📚 A-Level CIE Mathematics: Formula Handbook | A-Level CIE 数学：公式汇总手册
Mastering A-Level CIE Mathematics requires a solid grasp of key formulas across Pure Mathematics, Mechanics, and Statistics. This handbook compiles essential equations, identities, and relationships you must know for the 9709 syllabus, enabling quick revision and exam confidence.
掌握 A-Level CIE 数学需要在纯数学、力学和统计中牢固掌握关键公式。本手册汇总了 9709 大纲必须掌握的核心方程、恒等式和关系式，帮助你快速复习并增强考试信心。
1. Algebraic Identities and Surds | 代数恒等式与根式
(a + b)² = a² + 2ab + b²
(a + b)² = a² + 2ab + b²
(a - b)² = a² - 2ab + b²
(a - b)² = a² - 2ab + b²
a² - b² = (a + b)(a - b)
a² - b² = (a + b)(a - b)
(a + b)³ = a³ + 3a²b + 3ab² + b³
(a + b)³ = a³ + 3a²b + 3ab² + b³
√ab = √a · √b, √a/b = √a / √b
√ab = √a · √b, √a/b = √a / √b
1 / (√a + √b) = (√a - √b) / (a - b) (rationalising)
1 / (√a + √b) = (√a - √b) / (a - b) （分母有理化）
2. Quadratics | 二次函数与方程
For ax² + bx + c = 0, the discriminant is Δ = b² - 4ac.
对于 ax² + bx + c = 0，判别式 Δ = b² - 4ac。
x = [ -b ± √b² - 4ac ] / (2a)
x = [ -b ± √b² - 4ac ] / (2a)
Sum of roots: α + β = -b/a ; product: αβ = c/a.
根的和：α + β = -b/a ；根的积：αβ = c/a。
Completed square form: a(x - h)² + k, vertex (h, k).
完全平方形式：a(x - h)² + k，顶点为 (h, k)。
3. Functions and Logarithms | 函数与对数
f(x) = ax + b ⇒ inverse f⁻¹(x) = (x - b)/a.
f(x) = ax + b ⇒ 反函数 f⁻¹(x) = (x - b)/a。
Log rules: logₐ(xy) = logₐx + logₐy ; logₐ(x/y) = logₐx - logₐy ; logₐ(xⁿ) = n logₐx.
对数运算法则：logₐ(xy) = logₐx + logₐy ； logₐ(x/y) = logₐx - logₐy ； logₐ(xⁿ) = n logₐx。
logₐb = logₒb / logₒa (change of base)
logₐb = logₒb / logₒa （换底公式）
4. Coordinate Geometry | 坐标几何
Distance between (x₁, y₁) and (x₂, y₂): d = √[(x₂ - x₁)² + (y₂ - y₁)²].
两点 (x₁, y₁) 与 (x₂, y₂) 间的距离：d = √[(x₂ - x₁)² + (y₂ - y₁)²]。
Midpoint: M = ((x₁ + x₂)/2 , (y₁ + y₂)/2). Gradient m = (y₂ - y₁)/(x₂ - x₁).
中点：M = ((x₁ + x₂)/2 , (y₁ + y₂)/2)。斜率 m = (y₂ - y₁)/(x₂ - x₁)。
Line equation: y - y₁ = m(x - x₁), or ax + by + c = 0.
直线方程：y - y₁ = m(x - x₁) 或 ax + by + c = 0。
Circle centre (a, b) radius r: (x - a)² + (y - b)² = r².
圆心 (a, b)、半径 r 的圆：(x - a)² + (y - b)² = r²。
5. Trigonometry | 三角学
sin²θ + cos²θ = 1, tanθ = sinθ / cosθ
sin²θ + cos²θ = 1, tanθ = sinθ / cosθ
Reciprocals: secθ = 1/cosθ, cscθ = 1/sinθ, cotθ = 1/tanθ.
倒数关系：secθ = 1/cosθ，cscθ = 1/sinθ，cotθ = 1/tanθ。
Addition formulas: sin(A±B) = sinA cosB ± cosA sinB; cos(A±B) = cosA cosB ∓ sinA sinB; tan(A±B) = (tanA ± tanB) / (1 ∓ tanA tanB).
