📚 Year 11 CIE Additional Maths Quick Reference | CIE 进阶数学公式定理速查手册
This quick reference guide brings together all the essential formulae and theorems for the CIE Year 11 Additional Mathematics syllabus. Each section is presented with clear English statements followed by equivalent Chinese explanations, helping you navigate from basic algebra through to integration with confidence. Use it for your daily revision, last‑minute checks, and exam‑focused practice.
这份速查手册汇集了 CIE 十一年级进阶数学所有必备公式与定理。每个专题均采用英文说明紧接中文释义的方式呈现，帮助你从基础代数一直复习到积分。可以用于日常复习、考前速查和应试训练。
1. Algebraic Essentials | 代数基础
Mastery of indices and surds is the bedrock of higher‑level algebra. The laws of indices allow us to simplify expressions involving powers systematically.
熟练掌握指数与根式是高等代数的基础。指数法则使我们能够系统地化简含有幂的表达式。
For any real base a and rational exponents m, n the following hold true:
对于任意实数底 a 以及有理指数 m, n，以下法则成立：
- Product rule / 乘法法则: aᵐ × aⁿ = aᵐ⁺ⁿ
- Quotient rule / 除法法则: aᵐ ÷ aⁿ = aᵐ⁻ⁿ (a ≠ 0)
- Power of a power / 幂的乘方: (aᵐ)ⁿ = aᵐⁿ
- Power of a product / 积的乘方: (ab)ⁿ = aⁿ bⁿ
- Zero exponent / 零指数: a⁰ = 1 (a ≠ 0)
- Negative exponent / 负指数: a⁻ⁿ = 1/aⁿ
- Fractional exponent / 分数指数: a1/n = ⁿ√a and am/n = (ⁿ√a)ᵐ = ⁿ√aᵐ
Surds follow similar simplification rules. A surd is in its simplest form when the number under the root has no square factors.
根式遵循类似的化简规则。当根号下的数字不含平方因子时，这个根式就是最简形式。
√ab = √a × √b and √a/b = √a / √b
To rationalise a denominator such as 1/(√a + b), multiply numerator and denominator by its conjugate.
将分母有理化时，例如 1/(√a + b)，分子和分母同乘它的共轭根式。
2. Quadratic Functions | 二次函数
+ c appears throughout the syllabus. Its graph is a parabola whose concavity depends on the sign of a.
二次函数 f(x) = ax² + bx + c 贯穿整个考纲。其图像是一条抛物线，开口方向由 a 的符号决定。
Key algebraic forms:
重要的代数形式：
- General form / 一般式: y = ax² + bx + c
- Completed square form / 配方式: y = a(x - h)² + k, where vertex is (h, k)
- Factorised form / 因式分解式: y = a(x - p)(x - q), where p and q are roots
The quadratic formula gives the roots directly:
求根公式直接给出方程的根：
x = [-b ± √b² - 4ac] / (2a)
The discriminant Δ = b² - 4ac determines the nature of the roots:
判别式 Δ = b² - 4ac 决定了根的性质：
- Δ > 0: two distinct real roots / 两个不等实根
- Δ = 0: one repeated real root / 一个重实根
- Δ < 0: no real roots / 无实根
For completing the square, x² + bx becomes (x + b/2)² - (b/2)². This technique is essential for finding maximum or minimum values.
配方时，x² + bx 变为 (x + b/2)² - (b/2)²。这一技巧对于求最大值或最小值至关重要。
3. Equations and Inequalities | 方程与不等式
Solving equations often requires moving all terms to one side and factorising. Special attention is needed when dealing with absolute values and fractional equations.
解方程时常需要把所有项移到一边并进行因式分解。处理绝对值和分式方程时需要特别注意。
For simultaneous equations, one linear and one quadratic, substitute the linear expression into the quadratic. Always check for extraneous solutions.
对于一线性一二次的联立方程组，将线性表达式代入二次方程。需始终检验是否有多余的解。
Quadratic inequalities rely on a sign diagram. Solve the corresponding quadratic equation, plot the roots on a number line, and test the intervals.
