods nor Specialist.

Otherwise, a brilliant idea. I may recommend this as insurance against calculators dying because students forgot to charge them over the weekend… (surely that is not just my school)

2. Tungsten says:

The value of 1.96 is explicitly stated for both Methods and Specialist Mathematics.

1. John Friend says:

Yes it is. But there’s nothing about memorising this value. z = 2 was needed in Exam 1, but now z = 1.96 is needed. There is no consistency and no transparency.

@steve r: Back in the day (even during several years of CAS), the standard normal distribution tables were part of the Formula Sheet. Then they were deleted – I assume because they were considered obsolete. Which they were. Until some muppet decided that critical values were required to be for Exam 1, without telling anyone and without saying what values. The only ‘announcement’ of this ‘policy’, using z = 2, was in the 2016 Specialist Exam 1 and appeared again in the 2019 Exam 1 and some of the NHT exams. The secret switch to z = 1.96 happened in Q3(b) this year … How can you be sure of being correct to 1dp using z = 2, a whole number? If z = 2 was intended, why not give the same instruction (integer multiple of standard deviations) as used in all other years?

Why not just give the *#!*& critical values on the *#!*& Formula Sheet!?

On a related note, are students required to memorise the 68–95–99.7 rule? I told my Maths Methods classes to do so because you never know …

1. Vox pop says:

In Q5 I didn’t think it was clear in the final answers for speed in pt (a) and R in pt (b) whether your answer should include g or use a fractional equivalent?

Anyone care to speculate on what might be in P2? Little complex numbers, no statics or hypothesis testing in P1.

Overall I thought it was a pretty gentle paper.

2. Hvo says:

Yes, both Methods and Spec students are expected to know 68 -95-99.7 rule (I read somewhere in the VCAA report).

Overall, I felt disappointed over the inconsistency over the wording and the possibility of various correct answers. When doing the solution, I thought I should tell my students to ask for review if they felt hard done by.

1. marty says:

Thanks, Hvo. Were there questions other than the ones people have mentioned that concern you?

2. marty says:

Excuse my ignorance, and I can only pretend to care so much, but if you guys are arguing 1.96 is to be used in 3(b), what is intended in 3(a)?

1. John Friend says:

Since part (a) requires calculating , to quote anonymous, “using a 0.95 interval [but for z = 2] the answer is 0.975, and the question is to round up or down… given that 1.96 is expected knowledge for 0.95 interval, would it be expected to round up to 0.98 and not 0.97?”

The only way of answering part (a) is using z = 2 and then understanding that -2 is less than -1.96 and so is greater than and so rounding 0.975 UP is required. (Otherwise, which way to round is a toss of the coin). But, as I said earlier, maybe VCAA will remember its precedent from the 2019 NHT exam and accept rounded up or rounded down (but I wouldn’t count on it, and, in fact, rounding down is clearly wrong).

Part (b) wants an answer correct to 1 dp. It’s not obvious (to me, anyway) that using z = 2 (that is, data rounded to the nearest integer) will give the same answer, correct t 1 dp, as z = 1.96. So what choice is there?

To answer both parts properly requires knowing z = 1.96 is the more accurate critical value than z = 2. The question should have just said to use an integer multiple of standard deviations, as has been stated in previous years.

1. marty says:

I hate this stuff …

Thanks very much, John. May I ask to what extent “two decimal places” makes sense, given the mean and standard deviation are given to just two significant figures? (I could figure it out, but I refuse to do so.)

1. Tungsten says:

It seems to me that the number of significant figures in the values given might as well be assumed to be infinite (ie exact values) for the purposes of VCE exams. I cannot imagine anybody really caring about that.

1. marty says:

Thanks, Tungsten. I know no one cares and everyone assumes whatever. But I figured amongst all this crap about 1.96 and 2, and the number of decimal places, and rounding up and rounding down, it’s reasonable to ask whether the answer makes mathematical sense.

2. Red Five says:

It makes sense in historical context. VCAA has done this a lot on Specialist Paper 1 exams.

