Oxford University Mathematician takes Cambridge Entrance Exam (STEP Paper) PART 1

The exam taken by Tom is the STEP Paper 2 from 2021. The exam forms part of the entrance requirements for admission to the University of Cambridge to study Undergraduate Maths.

The exam is based on material covered in A-level Maths and AS-level Further Maths which are taken by 17-18 year old students in the UK as part of their high school education.

Produced by Dr Tom Crawford at the University of Oxford.

You can also follow Tom on Facebook, Twitter and Instagram @tomrocksmaths.

Get your Tom Rocks Maths merchandise here:

[fuminocci]

What I find most scary about this is the fact he spends half of the time explaining what he does and still has time to answer the questions

[Stand_Up_Guys]

"I have done 0 preparation for this" - are we ignoring the PhD

[toph]

I am head of a maths department at a northern grammar school and am one of the small team that teach A level further maths. I have wanted to offer STEP practice in school for a while and have wanted to get myself ‘STEP ready’ so that I can do that. This video has inspired me to actually give it a go thank you! Just now need to think about how I would go from capable of doing the questions, to teaching the thought processes that would lead me down each pathway. Eagerly awaiting the second part!

[farfetchdthegamer]

I’m a final year high school maths student, and the absolute joy I felt when I paused the video, worked out 2 i, and got it right, cannot be put into words.

[DrumsTheWord]

These exams are both terrifying and inspiring. It ensures that only the most polished students pass, but also that the more casual math fan (myself), will never fluke it. It was fascinating watching you work through these questions, Tom. Thanks for sharing!

[AdamBomb]

It's weird, this was one of the step papers that I did for my Cambridge application. I did absolutely terribly. But now, after doing almost 2 years of maths at Imperial, I can't imagine not being able to do those questions haha. Great video!

[dbn]

I am a American Registered Nurse. I have always struggled with Math. I find this so amazing someone can grasp all of this so easily Algebra was so hard for me. I passed and was a Nurse until I just retired.. l have a Grandson who loves math . Hopefully he will succeed at least half of what you have and he will be way smarter than his Granny . A great Uncle was an Engineer so that is where he gets it. UR coolest Professor I have ever seen.

[olayinkaanifowose]

Easy way to factorise the cubic at 17:13, if you notice that the -2u -1 part is -(2u +1), then you can factor that away from the remaining 4u^3 + 2u^2 to get 2u^2(2u +1), then you have 2u^2(2u +1) -(2u +1) which factors easily into (2u+1)(2u^2 -1)

[mmmTheMuffinMan]

Me looking at question one: "What the fuck"
Tom: "Question one looks reasonable"

[gideonding]

for a non-mathematician like me, this is like trying to read & decipher the cuneiform text from Sumerian tablets. but love your mind that can solve these equations. wonderful stuff. awesome vid!!

[richardfenton]

Sat the equivalent many years ago. Your thought processes reminded me of what I always stressed to my children...read all the questions through carefully first. Especially valuable in maths multipart questions, as you can get a sense of where the question is heading and spot shortcuts.

[tomcurtis]

something i didn't realise before my maths degree started.. the jump in thinking from A Level to these questions is the same as between lectures and the assignments you get every week😭
devastating reality

[susanneostermann]

hello there, german here, so sorry for my bad english.... stumbled over your channel yesterday. i passed my maths examen over 20 years ago, one of my favourite topics was spherical geometry, but algebra was nice, too.... never needed any of that afterwards, BUT: i had a very bad night last night, and watching you solving mathematical problems helped me over some hard hours. thanks for being so passionate and sharing it with us! 🙂

[johnnath]

I am a retied auditor (79 years old) with a lifelong passion for maths (A levels early 1960s, A grades in pure maths and applied maths). I have listened to your video up to question 1 (will view the rest later on). Think I have an easy solution for the final part of Qu. 1: We have 1 + tanxcot3x = 1+ tan2xtan4x, (cosxsin3x + sinxcos3x)/(cosxsin3x) = (cos2xcos4x + sin2xsin4x)/(cos2xcos4x), sin4x/(sin3xcosx) = cos2x/(cos2xcos4x) = 1/cos4x, 2sin4xcos4x = 2sin3xcosx = sin4x + sin2x = sin8x, sin2x = sin8x - sin4x = 2sin2xcos6x, sin2x(1 - 2cos6x) = 0 & sin2x = 0 or cos6x = 1/2.

[SuburbAllied]

I am a history teacher. Im glad for that, I had a hell of a time to remember over 400 different dates and years connected to equal amount of various historical persons, events and milestones through our history. Given the fact that all of these years and dates are in a chronological order, that sometimes even goes parallel with different events, and that many events happens on the same dates and years to which some of them are more or less connected to each other, and others are not but still happens in same time frame for various reasons - I never though history was so close to maths when I started my journey to become a history teacher for the upper grades in college and junior high school! Your video reminds me about that time, even though I never did solved equations.

[ranpancake]

Definitely worth the wait, looking forward to part two <3

[totientquotient]

Haha, I remember doing this paper 2 years ago. Good times.
Question 7 was a gem. Behind all the algebra is the neat result that the only points on the unit circle where you could take a step of unit length in the positive x-direction and still end up on the unit circle are precisely the points with polar angles 120 and 240 degrees.
Another approach (one that I remember fondly) for part (ii) would be to notice that S³=I <=> S³-I=0 <=> (S-I)(S²+S+I)=0.
Since S²=(a+d)S-I, we get S²+S+I=(a+d+1)S. Plugging this into the equation above, we see that (a+d+1)(S-I)S=0, which implies that (a+d+1)(S-I)=0 since S is invertible. But S≠I, so any non-zero multiple of S-I cannot be the zero matrix. Hence a+d+1=0, whence the result follows.

[anishsinghrawat-pool]

Solved Along with you professor It's Always fun solving questions from round the world exams (like the korean SAT, American SAT, STEP, MIT integration Bee etc.) along with Dr Crawford. Big Ups!!

[samayahone]

this is so much fun! ashame i couldnt catch the whole of the premiere but loved this! glad i could follow at least some of it.. :DD

[chilledvibes]

On Q1 ii) tan is also undefined at 3pi/2, therefore when x = 3pi/4, tan2x is also undefined so 3pi/4 is not a solution.