📚 Quick Reference Handbook of Formulae and Theorems: Year 10 AQA Further Mathematics | Year 10 AQA 进阶数学：公式定理速查手册
This bilingual quick reference handbook presents the core formulae, identities and theorems required for the Year 10 AQA Level 2 Certificate in Further Mathematics. Each section pairs concise English explanations with Chinese translations, followed by essential equations in a clear, centred format. Use it to consolidate understanding, support problem-solving and speed up revision.
这本中英双语速查手册整理了 Year 10 AQA 进阶数学（Level 2 Certificate）所必需的核心公式、恒等式与定理。每个小节都提供简洁的英文说明与对应的中文翻译，关键方程式以居中醒目的方式呈现，帮助你巩固理解、辅助解题并加速复习。
1. Algebra Fundamentals | 代数基础
wo numbers that multiply to c and add to b. The difference of two squares factorises as a² – b² = (a + b)(a – b).
乘法分配律用于展开括号：a(b + c) = ab + ac。对二次式 x² + bx + c 进行因式分解，需找到两个乘积为 c 且和为 b 的数。平方差公式为 a² – b² = (a + b)(a – b)。
The quadratic formula solves ax² + bx + c = 0.
二次求根公式用于求解 ax² + bx + c = 0。
x = (–b ± √b² – 4ac) / (2a)
The discriminant Δ = b² – ure of roots: two real roots if Δ > 0, one repeated root if Δ = 0, no real roots if Δ < 0.
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。
Completing the square rewrites x² + bx as (x + b/2)² – (b/2)². For a general quadratic ax² + bx + c, the completed form is a(x + b/(2a))² + (c – b²/(4a)).
配方将 x² + bx 写作 (x + b/2)² – (b/2)²。一般二次式 ax² + bx + c 的配方形式为 a(x + b/(2a))² + (c – b²/(4a))。
The factor theorem states (x – polynomial f(x) if and only if f(a) = 0. The remainder theorem: when f(x) is divided by (x – a), the remainder is f(a).
因式定理指出，(x – a) 是多项式 f(x) 的因式当且仅当 f(a) = 0。余数定理：f(x) 除以 (x – a)，余数为 f(a)。
2. Algebraic Fractions | 代数分式
To simplify an algebraic fraction, factorise the numerator and denominator and cancel common factors. For multiplication, multiply numerators and denominators, then simplify. For division, multiply by the reciprocal of the second fraction.
简化代数分式时，先将分子分母因式分解，再约去公因式。乘法将分子相乘、分母相乘后化简。除法则乘以第二个分式的倒数。
Addition and subtraction require a common denominator. Rewrite each fraction with the lowest common multiple of the denominators, then combine the numerators.
加减法需要通分：找出分母的最小公倍式，将各分式改写为同分母，再合并分子。
When solving an equation containing algebraic fractions, multiply every term by the common denominator to clear fractions, then solve the resulting polynomial equation. Always check that solutions do not make any original denominator zero.
解含分式的方程时，所有项乘以公分母以消去分母，然后解所得多项式方程。必须检验解是否会使原分母为零，若为零则舍去。
3. Functions | 函数
range is the set of possible outputs. fg(x) means apply g first, then f: fg(x) = f(g(x)).
函数 f(x) 将每个输入 x 映射到唯一的输出。定义域是可能的输入集合，值域是可能的输出集合。复合函数 fg(x) 表示先作用 g，再作用 f：fg(x) = f(g(x))。
e of f, and its range is the domain of f. Graphically, the inverse is a reflection in the line y = x.
反函数 f⁻¹(x) 逆转 f(x) 的作用。求反函数时，令 y = f(x)，交换 x 和 y，再解出 y。f⁻¹ 的定义域是 f 的值域，其值域是 f 的定义域。图像上，反函数关于直线 y = x 对称。
4. Coordinate Geometry | 坐标几何
For points A(x₁, y₁) and B(x₂, y₂), the gradient m = (y₂ – y₁)/(x₂ – x₁). The midpoint is ((x₁ + x₂)/2, (y₁ + y₂)/2). The distance AB = √((x₂ – x₁)² + (y₂ – y₁)²).
对于点 A(x₁, y₁) 和 B(x₂, y₂)，斜率 m = (y₂ – y₁)/(x₂ – x₁)，中点坐标为 ((x₁ + x₂)/2, (y₁ + y₂)/2)，距离 AB = √((x₂ – x₁)² + (y₂ – y₁)²)。
The equation of a straight line can be written as y = mx + c (slope-intercept form) or y – y₁ = m(x – nt-slope form). Parallel lines have equal gradients; perpendicular lines satisfy m₁ × m₂ = –1.
