📚 Edexcel A-Level Mathematics: Core Combined Practice 085 | Edexcel A-Level 数学：核心综合训练 085
This set brings together the most frequently examined core skills in the Edexcel A-Level Mathematics specification. It is designed as a quick but rigorous checkpoint for students who have completed the pure content and want to consolidate algebraic, trigonometric, logarithmic, calculus and vector techniques before moving on to timed past-paper practice. Each section below mixes concise revision notes with worked results, so you can revise actively rather than just rereading notes.
本套资料汇总了 Edexcel A-Level 数学大纲中最常考的核心技能。它适合已经完成纯数内容、希望在进入限时真题训练前巩固代数、三角、对数、微积分和向量技巧的学生。下面每一节都将简明扼要的复习要点与计算结果相结合，帮助你主动复习，而不只是重读笔记。
1. Laws of Indices and Surds | 指数律与根式
The laws of indices allow expressions such as am × an to be simplified. They underpin later work on logarithms, differentiation and integration. The key rules are used in almost every Edexcel pure mathematics paper, especially when rewriting algebraic fractions or solving equations.
指数律能够化简形如 am × an 的表达式。它们为后面的对数、微分和积分内容打下基础。这些关键法则几乎出现在 Edexcel 纯数学试卷的每一份中，尤其是在化简代数分式或解方程时。
am × an = am+n, am ÷ an = am−n, (am)n = amn, a−n = 1/an
When dealing with surds, write a number as the product of a square number and an integer: √72 = √(36 × 2) = 6√2. Rationalising the denominator often uses the difference of two squares: (a + b√c)(a − b√c) = a² − b²c.
处理根式时，可把一个数写成平方数与整数的乘积：√72 = √(36 × 2) = 6√2。分母有理化通常利用平方差公式：(a + b√c)(a − b√c) = a² − b²c。
- Simplify fully: (x3y−2)2 ÷ x4y3 = x6y−4 ÷ x4y3 = x2y−7 = x² / y⁷
- 完整化简：(x3y−2)2 ÷ x4y3 = x6y−4 ÷ x4y3 = x2y−7 = x² / y⁷
2. Quadratic Functions and the Discriminant | 二次函数与判别式
For a quadratic ax² + bx + c = 0, the discriminant Δ = b² − 4ac determines the nature of the roots. If Δ > 0 there are two distinct real roots; if Δ = 0 there is one repeated real root; if Δ < 0 there are no real roots but two complex roots. This is vital for questions on intersections of curves and lines.
对于二次方程 ax² + bx + c = 0，判别式 Δ = b² − 4ac 决定了根的性质。若 Δ > 0，有两个相异实根；若 Δ = 0，有一个重根；若 Δ < 0，没有实根，只有两个复数根。这在判断曲线与直线交点的问题中十分关键。
Δ = b² − 4ac
Completing the square transforms ax² + bx + c into a(x + p)² + q form, which gives the vertex coordinates (−p, q). This method is also used to solve quadratics and to prove inequalities.
配方法可将 ax² + bx + c 转化为 a(x + p)² + q 的形式，从而给出顶点坐标 (−p, q)。该方法也用于解二次方程和证明不等式。
3. Functions and Inverse Functions | 函数与反函数
A function f maps an input x to an output f(x). The inverse function f⁻¹ reverses this mapping, so that f⁻¹(f(x)) = x for all x in the domain of f. To find f⁻¹, write y = f(x), swap x and y, and solve for y; then replace y with f⁻¹(x).
函数 f 将输入 x 映射到输出 f(x)。反函数 f⁻¹ 则逆转这一映射，因此对于 f 定义域中的所有 x，都有 f⁻¹(f(x)) = x。求 f⁻¹ 的步骤为：写出 y = f(x)，交换 x 和 y，解出 y，再将 y 写成 f⁻¹(x)。
f(x) = 2x + 3, f⁻¹(x) = (x − 3)/2
The domain and range of a function are linked: the range of f becomes the domain of f⁻¹. Function composition (g ∘ f)(x) = g(f(x)) means apply f first, then g.
