📚 Year 11 WJEC Further Mathematics: Quick Reference Formula Handbook | Year 11 WJEC 进阶数学：公式定理速查手册
This quick reference handbook consolidates every essential formula and theorem required for the WJEC Level 2 Additional Mathematics (often taught as Further Mathematics in Year 11). It presents algebra, trigonometry, calculus, vectors, matrices, and more in a clear, bilingual layout designed for fast revision. Each section is paired with concise English and Chinese explanations, ensuring you can quickly locate and recall the relationships you need for both calculator and non-calculator papers.
这份速查手册整理了 WJEC Level 2 Additional Mathematics（通常作为 Year 11 进阶数学课程）所需的所有核心公式和定理。内容涵盖代数、三角学、微积分、向量、矩阵等领域，以清晰的中英对照形式呈现，便于快速复习。每个小节都配有简明的英文和中文解释，确保你能迅速定位并回忆考试中需要的关系式，无论是使用计算器还是非计算器试卷。
1. Algebra: Quadratic Formula and Discriminant | 代数：二次方程与判别式
x + c = 0, the solutions can be found using the quadratic formula. The discriminant Δ = b² – 4ac tells you about the nature of the roots without solving the equation completely.
对于任何标准形式的二次方程 ax² + bx + c = 0，可以使用二次求根公式求出解。判别式 Δ = b² – 4ac 能够直接揭示根的性质，无需完整求解方程。
x = (-b ± √b² – 4ac) / (2a)
xactly one repeated real root (a double root). When Δ < 0, the equation has no real roots, only complex ones.
若 Δ > 0，方程有两个不相等的实数根；若 Δ = 0，有一个重根；若 Δ < 0，方程无实数根。
2. Surds and Laws of Indices | 根式与指数定律
Simplifying expressions involving roots and powers is fundamental in further mathematics. The rules of indices allow you to manipulate terms with the same base efficiently, while surds often need to be rationalised when they appear in denominators.
化简含有根号和乘方的表达式是进阶数学的基础。指数定律使你能高效地处理相同底数的项，而当分母中出现根式时，通常需要有理化。
aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐᵐ
Fractional indices link roots and powers: a1/n = ⁿ√a, and am/n = (ⁿ√a negative index rule a⁻ⁿ = 1/aⁿ is used to move factors between numerator and denominator.
分数指数连接了根号和乘方：a1/n = ⁿ√a，am/n = (ⁿ√a)ᵐ。负指数规则 a⁻ⁿ = 1/aⁿ 用于在分子分母间移动因子。
√a × √b = √ab, √a / √b = √a/b (b ≠ 0)
To rationalise a denominator like 1/(√a + √b), multiply numerator and denominator by the conjugate (√a – √b).
要有理化形如 1/(√a + √b) 的分母，需要将分子分母同时乘以共轭根式 (√a – √b)。
3. Polynomials and the Factor Theorem | 多项式与因式定理
Polynomial division, factorisation, and the remainder theorem form a core part of algebraic manipulation in WJEC further maths. The factor theorem is a direct consequence of the remainder theorem and is used to find linear factors of polynomials.
多项式除法、因式分解和余式定理是 WJEC 进阶数学代数运算的核心内容。因式定理是余式定理的直接推论，用于寻找多项式的一次因式。
If f(a) = 0, then (x – a) is a factor of f(x).
To divide a polynomial f(x) by (x – p), synthetic division or long division can be used. The remainder when f(x) is divided by (x – p) equals f(p). For a cubic, once one factor is found, the remaining quadratic can be solved or factorised further.
可以用综合除法或长除法将多项式 f(x) 除以 (x – p)。f(x) 除以 (x – p) 所得余数等于 f(p)。对于三次多项式，找到一个一次因式后，剩下的二次式可继续求解或因式分解。
4. Binomial Expansion | 二项式展开
The binomial theorem expands expressions of the form (a + b)ⁿ for positive integer n. In further mathematics, you also extend this to (1 + x)ⁿ where n is rational and |x| < 1, leading to an infinite series.
二项式定理用于展开形如 (a + b)ⁿ 的正整数次幂。在进阶数学中，还会扩展到 n 为有理数且 |x| < 1 的 (1 + x)ⁿ，得到无穷级数。
(1 + x)ⁿ = 1 + nx + n(n – 1)/2! · x² + n(n – 1)(n – 2)/3! · x³ + …
For integer powers, the expansion has (n+1) terms and the general term is given by ⁿCᵣ aⁿ⁻ʳ bʳ. Pascal’s triangle provides a quick visual for the coefficients when n is small.