Does it really make though?

No, not really.

3. John Friend says:

Marty, nothing makes sense when it comes to VCAA.
An answer can never be more accurate that the least accurate datum used to calculate it. So how can you possibly give an answer correct to the accuracy asked for in the question when the least accurate datum has been rounded to the nearest integer? The whole thing is an absurd game with no transparency. The 2019 NHT Examination 1 Report proves it for this question.

On a related note of accuracy of answers, how can VCAA justify deducting a mark for giving answers like

(from Lami’s Theorem) instead of (from resolving forces) etc.

And don’t even get me started on things like wanting

3.3 rather than , but not always …

To penalise numerically correct answers given in a reasonable form (and even precise answers involving g) is just one gigantic absurdity. There is no mathematics in a lot of what VCAA does, just pedantry and sanctimony. And all this statscrap is fertile ground for it. The whole 2 versus 1.96, what to memorise and what value to use is an opaque minefield.

1. John Friend says:

And yes, NONE of this is mathematics. It’s just the sewerage that what small amount of mathematics there is floats in. Sewerage that is part of the joyless game otherwise known as VCE mathematics exams.

2. John Friend says:

Hi Tungsten, using the same accuracy during a calculation as that required in the final answer runs the risk of rounding error, leading to a final answer that is not correct to the required accuracy.

There is no way of knowing in advance if the final answer will be correct to 1 dp if z = 2.0 rather than 1.96 is used during the calculation.

PS – Thankyou for agreeing with me re: values on the formula sheet.

5. John Friend says:

@Marty: By the way, who is “youse all”?
Are they some poor sap from an ESL background struggling to do their best against the VCAA tide of unintelligible writing?
I hope it went well for them too.

6. Simon says:

I’ve got little to add to the discussion above. Overall I think it was a pretty straightforward exam with only a couple of the usual quirks. Here are my solutions – feel free to tear holes in them for me to fix!

2022 Sm VCAA Exam 1 – Solutions with no qns

1. marty says:

Thanks very much, Simon (and sorry to veto your questions + answers version, which would have been more helpful.)

2. John Friend says:

Question 2 – You shouldn’t use the same symbol as a dummy variable of integration and an integral terminal, I think it’s bad form. It’s unclear whether VCAA would penalise this or not. I would have x in the terminal and a different symbol for the dummy variable of integration, perhaps w. Personally, I’d avoid using the integral solution approach and simply get the anti-derivative and then solve for “C”.

Same for Question 9.

Question 3 (a) – I think the brevity of your solution will cause confusion for some students (and teachers). I did a double take at Var(X) = 4 Var(T) and thought for a moment you’d made the mistake I’ve seen so many times (and occasionally still see in some trial exams). I would make this all crystal clear:

define the random variable .
Only after doing this would I then calculate E(X) and Var(X):

where the are ‘clones’ of

therefore sd(X) = 3.

1) It’s not obvious to me that taking z = 2 rather than z = 1.96 will give the endpoints to the required accuracy. The standard error is 0.6 when using z = 2 and 0.588 when using z = 1.96. The latter calculation is not arithmetically onerous:

.

So in hindsight we see that there’s no difference in answer, to 1dp. But I don’t think we know this for sure beforehand (I sure don’t). And in fact:
(8.4, 9.6) (using the standard error of 0.6) is a little wider than the exact 95% CI, whereas
(8.5, 9.5) (using the more accurate standard error of 0.588 rounded down to 0.5) is a little narrower.

Now which of these is the ‘better’ approximation to the 95% CI …? After checking with a CAS, I see the former is ‘better’. But how would you know for sure in Exam 1? How would you know for sure which one VCAA wants …? Just do mathematical rounding and hope …?

2) The question asks “Find a 95% confidence interval for the mean time ….”

