📚 AS AQA Pure Mathematics PSM1 | AS AQA 纯数学 PSM1
This article provides a structured review of the core pure mathematics topics tested in the OxfordAQA International AS Mathematics (9660) Pure Mathematics PSM1 paper. It focuses on key techniques, common pitfalls, and exam-style reasoning.
本文系统梳理了牛津AQA国际AS数学（9660）纯数学PSM1试卷的核心考点，重点讲解关键技巧、常见错误以及考试型思维。
1. Algebraic Expressions | 代数表达式
You must be confident in manipulating algebraic expressions, including expanding brackets, factorising, and simplifying surds. Recognise the difference between exact values and decimal approximations.
你需要熟练掌握代数表达式的运算，包括去括号、因式分解以及化简根式。要区分精确值与十进制近似值。
- Surd rules: √ab = √a × √b, and √a/b = √a ÷ √b. Always simplify surds by removing square factors.
- 根式规则：√ab=√a×√b，√a/b=√a÷√b。化简根式时始终提取平方因数。
- Rationalising the denominator: multiply numerator and denominator by the conjugate when the denominator contains a surd.
- 分母有理化：当分母含有根式时，分子分母同乘共轭式。
- Index laws: aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ.
- 指数法则：aᵐ×aⁿ=aᵐ⁺ⁿ，aᵐ÷aⁿ=aᵐ⁻ⁿ，(aᵐ)ⁿ=aᵐⁿ。
Example: Express (√12 + √27) / √3 in the form a√b.
例：将 (√12+√27)/√3 化为 a√b 的形式。
Simplify each surd: √12 = 2√3, √27 = 3√3. Then (2√3 + 3√3)/√3 = 5√3/√3 = 5.
化简每个根式：√12=2√3，√27=3√3。于是 (2√3+3√3)/√3 = 5√3/√3 = 5。
2. Quadratics and Inequalities | 二次函数与不等式
Solving quadratic equations by factorising, completing the square, and the quadratic formula is essential. You also need to solve quadratic inequalities and understand the discriminant.
通过因式分解、配方法和求根公式解二次方程是基础。你还需要解二次不等式并理解判别式。
- Completing the square: x² + bx + c = (x + b/2)² − (b/2)² + c.
- 配方法：x²+bx+c = (x+b/2)² − (b/2)²+c。
- Discriminant: − 4ac. If Δ > 0, two distinct real roots; Δ = 0, one repeated root; Δ < 0, no real roots.
- 判别式：Δ=b²−4ac。若Δ>0，有两个不等实根；Δ=0，有一个重根；Δ<0，无实根。
- Inequality sign: For (x − p)(x − q) > 0, the solution is x < p or x > q (assuming p < q). For < 0, the solution is p < x < q.
- 不等式符号：对于(x−p)(x−q)>0，解为x<p或x>q（假设p<q）。对于<0，解为p<x<q。
The quadratic formula: x = (−b ± √b² − 4ac) / (2a)
求根公式：x = (−b ± √b²−4ac) / (2a)
When solving an inequality, always sketch the graph of the quadratic to avoid sign mistakes.
解不等式时，务必画出二次函数草图，避免符号错误。
3. Functions and Graphs | 函数与图像
Understand the domain and range of a function, composite functions, and inverse functions. Recognise transformations of graphs: translations and reflections.
理解函数的定义域与值域、复合函数和反函数。识别图像变换：平移和反射。
- Composite function: fg(x) = f(g(x)). Apply g first, then f.
- 复合函数：fg(x) = f(g(x))。先做g，再做f。
- Inverse function: -to-one functions have inverses. To find f⁻¹(x), swap x and y in y = f(x) and solve for y.
- 反函数：只有一一对应的函数才有反函数。求f⁻¹(x)时，在y=f(x)中将x与y互换，然后解出y。
- Translations: y = f(x) + a shifts vertically; y = f(x − a) shifts horizontally.
- 平移：y=f(x)+a垂直平移；y=f(x−a)水平平移。
- Reflections: y = −f(x) reflects in the x-axis; y = f(−x) reflects in the y-axis.
