📚 Year 10 Eduqas Maths: Formula & Theorem Quick Reference | Year 10 Eduqas 数学：公式定理速查手册
This quick-reference handbook brings together every essential formula and theorem you will meet in Year 10 Eduqas Mathematics. Use it alongside your class notes to check rules, plug in numbers and build confidence before assessments.
这本速查手册汇集了 Year 10 Eduqas 数学课程中你需要掌握的所有核心公式与定理。配合课堂笔记使用，既能快速核对法则、代入数值，也能在评估前帮你建立信心。
1. Number: Index Laws & Standard Form | 数：指数定律与标准形式
When multiplying powers with the same base, add the exponents: aᵐ × aⁿ = aᵐ⁺ⁿ.
同底数的幂相乘，指数相加：aᵐ × aⁿ = aᵐ⁺ⁿ。
When dividing powers with the same base, subtract the exponents: aᵐ ÷ aⁿ = aᵐ⁻ⁿ.
同底数的幂相除，指数相减：aᵐ ÷ aⁿ = aᵐ⁻ⁿ。
Raising a power to another power multiplies the exponents: (aᵐ)ⁿ = aᵐⁿ.
幂的乘方，指数相乘：(aᵐ)ⁿ = aᵐⁿ。
A number raised to a negative index is its reciprocal: a⁻ⁿ = 1/aⁿ.
负指数表示倒数：a⁻ⁿ = 1/aⁿ。
Fractional indices connect roots and powers: a1/n = ⁿ√a, and am/n = ⁿ√aᵐ.
分数指数联系根式与幂：a1/n = ⁿ√a，且 am/n = ⁿ√aᵐ。
Standard form writes a number as A × 10ⁿ where 1 ≤ A < 10 and n is an integer.
标准形式将一个数写成 A × 10ⁿ，其中 1 ≤ A < 10，n 为整数。
Example: 4 500 = 4.5 × 10³; 0.0032 = 3.2 × 10⁻³.
2. Algebra: Expanding & Factorising | 代数：展开与因式分解
Expand brackets by multiplying each term inside by the term outside: a(b + c) = ab + ac.
展开括号用外面的项乘括号内每一项：a(b + c) = ab + ac。
Expand double brackets using FOIL: (x + a)(x + b) = x² + (a + b)x + ab.
展开两个括号用首外内尾法：(x + a)(x + b) = x² + (a + b)x + ab。
ng is the reverse process: pull out the highest common factor, e.g. 6x² + 9x = 3x(2x + 3).
因式分解是展开的逆过程：提取最大公因式，例如 6x² + 9x = 3x(2x + 3)。
Factorising quadratics of the form x² + bx + c: find two numbers that multiply to c and add to b.
分解形如 x² + bx + c 的二次式：找出两个数，乘积为 c，和为 b。
If a ≠ 1 in ax² + bx + c, use the 'ac method' or systematic trial.
若二次项系数 a ≠ 1，使用“ac 方法”或系统尝试。
The difference of two squares: a² − b² = (a + b)(a − b).
平方差公式：a² − b² = (a + b)(a − b)。
3. Algebra: Quadratic Formula & Completing the Square | 代数：二次公式与配方法
r a quadratic equation ax² + bx + c = 0, the solutions are given by the quadratic formula:
对于二次方程 ax² + bx + c = 0，解由二次公式给出：
x = (−b ± √b² − 4ac) / 2a
re of the roots: Δ > 0 → two real roots, Δ = 0 → one repeated root, Δ < 0 → no real roots.
判别式 Δ = b² − 4ac 决定根的性质：Δ > 0 → 两个不等实根，Δ = 0 → 一个重根，Δ < 0 → 无实根。
Completing the square rewrites a quadratic as a(x + p)² + q:
配方法把二次式写成 a(x + p)² + q 的形式：
rt with x² + bx, add and subtract (b/2)²: x² + bx + (b/2)² − (b/2)² = (x + b/2)² − (b/2)².
从 x² + bx 开始，加上并减去 (b/2)²：x² + bx + (b/2)² − (b/2)² = (x + b/2)² − (b/2)²。
This form reveals the vertex of a parabola and makes it easy to solve equations.
这一形式能显示抛物线的顶点，也便于解方程。
4. Equations of Straight Lines | 直线方程
uation of a straight line is y = mx + c, where m is the gradient and c is the y-intercept.
直线的一般方程为 y = mx + c，其中 m 是斜率，c 是 y 轴截距。
Gradient between two points (x₁, y₁) and (x₂, y₂): m = (y₂ − y₁) / (x₂ − x₁).
