📚 AQA Maths: Formula Summary Handbook | AQA 数学：公式汇总手册
This handbook brings together the essential formulae you need for AQA A-Level Mathematics, covering pure, statistics and mechanics. Use it as a quick reference to reinforce your memory and build confidence before the exam.
这本手册汇总了 AQA A-Level 数学所需的全部核心公式，涵盖纯数学、统计和力学。你可以把它当作快速查阅工具，在考前巩固记忆、增强信心。
1. Algebraic Laws and Quadratic Formula | 代数法则与二次公式
Fluency with algebraic manipulation is fundamental. The laws of indices and the quadratic formula appear in many topics, from simplifying expressions to solving equations.
熟练掌握代数操作是基础。指数法则和二次公式出现在从化简表达式到求解方程的众多主题中。
a^m × a^n = a^(m+n)
(a^m)^n = a^(mn)
a^m / a^n = a^(m–n) (a ≠ 0)
a^0 = 1 (a ≠ 0)
The quadratic formula gives the roots of ax² + bx + c = 0. Remember to set the equation to zero first and check the discriminant b² – 4ac.
二次公式给出一元二次方程 ax² + bx + c = 0 的根。务必先将方程化为标准形式，并判断判别式 b² – 4ac 的值。
x = [–b ± √(b² – 4ac)] / (2a)
2. Coordinate Geometry and Straight Lines | 坐标几何与直线
Working with points, gradients and equations of lines is essential for both pure mathematics and mechanics problems. These formulae let you move easily between geometric and algebraic descriptions.
处理点、斜率和直线方程对纯数学和力学问题都至关重要。这些公式能让你在几何描述和代数表达之间轻松转换。
Distance: d = √[(x₂ – x₁)² + (y₂ – y₁)²]
Midpoint: M = ((x₁ + x₂)/2 , (y₁ + y₂)/2)
Gradient: m = (y₂ – y₁) / (x₂ – x₁)
Equation: y – y₁ = m(x – x₁) or y = mx + c
Parallel lines share the same gradient. For perpendicular lines, the product of their gradients equals –1, provided neither is vertical.
平行直线斜率相等。垂直直线的斜率乘积为 –1，前提是两直线均不垂直于 x 轴。
3. Sequences and Series: Arithmetic and Geometric | 数列与级数：等差与等比
Arithmetic and geometric sequences form recurring AQA exam questions. You must know how to find the nth term and sum of n terms, as well as the sum to infinity for convergent geometric series.
等差和等比数列是 AQA 考试中的常见题型。你必须掌握如何求第 n 项和前 n 项和，以及收敛等比级数的无穷和。
Arithmetic: uₙ = a + (n – 1)d
Sum: Sₙ = n/2 [2a + (n – 1)d] = n/2 (a + l)
Geometric: uₙ = ar^(n – 1)
Sum: Sₙ = a(1 – rⁿ) / (1 – r) for r ≠ 1
Sum to infinity: S∞ = a / (1 – r) (|r| < 1)
Always check that the series is geometric before applying the sum to infinity formula, and confirm that |r| is strictly less than 1.
使用无穷和公式前，务必确认该级数为等比级数且 |r| 严格小于 1。
4. Trigonometry: Basic Identities and Equations | 三角学：基本恒等式与方程
Trigonometric identities underpin the solution of equations and the manipulation of expressions involving sine, cosine and tangent. Both degrees and radians appear on the AQA specification.
三角恒等式是求解方程和化简表达式的基础。AQA 的考试规格中同时涉及角度制和弧度制。
sin²θ + cos²θ ≡ 1
tanθ ≡ sinθ / cosθ
Sine rule: a / sin A = b / sin B = c / sin C
Cosine rule: a² = b² + c² – 2bc cos A
Area of triangle: ½ ab sin C
Radians: π rad = 180°, arc length s = rθ, sector area A = ½ r²θ
The double angle formulae are particularly useful in integration and solving trigonometric equations.
倍角公式在积分和求解三角方程中尤其有用。
sin2θ = 2 sinθ cosθ
cos2θ = cos²θ – sin²θ = 2cos²θ – 1 = 1 – 2sin²θ
5. Exponentials and Logarithms | 指数与对数
Logarithms are the inverses of exponentials. You need to be able to switch between the two forms and apply the log laws to solve equations and differentiate/integrate.
