📚 Year 12 OCR Further Maths: Formula & Theorem Quick Reference Handbook | Year 12 OCR 进阶数学：公式定理速查手册
Whether you’re revising for a topic test or preparing for AS Further Mathematics examinations, having a compact collection of essential formulas and theorems is invaluable. This handbook brings together the key results from the Year 12 OCR Further Pure and applied topics, presented with clear reasoning and paired bilingual explanations. Every expression uses standard mathematical notation so you can quickly locate what you need, understand its meaning, and apply it with confidence.
无论你是在为单元测试复习，还是在为 AS 进阶数学考试作准备，一份精炼的公式与定理手册都极具价值。这本速查手册汇集了 Year 12 OCR 进阶纯数及应用主题中的核心结论，用清晰的思路和中英对照的说明呈现。所有表达式都使用标准的数学符号，你可以迅速找到所需内容，理解其含义，并自信地应用它们。
1. Complex Numbers – Basic Forms | 复数 —— 基本形式
A complex number can be written in Cartesian form as z = x + iy, where x = Re(z) is the real part and y = Im(z) is the imaginary part. The imaginary unit satisfies i² = −1. Equality of complex numbers means both real and imaginary parts are equal. The complex conjugate of z is z* = x − iy (or sometimes denoted as z̄).
复数可以写成笛卡尔形式 z = x + iy，其中 x = Re(z) 为实部，y = Im(z) 为虚部。虚数单位满足 i² = −1。两个复数相等意味着实部和虚部分别相等。 z 的共轭复数为 z* = x − iy（有时也记作 z̄）。
- Addition: (x₁ + iy₁) + (x₂ + iy₂) = (x₁ + x₂) + i(y₁ + y₂)
加法：(x₁ + iy₁) + (x₂ + iy₂) = (x₁ + x₂) + i(y₁ + y₂) - Multiplication: (x₁ + iy₁)(x₂ + iy₂) = (x₁x₂ − y₁y₂) + i(x₁y₂ + x₂y₁)
乘法：(x₁ + iy₁)(x₂ + iy₂) = (x₁x₂ − y₁y₂) + i(x₁y₂ + x₂y₁) - Division: multiply numerator and denominator by the conjugate of the denominator.
除法：分子分母同乘分母的共轭复数。 - Modulus: |z| = √(x² + y²), always non‑negative.
模： |z| = √(x² + y²)，始终非负。 - Argument: arg(z) = θ, where tan θ = y/x, taking care of the quadrant. The principal argument Arg(z) usually lies in (−π, π].
辐角：arg(z) = θ，其中 tan θ = y/x，需注意所在象限。主值 Arg(z) 通常落在 (−π, π]。
2. Modulus-Argument Form and de Moivre’s Theorem | 模–辐角形式与棣莫弗定理
Using the modulus r = |z| and argument θ, we write the modulus‑argument form: z = r (cos θ + i sin θ). Multiplication and division become neat in this form: if z₁ = r₁ (cos θ₁ + i sin θ₁) and z₂ = r₂ (cos θ₂ + i sin θ₂), then
利用模 r = |z| 和辐角 θ，我们可以写出模–辐角形式： z = r (cos θ + i sin θ)。在此形式下，乘法和除法变得非常简洁：若 z₁ = r₁ (cos θ₁ + i sin θ₁) 且 z₂ = r₂ (cos θ₂ + i sin θ₂)，则
- z₁z₂ = r₁r₂ (cos(θ₁+θ₂) + i sin(θ₁+θ₂))
- z₁/z₂ = (r₁/r₂) (cos(θ₁−θ₂) + i sin(θ₁−θ₂))
de Moivre’s theorem states that for any integer n, (r (cos θ + i sin θ))ⁿ = rⁿ (cos nθ + i sin nθ). This is a powerful tool for finding powers of complex numbers and for solving equations like zⁿ = 1, giving the nth roots of unity.
棣莫弗定理指出，对任意整数 n， (r (cos θ + i sin θ))ⁿ = rⁿ (cos nθ + i sin nθ)。这是求复数乘方以及解方程（如 zⁿ = 1，得到 n 次单位根）的有力工具。
The exponential form z = r eiθ is also used, where eiθ = cos θ + i sin θ, leading to the same algebraic rules.
指数形式 z = r eiθ 也很常用，其中 eiθ = cos θ + i sin θ，从而得到相同的代数规则。
3. Roots of Polynomial Equations | 多项式方程的根
For a polynomial P(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₀, the relationships between coefficients and roots (Vieta’s formulas) let you work with sums and products of roots without solving the equation directly.
对于多项式 P(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₀，系数与根之间的韦达定理使得你可以不用直接求解方程就能处理根的和与积。
- Quadratic ax² + bx + c = 0, roots α, β: sum α+β = −b/a, product αβ = c/a.
