📚 Year 11 Eduqas Maths: Essential Formulas & Theorems Quick Reference | Year 11 Eduqas 数学公式定理速查手册
This quick reference handbook compiles all the essential formulas, theorems and rules required for the Eduqas GCSE Mathematics examination at Year 11. Use it to revise key concepts across number, algebra, geometry, statistics and probability.
这本速查手册汇集了 Year 11 Eduqas 数学考试所需的所有重要公式、定理和法则。用于复习代数、几何、统计和概率等各个知识点。
1. Number – Key Rules | 数字 – 重要法则
BIDMAS / BODMAS: Operations must be carried out in order – Brackets, Indices, Division and Multiplication (left to right), Addition and Subtraction (left to right).
运算顺序：括号、指数、乘除（从左到右）、加减（从左到右）。
Prime factorisation: Every integer can be expressed as a product of prime factors. Use a factor tree to write numbers in index form, e.g. 60 = 2² × 3 × 5.
质因数分解：每个整数都可表示为质因数的乘积。用因数树写出指数形式，如 60 = 2² × 3 × 5。
Highest Common Factor (HCF): product of the lowest powers of common prime factors. Lowest Common Multiple (LCM): product of the highest powers of all prime factors present.
最大公因数 (HCF)：各数共有质因数最小幂次之积。最小公倍数 (LCM)：各数所有质因数最大幂次之积。
Index laws: aᵐ × aⁿ = aᵐ⁺ⁿ ; aᵐ ÷ aⁿ = aᵐ⁻ⁿ ; (aᵐ)ⁿ = aᵐⁿ ; a⁰ = 1 ; a⁻ⁿ = 1/aⁿ.
指数定律：同底数幂相乘，指数相加；相除指数相减；幂的幂指数相乘；任何非零数的 0 次幂为 1；负指数表示倒数。
Standard form: a × 10ⁿ where 1 ≤ a < 10 and n is an integer. Example: 6.2 × 10⁵.
标准形式：a × 10ⁿ，其中 1 ≤ a < 10，n 为整数。例如 6.2 × 10⁵。
Surds: √(ab) = √a × √b ; √(a/b) = √a / √b ; (√a)² = a. Rationalise denominators by multiplying by the conjugate or the surd itself.
根式：√(ab) = √a × √b ; √(a/b) = √a / √b ; (√a)² = a。通过乘以共轭或根号本身有理化分母。
Upper and lower bounds: When a measurement is given correct to a unit, the upper bound is the given value plus half the unit of accuracy; the lower bound minus half.
上界与下界：若测量值精确到某个单位，上界为给定值加上半个精度单位，下界为减去半个精度单位。
2. Fractions, Decimals & Percentages | 分数、小数与百分数
Fraction arithmetic: a/b + c/d = (ad + bc)/bd ; a/b × c/d = ac/bd ; a/b ÷ c/d = a/b × d/c.
分数运算：a/b + c/d = (ad + bc)/bd ; 乘法：ac/bd ; 除法：乘以倒数。
Convert fraction to decimal: divide numerator by denominator. Decimal to percentage: multiply by 100%. Percentage to fraction: write over 100 and simplify.
分数转小数：用分子除以分母。小数转百分数：乘以 100%。百分数转分数：写成分母 100 后化简。
Percentage change: % change = (difference ÷ original value) × 100%. Reverse percentage: if a price after a 20% reduction is £80, original = 80 ÷ 0.8.
百分数变化：变化百分比 = (差值 ÷ 原值) × 100%。逆向百分数：若降价 20% 后为 £80，原价 = 80 ÷ 0.8。
Simple interest: I = PRT / 100 where P = principal, R = annual rate %, T = time in years.
单利：利息 = 本金 × 年利率 (%) × 年数 / 100。
Compound interest: Amount = P(1 + r/100)ⁿ where r is the annual interest rate and n the number of years.
复利：本利和 = P(1 + r/100)ⁿ，r 为年利率，n 为年数。
Growth and decay: Multiply repeatedly by (1 ± percentage/100) for each time period. Exponential decay typical in half-life problems.
