📚 FP1 Module Revision Guide | 进阶数学 FP1 模块备考资料
FP1 is the first pure mathematics module in the A-Level Further Mathematics course. It introduces a wide range of new concepts and extends some familiar ideas from Core Mathematics, forming the foundation for later modules such as FP2 and FP3. This guide brings together the key topics, essential formulas, and proven exam strategies to help you revise effectively and perform with confidence.
FP1 是 A-Level 进阶数学课程的第一个纯数学模块。它引入了大量全新概念，并扩展了核心数学中的一些熟悉内容，为后续模块如 FP2 和 FP3 打下坚实的基础。本指南汇集了核心主题、必备公式以及经过验证的应试策略，帮助你高效复习，自信迎考。
1. Overview of FP1 Topics | FP1 主题概览
FP1 typically covers pure mathematics topics that build on your knowledge from AS Mathematics. While the exact content may vary slightly by exam board, the core themes remain consistent. The main areas include complex numbers, the roots of polynomial equations, series and sigma notation, proof by mathematical induction, matrices and linear transformations, numerical methods for solving equations, and parametric coordinate systems.
FP1 通常涵盖以 AS 数学知识为基础的纯数学主题。虽然不同考试局的具体内容可能稍有不同，但核心主题是一致的。主要领域包括复数、多项式方程的根、级数与求和符号、数学归纳法证明、矩阵与线性变换、求解方程的数值方法以及参数坐标系。
In the following sections, you will find a concise summary of each topic together with critical formulas and common pitfalls. The revision guide is designed to be used alongside past papers and your own notes, reinforcing the key ideas you need to demonstrate in the exam.
在以下各节中，你将看到每个主题的简明总结，以及关键公式和常见错误。本复习指南旨在与历年真题和你的个人笔记配合使用，巩固你在考试中需要展示的关键思路。
2. Complex Numbers & Argand Diagrams | 复数与阿尔冈图
A complex number is written as z = a + bi, where a and b are real numbers and i2 = −1. The real part is Re(z) = a and the imaginary part is Im(z) = b. The complex conjugate of z is z* = a − bi, which is the reflection of z in the real axis on an Argand diagram.
复数写作 z = a + bi，其中 a 和 b 是实数且 i2 = −1。实部为 Re(z) = a，虚部为 Im(z) = b。z 的共轭复数为 z* = a − bi，它在阿尔冈图中是 z 关于实轴的反射。
The modulus of z is |z| = √(a2 + b2), representing the distance from the origin to the point representing z. The argument arg(z) = θ is the angle measured from the positive real axis, usually taken in the interval (−π, π]. Pay close attention to the quadrant when calculating the argument.
z 的模长为 |z| = √(a2 + b2)，表示原点到代表 z 的点的距离。辐角 arg(z) = θ 是从正实轴开始测量的角度，通常取区间 (−π, π]。计算辐角时要特别注意象限。
Using Euler’s formula eiθ = cos θ + i sin θ, any complex number can be expressed in exponential form z = reiθ. De Moivre’s theorem states that (cos θ + i sin θ)n = cos nθ + i sin nθ, which is especially useful for finding powers and roots of complex numbers.
利用欧拉公式 eiθ = cos θ + i sin θ，任何复数都可以表示为指数形式 z = reiθ。棣莫弗定理给出 (cos θ + i sin θ)n = cos nθ + i sin nθ，这在求复数的乘方和方根时特别有用。
eiθ = cos θ + i sin θ | (cos θ + i sin θ)n = cos nθ + i sin nθ
When solving quadratic equations with real coefficients, a negative discriminant indicates a pair of complex conjugate roots. You must be able to express the roots in the form a ± bi and interpret them on the Argand diagram.
在求解实系数二次方程时，负判别式意味着有一对共轭复根。你必须能将根表示为 a ± bi 的形式，并在阿尔冈图上解释它们。
3. Roots of Polynomial Equations | 多项式方程的根
For a quadratic equation ax2 + bx + c = 0 with roots α and β, the relationships are α + β = −b/a and αβ = c/a. These are used to evaluate symmetric expressions such as α2 + β2 = (α + β)2 − 2αβ.
