TMUA Guidance
What Is The TMUA?
The Test of Mathematics for University Admissions (TMUA) is an assessment designed to evaluate the mathematical aptitude and problem-solving skills of students aspiring to pursue higher education in mathematics-intensive fields. Developed by Cambridge Assessment admissions testing, TMUA is recognized by several universities in the United Kingdom as part of their admissions process. The test goes beyond the standard curriculum, focusing on assessing the ability to apply mathematical concepts to real-world scenarios, think critically, and develop creative solutions to complex problems. The TMUA is split into two papers, each consisting of 20 multiple choice questions. The first paper focusses on applications of mathematical knowledge and the second primarily assesses mathematical reasoning.
The TMUA plays a crucial role in identifying candidates with exceptional mathematical potential, enabling universities to make well-informed admission decisions and select students who possess the analytical and logical thinking skills necessary for success in rigorous academic programs. For more information on the TMUA such as important registration dates and whether your course accepts it, visit https://www.admissionstesting.org/for-test-takers/test-of-mathematics-for-university-admission/.
What Content Will I Need To Know?
Both paper 1 and paper 2 require the fundamental pure mathematics knowledge taught at GCSE and AS level, however the TMUA specification is not limited to this – both papers also contain questions on Sequences and Series, require exposure to basic probability and combinatorics and demand familiarity with the modulus function, all of which are usually covered in the A2 pure/stats specification.
Additionally, TMUA paper 2 is based around the formal side of mathematics and contains many concepts that you probably haven’t seen or heard of before about theorems, proofs, propositional logic, and rigour. You might also see the occasional logic puzzle in paper 2.
Ultimately, paper 1 and paper 2 test different skills, and an exceptional performance in one paper shouldn’t be hindered by a low score in the other. Therefore, when approaching practice for the TMUA, it is crucial to tackle each skillset separately.
What Should I Do Before I Start To Attempt Questions?
Review the Content: Review the syllabus for TMUA Paper 1 to grasp the range of topics it covers. This may include calculus, algebra, number theory, sequences, series, and more. Make sure you have a strong foundation in these areas.
Conceptual Understanding: Focus on building a deeper understanding of concepts rather than memorizing formulas. Work through examples and solve A level questions to solidify your understanding. Most people cruise through AS and A level maths by memorising methods and formulas instead of understanding why those methods work. This might work for A level maths, but it certainly won’t be sufficient to achieve a high TMUA mark for the following reasons:
- Complex Problem Solving: The TMUA exam often presents complex and challenging mathematical problems that require deep understanding and critical thinking. Basic knowledge might not be enough to tackle these problems effectively.
- Application of Concepts: The exam doesn’t merely test your ability to recall formulas or methods. It requires you to apply mathematical concepts in novel and intricate ways. Without a deeper understanding, you might struggle to adapt your knowledge to unfamiliar scenarios.
- Analytical Thinking: Many TMUA questions assess your ability to analyse situations, break down problems into smaller parts, and formulate strategies to solve them. A superficial understanding might hinder your analytical skills and problem-solving strategies.
- Multiple Approaches: Some questions might have multiple valid approaches to solving them. A deeper understanding of underlying principles helps you choose the most efficient and elegant method, saving time and improving accuracy.
Go through the TMUA specification and whenever you encounter a formula, ask yourself if you truly understand why that formula works and how it’s derived. For example, you might be familiar with circle theorems from GCSE maths. Have you just memorised each circle theorem? Do you know how to prove each theorem? Another example is the binomial expansion formula. Not many people actually know why it works, and they just accept the fact that it does. Can you reason why the binomial expansion formula works using combinatorics (permutations and combinations)? When you encounter a topic that you don’t fully understand, dedicate time to gain a deeper understanding of that topic – watch YouTube videos, revisit your textbook or consult your teacher.
How Do I Prepare for TMUA Paper 1
For most people, preparing for TMUA paper 1 is easier than preparing for paper 2 as paper 1 concepts are more familiar to the average A level student. The biggest hurdle is the fact that you only have 8 sets of official TMUA papers to practice from (2017-2022 and the sample sets). The first thing you should do when you start preparing is to attempt the specimen paper 1 under timed conditions. This will give you a rough idea of where you are at and will also highlight topics that need improvement. It will also expose you to the harsh time restrictions of the TMUA. Remember, the TMUA is a time pressured exam. You only have an average of 3 minutes and 45 seconds to solve each question.
