With the new Project Maths programme being developed as we speak, diligent students might like to know which proofs are examinable under the new syllabus so they know which to look at.

It can be difficult to sift through the syllabi at projectmaths.ie but I have gone through them and here are the proofs required.

## Pilot Schools (such as Pobolscoil Corca Dhuibhne)

A Pilot School is one that has the new syllabus for Papers 1 *and* 2. The syllabus says that proofs using induction are required — however there are so many of these that it wouldn’t be a good idea to list them out. Instead understand induction and how it works.

### Paper 1

- De Moivre’s Theorem for .
- Prove that if and such that are the first term and common ratio of an infinite geometric series, that the sum to infinity of the series is given by:

### Paper 2

- Theorem 11: If three parallel lines cut off equal segments on some transversal line, then they will cut off equal segments on any other transversal.
- Theorem 12: Let be a triangle. If a line is parallel to and cuts in the ratio , then it also cuts in the same ratio (*see note below).
- Theorem 13: If triangles and are similar, then their sides are proportional:

. - Trigonometry Formula for angles . When side-lengths are involved, the proofs are required for a triangle with angles opposite sides .

## All Other Schools (such as the Tralee Schools)

“All Other Schools” are schools that only have the new syllabus for Paper 2. The syllabus says that proofs using induction are required — however there are so many of these that it wouldn’t be a good idea to list them out. Instead understand induction and how it works.

### Paper 1

- Prove the Factor Theorem for cubics.
- De Moivre’s Theorem for .
- Differentiate from first principles: , , , , , .
- Prove the sum, product and quotient rules for differentiation.
- Prove that for using induction.
- Prove that if and such that are the first term and common ratio of an infinite geometric series, that the sum to infinity of the series is given by:

### Paper 2

- Theorem 11: If three parallel lines cut off equal segments on some transversal line, then they will cut off equal segments on any other transversal.
- Theorem 12: Let be a triangle. If a line is parallel to and cuts in the ratio , then it also cuts in the same ratio.
- Theorem 13: If triangles and are similar, then their sides are proportional:

. - Trigonometry Formula for angles . When side-lengths are involved, the proofs are required for a triangle with angles opposite sides .

### Remark to Theorem 12

Incidentally the proof in your textbook is not the whole story. Suppose the line cuts at such that is *not *a fraction (e.g. ) — then the proof in the textbook does *not *apply. So technically if the examiner asks for a proof of Theorem 12 the proof in your textbook is only partial! However such are the inadequacies in the system, the textbook proof will of course be accepted and is the one you should learn.

Two correct proofs (which hold regardless of whether or not is rational/fraction or not) may be found here. My proof is given in the question but a David Mitra provides a much slicker (and standard as it turns out) proof. In a just world, were this proof produced you should be roundly applauded however, regrettably, there is a good chance the corrector doesn’t know the proof and ignorantly marks it as incorrect.

## 19 comments

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November 3, 2011 at 11:04 pm

PeterExcellent article! I found this on http://www.bulletinboard.ie when looking for grinds for my son! Really informative!

Thanks for this,

Peter

January 26, 2012 at 12:15 am

Roberto MendezJP Mc Carthy, could you help me. I ‘m 18 and live in Nicaragua and will be travelling to cork in May to sit the Leaving Cert Higher course. I have completed my Batchillerato here but I want to sit the Leaving. I am confused about the new Project Maths syllabus. I have read your take on it above and that was helpful to a point but it assumes the reader already has some idea of what it’s all about. I quite frankly don’t have a clue as I’m starting to study now for the exam in June. Is there a web site that gives a more detailed explanation of what is required to study ? Is there a math text book that goes with the new syllabus?

Thank you.

Roberto

January 26, 2012 at 10:10 am

J.P. McCarthyRoberto,

I would point you towards the following site: projectmaths.ie/. This site should have everything you need to know even if it is difficult to navigate.

You will most probably not be sitting the Leaving Cert in one of the initial 24 schools/pilot schools. Thus you will be sitting in a school “other than the initial schools”.

You can find up-to-date sample papers here and syllabus information here.

At its most basic level Project Maths is just the Leaving Cert with some stuff thrown out to make way for much more probability & statistics and geometry — with a greater emphasis on “real world applications”.

Regards,

J.P.

January 28, 2012 at 3:32 am

Roberto MendezJP, thank you for your advice. I think I have a handle on Project maths now but want to confirm it with you.

1. Paper 1 remains the same so I should cram previous Leaving exam hons maths papers 1 for the next four months.

2. In the old Paper 2 and therefore I should cram questions 6, 7 and 8 from the previous years as those style questions may/will reappear in this 2012 exam.

3. In this years Paper 2, questions 1 – 5 will be of the new project maths variety i.e Probability, stats and geometry of the “real world” type.

Roberto.

