Questions — AQA Paper 3 (123 questions)

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AQA Paper 3 2023 June Q11
11 A and B are mutually exclusive events.
Which one of the following statements must be correct?
Tick ( \(\checkmark\) ) one box. $$\begin{aligned} & P ( A \cup B ) = P ( A ) \times P ( B )
& P ( A \cup B ) = P ( A ) - P ( B )
& P ( A \cap B ) = 0
& P ( A \cap B ) = 1 \end{aligned}$$ □
□
□
□
\includegraphics[max width=\textwidth, alt={}, center]{6fba7e53-de46-460b-9bef-f1a6962f2e7d-19_2488_1716_219_153} \begin{center} \begin{tabular}{|l|l|} \hline \begin{tabular}{l}
AQA Paper 3 2023 June Q12
6 marks
12
12

  1. 12

  2. 12

  3. 12

  4. \end{tabular} &
    It is known that, on average, \(40 \%\) of the drivers who take their driving test at a local test centre pass their driving test.
    Each day 32 drivers take their driving test at this centre.
    The number of drivers who pass their test on a particular day can be modelled by the distribution B (32, 0.4)
    State one assumption, in context, required for this distribution to be used.
    [1 mark] \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)
    Find the probability that exactly 7 of the drivers on a particular day pass their test.
    [1 mark] \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)
    Find the probability that, at most, 16 of the drivers on a particular day pass their test.
    [1 mark] \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)
    Find the probability that more than 12 of the drivers on a particular day pass their test.
    [2 marks] \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)

    \hline \end{tabular} \end{center}
    12
  		5. Find the mean number of drivers per day who pass their test.[1 mark]
    12
  6. Find the standard deviation of the number of drivers per day who pass their test.
AQA Paper 3 2023 June Q13
13 There are two types of coins in a money box:
  • 20\% are bronze coins
  • 80\% are silver coins
Craig takes out a coin at random and places it back in the money box.
Craig then takes out a second coin at random.
13
  1. Find the probability that both coins were of the same type.
    13
  2. Find the probability that both coins are bronze, given that at least one of the coins is bronze.
    \includegraphics[max width=\textwidth, alt={}, center]{6fba7e53-de46-460b-9bef-f1a6962f2e7d-23_2488_1716_219_153} \begin{center} \begin{tabular}{|l|l|} \hline \begin{tabular}{l}
AQA Paper 3 2023 June Q14
1 marks
14
14

  1. \end{tabular} &
    The mass of aluminium cans recycled each day in a city may be modelled by a normal distribution with mean 24500 kg and standard deviation 5200 kg .
    State the probability that the mass of aluminium cans recycled on any given day is not equal to 24500 kg .
    [1 mark]

    \hline \end{tabular} \end{center} 14
  2. A member of the council claims that if a different sample of 24 days had been used the hypothesis test in part (b) would have given the same result. Comment on the validity of this claim.
AQA Paper 3 2023 June Q15
15
  1. A random sample of eight cars was selected from the Large Data Set. The masses of these cars, in kilograms, were as follows.
    \(\begin{array} { l l l l l l l l } 950 & 989 & 1247 & 1415 & 1506 & 1680 & 1833 & 2040 \end{array}\) It is given that, for the population of cars in the Large Data Set: $$\begin{aligned} \text { lower quartile } & = 1167
    \text { median } & = 1393
    \text { upper quartile } & = 1570 \end{aligned}$$ 15
    1. It was decided to remove any of the masses which fall outside the following interval. median \(- 1.5 \times\) interquartile range \(\leq\) mass \(\leq\) median \(+ 1.5 \times\) interquartile range Show that only one of the eight masses in the sample should be removed.
      15
  3. (ii) Write down the statistical name for the mass that should be removed in part (a)(i).
    15
  4. The table shows the probability distribution of the number of previous owners, \(N\), for a sample of cars taken from the Large Data Set.
    \(\boldsymbol { n }\)0123456 or more
    \(\mathbf { P } ( \boldsymbol { N } = \boldsymbol { n } )\)0.140.370.9 k0.250.4 k1.7 k0
    Find the value of \(\mathrm { P } ( 1 \leq N < 5 )\)
    15
  5. 15
    7. An expert team is investigating whether there have been any changes in \(\mathrm { CO } _ { 2 }\) emissions from all cars taken from the Large Data Set.
    The team decided to collect a quota sample of 200 cars to reflect the different years and the different makes of cars in the Large Data Set.
    Using your knowledge of the Large Data Set, explain how the team can collect this sample.
    \includegraphics[max width=\textwidth, alt={}]{6fba7e53-de46-460b-9bef-f1a6962f2e7d-29_2488_1716_219_153}
    \begin{center} \begin{tabular}{|l|} \hline \begin{tabular}{l}
AQA Paper 3 2023 June Q16
3 marks
16 A farm supplies apples to a supermarket.
The diameters of the apples, \(D\) centimetres, are normally distributed with mean 6.5 and standard deviation 0.73
\end{tabular}
\hline 16

