📚 Year 11 CCEA Further Maths Unit Test Mock Paper Walkthrough | Year 11 CCEA 进阶数学单元测试模拟卷解析
Welcome to this detailed walkthrough of a mock unit test for CCEA GCSE Further Mathematics. This paper covers key topics including algebraic manipulation, functions, trigonometry, vectors, calculus, matrices and sequences. Work through each question to reinforce your problem-solving techniques and build confidence for the real exam.
欢迎阅读本份 CCEA GCSE 进阶数学单元测试模拟卷的详细解析。试卷涵盖代数运算、函数、三角学、向量、微积分、矩阵和数列等核心主题。通过逐题解析，帮助巩固解题技巧，提升应考信心。
1. Simplifying Algebraic Fractions and Solving Rational Equations | 代数分式化简与有理方程求解
Q1. (a) Simplify fully (x² - 9)/(x² - 3x). (b) Solve 2/(x - 1) = 3/(x + 2).
Factorise the numerator x² - 9 as (x + 3)(x - 3). Factorise the denominator x² - 3x as x(x - 3).
将分子 x² - 9 因式分解为 (x + 3)(x - 3)；分母 x² - 3x 因式分解为 x(x - 3)。
Cancel the common factor (x - 3) provided x ≠ 3. The simplified expression is (x + 3)/x, where x ≠ 0, 3.
约去公因式 (x - 3)（注意 x ≠ 3）。化简后的表达式为 (x + 3)/x，其中 x ≠ 0, 3。
For (b), cross-multiply to obtain 2(x + 2) = 3(x - 1). Expand both sides: 2x + 4 = 3x - 3.
对于 (b) 部分，交叉相乘得到 2(x + 2) = 3(x - 1)。展开两边：2x + 4 = 3x - 3。
Rearrange: 4 + 3 = 3x - 2x, so x = 7. Check that x = 7 does not make any denominator zero.
移项：4 + 3 = 3x - 2x，因此 x = 7。检验 x = 7 不会使任何分母为零，解成立。
2. Completing the Square and Graphing a Quadratic Function | 配方法与二次函数图像
the form a(x + p)² + q. (b) State the coordinates of the minimum point and sketch the graph.
Factor out the coefficient of x² from the first two terms: f(x) = 2[x² - 4x] + 5.
提取 x² 的系数：f(x) = 2[x² - 4x] + 5。
quare inside the bracket: half of -4 is -2, square it to get 4. So x² - 4x = (x - 2)² - 4.
在括号内配方：-4 的一半是 -2，平方得 4。因此 x² - 4x = (x - 2)² - 4。
Substitute back: f(x) = 2[(x - 2)² - 4] + 5 = 2(x - 2)² - 8 + 5 = 2(x - 2)² - 3.
代回原式：f(x) = 2[(x - 2)² - 4] + 5 = 2(x - 2)² - 8 + 5 = 2(x - 2)² - 3。
² - 3. The minimum point is at (2, -3) because the coefficient 2 is positive, indicating a 'U'-shaped parabola opening upwards.
因此 f(x) = 2(x - 2)² - 3。最小值点为 (2, -3)，因为系数 2 为正，抛物线开口向上。
he y-intercept by setting x=0 gives f(0)=5, and the graph is symmetric about the line x=2.
绘制草图：标出顶点 (2, -3)，令 x = 0 得 y 截距为 5，图像关于直线 x = 2 对称。
3. Solving Trigonometric Equations in a Given Interval | 在给定区间内解三角方程
Q3. Solve 2 sin²θ - sinθ - 1 = 0 for 0° ≤ θ ≤ 360°.
Treat the equation as a quadratic in sinθ. Let y =
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