Year 11 CIE IGCSE Maths: Formula & Theorem Quick-Reference | 11年级 CIE IGCSE 数学：公式定理速查手册

📚 Year 11 CIE IGCSE Maths: Formula & Theorem Quick-Reference | 11年级 CIE IGCSE 数学：公式定理速查手册

This handbook gathers the most important formulas, identities and theorems you need for the Year 11 CIE IGCSE Mathematics examination. It is designed for rapid review – keep it with you and test yourself regularly. The IGCSE Extended syllabus is rich in algebraic manipulation, geometry, trigonometry and data handling, and accurate recall of these results will save you time and boost your confidence.

本速查手册汇集了 11 年级 CIE IGCSE 数学考试所需的最重要的公式、恒等式和定理。它专为快速复习而设计——随身携带，定期自测。IGCSE 拓展课程涵盖丰富的代数运算、几何、三角学与数据处理内容，准确记忆这些结论将为你节省时间并增强信心。

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1. Number and Operations | 数与运算

For any non‑zero number a, the index laws hold: a^m × aⁿ = a^m+n, a^m ÷ aⁿ = a^m−n, and (a^m)ⁿ = a^mn. Zero and negative indices are defined by a⁰ = 1 and a⁻ⁿ = 1 / aⁿ. Fractional indices link to roots: a^1/n = √[n]a and a^m/n = (√[n]a)^m.

对于任意非零数字 a，指数法则成立：a^m × aⁿ = a^m+n、a^m ÷ aⁿ = a^m−n 及 (a^m)ⁿ = a^mn。零指数与负指数定义为 a⁰ = 1、a⁻ⁿ = 1 / aⁿ。分数指数与根式关联：a^1/n = √[n]a 且 a^m/n = (√[n]a)^m。

Standard form writes a number as A × 10^k ltiply or divide numbers in standard form, handle the A parts and the powers of 10 separately.

标准形式将数字写作 A × 10^k，其中 1 ≤ A < 10 且 k 为整数。进行标准形式数的乘法或除法时，应分别处理 A 部分与 10 的幂。

Fractions, decimals and percentages are interchangeable. To increase a quantity by r% is principal, r is rate per annum, t is time in years. Compound interest: A = P(1 + r/100)^t.

分数、小数与百分数可相互转换。把一个量增加 r%，乘以 (1 + r/100)；减少 r%，乘以 (1 − r/100)。单利：I = Prt / 100，其中 P 为本金，r 为年利率，t 为年数。复利：A = P(1 + r/100)^t。

s are in the ratio a : b, one can be written as a/(a+b) of the total. Direct proportion: y ∝ x ⇒ y = kx. Inverse proportion: y ∝ 1/x ⇒ y = k/x.

比与比例：若两个量之比为 a : b，其中一个可表示为总量的 a/(a+b)。正比例：y ∝ x ⇒ y = kx。反比例：y ∝ 1/x ⇒ y = k/x。

Sets are represented using curly brackets { }. n(A) means the number of elements in set A. A ∪ B is the union, A ∩ B is the intersection, and A' is the complement.

集合使用花括号 { } 表示。n(A) 表示集合 A 中元素的个数。A ∪ B 为并集，A ∩ B 为交集，A' 为补集。

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2. Algebra Basics | 代数基础

roducts are: (a + b)² = a² + 2ab + b², (a − b)² = a² − 2ab + b², (a + b)(a − b) = a² − b².

展开括号使用分配律：a(b + c) = ab + ac。两个二项式相乘得 (a + b)(c + d) = ac + ad + bc + bd。三个特殊乘积为：(a + b)² = a² + 2ab + b²、(a − b)² = a² − 2ab + b²、(a + b)(a − b) = a² − b²。

f p + q = b and pq = c. The expression ax² + bx + c can be factorised by splitting the middle term or by grouping.

