📚 Year 11 Eduqas Further Maths: Formulae & Theorems Quick Reference Guide | Year 11 Eduqas 进阶数学：公式定理速查手册
This quick reference guide compiles all essential formulae and theorems for the Year 11 Eduqas Level 2 Certificate in Further Mathematics. Use it as a revision checklist to master algebra, trigonometry, calculus, matrices, vectors and coordinate geometry.
本速查手册汇总了 Year 11 Eduqas 进阶数学（Level 2 Certificate）所有核心公式与定理。可用作复习清单，帮助你掌握代数、三角、微积分、矩阵、向量和坐标几何。
1. Algebraic Manipulation & Solving Equations | 代数操作与解方程
Quadratic Formula: For ax² + bx + c = 0, the solutions are x = (-b ± √(b² – 4ac)) / (2a). The discriminant Δ = b² – 4ac tells you about the roots: Δ > 0 gives two distinct real roots; Δ = 0 gives one repeated real root; Δ < 0 gives no real roots.
二次公式：对于 ax² + bx + c = 0，解为 x = (-b ± √(b² – 4ac)) / (2a)。判别式 Δ = b² – 4ac：Δ > 0 有两个不同实根；Δ = 0 有一个重根；Δ < 0 无实根。
Simultaneous Equations: Solve a linear and a quadratic equation by substitution. Rearrange the linear equation to express one variable in terms of the other, substitute into the quadratic and solve.
联立方程：一个线性和一个二次方程通常用代入法：将线性方程改写为 y = … 或 x = …，代入二次方程后求解。
Algebraic Fractions: To add or subtract, find a common denominator: a/b + c/d = (ad + bc) / bd. Always simplify the final expression.
代数分式：加减法通分：a/b + c/d = (ad + bc) / bd，最后化简。
2. Polynomial Theorems: Factor & Remainder | 因式定理与余式定理
Remainder Theorem: When a polynomial f(x) is divided by (x − a), the remainder is f(a). For division by (ax − b), the remainder is f(b/a).
余式定理：多项式 f(x) 除以 (x − a) 的余数为 f(a)；除以 (ax − b) 的余数为 f(b/a)。
Factor Theorem: (x − a) is a factor of f(x) if and only if f(a) = 0. This is used to factorise cubic polynomials and solve equations.
因式定理：(x − a) 是 f(x) 的因式当且仅当 f(a) = 0。用于分解三次多项式并解方程。
Long Division & Synthetic Division: After identifying one factor, divide the polynomial to obtain a quadratic factor, which can then be factorised or solved.
长除法与综合除法：找到一个因式后，通过除法得到二次因式，再因式分解或求解。
3. Binomial Expansion | 二项式展开
Pascal’s Triangle & nCr: (a + b)ⁿ = aⁿ + (¹C₁) aⁿ⁻¹b + (¹C₂) aⁿ⁻²b² + … + bⁿ, where (¹Cr) = n! / [r! (n − r)!]. For n ≤ 5 you may use Pascal’s triangle directly.
帕斯卡三角形与组合数：(a + b)ⁿ 展开使用 nCr 系数，nCr = n! / [r! (n − r)!]。n ≤ 5 时可直用帕斯卡三角形。
Finding a Specific Term: The (r + 1)th term in the expansion of (a + b)ⁿ is (¹Cr) aⁿ⁻ʳ bʳ.
指定项的求法：(a + b)ⁿ 展开中第 (r + 1) 项为 (¹Cr) aⁿ⁻ʳ bʳ。
Applications: Use expansions to approximate powers, e.g. (1 + x)ⁿ ≈ 1 + nx for small x.
