Pre-U Edexcel Mathematics: Formula & Theorem Quick Reference Handbook | Pre-U Edexcel 数学：公式定理速查手册

📚 Pre-U Edexcel Mathematics: Formula & Theorem Quick Reference Handbook | Pre-U Edexcel 数学：公式定理速查手册

This quick reference handbook compiles the essential formulas and theorems needed for the Pre-U Edexcel Mathematics syllabus. Covering pure mathematics, statistics, and mechanics, it is designed to support revision and problem-solving with concise English and Chinese explanations.

这份速查手册汇编了 Pre-U Edexcel 数学课程必备的核心公式与定理。内容涵盖纯数学、统计和力学，并提供简洁的中英文配对解释，便于复习与解题参考。

1. Algebra Essentials | 代数基础

The quadratic formula gives the solutions of ax² + bx + c = 0 as x = ( –b ± √(b² – 4ac) ) / (2a).

一元二次方程 ax² + bx + c = 0 的求根公式为 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, and Δ < 0 gives no real roots.

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

If α and β are roots of ax² + bx + c = 0, then α + β = –b/a and αβ = c/a (Vieta’s formulas).

若 α 和 β 为 ax² + bx + c = 0 的根，则 α + β = –b/a，αβ = c/a（韦达定理）。

The Factor Theorem states that (x – a) is a factor of polynomial P(x) if and only if P(a) = 0.

因式定理：多项式 P(x) 含有因式 (x – a) 当且仅当 P(a) = 0。

The Remainder Theorem: when P(x) is divided by (x – a), the remainder is P(a).

余数定理：多项式 P(x) 除以 (x – a) 的余数为 P(a)。

The binomial expansion for (a + b)ⁿ is Σ(n choose k) aⁿ⁻ᵏ bᵏ, where (n choose k) = n! / (k!(n – k)!).

二项展开式 (a + b)ⁿ 为 Σ(k=0 to n) C(n,k) aⁿ⁻ᵏ bᵏ，其中 C(n,k) = n! / (k!(n – k)!)。

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2. Exponentials and Logarithms | 指数与对数

Index laws: aᵐ × aⁿ = aᵐ⁺ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1, a⁻ⁿ = 1/aⁿ.

指数运算法则：aᵐ × aⁿ = aᵐ⁺ⁿ，(aᵐ)ⁿ = aᵐⁿ，a⁰ = 1，a⁻ⁿ = 1/aⁿ。

The logarithm definition: if aˣ = b then logₐ b = x. The natural logarithm is ln x = logₑ x.

对数定义：若 aˣ = b，则 logₐ b = x。自然对数 ln x = logₑ x。

Log laws: logₐ (xy) = logₐ x + logₐ y, logₐ (x/y) = logₐ x – logₐ y, logₐ (xⁿ) = n logₐ x.

对数法则：logₐ (xy) = logₐ x + logₐ y，logₐ (x/y) = logₐ x – logₐ y，logₐ (xⁿ) = n logₐ x。

Change of base formula: logₐ b = logₓ b / logₓ a.

换底公式：logₐ b = logₓ b / logₓ a。

The exponential function eˣ and ln x are inverses: ln(eˣ) = x, e^(ln x) = x.

指数函数 eˣ 与 ln x 互为反函数：ln(eˣ) = x，e^(ln x) = x。

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3. Functions and Graphs | 函数与图形

A function f maps each input x to a unique output f(x). The domain is the set of allowed x-values, the range is the set of possible f(x).

函数 f 将每个输入 x 映射到唯一输出 f(x)。定义域为所有允许的 x 值，值域为所有可能的 f(x) 值。

Composite function: (f ∘ g)(x) = f(g(x)). Apply g first, then f.

复合函数：(f ∘ g)(x) = f(g(x))，先作用 g，再作用 f。

Inverse function f⁻¹ satisfies f(f⁻¹(x)) = x. Graph of f⁻¹ is a reflection of f in the line y = x.

反函数 f⁻¹ 满足 f(f⁻¹(x)) = x。f⁻¹ 的图像是 f 关于直线 y = x 的反射。

Graph transformations: y = f(x + a) shifts left by a; y = f(x) + a shifts up by a; y = f(ax) squashes horizontally by factor 1/a; y = a f(x) stretches vertically by factor a.

图形变换：y = f(x + a) 向左平移 a；y = f(x) + a 向上平移 a；y = f(ax) 水平方向压缩 1/a；y = a f(x) 竖直方向拉伸 a 倍。

Modulus function |x| gives the distance from zero. The graph y = |f(x)| reflects negative parts of f above the x-axis.

绝对值函数 |x| 表示到零的距离。y = |f(x)| 的图像将 f 的负值部分翻折到 x 轴上方。

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4. Trigonometry | 三角学

Radians: π radians = 180°. Arc length s = rθ, sector area A = ½ r²θ.

弧度制：π rad = 180°。弧长 s = rθ，扇形面积 A = ½ r²θ。

Fundamental identities: sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, 1 + cot²θ = csc²θ.

基本恒等式：sin²θ + cos²θ = 1，1 + tan²θ = sec²θ，1 + cot²θ = csc²θ。

Compound angle formulas: sin(A ± B) = sin A cos B ± cos A sin B, cos(A ± B) = cos A cos B ∓ sin A sin B, tan(A ± B) = (tan A ± tan B) / (1 ∓ tan A tan B).

