📚 A-Level AQA Mathematics: End-Term Revision Checklist | A-Level AQA 数学：期末复习提纲
A thorough revision checklist covering core pure mathematics, statistics and mechanics for the AQA A-level Mathematics (7357) specification. Use this guide to structure your revision, confirm key formulas and test your understanding of essential concepts.
一份全面覆盖 AQA A-level 数学（7357）核心纯数、统计与力学的复习提纲。用这份指南规划你的复习，巩固关键公式，并自测对各核心概念的理解。
1. Algebra & Functions Foundation | 代数与函数基础
Revise polynomial manipulation, the remainder and factor theorems, and how to compose and invert functions. Always check the domain and range of a transformed function.
复习多项式运算、余数定理与因式定理，以及如何对函数进行复合和求逆。始终注意变换后函数的定义域与值域。
The discriminant Δ = b² − 4ac of a quadratic ax² + bx + c = 0 determines the nature of its roots: Δ > 0 gives two distinct real roots, Δ = 0 gives one repeated root, and Δ < 0 gives no real roots.
二次方程 ax² + bx + c = 0 的判别式 Δ = b² − 4ac 决定根的性质：Δ > 0 有两个不等实根，Δ = 0 有一个重根，Δ < 0 无实根。
Manipulating inequalities: remember to reverse the inequality sign when multiplying or dividing by a negative number. Sketching quadratic and rational inequalities on a number line helps avoid mistakes.
处理不等式：当乘以或除以一个负数时，切记反转不等号方向。在数轴上画出二次不等式和有理不等式的示意图有助于避免错误。
Remainder theorem: f(x) ÷ (x − a) gives remainder f(a)
2. Coordinate Geometry & Conic Sections | 坐标几何与圆锥曲线
Work confidently with straight lines: y − y₁ = m(x − x₁), the distance formula √[(x₂ − x₁)² + (y₂ − y₁)²], and midpoint coordinates. Understand parallel and perpendicular gradients (m₁ m₂ = −1).
熟练处理直线：y − y₁ = m(x − x₁)，距离公式 √[(x₂ − x₁)² + (y₂ − y₁)²]，以及中点坐标。掌握平行与垂直斜率的关系（m₁ m₂ = −1）。
The equation of a circle with centre (a, b) and radius r is (x − a)² + (y − b)² = r². You may need to complete the square to find the centre and radius from a general equation.
圆心 (a, b)、半径为 r 的圆方程为 (x − a)² + (y − b)² = r²。你可能需要对方程进行配方，以便从一般式找出圆心和半径。
Intersection of a line and a circle often leads to a quadratic equation; use the discriminant to determine if the line is a tangent (Δ = 0), a secant (Δ > 0) or does not meet the circle (Δ < 0).
直线与圆相交常常导出一个二次方程；利用判别式判断直线是切线 (Δ = 0)、割线 (Δ > 0) 还是不相交 (Δ < 0)。
Circle: (x − a)² + (y − b)² = r²
3. Trigonometry | 三角学
Be fluent in both degrees and radians: π rad = 180°. Know exact values for sin, cos and tan at 0, π/6, π/4, π/3, π/2, etc. The graphs of y = sin x, y = cos x and y = tan x reveal symmetry and periodicity.
熟练掌握角度制与弧度制：π rad = 180°。熟记 0, π/6, π/4, π/3, π/2 等的 sin, cos, tan 精确值。y = sin x, y = cos x 和 y = tan x 的图像展示了对称性与周期性。
Key identities include sin²θ + cos²θ = 1 and tan θ = sin θ / cos θ. Double-angle formulas: sin 2θ = 2 sin θ cos θ, cos 2θ = cos²θ − sin²θ = 2 cos²θ − 1 = 1 − 2 sin²θ.
关键恒等式包括 sin²θ + cos²θ = 1 以及 tan θ = sin θ / cos θ。倍角公式：sin 2θ = 2 sin θ cos θ，cos 2θ = cos²θ − sin²θ = 2 cos²θ − 1 = 1 − 2 sin²θ。
Solving trigonometric equations: always consider the given interval and all quadrants using CAST. The sine rule a/sin A = b/sin B = c/sin C and cosine rule a² = b² + c² − 2bc cos A are essential for non-right-angled triangles.
