YOUR ULTIMATE GUIDE

# The Definitive Starter Guide To The TMUA (2023)

Written by: Matt Amalfitano-Stroud

The TMUA is one of the major hurdles in between you and getting your dream place at university. This guide is the first stepping stone on your journey to beating the exam, as you will learn everything you need to know about its format, the two papers and the questions you’ll find on exam day. Let’s get started!

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## TMUA BASICS: EVERYTHING YOU NEED TO KNOW

The first thing we need to take a look at is everything surrounding the TMUA. Let’s go over how you register, how it’s scored, what you need to bring and much, much more.

### What is the TMUA?

The **Test of Mathematics for University Admission (TMUA)** is a pretty self-explanatory title. But just to clarify, it’s a **150 minute long admissions test** used by a variety of universities primarily for their **Mathematics courses**.

The key thing to understand about the TMUA is that, for the most part, it is a **non-compulsory exam**. Most of the universities that use the TMUA give applicant’s the opportunity to apply for their courses **without submitting TMUA results**. In some cases, this may be because they **accept multiple admissions exams**, while others simply **do not make it a requirement**.

However, it is highly recommended that sit the exam and **aim to score highly**, as, in some cases such as **Durham University** and **Lancaster University**, achieving a certain score will likely **get you an offer or even a reduced offer**. Other’s, such as **LSE**, have clearly stated in the past that** the TMUA makes you a more competitive candidate**.

In recent years, **predicted grades** have become **less relevant** within university admissions, meaning **the TMUA is much more important** to making your application stand out (although the other parts of your application will still need to be strong).

The **key exception** to this is the **University of Cambridge**, who **require all applicants** (excluding mature students) to sit the TMUA for their **Computer Sciences** course.

### What is the Structure of the TMUA?

The TMUA is made up of **two separate papers** consisting of** 20 multiple-choice questions** each. You are given **75 minutes** to complete each paper. The questions on each paper are **fairly different** to each other:

**Paper 1** focuses on mathematics in its **purest form**. These questions are laid out simply and will see you **solving the problem that is presented to you**.

**Paper 2** focuses more on **mathematical reasoning **rather than pure knowledge. Many of the questions presented will require more than just solving a problem but to **logically assess a problem or another person’s wor**k using various techniques. Mathematical knowledge will still be essential here, but the question will be less about the solution and more about the **context of the question or solution given**.

This table summarises the TMUA Structure:

#### TMUA Structure

### Why is the TMUA used?

As previously stated, the TMUA is primarily used as a tool for **assess applicant’s mathematic abilities**, both in terms of pure maths and logical thinking.

As a non-essential exam for the most part, it can be assumed that **admissions teams will look more highly upon applicants who have sat the exam** and will generally be more likely to give an offer if a **high TMUA score** has been achieved.

Many of the courses that accept the TMUA **do not require applicants to interview**, meaning this exam will be their primary way of **understanding your abilities as an academic**. This will typically be considered **alongside your personal statement**, which will give them an idea of your **attitude and personality**.

### How much does the TMUA cost?

In order to sit the TMUA, **applicants will have to pay a fee** of either:

#### TMUA UK/EU Fee

£75

#### TMUA International Fee

£100

As well as this, you will need to pay an **additional fee of £35** if you wish to **appeal or enquire **about your results.

**Cambridge Assessments Admissions Testing** (CAAD) do offer a **TMUA bursary scheme** for applicants who meet **certain financial criteria**, so be sure to check if you are eligible if you are concerned about the registration fee.

### How to Register for the TMUA

Registration for the TMUA begins **September 1st**. Those who are attending **school or college** will first need to talk with the **exams officer**. They will inform you if your place of education is **registered as a testing centre**. If not, you are able to register for it become one via the **CAAD** website (this will need to be done by **September 16th**). You may also apply to sit the TMUA at an **alternate independent testing centre **if one is available near you.

If your school/college is already registered, your exams officer will next need to **register you for the TMUA. **For them to do this, you must provide the following:

**Personal Details (Name, DOB, etc)****Your UCAS Number****The names of the course you intend to apply for**

From there, the exams officer will then be able to get you register for the exam. Once complete, you will be given a **candidate number** which will be your **proof of registration.** You must not lose this number as it may be required on testing day. The deadline for registration to the TMUA is **September 30th**.

If you require a **modified paper**, you need to apply for this by **September 16th** and must be able to **provide medical evidence** proving the requirement. Any other **access arrangements** must be booked by **September 30th**, alongside the registration deadline.