二次不等式依赖于符号图。解出对应的二次方程，在数轴上标出根，然后检验各区间。
Absolute value equations such as |ax + b| = c lead to two linear equations: ax + b = c or ax + b = -c.
绝对值方程如 |ax + b| = c 可化为两个线性方程：ax + b = c 或 ax + b = -c。
For inequalities with absolute values, |f(x)| < k (k>0) is equivalent to -k < f(x) < k. Always express the final solution using set notation or interval notation.
对于绝对值不等式，|f(x)| < k (k>0) 等价于 -k < f(x) < k。最终解须用集合符号或区间符号表达。
4. Polynomials and Remainder Theorem | 多项式与余数定理
A polynomial P(x) can be divided by a linear divisor (x - a) using synthetic division or long division. The Remainder Theorem states that the remainder when P(x) is divided by (x - a) is P(a).
多项式 P(x) 可用综合除法或长除法除以线性因式 (x - a)。余数定理指出：P(x) 除以 (x - a) 的余数为 P(a)。
a) is a factor of P(x) if and only if P(a) = 0. This is used to factorise cubic and higher‑degree polynomials.
因式定理是余数定理的直接推论： (x - a) 是 P(x) 的因式当且仅当 P(a) = 0。这用于因式分解三次及更高次多项式。
To fully factorise a cubic, first find one factor by testing divisors of the constant term, then divide polyonmial to obtain a quadratic factor, which can be factorised further if possible.
要完全分解三次多项式，首先通过尝试常数项的因数找到一个因式，然后相除得到二次因式，可能时再继续分解。
5. Exponentials and Logarithms | 指数函数与对数
The exponential function y = aˣ tural exponential e ≈ 2.718 and natural logarithm ln are especially important in calculus.
指数函数 y = aˣ (a > 0, a ≠ 1) 与对数函数 y = logₐ x 互为反函数。自然指数 e ≈ 2.718 和自然对数 ln 在微积分中尤其重要。
The key logarithmic laws, valid for any base a > 0, a ≠ 1:
关键对数定律，对任意底数 a > 0, a ≠ 1 成立：
- Product / 乘积: logₐ(MN) = logₐ M + logₐ N
- Quotient / 商: logₐ(M/N) = logₐ M - logₐ N
- Power / 幂: logₐ (Mᵏ) = k logₐ M
- Change of base / 换底公式: logₐ b = (log_c b) / (log_c a)
To solve exponential equations, take logarithms on both sides. For example, 2ˣ = 5 becomes x ln 2 = ln 5, so x = ln 5 / ln 2.
解指数方程时，两边取对数。例如 2ˣ = 5 化为 x ln 2 = ln 5，从而 x = ln 5 / ln 2。
Remember that ln e = 1 and logₐ a = 1. The domain of logₐ x is x > 0.
记住 ln e = 1 且 logₐ a = 1。logₐ x 的定义域是 x > 0。
6. Straight Lines | 直线
The foundation of coordinate geometry lies in the distance, midpoint, and gradient formulae. For two points A(x₁, y₁) and B(x₂, y₂):
坐标几何的基础在于距离、中点和斜率公式。对于两点 A(x₁, y₁) 和 B(x₂, y₂)：
Distance AB = √[(x₂ - x₁)² + (y₂ - y₁)²]
Midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2)
Gradient m = (y₂ - y₁) / (x₂ - x₁)
The equation of a line can be written in various forms:
直线方程可以用多种形式表示：
- Gradient‑intercept / 斜截式: y = mx + c
- Point‑gradient / 点斜式: y - y₁ = m(x - x₁)
- General form / 一般式: ax + by + c = 0
-1. To find the point of intersection of two lines, solve their equations simultaneously.
平行直线斜率相等 (m₁ = m₂)，垂直直线满足 m₁ × m₂ = -1。求两直线交点时，联立它们的方程求解。
7. Circles | 圆
The circle with centre (h, k) and radius r has the standard equation:
圆心为 (h, k) 、半径为 r 的圆的标准方程是：
(x - h)² + (y - k)² = r²
neral form x² + y² + 2gx + 2fy + c = 0 represents a circle with centre (-g, -f) and radius √g² + f² - c, provided g² + f² - c > 0.
