CONTENTS (@30/1/23) 

STEP   

General Notes

STEP Exams - Tips (3 pages; 14/2/21)

STEP Exams - Preparation (3 pages; 4/6/20) 

STEP Exams - Methods (8 pages; 4/6/20) 

STEP Exams - Technique (7 pages; 4/6/20) 

Index of q'ns: Pure (25 pages; 1/7/21)

Index of q'ns: Mechanics (10 pages; 10/7/21)

Index of q'ns: Probability & Statistics (7 pages; 30/6/21)

[See Pure, Mechanics and Prob & Stats pages for topic-specific notes; especially "STEP" material.]

Exercises (with sol'ns)

Pure

General (6 pages, 6/11/22)

Algebra (8 pages, 6/11/22)

Complex Numbers (7 pages, 6/11/22)

Counting: See Probability & Statistics

Curve sketching (26 pages, 6/11/22)

Differential equations (2 pages, 6/11/22)

Differentiation (17 pages, 6/11/22)

Geometry (2 pages, 6/11/22)

Hyperbolic Functions (11 pages, 6/11/22)

Induction (2 pages, 6/11/22)

Inequalities (23 pages, 6/11/22)

Integers (8 pages, 6/11/22)

Integration (6 pages, 6/11/22)

Logarithms (15 pages, 6/11/22)

Polynomials (12 pages, 6/11/22)

Recurrence relations (10 pages, 6/11/22)

Series (3 pages, 6/11/22)

Transformations (11 pages, 6/11/22)

Trigonometry (32 pages, 6/11/22)

Vectors (13 pages, 6/11/22)

Mechanics

Collisions (13 pages, 6/11/22)

Forces (12 pages, 6/11/22)

Probability & Statistics

Counting (22 pages, 6/11/22)

Probability (9 pages, 6/11/22)

Statistics (14 pages, 6/11/22)

Solutions for Past Papers

2005, Paper 1: 1  3  4  5  11  12  13  14

2005, Paper 2: 1  4  6  10  11

2005, Paper 3: 2  3  7 8 13

2006, Paper 1: 1  2  3  4  5  6  7  8  9  10  11  12  13  14

2006, Paper 2: 2  4  6  10 12   

2006, Paper 3: 2 3  4  10  11  12  13  

2007, Paper 1: 1  2  3  4  5  7  12  

2007, Paper 2: 1  2  3  4  5

2007, Paper 3: 1  2  3  7  12

2008, Paper 1: 1  2  6  9  13  

2008, Paper 2: 1  3  4  5  6  7  10  

2008, Paper 3: 1  2  5  8  13  

2009, Paper 1: 1  2  9  12  13

2009, Paper 2: 5  8  10  11  13

2009, Paper 3: 1  4  6  7  8  11  

2010, Paper 1: 1  2  3  5  9  

2010, Paper 2: 4  5  7  12  13

2010, Paper 3: 2  3  8  9 12 

2011, Paper 1: 1  2  6  9  10

2011, Paper 2: 1  2  5  6  9  11  12

2011, Paper 3: 1  2  3  4  5  6  7  8  9  10  11  12  13

2012, Paper 1: 2  5  8  9  11  13

2012, Paper 2: 3  5  7  9  10  11  12  13

2012, Paper 3: 1  6  7  9  10  11  12  13

2013, Paper 1: 1 4 10 12 13

2013, Paper 2: 1 2 11 12 13

2013, Paper 3: 1 6 7 8 9

2014, Paper 1: 2 3 4 9 11

2014, Paper 2: 4 5 9 11 12

2014, Paper 3: 2 7 9 10 13

2015, Paper 1: 3 9 10 12 13

2015, Paper 2: 2 6 10 11 12

2015, Paper 3: 1 6 9 12 13

2016, Paper 1: 2 10 11 12 13

2016, Paper 2: 3 6 7 9 13

2016, Paper 3: 1 2 3 8 11

2017, Paper 1: 1 2 10 12 13

2017, Paper 2: 1 9 10 12 13

2017, Paper 3: 3 4 9 12 13

2018, Paper 1: 7 9 10 12 13

2018, Paper 2: 2 6 8 10 11 13

2018, Paper 3: 1 2 3 8 11

2019, Paper 1: 3 4 9 10 11

2019, Paper 2: 1 2 7 10 11

2019, Paper 3: 3 5 9 11 12

2020, Paper 2: 1 6 9 10 11 12

2020, Paper 3: 1 7 9 12

MAT 

Notes 

Notes on MAT paper (3 pages; 7/10/21)

