📚 Year 13 AQA Maths Formula & Theorem Quick Reference | AQA A-level 数学公式定理速查手册
This comprehensive quick-reference handbook covers the essential formulae, theorems, and standard results required for Year 13 AQA Mathematics, spanning pure mathematics, mechanics, and statistics. Each section presents the most frequently used relationships with clear notation, helping you to consolidate your revision and sharpen your problem‑solving skills.
这本全面的速查手册涵盖了 Year 13 AQA 数学考试所需的核心公式、定理和标准结论，横跨纯数学、力学与统计学。每一节以清晰的符号呈现最常用的关系，帮助你巩固复习并提升解题能力。
1. Algebraic Essentials | 代数基础
The laws of indices for all rational numbers:
aᵐ × aⁿ = aᵐ⁺ⁿ , aᵐ ÷ aⁿ = aᵐ⁻ⁿ , (aᵐ)ⁿ = aᵐⁿ
对所有有理数成立的指数律：
aᵐ × aⁿ = aᵐ⁺ⁿ , aᵐ ÷ aⁿ = aᵐ⁻ⁿ , (aᵐ)ⁿ = aᵐⁿ
The relationship between logarithms and exponentials: logₐ x = y ⟺ aʸ = x, with the natural logarithm ln x = logₑ x.
对数与指数的关系：logₐ x = y ⟺ aʸ = x，其中自然对数 ln x = logₑ x。
Key logarithm laws:
logₐ (xy) = logₐ x + logₐ y , logₐ (x/y) = logₐ x − logₐ y , logₐ (xⁿ) = n logₐ x
关键对数运算法则：
logₐ (xy) = logₐ x + logₐ y , logₐ (x/y) = logₐ x − logₐ y , logₐ (xⁿ) = n logₐ x
Partial fractions decomposition types: for distinct linear factors, repeated linear factors, and irreducible quadratic factors.
部分分式分解类型：不同线性因子、重复线性因子和不可约二次因子的分解。
Binomial expansion for rational n: (1 + x)ⁿ = 1 + nx + [n(n−1)/2!] x² + … , valid for |x| < 1.
有理指数 n 的二项展开式：(1 + x)ⁿ = 1 + nx + [n(n−1)/2!] x² + … ，在 |x| < 1 时成立。
2. Trigonometry and Radian Measure | 三角学与弧度制
Radian measure: π radians = 180°. Arc length s = rθ, sector area A = ½ r²θ.
弧度制：π 弧度 = 180°。弧长 s = rθ，扇形面积 A = ½ r²θ。
Fundamental identities:
sin²θ + cos²θ = 1 , 1 + tan²θ = sec²θ , 1 + cot²θ = cosec²θ
基本恒等式：
sin²θ + cos²θ = 1 , 1 + tan²θ = sec²θ , 1 + cot²θ = cosec²θ
Compound angle and double angle formulae:
sin(A ± B) = sin A cos B ± cos A sin B
cos(A ± B) = cos A cos B ∓ sin A sin B
tan(A ± B) = (tan A ± tan B)/(1 ∓ tan A tan B)
和角与倍角公式：
sin(A ± B) = sin A cos B ± cos A sin B
cos(A ± B) = cos A cos B ∓ sin A sin B
tan(A ± B) = (tan A ± tan B)/(1 ∓ tan A tan B)
Small‑angle approximations (θ in radians): sin θ ≈ θ, cos θ ≈ 1 − ½θ², tan θ ≈ θ.
小角近似（θ 以弧度为单位）：sin θ ≈ θ, cos θ ≈ 1 − ½θ², tan θ ≈ θ。
Derivatives of inverse trig functions:
d/dx (sin⁻¹ x) = 1/√(1−x²) , d/dx (cos⁻¹ x) = −1/√(1−x²) , d/dx (tan⁻¹ x) = 1/(1+x²)
反三角函数的导数：
d/dx (sin⁻¹ x) = 1/√(1−x²) , d/dx (cos⁻¹ x) = −1/√(1−x²) , d/dx (tan⁻¹ x) = 1/(1+x²)
3. Differentiation Techniques | 微分技巧
Standard derivatives:
| f(x) | f'(x) |
| xⁿ | n xⁿ⁻¹ |
| eˣ | eˣ |
| ln x | 1/x |
| sin x | cos x |
| cos x | −sin x |
| tan x | sec² x |
标准导数表。
The product rule:
d/dx (uv) = u’v + uv’
乘积法则：
d/dx (uv) = u’v + uv’
The quotient rule:
d/dx (u/v) = (u’v − uv’)/v²
商法则：
d/dx (u/v) = (u’v − uv’)/v²
The chain rule: if y = f(u) and u = g(x), then dy/dx = (dy/du) × (du/dx).
