TMUA stats — 280 questions, 82 methods

Glossary

Curated question: — Past-paper question with a worked solution and a tagged method.
Method: — Portable technique that pops up across years.
Worked solution: — Step-by-step solution written by us in our voice.
Past paper: — Official exam session published by the board.

These pages only count the curated set: questions with both a worked solution and at least one tagged method. Bulk past papers without that treatment are hidden.

1. Overview

Global profile of the TMUA corpus on 8 years (2016–2023).

280

Curated questions

8 years (2016–2023)

Methods

across 12 chapters

Past papers

9 sessions

Chapters covered

across the syllabus

Themes

Pure229
Reasoning48
Statistics3

Method types

Standard method66
Long technical9
Reflex pattern7

Questions per year

PureStatisticsReasoning

Stacked by chapter theme. Click a bar to dive into the session.

2. Top methods that keep coming back

Ranked by number of past-paper questions that hit each technique. Each card opens the full method file.

Substitute

u = (base)^{x}

to turn a disguised exponential equation into a polynomial in

u

Exponentials & Logarithms7 years·7 papers

hits

Integral-defined sequences and sums: find the inner integral in closed form, then collapse the sum

Integration3 years·6 papers

hits

To find the first wrong line of a purported proof, scan for the classic order-breaking operations

Logic & Proof6 years·6 papers

hits

To disprove a 'for all' claim, build a single counterexample that obeys the hypothesis but breaks the conclusion

Logic & Proof5 years·5 papers

hits

Treat 'necessary' and 'sufficient' as two separate one-way tests, and break whichever direction can fail

Logic & Proof4 years·4 papers

hits

Reduce a mixed

sin

cos

equation to a polynomial in one ratio via

sin^{2} + cos^{2} = 1

Trigonometry5 years·6 papers

hits

Apply the factor and remainder theorem by substituting the root, never by dividing

Algebra & Inequalities5 years·6 papers

hits

Turn a question about the nature or count of roots into a discriminant condition

Algebra & Inequalities5 years·6 papers

hits

Complete the square to read a circle's centre and radius (or a parabola's vertex)

Coordinate Geometry & Circles4 years·6 papers

hits

Rewrite a quotient, product or root as a sum of powers of

x

before differentiating

Differentiation5 years·6 papers

hits

Showing 10 of 82. See every method in the TMUA app →

Chapter weighting

280 questions distributed across 12 chapters. Bar colour = theme.

Logic & Proof
48 q10 methods100% solved
Top: To disprove a 'for all' claim, build a single counterexample that obeys the hypothesis but breaks the conclusion (7 hits)
Integration
34 q9 methods100% solved
Top: Integral-defined sequences and sums: find the inner integral in closed form, then collapse the sum (7 hits)
Algebra & Inequalities
32 q10 methods100% solved
Top: Turn a question about the nature or count of roots into a discriminant condition (6 hits)
Trigonometry
29 q8 methods100% solved
Top: Reduce a mixed $sin$ / $cos$ equation to a polynomial in one ratio via $sin^{2} + cos^{2} = 1$ (7 hits)
Functions & Graphs
26 q7 methods100% solved
Top: Turn every graph transformation into an algebraic substitution, and use which moves preserve the root count (6 hits)
Differentiation
23 q7 methods100% solved
Top: Rewrite a quotient, product or root as a sum of powers of $x$ before differentiating (6 hits)
Coordinate Geometry & Circles
22 q7 methods100% solved
Top: Complete the square to read a circle's centre and radius (or a parabola's vertex) (6 hits)
Sequences & Series
21 q8 methods100% solved
Top: When a sequence is defined by a recurrence, compute the first four or five terms before doing anything clever (6 hits)
Exponentials & Logarithms
19 q7 methods100% solved
Top: Substitute $u = (base)^{x}$ to turn a disguised exponential equation into a polynomial in $u$ (7 hits)
Proof & Number Theory
12 q4 methods100% solved
Top: To break a universal claim, test the explicit small cases and read off the counterexample (4 hits)
Combinatorics & Binomial
11 q4 methods100% solved
Top: Target one coefficient with the general term, never expand the whole thing (5 hits)
Probability & Statistics
3 q1 methods100% solved
Top: Turn means and percentages into absolute totals and integer counts (3 hits)

Browse by session

Open any year to see its top methods and chapter mix.

