Year 11 AQA Further Maths: Quick Reference Formula & Theorem Handbook | 公式定理速查手册

📚 Year 11 AQA Further Maths: Quick Reference Formula & Theorem Handbook | 公式定理速查手册

This quick reference handbook has been carefully compiled to cover the essential formulas, theorems and key results required for the AQA Level 2 Certificate in Further Mathematics. Use it to consolidate your knowledge, check rules during revision and ensure you are fully prepared for both calculator and non-calculator papers.

这份速查手册精心整理了 AQA 进阶数学（二级证书）考试中必备的公式、定理与关键结论。可用于巩固知识、复习核对，确保你在计算器与非计算器考试中都能从容应对。

1. Algebra and Binomial Expansion | 代数与二项式展开

The Factor Theorem: (x – a) is a factor of polynomial f(x) if and only if f(a) = 0. The Remainder Theorem: when f(x) is divided by (x – a), the remainder is f(a).

因式定理：多项式 f(x) 有因式 (x – a) 当且仅当 f(a) = 0。余数定理：f(x) 除以 (x – a) 的余数为 f(a)。

Binomial expansion for positive integer n uses the binomial coefficients ⁿCᵣ, often read from Pascal’s triangle. The general term is given by ⁿCᵣ aⁿ⁻ʳ bʳ.

正整数 n 的二项式展开使用二项式系数 ⁿCᵣ，常通过帕斯卡三角形读取。一般项为 ⁿCᵣ aⁿ⁻ʳ bʳ。

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

where ⁿCᵣ = n! / (r! (n – r)!). The coefficient ⁿCᵣ can also be obtained from Pascal’s triangle.

其中 ⁿCᵣ = n! / (r! (n – r)!)。系数 ⁿCᵣ 也可从帕斯卡三角形获得。

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2. Quadratics and the Discriminant | 二次方程与判别式

The general quadratic: ax² + bx + c = 0. Its solutions can be found by completing the square or using the quadratic formula:

一般二次方程：ax² + bx + c = 0。可通过配方法或二次公式求解：

x = (-b ± √(b² – 4ac)) / (2a)

The discriminant Δ = b² – 4ac determines the nature of the roots: Δ > 0 gives two distinct real roots, Δ = 0 gives one repeated real root, Δ < 0 gives no real roots.

判别式 Δ = b² – 4ac 决定根的性质：Δ > 0 有两个不等实根，Δ = 0 有一个重根，Δ < 0 无实根。

Completing the square: x² + bx = (x + b/2)² – (b/2)². For ax² + bx + c, factor out a first.

配方法：x² + bx = (x + b/2)² – (b/2)²。对于 ax² + bx + c，先提取 a。

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3. Inequalities | 不等式

Solving linear inequalities: perform operations as with equations, but reverse the inequality sign when multiplying or dividing by a negative number.

解线性不等式：操作同方程，但乘以或除以负数时需反转不等号方向。

For quadratic inequalities, sketch the graph to identify regions where f(x) > 0 or < 0. Always express final answers using set notation or interval notation as required.

对于二次不等式，画出函数图像以确定 f(x) > 0 或 < 0 的区域。最终答案通常用集合符号或区间表示。

Remember to use open or closed circles on the number line corresponding to strict or inclusive inequalities.

注意在数轴上，严格不等号用空心圆圈，包含等号用实心圆点。

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4. Functions and Transformations | 函数与图像变换

Composite function: (f ∘ g)(x) = f(g(x)). The order matters: apply g first, then f.

复合函数：(f ∘ g)(x) = f(g(x))。注意顺序：先作用 g，再作用 f。

Inverse function f⁻¹(x) : swap x and y in y = f(x) and solve for y. The graph of f⁻¹ is a reflection of f in the line y = x.

反函数 f⁻¹(x)：交换 y = f(x) 中的 x 和 y 并解出 y。f⁻¹ 的图像是 f 关于直线 y = x 的反射。

Transformations of y = f(x): y = f(x) + a is vertical translation by a; y = f(x + a) is horizontal translation by -a; y = a f(x) is vertical stretch factor a; y = f(ax) is horizontal stretch factor 1/a; y = -f(x) reflects in x-axis; y = f(-x) reflects in y-axis.

y = f(x) 的图像变换：y = f(x) + a 垂直平移 a；y = f(x + a) 水平平移 -a；y = a f(x) 垂直拉伸 a 倍；y = f(ax) 水平拉伸 1/a 倍；y = -f(x) 关于 x 轴反射；y = f(-x) 关于 y 轴反射。

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5. Coordinate Geometry: Straight Lines and Circles | 坐标几何：直线与圆

Equation of a straight line: y = mx + c, where m is gradient, c is y-intercept. Alternative form: y – y₁ = m(x – x₁).

直线方程：y = mx + c，m 为斜率，c 为 y 截距。也可用点斜式：y – y₁ = m(x – x₁)。

Midpoint of (x₁, y₁) and (x₂, y₂): ((x₁ + x₂)/2, (y₁ + y₂)/2). Distance: √((x₂ – x₁)² + (y₂ – y₁)²).

两点中点：((x₁+x₂)/2, (y₁+y₂)/2)。距离公式：√((x₂ – x₁)² + (y₂ – y₁)²)。

Equation of a circle with centre (a, b) and radius r: (x – a)² + (y – b)² = r². The general form x² + y² + 2gx + 2fy + c = 0 has centre (-g, -f) and radius √(g² + f² – c).

圆心 (a, b) 半径 r 的圆方程：(x – a)² + (y – b)² = r²。一般式 x² + y² + 2gx + 2fy + c = 0 的圆心为 (-g, -f)，半径为 √(g² + f² – c)。

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6. Trigonometry | 三角学

Exact trigonometric values for 0°, 30°, 45°, 60°, 90° must be memorised. For example, sin 30° = 1/2, cos 45° = 1/

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