I just received the following email from Mystery Student, Alex:

*Hi marty,*

*I’m currently taking Spec 3&4 and just had a couple of questions reading this post.*

*For testing linear dependence, you recommended using a ‘3×3 determinant’. I was just a bit confused, and I’m always looking for areas to improve my knowledge, blah blah blah.*

*Do you have any other areas that make questions more efficient that are glossed over by VCAA or textbooks?*

*Thanks a bunch 🙂*

I answered Alex briefly on the determinant question, but there are obviously readers much better informed than me about helpful tips and tricks for Specialist. And, in any case, such questions are best replied to by the crowd.

So, please make your suggestions in the comments below, including answering Alex’s specific question.

If the post takes off then I’ll perhaps try to categorise and summarise the suggestions in updates to the post. Also, if people think a companion Methods Tips post is worthwhile I’m happy to do that (although the worth of that is less obvious to me).

A few that immediately come to mind:

1. If then , with the value of determined by an initial condition. This can be stated without going through the integration / algebra.

2. can be used without doing substitution, which can be a time-saver when this appears as part of a more complicated integral involving partial fractions.

3. Viete’s formulas for the sum and product of roots for polynomials can be very useful for a range of questions that come up in complex numbers.

Thanks, SRK.

I would advise against doing these tricks. Who knows how stuffed the VCAA marking rubric is, and whether they’d accept these shortcuts.

Hi, Joe. Of course one has to be wary of VCAA’s arbitrary fussiness. But I think SRK knows what they are talking about, as do other SM teachers who read this blog.

If you or anyone thinks there a particular issue with a particular trick/shortcut then of course raise it and it can be discussed. But there are plenty of instances in SM where there are better and worse *approved* ways to do things.

Is there a list of approved VCAA techniques, or is it just insider knowledge gained after years of teaching and interacting with people who work at VCAA?

I think you just have to glean what you can from the (screwy) examination reports and folk wisdom.

I would be weary, yes, but I hope that VCAA’s marking schemes account for many different methods of tackling a question. Scouring the internet, I found a post some 13 years ago of this VCAA Maths top dog stating: “Where a particular approach/method is not required as part of a question, students can be rewarded for correct and relevant use of applicable mathematics.”

I do recall the second derivative test for SPs being some Houdini magic trick for us students not mentioned by VCAA at all, but still able to receive full marks. However, you may be right with ‘show that’ questions, etc, that they wouldn’t want you to use their trademarked method to which they partially enlighten us in the exam reports.

Your “weary” is a great Freudian slip.

No question that “show” and mark allocations are an indication that working is required. But definitely alternative methods are possible in some cases.

But don’t be guessing. The commenters here are pretty up on what VCAA will or will not accept. Deal with it trick by trick.

Just so you know, the second derivative test is actually in the VCAA Specialist Maths study design, under the subsection “Differential calculus and integral calculus.”

1) I remember this being taught by my teacher, really helpful. Doubt it could be used in questions requiring full working, or something like ‘show that the value of A=5’, etc.

2) I definitely see the use for a question about partial fractions, this was even mentioned in Methods, if I remember correctly. If the mark scheme had a given mark for ‘an appropriate u-sub’, I wonder how this would fare.

3) Looks really nifty from my brief squint at it. Would you know of a question where this could be applied?

On tip 1). See 2019 Exam 2, Question 3 a i. My recolllection, FWIW, from the “meet the assessors” presentation is that it was acceptable to simply state that , then substitute in the two boundary conditions and solve for . I understand teacher / student reservation about doing this sort of thing – VCAA have been notoriously pedantic about “show that” questions in some cases, and the lack of clarity / consistency about marking is immoral.

On tip 2). Firstly, See Red Five’s important point below. Secondly, yes, of course if the question demands a method, you must show that you’ve used it. I have also wondered about the extent to which this sort of method generalises – see 2011 Exam 1 Question 6, for example – it’s obvious that an antiderivative is , but one wonders if writing out the substitution was required for full marks (probably not, since it’s only 2 marks? but who knows…)

On tip 3). See 2017 Exam 1 Question 3. Observe that is also a root, and let be the remaining root. Then using the sum and product of roots we have and and we’re pretty much done.

Thanks very much, SRK. And I agree entirely on the immorality of the marking.

Remember to include the absolute value around the argument of your log in tip #2 if the domain is not restricted to

Otherwise, yes, all good. They seem more like common sense than tricks though.