对于二次方程 ax2 + bx + c = 0，根为 α 和 β，有关系 α + β = −b/a 和 αβ = c/a。这些用于计算对称表达式，例如 α2 + β2 = (α + β)2 − 2αβ。
For a cubic ax3 + bx2 + cx + d = 0 with roots α, β and γ, the three key identities are ∑α = −b/a, ∑αβ = c/a and αβγ = −d/a. For a quartic ax4 + bx3 + cx2 + dx + e = 0, the patterns extend: ∑α = −b/a, ∑αβ = c/a, ∑αβγ = −d/a and αβγδ = e/a.
对于三次方程 ax3 + bx2 + cx + d = 0，根为 α, β 和 γ，三个关键恒等式为 ∑α = −b/a，∑αβ = c/a 以及 αβγ = −d/a。对于四次方程 ax4 + bx3 + cx2 + dx + e = 0，规律扩展为：∑α = −b/a，∑αβ = c/a，∑αβγ = −d/a 和 αβγδ = e/a。
Quadratic: α + β = −b/a, αβ = c/a
Cubic: ∑α = −b/a, ∑αβ = c/a, αβγ = −d/a
Exam questions often ask you to find a new polynomial whose roots are related to those of a given polynomial, for example roots being α + k, kα, or α2. The strategy is to express the sum and product of the new roots in terms of the original symmetric sums, then construct the new equation.
考试中常见的问题是，给定一个多项式，求一个新多项式，其根与原多项式的根之间存在关系，例如根为 α + k, kα 或 α2。解题策略是用原来的对称和来表示新根的和与积，然后构建新的方程。
4. Series & Sigma Notation | 级数与求和符号
You must know three standard summation results for integers from 1 to n. They are often given in formula booklets, but memorising them saves time and reduces errors.
你必须牢记从 1 到 n 的整数的三个标准求和结果。它们通常在公式手册中有提供，但记住它们可以节省时间并减少错误。
∑r=1n r = n(n+1)/2
∑r=1n r2 = n(n+1)(2n+1)/6
∑r=1n r3 = n2(n+1)2/4
More complicated series, such as ∑ (2r+1)2 or ∑ (r+1)(r+3), can be expanded and broken into combinations of these standard forms. Always write out the first few terms to verify the pattern before applying the formulas.
更复杂的级数，例如 ∑ (2r+1)2 或 ∑ (r+1)(r+3)，可以展开并拆分为这些标准形式的组合。在应用公式之前，务必写出前几项来验证规律。
You may also be asked to use the method of differences to sum a series. This involves writing the general term as a difference, so that when terms are added, most cancel, leaving a simple expression.
你还可能被要求使用差分法对级数求和。这需要将一般项写成一个差分的形式，这样在各项相加时，大部分项会相消，只留下一个简单的表达式。
5. Proof by Induction | 归纳法证明
Mathematical induction is a rigorous method for proving that a statement P(n) is true for all positive integers n. The proof always follows four clear steps: base case, induction hypothesis, induction step, and conclusion.
数学归纳法是一种严格的方法，用于证明对于所有正整数 n，命题 P(n) 成立。证明总是遵循四个清晰的步骤：基础情形、归纳假设、归纳步骤和结论。
Step 1: Verify that the statement holds for the initial value, usually n = 1. Step 2: Assume P(k) is true for some integer k ≥ 1. Step 3: Show that P(k) leads to P(k+1) using algebraic manipulation. Step 4: State that by mathematical induction, P(n) is true for all positive integers n.
第一步：验证初始值（通常是 n = 1）时命题成立。第二步：假设对于某个整数 k ≥ 1，P(k) 为真。第三步：通过代数推导证明由 P(k) 可推出 P(k+1)。第四步：陈述由数学归纳法可知，P(n) 对所有正整数 n 成立。
Common induction problems in FP1 include summation identities, divisibility proofs (e.g., showing an expression is divisible by a number), matrix results (e.g., An), and recurrence relations. In divisibility proofs, write P(k+1) as a multiple of the divisor plus a term that uses the induction hypothesis.