应用：用展开式近似计算，如对小量 x，(1 + x)ⁿ ≈ 1 + nx。
4. Trigonometric Functions & Exact Values | 三角函数与精确值
Exact Values Table (sin θ, cos θ, tan θ):
精确值表：
| θ (°) | sin θ | cos θ | tan θ |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | ½ | √3/2 | 1/√3 |
| 45° | 1/√2 | 1/√2 | 1 |
| 60° | √3/2 | ½ | √3 |
| 90° | 1 | 0 | undefined |
Graphs of sin, cos and tan: sin θ and cos θ have period 360°, amplitude 1. tan θ has period 180° and vertical asymptotes at θ = 90°, 270°, etc.
三角函数图像：sin θ 和 cos θ 的周期为 360°，振幅 1。tan θ 的周期为 180°，在 θ = 90°, 270° 等处有垂直渐近线。
5. Trigonometric Identities & Equations | 三角恒等式与方程
Pythagorean Identity: sin²θ + cos²θ ≡ 1. This can be rearranged: sin²θ = 1 − cos²θ, cos²θ = 1 − sin²θ.
勾股恒等式：sin²θ + cos²θ ≡ 1，可变形为 sin²θ = 1 − cos²θ 或 cos²θ = 1 − sin²θ。
Tangent Identity: tanθ ≡ sinθ / cosθ.
正切恒等式：tanθ ≡ sinθ / cosθ。
Solving Trig Equations: If sinθ = k, first find the principal value α = sin⁻¹(k). Then, for 0° ≤ θ ≤ 360°, θ = α or θ = 180° − α (sin); for cos, θ = α or θ = 360° − α; for tan, θ = α or θ = α + 180°. Always adjust for the given interval.
解三角方程：若 sinθ = k，先求主值 α = sin⁻¹(k)。在 0°–360° 内，sin 的解为 θ = α 或 θ = 180° − α；cos 的解为 θ = α 或 θ = 360° − α；tan 的解为 θ = α 或 θ = α + 180°，并限制在给定区间内。
6. Differentiation Rules & Applications | 微分规则与应用
Basic Rule: If y = xⁿ, then dy/dx = nxⁿ⁻¹. [e.g. d/dx (x³) = 3x²].
基本法则：y = xⁿ ⇒ dy/dx = nxⁿ⁻¹。
Sum/Difference and Constant Multiple: d/dx [f(x) ± g(x)] = f ‘(x) ± g'(x); d/dx [k f(x)] = k f ‘(x).
和差与常数倍：d/dx [f(x) ± g(x)] = f ‘(x) ± g'(x)；d/dx [k f(x)] = k f ‘(x)。
Equation of a Tangent: Gradient m = dy/dx at the point (x₁, y₁). Tangent equation: y − y₁ = m(x − x₁).
切线方程：在 (x₁, y₁) 处斜率 m = dy/dx，切线为 y − y₁ = m(x − x₁)。
Stationary Points: Solve dy/dx = 0 to find x-coordinates. Nature: compute d²y/dx²; if > 0, minimum; if < 0, maximum; if = 0, could be a point of inflection (check sign change of dy/dx).
驻点：解 dy/dx = 0 得 x 坐标。用二阶导数 d²y/dx² 判断：> 0 为极小；< 0 为极大；= 0 可能为拐点，需检查一阶导数符号变化。
7. Integration – Basics & Area | 积分基础与面积
Indefinite Integration: ∫ xⁿ dx = xⁿ⁺¹ / (n + 1) + C, for n ≠ −1. Remember the constant of integration.
不定积分：∫ xⁿ dx = xⁿ⁺¹ / (n + 1) + C（n ≠ −1），勿忘积分常数。
Definite Integration and Area: ∫ₐᵇ f(x) dx = [F(x)]ₐᵇ = F(b) − F(a). This gives the signed area between the curve y = f(x), the x-axis and the lines x = a, x = b.
定积分与面积：∫ₐᵇ f(x) dx = [F(x)]ₐᵇ = F(b) − F(a)，表示曲线、x轴、x=a、x=b 所围的带号面积。
Area Between a Curve and a Line: Subtract the lower function from the upper function and integrate: ∫ (top − bottom) dx.
