📚 A-Level Mathematics: Introduction and Revision | A-Level 数学：导论与复习
Welcome to the opening chapter of the TutorHao AQA A-Level Mathematics revision series. This guide surveys the entire course structure, refreshes the algebra and coordinate geometry skills that appear in every paper, and presents a tried-and-tested revision method. Whether you are just starting Year 12 or sitting the final exams, this introduction establishes exactly where you are and what lies ahead.
欢迎阅读 TutorHao AQA A-Level 数学复习系列的开篇章节。本文纵览整个课程结构，重温每张试卷都会考查的代数与坐标几何技能，并介绍一套经过验证的复习方法。无论你刚刚开始 AS 阶段，还是即将参加最终考试，本导论都将帮你准确定位当前水平，并明确前行的方向。
1. Course Overview | 课程概览
The AQA A-Level Mathematics qualification (specification code 7357) is a linear course assessed entirely at the end of two years. The content is divided into two strands: Pure Mathematics and Applied Mathematics. Pure Mathematics accounts for two thirds of the course, while Applied Mathematics (Statistics and Mechanics) accounts for one third.
AQA A-Level 数学课程（规范编号 7357）为线性课程，全部考核在两年学习结束后进行。课程内容分为两大板块：纯数学与应用数学。纯数学占课程总量的三分之二，应用数学（统计与力学）占三分之一。
You will sit three two-hour papers, each marked out of 100:
你将参加三场时长两小时的考试，每场满分 100 分：
- Paper 1: Pure Mathematics 1 — 纯数学试卷一
- Paper 2: Pure Mathematics 2 — 纯数学试卷二
- Paper 3: Statistics and Mechanics — 统计与力学试卷
Papers 1 and 2 cover algebra, functions, trigonometry, calculus, sequences, exponentials, logarithms, and vectors. Paper 3 splits equally between statistical analysis (sampling, probability, hypothesis testing) and mechanics (kinematics, forces, Newton’s laws).
试卷一与试卷二覆盖代数、函数、三角学、微积分、数列、指数、对数与向量。试卷三平均分为统计分析（抽样、概率、假设检验）与力学（运动学、力、牛顿定律）两部分。
2. Assessment Objectives and Exam Structure | 考核目标与考试结构
AQA defines three assessment objectives (AOs) that decide how marks are allocated. AO1 tests mathematical knowledge and procedures (about 50% of total marks). AO2 tests reasoning, interpretation and communication (about 25%). AO3 tests problem-solving in unfamiliar contexts (about 25%).
AQA 规定了三个考核目标（AO），决定分数如何分配。AO1 考查数学知识与基本程序（约占总分 50%）。AO2 考查推理、解释与表达（约 25%）。AO3 考查在陌生情境中解决问题的能力（约 25%）。
Every exam paper begins with a formula booklet. You are expected to know which formula to select and when to apply it, rather than memorise every result blindly. However, core algebraic manipulations — expanding, factorising, completing the square, solving equations — must be automatic, because they are the scaffolding for all harder questions.
每份试卷开头都会附有公式手册。你需要具备判断选用哪个公式以及何时应用的能力，而不是盲目记忆所有结果。然而，核心代数运算——展开、因式分解、配方法、解方程——必须达到自动化的熟练程度，因为它们是解答一切难题的脚手架。
3. Algebra Essentials: Expanding and Factorising | 代数基础：展开与因式分解
Expanding brackets is the first skill to master. The distributive law states that a(b + c) = ab + ac. For two linear factors, apply each term of the first bracket to every term of the second:
展开括号是必须掌握的第一项技能。分配律指出 a(b + c) = ab + ac。对于两个线性因子，将第一个括号中的每一项与第二个括号中的每一项相乘：
(a + b)(c + d) = ac + ad + bc + bd
A classic example is (3x + 2)(x − 5). Expanding gives 3x² − 15x + 2x − 10 = 3x² − 13x − 10. The most common error is forgetting that −5 × 2 = −10 and that the x terms must be collected carefully.
