📚 A-Level Edexcel Further Maths: End-of-Term Revision Checklist | A-Level Edexcel 进阶数学：期末复习提纲
As the end of term approaches, A-Level Edexcel Further Maths students face the challenge of consolidating a vast array of advanced topics. This revision checklist breaks down the key concepts, essential techniques, and common pitfalls across Core Pure modules and popular applied options. Use it to structure your revision, identify weak areas, and approach your mocks with confidence.
随着学期末的临近，学习 A-Level Edexcel 进阶数学的学生需要整合大量高级课题。这份复习提纲将核心纯数模块和常见应用模块的关键概念、基本技巧和常见易错点一一梳理。用这份提纲来规划复习、找出薄弱环节，从容应对模拟考试。
1. Complex Numbers & Argand Diagrams | 复数与阿干特图
Complex numbers extend the real number system and are written as z = x + iy, where i² = –1. The modulus |z| = √(x² + y²) gives the distance from the origin, while the argument arg(z) = θ is the angle measured from the positive real axis. On an Argand diagram, addition and subtraction of complex numbers follow vector rules, multiplication rotates and scales, and division subtracts arguments.
复数扩展了实数系统，记作 z = x + iy，其中 i² = –1。模 |z| = √(x² + y²) 表示到原点的距离，辐角 arg(z) = θ 是从正实轴量起的角度。在阿干特图上，复数的加减遵循向量法则，乘法会旋转并缩放，除法对应辐角相减。
De Moivre’s theorem, (r(cos θ + i sin θ))ⁿ = rⁿ(cos nθ + i sin nθ), is fundamental for finding powers and roots. Euler’s formula eⁱᶿ = cos θ + i sin θ links exponentials to trigonometric functions. When solving equations, remember to express complex roots in polar form and find all nth roots, which are spaced evenly around a circle of radius r¹/ⁿ.
棣莫弗定理 (r(cos θ + i sin θ))ⁿ = rⁿ(cos nθ + i sin nθ) 是求幂和求根的基础。欧拉公式 eⁱᶿ = cos θ + i sin θ 将指数与三角函数联系起来。解方程时，记得用极坐标形式表达复数根，并求出所有 n 次方根，它们均匀分布在半径为 r¹/ⁿ 的圆上。
z = r eiθ
2. Roots of Polynomial Equations | 多项式方程的根
Relationships between roots and coefficients of polynomials are tested frequently. For a cubic equation ax³ + bx² + cx + d = 0 with roots α, β, γ, the sums are: Σα = –b/a, Σαβ = c/a, αβγ = –d/a. Similar symmetric sums exist for quartics. You must be able to derive new equations whose roots are functions of the original roots, such as α², 1/α, or α+β.
多项式根与系数的关系是常考内容。对于三次方程 ax³ + bx² + cx + d = 0，其根为 α, β, γ，则有 Σα = –b/a，Σαβ = c/a，αβγ = –d/a。四次方程也有类似的对称和式。你需要推导出新方程，其根为原根的函数，例如 α²、1/α 或 α+β。
A common technique is to substitute y = f(x) into the original polynomial to eliminate x. For example, if the new root y = 2α + 1, set x = (y – 1)/2 and substitute. Always check whether the transformation is one-to-one and consider potential repeated roots.
常用的方法是把 y = f(x) 代入原多项式消去 x。例如，若新根为 y = 2α + 1，则令 x = (y – 1)/2 并代入。务必检查变换是否一一对应，并注意可能的重根情况。
3. Series & Method of Differences | 级数与差分法
You need to know the standard sums: Σr = n(n+1)/2, Σr² = n(n+1)(2n+1)/6, Σr³ = n²(n+1)²/4. More complex series can be tackled using the method of differences. The idea is to express the general term as a difference f(r) – f(r+1), so that the sum telescopes.
需要熟记标准求和公式：Σr = n(n+1)/2，Σr² = n(n+1)(2n+1)/6，Σr³ = n²(n+1)²/4。更复杂的级数可用差分法处理，即把通项写成 f(r) – f(r+1) 的差式，从而让求和项前后相消。
For example, 1/(r(r+1)) can be split using partial fractions, and the sum from r=1 to n simplifies to 1 – 1/(n+1). Be careful with the limits and the final expression. Also practice deriving sums of polynomial series by expanding r(r+1)(r+2)… and using standard results.
