AQA · A-Level

数学

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A · 证明

Deduction, contradiction and induction: A single valid case that disproves a universal claim.
Complete cases and counterexamples: A proof that checks every possibility in a finite, complete set of cases.
Irrational roots and infinitely many primes: A proof that assumes the negation of a claim and derives an impossibility.

反例: 一个有效的反例，用以推翻全称命题。
穷举法证明: 一种在有限且完备的案例集合中检查所有可能性的证明。
反证法: 一种假设命题的否定并推导出矛盾的证明。

B · 代数与函数

Indices, surds and standard form: The power to which a base is raised.
Quadratics and inequalities: The quantity b²-4ac that determines the real roots of ax²+bx+c=0.
Domains, inverses and composition: The set of allowed inputs to a function.
Polynomial division, remainders and factors: The statement that x−a is a factor of polynomial P(x) exactly when P(a)=0.
Algebraic fractions and excluded values: An input for which an original denominator or divisor is zero.
Partial fractions with distinct linear factors: A simpler fraction in a sum that equals a rational expression on its original domain.
Repeated linear factors in partial fractions: A denominator factor occurring more than once, requiring a term for each power.
Linear and quadratic simultaneous equations: An ordered pair that satisfies every equation in a system at the same time.
Rational indices and real domains: An exponent written as a fraction of integers.
Surds and conjugate denominators: A paired expression with the sign between two terms reversed.
Completing the square and quadratic structure: A quadratic written as a multiple of a squared bracket plus a constant.
Equations quadratic in another expression: Replacing a repeated expression with a new variable to reveal a simpler equation.
Inequalities, fractions and solution sets: The values common to two solution sets, corresponding to AND.
Regions between linear and quadratic boundaries: The line or curve separating points that satisfy an inequality from those that do not.
Modulus graphs and piecewise solutions: The nonnegative distance of a real number from zero.
Reciprocal graphs and asymptotes: A line that a curve approaches in a limiting direction.
Inverse graphs and restricted branches: A function for which different inputs always give different outputs.
Composite functions and intermediate domains: Applying one function to the output of another, in a specified order.
Combined graph transformations and order: The factor multiplying the original horizontal coordinates when a graph is stretched or compressed.
Fitting and refining a function model: An observed output minus the output predicted by a model.
Polynomial sketches, repeated roots and signs: The number of times a root’s linear factor appears in a polynomial.

索引: 底数被提升的幂次。
判别式: b²-4ac 是决定 ax²+bx+c=0 实数根的判别式。
定义域: 函数的允许输入集合。
因式定理: 断言 x−a 是多项式 P(x) 的因式，当且仅当 P(a)=0。
使分母为零的值: 使原始分母或除数为零的输入。
部分分式: 和中的一个更简分式，其值等于原定义域上的有理表达式。
重因式: 出现多次的分母因式，需为每个幂次设置一项。
联立方程组解: 同时满足方程组中所有方程的有序数对。
有理指数: 以整数分数形式写出的指数。
共轭: 两个项之间符号相反的配对表达式。
配方法: 写成平方括号倍数加常数的二次式。
替换: 用新变量替换重复表达式，以揭示更简单的方程。
交点: 两个解集的公共值，对应逻辑“且”。
边界: 分隔满足不等式的点与不满足不等式的点的直线或曲线。
模: 实数到零的非负距离。
渐近线: 曲线在极限方向上趋近的直线或曲线。
一一映射时: 对于不同输入总是产生不同输出的函数。
构图: 按指定顺序将一个函数的输出作为另一个函数的输入。
水平伸缩因子: 当图像拉伸或压缩时，乘在原水平坐标上的因子。
剩余: 观测输出减去模型预测的输出。
重数: 多项式中根的线性因子出现的次数。

C · 解析几何

Coordinate geometry and tangents: The change in y divided by the corresponding change in x.
Parametric curves, domains and direction: A variable that determines both coordinates of a point.
Parametric modelling in motion and design: A model expressing two quantities using a common variable and stated assumptions.
General-form lines and contextual models: The form ax+by+c=0 for a straight line, with a and b not both zero.
Shifted circles, chords and tangents: The point at equal distance from every point on a circle.

