📚 A-Level Mathematics Core Topics Complete Review | A-Level数学核心考点全梳理
A-Level Mathematics demands a solid command of algebra, calculus, geometry, statistics, and mechanics. This structured guide condenses the essential theorems, formulas, and problem-solving techniques you must master before entering the exam hall.
A-Level数学要求考生熟练运用代数、微积分、几何、统计与力学知识。本篇文章系统浓缩考试必备的核心定理、公式与解题技巧，帮助你在考前建立完整的知识框架。
1. Algebra and Functions | 代数与函数
Quadratic equations in the form ax² + bx + c = 0 can be solved by factorisation, completing the square, or using the quadratic formula. The discriminant determines the nature of the roots.
形如 ax² + bx + c = 0 的二次方程可以通过因式分解、配方法或求根公式求解。判别式决定根的性质。
x = (−b ± √b² − 4ac) / 2a, Δ = b² − 4ac
If Δ > 0, there are two distinct real roots; if Δ = 0, there is one repeated real root; if Δ < 0, there are no real roots.
若 Δ > 0，方程有两个不等实根；若 Δ = 0，方程有一个重根；若 Δ < 0，方程无实根。
Function notation: the domain is the set of valid inputs, the range is the set of outputs. The inverse function f⁻¹(x) satisfies f(f⁻¹(x)) = x, and its graph is the reflection of f(x) in the line y = x.
函数记号：定义域是自变量的取值范围，值域是因变量的取值范围。反函数 f⁻¹(x) 满足 f(f⁻¹(x)) = x，其图像是 f(x) 关于直线 y = x 的对称图形。
Graph transformations: y = f(x + a) shifts left by a units, y = f(x) + a shifts up by a units, y = f(ax) is a horizontal stretch by factor 1/a, and y = af(x) is a vertical stretch by factor a.
图像变换：y = f(x + a) 向左平移 a 个单位，y = f(x) + a 向上平移 a 个单位，y = f(ax) 为水平方向拉伸 1/a 倍，y = af(x) 为竖直方向拉伸 a 倍。
Composite functions such as fg(x) mean applying g first, then f. Always check whether the output of g lies in the domain of f.
复合函数 fg(x) 表示先代 g 后代 f。解题时务必检查 g 的输出是否位于 f 的定义域内。
2. Polynomials and Inequalities | 多项式与不等式
The Factor and Remainder Theorems are powerful tools for handling higher-degree polynomials. Inequalities require the same algebraic fluency but with careful attention to sign changes.
因式定理与余数定理是处理高次多项式的有力工具。解不等式需要同等的代数能力，但必须特别注意符号变化。
Factor Theorem: if f(p) = 0, then (x − p) is a factor of f(x). This allows you to break down cubic and quartic expressions into linear and quadratic factors.
因式定理：若 f(p) = 0，则 (x − p) 是 f(x) 的一个因式。该定理可帮助你将三次、四次多项式分解为一次因式与二次因式的乘积。
Remainder Theorem: when f(x) is divided by (x − a), the remainder is f(a). This is faster than long division when only the remainder is needed.
余数定理：f(x) 除以 (x − a) 所得的余数等于 f(a)。当只需余数时，这比长除法更快。
For quadratic inequalities, rearrange to one side, find the critical values, then sketch a graph or test intervals to determine the solution set.
解二次不等式时，先将所有项移至一边，求出临界值，再通过图像草图或区间检验确定解集。
For rational inequalities, multiply through by the square of the denominator (always positive) to avoid reversing the inequality sign.
解分式不等式时，应乘以分母的平方（恒为正），以避免改变不等号方向。
3. Coordinate Geometry | 坐标几何
Coordinate geometry links algebra to shapes. Straight lines and circles are the most frequently tested objects in the pure mathematics papers.
坐标几何将代数与图形联系起来。直线与圆是纯数试卷中最高频的考点模型。
The gradient of a line through (x₁, y₁) and (x₂, y₂) is m = (y₂ − y₁) / (x₂ − x₁). The midpoint has coordinates ((x₁ + x₂)/2, (y₁ + y₂)/2).
过点 (x₁, y₁) 与 (x₂, y₂) 的直线斜率为 m = (y₂ − y₁) / (x₂ − x₁)。中点坐标为 ((x₁ + x₂)/2, (y₁ + y₂)/2)。
The equation of a line can be written as y = mx + c or ax + by + c = 0. Parallel lines have equal gradients; perpendicular lines satisfy m₁ × m₂ = −1.