两点 (x₁, y₁) 和 (x₂, y₂) 间的斜率：m = (y₂ − y₁) / (x₂ − x₁)。
Parallel lines have equal gradients: m₁ = m₂.
平行直线斜率相等：m₁ = m₂。
Perpendicular lines satisfy m₁ × m₂ = −1.
互相垂直的直线满足 m₁ × m₂ = −1。
The equation of a line through a given point (x₁, y₁) with gradient m: y − y₁ = m(x − x₁).
过已知点 (x₁, y₁) 且斜率为 m 的直线方程：y − y₁ = m(x − x₁)。
To find the midpoint of two points: M = ((x₁ + x₂)/2, (y₁ + y₂)/2).
两点间的中点：M = ((x₁ + x₂)/2, (y₁ + y₂)/2)。
5. Graphs: Transformations of Functions | 图形：函数变换
For a function f(x):
对于函数 f(x)：
f(x) + a is a vertical translation by a units upward.
f(x) + a 表示向上平移 a 个单位。
f(x + a) is a horizontal translation by −a units (left if a > 0).
f(x + a) 表示水平平移 −a 个单位（a > 0 时向左平移）。
−f(x) reflects the graph in the x-axis.
−f(x) 将图像关于 x 轴反射。
f(−x) reflects the graph in the y-axis.
f(−x) 将图像关于 y 轴反射。
af(x) with a > 1 is a vertical stretch by factor a; 0 < a < 1 compresses.
af(x)，a > 1 时沿 y 轴方向拉伸 a 倍；0 < a < 1 时压缩。
f(ax) with a > 1 is a horizontal compression by factor 1/a.
f(ax)，a > 1 时沿 x 轴方向压缩为原来的 1/a。
These transformations apply to linear, quadratic and trigonometric graphs.
这些变换适用于一次函数、二次函数和三角函数的图像。
6. Ratio, Proportion & Rates of Change | 比、比例与变化率
Simplify a ratio by dividing each part by their highest common factor.
通过将各部分除以它们的最大公因数来化简比。
To share a quantity in a given ratio, calculate the total number of parts, then divide.
按给定比例分配数量时，先求出总份数，再计算每一份的量。
Direct proportion: y is directly proportional to x if y ∝ x, i.e. y = kx where k is a constant.
正比例：若 y 与 x 成正比，记作 y ∝ x，即 y = kx，k 为常数。
Inverse proportion: y is inversely proportional to x if y ∝ 1/x, i.e. y = k/x.
反比例：若 y 与 x 成反比，记作 y ∝ 1/x，即 y = k/x。
Rates of change are interpreted as gradients of graphs; for a straight line, rate = Δy/Δx.
变化率可解释为图像斜率；对于直线，变化率 = Δy/Δx。
Speed = distance ÷ time; density = mass ÷ volume; pressure = force ÷ area.
速度 = 距离 ÷ 时间；密度 = 质量 ÷ 体积；压强 = 压力 ÷ 面积。
7. Geometry: Circle Theorems | 几何：圆定理
The angle at the centre is twice the angle at the circumference subtended by the same arc.
同弧所对的圆心角是圆周角的两倍。
Angles in the same segment are equal.
同一弓形内的圆周角相等。
The angle in a semicircle is a right angle (90°).
半圆内的圆周角是直角（90°）。
A radius meets a tangent at 90°.
半径与切线相交成 90°。
The perpendicular from the centre to a chord bisects the chord.
从圆心垂直于弦的直线平分该弦。
Opposite angles of a cyclic quadrilateral sum to 180°.
圆内接四边形的对角互补，和为 180°。
Tangents from an external point are equal in length.
从圆外一点引出的两条切线长度相等。
The alternate segment theorem: the angle between a tangent and a chord equals the angle in the alternate segment.
弦切角定理：切线与弦的夹角等于该弦所对的另一侧圆周角。
These theorems are often combined; always give a reason when solving a proof.
这些定理常组合运用；解题时务必注明理由。
8. Pythagoras' Theorem & Trigonometry | 毕达哥拉斯定理与三角学
In a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides:
在直角三角形中，斜边的平方等于两直角边的平方和：
a² + b² = c²
The three trigonometric ratios:
三个三角比：
sin θ = opposite / hypotenuse
sin θ = 对边 / 斜边
cos θ = adjacent / hypotenuse
cos θ = 邻边 / 斜边
tan θ = opposite / adjacent
tan θ = 对边 / 邻边
For any angle, sin² θ + cos² θ = 1.
