📚 Year 12 Cambridge Mathematics: Core Knowledge Review | 剑桥 Year 12 数学核心知识点梳理
Year 12 Cambridge AS Mathematics builds directly on IGCSE and introduces the foundational topics of Pure Mathematics. A clear understanding of algebra, functions, trigonometry, coordinate geometry and calculus is essential, not only for the AS exam but also for progression to A Level. This article brings together the core knowledge points you must master, with concise explanations and key formulas – a revision companion designed for Cambridge candidates.
剑桥 Year 12 AS 数学在 IGCSE 的基础上直接延伸，并导入纯数学的基础主题。扎实掌握代数、函数、三角学、坐标几何和微积分，不仅是应对 AS 考试的需要，也是顺利衔接 A Level 的关键。本文梳理了必须掌握的核心知识点，用简洁的解释和关键公式，为剑桥考生提供一份复习伴侣。
1. Algebraic Foundations | 代数基础
Manipulating algebraic expressions confidently is the first requirement. The laws of indices must be extended to rational exponents, and surd expressions should be simplified and rationalised.
熟练处理代数式是首要要求。指数律必须推广到有理指数，根式表达式需要化简与有理化。
Index rules for all rational m, n:
- am × an = am+n
- am ÷ an = am−n
- (am)n = amn
- a0 = 1, a−n = 1/an
- am/n = n√(am)
所有有理指数 m, n 的运算规则：同底数幂相乘指数相加；相除指数相减；幂的乘方指数相乘；零指数得 1，负指数变为倒数；分数指数表示为根式。
Surds like √8 should be simplified to 2√2. To rationalise a denominator such as 1/(√a + √b), multiply numerator and denominator by the conjugate √a − √b, exploiting the difference of two squares.
根式如 √8 应化简为 2√2。对分母 1/(√a + √b) 有理化时，分子分母同乘共轭根式 √a − √b，利用平方差公式化简。
Factorising quadratics, difference of squares and grouping are essential: x² − 9 = (x−3)(x+3); 6x² + x − 2 = (2x−1)(3x+2). Always check by expanding.
二次三项式的因式分解、平方差和分组分解法必须熟练：x² − 9 = (x−3)(x+3)；6x² + x − 2 = (2x−1)(3x+2)。始终通过展开验证。
2. Quadratic Functions | 二次函数
The general quadratic f(x) = ax² + bx + c can be rewritten in vertex form by completing the square: a(x + p)² + q, where the turning point is (−p, q).
一般二次函数 f(x) = ax² + bx + c 可通过配方法写成顶点式 a(x + p)² + q，其顶点为 (−p, q)。
Completing the square: for x² + bx, add and subtract (b/2)². For ax² + bx + c, factor a out of the x terms first.
配方法：对于 x² + bx，加减 (b/2)²；对于 ax² + bx + c，先提取系数 a。
The discriminant Δ = b² − 4ac determines the nature of roots:
- Δ > 0: two distinct real roots
- Δ = 0: one repeated real root
- Δ < 0: no real roots
判别式 Δ = b² − 4ac 决定根的性质：Δ > 0 有两个不等实根；Δ = 0 有一个重根；Δ < 0 没有实根。
Quadratic functions can model projectile motion and optimisation problems; always check the sign of a to decide whether the parabola opens upwards (a > 0) or downwards (a < 0).
二次函数可模拟抛射运动和优化问题；始终检查 a 的符号以判定抛物线开口向上 (a > 0) 还是向下 (a < 0)。
3. Equations and Inequalities | 方程与不等式
Solving quadratic equations can be done by factorising, completing the square, or applying the quadratic formula:
x = [−b ± √(b² − 4ac)] / (2a)
解二次方程可以用因式分解、配方法或求根公式。
Simultaneous equations often pair a linear and a quadratic. Substitute the linear expression into the quadratic, solve the resulting quadratic, and substitute back to find the other variable. Always check both pairs satisfy both original equations.
联立方程组通常包含一次与二次方程。将一次表达式代入二次方程，解所得二次方程，再回代求另一变量。养成检验两组解是否满足原方程的习惯。
When solving linear inequalities, treat them like equations but reverse the inequality sign when multiplying or dividing by a negative number.
解一元一次不等式时，方法与方程相同，但乘以或除以负数时需反转不等号。
For quadratic inequalities, sketch the graph and identify where the curve is above or below the x‑axis. Write the solution using interval notation or set notation, and show on a number line.
