Questions Paper 3 (350 questions)

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AQA Paper 3 2024 June Q10
5 marks Challenging +1.2
It is given that $$f'(x) = 5x^3 + x$$ Use differentiation from first principles to prove that $$f''(x) = 15x^2 + 1$$ [5 marks]
AQA Paper 3 2024 June Q11
10 marks Challenging +1.2
The curve \(C\) with equation $$y = (x^2 - 8x) \ln x$$ is defined for \(x > 0\) and is shown in the diagram below. \includegraphics{figure_11} The shaded region, \(R\), lies below the \(x\)-axis and is bounded by \(C\) and the \(x\)-axis. Show that the area of \(R\) can be written as $$p + q \ln 2$$ where \(p\) and \(q\) are rational numbers to be found. [10 marks]
AQA Paper 3 2024 June Q12
1 marks Easy -1.8
A random sample of 84 students was asked how many revision websites they had visited in the past month. The data is summarised in the table below.
Number of websitesFrequency
01
14
218
316
45
537
62
71
Find the interquartile range of the number of websites visited by these 84 students. Circle your answer. [1 mark] 3 \quad 4 \quad 19 \quad 42
AQA Paper 3 2024 June Q13
1 marks Easy -2.5
The shaded region on one of the Venn diagrams below represents \((A \cup C) \cap B\) Identify this Venn diagram. Tick (\(\checkmark\)) one box. [1 mark] \includegraphics{figure_13}
AQA Paper 3 2024 June Q14
5 marks Moderate -0.8
The annual cost of energy in 2021 for each of the 350 households in Village A can be modelled by a random variable \(X\) It is given that $$\sum x = 945\,000 \quad \sum x^2 = 2\,607\,500\,000$$ \begin{enumerate}[label=(\alph*)] \item Calculate the mean of \(X\). [1 mark] \item Calculate the standard deviation of \(X\). [2 marks] \item For households in Village B the annual cost of energy in 2021 has mean £3100 and standard deviation £325 Compare the annual cost of energy in 2021 for households in Village A and Village B. [2 marks]
AQA Paper 3 2024 June Q15
9 marks Moderate -0.8
It is given that $$X \sim \text{B}(48, 0.175)$$ \begin{enumerate}[label=(\alph*)] \item Find the mean of \(X\) [1 mark] \item Show that the variance of \(X\) is 6.93 [1 mark] \item Find P(\(X < 10\)) [1 mark] \item Find P(\(X \geq 6\)) [2 marks] \item Find P(\(9 \leq X \leq 15\)) [2 marks] \item The aeroplanes used on a particular route have 48 seats. The proportion of passengers who use this route to travel for business is known to be 17.5% Make two comments on whether it would be appropriate to use \(X\) to model the number of passengers on an aeroplane who are travelling for business using this route. [2 marks]
AQA Paper 3 2024 June Q16
4 marks Moderate -0.8
A medical student believes that, in adults, there is a **negative correlation** between the amount of nicotine in their blood stream and their energy level. The student collected data from a random sample of 50 adults. The correlation coefficient between the amount of nicotine in their blood stream and their energy level was \(-0.45\) Carry out a hypothesis test at the 2.5% significance level to determine if this sample provides evidence to support the student's belief. For \(n = 50\), the critical value for a one-tailed test at the 2.5% level for the population correlation coefficient is 0.2787 [4 marks]
AQA Paper 3 2024 June Q17
14 marks Moderate -0.8
In 2019, the lengths of new-born babies at a clinic can be modelled by a normal distribution with mean 50 cm and standard deviation 4 cm.
  1. This normal distribution is represented in the diagram below. Label the values 50 and 54 on the horizontal axis. [2 marks] \includegraphics{figure_17a}
  2. State the probability that the length of a new-born baby is less than 50 cm. [1 mark]
  3. Find the probability that the length of a new-born baby is more than 56 cm. [1 mark]
  4. Find the probability that the length of a new-born baby is more than 40 cm but less than 60 cm. [1 mark]
  5. Determine the length exceeded by 95% of all new-born babies at the clinic. [2 marks]
  6. In 2020, the lengths of 40 new-born babies at the clinic were selected at random. The total length of the 40 new-born babies was 2060 cm. Carry out a hypothesis test at the 10% significance level to investigate whether the mean length of a new-born baby at the clinic in 2020 has **increased** compared to 2019. You may assume that the length of a new-born baby is still normally distributed with standard deviation 4 cm. [7 marks]
AQA Paper 3 2024 June Q18
7 marks Easy -1.3
The Human Resources director in a company is investigating the graduate status and salaries of its employees. Event \(G\) is defined as the employee is a graduate. Event \(H\) is defined as the employee earns at least £40 000 a year. The director summarised the findings in the table of probabilities below.