Semantics? Perhaps.

Vieta’s formulas don’t seem to be as useful in VCE as they are in IB Mathematics – which says more about the way(s) in which the same idea can be examined quite differently by (very) different examination boards…

One man’s tip is another man’s trick.

Sometimes, the examiners’ report contains a few clues (although it does take a bit of guessing in regards to what it all means for the following year). The 2022 reports had the following which I found interesting (and others may have some more to say… hopefully…?)

1. Manipulation of the integrand. I’m thinking algebra skills with problems involving surface areas of revolution could be used to test this in 2023. Paper 1.

2. Implicit differentiation where expanding using a trig identity is not the most efficient method. Is there an instance where this would be more efficient? Paper 1.

3. Solving differential equations using definite integrals and the need for a dummy variable. Not sure this is relevant as there are other options. Paper 1.

4. Reading and responding to all aspects of a question. I’m definitely getting kinematics vibes here with the different acceleration formulas and questions such as “state the maximum value of … AND the time(s) at which it occurs”. Paper 2.

5. Considering limiting behaviour of a function. Feels more paper 1 type stuff, but was mentioned in the paper 2 report. Slope fields perhaps?

6. Working with random variables that are functions of other variables. Yeah… Nah. Recent posts on this blog about variables that are not independent but were assumed to be so… hope VCAA has learned the lesson and just doesn’t ask this. There hasn’t been too much on Type II errors in previous years, so maybe 2023 is when we will finally see one (links with the comment in paper 1 report about probability)

Of course, none of these are “tricks” and you should revise the entire course blah, blah, blah…

Thanks very much, RF.

1. Yes. I think the examination of the surface area generated by rotating a curve around an axis is a sure thing on Exam 1. More generally:

(a) For arclength be on the lookout for either a hidden perfect square or a linear inside the square root, and

(b) For surface area be on the lookout for the same things together with either using the ‘reverse chain rule’ or integration by parts. (But I think integration by parts will most likely be examined as a stand alone question rather than embedded in a surface area).

3. The ‘integral solution’ is almost certain to appear in Exam 2 where a CAS can be used to approximately evaluate what would probably be an intractable integral. Be on the look out for anti-derivatives that do not have an elementary ‘closed form’ but the approximate value of a definite integral is required. Kinematics is often a context where this will appear.

5. Use of phase diagrams. For example, see https://mathematicalcrap.com/2017/11/26/the-madness-of-crowd-models/

6. Type 2 errors have been on almost every exam (in disguise) since this pseudo-statistics stuff was first added in 2016. But only now has it been explicitly named. Do every stats question from every VCAA exam and you will see nothing new or different this year.

But the best thing to do is do every exam from 2016 and read every examination report. Nothing said in these posts includes secret inside information. It is all based on what is seen on exams and reports. And all the tips in the world don’t make up for a lack of experience or understanding.

Thanks, BiB. It’s exactly this folk wisdom that is the best guide.

Agree on all points. Will expand on just a couple:

1. With the disappearance of statics/free body diagrams, I do suspect that differential equations using one or more of the expressions for acceleration (such as ) may be more frequent, especially in paper 1.

2. Yes and no on the Type II error business. The probability of a Type II error has rarely been asked properly and it seems like the next logical step in the roll-out of all this stats-crap (for any VCAA exam setters reading, I LOVE the stats questions because I find them really easy and consider them to be really easy marks for well-prepared students, so please continue…)

3. Yes on integration by parts as a standalone question. Maybe even in the context where one of the parts is 1, such as .

One issue with surface area is the sample materials’ false assumption that a curve sweeps out a volume/region. Combined with VCAA’s expertise with ambiguous wording, this could lead to a significant screw-up, concerning whether the end disks are or are not to be included.

From what I’ve seen, my advice for this very reasonable concern would be to look for the suggestive word ‘curved’ in such questions:

“Find the area of the surface …”

I would not include the end disks for such wording. However, I can offer no advice for something like

“Find the area of the surface of the volume formed …”

except to say that I would include the end disks (because it’s [ham-fistedly] asking for the surface area of a solid).

It is absurd that we’re groping in the dark when it comes to trying to guess what the whims of VCAA might be in any given year. There is a lack of clarity and consistency everywhere you look. VCAA does not place authentic value on the meaning of words.

The only consolation is that every student does the same exam. If there are errors, try to think how VCAA would weasel out of admitting an error, answer the question accordingly and then move on. Don’t waste time stewing over it.