典型例子为 (3x + 2)(x − 5)。展开得 3x² − 15x + 2x − 10 = 3x² − 13x − 10。最常见的错误是忘记 −5 × 2 = −10，或者没有仔细合并 x 项。
Factorising reverses this process. Always look for a common factor first, then apply standard results:
因式分解是上述过程的逆运算。务必先寻找公因子，再套用标准公式：
- Difference of two squares: a² − b² = (a + b)(a − b) — 平方差公式
- Perfect square: a² ± 2ab + b² = (a ± b)² — 完全平方公式
- Quadratic trinomial: x² + bx + c = (x + p)(x + q), where p + q = b and pq = c — 二次三项式
For a general quadratic ax² + bx + c, factorise by grouping or by inspection. If the quadratic cannot be factorised with integer coefficients, the quadratic formula or completing the square is used instead.
对于一般二次式 ax² + bx + c，可采用分组法或观察法分解。若二次式无法用整数系数分解，则改用求根公式或配方法。
4. Surds | 根式
Surds are irrational numbers expressed with a radical sign, such as √2, 3√5 or 2√7. AQA requires you to simplify surds, multiply and divide them, and rationalise denominators.
根式是用根号表示的无理数，如 √2、3√5 或 2√7。AQA 要求你能够化简根式、进行乘除运算，并将分母有理化。
The two fundamental rules are the product rule and the quotient rule for square roots:
两条基本法则为平方根的乘法法则与除法法则：
√(ab) = √a × √b, a/b inside radical: √(a/b) = √a / √b
To simplify √72, look for the largest square factor: √72 = √(36 × 2) = √36 × √2 = 6√2. Conversely, 5√20 = 5 × √(4 × 5) = 5 × 2√5 = 10√5.
化简 √72 时，寻找最大的平方因子：√72 = √(36 × 2) = √36 × √2 = 6√2。同理，5√20 = 5 × √(4 × 5) = 5 × 2√5 = 10√5。
Rationalising the denominator removes the radical from the denominator. For 1/√3, multiply numerator and denominator by √3 to obtain √3/3. For expressions like 1/(2 + √5), multiply by the conjugate 2 − √5:
有理化分母可去除分母中的根号。对于 1/√3，将分子分母同乘 √3 得 √3/3。对于形如 1/(2 + √5) 的式子，乘以共轭式 2 − √5：
1/(2 + √5) = (2 − √5) / ((2)² − (√5)²) = (2 − √5) / (4 − 5) = √5 − 2
Notice that the denominator becomes −1, so the fraction simplifies neatly. In exams, an answer with a surd in the denominator is usually not accepted as fully simplified.
注意分母变为 −1，因此分数可简洁地化简。在考试中，分母含有根式的答案通常不被视为最简形式。
5. Indices and Exponential Notation | 指数与幂运算
The laws of indices govern every manipulation involving powers. You must know these six laws and apply them fluently with both positive and negative exponents.
指数法则支配所有涉及幂的运算。你必须熟练掌握以下六条法则，并能灵活运用于正负指数。
aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ
a⁰ = 1, a⁻ⁿ = 1/aⁿ, a^(1/n) = ⁿ√a, a^(m/n) = (ⁿ√a)ᵐ
Study these examples carefully:
请仔细研究以下示例：
- x³ × x⁵ = x⁸ — add the exponents — 指数相加
- y⁷ ÷ y² = y⁵ — subtract the exponents — 指数相减
- (2³)² = 2⁶ = 64 — multiply the exponents — 指数相乘
- 5⁻² = 1/25 — a negative exponent means reciprocal — 负指数表示倒数
- 16^(3/4) = (⁴√16)³ = 2³ = 8 — fraction exponent means root then power — 分数指数先开方再乘方
- 2x⁰ = 2 × 1 = 2 — any non-zero base to power 0 equals 1 — 任何非零数的 0 次幂等于 1
A frequent trap is applying the laws to terms that are added or subtracted. The rule aᵐ × aⁿ only works for multiplication of like bases, not for aᵐ + aⁿ.
