2023 VCE Maths Methods 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.
Consider the system of simultaneous linear equations below containing the parameter \( k \):
\( kx + 5y = k + 5 \)
\( 4x + (k + 1)y = 0 \)
The value(s) of \( k \) for which the system of equations has infinite solutions are
- A. \( k \in \{-5, 4\} \)
- B. \( k \in \{-5\} \)
- C. \( k \in \{4\} \)
- D. \( k \in \mathbb{R} \setminus \{-5, 4\} \)
- E. \( k \in \mathbb{R} \setminus \{-5\} \)
Which one of the following functions has a horizontal tangent at (0, 0)?
- A. \( y = x^{-1/3} \)
- B. \( y = x^{1/3} \)
- C. \( y = x^{2/3} \)
- D. \( y = x^{4/3} \)
- E. \( y = x^{3/4} \)
Suppose that \( \int_3^{10} f(x)\, dx = C \) and \( \int_7^{10} f(x)\, dx = D \). The value of \( \int_3^7 f(x)\, dx \) is
- A. \( C + D \)
- B. \( C + D - 3 \)
- C. \( C - D \)
- D. \( D - C \)
- E. \( CD - 3 \)
Let \( f(x) = \log_e x \), where \( x > 0 \) and \( g(x) = \sqrt{1 - x} \), where \( x < 1 \).
The domain of the derivative of \( (f \circ g)(x) \) is
- A. \( x \in \mathbb{R} \)
- B. \( x \in (-\infty, 1] \)
- C. \( x \in (-\infty, 1) \)
- D. \( x \in (0, \infty) \)
- E. \( x \in (0, 1) \)
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. Sketch the graph of \( f(x) = 2-\frac{3}{x - 1} \) on the axes below, labelling all asymptotes with their equations and axial intercepts with their coordinates. 3 marks
b. Find the values of \( x \) for which \( f(x) \leq 1 \). 1 mark
The graph of \( y = x + \frac{1}{x} \) is shown over part of its domain.
Use two trapeziums of equal width to approximate the area between the curve, the x-axis and the lines \( x = 1 \) and \( x = 3 \). 2 marks
End of examination questions
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