Top 200 Vocabulary Word List for the Test of Mathematics for University Admission (TMUA) | Must Know Words for Common Examination Instructions

The 200-word examination instruction list

TMUA instructions ask you to apply mathematics and distinguish claims that follow from claims that only look plausible. Necessary and sufficient describe different directions of a condition. A counterexample can defeat a universal claim. Read the domain and quantifier before testing an answer.

The numbered bank combines question directions, response formats and supporting terms. Check the official instructions for the exact item and version you are taking.

Exam requirements · Worked examples · Fencing Method

TMUA papers, marking and logical conditions

No.	Instruction or term	What it asks you to do
1	Applications of Mathematical Knowledge	Use mathematical techniques to solve the type of problem asked in Paper 1.
2	Mathematical Reasoning	Use logical deduction, reasoning and proof in the Paper 2 style.
3	Paper 1	Practise solving problems using the content in the TMUA specification.
4	Paper 2	Practise checking statements, implications and arguments as well as solving problems.
5	Multiple-choice paper	Choose the option that satisfies the conditions in the item.
6	No calculator	Use mathematical reasoning and arithmetic without a calculator.
7	Correct answer only	Choose the correct option under the stated marking rules.
8	No penalty for wrong answers	Check the official marking rule and give an answer to every question you can.
9	Necessary condition	Identify what must hold whenever the stated outcome holds.
10	Sufficient condition	Identify what guarantees the stated outcome.
11	Necessary and sufficient	Establish the implication in both directions.
12	If	Identify the hypothesis and what it guarantees in the stated rule.
13	Only if	Identify the condition that must hold when the stated outcome holds.
14	If and only if	Check that each statement implies the other.
15	Converse	Swap the hypothesis and conclusion and test the resulting statement separately.
16	Contrapositive	Reverse the implication and negate both statements to produce an equivalent implication.
17	Negation	State precisely what would make the original proposition false.
18	Implication	Read which statement guarantees the other.
19	Equivalence	Check both directions or transform each statement into the other.
20	For all	Check that the statement holds across every value in the specified domain.
21	There exists	Find at least one permitted value or case that works.
22	Exactly one	Establish that there is a solution and that a second one cannot occur.
23	At least one	Find one permitted example to establish existence.
24	At most one	Show there cannot be two or more valid cases.

Proof, counterexamples and solution checks

No.	Instruction or term	What it asks you to do
25	Counterexample	Give one permitted case where the proposed general statement fails.
26	Direct proof	Use valid steps from the assumptions to establish the conclusion.
27	Proof by contradiction	Assume the claim fails, then show that this contradicts an established fact.
28	Proof by cases	Cover all possible cases and establish the conclusion for each.
29	False converse	Recognise when a conclusion wrongly assumes an implication works in reverse.
30	Invalid step	Find the specific transformation or inference that does not follow.
31	Extraneous solution	Identify a value introduced by a manipulation that fails the original condition.
32	Lost solution	Check whether dividing or simplifying excluded a value that could work.
33	Domain restriction	Keep only the inputs for which the original expression or situation is defined.
34	Range of values	Find all possible outputs under the stated restrictions.
35	Excluded endpoint	Use a strict boundary where the endpoint itself is not allowed.
36	Included endpoint	Use a non-strict boundary where the endpoint is allowed.
37	Parameter	Use the fixed or adjustable quantity that defines a family of cases.
38	Existence	Show that at least one object or value meets the conditions.
39	Uniqueness	Show that only one permitted object or value meets the conditions.
40	Assume without loss of generality	Check that the cases are symmetric before simplifying to one representative case.
41	Equivalent transformation	Use an operation that preserves the solution set under the stated conditions.
42	Squaring both sides	Check signs and verify candidate answers in the original statement afterwards.
43	Divide by a variable	Check whether the divisor can be zero before taking the step.
44	Change of sign	Use the sign changes and stated assumptions relevant to the inference.
45	Check all cases	Test each permitted case against the original question.
46	Quantifier	Check whether the claim applies to all, some, none or exactly one of the allowed cases.
47	Disjunction	Read the alternatives and check whether the wording permits both to hold.
48	Conjunction	Check that both required statements hold together.
49	Mutually exclusive cases	Use cases that cannot occur together when the problem calls for a partition.
50	Exhaustive cases	Check that the cases cover every allowed possibility.
51	Open interval	Exclude both endpoints when the interval is open at both ends.
52	Closed interval	Include both endpoints when the interval is closed.
53	Piecewise function	Apply the rule for the part of the domain containing the input.
54	Monotonic	Check whether the function moves in one direction across the stated interval.
55	Increasing function	Compare outputs for ordered inputs and distinguish strict increase where required.
56	Decreasing function	Check whether outputs fall as inputs rise within the stated domain.
57	Discriminant	Use the sign of the quadratic discriminant to reason about real roots.
58	Repeated root	Identify a root with multiplicity greater than one.
59	Distinct roots	Count different values of the roots rather than multiplicities.
60	Intersection points	Find the coordinates at which the stated graphs share the same point.
61	Tangent condition	Use the relevant contact or gradient relationship to establish tangency.
62	Gradient of the tangent	Find the derivative at the named point when that method applies.
63	Sign chart	Determine where the expression is positive, negative or zero.
64	Absolute value	Use the distance from zero or separate cases based on the expression’s sign.
65	Test the boundary	Check the endpoints and any values where the relevant rule changes.

