📚 IGCSE CCEA Further Maths: Formula & Theorem Quick Reference | IGCSE CCEA 进阶数学：公式定理速查手册
This quick-reference handbook collects the essential formulae, identities and theorems for CCEA IGCSE Further Mathematics. Use it before mocks and final papers to check recall, practise substitutions and build speed under exam conditions. Each section is paired in English and Chinese so you can switch between languages while revising.
本手册汇总 CCEA IGCSE 进阶数学的核心公式、恒等式与定理。适用于模拟考试和正式考试前的快速复习，帮助检查记忆、练习代入并在考试中提高速度。每个小节均以中英双语对照，方便你随时切换语言进行复习。
1. Algebraic Manipulation and Surds | 代数运算与根式
For CCEA Further Maths, you should manipulate algebraic fractions, expand brackets and simplify surds fluently. The key identities are the difference of two squares and the perfect-square expansions. You must also be able to rationalise denominators such as 1/√a and 1/(a + √b).
在 CCEA 进阶数学中，你需要熟练处理代数分式、展开括号并化简根式。核心恒等式包括平方差公式和完全平方展开式。你还必须掌握分母有理化，例如 1/√a 与 1/(a + √b) 的化简。
a² − b² = (a − b)(a + b)
(a + b)² = a² + 2ab + b², (a − b)² = a² − 2ab + b²
√a × √b = √(ab) for a, b ≥ 0
1/√a = √a/a, 1/(a + √b) = (a − √b)/(a² − b)
When simplifying surds, always look for square factors under the root. For example, √48 = √(16 × 3) = 4√3. In rationalising a binomial denominator, multiply top and bottom by the conjugate, not by the denominator itself.
化简根式时，始终寻找根号下的平方因子。例如 √48 = √(16 × 3) = 4√3。在对二项式分母进行有理化时，分子分母要同时乘以共轭式，而不是原分母。
2. Quadratic Equations and Inequalities | 二次方程与不等式
The quadratic formula solves ax² + bx + c = 0. The discriminant determines the nature of the roots: two distinct real roots, one repeated real root, or no real roots. The sum and product of roots are often quicker than finding the roots separately.
二次方程求根公式用于解 ax² + bx + c = 0。判别式决定根的性质：两个不同实根、一个重根或没有实根。根的和与积通常比单独求根更快。
x = [−b ± √(b² − 4ac)] / 2a
D = b² − 4ac
α + β = −b/a, αβ = c/a
For quadratic inequalities, rearrange to make one side zero and find the critical values where the expression equals zero. If the leading coefficient is positive, the graph is a U-shape, so ax² + bx + c > 0 outside the interval between the roots and < 0 inside the interval.
解二次不等式时，先将一边移项为零，再求出表达式等于零的临界值。如果首项系数为正，图像为 U 形，因此 ax² + bx + c > 0 的解集在两根区间之外，而 < 0 的解集在两根之间。
3. Coordinate Geometry of Straight Lines and Circles | 直线与圆的坐标几何
Straight-line problems require gradient, distance and midpoint. Parallel lines have equal gradients, while perpendicular lines have gradients whose product is −1. Circle geometry links the centre, radius and tangent through the perpendicular radius property.
直线问题需要掌握斜率、距离和中点。平行直线斜率相等，垂直直线的斜率乘积为 −1。圆的几何问题通过半径与切线垂直这一性质将圆心、半径和切线联系起来。
m = (y₂ − y₁)/(x₂ − x₁), d = √[(x₂ − x₁)² + (y₂ − y₁)²]
midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2)
m₁ = m₂ for parallel; m₁m₂ = −1 for perpendicular
y − y₁ = m(x − x₁)
(x − a)² + (y − b)² = r²
To find a tangent to a circle at a point, first find the gradient of the radius to that point, then use the negative reciprocal as the tangent gradient. The normal to the circle is along the radius, so it has the same gradient as the radius.
求圆上一点的切线时，先求该点半径的斜率，再取其负倒数作为切线斜率。圆的法线沿半径方向，因此法线斜率与半径斜率相同。
4. Polynomials and Factor Theorem | 多项式与因式定理
The remainder theorem and factor theorem are the core tools for factorising cubics and higher-degree polynomials. They convert a division problem into a quick substitution problem. A cubic can often be factorised once one linear factor is known.
余数定理和因式定理是分解三次及更高次多项式的核心工具。它们将除法问题转化为快速代入问题。一旦找到一个一次因式，三次多项式通常就可以继续分解。
Remainder theorem: f(a) is the remainder when f(x) is divided by (x − a)
Factor theorem: (x − a) is a factor of f(x) ⇔ f(a) = 0
To factorise a cubic, test small integer values such as ±1, ±2, ±3 by substitution. Once a factor is found, divide the polynomial by that factor using long division or comparing coefficients. Then factorise the remaining quadratic.
