Year 10 CIE Maths: Formula & Theorem Quick Reference | CIE 数学公式定理速查手册

📚 Year 10 CIE Maths: Formula & Theorem Quick Reference | CIE 数学公式定理速查手册

This handy reference guide brings together the essential formulas and theorems you need for Year 10 CIE IGCSE Mathematics. It covers arithmetic, algebra, geometry, trigonometry, vectors, statistics and probability in a clear, bilingual format to support your revision.

本速查手册汇集了 Year 10 CIE IGCSE 数学必备的公式与定理，涵盖算术、代数、几何、三角、向量、统计与概率，并以清晰的双语对照形式帮助你轻松复习。

1. Arithmetic & Number Properties | 算术与数的性质

The order of operations is Parentheses, Exponents, Multiplication/Division, Addition/Subtraction (PEMDAS).

运算顺序为括号、指数、乘除、加减（PEMDAS）。

Percentage increase = (increase ÷ original) × 100%.

百分比增加 = （增加量 ÷ 原值）× 100%。

Percentage decrease = (decrease ÷ original) × 100%.

百分比减少 = （减少量 ÷ 原值）× 100%。

r × t, where P is principal, r is rate per annum (as a decimal), t is time in years.

单利 I = P × r × t，其中 P 为本金，r 为年利率（小数），t 为年数。

Compound interest A = P(1 + r/n)^nt for Core; Extended may use A = P(1 + r/100)ⁿ when interest is compounded annually.

复利公式 Core 阶段用 A = P(1 + r/n)^nt；若每年复利一次，扩展课程常用 A = P(1 + r/100)ⁿ。

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2. Algebraic Identities & Manipulation | 代数恒等式与运算

(a + b)² = a² + 2ab + b²

(a + b)² = a² + 2ab + b²

(a – b)² = a² – 2ab + b²

(a – b)² = a² – 2ab + b²

(a + b)(a – b) = a² – b²

(a + b)(a – b) = a² – b²

Factorising a quadratic: if x² + bx + c = (x + p)(x + q), then p + q = b and pq = c.

二次三项式因式分解：若 x² + bx + c = (x + p)(x + q)，则 p + q = b 且 pq = c。

Index laws: aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1, a⁻ⁿ = 1/aⁿ.

指数法则：aᵐ × aⁿ = aᵐ⁺ⁿ，aᵐ ÷ aⁿ = aᵐ⁻ⁿ，(aᵐ)ⁿ = aᵐⁿ，a⁰ = 1，a⁻ⁿ = 1/aⁿ。

Simplifying surds: √ab = √a × √b, √a/b = √a/√b.

根式化简：√ab = √a × √b，√a/b = √a/√b。

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3. Solving Linear Equations | 解一元一次方程

A linear equation in one variable can be written as ax + b = c. Solve by isolating x.

一元一次方程形如 ax + b = c，通过移项求解 x。

The solution is x = (c – b)/a, provided a ≠ 0.

解为 x = (c – b)/a，其中 a ≠ 0。

For simultaneous linear equations, use elimination or substitution to find the unique pair (x, y).

解二元一次方程组时，使用消元法或代入法求出唯一解 (x, y)。

When graphing, the intersection point of the two lines gives the solution.

图像法中，两条直线的交点坐标即为方程组的解。

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4. Quadratic Equations | 二次方程

The standard form is ax² + bx + c = 0, a ≠ 0.

标准形式为 ax² + bx + c = 0，a ≠ 0。

The quadratic formula:

二次方程求根公式：

x = [-b ± √(b² – 4ac)] / 2a

The discriminant Δ = b² – 4ac determines the nature of roots:

判别式 Δ = b² – 4ac 决定根的性质：

inct real roots; if Δ = 0, one real repeated root; if Δ < 0, no real roots.

Δ > 0 有两个不等实根；Δ = 0 有一个重实根；Δ < 0 无实根。

Completing the square: x² + bx = (x + b/2)² – (b/2)².

配方法：x² + bx = (x + b/2)² – (b/2)²。

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5. Ratio, Proportion & Rates | 比、比例与速率

A ratio a : b can be simplified by dividing both parts by their highest common factor.

比 a : b 可通过同除以最大公因数来化简。

Direct proportion: y = kx, where k is the constant of proportionality.

正比例：y = kx，其中 k 为比例常数。

Inverse proportion: y = k/x, or xy = k.

反比例：y = k/x，即 xy = k。

Speed = distance ÷ time. Average speed = total distance ÷ total time.

速率 = 距离 ÷ 时间。平均速率 = 总距离 ÷ 总时间。

Density = mass ÷ volume; Pressure = force ÷ area.

密度 = 质量 ÷ 体积；压强 = 力 ÷ 面积。

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6. Coordinate Geometry | 坐标几何

Distance between two points (x₁, y₁) and (x₂, y₂): d = &radic;[(x₂ – x₁)² + (y₂ – y₁)²].

两点间距离公式：d = &radic;[(x₂ – x₁)² + (y₂ – y₁)²]。

Midpoint of a segment: M = ((x₁ + x₂)/2, (y₁ + y₂)/2).

线段中点公式：M = ((x₁ + x₂)/2, (y₁ + y₂)/2)。

Gradient (slope) of a line: m = (y₂ – y₁)/(x₂ – x₁).

直线斜率：m = (y₂ – y₁)/(x₂ – x₁)。

Equation of a straight line: y = mx + c, where m is the gradient and c is the y-intercept.

直线方程：y = mx + c，其中 m 为斜率，c 为 y 轴截距。

Parallel lines have equal gradients; perpendicular lines satisfy m₁ × m₂ = -1.

平行线斜率相等；垂直线斜率满足 m₁ × m₂ = -1。

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7. Geometry: Angles & Polygons | 几何：角与多边形

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.

