AQA Paper 3 (Paper 3) Specimen

1 The graph of \(y = x ^ { 2 } - 9\) is shown below.
\includegraphics[max width=\textwidth, alt={}, center]{be9ad375-d3cd-488c-9e3b-a4cd035d0d1d-02_335_593_767_744} Find the area of the shaded region.
Circle your answer.
[0pt] [1 mark]
\(- 18 - 6618\)

2 A wooden frame is to be made to support some garden decking. The frame is to be in the shape of a sector of a circle. The sector \(O A B\) is shown in the diagram, with a wooden plank \(A C\) added to the frame for strength. \(O A\) makes an angle of \(\theta\) with \(O B\).
\includegraphics[max width=\textwidth, alt={}, center]{be9ad375-d3cd-488c-9e3b-a4cd035d0d1d-02_419_627_1695_847} 2

1. Show that the exact value of \(\sin \theta\) is \(\frac { 4 \sqrt { 14 } } { 15 }\)
   [0pt] [3 marks] 2
2. Write down the value of \(\theta\) in radians to 3 significant figures.
   2
3. Find the area of the garden that will be covered by the decking.

3 A circular ornamental garden pond, of radius 2 metres, has weed starting to grow and cover its surface. As the weed grows, it covers an area of \(A\) square metres. A simple model assumes that the weed grows so that the rate of increase of its area is proportional to \(A\). 3

1. Show that the area covered by the weed can be modelled by
   where \(B\) and \(k\) are constants and \(t\) is time in days since the weed was first noticed.
   [0pt] [4 marks] $$A = B \mathrm { e } ^ { k t }$$ 3
2. When it was first noticed, the weed covered an area of \(0.25 \mathrm {~m} ^ { 2 }\). Twenty days later the weed covered an area of \(0.5 \mathrm {~m} ^ { 2 }\) 3
3. 1. State the value of \(B\).
      [0pt] [1 mark] 3
4. (ii) Show that the model for the area covered by the weed can be written as $$A = 2 ^ { \frac { t } { 20 } - 2 }$$ [4 marks]
   Question 3 continues on the next page 3
5. (iii) How many days does it take for the weed to cover half of the surface of the pond?
   [0pt] [2 marks]
   3
6. State one limitation of the model.
   3
7. Suggest one refinement that could be made to improve the model.
   \(4 \quad \int _ { 1 } ^ { 2 } x ^ { 3 } \ln ( 2 x ) \mathrm { d } x\) can be written in the form \(p \ln 2 + q\), where \(p\) and \(q\) are rational numbers. Find \(p\) and \(q\).
   [0pt] [5 marks]

5

1. Find the first three terms, in ascending powers of \(x\), in the binomial expansion of \(( 1 + 6 x ) ^ { \frac { 1 } { 3 } }\)
   5
2. Use the result from part (a) to obtain an approximation to \(\sqrt [ 3 ] { 1.18 }\) giving your answer to 4 decimal places.
   5
3. Explain why substituting \(x = \frac { 1 } { 2 }\) into your answer to part (a) does not lead to a valid approximation for \(\sqrt [ 3 ] { 4 }\).

6 Find the value of \(\int _ { 1 } ^ { 2 } \frac { 6 x + 1 } { 6 x ^ { 2 } - 7 x + 2 } \mathrm {~d} x\), expressing your answer in the form
\(m \ln 2 + n \ln 3\), where \(m\) and \(n\) are integers.
[0pt] [8 marks]

7 The diagram shows part of the graph of \(y = \mathrm { e } ^ { - x ^ { 2 } }\)
\includegraphics[max width=\textwidth, alt={}, center]{be9ad375-d3cd-488c-9e3b-a4cd035d0d1d-10_376_940_607_392} The graph is formed from two convex sections, where the gradient is increasing, and one concave section, where the gradient is decreasing. 7