\(H\)\(H'\)
\(G\)0.210.18
\(G'\)0.070.54
\begin{enumerate}[label=(\alph*)] \item An employee is selected at random.
  1. Find P(\(G\)) [1 mark]
  2. Find P[\((G \cap H)'\)] [2 marks]
  3. Find P(\(H | G'\)) [2 marks]
\item Determine whether the events \(G\) and \(H\) are independent. Fully justify your answer. [2 marks]
AQA Paper 3 2024 June Q19
9 marks Standard +0.3
It is known that 80% of all diesel cars registered in 2017 had carbon monoxide (CO) emissions less than 0.3 g/km. Talat decides to investigate whether the proportion of diesel cars registered in 2022 with CO emissions less than 0.3 g/km has **changed**. Talat will carry out a hypothesis test at the 10% significance level on a random sample of 25 diesel cars registered in 2022.
    1. State suitable null and alternative hypotheses for Talat's test. [1 mark]
    2. Using a 10% level of significance, find the critical region for Talat's test. [5 marks]
    3. In his random sample, Talat finds 18 cars with CO emissions less than 0.3 g/km. State Talat's conclusion in context. [1 mark]
  2. Talat now wants to use his random sample of 25 diesel cars, registered in 2022, to investigate whether the proportion of diesel cars in England with CO emissions more than 0.5 g/km has changed from the proportion given by the Large Data Set. Using your knowledge of the Large Data Set, give **two** reasons why it is not possible for Talat to do this. [2 marks]
AQA Paper 3 Specimen Q1
1 marks Easy -1.8
The graph of \(y = x^2 - 9\) is shown below. \includegraphics{figure_1} Find the area of the shaded region. Circle your answer. [1 mark] \(-18\) \quad\quad \(-6\) \quad\quad \(6\) \quad\quad \(18\)
AQA Paper 3 Specimen Q2
6 marks Moderate -0.3
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 \(OAB\) is shown in the diagram, with a wooden plank \(AC\) added to the frame for strength. \(OA\) makes an angle of \(\theta\) with \(OB\). \includegraphics{figure_2}
  1. Show that the exact value of \(\sin\theta\) is \(\frac{4\sqrt{14}}{15}\) [3 marks]
  2. Write down the value of \(\theta\) in radians to 3 significant figures. [1 mark]
  3. Find the area of the garden that will be covered by the decking. [2 marks]
AQA Paper 3 Specimen Q3
13 marks Moderate -0.3
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 \(A\).
  1. Show that the area covered by the weed can be modelled by $$A = Be^{kt}$$ where \(B\) and \(k\) are constants and \(t\) is time in days since the weed was first noticed. [4 marks]
  2. When it was first noticed, the weed covered an area of 0.25 m². Twenty days later the weed covered an area of 0.5 m²
    1. State the value of \(B\). [1 mark]
    2. Show that the model for the area covered by the weed can be written as $$A = 2^{\frac{t}{20} - 2}$$ [4 marks]
    3. How many days does it take for the weed to cover half of the surface of the pond? [2 marks]
  3. State one limitation of the model. [1 mark]
  4. Suggest one refinement that could be made to improve the model. [1 mark]
AQA Paper 3 Specimen Q4
5 marks Standard +0.3
\(\int_1^2 x^3 \ln(2x) dx\) can be written in the form \(p\ln 2 + q\), where \(p\) and \(q\) are rational numbers. Find \(p\) and \(q\). [5 marks]
AQA Paper 3 Specimen Q5
11 marks Moderate -0.3
  1. Find the first three terms, in ascending powers of \(x\), in the binomial expansion of \((1 + 6x)^{\frac{1}{3}}\) [2 marks]
  2. Use the result from part (a) to obtain an approximation to \(\sqrt[3]{1.18}\) giving your answer to 4 decimal places. [2 marks]
  3. Explain why substituting \(x = \frac{1}{2}\) into your answer to part (a) does not lead to a valid approximation for \(\sqrt[3]{4}\). [1 mark]
AQA Paper 3 Specimen Q6
8 marks Challenging +1.2
Find the value of \(\int_1^2 \frac{6x + 1}{6x^2 - 7x + 2} dx\), expressing your answer in the form \(m\ln 2 + n\ln 3\), where \(m\) and \(n\) are integers. [8 marks]
AQA Paper 3 Specimen Q7
12 marks Standard +0.8
The diagram shows part of the graph of \(y = e^{-x^2}\) \includegraphics{figure_7} The graph is formed from two convex sections, where the gradient is increasing, and one concave section, where the gradient is decreasing.