直线方程可写作斜截式 y = mx + c，或点斜式 y – y₁ = m(x – x₁)。平行线斜率相等；垂直线满足 m₁ × m₂ = –1。
The equation of a circle with centre (a, b) and radius r is (x – a)² + (y – int is perpendicular to the radius at that point. To find the equation of a tangent, use the gradient of the radius to determine the perpendicular gradient, then apply the point-slope form.
圆心 (a, b)、半径 r 的圆方程为 (x – a)² + (y – b)² = r²。圆上某点的切线与该点处的半径垂直。求切线方程时，先由半径斜率得出垂线的斜率，再使用点斜式。
5. Differentiation | 微分
ower rule extends to sums and constant multiples: if y = a f(x) + b g(x), then dy/dx = a f '(x) + b g'(x).
y = xⁿ 的导数是 dy/dx = nxⁿ⁻¹。这一幂法则适用于和与常数倍：若 y = a f(x) + b g(x)，则 dy/dx = a f '(x) + b g'(x)。
d/dx (xⁿ) = nxⁿ⁻¹
nt is given by dy/dx evaluated at that point. The equation of the tangent at (x₁, y₁) is y – y₁ = m(x – x₁) with m = dy/dx. The normal is perpendicular to the tangent, so its gradient is –1/m (provided m ≠ 0).
曲线在某点的斜率由该点的导数值给出。点 (x₁, y₁) 处的切线方程为 y – y₁ = m(x – x₁)，其中 m = dy/dx。法线与切线垂直，其斜率为 –1/m（若 m ≠ 0）。
e d²y/dx²: if d²y/dx² > 0, it is a local minimum; if d²y/dx² < 0, a local maximum; if d²y/dx² = 0, the test is inconclusive.
驻点出现在 dy/dx = 0 处。判断驻点类型可检查两侧导数的符号（一阶导数检验）或使用二阶导数：若 d²y/dx² > 0，为局部极小值；d²y/dx² < 0，为局部极大值；若 d²y/dx² = 0，则该检验无法判定。
6. Integration | 积分
Integration reverses differentiation. The indefinite integral of xⁿ (n ≠ – where C is the constant of integration. The integral of a sum is the sum of the integrals.
积分是微分的逆运算。xⁿ（n ≠ –1）的不定积分为 ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C，其中 C 为积分常数。函数和的积分等于积分的和。
∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ –1)
= b. Evaluate it by substituting the limits into the antiderivative F(x): [F(x)]ₐᵇ = F(b) – F(a).
定积分 ∫ₐᵇ f(x) dx 表示曲线 y = f(x)、x 轴以及直线 x = a 和 x = b 围成的带符号面积。计算时先求原函数 F(x)，再代入上下限：[F(x)]ₐᵇ = F(b) – F(a)。
To find the area enclosed between a curve and a line, calculate the definite integral of the difference of the functions, usually (upper curve − lower curve) between the intersection points. Areas below the x-axis will be negative if evaluated simply; take the absolute value when finding total area.
求曲线与直线围成的面积时，计算两函数差的定积分，通常为（上方曲线 − 下方曲线）在交点之间的积分。位于 x 轴下方的面积若直接积分会得负值，求总面积时须取绝对值。
7. Matrices | 矩阵
tries zero. The identity matrix I (2 × 2) is [[1,0],[0,1]]; for any matrix A, AI = IA = A.
矩阵是一个数字矩形阵列。具有 m 行 n 列的矩阵称为 m × n 阶矩阵。零矩阵 O 的所有元素为零。单位矩阵 I（2 × 2）为 [[1,0],[0,1]]；对任意矩阵 A，有 AI = IA = A。
Matrix addition: add corresponding elements. Multiplication AB is defined only if the number of columns in A equals the number of rows in B. Multiply rows of A by columns of B. Matrix multiplication is not commutative: generally AB ≠ BA.
矩阵加法：对应元素相加。乘法 AB 仅在 A 的列数等于 B 的行数时有定义，用 A 的行与 B 的列相乘。矩阵乘法不满足交换律：一般 AB ≠ BA。
The determinant of a 2×2 matrix M = [[a,b],[c,d]] is det M = ad – bc. The inverse M⁻¹ exists only if det M ≠ 0, given by:
2×2 矩阵 M = [[a,b],[c,d]] 的行列式为 det M = ad – bc。仅当 det M ≠ 0 时逆矩阵存在，公式为：
M⁻¹ = 1/(ad – bc) [ [d, –b], [–c, a] ]
Common transformation matrices in 2D include reflection in the x-axis [[1,0],[0,–1]], reflection in the line y = x [[0,1],[1,0]], rotation by 90° anticlockwise [[0,– mbine transformations by multiplying matrices, with the first transformation on the right. Published by TutorHao | Year 10 进阶数学 Revision Series | aleveler.com
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