Year 11 Cambridge Additional Mathematics: Quick Reference to Formulas & Theorems | Year 11 剑桥进阶数学：公式定理速查手册

📚 Year 11 Cambridge Additional Mathematics: Quick Reference to Formulas & Theorems | Year 11 剑桥进阶数学：公式定理速查手册

This comprehensive guide covers all the key formulas, theorems, and properties required for the Cambridge IGCSE Additional Mathematics (0606) syllabus. It is designed as a revision aid to help Year 11 students quickly locate and review the essential mathematical content. Each section is presented in both English and Chinese to support bilingual learning and deepen conceptual understanding.

本综合指南涵盖了剑桥 IGCSE 进阶数学 (0606) 教学大纲所需的所有关键公式、定理和性质，旨在作为一份复习辅助资料，帮助 Year 11 学生快速查找和回顾核心数学内容。每个部分均以中英双语呈现，以支持双语学习并加深概念理解。

1. Algebra & Surds | 代数与根式

To simplify a surd, factor out the largest perfect square: √a²b = a√b, where a, b > 0. For example, √72 = √36×2 = 6√2.

化简二次根式时，提取最大的完全平方因子：√a²b = a√b，其中 a, b > 0。例如 √72 = √36×2 = 6√2。

Rationalising the denominator involves multiplying numerator and denominator by the conjugate of the denominator. For 1/(a + √b), multiply by (a - √b)/(a - √b) to obtain (a - √b)/(a² - b).

有理化分母是将分子和分母同时乘以分母的共轭式。对于 1/(a + √b)，乘以 (a - √b)/(a - √b) 可得 (a - √b)/(a² - b)。

aws of indices: aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1 (a ≠ 0), a⁻ⁿ = 1/aⁿ, a^1/n = ⁿ√a.

指数法则：aᵐ × aⁿ = aᵐ⁺ⁿ，aᵐ ÷ aⁿ = aᵐ⁻ⁿ，(aᵐ)ⁿ = aᵐⁿ，a⁰ = 1（a ≠ 0），a⁻ⁿ = 1/aⁿ，a^1/n = ⁿ√a。

2)² - (b/2)². This technique is used to solve quadratic equations and to find the vertex of a parabola.

配方法：x² + bx = (x + b/2)² - (b/2)²。此方法用于解二次方程和求抛物线的顶点。

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2. Quadratic Functions & Inequalities | 二次函数与不等式

tic equation is ax² + bx + c = 0, with solutions given by the quadratic formula: x = [-b ± √b² - 4ac Δ > 0, two distinct real roots; if Δ = 0, one repeated real root; if Δ < 0, no real roots.

一般二次方程为 ax² + bx + c = 0，其解由求根公式给出：x = [-b ± √b² - 4ac] / (2a)。判别式 Δ = b² - 4ac 决定根的性质：若 Δ > 0，有两个不同实根；若 Δ = 0，有一个重实根；若 Δ < 0，无实根。

ots αβ = c/a. A quadratic with given roots α and β can be written as x² - (α+β)x + αβ = 0.

根的和 α + β = -b/a，根的积 αβ = c/a。已知根 α 和 β 的二次方程可写为 x² - (α+β)x + αβ = 0。

For quadratic inequalities ax² + bx + c > 0 or < 0, identify the roots of the corresponding equation, sketch the graph, and select the intervals that satisfy the inequality. Remember that if a > 0, the parabola opens upwards.

对于二次不等式 ax² + bx + c > 0 或 < 0，先求出对应方程的根，画出草图，然后选取满足不等式的区间。注意若 a > 0，抛物线开口向上。

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3. Polynomials & Factor Theorem | 多项式与因式定理

nd only if P(k) = 0. Conversely, if P(k) = 0, then (x - k) divides the polynomial exactly.

因式定理指出，(x - k) 是多项式 P(x) 的因式当且仅当 P(k) = 0。反之，若 P(k) = 0，则 (x - k) 整除该多项式。

The Remainder Theorem: When polynomial P(x) is divided by (x - a), the remainder is P(a). This provides a quick way to evaluate remainders without long division.