解二次不等式时，画出示意草图，判断曲线在 x 轴上方还是下方的区间。用区间或集合表示解集，并在数轴上标出。
4. Coordinate Geometry: Straight Lines | 坐标几何：直线
The distance between two points (x₁, y₁) and (x₂, y₂) is √[(x₂−x₁)² + (y₂−y₁)²]; the midpoint is ((x₁+x₂)/2, (y₁+y₂)/2).
两点 (x₁, y₁) 与 (x₂, y₂) 的距离为 √[(x₂−x₁)² + (y₂−y₁)²]；中点坐标为 ((x₁+x₂)/2, (y₁+y₂)/2)。
The gradient m of a straight line through these points is (y₂−y₁)/(x₂−x₁). A line with gradient m and y‑intercept c has equation y = mx + c. The point–gradient form is y − y₁ = m(x − x₁).
过两点的直线斜率 m = (y₂−y₁)/(x₂−x₁)。斜率为 m、y 截距为 c 的直线方程为 y = mx + c。点斜式为 y − y₁ = m(x − x₁)。
Parallel lines have equal gradients (m₁ = m₂). Perpendicular lines satisfy m₁ × m₂ = −1. Use these conditions to find equations of lines parallel or perpendicular to a given line through a specified point.
平行直线斜率相等 (m₁ = m₂)；垂直直线满足 m₁ × m₂ = −1。利用这些条件可求过特定点且与已知直线平行或垂直的直线方程。
5. Circles | 圆
The standard equation of a circle with centre (a, b) and radius r is (x − a)² + (y − b)² = r².
以 (a, b) 为圆心、半径为 r 的圆的标准方程为 (x − a)² + (y − b)² = r²。
When the equation is given in expanded form x² + y² + Dx + Ey + F = 0, complete the square for x and y to find the centre and radius. For example, x² + y² − 4x + 6y − 3 = 0 becomes (x−2)² + (y+3)² = 16, centre (2, −3), radius 4.
若给出的方程为 x² + y² + Dx + Ey + F = 0，需对 x 和 y 分别配方来求圆心和半径。例如 x² + y² − 4x + 6y − 3 = 0 经配方得 (x−2)² + (y+3)² = 16，圆心 (2, −3)，半径 4。
The tangent to a circle at a point P has gradient equal to the negative reciprocal of the gradient of the radius OP. Use this to find the equation of the tangent, or to determine the coordinates of a point where a line touches the circle.
圆上一点 P 的切线斜率等于半径 OP 斜率的负倒数。利用这个性质可求切线方程，或确定直线与圆相切的切点坐标。
6. Binomial Expansion | 二项式展开
For a positive integer n, the binomial expansion is (a + b)ⁿ = Σ [C(n, k) aⁿ⁻ᵏ bᵏ] from k=0 to n, where C(n, k) = n! / [k!(n−k)!].
对正整数 n，二项式展开为 (a + b)ⁿ = Σ [C(n, k) aⁿ⁻ᵏ bᵏ] (k 从 0 到 n)，其中 C(n, k) = n! / [k!(n−k)!]。
When expanding (1 + x)ⁿ, the expansion simplifies to 1 + nx + [n(n−1)/2!] x² + [n(n−1)(n−2)/3!] x³ + … . This form is especially useful for approximations when x is small.
展开 (1 + x)ⁿ 时，展开式简化为 1 + nx + [n(n−1)/2!] x² + [n(n−1)(n−2)/3!] x³ + …。当 x 很小时，这个形式常用于近似计算。
You may be asked to find a specific coefficient in an expansion, or to expand a product of binomials. Set up the terms carefully and collect like powers.
考试可能要求找出展开式中某一特定项的系数，或展开两个二项式的乘积。仔细罗列各项，合并同次幂即可。
7. Trigonometry | 三角学
Exact values for sin, cos, tan of 0°, 30°, 45°, 60°, 90° must be memorised. They are derived from the special triangles 45°-45°-90° and 30°-60°-90°.
- sin 30° = 1/2, cos 60° = 1/2, tan 45° = 1
- sin 45° = cos 45° = 1/√2 = √2/2
- sin 60° = cos 30° = √3/2
特殊角 0°, 30°, 45°, 60°, 90° 的 sin, cos, tan 精确值必须记住，它们源于 45° 和 30°-60° 的直角三角板。
The sine rule (a/sin A = b/sin B = c/sin C) is used for non‑right‑angled triangles when you know either two angles and a side, or two sides and a non‑included angle (but beware the ambiguous case).
正弦定理 a/sin A = b/sin B = c/sin C 用于非直角三角形，适用情形为已知两角一边，或两边及一个非夹角（但需注意多解情况）。
The cosine rule (a² = b² + c² − 2bc cos A) applies when you know three sides or two sides and the included angle.
