📚 Year 11 Edexcel Further Maths: Core Concepts Summary | Year 11 Edexcel 进阶数学：核心知识点梳理
This article summarises the essential topics in the Year 11 Edexcel Further Maths syllabus (IGCSE Further Pure Mathematics 9-1). It aims to help you consolidate your revision and tackle exam-style questions with confidence.
本文梳理了 Year 11 Edexcel 进阶数学（IGCSE Further Pure Mathematics 9-1）的核心知识点，帮助同学们巩固复习，从容应对各类考试题型。
1. Number and Surds | 数与根式
Simplify surds by extracting square factors: for example, √48 = √16×3 = 4√3. Always express surds in their simplest form.
化简根式时要提取平方因子，例如 √48 = √16×3 = 4√3。务必把根式化成最简形式。
Rationalising a denominator such as 1/(√a + b) requires multiplying numerator and denominator by the conjugate (√a - b). This eliminates the surd from the denominator.
有理化分母如 1/(√a + b)，需要分子分母同乘共轭式 (√a - b)，从而消去分母中的根号。
Laws of indices must be applied fluently: aᵐ × aⁿ = aᵐ⁺ⁿ, (aᵐ)ⁿ = aᵐⁿ, aᵐ/ⁿ = ⁿ√aᵐ, and a⁻ⁿ = 1/aⁿ.
指数定律需熟练运用：aᵐ × aⁿ = aᵐ⁺ⁿ，(aᵐ)ⁿ = aᵐⁿ，aᵐ/ⁿ = ⁿ√aᵐ，以及 a⁻ⁿ = 1/aⁿ。
2. Algebra: Polynomials, Equations and Inequalities | 代数：多项式、方程与不等式
is a factor. The Remainder Theorem gives the remainder when f(x) is divided by (x - a), which is simply f(a).
因子定理指出，若多项式 f(x) 满足 f(a) = 0，则 (x - a) 是它的因式。余数定理给出 f(x) 除以 (x - a) 的余数为 f(a)。
To solve a cubic equation, first use the factor theorem to find one linear factor, then perform polynomial division or equate coefficients to reduce the cubic to a quadratic factor.
解三次方程时，先用因子定理找出一个一次因式，再通过多项式除法或比较系数法将三次方程降为二次方程求解。
When simplifying algebraic fractions, always factorise numerators and denominators first and cancel common factors. Be careful with addition and subtraction – find a common denominator.
化简代数分式时，务必先对分子分母因式分解，然后约去公因式。加减运算时要注意先通分，找公分母。
For simultaneous equations where one is linear and the other quadratic, substitute the linear expression into the quadratic to obtain a single equation in one variable.
对于联立方程组，若一个是一次、另一个是二次，通常将一次表达式代入二次方程，得到关于单一变量的方程。
(x - 2)(x - 3) = 0, then testing intervals or sketching the parabola. Solution: 2 < x < 3.
解二次不等式如 x² - 5x + 6 < 0，可先找出临界值 (x - 2)(x - 3) = 0，然后通过区间测试或抛物线草图得到解集：2 < x < 3。
ves two distinct real roots, Δ = 0 gives one repeated root, and Δ < 0 gives no real roots.
判别式 Δ = b² - 4ac 决定方程根的性质：Δ > 0 有两个不等实根，Δ = 0 有一个重根，Δ < 0 无实数根。
he square for x² + bx is done by adding (b/2)². The vertex of y = a(x - h)² + k is (h, k).
配方法中，x² + bx 需加上 (b/2)² 构成完全平方。二次函数 y = a(x - h)² + k 的顶点坐标为 (h, k)。
3. Coordinate Geometry | 坐标几何
x₁). Parallel lines have equal gradients, while perpendicular lines satisfy m₁ × m₂ = -1.
经过两点 (x₁, y₁) 和 (x₂, y₂) 的直线斜率 m = (y₂ - y₁)/(x₂ - x₁)。平行线斜率相等，垂直线斜率满足 m₁ × m₂ = -1。
he equation of a line can be written as y - y₁ = m(x - x₁) or in the form ax + by + c = 0.
