This is a brief post, intended as a place for discussion of the 2021 NHT Specialist Mathematics exams, here and here. See also the accompanying 2021 NHT Methods exams post, here.

We’ve made a few remarks, below, on specific questions, and see this WitCH, but we haven’t gone carefully through the exams. We’ll update with further remarks if and when commenters raise substantive issues with the questions, or either exam as a whole.

**UPDATE (10/09/21)**

The exam reports are out, here and here. We haven’t looked and don’t intend to look, except for Q10 on Exam 1. The suggested solution is, well, rooted. See here.

**EXAM 1 **(Here)

**Q2**. It is interesting that coefficient of friction is in the curriculum even when it isn’t. (It should be.) Both (a) and (b) are poorly worded, and a diagram is really required for (b). (01/09/21) An awful (and common) notational confusion, whether *P* represents the force (vector) or the magnitude of that force; if, as stated, the latter, than *P* cannot “act horizontally”, etc.

**Q3. **This one’s a big problem. Requiring a student to know the mean and standard for the binomial distribution (specific Methods material) in a Specialist exam is simply ridiculous. in private conversation with a teacher, VCAA has attempted to justify this with reference to the Study Design (p 81):

* Together [knowledge and skills from Specialist 12, Methods 12 and Methods34] cover the assumed knowledge and skills for Specialist Mathematics, which are drawn on as applicable in the development of content from the areas of study and key knowledge and skills for the outcomes. *(emphasis added)

This is nonsense. The specific binomial details are not required for the development of anything in Specialist. It is absurd to suggest that this passage gives license to the exam question, and VCAA should admit that they screwed up (again).

** Q4. Q5. **The “show that” in (a) is woosy. In (b), to request an answer in the form

is absurd and hilarious.

**Q6. **Poorly worded.

**Q7. **Hmm.

*If its velocity is 4 ms ^{‒}^{1} when x = 0, find its velocity, where v > 0, in metres per second, when x = 2.*

It takes a conscious effort to write that badly. (01/09/21) Plus, the quantity *v* is never defined in the question.

**Q9.** A debacle. See here.

**EXAM 2 **(Here)

**MCQ2. **An interesting and tricky question, presumably destroyed by CAS.

**MCQ3.** A nice question, but with the options in gratuitously non-obvious form.

**Q2.** Part (b)(ii) has infinitely many correct answers.

**Q5. **A berry that, when it falls on an inclined plane, neither bounces nor splatters nor rolls, but just slides? Do not eat.

**Q6. **More discrete probability. Not egregious in the manner of Q3 on Exam 1, but it shouldn’t be here.

Haven’t really looked at Exam 2, but on Exam 1:

2. The use of “P” here is a bit confusing. Initially “P” is used to refer to the magnitude of the force, whereas in part a), it appears that “P” is used to refer to the force. (I’m not sure what it means for a magnitude to “act horizontally”).

“a horizontal resistance force …” already mentioned by Marty, but this can all be avoided just by stating that the coefficient of friction is 0.25. I suspect that coefficients of friction are widely taught anyway, even if just because it appears on so many past exam papers.

The inclusion of “in terms of g” is, I think, intended to clarify to students that they don’t need to do any additional calculation involving g = 9.8, but the fact that this is even required is pretty damning (or an admission of guilt for past sins).

5. Why not just something like “find y as a function of x”?

6, Not quite seeing Marty’s objection to this, since the intended interpretation of the question is pretty obvious, but I do agree that the run-on sequence of prepositional phrases (“to the graph of the relation at (1,1)”) is a bit cringey. What else am I missing?

7. I don’t see the need for the brackets in “acceleration is (5 + 6x) m/s^2”.

Re: Q6. The wording is terrible. It’s like VCAA want to use one long sentence to say everything. We’re being monologued at. It is ‘stream-of-consciousness’ writing and is abominable. Here is how it should be worded:

Let a graph be defined by the relation where . The equation of the tangent to the graph at the point (1, 1) is . Find the values of a and b.

Break up the ‘sentence to infinity’ into a couple of short, clear sentences. VCAA have a special talent for NOT doing that.

Thanks, SRK. I’ll update on the vector/magnitude thing. The others are irritants, but probably not enough to flag. With Q6, yes it’s just the awful sentence structure. It’s not generally easy to write a maths question succinctly and clearly. But, Jesus, it’s not *that* hard.

