Year 11 CCEA Further Mathematics: Quick Reference Formula Handbook | CCEA 进阶数学：公式定理速查手册

📚 Year 11 CCEA Further Mathematics: Quick Reference Formula Handbook | CCEA 进阶数学：公式定理速查手册

This quick reference handbook brings together the most important formulae and theorems required for the Year 11 CCEA Further Mathematics specification. It covers pure mathematics topics including algebra, sequences, coordinate geometry, trigonometry, calculus, matrices and vectors. The compact bilingual format is designed to help you revise efficiently and locate key results at a glance.

本速查手册汇集了 Year 11 CCEA 进阶数学课程中最核心的公式与定理，涵盖代数、数列、坐标几何、三角学、微积分、矩阵和向量等纯数主题。中英对照的紧凑编排旨在帮助你高效复习，随时查阅关键结论。

---

1. Algebra and Functions | 代数与函数

² − b² = (a − b)(a + b). For completing the square, x² + bx + c = (x + b/2)² − (b/2)² + c.

因式分解与多项式展开是基本技能。常见的因式分解包括平方差公式：a² − b² = (a − b)(a + b)。配方法则：x² + bx + c = (x + b/2)² − (b/2)² + c。

t real roots occur when Δ > 0, one repeated root when Δ = 0, and no real roots when Δ < 0.

二次方程 ax² + bx + c = 0 的判别式为 Δ = b² − 4ac。当 Δ > 0 时有两个不同实根，Δ = 0 时有一个重根，Δ < 0 时无实根。

---

2. Quadratic Equations | 二次方程

The solution of ax² + bx + c = 0 is given by the quadratic formula:

二次方程 ax² + bx + c = 0 的求根公式为：

x = [ −b ± √b² − 4ac ] / 2a

If the roots are α and β, then sum of roots α + β = −b/a and product of roots αβ = c/a.

若两根为 α 和 β，则根的和 α + β = −b/a，根的积 αβ = c/a。

form a quadratic equation from given roots: x² − (sum of roots)x + (product of roots) = 0.

已知两根求二次方程：x² − (根的和)x + (根的积) = 0。

---

3. Sequences and Series | 数列与级数

Arithmetic sequence: nᵗʰ term u_n = a + (n − 1)d. Sum of first n terms: S_n = n/2 [2a + (n − 1)d] = n/2 (a + l), where l = last term.

等差数列：第 n 项 u_n = a + (n − 1)d。前 n 项和：S_n = n/2 [2a + (n − 1)d] = n/2 (a + l)，其中 l 为末项。

Geometric sequence: nᵗʰ term u_n = arⁿ⁻¹. Sum of first n terms (r ≠ 1): S_n = a(1 − rⁿ)/(1 − r). Infinite sum converges when |r| < 1: S_∞ = a/(1 − r).

等比数列：第 n 项 u_n = arⁿ⁻¹。前 n 项和（r ≠ 1）：S_n = a(1 − rⁿ)/(1 − r)。当 |r| < 1 时无穷级数收敛：S_∞ = a/(1 − r)。

---

4. Coordinate Geometry: Straight Lines & Circles | 坐标几何：直线与圆

y = mx + c. Parallel lines have equal gradients; perpendicular lines satisfy m₁ × m₂ = −1.

两点 (x₁, y₁) 与 (x₂, y₂) 间的斜率：m = (y₂ − y₁)/(x₂ − x₁)。直线方程：y − y₁ = m(x − x₁) 或 y = mx + c。平行线斜率相等；垂直线满足 m₁ × m₂ = −1。

): ( (x₁ + x₂)/2 , (y₁ + y₂)/2 ). Distance between two points: √[(x₂ − x₁)² + (y₂ − y₁)²].

中点坐标：( (x₁ + x₂)/2 , (y₁ + y₂)/2 )。两点间距离：√[(x₂ − x₁)² + (y₂ − y₁)²]。

− a)² + (y − b)² = r². General form: x² + y² + 2gx + 2fy + c = 0, centre (−g, −f), radius √g² + f² − c.

