📚 AQA A-Level Maths Unit 4 January 2022: Pure 4 Past Paper Guide | AQA 数学 A Level 单元4 2022年1月真题卷复习
This revision guide focuses on the key topics and question styles that typically appear in the AQA A-level Maths Unit 4 paper, especially the January 2022 Pure 4 or legacy Core 4 style paper. It covers advanced binomial expansion, partial fractions, parametric equations, vectors, differential equations, implicit differentiation, integration by substitution and trigonometric integration. Worked examples are provided to help you build speed and accuracy before the real exam.
本篇复习指南聚焦 AQA 数学 A Level 单元4 试卷——特别是 2022年1月 Pure 4 或旧版 Core 4 风格的试卷——中的核心考点与常见题型。内容涵盖高阶二项展开、部分分式、参数方程、向量、微分方程、隐函数求导、换元积分和三角积分。文中配有例题示范，帮助你在正式考试前提升解题速度与准确率。
1. Understanding Unit 4 (Pure 4) | 认识单元4（纯数学核心4）
AQA Unit 4 in the January 2022 series focuses on advanced pure mathematics. For most candidates this is the Pure 4 paper, testing higher-level algebra, calculus, vectors and differential equations. You are expected to carry out multi-step solutions quickly and to communicate your working clearly.
2022年1月的 AQA 单元4 试卷聚焦高阶纯数学。对多数考生而言，它对应 Pure 4 试卷，考查高等代数、微积分、向量与微分方程等内容。考试要求你能够快速完成多步骤解题，并清晰呈现推理过程。
Typical question blocks include a short opening part that develops a technique, followed by an application to an unfamiliar context. You should revise both standard methods and common problem-solving routes.
典型题目通常先设置一个小问来训练某种技巧，再将该技巧应用到较陌生的情境中。复习时既要掌握标准方法，也要熟悉常见的解题路径。
2. Binomial Expansion for Rational Powers | 有理数次幂的二项展开
In this paper, you may be asked to expand (1 + ax)ⁿ for rational n. Use the general binomial expansion:
试卷中可能出现有理数次幂的二项展开。使用一般二项展开式：
(1 + x)ⁿ = 1 + n x + [n(n - 1)/2!] x² + [n(n - 1)(n - 2)/3!] x³ + ...
This expansion is valid only when |x| < 1. For example, expand (1 + 3x)⁻¹ up to the term in x³:
该展开仅在 |x| < 1 时有效。例如，将 (1 + 3x)⁻¹ 展开到 x³ 项：
(1 + 3x)⁻¹ = 1 - 3x + 9x² - 27x³ + ...
Always state the validity condition. Here |3x| < 1, so |x| < 1/3. This condition is often worth a mark on its own.
务必注明有效条件。本例中 |3x| < 1，因此 |x| < 1/3。这个条件常常单独占分，不能遗漏。
3. Partial Fractions and Integration | 部分分式与积分
Partial fractions often appear before integration. Split an algebraic fraction such as 1/[(x + 1)(x - 2)] into A/(x + 1) + B/(x - 2), then integrate each term to obtain natural logarithm results.
部分分式常与积分结合出现。将代数分式 1/[(x + 1)(x - 2)] 分解为 A/(x + 1) + B/(x - 2)，再逐项积分，即可得到自然对数形式的结果。
the above fraction, solving A(x - 2) + B(x + 1) = 1 gives A = -1/3 and B = 1/3. Therefore:
对于上述分式，由 A(x - 2) + B(x + 1) = 1 解得 A = -1/3，B = 1/3。因此：
∫ 1/[(x + 1)(x - 2)] dx = -(1/3) ln|x + 1| + (1/3) ln|x - 2| + C
is tests your ability to solve simultaneous equations and apply ∫ 1/(ax + b) dx = (1/a) ln|ax + b| + C. Remember the modulus signs around the logs.
这类题考查解联立方程以及公式 ∫ 1/(ax + b) dx = (1/a) ln|ax + b| + C 的应用。注意对数中的绝对值符号不能省略。
4. Parametric Equations and Tangents | 参数方程与切线
For parametric equations x = f(t), y = g(t), the gradient is given by:
对于参数方程 x = f(t), y = g(t)，斜率由下式给出：
dy/dx = (dy/dt) ÷ (dx/dt)
arameter. Example: x = t², y = 2t at t = 1. Then dx/dt = 2t and dy/dt = 2, so dy/dx = 1/t.
常见考题是求某一参数值处的切线方程。例如 x = t², y = 2t，在 t = 1 处 dx/dt = 2t，dy/dt = 2，因此 dy/dx = 1/t。
1 and the point is (1, 2). The tangent is y - 2 = 1(x - 1), which simplifies to y = x + 1.
t = 1 时斜率为 1，点为 (1, 2)。切线方程为 y - 2 = 1(x - 1)，化简得 y = x + 1。
5. Vectors: Lines and Intersections | 向量：直线与交点
intersect. Set the position vectors equal and solve the simultaneous equations for λ and μ. If the third equation is consistent, the lines meet.
