Cambridge Entrance Examination – Mathematics for Natural Sciences (1981)

I thought I’d take 5 minutes this lunchtime to add another item to the collection of old examination papers I’ve been posting, as someone asked me about this type of examination via a comment recently. This is the Mathematics paper I took way back in November 1981 for entry the following October to do Natural Sciences. I also took papers in Physics and Chemistry, as well as a General paper. Looking at this after a gap of over 30 years it looks pretty tough. One thing I should point out, though, is that the timing of the paper required us to come back after A-levels for an extra term (“the seventh term”)  at my school, the Royal Grammar School in Newcastle,  to form the “Third Year Sixth” who were all Oxbridge candidates. We were then intensively coached for the entrance examination. You will notice, for example, a couple of questions on this paper relating to group theory, which wasn’t on the A-level syllabus but which we were taught specifically for this examination. Some schools couldn’t offer this specialist teaching so pupils from them were significantly disadvantaged by this form of selection. As it happens, I answered both the (relatively easy) questions on group theory and got in to Cambridge…

Comments on the content and/or difficulty are welcome through the box below!


46 Responses to “Cambridge Entrance Examination – Mathematics for Natural Sciences (1981)”

  1. Bryn Jones Says:

    Ah. The Oxbridge entrance examinations. An issue about which I have opinions.

    My belief is that the Oxbridge entrance exams acted (possibly inadvertently) to discriminate strongly against students who attended certain types of schools.

    A majority of state schools did not have the resources to prepare students for the entrance examinations, or in some cases to offer any support. Applicants to university from most schools applied in the fourth term of their A-level studies, and those who sat the Oxbridge entrance examinations would do so some months before completing their A-level studies. Despite this, the entrance examinations included material that would not be taught in most A level courses. In the case of the mathematics questions in the example here, much of the material is beyond the basic mathematics A level, and some beyond further mathematics. Many schools simply did not have the contact with the Oxbridge system to know how to support their students in preparing for the exams.

    In contrast, a minority of schools understood the system, and had the staff resources to support their students through the process.

    The whole system of university entrance examinations discriminated against a majority of potential candidates. It was a disgrace.

    • Anton Garrett Says:

      Don’t blame Oxbridge. I wanted to do “4th term application” but my headmaster told my parents it was “not school policy”. Frankly I suspect this (grammar) school wanted the fees, whether from parents or the LEA, for another term after A-level.

      Oxbridge were well aware of which applicants were 4th term after O-level and which were 7th term, and took it into account. I know because I was marking the entrance exam another 10 years on.

      • telescoper Says:

        If you look at the rubric you will see that the Section A/Section B structure is intended to allow pre A-level students a wider choice of easier questions. There’s still a question on groups in Section A, though. I did a few of these questions ast night while watching the football. I think Section A is do-able even for someone old and befuddled like me. I may try Section B some time, but not without trepidation…

    • telescoper Says:

      I wonder if you marked my paper?

      • Anton Garrett Says:

        I did it only for one year, and for College Research Fellows like myself it was paid work for which you could ask. The pay for what it turned out to be was lousy. But I’m glad I tasted it from the point of view of the setter rather than the candidate. The marking was boring and I gave myself a small bar of chocolate on the hour and a cup of tea on the half-hour. Initially I felt the responsibility of weighing people’s futures, but after seeing the patterns of mistakes that people made I was able to speed up without compromise. I did check that the candidates I had marked first, while I was still learning the pattern, got from me the same marks that they would have got if they had been lower down the pile of scripts.

        I think I marked the papers taken at the end of 1983, but not more than a year either way, so you (Peter) were already in. Certainly I don’t recall any papers diverting to discuss the merits of Newcastle United Football Club or jazz music.

    • Bryn Jones Says:

      Yes, I very much blame Oxbridge. It is impossible for a single set of examinations to cater for circumstances as different as 7th term candidates who have had many weeks of dedicated coaching, and 4th term candidates who have only glanced at a few past papers in the time they happened to find in the evening after completing their normal homework and who have only studied 60% of the A-level syllabus.

      Yes, it is possible for examiners to make highly laudable efforts to allow for the different circumstances, but I believe the difficulties faced by candidates who received no coaching from schools that had little understanding of university entrance examinations were so great that they would be disadvantaged however much sympathy was shown to them.

