📚 AQA Maths Topic Test: Core Revision Guide | AQA 数学主题测试：核心复习指南
Topic tests are one of the most effective ways to check your understanding before the full AQA A-level Mathematics exam. This guide walks through the key areas that appear again and again in AQA topic tests, with clear formulas, common question types and practical exam tips.
主题测试是在 AQA A-level 数学完整考试前检查你理解情况的最有效方法之一。本指南将带你梳理 AQA 主题测试中反复出现的关键领域，包括清晰公式、常见题型和实用考试技巧。
1. What Is an AQA Maths Topic Test? | 什么是 AQA 数学主题测试？
An AQA maths topic test is a short, focused assessment covering one specification topic, such as differentiation or hypothesis testing. It usually lasts 20 to 40 minutes and contains past-paper style questions with a range of demand from routine to problem solving.
AQA 数学主题测试是一种简短、集中的评估，覆盖一个考试大纲主题，例如微分或假设检验。它通常持续 20 到 40 分钟，包含历年真题风格的问题，难度从常规计算到问题解决不等。
Topic tests do not usually count towards your final grade, but they are designed to expose weak areas early. Treat every topic test as a timed exam so that you build both accuracy and speed before the real A-level papers.
主题测试通常不计入你的最终成绩，但它们的目的是尽早暴露薄弱环节。要把每一次主题测试当作限时考试来对待，这样你才能在真正的 A-level 考试前同时提高准确性和速度。
AQA topic tests often mix pure maths, statistics and mechanics content where appropriate. Use the specification checklist to know exactly which sub-topics can be tested, and revise the exact wording of command words such as 'show that' or 'hence'.
AQA 主题测试通常在适当的地方混合纯数、统计和力学内容。使用考试大纲检查表来准确了解哪些子主题可能被测试，并复习指令词的确切含义，例如 'show that'（证明）或 'hence'（因此）。
2. Algebra and Functions | 代数与函数
Algebra is the foundation of every AQA maths topic test. You must be able to factorise quadratics, complete the square, solve simultaneous equations and manipulate surds and indices without hesitation.
代数是每个 AQA 数学主题测试的基础。你必须能够熟练地进行二次式因式分解、配方、解联立方程以及处理根式和指数，不能犹豫。
rmula. You also need the discriminant Δ = b² − 4ac to determine how many real roots exist.
对于任意二次方程 ax² + bx + c = 0，其解由二次公式给出。你还需要判别式 Δ = b² − 4ac 来判断存在多少个实根。
x = (−b ± √b² − 4ac) / (2a) Δ = b² − 4ac
Function notation is heavily examined in AQA topic tests. Make sure you can form composite functions f(g(x)), find inverse functions and state the domain and range of a function correctly.
函数符号在 AQA 主题测试中考查很多。确保你能构造复合函数 f(g(x))、求反函数，并正确写出函数的定义域和值域。
= logₐ(x/y) and n logₐ x = logₐ(xⁿ). Always check that the base and argument are positive.
对数与指数也出现在代数部分。关键规则是 logₐ x + logₐ y = logₐ(xy)，logₐ x − logₐ y = logₐ(x/y) 以及 n logₐ x = logₐ(xⁿ)。始终检查底数和真数是否为正。
Common errors in this topic include forgetting to reverse the inequality when multiplying or dividing by a negative number, and incorrectly simplifying algebraic fractions. Write each line of working neatly to avoid sign slips.
这个主题的常见错误包括当乘以或除以负数时忘记反转不等号，以及错误地化简代数分式。每行步骤要写清楚，避免符号错误。
3. Coordinate Geometry | 坐标几何
Coordinate geometry topic tests focus on straight lines, circles and parametric equations. You need to recall the gradient formula, the midpoint formula and the distance formula under time pressure.
坐标几何主题测试侧重于直线、圆和参数方程。你需要在时间压力下记住斜率公式、中点公式和距离公式。
(x₁, y₁) is y − y₁ = m(x − x₁). The distance between two points is √((x₂−x₁)² + (y₂−y₁)²).
