Core Topics in Cambridge Year 10 Further Mathematics | 剑桥Year 10进阶数学核心知识点梳理

📚 Core Topics in Cambridge Year 10 Further Mathematics | 剑桥Year 10进阶数学核心知识点梳理

Cambridge Year 10 Further Mathematics (often based on the IGCSE Additional Mathematics 0606 syllabus) introduces students to more advanced algebraic, trigonometric and calculus concepts. This article provides a structured revision guide covering the essential topics, with clear explanations and examples to help you build a strong foundation.

剑桥 Year 10 进阶数学（通常基于 IGCSE 附加数学 0606 大纲）引领学生接触更深入的代数、三角与微积分概念。本文将按主题系统梳理核心知识点，通过清晰讲解与示例帮助你打下扎实基础。

1. Functions | 函数

function must be a one-to-one or many-to-one mapping; one-to-many relations are not functions. The notation f: x → f(x) or f(x) = … is used.

函数 f 将每个输入 x 映射到唯一输出 f(x)。定义域是所有允许输入的集合，值域是所有可能输出的集合。函数必须是一对一或多对一映射；一对多的关系不是函数。常用记号 f: x → f(x) 或 f(x) = … 表示。

range of f. Composite functions such as gf(x) = g(f(x)) apply one function after another.

反函数 f⁻¹(x) 仅在 f 为一一映射时存在。求法：写 y = f(x)，交换 x 和 y，再解出 y。f⁻¹ 的定义域就是 f 的值域。复合函数如 gf(x) = g(f(x)) 表示先作用 f 再作用 g。

Understanding the graph of a function helps visualise its behaviour. The vertical line test confirms if a graph represents a function, while the horizontal line test checks for invertibility.

理解函数图像有助于直观分析其性质。垂直线检验可判断图像是否表示函数，水平线检验则用于判断是否可逆。

---

2. Quadratic Functions | 二次函数

raph is a parabola. If a > 0, the parabola opens upwards and has a minimum point; if a < 0, it opens downwards and has a maximum point.

二次函数的一般形式为 f(x) = ax² + bx + c (a ≠ 0)。其图像为抛物线。若 a > 0，抛物线开口向上且有最小值点；若 a < 0，则开口向下且有最大值点。

vertical line x = h. The discriminant Δ = b² − 4ac determines the number of x-intercepts: Δ > 0 gives two distinct real roots, Δ = 0 gives one repeated root, and Δ < 0 gives no real roots.

顶点（转折点）可通过配方法得到：函数写成 a(x − h)² + k 后，顶点为 (h, k)。对称轴是垂直线 x = h。判别式 Δ = b² − 4ac 决定 x 轴截距的个数：Δ > 0 有两个不同实根，Δ = 0 有一个重根，Δ < 0 无实根。

The quadratic formula x = [−b ± √b² − 4ac] / 2a gives the roots explicitly. Knowing the vertex and intercepts allows you to sketch the graph accurately.

求根公式 x = [−b ± √b² − 4ac] / 2a 可直接给出根。掌握顶点和截距后便可精确绘制图像。

---

3. Equations, Inequalities and Graphs | 方程、不等式与图像

Solving equations involving quadratics, surds or rational expressions requires careful algebraic manipulation. Always check for extraneous solutions when squaring both sides. Simultaneous equations, linear and quadratic, can be solved by substitution.

求解含二次式、根式或有理式的方程需要细心的代数变形。两边平方时务必检验增根。线性与二次联立方程组可通过代入法求解。

Inequalities like ax² + bx + c > 0 are solved by sketching the related quadratic graph to identify intervals where the expression is positive. Critical values come from solving the equality. For rational inequalities, multiply both sides by the square of the denominator to preserve the inequality sign.

像 ax² + bx + c > 0 这样的不等式可通过画出相应二次图像来求解，确定表达式取正值的区间。临界值来自解对应的方程。对于有理不等式，两边乘以分母的平方以保持不等号方向。

The modulus function |x| gives the distance from zero. Equations such as |x − 3| = 5 lead to two cases: x − 3 = 5 or x − 3 = −5. Modulus graphs show V-shaped reflections.

绝对值函数 |x| 给出到零点的距离。方程 |x − 3| = 5 分为两种情况：x − 3 = 5 或 x − 3 = −5。绝对值图像呈 V 形反射。

---

4. Indices and Surds | 指数与根式

es are fundamental: aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1, a⁻ⁿ = 1/aⁿ, and a^1/n = n√a. These apply to rational exponents as well.

