Paper I

There are eight questions on Paper I each worth 50 marks. The most popular of these are the two algebra questions (Questions 1 & 2), the matrices and complex numbers question (Question 3), the two differentiation questions (6 & 7) and the integration question (Question 8). Questions 4 and 5 are less popular. Question 4 is usually centred on sequences and series, but be wary. Question 5 is an algebra question centred, for the main part, on logs, binomial theorem and proof by induction.

With regard to the algebra questions, you are almost certain to be asked about both cubics (factors, solving and cubic related identities) and quadratics (the b² - 4ac criteria, the α ,β story and comparison of coefficients). Bear in mind that quadratics and cubics are central themes on your course and questions are always asked on both. Remember also that it is quite some time since abstract inequalities have appeared in Q2 on Paper 1. The last time was as Question 2(C) 2000, so they are well due their turn as are indices in the form of U_n terms (see for example Q2 1999) which have also not been seen as part of Q1 or Q2 for quite some time. Some aspects of these did appear in Q4 2005. Equations with indices are due (see for example Q2C 2003).

Question 3 is on matrices and complex numbers. Make sure you know how to convert complex numbers to polar form and all of the applications of these numbers in such form. In view of the tendency towards the mixing of questions, be aware of the possible inclusion of the binomial expansion, Pascal's triangle and De Moivre's Theorem in the solving of trigonometric identities.

For the differentiation questions, make sure that you understand rates of change and, in particular, relative rates of change using the chain rule. Relative rates of change are long overdue and can prove difficult (see Q6C 1996). Turning points and/or points of inflection are invariably on the paper so do not go into the exam without knowing the criteria for these and do know how to diagnose the number of real roots of a cubic equation using turning points. Make sure you can prove the rules of differentiation. Watch out for differentiations of functions like y = 2^x and y = X^x. If you want more to chew on, ask yourself if you can prove that the derivative of a constant is zero or that the derivative of e^x is e^x. Do not forget that continuous periodic functions are still on the course and you should be able to find the period and range of these functions. Since the maths course was changed in 1994, the period and range of a function have never been asked on an honours paper.

With regard to the integration question, remember the circular substitution has come up only once since 1994 (Q6(C),1999) and volumes of integration are long overdue on this paper. If there is a question on volume or area of integration, it is a very good idea to draw a proper sketch of the volume or area in question, find all points of intersection and state in English or pictorially first the volume or area you are trying to calculate before actually stating it in terms of the integrals you will use to do the calculation. Watch out for questions involving area calculations between a curve and the y-axis.

Paper II

Paper II comes in two sections – Section A and Section B. Section A has seven questions, each worth 50 marks and five of which must be attempted. Section B is the option question. There are four questions in this section, each worth 50 marks and only one of these questions must be attempted.

With regard to Section A the most popular selections are Circle (Q1), Vectors (Q2), Line (Q3), the two trigonometry questions (Q4 and Q5). Some students tend to avoid the questions on probability and statistics and this is probably symptomatic of the fact that students either excel at these topics or do not get the hang of them at all.