    1. \hline [1 mark]
      \hline 16
  2. (ii) Find \(\mathrm { P } ( D > 7 )\)
    \hline [1 mark]
    \hline
    16
  3. (iii) The supermarket only accepts apples with diameters between 5 cm and 8 cm . Find the proportion of apples that the supermarket accepts.
    4. [1 mark] \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)

    \hline \end{tabular} \end{center} 16
  4. The farm also supplies plums to the supermarket. These plums have diameters that are normally distributed.
    It is found that \(60 \%\) of these plums have a diameter less than 5.9 cm .
    It is found that \(20 \%\) of these plums have a diameter greater than 6.1 cm .
    Find the mean and standard deviation of the diameter, in centimetres, of the plums supplied by the farm.
AQA Paper 3 2023 June Q17
17 A council found that \(70 \%\) of its new local businesses made a profit in their first year. The council introduced an incentive scheme for its residents to encourage the use of new local businesses. At the end of the scheme, a random sample of 25 new local businesses was selected and it was found that 21 of them had made a profit in their first year. Using a binomial distribution, investigate, at the \(2.5 \%\) level of significance, whether there is evidence of an increase in the proportion of new local businesses making a profit in their first year.
\includegraphics[max width=\textwidth, alt={}, center]{6fba7e53-de46-460b-9bef-f1a6962f2e7d-33_2488_1719_219_150} Question number Additional page, if required.
Write the question numbers in the left-hand margin.
AQA Paper 3 2024 June Q1
1 marks
1 Each of the series below shows the first four terms of a geometric series. Identify the only one of these geometric series that is convergent.
[0pt] [1 mark] Tick \(( \checkmark )\) one box.
\(0.1 + 0.2 + 0.4 + 0.8 + \ldots\)
\includegraphics[max width=\textwidth, alt={}, center]{deec0d32-b031-4227-bc80-7150a0acbc94-02_113_113_858_927}
\(1 - 1 + 1 - 1 + \ldots\)
\includegraphics[max width=\textwidth, alt={}, center]{deec0d32-b031-4227-bc80-7150a0acbc94-02_117_117_1014_927}
\(128 - 64 + 32 - 16 + \ldots\)
\includegraphics[max width=\textwidth, alt={}, center]{deec0d32-b031-4227-bc80-7150a0acbc94-02_118_117_1169_927}
\(1 + 2 + 4 + 8 + \ldots\) □
AQA Paper 3 2024 June Q2
2 The quadratic equation $$4 x ^ { 2 } + b x + 9 = 0$$ has one repeated real root. Find \(b\) Circle your answer.
\(b = 0\)
\(b = \pm 12\)
\(b = \pm 13\)
\(b = \pm 36\)
AQA Paper 3 2024 June Q4
4 A curve has equation \(y = x ^ { 4 } + 2 ^ { x }\) Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
AQA Paper 3 2024 June Q5
5 The diagram below shows a sector of a circle \(O A B\). The chord \(A B\) divides the sector into a triangle and a shaded segment. Angle \(A O B\) is \(\frac { \pi } { 6 }\) radians.
The radius of the sector is 18 cm .
\includegraphics[max width=\textwidth, alt={}, center]{deec0d32-b031-4227-bc80-7150a0acbc94-06_467_428_614_790} Show that the area of the shaded segment is $$k ( \pi - 3 ) \mathrm { cm } ^ { 2 }$$ where \(k\) is an integer to be found.
\includegraphics[max width=\textwidth, alt={}, center]{deec0d32-b031-4227-bc80-7150a0acbc94-07_2491_1753_173_123}
AQA Paper 3 2024 June Q6
6
  1. Find \(\int \left( 6 x ^ { 2 } - \frac { 5 } { \sqrt { x } } \right) \mathrm { d } x\) 6
  2. The gradient of a curve is given by $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 6 x ^ { 2 } - \frac { 5 } { \sqrt { x } }$$ The curve passes through the point \(( 4,90 )\). Find the equation of the curve.
AQA Paper 3 2024 June Q7
