📚 IGCSE OCR Further Maths: Formula & Theorem Quick Reference | IGCSE OCR 进阶数学：公式定理速查手册
This quick reference guide covers the essential formulas, identities, and theorems required for the OCR IGCSE Further Mathematics syllabus. Every key relationship is presented with its mathematical expression using Unicode symbols, accompanied by a concise explanation in English and Chinese. Use this handbook for rapid revision, problem-solving checks, and exam preparation. Mastering these formulas will give you the confidence to tackle advanced algebra, calculus, vectors, matrices, and trigonometry questions effectively.
本速查手册涵盖OCR IGCSE进阶数学大纲所要求的核心公式、恒等式和定理。每个关键关系式均以 Unicode 符号呈现数学表达式，并配以简洁的中英文解释。使用本手册进行快速复习、解题核对和备考。掌握这些公式将使你有信心高效应对高级代数、微积分、向量、矩阵和三角学问题。
1. Algebraic Identities and Factorisation | 代数恒等式与因式分解
The difference of two squares is a fundamental pattern: a² − b² = (a − b)(a + b). This identity is used extensively to factorise quadratic expressions and to rationalise denominators.
平方差公式是一个基本模式：a² − b² = (a − b)(a + b)。这个恒等式被广泛用于因式分解二次式和有理化分母。
Perfect square trinomials expand as (a + b)² = a² + 2ab + b² and (a − b)² = a² − 2ab + b². They are critical for completing the square.
完全平方式展开为 (a + b)² = a² + 2ab + b² 和 (a − b)² = a² − 2ab + b²。它们对配方法至关重要。
Sum and difference of cubes: a³ + b³ = (a + b)(a² − ab + b²) and a³ − b³ = (a − b)(a² + ab + b²). These appear in polynomial factorisation and can simplify complex rational expressions.
立方和与立方差：a³ + b³ = (a + b)(a² − ab + b²) 和 a³ − b³ = (a − b)(a² + ab + b²)。它们出现在多项式因式分解中，并能简化复杂的有理表达式。
2. Remainder and Factor Theorems | 余数定理与因式定理
For a polynomial f(x), the Remainder Theorem states: when f(x) is divided by (x − a), the remainder is f(a). This allows evaluation without performing full long division.
对于多项式 f(x)，余数定理指出：当 f(x) 除以 (x − a) 时，余数为 f(a)。这样无需执行完整的长除法即可求值。
The Factor Theorem is a special case: (x − a) is a factor of f(x) if and only if f(a) = 0. This theorem is used to find roots and factorise higher‑degree polynomials.
因式定理是一个特例：(x − a) 是 f(x) 的因式当且仅当 f(a) = 0。该定理用于求根和分解高次多项式。
When using these theorems, always test integer factors of the constant term first. Synthetic division or equating coefficients can then break down the polynomial entirely.
使用这些定理时，始终先测试常数项的整数因子。然后可采用综合除法或系数比较法完全分解多项式。
3. Binomial Expansion | 二项式展开
The binomial theorem for a positive integer exponent n states: (1 + x)ⁿ = 1 + nC₁ x + nC₂ x² + … + nCᵣ xʳ + … + xⁿ, where nCᵣ = n! / [r!(n − r)!]. This expansion is valid for rational n as an infinite series when |x| < 1.
对于正整数指数 n 的二项式定理为：(1 + x)ⁿ = 1 + nC₁ x + nC₂ x² + … + nCᵣ xʳ + … + xⁿ，其中 nCᵣ = n! / [r!(n − r)!]。当 |x| < 1 时，该展开式对有理数 n 可表为无穷级数。
For (a + bx)ⁿ, factor aⁿ out to write as aⁿ (1 + (b/a)x)ⁿ. Then expand the bracket using the binomial series. Remember that the term independent of x is aⁿ.
对于 (a + bx)ⁿ，提取 aⁿ 写成 aⁿ (1 + (b/a)x)ⁿ。然后用二项级数展开括号。切记与 x 无关的项是 aⁿ。
Use the general term T_{r+1} = ⁿCᵣ aⁿ⁻ʳ (b)ʳ to find a specific coefficient without expanding the entire series. This is especially useful for finding middle terms or specific powers.
