Year 8 Edexcel Further Pure Maths Mock Exam Walkthrough | 八年级Edexcel进阶数学单元测试模拟卷解析

📚 Year 8 Edexcel Further Pure Maths Mock Exam Walkthrough | 八年级Edexcel进阶数学单元测试模拟卷解析

Welcome to this detailed walkthrough of a Year 8 Edexcel Further Pure Maths mock unit test. We will go through a selection of typical exam-style questions, breaking down each solution step by step. Use this article to sharpen your problem-solving skills, avoid common mistakes, and build confidence for the real test.

欢迎来到八年级Edexcel进阶数学单元测试模拟卷的详细解析。我们将通过一组典型考题，逐题拆解求解过程。请利用本文打磨你的解题技巧，避免常见错误，为正式测试树立信心。

1. Solving Quadratic Equations by Factorising | 因式分解法解二次方程

Question: Solve x² − 7x + 12 = 0.

题目：解方程 x² − 7x + 12 = 0。

Step 1: Look for two numbers whose product is +12 and whose sum is −7. The pair is −3 and −4.

步骤1：寻找乘积为 +12 且和为 −7 的两个数。符合条件的数是 −3 和 −4。

Step 2: Rewrite the quadratic as (x − 3)(x − 4) = 0.

步骤2：将二次式写为 (x − 3)(x − 4) = 0。

Step 3: Set each bracket equal to zero: x − 3 = 0 gives x = 3, and x − 4 = 0 gives x = 4.

步骤3：令每个因式等于零：由 x − 3 = 0 得 x = 3，由 x − 4 = 0 得 x = 4。

Always check by expanding: (x − 3)(x − 4) = x² − 7x + 12. Both solutions are valid.

检验：将因式乘开得 x² − 7x + 12，解正确。两个解都成立。

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2. Algebraic Fractions and Simplification | 代数分式化简

Question: Simplify (x² − 4) ÷ (x − 2), where x ≠ 2.

题目：化简 (x² − 4) ÷ (x − 2)，其中 x ≠ 2。

Step 1: Recognise the numerator as a difference of two squares: x² − 4 = (x − 2)(x + 2).

步骤1：识别分子为平方差：x² − 4 = (x − 2)(x + 2)。

Step 2: Cancel the common factor (x − 2) from numerator and denominator.

步骤2：约去分子与分母的公因式 (x − 2)。

Step 3: The simplified expression is x + 2, provided x ≠ 2 to avoid division by zero.

步骤3：化简结果为 x + 2，注意 x ≠ 2 以避免除零。

This technique is crucial when working with rational functions in further algebra.

在进阶代数中处理有理函数时，这一技巧至关重要。

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3. Inequalities and Number Lines | 不等式与数轴表示

Question: Solve the inequality 3x − 5 > 7 and illustrate your answer on a number line.

题目：解不等式 3x − 5 > 7，并在数轴上表示解集。

Step 1: Add 5 to both sides: 3x > 12.

步骤1：两边同时加 5，得 3x > 12。

Step 2: Divide through by 3: x > 4. Remember, the inequality sign does not flip when dividing by a positive number.

步骤2：两边除以 3，得 x > 4。除以正数时不等号方向不变。

Step 3: On a number line, draw an open circle at 4 and shade the arrow to the right, indicating all values greater than 4.

步骤3：在数轴上，在 4 处画空心圆，并向右画箭头，表示所有大于 4 的值。

Open circle means 4 itself is not included. A closed circle would be used for ≥ or ≤.

空心圆表示 4 本身不包含在内。若是 ≥ 或 ≤，则使用实心圆。

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4. Straight Line Graphs: Gradient and Intercept | 直线图像：斜率与截距

traight line passing through A(1, 3) and B(3, 7). Give your answer in the form y = mx + c.

题目：求经过点 A(1, 3) 和 B(3, 7) 的直线方程，结果写成 y = mx + c 的形式。

Calculate the gradient m using m = (y₂ − y₁) / (x₂ − x₁) = (7 − 3) / (3 − 1) = 4 / 2 = 2.

步骤1：计算斜率 m，m = (y₂ − y₁) / (x₂ − x₁) = (7 − 3) / (3 − 1) = 4 / 2 = 2。

Step 2: Substitute one point, say A(1, 3), into y = 2x + c: 3 = 2×1 + c, so c = 1.

步骤2：代入一点，例如 A(1, 3)，到 y = 2x + c 中：3 = 2×1 + c，得 c = 1。

Step 3: The equation of the line is y = 2x + 1.

步骤3：直线方程为 y = 2x + 1。

You can verify using point B: when x = 3, y = 2×3 + 1 = 7, which matches perfectly.

可用点 B 验证：当 x = 3 时，y = 2×3 + 1 = 7，结果吻合。

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5. Simultaneous Equations by Substitution | 代入法解联立方程

Question: Solve the simultaneous equations: y = 2x + 1 and 3x + y = 11.

题目：解联立方程组：y = 2x + 1，3x + y = 11。

eady the subject of the first equation, substitute it into the second: 3x + (2x + 1) = 11.

步骤1：由于第一个方程已将 y 表示出来，直接代入第二个方程：3x + (2x + 1) = 11。

Step 2: Simplify to solve for x: 5x + 1 = 11 → 5x = 10 → x = 2.

步骤2：化简求 x：5x + 1 = 11 → 5x = 10 → x = 2。

p 3: Substitute x = 2 back into y = 2x + 1: y = 2×2 + 1 = 5. The solution is x = 2, y = 5.

步骤3：将 x = 2 代回 y = 2x + 1：y = 2×2 + 1 = 5。解为 x = 2，y = 5。

It is good practice to check both original equations: 3×2 + 5 = 11, which is correct.

