1
Learn the method
Nineteen teaching chapters. Every Paper 1 question family and every Paper 2 reasoning type gets its own chapter, its own technique and its own set of traps, with the method the official solutions themselves use.
TMUA 9.0 Blueprint · Paperback
The complete TMUA course, in one book.
TMUA 9.0 Blueprint is 510 pages of teaching, practice and worked solutions: 39 chapters, 500 original questions with the difficulty band printed beside each one, a full course in Paper 2's logic and proof, three full mock exams, and a worked solution for every single question in the book.
Paperback. Price, delivery and returns are set by Amazon.
Why this book exists
The past papers are the place to start. They run out fast.
UAT-UK has published eight sittings of past papers and a specimen — 360 questions, every one written for the paper exam that ended in 2024. Use all of them. Then notice that they come with worked answers rather than teaching, that no paper from the on-screen era has been released, and that you only get to sit each one cold once.
You only meet each paper cold once
Learn the test on the past papers and you have nothing left to check yourself against when it would count. This book is the practice you do first, so the official papers can be what they are best at: a readiness check under real conditions.
Half the exam is taught nowhere
Paper 2 is logic and proof — necessary and sufficient, quantifiers and negation, counterexample, error-spotting — and almost nobody meets it at school. Part 2 of the book is a course in it, from nothing.
The real paper never says how hard a question is
There are no difficulty labels on the day. Every practice question in this book prints its band beside it, on a scale fitted to all 360 official questions, so a question that fights you tells you where to practice rather than just costing you time.
How the book is built
Learn the method, drill it until it is quick, then sit it under the clock.
Two papers, three stages, in the order you would actually work through them. Start at chapter one if you are new to the test, or sit the diagnostic in chapter 7 and let it set your order.
2
Drill it until it is quick
140 Paper 1 and 140 Paper 2 questions, graded against the measured difficulty of the real papers and banded on the page, plus a 20-question diagnostic and 120 more across the three mocks.
3
Sit it under the clock
Three full mock exams, six papers, each built question by question to the family mix and difficulty ramp of a real sitting. 75 minutes a paper, back to back, no calculator, working on a whiteboard.
The part most books skip
Every question worked through, not just answered.
An answer key tells you that you were wrong. A worked solution tells you where the working went wrong, which is the only part that changes what you do next time.
All 500 questions are worked in full, in the shape the official solutions use: the answer, the method that actually runs, then a line on each wrong option that was built to catch a particular slip, saying which one. Most wrong options on the TMUA are reachable by one predictable mistake, and knowing which one you made is worth more than knowing you made it.
- The full method for the correct answer, step by step
- A line on every wrong option — the slip it was built to catch, and why it looked right
- A coverage table showing exactly which question family each question tests
Difficulty bands
Every question tells you how hard it is.
The real exam never labels a question, but it does get harder as it goes, and the mix is measured: about a quarter easy, half medium and a quarter hard on Paper 1, with Paper 2 heavier at the top. Every practice question in the book prints its band beside it, on a scale fitted to all 360 official past-paper questions. Three from the book, worked at their bands.
Easy
Nothing to spot. The topic names the method, and the working is two lines.
Paper 1 practice set · Question 8
The equation x2 + px + q = 0 has roots 22 and 32. What is p + q?
- A−13
- B0
- C23
- D36
- E49
Worked solution
- 1The roots are 4 and 9, so p = −(4 + 9) = −13.
- 2q = 4 × 9 = 36, so p + q = 23.
The correct answer is C.
Medium
One non-obvious move, then a routine method. Half of each paper.
The diagnostic · Question 4
For real values of x, 25x−1 − 630 · 5x−1 + 3125 = 0. What is the positive difference between the two values of x?
- A1
- B3
- C5
- D7
- E620
Worked solution
- 1See 25x−1 as (5x−1)2 and substitute u = 5x−1.
- 2Solve u2 − 630u + 3125 = 0: u = 625 or u = 5.
- 3So x − 1 = 4 or x − 1 = 1, giving x = 5 or 2. The difference is 3.
The correct answer is B.
Hard
The whole problem has to be re-seen before any algebra helps.
Chapter 5 · Paper 1, hard
The curve C has equation y = x5 − 5x3 + 1. It is reflected in the x-axis, and the reflected curve is then translated vertically downwards until it first has exactly two x-axis intersections. What is the equation of the image curve?