对数是指数的逆运算。你需要能在两种形式间灵活转换，并运用对数法则来解方程以及进行微积分运算。
a^x = e^(x ln a)
logₐ x = ln x / ln a
logₐ(xy) = logₐ x + logₐ y
logₐ(x / y) = logₐ x – logₐ y
logₐ(x^k) = k logₐ x
When differentiating, remember that e^x and ln x give the simplest results, while other bases require an extra ln factor.
求导时，请记住 e^x 和 ln x 会得出最简洁的结果，以其他数为底时会多出一个 ln 系数。
d/dx (e^x) = e^x
d/dx (ln x) = 1/x
6. Differentiation: Rules and Derivatives | 微分：求导法则与导数
Differentiation allows you to find gradients, rates of change and stationary points. Master the basic rules and standard derivatives before tackling chain, product and quotient applications.
微分能让你求出斜率、变化率以及驻点。在处理链式、乘积和商法则之前，先掌握基本法则和常见函数的导数。
Power rule: d/dx (xⁿ) = n xⁿ⁻¹
Constant multiple: d/dx [k f(x)] = k f ‘(x)
Sum/Difference: d/dx [f(x) ± g(x)] = f ‘(x) ± g'(x)
Product rule: d/dx (u v) = u’ v + u v’
Quotient rule: d/dx (u / v) = (u’ v – u v’) / v²
Chain rule: dy/dx = dy/du × du/dx
The derivatives of trigonometric and exponential functions must be memorised accurately.
三角和指数函数的导数必须准确记忆。
d/dx (sin x) = cos x
d/dx (cos x) = – sin x
d/dx (tan x) = sec² x
d/dx (a^x) = a^x ln a
If a curve is defined parametrically by x = f(t) and y = g(t), the gradient is found using dy/dx = (dy/dt) / (dx/dt).
如果曲线由参数方程 x = f(t)、y = g(t) 定义，斜率由 dy/dx = (dy/dt) / (dx/dt) 求得。
7. Integration: Techniques and Standard Integrals | 积分：技巧与标准积分
Integration is the reverse of differentiation. You will need standard results, the methods of substitution and integration by parts, and the ability to apply integrals to areas and volumes.
积分是微分的逆运算。你需要掌握标准结果、换元法和分部积分法，并能将积分应用于面积和体积的计算。
∫ xⁿ dx = xⁿ⁺¹ / (n+1) + C (n ≠ –1)
∫ (1/x) dx = ln |x| + C
∫ e^x dx = e^x + C
∫ sin x dx = – cos x + C
∫ cos x dx = sin x + C
∫ sec² x dx = tan x + C
Integration by parts reverses the product rule: ∫ u (dv/dx) dx = uv – ∫ v (du/dx) dx. Choose u and dv carefully to simplify the integral.
分部积分法源于乘积法则的反向运用：∫ u (dv/dx) dx = uv – ∫ v (du/dx) dx。合理选择 u 和 dv 能简化积分。
Substitution: ∫ f(g(x)) g'(x) dx = ∫ f(u) du, where u = g(x)
A definite integral from a to b is evaluated as F(b) – F(a), where F'(x) = f(x). The area between a curve and the x‑axis is given by ∫_a^b y dx.
从 a 到 b 的定积分由 F(b) – F(a) 计算，其中 F'(x) = f(x)。曲线与 x 轴之间的面积表示为 ∫_a^b y dx。
Volume of revolution about x-axis: V = π ∫_a^b y² dx
8. Vectors in 2D and 3D | 二维与三维向量
Vectors describe both magnitude and direction. The AQA course covers vector addition, scalar multiplication, the dot product and vector equations of lines.
向量既可以描述大小也能表示方向。AQA 课程涵盖向量加法、数乘、点积以及直线方程的向量形式。
2D vector: v = x i + y j, |v| = √(x² + y²)
3D vector: v = x i + y j + z k, |v| = √(x² + y² + z²)
Dot product: a · b = |a||b| cos θ = x₁x₂ + y₁y₂ (+ z₁z₂ in 3D)
Angle: cos θ = (a · b) / (|a||b|)
Two vectors are perpendicular if and only if their dot product is zero. The vector equation of a line is r = a + t b, where a is a point on the line and b is a direction vector.
两个向量垂直的充要条件是点积为零。直线的向量方程为 r = a + t b，
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导Cancel reply