分解三次多项式时，可代入 ±1、±2、±3 等小整数进行检验。找到一个因式后，用长除法或比较系数法除以该因式。剩下的二次式再继续因式分解。
5. Binomial Expansion | 二项式展开
The binomial expansion gives a systematic way to expand expressions of the form (1 + x)ⁿ or (a + b)ⁿ. For positive integer n the expansion is finite; for rational n it is infinite and only valid when |x| < 1 in the form (1 + x)ⁿ.
二项式展开提供了展开 (1 + x)ⁿ 或 (a + b)ⁿ 的系统方法。当 n 为正整数时，展开式是有限的；当 n 为有理数时，展开式为无穷级数，且对于 (1 + x)ⁿ 形式仅在 |x| < 1 时有效。
(1 + x)ⁿ = 1 + n x + [n(n − 1)/2!] x² + [n(n − 1)(n − 2)/3!] x³ + …
(a + b)ⁿ = Σ nCr aⁿ⁻ʳ bʳ
nCr = n! / [r!(n − r)!]
Remember that nC0 = 1 and nC1 = n. For a term involving (a + bx)ⁿ, rewrite it as aⁿ(1 + (b/a)x)ⁿ before using the standard expansion. Always state the range of validity when the index is not a positive integer.
记住 nC0 = 1，nC1 = n。对于 (a + bx)ⁿ 形式的项，先写成 aⁿ(1 + (b/a)x)ⁿ，再使用标准展开式。当指数不是正整数时，务必说明展开式的有效范围。
6. Sequences and Series | 数列与级数
Arithmetic and geometric sequences appear regularly in CCEA Further Maths. You need both the nth term and the sum of the first n terms. For geometric series, also know the sum to infinity and its convergence condition.
等差数列和等比数列在 CCEA 进阶数学中经常出现。你需要掌握第 n 项和前 n 项和。对于等比级数，还要掌握无穷级数的求和公式及其收敛条件。
Arithmetic: uₙ = a + (n − 1)d
Sₙ = n/2 [2a + (n − 1)d] = n/2 (a + l)
Geometric: uₙ = arⁿ⁻¹
Sₙ = a(1 − rⁿ)/(1 − r), r ≠ 1
S∞ = a/(1 − r), |r| < 1
In sequence questions, identify the type first: if the difference between consecutive terms is constant, use arithmetic; if the ratio is constant, use geometric. When finding the sum to infinity, check that |r| < 1 before applying the formula.
在数列问题中，首先要判断类型：若相邻项之差为常数，则用等差数列；若相邻项之比为常数，则用等比数列。求无穷级数的和时，先检查 |r| < 1 再代入公式。
7. Exponentials and Logarithms | 指数与对数
Logarithms are the inverse of exponentials and are essential for solving equations where the unknown is in the power. You must know the three logarithm laws and the change of base formula. The natural logarithm ln is built on base e.
对数是指数的逆运算，对于求解未知数在指数位置的方程至关重要。你必须掌握三条对数运算法则和换底公式。自然对数 ln 以 e 为底。
logₐ x = y ⇔ aʸ = x
logₐ(xy) = logₐ x + logₐ y
logₐ(x/y) = logₐ x − logₐ y
logₐ xⁿ = n logₐ x
log_b x = logₐ x / logₐ b
To solve an equation like 2ˣ = 7, take logs of both sides: x log 2 = log 7, so x = log 7 / log 2. For eˣ equations, use ln: if eˣ = 5, then x = ln 5. Remember ln e = 1 and eˡⁿ ˣ = x.
解 2ˣ = 7 这类方程时，两边取对数：x log 2 = log 7，因此 x = log 7 / log 2。对于 eˣ 方程，使用 ln：若 eˣ = 5，则 x = ln 5。记住 ln e = 1，eˡⁿ ˣ = x。
8. Trigonometry | 三角学
Trigonometry in CCEA Further Maths includes radian measure, arc length, sector area, the sine and cosine rules, and the fundamental identities. You must also know exact values for standard angles and be able to solve trigonometric equations in radians or degrees.
CCEA 进阶数学的三角学内容包括弧度制、弧长、扇形面积、正弦定理、余弦定理和基本恒等式。你还必须熟悉标准角的精确值，并能以弧度或角度为单位解三角方程。
π rad = 180°
l = rθ, A sector = ½ r²θ
sin²θ + cos²θ = 1, tan θ = sin θ / cos θ
a/sin A = b/sin B = c/sin C
a² = b² + c² − 2bc cos A
Area of triangle = ½ ab sin C
When solving trigonometric equations, sketch the graph or use the cast diagram to find all solutions in the required interval. Do not cancel sin θ or cos θ from both sides unless you separately consider the case where it equals zero.