- Ay = x5 − 5x3 − 6√3
- By = −x5 + 5x3 − 6√3
- Cy = x5 − 5x3
- Dy = −x5 + 5x3
- Ey = x5 − 5x3 + 6√3
Worked solution
- 1Reflect in the x-axis: y = −x5 + 5x3 − 1.
- 2Translating down by d means the curve y = −x5 + 5x3 meets the line y = 1 + d exactly twice.
- 3Differentiate: stationary points at x = 0 (an inflection) and x = ±√3.
- 4A horizontal line meets the curve twice only through a turning value; the local maximum is 6√3.
- 5The line rises from y = 1, so the first turning value it reaches is 6√3. The image is y = −x5 + 5x3 − 6√3.
The correct answer is B.
Easy: two stepsMedium: three stepsHard: five steps
Measured across all 360 official questions: Paper 1 is 23% easy, 53% medium and 24% hard; Paper 2 is 19%, 51% and 31%. Chapter 5 sets out what the scale measures — how much you have to reformulate before you can start, how many cases you carry, how much manipulation is involved, how precise you have to be, and how many layers deep the argument runs. The bands are applied by the same measurement to every question we write.
Paper 2
The half of the exam nobody has taught you.
Paper 2 asks you to decide whether an argument holds, whether one claim follows from another, and where a proof goes wrong. That is a real branch of mathematics with its own rules, and almost nobody meets it at school.
So Part 2 is a course in the ordinary sense, taught from nothing: an introduction to the paper, seven chapters on the kinds of reasoning it tests, one on its computational questions, a field guide to the answer formats Paper 2 uses and Paper 1 never does, and its own 140-question graded practice set. It closes with eight named shortcuts taken from UAT-UK's own worked answers.
- The language of logic: statements, conditionals and what follows from what
- Necessary and sufficient, quantifiers and negation, proof and counterexample
- Sophisticated chains, invented definitions, and error-spotting in purported proofs
What is inside
Five parts, 510 pages, nothing filler.
Written to be worked through from the first page to the last, and indexed well enough to dip into if you would rather not.
- I
The test itself
What the TMUA is and who requires it, what your score means in percentile terms, what the real exam screen looks like, how hard the real papers are, and how to pace 75 minutes with no calculator and a whiteboard. Then a 20-question diagnostic that shows you where you are weak before you spend a month practicing the wrong thing.
- II
Paper 1
One chapter for each of the nine question families, in specification order: algebra and functions, sequences and series, coordinate geometry, trigonometry, exponentials and logarithms, differentiation, integration, graphs and transformations, and the GCSE tail. Each shows the shapes the questions come in, the traps built into the options, and the shortcuts that cut them down. Closes with a 140-question graded practice set.
- III
Paper 2
A full course in logic and proof, taught from nothing: the language of logic, necessary and sufficient conditions, quantifiers and negation, proof and counterexample, what follows from what, sophisticated chains and invented definitions, error-spotting in purported proofs, and a field guide to the answer formats Paper 2 uses and Paper 1 never does. Closes with a 140-question graded practice set and a chapter of eight named shortcuts taken from UAT-UK's own worked answers.
- IV
Three full mock exams
Six papers, each built question by question to the structure of a real sitting: family mix, difficulty ramp, option counts and answer formats. Sit them timed and back to back, so a mock costs you the same two and a half hours the real thing will.
- V
Everything you need afterwards
Answer keys, full worked solutions for all 500 questions, a coverage table showing which question family each question tests, and the formulae you must memorise because there is no formulae booklet, with a methods sheet for each paper alongside them.
The book by numbers
The whole thing, in eight numbers.
500
Original practice questions, each with its band
500
Full worked solutions
3
Complete mock exams, six papers
39
Chapters
140
Paper 1 practice set questions
140
Paper 2 practice set questions
10
Chapters of logic and proof
510
Pages
Buy the book on AmazonPaperback. Price, delivery and returns are set by Amazon.
Book, platform, or both
They do different jobs. Neither is the proper one.
Some people think better with a book open and a pen in their hand. Some need the clock running and the real interface in front of them. Most end up wanting a bit of each, which is why these are built to sit alongside one another rather than compete.
