VCE Maths Methods Functions Mini Test 6
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.
Let \(a \in (0, \infty)\) and \(b \in R\).
Consider the function \(h: [-a, 0) \cup (0, a] \to R, h(x) = \frac{a}{x} + b\).
The range of \(h\) is
- A. \([b-1, b+1]\)
- B. \((b-1, b+1)\)
- C. \((-\infty, b-1) \cup (b+1, \infty)\)
- D. \((-\infty, b-1] \cup [b+1, \infty)\)
- E. \([b-1, \infty)\)
Which one of the following statements is true for \(f: R \to R, f(x) = x + \sin(x)\)?
- A. The graph of \(f\) has a horizontal asymptote
- B. There are infinitely many solutions to \(f(x) = 4\)
- C. \(f\) has a period of \(2\pi\)
- D. \(f'(x) \ge 0\) for \(x \in R\)
- E. \(f'(x) = \cos(x)\)
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.
a. Let \(f: \mathbb{R}\setminus\{\frac{1}{3}\} \to \mathbb{R}\), \(f(x) = \frac{1}{3x-1}\).
Find the rule of \(f^{-1}\). 2 marks
b. State the domain of \(f^{-1}\). 1 mark
Let \(P\) be a point on the straight line \(y = 2x - 4\) such that the length of \(OP\), the line segment from the origin \(O\) to \(P\), is a minimum.
a. Find the coordinates of \(P\). 3 marks
b. Find the distance \(OP\). Express your answer in the form \( \frac{a\sqrt{b}}{b} \), where \(a\) and \(b\) are positive integers. 2 marks
End of examination questions
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