2018 VCE Maths Methods Mini Test 11
Number of marks: 8
Reading time: 1 minute
Writing time: 12 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 graphs \(f: R \to R, f(x) = \cos\left(\frac{\pi x}{2}\right)\) and \(g: R \to R, g(x) = \sin(\pi x)\) are shown in the diagram below.
An integral expression that gives the total area of the shaded regions is
- A. \(\int_0^3 \left(\sin(\pi x) - \cos\left(\frac{\pi x}{2}\right)\right)dx\)
- B. \(2\int_\frac{5}{3}^{3} \left(\sin(\pi x) - \cos\left(\frac{\pi x}{2}\right)\right)dx\)
- C. \(\int_0^\frac{1}{3} \left(\cos\left(\frac{\pi x}{2}\right) - \sin(\pi x)\right)dx - 2\int_\frac{1}{3}^1 \left(\cos\left(\frac{\pi x}{2}\right) - \sin(\pi x)\right)dx - \int_\frac{5}{3}^3 \left(\cos\left(\frac{\pi x}{2}\right) - \sin(\pi x)\right)dx\)
- D. \(2\int_0^\frac{5}{3} \left(\cos\left(\frac{\pi x}{2}\right) - \sin(\pi x)\right)dx - 2\int_\frac{5}{3}^{3} \left(\cos\left(\frac{\pi x}{2}\right) - \sin(\pi x)\right)dx\)
- E. \(\int_0^\frac{1}{3} \left(\cos\left(\frac{\pi x}{2}\right) - \sin(\pi x)\right)dx + 2\int_\frac{1}{3}^1 \left(\sin(\pi x) - \cos\left(\frac{\pi x}{2}\right)\right)dx + \int_\frac{5}{3}^1 \left(\cos\left(\frac{\pi x}{2}\right) - \sin(\pi x)\right)dx\)
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 a part of the graph of \(y = x\sin(x)\), as shown below.
a.
i. Given that \( \int (x\sin(x))dx = \sin(x) - x\cos(x) + c \), evaluate \( \int_{n\pi}^{(n+1)\pi} (x\sin(x))dx \) when \(n\) is a positive even integer or 0. Give your answer in simplest form. 2 marks
ii. Given that \( \int (x\sin(x))dx = \sin(x) - x\cos(x) + c \), evaluate \( \int_{n\pi}^{(n+1)\pi} (x\sin(x))dx \) when \(n\) is a positive odd integer. Give your answer in simplest form. 1 mark
b. Find the equation of the tangent to \( y = x\sin(x) \) at the point \( \left(-\frac{5\pi}{2}, \frac{5\pi}{2}\right) \). 2 marks
c. Not in the current curriculum.
d. Let \( f: [0, 3\pi] \rightarrow \mathbb{R}, f(x) = (3\pi - x)\sin(x) \) and \( g: [0, 3\pi] \rightarrow \mathbb{R}, g(x) = (x - 3\pi)\sin(x) \).
The line \(l_1\) is the tangent to the graph of \(f\) at the point \( \left(\frac{\pi}{2}, \frac{5\pi}{2}\right) \) and the line \(l_2\) is the tangent to the graph of \(g\) at \( \left(\frac{\pi}{2}, -\frac{5\pi}{2}\right) \), as shown in the diagram below.
Find the total area of the shaded regions shown in the diagram above. 2 marks
End of examination questions
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