VCE Maths Methods Integral Calculus Mini Test 3
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 value of \(k\) for which the average value of \( y = \cos\left(kx - \frac{\pi}{2}\right) \) over the interval [0, π] is equal to the average value of \( y= \sin(x)\) over the same interval is
- A. \( \frac{1}{6} \)
- B. \( \frac{1}{5} \)
- C. \( \frac{1}{4} \)
- D. \( \frac{1}{3} \)
- E. \( \frac{1}{2} \)
Let \(f'(x) = \frac{2}{\sqrt{2x-3}}\)
If \(f(6) = 4\), then
- A. \(f(x) = 2\sqrt{2x-3}\)
- B. \(f(x) = \sqrt{2x-3} - 2\)
- C. \(f(x) = 2\sqrt{2x-3} - 2\)
- D. \(f(x) = \sqrt{2x-3} + 2\)
- E. \(f(x) = \sqrt{2x-3}\)
Suppose that \( \int_3^{10} f(x)\, dx = C \) and \( \int_7^{10} f(x)\, dx = D \). The value of \( \int_3^7 f(x)\, dx \) is
- A. \( C + D \)
- B. \( C + D - 3 \)
- C. \( C - D \)
- D. \( D - C \)
- E. \( CD - 3 \)
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 \( f : (-\infty, 1] \rightarrow \mathbb{R} \), \( f(x) = x^2 - 2x \). Part of the graph of \( y = f(x) \) is shown below.
a. State the range of \( f \). 1 mark
b. Sketch the graph of the inverse function \( y = f^{-1}(x) \) on the axes above. Label any endpoints and axial intercepts with their coordinates. 2 marks
c. Determine the equation and the domain for the inverse function \( f^{-1} \). 2 marks
d. Calculate the area of the regions enclosed by the curves of \( f \), \( f^{-1} \) and \( y = -x \). 2 marks
End of examination questions
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