📚 Year 11 CIE Additional Maths: Formula & Theorem Quick Reference | CIE 进阶数学：公式定理速查手册
This handbook collects all the essential formulas, identities, and theorems needed for the Year 11 CIE Additional Mathematics syllabus. It is designed as a fast reference so you can quickly recall key results in algebra, trigonometry, calculus, vectors, and coordinate geometry while revising or solving problems.
本手册汇集了 Year 11 CIE 进阶数学课程所需的所有核心公式、恒等式和定理。它旨在提供快速参考，让你在复习或解题时能够迅速回忆代数、三角、微积分、向量和坐标几何中的关键结论。
1. Surds and Indices | 根式与指数
A surd is an irrational root such as √2. The fundamental property is √ab = √a × √b for a, b ≥ 0.
根式是无理数的根式，例如 √2。基本性质为：对于 a, b ≥ 0，有 √ab = √a × √b。
To rationalise a denominator like 1/(√a + √b), multiply numerator and denominator by the conjugate √a - √b.
要将形如 1/(√a + √b) 的分母有理化，可将分子分母同乘共轭根式 √a - √b。
The index laws for all real exponents are: aᵐ × aⁿ = aᵐ⁺ⁿ and (aᵐ)ⁿ = aᵐⁿ.
适用于所有实数指数的指数律为：aᵐ × aⁿ = aᵐ⁺ⁿ 以及 (aᵐ)ⁿ = aᵐⁿ。
Zero and negative indices: a⁰ = 1 (a ≠ 0), a⁻ⁿ = 1/aⁿ.
零指数与负指数：a⁰ = 1（a ≠ 0），a⁻ⁿ = 1/aⁿ。
Fractional indices link roots and powers: a1/n = √[n]a and am/n = (√[n]a)ᵐ.
分数指数连接根式与幂：a1/n = ⁿ√a，am/n = (ⁿ√a)ᵐ。
2. Quadratic Functions | 二次函数
² + bx + c. Its graph is a parabola; if a > 0 it opens upward, if a < 0 it opens downward.
一般二次函数为 f(x) = ax² + bx + c。其图像为抛物线；若 a > 0 则开口向上，若 a < 0 则开口向下。
Completing the square rewrites the expression as a(x + p)² + q, where the vertex is at (-p, q).
配方法将表达式写成 a(x + p)² + q 的形式，其中顶点坐标为 (-p, q)。
The quadratic formula for the roots of ax² + bx + c = 0 is x = [-b ± √b² - 4ac] / (2a).
方程 ax² + bx + c = 0 的求根公式为 x = [-b ± √b² - 4ac] / (2a)。
wo distinct real roots, Δ = 0 gives one repeated real root, and Δ < 0 gives no real roots.
判别式 Δ = b² - 4ac 决定根的性质：Δ > 0 有两个不等实根，Δ = 0 有一个重实根，Δ < 0 无实根。
+ c = 0 with roots α and β, the sum of roots is α + β = -b/a and the product is αβ = c/a.
对于根为 α 和 β 的方程 ax² + bx + c = 0，根的和为 α + β = -b/a，根的积为 αβ = c/a。
3. Equations and Inequalities | 方程与不等式
A quadratic inequality such as ax² + bx + c > 0 is solved by sketching the parabola and identifying the intervals where the curve is above the x-axis.
求解二次不等式 ax² + bx + c > 0，可通过绘制抛物线并找出曲线位于 x 轴上方的区间。
For modulus inequalities like |x - a| < b, rewrite as -b < x - a < b and solve the compound inequality.
对于绝对值不等式 |x - a| < b，改写为 -b < x - a < b 并求解复合不等式。
Equations involving fractions are cleared by multiplying through by the lowest common denominator, always checking for extraneous solutions.
涉及分式的方程两边同乘最简公分母以消去分母，并始终检查是否有增根。
Simultaneous equations can be solved by substitution, elimination, or graphically to find the intersection points of the corresponding curves.
联立方程可通过代入法、消元法或图形法求解，求出相应曲线的交点坐标。
4. Polynomials and Factor Theorem | 多项式与因式定理
The Remainder Theorem states: when a polynomial p(x) is divided by (x - a), the remainder is p(a).