因式分解是展开的逆运算。首先寻找公因式：ab + ac = a(b + c)。形如 x² + bx + c 的二次式若能找到 p、q 满足 p + q = b 且 pq = c，则可分解为 (x + p)(x + q)。ax² + bx + c 可通过拆分中间项或分组进行分解。

verses the effect of f; it is found by writing y = f(x), swapping x and y, then solving for y.

若函数写作 f(x)，f(a) 表示将 x = a 代入表达式。复合函数满足 fg(x) = f(g(x))。反函数 f⁻¹ 的作用与 f 相反；可通过设 y = f(x)，交换 x 与 y 后解出 y 求得。

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3. Equations and Inequalities | 方程与不等式

A linear equation in one unknown can be solved by isolating the variable on one side. Equations with fractions are cleared by multiplying both sides by the lowest common denominator.

一元一次方程可通过将变量移到一侧并求解。含分数的方程应在等式两边同乘以最小公分母以去除分母。

The quadratic formula gives the roots of ax² + bx + c = 0 (a ≠ 0):

x = [−b ± √b² − 4ac] / 2a

二次方程求根公式给出 ax² + bx + c = 0 (a ≠ 0) 的根：x = [−b ± √b² − 4ac] / 2a。

repeated real root, Δ < 0 gives no real roots. Completing the square rewrites ax² + bx + c as a(x + p)² + q, revealing the turning point (−p, q).

判别式 Δ = b² − 4ac 决定根的性质：Δ > 0 表明有两个不等的实根，Δ = 0 有一个重实根，Δ < 0 无实根。配方法将 ax² + bx + c 写成 a(x + p)² + q 的形式，从而得到顶点 (−p, q)。

a number line, with an open circle for < or > and a filled circle for ≤ or ≥. Quadratic inequalities are solved by sketching the corresponding parabola.

解不等式与解方程类似，但当乘或除以负数时不等式方向反转。解集可用数轴表示，< 或 > 使用空心圆，≤ 或 ≥ 使用实心圆。二次不等式通过绘制对应抛物线求解。

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4. Simultaneous Equations and Sequences | 联立方程与数列

Two linear equations in x and y can be solved by elimination (adding or subtracting to remove one variable) or by substitution. A linear–quadratic system is best solved by substitution: replace y in the quadratic equation with the linear expression.

关于 x、y 的两个一次方程可通过消元法（加减消去一个变量）或代入法求解。一次与二次联立方程组最好用代入法：用一次表达式代换二次方程中的 y。

An arithmetic sequence has a common difference d. Its nth term is u_n = a + (n − 1)d, where a is the first term. The sum of the first n terms is S_n = n/2 [2a + (n − 1)d] = n/2 (a + l), where l is the last term.

等差数列的公差为 d。第 n 项为 u_n = a + (n − 1)d，其中 a 为首项。前 n 项和 S_n = n/2 [2a + (n − 1)d] = n/2 (a + l)，l 为末项。

A geometric sequence has a common ratio r. Its nth term is u_n = arⁿ⁻¹. The sum of the first n terms is S_n = a(1 − rⁿ) / (1 − r) for r ≠ 1. An infinite geometric series converges to a/(1 − r) when |r| < 1.

等比数列的公比为 r。第 n 项 u_n = arⁿ⁻¹。前 n 项和 S_n = a(1 − rⁿ) / (1 − r)，其中 r ≠ 1。当 |r| < 1 时，无穷等比级数收敛于 a/(1 − r)。

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5. Coordinate Geometry | 坐标几何

Distance between two points (x₁, y₁) and (x₂, y₂):

d = √[(x₂ − x₁)² + (y₂ − y₁)²]

两点 (x₁, y₁) 与 (x₂, y₂) 间的距离：d = √[(x₂ − x₁)² + (y₂ − y₁)²]。

Midpoint coordinates:

M = ((x₁ + x₂)/2, (y₁ + y₂)/2)

中点坐标：M = ((x₁ + x₂)/2, (y₁ + y₂)/2)。

g the points is m = (y₂ − y₁) / (x₂ − x₁). A line with gradient m and intercept c on the y‑ lines are parallel if their gradients are equal; perpendicular lines satisfy m₁ × m₂ = −1.

