📚 Year 9 CIE Further Maths: Formulae & Theorems Quick Reference Guide | Year 9 CIE 进阶数学：公式定理速查手册
This quick reference guide consolidates the essential formulae and theorems you need to master for Year 9 CIE Further Mathematics. Use it to reinforce your memory and speed up problem-solving for examinations and school assessments.
本速查手册浓缩了 Year 9 CIE 进阶数学必须掌握的核心公式与定理。用它来强化记忆，提升解题速度，从容应对考试和校内测评。
1. Algebraic Identities & Expansion | 代数恒等式与展开
2ab + b². This identity is fundamental for expanding and factorising quadratic expressions.
二项式的平方：(a + b)² = a² + 2ab + b²。这个恒等式是展开和因式分解二次表达式的基础。
l with a minus sign: (a − b)² = a² − 2ab + b². Notice the sign of the middle term changes.
带负号的二项式平方：(a − b)² = a² − 2ab + b²。注意中间项的符号变了。
two squares: (a + b)(a − b) = a² − b². This is one of the most frequently tested patterns.
平方差公式：(a + b)(a − b) = a² − b²。这是考试中最常出现的模式之一。
(a + b)³ = a³ + 3a²b + 3ab² + b³. Though less common, it appears in higher-tier questions.
立方展开：(a + b)³ = a³ + 3a²b + 3ab² + b³。虽不常直接考，但在高阶题中会用到。
2. Quadratic Equations & the Discriminant | 二次方程与判别式
utions can be found by factorising, completing the square, or using the quadratic formula.
标准二次方程为 ax² + bx + c = 0，其中 a ≠ 0。可通过因式分解、配方法或二次公式求解。
Quadratic formula: x = [ −b ± √b² − 4ac ] / (2a). Always rearrange to standard form before applying.
二次公式：x = [ −b ± √b² − 4ac ] / (2a)。使用前务必整理成标准形式。
tinct real roots; Δ = 0, one repeated real root; Δ < 0, no real roots (two complex roots).
判别式 Δ = b² − 4ac 决定根的性质：Δ > 0，有两个不等实根；Δ = 0，有一个重实根；Δ < 0，无实根。
α and β, sum = −b/a, product = c/a. Useful when solving without finding roots explicitly.
根的和与积：若根为 α 和 β，则和 = −b/a，积 = c/a。无需具体求根即可使用。
3. Linear Equations & Straight Line Graphs | 一次方程与直线图
ine in slope-intercept form: y = mx + c, where m is the gradient and c is the y-intercept.
斜截式直线方程：y = mx + c，其中 m 是斜率，c 是 y 轴截距。
etween two points (x₁, y₁) and (x₂, y₂): m = (y₂ − y₁) / (x₂ − x₁). The line is steeper if |m| is larger.
两点 (x₁, y₁) 和 (x₂, y₂) 之间的斜率：m = (y₂ − y₁) / (x₂ − x₁)。|m| 越大，直线越陡。
lope form: y − y₁ = m(x − x₁). This is effective when you know one point and the gradient.
点斜式：y − y₁ = m(x − x₁)。已知一点和斜率时，用这个形式最方便。
adients: m₁ = m₂. Perpendicular lines satisfy m₁ × m₂ = −1 (provided neither is vertical).
平行线斜率相等：m₁ = m₂。垂直线满足 m₁ × m₂ = −1（前提是不存在垂直斜率）。
4. Inequalities & Number Line Representation | 不等式与数轴表示
Linear inequalities are solved similarly to equations, but multiplying or dividing by a negative number reverses the inequality sign.
解一次不等式与解方程类似，但乘或除以负数时必须反转不等号方向。
tween a and b. Use open circles for < or > and closed circles for ≤ or ≥ on a number line.
复合不等式：a < x < b 表示 x 介于 a 和 b 之间。在数轴上开区间用空心圆圈，闭区间用实心圆圈。
Quadratic inequalities: sketch the parabola and find intervals where the function is above or below the x-axis. Always test intervals.
二次不等式：画出抛物线，找出函数在 x 轴上或下的区间。务必检验各个区间。
5. Coordinate Geometry – Distance, Midpoint & Circle | 坐标几何—距离、中点与圆
Distance between two points: d = √[(x₂ − x₁)² + (y₂ − y₁)²]. Derived from Pythagoras' theorem.
两点间距离：d = √[(x₂ − x₁)² + (y₂ − y₁)²]。由勾股定理推出。
e segment: M = ( (x₁ + x₂)/2 , (y₁ + y₂)/2 ). Very useful in geometry and vector problems.
线段中点：M = ( (x₁ + x₂)/2 , (y₁ + y₂)/2 )。在几何和向量题中非常实用。
adius r: (x − h)² + (y − k)² = r². For the unit circle centred at the origin: x² + y² = 1.
圆心 (h, k)、半径 r 的圆方程：(x − h)² + (y − k)² = r²。以原点为圆心的单位圆：x² + y² = 1。
6. Perimeter, Area & Volume of Common Shapes | 常见图形的周长、面积与体积
Rectangle: Area = l × w, Perimeter = 2(l + w). Triangle: Area = ½ × base × height.
矩形：面积 = l × w，周长 = 2(l + w)。三角形：面积 = ½ × 底 × 高。
rea = ½ (a + b) h, where a and b are the parallel sides and h is the perpendicular height.
梯形：面积 = ½ (a + b) h，其中 a 和 b 为平行边，h 为垂直高。
or πd, Area = πr². Arc length = (θ/360) × 2πr, sector area = (θ/360) × πr² (θ in degrees).
圆：周长 = 2πr 或 πd，面积 = πr²。弧长 = (θ/360) × 2πr，扇形面积 = (θ/360) × πr²（θ 以度计）。
gth. Cylinder: Volume = πr²h, curved surface area = 2πrh, total surface area = 2πr(r + h).
