**
New South Wales Higher School Certificate Mathematics Extension 2
**

(Online since January 1, 2001)

Mediocrity is something you can buy. Excellence is something you can download from the internet for free!

** August 4, 2023 **

Here is my article Variations on a Theme of Toloza which explores a generalisation of Toloza's formula for \(\pi\)

Here is my article on Toloza's formula for \(\pi\) as a pairwise alternating sum of reciprocals of binomial coefficients derived from results of the 1957 Leaving Certificate and 2014 HSC exams.

https://www.angelfire.com/ab7/fourunit/binomial-formula-for-pi.pdf

** August 2, 2023 **

Here is my article on the formula for e as an infinite product of binomial coefficients

https://www.angelfire.com/ab7/fourunit/e_binomial_product.pdf

This website has now moved to https://4unitmaths.com/me2.html

It has been purported in some textbooks that a model for projectile motion with quadratic drag could be \(R_x=-k\dot x^2\) and \(R_y=-k\dot y^2\).

Consider the following proposition.

If the model is appropriate then the motion is vertical, horizontal or stationary.

Now consider the contrapositive statement:

If the motion is not vertical, horizontal nor stationary then the model is not appropriate.

Most projectile motions are not vertical, horizontal nor stationary so you can draw your own conclusions from that.

** May 19, 2021 **

Corrections to Insight Mathematics Year 12 Standard 2 Chapter 9

** May 18, 2021 **

Corrections to Maths Quest Year 12 Standard 2 Chapter 9

** May 17, 2021 **

Corrections to Standard 2 Year 12 Excel Chapter 7

** April 17, 2021 **

Corrections to Terry Lee's NFM Chapter 11

** April 16, 2021 **

Corrections to MIF Advanced Year 12 Chapter 10

** April 15, 2021 **

Corrections to Cambridge Advanced Year 12 Section 10D

** April 10, 2021 **

Corrections to New Senior Mathematics Additional Sections 20.4 and 20.5

** March 10, 2021
**

Textbook list updated:

http://4unitmaths.com/texts.pdf

This now has the following titles coming out in 2021:

Decode, HSC Mathematics 4 Unit Combined - Expected October, 2021

Green, J. and Hunter, J., A+ HSC Year 12 Mathematics Extension 2 Study Notes - Expected September, 2021

Green, J. and Hunter, J., A+ HSC Year 12 Mathematics Extension 2 Practice Exams - Expected September, 2021

** May 17, 2020**

Ellipse of Best Fit Around the Mona Lisa's Face

**
March 8, 2020
**

Alternative solutions to Cambridge Extension 1 Year 12 Exercise 15E

These use a method outside the syllabus, namely the Penrose inverse. One of the questions in the textbook also has an incorrect answer (15E Q1). This is corrected in these solutions.

**
January 2, 2020
**

Alternative solutions to Cambridge Extension 2 Exercises 5F Q15-19, 5G Q16 and 5H Q18

These use some methods outside the syllabus, namely the scalar triple product and the Penrose inverse. One of the questions in the textbook also has an incorrect answer (5F Q17a). This is corrected in these solutions.

**
January 1, 2020
**

**
A 2020 View of Fermat's Last Theorem
**

As we approach the first anniversary of Jean-Pierre Wintenberger's death on 23 Jan 2019, Ken Ribet is giving a lecture at the JMM 2020 on 16 Jan 2020 about the possibility of simplifying the proof of Fermat's Last Theorem. This is 25 years after it was proved as a corollary of the proof of the semistable case of the Taniyama conjecture by Andrew Wiles.

A summary of the lecture is in the January 2020 Notices of the American Mathematical Society at https://www.ams.org/journals/notices/202001/rnoti-p82.pdf

After Wiles' proof, the full Taniyama conjecture was proved in 2001 by Christophe Breuil, Brian Conrad, Fred Diamond and Richard Taylor.

However as early as 1975 there has been another way to prove Fermat's Last Theorem via Serre's modularity conjecture. This asserts that an odd, irreducible, two-dimensional Galois representation over a finite field arises from a modular form.

This conjecture was proved by Chandrashekhar Khare and Jean-Pierre Wintenberger in 2009.

Unfortunately however this way isn't much simpler than Wiles' method.