### When and where is the TMUA sat?

The testing date for the TMUA is **October 18th in 2023**. This is **two weeks earlier** than previous year’s testing dates of early November, meaning applicants have lost two weeks of valuable preparation time. Therefore, it is incredibly important to **begin preparations as soon as possible** (using **TMUA.Ninja** fast-tracks your progress and get you ready in time).

As previously stated, you should be able to sit the TMUA at your **school or college**. However, if your school **isn’t registered as a TMUA testing centre** or you’re not attending a school or college, you can sit the assessment at an **approved test centre**.

### Who has to sit the TMUA?

The TMUA is accepted by **10 universities**. As previously stated, the only one of these to require the TMUA is **Cambridge** for their **Computer Sciences** course. The rest of these university courses will **consider (but do not require) your TMUA score** alongside the rest of your application:

#### TMUA Universities

- Mathematics (BSc)
- Mathematical Sciences (BSc)
- Mathematics with Statistics (BSc)
- Mathematics with Computer Science (BSc)
- Mathematics with Actuarial Science (BSc)
- Mathematics with Finance (BSc)
- Mathematics with Language (several) (BSc)
- Mathematics, Operational Research, Statistics and Economics (BSc)
- Mathematical Physics (Ma)

### Key TMUA Dates

As we’ve just seen, the TMUA will be sat on **October 18th in 2023**, but there are various other important dates to keep in mind, so be sure to remember these:

Applicants Deadlines 2023 | |
---|---|

Registration Opens | 1st September 2023 |

Modified Papers Registration /Apply to Become a Testing Centre Deadlines | 15th September 2023 |

Registration Closes | 29th September 2023 |

TMUA Testing Date | 18th October 2023 |

TMUA Results Released | 1st December 2023 |

But when should you start preparing? We would recommend starting **six-months in advance!** You can see how we suggest you use your time with our **TMUA Preparation Timeline**.

### What do I Need for the TMUA?

The TMUA is a written assessment, meaning you will be answering everything on **paper**. On the day, you’re going to need take a **soft (HB) pencil** with you for both papers. You will be issued with a separate **answer sheet** for each section on which to indicate your answers.**Calculators are not permitted in the TMUA, along with any other supporting materials**.

However, **taking a watch** with you is always a good idea. Try to bring one that can **show you the time in seconds**, as this will allow you to have a **much more accurate idea of the time you’re spending on a question**.

You will also be required to **bring a form of ID** to the testing centre.

### How Hard is the TMUA?

At first glance, the TMUA may seem like a fairly easy test. You’re given **two and a half hours** to answer **40 multiple choice questions**, all of which are focused on mathematics. If you have **effectively prepared** and are confident with you mathematical abilities, then the TMUA should be **more than possible to succeed in**.

That’s not to say that the TMUA is easy though, as these questions will be testing you on **advanced level maths**. You’re going to need to be confident with maths up to **A-Level difficulty** (or equivalent), so some of the topics covered will be fairly new to you. **Everything covered will be in your school curriculum however**.

Be aware that** Paper 2 may trip you up** if you haven’t properly practiced it. These questions rely more on **mathematical reasoning** than just pure mathematics, so the **logic-based questions** may catch you off guard.

### Can I resit the TMUA?

Students **cannot attempt the test twice** during the same admissions cycle. If you apply again in another admissions cycle, you must take the test again.

Some universities do accept **alternate admissions tests**, including the **MAT and STEP**, so be sure to take one of these exams if you’re unhappy with your TMUA score.

### How is the TMUA scored?

In the TMUA, **each question is worth** **one mark**. Your raw marks out of 40 will be **how many questions you got correct**. There is **no negative marking** so incorrect answers will not deduct marks from your total.

