VCE Maths Methods Logs & Exponentials Mini Test 3
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.
Given that \(\log_2(n+1) = x\), the values of \(n\) for which \(x\) is a positive integer are
- A. \(n = 2k, k \in Z^+\)
- B. \(n = 2^k - 1, k \in Z^+\)
- C. \(n = 2k - 1, k \in Z^+\)
- D. \(n = 2k-1, k \in Z^+\)
- E. \(n = 2k, k \in Z^+\)
The graph of \( y = \log_e(x) + \log_e(2x) \), where \( x > 0 \), is identical, over the same domain, to the graph of
- A. \( y = 2\log_e\left(\frac{1}{2}x\right) \)
- B. \( y = 2\log_e(2x) \)
- C. \( y = \log_e(2x^2) \)
- D. \( y = \log_e(3x) \)
- E. \( y = \log_e(4x) \)
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 the equation \(2\log_2(x+5) - \log_2(x+9) = 1\). 3 marks
Consider the functions \(f: \mathbb{R} \to \mathbb{R}\), \(f(x) = 3+2x-x^2\) and \(g: \mathbb{R} \to \mathbb{R}\), \(g(x) = e^x\).
a. State the rule of \(g(f(x))\). 1 mark
b. State the rule of \(f(g(x))\). 1 mark
c. Solve \(f(g(x)) = 0\). 2 marks
d. State the number of solutions to \(g(f(x)) + f(g(x)) = 0\). 1 mark
End of examination questions
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