IGCSE Cambridge Further Pure Maths: Formula & Theorem Quick Reference Handbook | IGCSE Cambridge 进阶数学：公式定理速查手册

📚 IGCSE Cambridge Further Pure Maths: Formula & Theorem Quick Reference Handbook | IGCSE Cambridge 进阶数学：公式定理速查手册

This quick reference guide collects the essential formulas, identities and key theorems required for the Cambridge IGCSE Further Pure Mathematics syllabus. It is designed to help you consolidate knowledge, speed up exam revision and avoid common mistakes. Each section presents English explanations followed immediately by their Chinese counterparts.

本书速查手册汇集了剑桥 IGCSE 进阶数学课程所必需的核心公式、恒等式和关键定理，旨在帮助你巩固知识、提高考前复习效率并避开常见错误。每一小节均采用英文解释与中文解释紧密配对的形式呈现。

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1. Algebraic Identities and Polynomials | 代数恒等式与多项式

Three fundamental identities appear throughout algebra: the square of a sum, the square of a difference and the difference of two squares.

代数中最基本的三个恒等式为：和的平方、差的平方以及平方差。

(a + b)² = a² + 2ab + b²

(a - b)² = a² - 2ab + b²

a² - b² = (a + b)(a - b)

The Remainder Theorem states that when a polynomial P(x) is divided by (x - a), the remainder is P(a). If the remainder is zero, then (x - a) is a factor – this is the Factor Theorem.

余式定理指出，当多项式 P(x) 除以 (x - a) 时，余数为 P(a)。若余数为零，则 (x - a) 是 P(x) 的因式——这就是因式定理。

e sum and product of the roots of ax² + bx + c = 0 are given by -b/a and c/a respectively.

二次方程 ax² + bx + c = 0 的根之和为 -b/a，根之积为 c/a。

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2. Binomial Theorem | 二项式定理

integer n, the expansion of (a + b)ⁿ is given by the sum of terms of the form C(n, r) aⁿ⁻ʳ bʳ, where C(n, r) = n! / [r!(n - r)!].

对于正整数 n，(a + b)ⁿ 的展开式由形如 C(n, r) aⁿ⁻ʳ bʳ 的各项之和给出，其中 C(n, r) = n! / [r!(n - r)!]。

(a + b)ⁿ = Σ [C(n, r) aⁿ⁻ʳ bʳ] for r = 0 to n

The (r+1)th term in the expansion is Tᵣ₊₁ = C(n, r) aⁿ⁻ʳ bʳ.

展开式中的第 (r+1) 项为 Tᵣ₊₁ = C(n, r) aⁿ⁻ʳ bʳ。

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3. Exponentials and Logarithms | 指数与对数

Fundamental laws of indices: aᵐ × aⁿ = aᵐ⁺ⁿ, (aᵐ)ⁿ = aᵐⁿ, and a⁻ⁿ = 1 / aⁿ.

基本指数法则：aᵐ × aⁿ = aᵐ⁺ⁿ，(aᵐ)ⁿ = aᵐⁿ，以及 a⁻ⁿ = 1 / aⁿ。

The logarithm of a number y to base a is defined as logₐ y = x if aˣ rules: logₐ (xy) = logₐ x + logₐ y, logₐ (x/y) = logₐ x - logₐ y, and logₐ xᵏ = k logₐ x.

以 a 为底 y 的对数定义为：若 aˣ = y，则 logₐ y = x。关键运算法则：logₐ (xy) = logₐ x + logₐ y，logₐ (x/y) = logₐ x - logₐ y，logₐ xᵏ = k logₐ x。

The change of base formula is logₐ b = logₓ b / logₓ a, often taken with natural logs.

换底公式为 logₐ b = logₓ b / logₓ a，经常使用自然对数进行计算。

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4. Equations and Inequalities | 方程与不等式

two distinct real roots if Δ > 0, one repeated root if Δ = 0, and no real roots if Δ < 0.

