TMUA 2021: How the Papers Were Set and Scored
TMUA 2021 refers to the 2021 sitting of the Test of Mathematics for University Admission, delivered that year by Cambridge Assessment Admissions Testing (CAAT), made up of Paper 1 and Paper 2 and reported on a scale that runs from 1 (low) to 9 (high) to one decimal place. That year's official Explanation of Results states that the scale is designed so that approximately one third of candidates achieve overall scores higher than 6.5, with high scores capped at 9.0. The 2021 cycle was also the point at which the University of Cambridge appeared on the official list of courses accepting TMUA, for G400 Computer Science only. This page covers the shape of the papers, how the reported scores should be read, the difficulty and topic profile of the year as catalogued by FrontierVUE, and how to work through both papers on the platform.
TMUA 2021 at a Glance
- Papers sat
- Two papers sat back to back: Paper 1 'Applications of Mathematical Knowledge' and Paper 2 'Mathematical Reasoning'
- Length and question count
- 2 hours 30 minutes in total, each paper 75 minutes and 20 multiple-choice questions
- Score scale
- 1 (low) to 9 (high), reported to one decimal place; Paper 1 and Paper 2 were each reported separately alongside the overall score
- Official scale anchor
- The scale is designed so that approximately one third of candidates achieve overall scores higher than 6.5; high scores are capped at 9.0
- Delivery body and sittings
- Cambridge Assessment Admissions Testing (CAAT); in the CAAT era TMUA ran once a year, in the autumn
- Difficulty (FrontierVUE catalogue)
- Paper 1: of 20 questions held, 4 rated easy, 12 medium, 4 hard. Paper 2: of 20 questions held, 6 rated easy, 10 medium, 4 hard
The paper structure, score scale and scale anchor come from the official Explanation of Results for TMUA 2021 and the official pages of that period; the difficulty line reflects FrontierVUE's own catalogue ratings, not an official assessment of the papers. TMUA has been delivered by UAT-UK since 2024 under a revised scaling process, and the awarding body states that scores from 2024 onwards should not be compared directly with earlier ones. FrontierVUE has no affiliation with, endorsement from, or partnership with UAT-UK, Cambridge Assessment or the University of Cambridge.
01
What TMUA 2021 consisted of, and how results were reported
TMUA 2021 was delivered by Cambridge Assessment Admissions Testing (CAAT) as two papers sat back to back: Paper 1, Applications of Mathematical Knowledge, and Paper 2, Mathematical Reasoning. The test ran 2 hours 30 minutes in total, each paper 75 minutes with 20 multiple-choice questions. That timing has been in place since 2017; the first version, in 2016, ran 2 hours in total with 60 minutes per paper. Results were reported on a scale from 1 (low) to 9 (high) to one decimal place, and in the CAAT era Paper 1 and Paper 2 were each reported separately alongside the overall score, which is why that year's Explanation of Results carries three distribution charts.
Papers
Two: Paper 1 and Paper 2
Paper 1 was titled Applications of Mathematical Knowledge on the official pages by this year at the latest; earlier in the CAAT era it was called Mathematical Thinking
Total length
2 hours 30 minutes
Per paper
75 minutes and 20 multiple-choice questions
Score scale
1 (low) to 9 (high), to one decimal place
How many scores
An overall score plus separately reported Paper 1 and Paper 2 scores
Sittings
One autumn sitting per year throughout the CAAT era
The two sittings per year, in October and January, only arrived with UAT-UK from 2024
One change stands out on the university side this year: the University of Cambridge appeared on the official list of courses accepting TMUA, covering G400 Computer Science only. For the two cycles before that, Cambridge Computer Science ran the same test under a separate name, CTMUA (Cambridge Test of Mathematics for University Admission), defined officially as a pre-interview admissions test for that course; the CTMUA section was later retired and its URL has redirected since February 2022. From the 2022 sitting Cambridge added L100 Economics, and the official wording made TMUA compulsory for Cambridge applicants, except for mature students applying to one of the mature Colleges for Economics.