Exam 1 Q7. But you only need to find its velocity when x = 2, if its velocity is 4 ms‒1 when x = 0.

I agree with Marty that the writing is bad. Poor writing is one of VCAA’s many faults. It should have been worded something like:

“A particle ….

When x = 0 it’s velocity is 4 m/s. Find its velocity v > 0 when x = 2. Give your answer in m/s.

Furthermore, the writing deserves an *extra* whack …

Nowhere in the question is the symbol defined. We must (reasonably) assume it means velocity. VCAA does not give students the same courtesy of making such assumptions – undefined symbols get penalised regardless of how reasonable the assumption is. The classic example of this hypocrisy is the typical VCAA question “Find the derivative of f(x)” and a student writes

without defining what is.

When you set yourself up as a smug, sanctimonious, pedantic, know-all shithead, you must expect to be judged by the same standards.

Thanks, John. Very good point about v. I will add.

Re: Exam 1.

I will make only two comments (for the moment):

1) Q3 is FUBAR. The writer(s), vettor(s) and VCAA are a total disgrace. Teachers will (should) go medieval on VCAA if this ever happens in the November exam.

2) I have seen Q9 on at least two different commercial trial exams: Insight 2020 and TSFX 2015 (amended for the current Stupid Design). The similarity is astonishing but not for one moment am I suggesting plagiarism. It is clearly a very popular arclength question.

Re: Exam 2. I will make only three comments (for the moment, at least):

1) Re: MCQ 2. Yes, using a ‘slider’ or a ‘Manipulate’ command in a CAS enables the behaviour of the graph to be easily observed as the value of is changed. Each option can be trivially tested and Option C quickly identified as the correct answer. It would make a nice non-CAS question.

The similar MCQ 3 is also completely trivialised with a CAS.

2) Re: Q6. Agreed. Not as egregious but still a pointless dirty trick that also wastes 4 marks that would have been better used testing explicit Specialist Maths content.

3) Re: Q5. Yes, a suspect berry that rolls down a poorly located plank in a completely unrealistic manner. VCAA and their stupid ‘non-real life’ contexts.

Both exams smell like they were written by a Maths Methods teacher(s) with limited understanding of the Specialist Maths curriculum in particular and limited mathematical expertise in general.

JF, I think the arc length question on Exam 1 is just a classic. I remember a little while ago looking through a bunch of mainstream first-year calculus textbooks (along the lines of Stewart, etc.) to see what sorts of questions they had on arc length (since the Cambridge book is a bit light on for good ones). They all had pretty much the same set of questions – including ln(sec(x)). I think it’s just the nature of the question that there are very few cases that are neither completely contrived nor impossible to calculate exactly.

If I’m to believe what you say, then I have to believe that the writers actually consult first year calculus textbooks …. And can understand what they read. You’re asking me to believe a lot, SRK 😉

And even if this is true, it’s those previous trial exams that made the first year calculus textbooks question ‘operational’ within a Specialist Maths context … It’s gross carelessness on VCAA’s part not to have checked the commercial trial exams. Particularly a 2020 trial exam …

I think the point I was trying to make is that there seems to be a fairly limited pool of arc length questions that are reasonable to ask in a non-calculator setting, and the “standard” questions are probably well known to people who write exams / trial exams, even if they’re not being directly quoted from a specific textbook.

On the point about whether VCAA should check whether their exam questions have appeared in commercial trial exams, I note that quite a few of the arc length questions from past exams appear in the Cambridge textbook. See: 2016 Q7 (Cambridge, 8E: Q6); 2017 Q7 (8E: Q8); 2017 NHT Q11 (8E: Q7).

The attached may be of interest.

Finding curves with computable arc length

And also this attachment.

Arclength and Pythagorean Triples

And maybe this attachment (a bit advanced).

Summarising and presenting this material was on my list of MAV Conference Presentations ideas. It never eventuated because I refuse to pay for the ‘privilege’ of presenting (besides, the MAV never seemed overly keen for presentations that had significant mathematical content). What a fucking nerve – the MAV *sells* a product (the conference content) and requires the people who provide the content of the product to pay for providing the content!! Barnum would be proud – “This way to the egress!”

(But I wonder if the VCAA Assessors get charged for presenting …??)

