Mathematics Exam Prep: Core Formulas and Application Tips | 数学备考：核心公式整理与运用技巧

📚 Mathematics Exam Prep: Core Formulas and Application Tips | 数学备考：核心公式整理与运用技巧

Mastering core formulas is the foundation of success in any mathematics examination. However, memorising a formula is not enough — understanding when and how to apply it under time pressure makes the real difference. This article compiles the most frequently tested formulas across key topics and presents practical strategies to use them effectively in your exam.

掌握核心公式是任何数学考试取得成功的基石。然而，仅仅记住公式远远不够——在时间压力下准确判断何时使用、如何应用，才是真正的差距所在。本文整理了各大高频考点的核心公式，并提供实用的运用策略，帮助你在考试中高效得分。

1. Quadratic Equations and the Discriminant | 二次方程与判别式

The quadratic formula solves equations of the form ax² + bx + c = 0. 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 exactly one repeated real root; if Δ < 0, there are no real roots but two complex conjugate roots.

二次方程的标准形式为 ax² + bx + c = 0，其求根公式为 x = (−b ± √(b² − 4ac)) / 2a。判别式 Δ = b² − 4ac 决定根的性质：Δ > 0 时有两个不等实根；Δ = 0 时有一个重根；Δ < 0 时无实根，但有一对共轭复根。

x = (−b ± √(b² − 4ac)) / 2a, Δ = b² − 4ac

A common exam trick involves using the sum and product of roots. If α and β are the roots, then α + β = −b/a and αβ = c/a. These relations let you construct quadratic equations from given roots or evaluate symmetric expressions like α² + β² without solving for the roots.

考试中常用到根与系数的关系：若 α 和 β 为方程的根，则 α + β = −b/a，αβ = c/a。利用这些关系，你可以从已知根构造方程，或直接求 α² + β² 等对称式子的值，而无需真正解出根。

2. Sequences and Series | 数列与级数

Arithmetic sequences have 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]. Geometric sequences multiply by a common ratio r, with nth term uₙ = arⁿ⁻¹ and finite sum Sₙ = a(1 − rⁿ)/(1 − r).

等差数列的公差恒定，通项为 uₙ = a + (n − 1)d，前 n 项和为 Sₙ = n/2 [2a + (n − 1)d]。等比数列每一项乘以固定的公比 r，通项为 uₙ = arⁿ⁻¹，有限项和为 Sₙ = a(1 − rⁿ)/(1 − r)。

Sₙ = n/2 [2a + (n − 1)d] (Arithmetic) | Sₙ = a(1 − rⁿ)/(1 − r) (Geometric)

The infinite geometric series converges only when |r| < 1, giving S∞ = a/(1 − r). In exams, always check this convergence condition first before applying the infinite sum formula. For arithmetic series, the average of the first and last terms multiplied by n gives the same sum — a quick mental-check tool.

无穷等比级数仅在 |r| < 1 时收敛，此时 S∞ = a/(1 − r)。考试中，务必先验证收敛条件再使用无穷求和公式。对于等差数列，首末项平均值乘以 n 即可得到总和——这是快速验算的实用技巧。

3. Exponential and Logarithmic Functions | 指数函数与对数函数

The laws of logarithms are essential tools for solving exponential equations. Key identities include log(xy) = log x + log y, log(x/y) = log x − log y, and log(xⁿ) = n log x. The change of base rule states logₐ b = log b / log a, which allows conversion between different bases.

对数法则在解指数方程时至关重要，核心恒等式包括：log(xy) = log x + log y、log(x/y) = log x − log y、log(xⁿ) = n log x。换底公式 logₐ b = log b / log a 允许在不同底数之间灵活转换。

log(xy) = log x + log y, log(xⁿ) = n log x, logₐ b = log b / log a

When solving equations involving eˣ or ln x, apply the natural logarithm to both sides to bring powers down. Remember that ln(e) = 1 and e^(ln x) = x. Exponential growth and decay problems often require the form y = Aeᵏᵗ — identify A as the initial value and k as the growth or decay rate.