增长与衰减：每个时段乘以 (1 ± 百分比/100)。半衰期问题中常见指数衰减。
3. Ratio & Proportion | 比与比例
Simplify ratios by dividing all parts by their highest common factor. Dividing a quantity in a given ratio: find the value of one share by dividing the total by the sum of the ratio parts.
通过除以各部分的最大公因数化简比。按给定比例分配数量：用总量除以比例之和求出每份，再乘以相应份数。
Direct proportion: y = kx, where k is the constant of proportionality. The graph is a straight line through the origin.
正比例：y = kx，k 为比例常数。图像是过原点的直线。
Inverse proportion: y = k/x, giving a rectangular hyperbola. As x increases, y decreases.
反比例：y = k/x，图像为反比曲线。x 增大时 y 减小。
Scale factors: Enlargement by scale factor k multiplies all lengths by k. Area multiplies by k², volume by k³.
比例因子：以因子 k 放大，所有长度乘以 k，面积乘以 k²，体积乘以 k³。
Best buys and exchange rates: Compare unit price or convert using the given rate. Currency conversion: multiply by the exchange rate.
最佳购买与汇率：比较单价或用给定汇率换算。货币换算：乘以汇率。
4. Algebraic Expressions & Equations | 代数表达式与方程
Expanding brackets: a(b + c) = ab + ac. Double brackets: (x + a)(x + b) = x² + (a + b)x + ab.
展开括号：a(b + c) = ab + ac。双括号：(x + a)(x + b) = x² + (a + b)x + ab。
Factorising: take out the common factor; difference of two squares: a² – b² = (a – b)(a + b). Quadratic trinomial: find two numbers that multiply to ac and add to b for ax² + bx + c.
因式分解：提取公因式；平方差公式：a² – b² = (a – b)(a + b)。二次三项式：对 ax² + bx + c，寻找乘积为 ac 和为 b 的两数。
Solving linear equations: isolate the variable using inverse operations. For inequalities, reverse the inequality sign when multiplying or dividing by a negative number.
解线性方程：用逆运算隔离变量。解不等式时，乘除负数需反转不等号。
Quadratic formula: For ax² + bx + c = 0, x = (-b ± √(b² – 4ac)) / 2a. The discriminant b² – 4ac determines the nature of roots.
二次公式：对于 ax² + bx + c = 0，x = (-b ± √(b² – 4ac)) / 2a。判别式 b² – 4ac 决定根的性质。
Completing the square: x² + bx + c = (x + b/2)² + c – (b/2)². Useful for sketching graphs and solving equations.
配方法：x² + bx + c = (x + b/2)² + c – (b/2)²。常用于画图表和求解方程。
Simultaneous equations: solve by elimination or substitution. Graphical solution is the intersection point of the lines.
联立方程：用消元法或代入法求解。图解法中解为直线交点。
5. Sequences | 数列
Linear sequence (arithmetic): nth term = a + (n – 1)d, where a is the first term and d the common difference.
线性数列（等差数列）：第 n 项 = a + (n – 1)d，a 为首项，d 为公差。
Quadratic sequence: the second difference is constant. The nth term can be written as an² + bn + c. Find a from half the second difference, then solve for b and c.
二次数列：二阶差为常数。第 n 项可写成 an² + bn + c。a 为二阶差的一半，再解 b 和 c。
Geometric sequence: each term is multiplied by a constant ratio r. nth term = arⁿ⁻¹.
等比数列：每一项乘以固定公比 r。第 n 项 = arⁿ⁻¹。
Fibonacci sequence: each term is the sum of the two preceding terms: uₙ = uₙ₋₁ + uₙ₋₂.
斐波那契数列：每一项是前两项之和：uₙ = uₙ₋₁ + uₙ₋₂。
Other sequences: recognisable by patterns, e.g. triangular numbers Tₙ = n(n+1)/2, square numbers n².
