📚 Year 10 OCR Maths: Formula & Theorem Quick Reference Handbook | Year 10 OCR 数学：公式定理速查手册
This quick reference handbook brings together all the essential formulas, identities and theorems covered in the Year 10 OCR mathematics curriculum. Use it to review key facts before tests, to support your homework, or to build a solid foundation for GCSE success. Each section pairs a concise English explanation with a corresponding Chinese translation so you can master both the concept and the terminology.
这本速查手册汇集了 Year 10 OCR 数学课程中的所有核心公式、恒等式和定理。你可以在考试前复习关键知识点，辅助完成作业，或为 GCSE 的成功打下坚实基础。每个小节都配有简洁的英文解释和对应的中文翻译，让你同时掌握概念和术语。
1. Algebraic Manipulation and Key Identities | 代数运算与重要恒等式
The distributive law tells us how to multiply a single term by a bracket: a(b + c) = ab + ac. It works for subtraction as well: a(b – c) = ab – ac.
分配律告诉我们单项式乘以括号的规则：a(b + c) = ab + ac。减法同样适用：a(b – c) = ab – ac。
When we expand two brackets, we multiply every term in the first bracket by every term in the second: (a + b)(c + d) = ac + ad + bc + bd. A special case is the difference of two squares: (a + b)(a – b) = a² – b².
展开两个括号时，要用第一个括号里的每一项乘以第二个括号里的每一项：(a + b)(c + d) = ac + ad + bc + bd。一个特例是平方差公式：(a + b)(a – b) = a² – b²。
The perfect square identities help factorise or expand quickly: (a + b)² = a² + 2ab + b² and (a – b)² = a² – 2ab + b². Recognising these patterns makes solving quadratic equations much easier.
完全平方恒等式有助于快速因式分解或展开：(a + b)² = a² + 2ab + b² 以及 (a – b)² = a² – 2ab + b²。识别这些模式能让解二次方程变得简单许多。
To factorise a quadratic expression like x² + bx + c, look for two numbers that multiply to give c and add to give b. If the coefficient of x² is not 1, use the grouping method or the ‘ac’ method.
因式分解形如 x² + bx + c 的二次式时，需找出乘积为 c、和为 b 的两个数。如果 x² 的系数不是 1，则可使用分组法或 ‘ac’ 方法。
2. Linear Equations and Inequalities | 一次方程与不等式
A linear equation can always be written in the form ax + b = 0. To solve it, perform the same operation on both sides of the equals sign until the variable is isolated. Always check your solution by substituting it back into the original equation.
一次方程总可以写成 ax + b = 0 的形式。求解时，在等号两边进行相同的运算，直至分离出未知数。最后务必将解代回原方程进行检验。
When rearranging inequalities, remember that multiplying or dividing both sides by a negative number reverses the inequality sign: if x > 3, then -x < -3.
处理不等式变形时，切记两边同乘或除以负数会反转不等号方向：若 x > 3，则 -x < -3。
To solve simultaneous linear equations, use the elimination method (add or subtract to remove one variable) or the substitution method (rearrange one equation and substitute into the other). Graphical solutions correspond to the point where the two lines intersect.
求解联立一次方程组时，可用消元法（相加或相减消去一个未知数）或代入法（将一个方程变形后代入另一个方程）。图像解则对应两条直线交点的坐标。
3. Quadratic Equations and Their Graphs | 二次方程及其图像
Quadratic formula: x = [-b ± √(b² – 4ac)] / (2a)
二次公式：x = [-b ± √(b² – 4ac)] / (2a)
The expression under the square root, b² – 4ac, is called the discriminant (Δ). If Δ > 0, the equation has two distinct real roots; if Δ = 0, there is exactly one real root (repeated); if Δ < 0, there are no real roots.
根号下的式子 b² – 4ac 称为判别式 (Δ)。若 Δ > 0，方程有两个相异实根；若 Δ = 0，有一个实根（重根）；若 Δ < 0，则没有实根。
Every quadratic graph is a parabola. For y = ax² + bx + c, the axis of symmetry is the vertical line x = -b/(2a), and the turning point occurs at that x-value. When a > 0, the parabola opens upwards (∪); when a < 0, it opens downwards (∩).