I can only advise on what I’d do given what I see.

But I think you’ve gotten your blogs mixed up. I expect a withering response to be here: https://mathematicalcrap.com/2023/09/29/amts-gender-fetishism/#comment-23629

Ah indeed. Thanks.

One would hope that a question setter/checker was alert to the difference between an open surface and a closed surface. BiB mentions below that a reasonable person would/should take “curved” to mean an open surface, but again, VCAA…?

Textbooks, and I’ve seen a few examples, are pretty good on this.

VCAA…? Guess we wait and see.

Again, the sample materials screw this up.

So either the exams get the wording right and in doing so send the message that the sample materials screwed up. Or the exam wording is consistent with the sample materials and there is a total denial that any screw-up exists.

Correct.

Despite that hope, I think we can all agree that some question checkers are comatose at the wheel.

I assume you’re referring to university level textbooks …?

IB textbooks, mostly out of print now unfortunately, but also some 1st year calculus textbooks I’ve collected (including some I have from Marty – thanks again!).

What can I say… I like to read…

But getting back to the point – I think there are a few tricks/tips that could be relevant here about how to read a VCAA question; I’m just not 100% sure how to distil the essence of these ideas for the moment.

Thanks, RF. It may be worth simply throwing the ideas in, without too much concern for distillation.

The main ‘tip’ is how to interpret a “Show that …” question.

From VCAA Maths Methods 2021 Examination 2 Report:

“A reminder that ‘show that’ questions require a reasoned argument. The answer is given and students are required to provide a detailed progression to the answer.”

In other words,

“Show that the solutions to are x = 3 and x = 2″

means solving showing all the tedious steps, NOT substituting x = 3 and x = 2 into and getting zero.

Insanity at its finest but you have to play the game.

As a bonus tip, a “Show that …” question usually gives a result that is needed to answer later parts of the question.

“(a) Show that the solutions to are x = 3 and x = 2.

(b) Sketch a graph of . Label all axes intercepts with their coordinates.”

On this point (which I believe you have given a fair account of VCAA’s opinion) I think the whole system is absurd.

Substituting into is a valid way to SHOW THAN is a solution.

If the question were to say “show that and are the ONLY real solutions (actually, in Methods they wouldn’t bother with the qualifier “real”) then fine, expect students to factorise and solve, but this is often not what questions ask.

As to the mark allocations for these and the associated examiner “comments”…

The rest of your point is quite valid and you can do the rest of the question without managing the SHOW THAT component, so don’t be put off by a difficult SHOW THAT at the start of a question (quite common in Complex Numbers Exam 2 Section B…)

VCAA has its own special and absurd meaning for “Show”. So it’s not good enough to understand the mathematics, you also have to understand VCAA Newspeak.

I usually find it best to treat a “Show that …” question as a “Find …” question and then proceed to write a solution that even a pinhead could follow. So

“Show that the solutions to are x = 3 and x = 2″

becomes “Find the solutions to ” and then proceed with the solution for a pinhead.

What I find comedic (in a gallows humor way) is that VCAA Newspeak is not consistent. For example, I quoted advice about “Show that …” from the VCAA Maths Methods 2021 Examination 2 Report. However, if we consult the VCAA Maths Methods 2022 Examination 1 Report, we discover the following:

“Students are reminded that it is not acceptable to work both sides of a ‘show that’ question at once; rather they should start with one side and show that a clear progression of mathematical calculations results in the same expression that is on the other side.”

Do you see the inconsistency? If the 2022 ‘advice’ is trusted, then we can start with either side and get full credit. For example, under this ‘advice’ we should get full credit for substituting x = 3 and x = 2 into and getting zero.

So which ‘advice’ to follow? 2021 or 2022? Personally, I think it’s safer to follow the older 2021 advice.

Of course, things could be easily settled if VCAA defined its Newspeak language in the Study Design. You might not like the definitions but at least there would be clarity for everyone. But this never happens.

As for what the VCAA Newspeak definition of “Prove” is …. and whether its definition is malleable according to context (for example “Prove the identity …” versus “Prove that if is even then x is even”), an undiscovered country full of potentially unpleasant surprises awaits us all.

Hmmm… that 2022 advice directly contradicts what was said at MAVCON a few years earlier at a “meet the assessors” session…

Pity there is no recording of said session.