其他数列：如三角形数 Tₙ = n(n+1)/2，平方数 n²。
6. Graphs & Functions | 图形与函数
Straight-line graph: y = mx + c, where m is the gradient and c the y-intercept. m = (y₂ – y₁) / (x₂ – x₁).
直线图：y = mx + c，m 为斜率，c 为 y 截距。斜率 = (y₂ – y₁) / (x₂ – x₁)。
Parallel lines have equal gradients. Perpendicular lines: the product of their gradients is –1, i.e. m₁ × m₂ = –1.
平行线斜率相等。垂直线斜率之积为 –1，即 m₁ × m₂ = –1。
Quadratic graph: y = ax² + bx + c is a parabola. If a > 0, it is upright ∪; if a < 0, ∩. The turning point can be found by completing the square.
二次函数图：y = ax² + bx + c 为抛物线。a > 0 开口向上；a < 0 开口向下。顶点可通过配方法求得。
Cubic, reciprocal and exponential graphs: y = x³, y = 1/x, y = kˣ shape features should be known.
三次、反比和指数图：应掌握 y = x³, y = 1/x, y = kˣ 的大致形状。
Transformations: f(x) + a shifts vertically; f(x + a) shifts horizontally left by a; –f(x) reflects in x-axis; f(–x) reflects in y-axis; af(x) stretches vertically by scale factor a; f(ax) stretches horizontally by 1/a.
图形变换：f(x) + a 纵向平移；f(x + a) 向左平移 a；–f(x) 以 x 轴反射；f(–x) 以 y 轴反射；af(x) 纵向拉伸 a 倍；f(ax) 横向压缩 1/a。
7. Angles & Lines | 角与线
Angles around a point sum to 360°. Angles on a straight line sum to 180°. Vertically opposite angles are equal.
绕一点一周的角度和为 360°。直线上的邻角和为 180°。对顶角相等。
Parallel lines: Alternate angles are equal; corresponding angles are equal; interior (co-interior) angles sum to 180°.
平行线：内错角相等；同位角相等；同旁内角之和为 180°。
Bearings are measured clockwise from north, always given as three figures.
方位角从正北顺时针测量，始终用三位数字表示。
Polygon angles: Sum of exterior angles of any convex polygon = 360°. Sum of interior angles = (n – 2) × 180°, where n is the number of sides. Each interior angle of a regular polygon = (n – 2) × 180° / n.
多边形角：任何凸多边形的外角和为 360°。内角和 = (n – 2) × 180°。正多边形每个内角 = (n – 2) × 180° / n。
8. Triangles & Polygons | 三角形与多边形
Types of triangles: equilateral (3 equal sides, 60° each), isosceles (2 equal sides, base angles equal), scalene, right-angled.
三角形类别：等边三角形（三边相等，每角 60°），等腰三角形（两腰相等，底角相等），不等边三角形，直角三角形。
Pythagoras’ theorem: In a right-angled triangle, a² + b² = c², where c is the hypotenuse.
勾股定理：在直角三角形中，两直角边的平方和等于斜边的平方：a² + b² = c²。
Trigonometric ratios: sin θ = opposite / hypotenuse, cos θ = adjacent / hypotenuse, tan θ = opposite / adjacent. Remember SOH CAH TOA.
三角比：sin θ = 对边 / 斜边，cos θ = 邻边 / 斜边，tan θ = 对边 / 邻边。记忆口诀 SOH CAH TOA。
Sine rule: a / sin A = b / sin B = c / sin C. Use for non‑right triangles when given two angles and a side, or two sides and a non‑included angle.
正弦定理：a / sin A = b / sin B = c / sin C。用于已知两角一边或两边一对角的非直角三角形。
Cosine rule: a² = b² + c² – 2bc cos A. Use for two sides and the included angle, or three sides.
余弦定理：a² = b² + c² – 2bc cos A。用于已知两边夹角或三边的情况。
Area of a triangle: ½ ab sin C, where a and b are two sides and C the included angle.
三角形面积：½ ab sin C，a、b 为两边，C 为夹角。
Similar shapes
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