Oxford University Mathematician takes Cambridge Entrance Exam (STEP Paper) PART 1
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- Опубликовано: 10 сен 2024
- Oxford Mathematician Dr Tom Crawford completes the STEP Exam which is used by Cambridge University as part of the admissions process to study Undergraduate Mathematics. Part 2 is here: • Oxford University Math...
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.
You can download the test for yourself here: tomrocksmaths....
And the mark scheme is here: tomrocksmaths....
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.
Tom is an Early-Career Teaching and Outreach Fellow at St Edmund Hall: www.seh.ox.ac....
For more maths content check out Tom's website tomrocksmaths....
You can also follow Tom on Facebook, Twitter and Instagram @tomrocksmaths.
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Get your Tom Rocks Maths merchandise here:
beautifulequat...
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
That's not scary, he's Rollin on it...he's smashing it.
Yea..he's doing alot better than be on that...whew.
'Relatives and stuff"
It's started to scare me when I studying that stuff
Not as far as he is on it. I over think it and failed
My final bad..I have to retake it...
@Thawne ha haa...well maybe it is scary...because I'm not that good at it...but maybe he studies alot more...and doesent have to deal with all the other things collage students have to deal with....I'm not sure....
If you were calling me boomer....thanks..but I'm not that old
"I have done 0 preparation for this" - are we ignoring the PhD
yep!
Tbf a phd is specialised and could not have anything to Do with this, also the higher in maths you go I believe the worse at the basic stuff you get
apparently!
@@ashedaniels1087 Not so much that you get worse at the basic stuff, but some of the stuff taught at gcse/a-level especially in mech/stats is pretty specific and not something that a lot of pure mathematicians will looks back on. Also deriving formulae and so forth isnt something you typically bother to do once youve learned them.
@@ashedaniels1087 I think you’re taking the joke a bit too seriously
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!
What does STEP mean (I have poor grades of Maths + Further Maths at A Level and a applied statistics module as part of a business degree (which I got a 2.1 in , what that says about A Level knowledge and degree grading I have no idea -maybe just being two years older makes a difference )
@@highpath4776 step is the exam he’s taking in this video. It’s just the name of the exam
@@highpath4776 Sixth Term Examination Paper - i.e. you sit it at the end of two years (6 terms) of A-level study
@@highpath4776hehe. I got an A in both maths and further maths a level, and got a 2.2 at uni in economics from Cambridge.
Lots of stuff makes a difference. Your situation and emotional health in the relevant years.
Degrees aren’t that hard.. somehow instil got a 2.2.. I chose a stupid paper in my final year.. and had a lot of obstacles at that time.
During school I had all the environment in my favour (boarding school, and dad not interfering with my life.)
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.
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.
U literally sub in stuff into an equation lol
@@Pudgy973 Well yes obviously you just sub stuff into an equation, I was talking about the fact that it’s nice that there’s a quarsion like that on an otherwise stinking hard paper.
@@farfetchdthegamer3810 Most of the questions are like that. The first part is usually something easy or obvious. You may occasionally need a second or two to think about it, but it's generally straightforward stuff.
And then the questions devolve.
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)
Woah clever, thanks for sharing your insight
Nice spot!
You fooking wot mate.
You can also group the 4u^3 and 2u^2, and group the -2u and -1, which is essentially what you did here but maybe easier to think about for a lot of peeps.
I like to say that the first 2 guys are just like the last 2 guys but multiplied by u2 😂
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!
How can it be terrifying? It is standard material for a 14 year old in any decent education system.
@@gswombat
No it isn't.
@@A_Lemon_Party It is in my country.
@@gswombat
Sure.
@@A_Lemon_Party It is. Just because you come from poor quality education system does not mean all such systems are as bad as yours.
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!
Exact same situation but the natural science paper!
I could have tried for Oxford / Cambridge but they didnt really do the undergrad course I wanted, as it was I did a business degree and spent six months in oxford working for a commercial business there , I still never went to the colleges - too intense work really , would have liked to find some stats and computer time to copy over the data I was analysing - could have been interesting.
@@jamesrowntree I assume you took it back when the second part was long-answer? Now the whole thing is multiple choice, and to be honest, I find the complexity of the questions very easy. The difficult part is the non-calc aspect, but mostly the sheer speed at which you have to do it.
None of the questions were anywhere near as hard as the Physics or Chemistry olympiads, which actually didn't make me very happy as I felt it was more of a speed test. Luckily a lot of the admission is based on the interview, which is a fair bit harder.
