### What is STEP?

STEP (Sixth Term Examination Paper) is an examination used by Cambridge University as part of its procedure for admitting students to study mathematics. Applicants are interviewed in December, and may then be offered a place conditional on the results of their public examinations (A-level, International Baccalaureate, etc) and STEP. The examinations are sat in June and offers are conﬁrmed in August when all the examination results are available.

STEP is used for conditional offers not just by Cambridge, but (at the time of writing) also by Warwick University 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 the past papers even if they do not take the examination. In 2015, 4322 scripts were marked, only about 1000 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 some of the questions in this book were drawn. At that time, there were STEPs in many subjects but by 2001 only the mathematics papers remained. The examination has been more or less stable over nearly 30 years: it has not been blown about by the various fads in the public examinations systems that came and went during that time.

There are three STEPs, called papers I, II and III. Each paper has thirteen questions, including three on mechanics 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, 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, 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 syllabuses for these areas consist of material that a student with a particular interest might have covered. It has to be said, though, that the statistics questions are very likely to require knowledge of probability rather than statistics (for example, there are very few questions on statistical tests of given data). This is because the 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 this level.

### 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 other
mathematics-based courses. There is strong evidence that success in STEP correlates very well with
university examination results.
^{1} - It acts as preparation for the university course, because the style of mathematics found in STEP questions 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 are unlikely to thrive on a difficult university mathematics course.

### STEP vs A-level

A-level^{2}
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

And here, for comparison, is a STEP question, which requires both competence in basic mathematical techniques and mathematical intuition. Note that help is given for the ﬁrst integral, so that everyone starts at the same level. Then, for the second integral, candidates have to show that they understand why the substitution used in the ﬁrst part worked, and how it can be adapted.

Use the substitution $x=2-cos\mathit{\theta}$ to evaluate the integral

$${\int}_{3\u22152}^{2}{\left(\frac{x-1}{3-x}\right)}^{\phantom{\rule{0.3em}{0ex}}\frac{1}{2}}\phantom{\rule{0.3em}{0ex}}\mathrm{d}x.$$

Show that, for $a<b$,

$${\int}_{p}^{q}{\left(\frac{x-a}{b-x}\right)}^{\phantom{\rule{0.3em}{0ex}}\frac{1}{2}}\phantom{\rule{0.3em}{0ex}}\mathrm{d}x=\frac{\left(b-a\right)\left(\pi +3\surd 3-6\right)}{12},$$where $p=\left(3a+b\right)\u22154$ and $q=\left(a+b\right)\u22152$.

The differences between STEP and A-level are:

- 1.
- STEP questions are much longer. Candidates completing four questions in three hours will almost certainly 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 mathematical techniques).
- 5.
- 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 to match the difficulty of the question with the ability of the candidates. For example, you could reasonably have the question ‘Was Henry VIII a good king?’ on a lower-school history paper, an A-level paper, or as a PhD topic. The answers would (or should) differ according to the level. On mathematics examination papers, the question has to be tailored to the level in order to discriminate between the candidates: if it is too easy, nearly all candidates will score very high marks; if it is too hard, nearly all candidates will make little progress on any of the questions.

### Setting STEP

STEP 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 the STEP coordinator
(me!), who tries to enforce uniformity of difficulty, checks suitability of material and style, and
tries to reduce overlap between the papers. Examiners then produce a second draft, based on the
coordinator’s suggestions. The second drafts are agreed with the coordinator and then circulated to 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 into
account 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 by a team of students. At each stage, the drafts are produced camera-ready, using a
special mathematical word-processing package called LaTeX (which is also used to typeset this
book).^{3}