棱柱：体积 = 截面积 × 长。圆柱：体积 = πr²h，侧面积 = 2πrh，总表面积 = 2πr(r + h)。
7. Pythagoras' Theorem & Right-Angled Trigonometry | 勾股定理与直角三角形三角学
hypotenuse c: a² + b² = c². Always check if the triangle is right-angled before applying.
在直角三角形中，直角边 a、b，斜边 c：a² + b² = c²。使用前必须确认三角形为直角三角形。
/ hypotenuse, cos θ = adjacent / hypotenuse, tan θ = opposite / adjacent. Use SOH CAH TOA.
三角函数比：sin θ = 对边 / 斜边，cos θ = 邻边 / 斜边，tan θ = 对边 / 邻边。记住 SOH CAH TOA 口诀。
Exact trig values for common angles: sin 30° = ½, cos 45° = √2/2, tan 60° = √3 etc. Must be memorised for non-calculator papers.
常见角度的精确三角值：sin 30° = ½，cos 45° = √2/2，tan 60° = √3 等。非计算器试卷必须熟记。
Angle of elevation and depression are measured from the horizontal line. They are equal in alternate parallel line situations.
仰角和俯角从水平线开始计量。在平行线交替情况下二者相等。
8. Bearings & Scale Drawing | 方位角与比例绘图
A bearing is an angle measured clockwise from the north direction, always written as a three-digit figure (e.g., 045°, 210°).
方位角是从正北方向顺时针度量的角度，始终用三位数字表示（如 045°、210°）。
When solving bearing problems, draw a clear north line at each point and use parallel line angle properties and trigonometry.
解决方位角问题时，在每个点画出清晰的北方向，利用平行线角度性质和三角学求解。
awing: scale = diagram length : actual length. Ensure consistent units before calculating.
比例绘图：比例尺 = 图上长度 : 实际长度。计算前确保单位一致。
9. Sequences & Series | 数列与级数
t term and d is the common difference. Sum of the first n terms: Sₙ = n/2 [2a + (n − 1)d].
等差数列：第 n 项 = a + (n − 1)d，其中 a 为首项，d 为公差。前 n 项和：Sₙ = n/2 [2a + (n − 1)d]。
e r is the common ratio. Sum of the first n terms (for r > 1): Sₙ = a(rⁿ − 1)/(r − 1). For |r| < 1, infinite sum converges to a/(1 − r).
等比数列：第 n 项 = arⁿ⁻¹，r 为公比。当 r > 1 时，前 n 项和：Sₙ = a(rⁿ − 1)/(r − 1)。若 |r| < 1，无穷级数和收敛于 a/(1 − r)。
Linear sequences from patterns: identify the constant difference to find the nth term formula. Quadratic sequences need a second difference being constant.
从图形模式得出的线性数列：通过常数差求第 n 项公式。二次数列则需二阶差为常数。
10. Transformations & Vectors | 变换与向量
Translation is sliding a shape without rotation. It is described by a column vector (x over y) which indicates movement right/left and up/down.
平移是图形不旋转地滑动。用列向量（上 x 下 y）表示，指示向右/左和向上/下的移动量。
e specified (e.g., y = x, x-axis). Each point and its image are equidistant from the line.
反射：需指定对称轴（例如 y = x，x 轴）。每个点与其像点到对称轴距离相等。
Rotation: define centre, angle and direction (clockwise/anticlockwise). Enlargement: centre and scale factor. Negative scale factors produce an inverted image.
旋转：需确定中心、角度和方向（顺时针/逆时针）。放大：需确定中心和比例因子。负比例因子产生倒立像。
osition vector, addition, subtraction and scalar multiplication are essential. Magnitude = √x² + y².
向量有大小和方向。位置向量、向量的加减与数乘是重点。向量的大小 = √x² + y²。
11. Probability – Compound Events & Tree Diagrams | 概率—复合事件与树形图
× P(B) if A and B are independent. For mutually exclusive events, P(A or B) = P(A) + P(B).
若 A 与 B 独立，则 P(A and B) = P(A) × P(B)。对于互斥事件，P(A or B) = P(A) + P(B)。
Tree diagrams multiply along branches and add across outcomes. Probabilities on each set of branches sum to 1.
树形图：沿分支相乘，不同结果相加。每组分支上的概率之和为 1。
Conditional probability: P(A|B) = P(A and B) / P(B). The notation | means 'given that'. Always adjust the sample space.
条件概率：P(A|B) = P(A and B) / P(B)。符号 | 表示“在...已发生的条件下”。务必调整样本空间。
pected frequency = probability × number of trials. Useful for checking long-term outcomes.
期望频数 = 概率 × 试验次数。用于检验长期结果。
12. Statistics – Averages, Dispersion & Charts | 统计—平均数、离散度与图表
er of values. Median is the middle value when data are ordered; mode is the most frequent.
平均数 = 所有值之和 ÷ 数据个数。中位数是排序后中间的值；众数是出现频率最高的值。
m. It is a measure of spread. Interquartile range (IQR) = upper quartile − lower quartile.
极差 = 最大值 − 最小值，是离散度的一个度量。四分位距 (IQR) = 上四分位数 − 下四分位数。
artiles. Histograms use frequency density = frequency ÷ class width (for unequal classes).
累积频数图有助于找中位数和四分位数。直方图（组距不等时）使用频数密度 = 频数 ÷ 组距。
Scatter graphs show correlation, and a line of best fit can be drawn by eye. Outliers should be identified and can affect the mean significantly.
散点图显示相关性，可以凭视觉画出最佳拟合线。异常值需要识别，它们会显著影响均值。
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