利用通项公式 T_{r+1} = ⁿCᵣ aⁿ⁻ʳ (b)ʳ 来求特定系数而无需展开整个级数。这特别有利于求中间项或特定幂次。
4. Sequences and Series | 数列与级数
An arithmetic sequence has a common difference d: uₙ = a + (n − 1)d. The sum of the first n terms is Sₙ = n/2 [2a + (n − 1)d] or Sₙ = n/2 (a + l), where l is the last term.
等差数列有公差 d：uₙ = a + (n − 1)d。前 n 项和为 Sₙ = n/2 [2a + (n − 1)d] 或 Sₙ = n/2 (a + l)，其中 l 为末项。
A geometric sequence has a common ratio r: uₙ = arⁿ⁻¹. The sum of the first n terms is Sₙ = a(1 − rⁿ)/(1 − r) for r ≠ 1. An infinite geometric series converges to a/(1 − r) when |r| < 1.
等比数列有公比 r：uₙ = arⁿ⁻¹。前 n 项和为 Sₙ = a(1 − rⁿ)/(1 − r)，r ≠ 1。当 |r| < 1 时，无穷等比级数收敛于 a/(1 − r)。
Sigma notation is used to sum series: Σ k from k=1 to n equals n(n+1)/2, Σ k² = n(n+1)(2n+1)/6, Σ k³ = [n(n+1)/2]². These standard results help evaluate polynomial summations.
Σ 符号用于级数求和：Σ k (k从1到n) 等于 n(n+1)/2，Σ k² = n(n+1)(2n+1)/6，Σ k³ = [n(n+1)/2]²。这些标准结果有助于计算多项式求和。
5. Functions and Graphs | 函数与图像
A function f(x) maps each input to exactly one output. The domain is the set of allowed x‑values; the range is the set of corresponding f(x) values. Composite functions fg(x) = f(g(x)) are evaluated by applying g first, then f.
函数 f(x) 将每个输入映射到唯一输出。定义域是允许的 x 值集合；值域是对应 f(x) 值的集合。复合函数 fg(x) = f(g(x)) 通过先应用 g 后应用 f 来求值。
The inverse function f⁻¹(x) is found by swapping x and y in y = f(x) and solving for y; it exists only if f is one‑to‑one. The graphs of f and f⁻¹ are reflections in the line y = x.
反函数 f⁻¹(x) 通过对 y = f(x) 中的 x 和 y 交换并解出 y 求得；只有 f 是一对一映射时才存在。f 和 f⁻¹ 的图像关于直线 y = x 对称。
Transformations of graphs: y = f(x) + a translates vertically, y = f(x + a) translates horizontally to the left by a. y = −f(x) reflects in the x‑axis, y = f(−x) reflects in the y‑axis. y = a f(x) stretches vertically by scale factor a.
图像变换：y = f(x) + a 垂直平移，y = f(x + a) 向左水平平移 a 个单位。y = −f(x) 关于 x 轴反射，y = f(−x) 关于 y 轴反射。y = a f(x) 垂直拉伸 a 倍。
6. Differentiation | 微分
The derivative f ‘(x) = lim_{h→0} [f(x+h) − f(x)]/h. For power functions, if f(x) = xⁿ then f ‘(x) = nxⁿ⁻¹. This rule extends to sums and constant multiples: d/dx (c·u) = c·du/dx.
导数 f ‘(x) = lim_{h→0} [f(x+h) − f(x)]/h。对于幂函数，若 f(x) = xⁿ，则 f ‘(x) = nxⁿ⁻¹。此规则可推广到和与常数倍：d/dx (c·u) = c·du/dx。
The gradient of a curve at a point is the value of the derivative at that x‑coordinate. The equation of a tangent at (a, f(a)) is y − f(a) = f ‘(a)(x − a), and the normal has gradient −1/f ‘(a).
