Roadmap
Further Pure Mathematics 1
Roots of polynomial equations
1.1 - Roots and coefficients
1.1 - Root relations for quartics
1.1 - Symmetric expressions in roots
1.1 - Shifting and scaling roots
1.1 - Reciprocals and squares of roots
Rational functions and graphs
1.2 - Rational functions and asymptotes
1.2 - Sketching rational curves
1.2 - Finding ranges with a discriminant
1.2 - Reciprocal and square-root graphs
1.2 - Modulus graphs and inequalities
Summation of series
1.3 - Sums of powers
1.3 - The method of differences
1.3 - Sums to infinity
Matrices and transformations
1.4 - Calculating with matrices
1.4 - Matrix algebra and powers
1.4 - Determinants and inverses of two-by-two matrices
1.4 - Determinants and inverses of three-by-three matrices
1.4 - Inverting matrix products
1.4 - Matrices for plane transformations
1.4 - Composing and reversing transformations
1.4 - Invariant points and lines
Polar coordinates
1.5 - Polar and Cartesian coordinates
1.5 - Sketching polar curves
1.5 - Areas in polar coordinates
Vectors and planes
1.6 - The vector product
1.6 - Equations of planes
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1.6 - Lines meeting planes
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1.6 - Angles and distances involving planes
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1.6 - The line where two planes meet
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1.6 - The shortest distance between skew lines
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Proof by induction
1.7 - Proof by induction for sums
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1.7 - Proof by induction for sequences
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1.7 - Proof by induction for divisibility
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1.7 - Proof by induction for matrix powers
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Further Pure Mathematics 2
Hyperbolic functions
2.1 - Hyperbolic functions and graphs
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2.1 - Hyperbolic identities
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2.1 - Inverse hyperbolic functions
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Matrix systems and eigenvectors
2.2 - Three equations as a matrix equation
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2.2 - Singular systems and three planes
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2.2 - Eigenvalues and eigenvectors
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2.2 - Eigenvectors in three dimensions
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2.2 - Diagonalising a matrix
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2.2 - Using the characteristic equation
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Further differentiation
2.3 - Differentiating inverse and hyperbolic functions
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2.3 - Second derivatives in other forms
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2.3 - Maclaurin expansions
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Further integration
2.4 - Hyperbolic and inverse-form integrals
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2.4 - Trigonometric and hyperbolic substitutions
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2.4 - Choosing an integration method
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2.4 - Reduction formulae
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2.4 - Bounding sums with integrals
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2.4 - Lengths of curves
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2.4 - Surface areas of revolution
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Complex numbers
2.5 - De Moivre's theorem
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2.5 - Multiple-angle identities
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2.5 - Trigonometric powers and sums
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2.5 - Complex roots and roots of unity
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Differential equations
2.6 - First-order linear differential equations
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2.6 - Complementary functions
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2.6 - Finding particular integrals
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2.6 - Substitutions in differential equations
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2.6 - Differential equations in context
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Further Mechanics
Projectiles
3.1 - Modelling projectile motion
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3.1 - Flight time, range and height
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3.1 - The equation of a trajectory
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Rigid-body equilibrium
3.2 - Moments of forces
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3.2 - Centres of mass and symmetry
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3.2 - Centres of mass of composite bodies
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3.2 - Equilibrium of a rigid body
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3.2 - Toppling and sliding
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Circular motion
3.3 - Horizontal circular motion
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3.3 - Vertical circular motion
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3.3 - Completing a vertical circle
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Elastic strings and springs
3.4 - Forces in elastic strings and springs
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3.4 - Elastic potential energy
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3.4 - Elastic systems in motion
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Variable forces
3.5 - Variable forces depending on time or velocity
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3.5 - Variable forces depending on displacement
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Momentum and impacts
3.6 - Direct impacts and restitution
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3.6 - Direct impact of two spheres
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3.6 - Oblique impact with a fixed surface
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3.6 - Oblique impact of smooth spheres
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Further Probability and Statistics
Continuous random variables
4.1 - Piecewise probability densities
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4.1 - Cumulative distribution functions
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4.1 - Distributions of transformed variables
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Inference for means
4.2 - The one-sample t-test
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4.2 - Pooled variance and the two-sample t-test
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4.2 - Paired-sample t-tests
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4.2 - Comparing means with a normal distribution
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4.2 - Confidence intervals for one mean
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4.2 - Confidence intervals for differences of means
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Chi-squared tests
4.3 - Fitting distributions to frequency data
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4.3 - Chi-squared goodness of fit
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4.3 - Chi-squared tests of independence
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Non-parametric tests
4.4 - Non-parametric tests and the sign test
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4.4 - Wilcoxon signed-rank tests
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4.4 - Non-parametric tests for paired data
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4.4 - The Wilcoxon rank-sum test
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4.4 - Normal approximations for rank tests
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Probability generating functions
4.5 - Probability generating functions
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4.5 - Means and variances from PGFs
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4.5 - PGFs of independent sums
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