@@matthewrayner571 My exact thoughts, the questions are mostly trivial and it's about speed and that annoyed me. I stupidly didn't do enough practice when I applied because I thought I could wing it and get an interview where I knew I would perform well. As it was I screwed up because it turns out most people can't do that and get enough questions right in the time available, didn't get an invite, but thats probably for the best anyway because I sort of knew at the time but lied to myself about the fact I didn't really want to go there. The Physics Olympiads were by contrast pretty fun, I ended up going with questions on the James Webb telescope and something else which I can't quite remember but also related to astronomy. Much more interesting than doing tons of past papers for an entrance exam imo.
@@jamesrowntree exact same bro, you at UCL? I'm here doing natsci 😂😂
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.
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!!
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
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.
Me looking at question one: "What the fuck"
Tom: "Question one looks reasonable"
Same
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! 🙂
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.
Impressive!
Bloody hell
yeah it was a tough paper but not too bad. I did 2 3 and 8 fully but the rest were a mash of partials of 1 4 and 7. Great qyestions but they required some intuition
@@niampatel9115 so ur doing maths at cambridge I'm assuming?
@@siamsama2581 nope - I never got an offer as I messed up my interview. However I knew how useful step was so sat it anyways. I’m now in my second year at St Andrews doing maths. I did get the grades needed for Cambridge so at least I showed what I could do
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.
Everything is connected.....Life teaches us that. You don't need to be a Uni grad to learn. Usually, Uni ruins life learning skills. Look at Henry Ford, who couldn't virtually read or write and know basic info yet changed the world with his V8 engine. Education is not the name of the game, persistence is.
History is more important than mathematics as recent events have proved
Watch Tom take more high school exams on the designated playlist here: ruclips.net/p/PLMCRxGutHqfm3t0IVJabEab6OasV9WLrl
If you are mathematician then why don't you crack the code for SEO and RUclips algorithm
Maybe he would crack the code if you are willing to pay for his effort
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!!
I never saw that emoji before....
Definitely worth the wait, looking forward to part two
I adored pure maths as an A level student. However I became a dentist.. and 30 years later can’t remember any of my beloved maths. But still love watching the mathamagicians. Lol.. my daughter is doing a msc in maths!!
Wow!! I didn't know a Math professor can be this cool!
Your command of pure mathematics is exemplary!!!!
It would nice to see you perform "applied" mathematics, in whatever applied discipline you choose (economics, engineering, etc.) with the aid of an "antiquated" computational device; specifically a "slide rule", with no time limit.
HINT: A solution is provided when two logathrimic scales intersect. All that is needed is using the appropriate logathrimic scales.
I’ll be sitting this exam in a few months 😓 This video came at a great time.
Hey, Best of luck!
Good Luck bro
Good luck!
Same
Good luck - you got this!!
I took this exact paper alone in a room for 3 hours, love watching you do them back
Wow, my Cambridge physics interview in 1992 was so much easier than this. No test like this! No wonder the maths students I knew all had a mad faraway look in their eyes :)
At my interview, I just talked (convincingly...? who knows) about kinetic energy and silly putty. I was 16 and just bloody loved physics. They took me on anyway, the mad fools.
Great stuff Tom. Utter kudos for showing your thoughts, doubts and mistakes on the way to an answer. Academia should be for people who care about all those things x
Perfect sir for online classes that anyone remotely can complete graduation
Hi! I’m a student from Singapore and it would be interesting to see you try an O level a- math paper(this paper is meant for 16 year olds) :D interesting video btw, quite enjoyed it!
Agree with this!!
I’m not sure,the trigonometry questions are of the same level as I encounter here in Singapore, but there are some things that I’ve simply not learnt yet, so I wouldn’t be a fair judge on whether it’s the equivalent. If you really do want to compare O levels with another exam, then your best guess would be the GCSEs in the UK I think. However, the GCSE paper to me looks easy, way easier than some of the other math questions I’ve come across here. Sure, maybe that’s because of my inclination towards math in general, but the questions there are actually not all that difficult to comprehend.
However GCE O levels from what I heard was the old system used in the UK, focused on a different aspect of learning and comprehension. Not able to remember specifics though.
I think this is required viewing for anyone wanting to sit any A level maths exams.
He shows his thinking process in solving the problems. Also notice that he may have forgotten a lot of his analysis/algebra but he remembered his baics and can therefore derive lots of things from first principle.
