2022 VCE Maths Methods Mini Test 5
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.
Which of the pairs of functions below are not inverse functions?
- A. \( \begin{cases} f(x) = 5x + 3 \quad x \in \mathbb{R} \\ g(x) = \frac{x - 3}{5} \quad x \in \mathbb{R} \end{cases} \)
- B. \( \begin{cases} f(x) = \frac{2}{3}x + 2 \quad x \in \mathbb{R} \\ g(x) = \frac{3}{2}x - 3 \quad x \in \mathbb{R} \end{cases} \)
- C. \( \begin{cases} f(x) = x^2 \quad x < 0 \\ g(x) = \sqrt{x} \quad x > 0 \end{cases} \)
- D. \( \begin{cases} f(x) = \frac{1}{x} \quad x \ne 0 \\ g(x) = \frac{1}{x} \quad x \ne 0 \end{cases} \)
- E. \( \begin{cases} f(x) = \log_e(x) + 1 \quad x > 0 \\ g(x) = e^{x - 1} \quad x \in \mathbb{R} \end{cases} \)
The graph of \( y = f(x) \) is shown below.
The graph of \( y = f'(x) \), the first derivative of \( f(x) \) with respect to \( x \), could be
If \( \int_0^b f(x)\,dx = 10 \) and \( \int_0^a f(x)\,dx = -4 \), where \( 0 < a < b \), then \( \int_a^b f(x)\,dx \) is equal to
- A. –6
- B. –4
- C. 0
- D. 10
- E. 14
Let \( f : [0, \infty) \to \mathbb{R}, \ f(x) = \sqrt{2x + 1} \).
The shortest distance, \( d \), from the origin to the point \( (x, y) \) on the graph of \( f \) is given by
- A. \( d = x^2 + 2x + 1 \)
- B. \( d = x^2 + \sqrt{2x + 1} \)
- C. \( d = \sqrt{x^2 - 2x + 1} \)
- D. \( d = x + 1 \)
- E. \( d = 2x + 1 \)
An organisation randomly surveyed 1000 Australian adults and found that 55% of those surveyed were happy with their level of physical activity.
An approximate 95% confidence interval for the percentage of Australian adults who were happy with their level of physical activity is closest to
- A. (4.1, 6.9)
- B. (50.9, 59.1)
- C. (52.4, 57.6)
- D. (51.9, 58.1)
- E. (45.2, 64.8)
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.
a. Solve \( 10^{3x - 13} = 100 \) for \( x \). 2 marks
b. Find the maximal domain of \( f \), where \( f(x) = \log_e(x^2 - 2x - 3) \). 3 marks
End of examination questions
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