A-Level Maths Support Pack 1: Comprehensive Contents Guide | A-Level数学支持包1：内容全面指南

📚 A-Level Maths Support Pack 1: Comprehensive Contents Guide | A-Level数学支持包1：内容全面指南

Welcome to this detailed breakdown of the contents of Support Pack 1 for AQA A-Level Mathematics. This pack focuses on the core algebraic and analytical skills that form the foundation of the entire course. Understanding these topics is essential for success in both pure mathematics and applied modules.

欢迎浏览AQA A-Level数学支持包1的内容详细解析。本支持包聚焦于构成整个课程基础的核心代数与分析技能。理解这些主题对于纯数学与应用模块的成功至关重要。

1. Algebraic Expressions | 代数表达式

This section revises basic algebraic manipulation, which is assumed knowledge for A-Level. You will learn to simplify expressions, use index laws, and handle surds confidently.

本节复习基础代数运算，这是A-Level的预备知识。你将学习化简表达式、运用指数法则，并自信地处理无理数。

Use the index laws including fractional and negative powers: a^m × aⁿ = a^m+n, a^m ÷ aⁿ = a^m−n, (a^m)ⁿ = a^mn.

运用包含分数与负指数的指数法则：a^m × aⁿ = a^m+n，a^m ÷ aⁿ = a^m−n，(a^m)ⁿ = a^mn。
Simplify surds, e.g. √12 = 2√3, and rationalise denominators: 1/√2 = √2/2.

化简无理数，例如√12 = 2√3，并有理化分母：1/√2 = √2/2。
Expand brackets and factorise expressions, including grouping and the difference of two squares: x² − y² = (x + y)(x − y).

展开括号并因式分解表达式，包括分组法和平方差：x² − y² = (x + y)(x − y)。

2. Quadratics | 二次方程

Quadratics are everywhere in A-Level maths. This section covers solving quadratic equations, completing the square, and using the discriminant.

二次函数在A-Level数学中无处不在。本节涵盖求解二次方程、配方法以及判别式的使用。

Solve by factorisation, by the quadratic formula x = (−b ± √(b² − 4ac)) / 2a, or by completing the square.

通过因式分解、二次公式x = (−b ± √(b² − 4ac)) / 2a或配方法求解。
Complete the square: ax² + bx + c = a(x + b/2a)² + (c − b²/4a).

配方：ax² + bx + c = a(x + b/2a)² + (c − b²/4a)。
Use the discriminant Δ = b² − 4ac to determine the number of real roots: Δ > 0 two distinct, Δ = 0 one repeated, Δ < 0 none.

使用判别式Δ = b² − 4ac判断实根个数：Δ > 0两个不同实根，Δ = 0一个重根，Δ < 0无实根。

3. Equations and Inequalities | 方程与不等式

This topic extends solving techniques to simultaneous equations and linear/quadratic inequalities. These skills are vital for modelling problems.

本主题将求解技术扩展到联立方程以及线性/二次不等式。这些技能对于建模问题至关重要。

Solve simultaneous equations by substitution and elimination, including one linear and one quadratic equation.

通过代入消元法求解联立方程，包括一个线性方程与一个二次方程的情形。
Represent linear inequalities on a number line and solve compound inequalities such as 2 < 3x + 1 ≤ 7.

在数轴上表示线性不等式，并求解复合不等式如2 < 3x + 1 ≤ 7。
Solve quadratic inequalities by sketching the graph or sign diagrams: e.g. x² − 5x + 6 > 0 ⇒ x < 2 or x > 3.

通过画图或符号表求解二次不等式：例如x² − 5x + 6 > 0 ⇒ x < 2 或 x > 3。

4. Graphs and Transformations | 图形与变换

Graphical understanding is central to A-Level pure mathematics. This section covers cubic, quartic and reciprocal graphs, plus transformations.

图形理解是A-Level纯数学的核心。本节涵盖三次、四次和倒数函数图形，以及变换。

Sketch graphs of y = ax³ + bx² + cx + d, identifying roots, intercepts and turning points.

绘制y = ax³ + bx² + cx + d的草图，识别根、截距和拐点。
Recognise reciprocal graphs: y = k/x and y = k/x², noting vertical and horizontal asymptotes.

识别倒数图形：y = k/x和y = k/x²，注意垂直和水平渐近线。
Apply transformations: y = f(x) + a (translation up), y = f(x + a) (translation left), y = −f(x) (reflection in x-axis), y = f(−x) (reflection in y-axis), y = af(x) (vertical stretch).

应用变换：y = f(x) + a（上移），y = f(x + a)（左移），y = −f(x)（关于x轴对称），y = f(−x)（关于y轴对称），y = af(x)（垂直伸缩）。

5. Straight Line Graphs | 直线图

Understanding straight lines and gradients prepares you for coordinate geometry in later topics. You will also learn about parallel and perpendicular lines.

理解直线和斜率为你后续的坐标几何做准备。你还将学习平行线与垂直线。

The equation y = mx + c, with gradient m and y-intercept c. Also the general form ax + by + c = 0.

方程y = mx + c，其中m为斜率，c为y截距。还有一般式ax + by + c = 0。
Gradient of a line through (x₁, y₁) and (x₂, y₂) is (y₂ − y₁)/(x₂ − x₁).

通过点(x₁, y₁)和(x₂, y₂)的直线斜率为(y₂ − y₁)/(x₂ − x₁)。
Parallel lines have equal gradients; perpendicular lines satisfy m₁ × m₂ = −1.

平行线斜率相等；垂直线满足m₁ × m₂ = −1。

6. Circles | 圆

Coordinate geometry of circles is a new topic at A-Level. You need the general equation and properties of chords and tangents.

