📚 Year 11 CCEA Maths Formula & Theorem Quick Reference Guide | CCEA Year 11 数学公式定理速查手册
This quick reference guide brings together all the essential formulas and theorems you will need for the CCEA Year 11 Mathematics course. Keeping these relationships at your fingertips will help you move confidently through algebra, coordinate geometry, trigonometry, statistics, and beyond.
这份速查手册汇集了 CCEA Year 11 数学课程所需的全部核心公式和定理。熟悉这些关系式，可以帮助你自信地应对代数、坐标几何、三角学和统计等各个模块。
1. Laws of Indices | 指数定律
When multiplying powers with the same base, add the exponents: aᵐ × aⁿ = aᵐ⁺ⁿ.
同底数幂相乘，指数相加：aᵐ × aⁿ = aᵐ⁺ⁿ。
When dividing, subtract the exponents: aᵐ ÷ aⁿ = aᵐ⁻ⁿ (provided the base is non‑zero).
相除则指数相减：aᵐ ÷ aⁿ = aᵐ⁻ⁿ（底数不为零）。
Raising a power to another power multiplies the exponents: (aᵐ)ⁿ = aᵐⁿ.
幂的乘方，指数相乘：(aᵐ)ⁿ = aᵐⁿ。
The power of a product distributes: (ab)ⁿ = aⁿ bⁿ.
积的乘方等于各因式分别乘方：(ab)ⁿ = aⁿ bⁿ。
A negative exponent gives the reciprocal: a⁻ⁿ = 1 / aⁿ.
负指数表示倒数：a⁻ⁿ = 1 / aⁿ。
Fractional exponents represent roots: a1/n = ⁿ√a (the nth root of a), and more generally am/n = ⁿ√aᵐ = (ⁿ√a)ᵐ.
分数指数表示方根：a1/n = ⁿ√a（a 的 n 次方根），更一般地有 am/n = ⁿ√aᵐ = (ⁿ√a)ᵐ。
2. Expanding and Factorising | 展开与因式分解
Single bracket expansion multiplies each term inside: a(b + c) = ab + ac.
单项式乘以括号，每项相乘：a(b + c) = ab + ac。
Double brackets use the FOIL method: (x + p)(x + q) = x² + (p+q)x + pq.
双括号展开用首外内尾法则：(x + p)(x + q) = x² + (p+q)x + pq。
Difference of two squares: a² − b² = (a + b)(a − b).
平方差公式：a² − b² = (a + b)(a − b)。
Factorising a quadratic ax² + bx + c with a=1 looks for two numbers that multiply to c and add to b.
当 a=1 时，将二次三项式 ax² + bx + c 因式分解，只需找出两数，积为 c，和为 b。
For a ≠ 1, split the middle term or use the grouping method.
当 a 不为 1 时，用十字相乘法或分组法分解。
3. Quadratic Equations and the Discriminant | 二次方程与判别式
The standard quadratic equation is ax² + bx + c = 0 (a ≠ 0). Its solutions are given by the quadratic formula:
标准二次方程为 ax² + bx + c = 0（a ≠ 0）。求根公式为：
x = ( −b ± √b² − 4ac ) / (2a)
The expression inside the square root, Δ = b² − 4ac, is the discriminant. It tells us the nature of the roots:
根号内的表达式 Δ = b² − 4ac 称为判别式，用来判断根的性质：
Δ = 0, there is one repeated real root. If Δ < 0, there are no real roots (complex only).
当 Δ > 0 时，有两个不等实根；Δ = 0 时，有两个相等实根（重根）；Δ < 0 时，没有实数根。
The sum of the roots is −b/a and the product is c/a.
两根之和为 −b/a，积为 c/a。
Completing the square rewrites x² + bx as (x + b/2)² − (b/2)², useful for graphing and solving.
配方法将 x² + bx 写成 (x + b/2)² − (b/2)²，用于绘图和解方程。
4. Straight Lines and Coordinate Geometry | 直线与坐标几何
The gradient (slope) between two points (x₁, y₁) and (x₂, y₂) is:
两点 (x₁, y₁) 与 (x₂, y₂) 间的斜率（梯度）为：
m = (y₂ − y₁) / (x₂ − x₁)
The equation of a straight line can be written as y = mx + c, where m is the gradient and c is the y‑intercept.
