Year 11 CIE Formula & Theorem Quick Reference | Year 11 CIE 公式定理速查手册

📚 Year 11 CIE Formula & Theorem Quick Reference | Year 11 CIE 公式定理速查手册

Welcome to your essential quick-reference guide for Year 11 CIE IGCSE Mathematics, Physics and Chemistry. This bilingual handbook presents the must-know formulae, theorems and key relationships in a clear, side-by-side English–Chinese format. Whether you are revising for mocks or preparing for the final exams, use these paired summaries to strengthen your recall and deepen your understanding.

欢迎使用 Year 11 CIE IGCSE 数学、物理和化学核心公式定理速查手册。本双语手册以清晰的英中对照方式，列出必背公式、核心定理与重要关系。不论你正在准备模拟考还是冲刺大考，都可以借助这些配对总结强化记忆、加深理解。

1. Algebraic Identities | 代数恒等式

The square of a binomial and the difference of two squares appear frequently in factorisation and equation solving. Memorise them in both expanded and factorised forms to speed up your work.

二项式的平方以及平方差公式在因式分解和解方程中出现频率极高。熟记展开式和因式形式能大幅提高解题速度。

(a + b)² = a² + 2ab + b²
(a – b)² = a² – 2ab + b²
a² – b² = (a + b)(a – b)

(a + b)² ≡ a² + 2ab + b²	(a + b)² ≡ a² + 2ab + b²
(a – b)² ≡ a² – 2ab + b²	(a – b)² ≡ a² – 2ab + b²
a² – b² ≡ (a + b)(a – b)	a² – b² ≡ (a + b)(a – b)

2. Quadratic Formula & Discriminant | 二次方程求根公式与判别式

For any quadratic equation ax² + bx + c = 0 (a ≠ 0), the solutions are given by the quadratic formula. The discriminant Δ = b² – 4ac determines the nature of the roots without solving the equation.

对于任意二次方程 ax² + bx + c = 0 (a ≠ 0)，其解由求根公式给出。判别式 Δ = b² – 4ac 在不求解方程的情况下即可判断根的性质。

x = [ –b ± √(b² – 4ac) ] ÷ (2a)

Δ > 0: two distinct real roots / 两个不等实根
Δ = 0: one repeated real root / 一个重根
Δ < 0: no real roots / 无实根

3. Pythagoras’ Theorem | 勾股定理

In a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. This theorem is the foundation of trigonometry and coordinate geometry distance calculations.

在直角三角形中，斜边的平方等于两直角边的平方和。该定理是三角学和坐标几何中距离计算的基础。

c² = a² + b² (where c is the hypotenuse / c 为斜边)

Always label the right angle carefully and identify the hypotenuse before applying the formula. The converse also holds: if c² = a² + b², the triangle is right-angled.

应用公式前务必标出直角并确定斜边。其逆命题也成立：若 c² = a² + b²，则该三角形为直角三角形。

4. Trigonometric Ratios | 三角函数比

For any acute angle θ in a right-angled triangle, the three primary trigonometric ratios are defined in terms of the opposite, adjacent and hypotenuse sides.

对于直角三角形中任一锐角 θ，三个基本的三角函数比由对边、邻边和斜边的比值定义。

sin θ = opposite / hypotenuse	sin θ = 对边 ÷ 斜边
cos θ = adjacent / hypotenuse	cos θ = 邻边 ÷ 斜边
tan θ = opposite / adjacent	tan θ = 对边 ÷ 邻边

Common values to remember: sin 30° = ½, cos 60° = ½, tan 45° = 1. These exact values are often tested without a calculator.

需要记住的常见值：sin 30° = ½，cos 60° = ½，tan 45° = 1。这些精确值常在不允许使用计算器时考查。

5. Sine and Cosine Rules | 正弦定理与余弦定理

For non-right-angled triangles, the sine rule links side lengths to the sines of opposite angles, while the cosine rule generalises Pythagoras’ theorem.

对于非直角三角形，正弦定理将边长与对角的正弦值联系起来，而余弦定理是勾股定理的推广。

Sine rule: a / sin A = b / sin B = c / sin C

正弦定理：a ÷ sin A = b ÷ sin B = c ÷ sin C

Cosine rule: a² = b² + c² – 2bc cos A

余弦定理：a² = b² + c² – 2bc cos A

Use the sine rule when you know two angles and a side, or two sides and a non-included angle. Apply the cosine rule when you know two sides and the included angle, or all three sides.

当已知两角一边，或两边一对角时使用正弦定理；当已知两边及其夹角，或三边时使用余弦定理。

6. Area of a Triangle | 三角形面积公式

The most flexible formula for triangular area in CIE exams uses two sides and the included angle. This is especially useful when the perpendicular height is not given.

在 CIE 考试中，最灵活的三角形面积公式使用两条边及其夹角。这在未给出垂直高度时特别有用。

Area = ½ × a × b × sin C

面积 = ½ × a × b × sin C

Also recall the standard formula Area = ½ × base × height, which is a special case of the above when the angle is 90°.

同时不要忘记基本公式面积 = ½ × 底 × 高，它实际上是上述公式在夹角为 90° 时的特例。

7. Speed, Density and Pressure | 速度、密度与压强

These three compound measures are linked by a common structure: rate = amount ÷ change in another quantity. Learn the triangle method for rearranging each one.

