Building a Smart Study Plan for Paper Two

Trigonometry is the most important part of this paper for marks. But we will start on the likelihood side, with its centrepiece: probability.

One thing to know about this marks data: the past paper questions are categorised as probability, but that label has two topics hiding inside it. Binomials and permutations and combinations do not generally come up on their own — they usually appear in the context of a probability question. So when you study for the probability marks, you are really studying a chain of three topics.

Permutations and combinations are the maths of counting. A permutation counts the ways things can be arranged, where order matters; a combination counts the ways things can be chosen, where it does not. Probability leans on this constantly, because a probability is favourable outcomes divided by total outcomes — and counting those outcomes is exactly this skill.

The binomial topic gives you the "n choose k" numbers — the binomial coefficients from expanding brackets like (x + y)ⁿ. In Leaving Cert probability they power the classic repeated-trials question: the chance of exactly so many successes in so many attempts. The coefficient counts the ways those successes can be arranged among the attempts — combinations again, working inside probability.

That is why the study order inside this block runs binomials first, then permutations and combinations, and then probability itself.

After probability comes statistics.

Probability and statistics do not really overlap in their ideas. You will probably not need to know much about one to answer the other. But here is the catch from the past papers: one question can have two parts, where one part is a statistics question and the other is a probability question — sometimes even using the same data given at the start.

Beyond the probability chain, the likelihood block needs very little: some algebra to get going, and after that you mostly just crunch numbers and learn the concepts. And it does not lead anywhere else — no later topic builds on it. It is a fairly self-contained unit.

Now the most important topic. Trigonometry matters for two reasons. First, it is worth a huge amount of marks on its own. Second, it feeds into the other spatial topics: the line, the circle, and the axioms and theorems. Questions from those topics will not always use trigonometry — but they often will. You will often have to work with the angles made by two lines crossing, or the angles inside a circle, and that needs some trigonometry.

You may notice that trigonometry is also on Paper One. Roughly speaking, the two papers use it in different ways. Paper One is more symbolic trigonometry: working with sine and cosine functions and doing maths on them. Paper Two is more spatial: working out angles from points and lines, and working with them that way.

There are not many prerequisites, but the symbolic trigonometry from Paper One is worth learning first — then you can apply it to the spatial questions here. One thing to note: on Paper One, trigonometry also leaks into other topics in small ways. In differentiation, for example, you may have to differentiate a trigonometric function, get another trigonometric function back, and work with the result. But on the whole, trigonometry is fairly self-contained on Paper One. It is here on Paper Two that it spreads out and matters for other topics.

Next is co-ordinate geometry of the line. The line has a lot of value on its own, but it is also hugely important as a prerequisite for the circle questions.

Students usually find it easier than the circle questions, so it is a good place for everyone to pick up some marks without a ton of study. You will be dealing with things like where lines cross, or the distance between lines or points — and you will need to rely on some trigonometry.

Next is the circle — its full name is co-ordinate geometry of the circle. Like all the topics on Paper Two, it comes up every year. It is worth a good amount of marks, and the questions are usually quite similar from year to year.

One thing to note: this topic is closely tied to the line. They are linked in two ways. First, the concepts relate to each other. Second, questions from both topics are often bundled together.

Last of the spatial topics is the axioms and theorems — as you can see, they have come up in every past paper so far. In general, you will need them to answer questions about lines and circles.

You will not usually be asked to state a theorem or an axiom, or to prove one. Instead, they are tools. You use them to solve bigger problems with lines and circles.

They do need some study, but you will find a lot of them intuitive. You just need to see what tools are available — after that, you will probably pick them up quite quickly, and sharpen the skill by using them in past papers. So you probably should not spend a lot of time trying to memorise the theorems.