15-RationalArcLength

Thanks Marty so much for putting incredible amount of efforts in organizing these posts and it is great to see uncountable and valuable mathematical discussions here from everyone.

Re: “A berry does not bounce nor deform nor splash…Never eat it.”

This is really a “rigid” berry! Can we say it has some properties of manifold?

John, you revealed another small corner of your mathematical wealth treasury by sharing some good arc length stuff with us. Greatly appreciated. If only you (or Marty) present again on MAV conference…It‘s been so long…Six years?

I guess, though, the chance of eventualising that might be statistically insignificant…

SRK:

You have acute awareness of similarities between textbook questions and exam questions…People always expect some sort of heavily-algebraic arc length question on SM exam 1s in the past……As teachers we are always keen to see something new but …never know. Writing an elegant exam question really needs strong improvisational skills plus a little spark of thoughts…

Getting my head around with limiting equilibrium, tension and rate of changing angle theta with respect to time is not very easy, and here I got more inspirations and enlightenments…Very nice.

Nothing else assumed from Methods 34 except for normal distribution? Well, well…Hopefully. I will be “very surprised” if there are any other discrete/continuous r.vs involved in this November SM exams.

In fact I won’t be too astonished if something similar to old exam questions 20 years ago or so would appear on November exams…New bottle, old wines…Like MCQ3 in this NHTSM2, it directly comes from (correct me if mistaken) 2001 SM exam 1…

Hi Lancelot. Kind words for everybody as always.

Speaking for myself, a *significant* change in the current MAV mindset is necessary before I’d consider presenting again. To present properly requires a lot of work including a huge amount of time preparing the paper for the Conference Proceedings Book. It’s beyond the pale in my view that I (or anyone) providing genuine educational content would have to pay for doing this.

Fun fact: In the ‘olden days’ Presenters got ‘paid’:

1) free attendance on each day of presenting,

2) a small gift (it’s the thought that counts), and

3) access to the Presenters Lounge which also provided decent food.

I assume that under the current MAV business model all this is considered an unnecessary expense. OK, fair enough for 2) and 3). But to require presenters pay for providing the product …

I would presume that at a minimum Marty would want to see a significant similar change of mindset. Except he was also very poorly treated and I think as part of the minimum a formal apology is warranted. It’s a real shame because Marty was always a strong attraction and had the theatre packed – he was always entertaining, thought provoking, educational and took no prisoners. The Conference is the poorer for his absence.

Change in any organisation starts from the top. So I assume ‘infomercials’ will continue to be .

On another disappointing note, I wonder how much teachers really care about things like unreasonable statistics content (such as the binomial distribution) appearing on the Specialist Exam 1 and the potential for this shit (or even worse) to happen in future exams. The reaction in the wider teaching community has been disappointingly under-whelming. It seems to me that most teachers only start bleating about the stable door the horse has bolted. This sort of apathy is why VCAA and others gets away with so much shit. (On the whole, I noted a similar apathy in response to the Daft Mathematics Stupid Design and the Daft ACARA Curricsputum document) I wonder if it’s a variation on the ‘By-stander Effect’ …

Hi Lancelot.

Re: Like MCQ3 in this NHTSM2, it directly comes from (correct me if mistaken) 2001 SM exam 1…

It rings a bell but I couldn’t find it. What question on the 2001 SM Exam 1?

It wouldn’t surprise me if old questions were being copied. Why not, when you don’t have the expertise to write your own? Forgetting (or not understanding) how trivial many of these questions (such as 2021 NHT MCQ Q3) are using contemporary CAS technology … (particularly ). The mind-set of the VCAA exam writers still seems firmly planted in the pre-CAS era.

Re: I will be “very surprised” if there are any other discrete/continuous r.vs involved in this November SM exams.

Well, as I remarked earlier, even VCAA cannot be so dumb (could they?) as to include Methods content that was not on the Stupid Design in 2020 – such as continuous random variables that are not normal. I’m sure the NHT teachers were very surprised when “discrete …r.vs” appeared … But now that we are fore-warned of this VCAA idiocy, why would it surprise you if it appeared in November?

Thanks for the kind words, Lancelot. I last presented at MAV in 2016. I would definitely present at MAV again, or pretty much anywhere, if invited. But of course MAV and such groups won’t invite me, because I refuse to play the role of performing monkey.