- 反射：y=−f(x)关于x轴反射；y=f(−x)关于y轴反射。
Always state the domain of an inverse function, which equals the range of the original function.
始终写出反函数的定义域，它等于原函数的值域。
4. Coordinate Geometry | 坐标几何
Lines, circles, and the distance formula appear frequently. You must be able to find midpoints, gradients, parallel and perpendicular lines, and equations of circles.
直线、圆和距离公式经常出现。你必须能求中点、斜率、平行与垂直直线，以及圆的方程。
- Gradient: m = (y₂ − y₁) / (x₂ − x₁).
- 斜率：m = (y₂−y₁)/(x₂−x₁)。
- Parallel lines: m₁ = m₂. Perpendicular lines: m₁m₂ = −1.
- 平行线：m₁=m₂。垂直线：m₁m₂=−1。
- Equation of a circle: (x − a)² + (y − b)² = r², where (a, b) is the centre and r is the radius.
- 圆的方程：(x−a)²+(y−b)²=r²，其中(a,b)为圆心，r为半径。
- Distance between two points: √((x₂ − x₁)² + (y₂ − y₁)²).
- 两点间距离：√((x₂−x₁)²+(y₂−y₁)²)。
Midpoint: ((x₁ + x₂)/2, (y₁ + y₂)/2)
中点：((x₁+x₂)/2, (y₁+y₂)/2)
When finding the equation of the tangent to a circle, use the fact that the radius is perpendicular to the tangent.
求圆的切线方程时，利用半径垂直于切线的性质。
5. Sequences and Series | 数列与级数
Arithmetic and geometric sequences are tested in AS. You need to use the nth term formulas and sums of finite series.
等差数列和等比数列在AS中会考到。你需要使用通项公式和有限项求和公式。
- Arithmetic nth term: uₙ = a + (n − 1)d.
- 等差数列通项：uₙ=a+(n−1)d。
- Arithmetic sum: Sₙ = n/2 [2a + (n − 1)d] = n/2 (a + l), where l is the last term.
- 等差数列求和：Sₙ=n/2[2a+(n−1)d] = n/2(a+l)，其中l为末项。
- Geometric nth term: uₙ = arⁿ⁻¹.
- 等比数列通项：uₙ=arⁿ⁻¹。
- Geometric sum: Sₙ = a(1 − rⁿ) / (1 − r), r ≠ 1.
- 等比数列求和：Sₙ=a(1−rⁿ)/(1−r)，r≠1。
For the sum to infinity of a geometric series, you must have |r| < 1. Then S∞ = a / (1 − r).
等比级数求和存在需满足|r|<1。此时S∞=a/(1−r)。
6. Trigonometry | 三角函数
AS trigonometry focuses on exact values, solving equations, and the sine and cosine rules for triangles. Radians may be introduced, so be comfortable converting.
AS三角学重点为精确值、解三角方程以及三角形的正弦定理和余弦定理。弧度制可能引入，所以应熟练转换。
- Exact values: sin 30° = ½, cos 30° = √3/2, tan 30° = 1/√3; sin 45° = √2/2, cos 45° = √2/2, tan 45° = 1; sin 60° = √3/2, cos 60° = ½, tan 60° = √3.
- 精确值：sin30°=½，cos30°=√3/2，tan30°=1/√3；sin45°=√2/2，cos45°=√2/2，tan45°=1；sin60°=√3/2，cos60°=½，tan60°=√3。
- Sine rule: a/sin A = b/sin B = c/sin C.
- 正弦定理：a/sin A = b/sin B = c/sin C。
- Cosine rule: a² = b² + c² − 2bc cos A.
- 余弦定理：a²=b²+c²−2bc cos A。
- Identities: tan θ = sin θ / cos θ; sin²θ + cos²θ = 1.
- 恒等式：tanθ=sinθ/cosθ；sin²θ+cos²θ=1。
5 in degrees, remember the symmetry of the sine graph: additional solutions may lie in the second quadrant.
解形如sinθ=0.5的方程（角度制）时，记住正弦图像的对称性：另一个解可能在第二象限。
7. Exponentials and Logarithms | 指数与对数
Logarithms are the inverse of exponentials. You must know the laws of logs and how to solve exponential equations using logarithms.
