There are two papers, each of two-and-ahalf hours duration. Paper 1 is made up of eight questions, and you must do six of these. Paper 2 is split into two sections: section A (answer five out of seven questions) and section B, the option section (answer one out of four questions).
Division of questions
The three-part division of questions means that consistent work over the course is usually well rewarded. Sixty per cent of the marks (or a C2) is possible on the (a) and (b) parts alone, and the higher grades go to those who answer a number of part (c)'s correctly. Concentrate your revision, therefore, on the first two parts, especially as a good understanding of these will give you a good grounding in the concepts you need for answering part (c) of each question.
Timing
Timing is one of the keys to success. Each paper is 150 minutes long, allowing you 10 minutes to read the paper and 20 to 23 minutes for each question. Try to get used to doing three questions in an hour, as this will help you to pace yourself in an exam without keeping your eyes glued to the clock. Also, it is important to get down to work quickly, as "minutes are marks". It is of more benefit to move to the sixth question and complete parts (a) and (b) than it is to waste time struggling on the last line of part (c) in the fifth question.
Paper 1
(answer six from eight questions)
Algebra
Questions 1 and 2
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. The key topics to be familiar with are:
Simplifying expressions
- Fractions, common denominator
Simultaneous equations
- Linear in three unknowns and 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
Matrices and complex numbers
Question 3
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 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
Calculus
Questions 6, 7 and 8
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
- 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, areas of increase and decrease, asymptotes, turning points, local maximums, minimums and points of inflection
- Rates of change
- Approximating the root using the Newton
-Raphson formula
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
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 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. Students should also be familiar with the derivation and proof of the properties of the binomial 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
Most 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.
Paper 2
Section A
(answer five from seven questions)
Co-ordinate geometry
Questions 1 and 3
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), and the internal and external divisors of a line segment. 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 both the equation of a tangent to a circle at a point on the circle and from a point outside the circle. 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.
Vectors
Question 2
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.
Trigonometry
Questions 4 and 5
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, typically three dimensions, using the sine and cosine rules
Probability and statistics
Questions 6 and 7
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 or not order is important or not.
Probability:
An extension to counting, which divides the number of acceptable outcomes by the total number of possible outcomes.
Difference equations:
The solving of a second order recursive equation using the associated characteristic quadratic equation.
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).
Further calculus
Question 8
This requires some additional theory and divides neatly into four concise units:
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.
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.
MacLaurin series expansion:
Taylor's Theorem in the log tables can be adapted for this formula. Simply replace the h by x and the x by 0.
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.
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.