This suggests that 95% CI is NOT unique. Given that we’re limiting ourselves to the Wald CI (see

GitS 2 : John Friend – A lack of Confidence

for a discussion on some different types of confidence intervals – yep, I’ll never miss a plug) the 95% Wald CI unique. The wording of the question is erroneous (*). In the past there has been more appropriate wording (yep, it’s all relative) (2016 Exam 1 Q2, 2016 Exam 2 Q19):

“Find an approximate 95% confidence interval for …” And many trial exams have additionally said “Give the endpoints correct to one decimal place.” rather than give your answer … correct to one decimal place.”

* Erroneous, sloppy and amateurish.

1. Tungsten says:

An exam assessor has told me that Simon’s integral approach with the same x in a terminal and dx should be fine, so I would be surprised if it were penalised. My impression is that it is used by a not negligible number of students.

I am interested in the second line of Simon’s working for question 2, where the integral on the right has terminals 2->x before the dx’s “cancel”. It is a similar question with integration by substitution. Could anybody explain whether/why the terminals would be kept the same at this step? I usually avoid the issue by just going straight to integral of dy alone (in this case).

1. marty says:

Definitely Simon’s double use of x in Q2 is wrong, and should be avoided. (That doesn’t mean students should be penalised for doing it.)

As for Simon’s second line, you have f(x) = g(x), and so . I wouldn’t express the working this way, and would do what I think you’re indicating, but Simon’s is a nicely legal way to do it.

1. marty says:

Thanks, Anonymous (presumably Simon).

I understand why you want to avoid introducing the dummy variable t. Still, what you’ve written is plain wrong.

I also understand your reason for making the integration by substitution explicit. That’s a judgment call, and my own feeling is it’s not worth it, but I take the point.

2. Simon says:

Re the integration in Q2 & 9: I actually prefer the double use of x like this (obviously, since I did it). I think the cognitive overhead of an extra symbol is more than the slight chance of confusion re its meaning. The meaning of a variable upper limit and the integration variable is so close that using the same symbol should be fine. And an indefinite integral implicitly has the x in the upper limit anyway…

The reason I write the solution to the DE this way is that it reminds me I really am doing an integral by substitution and that the change of variables needs to be 1:1 on the integration domain. Too easy to make errors otherwise (eg we used to have a question on a 3rd year classical mechanics assignment that caught out 90% of students each year with this issue; or the spring problem later in this exam).

As for what is better pedagogically… I’m not sure

1. marty says:

Thanks, Simon.

I understand why you want to avoid introducing the dummy variable t. Still, what you’ve written is wrong; it may not confuse your students now but it is sowing the seeds for confusion later.

I also understand your reason for making the integration by substitution explicit. That’s a judgment call, and my own feeling is it’s not worth it at VCE (except once or twice in class), intrinsically and also because it requires the choice of “t or double-use of x”, but I see the point.

1. Simon says:

Thanks Marty,

It is notational choice. So it is either more or less useful in both clearly and accurately expressing an idea, I’m not sure if it could be called “wrong”. What do you see these seeds of confusion growing into?

Yeah – making the change of variables explicit in solving such a DE in VCE is probably not worth it. And I do only demonstrate it that way a couple of times before going to the lazy way of “splitting” the infinitesimals. However, it is inevitable that one day VCAA will slip up and include a DE where you need to be careful, then who will be laughing… 😁

1. marty says:

I’m sure it could be called wrong. You’re using x in the same expression to mean two entirely different things. Hence the seeds.

1. Simon says:

Not convinced yet. You already use x in two different ways in an integral: in the integrand and attached to a ‘d’ as a infinitesimal. Having it attached to the integral sign and understanding the scope of its definition doesn’t seem too hard.

And it makes more sense when the symbol has some physical meaning, such as time in kinematics: .

In other places (multiparameter functions and/or multiple integrals) it can lead to confusion between the bound and free variables, so should probably be avoided… Anyway, probably enough said about this.

1. marty says:

It’s wrong.

2. Simon says:

Thanks for the detailed feedback John!

I’ve responded re the integration in Q2 and Q9 above. Happy to be convinced what I do is bad practice.