斜率: y的变化量除以相应的x变化量。
参数: 确定点两个坐标的变量。
参数模型: 使用公共变量和既定假设表达两个量的模型。
一般形式: 直线形式 ax+by+c=0，其中 a 和 b 不全为零。
心: 到圆上每一点距离相等的点。

D · 数列与级数

Sequences, series and recurrence: The constant multiplier between consecutive terms of a geometric sequence.
Binomial expansion and valid approximations: The number of ways to choose a specified number of objects from a set.
Factorials, combinations and binomial coefficients: A selection in which order does not distinguish different choices.
Increasing, decreasing and periodic recurrences: A sequence whose terms repeat after a fixed positive number of steps.
Sigma notation, indices and changed limits: The variable that takes each integer value between a sum’s stated limits.
Arithmetic series and inverse problems: The fixed amount added to each term to obtain the next.
Geometric sums, alternating ratios and convergence: The fixed factor multiplying each term to obtain the next.
Repeated-deposit sequence models and timing: An amount added to an account at a stated time.

公比: 将每一项乘以 ⟦number⟧ 以获得下一项的固定因子。
二项系数: 从集合中选取指定数量对象的方法数。
合在一起: 顺序不区分不同选择的选取方式。
周期数列: 在固定正整数步数后重复出现的数列。
求和指标: 取求和指定界限间每个整数值的变量。
公差: 为获得下一项而加到每一项上的固定数值。
存款: 在指定时间存入账户的金额。

E · 三角学

Right triangles and non-right triangles: The side opposite the right angle in a right-angled triangle.
Radians, identities and trigonometric equations: The angle subtended by an arc whose length equals the radius.
Radian measure, arcs, sectors and segments: The region bounded by two radii and their circular arc.
Small-angle approximations and their limits: A nearby simpler value used with a stated accuracy limit.
All-angle definitions, exact values and periodic graphs: The circle of radius one centred at the coordinate origin.
Reciprocal trigonometric functions and excluded angles: The reciprocal of cosine wherever cosine is nonzero.
Inverse trigonometric branches and principal values: The single inverse output chosen from a stated restricted interval.
Reciprocal identities and one-sided proof chains: An equality true for every input in its stated common domain.
Compound-angle formulae and geometric proofs: An angle expressed as the sum or difference of two angles.
Double-angle identities, quadrant signs and squared forms: An angle equal to twice the original angle.
Auxiliary-angle forms, ranges and interval solutions: The nonnegative size of an oscillation about its centre line.
Quadratic trigonometric equations and complete root sets: A possible solution that still needs checking in the original equation and interval.
Multiple-angle equations and transformed intervals: The interval for a substituted angle after applying the same input change to its endpoints.
Trigonometric components and motion models: The signed projection of a vector along a chosen coordinate direction.
Sine rule, cosine rule and triangle area: The angle between the two named sides.
The ambiguous sine-rule case and valid triangle counts: Given triangle data that can produce two different valid triangles.
Key-point sketches and transformed tangent domains: A line approached by a graph while its values grow without bound or tend toward a limit.

斜边: 直角三角形中直角所对的边。
弧度: 弧长等于半径所对的圆心角。
扇形: 由两条半径及其圆弧所围成的区域。
近似: 在指定精度限制下使用的邻近更简单的值。
单位圆: 以坐标原点为圆心、半径为一的圆。
割线: 余弦非零处的倒数。
主值: 从给定受限区间中选取的单个逆输出值。
身份认同: 在其给定的公共定义域内对每个输入均成立的等式。
倍角公式: 表示为两个角之和或之差的角。
倍角: 等于原角两倍的角。
振幅: 关于其中心线振荡的非负大小。
候选人: 一个仍需在原方程和区间中进行检验的可能解。
⟦number⟧变换区间: 对端点施加相同输入变换后，代入角的取值区间。
分量: 向量沿选定坐标方向的有符号投影。
夹角: 两条指定边之间的夹角。
二义性情况 / 模糊情形: 给定三角形数据可生成两种不同有效三角形的情况。
渐近线: 当函数值无界增长或趋向某极限时，曲线趋近的直线。

