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

1. De Moivre’s Theorem for $n\in\mathbb{N}$.
2. Prove that if $a$ and $r$ such that $|r|<1$ are the first term and common ratio of an infinite geometric series, that the sum to infinity of the series is given by:
$\frac{a}{1-r}.$

### Paper 2

1. 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.
2. Theorem 12: Let $\Delta ABC$ be a triangle. If a line $L$ is parallel to $BC,$ and cuts $[AB]$ in the ratio $s:t$, then it also cuts $[AC]$ in the same ratio (*see note below).
3. Theorem 13: If triangles $\Delta ABC$ and $\Delta A'B'C'$ are similar, then their sides are proportional:
$\frac{|AB|}{|A'B'|}=\frac{|BC|}{|B'C'|}=\frac{|AC|}{|A'C'|}$.
4. Trigonometry Formula for angles $A,\,B,\,C$. When side-lengths are involved, the proofs are required for a triangle with angles $A,\,B,\,C$ opposite sides $a,\,b,\,c$.
$\cos^2A+\sin^2A=1$
$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$
$a^2=b^2+c^2-2bc\cos A$
$\cos(A-B)=\cos A\cos B+\sin A\sin B$
$\cos(A+B)=\cos A\cos B-\sin A\sin B$
$\cos2A=\cos^2A-\sin^2A$
$\sin(A+B)=\sin A\cos B+\sin B\cos A$
$\tan(A+B)=\frac{\tan A+\tan B}{1-\tan A\tan B}$

## 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

1. Prove the Factor Theorem for cubics.
2. De Moivre’s Theorem for $n\in\mathbb{Z}$.
3. Differentiate from first principles: $x^2$$x^3$$\sin x$$\cos x$$\sqrt{x}$, $1/x$.
4. Prove the sum, product and quotient rules for differentiation.
5. Prove that $\frac{d}{dx}x^n=nx^{n-1}$ for $n\in\mathbb{N}$ using induction.
6. Prove that if $a$ and $r$ such that $|r|<1$ are the first term and common ratio of an infinite geometric series, that the sum to infinity of the series is given by:
$\frac{a}{1-r}.$

### Paper 2

1. 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.
2. Theorem 12: Let $\Delta ABC$ be a triangle. If a line $L$ is parallel to $BC,$ and cuts $[AB]$ in the ratio $s:t$, then it also cuts $[AC]$ in the same ratio.
3. Theorem 13: If triangles $\Delta ABC$ and $\Delta A'B'C'$ are similar, then their sides are proportional:
$\frac{|AB|}{|A'B'|}=\frac{|BC|}{|B'C'|}=\frac{|AC|}{|A'C'|}$.
4. Trigonometry Formula for angles $A,\,B,\,C$. When side-lengths are involved, the proofs are required for a triangle with angles $A,\,B,\,C$ opposite sides $a,\,b,\,c$.
$\cos^2A+\sin^2A=1$
$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$
$a^2=b^2+c^2-2bc\cos A$
$\cos(A-B)=\cos A\cos B+\sin A\sin B$
$\cos(A+B)=\cos A\cos B-\sin A\sin B$
$\cos2A=\cos^2A-\sin^2A$
$\sin(A+B)=\sin A\cos B+\sin B\cos A$
$\tan(A+B)=\frac{\tan A+\tan B}{1-\tan A\tan B}$

### Remark to Theorem 12

Incidentally the proof in your textbook is not the whole story. Suppose the line $L$ cuts $AB$ at $D$ such that $|AD|/|DC|$ is not a fraction (e.g. $\sqrt{2}$) — 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 $|AD|/|DC|$ 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.