📚 AQA AS Maths Unit 1 (January 2021): Complete Revision Guide | AQA AS 数学 Unit 1（2021 年 1 月）：完整复习指南
The January 2021 AQA AS Mathematics Unit 1 paper tested the core pure topics that form the foundation of the AS course. This guide summarises the key question types, common pitfalls and exam-ready strategies, using Jan 2021-style examples to sharpen your skills.
2021 年 1 月 AQA AS 数学 Unit 1 试卷考查了构成 AS 课程基础的核心纯数专题。本指南总结关键题型、常见陷阱和备考策略，并结合 2021 年 1 月风格的例题帮助你提升解题能力。
1. Understanding the January 2021 Unit 1 Paper | 了解 2021 年 1 月 Unit 1 试卷
The AQA AS Mathematics Unit 1 paper typically focuses on pure mathematics: algebra, coordinate geometry, trigonometry, differentiation, integration, exponentials and logarithms. In January 2021, questions ranged from short manipulative tasks to multi-step problem-solving exercises that required clear logical presentation.
AQA AS 数学 Unit 1 试卷通常聚焦纯数学：代数、坐标几何、三角学、微分、积分、指数与对数。2021 年 1 月的题目从简短的运算题到多步骤问题解决题都有，要求清晰的逻辑表达。
Most questions begin with a routine calculation, such as differentiating a polynomial or solving a trigonometric equation, and then ask you to apply the result to a real-world or geometric context. This structure means that method marks are available even if the final answer is not reached.
大多数题目先进行常规计算，例如对多项式求导或解三角方程，然后要求你将结果应用到实际问题或几何情境中。这种结构意味着即使没有算出最终答案，也可以获得方法分。
2. Mark Allocation and Timing | 分值分配与时间管理
An AS Unit 1 paper usually carries 80 marks and lasts 90 minutes, giving roughly 1.1 minutes per mark. Use this to plan your time: spend about 45 seconds on a 1-mark question and up to 5 minutes on a 5-mark question.
AS Unit 1 试卷通常为 80 分、90 分钟，大约每分 1.1 分钟。利用这个比例规划时间：1 分题约花 45 秒，5 分题最多花 5 分钟。
Short-answer questions often test a single skill, such as using the factor theorem or finding the equation of a tangent. Longer questions link two or more topics, for example integrating a function and then finding the area under a curve.
简答题通常考查单一技能，例如使用因式定理或求切线方程。长题目会把两个或多个专题联系起来，例如先对函数积分，再求曲线下的面积。
Always write down intermediate steps clearly. Even if you make an arithmetic slip, a correct method can secure most of the marks. In January 2021-style scripts, examiners credited valid alternative approaches as long as the notation was unambiguous.
始终清楚地写出中间步骤。即使出现计算失误，正确的方法也能获得大部分分数。在 2021 年 1 月风格的评分中，只要符号明确，有效的替代方法同样给分。
3. Core Topic 1: Algebraic Manipulation and Polynomials | 核心专题 1：代数运算与多项式
e factor theorem is a high-frequency skill: if f(a) = 0, then (x - a) is a factor of f(x).
考试会要求化简有理式、展开括号和因式分解多项式。因式定理是高频技能：如果 f(a) = 0，那么 (x - a) 是 f(x) 的因式。
gives 2x² - x - 3, which factorises as (2x - 3)(x + 1). So f(x) = (x - 2)(2x - 3)(x + 1).
例如，设 f(x) = 2x³ - 5x² - x + 6。因为 f(2) = 16 - 20 - 2 + 6 = 0，所以 (x - 2) 是一个因式。除以 (x - 2) 得 2x² - x - 3，进一步分解为 (2x - 3)(x + 1)。因此 f(x) = (x - 2)(2x - 3)(x + 1)。
sions should be handled automatically by inspection or by the quadratic formula: x = (-b ± √b² - 4ac) / 2a. In January 2021-style questions, you may be told to leave surds in exact form.
二次式应能通过观察或用求根公式 x = (-b ± √b² - 4ac) / 2a 迅速处理。在 2021 年 1 月风格的题目中，可能会要求将根式保留为精确形式。
4. Core Topic 2: Coordinate Geometry and Circles | 核心专题 2：坐标几何与圆
a)² + (y - b)² = r². For a circle centred at the origin, this simplifies to x² + y² = r².