二次方程的判别式 Δ = b² - 4ac 决定根的性质：若 Δ > 0，有两个相异实根；若 Δ = 0，有一个重根；若 Δ < 0，无实根。

To solve a quadratic inequality, sketch the graph of the quadratic and identify where it lies above or below the x‑axis, then express the solution using interval notation.

解二次不等式时，先画出对应二次函数的图像，找出图像位于 x 轴上方或下方的区间，再用区间记号表示解集。

For linear inequalities, the same operations apply as for equations, but multiplying or dividing by a negative number reverses the inequality sign.

对于线性不等式，运算方法与方程相同，但当两边同乘或同除一个负数时，必须将不等号反向。

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5. Coordinate Geometry and Graphs | 坐标几何与图像

distance AB is √[(x₂ - x₁)² + (y₂ - y₁)²] and the midpoint is ((x₁ + x₂)/2, (y₁ + y₂)/2).

已知两点 A(x₁, y₁) 和 B(x₂, y₂)，距离 AB = √[(x₂ - x₁)² + (y₂ - y₁)²]，中点坐标为 ((x₁ + x₂)/2, (y₁ + y₂)/2)。

o lines are parallel if their gradients are equal; they are perpendicular if m₁ × m₂ = -1.

过 A、B 两点的直线的斜率为 m = (y₂ - y₁) / (x₂ - x₁)。若两直线平行，则斜率相等；若两直线垂直，则 m₁ × m₂ = -1。

². A straight line is a tangent to a circle if the discriminant of the combined quadratic is zero.

圆心为 (h, k)、半径为 r 的圆的方程为 (x - h)² + (y - k)² = r²。若直线与圆联立所得二次方程的判别式为零，则该直线为圆的切线。

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6. Sequences and Series | 数列与级数

= a + (n - 1)d and the sum of the first n terms is Sₙ = n/2 [2a + (n - 1)d] = n/2 (a + l).

等差数列中，第 n 项 uₙ = a + (n - 1)d，前 n 项和 Sₙ = n/2 [2a + (n - 1)d] = n/2 (a + l)。

term is uₙ = arⁿ⁻¹ and the sum of the first n terms is Sₙ = a(1 - rⁿ) / (1 - r) for r ≠ 1.

等比数列中，第 n 项 uₙ = arⁿ⁻¹，前 n 项和 Sₙ = a(1 - rⁿ) / (1 - r)，其中 r ≠ 1。

A geometric series with |r| < 1 converges to an infinite sum S∞ = a / (1 - r).

当公比满足 |r| < 1 时，等比级数收敛，其无穷和为 S∞ = a / (1 - r)。

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7. Trigonometry | 三角学

e and arc length: arc length s = rθ, and the area of a sector is ½ r²θ, with θ in radians.

弧度制将角度与弧长联系起来：弧长 s = rθ，扇形面积为 ½ r²θ，其中 θ 以弧度为单位。

ty is sin²θ + cos²θ = 1. From this, 1 + tan²θ = sec²θ and 1 + cot²θ = cosec²θ are derived.

基本的毕达哥拉斯恒等式为 sin²θ + cos²θ = 1。由此可推导出 1 + tan²θ = sec²θ 以及 1 + cot²θ = cosec²θ。

= c / sin C. The cosine rule: a² = b² + c² - 2bc cos A. Area of triangle ABC = ½ ab sin C.

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 ABC 的面积 = ½ ab sin C。

Compound-angle formulas: sin(A ± B) = sin A cos B ± cos A sin B; cos(A ± B) = cos A cos B ∓ sin A sin B; tan(A ± B) = (tan A ± tan B) / (1 ∓ tan A tan B).

和角公式：sin(A ± B) = sin A cos B ± cos A sin B；cos(A ± B) = cos A cos B ∓ sin A sin B；tan(A ± B) = (tan A ± tan B) / (1 ∓ tan A tan B)。

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8. Vectors in Two Dimensions | 二维向量

an be represented as a column matrix or in terms of i and j: v = xi + yj. The magnitude is |v| = √x² + y².