链式法则：若 y = f(u) 且 u = g(x)，则 dy/dx = (dy/du) × (du/dx)。
Parametric differentiation: given x = x(t), y = y(t), then dy/dx = (dy/dt) / (dx/dt).
参数微分：给定 x = x(t), y = y(t)，则 dy/dx = (dy/dt) / (dx/dt)。
Implicit differentiation allows finding derivatives without explicit rearrangement.
隐函数微分法允许不显式解出 y 而求导。
4. Integration and Its Applications | 积分及其应用
Standard integrals:
| f(x) | ∫ f(x) dx |
| xⁿ (n ≠ −1) | xⁿ⁺¹/(n+1) + C |
| 1/x | ln|x| + C |
| eˣ | eˣ + C |
| sin x | −cos x + C |
| cos x | sin x + C |
| sec² x | tan x + C |
标准积分表。
Integration by parts:
∫ u dv/dx dx = uv − ∫ v du/dx dx
分部积分法：
∫ u dv/dx dx = uv − ∫ v du/dx dx
Integration by substitution: ∫ f(g(x)) g'(x) dx = ∫ f(u) du, where u = g(x).
代换积分法：∫ f(g(x)) g'(x) dx = ∫ f(u) du，其中 u = g(x)。
Definite integrals: area under curve = ∫ₐᵇ y dx, volume of revolution (about x‑axis) = π ∫ₐᵇ y² dx.
定积分：曲线下方面积 = ∫ₐᵇ y dx，绕 x 轴旋转形成的体积 = π ∫ₐᵇ y² dx。
Solving first‑order differential equations by separating variables: ∫ 1/g(y) dy = ∫ f(x) dx.
分离变量法解一阶微分方程：∫ 1/g(y) dy = ∫ f(x) dx。
5. Numerical Methods | 数值方法
The Newton‑Raphson iteration for solving f(x)=0:
xₙ₊₁ = xₙ − f(xₙ)/f'(xₙ)
求解 f(x)=0 的牛顿‑拉弗森迭代：
xₙ₊₁ = xₙ − f(xₙ)/f'(xₙ)
Fixed‑point iteration rearranges the equation to x = g(x) and uses xₙ₊₁ = g(xₙ). Convergence requires |g'(x)| < 1 near the root.
不动点迭代将方程改写为 x = g(x)，并使用 xₙ₊₁ = g(xₙ)，收敛要求在根附近 |g'(x)| < 1。
The trapezium rule for approximating a definite integral:
∫ₐᵇ y dx ≈ (h/2)[y₀ + 2(y₁ + y₂ + … + yₙ₋₁) + yₙ] , where h = (b−a)/n.
梯形法则近似定积分：
∫ₐᵇ y dx ≈ (h/2)[y₀ + 2(y₁ + y₂ + … + yₙ₋₁) + yₙ]，其中 h = (b−a)/n。
Numerical methods are used when exact analytic solutions are impossible; careful choice of initial value is essential.
当无法获得精确解析解时使用数值方法，慎重选择初值至关重要。
6. Vectors in 3D | 三维向量
The dot (scalar) product:
a · b = |a||b| cosθ = a₁b₁ + a₂b₂ + a₃b₃
点积（数量积）：
a · b = |a||b| cosθ = a₁b₁ + a₂b₂ + a₃b₃
The angle between two vectors: cosθ = (a·b) / (|a||b|). Vectors are perpendicular when a·b = 0.
两向量夹角：cosθ = (a·b) / (|a||b|)。当 a·b = 0 时两向量垂直。
Vector equation of a straight line: r = a + λd, where a is a position vector and d is the direction vector.
直线的向量方程：r = a + λd，其中 a 为位置向量，d 为方向向量。
Equation of a plane (scalar product form): r · n = d, or (r − a) · n = 0, where n is a normal vector.
平面方程（点积形式）：r · n = d 或 (r − a) · n = 0，其中 n 为法向量。
The shortest distance from a point to a line or plane often uses dot product projections.
点到直线或平面的最短距离常利用点积投影计算。
7. Sequences, Series and Binomial Expansion | 数列、级数与二项展开
Arithmetic sequence: nth term uₙ = a + (n−1)d, sum Sₙ = n/2 [2a + (n−1)d] = n/2 (a + l).
等差数列：第 n 项 uₙ = a + (n−1)d，前 n 项和 Sₙ = n/2 [2a + (n−1)d] = n/2 (a + l)。
Geometric sequence: nth term uₙ = arⁿ⁻¹, sum Sₙ = a(1−rⁿ)/(1−r) for r ≠ 1. Sum to infinity S∞ = a/(1−r) when |r| < 1.
