Core Pre-U Cambridge Mathematics: A Comprehensive Review of Key Topics | 剑桥 Pre-U 数学：核心知识点梳理

📚 Core Pre-U Cambridge Mathematics: A Comprehensive Review of Key Topics | 剑桥 Pre-U 数学：核心知识点梳理

The Cambridge Pre-U Mathematics course is a rigorous qualification designed to stretch the most able students, providing a deep and coherent understanding of pure mathematics along with applications in statistics and mechanics. This article revisits the essential core topics, offering a structured revision guide that highlights key definitions, theorems, and techniques required for success in the examination.

剑桥 Pre-U 数学课程是一门富有挑战性的资格认证，旨在激发最优秀学生的潜能，帮助他们对纯数学及其在统计和力学中的应用建立深入且连贯的理解。本文重新梳理了核心知识点，提供一份结构化的复习指南，突出考试成功所必需的关键定义、定理和技巧。

1. Algebra and Equations | 代数与方程

The quadratic equation ax² + bx + c = 0 has solutions given by the formula x = (-b ± √(b² – 4ac)) / (2a). The discriminant Δ = b² – 4ac determines the nature of the roots: two distinct real roots if Δ > 0, one repeated real root if Δ = 0, and a conjugate pair of complex roots if Δ < 0.

二次方程 ax² + bx + c = 0 的解由公式 x = (-b ± √(b² – 4ac)) / (2a) 给出。判别式 Δ = b² – 4ac 决定了根的性质：若 Δ > 0，有两个相异实根；若 Δ = 0，有一个重实根；若 Δ < 0，有一对共轭复根。

Polynomial division together with the Factor Theorem enables root-finding and factorisation. If f(a) = 0, then (x – a) is a factor of f(x). The Remainder Theorem states that when f(x) is divided by (x – a), the remainder is f(a). Partial fractions decompose rational expressions into simpler forms, handling distinct linear factors, repeated factors, and irreducible quadratic factors.

多项式除法与因式定理共同帮助求根和因式分解。若 f(a) = 0，则 (x – a) 是 f(x) 的一个因式。余数定理指出，当 f(x) 除以 (x – a) 时，余数为 f(a)。部分分式将有理式分解为更简单的形式，可处理相异线性因式、重复因式以及不可约二次因式。

The laws of indices and logarithms are foundational: a^m × aⁿ = a^m+n, a^m ÷ aⁿ = a^m-n, (a^m)ⁿ = a^mn, a⁰ = 1, a^-n = 1/aⁿ. For logarithms: log_a(xy) = log_ax + log_ay, log_a(x/y) = log_ax – log_ay, log_ax^k = k log_ax. The change-of-base formula is log_ba = log_ca / log_cb.

指数与对数的法则是基础：a^m × aⁿ = a^m+n, a^m ÷ aⁿ = a^m-n, (a^m)ⁿ = a^mn, a⁰ = 1, a^-n = 1/aⁿ。对数法则为：log_a(xy) = log_ax + log_ay, log_a(x/y) = log_ax – log_ay, log_ax^k = k log_ax。换底公式为 log_ba = log_ca / log_cb。

Simultaneous equations in two variables may be linear or include a non-linear second equation. Substitution and elimination are standard approaches. Equations involving absolute values, such as |ax + b| = c, require considering both positive and negative branches, leading to two separate linear equations.

二元方程组可以是线性的，也可以是包含非线性方程的混合型。代入法和消元法是标准解法。含有绝对值的方程，例如 |ax + b| = c，需要同时考虑正负两种情形，进而化为两个独立的线性方程。

x = (-b ± √(b² – 4ac)) / (2a)

2. Functions and Graphs | 函数与图像

A function maps elements from its domain to exactly one element in its codomain. The range is the set of all output values. A function is one-to-one (injective) if f(a) = f(b) implies a = b, and onto (surjective) if its range equals the codomain. The inverse function f^-1 exists if and only if f is bijective.

函数将其定义域中的每个元素映射到陪域中唯一的元素。值域是所有输出值的集合。若 f(a) = f(b) 蕴含 a = b，则函数为单射；若值域等于陪域，则为满射。反函数 f^-1 存在的充要条件是 f 为双射。

Composite functions (f ∘ g)(x) = f(g(x)) are defined when the range of g is a subset of the domain of f. Graph transformations include translations by vector (a, b): y = f(x – a) + b; stretches: horizontal by factor 1/a as y = f(ax), vertical by factor a as y = a f(x); and reflections in axes: y = -f(x) reflects in the x-axis, y = f(-x) reflects in the y-axis.

复合函数 (f ∘ g)(x) = f(g(x)) 在 g 的值域包含于 f 的定义域时定义。图像变换包括平移：按向量 (a, b) 平移为 y = f(x – a) + b；伸缩：水平方向因子 1/a 对应 y = f(ax)，垂直方向因子 a 对应 y = a f(x)；以及轴对称：y = -f(x) 关于 x 轴对称，y = f(-x) 关于 y 轴对称。

The modulus function |x| yields the absolute value, and graphs involving |f(x)| or f(|x|) require piecewise consideration. An even function satisfies f(-x) = f(x) (symmetry about y-axis), while an odd function satisfies f(-x) = -f(x) (rotational symmetry about the origin). Periodic functions such as sin x repeat every 2π.

模函数 |x| 给出绝对值，涉及 |f(x)| 或 f(|x|) 的图像需要分段考虑。偶函数满足 f(-x) = f(x)（关于 y 轴对称），奇函数满足 f(-x) = -f(x)（关于原点旋转对称）。周期函数如 sin x 以 2π 为周期重复。

f(-x) = f(x) even, f(-x) = -f(x) odd

3. Trigonometry | 三角学

Radian measure links angle to arc length: for angle θ in radians, arc length = rθ and sector area = ½ r²θ. The fundamental identity is sin²θ + cos²θ = 1, and its derived forms: 1 + tan²θ = sec²θ and 1 + cot²θ = cosec²θ.

弧度制将角度与弧长联系起来：对于 θ 弧度，弧长 = rθ，扇形面积 = ½ r²θ。基本恒等式为 sin²θ + cos²θ = 1，衍生形式为 1 + tan²θ = sec²θ 和 1 + cot²θ = cosec²θ。

Compound-angle and double-angle formulas are essential: sin(A ± B) = sinA cosB ± cosA sinB, cos(A ± B) = cosA cosB ∓ sinA sinB, tan(A ± B) = (tanA ± tanB) / (1 ∓ tanA tanB). Double-angle: sin 2θ = 2 sinθ cosθ, cos 2θ = cos²θ – sin²θ = 2cos²θ – 1 = 1 – 2sin²θ.

和角与倍角公式至关重要：sin(A ± B) = sinA cosB ± cosA sinB, cos(A ± B) = cosA cosB ∓ sinA sinB, tan(A ± B) = (tanA ± tanB) / (1 ∓ tanA tanB)。倍角公式：sin 2θ = 2 sinθ cosθ, cos 2θ = cos²θ – sin²θ = 2cos²θ – 1 = 1 – 2sin²θ。

Published by TutorHao | Pre-U Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）