2017 VCE Maths Methods Mini Test 5
Number of marks: 9
Reading time: 2 minutes
Writing time: 13 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.
The equation \((p - 1)x^2 + 4x = 5 - p\) has no real roots when
- A. \(p^2 - 6p + 6 < 0\)
- B. \(p^2 - 6p + 1 > 0\)
- C. \(p^2 - 6p - 6 < 0\)
- D. \(p^2 - 6p + 1 < 0\)
- E. \(p^2 - 6p + 6 > 0\)
If \(y = a^{b-4x} + 2\), where \(a > 0\), then \(x\) is equal to
- A. \(\frac{1}{4}(b - \log_a(y-2))\)
- B. \(\frac{1}{4}(b - \log_a(y+2))\)
- C. \(b - \log_a(\frac{1}{4}(y+2))\)
- D. \(\frac{b}{4} - \log_a(y-2)\)
- E. \(\frac{1}{4}(b+2 - \log_a(y))\)
The average rate of change of the function with the rule \(f(x) = x^2 - 2x\) over the interval \([1, a]\), where \(a > 1\), is 8.
The value of \(a\) is
- A. 9
- B. 8
- C. 7
- D. 4
- E. \(1 + \sqrt{2}\)
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.
For Jac to log on to a computer successfully, Jac must type the correct password. Unfortunately, Jac has forgotten the password. If Jac types the wrong password, Jac can make another attempt. The probability of success on any attempt is \(\frac{2}{5}\). Assume that the result of each attempt is independent of the result of any other attempt. A maximum of three attempts can be made.
a. What is the probability that Jac does not log on to the computer successfully? 1 mark
b. Calculate the probability that Jac logs on to the computer successfully. Express your answer in the form \(\frac{a}{b}\), where \(a\) and \(b\) are positive integers. 1 mark
c. Calculate the probability that Jac logs on to the computer successfully on the second or on the third attempt. Express your answer in the form \(\frac{c}{d}\), where \(c\) and \(d\) are positive integers. 2 marks
Let \((\tan(\theta)-1)(\sin(\theta)-\sqrt{3}\cos(\theta))(\sin(\theta)+\sqrt{3}\cos(\theta)) = 0\).
a. State all possible values of \(\tan(\theta)\). 1 mark
b. Hence, find all possible solutions for \((\tan(\theta)-1)(\sin^2(\theta)-3\cos^2(\theta)) = 0\), where \(0 \le \theta \le \pi\). 2 marks
End of examination questions
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