📚 Year 12 AQA Further Maths: Formula & Theorem Quick Reference Handbook | AQA 进阶数学 Year 12 公式定理速查手册
This quick reference handbook brings together the essential formulae and theorems you will encounter in the first year of the AQA Further Mathematics course. It covers core pure topics along with selected mechanics and statistics concepts that appear in Year 12, organised for rapid revision. Keep this guide handy while tackling problem sets, and use it to check your recall before assessments.
本速查手册汇集了你在 AQA 进阶数学课程第一年将遇到的核心公式与定理。内容涵盖 Year 12 的纯数重点以及部分力学和统计知识，并按照便于快速复习的方式编排。建议你在练习习题时常备此册，并在评估前用它检验记忆效果。
1. Algebra & Factorisation | 代数与因式分解
Polynomial identities and the factor theorem form the backbone of algebraic manipulation in Further Maths. You must be able to factorise cubics and quartics and work confidently with the remainder theorem. The relationship between roots and coefficients is also essential.
多项式恒等式与因式定理是进阶数学中代数运算的基石。你必须能对三次和四次多项式进行因式分解，并能熟练应用余式定理。根与系数之间的关系同样重要。
| Formula / Theorem (English) | 中文说明 |
|---|---|
| If f(a) = 0 then (x − a) is a factor of f(x) (Factor Theorem) | 若 f(a) = 0，则 (x − a) 是 f(x) 的因式（因式定理） |
| For cubic ax³ + bx² + cx + d = 0 with roots α, β, γ: α + β + γ = −b/a, αβ + βγ + γα = c/a, αβγ = −d/a | 三次方程 ax³ + bx² + cx + d = 0 的根 α, β, γ 满足： 和 = −b/a，两两积之和 = c/a，积 = −d/a |
2. Quadratics & Functions | 二次方程与函数
Although quadratics are central to GCSE work, Year 12 extends your skills to discriminants, completing the square for curve sketching, and the manipulation of composite and inverse functions. You will also work with the modulus function and its graph transformations.
尽管二次方程是 GCSE 的核心内容，Year 12 将你的能力拓展到判别式、用于曲线绘图的配方法，以及复合函数与反函数的运算。你还将学习模函数及其图像变换。
| Formula / Rule | 中文 |
|---|---|
| Discriminant Δ = b² − 4ac for ax² + bx + c = 0 | 一元二次方程 ax² + bx + c = 0 的判别式 Δ = b² − 4ac |
| If fg(x) = f(g(x)), domain of fg is x ∈ domain of g such that g(x) ∈ domain of f | 复合函数 fg(x) = f(g(x)) 的定义域：x 属于 g 的定义域且 g(x) 属于 f 的定义域 |
| |x| = x for x ≥ 0, |x| = −x for x < 0 | 绝对值函数：当 x ≥ 0 时 |x| = x；当 x < 0 时 |x| = −x |
3. Exponentials & Logarithms | 指数与对数
The natural exponential function eˣ ithms and able to solve equations where the unknown appears as a power. The relationship aˣ = ex ln a is particularly useful.
自然指数函数 eˣ 与自然对数 ln x 是增长与衰减问题以及微积分的核心。你必须熟练掌握对数运算律，并能够求解未知数出现在指数位置的方程。恒等式 aˣ = ex ln a 尤其有用。
| Formula / Law | 中文 |
|---|---|
| logₐ(xy) = logₐx + logₐy | 积的对数等于对数之和 |
| logₐ(x/y) = logₐx − logₐy | 商的对数等于对数之差 |
| logₐ(xⁿ) = n logₐx | 幂的对数等于指数乘以底数的对数 |
| aˣ = ex ln a | 指数换底公式 |
4. Trigonometry | 三角函数
In Year 12 Further Maths you use radian measure exclusively and work with the reciprocal trigonometric functions sec, cosec and cot. The Pythagorean identities, compound-angle formulae and the small-angle approximations are tested regularly, especially in calculus and vector contexts.