It’s a matter of semantics – The end disks are surfaces and so, I assume, their contributions can be excluded from surface. But we shouldn’t have to guess or hope or assume. It’s not hard to word things correctly. But VCAA has a history of doubling down on its denials of errors, even in the face of the blindingly obvious.

1. And don’t forget that the straight line motion formulae for constant acceleration are back in the SD and appear on the Formula Sheet. Such formulae can make life easier and remove the need to set up and solve a DE provided the acceleration is constant. I wouldn’t be surprised to see one.

2. I agree. Every year (except one) the following generic question(s) is asked:

Find, correct to [however many] decimal places, the [minimum/maximum] value of [sample mean] such that Ho is [rejected/accepted].

Probabilities involving this critical value and a ‘true mean’ are then sometimes asked for and these are what we can now refer to in Specialist Maths as the probability of a type 2 error.

It’s probably worth adding that the probability of a type 1 error is always the level of significance (an easy 1 mark that is also easy to over-think and get wrong).

3. I suspect that if it was “… in the context where one of the parts is 1 [above] …” then it would be a “Show, using integration by parts, that …”

Anyway, now we’re entering the realm of prediction rather than ‘tips’ …

An interesting point (whether you realised it or not…) studying the formula sheet for a few clues is a good tip. Particularly, key formulas you may need which are NOT given on the sheet (in Methods, average value of a function comes to mind as the most common example).

I’m not holding my breath that one day VCAA will explain why a formula sheet is required for Exam 2.

VCAA can be politically correct with the names of people in their dumb-ass ‘real life contexts’ but it can’t do one simple thing that reduces waste.

You could always send VCAA an email…

Are you related to Elliot Goblet?

Not sure what you mean by the question…

Elliot Goblet is a comedy character created by the Australian comedian Jack Levi. The character is known for the deadpan delivery of one-line jokes.

Ah. In that case, no, but thanks for the compliment (or whatever it was).

In all seriousness though, I did try out the new VCAA Mathematics head by sending a query email about a line in the study design that I felt was a bit… unclear.

I got a reply within a week that clarified the situation a little bit while also ruling nothing out. On balance I am grateful for the reply but not sure I learned anything.

If you’re not sure you learned (i.e. didn’t learn) anything, why are you grateful for the reply?

What was the query?

I’m grateful because someone at VCAA took the time to reply to my email and give a response that did not presume I was an idiot. I’m not sure that I learned anything, but on reflection this means that I possibly hadn’t missed anything, which is a great relief. Hence, I’m somewhat glad I sent the email (even though it was on behalf of another teacher who did not want to contact VCAA themselves).

The query was about the following line from the Study Design:

“…effect of variation in the value(s) of defining parameters on the graph of a given probability mass function for a discrete random variable…”

I was asking if it was referring to the effect of transforming a variable, X by the linear transformation and how and were related to and respectively.

Michael MacNeil sent me the following reply:

“The study design dot-point to which you refer pertains to the parameters governing the generation of the distribution rather than linear transforms applied to the sample space. In this instance, and historically (back to the 2006 study design), this is looking at the number of trials and probability of success for the binomial distribution, and how variations in both of these affect the shape of the distribution (i.e. as n gets large, the distribution takes on a bell-shaped envelope, shifting p from 0.5 introduces skewness). For continuous random variables, the normal distribution is the example where the defining parameters are mean and variance.”

Of course, none of this is relevant to this discussion – so Marty, you are welcome to delete this comment.

Thanks, RF. Either I’m getting better at grabbing a hold of smoke, or you got an actual answer from VCAA. In this case I think it’s the latter – the way I read it, you got a definitive and reasonably clear answer to your question:

“…effect of variation in the value(s) of defining parameters on the graph of a given probability mass function for a discrete random variable…”

means the effect of the defining parameters n and p on the ‘shape’ of the binomial distribution (*).

(*) The funny thing is that ever since probability was butchered by the explicit deletion of the hypergeometric distribution (it still lurks – unnamed – in the shadows of sampling) and the the Poisson distribution, the only possible current pmf is that of the binomial distribution. It wouldn’t have hurt the VCAA to have mentioned it in the SD as an example of what the statement meant.

And for those who might be wondering, pmf is an acronym for probability mass function. It’s the discrete analogue of a pdf.

I believe that if Y = aX + b, the relationship between sd(X) and sd(Y) is not in the Study Design for Methods (it is for Specialist). Neither I think is the relationship between E(X) and E(Y) but I think many teachers teach it and the proof is pretty simple.