一般式 x² + y² + 2gx + 2fy + c = 0 表示圆心为 (-g, -f) 、半径为 √g² + f² - c 的圆，其中要求 g² + f² - c > 0。
To find the equation of the tangent to a circle at a given point, use the fact that the radius is perpendicular to the tangent, or apply the 'short cut' method of replacing x², y², x, y in the circle's equation.
求圆上给定点处的切线方程时，可以利用半径垂直于切线的性质，或采用替换法将圆方程中的 x², y², x, y 用对应表达式替换。
When solving a line–circle intersection, substitute the linear equation into the circle's equation to form a quadratic. The discriminant of this quadratic indicates whether the line cuts, touches, or misses the circle.
解直线与圆的交点时，将直线方程代入圆方程得到二次式。其判别式可判断直线是相交、相切还是相离。
8. Functions and Graphs | 函数与图像
A function f maps each element of the domain to exactly one element of the codomain. The vertical line test is used to check if a graph represents a function.
函数 f 将定义域中的每个元素映射到值域中的唯一元素。竖直线检验可用于判断图像是否表示函数。
The concepts of domain, range, and composition are vital. The composite function f(g(x)) is defined when the range of g is a subset of the domain of f. The inverse function f⁻¹(x) exists only if f is one‑to‑one.
定义域、值域和复合函数的概念至关重要。复合函数 f(g(x)) 在 g 的值域包含于 f 的定义域时有定义。反函数 f⁻¹(x) 仅当 f 是一一映射时才存在。
To sketch graphs of y = |f(x)|, reflect any part of y = f(x) that lies below the x‑axis in the x‑ r transformations, y = a f(b(x - c)) + d involves stretches, reflections and translations.
画 y = |f(x)| 的图像时，将原图像位于 x 轴下方的部分关于 x 轴翻折。对于变换 y = a f(b(x - c)) + d，涉及伸缩、反射和平移。
Recognise the shapes of key functions: y = aˣ = 1/x, and y = xⁿ for integer n. Their asymptotes and intercepts are frequently examined.
要能识别重要函数的图像形状： y = aˣ、 y = logₐ x、 y = 1/x 和 y = xⁿ (n 为整数)。它们的渐近线和截距常出现在考题中。
9. Trigonometric Ratios | 三角比
The three basic trigonometric ratios in a right‑angled triangle are sine, cosine and tangent. For an angle θ:
直角三角形中的三个基本三角比是正弦、余弦和正切。对于角 θ：
sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent
Exact values for special angles (0°, 30°, 45°, 60°, 90°) must be memorised. These can be derived from the two special triangles: isosceles right triangle (45°) and half‑equilateral triangle (30°‑60°).
必须熟记特殊角（0°、30°、45°、60°、90°）的精确值。它们可由两个特殊三角形导出：等腰直角三角形（45°）和半个等边三角形（30°‑60°）。
Angles of elevation and depression are measured from the horizontal. In three‑dimensional problems, look for right‑angled triangles in different planes and apply Pythagoras’ theorem and trigonometry.
仰角和俯角从水平线量起。在三维问题中，要在不同平面找出直角三角形，并运用勾股定理和三角比。
Bearings are measured clockwise from North. Always express bearings as three‑digit numbers, e.g. 045°.
方位角从正北顺时针测量。始终用三位数字表示方位角，如 045°。
10. Trigonometric Identities and Equations | 三角恒等式与方程
The fundamental identities link the three ratios and are essential for simplifying expressions and solving equations.
基本恒等式将三个三角比联系起来，它们对于化简表达式和求解方程至关重要。
tan θ = sin θ / cos θ
sin²θ + cos²θ = 1
nd 1 + tan²θ = sec²θ, but the ones above are sufficient for the Additional Maths syllabus.
由它们可推出另外两个恒等式： 1 + cot²θ = csc²θ 和 1 + tan²θ = sec²θ，但进阶数学考纲仅要求掌握上述两个。
To solve trigonometric equations, first rearrange to obtain a single trigonometric function on one side. Then find the principal value and use the periodic properties of the trig functions to generate all solutions within the given domain. Always consider the CAST diagram or the graphs of sine, cosine and tangent.
