📚 A-Level WJEC Further Mathematics: Formula & Theorem Quick Reference | A-Level WJEC 进阶数学：公式定理速查手册
This quick reference collects the essential pure mathematics formulas and theorems needed for the WJEC A-Level Further Mathematics specification. Use it to reinforce your revision and keep core results at your fingertips.
本速查手册汇集了 WJEC A-Level 进阶数学纯数学部分的核心公式与定理，方便你在复习中随时查阅、巩固重点知识。
1. Complex Numbers | 复数
A complex number takes the form z = x + iy, where x, y ∈ ℝ and i² = −1. The complex conjugate is z* = x − iy; its modulus is |z| = √(x² + y²) and its argument arg(z) = θ satisfies tan θ = y/x, with the quadrant chosen carefully.
|z| = √(x² + y²), arg(z) = arctan(y/x)
复数可写作 z = x + iy，其中 x, y 为实数，i² = −1。共轭复数 z* = x − iy；模 |z| = √(x² + y²)，辐角 arg(z) = θ 满足 tan θ = y/x，并需选择正确象限。
De Moivre’s theorem states that for any integer n, (cos θ + i sin θ)ⁿ = cos nθ + i sin nθ. Using the polar form z = r(cos θ + i sin θ) we obtain zⁿ = rⁿ(cos nθ + i sin nθ).
(cos θ + i sin θ)ⁿ = cos nθ + i sin nθ
De Moivre 定理指出，对于任意整数 n，有 (cos θ + i sin θ)ⁿ = cos nθ + i sin nθ。利用极坐标形式 z = r(cos θ + i sin θ) 即可求得 zⁿ = rⁿ(cos nθ + i sin nθ)。
The n‑th roots of a complex number w are given by z = r^{1/n} [cos((θ+2kπ)/n) + i sin((θ+2kπ)/n)] for k = 0, 1, …, n−1. Euler’s formula e^{iθ} = cos θ + i sin θ offers an elegant link.
复数 w 的 n 次方根为 z = r^{1/n} [cos((θ+2kπ)/n) + i sin((θ+2kπ)/n)]，k = 0, 1, …, n−1。欧拉公式 e^{iθ} = cos θ + i sin θ 给出了简洁的表达。
2. Roots of Polynomial Equations | 多项式方程的根
For a quadratic equation ax² + bx + c = 0 with roots α and β, Vieta’s formulas give α+β = −b/a and αβ = c/a. For cubic and quartic equations analogous sum‑product relations exist.
α + β = −b/a, αβ = c/a
对于二次方程 ax² + bx + c = 0，其根为 α, β，由韦达定理可得 α+β = −b/a，αβ = c/a。三次与四次方程也有对应的和积关系。
If a polynomial has real coefficients, any non‑real roots occur in conjugate pairs. Given a root α, its conjugate α* is also a root. This fact is essential when constructing equations with given roots.
若实系数多项式有非实数根，则它们一定以共轭对出现。若 α 为根，其共轭 α* 亦为根。这一性质常用于构造满足指定根的方程。
The substitution method can transform the sum of powers Σα², Σα³ etc. into symmetric combinations, enabling the evaluation of new roots’ relationships without solving the original equation.
替换法可将根的幂和 Σα², Σα³ 等化为对称组合，无需解原方程即可求得新根的关系。
3. Matrices and Linear Transformations | 矩阵与线性变换
For a 2×2 matrix M = [[a, b], [c, d]], the determinant is det M = ad − bc. M is invertible if and only if det M ≠ 0, with inverse M⁻¹ = (1/det M) [[d, −b], [−c, a]].
det M = ad − bc, M⁻¹ = (1/det M)[[d, −b], [−c, a]]
对 2×2 矩阵 M = [[a, b], [c, d]]，行列式为 det M = ad − bc。当且仅当 det M ≠ 0 时 M 可逆，逆矩阵为 M⁻¹ = (1/det M) [[d, −b], [−c, a]]。
A matrix represents a linear transformation. Common transformations include rotation by angle θ (matrix [[cos θ, −sin θ], [sin θ, cos θ]]), reflection in a line through the origin, and scaling. The unit square method helps identify the transformation.