最好代入原方程组检验：3×2 + 5 = 11，结果正确。

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6. Sequences: Finding the nth Term | 数列：求第 n 项

Question: Find the nth term of the arithmetic sequence: 5, 9, 13, 17, ...

题目：求等差数列 5, 9, 13, 17, ... 的第 n 项。

Step 1: Determine the common difference d: 9 − 5 = 4, so d = 4.

步骤1：确定公差 d：9 − 5 = 4，即 d = 4。

nth term of an arithmetic sequence is a + (n − 1)d, where a is the first term. Here a = 5.

步骤2：等差数列第 n 项公式为 a + (n − 1)d，其中 a 为首项。此处 a = 5。

Step 3: Substitute the values: nth term = 5 + (n − 1)×4 = 5 + 4n − 4 = 4n + 1.

步骤3：代入数值：第 n 项 = 5 + (n − 1)×4 = 5 + 4n − 4 = 4n + 1。

= 1, term = 5; n = 2, term = 9; correct. This linear expression models the whole sequence.

验证：n = 1 时项为 5；n = 2 时项为 9；正确。该一次式刻画了整个数列。

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7. Indices and Powers | 指数与幂运算

Question: Simplify (3x²y)² ÷ (xy²).

题目：化简 (3x²y)² ÷ (xy²)。

1: Apply the power to each factor inside the brackets: (3x²y)² = 3² × (x²)² × y² = 9x⁴y².

步骤1：将指数分配到括号内每个因子：(3x²y)² = 3² × (x²)² × y² = 9x⁴y²。

a fraction and subtract exponents for like bases: 9x⁴y² ÷ (x¹y²) = 9 x⁴⁻¹ y²⁻² = 9 x³ y⁰.

步骤2：将除法写成分式，底数相同则指数相减：9x⁴y² ÷ (x¹y²) = 9 x⁴⁻¹ y²⁻² = 9 x³ y⁰。

Step 3: Since y⁰ = 1, the final simplified expression is 9x³.

步骤3：因为 y⁰ = 1，最终化简结果为 9x³。

Remember: any non-zero base to the power of zero equals 1. Always simplify step by step to avoid errors.

记住：任意非零底数的零次幂等于 1。务必逐步化简以避免错误。

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8. Function Notation and Evaluation | 函数记号与求值

Question: Given the function f(x) = x² − 3x, evaluate f(4) and f(−2).

题目：已知函数 f(x) = x² − 3x，计算 f(4) 和 f(−2)。

Step 1: Substitute x = 4 into the function: f(4) = (4)² − 3×4 = 16 − 12 = 4.

步骤1：将 x = 4 代入函数：f(4) = (4)² − 3×4 = 16 − 12 = 4。

ep 2: Substitute x = −2: f(−2) = (−2)² − 3×(−2). Remember that (−2)² = 4 and −3×(−2) = +6.

步骤2：将 x = −2 代入：f(−2) = (−2)² − 3×(−2)。注意 (−2)² = 4，且 −3×(−2) = +6。

Step 3: Simplify: 4 + 6 = 10, so f(−2) = 10.

步骤3：化简：4 + 6 = 10，因此 f(−2) = 10。

Function notation is a way of showing that the output depends on the input. Always use brackets carefully when substituting negative numbers.

函数记号表示输出依赖于输入。代入负数时务必小心使用括号。

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9. Factorising Quadratics with a Coefficient >1 | 二次项系数大于 1 的因式分解

Question: Factorise fully 2x² + 7x + 3.

题目：将 2x² + 7x + 3 彻底因式分解。

o numbers that multiply to 6 and add to the middle coefficient 7. The numbers are 6 and 1.

步骤1：将 x² 系数 a = 2 与常数项 c = 3 相乘：2 × 3 = 6。寻找乘积为 6 且和为中间系数 7 的两个数，分别是 6 和 1。

Step 2: Split the middle term using these numbers: 2x² + 6x + 1x + 3.

步骤2：用找到的数拆解中间项：2x² + 6x + 1x + 3。

cond pair x + 3 = 1(x + 3). Factor out the common binomial (x + 3) to get (2x + 1)(x + 3).

步骤3：分组分解：第一组 2x² + 6x = 2x(x + 3)；第二组 x + 3 = 1(x + 3)。提取公因式 (x + 3) 得 (2x + 1)(x + 3)。

Expand to check: (2x + 1)(x + 3) = 2x² + 6x + x + 3 = 2x² + 7x + 3. Correct.

展开检验：(2x + 1)(x + 3) = 2x² + 6x + x + 3 = 2x² + 7x + 3，正确。

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10. Length of a Line Segment | 线段长度

Question: Find the exact distance between points P(2, 5) and Q(6, 9).

题目：求点 P(2, 5) 和点 Q(6, 9) 之间的精确距离。

Step 1: Calculate the differences in x and y: Δx = 6 − 2 = 4, Δy = 9 − 5 = 4.

步骤1：计算横纵坐标差：Δx = 6 − 2 = 4，Δy = 9 − 5 = 4。

Step 2: Use the distance formula: d = √((Δx)² + (Δy)²) = √4² + 4² = √16 + 16 = √32.

步骤2：应用距离公式：d = √((Δx)² + (Δy)²) = √4² + 4² = √16 + 16 = √32。

Step 3: Simplify √32 by extracting the largest square factor: √32 = √16 × 2 = √16 × √2 = 4√2.

步骤3：化简 √32，提取最大平方因子：√32 = √16 × 2 = √16 × √2 = 4√2。

The exact distance is 4√2 units. Leaving the answer in surd form shows precision in further maths.

精确距离为 4√2 单位。在进阶数学中保留根式可体现精确性。

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