MATHEMATICS REVISION: what you need to know

What we expect you to know very well is a selection of topics from the A-level syllabus, as follows. If youwonder why such key topics as differentiation and integration and some others are not included, this is becausewe begin by studying these in-depth, so will go over the basics in class. The following are the A-level topicswhich we won’t be revising in October, hence the need for you to refamiliarize yourself.

They go under two broad headings:

1. Algebra/GeometryQuadratic Equations. Algebraic Fractions. Factorization, completing the square. Simultaneous equations.

Inequalities. Manipulation of surds. Polynomials: factorization, division, remainder/factor theorem. Indicesand their laws. Exponential/logarithm function and their graphs. Laws of Logarithms. Solution of ax = b.Modulus function and graph. Function and Inverse, and their graphs. Partial Fractions with/without repeatedlinear factors. Binomial expansion. Factorial notation.

2. TrigonometrySine/cosine law. Area of triangle. Degree vs. Radian. Arc length, area of sector of circle. Sine/Cosine/

Tangent and their reciprocal functions: graphs, symmetries, periodicity. Trigonometric identities, including:

sinx

cosx= , sin2 x+ cos2 x = , sec2 x = 1 + . . . , sin 2x = , sin(A+B) =

Solution of simple trigonometric equations, e.g. 2 sin2 x+ 5 cosx = 4. Inverse functions arcsin,arcos and arctan,their graphs. Sin/cos/tan of special angles: π/2, π/3, π/4, π/6; similarly for other quadrants.

Please note that the above is not an exhaustive list, but for each topic indicates the level of sophistication ex-pected. Here is the OCR board A-level syllabus:

OCR-Mathematics

On Pgs 34-46 is the key information: the topics for the Core modules C1-C4, which you should know very well.Other exam boards will have a syllabus which varies only slightly from this one.

The remaining topics, especially differentiation and integration will be covered in the first term, but we will goover the basics quickly and deepen them. It is recommended that you revise these as well.

We begin the first term with a topic not covered in A-level mathematics: complex numbers. If you’ve neverseen it before, it might be beneficial for you to have a look at complex numbers before you arrive. The way wedo mathematics at university differs from the style you are used to in that we go through material quite quickly.In the first term a lot of it is revision, but old topics are deepened, and when new topics are covered, we expectyou to absorb them rapidly, which can be a challenge. Some preparation in September will help you with thetransition to university in the first term.

PTO

1. Here is a very useful self-study resource: Mathematics Materials . Check out the HELM pages there.This is ”Helping Engineers Learn Mathematics” and includes an introduction to all the mathematics youwill need, and some fairly high-level material as well, covering everything from the most basic late-GCSEmaterial, through the entire A-Level Maths and Further Maths contents, and most of the Mathematics youwill do at university. It covers a great deal of material and you will be able to use it throughout your firstyear (and part of your second year) for extra practice in Mathematics. To prepare for October, here’s a listof topics and subtopics you should look at, with reference to the relevant workbooks(WKB):

(a) Algebra: Rational expressions (partial fractions, polynomial division, graphs: WKB 1,2,3.

(b) Trigonometry: all topics, WKB 4.

(c) Functions and Graphs: General (composite, inverse, odd/even), Particular( modulus, Heaviside),Hyperbolic and their inverses. WKB 2,6.

(d) Differentiation: Techniques (Product/Quotient/Chain rule, miscellaneous examples, implicit func-tions), Applications (Stationary points, points of inflexion). WKB 11,12.

(e) Integration: Techniques( by parts, by substitution, partial fractions, improper integrals), Applications(Area between curves). WKB 13,14.

(f) Permutations and Combinations: Binomial Theorem. WKB 16.

(g) Complex Numbers: Arithmetic, Modulus and Argument, De Moivre’s Theorem: all subtopics in eachcase. Highly recommended. WKB 10.

Each workbook is a series of booklets for subtopics, but the index for the whole workbook is always inthe first booklet so you can use this to navigate. Familarize yourself with this marvellous resource, it willprove to be useful over the next two years, not just months.

2. Another resource is Metric Maths . It is excellent for revision and higher-level topices and we wouldencourage you to spend some time here. All the above topics, under the subheadings as listed, are availableon Metric Maths, an online tool which includes some theoretical background for all the topics covered, and,crucially, a large supply of questions for you to try, enter the answer online and get immediate feedback.This last part is what makes the resource very attractive: as much practice as you want or can handle!

You may notice that, as with HELM, Metric covers topics that go all the way into second-year universitymaths: get used to the format now and benefit throughout your years of study.