📚 AQA AS Maths Unit 2 June 2022: Question Paper Analysis and Key Skills | AQA AS 数学第2单元 2022年6月试卷解析与核心技巧
This article reviews the AQA AS Mathematics Unit 2 June 2022 question paper. It focuses on the main pure mathematics and statistics topics, the style of questions that appeared, and the most effective strategies for revision and for gaining method marks in the exam.
本文回顾 AQA AS 数学第2单元 2022年6月试卷，重点梳理纯数与统计的核心考点、常见题型，并给出复习和获取方法分的高效策略。试卷通常将纯数与应用统计混合，要求学生在 1 小时 30 分钟内完成 80 分的题目。
1. Paper Structure and Mark Allocation | 试卷结构与分值分布
The June 2022 Unit 2 paper is 1 hour 30 minutes long and carries 80 marks. It includes pure mathematics content together with statistics. Questions are arranged in a mixture of short structured parts and longer problem-solving tasks.
2022年6月第2单元考试时长为1小时30分钟，满分80分。试卷包括纯数学和统计内容。题目由短小的结构化小题和较长的解决问题型大题混合组成。
The mark scheme rewards clear notation, correct substitution into formulas, and valid conclusions written in context. Even if a final answer is wrong, a correctly stated method can still earn several marks.
评分方案奖励清晰的符号、正确的公式代入以及在题目背景下给出的有效结论。即使最终答案错误，只要方法正确，仍然可以获得若干方法分。
| Part | Typical content | Common marks |
|---|---|---|
| Pure | Polynomials, binomial expansion, trigonometry, differentiation, integration, logs | About 50-60 |
| Statistics | Data, probability, binomial distribution, hypothesis testing | About 20-30 |
2. Polynomials and the Factor Theorem | 多项式与因式定理
A common June 2022 style question asks you to use the factor theorem to show that (x – a) is a factor of a cubic, and then to fully factorise the polynomial. You must write f(a) = 0 explicitly to secure the first method mark.
2022年6月试卷中常见的题型是要求利用因式定理证明 (x – a) 是三次多项式的一个因式，然后对多项式进行完全分解。你必须明确写出 f(a)=0，才能确保拿到第一个方法分。
For example, if f(x) = x³ – 4x² + x + 6, showing f(2) = 8 – 16 + 2 + 6 = 0 proves (x – 2) is a factor. After polynomial division, you can write f(x) = (x – 2)(x² – 2x – 3) = (x – 2)(x – 3)(x + 1).
例如，若 f(x)=x³-4x²+x+6，写出 f(2)=8-16+2+6=0 即可证明 (x-2) 是因式。进行多项式除法后，可写成 f(x)=(x-2)(x²-2x-3)=(x-2)(x-3)(x+1)。
Always set out your division or use comparing coefficients clearly. If the question says “hence solve”, use your factorised form to write down all three roots.
务必清晰地展示长除法或待定系数法。如果题目说“hence solve”，就要利用因式分解后的形式写出全部三个根。
3. Binomial Expansion | 二项式展开
Unit 2 June 2022 tested binomial expansion, often with a bracket of the form (a + bx)ⁿ for a positive integer n. You should first take out the factor a to write it as aⁿ(1 + (b/a)x)ⁿ, then apply the standard expansion.
第2单元 2022年6月考查了二项式展开，常见形式是 (a+bx)ⁿ，其中 n 为正整数。应先提取公因子 a，写成 aⁿ(1+(b/a)x)ⁿ，再应用标准展开式。
The general form for a positive integer n is:
正整数 n 的一般展开式为：
(1 + x)ⁿ = 1 + n x + n(n-1)/2! x² + n(n-1)(n-2)/3! x³ + …
Remember to multiply every term by aⁿ after expanding. If a question asks for the first four terms of (2 – 3x)⁵, write 2⁵(1 – (3/2)x)⁵ and then expand up to the x³ term.
展开后记得将每一项都乘以 aⁿ。若题目要求 (2-3x)⁵ 的前四项，可写成 2⁵(1-(3/2)x)⁵，然后展开到 x³ 项为止。
For positive integer powers the expansion is finite and valid for all x, so no validity statement is needed at AS level. However, you must simplify coefficients carefully.
对于正整数次幂，展开式是有限项，对所有 x 都成立，因此 AS 阶段不需要写有效性范围。但必须仔细化简每一项系数。
4. Trigonometry and Equations | 三角学与三角方程
Trigonometric questions in June 2022 often asked students to solve equations such as 2sinθ = 1 for 0° ≤ θ ≤ 360°, or to use sin²θ + cos²θ = 1 to convert between functions.