解三角方程：始终结合给定区间，并借助 CAST 规则考虑所有象限。正弦定理 a/sin A = b/sin B = c/sin C 和余弦定理 a² = b² + c² − 2bc cos A 对非直角三角形至关重要。
| θ (rad) | sin θ | cos θ | tan θ |
|---|---|---|---|
| 0 | 0 | 1 | 0 |
| π/6 | 1/2 | √3/2 | 1/√3 |
| π/4 | 1/√2 | 1/√2 | 1 |
| π/3 | √3/2 | 1/2 | √3 |
| π/2 | 1 | 0 | undefined |
4. Exponentials & Logarithms | 指数与对数函数
The natural exponential function y = eˣ and the natural logarithm y = ln x are inverse functions. Recall that ln(eˣ) = x and e^(ln x) = x for x > 0.
自然指数函数 y = eˣ 与自然对数函数 y = ln x 互为反函数。记住对于 x > 0，有 ln(eˣ) = x 且 e^(ln x) = x。
Laws of logarithms: logₐ (xy) = logₐ x + logₐ y; logₐ (x/y) = logₐ x − logₐ y; logₐ (xⁿ) = n logₐ x. These are used to solve exponential equations and to linearise data.
对数运算定律：logₐ (xy) = logₐ x + logₐ y；logₐ (x/y) = logₐ x − logₐ y；logₐ (xⁿ) = n logₐ x。这些定律用于求解指数方程并对数据进行线性化。
When modelling with exponentials, the forms y = A eᵏˣ and y = A bˣ are interchangeable. Using logarithms transforms an exponential graph into a straight line, allowing constants to be found from gradient and intercept.
用指数函数建模时，形式 y = A eᵏˣ 与 y = A bˣ 可以相互转换。利用对数可将指数图像转化为直线，从而通过斜率和截距求出常数。
e^(ln x) = x ; ln(eˣ) = x
5. Differentiation Techniques & Applications | 微分技巧与应用
Master standard derivatives: d/dx (xⁿ) = n xⁿ⁻¹, d/dx (eˣ) = eˣ, d/dx (ln x) = 1/x, d/dx (sin x) = cos x, d/dx (cos x) = −sin x, d/dx (tan x) = sec² x.
掌握基本导数：d/dx (xⁿ) = n xⁿ⁻¹，d/dx (eˣ) = eˣ，d/dx (ln x) = 1/x，d/dx (sin x) = cos x，d/dx (cos x) = −sin x，d/dx (tan x) = sec² x。
Apply the chain rule (dy/dx = dy/du × du/dx), product rule and quotient rule accurately. Implicit differentiation handles equations where y is not explicitly expressed, such as x² + y² = r².
准确运用链式法则 (dy/dx = dy/du × du/dx)、乘积法则和商法则。隐函数微分用于处理无法将 y 显式表达的方程，例如 x² + y² = r²。
Find equations of tangents and normals at a point. Use first and second derivatives to locate stationary points and classify maxima, minima and points of inflection. Optimisation problems often involve modelling a real-world quantity and differentiating to find an extremum.
求一点处的切线与法线方程。利用一阶导数和二阶导数找出驻点，并判断极大值、极小值和拐点。优化问题通常需对实际量建模，并通过微分求极值。
d/dx [ln(x)] = 1/x
6. Integration Techniques & Applications | 积分技巧与应用
Integration is the reverse of differentiation. Know the basic integrals: ∫ xⁿ dx = xⁿ⁺¹/(n+1) + c (n ≠ −1), ∫ 1/x dx = ln|x| + c, ∫ eˣ dx = eˣ + c, ∫ sin x dx = −cos x + c, ∫ cos x dx = sin x + c.
积分是微分的逆运算。熟记基本积分：∫ xⁿ dx = xⁿ⁺¹/(n+1) + c (n ≠ −1)，∫ 1/x dx = ln|x| + c，∫ eˣ dx = eˣ + c，∫ sin x dx = −cos x + c，∫ cos x dx = sin x + c。
Definite integrals compute the area between a curve and the x-axis. Area between two curves requires careful subtraction: ∫ [f(x) − g(x)] dx over the interval. Remember to take absolute values if the curve crosses the axis.