Algebra, graphs and mathematical reasoning

No.	Instruction or term	What it asks you to do
66	Expand and simplify	Multiply out the brackets and combine like terms.
67	Make the subject	Rearrange the formula so the named quantity is isolated.
68	Solve the inequality	Find the set of values that make it true and keep the correct inequality direction.
69	Find all solutions	Include every value that meets the stated equation and domain.
70	Find the nth term	Write the expression that gives the term at position n.
71	Write in standard form	Express the number as a value from 1 up to but excluding 10 times a power of ten.
72	Give an exact value	Use a form that preserves the full mathematical value.
73	Give your answer to one decimal place	Round to one digit to the right of the decimal point.
74	Give your answer to two significant figures	Keep two significant digits in the rounded result.
75	Round to the nearest integer	Choose the closest whole-number value under the rounding rule.
76	Construct	Use the stated tools and method to make an accurate geometric drawing.
77	Sketch	Show the requested shape and key features without requiring exact scale unless stated.
78	Plot	Mark the points at the given coordinates or values.
79	Measure	Take the required reading from a diagram or scale when the task permits it.
80	Not drawn accurately	Use the supplied mathematical facts rather than the apparent appearance of the diagram.
81	Draw to scale	Use the scale given to make a drawing with the required proportions.
82	Identify a graph transformation	Determine how the expression changes the position or shape of the graph.
83	Locus	Show the set of points satisfying the specified condition.
84	Error interval	Give the range of possible original values using the correct endpoint signs.
85	Upper bound	Identify the upper limit of values consistent with the stated precision.
86	Lower bound	Identify the lower limit of values consistent with the stated precision.
87	Estimate from a graph	Use the scale and plotted information to obtain a sensible approximate value.
88	Compare the distributions	Use corresponding measures of location or spread as the question asks.
89	Assume	Use the stated model or simplifying condition in the calculation.
90	Explain a limitation	Identify the assumption or data issue that restricts the conclusion.
91	Use the formula	Substitute the appropriate values with consistent units.
92	Give units	Attach the required measurement unit to the final result.
93	Independent events	Check whether one event changes the probability of the other.
94	Use the frequency	Use the count for the named value or interval rather than a percentage unless requested.
95	Change to	Express the value in the requested form or unit.
96	Does the premise imply the claim	Check for a valid inference or a counterexample.
97	Do not use a graphical method	Use an algebraic method or interpretation as the task directs.
98	Estimate the mean from grouped data	Use class midpoints and frequencies to estimate the total and divide by the total frequency.
99	Find a counterexample	Give a permitted value or case for which the general statement fails.
100	One possible answer	Give one value or example meeting all of the stated conditions.
101	Integer solutions	Include only the whole-number values, including negative values or zero where allowed.
102	Prime factors	Use factors that are prime numbers and preserve their multiplicities.
103	Product of prime factors	Write the number as the product of the prime factors that compose it.
104	Highest common factor	Find the largest positive whole number that divides each named number.
105	Lowest common multiple	Find the smallest positive whole number that is a multiple of all named numbers.
106	Perpendicular	Use the right-angle relationship in the construction or reasoning.
107	Enlargement scale factor	Use the multiplier of corresponding lengths from the centre of enlargement.
