AQA AS Further Maths Unit 2 (January 2020) - Paper Review and Revision Guide | AQA AS 进阶数学 Unit 2（2020年1月）试卷回顾与备考指南

📚 AQA AS Further Maths Unit 2 (January 2020) – Paper Review and Revision Guide | AQA AS 进阶数学 Unit 2（2020年1月）试卷回顾与备考指南

The January 2020 AQA Further Mathematics Unit 2 paper assesses your understanding of key further pure topics. This article breaks down each section of the paper, explains the core mathematical concepts, and offers revision advice based on the typical questions asked.

2020年1月AQA进阶数学Unit 2试卷重点考查你对进阶纯数学核心主题的理解。本文逐一拆解试卷的各个部分，解释核心数学概念，并根据典型题目提供备考建议。

1. Understanding the Paper | 了解试卷

Unit 2 is one of the two written papers for AS Further Mathematics. The paper is 1 hour 30 minutes long and carries 75 marks. It consists of a mixture of short, medium and extended response questions. The topics covered are those from the further pure curriculum: inequalities, roots of polynomials, complex numbers, inverse trigonometric functions, hyperbolic functions, further calculus, polar coordinates and numerical methods.

Unit 2是AS进阶数学的两场笔试之一。考试时长1小时30分钟，满分75分。试卷包含简答题、中等题和扩展题。考查内容来自进阶纯数学课程：不等式、多项式根、复数、反三角函数、双曲函数、进阶微积分、极坐标和数值方法。

Paper code: 7366/2 (AS Further Mathematics).
Calculator allowed, but no formula sheet.
All questions are compulsory.

试卷代码：7366/2（AS进阶数学）。
允许使用计算器，但无公式表。
所有题目均为必做题。

2. Inequalities | 不等式

Questions in this section involve solving rational inequalities and quadratic inequalities. You must be able to find the range of x satisfying the inequality, including cases with signs.

本节的题目涉及求解分式不等式和二次不等式。你必须能够确定满足不等式的x的取值范围，包括符号变化的情况。

For example, solve x/(x−2) ≥ 3.

例如，求解 x/(x−2) ≥ 3。

x/(x−2) − 3 ≥ 0 → (x−6)/(x−2) ≥ 0

The critical values are 2 and 6. Using a sign table gives x ≤ 2 or x ≥ 6, but since x = 2 is undefined, the solution is x < 2 or x ≥ 6.

临界值为2和6。使用符号表可得x ≤ 2或x ≥ 6，但由于x=2处无定义，解为x < 2或x ≥ 6。

3. Roots of Polynomial Equations | 多项式方程的根

You are expected to find the roots of cubic and quartic equations, particularly when complex roots occur in conjugate pairs. Questions often ask you to use the given root to find the other roots and factorise the polynomial.

你需要求解三次和四次方程的根，尤其是当出现共轭复根时。题目通常会给出一个根，要求你求出其他根并对多项式进行因式分解。

For a cubic with roots α, β, γ:

对于三次方程，根为α, β, γ：

Σα = −b/a, Σαβ = c/a, αβγ = −d/a

If α is complex, then its conjugate ᾱ is also a root. Use the sum and product of roots to find the real root.

如果α是复数，则其共轭ᾱ也是根。利用根的和与积可求出实根。

4. Complex Numbers | 复数

This unit revisits complex numbers with added focus on modulus-argument form, De Moivre’s theorem, and polynomial identities.

本节重新审视复数，并重点考察模-辐角形式、棣莫弗定理以及多项式恒等式。

Expressing complex numbers in exponential form:

用指数形式表示复数：

z = re^(iθ), where r = |z|, θ = arg(z)

A typical question may ask: “Given that z = 1 + √3 i, find z⁵ in the form a + bi.”

典型问题也许是：“已知 z = 1 + √3 i，求 z⁵，结果写成 a + bi 的形式。”

|z| = 2, arg(z) = π/3, so z⁵ = 2⁵ cos(5π/3) + i 2⁵ sin(5π/3) = 16 − 16√3 i

Remember to convert to radians and simplify trigonometric values.

注意使用弧度制，并化简三角函数值。

5. Inverse Trigonometric Functions | 反三角函数

You need to know the domains and ranges of arcsin, arccos and arctan, and use their derivatives and integrals.

你需要了解arcsin、arccos和arctan的定义域与值域，并使用它们的导数与积分。

The derivatives are:

它们的导数为：

d/dx(arcsin x) = 1/√(1−x²), d/dx(arctan x) = 1/(1+x²)

A common exam question asks to differentiate an expression like y = arcsin(x/3) or arctan(√x). Use the chain rule and simplify.