Wanting a performing monkey is OK (I’m one of those most of the time) …. Wanting a monkey is what’s objectionable ….

Maybe if you get invited, you don’t have to pay to present.

Of course. I would never pay to speak.

Thanks for the references, JF. I am familiar with the result in the first paper you posted, and have used it to construct arc length examples / questions for my students. The last paper will take a bit more time to get through – a lockdown project, I guess.

Thanks, John. Nice articles. I think SRK’s point still stands, that in practice there are just not so many easy-reach arclength problems. That was always my feeling when setting assignments/exams at uni. I know that there are reasons in this case to suspect that VCAA is not as innocent as all that. But it is not easy to conclude that they are as guilty as all that.

And on the topic of arclength (and hence the related topic of volume), this attachment might be of interest.

Supersolids-Solids having finite volume and infinite surfaces

Further to Exam 1 Q3 and the mealy-mouthed excuses given by VCAA:

The Mathematics Stupid Design was amended last year, so going on VCAA’s own mealy-mouthed excuse, there will/should be nothing on the Specialist Exams this year that requires knowledge of continuous random variables except for the normal random variables. Because continuous random variables were deleted from the Maths Methods Stupid Design in 2020. So, in particular, there should be no questions (like the 2019 NHT Exam 2 MCQ 18) that require using the Central Limit Theorem.

If this proves NOT to be the case, then, as The Thing would say: “It’s clobberin’ time!”

https://i.pinimg.com/originals/3d/0a/3f/3d0a3fc3f9b67f7f4910ba07b5023a9f.jpg

A clarifying statement from VCAA would be very useful, but also very unexpected.

Thanks, John. In brief reply, Q3 on Exam 1 is obviously a screw up, but I don’t think it should be taken as a sign of (that particular) idiocy being likely to appear on a standard VCE exam. VCAA knows that people would, rightly, be screaming if that kind of shit was pulled on the main exams. So, in brief, I think VCAA knows exactly that Q3 is a screw up, but simply don’t have the integrity to admit it. Of course not admitting it is a further screw up. But I don’t think it means that binomial is now considered by VCAA to be fair game for Specialist.

I’m sorry if this is off-topic, but as a foreigner… these exams seem overall worse than VCAA exams, is that right? Is there a reason?

You know that NHT or whoever it is could just pay someone decent (a careful math academic or math teacher) a few hundred dollars and they would go through and notice most of these problems. I guess they don’t care? Or they don’t realise that this is an option?

Hi, Glen. It’s not off-topic at all. There’s really two separate points: NHT, and general vetting.

1) The NHT exams are typically taken by three men and a dog. As such, no one really cares about them, inside VCAA or out. EXCEPT that they are unofficially precedent for standard VCAA exams.

2) In general, the VCAA exams, at least the standard ones, *do* get vetted, and to at least some extent, some times, by “mathematicians”. But these “mathematicians” are selected by their level of compliance, not competence. And/Plus the underlying curriculum, and the underlying culture, is so corrupted, so devoid of mathematical (or any) sense, how would even a competent mathematician be able to vet anything? Look at any VCAA exam. Where would you begin? Sure, you could remove some howlers, but the whole thing is always one monster howler.

And, as I keep saying, it is all ritual. No one cares.

Talking in specifics, if I was asked to vet an exam, I would take a close look at the curriculum and what was supposed to be tested as well, not just the exam itself. That’s what I mean by “careful math academic”.

Interesting that there is a vetting process at all, even a farcical one.

Your last point has me interested. If I was to vet one of these exams, I would guess that any comments on the curriculum would be irrelevant/ignored. But is your point that it is impossible to write good questions based on the curriculum as it stands? That seems hard to believe.

Or is your point that the vetting is limited to only correcting the questions as given, not to scrapping entire questions and proposing better ones?

If I *were* to vet one …

My points are

a) It is very difficult to write good questions based upon the curriculum.

b) No one cares.

Re: b) Most people care. Some (most of the most) just don’t care enough.

You missed c): Pay peanuts and all you’ll get are monkeys (trained, not performing).

Po-tay-to po-tah-to.

I also forgot

(d): Most teachers are unaware how shitty it all is.

I finally had the courage to look at Exam 2. Did anyone else notice the breakthrough achieved with Q5? It seems that the berry, when hitting the plank, suddenly loses the momentum component perpendicular to that plank. But it does not deform or bounce!! This is evidence of new physics. Does the energy lost go down a wormhole or is this the effect of unseen dimensions?