对顶角相等。

In a triangle, the sum of interior angles is 180°.

三角形内角和为 180°。

Pythagoras' theorem: in a right-angled triangle, c² = a² + b², where c is the hypotenuse.

勾股定理：直角三角形中，斜边 c² = a² + b²。

Sum of interior angles of an n-sided polygon = (n – 2) × 180°.

n 边形内角和 = (n – 2) × 180°。

Exterior angle of a regular n-sided polygon = 360&deg; ÷ n.

正 n 边形外角 = 360&deg; ÷ n。

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8. Mensuration: Perimeter, Area & Volume | 测量：周长、面积与体积

Shape	Perimeter/Circumference	Area
Rectangle	2(l + w)	l × w
Triangle	a + b + c	½ × b × h
Circle	C = 2πr or πd	A = πr²
Parallelogram	2(a + b)	b × h
Trapezium	Sum of four sides	½ (a + b) h

Volume of cuboid: V = l × w × h.

长方体体积：V = l × w × h。

Volume of prism: V = area of cross-section × length.

柱体体积：V = 底面积 × 高。

Volume of cylinder: V = &pi;r²h. Curved surface area = 2&pi;rh.

圆柱体积：V = &pi;r²h；侧面积 = 2&pi;rh。

Volume of pyramid: V = (1/3) × base area × height.

棱锥体积：V = (1/3) × 底面积 × 高。

Volume of cone: V = (1/3)&pi;r²h; curved surface area = πrl, where l is slant height.

圆锥体积：V = (1/3)&pi;r²h；侧面积 = πrl，l 为斜高。

Volume of sphere: V = (4/3)&pi;r³; surface area = 4&pi;r².

球体积：V = (4/3)&pi;r³；表面积 = 4&pi;r²。

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9. Trigonometry | 三角学

otenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent (SOH CAH TOA).

直角三角形中：sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边（SOH CAH TOA）。

Exact values for 30°, 45° and 60°:

30°、45°、60° 的精确值：

θ	sin θ	cos θ	tan θ
30°	1/2	√3/2	1/√3
45°	1/√2	1/√2	1
60°	√3/2	1/2	√3

For any triangle (Extended): sine rule: a/sin A = b/sin B = c/sin C.

任意三角形（扩展课程）正弦定理：a/sin A = b/sin B = c/sin C。

Cosine rule (Extended): a² = b² + c² – 2bc cos A.

余弦定理（扩展课程）：a² = b² + c² – 2bc cos A。

Area of a triangle (Extended): A = ½ ab sin C.

三角形面积（扩展课程）：A = ½ ab sin C。

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10. Vectors & Transformations | 向量与变换

A vector can be written as a column matrix [x; y] or as xi + yj.

向量可表示为列矩阵 [x; y] 或 xi + yj。

The magnitude of vector v(a, b) is |v| = &radic;(a² + b²).

向量 v(a, b) 的模长 |v| = &radic;(a² + b²)。

Adding vectors: [a; b] + [c; d] = [a+c; b+d].

向量加法：[a; b] + [c; d] = [a+c; b+d]。

Multiplying a vector by scalar k: k[a; b] = [ka; kb].

向量数乘：k[a; b] = [ka; kb]。

Translation by vector [a; b] moves a point (x, y) to (x+a, y+b).

按向量 [a; b] 平移将点 (x, y) 移至 (x+a, y+b)。

Reflection in the x-axis: (x, y) → (x, –y). Reflection in the y-axis: (x, y) → (–x, y). Reflection in y = x: (x, y) → (y, x).

关于 x 轴对称反射：(x, y) → (x, –y)；关于 y 轴：(x, y) → (–x, y)；关于 y = x：(x, y) → (y, x)。

Rotation 90° anticlockwise about origin: (x, y) → (–y, x); 180°: (x, y) → (–x, –y).

绕原点逆时针旋转 90°：(x, y) → (–y, x)；180°：(x, y) → (–x, –y)。

Enlargement with centre origin and scale factor k: (x, y) → (kx, ky).

以原点为中心、缩放因子 k 的放大：(x, y) → (kx, ky)。

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11. Statistics | 统计

Mean = (sum of all values) ÷ (number of values).

平均数 = 所有数值之和 ÷ 数据个数。

For a frequency table, mean = Σ (f × x) / Σ f.

对于频数表，平均数 = Σ (f × x) / Σ f。

Median is the middle value when data are ordered; for n values, median position = (n+1)/2.

中位数是排序后中间的值；n 个数据，中位数位置 = (n+1)/2。

Mode is the most frequent value.

众数是出现次数最多的值。

Range = maximum value – minimum value.

极差 = 最大值 – 最小值。

In cumulative frequency graphs, the median, quartiles and percentiles can be estimated.

累积频数图中可估计中位数、四分位数和百分位数。

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12. Probability | 概率

ent A, P(A) = (number of favourable outcomes) / (total number of equally likely outcomes).

事件 A 的概率 P(A) = 有利结果数 / 所有等可能结果总数。

P(A) always satisfies 0 ≤ P(A) ≤ 1.

P(A) 总满足 0 ≤ P(A) ≤ 1。

Complementary events: P(not A) = 1 – P(A).

互斥事件：P(非 A) = 1 – P(A)。

For independent events A and B: P(A and B) = P(A) × P(B).

对于独立事件 A 和 B：P(A 且 B) = P(A) × P(B)。

For mutually exclusive events A and B: P(A or B) = P(A) + P(B).

对于互斥事件 A 和 B：P(A 或 B) = P(A) + P(B)。

Tree diagrams show all possible outcomes and are used to find combined probabilities.

树状图列出所有可能结果，用于计算组合概率。

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