常见的考题要求对 y = arcsin(x/3) 或 arctan(√x) 求导。使用链式法则并化简。

6. Hyperbolic Functions | 双曲函数

Hyperbolic functions are defined in terms of exponential functions. You must be comfortable with their properties and identities.

双曲函数是用指数函数定义的。你需要熟练掌握它们的性质与恒等式。

Definitions:

定义：

sinh x = (eˣ − e⁻ˣ)/2, cosh x = (eˣ + e⁻ˣ)/2

The most useful identity is cosh²x − sinh²x = 1. A question may ask you to prove that (cosh x + sinh x)ⁿ = cosh(nx) + sinh(nx).

最重要的恒等式是 cosh²x − sinh²x = 1。题目可能要求证明 (cosh x + sinh x)ⁿ = cosh(nx) + sinh(nx)。

You should also know how to solve equations involving hyperbolic functions, for example cosh x = 3, by converting to exponentials and solving a quadratic in eˣ.

你还应知道如何求解涉及双曲函数的方程，例如 cosh x = 3，转换为指数形式后求解关于eˣ的二次方程。

7. Further Calculus | 进阶微积分

Integration questions may involve partial fractions, substitution, or reduction formulas. Differentiation may require implicit differentiation of hyperbolic or inverse trigonometric functions.

积分题可能涉及部分分式、换元法或递推公式。求导题可能需要隐式求导，对象可能是双曲函数或反三角函数。

Example integration:

积分示例：

∫ 1/(x²−4) dx = ∫ 1/((x−2)(x+2)) dx = 1/4 ln|(x−2)/(x+2)| + C

For an integral like ∫ 1/(1+x²) dx, the answer is arctan x + C. Recognise these standard results quickly.

对于 ∫ 1/(1+x²) dx，其结果为 arctan x + C。要快速识别这些标准表达式。

8. Polar Coordinates | 极坐标

Polar coordinates questions often ask you to convert between cartesian and polar forms, sketch simple curves, and find areas enclosed by polar curves.

极坐标题目通常要求你完成直角坐标与极坐标的转换、绘制简单曲线，以及求极坐标曲线围成的面积。

Conversion formulas:

转换公式：

x = r cos θ, y = r sin θ, r² = x² + y², tan θ = y/x

Area formula: A = ½ ∫ θ₁ to θ₂ r² dθ.

面积公式：A = ½ ∫₍θ₁₎^(θ₂) r² dθ。

Common polar curves include circles like r = 2a cos θ, and cardioids like r = a(1 + cos θ). Sketching requires knowledge of symmetry.

常见的极坐标曲线包括 r = 2a cos θ 这样的圆，以及 r = a(1 + cos θ) 这样的心形线。绘制时需了解对称性。

9. Numerical Methods | 数值方法

Numerical methods appear in this paper as interval bisection, Newton-Raphson iteration, or fixed-point iteration. You may need to show that a root exists, or perform iterations to a given accuracy.

试卷中的数值方法包括二分法、牛顿-拉弗森迭代或不动点迭代。你可能需要证明根的存在，或迭代到指定精度。

Newton-Raphson formula:

牛顿-拉弗森公式：

xₙ₊₁ = xₙ − f(xₙ)/f′(xₙ)

For example, to solve x³ − 2x − 5 = 0 with x₀ = 2, compute successive approximations until they converge.

例如，求解 x³ − 2x − 5 = 0，取初始值 x₀ = 2，逐步迭代直到收敛。

Familiarise yourself with the required notation and ensure you never round prematurely.

要熟悉所需符号，并且切勿过早四舍五入。

10. Common Pitfalls and Exam Tips | 常见误区与应试技巧

Many students lose marks by forgetting to include the modulus sign in antiderivatives of 1/x, or by missing the constant of integration. Always check whether the domain of an inverse function is restricted.

许多学生因忘记在1/x的原函数中添加绝对值符号，或漏写积分常数而失分。务必检查反函数的定义域是否受限。

Other common errors:

其他常见错误：

Incorrectly finding arguments of complex numbers: always use a sketch of the quadrant.
Forgetting that polar curves are often symmetric about the initial line.
Using degrees instead of radians in calculus.

求复数辐角时出错：务必用草图查象限。
忘记极坐标曲线通常关于极轴对称。
在微积分中使用角度制而不是弧度制。

Revision strategy: practice past papers under timed conditions, and create a formula sheet to memorise. The January 2020 paper is an excellent resource to simulate real exam pressure.

备考策略：在限时条件下练习历年真题，并制作公式表帮助记忆。2020年1月的试卷是模拟真实考试压力的绝佳资源。

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AQA AS Further Maths Unit 2 (January 2020) – Paper Review and Revision Guide | AQA AS 进阶数学 Unit 2（2020年1月）试卷回顾与备考指南