📚 WJEC Year 11 Further Mathematics: Quick Reference Formula & Theorem Handbook | Year 11 WJEC 进阶数学：公式定理速查手册
Welcome to your essential quick reference handbook for WJEC Year 11 Further Mathematics. This guide compiles all the crucial formulas, identities, and theorems you need to memorise and apply with confidence. Use it as a daily revision companion to consolidate your understanding and sharpen your problem‑solving skills.
欢迎使用 WJEC Year 11 进阶数学必备速查手册。本指南汇总了所有需要牢记并熟练运用的关键公式、恒等式与定理。将其作为日常复习伙伴，可帮助你巩固理解、提升解题能力。
1. Algebraic Laws and Polynomials | 代数法则与多项式
Algebra is built on fundamental laws: commutative, associative, and distributive. When expanding polynomials, the distributive law plays a key role. Memorising basic identities saves time in algebraic manipulation.
代数建立在基本定律之上：交换律、结合律与分配律。展开多项式时，分配律起关键作用。熟记基本恒等式可以在代数运算中节省时间。
(a+b)² = a² + 2ab + b²
(a−b)² = a² − 2ab + b²
(a+b)(a−b) = a² − b²
ently, the Factor Theorem tells us that (x−a) is a factor of f(x) if and only if f(a) = 0.
余式定理指出，当多项式 f(x) 除以 (x−a) 时，余式为 f(a)。由此，因式定理告诉我们，(x−a) 是 f(x) 的因式当且仅当 f(a) = 0。
2. Quadratic Functions and Equations | 二次函数与方程
≠ 0), the discriminant is Δ = b² − 4ac. The solutions are given by the quadratic formula:
对于标准形式 ax² + bx + c = 0 (a ≠ 0) 的二次方程，判别式为 Δ = b² − 4ac。其解由求根公式给出：
x = [−b ± √b² − 4ac] / (2a)
x of the parabola. The sum and product of roots α and β satisfy α + β = −b/a and αβ = c/a.
完全平方可将二次式改写为 a(x−h)² + k，其中 (h,k) 为抛物线的顶点。两根 α 与 β 的和与积满足 α + β = −b/a 及 αβ = c/a。
3. Inequalities and Modulus | 不等式与绝对值
Solving linear inequalities mirrors solving equations, but multiplying or dividing by a negative number reverses the inequality sign. For quadratic inequalities, sketch the graph to identify intervals where the expression is positive or negative.
解线性不等式与解方程类似，但乘以或除以负数时需反转不等号。对于二次不等式，通过绘制草图确定表达式为正或负的区间。
The modulus (absolute value) is defined as |x| = x if x ≥ 0, and |x| = −x if x < 0. Key results: |x| < a ⇒ −a < x < a, and |x| > a ⇒ x < −a or x > a.
模（绝对值）定义为：若 x ≥ 0，则 |x| = x；若 x < 0，则 |x| = −x。关键结论：|x| < a ⇒ −a < x < a；|x| > a ⇒ x < −a 或 x > a。
4. Indices and Surds | 指数与根式
Mastery of the laws of indices is essential for simplifying expressions and solving equations. The same laws underpin work with logarithms and calculus.
掌握指数定律对于化简表达式和求解方程至关重要。这些定律也是学习对数与微积分的基础。
- aᵐ × aⁿ = aᵐ⁺ⁿ
- aᵐ × aⁿ = aᵐ⁺ⁿ
- aᵐ ÷ aⁿ = aᵐ⁻ⁿ
- aᵐ ÷ aⁿ = aᵐ⁻ⁿ
- (aᵐ)ⁿ = aᵐⁿ
- (aᵐ)ⁿ = aᵐⁿ
- a⁰ = 1 (a ≠ 0)
- a⁰ = 1 (a ≠ 0)
- a⁻ⁿ = 1/aⁿ
- a⁻ⁿ = 1/aⁿ
- a1/n = √[n]a
- a1/n = n次方根号a
With surds, always simplify radicals where possible and rationalise the denominator. For example, 1/√a = √a / a, and 1/(√a + √b) rationalises using the difference of two squares.
处理根式时，应尽可能化简根号并使分母有理化。例如，1/√a = √a / a；对于 1/(√a + √b)，可利用平方差公式有理化。
5. Logarithms | 对数
If aˣ base e ≈ 2.718. Logarithms reverse the operation of exponentiation and obey strict rules.
若 aˣ = b，则 x = logₐ b。自然对数 ln x 以 e ≈ 2.718 为底。对数是指数运算的逆运算，并遵循严格的运算法则。
logₐ (xy) = logₐ x + logₐ y
logₐ (x/y) = logₐ x − logₐ y
logₐ xᵏ = k logₐ x
The change‑of‑ c a. This is particularly useful when evaluating unfamiliar logarithms using a calculator.