直线方程可写作 y = mx + c 或 ax + by + c = 0。两直线平行等价于斜率相等；两直线垂直等价于 m₁ × m₂ = −1。
A circle with centre (a, b) and radius r has equation (x − a)² + (y − b)² = r². The general form x² + y² + 2gx + 2fy + c = 0 has centre (−g, −f).
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)。
A tangent to a circle is perpendicular to the radius at the point of contact. Use this to find tangent equations or to prove tangency.
圆的切线在切点处垂直于半径。利用这一性质可以求切线方程或证明相切关系。
4. Sequences and Series | 数列与级数
Arithmetic and geometric sequences appear in every A-Level examination. You must know both the nth-term formula and the sum formulas, and know when each applies.
等差数列与等比数列在每次A-Level考试中都会出现。你不仅要掌握通项公式与求和公式，还要明确各自的适用条件。
Arithmetic: uₙ = a + (n−1)d, Sₙ = n/2 [2a + (n−1)d]
Geometric: uₙ = arⁿ⁻¹, Sₙ = a(1 − rⁿ)/(1 − r), S∞ = a/(1 − r) for |r| < 1
In an arithmetic sequence, the difference d is constant. The sum formula can also be written as Sₙ = n/2 (a + l), where l is the last term.
等差数列中公差 d 为常数。求和公式也可写作 Sₙ = n/2 (a + l)，其中 l 为末项。
In a geometric sequence, the ratio r is constant. The infinite sum exists only when |r| < 1, otherwise the series diverges.
等比数列中公比 r 为常数。无穷项和仅在 |r| < 1 时存在，否则级数发散。
The binomial expansion states that (a + b)ⁿ = aⁿ + nC1 aⁿ⁻¹b + nC2 aⁿ⁻²b² + ... + bⁿ, where nCr are binomial coefficients.
二项式定理指出 (a + b)ⁿ = aⁿ + nC1 aⁿ⁻¹b + nC2 aⁿ⁻²b² + ... + bⁿ，其中 nCr 为二项式系数。
When n is not a positive integer, the binomial expansion only converges when |x| < 1, giving an infinite series.
当 n 不是正整数时，二项展开只在 |x| < 1 时收敛，此时展开为无穷级数。
5. Trigonometry | 三角函数
Trigonometry is a major topic in both pure mathematics and mechanics. You must be fluent in radians, identities, and equation-solving techniques.
三角函数是纯数与应用数学中的重点内容。你必须熟练掌握弧度制、恒等式以及方程求解方法。
sin²θ + cos²θ = 1, tanθ = sinθ / cosθ, 1 + tan²θ = sec²θ
In radians, arc length is s = rθ and sector area is A = ½r²θ. Convert between degrees and radians using π rad = 180°.
在弧度制中，弧长公式为 s = rθ，扇形面积公式为 A = ½r²θ。利用 π rad = 180° 进行度与弧度的换算。
To solve a trigonometric equation, first reduce it to a single function using identities, then find all solutions in the required interval, adding periods of 2π or π as needed.
解三角方程时，先用恒等式将方程化为单一三角函数，再求指定区间内的所有解，必要时加上 2π 或 π 的周期。
Double-angle formulas are essential: sin2θ = 2sinθcosθ and cos2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ.
二倍角公式至关重要：sin2θ = 2sinθcosθ，cos2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ。
For expressions of the form a sinθ + b cosθ, rewrite as R sin(θ ± α) using R = √a² + b². This simplifies max-min problems and equation solving.
对于 a sinθ + b cosθ 形式的表达式，可化为 R sin(θ ± α)，其中 R = √a² + b²。这一技巧能简化最值问题与方程求解。
6. Differentiation | 微分
Differentiation measures rates of change and determines the shape of curves. Master the standard rules before attempting exam questions.
微分用于度量变化率并确定曲线形态。解答试题前，必须先掌握基本求导法则。
d/dx (xⁿ) = nxⁿ⁻¹, d/dx (eˣ) = eˣ, d/dx (ln x) = 1/x
The product rule states d/dx (uv) = u dv/dx + v du/dx. The quotient rule states d/dx (u/v) = (v du/dx − u dv/dx) / v².