For your final result, your raw marks are **converted and placed into a scale from 1.0 to 9.0.** These are the scores that your **prospective universities** will see. Below you will find the **Score Conversion Tables** for both papers of the TMUA, as well as the **overall conversion** out of 40:

#### TMUA Paper 1 Score Conversion Chart

Paper 1 Raw Score | Paper 1 Grade |
---|---|

20 | 9.0 |

19 | 9.0 |

18 | 8.3 |

17 | 7.9 |

16 | 7.6 |

15 | 7.3 |

14 | 7.0 |

13 | 6.8 |

12 | 6.6 |

11 | 6.1 |

10 | 5.6 |

9 | 5.0 |

8 | 4.5 |

7 | 3.9 |

6 | 3.3 |

5 | 2.6 |

4 | 1.9 |

3 | 1.0 |

2 | 1.0 |

1 | 1.0 |

0 | 1.0 |

#### TMUA Paper 2 Score Conversion Chart

Paper 2 Raw Score | Paper 2 Grade |
---|---|

20 | 9.0 |

19 | 9.0 |

18 | 8.3 |

17 | 7.8 |

16 | 7.5 |

15 | 7.2 |

14 | 6.9 |

13 | 6.7 |

12 | 6.3 |

11 | 5.7 |

10 | 5.1 |

9 | 4.5 |

8 | 3.9 |

7 | 3.3 |

6 | 2.7 |

5 | 1.9 |

4 | 1.1 |

3 | 1.0 |

2 | 1.0 |

1 | 1.0 |

0 | 1.0 |

#### TMUA Overall Score Conversion Chart

OVERALL RAW SCORE | 40 | 39 | 38 | 37 | 36 | 35 | 34 | 33 | 32 | 31 | 30 | 29 | 28 | 27 | 26 | 25 | 24 | 23 | 22 | 21 | 20 | 19 | 18 | 17 | 16 | 15 | 14 | 13 | 12 | 11 | 10 | 9 | 8 | 7 | 6 | 5 | 4 | 3 | 2 | 1 | 0 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

OVERALL GRADE | 9.0 | 9.0 | 9.0 | 8.6 | 8.3 | 8.1 | 7.9 | 7.7 | 7.5 | 7.4 | 7.2 | 7.1 | 7.0 | 6.8 | 6.7 | 6.6 | 6.5 | 6.2 | 5.9 | 5.6 | 5.4 | 5.1 | 4.8 | 4.5 | 4.2 | 3.9 | 3.6 | 3.3 | 3.0 | 2.6 | 2.3 | 1.9 | 1.5 | 1.1 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 |

### How do the universities get my results?

As the TMUA is **non-compulsory** for most universities, **your results will not be automatically sent to them**. Instead, you will need to **submit them yourself** once you have received them on November 29th. Here is the process of doing so:

- At the end of your test, you will receive a
**Confidential Results Information sheet**. This contains**login details**for you account on**Metritests**. This is where you will be able to access your**TMUA results**when they are published. - Under the
**TMUA test option**, you will need click “**Share Results**” in order to begin the process. - You will then need to
**select the universities**you wish to send the results to, as well as the degree. Once you have saved this option on the site, the results will be**immediately available for the universities to see**.

Please note that sending the results is **not a requirement after you have completed the TMUA**. If, for whatever reason, you decide you don’t want your chosen universities to see your results, then simply **do not complete the process** outlined. The rest of your application **will not be affected** by this.

If you’re applying to the **University of Cambridge**, your **results will be automatically sent** to the university. There is **no option to withhold your results**. We have a full guide about **TMUA Scoring and Results** that will show you what your score means and how it is used.

**It’s never too early to begin your TMUA preparation, so start now with TMUA.Ninja!**

With our TMUA Preparation Platform, you can get access to **100’s of practice questions** for everything covered in the TMUA. Sign up now **for free** to begin your TMUA prep.

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## TMUA PAPER 1

With the basics out of the way, let’s take look at the first paper in the TMUA, which is all about pure mathematics.

**Paper 1** of the TMUA is (obviously) is the first half of the TMUA. Here, you will find **20 multiple-choice questions** testing your **pure mathematics skills** in a variety of disciplines. While the questions are relatively **straight-forward format-wise**, they are far from easy!

When we say that they’re straight-forward, we mean that you’re going to **solving mathematical problems** and not too much else.You’re goal here is to **find the solution** to the mathematical problem presented to you, which is what you’d typically expect from a maths exam at this level.

You can see below exactly what to expect from a page on the TMUA:

### TMUA Paper 1 Example Page

Each page contains just **one question**, which means you’ll have plenty of space for **rough work**. Remember that **nothing on the question paper will be marked**, so you can use the space provided for any kind of calculations that you need (which we highly encourage you to make the most of).

### TMUA Paper 1 Specification

Next up, it’s time to look at the **exam specification for the TMUA**. This is the official list of **every topic** that will potentially appear in theTMUA. The specification is split into **two sections**, Section 1 and Section 2. Section 1 covers all of the **pure mathematical knowledge** that you’re going to need to know when sitting the TMUA and is required for **both papers**.