  1. Find the values of \(x\) for which the graph is concave. [4 marks]
  2. The finite region bounded by the \(x\)-axis and the lines \(x = 0.1\) and \(x = 0.5\) is shaded. \includegraphics{figure_7b} Use the trapezium rule, with 4 strips, to find an estimate for \(\int_{0.1}^{0.5} e^{-x^2} dx\) Give your estimate to four decimal places. [3 marks]
  3. Explain with reference to your answer in part (a), why the answer you found in part (b) is an underestimate. [2 marks]
  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. [3 marks]
AQA Paper 3 Specimen Q8
2 marks Easy -2.0
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."
  1. What is the population for this investigation? Circle your answer. [1 mark]
    Edna and RalphThe 2065 households in the villageThe energy intake for the village from eating outThe 100 households selected
  2. What is the sampling method suggested by Ralph? Circle your answer. [1 mark]
    OpportunityRandom numberContinuous random variableSimple random
AQA Paper 3 Specimen Q9
3 marks Moderate -0.8
A survey has found that, of the 2400 households in Growmore, 1680 eat home-grown fruit and vegetables.
  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. [2 marks]
  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. [1 mark]
AQA Paper 3 Specimen Q10
7 marks Moderate -0.8
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.
Sample sizeCorrelation coefficient
A38270.088
B37350.246
C240.400
D1250-1.183
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
    1. State appropriate hypotheses for Shona to use in her test. [1 mark]
    2. Determine if there is sufficient evidence to reject the null hypothesis. Fully justify your answer. [1 mark]
  2. Shona's teacher tells her to remove calculation D from the table as it is incorrect. Explain how the teacher knew it was incorrect. [1 mark]
  3. 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. [1 mark]
  4. Apart from CO2 and CO emissions, state one other type of emission that Shona could investigate using the Large Data Set. [1 mark]
  5. Wesley claims that calculation C shows that a heavier car causes higher CO2 emissions. Give two reasons why Wesley's claim may be incorrect. [2 marks]
AQA Paper 3 Specimen Q11
3 marks Moderate -0.8
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 $$P(N = n) = \begin{cases} \frac{3}{4}\left(\frac{1}{4}\right)^{n-1} & \text{for } n = 1, 2 \\ k & \text{for } n = 3 \end{cases}$$
  1. Find the value of \(k\). [1 mark]
  2. Find the probability that a customer, visiting Terence's shop during the spring, will find at least 2 checkouts staffed. [2 marks]
AQA Paper 3 Specimen Q12
10 marks Standard +0.8
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.
  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. [6 marks]
  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. [4 marks]
AQA Paper 3 Specimen Q13
8 marks Moderate -0.8
In the South West region of England, 100 households were randomly selected and, for each household, the weekly expenditure, \(£X\), per person on food and drink was recorded. The maximum amount recorded was £40.48 and the minimum amount recorded was £22.00 The results are summarised below, where \(\bar{x}\) denotes the sample mean. $$\sum x = 3046.14 \quad\quad \sum (x - \bar{x})^2 = 1746.29$$
    1. Find the mean of \(X\) Find the standard deviation of \(X\) [2 marks]
    2. Using your results from part (a)(i) and other information given, explain why the normal distribution can be used to model \(X\). [2 marks]
    3. Find the probability that a household in the South West spends less than £25.00 on food and drink per person per week. [1 mark]
  2. For households in the North West of England, the weekly expenditure, \(£Y\), per person on food and drink can be modelled by a normal distribution with mean £29.55 It is known that \(P(Y < 30) = 0.55\) Find the standard deviation of \(Y\), giving your answer to one decimal place. [3 marks]
AQA Paper 3 Specimen Q14
11 marks Standard +0.3
A survey during 2013 investigated mean expenditure on bread and on alcohol. The 2013 survey obtained information from 12 144 adults. The survey revealed that the mean expenditure per adult per week on bread was 127p.
  1. For 2012, it is known that the expenditure per adult per week on bread had mean 123p, and a standard deviation of 70p.
    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. [5 marks]
    2. 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. [2 marks]
  2. 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. \(H_0 : \mu = 307\) \(H_1 : \mu \neq 307\) This test resulted in the null hypothesis, \(H_0\), being rejected. State, with a reason, whether the test result supports the following statements:
    1. the mean UK expenditure on alcohol per adult per week increased by 17p from 2009 to 2013; [2 marks]
    2. the mean UK consumption of alcohol per adult per week changed from 2009 to 2013. [2 marks]
AQA Paper 3 Specimen Q15
6 marks Standard +0.8
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 \(P(B|A) = 0.25\) and \(P(A|B) = 0.1\) 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. [6 marks]