澳大利亚私立中学九年级综合能力测试卷（高难度）
考试时间：120分钟 总分：100分 考试说明：本试卷涵盖代数、线性关系、二次函数及问题解决四大核心模块，注重知识综合运用与逻辑推理能力考查，所有解答需附详细步骤及验证过程。
Section 1: Algebra（代数，20分）
1.Solve the equation: \(\frac{3x - 2}{4} - \frac{2x + 5}{3} = \frac{x - 1}{6} + 2\)（6分）2.Literal equation manipulation: Given \(A = \frac{1}{2}h(b_1 + b_2)\) (area of a trapezoid), rearrange to make \(b_2\) the subject. Then find \(b_2\) when \(A = 48\), \(h = 6\), \(b_1 = 5\)（7分）3.Worded problem: A train leaves Station A at 10:00 AM at speed \(v\) km/h. Another train leaves Station B (300 km east of A) at 11:00 AM, traveling west at \((v + 20)\) km/h. They meet at 1:00 PM. Find \(v\)（7分）
Section 2: Linear Relations（线性关系，25分）
4. Given points \(P(2, 5)\) and \(Q(6, 13)\)（25分）
1.(a) Find the gradient of line \(PQ\)（5分）
2.(b) Find the equation of line \(PQ\)（5分）
3.(c) Find the equation of line \(l\) (perpendicular to \(PQ\) and passing through the midpoint of \(PQ\))（5分）
4.(d) Calculate the distance between \(P\) and \(Q\)（5分）
5.(e) Solve the simultaneous equations of \(PQ\) and line \(l\)（5分）
Section 3: Quadratics（二次函数，30分）
5. For \(y = 3x^2 - 16x - 12\)（15分）
1.(a) Factorise completely over \(\mathbb{R}\)（5分）2.(b) Solve \(3x^2 - 16x - 12 = 0\) using the factorisation（5分）
3.(c) Sketch the parabola, labeling x-intercepts, y-intercept, vertex, and axis of symmetry（5分）
6. A ball’s height is \(h = -5t^2 + 15t + 2\) (meters, \(t\) = seconds)（15分）
1.(a) Find the time of maximum height（5分）
2.(b) Find the maximum height（5分）
3.(c) Find the time when the ball hits the ground (to 2 decimal places)（5分）
Section 4: Problem Solving（问题解决，25分）
7. A rectangular plot is fenced on 3 sides (one side against a wall) with 100 meters of fencing. Let \(x\) = length parallel to the wall, \(y\) = length perpendicular to the wall（25分）
1.(a) Express \(y\) in terms of \(x\)（5分）
2.(b) Show area \(A = \frac{1}{2}x(100 - x)\)（6分）3.(c) Find dimensions that maximise area, and the maximum area（7分）
4.(d) If a 4m × 4m square flower bed is placed in the plot, does the maximum area change? Justify（7分）
Detailed Solutions & Verification（详细解答与验证）
Section 1: Algebra（代数）
1. Solve \(\frac{3x - 2}{4} - \frac{2x + 5}{3} = \frac{x - 1}{6} + 2\)Step 1: Eliminate denominators by multiplying all terms by the least common multiple (LCM) of 4, 3, 6, which is 12:
\(3(3x - 2) - 4(2x + 5) = 2(x - 1) + 24\)
Step 2: Expand each term and simplify the left and right sides:
\(9x - 6 - 8x - 20 = 2x - 2 + 24\)
\(x - 26 = 2x + 22\)
Step 3: Rearrange terms to solve for \(x\):
\(x - 2x = 22 + 26\)
\(-x = 48 \implies x = -48\)
Verification: Substitute \(x = -48\) into the original equation. Left side: \(\frac{3\times(-48)-2}{4} - \frac{2\times(-48)+5}{3} = \frac{-146}{4} - \frac{-91}{3} \approx -36.5 + 30.333 = -6.167\); Right side: \(\frac{-48 - 1}{6} + 2 = \frac{-49}{6} + 2 \approx -8.167 + 2 = -6.167\). Both sides are equal, so \(x = -48\) is correct.2. Rearrange \(A = \frac{1}{2}h(b_1 + b_2)\) for \(b_2\) and find its valueStep 1: Multiply both sides by 2 to eliminate the fraction:
\(2A = h(b_1 + b_2)\)
Step 2: Divide both sides by \(h\) to isolate the term containing \(b_2\):