直线方程可写作点斜式 y - y₁ = m(x - x₁)，也可化为一般式 ax + by + c = 0。
b)² = r². To find where a line intersects a circle, solve their equations simultaneously.
以 (a, b) 为圆心、r 为半径的圆方程为 (x - a)² + (y - b)² = r²。求直线与圆的交点可联立方程求解。
A tangent to a circle at a given point is perpendicular to the radius at that point. The normal to a curve at a point is perpendicular to the tangent.
圆上某点的切线与过该点的半径互相垂直。曲线上某点的法线与该点切线垂直。
4. Calculus | 微积分
x = nxⁿ⁻¹. The derivative can be used to find equations of tangents and normals to curves.
微分可求出梯度函数：对 y = xⁿ，有 dy/dx = nxⁿ⁻¹。利用导数可求曲线的切线与法线方程。
0. Use the second derivative d²y/dx² to classify them: positive for minimum, negative for maximum, zero suggests point of inflection (check sign change).
驻点满足 dy/dx = 0。利用二阶导数 d²y/dx² 判别：d²y/dx² > 0 为极小点，< 0 为极大点；若为零，需检查一阶导数符号变化判断拐点。
+1) + C (n ≠ -1). The constant of integration C must be included for indefinite integrals.
积分是微分的逆运算：∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ -1)。不定积分必须加积分常数 C。
en by the definite integral ∫ₐᵇ f(x) dx. Evaluate the integral at the limits and subtract.
曲线 y = f(x) 与 x = a、x = b 及 x 轴围成的面积可用定积分 ∫ₐᵇ f(x) dx 计算，代入上下限相减。
velocity v = ds/dt and acceleration a = dv/dt = d²s/dt². Integration can recover s from v.
运动学中，若位移 s = f(t)，则速度 v = ds/dt，加速度 a = dv/dt = d²s/dt²。通过积分可从速度得到位移。
5. Trigonometry | 三角学
measure: π radians = 180°. Arc length s = rθ and sector area A = ½ r²θ, with θ in radians.
弧度制：π 弧度 = 180°。弧长公式 s = rθ，扇形面积 A = ½ r²θ，其中 θ 必须以弧度为单位。
/2 and their multiples. The graphs of y = sin x, y = cos x and y = tan x show periodicity.
任意角的三角函数：需熟记 0, π/6, π/4, π/3, π/2 等特殊角的精确值，并理解正弦、余弦、正切曲线的周期性。
for 0 ≤ x ≤ 2π. Use the CAST diagram or graph to find all solutions in the given interval.
解简单三角方程，如 sin x = 0.5 在 0 ≤ x ≤ 2π 内求解，借助 CAST 图或图像找出指定区间内的所有根。
, and sin² θ + cos² θ = 1. These can be used to prove other identities or solve equations.
核心恒等式：tan θ = sin θ / cos θ 以及 sin² θ + cos² θ = 1。这些恒等式常用于证明三角恒等式或解方程。
6. Logarithms and Exponentials | 对数与指数函数
The logarithm logₐ x is the exponent to which base a must be raised to give x. The natural logarithm ln x uses base e.
对数 logₐ x 表示以 a 为底得到 x 所需的指数。自然对数 ln x 的底是 e。
gₐ x - logₐ y; logₐ (xⁿ) = n logₐ x. These are essential for condensing or expanding logs.
对数运算律：logₐ (xy) = logₐ x + logₐ y；logₐ (x/y) = logₐ x - logₐ y；logₐ (xⁿ) = n logₐ x。这是合并或展开对数的基础。
To solve an exponential equation like 32x = 5x-1, take logs on both sides: 2x ln 3 = (x-1) ln 5, then solve for x.
解指数方程如 32x = 5x-1，可两边取对数得 2x ln 3 = (x-1) ln 5，然后解出 x。
7. Sequences and Series | 数列与级数
s the common difference. Sum of first n terms: Sₙ = n/2 (2a + (n-1)d) or Sₙ = n/2 (a + L).