7 The graphs with equations $$y = 2 + 3 x - 2 x ^ { 2 } \text { and } x + y = 1$$ are shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{deec0d32-b031-4227-bc80-7150a0acbc94-10_791_721_550_719} The graphs intersect at the points \(A\) and \(B\)
7
  1. On the diagram above, shade and label the region, \(R\), that is satisfied by the inequalities $$0 \leq y \leq 2 + 3 x - 2 x ^ { 2 }$$ and $$x + y \geq 1$$ 7
  2. Find the exact coordinates of \(A\)
AQA Paper 3 2024 June Q8
2 marks
8 The temperature \(\theta ^ { \circ } \mathrm { C }\) of an oven \(t\) minutes after it is switched on can be modelled by the equation $$\theta = 20 \left( 11 - 10 \mathrm { e } ^ { - k t } \right)$$ where \(k\) is a positive constant.
Initially the oven is at room temperature.
The maximum temperature of the oven is \(T ^ { \circ } \mathrm { C }\)
The temperature predicted by the model is shown in the graph below.
\includegraphics[max width=\textwidth, alt={}, center]{deec0d32-b031-4227-bc80-7150a0acbc94-12_750_1319_870_424} 8
  1. Find the room temperature.
    8
  2. Find the value of \(T\)
    [0pt] [2 marks]
    Question 8 continues on the next page 8
  3. The oven reaches a temperature of \(86 ^ { \circ } \mathrm { C }\) one minute after it is switched on. 8
    1. Find the value of \(k\).
      8
  5. (ii) Find the time it takes for the temperature of the oven to be within \(1 ^ { \circ } \mathrm { C }\) of its maximum.
    \includegraphics[max width=\textwidth, alt={}, center]{deec0d32-b031-4227-bc80-7150a0acbc94-15_2493_1759_173_119} \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{deec0d32-b031-4227-bc80-7150a0acbc94-16_805_869_459_651}
    \end{figure} The centre of the circle is \(P\) and the circle intersects the \(y\)-axis at \(Q\) as shown in Figure 1. The equation of the circle is $$x ^ { 2 } + y ^ { 2 } = 12 y - 8 x - 27$$
AQA Paper 3 2024 June Q9
9
  1. Express the equation of the circle in the form $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = k$$ where \(a , b\) and \(k\) are constants to be found.
    9
  2. State the coordinates of \(P\) 9
  3. Find the \(y\)-coordinate of \(Q\)
    \section*{Question 9 continues on the next page} 9
  4. The line segment \(Q R\) is a tangent to the circle as shown in Figure 2 below. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{deec0d32-b031-4227-bc80-7150a0acbc94-18_885_1180_456_495}
    \end{figure} The point \(R\) has coordinates \(( 9 , - 3 )\).
    Find the angle QPR
    Give your answer in radians to three significant figures.
    It is given that $$f ( x ) = 5 x ^ { 3 } + x$$ Use differentiation from first principles to prove that $$f ^ { \prime } ( x ) = 15 x ^ { 2 } + 1$$
AQA Paper 3 2024 June Q11
10 marks
11 The curve \(C\) with equation $$y = \left( x ^ { 2 } - 8 x \right) \ln x$$ is defined for \(x > 0\) and is shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{deec0d32-b031-4227-bc80-7150a0acbc94-20_862_632_502_767} 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.
[0pt] [10 marks]
\section*{END OF SECTION A}
AQA Paper 3 2024 June Q12
1 marks
12 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.
[0pt] [1 mark]
341942 Identify this Venn diagram. Tick ( ✓ ) one box.
\includegraphics[max width=\textwidth, alt={}, center]{deec0d32-b031-4227-bc80-7150a0acbc94-23_506_501_584_374}
\includegraphics[max width=\textwidth, alt={}, center]{deec0d32-b031-4227-bc80-7150a0acbc94-23_117_111_580_897}
\includegraphics[max width=\textwidth, alt={}, center]{deec0d32-b031-4227-bc80-7150a0acbc94-23_508_504_580_1203}