余式定理指出：当多项式 p(x) 除以 (x - a) 时，余式为 p(a)。
The Factor Theorem follows: (x - a) is a factor of p(x) if and only if p(a) = 0.
因式定理如下：(x - a) 是 p(x) 的因式当且仅当 p(a) = 0。
To factorise a cubic, first find one linear factor using the factor theorem, then perform polynomial division or equate coefficients to obtain the remaining quadratic factor.
要对三次多项式因式分解，先用因式定理找出一个线性因式，然后进行多项式除法或比较系数以得到剩余的二次因式。
A polynomial of degree n can have at most n real roots, and its graph will cross or touch the x-axis at most n times.
n 次多项式最多有 n 个实根，其图像与 x 轴的交点或切点最多 n 处。
5. Binomial Expansion | 二项式展开
The binomial coefficient nCr = n! / (r!(n - r)!) counts the number of ways to choose r objects from n.
二项式系数 nCr = n! / (r!(n - r)!) 表示从 n 个物体中选取 r 个的方案数。
For a positive integer n, the expansion is (a + b)ⁿ = Σ (nCr) aⁿ⁻ʳ bʳ, where r runs from 0 to n.
对于正整数 n，展开式为 (a + b)ⁿ = Σ (nCr) aⁿ⁻ʳ bʳ，其中 r 从 0 到 n。
The term independent of x in an expansion can be found by setting the power of x to zero and solving for r.
展开式中与 x 无关的项可通过将 x 的指数设为零并解出 r 来求得。
For (1 + x)ⁿ with |x| < 1 and n not a positive integer, the expansion is valid as an infinite series: 1 + nx + n(n-1)x²/2! + ... .
当 |x| < 1 且 n 不是正整数时，(1 + x)ⁿ 可展开为无穷级数：1 + nx + n(n-1)x²/2! + ... 。
6. Exponential and Logarithmic Functions | 指数与对数函数
The natural exponential function is y = eˣ and its inverse is the natural logarithm y = ln x, where ln(eˣ) = x and eln x = x.
自然指数函数为 y = eˣ，其反函数为自然对数 y = ln x，满足 ln(eˣ) = x 且 eln x = x。
≠ 1: logₐ (xy) = logₐ x + logₐ y, logₐ (x/y) = logₐ x - logₐ y, and logₐ (xᵏ) = k logₐ x.
对数运算律对任意底数 a > 0, a ≠ 1 成立：logₐ (xy) = logₐ x + logₐ y，logₐ (x/y) = logₐ x - logₐ y，logₐ (xᵏ) = k logₐ x。
Change of base formula: logₐ b = log b / log a (using any common base).
换底公式：logₐ b = log b / log a（可使用任意常用底数）。
Exponential growth and decay are modeled by A = A₀ ekt; solving for t often requires taking natural logarithms of both sides.
指数增长与衰减用模型 A = A₀ ekt；求解 t 时常需对方程两边取自然对数。
7. Trigonometry: Radians, Arc, Sector | 三角学：弧度、弧长、扇形
s are measured in radians: π rad = 180°. To convert degrees to radians, multiply by π/180.
角度用弧度度量：π 弧度 = 180°。将角度转换为弧度，乘以 π/180。
circle of radius r and angle θ (in radians), arc length s = rθ and sector area A = ½ r² θ.
对于半径为 r、圆心角为 θ（弧度）的圆，弧长 s = rθ，扇形面积 A = ½ r² θ。
gent of common angles like π/6, π/4, π/3 should be known exactly: sin(π/6) = ½, cos(π/6) = √3/2, tan(π/4) = 1, etc.
应熟记常用角 π/6, π/4, π/3 的正弦、余弦和正切值：sin(π/6) = ½, cos(π/6) = √3/2, tan(π/4) = 1 等。
Trigonometric graphs for y = a sin(bx) + c have amplitude |a|, period 2π/b, and vertical shift c.
三角函数 y = a sin(bx) + c 的图像具有振幅 |a|、周期 2π/b 和垂直平移 c。
8. Trigonometric Identities and Equations | 三角恒等式与方程
is sin² θ + cos² θ = 1, from which we derive 1 + tan² θ = sec² θ and 1 + cot² θ = csc² θ.
