FM - Pure Notes & Exercises [See also STEP/MAT/TMUA pages]
(recently added material is in green - where changes are significant)
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General
Exam Technique (12 pages; 8/5/24)
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; 12/7/24)
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 Fns 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;15/6/24)
Integration theory (7 pages; 7/11/24)
Integration: Arc lengths & Surface areas (3 pages; 10/5/17)
Improper integrals (3 pages; 20/9/18)
Integration - Part 1(E) Exs (1 page; 7/11/17); Sol'ns (8 pages; 7/11/17)
Integration - Part 2 (H) Exs (3 pages; 20/6/24); Sol'ns (24 pages; 20/6/24)
Integration - Part 3 (Hyp) Exs (2 pages; 6/2/18); Sol'ns (12 pages; 17/4/20)
Integration Exs (Overview)
General 2 5
Improper Integrals 1
Arc Length6
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 56 57 58
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; 12/11/24)
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)
Cubics: See STEP/MAT/TMUA pages
Roots of polynomials (6 pages; 15/2/17)
Polynomials Exs 1 2 3 4 5 6 7 8
Proof - See STEP page, and Induction
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 26
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