📚 Mastering Core Formulas for A-Level Maths | 基础公式的掌握与应用
A-Level Mathematics is not about memorising every formula blindly — it is about knowing which formula to apply, when to apply it, and how to manipulate it under exam pressure. This guide breaks down the essential formulas across key topics, with strategies for active retention and practical application.
A-Level 数学不是盲目地记住所有公式，而是要知道在考试压力下选择哪个公式、何时应用、如何变形。本指南将核心主题中的必备公式进行拆解，并提供主动记忆和实际应用的策略。
1. Algebra: Quadratic Equations and Discriminants | 代数：二次方程与判别式
The quadratic formula is the backbone of algebraic manipulation. For any quadratic equation ax² + bx + c = 0, the solutions are given by x = (−b ± √(b² − 4ac)) / 2a. The discriminant, Δ = b² − 4ac, determines the nature of the roots: Δ > 0 gives two distinct real roots, Δ = 0 gives one repeated root, and Δ < 0 gives no real roots.
二次公式是代数变形的基石。对于任意二次方程 ax² + bx + c = 0，其解为 x = (−b ± √(b² − 4ac)) / 2a。判别式 Δ = b² − 4ac 决定根的性质：Δ > 0 有两个不等实根，Δ = 0 有一对重根，Δ < 0 没有实根。
In exam questions, the discriminant is often used in disguise — for example, when a line is tangent to a curve, setting the discriminant to zero is the fastest way to find the unknown parameter. You should also be comfortable completing the square: ax² + bx + c = a(x + b/2a)² + (c − b²/4a), which reveals the vertex of a parabola directly.
在考试题中，判别式常常以隐蔽的形式出现——例如，当一条直线与曲线相切时，令判别式为零是求未知参数最快的方法。你还应熟练配方法：ax² + bx + c = a(x + b/2a)² + (c − b²/4a)，它直接揭示抛物线的顶点。
x = (−b ± √(b² − 4ac)) / 2a, Δ = b² − 4ac
2. Functions: Transformations and Inverse Functions | 函数：变换与反函数
Function transformations follow a predictable pattern. For a function f(x), the graph of f(x) + a moves up by a, f(x + a) moves left by a, −f(x) reflects in the x-axis, and f(−x) reflects in the y-axis. A horizontal stretch by factor 1/k is achieved by f(kx), while a vertical stretch by factor k is achieved by kf(x).
函数变换遵循可预测的模式。对于函数 f(x)，f(x) + a 的图像向上平移 a，f(x + a) 向左平移 a，−f(x) 关于 x 轴对称翻折，f(−x) 关于 y 轴对称翻折。水平拉伸因子 1/k 由 f(kx) 实现，而垂直拉伸因子 k 由 kf(x) 实现。
The inverse function f⁻¹(x) exists only if f is one-to-one. To find the inverse, swap x and y, then rearrange. The domain of f becomes the range of f⁻¹, and vice versa. A common exam trap is forgetting to state the domain restriction when finding an inverse of a quadratic — you must restrict the domain so the function becomes one-to-one.
反函数 f⁻¹(x) 仅当 f 是一一对应时才存在。求反函数时，交换 x 和 y，再重新整理。f 的定义域成为 f⁻¹ 的值域，反之亦然。一个常见的考试陷阱是在求二次函数的反函数时忘记说明定义域限制——你必须限制定义域使函数成为一一对应。
y = f(x) → x = f⁻¹(y); domain(f) = range(f⁻¹)
3. Trigonometry: Identities and Solving Equations | 三角学：恒等式与解方程
Trigonometry is a language of its own, and fluency requires memorising not just the definitions but the relationships between the six ratios. The Pythagorean identity sin²θ + cos²θ = 1 is the most recycled identity in A-Level papers. From it, you can derive 1 + tan²θ = sec²θ and cot²θ + 1 = csc²θ — these are essential when converting expressions into a solvable form.
三角学有其独特的语言，掌握它不仅需要记住定义，还需要理解六个比值之间的关系。毕达哥拉斯恒等式 sin²θ + cos²θ = 1 是 A-Level 试卷中使用频率最高的恒等式。由此可推导出 1 + tan²θ = sec²θ 和 cot²θ + 1 = csc²θ——在将表达式转换为可解形式时这些至关重要。
Double-angle formulas appear whenever you see 2θ: sin(2θ) = 2sinθcosθ, cos(2θ) = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ, and tan(2θ) = 2tanθ / (1 − tan²θ). These are indispensable for integration and for solving trigonometric equations over a given interval. When solving equations, always sketch the graph or use the CAST diagram to check how many solutions exist in the required range.
