AQA OxfordAQA 9660 MA01 Pure Mathematics 1: June 2023 Exam Revision | AQA OxfordAQA 9660 MA01 纯数学一：2023年6月考试复习

📚 AQA OxfordAQA 9660 MA01 Pure Mathematics 1: June 2023 Exam Revision | AQA OxfordAQA 9660 MA01 纯数学一：2023年6月考试复习

This revision guide focuses on the core skills assessed in AQA OxfordAQA 9660 MA01 Pure Mathematics 1, as seen in the June 2023 written paper (WRE). It breaks down the main topic areas, common question styles, and key techniques you need to master.

本复习指南聚焦 AQA OxfordAQA 9660 MA01 纯数学一的考查核心技能，内容基于 2023 年 6 月笔试真题（WRE）。我们拆解主要知识板块、常见题型以及你必须掌握的关键解题方法。

1. Algebra and Surds | 代数与根式

Simplifying surds, rationalising denominators, and expanding brackets are nearly always tested. For example, to rationalise 1/(√2 + 1), multiply numerator and denominator by the conjugate √2 - 1.

化简根式、分母有理化以及展开括号几乎是必考内容。例如，要对 1/(√2 + 1) 分母有理化，分子分母同乘共轭根式 √2 - 1。

x = [-b ± √b² - 4ac] / (2a)

real roots, if Δ = 0 there is one repeated root, and if Δ < 0 there are no real roots.

配方法和求根公式至关重要。判别式 Δ = b² - 4ac 说明实根个数：若 Δ > 0 有两个不同实根，若 Δ = 0 有一个重根，若 Δ < 0 无实根。

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2. Functions and Graphs | 函数与图像

write y = f(x), swap x and y, then rearrange; remember to state the domain of the inverse.

你必须理解定义域、值域、一一对应函数和反函数。求反函数时，先写 y = f(x)，交换 x 和 y 后重新整理；记得写出反函数的定义域。

y = f(x) → y = f(x + a), y = f(x) + a, y = af(x), y = f(ax)

Transformations include vertical shifts by adding a constant, horizontal shifts by changing x to x + a, vertical stretches by multiplying the whole function, and horizontal stretches by multiplying x inside the function. Be precise about the order when two transformations are combined.

图像变换包括：加常数实现竖直平移，将 x 换成 x + a 实现水平平移，整体乘常数实现竖直拉伸，以及在函数内部乘常数实现水平拉伸。当两种变换组合时，要注意顺序。

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3. Coordinate Geometry | 坐标几何

form y - y₁ = m(x - x₁) is often the fastest way to write the equation of a straight line.

过 (x₁, y₁) 和 (x₂, y₂) 的直线斜率为 (y₂ - y₁)/(x₂ - x₁)。点斜式 y - y₁ = m(x - x₁) 通常是写出直线方程最快的方法。

m₁ × m₂ = -1

+ y² + 2gx + 2fy + c = 0 into (x - a)² + (y - b)² = r², giving centre (a, b) and radius r.

垂直直线的斜率乘积为 -1，因此若一条直线斜率为 m，与之垂直的直线斜率为 -1/m。对于圆，配方法把 x² + y² + 2gx + 2fy + c = 0 化为 (x - a)² + (y - b)² = r²，得到圆心 (a, b) 和半径 r。

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4. Sequences and Series | 数列与级数

difference. The sum of the first n terms is Sₙ = n/2 [2a + (n - 1)d] or Sₙ = n/2 (a + l).

等差数列中，第 n 项为 uₙ = a + (n - 1)d，其中 a 为首项，d 为公差。前 n 项和为 Sₙ = n/2 [2a + (n - 1)d] 或 Sₙ = n/2 (a + l)。

uₙ = arⁿ⁻¹, Sₙ = a(1 - rⁿ)/(1 - r)

ⁿ⁻¹ and Sₙ = a(1 - rⁿ)/(1 - r) for r ≠ 1. An infinite geometric series converges only when |r| < 1, and its sum to infinity is S∞ = a/(1 - r).