矩阵代表一个线性变换。常见变换包括旋转 θ 角（矩阵 [[cos θ, −sin θ], [sin θ, cos θ]]）、关于过原点直线的反射以及缩放。单位正方形法有助于识别变换类型。
The image of a point (x, y) under transformation M is obtained by matrix multiplication. Invariant lines and points can be found by solving Mv = v or by setting the transformed line equation proportional to the original.
点 (x, y) 在变换 M 下的像通过矩阵乘法得到。不变直线与不变点可由 Mv = v 或将变换后的直线方程设为与原直线成比例来求解。
4. Summation of Series | 级数求和
Standard results often used are Σᵣ₌₁ⁿ r = ½ n(n+1), Σᵣ₌₁ⁿ r² = ⅙n(n+1)(2n+1), and Σᵣ₌₁ⁿ r³ = ¼n²(n+1)². These can be proved by induction and are assumed in WJEC examinations.
Σ r = n(n+1)/2, Σ r² = n(n+1)(2n+1)/6, Σ r³ = n²(n+1)²/4
常用的标准结果为 Σᵣ₌₁ⁿ r = ½ n(n+1)，Σᵣ₌₁ⁿ r² = ⅙n(n+1)(2n+1)，Σᵣ₌₁ⁿ r³ = ¼n²(n+1)²。它们可用归纳法证明，在 WJEC 考试中可直接引用。
To sum a more complicated series such as Σ (3r² − 2r + 1), split it using linearity: 3Σ r² − 2Σ r + Σ 1. Always express the final answer as a fully factorised polynomial in n.
对于更复杂的级数如 Σ (3r² − 2r + 1)，可利用线性性质拆分为 3Σ r² − 2Σ r + Σ 1。最终答案应化为 n 的完全因式分解形式。
For an infinite geometric series with first term a and common ratio |r| < 1, the sum to infinity is S∞ = a/(1−r). Convergence requires |r| < 1.
对于首项为 a、公比满足 |r| < 1 的无穷等比级数，其无穷和为 S∞ = a/(1−r)。收敛条件为 |r| < 1。
5. Proof by Induction | 数学归纳法
A typical induction proof has four steps: (i) Basis – verify the statement for n = 1 (or the smallest applicable n). (ii) Assumption – assume the statement true for n = k. (iii) Inductive step – show that the truth for n = k implies truth for n = k+1. (iv) Conclusion – state that by mathematical induction the statement holds for all n ∈ ℕ.
Basis → Assumption → Inductive Step → Conclusion
标准归纳法包含四步：(i) 奠基 —— 证明 n = 1（或最小适用 n）时命题成立。(ii) 假设 —— 假设 n = k 时命题成立。(iii) 递推 —— 由 n = k 成立推导 n = k+1 成立。(iv) 结论 —— 由数学归纳法知命题对所有自然数 n 成立。
Induction is used to prove summation formulas, divisibility results (e.g. “3ⁿ − 1 is divisible by 2”), matrix powers, and inequalities. Always write the induction hypothesis clearly and manipulate the (k+1)‑th case to exploit it.
归纳法常用于证明求和公式、整除性（如“3ⁿ − 1 能被 2 整除”）、矩阵幂以及不等式。务必清晰写出归纳假设，并在处理 (k+1) 情形时加以应用。
6. Hyperbolic Functions | 双曲函数
The hyperbolic functions are defined as sinh x = (eˣ − e⁻ˣ)/2, cosh x = (eˣ + e⁻ˣ)/2, and tanh x = sinh x / cosh x. Their graphs and properties mirror those of trigonometric functions, with key identities such as cosh²x − sinh²x = 1.
cosh²x − sinh²x = 1
双曲函数定义为 sinh x = (eˣ − e⁻ˣ)/2，cosh x = (eˣ + e⁻ˣ)/2，tanh x = sinh x / cosh x。它们的图像与性质与三角函数类似，核心恒等式为 cosh²x − sinh²x = 1。
Derivatives: d/dx sinh x = cosh x, d/dx cosh x = sinh x, d/dx tanh x = sech²x. Integration formulas follow directly, for example ∫ sinh x dx = cosh x + C.