倍角公式在出现 2θ 时必不可少：sin(2θ) = 2sinθcosθ，cos(2θ) = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ，tan(2θ) = 2tanθ / (1 − tan²θ)。这些在积分和求解指定区间内的三角方程中不可或缺。解方程时，始终画图或使用 CAST 象限图来检查在所需范围内有多少个解。
sin²θ + cos²θ = 1; sin(2θ) = 2sinθcosθ; cos(2θ) = 2cos²θ − 1
4. Differentiation: Rules and Applications | 微分：法则与应用
Differentiation measures the rate of change. The power rule d/dx(xⁿ) = nxⁿ⁻¹ is the first formula you learn, but the product rule and quotient rule are equally critical. For two functions u(x) and v(x): d/dx(uv) = u·dv/dx + v·du/dx, and d/dx(u/v) = (v·du/dx − u·dv/dx) / v².
微分衡量变化率。幂法则 d/dx(xⁿ) = nxⁿ⁻¹ 是第一个学习的公式，但乘积法则和商法则同样关键。对于两个函数 u(x) 和 v(x)：d/dx(uv) = u·dv/dx + v·du/dx，d/dx(u/v) = (v·du/dx − u·dv/dx) / v²。
The chain rule connects composite functions: dy/dx = dy/du × du/dx. This single rule powers everything from differentiating e^(3x) to differentiating (x² + 1)⁵. Applications of differentiation include finding stationary points, determining their nature via the second derivative, and solving optimisation problems — the classic “maximum volume” or “minimum cost” question that appears on nearly every paper.
链式法则连接复合函数：dy/dx = dy/du × du/dx。这一条法则支撑着从 e^(3x) 的微分到 (x² + 1)⁵ 的微分的一切运算。微分的应用包括求驻点、通过二阶导数判断其性质，以及解决优化问题——几乎每张试卷都会出现的经典”最大体积”或”最小成本”问题。
d/dx(xⁿ) = nxⁿ⁻¹; d/dx(uv) = u·v’ + v·u’; dy/dx = dy/du × du/dx
5. Integration: Techniques and Definite Integrals | 积分：技巧与定积分
Integration is the reverse process of differentiation. The fundamental rule is ∫xⁿ dx = xⁿ⁺¹/(n+1) + C, provided n ≠ −1. The special case ∫1/x dx = ln|x| + C is a common stumbling block — students forget the absolute value or forget the + C entirely.
积分是微分的逆过程。基本法则为 ∫xⁿ dx = xⁿ⁺¹/(n+1) + C，前提是 n ≠ −1。特殊情况 ∫1/x dx = ln|x| + C 是一个常见的绊脚石——学生往往忘记绝对值符号或完全忘记 + C。
Definite integrals are evaluated between limits: ∫ₐᵇ f(x) dx = [F(x)]ₐᵇ = F(b) − F(a), which gives the net area under a curve from x = a to x = b. To find the area between a curve and the x-axis, integrate with respect to x; for area between a curve and the y-axis, integrate with respect to y. When the region is bounded by two curves, subtract the lower curve from the upper curve before integrating.
定积分在上下限之间求值：∫ₐᵇ f(x) dx = [F(x)]ₐᵇ = F(b) − F(a)，它给出从 x = a 到 x = b 曲线下的净面积。求曲线与 x 轴之间的面积时，对 x 积分；求曲线与 y 轴之间的面积时，对 y 积分。当区域由两条曲线围成时，先取上曲线减下曲线再积分。
For integration by substitution, identify a function and its derivative inside the integrand. For integration by parts, use ∫u dv = uv − ∫v du — choosing u and dv strategically (often using the LIATE rule: Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential) makes the integral dramatically simpler.
