2023 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.
A box contains \( n \) green balls and \( m \) red balls. A ball is selected at random, and its colour is noted. The ball is then replaced in the box. In 8 such selections, where \( n \ne m \), what is the probability that a green ball is selected at least once?
- A. \( 8 \left( \frac{n}{n + m} \right) \left( \frac{m}{n + m} \right)^7 \)
- B. \( 1 - \left( \frac{n}{n + m} \right)^8 \)
- C. \( 1 - \left( \frac{m}{n + m} \right)^8 \)
- D. \( 1 - \left( \frac{n}{n + m} \right) \left( \frac{m}{n + m} \right)^7 \)
- E. \( 1 - 8 \left( \frac{n}{n + m} \right) \left( \frac{m}{n + m} \right)^7 \)
\[ f(x) = \begin{cases} \tan\left(\frac{x}{2}\right) & 4 \leq x < 2\pi \\ \sin(ax) & 2\pi \leq x \leq 8 \end{cases} \]
The value of \( a \) for which \( f \) is continuous and smooth at \( x = 2\pi \) is- A. −2
- B. \( -\sqrt{2} \)
- C. \( -\frac{1}{2} \)
- D. \( \frac{1}{2} \)
- E. 2
A continuous random variable \( X \) has the following probability density function:
\[ g(x) = \begin{cases} \frac{x - 1}{20} & 1 \leq x < 6 \\ \frac{9 - x}{12} & 6 \leq x \leq 9 \\ 0 & \text{elsewhere} \end{cases} \]
The value of \( k \) such that \( \Pr(X < k) = 0.35 \) is
- A. \( \sqrt{14} - 1 \)
- B. \( \sqrt{14} + 1 \)
- C. \( \sqrt{15} - 1 \)
- D. \( \sqrt{15} + 1 \)
- E. \( 1 - \sqrt{15} \)
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.
Consider \( f : (-\infty, 1] \rightarrow \mathbb{R} \), \( f(x) = x^2 - 2x \). Part of the graph of \( y = f(x) \) is shown below.
a. State the range of \( f \). 1 mark
b. Sketch the graph of the inverse function \( y = f^{-1}(x) \) on the axes above. Label any endpoints and axial intercepts with their coordinates. 2 marks
c. Determine the equation and the domain for the inverse function \( f^{-1} \). 2 marks
d. Calculate the area of the regions enclosed by the curves of \( f \), \( f^{-1} \) and \( y = -x \). 2 marks
End of examination questions
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