📚 Year 11 CCEA Maths: Formula & Theorem Quick Reference Guide | Year 11 CCEA 数学：公式定理速查手册
This quick reference guide is your go-to resource for the essential formulas and theorems in the Year 11 CCEA Mathematics specification. Use it alongside past papers and revision to reinforce key concepts and build exam confidence.
这份速查手册是 Year 11 CCEA 数学课程中核心公式和定理的必备资料。配合过往试题和复习使用，可以巩固关键概念，增强考试信心。
Each section below pairs the English statement of a rule with its Chinese explanation, so you can revise bilingually and check that you understand both the formula and the condition under which it may be used. Work through one section at a time, then test yourself with a past-paper question on that topic.
以下每个小节都把英文规则表述与中文解释配对呈现，方便你双语复习，并确认自己既记住了公式，也理解它的适用条件。建议一次只攻克一个小节，随后用一道真题检验掌握程度。
1. Number Properties | 数的性质
Every integer greater than 1 can be expressed as a unique product of prime numbers. Prime factorisation is the foundation for finding HCF and LCM.
每个大于 1 的整数都可以唯一地写为质数的乘积。质因数分解是求最大公因数 (HCF) 和最小公倍数 (LCM) 的基础。
To find the HCF, take the smallest power of each common prime factor. For the LCM, take the highest power of every prime factor that appears in any of the numbers.
求 HCF 时，取每个公共质因数的最小次幂。求 LCM 时，取所有出现过的质因数的最高次幂。
60 = 2² × 3 × 5, 72 = 2³ × 3² ⇒ HCF(60, 72) = 2² × 3 = 12, LCM(60, 72) = 2³ × 3² × 5 = 360
The laws of indices make it easier to handle expressions with powers. Multiplying adds the exponents, dividing subtracts them, and raising a power to another power multiplies the exponents.
指数法则可以简化含幂的表达式。乘法将指数相加，除法将指数相减，幂的乘方则将指数相乘。
aᵐ × aⁿ = aᵐ⁺ⁿ | aᵐ ÷ aⁿ = aᵐ⁻ⁿ | (aᵐ)ⁿ = aᵐⁿ | a⁰ = 1 | a⁻ⁿ = 1 ÷ aⁿ
Fractional indices connect powers with roots: a1/2 = √a and a1/3 = ∛ procal of the positive power, which is a common source of marks in CCEA calculator papers.
分数指数把幂与根号连接起来：a1/2 = √a，a1/3 = ∛a。一个数的负指数结果等于其正指数幂的倒数，这是 CCEA 计算器试卷中常见的得分点，也是最容易失分的地方之一。
2. Fractions, Decimals, Percentages and Ratio | 分数、小数、百分数与比
To add or subtract fractions, rewrite them over a common denominator, then combine the numerators. To multiply, multiply numerators together and denominators together. To divide, multiply by the reciprocal of the second fraction.
分数加减时，先通分，再合并分子。乘法时分子乘分子、分母乘分母。除法时乘以第二个分数的倒数。
a/b + c/d = (ad + bc) ÷ bd | a/b ÷ c/d = a/b × d/c
Percentage change is calculated by dividing the actual change by the original amount and multiplying by 100. Repeated percentage change such as compound interest uses a multiplier raised to the power of the number of periods.
百分数变化等于实际变化量除以原值再乘以 100。像复利这样的重复百分比变化，则用乘数的时间期数次幂来计算。
Percentage change = (change ÷ original) × 100 | A = P(1 + r ÷ 100)ⁿ
Ratio questions in CCEA papers usually give a total to be shared or a difference between parts. Find the value of one part first, then multiply by the number of parts required. For direct proportion the quantities increase together; for inverse proportion, one quantity increases as the other decreases.
CCEA 试题中的比例题通常给出需要按比分配的总量，或给出各部分之间的差。先求一份的值，再乘以所需的份数。正比关系下两个量同增同减；反比关系下一个量增大时另一个量减小。
Direct: y = kx | Inverse: y = k ÷ x | y ∝ x² | y ∝ √x
3. Standard Form, Surds and Approximation | 标准形式、根式与近似
n is an integer. Large numbers have a positive index, small numbers have a negative index.
标准形式把一个数写成 A × 10ⁿ，其中 1 ≤ A < 10，n 为整数。大数的指数为正，小数的指数为负。
When multiplying in standard form, multiply the A values and add the indices. When dividing, divide the A values and subtract the indices. Always adjust the answer if A is no longer between 1 and 10.
标准形式乘法时，A 值相乘、指数相加。除法时，A 值相除、指数相减。若结果中 A 不再介于 1 与 10 之间，需要重新调整。
(A × 10ᵐ) × (B × 10ⁿ) = (A × B) × 10ᵐ⁺ⁿ | (A × 10ᵐ) ÷ (B × 10ⁿ) = (A ÷ B) × 10ᵐ⁻ⁿ
Surds obey the rules √ab = √a × √b and √a ÷ b = √a ÷ √b. To rationalise a denominator, multiply the numerator and denominator by the surd in the denominator, so the denominator becomes a whole number.
根式满足 √ab = √a × √b 与 √a ÷ b = √a ÷ √b。有理化分母时，分子与分母同乘分母中的根式，使分母变成整数。
Rounding to a given number of decimal places or significant figures produces an error. For a value rounded to a unit of size u, the upper bound is the stated value plus u ÷ 2 and the lower bound is the stated value minus u ÷ 2.
按指定小数位或有效数字修约会带来误差。若某值以单位 u 修约，则上限为给定值加 u ÷ 2，下限为给定值减 u ÷ 2。
4. Algebraic Manipulation and Factorising | 代数变形与因式分解
Expanding a double bracket means multiplying every term in the first bracket by every term in the second. The three identities worth memorising are the square of a sum, the square of a difference and the difference of two squares.
