📚 Year 11 OCR Mathematics: Ultimate Formula & Theorem Quick Reference Guide | Year 11 OCR 数学：公式定理速查手册
This quick-reference handboook compiles every essential formula, identity, and theorem needed for the Year 11 OCR GCSE Mathematics course. Whether you are consolidating Higher-tier topics or revising core concepts, this guide organises the syllabus into twelve manageable sections, with bilingual explanations to reinforce your understanding and recall under exam pressure.
这本速查手册收录了 Year 11 OCR GCSE 数学课程中所有必备公式、恒等式与定理。无论你是在巩固高阶内容还是复习基础概念，本指南将考纲划分为十二个易于掌握的部分，并以双语讲解加深理解，帮助你在考试压力下快速回忆。
1. Number Operations and Surds | 数运算与根式
When simplifying surds, remember that √ab = √a × √b and √a/b = √a / √b, provided a and b are positive. Always express surds in their simplest form, e.g. √50 = 5√2.
化简根式时请记住：√ab = √a × √b，√a/b = √a / √b，其中 a、b 为正数。务必将根式化为最简形式，例如 √50 = 5√2。
Rationalising the denominator removes a surd from the bottom of a fraction. For a single surd denominator of the form √b, multiply numerator and denominator by √b. For a binomial such as a + √b, use its conjugate a – √b.
分母有理化可消去分数下方的根号。对于形如 √b 的单项根式分母，将分子分母同乘 √b；对于形如 a + √b 的二项式分母，则使用其共轭式 a – √b 相乘。
ⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1, and a⁻ⁿ = 1/aⁿ. Fractional indices link roots and powers: a1/n = ⁿ√a, am/n = (ⁿ√a)ᵐ = ⁿ√aᵐ.
整数及分数指数运算法则是重要工具。核心法则：aᵐ × aⁿ = aᵐ⁺ⁿ，aᵐ ÷ aⁿ = aᵐ⁻ⁿ，(aᵐ)ⁿ = aᵐⁿ，a⁰ = 1，a⁻ⁿ = 1/aⁿ。分数指数将根式与幂联系起来：a1/n = ⁿ√a，am/n = (ⁿ√a)ᵐ = ⁿ√aᵐ。
and n is an integer. Calculations with standard form rely heavily on the index laws above.
标准形式将极大或极小的数字表示为 a × 10ⁿ，其中 1 ≤ a < 10 且 n 为整数。标准形式下的计算高度依赖上述指数法则。
√ab = √a × √b
aᵐ × aⁿ = aᵐ⁺ⁿ
2. Algebraic Manipulation & Identities | 代数运算与恒等式
brackets (x + a)(x + b), the result is x² + (a + b)x + ab. When dealing with trinomials, systematically multiply each term in the first bracket by every term in the second.
展开括号运用分配律：a(b + c) = ab + ac。对于双重括号 (x + a)(x + b)，结果为 x² + (a + b)x + ab。处理三项式时，需将第一个括号的每一项与第二个括号的每一项相乘。
that multiply to c and add to b. The difference of two squares is instantly factorable: a² – b² = (a + b)(a – b).
因式分解则提取公因子或逆向展开。对于二次式 x² + bx + c，寻找两个相乘得 c、相加得 b 的数字。平方差公式可直接分解：a² – b² = (a + b)(a – b)。
Algebraic identities hold true for all values of the variables. For example, (a + b)² ≡ a² + 2ab + b² and (a – b)² ≡ a² – 2ab + b². These are frequently used when completing the square or simplifying rational expressions.
代数恒等式对所有变量取值均成立。例如 (a + b)² ≡ a² + 2ab + b²，以及 (a – b)² ≡ a² – 2ab + b²。这些恒等式在配方或化简有理式时经常使用。
To complete the square for x² + bx + c, write it as (x + b/2)² – (b/2)² + c. This form reveals the vertex of a parabola and helps solve equations when factorisation is not straightforward.
为 x² + bx + c 配方时，可写成 (x + b/2)² – (b/2)² + c。该形式揭示抛物线的顶点，并在因式分解不易时帮助解方程。
(a ± b)² ≡ a² ± 2ab + b²
a² – b² ≡ (a + b)(a – b)
3. Linear Equations & Inequalities | 一元一次方程与不等式
A linear equation can be solved by performing the same operation on both sides to isolate the unknown. When the unknown appears on both sides, collect like terms first. Always check your solution by substituting back into the original equation.
解一元一次方程时，可对等式两边施行相同的运算以分离未知数。若两边均含未知数，先合并同类项。务必代回原方程检验解。
Linear inequalities are solved similarly to equations, but with one critical difference: multiplying or dividing by a negative number reverses the inequality sign. On a number line, open circles indicate strict inequalities (< or >) and closed circles indicate ≤ or ≥.
