📚 Year 10 Edexcel Further Maths: Core Concepts Review | 10年级爱德思进阶数学：核心知识点梳理
Year 10 Edexcel Further Mathematics builds on the standard GCSE course by introducing more advanced algebra, trigonometry, calculus and matrix work. This revision guide covers the essential topics you need to master to feel confident in problem-solving and to prepare for further study at A Level.
10年级爱德思进阶数学在普通GCSE课程的基础上引入了更深入的代数、三角学、微积分和矩阵等内容。本复习指南涵盖了你需要掌握的核心知识点，助你在解题时充满信心，并为将来A Level的学习做好准备。
1. Surds and Indices | 根式与指数
A surd is an irrational number written in root form, such as √2 or ³√7. Leaving answers as surds keeps them exact and avoids rounding inaccuracies.
根式是以根号形式表示的无理数，例如√2或³√7。将答案保留为根式可以保证精确，避免舍入误差。
To simplify a surd, look for the largest square factor: √72 = √36 × 2 = 6√2. When adding or subtracting surds, first simplify each term so that you can combine like radicals.
化简根式时，要找出最大的平方因子：√72 = √36 × 2 = 6√2。进行根式加减时，先化简每一个项，再合并同类根式。
The key index laws for rational exponents are: am × an = am+n, (am)n = amn and a-n = 1 / an. Fractional powers represent roots: a1/2 = √a, a1/3 = ³√a.
有理指数的主要法则是：am × an = am+n、(am)n = amn 以及 a-n = 1 / an。分数指数表示方根：a1/2 = √a，a1/3 = ³√a。
√a × √b = √ab ; √a / √b = √a/b
√a × √b = √ab ; √a / √b = √a/b
2. Quadratic Functions and Inequalities | 二次函数与不等式
> 0 gives two distinct real roots, Δ = 0 one repeated real root, and Δ < 0 no real roots.
二次函数的形式为 f(x) = ax² + bx + c。判别式 Δ = b² − 4ac 决定了根的性质：Δ > 0 有两个不等实根，Δ = 0 有一个重根，Δ < 0 无实根。
Solving quadratic inequalities such as x² − 5x + 6 > 0 requires finding critical values and sketching a graph or using a sign table. Write the solution in interval form, e.g. x < 2 or x > 3.
解二次不等式（例如 x² − 5x + 6 > 0）需要求出临界值，并通过绘制草图或使用符号表来确定区间。答案用区间表示，如 x < 2 或 x > 3。
x + p)² + q, which immediately reveals the vertex (−p, q) and the line of symmetry x = −p.
配方法可将 ax² + bx + c 写成 a(x + p)² + q 的形式，从而直接得到顶点 (−p, q) 和对称轴 x = −p。
3. Polynomials and Algebraic Fractions | 多项式与代数分式
A polynomial is an expression consisting of variables and coefficients, such as 2x³ − 5x² + x − 7. The degree is the highest exponent. Adding and subtracting polynomials involves combining like terms.
多项式是由变量和系数组成的表达式，例如 2x³ − 5x² + x − 7。次数是最高指数。对多项式进行加减运算只需合并同类项。
lynomial P(x) if and only if P(a) = 0. This is essential for factorising cubics and higher‑order polynomials.
因式定理指出，当且仅当 P(a) = 0 时，(x − a) 是多项式 P(x) 的一个因式。这条定理对分解三次及更高次多项式至关重要。
When simplifying algebraic fractions, factorise numerator and denominator fully and cancel common factors. Remember to state restrictions (values that make the denominator zero).
化简代数分式时，要将分子和分母彻底因式分解，然后约去公因式。务必注明限制条件（使分母为零的值）。
4. Simultaneous Equations | 联立方程
ne equation so that coefficients of one variable match, then add or subtract to eliminate.
线性联立方程可通过消元法或代入法求解。对于 2x + 3y = 8 和 4x − y = 1 这样的方程组，可将其中一个方程乘以某数使得某一变量的系数相等，然后通过加减消元。
When one equation is linear and the other is quadratic, use substitution. Replace one variable from the linear equation into the quadratic, solve the resulting quadratic, and then find the corresponding values of the other variable.
当一个方程为线性、另一个为二次方程时，应使用代入法。从线性方程中解出一个变量，代入二次方程，解出所得的一元二次方程，再代回求出另一个变量的值。
Always check your solutions by substituting both x and y back into the original equations. Be careful with signs during expansion.
