📚 IGCSE CCEA Further Mathematics: Formula & Theorem Quick Reference Handbook | IGCSE CCEA 进阶数学：公式定理速查手册
This comprehensive quick-reference handbook presents the essential formulas and theorems required for the CCEA IGCSE Further Mathematics examination. It is designed as a rapid revision tool, consolidating key results from algebra, calculus, trigonometry, coordinate geometry, vectors, and sequences. Candidates can use this guide to refresh memory, check conventions, and build confidence before tackling past papers.
这本速查手册汇集了 CCEA IGCSE 进阶数学考试所需的所有核心公式与定理，旨在作为快速复习工具，整合代数、微积分、三角学、坐标几何、向量和数列中的关键结论。考生可借助本手册巩固记忆、核对符号，并在练习真题前建立信心。
1. Surds and Indices | 根式与指数
The laws of indices lay the foundation for algebraic manipulation. For a real base a > 0 and rational exponents m, n: multiplication adds exponents, division subtracts them, and a power raised to a power multiplies the exponents. Zero and negative indices are covered by a⁰ = 1 (a ≠ 0) and a⁻ⁿ = 1/aⁿ. Fractional exponents link powers to roots: a^(1/n) = ⁿ√a and a^(m/n) = (ⁿ√a)ᵐ = ⁿ√(aᵐ).
指数法则是代数运算的基础。对于实数底数 a > 0 和有理指数 m, n：乘法指数相加，除法指数相减，幂的幂则指数相乘。零指数和负指数遵循 a⁰ = 1（a ≠ 0）以及 a⁻ⁿ = 1/aⁿ。分数指数将幂与根式联系起来：a^(1/n) = ⁿ√a，a^(m/n) = (ⁿ√a)ᵐ = ⁿ√(aᵐ)。
Surds can be simplified by extracting perfect square factors. For positive radicands, √(ab) = √a × √b and √(a/b) = √a / √b. When rationalising denominators, multiply by a conjugate: 1/√a = √a / a, and 1/(√a + √b) = (√a − √b) / (a − b). Similar techniques apply for binomial denominators containing surds.
根式可通过提取完全平方因子简化。对于正被开方数，√(ab) = √a × √b，√(a/b) = √a / √b。分母有理化时乘以共轭根式：1/√a = √a / a，以及 1/(√a + √b) = (√a − √b) / (a − b)。对含有根式的二项分母也适用类似技巧。
2. Quadratic Functions and Equations | 二次函数与方程
The quadratic formula gives the roots of ax² + bx + c = 0:
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 one repeated real root, and Δ < 0 gives no real roots. The sum of the roots is α + β = −b/a and the product is αβ = c/a.
判别式 Δ = b² − 4ac 决定了根的性质：Δ > 0 有两个不等实根，Δ = 0 有一个重根，Δ < 0 无实根。根的和为 α + β = −b/a，积为 αβ = c/a。
Completing the square rewrites the quadratic in vertex form a(x − h)² + k, where the vertex is (h, k). This is used to find maximum or minimum values and to solve quadratic inequalities. To solve an inequality, sketch the graph to see where the curve lies above or below the x‑axis.
配方法将二次式写成顶点式 a(x − h)² + k，其中顶点为 (h, k)。此法可用于求最大值或最小值，也可解二次不等式。解不等式时，可通过画图观察曲线位于 x 轴上方或下方的区域。
3. Polynomials and the Factor Theorem | 多项式与因式定理
The Remainder Theorem states: when a polynomial f(x) is divided by (x − a), the remainder is r = f(a). The Factor Theorem follows: (x − a) is a factor of f(x) if and only if f(a) = 0. These tools identify roots and simplify polynomial division. Long division or synthetic division can reduce a cubic or higher-degree polynomial after one factor has been found.