在 Year 12 进阶数学中，你将完全使用弧度制，并处理倒数三角函数 sec、cosec 和 cot。勾股恒等式、复合角公式以及小角近似在考试中频繁出现，尤其是在微积分和向量问题中。
| Identity / Approximation | 中文 |
|---|---|
| sin²θ + cos²θ = 1 | 正弦余弦平方和为 1 |
| 1 + tan²θ = sec²θ, 1 + cot²θ = cosec²θ | 正切与正割、余切与余割的勾股恒等式 |
| sin(A±B) = sinAcosB ± cosAsinB | 正弦的复合角公式 |
| cos(A±B) = cosAcosB ∓ sinAsinB | 余弦的复合角公式 |
| tan(A±B) = (tanA ± tanB)/(1 ∓ tanAtanB) | 正切的复合角公式 |
| For small θ (radians): sinθ ≈ θ, cosθ ≈ 1 − θ²/2, tanθ ≈ θ | 小角近似（θ 取弧度） |
5. Differentiation | 微分
Building on GCSE and single maths, you now differentiate exponentials, logarithms, trigonometric and inverse trigonometric functions. The rules – chain, product and quotient – must be applied fluently. Parametric differentiation and the concept of the second derivative for rates of change are also introduced.
在 GCSE 和单数学的基础上，你现在将对指数函数、对数函数、三角函数及反三角函数进行微分。链式法则、乘法法则和商法则必须熟练应用。参数微分以及用于变化率的二阶导数概念也被引入。
| Derivative (English) | 导数公式（中文） |
|---|---|
| d/dx (xⁿ) = nxⁿ⁻¹ | 幂函数的导数 |
| d/dx (eˣ) = eˣ | eˣ 的导数等于 eˣ |
| d/dx (ln x) = 1/x | 自然对数函数的导数 |
| d/dx (sin x) = cos x, d/dx (cos x) = −sin x | 正弦与余弦的导数 |
| d/dx (tan x) = sec² x | 正切函数的导数为正割平方 |
| Chain rule: dy/dx = dy/du × du/dx | 链式法则：y 关于 x 的导数等于 y 关于 u 的导数乘以 u 关于 x 的导数 |
| Product rule: d/dx (uv) = u'v + uv' | 乘法法则：前微后不微加上前不微后微 |
| Quotient rule: d/dx (u/v) = (u'v − uv')/v² | 商法则：(分子的导数乘以分母减去分子乘以分母的导数) 除以分母的平方 |
6. Integration | 积分
Integration is the reverse of differentiation, but Year 12 also introduces definite integration, areas under curves and the trapezium rule. You will use standard integrals, integration by substitution and, later, by parts. The fundamental theorem of calculus links differentiation and integration.
积分是微分的逆运算，但 Year 12 还介绍了定积分、曲线下面积以及梯形法则。你将使用标准积分、代入积分法，并在稍后学习分部积分法。微积分基本定理将微分与积分联系起来。
| Integral / Rule | 中文 |
|---|---|
| ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, n ≠ −1 | 幂函数的基本积分 |
| ∫ eˣ dx = eˣ + C | 指数函数的积分 |
| ∫ 1/x dx = ln|x| + C | 1/x 的积分为自然对数 |
| ∫ sin x dx = −cos x + C, ∫ cos x dx = sin x + C | 三角函数的积分 |
| Trapezium rule: ∫ₐᵇ y dx ≈ ½h[(y₀ + yₙ) + 2(y₁ + … + yₙ₋₁)], h = (b−a)/n | 梯形法则求定积分近似值 |
| Area between curve and x-axis: A = ∫ₐᵇ y dx (y ≥ 0) | 曲线与 x 轴之间的面积（y ≥ 0） |
7. Vectors | 向量
Vector methods are used extensively in pure maths mechanics. You need to be comfortable with position vectors, magnitude, direction, and the scalar (dot) product. In mechanics, vectors describe forces, velocities and displacements in two dimensions.
向量方法在纯数学和力学中广泛应用。你需要熟练掌握位置向量、大小、方向以及数量积（点积）。在力学中，向量用于描述二维情况下的力、速度和位移。
| Formula / Concept | 中文 |
|---|---|
| Magnitude: |a| = √x² + y² for a = xi + yj | 向量 a = xi + yj 的模：√x² + y² |
| Dot product: a·b = |a||b|cosθ = x₁x₂ + y₁y₂ | 数量积的定义与坐标计算公式 |
| If a·b = 0 ⇒ vectors are perpendicular | 两向量垂直当且仅当点积为零 |
| Position vector AB = b − a | 由点 A 到点 B 的向量等于终点位置向量减去起点位置向量 |
8. Sequences & Series | 数列与级数
Arithmetic and geometric series are extended with sigma notation and the sum to infinity for convergent geometric series. You also encounter recurrence relations and may study the method of differences for summation.
