📚 Year 11 CAIE Additional Mathematics: Formula & Theorem Quick Reference | Year 11 CAIE 进阶数学：公式定理速查手册
This quick reference handbook is designed for Year 11 students taking the CAIE Additional Mathematics (0606) course. It consolidates the essential formulas, identities, and theorem statements that you must recall for your examinations. Use it alongside your revision to strengthen fluency and accuracy in applying abstract concepts to both routine and problem-solving contexts.
这本速查手册专为学习CAIE附加数学（0606）课程的11年级学生设计。它汇总了考试必须熟记的核心公式、恒等式和定理陈述。你可以结合复习使用，提升将抽象概念运用于常规题型及问题解决情境的流畅度与准确度。
1. Functions & Inverse Functions | 函数与反函数
A function f maps each input x from the domain to exactly one output f(x). The domain is the set of all permissible x-values; the range is the set of all resulting y-values. To find the inverse f⁻¹(x), write y = f(x), swap x and y, then solve for y. The inverse exists only if f is one-to-one. The graphs of f and f⁻¹ are reflections in the line y = x.
函数f将定义域中的每个输入x映射到唯一的输出f(x)。定义域是所有允许的x值集合；值域是所有得到的y值集合。求反函数f⁻¹(x)时，先设y = f(x)，交换x与y然后解出y。仅当f是一一映射时反函数才存在。f与f⁻¹的图像关于直线y = x对称。
Composite functions: (gf)(x) = g(f(x)). The domain of gf is x ∈ domain of f such that f(x) ∈ domain of g. The range of gf is found by applying g to the range of f restricted to the relevant domain.
复合函数：(gf)(x) = g(f(x))。gf的定义域为满足x ∈ f的定义域且f(x) ∈ g的定义域的x值。gf的值域是将g作用于被限制的f的值域得到的。
Modulus function |x| gives the absolute value. The equation |ax + b| = c leads to two linear equations; the inequality |ax + b| ≤ c is equivalent to −c ≤ ax + b ≤ c.
绝对值函数|x|给出非负值。方程|ax + b| = c拆分为两个一次方程；不等式|ax + b| ≤ c等价于−c ≤ ax + b ≤ c。
2. Quadratic Functions & Inequalities | 二次函数与不等式
The general form of a quadratic is f(x) = ax² + bx + c, where a ≠ 0. Completing the square rewrites it as f(x) = a(x − h)² + k. The vertex (turning point) is at (h, k). For a > 0 the parabola opens upwards and the vertex is a minimum; for a < 0 it opens downwards with a maximum. The discriminant Δ = b² − 4ac determines the nature of the roots: Δ > 0 gives two distinct real roots, Δ = 0 gives one repeated real root, Δ < 0 gives no real roots.
二次函数的一般形式为f(x) = ax² + bx + c，其中a ≠ 0。配方法将其改写成f(x) = a(x − h)² + k。顶点（转折点）为(h, k)。当a > 0时抛物线开口向上，顶点为最小值点；当a < 0时开口向下，顶点为最大值点。判别式Δ = b² − 4ac决定了根的性质：Δ > 0有两个不同实根，Δ = 0有一个重根，Δ < 0没有实根。
Solving quadratic inequalities such as (x − p)(x − q) > 0 requires sketching the parabola. For a > 0, the inequality (x − p)(x − q) > 0 is satisfied when x < p or x > q (outside the roots); (x − p)(x − q) < 0 holds for p < x < q (between the roots).
解二次不等式如(x − p)(x − q) > 0需要画抛物线草图。当a > 0时，(x − p)(x − q) > 0在x < p或x > q时成立（在两根之外）；(x − p)(x − q) < 0在p < x < q时成立（在两根之间）。
3. Indices, Surds & Logarithms | 指数、根式与对数
Laws of indices: aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1 (a ≠ 0), a⁻ⁿ = 1/aⁿ, a^{m/n} = ⁿ√(aᵐ) = (ⁿ√a)ᵐ. Surds can be simplified by extracting perfect squares: √(a²b) = a√b, and rationalising denominators such as 1/(√a + √b) by multiplying by the conjugate (√a − √b).