直线方程可写为 y = mx + c，其中 m 是斜率，c 是 y 轴截距。
Another useful form is the point–gradient formula: y − y₁ = m(x − x₁).
另一种有用的形式是点斜式：y − y₁ = m(x − x₁)。
The midpoint of a segment is ( (x₁+x₂)/2 , (y₁+y₂)/2 ).
线段中点坐标为 ( (x₁+x₂)/2 , (y₁+y₂)/2 )。
The distance between the two points is d = √((x₂−x₁)² + (y₂−y₁)²) (Pythagoras' theorem applied).
两点间距离为 d = √((x₂−x₁)² + (y₂−y₁)²)（本质是勾股定理应用）。
Parallel lines have equal gradients; perpendicular lines have gradients that multiply to −1 (i.e. m₁ × m₂ = −1).
平行线斜率相等；垂直线斜率乘积为 −1（即 m₁ × m₂ = −1）。
5. Solving Simultaneous Equations | 解联立方程
For two linear equations, elimination or substitution can find the unique intersection point. Always check your solution in both original equations.
对于两个线性方程，可用消元法或代入法求出唯一交点。务必代入原方程验算。
One linear and one quadratic equation are solved by substitution, leading to a quadratic in one variable.
一次与二次方程联立时，代入后会得到关于单个变量的二次方程，再求解。
Graphical solutions correspond to the points where the line and curve intersect.
图解法中，解对应直线与曲线的交点坐标。
6. Ratio, Proportion and Rates of Change | 比例、比率与变化率
A ratio a : b compares two quantities. To share an amount in a given ratio, divide the total by the sum of the parts and multiply.
比例 a : b 用于比较两个量。按比例分配时，先将总量除以总份数，再乘以各份额。
Direct proportion means y = kx for some constant k. Its graph is a straight line through the origin.
正比例关系为 y = kx（k 为常数），图像是过原点的直线。
Inverse proportion means y = k / x. The graph is a rectangular hyperbola (never touches the axes).
反比例关系为 y = k / x，图像是一条双曲线（永不触及坐标轴）。
Rates of change often involve interpreting the gradient of a graph, for example speed from a distance–time graph.
变化率常通过图像斜率来解读，例如距离–时间图的斜率表示速度。
7. Compound Interest and Depreciation | 复利与折旧
Compound interest is calculated with the formula A = P(1 + r/100)ⁿ, where P is the principal, r is the annual interest rate, and n is the number of compounding periods.
复利计算公式为 A = P(1 + r/100)ⁿ，其中 P 为本金，r 为年利率，n 为计息期数。
If interest is compounded more than once a year, adjust the rate and number of periods accordingly: r becomes r/k and n becomes k × years, where k is the number of compounding periods per year.
若一年内多次计复利，需相应调整利率与期数：r 变为 r/k，n 变为 k × 年数，其中 k 为每年计息次数。
Depreciation by a fixed percentage uses the same model with a negative rate: A = P(1 − r/100)ⁿ.
按固定百分比折旧时，公式相同但用负增长率：A = P(1 − r/100)ⁿ。
Recognise growth and decay graphs: exponential curves that rise or fall towards an asymptote.
理解增长和衰减的图像：指数曲线上升或下降并趋近于渐近线。
8. Perimeter, Area and Volume | 周长、面积和体积
Area of a triangle: A = ½ × base × height = ½ bh. For any triangle with two sides and included angle: ½ ab sin C.
三角形面积：S = ½ × 底 × 高 = ½ bh。已知两边及其夹角时：S = ½ ab sin C。
Area of a trapezium: A = ½ (a + b)h, where a and b are the parallel sides.
梯形面积：S = ½ (a + b)h，其中 a、b 为平行边。
Circle: circumference C = 2πr = πd, area A = πr².