| 2020 | 2021 | |
|---|---|---|
| Scale and anchor | 1 (low) to 9 (high), to one decimal place; approximately one third of candidates above 6.5, capped at 9.0 | Word for word the same: 1 (low) to 9 (high), to one decimal place; approximately one third above 6.5, capped at 9.0 |
| Official distributions | The 2020 Explanation of Results gives three bar charts, covering the overall score and Papers 1 and 2, and attaches no numeric table | The 2021 Explanation of Results likewise gives three bar charts and attaches no numeric table |
| How Cambridge used it | Cambridge Computer Science sat the same test under the CTMUA name | Listed on the official TMUA list, for G400 Computer Science only |
| Context of the year | A pandemic year: TMUA still ran and CAAT still issued an Explanation of Results, a document that records no special arrangements | By this year at the latest, the official pages titled Paper 1 Applications of Mathematical Knowledge |
Both years sit within the CAAT era and share one scaling approach, so they can be read side by side. Scores from the UAT-UK era, from 2024 onwards, cannot be compared with them directly; the next section explains why.
Two rules that candidates ask about most are stated plainly on the current official TMUA page: no calculator or dictionary may be used, and scores are based on the number of correct answers with no marks lost for wrong ones, so every question is worth attempting. Note that this wording is taken from the current UAT-UK page rather than from the 2021 documents themselves.
02
How to read a TMUA 2021 score
The 2021 Explanation of Results makes two firm statements about scores: results are reported on a scale that runs from 1 (low) to 9 (high) to one decimal place, and the scale is designed so that approximately one third of candidates achieve overall scores higher than 6.5, with high scores capped at 9.0. The document then presents three distribution bar charts, for the overall score and for Papers 1 and 2, and attaches no numeric table. Any claim of the form 'roughly this percentage of candidates scored in that band' therefore comes from reading a chart by eye rather than from this document, and is worth ignoring.
Range and precision of the scale
The official wording: results for TMUA are reported on a scale that runs from 1 (low) to 9 (high), with scores being reported to one decimal place. A 2021 TMUA result therefore looks like 6.5, not a grade band and not a raw mark.
The anchor: what 6.5 means
The official wording: the scale has been designed so that approximately one third of candidates will achieve overall scores higher than 6.5, and high scores are capped at 9.0. That sentence is the reference point the 2021 Explanation of Results provides: above 6.5 places a candidate in that approximate one third, and 9.0 is the ceiling.
How many scores are reported
In the CAAT era Paper 1 and Paper 2 were each reported separately alongside the overall score, which is why the 2021 document contains three distributions. This differs from the UAT-UK era, where the two papers are equated and scaled together to produce a single overall score.
What the 2021 Explanation of Results does not include
The 2021 Explanation of Results sets out the scale and three distribution charts; it includes no numeric distribution table and no percentile conversion chart. To place a score, the basis this document provides is its statement that approximately one third of candidates score above 6.5.
Reading 2021 against 2020
The 2020 Explanation of Results carries the same description of the scale and the same anchor sentence, word for word, and both years sit within one CAAT-era approach, so they can be read together. One further detail: the sentence in the earlier CAAT-era documents (2016 to 2019) about candidates who do not respond to any items receiving a score of 0 no longer appears from 2020.
Comparing across eras is the single easiest mistake to make with TMUA scores. TMUA has been delivered by UAT-UK since 2024 under a revised scaling process, and the official wording is explicit: this scaling process has been revised for 2024/25, and scores should not be compared directly with TMUA scores from earlier years. A 6.5 from 2021 and a 6.5 from 2024 onwards are therefore marks on different rulers, and joining the two eras into a single trend line does not hold. In the UAT-UK era a single overall score is reported, with Papers 1 and 2 equated and scaled together, and the scale is anchored to the October cohort: the median is fixed at 4.5 and roughly 10% of candidates score above 7.0, with the floor at 1.0 and the cap at 9.0.
One more institutional difference from the CAAT era is worth remembering: candidates could decide for themselves whether to share their results with universities after receiving them, which the official pages described as a risk-free opportunity. That is no longer the case under UAT-UK, where results are sent automatically through the UCAS match to any institution whose course requires the test, and candidates cannot choose who sees their score. The 2021 experience of receiving a score therefore does not transfer directly to today.
