2022 VCE Maths Methods Mini Test 7
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 \( \frac{d}{dx} \left( x \cdot \sin(x) \right) = \sin(x) + x \cdot \cos(x) \), then \( \frac{1}{k} \int x \cos(x)\, dx \) is equal to
- A. \( k \left( x \cdot \sin(x) - \int \sin(x)\, dx \right) + c \)
- B. \( \frac{1}{k} \cdot x \cdot \sin(x) - \int \sin(x)\, dx + c \)
- C. \( \frac{1}{k} \left( x \cdot \sin(x) - \int \sin(x)\, dx \right) + c \)
- D. \( \frac{1}{k} (x \cdot \sin(x) - \sin(x)) + c \)
- E. \( \frac{1}{k} \left( \int x \cdot \sin(x)\, dx - \int \sin(x)\, dx \right) + c \)
A bag contains three red pens and \( x \) black pens. Two pens are randomly drawn from the bag without replacement.
The probability of drawing a pen of each colour is equal to
- A. \( \frac{6x}{(2 + x)(3 + x)} \)
- B. \( \frac{3x}{(2 + x)(3 + x)} \)
- C. \( \frac{x}{2 + x} \)
- D. \( \frac{3 + x}{(2 + x)(3 + x)} \)
- E. \( \frac{3 + x}{5 + 2x} \)
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.
The graph of \( y = f(x) \), where \( f : [0, 2\pi] \rightarrow \mathbb{R},\ f(x) = 2\sin(2x) - 1 \), is shown below.
a. On the axes above, draw the graph of \( y = g(x) \), where \( g(x) \) is the reflection of \( f(x) \) in the horizontal axis. 2 marks
b. Find all values of \( k \) such that \( f(k) = 0 \) and \( k \in [0, 2\pi] \). 3 marks
c. Let \( h : D \rightarrow \mathbb{R},\ h(x) = 2\sin(2x) - 1 \), where \( h(x) \) has the same rule as \( f(x) \) with a different domain. The graph of \( y = h(x) \) is translated \( a \) units in the positive horizontal direction and \( b \) units in the positive vertical direction so that it is mapped onto the graph of \( y = g(x) \), where \( a, b \in (0, \infty) \).
i. Find the value for \( b \). 1 mark
ii. Find the smallest positive value for \( a \). 1 mark
iii. Hence, or otherwise, state the domain, \( D \), of \( h(x) \). 1 mark
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