108	Congruence condition	Choose the condition that establishes that the triangles have the same shape and size.
109	Frequency polygon	Plot the frequencies at the class midpoints and join the points as instructed.
110	Relative frequency estimate	Use the observed proportion as the estimated probability when the task asks for it.
111	Conditional probability	Use the sample space that remains once the stated condition is known.
112	Combined mean	Use total values and sample sizes to find the mean across the groups.
113	Repeated percentage change	Apply each period’s multiplier to the result from the previous period.
114	Optimise over the domain	Check the allowed values before deciding which output is largest or smallest.
115	Check the proposed working	Use the shown steps to judge whether each follows from the previous one.
116	Diagrams are not accurately drawn	Base calculations on the provided labels and established geometry.
117	Express as a fraction	Give the result in fractional form.
118	Give an upper bound	Find the upper endpoint for possible values under the stated rounding rule.
119	Give a lower bound	Find the lower endpoint for possible values under the stated rounding rule.
120	Iteration	Repeat the given update rule with the stated starting value.
121	Write an inequality	Translate the condition into an inequality with the correct sign and direction.
122	Draw a locus	Mark the set of points satisfying the stated geometric rule.
123	Perpendicular gradient	Use the condition for perpendicular nonvertical lines where the stated assumptions apply.
124	Measure the bearing	Give the clockwise angle from north using the convention specified.
125	Cumulative frequency	Keep a running total through the ordered values or intervals.
126	Estimate the median	Use the midpoint of the ordered data or the appropriate position on a grouped display.
127	Relative frequency	Divide the observed count for the event by the number of trials.
128	Expected frequency	Multiply the probability by the number of trials when the stated model applies.
129	Determine the graph shift	Identify the direction and size of the translation from the new expression.
130	Explain one difference	Identify one matched contrast and explain its relevance or basis if required.
131	Calculate a rate	Divide the relevant change or quantity by the appropriate time or other denominator.
132	Calculate the percentage change	Divide the change by the starting value and multiply by 100.
133	To scale	Use the stated relationship between drawing distances and actual distances.
134	Not to scale	Use the stated measurements and relationships when calculating.
135	Exact value	Leave the answer in a form that preserves its full value.
136	Decimal places	Count the required digits to the right of the decimal point.
137	Nearest whole number	Round to the closest integer.
138	In terms of	Express the result using the named quantity or symbol.
139	Simplify	Rewrite the expression in an equivalent simpler form.
140	Factorise	Write the expression as a product of factors.
141	Solve	Find the values that satisfy the equation or inequality.
142	Substitute	Replace a variable with the given value or expression.
143	Coordinate	Give the position using the specified ordered values.
144	Line of best fit	Draw a line representing the overall pattern of plotted data.
145	Independent variable	Identify what the investigator changes deliberately.
146	Dependent variable	Identify the measured outcome of the investigation.
147	Control variable	Identify what should be kept constant for a fair comparison.
148	Measurement	Read or calculate a quantity and report the appropriate unit.
149	Estimate from the graph	Use the plotted trend to obtain a sensible approximate value.
150	Geometric reasoning	Use the stated properties to decide which conclusions follow.