等比数列：第 n 项 uₙ = arⁿ⁻¹，前 n 项和 Sₙ = a(1−rⁿ)/(1−r)（r ≠ 1）。无穷级数和 S∞ = a/(1−r)（|r| < 1）。
The binomial series for rational powers: (1+x)ⁿ = 1 + nx + n(n−1)x²/2! + n(n−1)(n−2)x³/3! + … , |x| < 1.
有理幂二项级数：(1+x)ⁿ = 1 + nx + n(n−1)x²/2! + n(n−1)(n−2)x³/3! + … ，|x| < 1。
Expansion of (a+bx)ⁿ: first write as aⁿ(1 + (b/a)x)ⁿ and apply the series, then identify the range of validity.
(a+bx)ⁿ 的展开：先写成 aⁿ(1 + (b/a)x)ⁿ 并应用级数，再确定收敛范围。
Series expansions of standard functions (Maclaurin): eˣ = Σ xʳ/r! , sin x = Σ (−1)ʳ x²ʳ⁺¹/(2r+1)! , cos x = Σ (−1)ʳ x²ʳ/(2r)! .
标准函数的麦克劳林级数：eˣ = Σ xʳ/r!，sin x = Σ (−1)ʳ x²ʳ⁺¹/(2r+1)!，cos x = Σ (−1)ʳ x²ʳ/(2r)!。
8. Parametric Equations and Polar Coordinates | 参数方程与极坐标
For parametric curves x = f(t), y = g(t), the gradient is dy/dx = (g'(t))/(f'(t)).
对于参数曲线 x = f(t), y = g(t)，梯度为 dy/dx = (g'(t))/(f'(t))。
The area under a parametric curve: ∫ y dx = ∫ g(t) f'(t) dt, with appropriate limits.
参数曲线下方面积：∫ y dx = ∫ g(t) f'(t) dt，需适当地改变积分限。
Polar curves r = f(θ): conversion formulas x = r cosθ, y = r sinθ. The area of a polar sector is A = ½ ∫ r² dθ.
极坐标曲线 r = f(θ) ：转换公式 x = r cosθ, y = r sinθ。极扇形的面积 A = ½ ∫ r² dθ。
The gradient of a polar curve: dy/dx = (r’ sinθ + r cosθ) / (r’ cosθ − r sinθ).
极坐标曲线的梯度：dy/dx = (r’ sinθ + r cosθ) / (r’ cosθ − r sinθ)。
Common polar curves include cardioids r = a(1+cosθ) and roses r = a sin(nθ).
常见极坐标曲线包括心脏线 r = a(1+cosθ) 和玫瑰线 r = a sin(nθ)。
9. Mechanics – Kinematics and Forces | 力学 – 运动学与力
The SUVAT equations for constant linear acceleration:
v = u + at, s = ut + ½at², s = ½(u+v)t, v² = u² + 2as, s = vt − ½at²
匀加速直线运动 SUVAT 方程：
v = u + at, s = ut + ½at², s = ½(u+v)t, v² = u² + 2as, s = vt − ½at²
Newton’s second law: F = ma, where F is the resultant force in the direction of acceleration.
牛顿第二定律：F = ma，其中 F 为沿加速度方向的合力。
Momentum p = mv. The impulse‑momentum principle: Impulse = F Δt = Δp.
动量 p = mv。冲量‑动量原理：冲量 = F Δt = Δp。
For an object in equilibrium, resultant force is zero and resultant moment about any point is zero.
处于平衡状态的物体，合力为零且对任意点的合力矩为零。
The coefficient of friction μ satisfies F ≤ μR, with F = μR at the point of limiting friction.
摩擦系数 μ 满足 F ≤ μR，在极限摩擦时 F = μR。
10. Mechanics – Energy and Projectiles | 力学 – 能量与抛体
Work done by a constant force: W = F s cosθ, where θ is the angle between force and displacement.
恒力做功：W = F s cosθ，其中 θ 为力与位移的夹角。
Kinetic energy KE = ½ mv², gravitational potential energy GPE = mgh.
动能 KE = ½ mv²，重力势能 GPE = mgh。
The work‑energy principle: net work done = change in kinetic energy.
功能原理：合力所做的功等于动能的变化量。
Power P = Fv for a force moving at velocity v in its direction.
功率：力在其方向上以速度 v 运动时 P = Fv。
Projectile motion under gravity (no air resistance):
Horizontal: x = uₓ t, Vertical: y = uᵧ t − ½ gt²
Time of flight t = 2u sinα / g, Range R = u² sin2α / g, Max height H = u² sin²α/(2g)
抛体运动（无空气阻力，重力作用）：
水平：x = uₓ t, 垂直：y = uᵧ t − ½ gt²
飞行时间 t = 2u sinα / g, 射程 R = u² sin2α / g, 最大高度 H = u² sin²α/(2g)
Velocity components: uₓ =
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