Once you have attempted specimen paper 1, do not waste the other paper 1s just yet. Save them for timed mocks closer to the date of the exam. Instead, you can attempt the Oxford MAT papers (only the multiple-choice section). The papers can be found here https://www.maths.ox.ac.uk/study-here/undergraduate-study/maths-admissions-test. These questions are very similar to TMUA paper 1 questions and require the same level of problem solving, so is therefore a great resource to make use of. Make sure you do these questions timed – In your first few MAT papers aim to answer all 10 questions in an hour and as you progress, reduce this time to 45 minutes.
If you have exhausted all MAT papers, you can purchase this book and attempt the multiple-choice questions. The book has over 200 TMUA style questions and is a great resource. However, it is only recommended to purchase this book if you have used up every other resource. https://www.amazon.co.uk/STEP-MAT-TMUA-University-Mathematics/dp/1398323314/ref=asc_df_1398323314/?tag=googshopuk-21&linkCode=df0&hvadid=500952624607&hvpos=&hvnetw=g&hvrand=4867595797241122171&hvpone=&hvptwo=&hvqmt=&hvdev=c&hvdvcmdl=&hvlocint=&hvlocphy=9046111&hvtargid=pla-1446150449272&psc=1&th=1&psc=1
After finishing every MAT multiple choice question from 2007 to 2022, you can attempt the remaining TMUA paper 1s under mock conditions. Make sure you dedicate an hour and 15 minutes to complete each paper and ensure that there are no disturbances.
How Do I Prepare For TMUA Paper 2
Paper 2 is much harder to prepare for as it contains topics you probably won’t be familiar with. Luckily, Cambridge Assessment Admissions Testing have released a 70-page document to help you understand the terminology used in paper 2 and the types of questions you will be asked.
Unlike paper 1, these types of questions cannot be found in other admissions exams such as the MAT, so instead you must resort to other sources such as YouTube videos and logic puzzles as well as completing the TMUA paper 2s under mock conditions. Here is a list of the niche areas of paper 2 and how you can get better at each one.
Propositional Logic
Understand Fundamentals: Begin by thoroughly understanding the basics of propositional logic, including logical operators (AND, OR, NOT, IMPLIES, IF AND ONLY IF), truth tables, and the concept of propositions.
Understand Key terms: Learn what is meant by the contrapositive, converse, negation, necessity, sufficiency. This is crucial for success in paper 2.
Identify Patterns: Look for patterns and common structures in logical expressions. Identifying these patterns can help you anticipate the outcomes of logical operations.
Work on Proofs: Practice constructing proofs using propositional logic. Proofs involve demonstrating the validity of logical arguments, which sharpens your reasoning skills.
Apply Logic: Look for opportunities to apply propositional logic in real-life scenarios. This can help you see the practical implications of the concepts and will also ultimately make it easier to get your head around propositional logic. Which one of these statements is correct?
- “If there are clouds in the sky, it is raining”.
- “If it is raining, there are clouds in the sky”.
This is an example of propositional logic applied to a real-life scenario.
Another test to see if you have grasped the idea of the contrapositive and propositional logic is the following puzzle:
You are shown a set of four cards placed on a table, each of which has a number on one side and a colour on the other. The visible faces of the cards show 3, 8, blue and red. Which card(s) must you turn over in order to test that if a card shows an even number on one face, then its opposite face is blue?
Answers: In the real-life example about clouds and rain, the second statement is correct. In the card puzzle, most people can immediately identify that the 8 card must be turned over to test if the statement is true, however most people incorrectly state that you also have to turn over the blue card. The correct answer to this puzzle is that you must turn over the red card and the 8 card, because we are testing “if a card shows an even number on one side then the other side is blue”. We are not testing “if a card shows blue on one side then the other side is even”.
If you are struggling to differentiate between the two statements, then it is important to review the Cambridge logic document once again. If you didn’t get the correct answer to the puzzle (most people do not answer it correctly the first time), it means you have not fully understood what the contrapositive of a statement is and should spend some more time looking at what it is.
Identifying Errors In Proof
Another common question asked in TMUA paper 2 is to identify an error in a proof. The question will show you multiple lines of reasoning and you have to identify which line, if any, contains the error in the proof. An example of this type of question follows:
Proof that 2 = 1. Find the first line where an error occurs.
- a = b
- a^2 = ab (Multiply both sides by a)
- a^2 – b^2 = ab – b^2 (subtract b^2 from both sides)
- (a-b)(a+b) = b(a-b) (factorise both sides)
- (a+b) = b (divide both sides by a-b)
- (b+b) = b (since a=b, substitute the a on the left-hand side for a b)
- 2b = b (add the two bs on the left-hand side)
- 2 = 1 (divide both sides by b)
Answer: The first error occurs on line five. We are dividing by (a-b), however a=b so we are dividing by zero.
