📚 Pure Mathematics 1 Coursebook: Core Topics and Exam Skills | 纯数学1教材：核心主题与应试技巧
This revision guide walks through the main content areas of the AS Level Pure Mathematics 1 coursebook. It is designed to help you recall key methods, avoid common errors, and apply formulas confidently in exam-style questions.
本复习指南梳理 AS Level 纯数学 1 教材的主要板块，帮助你回忆关键方法、避开常见错误，并在考试题型中自信地使用公式。
1. Quadratics | 二次函数
Quadratics appear in factorisation, completing the square, the discriminant, and quadratic inequalities. You must be able to move between the general form ax² + bx + c, the completed square form a(x - h)² + k, and the factorised form a(x - p)(x - q).
二次函数考查因式分解、配方法、判别式以及二次不等式。你必须能在一般式 ax² + bx + c、配方式 a(x - h)² + k 和乘积式 a(x - p)(x - q) 之间熟练转换。
eal roots, if Δ = 0 there is one repeated real root, and if Δ < 0 there are no real roots.
判别式 Δ = b² - 4ac 可以判断根的性质：Δ > 0 时有两个不同实根，Δ = 0 时有一个二重实根，Δ < 0 时没有实根。
ax² + bx + c = 0 ⇒ x = (-b ± √b² - 4ac) / (2a)
values x = 2 and x = 3 split the number line, and a quick sign test gives x < 2 or x > 3.
解二次不等式 x² - 5x + 6 > 0 时，先因式分解得到 (x - 2)(x - 3) > 0。临界值 x = 2 和 x = 3 将数轴分段，快速判断符号后解为 x < 2 或 x > 3。
2. Functions | 函数
A function f assigns exactly one output f(x) to each input x. The domain is the set of allowed inputs, and the range is the set of possible outputs produced by the domain.
函数 f 将每个输入 x 对应到唯一输出 f(x)。定义域是允许的输入集合，值域是定义域对应产生的输出集合。
y, then solve for y. A composite function such as gf(x) means apply f first, then apply g.
反函数 f⁻¹ 只有在函数是一一对应时才存在。求反函数的方法是：令 y = f(x)，交换 x 与 y，再解出 y。复合函数 gf(x) 表示先作用 f，再作用 g。
- y = f(x) + a — vertical translation by a | 垂直平移 a
- y = f(x + a) — horizontal translation left by a | 向左平移 a
- y = -f(x) — reflection in the x-axis | 关于 x 轴对称
- y = f(-x) — reflection in the y-axis | 关于 y 轴对称
- y = af(x) — vertical stretch scale factor a | 垂直拉伸 a 倍
- y = f(ax) — horizontal stretch scale factor 1/a | 水平拉伸 1/a 倍
3. Coordinate Geometry | 坐标几何
(y₂ - y₁)/(x₂ - x₁). The equation of a straight line can be written as y - y₁ = m(x - x₁).
对于直线，两点 (x₁, y₁) 与 (x₂, y₂) 之间的斜率为 m = (y₂ - y₁)/(x₂ - x₁)。直线方程可以写成 y - y₁ = m(x - x₁)。
two points is √((x₂ - x₁)² + (y₂ - y₁)²), and the midpoint is ((x₁ + x₂)/2, (y₁ + y₂)/2).
互相垂直的直线斜率之积为 -1，即 m₁ × m₂ = -1。两点间距离为 √((x₂ - x₁)² + (y₂ - y₁)²)，中点坐标为 ((x₁ + x₂)/2, (y₁ + y₂)/2)。
(y - b)² = r². The general form x² + y² + 2gx + 2fy + c = 0 has centre (-g, -f) and radius √g² + f² - c.
圆心为 (a, b)、半径为 r 的圆方程为 (x - a)² + (y - b)² = r²。一般式 x² + y² + 2gx + 2fy + c = 0 的圆心为 (-g, -f)，半径为 √g² + f² - c。
A tangent to a circle is perpendicular to the radius at the point of contact. Completing the square is often needed to convert a circle equation into centre-radius form.
圆的切线与切点处半径垂直。考试中经常需要用配方法将圆方程转化为圆心-半径形式。
4. Circular Measure | 弧度制
radius. The conversion is π rad = 180°, so to change degrees to radians multiply by π/180.
弧度制用弧长除以半径来度量角。换算关系为 π rad = 180°，因此角度转弧度要乘以 π/180。
, where θ is measured in radians. The area of a segment is the sector area minus the area of the triangle formed by the two radii and the chord.
半径为 r 的圆中，弧长公式为 s = rθ，扇形面积公式为 A = ½r²θ，其中 θ 必须以弧度为单位。弓形面积等于扇形面积减去两条半径和弦构成的三角形面积。
s = rθ A = ½r²θ = ½rs
5. Trigonometry | 三角函数
You must know the exact values of sin θ, cos θ, and tan θ for 0°, 30°, 45°, 60°, and 90°. In radians these are 0, π/6, π/4, π/3, and π/2.
你必须熟记 0°、30°、45°、60° 和 90° 时 sin θ、cos θ、tan θ 的精确值。用弧度表示为 0、π/6、π/4、π/3 和 π/2。
| θ | 0° | 30° | 45° | 60° | 90° |
|---|---|---|---|---|---|
| sin θ | 0 | 1/2 | √2/2 | √3/2 | 1 |
| cos θ | 1 | √3/2 | √2/2 | 1/2 | 0 |
| tan θ | 0 | 1/√3 | 1 | √3 | undefined |
os θ. To solve trig equations, use the quadrant rule or draw the graph over the required interval.
核心恒等式为 sin²θ + cos²θ = 1，比值恒等式为 tan θ = sin θ / cos θ。解三角方程时要使用象限规则或画出指定区间内的函数图像。
2π gives θ = π/6 and θ = 5π/6, because sine is positive in the first and second quadrants.