20162p

32questions

100% solved · 37 methods

20172p

31questions

100% solved · 34 methods

20182p

29questions

100% solved · 31 methods

20192p

33questions

100% solved · 35 methods

20202p

31questions

100% solved · 38 methods

20212p

34questions

100% solved · 34 methods

20222p

34questions

100% solved · 31 methods

20232p

31questions

100% solved · 31 methods

Specimen2p

25questions

100% solved · 31 methods

Appendix — Year × method heatmap

Each cell = number of questions in that year that hit the method. Top 25 methods plotted.

Method	2016	2017	2018	2019	2020	2021	2022	2023	Specimen
Substitute $u = (base)^{x}$ to turn a disguised exponential equation into a polynomial in $u$ Exponentials & Logarithms	1		1	1	1	1		1	1
Integral-defined sequences and sums: find the inner integral in closed form, then collapse the sumIntegration		2				3		2
To find the first wrong line of a purported proof, scan for the classic order-breaking operationsLogic & Proof		1	2	1	1	1			1
To disprove a 'for all' claim, build a single counterexample that obeys the hypothesis but breaks the conclusionLogic & Proof		1	1		2	2			1
Treat 'necessary' and 'sufficient' as two separate one-way tests, and break whichever direction can failLogic & Proof				2		3	1	1
Reduce a mixed $sin$ / $cos$ equation to a polynomial in one ratio via $sin^{2} + cos^{2} = 1$ Trigonometry	2				1	1	2		1
Apply the factor and remainder theorem by substituting the root, never by dividingAlgebra & Inequalities	1	1	1	1					2
Turn a question about the nature or count of roots into a discriminant conditionAlgebra & Inequalities			1	1	1			2	1
Complete the square to read a circle's centre and radius (or a parabola's vertex)Coordinate Geometry & Circles		1				2	2		1
Rewrite a quotient, product or root as a sum of powers of $x$ before differentiatingDifferentiation	1	2	1		1				1
Turn every graph transformation into an algebraic substitution, and use which moves preserve the root countFunctions & Graphs	1		1	1	1		1	1
Rewrite the integrand as a sum of powers of $x$ before integratingIntegration	1	1	1	1		1		1
For a 'which of I, II, III must be true' item, confirm the forced ones then kill each survivor with one targeted counterexampleLogic & Proof				2	1	1	1		1
Settle a polynomial's real-root count from its shape, not by solvingLogic & Proof		1	1		1	2	1
Mind the direction of an implication: the contrapositive keeps the truth value, the converse and inverse do notLogic & Proof					1	1	2	2
Solve for the ratio, then discard the extraneous root using the stated constraintSequences & Series		1		1	1	1		1	1
When a sequence is defined by a recurrence, compute the first four or five terms before doing anything cleverSequences & Series	2	1		1			1		1
Recognise a disguised geometric series, extract its first term and ratio, then apply $\frac{a}{1 - r}$ Sequences & Series	2	1						3
Target one coefficient with the general term, never expand the whole thingCombinatorics & Binomial			2	1	1			1
Perpendicular means negative-reciprocal gradient, and a perpendicular bisector is that gradient through the midpointCoordinate Geometry & Circles		2				1		1	1
Optimise a gradient or stationary feature that depends on a parameter: differentiate in $x$ , then optimise over the parameterDifferentiation	1	1	1	1			1
Count roots and control root number from the signs of the stationary values, never by solvingFunctions & Graphs	1		2	1					1
Match a graph to its equation by testing a few discriminating features, not by plottingFunctions & Graphs		1		1	1		1		1
Complete the square to read off the vertex, the axis of symmetry, the optimum and tangency in one moveFunctions & Graphs		1			2			2
For a total (unsigned) area, split at every root and add the pieces in modulusIntegration	1		1	1		1	1

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Last updated 2026-06-27 · Source: official past papers, classified by The Maths Tailor.