You can get better at these questions by looking out for some common errors. These will become evident to you when you complete all TMUA paper 2s, but some common ones to look out for are:
- Dividing by zero
- Introduction of Extraneous solutions (common when dealing with logarithms and absolute values)
- Disregarding Valid Solutions (common when dealing with trigonometry)
- Making incorrect assumptions
Here are some general strategies to deal with these types of questions:
- Understand the Proof Structure: Familiarize yourself with the common structure of mathematical proofs. Know how a proof is typically organized, including the use of assumptions, statements, and logical connections.
- Review Proof Techniques: Study different proof techniques, such as direct proof, proof by contradiction, proof by contrapositive, and induction. Understand the flow of reasoning in each technique.
- Break Down the Proof: When presented with a proof, break it down line by line. Understand the purpose of each step and how it contributes to the overall argument.
- Check Assumptions: Examine the assumptions or premises provided in the proof. Ensure they are valid and accurate.
- Verify Logical Steps: Evaluate each logical step in the proof. Ensure that the statements are logically connected and that each inference is valid.
- Look for Invalid Deductions: Pay close attention to statements that might be unsupported or incorrectly deduced from previous steps. These are often where errors occur.
- Use Counterexamples: If a statement seems incorrect, consider counterexamples that would make it false. Counterexamples can help you identify flawed reasoning.
- Consider Special Cases: Think about whether the proof holds true for specific cases. If it doesn’t, this could indicate an incorrect step.
- Focus on Negations: Logical errors often involve misinterpreting negations. Pay close attention to how negations are used and make sure they’re consistent.
- Practice Regularly: Solve a variety of proof dissection questions from different sources. There are many on YouTube such as https://www.youtube.com/watch?v=SGUZ-8u1OxM
- Develop a Critical Eye: Train yourself to be critical of each step in the proof. Don’t take anything for granted; question the validity of every statement.
Once you have mastered the concepts in the Cambridge logic document, paper 2 questions will be a breeze.
Commonly Occurring Questions/Concepts
As stated earlier, these common questions and the methods to solve them are not exhaustive. Ultimately the TMUA is about problem solving under time pressure and applying methods to novel scenarios. You might need to use a completely different method to solve a certain type of question, but these are the common methods that work in most cases.
Type of Question | How to solve | Number of occurrences (2017-2022) |
Identify the statements that must be true based on a given condition (necessary statements) | Try algebraic manipulation, rearranging, finding counterexamples. If the condition involves an equation, consider drawing a graph. If the condition involves inequalities, consider adding/multiplying them (subtraction & division doesn’t work for inequalities). Work through the statements logically and systematically, don’t take shortcuts and try to express the solution in an algebraic form. | 22 |
Finding a counterexample to a statement | Write out the negated version of the statement then substitute all the examples and see which one satisfies the negated statement. You can also use a table to make this easier. | 11 |
Finding errors in a proof / attempt to solve an equation | Work through each line one at a time and try to provide a counterexample. If an incorrect solution is found, plug that solution into each statement and see where it starts to become valid. | 11 |
Integrate / differentiate an expression | Simplify a problem into individual terms, then integrate each term separately. | 10 |
Finding the roots or sum of roots of a polynomial | Make use of the factor theorem and remainder theorem. You can also use the sum and product of roots formula (this is taught in further maths) however this is not necessary. Use the quadratic formula and keep an eye out for quadratics in disguise. | 10 |
Coefficient of a binomial or multinomial expansion | If combinatorics is intuitive to you, then using combinatorics will usually be a much faster method than using the formula you learn at school (especially when you have to expand more than two terms). Take note of negative signs, be careful with the algebraic manipulation. You can use combinatorics in the form of “choosing” one variable or the other for trinomial expansions. | 8 |
Find the minimum / maximum value of a quadratic expression | Complete the square of the quadratic, spotting hidden quadratics if necessary and expanding a factorised form if necessary. | 8 |
Graph transformations | Sketch the graph. You can also substitute values of x and y with the transformation. | 7 |
Solving simultaneous equations with logarithms and/or indices | Use log laws and index laws to remove the log and index expressions, then solve the simultaneous equations normally. Substitute the variables with other variables and solve the logarithm/index at the end. Refer to the definition of a logarithm and what it actually means. This is a useful tool to simplify expressions. | 7 |
Coordinate geometry | Always sketch graphs, even if you think you won’t need to. Coordinate geometry includes finding gradients, midpoints, normals, areas between lines. | 5 |
