Year 10 Eduqas Further Maths: Formula & Theorem Quick Reference Handbook | Year 10 Eduqas 进阶数学：公式定理速查手册

📚 Year 10 Eduqas Further Maths: Formula & Theorem Quick Reference Handbook | Year 10 Eduqas 进阶数学：公式定理速查手册

This quick reference handbook provides a concise summary of all essential formulas, theorems and key concepts you need to master in Year 10 Eduqas GCSE Further Mathematics. Use it to reinforce your understanding and as a last-minute revision aid before tests and exams.

本速查手册为你总结了 Year 10 Eduqas 进阶数学中所有必须掌握的公式、定理和关键概念。它既能帮助你巩固理解，也可作为考前快速复习的得力工具。

1. Quadratic Equations and the Discriminant | 二次方程与判别式

uation is ax² + bx + c = 0, where a ≠ 0. Its solutions are given by the quadratic formula:

二次方程的标准形式为 ax² + bx + c = 0，其中 a ≠ 0。求根公式为：

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 两个不相等的实根
• Δ = 0 → one repeated real root 一个重实根
• Δ < 0 → no real roots 没有实根

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2. Polynomial Factor and Remainder Theorems | 多项式因式定理与余式定理

Remainder Theorem: When a polynomial f(x) is divided by (x − a), the remainder is f(a).

余式定理：当多项式 f(x) 除以 (x − a) 时，余数等于 f(a)。

Factor Theorem: (x − a) is a factor of f(x) if and only if f(a) = 0.

因式定理：(x − a) 是 f(x) 的因式当且仅当 f(a) = 0。

These theorems are used together with polynomial long division to factorise higher-degree polynomials.

这两个定理常与多项式长除法结合使用，用于分解高次多项式。

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3. Binomial Expansion | 二项式展开

For a positive integer n, the expansion of (a + b)ⁿ is given by:

对于正整数 n，(a + b)ⁿ 的展开式为：

(a + b)ⁿ = ⁿC₀ aⁿ + ⁿC₁ aⁿ⁻¹b + ⁿC₂ aⁿ⁻²b² + … + ⁿCₙ bⁿ

where the binomial coefficient ⁿCᵣ = n! / [r!(n − r)!]. Pascal's triangle gives a quick way to find these coefficients.

其中二项式系数 ⁿCᵣ = n! / [r!(n − r)!]。帕斯卡三角可以快速得到这些系数。

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4. Coordinate Geometry: Straight Lines | 坐标几何：直线

Distance between points (x₁, y₁) and (x₂, y₂):

两点间距离：

d = √[(x₂ − x₁)² + (y₂ − y₁)²]

Midpoint M:

中点 M：

M = ( (x₁ + x₂)/2 , (y₁ + y₂)/2 )

Gradient (slope) m:

斜率 m：

m = (y₂ − y₁) / (x₂ − x₁)

a line: y = mx + c (gradient-intercept form) or y − y₁ = m(x − x₁) (point-gradient form).

直线方程：y = mx + c（斜截式）或 y − y₁ = m(x − x₁)（点斜式）。

Parallel lines have equal gradients. Perpendicular lines satisfy m₁ × m₂ = −1.

平行线斜率相等。垂直线满足 m₁ × m₂ = −1。

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5. Coordinate Geometry: Circles | 坐标几何：圆

Standard equation of a circle with centre (a, b) and radius r:

圆的标准方程，圆心 (a, b)，半径 r：

(x − a)² + (y − b)² = r²

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

一般式：x² + y² + 2gx + 2fy + c = 0，圆心 (−g, −f)，半径 √g² + f² − c。

quals the radius, or if solving simultaneously yields a quadratic with discriminant Δ = 0.

直线与圆相切的条件是：圆心到直线的垂直距离等于半径，或联立方程所得二次式的判别式 Δ = 0。

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6. Trigonometric Basics | 三角学基础

Exact values for key angles:

特殊角的精确值：

θ	30°	45°	60°
sin θ	½	√2/2	√3/2
cos θ	√3/2	√2/2	½
tan θ	1/√3	1	√3

In a right‑ n = opposite/hypotenuse, cos = adjacent/hypotenuse, tan = opposite/adjacent (SOH CAH TOA).

在直角三角形中：sin = 对边/斜边，cos = 邻边/斜边，tan = 对边/邻边（SOH CAH TOA）。

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7. Trigonometric Identities, Sine and Cosine Rules | 三角恒等式、正弦定理与余弦定理

Fundamental identity: sin²θ + cos²θ = 1; also tanθ = sinθ / cosθ.

