Pure Notes & Exercises [See also STEP/MAT/TMUA pages]

(recently added material is in green) 

 

 

General

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

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

Reference (Pure) (7 pages; 13/1/23)

Clarifications (Pure) (3 pages; 2/9/23)

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

 

 

Algebra

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

 

 

Circles

Circle Theorems (11 pages; 7/8/17)

 

 

Complex Numbers

Complex Numbers - Part 1 (12 pages; 4/6/23)

Complex Numbers - Part 2 (12 pages; 4/6/23)

Complex Numbers - Part 3 (12 pages; 4/6/23)

Complex Numbers Exs 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

 

 

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

Counting Exs 1 2 3 4 5 6 7

 

 

Cubics - see Polynomials

 

Curve Sketching - see Functions

 

 

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 Exs 1 2

 

 

 

Differentiation

Turning Points & Points of Inflexion (7 pages; 5/6/23)

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

Differentiation Exs 1 2 3 4 5 6 

 

 

 

Ellipses - see Conics

 

 

Functions 

Mappings & Functions (3 pages; 2/6/23)

Convex & Concave Functions (2 pages; 2/6/23)

Exponential Functions (2 pages; 2/6/23)

Inverse Functions (3 pages; 2/6/23)

Modulus Function (2 pages; 2/6/23)

Rational Functions (5 pages; 2/6/23)

y squared = f(x) (2 pages; 2/6/23)

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

Curve Sketching Exs  1 2 3 4 5 6 7 8  9 10

 

 

 

Geometry

Geometry Exs 1 2 3 4 5 6

 

 

Graphs: See Functions

 

 

Groups

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

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

 

 

 

Induction

Induction (3 pages; 2/6/23)

Induction Exs 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

 

 

Inequalities

Inequalities (4 pages; 3/6/23)

Inequalities Exs 1 2 3 4

 

 

Integration

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

Integration theory (6 pages; 4/6/23)

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

Improper integrals (3 pages; 20/9/18)

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 (Overview)

  General 2 5

  Improper Integrals 1   

  Arc Length 6

  Volume of Revolution / Surface Area 3 4 7 8

 

 

Linear Interpolation

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

Linear Interpolation Exs 1

 

 

Logarithms

Logarithms (5 pages; 5/6/23)

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

Logarithms Exs 1 2 3 4 5

 

 

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 53

  Inverses (Exs) 19 22

  Shears (Exs) 31 32 33 34 54

  Simultaneous Eq'ns (Exs) 3 35 36 37 38 39 55

  Transformations (Exs) 1 2 17 40 41 44 50

 

 

Numbers

Surds Exs 1 2 3

Algebraic & Transcendental Numbers (1 page; 5/6/23)

Number Theory (18 pages; 20/2/24)

 

 

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

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

Parametric Eq'ns Exs 1 

 

 

Partial Fractions

Partial Fractions (10 pages; 8/8/19)

Partial Fractions Exs 1

 

 

Polar Curves

Polar Curves (21 pages; 2/11/20)

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

Polar Curves Exs 1 2 3 4 5

 

 

Polynomials 

Quadratics (4 pages; 5/6/23)

Quadratics Exs 1 2 3 4 5

Cubics (3 pages; 5/6/23)

Cubics Exs [3 pages, 29/7/16] Sol'ns (10 pages, 29/7/16) 

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

Polynomials Exs 1 2 3 4 5 6 7 8

   

 

 

Proof  (see also Induction)

Proof (4 pages; 16/1/23)

Proof Exs 1 2 3 4 5 6 7

 

 

Quadratics - see Polynomials

 

 

Recurrence Relations

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

 

 

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)

Sequences & Series Exs 1 2 3 4 5 6 7

 

 

Transformations: see Functions

 

 

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 2 3 4 5 6 7 8 9 10 11

 

 

Vectors

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

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 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25