Year 13 OCR Further Maths Formula & Theorem Quick Reference | OCR 进阶数学公式定理速查手册

📚 Year 13 OCR Further Maths Formula & Theorem Quick Reference | OCR 进阶数学公式定理速查手册

This quick-reference handbook brings together the most essential formulae and theorems required for the Year 13 OCR A Level Further Mathematics course. It covers the compulsory Core Pure 2 content together with key results from popular optional modules such as Further Mechanics and Further Statistics. Use it for rapid revision, self-testing, and examination readiness.

本速查手册汇集了 Year 13 OCR A Level 进阶数学所需的最核心公式与定理。它涵盖必修的 Core Pure 2 内容以及进阶力学、进阶统计等常见选修模块中的关键结论，可帮助你快速复习、自我检测和从容应考。

1. Complex Numbers & De Moivre’s Theorem | 复数与棣莫弗定理

A complex number is written as z = x + iy with modulus |z| = √(x² + y²) and argument θ = arctan(y/x) (adjusted for the quadrant). The polar form is z = r(cosθ + i sinθ), where r = |z|. For multiplication in polar form, z₁z₂ = r₁r₂[cos(θ₁+θ₂) + i sin(θ₁+θ₂)]; for division, z₁/z₂ = (r₁/r₂)[cos(θ₁−θ₂) + i sin(θ₁−θ₂)].

复数记作 z = x + iy，模为 |z| = √(x² + y²)，辐角为 θ = arctan(y/x)（需根据象限调整）。其极坐标形式为 z = r(cosθ + i sinθ)，其中 r = |z|。极坐标下的乘法：z₁z₂ = r₁r₂[cos(θ₁+θ₂) + i sin(θ₁+θ₂)]；除法：z₁/z₂ = (r₁/r₂)[cos(θ₁−θ₂) + i sin(θ₁−θ₂)]。

De Moivre’s theorem states that (cosθ + i sinθ)ⁿ = cos(nθ) + i sin(nθ) for any integer n. It is particularly powerful for finding powers of complex numbers and for expressing cos(nθ) and sin(nθ) as polynomials in cosθ and sinθ.

棣莫弗定理指出 (cosθ + i sinθ)ⁿ = cos(nθ) + i sin(nθ) 对任意整数 n 均成立。该定理在求复数乘幂以及将 cos(nθ) 和 sin(nθ) 表示为 cosθ、sinθ 的多项式时尤为有用。

The n-th roots of a complex number: if z = r(cosθ + i sinθ), then its n distinct n-th roots are given by z^{1/n} = r^{1/n}[cos((θ+2kπ)/n) + i sin((θ+2kπ)/n)] for k = 0,1,…,n−1. The n-th roots of unity (where z=1) are ω = cos(2kπ/n) + i sin(2kπ/n), summing to zero and satisfying 1 + ω + ω² + … + ω^{n-1} = 0.

复数的 n 次方根：若 z = r(cosθ + i sinθ)，则 n 个相异的 n 次方根为 z^{1/n} = r^{1/n}[cos((θ+2kπ)/n) + i sin((θ+2kπ)/n)]，k = 0,1,…,n−1。n 次单位根（z=1）为 ω = cos(2kπ/n) + i sin(2kπ/n)，其和为零，且满足 1 + ω + ω² + … + ω^{n-1} = 0。

2. Matrices, Determinants, and Eigenvalues | 矩阵、行列式与特征值

For a 2×2 matrix A = [[a,b],[c,d]], the determinant is det(A) = ad − bc, and if non-zero the inverse is A⁻¹ = (1/det(A)) [[d,−b],[−c,a]]. For a 3×3 matrix, the determinant can be computed using the rule of Sarrus or cofactor expansion. A matrix is singular iff its determinant is zero.

对于 2×2 矩阵 A = [[a,b],[c,d]]，行列式为 det(A) = ad − bc，若非零则逆矩阵为 A⁻¹ = (1/det(A)) [[d,−b],[−c,a]]。对 3×3 矩阵可使用萨鲁斯法则或余子式展开求行列式。矩阵是奇异的当且仅当其行列式为零。

The eigenvalue equation is Av = λv. Eigenvalues λ are found by solving the characteristic equation det(A − λI) = 0. For each λ the corresponding eigenvectors are non-zero solutions to (A − λI)v = 0. If a 3×3 matrix has three linearly independent eigenvectors, it can be diagonalised: P⁻¹AP = D, where P is the matrix of eigenvectors and D the diagonal matrix of eigenvalues. The Cayley–Hamilton theorem states that a square matrix satisfies its own characteristic equation, i.e. if p(λ) = 0, then p(A) = 0. This can be used to compute powers of matrices.

特征方程为 Av = λv。特征值 λ 由求解特征方程 det(A − λI) = 0 获得。对每个 λ，相应的特征向量是 (A − λI)v = 0 的非零解。若 3×3 矩阵拥有三个线性无关的特征向量，则可对角化：P⁻¹AP = D，其中 P 为特征向量矩阵，D 为特征值构成的对角矩阵。凯莱–哈密顿定理指出方阵满足自身的特征方程，即若 p(λ)=0 则 p(A)=0。此定理常用于计算矩阵的乘幂。

3. Hyperbolic Functions | 双曲函数

The hyperbolic functions are defined by sinh x = (eˣ − e⁻ˣ)/2, cosh x = (eˣ + e⁻ˣ)/2, and tanh x = sinh x / cosh x. The fundamental identity is cosh²x − sinh²x = 1. Double-angle formulae include sinh(2x) = 2 sinh x cosh x and cosh(2x) = cosh²x + sinh²x = 2cosh²x − 1 = 2sinh²x + 1. Their inverses, for example arsinh x = ln(x + √(x²+1)), have standard derivative forms.

双曲函数定义为 sinh x = (eˣ − e⁻ˣ)/2，cosh x = (eˣ + e⁻ˣ)/2，<

Published by TutorHao | Year 13 进阶数学 Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）