MAT - Preparation (1 page; 7/10/21)

MAT - Exam Technique (Summary) (1 page; 7/10/21)

MAT Exam Technique (12 pages; 29/9/21) 

Important Ideas (17 pages; 7/10/21)

MAT - Extra Material (2 pages; 28/9/21)

MAT - Index of Q'ns (14 pages; 4/11/20)

Exercises (with sol'ns)

General (4 pages; 3/11/22)

Algebra (4 pages; 3/11/22)

Counting (20 pages; 3/11/22)

Curve Sketching (6 pages; 4/11/22)

Differentiation (2 pages; 2/11/22)

Geometry (2 pages; 3/11/22)

Inequalities (19 pages; 4/11/22)

Integers (8 pages; 3/11/22)

Integration (2 pages; 2/11/22)

Logarithms (7 pages; 4/11/22)

Polynomials (10 pages; 5/11/22)

Recurrence relations (10 pages; 4/11/22)

Series (3 pages; 2/11/22)

Transformations (9 pages; 4/11/22)

Trigonometry (16 pages; 4/11/22)

Solutions for Past Papers

Specimen 1: MC 3 4 5

Specimen 2: MC 2 4

2007: MC 4 5

2008: MC 2 3 5

2009: MC 2 6

2010: MC 3

2011: MC 2

2012: MC 5 7

2013: MC 2 3 4 5

2014: MC 2 3 4 5

2015: MC 2 3 4 5

2016: MC

2017: MC 2 6 7

2018: MC

2019: MC 2 3 5

2020: MC 2 4 6

2021: MC 2

TMUA

Notes 

TMUA - Exam Technique (8 pages; 29/9/21)

TMUA - Extra Material (1 page; 28/9/21)

Important Ideas (17 pages; 28/9/21)

Exercises (with sol'ns)

General (4 pages; 5/11/22)

Algebra (4 pages; 5/11/22)

Counting (20 pages; 5/11/22)

Curve Sketching (6 pages; 5/11/22)

Differentiation (2 pages; 5/11/22)

Geometry (2 pages; 5/11/22)

Inequalities (19 pages; 5/11/22)

Integers (8 pages; 5/11/22)

Integration (2 pages; 5/11/22)

Logarithms (7 pages; 5/11/22)

Polynomials (10 pages; 5/11/22)

Recurrence relations (10 pages; 5/11/22)

Series (3 pages; 5/11/22)

Transformations (9 pages; 5/11/22)

Trigonometry (16 pages; 5/11/22)

Solutions for Past Papers

Specimen Paper 1

Practice (aka 2016) Paper 1

2020 [also used for Computer-based Demo] Paper 1

2020 [also used for Computer-based Demo] Paper 2

2021 Paper 2

Pure Notes & Exercises

Further Maths Exam Technique (14 pages; 10/12/19) 

Pure Topics by Exam Board (22 pages; 7/8/19)

Clarifications - Pure (3 pages; 13/1/23)

Reference (Pure) (7 pages; 4/1/21)

Prerequisite Pure Topics (for Further Maths and STEP) (4 pages; 6/12/21)

Algebra - Introduction (3 pages; 25/8/20)

Algebra Methods (5 pages; 23/1/18)

Algebra Exs 1 2 3

Circle Theorems (11 pages; 7/8/17)

Complex Numbers [Important Ideas] (2 pages; 22/10/20)

Complex Numbers - Part 1 (12 pages; 11/12/15)

Complex Numbers - Part 2 (11 pages; 18/9/19)

Complex Numbers - Part 3 (11 pages; 18/9/19)

Complex Number Methods (STEP) (9 pages; 28/2/15)

Complex Numbers Exs 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Conics

Conics (4 pages; 17/8/19)

Parabolas (10 pages; 16/7/19)

Parabolas Exs 1 2 3 4 5

Ellipses (7 pages; 18/8/19)

Ellipses Exs 1 2 3

Hyperbolas (13 pages; 11/9/19)