2022年6月的三角题常要求求解如 2sinθ=1（0°≤θ≤360°）这样的方程，或使用 sin²θ+cos²θ=1 在正弦和余弦之间进行转换。
Use the CAST diagram or the graphs of sinθ and cosθ to make sure you identify all possible solutions in the given interval. For 2sinθ = 1, sinθ = 1/2 gives θ = 30° and θ = 150°.
使用 CAST 图或 sinθ、cosθ 的图像，确保找出给定区间内的所有解。对于 2sinθ=1，sinθ=1/2，得到 θ=30° 和 θ=150°。
sin²θ + cos²θ = 1
Always show the intermediate step where you isolate sinθ or cosθ before giving the angles. If the equation is quadratic in sinθ, factorise it first, then reject impossible values such as sinθ = 2.
在给出角度之前，始终展示分离 sinθ 或 cosθ 的中间步骤。如果方程是 sinθ 的二次式，先因式分解，然后舍去不可能的值，例如 sinθ=2。
5. Differentiation and Integration | 微分与积分
Pure questions on Unit 2 routinely test polynomial differentiation and integration. You may be asked to find the gradient of a tangent, locate stationary points, or find an indefinite integral and use a point to determine the constant of integration.
第2单元的纯数题常规考查多项式微分和积分。你可能需要求切线的梯度、判断驻点，或求不定积分，并利用已知点确定积分常数。
d/dx (xⁿ) = n xⁿ⁻¹
∫ xⁿ dx = xⁿ⁺¹/(n+1) + c, n ≠ -1
For example, if y = 3x⁴ – 5x² + 7, then dy/dx = 12x³ – 10x. If the gradient at x = 1 is required, substitute x = 1 to get 2.
例如，若 y=3x⁴-5x²+7，则 dy/dx=12x³-10x。若要求 x=1 处的梯度，代入 x=1 得到 2。
For integration, remember to include +c for indefinite integrals. If a curve passes through (2, 10), substitute these values to find the particular value of c.
对于积分，不定积分务必加上 +c。若曲线经过点 (2,10)，代入这些值即可求出具体的 c 值。
6. Exponentials and Logarithms | 指数与对数
June 2022 questions included solving exponential equations and applying logarithmic rules. You are expected to know that ln eˣ = x and e^(ln x) = x, and to use log laws to simplify or solve equations.
2022年6月的试题包括求解指数方程和应用对数运算法则。你需要掌握 ln eˣ=x 和 e^(ln x)=x，并能利用对数法则化简或解方程。
logₐ x + logₐ y = logₐ(xy)
logₐ x – logₐ y = logₐ(x/y)
logₐ xⁿ = n logₐ x
To solve 3e²ˣ = 15, divide both sides by 3, then take natural logs: 2x = ln5, so x = ln5/2. Always give an exact answer, then a decimal if requested.
求解 3e²ˣ=15 时，两边先除以 3，再取自然对数：2x=ln5，所以 x=ln5/2。务必先给出精确答案，若题目要求再给出小数近似。
In modelling questions, set up an equation of the form P = A eᵏᵗ, substitute known values, and use logs to find the unknown constant k or time t.
在建模题中，建立 P=Aeᵏᵗ 形式的方程，代入已知值，并使用对数求出未知常数 k 或时间 t。
7. Statistics: Data and Measures of Spread | 统计：数据与离散程度
Statistics questions in the June 2022 Unit 2 paper asked candidates to interpret box plots, find the median, quartiles, range and interquartile range, and compare two data sets in context.
2022年6月第2单元的统计题要求考生解释箱线图，求中位数、四分位数、极差和四分位距，并在具体背景下比较两组数据。
IQR = Q₃ – Q₁
When comparing distributions, make two comments: one about location, such as the median, and one about spread, such as the IQR or range. Always refer to the context of the question, for example “on average, boys spent 20 minutes longer on homework”.
比较分布时，需要给出两方面的评论：一是集中趋势，例如中位数；二是离散程度，例如四分位距或极差。始终结合题目背景，例如“平均而言，男生做作业多花20分钟”。
For grouped frequency data, use the mid-interval values to estimate the mean. Show your full calculation with Σfx and Σf written clearly.
对于分组频率数据，使用组中值来估计平均数。清晰展示 Σfx 和 Σf 的计算过程。
8. Probability and Venn Diagrams | 概率与韦恩图
Venn diagram questions appeared frequently in this paper. The key formula for two events is P(A∪B) = P(A) + P(B) – P(A∩B). Fill in the regions of the diagram first, then answer probability questions using the total probability being 1.
