A-Level OCR Maths: Formula Summary Handbook | A-Level OCR 数学：公式汇总手册

📚 A-Level OCR Maths: Formula Summary Handbook | A-Level OCR 数学：公式汇总手册

This handbook brings together the essential formulae required for the OCR A-Level Mathematics specification (H240). It covers Pure Mathematics, Mechanics and Statistics, organised by topic for quick reference during revision. All key results are stated without proof, and variable meanings are clearly defined. Use this as your one‑stop checklist when practising past papers.

这份手册汇集了 OCR A-Level 数学（H240）所必需的核心公式，涵盖纯数、力学与统计三大领域，按主题分类编排，方便复习时快速查阅。所有关键结果均直接列出，变量含义明确，可当作刷真题时的速查清单使用。

1. Algebraic Laws and Indices | 代数法则与指数

For any real numbers a, b and positive integers m, n, the laws of indices are a cornerstone of algebraic manipulation.

对于任意实数 a, b 和正整数 m, n，指数法则是代数运算的基石。

aᵐ × aⁿ = aᵐ⁺ⁿ (multiplying same base / 同底数幂相乘)；
aᵐ ÷ aⁿ = aᵐ⁻ⁿ (dividing same base / 同底数幂相除)；
(aᵐ)ⁿ = aᵐⁿ (power of a power / 幂的乘方)；
(ab)ⁿ = aⁿ bⁿ (power of a product / 积的乘方)；
a⁰ = 1 (a ≠ 0 / 任何非零数的零次幂为 1)；
a⁻ⁿ = 1 / aⁿ (negative index / 负指数)；
a^(m/n) = ⁿ√(aᵐ) (fractional index / 分数指数)。

The quadratic formula for solving ax² + bx + c = 0 (a ≠ 0) is essential.

解一元二次方程 ax² + bx + c = 0（a ≠ 0）的求根公式至关重要。

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 a repeated real root, Δ < 0 gives two complex roots.

判别式 Δ = b² – 4ac 决定根的性质：Δ > 0 有两个不等实根，Δ = 0 有一个重实根，Δ < 0 有两个共轭复根。

Completing the square transforms a quadratic into the form a(x + p)² + q, where p = b/(2a) and q = – (b² – 4ac)/(4a).

配方法将二次式化为 a(x + p)² + q，其中 p = b/(2a)，q = – (b² – 4ac)/(4a)。

2. Quadratic Functions and Polynomials | 二次函数与多项式

The sum and product of roots α, β of ax² + bx + c = 0 are:

方程 ax² + bx + c = 0 的两根 α, β 之和与积为：

α + β = -b/a, αβ = c/a

For a cubic ax³ + bx² + cx + d = 0 with roots α, β, γ, the relationships extend to:

对于三次方程 ax³ + bx² + cx + d = 0，根 α, β, γ 满足：

α + β + γ = -b/a, αβ + βγ + γα = c/a, αβγ = -d/a

These symmetric sums are very useful for forming new equations or simplifying polynomial expressions.

这些对称和常用于构造新方程或化简多项式表达式。

Factor theorem: (x – p) is a factor of f(x) if and only if f(p) = 0. Remainder theorem: when f(x) is divided by (x – p), the remainder is f(p).

因式定理：(x – p) 是 f(x) 的因式当且仅当 f(p) = 0。余式定理：f(x) 除以 (x – p) 的余式为 f(p)。

3. Functions and Transformations | 函数与变换

For a given function y = f(x), the basic graph transformations are:

对于给定函数 y = f(x)，基本图像变换为：

y = f(x) + a : translation by vector (0, a) / 沿 y 轴平移 a；
y = f(x + a) : translation by vector (-a, 0) / 沿 x 轴平移 -a；
y = a f(x) : vertical stretch by factor a / 垂直方向伸缩 a 倍；
y = f(ax) : horizontal stretch by factor 1/a / 水平方向伸缩 1/a 倍；
y = -f(x) : reflection in x‑axis / 关于 x 轴对称；
y = f(-x) : reflection in y‑axis / 关于 y 轴对称。

Composite function (g ∘ f)(x) = g(f(x)) means applying f first, then g. The domain and range must be considered carefully.

复合函数 (g ∘ f)(x) = g(f(x)) 表示先作用 f 再作用 g，需仔细考虑定义域与值域。

The modulus function |x| is defined as: |x| = x for x ≥ 0, and |x| = -x for x < 0. Solving |ax + b| = c yields two cases.

