2020 QCAA Maths Methods Paper 1 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.
If the probability of success in a Bernoulli trial is 0.30, the variance is
- (A) 0.70
- (B) 0.46
- (C) 0.30
- (D) 0.21
The life expectancy (in years) of an electronic component can be represented by the probability density function
\[ p(x) = \begin{cases} \frac{1}{x^2}, & x \ge 1 \\ 0, & \text{otherwise} \end{cases} \]The probability that the component lasts between 1 and 10 years is
- (A) 0.010
- (B) 0.100
- (C) 0.900
- (D) 0.990
A test includes six multiple choice questions. Each question has four options for the answer. If the answers are guessed, the probability of getting at most two questions correct is represented by
- (A) \(\binom{6}{0}0.25^0 \times 0.75^6 + \binom{6}{1}0.25^1 \times 0.75^5\)
- (B) \(\binom{6}{0}0.25^0 \times 0.75^6 + \binom{6}{1}0.25^1 \times 0.75^5 + \binom{6}{2}0.25^2 \times 0.75^4\)
- (C) \(1 - \left( \binom{6}{0}0.25^0 \times 0.75^6 + \binom{6}{1}0.25^1 \times 0.75^5 \right)\)
- (D) \(1 - \left( \binom{6}{0}0.25^0 \times 0.75^6 + \binom{6}{1}0.25^1 \times 0.75^5 + \binom{6}{2}0.25^2 \times 0.75^4 \right)\)
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) \(2\ln(x^2+2x) + c\)
Two types of material (A and B) are being tested for their ability to withstand different temperatures. A random selection of both materials was subjected to extreme temperature changes and then classified according to their condition after they were removed from the testing facility. The results are shown in the table.
| Material | |||
|---|---|---|---|
| A | B | Total | |
| Broke completely | 25 | 43 | 68 |
| Showed defects | 35 | 38 | 73 |
| Remained intact | 35 | 24 | 59 |
| Total | 95 | 105 | 200 |
An approximate 95% confidence interval for the probability that material A will break completely or show defects is given by
\[ \left( c - 1.96\sqrt{\frac{c(1-c)}{n}}, c + 1.96\sqrt{\frac{c(1-c)}{n}} \right) \]The values of \(c\) and \(n\) are
- (A) \(\frac{60}{95}\) and 95
- (B) \(\frac{60}{200}\) and 95
- (C) \(\frac{140}{200}\) and 95
- (D) \(\frac{60}{200}\) and 200
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.
An object is moving in a straight line from a fixed point. The object is at the origin initially.
The acceleration \(a\) (in m s\(^{-2}\)) of the object is given by
\[ a(t) = \pi \cos(\pi t) \quad t \ge 0, \text{where } t \text{ is time in seconds.} \]
The velocity at \(t = 1\) is 0.5 m s\(^{-1}\).
a) Determine the initial acceleration. [1 mark]
b) Determine the initial velocity. [2 marks]
c) Determine the displacement after one second. [2 marks]
END OF PAPER
