2018 VCE Maths Methods Mini Test 7
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.
The graph of \(y = \tan(ax)\), where \(a \in R^+\), has a vertical asymptote \(x = 3\pi\) and has exactly one \(x\)-intercept in the region \((0, 3\pi)\).
The value of \(a\) is
- A. \(\frac{1}{6}\)
- B. \(\frac{1}{3}\)
- C. \(\frac{1}{2}\)
- D. \(1\)
- E. \(2\)
The discrete random variable \(X\) has the following probability distribution.
| \(x\) | 0 | 1 | 2 | 3 | 6 |
| \(\Pr(X=x)\) | \(\frac{1}{4}\) | \(\frac{9}{20}\) | \(\frac{1}{10}\) | \(\frac{1}{20}\) | \(\frac{3}{20}\) |
Let \(\mu\) be the mean of \(X\).
\(\Pr(X < \mu)\) is
- A. \(\frac{1}{2}\)
- B. \(\frac{1}{4}\)
- C. \(\frac{17}{20}\)
- D. \(\frac{4}{5}\)
- E. \(\frac{7}{10}\)
In a particular scoring game, there are two boxes of marbles and a player must randomly select one marble from each box. The first box contains four white marbles and two red marbles. The second box contains two white marbles and three red marbles. Each white marble scores \(-2\) points and each red marble scores \(+3\) points. The points obtained from the two marbles randomly selected by a player are added together to obtain a final score.
What is the probability that the final score will equal \(+1\)?
- A. \(\frac{2}{3}\)
- B. \(\frac{1}{5}\)
- C. \(\frac{2}{5}\)
- D. \(\frac{2}{15}\)
- E. \(\frac{8}{15}\)
Two events, \(A\) and \(B\), are independent, where \(\Pr(B) = 2\Pr(A)\) and \(\Pr(A \cup B) = 0.52\)
\(\Pr(A)\) is equal to
- A. \(0.1\)
- B. \(0.2\)
- C. \(0.3\)
- D. \(0.4\)
- E. \(0.5\)
A probability density function, \(f\), is given by
\( f(x) = \begin{cases} \frac{1}{12}(8x-x^3) & 0 \le x \le 2 \\ 0 & \text{elsewhere} \end{cases} \)The median, \(m\), of this function satisfies the equation
- A. \(-m^4 + 16m^2 - 6 = 0\)
- B. \(-m^4 + 4m^2 - 6 = 0\)
- C. \(m^4 - 16m^2 = 0\)
- D. \(m^4 - 16m^2 + 24 = 0.5\)
- E. \(m^4 - 16m^2 + 24 = 0\)
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.
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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