📚 OCR Year 13 Further Maths Formula & Theorem Quick Reference | OCR 13年级进阶数学公式定理速查手册
This quick reference guide compiles the essential formulae and theorems from the OCR Year 13 Further Mathematics syllabus, covering Pure Core topics such as complex numbers, matrices, eigenvectors, vectors, hyperbolic functions, Maclaurin series, polar coordinates, differential equations and proof by induction. Every result below is written in the notation OCR examiners expect, so you can use the page as a single revision companion in the final weeks before the exam.
本速查手册汇编了 OCR 13 年级进阶数学教学大纲中的核心公式与定理，涵盖复数、矩阵、特征向量、向量、双曲函数、麦克劳林级数、极坐标、微分方程以及归纳法证明等纯数内容。以下每个结论都采用 OCR 考官认可的书写记号，可直接作为考前最后几周的唯一复习参考。
1. Complex Numbers: Modulus-Argument and Exponential Form | 复数：模-辐角形式与指数形式
ro complex number z = x + iy can be written in modulus-argument form, where the modulus is |z| = r = √x² + y² defined modulo 2π, so we normally quote the principal argument in the range −π < θ ≤ π.
任何非零复数 z = x + iy 都可以写成模-辐角形式，其中模为 |z| = r = √x² + y²，辐角 arg(z) = θ 是从正实轴逆时针量取的角度。辐角只在模 2π 的意义下唯一，因此通常取主辐角范围 −π < θ ≤ π。
z = r(cos θ + i sin θ) = r eⁱθ
Euler' etween trigonometric and exponential notation, and it is the fastest route to multiplication and division. Multiplying two complex numbers adds their arguments and multiplies their moduli; dividing subtracts the arguments and divides the moduli.
欧拉公式 eⁱθ = cos θ + i sin θ 把三角形式与指数形式互相转化，是处理乘除法最快的途径。两个复数相乘时模相乘、辐角相加；相除时模相除、辐角相减。
z₁z₂ = r₁r₂ eⁱ⁽θ¹⁺θ²⁾ and z₁ ÷ z₂ = (r₁ ÷ r₂) eⁱ⁽θ¹⁻θ²⁾
The conjugate is z* = x − iy = r(cos θ − i sin θ) = r e⁻ⁱθ, and the quotient z z* = |z| t, its two roots form a conjugate pair, so if a + bi is a root then a − bi is also a root.
共轭复数为 z* = x − iy = r(cos θ − i sin θ) = r e⁻ⁱθ，并且乘积 z z* = |z|² 恒为实数。当实系数二次方程的判别式为负时，两个根构成共轭对：若 a + bi 是根，则 a − bi 也是根。
2. De Moivre's Theorem and Roots of Unity | 棣莫弗定理与单位根
De Moivre's theorem states that for any integer n, the n-th power of a unit-modulus complex number simply multiplies the angle by n. This single result underpins powers, roots, trigonometric identities and many summation questions.
棣莫弗定理指出，对任意整数 n，单位模复数的 n 次幂只需把角度乘以 n。这一结论支撑了求幂、求根、三角恒等式以及大量求和问题。
(cos θ + i sin θ)ⁿ = cos nθ + i sin nθ
Combining De Moivre with the exponential form gives the general power of any complex number, and the same idea reversed gives the n distinct n-th roots, which are equally spaced around a circle of radius r1/n.
将棣莫弗定理与指数形式结合，即可得到任意复数的任意次幂；反过来使用同一思路，可以得到 n 个互不相同的 n 次方根，它们均匀分布在半径为 r1/n 的圆上。
zⁿ = rⁿ eⁱⁿθ = rⁿ(cos nθ + i sin nθ)
z1/n = r1/n eⁱ⁽θ⁺²ᵏᵖⁱ⁾ᐟⁿ, k = 0, 1, 2, …, n − 1
n exam technique is to write z + 1 ÷ z = 2 cos θ and zⁿ + 1 ÷ zⁿ = 2 cos nθ, then expand (z + 1 ÷ z)ⁿ by the binomial theorem to express cos nθ as a polynomial in cos θ.
单位根（即 zⁿ = 1 的解）模均为 1，辐角为 2kπ/n；它们的和恒为 0，乘积为 (−1)ⁿ⁻¹。考试中最常见的手法是把 z + 1 ÷ z 写成 2 cos θ、把 zⁿ + 1 ÷ zⁿ 写成 2 cos nθ，再用二项式定理展开 (z + 1 ÷ z)ⁿ，从而将 cos nθ 表示为 cos θ 的多项式。
- To find roots geometrically, plot them on an Argand diagram and use symmetry about the real axis.
要在阿尔冈图上求根，可先描点，再利用关于实轴的对称性。 - To sum a series such as Σ cos kθ, treat it as the real part of a geometric series in eⁱθ.
求和如 Σ cos kθ 时，把它看作以 eⁱθ 为公比的等比级数的实部。 - Never forget the +2kπ term, otherwise you will lose marks for missing roots.