对于换元积分法，找出被积函数中的函数及其导数。对于分部积分法，使用 ∫u dv = uv − ∫v du——策略性地选择 u 和 dv（通常使用 LIATE 规则：对数、反三角、代数、三角、指数）可大大简化积分。
∫xⁿ dx = xⁿ⁺¹/(n+1) + C (n≠−1); ∫ₐᵇ f(x) dx = F(b) − F(a)
6. Exponential and Logarithmic Functions | 指数与对数函数
The exponential function eˣ has a unique property: it is its own derivative. The general exponential function aˣ differentiates as aˣ ln(a), and its integral is aˣ / ln(a) + C. The natural logarithm ln x is defined as the inverse of eˣ, and its derivative is 1/x.
指数函数 eˣ 有一个独特性质：它是自身的导数。一般指数函数 aˣ 的导数为 aˣ ln(a)，其积分为 aˣ / ln(a) + C。自然对数 ln x 定义为 eˣ 的反函数，其导数为 1/x。
The three logarithm laws are non-negotiable: ln(ab) = ln a + ln b, ln(a/b) = ln a − ln b, and ln(aⁿ) = n ln a. These laws convert multiplication into addition and powers into products, which is exactly why logarithms are used to solve exponential equations like 2ˣ = 100. The change of base formula, logₐb = log b / log a, is essential when working with bases other than e or 10.
三大对数法则不可妥协：ln(ab) = ln a + ln b，ln(a/b) = ln a − ln b，ln(aⁿ) = n ln a。这些法则将乘法转化为加法、幂转化为乘积，这正是对数用于解指数方程（如 2ˣ = 100）的原因。换底公式 logₐb = log b / log a 在处理 e 或 10 以外的底数时至关重要。
Modelling questions involving exponential growth and decay — such as radioactive decay or population growth — follow the form N = N₀e^(kt). Being able to rearrange for k or t is a key skill. Always check whether k is positive (growth) or negative (decay).
涉及指数增长和衰减的建模题——如放射性衰变或人口增长——遵循 N = N₀e^(kt) 的形式。能够解出 k 或 t 是关键技能。始终检查 k 是正（增长）还是负（衰减）。
d/dx(eˣ) = eˣ; ln(ab) = ln a + ln b; logₐb = log b / log a
7. Vectors: Magnitude, Dot Product, and Geometry | 向量：模长、点积与几何
Vectors combine magnitude and direction. The magnitude of a vector a = (x, y) is |a| = √(x² + y²), and in three dimensions |a| = √(x² + y² + z²). A unit vector in the direction of a is a / |a|.
向量兼具大小和方向。向量 a = (x, y) 的模为 |a| = √(x² + y²)，在三维空间中 |a| = √(x² + y² + z²)。a 方向上的单位向量为 a / |a|。
The dot product is the bridge between algebra and geometry in vector problems. For two vectors a = (a₁, a₂) and b = (b₁, b₂), a · b = a₁b₁ + a₂b₂. Geometrically, a · b = |a||b|cosθ, where θ is the angle between the vectors. This means two vectors are perpendicular exactly when their dot product is zero.
点积是向量问题中代数与几何之间的桥梁。对于两个向量 a = (a₁, a₂) 和 b = (b₁, b₂)，a · b = a₁b₁ + a₂b₂。几何上，a · b = |a||b|cosθ，其中 θ 是两向量之间的夹角。这意味着两个向量垂直当且仅当它们的点积为零。
In geometry problems, you will often be asked to find the angle between two lines, between a line and a plane, or determine whether lines intersect. The vector equation of a line r = a + tb, where a is a point on the line and b is the direction vector, is the starting point for most of these questions.
在几何问题中，常需要求两直线之间的夹角、直线与平面之间的夹角，或判断直线是否相交。直线的向量方程为 r = a + tb，其中 a 是直线上一点，b 是方向向量，这是大多数此类问题的起点。
a · b = |a||b|cosθ; a · b = a₁b₁ + a₂b₂ + a₃b₃
8. Coordinate Geometry: Circles and Lines | 坐标几何：圆与直线
The equation of a straight line takes several forms: y = mx + c (gradient-intercept form), y − y₁ = m(x − x₁) (point-gradient form), and ax + by + c = 0 (general form). The gradient m = (y₂ − y₁)/(x₂ − x₁) measures steepness, and two lines are perpendicular if m₁ × m₂ = −1.