等差数列与等比数列在引入总和符号后得到拓展，并涉及收敛等比数列的无穷和。你还将遇到递推关系，并可能学习用于求和的差分法。
| Formula | 中文 |
|---|---|
| Arithmetic nth term: uₙ = a + (n−1)d | 等差数列通项公式 |
| Arithmetic sum: Sₙ = n/2 [2a + (n−1)d] | 等差数列前 n 项和 |
| Geometric nth term: uₙ = arⁿ⁻¹ | 等比数列通项公式 |
| Geometric sum: Sₙ = a(1−rⁿ)/(1−r) for r ≠ 1 | 等比数列前 n 项和 |
| Sum to infinity: S∞ = a/(1−r) for |r| < 1 | 无穷等比级数求和（|r| < 1） |
9. Binomial Expansion | 二项展开式
The general binomial theorem is valid for any rational exponent, provided the expansion is written in the form (1 + x)ⁿ with |x| < 1. You must be able to expand and state the range of validity. The formula for (a + bx)ⁿ requires careful manipulation.
广义二项式定理适用于任何有理指数，前提是将式子写成 (1 + x)ⁿ 且满足 |x| < 1。你必须能够进行展开并说明其成立范围。对于 (a + bx)ⁿ 形式，需要先进行变形。
(1 + x)ⁿ = 1 + nx + [n(n−1)/2!] x² + [n(n−1)(n−2)/3!] x³ + … for |x| < 1
(1 + x)ⁿ = 1 + nx + [n(n−1)/2!] x² + [n(n−1)(n−2)/3!] x³ + … 对于 |x| < 1
| Step | 步骤 |
|---|---|
| Write (a + bx)ⁿ = aⁿ (1 + (b/a)x)ⁿ and then expand | 提取 aⁿ 化为 (1 + (b/a)x)ⁿ 后再展开 |
10. Numerical Methods | 数值方法
When equations cannot be solved algebraically, numerical methods locate roots. The sign-change rule confirms a root exists in an interval. Iterative formulae, like the Newton‑Raphson method, are used to home in on solutions. You must be able to rearrange equations and use iterative processes accurately.
当方程无法解析求解时，数值方法可用来寻找根。符号改变法则能确认区间内存在根。像牛顿-拉弗森迭代公式等迭代方法用于逼近解。你必须能够对方程进行重排并准确使用迭代过程。
| Method / Formula | 中文 |
|---|---|
| If f(a) and f(b) have opposite signs, there is at least one root in [a, b] | 若 f(a) 与 f(b) 异号，则区间 [a, b] 内至少存在一个根 |
| Newton‑Raphson: xₙ₊₁ = xₙ − f(xₙ)/f'(xₙ) | 牛顿-拉弗森迭代公式 |
| Rearrangement: write f(x) = 0 as x = g(x), then iterate xₙ₊₁ = g(xₙ) | 迭代法：将方程改写为 x = g(x)，再用 xₙ₊₁ = g(xₙ) 迭代 |
11. Probability & Statistics Fundamentals | 概率与统计基础
Year 12 statistics includes probability laws, conditional probability using Venn diagrams and tree diagrams, and the binomial distribution as a discrete probability model. You also cover measures of central tendency and spread for data sets.
Year 12 统计包括概率运算律、借助文氏图和树状图的条件概率，以及作为离散概率模型的二项分布。你还会学习数据集的集中趋势与离散程度量数。
| Formula / Rule | 中文 |
|---|---|
| P(A ∪ B) = P(A) + P(B) − P(A ∩ B) | 加法公式 |
| P(A|B) = P(A ∩ B)/P(B) | 条件概率定义 |
| For independent events: P(A ∩ B) = P(A)P(B) | 独立事件的乘法公式 |
| Binomial probability: P(X = r) = ⁿCᵣ pʳ (1−p)ⁿ⁻ʳ | 二项分布概率公式 |
| Mean of binomial: μ = np, Variance: σ² = np(1−p) | 二项分布的均值与方差 |
12. Kinematics in One Dimension | 一维运动学
Mechanics in Year 12 begins with constant acceleration formulae (SUVAT) and the interpretation of displacement–time and velocity–time graphs. You will apply vector notation to motion in a straight line and understand the relationships between displacement, velocity and acceleration through differentiation and integration.
Year 12 力学从匀加速运动公式 (SUVAT) 以及位移-时间图、速度-时间图的解读开始。你将把向量符号应用于直线运动，并通过微分与积分理解位移、速度和加速度之间的关系。
| SUVAT Equation | 中文 |
|---|---|
| v = u + at | 速度-时间关系 |
| s = ut + ½at² | 位移-时间关系 |
| s = ½(u + v)t | 平均速度求位移 |
| v² = u² + 2as | 速度-位移关系 |
| Velocity is ds/dt, acceleration is dv/dt | 速度为位移的导数，加速度为速度的导数 |
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