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2018 VCAA Maths Methods Exam 1
Adapted for Year 11. Unit 1 & 2.
This is the full VCE Maths Methods Exam with worked solutions. You can also try Mini-Tests, which are official VCAA exams split into short tests you can do anytime.
Number of marks: 21
Reading time: 10 minutes
Writing time: 30 minutes
Instructions
• Answer all questions in the spaces provided.
• Write your responses in English.
• In all 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. If \( y = (-3x^3 + x^2 - 64)^3 \), find \( \frac{dy}{dx} \). 1 mark
b. Not applicable for Year 11 students. 2 marks
The derivative with respect to \(x\) of the function \( f: (1, \infty) \rightarrow \mathbb{R} \) has the rule \( f'(x) = \frac{1}{2} - \frac{1}{(2x-2)} \).
Given that \( f(2) = 0 \), find \( f(x) \) in terms of \(x\). 3 marks
Let \( f: [0, 2\pi] \rightarrow \mathbb{R}, f(x) = 2\cos(x) + 1 \).
a. Solve the equation \( 2\cos(x) + 1 = 0 \) for \( 0 \le x \le 2\pi \). 2 marks
b. Sketch the graph of the function \(f\) on the axes below. Label the endpoints and local minimum point with their coordinates. 3 marks
Let \( f: (2, \infty) \rightarrow \mathbb{R} \), where \( f(x) = \frac{1}{(x-2)^2} \).
State the rule and domain of \( f^{-1} \). 3 marks
Two boxes each contain four stones that differ only in colour.
Box 1 contains four black stones.
Box 2 contains two black stones and two white stones.
A box is chosen randomly and one stone is drawn randomly from it.
Each box is equally likely to be chosen, as is each stone.
a. What is the probability that the randomly drawn stone is black? 2 marks
b. It is not known from which box the stone has been drawn.
Given that the stone that is drawn is black, what is the probability that it was drawn from Box 1? 2 marks
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
End of examination questions
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