📚 Year 10 CAIE Mathematics: Formula & Theorem Quick Reference Handbook | Year 10 CAIE 数学：公式定理速查手册
This quick reference handbook covers the essential formulas and theorems you need for Year 10 CAIE Mathematics (IGCSE 0580/0980). Use it to revise key concepts across algebra, geometry, trigonometry, probability, statistics and more. Each section presents formulas in clear, concise statements so you can quickly locate what you need.
这本速查手册涵盖了 Year 10 CAIE 数学（IGCSE 0580/0980）所需的核心公式与定理。可用于复习代数、几何、三角学、概率、统计等重点内容。每个部分以清晰简明的陈述呈现公式，方便你快速查找所需知识。
1. Arithmetic and Number | 算术与数
The product of a/aⁿ is aᵐ⁺ⁿ (where a ≠ 0). When multiplying powers with the same base, add the exponents.
同底数幂相乘，指数相加：aᵐ × aⁿ = aᵐ⁺ⁿ （a ≠ 0）。
aᵐ ÷ aⁿ = aᵐ⁻ⁿ. For division, subtract the exponents.
同底数幂相除，指数相减：aᵐ ÷ aⁿ = aᵐ⁻ⁿ。
(aᵐ)ⁿ = aᵐⁿ. When raising a power to another power, multiply the exponents.
幂的乘方，指数相乘：(aᵐ)ⁿ = aᵐⁿ。
m: a × 10ⁿ where 1 ≤ a < 10 and n is an integer. Use for very large or very small numbers.
标准形式：a × 10ⁿ，其中 1 ≤ a < 10 且 n 为整数，用于表示极大或极小的数。
Percentage change = (change ÷ original value) × 100%. Reverse percentages: original = new ÷ (1 ± r%).
百分比变化 =（变化量 ÷ 原值）× 100%。反向百分比：原值 = 新值 ÷（1 ± r%）。
2. Algebra: Expanding, Factorising and Simplifying | 代数：展开、因式分解与化简
a(b + c) = ab + ac. Expanding products of binomials: (x + a)(x + b) = x² + (a + b)x + ab.
分配律：a(b + c) = ab + ac。二项式乘积展开：(x + a)(x + b) = x² + (a + b)x + ab。
Difference of two squares: a² – b² = (a – b)(a + b). Recognise this pattern for quick factorisation.
平方差公式：a² – b² = (a – b)(a + b)。识别此模式可快速因式分解。
Factorising quadratics: x² + bx + c = (x + p)(x + q) where p + q = b and pq = c.
二次三项式因式分解：x² + bx + c = (x + p)(x + q)，其中 p + q = b 且 pq = c。
Algebraic fractions: add/subtract by finding a common denominator; multiply numerators and denominators separately; divide by multiplying by the reciprocal.
代数分式：加减时先通分；乘法分别乘分子分母；除法乘以倒数。
3. Linear Equations and Inequalities | 线性方程与不等式
rform the same operation on both sides to isolate the variable. E.g., 2x + 5 = 11 → x = 3.
解线性方程：在等号两边执行相同运算以分离变量。例如 2x + 5 = 11，得 x = 3。
Inequalities: treat like equations, but if multiplying or dividing by a negative number, reverse the inequality sign. a < b is equivalent to -a > -b.
不等式：解法类似方程，但乘以或除以负数时，不等号方向反转。a < b 等价于 -a > -b。
Compound inequalities: '3 < x ≤ 7' eans x is greater than 3 and at most 7. Express on a number line with open/closed circles.
复合不等式：如 3 < x ≤ 7 表示 x 大于 3 且不大于 7。在数轴上用空心/实心圆圈表示。
Solving simultaneous linear equations: elimination (add/subtract to remove one variable) or substitution.
解联立线性方程：消元法（加减消去一个变量）或代入法。
4. Quadratic Equations | 二次方程
General form: ax² + bx + c = 0, where a ≠ 0.
一般式：ax² + bx + c = 0，其中 a ≠ 0。
Factorisation method: set each factor to zero. If (x – p)(x – q) = 0, then x = p or x = q.
因式分解法：令每个因式为零。若 (x – p)(x – q) = 0，则 x = p 或 x = q。
Quadratic formula:
二次公式：
x = [–b ± √b² – 4ac] ÷ (2a)
Discriminant Δ = b² – 4ac. Two distinct real roots if Δ > 0, one repeated root if Δ = 0, no real roots if Δ < 0.
判别式 Δ = b² – 4ac。若 Δ > 0 有两个不等实根，Δ = 0 有一个重根，Δ < 0 无实根。
Completing the square: rewrite x² + bx as (x + b/2)² – (b/2)². Useful for finding the vertex of a parabola.
配方法：将 x² + bx 改写为 (x + b/2)² – (b/2)²，常用于求抛物线顶点。
5. Sequences | 数列
Arithmetic sequence: nth term = a + (n – 1)d, where a = first term, d = common difference.