韦恩图题在该试卷中出现频繁。两个事件的关键公式是 P(A∪B)=P(A)+P(B)-P(A∩B)。先填完韦恩图各区域，再利用总概率为1来回答概率问题。
P(A∪B) = P(A) + P(B) – P(A∩B)
P(A|B) = P(A∩B) / P(B)
For conditional probability, use the formula P(A|B) = P(A∩B) / P(B). Make sure you divide by the correct event, and interpret the result in context if the question asks.
对于条件概率，使用公式 P(A|B)=P(A∩B)/P(B)。确保除数是正确的事件，若题目要求，还需结合背景解释结果。
9. Binomial Distribution | 二项分布
The binomial distribution is a regular feature of Unit 2. You write X~B(n,p), where n is the number of trials and p is the probability of success. Use the formula to find probabilities.
二项分布是第2单元的常考内容。记作 X~B(n,p)，其中 n 是试验次数，p 是每次成功的概率。使用公式计算概率。
P(X = r) = ⁿCᵣ p^r (1 – p)^(n-r)
Identify n and p from the question wording. For example, “8 independent trials with probability 0.3” gives X~B(8, 0.3). To find P(X ≥ 2), use P(X ≥ 2) = 1 – P(X ≤ 1).
从题目表述中找出 n 和 p。例如“8次独立试验，每次成功概率为0.3”对应 X~B(8,0.3)。求 P(X≥2) 时，可用 P(X≥2)=1-P(X≤1) 来计算。
State the assumptions for a binomial model: fixed number of trials, two possible outcomes, independent trials, and constant probability. These statements can earn marks.
要陈述二项分布的假设条件：试验次数固定、每次只有两个结果、试验相互独立、每次概率不变。写出这些条件可以得到分数。
10. Hypothesis Testing | 假设检验
Hypothesis testing questions require a clear structure. You must state H₀ and H₁, give the test statistic, find the p-value or critical region, and write a conclusion in the context of the question.
假设检验题要求结构清晰。必须写出 H₀ 和 H₁、给出检验统计量、求出 p值或临界区域，并在题目背景下写出结论。
H₀: p = 0.5
H₁: p > 0.5
At a 5% significance level, if the p-value is less than 0.05, reject H₀ and say there is sufficient evidence to support the alternative. If not, do not reject H₀ and say there is insufficient evidence.
在5%显著性水平下，如果p值小于0.05，就拒绝 H₀，并说明有足够证据支持备择假设。如果不是，则不拒绝 H₀，说明证据不足。
In June 2022 many students lost marks by writing vague conclusions. Always include the probability and a contextual phrase, for example “there is sufficient evidence at the 5% level that the proportion of defective items has increased”.
2022年6月许多考生因结论表述模糊而丢分。务必包含概率和背景短语，例如“在5%显著性水平下，有足够证据表明次品率上升了”。
11. Exam Technique and Common Mistakes | 考试技巧与常见错误
Show full working for every question. On AQA papers, final answers without working may lose method marks even if correct. Write down formulas before substituting numbers.
每道题都要展示完整步骤。在 AQA 试卷中，仅写最终答案而没有过程，即使答案正确也可能丢失方法分。代入数字前先写出公式。
- Check calculator mode: radians or degrees.
- Label axes on graphs and include units where needed.
- For statistical tests, quote the p-value and compare it to the significance level.
- For integration, always write +c for indefinite integrals.
- Read the question carefully: exact answer or decimal answer?
检查计算器是在弧度制还是角度制。坐标轴要标注，必要时带上单位。统计检验要引用p值并与显著性水平比较。不定积分要写 +c。仔细读题：要求精确答案还是小数答案？
Manage time carefully. Aim to spend about 1 minute per mark, leaving time to check the applied statistics questions, which often need written conclusions.
合理分配时间。平均每分用1分钟，留出时间检查应用统计题，这类题通常需要写出结论性文字。
12. Summary and Revision Checklist | 总结与复习清单
Use the June 2022 Unit 2 question paper as a timed mock, then mark it against the official AQA mark scheme. Identify which topics caused the most lost marks and revise those areas first.
将2022年6月第2单元试卷作为限时模拟卷，然后对照 AQA 官方评分方案自行批改。找出失分最多的主题，并优先复习这些薄弱环节。
| Checklist item | Confident? |
|---|---|
| Factor theorem and cubic factorisation | ☐ |
| Binomial expansion of (a + bx)ⁿ | ☐ |
| Solving trigonometric equations with CAST | ☐ |
| Differentiation and integration of polynomials | ☐ |
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