2021 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.
The tangent to the graph of \( y = x^3 - ax^2 + 1 \) at \( x = 1 \) passes through the origin.
The value of \(a\) is
- A. \( \frac{1}{2} \)
- B. 1
- C. \( \frac{3}{2} \)
- D. 2
- E. \( \frac{5}{2} \)
The graph of the function \(f\) is shown below.
The graph corresponding to \(f'\) is
Let \( g(x) = x + 2 \) and \( f(x) = x^2 - 4 \).
If h is the composite function given by \( h: [-5, -1) \to R, h(x) = f(g(x)) \), then the range of \(h\) is
- A. (-3, 5]
- B. [-3, 5)
- C. (-3, 5)
- D. (-4, 5]
- E. [-4, 5]
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 \( f : \mathbb{R} \rightarrow \mathbb{R}, f(x) = x^2 - 4 \) and \( g : \mathbb{R} \rightarrow \mathbb{R}, g(x) = 4(x-1)^2 - 4 \).
a. The graphs of \(f\) and \(g\) have a common horizontal axis intercept at \((2, 0)\).
Find the coordinates of the other horizontal axis intercept of the graph of \(g\). 2 marks
b. Let the graph of \(h\) be a transformation of the graph of \(f\) where the transformations have been applied in the following order:
- • dilation by a factor of \( \frac{1}{2} \) from the vertical axis (parallel to the horizontal axis)
- • translation by two units to the right (in the direction of the positive horizontal axis)
A random variable \(X\) has the probability density function \(f\) given by \( f(x) = \begin{cases} \frac{k}{x^2} & 1 \le x \le 2 \\ 0 & \text{elsewhere} \end{cases} \) where \(k\) is a positive real number.
a. Show that \(k = 2\). 1 mark
b. Find \(E(X)\). 2 marks
End of examination questions
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