IGCSE AQA Further Maths Unit Test Mock Paper Solutions | IGCSE AQA 进阶数学单元测试模拟卷解析

📚 IGCSE AQA Further Maths Unit Test Mock Paper Solutions | IGCSE AQA 进阶数学单元测试模拟卷解析

This article walks through a full unit test mock paper for AQA IGCSE Further Mathematics, providing step-by-step solutions and bilingual explanations for each question. Covering algebra, functions, coordinate geometry, calculus, matrices, vectors and trigonometry, these model answers highlight key techniques and common pitfalls. Use this walkthrough to reinforce your understanding, sharpen problem-solving speed, and build confidence before your next assessed task or examination.

本文详细讲解一套为 AQA IGCSE 进阶数学设计的单元测试模拟卷，逐题给出分步解析与中英对照说明。内容涵盖代数、函数、坐标几何、微积分、矩阵、向量和三角学，模型答案突出关键方法与常见误区。通过这份解析，你可以巩固理解、提升解题速度，并在下一次评估或考试前树立信心。

1. Exponential Equation | 指数方程

Question: Solve 4^x+1 = 8^2x−3.

题目： 解方程 4^x+1 = 8^2x−3。

Rewrite 4 as 2² and 8 as 2³ so that both sides have base 2.

将 4 写成 2²，8 写成 2³，使两边都使用底数 2。

Left side: (2²)^x+1 = 2^2(x+1) = 2^2x+2. Right side: (2³)^2x−3 = 2^3(2x−3) = 2^6x−9.

左边：(2²)^x+1 = 2^2(x+1) = 2^2x+2。右边：(2³)^2x−3 = 2^3(2x−3) = 2^6x−9。

Equate the exponents: 2x+2 = 6x−9.

令指数相等：2x+2 = 6x−9。

Solve: 2x+2 = 6x−9 → 2+9 = 6x−2x → 11 = 4x → x = 11/4.

求解：2x+2 = 6x−9 → 2+9 = 6x−2x → 11 = 4x → x = 11/4。

Check: 4^11/4+1 = 4^15/4 = (2²)^15/4 = 2^15/2; 8^2(11/4)−3 = 8^11/2−3 = 8^11/2−6/2 = 8^5/2 = (2³)^5/2 = 2^15/2. Correct.

检验：4^11/4+1 = 4^15/4 = (2²)^15/4 = 2^15/2；8^2(11/4)−3 = 8^11/2−3 = 8^11/2−6/2 = 8^5/2 = (2³)^5/2 = 2^15/2。正确。

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2. Composite Functions | 复合函数

Question: f(x)=3x−2, g(x)=2x+5. Find fg(x) and solve fg(x)=1.

题目： f(x)=3x−2，g(x)=2x+5。求 fg(x) 并解方程 fg(x)=1。

fg(x) means f(g(x)). Substitute g(x) into f: f(g(x)) = 3(2x+5) − 2.

fg(x) 表示 f(g(x))。将 g(x) 代入 f：f(g(x)) = 3(2x+5) − 2。

Simplify: 3(2x+5) − 2 = 6x + 15 − 2 = 6x + 13.

化简：3(2x+5) − 2 = 6x + 15 − 2 = 6x + 13。

Now solve 6x + 13 = 1 → 6x = 1 − 13 → 6x = −12 → x = −2.

现在解 6x + 13 = 1 → 6x = 1 − 13 → 6x = −12 → x = −2。

Check: g(−2)=1, f(1)=3(1)−2=1. Works.

检验：g(−2)=1，f(1)=3(1)−2=1。成立。

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3. Coordinate Geometry: Distance, Midpoint and Perpendicular Bisector | 坐标几何：距离、中点与垂直平分线

Question: Points A(2,5) and B(−4,1). Find (i) length AB, (ii) midpoint of AB, (iii) equation of the perpendicular bisector of AB.

题目： 点 A(2,5) 和 B(−4,1)。求 (i) AB 的长度，(ii) AB 的中点，(iii) AB 的垂直平分线方程。

Length AB = √[(x₂−x₁)² + (y₂−y₁)²] = √[(−4−2)² + (1−5)²] = √[(−6)² + (−4)²] = √[36+16] = √52 = 2√13.

AB 的长度 = √[(x₂−x₁)² + (y₂−y₁)²] = √[(−4−2)² + (1−5)²] = √[(−6)² + (−4)²] = √[36+16] = √52 = 2√13。

Midpoint M = ((x₁+x₂)/2, (y₁+y₂)/2) = ((2+(−4))/2, (5+1)/2) = (−2/2, 6/2) = (−1,3).