每一个二次方程的图像都是一条抛物线。对于 y = ax² + bx + c，对称轴是竖直线 x = -b/(2a)，转折点也出现在该 x 值处。当 a > 0 时抛物线开口向上；当 a < 0 时开口向下。
To complete the square, rewrite ax² + bx + c as a(x + p)² + q. This form reveals the vertex (-p, q) immediately and is essential for sketching graphs and solving equations.
配方法即把 ax² + bx + c 写成 a(x + p)² + q 的形式。这种形式直接给出顶点 (-p, q)，对绘制图像和求解方程至关重要。
4. Sequences | 数列
An arithmetic sequence has a constant difference between consecutive terms. The nth term is given by uₙ = a + (n – 1)d, where a is the first term and d is the common difference.
等差数列相邻两项的差为常数。第 n 项公式为 uₙ = a + (n – 1)d，其中 a 为首项，d 为公差。
A geometric sequence has a constant ratio between consecutive terms. The nth term is uₙ = arⁿ⁻¹, where a is the first term and r is the common ratio.
等比数列相邻两项的比值为常数。第 n 项公式为 uₙ = arⁿ⁻¹，其中 a 为首项，r 为公比。
Sequences can also be defined recursively, giving the first term and a rule to find the next. For example, the Fibonacci-style sequence u₁ = 1, u₂ = 1, uₙ₊₂ = uₙ₊₁ + uₙ generates 1, 1, 2, 3, 5, 8, …
数列也可以通过递推关系定义，即给出首项和推导下一项的规则。例如，类斐波那契数列 u₁ = 1, u₂ = 1, uₙ₊₂ = uₙ₊₁ + uₙ 生成 1, 1, 2, 3, 5, 8, …
5. Coordinate Geometry and Straight Lines | 坐标几何与直线
Gradient formula: m = (y₂ – y₁) / (x₂ – x₁)
斜率公式：m = (y₂ – y₁) / (x₂ – x₁)
The distance between two points (x₁, y₁) and (x₂, y₂) is √[(x₂ – x₁)² + (y₂ – y₁)²]. The midpoint has coordinates ((x₁ + x₂)/2, (y₁ + y₂)/2).
两点 (x₁, y₁) 与 (x₂, y₂) 间的距离为 √[(x₂ – x₁)² + (y₂ – y₁)²]。中点坐标为 ((x₁ + x₂)/2, (y₁ + y₂)/2)。
The equation of a straight line can be written as y = mx + c, where m is the gradient and c is the y-intercept. An alternative form is y – y₁ = m(x – x₁), which is useful when you know a point on the line and the gradient.
直线方程可写为 y = mx + c，其中 m 为斜率，c 为 y 轴截距。另一种形式是 y – y₁ = m(x – x₁)，当已知直线上一点和斜率时十分有用。
Two lines are parallel if their gradients are equal (m₁ = m₂). Two lines are perpendicular if the product of their gradients is -1 (m₁ × m₂ = -1).
若两直线斜率相等 (m₁ = m₂)，则它们平行。若两直线斜率之积为 -1 (m₁ × m₂ = -1)，则它们互相垂直。
6. Pythagoras’ Theorem and Trigonometry | 勾股定理与三角学
Pythagoras’ theorem: a² + b² = c² (c is the hypotenuse)
勾股定理：a² + b² = c²（c 为斜边）
In a right-angled triangle, the trigonometric ratios are defined as sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent. SOH CAH TOA is a helpful memory aid.
在直角三角形中，三角比的定义为 sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。SOH CAH TOA 是帮助记忆的口诀。
You should memorise the exact values: sin 30° = ½, sin 45° = √2/2, sin 60° = √3/2; cos 30° = √3/2, cos 45° = √2/2, cos 60° = ½; tan 30° = √3/3, tan 45° = 1, tan 60° = √3.