切勿漏掉 +2kπ 项，否则会因根缺失而失分。
3. Matrices: Determinants, Inverses and Transformations | 矩阵：行列式、逆矩阵与线性变换
the inverse have compact closed forms that you should be able to write down instantly. A matrix is singular precisely when its determinant is zero, and only non-singular matrices have inverses.
对 2 × 2 矩阵，行列式与逆矩阵都有简洁的封闭形式，应做到一写即出。当且仅当行列式为零时矩阵为奇异矩阵，只有非奇异矩阵才存在逆矩阵。
; c d] = ad − bc and [a b; c d]⁻¹ = (1 ÷ (ad − bc)) [d −b; −c a]
minant by expanding along a row or column using cofactors, and the inverse is the transpose of the cofactor matrix divided by the determinant. In exams, elimination with row operations is usually quicker than the adjugate formula.
对 3 × 3 矩阵，可沿某一行或某一列用代数余子式展开求行列式，逆矩阵等于余子式矩阵的转置除以行列式。考试中，用行变换消元通常比伴随矩阵公式更快。
Matrices also act as linear transformations of the plane. The determinant gives the area scale factor (a negative determinant means the orientation is reversed), and lines through the origin that are mapped onto themselves are invariant lines.
矩阵也可视为平面的线性变换。行列式给出面积缩放因子（行列式为负表示定向翻转），而映射到自身的过原点直线称为不变直线。
| Transformation / 变换 | Matrix / 矩阵 | Area factor / 面积因子 |
| Rotation by θ / 旋转 θ | [cos θ −sin θ; sin θ cos θ] | 1 |
| Reflection in y = x tan θ / 关于 y = x tan θ 反射 | [cos 2θ sin 2θ; sin 2θ −cos 2θ] | −1 |
| Enlargement scale k / 放大 k 倍 | [k 0; 0 k] | k² |
| Shear parallel to x-axis / 沿 x 轴剪切 | [1 k; 0 1] | 1 |
istent (no solutions) or have infinitely many solutions, and you must investigate the geometry rather than blindly quoting an inverse.
把三元一次方程组写成 Mx = b 后，只要 det M ≠ 0，解为 x = M⁻¹b。若 det M = 0，方程组要么无解（不相容），要么有无穷多解，此时必须讨论几何意义，而不能机械地套用逆矩阵。
4. Eigenvalues, Eigenvectors and Cayley-Hamilton | 特征值、特征向量与凯莱-哈密顿定理
An eigenvector of a square matrix M is a non-zero vector v whose direction is unchanged by the transformation; the scale factor λ is the corresponding eigenvalue. Eigenvalues are found from the characteristic equation, obtained by setting the determinant of M − λI to zero.
方阵 M 的特征向量是指方向在变换下保持不变的向量 v（非零），对应的伸缩倍数 λ 称为特征值。特征值由特征方程给出：令 M − λI 的行列式为零。
Mv = λv ⇔ (M − λI)v = 0 ⇔ det(M − λI) = 0
+ det M = 0, which is exactly the Cayley-Hamilton equation satisfied by the matrix itself.
对 2 × 2 矩阵，该方程化为 λ² − (trace M)λ + det M = 0，这正是矩阵自身所满足的凯莱-哈密顿方程。
M² − (trace M)M + (det M)I = O
Cayley-Hamilton lets you reduce high powers of a matrix, and it also gives a neat way of finding M⁻¹ without computing cofactors: rearrange the identity and multiply by M⁻¹.
凯莱-哈密顿定理可用于化简矩阵的高次幂，也能在不求余子式的情况下巧求 M⁻¹：把恒等式变形后两边乘以 M⁻¹ 即可。
genvectors as its columns. Powers then follow immediately, since Mⁿ = PDⁿP⁻¹ and powers of a diagonal matrix act entry by entry.
若矩阵有 n 个线性无关的特征向量，则可对角化为 M = PDP⁻¹，其中 D 是以特征值为对角元的对角矩阵，P 的列向量为特征向量。此时幂运算立刻简化：Mⁿ = PDⁿP⁻¹，而对角矩阵的幂只需逐个元素取幂。
- Symmetric matrices always have real eigenvalues and perpendicular eigenvectors.
实对称矩阵的特征值恒为实数，特征向量互相垂直。 - An orthogonal matrix satisfies MᵀM = I and therefore preserves lengths and angles.
正交矩阵满足 MᵀM = I，因此保持长度与夹角不变。 - Always normalise eigenvectors when a question asks for a unit eigenvector.
当题目要求单位特征向量时，务必对特征向量作归一化。
5. Vectors: Scalar Product, Lines and Planes | 向量：点积、直线与平面
The scalar product measures how much two vectors point in the same direction. It gives the angle between them and a test for perpendicularity, both of which appear constantly in OCR vector questions.
