2021 QCE Maths Methods Paper 1 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.
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 graph of \(y = f(x)\), where \(f(x)\) is the quadratic function \(f(x) = ax^2 + bx + 4\), is shown. Two regions of the area between the graph of \(y = f(x)\) and the \(x\)-axis are shaded.
Region \(P\) has an area of \(\frac{13}{6}\) units\(^2\) and Region \(Q\) has an area of \(\frac{43}{6}\) units\(^2\).
Determine the values of \(a\) and \(b\).
A firm aims to have 95% confidence in estimating the proportion of office workers who respond to an email in less than an hour to within \(\pm 0.05\).
A survey has never been undertaken before, so no past data is available.
The firm believes that if the proportion is 0.5, then this will result in the largest variability in the sample proportion.
Based on this, determine the sample size needed using the approximate value of \(z = 2\) for the 95% confidence interval.
Justify the choice of 0.5 for the proportion.
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