Find number of roots of a polynomial | Sketch the graph of the polynomial and differentiate to find the x coordinates of the turning points and their respective y coordinates to know where the x-axis is relative to the graph. | 5 |
Find the ith term or sum of a recursive sequence | Find a pattern by brute-forcing the sequence going up from 1 and extend that pattern using modular arithmetic. For example, you find that the first 8 terms of a sequence are 4,5,2,4,5,2,4,5, and you are asked for the 2023rd term. You can see that the sequence repeats every 3 terms, and you can also work out that 2023 has a remainder of 1 when divided by 3. Since the pattern indicates that all terms with a remainder of 1 when divided by 3 have the value 4, you can deduce that the 2023rd term is 4. | 5 |
Conditions for which a given statement is true | Plug in examples of each of the options and see which option leads to the statement being satisfied. If it involves graphs, sketch a diagram. | 5 |
Area of graph in specified region | Sketch a graph and split the areas. Find any x-intercepts, then integrate over the separate regions. Remember, finding areas of graphs is not the same as integration; finding areas of graphs is unsigned whereas integration is signed. | 5 |
Find the negation of a statement | Split the statement into sections and negate each section individually. | 4 |
Manipulation of logarithmic expressions | Use Logarithm and Indices laws. Read the options to make sure the log base is correct. Keep an eye out for hidden quadratics. | 4 |
Comparing terms of an arithmetic and geometric series | Equate individual terms to get simultaneous equations in terms of the starting terms, the common difference of the arithmetic sequence and the common ratio of the geometric sequence. Eliminate the unknowns. | 4 |
Solving inequalities | Take critical points and do case-by-case analysis. You can use a table and in most cases, sketching a graph will help. | 4 |
Finding number of solutions of a trigonometric equation between a certain interval | Draw the graphs to find the number of intercepts with the x-axis. Use the symmetry and periodic nature of trig functions to help. Be familiar with common trig values (0,30,45,60,90). | 4 |
Determining whether the trapezium rule gives an overestimate/underestimate | Be familiar with the second derivative concavity conditions for an underestimate/overestimate. Sketch the graph and draw the trapezium of each option. | 4 |
Values of a constant where a quadratic has a certain number of roots | Sketch the quadratic if necessary. Use the discriminant. | 4 |
Trigonometric graphs and ranges where a condition is met | Use the symmetry and periodic nature of trig functions. Sketch the graphs and directly compare the range or split into cases and analyse case by case. | 3 |
Find the gradient of a curve at a point, or the range where the gradient is increasing | Simplify the expression then differentiate, plugging in the value of x. | 3 |
Find length, area, or angle of a 2D diagram | Keep an eye out for similar triangles and make use of all the information given. Sketch a diagram if necessary and make use of trigonometric rules. | 3 |
Finding a sequence of numbers/letters given constraints | You can use algebra, or recursive logic to try to maximise a value while satisfying the constraints. If you are dealing with letters, you can consider writing out the possibilities in a grid. | 3 |
Find the expression of a polynomial given the value at certain points. | Make use the remainder theorem to find simultaneous equations and solve. | 3 |
Solving questions about the modulus function. | You must sketch a graph for every modulus function question. Break down function into critical points and find the intercepts of the ranges. | 3 |
Solving trigonometric equations within a certain interval | Use relevant trigonometric identities to make the terms the same. A trick is to draw a triangle to find the trigonometric values. A common error is not ensuring that the signs are correct for that quadrant. | 3 |
Choose the correct sketch of a graph | Consider critical points, plug them into the equation and see which points are on the graph. You can also complete the square to find the coordinates of the vertex. You can also consider what happens when x is very large or very small and the behaviour of the graph near the origin. | 3 |
True/false statements based on triangle or quadrilateral properties. | Know the quadrilateral properties and try to find counterexamples. Understand the properties of triangles that uniquely define a triangle. | 3 |
Solving simultaneous quadratic / linear equations. | Substitution of linear equation into the quadratic equation. | 2 |
Area and volume of shapes | Know basic formulas of surface area and volumes (necessary ones will be in the specification). Find expressions and be confident with the algebraic manipulation. | 2 |
Find the largest/smallest value in the options | Approximate each option and then compare. You can also do pairwise comparisons to eliminate options. | 2 |
Determine true and false statements | Case by case analysis. Use a table to make it easier. | 2 |
Find (integer) solutions that meet a given constraint | Find a single expression that meets the constraint, then list out all the cases. | 2 |
Sum to infinity of a geometric progression | Find a and r, then use the formula to find the sum to infinity. Make sure |r| is less than 1. | 2 |