      I recall some of my fellow A-level students sitting the Cambridge exam in my comprehensive school. Of the half dozen or so who sat the paper, none got conditional offers: all were rejected. One of them went on to get four A-grade A levels. The door was closed to them. It was the system that was wrong.

      • Anton Garrett Says:

        “It is impossible for a single set of examinations to cater for circumstances as different as 7th term candidates who have had many weeks of dedicated coaching, and 4th term candidates who have only glanced at a few past papers in the time they happened to find in the evening after completing their normal homework and who have only studied 60% of the A-level syllabus.”

        Here is how to do the impossible. Take the average mark of 4th term entrants, the average mark of 7th term entrants, and renormalise each group by their average.

      • Bryn Jones Says:

        But splitting the candidates into two groups would only provide information about the relative performance within in each group, and in the case of 4th-term candidates, in an examination for which they are mostly very poorly prepared.

        This would provide no information about the relative merits of the two groups. It would provide no valid information about how many of the 4th-term group should be given conditional offers of university places relative to the 7th-term group.

      • Anton Garrett Says:

        It might have been better for 4th term and 7th term candidates to take the same exam under the same rules per paper. Then you can renormalise each candidate’s score by the average of the group he was in, either the 4th term or the 7th term group – and compare all candidates. But why do you suppose that Oxbridge is incapable of making genuine allowance for the difference, or of not wishing to?

      • Bryn Jones Says:

        Right. That may have taken out some of the worst injustice in comparing the two groups. However, significant biases would have remained within the 4th-term candidates between those who had received some dedicated coaching for the examination and those who had none.

        I recall that no coaching of any kind was available for university entrance examinations in my old school because of a lack of resources. Candidates were lent some past papers and had to work through them in their own time, if they could squeeze the work between their other studies. They competed against candidates who had received coaching and had been led through past papers.

        I suppose that Oxbridge was (and remains) incapable of making allowance for difference because about 50% of the British students admitted to Oxbridge undergraduate courses attended private schools, whereas only 7% of the general population did. Of those admitted from state-funded schools, a majority came from remaining grammar schools, church schools or other specialist schools. There was (and remains) a strong bias within the system.

        The only fair basis for assessing whether candidates are worthy of admission to a university is a system of examinations understood by all and for which all candidates are taught – and a system which is capable of distinguishing between excellent and outstanding candidates. Any system of specialist examinations will favour those who are given specialist coaching and have support from teachers who understand the system.

      • Anton Garrett Says:

        “However, significant biases would have remained within the 4th-term candidates between those who had received some dedicated coaching for the examination and those who had none.”

        Obviously, but what is Oxbridge supposed to do about that?

        I hope we can agree that the task of Oxbridge is to pick those who will get the best degrees after 3 years, regardless of any other factor. That is not identical with those who do best on the entrance exam, but there is a correlation, and Oxbridge genuinely attempted to take account of the imperfection in the correlation. I don’t see what else it could have done at the time.

      • Bryn Jones Says:

        Oxbridge should have done what all other British universities did at the time: judge people on a universal system of examinations that all candidates for university took on (broadly) equal terms: A levels.

        It is the task of all universities to pick those people who will get the best degrees.

        That the entrance examinations selected a cohort of undergraduates with a majority from private schools shows that it was a total failure. It simply did not function.

      • Bryn Jones Says:

        (Something went wrong with my first attempt to post a comment. Perhaps Peter would care to edit it.)

      • telescoper Says:

        Could your friend also have applied for post A-level entry?

      • Bryn Jones Says:

        I presume my school friend could have applied to Cambridge again the following year. However, he may not have wanted to do so. He may have thought it not worth taking a year out of his career given that many other British universities provide an excellent education. He may not have realised that it was possible. He may have thought that the prejudice within the Oxbridge system against applicants from comprehensive schools might have disadvantaged him the following year. I can’t say.

    • telescoper Says:

      As far as I can see it, the entrance examination was there to try to select candidates who would actually be capable of coping with the courses being taught at Cambridge. They would argue that it was unfair to let lots of students in who were likely to drop out. The choice was then watering down the degree content, or being highly selective. Choosing the latter led to an automatic bias towards students from schools that provided adequate preparation, mostly private ones. I wouldn’t put all the blame on Oxbridge for the system as was, but I am not sorry that the entrance examinations were scrapped.