03
Topic emphasis and difficulty in 2021 (FrontierVUE catalogue)
Everything below reflects the FrontierVUE catalogue: the platform's own difficulty ratings and syllabus tags for the Paper 1 and Paper 2 questions of 2021. These are not official paper statistics and not an official verdict on the difficulty of the year. The 2021 Explanation of Results gives the scale and three distribution charts, and includes no per-question difficulty or topic breakdown.
20 questions
Held from Paper 1: 4 rated easy, 12 medium, 4 hard
20 questions
Held from Paper 2: 6 rated easy, 10 medium, 4 hard
4 questions
Each paper carries 4 hard-rated questions, an even split between the two
The two papers do not share one profile. Paper 1 is dominated by medium-rated questions (12 of them) with fewer easy ones; Paper 2 carries more easy questions (6) alongside 10 medium. The two papers carry the same number of hard-rated questions, 4 each. The practical implication: what separated candidates this year was consistency and speed on medium questions rather than a handful of exotic ones, so the rhythm of 20 questions in 75 minutes has to be rehearsed.
Necessary and sufficient conditions
5
Recognising and sketching standard graphs
5
Lines, quadratics, cubics, trigonometric, logarithmic, exponential, square root and modulus functions
Forming and solving equations and simultaneous equations
4
Including linear with quadratic pairs, and turning a described situation into algebra
Quadratics, the discriminant and completing the square
4
Applications of differentiation
4
Gradients, tangents and normals, stationary points, strictly increasing and decreasing functions
Definite and indefinite integration of powers
4
Including expressions that must be simplified before integrating
Straight lines: gradient, intercept, parallel and perpendicular
3
Graphs, symmetry and periodicity of sine, cosine and tangent
3
Methods of proof
3
Direct deduction, exhaustion, contradiction, and disproof by counterexample
Roots, intercepts and turning points of quadratics
2
Ranked by tag frequency, the weight of 2021 falls in three places. First, logical judgements about necessary and sufficient conditions, which appear 5 times and sit among the most frequent tags. Second, recognising and sketching standard graphs, also 5. Third, the algebra of quadratics and straight lines: completing the square, the discriminant, gradients and intercepts together account for a substantial share. Proof methods appear 3 times, in keeping with the Mathematical Reasoning brief of Paper 2. A sensible order of revision is graphs and completing the square first, then focused work on necessary and sufficient conditions, and only then full papers under time.
04
How to practise the 2021 papers on FrontierVUE
01Step 1
Open the TMUA past paper list
Go to the past paper list at /papers/tmua.
02Step 2
Find the two 2021 papers by year
Locate Paper 1 and Paper 2 under 2021; both are in the bank.
03Step 3
Sit each paper to the official clock
Work in the timed interface at the real pace of 20 multiple-choice questions in 75 minutes, with nothing to hand.
04Step 4
Review the solutions and error analysis
After submitting you get worked solutions and an error breakdown; follow the topic tags back into targeted practice on the weak areas.
16 papers
TMUA past papers held, 320 questions across the 16, plus 2 official specimen papers (FrontierVUE catalogue)
1102 questions
TMUA questions on the site in total, including mock papers and the topic bank, not the past-paper count (FrontierVUE catalogue)
20 questions
Held from each 2021 paper, matching the official question count
For method alongside the papers, the guides under /exams/tmua/guides/ are built to pair with them: use a guide to settle the approach to one question type, then return to the full 2021 papers to test it against the clock.
TMUA guides
/exams/tmua/guides/
paper-1What Paper 1 asks for and how to work through it
paper-2The reasoning and proof demands of Paper 2
logic-and-proofNecessary and sufficient conditions and the methods of proof, matching this year's most frequent tags
syllabusWhat the syllabus covers, for locating your own gaps
scoringThe score scale and how results are reported
time-managementHow to spend the 75 minutes of a single paper
Tap a branch to unfold
When you work through the 2021 papers, keep their era in mind: this was a CAAT-era multiple-choice sitting whose results came as an overall score plus separate Paper 1 and Paper 2 scores. From 2024, under UAT-UK, the test is computer based, reports a single overall score, runs in October and January, and uses a revised scale. The 2021 papers remain sound material for building mathematical thinking and pace, but the feel of a score from that year is not a benchmark for a score today.