曲线在某点的梯度是该点 x 坐标处导数的值。在 (a, f(a)) 处的切线方程为 y − f(a) = f ‘(a)(x − a)，法线斜率为 −1/f ‘(a)。
Second derivative d²y/dx² measures the rate of change of the gradient. Stationary points occur where f ‘(x) = 0: they are maxima (f ”(x) < 0), minima (f ''(x) > 0), or points of inflection. Kinematics uses differentiation: v = ds/dt, a = dv/dt.
二阶导数 d²y/dx² 度量梯度的变化率。驻点出现在 f ‘(x) = 0 处：它们可以是极大值点 (f ”(x) < 0)、极小值点 (f ''(x) > 0) 或拐点。运动学中使用微分：v = ds/dt，a = dv/dt。
7. Integration | 积分
Integration reverses differentiation: ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C for n ≠ −1. The constant of integration C represents the family of antiderivatives. Definite integrals compute the area under a curve between two limits.
积分是微分的逆运算：∫ xⁿ dx = xⁿ⁺¹/(n+1) + C，n ≠ −1。积分常数 C 表示原函数族。定积分计算曲线下在两个界限之间的面积。
The area between a curve y = f(x) and the x‑axis from x = a to x = b is given by ∫ₐᵇ f(x) dx. Areas below the x‑axis are negative, so total enclosed area requires splitting the interval at intercepts.
曲线 y = f(x) 与 x 轴之间从 x = a 到 x = b 的面积为 ∫ₐᵇ f(x) dx。x 轴下方的面积为负，因此总围成面积需要在截点处分割区间。
Kinematics applications: displacement s = ∫ v dt, velocity v = ∫ a dt. Use initial conditions to find the constant C. Problems often require combining differentiation and integration over a given time interval.
运动学应用：位移 s = ∫ v dt，速度 v = ∫ a dt。使用初始条件求出常数 C。问题往往要求在给定时间区间内结合微分和积分进行求解。
8. Matrices | 矩阵
A 2×2 matrix is written as ( a b \ c d ). Addition and subtraction are element‑wise. Scalar multiplication multiplies every element by the scalar. The zero matrix O is the additive identity.
2×2 矩阵写为 ( a b \ c d )。加法和减法按元素进行。标量乘法将标量乘以每个元素。零矩阵 O 是加法单位元。
Matrix multiplication: to multiply two 2×2 matrices, take the dot product of rows of the first with columns of the second. Multiplication is associative but not commutative: AB ≠ BA generally.
矩阵乘法：要乘两个 2×2 矩阵，取第一个矩阵的行与第二个矩阵的列的点积。乘法满足结合律但不满足交换律：一般而言 AB ≠ BA。
The determinant of a 2×2 matrix M = ( a b \ c d ) is det M = ad − bc. If det M ≠ 0, the inverse M⁻¹ = 1/(ad−bc) ( d −b \ −c a ). A singular matrix has det = 0 and no inverse. Transformations: matrices can represent rotations, reflections, enlargements, and shears.
2×2 矩阵 M = ( a b \ c d ) 的行列式为 det M = ad − bc。若 det M ≠ 0，逆矩阵为 M⁻¹ = 1/(ad−bc) ( d −b \ −c a )。奇异矩阵的行列式为 0，且没有逆矩阵。变换：矩阵可表示旋转、反射、缩放和切变。
9. Vectors in Two Dimensions | 二维向量
A vector is a quantity with magnitude and direction. It can be written as a column vector ( x \ y ) or as xi + yj. The magnitude |v| = √(x² + y²). Unit vectors have magnitude 1; to find a unit vector divide a vector by its magnitude.
向量是具有大小和方向的量。可写为列向量 ( x \ y ) 或 xi + yj。模长 |v| = √(x² + y²)。单位向量的模长为 1；要得到单位向量，将向量除以其模长。
Vector addition: add corresponding components. Scalar multiplication scales both components. The position vector of a point A is OA; the vector AB = OB − OA.
向量加法：对应分量相加。标量乘法缩放两个分量。点 A 的位置向量为 OA；向量 AB = OB − OA。
The dot product or scalar product: a·b = |a||b| cos θ = x₁x₂ + y₁y₂. It is used to test perpendicular vectors (a·b = 0) and to find the acute angle between two vectors.
