2020 VCE Maths Methods Mini Test 9
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 \(f(x) = -\log_e(x+2)\).
A tangent to the graph of \(f\) has a vertical axis intercept at \((0, c)\).
The maximum value of \(c\) is
- A. -1
- B. \(-1 + \log_e(2)\)
- C. \(-\log_e(2)\)
- D. \(-1 - \log_e(2)\)
- E. \(\log_e(2)\)
Let \(a \in (0, \infty)\) and \(b \in R\).
Consider the function \(h: [-a, 0) \cup (0, a] \to R, h(x) = \frac{a}{x} + b\).
The range of \(h\) is
- A. \([b-1, b+1]\)
- B. \((b-1, b+1)\)
- C. \((-\infty, b-1) \cup (b+1, \infty)\)
- D. \((-\infty, b-1] \cup [b+1, \infty)\)
- E. \([b-1, \infty)\)
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.
Consider the function \(f(x) = x^2 + 3x + 5\) and the point \(P(1, 0)\). Part of the graph of \(y = f(x)\) is shown below.
a. Show that point \(P\) is not on the graph of \(y = f(x)\). 1 mark
b. Consider a point \(Q(a, f(a))\) to be a point on the graph of \(f\).
i. Find the slope of the line connecting points \(P\) and \(Q\) in terms of \(a\). 1 mark
ii. Find the slope of the tangent to the graph of \(f\) at point \(Q\) in terms of \(a\). 1 mark
iii. Let the tangent to the graph of \(f\) at \(x=a\) pass through point \(P\).
Find the values of \(a\). 2 marks
iv. Give the equation of one of the lines passing through point \(P\) that is tangent to the graph of \(f\). 1 mark
c. Find the value, \(k\), that gives the shortest possible distance between the graph of the function of \(y = f(x-k)\) and point \(P\). 2 marks
End of examination questions
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