曲线与直线间的面积：用上减下积分：∫ (上函数 − 下函数) dx。
Trapezium Rule (optional but useful): ∫ₐᵇ f(x) dx ≈ h/2 [y₀ + y&subn; + 2(y₁ + y₂ + … + y&subn;−₁)], where h = (b − a)/n.
梯形法则（拓展）：∫ₐᵇ f(x) dx ≈ h/2 [y₀ + y&subn; + 2(y₁ + … + y&subn;−₁)]，h = (b − a)/n。
8. Matrices: Operations & Inverses | 矩阵运算与逆矩阵
Order and Multiplication: An m × n matrix has m rows, n columns. Two matrices can be multiplied if the number of columns of the first equals the number of rows of the second.
阶与乘法：m × n 矩阵有 m 行 n 列。相乘要求左矩阵列数等于右矩阵行数。
Determinant of a 2×2 Matrix: If A = [[a, b], [c, d]], then det(A) = ad − bc.
2 × 2 矩阵行列式：A = [[a, b], [c, d]]，det(A) = ad − bc。
Inverse: If det(A) ≠ 0, A⁻¹ = 1/det(A) [[d, −b], [−c, a]].
逆矩阵：若 det(A) ≠ 0，A⁻¹ = 1/det(A) [[d, −b], [−c, a]]。
Solving Simultaneous Equations Using Matrices: Write equations as AX = B, then X = A⁻¹B.
用矩阵解联立方程组：写成 AX = B 形式，则 X = A⁻¹B。
Matrix Transformations: Common transformations of the unit square: rotation by θ: [[cosθ, −sinθ], [sinθ, cosθ]]; reflection in x-axis: [[1, 0], [0, −1]]; enlargement scale factor k: [[k, 0], [0, k]].
矩阵变换：单位正方形的常见变换：旋转 θ：[[cosθ, −sinθ], [sinθ, cosθ]]；x 轴反射：[[1, 0], [0, −1]]；放大 k 倍：[[k, 0], [0, k]]。
9. Vectors in 2D | 二维向量
Column and Component Forms: A vector a can be written as a = xi + yj = (x, y) or in column form [[x], [y]].
列向量与分量：向量 a = xi + yj = (x, y)，也可写成列向量 [[x], [y]]。
Magnitude: |a| = √(x² + y²).
模长：|a| = √(x² + y²)。
Addition and Scalar Multiplication: a + b = (x₁+x₂)i + (y₁+y₂)j; ka = (kx)i + (ky)j.
加法与标量乘法：a + b = (x₁+x₂)i + (y₁+y₂)j；ka = (kx)i + (ky)j。
Parallel Vectors: Vectors a and b are parallel if b = ka for some scalar k ≠ 0.
平行向量：若存在 k ≠ 0 使 b = ka，则 a 与 b 平行。
10. Sequences & the nth Term | 数列与通项公式
Linear Sequences: nth term = an + b, where a is the first difference. Find a from the difference and b by substituting n = 1.
线性数列：通项 u&subn; = an + b，其中 a 是一阶差分。通过差分和首项确定 a, b。
Quadratic Sequences: nth term = an² + bn + c. The second difference is constant and equal to 2a. Use the first few terms to set up equations or use a general method.
二次数列：通项 u&subn; = an² + bn + c。二阶差分为常数 2a，再通过前几项列方程求解 b, c。
Limiting Value: Some sequences approach a limit L. For a sequence defined by a recurrence relation, set u&subn;+1; = u&subn; = L and solve for L, provided the sequence converges.
极限值：有些数列趋于极限 L。若数列由递推关系定义，可令 u&subn;+1; = u&subn; = L 求解（需保证收敛）。
11. Coordinate Geometry: Lines & Circles | 坐标几何：直线与圆
Gradient and Straight Line: Gradient m = (y₂ − y₁) / (x₂ − x₁). Equation: y − y₁ = m(x − x₁) or y = mx + c.
斜率与直线：
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