VCE Maths Methods Logs & Exponentials Mini Test 1
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 asymptote(s) of the graph of \( y = \log_e(x + 1) - 3 \) are
- A. \( x = -1\) only
- B. \( x = 1\) only
- C. \( y = -3\) only
- D. \( x = -1\) and \( y = -3 \)
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. \( g(x) = e^{x(x - 3)} \)
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.
Solve \( 2\log_3(x - 4) + \log_3(x) = 2 \) for \( x \). 4 marks
The function \(f: \mathbb{R} \to \mathbb{R}\), \(f(x)\) is a polynomial function of degree 4. Part of the graph of \(f\) is shown below.
The graph of \(f\) touches the \(x\)-axis at the origin.
a. Find the rule of \(f\). 1 mark
Let \(g\) be a function with the same rule as \(f\).
Let \(h: D \to \mathbb{R}\), \(h(x) = \log_e(g(x)) - \log_e(x^3+x^2)\), where \(D\) is the maximal domain of \(h\).
b. State \(D\). 1 mark
c. State the range of \(h\). 2 marks
End of examination questions
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