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for very weak concentrations (~10-6 M) the baseline concentration of H3O+ ions in pure water becomes significant, and must be taken into account use this formula, [H3O+] = Ca + Kw/[H3O+] where Ca is the concentration of the acid, and Kw is the dissociation constant of water...

Re: 2012 HSC MX2 Marathon texshop (i.e. latex) with pstricks (and pst-plot)

Re: 2012 HSC MX2 Marathon lol :D question: graph x^3 = 1 + xy^2 (perhaps a tad more accessible than that ^)

an increase in pressure will shift the equilibrium: CO2(g) ⇌ CO2(aq) to the right to reduce the gas pressure this will push the equilibrium: CO2(aq) + H2O(l) ⇌ H2CO3(aq) to the right (i'm not entirely sure if this will drive the dissolution reaction to the right to minimise the change in...

Re: 2012 HSC MX2 Marathon \\\textup{Let } z = r\, \mathrm{cis}(\theta).\\\textup{Show that } |z+1|\times|z-1| = 1 \textup{ implies } r^2 = 2\cos(2\theta).

Re: 2012 HSC MX2 Marathon centre: (1,√3) radius: 2√3. \\\textup{Let }z_{1}=3\left (\cos\left (\frac{\pi}{3} \right )+i\sin\left (\frac{\pi}{3} \right )\right ) \textup{ and } z_{2}=4\left (\cos\left (\frac{\pi}{6} \right )+i\sin\left (\frac{\pi}{6} \right )\right ).\\\\\textup{Determine the...

Re: 2012 HSC MX2 Marathon 1984 I think – first year with standard integrals

Re: 2012 HSC MX2 Marathon not long this will help with part i.

Re: 2012 HSC MX2 Marathon that makes it easier! (a) \begin{align*}\sin(\theta) + i\cos(\theta) &= \frac{\left (\sin(\theta) + i\cos(\theta) \right ) \left ( 1 + \sin(\theta) - i\cos(\theta) \right )}{1 + \sin(\theta) - i\cos(\theta)}\\&= \frac{\sin(\theta) + \sin^2(\theta) -...

Re: 2012 HSC MX2 Marathon correct! \begin{align*}\textup{(a)} &\textup{ It is given that } A,B>0 \textup{ and } n \textup{ is a positive integer.}\\i. &\textup{ Find } \frac{A^{n+1} - A^{n}B + B^{n+1} - B^{n}A}{A-B}, \textup{ and deduce }\\& A^{n+1} + B^{n+1} \geq A^{n}B + B^{n}A.\\ii...

Re: 2012 HSC MX2 Marathon oh.. well anyway: \begin{align*}S_{\infty}&=\frac{a}{1-r}\\&=\frac{1}{1-(-x)}\\&=\frac{1}{1+x}, \textup{ for } |x| <1.\end{align*} \begin{align*}\therefore \frac{1}{1+x} &= 1 - x + x^2 - x^3 + x^4 - \cdots, \textup{ for } |x| <1\\\int \frac{1}{1+x} \, \mathrm{d}x &=...

Re: 2012 HSC MX2 Marathon cool (although you should probably include limit swap step: -\int_{\infty}^{0} \frac{\log \left (\frac{a^2}{u} \right )}{{\frac{a^4}{u^2} + a^2}}\times\frac{a^2}{u^2} \, {\mathrm{d} u} = \int^{\infty}_{0} \frac{\log \left (\frac{a^2}{u} \right )}{{\frac{a^4}{u^2} +...

btw I posted another question in the marathon oh and how was scischool?

\begin{align*}z&=r\textup{cis}(\theta), |z|=1 \Rightarrow r=1\\\therefore z &= \textup{cis}(\theta)\\z^n &= \textup{cis}(n\theta)\\&=\cos(n\theta) + i\sin(n\theta)\\\frac{1}{z^n} &= \textup{cis}(-n\theta)\\&=\cos(-n\theta) + i\sin(-n\theta)\\&=\cos(n\theta) - i\sin(n\theta)\\\end{align*}...

Re: 2012 HSC MX2 Marathon ok I'll stop lol – until tomorrow \begin{align*}&\textup{Let } I = \int^{\infty}_{0} \frac{\log(x)}{x^2 + a^2} \, \mathrm{d} x, \, a>0.\\&\textup{By using the substitution } x =\frac{a^2}{u} \textup{ (or otherwise), show that}\\&I=\frac{\pi}{2a}\log(a).\end{align*}

Re: 2012 HSC MX2 Marathon \textup{By writing } \cos^2(x) \textup{ as } \left ( \frac{e^{ix} + e^{-ix}}{2} \right )^2, \textup{ find } \int \cos^2(x) \, \mathrm{d} x. \textup{ (treat }i\textup{ as a constant)}

Re: 2012 HSC MX2 Marathon \\\textup{The exponential function can be defined as: } e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots,\\ \textup{and the sine function as: } \sin (x)= x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots. \begin{align*}\textup{(a) } &\textup{Show that:}\\\textit{i...

both are accepted by the board of studies. I prefer this one because it's simpler C6H10(l) + Br2(aq) –> C6H10Br2(l) edit: take a look at this for more info

During free radical polymerisation the radical end often curls back on part of the existing chain, exposing a hydrogen to an electron. The hydrogen atom is subsequently captured, leaving the radical in the middle of the chain. This is called ‘backbiting’, and occurs about every 50 atoms, causing...

there are only three: - pentane - 2-methylbutane - 2,2-dimethylpropane (neopentane) I think this is prelim stuff, so you can't be directly examined on it (however it's assumed knowledge)