中点 M = ((x₁+x₂)/2, (y₁+y₂)/2) = ((2+(−4))/2, (5+1)/2) = (−2/2, 6/2) = (−1,3)。

Gradient of AB = (1−5)/(−4−2) = (−4)/(−6) = 2/3. Perpendicular gradient m_perp = −3/2.

AB 的斜率 = (1−5)/(−4−2) = (−4)/(−6) = 2/3。垂直斜率 m_perp = −3/2。

Equation through M(−1,3): y−3 = (−3/2)(x+1). Multiply by 2: 2y−6 = −3x−3 → 3x+2y−3=0.

过 M(−1,3) 的方程：y−3 = (−3/2)(x+1)。乘以 2：2y−6 = −3x−3 → 3x+2y−3=0。

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4. Differentiation: Power Rule | 求导：幂法则

Question: Differentiate y = 2x⁵ − 3/√x + 4x⁻² with respect to x.

题目： 对 y = 2x⁵ − 3/√x + 4x⁻² 关于 x 求导。

Rewrite terms: y = 2x⁵ − 3x⁻½ + 4x⁻².

重写各项：y = 2x⁵ − 3x⁻½ + 4x⁻²。

Apply power rule dy/dx = n·a xⁿ⁻¹:

应用幂法则 dy/dx = n·a xⁿ⁻¹：

Derivative of 2x⁵ = 2·5 x⁵⁻¹ = 10x⁴.

2x⁵ 的导数 = 2·5 x⁵⁻¹ = 10x⁴。

Derivative of −3x⁻½ = −3·(−½) x⁻½⁻¹ = (3/2) x⁻³⁄₂ = 3/(2x³⁄₂).

−3x⁻½ 的导数 = −3·(−½) x⁻½⁻¹ = (3/2) x⁻³⁄₂ = 3/(2x³⁄₂)。

Derivative of 4x⁻² = 4·(−2) x⁻³ = −8x⁻³ = −8/x³.

4x⁻² 的导数 = 4·(−2) x⁻³ = −8x⁻³ = −8/x³。

Full answer: dy/dx = 10x⁴ + 3/(2x³⁄₂) − 8/x³.

最终答案：dy/dx = 10x⁴ + 3/(2x³⁄₂) − 8/x³。

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5. Integration: Indefinite Integral | 积分：不定积分

Question: Find ∫ (6x² − 4x + 1) dx.

题目： 求 ∫ (6x² − 4x + 1) dx。

Use the reverse power rule: ∫ axⁿ dx = a·xⁿ⁺¹/(n+1) + C.

使用反向幂法则：∫ axⁿ dx = a·xⁿ⁺¹/(n+1) + C。

∫ 6x² dx = 6·x²⁺¹/(2+1) = 6x³/3 = 2x³.

∫ 6x² dx = 6·x²⁺¹/(2+1) = 6x³/3 = 2x³。

∫ −4x dx = −4·x¹⁺¹/(1+1) = −4x²/2 = −2x².

∫ −4x dx = −4·x¹⁺¹/(1+1) = −4x²/2 = −2x²。

∫ 1 dx = x.

∫ 1 dx = x。

Combine: ∫ (6x² − 4x + 1) dx = 2x³ − 2x² + x + C, where C is the constant of integration.

合并：∫ (6x² − 4x + 1) dx = 2x³ − 2x² + x + C，其中 C 为积分常数。

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6. Matrices: Multiplication and Inverse | 矩阵：乘法与逆矩阵

Question: A = ⎛ 3 1 ⎞ B = ⎛ 0 2 ⎞. Find AB and A⁻¹.

题目： A = ⎛ 3 1 ⎞ B = ⎛ 0 2 ⎞。求 AB 和 A⁻¹。

Matrix multiplication AB: (first row)×(columns of B). AB = ⎛ 3×0+1×4 3×2+1×1 ⎞ = ⎛ 4 7 ⎞.

矩阵乘法 AB：(第一行)×(B 的列)。AB = ⎛ 3×0+1×4 3×2+1×1 ⎞ = ⎛ 4 7 ⎞。

Second row: 2×0+(−1)×4 = −4, 2×2+(−1)×1 = 3. So AB = ⎛ 4 7 ⎞.

第二行：2×0+(−1)×4 = −4，2×2+(−1)×1 = 3。所以 AB = ⎛ 4 7 ⎞。

For A⁻¹, determinant D = 3·(−1) − 1·2 = −3−2 = −5 ≠ 0, so inverse exists.