FP1 中常见的归纳问题包括求和恒等式、整除性证明（例如证明一个表达式能被某数整除）、矩阵结果（例如 An）以及递推关系。在整除性证明中，将 P(k+1) 写成除数的倍数加上利用归纳假设的一项。
6. Matrices & Linear Transformations | 矩阵与线性变换
A matrix is a rectangular array of numbers; in FP1 you mostly work with 2 × 2 matrices. You need to be comfortable with addition, subtraction, multiplication by a scalar, and matrix multiplication. Remember that matrix multiplication is not commutative: AB ≠ BA in general.
矩阵是一个矩形数字阵列；在 FP1 中你主要处理 2 × 2 矩阵。你需要熟练掌握加法、减法、与标量的乘法以及矩阵乘法。要记住矩阵乘法不满足交换律：一般情况 AB ≠ BA。
The determinant of a 2 × 2 matrix M = [a b; c d] is det(M) = ad − bc. A matrix is singular if its determinant is zero and non-singular otherwise. The inverse of a non-singular matrix is M−1 = (1/det M) [d −b; −c a].
2 × 2 矩阵 M = [a b; c d] 的行列式为 det(M) = ad − bc。行列式为零的矩阵是奇异的，否则是非奇异的。非奇异矩阵的逆矩阵为 M−1 = (1/det M) [d −b; −c a]。
det(M) = ad − bc | M−1 = 1/(ad−bc) [d, −b; −c, a]
Matrices are used to represent linear transformations such as rotations, reflections, stretches and shears. The image of a point (x, y) is found by multiplying the transformation matrix by the column vector [x; y]. You must know the standard matrices for reflection in the coordinate axes and lines y = ±x, rotation through angle θ, and stretches parallel to the axes.
矩阵用于表示线性变换，如旋转、反射、拉伸和剪切。将变换矩阵乘以列向量 [x; y] 即可得到点 (x, y) 的像。你必须知道关于坐标轴和直线 y = ±x 的反射、旋转 θ 角以及平行于坐标轴的拉伸的标准矩阵。
Combined transformations correspond to multiplying matrices in reverse order. If transformation A is followed by transformation B, the overall matrix is BA. Invariant points and invariant lines can be found by solving Mx = x or Mx = λx.
复合变换对应于按逆序相乘矩阵。如果先施行变换 A 再施行变换 B，总体矩阵为 BA。通过求解 Mx = x 或 Mx = λx 可以找出不变点和不变直线。
7. Numerical Methods | 数值方法
Numerical methods are used to find approximate roots of equations f(x) = 0 when algebraic solutions are impossible or difficult. You must be able to apply interval bisection, linear interpolation, and the Newton-Raphson method.
当代数解法不可能或很困难时，数值方法被用来寻找方程 f(x) = 0 的近似根。你必须能够应用区间二分法、线性插值法和牛顿-拉弗森方法。
Interval bisection repeatedly halves an interval [a, b] where f(a) and f(b) have opposite signs. The midpoint c = (a+b)/2 is calculated, and the subinterval containing a sign change is retained. The process continues until the desired accuracy is reached.
区间二分法反复将区间 [a, b] 减半，其中 f(a) 和 f(b) 异号。计算中点 c = (a+b)/2，保留含有符号变化的子区间。继续这一过程直到达到所需的精度。
Linear interpolation uses similar triangles to estimate where the chord between (a, f(a)) and (b, f(b)) crosses the x-axis. The formula x ≈ a − (b−a)f(a) / (f(b)−f(a)) gives an improved approximation. This is also a required skill for the exam.
线性插值法利用相似三角形来估算过 (a, f(a)) 和 (b, f(b)) 的弦与 x 轴的交点。公式 x ≈ a − (b−a)f
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