This is an indirect segue to my favourite theorem name. When you calculate E(X^2) you are using the Law of the Unconscious Statistician (LOTUS).

Be careful RF or you might become the go to guy for sending questions to VCAA for teachers who can’t for various reasons.

Hi BiB, thank you, the comments have been extremely helpful.

2. I always have trouble with the question you outlined above. Specifically, with Pr(Z>c)>0.05, I never know how the inequality sign flips when it is just c. As in, does it become c>1.644 or c<1.644? Any help is greatly appreciated.

It depends what accuracy the question specifies.

If you want the value of c, correct to three decimal places, such that Pr(Z > c) > 0.05 then you first need to get the value of c to more accuracy and then check whether to round up or down:

c = 1.64485.

Check c = 1.645 (round up): Pr(Z > 1.645) = 0.0499849

Check c = 1.644 (round down): Pr( Z > 1.644) = 0.0500881

So the correct value of c is found, perhaps counter-intuitively, by rounding down.

This is not to be confused with ” Find the value of c, correct to three decimal places, such that Pr(Z > c) = 0.05.”

In this case I would always use mathematical rounding and, in this case, give c = 1.645 as the answer.

The distinctions between Pr(Z > c) > 0.05 and Pr(Z > c) = 0.05 (and indeed Pr(Z > c) < 0.05) are subtle but important.

I offer no assurance that the VCAA understands any of this. From what I saw on the examination report for 2018 Exam 2 Section B Question 6 part (e), they don't. But for all other years the required rounding to get the answer has been consistent with intuitive mathematical rounding (I don't know whether that's by dumb luck or good planning).

You should do all the questions, understand their solution (no need for fancy expensive commercially available solutions, good but terse solutions are available for free on-line from itute), and read the reports.

These are difficult to explain without a diagram, so draw one and shade the critical region.

Now, if you choose one side (round up or round down) then the rounded value will either be in the critical region or not.

Choose the value that is in the critical region. Always.

If that makes no sense, I apologize – it can take a bit of thinking to see the logic here, but I promise there is a correct answer in these situations.

Pictures/diagrams are always an excellent idea. They make the situation so much clearer and will usually be considered part of the method for getting the answer and therefore worth a mark. A good picture (include appropriate labelling) really is worth a thousand words.

DI method for integration by parts!

really quick and easy to use, although vcaa probably wont accept it as method if it a 4 marker

I use it as a checking technique for exam 1

What on earth is the DI method?

A short cut:

https://www.wikihow.com/Integrate-Using-the-Tabular-Method

Good for checking answers, not recommended for use as a solution in a VCAA exam.

The “DI method” got a lot of traction over the last few years when Red Pen Black Pen started popularising it (https://www.youtube.com/watch?v=2I-_SV8cwsw) – I noticed Michael Penn using it after that point too (https://www.youtube.com/@MichaelPennMath).

Personally, I don’t like it – too much memory required. I prefer actually trying to write one term as a derivative then moving it with the chain rule – makes more sense to me – less likely to screw it up and only marginally more writing.

A couple of things I heard from others experienced colleagues:

1. Implicit differentiation questions are on the paper 1s almost every year…sometimes students use fancy method like partial derivatives to get the dy/dx expressions…while avoiding potentials to make mistakes in using product and/or chain rules.

The formula is dy/dx= – Fx/Fy, where F(x,y) is a two variable function…

A quick demo is here: https://math.stackexchange.com/questions/1021988/calculus-fracdyd-x-using-partial-derivatives

The risk of doing this is – if incorrectly executed, little or no mark will be awarded.

2. Integration factors can be useful in solving separable DEs… and it is not uncommon for the kids who take UMEP to use it.

3. In exam 2, particular Section B, if a student used “desolve” command or similar with their CAS technology they should spell out the “General solution” first and show an attempt to find integration constant… This is particularly important if a question is 2 or 3 marks…

4. When variable acceleration is assessed, DO NOT use “suvat”. Sounds silly but fatal if the kids applied constant acceleration formula inappropriately… Also, almost all similar questions in the past were targeted to the use of a = v*dv/dx or a =1/2d(v^2)/dx. Rarely was x”(t) assesses although in one or two relevant questions could it be applied (in a convoluted way, such as 2nd order homogeneous DE and characteristic equation stuff…)

On the other hand if the use of “suvat” is obvious and more efficient in a question then use of calculus is definitely OK but student must, for example, get correct integration constant but subbing in suitable values…

5. When finding the anti-derivative for some functions, the result of ln|g(x)|+c is often misused…If the anti derivative is wrong, any following steps will be unfruitful. And for some simple definite integral problems, it is not always necessary to make a substitution. If a substitution is attempted, the new terminals must match their substitution u=g(x)…

6. With the new study design, correct vector notations should be enforced, especially the use of dot product and cross product. Misuse of symbols would cost student marks.

