2025 VCE Maths Methods Mini Test 7
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.
Consider the function \(h(x)=a\log_{e}(bx)\), where \(a, b\in \mathbb{R}\backslash\{0\}\).
Given that its derivative \(h^{\prime}(x)\) has range \((0,\infty)\), which of the following must be true?
- A. \(a>0\) only
- B. \(a>0\) and \(b<0\)
- C. \(a>0\) and \(b>0\)
- D. \(ab>0\)
Given that \(f:\mathbb{R}\rightarrow \mathbb{R}\) satisfies
\(\int_{1}^{2}f(x)dx>\int_{1}^{3}f(x)dx\)
the graph of \(y=f(x)\) could be
Consider the following graphs, which represent probability mass functions.
Which pair of these probability mass functions has the same mean?
- A. I and II
- B. I and IV
- C. II and III
- D. II and IV
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.
a. Solve \( e^{2x} - 8e^x + 7 = 0 \) for \( x \). 2 marks
b. Let \( g(x) = e^{2x} - 8e^x + 7 \), where \( x \in \mathbb{R} \). The function \( g(x) \) has exactly one stationary point, a local minimum. Find the largest value of \( a \) such that when \( g \) is restricted to the domain \( (-\infty, a] \) it has an inverse function. 2 marks
Consider the binomial random variable \( X \sim \text{Bi}\left(6, \frac{1}{4}\right) \).
a. Find \( \text{var}(X) \). 1 mark
b. Determine \( \Pr(X \ge 5) \). Give your answer in the form \( \frac{a}{2^b} \), where \( a, b \in \mathbb{Z} \). 2 marks
End of examination questions
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