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Re: HSC 2013 4U Marathon $Find a recurrence formula for$ I(m,k,n) = \int_0^1 x^m (1-x^k)^n \ dx $Hence prove$ \left( \frac{k}{m+n+1} \right)^n < \frac{1}{n!} \int_0^1 x^m (1-x^k)^n \ dx < \left( \frac{k}{m+1} \right)^n

If there ever was a BOS trial for other subjects I doubt it would be as good as the Maths ones Mainly due to syllabus constraints I think

Don't motivate him to make it harder!

Just so nobody else asks, they aren't good marks (except 1)

What made you scrap this question? Was it too easy for an inequality? (and instead you want to crush us with an impossible one on Thursday?)

No space in your inbox And 4u as well if you can do that Carrot

messaged

Why not? The lines XT and XM and XN lie on the same plane or 'ground level'. If angle PXT is 90 then angle PXM is 90 and PXN is 90

For the second step, call the point X below P. (the one 450 m below it) Call the points where the tangents intersect the circle M and N. We know that MX = NX, because tangents from external point are equal. We know that angle PXM = angle PXN = 90 degrees, since X is directly beneath P...

Thanks for the link, I'll get onto once I put in the effort to dig up those old certificates

ohhh I see, yeah I didn't do it lol This is the second time I've confused ICAS with this UNSW competition, the first time resulted in me not knowing of the existence of the exam.

I did haha, I typically do not do well in ICAS, I don't know if results have been released bar the top list though. So I may have gotten a Participation for all I know.

This is very humbling but please note that I have not succeeded in maths outside of my high school :) One example of this fact is that I have not made that UNSW competition list that Realise made it into.

Re: MX2 Integration Marathon $i)$ \ \ I_n = \int_{-1}^0 \frac{x^n}{\sqrt{x+1}} \ dx $ii) Find in the form of an integral$ \sum_{n=0}^{\infty} \frac{(-1)^n}{ \binom{2n}{n}} $iii) Evaluate this integral using Wolfram Alpha (or suffer evaluating it), note that$ \tanh^{-1} x = \frac{1}{2}...

Re: MX2 Integration Marathon \int_{a}^b (x-a)^m (x-b)^n \ dx

Sy123: Extension 2 Only

Re: HSC 2013 4U Marathon Very true, my bad.

Re: HSC 2013 4U Marathon $You are given that for a certain radius of convergence, a function$ \ \ f(x) \ \ $which is infinitely differentiable at$ \ \ x=0 f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)} (0)}{n!}x^n $i) Show that$ \sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} +...

Re: MX2 Integration Marathon You may have made a silly mistake, because: I_n - I_{n-2} = \int_0^{\pi} \frac{\sin(nx) - \sin(n-2)x}{\sin x} \ dx = \int_0^{\pi} \frac{2\sin \left(\frac{nx - (n-2)x}{2} \right ) \cos \left(\frac{nx + (n-2)x}{2} \right)}{\sin x} \ dx I_n -I_{n-2} =...