连接两点的直线斜率为 m = (y₂ − y₁) / (x₂ − x₁)。斜率为 m、y 轴截距为 c 的直线方程为 y = mx + c。已知点 (x₁, y₁) 与斜率 m，直线方程为 y − y₁ = m(x − x₁)。两条直线平行当且仅当它们的斜率相等；垂直直线的斜率满足 m₁ × m₂ = −1。

e equation x² + y² + 2gx + 2fy + c = 0 represents a circle with centre (−g, −f) and radius √g² + f² − c ph of y = ax² + bx + c, find the axis of symmetry x = −b/(2a), the turning point and the y‑intercept (0, c).

方程 x² + y² + 2gx + 2fy + c = 0 表示圆心为 (−g, −f)、半径为 √g² + f² − c 的圆。要画出 y = ax² + bx + c 的图像，应先求出对称轴 x = −b/(2a)、顶点和 y 截距 (0, c)。

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6. Euclidean Geometry | 欧氏几何

Angles around a point sum to 360°; on a straight line to 180°. Vertically opposite angles are equal. Alternate angles, corresponding angles and interior angles on parallel lines satisfy the parallel line rules.

一点周围的角之和为 360°；直线上的相邻角之和为 180°。对顶角相等。平行线中的内错角、同位角与同旁内角遵循平行线法则。

xterior angle of a triangle equals the sum of the two opposite interior angles. Pythagoras' theorem for a right‑angled triangle with hypotenuse c: a² + b² = c².

三角形内角和为 180°。三角形的一个外角等于两个不相邻的内角之和。勾股定理：对于斜边为 c 的直角三角形，a² + b² = c²。

Polygon angle sum: sum of interior angles of an n‑sided polygon = (n − 2) × 180°. Each exterior angle of a regular n‑sided polygon = 360°/n.

多边形内角和：n 边形的内角和 = (n − 2) × 180°。正 n 边形的每个外角 = 360°/n。

Circle theorems: angle at centre is twice angle at circumference; angle in a semicircle is 90°; angles in the same segment are equal; opposite angles of a cyclic quadrilateral sum to 180°; tangent is perpendicular to radius at the point of contact.

圆定理：圆心角是圆周角的两倍；半圆上的圆周角为 90°；同弧上的圆周角相等；圆内接四边形对角之和为 180°；切线在切点处垂直于半径。

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7. Mensuration | 测量

Perimeter and area of basic shapes (where b is base, h is height, r is radius):

• Rectangle: P = 2(l + w), A = lw
• Parallelogram: A = b × h
• Triangle: A = ½ × b × h
• Trapezium: A = ½(a + b)h, where a and b are parallel sides
• Circle: Circumference C = 2πr, area A = πr²
• Sector of a circle: arc length = (θ/360) × 2πr, sector area = (θ/360) × πr²

基本图形的周长与面积（b 为底，h 为高，r 为半径）：矩形 P = 2(l + w)，A = lw；平行四边形 A = b × h；三角形 A = ½ × b × h；梯形 A = ½(a + b)h，其中 a、b 为平行边；圆周长 C = 2πr，面积 A = πr²；扇形弧长 = (θ/360) × 2πr，扇形面积 = (θ/360) × πr²。

Surface area and volume of solids:

• Cuboid: V = lwh, surface area = 2(lw + lh + wh)
• Prism: V = area of cross‑section × length
• Cylinder: V = πr²h, curved surface area = 2πrh, total surface area = 2πr(h + r)
• Pyramid: V = ⅓ × base area × vertical height
• Cone: V = ⅓πr²h, curved surface area = πrl (l = slant height), total surface area = πr(l + r)
• Sphere: V = ⁴⁄₃πr³, surface area = 4πr²

