Year 11 Cambridge Further Maths Quick Reference Formula and Theorem Handbook | Year 11 剑桥进阶数学：公式定理速查手册

📚 Year 11 Cambridge Further Maths Quick Reference Formula and Theorem Handbook | Year 11 剑桥进阶数学：公式定理速查手册

This concise handbook gathers the essential formulas and theorems for the Year 11 Cambridge Further Mathematics course (IGCSE Additional Mathematics 0606 / early A Level Pure). Use it to revise key algebra, coordinate geometry, trigonometry, calculus and vector techniques that underpin problem‑solving and examination success.

本手册汇总了 Year 11 剑桥进阶数学（IGCSE Additional Mathematics 0606 或早期 A Level 纯数）的核心公式与定理，涵盖代数、坐标几何、三角学、微积分及向量等关键内容，便于随时查阅，助力备考与解题。

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1. Algebra and Identities | 代数与恒等式

expansion and perfect squares are used frequently: (a ± b)² = a² ± 2ab + b², and (a + b)(a – b) = a² – b².

二次展开和完全平方公式频繁出现：(a ± b)² = a² ± 2ab + b², 以及 (a + b)(a – b) = a² – b²。

When factorising sums or differences of cubes, remember a³ + b³ = (a + b)(a² – ab + b²) and a³ – b³ = (a – b)(a² + ab + b²).

立方和与立方差分解时，记住 a³ + b³ = (a + b)(a² – ab + b²) 以及 a³ – b³ = (a – b)(a² + ab + b²)。

The Binomial Theorem for positive integer n: (1 + x)ⁿ = 1 + nx + [n(n–1)/2!] x² + … + [n(n–1)…(n–r+1)/r!] xʳ. For (a + b)ⁿ, factor out aⁿ to obtain aⁿ(1 + b/a)ⁿ.

正整数 n 的二项式定理：(1 + x)ⁿ = 1 + nx + [n(n–1)/2!] x² + … + [n(n–1)…(n–r+1)/r!] xʳ。对于 (a + b)ⁿ，提取 aⁿ 得到 aⁿ(1 + b/a)ⁿ。

(a + b)² = a² + 2ab + b²

a³ – b³ = (a – b)(a² + ab + b²)

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2. Quadratic Functions | 二次函数

The general form is f(x) = ax² + bx + c. The discriminant Δ = b² – s: Δ > 0 gives two distinct real roots, Δ = 0 one real repeated root, Δ < 0 no real roots.

一般式为 f(x) = ax² + bx + c。判别式 Δ = b² – 4ac 决定根的性质：Δ > 0 两个不等实根，Δ = 0 一个重实根，Δ < 0 无实根。

The quadratic formula solves ax² + bx + c = 0: x = [–b ± √b² – 4ac] / (2a).

求根公式为 ax² + bx + c = 0 的解：x = [–b ± √b² – 4ac] / (2a)。

Sum and product of roots (α + β) = –b/a, αβ = c/a. The completed square form is a(x – h)² + k, with vertex (h, k).

根的和与积：α + β = –b/a, αβ = c/a。完全平方形式为 a(x – h)² + k，顶点坐标为 (h, k)。

x = [–b ± √b² – 4ac] / (2a)

α + β = –b/a, αβ = c/a

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3. Equations and Inequalities | 方程与不等式

Simultaneous equations can be solved by substitution or elimination. For linear–quadratic pairs, substitute the linear expression into the quadratic and use the discriminant to check for intersections.

联立方程可通过代入法或消元法求解。对于一次与二次联立，将线性表达式代入二次方程，并利用判别式检验交点个数。

When solving quadratic inequalities, e.g. x² – 5x + 6 > 0, factorise to (x – 2)(x – 3) > 0 and use a sign diagram. The solution is x < 2 or x > 3.

解二次不等式时，如 x² – 5x + 6 > 0，因式分解为 (x – 2)(x – 3) > 0 并利用符号图，解集为 x < 2 或 x > 3。

For modulus equations |ax + b| = c, rewrite as ax + b = ±c. For inequalities |ax + b| < c, solve –c < ax + b < c; for > c solve two separate inequalities.

绝对值方程 |ax + b| = c 转化为 ax + b = ±c。绝对值不等式 |ax + b| < c 转化为 –c < ax + b < c；> c 则拆成两个不等式。

|2x – 1| ≤ 5 ⇒ –5 ≤ 2x – 1 ≤ 5

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4. Exponentials and Logarithms | 指数与对数

Laws of exponents: aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1, a⁻ⁿ = 1/aⁿ.