对数是指数的逆运算。你必须知道对数法则以及如何使用对数求解指数方程。
- Definition: logₐ x = b means aᵇ = x.
- 定义：logₐ x=b 等价于 aᵇ=x。
- Laws: logₐ x + logₐ y = logₐ(xy); logₐ x − logₐ y = logₐ(x/y); n logₐ x = logₐ(xⁿ).
- 法则：logₐ x+logₐ y=logₐ(xy)；logₐ x−logₐ y=logₐ(x/y)；n logₐ x=logₐ(xⁿ)。
- Special values: logₐ 1 = 0, logₐ a = 1.
- 特殊值：logₐ 1=0，logₐ a=1。
- Change of base: logₐ b = logₖ b / logₖ a.
- 换底公式：logₐ b=logₖ b/logₖ a。
To solve 2ˣ = 5, take logs of both sides: x = log₁₀5 / log₁₀2.
解2ˣ=5时，两边取对数：x=log₁₀5/log₁₀2。
8. Differentiation | 微分
Differentiation is a major topic in PSM1. You need to differentiate polynomials and use derivatives to find gradients, tangents, and normals.
微分是PSM1的主要话题。你需要对多项式求导，并利用导数求斜率、切线和法线。
- Power rule: d/dx (kxⁿ) = knxⁿ⁻¹.
- 幂法则：d/dx(kxⁿ)=knxⁿ⁻¹。
- Constant: d/dx (c) = 0.
- 常数：d/dx(c)=0。
- Gradient of tangent: substitute x into dy/dx.
- 切线的斜率：将x代入dy/dx。
- Normal gradient: mnormal = −1 / m_tangent.
- 法线斜率：m法线 = −1/m切线。
If y = xⁿ, then dy/dx = nxⁿ⁻¹.
若 y=xⁿ，则 dy/dx = nxⁿ⁻¹。
Always expand brackets before differentiating unless you can use the chain rule, which is not required at AS.
在求导前先展开括号，除非你能使用链式法则（AS不要求）。
9. Integration | 积分
Integration is the reverse of differentiation. You must be able to integrate polynomials and evaluate definite integrals to find areas.
积分是微分的逆运算。你必须能对多项式积分，并计算定积分求面积。
- Indefinite integral: ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, n ≠ −1.
- 不定积分：∫ xⁿ dx = xⁿ⁺¹/(n+1)+C，n≠−1。
- Definite integral: ∫ₐᵇ f'(x) dx = [f(x)]ₐᵇ = f(b) − f(a).
- 定积分：∫ₐᵇ f'(x) dx = [f(x)]ₐᵇ = f(b)−f(a)。
- Area under curve: A = ∫ₐᵇ y dx.
- 曲线下的面积：A=∫ₐᵇ y dx。
If the curve goes below the x-axis, the integral gives a negative value. Take the absolute value for the area, or split the interval at the roots.
若曲线在x轴下方，积分得负值。求面积时取绝对值，或在根处拆分区间。
10. Exam Tips and Common Mistakes | 考试技巧与常见错误
PSM1 rewards clear working and correct use of notation. Many students lose marks by skipping steps or misreading the question.
PSM1看重清晰的步骤和正确的符号书写。许多学生因跳步或读错题而丢分。
- Show all working: Method marks are awarded even if the final answer is wrong.
- 写出所有步骤：即使最终答案错误，方法分也会给。
- Check domain/range: When giving inverse functions, always state the domain.
- 检查定义域/值域：给出反函数时，必须写明定义域。
- Use radians if specified: In trigonometry, set your calculator correctly.
- 若指定弧度则用弧度：在三角题中，正确设置计算器模式。
- Revise surds and indices: These often appear in the first few questions and can be simple marks.
- 复习根式和指数：它们常出现在前几题，是容易拿到的分。
Manage your time: PSM1 usually has around 10–12 questions. Attempt every part; partial credit is available.
管理好时间：PSM1通常有10–12道题。尽量作答每一部分，部分分数是可以得到的。
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