      • John Peacock Says:

        Unlike Peter, I do think it was a pity that the entrance exam disappeared. This is because it attempted to address Bryn Jones’s critique about an advantage being given to well-drilled public school types, by asking questions that were far from fomulaic fodder: rather than needing a detailed knowledge of all parts of the syllabus, you needed to think. I vividly recall “estimate the power developed by a grasshopper as it jumps” from one of the past physics papers. I’d never seen anything like this before, nor did I encounter such freedom anywhere in the rest of my time at school.

        I think this careful design went a good way to closing the state/non-state gap, and much of the remaining gap was closed in other ways. For example, the papers were marked centrally, and marks returned to the college to which the application was made. But for 4th-term candidates, the exam scripts were also returned, so anyone who turned in a partially brilliant performance due to incomplete preparation could be picked up and judged more on potential. And I’d be astonished if Cambridge academics lacked the sense to make greater allowance for such variance in state school applicants. They undoubtedly do so now (see admissions committees have data on school background and expect a higher standard from candidates from “good” schools.

        So although nothing is perfect, I really can’t see that Bryn Jones is correct in criticising the system so strongly. The twofold problem is that too few able state pupils apply and that some of those who do have such a poor foundation that it can be predicted that they would not cope. I think the days of the entrance exam were better in both these respects, since it was possible to get an objective advanced taste of what the intellectual atmosphere would be. And it gave the university better data than A-level grades; they should bring it back.

      • Bryn Jones Says:

        But the consequences of the entrance examinations show that there were indeed advantages for well-drilled public school types: half of those people admitted to Oxbridge courses attended private schools. That in itself is evidence that system was biased.

        We might imagine this was because privately educated candidates were trained to think in the right way to perform well in the entrance examinations.

        Of course Oxbridge argues that too few candidates applied from state school (and still do). That may well be true, but relatively few state-school candidates applied (and still apply) to Oxbridge because they knew the system is biased against them. They have more important priorities than waste their time throwing themselves on a discriminatory system.

        State school pupils apply to other leading universities, are admitted in large numbers and do very well in their studies.

      • telescoper Says:

        That’s the real point behind Oxbridge admissions exams. If Oxbridge really does take the most capable school kids, why don’t they all get first class degrees?

        The evidence suggests that either the entrance examination didn’t really select the right people or that Oxbridge teaching is basically incompetent….

        I don’t think A levels are good enough to select on either. What we should have is a national university entrance examination, set by universities rather than by exam boards.

      • John Peacock Says:

        Bryn: where is the evidence for your assertion that Oxbridge is biased against state school pupils? The fact that 50% of the intake comes from private schools does not prove this. The Grauniad article I linked to paints a picture of people trying as hard as they can to admit state pupils who could actually cope with the course. I don’t believe you or I would react very differently given the set of people who apply. The solution is for more good state pupils to apply. The reason they don’t could well be because they have a fear that the system is prejudiced against them. If this is really true, let’s see some proof; otherwise, it’s not a helpful message to spread.

      • Anton Garrett Says:

        Private schools give a better education and it is hard to catch up later on, so Oxbridge is not being discriminatory. It is not Oxbridge’s role to engage in social engineering that over-corrects for the inadequacies of the public education system at the expense of people who slave their guts out and sacrifice their standard of living to give their children a good start. That some people cannot afford even that is a tragedy but is not Oxbridge’s fault.

      • John Peacock Says:

        Anton: we’re not discussing social engineering, but it was suggested above (and I agree) that Cambridge ought to select the students whose achievement it expects will be the greatest at the end of study. It’s rather obvious that this objective will not be attained if you select purely on performance at entry; rather, you should correct your prediction of the final outcome based on your knowledge of the level of schooling received. This is a simple Bayesian approach, and they like Bayes at Cambridge. It’s my understanding that such adjustments are made, as one would hope and expect.