指数运算法则：aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1, a⁻ⁿ = 1/aⁿ。

Logarithms are the inverse of exponentials: if y = aˣ then x = logₐ y. Common bases: logₐ a = 1, logₐ 1 = 0. Natural log ln x = logₑ x.

对数是指数的逆运算：若 y = aˣ 则 x = logₐ y。常用性质：logₐ a = 1，logₐ 1 = 0。自然对数 ln x = logₑ x。

Laws of logs: logₐ (xy) = logₐ x + logₐ y, logₐ (x/y) = logₐ x – logₐ y, logₐ xⁿ = n logₐ x. Change of base: logₐ b = log_c b / log_c a.

对数法则：logₐ (xy) = logₐ x + logₐ y, logₐ (x/y) = logₐ x – logₐ y, logₐ xⁿ = n logₐ x。换底公式：logₐ b = log_c b / log_c a。

ln (e³x) = 3 + ln x

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5. Coordinate Geometry and Lines | 坐标几何与直线

Distance between two points (x₁, y₁) and (x₂, y₂): d = √[(x₂ – x₁)² + (y₂ – y₁)²]. Midpoint: M = ((x₁+x₂)/2, (y₁+y₂)/2).

两点 (x₁, y₁) 与 (x₂, y₂) 的距离：d = √[(x₂ – x₁)² + (y₂ – y₁)²]。中点：M = ((x₁+x₂)/2, (y₁+y₂)/2)。

Gradient of a line: m = (y₂ – y₁)/(x₂ – x₁). The equation of a straight line can be written as y – y₁ = m(x – x₁) or y = mx + c, where c is the y‑intercept.

直线的斜率：m = (y₂ – y₁)/(x₂ – x₁)。直线方程可写为 y – y₁ = m(x – x₁) 或 y = mx + c，其中 c 为 y 轴截距。

Parallel lines have equal gradients; perpendicular lines satisfy m₁ × m₂ = –1. For lines ax + by + c = 0, the gradient is –a/b and the perpendicular gradient is b/a.

平行线斜率相等；垂直线的斜率满足 m₁ × m₂ = –1。对于直线 ax + by + c = 0，斜率为 –a/b，垂直斜率为 b/a。

d = √[(x₂ – x₁)² + (y₂ – y₁)²]

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6. Trigonometry | 三角学

For right‑ t/hypotenuse, tan θ = opposite/adjacent. The exact values for 30°, 45°, 60° must be known.

直角三角形中：sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。应牢记 30°、45°、60° 的精确值。

le: a/sin A = b/sin B = c/sin C, or sin A/a = sin B/b = sin C/c. Cosine rule: a² = b² + c² – 2bc cos A.

正弦定理：a/sin A = b/sin B = c/sin C，或 sin A/a = sin B/b = sin C/c。余弦定理：a² = b² + c² – 2bc cos A。

x and y = tan x, and transformations such as amplitude and period changes, are essential.

核心恒等式：tan θ = sin θ / cos θ，sin² θ + cos² θ = 1。三角函数 y = sin x、y = cos x 与 y = tan x 的图像，以及振幅、周期等变换，须熟练掌握。

sin 30° = ½, cos 45° = √2/2, tan 60° = √3

sin² θ + cos² θ ≡ 1

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7. Circular Measure and Radians | 弧度制

rt degrees to radians, multiply by π/180; to convert radians to degrees, multiply by 180/π.

弧度制：π 弧度 = 180°。角度转弧度乘以 π/180；弧度转角度乘以 180/π。

θ rad at centre: s = rθ. Area of sector: A = ½ r² θ. Area of segment: A_segment = ½ r² (θ – sin θ).

半径为 r 的圆，圆心角 θ 弧度对应的弧长：s = rθ。扇形面积：A = ½ r² θ。弓形面积：A_segment = ½ r² (θ – sin θ)。

When using these formulas, θ must be in radians. The chord length is 2r sin(θ/2).

使用这些公式时，θ 必须为弧度。弦长为 2r sin(θ/2)。

s = rθ, A_sector = ½ r²θ

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8. Functions | 函数

input x (domain) to exactly one output f(x) (range). The notation f: x → 2x + 3 is common.

函数将定义域内每个输入 x 对应到唯一的输出 f(x)（值域）。常用符号如 f: x → 2x + 3。

g; to find it, rearrange y = f(x) to make x the subject and swap variables. f⁻¹(f(x)) = x.

复合函数：(gf)(x) = g(f(x))。反函数 f⁻¹(x) 逆转映射；可由 y = f(x) 解出 x，再交换变量得到。满足 f⁻¹(f(x)) = x。

The modulus function |x| gives the absolute value. Its graph is a V‑ f(x) + a (vertical shift), y = f(x + a) (horizontal shift), y = af(x) (vertical stretch).