对于整数次幂，展开式共有 (n+1) 项，其通项公式为 ⁿCᵣ aⁿ⁻ʳ bʳ。当 n 较小时，杨辉三角可以直观地给出系数。
5. Trigonometry: Identities and Equations | 三角学：恒等式与方程
The trigonometric functions sine, cosine, and tangent are extended beyond right-angled triangles using the unit circle. Key identities must be memorised to simplify expressions and solve equations within a given interval.
利用单位圆，正弦、余弦和正切函数的定义从直角三角形拓展到任意角。必须熟记重要的三角恒等式，以便在给定区间内化简表达式和解方程。
sin²θ + cos²θ = 1, tanθ = sinθ / cosθ
For any triangle with sides a, b, c and opposite angles A, B, C, the sine rule and cosine rule connect sides and angles. The area can be found using ½ ab sinC.
对于任意三角形，边长 a, b, c 及其对角 A, B, C 满足正弦定理和余弦定理。三角形面积可用公式 ½ ab sinC 求得。
a/sinA = b/sinB = c/sinC
a² = b² + c² – 2bc cosA
Exact trigonometric values for 30°, 45°, 60°, and 90° must be known, often in surd form, and are frequently tested in non-calculator papers.
必须记住 30°、45°、60° 和 90° 等特殊角的精确三角函数值，通常以根式形式出现，经常在非计算器试卷中考查。
6. Coordinate Geometry: Straight Lines and Circles | 坐标几何：直线与圆
Coordinate geometry combines algebraic equations with geometric shapes. The gradient, distance, and midpoint formulas are the building blocks for working with straight lines, while the equation of a circle links an algebraic form to its centre and radius.
坐标几何将代数方程与几何图形联系起来。斜率、距离和中点公式是处理直线的基础，而圆的方程则通过代数形式表达其圆心和半径。
Distance d = √[(x₂ – x₁)² + (y₂ – y₁)²]
The equation of a straight line can be expressed as y = mx + c or y – y₁ = m(x – x₁), where m = (y₂ – y₁)/(x₂ – Parallel lines share the same gradient; products of gradients of perpendicular lines equal –1.
直线方程可写作 y = mx + c 或 y – y₁ = m(x – x₁)，其中 m = (y₂ – y₁)/(x₂ – x₁)。平行直线斜率相等；互相垂直的直线斜率乘积为 –1。
Circle: (x – a)² + (y – b)² = r² centre (a, b), radius r
The tangent to a circle at a point is perpendicular to the radius at that point. You can find the equation of a tangent using the gradient of the radius and the point of contact.
圆上一点处的切线垂直于该点处的半径。利用半径的斜率和切点坐标，可以求出切线方程。
7. Calculus: Differentiation | 微积分：微分
Differentiation gives the instantaneous rate of change and the gradient of a curve. WJEC further maths requires fluency in differentiating polynomials, trigonometric functions, exponentials, and logarithms, and applying these to find tangents, normals, and stationary points.
微分用于求瞬时变化率和曲线的斜率。WJEC 进阶数学要求熟练对多项式、三角函数、指数函数和对数函数求导，并应用于求切线、法线和驻点。
d/dx (xⁿ) = n xⁿ⁻¹
For basic functions, remember: d/dx (sinx) = cosx, d/dx (cosx) = –sinx, d/dx (eˣ) = eˣ ct rule, and quotient rule extend differentiation to composite and more complex functions.
基本函数的导数为：d/dx (sinx) = cosx，d/dx (cosx) = –sinx，d/dx (eˣ) = eˣ，d/dx (ln x) = 1/x。链式法则、乘积法则和商法则可将微分推广到复合函数和更复杂的函数。
Stationary points: set dy/dx = 0. Use second derivative d²y/dx² to test nature.
ximum; if d²y/dx² = 0, test by inspecting the sign of the first derivative on either side.
判断驻点性质：若二阶导数 d²y/dx² > 0，则该点为极小值点；若 d²y/dx² < 0，为极大值点；若 d²y/dx² = 0，则通过检查一阶导数在该点邻近的符号来判定。
8. Calculus: Integration | 微积分：积分
Integration is the reverse process of differentiation. It is used to find areas under curves and to recover functions from their derivatives. The indefinite integral always includes an arbitrary constant '+ c'.