FAQ
Frequently asked questions
How hard was TMUA 2021?
The 2021 Explanation of Results does not comment on the difficulty of the papers and sets out no boundaries, and UAT-UK states on its current results page that it does not provide any grade boundaries or a pass mark. The reference point that 2021 document does give is the scale anchor: the scale is designed so that approximately one third of candidates score above 6.5, with a cap at 9.0. In the FrontierVUE catalogue, Paper 1 breaks down as 4 easy, 12 medium and 4 hard out of 20, and Paper 2 as 6 easy, 10 medium and 4 hard out of 20. Medium-rated questions dominate and each paper carries 4 hard ones, which is the clearest available picture of the year, though it is a platform rating rather than an official assessment.
What counts as a good TMUA 2021 score?
The figure with a defined meaning for 2021 is 6.5: the 2021 Explanation of Results states that the scale is designed so that approximately one third of candidates achieve overall scores higher than 6.5, with high scores capped at 9.0. That document sets out no boundary, pass mark or percentile table, and UAT-UK states on its current results page that it does not provide any grade boundaries or a pass mark, so a claim that some score guarantees an outcome has no basis in either source. Each university decides how it uses the score and what it looks for, so go by the published requirements of the course you are applying to in that year.
Can a TMUA 2021 score be compared with scores from 2024 onwards?
No. TMUA has been delivered by UAT-UK since 2024 under a revised scaling process, and the official wording is that this scaling process has been revised for 2024/25, and scores should not be compared directly with TMUA scores from earlier years. Both eras report on a 1.0 to 9.0 scale to one decimal place, but they are calibrated differently: in the CAAT era, which includes 2021, the anchor was approximately one third of candidates above 6.5, with the overall score and both paper scores reported separately; under UAT-UK the two papers are equated and scaled together into a single overall score anchored to the October cohort, with the median at 4.5 and roughly 10% above 7.0. The two eras do not belong on one trend line.
Which universities used TMUA in 2021?
The clearest change in the 2021 cycle was the University of Cambridge appearing on the official list of courses accepting TMUA, for G400 Computer Science only; from the 2022 sitting Cambridge added L100 Economics. The CAAT-era list had stood at 9 universities from the 2019 cycle, when Bath was added, and reached 10 in 2023, the final CAAT year: Cardiff, Durham, Lancaster, LSE, Bath, Cambridge, Nottingham, Sheffield, Southampton and Warwick. The list was reshaped under UAT-UK and now stands at 7 universities (Imperial, Cambridge, Durham, LSE, Oxford, Warwick and UCL), with Cardiff, Lancaster, Sheffield, Southampton, Nottingham and Bath no longer among them, so the 2021 list is not a guide for applying today.
Sources
Official sources
- Explanation of Results for TMUA 2021 (official PDF, archived)
- Explanation of Results for TMUA 2020 (official PDF, archived), for same-era comparison
- The final CAAT-era list of courses accepting TMUA (2023, official PDF, archived)
- UAT-UK Explanation of Results, October 2024 and January 2025, containing the statement that scores are not comparable across eras
- UAT-UK official TMUA results page (current scoring and reporting)
- UAT-UK official TMUA test page (current format and rules)
By year
Other TMUA years
FrontierVUE is an independent practice platform. It is not affiliated with or endorsed by UAT-UK, Pearson, OCR, the University of Cambridge, Imperial College London, or any official admissions-test owner.
FrontierVUE 是独立的备考练习平台，与 UAT-UK、Pearson、OCR、剑桥大学、 帝国理工学院或任何官方入学考试主办方均无隶属或背书关系。
Practise TMUA 2021 online
Timed interface, worked solutions and per-question analytics. The two most recent past papers are free to try.