January 28, 2012 at 3:36 am

Roberto Mendez(I wrote that wrong)

2. In the old Paper 2 questions 6,7 and 8 remain fundamentally the same and therefore I should cram these questions from the previous years as those style questions may/will/should reappear in this 2012 exam?

January 28, 2012 at 3:19 pm

J.P. McCarthyRoberto,

First off I would never put “cram” and maths in the same sentence! I prefer the euphemism “look at” or even better “understand”.

Yes Paper 1 remains the same.

Paper 2 is new. See a sample.

What is new is further statistics and probability and there is more geometry than before. Questions 2, 8, 10 and 11 from the old paper are gone. However on the project maths page there are enough samples, pre-s and old exam papers (from the pilot schools) for you to practise without going to the old paper 2s.

Regards,

J.P.

April 14, 2012 at 5:43 pm

CathalHi J.P.

Thanks for posting the proofs.

I am familiar with all of the proofs apart from the infinite geometric series formula. I cant find this in any past papers or text books. I would be very grateful if you could tell me where to find it.

Thanks a million.

Regards

Cathal

May 3, 2012 at 10:32 am

J.P. McCarthyCathal,

Sorry for not getting back to you sooner. We have a number of ways of proving the infinite geometric series formula but your best bet is by deriving the expression for the first terms of a geometric series:

.

Now the sum of the infinite geometric series is given by

.

Now if then and we are left with

.

Regards,

J.P.

June 2, 2012 at 2:50 pm

coletteHi J.P is there any chance you could send me a link of where to find the proof of the volume of a cylinder? or can i simply find the volume of a cone and multiply my answer but three? would this be suffice if i was asked to prove that the volume of a cylinder is equal to pie(r squared)h?

June 5, 2012 at 8:26 am

J.P. McCarthyCollette,

Here http://en.wikibooks.org/wiki/Calculus/Volume#Example_1:_A_right_cylinder

I would be surprised if it is not in your text book.

Finding the volume of a cone and then multiplying by three would NOT suffice.

Regards,

June 5, 2012 at 12:49 am

Claire A. RushI saw somewhere that we need to know the Ordinary Level proofs, so I was just wondering what are the proofs required for the Honours Paper? You don’t need explain them all, just a list would be really useful, I can use my textbooks to figure them out 🙂 Also, do you know the constructions we need to know? Thanks and sorry for the question.

June 5, 2012 at 8:23 am

J.P. McCarthyClaire,

I presume you mean what are the OL level proofs?

Your best bet is to look at http://www.ncca.ie/en/Curriculum_and_Assessment/Post-Primary_Education/Project_Maths/Syllabuses_and_Assessment/LC_Maths_for_examination_in_2012.pdf (assuming you are not in a pilot school).

From looking at this mu impression is that there are no examinable proofs for the OL course.

If you asked me which constructions are examinable I would have said the circumcircle, incircle and centroid but this doesn’t seem to be the case (use CTRL-F to search the above pdf file for “CONSTRUCTION”). I don’t think I can give you definitive advice at this point.

If you asked this question three months ago I would say learn and understand ALL of the constructions: they will greatly help your geometry. The reason for this is that there is a duality between many constructions and theorems. We can believe the theorems by looking at constructions (and indeed inspire proofs) and various constructions are done by using theorems.

Regards,

J.P.

June 8, 2012 at 10:14 am

E. MurphyPlease tell me the infinite geometric series formula is not examinable. It’s not marked as a proof in Text and Tests 4?!

June 8, 2012 at 10:24 am

J.P. McCarthyE,

Looking at the syllabi it is examinable but unlikely to come up I would suggest.

your best bet is by deriving the expression for the first terms of a geometric series using induction:

.

Now the sum of the infinite geometric series is given by

.

Now if then and we are left with

.

Regards,

J.P.

June 8, 2012 at 11:09 pm

Claire A. RushI’m doing Honours, that’s why I was wondering what proofs we needed.

June 9, 2012 at 8:32 am

JohnnyHi,

I lookes through the ncca.ie website but I couldnt find a step-by-step explanation of theorems 11,12,13 for higher level. Any chance you could point me in the right direction? Thanks!

June 11, 2012 at 1:38 pm

J.P. McCarthyJohnny,

This is obviously too late but they really should be in your textbooks (unless ye don’t have a new text book for Project Maths).

Regards,

J.P.

June 9, 2012 at 9:46 pm

CorneliusHi,

I am just wondering about the proofs for the Trigonometry Formula like if we were asked to derive cos A + B, in the text books they say u simply take the formula for cos A – B and replace B with -B . but would u have to derive cos A -B first then if asked cos A + B??

Regards

Cornelius

June 11, 2012 at 1:37 pm

J.P. McCarthyCorneliuis,

This is obviously too late but yes you couldn’t assume say the result to prove the one.

You must prove the formula first.

Regards,

J.P.