      • Bryn Jones Says:

        Yes, it is true that a 50% intake from private schools does not
        alone prove that the Oxbridge admissions system is biased: we could conceive a society in which private education was so superior to state education that the small minority of children educated privately were far better prepared for university than those educated in the state sector. I do not, however, believe that is the case in Britain. Private education may in general be better, but many state-educated students achieve outstanding results, equal to the best from private schools.

        A central problem is that the Oxbridge admissions system rejected candidates from state school who went on to get better A level results than many let in. In my own cohort at school, one of my fellow students sat the Cambridge admissions examinations, was rejected, and then went on to achieve four A-grade A levels. Yet candidates from other schools were admitted with only three As, or with two As and a B.

        There should have been a level playing field, and the Oxbridge entrance examinations achieved something very far from that.

      • Anton Garrett Says:

        Bryn: Thank to the grammar schools my mother came from the poorest part of Sheffield to Girton College, Cambridge, where at a doubtless relatively austere but to her sumptous postwar matriculation dinner she found the young women next to her discussing the servant problem. She was from the serving classes. That’s why I don’t find it easy to recognise the Oxbridge of which you speak. When you use the word ‘prejudice’, are you suggesting any reason other than accident for its failure to identify with total accuracy the candidates who might get the best degrees after 3 years?

        NB Just about everyone has been taking a gap year for quite a while now.

      • telescoper Says:


        In some subjects a gap year is commonplace, but it’s actually rather rare in science subjects.

        I also got to Cambridge via a grammar school to which I had won a scholarship at 11+. I was grateful to have had the chance to get in, given that I was – and remain – the only person from my family ever to have gone to university.

        There was however a sizable cohort of upper-class twits at Magdalene which made me wonder if people like me were just token gestures…


      • Anton,

        Yes, many good candidates from grammar schools did get into Oxbridge. The grammar schools were able to prepare their pupils for entrance examinations, at the cost of rather poor educational provision to the majority of the secondary school pupils who went to secondary modern schools. Grammar schools had the resources to devote to this, and their teachers had sufficient contact with the Oxbridge system to know how to prepare their students.

        Society changed and it was deemed unacceptable to blight future lives by sending less-gifted children to dump schools. The number of universities increased greatly. It became no longer possible for the majority of schools to devote resources to understanding the particular entrance mechanisms of just two universities, and to provide coaching to candidates to just two.

        Assessment of candidates for university courses should have been on the basis of universally understood examinations, either A levels or some national equivalent, not through individual universities devising their own examination systems that were understood only by a small number of teachers who had the right contacts.


      • Anton Garrett Says:


        You wrote: “It became no longer possible for the majority of schools to devote resources to understanding the particular entrance mechanisms of just two universities”

        Nobody asked the schools to do that. They did it spontaneously because they wanted their best pupils to get what they believed was the best subsequent education.


      • Anton,

        Yes, some schools were indeed able to devote resources to the specialist task of preparing some pupils for Oxbridge entrance examinations. They were the best funded schools, either because they were private institutions funded by fees, or because they were grammar schools having greater funding (extending through to sixth form teaching) as a consequence of denying resources to the majority of children who went to secondary modern dump schools. They were also schools that employed teachers with an insider knowledge of the admissions process into just two universities.

        The issue is that candidates to Oxford or Cambridge universities were more likely to be successful if they happened to go to one of these schools. It was not an issue of the candidates ability alone.


      • Anton Garrett Says:

        Bryn: If, as youy wrote above, “many other British universities provide an excellent education”, why does the Oxbridge entrance issue matter at all?

      • Bryn Jones Says:

        The Oxbridge entrance mechanisms matter for several reasons.

        (i) People should have the ability to compete based on merit on equal terms for entry to whichever university they may choose.

        (ii) Many universities have national excellence, and some international excellence, in teaching in some particular academic subjects. This includes Oxford and Cambridge. Therefore from time to time a prospective student might regard Oxbridge as outstanding in his/her subject and would aspire to study that subject there, just as other students might judge some other university to be excellent in some other subjects.

        (iii) There is a rather widespread perception in British society that graduates of Oxford and Cambridge are universally excellent (regardless of their genuine talents). This can confer advantages in life after graduation, particularly in career competition. Rightly or very wrongly, an Oxbridge degree can confer advantages over degrees of equivalent quality from other universities.

