2019 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.
\(\int_0^{\frac{\pi}{3}} (a\sin(x) + b\cos(x))dx\) is equal to
- A. \(\frac{(2-\sqrt{3})a-b}{2}\)
- B. \(\frac{b-(2-\sqrt{3})a}{2}\)
- C. \(\frac{(2-\sqrt{3})a+b}{2}\)
- D. \(\frac{(2-\sqrt{3})b-a}{2}\)
- E. \(\frac{(2-\sqrt{3})b+a}{2}\)
Let \(f'(x) = 3x^2 - 2x\) such that \(f(4) = 0\).
The rule of \(f\) is
- A. \(f(x) = x^3 - x^2\)
- B. \(f(x) = x^3 - x^2 + 48\)
- C. \(f(x) = x^3 - x^2 - 48\)
- D. \(f(x) = 6x - 2\)
- E. \(f(x) = 6x - 24\)
A rectangular sheet of cardboard has a length of 80 cm and a width of 50 cm. Squares, of side length \(x\) centimetres, are cut from each of the corners, as shown in the diagram below.
A rectangular box with an open top is then constructed, as shown in the diagram below.
The volume of the box is a maximum when \(x\) is equal to
- A. 10
- B. 20
- C. 25
- D. \(\frac{100}{3}\)
- E. \(\frac{200}{3}\)
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.
The only possible outcomes when a coin is tossed are a head or a tail. When an unbiased coin is tossed, the probability of tossing a head is the same as the probability of tossing a tail.
Jo has three coins in her pocket; two are unbiased and one is biased. When the biased coin is tossed, the probability of tossing a head is \(\frac{1}{3}\).
Jo randomly selects a coin from her pocket and tosses it.
a. Find the probability that she tosses a head. 2 marks
b. Find the probability that she selected an unbiased coin, given that she tossed a head. 1 mark
a. Solve \(1 - \cos(\frac{x}{2}) = \cos(\frac{x}{2})\) for \(x \in [-2\pi, \pi]\). 2 marks
b. The function \(f: [-2\pi, \pi] \to \mathbb{R}\), \(f(x) = \cos(\frac{x}{2})\) is shown on the axes below.
Let \(g: [-2\pi, \pi] \to \mathbb{R}\), \(g(x) = 1 - f(x)\).
Sketch the graph of \(g\) on the axes above. Label all points of intersection of the graphs of \(f\) and \(g\), and the endpoints of \(g\), with their coordinates. 2 marks
End of examination questions
VCE is a registered trademark of the VCAA. The VCAA does not endorse or make any warranties regarding this study resource. Past VCE exams and related content can be accessed directly at www.vcaa.vic.edu.au