余式定理指出：当多项式 f(x) 除以 (x − a) 时，余数为 r = f(a)。因式定理接续而来：(x − a) 是 f(x) 的因式当且仅当 f(a) = 0。利用这些工具可求根并简化多项式除法。找出一个因式后，可用长除法或综合除法将三次或更高次多项式降阶。
For a cubic f(x) with integer coefficients, possible rational roots are of the form ±(factor of constant term) / (factor of leading coefficient). Testing these candidates with f(a) = 0 efficiently discovers factors. Once fully factorised, the graph’s intercepts and rough shape can be sketched.
对于整系数三次多项式 f(x)，可能的有理根形式为 ±(常数项因数)/(首项系数因数)。通过 f(a) = 0 检验这些候选值可高效发现因式。完全因式分解后，可描绘截距和大致形状。
4. Coordinate Geometry – Straight Lines and Circles | 坐标几何 – 直线与圆
The distance between two points (x₁, y₁) and (x₂, y₂) is d = √[(x₂ − x₁)² + (y₂ − y₁)²]. The midpoint is M = ((x₁ + x₂)/2, (y₁ + y₂)/2). The gradient of the line joining them is m = (y₂ − y₁) / (x₂ − x₁), provided x₂ ≠ x₁.
两点 (x₁, y₁) 与 (x₂, y₂) 之间的距离为 d = √[(x₂ − x₁)² + (y₂ − y₁)²]，中点为 M = ((x₁ + x₂)/2, (y₁ + y₂)/2)。连接这两点的直线斜率为 m = (y₂ − y₁)/(x₂ − x₁)，其中 x₂ ≠ x₁。
Straight line equations: y − y₁ = m(x − x₁) (point‑gradient form), y = mx + c (gradient‑intercept). Parallel lines share the same gradient, while perpendicular lines satisfy m₁ × m₂ = −1. A circle with centre (a, b) and radius r has equation (x − a)² + (y − b)² = r². The tangent at a point on the circle is perpendicular to the radius drawn to that point.
直线方程：y − y₁ = m(x − x₁)（点斜式），y = mx + c（斜截式）。平行直线斜率相等，垂直直线满足 m₁ × m₂ = −1。圆心 (a, b)、半径 r 的圆的方程为 (x − a)² + (y − b)² = r²。圆上一点处的切线与过该点的半径垂直。
5. Sequences and Series | 数列与级数
Arithmetic sequence: the nth term uₙ = a + (n − 1)d, where a is the first term and d the common difference. Sum of the first n terms: Sₙ = n/2 [2a + (n − 1)d] = n/2 (a + l), with l being the last term.
等差数列：第 n 项 uₙ = a + (n − 1)d，其中 a 为首项，d 为公差。前 n 项和：Sₙ = n/2 [2a + (n − 1)d] = n/2 (a + l)，l 为末项。
Geometric sequence: nth term uₙ = ar^(n−1), with common ratio r. Sum of the first n terms: Sₙ = a(1 − rⁿ) / (1 − r) for r ≠ 1. For |r| < 1, the sum to infinity exists: S∞ = a / (1 − r). These series appear in compound interest and population growth models.
等比数列：第 n 项 uₙ = ar^(n−1)，公比为 r。前 n 项和：当 r ≠ 1 时 Sₙ = a(1 − rⁿ)/(1 − r)。若 |r| < 1，无穷级数收敛且和为 S∞ = a/(1 − r)。这类级数出现在复利和人口增长模型中。
The binomial theorem expands (a + b)ⁿ = Σ_(k=0)^n ⁿCₖ aⁿ⁻ᵏ bᵏ, where the binomial coefficient ⁿCₖ = n! / (k!(n−k)!). For expressions of the form (1 + x)ⁿ with rational n and |x| < 1, the series expansion is 1 + nx + [n(n−1)/2!] x² + [n(n−1)(n−2)/3!] x³ + ... . This infinite expansion is crucial for approximations.