解含 eˣ 或 ln x 的方程时，两边取自然对数可将指数降为系数。注意 ln(e) = 1、e^(ln x) = x。指数增长与衰减问题通常以 y = Aeᵏᵗ 的形式出现——A 为初始值，k 为增长率或衰减率。

4. Trigonometric Identities | 三角恒等式

The Pythagorean identity sin²θ + cos²θ = 1 is the most frequently used trigonometric formula. From it, we derive tan²θ + 1 = sec²θ and cot²θ + 1 = csc²θ. The double-angle formulas, sin 2θ = 2 sin θ cos θ and cos 2θ = cos²θ − sin²θ, are indispensable for integration and equation solving.

毕达哥拉斯恒等式 sin²θ + cos²θ = 1 是最常用的三角公式。由它可推导出 tan²θ + 1 = sec²θ 与 cot²θ + 1 = csc²θ。倍角公式 sin 2θ = 2 sin θ cos θ 和 cos 2θ = cos²θ − sin²θ 在积分和方程求解中不可或缺。

sin²θ + cos²θ = 1, sin 2θ = 2 sin θ cos θ

For solving trigonometric equations, express everything in terms of sin and cos first. When faced with a sin θ + b cos θ, rewrite it as R sin(θ + α), where R = √(a² + b²) and tan α = b/a. This transformation simplifies many exam questions dramatically.

解三角方程时，先将所有项化为 sin 和 cos 的形式。遇到 a sin θ + b cos θ 的表达式，可改写为 R sin(θ + α)，其中 R = √(a² + b²)，tan α = b/a。这种变换能极大简化许多考题。

5. Differentiation Rules | 求导法则

The derivative of xⁿ is nxⁿ⁻¹, and differentiation is linear. The product rule states d/dx(uv) = u’v + uv’, and the quotient rule gives d/dx(u/v) = (u’v − uv’)/v². The chain rule, dy/dx = dy/du × du/dx, is the backbone for differentiating composite functions.

xⁿ 的导数为 nxⁿ⁻¹，且求导具有线性性。乘积法则为 d/dx(uv) = u’v + uv’，商法则为 d/dx(u/v) = (u’v − uv’)/v²。链式法则 dy/dx = dy/du × du/dx 是复合函数求导的核心工具。

d/dx(xⁿ) = nxⁿ⁻¹, (uv)’ = u’v + uv’, dy/dx = dy/du × du/dx

For exponential and logarithmic functions, remember d/dx(eˣ) = eˣ and d/dx(ln x) = 1/x. The derivatives of trigonometric functions — d/dx(sin x) = cos x and d/dx(cos x) = −sin x — are tested heavily in A-level papers. Practise recognising which rule applies before differentiating.

指数和对数函数的导数需牢记：d/dx(eˣ) = eˣ，d/dx(ln x) = 1/x。三角函数的导数——d/dx(sin x) = cos x、d/dx(cos x) = −sin x——在 A-level 试卷中考查频率极高。求导前先判断应使用哪一条法则，是提高准确率的关键。

6. Integration Essentials | 积分基础

Integration reverses differentiation. The power rule gives ∫xⁿ dx = xⁿ⁺¹/(n + 1) + C for n ≠ −1. Special cases include ∫1/x dx = ln|x| + C, ∫eˣ dx = eˣ + C, ∫sin x dx = −cos x + C, and ∫cos x dx = sin x + C.

积分是求导的逆运算。幂函数积分公式为 ∫xⁿ dx = xⁿ⁺¹/(n + 1) + C（n ≠ −1）。特殊情形包括：∫1/x dx = ln|x| + C、∫eˣ dx = eˣ + C、∫sin x dx = −cos x + C、∫cos x dx = sin x + C。

∫xⁿ dx = xⁿ⁺¹/(n + 1) + C (n ≠ −1), ∫1/x dx = ln|x| + C

Integration by substitution is the reverse of the chain rule. Spot expressions where the derivative of the inner function appears as a factor. Integration by parts, ∫u dv = uv − ∫v du, applies when the integrand is a product of two different types of functions — commonly polynomial × exponential or polynomial × trigonometric.