绝对值函数 |x| 给出非负值，图像为 V 形。函数变换：y = f(x) + a 垂直平移，y = f(x + a) 水平平移，y = af(x) 纵向伸缩。

f(x) = |x – 2| is a V‑shape with vertex at (2, 0)

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9. Differentiation | 微分

The derivative f ' ient of a curve. For y = xⁿ, dy/dx = n xⁿ⁻¹ (n ≠ 0). Constant term differentiates to zero.

导数 f '(x) 或 dy/dx 表示曲线斜率。对于 y = xⁿ，dy/dx = n xⁿ⁻¹（n ≠ 0）。常数项导数为零。

The gradient of the tangent at x = a is f '(a). The normal gradient is –1/f '(a). Equation of tangent: y – y₁ = m_T (x – x₁).

在 x = a 处的切线斜率为 f '(a)，法线斜率为 –1/f '(a)。切线方程：y – y₁ = m_T (x – x₁)。

Chain rule: if y = f(u) and u = g(x), then dy/dx = (dy/du) × (du/dx). Product rule: (uv)' = u'v + uv'. Quotient rule: (u/v)' = (u'v – uv') / v².

链式法则：若 y = f(u) 且 u = g(x)，则 dy/dx = (dy/du) × (du/dx)。乘积法则：(uv)' = u'v + uv'。商法则：(u/v)' = (u'v – uv') / v²。

occur where dy/dx = 0. Use first or second derivative test to classify maxima and minima.

二阶导数 d²y/dx² 描述凹凸性；驻点满足 dy/dx = 0。用一阶或二阶导数判别式区分极大值与极小值。

d/dx (xⁿ) = n xⁿ⁻¹

d/dx [ (3x + 1)² ] = 2(3x + 1) × 3 = 6(3x + 1)

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10. Integration | 积分

k f(x) dx = k ∫ f(x) dx. The fundamental integral ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, valid for n ≠ –1.

积分是微分的逆运算：∫ k f(x) dx = k ∫ f(x) dx。基本积分公式 ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C，其中 n ≠ –1。

Special cases: ∫ 1/x dx = ln |x| + C, ∫ eˣ dx = eˣ + C, ∫ cos kx dx = (1/k) sin kx + C, ∫ sin kx dx = –(1/k) cos kx + C.

特殊积分：∫ 1/x dx = ln |x| + C，∫ eˣ dx = eˣ + C，∫ cos kx dx = (1/k) sin kx + C，∫ sin kx dx = –(1/k) cos kx + C。

resents the signed area under y = f(x) from x = a to x = b. Area between curves: ∫ₐᵇ (f(x) – g(x)) dx, where f(x) is above g(x).

定积分 ∫ₐᵇ f(x) dx 表示曲线 y = f(x) 从 x = a 到 x = b 的带符号面积。两曲线间的面积：∫ₐᵇ (f(x) – g(x)) dx，前提 f(x) 在上方。

When integrating without limits, always add the constant of integration C. Area under the x‑axis is subtracted.

不定积分必须加积分常数 C。x 轴下方的面积需作减法处理。

∫ (2x³ + 4x) dx = ½ x⁴ + 2x² + C

Area = ∫₀² (x² + 1) dx

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11. Vectors in Two Dimensions | 平面向量

A vector has magnitude and direction. In component form: a = xi + yj, with magnitude |a| = √x² + y². The zero vector is 0i + 0j.

向量具有大小和方向。分量形式为 a = xi + yj，模为 |a| = √x² + y²。零向量为 0i + 0j。

Vector addition is performed component‑wise: a + b = (x₁+x₂)i + (y₁+y₂)j. Scalar multiplication: k a = kx i + ky j.

向量加法为对应分量相加：a + b = (x₁+x₂)i + (y₁+y₂)j。标量乘法：k a = kx i + ky j。

The position vector of point P(x, y) is p = xi + yj. The vector from A to B is AB = b – a. A unit vector in the direction of a is â = a / |a|.

点 P(x, y) 的位置向量为 p = xi + yj。从 A 到 B 的向量为 AB = b – a。a 方向上的单位向量为 â = a / |a|。

Dot product (scalar product): a • b = x₁ x₂ + y₁ y₂ = |a||b| cos θ, where θ is the angle between them. Two non‑zero vectors are perpendicular if a • b = 0.

点积（标量积）：a • b = x₁ x₂ + y₁ y₂ = |a||b| cos θ，其中 θ 是两向量的夹角。非零向量垂直当且仅当 a • b = 0。

a = 3i – 4j, |a| = √(3² + (–4)²) = 5

a • b = |a||b| cos θ

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