Section 1 is further spilt into **two ‘Parts’**, Part 1 and Part 2. Essentially, Part 1 covers **Advanced Maths** while Part 2 covers **Standard Maths** (which feels like the wrong order!). It feels strange to begin with the advanced knowledge required for the exam, so we’ll go out of the order of the specification and** begin with Part 2: **

#### TMUA Specification: Section 1, Part 2

- Use the four functions on whole numbers, integers, fractions, decimals and numbers in index form.
- Use terms relating to factors and squares.
- Use index laws.
- Understand equivalent fractions.
- Convert between fractions, percentages and decimals.
- Understand direct and indirect proportion.
- Use ratio notation.
- Use surds and Pi in exact calculations.
- Understand number operations and their uses.
- Calculate upper and lower bounds to contextual problems.
- Know and use approximation methods to a specified degree of accuracy.

- Distinguish the roles of different letter symbols.
- Manipulate algebraic terms.
- Use index laws in relation to algebra.
- Know and use (
*a*)^{b}=^{c}*a*.^{bc} - Set up and solve linear equations, including simultaneous equations.
- Factorise quadratics.
- Simplify rational expressions
- Set up and solve quadratic equations.
- Derive formulae, substitute into a formula and change the subject of a formula.
- Solve linear inequalities.
- Generate terms of a sequence and describe the nth term with linear expressions.
- Use Cartesian coordinates in 4 quadrants.
- Know and understand the equations of straight lines, including
*y*=*mx*+*c*. - Solve simultaneous equations with a linear and a quadratic.
- Recognise and interpret various graphs, including quadratic functions and cubic functions.
- Construct linear functions from actual problems, including graphs.

- Know and use properties of angles at various points.
- Understand and use various angle properties.
- Calculate and use the fumes of angles in polygons.
- Recognise and use reflectional and rotational symmetry in 2D shapes.
- Understand congruence and similarity.
- Use Pythagoras theorem in 2D and 3D.
- Understand on create geometrical proofs, include the use of circle theorems.
- Represent 3D shapes in 2D.
- Describe and transform 2D shapes with various techniques.
- Understand and use the standard trigonometric ratios of sin, cos and tan.

- Calculate the permitters and areas of shapes.
- Find the circumferences and areas of circles.
- Calculate volumes and surface areas from 3D shapes.
- Use vectors and the sum of two vectors.
- Use and interpret maps and scale drawings.
- Understand and use the effect of enlargement.
- Convert between two units of measurement.
- Know the SI prefixes in connection to SI units.
- Understand and use three-figure bearings and compound measures.

- Identify potential sources of bias.
- Identify flaws in various data collection methods.
- Group and understand discrete and continuous data.
- Extract data from tables and lists.
- Design and use two-way tables.
- Interpret various forms of chart, including bar charts and line graphs.
- Interpret cumulative frequency in various forms, including tables and graphs.
- Calculate and interpret averages in data, including mean, median and mode.
- Calculate average rates in combines samples/events.
- Interpret scatter diagrams and identify correlations with line of best fit.
- Make comparisons between data sets with statistical measures and graphical representations.

- Understand and use the vocabulary of probability and the probability scale.
- Understand and use the estimates and measures of probability, including relative frequency and theoretical models.
- List outcomes fro single and combine events.
- Identify different mutually exclusive outcomes.
- Construct and use Venn diagrams to solve union and intersection categorisation problems.
- Know when to add or multiple two probabilities.
- Understand the use of tree diagrams to represent outcomes of combined events.
- Compare experimental and theoretical probabilities.
- Understand repeated experiments with different outcomes.

None of this should be **particularly tricky** for you, so let’s move straight on to the more **advanced stuff**; Part 1:

#### TMUA Specification: Section 1, Part 1

- Know and use the laws of indices.
- Know and use the manipulation of surds.
- Understand and use quadratic functions, including graphs, use of discriminant and completing the square.
- Know how to sovle simultaneous equations via substitution.
- Know how to solve linear and quadratic inequalities.
- Know and understand the manipulation of polynomials e.g. expanding brackets, factorising.
- Know and use Factor Theorem and Remainder Theorem.

You can check out out **TMUA Algebra Guide** to learn even more about what you need to revise.

- Know and understand Arithmetic series and Geometric Series.
- Understand summing to a finite and infinite geometric series.
- Know and use binomial expansions.
- Know and use factorials.