点积衡量两个向量方向的一致程度，可用来求夹角以及判定垂直，这两类应用在 OCR 向量题中反复出现。
a · b = |a||b| cos θ = a₁b₁ + a₂b₂ + a₃b₃
A line can be written in vector form using a point a on the line and a direction vector b, or in Cartesian form by eliminating the parameter t. The direction vector of a line is simply the vector joining any two distinct points on it.
直线可用向量形式表示：取直线上一点 a 与方向向量 b；也可消去参数 t 化为笛卡尔形式。直线的方向向量就是连接线上任意两个不同点的向量。
r = a + t b ⇔ (x − a₁) ÷ b₁ = (y − a₂) ÷ b₂ = (z − a₃) ÷ b₃
t a in the plane. The angle between two planes equals the angle between their normals, and the angle between a line and a plane is found from the angle between the line's direction and the plane's normal, subtracted from 90°.
平面方程为 r · n = d，其中 n 为法向量，d = a · n（a 为平面内任意一点）。两平面的夹角等于两法向量的夹角；直线与平面的夹角则由直线方向与平面法向量的夹角取余角（90° 减去该角）得到。
The perpendicular distance from a point P to the plane through A with normal n is the projection of AP onto the unit normal. This formula replaces a great deal of messy algebra and is worth memorising.
点 P 到过 A 且法向量为 n 的平面的垂距，等于 AP 在单位法向量上的投影。该公式可替代大量繁琐代数运算，值得记忆。
distance = |(P − A) · n| ÷ |n|
For intersecting lines, set the two vector equations equal and solve the resulting simultaneous equations; if the system is inconsistent and the direction vectors are not parallel, the lines are skew.
求两直线交点时，令两条向量方程相等并解方程组；若方程组无解且方向向量不平行，则两直线异面（斜交）。
6. Hyperbolic Functions and Their Inverses | 双曲函数及其反函数
Hyperbolic functions are defined from exponentials, which means every hyperbolic identity can be derived from the definitions rather than memorised in isolation.
双曲函数由指数函数定义，因此每个双曲恒等式都可以从定义推出，而不必孤立地死记。
sinh x = (eˣ − e⁻ˣ) ÷ 2 cosh x = (eˣ + e⁻ˣ) ÷ 2 tanh x = sinh x ÷ cosh x
The fundamental identity mirrors the circular one but with a sign change, and the double-angle results follow the same pattern. Osborn's rule summarises this: take the corresponding trigonometric identity and change the sign of every term containing a product of two sines.
基本恒等式与圆函数类似但符号不同，倍角公式也遵循同样规律。Osborn 法则概括为：取对应的三角恒等式，凡是含有两个正弦之积的项都改变符号。
− sinh²x = 1 cosh 2x = cosh²x + sinh²x sinh 2x = 2 sinh x cosh x
The inverse hyperbolic functions can be written as logarithms, a form that is essential when you integrate expressions involving square roots.
反双曲函数可以写成对数形式，这一形式在积分含根号的表达式时不可或缺。
arsinh x = ln(x + √x² + 1) arcosh x = ln(x + √x² − 1), x ≥ 1
artanh x = ½ ln((1 + x) ÷ (1 − x)), |x| < 1
ative of cosh x is sinh x. For tanh x the derivative is sech²x, where sech x = 1 ÷ cosh x.
双曲函数求导不需要变号：sinh x 的导数是 cosh x，cosh x 的导数是 sinh x。tanh x 的导数为 sech²x，其中 sech x = 1 ÷ cosh x。
- sh x is the shape of a hanging cable and is always above the x-axis, with minimum value 1.
y = cosh x 的图像是悬链线形状，始终位于 x 轴上方，最小值为 1。 - Remember the domain restriction on arcosh and artanh; examiners award method marks for stating it.
牢记 arcosh 与 artanh 的定义域限制，写明这一点可获方法分。 - Integrals of the form 1 ÷ √x² + a² lead directly to inverse hyperbolic logarithms.
形如 1 ÷ √x² + a² 的积分可直接化为反双曲对数形式。
7. Differentiation Techniques and Maclaurin Series | 微分技巧与麦克劳林级数
Further Mathematics extends differentiation to implicit, parametric and inverse functions. For a parametric curve, the first derivative comes from the chain rule and the second derivative requires one extra step of differentiation with respect to the parameter.
进阶数学把微分推广到隐函数、参数方程与反函数。对参数曲线，一阶导数由链式法则给出，二阶导数则需要再对参数求导一次。
dy ÷ dx = (dy ÷ dt) ÷ (dx ÷ dt) d²y ÷ dx² = (d ÷ dt)(dy ÷ dx) ÷ (dx ÷ dt)
The Maclaurin series expands a function as an infinite polynomial built from its derivatives at zero. It is valid only where the series converges, and OCR questions frequently ask for a fixed number of terms plus a statement of the range of validity.
麦克劳林级数把函数展开为以零点处各阶导数为系数的
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