Pre-U CIE Mathematics: Quick Reference Formula and Theorem Handbook | Pre-U CIE 数学：公式定理速查手册

📚 Pre-U CIE Mathematics: Quick Reference Formula and Theorem Handbook | Pre-U CIE 数学：公式定理速查手册

This quick reference handbook covers essential formulas, theorems and standard results from the Cambridge Pre-U Mathematics syllabus (9777/9794). It is designed as a concise revision tool for recalling key algebraic, trigonometric, calculus, vector, complex number, probability, mechanics and further pure facts in one place. Use it alongside past-paper practice, and always check the exact wording and domain conditions required by the question.

本速查手册涵盖剑桥 Pre-U 数学教学大纲（9777/9794）中的核心公式、定理与标准结论，是一份精炼的复习工具，帮助你在一个文档中快速回顾代数、三角、微积分、向量、复数、概率、力学与进阶纯数的重要知识点。建议配合历年真题练习使用，并始终留意题目要求的具体条件与定义域。

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1. Algebraic Identities and Formulae | 代数恒等式与公式

The square of a sum and the square of a difference are the most frequently used expansions in Pre-U algebra. They appear in completing the square, solving equations and simplifying rational expressions.

和的平方与差的平方是 Pre-U 代数中最常用的展开式，常见于配方、解方程与化简有理式。

(a + b)² = a² + 2ab + b²

(a − b)² = a² − 2ab + b²

The difference of two squares factorises immediately, while the sum and difference of cubes require the standard factor formulae.

平方差可立即因式分解，立方和与立方差则需要标准公式。

a² − b² = (a − b)(a + b)

a³ + b³ = (a + b)(a² − ab + b²)

a³ − b³ = (a − b)(a² + ab + b²)

For a binomial cube, the signs alternate according to the sign inside the bracket.

二项式立方中，符号根据括号内的符号交替变化。

(a + b)³ = a³ + 3a²b + 3ab² + b³

(a − b)³ = a³ − 3a²b + 3ab² − b³

Completing the square rewrites a quadratic in vertex form, which is essential for finding the minimum or maximum value.

配方法将二次式改写为顶点形式，是求最小值或最大值的关键。

ax² + bx + c = a(x + b/(2a))² + c − b²/(4a)

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2. Indices, Surds and Logarithms | 指数、根式与对数

The laws of indices allow powers to be combined, and they extend directly to fractional and negative indices. You must be confident with the equivalence between roots and fractional powers.

指数律用于合并幂，并可直接推广到分数指数与负指数。必须熟练掌握根式与分数指数之间的等价关系。

aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ

a⁰ = 1, a⁻ⁿ = 1/aⁿ, a^1/n = ⁿ√a, a^m/n = ⁿ√aᵐ

Rationalising surds removes roots from the denominator. For a denominator of the form a ± √b, multiply by the conjugate a ∓ √b.

根式有理化用于去掉分母中的根号。若分母为 a ± √b，可乘以共轭式 a ∓ √b。

1/(a + √b) = (a − √b)/(a² − b)

The laws of logarithms convert multiplication into addition, division into subtraction and powers into coefficients. The change of base formula is frequently needed in equations and numerical work.

对数律将乘法转化为加法、除法转化为减法、幂转化为系数。换底公式在方程与数值计算中经常需要。

logₐ(xy) = logₐx + logₐy

logₐ(x/y) = logₐx − logₐy

logₐ(xⁿ) = n logₐx

logₐx = log_b x / log_b a

When solving logarithmic equations, always check that the arguments remain positive in the original equation.

解对数方程时，务必代回原方程检验真数是否为正。

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3. Quadratic Equations, Discriminant and Polynomials | 二次方程、判别式与多项式

The quadratic formula gives the roots of any quadratic equation, and the discriminant determines the nature of those roots without solving the equation fully.

二次求根公式给出任意二次方程的根，而判别式可在不完全求解的情况下判断根的性质。

x = (−b ± √b² − 4ac) / (2a)

Δ = b² − 4ac

d real root. If Δ < 0, there are no real roots, but there are two complex conjugate roots.

若 Δ > 0，有两个不同实根；若 Δ = 0，有一个重实根；若 Δ < 0，无实根，但有一对共轭复根。

Vieta's formulas relate the sum and product of the roots to the coefficients of the quadratic.

韦达定理将两根之和与两根之积同二次方程的系数联系起来。

α + β = −b/a, αβ = c/a

by (x − a), the remainder is P(a). The factor theorem is the special case where P(a) = 0.

余数定理指出，多项式 P(x) 除以 (x − a) 的余数为 P(a)。因式定理是 P(a) = 0 的特殊情形。

P(x) = (x − a)Q(x) + P(a)

For polynomial inequalities, first factorise completely, then use a sign diagram or sketch to determine the solution intervals.

解多项式不等式时，先完全因式分解，再用符号图或草图确定解区间。

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4. Partial Fractions and Sequences and Series | 部分分式与数列级数

Partial fractions reverse the process of adding algebraic fractions. The form of the decomposition depends on whether the denominator has distinct linear factors, repeated linear factors or an irreducible quadratic factor.

部分分式是有理式加法的逆过程。分解形式取决于分母具有不同线性因式、重复线性因式还是不可约二次因式。

px + q / [(x − a)(x − b)] = A/(x − a) + B/(x − b)

px + q / (x − a)² = A/(x − a) + B/(x − a)²

An arithmetic sequence has a constant common difference, while a geometric sequence has a constant common ratio. Their nth terms and sums are standard results.