定积分计算曲线与 x 轴之间的面积。两曲线间的面积需仔细相减：在区间上 ∫ [f(x) − g(x)] dx。若曲线穿越 x 轴，记得对面积取绝对值。
Techniques include integration by substitution and integration by parts: ∫ u dv = u v − ∫ v du. Differential equations of the form dy/dx = f(x)g(y) are solved by separating variables and integrating.
积分技巧包括换元积分法和分部积分法：∫ u dv = u v − ∫ v du。形如 dy/dx = f(x)g(y) 的微分方程可通过分离变量后积分求解。
∫ u dv = u v − ∫ v du
7. Sequences & Binomial Expansion | 数列与二项式展开
Arithmetic sequences: nth term uₙ = a + (n−1)d, sum Sₙ = n/2 (2a + (n−1)d) = n/2 (a + l). Geometric sequences: uₙ = arⁿ⁻¹, finite sum Sₙ = a(1 − rⁿ)/(1 − r) for r ≠ 1; infinite sum exists and equals a/(1 − r) when |r| < 1.
等差数列：第 n 项 uₙ = a + (n−1)d，求和 Sₙ = n/2 (2a + (n−1)d) = n/2 (a + l)。等比数列：uₙ = arⁿ⁻¹，有限项求和 Sₙ = a(1 − rⁿ)/(1 − r) (r ≠ 1)；当 |r| < 1 时无穷项和存在且为 a/(1 − r)。
The binomial expansion (1 + x)ⁿ = 1 + n x + [n(n−1)/2!] x² + … is valid for |x| < 1 when n is not a positive integer. The general term uses nCr = n! / [r! (n−r)!]. Applications include finding approximations and estimating square roots.
二项式展开 (1 + x)ⁿ = 1 + n x + [n(n−1)/2!] x² + … 在 n 不为正整数且 |x| < 1 时成立。通项用到 nCr = n! / [r! (n−r)!]。应用包括求近似值和估算平方根。
Sigma notation Σ is used to write series concisely. You may be required to find the sum of a sequence from given terms or to reconstruct the polynomial form.
Σ 符号用于简洁地书写级数。可能会要求你根据给定项求序列的和，或还原其多项式形式。
Sₙ = n/2 (2a + (n−1)d)
8. Vectors | 向量
Vectors represent both magnitude and direction. In 2D, vectors can be written as ai + bj or as column vectors. The magnitude |v| is √(a² + b²).
向量表示大小和方向。在二维中，向量可写作 ai + bj 或列向量。模长 |v| 为 √(a² + b²)。
The scalar (dot) product a·b = |a||b| cos θ is used to find the angle between two vectors. For coordinates a·b = a₁b₁ + a₂b₂ + a₃b₃.
数量积（点积）a·b = |a||b| cos θ 用于求两向量间的夹角。在坐标形式下 a·b = a₁b₁ + a₂b₂ + a₃b₃。
Vector equation of a line: r = a + λ b, where a is a point on the line and b is the direction vector. Two lines intersect if their position vectors can be equated for some values of the parameters.
直线的向量方程：r = a + λ b，其中 a 是直线上一点的位置向量，b 是方向向量。若存在参数值使两条直线的位置向量相等，则两直线相交。
a·b = a₁b₁ + a₂b₂ + a₃b₃
9. Statistics: Data & Probability | 统计：数据与概率
Interpret and construct box plots, histograms, cumulative frequency diagrams and scatter graphs. Outliers can be identified using the interquartile range (IQR) rule: Q1 − 1.5×IQR and Q3 + 1.5×IQR.
会解读和绘制箱形图、直方图、累积频率图及散点图。离群值可通过四分位距 (IQR) 法则识别：Q1 − 1.5×IQR 和 Q3 + 1.5×IQR。
Measures of central tendency (mean, median, mode) and measures of spread (variance, standard deviation) summarise data. The formula for variance is σ² = Σ(x − x̄)² / n or using Σx²/n − x̄².
集中量数（平均数、中位数、众数）和离散量数（方差、标准差）可以概括数据。方差公式为 σ² = Σ(x − x̄)² / n 或使用 Σx²/n − x̄²。
Probability rules: P(A ∪ B) = P(A) + P(B) − P(A ∩ B). Conditional probability P(A|B) = P(A ∩ B)/P(B). Tree diagrams help handle sequential events and rely on multiplying along branches.