等差数列：第 n 项 = a + (n – 1)d，其中 a 为首项，d 为公差。
Sum of first n terms of an arithmetic series: Sₙ = n/2 [2a + (n – 1)d] or Sₙ = n/2 (a + l), where l = last term.
等差数列前 n 项和：Sₙ = n/2 [2a + (n – 1)d] 或 Sₙ = n/2 (a + l)，l 为末项。
Linear sequences: term-to-term rule involves adding or subtracting a constant. Find the general term of a linear pattern.
线性数列：项与项之间加或减一个常数。求线性模式的通项公式。
d differences are constant. nth term = an² + bn + c, where a = half the second difference.
二次数列：二阶差恒定。第 n 项 = an² + bn + c，其中 a 为二阶差的一半。
6. Mensuration: Perimeter, Area and Volume | 量度：周长、面积与体积
Rectangle: Perimeter = 2(l + w), Area = l × w.
矩形：周长 = 2(长 + 宽)，面积 = 长 × 宽。
Triangle: Area = ½ × base × height. For right-angled triangle, apply Pythagoras' theorem.
三角形：面积 = ½ × 底 × 高。直角三角形可使用勾股定理。
a = base × vertical height. Trapezium: Area = ½(a + b)h, where a and b are parallel sides.
平行四边形：面积 = 底 × 垂直高。梯形：面积 = ½(a + b)h，a 和 b 为平行边。
mference = 2πr or πd, Area = πr². Arc length = (θ/360) × 2πr, sector area = (θ/360) × πr².
圆：周长 = 2πr 或 πd，面积 = πr²。弧长 = (θ/360) × 2πr，扇形面积 = (θ/360) × πr²。
th. Cylinder: Volume = πr²h, Curved surface area = 2πrh, Total surface area = 2πrh + 2πr².
棱柱：体积 = 横截面积 × 长。圆柱：体积 = πr²h，侧面积 = 2πrh，总表面积 = 2πrh + 2πr²。
se area × height. Cone: Volume = ⅓πr²h, Curved surface area = πrl, where l = slant height.
棱锥：体积 = ⅓ × 底面积 × 高。圆锥：体积 = ⅓πr²h，侧面积 = πrl，l 为斜高。
Sphere: Volume = ⁴⁄₃πr³, Surface area = 4πr².
球体：体积 = ⁴⁄₃πr³，表面积 = 4πr²。
7. Pythagoras’ Theorem and Trigonometry | 勾股定理与三角学
Right-angled triangle: a² + b² = c², where c is the hypotenuse. Use to find missing sides.
直角三角形：a² + b² = c²，c 为斜边。用于求未知边长。
Trigonometric ratios (sine, cosine, tangent):
三角函数（正弦、余弦、正切）：
sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent
Exact values for special angles: sin 30° = ½, cos 30° = √3/2, tan 30° = 1/√3; sin 45° = 1/√2, cos 45° = 1/√2, tan 45° = 1; sin 60° = √3/2, cos 60° = ½, tan 60° = √3.
特殊角精确值：sin 30° = ½，cos 30° = √3/2，tan 30° = 1/√3；sin 45° = 1/√2，cos 45° = 1/√2，tan 45° = 1；sin 60° = √3/2，cos 60° = ½，tan 60° = √3。
se for non-right-angled triangles when given two angles and any side, or two sides and a non-included angle.
正弦定理：a/sin A = b/sin B = c/sin C。适用于非直角三角形，已知两角和任一边，或两边及一对角。
Cosine rule: a² = b² + c² – 2bc cos A. Use when given two sides and the included angle, or three sides.
余弦定理：a² = b² + c² – 2bc cos A。适用于已知两边及其夹角，或三边求角。
Area of triangle using sine: Area = ½ ab sin C.
用正弦求三角形面积：面积 = ½ ab sin C。
8. Geometry: Angles and Circle Theorems | 几何：角与圆定理
Angles on a straight line sum to 180°. Angles around a point sum to 360°.
平角为 180°，周角为 360°。
Vertically opposite angles are equal. Corresponding angles on parallel lines are equal; alternate angles are equal; interior (co-interior) angles sum to 180°.
对顶角相等。平行线同位角相等，内错角相等，同旁内角互补（和为 180°）。
m = 180°. Exterior angle of a triangle equals the sum of the two opposite interior angles.
三角形内角和为 180°。三角形外角等于不相邻两内角之和。
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 point of contact; alternate segment theorem.
圆定理：圆心角是圆周角的两倍；半圆圆周角为 90°；同弧圆周角相等；圆内接四边形对角互补；切线垂直于过切点的半径；弦切角定理。
9. Probability | 概率
lity of an event = number of favourable outcomes / total number of outcomes. 0 ≤ P(E) ≤ 1.
