Paper 3 builds on Paper 1 and adds the topics that make A Level Math feel different: complex numbers, differential equations, partial fractions, binomial series and more demanding calculus. The examiner reports for it are shorter on general advice than the reports for the statistics and mechanics papers, but the same messages come back often. This guide gathers the points that apply to the whole paper. Each topic has its own guide.
This guide is based on a reading of Cambridge examiner reports for recent sessions. The advice is summarised in our own words.
Show working, and be careful where the answer is given
The paper's cover says that unsupported answers from a calculator earn no marks. In Paper 3 the reports add specific points:
- Complex numbers: the calculator should not be used. Working has to be shown to justify the result.
- Exact answers: where an exact answer is required, such as in a question on integration or a logarithm, decimal working is not appropriate.
- Given answers: in a question that asks you to obtain a given answer, take particular care to show full working. If you cannot reach the printed answer, find your error instead of adjusting your working to force it.
- Proofs: a proof question needs the extra detail that makes it a proof, such as quoting the identity or showing the combined fraction.
- Calculator use: the reports say the calculator should be limited to simple work, such as solving a quadratic equation or a pair of linear equations.
Choose the method before you start
One report says candidates should gauge the method that best suits the problem before they start, since many questions have several approaches, and should make sure that any formulae used are correct so the rest of the question can be completed. This is especially true of integration, where the choice between substitution, parts, partial fractions and an identity decides how long the question takes.
Presentation
- Set out your work so that one line follows another, not as disconnected statements around the page. The reports link a disorganised approach to sign errors, accidentally omitted terms and constants left outside brackets.
- Use brackets. Missing brackets appear in the reports for differentiation, logarithms, trigonometry, partial fractions and integration.
- Write clearly and do not overwrite one solution with another. The paper is scanned, and overwritten work is hard to read.
- Bring the right equipment: a pencil, a black pen, a ruler, a protractor and compasses.
Where the paper catches students out
Sketches and diagrams
Several reports remind students what a sketch must show. Standard trigonometric and logarithmic graphs should have their key features (such as where a logarithm crosses the x-axis, or the minimum of cosecant). An Argand diagram must use the same scale on both axes. In a numerical-solution question, the sketch should also show why the equation has a single root.
Logarithms and indices
The reports ask students to know all the laws of logarithms and the laws of indices, and to be confident about when to use brackets. The most common error across several topics is treating the logarithm of a sum as a sum of logarithms.
The chain rule
The chain rule appears in the key messages and in almost every calculus topic. Reports say students should know exactly what they are differentiating with respect to, and should not simply add dy/dx to every term.
Significant figures and decimal places
The reports say candidates need to understand the difference. Check what the question asks for, and what "exact" means.
Notation for vector lines
Candidates are expected to use correct notation for the vector equation of a line, starting with r = …, and to know that parallel lines need direction vectors that are multiples of each other, not necessarily equal.
Quick sketches for arguments
When a question involves the argument of a complex number, a quick sketch on the Argand diagram shows which of the two trigonometric answers is correct.
Areas the examiners flagged
In one set of examiner comments for this paper, the areas listed as needing improvement included solving an equation that requires logarithms, sketching standard trigonometric and logarithmic functions, long division of a quartic by a quadratic, and integrating trigonometric functions such as sin 2x cos x, tan x and tan x sec²x. The same report asked candidates to prepare all topics on the syllabus.
Before you move on
After every Paper 3 worksheet, ask yourself:
- Did I choose the method before I began?
- Did I write each step on its own line, with brackets?
- Did I follow the given substitution or the given instruction?
- Is my answer exact when it should be?
- Did I draw the sketch with its features and scales?
Questions students ask
What do examiners say most often about Paper 3?
Show working and choose the method before you start. The cover says unsupported calculator answers earn no marks. In complex numbers the calculator should not be used, and where an exact answer is required, decimal working is not appropriate.
What should I do if I cannot reach the printed answer in a "show that" question?
Find your error instead of adjusting your working to force the answer. In questions that give the answer, take particular care to show full working.
Which Paper 3 mistake appears across several topics?
Treating the logarithm of a sum as a sum of logarithms. Reports also ask students to know all the laws of logarithms and indices and to use brackets with confidence.
What must a sketch show in Paper 3?
Key features of standard trigonometric and logarithmic graphs, such as where a logarithm crosses the x-axis. An Argand diagram needs the same scale on both axes. In a numerical-solution question, the sketch should also show why the equation has one root.
Which areas did examiners say needed improvement in Paper 3?
In one set of comments: solving equations that need logarithms, sketching standard trigonometric and logarithmic functions, long division of a quartic by a quadratic, and integrating functions such as sin 2x cos x, tan x and tan x sec²x. The report asked candidates to prepare all topics on the syllabus.
Practice
Choose Paper 3 and some topics to get a mixed set of past paper questions. Attempt each question first, and open the mark scheme only afterwards.