Further Pure 2

What further pure 2 covers

One of the eight optional papers of 9FM0, sat as Paper 4A. It is an Option 2 paper, which means it may be taken only in a matching pair with Further Pure 1, so a student sitting it sits FP1 as Paper 3. The content is the widest on the qualification: group theory, eigenvectors, number theory and the recurrence relations that close the algebra of sequences.

The main ideas

• The four group axioms, Cayley tables, the order of a group and of an element, cyclic groups and their generators.
• Subgroups, Lagrange's theorem, and isomorphism between groups of order up to eight.
• Reduction formulae derived by parts, then walked down to a base case that can be integrated directly.
• Arc length and the area of a surface of revolution, in cartesian, parametric and polar form.
• Eigenvalues and eigenvectors of 2 by 2 and 3 by 3 matrices, then diagonalisation and the Cayley-Hamilton theorem.
• Further loci in the Argand diagram, and transformations of the complex plane by squaring and by Mobius maps.
• The Euclidean algorithm and Bezout's identity, modular arithmetic and Fermat's little theorem, combinatorics, and recurrence relations.

The results it turns on

closure, associativity, an identity, and an inverse for every element

the four axioms a set and operation have to satisfy

the order of any subgroup divides the order of the group

Lagrange's theorem, which also applies to the order of an element

s = ∫√(1 + (dy/dx)²) dx

with parametric and polar equivalents in the booklet

det(M − λI) = 0

the characteristic equation, whose roots are the eigenvalues

P⁻¹MP = D, so Mⁿ = PDⁿP⁻¹

diagonalisation, which makes a high power cheap

a^p−1 ≡ 1 (mod p), for prime p not dividing a

Fermat's little theorem, which reduces a large exponent

Where it usually goes wrong

• Closure is the axiom most often skipped and the one a proposed set most often fails, so check it explicitly and by name.
• The eigenvector order in P has to match the eigenvalue order in D. A mismatch invalidates everything after it.
• The locus |z - a| = k|z - b| is a circle for k other than 1, and degenerates to the perpendicular bisector when k = 1, so state which case you are in.
• Cancelling a common factor in a congruence is safe when that factor is coprime to the modulus, and unreliable otherwise.

Where to start

The two group lessons first, as one block. Reduction formulae and arc length reuse parts and substitution, so take them after Further calculus. Eigenvalues before diagonalisation. The two complex plane lessons follow Core Pure complex numbers. Number theory, combinatorics and recurrence relations are independent of the rest and can be fitted anywhere.