指数律：aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1 (a ≠ 0), a⁻ⁿ = 1/aⁿ, a^{m/n} = ⁿ√(aᵐ) = (ⁿ√a)ᵐ。根式可通过提取完全平方化简：√(a²b) = a√b；分母有理化如1/(√a + √b)乘以共轭根式(√a − √b)。
Definition of logarithm: if aˣ = y then x = logₐy. Common logarithms log₁₀x and natural logarithms ln x = logₑx. Key properties: logₐ1 = 0, logₐa = 1, logₐ(xy) = logₐx + logₐy, logₐ(x/y) = logₐx − logₐy, logₐ(xⁿ) = n logₐx. Change of base: logₐb = logₐb / logₐa.
对数定义：若aˣ = y，则x = logₐy。常用对数log₁₀x，自然对数ln x = logₑx。关键性质：logₐ1 = 0, logₐa = 1, logₐ(xy) = logₐx + logₐy, logₐ(x/y) = logₐx − logₐy, logₐ(xⁿ) = n logₐx。换底公式：logₐb = logₐb / logₐa。
4. Polynomials & Remainder Theorem | 多项式与余式定理
For a polynomial P(x), the Remainder Theorem states: when P(x) is divided by (x − a), the remainder is P(a). The Factor Theorem follows: (x − a) is a factor of P(x) if and only if P(a) = 0. These theorems are used to factorise cubic or quartic polynomials and to solve polynomial equations.
对于多项式P(x)，余式定理指出：P(x)除以(x − a)的余数为P(a)。因式定理则表明：(x − a)是P(x)的因式当且仅当P(a) = 0。这些定理用于因式分解三次或四次多项式并解多项式方程。
Synthetic division or long division can reduce the degree. If a, b, c are roots of the cubic x³ + px² + qx + r = 0, then sum of roots α + β + γ = −p, sum of pairwise products αβ + βγ + γα = q, product αβγ = −r. For a quartic x⁴ + px³ + qx² + rx + s = 0, similar symmetric sums apply.
综合除法或长除法可以降低次数。若α, β, γ是三次方程x³ + px² + qx + r = 0的根，则根之和α + β + γ = −p，两两乘积之和αβ + βγ + γα = q，根之积αβγ = −r。对于四次方程x⁴ + px³ + qx² + rx + s = 0也有类似的对称求和关系。
5. Binomial Theorem | 二项式定理
For a positive integer n, the expansion is (a + b)ⁿ = Σₖ₌₀ⁿ (ⁿCₖ) aⁿ⁻ᵏ bᵏ, where ⁿCₖ = n! / [k!(n − k)!]. The term independent of x can be found by setting the power of x to zero. When n is rational and |x| < 1, the binomial series (1 + x)ⁿ = 1 + nx + n(n−1)x²/2! + n(n−1)(n−2)x³/3! + … is used for approximation and expansion of rational functions. The expansion is valid within its radius of convergence |x| < 1.
对于正整数n，展开式为(a + b)ⁿ = Σₖ₌₀ⁿ (ⁿCₖ) aⁿ⁻ᵏ bᵏ，其中ⁿCₖ = n! / [k!(n − k)!]。常数项可通过令x的指数为零求得。当n为有理数且|x| < 1时，可用二项式级数(1 + x)ⁿ = 1 + nx + n(n−1)x²/2! + n(n−1)(n−2)x³/3! + … 进行近似计算和有理函数展开。该展开在收敛半径|x| < 1内有效。
6. Trigonometry | 三角学
Basic trigonometric ratios in a right-angled triangle: sin θ = opp/hyp, cos θ = adj/hyp, tan θ = opp/adj. Special angles and their exact values must be known: sin 30° = ½, sin 45° = 1/√2, sin 60° = √3/2, cos 30° = √3/2, cos 45° = 1/√2, cos 60° = ½, tan 30° = 1/√3, tan 45° = 1, tan 60° = √3. The signs of the trigonometric functions in the four quadrants can be remembered with “All Students Take Calculus” (A→All positive, S→Sine positive, T→Tangent positive, C→Cosine positive).
