📚 Year 10 Edexcel Further Mathematics: Quick Reference Formula & Theorem Handbook | 爱德思 Year 10 进阶数学 公式定理速查手册
This handbook provides a concise reference for the essential formulas and theorems covered in the Year 10 Edexcel Further Mathematics course. You can use it to quickly review key results, consolidate your understanding and prepare effectively for exams. Each section presents definitions and formulas in both English and Chinese, followed by worked-style statements and practice hints.
本手册是爱德思 Year 10 进阶数学课程核心公式与定理的速查指南，帮助同学们快速回顾重要结果、巩固理解并高效备考。每个小节均以中英双语呈现定义与公式，并配有应用提示，方便自查与记忆。
1. Quadratic Equations and Functions | 二次方程与函数
x² + bx + c with a ≠ 0. The curve y = f(x) is a parabola with a vertical line of symmetry.
二次函数的标准形式为 f(x) = ax² + bx + c 且 a ≠ 0。其图像 y = f(x) 是一条抛物线，具有一条竖直对称轴。
Quadratic formula: For ax² + bx + c = 0, the roots are given by x = (−b ± √b² − 4ac) / (2a). The expression b² − 4ac is called the discriminant, denoted Δ.
求根公式：对于 ax² + bx + c = 0，根为 x = (−b ± √b² − 4ac) ÷ (2a)。其中 b² − 4ac 称为判别式，记作 Δ。
peated real root; if Δ < 0, the roots are complex (not on the Further Maths Y10 pure syllabus, but important for extension).
当 Δ > 0 时，方程有两个不等实根；当 Δ = 0 时，有一个重实根；当 Δ < 0 时，根为复数（虽不属 Year 10 纯数核心，但可作为拓展理解）。
ct of roots: If α and β are the roots of ax² + bx + c = 0, then α + β = −b/a and αβ = c/a.
根与系数的关系：若 α 和 β 是 ax² + bx + c = 0 的两根，则 α + β = −b/a，αβ = c/a。
= a(x + b/(2a))² + (c − b²/(4a)). The vertex of the parabola is at (−b/(2a), c − b²/(4a)).
配方法：ax² + bx + c = a(x + b/(2a))² + (c − b²/(4a))。抛物线的顶点坐标为 (−b/(2a), c − b²/(4a))。
2. Sequences and Series | 数列与级数
consecutive terms. Its nth term is given by uₙ = a + (n − 1)d, where a is the first term.
等差数列中，相邻两项的差 d 为常数。第 n 项公式为 uₙ = a + (n − 1)d，其中 a 为首项。
arithmetic series: Sₙ = n/2 [2a + (n − 1)d] or Sₙ = n/2 (a + l), where l is the last term.
等差数列前 n 项和：Sₙ = n/2 [2a + (n − 1)d] 或 Sₙ = n/2 (a + l)，其中 l 为末项。
ometric sequence has a constant ratio r between consecutive terms. The nth term is uₙ = arn−1.
等比数列中，相邻两项的比值 r 为常数。第 n 项为 uₙ = arn−1。
a geometric series: Sₙ = a(1 − rⁿ) / (1 − r) for r ≠ 1. The sum to infinity exists only if |r| < 1 and is given by S∞ = a / (1 − r).
等比数列前 n 项和：当 r ≠ 1 时，Sₙ = a(1 − rⁿ) / (1 − r)。无穷等比级数仅在 |r| < 1 时收敛，其和为 S∞ = a / (1 − r)。
3. Coordinate Geometry | 坐标几何
o lines are parallel if their gradients are equal; they are perpendicular if m₁ × m₂ = −1.
连接两点 (x₁, y₁) 和 (x₂, y₂) 的直线的斜率（梯度）为 m = (y₂ − y₁) / (x₂ − x₁)。两直线平行时斜率相等；垂直时满足 m₁ × m₂ = −1。
ween two points: d = √[(x₂ − x₁)² + (y₂ − y₁)²]. Midpoint: M = ((x₁ + x₂)/2, (y₁ + y₂)/2).
两点间距离：d = √[(x₂ − x₁)² + (y₂ − y₁)²]；中点坐标：M = ((x₁ + x₂)/2, (y₁ + y₂)/2)。
Equation of a straight line: y − y₁ = m(x − x₁) is the point‑gradient form. General form: ax + by + c = 0.
直线方程的点斜式：y − y₁ = m(x − x₁)。一般式：ax + by + c = 0。
dius r is (x − h)² + (y − k)² = r². If the circle passes through the origin, h² + k² = r².
圆心为 (h, k)、半径为 r 的圆方程为 (x − h)² + (y − k)² = r²。若过原点，则 h² + k² = r²。
4. Exponential and Logarithmic Functions | 指数函数与对数函数
The function f(x) = aˣ r a > 0, a ≠ 1 is called an exponential function. The natural exponential function is eˣ, where e ≈ 2.71828. The gradient of the curve y = eˣ at x = 0 is 1.
