📚 AS Mathematics Complete Revision Guide | AS 数学完整复习指南
AS Mathematics forms the foundation of advanced study in algebra, calculus, trigonometry and geometry. This revision guide gathers the core skills and formulas you must master, with exam-style insights to help you secure top marks.
AS 数学是代数、微积分、三角学与几何等高等学习的基石。本复习指南汇总了你必须掌握的核心技能与公式，并结合考试要点，助你冲击高分。
1. Algebra & Quadratics | 代数与二次函数
The quadratic expression can be written in three useful forms: expanded form ax² + bx + c, factorised form a(x − p)(x − q), and completed square form a(x − h)² + k. Each form reveals different features of the graph.
二次表达式有三种常用形式：展开式 ax² + bx + c、因式分解式 a(x − p)(x − q) 与配方式 a(x − h)² + k。每种形式都会揭示图像的不同特征。
To complete the square for x² + bx, add and subtract (b/2)². For example: x² + 6x + 8 = (x + 3)² − 9 + 8 = (x + 3)² − 1. The vertex of the parabola is (−3, −1).
对 x² + bx 配方时，需加减 (b/2)²。例如：x² + 6x + 8 = (x + 3)² − 9 + 8 = (x + 3)² − 1。抛物线顶点为 (−3, −1)。
The quadratic formula gives the roots of ax² + bx + c = 0:
x = (−b ± √(b² − 4ac)) / 2a
The discriminant Δ = b² − 4ac determines the nature of the roots: if Δ > 0, there are two distinct real roots; if Δ = 0, there is one repeated root; if Δ < 0, there are no real roots.
判别式 Δ = b² − 4ac 决定根的性质：若 Δ > 0，有两个不等实根；若 Δ = 0，有一个二重根；若 Δ < 0，没有实根。
When solving quadratic inequalities, sketch the parabola and read the x-values above or below the x-axis. For (x − 2)(x + 3) ≤ 0, the solution is −3 ≤ x ≤ 2.
解二次不等式时，画出抛物线草图，读取 x 轴上方或下方的 x 值。对于 (x − 2)(x + 3) ≤ 0，解集为 −3 ≤ x ≤ 2。
2. Coordinate Geometry | 坐标几何
The gradient of a line through points (x₁, y₁) and (x₂, y₂) is m = (y₂ − y₁) / (x₂ − x₁). Parallel lines have equal gradients, while perpendicular lines satisfy m₁ × m₂ = −1.
经过点 (x₁, y₁) 和 (x₂, y₂) 的直线斜率为 m = (y₂ − y₁) / (x₂ − x₁)。平行直线斜率相等，垂直直线满足 m₁ × m₂ = −1。
The equation of a straight line can be written as y = mx + c or y − y₁ = m(x − x₁). The midpoint of two points is ((x₁ + x₂)/2, (y₁ + y₂)/2), and the distance between them is √((x₂ − x₁)² + (y₂ − y₁)²).
直线方程可写作 y = mx + c 或 y − y₁ = m(x − x₁)。两点的中点为 ((x₁ + x₂)/2, (y₁ + y₂)/2)，两点间距离为 √((x₂ − x₁)² + (y₂ − y₁)²)。
The general equation of a circle with centre (a, b) and radius r is (x − a)² + (y − b)² = r². Expanding and rearranging gives the alternative form x² + y² + 2gx + 2fy + c = 0, where the centre is (−g, −f).
圆心为 (a, b)、半径为 r 的圆的标准方程为 (x − a)² + (y − b)² = r²。展开整理可得另一形式 x² + y² + 2gx + 2fy + c = 0，其中圆心为 (−g, −f)。
To find a tangent or normal to a circle at a given point, first determine the gradient of the radius to that point. The tangent is perpendicular to the radius.
求圆上某点处的切线与法线时，先求出该点半径的斜率。切线垂直于半径。
3. Circular Measure | 弧度制与扇形的度量
Radians are the natural unit for angles in advanced mathematics. A full circle of 360° equals 2π radians, so 180° = π rad and 90° = π/2 rad.
弧度是高等数学中角度的自然单位。一个完整圆周 360° 等于 2π 弧度，因此 180° = π rad，90° = π/2 rad。
To convert degrees to radians, multiply by π/180. To convert radians to degrees, multiply by 180/π. For example, 60° = π/3 and 1.5 rad ≈ 85.9°.
度转弧度需乘以 π/180，弧度转度需乘以 180/π。例如，60° = π/3，1.5 rad ≈ 85.9°。
For a circle of radius r and an angle θ measured in radians, the arc length s = rθ and the sector area A = ½r²θ.
对半径为 r 的圆，若圆心角 θ 以弧度为单位，则弧长 s = rθ，扇形面积 A = ½r²θ。
These formulas are linear in θ, which is why radian measure simplifies many calculations compared with degrees.
