📚 Year 12 AQA Maths: Core Topic Summary | Year 12 AQA 数学：核心知识点梳理
Year 12 AQA Mathematics builds the fundamental toolkit you need for the full A-level. The pure mathematics core dominates the first year, covering proof, algebra, coordinate geometry, sequences, trigonometry, exponentials, logarithms, differentiation, integration, vectors and the binomial expansion. A strong command of these topics ensures you can tackle applied units and the more demanding Year 13 content with confidence.
Year 12 AQA 数学为你构建完成完整 A-level 所需的基础工具包。第一年的纯数学核心占据主导地位，涵盖证明、代数、坐标几何、数列、三角学、指数、对数、微分、积分、向量以及二项式展开。熟练掌握这些主题将确保你能够自信地应对应用单元以及更具挑战性的 Year 13 内容。
1. Proof | 证明
Mathematical proof is a logical argument that demonstrates the truth of a statement. In Year 12 you learn three main types: deduction (direct proof), exhaustion, and contradiction. For example, to prove that the sum of two even numbers is even, you can let the numbers be 2m and 2n, add them, and factorise to show the result is a multiple of 2.
数学证明是一种用来展示命题真实性的逻辑论证。在 12 年级中，你主要学习三种类型：演绎法（直接证明）、穷举法和反证法。例如，为了证明两个偶数的和为偶数，你可以设这两个数为 2m 和 2n，将它们相加并因式分解，以表明结果是 2 的倍数。
Proof by contradiction is especially powerful: assume the statement is false, then deduce an impossible result, such as 1 = 0 or a rational number equalling an irrational one. Common exam proofs include irrationality of √2 and infiniteness of primes.
反证法尤其强大：假设命题为假，然后推导出不可能的结果，比如 1 = 0 或有理数等于无理数。考试中常见的证明包括 √2 的无理性以及素数有无穷多个。
2. Algebra and Functions | 代数与函数
This topic extends GCSE algebra to polynomials of higher degree, surds, indices and functions. You must be confident with factorising cubic and quartic expressions, using the factor theorem and algebraic division. The remainder theorem states that when a polynomial f(x) is divided by (x – a), the remainder is f(a).
该主题将 GCSE 代数拓展到更高次的多项式、根式、指数和函数。你必须能够熟练地对三次和四次表达式进行因式分解，并会使用因式定理和多项式除法。余数定理指出，当多项式 f(x) 除以 (x – a) 时，余数为 f(a)。
Key function concepts include domain, range, composite and inverse functions. You sketch graphs of y = f(x) and apply transformations such as y = f(x) + a, y = f(x + a), y = a f(x) and y = f(ax). Always label the coordinates of any intersections with the axes and the equations of asymptotes for rational functions.
关键的函数概念包括定义域、值域、复合函数和反函数。你需要画出 y = f(x) 的草图，并应用变换，如 y = f(x) + a、y = f(x + a)、y = a f(x) 以及 y = f(ax)。对于有理函数，务必标注与坐标轴的交点以及渐近线方程。
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The domain of a function is the set of input values; the range is the set of output values. The inverse function f-1(x) reflects the graph in the line y = x.
函数的定义域是输入值的集合；值域是输出值的集合。反函数 f-1(x) 的图像关于直线 y = x 对称。
3. Coordinate Geometry | 坐标几何
You review straight-line graphs y = mx + c, where m is the gradient and c is the y-intercept. The equation of a line through (x₁, y₁) with gradient m is y – y₁ = m(x – x₁). Parallel lines have equal gradients; perpendicular lines satisfy m₁ × m₂ = –1.
你复习直线图像 y = mx + c，其中 m 是斜率，c 是 y 轴截距。经过点 (x₁, y₁) 且斜率为 m 的直线方程为 y – y₁ = m(x – x₁)。平行线斜率相等；垂直线满足 m₁ × m₂ = –1。
The circle equation (x – a)² + (y – b)² = r² has centre (a, b) and radius r. Completing the square on x and y allows you to find the centre and radius from an expanded form. You use the discriminant of a quadratic to determine whether a line is tangent to a circle.