例如，1/(r(r+1)) 可用部分分式拆分，从 r=1 到 n 求和后简化为 1 – 1/(n+1)。注意求和限和最终表达式。还要练习通过展开 r(r+1)(r+2)… 并利用标准结果来推导多项式级数的和。
4. Matrices & Linear Transformations | 矩阵与线性变换
Matrices represent linear transformations including rotations, reflections, stretches, shears, and enlargements. You should be confident finding the determinant and inverse of a 2×2 and 3×3 matrix. The inverse of A is (1/det A) adj A, where adj A is the adjugate. Singular matrices (det = 0) have no inverse and map the plane to a line or a point.
矩阵表示线性变换，包括旋转、反射、拉伸、剪切和缩放。你应熟练掌握求 2×2 和 3×3 矩阵的行列式和逆矩阵。A 的逆矩阵为 (1/det A) adj A，其中 adj A 是伴随矩阵。奇异矩阵（det = 0）没有逆矩阵，它们会把平面映射为一条直线或一个点。
For 3×3 matrices, solve linear equations using the inverse or by row operations. Eigenvalues and eigenvectors are not in Core Pure but appear in some applied modules; however, understanding invariant lines and planes from the transformation matrix M is essential: solve Mv = λv for invariant lines through the origin.
对于 3×3 矩阵，可用逆矩阵或行变换求解线性方程组。特征值与特征向量虽然不在核心纯数中，但在某些应用模块里出现；但理解变换矩阵 M 下的不变线和不变面非常重要：通过解 Mv = λv 可求得过原点的不变线。
det(A) = ad – bc for A = ⟨a b; c d⟩
5. Proof by Induction | 归纳法证明
Proof by induction involves a base case, an inductive hypothesis, and the inductive step. It is frequently used to prove series summation formulas, divisibility statements, and matrix powers. Always state the proposition P(n) clearly, check n=1 (or the starting integer), assume P(k), and then prove P(k+1).
归纳法证明包括基础情形、归纳假设和归纳步骤。常用于证明级数求和公式、整除性命题和矩阵的幂。要清晰地写出命题 P(n)，验证 n=1（或起始整数），假设 P(k) 成立，然后证明 P(k+1)。
When proving divisibility, express the target expression in terms of the assumed one. For matrices, write Mᵏ⁺¹ = Mᵏ M and substitute the assumed form. Don’t forget to write a concluding sentence that ties the inductive step back to the principle of mathematical induction.
证明整除性时，要将目标表达式用假设的式子表示出来。对于矩阵，写出 Mᵏ⁺¹ = Mᵏ M 并代入所假设的形式。别忘了写出总结句，将归纳步骤与数学归纳法原理联系起来。
6. Hyperbolic Functions | 双曲函数
Hyperbolic functions are defined as: sinh x = (eˣ – e⁻ˣ)/2, cosh x = (eˣ + e⁻ˣ)/2, and tanh x = sinh x / cosh x. They satisfy identities analogous to trigonometric ones, such as cosh² x – sinh² x = 1, and sinh 2x = 2 sinh x cosh x. Their graphs show that sinh is an odd function and cosh is even.
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。它们满足类似于三角函数的恒等式，例如 cosh² x – sinh² x = 1，sinh 2x = 2 sinh x cosh x。从图像上看，sinh 是奇函数，cosh 是偶函数。
Inverse hyperbolic functions can be expressed as logarithms: arsinh x = ln(x + √(x²+1)), arcosh x = ln(x + √(x²–1)) for x ≥ 1, artanh x = ½ ln((1+x)/(1–x)) for |x| < 1. Differentiating these inverse functions yields rational forms, which are essential for integration.
反双曲函数可用对数表示：arsinh x = ln(x + √(x²+1))，arcosh x = ln(x + √(x²–1))（x ≥ 1），artanh x = ½ ln((1+x)/(1–x))（|x| < 1）。对这些反函数求导会得到有理式，这是在积分中必不可少的。
cosh² x – sinh² x = 1
7. Further Calculus & Polar Coordinates | 进阶微积分与极坐标
Key integration techniques include reduction formulae, use of partial fractions, and integration using trigonometric and hyperbolic substitutions. For example, ∫ dx/√(a²+x²) can be solved with the substitution x = a sinh u. Reduction formulas help tackle integrals of the form ∫ sinⁿ x dx by relating Iₙ to Iₙ₋₂.
关键的积分方法包括递推公式、部分分式以及三角和双曲代换。例如，∫ dx/√(a²+x²) 可用代换 x = a sinh u 求解。递推公式通过将 Iₙ 与 Iₙ₋₂ 相联系，来处理形如 ∫ sinⁿ x dx 的积分。
Polar coordinates (r, θ) describe curves where r is a function of θ. The area enclosed by a polar curve is A = ½ ∫ r² dθ. For tangents at the pole, find θ where r=0. Common shapes include cardioids r = a(1+cos θ) and roses r = a cos nθ. Know how to convert between Cartesian and polar forms.