等比数列中，uₙ = arⁿ⁻¹，Sₙ = a(1 - rⁿ)/(1 - r)（r ≠ 1）。无穷等比级数仅在 |r| < 1 时收敛，其无穷和为 S∞ = a/(1 - r)。

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5. Trigonometry | 三角学

AST diagrams or graph sketches to solve equations such as sin θ = 0.5 in a given interval.

必须掌握 30°、45°、60° 的正弦、余弦和正切精确值，可由标准 45-45-90 和 30-60-90 三角形快速推导。使用 CAST 图或函数图像草图解方程，例如 sin θ = 0.5 在给定区间内的解。

sin² θ + cos² θ = 1, sin θ / cos θ = tan θ

rule a/sin A = b/sin B = c/sin C and the cosine rule a² = b² + c² - 2bc cos A are needed.

恒等式 sin θ / cos θ = tan θ 和 sin² θ + cos² θ = 1 是化简表达式与解较难方程的核心。对于非直角三角形，需要正弦定理 a/sin A = b/sin B = c/sin C 和余弦定理 a² = b² + c² - 2bc cos A。

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6. Exponentials and Logarithms | 指数与对数

ₐ(x/y), and k logₐ x = logₐ(xᵏ). The change of base formula is logₐ x = log_b x / log_b a.

指数与对数法则是基础：logₐ x + logₐ y = logₐ(xy)，logₐ x - logₐ y = logₐ(x/y)，k logₐ x = logₐ(xᵏ)。换底公式为 logₐ x = log_b x / log_b a。

x log 2 = log 7 → x = log 7 / log 2

To solve equations like 2ˣ For natural logarithms, ln e = 1, and the functions e and ln cancel as inverse operations.

解 2ˣ = 7 这类方程时，两边取对数：x log 2 = log 7，所以 x = log 7 / log 2。对于自然对数，ln e = 1，e 与 ln 互为逆运算可以相互抵消。

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7. Differentiation | 微分

ituting the x-value into dy/dx. The tangent equation is y - y₁ = m(x - x₁) with m = dy/dx.

对于 y = xⁿ，导数为 dy/dx = nxⁿ⁻¹。曲线上某点的斜率代入 x 值到 dy/dx 得到。切线方程为 y - y₁ = m(x - x₁)，其中 m = dy/dx。

dy/dx = 0 at stationary points

um, if d²y/dx² < 0 it is a local maximum, and if d²y/dx² = 0 further testing is needed.

驻点出现在 dy/dx = 0 处。使用二阶导数 d²y/dx² 或 dy/dx 的符号变化判断：若 d²y/dx² > 0 为局部极小值，若 d²y/dx² < 0 为局部极大值，若 d²y/dx² = 0 则需要进一步检验。

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8. Integration | 积分

≠ -1. Definite integrals ∫ₐᵇ f(x) dx give the exact area between the curve and the x-axis.

积分是微分的逆运算。对于 y = xⁿ，∫ xⁿ dx = xⁿ⁺¹/(n + 1) + C（n ≠ -1）。定积分 ∫ₐᵇ f(x) dx 给出曲线与 x 轴之间的精确面积。

Area = F(b) - F(a), where F'(x) = f(x)

re F is the antiderivative. Watch out for regions below the x-axis, where the integral is negative and must be adjusted or split into separate parts.

求曲线下 x = a 到 x = b 的面积时，计算 F(b) - F(a)，其中 F 是原函数。注意 x 轴下方的区域积分值为负，需调整或分段计算。

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9. Proof and Problem Solving | 证明与问题解决

Simple proof by deduction, by exhaustion, or by contradiction appears frequently in the written paper. Make every step clear and use logical connectives such as 'if...then' and 'therefore' to show the flow of reasoning.

演绎法、穷举法或反证法的简单证明经常出现在笔试中。每一步要清晰，并使用 '如果...那么'、'因此' 等逻辑连接词来展示推理过程。

Let n = 2k or n = 2k + 1

Word problems often combine several Pure

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