\includegraphics[max width=\textwidth, alt={}, center]{deec0d32-b031-4227-bc80-7150a0acbc94-23_117_120_580_1710}
\includegraphics[max width=\textwidth, alt={}, center]{deec0d32-b031-4227-bc80-7150a0acbc94-23_508_501_1135_374}
\includegraphics[max width=\textwidth, alt={}, center]{deec0d32-b031-4227-bc80-7150a0acbc94-23_112_111_1133_897}
\includegraphics[max width=\textwidth, alt={}, center]{deec0d32-b031-4227-bc80-7150a0acbc94-23_505_506_1133_1201}
\includegraphics[max width=\textwidth, alt={}, center]{deec0d32-b031-4227-bc80-7150a0acbc94-23_112_109_1133_1717} Turn over for the next question
AQA Paper 3 2024 June Q14
14 The annual cost of energy in 2021 for each of the 350 households in Village A can be modelled by a random variable \(\pounds X\) It is given that $$\sum x = 945000 \quad \sum x ^ { 2 } = 2607500000$$ 14
  1. Calculate the mean of \(X\). 14
  2. Calculate the standard deviation of \(X\).
    14
  3. For households in Village B the annual cost of energy in 2021 has mean \(\pounds 3100\) and standard deviation £325 Compare the annual cost of energy in 2021 for households in Village A and Village B.
AQA Paper 3 2024 June Q15
7 marks
15 It is given that $$X \sim \mathrm {~B} ( 48,0.175 )$$ 15
  1. Find the mean of \(X\)
    [0pt] [1 mark] 15
  2. Show that the variance of \(X\) is 6.93
    [0pt] [1 mark] 15
  3. Find \(\mathrm { P } ( X < 10 )\)
    [0pt] [1 mark] 15
  4. \(\quad\) Find \(\mathrm { P } ( X \geq 6 )\)
    [0pt] [2 marks]
    15
  5. \(\quad\) Find \(\mathrm { P } ( 9 \leq X \leq 15 )\)
    [0pt] [2 marks] L
    15
  6. 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.
AQA Paper 3 2024 June Q16
16 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
AQA Paper 3 2024 June Q17
17 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 . 17
  1. This normal distribution is represented in the diagram below. Label the values 50 and 54 on the horizontal axis. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{deec0d32-b031-4227-bc80-7150a0acbc94-29_375_531_644_817} \captionsetup{labelformat=empty} \caption{Length (cm)}
    \end{figure} 17
  2. State the probability that the length of a new-born baby is less than 50 cm .
    17
  3. Find the probability that the length of a new-born baby is more than 56 cm .
    17
  4. Find the probability that the length of a new-born baby is more than 40 cm but less than 60 cm .
    17
  5. Determine the length exceeded by 95\% of all new-born babies at the clinic.
    17
  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 .
AQA Paper 3 2024 June Q18
4 marks
18
  1. (ii)
    [0pt] [2 marks]
    \end{tabular}}
    \hline \end{tabular} \end{center}
AQA Paper 3 2024 June Q19
2 marks
19 It is known that 80\% of all diesel cars registered in 2017 had carbon monoxide (CO) emissions less than \(0.3 \mathrm {~g} / \mathrm { km }\). Talat decides to investigate whether the proportion of diesel cars registered in 2022 with CO emissions less than \(0.3 \mathrm {~g} / \mathrm { km }\) 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. 19
    1. State suitable null and alternative hypotheses for Talat's test. 19
  2. (ii) Using a 10\% level of significance, find the critical region for Talat's test.
    19
  3. (iii) In his random sample, Talat finds 18 cars with CO emissions less than \(0.3 \mathrm {~g} / \mathrm { km }\). State Talat's conclusion in context. 19
  4. 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 \(0.5 \mathrm {~g} / \mathrm { km }\) 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.
    [0pt] [2 marks]