绝对值函数 |x| 定义为：x ≥ 0 时 |x| = x，x < 0 时 |x| = -x。解方程 |ax + b| = c 需分两种情形。

4. Trigonometry | 三角学

For a right‑angled triangle, the basic ratios are:

在直角三角形中，基本比值定义为：

sin θ = opposite / hypotenuse, cos θ = adjacent / hypotenuse, tan θ = opposite / adjacent

Exact values for angles 0°, 30°, 45°, 60°, 90° must be memorised. For instance, sin 30° = 1/2, cos 45° = √2/2, tan 60° = √3.

必须熟记 0°、30°、45°、60°、90° 的精确值，如 sin 30° = 1/2，cos 45° = √2/2，tan 60° = √3。

Key identities derived from the unit circle:

由单位圆导出的关键恒等式：

tan θ = sin θ / cos θ
sin² θ + cos² θ = 1
sin(90° – θ) = cos θ, cos(90° – θ) = sin θ

For all triangles, the sine rule and cosine rule are essential tools.

正弦定理与余弦定理适用于所有三角形。

Sine rule: a / sin A = b / sin B = c / sin C

Cosine rule: a² = b² + c² – 2bc cos A

Area of a triangle: (1/2)ab sin C, or Heron’s formula s = (a+b+c)/2, area = √[s(s-a)(s-b)(s-c)].

三角形面积公式：½ ab sin C，或海伦公式 s = (a+b+c)/2，面积 = √[s(s-a)(s-b)(s-c)]。

Trigonometric equations often require the use of quadrant diagrams or CAST to find all solutions in a given interval.

解三角方程常需借助象限图或 CAST 法找出给定区间内的所有解。

5. Exponentials and Logarithms | 指数与对数

The function y = aˣ (a > 0) is an exponential. The natural exponential y = eˣ has special importance where e ≈ 2.71828.

函数 y = aˣ（a > 0）称为指数函数。自然指数 y = eˣ 尤为重要，其中 e ≈ 2.71828。

The logarithm is the inverse operation: y = logₐ x ⇔ aʸ = x. The natural logarithm is ln x = logₑ x.

对数为指数运算的逆：y = logₐ x ⇔ aʸ = x。自然对数记为 ln x = logₑ x。

Key laws of logs:

对数运算法则：

logₐ x + logₐ y = logₐ (xy)
logₐ x – logₐ y = logₐ (x/y)
n logₐ x = logₐ (xⁿ)
logₐ a = 1, logₐ 1 = 0
Change of base: logₐ x = log_b x / log_b a

When solving exponential equations, taking ln of both sides is often the first step. For example, e²ˣ = 5 becomes 2x = ln 5, so x = (ln 5)/2.

解指数方程时，通常先两侧取自然对数。例如 e²ˣ = 5 → 2x = ln 5 → x = (ln 5)/2。

The derivative of eˣ is eˣ, and d/dx (ln x) = 1/x. These are crucial for calculus applications.

eˣ 的导数为 eˣ，ln x 的导数为 1/x，这两条是微积分应用的基础。

6. Sequences and Series | 数列与级数

An arithmetic sequence has a common difference d: uₙ = a + (n-1)d, where a = u₁.

等差数列有公差 d：uₙ = a + (n-1)d，其中 a = u₁。

Sum of the first n terms of an arithmetic series:

等差数列前 n 项和：

Sₙ = n/2 [2a + (n-1)d] = n/2 (a + l)

where l = last term = a + (n-1)d.

其中 l = 末项 = a + (n-1)d。

A geometric sequence has a common ratio r: uₙ = a rⁿ⁻¹, a ≠ 0.

等比数列有公比 r：uₙ = a rⁿ⁻¹，a ≠ 0。

Sum of the first n terms (r ≠ 1):

前 n 项和（r ≠ 1）：

Sₙ = a(1 – rⁿ) / (1 – r)

Sum to infinity of a convergent geometric series (|r| < 1):

收敛的无穷等比级数求和（|r| < 1）：

S∞ = a / (1 – r)

The binomial expansion of (1 + x)ⁿ for rational n, valid for |x| < 1, is:

二项展开式 (1 + x)ⁿ（n 为有理数，|x| < 1）：

(1 + x)ⁿ = 1 + nx + [n(n-1)/2!] x² + [n(n-1)(n-2)/3!] x³ + …

For positive integer n, the expansion terminates and is given by the binomial theorem.

