📚 Edexcel AS Maths Practice Paper (Pure) | 爱德思AS数学模拟试卷（纯数）
This article provides a full exam-style practice paper for the Edexcel AS Mathematics Paper 1 (Pure Mathematics), complete with detailed, step-by-step bilingual solutions. The paper covers essential AS topics, including algebra, coordinate geometry, binomial expansion, trigonometry, differentiation, integration, and exponentials. Each question is followed by a worked solution to help you check your reasoning and common pitfalls.
本文提供一套完整的爱德思AS数学试卷1（纯数学）的模拟考题，并配有详细的双语解答步骤。试卷涵盖代数、坐标几何、二项式展开、三角学、微分、积分、指数与对数等AS必考内容。每道题后附有详解，帮助你检验思路、避开常见失分点。
1. Algebraic Simplification | 代数化简
Question
Simplify (a) (8x⁶)¹/³ (2 marks), (b) (√x + 2√x)² , expressing your answer as a single term in x (2 marks). (c) Rationalise the denominator of 1/(2 – √3) (1 mark).
化简 (a) (8x⁶)¹/³ (2分), (b) (√x + 2√x)²，结果表示为x的单项式 (2分)。(c) 将 1/(2 – √3) 有理化分母 (1分)。
Worked Solution | 解答
(a) (8x⁶)¹/³ = 8¹/³ × (x⁶)¹/³ = 2 × x² = 2x².
(a) (8x⁶)¹/³ = 8¹/³ × (x⁶)¹/³ = 2 × x² = 2x².
(b) (√x + 2√x)² = (3√x)² = 9x.
(b) (√x + 2√x)² = (3√x)² = 9x.
(c) 1/(2 – √3) = (2 + √3)/((2 – √3)(2 + √3)) = (2 + √3)/(4 – 3) = 2 + √3.
(c) 1/(2 – √3) = (2 + √3)/((2 – √3)(2 + √3)) = (2 + √3)/(4 – 3) = 2 + √3.
2. Quadratics and the Discriminant | 二次函数与判别式
Question
ots. (a) Find the range of possible values of k (3 marks). (b) State the values of k for which the equation has exactly one real root (1 mark). (c) Verify your answer to part (b) by factorisation (2 marks).
方程 x² + kx + 9 = 0 无实数根。(a) 求k的可能取值范围 (3分)。(b) 写出使方程恰好有一个实数根的k值 (1分)。(c) 通过因式分解验证(c)中的答案 (2分)。
Worked Solution | 解答
(a) Discriminant Δ = k² – 36. For no real roots, Δ < 0 ⇒ k² < 36 ⇒ –6 < k < 6.
(a) 判别式 Δ = k² – 36。无实根时 Δ < 0 ⇒ k² < 36 ⇒ –6 < k < 6。
(b) One real root (equal roots) occurs when Δ = 0 ⇒ k = 6 or k = –6.
(b) 一个实数根（重根）发生在 Δ = 0，所以 k = 6 或 k = –6。
(c) When k = 6, equation is x² + 6x + 9 = (x + 3)² = 0, root x = –3 (repeated). When k = –6, equation is x² – 6x + 9 = (x – 3)² = 0, root x = 3 (repeated).
(c) 当 k = 6，方程为 x² + 6x + 9 = (x + 3)² = 0，重根 x = –3。当 k = –6，方程为 x² – 6x + 9 = (x – 3)² = 0，重根 x = 3。
3. Coordinate Geometry | 坐标几何
Question
of the perpendicular bisector of AB, giving your answer in the form ax + by = c (4 marks).
已知点 A(1, 2) 和 B(5, 8)：(a) 求 AB 的中点 M (2分)。(b) 求 AB 的垂直平分线的方程，答案写成 ax + by = c 的形式 (4分)。
Worked Solution | 解答
(a) Midpoint M = ((1+5)/2, (2+8)/2) = (3, 5).
(a) 中点 M = ((1+5)/2, (2+8)/2) = (3, 5)。
(b) Gradient of AB = (8 – 2)/(5 – 1) = 6/4 = 3/2. Gradient of perpendicular bisector = –2/3.
(b) AB 的斜率 = (8 – 2)/(5 – 1) = 6/4 = 3/2。垂直平分线斜率 = –2/3。
Equation through M: y – 5 = –2/3 (x – 3). Multiply by 3: 3y – 15 = –2x + 6 ⇒ 2x + 3y = 21.
经过 M 的方程：y – 5 = –2/3 (x – 3)。两边乘3：3y – 15 = –2x + 6 ⇒ 2x + 3y = 21。
4. Binomial Expansion | 二项式展开
Question
(a) Expand (1 + 2x)⁵ in ascending powers of x up to and including the term in x³ (3 marks). (b) Hence estimate (1.01)⁵, giving your answer to 4 decimal places (2 marks).
(a) 将 (1 + 2x)⁵ 按x的升幂展开至x³项
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