这些公式对 θ 是线性的，因此与度数相比，弧度制能极大简化许多计算。
4. Trigonometry | 三角学
For any angle θ, the fundamental identities are: sin²θ + cos²θ = 1 and tanθ = sinθ / cosθ. These are used constantly to simplify expressions and solve equations.
对任意角 θ，基本恒等式为：sin²θ + cos²θ = 1 以及 tanθ = sinθ / cosθ。这些恒等式常用于化简表达式和解方程。
You must know the exact values for common angles: sin30° = ½, cos30° = √3/2, tan45° = 1, sin60° = √3/2, cos60° = ½. In radians, 30° = π/6, 45° = π/4, 60° = π/3.
你必须牢记常见角的精确值：sin30° = ½，cos30° = √3/2，tan45° = 1，sin60° = √3/2，cos60° = ½。在弧度制中，30° = π/6，45° = π/4，60° = π/3。
When solving trig equations such as 2cosθ = √3 for 0 ≤ θ < 2π, first find the acute angle θ = π/6, then use the ASTC quadrant rule to locate all solutions: θ = π/6 and θ = 11π/6.
解三角方程（如 0 ≤ θ < 2π 内求 2cosθ = √3）时，先求锐角 θ = π/6，再利用ASTC象限规则找出所有解：θ = π/6 与 θ = 11π/6。
Remember that tanθ repeats every π radians, while sinθ and cosθ repeat every 2π radians. Always check the required domain before giving your final answer set.
注意 tanθ 的周期为 π 弧度，sinθ 与 cosθ 的周期为 2π 弧度。给出最终解集前务必核对题目要求的定义域。
5. Differentiation | 微分
Differentiation measures the instantaneous rate of change. The power rule states: if y = xⁿ, then dy/dx = nxⁿ⁻¹. This rule also works for negative and fractional powers.
微分度量瞬时变化率。幂法则为：若 y = xⁿ，则 dy/dx = nxⁿ⁻¹。该法则对负幂与分数幂同样适用。
The derivative of a constant is zero, and for y = kxⁿ we have dy/dx = knxⁿ⁻¹. The derivative of a sum is the sum of the derivatives: d/dx (u + v) = du/dx + dv/dx.
常数的导数为零；若 y = kxⁿ，则 dy/dx = knxⁿ⁻¹。和的导数等于导数的和：d/dx (u + v) = du/dx + dv/dx。
At a point on the curve y = f(x), the value of dy/dx gives the gradient of the tangent. The normal is perpendicular to the tangent, so its gradient is −1/(dy/dx).
在曲线 y = f(x) 上的某点处，dy/dx 的值即为切线的斜率。法线与切线垂直，因此其斜率为 −1/(dy/dx)。
Stationary points occur where dy/dx = 0. Use the second derivative d²y/dx² to classify: positive means a local minimum, negative means a local maximum, and zero means further investigation is needed.
驻点出现在 dy/dx = 0 处。用二阶导数 d²y/dx² 判断其性质：正值对应局部极小值，负值对应局部极大值，为零则需进一步判断。
6. Integration | 积分
Integration is the reverse of differentiation. The general power rule for integration is: ∫xⁿ dx = xⁿ⁺¹/(n+1) + C, for n ≠ −1. Always include the constant of integration C for indefinite integrals.
积分是微分的逆运算。积分的一般幂法则为：∫xⁿ dx = xⁿ⁺¹/(n+1) + C，其中 n ≠ −1。求不定积分时务必加上积分常数 C。
To integrate (ax + b)ⁿ where n ≠ −1, use ∫(ax + b)ⁿ dx = (ax + b)ⁿ⁺¹ / (a(n+1)) + C. This is a very common exam pattern.
积分形如 (ax + b)ⁿ（n ≠ −1）的式子时，使用 ∫(ax + b)ⁿ dx = (ax + b)ⁿ⁺¹ / (a(n+1)) + C。这是非常常见的考试题型。
The definite integral ∫ₐᵇ f(x) dx gives the signed area between the curve and the x-axis from x = a to x = b. Evaluate the antiderivative at b, then subtract its value at a.
定积分 ∫ₐᵇ f(x) dx 给出曲线与 x 轴之间从 x = a 到 x = b 的有向面积。先求原函数在 b 处的值，再减去在 a 处的值。
When the curve lies below the x-axis, the definite integral is negative. To find the physical area, take the absolute value or split the integration at the x-intercepts.
当曲线位于 x 轴下方时，定积分为负。要求实际面积，需取绝对值，或在 x 轴截点处分段积分。
7. Arithmetic & Geometric Series | 等差与等比数列
An arithmetic sequence has a constant common difference d. The nth term is uₙ = a + (n − 1)d, and the sum of the first n terms is Sₙ = n/2 [2a + (n − 1)d].