和角公式：sin(A±B) = sinA cosB ± cosA sinB；cos(A±B) = cosA cosB ∓ sinA sinB；tan(A±B) = (tanA ± tanB) / (1 ∓ tanA tanB)。
sinθ cosθ ; cos2θ = cos²θ - sin²θ = 2cos²θ - 1 = 1 - 2sin²θ ; tan2θ = 2tanθ / (1 - tan²θ).
倍角公式：sin2θ = 2sinθ cosθ ；cos2θ = cos²θ - sin²θ = 2cos²θ - 1 = 1 - 2sin²θ ；tan2θ = 2tanθ / (1 - tan²θ)。
Radians: π rad = 180°, arc length s = rθ, sector area A = ½ r²θ.
弧度制：π rad = 180°，弧长 s = rθ，扇形面积 A = ½ r²θ。
6. Sequences and Series | 数列与级数
Arithmetic nth term: uₙ = a + (n-1)d. Sum: Sₙ = n/2 [2a + (n-1)d].
等差数列第 n 项：uₙ = a + (n-1)d。求和：Sₙ = n/2 [2a + (n-1)d]。
Geometric nth term: uₙ = arⁿ⁻¹. Sum: Sₙ = a(1 - rⁿ)/(1 - r) for r ≠ 1.
等比数列第 n 项：uₙ = arⁿ⁻¹。求和：Sₙ = a(1 - rⁿ)/(1 - r)，r ≠ 1。
Sum to infinity: S∞ = a / (1 - r), valid for |r| < 1.
无穷等比级数求和：S∞ = a / (1 - r)，要求 |r| < 1。
Binomial expansion: (1 + x)ⁿ = 1 + nx + n(n-1)x²/2! + ... for |x| < 1, n rational. General term: C(n,r) an-r br.
二项式展开：(1 + x)ⁿ = 1 + nx + n(n-1)x²/2! + ... ，|x| < 1，n 为有理数。通项：C(n,r) an-r br。
7. Differentiation | 微分
d/dx (xⁿ) = n xⁿ⁻¹
d/dx (xⁿ) = n xⁿ⁻¹
d/dx (eˣ) = eˣ, d/dx (ln x) = 1/x, d/dx (sin x) = cos x, d/dx (cos x) = -sin x, d/dx (tan x) = sec²x.
d/dx (eˣ) = eˣ，d/dx (ln x) = 1/x，d/dx (sin x) = cos x，d/dx (cos x) = -sin x，d/dx (tan x) = sec²x。
Chain rule: dy/dx = dy/du · du/dx.
链式法则：dy/dx = dy/du · du/dx。
Product rule: d/dx (uv) = u'v + uv'.
乘积法则：d/dx (uv) = u'v + uv'。
Quotient rule: d/dx (u/v) = (u'v - uv') / v².
商法则：d/dx (u/v) = (u'v - uv') / v²。
ic: dy/dx = (dy/dt) / (dx/dt). Implicit differentiation: differentiate both sides w.r.t x.
参数微分：dy/dx = (dy/dt) / (dx/dt)。隐函数微分：对等式两边关于 x 求导。
8. Integration | 积分
∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, n ≠ -1
∫ xⁿ dx = xⁿ⁺¹/(n+1) + C，n ≠ -1
∫ 1/x dx = ln|x| + C, ∫ eˣ dx = eˣ + C, ∫ sin x dx = -cos x + C, ∫ cos x dx = sin x + C.
∫ 1/x dx = ln|x| + C，∫ eˣ dx = eˣ + C，∫ sin x dx = -cos x + C，∫ cos x dx = sin x + C。
Substitution: ∫ f(g(x))g'(x) dx = ∫ f(u) du, where u = g(x).
换元积分：∫ f(g(x))g'(x) dx = ∫ f(u) du，其中 u = g(x)。
Integration by parts: ∫ u dv = uv - ∫ v du.
分部积分：∫ u dv = uv - ∫ v du。
ral: ∫ₐᵇ f(x) dx = F(b) - F(a). Area between curves, volume of revolution V = π ∫ₐᵇ y² dx.
