AQA Further Core Maths A-Level

Sum of Series

1. Proof of the sum of the series ∑r
2. Proof of the sum of the series ∑r²
3. Proof of the sum of the series ∑r³
4. Proof for other series

Various Types of Proofs

1. Divisibility and multiple test proofs
2. Matrix proofs

Complex Numbers

1. Real and imaginary numbers
2. Addition, subtraction and multiplying complex numbers and simplifying powers of i
3. Complex conjugates
4. Division of a complex number by a complex number
5. Argand diagrams
6. Modulus and argument of a complex number
7. Solving problems with complex numbers
8. Exponential Form (Euler's relation)
9. Multiplication and division rules for mod and argument of two complex numbers
10. Complex numbers specimen questions (mixed exam boards)

De Moivre’s Theorem

1. De Moivre's theorem
2. Expressing sin(nθ) and cos(nθ) in terms of sinθ and cosθ
3. Expressing sinⁿθ and cosⁿθ in terms of sin(kθ) and cos(kθ)
4. nth roots of a complex number

Loci in the Complex Plane

1. The locus of a point moving in a circle

Matrices

1. Introduction and dimension of a matrix
2. Addition and subtraction and multiplying a matrix by a scalar
3. Matrix multiplication
4. Identity and Inverse of a 2x2 matrix

Simultaneous Equations by Matrix Methods

1. Solving simultaneous linear equations in 2 unknowns

Matrix Linear Transformations

1. Linear transformations - rotations
2. Linear transformations - reflections
3. Linear transformations - enlargement
4. How well do you know your transformations?
5. Combinations of transformations

Factorising determinants

Eigenvalues and Eigenvectors

Roots of Polynomial Equations

Standard Summations

1. Sum of the first n natural numbers ∑r and the results for ∑a and ∑(ar+b)
2. Sum of the squares of the first n natural numbers ∑r²
3. Sum of the cubes of the first n natural numbers ∑r³
4. Using known formulae to sum more complex series

Method of Differences

1. Method of differences

Maclaurin’s Series

Limits

Fractional Inequalities

1. Fractional inequalities

Modulus Inequalities

1. Modulus inequalities fractional type

Graphs of Rational Functions

Conic Sections

Parabola

The Ellipse

The Hyperbola

Applications of Integration – Volumes of revolution

1. Volume of revolution about the x-axis

Mean Value of a Function

Integrals involving Partial fractions

1. Integrals involving partial fractions

Differentiating Inverse Trigonometric Functions

Standard Integrals Involving Inverse Trigonometric Functions

Applications of Integration – Arc length

Applications of Integration – Surface area of revolution

Deriving and using reduction formulae

Improper Integrals

Scalar Product (Dot Product)

1. Scalar product
2. Perpendicular vectors

Vector Equations of Lines

1. Vector equation of a line
2. Angle between two lines
3. Parallel lines
4. Intersecting and skew lines
5. Closest point to a line and shortest distance from the origin

Lines

Vector Product

Planes

Polar Coordinates

Equations of Curves

Area Bounded by a Polar Curve

Hyperbolic Functions

1. Osborn's rule
2. Solving hyperbolic equations using hyperbolic identities

Differentiation of Hyperbolic Functions

Standard Integrals Involving Hyperbolic Functions

Exact Equations (Integrating Factors)

Equations of the form a d²y/dx² + b dy/dx + cy = 0

Equations of the form a d²y/dx² + b dy/dx + cy = f(x)

Numerical Integration