直角三角形中基本的三角比：sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。必须记住特殊角的精确值：sin 30° = ½, sin 45° = 1/√2, sin 60° = √3/2, cos 30° = √3/2, cos 45° = 1/√2, cos 60° = ½, tan 30° = 1/√3, tan 45° = 1, tan 60° = √3。四个象限中三角函数的符号可用口诀“All Students Take Calculus”记忆（第一象限全正，第二象限正弦正，第三象限正切正，第四象限余弦正）。
Fundamental identities: sin²θ + cos²θ ≡ 1, tan θ ≡ sin θ / cos θ, 1 + tan²θ ≡ sec²θ, 1 + cot²θ ≡ cosec²θ. Trigonometric equations such as sin θ = k are solved by finding the principal value and then using the CAST diagram or the general solutions: for sin θ = k, θ = nπ + (−1)ⁿα (radians) or θ = 180°n + (−1)ⁿα (degrees). For cos θ = k, θ = 2nπ ± α; for tan θ = k, θ = nπ + α.
基本恒等式：sin²θ + cos²θ ≡ 1, tan θ ≡ sin θ / cos θ, 1 + tan²θ ≡ sec²θ, 1 + cot²θ ≡ cosec²θ。解三角方程如sin θ = k的方法是先求出主值，再利用CAST图或通解：对于sin θ = k，θ = nπ + (−1)ⁿα（弧度制）或θ = 180°n + (−1)ⁿα（角度制）。对于cos θ = k，θ = 2nπ ± α；对于tan θ = k，θ = nπ + α。
7. Coordinate Geometry: Straight Lines | 坐标几何：直线
The gradient m of a line through points (x₁, y₁) and (x₂, y₂) is m = (y₂ − y₁)/(x₂ − x₁). The equation of a line can be written in slope-intercept form y = mx + c or point-slope form y − y₁ = m(x − x₁). Two lines with gradients m₁ and m₂ are parallel if m₁ = m₂, and perpendicular if m₁m₂ = −1. The midpoint of the segment joining the points is ((x₁ + x₂)/2, (y₁ + y₂)/2). The distance between the points is √[(x₂ − x₁)² + (y₂ − y₁)²].
过点(x₁, y₁)和(x₂, y₂)的直线斜率m = (y₂ − y₁)/(x₂ − x₁)。直线方程可写成斜截式y = mx + c或点斜式y − y₁ = m(x − x₁)。若两直线斜率分别为m₁和m₂，当m₁ = m₂时两直线平行，当m₁m₂ = −1时垂直。两点间线段的中点坐标为((x₁ + x₂)/2, (y₁ + y₂)/2)。两点距离为√[(x₂ − x₁)² + (y₂ − y₁)²]。
8. Circles | 圆
The standard equation of a circle with centre (a, b) and radius r is (x − a)² + (y − b)² = r². The general form x² + y² + 2gx + 2fy + c = 0 has centre (−g, −f) and radius √(g² + f² − c). The circle exists if g² + f² − c > 0.
圆的标准方程为(x − a)² + (y − b)² = r²，其中圆心为(a, b)，半径为r。一般式x² + y² + 2gx + 2fy + c = 0的圆心为(−g, −f)，半径为√(g² + f² − c)。圆存在当且仅当g² + f² − c > 0。
The tangent to a circle at a point P is perpendicular to the radius at P. The length of the tangent from an external point (x₁, y₁) to the circle (x − a)² + (y − b)² = r² is √[(x₁ − a)² + (y₁ − b)² − r²]. Intersection between a line and a circle is found via simultaneous equations; the discriminant of the resulting quadratic determines the number of intersection points (Δ > 0 → two points, Δ = 0 → tangent, Δ < 0 → no intersection).