一个常见的陷阱是将法则误用于加减项。法则 aᵐ × aⁿ 只适用于同底数幂的乘法，而不能用于 aᵐ + aⁿ。
6. Quadratic Equations | 二次方程
A quadratic equation has the general form ax² + bx + c = 0, with a ≠ 0. AQA requires three methods of solution, and you must choose the most efficient one for each question.
二次方程的一般形式为 ax² + bx + c = 0，其中 a ≠ 0。AQA 要求掌握三种解法，并能针对不同题目选择最高效的方法。
Method 1 — factorisation. When the quadratic factorises neatly, it is the fastest method. For x² − 5x + 6 = 0, we find (x − 2)(x − 3) = 0, so x = 2 or x = 3.
方法一——因式分解法。当二次式可以简洁分解时，这是最快的方法。对于 x² − 5x + 6 = 0，可分解为 (x − 2)(x − 3) = 0，故 x = 2 或 x = 3。
Method 2 — completing the square. Rewrite x² + bx + c as (x + b/2)² − (b/2)² + c. This method is essential for finding the turning point of a parabola and for solving equations that do not factorise.
方法二——配方法。将 x² + bx + c 改写为 (x + b/2)² − (b/2)² + c。此法对于求抛物线的顶点以及解无法因式分解的方程至关重要。
Method 3 — the quadratic formula. This always works. For ax² + bx + c = 0:
方法三——求根公式。此方法永远有效。对于 ax² + bx + c = 0：
x = (−b ± √(b² − 4ac)) / 2a
The quantity b² − 4ac is called the discriminant, denoted Δ. It determines the nature of the roots:
b² − 4ac 称为判别式，记作 Δ。它决定根的性质：
- Δ > 0: two distinct real roots — 两个不等实根
- Δ = 0: one repeated real root — 一个二重实根
- Δ < 0: no real roots (two complex roots) — 无实根（两个复数根）
For example, 2x² + 3x + 5 = 0 has Δ = 9 − 40 = −31 < 0, so it has no real solutions. In contrast, x² − 6x + 9 = 0 has Δ = 0, giving the repeated root x = 3.
例如，2x² + 3x + 5 = 0 的判别式 Δ = 9 − 40 = −31 < 0，因此无实数解。而 x² − 6x + 9 = 0 的判别式 Δ = 0，给出二重根 x = 3。
7. Simultaneous Equations and Inequalities | 联立方程与不等式
Two linear simultaneous equations can be solved by elimination or substitution. Elimination involves aligning coefficients and adding or subtracting the equations. Substitution involves rearranging one equation and inserting it into the other.
两个线性联立方程可用消元法或代入法求解。消元法通过对齐系数后相加或相减方程实现；代入法则先重排一个方程，再将其代入另一个方程。
When one equation is linear and the other is quadratic, substitution is mandatory. Solve y = 2x + 1 together with y = x² − 4x + 5:
当一个方程为线性、另一个为二次时，必须采用代入法。联立求解 y = 2x + 1 与 y = x² − 4x + 5：
2x + 1 = x² − 4x + 5 → x² − 6x + 4 = 0
Using the quadratic formula, x = (6 ± √(36 − 16)) / 2 = (6 ± √20) / 2 = 3 ± √5. Substitute back to find the corresponding y-values. The discriminant of the resulting quadratic tells us whether the line intersects the curve, touches it, or misses it entirely.
