📚 Year 10 CCEA Further Mathematics: Formula & Theorem Quick Reference Handbook | CCEA 进阶数学：公式定理速查手册
This quick reference handbook provides all the essential formulas and theorems you need for Year 10 CCEA Further Mathematics. Each section is carefully structured to help you revise and apply key concepts efficiently. Keep this guide handy for exam preparation and daily study.
本速查手册涵盖了您在学习 CCEA 进阶数学（Year 10）时所需的所有核心公式与定理。每个部分都经过精心编排，助您高效复习并灵活运用关键概念。请将本手册置于手边，随时用于备考和日常学习。
1. Algebraic Fundamentals | 代数基础
Index laws are the backbone of algebraic manipulation. When multiplying like bases, add the exponents; when raising a power to another power, multiply them. A base raised to zero equals one, and a negative exponent indicates a reciprocal.
指数定律是代数运算的基石。同底数幂相乘，指数相加；幂的乘方，指数相乘。任何非零底的零次幂等于 1，而负指数表示取倒数。
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Product of powers: aᵐ × aⁿ = aᵐ⁺ⁿ
同底数幂相乘：aᵐ × aⁿ = aᵐ⁺ⁿ
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Power of a power: (aᵐ)ⁿ = aᵐⁿ
幂的乘方：(aᵐ)ⁿ = aᵐⁿ
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Zero exponent: a⁰ = 1 (a ≠ 0)
零指数：a⁰ = 1 (a ≠ 0)
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Negative exponent: a⁻ⁿ = 1 / aⁿ
负指数：a⁻ⁿ = 1 / aⁿ
Two essential factorisations involve common factors and the difference of two squares. Always look for a common factor first, then check for the pattern a² - b².
两项重要的因式分解是提取公因式和平方差公式。首先寻找公因式，再检查是否符合 a² − b² 的形式。
a² − b² = (a − b)(a + b)
2. Quadratic Equations | 二次方程
riminant Δ = b² − 4ac. The discriminant reveals the nature of the roots: two distinct real roots if Δ > 0, one repeated root if Δ = 0, and no real roots if Δ < 0.
二次方程的标准形式为 ax² + bx + c = 0，其中 a ≠ 0。可使用求根公式求解，其中判别式 Δ = b² − 4ac 决定了根的性质：若 Δ > 0 则有两个不等实根，Δ = 0 有一个重根，Δ < 0 则无实根。
x = [−b ± √b² − 4ac] / (2a)
The sum and product of the roots (α, β) are linked to the coefficients without needing to solve the equation. This is known as Vieta's formulas.
两根之和与两根之积（α, β）与系数直接相关，无需解方程即可求得，这被称为韦达定理。
α + β = −b/a
αβ = c/a
3. Exponentials and Logarithms | 指数与对数
arithm answers the question: to what power must a base be raised to produce a given number? If aˣ are 10 (common log, written log₁₀ or simply log) and e ≈ 2.718 (natural log, written ln).
对数回答的是：底数需要被提升到多少次幂才能得到给定的数？若 aˣ = N，则 logₐ N = x。常用底数有 10（常用对数，记为 log 或 log₁₀）和 e ≈ 2.718（自然对数，记为 ln）。
The three fundamental logarithm laws mirror the index laws:
三条基本的对数定律与指数定律对应：
logₐ (MN) = logₐ M + logₐ N
logₐ (M/N) = logₐ M − logₐ N
logₐ (Mᵏ) = k logₐ M
To change the base of a logarithm, use the change-of-base formula.
如需转换对数的底数，可使用换底公式。
logₐ b = logₓ b / logₓ a
4. Functions and Graphs | 函数与图像
The composite function fg(x) means apply g first, then f. The notation is fg(x) = f(g(x)).
函数 f(x) 取一个输入 x 并产生唯一输出。复合函数 fg(x) 表示先应用 g，再应用 f，记作 fg(x) = f(g(x))。
d it, write y = f(x), swap x and y, then solve for y. The domain of f⁻¹ is the range of f.
反函数 f⁻¹(x) 逆转 f(x) 的作用。求反函数的方法是：令 y = f(x)，交换 x 和 y，再解出 y。反函数的定义域是原函数的值域。
fficiently. The table below summarises the four basic transformations applied to y = f(x).
图像变换用于快速绘制相关曲线。下表总结了应用于 y = f(x) 的四种基本变换。
| Transformation | Effect | 变换 | 效果 |
|---|---|---|---|
| f(x + a) | Shift left by a | f(x + a) | 向左平移 a 单位 |
| f(x) + a | Shift up by a | f(x) + a | 向上平移 a 单位 |
| af(x) (a > 0) | Vertical stretch by factor a | af(x) (a > 0) | 垂直拉伸 a 倍 |
| f(ax) (a > 0) | Horizontal stretch by factor 1/a | f(ax) (a > 0) | 水平压缩至 1/a 倍 |
5. Trigonometry | 三角学
The sine and cosine rules allow you to solve any triangle, even when it is not right-angled. The sine rule connects side lengths to the sines of opposite angles. Use it when you know two angles and one side, or two sides and a non-included angle.
正弦定理与余弦定理可用于求解任意三角形，即使不是直角三角形。正弦定理将边长与对角的正弦联系起来。当已知两角一边或两边一角（非夹角）时使用。
a / sin A = b / sin B = c / sin C
The cosine rule generalises Pythagoras' theorem. It is useful when you know three sides or two sides and the included angle.