积分是微分的逆过程，用于计算曲线下的面积以及从导数还原函数。不定积分结果始终包含任意常数 '+ c'。
∫ xⁿ dx = xⁿ⁺¹ / (n+1) + c, n ≠ –1
Key integrals you must recognise instantly: ∫ sinx dx = –cosx + c, ∫ cosx dx = sinx + c, ∫ eˣ dx = eˣ + c, and ∫ 1/x dx = ln|x| + c. The constant of integration is essential.
必须能迅速辨认的核心积分：∫ sinx dx = –cosx + c，∫ cosx dx = sinx + c，∫ eˣ dx = eˣ + c，∫ 1/x dx = ln|x| + c。积分常数至关重要。
Definite integral: Area = ∫ₐᵇ f(x) dx = F(b) – F(a)
When using a definite integral to find an area, take care if the curve crosses the x‑axis; areas below the axis give a negative contribution and must be handled separately by splitting the integral.
使用定积分求面积时，若曲线穿过 x 轴，需特别小心；x 轴以下部分会给出负值贡献，必须通过分割积分区间分别计算。
9. Vectors | 向量
Vectors describe quantities that have both magnitude and direction. In two dimensions, a vector can be represented as a column vector or in component form. Vector operations underpin many geometry and mechanics problems.
向量用于描述既有大小又有方向的量。在平面中，向量可以用列向量或分量形式表示。向量运算为许多几何和力学问题奠定基础。
Vector a = (a₁, a₂) Magnitude |a| = √a₁² + a₂²
roduct (scalar product) a·b = a₁b₁ + a₂b₂ is closely related to the angle between vectors.
向量加法和数乘按分量进行：a + b = (a₁+b₁, a₂+b₂)，k a = (k a₁, k a₂)。点乘（数量积）a·b = a₁b₁ + a₂b₂ 与两向量之间的夹角密切相关。
a·b = |a||b| cosθ
dicular. The position vector of a point gives its coordinates relative to the origin; the vector AB is found by b – a.
若 a·b = 0，则向量互相垂直。点的位置向量给出了它相对于原点的坐标；向量 AB 可由 b – a 得到。
10. Matrices | 矩阵
Matrices are rectangular arrays of numbers used to represent linear transformations such as rotations, reflections, and enlargements. Multiplying a matrix by a vector applies the corresponding transformation to that position vector.
矩阵是由数字组成的矩形阵列，用于表示线性变换，例如旋转、反射和缩放。用一个矩阵乘以一个向量，即对该位置向量施加相应的变换。
2 × 2 matrix A = [a b; c d] Determinant det(A) = ad – bc
the transformation is singular and collapses area to zero. The inverse matrix, when it exists, reverses the transformation.
行列式决定了矩阵是否可逆。若 det(A) = 0，该变换是奇异的，会将面积收缩为零。逆矩阵（若存在）则用于还原变换。
A⁻¹ = 1/(ad – bc) · [d –b; –c a]
Matrix multiplication is not commutative: AB ≠ BA in general. To multiply matrices, the number of columns in the first matrix must equal the number of rows in the second.
矩阵乘法一般不满足交换律：一般情况下 AB ≠ BA。进行矩阵乘法时，第一个矩阵的列数必须等于第二个矩阵的行数。
11. Sequences and Series | 数列与级数
Sequences and series appear regularly in WJEC further mathematics, particularly arithmetic and geometric progressions. Understanding the nth term and the sum of the first n terms is essential for problem solving.
数列和级数经常出现在 WJEC 进阶数学中，特别是等差数列和等比数列。掌握第 n 项以及前 n 项和的公式对解题至关重要。
Arithmetic: nth term = a + (n – 1)d, Sₙ = n/2 [2a + (n – 1)d] = n/2 (a + l)
term a and common ratio r, the nth term is arⁿ⁻¹. The sum of the first n terms is Sₙ = a(1 – rⁿ)/(1 – r) for r ≠ 1. When |r| < 1, the sum to infinity converges.
对于首项为 a、公比为 r 的等比数列，第 n 项为 arⁿ⁻¹。前 n 项和当 r ≠ 1 时为 Sₙ = a(1 – rⁿ)/(1 – r)。若 |r| < 1，无穷级数收敛。
S₅ = a / (1 – r), |r| < 1
Be sure to check the common ratio carefully; the sum to infinity formula only applies when the series converges. Some problems require forming and solving equations using these sum formulas.
务必仔细检查公比；无穷级数求和公式仅在级数收敛时成立。有些题目需要利用这些求和公式建立并求解方程。
Published by TutorHao | Further Mathematics Revision Series | aleveler.com
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