Year 10 WJEC Further Mathematics: Quick Reference Handbook of Formulas and Theorems | Year 10 WJEC 进阶数学：公式定理速查手册

📚 Year 10 WJEC Further Mathematics: Quick Reference Handbook of Formulas and Theorems | Year 10 WJEC 进阶数学：公式定理速查手册

This quick reference handbook compiles all essential formulas and theorems required for the Year 10 WJEC Level 2 Additional Mathematics course (also relevant for Further Mathematics preparation). Use it as a handy revision tool to check definitions, identities, and key results across Pure Mathematics, Mechanics, and Statistics.

这本速查手册整理了 Year 10 WJEC Level 2 附加数学（亦适用于进阶数学备考）所需的所有核心公式和定理。你可以将其用作便捷的复习工具，快速查阅纯数学、力学和统计中的定义、恒等式与关键结论。

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1. Algebra Essentials | 代数基础

Product of powers: aᵐ × aⁿ = aᵐ⁺ⁿ

幂的乘法：aᵐ × aⁿ = aᵐ⁺ⁿ

Quotient of powers: aᵐ ÷ aⁿ = aᵐ⁻ⁿ

幂的除法：aᵐ ÷ aⁿ = aᵐ⁻ⁿ

Power of a power: (aᵐ)ⁿ = aᵐⁿ

幂的乘方：(aᵐ)ⁿ = aᵐⁿ

Power of a product: (ab)ⁿ = aⁿ bⁿ

积的乘方：(ab)ⁿ = aⁿ bⁿ

Zero exponent: a⁰ = 1 (a ≠ 0)

零指数：a⁰ = 1（a ≠ 0）

Negative exponent: a⁻ⁿ = 1 / aⁿ

负指数：a⁻ⁿ = 1 / aⁿ

Fractional exponent: a^m/n = ⁿ√aᵐ

分数指数：a^m/n = ⁿ√aᵐ

Difference of squares: a² – b² = (a + b)(a – b)

平方差公式：a² – b² = (a + b)(a – b)

Perfect square trinomial: a² ± 2ab + b² = (a ± b)²

完全平方式：a² ± 2ab + b² = (a ± b)²

Rationalising denominator: 1/√a = √a / a; 1/(√a + √b) = (√a – √b)/(a – b)

分母有理化：1/√a = √a / a；1/(√a + √b) = (√a – √b)/(a – b)

Expanding brackets: (x + a)(x + b) = x² + (a + b)x + ab

展开括号：(x + a)(x + b) = x² + (a + b)x + ab

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

Standard form: ax² + bx + c = 0, a ≠ 0

标准形式：ax² + bx + c = 0，a ≠ 0

Quadratic formula: x = [ -b ± √b² – 4ac ] / (2a)

求根公式：x = [ -b ± √b² – 4ac ] / (2a)

Discriminant: Δ = b² – 4ac. Δ > 0: two distinct real roots; Δ = 0: one repeated real root; Δ < 0: no real roots.

判别式：Δ = b² – 4ac。Δ > 0：两个相异实根；Δ = 0：一个重实根；Δ < 0：无实根。

Sum and product of roots: if roots are α and β, then α + β = –b/a, αβ = c/a

根的和与积：若根为 α 和 β，则 α + β = –b/a，αβ = c/a

Completing the square: x² + bx = (x + b/2)² – (b/2)²

配方法：x² + bx = (x + b/2)² – (b/2)²

Vertex form: y = a(x – h)² + k, vertex (h, k); for y = ax² + bx + c, vertex x = –b/(2a), y = (4ac – b²)/(4a)

顶点式：y = a(x – h)² + k，顶点为 (h, k)；对于 y = ax² + bx + c，顶点横坐标 x = –b/(2a)，纵坐标 y = (4ac – b²)/(4a)

Sketching: a > 0: ∪ shape; a < 0: ∩ shape. y-intercept = c.

草图：a > 0 呈 ∪ 形；a < 0 呈 ∩ 形。y 轴截距为 c。

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3. Coordinate Geometry of Straight Lines | 直线坐标几何

Gradient between two points (x₁, y₁) and (x₂, y₂): m = (y₂ – y₁) / (x₂ – x₁)

两点间的斜率 (x₁, y₁) 和 (x₂, y₂)：m = (y₂ – y₁) / (x₂ – x₁)

Midpoint: ( (x₁ + x₂)/2 , (y₁ + y₂)/2 )

中点坐标：( (x₁ + x₂)/2 , (y₁ + y₂)/2 )

Distance between two points: d = √[(x₂ – x₁)² + (y₂ – y₁)²]

两点间距离：d = √[(x₂ – x₁)² + (y₂ – y₁)²]

Equation of a straight line: y – y₁ = m(x – x₁); also y = mx + c and ax + by + c = 0

