2024 QCE Maths Methods Paper 2 Mini Test 2

External Assessment Paper 2 — Technology-active

Number of marks: 9

Perusal time: 1 minute

Writing time: 15 minutes

Section 1

Instructions
• This section has 10 questions and is worth 10 marks.
• Use a 2B pencil to fill in the A, B, C or D answer bubble completely.
• Choose the best answer for Questions 1 10.
• If you change your mind or make a mistake, use an eraser to remove your response and fill in the new answer bubble completely.

QUESTION 1 [2024 Paper 2 Q2]

Calculate the expected value of a continuous random variable \(X\) with the probability density function

p(x) = \begin{cases} \frac{1}{4}x^2, & 0 \le x \le \sqrt[3]{12} \\ 0, & \text{otherwise} \end{cases}

(A) 1.72
(B) 1.15
(C) 1.00
(D) 0.11

Section 2

Instructions
• Write using black or blue pen.
• Questions worth more than one mark require mathematical reasoning and/or working to be shown to support answers.
• If you need more space for a response, use the additional pages at the back of this book.
– On the additional pages, write the question number you are responding to.
– Cancel any incorrect response by ruling a single diagonal line through your work.
– Write the page number of your alternative/additional response, i.e. See page …
– If you do not do this, your original response will be marked.
• This section has nine questions and is worth 45 marks.

QUESTION 2 (8 marks) [2024 Paper 2 Q13]

The number of termites in a particular nest can be modelled by \(N(t) = \frac{A}{2+e^{-t}}\), where \(A\) is a constant and \(t\) represents time (months) since the nest first became a visible mound above ground level.
It is estimated that when the mound first became visible, the population was \(3 \times 10^5\) termites.

a) Determine the value of \(A\). [1 mark]

b) Determine the number of termites in the nest half a year after the mound became visible. [2 marks]

c) Determine the time in months after the mound became visible for the initial population to increase by 130 000 termites. Express the time as a decimal. [2 marks]

d) Develop a formula for the rate of change in the number of termites at any time after the mound became visible. Express your formula as a fraction. [2 marks]

e) Determine the rate of change in the number of termites five months after the mound became visible. [1 mark]

END OF PAPER

QCE is a registered trademark of the QCAA. The QCAA does not endorse or make any warranties regarding this study resource. Past QCE exams and related content can be accessed directly at www.qcaa.qld.edu.au/