换元积分法是链式法则的逆向运用。若被积函数中含有某内层函数的导数因子，即可尝试换元。分部积分公式 ∫u dv = uv − ∫v du 适用于两种不同类型函数相乘的情形——常见于多项式 × 指数函数或多项式 × 三角函数。

7. Coordinate Geometry | 解析几何

The equation of a straight line is y = mx + c, where m is the gradient and c is the y-intercept. The gradient between two points (x₁, y₁) and (x₂, y₂) is m = (y₂ − y₁)/(x₂ − x₁). Perpendicular gradients multiply to give −1, while parallel lines share the same gradient.

直线方程的标准形式为 y = mx + c，其中 m 为斜率，c 为 y 轴截距。两点 (x₁, y₁) 和 (x₂, y₂) 间的斜率为 m = (y₂ − y₁)/(x₂ − x₁)。垂直直线的斜率乘积为 −1，平行直线的斜率相同。

m = (y₂ − y₁)/(x₂ − x₁), (x − a)² + (y − b)² = r²

The circle with centre (a, b) and radius r has equation (x − a)² + (y − b)² = r². The midpoint formula, ((x₁ + x₂)/2, (y₁ + y₂)/2), and the distance formula, √((x₂ − x₁)² + (y₂ − y₁)²), are foundational tools. When finding the intersection of a line and a circle, substitute the linear equation into the circle equation and solve the resulting quadratic.

圆心为 (a, b)、半径为 r 的圆方程为 (x − a)² + (y − b)² = r²。中点公式 ((x₁ + x₂)/2, (y₁ + y₂)/2) 与距离公式 √((x₂ − x₁)² + (y₂ − y₁)²) 是最基础的工具。求直线与圆的交点时，将直线方程代入圆方程并解所得二次方程即可。

8. Vectors | 向量

The dot product of two vectors a and b satisfies a · b = |a||b| cos θ, where θ is the angle between them. Two vectors are perpendicular if and only if a · b = 0. In component form, if a = (a₁, a₂, a₃) and b = (b₁, b₂, b₃), then a · b = a₁b₁ + a₂b₂ + a₃b₃.

向量 a 和 b 的点积满足 a · b = |a||b| cos θ，其中 θ 为两向量夹角。两向量垂直的充要条件是 a · b = 0。在分量形式下，若 a = (a₁, a₂, a₃)，b = (b₁, b₂, b₃)，则 a · b = a₁b₁ + a₂b₂ + a₃b₃。

a · b = |a||b| cos θ, a · b = a₁b₁ + a₂b₂ + a₃b₃

The magnitude of a vector is |a| = √(a₁² + a₂² + a₃²). To find the angle between two vectors, rearrange the dot product formula. Position vectors track points relative to the origin, while direction vectors describe the direction of a line. The vector equation of a line is r = a + λb, where a is a position vector and b is the direction vector.

向量的模为 |a| = √(a₁² + a₂² + a₃²)。求两向量夹角时，只需重排点积公式即可。位置向量描述点相对于原点的坐标，方向向量描述直线的方向。直线的向量方程为 r = a + λb，其中 a 为位置向量，b 为方向向量。

9. Statistics and Probability | 统计与概率

The mean of a data set is x̄ = Σx/n, and the variance is σ² = Σ(x − x̄)²/n = Σx²/n − x̄². The standard deviation is the square root of variance. For grouped data, use midpoints and frequencies: x̄ = Σfx/Σf, and σ² = Σfx²/Σf − x̄².

数据集的均值为 x̄ = Σx/n，方差为 σ² = Σ(x − x̄)²/n = Σx²/n − x̄²，标准差为方差的平方根。对于分组数据，使用组中值和频数计算：x̄ = Σfx/Σf，σ² = Σfx²/Σf − x̄²。

x̄ = Σx/n, σ² = Σ(x − x̄)²/n

The binomial distribution, X ~ B(n, p), has mean np and variance np(1 − p), with probabilities given by P(X = k) = C(n, k)pᵏ(1 − p)ⁿ⁻ᵏ. Conditional probability follows P(A|B) = P(A ∩ B)/P(B). For independent events, P(A ∩ B) = P(A)P(B); for mutually exclusive events, P(A ∪ B) = P(A) + P(B).