### STEP Questions

STEP questions do not fall into any one category. Typically, there will be a range of types on each of the papers. 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 are asked 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 part by 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 a certain 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 parameter are not given in the question, and candidate has to identify them for herself or himself.
- Another good type of question requires candidates to do some preliminary special-case work and then prove a general result.
- In another type, candidates have to show that they can understand and use new notation or a new theorem.
- 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 all the 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 elementary mathematics with an unusual slant. Such questions try to test capacity for clear and logical thought without using much mathematical knowledge (like the calendar question mentioned in the section on preparation below, or questions concerning islands populated by toads ‘who always tell the truth’ and frogs ‘who always ﬁb’).
- Some questions are devised to check that you do not simply apply routine methods blindly. For example, a function might have a maximum value at the end of the interval upon which it is deﬁned, even though its derivative might be non-zero there. Finding the maximum in such a case is not simply a question of routine differentiation.
- There are always questions speciﬁcally on integration or differentiation, and many others (including mechanics and probability) that use calculus as a means to an end.
- Graph-sketching is regarded by mathematicians as a fundamental skill and there are nearly always questions that require a sketch.
- Basic ideas from analysis, such as
$$0\le \mathrm{f}\phantom{\rule{1.00006pt}{0ex}}\left(x\right)\le k\Rightarrow 0\le {\int}_{a}^{b}\mathrm{f}\phantom{\rule{1.00006pt}{0ex}}\left(x\right)\mathrm{d}x\le \left(b-a\right)k$$
or

$${\mathrm{f}\phantom{\rule{1.00006pt}{0ex}}}^{\prime}\left(x\right)>0\Rightarrow \mathrm{f}\phantom{\rule{1.00006pt}{0ex}}\left(b\right)>\mathrm{f}\phantom{\rule{1.00006pt}{0ex}}\left(a\right)\text{for}ba$$or the relationship between an integral and a sum often come up, though knowledge of such results is never assumed — candidates may be told that they can use the result without proof, or a sketch ‘proof’ may be asked for.

- As mentioned above, questions on statistical tests are rare, because questions that require real understanding (rather than ‘cookbook’ methods) tend to be too difficult. More often, the questions in the Probability and Statistics section are about probability.
- The mechanics questions normally require a ﬁrm understanding of the basic principles (when to apply conservation of momentum and energy, for example) and may well involve a differential equation. Projectile questions are often set, but are never routine.

### Advice to candidates

#### First appearances

I am often asked whether STEP is ‘difficult’. Of course, it depends on what is meant by ‘difficult’; it is not difficult compared with the mathematics I do every day. But to be on the safe side, I always answer ‘yes’ before explaining further.

Your ﬁrst impression on looking at a STEP paper is likely to be that it does indeed look very difficult. Don’t be discouraged! Its difficult appearance is largely due to it being very different in style from what you are used to.

At the time of writing, a typical A-level examination lasts 90 minutes and contains 10 compulsory questions. That is 9 minutes per question. If you are considering studying mathematics at a top university, it is likely that you will manage to do them all and get them nearly all right in the time available. A STEP examination lasts 3 hours, and you are only supposed to do six questions in three hours. You are very likely to get a grade 1 if you manage four questions (not necessarily complete); that means that each question is designed to take 45 minutes. If you compare a 9 minute question with a 45 minute question, of course the 45 minute question looks very hard!

You may be put off by the number of subjects covered on the paper. You should not be. STEP is supposed to provide sufficient questions for all candidates, no matter which mathematics syllabus (at the appropriate level) they have covered. It would be a very exceptional candidate who had the knowledge required to do all the questions. And there is plenty of choice (6 questions out of 13).

Once you get used to the idea that STEP is very different from A-level, it becomes much less daunting.

#### Preparation

The best preparation for STEP (apart, of course, from working through this excellent book) is to work slowly through
old papers.^{4}
Hints and answers are available for some years, but you should use these with discretion: doing a question with
hints and answers in front of you is nothing like doing it yourself, and you may well miss the whole point of the
question (which is to make you think about mathematics). In general, thinking about the problem is much more
important than getting the answer.