Hyperbolas Exs 1 2 3 4 5 6

Counting Exs 1 2 3 4 5 6 7

Cubics - see Polynomials

Curve Sketching

Curve Sketching (STEP) (5 pages; 20/1/23)

Graphs Exs

  Sketches (Exs) 1 2 3 4 7 8 12

  Transformations (Exs) 5 6 9 10 11 

Differential Equations (see also Mechanics: Oscillations)

Models (4 pages, 19/2/20)

1st order (3 pages, 23/2/19)

2nd order (linear, constant coeffs) (9 pages; 28/5/20)

Approximate methods (9 pages; 27/2/21)

Linear systems (7 pages; 9/3/21)

Differential Eq'ns (STEP) (5 pages; 18/11/18)

Differential Eq'ns Exs 1 2

Differentiation

Differentiation [Important Ideas] (1 page; 22/10/20)

Turning Points & Points of Inflexion [Important Ideas] (2 pages; 4/1/21)

Differentiation - Misc (4 pages; 3/9/18)

Turning points (3 pages; 7/4/20)

Points of inflexion (3 pages; 24/10/18)

Differentiation Exs 1 2 3 4 5 6 

Ellipses - see Conics

Functions 

Functions [Important Ideas] (2 pages; 22/10/20)

Function Terminology (5 pages; 12/1/23)

Convexity & concavity (3 pages; 29/8/18)

Sketching rational functions (5 pages; 11/12/19)

y squared = f(x) (2 pages; 29/8/18)

Modulus function (3 pages; 4/9/18)

Transformations [Important Ideas] (3 pages; 22/10/20)

Transformations (STEP) (6 pages; 12/12/19)

Functions - Misc (7 pages; 4/9/18)

Geometry Exs 1 2 3 4 5 6

Graphs - see Curve Sketching

Groups - Part 1 (10 pages; 22/10/16)

Groups - Part 2 (12 pages; 23/10/16)

Groups Exs 1 2 3 4

Hyperbolas - see Conics

Hyperbolic Functions (10 pages; 26/7/16)

Hyperbolic Functions Exs 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Induction [Important Ideas] (4 pages; 22/10/20)

Induction (4 pages; 4/10/20)

Induction Exs [3 pages] Sol'ns [23 pages]

Inequalities [Important Ideas] (2 pages; 22/10/20)

Inequalities (7 pages; 5/9/18)

Inequalities Exs 1 2 3 4 5 6 7 8 9 10 11 12

Integers [Important Ideas] (1 page; 22/10/20)

Integers Exs 1 2 3 4 5

Integration methods (11 pages; 14/4/21)

Integration Ideas (STEP) (4 pages; 14/4/21)

Integration theory (5 pages; 28/9/18)

Integration: Arc lengths & Surface areas (3 pages; 10/5/17)

Improper integrals (3 pages; 20/9/18)

Fundamental Thm of Calculus (1 page; 10/7/21)

Integration - P1 (E) Exs (1 page; 7/11/17); Sol'ns (8 pages; 7/11/17)

Integration - P2 (H) Exs (3 pages; 6/4/21); Sol'ns (23 pages; 6/4/21)

Integration - P3 (Hyperbolic) Exs (2 pages; 6/2/18); Sol'ns (12 pages; 17/4/20)

Integration Exs 2 5 6

   Improper Integrals (Exs) 1   

   Volume of Revolution / Surface Area (Exs) 3 4 7 

Linear Interpolation (5 pages; 7/2/16)

Linear Interpolation Exs 1

Logarithms [Important Ideas] (2 pages; 22/10/20)

Logarithms (4 pages; 13/2/17)

Modelling Curves using Logs (4 pages; 7/8/17)

Logarithms Exs 1 2 3 4 5 6 7 8 9

Matrices

Determinants & Inverses (14 pages; 3/6/19)

Transformations (9 pages; 1/10/19)

Invariant points and lines - introduction (4 pages; 16/4/20)

Invariant points & lines - conditions (12 pages; 16/4/20) 

Shears (8 pages; 12/8/18) 

Eigenvectors (10 pages; 4/9/18)

Eigenvectors & Invariance (6 pages; 16/4/20) 