圆上一点P处的切线与该点处的半径垂直。从圆外点(x₁, y₁)到圆(x − a)² + (y − b)² = r²的切线长为√[(x₁ − a)² + (y₁ − b)² − r²]。直线与圆的交点通过联立方程求解；所得二次方程的判别式决定交点个数（Δ > 0 → 两点，Δ = 0 → 相切，Δ < 0 → 无交点）。
9. Vectors in Two Dimensions | 二维向量
A vector can be expressed as a column vector v = (x, y) or in terms of unit vectors i and j: v = xi + yj. The magnitude of v is |v| = √(x² + y²). A unit vector in the direction of v is v/|v|.
向量可表示为列向量v = (x, y)或用单位向量i、j表示：v = xi + yj。向量v的模为|v| = √(x² + y²)。与v同方向的单位向量为v/|v|。
Addition and subtraction are done component-wise: a + b = (x₁ + x₂)i + (y₁ + y₂)j. The position vector of a point A relative to an origin O is OA. The vector AB = OB − OA. If points A, B, C are collinear, then AC = kAB for some scalar k.
向量的加减法按分量进行：a + b = (x₁ + x₂)i + (y₁ + y₂)j。点A相对于原点O的位置向量是OA。向量AB = OB − OA。如果三点A, B, C共线，则存在标量k使得AC = kAB。
10. Differentiation: Rules & Techniques | 微分：法则与技巧
The derivative of xⁿ is nxⁿ⁻¹ for any real n. Basic 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. The derivative of a constant is zero.
xⁿ的导数为nxⁿ⁻¹（对任意实数n均成立）。基本导数：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。常数的导数为零。
Rules: Product Rule: d/dx [u v] = u’v + u v’. Quotient Rule: d/dx [u/v] = (u’v − u v’) / v². Chain Rule: if y = f(u) and u = g(x), then dy/dx = (dy/du)(du/dx). This is used for composite functions such as y = (3x + 1)⁵: dy/dx = 5(3x + 1)⁴ × 3.
运算法则：乘法法则 d/dx [u v] = u’v + u v’。除法法则 d/dx [u/v] = (u’v − u v’) / v²。链式法则：若y = f(u)且u = g(x)，则dy/dx = (dy/du)(du/dx)。这一法则用于复合函数，例如y = (3x + 1)⁵：dy/dx = 5(3x + 1)⁴ × 3。
11. Applications of Differentiation | 微分的应用
The derivative dy/dx gives the gradient of the tangent to a curve at a point. The gradient of the normal is −1/(dy/dx). Stationary points occur where dy/dx = 0. Their nature is determined by the second derivative test: if d²y/dx² > 0, the point is a minimum; if d²y/dx² < 0, it is a maximum; if d²y/dx² = 0, further investigation is needed (or use the first derivative test).
导数dy/dx给出曲线在某点处切线的斜率。法线的斜率为−1/(dy/dx)。驻点发生在dy/dx = 0处。驻点的性质通过二阶导数判别：若d²y/dx² > 0，为极小值点；若d²y/dx² < 0，为极大值点；若d²y/dx² = 0，则需进一步判断（或使用一阶导数符号法）。
Rates of change problems involve using the chain rule: dV/dt = (dV/dr)(dr/dt). Small increments: δy ≈ (dy/dx) δx, which allows approximating the change in y for a small change in x.
变化率问题需利用链式法则：dV/dt = (dV/dr)(dr/dt)。小增量近似：δy ≈ (dy/dx) δx，可用于估计x微小变化时y的相应变化。
12. Integration & Area Under a Curve | 积分与曲线下方面积
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