F · 指数函数与对数

Exponentials, logarithms and modelling: The time for a decaying quantity to fall to half its initial value.
Exponential graphs and logarithmic inverses: The exponent to which a stated valid base must be raised to produce a positive input.
Logarithm laws, negative powers and equation domains: The input inside a logarithm, which must be positive for a real logarithm.
Logarithmic graphs for power and exponential fits: The transformed vertical value when the chosen horizontal variable is zero.
Proportional rates, half-life and model limits: A rate of change equal to a constant multiplied by the current amount.

半衰期: 衰减量降至初始值一半所需的时间。
对数: 使给定有效底数产生正输入的幂次。
参数: 对数内的输入值，实数对数要求其必须为正。
截距: 当所选水平变量为零时的变换后垂直值。
⟦number⟧比例速率: 变化率等于常数乘以当前量的情形。

G · 微分

Derivatives and stationary points: The instantaneous rate of change, also the gradient of a tangent.
Chain, product, quotient and implicit differentiation: The rule that multiplies the outer derivative by the inner derivative for a composite function.
Secants, limits and first-principles power derivatives: The change in function output divided by a nonzero change in input.
First-principles sine and cosine derivatives: A small-angle limit used with angles measured in radians.
Gradient graphs, concavity and inflection tests: A point where a continuous curve changes its direction of concavity.
Rational-power derivatives and valid domains: An exponent that can be written as a fraction of integers, interpreted with a valid real root.
Exponential, logarithmic and trigonometric derivatives: The derivative of the changing input inside a composite function.
Product and quotient rules with retained exclusions: The rule for differentiating a ratio of two changing functions when the denominator is nonzero.
Chain rules for nested functions and restricted inputs: The outer derivative evaluated at the inner output, multiplied by the derivative of that inner input.
Inverse derivatives at corresponding points: The rate of the inverse output with respect to its input, evaluated at the corresponding original point.
Connected rates, geometry and time units: Rates linked by a relationship between quantities changing with the same time variable.
Implicit first derivatives and tangent domains: An equation relating x and y without isolating one as an explicit function of the other.
Parametric first derivatives and vertical tangents: The ratio of the y-rate to the x-rate along a parametrised curve, when the x-rate is nonzero.
Tangent and normal equations and gradient conditions: A line through the curve point perpendicular to its local tangent.
Optimisation with constraints, boundaries and rejected roots: The allowed input values after the original physical or mathematical constraints are applied.
Constructing differential equations from rate laws: A relationship involving an unknown function and one or more of its derivatives.

导数: 瞬时变化率，亦即切线的斜率。
链法则: 对于复合函数，将外层函数的导数乘以内层函数的导数的规则。
差商: 函数输出的变化量除以非零的输入变化量。
⟦number⟧弧度限制: 用于弧度制角度的小角度极限。
拐点: 连续曲线改变其凹凸性方向的点。
⟦number⟧有理指数: 可写为整数分数的指数，并解释为有效的实根。
⟦number⟧内部速率: 复合函数内部变化输入量的导数。
商法则: 当分母不为零时，两个变化函数之比的求导法则。
⟦number⟧复合导数: 在外层输出处计算外层导数，再乘以该内层输入量的导数。
⟦number⟧反函数导数: 逆函数的输出相对于其输入的变化率，在对应的原始点上计算。
⟦number⟧关联变化率: 由随同一时间变量变化的量之间的关系所联系的比率。
⟦number⟧隐式关系: 关联x和y但不将其中一个表示为另一个的显式函数的方程。
number⟧参数梯度: 沿参数化曲线，当x的变化率非零时，y的变化率与x的变化率之比。
⟦number⟧法线: 通过曲线上某点且垂直于该点局部切线的直线。
⟦number⟧可行域: 应用原始物理或数学约束后允许的输入值。
微分方程: 涉及未知函数及其一个或多个导数的关系。