乘法法则：d/dx (uv) = u dv/dx + v du/dx。除法法则：d/dx (u/v) = (v du/dx − u dv/dx) / v²。
The chain rule d/dx f(g(x)) = f'(g(x)) g'(x) is used for composite functions. It is often written as dy/dx = dy/du × du/dx.
链式法则 d/dx f(g(x)) = f'(g(x)) g'(x) 用于复合函数求导，常写作 dy/dx = dy/du × du/dx。
The gradient of a tangent at x = a is f'(a). The normal is perpendicular to the tangent, so its gradient is −1 / f'(a).
切线在 x = a 处的斜率为 f'(a)。法线与切线垂直，故其斜率为 −1 / f'(a)。
Stationary points satisfy f'(x) = 0. Use the second derivative f''(x) to classify: f''(x) < 0 gives a maximum, f''(x) > 0 gives a minimum.
驻点满足 f'(x) = 0。利用二阶导数 f''(x) 判断类型：f''(x) < 0 为极大值点，f''(x) > 0 为极小值点。
7. Integration | 积分
Integration is the reverse of differentiation and is used to calculate areas, volumes, and accumulated change. Know the standard integrals and the trapezium rule.
积分是微分的逆运算，用于计算面积、体积与累积变化量。要熟记基本积分公式与梯形法则。
∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ −1), ∫ 1/x dx = ln|x| + C
The definite integral ∫ₐᵇ f(x) dx represents the signed area between the curve and the x-axis from x = a to x = b. A negative result means the curve lies below the axis.
定积分 ∫ₐᵇ f(x) dx 表示曲线与 x 轴之间从 x = a 到 x = b 的有向面积。若结果为负，说明曲线在轴下方。
To integrate a composite expression such as f'(x)[f(x)]ⁿ, use the reverse chain rule: ∫ (f(x))ⁿ f'(x) dx = (f(x))ⁿ⁺¹/(n+1) + C.
求复合函数积分如 ∫ (f(x))ⁿ f'(x) dx 时，使用逆链式法则，结果为 (f(x))ⁿ⁺¹/(n+1) + C。
When an exact integral is difficult, the trapezium rule approximates ∫ₐᵇ f(x) dx using straight-line segments. More strips give greater accuracy.
当定积分难以精确计算时，可用梯形法则以直线段近似曲线。分割条数越多，精度越高。
The area between two curves y = f(x) and y = g(x) is given by ∫ |f(x) − g(x)| dx over the overlap interval.
两曲线 y = f(x) 与 y = g(x) 之间的面积等于在公共区间内对 |f(x) − g(x)| 的积分。
8. Exponentials and Logarithms | 指数与对数
Exponential functions and logarithms model growth, decay, and many natural processes. The laws of logarithms are indispensable for simplifying equations.
指数函数与对数可用于描述增长、衰减及许多自然过程。对数运算法则是化简方程不可或缺的工具。
logₐ(xy) = logₐx + logₐy, logₐ(x/y) = logₐx − logₐy, logₐ(xᵏ) = k logₐx
To solve an exponential equation, take natural logs on both sides. For example, if eᵏˣ = A, then kx = ln A.
解指数方程时，两边同时取自然对数。例如若 eᵏˣ = A，则 kx = ln A。
The change of base formula logₐb = ln b / ln a allows conversion between different logarithmic bases.
换底公式 logₐb = ln b / ln a 可将对数在不同底数之间转换。
Exponential growth and decay follow N = N₀eᵏᵗ. If k > 0, the quantity increases; if k < 0, it decreases toward zero.
指数增长与衰减遵循 N = N₀eᵏᵗ。若 k > 0，量值增加；若 k < 0，量值向零衰减。
For modelling linearised data, taking logs of both sides turns y = axⁿ into ln y = ln a + n ln x, enabling the use of straight-line graphs.
数据建模中，对等式两边取对数可将 y = axⁿ 化为 ln y = ln a + n ln x，从而使用直线图进行分析。
9. Vectors | 向量
Vectors describe quantities with both magnitude and direction. In two dimensions, examine positions, displacements, and angle relationships using scalar products.
向量描述既有大小又有方向的量。在二维空间中，利用数量积研究位置、位移与角度关系。
|v| = √x² + y², a·b = |a||b| cosθ = x₁x₂ + y₁y₂
The unit vector in the direction of v is v / |v|. It has magnitude 1 and is often denoted as a hat symbol.
向量 v 方向上的单位向量为 v / |v|，其模长为 1，常以尖帽符号表示。
The position vector of a point is its displacement from the origin. The vector from A to B is given by AB = OB − OA.
