Year 11 Eduqas Further Maths Formula & Theorems Quick Reference | Year 11 Eduqas 进阶数学：公式定理速查手册

📚 Year 11 Eduqas Further Maths Formula & Theorems Quick Reference | Year 11 Eduqas 进阶数学：公式定理速查手册

This quick reference guide gathers the essential formulas, theorems and algebraic techniques required for the Eduqas Level 2 Certificate in Further Mathematics. Each section presents key results with a short explanation in English followed by its Chinese equivalent, helping you to memorise and apply the concepts efficiently.

本速查手册汇总了 Eduqas 进阶数学（Further Mathematics）等级 2 证书考试所需的核心公式、定理和代数技巧。每个小节用英文给出要点说明，紧跟着对应的中文解释，帮助你高效记忆并灵活应用。

1. Algebraic Expansion & Factorisation | 代数展开与因式分解

+b)(a−b) = a² − b². These underpin completing the square and many simplification problems.

熟记三个基本二项式展开：(a+b)² = a² + 2ab + b²，(a−b)² = a² − 2ab + b²，以及平方差公式 (a+b)(a−b) = a² − b²。它们是完全平方配方法和许多化简题的基础。

ic ax²+bx+c where a=1, find two numbers that multiply to c and add to b. When a≠1, use the 'ac' then split the middle term and factor by grouping. Always check for a common factor first.

对首项系数 a=1 的二次式 ax²+bx+c，找两个数积为 c 和为 b；若 a≠1，使用“ac 法”：找两个数积为 ac 和为 b，分解中项后分组提取公因式。注意先提取公因子。

Completing the square rewrites x² + bx + c as (x + b/2)² − (b/2)² + c.

配方法将 x² + bx + c 变形为 (x + b/2)² − (b/2)² + c。

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2. Surds & Indices | 根式与指数

Surd simplification relies on √ab = √a × √b and √a/b = √a / √b. Always express surds in simplest form, e.g. √48 = √16×3 = 4√3. Rationalising denominators: for 1/√a multiply numerator and denominator by √a; for 1/(a+√b) use the conjugate a−√b.

根式化简依据 √ab = √a × √b 和 √a/b = √a / √b。始终化为最简根式，如 √48 = 4√3。分母有理化：1/√a 型分子分母同乘 √a；1/(a+√b) 型则用共轭根式 a−√b。

ᵐ ÷ xⁿ = xᵐ⁻ⁿ, (xᵐ)ⁿ = xᵐⁿ. Negative and fractional indices follow x⁻ⁿ = 1/xⁿ and x¹/ⁿ = ⁿ√x, xᵐ/ⁿ = ⁿ√xᵐ or (ⁿ√x)ᵐ.

指数运算法则：xᵐ × xⁿ = xᵐ⁺ⁿ，xᵐ ÷ xⁿ = xᵐ⁻ⁿ，(xᵐ)ⁿ = xᵐⁿ。负指数与分数指数定义为 x⁻ⁿ = 1/xⁿ，x¹/ⁿ = ⁿ√x，xᵐ/ⁿ = ⁿ√xᵐ = (ⁿ√x)ᵐ。

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3. Quadratic Functions & the Discriminant | 二次函数与判别式

The quadratic formula solves ax²+bx+c = 0:

x = (−b ± √b² − 4ac) / (2a)

二次公式解 ax²+bx+c = 0：x = (−b ± √b² − 4ac) / (2a)。

roots (complex). For the graph y = ax²+bx+c, if a>0 the parabola opens upwards and has a minimum point; a<0 opens downwards with a maximum.

判别式 Δ = b²−4ac 决定根的性质：Δ > 0 有两个不等实根，Δ = 0 有一个重根，Δ < 0 无实根。对二次函数图像 y = ax²+bx+c，a>0 开口向上有最小值点；a<0 开口向下有最大值点。

square form y = a(x−h)² + k are (h, k). Alternatively, x-coordinate of vertex is −b/(2a).

由配方式 y = a(x−h)² + k 得顶点 (h, k)；也可用公式 x = −b/(2a) 求顶点横坐标。

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4. Polynomials: Remainder & Factor Theorems | 多项式：余数定理与因式定理

. The Factor Theorem is a special case: (x−a) is a factor of p(x) if and only if p(a) = 0.

