QUESTION BY QUESTION: The bane of most students when they first meet it in second year, algebra is integral to the understanding of mathematics, as most solutions will boil down to the successful algebraic manipulation of equations.

PAPER 1

Questions 1 and 2 (Algebra)

The key topics to be familiar with are:

Simplifying Expressions - Fractions, common denominator

Simultaneous Equations - Linear in three unknowns or one linear and one non-linear in two unknowns

Solving Equations - Quadratic - by factoring and use of formula. Cubic - using the Factor Theorem and long division. Equations involving inequalities and the modulus

Quadratic Roots - Properties, Sum = -b/a and product = c/a

Question 3 (Matrices and complex numbers)

This is a popular question with most students. The part of the question relating to matrices is quite straightforward. The complex number part is usually broken into a number of different parts, allowing the student to gain up to 40 marks reasonably quickly. The main areas are as follows:

Matrices

Properties: Addition, subtraction and multiplication of matrices with emphasis on the non-commutative property of matrix multiplication; Equality of matrices; Existence and calculation of the inverse of a matrix

Complex numbers

Properties: Addition, subtraction and multiplication; Division of complex numbers, using the conjugate; Properties of the conjugate and the modulus; The Conjugate Root Theorem, its proof and application to solving polynomials with Real coefficients; Equality of two complex numbers and how this may be applied to the finding of the square root of a complex number

De Moivre's theorem: Polar form using argument and modulus; Statement and proof of De Moivre's theorem; Derivation of trigonometric formula; Calculation of higher powers and roots of a complex number

Questions 4 and 5

These questions cover a variety of different topics, the main ones of which are the following:

Sequences and Series

Students should know the formulae and definitions of arithmetic and geometric sequences and series and be able to handle telescoping series using the method of partial fractions. The use of infinite geometric series in the changing of recurring decimals to fractional form is also required.

Binomial Theorem

Most often the questions require a working knowledge of the general term in the expansion, looking for a particular term or coefficient.

Equations (logarithms and indices)

In recent years, equations involving logs have appeared regularly in question 5. Students need to be familiar with the definition and properties of logarithms and be able to manipulate the equations accordingly. Remember always to check the validity of your solution, not forgetting that you can't get the log of a negative number. Typically, there will be two solutions but one won't make sense. Questions involving indices have appeared either in question 5 or as part of the algebra questions. Normally a change of variable is required, resulting in a quadratic equation that is then solved, followed by a back substitution to yield the final result.

Induction

Many students avoid this, but it is worth noting that the marking scheme on induction is very student-friendly. You can get substantial marks for each step, i.e. substituting n=1, n=k and n=k+1.

Questions 6, 7 and 8 (Calculus)

Calculus is one of the most useful branches of mathematics. It is used in engineering, business, and wherever optimisation or rate of change is sought. Two topics are examined here: differentiation (questions 6 and 7), and integration (question 8). The key topics in these are as follows:

Differentiation

Limits - Finite, Infinite and Trigonometric limits

Methods - Use of First Principles to prove the six results given in the syllabus; Proof of first derivatives of sums, products and quotients; Differentiation by rule, using the Product, Quotient and Chain rule; Implicit Differentiation; Parametric Differentiation

Application - Slope of a curve and the equation of the tangent to the curve; Curve sketching, regions where the graph is increasing or decreasing, asymptotes, turning points, local maximums, minimums and points of inflection; Rates of change; Newton-Raphson method for finding approximate roots of cubic equations. (Learn the formula - it is not always given in the question.)

Integration

Methods - Basic integrals; Indefinite and Definite integrals - Use of substitution (remember to change limits when you change variables)

Application - Area under a curve - Volume of rotation of a curve about the x or y-axis, including the proofs of the formulae for the volume of a sphere, cylinder and cone

PAPER 2

Section A

Questions 1 and 3 (Co-ordinate geometry)

The co-ordinate geometry questions are based on the line and the circle. Neither of these questions can be attempted without total and instant knowledge of the required formulae.

The Line formulae essential checklist is: slope, mid-point, distance, equation of a line, area of a triangle, perpendicular distance from a point to a line (and derivation of such). Conditions for two lines to be parallel or perpendicular are also required, as is the derivation of the tan of the angle between two lines.

In addition, transformation geometry has become a large part of the line question and students should know how to use this to find the image of a point, a line and a line segment using parametric form.

Questions on the circle usually involve three conditions, which give rise to three equations, which can then be solved simultaneously. Also, you should know how to find the equation of a tangent to a circle at a point on the circle (and derivation of same). Finally, you should be able to take the equation of a circle in parametric form and transform it into Cartesian form, clearly indicating the centre and radius.

Question 2 (Vectors)

This question is divided into two main parts. The first requires the expression of one vector in terms of another using a diagram (with pathways) from one point to another. The second expresses vectors on a plane using a pair of orthonormal vectors i and j . Students need to be able to add, subtract and set equal vectors so expressed and also find the related perpendicular vector. Vector multiplication is defined by means of the dot product, which yields a scalar and may be used to find the angle between two vectors.

Questions 4 and 5 (Trigonometry)

A good understanding of trigonometry is important not only for these two questions but also for many other questions, most notably those on calculus on paper 1. Before beginning this question open page 9 of the log tables, as it contains most of the information you'll need.

The questions come in four main types:

trigonometric limits, usually asked as a part (a);

proving identities and manipulation of formulae;

solving equations involving trigonometric functions;

solving triangles using the sine and cosine rules.

Questions 6 and 7 (Probability and Statistics)

These questions are based on four main concepts:

Counting: Involves permutations and combinations, and looks at the number of ways a certain objective can be achieved and whether the order in which items are selected is significant.

Probability: Builds on the idea of counting, and is defined as the number of outcomes of interested divided by the number of possible outcomes.

Difference equations: The solving of such equations using the associated characteristic quadratic equations and the proof of the theorem which justifies this approach.

Statistics: Involves the calculation of the mean and standard deviation. Tends to be more theoretical rather than simple calculation.

Section B - Option topics

For their option question, more than 90 per cent of students take question 8 (further calculus); a small number take question 9 (further probability); and the smallest percentage opts for question 10 (groups) or question 11 (further geometry).

Question 8 (Further calculus)

This requires very little additional theory and divides neatly into four concise units.

Maximum and minimum type questions: Read the question carefully to determine what it is you want to optimise. Write out what information you are given and draw a picture if necessary. Use what you're given to get what you want in terms of one variable. Differentiate, set equal to zero, and solve. (Does the significance (max or min?) of the first derivative being equal to zero for certain x need to be tested by examining the second derivative?)

Integration by parts: a formula driven substitution for dealing with particular integrals. This formula can be found in the log tables and the only decision needed is to decide which part of the integrand is u and which part is dv (formula given on page 42 of tables). A maximum of two steps is required.

General rule: Let u = LIATE (in that order) and dv = (what's left) dx where L = Logs, I = Inverse trig functions, A = Algebraic powers of x, T = Trigonometric functions, and E = Exponential functions.

The ratio test: A formula used to test the convergence of infinite series and possibly to find the range of values of x for which a series is convergent.

Maclaurin series expansion: Taylor's Theorem in the maths tables (page 42) can be adapted for this formula. Simply replace the h by x and the x by 0.