基本恒等式：sin²θ + cos²θ = 1；同时 tanθ = sinθ / cosθ。

Sine rule (for any triangle):

正弦定理（适用于任意三角形）：

a / sin A = b / sin B = c / sin C

Cosine rule:

余弦定理：

a² = b² + c² − 2bc cos A

cos A = (b² + c² − a²) / (2bc)

Area of a triangle: Area = ½ ab sin C.

三角形面积：面积 = ½ ab sin C。

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8. Vectors | 向量

A vector can be written as a column vector (x y)ᵀ or in i, j notation: xi + yj.

向量可表示为列向量 (x y)ᵀ 或用 i, j 分量形式：xi + yj。

Magnitude (length) of a vector v = xi + yj: |v| = √x² + y².

向量 v = xi + yj 的模（长度）：|v| = √x² + y²。

Vectors are added and subtracted component‑wise. Scalar multiplication multiplies each component.

向量加减法按对应分量进行；标量乘法将每个分量分别乘以该标量。

Two vectors are parallel if one is a scalar multiple of the other: v = ku.

若两个向量成标量倍数关系 (v = ku)，则它们平行。

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9. Matrix Operations | 矩阵运算

For 2×2 matrices, addition and subtraction are done element‑wise. Scalar multiplication multiplies every element.

对于 2×2 矩阵，加减法按元素对应位置进行；标量乘法用该标量乘以矩阵中的每个元素。

Matrix multiplication: A (m×n) × B (n×p) gives an m×p matrix. In general, AB ≠ BA.

矩阵乘法：A (m×n) × B (n×p) 得到 m×p 矩阵。一般情况下 AB ≠ BA。

Identity matrix I = [[1,0],[0,1]]; AI = IA = A.

单位矩阵 I = [[1,0],[0,1]]；满足 AI = IA = A。

Determinant of A = [[a,b],[c,d]]: det A = ad − bc.

矩阵 A = [[a,b],[c,d]] 的行列式：det A = ad − bc。

Inverse of A exists if det A ≠ 0:

当 det A ≠ 0 时，逆矩阵存在：

A⁻¹ = (1 / det A) [[d, −b],[−c, a]]

To solve a system AX = B, multiply both sides by A⁻¹: X = A⁻¹B.

解线性方程组 AX = B 时，两边同乘以 A⁻¹ 得：X = A⁻¹B。

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10. Sequences and Series | 序列与级数

Arithmetic sequence: nth term uₙ = a + (n−1)d.

等差数列：第 n 项 uₙ = a + (n−1)d。

Sum of first n terms: Sₙ = n/2 [2a + (n−1)d] = n/2 (a + l), where l is the last term.

前 n 项和：Sₙ = n/2 [2a + (n−1)d] = n/2 (a + l)，其中 l 是末项。

Geometric sequence: nth term uₙ = arⁿ⁻¹.

等比数列：第 n 项 uₙ = arⁿ⁻¹。

Sum of first n terms (r ≠ 1): Sₙ = a(1 − rⁿ) / (1 − r).

前 n 项和（r ≠ 1）：Sₙ = a(1 − rⁿ) / (1 − r)。

For an infinite geometric series with |r| < 1, the sum to infinity is S∞ = a / (1 − r).

当 |r| < 1 时，无限等比级数的和为 S∞ = a / (1 − r)。

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11. Introduction to Differentiation | 微分初步

The power rule: if y = xⁿ, then dy/dx = nxⁿ⁻¹.

幂法则：如果 y = xⁿ，则 dy/dx = nxⁿ⁻¹。

The derivative of a constant is zero. The derivative of a sum (or difference) is the sum (or difference) of the derivatives.

常数的导数为零。和（或差）的导数等于导数的和（或差）。

At a point x = a, the gradient of the tangent to the curve y = f(x) is f'(a). The normal gradient is −1 / f'(a), provided f'(a) ≠ 0.

在 x = a 处，曲线 y = f(x) 的切线斜率为 f'(a)。法线斜率为 −1 / f'(a)（假设 f'(a) ≠ 0）。

ionary points occur where dy/dx = 0. These can be used to find maximum and minimum values.

令 dy/dx = 0 可求得驻点，驻点可用于寻找最大值与最小值。

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