对于 A⁻¹，行列式 D = 3·(−1) − 1·2 = −3−2 = −5 ≠ 0，因此逆矩阵存在。

A⁻¹ = (1/D) ⎛ −1 −1 ⎞ = (−1/5) ⎛ −1 −1 ⎞ = ⎛ 1/5 1/5 ⎞.

A⁻¹ = (1/D) ⎛ −1 −1 ⎞ = (−1/5) ⎛ −1 −1 ⎞ = ⎛ 1/5 1/5 ⎞。

Check: A·A⁻¹ = I. Works.

验证：A·A⁻¹ = I。成立。

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7. Vectors: Magnitude, Dot Product and Angle | 向量：模、点积与夹角

Question: a = 3i − 4j, b = −2i + 5j. Find |a|, a·b, and the acute angle between a and b.

题目： a = 3i − 4j，b = −2i + 5j。求 |a|、a·b 以及 a 与 b 之间的锐角。

|a| = √(3² + (−4)²) = √9+16 = √25 = 5.

|a| = √(3² + (−4)²) = √9+16 = √25 = 5。

a·b = 3·(−2) + (−4)·5 = −6 − 20 = −26.

a·b = 3·(−2) + (−4)·5 = −6 − 20 = −26。

|b| = √((−2)² + 5²) = √4+25 = √29.

|b| = √((−2)² + 5²) = √4+25 = √29。

cos θ = (a·b) / (|a||b|) = −26 / (5√29 ; acute supplement is relevant for direction analysis. The acute angle α satisfies cos α = |cos θ| = 26/(5√29). Use calculator: α ≈ cos⁻¹(26/(5×5.385)) ≈ cos⁻¹(26/26.925) ≈ cos⁻¹(0.9656) ≈ 15.1°.

cos θ = (a·b) / (|a||b|) = −26 / (5√29)。由于 cos θ 为负，θ 为钝角；对于方向分析，锐角 α 满足 cos α = |cos θ| = 26/(5√29)。使用计算器：α ≈ cos⁻¹(26/(5×5.385)) ≈ cos⁻¹(26/26.925) ≈ cos⁻¹(0.9656) ≈ 15.1°。

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8. Trigonometric Equation | 三角方程

Question: Solve cos 2x = ½ for 0° ≤ x ≤ 180°.

题目： 解 cos 2x = ½，其中 0° ≤ x ≤ 180°。

Basic angle for cos α = ½ is α = 60° (and α = 300°, ... but focus on 0°–360° for 2x).

cos α = ½ 的基本角为 α = 60°（还有 α = 300° 等，但我们关注 2x 在 0°–360° 范围内）。

Since 0° ≤ x ≤ 180°, then 0° ≤ 2x ≤ 360°. So 2x = 60° or 2x = 360°−60° = 300°.

由于 0° ≤ x ≤ 180°，因此 0° ≤ 2x ≤ 360°。所以 2x = 60° 或 2x = 360°−60° = 300°。

Divide by 2: x = 30°, or x = 150°.

除以 2：x = 30° 或 x = 150°。

Check: cos(60°)=½, cos(300°)=½. Both valid.

验证：cos(60°)=½，cos(300°)=½。均有效。

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9. Common Mistakes and Exam Tips | 常见错误与考试技巧

When tackling IGCSE Further Maths papers, watch out for algebraic slips when rewriting exponents or fractions. Always verify your base conversions and check cancellations. For composite functions, be systematic about the order: apply the inner function first. With coordinate geometry, a sketch can help avoid sign errors in gradients and midpoints. In calculus, remember to add the constant of integration for indefinite integrals and double-check your power rule signs for negative exponents. For matrices, confirm the determinant is non‑zero before computing the inverse, and set out multiplication row‑by‑column carefully. In vectors, the dot product yields a scalar; use absolute values for the acute angle when the dot product is negative. In trigonometry, consider the full domain after scaling the angle (e.g., 2x), and check all possible quadrants. Regular practice under timed conditions, plus careful checking of units and domain restrictions, will boost both accuracy and speed.

在做 IGCSE 进阶数学试卷时，要留意改写指数或分数时的代数错误。一定要验证底数转换并检查约分。对于复合函数，严格按照顺序操作：先代入内层函数。坐标几何中，作一个简图有助于避免斜率和中点中的符号错误。微积分方面，记住不定积分要加积分常数，并仔细检查负指数幂法则的符号。矩阵中，先确认行列式非零再计算逆矩阵，并细心排列行乘以列。向量里，点积得到的是一个标量；当点积为负时，求锐角要用绝对值。三角学中，在放缩角度（如 2x）后要考虑完整定义域并检查所有可能的象限。在限时条件下定期练习，加上仔细检查单位和定义域限制，将会同时提升准确度和速度。

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