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.


Question 1 [2022 Exam 2 Section A Q7]

The graph of \( y = f(x) \) is shown below.

Graph of y = f(x)

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

Options for f prime graph
Correct Answer: E
Click here for full solution
Question 2 [2022 Exam 2 Section A Q19]

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. \( \frac{a + b - \sqrt{a^2 - 2ab + b^2}}{6} \)
Correct Answer: D
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Question 3 [2021 Exam 2 Section A Q7]

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. \( \frac{5}{2} \)
Correct Answer: B
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Question 4 [2020 Exam 2 Section A Q6]

Part of the graph of \(y = f'(x)\) is shown below.

Graph of the derivative function f'(x)

The corresponding part of the graph of \(y = f(x)\) is best represented by

Graphs for options A, B, C, D, and E
Correct Answer: C
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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.


Question 1 [2021 Exam 1 Q9]

Consider the unit circle \( x^2 + y^2 = 1 \) and the tangent to the circle at the point \(P\), shown in the diagram below.

Unit circle with tangent at P and point A(2,0)

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.

Triangle OAP' with angle theta

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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