定积分：∫ₐᵇ f(x) dx = F(b) - F(a)。曲线间的面积，旋转体体积 V = π ∫ₐᵇ y² dx。
9. Vectors | 向量
Position vector: r = xi + yj + zk. Magnitude |r| = √x² + y² + z².
位置向量：r = xi + yj + zk。模长 |r| = √x² + y² + z²。
Dot product: a · b = |a||b| cos θ = x₁x₂ + y₁y₂ + z₁z₂.
点积：a · b = |a||b| cos θ = x₁x₂ + y₁y₂ + z₁z₂。
Angle between vectors: cos θ = (a · b) / (|a||b|).
向量夹角：cos θ = (a · b) / (|a||b|)。
a × b = (y₁z₂ - z₁y₂)i - (x₁z₂ - z₁x₂)j + (x₁y₂ - y₁x₂)k; direction perpendicular to both.
叉积（CIE P3）：a × b = (y₁z₂ - z₁y₂)i - (x₁z₂ - z₁x₂)j + (x₁y₂ - y₁x₂)k；方向垂直于两向量。
Vector equation of a line: r = a + tb. Scalar product form of a plane: r · n = a · n.
直线向量方程：r = a + tb。平面点积形式：r · n = a · n。
10. Complex Numbers | 复数
z = x + iy, where i² = -1. Real part Re(z) = x, imaginary part Im(z) = y.
z = x + iy，其中 i² = -1。实部 Re(z) = x，虚部 Im(z) = y。
Modulus: |z| = √x² + y². Argument: θ = arg(z) = tan⁻¹(y/x).
模：|z| = √x² + y²。辐角：θ = arg(z) = tan⁻¹(y/x)。
Polar form: z = r(cos θ + i sin θ). Exponential form: z = r eiθ.
极坐标形式：z = r(cos θ + i sin θ)。指数形式：z = r eiθ。
De Moivre's theorem: (cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ).
棣莫弗定理：(cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ)。
Roots of unity: zⁿ = 1 ⇒ z = cos(2kπ/n) + i sin(2kπ/n), k = 0,1,...,n-1.
单位根：zⁿ = 1 ⇒ z = cos(2kπ/n) + i sin(2kπ/n)，k = 0,1,...,n-1。
11. Mechanics - Kinematics | 力学 - 运动学
Constant acceleration (SUVAT) equations:
匀加速运动公式（SUVAT）：
v = u + at
v = u + at
s = ut + ½ at²
s = ut + ½ at²
s = (u + v)t / 2
s = (u + v)t / 2
v² = u² + 2as
v² = u² + 2as
re s = displacement, u = initial velocity, v = final velocity, a = acceleration, t = time.
其中 s = 位移，u = 初速度，v = 末速度，a = 加速度，t = 时间。
Using calculus: v = ds/dt, a = dv/dt = d²s/dt². s = ∫ v dt, v = ∫ a dt.
利用微积分：v = ds/dt，a = dv/dt = d²s/dt²。s = ∫ v dt，v = ∫ a dt。
12. Probability & Statistics | 概率与统计
Probability rules: P(A∪B) = P(A) + P(B) - P(A∩B). For independent events, P(A∩B) = P(A)P(B).
概率法则：P(A∪B) = P(A) + P(B) - P(A∩B)。若事件独立，P(A∩B) = P(A)P(B)。
Conditional probability: P(A|B) = P(A∩B)/P(B).
条件概率：P(A|B) = P(A∩B)/P(B)。
Permutations: ⁿPᵣ = n!/(n-r)!. Combinations: ⁿCᵣ = n!/(r!(n-r)!).
排列：ⁿPᵣ = n!/(n-r)!。组合：ⁿCᵣ = n!/(r!(n-r)!)。
Binomial distribution: X ~ B(n, p), P(X=r) = ⁿCᵣ pʳ (1-p)ⁿ⁻ʳ. Mean μ = np, variance σ² = np(1-p).
二项分布：X ~ B(n, p)，P(X=r) = ⁿCᵣ pʳ (1-p)ⁿ⁻
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