Manipulation of indices | Know your indices rules. You can break down the base into products of prime factors. | 2 |
Determining the signs of the constants for a geometric sequence | Use the constraints to express the constants in simultaneous equations. | 2 |
Finding a and d in an arithmetic sequence | Represent the conditions of the question algebraically and then manipulate. | 2 |
Possible triangles given certain constraints | Draw a diagram, with an arc representing a length. Then use trigonometric rules. | 2 |
Find the sum of a trigonometric series | Use trigonometric identities to convert the series into values that are easy to calculate. Can also write out a few values to find out the pattern. | 2 |
Probability of a certain event happening | Consider using symmetry. | 2 |
Find area between curves | Find the intercepts of the curves, then integrate over the intercepts. | 2 |
Visual logic puzzles | Work through the problem logically, using diagrams to help. | 2 |
Finding the mean of a set of data | The mean is the total value divided by the number of elements. For the same elements, the total value is the same. Can use algebraic substitution, then list out all of the possibilities in a table. | 2 |
Changing the subject | Note the constraints given in the question. | 1 |
Finding the radius of a circle | Complete the square on x and y to express the equation in the form (x-a)^2+(y-b)^2=r^2 | 1 |
Finding the maximum/minimum of an approximation | Use the appropriate approximated values, which can be found by observing the equation that links the expressions together. | 1 |
Possible orders of items given constraints | Factorial to find total, then divide by the pairs that have a certain order already. | 1 |
Number of local maximas/minimas of a polynomial | Sketch a graph of that degree and consider the possible scenarios. | 1 |
Find unknowns in a polynomial expansion | Expand the polynomial and compare coefficients. | 1 |
Composite fractions/percentages | Represent the unknown value as x, then substitute the unknown value into the constraints. | 1 |
Finding intercepts and unions of items and conditions | Use a tree diagram or a two-way table. | 1 |
Shortest distance on a 3D shape | Unpack the 3D shape into its 2D net and use trigonometry to find lengths and angles. | 1 |
Ordering steps of a proof | Use the options to limit the cases, then go through case by case. This should be solved by process of elimination. | 1 |
Find the range for which a given set of constraints is true | Break each constraint down into its simplest form, then draw a number line and find the intercept. | 1 |
Correct equation of a graph | Substitute the values of critical points and match them to the graph. Use process of elimination to eliminate any obviously incorrect options. | 1 |
Find point on derivative graph that corresponds to a point on the actual function | Know the characteristics of special points on a graph, and what they mean for its first and second derivative. | 1 |
Minimise distance between two points on a graph | Express the distance between the two points in an algebraic form, then minimise the value of that algebraic form using completing the square. | 1 |
Find value of unknown where a polynomial does not intersect the x-axis | The discriminant must be negative. You can also sketch the polynomial. | 1 |
Finding distance between circles | Find the distance between the centres of the circles then subtract the radius. | 1 |
Solving trigonometric simultaneous equations | Substitute the trigonometric function with other variables then solve normally. | 1 |
Minimise distance between a quadratic and a line | Minimum distance is when the gradient of the curve is equal to the gradient of the line. Find the point on the curve with this property and use the shortest distance formula to find the shortest distance between that point and the line. | 1 |
Determine whether a condition is necessary or sufficient for a given statement | Try to find counterexamples. If a counterexample can’t be found, try to think of a general proof. | 1 |
Find the value of an expression | Simplify as much as possible, spot patterns in the question. | 1 |
Find the equation of a quadratic with related roots | Use the sum and product of roots formula to find equations (this is taught in further maths and isn’t in TMUA spec but is worth learning for the TMUA). Determine what needs to be found for the sum and product of roots for the new equation, and find them using the old equations. | 1 |
Find the angle of a tangent between two circles | Use Pythagoras theorem. When you have two circles, you always know the radius of each circle as well as the distance between the circles, which can be used to solve for other unknowns. Draw a diagram to get a clearer idea. | 1 |
Lowest positive integer for which a quadratic is positive | Plug in special values into the equation. You can also use binary search if function is monotonic in the positive x region. | 1 |
Converting between degrees and radians | Be careful when converting and double check. Work out the answer before checking each of the options. | 1 |
Find maximum/minimum number of true statements | Work through all of the cases separately and systematically. Don’t cut corners. | 1 |
Find the true statement given that exactly one of the statements are true | Express each statement as if…then… statements, then find contrapositives and cancel them out. You can replace statements with mathematical symbols. | 1 |