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Re: HSC 2016 4U Marathon Yeah but most 4U students wouldn't understand the meaning of this of course. (Nor are they expected to.)

Yeah

You'd need to do REALLY well in 2U to beat a sufficiently good 4U mark (and once you reach a certain stage X, a 4U mark exceeding X is generally better for your ATAR than any 2U mark. Not sure exactly what X is, check out some ATAR calculators). Also, if you do 4U, then 3U will count for two...

$\noindent 1) We have $f(x) = x^{3}-3x$. It follows from the \textsl{inverse function theorem} that$ $$\begin{align*}\left(f^{-1}\right)^{\prime}\left(-\frac{11}{8}\right) &= \frac{1}{f^{\prime}\left(\frac{1}{2}\right)}\\ &= \frac{1}{3\left(\frac{1}{2}\right)^{2}-3}\\ &= \frac{8}{3 - 24}\\ &= -...

How did people find it this year (Ext. 2 paper)?

No, that's very good. :p

Basically, if you think of these things in terms of the rate parameter and the MTBF (mean time between failures), you'll be less likely to make a mistake. The conventions are really just notational things, but the rate parameter and the MTBF will be intrinsic to the system in question. So if...

Just depending on the situation, one may be more natural than the other. $\noindent Convention 1: $T \sim \mathrm{Exp}(\lambda)$ means pdf is $f_{T} (t) = \lambda e^{-\lambda t}$ (for $t > 0$). In this convention, $\lambda$ is the ``rate'' of occurrence of the underlying Poisson process (e.g...

The expected time to the next claim is 10 years (inverse of the rate parameter in the Poisson distribution). Depending on which convention of the Exponential you're using, the time to next claim is either Exp(10) or Exp(0.1). I think you may have been confused between the two conventions...

$\noindent To show the $\sin \left(\tan^{-1} \xi \right) = \frac{\xi}{\sqrt{1 + \xi^{2}}}$ result, one way you can do it is by drawing up a right-angled triangle and using Pythagoras' theorem and right-angled trigonometry. (This proves it for positive $\xi$ at least. The result for negative...

$\noindent We need to assume the sine part is positive for the equation to make sense, so we'll show that $|z+1| = 2 |\sin (\mathrm{Arg}(z-1))|$.$ $\noindent Let $z$ be a complex number on the unit circle with $z\neq 1$ (so that $\mathrm{Arg}(z-1)$ makes sense). So $z = \cos \theta + i \sin...

Basically with HSC Physics projectiles, they generally expect you to assume there's no air resistance unless explicitly stated otherwise.

Which first part? He probably provided the differential equation to give a taste of Picard iteration. (But yeah, don't need it for the actual question.)

Re: Year 12 Mathematics 3 Unit Cambridge Question & Answer Thread $\noindent Recall that the binomial coefficients are symmetric about the central binomial coefficient $\binom{2n}{n}$. Hence if we take the sum of coefficients from $r=0$ to $n-1$, it's half what we'd get if we took the sum of...

Note that you could always show this by direct computation, since it suffices to just check five cases.

You don't need to "test" anything here, you can just say that since h is a quadratic function of t with negative leading coefficient, the stationary point is a maximum. Anyway, for this question, it sounds like they're letting you assume it's a maximum and are just asking for when this occurs.

Re: HSC 2016 4U Marathon $\noindent Let $P(z) = az^{2} + bz + 1$ ($a\neq 0$) be a quadratic with real coefficients.$ $\noindent a) Find necessary and sufficient conditions on the coefficients $a$ and $b$ in order for $P(z)$ to have a root on the unit circle (i.e. a root $\alpha \in \mathbb{C}$...

0.0006 is one significant figure. For 150, there's no confusion about whether it's one or two, the only confusion is whether it's two or three. I think you should consider it as two significant figures, but as Wikipedia says, "In a number without a decimal point, trailing zeros may or may not be...

No.

Re: HSC 2016 3U Marathon Yeah, theta is maximised when P = T.