Thanks very much, AST.

Re: 6. I would hope this has always been the case. In other words, if a and b are vectors and the dot product a.b is part of the working, things like ab and axb should ALWAYs have been penalised. The VCAA are hypocritical pedants when it comes to notation, but on this occasion the penalty is warranted.

Re: 2. Can you give an example from a VCAA exam of where the integration factor is useful and UMEP students used it rather than the ‘prescribed’ technique?

Re: 4. The return of the ‘SUVAT’ formulae is hilarious. In my opinion they should never have been removed from the study design. I’ve heard they were removed because students kept using them for non-uniform acceleration questions (a pathetic reason for removal if true). I never understood their removal given that many students who do Specialist also do Physics where ‘SUVAT’ is routine.

In general, I find many ‘UMEP tips’ (and the ‘tabular method’ as a short cut for integration by parts) are given by tutors (particularly commercial tutoring companies) in an implicit attempt to show how wonderful they are and how they provide an ‘edge’ for their lucky students. I often see ‘UMEP tips’ used (particularly by weak students), to the detriment of the student. A little knowledge is a dangerous thing. Indeed, it should be made crystal clear that

“The risk of doing this is – if incorrectly executed, little or no mark will be awarded.”

I usually get protest and pushback from students when I point out the above caveat emptor, at which point I simply say to the student “Why did you even ask me? Do whatever you damn well want, what would I know”.

Re: 5. I usually advocate making the explicit substitution. It costs nothing, may earn a mark, and usually avoids the dumb mistake arising from ignorant use of ‘short-cut’ formulae (again, usually by weak students).

Two (potential) examples – if I’ve not mistaken:

2016 SM2 ERQ3a – the concentration problem and the 3 marks DE, and

2021 SM1 Q7a, the same question RF just mentioned.

I guess the 2016 one was more relevant because the original DE has the particular form with one side = 0, which might induce a handful of kids thinking about the integrating factor method (a valid method if not stuffing up +/-).

These commercial companies may find their ways of promoting themselves – indeed detrimental if they don’t disclaim/inform any potential risks to their clients about these “investments strategies”.

I don’t disagree that both (potential) examples can be solved using the Integrating Factor (IF) method. What I was asking was:

a) Is that approach the most efficient approach? (In my opinion it definitely is not).

b) Are you personally aware of any student, UMEP or otherwise, who used the IF method? (In my opinion, using the method is foolish).

In my opinion the two examples given can be answered much more efficiently and safely using ‘separation of variables’.

1. At VCE level, Integrating Factor is not the most efficient approach.

2. Only handful of kids use them – a couple of them have decent maths skills and mathematical acumen, the rest just want to show their “mathematical muscles” to some extent.

I understand why “the rest” want to do it (and foolhardiness attracts its own reward). But why do the “couple of them [with] decent maths skills and mathematical acumen” do it? Don’t they have the common sense to realise it’s unsafe and inefficient?

They are all children playing with loaded guns. Where are “the rest” getting their guns from?

I had a read about the partial derivative method for implicit differentiation of functions. Interesting. Very interesting.

Not going to teach it to SM students though. Too much could go horribly wrong.

Agree with BiB on the IFs in DEs. The DEs in SM are (in my opinion) not the sort of DEs for which an IF is really much help and if you were to spend time looking for this, you may miss the more straightforward path to solution.

(Not that IFs are that difficult, but when time is at a premium as is the case in SM exams…)

Also agree about the dangers of partial derivatives, although the idea looks interesting in and of itself…

Just looking through the 2021 SM Paper 1, specifically Q7 and found a perennial bug-bear…

If you are referring to the set of non-negative integers, you cannot (according to the answer given by the examiners report) simply write and must write

In my opinion I‘d stick to

Z+ U {0}

instead of N U {0}, if I were to express non-negative integer set.

It is true to say – focusing on “fancy method” is at the cost of losing a more efficient approach. In this case separating and integrating.