二次方程 ax² + bx + c = 0，根 α, β：和 α+β = −b/a，积 αβ = c/a。 - Cubic ax³ + bx² + cx + d = 0, roots α, β, γ: Σα = α+β+γ = −b/a, Σαβ = αβ+βγ+γα = c/a, αβγ = −d/a.
三次方程，根 α, β, γ： Σα = α+β+γ = −b/a， Σαβ = αβ+βγ+γα = c/a， αβγ = −d/a。 - Quartic ax⁴ + bx³ + cx² + dx + e = 0, roots α, β, γ, δ: Σα = −b/a, Σαβ = c/a, Σαβγ = −d/a, αβγδ = e/a.
四次方程，根 α, β, γ, δ： Σα = −b/a， Σαβ = c/a， Σαβγ = −d/a， αβγδ = e/a。
These formulas are particularly useful when finding other related polynomials or working with symmetric functions of the roots.
当需要求相关多项式或处理根的对称函数时，这些公式尤其有用。
4. Matrices – Size, Operations and Determinants | 矩阵 —— 阶数、运算与行列式
An m×n matrix has m rows and n columns. Addition and subtraction are performed element‑wise and require matrices of the same order. Matrix multiplication AB is defined only when the number of columns of A equals the number of rows of B. The product AB has the same number of rows as A and the same number of columns as B.
一个 m×n 矩阵有 m 行和 n 列。加法和减法逐元素进行，且要求矩阵同型。矩阵乘法 AB 仅在 A 的列数等于 B 的行数时有定义。乘积 AB 的行数与 A 的行数相同，列数与 B 的列数相同。
The determinant of a 2×2 matrix A = [[a, b], [c, d]] is
det A = ad − bc
For a 3×3 matrix, the determinant can be computed by expansion along any row or column. The standard sign pattern for cofactors is
| + − + |
| − + − |
| + − + |
对 3×3 矩阵，行列式可以按任意一行或一列展开计算。余子式的标准符号规律如上所示。
If det A = 0, the matrix is singular and has no inverse; otherwise it is non‑singular and an inverse exists.
若 det A = 0，矩阵是奇异的，没有逆矩阵；否则是非奇异的，逆矩阵存在。
5. Inverse of a Matrix and Linear Transformations | 逆矩阵与线性变换
For a non‑singular 2×2 matrix A = [[a, b], [c, d]], the inverse is given by
A⁻¹ = (1 / det A) [[d, −b], [−c, a]]
对于非奇异的 2×2 矩阵，逆矩阵如上所示。
A linear transformation T in two dimensions maps (x, y) to (x’, y’) via a matrix M: [x’ y’]T = M [x y]T. Common transformations include:
二维线性变换 T 通过矩阵 M 将 (x, y) 映射为 (x’, y’)： [x’ y’]T = M [x y]T。常见的变换包括：
- Enlargement scale factor k: [[k, 0], [0, k]]
放大因子 k： [[k, 0], [0, k]] - Rotation by angle θ anticlockwise: [[cos θ, −sin θ], [sin θ, cos θ]]
逆时针旋转 θ 角： [[cos θ, −sin θ], [sin θ, cos θ]] - Reflection in the line y = (tan θ)x: [[cos 2θ, sin 2θ], [sin 2θ, −cos 2θ]]
关于直线 y = (tan θ)x 的反射： [[cos 2θ, sin 2θ], [sin 2θ, −cos 2θ]] - Shear parallel to x‑axis, factor k: [[1, k], [0, 1]]
沿 x 轴方向的错切，因子 k： [[1, k], [0, 1]]
Invariant points satisfy M x = x; invariant lines have direction vectors that are eigenvectors of M.
不变点满足 M x = x；不变线的方向向量是 M 的特征向量。
6. Scalar (Dot) Product and Vector (Cross) Product | 标量积（点积）与向量积（叉积）
For vectors a = (a₁, a₂, a₃) and b = (b₁, b₂, b₃):
- Scalar (dot) product: a·b = a₁b₁ + a₂b₂ + a₃b₃ = |a||b| cos θ, where θ is the angle between them. It is a scalar.
- Vector (cross) product: a × b is a vector perpendicular to both a and b. Its magnitude is |a × b| = |a||b| sin θ, giving the area of the parallelogram spanned by a and b. In component form, a × b = (a₂b₃ − a₃b₂, a₃b₁ − a₁b₃, a₁b₂ − a₂b₁).
对向量 a = (a₁, a₂, a₃) 和 b = (b₁, b₂, b₃)：
- 标量积（点积）：a·b = a₁b₁ + a₂b₂ + a₃b₃ = |a||b| cos θ，其中 θ 是两向量间的夹角。结果是个标量。
- 向量积（叉积）：a × b 是一个同时垂直于 a 和 b 的向量。其模为 |a × b| = |a||b| sin θ，等于以 a 和 b 为邻边的平行四边形面积。坐标形式：a × b = (a₂b₃ − a₃b₂, a₃b₁ − a₁b₃, a₁b₂ − a₂b₁)。
The triple scalar product a·(b × c) = b·(c × a) = c·(a × b) gives the volume of the parallelepiped formed by the three vectors.