指数法则是基础：aᵐ × aⁿ = aᵐ⁺ⁿ，aᵐ ÷ aⁿ = aᵐ⁻ⁿ，(aᵐ)ⁿ = aᵐⁿ，a⁰ = 1，a⁻ⁿ = 1/aⁿ，以及 a^1/n = n√a。这些法则同样适用于有理指数。

Surds are irrational roots such as √2. Simplify surds by extracting square factors: √18 = 3√2. Rationalise denominators when a surd appears: 1/√a → √a/a, and 1/(a + √b) → (a − √b)/(a² − b). Always express answers in simplest surd form.

根式是无理根，如 √2。通过提取平方因子化简：√18 = 3√2。分母含根式时进行有理化：1/√a → √a/a，1/(a + √b) → (a − √b)/(a² − b)。答案始终用最简根式表示。

Working with fractional and negative indices enables you to handle expressions like ⁴√x³ = x^3/4. These skills are essential for calculus and algebraic manipulation later on.

掌握分数指数和负指数有助于处理如 ⁴√x³ = x^3/4 的式子。这些技能对后续微积分和代数运算至关重要。

---

5. Polynomials and Factor Theorem | 多项式与因式定理

, the remainder is P(a). The Factor Theorem follows: (x − a) is a factor of P(x) if and only if P(a) = 0.

多项式 P(x) 可用长除法或综合除法除以 (x − a)。余式定理指出，P(x) 除以 (x − a) 所得余数为 P(a)。因式定理由此得出：(x − a) 是 P(x) 的因式当且仅当 P(a) = 0。

To factorise a cubic polynomial, first find one linear factor by testing small integer values. Then divide to obtain a quadratic factor, which can be factorised further or solved using the quadratic formula. This method is crucial for solving higher-degree equations.

分解三次多项式时，先通过试验小整数值找到一个一次因式。接着做除法得到二次因式，再将其进一步分解或用求根公式求解。此方法是解高次方程的关键。

e roots r₁, r₂, r₃ satisfy P(r) = 0. Repeated roots indicate the graph touches the x-axis.

始终将多项式写成因式乘积形式：P(x) = (x − r₁)(x − r₂)(x − r₃)。根 r₁, r₂, r₃ 满足 P(r) = 0。重根表明图像与 x 轴相切。

---

广告 / Sponsored

6. Binomial Expansion | 二项式展开

The Binomial Theorem for positive integer n gives (a + b)ⁿ = Σ (nCr) aⁿ⁻ʳ bʳ, where nCr = n! / [r!(n−r)! the binomial coefficient, also written as C(n, r) or (n r). The expansion has (n+1) terms.

对于正整数 n，二项式定理为 (a + b)ⁿ = Σ (nCr) aⁿ⁻ʳ bʳ，其中 nCr = n! / [r!(n−r)!] 是二项式系数，也可写作 C(n, r) 或 (n r)。展开式共有 (n+1) 项。

Pascal' find coefficients for small n. For example, (1 + x)⁴ = 1 + 4x + 6x² + 4x³ + x⁴. Each term's power of x increases, while coefficients are symmetric.

帕斯卡三角可快速得出小 n 值的系数。例如 (1 + x)⁴ = 1 + 4x + 6x² + 4x³ + x⁴。各项 x 的幂次递增，系数呈对称分布。

)²(−3) + 3(2x)(−3)² + (−3)³. Simplify step by step. You might also need to find a specific term, like the term in xᵏ, by setting r appropriately.

当项中含有系数时需小心处理：(2x − 3)³ = (2x)³ + 3(2x)²(−3) + 3(2x)(−3)² + (−3)³。逐步化简。也可能需要求特定项，如 xᵏ 项，通过设定合适的 r 值即可。

---

7. Trigonometry | 三角学

gth s = rθ and sector area A = ½ r²θ. Conversion between degrees and radians is essential.

角度可用度数或弧度度量；180° = π 弧度。弧度制使弧长公式 s = rθ 和扇形面积 A = ½ r²θ 变得简洁。度与弧度的互换是基本功。

dentities sin²θ + cos²θ = 1 and tan θ = sin θ / cos θ are used frequently. Quadrant signs (All, Sine, Tan, Cos) help determine signs of trig ratios.

基本三角函数 sin θ、cos θ、tan θ 在单位圆上定义。需熟记 0°、30°、45°、60°、90°（或弧度）的精确值。恒等式 sin²θ + cos²θ = 1 和 tan θ = sin θ / cos θ 经常使用。象限符号（全正、Sin、Tan、Cos）有助于判断三角比的符号。

inear forms like 2 sin x − 1 = 0, solve directly. For equations like sin 2x = 0.5, consider the multiple angle first.

