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$\noindent It is explained here with an example:$ https://en.wikipedia.org/wiki/Gay%E2%80%93Lussac_law#Law_of_combining_volumes

Re: HSC 2016 3U Marathon $\noindent \textbf{NEW QUESTION}$ $\noindent Let $n$ and $k$ be positive integers with $k\leqslant n$. Suppose there are a total of $n$ marbles in an urn and exactly one of these is red, and the others are all green. Alice randomly selects $k$ marbles from the urn...

In general there's no way to work these out purely by logic. Scientists observe what happens and record these observations. So you basically need to memorise the chemical reactions, as they are just the way nature is rather than something that can be deduced purely by logic (generally speaking).

Aside: for Question 2 of the 2015 4U BOS Trial, it should technically have specified that we are talking about the conic section being centred at the origin (or same point on the x-axis), since otherwise we can have hyperbolae that are "equi-eccentric" (same eccentricity) but possessing...

Correct. Some / lots of the Q's in the Advanced Level Marathon are achievable using HSC methods, it's just that in the HSC, they'd guide you through the steps by making them parts of the question, whereas in the Marathons, there's usually little guidance, so it's not obvious what to do. Best off...

No, the HSC papers are much easier. You don't need to worry about that thread much to do well in the HSC. Read this post: (Link to that post: http://community.boredofstudies.org/14/mathematics-extension-2/331666/hsc-2015-4u-marathon-advanced-level-21.html#post7090304 )

Re: Extracurricular Integration Marathon $\noindent For the missing sign, your first line would have the R.H.S. have a $2\ln (\cos x)$ in the integrand, but the original integral had $\sec$, not $\cos$, so you need to put in a minus sign in your first line to make it equal a $\ln (\sec x)$, as...

What type of game would the games be? 10-minutes? Bullet?

You might want to check out this post by Fizzy_Cyst: http://community.boredofstudies.org/263/space/335513/features-aether-model-hsc-q.html#post6925182

Due to the nature of school maths education, the general populace seem to have no clue what maths is. They probably think it's just about doing multiplication and addition, so no wonder they see no beauty in it. For example, in that Project video with the Australian student who helped come up...

Well a lot of high school maths is very tedious or repetitive (e.g. some people hate exercises on using quadratic formula or factorising things etc.), so most people don't see the beauty in maths, due to its nature in high school maths class. There is a good article about this here...

Re: HSC 2016 4U Marathon Maybe he just wants help with it. If that's the case, it might be better to create a thread for it.

Re: HSC 2016 3U Marathon There's a way to do it via trig. identities, wu345 posted a picture of his solution in a previous post.

Re: HSC 2016 4U Marathon Well technically speaking, it's not just the fact that the product is a real number that is significant, but rather that this product is equal to the modulus-squared (of the ω), but correct method, well done :).

$\noindent For the curve in your post, let us consider the curve $x^x y^y = c$, where $c>0$ is a constant and $x,y>0$. First of all, note that the minimum value of the function $\phi (t):= t\ln t$ is $\phi _{\text{min.}}=-\mathrm{e}^{-1}$ (occurring at and only at $t=\mathrm{e}^{-1}$). This is...

Re: HSC 2016 4U Marathon Should there be i's in the exponents (i.e. complex exponentials)? Otherwise there's not much point thinking of them as vectors, is there? (Haven't thought much about the Q.)

$\noindent Haha yeah :-). But you wrote $\frac{5\pi}{4}$ instead of $\frac{9\pi}{4}$. :$\mathcal{P}$$

$\noindent One very simple way is to first simply sketch the shape of the cosine graph like usual, without labelling the $x$-axis values yet (just draw in the first cycle of it). Then once we've drawn the shape in, we focus on getting the $x$-axis right.$ $\noindent Since the period is $3\pi$...

Re: MX2 2016 Integration Marathon $\noindent We have $ $$\begin{align*}\mathcal{I}:= \int \frac{\cos x}{2-\cos x }\text{ d}x &= -\int \frac{\cos x}{\cos x -2}\text{ d}x \\ &= -\int \frac{\cos x-2+2}{\cos x -2}\text{ d}x \\&= -\int \left(1+\frac{2}{\cos x -2} \right)\text{ d}x \\ &= -x -...

Yeah, only the first person to click the link and join can play. After that, people who click the link become spectators.