直线方程有几种形式：y = mx + c（斜截式）、y − y₁ = m(x − x₁)（点斜式）和 ax + by + c = 0（一般式）。斜率 m = (y₂ − y₁)/(x₂ − x₁) 衡量陡峭程度，两条直线垂直当且仅当 m₁ × m₂ = −1。
The equation of a circle with centre (a, b) and radius r is (x − a)² + (y − b)² = r². Expanding this gives x² + y² + 2gx + 2fy + c = 0, where the centre is (−g, −f) and the radius is √(g² + f² − c). Knowing how to complete the square to recover the centre and radius is a routine requirement in exam questions.
圆心为 (a, b)、半径为 r 的圆的方程为 (x − a)² + (y − b)² = r²。展开可得 x² + y² + 2gx + 2fy + c = 0，其中圆心为 (−g, −f)，半径为 √(g² + f² − c)。会通过配方法恢复圆心和半径是考试题中的常规要求。
For questions asking whether a line is tangent to a circle, substitute the line equation into the circle equation, then use the discriminant: Δ = 0 means tangent, Δ > 0 means two intersections, Δ < 0 means no intersection.
对于判断直线是否与圆相切的问题，将直线方程代入圆方程，然后使用判别式：Δ = 0 表示相切，Δ > 0 表示两个交点，Δ < 0 表示无交点。
(x − a)² + (y − b)² = r²; m₁ × m₂ = −1 ⇔ perpendicular
9. Sequences and Series: Arithmetic and Geometric | 数列与级数：等差与等比
An arithmetic sequence has a constant difference d between consecutive terms. The nth term is aₙ = a + (n−1)d, and the sum of the first n terms is Sₙ = (n/2)(2a + (n−1)d) or equivalently Sₙ = (n/2)(a + L), where L is the last term.
等差数列的相邻项之差 d 为常数。第 n 项为 aₙ = a + (n−1)d，前 n 项和为 Sₙ = (n/2)(2a + (n−1)d)，等价地 Sₙ = (n/2)(a + L)，其中 L 为末项。
A geometric sequence has a constant ratio r. The nth term is aₙ = arⁿ⁻¹, and the sum of the first n terms is Sₙ = a(1 − rⁿ)/(1 − r) for r ≠ 1. The sum to infinity exists only when |r| < 1 and is given by S∞ = a/(1 − r).
等比数列的公比 r 为常数。第 n 项为 aₙ = arⁿ⁻¹，前 n 项和为 Sₙ = a(1 − rⁿ)/(1 − r)（r ≠ 1）。无穷级数之和仅在 |r| < 1 时存在，且 S∞ = a/(1 − r)。
A key exam trap is confusing Sₙ with S∞: when a problem says “the sum to infinity” but the ratio has a magnitude greater than 1, the sum does not converge — you should identify the sequence as divergent.
一个关键的考试陷阱是混淆 Sₙ 和 S∞：当题目说”无穷项之和”但公比的绝对值大于 1 时，该级数不收敛——你应该判断该数列为发散。
aₙ = a + (n−1)d; Sₙ = a(1 − rⁿ)/(1 − r); S∞ = a/(1 − r), |r| < 1
10. Binomial Expansion | 二项式展开
The binomial theorem states that (a + b)ⁿ = Σₖ₌₀ⁿ C(n, k) aⁿ⁻ᵏ bᵏ, where C(n, k) = n! / (k!(n−k)!) is the binomial coefficient. For (1 + x)ⁿ where n is a positive integer, the expansion is 1 + nx + n(n−1)x²/2! + n(n−1)(n−2)x³/3! + … + xⁿ.
二项式定理指出 (a + b)ⁿ = Σₖ₌₀ⁿ C(n, k) aⁿ⁻ᵏ bᵏ，其中 C(n, k) = n! / (k!(n−k)!) 为二项式系数。对于 n 为正整数的 (1 + x)ⁿ，展开式为 1 + nx + n(n−1)x²/2! + n(n−1)(n−2)x³/3! + … + xⁿ。
For fractional or negative powers, the expansion of (1 + x)ⁿ is valid only when |x| < 1, and becomes an infinite series: 1 + nx + n(n−1)x²/2! + n(n−1)(n−2)x³/3! + … This is frequently used to find approximations of expressions like √(1.02) or 1/(1.01)³.
