2020 VCE Maths Methods Mini Test 7
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.
Part of the graph of a function \(f\), where \(a > 0\), is shown below.
The average value of the function \(f\) over the interval \([-2a, a]\) is
- A. 0
- B. \(\frac{a}{3}\)
- C. \(\frac{a}{2}\)
- D. \(\frac{3a}{4}\)
- E. \(a\)
A right-angled triangle, \(OBC\), is formed using the horizontal axis and the point \(C(m, 9-m^2)\), where \(m \in (0, 3)\), on the parabola \(y = 9 - x^2\), as shown below.
The maximum area of the triangle \(OBC\) is
- A. \(\frac{\sqrt{3}}{3}\)
- B. \(\frac{2\sqrt{3}}{3}\)
- C. \(\sqrt{3}\)
- D. \(3\sqrt{3}\)
- E. \(9\sqrt{3}\)
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.
Let \(f: [0, 2] \to \mathbb{R}\), where \(f(x) = \frac{1}{\sqrt{2}}\sqrt{x}\).
a. Find the domain and the rule for \(f^{-1}\), the inverse function of \(f\). 2 marks
The graph of \(y = f(x)\), where \(x \in [0, 2]\), is shown on the axes below.
b. On the axes above, sketch the graph of \(f^{-1}\) over its domain. Label the endpoints and point(s) of intersection with the function \(f\), giving their coordinates. 2 marks
c. Find the total area of the two regions: one region bounded by the functions \(f\) and \(f^{-1}\), and the other region bounded by \(f\), \(f^{-1}\) and the line \(x=1\). Give your answer in the form \(\frac{a-b\sqrt{b}}{6}\), where \(a, b \in Z^+\). 4 marks
End of examination questions
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