The Khare-Wintenberger proof is at

Khare, Chandrashekhar; Wintenberger, Jean-Pierre (2009), "Serre's modularity conjecture (I)", Inventiones Mathematicae, 178 (3): 485-504

(preprint: https://www.math.ucla.edu/~shekhar/papers/results.pdf )

and

Khare, Chandrashekhar; Wintenberger, Jean-Pierre (2009), "Serre's modularity conjecture (II)", Inventiones Mathematicae, 178 (3): 505-586

(preprint: https://www.math.ucla.edu/~shekhar/papers/proofs.pdf )

**
December 7, 2018
**

New largest known prime 2
^{82589933}
-1 discovered with 24862048 digits by Patrick Laroche. You can download all the digits here:
https://www.mersenne.org/primes/digits/M82589933.zip
. More info here
https://www.mersenne.org

**
December 1, 2018
**

**
August 1, 2018
**

Akshay Venkatesh has won a 2018 Fields Medal making him only the second Australian to do so. The first was Terence Tao in 2006.

Here are some articles on it:

SMH:

The Australian:

Quanta magazine (more detailed):

https://www.quantamagazine.org/fields-medalist-akshay-venkatesh-bridges-math-and-time-20180801/

**
July 10, 2018
**

**
January 29, 2018
**

The syllabuses have now been updated today.

New versions are available here:

In particular the Standard one replaces

"Students of the Mathematics Standard 1 and Mathematics Standard 2 courses study a common Year 11 course, Mathematics Standard Year 11, leading to the Mathematics Standard 1 Year 12 and Mathematics Standard 2 Year 12 courses. Schools have flexibility in providing alternate approaches to Mathematics Standard in Year 11 to address material essential for Mathematics Standard 1 in Year 12. This material is denoted by the symbol ◊. Students who follow the ◊ pathway in Year 11 Mathematics Standard will only be eligible for Mathematics Standard 1 in Year 12.�

with

"Students studying the Mathematics Standard syllabus undertake a common course in Year 11. For the Year 12 course students can elect to study either Mathematics Standard 1 or Mathematics Standard 2. Students who intend to study the Mathematics Standard 2 course in Year 12 must study all Mathematics Standard Year 11 course content. Students who intend to study the Mathematics Standard 1 course in Year 12 must have studied the content identified by the symbol ◊ which forms the foundation of course. This content is important for the development and consolidation of numeracy skills.�

In Advanced, the formula for variance which was in the Glossary has now been relocated in the body of the syllabus.

In Extension 1, The t-formulae in the recently published version of new Stage 6 Mathematics Extension 1 syllabus for implementation in 2019 that were presented in terms of θ will be changed to be expressed in terms of A

**
January 26, 2018
**

Eddie Woo is now the recipient of the 2018 Australian Local Hero Award.

iview link for Eddie Woo's 2018 Australia Day Address: http://iview.abc.net.au/programs/australia-day-address-eddie-woo/NS1856H001S00#pageloaded (note that it is still on facebook too but is now easier to find on iview)

**
January 17, 2018
**

Maths teacher Eddie Woo will give the 2018 Australia Day Address.

It will be on tv on ABC News 24 at 11.30pm on 23rd January, 2018 (or live stream on facebook at 12:30pm at https://www.facebook.com/AustraliaDay26 )

Official Announcement: https://www.nsw.gov.au/your-government/the-premier/media-releases-from-the-premier/star-maths-teacher-mr-eddie-woo-to-deliver-2018-australia-day-address/

**
December 26, 2017
**

New largest known prime 2
^{77232917}
-1 discovered with 23249425 digits by Jonathan Pace. You can download all the digits here:
http://www.mersenne.org/primes/digits/M77232917.zip
. More info here
https://www.mersenne.org

**
December 2, 2017
**

**
November 23, 2017
**

NEW SYLLABUSES RELEASED.

Advanced ; Extension 1 ; Extension 2

Also the Standard syllabus has been re-released and updated to include information on common content identified with paper clips. The new version can be redownloaded here

Oh well. It took a bit more than 2 weeks. More like 2 months!

**
September 16, 2017
**

At the MANSW Conference today a NESA representative announced that the final syllabuses will be released within the next 2 weeks - but an exact date was not given.

**
August 24, 2017
**

**Trigonometry discovered to have come from Babylonians 1500 years before the Greeks**

The origins of trigonometry is usually attributed to Hipparchus. Two mathematicians from UNSW have discovered that trigonometry came from Babylonians 1500 years earlier. Here is The Telegraph article on it: http://www.telegraph.co.uk/science/2017/08/24/3700-year-old-babylonian-tablet-rewrites-history-maths-could/

Here is the more thorough research article in Historia Mathematica:

http://www.sciencedirect.com/science/article/pii/S0315086017300691

**
July 19, 2017
**

According to NESA the syllabuses should be finished by "the middle of the year": http://educationstandards.nsw.edu.au/wps/portal/nesa/about/news/news-stories/news-stories-detail/timeframe-for-release-of-support-materials-for-new-stage-6-syllabuses

Well 12 noon today is the middle of the year and they still aren't finished.