用求根公式得 x = (6 ± √(36 − 16)) / 2 = (6 ± √20) / 2 = 3 ± √5。代回原方程求对应的 y 值。所得二次方程的判别式告诉我们直线与曲线相交、相切还是完全不相交。
For quadratic inequalities, first solve the corresponding equation, then sketch a graph or test intervals on a number line. For x² − 4 < 0, the roots are x = ±2, and the shaded region gives −2 < x < 2. Remember to reverse the inequality sign when multiplying or dividing by a negative number.
对于二次不等式，先解对应的方程，再画草图或在数轴上测试区间。对于 x² − 4 < 0，根为 x = ±2，阴影区域给出 −2 < x < 2。切记：乘以或除以负数时要反转不等号。
8. Coordinate Geometry | 坐标几何
Coordinate geometry links algebra to shape. Three results underpin every straight-line question in the course.
坐标几何将代数与图形联系起来。以下三个结论支撑着课程中所有直线相关的问题。
Midpoint: ((x₁ + x₂) / 2, (y₁ + y₂) / 2)
Gradient: m = (y₂ − y₁) / (x₂ − x₁)
Distance: d = √((x₂ − x₁)² + (y₂ − y₁)²)
Given the gradient m and a point (x₁, y₁), the equation of a line can be written in two equivalent forms: y − y₁ = m(x − x₁) or y = mx + c. The form y = mx + c is useful for reading the gradient and y-intercept directly, while the point-slope form is safer when you only know one point.
已知斜率 m 和一点 (x₁, y₁)，直线方程可写成两种等价形式：y − y₁ = m(x − x₁) 或 y = mx + c。形式 y = mx + c 便于直接读出斜率与 y 截距，而点斜式在只知道一个点时更为稳妥。
Parallel lines have equal gradients: m₁ = m₂. Perpendicular lines satisfy m₁ × m₂ = −1. For example, the line perpendicular to y = 3x + 2 has gradient −1/3. Vertical and horizontal lines are special cases: x = k is vertical with infinite gradient, and y = k is horizontal with gradient 0.
平行线的斜率相等：m₁ = m₂。垂直线的斜率满足 m₁ × m₂ = −1。例如，与 y = 3x + 2 垂直的直线斜率为 −1/3。垂直与水平线为特殊情况：x = k 为垂直线，斜率无穷大；y = k 为水平线，斜率为 0。
The equation of a circle also belongs to this topic: (x − a)² + (y − b)² = r², where (a, b) is the centre and r is the radius. Be able to complete the square to convert x² + y² + 2gx + 2fy + c = 0 into centre-radius form.
圆的方程也属于本专题：(x − a)² + (y − b)² = r²，其中 (a, b) 为圆心，r 为半径。务必掌握通过配方法将 x² + y² + 2gx + 2fy + c = 0 转换为圆心半径形式。
9. Essential Formula Table | 必背公式表
The table below summarises the formulas you are expected to recall instantly, without consulting the formula booklet. Revise this table weekly.
下表总结了你应该能够即时回忆、无需翻阅公式手册的核心公式。请每周复习此表。
| Formula | 公式 | Usage | 用途 |
| x = (−b ± √(b² − 4ac)) / 2a | Solving quadratic equations | 求解二次方程 |
| b² − 4ac | Discriminant, nature of roots | 判别式，判断根的性质 |
| aᵐ × aⁿ = aᵐ⁺ⁿ | Index law for products | 同底数幂相乘 |
| a⁻ⁿ = 1/aⁿ | Negative exponents | 负指数 |
| a^(m/n) = (ⁿ√a)ᵐ | Fractional exponents | 分数指数 |
| m = (y₂ − y₁) / (x₂ − x₁) | Gradient of a line | 直线斜率 |
| √((x₂ − x₁)² + (y₂ − y₁)²) | Distance between two points | 两点间距离 |
| m₁ × m₂ = −1 | Perpendicular gradients | 垂直直线斜率关系 |
| (x − a)² + (y − b)² = r² | Circle equation | 圆的标准方程 |
10. Common Mistakes to Avoid | 常见错误警示
Examiners report the same errors year after year.
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