Core instructions and response checks

No.	Instruction or term	What it asks you to do
151	Identify	Name or pick out the item the question specifies.
152	Select	Choose only the number of options requested.
153	Give	Supply the requested fact, example or result.
154	State	Express the required point directly and clearly.
155	Name	Supply the correct term, person or item.
156	List	Set out the requested items separately.
157	Define	Give the precise meaning used in the subject or question.
158	Describe	Report the relevant characteristics, events or pattern.
159	Explain	Connect the result to its cause, mechanism or reasoning.
160	Compare	Set the items side by side on the same relevant feature.
161	Assess	Judge the extent, value or strength of something using evidence.
162	Illustrate	Make an idea clear with a relevant example or diagram.
163	Demonstrate	Show how a fact, process or result can be established.
164	Determine	Work from the given information to establish an answer.
165	Check	Test the answer against the original instruction or condition.
166	Correct	Repair the specific error while preserving the intended meaning or result.
167	Sequence	Put stages into a meaningful order.
168	Based on	Ground the answer in the material named in the question.
169	Specific	Precise enough to identify a particular fact or example.
170	Conclusion	The result reached after considering the reasoning or evidence.
171	Assumption	A premise taken as given for the purpose of the question.
172	Condition	A requirement that must be satisfied.
173	Record	Write the result in the space and form provided.
174	Classify	Assign each item to a category using the stated rule.
175	Actual length	Convert the drawing measurement using the scale or ratio given.
176	Account for	Explain why the stated result or difference occurred.
177	Find in terms of x	Give an expression involving x rather than a decimal obtained by choosing a value for x.
178	Check the solution	Substitute the result into the original equation to verify it.
179	Find the exact value	Retain radicals, fractions, logarithms or π in exact form where appropriate.
180	Turning point	Read or find the maximum or minimum point of the graph when the task asks for it.
181	Identify the feature of the graph	Use the graph or equation to decide what the named point represents.
182	Find the area	Find a non-negative area and account for changes in sign or intersecting curves.
183	Set up an equation	Translate the given conditions into a mathematical relationship.
184	Resolve a ratio	Divide the total into parts using the stated ratio.
185	Possible outcome	Identify the event or value that the probability question refers to.
186	Expected value	Find the probability-weighted mean under the stated model.
187	Compare and contrast	Address similarities and differences for the named items.
188	Draw	Provide a labelled and accurate representation where required.
189	Variable	Identify a quantity or feature that can change.
190	Keep the comparison fair	Compare quantities on the same stated basis and units.
191	Rounding interval	Use the stated precision to find the range of values that could produce the rounded result.
192	Sample size	Check how many observations, individuals or trials contribute to the result.
193	Representative sample	Check whether the selected cases reflect the population of interest.
194	Rate of change	Find the change in one quantity per unit change in another.
195	Round	Adjust a number to the requested precision using the next digit.
196	One possible value	Supply a single value that satisfies the given conditions.
197	Equation	Use or form a statement that two expressions have equal value.
198	Inequality	Use or form the required greater-than or less-than relationship.
199	Constant term	Identify the part of the expression with no variable factor.
200	Unit rate	Calculate the amount corresponding to one unit of the other quantity.

What the examination requires

Two multiple-choice papers

UAT-UK lists Paper 1, Applications of Mathematical Knowledge, and Paper 2, Mathematical Reasoning. Each has 20 questions and 75 minutes. Paper 1 applies mathematics in unfamiliar situations. Paper 2 also tests reasoning and ideas from elementary logic. Use the official content specification to check the scope.

No calculator or dictionary

Practise both papers without those tools. UAT-UK says there is no penalty for wrong answers. Use the official practice interface to learn how to review questions, and track which items need a short counterexample and which require more sustained reasoning.