对多项式 p(x)，余数定理指出除以 (x−a) 的余数为 p(a)。因式定理为其特例：(x−a) 是 p(x) 的因式当且仅当 p(a)=0。

2, etc.) to find one root, then use polynomial division or equating coefficients to obtain a quadratic factor, which can be factorised further if possible. Always repeat the Factor Theorem on the quadratic.

分解三次式 ax³+bx²+cx+d 时，代入小整数试探求一个根，再用多项式除法或系数比较得到二次因式，最后对二次式继续分解。

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5. Binomial Expansion | 二项式展开

For positive integer n, the binomial theorem expands (a+b)ⁿ using Pascal's triangle or the formula:

ⁿCᵣ aⁿ⁻ʳ bʳ where ⁿCᵣ = n! / [r! (n−r)!]

正整数 n 时，可用杨辉三角或组合数 ⁿCᵣ = n! / [r! (n−r)!] 展开 (a+b)ⁿ = Σ ⁿCᵣ aⁿ⁻ʳ bʳ。

The expansion of (1+x)ⁿ is 1 + nx + [n(n−1)/2!]x² + ... + xⁿ. When asked for a particular term, set the power of the required variable and solve for r. Always simplify coefficients fully.

(1+x)ⁿ = 1 + nx + [n(n−1)/2!]x² + ... + xⁿ。求指定项时令变量指数对应求出 r，系数需化为最简。

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6. Coordinate Geometry: Lines & Circles | 坐标几何：直线与圆

₂−x₁)² + (y₂−y₁)²]. The midpoint M = ((x₁+x₂)/2, (y₁+y₂)/2). Gradient m = (y₂−y₁)/(x₂−x₁).

两点距离 d = √[(x₂−x₁)² + (y₂−y₁)²]，中点 M=( (x₁+x₂)/2, (y₁+y₂)/2 )，斜率 m = (y₂−y₁)/(x₂−x₁)。

Equation of a straight line: slope–intercept form y = mx + c; point– m(x−x₁). Parallel lines share the same gradient; perpendicular lines satisfy m₁ × m₂ = −1.

直线方程：斜截式 y = mx + c；点斜式 y−y₁ = m(x−x₁)。两直线平行则斜率相等；垂直则 m₁ × m₂ = −1。

find centre and radius from the general form x²+y²+2gx+2fy+c=0: centre = (−g,−f), radius = √g²+f²−c.

圆的标准方程：(x−a)²+(y−b)² = r²，圆心 (a,b) 半径 r。从一般方程 x²+y²+2gx+2fy+c=0 配方得圆心(−g,−f)、半径 √g²+f²−c。

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7. Trigonometry: Ratios, Identities & Equations | 三角学：比、恒等式与解方程

Exact values for 30°, 45°, 60° must be memorised: sin 30°=1/2, cos 30°=√3/2, tan 30°=1/√3; sin 45°=√2/2, cos 45°=√2/2, tan 45°=1; sin 60°=√3/2, cos 60°=1/2, tan 60°=√3. The unit circle and CAST diagram help determine signs in each quadrant.

必须熟记 30°,45°,60° 的精确值：sin30°=1/2, cos30°=√3/2, tan30°=1/√3 等。利用单位圆和 CAST 图判断各象限三角比的符号。

quations into a single trig function and solve for θ in the given interval, remembering all possible solutions from the CAST diagram.

核心恒等式：sin²θ+cos²θ=1，tanθ=sinθ/cosθ。解三角方程时化为单一三角比，在给定区间内利用CAST图求出所有解。

= b/sin B = c/sin C (sine rule); a² = b² + c² − 2bc cos A (cosine rule). Area = ½ab sin C.

正弦定理 a/sinA = b/sinB = c/sinC 与余弦定理 a² = b²+c² − 2bc cosA 适用于任意三角形；面积公式 = ½ab sinC。

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8. Differentiation | 微分

erences, differentiate term by term. Constant multiples are preserved: d/dx [k f(x)] = k f'(x). The derivative of a constant is zero.

xⁿ 的导数为 nxⁿ⁻¹。多项式逐项求导，常数倍可提出：d/dx [k f(x)] = k f'(x)。常数的导数为 0。

The gradient of the tangent to y=f(x) at x=a is f'(a). The equation of the tangent uses point (a, f(a)) and gradient m = f'(a). The normal has gradient −1/f'(a) provided f'(a) ≠ 0.