余弦定理是勾股定理的推广。当已知三边或两边及其夹角时使用。
a² = b² + c² − 2bc cos A
The area of any triangle can be calculated using two sides and the included angle. This formula is extremely efficient for non-right triangles.
任意三角形的面积可用两边及其夹角计算。此公式对于非直角三角形极为高效。
Area = ½ ab sin C
The two core trigonometric identities are derived from the unit circle. They are fundamental for solving equations and proving other relationships.
两个核心三角恒等式源自单位圆，是解方程与证明其它关系的基础。
sin²θ + cos²θ = 1
tan θ = sin θ / cos θ
6. Vectors | 向量
i + yj can be written as a column vector. Its magnitude (length) is found using Pythagoras' theorem.
向量既有大小又有方向。在二维空间中，向量 v = xi + yj 可表示为列向量，其模长（长度）用勾股定理计算。
|v| = √x² + y²
The dot product (scalar product) of two vectors is a number, not a vector. It helps to find the angle between two vectors.
两个向量的点积（标量积）结果是一个数而非向量，用于求两向量的夹角。
v · w = x₁x₂ + y₁y₂
The angle θ between two non-zero vectors v and w is given by the cosine formula.
两个非零向量 v 与 w 之间的夹角 θ 由余弦公式给出。
cos θ = (v · w) / (|v| |w|)
7. Matrices | 矩阵
, which are the core of CCEA Year 10 Further Mathematics. Matrix multiplication is not commutative; order matters.
本节聚焦于 2×2 矩阵，这是 CCEA Year 10 进阶数学的核心内容。矩阵乘法不满足交换律，顺序至关重要。
whether a matrix is invertible. If det(A) = 0, the matrix is singular and has no inverse.
行列式是一个数，它决定了矩阵是否可逆。若 det(A) = 0，则矩阵为奇异矩阵，不存在逆矩阵。
For A = [[a, b],[c, d]], det(A) = ad − bc
rix is obtained by swapping the diagonal elements, changing the signs of the off-diagonal elements, and multiplying by the reciprocal of the determinant.
可逆 2×2 矩阵的逆矩阵可通过交换主对角线元素、改变副对角线元素符号并乘上行列式的倒数得到。
A⁻¹ = (1 / (ad − bc)) [[d, −b],[−c, a]]
8. Calculus: Differentiation | 微积分：导数
Differentiation finds the gradient of a curve at any point. The derivative of xⁿ is nxⁿ⁻¹, and the process is linear, meaning derivatives can be taken term by term.
微分用于求曲线上任意一点的斜率。xⁿ 的导数为 nxⁿ⁻¹，求导运算具有线性性，可以逐项求导。
d/dx (xⁿ) = n xⁿ⁻¹
e equation of a tangent at a point (x₁, y₁), first differentiate to get the gradient m = f'(x₁), then use y − y₁ = m(x − x₁).
求点 (x₁, y₁) 处的切线方程时，先求导得到梯度 m = f'(x₁)，再使用点斜式 y − y₁ = m(x − x₁)。
The second derivative, f''(x), tells us about the concavity of the graph and helps classify stationary points as maxima, minima, or points of inflection.
二阶导数 f''(x) 揭示了图像的凹凸性，并有助于将驻点区分为极大值、极小值或拐点。
9. Calculus: Integration | 微积分：积分
a constant C. This is the power rule for integration, valid for all real n except n = −1.
积分是微分的逆运算。不定积分给出相差一个常数 C 的函数族。下面是幂函数的积分法则，除 n = −1 外对所有实数 n 成立。
∫ xⁿ dx = xⁿ⁺¹ / (n+1) + C, n ≠ −1
A definite integral calculates the exact area under a curve between two limits a and b. It is evaluated using the Fundamental Theorem of Calculus.
定积分用于计算曲线在上下限 a 和 b 之间的准确面积，可通过微积分基本定理求出。
∫ₐᵇ f(x) dx = F(b) − F(a)
When the curve falls below the x-axis, the integral yields a negative value. To find total area, split the interval and take absolute values.
当曲线位于 x 轴下方时，定积分结果为负值。若要求总面积，需分段积分并取绝对值。
10. Sequences and Series | 序列与级数
An arithmetic sequence has a constant difference d between consecutive terms. The nth term and the sum of the first n terms are given here.
等差数列相邻两项之差 d 恒定。第 n 项（通项）与前 n 项之和公式如下。
uₙ = a + (n − 1)d
Sₙ = n/2 [2a + (n − 1)d] = n/2 (a + l)
In a geometric sequence, each term is multiplied by a constant ratio r. The formulas cover finite and infinite sums.
等比数列中，每一项乘以固定公比 r 得到后一项。以下公式涵盖有限和与无穷和。
uₙ = a rⁿ⁻¹
Sₙ = a(1 − rⁿ) / (1 − r), r ≠ 1
If |r| < 1, the infinite geometric series converges to a finite sum. This result is extremely useful in recurring decimal conversions and modelling.
若 |r| < 1，无穷等比级数收敛至一个有限和。这一结果在循环小数转化与建模中极为有用。
S∞ = a / (1 − r), |r| < 1
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