等差数列：第 n 项 Uₙ = a + (n-1)d，a 为首项，d 为公差。前 n 项和 Sₙ = n/2 (2a + (n-1)d) 或 Sₙ = n/2 (a + L)。
arⁿ⁻¹, where r is the common ratio. Sum of first n terms: Sₙ = a(1 - rⁿ)/(1 - r) if r ≠ 1.
等比数列：第 n 项 Uₙ = arⁿ⁻¹，r 为公比。前 n 项和 Sₙ = a(1 - rⁿ)/(1 - r)，r ≠ 1。
Sum to infinity of a geometric series exists when |r| d is given by S∞ = a/(1 - r). This is useful for recurring decimals and convergent series.
当 |r| < 1 时，等比级数的无穷和 S∞ = a/(1 - r)。该公式在循环小数和收敛级数问题中很有用。
8. Binomial Expansion | 二项式展开
For negative or fractional powers n, (1 + ax)ⁿ can be expanded as 1 + n(ax) + [n(n-1)/2!](ax)² + [n(n-1)(n-2)/3!](ax)³ + ... provided |ax| < 1.
当 n 为负数或分数时，(1 + ax)ⁿ 可展开为 1 + n(ax) + [n(n-1)/2!](ax)² + [n(n-1)(n-2)/3!](ax)³ + ...，但必须满足 |ax| < 1。
For example, (1 - 3x)⁻¹ expands to 1 + 3x + 9x² + 27x³ + ... ; the expansion is valid for |3x| < 1, i.e. |x| < 1/3.
例如 (1 - 3x)⁻¹ 展开为 1 + 3x + 9x² + 27x³ + ...，展开的有效范围是 |3x| < 1，即 |x| < 1/3。
Always simplify the expression into the form (1 + u)ⁿ before expanding, and state the validity condition clearly.
展开前务必将表达式化为 (1 + u)ⁿ 的标准形式，并明确写出有效性的范围。
9. Vectors | 向量
can be expressed in component form a = xi + yj or as column vectors. The magnitude of a = √x² + y².
向量可用分量形式 a = xi + yj 或列向量表示。向量 a 的模为 |a| = √x² + y²。
ed component-wise; vectors are parallel if one is a scalar multiple of the other: a = k b.
向量的加减与数乘均按分量运算；若一个向量是另一个的标量倍数，即 a = k b，则两向量平行。
rigin. The vector from point A to B is AB = OB - OA, where OA and OB are position vectors.
位置向量以原点为起点。从点 A 到点 B 的向量 AB = OB - OA，其中 OA、OB 为位置向量。
Collinear points lie on the same straight line, meaning the vectors between them are parallel and share a common point.
共线点位于同一直线上，即点间的向量互相平行且具有公共点。
10. Matrices | 矩阵
A 2x2 matrix transforms points in the plane. Matrix multiplication is associative but not commutative: AB is generally not equal to BA.
2×2 矩阵可实现平面上的点变换。矩阵乘法满足结合律但不满足交换律，即 AB 通常不等于 BA。
tity matrix I = [[1,0],[0,1]] leaves points unchanged under multiplication: A I = I A = A.
单位矩阵 I = [[1,0],[0,1]] 在乘法中不改变矩阵：A I = I A = A。
The determinant of
| a | b |
| c | d |
矩阵
| a | b |
| c | d |
The inverse of a 2x2 matrix is A⁻¹ = 1/(ad - bc)
| d | -b |
| -c | a |
当行列式不为零时，2×2 矩阵的逆矩阵为 A⁻¹ = 1/(ad - bc)
| d | -b |
| -c | a |
Common transformation matrices include reflections (
| 1 | 0 |
| 0 | -1 |
| 0 | -1 |
| 1 | 0 |
| k | 0 |
| 0 | 1 |
常见变换矩阵包括反射（如 x 轴反射
| 1 | 0 |
| 0 | -1 |
| 0 | -1 |
| 1 | 0 |
| k | 0 |