向量可用列矩阵或单位向量 i 和 j 表示：v = xi + yj。其模长为 |v| = √x² + y²。

The dot product of vectors a and b is a·b = |a||b| cos θ. In component form, a·b = x₁x₂ + y₁y₂.

向量 a 与 b 的点积为 a·b = |a||b| cos θ。坐标形式下，a·b = x₁x₂ + y₁y₂。

Two non‑zero vectors are perpendicular if and only if their dot product is zero.

两个非零向量互相垂直的充要条件是它们的点积为零。

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9. Matrices and Transformations | 矩阵与变换

et M = ad - bc. The inverse matrix, when det M ≠ 0, is M⁻¹ = (1/det M) [[d, -b], [-c, a]].

对于 2×2 矩阵 M = [[a, b], [c, d]]，行列式为 det M = ad - bc。当 det M ≠ 0 时，逆矩阵为 M⁻¹ = (1/det M) [[d, -b], [-c, a]]。

Geometric transformations can be represented by matrices. A rotation through angle θ anticlockwise about the origin is given by [[cos θ, -sin θ], [sin θ, cos θ]].

几何变换可用矩阵表示。绕原点逆时针旋转角度 θ 的变换矩阵为 [[cos θ, -sin θ], [sin θ, cos θ]]。

Reflection in the x‑ the line y = x uses [[0, 1], [1, 0]]. An enlargement scale factor k uses [[k, 0], [0, k]].

关于 x 轴的反射用矩阵 [[1, 0], [0, -1]]；关于直线 y = x 的反射用矩阵 [[0, 1], [1, 0]]。缩放因子为 k 的放大变换用矩阵 [[k, 0], [0, k]]。

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10. Differentiation | 微分

The derivative of xⁿ is n xⁿ⁻¹. This basic rule extends to sums and constant multiples.

xⁿ 的导数为 n xⁿ⁻¹。这一基本法则可推广至和与常数倍的情形。

Derivatives of key functions:

f(x)	f'(x)
sin x	cos x
cos x	-sin x
tan x	sec² x
eˣ	eˣ
ln x	1/x

常用函数的导数：如上表所示。

The product rule: d/dx (uv) = u' v + u v'. The quotient rule: d/dx (u/v) = (u' v - u v') / v². The chain rule: dy/dx = dy/du × du/dx.

乘积法则：d/dx (uv) = u' v + u v'。商法则：d/dx (u/v) = (u' v - u v') / v²。链式法则：dy/dx = dy/du × du/dx。

At a stationary point, f' sign test to determine whether the point is a local maximum, minimum or an inflection point.

在驻点处，f'(x) = 0。利用二阶导数或一阶导数的符号变化可以判断该点是局部极大值、局部极小值还是拐点。

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11. Integration | 积分

Integration is the reverse of differentiation: ∫ f'(x) dx = f(x) + C. The general power rule: ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, for n ≠ -1.

积分是微分的逆运算：∫ f'(x) dx = f(x) + C。一般幂函数的积分：∫ xⁿ dx = xⁿ⁺¹/(n+1) + C（n ≠ -1）。

Fundamental integrals include:

基本积分包括：

f(x)	∫ f(x) dx
eˣ	eˣ + C
1/x	ln |x| + C
sin x	-cos x + C
cos x	sin x + C

definite integral ∫ₐᵇ f(x) dx represents the signed area between the curve y = f(x), the x‑axis and the lines x = a and x = b.

定积分 ∫ₐᵇ f(x) dx 表示曲线 y = f(x)、x 轴以及直线 x = a、x = b 所围成的带符号面积。

The area between two curves y = f(x) and y = g(x) from a to b is ∫ₐᵇ [f(x) - g(x)] dx.

两条曲线 y = f(x) 与 y = g(x) 在 a 到 b 之间的面积为 ∫ₐᵇ [f(x) - g(x)] dx。

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