点积或标量积：a·b = |a||b| cos θ = x₁x₂ + y₁y₂。用于检验垂直向量 (a·b = 0) 以及求两向量间的锐角。
10. Coordinate Geometry and Straight Lines | 坐标几何与直线
The gradient m of the line through (x₁, y₁) and (x₂, y₂) is (y₂ − y₁)/(x₂ − x₁). Parallel lines have equal gradients; perpendicular lines satisfy m₁ × m₂ = −1.
经过 (x₁, y₁) 和 (x₂, y₂) 的直线的梯度 m 为 (y₂ − y₁)/(x₂ − x₁)。平行线具有相等的梯度；垂直线满足 m₁ × m₂ = −1。
Equation forms: y − y₁ = m(x − x₁) (point–gradient), y = mx + c (slope–intercept), ax + by + c = 0 (general form). The midpoint of two points is ((x₁+x₂)/2, (y₁+y₂)/2); the distance between them is √[(x₂−x₁)² + (y₂−y₁)²].
直线方程形式：y − y₁ = m(x − x₁)（点斜式），y = mx + c（斜截式），ax + by + c = 0（一般式）。两点的中点为 ((x₁+x₂)/2, (y₁+y₂)/2)；两点间的距离为 √[(x₂−x₁)² + (y₂−y₁)²]。
The circle equation is (x − h)² + (y − k)² = r², centre (h, k). You can complete the square to find the centre and radius from the expanded form x² + y² + 2gx + 2fy + c = 0, where centre is (−g, −f) and r = √(g² + f² − c).
圆的方程为 (x − h)² + (y − k)² = r²，圆心为 (h, k)。你可以通过配方法从展开式 x² + y² + 2gx + 2fy + c = 0 中找出圆心 (−g, −f) 和半径 r = √(g² + f² − c)。
11. Trigonometry and Identities | 三角学与恒等式
In a right‑angled triangle, sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent. The exact values for 0°, 30°, 45°, 60°, 90° must 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。必须牢记 0°、30°、45°、60°、90° 的精确值。
The Pythagorean identity: sin²θ + cos²θ ≡ 1. From this, tan²θ + 1 ≡ sec²θ and 1 + cot²θ ≡ cosec²θ. These are used to simplify expressions and solve equations.
勾股恒等式：sin²θ + cos²θ ≡ 1。由此可得 tan²θ + 1 ≡ sec²θ 和 1 + cot²θ ≡ cosec²θ。这些恒等式用于化简表达式和解方程。
The sine rule: a/sin A = b/sin B = c/sin C. The cosine rule: a² = b² + c² − 2bc cos A. Area of a triangle: ½ ab sin C. These apply to any triangle, not just right‑angled ones.
正弦定理：a/sin A = b/sin B = c/sin C。余弦定理：a² = b² + c² − 2bc cos A。三角形面积：½ ab sin C。这些适用于任意三角形，而不仅仅是直角三角形。
12. Polynomial Equations and Inequalities | 多项式方程与不等式
Quadratic formula for ax² + bx + c = 0: x = [−b ± √(b² − 4ac)] / (2a). The discriminant Δ = b² − 4ac determines the nature of roots: Δ > 0 two distinct real roots, Δ = 0 double root, Δ < 0 no real roots.
二次方程 ax² + bx + c = 0 的求根公式：x = [−b ± √(b² − 4ac)] / (2a)。判别式 Δ = b² − 4ac 决定根的性质：Δ > 0 两个不等实根，Δ = 0 重根，Δ < 0 无实根。
To solve polynomial equations, factorise fully using the Factor Theorem and then set each factor to zero. For inequalities, sketch a sign diagram or use critical values to determine intervals where the polynomial is positive or negative.
解多项式方程时，利用因式定理完全因式分解，然后令每个因式为零。对于不等式，绘制符号图或使用临界值确定多项式为正或负的区间。
Simultaneous equations involving one linear and one quadratic are solved by substitution. The resulting quadratic can yield two solutions, representing the intersection points of a line and a curve.
涉及一个线性和一个二次的联立方程通过代入法求解。得到的二次方程可给出两个解，分别代表直线与曲线的交点。
Published by TutorHao | Further Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导