Year 11 WJEC Maths: Formula & Theorems Quick Reference | Year 11 WJEC 数学：公式定理速查手册

📚 Year 11 WJEC Maths: Formula & Theorems Quick Reference | Year 11 WJEC 数学：公式定理速查手册

This quick reference guide covers all the essential formulas, rules and theorems you need for the Year 11 WJEC GCSE Mathematics exam. Keep it with you during revision to check facts quickly and build confidence before your papers.

本速查手册涵盖 Year 11 WJEC GCSE 数学考试所需的所有基本公式、法则与定理。复习时随身携带，可随时查阅要点，增强考前信心。

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1. Algebra Essentials | 代数基础

rackets like (x + a)(x + b), use FOIL: First, Outer, Inner, Last, then collect like terms.

展开括号：将括号内的每一项与括号外的项相乘。单项括号如 a(b + c) = a × b + a × c；双括号如 (x + a)(x + b) 使用 FOIL 法则：首项、外项、内项、尾项依次相乘，再合并同类项。

that multiply to c and add to b: x² + bx + c = (x + p)(x + q) where p + q = b and pq = c.

因式分解是展开的逆运算。先提取各项的最高公因式 (HCF)。对于形如 x² + bx + c 的二次式，找出两个数相乘得 c 且相加得 b：x² + bx + c = (x + p)(x + q)，满足 p + q = b 且 pq = c。

b² = (a + b)(a − b). Spotting these patterns saves time in both expanding and factorising.

值得记住的特殊展开式：(a + b)² = a² + 2ab + b²；(a − b)² = a² − 2ab + b²；平方差 a² − b² = (a + b)(a − b)。识别这些模式能加快展开与因式分解的速度。

Solving linear equations: isolate the unknown by performing the same operation on both sides. Add, subtract, multiply or divide as needed, always keeping the equation balanced. Substitute your solution back into the original to check.

解一元一次方程：对等式两边执行相同的运算来分离未知数。根据需要进行加、减、乘、除，始终保持方程平衡。将解代回原方程进行验证。

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2. Quadratic Equations | 二次方程

For any quadratic equation ax² + bx + c = 0, the quadratic formula gives the solutions:

x = [−b ± √b² − 4ac] / (2a)

对任意二次方程 ax² + bx + c = 0，求根公式为：

x = [−b ± √b² − 4ac] / (2a)

es two distinct real roots; Δ = 0 gives one repeated real root; Δ < 0 gives no real roots.

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。

b/2)²: x² + bx = (x + b/2)² − (b/2)². This form helps find the vertex and solve equations.

配方法将二次式写为 a(x + p)² + q 的形式。对于 x² + bx，添加并减去 (b/2)²：x² + bx = (x + b/2)² − (b/2)²。这种形式有助于求出顶点坐标和求解方程。

The graph of a quadratic is a parabola. If a > 0 the parabola opens upwards (∪); if a < 0 it opens downwards (∩ s of symmetry is x = −b/(2a), and the turning point can be found by completing the square.

二次函数的图像是一条抛物线。a > 0 时开口向上 (∪)；a < 0 时开口向下 (∩)。对称轴为 x = −b/(2a)，通过配方法可求得顶点坐标。

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3. Pythagoras & Trigonometry | 勾股定理与三角学

Pythagoras’ theorem: in a right‑ dentify the hypotenuse (longest side) opposite the right angle before substituting values.

勾股定理：在直角三角形中，斜边为 c，有 a² + b² = c²。代入数值前，务必先确认斜边（最长边）位于直角对侧。

Trigonometric ratios in right‑ θ = adjacent / hypotenuse, tan θ = opposite / adjacent. Use SOH CAH TOA to remember them.

直角三角形中的三角函数比：sin θ = 对边 / 斜边，cos θ = 邻边 / 斜边，tan θ = 对边 / 邻边。可用 SOH CAH TOA 帮助记忆。

A = b / sin B = c / sin C. Use when you know two angles and a side, or two sides and a non‑included angle. The reciprocal form sin A / a = sin B / b = sin C / c also works.

正弦定理（任意三角形）：a / sin A = b / sin B = c / sin C。适用于已知两角一边，或已知两边及一个非夹角。其倒数形式 sin A / a = sin B / b = sin C / c 同样成立。

c² − a²) / (2bc). Use when you have two sides and the included angle, or all three sides.

余弦定理：a² = b² + c² − 2bc cos A。求角度时变形为 cos A = (b² + c² − a²) / (2bc)。已知两边及其夹角，或已知三边时使用。

trigonometry: Area = ½ a b sin C, where a and b are two sides and C is the included angle.

用三角函数求三角形面积：面积 = ½ a b sin C，其中 a、b 为两边，C 为其夹角。

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4. Mensuration Formulas | 测量公式

b × h; trapezium = ½ (a + b) h; circle = π r². For a circle, circumference C = 2π r = π d.

面积公式：矩形 = 长 × 宽；三角形 = ½ × 底 × 高；平行四边形 = 底 × 高；梯形 = ½ (a + b) × 高；圆 = π r²。圆的周长 C = 2π r = π d。

rea of a sector: (θ / 360) × π r², where θ is the angle at the centre measured in degrees.

扇形弧长：(θ / 360) × 2π r。扇形面积：(θ / 360) × π r²，其中 θ 为圆心角的度数。

Volume formulas: cuboid = l × w × h; prism = area of cross‑section × length; cylinder = π r² h. Remember that a cylinder is a circular prism.

体积公式：长方体 = 长 × 宽 × 高；棱柱 = 横截面积 × 长；圆柱体 = π r² h。注意圆柱体可视为圆形棱柱。

ed surface of cone = π r l (where l is slant height); curved surface of cylinder = 2π r h.

圆锥体积 = ⅓ π r² h；球体积 = (4/3) π r³。表面积：球 = 4π r²；圆锥侧面积 = π r l（l 为斜高）；圆柱侧面积 = 2π r h。

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5. Compound Measures | 复合测量

Time = Distance ÷ Speed. Always check that units match, e.g. m/s with metres and seconds.

速度 = 路程 ÷ 时间。平均速度公式可变形为：路程 = 速度 × 时间，时间 = 路程 ÷ 速度。务必保持单位一致，例如使用米和秒得到 m/s。

on rearrangement is Mass = Density × Volume. Water has a density of 1 g/cm³ or 1000 kg/m³.

密度 = 质量 ÷ 体积。常见变形为：质量 = 密度 × 体积。水的密度为 1 g/cm³ 或 1000 kg/m³。

÷ Area. Force = Pressure × Area. Pressure is usually measured in N/m² (pascals) or N/cm².

压强 = 力 ÷ 面积。力 = 压强 × 面积。压强常用单位是 N/m²（帕斯卡）或 N/cm²。

to base units first. 1 km = 1000 m, 1 hour = 3600 s, 1 tonne = 1000 kg, 1 cm² = 0.0001 m².

单位换算：计算复合量时，先将所有量换算为基本单位。1 km = 1000 m，1 小时 = 3600 s，1 吨 = 1000 kg，1 cm² = 0.

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