余式定理：多项式 P(x) 除以 (x - a) 时，余数为 P(a)。这提供了一种无需长除法即可快速计算余数的方法。

Polynomial long division is used to factorise higher-degree polynomials once one factor has been identified. The quotient can be factorised further if possible.

多项式长除法用于在找到一个因式后对高次多项式进行因式分解。所得的商可以继续分解（如可能）。

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4. Exponential and Logarithmic Functions | 指数与对数函数

The exponential function is y = aˣ Its inverse is the logarithmic function y = logₐ x. The natural exponential function y = eˣ has base e ≈ 2.71828, and its inverse is the natural logarithm y = ln x.

指数函数为 y = aˣ，其中 a > 0，a ≠ 1。其反函数是对数函数 y = logₐ x。自然指数函数 y = eˣ 的底 e ≈ 2.71828，其反函数为自然对数 y = ln x。

Also, logₐ a = 1, logₐ 1 = 0, and the change of base formula: logₐ b = log_c b / log_c a.

重要对数法则：logₐ (mn) = logₐ m + logₐ n；logₐ (m/n) = logₐ m - logₐ n；logₐ (mⁿ) = n logₐ m。此外，logₐ a = 1，logₐ 1 = 0，以及换底公式：logₐ b = log_c b / log_c a。

lving exponential equations often involves taking logarithms on both sides. For example, 2ˣ = 5 becomes x = log₂ 5 or x = ln 5 / ln 2.

解指数方程时常对方程两边取对数。例如 2ˣ = 5 可化为 x = log₂ 5 或 x = ln 5 / ln 2。

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

₂, y₂) is given by d = √[(x₂ - x₁)² + (y₂ - y₁)²]. The midpoint is ((x₁+x₂)/2, (y₁+y₂)/2).

两点 (x₁, y₁) 和 (x₂, y₂) 之间的距离公式为 d = √[(x₂ - x₁)² + (y₂ - y₁)²]。中点坐标为 ((x₁+x₂)/2, (y₁+y₂)/2)。

- y₁)/(x₂ - x₁). A horizontal line has gradient 0; a vertical line has undefined gradient.

过两点的直线梯度（斜率）为 m = (y₂ - y₁)/(x₂ - x₁)。水平线的梯度为 0；竖直线的梯度无定义。

ept form y = mx + c; point-gradient form y - y₁ = m(x - x₁); general form ax + by + c = 0.

直线方程的形式：斜截式 y = mx + c；点斜式 y - y₁ = m(x - x₁)；一般式 ax + by + c = 0。

nts, m₁ = m₂. Perpendicular lines satisfy m₁ × m₂ = -1, provided neither line is vertical.

平行线的梯度相等，m₁ = m₂。垂直线的梯度满足 m₁ × m₂ = -1，前提是两条线均非竖直。

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6. Trigonometry | 三角学

are: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent.

在直角三角形中，基本三角比为：sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。

we know two angles and one side (AAS) or two sides and a non-included angle (SSA, ambiguous case).

适用于任意三角形 ABC 的正弦定理：a/sin A = b/sin B = c/sin C。当已知两角一边（AAS）或两边及非夹角（SSA，可能存在歧义）时使用。

e (SAS) or all three sides (SSS). The angle can be found via cos A = (b² + c² - a²)/(2bc).

余弦定理：a² = b² + c² - 2bc cos A。适用于已知两边及其夹角（SAS）或三边长度（SSS）。求角时可改写为 cos A = (b² + c² - a²)/(2bc)。

, where a and b are two sides and C is the included angle. Also, Area = ½ × base × height.

三角形面积公式：½ ab sin C，其中 a 和 b 为两边，C 为夹角。此外，面积 = ½ × 底 × 高。

² θ + cos² θ = 1. These are fundamental for simplifying expressions and solving equations.

三角恒等式：tan θ = sin θ / cos θ；sin² θ + cos² θ = 1。它们是化简表达式和解方程的基础。

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7. Circular Measure & Radians | 弧度制与圆度量

e by an arc equal in length to the radius. Conversion: π radians = 180°, so 1 rad ≈ 57.3°.

一弧度是指圆心角所对的弧长等于半径时所对应的角。换算：π 弧度 = 180°，因此 1 弧度 ≈ 57.3°。

radians. Sector area: A = ½ r² θ. These are used extensively in circular measure problems.