点的位置向量是它相对于原点的位移。从 A 指向 B 的向量为 AB = OB − OA。
If two vectors are perpendicular, their scalar product is zero: a·b = 0. This is the quickest test for perpendicularity.
若两向量垂直，则它们的数量积为零：a·b = 0。这是判定垂直的最快捷方法。
The cosine of the angle between two vectors is cosθ = (a·b) / (|a||b|). This formula is also used to project one vector onto another.
两向量夹角的余弦为 cosθ = (a·b) / (|a||b|)。此公式也用于计算向量投影。
10. Statistics and Probability | 统计与概率
Statistics and probability are core components of the A-Level Mathematics curriculum. You need to summarise data correctly and apply probability rules with precision.
统计与概率是A-Level数学课程的核心部分。你需要正确概括数据特征，并精确应用概率运算法则。
The mean of a data set is x̄ = Σx / n. The variance is σ² = Σ(x − x̄)² / n, and the standard deviation is √σ². Alternative formula: σ² = Σx²/n − x̄².
数据集的均值为 x̄ = Σx / n。方差为 σ² = Σ(x − x̄)² / n，标准差为 √σ²。等价公式：σ² = Σx²/n − x̄²。
The addition rule states P(A ∪ B) = P(A) + P(B) − P(A ∩ B). For mutually exclusive events, P(A ∩ B) = 0.
加法法则：P(A ∪ B) = P(A) + P(B) − P(A ∩ B)。若 A、B 互斥，则 P(A ∩ B) = 0。
Conditional probability satisfies P(A | B) = P(A ∩ B) / P(B). Two events are independent exactly when P(A ∩ B) = P(A) × P(B).
条件概率满足 P(A | B) = P(A ∩ B) / P(B)。两事件相互独立当且仅当 P(A ∩ B) = P(A) × P(B)。
The correlation coefficient r measures the strength of a linear relationship in bivariate data. Values close to +1 or −1 indicate a strong relationship.
相关系数 r 度量双变量数据线性相关的强度。数值越接近 +1 或 −1，线性关系越强。
11. Mechanics | 力学
Mechanics applies mathematics to physical systems. The SUVAT equations and Newton's second law form the backbone of kinematics and dynamics questions.
力学将数学应用于物理系统。运动学中的 SUVAT 方程与牛顿第二定律是运动学与动力学问题的核心。
v = u + at, s = ut + ½at², v² = u² + 2as
SUVAT equations apply only when acceleration is constant. Choose the equation that contains the three known variables and the one unknown you need.
SUVAT 方程组仅适用于匀加速运动。选择同时包含三个已知量与一个待求量的方程进行求解。
Newton's second law is F = ma. When multiple forces act, first resolve them into components, then sum forces in each direction separately.
牛顿第二定律为 F = ma。当多个力作用时，先将各力分解，再分别对每个方向求和。
The weight of an object is W = mg, acting vertically downward. Friction acts parallel to the surface and opposes motion.
物体的重力为 W = mg，方向竖直向下。摩擦力沿接触面方向作用且阻碍运动。
For connected particles, treat the system as a whole to find acceleration, then isolate one particle to find internal forces such as tension.
对于连接体问题，先把系统看作整体求加速度，再隔离单个物体求内力如绳张力。
12. Exam Strategy | 应试策略
Strong exam technique can raise your grade significantly. Careful reading, clear working, and smart time allocation are just as important as knowing the mathematics.
优秀的应试技巧可以显著提升成绩。仔细审题、规范书写与合理分配时间，与掌握数学知识同样重要。
Read each question twice. Identify the topic, underline given data, and write down the relevant formula before substituting numbers.
每道题至少读两遍。确定考点、划出已知数据，并在代入数值前先写出对应公式。
Show every step of your working. A-Level mark schemes award method marks even when the final answer is incorrect.
展示全部解答过程。A-Level评分标准中，即使最终结果有误，过程分仍然有效。
Check units and significant figures, especially in mechanics where answers often require degrees, metres per second, or newtons.
注意单位与有效数字，尤其是在力学题中，答案常需以角度、米每秒或牛顿为单位。
Time management: allocate minutes roughly proportional to marks. If a question takes too long, leave it and return later.
时间管理：分配时间应与分值大致成比例。若某题耗时过长，先跳过并留待复查。
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