        (iv) The practice of recruiting students preferentially from private schools means that the Oxbridge student body comes preferentially from already high-achieving families, with the consequent connections that can provide useful knowledge in helping to establish careers. Becoming part of this student body can provide an individual with networking opportunities that will be useful later in life.

        (v) Government teaching funding for Oxbridge has been more generous than other universities for generations, justified by government on an argument that teaching at those two universities is carried out both by central university departments and by colleges. The two universities and their colleges, due to their ages, have built up large endowments that are partly used for funding academic activities. This means that there are funds within Oxbridge that other universities do not have. This funding can, for example, employ people who choose to embark on academic careers, providing advantages to people within the Oxbridge system. (It is not clear to me whether this preferential state funding has gone to the wall with the recent changes in state funding and student fees.)

  2. John Peacock Says:

    Peter: thanks for the nostalgia trip. I must have taken a similar paper, although I don’t think I kept a copy. Normally, on progression through an education system, you think “this stuff I’m doing now is tough, but last year’s material looks easy now”. But that doesn’t apply here – I hope I could still do it all (would I have the courage to try…?), but it would clearly require careful thought.

    It is somewhat incredible today to think that they could have got away with setting such a challenge. Although you said it was for post-sixth year, plenty of candidates took the option of sitting the entrance pre-A level, which I did aged 17 in Nov 1973. This meant you had to mug up on half the A-level syllabus unaided and then sit a paper that was objectively far too hard. But since it never occurred to me to complain that this was unreasonable, I just made the best of it. I’m sure this experience of being thrown in the deep end at an early age was hugely beneficial. I does make you wonder whether we shouldn’t practice this “being cruel to be kind” philosophy more today. But more likely the planned expansion of maths teaching up till age 18 for all will make papers like this even less likely to be encountered.

    • telescoper Says:

      You have 3 hours. Time starts now.

    • Yes, that was me in 1975. I took it 4th term, got no coaching from my comprehensive school and just did it, as I had no idea how hard it was supposed to be. It was just a case of having a go. Happily, there was no pressure, as obviously nobody from my school ever went to Cambridge, so there were no expectations.

  3. telescoper Says:

    I have the physics paper too. I’ll put it up when I get time to scan it.

  4. The relevant statistic in determining the existence of bias is of course not what proportion of successful applicants are from which type of school, but what proportion of applicants from each type of school are successful. (This assumes of course that students from private schools are not inherently more capable than those from state schools, which I do not believe they are.)

    I don’t know about Cambridge, but Oxford certainly makes these statistics publicly available. See

    Applicants from independent schools are still more likely to get accepted, though this effect may not be quite as large as you thought. On the other hand, applicants from independent schools are more likely to apply to undersubscribed courses (such as, I presume, theology or fine art).

    • PS: I was wrong about Fine Art – it is one of the more over-subscribed courses. Classics and Theology are the two least popular ones.

  5. Also, all physics applicants to Oxford currently have to do a standard written exam. The material covered by the exam does not include anything beyond the A-level syllabus. In fact I think it doesn’t even include Further Maths modules, since having done Further Maths is only desirable and not a necessary criterion for entry. Therefore this is about as close to the style of examination advocated by Bryn Jones as possible.

    Applicants are also interviewed at individual colleges (3 interviews each). Department statistics show that first-year examination results and final degree classifications are much more tightly correlated with test scores than with interview marks, therefore interviews are weighted down.

    In my experience, a lot of effort is made to assist state-school candidates – and more women. Certain colleges also prefer to accept lower-ranked candidates from UK comprehensives than very highly ranked overseas applicants.

    Despite this the success rate for both state-school applicants and women is anomalously low. Personally I suspect the blame for this should lie with the schools rather than the university.

    • Bryn Jones Says:

      I’m not familiar with the current entrance examinations used by Oxford, and therefore do not wish to comment directly on them, but I understand that they are substantially different to the old system.

      My view is that the old system of entrance exams was fundamentally unfair, and was probably unnecessary given that A levels once provided a very good, broadly fair, assessment of academic ability, even for the most gifted students (though in most cases only after the university application).

      A better system than entrance examinations set by any one university would be either (i) examination assessment based only on A level results (with an A level system that was capable of grading different abilities among the very brightest students) or a successor to A level; or (ii) a common set of entrance examinations used by all interested universities (and which would be universally understood across the A-level education sector).

