2024 VCE Maths Methods Mini Test 3
Number of marks: 12
Reading time: 2 minutes
Writing time: 18 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.
Consider the functions \( f : (1, \infty) \to \mathbb{R}, f(x) = x^2 - 4x \) and \( g: \mathbb{R} \to \mathbb{R}, g(x) = e^{-x} \)
The range of the composite function \( g(f(x)) \) is
- A. \( (0, e^3) \)
- B. \( (0, e^3] \)
- C. \( (0, e^4) \)
- D. \( (0, e^4] \)
Consider the function \( f(x) = \frac{2x+1}{3-x} \), with domain \( x \in \mathbb{R} \setminus \{3\} \).
The inverse of \( f \) is
- A. \( f^{-1}(x) = \frac{3x - 1}{x + 2} \) with domain \( x \in \mathbb{R} \setminus \{3\} \)
- B. \( f^{-1}(x) = 3 - \frac{7}{x + 2} \) with domain \( x \in \mathbb{R} \setminus \{-2\} \)
- C. \( f^{-1}(x) = 3 + \frac{5}{x + 2} \) with domain \( x \in \mathbb{R} \setminus \{-2\} \)
- D. \( f^{-1}(x) = \frac{1 - 3x}{x + 2} \) with domain \( x \in \mathbb{R} \setminus \{-2\} \)
A fair six-sided die is repeatedly rolled. What is the minimum number of rolls required so that the probability of rolling a six at least once is greater than 0.95?
- A. 15
- B. 16
- C. 17
- D. 18
Some values of the functions \( f: \mathbb{R} \to \mathbb{R} \) and \( g: \mathbb{R} \to \mathbb{R} \) are shown below.
| x | 1 | 2 | 3 |
|---|---|---|---|
| f(x) | 0 | 4 | 5 |
| g(x) | 3 | 4 | -5 |
The graph of the function \( h(x) = f(x) - g(x) \) must have an x-intercept at
- A. (2, 0)
- B. (3, 0)
- C. (4, 0)
- D. (5, 0)
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 \( g : \mathbb{R} \setminus \{-3\} \rightarrow \mathbb{R},\ g(x) = \frac{1}{x + 3} - 2 \).
a. On the axes below, sketch the graph of \( y = g(x) \), labelling all asymptotes with their equations and axis intercepts with their coordinates. 2 marks
b. Determine the area of the region bounded by the line \( x = -2 \), the x-axis, the y-axis and the graph of \( y = g(x) \). 3 marks
Let \( X \) be a binomial random variable where \( X \sim \text{Bi}\left(4, \frac{9}{10}\right) \).
a. Find the standard deviation of \( X \). 1 mark
b. Find \( \Pr(X < 2) \). 2 marks
End of examination questions
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