Q3a – yeah, I could have made that clearer. What you said is fair.
Q3b – I agree – it feels like the exam could have had a couple more people work though it to iron out things like these numerical issues and get more precise wording. When I taught CI intervals I did refer to your notes and the different choices / approximations. But I think the Wald CI interval is probably fine for a first bite at such statistics.

7. Terry Mills says:

These days, which is the preferred notation? or

1. John Friend says:

I think there is a well established convention that . Your latter notation is clearly ambiguous at best.

What I find interesting/strange is that is not in common usage, or any usage for that matter.

1. Tungsten says:

I have seen used several times, but certainly not in a VCE context. One issue might be the ambiguity with iteration, such as where might be confused for , although the power meaning is universal in all that I have seen.

1. Terry Mills says:

When stating a law of the iterated logarithm, authors would usually write .

2. Red Five says:

Depends, are you squaring the angle or the ratio?

8. marty says:

Thanks, everyone. I’ve updated the post with my own (minimal) thoughts.

1. John Friend says:

Thanks, Marty.

Re: “Q2 … The instruction to “Give your answer in the form y = f(x)” is poor wording.”

I’ll defend VCAA here – it obviously doesn’t want the answer left in implicit form and I support that. I therefore don’t see any alternative to asking “Give your answer in the form y = f(x).”

Re: “Q4. Not a good question. The partial fractions are simple enough to do by inspection, or by the wrong set-up.”

Yes, the correct form can be found using the “wrong set-up”. In which case the student would get zero due to the policy “Correct answers found using incorrect mathematics get zero.” I commented earlier that the coefficient of the x^2 term in the numerator could be changed to 4 to force the use of partial fractions but with minimal additional algebra required. Personally, for 4 marks I’d have preferred a numerator with degree equal to or higher than the degree of the denominator. I also thought giving the denominator in factorised form was unnecessary – unless VCAA was worried that some students wouldn’t be able to (or think to) correctly factorise the denominator. Should VCAA be worried about this …?

Re: “Q7. The computations were pretty heavy, and didn’t test much for the effort. The final required form takes work, is unnatural, and there are infinitely many answers of the required form.”

I didn’t think the computations were too arduous, particularly if the point was substituted making dy/dx the subject. And compared with the ridiculous 2019 Exam 1 Question 10, the computational effort is banal. I agree that after the implicit differentiation is done, what’s tested is hardly worth the effort.
Since it’s specified that , I don’t think “there are infinitely many answers of the required form”. I can see only four:

(probably the intended answer), , and .

Any of the last three would only be given by a brave smart-ass. In fairness, I understand why VCAA specifies a form for some answers, but in another comment (for the form of answer asked for in Question 10 (b)) I’ve noted that VCAA doesn’t do this is in a good way.

Re: “Q10. Part (a) seems a pointlessly easy trig graph.”

I will surmise that the point was:
1) To provide a (unnecessary) prop for part (b).
2) To tick a box in some ‘coverage of content’ rubric.
3) To give some marks to the lame and infirm.

It would have been much better to have made it worth 1 mark, taken a mark from part (b) and inserted a new part (b) worth 3 marks that asked for the value of . (In fact, I would have asked that the value of be found using an appropriate double angle formula – just to make it more interesting than using the compound angle formula). Then, requiring tan(π/12) has been forewarned and is no longer “surprising”. (I’d have used a different word to “surprising” but children read these comments).

1. marty says:

Thanks, John. Obviously none of these are major concerns, but I’ll give brief replies.

Q4. Yes, I might have made the more important point, that the question is too easy. But it is also a bad idea to write a question where one can obtain the correct answer by an implausible but incorrect method.

Q7. Ah, you’re right. I’ll amend (although of course my point remains). VCAA shouldn’t have to specify the form of the answer. This is not (or at least was not) standard at university. It should just be understood by everyone that the answer be given in some reasonably simplified and/or useful and/or standard form.

Q10. No real disagreement.

9. steve r says:

Hi,