Pre-U WJEC Mathematics: Formula and Theorem Quick Reference | Pre-U WJEC 数学：公式定理速查手册

📚 Pre-U WJEC Mathematics: Formula and Theorem Quick Reference | Pre-U WJEC 数学：公式定理速查手册

This quick reference guide distils the essential formulas, theorems, and identities required for the WJEC Pre-U in Mathematics. Whether you are revising Pure Mathematics, Mechanics, or Statistics, these carefully selected results will help you navigate the syllabus with speed and confidence. Each section pairs a concise statement in English with its Chinese translation, enabling bilingual learners to reinforce understanding across languages.

本速查手册浓缩了 WJEC Pre-U 数学考试所需的核心公式、定理与恒等式。无论你在复习纯数学、力学还是统计，这些精心选取的结果都能帮助你快速、自信地掌握考纲要点。每个小节均以英文阐述后紧跟中文译文，助力双语学习者加深理解。

1. Algebraic Laws and Surds | 代数法则与根式

= a + (b + c), and a(b + c) = ab + ac. These properties permit rearrangement and expansion without altering value.

交换律、结合律与分配律是一切代数运算的基础。对任意实数 a, b, c：a + b = b + a，ab = ba，(a + b) + c = a + (b + c)，且 a(b + c) = ab + ac。这些性质保证重新排列和展开不改变量值。

When simplifying surds, the key rules are √ab = √a × √b and √a/b = √a / √b for a, b ≥ 0. Rationalising the denominator often uses the conjugate: 1/(√a + √b) = (√a − √b)/(a − b).

化简根式时关键规则为 √ab = √a × √b 以及 √a/b = √a / √b（a, b ≥ 0）。分母有理化通常使用共轭根式：1/(√a + √b) = (√a − √b)/(a − b)。

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2. Quadratic Equations and Functions | 二次方程与函数

ation ax² + bx + c = 0, a ≠ 0, has discriminant Δ = b² − 4ac. Roots are given by x = [−b ± √b² − 4ac t real roots; Δ = 0 gives one repeated real root; Δ < 0 gives two complex conjugate roots.

标准二次方程 ax² + bx + c = 0（a ≠ 0）的判别式为 Δ = b² − 4ac。根由公式 x = [−b ± √b² − 4ac] / (2a) 给出。根的性质取决于 Δ：Δ > 0 有两个不等实根；Δ = 0 有一个重实根；Δ < 0 有一对共轭复根。

(c − b²/(4a)). The vertex of the parabola y = ax² + bx + c lies at (−b/(2a), f(−b/(2a))).

配方法将 ax² + bx + c 化为 a(x + b/(2a))² + (c − b²/(4a))。抛物线 y = ax² + bx + c 的顶点坐标为 (−b/(2a), f(−b/(2a)))。

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3. Coordinate Geometry: Straight Lines and Circles | 坐标几何：直线与圆

− x₁). The equation y = mx + c represents a line with slope m and y-intercept c. The point– . For two lines with gradients m₁ and m₂: parallel if m₁ = m₂; perpendicular if m₁m₂ = −1.

经过 (x₁, y₁) 和 (x₂, y₂) 的直线斜率 m = (y₂ − y₁)/(x₂ − x₁)。方程 y = mx + c 表示斜率为 m、y 轴截距为 c 的直线。点斜式为 y − y₁ = m(x − x₁)。两直线斜率为 m₁ 和 m₂ 时：平行当 m₁ = m₂；垂直当 m₁m₂ = −1。

(y − b)² = r². The general form x² + y² + 2gx + 2fy + c = 0 has centre (−g, −f) and radius √g² + f² − c.

圆心 (a, b) 半径 r 的圆方程为 (x − a)² + (y − b)² = r²。一般形式 x² + y² + 2gx + 2fy + c = 0 的圆心为 (−g, −f)，半径为 √g² + f² − c。

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4. Trigonometry: Basic Ratios and Graphs | 三角学：比值与图像

potenuse, tan θ = opposite/adjacent. Exact values for standard angles should be memorised:

对于直角三角形，sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。必须牢记特殊角的精确值：

θ (degrees)	sin θ	cos θ	tan θ
0°	0	1	0
30°	½	√3/2	1/√3
45°	1/√2	1/√2	1
60°	√3/2	½	√3
90°	1	0	undefined

The tangent graph has period 180° (π radians) and vertical asymptotes at θ = 90° + 180°k.

正弦和余弦图像以 360°（2π 弧度）为周期。正切图像以 180°（π 弧度）为周期，且在 θ = 90° + 180°k 处有铅垂渐近线。

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5. Trigonometric Identities and Equations | 三角恒等式与方程

), and sin²θ + cos²θ = 1. Derived forms include 1 + tan²θ = sec²θ and 1 + cot²θ = cosec²θ.

