Exam-style Practice: Paper 2: Pure Mathematics | 考试风格练习：卷二纯数学

📚 Exam-style Practice: Paper 2: Pure Mathematics | 考试风格练习：卷二纯数学

This article provides exam-style practice questions and fully worked solutions for the Edexcel A-Level Mathematics Paper 2 (Pure Mathematics). The topics covered reflect the most frequently tested areas in recent examination series, including algebraic proof, inequalities, coordinate geometry, sequences and series, trigonometry, logarithms, calculus, numerical methods and vectors.

本文围绕爱德思 A-Level 数学卷二（纯数学）提供考试风格的练习题与完整解答。所选主题对应近年考试中最常出现的考查范围，包括代数证明、不等式、坐标几何、数列与级数、三角学、对数、微积分、数值方法与向量。

Before reading the solutions, attempt every question independently under timed conditions. This will help you identify weaknesses and build exam stamina.

在阅读解答之前，请在限时条件下独立完成每一道题。这会帮助你发现薄弱环节并增强考场耐力。

1. Algebraic Proof | 代数证明

Question: Prove that the sum of an even integer and an odd integer is always odd.

题目：证明一个偶数与一个奇数之和总是一个奇数。

Let the even integer be 2n and the odd integer be 2m + 1, where n and m are integers.

设偶数为 2n，奇数为 2m + 1，其中 n 和 m 是整数。

Their sum is 2n + (2m + 1) = 2(n + m) + 1. Since n + m is an integer, 2(n + m) + 1 is one more than a multiple of 2, so it is odd. The statement is therefore proven.

它们的和为 2n + (2m + 1) = 2(n + m) + 1。由于 n + m 是整数，2(n + m) + 1 比 2 的倍数多 1，因此它是奇数。命题得证。

Exam tip: In proof questions, always define your variables clearly and justify each algebraic step. Marks are often awarded for a clear logical structure.

考试提示：在证明题中，务必清晰定义变量，并说明每一步代数变形的依据。阅卷常按清晰的逻辑结构给分。

2. Quadratic Inequalities | 二次不等式

Question: Solve the inequality x² − 5x + 6 > 0.

题目：解不等式 x² − 5x + 6 > 0。

First, factorise the quadratic expression:

首先对二次表达式进行因式分解：

x² − 5x + 6 = (x − 2)(x − 3)

The critical values are x = 2 and x = 3. These divide the real line into three intervals: x < 2, 2 < x < 3 and x > 3.

临界值为 x = 2 和 x = 3。这两点将实数轴划分为三个区间：x < 2、2 < x < 3 和 x > 3。

• For x < 2, both factors are negative, so the product is positive.
• 当 x < 2 时，两个因式均为负，乘积为正。
• For 2 < x < 3, the factors have opposite signs, so the product is negative.
• 当 2 < x < 3 时，两个因式符号相反，乘积为负。
• For x > 3, both factors are positive, so the product is positive.
• 当 x > 3 时，两个因式均为正，乘积为正。

x < 2 或 x > 3

Exam tip: Sketching the parabola y = x² − 5x + 6 is the quickest way to see the solution region. The graph is a ‘smile’ curve crossing the x-axis at 2 and 3.

考试提示：画出抛物线 y = x² − 5x + 6 是确定解区间最快的方法。该图像是开口向上的曲线，与 x 轴交于 2 和 3 两点。

3. Coordinate Geometry: Circle | 坐标几何：圆

Question: The point A(3, 4) lies on a circle with centre C(1, 2). Find the equation of the circle.

题目：点 A(3, 4) 位于圆心为 C(1, 2) 的圆上。求该圆的方程。

The radius r is the distance from the centre C to the point A:

半径 r 为圆心 C 到点 A 的距离：

r = √[(3 − 1)² + (4 − 2)²] = √(4 + 4) = √8

The equation of a circle with centre (a, b) and radius r is (x − a)² + (y − b)² = r². Substituting a = 1, b = 2 and r² = 8:

圆心为 (a, b)、半径为 r 的圆方程为 (x − a)² + (y − b)² = r²。代入 a = 1、b = 2 和 r² = 8：

(x − 1)² + (y − 2)² = 8

Exam tip: If the centre is the midpoint of the diameter, first find the midpoint to obtain the centre, then use the distance formula to find the radius.

Published by TutorHao | A-Level Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）