圆心为 (a, b)、半径为 r 的圆的方程是 (x - a)² + (y - b)² = r²。对于圆心在原点的圆，方程简化为 x² + y² = r²。
3, 4). The radius from (0, 0) to (3, 4) has gradient 4/3, so the tangent gradient is the negative reciprocal, -3/4.
2021 年 1 月风格常见任务是：求圆 x² + y² = 25 在点 (3, 4) 处的切线。从 (0, 0) 到 (3, 4) 的半径斜率为 4/3，因此切线斜率是其负倒数 -3/4。
ways check that the tangent line is perpendicular to the radius by verifying the product of gradients equals -1.
利用 y - 4 = -3/4 (x - 3) 并化简得 4y - 16 = -3x + 9，即 3x + 4y = 25。始终通过验证两条直线斜率之积等于 -1 来检查切线是否与半径垂直。
5. Core Topic 3: Trigonometry and Identities | 核心专题 3：三角学与恒等式
You must use exact values for sin, cos and tan at 0°, 30°, 45°, 60° and 90°. In January 2021, candidates were often required to switch between degrees and radians, especially in calculus questions where radians are essential.
必须熟练使用 0°、30°、45°、60° 和 90° 的 sin、cos、tan 精确值。2021 年 1 月考试中，考生经常需要在角度制与弧度制之间转换，尤其在微积分题中弧度制必不可少。
cos x for 0° ≤ x ≤ 360°. Dividing both sides by cos x (where cos x ≠ 0) gives tan x = 1/2.
两个基本恒等式是 sin²θ + cos²θ = 1 和 tan θ = sin θ / cos θ。典型方程如 2 sin x = cos x，其中 0° ≤ x ≤ 360°。两边除以 cos x（cos x ≠ 0）得 tan x = 1/2。
The principal solution is 26.6°, and since tangent has period 180°, the other solution in the interval is 206.6°. Always sketch the tangent graph or use the quadrants to avoid losing solutions.
主解为 26.6°，由于正切函数的周期为 180°，区间内的另一个解是 206.6°。始终画出正切图像或使用象限判断，以免漏解。
6. Core Topic 4: Differentiation and Tangents | 核心专题 4：微分与切线
points. If y = xⁿ, then dy/dx = nxⁿ⁻¹, and the derivative of a sum is taken term by term.
AS 阶段的微分主要针对多项式，并简单应用于斜率、切线、法线和驻点。如果 y = xⁿ，则 dy/dx = nxⁿ⁻¹，和的导数逐项求导。
tting 3x² - 3 = 0 gives x = 1 or x = -1. These are the x-coordinates of stationary points.
考虑 y = x³ - 3x + 2。则 dy/dx = 3x² - 3。令 3x² - 3 = 0 得 x = 1 或 x = -1。这些是驻点的 x 坐标。
= -1, so (-1, 4) is a local maximum. This sign test is a common Jan 2021-style mark point.
二阶导数 d²y/dx² = 6x 在 x = 1 处为正，因此 (1, 0) 是局部极小点；在 x = -1 处为负，因此 (-1, 4) 是局部极大点。这种符号判断是 2021 年 1 月风格常见的得分点。
7. Core Topic 5: Integration and Area | 核心专题 5：积分与面积
xⁿ⁺¹ / (n + 1) + C, where C is the constant of integration and n ≠ -1. Definite integrals have limits and do not require C.
积分是微分的逆运算。对于多项式，∫ xⁿ dx = xⁿ⁺¹ / (n + 1) + C，其中 C 是积分常数且 n ≠ -1。定积分带有上下限，不需要 C。
d two vertical lines. For example, ∫₀² (3x² - 2) dx = [x³ - 2x]₀² = (8 - 4) - (0 - 0) = 4.
2021 年 1 月风格常见题要求计算曲线、x 轴与两条竖直线之间的面积。例如，∫₀² (3x² - 2) dx = [x³ - 2x]₀² = (8 - 4) - (0 - 0) = 4。
If the curve lies below the x-axis, the definite integral gives a negative value. In that case, take the absolute value or split the area into separate regions to find the true geometrical area.
如果曲线位于 x 轴下方，定积分会得出负值。此时应取绝对值或将面积分成不同区域，以求出真正的几何面积。
8. Core Topic 6: Exponentials and Logarithms | 核心专题 6：指数与对数
The natural logarithm ln x is the inverse of eˣ, so ln(eˣ) = x and eln x = x. These properties allow you to solve equations such as 3e2x = 27.