立体的表面积与体积：长方体 V = lwh，表面积 = 2(lw + lh + wh)；棱柱 V = 横截面积 × 长；圆柱 V = πr²h，侧面积 = 2πrh，全面积 = 2πr(h + r)；棱锥 V = ⅓ × 底面积 × 垂直高；圆锥 V = ⅓πr²h，侧面积 = πrl（l 为母线长），全面积 = πr(l + r)；球 V = ⁴⁄₃πr³，表面积 = 4πr²。

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8. Trigonometry | 三角学

In a right‑angled triangle, the three basic ratios are:

sin θ = opposite / hypotenuse, cos θ = adjacent / hypotenuse, tan θ = opposite / adjacent

在直角三角形中，三个基本比为：sin θ = 对边 / 斜边，cos θ = 邻边 / 斜边，tan θ = 对边 / 邻边。

Exact values for common angles should be memorised:

θ	sin θ	cos θ	tan θ
0°	0	1	0
30°	½	√3/2	1/√3
45°	1/√2	1/√2	1
60°	√3/2	½	√3
90°	1	0	undefined

常见角的精确值应熟记：0°、30°、45°、60°、90° 的正弦、余弦和正切值如上表所示。

For non‑right‑angled triangles, the sine rule and cosine rule apply:

a / sin A = b / sin B = c / sin C (sine rule)

a² = b² + c² − 2bc cos A (cosine rule)

对于任意三角形，可运用正弦定理与余弦定理：a / sin A = b / sin B = c / sin C；a² = b² + c² − 2bc cos A。

e found using Area = ½ ab sin C. The ambiguous case of the sine rule occurs when using SSA – two possible angles can exist (θ and 180° − θ).

任意三角形的面积可用公式 Area = ½ ab sin C 计算。使用正弦定理时会出现模糊情况（已知两边及一对角），可能有两个角满足条件（θ 和 180° − θ）。

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9. Vectors and Transformations | 向量与变换

A vector has magnitude and direction. It can be represented as a column vector (x, y)^T or as xi + yj. The magnitude of v = (x, y) is |v| = √x² + y². Vectors are added component‑wise: (x₁, y₁) + (x₂, y₂) = (x₁ + x₂, y₁ + y₂). Multiplying by a scalar k gives (kx, ky).

向量既有大小也有方向。它可表示为列向量 (x, y)^T 或 xi + yj。向量 v = (x, y) 的模为 |v| = √x² + y²。向量分量相加：(x₁, y₁) + (x₂, y₂) = (x₁ + x₂, y₁ + y₂)；数乘 k 得 (kx, ky)。

s AB = OB − OA = b − a. Two vectors are parallel if one is a scalar multiple of the other.

点 P 相对于原点 O 的位置向量为 OP = p。从 A 到 B 的向量为 AB = OB − OA = b − a。如果一个向量是另一个向量的数乘，则两向量平行。

can be described by matrices. The following 2×2 matrices represent basic transformations:

• Reflection in the x‑axis: [[1, 0], [0, −1]]
• Reflection in the y‑axis: [[−1, 0], [0, 1]]
• Reflection in the line y = x: [[0, 1], [1, 0]]
• Rotation 90° anticlockwise about O: [[0, −1], [1, 0]]
• Enlargement scale factor k, centre O: [[k, 0], [0, k]]

平面坐标中的变换可用矩阵描述。以下 2×2 矩阵表示基本变换：x 轴反射：[[1, 0], [0, −1]]；y 轴反射：[[−1, 0], [0, 1]]；直线 y = x 反射：[[0, 1], [1, 0]]；绕原点逆时针旋转 90°：[[0, −1], [1, 0]]；中心为原点、比例因子为 k 的放大：[[k, 0], [0, k]]

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