Matrices Exs

  General (Exs) 16 21 25

  Determinants (Exs) 18 23 24 26

  Eigenvectors (Exs) 4 5 6 7 8 9 10 11 12 13 14 15 47 48 51

  Invariant Points & Lines (Exs) 27 28 29 30 42 43 45 46 49 52

  Inverses (Exs) 19 22

  Shears (Exs) 31 32 33 34

  Simultaneous Eq'ns (Exs) 3 35 36 37 38 39

  Transformations (Exs) 1 2 17 40 41 44 50

Number Theory (17 pages; 7/10/18)

Numerical Methods

Calculator method (1 page; 5/10/20)

Errors (10 pages; 27/3/20)

Sol'n of Eq'ns: Bisection method (2 pages; 22/10/18)

Sol'n of Eq'ns: Method of False Position (4 pages; 22/10/18)

Sol'n of Eq'ns: Newton-Raphson (4 pages; 22/10/18)

Sol'n of Eq'ns: Secant method (2 pages; 12/6/20)

Sol'n of Eq'ns: Fixed Point (16 pages; 27/3/20)

Sol'n of Eq'ns: Comparison of methods (2 pages; 22/10/18)

Sol'n of Eq'ns: Convergence (3 pages; 12/6/20)

Numerical Integration (20 pages; 31/3/20)

Numerical Differentiation (8 pages; 31/3/20)

Approximating F'ns: Newton's Forward Difference (9 pages; 12/6/20)

Approximating F'ns: Lagrange's method (2 pages; 27/3/20)

Differential Eq'ns:Approximate methods (9 pages; 27/2/21)

Numerical Methods Exs

  Convergence (Exs) 2

  Integration (Exs) 3 4 5 6 7

  Sol'n of Eq'ns (Exs) 1

Parabolas - see Conics

Parametric equations (7 pages; 5/5/21)

Parametric Eq'ns Exs 1 

Partial Fractions (10 pages; 8/8/19)

Partial Fractions Exs 1

Polar Curves (21 pages; 2/11/20)

Polar Curves - Summary (11 pages; 2/11/20)

Polar Curves Exs 1 2 3 4

Polynomials 

Polynomials [Important Ideas] (2 pages; 4/1/21)

Roots of polynomials (6 pages; 15/2/17)

Quadratic Functions [Important Ideas] (2 pages; 22/10/20)

Quadratics - factorising (3 pages; 21/1/17)

Quadratics Exs 1 2 3 4 5

Cubics [Important Ideas] (3 pages; 24/10/20)

Cubics (STEP) (4 pages; 11/12/19)

Cubics Qns [3 pages] Sol'ns [10 pages] 

Polynomials Exs 4 5

   Factorisations (Exs) 1 2 3

   Relations between roots (Exs) 6 7 8 9 10 11 12 13

Proof (4 pages; 16/1/23)

Proof Exs 1 2 3 4 5 6 7

Quadratics - see Polynomials

Recurrence relations (16 pages; 5/9/18)

Recurrence relations Exs 1

Sequences & Series

Series [Important Ideas] (3 pages; 11/3/21)

Arithmetic Sequences & Series (2 pages; 23/1/17)

Arithmetic Series Exs 1 2 3 4

Method of Differences (3 pages; 6/10/20)

Maclaurin Series (7 pages; 23/7/21)

Maclaurin Series Exs 1 2 3 4 5 6

Taylor Series (5 pages; 23/7/21)

Series Exs 5 6 7 8

   Method of Differences (Exs) 1 2 3 4

Surds Exs 1 2 3

Trigonometry

Basics (2 pages; 15/4/21)

Relations (3 pages; 15/4/21)

Sine & Cosine Rules (5 pages; 15/4/21)

Radians (4 pages; 15/4/21)

Sol'n of eq'ns (2 pages; 15/4/21)

Compound Angle Formulae (4 pages; 15/4/21)

Small Angle Approximations (2 pages; 15/4/21) 

Inverse Functions (5 pages; 15/4/21)

Factor Formulae (1 page; 15/4/21)

t=tan(x/2) substitution (4 pages; 24/3/20)

