QCAA Maths Methods Paper 1 Integral Calculus Mini Test 5
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 \(\int_{1}^{3} (2x+3)dx\)
- (A) 2
- (B) 4
- (C) 14
- (D) 16
The graph of \(f''(x)\) is shown.
Which of the following could be the graph of \(f'(x)\)?
Determine \(\int 10e^{4x} dx\)
- (A) \(\frac{10e^{4x+1}}{4x+1} + c\)
- (B) \(40e^{4x} + c\)
- (C) \(\frac{5}{2}e^{4x} + c\)
- (D) \(2e^{5x} + c\)
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.
At the end of the first stage of its growth cycle, a species of tree has a height of 5 metres and a trunk radius of 15 cm.
In the second stage of its growth cycle, the tree stays at this height for the next 10 years. However, the growth rate of the trunk radius (in cm per year) varies over the 10 years and is given by the function \[ r(t) = 1.5 + \sin\left(\frac{\pi t}{5}\right) \] Assume the density (mass per unit volume) of the tree trunk is approximately 1 g/cm\(^3\) and the tree trunk is in the shape of a cylinder.
Determine the ratio of the trunk's mass at the end of the second stage to its mass at the end of the first stage.
END OF PAPER