经过点 (x₁, y₁) 且斜率为 m 的直线方程为 y − y₁ = m(x − x₁)。两点之间的距离为 √((x₂−x₁)² + (y₂−y₁)²)。
Distance = √((x₂−x₁)² + (y₂−y₁)²) Circle: (x − a)² + (y − b)² = r²
0 to the standard form (x − a)² + (y − b)² = r². The centre is (a, b) and the radius is r.
对于圆，你必须能够通过配方将一般式 x² + y² + 2gx + 2fy + c = 0 转化为标准式 (x − a)² + (y − b)² = r²。圆心为 (a, b)，半径为 r。
find a Cartesian equation, or to find the gradient dy/dx using dy/dx = (dy/dt) / (dx/dt).
参数方程在 AQA 主题测试中也很常见。你可能被要求消去参数 t 来求笛卡尔方程，或者用 dy/dx = (dy/dt) / (dx/dt) 来求斜率 dy/dx。
A typical topic test question might ask you to find the tangent or normal to a circle at a given point. The radius is perpendicular to the tangent, so use the negative reciprocal of the radius gradient.
一个典型的主题测试题目可能会要求你求圆在给定点处的切线或法线。半径与切线垂直，因此使用半径斜率的负倒数。
4. Trigonometry | 三角学
Trigonometry in AQA topic tests covers radian measure, trigonometric identities, solution of trig equations and arc or sector problems. You must be fluent in both degrees and radians.
AQA 主题测试中的三角学涵盖弧度制、三角恒等式、三角方程求解以及弧长或扇形问题。你必须熟练使用角度制和弧度制。
cosθ. You also need the double angle formulas sin2θ = 2sinθcosθ and cos2θ = cos²θ − sin²θ.
基本恒等式是 sin²θ + cos²θ = 1 和 tanθ = sinθ/cosθ。你还需要二倍角公式 sin2θ = 2sinθcosθ 和 cos2θ = cos²θ − sin²θ。
sin²θ + cos²θ = 1 tanθ = sinθ/cosθ sin2θ = 2sinθcosθ
When solving trigonometric equations, always find all solutions in the given interval. Draw the CAST diagram or sketch the graph to avoid missing negative or reflected solutions.
解三角方程时，一定要找出给定区间内的所有解。使用 CAST 图或画出函数图像，避免漏掉负值或反射解。
180. The arc length is l = rθ and the sector area is A = ½r²θ, where θ must be in radians.
弧度制在 AQA 数学中至关重要。记住 π 弧度 = 180°，因此将角度转换为弧度要乘以 π/180。弧长为 l = rθ，扇形面积为 A = ½r²θ，其中 θ 必须以弧度为单位。
Common mistakes include using degree mode on a calculator when radians are required, and dividing by sinθ or cosθ without considering the zero case. Factorise instead of dividing when possible.
常见错误包括在需要弧度时把计算器设置在角度模式，以及除以 sinθ 或 cosθ 而没有考虑其为零的情况。尽可能使用因式分解而不是直接除以三角函数。
5. Sequences and Series | 数列与级数
Sequences and series questions test arithmetic progressions, geometric progressions and the binomial expansion. You need to know the standard formulas and be able to interpret word problems.
数列与级数问题考查等差数列、等比数列以及二项展开。你需要掌握标准公式，并能够解读文字应用题。
n/2(a + l), where a is the first term, d is the common difference and l is the last term.
对于等差数列，第 n 项为 uₙ = a + (n−1)d，前 n 项和为 Sₙ = n/2 [2a + (n−1)d] 或 Sₙ = n/2(a + l)，其中 a 是首项，d 是公差，l 是末项。
uₙ = a + (n−1)d Sₙ = n/2 [2a + (n−1)d] uₙ = arⁿ⁻¹
the nth term is uₙ = arⁿ⁻¹. The sum of the first n terms is Sₙ = a(1 − rⁿ)/(1 − r), and if |r| < 1 the sum to infinity is S∞ = a/(1 − r).