圆的方程 (x – a)² + (y – b)² = r² 的圆心为 (a, b)，半径为 r。对 x 和 y 进行配方可以从展开式中求出圆心和半径。你还可以利用二次方程的判别式来判断一条直线是否与圆相切。
4. Sequences and Series | 数列与级数
An arithmetic sequence has a first term a and a common difference d. The nth term is uₙ = a + (n – 1)d. The sum of the first n terms is Sₙ = n/2 [2a + (n – 1)d] or Sₙ = n/2 (a + l), where l is the last term.
等差数列有首项 a 和公差 d。第 n 项为 uₙ = a + (n – 1)d。前 n 项之和为 Sₙ = n/2 [2a + (n – 1)d] 或 Sₙ = n/2 (a + l)，其中 l 为末项。
Geometric sequences have a first term a and a common ratio r. The nth term is uₙ = a rn–1. The sum of the first n terms is Sₙ = a(1 – rn) / (1 – r) for r ≠ 1. For |r| < 1, the sum to infinity exists and equals S∞ = a / (1 – r).
等比数列有首项 a 和公比 r。第 n 项为 uₙ = a rn–1。前 n 项之和为 Sₙ = a(1 – rn) / (1 – r)（r ≠ 1）。当 |r| < 1 时，无穷级数之和存在且等于 S∞ = a / (1 – r)。
Sigma notation Σ is used to write series compactly. You need to find sums of simple series by using arithmetic or geometric formulas, or by splitting the sum into recognisable parts.
求和符号 Σ 用于简洁地表示级数。你需要通过使用等差或等比公式，或者将求和拆分成可识别的部分来求出简单级数的和。
5. Trigonometry | 三角学
Exact trigonometric values for 30°, 45°, 60° and related angles must be memorised. The sine and cosine rules apply to any triangle: a / sin A = b / sin B = c / sin C for the sine rule, and a² = b² + c² – 2bc cos A for the cosine rule. The area of a triangle is ½ ab sin C.
必须熟记 30°、45°、60° 及相关角度的精确三角值。正弦定理和余弦定理适用于任意三角形：正弦定理为 a / sin A = b / sin B = c / sin C，余弦定理为 a² = b² + c² – 2bc cos A。三角形的面积公式为 ½ ab sin C。
Trigonometric identities such as sin²θ + cos²θ = 1 and tanθ = sinθ / cosθ are used to solve equations and simplify expressions. You solve equations like sin x = k or cos 2x = 0.5 in a given interval, often using the quadrant rule or the graphs of sin, cos and tan.
诸如 sin²θ + cos²θ = 1 以及 tanθ = sinθ / cosθ 等三角恒等式用于解方程和简化表达式。你需要在给定区间内求解如 sin x = k 或 cos 2x = 0.5 的方程，通常需要利用象限规则或正弦、余弦、正切的图像。
You also work with radians as a measure of angle, where 180° = π rad. This is essential for later calculus of trigonometric functions.
你还会使用弧度作为角的度量单位，其中 180° = π 弧度。这对于后续的三角函数微积分至关重要。
6. Exponentials and Logarithms | 指数与对数
The exponential function y = ex and the natural logarithm y = ln x are inverses of each other. Key properties: ln 1 = 0, ln e = 1, and eln x = x for x > 0. The laws of logarithms allow you to manipulate expressions: logₐ (xy) = logₐ x + logₐ y, logₐ (x/y) = logₐ x – logₐ y, and logₐ (xk) = k logₐ x.
指数函数 y = ex 与自然对数 y = ln x 互为反函数。关键性质：ln 1 = 0，ln e = 1，且对于 x > 0 有 eln x = x。对数运算律让你能够处理表达式：logₐ (xy) = logₐ x + logₐ y，logₐ (x/y) = logₐ x – logₐ y，以及 logₐ (xk) = k logₐ x。
You solve equations by taking logs of both sides, e.g. 2x = 5 → x = log 5 / log 2. Knowledge of linearising data with y = abx and y = axn using logs is also required.