A third alternative could be …

But it’s ridiculous that is not OK and gets penalised. It’s another example of VCAA refusing to explicitly address an ambiguity in the study design. Another item on the long list of churlish insanities.

Here is the biggest part of the issue: I do not know if a student would be marked wrong for writing N as the glossary of terms provided on the VCAA website lists two different definitions of N.

The fact that the examiner’s report did not write N but wrote does not mean writing N is wrong.

Or does it?

What is definitely wrong is the uncertainty!

So far you’re 1 for 1 with the VCAA. Maybe you should try for 2 for 2 …

(This is going to be deliberately cryptic)

I have been corresponding with VCAA a bit too much (for my liking) so far this year and while I do have plans to ask about their definition of the set they call (personally, I prefer the double-strike, but that is a lesser matter) it will not be until at least July 2024 and perhaps even February 2025.

I’m trying to get off the VCAA radar, not center myself in it.

Understood, comrade!

Here’s a tip: Let x = a be a solution to f”(x) = 0. Then there is only a point of inflection (PI) at x = a. Further investigation is needed to determine if there is or is not an point of inflection at x = a.

1) If f”'(a) neq 0 then there is a PI at x = a (‘triple derivative test’).

If f”'(a) = 0 then no conclusion can be drawn and the sign test must be used. (It seems that there are some trial exam writers that do not know this).

2) If the signs of f”(x) to the left and right of x = a are different then there is a PI at x = a (‘sign test’).

If the signs are the same then there is not a PI at x = a.

Note:

If f(x) = x^4 then f”(0) = 0 and f”'(0) = 0 but there is no PI at x = 0.

If f(x) = x^3 then f”(0) = 0 and f”'(0) = 0 and there is a PI at x = 0.

In both cases the sign test must be used to draw a conclusion.

at is a classic example and should be known by all MM and SM students.

I would argue similarly for at for a slightly different reason but along the same lines.

is a measure of the concavity of at but a point of inflection requires this concavity to CHANGE.

VCAA has been known to ask students to SHOW that a POI occurs and a sign test is really the only option.

Open question for anyone… (personally I think it is just more evidence of how little VCAA cares about the NHT examiners’ reports, but others may have a different opinion)

To rationalise or not to rationalise, that is the question.

VCAA SM 2021 NHT Paper 1.

Q2 and Q9 both involve answers where the denominator is a surd. In Q2, the answer leaves the surd denominator, but in Q9 the denominator is rationalised.

I see no good reason why one is better than another (unless the question specifies the form of the final answer, which is a different matter).

Ideas? I’m sure a few SM students may like to know the answer before they sit they exams in a couple of weeks…

Indeed …

To rationalise or not rationalise, that is the question

Whether ’tis nobler in the mind to suffer

The slings and arrows of outrageous VCAA,

Or to take arms against a sea of bulltish …

Unless the Report specifically says that rationalising the denominator is required, I ignore it when the Report does it. I just assume it’s the usual fetish, done for no reason except for the sake of doing it.

I agree that doing it is generally no ‘better’ than not doing it. I tell my students to only do it when the question explicitly says to do it – “Give your answer in the form …”, and I recommend it when there’s a ‘useful’ simplification eg (so you’re effectively cancelling out a common factor and getting rid of the fraction).

Clearly there’s no value in doing things you don’t have to do. At best it wastes time. At worst it wastes time AND you lose your answer mark if you make a dumb mistake (always likely in an exam). VCAA still deducts marks even if you make a mistake in something you didn’t have to do.

But who really knows …? I don’t understand the fetish for doing it all the time (and I doubt a good reason could be given). The thing to remember is that exams and reports are written by people no better than us, not by mathematical deities (although they might try and have us believe that they are).

The “simplifying a fraction” argument did occur to me, but why not give the required form, like they do with other questions???

I really do suspect it is something of an NHT only thing.

Could be. But it’s also a fetish I see in answers for trial exams. (And VCAA exam writers get their gig somehow …)

I think the “required form” should always be stated in a question. Unless you don’t care. In which case such a policy should be transparent and/or the ‘default’ required form declared somewhere …

Ditto with finding the equation of a line – often a “required form” is stated in the question. But when it’s not, the ‘default’ “required form” is y = mx + c. This has not been declared anywhere – it’s secret policy disclosed only to assessors.

I often wonder how much of this secrecy is deliberate, how much is unintentional, and how much arises simply from not giving a stuff about making things clear.