向量题可能给出两条直线 r = a + λ b，要求判断它们是否相交。令两条直线的位置向量相等，解关于 λ 和 μ 的方程组；若第三式也成立，则两直线相交。
You may also need the scalar product to find angles:
你还可能需要使用数量积来求夹角：
a · b = |a| |b| cos θ
lines have direction vectors d₁ and d₂, the acute angle between them is found from cos θ = |d₁ · d₂| / (|d₁| |d₂|). Always give the acute angle unless the question asks otherwise.
若两条直线的方向向量分别为 d₁ 和 d₂，它们之间的锐角可由 cos θ = |d₁ · d₂| / (|d₁| |d₂|) 求得。除非题目另有要求，否则一律给出锐角。
6. First Order Differential Equations | 一阶微分方程
erential equations often use separation of variables. For dy/dx = f(x) g(y), rearrange to:
一阶微分方程常使用分离变量法。对于 dy/dx = f(x) g(y)，可变形为：
∫ 1/g(y) dy = ∫ f(x) dx
After integration, use any given conditions to find the constant. A typical application is a rate of change model, such as exponential growth or decay.
积分后利用给定条件求出常数。典型应用为变化率模型，例如指数增长或指数衰减。
Example: solve dy/dx = xy, y(0) = 3. Separating variables gives ∫ 1/y dy = ∫ x dx, so ln|y| = x²/2 + C. Applying y(0) = 3 gives C = ln 3, hence y = 3 ex²/2.
例如：求解 dy/dx = xy，y(0) = 3。分离变量得 ∫ 1/y dy = ∫ x dx，所以 ln|y| = x²/2 + C。代入 y(0) = 3 得 C = ln 3，因此 y = 3 ex²/2。
7. Implicit Differentiation | 隐函数求导
n y is not explicitly given. For x² + y² = 25, differentiate both sides with respect to x:
当 y 未显式给出时需使用隐函数求导。例如对 x² + y² = 25 两边关于 x 求导：
2x + 2y (dy/dx) = 0
e coordinates to find the tangent gradient at a point such as (3, 4), giving dy/dx = -3/4.
整理得 dy/dx = -x/y。代入坐标（如 (3, 4)）即可求出切线斜率，此处 dy/dx = -3/4。
Remember to use the product rule when differentiating mixed terms such as x y, and the chain rule for yⁿ terms. It is easy to lose marks by forgetting dy/dx after differentiating a function of y.
对 x y 这类混合项求导时要使用乘法法则，对 yⁿ 项求导时要使用链式法则。很多考生因在 y 的函数求导后漏写 dy/dx 而丢分。
8. Integration by Substitution | 换元积分法
Substitution is essential for integrals like ∫ x √2x + 1 dx. Let u = 2x + 1, so du/dx = 2 and x = (u - 1)/2. The integral becomes:
换元积分法对 ∫ x √2x + 1 dx 这类积分至关重要。设 u = 2x + 1，则 du/dx = 2，且 x = (u - 1)/2。积分变为：
∫ [(u - 1)/2] √u (du/2) = (1/4) ∫ (u³⁄² - u¹⁄²) du
Integrate term by term and substitute back to give the answer in terms of x. Do not forget to replace dx with du/2, and to change the limits if the question is a definite integral.
逐项积分后代回原变量，结果需用 x 表示。不要忘记将 dx 替换为 du/2；若为定积分，还需同步更换积分上下限。
9. Trigonometric Integration and Identities | 三角积分与恒等式
ric integrals often require identities. To integrate sin² x, use cos 2x = 1 - 2sin² x, so:
三角积分常需要恒等式。求 ∫ sin² x dx 时，利用 cos 2x = 1 - 2sin² x，得：
sin² x = (1 - cos 2x) / 2
2 - (sin 2x)/4 + C. Similar identities apply for cos² x and for products like sin x cos x.
因此 ∫ sin² x dx = x/2 - (sin 2x)/4 + C。类似恒等式适用于 cos² x 以及 sin x cos x 等乘积形式。
Know the double-angle and factor formulae, as questions sometimes combine trigonometric identities with integration by parts or substitution.
要熟悉二倍角公式与和差化积公式，因为有些题目会将三角恒等式与分部积分或换元积分结合考查。
10. Exam Technique and Common Errors | 考试技巧与常见错误
In the real January 2022 paper, questions are linked but earlier parts often provide hints. Show all working, keep exact values until the final answer, and check that any validity condition is stated.
在真实的 2022年1月试卷中，各小题互相联系，前面的结果常为后续提供提示。写出完整过程，保留精确值到最后一步，并检查是否注明了有效条件或定义域。
Common errors include dropping the constant of integration, forgetting the modulus sign in log integrals, using degrees instead of radians, and giving vector coordinates without all three components. Review your working line by line before moving on.
常见错误包括漏掉积分常数、省略对数积分中的绝对值符号、角度误用度数制而非弧度制，以及向量答案缺少三个坐标分量。进入下一题前应逐行检查解题过程。
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