Notice how the habit of carefulness allows correct solutions to develop without mental strain.
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
Loving the "root 3/2, root 2/2 and root 1/2" reminder.
I used to study physics but had to drop out of uni. Watching videos like this makes me miss all the maths I had to do. Might try this paper myself.
Definitely worth the wait, looking forward to part two
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.
This was the paper I took, and I have to say was very happy with the questions - were definitely on the easier side of things. With Q1, I couldn’t quite get the last part but everything else was easy. The question on the Lambert-W function and the floor function simultaneous equations were very easy, as well as the rotational matrices. I didn’t attempt the differential equation question, but it seemed easy to do in my spare time. The geometric question with the road was horrible, I really don’t think many people would have got that in timed conditions
This is the type of teacher I wanna be. Looks not so professional but is the real deal.
He is the good ending of logan paul
@@ivoturi 😭😭😭😭
Sir please take jee advanced exam
Too easy for him not worth it
Go ask your school teacher. He's prof 😁
@@Danny-ql2it True, I think JEE Advanced questions are trivial. STEP actually require a brain. JEE Advanced is number crunching and working under time pressure
Shut up
@@veerpatel6719 that's what makes the exam tough lol. Every sum is solvable only the time constraint makes it look tough
I don't understand anything
But i love watching all the videos about math. Don't why?
A different approach for the problem 7 (iii)
We know from (ii) that S^2=S*S=-(I+S).
We also know that det(S)=1, which means that S is invertible.
Hence S*S*S(-1)=-(I+S)*S(-1), hence S= -[S^(-1)+I]
We notice that [S^(-1)+I]*[S+I]=2I+S+S^(-1)=2I-[S^(-1)+I]+S^(-1)=I
Hence -S is the inverse of (S+I), which is also a rotation: that implies that S is a rotation.
Since [S+I]^(-1)=[S+I]^(T)=S^T+I (because the diagonal doesn't chenge), we get that S=-S^T-I
So we have that S+S^T=-I
Hence b+c=0 and 2a=-1 and 2d=-1, it follows that a=d=-1/2 and c=-b
Since det(S)=1=1/4+b^2, we conclude that b=+-sqrt(3)/2
Dude, as a Doctor in Aerospace Propulsion myself, I cannot appreciate you much. You are aceing, I have now turned more towards applied mathematics due to my work requirements. We don't even do this kind of maths anymore in the Industry😂😂😂. It's a pleasure to work for Rolls-Royce Aerospace and defence in the UK.
1:23:00 A slicker way to prove a+d = -1 assuming
[1]: S ≠ I,
[2]: S² = (a+d)S - I,
[3]: S³ = I,
is to multiply [2] by S², then reduce the exponents of S via [3] to obtain
S = (a+d)I - S².
Solving this for S² yields
[4]: S² = (a+d)I - S.
The LHSs of both [2] and [4] being the same, equate their RHSs:
(a+d)S - I = (a+d)I - S.
Combining like terms and factoring,
(a+d+1)(S-I) = 0.
But S-I ≠ 0 via [1]. Therefore a+d = -1. []
Nice! I remember thinking this as I was editing the video
These are a good warm up for problem solving techniques. The whole thing should be answered so people get a good taste of the mathematical techniques, the disciplines and the strategies to tackle the problems. With a good training this are not hard but they DO requires exhaustive training and education on all of these fields.
The preparation should be in high school with doctoral professors and a good environment practicing the theory, problems and proof techniques. This is not hard if it is addressed in ORDER, with DISCIPLINE and a good school programme.
For me mathematics is my food. I have to dedicate at least 5 to six hours solving problems, proving theorems and most importantly what is GOING ON and WHY math works the way it does.
:D
I know, the STEP questions are surprisingly well-written. They are a very good starting point if someone is trying to get into competitive math, or pure math.
Also late comment but something that even the STEP mark scheme missed:
In the last part of question 1, as we have tan terms, there are values for x for which it is undefined. You first split the tan terms as sin/cos, so we have cos(nx) terms in the denominator for n = 1,2,3,4. we need to make sure these are never 0, else the expression is undefined and equality doesn’t hold.
you’ll find that the values x = pi/4 , pi/2 , 3pi/4 have cos(nx) = 0 for at least one of the n and so they must be discarded from the full solution set.