等差数列具有恒定公差，等比数列具有恒定公比。它们的通项与前 n 项和都是标准结论。

aₙ = a + (n − 1)d, Sₙ = n/2 [2a + (n − 1)d]

aₙ = arⁿ⁻¹, Sₙ = a(1 − rⁿ)/(1 − r), r ≠ 1

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An infinite geometric series converges only when the common ratio satisfies |r| < 1, and its sum is then a/(1 − r).

无穷等比级数仅当公比满足 |r| < 1 时收敛，其和为 a/(1 − r)。

S∞ = a/(1 − r), |r| < 1

The method of differences is useful for series whose terms can be written as a difference of consecutive expressions.

差分法适用于通项可写成相邻表达式之差的级数。

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5. Binomial Theorem and Binomial Series | 二项式定理与二项级数

For a positive integer n, the binomial theorem expands (a + b)ⁿ as a finite sum involving binomial coefficients.

对于正整数 n，二项式定理将 (a + b)ⁿ 展开为含二项式系数的有限和。

(a + b)ⁿ = Σ (r = 0 to n) [ⁿCᵣ aⁿ⁻ʳ bʳ]

ⁿCᵣ = n! / [r!(n − r)!]

The general term is useful when a particular power or coefficient is required without full expansion.

当需要某一特定幂次或系数而不必完全展开时，通项公式非常有用。

Tᵣ₊₁ = ⁿCᵣ aⁿ⁻ʳ bʳ

For any real n, the binomial series expansion of (1 + x)ⁿ is valid when |x| < 1.

对于任意实数 n，二项级数展开 (1 + x)ⁿ 在 |x| < 1 时成立。

(1 + x)ⁿ = 1 + nx + n(n − 1)x²/2! + n(n − 1)(n − 2)x³/3! + ...

When expanding (a + x)ⁿ, first factor out a to obtain aⁿ(1 + x/a)ⁿ, then apply the standard series with the condition |x/a| < 1.

展开 (a + x)ⁿ 时，先提取 a 得到 aⁿ(1 + x/a)ⁿ，再用标准级数，条件为 |x/a| < 1。

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6. Trigonometry: Identities and Equations | 三角恒等式与方程

Pythagorean identities are the foundation of trigonometric simplification. Reciprocal and quotient identities allow every trigonometric function to be expressed in terms of sine and cosine.

毕达哥拉斯恒等式是三角化简的基础。倒数关系与商数关系可将所有三角函数用正弦和余弦表示。

sin²θ + cos²θ = 1

1 + tan²θ = sec²θ, 1 + cot²θ = cosec²θ

tanθ = sinθ/cosθ, cotθ = cosθ/sinθ

sin(A ± B), cos(A ± B) and tan(A ± B). The double-angle formulae follow by setting B = A.

和角公式用于展开 sin(A ± B)、cos(A ± B) 与 tan(A ± B)。令 B = A 即可得到倍角公式。

sin(A ± B) = sinA cosB ± cosA sinB

cos(A ± B) = cosA cosB ∓ sinA sinB

tan(A ± B) = (tanA ± tanB)/(1 ∓ tanA tanB)

sin2θ = 2sinθcosθ

cos2θ = cos²θ − sin²θ = 1 − 2sin²θ = 2cos²θ − 1

To solve trigonometric equations, find the principal value and then use the general solution or the symmetry of the graph over the required interval.

解三角方程时，先求主值，再根据通解或指定区间内函数图像的对称性写出全部解。

sinθ = sinα → θ = nπ + (−1)ⁿα

cosθ = cosα → θ = 2nπ ± α

tanθ = tanα → θ = nπ + α

The sine rule, cosine rule and area formula apply to any triangle, not only right-angled triangles.

正弦定理、余弦定理与面积公式适用于任意三角形，而不仅限于直角三角形。

a/sinA = b/sinB = c/sinC = 2R

a² = b² + c² − 2bc cosA

Area = ½ab sinC

Expressions of the form a sinθ + b cosθ can be written as R sin(θ + α), where R = √a² + b² and tanα = b/a.

形如 a sinθ + b cosθ 的表达式可写成 R sin(θ + α)，其中 R = √a² + b²，tanα = b/a。

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7. Differentiation and Integration | 微分与积分

Standard derivatives should be memorised, including the derivatives of trigonometric, exponential and logarithmic functions.

标准导数需要熟记，包括三角函数、指数函数与对数函数的导数。

d/dx (xⁿ) = nxⁿ⁻¹

d/dx (eˣ) = eˣ, d/dx (ln x) = 1/x

d/dx (sin x) = cos x, d/dx (cos x) = −sin x, d/dx (tan x) = sec²x

The product rule, quotient rule and chain rule are essential for differentiating composite and combined functions.

乘积法则、商法则与链式法则是求复合函数与组合函数导数的关键。

(uv)' = u'v + uv'

(u/v)' = (u'v − uv')/v²

dy/dx = dy/du × du/dx

Standard integrals reverse the differentiation results. Remember the constant of integration for indefinite integrals.

标准积分是导数结果的逆运算。不定积分不要忘记积分常数。

∫ xⁿ dx = xⁿ⁺¹/(n + 1) + C, n ≠ −1

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