一次不等式的解集表示在数轴上：用空心圆圈表示严格不等号（< 或 >），实心圆圈表示包含等号（≤ 或 ≥）。
Inequalities can also be displayed using set‑builder notation, e.g. {x : x > 3} or in interval notation. For combined inequalities such as –2 < x ≤ 4, the solution set includes all values greater than –2 and up to and including 4.
不等式也可用构建集合记号表示，如 {x : x > 3}，或使用区间记号。对于联立不等式如 –2 < x ≤ 4，解集包含所有大于 –2 且小于等于 4 的数。
If a < b and c > 0 then ac < bc
If a < b and c < 0 then ac > bc
4. Quadratic Expressions & Equations | 二次表达式与方程
actorising, completing the square, or using the quadratic formula. The discriminant Δ = b² – istinct real roots if Δ > 0, one repeated root if Δ = 0, and no real roots if Δ < 0.
二次方程 ax² + bx + c = 0 可用因式分解、配方法或求根公式求解。判别式 Δ = b² – 4ac 决定根的性质：Δ > 0 时有两个不等实根，Δ = 0 时有一个重根，Δ < 0 时无实根。
The quadratic formula is mandatory for any quadratic that does not factorise neatly:
当二次式不能顺利分解时，必须使用二次求根公式：
x = (–b ± √b² – 4ac) / 2a
The graph of y = ax² + bx + c is a parabola. Its vertex has x-coordinate – smiles (minimum point); if a < 0 it frowns (maximum point). The y-intercept is (0, c).
y = ax² + bx + c 的图像为抛物线。顶点横坐标为 –b/(2a)；当 a > 0 时抛物线开口向上（最小值点），a < 0 时开口向下（最大值点）。y 轴截距为 (0, c)。
Factorising a quadratic ax² + bx + c when a ≠ 1 often uses the 'ac method': find two numbers that multiply to ac and add to b, then split the middle term and factor by grouping.
当 a ≠ 1 时分解二次三项式常使用“ac 法”：寻找相乘为 ac、相加为 b 的两个数，然后拆分中项并分组分解。
5. Simultaneous Equations & Graphs | 联立方程与图像
Two linear equations can be solved simultaneously by elimination, substitution, or graphically. The solution is the point of intersection of the two straight lines. If the lines are parallel, there is no solution; if they are the same line, there are infinitely many solutions.
两个线性方程可用消元法、代入法或图像法联立求解。解即为两条直线交点的坐标。若两直线平行则无解；若重合则有无穷多解。
When one equation is quadratic and the other linear, substitution yields a quadratic which can have zero, one, or two intersection points. Always substitute the linear expression into the nonlinear equation.
若一个方程为二次，另一个为一次，通过代入得到一个二次方程。该方程组可能有零个、一个或两个交点。务必将线性表达式代入非线性方程。
ions and reading intersection points. For instance, the solutions of f(x) = g(x) are the x‑coordinates where the graphs of y = f(x) and y = g(x) meet.
方程图像解法是绘制相应函数并读取交点。例如 f(x) = g(x) 的解即为 y = f(x) 与 y = g(x) 图像交点的横坐标。
Simultaneous inequalities define a region on the coordinate plane. Solid lines are used for ≤ or ≥, dashed lines for < or >; shade the region that satisfies all constraints.
联立不等式在坐标系上界定一个区域。实线表示包含等号（≤ 或 ≥），虚线表示严格不等号（< 或 >）；对满足所有约束的区域进行着色。
6. Sequences | 数列
An arithmetic sequence has a constant common difference d. The nth term is uₙ = a + (n – 1)d, where a is the first term. The sum of the first n terms is Sₙ = n/2 [2a + (n – 1)d] or Sₙ = n/2 (a + l), where l is the last term.
等差数列具有固定的公差 d。第 n 项为 uₙ = a + (n – 1)d，其中 a 为首项。前 n 项和为 Sₙ = n/2 [2a + (n – 1)d] 或 Sₙ = n/2 (a + l)，l 为末项。
ratio r. The nth term is uₙ = arⁿ⁻¹. The sum of the first n terms (for r ≠ 1) is Sₙ = a(1 – rⁿ) / (1 – r). For |r| < 1, an infinite sum exists: S∞ = a / (1 – r).
等比数列具有固定的公比 r。第 n 项为 uₙ = arⁿ⁻¹。当 r ≠ 1 时前 n 项和为 Sₙ = a(1 – rⁿ) / (1 – r)。当 |r| < 1 时，无穷级数和存在：S∞ = a / (1 – r)。
Sequences may also be defined by a term‑to‑ es. Recognising patterns in differences helps identify whether a sequence is linear or quadratic.
