STEP Past Papers: Where to Find Them and How to Use Them

STEP past papers are the main preparation resource for the Sixth Term Examination Papers in mathematics, which Cambridge uses in its conditional offers for Mathematics and which some other universities, including Warwick and Imperial, also use. OCR, which sets and runs STEP, publishes the recent papers and examiner reports free, and the University of Cambridge adds a searchable question database, worked solutions and a free support programme. Below are the location of each resource, the structure of the papers and a plan for working through them between the summer of Year 12 and the June exams.

Where to find official STEP past papers

Every resource below is free. Students should download papers from OCR and Cambridge directly, since copies on third-party sites are sometimes incomplete or mislabelled.

The OCR Preparing for STEP page holds the papers, the solutions and the specification. The Cambridge STEP page links to the database, the support programme and the free download of Dr Stephen Siklos's book, and Cambridge recommends that link over copies on other sites.

How a STEP paper is structured

STEP 2 and STEP 3 have the same structure. Each paper lasts three hours and has 12 questions, of which 8 are pure mathematics, 2 are mechanics and 2 are probability and statistics. Each question is marked out of 20, and although every answer is marked, only the six highest-scoring answers count, so the maximum on a paper is 120.

Calculators and formula books are not allowed. In the answer booklet, candidates shade a circle on each page to show which question they are answering, and OCR publishes a sample answer book on the Preparing for STEP page so that this is familiar before the exam.

STEP 1 is no longer set. Its papers remain useful practice, and Cambridge notes that its content was roughly equivalent to A-level Mathematics, which makes the old STEP 1 papers a good starting point before moving on to STEP 2.

Featured tutors

Ellis A.

Master Tutor

Computer Science

I'm Head of Computer Science at a secondary school in Cornwall, where I lead staff CPD, after seven years in technology consulting. I teach GCSE and A-Level Computer Science, and I'm known for explaining the algorithms pupils find hardest in plain terms, little and often.

Enquire about this tutor

Michael M.

Master Tutor

EconomicsBusiness StudiesOxbridge Preparation

I taught Economics for nine years, latterly as Head of Economics at Winchester College, and I examine for AQA and Edexcel. I teach Economics and Business Studies to A-Level and prepare Oxbridge and UCAS applicants, working from Rosenshine's principles of direct instruction.

Enquire about this tutor

Michael O.

Master Tutor

MathsEconomicsPhysicsChemistry

I have a First in Economics from York and an MSc from UCL, and have tutored for seven years, including SEN students. I teach maths, economics, physics, chemistry and biology to GCSE, A-Level, IB and AP. I build strong foundations so advanced questions feel intuitive.

Enquire about this tutor

See all tutors

Which past papers still match the specification

The STEP specification was revised for the papers set from 2019 onwards. Cambridge says the style of the questions did not change, so most older questions remain good preparation. A small number of older questions use mathematics that is no longer examined, and students should check any unfamiliar topic against the current STEP specification before spending time on it.

Two points help when choosing papers:

• Use the recent papers last. Papers from the current specification are the closest match to the real exam, so they work best as timed mocks in the final weeks.
• Use the database for topic practice. Filtering the Cambridge database by topic gives a run of questions on, for example, integration or vectors, which suits the early months when the aim is to build technique.

An original STEP-style question and walkthrough

The question below was written for this guide in the style of a STEP 2 pure question. It shows how a STEP question builds: early parts give the method, and later parts expect the student to see how to use it.

"

Let f(x) = x³ − 3x + 1.
(i) Show that the equation f(x) = 0 has three real roots, all lying between −2 and 2.
(ii) By substituting x = 2cos θ, show that the roots are 2cos(2π/9), 2cos(4π/9) and 2cos(8π/9).
(iii) Hence find the value of cos(2π/9) cos(4π/9) cos(8π/9).
(iv) Find also the value of cos²(2π/9) + cos²(4π/9) + cos²(8π/9).

Part (i). Evaluating f at a few points gives f(−2) = −1, f(0) = 1, f(1) = −1 and f(2) = 3. The sign changes on the intervals from −2 to 0, from 0 to 1 and from 1 to 2, and since f is continuous and a cubic has at most three roots, there is exactly one root in each interval.

Part (ii). With x = 2cos θ, the equation becomes 8cos³θ − 6cos θ + 1 = 0, which is 2(4cos³θ − 3cos θ) + 1 = 0. The triple-angle identity turns this into 2cos 3θ + 1 = 0, so cos 3θ = −1/2. Taking 3θ = 2π/3, 4π/3 and 8π/3 gives θ = 2π/9, 4π/9 and 8π/9, and these give three different values of 2cos θ, so they are the three roots.

Part (iii). For a cubic x³ + bx² + cx + d, the product of the roots is −d, which here is −1. So 8 cos(2π/9) cos(4π/9) cos(8π/9) = −1, and the product of the cosines is −1/8.

Part (iv). The sum of the squares of the roots equals the square of their sum minus twice the sum of their products in pairs. The roots sum to 0 and their pairwise products sum to −3, so the squares sum to 0 − 2(−3) = 6. Dividing by 4 gives 3/2.

A strong answer states why each step holds, such as the continuity argument in part (i) and the check in part (ii) that the three values are different. Examiner reports often comment on answers that reach the right result with a gap in the reasoning, and marks are lost there.

How to use STEP past papers in a preparation plan

Cambridge describes STEP as aimed at roughly the top 5% of A-level Mathematics candidates and suggests that students adjust their expectations accordingly. A first STEP question can take well over an hour, and that is normal. The plan below runs from the summer of Year 12 to the June exams and assumes A-level Mathematics and Further Mathematics are being studied alongside.

Three habits make past papers more useful:

• Read the examiner report after each paper. OCR's reports say which questions were popular and where answers lost marks, which is often more useful than the solution alone.
• Choose questions on purpose. Only six answers count, so a student who knows which topics they handle best can decide on their six quickly in the exam.
• Keep a methods log. Recording the key idea of each question, such as a substitution or a symmetry argument, builds a reference list to revise from in the final weeks.

Grades, results and what Cambridge sees

STEP grades run from S, the highest, through 1, 2 and 3 to U. Grade boundaries are set each year after marking and are published in OCR's explanation of results, so a mark that earned a grade 1 in one year may not do so in the next. A Cambridge offer usually includes grades in STEP 2 and STEP 3, and college staff can see the marks and the scripts themselves, which lets them take account of a near miss. Dates, fees and the latest grade boundaries are in the key facts on this page, and the STEP exam guide covers registration and results in more detail.

Related guides

• STEP maths exam guide
• Applying for maths at Cambridge
• TMUA past papers
• How to prepare for the TMUA
• Oxford mathematics: the application and the TMUA
• University admissions tests in the UK: which test for which course
• Super-curricular activities for maths

University admissions support from Taylor Tuition

Students preparing for STEP often want someone to check their written solutions and help them plan the months before the exams. Our tutors are graduates of Russell Group universities, including Oxford and Cambridge, and the tutor pool includes former heads of department and school leaders, experienced exam-board examiners, teachers from independent and boarding schools, and educational consultants with independent-school admissions experience.

Sessions run online for families across the UK and internationally, and in person in central London. Fees start from £50 per 50-minute session and depend on tutor level and subject, there is no registration fee and the initial consultation is free. Tutors who work with children hold an enhanced DBS check, and families receive progress reports after sessions.

Read more about our A-level Further Maths tutors and admissions test tutors, or send an enquiry and we reply with tutor recommendations.