VCE Maths Methods Integral 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.
If \(\int_1^8 f(x)dx = 5\), then \(\int_0^2 f(2(x+2))dx\) is equal to
- A. 12
- B. 10
- C. 8
- D. \(\frac{1}{2}\)
- E. \(\frac{5}{2}\)
Part of the graph of a function \(f\), where \(a > 0\), is shown below.
The average value of the function \(f\) over the interval \([-2a, a]\) is
- A. 0
- B. \(\frac{a}{3}\)
- C. \(\frac{a}{2}\)
- D. \(\frac{3a}{4}\)
- E. \(a\)
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.
a. Let \( g : \left( \frac{3}{2}, \infty \right) \rightarrow \mathbb{R},\ g(x) = \frac{3}{2x - 3} \).
Find the rule for an antiderivative of \( g(x) \). 1 mark
b. Evaluate \( \int_{0}^{1} f(x)\left(2f(x) - 3\right) dx \), where \( \int_{0}^{1} \left[f(x)\right]^2 dx = \frac{1}{5} \) and \( \int_{0}^{1} f(x)\, dx = \frac{1}{3} \). 3 marks
a. Evaluate \( \int_{0}^{\frac{\pi}{3}} \sin(x) \, dx \). 1 mark
b. Hence, or otherwise, find all values of \( k \) such that \[ \int_{0}^{\frac{\pi}{3}} \sin(x) \, dx = \int_{k}^{\frac{\pi}{2}} \cos(x) \, dx, \] where \( -3\pi < k < 2\pi \). 3 marks
End of examination questions
VCE is a registered trademark of the VCAA. The VCAA does not endorse or make any warranties regarding this study resource. Past VCE exams and related content can be accessed directly at www.vcaa.vic.edu.au
