2023 VCE Maths Methods Mini Test 11
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.
Find all values of \( k \), such that the equation
\( x^2 + (4k + 3)x + 4k^2- \frac{9}{4} = 0 \)
has two real solutions for \( x \), one positive and one negative.
- A. \( k > -\frac{3}{4} \)
- B. \( k \geq -\frac{3}{4} \)
- C. \( k > \frac{3}{4} \)
- D. \( -\frac{3}{4} < k < \frac{3}{4} \)
- E. \( k < -\frac{3}{4} \text{ or } k > \frac{3}{4} \)
Let \( f(x) = \log_e\left(x + \frac{1}{\sqrt{2}}\right) \).
Let \( g(x) = \sin(x) \) where \( x \in (-\infty, 5) \).
The largest interval of \( x \) values for which \( (f \circ g)(x) \) and \( (g \circ f)(x) \) both exist is
- A. \( \left( -\frac{1}{\sqrt{2}}, \frac{5\pi}{4} \right) \)
- B. \( \left[ -\frac{1}{\sqrt{2}}, \frac{5\pi}{4} \right) \)
- C. \( \left( -\frac{\pi}{4}, \frac{5\pi}{4} \right) \)
- D. \( \left[ -\frac{\pi}{4}, \frac{5\pi}{4} \right) \)
- E. \( \left[ -\frac{\pi}{4}, -\frac{1}{\sqrt{2}} \right] \)
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. 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
Let \( \hat{P} \) be the random variable that represents the sample proportion of households in a given suburb that have solar panels installed.
From a sample of randomly selected households in a given suburb, an approximate 95% confidence interval for the proportion \( p \) of households having solar panels installed was determined to be (0.04, 0.16).
a. Find the value of \( \hat{p} \) that was used to obtain this approximate 95% confidence interval. 1 mark
b. Find the size of the sample from which this 95% confidence interval was obtained.
Use \( z = 2 \) to approximate the 95% confidence interval. 2 marks
c. A larger sample of households is selected, with a sample size four times the original sample. The sample proportion of households having solar panels installed is found to be the same.
By what factor will the increased sample size affect the width of the confidence interval? 1 mark
End of examination questions
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