- Know and use equations for circles.
- Know and use equations for straight lines.
- Know and understand circle properties.

- Know the solution of trigonometric identities.
- Know and understand the values of sin, cost, tan for 0, 30, 45, 60 and 90 degrees.
- Understand and interpret Sine, Cosine, Tangent graphs, symmetries, periodicities.

- Know and understand the graph of
*y*=*a*series.^{x} - Solve equations of the form
*a*=^{x}*b*. - Understand and use the Law of Logarithms:
*a*^{b }*= c <-> b = log*_{a}*c**log*_{a}*x + log*_{a}*y = log*_{a}*(xy)**log*_{a}*x – log*_{a}*y = log*_{a}*(x/y)**k log*_{a}*x = log*_{a}*(x*^{k}*)**log*_{a}*1/x = – log*_{a}*x**log*_{a}*a = 1*

- Know and understand first order and second order derivatives.
- Know and use differentiation of functions like
*y*=*x*.^{n} - Have familiarity with notation.

- Know and use definite and indefinite integrals for
*y*=*x*^{n } - Solve Differential Equations in the form:
*dy*/*dx*=*f*(*x*). - Understand the Fundamental Theorem of Calculus and its application.
- Combining integrals with either equal or contiguous ranges
- Solve differential equations in the form of
*dy*/*dx*=*f*(*x*). - Know and use the trapezium rule.

- Understand sketching of common functions including lines, quadratics, cubics, trigonometric functions, logarithmic functions and exponential functions.
- Know and use the graph of series
*y*=*a*^{x}. - Know and use the manipulation of functions with simple transformations.
- Use geometric interpretation of algebraic solutions of equations.

That’s a fair amount to take in, but **none of this should be new to you**. Remember that the TMUA deals with content knowledge up to **A-Level difficulty**, so everything in the specification should be covered within your **school’s curriculum**. Just bear in mind the order your school teaches in may mean **some topics are not covered before you sit the exam** in October. If this is the case, you will need skip forward a bit and **teach the topic to yourself**, although **asking your teacher for help** is always a good idea!

If you want to **learn more** about this part of the exam, want to try out some practice questions or are just looking for some preparation tips, be sure to check out our **TMUA Paper 1 Guide**.

### TMUA Paper 1 Practice Questions

With all that in mind, it’s time to take a look at some **TMUA Practice Questions** to put your knowledge to the test:

#### TMUA Paper 1 Practice Question 1

**Given that, in the expansion of (3 x + b)^{7}, the coefficient of x^{4} is the same as the coefficient of x² in (3b + x)^{4}, find the positive constant b.**

A) 2/105

B) 105/2

C) 107/3

D) 3/107

E) 109/4

**The correct answer is A.**

**In the first expansion, the coefficient of x^{4} is given by (^{7}_{4})3^{4}b^{3} = 2835b³ and in the second, the coefficient of x² is (^{4}_{2})(3b)^{2} = 54b², and equating, we see that b = 54/2835 = 2/105. **

#### TMUA Paper 1 Practice Question 2

Find the distance between the intersection points of *y* = |*x* – 4| and *y* = 5 – |*x* – 1|.

A) There are no intersections

B) 4

C) √34

D) 5

E) 3

F) √41

**The correct answer is C.**

**Knowing that the highest point of y = 5 – |x – 1| will be at x = 1 tells you that point is at (1,5), which is above the line y = |x – 4| as it is at (1,3) there. This tells you there are two intersection points, with a quick sketch.**

**While this may or may not be to scale, it doesn’t matter; we can see from the sketch that there is one intersection when both are in their first half, and one in their second half, whether the apex or base of either two graphs should be closer or further apart is irrelevant. Thus, we solve 5 – ( x -1) = x – 4 (gives x =5) and 5 + x – 1 = 4 – x (gives x = 0).**

**These points are (5,1) and (0,4) ****respectively and using Pythagoras’ Theorem tells us their distance is √(5 – 0)² + (1 – 4)² = √25 + 9 = √34.**

#### TMUA Paper 1 Practice Question 3

**Every day, a man decides whether to walk his dog with probability 1/3 that he does take it for a walk, independently of choices made on other days. After 4**** days, what is the probability that he walked his dog at least once every 2 days?**