基本恒等式：tan θ = sin θ / cos θ（cos θ ≠ 0），以及 sin²θ + cos²θ = 1。由此可导出 1 + tan²θ = sec²θ 以及 1 + cot²θ = cosec²θ。

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 ∓ A cos A; cos 2A = cos²A − sin²A = 2cos²A − 1 = 1 − 2sin²A; tan 2A = 2 tan A / (1 − tan²A).

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)。倍角公式：sin 2A = 2 sin A cos A；cos 2A = cos²A − sin²A = 2cos²A − 1 = 1 − 2sin²A；tan 2A = 2 tan A / (1 − tan²A)。

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6. Differentiation Rules | 微分法则

For f(x) = xⁿ, n rational, f '(x) = n xⁿ⁻¹. Constant multiple rule: d/dx [c f(x)] = c f '(x). Sum rule: d/dx [f(x) ± g(x)] = f '(x) ± g'(x).

对 f(x) = xⁿ（n 为有理数），f '(x) = n xⁿ⁻¹。常数倍法则：d/dx [c f(x)] = c f '(x)。和差法则：d/dx [f(x) ± g(x)] = f '(x) ± g'(x)。

The chain rule: dy/dx = dy/du × du/dx. Product rule: d/dx [u v] = u' v + u v'. Quotient rule: d/dx [u/v] = (u' v − u v') / v².

链式法则：dy/dx = dy/du × du/dx。乘法法则：d/dx [u v] = u' v + u v'。除法法则：d/dx [u/v] = (u' v − u v') / v²。

ard functions: d/dx (sin x) = cos x, d/dx (cos x) = −sin x, d/dx (tan x) = sec² x, d/dx (eˣ) = eˣ, d/dx (ln x) = 1/x, d/dx (aˣ) = aˣ ln a.

标准函数导数：d/dx (sin x) = cos x，d/dx (cos x) = −sin x，d/dx (tan x) = sec² x，d/dx (eˣ) = eˣ，d/dx (ln x) = 1/x，d/dx (aˣ) = aˣ ln a。

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7. Integration Techniques | 积分技巧

gration as the reverse of differentiation: ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, n ≠ −1. ∫ 1/x dx = ln|x| + C. For definite integrals, ∫ₐᵇ f(x) dx = F(b) − F(a).

积分作为微分的逆运算：∫ xⁿ dx = xⁿ⁺¹/(n+1) + C，n ≠ −1。∫ 1/x dx = ln|x| + C。对定积分，∫ₐᵇ f(x) dx = F(b) − F(a)。

Standard integrals: ∫ eˣ dx = eˣ x + C; ∫ cos x dx = sin x + C; ∫ sec² x dx = tan x + C; ∫ 1/(1+x²) dx = arctan x + C; ∫ 1/√1−x² dx = arcsin x + C.

标准积分：∫ eˣ dx = eˣ + C；∫ sin x dx = −cos x + C；∫ cos x dx = sin x + C；∫ sec² x dx = tan x + C；∫ 1/(1+x²) dx = arctan x + C；∫ 1/√1−x² dx = arcsin x + C。

Integration by substitution: ∫ f(g(x)) g'(x) dx = ∫ f(u) du where u = g(x). Integration by parts: ∫ u dv = u v − ∫ v du.

换元积分法：设 u = g(x)，则 ∫ f(g(x)) g'(x) dx = ∫ f(u) du。分部积分法：∫ u dv = u v − ∫ v du。

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8. Exponentials and Logarithms | 指数与对数

Laws of indices: aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ / aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1, a⁻ⁿ = 1/aⁿ, a¹/ⁿ = ⁿ√a.

指数法则：aᵐ × aⁿ = aᵐ⁺ⁿ，aᵐ / aⁿ = aᵐ⁻ⁿ，(aᵐ)ⁿ = aᵐⁿ，a⁰ = 1，a⁻ⁿ = 1/aⁿ，a¹/ⁿ = ⁿ√a。

The natural logarithm ln x = logₑ x where e ≈ 2.71828. Conversion: logₐ x = (ln x)/(ln a).

对数法则：logₐ(xy) = logₐ x + logₐ y；logₐ(x/y) = logₐ x − logₐ y；logₐ(xⁿ) = n logₐ x；logₐ 1 = 0；logₐ a = 1。自然对数 ln x = logₑ x，其中 e ≈ 2.71828。换底公式：logₐ x = (ln x)/(ln a)。

The exponential function eˣ and ln x are inverses: e^{ln x} = x for x > 0; ln(eˣ) = x. Growth and decay models use y = A e^kt.