概率法则：P(A ∪ B) = P(A) + P(B) − P(A ∩ B)。条件概率 P(A|B) = P(A ∩ B)/P(B)。树状图有助于处理序贯事件，并依赖于沿分支相乘。
Variance: σ² = Σ(x − x̄)² / n
10. Statistical Distributions & Hypothesis Testing | 统计分布与假设检验
Binomial distribution X ~ B(n, p): P(X = r) = nCr pʳ (1−p)ⁿ⁻ʳ. Mean = np, variance = np(1−p). Use binomial cumulative tables or calculator functions where appropriate.
二项分布 X ~ B(n, p)：P(X = r) = nCr pʳ (1−p)ⁿ⁻ʳ。均值 = np，方差 = np(1−p)。适当时使用二项累积分布表或计算器函数。
Normal distribution X ~ N(μ, σ²): standardise using Z = (X − μ)/σ. The total area under the standard normal curve is 1; use tables to find probabilities. You may need to find unknown μ or σ given probabilities.
正态分布 X ~ N(μ, σ²)：用 Z = (X − μ)/σ 标准化。标准正态曲线下总面积为 1；用表求概率。可能会要求在给定概率下求未知的 μ 或 σ。
Hypothesis testing: state null and alternative hypotheses (H₀ and H₁), choose a significance level, identify the test statistic and critical region, and interpret the p-value. A result is statistically significant if p < significance level or if the test statistic falls in the critical region.
假设检验：陈述原假设和备择假设 (H₀ 与 H₁)，选择显著性水平，确定检验统计量和拒绝域，并解读 p 值。若 p < 显著性水平或检验统计量落入拒绝域，则结果具有统计显著性。
Z = (X − μ) / σ
11. Mechanics: Kinematics & Linear Motion | 力学：运动学与直线运动
For constant acceleration, use the suvat equations: v = u + at, s = ut + ½ at², s = ½ (u + v)t, v² = u² + 2as, s = vt − ½ at². Define positive direction clearly before solving.
对于匀加速运动，使用 suvat 方程：v = u + at, s = ut + ½ at², s = ½ (u + v)t, v² = u² + 2as, s = vt − ½ at²。求解前须清晰定义正方向。
Displacement–time and velocity–time graphs: the gradient of a displacement–time graph gives velocity; the gradient of a velocity–time graph gives acceleration. Area under a velocity–time graph gives displacement.
位移–时间图和速度–时间图：位移–时间图的斜率表示速度；速度–时间图的斜率表示加速度。速度–时间图下的面积表示位移。
Projectiles: treat horizontal and vertical motion separately. Horizontal velocity is constant; vertical motion uses suvat with a = ±g (9.8 m s⁻²). The time of flight is determined by the vertical motion.
抛体运动：将水平与竖直运动分开处理。水平速度恒定；竖直运动使用加速度 a = ±g (9.8 m s⁻²) 的 suvat 方程。飞行时间由竖直运动决定。
v² = u² + 2as
12. Mechanics: Newton’s Laws & Equilibrium | 力学：牛顿定律与平衡
Newton’s second law: F = ma, where F is the resultant force. Resolve forces into components along perpendicular directions, often parallel and perpendicular to an inclined plane.
牛顿第二定律：F = ma，其中 F 为合力。通常将力沿相互垂直的方向（如平行和垂直于斜面）进行分解。
Friction: F_friction ≤ μ R, where R is the normal reaction and μ is the coefficient of friction. In limiting equilibrium, use F = μ R. For connected particles, write equations of motion for each mass and link via tension in inextensible strings.
摩擦力：F_friction ≤ μ R，其中 R 为法向反作用力，μ 为摩擦系数。在极限平衡时使用 F = μ R。对于连接质点，分别对每个质量列出运动方程，并通过不可伸长的绳子上的张力关联。
Moments: the turning effect of a force about a point is force × perpendicular distance. For equilibrium, the sum of clockwise moments equals the sum of anticlockwise moments, and resultant force is zero. Draw clear force diagrams.
力矩：力对一点的转动效应为力 × 垂直距离。平衡时，顺时针力矩之和等于逆时针力矩之和，且合力为零。务必画出清晰的受力图。
F = ma
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