A) 1/27

B) 7/27

C) 8/27

D) 11/27

E) 1/9

F) 1/3

**The correct answer is B.**

**For the man to walk his dog “at least once every two days”, we know he never had two consecutive days which he did not walk the dog. So, representing going for a walk as W and not going as N, the possible ways he spent his 4 days are WWWW, WWWN, WWNW, WNWW, NWWW, NWNW, NWWNWNWN. These are not all equally likely as each W has 1/3 chance of appearing, and each N has a 2/3 chance. Thus, the probability that each set of 4 days occurred is 1/81, 2/81, 2/81, 2/81, 2/81, 4/81, 4/81, 4/81.**

**As we only want to know the probability any one of these occurred, we can sum them, to get (1+ 2 x 4 + 3 x 4)/81 = 21/81 = 7/27. **

#### TMUA Paper 1 Practice Question 4

**When is (sin2 x)/(1 – cos2^{x})(3x -2) ≥ 0 in the range of 0 < x < π?**

A) log_{3}2 ≤ *x* ≤ (π/2)

B) log_{3}2 < *x* < (π/2)

C) (π/2) < *x* ≤ log_{3}2

D) 0 < *x* < log_{3}2

E) (π/2) ≤ *x* ≤ log_{3}2

F) 0 < *x* ≤ log_{3}2

**The correct answer is A.**

**Using trigonometric identities, cos2 x = 1 – 2sin²x, sin2x = 2sinxcosx.**

**This means that sin2 x/(1-cos2x) = cosx/sinx.**

**We then know the critical values of x are when cosx is equal to 0(x = nπ/2). The other part of the expression has only 1 critical value at x = log_{3}2. All that’s left to determine is whether they are positive or negative around these values.**

**(3^{x} – 2) is increasing, so it must be below the x-axis before x == log_{3}2 and positive after. We only need to consider π/2 which is the only critical value in range for this problem, and both sinx and cosx are positive at x = 0, so the graph must be positive before x = π/2 and negative after. We are looking for where they have the same sign, and we know that log_{3}2 < 1(as log_{3}3 = 1) so it must be log_{3}2 ≤ x ≤ (π/2).**

**There’s a lot to revise for the TMUA, so why not get some help from TMUA.Ninja?**

With our **TMUA Preparation Platform**, you can get access to **guides and practice questions** for **all content** of the TMUA. Click below to find out how you can **optimise your TMUA revision**.

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## TMUA PAPER 2

Paper 2 of the TMUA features questions that you may not be as used to. You’re going to need to implement skills in mathematical logic to get a good score here, so let’s see what you need to know!

We’ve mention this a few times already, but it’s important to properly understand that **Paper 2 of the TMUA does not feature the same style of mathematical questions as Paper 1**. To properly understand why this is, let’s look at an **example of a typical page** in Paper 2:

### TMUA Paper 2 Example Page

The difference should be apparent immediately. Questions here are **not about simply solving a problem**, but using the solution to **answer questions within the context of the question**, or to **evaluate a problem and/or solution** and discuss it’s reasoning or its flaws.

This makes its sound like an entirely different subject than Paper 1, but **the difference isn’t as extreme as you think**. These are still undeniably **maths questions**, and as such you will be required to **use maths to solve a problem**. The key thing is that you will have **extra steps and extra things to consider **when doing so.

Even though the question may ask you to **disprove a mathematical argument** or **explain a problem**, you’re still going to be using the **core principles of mathematics** that you’ve been learning throughout the years.To understand this further, lets move on to the **specification** for Paper 2:

### TMUA Paper 2 Specification

First things first: there is **no new mathematical knowledge **that will be required for this paper. Everything that will be included here is covered in** Section 1** of the **TMUA Specification**. That will certainly make your revision a bit easier, as everything you need to know for both sections is covered there; at least in terms of pure maths.

As you could probably guess, the **different style of questions** for Paper 2 will require some **different types of skills and techniques**. These have all be **outlined in the specification **under **Section 2**, so let’s take a look at what you need to practice:

#### TMUA Specification: Section 2

##### The Logic of Argument

This is a concept that definitely isn’t unique to maths but **will be required for a lot of the questions** you’ll encounter during your revision and on testing day. Using this in mathematics may be **slightly easier** than a subject like politics however, as a Mathematica problem is **always going to have an objectively correct solution**.