形如 f(x) = aˣ（a > 0 且 a ≠ 1）的函数称为指数函数。自然指数函数为 eˣ，其中 e ≈ 2.71828。曲线 y = eˣ 在 x = 0 处的斜率为 1。
ey rules: logₐ (xy) = logₐ x + logₐ y, logₐ (x/y) = logₐ x − logₐ y, logₐ (xⁿ) = n logₐ x.
对数函数是指数函数的反函数。logₐ x 表示以 a 为底 x 的对数。特别地，ln x = logₑ x。基本法则：logₐ (xy) = logₐ x + logₐ y，logₐ (x/y) = logₐ x − logₐ y，logₐ (xⁿ) = n logₐ x。
logₐ b = logₓ b / logₓ a. The natural logarithm is useful for calculus: d/dx (ln x) = 1/x.
换底公式：logₐ b = logₓ b / logₓ a。自然对数在微积分中很重要，例如 d/dx (ln x) = 1/x。
5. Binomial Theorem | 二项式定理
binomial expansion of (a + b)ⁿ is given by Σ (k=0 to n) C(n, k) aⁿ⁻ᵏ bᵏ, where C(n, k) = n! / [k!(n−k)!] is the binomial coefficient.
对正整数 n，(a + b)ⁿ 的二项展开式为 Σ (k=0 到 n) C(n, k) aⁿ⁻ᵏ bᵏ，其中二项式系数 C(n, k) = n! / [k!(n−k)!]。
The first few expansions are useful for quick reference: (1 + x)ⁿ ≈ 1 + nx + n(n−1)/2! x² + n(n−1)(n−2)/3! x³ + … This infinite series is valid for |x| < 1 when n is not a positive integer.
常用开场式：(1 + x)ⁿ ≈ 1 + nx + n(n−1)/2! x² + n(n−1)(n−2)/3! x³ + … 当 n 不是正整数时，该无穷级数在 |x| < 1 时收敛。
rom k=0) is C(n, k) aⁿ⁻ᵏ bᵏ. Often used to approximate small changes or compound interest.
当 n 为正整数时，展开式共有 n+1 项。第 k 项（k 从 0 开始）为 C(n, k) aⁿ⁻ᵏ bᵏ。该定理常用于小变化近似或复利计算。
6. Matrices | 矩阵
[a b; c d] and B = [e f; g h], the sum A + B is obtained by adding corresponding entries.
矩阵是一个数表。对于 2 × 2 矩阵 A = [a b; c d] 和 B = [e f; g h]，加法 A + B 是对应元素相加。
Multiplication: A × B = [a b; c d] × [e f; g h] = [(ae+bg) (af+bh); (ce+dg) (cf+dh)]. In general, matrix multiplication is not commutative.
乘法：A × B = [a b; c d] × [e f; g h] = [(ae+bg) (af+bh); (ce+dg) (cf+dh)]。一般情况下矩阵乘法不满足交换律。
A) = ad − bc. The inverse A⁻¹ exists only if det(A) ≠ 0 and is given by A⁻¹ = 1/(ad−bc) [d −b; −c a].
2×2 矩阵的行列式：det(A) = ad − bc。当下满足 det(A) ≠ 0 时逆矩阵存在，且 A⁻¹ = 1/(ad−bc) [d −b; −c a]。
fies A × I = I × A = A. The zero matrix O has all entries 0 and acts as additive identity.
单位矩阵 I = [1 0; 0 1]，满足 A × I = I × A = A。零矩阵 O 的所有元素均为 0，是加法单位元。
7. Vectors | 向量
A vector quantity has magnitude and direction. In two dimensions, a vector can be written as a column vector [p; q] or as p i + q j, where i and j are unit vectors along the x‑ and y‑axes.
向量既有大小又有方向。在二维空间中，向量可表示为列向量 [p; q] 或 p i + q j，其中 i 和 j 分别为沿 x 轴和 y 轴的单位向量。
Magnitude (length) of a vector v = [x; y] is |v| = √x² + y². A unit vector in the same direction is v / |v|.
向量 v = [x; y] 的模（长度）为 |v| = √x² + y²。同方向的单位向量为 v / |v|。
ddition: [x₁; y₁] + [x₂; y₂] = [x₁+x₂; y₁+y₂]. Scalar multiplication: k [x; y] = [kx; ky].
向量加法：[x₁; y₁] + [x₂; y₂] = [x₁+x₂; y₁+y₂]。标量乘法：k [x; y] = [kx; ky]。
Dot (scalar) product: a · b = |a| |b| cos θ, and in components a · b = a₁b₁ + a₂b₂. It is zero if the vectors are perpendicular.