数列也可通过递推公式定义，如斐波那契数列 uₙ₊₂ = uₙ₊₁ + uₙ 并给定初值。通过观察差分级数有助于判断数列是线性还是二次型。
Arithmetic: uₙ = a + (n – 1)d
Geometric: uₙ = arⁿ⁻¹
7. Ratio, Proportion & Rates | 比、比例与率
Ratios compare quantities and can be simplified like fractions by dividing by common factors. The unitary method uses a value for one unit to scale up or down proportionally.
比用于比较数量，可像分数一样除以公因子化简。归一法先求出一个单位的值，再按比例放大或缩小。
e through the origin. Inverse proportion means y = k/x, producing a rectangular hyperbola.
正比例指 y = kx，其中 k 为比例常数。正比例关系的图像是过原点的直线。反比例指 y = k/x，其图像为双曲线。
Percentage change = (actual change ÷ original value) × 100% incipal and r is the annual interest rate. Depreciation works similarly with a minus sign.
百分数变化 = (实际变化 ÷ 原值) × 100%。对于复利计算，n 年后总额为 A = P(1 + r/100)ⁿ，其中 P 为本金，r 为年利率。贬值计算同理，但符号为减。
lume, pressure = force ÷ area. Rearranging these formulas enables any unknown to be found.
速度、密度与压强属于复合单位：速度 = 距离 ÷ 时间，密度 = 质量 ÷ 体积，压强 = 力 ÷ 面积。通过公式变形可求任何未知量。
y = kx (direct)
y = k/x (inverse)
8. Geometry & Angle Theorems | 几何与角度定理
Angles on a straight line sum to 180°, around a point sum to 360°. Vertically opposite angles are equal. Alternate angles, corresponding angles, and co‑interior angles arise when parallel lines are cut by a transversal: alternate angles are equal, corresponding angles are equal, co‑interior angles sum to 180°.
直线上的角之和为 180°，绕一点的周角为 360°。对顶角相等。平行线被截线所截时产生内错角、同位角与同旁内角：内错角相等，同位角相等，同旁内角互补（和为 180°）。
sum of the two opposite interior angles. For any polygon, the sum of interior angles is (n – 2) × 180°, and the sum of exterior angles is always 360°.
三角形内角和为 180°。外角等于两对内角之和。对于任意多边形，内角和 = (n – 2) × 180°，外角和始终为 360°。
Circle theorems are crucial: the angle at the centre is twice the angle at the circumference; angles in the same segment are equal; the angle in a semicircle is 90°; opposite angles of a cyclic quadrilateral sum to 180°; a tangent meets a radius at 90°; tangents from an external point are equal in length; the alternate segment theorem states that the angle between a tangent and a chord equals the angle in the alternate segment.
圆定理至关重要：圆心角是同一弧所对圆周角的两倍；同弓形上的圆周角相等；半圆上的圆周角是 90°；圆内接四边形对角互补；切线与半径相交成 90°；从圆外一点引出的两条切线长相等；弦切角定理指出切线与弦的夹角等于弦所对另一侧的圆周角。
Congruence and similarity conditions determine when shapes are identical or proportional. SSS, SAS, ASA, RHS are congruence tests; AA is sufficient for similarity.
全等与相似条件判断图形相同或成比例。SSS、SAS、ASA、RHS 是全等判据；AA 足以判定相似。
9. Pythagoras & Trigonometry | 毕达哥拉斯与三角学
In a right‑angled triangle, Pythagoras' he hypotenuse. This theorem is used to find missing side lengths and to verify if a triangle is right‑angled.
在直角三角形中，毕达哥拉斯定理指出 a² + b² = c²，其中 c 为斜边。该定理用于求未知边长或判断三角形是否为直角三角形。
site/adjacent. Memorise exact values for 0°, 30°, 45°, 60°, 90°: sin 30° = ½, sin 45° = 1/√2, sin 60° = √3/2; cos 30° = √3/2, cos 45° = 1/√2, cos 60° = ½; tan 30° = 1/√3, tan 45° = 1, tan 60° = √3.
直角三角形中的三角比：sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。牢记 0°、30°、45°、60°、90° 的精确值：sin 30° = ½，sin 45° = 1/√2，sin 60° = √3/2；cos 30° = √3/2，cos 45° = 1/√2，cos 60° = ½；tan 30° = 1/√3，tan 45° = 1，tan 60° = √3。
For non‑right‑ c/sin C, or its flipped form sin A/a = sin B/b = sin C/c. The cosine rule is a² = b² + c² – 2bc cos A, which generalises Pythagoras' theorem.