**Mathematical logic**, as it’s called in the context of the TMUA, will be used for **fairly simple scenarios** here, so it’s not going to be too complex. Essentially, this will be your method of **developing a correct statement to answer the question** you’ve been asked. It will typically take a form like these:

**If A then B **

**A if B **

**A only if B **

**A if and only if B**

So for example, **if x is odd, then x + 1 is even**. Keep in mind that you will also need to understand the **converse and contrapositive **of your statement.

On the other side of this, you will need to be able to **analyse statements **included within** **questions that will ask you to **prove or disprove them**.

Some of the key terms of revise here include **necessary, sufficient, for all, for some and there exists**.

##### Mathematical Proof

All of these questions are **multiple-choice**, so you’re not going to need to be writing up full statements providing the proof of your findings. However, some questions will still relate to this in someway, whether it’s due to the question providing **different forms of proof in its answers**, or potentially asking you to **disprove an argument due to missing or insubstantial proof**.

There are **different forms of proof** that you’re going to need to consider include:

- Direct deductive proof
- Proof by cases
- Proof by contradiction
- Disproof by counter example

These are the **key principles to mathematical proof** highlighted by the specification:

- Deducing implications from given statements
- Rearranging statement sequences to give a correct order
- Using sophisticated chains of reasoning to solve

These will all be **important techniques** to practice when solving these questions.

##### Identifying Error in Proofs

This one is linked pretty closely to the last point, seeing as they both relate to **mathematical proof**. We touched on this briefly before, but some questions are going to require you to **review a given sequence of statements or work** in order to **determine errors** within them.

In the TMUA, this is referred to as **identifying errors in purported proofs**, as what you’re reviewing will be presented as **the proof to a solution** (basically the working out). Of course, **there will be an error** in there somewhere and your job could be one of various things.

You may be required to simply **identify the mistake within the proof**. However, the questions could potentially go further and ask you to **determine how the error was caused** in the first place, essentially **finding the reasoning **behind it. Of course, you may also be asked to **correct the mistake** and provide the actual solution to initial problem.

There are a few plenty of **simple mathematical errors** to keep an eye out for in purported proofs. A couple of examples given include the following statements:

**If ab = ac, then b = c**

**If sin𝐴 = sin𝐵, then 𝐴 = 𝐵**

These are pretty simple mistakes, but they may be **difficult to spot **within the **large sequences of statements** that are typically featured in Paper 2 questions.

It’s pretty difficult to **properly understand** the use of all these techniques without seeing them in context, as these aren’t the kind of skills you can “**revise**” in a traditional sense. Our **TMUA Paper 2 Guide** goes into more detail about **how you can tackle these questions effectively**. However, **practice is going to be the most important thing you can do to prepare**.

Outside of the TMUA, this is a **very uncommon style of question** which means practicing it can be tricky. You’re unfortunately **not going to be able to find many questions like this **outside of TMUA practice materials and past papers, as **neither the STEP or MAT exam** **have anything that compare** **to Paper 2**. Therefore, you need to be careful not to use up all of your resources too early into your preparation timeline.

It’s good to **take past papers in proper exam conditions**, but you won’t get as much out of this until you’ve **effectively revised your knowledge and skills** in a more casual setting. Of course, the best way to mitigate this problem is with **TMUA.Ninja**, which provides you with **100’s of additional practice questions** and an **easy to use exam simulator**, including **worked solutions** for every question included.

### TMUA Paper 2 Practice Questions

To get your practice started, let’s try out some **practice questions **and get a better idea of what to expect:

#### TMUA Paper 2 Practice Question 1

**Consider the statement: “If n is an integer and n² is divisible by 4, then n is divisible by 4”.**

**How many counterexamples are there to this in the range 50 ≤ n ≤ 100.**

A) 24

B) 25

C) 11

D) 12

E) 13

**The correct answer is E.**

**Clearly we need to count the number of multiples of two, but not of four, in the range. So 50,54,58,…,98. There are 13.**

#### TMUA Paper 2 Practice Question 2

**A Pythagorean triple is where 3 positive integers, a, b and c fit the equation a² + b² = c².**

**The 2 smallest integers of a Pythagorean triple are 20 and 21. Find the third integer. Comparing this to the smallest Pythagorean triple, what is the highest common factor of the sum of the integers in both equations and the integer formed by the two largest numbers plus the 2 smallest numbers from both equations.**

A) 1

B) 2

C) 3

D) 5

E) 7

F) 11

G) 13

H) 12

**The correct answer is A.**

**20² + 21² = 29²**

**The smallest triple is 3 ^{2} + 4^{2} = 5^{2}. If this is not known, it can easily be worked out by writing down all the square numbers until you see 2 that will add to make another one. **

