Exam structure and timing

Each paper lasts 150 minutes, with six questions to be answered and 300 marks available.

Allow 20-23 minutes per question, with 10 minutes at the start to read the paper.

In order to gain an overall pass mark, you need to obtain 240 marks out of a possible 600, or 120 marks on each paper.

It is vital to attempt six questions if you hope to maximise your grade, so move on to your sixth question and complete parts (a) and (b), if possible, rather than struggle over the last few lines of part (c) on your fifth question.

Try not to leave any blanks, as the examiner cannot give marks if there is nothing there.

Division of questions

Each question is divided into three parts, (a), (b) and (c), with 30 marks out of 50 going for parts (a) and (b). This in effect means that it is possible to achieve a C2 grade on the first two parts, with the higher grades being reached on the successful completion of part (c).

The key then is to concentrate your study initially on these first two parts and they will in turn provide you with a good foundation, which will allow you to attempt part (c).

Paper 1

(answer six from eight questions)

Arithmetic and money

Question 1

This question tests basic arithmetic skills. You can use your calculator, but also do a quick mental check on your answer. You need to be able to calculate the following items:

-Percentages and how they apply to profit, loss and compound interest questions

-Ratios and exchange rates involving the different currencies, which may be subject to bank charges and commissions

A popular question is the calculation of net take-home pay, subject to tax-free allowance and various tax bands.

Algebra

Questions 2 and 3

Most questions will eventually boil down to the solving or simplification of some equation. You must be proficient in the solving of equations that will appear in three basic forms:

-Simultaneous equations, both linear and non-linear, in two variables

-Rational equations in one variable, which will often 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.

In addition to this, some of the questions in previous years have examined the rules and manipulation of indices as a part (c).

Complex numbers

Question 4

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

Sequences and series

Question 5

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. 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.

Functions and differentiation

Questions 6, 7 and 8

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 as a function of time.

Graphing of functions also play an important part in this question and you may be asked to construct a table and hence graph a given function. 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

(answer five from seven questions)

Mensuration

Question 1

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 and spheres. 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 + 4(evens) + 2(odds)

Co-ordinate geometry

Questions 2 and 3

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 or y-axis.

You need to be able to write the equation for a circle in standard form and to read off the centre and radius. The finding of the tangent to a circle at a point on the circle is often asked, as is the finding of the image of a circle under a central or axial symmetry or translation.

Geometry theorems

Question 4

One of the 10 theorems you have learned will be examined in question 4. 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.

Trigonometry

Question 5

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 also be familiar with trigonometry of the triangle and circle (note: both the sine rule and cosine rule are on page 9 of the maths tables).

Probability

Question 6

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.

Statistics

Question 7

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 and 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

(answer one from four questions)

Further geometry

Question 8

In this option topic there are five different and more challenging theorems whose proofs 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.

Vectors

Question 9

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, subtract and multiply these vectors.

Further series

Question 10

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.

Linear programming

Question 11

You need to be able to translate the "English" into two linear equations, which are then plotted and the point of intersection is found using simultaneous equations.

The equations themselves will be in the form of an inequality and the correct side of the individual lines must be found which then gives a region in the first quadrant. An item, typically profit or income, is then optimised over this region by substituting the co-ordinates of the corners into the equation.