点积（标量积）：a · b = |a| |b| cos θ，用分量计算时 a · b = a₁b₁ + a₂b₂。若两向量垂直，点积为零。
8. Trigonometry | 三角学
In a right‑ θ = opposite / adjacent. The exact values for 0°, 30°, 45°, 60°, 90° should be memorised.
在直角三角形中，sin θ = 对边 / 斜边，cos θ = 邻边 / 斜边，tan θ = 对边 / 邻边。0°、30°、45°、60°、90° 的精确值必须牢记。
Key identity: sin² θ + cos² θ ≡ es: 1 + tan² θ = sec² θ, 1 + cot² θ = cosec² θ. (sec = 1/cos, cosec = 1/sin, cot = 1/tan).
核心恒等式：sin² θ + cos² θ ≡ 1。由此可导出 1 + tan² θ = sec² θ 和 1 + cot² θ = cosec² θ。（注：sec = 1/cos，cosec = 1/sin，cot = 1/tan）。
r any triangle). Cosine rule: a² = b² + c² − 2bc cos A, and similarly for the other sides.
正弦定理：a / sin A = b / sin B = c / sin C（对任意三角形均成立）。余弦定理：a² = b² + c² − 2bc cos A，其他边同理可推。
le: Area = ½ ab sin C. The ambiguous case of the sine rule arises when two sides and a non‑included acute angle are given.
已知两边及其夹角时的三角形面积公式：Area = ½ ab sin C。正弦定理中若已知两边及其中一边对角（锐角），可能出现多解情况。
9. Introduction to Calculus | 微积分入门
Differentiation from first principles: for f(x), the derivative is f' (x+h) − f(x)] / h. The derivative gives the gradient of the tangent to the curve y = f(x).
根据第一原理求导：f'(x) = lim_(h→0) [f(x+h) − f(x)] / h。导数表示曲线 y = f(x) 的切线斜率。
Power rule: If f(x) = xⁿ, then f'(x) = n xⁿ⁻¹. Constant multiple and sum rules: d/dx [c f(x)] = c f'(x), d/dx [f(x) ± g(x)] = f'(x) ± g'(x).
幂函数求导：若 f(x) = xⁿ，则 f'(x) = n xⁿ⁻¹。常数倍法则和加减法则：d/dx [c f(x)] = c f'(x)，d/dx [f(x) ± g(x)] = f'(x) ± g'(x)。
of differentiation: ∫ xⁿ dx = xⁿ⁺¹ / (n+1) + C, for n ≠ −1. The special case ∫ 1/x dx = ln |x| + C is often covered in extension.
不定积分是微分的逆运算：∫ xⁿ dx = xⁿ⁺¹ / (n+1) + C，n ≠ −1。特例 ∫ 1/x dx = ln |x| + C 常在拓展部分出现。
Definite integral ∫ₐᵇ f(x) dx gives the signed area between the curve and the x‑axis from x = a to x = b. The fundamental theorem: ∫ₐᵇ f(x) dx = F(b) − F(a), where F'(x) = f(x).
定积分 ∫ₐᵇ f(x) dx 表示曲线与 x 轴在 x = a 到 x = b 之间的带符号面积。微积分基本定理：∫ₐᵇ f(x) dx = F(b) − F(a)，其中 F'(x) = f(x)。
10. Functions and Transformations | 函数与图像变换
rse function f⁻¹ reverses the mapping; its graph is the reflection of y = f(x) in the line y = x.
函数 f 将每一个输入 x 映射到唯一输出 f(x)。定义域为允许输入的集合，值域为可能输出的集合。反函数 f⁻¹ 逆转该映射，其图像是 y = f(x) 关于直线 y = x 的对称图形。
(up if a>0). y = f(x + a) translates the graph −a units horizontally (left if a>0).
图像 y = f(x) 的变换：y = f(x) + a 将图像向上平移 a 个单位（a>0 时）；y = f(x + a) 将图像向左平移 a 个单位。
a f(x) stretches the graph vertically by factor a (if a>1, it is a stretch away from x‑axis; if 0<a<1, a compression). y = f(ax) stretches horizontally by factor 1/a.
拉伸变换：y = a f(x) 将图像沿竖直方向拉伸 a 倍（a>1 时为拉伸，0<a<1 时为压缩）；y = f(ax) 将图像沿水平方向拉伸 1/a 倍。
Reflections: y = −f(x) reflects the graph in the x‑axis; y = f(−x) reflects in the y‑ sformations should be applied in the correct order, usually vertical transformations last.
对称变换：y = −f(x) 关于 x 轴对称；y = f(−x) 关于 y 轴对称。复合变换须注意顺序，通常先执行水平变换，最后执行竖直变换。
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