VCE Maths Methods Diff Calculus Mini Test 4
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 \( y = f(x) \) is shown below.

The graph of \( y = f'(x) \), the first derivative of \( f(x) \) with respect to \( x \), could be

A box is formed from a rectangular sheet of cardboard, which has a width of \( a \) units and a length of \( b \) units, by first cutting out squares of side length \( x \) units from each corner and then folding up to form an open-top container.
The maximum volume of the box occurs when \( x \) is equal to
- A. \( \frac{a - b + \sqrt{a^2 - ab + b^2}}{6} \)
- B. \( \frac{a + b + \sqrt{a^2 - ab + b^2}}{6} \)
- C. \( \frac{a - b - \sqrt{a^2 - ab + b^2}}{6} \)
- D. \( \frac{a + b - \sqrt{a^2 - ab + b^2}}{6} \)
- E. \( \frac{a + b - \sqrt{a^2 - 2ab + b^2}}{6} \)
The tangent to the graph of \( y = x^3 - ax^2 + 1 \) at \( x = 1 \) passes through the origin.
The value of \(a\) is
- A. \( \frac{1}{2} \)
- B. 1
- C. \( \frac{3}{2} \)
- D. 2
- E. \( \frac{5}{2} \)
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 unit circle \( x^2 + y^2 = 1 \) and the tangent to the circle at the point \(P\), shown in the diagram below.

a. Show that the equation of the line that passes through the points \(A\) and \(P\) is given by \( y = -\frac{x}{\sqrt{3}} + \frac{2}{\sqrt{3}} \). 2 marks
b. Not applicable to current curriculum.
c. For \( 0 < q \le 1 \), let \(P'\) be the point of intersection of the graph of \(h\) with the unit circle, where \(P'\) is always the point of intersection that is closest to \(A\), as shown in the diagram below.

Let \(g\) be the function that gives the area of triangle \(OAP'\) in terms of \( \theta \).
i. Define the function \(g\). 2 marks
ii. Determine the maximum possible area of the triangle \(OAP'\). 2 marks
End of examination questions
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