二项式定理将 (a + b)ⁿ 展开为 Σ_(k=0)^n ⁿCₖ aⁿ⁻ᵏ bᵏ，其中二项系数 ⁿCₖ = n!/(k!(n−k)!)。对于 (1 + x)ⁿ 形式且 n 为有理数、|x| < 1 的情形，其无穷级数展开为 1 + nx + [n(n−1)/2!] x² + [n(n−1)(n−2)/3!] x³ + ...，这在近似计算中十分重要。
6. Exponential and Logarithmic Functions | 指数与对数函数
The natural exponential function y = eˣ has derivative eˣ and is its own integral. Its inverse is the natural logarithm, x = ln y. Key log laws: ln(pq) = ln p + ln q, ln(p/q) = ln p − ln q, and ln(pʳ) = r ln p. These enable the simplification of exponential equations.
自然指数函数 y = eˣ 的导数为自身，积分亦然。其反函数为自然对数 x = ln y。关键对数法则：ln(pq) = ln p + ln q，ln(p/q) = ln p − ln q，ln(pʳ) = r ln p。这些法则可用于化简指数方程。
Change of base: logₐ b = log_c b / log_c a, linking logs of different bases. Exponential growth and decay models take the form y = A e^(kt); k > 0 implies growth, k < 0 decay. Solving equations like e²ˣ = 5 involves taking natural logs and isolating the variable.
换底公式：logₐ b = log_c b / log_c a，连接不同底数的对数。指数增长与衰减模型为 y = A e^(kt)；k > 0 表示增长，k < 0 表示衰减。解 e²ˣ = 5 这类方程时，可两边取自然对数并分离变量。
7. Trigonometry | 三角学
Radian measure: π rad = 180°. The arc length of a sector is s = rθ and the area is A = ½ r²θ, with θ in radians. Exact trigonometric values should be memorised for 0°, 30°, 45°, 60°, 90° and their radian equivalents.
弧度制：π rad = 180°。扇形的弧长 s = rθ，面积 A = ½ r²θ，其中 θ 单位为弧度。需熟记 0°、30°、45°、60°、90° 及其弧度对应值的精确三角比。
Fundamental identities: sin²θ + cos²θ = 1 and tanθ = sinθ / cosθ. The sine rule a / sin A = b / sin B = c / sin C is used with non‑right triangles. The cosine rule a² = b² + c² − 2bc cos A gives a side or an angle. Area of a triangle: ½ ab sin C. The CAST diagram helps identify the sign of trig ratios in each quadrant. Solving equations such as sin(2θ − 30°) = 0.5 requires adjusting the interval and finding all solutions within the given domain.
基本恒等式：sin²θ + cos²θ = 1 以及 tanθ = sinθ / cosθ。正弦定理 a / sin A = b / sin B = c / sin C 用于非直角三角形。余弦定理 a² = b² + c² − 2bc cos A 可求边或角。三角形面积：½ ab sin C。CAST 图帮助判断各象限三角比的符号。解 sin(2θ − 30°) = 0.5 这类方程时需调整区间并求出给定范围内的所有解。
8. Differentiation | 微分
The power rule, d/dx (xⁿ) = n xⁿ⁻¹, is the foundation of differentiation. Constant multiple and sum rules apply: d/dx [c·f(x)] = c·f'(x) and (f ± g)’ = f’ ± g’. The chain rule, product rule, and quotient rule are necessary for composite, product, and rational functions:
Chain: dy/dx = dy/du × du/dx; Product: (uv)’ = u’v + uv’; Quotient: (u/v)’ = (u’v − uv’) / v²
Standard derivatives include d/dx (eˣ) = eˣ, d/dx (ln x) = 1/x, d/dx (sin x) = cos x, d/dx (cos x) = −sin x, and d/dx (tan x) = sec² x. The second derivative f”(x) tests the nature of stationary points: f”(x) > 0 indicates a local minimum, f”(x) < 0 a local maximum, and f''(x) = 0 may require further investigation.
幂函数求导法则 d/dx (xⁿ) = n xⁿ⁻¹ 是微分的基础。常数倍和加减法则：d/dx [c·f(x)] = c·f'(
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