1. Find the values of \(x\) for which the graph is concave. 7
2. The finite region bounded by the \(x\)-axis and the lines \(x = 0.1\) and \(x = 0.5\) is shaded.
   \includegraphics[max width=\textwidth, alt={}, center]{be9ad375-d3cd-488c-9e3b-a4cd035d0d1d-11_372_937_584_355} Use the trapezium rule, with 4 strips, to find an estimate for \(\int _ { 0.1 } ^ { 0.5 } e ^ { - x ^ { 2 } } d x\) Give your estimate to four decimal places.
   [0pt] [3 marks]
   7
3. Explain with reference to your answer in part (a), why the answer you found in part (b) is an underestimate.
   [0pt] [2 marks] 7
4. By considering the area of a rectangle, and using your answer to part (b), prove that the shaded area is 0.4 correct to 1 decimal place.
   [0pt] [3 marks]
   \section*{END OF SECTION A
   TURN OVER FOR SECTION B}

8 Edna wishes to investigate the energy intake from eating out at restaurants for the households in her village. She wants a sample of 100 households.
She has a list of all 2065 households in the village.
Ralph suggests this selection method.
"Number the households 0000 to 2064. Obtain 100 different four-digit random numbers between 0000 and 2064 and select the corresponding households for inclusion in the investigation." 8

1. What is the population for this investigation?
   Circle your answer.
   [0pt] [1 mark] Edna and Ralph
   The 2065
   The energy households intake for the The 100 in the village village from households eating out selected 8
2. What is the sampling method suggested by Ralph?
   Circle your answer.
   [0pt] [1 mark] Opportunity Random number Continuous random variable Simple random

9 A survey has found that, of the 2400 households in Growmore, 1680 eat home-grown fruit and vegetables. 9

1. Using the binomial distribution, find the probability that, out of a random sample of 25 households in Growmore, exactly 22 eat home-grown fruit and vegetables.
   [0pt] [2 marks]
   9
2. Give a reason why you would not expect your calculation in part (a) to be valid for the 25 households in Gifford Terrace, a residential road in Growmore.
   Shona calculated four correlation coefficients using data from the Large Data Set.
   In each case she calculated the correlation coefficient between the masses of the cars and the CO2 emissions for varying sample sizes. A summary of these calculations, labelled A to D, are listed in the table below.

   \cline { 2 - 3 } \multicolumn{1}{c|}{}	Sample size	Correlation coefficient
   A	3827	0.088
   B	3735	0.246
   C	24	0.400
   D	1250	- 1.183

   Shona would like to use calculation A to test whether there is evidence of positive correlation between mass and CO2 emissions. She finds the critical value for a one-tailed test at the 5\% level for a sample of size 3827 is 0.027

10

1. 1. State appropriate hypotheses for Shona to use in her test. 10
2. (ii) Determine if there is sufficient evidence to reject the null hypothesis.
   Fully justify your answer.
   [0pt] [1 mark] 10
3. Shona's teacher tells her to remove calculation \(D\) from the table as it is incorrect.
   Explain how the teacher knew it was incorrect.
   [0pt] [1 mark] 10
4. Before performing calculation B, Shona cleaned the data. She removed all cars from the Large Data Set that had incorrect masses. Using your knowledge of the large data set, explain what was incorrect about the masses which were removed from the calculation.
   [0pt] [1 mark] 10
5. Apart from CO 2 and CO emissions, state one other type of emission that Shona could investigate using the Large Data Set. 10
6. Wesley claims that calculation C shows that a heavier car causes higher CO 2 emissions. Give two reasons why Wesley's claim may be incorrect.

11 Terence owns a local shop. His shop has three checkouts, at least one of which is always staffed. A regular customer observed that the probability distribution for \(N\), the number of checkouts that are staffed at any given time during the spring, is $$\mathrm { P } ( N = n ) = \left\{ \begin{array} { c c } \frac { 3 } { 4 } \left( \frac { 1 } { 4 } \right) ^ { n - 1 } & \text { for } n = 1,2
k & \text { for } n = 3 \end{array} \right.$$ 11

1. Find the value of \(k\).
   [0pt] [1 mark]
   11
2. Find the probability that a customer, visiting Terence's shop during the spring, will find at least 2 checkouts staffed.
   [0pt] [2 marks]