圆的标准方程：圆心 (a, b)，半径 r，(x − a)² + (y − b)² = r²。一般式：x² + y² + 2gx + 2fy + c = 0，圆心 (−g, −f)，半径 √g² + f² − c。

---

5. Trigonometry | 三角学

In a right‑ cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent. Pythagoras’ theorem: a² + b² = c².

直角三角形中：sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。勾股定理：a² + b² = c²。

Sine rule (non‑ = b/sin B = c/sin C. Cosine rule: a² = b² + c² − 2bc cos A. Area of triangle: ½ ab sin C.

正弦定理（非直角）：a/sin A = b/sin B = c/sin C。余弦定理：a² = b² + c² − 2bc cos A。三角形面积：½ ab sin C。

+ cos²θ = 1. tan θ = sin θ / cos θ. Exact values for 0°, 30°, 45°, 60°, 90° are essential.

关键恒等式：sin²θ + cos²θ = 1。tan θ = sin θ / cos θ。务必熟记 0°、30°、45°、60°、90° 的精确值。

---

6. Differentiation | 微分

The derivative of xⁿ is d/dx (xⁿ) = nxⁿ⁻¹. Sum/difference rule: d/dx [f(x) ± g(x)] = f'(x) ± g'(x). Constant multiple rule: d/dx [k·f(x)] = k·f'(x).

xⁿ 的导数为 d/dx (xⁿ) = nxⁿ⁻¹。和差法则：d/dx [f(x) ± g(x)] = f'(x) ± g'(x)。常数倍法则：d/dx [k·f(x)] = k·f'(x)。

At a stationary point f'(x) = 0. Use second derivative f''(x) to classify: f''(x) > 0 → minimum; f''(x) < 0 → maximum; f''(x) = 0 → point of inflection (inconclusive).

驻点处 f'(x) = 0。用二阶导数分类：f''(x) > 0 → 极小值点；f''(x) < 0 → 极大值点；f''(x) = 0 → 拐点（需进一步判断）。

Equation of tangent at x = a: y − f(a) = f'(a)(x − a). Equation of normal: y − f(a) = −1/f'(a) (x − a).

x = a 处的切线方程：y − f(a) = f'(a)(x − a)。法线方程：y − f(a) = −1/f'(a) (x − a)。

---

广告 / Sponsored

7. Integration | 积分

C, for n ≠ −1. Also ∫ [f(x) ± g(x)] dx = ∫ f(x) dx ± ∫ g(x) dx; ∫ k·f(x) dx = k ∫ f(x) dx.

积分是微分的逆运算。∫ xⁿ dx = xⁿ⁺¹/(n+1) + C，当 n ≠ −1。同样 ∫ [f(x) ± g(x)] dx = ∫ f(x) dx ± ∫ g(x) dx；∫ k·f(x) dx = k ∫ f(x) dx。

Definite integral from a to b: ∫ₐᵇ f(x) dx = F(b) − F(a), where F'(x) = f(x). It gives the signed area between the curve and the x‑axis.

从 a 到 b 的定积分：∫ₐᵇ f(x) dx = F(b) − F(a)，其中 F'(x) = f(x)。它表示曲线与 x 轴之间的带符号面积。

Area bounded by curve y = f(x) and x‑axis between x = a and x = b is given by ∫ₐᵇ |f(x)| dx if the curve dips below the axis.

曲线 y = f(x) 与 x 轴在 x = a 到 x = b 之间所围面积若曲线有部分在轴下方，则使用 ∫ₐᵇ |f(x)| dx 计算总面积。

---

8. Matrices | 矩阵

A 2×2 matrix A =
  | a b |
  | c d |
has determinant det A = ad − bc. If det A ≠ 0, the inverse is A⁻¹ = 1/(ad − bc) | d −b |
                                            | −c a |.

二阶矩阵 A = [[a, b], [c, d]] 的行列式为 det A = ad − bc。若 det A ≠ 0，其逆矩阵为 A⁻¹ = 1/(ad − bc) [[d, −b], [−c, a]]。

Matrix multiplication is associative but not commutative. Identity matrix I = |1 0|
                                |0 1|.