解三角方程：分离出三角比，求出主值，再利用对称性求出给定区间内的所有解。对于 2 sin x − 1 = 0 这类线性形式可直接求解。对于 sin 2x = 0.5 这类倍角方程，需先考虑倍角的范围。

---

8. Introduction to Differentiation | 微分初步

Differentiation finds the gradient of a curve. The derivative f' ed as the limit of [f(x+h) − f(x)] / h as h → 0. For power functions, if f(x) = xⁿ, then f'(x) = nxⁿ⁻¹. The rule extends to constant multiples, sums and differences.

微分用于求曲线的梯度。导数 f'(x) 定义为 [f(x+h) − f(x)] / h 当 h → 0 时的极限。对于幂函数，若 f(x) = xⁿ，则 f'(x) = nxⁿ⁻¹。该法则可推广到常数倍、和与差。

resents the instantaneous rate of change. At a point, the gradient of the tangent equals f'(x). The normal to the curve has gradient −1 / f'(x). You can find the equation of a tangent using y − y₁ = m (x − x₁).

导数表示瞬时变化率。曲线上一点处切线的斜率等于 f'(x)。法线的斜率为 −1 / f'(x)。可用 y − y₁ = m(x − x₁) 求切线方程。

Stationary points occur where f'(x) = 0. Use the second derivative f''(x) to classify them: f''(x) > 0 indicates a local minimum, f''(x) < 0 a local maximum. If f''(x) = 0, further testing is needed.

驻点出现在 f'(x) = 0 处。用二阶导数 f''(x) 进行分类：f''(x) > 0 表示局部极小值，f''(x) < 0 表示局部极大值。若 f''(x) = 0，则需要进一步检验。

---

9. Introduction to Integration | 积分初步

ers, ∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + C, where n ≠ −1. The constant of integration C is essential.

积分是微分的逆运算。不定积分 ∫ f(x) dx 给出原函数族。对于幂函数，∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + C，其中 n ≠ −1。积分常数 C 必不可少。

limits into the antiderivative: F(b) − F(a). Areas below the x-axis are negative, so interpret carefully.

定积分 ∫ₐᵇ f(x) dx 计算曲线 y = f(x) 与 x 轴在 x = a 到 x = b 之间的有号面积。将上下限代入原函数计算：F(b) − F(a)。x 轴下方的面积为负，解释时需留意。

To find the area between a curve and a line, integrate the difference. Kinematics problems connect integration with motion: velocity is the derivative of displacement, and acceleration is the derivative of velocity. Integrating acceleration gives velocity.

求曲线与直线之间的面积时，对差函数积分。运动学问题将积分与运动联系起来：速度是位移的导数，加速度是速度的导数。对加速度积分可得速度。

---

10. Vectors in Two Dimensions | 二维向量

It can be represented in column form (x y) or using i, j notation: xi + yj. The magnitude |v| = √x² + y² gives the length of the vector. A unit vector has magnitude 1 and is found by dividing the vector by its magnitude.

向量具有大小和方向。可用列向量形式 (x y) 或 i、j 记号表示：xi + yj。模 |v| = √x² + y² 给出向量的长度。单位向量模为 1，通过将向量除以其模得到。

Vector addition and subtraction are done component-wise. Multiplication by a scalar changes the magnitude but not the direction (unless negative). The position vector of a point describes its location relative to the origin.

向量加减按分量进行。数乘改变大小但不改变方向（除非乘数为负）。点的位置向量描述其相对于原点的位置。

(dot) product of a = x₁i + y₁j and b = x₂i + y₂j is a·b = x₁x₂ + y₁y₂. It is also equal to |a||b| θ is the angle between the vectors. This helps test for perpendicularity: a·b = 0 means a ⟂ b.

向量 a = x₁i + y₁j 与 b = x₂i + y₂j 的数量积（点积）为 a·b = x₁x₂ + y₁y₂。它也等于 |a||b| cos θ，其中 θ 是两向量夹角。这可用于检验垂直：a·b = 0 意味着 a ⟂ b。

Vectors are used to solve geometrical problems involving parallel lines, midpoints, and ratios. Expressing vectors in terms of known base vectors and applying these rules forms a powerful tool for proofs.

向量可用于解决涉及平行线、中点和比例的几何问题。用已知基向量表示向量并应用这些法则，是进行证明的有力工具。

---

Published by TutorHao | Further Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）