QCAA Maths Methods Paper 2 Integral Calculus Mini Test 7
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.
\(\int_{a}^{5a} \frac{1}{x+a} dx\), \(a \neq 0\) is
- (A) 1.7918
- (B) 1.6094
- (C) 1.3863
- (D) 1.0986
A marble moves in one direction in a straight line with velocity \(v = 2\ln(t+1)\) (in metres per second) where \(t\) is time (in seconds) since the marble passed through the origin. Determine the distance from the origin the marble has rolled after 4 seconds.
- (A) 0.40 m
- (B) 1.60 m
- (C) 3.22 m
- (D) 8.09 m
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.
A field is divided into five sections as shown. The width of each section is 1 metre. The perpendicular height, in metres, of each section is given in the diagram. The area of the field was approximated using the trapezoidal rule and found to be 11.12 m\(^2\).
a) Determine the height marked \(x\) on the diagram. [2 marks]
b) Determine the area of the field, given the shape of the field is modelled by the function
\[ f(x) = \frac{4x}{x^2+1} + 1, \quad 0 \le x \le 5 \] [1 mark]The velocity function of an object in m s\(^{-1}\) is given by \( v(t) = \cos\left(6t+\frac{\pi}{2}\right) + 2 \), \( 0 \le t \le 5 \). Initially, the object is at the origin.
a) Determine the displacement function. [2 marks]
b) What is the displacement of the object from the origin, in metres (m), after three seconds? [2 marks]
END OF PAPER