Should you try to learn up areas of mathematics that are not in your syllabus in preparation for STEP? The important thing to know is that it is much better to be very good at your syllabus than to have have a sketchy knowledge of lots of additional topics: depth rather than breadth is what matters. It may conceivably be worth your while to round out your knowledge of a topic you have already studied to ﬁt in with the STEP syllabus; it is probably not worth your while to learn a new topic for the purposes of the exam, though I can think of a couple of exceptions:

- Hyperbolic functions obey simple rules similar to trigonometric functions (in fact, they are the same functions in the complex plane). If you haven’t come across them, you can easily master them in a short time and this will open the door to many questions that would otherwise have been inaccessible to you.
- de Moivre’s theorem (relating to complex numbers) is also very straightforward, though questions requiring de Moivre are less common.

It is worth emphasising that there is no ‘hidden agenda’: a candidate who does two complete probability questions and two complete mechanics questions will obtain the same mark and grade as one who does four complete pure questions.

Just as the examiners have no hidden agenda concerning syllabus, so they have no hidden agenda concerning your
method of answering the question. If you can get to the end of a question correctly you will get full marks whatever
method you use.^{5}
Some years ago one of the questions asked candidates to ﬁnd the day of the week of a given date (say, the 5th of
June 1905). A candidate who simply counted backwards day by day from the date of the exam
would have received full marks for that question (but would not have had time to do any other
questions).

You may be worried that the examiners expect some mysterious thing called rigour. Do not worry: STEP is an exam for schools, not universities, and the examiners understand the difference. Nevertheless, it is extremely important that you present ideas clearly, and show working at all stages.

#### Presentation

You should set out your answer legibly and logically (don’t scribble down the ﬁrst thought that comes into your head) – this not only helps you to avoid silly mistakes but also signals to the examiner that you know what you are doing (which can be effective even if you haven’t the foggiest idea what you are doing).

Examiners are not as concerned with neatness as you might fear. However if you receive
complaints from your teachers that your answers are difficult to follow then you should
listen.^{6}
Remember that more space usually means greater legibility. Try writing on alternate lines (this leaves a blank
line for corrections).

Try to read your answers with a hostile eye. Have you made it clear when you have come to the end of a particular argument? Try underlining your conclusions. Have you explained what you are trying to do? For example, if a question asks ‘Is $A$ true?’ try beginning your answer by writing ‘$A$ is true’ — if you think that it is true — so that the examiner knows which way your argument leads. If you used some idea (for example, integration by substitution), did you tell the examiner that this is what you were doing?

#### What to do if you cannot get started on a problem

Try the following, in order.

- Reread the question to check that you understand what is wanted.
- Reread the question to look for clues – the way it is phrased, or the way a formula is written, or other relevant parts of the question. (You may think that the setters are trying to set difficult questions or to catch you out. Usually, nothing could be further from the truth: they are probably doing all in their power to make it easy for you by trying to tell you what to do).
- Try to work out exactly what it is that you don’t understand.
- Simplify the notation – e.g. by writing out sums explicitly.
- Look at special cases (choose special values which simplify the problem) in order to try to understand why a result is true.
- Write down your thoughts – in particular, try to express the exact reason why you are stuck.
- Go on to another question and go back later.
- If you are preparing for the examination (but not in the actual examination!) take a short break.
^{7} - Discuss it with a friend or teacher (again, better not do this in the actual examination) or consult
the hints and answers, but make sure you still think it through yourself.
BUT REMEMBER: following someone else’s solution is not remotely the same thing as doing the problem yourself. Once you have seen someone else’s solution to a problem, then you are deprived, for ever, of much of the beneﬁt that could have come from working it out yourself.

Even if, ultimately, you get stuck on a particular problem, you derive vastly more beneﬁt from seeing a solution to something with which you have already struggled, than by simply following a solution to something to which you’ve given very little thought.

#### What if a problem isn’t coming out?

If you have got started but the answer doesn’t seem to be coming out, then try the following.

- Check your algebra. In particular, make sure that what you have written works in special cases. For
example: if you have written the series for $log\left(1+x\right)$
as
$$1-x+\frac{1}{2}{x}^{2}-\frac{1}{3}{x}^{3}+\cdots \phantom{\rule{0.3em}{0ex}}$$
then a quick check will reveal that it doesn’t work for $x=0$; clearly, the 1 should not be there.