始终将求得的 x 和 y 值代回原方程进行验算。展开时注意符号。
5. Coordinate Geometry | 坐标几何
(y₂ − y₁)/(x₂ − x₁). The equation can be written in the form y = mx + c, where c is the y‑intercept, or using y − y₁ = m(x − x₁).
经过点 (x₁, y₁) 和 (x₂, y₂) 的直线斜率 m = (y₂ − y₁)/(x₂ − x₁)。直线方程可写为 y = mx + c（c 为 y 轴截距），或使用点斜式 y − y₁ = m(x − x₁)。
e gradient. The gradient of a perpendicular line is the negative reciprocal: m₁ × m₂ = −1.
平行直线斜率相等。垂直直线的斜率互为负倒数：m₁ × m₂ = −1。
((x₁ + x₂)/2, (y₁ + y₂)/2). The distance between two points is √[(x₂ − x₁)² + (y₂ − y₁)²].
线段的中点坐标为 ((x₁ + x₂)/2, (y₁ + y₂)/2)。两点间距离为 √[(x₂ − x₁)² + (y₂ − y₁)²]。
6. Trigonometry | 三角学
Sine, cosine and tangent are defined for angles in right‑angled triangles. In Further Maths, you extend these definitions to angles of any size using the unit circle and the CAST diagram to determine signs.
正弦、余弦和正切最初在直角三角形中定义。在进阶数学中，你需要利用单位圆和 CAST 图将这些定义推广到任意角，并判断符号。
gonometric values for 30°, 45° and 60° must be memorised, e.g. sin 30° = 1/2, cos 45° = 1/√2. Recognising these helps simplify expressions without a calculator.
必须熟记 30°、45° 和 60° 的精确三角函数值，如 sin 30° = 1/2, cos 45° = 1/√2。熟悉这些值有助于在没有计算器的情况下化简表达式。
sin²θ + cos²θ = 1 ; tan θ = sin θ / cos θ
sin²θ + cos²θ = 1 ； tan θ = sin θ / cos θ
The sine rule and cosine rule are used for non‑right‑angled triangles: a/sin A = b/sin B = c/sin C; a² = b² + c² − 2bc cos A.
对于非直角三角形，使用正弦定理和余弦定理：a/sin A = b/sin B = c/sin C；a² = b² + c² − 2bc cos A。
7. Sequences and Series | 数列与级数
An arithmetic sequence has a constant difference d. The nth term is un = a + (n − 1)d, and the sum of the first n terms is Sn = n/2 [2a + (n − 1)d] = n/2 (a + l).
等差数列具有恒定的公差 d。其第 n 项为 un = a + (n − 1)d，前 n 项和为 Sn = n/2 [2a + (n − 1)d] = n/2 (a + l)。
A geometric sequence has a constant ratio r. The nth term is un = arⁿ⁻¹. The sum of the first n terms (for r ≠ 1) is Sn = a(1 − rⁿ)/(1 − r).
等比数列具有恒定的公比 r。第 n 项为 un = arⁿ⁻¹。当 r ≠ 1 时，前 n 项和为 Sn = a(1 − rⁿ)/(1 − r)。
write series compactly. For example, Σ(2k+1) from k=1 to 5 represents the sum 3+5+7+9+11.
用 Σ 符号可以简洁地表示级数之和。例如 Σ(2k+1) 从 k=1 到 5 表示 3+5+7+9+11。
8. Introduction to Calculus – Differentiation | 微积分入门 – 微分
x = nxⁿ⁻¹. This rule works for any real power n, including negative and fractional powers.
微分用于求曲线的斜率。对 y = xⁿ，导数为 dy/dx = nxⁿ⁻¹。该法则适用于任意实数指数 n，包括负指数和分数指数。
differentiate a sum, differentiate term by term: if y = 3x² + 4x − 1, then dy/dx = 6x + 4.
对多项式逐项求导，例如 y = 3x² + 4x − 1，则 dy/dx = 6x + 4。
The gradient at a specific point is found by substituting the x‑ /dx. The equation of a tangent at that point can then be written using y − y₁ = m(x − x₁).
在某点的梯度可通过将 x 坐标代入 dy/dx 求得。该点切线的方程可用点斜式 y − y₁ = m(x − x₁) 写出。
if d²y/dx² > 0 it is a minimum; if < 0 a maximum; if = 0 further investigation is needed.
