VCE Maths Methods Basic Skills Mini Test
Number of marks: 13
Reading time: 2 minutes
Writing time: 20 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.
Some values of the functions \( f: \mathbb{R} \to \mathbb{R} \) and \( g: \mathbb{R} \to \mathbb{R} \) are shown below.
| x | 1 | 2 | 3 |
|---|---|---|---|
| f(x) | 0 | 4 | 5 |
| g(x) | 3 | 4 | -5 |
The graph of the function \( h(x) = f(x) - g(x) \) must have an x-intercept at
- A. (2, 0)
- B. (3, 0)
- C. (4, 0)
- D. (5, 0)
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. \( k \in \mathbb{R} \setminus \{-5\} \)
Let \(f\) and \(g\) be functions such that \(f(2) = 5\), \(f(3) = 4\), \(g(2) = 5\), \(g(3) = 2\) and \(g(4) = 1\).
The value of \(f(g(3))\) is
- A. 1
- B. 2
- C. 3
- D. 4
- E. 5
End of Section A
Section B – Calculator Allowed
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 rectangular sheet of cardboard has a width of \(h\) centimetres. Its length is twice its width. Squares of side length \(x\) centimetres, where \(x > 0\), are cut from each of the corners, as shown in the diagram below.
The sides of this sheet of cardboard are then folded up to make a rectangular box with an open top, as shown in the diagram below. Assume that the thickness of the cardboard is negligible and that \(V_{box} > 0\).
A box is to be made from a sheet of cardboard with \(h = 25\) cm.
a. Show that the volume, \(V_{box}\), in cubic centimetres, is given by \(V_{box}(x) = 2x(25 – 2x)(25 – x)\). 1 mark
b. State the domain of \(V_{box}\). 1 mark
End of Section B
Section C – 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.
Let \(P\) be a point on the straight line \(y = 2x - 4\) such that the length of \(OP\), the line segment from the origin \(O\) to \(P\), is a minimum.
a. Find the coordinates of \(P\). 3 marks
b. Find the distance \(OP\). Express your answer in the form \( \frac{a\sqrt{b}}{b} \), where \(a\) and \(b\) are positive integers. 2 marks
Consider the simultaneous linear equations
\( 3k x - 2y = k + 4 \)
\( (k - 4)x + ky = -k \)
where \( x, y \in \mathbb{R} \) and \( k \) is a real constant.
Determine the value of \( k \) for which the system of equations has no real solution. 3 marks
End of examination questions
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