曲线在 x=a 处切线的斜率为 f'(a)。切线方程用点 (a, f(a)) 及斜率 f'(a)；法线斜率则为 −1/f'(a)（f'(a)≠0）。

Stationary points occur where f'(x)=0. Determine their nature using the second derivative: f''(x)>0 → minimum, f''(x)<0 → maximum. If f''(x)=0, test sign of f'(x) on either side.

令 f'(x)=0 得驻点；用二阶导数判定：f''(x)>0 为极小点，f''(x)<0 为极大点。若 f''(x)=0，则检查一阶导数左右的符号变化。

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9. Integration | 积分

r n≠−1. The constant of integration C must always be included in indefinite integrals. Integrate term by term.

积分是微分的逆运算：∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n≠−1)。不定积分必须加积分常数 C，多项式逐项积分。

F(b)−F(a), where F is an antiderivative of f. Areas below the x-axis give negative contributions; take absolute value or adjust limits when finding total enclosed area.

定积分 ∫ₐᵇ f(x) dx = F(b)−F(a) 计算曲线 y=f(x) 与 x 轴在 a 到 b 之间的有向面积。x 轴下方区域贡献为负，求总面积时需分段取绝对值。

Given a derivative f'(x) and a point on the original curve, integrate to find f(x) and use the point to determine C.

已知导数 f'(x) 及曲线上一点，积分后代入该点坐标确定常数 C。

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10. Exponentials & Logarithms | 指数函数与对数

The exponential function eˣ has derivative eˣ and integral eˣ + C. The natural logarithm ln x is its inverse, defined for x>0: y = eˣ ⇔ x = ln y. Key log laws: ln(ab) = ln a + ln b, ln(a/b) = ln a − ln b, ln aⁿ = n ln a.

指数函数 eˣ 的导数等于自身，积分也为 eˣ+C。自然对数 ln x 为其反函数，定义域 x>0。对数运算法则：ln(ab)=ln a+ln b，ln(a/b)=ln a−ln b，ln aⁿ=n ln a。

Solve exponential equations by taking logs: e²ˣ=5 ⇒ or the change-of-base formula logₐ b = ln b / ln a. Remember to check domain restrictions.

解指数方程时两边取对数：e²ˣ=5 ⇒ 2x=ln5。不同底数时可利用换底公式 logₐ b = ln b / ln a，并注意定义域限制。

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11. Sequences & Series | 数列与级数

Sum of the first n terms: Sₙ = n/2 [2a + (n−1)d] = n/2 (a + l), where l is the last term.

等差数列公差为 d：第 n 项 uₙ = a + (n−1)d，首项 a。前 n 项和 Sₙ = n/2 [2a + (n−1)d] = n/2 (a+l)，l 为末项。

2, Σᵢ₌₁ⁿ i² = n(n+1)(2n+1)/6. Break sums into components to simplify polynomial sequences.

Σ 记号简洁表示求和。常用结果：∑1=n，∑i=n(n+1)/2，∑i²=n(n+1)(2n+1)/6。可将多项式数列拆分为已知求和公式的组合。

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12. Matrices | 矩阵

A 2×2 matrix is written as:

a	b
c	d

2×2 矩阵记作 [[a,b],[c,d]]。

try by the scalar. Matrix multiplication is not commutative: AB ≠ BA in general. For A=(aᵢⱼ) and B=(bⱼₖ), the product (AB)ᵢₖ = Σⱼ aᵢⱼ bⱼₖ.

加法与减法逐元对应进行；标量乘法将每个元乘以该标量。矩阵乘法不满足交换律。乘法时 (AB) 的第 i 行第 k 列元素为 A 的第 i 行与 B 的第 k 列对应元素乘积之和。

, given by M⁻¹ = (1/det(M)) [[d,−b],[−c,a]]. M M⁻¹ = I, the identity matrix [[1,0],[0,1]].

2×2 矩阵 M 的行列式 det(M)=ad−bc。若 det(M)≠0 则逆矩阵 M⁻¹ = (1/det(M)) [[d,−b],[−c,a]]，满足 M M⁻¹=I (单位阵)。

Matrices represent linear transformations: rotations, reflections, enlargements. Combining transformations corresponds to multiplying their matrices in the correct order (first transformation on the right).

矩阵可表示线性变换（旋转、反射、缩放）。复合变换的矩阵按变换先后从右向左相乘。

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