In print
The book
Better when you are learning the method and want to work at your own pace.
- Everything in one place, in a deliberate order, cover to cover
- Nothing to log into and no screen — it works on a train and in a study period
- Annotate it, work the same question twice, and see the band beside every one
- The teaching chapters: the technique, the traps and a worked example for every question family
- Worked solutions a few pages away from the question, with a line on every wrong option
On screen
TMUA Ghost, the platform
Better when you know the method and need volume, timing and the real conditions.
- A test player built to match the Pearson VUE one, so the exam screen is familiar before the day
- Every question on the clock, at the pace the real paper sets
- Peer statistics on every question: how long it took everyone else, and which wrong option caught them
- Full-length mocks marked instantly, with an estimated score on the 1.0 to 9.0 scale
- A free account, no card, for as long as you want it
There is a code inside the book that brings the price of a paid plan down for readers. You do not need it to start: the free account is free either way.
Before you buy anything
Try five questions.
Marked as you go, with a line on why. A free account adds the full worked solution on each; the book gives you 500 more in print, in the order they are best learned.
Try it now · no sign-up
5 original TMUA questions, written to the current spec. See how you do before you commit to anything.
1 / 5
Paper 1IntegrationMedium
Evaluate
Tap an answer to see if you got it.Score 0 / 5
Who it is for
Anyone sitting the TMUA, and anyone helping them.
The TMUA is sat by applicants for mathematics, computer science, economics and related courses at Cambridge, Oxford, Imperial, LSE, Warwick, Durham and UCL. The book covers both papers in full: Paper 1's applications of mathematical knowledge and Paper 2's mathematical reasoning, including the logic and proof that school does not teach.
If you are starting from scratch, start at chapter one. If you already know the test and want volume, sit the diagnostic and go where it sends you. Parents and teachers get the same thing the candidate does — the worked solutions are written to be read by someone who has never sat the test.
Common questions
Straight answers, no fluff.
Is this an official UAT-UK book?
No. GhostPrep is independent and is not affiliated with, endorsed by or connected to UAT-UK, Pearson VUE or any of the universities that use the TMUA. Every question in the book was written for the book — none is copied from the official papers. The teaching is grounded in the nine published past papers and their official worked answers, and where the public evidence runs out, the book says so rather than guessing.
Do I need the platform as well as the book?
No. The book is complete on its own — teaching, practice and worked solutions for both papers. The platform adds the things paper cannot do: the clock, a test player built to match the Pearson VUE one, a far larger bank of banded questions, peer statistics on every question, and instant marking. The three mocks in the book are on there too, so you can sit them on screen once you have sat them on paper. Which suits you is a question about how you learn, not about which is better.
What does the difficulty band on each question mean?
Easy, medium or hard, on a scale we built by scoring all 360 official past-paper questions against the examiners' own worked answers. It measures what the working demands — how much reformulation, how many cases, how much manipulation, how much precision, how many layers — rather than how familiar the topic is. The real paper never labels a question, but it does get harder as it goes, and the bands tell you where a practice question would sit on that ramp.
What format is it, and how long is it?
A large-format A4 paperback, 510 pages, with a greyscale interior. It is a big book — the format is deliberate, because a practice set laid out at exam width and a worked solution you can annotate both need the room.
How much preparation time does it represent?
Working both courses and both practice sets properly is a term's work at a sensible pace, and each mock is two and a half hours on top. It is built to be started months out, not the week before, though the diagnostic in chapter 7 means a late start can go straight to the chapters it needs and the mocks.
How do I buy it, and what if something is wrong with it?
It is sold on Amazon, which handles payment, delivery and returns under its own policy. The price, the delivery estimate and the returns window are all set by Amazon rather than by us. If there is a problem with the content of the book itself, contact us directly and we will sort it.
Which version of the TMUA does it cover?
The test as UAT-UK sets it now: on screen at a Pearson VUE centre, two 75-minute papers of 20 questions each, no calculator, one score on the 1.0 to 9.0 scale. The mocks follow the family mix, difficulty ramp, option counts and answer formats of a real sitting, and the chapter on the exam screen shows what you will actually see on the day.
TMUA · The textbook
500 questions. 500 worked solutions.
Buy the book, start free on the platform, or do both. They were built to go together.