Trigonometry Exs 1 3 11 16

   Equations (Exs) 2 12 13 17 19 20

   Identities (Exs) 4 5 6 8 14 15 22 23 24 25

   Sketches (Exs) 7 18

   Triangles (Exs) 9 10 21 

Vectors

Vectors [Important ideas] (18 pages; 21/1/21)

Vector Methods (STEP) (5 pages; 26/3/21)

Vectors - Introduction (6 pages; 4/8/18)

Scalar Product (4 pages; 19/9/20)

Vectors - Lines (6 pages; 18/9/20)

Vectors - Planes (7 pages; 10/2/20)

Configuration of 3 planes (11 pages; 1/8/21)

Vectors - Angles (5 pages; 4/8/18)

Vectors - Intersections (3 pages; 4/8/18)

Vectors - Shortest distances (14 pages; 4/8/18)

Vector Product (8 pages; 4/8/18)

Triple Products (4 pages; 4/8/18)

Vectors: Misc (1 page; 4/4/18)

Vectors Exs 2 4 14 16 21 24 25 26 27 28 29 30

   Lines (Exs) 5 6

   Planes (Exs) 7 9 11 15 34

   Lines & Planes (Exs) 1 8 10 17 20 23

   Shortest Distance (Exs) 3 12 13 18 19 22 32 33

   Vector Product (Exs) 31

Miscellaneous [Important Ideas] (1 page; 22/10/20)

Misc. Problems 1 

Tests 1  

Mechanics Notes & Exercises 

Exam Board Topics - Mechanics (8 pages; 9/7/21)

Useful Results (2 pages; 5/5/20)

Centre of Mass - Contents (2 pages; 7/8/17)

Centre of Mass - Part 1 (13 pages; 7/8/17)

Centre of Mass - Part 2 (12 pages; 7/8/17)

Centre of Mass Exs 1 2 3 4 5 

Circular Motion [Important Ideas] (2 pages; 25/10/20)

Circular Motion - Constant Speed (9 pages; 25/10/20)

Circular Motion - Variable Speed (13 pages; 30/9/15)

Circular Motion Exs 1 2

Energy: See Work & Energy

Forces

Forces [Important Ideas] (18 pages; 14/1/21)

Equilibrium [Important Ideas] (11 pages; 12/11/20)

Frameworks (9 pages; 25/3/21)

Bending moment & shearing force (5 pages; 25/3/21)

Fictitious forces (5 pages; 25/10/20) 

Systems of forces (5 pages; 14/11/18)

Toppling (2 pages; 13/11/20) 

Forces Exs 1 2 3 4 5 6 7 8 9 10 11   

Friction: see also Forces

Friction Exs 1 2 3 4

Hooke's Law (9 pages; 10/3/21)

Hooke's Law Exs 1 2 3 4

Impulse & Momentum

Collisions - Important Ideas (4 pages; 22/4/21)

Impulse & Momentum (14 pages; 19/2/17)

Collisions - reversal of direction (6 pages; 24/6/20) 

Oblique impacts (4 pages; 23/8/18)

Oblique impact with plane Exs

Impulses & connected particles (13 pages; 28/8/18) 

Impulse & Momentum Exs 1 2 3 4 5 6 7 8 9 10 11 12 13 

Kinematics

Kinematics (7 pages; 7/8/17)

Kinematics Exs 1 2

Moments of Inertia: see Rotating Bodies

Oscillations

Differential equations: Oscillations (11 pages; 29/5/20)

Oscillations Exs 1 2 3 4

Projectiles (11 pages; 10/3/21)

Rotating Bodies

Rotating Bodies (5 pages; 19/8/19) 

Rotating Bodies - Examples (6 pages; 16/3/18) 

Moments of Inertia (7 pages; 23/3/18) 

Rolling wheel - Point on circumference (5 pages; 1/2/21) 

Rolling wheel - Friction (6 pages; 15/2/21) 

Rotating Bodies Ex 1

Suvat equations (5 pages; 24/1/17)

Work & Energy

Work & Energy (4 pages; 21/11/20)

Gravitational Potential Energy (3 pages;15/2/19) 

Work & Energy Exs 1 2 3 4 

Probability & Statistics Notes & Exercises

Probability & Statistics [Important Ideas] (10 pages; 28/6/21)

Exam Board topics - Prob. & Stats (11 pages; 30/6/21)

Useful Results (4 pages; 9/7/21)