等差数列的公差 d 恒定。第 n 项为 uₙ = a + (n − 1)d，前 n 项和为 Sₙ = n/2 [2a + (n − 1)d]。
For example, the sum of the first 20 terms of the series 3, 7, 11, 15, … is S₂₀ = 20/2 [2(3) + 19(4)] = 10 × 82 = 820.
例如，数列 3, 7, 11, 15, … 的前 20 项和为 S₂₀ = 20/2 [2(3) + 19(4)] = 10 × 82 = 820。
A geometric sequence has a constant common ratio r. The nth term is uₙ = arⁿ⁻¹, and the sum of the first n terms is Sₙ = a(1 − rⁿ) / (1 − r).
等比数列的公比 r 恒定。第 n 项为 uₙ = arⁿ⁻¹，前 n 项和为 Sₙ = a(1 − rⁿ) / (1 − r)。
If |r| < 1, the infinite geometric series converges to the sum to infinity S∞ = a / (1 − r). This formula only exists when −1 < r < 1.
若 |r| < 1，无穷等比级数收敛，其无穷和为 S∞ = a / (1 − r)。该公式仅在 −1 < r < 1 时成立。
8. Exponentials & Logarithms | 指数与对数
The exponential function y = eˣ and the natural logarithm y = ln x are inverse functions. This means ln(eˣ) = x and e^(ln x) = x.
指数函数 y = eˣ 与自然对数 y = ln x 互为反函数。因此 ln(eˣ) = x，e^(ln x) = x。
The three laws of logarithms are essential: log(ab) = log a + log b, log(a/b) = log a − log b, and log aⁿ = n log a. These hold for any valid base.
对数的三大法则至关重要：log(ab) = log a + log b，log(a/b) = log a − log b，以及 log aⁿ = n log a。这些法则对任意有效底数均成立。
To solve aˣ = b, take logarithms of both sides: x = log b / log a. For example, solving 2ˣ = 10 gives x = log10 / log2 ≈ 3.322.
解方程 aˣ = b 时，两边取对数：x = log b / log a。例如，解 2ˣ = 10 得 x = log10 / log2 ≈ 3.322。
Exponential growth and decay are modelled by N = N₀e^(kt). When k > 0, this describes growth; when k < 0, it describes decay. Common contexts include radioactivity, population and cooling.
指数增长与衰减可用 N = N₀e^(kt) 建模。当 k > 0 时表示增长，当 k < 0 时表示衰减。常见情形包括放射性、人口与冷却问题。
9. Vectors | 向量
A vector in two dimensions can be written as a column vector [x y], or in component form xi + yj. The magnitude is |a| = √(x² + y²), which represents the length of the vector.
二维向量可写成列向量 [x y]，或分量形式 xi + yj。其模长为 |a| = √(x² + y²)，代表向量的长度。
The unit vector in the direction of a is â = a / |a|. For example, the unit vector of (3, 4) is (3/5, 4/5).
向量 a 方向上的单位向量为 â = a / |a|。例如，(3, 4) 的单位向量为 (3/5, 4/5)。
To add or subtract vectors, combine corresponding components. To multiply a vector by a scalar k, multiply every component by k. A vector from A to B is AB = OB − OA, the difference of the position vectors.
向量的加法与减法按对应分量进行。用标量 k 乘以向量时，每个分量都乘以 k。从 A 到 B 的向量为 AB = OB − OA，即位置向量之差。
Two vectors are parallel if one is a non-zero scalar multiple of the other. The midpoint of AB has position vector (OA + OB)/2.
若一个向量是另一个向量的非零标量倍数，则两向量平行。AB 的中点的位置向量为 (OA + OB)/2。
10. Exam Strategy & Common Mistakes | 考试策略与常见错误
Losing negative signs is the most frequent algebraic error. Always rewrite subtraction carefully and check each line of working before moving on.
丢失负号是最常见的代数错误。务必仔细改写减法步骤，并在继续之前检查每一行运算。
For indefinite integrals, forgetting the constant +C loses marks every year. For definite integrals, forgetting to subtract the lower limit is another common slip.
对于不定积分，忘记常数 +C 每年都会导致失分。对于定积分，忘记代入下限并相减是另一个常见失误。
When solving trigonometric equations, ensure your calculator is in the correct mode: radians for radian questions and degrees for degree questions. Also verify that every solution lies within the given interval.
解三角方程时，确保计算器处于正确模式：弧度制题目用弧度，角度制题目用度。同时核对每个解是否都在给定区间内。
Finally, always show clear method. In AS Mathematics, method marks are generous: even if your final answer is wrong, a correct structured approach can still earn most of the credit.
最后，务必清晰展示解题步骤。在 AS 数学中，方法分通常很慷慨：即使最终答案有误，正确规范的思路仍能获得大部分分数。
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