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Re: HSC 2016 4U Marathon First of all, "factors" are by definition integers (which implies they are rational). To prove a positive integer N ≥ 2 is composite, we are trying to show that it can be written as N = ab, where a and b are both integers with 1 < a, b < N (i.e. both strictly between...

General Maths (the maths subject the OP did) is quite useless for maths-based courses though.

$Yes, you can use the cosine rule if you want. Also, for the solution's method, if $a=\mathrm{cis} \left( \frac{\pi}{3}\right)$, note that then $a^3 = \mathrm{cis} \left(\pi\right)= -1$ and $a\neq -1$, so $a^3 + 1 = 0 \Rightarrow (a+1)(a^2 - a + 1) = 0 \Rightarrow a^2 - a + 1 = 0$, as $a\neq...

Relatively irrelevant, I would guess.

Certain differential equations have standard methods for solving them (e.g. linear first order differential equations, second order differential equations with constant coefficients, exact differential equations, etc.). Other ones have no known methods of analytic solution or can't be solved...

Re: HSC 2016 4U Marathon - Advanced Level I think it essentially can have any speed and the answer is not impacted, because the path traced out is the same (I think we are assuming a perfect mathematical universe with no friction or loss of energy in any way, and perfect reflection off the...

$No, not necessarily. That link (\url{https://studentsonline.bostes.nsw.edu.au/go/exams/illness\_misadventure/}) says:$ $\textit{``If you cannot attend an exam because of illness or misadventure, notify your school principal immediately. You should never risk harm in order to attend an exam, or...

Re: HSC 2016 4U Marathon - Advanced Level $I think he's pretty much saying that if $p+\sqrt{q}$ is a root of the polynomial he gave, so is $p-\sqrt{q}$.$

$Note that we only care about the cases where $0\leq \theta \leq 2\pi$, so we could state at the start that this is the domain of our function $A(\theta)$, and so we don't need to worry about values of $\theta$ outside this domain. We just want to maximise $A(\theta)$ on this domain.$

$By the way, note that $\frac{\mathrm{d}A}{\mathrm{d}\theta}$ is not 0 when $\theta = -2$; rather, it is undefined there, as the denominator would be 0. It also approaches $\infty$ in magnitude as $\theta \to -2$.$

$What do you mean ``common factors''? Factors are defined for integers only. It does not seem clear at all to me that $\frac{\pi}{\mathrm{e}}$ is irrational. In fact, it is an open problem according to the link posted above by Paradoxica.$

Re: HSC 2016 4U Marathon Just because two functions satisfy the same second-order ODE, it doesn't mean they are identical.

Just use Tsiolkovsky.

If you're ranked first, it's irrelevant for the moderation of your internal marks how the other people in your cohort perform. You'll just get the cohort's best external mark as your internal (i.e. assessment) mark.

Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread $Let the desired points be $X=\left(10t,5t^2\right)$ (we need to find the values of $t$, and you should end up with two values for it). We know that the slope of the tangent at $X$ is $t$ (easy to prove; you should know how to...

Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread $So firstly, you left out a $p$ in your last factor \Big{(}should have been a $\left( 2p + p^3\right)$\Big{)}. And secondly, the formula to use should have been $\mathcal{A} = \frac{1}{2}\left| 2a + ap^2 \right| \left|2ap + ap^3...

Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread $The area $\mathcal{A}$ is given by$ $$\begin{align*}\mathcal{A} &= \left |\frac{1}{2}\times \left(2a + ap^2 \right)\times \left( 2ap + ap^3 \right) \right| \\ &=\left| \frac{1}{2}a^2 \left(2+p^2...

Re: HSC 2016 2U Marathon "Domain" refers to the set of all input values where the function is defined. For the absolute value function you posted, the maximal domain the set of all real numbers (\mathbb{R}).

Most of the important mathematical areas for computer science are from discrete mathematics (if you know what that is). Here's a textbook / notes written by two Princeton academics called Mathematics for Computer Science: http://www.cs.princeton.edu/courses/archive/spr10/cos433/mathcs.pdf If...

$Let the father's age today be $f$ years and the son's age today be $s$ years.$ $We are given that:$ $$f=s^2 $ \ \ (1)$ $$f + 20 = 3(s+20)$ \ \ (2)$ $Substitute the expression for $f$ from (1) into (2):$ $$\begin{align*} s^2 + 20 &= 3(s+20) \\ \Rightarrow s^2 + 20 &= 3s + 60\\...