二项分布 X ~ B(n, p) 的均值为 np，方差为 np(1 − p)，概率公式为 P(X = k) = C(n, k)pᵏ(1 − p)ⁿ⁻ᵏ。条件概率公式为 P(A|B) = P(A ∩ B)/P(B)。独立事件满足 P(A ∩ B) = P(A)P(B)；互斥事件满足 P(A ∪ B) = P(A) + P(B)。

10. Formula Application Strategies | 公式运用策略

The first and most critical strategy is to read the question fully before writing anything. Identify which topic the question tests, then recall the relevant formula sheet section mentally. Write down every given value with its symbol — this prevents careless substitution errors and reveals which formula naturally links the givens to the unknown.

第一条也是最重要的策略：动笔前完整读题。判断题目考查哪个知识模块，然后在脑中定位公式表中相应的部分。将每个已知量连同其符号逐一写下——这既能防止代入时的粗心错误，也能帮你看清哪条公式天然地将已知量与未知量联系起来。

Check units and rounding requirements before calculating — examiners reward precision in the final answer.

计算前确认单位与保留位数的要求——阅卷人非常看重最终答案的精确度。
For multi-step problems, write intermediate results clearly and label them; partial credit often depends on visible working.

多步骤题目中，清晰写出中间结果并加注标签；步骤分往往取决于可见的演算过程。
Use estimation as a sanity check. If your computed answer is wildly different from your estimate, revisit the formula.

用估算做合理性检查。若计算结果与估算相差甚远，回头重新审视所用公式。

Time management is equally important. Allocate roughly one minute per mark. If a question requires a formula you cannot recall, move on and return later — the exam paper often contains other questions that provide clues through similar structures or contexts.

时间管理同样关键。大致按每分钟一分的节奏分配时间。如果某题所需公式一时想不起来，先跳过再回头——试卷中其他题目往往通过相似结构或场景提供线索。

11. Common Pitfalls and Shortcuts | 常见陷阱与速算技巧

A frequent mistake in differentiation and integration is forgetting the constant C in indefinite integrals, or omitting the + C when the question specifies ‘general solution’. Another classic error is applying the product rule to simple products like x² × x³ instead of simplifying first. Always simplify expressions before differentiating or integrating where possible.

不定积分漏写常数 C 是最常见的失分点，尤其是题目明确要求“通解”的时候。另一个经典错误是对 x² × x³ 这类可直接化简的乘积强行套用乘积法则。尽量在求导或积分前先化简表达式。

When using the quadratic formula, check that a, b, and c carry their correct signs. A missing negative sign is the single most common algebraic error.

使用求根公式时，务必核对 a、b、c 的符号。漏写负号是最常见的代数错误。
In probability questions, verify whether events are independent or mutually exclusive before choosing your formula.

概率题中，选择公式前先确认事件是独立还是互斥。
For trigonometric equations, check the domain and list all solutions within the specified range — partial solutions lose marks.

解三角方程时，注意定义域并在指定范围内列出全部解——遗漏解会扣分。
Memorise common exact values: sin 30° = 1/2, sin 45° = √2/2, sin 60° = √3/2, and cos 0° = 1. These save time and reduce calculator dependence.

牢记常用精确值：sin 30° = 1/2、sin 45° = √2/2、sin 60° = √3/2、cos 0° = 1。它们能节省时间、减少对计算器的依赖。

Perhaps the most powerful shortcut of all is regular, repeated practice. Formulas become genuinely useful only after you have applied them in varied contexts. Build a personal formula card from past paper questions — note not just the formula itself but the question type that triggers its use. This active recall process transforms passive knowledge into exam-ready skill.

所有速算技巧中，最强大的一条是反复练习。公式只有在多样化的情境中应用过，才能真正为你所用。从历年真题中整理一张个人公式卡——不仅记录公式本身，更记录触发使用该公式的题型特征。这种主动回忆的过程能将被动知识转化为考场上的实战能力。

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Mathematics Exam Prep: Core Formulas and Application Tips | 数学备考：核心公式整理与运用技巧