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Re: MX2 2015 Integration Marathon \int_0^{\pi/2}\log(\sin 2x)\, dx\\ \\ = \frac{1}{2}\int_0^{\pi}\log(\sin x)\, dx \\ \\ = \frac{1}{2}\int_0^{\pi/2}\log(\sin x)\,dx+\frac{1}{2}\int_{\pi/2}^{\pi}\log(\sin x)\, dx\\ \\ = \frac{1}{2}(I+\int_0^{\pi/2}\log(\sin(\pi-x))\, dx)\\ \\...

Re: MX2 2015 Integration Marathon You are mostly right, not the last line though. Will fix up my eqn which isn't displaying for some reason.

Re: MX2 2015 Integration Marathon I=\int_0^{\pi/2}\log(\sin x)\, dx\\ \\ = \int_0^{\pi/2}\log(\cos x)\, dx\\ \\\Rightarrow 2I=\int_0^{\pi/2}\log(\sin x\cos x)\, dx\\ \\ = \int_0^{\pi/2}\log(\sin 2x)\, dx-\frac{\pi}{2}\log(2)\\ \\ = I-\frac{\pi}{2}\log(2)\\ \\ \Rightarrow I=-\frac{\pi}{2}\log(2).

why has this thread dragged out so long lol.

Re: 2015 permutation X2 marathon Yep: 1/3 and 13/27 are the correct numbers. The role of the friday information I find the most interesting, as if you replace it by a random property with probability q, then the answer is near 1/3 for large q and near 1/2 for small q (like the q=1/7 in this...

Re: 2015 permutation X2 marathon The first part is not 1/2, although as braintic said, it would be if the statement was something like "my eldest child is a girl born on a Friday". Part of the difficulty in this classic question is posed by potentially ambiguous wording. Consider the...

Re: 2015 permutation X2 marathon A woman has two children, one of whom is a girl born on a Friday. What is the probability that her other child is also female? Bonus: Consider this same problem with "born on a Friday" replaced by another random condition that holds with probability q. Explain...

An oldish video, but for anyone who wants to procrastinate / learn more about the melee scene: https://www.youtube.com/playlist?list=PLoUHkRwnRH-IXbZfwlgiEN8eXmoj6DtKM.

Re: HSC 2015 4U Marathon For this to make sense you would need to: a) Define what you mean by "log(z)", and "e^z". The former is subtle as it needs what is called a branch cut. b) Properly define differentiation and integration of functions from the real line to the complex plane and from the...

Re: HSC 2015 4U Marathon You would have to include a definition of the exponential function applied to complex numbers or what it means to raise things to complex powers, otherwise the LHS is a meaningless string of characters. Eg/ Try a logically similar question. Prove that Sean(x)=cot(7x).

Well my point is also that pedantry is not a linear scale. Some markers dislike geometric methods in questions that aren't specifically about geometry for example, some are especially anal about the formatting of Euclidean geometry proofs, some are more tolerant than others about ambiguous...

"Correct" is a slightly subjective term when it comes to working out. There are often multiple ways of doing things, as well as the student's choice of how much detail to explain his reasoning in. Different schools / even different staff members at the same school will generally have different...

Haven't thought about this sort of stuff (and in particular this problem) for ages. I suspect it would be quite hard if not impossible to get a closed form for the pdf. Otherwise summing a series involving the pdf would be the natural way to find the mean. Given that the exploitation of...

It's very similar to the logical processes in choosing substitutions for indefinite integration. After all, integrating f is nothing more than solving the differential equation: u' = f for u. You should first learn learn to recognise basic forms, and ways of classifying these things...

Re: MX2 2015 Integration Marathon Good stuff glittergal, dyadic decomposition is exactly what I wanted someone to come up with for estimating that integral. (And your exponent is the correct one). :).

The fundamental theorem of algebra asserts that any non-constant polynomial with complex coefficients has a complex root. It is notorious as being quite hard to prove for what it is, typically requiring some machinery at least as advanced as complex analysis / vector calculus / algebraic...

Whilst regulation is ideal on the tax and working with children side of things, I don't see how it is a good thing to tie the hands of private tutors in terms of what they can teach. A major reason why anyone considers spending money on tutoring worthwhile is that you can choose a more...

Think about the material critically when first learning theory, ask yourself lots of dumb questions and try your best to resolve them. Be patient and don't flip to answers when you are semi-sure of something, reach the level of confidence where you are almost certain of your answer's...

(If rigorous courses in analysis are actually a requirement that cannot be avoided, this surprises me a little. Very few teachers I have met seem to have retained this knowledge if they ever learned it!)

Unfortunately, I think you can get away with doing very little actual mathematics beyond high school level, hence the masses of teachers of subpar quality. (I once sat in a course with some people training to become mathematics teachers at ACU, and the quality of the students seemed fairly...