自然对数 ln x 是 eˣ 的逆运算，因此 ln(eˣ) = x，且 eln x = x。利用这些性质可以求解 3e2x = 27 这类方程。
Dividing by 3 gives e2x logarithmic form is preferred in assessments unless a decimal approximation is requested.
两边除以 3 得 e2x = 9。两边取自然对数得 2x = ln 9，因此 x = ½ ln 9 = ln 3。除非题目要求近似小数，否则评估中优先使用精确对数形式。
ential functions model growth and decay. For example, if a population is given by P = 500e0.2t, then the initial population is 500, and the rate of change is dP/dt = 100e0.2t.
指数函数用于建立增长与衰减模型。例如，若人口数量为 P = 500e0.2t，则初始人口为 500，变化率为 dP/dt = 100e0.2t。
9. Worked Problem: A Jan 2021-Style Application | 例题精讲：2021 年 1 月风格应用题
. (b) Factorise f(x) fully. (c) Find the coordinates of the stationary points on y = f(x).
设 f(x) = x³ - 6x² + 9x - 4。(a) 证明 (x - 1) 是 f(x) 的因式。(b) 将 f(x) 彻底因式分解。(c) 求曲线 y = f(x) 上驻点的坐标。
re by the factor theorem, (x - 1) is a factor. Division gives f(x) = (x - 1)(x² - 5x + 4).
对于 (a)，代入 x = 1：f(1) = 1 - 6 + 9 - 4 = 0。因此根据因式定理，(x - 1) 是一个因式。除法得 f(x) = (x - 1)(x² - 5x + 4)。
adratic factorises as (x - 1)(x - 4), so f(x) = (x - 1)²(x - 4). For (c), differentiate: f '(x) = 3x² - 12x + 9 = 3(x² - 4x + 3) = 3(x - 1)(x - 3).
二次式分解为 (x - 1)(x - 4)，因此 f(x) = (x - 1)²(x - 4)。对于 (c)，求导：f '(x) = 3x² - 12x + 9 = 3(x² - 4x + 3) = 3(x - 1)(x - 3)。
Set f ' (1) = 0 and f(3) = 27 - 54 + 27 - 4 = -4. So the stationary points are (1, 0) and (3, -4).
令 f '(x) = 0 得 x = 1 或 x = 3。对应的 y 值为 f(1) = 0 和 f(3) = 27 - 54 + 27 - 4 = -4。因此驻点为 (1, 0) 和 (3, -4)。
10. Common Mistakes and How to Avoid Them | 常见错误与规避方法
Many candidates lose marks by forgetting the constant of integration. Every indefinite integral must end with + C unless it is a definite integral with limits.
许多考生因忘记积分常数而失分。除非是带上下限的定积分，否则每个不定积分都必须以 + C 结尾。
Angle units cause frequent errors. Before solving a trigonometric equation, check whether the interval is in degrees or radians, and set your calculator to the correct mode.
角度单位常导致错误。解三角方程前，先检查区间是用角度制还是弧度制，并将计算器设置为正确模式。
Expanding brackets incorrectly is another hazard. Write each term systematically, and after simplification compare the number of terms with the original product to catch missed signs.
错误展开括号是另一个陷阱。应系统地写出每一项，化简后将项数与原始乘积进行比较，以发现遗漏的符号。
11. Final Revision Checklist | 最终复习清单
Before the exam, make sure you can recall the quadratic formula, the factor theorem, the equation of a circle, the basic trigonometric identities, and the power rule for differentiation and integration.
考试前，确保你能熟练回忆求根公式、因式定理、圆的方程、基本三角恒等式，以及微积分的幂法则。
Practise past-paper questions under timed conditions, then mark your own work using the official mark scheme. Write down your mistakes in a log and revisit those topics the next day.
在限时条件下练习历年真题，然后对照官方评分标准自行评分。把错误记录在错题本中，第二天重新复习这些专题。
Finally, check your calculator settings, bring spare pens and read each question twice. Underline key information such as 'exact value', 'leave in terms of π' or 'show that' to avoid unnecessary approximation errors.
最后，检查计算器设置，准备备用笔，每道题读两遍。用下划线标出关键信息，如“精确值”“保留 π”“证明”等，以避免不必要的近似误差。
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