指数函数 eˣ 与 ln x 互为反函数：当 x > 0 时有 e^{ln x} = x；ln(eˣ) = x。增长和衰减模型常使用 y = A e^kt。

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9. Vectors in 2D and 3D | 平面与空间向量

A vector is a quantity with magnitude and direction. In 2D, a = a₁ i + a₂ j; magnitude |a| = √a₁² + a₂². In 3D, a = a₁ i + a₂ j + a₃ k; |a| = √a₁² + a₂² + a₃². A unit vector in the direction of a is â = a / |a|.

向量是具有大小和方向的量。在二维中，a = a₁ i + a₂ j；模长 |a| = √a₁² + a₂²。在三维中，a = a₁ i + a₂ j + a₃ k；|a| = √a₁² + a₂² + a₃²。a 方向的单位向量为 â = a / |a|。

The scalar (dot) product: a · b = |a||b| ). If a · b = 0, vectors are perpendicular. The angle between vectors is cos θ = (a · b)/(|a||b|).

标量积（点积）：a · b = |a||b| cos θ = a₁b₁ + a₂b₂（三维加 a₃b₃）。若 a · b = 0 则向量垂直。向量夹角由 cos θ = (a · b)/(|a||b|) 给出。

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10. Sequences and Series | 数列与级数

1)d. Sum of first n terms: Sₙ = n/2 [2a + (n−1)d] = n/2 (a + l), where l is the last term.

等差数列首项为 a，公差为 d。第 n 项：uₙ = a + (n−1)d。前 n 项和：Sₙ = n/2 [2a + (n−1)d] = n/2 (a + l)，其中 l 为末项。

n ratio r (r ≠ 0). nth term: uₙ = a rⁿ⁻¹. Sum of first n terms: Sₙ = a(1 − rⁿ)/(1 − r) for |r| ≠ 1. Infinite sum: S∞ = a/(1 − r) provided |r| < 1.

等比数列首项为 a，公比为 r（r ≠ 0）。第 n 项：uₙ = a rⁿ⁻¹。前 n 项和：当 |r| ≠ 1 时 Sₙ = a(1 − rⁿ)/(1 − r)。无穷级数和：当 |r| < 1 时 S∞ = a/(1 − r)。

The binomial expansion for (1 + x)ⁿ, n rational and |x| < 1, is (1 + x)ⁿ = 1 + n x + n(n−1)/2! x² + n(n−1)(n−2)/3! x³ + … . Factorial notation: n! = n × (n−1) × … × 1.

二项展开式 (1 + x)ⁿ，n 为有理数且 |x| < 1，则 (1 + x)ⁿ = 1 + n x + n(n−1)/2! x² + n(n−1)(n−2)/3! x³ + … 。阶乘记号：n! = n × (n−1) × … × 1。

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11. Probability Fundamentals | 概率基础

r a sample space S, probability P(A) = n(A)/n(S) provided outcomes are equally likely. P(A') = 1 − P(A). For mutually exclusive events A and B, P(A ∪ B) = P(A) + P(B).

对于样本空间 S，若所有结果等可能发生，则概率 P(A) = n(A)/n(S)。P(A') = 1 − P(A)。若事件 A 与 B 互斥，则 P(A ∪ B) = P(A) + P(B)。

Conditional probability: P(A|B) = P(A ∩ B) / P(B), P(B) > 0. 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)，其中 P(B) > 0。若 P(A ∩ B) = P(A)P(B)，则事件 A 与 B 相互独立。

: E(X) = Σ x P(X = x). Variance: Var(X) = E(X²) − [E(X)]². Standard deviation σ = √Var(X).

离散随机变量期望：E(X) = Σ x P(X = x)。方差：Var(X) = E(X²) − [E(X)]²。标准差 σ = √Var(X)。

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12. Kinematics in One Dimension | 一维运动学

locity v, displacement s, time t: v = u + at, s = ut + ½at², v² = u² + 2as, s = ½(u + v)t.

对于恒定加速度 a，初速度 u，末速度 v，位移 s，时间 t：v = u + at，s = ut + ½at²，v² = u² + 2as，s = ½(u + v)t。

ation and integration: v = ds/dt, a = dv/dt = d²s/dt². Conversely, s = ∫ v dt, v = ∫ a dt.

位移、速度和加速度通过微积分相联系：v = ds/dt，a = dv/dt = d²s/dt²。反过来，s = ∫ v dt，v = ∫ a dt。

sually g = 9.8 m s⁻²). Use the same kinematic equations with appropriate sign conventions.

在重力作用下的竖直运动，加速度 a = −g（通常 g = 9.8 m s⁻²）。使用相同的运动学方程，注意符号正负约定。

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