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Re: MATH1231/1241/1251 SOS Thread $\noindent In general, any $n\times n$ real matrix with $n$ distinct eigenvalues is diagonalisable (which is equivalent to saying there is a basis for $\mathbb{R}^n$ consisting entirely of eigenvectors of $A$ (such a basis is called an \textsl{eigenbasis}))...

Re: HSC 4U Integration Marathon 2017 Find \int \frac { x^{ \frac { 2 }{ 3 } } }{ \left( 1+x^{ \frac { 3 }{ 5 } } \right) ^{ 2 } }\, \mathrm{d}x. (Courtesy of leehuan here: http://community.boredofstudies.org/14/mathematics-extension-2/344856/mx2-2016-integration-marathon-41.html#post7183192 )

Re: HSC 4U Integration Marathon 2017 The LHS integrates the PDF of a Gamma(n,beta) r.v. (where beta = 1/lambda) from 0 to t and the RHS is the probability a Poisson(lambda*t) r.v. is bigger than or equal to n. The identity thus shows that in a Poisson process, the number of events occurring in...

Re: HSC 4U Integration Marathon 2017 $\noindent Let $\lambda , t > 0$. Show that $\int_{0}^{t} \frac{\lambda^{n}}{(n-1)!}\tau^{n-1} e^{-\lambda \tau}\, \mathrm{d}\tau = 1 - \sum_{k=0}^{n-1} e^{-\lambda t}\frac{\left(\lambda t\right)^{k}}{k!}$, for $n=1,2,3,\ldots$.$

Re: HSC 4U Integration Marathon 2017 $\noindent Find $\int _{0}^{\frac{\pi}{2}} \frac{\cos x}{\cos x +\sin x}\, \mathrm{d}x$.$

Re: HSC 4U Integration Marathon 2017 Why that reaction haha? Is there some story behind this integral?

That part was just a general statement.

Ah right, I don't recall the specifics of the question, I just skimmed through the paper and remember seeing something that looked like Mean Value Theorem in it.

Don't really need to measure any slopes. The reason I thought it was (A) was that f''(1) looked like 0 (inflexion), so f''(1) had to be smallest. (This narrows it down to between (A) and (B).) Assuming this then, it is also clear that f"(x) >= 0 for 0 < x < 1. So the slope at the right endpoint...

Y uses the -GMm/r formula for E_p, which is a result of assuming gravity is varying (namely GM/r^2 at radial distance r). $\noindent That is, $-\frac{GMm}{r} = -\int_{\infty}^{r} -m\frac{GM}{\rho^2}\, \mathrm{d}\rho$.$ (To recall the mathematical definition of potential energy and how it...

The model X is assuming constant gravity, whilst Y takes into account changing gravity.

tywebb posted a copy: http://community.boredofstudies.org/1188/mathematics-extension-1/356063/ext-1-paper.html .

We should always check our answers to see if they are reasonable. Note that (a – a^2)/4 is going to be negative a lot of the time (e.g. if a = 2). But "a" was certainly allowed to be any positive number, so we'd be getting a negative focal length, a problem. (And it's pretty clear that the...

It can make 3U become a (at least relative) piece of cake.

tywebb posted it here: http://community.boredofstudies.org/1188/mathematics-extension-1/356063/ext-1-paper.html#post7208365 .

Do you think they intentionally made a bomb Q. because of circumstances at the time? I think it was just a coincidence.

Would be cool if they did a projectile Q involving bouncing down a staircase or something (could easily throw in an induction with that too) :). (Maybe something like this would only be considered in BOS Trials though, not HSC.) And haha, never consciously noticed that they did a bomb question...

Ah OK (thought your comment was meant as a reference to the projectiles Q.).

Not raw-wise.

That projectile question could've easily been made more tedious by making the projectile hit the wall at a point in its trajectory that was not the max. height, so that we'd have needed to calculate the angle it hits the wall at, and use this (and the law of reflection) to obtain the initial...