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Further Mathematics · 9231

Further Pure Mathematics 2

01Hyperbolic functions

Definitions and graphs2

Understanding sinh and cosh Tanh and the reciprocal functions

Identities and equations3

The basic hyperbolic identities Addition and double-input identities Solving hyperbolic equations

Inverse functions3

Understanding inverse hyperbolic functions Deriving logarithmic forms Inverse sech, cosech and coth

Graph transformations1

Graph transformations and mixed questions

02Matrices 2

Eigenvalues and eigenvectors4

Understanding eigenvectors Finding eigenvalues and eigenvectors Three-by-three matrices Eigenvalues of related matrices

Diagonalisation and powers3

Understanding diagonalisation Calculating matrix powers The Cayley–Hamilton theorem

Systems of equations2

Solving three linear equations Solutions, parameters and planes

03Differentiation

Second derivatives2

Implicit second derivatives Parametric second derivatives

Hyperbolic and inverse derivatives3

Hyperbolic derivatives Inverse trigonometric derivatives Inverse hyperbolic derivatives

Maclaurin series3

Building a Maclaurin polynomial Using known series Series from derivative relations

04Integration

Integration methods4

Hyperbolic integrals Integrals with square roots Integrals with quadratic denominators Parts and further substitutions

Reduction formulae2

Trigonometric reduction formulae Other reduction formulae

Arc length and surface area3

Cartesian and parametric arc length Polar arc length Surface area of revolution

Sums and integrals2

Integral bounds for sums Limits of sums

05Complex numbers

Complex forms and powers2

Complex numbers as lengths and angles De Moivre's theorem

Trigonometric identities2

Multiple angles and polynomial equations Powers of sine and cosine

Complex roots2

Roots of unity Complex roots and rational powers

Trigonometric sums2

Finite trigonometric sums Weighted and binomial sums

06Differential equations

First-order linear equations2

Why integrating factors work Choosing an integrating factor

Homogeneous second-order equations3

The auxiliary equation Repeated roots Complex roots and oscillation

Particular integrals2

Complementary functions and particular integrals When a PI trial is already in the CF

Substitutions3

First-order substitutions Substituting for the dependent variable Changing the independent variable

Initial conditions and models1

Initial conditions and models

Revision Notes Practice Questions Topical Past Paper Questions Past Papers

Resources

Revision Notes Practice Questions Topical Past Paper Questions Past Papers

Further Pure Mathematics 2/Integration

Polar arc length

Polar arc length

Practise Integration

Contents

Further Mathematics · 9231

Further Pure Mathematics 2

01Hyperbolic functions

Definitions and graphs2

Understanding sinh and cosh Tanh and the reciprocal functions

Identities and equations3

The basic hyperbolic identities Addition and double-input identities Solving hyperbolic equations

Inverse functions3

Understanding inverse hyperbolic functions Deriving logarithmic forms Inverse sech, cosech and coth

Graph transformations1

Graph transformations and mixed questions

02Matrices 2

Eigenvalues and eigenvectors4

Understanding eigenvectors Finding eigenvalues and eigenvectors Three-by-three matrices Eigenvalues of related matrices

Diagonalisation and powers3

Understanding diagonalisation Calculating matrix powers The Cayley–Hamilton theorem

Systems of equations2

Solving three linear equations Solutions, parameters and planes

03Differentiation

Second derivatives2

Implicit second derivatives Parametric second derivatives

Hyperbolic and inverse derivatives3

Hyperbolic derivatives Inverse trigonometric derivatives Inverse hyperbolic derivatives

Maclaurin series3

Building a Maclaurin polynomial Using known series Series from derivative relations

04Integration

Integration methods4

Hyperbolic integrals Integrals with square roots Integrals with quadratic denominators Parts and further substitutions

Reduction formulae2

Trigonometric reduction formulae Other reduction formulae

Arc length and surface area3

Cartesian and parametric arc length Polar arc length Surface area of revolution

Sums and integrals2

Integral bounds for sums Limits of sums

05Complex numbers

Complex forms and powers2

Complex numbers as lengths and angles De Moivre's theorem

Trigonometric identities2

Multiple angles and polynomial equations Powers of sine and cosine

Complex roots2

Roots of unity Complex roots and rational powers

Trigonometric sums2

Finite trigonometric sums Weighted and binomial sums

06Differential equations

First-order linear equations2

Why integrating factors work Choosing an integrating factor

Homogeneous second-order equations3

The auxiliary equation Repeated roots Complex roots and oscillation

Particular integrals2

Complementary functions and particular integrals When a PI trial is already in the CF

Substitutions3

First-order substitutions Substituting for the dependent variable Changing the independent variable

Initial conditions and models1

Initial conditions and models

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