余弦定理 a² = b² + c² − 2bc cos A 适用于已知三边或两边及其夹角的情形。
To solve trigonometric equations in the range 0° to 360°, use the quadrant diagram (ASTC) to find all principal solutions before generating the general solution. Basic identities like sin²θ + cos²θ = 1 and tanθ = sinθ/cosθ are indispensable for simplifying expressions and proving identities.
在 0° 到 360° 范围内解三角方程时，用象限图 (ASTC) 先找出主要解，再生成所有解。基本恒等式 sin²θ + cos²θ = 1 和 tanθ = sinθ/cosθ 是化简与证恒等式的必备工具。
8. Vectors | 向量
A vector in two dimensions can be written as a column vector (＜上标 x 下标 y＞) or using unit vectors i and j: v = xi + yj. The magnitude |v| = √(x² + y²).
平面中的向量可写为列向量或使用单位向量 i 和 j 表示：v = xi + yj。向量的大小（模）为 |v| = √(x² + y²)。
Vector addition and scalar multiplication are performed component‑wise. The position vector of a point P is the vector from the origin to P: OP = p.
向量的加法与数乘按分量进行。点 P 的位置向量是从原点指向 P 的向量：OP = p。
If AB = b − a, then the vector from A to B is the difference of their position vectors. Vectors are parallel if one is a scalar multiple of the other.
向量 AB = b − a，即两点位置向量之差。若一个向量是另一向量的标量倍数，则两向量平行。
Vectors are used to solve geometric problems involving midpoints, ratios and collinearity. For example, points A, B, C are collinear if AB = k BC for some scalar k.
向量可用于解决涉及中点、比例和共线的几何问题。例如，若存在标量 k 使 AB = k BC，则 A, B, C 共线。
9. Differentiation | 微分
The derivative f'(x) is defined as the limit of the difference quotient: f'(x) = limh→0 [f(x+h) − f(x)] / h. For a power function f(x) = xⁿ, the derivative is f'(x) = n xⁿ⁻¹.
导数 f'(x) 定义为差商的极限：f'(x) = limh→0 [f(x+h) − f(x)] / h。对幂函数 f(x) = xⁿ，导数为 f'(x) = n xⁿ⁻¹。
Basic differentiation rules:
- Constant rule: d/dx (c) = 0
- Power rule: d/dx (xⁿ) = n xⁿ⁻¹
- Sum/difference rule: d/dx (u ± v) = u’ ± v’
基本求导法则：常数导数为 0；幂函数求导降幂；和差函数分别求导。
The gradient of a curve at a point is given by the derivative. The equation of the tangent is y − y₀ = f'(x₀)(x − x₀); the normal is perpendicular to the tangent, so its gradient is −1/f'(x₀).
曲线上某点切线的斜率等于导数。切线方程为 y − y₀ = f'(x₀)(x − x₀)；法线与切线垂直，斜率为 −1/f'(x₀)。
Stationary points occur where f'(x) = 0. Use the second derivative or a sign chart to classify: if f”(x) > 0, it’s a local minimum; if f”(x) < 0, a local maximum; if f''(x) = 0, further investigation is needed.
驻点发生在 f'(x) = 0 处。用二阶导数或符号表判定类型：f”(x) > 0 为局部极小；f”(x) < 0 为局部极大；f''(x) = 0 时需进一步讨论。
10. Integration | 积分
Integration is the reverse process of differentiation. The indefinite integral of xⁿ (n ≠ −1) is ∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + c, where c is the constant of integration.
积分是微分的逆运算。xⁿ (n ≠ −1) 的不定积分为 ∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + c，c 为积分常数。
To find the constant c, use an initial condition or a point through which the original function passes. Never forget the ‘+c’ in indefinite integration.
利用已知函数值或初始条件可求出常数 c。不定积分中务必加上 ‘+c’。
A definite integral ∫ₐᵇ f(x) dx represents the exact area between the curve y = f(x) and the x‑axis from x = a to x = b, but areas below the axis are subtracted in the raw evaluation. For total area, split the interval where the sign changes and take absolute values.
定积分 ∫ₐᵇ f(x) dx 表示曲线 y = f(x) 与 x 轴在 [a, b] 之间的净面积，x 轴下方部分会被减去。求总面积时，需在符号改变处分段并取绝对值相加。
When finding the area between a curve and a line, integrate the difference of the functions: ∫ (upper − lower) dx.
求曲线与直线围成的面积时，对上方函数减去下方函数的差值积分：∫ (上 − 下) dx。
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