对于分数或负指数幂，(1 + x)ⁿ 的展开仅在 |x| < 1 时有效，且变为无穷级数：1 + nx + n(n−1)x²/2! + n(n−1)(n−2)x³/3! + … 这常用于求 √(1.02) 或 1/(1.01)³ 等表达式的近似值。
When expanding (a + bx)ⁿ, always factor out aⁿ first: (a + bx)ⁿ = aⁿ(1 + bx/a)ⁿ. Then apply the standard expansion with x replaced by bx/a. This is a routine technique that saves time and reduces sign errors.
展开 (a + bx)ⁿ 时，先提出 aⁿ：(a + bx)ⁿ = aⁿ(1 + bx/a)ⁿ。然后用 bx/a 替换 x 应用标准展开式。这是一种常规技巧，既节省时间又减少符号错误。
(a + b)ⁿ = Σ C(n, k) aⁿ⁻ᵏ bᵏ; (1 + x)ⁿ = 1 + nx + n(n−1)x²/2! + …
11. Statistics: Mean, Variance, and Probability | 统计学：均值、方差与概率
The mean of a data set is x̄ = Σx/n for raw data, and x̄ = Σ(fx)/Σf for grouped data. The variance is Var = Σ(x − x̄)²/n, more efficiently calculated as E(X²) − (E(X))². The standard deviation is the square root of the variance.
数据集的均值为 x̄ = Σx/n（原始数据），分组数据为 x̄ = Σ(fx)/Σf。方差为 Var = Σ(x − x̄)²/n，更高效地计算为 E(X²) − (E(X))²。标准差是方差的平方根。
For probability, the addition rule P(A ∪ B) = P(A) + P(B) − P(A ∩ B) handles overlapping events, while the multiplication rule P(A ∩ B) = P(A) × P(B|A) governs conditional probability. Two events are independent if P(A ∩ B) = P(A) × P(B).
对于概率，加法法则 P(A ∪ B) = P(A) + P(B) − P(A ∩ B) 处理重叠事件，而乘法法则 P(A ∩ B) = P(A) × P(B|A) 支配条件概率。两个事件独立当且仅当 P(A ∩ B) = P(A) × P(B)。
The binomial distribution X ~ B(n, p) has mean np and variance np(1 − p). The normal distribution X ~ N(μ, σ²) is symmetric about μ, and standardisation uses Z = (X − μ)/σ to convert to the standard normal N(0, 1). Continuity corrections are needed when approximating a binomial distribution with a normal distribution.
二项分布 X ~ B(n, p) 的均值为 np，方差为 np(1 − p)。正态分布 X ~ N(μ, σ²) 关于 μ 对称，标准化使用 Z = (X − μ)/σ 转换到标准正态 N(0, 1)。在用正态分布近似二项分布时需要进行连续性修正。
Var = E(X²) − (E(X))²; Z = (X − μ)/σ
12. Proof and Exam Strategy | 证明与考试策略
Proof by contradiction, proof by exhaustion, and proof by inspection are the three main methods tested at A-Level. In proof by contradiction, you assume the statement is false and derive a logical impossibility. For example, proving that √2 is irrational starts by assuming √2 = a/b in lowest terms and deriving a contradiction.
反证法、穷举法和直接检验法是 A-Level 测试的三种主要证明方法。在反证法中，你假设命题为假并推导出逻辑上的不可能性。例如，证明 √2 为无理数，先假设 √2 = a/b 为最简分数，再推导出矛盾。
For your exam strategy, keep a formula sheet organised by topic. Before each practice paper, write out every formula from memory — this active recall is far more effective than passive reading. When solving, highlight which formula applies before substituting numbers. Mark questions that require multiple formula applications; these multi-step problems are where marks are lost.
对于考试策略，按主题整理公式表。每次练习试卷前，凭记忆写出所有公式——这种主动回忆远优于被动阅读。解题时，在代入数值前先标出适用哪个公式。标记需要多次应用公式的题目；这些多步骤问题正是失分所在。
Finally, memorise the examiner’s language: “hence” means you must use the previous result, “exact value” means leave your answer in surd or fractional form, and “to 3 significant figures” requires a specific degree of accuracy. Interpreting these keywords correctly is a formula for marks in itself.
最后，牢记考官的用词：”hence”（由此）意味着你必须使用前一个结果，”exact value”（精确值）意味着答案以根式或分数形式保留，”to 3 significant figures”（保留三位有效数字）要求特定的精度。正确理解这些关键词本身就是得分公式。
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