QCAA Maths Methods Differential Calculus Mini Test 8
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.
The second derivative of the function \(f(x)\) is given by \(f''(x) = \frac{2x}{1+x^2}\).
The interval on which the graph of \(f(x)\) is concave up is
- (A) \(x < 0\)
- (B) \(x \le 0\)
- (C) \(x > 0\)
- (D) \(x \ge 0\)
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 volume of water in a tank is represented by a function of the form
\[ V(t) = Ae^{kt} \text{, where } V \text{ is in litres and } t \text{ is in minutes.} \]
Initially, the volume is 100 litres and it is decreasing by 50 litres per minute.
Determine the time at which the volume is decreasing at the rate of \( \frac{50}{7} \) litres per minute.
Express your answer in the form \(\ln(a)\).
Determine the derivative of each of the following with respect to \(x\).
a) \( y = \frac{1}{\sin(x)} \) [1 mark]
b) \( y = x^2 \times e^{-x} \)
Express your answer in factorised form. [2 marks]
END OF PAPER