和差角公式：sin(A ± B) = sin A cos B ± cos A sin B，cos(A ± B) = cos A cos B ∓ sin A sin B，tan(A ± B) = (tan A ± tan B) / (1 ∓ tan A tan B)。

Double angle formulas: sin 2θ = 2 sin θ cos θ, cos 2θ = cos²θ – sin²θ = 2cos²θ – 1 = 1 – 2sin²θ, tan 2θ = 2 tan θ / (1 – tan²θ).

倍角公式：sin 2θ = 2 sin θ cos θ，cos 2θ = cos²θ – sin²θ = 2cos²θ – 1 = 1 – 2sin²θ，tan 2θ = 2 tan θ / (1 – tan²θ)。

Sine rule: a / sin A = b / sin B = c / sin C. Cosine rule: a² = b² + c² – 2bc cos A.

正弦定理：a / sin A = b / sin B = c / sin C。余弦定理：a² = b² + c² – 2bc cos A。

Area of a triangle: ½ ab sin C.

三角形面积：½ ab sin C。

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5. Calculus | 微积分

First principles definition of derivative: f'(x) = limₕ→₀ (f(x + h) – f(x)) / h.

导数定义：f'(x) = limₕ→₀ (f(x + h) – f(x)) / h。

Key derivative rules: Power rule d/dx (xⁿ) = nxⁿ⁻¹. Constant multiple rule d/dx (c f) = c f’. Sum rule d/dx (f ± g) = f’ ± g’.

导数基本法则：幂法则 d/dx (xⁿ) = nxⁿ⁻¹；常数倍法则 d/dx (c f) = c f’；和差法则 d/dx (f ± g) = f’ ± g’。

Product rule: d/dx (uv) = u’v + uv’. Quotient rule: d/dx (u/v) = (u’v – uv’) / v². Chain rule: dy/dx = (dy/du) × (du/dx).

乘积法则：d/dx (uv) = u’v + uv’；商法则：d/dx (u/v) = (u’v – uv’) / v²；链式法则：dy/dx = (dy/du) × (du/dx)。

Derivatives of standard functions:

标准函数导数：

f(x) (English)	f'(x)	f(x) (中文)	f'(x)
sin x	cos x	sin x	cos x
cos x	–sin x	cos x	–sin x
tan x	sec² x	tan x	sec² x
eˣ	eˣ	eˣ	eˣ
ln x	1/x	ln x	1/x

Integration as reverse of differentiation: ∫ f'(x) dx = f(x) + C. Basic integrals include ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ –1), ∫ cos x dx = sin x + C, ∫ eˣ dx = eˣ + C, ∫ 1/x dx = ln|x| + C.

积分是微分的逆运算：∫ f'(x) dx = f(x) + C。基本积分有 ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ –1)，∫ cos x dx = sin x + C，∫ eˣ dx = eˣ + C，∫ 1/x dx = ln|x| + C。

Definite integral ∫ₐᵇ f(x) dx gives the signed area under curve y = f(x) from a to b.

定积分 ∫ₐᵇ f(x) dx 表示曲线 y = f(x) 从 a 到 b 所围成的带号面积。

The trapezium rule approximates the area under a curve: ∫ₐᵇ f(x) dx ≈ ½ h [y₀ + 2(y₁ + y₂ + … + yₙ₋₁) + yₙ], where h = (b – a)/n.

梯形法则近似曲线下的面积：∫ₐᵇ f(x) dx ≈ ½ h [y₀ + 2(y₁ + y₂ + … + yₙ₋₁) + yₙ]，其中 h = (b – a)/n。

Separation of variables for solving differential equations: dy/dx = g(x)h(y) ⇒ ∫ 1/h(y) dy = ∫ g(x) dx.

分离变量法解微分方程：若 dy/dx = g(x)h(y)，则 ∫ 1/h(y) dy = ∫ g(x) dx。

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6. Vectors | 向量

A vector has magnitude and direction. Position vector of a point A relative to origin O is OA. In component form, a = a₁i + a₂j + a₃k.

向量具有大小和方向。点 A 相对于原点 O 的位置向量为 OA。分量形式：a = a₁i + a₂j + a₃k。

Magnitude of a is |a| = √(a₁² + a₂² + a₃²). A unit vector has magnitude 1, given by â = a / |a|.

向量 a 的大小为 |a| = √(a₁² + a₂² + a₃²)。单位向量大小为 1，可表示为 â = a / |a|。

Scalar (dot) product: a · b = |a||b| cos θ = a₁b₁ + a₂b₂ + a₃b₃. If a · b = 0, vectors are perpendicular.

标量积（点乘）：a · b = |a||b| cos θ = a₁b₁ + a₂b₂ + a₃b₃。若 a · b = 0，则两向量垂直。

Vector (cross) product: a × b = |a||b| sin θ n̂, where n̂ is perpendicular to both a and b. In components: a × b = (a₂b₃ – a₃b₂)i – (a₁b₃ – a₃b₁)j + (a₁b₂ – a₂b₁)k.

向量积（叉乘）：a × b = |a||b| sin θ n̂，n̂ 垂直于 a 和 b。分量式计算：a × b = (a₂b₃ – a₃b₂)i – (a₁b₃ – a₃b₁)j + (a₁b₂ – a₂b₁)k。

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

直线方程向量形式：r = a + t d，其中 a 为直线上一点，d 为方向向量。

Equation of a plane: r = a + s u + t v (parametric) or r · n = a · n (Cartesian form).

平面方程：参数形式 r = a + s u + t v，或笛卡尔形式 r · n = a · n。

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7. Sequences and Series | 序列与级数

Arithmetic progression (AP): nth term uₙ = a + (n – 1)d, sum of first n terms Sₙ = n/2 [2

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