函数的定义域和值域相互关联：f 的值域成为 f⁻¹ 的定义域。复合函数 (g ∘ f)(x) = g(f(x)) 表示先作用 f，再作用 g。
4. Coordinate Geometry: Lines and Circles | 坐标几何：直线与圆
The gradient of a line through two points (x₁, y₁) and (x₂, y₂) is m = (y₂ − y₁)/(x₂ − x₁). The equation of a line with gradient m passing through (x₁, y₁) is y − y₁ = m(x − x₁). Parallel lines have equal gradients, while perpendicular lines satisfy m₁m₂ = −1.
经过两点 (x₁, y₁) 和 (x₂, y₂) 的直线斜率为 m = (y₂ − y₁)/(x₂ − x₁)。过点 (x₁, y₁) 且斜率为 m 的直线方程为 y − y₁ = m(x − x₁)。平行直线斜率相等，垂直直线的斜率满足 m₁m₂ = −1。
The equation of a circle with centre (a, b) and radius r is (x − a)² + (y − b)² = r². To find intersections with a line, substitute the linear equation into the circle equation and solve the resulting quadratic.
以 (a, b) 为圆心、r 为半径的圆的方程为 (x − a)² + (y − b)² = r²。求圆与直线的交点时，可将直线方程代入圆的方程，解所得的二次方程。
d = √((x₂ − x₁)² + (y₂ − y₁)²)
5. Trigonometric Identities and Equations | 三角恒等式与方程
The two most useful identities in A-Level Mathematics are the Pythagorean identity and the tangent identity: sin²θ + cos²θ = 1 and tanθ = sinθ / cosθ. These allow expressions to be rearranged into solvable forms. The exact values for 0°, 30°, 45°, 60° and 90° must be memorised or quickly derived from triangles.
A-Level 数学中最常用的两个恒等式是毕达哥拉斯恒等式和正切恒等式：sin²θ + cos²θ = 1 以及 tanθ = sinθ / cosθ。它们可将表达式变形为可解形式。0°、30°、45°、60° 和 90° 的精确值必须熟记，或能由特殊三角形快速推出。
sin²θ + cos²θ = 1, sin 2θ = 2 sinθ cosθ
When solving equations such as 2 sin²θ + 3 sinθ − 2 = 0, substitute u = sinθ to obtain a quadratic in u, solve for u, and then find θ in the required interval. Always check whether solutions are in degrees or radians.
解形如 2 sin²θ + 3 sinθ − 2 = 0 的方程时，可令 u = sinθ，得到关于 u 的二次方程，解出 u 后再在指定区间内求 θ。务必注意最终答案使用角度制还是弧度制。
6. Exponentials and Logarithms | 指数函数与对数
The natural exponential function y = eˣ has derivative and integral both equal to eˣ. Its inverse is the natural logarithm y = ln x. The laws of logarithms convert multiplication into addition, division into subtraction, and powers into multiplication.
自然指数函数 y = eˣ 的导数和积分都等于 eˣ。它的反函数是自然对数 y = ln x。对数法则可将乘法转化为加法，除法转化为减法，幂转化为乘法。
logₐ(xy) = logₐx + logₐy, logₐ(x/y) = logₐx − logₐy, logₐ(xⁿ) = n logₐx
To solve an exponential equation such as 3²ˣ⁺¹ = 5, take logs of both sides: (2x + 1)ln 3 = ln 5, then solve for x. Exponential growth and decay models often appear as P = Aekt or P = Ae−kt.
