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Re: Extracurricular Integration Marathon $\noindent In this question the following result previously found will be used \int^\infty_{-\infty} \frac{\cos ax - \cos bx}{x^2} \, dx = \pi (b - a), \quad \{a,b\} \in \mathbb{R}^+ $\noindent Integrating by parts we have$...

Re: Extracurricular Integration Marathon $\noindent \textbf{New Question} $\noindent Evaluate $\int^{\frac{\pi}{2}}_0 \ln \left (\frac{1 + \sin x}{1 - \sin x} \right ) \frac{\cos^2 x}{\sin x} \, dx.

Re: Extracurricular Integration Marathon $\noindent In order to evaluate this integral we will convert it into a double integral. Observing that$ \int^b_a \sin (ux) \, du = \frac{\cos (ax) - \cos (bx)}{x}, $ the integral can be rewritten as$...

Re: Extracurricular Integration Marathon $\noindent In our solution we will make use of the following two Maclaurin series expansions \tan^{-1} t = \sum^\infty_{n = 0} \frac{(-1)^n t^{2n + 1}}{2n + 1}, \, |t| < 1 $ and $\,\sinh^{-1} t = \sum^\infty_{n = 0} \frac{(-1)^n (2n)! t^{2n +...

Re: HSC 2017 MX2 Integration Marathon $\noindent Ask yourself the question what is the integral $\int \frac{dt}{t}? $\noindent Of course you will immediately say it is the natural logarithm! Well this is just a convenient way of papering over the fact that until we expanded our function...

Re: HSC 2017 MX2 Integration Marathon $\noindent When we say an integral cannot be integrated in \textit{elementary terms} it means no primitive for the function can be found in terms of the so-called elementary functions. The elementary functions are polynomials, roots, exponential...

$\noindent \textbf{Proof for the second part}$ $\noindent Squaring the equation $|z - z_1| = k|z - z_2|$ and making use of the result $w \bar{w} = |w|^2$ where $w \in \mathbb{C}$ we have$ \begin{align*}|z - z_1|^2 &= k^2|z - z_2|^2\\(z - z_1)\overline{(z - z_1)} &= k^2(z - z_2)\overline{(z...

$\noindent The general equation of a circle in the complex plane should be $ \alpha z \bar{z} + \beta z + \bar{\beta} \bar{z} + \gamma = 0. $\noindent \textbf{Proof for first part}$ $\noindent As should be well-known, the complex numbers $z$ satisfied by the set of points given by$...

$\noindent Recall a continuous function $f$ is said to be increasing if $f'(x) > 0$ (namely, it derivative is positive) for all values of $x$ on some interval. $\noindent Now let $f(x) = \sin x - x + \frac{x^3}{4}$. So$ \begin{align*}f'(x) &= \cos x - 1 + \frac{3x^2}{4}\\f''(x) &= -\sin x +...

Re: HSC 2017 MX2 Integration Marathon $\noindent While you may not know a particular method explicitly, it is possible in a Question 16 type of question in the HSC you could be \textit{guided} through using a previously unknown technique or method.

Re: HSC 2017 MX2 Integration Marathon $\noindent You might like to try out the rule on the following question. And just for fun, let us find this integral in three different ways with a few hints provided along the way.$ $\noindent Find $\int \frac{x^2}{(x \sin x + \cos x)^2} \, dx $ as...

Re: Extracurricular Integration Marathon $\noindent We begin with a few preliminary results that I will make use of in my derivation.$ \int^\infty_0 e^{-x^2} \, dx = \frac{\sqrt{\pi}}{2}$ (the Gaussian integral)$ \int^\infty_0 e^{-2x^2} \, dx = \frac{\sqrt{\pi}}{2\sqrt{2}}$ (Substitute $u =...

Re: International Baccalaureate Marathon $Not sure if you know about the \textit{vector cross product}, but if you do one of the geometric interpretations for the magnitude of the cross product between two vectors corresponds to the area of a triangle.$ $If two sides of a triangle are...

Hardly, there is simply too much money at stake. Just wait and see. The first foolish author/publisher to bring their text to market will have their material ruthlessly and unashamedly pillaged!

Couldn't agree more. The new HSC physics and chemistry syllabuses are to be highly commended. Compared to the past 16 years of misery that has been intentionally inflicted on the students of NSW it makes for a long overdue change back to the sensible. Those who actually wish to study physics or...

Overall the Mathematics Advanced and Extension 1 and 2 syllabuses seem pretty disappointing. Matrices and determinants, which was slatted for inclusion in 2 Unit, seems to have been entirely removed, and the so-called "Advanced Calculus Skills" in 4 Unit is a very weak imitation of the current...

A few slots are still available for anyone else who may be interested.

Are offering tuition in Preliminary/HSC Mathematics (2U), Mathematics Extension 1 (3 Unit), and Mathematics Extension 2 (4 Unit). Are also available for tutoring in First-Year University Mathematics. About Me My name is Sean and I have 15+ years teaching experience at the secondary and...

By definition, all 4 Unit students are 3 Unit students already so would naturally be allowed to choose Statistics in Year 12 if they so wished.

Or perhaps statistics could be added as an additional unit at HSC level only open to students studying either 2 or 3 Unit. In this way you would only be drawing on your existing pool of 2 and 3 Unit students rather than Statistics having to compete against them for students. It would be quite a...