2024 VCE Maths Methods Mini Test 9

Number of marks: 11

Reading time: 2 minutes

Writing time: 16 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.

Question 1 [2024 Exam 2 Section A Q17]

Consider the algorithm below which prints the roots of the cubic polynomial \( f(x) = x^3 - 2x^2 - 9x + 18 \).

\begin{aligned} &\texttt{define } f(x) \\ &\quad \texttt{return } (x^3 - 2x^2 - 9x + 18) \\ &\texttt{c } \gets f(0) \\ &\texttt{if } c < 0 \texttt{ then} \\ &\quad \texttt{c } \gets -c \\ &\texttt{end if} \\ &\texttt{while } c > 0 \texttt{ do} \\ &\quad \texttt{if } f(c) = 0 \texttt{ then} \\ &\quad\quad \texttt{print c} \\ &\quad \texttt{end if} \\ &\quad \texttt{if } f(-c) = 0 \texttt{ then} \\ &\quad\quad \texttt{print -c} \\ &\quad \texttt{end if} \\ &\quad \texttt{c } \gets c - 1 \\ &\texttt{end while} \end{aligned}

Question 2 [2024 Exam 2 Section A Q18]

Find the value of \( x \) which maximises the area of the trapezium shown below.

A. 10
B. 5\(\sqrt{2}\)
C. 7
D. \(\sqrt{10}\)

Question 3 [2024 Exam 2 Section A Q19]

Consider the normal random variable \( X \) such that \( \Pr(X < 10) = 0.2 \) and \( \Pr(X > 18) = 0.2 \).
The value of \( \Pr(X < 12) \) is closest to

A. 0.134
B. 0.297
C. 0.337
D. 0.365

Question 4 [2024 Exam 2 Section A Q20]

The function \( f: \mathbb{R} \to \mathbb{R} \) has average value \( k \) on the interval [0, 2] and satisfies \( f(x) = f(x + 2) \) for all \( x \in \mathbb{R} \).
The value of the definite integral \( \int_2^6 f(x)\,dx \) is

A. 2k
B. 3k
C. 4k
D. 6k

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.

Question 1 [2024 Exam 1 Q8]

Let \( g : \mathbb{R} \rightarrow \mathbb{R}, \quad g(x) = \sqrt[3]{x - k}+ m, \quad \text{where } k \in \mathbb{R} \setminus \{0\} \text{ and } m \in \mathbb{R} \).

Let the point \( P \) be the y-intercept of the graph of \( y = g(x) \).

a. Find the coordinates of \( P \), in terms of \( k \) and \( m \). 1 mark

b. Find the gradient of \( g \) at \( P \), in terms of \( k \). 2 marks

c. Given that the graph of \( y = g(x) \) passes through the origin, express \( k \) in terms of \( m \). 1 mark

d. Let the point \( Q \) be a point different from the point \( P \), such that the gradient of \( g \) at points \( P \) and \( Q \) are equal.
Given that the graph of \( y = g(x) \) passes through the origin, find the coordinates of \( Q \) in terms of \( m \). 3 marks

End of examination questions

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