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$\noindent (The question should say it touches the coordinate \textit{axes} (plural), i.e. it is tangent to both the $x$-axis and the $y$-axis.)$ $\noindent We will generalise it to an ellipse $\mathcal{E}$ with semi-major axis $a>0$ and semi-minor axis $b$, where $0<b\leq a$ (for this...

Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread $\noindent For the first 10 seconds, the car is accelerating at a constant rate of $2\text{ m}/\text{s}^2$ (so it is constantly picking up speed during this time). Since $\Delta v = a \Delta t$, where $a$ is the constant...

Re: MX2 2016 Integration Marathon Yeah.

Re: MX2 2016 Integration Marathon $\noindent We can use a Drsoccerball-like substitution of $x =\tan \frac{\theta}{2}, \theta \in (-\pi, \pi)$ $\Big{(}$so the new limits of integration become from $0$ to $\frac{2\pi}{3} \Big{)}$, it's just that we need to make sure we use the fact that...

It is actually possible to rote learn some things in HSC maths.

Re: HSC Chemistry Marathon 2016 $\noindent In general, if given a half-life, it's best to use exponentials with base 2 rather than base e. (This is the whole point of using a \textit{half-life} essentially.) If a substance undergoes radioactive decay with half-life of $\tau$ (in some units of...

Re: HSC Chemistry Marathon 2016 $\noindent Yes there's a much faster way $\ddot{\smile}$. Note that the half-life is 8 days, so if the original amount is $M_0$, then after 8 days, the amount remaining is $\frac{1}{2} M_0$. After 16 days, the amount remaining is $\frac{1}{2^2} M_0$...

$\noindent Basically, the locus of $z$ for $\arg (z-z_0) - \arg (z -z_1) = \alpha$ (where $\alpha$ is fixed with $0\leq \alpha \leq \pi$ and $z_0, z_1$ are fixed complex numbers) is one you should be familiar with. In fact, the shapes are different depending on whether $\alpha = 0$, $\alpha...

$\noindent This says that $\arg \left(\frac{z-2}{z+2}\right) =\frac{\pi}{3}$. So $\arg \left(z-2\right) -\arg \left(z+2\right) =\frac{\pi}{3}$. So the locus is the major arc of a circle starting at the point $z=2$ in the complex plane and going counter-clockwise to the point $z=-2$, with the...

Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread Yes, that's a crucial relationship to know. For part (i), use the factorial definition of the binomial coefficients and take common denominators and simplify etc. (you'll need to use that relation).

$\noindent $\Big{(}$If in the original question $n=-1$, then the integral becomes the integral of $\tan x$ from $0$ to $\frac{\pi}{2}$, which does not converge.$\Big{)}$$

$\noindent We have$ $$\begin{align*}\int _0 ^1 u^n \text{ d}u &= \left[\frac{u^{n+1}}{n+1} \right]_0 ^1 \quad (\text{using the power rule for integration, assuming }n\neq -1) \\ &= \frac{1^{n+1}}{n+1} - \frac{0^{n+1}}{n+1} \\ &= \frac{1}{n+1} - 0 \\ &= \frac{1}{n+1}. \end{align*}$$

Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread $\noindent Here's how to do the induction question.$ $\noindent \textbf{Proposition.} For any integer $n\geq 3$, we have $\sum _{j=3}^{n}\binom{j-1}{2} = \binom{n}{3}$.$ $\noindent \textbf{Proof by induction.}$...

$\noindent Let $u=\cos x$, $\mathrm{d}u=-\sin x \text{ d}x$. When $x=0,u=1$ and when $x=\frac{\pi}{2},u=0$, so we have$ $$\begin{align*}\int _0 ^{\frac{\pi}{2}} \sin x \cos ^n x \text{ d}x &= \int _1 ^ 0 - u^n \text{ d}u \\ &= \int_0 ^1 u^n \text{ d}u \\ &= \frac{1}{n+1}.\end{align*}$$

Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread $\noindent The relationship is $(r+1)! =(r+1)r!$ for any integer $r \geq 0$.$

Re: HSC 2016 2U Marathon $\noindent We can't do the working out quite how you did it unfortunately, because there's a variable in both the base and the exponent. When we let $u$ equal something, it should get rid of all $x$'s from the expression. Like how if we had $y=\ln (\cos x)$, to find...

Re: HSC 2016 2U Marathon $\noindent Nested powers are done from the top down. So something like $a^{{b}^{c}}$ means $a^{\left(b^{c}\right)}$. So this is distinct from $a^{bc}$; instead, $a^{bc}$ refers to $\left(a^{b}\right)^{c}$, which is different.$

You pretty much need to know about subshells to explain the pattern, and I don't think subshells are in the Prelim (or even HSC) Chemistry syllabus.

Re: HSC 2016 2U Marathon Regarding how the marathon is meant to work, I think we're meant to wait for the last unanswered Q., to be answered, then the person who answered that posts a new Q. Here's a new question if you don't want to do the ones about averages: $\noindent Let $n$ be a positive...

Re: HSC 2016 2U Marathon $\noindent In fact, $\frac{\mathrm{d}}{\mathrm{d}x} \left(\ln (-x)\right) =\frac{1}{x}$. More generally, if $k$ is a non-zero constant, then $\frac{\mathrm{d}}{\mathrm{d}x}\left( \ln (kx)\right) =\frac{1}{x}$.$ (Oh, by the way, the function is actually ln (lowercase...