圆的坐标几何是A-Level的新主题。你需要掌握圆的一般方程以及弦与切线的性质。

The equation of a circle centre (a, b) radius r: (x − a)² + (y − b)² = r².

圆心(a, b)半径r的圆方程：(x − a)² + (y − b)² = r²。
Expand to general form: x² + y² + 2gx + 2fy + c = 0, with centre (−g, −f) and radius √(g² + f² − c).

展开为一般式：x² + y² + 2gx + 2fy + c = 0，圆心(−g, −f)，半径√(g² + f² − c)。
The perpendicular from the centre to a chord bisects the chord. The tangent at a point is perpendicular to the radius.

圆心到弦的垂线平分弦。切线与半径垂直。

7. Trigonometry | 三角函数

This section revises exact values and trigonometric identities, essential for solving equations and modelling periodic phenomena.

本节复习精确值与三角恒等式，对于求解方程和建模周期现象至关重要。

Know exact values for angles 0°, 30°, 45°, 60°, 90°. For example sin 30° = ½, cos 45° = √2/2, tan 60° = √3.

掌握0°、30°、45°、60°、90°角的精确值。例如sin 30° = ½，cos 45° = √2/2，tan 60° = √3。
Use the fundamental identity sin² θ + cos² θ = 1, and tan θ = sin θ / cos θ.

使用基本恒等式sin² θ + cos² θ = 1，以及tan θ = sin θ / cos θ。
Solve trigonometric equations in a given interval, e.g. 2 cos θ − 1 = 0 for 0° ≤ θ < 360° ⇒ θ = 60°, 300°.

在给定区间内求解三角方程，例如2 cos θ − 1 = 0，0° ≤ θ < 360° ⇒ θ = 60°，300°。

8. Exponentials and Logarithms | 指数与对数

Exponential growth and logarithms are key to many real-world applications, from compound interest to radioactive decay. This section defines the natural exponential and its inverse.

指数增长和对数是许多实际应用的关键，从复利到放射性衰变。本节定义自然指数及其反函数。

The exponential function y = e^x, and its inverse the natural logarithm y = ln x.

指数函数y = e^x，及其反函数自然对数y = ln x。
Laws of logarithms: ln(ab) = ln a + ln b, ln(a/b) = ln a − ln b, ln(aⁿ) = n ln a.

对数法则：ln(ab) = ln a + ln b，ln(a/b) = ln a − ln b，ln(aⁿ) = n ln a。
Solve equations involving exponentials by taking logs, e.g. 3^x = 20 ⇒ x = ln 20 / ln 3.

通过取对数求解指数方程，例如3^x = 20 ⇒ x = ln 20 / ln 3。

9. Differentiation | 微分

Differentiation is the first step into calculus. You will learn the power rule, gradients of tangents, and stationary points.

微分是进入微积分的第一步。你将学习幂法则、切线斜率和驻点。

If y = xⁿ, then dy/dx = nxⁿ⁻¹. This works for positive, negative and fractional n.

若y = xⁿ，则dy/dx = nxⁿ⁻¹。这对于正、负和分数n均适用。
Find the gradient of a curve at a point by substituting x into dy/dx.

将x代入dy/dx，求出曲线在某一点的斜率。
Determine stationary points by setting dy/dx = 0, and classify them using the second derivative or sign table.

令dy/dx = 0求驻点，并通过二阶导数或符号表进行分类。

10. Integration | 积分

Integration reverses differentiation. This section covers indefinite and definite integrals, including finding areas under curves.

积分是微分的逆运算。本节涵盖不定积分和定积分，包括求曲线下的面积。

The power rule for integration: ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, for n ≠ −1.

积分幂法则：∫ xⁿ dx = xⁿ⁺¹/(n+1) + C，其中n ≠ −1。
Evaluate definite integrals and interpret as the signed area between the curve and the x-axis.

计算定积分并将其解释为曲线与x轴之间的有向面积。
Find the area enclosed by a curve and a line by integrating the difference between their y-values.

通过积分两条曲线y值的差，求曲线与直线围成的面积。

11. Vectors | 向量

Vectors are fundamental in mechanics and later pure topics. This section introduces vector notation, magnitude and displacement vectors.

向量是力学和后续纯数学主题的基础。本节介绍向量符号、模长和位移向量。

Use column vectors and basis notation i, j, k. For example v = 3i + 2j.

使用列向量和基底记号i、j、k。例如v = 3i + 2j。
Calculate the magnitude: |v| = √(x² + y²). The direction angle θ satisfies tan θ = y/x.

计算模长：|v| = √(x² + y²)。方向角θ满足tan θ = y/x。
Add, subtract and scale vectors geometrically and algebraically.

几何与代数地完成向量的加减与缩放。

12. Revision Strategies | 复习策略

Support Pack 1 is designed to refresh your foundation. Regular practice and self-testing are essential to retain these core skills.

支持包1旨在唤醒你的基础。定期练习与自我测试对于保留这些核心技能至关重要。

Draw out a topic checklist and tick off each skill after completing practice questions.

制作主题清单，并在完成练习后勾选每一项技能。
Use past exam questions to familiarise yourself with AQA style and common mark schemes.

使用历年真题熟悉AQA风格和常见评分方案。
Identify weak areas early and revisit the relevant section in this pack before moving on to new material.

及早找出薄弱环节，并在学习新内容前重新查阅本支持包的相关部分。

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A-Level Maths Support Pack 1: Comprehensive Contents Guide | A-Level数学支持包1：内容全面指南