展开双重括号，就是把第一个括号中的每一项与第二个括号中的每一项相乘。值得记牢的三个恒等式是完全平方（和）、完全平方（差）以及平方差。
(a + b)² = a² + 2ab + b² | (a − b)² = a² − 2ab + b² | a² − b² = (a + b)(a − b)
Factorising reverses expansion. Always remove the highest common factor first, then look for a difference of two squares or a quadratic trinomial. For x² + bx + c, find two numbers that multiply to c and add to b.
因式分解是展开的逆过程。先提取最大公因式，再看是否为平方差或二次三项式。对于 x² + bx + c，需找两个数，其积为 c、和为 b。
Algebraic fractions are simplified by factorising the numerator and denominator and cancelling common factors. Rearranging a formula means isolating the required variable by performing the same inverse operation on both sides.
代数分式化简的方法是：把分子与分母分别因式分解，再约去公因式。公式变形则是在两边施加相同的逆运算，把目标变量单独留在一边。
5. Quadratic Equations and the Discriminant | 一元二次方程与判别式
olving. Factorising is the quickest method when the factors are integers, and completing the square is useful when the answer must be given in surd form.
解一元二次方程前，必须先整理成 ax² + bx + c = 0 的形式。当因式为整数时，因式分解最快；当答案需写成根式形式时，配方法最方便。
x = (−b ± √b² − 4ac) ÷ 2a
The quadratic formula works for every quadratic equation, including those that do not factorise. Substitute the values of a, b and c carefully, and remember that the whole numerator is divided by 2a.
求根公式对任何一元二次方程都适用，包括无法因式分解的情形。代入 a、b、c 时要格外小心，并记住整个分子都要除以 2a。
ts when Δ < 0. Completing the square rewrites the expression as (x + p)² + q, which also gives the coordinates of the turning point.
判别式 Δ = b² − 4ac 决定方程实根的个数：Δ > 0 时有两个不同实根，Δ = 0 时有一个重根，Δ < 0 时无实根。配方把表达式改写成 (x + p)² + q，同时也给出抛物线的顶点坐标。
Δ = b² − 4ac | Turning point of y = (x + p)² + q is (−p, q)
6. Linear Equations, Inequalities and Straight Lines | 线性方程、不等式与直线
To solve a linear equation, remove brackets, collect the variable terms on one side and the numbers on the other, then divide by the coefficient. Every step must be applied to both sides.
解线性方程时，先去括号，把含未知数的项集中到一边、常数项集中到另一边，最后除以系数。每一步都必须对等式两边同时进行。
can be shown on a number line with a filled circle for ≤ or ≥ and an open circle for < or >.
不等式的解法相同，但两边同时乘以或除以负数时，不等号方向必须改变。解可以在数轴上表示：≤ 与 ≥ 用实心圆点，< 与 > 用空心圆点。
adient and c is the y-intercept. The gradient between two points is the change in y divided by the change in x.
直线方程通常写成 y = mx + c，其中 m 是斜率，c 是 y 轴截距。两点之间的斜率等于 y 的变化量除以 x 的变化量。
m = (y₂ − y₁) ÷ (x₂ − x₁) | Midpoint = ((x₁ + x₂) ÷ 2, (y₁ + y₂) ÷ 2)
Parallel lines have equal gradients. Perpendicular lines have gradients whose product is −1, so the perpendicular gradient is the negative reciprocal.
平行直线的斜率相等。垂直直线的斜率之积为 −1，因此垂直方向的斜率是原斜率的负倒数。
7. Sequences and Proportion | 数列与比例
In an arithmetic sequence each term differs from the previous one by a constant amount d. In a geometric sequence each term is obtained by multiplying the previous term by a constant ratio r.
等差数列中，每一项与前一项的差是常数 d。等比数列中，每一项都是前一项乘以固定的公比 r。
Arithmetic: nth term = a + (n − 1)d, Sₙ = n ÷ 2 × (2a + (n − 1)d)
Geometric: nth term = a × rⁿ⁻¹
Special sequences are worth recognising quickly: square numbers 1, 4, 9, 16, …; triangular numbers 1, 3, 6, 10, …; and the Fibonacci sequence 1, 1, 2, 3, 5, 8, … where each term is the sum of the two before it. Quadratic sequences have a constant second difference, and the coefficient of n² is half of that second difference.
有几类特殊数列值得快速识别：平方数 1, 4, 9, 16, …；三角形数 1, 3, 6, 10, …；以及斐波那契数列 1, 1, 2, 3, 5, 8, …，其中每一项等于前两项之和。二阶差为常数的数列是二次数列，且 n² 的系数等于该二阶差的一半。
Proportion problems can be solved by establishing the constant of proportionality k, or by using the unitary method. In inverse proportion the product of the two quantities stays constant, so doubling one quantity halves the other.
比例问题可以通过确定比例常数 k 求解，也可以用单位法。反比关系中两个量的乘积保持恒定，因此一个量翻倍时另一个量减半。
8. Angles, Polygons and Circle Theorems | 角、多边形与圆定理
Angles on a straight line add up to 180°, angles around a point add up to 360°, and the interior angles of a triangle add up to 180°. Vertically opposite angles are equal.
同一直线上的角之和为 180°，围绕一点的角之和为 360°，三角形内角和为 180°。对顶角相等。
When two parallel lines are cut by a transversal, corresponding angles are equal, alternate angles are equal, and co-interior angles add up to 180°. Always quote the reason when a question asks you to justify an answer.
两条平行线被一条截线所截时，同位角相等、内错角相等、同旁内角之和为 180°。当题目要求说明理由时，必须写明所用定理的名称。
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