**Sum of the integers in both equations – 20 + 21 + 29 + 3 + 4 + 5 = 82.**

**Product of the two largest numbers minus the 2 smallest numbers from both equations = (29 +21) + (3+4) = 57.**

**HCF of 57 and 82:**

**82 factors: 1, 2, 41, 82**

**57 factors: 1, 3, 19, 57**

**HCF = 1**

#### TMUA Paper 2 Practice Question 3

**Consider the graphs of the two functions y = mx + 10 and y = log_{2}x. Which of the following correctly identifies the sufficiency and necessity of:**

*m*< 0*m*> 10

**in relation to the statement that there are intersection points of the two graphs?**

A) Both necessary and sufficient.

B) Both necessary.

C) Both sufficient.

D) 1 is sufficient, the other is necessary.

E) Only 1 is necessary.

F) Only 1 is sufficient.

G) Neither is necessary or sufficient.

**The correct answer is F.**

**We want the two graphs to intersect but equating the expressions won’t get us very far. Instead, a sketch of both graphs is more illuminating. The line intersects the y-axis at 10, which is much higher than the other line. The gradient of log _{2}x is decreasing, and as the gradient of the straight line is constant, if it begins large enough, they will never meet. 10 is one such value and so anything larger will also never meet the curve. However, there are also slightly smaller values which would also work, so it is not necessary. For m negative, the line must proceed downwards from(0,10). One can see that, as eventually log_{2}x = 10, even an almost flat decreasing straight line would intersect the curve. Thus, no matter the value of m, if it is negative, the lines must cross. So, 1 is sufficient, 2 is neither.**

#### TMUA Paper 2 Practice Question 4

**Consider the following proof of the alternate segment theorem:**

**Let a tangent touch a circle at***A*and let*B*,*C*be two points on the circumference of the circle. Let the angle between the line*CA*and the tangent b α°.**By drawing radii from***C*&*A*to the centre of the circle, which we will call*O*, we know that the angle between*OA*and the tangent is 90°.**This means the angle***BAO*is (90 – α)°.**Since the triangle***OBA*is isosceles, the angle*OBA*is also (90 – α)°.**Then the angle***AOB*is 180° – (180 – 2α)° = 2α°**This means the***angle ACB is*(2α/2)° = α°.

**Where is the first error?**

A) The proof is correct

B) Line 1

C) Line 2

D) Line 3

E) Line 4

F) Line 5

**The correct answer is C. **

**By drawing the circle and tangent, the error becomes relatively clear. The angle that would be equal to α using the alternate segment theorem would be ABC, not ACB as is calculated in the question, so the result is clearly false. To find the error, notice that angle BAO has nothing to do with α, and it is in fact angle CAO that is(90 – α)º. So, the error is on line III.**

These practice questions should highlight **how different these questions can be to what you’re used to**, but should also show you the **similarities** and ease your concerns about them. Remember; **it’s still maths**, it’s just **presented a bit differently** and requires a **slightly different viewpoint**. As long as your **maths knowledge** is on point, you shouldn’t face any major problems (you still need to practice, of course!)

### TMUA Final Tips

Here are a few **final tips** to help you with your preparation and exam techniques:

- Plan your revision ahead of time, the benefits will become obvious as soon as you start!
- Gather as many resources as possible. TMUA resources currently aren’t the easiest to find so don’t leave this to the last minute (TMUA.Ninja is most likely the best option!).
- Learn your equations off by heart, these will be essential for many of the questions included.
- As you get closer to the exam, you definitely need to take regular mock exams with past papers or practice questions. They will help you get used to the time limits of the exam.
- In the exam, don’t be afraid to leave particularly difficult questions for the end. With only 20 questions to do in 75 minutes, you’re more than likely going to have enough time come back to them.
- Be sure to leave an answer for every question in the test, even if it’s just a guess. There’s no negative marking so you need to give yourself the chance to get the mark.

So, that completes the **first step of your TMUA preparation**. You now have a good understanding of what the TMUA actually is and what it’s going to test you on, so the next step is to **begin your revision**. We would recommend checking out more of our **TMUA guides** or trying out our **free TMUA.Ninja demo**, which will let you trial our extensive suite of tutorials, practice questions and past papers! Whatever you decide to do, be sure to **revise consistently** up until testing day to give yourself the best chance of success!

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