Year 10 OCR Further Maths: Quick Reference Handbook of Formulas and Theorems | Year 10 OCR 进阶数学：公式定理速查手册

📚 Year 10 OCR Further Maths: Quick Reference Handbook of Formulas and Theorems | Year 10 OCR 进阶数学：公式定理速查手册

This quick reference handbook compiles the key formulas and theorems required for the Year 10 OCR Further Mathematics course. Use it as a rapid revision aid to reinforce your understanding and boost your exam confidence.

本速查手册汇编了Year 10 OCR进阶数学课程所需的关键公式和定理，可作为快速复习工具，巩固理解、提升应试信心。

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

The standard form of a quadratic equation is ax² + bx + c = 0, where a ≠ 0.

二次方程的标准形式为 ax² + bx + c = 0，其中 a ≠ 0。

The solutions are given by the quadratic formula:

解由求根公式给出：

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。

If Δ > 0, the equation has two distinct real roots.

若 Δ > 0，方程有两个不相等的实数根。

If Δ = 0, the equation has two equal real roots (a repeated root).

若 Δ = 0，方程有两个相等的实数根（重根）。

If Δ < 0, the equation has no real roots (two complex roots).

若 Δ < 0，方程无实数根（有两个复数根）。

of symmetry is the line x = -b/(2a). The coordinates of the vertex are (-b/(2a), -Δ/(4a)).

抛物线 y = ax² + bx + c 的顶点位于 x = -b/(2a)，对称轴为直线 x = -b/(2a)。顶点坐标为 (-b/(2a), -Δ/(4a))。

ting the square rewrites the quadratic as a(x + p)² + q, where p = b/(2a) and q = -Δ/(4a).

配方可将二次式改写为 a(x + p)² + q，其中 p = b/(2a)，q = -Δ/(4a)。

The product and sum of roots are linked to coefficients (see next section).

根的和与积与系数相关（见下节）。

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2. Roots and Coefficients (Vieta's Formulas) | 根与系数的关系（韦达定理）

For a quadratic equation ax² + bx + c = 0 with roots α and β:

对于二次方程 ax² + bx + c = 0，设根为 α 和 β，有：

α + β = -b/a, αβ = c/a

For a cubic equation ax³ + bx² + cx + d = 0 with roots α, β, γ:

对于三次方程 ax³ + bx² + cx + d = 0，设根为 α, β, γ，有：

α + β + γ = -b/a
αβ + αγ + βγ = c/a
αβγ = -d/a

These relationships are extremely useful for forming equations from known roots or finding symmetric functions of roots without solving the equation.

这些关系对于由已知根构造方程、或在不解方程的情况下求根的对称函数极为有用。

For example, α² + β² can be expressed as (α + β)² - 2αβ.

例如，α² + β² 可表示为 (α + β)² - 2αβ。

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3. Inequalities | 不等式

Solving linear inequalities follows the same rules as equations, except that multiplying or dividing by a negative number reverses the inequality sign.

解线性不等式遵循与方程相同的规则，但当乘或除以负数时，不等号方向需反转。

> 0 (with a > 0), find the roots of ax² + bx + c = 0 and sketch the graph. The solution is 'outside' maller root or x > larger root. For ax² + bx + c < 0, the solution lies between the roots.

对于二次不等式 ax² + bx + c > 0（a > 0），先求对应方程的根并画草图。解集在根的两侧：x < 较小根 或 x > 较大根。若 ax² + bx + c < 0，解集在两根之间。

Absolute value inequalities can be rewritten without modulus:

绝对值不等式可去掉绝对值号改写：

|x| < k ⇔ -k < x < k
|x| > k ⇔ x < -k or x > k (for k > 0)

e usually expressed using interval notation or set-builder notation, e.g. {x: -2 ≤ x < 5}.

解通常用区间或集合表示，例如 {x: -2 ≤ x < 5}。

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4. Polynomials and Algebraic Division | 多项式与代数除法

The Remainder Theorem states: when a polynomial f(x) is divided by (x - a), the remainder is f(a).

余数定理：多项式 f(x) 除以 (x - a) 时，余数为 f(a)。

f(a) = 0, then (x - a) is a factor of f(x). Conversely, if (x - a) is a factor, f(a) = 0.

因式定理为特例：若 f(a) = 0，则 (x - a) 是 f(x) 的因式。反之，若 (x - a) 是因式，则 f(a) = 0。

Polynomial long division allows you to divide a polynomial by another polynomial of lower degree, helping to factorise cubics or higher-degree polynomials once one factor is known.

多项式长除法可将多项式除以较低次的多项式，一旦知道一个因式，可用来分解三次或更高次多项式。

An alternative method is synthetic division, which works efficiently for divisors of the form (x - a).

另一种方法是综合除法，对于形式为 (x - a) 的除式尤其高效。

Always write the polynomial in descending powers of x, including zero coefficients for any missing terms.

始终将多项式按 x 的降幂排列，缺失项补系数为零。

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5. Binomial Expansion | 二项式展开

For a positive integer n, the binomial theorem gives:

对于正整数 n，二项式定理给出：

(a + b)ⁿ = Σ_r=0ⁿ nCr · a^n-r · b^r

where nCr = n! / (r!(n - r)!) is the binomial coefficient, also written as C(n, r) or shown in Pascal's triangle.

其中 nCr = n! / (r!(n - r)!) 是二项式系数，也可写作 C(n, r)，或通过帕斯卡三角形求得。

The general term (the (r+1)th term) is:

通项（第 r+1 项）为：

T_r+1 = nCr · a^n-r · b^r

For the special case (1 + x)ⁿ:

特殊情况 (1 + x)ⁿ：

(1 + x)ⁿ = 1 + nC1 x + nC2 x² + ... + nCr x^r + ... + xⁿ

Pascal' um of the coefficients in the expansion of (a + b)ⁿ is 2ⁿ (obtained by setting a = b = 1).

帕斯卡三角形可快速得到较小 n 的系数。(a + b)ⁿ 展开式的系数之和为 2ⁿ（令 a = b = 1 即得）。

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6. Exponentials and Logarithms | 指数与对数

Laws of exponents (for a > 0):

指数法则（a > 0）：

a^m × aⁿ = a^m+n,  (a^m)ⁿ = a^mn,  a^m / aⁿ = a^m-n,  a⁰ = 1,  a^-n = 1/aⁿ.

对数定义：若 a^x = N，则 x = logₐ N，其中 a > 0，a ≠ 1，N > 0。

ural logarithm ln x is logₑ x with base e ≈ 2.71828. The natural exponential function is e^x. Published by TutorHao | Year 10 进阶数学 Revision Series | aleveler.com

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