The strange properties of the berry have been noted, and a suitable warning have been posted.

At least when I write SACs, the UNreality I deliberately write is Blues Brothers unreality, not an earnest and misguided VCAA-esque attempt to imitate ‘real’ reality. I don’t know why VCAA can’t just bite the bullet and set all their questions in a Blues Brothers universe where everyone knows that your tongue is planted firmly in your cheek and reality is suspended. An example from my recent SAC 2:

Oliver has learned that a criminal mastermind known as The Calculator has stolen a shipment of weapons of maths instruction and is planning to re-purpose them for use in corrupting mathematics education in Star City. As Calculus Arrow, Oliver is determined to take down The Calculator and neutralise this threat.

Question 1 (21 marks)

One of the stolen weapons is a function with the equation . This function has many dangerous properties so Calculus Arrow decides to study it over a controlled domain and carefully probe for weaknesses.

etc.

Calculus Arrow can find no weakness in the function so he decides to integrate it into one of his specialist arrows. Shooting this arrow will undo any effects the function might have if The Calculator uses it. But the integration is very tricky and dangerous. Calculus Arrow needs the help of his friend [redacted], also known as the vigilante Spartan.

Calculus Arrow and [redacted] are in the Calculus Cave pondering over the integration of . [redacted] finally decides a substitution is needed. Calculus Arrow agrees.

etc.

Jesus. You’re asking for the return of Tasmania Fucking Jones? The cute scenarioising on top of the idiocy was intolerable.

I don’t want cute, I want cool Marvel Universe. DC Universe. I want epic contexts so outrageous that you just have to buckle up and laugh. And all the while the maths is kept real.

Tasmania Jones was a try-hard wanker. I want Calculus Arrow. I want Die Hard: With a Random Variable. I want The Functions of the Galaxy. I want Bill and Ted’s Excellent Mathematical Adventure. I want Lethal Function. I want Survivor: Integration Island. I want Trigonometry Widow. Get me Galaxian Highscore and his sidekick Max Hardcase. I want …

The ‘non-real life’ contexts VCAA use are lame, contrived and dubious. If we must have contexts, let’s embrace contexts so ridiculous that they’re awesome. I want larger-than-real-life contexts.

I don’t want cute wanker Tasmania Jones. I want the fucking awesome Iron Logarithm.

And perhaps some dogs for which the number of spots follows some type of sampling distribution…?

Your mask is wearing thin super-hero…

RF, it’s funny you should mention dogs (and allude to my not-so-secret identity in passing!). For those who came in late ….

Here’s the preamble to a question from a SAC a few years ago:

Question 1

Canine Bio-Engineering is a bio-tech company that has bred a genetically modified Dalmation dog (canis familiaris minima maculosa) for spot-intolerant people who want a Dalmation dog but don’t want it to have too many spots.

The number X of spots on the modified Dalmation dog is a discrete random variable with probability distribution as shown in the following table:

By the way, a competitor company bred a genetically modified zebra …

Not to be outdone, a competitor bio-tech company (yes, it’s a dog eat dog world we live in) has bred a genetically modified zebra (equus quagga minima clavatus) for stripe-intolerant people who want a pet zebra but don’t want it to have too many stripes. The number Y of stripes on the modified zebra is a discrete random variable with probability distribution as shown in the following table:

But RF, you’re 10 years behind the curve. Put on your game face and suit up!

PS – Who could forget Harry Potter and the Chamber of SACs …

In my defense, I spent most of my teaching career in the IB world. SACs are a recent issue I have had to contend with.

Some of the SACs I’ve written though are groan-worthy as well – the rate of cooling to work out what time the Specialist teacher was killed on the morning of the SAC – the rate at which the pool of blood was growing…

My point (albeit a bit obscured) is that writing decent questions doesn’t have to be at the expense of telling a story – so long as the story is clearly tongue-in-cheek and so not an essential part of the problem.

VCAA – if you’re listening…

Re: In my defense, I spent most of my teaching career in the IB world. SACs are a recent issue I have had to contend with.

That would have been very jarring.

To be totally fair, the IB has made a few mis-steps in recent times: getting rid of matrices from the SL course, abandoning half-decent portfolio tasks for single, “must show personal engagement” investigations and the inevitable creep of the CAS machines.