直线方程：点斜式 y – y₁ = m(x – x₁)；斜截式 y = mx + c；一般式 ax + by + c = 0

Parallel lines: m₁ = m₂

平行直线：斜率相等，m₁ = m₂

Perpendicular lines: m₁ × m₂ = –1

互相垂直直线：斜率积为 –1，m₁ × m₂ = –1

Intercepts: x-intercept when y = 0; y-intercept when x = 0

截距：x 轴截距时 y = 0；y 轴截距时 x = 0

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4. Sequences & Series | 数列与级数

Arithmetic progression (AP): nth term u_n = a + (n – 1)d

等差数列：第 n 项 u_n = a + (n – 1)d

Sum of first n terms of AP: S_n = n/2 [2a + (n – 1)d] = n/2 (a + l), where l is last term

等差数列前 n 项和：S_n = n/2 [2a + (n – 1)d] = n/2 (a + l)，其中 l 为末项

Geometric progression (GP): nth term u_n = arⁿ⁻¹

等比数列：第 n 项 u_n = arⁿ⁻¹

Sum of first n terms of GP: S_n = a(1 – rⁿ) / (1 – r), for r ≠ 1

等比数列前 n 项和：S_n = a(1 – rⁿ) / (1 – r)，r ≠ 1

Sum to infinity of GP: S∞ = a / (1 – r), valid for |r| < 1

等比级数无穷项和：S∞ = a / (1 – r)，当 |r| < 1 时成立

Common difference / ratio: d = u_n – u_n-1; r = u_n / u_n-1

公差与公比：d = u_n – u_n-1；r = u_n / u_n-1

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5. Binomial Expansion | 二项式展开

Binomial theorem for positive integer n: (a + b)ⁿ = Σ [nCr aⁿ⁻ʳ bʳ] from r=0 to n, where nCr = n! / (r!(n – r)!)

正整数指数的二项式定理：(a + b)ⁿ = Σ [nCr aⁿ⁻ʳ bʳ]（r 从 0 到 n），其中 nCr = n! / (r!(n – r)!)

Pascal's triangle relation: nCr + nC(r+1) = (n+1)C(r+1)

帕斯卡三角关系：nCr + nC(r+1) = (n+1)C(r+1)

Expansion of (1 + x)ⁿ: 1 + nx + [n(n – 1)/2!] x² + [n(n – 1)(n – 2)/3!] x³ + … + xⁿ

(1 + x)ⁿ 的展开：1 + nx + [n(n – 1)/2!] x² + [n(n – 1)(n – 2)/3!] x³ + … + xⁿ

Special case: nCr = nC(n – r)

特殊性质：nCr = nC(n – r)

Term independent of x: find r such that power of x is zero

常数项：找到使 x 的幂为零的 r

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6. Trigonometry | 三角学

Exact trigonometric values: sin 0°=0, sin 30°=1/2, sin 45°=1/√2, sin 60°=√3/2, sin 90°=1; cos 0°=1, cos 30°=√3/2, cos 45°=1/√2, cos 60°=1/2, cos 90°=0; tan 0°=0, tan 30°=1/√3, tan 45°=1, tan 60°=√3, tan 90° undefined.

特殊角精确值：sin 0°=0, sin 30°=1/2, sin 45°=1/√2, sin 60°=√3/2, sin 90°=1；cos 0°=1, cos 30°=√3/2, cos 45°=1/√2, cos 60°=1/2, cos 90°=0；tan 0°=0, tan 30°=1/√3, tan 45°=1, tan 60°=√3, tan 90° 无定义。

Pythagorean identity: sin²θ + cos²θ = 1

毕达哥拉斯恒等式：sin²θ + cos²θ = 1

Tangent identity: tanθ = sinθ / cosθ

正切恒等式：tanθ = sinθ / cosθ

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

正弦定理：a / sin A = b / sin B = c / sin C

Cosine rule: a² = b² + c² – 2bc cos A; cos A = (b² + c² – a²) / (2bc)

余弦定理：a² = b² + c² – 2bc cos A；cos A = (b² + c² – a²) / (2bc)

Area of a triangle: Area = ½ ab sin C

三角形面积：面积 = ½ ab sin C

Ambiguous case: when given two sides and a non-included angle, there may be two possible triangles.

模糊情况：已知两边和其中一边对角时，可能存在两个三角形。

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7. Vectors | 向量

Representation: vector v = xi + yj or column vector [x, y]

向量表示：v = xi + yj 或列向量 [x, y]

Magnitude (modulus): |v| = √x² + y²

模长：|v| = √x² + y²

Addition and subtraction: (x₁i + y₁j) ± (x₂i + y₂j) = (x₁ ± x₂)i + (y₁ ± y₂)j

加减法：(x₁i + y₁j) ± (x₂i + y₂j) = (x₁ ± x₂)i + (y₁ ± y₂)j

Scalar multiplication: k(xi + yj) = kxi + kyj

标量乘法：k(xi + yj) = kxi + kyj

Position vector: vector from origin O to point P is OP

位置向量：从原点 O 到点 P 的向量为 OP

Unit vector: a vector with magnitude 1 in the direction of v: v / |v|

单位向量：方向与 v 相同、长度为1的向量：v / |v|

Parallel vectors: v is parallel to w if v = kw

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