The final answer for the solution set is:
x ∈ {0, pi/18, 5pi/18, 7pi/18, 11pi/18, 13pi/18, 17pi/18, pi}
it’s really subtle and again the STEP mark scheme missed it, so they probably marked it as fully right if u included pi/4, pi/2 and 3pi/4 as solutions (but these are actually wrong, u can see by plugging into the original equation)
To do that matrix question equation you can just do det(S+xI)=1
That 1st question has a very easy and smart solution
The matrix question is an interesting one. It's solvable rapidly by elementary methods and it's very likely what STEP takers are supposed to use, but all the expansions makes that a rather crass solution. Part ii is general enough that it should be possible to prove this without resorting to writing down expansions. I spent a couple of hours on that and I got a solution, which is a bit less elementary.
For part ii, we know that det(S) = 1 from det(S^3) = 1 and S being real. Consider now the characteristic polynomial of S, which, because S is 2x2 is simply
x^2 - tr(S) x + det(S) = 0.
This becomes x^2 - tr(S) x + 1 = 0 because det(S) = 1. Hamilton-Cayley allows us to replace x by S (and 1 by I):
S^2 - tr(S) S+ I = 0.
therefore, S^2 + I = tr(S) S = (a + d) S as needed.
Then the interesting part is proving that (a + d) = -1. We know that S has two (possibly, and actually necessarily complex) eigenvalues by the fundamental theorem of algebra. Call them x1 and x2. We know that x1x2 = det(S) = 1. We also know (x1 + x2) = tr(S) = a + d. Since S^3 = I, x_1^3 = x_2^3 = 1. Therefore, x_1 = 1, e^(2πi/3) or e^(4πi/3). If x_1 = 1, then x_2 = 1 since x_1x_2 = 1. If x_1 = e^(2πi/3) then x_2 = e^(4πi/3) to get x_1x_2 = 1. Suppose x_1 = x_2 = 1. Then a + d = 2, and from the previous identity:
S^2 + I = 2S ---> (S - I)^2 = 0 --> S = I or -I. -I won't give S^3 = I, and I is forbidden. Thus a + d ≠ 2, leaving us with eigenvalues e^(2πi/3) and e^(4πi/3), so a + d = -1.
This looks really fun! I'd probably pick 2, 4, 5, 7, 8, 11 (and 1, 3, 6 look doable but tedious so they'd be good backups)
I think bprp would definitely enjoy 4 :D
Did this paper earlier today and 8 felt a lot more tedious, parts ii) onwards were fine but that initial differentiation was pretty painful with how much product rule and algebra you have to do, but I did this paper timed and this was the last question I attempted so maybe I was just tired haha
yeah generally you aren't gonna answer all 6 questions in the 3 hours, 4 full solutions guarantees you a high grade. I did the paper back in 2021 and q3 in particular was surprisingly elegant - lots of deductions needed and i found it really fun. 2 was more tedious but i like algebra so couldn;t complain.
@@niampatel9115 Which questions did you do overall, any applied?
@@squeezy8414 2,3,7,8,1,5. No applied. Generally the mech is awful unless it’s projectiles or collisions. And stats sometimes js nice but it’s not my strong point so I left it
Think I’m trying to say thank you for making maths cool..the kids nowadays need to realise maths is GREAT!
You have a really good intuition when solving problems :) 👍
For sure I would have "probably :)" skiped the probabilty question too 🙃
After a while thinking about it I figured out a kind of "brute force" method for finding a factorisation for 4(u^3)+2(u^2)-2u-1 but your "method of guessing" is way better!
It’s toooo easy for Korean high school students. I am a Korean who taught mathematics for korean high school students. Now, i am a mechanical engineer. As for me, it’s too simple problems for me. Anyway, thanks a lot for uploading this video……
Something I love about question 7 is you can explain part ii) by simply explaining what the determinant represents in terms of change in area after a transformation. Since applying S three times doesn't change the area of a shape, applying S once cannot do so either, therefore the determinant of S has to be one.
I remember when I was taking step and the matrix questions always scared me, and now at uni I actually quite like linear algebra (while still finding it just as difficult). For q7 ii) if we are given S^3- I = 0 we have that the minimal polynomial of S must divide x^3-1, which implies the minimal polynomial is x^2 + X +1 as S is not I and we're working over the reals. Since the minimal polynomial has degree 2, it must also be the characteristic polynomial and hence the trace is -1.
This is a lovely solution, I didn't think of that at all. I did do something similar --> S needs to satisfy its own characteristic equation (Hamilton-Cayley) so you get the S^2 equation that way. After that it's a matter of showing a + d = 2 or -1, which can be done by arguing about the possible eigenvalues of S.