对于任意角，有 sin² θ + cos² θ = 1。
Exact trigonometric values to remember:
要求记忆的特殊角精确值：
| 0° | 30° | 45° | 60° | 90° | |
|---|---|---|---|---|---|
| sin | 0 | ½ | 1/√2 | √3/2 | 1 |
| cos | 1 | √3/2 | 1/√2 | ½ | 0 |
| tan | 0 | 1/√3 | 1 | √3 | undefined |
Use right-angled trigonometry to find missing sides and angles, and apply it to bearings and 3-D problems.
利用直角三角比求未知边和角，可以应用于方位角与三维问题。
9. Perimeter, Area & Volume | 周长、面积与体积
Triangle: Area = ½ × base × height.
三角形：面积 = ½ × 底 × 高。
Parallelogram: Area = base × perpendicular height.
平行四边形：面积 = 底 × 高。
Trapezium: Area = ½ × (a + b) × h, where a and b are parallel sides.
梯形：面积 = ½ × (上底 + 下底) × 高。
Circle: Circumference = 2πr or πd; Area = πr².
圆：周长 = 2πr 或 πd；面积 = πr²。
Arc length = (θ/360) × 2πr; Sector area = (θ/360) × πr² (θ in degrees).
弧长 = (θ/360) × 2πr；扇形面积 = (θ/360) × πr²（θ 为角度）。
Prism volume: cross-sectional area × length.
棱柱体积：横截面积 × 长度。
Cylinder: Volume = πr²h; Curved surface area = 2πrh; Total surface area = 2πrh + 2πr².
圆柱：体积 = πr²h；侧面积 = 2πrh；总表面积 = 2πrh + 2πr²。
Pyramid volume = ⅓ × base area × height; Cone volume = ⅓πr²h; Sphere volume = ⁴⁄₃πr³.
棱锥体积 = ⅓ × 底面积 × 高；圆锥体积 = ⅓πr²h；球体积 = ⁴⁄₃πr³。
ace area of a sphere = 4πr²; curved surface area of a cone = πrl, where l is slant height.
球表面积 = 4πr²；圆锥侧面积 = πrl，l 为母线长。
10. Statistics: Averages & Spread | 统计：平均数与离散程度
Mean = sum of all values ÷ number of values.
平均数 = 所有数值之和 ÷ 数据个数。
Median is the middle value when data are ordered; for an even count, it is the mean of the two middle values.
中位数是排序后中间的数；若个数为偶数，则取中间两个数的平均数。
Mode is the most frequent value; range = maximum − minimum.
众数是出现次数最多的值；极差 = 最大值 − 最小值。
For grouped data, use mid-interval values to estimate the mean:
对于分组数据，用组中值估算平均数：
Estimated mean = Σ(f × midpoint) / Σf
Quartiles divide ordered data into four equal parts: lower quartile Q1, median Q2, upper quartile Q3.
四分位数把有序数据分成四等份：下四分位数 Q1、中位数 Q2、上四分位数 Q3。
Interquartile range (IQR) = Q3 − Q1, a measure of spread not affected by outliers.
四分位距 IQR = Q3 − Q1，是一种不受异常值影响的离散度量。
Box plots display minimum, Q1, median, Q3 and maximum.
箱形图展示最小值、Q1、中位数、Q3 和最大值。
11. Probability: Rules & Tree Diagrams | 概率：规则与树状图
of an event = number of favourable outcomes ÷ total number of outcomes, if equally likely.
若结果等可能，事件概率 = 有利结果数 ÷ 总结果数。
0 (impossible) to 1 (certain); sum of all mutually exclusive and exhaustive outcomes = 1.
概率范围：0（不可能）到 1（必然）；互斥且完备事件组概率之和为 1。
For mutually exclusive events A and B: P(A or B) = P(A) + P(B).
互斥事件 A 和 B：P(A 或 B) = P(A) + P(B)。
For independent events: P(A and B) = P(A) × P(B).
独立事件：P(A 和 B) = P(A) × P(B)。
Conditional probability: P(A given B) = P(A and B) / P(B).
条件概率：P(A 给定 B) = P(A 与 B) / P(B)。
Tree diagrams display sequences of events: multiply along branches, add probabilities of different outcomes.
树状图展示事件顺序：沿着分支相乘，不同结果概率相加。
Remember to update probabilities for dependent events on the second set of branches.
注意相关事件在第二层分支上要更新概率。
Expected frequency = probability × number of trials.
期望频数 = 概率 × 试验次数。
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