换底公式可将任意底数转换为新底数 c：logₐ b = log_c b / log_c a。当使用计算器求不熟悉的对数时，这一公式尤为有用。
6. Sequences and Series | 数列与级数
the first n terms is Sₙ = n/2 [2a + (n−1)d] or Sₙ = n/2 (a + l), where l is the last term.
对于首项为 a、公差为 d 的等差数列，第 n 项为 uₙ = a + (n−1)d。前 n 项和为 Sₙ = n/2 [2a + (n−1)d] 或 Sₙ = n/2 (a + l)，其中 l 为末项。
nth term is uₙ = arⁿ⁻¹. The sum of the first n terms (r ≠ 1) is Sₙ = a(1−rⁿ) / (1−r). For |r| < 1, the sum to infinity exists: S∞ = a / (1−r).
等比数列首项为 a、公比为 r，其第 n 项为 uₙ = arⁿ⁻¹。前 n 项和 (r ≠ 1) 为 Sₙ = a(1−rⁿ) / (1−r)。当 |r| < 1 时，无穷项和存在：S∞ = a / (1−r)。
7. Binomial Expansion | 二项式展开
For a positive integer n, (a + b)ⁿ expands using binomial coefficients. The (r+1)th term is given by Tr+1 = C(n,r) aⁿ⁻ʳ bʳ, where C(n,r) = n! / [r!(n−r)!].
对于正整数 n，(a + b)ⁿ 利用二项式系数展开。第 (r+1) 项由 Tr+1 = C(n,r) aⁿ⁻ʳ bʳ 给出，其中 C(n,r) = n! / [r!(n−r)!]。
When n is rational and |x| < 1, the expansion of (1+x)ⁿ is infinite: (1+x)ⁿ = 1 + nx + [n(n−1)/2!] x² + ... Always state the range of validity. Pascal's triangle helps generate coefficients quickly for small integer n.
当 n 为有理数且 |x| < 1 时，(1+x)ⁿ 的展开为无穷级数：(1+x)ⁿ = 1 + nx + [n(n−1)/2!] x² + ... 务必注明有效范围。对于较小的整数 n，帕斯卡三角形可快速生成系数。
8. Trigonometry | 三角学
In a right‑ potenuse, and tanθ = opposite/adjacent. Exact values for special angles must be memorised.
在直角三角形中，sinθ = 对边/斜边，cosθ = 邻边/斜边，tanθ = 对边/邻边。特殊角的精确值必须牢记。
| θ (degrees) | 0° | 30° | 45° | 60° | 90° |
|---|---|---|---|---|---|
| sin | 0 | ½ | 1/√2 | √3/2 | 1 |
| cos | 1 | √3/2 | 1/√2 | ½ | 0 |
| tan | 0 | 1/√3 | 1 | √3 | undefined |
c/sinC, and a² = b² + c² − 2bc cosA. Area of a triangle = ½ab sinC. Radians: π rad = 180°.
基本恒等式：sin²θ + cos²θ = 1。正弦与余弦定理适用于任意三角形：a/sinA = b/sinB = c/sinC，以及 a² = b² + c² − 2bc cosA。三角形面积 = ½ab sinC。弧度制：π rad = 180°。
9. Differentiation | 微分
, d/dx (xⁿ) = n xⁿ⁻¹. The rules of differentiation are additive, and constants factor out.
微分用于度量瞬时变化率。对于幂函数，d/dx (xⁿ) = n xⁿ⁻¹。微分法则具有可加性，常数因子可提出。
Product Rule: d/dx (uv) = u'v + uv'
Quotient Rule: d/dx (u/v) = (u'v − uv') / v²
Chain Rule: dy/dx = dy/du × du/dx
Standard derivatives: d/dx (sin x) = cos x, d/dx (cos x) = −sin x, d/dx (eˣ) = eˣ, d/dx (ln x) = 1/x. The second derivative f''(x) indicates concavity and determines the nature of stationary points.
标准导数：d/dx (sin x) = cos x，d/dx (cos x) = −sin x，d/dx (eˣ) = eˣ，d/dx (ln x) = 1/x。二阶导数 f''(x) 反映凹凸性，并用于判定驻点的性质。
10. Integration | 积分
he indefinite integral adds an arbitrary constant C: ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, for n ≠ −1.
积分是微分的逆运算。不定积分需加上任意常数 C：∫ xⁿ dx = xⁿ⁺¹/(n+1) + C，其中 n ≠
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