解指数方程 3²ˣ⁺¹ = 5 时，可对方程两边取对数：(2x + 1)ln 3 = ln 5，然后解出 x。指数增长与衰减模型常以 P = Aekt 或 P = Ae−kt 的形式出现。
7. Differentiation Rules | 微分法则
The derivative of xⁿ is n xⁿ⁻¹. This power rule combines with the sum rule to differentiate polynomials. The chain rule, product rule and quotient rule handle composite, multiplied, and divided functions respectively.
xⁿ 的导数是 n xⁿ⁻¹。这一幂法则与加法法则结合，可对多项式求导。链式法则、乘积法则和商法则分别处理复合函数、相乘函数和相除函数。
dy/dx = dy/du × du/dx, d(uv)/dx = u’v + uv’, d(u/v)/dx = (u’v − uv’) / v²
For example, if y = (3x² + 1)⁵, let u = 3x² + 1, then dy/dx = 5u⁴ × 6x = 30x(3x² + 1)⁴. Differentiation is also used to find gradients of tangents, normal lines, and rates of change in applied contexts.
例如，若 y = (3x² + 1)⁵，令 u = 3x² + 1，则 dy/dx = 5u⁴ × 6x = 30x(3x² + 1)⁴。微分还可用于求切线和法线的斜率，以及应用情境中的变化率。
8. Integration Techniques | 积分技巧
Integration reverses differentiation. The basic power rule for integration is ∫ xⁿ dx = xⁿ⁺¹/(n + 1) + C for n ≠ −1. The special case ∫ x⁻¹ dx = ln|x| + C must be remembered separately.
积分是微分的逆运算。幂函数的基本积分法则为 ∫ xⁿ dx = xⁿ⁺¹/(n + 1) + C（n ≠ −1）。特殊情况 ∫ x⁻¹ dx = ln|x| + C 需要单独记忆。
∫ xⁿ dx = xⁿ⁺¹/(n + 1) + C, ∫ eˣ dx = eˣ + C, ∫ 1/x dx = ln|x| + C
Definite integrals evaluate the signed area under a curve between two limits. To calculate area between a curve and a line, integrate the difference of the two functions and use the limits of the intersection points.
定积分计算曲线与 x 轴之间在两个界限内的带符号面积。计算曲线与直线所夹面积时，应积分两个函数之差，并以交点作为积分上下限。
9. Sequences and Binomial Expansion | 数列与二项展开
Arithmetic sequences have a common difference d, with nth term uₙ = a + (n − 1)d and sum Sₙ = n/2 [2a + (n − 1)d]. Geometric sequences have a common ratio r, with nth term uₙ = arⁿ⁻¹ and sum to infinity S∞ = a / (1 − r) when |r| < 1.
等差数列有公差 d，第 n 项为 uₙ = a + (n − 1)d，前 n 项和为 Sₙ = n/2 [2a + (n − 1)d]。等比数列有公比 r，第 n 项为 uₙ = arⁿ⁻¹，当 |r| < 1 时无穷项和为 S∞ = a / (1 − r)。
For a positive integer n, the binomial expansion is (a + b)ⁿ = Σ from k=0 to n of C(n,k) aⁿ⁻ᵏ bᵏ. The term independent of x is often found by setting the power of x in a general term equal to zero.
对于正整数 n，二项展开式为 (a + b)ⁿ = Σ（k 从 0 到 n）C(n,k) aⁿ⁻ᵏ bᵏ。求与 x 无关的项时，通常令一般项中 x 的指数等于零。
(1 + x)ⁿ = 1 + nx + n(n−1)/2! x² + n(n−1)(n−2)/3! x³ + …
10. Vectors in Two Dimensions | 二维向量
A vector has magnitude and direction, while a scalar has only magnitude. Vectors can be written in column form or as ai + bj. Addition, subtraction and multiplication by a scalar follow simple component-wise rules.
向量既有大小又有方向，而标量只有大小。向量可写成列向量形式或 ai + bj 形式。向量的加法、减法和数乘遵循逐分量运算的简单法则。
|ai + bj| = √(a² + b²), unit vector = (ai + bj) / |ai + bj|
The position vector of a point relative to an origin O is the vector that connects O to the point. In kinematics, vector quantities such as velocity and acceleration can be integrated or differentiated component-wise to link displacement, velocity and acceleration.
点相对于原点 O 的位置向量是从 O 指向该点的向量。在运动学中，速度、加速度等向量可逐分量积分或微分，从而建立位移、速度与加速度之间的联系。
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