That's not necessarily a bad thing provided that they commence new processes in order to create a better syllabus than the one we have now. Evidently if they persist with the current draft it will be an act of vandalism and result in a worse syllabus.

**
June 19, 2017
**

2017 SMH HSC Maths Study Guide

**
May 22, 2017
**

Bill Pender's submissions:

http://4unitmaths.com/Pender-Submission-2016.pdf

http://4unitmaths.com/Pender-Submission-2017.pdf

http://4unitmaths.com/pender-comment-may-2017.pdf

(reproduced with permission).

**
May 5, 2017
**

At a PD today at the AIS Head Office at 99 York St, Sydney, NESA said they will release the new calculus courses "by the middle of the year", which is Midday, July 2, 2017. The Standard syllabus will be updated and re-released at the same time to include information about common content. They will also instigate a 5-year syllabus review cycle.

**
April 22, 2017
**

There is an article about the delayed syllabus implementation in the Sydney Morning Herald at http://www.smh.com.au/national/education/release-of-the-new-advanced-hsc-maths-syllabuses-to-be-delayed-until-2019-20170421-gvpkn0.html

But unfortunately their title is wrong. The title of the article is "Release of the new advanced HSC maths syllabuses to be delayed until 2019".

But if you refer to the official statement from NESA they clearly state that the syllabuses will be released later this year. That's 2017, Not 2019.

So although it is correct to say the implementation is delayed till 2019, it is not correct to say that they will be released in 2019. The author was informed of this error but they have not corrected it. So I am correcting it here in case teachers might see the SMH article and think they won't get the syllabuses till 2019. According to NESA (who are the authority in this matter, not the SMH) they should get them this year.

**
April 21, 2017
**

NESA have decided to delay the Mathematics Advanced, Extension 1 and Extension 2 courses by another year, as predicted back in March. They have however not delayed Mathematics Standard which is to start in year 11 next year:

**
March 16, 2017
**

**Possible scenario going forward with the new syllabuses.**

Support materials are supposed to be released along with the new syllabuses. Such material has not yet been released. There was a ministerial statement in 2011 which specifies that syllabus materials be in schools 1 year prior to implementation:

Although we now have a new minister of Education, he has not rescinded the 2011 ministerial statement and hence it remains valid.

So NESA aren�t just running out of time. They already HAVE run out of time to get all the syllabus materials to schools in a manner compliant with the ministerial statement.

Hence it has been proposed to delay implementation another year for the calculus courses:

https://www.mansw.nsw.edu.au/documents/item/218.pdf

Hence under this scenario we might see Mathematics Standard being implemented in year 11 in 2018 and the calculus courses in 2019 instead.

It is possible that NESA won�t accept this proposal - but that would then not be compliant with the 2011 ministerial statement.

**
March 14, 2017
**

Response to NESA's Further Consultation on Stage 6 Mathematics Advanced and Extension syllabuses

**
February 21, 2017
**

New Mathematics Standard Syllabus Released: https://syllabus.bostes.nsw.edu.au/assets/mathematics_standard/mathematics-standard-stage-6-syllabus-2017.pdf

This will be for implementation for Year 11 in 2018 and Year 12 in 2019.

Also Draft 2 of the Mathematics Advanced, Mathematics Extension 1 and Mathematics Extension 2 were also released:

Mathematics Advanced Draft 2: https://syllabus.bostes.nsw.edu.au/assets/mathematics_advanced/mathematics-advanced-stage-6-syllabus-2017.pdf

Mathematics Extension 1 Draft 2: https://syllabus.bostes.nsw.edu.au/assets/mathematics_extensio/mathematics-extension-1-stage-6-syllabus-2017.pdf

Mathematics Extension 2 Draft 2: https://syllabus.bostes.nsw.edu.au/assets/mathematics_extentio/mathematics-extension-2-stage-6-syllabus-2017.pdf

These drafts are for consultation till March 14, 2017 : http://educationstandards.nsw.edu.au/wps/portal/nesa/about/news/news-stories/news-stories-detail/further-consultation-maths-calculus-based-hsc-syllabuses

**
February 20, 2017
**

The new Senior syllabus will be released tomorrow at the NESA website at http://educationstandards.nsw.edu.au/wps/portal/nesa/home/

Here are some articles on it in the Sydney Morning Herald and The Australian:

**
February 19, 2017
**

3 new articles related to the proof of Fermat's Last Theorem from the March 2017 issue of the Notices of the American Mathematical Society:

1. Ad Honorem Sir Andrew J. Wiles: http://www.ams.org/publications/journals/notices/201703/rnoti-p197.pdf

2. Interview with New AMS President Kenneth A. Ribet: http://www.ams.org/publications/journals/notices/201703/rnoti-p229.pdf

3. What is an Elliptic Curve?: http://www.ams.org/publications/journals/notices/201703/rnoti-p241.pdf

**
January 10, 2017
**

State maths revamp under attack - incoherent, rushed and appalling, say critics of syllabus: http://www.smh.com.au/national/education/new-hsc-mathematics-syllabus-splits-education-leaders-20170109-gto69c.html

**
January 1, 2017
**

Last year the University of New South Wales held a professional development day for teachers pertaining to new topics from the draft syllabus and set up a publicly accessible moodle page with downloadable resources at http://moodle.telt.unsw.edu.au/course/view.php?id=26796

**
December 16, 2016
**

Version 8.3 of the Australian curriculum was released today.

F-10: http://www.australiancurriculum.edu.au/download/f10

Senior: http://www.australiancurriculum.edu.au/Download/SeniorSecondary

**
December 3, 2016
**

**
October 20, 2016
**

The draft syllabus isn't good enough and they need to start again: http://www.smh.com.au/national/education/hsc-maths-teachers-in-revolt-over-proposed-new-syllabus-20161019-gs6eyf.html

But I bet they won't.

**
August 31, 2016
**

**
July 20, 2016
**

New draft syllabuses released at http://www.boardofstudies.nsw.edu.au/syllabuses/curriculum-development/senior-years.html

Consultation ends August 31, 2016

**
July 12, 2016
**

**
July 4, 2016
**

Question 8a in the 2002 Extension 2 HSC exam can be extended to prove the "Basel problem"

\(\frac{1}{1^{2}}+\frac{1}{2^{2}}+\frac{1}{3^{2}}+\dots=\frac{\pi^2}{6}\)

The original question with 4 parts is at http://www.boardofstudies.nsw.edu.au/hsc_exams/hsc2002exams/pdf_doc/mathemat_ext2_02.pdf

Here is the solution to the original question my mojako: http://4unitmaths.com/2002-8a-mojako.pdf

Here is the extension with 4 more parts proving the Basel problem as a pdf file: http://4unitmaths.com/2002-8a-extended.pdf

Here is the solution to the extended parts: http://4unitmaths.com/2002-8a-extended-sol.pdf

Historical note: This was first proposed in 1644 by Pietro Mengoli and then solved by Euler in 1734. But Euler did not use this method. In fact there are several methods by which it can be proved. The method in this HSC exam together with the extension was first used by Augustin Louis Cauchy in 1821 in a book called Cours d'Analyse.

**
June 30, 2016
**

Version 8.2 of the Australian curriculum was released today.

**
February 16, 2016
**

New writing briefs for senior maths HSC:

**
December 16, 2015
**

Version 8.1 of the Australian curriculum was released today.

It seems though that only the F-10 was really updated for mathematics. The senior seems to still be a mixture of versions 7.5 and 8.0.

There are 2 new documents for F-10 called Sequence of content and Sequence of achievement:

http://www.acara.edu.au/verve/_resources/Mathematics_-_Sequence_of_content.pdf

http://www.acara.edu.au/verve/_resources/Mathematics_Sequence_of_achievement.pdf

**
December 5, 2015
**

**
November 9, 2015
**

Mysterious calculator exercise

**
November 8, 2015
**

New book Prime Numbers and the Riemann Hypothesis to be published in hardcover and paperback at http://www.amazon.com/Prime-Numbers-Riemann-Hypothesis-Barry/dp/1107101921/ but is available now from one of the author's own websites as a pdf at http://wstein.org/rh/rh.pdf

**
November 7, 2015
**

Polynomials questions from past Leaving Certificate papers 1920-1955

**
November 6, 2015
**

The new reference sheet has now been published a few days early: http://www.boardofstudies.nsw.edu.au/syllabus_hsc/pdf_doc/maths-ref-sheet.pdf . This will be used in next year's HSC exams for 2 unit, Ext. 1 and Ext. 2.

**
November 5, 2015
**

Schoolboys Ivan Zelich and Xuming Liang make a new theorem called the Liang-Zelich theorem.