导数公式：d/dx sinh x = cosh x，d/dx cosh x = sinh x，d/dx tanh x = sech²x。相应的积分公式可直接写出，例如 ∫ sinh x dx = cosh x + C。
The inverse hyperbolic functions can be expressed as logarithms: arsinh x = ln(x + √(x²+1)), arcosh x = ln(x + √(x²−1)) (x ≥ 1), artanh x = ½ ln((1+x)/(1−x)) (|x| < 1). These are useful in integration.
反双曲函数可表示为对数形式：arsinh x = ln(x + √(x²+1))，arcosh x = ln(x + √(x²−1)) (x ≥ 1)，artanh x = ½ ln((1+x)/(1−x)) (|x| < 1)。这些形式在积分中非常有用。
7. Polar Coordinates | 极坐标
In polar coordinates a point is given by (r, θ) where r is the distance from the origin and θ the angle measured from the initial line. Conversion to Cartesian: x = r cos θ, y = r sin θ; conversely r = √(x² + y²), θ = arctan(y/x) with appropriate quadrant.
在极坐标中，点表示为 (r, θ)，r 为到原点的距离，θ 为从极轴测量的角度。与直角坐标的转换：x = r cos θ, y = r sin θ；反之 r = √(x² + y²)，θ = arctan(y/x) 并选择正确象限。
For a curve r = f(θ), the gradient of the tangent is given by dy/dx = (r sin θ + r′ cos θ) / (r cos θ − r′ sin θ), where r′ = dr/dθ. The area enclosed by a polar curve from θ = α to θ = β is A = ½ ∫ₐᵝ r² dθ.
对于曲线 r = f(θ)，切线斜率为 dy/dx = (r sin θ + r′ cos θ) / (r cos θ − r′ sin θ)，其中 r′ = dr/dθ。极曲线在 θ = α 到 θ = β 之间围成的面积为 A = ½ ∫ₐᵝ r² dθ。
Common polar curves include the cardioid r = a(1 + cos θ) and the rose r = a sin nθ. Symmetry properties can often reduce the integration range.
常见极曲线有心形线 r = a(1 + cos θ) 和玫瑰线 r = a sin nθ。利用对称性常可简化积分区间。
8. Further Calculus: Arc Length and Surface Area | 进阶微积分：弧长与表面积
For a curve defined by y = f(x) from x = a to x = b, the arc length is s = ∫ₐᵇ √(1 + (dy/dx)²) dx. If the curve is given parametrically by (x(t), y(t)), then s = ∫ₜ₁ᵗ² √((dx/dt)² + (dy/dt)²) dt.
对于由 y = f(x) 定义的曲线从 x = a 到 x = b，弧长公式为 s = ∫ₐᵇ √(1 + (dy/dx)²) dx。若曲线由参数方程 (x(t), y(t)) 给出，则 s = ∫ₜ₁ᵗ² √((dx/dt)² + (dy/dt)²) dt。
The area of a surface of revolution when a curve is rotated about the x‑axis is S = ∫ 2π y ds, where ds is the arc‑length element. For y = f(x), S = 2π ∫ₐᵇ y √(1 + (dy/dx)²) dx.
曲线绕 x 轴旋转一周所得旋转面的表面积为 S = ∫ 2π y ds，其中 ds 为弧长元素。对 y = f(x
Published by TutorHao | A-Level 进阶数学 Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导