Title: InkMaths / Further Pure 2: groups, matrices and number theory
On every note of this deck, unless a note says otherwise:
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Front: STANDARD RESULT — LEARN IT
Arc length:
cartesian, then polar = ?
Back: ∫ √1 + (dydx)^2 dx, ∫ √r^2 + (drdθ)^2 dθ

Front: CHECK YOURSELF
A group has order 8. Explain why it cannot contain an element of order 3.
Back: The powers of such an element would form a subgroup of order 3, and 3 does not divide 8, contradicting Lagrange's theorem.

Front: DEFINE
Define: Bezout's identity.
Back: The highest common factor of a and b can be written as ax + by for integers x and y, found by unwinding the Euclidean algorithm. When that factor is 1, x is the inverse of a modulo b.

Front: SPOT THE ERROR
Say what is wrong with this claim:
Permutations and combinations count the same thing, so either formula will do.
Back: Permutations count arrangements and combinations count selections; they differ by exactly the r! orderings of each selection. Choosing 11 from 21 gives 352716 selections but almost forty million times as many ordered line-ups.

Front: STANDARD RESULT — LEARN IT
Inversion:
w = 1z: image of a line missing 0?
Back: a circle through 0, and the map sends that circle back to a line

Front: DEFINE
Define: Cayley-Hamilton theorem.
Back: Every square matrix satisfies its own characteristic equation, which turns any power of it into a linear combination of lower powers and gives the inverse without cofactors.

Front: DEFINE
Define: group.
Back: A set with a binary operation satisfying closure, associativity, the existence of an identity, and an inverse for every element. Commutativity is not required.

Front: STANDARD RESULT — LEARN IT
Surface of revolution:
about the x-axis, S = ?
Back: 2π ∫ y ds = 2π ∫ y √1 + (dydx)^2 dx

Front: STANDARD RESULT — LEARN IT
Reduction formula:
I_n = ∫_0^π/2 sin ^n x dx: n I_n = ?
Back: (n - 1) I_n-2, with I_0 = π2, I_1 = 1

Front: DEFINE
Define: Lagrange's theorem.
Back: The order of any subgroup of a finite group divides the order of the group, so the order of every element divides it too.

Front: STANDARD RESULT — LEARN IT
Diagonalisation:
M^n = ?
Back: P D^n P^-1, P of eigenvectors, D of eigenvalues in step

Front: STANDARD RESULT — LEARN IT
Eigenvalues:
2 by 2 characteristic equation = ?
Back: λ^2 - (trace)λ + det = 0

Front: STANDARD RESULT — LEARN IT
Fermat's little theorem:
p prime, a not a multiple of p: a^p-1 ≡ ?
Back: 1 (mod p), so exponents reduce modulo p - 1

Front: STANDARD RESULT — LEARN IT
Recurrence relations:
u_n+2 + b u_n+1 + c u_n = 0: u_n = ?
Back: A m_1^n + B m_2^n, repeated root needs (A + Bn)m^n

Front: DEFINE
Define: Apollonius locus.
Back: The set of points whose distances from two fixed points are in a fixed ratio other than 1. It is a circle, not a line; equal distances give the perpendicular bisector instead.