Paper 1 (Pure Mathematics 1): what examiners say, session after session

Paper 1 is the paper where most students first meet A Level Math, and the examiner reports for it are among the most consistent. The same handful of messages appear year after year. This guide gathers the points that apply to the whole paper. Each topic has its own guide with the mistakes for that topic.

This guide is based on a reading of Cambridge examiner reports for recent sessions. The advice is summarised in our own words.

The headline message: show the method

The paper's front page says that unsupported calculator answers will not earn marks, and the report for Paper 1 repeats it in nearly every recent session. In Paper 1 it matters most in three places:

• Quadratic equations. Show the factorisation, the formula with the numbers substituted, or the completed square. Check that your brackets expand to the original quadratic.
• Trigonometric equations. Show the step from the equation to the angle, not only the angles themselves.
• Simultaneous equations and definite integrals. Reports describe students who used an equation solver or the calculator's integral function and wrote down only the result. No method marks can be given.

Read each part in the light of the ones before it

• If a part says "hence", it expects you to use the previous result. Students who restarted from scratch lost marks and time.
• Even without the word, a question often builds from one part to the next. Reports encourage students to ask why an earlier part was set. For example, completing the square in one part is often there to help find the range, the minimum or the inverse in a later part.
• Information in the introduction applies to the whole question. Information given in one part applies to that part only. Reports mention students who carried a condition from part (a) into part (c) by mistake.

Where the paper catches students out

Reading the reports side by side shows a few places where marks go missing again and again.

Describing transformations

This is probably the most repeated presentation point. A translation should be given as a column vector or with a clear direction. A stretch should be called a stretch (not a squeeze or a compression), with the scale factor and the direction. The order in which transformations are applied matters too. See the guide on function transformations.

Domain, range and inverse functions

Range and domain are confused, the wrong inequality sign is used, or the range is written in terms of x. When an inverse is found by taking a square root, the correct sign has to be chosen, and the reports note that this is linked to the domain of the original function. See the guide on functions.

All the solutions

Missed solutions appear in quadratics, trigonometry, circles and calculus. Students take only the positive square root, divide through by a common factor and lose the solution zero, or list only some of the angles in the given range.

Algebra and signs

Sign errors and slips in expansion appear in almost every topic. Typical examples are squaring a bracket and getting two terms, not three, simplifying (ax)ⁿ to axⁿ, and losing a negative sign when substituting. The advice in the reports is consistent: use brackets when substituting, set out your working in order, and check.

Calculus habits

Constants of integration are forgotten, the chain rule is not applied, and the derivative is set equal to the value instead of to zero at a stationary point. For definite integrals, limits are mixed up, or the answer is not checked against the diagram. See the guides on differentiation and integration.

Sketch and draw

Reports recommend a sketch on many Paper 1 questions even when the question does not ask for one. In coordinate geometry and circles, a diagram usually prevents the assumption that a triangle is isosceles or that a line is horizontal. In circular measure, labelled angles prevent students from using the wrong angle or the wrong radius. Diagrams in the paper are often not drawn to scale.

Accuracy

• Keep at least 4 significant figures in your working for a final answer to 3 significant figures.
• Give angles in degrees to 1 decimal place and angles in radians to 3 significant figures.
• When an exact answer is asked for, stay in exact form. In one question on a sector and a triangle, for example, students lost marks by finding decimal angles on the calculator when the question asked for an exact area.
• Know the exact trigonometric values for common angles in both degrees and radians. The reports describe this as particularly useful for this paper.

See the guide on accuracy for the full set of reminders.

Know the useful short methods

Some questions that students find long have shorter methods, and the reports note that successful candidates tend to use them:

• Completing the square to find a range, a minimum or an inverse, rather than differentiating.
• Using the general term in a binomial expansion, not writing out every term.
• Using the discriminant for tangents and intersections.

Before you move on

After every Paper 1 worksheet, ask yourself:

1. Did I show the method for every equation I solved?
2. Did I use "hence" parts properly?
3. Did I give the answer in the form asked for?
4. Did I find all the solutions in the given range?
5. Did I lose a sign somewhere?

Questions students ask

What do examiners say most often about Paper 1?

Show the method. The front page says unsupported calculator answers will not earn marks, and the Paper 1 report repeats it in nearly every recent session. It matters most for quadratic equations, trigonometric equations, simultaneous equations and definite integrals.

Where do students lose marks in Paper 1 again and again?

In transformations (vague wording and wrong order), domain and range, missed solutions, algebra and sign slips, and calculus habits such as forgetting the constant of integration or not applying the chain rule.

What does "hence" mean in a Paper 1 question?

It tells you to use the previous result. Students who restarted from scratch lost marks and time. Even without the word, a question often builds from one part to the next, so ask why an earlier part was set.

Does information from one part of a question apply to the whole question?

Information in the introduction applies to the whole question. Information given in one part applies to that part only. Reports mention students who carried a condition from part (a) into part (c) by mistake.

How should I describe a transformation in Paper 1?

Give a translation as a column vector or with a clear direction. Call a stretch a stretch, not a squeeze or compression, and give the scale factor and the direction. The order of the transformations matters too.