^{8}A note on the subject of algebra. In many of the problems in this book, the algebra is quite stiff: you have to go through many lines of calculation before you get to an expression recognisably close to your target. Really, the only way to manage this efficiently is to check each line carefully before going on the next line. Otherwise, you can waste hours.

^{9} - Make sure that what you have written makes sense. For example, in a problem which is dimensionally
consistent, you cannot add $x$
(with dimension length, say) to ${x}^{2}$
or to $expx$
(which itself does not make sense — the argument of $exp$
has to be dimensionless). Even if there are no dimensions in the problem, it is often possible to
mentally assign dimensions and hence enable a quick check.
Be wary of applying familiar processes to unfamiliar objects (very easy to do when you are feeling at sea): for example, it is all too easy, if you are not sure where your solution is going, to solve the vector equation $a.x=1$ by dividing both sides by vector $a$; a bad idea.

- Analyse exactly what you are being asked to do. Try to understand the hints, explicit and implicit. Remember to distinguish between terms such as explain/prove/deﬁne/etc. (There is essentially no difference between ‘prove’ and ‘show’: the former tends to be used in more formal situations, but if you are asked to ‘show’ something, a proper proof is required.)
- Remember that different parts of a question are often linked. There may be guidance in the notation and choice of names of variables in the question.
- If you get irretrievably stuck in the exam, state in words what you are trying to do and move on (at A-level, you don’t get credit for merely stating intentions, but STEP examiners are generally grateful for any sign of intelligent life).

#### What to do after completing a question

It is a natural instinct to consider that you have ﬁnished with a question once you have got to an answer. However this instinct should be resisted both from a general mathematical point of view and from the much narrower view of preparing for an examination. Instead, when you have completed a question you should stop for a few minutes and think about it. Here is a check list for you to run through.

- Look back over what you have done, checking that the arguments are correct and making sure that they work for any special cases you can think of. It is surprising how often a chain of completely spurious arguments and gross algebraic blunders leads to the given answer.
- Check that your answer is reasonable. For example if the answer is a probability $p$
then you should check that $0\le p\le 1$.
If your answer depends on an integer $n$,
does it behave as it should when $n\to \infty $?
Is it dimensionally correct?
If, in the exam, you ﬁnd that your answer is not reasonable, but you don’t have time to do anything about it, then write a brief phrase showing that you understand that your answer is unreasonable (e.g. ‘This is wrong because mass must be positive’).

- Check that you have used all the information given. In many ways the most artiﬁcial aspect of examination questions is that you are given exactly the amount of information required to answer the question. If your answer does not use everything you are given then either it is wrong or you are faced with a very unusual examination question.
- Check that you have understood the point of the question. It is, of course, the case that not all exam questions have a point, but many do. What idea did the examiners want you to have? Which techniques did they want you to demonstrate? Is the result of the question interesting in some way? Does it generalise? If you can see the point of the question would your working show the point to someone who did not know it in advance?
- Make sure that you are not unthinkingly applying mathematical tools which you do not fully
understand.
^{10} - It is good preparation for the examination to try to see how the problem ﬁts into the wider context and see if there is a special point which it is intended to illustrate. You may need help with this.
- As preparation for the examination, make sure that you actually understand not only what you have done, but also why you have done it that way rather than some other way. This is particularly important if you have had to use a hint or solution.
- In the examination, check that you have given the detail required. There often comes a point in a
question where, if we could show that $A$
implies $B$,
then the result follows. If, after a lot of thought, you suddenly see that $A$
does indeed imply $B$
the natural thing to do is to write triumphantly ‘But $A$
implies $B$
so the result follows’
^{11}. Unfortunately unscrupulous individuals (not you, of course) who have no idea why $A$ should imply $B$ (apart from the fact that it would complete the question) could, and do, write the exactly the same thing. Go back through the major points of the question making sure that you have not made any major unexplained leaps.