Roadmap
Pure Mathematics 1
Quadratics
1.1 - Completing the square
1.1 - Solving quadratic equations
1.1 - Counting real roots
1.1 - Solving quadratic inequalities
1.1, 1.3 - Linear and quadratic intersections
1.1 - Equations in quadratic form
Functions
1.2 - Functions, domains and ranges
1.2 - Combining functions
1.2 - Reversing a function
1.2 - Moving and stretching graphs
Coordinate geometry
1.3 - Equations of straight lines
1.3 - Equations of circles
1.3 - Lines meeting circles
Circular measure
1.4 - Measuring angles in radians
1.4 - Areas and perimeters of sectors
Trigonometry
1.5 - Exact values and reference angles
1.5 - Sine, cosine and tangent graphs
1.1, 1.5 - Trigonometric identities and equations
Sequences and series
1.6 - Expanding integer powers
1.6 - Arithmetic progressions
1.6 - Geometric progressions
1.6 - Sums to infinity
Differentiation
1.7 - From chords to a tangent
1.7 - Differentiating powers
1.7 - Differentiating a function of a function
1.7 - Tangents and normals
1.7 - Rates of change
1.7 - Stationary points and optimisation
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Integration
1.8 - Integration as reverse differentiation
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1.8 - Finding a curve from its gradient
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1.8 - Definite integrals
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1.8 - Areas between curves
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1.8 - Volumes of revolution
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Advanced pure mathematics
Algebra
2.1, 3.1 - Modulus graphs and equations
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2.1, 3.1 - Dividing polynomials
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2.1, 3.1 - Factor and remainder theorems
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Logarithms and exponentials
2.2, 3.2 - Logarithms and their laws
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2.2, 3.2 - The functions exp and ln
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2.2, 3.2 - Straightening curved relationships
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Advanced trigonometry
2.3, 3.3 - Reciprocal trigonometric functions
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2.3, 3.3 - Compound-angle formulae
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2.3, 3.3 - Double-angle identities
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2.3, 3.3 - Combining sine and cosine
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Advanced differentiation
2.4, 3.4 - Differentiating exponentials and logarithms
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2.4, 3.4 - Differentiating trigonometric functions
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2.4, 3.4 - Product and quotient rules
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2.4, 3.4 - Differentiating parametric curves
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2.4, 3.4 - Implicit differentiation
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Advanced integration
2.5, 3.5 - Exponential and trigonometric integrals
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2.5, 3.5 - Using identities to integrate
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Numerical equations
2.6, 3.6 - Locating a root
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2.6, 3.6 - Iterating towards a root
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Pure Mathematics 3 extensions
Algebraic extensions
3.1 - Partial fractions
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3.1 - Binomial series with rational powers
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Calculus extensions
3.4, 3.5 - Inverse tangent in calculus
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3.5 - Recognising a logarithmic integral
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3.5 - Integrating partial fractions
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3.5 - Integration by parts
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3.5 - Integration by substitution
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Vectors in space
3.7 - Vectors and coordinates
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3.7 - Vector equations of lines
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3.7 - Angles and perpendiculars with vectors
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Differential equations
3.8 - Solving separable differential equations
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3.8 - Interpreting differential-equation models
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Complex numbers
3.9 - Calculating with complex numbers
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3.9 - Argand diagrams and polar form
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3.9 - Rotations and square roots
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3.9 - Loci in an Argand diagram
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Pure Mathematics 2 only
Numerical integration
2.5 - The trapezium rule
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Mechanics
Forces and equilibrium
4.1, 4.4 - Modelling forces
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4.1 - Components and resultants
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4.1 - Friction and limiting equilibrium
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Motion in a straight line
4.2 - Displacement and velocity graphs
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4.2 - Constant acceleration
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4.2 - Motion with calculus
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Newton's laws
4.4 - Forces causing acceleration
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4.4 - Connected particles
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Momentum
4.3 - Conservation of momentum
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Energy and power
4.5 - Work and energy
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4.5 - Conservation of energy
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4.5 - Power and motion
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Probability and Statistics 1
Representing data
5.1 - Choosing a data display
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5.1 - Cumulative frequency and quartiles
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5.1 - Mean and standard deviation
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Counting arrangements
5.2 - Permutations and combinations
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5.2 - Arrangements with restrictions
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Probability
5.3 - Combining events
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5.3 - Conditional probability
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Discrete distributions
5.4 - Discrete random variables
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5.4 - The binomial distribution
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5.4 - Waiting for the first success
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Normal distribution
5.5 - Normal probabilities
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5.5 - Finding normal cut-offs and parameters
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5.5 - Approximating a binomial distribution
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Probability and Statistics 2
Poisson models
6.1 - The Poisson distribution
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6.1 - Approximating with Poisson and normal models
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Combining random variables
6.2 - Means and variances of combinations
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6.2 - Normal and Poisson combinations
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Continuous distributions
6.3 - Probability density functions
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6.3 - Mean, variance and percentiles
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Sampling and estimation
6.4 - Random samples and unbiased estimates
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6.4 - The distribution of a sample mean
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6.4 - Confidence intervals for a mean
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6.4 - Confidence intervals for a proportion
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Hypothesis testing
6.5 - Hypotheses and critical regions
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6.5 - Binomial and Poisson tests
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6.5 - Testing a population mean
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6.5 - Type I and Type II errors
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