对于斜三角形，使用正弦定理：a/sin A = b/sin B = c/sin C，或其倒置形式 sin A/a = sin B/b = sin C/c。余弦定理为 a² = b² + c² – 2bc cos A，是毕达哥拉斯定理的一般形式。
sin C. This formula is especially useful when two sides and the included angle are known.
用三角学求三角形面积：面积 = ½ ab sin C。当已知两边及其夹角时该公式尤为实用。
a² + b² = c²
sin θ = opposite / hypotenuse
a/sin A = b/sin B = c/sin C
10. Area, Volume & Surface Area | 面积、体积与表面积
b)h. For a sector of a circle, arc length = (θ/360) × 2πr, area of sector = (θ/360) × πr².
关键平面图形：圆面积 = πr²，周长 = 2πr 或 πd。梯形面积 = ½(a + b)h。扇形弧长 = (θ/360) × 2πr，扇形面积 = (θ/360) × πr²。
3D shapes: volume of a prism = area of cross‑ ace area = πrl, where l is the slant height. Sphere volume = 4/3 πr³, surface area = 4πr².
立体图形：棱柱体积 = 横截面积 × 长。圆柱体积 = πr²h，表面积 = 2πrh + 2πr²。棱锥或圆锥体积 = ⅓ × 底面积 × 高。圆锥侧面积 = πrl，其中 l 为斜高。球体积 = 4/3 πr³，表面积 = 4πr²。
For composite solids, break the shape into simpler parts, calculate each volume or surface area separately, and add or subtract as needed. Careful attention must be paid to hidden surfaces when working with combined 3D objects.
对于组合体，将其拆分为简单部分，分别计算各部分的体积或表面积，再根据需要相加或相减。处理复合立体时，需注意隐藏面。
Similar shapes have volumes and surface areas proportional to the cube and square of the linear scale factor, respectively. If the length scale factor is k, area scale factor is k², volume scale factor is k³.
相似形体的体积比等于线性比例因子的立方，面积比等于线性比例因子的平方。若长度比是 k，则面积比是 k²，体积比是 k³。
Circle: C = 2πr, A = πr²
Sphere: V = 4/3 πr³, S = 4πr²
11. Probability | 概率
umber of favourable outcomes / total number of possible outcomes, assuming all outcomes are equally likely. Probabilities lie between 0 and 1.
若所有结果等可能，事件概率为 P(A) = 有利结果数 / 所有可能结果总数。概率值介于 0 与 1 之间。
The complement rule: P(not A) = 1 – events A and B, P(A or B) = P(A) + P(B). For independent events, P(A and B) = P(A) × P(B).
互补规则：P(非 A) = 1 – P(A)。互斥事件 A 与 B 满足 P(A 或 B) = P(A) + P(B)。独立事件满足 P(A 且 B) = P(A) × P(B)。
Tree diagrams visually show combined events. Multiply along branches for 'and' probabilities; add outcomes at the ends for 'or' probabilities. Conditional probability P(A|B) = P(A and B) / P(B), where the sample space is reduced to event B.
树状图直观显示复合事件。沿分支相乘得“且”概率；将终点结果相加得“或”概率。条件概率 P(A|B) = P(A 且 B) / P(B)，此时的样本空间缩小为事件 B。
Venn diagrams and two‑way tables are alternative ways to organise outcomes and apply the addition rule for non‑mutually exclusive events: P(A or B) = P(A) + P(B) – P(A and B).
范恩图与双向表是整理结果的替代方法，用于处理非互斥事件的加法规则：P(A 或 B) = P(A) + P(B) – P(A 且 B)。
P(A|B) = P(A ∩ B) / P(B)
P(A ∪ B) = P(A) + P(B) – P(A ∩ B)
12. Statistics | 统计
it is the mean of the two central values. Mode is the most frequent value. Range = maximum – minimum.
集中趋势的度量：平均数 = 数据总和 ÷ 数据个数。中位数是排序后居中的数值；若数据个数为偶数，则取居中两数的平均数。众数是出现频率最高的值。极差 = 最大值 – 最小值。
quency; the estimated mean uses midpoints of class intervals: mean ≈ Σ(f × midpoint) / Σf.
对于连续分组数据，次数密度最高的组即为众数组。中位数所在组通过累计频率来寻找；估算平均数使用组中值：平均数 ≈ Σ(f × 组中值) / Σf。
Scatter graphs show correlation between two variables. A line of best fit can be drawn by eye and used to make predictions. Interpretation must distinguish between correlation and causation.
散点图展示两变量间的相关关系。最佳拟合线可凭目测画出
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