Youtube video: https://www.youtube.com/watch?v=XxuO5W23Z60

Embed:

Daily Mail article: http://www.dailymail.co.uk/news/article-3304802/Meet-boy-geniuses-developed-math-theorem-calculates-problems-faster-computer-despite-high-school.html

Proof of theorem: http://ijgeometry.com/wp-content/uploads/2015/10/1.pdf

**
October 26, 2015
**

Terry Lee has updated his website today to include solutions to the 2015 Ext. 1 and 2 HSC exams which also concluded today: http://extension2.com

The papers themselves are (or will be) available at http://www.boardofstudies.nsw.edu.au/hsc_exams/2015/

**
October 18, 2015
**

Version 8.0 of the Australian Curriculum was released today.

**
October 16, 2015
**

The draft writing briefs are online now a few days early:

General 1 and 2 draft writing brief

2 unit/ ext. 1 and 2 draft writing brief

**
October 9, 2015
**

Draft Writing Briefs for new senior syllabuses will be online for consultation beginning on October 19, 2015: http://news.boardofstudies.nsw.edu.au/index.cfm/2015/10/9/Consultation-on-Senior-Years-English-Mathematics-Science-and-History-syllabuses

**
October 2, 2015
**

A new reference sheet will be published on the BOSTES website on November 9 following the completion of the 2015 HSC which will be used in HSC exams for calculus courses beginning in 2016: http://news.boardofstudies.nsw.edu.au/index.cfm/2015/10/2/Development-of-new-reference-sheet-for-HSC-Mathematics-examinations-from-2016

**
October 1, 2015
**

El Capitan has been released. Here are instructions for making a bootable usb: http://users.tpg.com.au/nanahcub/elcapitan.html

**
September 18, 2015
**

During the Education Council meeting today it was announced that an updated version of the F-10 National curriculum will be published in mid-October: http://www.scseec.edu.au/site/DefaultSite/filesystem/documents/Communiques%20and%20Media%20Releases/2015%20Communiques/Education%20Council%20Communique%20-%2018%20September%202015.pdf

**
July 11, 2015
**

International Mathematical Olympiad 2015

**
March 7, 2015
**

New youtube of Terry Tao:

(link: https://www.youtube.com/watch?v=P3dLcTkJRr0 )

**
Teaching Resources
**

4 unit Syllabus (from boardofstudies server)

2 and 3 unit Syllabus (from boardofstudies server)

Terry Lee's HSC Solutions website

Youtube video for the new VERIFY function on the CASIO fx-82AU PLUS II calculator

Vectors summary ; Matrices summary

Alternative solution to James Ruse 2011 Trial Question 8b

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

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

Professional mathematics versus amateur mathematics

On December 25, 2011 the largest known twin primes were found by Timothy D. Winslow. They are 3756801695685x2
^{666669}
1 both of which have 200,700 digits.

They are at http://4unitmaths.com/tp1.pdf and http://4unitmaths.com/tp2.pdf .

Summary of the proof of Fermat's Last Theorem

Online video on Fermat's Last Theorem: msri

Bill Pender's Harder 3 unit inservice

English translation of Hilbert's radio address

International Mathematical Olympiads

1916 LC, 1989 HSC and 2001 HSC

Another proof of the irrationality of e

Alternative solution to 2003 HSC Q3(a)(iv)The General Conic and Dandelin Spheres

Proof of the Fundamental Theorem of Algebra

**
University Mathematics
**

Proof of Fermat's Last Theorem

Proof of the Taniyama-Shimura-Weil Conjecture

Beal Prize for $1,000,000 for proving (or disproving) the Beal Conjecture, i.e., that the only solutions to the equation \(A^x + B^{\ y} = C^{\ z}\), when \(A\), \(B\), \(C\), are positive integers, and \(x\), \(y\) and \(z\) are positive integers greater than 2, are those in which \(A\), \(B\) and \(C\) have a common factor

Proof of Poincare conjecture - part 1

Proof of Poincare conjecture - part 2

Proof of Poincare conjecture - part 3

The Riemann Hypothesis - Part 1

The Riemann Hypothesis - Part 2

Lagarias Equivalence to the Riemann Hypothesis

Birch and Swinnerton-Dyer conjecture

The ABC Conjecture and Mochizuki's proposed proof

**
Online LaTeX editors
**

Too many philistines are using Word. They should stop being philistines and start using LaTeX.

For web browsers (nothing needs to be installed): Verbosus ; ShareLaTeX

iPad app: TeX Touch . Files created in this app can be compiled via the TeX Cloud .

**
Forums
**

http://community.boredofstudies.org/14/mathematics-extension-2/

http://www.artofproblemsolving.com/community

**
Other websites
**

Higher School Certificate Online

The American Mathematical Society

Last modified on August 4, 2023 by

Derek Buchanan

Copyleft: Derek Buchanan, 2001