Year 11 OCR Maths: Core Topics Summary | Year 11 OCR 数学：核心知识点梳理

📚 Year 11 OCR Maths: Core Topics Summary | Year 11 OCR 数学：核心知识点梳理

Year 11 is the crucial GCSE year, and OCR Mathematics covers a broad range of core topics that form the foundation for higher study. This article summarises the essential knowledge you need to revise, from number skills to advanced algebra, geometry, statistics and probability.

Year 11 是关键的 GCSE 学年，OCR 数学涵盖了广泛的核心主题，为更高层次的学习奠定基础。本文梳理了你需要复习的核心知识点，从数字技巧到高级代数、几何、统计与概率。

1. Surds and Indices | 根式与指数

Understand the laws of indices: aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1, a⁻ⁿ = 1/aⁿ, and a^{1/n} = ⁿ√a. For fraction indices, a^{m/n} = (ⁿ√a)ᵐ.

掌握指数法则：aᵐ × aⁿ = aᵐ⁺ⁿ，aᵐ ÷ aⁿ = aᵐ⁻ⁿ，(aᵐ)ⁿ = aᵐⁿ，a⁰ = 1，a⁻ⁿ = 1/aⁿ，以及 a^{1/n} = ⁿ√a。对于分数指数，a^{m/n} = (ⁿ√a)ᵐ。

Simplify surds: √(ab) = √a × √b, √(a/b) = √a / √b. Rationalise denominators, e.g. 1/√a → √a/a, or for expressions like 1/(√a + √b) multiply by the conjugate.

化简根式：√(ab) = √a × √b，√(a/b) = √a / √b。分母有理化，例如 1/√a → √a/a，或对于 1/(√a + √b) 这样的表达式，乘以共轭式。

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2. Standard Form | 标准形式

Write numbers in the form A × 10ⁿ where 1 ≤ A < 10 and n is an integer. Move the decimal point to create A, and n is the number of places moved (positive for large numbers, negative for small).

将数字写成 A × 10ⁿ 的形式，其中 1 ≤ A < 10，n 为整数。移动小数点得到 A，移动的位数即为 n（大数为正，小数为负）。

Perform calculations: multiply numbers by multiplying A values and adding exponents; divide by dividing A values and subtracting exponents. Add or subtract only if powers are equal; adjust first.

进行计算：相乘时，将 A 值相乘、指数相加；相除时，将 A 值相除、指数相减。只有当幂次相同时才能直接加减；否则需先调整。

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3. Expanding and Factorising | 展开与因式分解

Expand brackets using the distributive law: a(b + c) = ab + ac. For double brackets, (x + a)(x + b) = x² + (a+b)x + ab. For triple brackets, expand two first, then multiply by the third.

使用分配律展开括号：a(b + c) = ab + ac。对于双括号，(x + a)(x + b) = x² + (a+b)x + ab。对于三个括号，先展开两个，再乘以第三个。

Factorise expressions by finding common factors: e.g. 6x² + 9x = 3x(2x + 3). Factorise quadratics: x² + bx + c = (x + p)(x + q) where p+q = b and pq = c. For ax² + bx + c, use splitting the middle term or trial. Recognise difference of two squares: a² – b² = (a+b)(a-b).

因式分解先提取公因式：如 6x² + 9x = 3x(2x + 3)。分解二次式：x² + bx + c = (x + p)(x + q)，其中 p+q = b，pq = c。对于 ax² + bx + c，使用拆项法或尝试。识别平方差公式：a² – b² = (a+b)(a-b)。

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4. Quadratic Equations | 二次方程

Solve quadratic equations by factorising, completing the square, or using the quadratic formula: x = [-b ± √(b² – 4ac)] / 2a.

通过因式分解、配方法或二次公式 x = [-b ± √(b² – 4ac)] / 2a 解二次方程。

The discriminant Δ = b² – 4ac tells the nature of 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 决定了根的性质：Δ > 0 有两个不等实根；Δ = 0 有一个重根；Δ < 0 没有实根。

Complete the square: write x² + bx + c as (x + b/2)² – (b/2)² + c. This helps find the vertex of a parabola.

配方法：将 x² + bx + c 写成 (x + b/2)² – (b/2)² + c。这有助于求抛物线的顶点。

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5. Simultaneous Equations | 联立方程组

Solve linear simultaneous equations by elimination or substitution. Align coefficients and add/subtract to eliminate one variable. Solve the resulting one-variable equation, then substitute back.

用消元法或代入法解线性方程组。对齐系数，相加或相减以消去一个变量。解出单变量方程，再回代。

For one linear and one quadratic equation, use substitution: replace y from the linear into the quadratic, then solve the resulting quadratic equation. Check for extraneous solutions.

对于一线性一二次方程组，使用代入法：将线性方程中的 y 代入二次方程，然后解所得二次方程。检查是否有增根。

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6. Inequalities | 不等式

Solve linear inequalities like equations, but reverse the inequality sign when multiplying or dividing by a negative number. Represent solutions on a number line with open/closed circles.

像解方程一样解线性不等式，但当乘以或除以负数时，不等式方向反转。在数轴上用空心/实心圆表示解。

For quadratic inequalities, solve the corresponding equation, sketch the graph, and determine intervals where the quadratic is above or below zero. Write solution sets using inequality or set notation.

对于二次不等式，解对应方程，绘制图像草图，然后确定二次式大于或小于零的区间。用不等式或集合符号表示解集。

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7. Circle Theorems | 圆定理

Key theorems: Angle at centre = 2 × angle at circumference. Angle in a semicircle is 90°. Angles in the same segment are equal. Opposite angles in a cyclic quadrilateral sum to 180°.

核心定理：圆心角 = 2 × 圆周角。半圆所对圆周角为 90°。同弧上的圆周角相等。圆内接四边形对角互补，和为 180°。

Tangent-radius property: tangent to a circle is perpendicular to the radius at the point of contact. Alternate segment theorem: angle between tangent and chord equals angle in the alternate segment.

Published by TutorHao | Year 11 Mathematics Revision Series | aleveler.com

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