VCE Maths Methods Diff Calculus Mini Test 6
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.
A rectangular sheet of cardboard has a length of 80 cm and a width of 50 cm. Squares, of side length \(x\) centimetres, are cut from each of the corners, as shown in the diagram below.
A rectangular box with an open top is then constructed, as shown in the diagram below.
The volume of the box is a maximum when \(x\) is equal to
- A. 10
- B. 20
- C. 25
- D. \(\frac{100}{3}\)
- E. \(\frac{200}{3}\)
Part of the graph of \(y = f(x)\) is shown below.
The corresponding part of the graph of \(y = f'(x)\) is best represented by
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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