H · 积分

Integrals, area and accumulation: A function whose derivative equals the given integrand.
Substitution, parts and partial fractions: An integration method based on the derivative of a product.
Differential equations and numerical solutions: A specified value that selects a particular solution from a family.
The Fundamental Theorem and accumulated change: A function whose value is the signed integral from a fixed starting point to a variable endpoint.
Integration as the limit of rectangle sums: A sum of signed rectangle contributions that approaches an integral as the partition becomes finer.
Elementary antiderivatives, coefficients and domains: A function whose derivative is the specified integrand on the stated interval.

原函数: 在指定区间上其导数为指定被积函数的函数。
分部积分: 一种基于乘积导数的积分方法。
初始条件: 从一族解中选定特定解的指定值。
累积函数: 其值为从固定起点到可变终点的定号积分的函数。
黎曼和: 随着划分变细，定号矩形贡献之和趋近于积分。

I · 数值方法

Root finding and numerical integration: A repeated update in which each new approximation is calculated from the previous one.

迭代的示例: 一种重复更新过程，其中每个新近似值均从前一个近似值计算得出。

J · 向量

Vectors and transformation geometry: The vector sum representing the combined displacement or force.
Reflections, rotations and enlargements: The point from which each point's displacement is multiplied by a scale factor.

合力: 表示合位移或合力的向量和。
放大中心: 每个点的位移按缩放因子进行乘法的原点。

K · 统计抽样

Sampling and a large data set: The list or population definition from which a sample is selected.

抽样框: 从中选取样本的列表或总体定义。

L · 数据呈现与解读

Data summaries, histograms and interpretation: Frequency divided by class width, used as histogram height.
Regression, financial models and residuals: The observed value minus the value predicted by a fitted model.

频率密度: 频率除以组距，用作直方图的高度。
剩余: 观测值减去拟合模型预测的值。

M · 概率

Probability, trees and conditional reasoning: The probability of an event after restricting the sample space to a stated condition.

条件概率: 将样本空间限制在特定条件后某事件发生的概率。

N · 统计分布

Binomial, normal and Poisson models: The probability-weighted mean of a random variable.

期望值: 随机变量的概率加权均值。

O · 统计假设检验

Hypothesis testing and contextual conclusions: The chosen probability threshold for rejecting a null hypothesis.

显著性水平: 用于拒绝零假设所选定的概率阈值。

P · 力学中的量与单位

Accuracy, bounds and compound measures: The smallest possible value consistent with a stated rounding rule.
Kinematics and Newton's laws: The vector sum of all external forces acting on the modelled object.

下界: 符合给定舍入规则的最小可能值。
合力: 作用于被建模对象的所有外力的向量和。

Q · 运动学

Kinematics and Newton's laws: The vector sum of all external forces acting on the modelled object.
Integrals, area and accumulation: A function whose derivative equals the given integrand.

合力: 作用于被建模对象的所有外力的向量和。
原函数: 其导数等于给定被积函数的函数。

R · 力与牛顿定律

Kinematics and Newton's laws: The vector sum of all external forces acting on the modelled object.

合力: 作用于被建模对象的所有外力的向量和。

S · 力矩

Moments, equilibrium and centres of mass: Force multiplied by the perpendicular distance from its line of action to a pivot.

时刻: 力与其作用线到支点的垂直距离之积。

备考指南

Paper 1 covers A–I. Paper 2 adds J and P–S to Paper 1 content. Paper 3 adds K–O to Paper 1 content.
All students study pure, mechanics and statistics. Use the AQA large data set and the 7357 formula booklet.
Calculators need statistical distribution functions; symbolic algebra/differentiation/integration facilities are prohibited under the examination rules.

仍需教学覆盖内容

Official question-bank boundaries, scheme alignment and all objective-level teaching coverage require the recorded review; no practice registry promotion.
The authored diagnostic assessments are not full-length qualification mocks.