Read the logic and proof guidance

Identify the domain, quantifier and direction of implication before assessing a statement. When a question asks whether a proof is valid, find the exact step where the reasoning succeeds or fails. When a choice claims something is true for all values, check whether any permitted counterexample defeats it.

Check the course and application cycle

UAT-UK publishes test and sitting information, while each university or course sets its own admissions requirement. Check both for your entry year. The provider reports an overall score and does not set a universal pass or fail result for the test.

• Identify the action before deciding on an answer.
• Check the source, situation or condition the instruction refers to.
• Use the requested form and respect any limits.

Worked examples of common examination instructions

These original examples show how a change in the instruction changes what belongs in the answer.

Keep the direction of implication

For integers, being divisible by 6 guarantees divisibility by 3. Divisibility by 3 is necessary for divisibility by 6. It is insufficient on its own, since 9 is divisible by 3 and not by 6. Read which condition the question calls necessary.

Negate a universal statement

The negation of ‘Every integer in the set is even’ is ‘At least one integer in the set is not even.’ A single odd member is enough to disprove the original statement. An answer claiming every member is odd is much stronger than the negation.

Find the step that could lose a solution

From x² = 3x, dividing both sides by x gives x = 3 only if x is nonzero. The original equation also allows x = 0. Factoring gives x(x − 3) = 0 and keeps both cases visible. Check the denominator before dividing.

Check an endpoint before choosing an option

An original question defines f(x) = x² for 0 < x ≤ 3. The range is 0 < f(x) ≤ 9. Nine is attained at x = 3. Zero can be approached but is outside the permitted outputs. Keep the inequality signs from the domain and check which endpoints the function reaches.

Use a counterexample to test a claim about signs

If a statement says a² > a for every positive real a, take a = 1/2. Its square is 1/4, so the claim fails. Testing one value larger than one can produce a true case without proving the general statement.

Use the Fencing Method to practise instructions

Start with an accurate short answer. Add one specific detail and then the relationship, evidence or explanation the task asks for. Use the same instruction in a fresh example.

1. Being divisible by six guarantees divisibility by three.
2. For integers, divisibility by six implies divisibility by three.
3. Divisibility by three is necessary for divisibility by six, and nine shows it is insufficient.
4. Use a different implication. State the domain, identify the direction and look for a counterexample to the reverse.

Read the eduKate Fencing Method for Vocabulary.

Practise and check your answers

Keep the domain beside your working

Write the allowed values before solving an equation or testing a claim. Circle any step that divides by an expression, squares both sides or uses a boundary. Check whether the step preserves the original possibilities. Then substitute candidate values in the original statement.

Practise a proof check and a counterexample check

For one practice item, explain why each step follows. For another, try to defeat the claim with an endpoint, zero, a small positive number or another allowed special case. Use the question’s quantifier to decide when an example is enough and when a general argument is required.

Practice 1. What does exactly one ask you to establish?

Show that at least one case works and that no second case can work.

Practice 2. A solution squares both sides. What should you do at the end?

Substitute the candidate values into the original statement and reject any that fail it.

Practice 3. How do you disprove a claim about all real numbers?

Find one real value allowed by the question that makes the claim false.

Practice 4. Does a true converse automatically follow from a true statement?

No. Test the reverse implication as a separate statement.

Frequently asked questions

Does the TMUA require written proof responses?

Both papers use multiple choice. You can still write intermediate reasoning in practice to decide whether a proposed statement or proof is valid.

Do wrong answers lose marks?

UAT-UK says wrong answers carry no penalty.

Does every mathematics course require the TMUA?

Check the official course entry requirements for the programme and year you are applying to.

Official sources and next reading

• UAT-UK TMUA test format
• UAT-UK official preparation materials
• TMUA content specification

Related guide in this series · AQA GCSE Mathematics examination instructions

Read eduKateSG’s 200-word vocabulary guide for more ways to group and practise vocabulary.

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