当 n 为正整数时，展开式为有限项，由二项式定理给出。

7. Calculus: Differentiation | 微积分：微分

Derivative from first principles:

导数的第一原理定义：

f'(x) = lim_{h→0} [f(x+h) – f(x)] / h

Standard derivatives (where k is a constant):

标准导数（k 为常数）：

d/dx (xⁿ) = n xⁿ⁻¹ (for any real n / n 为任意实数)
d/dx (eᵏˣ) = k eᵏˣ
d/dx (ln x) = 1/x
d/dx (sin x) = cos x, d/dx (cos x) = -sin x
d/dx (tan x) = sec² x
d/dx (sec x) = sec x tan x, d/dx (cosec x) = -cosec x cot x, d/dx (cot x) = -cosec² x

Product rule: if y = u v, then dy/dx = u’ v + v’ u.

乘法法则：若 y = u v，则 dy/dx = u’ v + v’ u。

Quotient rule: if y = u/v, then dy/dx = (v u’ – u v’) / v².

除法法则：若 y = u/v，则 dy/dx = (v u’ – u v’) / v²。

Chain rule: if y = f(g(x)), then dy/dx = f'(g(x)) × g'(x).

链式法则：若 y = f(g(x))，则 dy/dx = f'(g(x)) × g'(x)。

For parametric equations x = f(t), y = g(t), dy/dx = (dy/dt) / (dx/dt).

参数方程求导：x = f(t), y = g(t)，dy/dx = (dy/dt) / (dx/dt)。

8. Calculus: Integration | 微积分：积分

Integration is the reverse of differentiation. The indefinite integral includes an arbitrary constant c.

积分是微分的逆运算，不定积分包含任意常数 c。

Standard integrals:

标准积分公式：

∫ xⁿ dx = xⁿ⁺¹/(n+1) + c, n ≠ -1
∫ 1/x dx = ln|x| + c
∫ eᵏˣ dx = (1/k) eᵏˣ + c
∫ cos x dx = sin x + c, ∫ sin x dx = -cos x + c
∫ sec² x dx = tan x + c, ∫ cosec² x dx = -cot x + c
∫ sec x tan x dx = sec x + c, ∫ cosec x cot x dx = -cosec 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.

定积分 ∫ₐᵇ f(x) dx 表示曲线与 x 轴之间从 a 到 b 的有号面积。

Area between two curves y = f(x) and y = g(x) from a to b is ∫ₐᵇ [f(x) – g(x)] dx, where f(x) ≥ g(x) on [a,b].

两条曲线之间的面积：若在 [a,b] 上 f(x) ≥ g(x)，则面积 = ∫ₐᵇ [f(x) – g(x)] dx。

Fundamental Theorem: ∫ₐᵇ f'(x) dx = f(b) – f(a)

Integration by substitution (reverse chain rule) is used for composite functions; for definite integrals, remember to change limits.

换元积分法（逆链式法则）用于复合函数；计算定积分时务必同时变换上下限。

Integration by parts: ∫ u dv/dx dx = uv – ∫ v du/dx dx, based on the product rule.

分部积分法：∫ u dv/dx dx = uv – ∫ v du/dx dx，来源于乘法法则。

9. Vectors | 向量

A vector in component form is expressed using i, j, k: a = a₁ i + a₂ j + a₃ k.

向量的分量形式用 i, j, k 表示：a = a₁ i + a₂ j + a₃ k。

Magnitude (length) of a: |a| = √(a₁² + a₂² + a₃²).

模长：|a| = √(a₁² + a₂² + a₃²)。

Dot (scalar) product: a · b = a₁b₁ + a₂b₂ + a₃b₃ = |a||b| cos θ, where θ is the angle between them.

数量积（点乘）：a · b = a₁b₁ + a₂b₂ + a₃b₃ = |a||b| cos θ，θ 为两向量夹角。

Vectors are perpendicular (orthogonal) if a · b = 0.

若 a · b = 0，则两向量垂直。

Equation of a straight line in vector form: r = a + td, where a is a position vector on the line and d is the direction vector.

直线向量方程：r = a + td，其中 a 为直线上一点的位置向量，d 为方向向量。

For two lines with direction vectors d₁ and d₂, the acute angle θ satisfies cos θ = |d₁ · d₂| / (|d₁||d₂|).

两直线夹角（锐角）θ 满足 cos θ = |d₁ · d₂| / (|d₁||d₂|)。

10. Statistics | 统计学

For a data set x₁, x₂, …, xₙ, the mean is x̄ = (Σ xᵢ)/n. The variance measures spread:

对于数据集 x₁, x₂, …, xₙ，均值 x̄ = (Σ xᵢ)/n。方差衡量离散程度：

Variance σ² = Σ(xᵢ – x̄)² / n

The standard deviation σ is the square root of variance. The formula Σ xᵢ² / n – x̄² is computationally convenient.

标准差 σ 为方差的平方根。计算时常用公式 Σ xᵢ² / n – x̄²。

For grouped data with frequencies fᵢ, midpoints xᵢ, mean = Σ fᵢ xᵢ / Σ fᵢ, variance = Σ fᵢ xᵢ² / Σ fᵢ – mean².