1:21:29 at this point u can take the determinant on both sides and its done since det I = det S = 1
... You get a+b equals -1 or 2
But 2 is Not possible because this would imply I=S
In the first problem, didn't you miss x=0 as a solution, since sin(0) is also an x intercept, and 0 is within the valid range of x?
Yes, he did.
Have you ever had a look at the Indian test: Joint Entrance Examination (Advanced)?
On the website there is an archive with the tests from the past years.
Yes its one the easiest test out there
Dammit I was hoping it would be an older paper. I'm saving the newer ones until closer to the date so I'll come back once I've done the paper :)
Lovely guy with a great approach to getting the subject online. But please try to avoid the hands on head captions as you clearly dont find these tests too demanding.
Yes, he finally did it!!!!!
30 minutes for just question 1
BRUTAL!!
Thanks for the honesty of this lived work.
I guess a mistake at 39:12 :
3Pi/4 is not included in the solutions, since tan(2 * 3Pi/4) is not defined.
0 is included in the solutions.
There are indeed 8 solutions, but one less and one more ;-) .
I love this section.❤
Nice. He did exactly what i would have done. Nice work
16.20 instead of factorizing and loosing 10 hours couldn't you group 2u^2(2u+1) - (2u+1)=0 ???
Yes, good spot!
Best content while learning
Great video once again! :)
You don't have to remember the cos A+B identities, you just multiple 2 complex numbers together , (1, theta) , (1, phi)
On 1:24:26 i do not know if i did something wrong but applying determinant to both sides of the equation, I get, provided that det|S|=1 and det|I|=1, the equation comes to 1+(a+d) = (a+d)^2 -1 . Then, after substituting a+d=x, it lead to a second degree equation which is x^2-x-2=0 whose solutions are x=2 and x=-1. Thus, a+d=2 and a+d=-1. Have i done anything incorrectly? Is this wrong? How i discard the a+d=2 solution?
Thanks :)
The a+d=2 solution leads to S = I, whereas you know S isn't the identity matrix.
Interesting video, but I must confess I resolved Q1 in less than 10 mins. Wating now impatiently for Q2 to come up.
I wonder how long it would take a maths prodigy like Terrence Tao to ace this paper?
Bro you forgot x=0 as a solution for tan(x) question. Also, 3pi/4 is also not a solution as when substituted into tan(2x), it is undefined.
Actually, we don't have to go through a cubic formula to solve for x in the first question.
We had:
2cos2x ( cos4x + cos 2x)=1
If we expand it we get:
2cos2x . cos4x +2(cos2x)^2 = 1.
But we know from 12:06 that 2(cos2x)^2 - 1 = cos4x. So we simply move the 1 from the right of our equation to the left and this will get us:
2cos2x cos4x + cos4x = 0.
At this point, we factorise by cos4x and the rest is simple calculation!
Right
Nice!
I've stumbled upon this. How do I select the English translation? Mine seems to be stuck in a foreign language. ;-)
I'm in Algebra 1001 in college, and I have to say I don't understand anything that he is saying, but it is interesting nonetheless. More so that I don't understand what any of the signs mean.
Hello Tom Rocks Maths,
I'm currently taking my GCSEs in year 11 and I am just wondering: is this what is going top be taught to me if I choose maths and further maths at A level?
If so that seems daunting but equally inspiring to have to learn seemingly a lot in two years.
This makes me want to revise trig functions so I make sure I can be on track to do questions like these in the future!
Not Tom here but I'm currently giving my A2s and I just wanna say you'll do great- Revise trig problems and precalculus fundamentals, as is with all the subjects I know in A-levels, if your fundamentals are deeply understood, you'll have a much easier time digesting all the information in. I'm currently struggling with Chemistry since I didn't focus much on the fundamentals since my GCSEs and AS levels but relearning and relieving one small headache at a time~ Goodluck~
What I would say is I got a 9 and maths and a 9 in further maths and I deeply regret taking maths a level
@@freddxszn7845 is it very tough?
Again not Tom, but I did A Level maths and this is the equivalent of being thrown into the deep end of a pool with all your limbs cut off. You would be given very basic examples to start with and build up confidence as you went along (like most other things in maths). You eventually realise that most calculus is just very simple calculus wearing a funny hat. Complex problems break down into smaller, more manageable pieces. I recommend taking maths (I can't speak for further maths but I heard nothing but horrors from my peers). Maths is all about learning different rules and applying those rules (and their exceptions) to different examples. It's like chemistry with less hexagons!
