Dr Tom Crawford – a mathematician at both the University of Oxford and the University of Cambridge – attempts the Scottish Advanced Higher Maths Exam taken by high school students in Scotland.
You can download the exam taken by Tom here: https://tomrocksmaths.com/wp-content/uploads/2026/04/2024-ah-maths-p1.pdf
And the official mark scheme is here: https://tomrocksmaths.com/wp-content/uploads/2026/04/2024-ah-maths-p1-mark-scheme.pdf
Watch more in the ‘Exam’ series here: https://www.youtube.com/playlist?list=PLMCRxGutHqfm3t0IVJabEab6OasV9WLrl
Dr Tom Crawford teaches undergraduate Maths at the University of Oxford and the University of Cambridge.
Edited by Ryd Cook http://rydcook.com/
Produced by Dr Tom Crawford at the University of Oxford. Tom is Public Engagement Lead at the Oxford University Department of Continuing Education: https://www.conted.ox.ac.uk/profiles/tom-crawford
For more maths content check out Tom’s website https://tomrocksmaths.com/
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26 Comments
As someone who's currently averaging ~98% in Advanced Higher Mathematics, I really think Scotland's education system is lagging behind England's in terms of mathematical rigour. There are definitely other things to consider, but ultimately our Maths course content and exam difficulty is much easier than England's in my opinion. Although I will say that paper 2 is longer and has some much more challenging questions (but ultimately not much creative thinking).
Wish A level Maths papers were this straightforward 😭
Большое спасибо вам, Том❤
удачи в развитии канала, вы отлично популяризируете математику и прививаете к ней любовь!
Watch more in the 'Exam' series here: https://www.youtube.com/playlist?list=PLMCRxGutHqfm3t0IVJabEab6OasV9WLrl
If you're wondering, paper 2 is just about the same when it comes to question structure, with there rarely being any focus on a real life story and more just "Hey! Solve this second order non-homogenous differential equation please!" It just has more multi-part questions.
There do exist questions that involve lots of stories in them, but those are typically reserved for the National 5 and Higher Applications of Maths exams, which are entirely separate from the normal Maths exams (Higher Applications of Maths requires you to know how to use R Studio)
please try the second round of the turkish ayt exam this exam is much more detailed than tyt mathematics exam
P.O.I is defined as this unfortunately. The alternate definition is looking at the third derivative non-zero
Back in the 90s when this was "6th year studies" it was expected that you would have 10-15 minutes at the end of the exam to check your answers for errors and that you have actually answered the questions asked.
Great video! It would be awesome if you could also solve the AYT exam YKS round 2 (Turkey)
@TomRocksMaths try doing a Maths D or Maths C (entrance exam for ENS(D) or polytechnique (C))
As someone getting ready to sit there Advanced higher maths exam in six weeks time this video was really enjoyable to watch .As someone who applied to Oxford this year , I fully agree with the lack of "story telling " and information within SQA AH maths questions and I think it really prevents Scottish pupils from being able to develop the skills needed to pick out valuable information within passages to answer questions. I definitely saw this when It came to answering problem solving questions within the TSA test and although its simple maths it was often a struggle due to the large amount of information and context in the questions ,I would like to see the SQA move to trying to make the maths exams a bit more challenging with a bigger focus on trying to solve problems in real life applications instead of it feeling like just regurgitating solutions learnt over the year.Would you consider making a video on the year 12 QCAA Specialist maths exam?really enjoy your content .
Great video Tom!
I missed those! Loved this one, no bs, just showing you know your maths.
My understanding of point of inflection is that it's a point where the curve switches from curving left to curving right or vice versa (left and right from the perspective of something following along the curve).
Like all (non-boundary) local maxima and minima have first derivative zero, but not all points where the first derivative is zero are maxima or minima, all points of inflection have second derivative zero, but not all points where the second derivative is zero are points of inflection.
My recollection from being taught this at school (in England) roughly 30 years ago is that points of inflection weren't actually explained; they were just listed as the third possible type of stationary point – minimum, maximum, or point of inflection. If the second derivative is non-zero, then you have an easily identified minimum or maximum; if the second derivative is zero, then it could be any of the three. For example, y=x^3, y=x^4 and y=-x^4 each have a stationary point at x=0, and y''(0)=0 in each case, but they're respectively a point of inflection, minimum, and maximum.
Going beyond what I was taught at the time to what I've learned since: if the second derivative is zero and the third is non-zero at a point, then you have an easily identified point of inflection; if the third derivative is also zero, then you might have a point of inflection – you can either consider higher derivatives (if the first non-zero derivative above the second is an odd one – third, fifth, etc – then it is a point of inflection; if it's even, then it's not) or the nearby values of the second derivative (if they change sign, it's a point of inflection).
i have a numeracy exam next month and was wondering if you could do a video of you completing the gcse numeracy exam any exam board is okay but preferably WJEC please keep up the good work love your videos man 🙂
Can you try the greek panhellenic exams?
Point of inflection has been confused with stationary point of inflection.
A necessary (but not sufficient) condition for P.O.I is f''(x)=0. But then you must also show f''(x + epsilon) and f''(x – epsilon) have different sign. Without this last part one could still have a maximum or minimum, e.g. f(x) = x^4.
For reference this is also the definition of P.O.I in a-level.
Thanks for great video 🙂
Thank you for the video! It would be very interesting to see you doing Northern Irish Further Maths or Maths A-level papers (CCEA exam board).
From AB. Hello TRM. If you slowed down you wouldn't make the mistakes that later on in a question you've realised you have done and spend a lot more time to correct them. It's not a speed test. It's a test to let the examiner know you understand the fundamentals. You will always be given enough time to answer them all. Calm down and let your knowledge do the talking. You know this. Love to all. 62 year old AB who took undergraduate maths at Warwick
To invert the matrix M augment with the identity matrix, I,e form [M | I]. Use row operations to make [I | B]. Then B is the inverse of M. There is a formula but using it just tests memory not understanding.
for p5 f''(x)=0 is necessary but not sufficient for point of inflection, a most obvious counterexample is f(x)=x^4
To comment on the point of inflection thing: I have always considered the point of inflection to be the point at which a curve changes concavity. The first derivative gives us the change in gradient, the second derivative gives us the change in concavity
Tom is always a joy ! The Singapore A levels H2 Further Math is a killer as well https://www.alevel-maths.com/further-maths https://www.seab.gov.sg/files/A%20Level%20Syllabus%20Sch%20Cddts/2025/9649_y25_sp_2.pdf
The way I learnt a point of inflection in high school was a change in concavity of a function. Where a stationary point of inflection is the special case you spoke of where both derivative and second derivative =0
I think you should try the second stage of the YKS, the AYT, especially the one from 2021.
Maths Teacher here! I usually teach the students that the point of inflection is when the function goes from being concave to convex or vice versa. So finding it would involve setting second derivative to 0 and checking either side 🙂