基本恒等式为 sin² θ + cos² θ = 1，由此可得 1 + tan² θ = sec² θ 和 1 + cot² θ = csc² θ。
Compound angle formulas: sin(A ± B) = sin A cos B ± cos A sin B; cos(A ± B) = cos A cos B ∓ sin A sin B; tan(A ± B) = (tan A ± tan B) / (1 ∓ tan A tan B).
和角公式：sin(A ± B) = sin A cos B ± cos A sin B；cos(A ± B) = cos A cos B ∓ sin A sin B；tan(A ± B) = (tan A ± tan B) / (1 ∓ tan A tan B)。
cos 2A = cos² A - sin² A = 2 cos² A - 1 = 1 - 2 sin² A, tan 2A = (2 tan A) / (1 - tan² A).
倍角公式：sin 2A = 2 sin A cos A，cos 2A = cos² A - sin² A = 2 cos² A - 1 = 1 - 2 sin² A，tan 2A = (2 tan A) / (1 - tan² A)。
Trigonometric equations are solved by using identities to reduce them to a single trig function, then finding all solutions in the required interval.
求解三角方程时，利用恒等式将其化为单一三角函数，然后求出给定区间内的所有解。
9. Coordinate Geometry: Lines and Circles | 坐标几何：直线与圆
₁) and (x₂, y₂) is √[(x₂ - x₁)² + (y₂ - y₁)²]; the midpoint is ((x₁ + x₂)/2, (y₁ + y₂)/2).
两点 (x₁, y₁) 和 (x₂, y₂) 之间的距离为 √[(x₂ - x₁)² + (y₂ - y₁)²]；中点为 ((x₁ + x₂)/2, (y₁ + y₂)/2)。
Two lines are parallel if their gradients are equal; they are perpendicular if m₁ m₂ = -1.
过两点的直线斜率为 m = (y₂ - y₁) / (x₂ - x₁)。两直线平行若斜率相等；垂直若 m₁ m₂ = -1。
of a straight line: y - y₁ = m(x - x₁) or y = mx + c. The general form is ax + by + c = 0.
直线方程：y - y₁ = m(x - x₁) 或 y = mx + c。一般式为 ax + by + c = 0。
y - b)² = r². The expanded form x² + y² + 2gx + 2fy + c = 0 has centre (-g, -f) and radius √g² + f² - c.
圆心为 (a, b)、半径为 r 的圆的方程为 (x - a)² + (y - b)² = r²。展开式 x² + y² + 2gx + 2fy + c = 0 中，圆心为 (-g, -f)，半径为 √g² + f² - c。
10. Differentiation | 微分
The derivative of xⁿ is n xⁿ⁻¹ for any real constant n. The derivative of a constant is zero.
xⁿ 的导数为 n xⁿ⁻¹（n 为任意实常数）。常数的导数为零。
d derivatives: d/dx (sin x) = cos x, d/dx (cos x) = -sin x, d/dx (tan x) = sec² x, d/dx (eˣ) = eˣ, d/dx (ln x) = 1/x.
标准导数：d/dx (sin x) = cos x，d/dx (cos x) = -sin x，d/dx (tan x) = sec² x，d/dx (eˣ) = eˣ，d/dx (ln x) = 1/x。
Product rule: if y = u v, then dy/dx = u (dv/dx) + v (du/dx).
乘法法则：若 y = u v，则 dy/dx = u (dv/dx) + v (du/dx)。
Quotient rule: if y = u/v, then dy/dx = (v du/dx - u dv/dx) / v².
除法法则：若 y = u/v，则 dy/dx = (v du/dx - u dv/dx) / v²。
Chain rule: if y = f(g(x)), then dy/dx = f'(g(x)) g'(x). In function notation, dy/dx = (dy/du)(du/dx).
链式法则：若 y = f(g(x))，则 dy/dx = f'(g(x)) g'(x)。用函数记号表示为 dy/dx = (dy/du)(du/dx)。
d derivative test to determine if the point is a maximum, minimum, or point of inflection.
平稳点出现在 dy/dx = 0 处。用一阶或二阶导数检验以确定该点是极大值、极小值还是拐点。
11. Integration | 积分
Integration
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