Year 13 AQA Maths: Unit Test Mock Paper Walkthrough | AQA 数学 Year 13 单元测验模拟卷解析

📚 Year 13 AQA Maths: Unit Test Mock Paper Walkthrough | AQA 数学 Year 13 单元测验模拟卷解析

This article provides a full, step-by-step walkthrough of a Year 13 AQA Mathematics unit test mock paper. The paper is built around core pure topics: inverse trigonometric differentiation, implicit differentiation, trigonometric integration, parametric equations, first-order differential equations and 3D vectors. Every question is solved in detail, common pitfalls are highlighted, and mark scheme tips are shared so you can see exactly how examiners award marks. Use this as both a revision tool and an exam-technique workout.

本文为 AQA 数学 Year 13 单元测验模拟卷提供逐题精讲。试卷紧扣纯数核心考点：反三角函数求导、隐函数微分、三角积分、参数方程、一阶微分方程和三维向量。每道题都给出详细求解步骤，指出常见错误，并分享评分标准要点，帮助你透彻理解知识点并掌握得分技巧。

1. Mock Paper Overview | 模拟卷概览

This mock paper is designed as a 50‑minute unit test worth 50 marks. It contains five questions that sample the key pure mathematics content for Year 13 AQA: differentiation of inverse trig functions and implicit differentiation, integration using trigonometric identities, parametric differentiation and area, solving a separable differential equation, and vector geometry in three dimensions. The questions are styled to match the phrasing and demand of real AQA papers, mixing straightforward manipulations with multi-step problem solving.

本模拟卷按 50 分钟、50 分的单元测验设计。共 5 道大题，覆盖 Year 13 AQA 纯数的核心内容：反三角函数与隐函数求导、三角恒等式积分、参数方程求导与面积、可分离变量的一阶微分方程、以及三维向量几何。题目风格贴近真实 AQA 考题，既有直接计算，也有多步骤的综合求解。

2. Q1: Inverse Trig and Implicit Differentiation | 反三角函数与隐函数求导

Question 1 (10 marks)
(a) Given f(x) = arctan(3x), find f'(x). [3 marks]
(b) The curve C has equation x³ + y³ = 6xy. Find dy/dx in terms of x and y. [7 marks]

Solution (a)

We use the standard result d/dx [arctan(u)] = u’/(1+u²). Let u = 3x, so u’ = 3. Then f'(x) = 3/(1+(3x)²) = 3/(1+9x²).

利用公式 d/dx [arctan(u)] = u’/(1+u²)。令 u = 3x，则 u’ = 3。因此 f'(x) = 3/(1+9x²)。

Solution (b)

Differentiate both sides of x³ + y³ = 6xy with respect to x, remembering that y is a function of x. This gives 3x² + 3y²(dy/dx) = 6y + 6x(dy/dx). Collect the dy/dx terms on one side: 3y²(dy/dx) – 6x(dy/dx) = 6y – 3x². Factorise: dy/dx (3y² – 6x) = 6y – 3x². Finally, divide through: dy/dx =

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