这三个复合量具有共同的结构：率 = 某一量 ÷ 另一量的变化。利用三角形法可以轻松变换每一个公式。

Average speed = total distance ÷ total time	平均速度 = 总路程 ÷ 总时间
Density = mass ÷ volume	密度 = 质量 ÷ 体积
Pressure = force ÷ area	压强 = 力 ÷ 面积

Units must be consistent: speed in m/s or km/h, density in g/cm³ or kg/m³, pressure in Pa or N/m². Always convert units before substituting numbers.

单位必须统一：速度用 m/s 或 km/h，密度用 g/cm³ 或 kg/m³，压强用 Pa 或 N/m²。代入数字前务必先换算单位。

8. Forces and Newton’s Second Law | 力与牛顿第二定律

The net force acting on a body is directly proportional to its acceleration, with mass as the constant of proportionality. This is the single most important equation in CIE Physics.

作用在物体上的合力与其加速度成正比，比例常数即为质量。这是 CIE 物理中最重要的一个方程。

F = m × a (resultant force = mass × acceleration / 合力 = 质量 × 加速度)

For objects in free fall near the Earth’s surface, weight W = m × g, where g = 9.8 m/s² (often approximated as 10 m/s² in CIE papers).

对于地球表面附近的自由落体，重力 W = m × g，其中 g = 9.8 m/s²（CIE 试卷中常近似为 10 m/s²）。

9. Energy, Work and Power | 能量、功与功率

Work done by a force is the product of the force and the distance moved in the direction of the force. Power is the rate at which work is done or energy is transferred.

力所做的功等于力与沿力方向移动距离的乘积。功率是做功或能量转换的速率。

Work = F × d (force × distance in the direction of the force / 力 × 沿力方向的距离)

Gravitational potential energy: Ep = m × g × h

重力势能：Ep = m × g × h

Kinetic energy: Ek = ½ × m × v²

动能：Ek = ½ × m × v²

Power: P = W ÷ t (or P = E ÷ t)

功率：P = W ÷ t (或 P = E ÷ t)

In conservative systems, the total mechanical energy (Ek + Ep) remains constant if no external work is done. This principle underpins many roller-coaster and pendulum problems.

在保守系统中，若无外力做功，总机械能（Ek + Ep）保持不变。这一原理是许多过山车和摆锤问题的基础。

10. Electrical Quantities | 电学量基本公式

Ohm’s law, electrical power and the relationships linking charge, current and voltage form the core of CIE circuit analysis. Know these three equations in all rearrangements.

欧姆定律、电功率以及电荷、电流与电压之间的关系构成了 CIE 电路分析的核心。请熟练掌握以下三个公式的各种变形。

V = I × R (voltage = current × resistance / 电压 = 电流 × 电阻)

P = I × V (power = current × voltage / 功率 = 电流 × 电压)

Q = I × t (charge = current × time / 电荷量 = 电流 × 时间)

By combining these, you can also derive P = I²R and P = V²/R, which are useful when comparing components in series or parallel.

将这些公式组合可推出 P = I²R 和 P = V²/R，这在比较串联或并联电路元件时非常有用。

11. Moles and Concentration | 物质的量、摩尔与浓度

In CIE Chemistry, the mole is the central unit for quantifying substance. Master the conversion between mass, moles and molar mass, and the definition of concentration.

在 CIE 化学中，摩尔是量化物质的核心单位。必须掌握质量、物质的量和摩尔质量之间的换算，以及浓度的定义。

n = m ÷ M (moles = mass ÷ molar mass / 物质的量 = 质量 ÷ 摩尔质量)

concentration = n ÷ V (mol/dm³) where V is volume in dm³

浓度 = n ÷ V (mol/dm³)，其中 V 单位为 dm³

At room temperature and pressure (RTP), 1 mole of any gas occupies 24 dm³. Use this molar volume for gas stoichiometry: volume of gas = n × 24 dm³.

在常温常压下 (RTP)，1 摩尔任何气体的体积为 24 dm³。在气体化学计量中使用此摩尔体积：气体体积 = n × 24 dm³。

12. Titration and Stoichiometry | 滴定与化学计量

Titration calculations in CIE Chemistry rely on the balanced equation to find the mole ratio between reactants. Always work systematically from moles of the known to moles of the unknown.

CIE 化学中的滴定计算依赖于配平方程式以确定反应物间的摩尔比。解题时务必系统地从已知物的量推算未知物的量。

Key steps: (1) Calculate moles of the standard solution using n = c × V (in dm³). (2) Use the mole ratio from the equation to find moles of the unknown. (3) Convert to required units (concentration, mass, etc.).

关键步骤：(1) 利用 n = c × V (V 用 dm³) 计算标准溶液的物质的量；(2) 根据方程式中的摩尔比求出未知物的物质的量；(3) 换算为要求的单位（浓度、质量等）。

Keep a clear table of volumes and concentrations to avoid confusing the rough titre with the concordant results. Concordancy is typically within 0.1 cm³.

制作清晰的体积和浓度表格，避免将粗略滴定值与符合要求的结果混淆。符合要求的结果通常在 0.1 cm³ 以内。

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Year 11 CIE Formula & Theorem Quick Reference | Year 11 CIE 公式定理速查手册