📚 AQA A Level Maths June 2018 Pure Mathematics Paper: Key Topics and Exam Solutions | AQA A Level 数学 2018年6月纯数学试卷：核心考点与解题解析
The AQA A Level Mathematics June 2018 Pure Mathematics paper (7357/1) was the first sitting of the new linear specification. It tested a broad range of pure topics including algebra, functions, trigonometry, exponentials, calculus, proof, vectors and sequences. This article breaks down the main question types, common pitfalls and revision priorities.
AQA A Level 数学 2018 年 6 月纯数学试卷（7357/1）是新线性大纲的首次考试。试卷涵盖了代数、函数、三角、指数、微积分、证明、向量和数列等广泛的纯数学主题。本文拆解主要题型、常见错误与复习重点。
1. Paper Overview and Structure | 试卷概况与结构
The June 2018 Pure Mathematics paper lasted 2 hours and carried 100 marks. It contained around 15 to 17 compulsory questions, ranging from short skill checks to multi-step problem solving. No calculator restrictions applied under the new AQA specification, but exact values and algebraic manipulation were still essential throughout the paper.
2018 年 6 月纯数学试卷考试时间为 2 小时，满分 100 分。试卷包含约 15 至 17 道必答题，从简短的技能考查到多步问题解决。新 AQA 大纲下纯数学试卷允许使用计算器，但精确值与代数运算在全卷中仍然至关重要。
The paper was designed to assess fluency in basic techniques as well as the ability to apply those techniques in unfamiliar contexts. Later questions often linked algebra, calculus and geometry, so candidates needed to switch quickly between different areas of pure mathematics.
试卷旨在考查基本技能的熟练程度以及在陌生情境中应用这些技能的能力。后面的题目常常将代数、微积分和几何结合起来，因此考生需要能够在纯数学的不同领域之间快速切换。
2. Algebra and Surds | 代数与根式
Early questions in the paper often required simplifying surds, rationalising denominators and manipulating indices. For example, an expression such as (√5 + 2)/(√5 − 2) can be rationalised by multiplying both the numerator and the denominator by the conjugate √5 + 2. This process removes the radical from the denominator and gives an answer in the form a + b√c.
试卷前面的题目通常要求化简根式、分母有理化以及处理指数。例如，形如 (√5 + 2)/(√5 − 2) 的表达式可通过将分子和分母同时乘以共轭根式 √5 + 2 来进行有理化。这一过程可以消去分母中的根号，并将答案写成 a + b√c 的形式。
(√5 + 2)/(√5 − 2) × (√5 + 2)/(√5 + 2) = (9 + 4√5)/1 = 9 + 4√5
Common errors included incorrect expansion of brackets or forgetting to simplify the final surd. It is important to check that the denominator becomes a rational number and that the final expression is in its simplest form.
常见错误包括括号展开不正确或忘记化简最终根式。务必检查分母是否已变为有理数，并确保最终表达式为最简形式。
3. Quadratics and the Discriminant | 二次函数与判别式
ant is given by Δ = b² − 4ac. The sign of Δ tells us how many real roots the equation has.
二次函数题通常要求完成平方、求顶点，或使用判别式判断实根个数。对于二次函数 f(x) = ax² + bx + c，判别式为 Δ = b² − 4ac。判别式的符号可以说明方程有多少个实根。
Δ = b² − 4ac
A sketch of the parabola helps identify the region where the curve lies below the x-axis.
若 Δ > 0，则有两个不同实根；若 Δ = 0，则有一个重根；若 Δ < 0，则无实根。在 2018 年 6 月试卷中，考生还需解释形如 x² + 6x + 5 ≤ 0 的不等式。画出抛物线草图有助于确定曲线位于 x 轴下方的区间。
4. Functions and Transformations | 函数与图像变换
= (x − 3)/2. The order matters in a composite function: fg(x) means apply g first, then f.
函数题包括求反函数、构造复合函数以及描述图像变换。若 f(x) = 2x + 3，则通过将 y = 2x + 3 改写为以 x 为主语，可得反函数 f⁻¹(x) = (x − 3)/2。复合函数中顺序很重要：fg(x) 表示先作用 g，再作用 f。
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