That said, the HL course put a lot of emphasis on proper mathematics and the examiners wrote really, really good questions.

The HL Statistics optional module is a really rich source of ideas Specialist teachers could easily use for some half-decent SAC starting points and I suspect with the introduction of Graph Theory from 2023, I might revisit the Networks module exam questions as well for inspiration.

Yes, their exams also had stories on occasion, but the questions were fundamentally mathematics questions.

SO IT CAN BE DONE.

The entire world is getting dumber. Just, Australia is getting dumberer.

Yes, I have all the IB exams and the old Portfolio tasks. You are right, of course.

Marty, Dumb and Dumber … What a great context for a SAC. I’ll make sure I use it when I’m next audited – because I know exactly who the starring characters will be in the storyline.

I’d go Fear and Loathing in Las Vegas, or Eraserhead, but that’s probably just me.

Eraserhead – I remember seeing it at Melb Uni. Very strange movie. I’m not sure it would resonate with the post-2020 cohort of students …

Hmmm … ‘Fear and Loathing in Mathematical Methods’ has possibilities.

If we’re going down path, I’d suggest Zero Effect … (whose title comes close to predicting some of my recent SAC results).

We should probably continue going down this particular rabbit hole at SACs of Shit ( https://mathematicalcrap.com/2020/05/03/sacs-of-shit/ ) … (Unless we’ve hit rock-bottom here lol)

Hahaha too funny.

For those interested, the Examiners “Reports” for Specialist NHT 2021 are now available on the VCAA website.

I’m guessing that means Methods will follow soon.

Top of page 1 on the Paper 1 report: “…these are not intended to be exemplary or complete responses”

(Probably says the same thing on all reports, I’ve only just noticed it.

Too funny, except it wasn’t intended as a joke.

The report solutions are most definitely exemplary.

Indeed. Exemplary of what, though? That’s the (rhetorical) question …

OK – here’s my question (VCAA if you’re listening…)

Since the NHT reports contain no statistical reports of what students scored (on average) for the questions, how many students chose each MC answer, what students did well on and what they did not, these are essentially just a set of answers. I would assume that answers are written BEFORE the exam is actually given to students…

…SO WHY THE LONG DELAY IN PUBLISHING THEM?

That’s a very good question.

Re: Q3 and the inclusion of Maths Methods probability content not specifically on the Specialist Maths Stupid Design.

Writers of trial exams for commercial organisations often take their lead from the VCAA exams. For example, one such organisation demands that the writer very closely imitate the questions asked on the previous year’s exam, showing very little insight.

And so my school buys a bunch of Specialist Maths trial exams ( my unequivocal advice to). I have just started taking a look at some of them – several have included Maths Methods probability style questions. For example:

a) Trial Exam 1 from one hefty organisation gives a pdf and asks for the variance to be calculated.

b) Trial Exam 2 from another organisation (by the way, a favourite band of mine is TISM) has a question with many defects. Within this disaster zone:

i) it gives a pdf and asks for an expected value of something

ii) that may or may not be a random variable (the symbol is not defined, so it’s anyone’s guess).

Monkey see, monkey do. Such are the consequences of

1) VCAA incompetence,

2) VCAA incompetence-denialists,

3) Gormless trial exam writers.

4) Decision-makers in schools who insist on buying trial exams when there is absolutely no need to. As I’ve said before, most schools have access to at least 100 exams including VCAA exams, commercial trial exams previously purchased, and trial exams freely available on the internet. It’s total stupidity to be buying trial exams year after year!

And don’t get me started about the stupid writers of statistics questions who have no clue about statistics. (One neat writer of a Trial Exam 2 thinks that the distribution of sample means from a normal population is only normal …, getting back to a favourite band of mine, another organisations thinks that you construct confidence intervals for sample means …) So much shit!!

And, This Is Serious Mum, there’s at least one organisation whose writer(s) and vettor(s) think that if X and Y are independent random variables, then Var(aX + bY) = a Var(X) + b Var(Y). How does this shit ever see the light of day, let alone that people actually to let it see the light of day!??

The worst part is that there are plenty of teachers who swallow this shit and come back for more. I know of many schools that simply copy and paste from commercial trial exams when ‘writing’ their Probability SAC 3 and simply use the ‘solutions’ to mark them.