Advanced Problems In Mathematics - Colmanweb

STEPWhat is STEP?STEP (Sixth Term Examination Paper) is an examination used by Cambridge University as part of itsprocedure for admitting students to study mathematics. Applicants are interviewed in December, andmay then be offered a place conditional on the results of their public examinations (A-level, InternationalBaccalaureate, etc) and STEP. The examinations are sat in June and offers are confirmed in August whenall the examination results are available.STEP is used for conditional offers not just by Cambridge, but (at the time of writing) also by WarwickUniversity for almost all of its Mathematics offers, and to a lesser extent by some other English universities. Many other university mathematics departments recommend that their applicants practise on thepast papers even if they do not take the examination. In 2015, 4322 scripts were marked, only about1000 of which were written by students holding an offer from Cambridge.The first STEPs were taken in 1987, and there was were specimen papers before that from which someof the questions in this book were drawn. At that time, there were STEPs in many subjects but by 2001only the mathematics papers remained. The examination has been more or less stable over nearly 30years: it has not been blown about by the various fads in the public examinations systems that cameand went during that time.There are three STEPs, called papers I, II and III. Each paper has thirteen questions, including three onmechanics and two on probability/statistics. Candidates are assessed on six questions only.The pure mathematics question in Papers I and II are based the core A-level Mathematics syllabus, withsome minor additions, which is listed at the end of this book. The pure mathematics questions in PaperIII are based on a ‘typical’ Further Mathematics mathematics A-level syllabus (at the time of writing,there is not even a partial core for Further Mathematics A-levels).There is also no core (at the time of writing) for A-level mechanics and statistics, so the STEP syllabusesfor these areas consist of material that a student with a particular interest might have covered. It has tobe said, though, that the statistics questions are very likely to require knowledge of probability ratherthan statistics (for example, there are very few questions on statistical tests of given data). This is becausethe underlying theory of statistics is quite difficult, and therefore unsuitable for examining at this level,whereas the application of statistical tests is rather routine and again unsuitable for examination at thislevel.What is the purpose of STEP?From the point of view of admissions to a university mathematics course, STEP has three purposes. It is used as a hurdle for entrance to university mathematics courses, and sometimes for othermathematics-based courses. There is strong evidence that success in STEP correlates very well

2 Advanced Problems in Mathematicswith university examination results.1 It acts as preparation for the university course, because the style of mathematics found in STEPquestions is similar to that of undergraduate mathematics. It tests motivation. It is important to prepare for STEP (by working through old papers, for example), which can require considerable dedication. Those who are not willing to make the effort areunlikely to thrive on a difficult university mathematics course.STEP vs A-levelA-level2 tests mathematical knowledge and technique by asking you to tackle fairly stereotyped problems. STEP asks you to apply the same knowledge and technique to problems that are, ideally, unfamiliar.Here is an A-level question, in which you follow the instructions in the question:By using the substitution u 2x 1, or otherwise, find 2xdx.(2x 1)2And here, for comparison, is a STEP question, which requires both competence in basic mathematicaltechniques and mathematical intuition. Note that help is given for the first integral, so that everyonestarts at the same level. Then, for the second integral, candidates have to show that they understandwhy the substitution used in the first part worked, and how it can be adapted.Use the substitution x 2 cos θ to evaluate the integral 23/2Show that, for a b, qp(x ab x(x 13 x)12dx )12dx. (b a)(π 3 3 6),12where p (3a b)/4 and q (a b)/2.The differences between STEP and A-level are:1.STEP questions are much longer. Candidates completing four questions in three hours will almostcertainly get a grade 1.2.STEP questions are much less routine.3.STEP questions may require considerable dexterity in performing mathematical manipulations.4.Individual STEP questions may require knowledge of several different areas of mathematics (especially the mechanics and statistics questions, which will often require advanced pure mathematicaltechniques).1Recent studies comparing rank in STEP with rank in first-year Cambridge mathematics examinations reveal a Spearman correlation coefficient of 0.63, which is very high in comparison with other predictors of university examination results.2I use the term ‘A-level’ here as a shorthand for a typical school mathematics examination. The particular examinations you takemay well be very different in style and format but, even if that is the case, I am sure some of what follows will strike a chordwith you.

