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You are certainly correct, although morally this is really equivalent to IBP since that result come from the product rule and integration by parts is simply a formalisation of the 'reverse product rule'. Sometimes showing more steps can be a good idea as it means we are more certain to receive method marks and, even if the shortcut is mathematically correct, it can also reduce the risk of unfriendly marking errors! - Rowan

Thank you for your very kind comment -- clarity is certainly the goal! - Rowan

yeah but for university admissions, unlikely to expect students to know about this stuff.

Yes! You can read about our MAT course here www.vantageadmissions.co.uk/course/mathematics-admissions-test . Our STEP course is slightly changing for 2025 (lessons will be pre-recorded rather than live to afford better flexibility, with the weekly live lessons then replaced with office hours) but the basic format and price should be the same when our STEP 2025 programme launches in early December www.vantageadmissions.co.uk/course/sixth-term-examination-paper . I am happy to discuss further in a free consultation which can be booked on our website. - Rowan

Yes - stay tuned!

Might be worth it to know the FTC proof as well (as long as you can explain pretty well where FTC comes from).

@@-taehyun Whats the FTC proof please teach me bro

@@宻 Fundamental theorem of Calculus i.e integral of f(x) from a to b is F(b) - F(a) where F'(x) = f(x). if you look at f(a+b-x) and use a substitution u = a+b-x you will find that by swapping limits and evaluating by our previous rule (FTC) that you get F(b)-F(a), the same result as evaluating the integral of f(x). Bit hard to explain through a youtube comment i'll latex up a version and send it through later.

How do you know someone is preparing for IIT? They'll tell you

Certainly stay tuned for more! - Rowan

You are of course completely correct that k needn't be an integer. I suspect this restriction was given so that trying k=1 first would be an accessible option; once seeing that this game worked for that case, students would then hopefully realise that the fact the power was 1 was not actually important and the argument worked more generally. - Rowan

Yes, you're absolutely right -- any tricks which are ultimately based on graphical symmetries (for instance, the trigonometric identities we noted are equivalent to the fact that sine and cosine are each other's reflection in x = pi/4) are very important for these interviews and are well-worth getting used to! -Rowan

how do i learn those?

and are there any other niche rules like kings,queens, jacks rule i should know for interview preparation in general? many thaknks

@ kings rule, jacks rule, odd and even functions, principle of inclusion exclusion (then the derangement formula which can be derived from this) if i think of any more i will drop them in the comment. you can google to find out more about these

@@宻 As well as these sorts of tricks with names, as far as integration is concerned it's really important when preparing for maths interviews to build a rock-solid grasp on the more standard techniques. For example: think carefully about how you might spot a good substitution beyond simply substituting the 'complicated' bit of the function (examples: trigonometric substitutions motivated by trig identities, u =f(x) substitutions which proceed more smoothly since f'(x) is a factor in the integrand and then the f'(x)dx nicely becomes du); think about how you might spot a good application of integration by parts (LIATE is a useful acronym here, and you should also think about tricks like writing f(x) as 1 * f(x) to allow IBP, and 'partitioning powers' e.g. in x^2 e^x^2 letting one of the x's from x^2 go with e^{x^2} as dv/dx so that you can integrate it). Also, the trick used in this question where we essentially proved that the integral solved an equation, I = pi/2 - I, rather than 'solving' in the traditional sense, is used very broadly beyond just 'king's rule'. - Rowan

When we talk about ‘expected’ in the sense of a random variable, we don’t mean in a deterministic sense of anticipating a precise value, but rather the definition as a sum given in the video. One way to think about it intuitively is the mean average result you would get if you repeated the experiment a very large number of times. There are indeed cases where this notion does not align well with intuition - e.g. some random variables may have an infinite expectation. See, for instance, the St. Petersburg Paradox. Luckily, in this case, the notion of expectation actually works very well!

Congratulations! Rest assured the S,1 offer will not have been because of any lack of interview performance on your part. Rather, several colleges seem to have decided to make S,1 offers across the board recently and especially this year. For instance, we had multiple students receive offers at Pembroke, all S,1. Traditionally S,1 offers were mostly made to reapplicants and almost always such students have still been admitted if receiving a 1,1. So, certainly aim for the S, but in all likelihood you will be accepted with 1s too. It’s not too late to join our STEP 2024 programme, so don’t hesitate to get in touch if you would like to hear more! - Rowan

I don’t think Oxbridge has better ‘teaching’ but rather it kind of teaches/focuses on different things. The entrance exams are there to try to ensure that will work for the students given the time constraints

an oxbridge degree would be worthless if they lowered the standards to the point where the average joe could ace it. the courses are intense, and competitive, and as such they have to have a rigorous admissions system. if you don't like that then just don't apply.

Well, concepts cannot be explained if someone doesn't have the pre requisite knowledge about the concept. For example, you won't be able to explain what prime numbers are to someone who doesn't know maths. The high expectations are so that they get students who has already a lot of background knowledge and can therefore connect with the lecturer much more easily than someone getting a B or C for example. Uni content at those universities is very intense and if the lecturer has to explain everything, then it would take ages to finish the degree. Getting someone who is appropriate for a task isn't unfair, given the task could have done by any human being.

Thank you very much for spotting this - you are absolutely right that two of the three triangles have that centre angle mislabelled. This will be corrected in an annotation.