圆：周长 C = 2πr = πd，面积 S = πr²。
Arc length (angle θ in degrees): L = (θ/360) × 2πr; sector area: A = (θ/360) × πr².
弧长（角度制）：弧长 = (θ/360) × 2πr；扇形面积：S = (θ/360) × πr²。
Prism volume: V = area of cross‑section × length.
棱柱体积：V = 横截面积 × 长。
Cylinder: V = πr²h, curved surface area C = 2πrh, total surface area A = 2πr(r + h).
圆柱：V = πr²h，侧面积 Sₛ = 2πrh，全面积 S = 2πr(r + h)。
Cone: V = ⅓πr²h, curved surface area C = πrl (where l is slant height).
圆锥：V = ⅓πr²h，侧面积 Sₛ = πrl（l 为母线长）。
Sphere: V = ⁴⁄₃πr³, surface area A = 4πr².
球：V = ⁴⁄₃πr³，表面积 S = 4πr²。
9. Pythagoras and Basic Trigonometry | 勾股定理与基本三角
In a right‑angled triangle, Pythagoras' theorem states a² + b² = c², where c is the hypotenuse.
在直角三角形中，勾股定理表述为 a² + b² = c²，其中 c 为斜边。
The three primary trigonometric ratios for an acute angle θ:
锐角 θ 的三个基本三角比：
sin θ = opposite / hypotenuse
cos θ = adjacent / hypotenuse
tan θ = opposite / adjacent
Use SOH CAH TOA to remember them. To find an unknown angle, apply the inverse functions sin⁻¹, cos⁻¹, tan⁻¹.
用 SOH CAH TOA 记忆这些关系。求角时，使用反函数 sin⁻¹、cos⁻¹、tan⁻¹。
Know the exact values for key angles: sin 30° = ½, sin 45° = √2/2, sin 60° = √3/2; cos 30° = √3/2, cos 45° = √2/2, cos 60° = ½; tan 30° = 1/√3, tan 45° = 1, tan 60° = √3.
牢记特殊角的精确值：sin 30° = ½, sin 45° = √2/2, sin 60° = √3/2；cos 30° = √3/2, cos 45° = √2/2, cos 60° = ½；tan 30° = 1/√3, tan 45° = 1, tan 60° = √3。
10. Sine and Cosine Rules | 正弦定理与余弦定理
For any triangle (not necessarily right‑angled), the Sine Rule links sides and opposite angles:
对于任意三角形（不一定为直角三角形），正弦定理关联边与对角：
a / sin A = b / sin B = c / sin C
It is used when two angles and one side are known, or when two sides and a non‑included angle are given (watch for the ambiguous case).
当已知两角与一边，或两边及非夹角时使用（注意可能存在的多解情况）。
The Cosine Rule extends Pythagoras when you know two sides and the included angle, or all three sides:
余弦定理为已知两边及夹角，或三边时提供工具：
a² = b² + c² − 2bc cos A
Equivalently, the angle version finds an angle: cos A = (b² + c² − a²) / (2bc).
求角时可变形为：cos A = (b² + c² − a²) / (2bc)。
The area of any triangle can be found using A = ½ ab sin C.
任意三角形面积可用 S = ½ ab sin C。
11. Circle Theorems | 圆的定理
The angle at the centre is twice the angle at the circumference standing on the same arc.
圆心角等于同弧所对圆周角的二倍。
Angles in the same segment (subtended by the same chord) are equal.
同弓形内的圆周角（同一弦所对的圆周角）相等。
The angle in a semicircle is a right angle (90°).
半圆所对的圆周角为直角（90°）。
Opposite angles of a cyclic quadrilateral sum to 180°.
圆内接四边形对角互补，和为 180°。
The perpendicular from the centre to a chord bisects the chord.
圆心到弦的垂直线平分该弦。
A tangent is perpendicular to the radius at the point of contact.
切线与过切点的半径垂直。
The alternate segment theorem: the angle between a tangent and a chord equals the angle in the alternate segment.
弦切角定理：切线与弦的夹角等于其异侧圆周角。
12. Statistics and Probability | 统计
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