例如，在 0 ≤ θ ≤ 2π 内解 sin θ = 1/2，可得 θ = π/6 和 θ = 5π/6，因为正弦在第一象限和第二象限均为正。
6. Series | 数列与级数
r a positive integer n is (a + b)ⁿ = aⁿ + nC1 aⁿ⁻¹b + nC2 aⁿ⁻²b² + ... + bⁿ, where nCr = n! / (r!(n - r)!). Pascal's triangle gives the coefficients quickly for small n.
正整数 n 的二项展开式为 (a + b)ⁿ = aⁿ + nC1 aⁿ⁻¹b + nC2 aⁿ⁻²b² + ... + bⁿ，其中 nCr = n! / (r!(n - r)!)。当 n 较小时，帕斯卡三角可以快速给出系数。
d] = n/2 (a + l), where a is the first term, d the common difference, and l the last term.
等差数列的第 n 项为 aₙ = a + (n - 1)d，前 n 项和为 Sₙ = n/2 [2a + (n - 1)d] = n/2 (a + l)，其中 a 是首项，d 是公差，l 是末项。
on has nth term aₙ = arⁿ⁻¹ and sum Sₙ = a(1 - rⁿ)/(1 - r), where r is the common ratio. If |r| < 1, the infinite sum converges to S∞ = a/(1 - r).
等比数列的第 n 项为 aₙ = arⁿ⁻¹，前 n 项和为 Sₙ = a(1 - rⁿ)/(1 - r)，其中 r 是公比。如果 |r| < 1，则无穷级数收敛，其和为 S∞ = a/(1 - r)。
7. Differentiation | 微分
ntiation gives the gradient function of a curve. The derivative from first principles is f' lim h→0 [f(x + h) - f(x)] / h. For practical work, use the power rule: d/dx (xⁿ) = n xⁿ⁻¹.
微分给出曲线的梯度函数。从第一性原理出发，导数为 f'(x) = lim h→0 [f(x + h) - f(x)] / h。实际做题时使用幂函数求导法则：d/dx (xⁿ) = n xⁿ⁻¹。
sum of the derivatives, and constants multiply through. If y = axⁿ, then dy/dx = an xⁿ⁻¹.
和的导数等于导数的和，常数可以提出。如果 y = axⁿ，则 dy/dx = an xⁿ⁻¹。
gradient is -1/(dy/dx). The equation of the tangent or normal is then y - y₁ = m(x - x₁).
曲线上某点的切线斜率就是 dy/dx。法线与切线垂直，因此法线斜率为 -1/(dy/dx)。切线或法线方程可以写成 y - y₁ = m(x - x₁)。
Stationary points occur where f'(x) = 0. Use the second derivative: if f''(x) > 0 the point is a minimum, if f''(x) < 0 it is a maximum, and if f''(x) = 0 further investigation is needed.
驻点出现在 f'(x) = 0 处。用二阶导判断：若 f''(x) > 0 则为极小值点，若 f''(x) < 0 则为极大值点，若 f''(x) = 0 则需要进一步判断。
8. Integration | 积分
/(n + 1) + c, for n ≠ -1. The constant c must always be included for indefinite integrals.
积分是微分的逆运算。基本公式为 ∫ xⁿ dx = xⁿ⁺¹/(n + 1) + c，其中 n ≠ -1。不定积分必须始终加上常数 c。
ative of f. It gives the signed area between the curve and the x-axis from x = a to x = b.
定积分的计算方式为 ∫ₐᵇ f(x) dx = F(b) - F(a)，其中 F 是 f 的一个原函数。它表示曲线与 x 轴在 x = a 到 x = b 之间所夹的带符号面积。
If the curve lies below the x-axis, the definite integral is negative. To find the actual area, integrate separately above and below the axis and take absolute values, or integrate the difference between two curves.
如果曲线位于 x 轴下方，定积分为负。求实际面积时需要分别在 x 轴上方和下方积分并取绝对值，或者对两条曲线的差进行积分。
9. Exam Technique and Common Errors | 考试技巧与常见错误
Always show your method clearly. Marks are awarded for correct substitution, rearrangement, and final accuracy, so even if the final answer is wrong, clear working can still earn method marks.
始终清晰地展示解题过程。评分会给在正确代入、变形和最终结果上，因此即使最终答案错误，清晰的步骤仍然可以获得方法分。
Use radians in circular measure and in calculus when the variable is an angle. Check whether your calculator is in degree or radian mode before starting a trig or circular measure question.
在弧度制和含角度变量的微积分题中要使用弧度制。开始做三角或弧度制题目前，先检查计算器处于角度模式还是弧度模式。
Common errors include forgetting the constant c in integration, writing √9 uality signs incorrectly, and ignoring domain restrictions when finding inverse functions.
常见错误包括：积分时忘记常数 c；解 x² = 9 时把 √9 只写成 3 而漏掉 ±3；不等式变号错误；求反函数时忽略定义域限制。
10. Formula Checklist and Study Plan | 公式清单与学习计划
Before your assessment, make sure you can recall the quadratic formula and discriminant, the equation of a circle, arc length and sector area, exact trig values, AP and GP formulas, the power rule, and the integration formula.
考试前请确认自己能熟练回忆：二次方程求根公式与判别式、圆的方程、弧长与扇形面积、三角精确值、等差数列与等比数列公式、幂函数求导法则以及积分基本公式。
A useful study plan is to focus on one topic per session, complete past-paper questions under timed conditions, and keep a mistake log. Revisit the mistake log weekly so errors are not repeated in the real exam.
一个有效的学习计划是：每次集中复习一个主题，在限时条件下完成历年真题，
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