2020 VCE Maths Methods Mini Test 1
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.
Let \(f\) and \(g\) be functions such that \(f(-1) = 4\), \(f(2) = 5\), \(g(-1) = 2\), \(g(2) = 7\) and \(g(4) = 6\).
The value of \(g(f(-1))\) is
- A. 2
- B. 4
- C. 5
- D. 6
- E. 7
Let \(p(x) = x^3 - 2ax^2 + x - 1\), where \(a \in R\). When \(p\) is divided by \(x + 2\), the remainder is 5.
The value of \(a\) is
- A. 2
- B. \(-\frac{7}{4}\)
- C. \(-\frac{1}{2}\)
- D. \(\frac{3}{2}\)
- E. -2
Let \(f'(x) = \frac{2}{\sqrt{2x-3}}\)
If \(f(6) = 4\), then
- A. \(f(x) = 2\sqrt{2x-3}\)
- B. \(f(x) = \sqrt{2x-3} - 2\)
- C. \(f(x) = 2\sqrt{2x-3} - 2\)
- D. \(f(x) = \sqrt{2x-3} + 2\)
- E. \(f(x) = \sqrt{2x-3}\)
The solutions of the equation \(2\cos\left(2x - \frac{\pi}{3}\right) + 1 = 0\) are
- A. \(x = \frac{\pi(6k-2)}{6}\) or \(x = \frac{\pi(6k-3)}{6}\), for \(k \in Z\)
- B. \(x = \frac{\pi(6k-2)}{6}\) or \(x = \frac{\pi(6k+5)}{6}\), for \(k \in Z\)
- C. \(x = \frac{\pi(6k-1)}{6}\) or \(x = \frac{\pi(6k+2)}{6}\), for \(k \in Z\)
- D. \(x = \frac{\pi(6k-1)}{6}\) or \(x = \frac{\pi(6k+3)}{6}\), for \(k \in Z\)
- E. \(x = \pi\) or \(x = \frac{\pi(6k+2)}{6}\), for \(k \in Z\)
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. Let \(y = x^2\sin(x)\).
Find \(\frac{dy}{dx}\). 1 mark
b. Evaluate \(f'(1)\), where \(f: \mathbb{R} \to \mathbb{R}\), \(f(x) = e^{x^2 - x + 3}\). 2 marks
A car manufacturer is reviewing the performance of its car model X. It is known that at any given six-month service, the probability of model X requiring an oil change is \(\frac{17}{20}\), the probability of model X requiring an air filter change is \(\frac{3}{20}\) and the probability of model X requiring both is \(\frac{1}{20}\).
a. State the probability that at any given six-month service model X will require an air filter change without an oil change. 1 mark
b. The car manufacturer is developing a new model, Y. The production goals are that the probability of model Y requiring an oil change at any given six-month service will be \(\frac{m}{m+n}\), the probability of model Y requiring an air filter change will be \(\frac{n}{m+n}\) and the probability of model Y requiring both will be \(\frac{1}{m+n}\), where \(m, n \in Z^+\).
Determine \(m\) in terms of \(n\) if the probability of model Y requiring an air filter change without an oil change at any given six-month service is 0.05. 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