矩阵乘法满足结合律但不满足交换律。单位矩阵 I = [[1, 0], [0, 1]]。

Transformations can be expressed as matrices: reflection, rotation, enlargement, shear. For example, rotation by 90° anticlockwise about origin: |0 −1|
                                                                      |1 0|.

几何变换可用矩阵表示：如绕原点逆时针旋转 90° 的矩阵是 [[0, −1], [1, 0]]。

---

9. Vectors | 向量

A vector is represented as v = xi + yj or as a column vector | x |
                                             | y |.
Magnitude: |v| = √x² + y².

向量可表示为 v = xi + yj 或列向量 [[x], [y]]。模长为 |v| = √x² + y²。

₁j) + (x₂i + y₂j) = (x₁ + x₂)i + (y₁ + y₂)j. Scalar multiplication: k·xi + yj = kxi + kyj.

加减法：(x₁i + y₁j) + (x₂i + y₂j) = (x₁ + x₂)i + (y₁ + y₂)j。数乘：k·(xi + yj) = kxi + kyj。

to B: AB = OB − OA. Collinear points have vectors that are scalar multiples of each other.

从 A 到 B 的位置向量：AB = OB − OA。共线点对应的向量互相成标量倍数。

---

10. Exponentials and Logarithms | 指数与对数

Laws of indices: aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1, a⁻ⁿ = 1/aⁿ.

指数律：aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1, a⁻ⁿ = 1/aⁿ。

Definition of logarithm: logₐ b = c ⇔ aᶜ . Common logarithm base 10: log₁₀ x is often written as log x. Natural log: ln x = logₑ x.

对数定义：logₐ b = c ⇔ aᶜ = b。常用对数：log₁₀ x 常简写为 log x。自然对数：ln x = logₑ x。

Log laws: logₐ (XY) = logₐ X + logₐ Y, logₐ (X/Y) = logₐ X − logₐ Y, logₐ (Xⁿ) = n logₐ X.

对数法则：logₐ (XY) = logₐ X + logₐ Y, logₐ (X/Y) = logₐ X − logₐ Y, logₐ (Xⁿ) = n logₐ X。

Solving exponential equations often requires taking logs of both sides, e.g. 2ˣ = 5 → x = log 5 / log 2.

解指数方程常需两边取对数，如 2ˣ = 5 → x = log 5 / log 2。

---

11. Binomial Expansion | 二项式展开

For positive integer n, (a + b)ⁿ = Σ (nCr) aⁿ⁻ʳ bʳ, where r runs from 0 to n. nCr = n! / [r! (n−r)!]. Pascal’s triangle provides the coefficients quickly for small n.

对正整数 n，(a + b)ⁿ = Σ (nCr) aⁿ⁻ʳ bʳ，r 从 0 到 n。nCr = n! / [r! (n−r)!]。对于较小的 n，可用杨辉三角快速得到系数。

When the first term is 1, (1 + x)ⁿ ≈ 1 + nx for small |x|, a linear approximation.

当首项为 1 时，(1 + x)ⁿ ≈ 1 + nx（当 |x| 很小时），这是一种线性近似。

---

12. Inequalities | 不等式

Solving linear inequalities follows the same rules as equations, except multiplying or dividing by a negative number reverses the inequality sign. For example, −2x < 4 ⇒ x > −2.

解一元一次不等式遵循与方程相同的规则，但是乘或除以负数不等号方向要改变。例如 −2x < 4 ⇒ x > −2。

Quadratic inequalities: solve the corresponding quadratic equation first, then test intervals on a sign diagram. For x² − 5x + 6 > 0, solutions are x < 2 or x > 3.

二次不等式：先解对应的二次方程，再用符号表测试区间。如 x² − 5x + 6 > 0 的解为 x < 2 或 x > 3。

When expressing solution sets, use set notation or interval notation and represent on a number line. Solid dots for included endpoints, open dots for excluded endpoints.

写出解集时可使用集合符号或区间符号，并在数轴上表示。包含端点用实心圆点，不包含端点用空心圆点。

Published by TutorHao | Further Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）