驻点出现在 dy/dx = 0 处。用二阶导数 d²y/dx² 进行判别：若 d²y/dx² > 0 为极小值点；若 < 0 为极大值点；若 = 0 则需进一步分析。
9. Introduction to Calculus – Integration | 微积分入门 – 积分
gral of xⁿ (n ≠ −1) is ∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + c, where c is the constant of integration.
积分是微分的逆运算。xⁿ 的不定积分（n ≠ −1）为 ∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + c，其中 c 为积分常数。
A definite integral ∫ab f(x) dx calculates the signed area between the curve y = f(x) and the x‑ rom x = a to x = b. Evaluate by finding the antiderivative F(x) and computing F(b) − F(a).
定积分 ∫ab f(x) dx 用于计算曲线 y = f(x) 与 x 轴之间从 x = a 到 x = b 的带符号面积。先求出原函数 F(x)，再计算 F(b) − F(a)。
Remember to split the area if the curve crosses the x‑axis, otherwise the integral will subtract parts below the axis. Always sketch the curve first.
若曲线穿过 x 轴，必须将积分区间分段，否则积分会把轴下方部分计为负面积。解题时最好先画出曲线草图。
10. Matrix Transformations | 矩阵变换
used to represent transformations such as rotations, reflections, enlargements and shears.
矩阵是一个矩形的数字阵列。在进阶数学中，主要使用 2 × 2 矩阵来表示平面上的变换，如旋转、反射、放大和剪切。
To apply a transformation to a point, write the coordinates as a column vector and multiply on the left by the transformation matrix. For a reflection in the y‑axis, the matrix is [[−1,0],[0,1]].
对点施加变换时，将坐标写成列向量，左乘变换矩阵。关于 y 轴的反射矩阵为 [[−1,0],[0,1]]。
],[c,d]] is ad − bc. If the determinant is zero, the transformation is singular (collapses the plane); otherwise it is non‑singular and has an inverse.
2 × 2 矩阵 [[a,b],[c,d]] 的行列式为 ad − bc。若行列式为零，变换是奇异的（会压扁平面）；否则为非奇异，并存在逆矩阵。
Combining transformations is achieved by multiplying the corresponding matrices. Note that the order matters: AB means applying B first, then A.
复合变换通过将相应矩阵相乘来实现。注意顺序至关重要：AB 表示先实施 B 变换，再实施 A 变换。
11. Functions and Graphs | 函数与图像
u must be able to handle composite functions fg(x) = f(g(x)) and inverse functions f⁻¹(x).
函数记号 f(x) 取代了 y。定义域是输入值的集合，值域是输出值的集合。进阶数学要求能够处理复合函数 fg(x) = f(g(x)) 和反函数 f⁻¹(x)。
swap x and y, then solve for y. The graph of f⁻¹ is the reflection of f in the line y = x.
求反函数时，令 y = f(x)，交换 x 和 y，然后解出 y。反函数 y = f⁻¹(x) 的图像是原函数关于直线 y = x 的反射。
Transformations of graphs include translations f(x − a), f(x) + b, stretches af(x), f(kx), and reflections −f(x), f(−x). Be precise with the order when combining transformations.
函数图像的变换包括平移 f(x − a) 和 f(x) + b、伸缩 af(x) 和 f(kx)，以及反射 −f(x) 和 f(−x)。复合变换时应注意先后顺序。
12. Vectors | 向量
A vector has magnitude and direction. It can be represented as a column vector [[x],[y]] or using i, j notation. The zero vector is [[0],[0]].
向量具有大小和方向。可用列向量 [[x],[y]] 或 i、j 记号表示。零向量为 [[0],[0]]。
Vectors can be added and subtracted component‑wise. Scalar multiplication changes the magnitude but not the direction (unless the scalar is negative).
向量可通过分量对应相加减。标量乘法会改变向量的大小，但不改变方向（标量为负时反向）。
The magnitude of a vector v = [[x],[y]] is |v| = √x² + y². A unit vector has magnitude 1 and is found by dividing each component by the magnitude.
向量 v = [[x],[y]] 的模为 |v| = √x² + y²。单位向量的模为 1，可将各分量除以模来求得。
Vector geometry problems often involve finding position vectors, translating shapes, or proving collinearity (points lie on a straight line) using scalar multiples.
向量几何题常涉及求位置向量、平移图形，或利用标量倍数证明共线性（点共线）。
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