FP1Further Proof

Further Mathematics extends the proof techniques from A Level Mathematics with more sophisticated methods:

• Proof by induction — a two-step process: (1) prove the base case, (2) assume true for some n=k and prove for n=k+1. Induction is particularly powerful for proving results about sequences, series, divisibility, matrix powers, and recurrence relations.
• Strong induction — assumes the statement is true for all values up to k, not just k itself. This is needed when the inductive step for k+1 depends on multiple previous values.
• Proof by contradiction — continues from A Level with more complex examples involving irrationality, infinite primes, and properties of special numbers.

FP2Complex Numbers

Complex numbers extend the real number system by introducing i where i^2 = -1. Every complex number can be written as z = x + iy (Cartesian form) or z = r(\cos\theta + i\sin\theta) (modulus-argument form).

The Argand diagram represents complex numbers geometrically as points in a plane, with the real part on the horizontal axis and the imaginary part on the vertical axis. Addition corresponds to vector addition; multiplication corresponds to scaling and rotation.

De Moivre's theorem states that (r(\cos\theta + i\sin\theta))^n = r^n(\cos n\theta + i\sin n\theta). This is used to find powers and roots of complex numbers. The nth roots of unity lie equally spaced around the unit circle, forming a regular n-gon.

Loci on the Argand diagram — the equation |z - a| = r represents a circle centre a radius r; |z - a| = |z - b| represents the perpendicular bisector of the line joining a and b; \arg(z - a) = \theta represents a half-line.

FP3Further Algebra

Relations between roots and coefficients — for a polynomial a_n x^n + \cdots + a_0 = 0 with roots \alpha_1, \alpha_2, \ldots, \alpha_n:

• Sum of roots: \sum \alpha_i = -\frac{a_{n-1}}{a_n}
• Sum of products of roots taken two at a time: \sum \alpha_i\alpha_j = \frac{a_{n-2}}{a_n}
• Product of roots: \prod \alpha_i = (-1)^n \frac{a_0}{a_n}

These relationships let you form new equations whose roots are related to the original roots (e.g. \alpha^2, \frac{1}{\alpha}, \alpha + \beta) without solving the original equation.

Method of differences — if a series term can be written as f(r) - f(r+1) or similar, most terms cancel when summed, leaving only boundary terms. This is particularly useful for rational and trigonometric series.

FP4Further Functions

Inverse trigonometric functions (\arcsin, \arccos, \arctan) require restricted domains to be well-defined. Know their domains, ranges, and graphs.

Hyperbolic functions are defined using exponentials:

• \sinh x = \frac{e^x - e^{-x}}{2}
• \cosh x = \frac{e^x + e^{-x}}{2}
• \tanh x = \frac{\sinh x}{\cosh x}

Hyperbolic functions satisfy identities analogous to trigonometric ones (e.g. \cosh^2 x - \sinh^2 x = 1). The inverse hyperbolic functions have logarithmic forms: \text{arsinh } x = \ln(x + \sqrt{x^2+1}).

Differentiation and integration of hyperbolic and inverse hyperbolic functions follow standard patterns and are required in Further Calculus.

FP5Matrices

Matrix operations — addition (element-wise), multiplication (rows by columns), and scalar multiplication. Matrix multiplication is associative but not commutative.

Determinants — for 2\times 2 and 3\times 3 matrices. A matrix is singular if its determinant is zero. The determinant of a product equals the product of determinants.

Inverse matrices — a square matrix has an inverse if and only if it is non-singular. For 2\times 2: if A = \begin{pmatrix} a & b \ c & d \end{pmatrix}, then A^{-1} = \frac{1}{\det A}\begin{pmatrix} d & -b \ -c & a \end{pmatrix}.

Transformations — matrices represent linear transformations: reflections, rotations, enlargements, stretches, and shears. The composition of transformations corresponds to matrix multiplication.

Eigenvalues and eigenvectors — a non-zero vector \mathbf{v} is an eigenvector of matrix A with eigenvalue \lambda if A\mathbf{v} = \lambda\mathbf{v}. Eigenvalues are found by solving \det(A - \lambda I) = 0 (the characteristic equation).

FP6Further Vectors

Vector (cross) product — \mathbf{a} \times \mathbf{b} is a vector perpendicular to both \mathbf{a} and \mathbf{b}, with magnitude |\mathbf{a}||\mathbf{b}||\sin\theta|. In component form, use the determinant method with unit vectors \mathbf{i}, \mathbf{j}, \mathbf{k}.

Scalar triple product — \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) gives the volume of the parallelepiped formed by the three vectors. If it equals zero, the vectors are coplanar.

Lines in 3D — parametric form \mathbf{r} = \mathbf{a} + \lambda\mathbf{d} and symmetric form. Planes — cartesian form ax + by + cz = d or vector form \mathbf{r} \cdot \mathbf{n} = k.

Finding intersections, angles between lines and planes, and shortest distances are standard problems. The angle between a line and a plane is the complement of the angle between the line and the normal to the plane.

FP7Further Calculus

Reduction formulas express an integral I_n in terms of I_{n-1} or I_{n-2}. Typically derived using integration by parts. They are useful for evaluating integrals involving powers of trigonometric functions.

Arc length of a curve y = f(x) from a to b is \int_a^b \sqrt{1 + (\frac{dy}{dx})^2}\,dx. For parametric curves, use \int \sqrt{(\frac{dx}{dt})^2 + (\frac{dy}{dt})^2}\,dt.

Surface area of revolution — rotating y = f(x) about the x-axis: S = 2\pi \int_a^b y\sqrt{1+(\frac{dy}{dx})^2}\,dx.

Mean value of a function over [a,b] is \frac{1}{b-a}\int_a^b f(x)\,dx.

FP8Polar Coordinates

In polar coordinates, a point is described by (r, \theta) where r is the distance from the origin and \theta is the angle from the initial line. The conversion formulas are x = r\cos\theta, y = r\sin\theta.

Common polar curves include: cardioids (r = a(1 + \cos\theta)), roses (r = a\cos n\theta), and spirals (r = a\theta). Sketching these requires understanding symmetry and key points.

The area enclosed by a polar curve is \frac{1}{2}\int_{\alpha}^{\beta} r^2\,d\theta. The arc length in polar form is \int \sqrt{r^2 + (\frac{dr}{d\theta})^2}\,d\theta.

To find tangents parallel or perpendicular to the initial line, find where \frac{dy}{d\theta} = 0 (parallel) or \frac{dx}{d\theta} = 0 (perpendicular).

FP9Further Differential Equations

First-order linear equations — equations of the form \frac{dy}{dx} + P(x)y = Q(x) are solved using an integrating factor \mu = e^{\int P\,dx}. Multiply through by \mu to make the left side a perfect derivative.

Second-order linear equations with constant coefficients — of the form a\frac{d^2y}{dx^2} + b\frac{dy}{dx} + cy = f(x). The solution has two parts:

• Complementary function — solve the auxiliary equation am^2 + bm + c = 0. If roots are real and distinct: Ae^{m_1 x} + Be^{m_2 x}. If repeated: (A + Bx)e^{mx}. If complex: e^{\alpha x}(A\cos\beta x + B\sin\beta x).
• Particular integral — guess a form similar to f(x) and substitute to find constants. If your guess matches the complementary function, multiply by x (or x^2 if necessary).