三向量的标量三重积 a·(b × c) = b·(c × a) = c·(a × b) 给出以这三个向量为棱的平行六面体的体积。
7. Equations of Lines and Planes in 3D | 三维空间中直线与平面的方程
A line in 3D can be expressed in vector form as:
r = a + λ b
where a is a point on the line, b is a direction vector, and λ is a scalar parameter.
其中 a 是直线上一点，b 是方向向量，λ 是标量参数。
The Cartesian equation of a plane is ax + by + cz = d, where n = (a, b, c) is a normal vector to the plane. The vector equation of a plane is:
r·n = a·n
where a is a known point on the plane. An alternative parametric form is r = a + λ u + µ v, with u and v being direction vectors parallel to the plane.
平面的笛卡尔方程为 ax + by + cz = d，其中 n = (a, b, c) 是平面的法向量。平面的向量方程为：r·n = a·n，这里 a 是平面上的一个已知点。另一种参数形式为 r = a + λ u + µ v，其中 u 和 v 是平行于平面的两个方向向量。
The perpendicular distance from a point P with position vector p to the plane r·n = d is given by:
distance = |p·n − d| / |n|
点 P（位置向量为 p）到平面 r·n = d 的垂直距离如上式所示。
8. Further Integration Techniques | 进阶积分技巧
Integrating by substitution is a key method. For a substitution x = g(u), the integral becomes ∫ f(x) dx = ∫ f(g(u)) g'(u) du. A particularly useful substitution for integrals involving √(a² − x²) is x = a sin θ; for √(x² + a²) try x = a tan θ; for √(x² − a²) try x = a sec θ.
换元积分是一个核心方法。对于代换 x = g(u)，积分变为 ∫ f(x) dx = ∫ f(g(u)) g'(u) du。处理含有 √(a² − x²) 的积分时，常用 x = a sin θ；含有 √(x² + a²) 时尝试 x = a tan θ；含有 √(x² − a²) 时尝试 x = a sec θ。
Integration by parts follows the formula ∫ u dv = uv − ∫ v du. Choose u according to the LIATE rule (Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential) to simplify.
分部积分法遵循公式 ∫ u dv = uv − ∫ v du。依据 LIATE 原则（对数、反三角、代数、三角、指数）选取 u 以简化计算。
Improper integrals are evaluated as limits: ∫ₐ^∞ f(x) dx = limb→∞ ∫ₐ^b f(x) dx, and similarly for integrals with unbounded integrands.
反常积分通过极限计算：∫ₐ^∞ f(x) dx = limb→∞ ∫ₐ^b f(x) dx，对被积函数无界的情形也是如此。
9. Arc Length and Surface Area of Revolution | 弧长与旋转体表面积
For a curve defined by y = f(x) from x = a to x = b, the arc length s is
s = ∫ₐ^b √(1 + (dy/dx)²) dx
对于由 y = f(x) 定义、从 x = a 到 x = b 的曲线，弧长 s 如上式。
When the same curve is rotated about the x‑axis, the surface area of the resulting solid of revolution is
A = 2π ∫ₐ^b y √(1 + (dy/dx)²) dx
如果将此曲线绕 x 轴旋转，所得旋转体的表面积如上式。
For a curve given parametrically by (x(t), y(t)), t ∈ [t₁, t₂], the arc length becomes
s = ∫ₜ₁^ₜ₂ √((dx/dt)² + (dy/dt)²) dt
对于由参数方程 (x(t), y(t)) 给出的曲线，弧长公式如上。
The surface area when rotated about the x‑axis in parametric form is
A = 2π ∫ₜ₁^ₜ₂ y(t) √((dx/dt)² + (dy/dt)²) dt
绕 x 轴旋转的表面积（参数形式）如上。
10. Reduction Formulae | 递推公式
Reduction formulae are used to express an integral of a certain type in terms of a simpler, or lower‑order, integral. Typically, they involve integration by parts. For example, if Iₙ = ∫ sinⁿ x dx, then a standard reduction formula is
Iₙ = −(1/n) sinⁿ⁻¹ x cos x + ((n−1)/n) Iₙ₋₂
递推公式用于将某类积分表示为更简单或更低阶的积分，通常借助分部积分。例如，若 Iₙ = ∫ sinⁿ x dx，则一个标准的递推公式如上。
Similarly for Iₙ = ∫ cosⁿ x dx, we obtain
Iₙ = (1/n) cosⁿ⁻¹ x sin x + ((n−1)/n) Iₙ₋₂
类似地，对于 Iₙ = ∫ cosⁿ x dx，可得到以上的递推公式。
Reduction formulae are extremely helpful for definite integrals where limits are 0 and π/2, making the boundary terms vanish and linking even/odd n cases.
递推公式对于上下限为 0 和 π/2 的定积分尤其有帮助，此时边界项消失，并将偶数和奇数 n 的情况联系起来。
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