分组数据：设组中值为 xᵢ，频数为 fᵢ，均值 = Σ fᵢ xᵢ / Σ fᵢ，方差 = Σ fᵢ xᵢ² / Σ fᵢ – 均值²。

Probability: P(A ∪ B) = P(A) + P(B) – P(A ∩ B). For mutually exclusive events, P(A ∩ B) = 0.

概率加法公式：P(A ∪ B) = P(A) + P(B) – P(A ∩ B)。若互斥，则 P(A ∩ B) = 0。

Conditional probability: P(A|B) = P(A ∩ B) / P(B). Events A and B are independent if P(A ∩ B) = P(A) P(B).

条件概率：P(A|B) = P(A ∩ B) / P(B)。若 A、B 独立，则 P(A ∩ B) = P(A) P(B)。

For a binomial distribution X ~ B(n, p):

二项分布 X ~ B(n, p)：

P(X = r) = (ⁿCᵣ) pʳ (1-p)ⁿ⁻ʳ

Mean μ = np, variance σ² = np(1-p).

均值 μ = np，方差 σ² = np(1-p)。

Normal distribution X ~ N(μ, σ²). Standardisation: Z = (X – μ)/σ ~ N(0,1). Use the standard normal table for probabilities.

正态分布 X ~ N(μ, σ²)。标准化：Z = (X – μ)/σ ~ N(0,1)。查标准正态分布表求概率。

11. Mechanics: Kinematics | 力学：运动学

For motion in a straight line with constant acceleration a, the SUVAT equations link displacement (s), initial velocity (u), final velocity (v), acceleration (a) and time (t):

匀加速直线运动中，SUVAT 方程关联位移 (s)、初速度 (u)、末速度 (v)、加速度 (a) 和时间 (t)：

v = u + a t
s = u t + ½ a t²
s = ½ (u + v) t
v² = u² + 2 a s
s = v t – ½ a t²

These apply only when acceleration is constant; direction must be considered by choosing a positive sense.

这些方程仅在加速度恒定时成立；使用前必须规定正方向。

Velocity is the rate of change of displacement, v = ds/dt. Acceleration is a = dv/dt = d²s/dt².

速度是位移的变化率 v = ds/dt，加速度 a = dv/dt = d²s/dt²。

For motion under gravity near the Earth’s surface, a = g = 9.8 m s⁻² (downwards). Projectile motion separates horizontal and vertical components: horizontal velocity u cos θ is constant; vertical motion uses u sin θ and constant a = -g.

地表附近的抛体运动中，加速度 a = g = 9.8 m s⁻²，方向向下。水平速度 u cos θ 为常数，竖直运动初速 u sin θ，加速度恒为 -g。

12. Mechanics: Forces and Newton’s Laws | 力学：力与牛顿定律

Newton’s laws form the basis of mechanics:

牛顿定律构成力学基础：

First law: An object remains at rest or in uniform motion unless acted upon by a resultant force. / 若无合力作用，物体保持静止或匀速直线运动。
Second law: F = m a, where F is resultant force, m mass, a acceleration. / 合力 F = m a。
Third law: Action and reaction are equal in magnitude and opposite in direction. / 作用力与反作用力大小相等、方向相反。

Weight of an object is W = m g, where g = 9.8 m s⁻² (unless otherwise stated).

重力 W = m g，g 通常取 9.8 m s⁻²。

Friction: the maximum static friction is F ≤ μ R, where R is the normal reaction and μ the coefficient of friction. For kinetic (dynamic) friction, F = μₖ R.

摩擦力：最大静摩擦 F ≤ μ R，动摩擦 F = μₖ R，R 为法向反作用力。

Resolving forces on an inclined plane of angle θ: component of weight down the slope = mg sin θ, component perpendicular to the slope = mg cos θ.

斜面上分解重力：沿斜面向下分力 mg sin θ，垂直斜面分力 mg cos θ。

Tension in a light, inextensible string is constant throughout its length. For connected particles, treat the whole system to find acceleration, then analyse individual particles to find tension.

轻绳中张力处处相等。处理连接体问题时，先整体求加速度，再隔离分析求绳中张力。

Momentum p = m v. Impulse = change in momentum = F Δt = m v – m u. Momentum is conserved in collisions when no external resultant force acts.

动量 p = m v。冲量 = 动量变化 = F Δt = m v – m u。碰撞中若无外部合力，动量守恒。

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A-Level OCR Maths: Formula Summary Handbook | A-Level OCR 数学：公式汇总手册