解三角方程时，先整理使方程一边只含一个三角函数。然后求出主值，并利用三角函数的周期性在给定范围内找出所有解。始终要参考CAST图或正弦、余弦、正切的图像。
The sine rule and cosine rule are used for non‑right‑angled triangles:
正弦定理和余弦定理用于非直角三角形：
a/sin A = b/sin B = c/sin C (Sine Rule)
a² = b² + c² - 2bc cos A (Cosine Rule)
ab sin C. Use the cosine rule when given three sides or two sides and the included angle.
三角形面积可用公式 面积 = ½ ab sin C 计算。已知三边或两边及夹角时使用余弦定理。
11. Differentiation | 微分法
Differentiation gives the gradient of a curve at any point. The derivative of y with respect to x is denoted by dy/dx or f'(x).
微分可求出曲线上任意点的斜率。y 关于 x 的导数记作 dy/dx 或 f'(x)。
The basic derivatives to learn by heart:
需要牢记的基本导数：
| Function f(x) | Derivative f'(x) |
| xⁿ | n xⁿ⁻¹ |
| eˣ | eˣ |
| ln x | 1/x |
| sin x | cos x |
| cos x | -sin x |
The chain rule, product rule and quotient rule enable differentiation of composite, product and quotient functions respectively.
链式法则、乘法法则和除法法则分别用于复合函数、乘积和商的微分。
- Chain rule / 链式法则: dy/dx = dy/du × du/dx
- Product rule / 乘法法则: if y = u v, then dy/dx = u dv/dx + v du/dx
- Quotient rule / 除法法则: if y = u/v, then dy/dx = (v du/dx - u dv/dx)/v²
ionary points occur where dy/dx = 0. Use the second derivative test to classify them: if f''(a) > 0, it is a minimum; if f''(a) < 0, it is a maximum; if f''(a) = 0, check the gradient either side.
驻点出现在 dy/dx = 0 处。用二阶导数判定其性质：若 f''(a) > 0 则为极小值点；若 f''(a) < 0 则为极大值点；若 f''(a) = 0，则需检查该点两侧的斜率。
tions: velocity v = ds/dt, acceleration a = dv/dt = d²s/dt², where s denotes displacement.
运动学应用：速度 v = ds/dt，加速度 a = dv/dt = d²s/dt²，其中 s 表示位移。
12. Integration | 积分法
on f(x) is written as ∫ f(x) dx and gives a family of functions differing by a constant C.
积分是微分的逆运算。函数 f(x) 的不定积分记为 ∫ f(x) dx，得到相差一个常数 C 的一族函数。
Standard integrals to memorise:
需要记忆的标准积分：
| Integrand | Integral ∫ f(x) dx |
| xⁿ (n ≠ -1) | xⁿ⁺¹/(n+1) + C |
| 1/x | ln |x| + C |
| eˣ | eˣ + C |
| cos x | sin x + C |
| sin x | -cos x + C |
+1) + C, provided n ≠ -1. The same pattern applies to trigonometric functions of the form sin(ax+b) and cos(ax+b).
对于线性函数的幂， ∫ (ax + b)ⁿ dx = (1/a) × (ax + b)ⁿ⁺¹/(n+1) + C，其中 n ≠ -1。同样模式适用于形如 sin(ax+b) 和 cos(ax+b) 的三角函数。
area under the curve y = f(x) between x = a and x = b provided the curve lies above the x‑axis. If the curve crosses the axis, split the integral into separate sections.
定积分 ∫ₐᵇ f(x) dx 给出曲线 y = f(x) 下方在 x=a 与 x=b 之间的面积，前提是曲线在 x 轴上方。如果曲线穿过轴，需将积分分段计算。
or kinematics, displacement is obtained by integrating velocity, and velocity by integrating acceleration.
两曲线 y = f(x) 与 y = g(x) 之间的面积用 ∫ₐᵇ [f(x) - g(x)] dx 计算，其中在该区间上 f(x) ≥ g(x)。在运动学中，位移由速度积分得到，速度由加速度积分得到。
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