2023 QCE Maths Methods Paper 1 Mini Test 3
External Assessment Paper 1 — Technology-free
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.
Determine the mean of the continuous random variable \(X\) with the probability density function
\[ f(x) = \begin{cases} \frac{1}{8}x, & 0 \le x \le 4 \\ 0, & \text{otherwise} \end{cases} \]- (A) \(\frac{1}{8}\)
- (B) \(\frac{3}{8}\)
- (C) \(\frac{1}{2}\)
- (D) \(\frac{8}{3}\)
A sample of size \(n\) was used to estimate a population proportion. An approximate margin of error of 3% was calculated using \(z = 1.96\). Given the sample proportion was 0.6, determine \(n\).
- (A) \(n = \frac{\left(\frac{0.03}{1.96}\right)^2}{0.24}\)
- (B) \(n = \frac{0.24}{\left(\frac{0.03}{1.96}\right)^2}\)
- (C) \(n = \frac{\left(\frac{0.03}{1.96}\right)^2}{2.4}\)
- (D) \(n = \frac{2.4}{\left(\frac{0.03}{1.96}\right)^2}\)
Determine \(\int_0^3 \pi \sin\left(\frac{\pi}{3}x\right)dx\).
- (A) \(3\)
- (B) \(6\)
- (C) \(-3\)
- (D) \(-6\)
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.
The rate of change in the number of bacteria in a science experiment is represented by \(\frac{dP}{dt} = e^{2t}\), \(t \ge 0\), where \(t\) represents the time (hours) since starting the experiment and \(P\) represents the number of bacteria present (thousands). Initially there are 60 000 bacteria present, i.e. \(P(0) = 60\).
a) Determine the equation for \(P(t)\). [2 marks]
b) Determine the change in the number of bacteria during the third hour. Express your answer in terms of \(e\). [2 marks]
c) Determine how long it will take for the number of bacteria present to double after starting the experiment. [2 marks]
END OF PAPER