事件概率 = 有利结果数 / 所有可能结果数。0 ≤ P(E) ≤ 1。
Sum of probabilities of all outcomes = 1. P(not A) = 1 – P(A).
所有结果概率之和为 1。P(非 A) = 1 – P(A)。
ed frequency = probability × number of trials. Use for experimental probability estimates.
期望频数 = 概率 × 试验次数。用于估计实验概率。
clusive events, P(A or B) = P(A) + P(B). For independent events, P(A and B) = P(A) × P(B).
组合事件概率：互斥事件 P(A 或 B) = P(A) + P(B)。独立事件 P(A 且 B) = P(A) × P(B)。
Tree diagrams: multiply along branches for 'and', add probabilities of relevant branches for 'or'. Conditional probability: P(A given B) = P(A and B) / P(B).
树状图：沿分支相乘表示“且”，将相关分支概率相加表示“或”。条件概率：P(A|B) = P(A 且 B) / P(B)。
10. Statistics | 统计
of values. Median = middle value when ordered. Mode = most frequent value. Range = maximum – minimum.
平均数 = 数据总和 ÷ 数据个数。中位数 = 排序后中间的值。众数 = 出现频率最高的值。极差 = 最大值 – 最小值。
For grouped data, estimate mean using midpoints: mean ≈ Σ(f × midpoint) / Σf.
对于分组数据，用组中值估算平均数：平均数 ≈ Σ(f × 组中值) / Σf。
ve frequency helps locate median and quartiles. Interquartile range (IQR) = upper quartile – lower quartile.
频数表：累积频数用于确定中位数和四分位数。四分位距 (IQR) = 上四分位数 – 下四分位数。
Scatter diagrams: correlation (positive, negative, none). Line of best fit by eye; use to make predictions within the data range.
散点图：相关性（正、负、无）。目测画最佳拟合线；用于数据范围内预测。
Pie charts: angle for a sector = (category frequency ÷ total frequency) × 360°.
饼图：扇区角度 =（类别频数 ÷ 总频数）× 360°。
11. Functions and Graphs | 函数与图像
pression. f(a) means substitute x = a. For inverse function, swap x and y and solve for y.
函数记号：f(x) = 表达式。f(a) 表示代入 x = a。反函数：交换 x 与 y 并解出 y。
Gradient of a straight line: m = (y₂ – y₁)/(x₂ – x₁). Equation of a line: y = mx + c (slope-intercept form) or y – y₁ = m(x – x₁) (point-slope form).
直线斜率：m = (y₂ – y₁)/(x₂ – x₁)。直线方程：y = mx + c（斜截式）或 y – y₁ = m(x – x₁)（点斜式）。
Parallel lines have equal gradients. Perpendicular lines: product of gradients = –1 (m₁ × m₂ = –1).
平行线斜率相等。垂直线斜率乘积为 –1（m₁ × m₂ = –1）。
Graph transformations: f(x) + a shifts up by a; f(x + a) shifts left by a; -f(x) reflects in x-axis; f(–x) reflects in y-axis; kf(x) stretches vertically by factor k.
图像变换：f(x) + a 向上平移 a；f(x + a) 向左平移 a；–f(x) 关于 x 轴反射；f(–x) 关于 y 轴反射；kf(x) 垂直方向拉伸 k 倍。
Distance between two points: √[(x₂ – x₁)² + (y₂ – y₁)²]. Midpoint: ((x₁ + x₂)/2, (y₁ + y₂)/2).
两点间距离：√[(x₂ – x₁)² + (y₂ – y₁)²]。中点：((x₁ + x₂)/2, (y₁ + y₂)/2)。
12. Vectors and Transformations | 向量与变换
esented as a column (x component top, y component bottom) or as a bold letter. Magnitude = √x² + y².
向量表示为列向量 (x 在上，y 在下) 或粗体字母。模长 = √x² + y²。
Vector addition: add corresponding components. Scalar multiplication: multiply each component by the scalar.
向量加法：对应分量相加。标量乘法：每个分量乘以标量。
Translation: moving a shape by a vector without rotation or reflection. Reflection: mirror image in a given line. Rotation: turning about a centre through an angle. Enlargement: scale factor k about a centre; if k > 1, shape gets larger; if 0 < k < 1, shape gets smaller; negative k produces an inverted image.
平移：沿向量移动图形，无旋转、无翻转。反射：关于给定直线的镜像。旋转：绕中心旋转一定角度。放大：以缩放因子 k 关于中心放大；k > 1 形状变大，0 < k < 1 形状变小，负 k 产生倒立图像。
Combine transformations: carry out one after the other; order often matters. Use matrix multiplication (extension) or tracing paper for rotations/reflections.
复合变换：依次执行；顺序通常重要。可用矩阵乘法（拓展）或描图纸辅助旋转/反射。
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