2023 VCE Maths Methods 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.
Let \( X \) be a normal random variable with mean 100 and standard deviation 20.
Let \( Y \) be a normal random variable with mean 80 and standard deviation 10.
Which of the diagrams below best represents the probability density functions for \( X \) and \( Y \), plotted on the same set of axes?
Let \( f(x) = e^{x - 1} \).
Given that the product function \( f(x) \times g(x) = e^{(x - 1)^2} \), the rule for the function \( g \) is:
- A. \( g(x) = e^{x - 1} \)
- B. \( g(x) = e^{(x - 2)(x - 1)} \)
- C. \( g(x) = e^{(x + 2)(x - 1)} \)
- D. \( g(x) = e^{x(x - 2)} \)
- E. \( g(x) = e^{x(x - 3)} \)
A cylinder of height \( h \) and radius \( r \) is formed from a rectangular metal sheet of length \( x \) and width \( y \), by cutting
along the dashed lines shown below.
The volume of the cylinder in terms of \( x \) and \( y \) is
- A. \( \pi x^2 y \)
- B. \( \frac{\pi x y^2 - 2 y^3}{4\pi^2} \)
- C. \( \frac{2 y^3 - \pi x y^2}{4\pi^2} \)
- D. \( \frac{\pi x y - 2 y^2}{2\pi} \)
- E. \( \frac{2 y^2 - \pi x y}{2\pi} \)
Consider the function \( f : [-a\pi, a\pi] \to \mathbb{R} \), \( f(x) = \sin(ax) \), where \( a \) is a positive integer.
The number of local minima in the graph of \( y = f(x) \) is always equal to
- A. 2
- B. 4
- C. \( a \)
- D. \( 2a \)
- E. \( a^2 \)
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 shapes of two walking tracks are shown below.
Track 1 is described by the function \( f(x) = a - x(x - 2)^2 \).
Track 2 is defined by the function \( g(x) = 12x + bx^2 \).
The unit of length is kilometres.
a. Given that \( f(0) = 12 \) and \( g(1) = 9 \), verify that \( a = 12 \) and \( b = -3 \). 1 mark
b. Verify that \( f(x) \) and \( g(x) \) both have a turning point at \( P \).
Give the coordinates of \( P \). 2 marks
c. A theme park is planned whose boundaries will form the triangle \( \triangle OAB \) where \( O \) is the origin, \( A \) is at \( (k, 0) \) and \( B \) is at \( (k, g(k)) \), as shown below, where \( k \in (0, 4) \).
Find the maximum possible area of the theme park, in km\(^2\). 3 marks
End of examination questions
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