Exam (elaborations) MA 2

I have set out below the answers to the questions you will want to ask before getting down to work. If I have forgotten anything, let me know and I will include it in the next of what I hope will be many editions. Is this booklet suitable for me? The booklet is intended for anyone taking A-level mathematics or the equivalent (Baccalaureat or CSYS, for example) who wants to try something more challenging than A-level style questions. However, many questions involve topics which occur in Further Mathematics syllabuses. Furthermore, even if you are taking Further Mathematics there are sure to be some questions on topics which are not in your particular syllabus since syllabuses differ widely between examination boards. You will therefore have to be selective in your choice of questions. What is the purpose of this booklet? There is considerable feeling amongst university teachers that the present A-levels, though excellent in many ways, fail to equip students with some of the techniques required for a university mathematics course, or indeed for courses such as engineering, physical sciences and economics which depend on mathematics. The first purpose of this booklet is to offer an opportunity for such students to look into a new world of mathematics. The second purpose is to give support to students who are planning to take STEP. It will be especially useful for students in schools which cannot, for lack of time or staff, supply high level teaching in mathematics. What is STEP? STEP is an examination administered by Cambridge Assessment. It is used as a basis for conditional offers in Mathematics by Cambridge and Warwick universities, but the papers are taken by many students who do not hold offers from these universities‘. Anyone can enter and it is my view that good mathematicians should enter. STEP should be regarded as a challenge in the same spirit as the national mathematics competitions and olympiads which are so popular. However, STEP provides a very different sort of challenge from these competitions, since it concentrates on in-depth problems in mainstream (e.g. A-level) mathematics. Where did the questions come from? Mostly from old STEP papers. In fact, the great majority are questions I set myself when I was a STEP examiner. Others were set by my colleagues Tom K¨orner, Michael Potter, Dennis Barden and the Rae Mitchell. However, some of the questions have a long history (maybe all of them: it is sometimes said that there is no such thing as a new mathematics question), so I do not think any of us would claim any special credit for our efforts. I chose mostly my own questions not because I think that they are the best and not because I can do them. (At least, not just for these reasons.) I chose them because I have retained a clear idea of what I was looking for when I set them and this makes the discussion and solution much easier to provide. Will I find the questions difficult? I hope so. If I have provided a set of 43 questions that you can romp through, then I have wasted my time. Do not be discouraged if you are struggling with a question; the most worthwhile things are learnt the hard way. It is important that you adjust your sights. A typical A-level question may take, say, ten minutes and consist of a single guided step (‘Use the substitution x = · · · to evaluate the following integral’). The questions in this booklet are multi-step, and often you have to work through many steps without guidance. If you do four STEP questions reasonably well in the three-hour examination you are doing well. You must keep this comparison firmly in your mind.