QCAA Maths Methods Paper 1 Integral Calculus Mini Test 6
External Assessment Paper 1 — Technology-free
Number of marks: 10
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 \(\int \frac{x+1}{x^2 + 2x} dx\)
- (A) \(\ln \left( \frac{1}{2x+2} \right) + c\)
- (B) \(\ln(2x+2) + c\)
- (C) \(\frac{1}{2}\ln(x^2+2x) + c\)
- (D) \(2\ln(x^2+2x) + c\)
Determine \(2 \int (4x + 6)^3 dx\)
- (A) \(16(4x+6)^4 + c\)
- (B) \(8(4x+6)^4 + c\)
- (C) \(\frac{(4x+6)^4}{2} + c\)
- (D) \(\frac{(4x+6)^4}{8} + c\)
Determine \(\int \frac{e^x+1}{e^x}dx\)
- (A) \(x - e^{-x} + c\)
- (B) \(x + e^{-x} + c\)
- (C) \(1 + xe^{-x} + c\)
- (D) \(x + xe^{-x} + c\)
The graphs of \(f(x) = e^x\) and \(g(x) = x^2 - 1\) are shown.
The area of the shaded section bounded by these graphs between the lines \(x = 0\) and \(x = 1\) is
- (A) \(1-e\)
- (B) \(e-2\)
- (C) \(e - \frac{5}{3}\)
- (D) \(e - \frac{1}{3}\)
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