弧长公式：s = rθ，其中 r 为半径，θ 为弧度制角度。扇形面积：A = ½ r² θ。这些在弧度制问题中被广泛使用。

segment, the area of a segment = area of sector - area of triangle = ½ r² θ - ½ r² sin θ.

对于弦和弓形，弓形面积 = 扇形面积 - 三角形面积 = ½ r² θ - ½ r² sin θ。

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8. Sequences & Series | 数列与级数

e sum of the first n terms Sₙ = n/2 [2a + (n-1)d] = n/2 (a + l), where l is the last term.

等差数列 (AP) 具有公差 d：第 n 项 uₙ = a + (n-1)d，其中 a 为首项。前 n 项和 Sₙ = n/2 [2a + (n-1)d] = n/2 (a + l)，其中 l 为末项。

e sum of the first n terms Sₙ = a(1 - rⁿ)/(1 - r) for r ≠ 1. The sum to infinity exists if |r| < 1 and is S∞ = a/(1 - r).

等比数列 (GP) 具有公比 r：第 n 项 uₙ = arⁿ⁻¹。前 n 项和 Sₙ = a(1 - rⁿ)/(1 - r)，r ≠ 1。无穷项和当 |r| < 1 时存在，S∞ = a/(1 - r)。

The binomial expansion of (1 + x)ⁿ for negative or fractional n (where |x| < 1) yields an infinite series: (1 + x)ⁿ = 1 + nx + n(n-1)x²/2! + n(n-1)(n-2)x³/3! + ... .

对于负指数或分数指数 n（|x| < 1），(1 + x)ⁿ 的二项展开式得到一个无穷级数：(1 + x)ⁿ = 1 + nx + n(n-1)x²/2! + n(n-1)(n-2)x³/3! + ... 。

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9. Binomial Expansion | 二项展开

, the binomial theorem states: (a + b)ⁿ = Σ (from k=0 to n) [nCk · aⁿ⁻ᵏ bᵏ], where nCk = n! / [k!(n-k)!]. The coefficients can be found from Pascal's triangle.

对于正整数 n，二项式定理为：(a + b)ⁿ = Σ (k=0→n) [nCk · aⁿ⁻ᵏ bᵏ]，其中 nCk = n! / [k!(n-k)!]。系数可由帕斯卡三角形求得。

The term independent of x in an expansion is the constant term; it can be found by setting the power of x to zero. Similarly, to find the coefficient of a specific power, equate the exponent to that value and solve for k.

展开式中不含 x 的项即常数项；可通过令 x 的指数为零求得。同理，求特定次幂的系数时，令指数等于该值并解出 k。

When n is not a positive integer, the expansion is valid only for |x| < 1 if the binomial is of the form (1 + x)ⁿ, and it becomes an infinite series.

当 n 不是正整数时，若二项式为 (1 + x)ⁿ 形式，展开式仅在 |x| < 1 时有效，并且变为无穷级数。

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10. Vectors | 向量

A vector quantity has both magnitude and direction. In component form, a vector is written as v = (x, y) or xi + yj. The zero vector is (0, 0).

向量具有大小和方向。在分量形式中，向量写作 v = (x, y) 或 xi + yj。零向量为 (0, 0)。

Magnitude of a vector: |v| = √x² + y². A unit vector has magnitude 1; the unit vector in the direction of v is v/|v|.

向量的模：|v| = √x² + y²。单位向量的模为 1；沿 v 方向的单位向量为 v/|v|。

) = (x₁ + x₂, y₁ + y₂). Subtraction is similar. Scalar multiplication: k(x, y) = (kx, ky).

向量加法：(x₁, y₁) + (x₂, y₂) = (x₁ + x₂, y₁ + y₂)。减法类似。数乘：k(x, y) = (kx, ky)。

The dot product of two vectors a and b is a·b = x₁ x₂ + y₁ y₂. It is also equal to |a||b| cos θ, where θ is the angle between them. Two vectors are perpendicular if a·b = 0.