你需要牢记精确值：sin 30° = ½, sin 45° = √2/2, sin 60° = √3/2；cos 30° = √3/2, cos 45° = √2/2, cos 60° = ½；tan 30° = √3/3, tan 45° = 1, tan 60° = √3。
Angles of elevation and angles of depression are measured from the horizontal. They appear in many real-life trigonometry problems. Drawing a clear diagram will always help you set up the correct ratio.
仰角和俯角都是相对于水平线测量的。它们出现在许多实际三角学问题中。绘制清晰的示意图总能帮助你列出正确的比例。
An important trigonometric identity is sin² θ + cos² θ = 1. This relationship holds for any angle and can be used to find one ratio when the other is known.
一个重要三角恒等式是 sin² θ + cos² θ = 1。这个关系对任意角都成立，可用于在已知一种比值时求另一种比值。
7. Perimeter, Area, Surface Area and Volume | 周长、面积、表面积与体积
Circles: circumference = 2πr, area = πr². For a sector with angle θ°, arc length = (θ/360) × 2πr and sector area = (θ/360) × πr².
圆：周长 = 2πr，面积 = πr²。对于圆心角为 θ° 的扇形，弧长 = (θ/360) × 2πr，扇形面积 = (θ/360) × πr²。
Triangle area = ½ × base × height; parallelogram area = base × height; trapezium area = ½(a + b)h, where a and b are the parallel sides.
三角形面积 = ½ × 底 × 高；平行四边形面积 = 底 × 高；梯形面积 = ½(a + b)h，其中 a 和 b 为平行边。
Prism volume = area of cross-section × length. Pyramid volume = ⅓ × base area × height. Cone volume = ⅓πr²h; curved surface area = πrl, where l is slant height. Sphere volume = ⁴⁄₃πr³; surface area = 4πr².
棱柱体积 = 横截面积 × 长。棱锥体积 = ⅓ × 底面积 × 高。圆锥体积 = ⅓πr²h；侧面积 = πrl，其中 l 是母线长。球体积 = ⁴⁄₃πr³；表面积 = 4πr²。
8. Angles, Polygons and Circle Theorems | 角、多边形与圆定理
The sum of interior angles of an n-sided polygon is (n – 2) × 180°. The sum of exterior angles (one at each vertex) is always 360°, regardless of the number of sides.
n 边形的内角和为 (n – 2) × 180°。外角和（每个顶点取一个外角）恒为 360°，与边数无关。
When a transversal crosses parallel lines, corresponding angles are equal, alternate interior angles are equal, and co-interior (allied) angles sum to 180°. Vertically opposite angles are also equal.
当一条截线与平行线相交时，同位角相等，内错角相等，同旁内角之和为 180°。对顶角也相等。
Circle theorems: The angle at the centre is twice the angle at the circumference standing on the same arc. The angle in a semicircle is always 90°. Angles in the same segment are equal. The sum of opposite angles in a cyclic quadrilateral is 180°.
圆定理：圆心角等于同一弧所对的圆周角的两倍。半圆上的圆周角恒为 90°。同弧所对的圆周角相等。圆内接四边形对角互补，和为 180°。
9. Probability | 概率
The probability of an event A, P(A) = number of favourable outcomes / total number of possible outcomes. All probabilities lie between 0 and 1 inclusive.
事件 A 的概率 P(A) = 有利结果数 / 所有可能结果总数。所有概率值都在 0 到 1 之间（含端点）。
For mutually exclusive events, P(A or B) = P(A) + P(B). If events are not mutually exclusive, subtract the intersection: P(A or B) = P(A) + P(B) – P(A and B).
对于互斥事件，P(A 或 B) = P(A) + P(B)。若事件不互斥，则需减去交集部分：P(A 或 B) = P(A) + P(B) – P(A 且 B)。
For independent events, P(A and B) = P(A) × P(B). Tree diagrams help organise the probabilities of sequential events; multiply along
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