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2023 VCE Maths Methods Mini Test 5
Adapted for Year 11. Unit 1 & 2.
Number of marks: 8
Reading time: 2 minutes
Writing time: 12 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. \( 1 - 8 \left( \frac{n}{n + m} \right) \left( \frac{m}{n + m} \right)^7 \)
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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