极坐标 (r, θ) 描述的是 r 随 θ 变化的曲线。极坐标曲线围成的面积为 A = ½ ∫ r² dθ。对于极点处的切线，可求出使得 r=0 的 θ 值。常见图形有心形线 r = a(1+cos θ) 和玫瑰线 r = a cos nθ。要学会直角坐标与极坐标的互化。
Area = ½ ∫ r² dθ
8. Differential Equations | 微分方程
First-order differential equations include separable, linear, and those that can be solved using an integrating factor. The integrating factor method for dy/dx + P(x)y = Q(x) uses I.F. = e^(∫ P dx). For second-order homogeneous ODEs with constant coefficients, the auxiliary equation ar² + br + c = 0 determines the form of the general solution.
一阶微分方程包括可分离变量型、线性型和可用积分因子求解的方程。对于 dy/dx + P(x)y = Q(x) 的积分因子法，使用 I.F. = e^(∫ P dx)。对于常系数二阶齐次常微分方程，辅助方程 ar² + br + c = 0 决定了通解的形式。
If the roots are real and distinct, y = Ae^(r₁x) + Be^(r₂x); if repeated, y = (A + Bx)e^(rx); if complex conjugate α ± iβ, y = e^(αx)(A cos βx + B sin βx). Non-homogeneous equations require finding a particular integral by trial functions. Always find the complementary function first.
若为不等实根，y = Ae^(r₁x) + Be^(r₂x)；若为重根，y = (A + Bx)e^(rx)；若为共轭复根 α ± iβ，y = e^(αx)(A cos βx + B sin βx)。非齐次方程需要用试探函数求特解。一定要先求出余函数。
d²y/dx² + a dy/dx + b y = 0 → ar² + br + c = 0
9. Further Vectors | 进阶向量
In Core Pure, vector work extends to lines and planes in 3D. The vector equation of a line is r = a + λb, where a is a point on the line and b is the direction vector. A plane can be expressed as r·n = d, or r = a + λu + μv. Scalar product and cross product are used for finding angles, distances, and intersections.
在核心纯数中，向量内容扩展到三维空间中的直线与平面。直线的向量方程为 r = a + λb，其中 a 是直线上一点，b 是方向向量。平面可表示为 r·n = d，或 r = a + λu + μv。点积和叉积用于求夹角、距离和交点。
Be able to find the shortest distance from a point to a line and from a point to a plane. The distance from point P to plane r·n = d is |(a – p)·n| / |n|, where a is any point on the plane. When two planes intersect, their line of intersection can be found by solving simultaneously.
要会求点到直线和点到平面的最短距离。点 P 到平面 r·n = d 的距离为 |(a – p)·n| / |n|，其中 a 为平面上任一点。当两平面相交时，通过联立可求出交线。
Distance = |(a – p)·n| / |n|
10. Applied Modules – A Quick Guide | 应用模块速览
Edexcel Further Maths allows you to choose two applied modules from Further Mechanics 1 & 2, Decision 1 & 2, Further Statistics 1 & 2, or Further Pure 3. Each module demands specific techniques. Further Mechanics 1 covers momentum, impulse, work-energy principle, and elastic collisions; knowing the restitution law e = (speed of separation)/(speed of approach) is crucial.
Edexcel 进阶数学允许选择两个应用模块，如进阶力学 1 & 2、决策数学 1 & 2、进阶统计 1 & 2 或进阶纯数 3。每个模块都有专门的方法。进阶力学 1 涵盖动量、冲量、功能原理和弹性碰撞；掌握恢复系数 e = (分离速度)/(接近速度) 至关重要。
Decision Mathematics 1 involves algorithms on graphs: Kruskal’s, Prim’s, Dijkstra’s, and linear programming. You must be able to formulate problems, find critical paths, and apply the simplex method. Further Statistics 1 tests geometric and negative binomial distributions, the central limit theorem, and hypothesis testing with Type I/II errors.
决策数学 1 涉及图的算法：Kruskal、Prim、Dijkstra 和线性规划。你必须能构建问题、找到关键路径并应用单纯形法。进阶统计 1 则考查几何分布、负二项分布、中心极限定理以及包含第一类和第二类错误的假设检验。
Whichever combination you have chosen, focus on the standard exam question styles. Practise setting out logical reasoning, interpreting contexts, and linking conclusions back to the problem. Applied modules often carry many marks for method, so show every step clearly.
无论你选择了哪种组合，都要重点练习考试的标准题型。培养逻辑推理、语境解读以及将结论联系回原题的能力。应用模块在方法上通常占分很多，因此务必每步都写清楚。
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