QUESTION BY QUESTION: This question tests basic arithmetic skills. You can use your calculator, but also do a quick mental check on your answer.

PAPER 1

Question 1 (Arithmetic and money)

You need to be able to calculate the following items:

- percentages and how they apply to profit or loss, compound interest and percentage-error questions;

- ratios and exchange rates involving the euro and other currencies, which may be subject to bank charges and commissions;

- calculations involving scientific notation;

- calculations involving speed, time and fuel consumption.

Questions 2 and 3 (Algebra)

Most questions take the form of the solving or simplification of an equation or inequality. You must be proficient in the solving of equations that will appear in three basic forms:

- simultaneous equations, either both linear or one linear and one non-linear, in two variables;

- fractional equations in one variable, which will need to be simplified before solving;

- cubic equations, which will require first "subbing in", followed by long division to find the other roots and factors. Rules of indices and manipulation of formulae questions will appear also.

Question 4 (Complex numbers)

The questions from year to year are very similar. Each year, this single question manages to examine almost every aspect of complex numbers on the course.

You must be familiar with the following:

- operations with complex numbers (addition, subtraction, multiplication, division and equality);

- Argand diagram and modulus;

- complex equations.

Question 5 (Sequences and series)

Like most other questions on the first paper, a certain proficiency with algebra is required to answer many of the harder questions on sequences and series.

This is primarily a formula-driven question. It is essential to know how to find the common difference (d) and the common ratio (r) in the sentence. You must familiarise yourself with the formulae for the nth term and the sum of the first n terms for both arithmetic and geometric sequences and series.

Questions 6, 7 and 8 (Functions and differentiation)

The final three questions on this paper are based on differentiation and its application to functions.

You need to be able to differentiate basic functions, up to quadratics, using first principles and be able to differentiate more complex functions using the product, quotient and chain rules.

Applications of differentiation will include:

- finding the slope and equation of a tangent to a curve;

- finding the turning points of a curve in the form of maximum and minimum points;

- using differentiation to find the speed and acceleration of a particle given its displacement (distance travelled) as a function of time.

Graphing of functions also play an important part in this question and you may be asked to graph a function (or possibly two functions using the same axes) by first constructing a table (or tables). For greater accuracy and clarity, always use the graph paper provided. A graph of a periodic function may also be given, and from this the period and range can be found.

PAPER 2

Section A

Question 1 (Mensuration)

This is commonly referred to as the area and volume question. The majority of the formulae required may be found in the opening pages of the log tables.

The questions often involve compound bodies consisting of cones, cylinders, spheres and hemispheres.

Part (b) might ask you, for example, to use Simpson's rule to find the area of an irregular shape. The formula can also be found in the tables but it is perhaps easier to number the vertical measurements and use the more friendly form of the rule:

Area = h/3 [ first + last + 2 (odds)

+ 4(evens) ]

Questions 2 and 3 (Co-ordinate geometry)

The co-ordinate geometry questions are based on the line and the circle and you must know the required formulae by heart. Note that they are not given in the exam.

For the line an essential checklist is:

- slope, mid point, distance, equation of a line and area of a triangle;

- the condition for lines to be parallel or perpendicular is required, as is the intersection of two lines (using simultaneous equations) and the intersection of a line and the x and/or y-axis;

- finding the slope of the line from the equation of the line.

You need to be able to write the equation for a circle in standard form and to read off the centre and radius.

A common question is to find the equation of a tangent to a circle.

You will need to be able to find the point(s) of intersection of a line and a circle.

You should be able to test to see if a point lies inside or outside a circle.

Question 4 (Geometry theorems)

You will be asked to prove one of the 10 theorems listed in the syllabus.

One other part of this question will be a very simple application of one of these theorems, no difficult "cuts", as they are known.

In another part of this question you will use a couple of these theorems to find lengths and areas of images under the type of transformation known as an enlargement.

Question 5 (Trigonometry)

A number of the key formulae that are required are given in the maths tables; some others have to be learned.

In practice, you will find that a lot of the work you do in answering questions on trigonometry involves a calculator.

So you should learn how to use the trigonometric buttons on your calculator long before the Leaving Cert exam.

Trigonometric functions are used to calculate unknown sides and angles, first in right-angled triangles, then in any triangle - you must know how to apply the sine rule and the cosine rule.

You may be required to find the area of a triangle (see page six of the tables).

Question 6 (Probability)

There is no algebra involved, just adding, multiplying and dividing whole numbers.

Many of these calculations can be performed on a calculator.

This is very often a short enough question.

There are usually parts on arrangements, choosing and probability itself. All of these ideas involve counting techniques, i.e. counting the number of outcomes that are possible when an experiment is performed. The trick is to analyse the experiment carefully and logically, before writing any numbers down on paper.

Make sure you can apply the Principle of Counting (e.g. if you can make a journey outwards in three different ways and the journey back in two different ways, then you can make the entire journey in 3 x 2 = 6 different ways.)

Permutations and combinations. (Is the order in which items are selected significant? If YES, it is a permutation; if NO, it is a combination.)

Probability: defined as the number of outcomes of interest divided by the number of possible outcomes.

Question 7 (Statistics)

There is seldom a year when a graph of some sort is not required. Many other parts of questions can be tackled by constructing standard tables.

Note the following terms.

- The histogram and the cumulative frequency curve are used to display data.

- The mean and the median are two ways of measuring the average.

- The standard deviation and the interquartile range measure the spread, or dispersion, of the data.

Section B

(Option Topics)

Question 8 (Further geometry)

In this option topic there are four different and more challenging theorems (the first with a corollary), the proofs of which have to be learned, and which may be examined.

Cuts of any standard, easy or hard, can be asked. To be able to tackle these, you have to have all the theorems at your fingertips, and be particularly good at deciding

which theorem to use in any given question.

Question 9 (Vectors)

In this question, there will be parts on general vectors, i.e. vectors without any reference to a co-ordinate system. There will also be questions on vectors in the familiar ij plane from co-ordinate geometry.

You must be able to add and subtract vectors, and multiply a vector by a scalar and multiply vectors using the dot (scalar) product.

Question 10 (Further series)

You have studied arithmetic and geometric sequences and series as abstract mathematical quantities for question 5 on Paper 1. But they do have a number of practical applications.

In this option topic, you are required to deal with the applications of these series.

You are also required to study the binomial series for this question.

Question 11 (Linear programming)

You are required to express the information given as two linear inequalities, which are then plotted resulting in a region in the first quadrant.

Note the co-ordinates of the vertices of this region - use simultaneous equations in the case of one vertex (corner).

An item, typically profit or income, is then maximised over this region by substituting the co-ordinates of the vertices (corners) into the profit expression.

Given a region marked on a diagram, you may be required to write down a number of inequalities which define the region.