VCE Maths Methods Integral Calculus Mini Test 9
Number of marks: 10
Reading time: 2 minutes
Writing time: 15 minutes
Section A – Calculator Allowed
Instructions
• Answer all questions in pencil on your Multiple-Choice Answer Sheet.
• Choose the response that is correct for the question.
• A correct answer scores 1; an incorrect answer scores 0.
• Marks will not be deducted for incorrect answers.
• No marks will be given if more than one answer is completed for any question.
• Unless otherwise indicated, the diagrams in this book are not drawn to scale.
If \(\int_1^{12} g(x)dx = 5\) and \(\int_{12}^{5} g(x)dx = -6\), then \(\int_1^{5} g(x)dx\) is equal to
- A. \(-11\)
- B. \(-1\)
- C. \(1\)
- D. \(3\)
- E. \(11\)
The graphs of \(f: [0, \frac{\pi}{2}] \to R, f(x) = \cos(x)\) and \(g: [0, \frac{\pi}{2}] \to R, g(x) = \sqrt{3}\sin(x)\) are shown below.
The graphs intersect at \(B\).
The ratio of the area of the shaded region to the area of triangle \(OAB\) is
- A. \(9:8\)
- B. \(\sqrt{3}-1 : \frac{\sqrt{3}\pi}{8}\)
- C. \(8\sqrt{3}-3 : 3\pi\)
- D. \(\sqrt{3}-1 : \frac{\sqrt{3}\pi}{4}\)
- E. \(1 : \frac{\sqrt{3}\pi}{8}\)
End of Section A
Section B – No Calculator
Instructions
• Answer all questions in the spaces provided.
• Write your responses in English.
• In questions where a numerical answer is required, an exact value must be given unless otherwise specified.
• In questions where more than one mark is available, appropriate working must be shown.
• Unless otherwise indicated, the diagrams in this book are not drawn to scale.
The derivative with respect to \(x\) of the function \( f: (1, \infty) \rightarrow \mathbb{R} \) has the rule \( f'(x) = \frac{1}{2} - \frac{1}{(2x-2)} \).
Given that \( f(2) = 0 \), find \( f(x) \) in terms of \(x\). 3 marks
Let \(f: \mathbb{R}\setminus\{1\} \to \mathbb{R}\), \(f(x) = \frac{2}{(x-1)^2} + 1\).
a.
i. Evaluate \(f(-1)\). 1 mark
ii. Sketch the graph of \(f\) on the axes below, labelling all asymptotes with their equations. 2 marks
b. Find the area bounded by the graph of \(f\), the \(x\)-axis, the line \(x=-1\) and the line \(x=0\). 2 marks
End of examination questions
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