两向量 a 与 b 的点积为 a·b = x₁ x₂ + y₁ y₂，也等于 |a||b| cos θ，其中 θ 是它们的夹角。若 a·b = 0，则两向量垂直。

In geometry, the vector equation of a straight line is r = a + td, where a is a point on the line, d is the direction vector, and t is a scalar parameter.

在几何中，直线的向量方程为 r = a + td，其中 a 是直线上一点，d 是方向向量，t 为标量参数。

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11. Differentiation | 微分

any real n. This rule is the foundation of differentiation. For a constant c, d(c)/dx = 0.

对于任意实数 n，y = xⁿ 的导数为 dy/dx = n xⁿ⁻¹。该法则是微分的基础。常数的导数为零：d(c)/dx = 0。

Derivative of sum/difference: d/dx [f(x) ± g(x)] = f'(x) ± g'(x). Scalar multiple: d/dx [k f(x)] = k f'(x).

和差法则：d/dx [f(x) ± g(x)] = f'(x) ± g'(x)。数乘法则：d/dx [k f(x)] = k f'(x)。

a point is given by the derivative. The equation of the tangent at (x₁, y₁) is y - y₁ = f'(x₁)(x - x₁). The normal is perpendicular to the tangent, so its gradient is -1/f'(x₁).

曲线在某点的梯度由导数给出。在 (x₁, y₁) 处的切线方程为 y - y₁ = f'(x₁)(x - x₁)。法线垂直于切线，故其梯度为 -1/f'(x₁)。

etric functions: d/dx (sin x) = cos x; d/dx (cos x) = -sin x; d/dx (tan x) = sec² x. For eˣ, d/dx (eˣ) = eˣ, and for ln x, d/dx (ln x) = 1/x.

三角函数的导数：d/dx (sin x) = cos x；d/dx (cos x) = -sin x；d/dx (tan x) = sec² x。对 eˣ，d/dx (eˣ) = eˣ；对 ln x，d/dx (ln x) = 1/x。

Second derivative d²y/dx² determines the nature of stationary points: if f''(x) > 0, the point is a local minimum; if f''(x) < 0, it's a local maximum; if f''(x) = 0, further testing is needed.

二阶导数 d²y/dx² 用于判断驻点性质：若 f''(x) > 0，为局部极小点；若 f''(x) < 0，为局部极大点；若 f''(x) = 0，需进一步检验。

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12. Integration | 积分

n. The indefinite integral ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, for n ≠ -1. For n = -1, ∫ 1/x dx = ln |x| + C.

积分是微分的逆过程。不定积分 ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C，其中 n ≠ -1。当 n = -1 时，∫ 1/x dx = ln |x| + C。

ndamental theorem of calculus: For a definite integral, ∫ₐᵇ f(x) dx = F(b) - F(a), where F'(x) = f(x). This computes the area under the curve y = f(x) between x = a and x = b.

微积分基本定理：对于定积分，∫ₐᵇ f(x) dx = F(b) - F(a)，其中 F'(x) = f(x)。它用于计算曲线 y = f(x) 在 x = a 与 x = b 之间的面积。

ded for total area. Area between two curves y = f(x) and y = g(x) from x = a to x = b is ∫ₐᵇ [f(x) - g(x)] dx, assuming f(x) ≥ g(x).

曲线与 x 轴之间的面积：轴上方面积为正；下方面积为负，因此求总面积时需取绝对值或分割区间。两条曲线 y = f(x) 和 y = g(x) 在 x = a 到 x = b 之间的面积为 ∫ₐᵇ [f(x) - g(x)] dx，假设 f(x) ≥ g(x)。

tegration of trigonometric functions: ∫ cos x dx = sin x + C; ∫ sin x dx = -cos x + C; ∫ eˣ dx = eˣ + C; ∫ 1/x dx = ln|x| + C.

三角函数的积分：∫ cos x dx = sin x + C；∫ sin x dx = -cos x + C；∫ eˣ dx = eˣ + C；∫ 1/x dx = ln|x| + C。

then v(t) = ds/dt, and acceleration a(t) = dv/dt. Conversely, s = ∫ v dt, and v = ∫ a dt.

运动学应用：若位移 s(t) 的导数为速度 v(t)，则 v(t) = ds/dt，加速度 a(t) = dv/dt。反之，s = ∫ v dt，v = ∫ a dt。

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