你需要通过对等式两边取对数来解方程，例如 2x = 5 → x = log 5 / log 2。此外，还需要掌握如何利用对数将 y = abx 和 y = axn 数据线性化的知识。
7. Differentiation | 微分
Differentiation gives the gradient of a curve at a point. From first principles, the derivative is f'(x) = limh→0 [f(x+h) – f(x)] / h. In practice you use standard results:
微分给出了曲线在某一点的斜率。从第一性原理出发，导数为 f'(x) = limh→0 [f(x+h) – f(x)] / h。在实际应用中，你会用到标准结果：
| f(x) | f'(x) |
|---|---|
| xn | n xn-1 |
| ex | ex |
| ln x | 1/x |
| sin x | cos x |
| cos x | – sin x |
The chain rule, product rule and quotient rule extend your ability to differentiate composite, multiplied or divided functions. The second derivative f”(x) helps determine the nature of stationary points. You also find equations of tangents and normals using the derivative as the gradient at a point.
链式法则、乘积法则和商法则将你的微分能力拓展到复合、相乘或相除的函数。二阶导数 f”(x) 有助于判断驻点的性质。你还要利用某点的导数作为斜率，来求该点的切线和法线方程。
8. Integration | 积分
Integration is the reverse process of differentiation. The indefinite integral of xn (n ≠ –1) is ∫ xn dx = (1/(n+1)) xn+1 + c. You also integrate standard functions: ∫ ex dx = ex + c, ∫ (1/x) dx = ln |x| + c, ∫ cos x dx = sin x + c, ∫ sin x dx = – cos x + c.
积分是微分的逆过程。xn（n ≠ –1）的不定积分为 ∫ xn dx = (1/(n+1)) xn+1 + c。你还要积分标准函数：∫ ex dx = ex + c，∫ (1/x) dx = ln |x| + c，∫ cos x dx = sin x + c，∫ sin x dx = – cos x + c。
Definite integrals have limits and yield a number; they can be evaluated as F(b) – F(a) where F'(x) = f(x). The area between a curve y = f(x) and the x-axis from x = a to x = b is given by ∫ab f(x) dx, provided the region is above the axis. For areas below the axis, the integral gives a negative answer, so you work with absolute values or split the region.
定积分带有上下限，其结果为数值；可利用 F(b) – F(a) 计算，其中 F'(x) = f(x)。曲线 y = f(x) 与 x 轴之间从 x = a 到 x = b 的区域面积由 ∫ab f(x) dx 给出，前提是该区域位于 x 轴上方。对于 x 轴下方的区域，积分结果为负，因此你需要使用绝对值或将区域分割处理。
9. Vectors | 向量
Vectors in Year 12 are primarily two-dimensional. A vector can be expressed as a column vector (x)(y) or in i, j notation as x i + y j. The magnitude of vector v = a i + b j is |v| = √(a² + b²). A unit vector has magnitude 1; the unit vector in the direction of v is v / |v|.
12 年级的向量主要是二维的。向量可以表示为列向量 (x, y) 或以 i, j 符号表示为 x i + y j。向量 v = a i + b j 的模长为 |v| = √(a² + b²)。单位向量的模长为 1；沿 v 方向的单位向量为 v / |v|。
You add and subtract vectors by combining i components and j components separately. Scalar multiplication changes the magnitude but not the direction (unless the scalar is negative, which reverses it). Position vectors, velocity vectors and geometric problems using vectors are common exam questions.
你可以通过分别组合 i 分量与 j 分量来对向量进行加减。标量乘法只会改变向量的模长而不会改变方向（除非标量为负，此时方向反转）。位置向量、速度向量以及运用向量解决几何问题是考试中的常见题型。
10. Binomial Expansion | 二项式展开
For a positive integer n, the expansion of (a + b)n uses binomial coefficients from Pascal’s triangle or the formula: (a + b)n = Σr=0n nCr an–r br, where nCr = n! / (r!(n – r)!). You usually set a = 1 or a constant to find specific terms.
对于正整数 n，(a + b)n 的展开式使用帕斯卡三角形中的二项式系数，或者使用公式：(a + b)n = Σr=0n nCr an–r br，其中 nCr = n! / (r!(n – r)!)。你通常令 a = 1 或取某个常数以求出指定的项。
The expansion can be used to approximate expressions like (1 + x)n for small x, by taking the first few terms. You must state the range of validity (|x| < 1 for infinite series expansions in later work, but Year 12 typically focuses on finite expansions).
通过取前几项，展开式可用于近似如 (1 + x)n 这类在小 x 时的表达式。你必须注明有效范围（后续内容中，对于无穷级数展开要求 |x| < 1，但 12 年级通常侧重于有限项展开）。
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