解三角方程时，画出函数图像或使用象限图来找到指定区间内的所有解。不要随便从等式两边约去 sin θ 或 cos θ，除非你单独考虑它等于零的情况。
9. Differentiation | 微分
Differentiation gives the gradient function of a curve. The main techniques are the power rule, the chain rule, the product rule and the quotient rule. Stationary points are found where the first derivative is zero, and their nature is tested by the second derivative.
微分给出曲线的导数函数。主要方法包括幂函数求导、链式法则、乘法法则和除法法则。令一阶导数为零即可找到驻点，驻点的性质可通过二阶导数判断。
d/dx xⁿ = n xⁿ⁻¹
d/dx eˣ = eˣ, d/dx ln x = 1/x, d/dx sin x = cos x, d/dx cos x = −sin x
Chain rule: dy/dx = dy/du × du/dx
Product rule: d/dx(uv) = u′v + uv′
Quotient rule: d/dx(u/v) = (u′v − uv′)/v²
At a stationary point, dy/dx = 0. If d²y/dx² > 0 the point is a minimum; if d²y/dx² < 0 it is a maximum; if d²y/dx² = 0 the test is inconclusive and you should check the gradient on either side. The tangent gradient at a point is the derivative value, and the normal gradient is −1/m.
在驻点处，dy/dx = 0。若 d²y/dx² > 0，该点为极小值点；若 d²y/dx² < 0，该点为极大值点；若 d²y/dx² = 0，则无法判定，需要检查两侧的导数符号。曲线上某点的切线斜率等于该点的导数值，法线斜率为 −1/m。
10. Integration | 积分
Integration is the reverse of differentiation and is used to find areas under curves and to recover functions from their derivatives. The power rule for integration breaks down when n = −1, in which case the result is a natural logarithm.
积分是微分的逆运算，用于求曲线下的面积以及由导数还原原函数。积分中的幂函数法则在 n = −1 时失效，此时结果为自然对数。
∫ xⁿ dx = xⁿ⁺¹/(n + 1) + c, n ≠ −1
∫ 1/x dx = ln |x| + c
∫ eˣ dx = eˣ + c
∫ sin x dx = −cos x + c, ∫ cos x dx = sin x + c
∫ (ax + b)ⁿ dx = (ax + b)ⁿ⁺¹ / [a(n + 1)] + c, n ≠ −1
For a definite integral, substitute the limits into the antiderivative and subtract: ∫ₐᵇ f(x)dx = F(b) − F(a). The area between a curve and the x-axis is given by the definite integral, but you must split the interval where the curve crosses the axis and take absolute values of each part.
对于定积分，将上下限代入原函数并相减：∫ₐᵇ f(x)dx = F(b) − F(a)。曲线与 x 轴之间的面积由定积分给出，但如果曲线穿过 x 轴，则必须在交点处拆分区间，并对每一部分取绝对值。
11. Vectors | 向量
Vectors describe both magnitude and direction. In CCEA Further Maths, you will calculate magnitudes, find unit vectors, use the scalar product to find angles, and write vector equations of lines. Parallel and perpendicular conditions are tested frequently.
向量同时描述大小和方向。在 CCEA 进阶数学中，你需要计算模长、求单位向量、利用数量积求角度，并写出直线的向量方程。平行与垂直的判定条件是常见考点。
a = x i + y j + z k, |a| = √(x² + y² + z²)
unit vector â = a / |a|
a · b = x₁x₂ + y₁y₂ + z₁z₂ = |a||b| cos θ
cos θ = (a · b) / (|a||b|)
perpendicular: a · b = 0; parallel: a = k b
A vector line can be written as r = a + t b, where a is a position vector on the line and b is the direction vector. To prove two vectors are perpendicular, show their scalar product is zero; to prove parallel, show one is a scalar multiple of the other.
向量直线可写为 r = a + t b，其中 a 为直线上一点的位置向量，b 为方向向量。证明两向量垂直时，需说明它们的数量积为零；证明平行时，需说明一个向量是另一个向量的标量倍。
12. Matrices | 矩阵
Matrices are used to represent transformations and to solve simultaneous equations. For 2 × 2 matrices, the determinant and inverse are the key tools. Matrix multiplication is not commutative, so the order of multiplication matters.
矩阵用于表示线性变换以及求解联立方程。对于 2 × 2 矩阵，行列式和逆矩阵是关键工具。矩阵乘法不满足交换律，因此乘法顺序非常重要。
det A = ad − bc for A = [[a, b], [c, d]]
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