Binomial distribution (7 pages; 16/2/17)

Chi-Squared Test

Contingency Tables & Goodness of Fit (8 pages; 12/5/17)

Contingency Tables & Goodness of Fit - Important Ideas (3 pages; 9/7/21)

Chi-Squared Exs

  Contingency Tables (Exs) 2a 2b

  Goodness of Fit (Ex) 1

Conditional Probability (2 pages; 1/2/23)

Confidence Intervals - Types (3 pages; 16/4/16)

Correlation & Regression

Correlation (12 pages; 28/6/21)

Rank Correlation (5 pages; 23/8/19)

Regression (8 pages; 23/8/19)

Correlation Exs 1 2

Counting: See Probability & Counting

Discrete Random Variables

Discrete Random Variables (8 pages; 4/2/23)

Discrete Random Variables Exs 1 2 3 4

Linear Coding (2 pages; 8/7/21)

Geometric distribution (3 pages; 12/11/18)

Geometric Distribution Exs 1 2

Hypothesis Tests - General (3 pages; 22/5/20)

Hypothesis Tests - Types (8 pages; 24/1/20)

Normal distribution (9 pages; 21/2/17)

Normal Distribution Exs 1 2

Poisson distribution (7 pages; 10/2/21)

Approximations to Binomial & Poisson (4 pages; 30/1/17)

Poisson Distribution Exs 1 2 3 4

Probability & Counting

Probability & Counting Approaches [STEP] (12 pages; 4/2/23)

Selections (6 pages; 14/7/21)

Probability Exs 1 2 3 4 5 6 7

Probability Generating Functions (5 pages; 16/4/21)

Probability Generating Function Exs 1 2

Sampling (4 pages; 7/8/17)

Uniform distribution (Discrete) (2 pages; 20/3/20)

Uniform Distribution (Discrete) Ex 1

Variance

Sample variance (3 pages; 8/7/21)

Discrete (aka Decision) Maths Notes & Exercises

Topics by exam board (9 pages; 30/7/19)

Algorithms 

Bin Packing Algorithms (3 pages; 7/8/18)

Maximum Matching algorithm (4 pages; 6/9/13)

Algorithms Exs 1 7 

   Bin Packing (Exs) 3 5 6

   Sorting (Exs) 2 4     

Critical Path Analysis - Part 1 (12 pages; 22/1/20)

Critical Path Analysis - Part 2 (14 pages; 7/11/20)

Critical Path Analysis Exs 1a 1b 2 

Dijkstra's algorithm: See Shortest Paths

Floyd's algorithm: See Shortest Paths

Game Theory Exs 1 2 3

Graphs & Networks

Graphs Glossary (3 pages; 15/1/14)

Directed Edges (incl. Incidence & Distance matrices) (5 pages; 19/11/19)

Planarity Algorithm (5 pages; 30/6/20)

Graphs Ex 2

   Kuratowski's Thm (Ex) 1

Planarity algorithm Ex 1

Hungarian Algorithm  (9 pages; 28/8/17)

Kuratowski's Thm: See Graphs & Networks

Linear Programming & Simplex method

Linear Programming (9 pages; 21/11/19)

Formulating network problems as LP problems (7 pages; 21/1/20)

Simplex method - Part 1 (15 pages; 31/1/18)

Simplex method - Part 2 (16 pages; 31/1/18)

Linear Programming Exs 1 2a 2b

   Formulate as LP problem (Exs) 3 4 5 6 7

   Simplex (Exs) 2c 8 9a 9b

Minimum Connector Problem (6 pages; 21/11/20) 

Minimum Connector Exs 1 2

Network Flows - Introduction (5 pages; 16/1/21)

Networks: See Graphs & Networks

Planarity algorithm: See Graphs & Networks

Route Inspection Problem (5 pages; 12/9/17)

Route Inspection Exs 1 2

Shortest Paths

Dijkstra's Algorithm (5 pages; 19/11/20)

Floyd's Algorithm (21 pages; 17/6/20)

Dijkstra's algorithm Ex 1

Floyd's Algorithm Exs 1 2

Simplex: See Linear Programming & Simplex method

Travelling Salesman problem (10 pages; 29/6/20)

Travelling Salesman problem Ex 1