New South Wales Higher School Certificate Mathematics Extension 2

New South Wales Higher School Certificate Mathematics Extension 2

(Online since January 1, 2001)

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Practice papers: 200 papers ; 124 papers

New

January 4, 2012

The 2011 James Ruse paper asked students to

Prove that $\tan^{-1}3+\tan^{-1}5+\tan^{-1}({4\over7})=\pi$

and they provided the following solution:

$0<\tan^{-1}5<{\pi\over2}$

$0<\tan^{-1}3<{\pi\over2}$

$\therefore0<\tan^{-1}3+\tan^{-1}5<\pi$

$\tan(\tan^{-1}3+\tan^{-1}5)={3+5\over1-15}=-{4\over7}$

$\therefore\tan^{-1}3+\tan^{-1}5=\tan^{-1}(-{4\over7})+n\pi$

But as shown above

$0<\tan^{-1}3+\tan^{-1}5<\pi$

$\therefore n=1$

$\tan^{-1}3+\tan^{-1}5=\tan^{-1}(-{4\over7})+\pi$

$\tan^{-1}3+\tan^{-1}5=-\tan^{-1}({4\over7})+\pi$

$\therefore\tan^{-1}3+\tan^{-1}5+\tan^{-1}({4\over7})=\pi$

This can however be generalised to the following:

Show that for $x>\sqrt{y^2+1},\ \tan^{-1}(x-y)+\tan^{-1}(x+y)+\tan^{-1}{2x\over x^2-y^2-1}=\pi$

and James Ruse's result follows by letting $x=4,\ y=1$

Alternative Solution:

Suppose $x>\sqrt{y^2+1}$

Then

$\tan\alpha={1\over1/(x-y)}=x-y$

$\tan\beta={1\over1/(x+y)}=x+y$

$\begin{aligned}\& \tan\gamma&=\textstyle\tan\big(\tan^{-1}{1/(x-y)\over1}+\tan^{-1}{1/(x+y)\over1}\big)\\ &={{1\over x-y}+{1\over x+y}\over1-{1\over x-y}\cdot{1\over x+y}}\\ &=\textstyle{2x\over x^2-y^2-1}\end{aligned}$

$\therefore\alpha+\beta+\gamma=\tan^{-1}(x-y)+\tan^{-1}(x+y)+\tan^{-1}{2x\over x^2-y^2-1}=\pi$ (Angle sum of triangle)

Letting $x=4,\ y=1$

$\tan^{-1}3+\tan^{-1}5+\tan^{-1}{4\over7}=\pi$

December 3, 2011

2011 Putnam

More available at http://amc.maa.org/a-activities/a7-problems/putnamindex.shtml

July 19, 2011

2011 IMO

Teaching Resources

Syllabus (from boardofstudies server)

Syllabus Issues

Solution to 2010 Mathematics Extension 2 HSC exam Question 8

Link Between 1995 and 2010 HSC Exams Leads To Generalised Wallis Product (preprint) - another version appeared in MANSW's Reflections, Vol. 36, No. 4, 2011, pp. 22-23

Parabola Magazine Online

Barbarians at the Helm, by Derek Buchanan

How NOT to find the surface area of revolution, by Derek Buchanan

Johan Wastlund's Elementary Proof of the Wallis Product Formula for pi

Yet another proof of the irrationality of e

DON'T BAN YOUTUBE!

Three Unit Notes 1

Three Unit Notes 2

Alf van der Poorten's 28 online Number Theory lectures (28 mp4's)

Professional mathematics versus amateur mathematics

Sixty 4 unit lectures

Mathematics Extension 1 website

A new Mersenne prime, 2^43,112,609-1 has been found on August 23, 2008 on Edson Smith's computer and is the largest known prime to date and has 12,978,189 digits 316,470,269,...,511 which you can download at http://prime.isthe.com/no.index/chongo/merdigit/long-m43112609/prime-c.html It is the first discovery of a prime with more than 10,000,000 digits and hence the $100,000 prize was awarded on October 22, 2009. There is another prize for the first discovery of a prime with more than 100,000,000 digits for $150,000. More info on these prizes are at http://www.eff.org/awards/coop More info on this discovery is at http://www.mersenne.org

Also, on July 25, 2009 the largest known twin primes were found by SunGard Availability Services. They are 65516468355x2³³³³³³ą1 both of which have 100355 digits, the smaller of which is at http://www.primegrid.com/download/tm333333.pdf. Add 2 for the larger one (i.e., replace the last 2 digits, 59, by 61).

Summary of the proof of Fermat's Last Theorem

Online videos on Fermat's Last Theorem: uktv on google video (or youtube) ; msri

Bill Pender's Harder 3 unit inservice

pisquaredonsix.pdf

Wiles' online lecture

Clay Meeting online

Tate's online lecture

Atiyah's online lecture

David Hilbert's radio address

English translation of Hilbert's radio address

SMH HSC Survival Guide 2009

Assignments