Exam timetable
The timetable for the 2024 examinations is available here.
Exam rubrics
Exam rubrics for the 2024 examinations will be available here.
Past exam papers
All papers are copyright by the University of Cambridge and may not be reproduced without permission.
The papers are stored as PDF files. The PDF files can be viewed and printed using the Adobe Acrobat viewer. This program is available free of charge from the Adobe web site.
Papers are available for past years from 2001, with the exception of 2020. Click on the links below to select the year you wish to view:
- 2024
- 2023
- 2022
- 2021
- 2019
- 2018
- 2017
- 2016
- 2015
- 2014
- 2013
- 2012
- 2011
- 2010
- 2009
- 2008
- 2007
- 2006
- 2005
- 2004
- 2003
- 2002
- 2001
Essay
The essay is a key component of Part III and counts for 3 units of examination credit, equivalent to a 24-lecture course. Note that, since there is a maximum of 16 units of credit that can be obtained from examination papers for lecture courses, it is essential to submit an essay in order to achieve a total amount of examination credit in the range 17-19 units. (See Sections 9 and 10 of the Part III Handbook.)
Experience has shown that the great majority of Part III students find that working on their essay is an enjoyable change from learning from lectures and is valuable training for research as well as a range of other careers.
A list of essay titles approved by the Part III Examiners will be announced by the end of the fourth week of the Michaelmas term and will appear together with more detailed descriptions in the Essay Booklet:
This booklet also gives details of the process of allocating essay titles to students, based on their preferences, and includes general guidelines and instructions about writing the essay, including advice concerning plagiarism.
Additional essay titles may be approved by the Part III Examiners and will be added to the booklet not later than 1 March.
Essays and grade descriptors
Just as with written examination papers, the Assessor awards a numerical mark out of a maximum of 100 to each essay and in addition assigns a ‘quality mark’ (see Appendix IV of the Part III Handbook). The Faculty Board has specified that the minimum performance deserving of a distinction on a paper or an essay is associated with α-, while the minimum performance deserving of a pass is associated with β-.
The Faculty Board does not necessarily expect the mark distribution for essays to be the same as that for written examinations. Indeed, in recent years for many students the essay mark has been amongst their highest marks across all examination papers, both because of the typical amount of effort they have devoted to the essay and the different skill set being tested (compared to a time-limited written examination). The Faculty Board wishes that the hard work and talent thus exhibited should be properly rewarded.
There is no prescribed length for the essay in the University Ordinances and the Faculty Board recognises that the length of an essay is only a weak reflection of the quantity of work involved and bears no relation to the quality of the work done. However, it is anxious to prevent the essay absorbing too much of the candidate’s time. It is therefore perfectly content if a topic is set for which an excellent essay requires about 5000 words and would normally be unhappy if a topic were set for which an excellent essay required more than about 8000 words. In order to provide greater clarity, the Faculty Board now requires that all candidates use a standard LaTeX template and has agreed that the expected length of an essay should normally then be between 20 and 30 pages and should only exceptionally be more than 35 pages.
In light of remarks above, as well as the comments of both internal and external Examiners over the years, the Faculty Board considers the following descriptors of the broad grade ranges for an essay to be appropriate. The Board trusts that these guidelines prove useful in guiding the judgement of the inevitably large number of Assessors marking essays, and thereby strengthen the mechanisms by which all essays are assessed uniformly. They are intended to be neither prescriptive nor comprehensive, but rather general guidance consistent with long-standing practice within the Faculty.
An Essay of α-Grade Standard (α-, α, α+)
Typical characteristics expected of an essay of α-grade standard include:
- Demonstration of clear mastery of the underlying mathematical content of the essay.
- Demonstration of thorough understanding and cogent synthesis of advanced mathematical concepts.
- A well-structured and well-written essay of appropriate length (5000-8000 words) with
- few grammatical or presentational issues;
- a clear introduction demonstrating an appreciation of the context of the central topic of the essay;
- a coherent presentation of that central topic;
- a final section which draws the essay to a clear and comprehensible end, summarising well the key points while suggesting possible future work.
An essay of α-grade standard would be consistent with the quality expected of an introductory chapter of a PhD thesis from a leading mathematics department. A more elegant presentation and synthesis than that presented in the underlying papers, perhaps in the form of a shorter or more efficient proof of some mathematical result would be one possible characteristic of an essay of α-grade standard. Furthermore, it would be expected that an essay containing publishable results would be of α+ standard, but, for the avoidance of doubt, publishable results are not necessary for an essay to be of α+ standard. A mark in the α+ range should be justified by an explicit additional statement from the Assessor highlighting precisely which aspects of the essay are of particularly distinguished quality.
An Essay of β-Grade Standard (β-, β, β+)
Essays of β-grade standard encompass a wide range, but all should demonstrate understanding and synthesis of mathematical concepts at the level expected for a pass mark in a Part III lecture course.
Typical characteristics expected of an essay of β+ standard include:
- Demonstration of good mastery of most of the underlying mathematical content of the essay.
- A largely well-structured essay of appropriate length (5000-8000 words) with
- some minor, grammatical or presentational issues;
- an introduction demonstrating an appreciation of at least some context of the central topic of the essay;
- a reasonable presentation of that central topic;
- a final section which draws the entire essay to a comprehensible end, summarising the key points.
Such essays would not typically exhibit extensive reading beyond the suggested material in the essay description, or original content.
Typical positive characteristics of an essay of β- (pass) standard include:
- Demonstration of understanding of some of the underlying mathematical content of the essay;
- An essay consistent with the quality expected of an upper-second-class final-year project from a leading mathematics department;
while negative characteristics might include some non-trivial flaws in presentation, for example:
- An inappropriate length;
- Repetition or lack of clarity;
- Lack of a coherent structure;
- The absence of either an introduction or conclusion.
For the avoidance of doubt, a key aspect of the essay is that the important mathematical content is presented clearly in (at least close to) the suggested length. An excessively long essay is likely to be of (at best) pass standard.