12 During the 2006 Christmas holiday, John, a maths teacher, realised that he had fallen ill during 65\% of the Christmas holidays since he had started teaching. In January 2007, he increased his weekly exercise to try to improve his health.
For the next 7 years, he only fell ill during 2 Christmas holidays. 12

1. Using a binomial distribution, investigate, at the \(5 \%\) level of significance, whether there is evidence that John's rate of illness during the Christmas holidays had decreased since increasing his weekly exercise.
   [0pt] [6 marks] 12
2. State two assumptions, regarding illness during the Christmas holidays, that are necessary for the distribution you have used in part (a) to be valid. For each assumption, comment, in context, on whether it is likely to be correct.
   [0pt] [4 marks]

13 In the South West region of England, 100 households were randomly selected and, for each household, the weekly expenditure, \(\pounds X\), per person on food and drink was recorded. The maximum amount recorded was \(\pounds 40.48\) and the minimum amount recorded was £22.00 The results are summarised below, where \(\bar { X }\) denotes the sample mean. $$\sum x = 3046.14 \quad \sum ( x - \bar { x } ) ^ { 2 } = 1746.29$$ 13

1. 1. Find the mean of \(X\)
      Find the standard deviation of \(X\)
      [0pt] [2 marks] 13
2. (ii) Using your results from part (a)(i) and other information given, explain why the normal distribution can be used to model \(X\).
   [0pt] [2 marks] 13
3. (iii) Find the probability that a household in the South West spends less than \(\pounds 25.00\) on food and drink per person per week.
   13
4. For households in the North West of England, the weekly expenditure, \(\pounds Y\), per person on food and drink can be modelled by a normal distribution with mean \(\pounds 29.55\) It is known that \(\mathrm { P } ( Y < 30 ) = 0.55\)
   Find the standard deviation of \(Y\), giving your answer to one decimal place.
   [0pt] [3 marks]

14 A survey during 2013 investigated mean expenditure on bread and on alcohol.
The 2013 survey obtained information from 12144 adults.
The survey revealed that the mean expenditure per adult per week on bread was 127p.
14

1. For 2012, it is known that the expenditure per adult per week on bread had mean 123p, and a standard deviation of 70p. 14
2. 1. Carry out a hypothesis test, at the \(5 \%\) significance level, to investigate whether the mean expenditure per adult per week on bread changed from 2012 to 2013. Assume that the survey data is a random sample taken from a normal distribution.
      [0pt] [5 marks] 14
3. (ii) Calculate the greatest and least values for the sample mean expenditure on bread per adult per week for 2013 that would have resulted in acceptance of the null hypothesis for the test you carried out in part (a)(i). Give your answers to two decimal places.
   [0pt] [2 marks] 14
4. The 2013 survey revealed that the mean expenditure per adult, per week on alcohol was 324p. The mean expenditure per adult per week on alcohol for 2009 was 307p.
   A test was carried out on the following hypotheses relating to mean expenditure per adult per week on alcohol in 2013.
   \(\mathrm { H } _ { 0 } : \mu = 307\)
   \(\mathrm { H } _ { 1 } : \mu \neq 307\)
   This test resulted in the null hypothesis, \(\mathrm { H } _ { 0 }\), being rejected.
   State, with a reason, whether the test result supports the following statements:
   14
5. 1. the mean UK expenditure on alcohol per adult per week increased by 17 p from 2009 to 2013; 14
6. (ii) the mean UK consumption of alcohol per adult per week changed from 2009 to 2013.

15 A sample of 200 households was obtained from a small town.
Each household was asked to complete a questionnaire about their purchases of takeaway food.
\(A\) is the event that a household regularly purchases Indian takeaway food.
\(B\) is the event that a household regularly purchases Chinese takeaway food.
It was observed that \(\mathrm { P } ( B \mid A ) = 0.25\) and \(\mathrm { P } ( A \mid B ) = 0.1\)
Of these households, 122 indicated that they did not regularly purchase Indian or Chinese takeaway food.
A household is selected at random from those in the sample.
Find the probability that the household regularly purchases both Indian and Chinese takeaway food.
[0pt] [6 marks]