对于等比数列，第 n 项为 uₙ = arⁿ⁻¹。前 n 项和为 Sₙ = a(1 − rⁿ)/(1 − r)，如果 |r| < 1，无穷和为 S∞ = a/(1 − r)。
ansion appears frequently in AQA topic tests. For a positive integer n, (a+b)ⁿ = Σ ₙCᵣ aⁿ⁻ʳ bʳ, where ₙCᵣ = n! / [r!(n−r)!]. You must also be able to expand (1+ax)ⁿ for rational n using the series form.
二项展开在 AQA 主题测试中经常出现。对于正整数 n，(a+b)ⁿ = Σ ₙCᵣ aⁿ⁻ʳ bʳ，其中 ₙCᵣ = n! / [r!(n−r)!]。你还必须能够使用级数形式展开有理数 n 时的 (1+ax)ⁿ。
Word problems often ask you to form an equation from a real-life scenario, such as savings growth or population change. Define the first term and common ratio clearly before substituting into the formula.
文字应用题通常要求你从现实情境中建立方程，例如储蓄增长或人口变化。在代入公式之前，要清晰地定义首项和公比。
6. Differentiation | 微分
Differentiation is one of the most heavily weighted pure topics in AQA maths. You must be able to differentiate polynomial, trigonometric, exponential and logarithmic functions confidently.
微分是 AQA 数学中权重最高的纯数主题之一。你必须能够自信地对多项式函数、三角函数、指数函数和对数函数求导。
e basic rule is d/dx xⁿ = nxⁿ⁻¹. You also need d/dx sinx = cosx, d/dx cosx = −sinx, d/dx eˣ = eˣ and d/dx lnx = 1/x.
基本规则是 d/dx xⁿ = nxⁿ⁻¹。你还需要记住 d/dx sinx = cosx，d/dx cosx = −sinx，d/dx eˣ = eˣ 以及 d/dx lnx = 1/x。
d/dx xⁿ = nxⁿ⁻¹ d/dx eˣ = eˣ d/dx lnx = 1/x
For composite functions, use the chain rule: if y = f(g(x)), then dy/dx = f'(g(x))g'(x). The product rule is d/dx(uv) = u'v + uv' and the quotient rule is d/dx(u/v) = (u'v − uv')/v².
对于复合函数，使用链式法则：如果 y = f(g(x))，则 dy/dx = f'(g(x))g'(x)。乘积法则是 d/dx(uv) = u'v + uv'，商法则是 d/dx(u/v) = (u'v − uv')/v²。
e or a sign test to classify them as maxima, minima or points of inflection. Many topic tests combine this with an optimisation problem.
驻点通过令 dy/dx = 0 来求得。使用二阶导数或符号检验来判断它们是极大值点、极小值点还是拐点。许多主题测试会将其与优化问题结合起来考查。
AQA often asks you to find the equation of a tangent or normal at a point on a curve. The gradient of the tangent is dy/dx at that point, and the normal gradient is its negative reciprocal.
AQA 经常要求你求出曲线上某一点处切线或法线的方程。切线的斜率是该点的 dy/dx，而法线斜率是其负倒数。
7. Integration | 积分
Integration is the reverse of differentiation and is tested both as an algebraic skill and in area problems. You must be able to integrate powers, exponentials, trigonometric functions and rational functions.
积分是微分的逆运算，既作为代数技能考查，也用于面积问题。你必须能够对幂函数、指数函数、三角函数和有理函数进行积分。
The basic integral is ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C for n ≠ −1. You also need ∫ 1/x dx = ln|x| + C, ∫ eˣ dx = eˣ + C and ∫ sinx dx = −cosx + C.
基本积分是 ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C（n ≠ −1）。你还需要记住 ∫ 1/x dx = ln|x| + C，∫ eˣ dx = eˣ + C 以及 ∫ sinx dx = −cosx + C。
∫ xⁿ dx = xⁿ⁺¹/(n+1) + C ∫ 1/x dx = ln|x| + C ∫ eˣ dx = eˣ + C
Definite integrals are used to find the area under a curve. The area between a curve y = f
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