      I still have fundamental concerns about present-day Oxbridge admissions statistics. They show a large under-representation of state-schools in application numbers. Wherever the reason for that lies, the two universities concerned have the responsibility to engage with schools, teachers and potential university applicants to convince them the admissions process is meritocratic. That is a duty for institutions that receive any kind of funding from the taxes of the general population. And there are many other excellent universities in Britain that provide an outstanding education without any biases based on applicant background, perceived or real.

      • I agree with all of that. Oxford and Cambridge do appear to put quite a lot of effort into encouraging applicants from state schools – whether this is effective, or enough, I don’t know.

        Something that I do wonder about though is the role of assumed cultural stereotypes on behaviour. Specifically I mean that state-school applicants perhaps avoid Oxford because they assume that even if they were to get in, they would have to spend all their time socialising with “posh” people. They’d rather go to a university with more “real” people, whatever that means.

        This is certainly a position I’ve have heard plenty of anecdotal evidence for. I can imagine a lot of young people are put off applying because of the perceived effect on their social life rather than because they think they lack the ability to get in (teenagers are often very sure of their innate abilities, like that young woman who wrote the spoof rejection letter).

        It is also a sad, ignorant view – as everyone who has been to any university knows, you meet a far wider range of people there, with a wider range of personalities, than you imagined existed before you went.

      • Anton Garrett Says:

        Bryn: If there are “many other excellent universities in Britain that provide an outstanding education” then what’s the problem? Oxbridge cannot be responsible for the fact that some employers prefer their degrees to others.

        Furthermore, since NuLabour started kicking Oxbridge in the teeth by social engineering using financial means, Oxbridge now actively discriminates *against* private schools. This is an open secret among college admissions tutors. It is easy to grumble about the super-rich but there aren’t many of them, and what about the people who have made huge sacrifice to give their children a better education than the State currently provides (which could easily be improved at no cost because the teacher training colleges actively harm good teaching nowadays)? People in Africa slog their guts out to pay for better education for their children and are lauded in the West for it, but we kick our own people who do the same in the teeth. NuLabour had two keywords: they were for ‘excellence’ and against ‘elitism’, but since only a minority are excellent you have to choose one or the other and they always chose to shaft excellence.

      • Bryn Jones Says:

        Anton, I listed several reasons above why access to Oxford and Cambridge should be meritocratic.

        The former Labour government set up the Office for Fair Access. While such a development was in my view very favourable, I’m not convinced that the OFFA has had the drive to analyse current barriers to university participation, or to understand all the factors operating and how they affect different sectors within the university system.

        Of course private schools will complain that factors act against them. They would do that at all times except when everything operates in their favour. What are needed are university admission systems that are fully meritocratic: applicants are assessed objectively on their abilities and achievements, with no preferential opportunities for applicants from some particular backgrounds to prove their potentials.

    • Anton Garrett Says:

      Well, it certainly isn’t meritocratic now. See my comment immediately above about admission tutors.

  6. “I have the physics paper too. I’ll put it up when I get time to scan it.”

    Ok, so you have the paper, great. Please, please post it 🙂
    If you can find any STEP physics papers it would be interesting to compare. STEP was from 1987 to 2002, I took it in 2001.

  7. […] response to a request to a while ago when I posted the Mathematics paper, here is the Physics paper I took as part of the Cambridge Entrance  Examinations way back in […]

  8. […] and the Foreign Service Exam.  Popham urges us to not teach to the test but if the test were a work of staggering genius which could really measure what we thought students should all know and measured it in a way that […]

  9. Sebastian Says:

    Thank you for posting these, producing a mix of nostalgia and rekindled anxiety. I fortunately did not do this maths paper but I remember the physics paper only too well as being 3 hours of headless panic flitting from one question to another – I still have nightmares. Presumably mine was a common reaction since at least I ended up with an acceptable ‘B’ grade. I also did Chemistry and the General Paper plus Biology for Medicine entry. General paper was fabulous with such abstract essays as ‘Is music a branch of mathematics?’
    Hard exams but they led to 6 great years up at Corpus. I cannot comment too much on the admissions process now but the ratio of perspiration:inspiration needed seems to have changed.

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