@@The_Unholy_Nerd That sounds great! it seems to me like you are saying that there are few new concepts learned but all have incredible depth (I have heard about all of the different rules for differentiation and such which is an clearly great increase of depth of something i know basically.)
Thinking of this overall, it seems like further maths at A-Level will be like UKMT maths competitions where you have to use the rules that you have learned to the utmost. To what extent is my analogy true?
No one can beat jee advanced maths
Are these questions in increasing difficulty?
@kennethha8173
Not really, It depends on what you are better or worse at.
It usually starts with Algebra/Trig. For me I am much better at Geometry than Algebra. Geometry is usually at the end with Calculus if I remember correctly.
What's the math chair at Oxford called? And what's the origin of the word "Lucasian"?
Hello Dr Crawford. Thank you for the video. I was wondering if you are planning to do any more step papers. I thought your explanation of these questions were very insightful
Not for now - I'm trying to do as many different exams as I an so that I can compare their difficulty level.
@@TomRocksMaths What do you mean to convey with the word "professor" in the title? Is the university aware that you are marketing yourself in this way?
On Q1 i) why can you take y=2x and cos(y)? It seems to me you're 'inventing' extra solutions by taking twice the domain
Dr Crawford ,I think if you take an IQ test lots of people will be interested ,including me.
Sin 4x: x = 0 too.
You're smart Dr Tom Crawford
Amazing!
I have no idea what's going on but still watched 1 hour 36 minutes and 54 seconds of this video.
In the first question, how do you go from Cos(a-3a) = Cos(+2a)??? @5:20 time marker. I understood everything before and after, i just don't get how it's not Cos(-2a)?
The cosine of an angle and the cosine of the negative of that angle are the same. I think thats why.
35:26 forgot 0 as a suitable solution, as it is included in the inequality
I bet this video was 1. A reassurance for Oxford to extend his contract. And 2. A good enough reason for Oxford to give Tom a pay raise
I've just done this paper for revision, and I have a question about number 5. I did it using integration by part, which resulted in one of the terms being x/(x-a) instead of a/(x-a). This lead to different values for the constants and equations, but the overall graphs and stationary points seemed to be identical. I was wondering if this loses any marks or is still a valid solution, as it's not mentioned in the mark scheme? Thank you!
I found this result: -ln(x-a) -x/(x-a)
I was searching in the comments if anyone found something similar or different than a/(x-a)
Yes, this is valid: x/(x-a) = a/(x-a) + 1, so you've just added on an extra constant that makes no difference.
@@carminoski Yes, this is valid: x/(x-a) = a/(x-a) + 1, so you've just added on an extra constant that makes no difference.
this is so easy for me I'm in 12th grade in CBSE
Ok. Go focus on your Boards now
cos a cos 3 a= 1/2(cos 4a +cos 2a)
Dr. Tom, please try OMMS Oxford or Part III Cambridge Papers.
don't forget the '0'. 'x=0' should be the 8th solution.
the answer '3pi/4' should also be crossed out at 39:07 , because tan(2* 3pi/4) is also meaningless.
Good point, even the STEP mark schemes missed this
missed x=0 in the first question in sin4x=0 but I wouldnt even come close to that part of the solution so congrats
If I was an 'ethnic' what would I get in reduced % professor considering my disadvantage?
16:40 I dont get it... how did he factor that in his head 💀. I've always thought of using factor theorem but then getting 1/sqrt 2 by guessing was never going to be possible. Unless it's somehow linked to cos (45°) being 1/sqrt 2.
4u^3 + 2u^2 -2u - 1
= 4u^3 + 2u^2 - (2u+1)
= 2u^2(2u +1) -(2u+1)
= (2u+1)(2u^2 -1)
:) hopefully it makes sense
All those problems are merely about basic concepts of maths, without anything complex
I found the biggest stumbling block for me in mathematics was understanding the complexities of the 'Gusintu's theorem - as in I Gusintu 2 and 2 Gustintu 4!!!
I wonder if you could pass an taiwanese highschool test without preparation - not that it's difficult but it's just the amount that you need to have available at once (math, physics, chemistry, biology, english).
All the mathematical topics within the tests are basics in my work, but even there I wouldn't know some of the formulas without looking them up (re-memorizing them).
Those who properly pass those tests should have no problem with the Cambridge Entrance Exam I'd say.
Which app are you using to write notes on iPad?
It’s called “paper”