Stephen Siklos 35.The marks available for each part of the question are not disclosed on the paper.6.There is a lot of choice on STEP papers (6 questions out of 13).7.Calculators are not permitted in STEP examinations.These difference matter, because in mathematics more than in any other subject it is very important tomatch the difficulty of the question with the ability of the candidates. For example, you could reasonablyhave the question ‘Was Henry VIII a good king?’ on a lower-school history paper, an A-level paper, or asa PhD topic. The answers would (or should) differ according to the level. On mathematics examinationpapers, the question has to be tailored to the level in order to discriminate between the candidates: if itis too easy, nearly all candidates will score very high marks; if it is too hard, nearly all candidates willmake little progress on any of the questions.Setting STEPSTEP is produced under the auspices of the Cambridge Assessment examining board. The setting procedure starts 30 months before the date of the examination, when the three examiners (one for each paper,from schools or universities) are asked to produce a draft paper. The first drafts are then vetted by theSTEP coordinator (me!), who tries to enforce uniformity of difficulty, checks suitability of material andstyle, and tries to reduce overlap between the papers. Examiners then produce a second draft, based onthe coordinator’s suggestions. The second drafts are agreed with the coordinator and then circulatedto three moderators (normally school teachers), and to the other examiners, who produce written comments and discuss the drafts in a two-day meeting. The examiners then produce third drafts, taking intoaccount the consensus at the meetings. These drafts are sent to a vetter, who works through the papers,pointing out mistakes and infelicities. The resulting draft is checked by a second vetter and finally bya team of students. At each stage, the drafts are produced camera-ready, using a special mathematicalword-processing package called LaTeX (which is also used to typeset this book).3STEP QuestionsSTEP questions do not fall into any one category. Typically, there will be a range of types on each of thepapers. Here are some thoughts, in no particular order. My favourite sort of question is in two (or maybe more) parts: in the first part, candidates areasked to perform some unfamiliar task and are told how to do it (integration using a given substitution, or expressing a quartic as the algebraic sum of two squares, for example); for the later parts,candidates are expected to demonstrate that they have understood and learned from the first partby applying the method to a new and perhaps more complicated task. Another favourite of mine is the question which has different answers according to the value of acertain number (or parameter). A common example involves sketching a graph whose shape depends on whether a parameter is positive or negative. Ideally, the different values of the parameterare not given in the question, and candidate has to identify them for herself or himself.3It is freely available, so you might like to try it out. You will probably find it good fun to use, but quite time-consuming and notreally suitable for writing out the solutions of STEP problems.

4 Advanced Problems in Mathematics Another good type of question requires candidates to do some preliminary special-case work andthen prove a general result. In another type, candidates have to show that they can understand and use new notation or a newtheorem. Questions with several unrelated parts (for example, three integrals using different techniques)are generally avoided; but if they occur, there tends to be a ‘sting in the tail’ involving putting allthe parts together in some way. Some questions do not rely on any part of the syllabus: instead they might require ‘common sense’,involving counting or seeing patterns, or they might involve some aspect of more elementarymathematics with an unusual slant. Such questions try to test capacity for clear and logical thoughtwithout using much mathematical knowledge (like the calendar question mentioned in the sectionon preparation below, or questions concerning islands populated by toads ‘who always tell thetruth’ and frogs ‘who always fib’). Some questions are devised to check that you do not simply apply routine methods blindly. Forexample, a function might have a maximum value at the end of the interval upon which it isdefined, even though its derivative might be non-zero there. Finding the maximum in such a caseis not simply a question of routine differentiation. There are always questions specifically on integration or differentiation, and ma

The pure mathematics question in Papers I and II are based the core A-level Mathematics syllabus, with some minor additions, which is listed at the end of this book. The pure mathematics questions in Paper III are based on a ‘typical’ Further Mathematics mathematics A-level syllabus (at the time of writing,