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AQA Paper 3 2024 June Q11
10 marks Challenging +1.2
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 Easy -3.0
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
5 marks Easy -1.2
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
9 marks Easy -1.2
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
4 marks Moderate -0.8
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
15 marks Moderate -0.3
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
7 marks Moderate -0.8
18
  1. (ii)
    [0pt] [2 marks]
    \end{tabular}}
    \hline \end{tabular} \end{center}
AQA Paper 3 2024 June Q19
9 marks Standard +0.3
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]
Edexcel AS Paper 2 2018 June Q1
3 marks Moderate -0.8
  1. A company is introducing a job evaluation scheme. Points ( \(x\) ) will be awarded to each job based on the qualifications and skills needed and the level of responsibility. Pay ( \(\pounds y\) ) will then be allocated to each job according to the number of points awarded.
Before the scheme is introduced, a random sample of 8 employees was taken and the linear regression equation of pay on points was \(y = 4.5 x - 47\)
  1. Describe the correlation between points and pay.
  2. Give an interpretation of the gradient of this regression line.
  3. Explain why this model might not be appropriate for all jobs in the company.
Edexcel AS Paper 2 2018 June Q2
4 marks Moderate -0.3
  1. A factory buys \(10 \%\) of its components from supplier \(A , 30 \%\) from supplier \(B\) and the rest from supplier \(C\). It is known that \(6 \%\) of the components it buys are faulty.
Of the components bought from supplier \(A , 9 \%\) are faulty and of the components bought from supplier \(B , 3 \%\) are faulty.
  1. Find the percentage of components bought from supplier \(C\) that are faulty. A component is selected at random.
  2. Explain why the event "the component was bought from supplier \(B\) " is not statistically independent from the event "the component is faulty".
Edexcel AS Paper 2 2018 June Q3
7 marks Moderate -0.3
  1. Naasir is playing a game with two friends. The game is designed to be a game of chance so that the probability of Naasir winning each game is \(\frac { 1 } { 3 }\) Naasir and his friends play the game 15 times.
    1. Find the probability that Naasir wins
      1. exactly 2 games,
      2. more than 5 games.
    Naasir claims he has a method to help him win more than \(\frac { 1 } { 3 }\) of the games. To test this claim, the three of them played the game again 32 times and Naasir won 16 of these games.
  2. Stating your hypotheses clearly, test Naasir's claim at the \(5 \%\) level of significance.
Edexcel AS Paper 2 2018 June Q4
8 marks Moderate -0.8
  1. Helen is studying the daily mean wind speed for Camborne using the large data set from 1987. The data for one month are summarised in Table 1 below.
\begin{table}[h]
Windspeed\(\mathrm { n } / \mathrm { a }\)67891112131416
Frequency13232231212
\captionsetup{labelformat=empty} \caption{Table 1}
\end{table}
  1. Calculate the mean for these data.
  2. Calculate the standard deviation for these data and state the units. The means and standard deviations of the daily mean wind speed for the other months from the large data set for Camborne in 1987 are given in Table 2 below. The data are not in month order. \begin{table}[h]
    Month\(A\)\(B\)\(C\)\(D\)\(E\)
    Mean7.588.268.578.5711.57
    Standard Deviation2.933.893.463.874.64
    \captionsetup{labelformat=empty} \caption{Table 2}
    \end{table}
  3. Using your knowledge of the large data set, suggest, giving a reason, which month had a mean of 11.57 The data for these months are summarised in the box plots on the opposite page. They are not in month order or the same order as in Table 2.
    1. State the meaning of the * symbol on some of the box plots.
    2. Suggest, giving your reasons, which of the months in Table 2 is most likely to be summarised in the box plot marked \(Y\). \includegraphics[max width=\textwidth, alt={}, center]{2edcf965-9c93-4a9b-9395-2d3c023801af-11_1177_1216_324_427}
Edexcel AS Paper 2 2018 June Q5
8 marks Moderate -0.3
5. A biased spinner can only land on one of the numbers \(1,2,3\) or 4 . The random variable \(X\) represents the number that the spinner lands on after a single spin and \(\mathrm { P } ( X = r ) = \mathrm { P } ( X = r + 2 )\) for \(r = 1,2\) Given that \(\mathrm { P } ( X = 2 ) = 0.35\)
  1. find the complete probability distribution of \(X\). Ambroh spins the spinner 60 times.
  2. Find the probability that more than half of the spins land on the number 4 Give your answer to 3 significant figures. The random variable \(Y = \frac { 12 } { X }\)
  3. Find \(\mathrm { P } ( Y - X \leqslant 4 )\)
Edexcel AS Paper 2 2018 June Q6
4 marks Moderate -0.8
  1. A man throws a tennis ball into the air so that, at the instant when the ball leaves his hand, the ball is 2 m above the ground and is moving vertically upwards with speed \(9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
The motion of the ball is modelled as that of a particle moving freely under gravity and the acceleration due to gravity is modelled as being of constant magnitude \(10 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) The ball hits the ground \(T\) seconds after leaving the man's hand.
Using the model, find the value of \(T\).
Edexcel AS Paper 2 2018 June Q7
7 marks Moderate -0.3
  1. A train travels along a straight horizontal track between two stations, \(A\) and \(B\).
In a model of the motion, the train starts from rest at \(A\) and moves with constant acceleration \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for 80 s .
The train then moves at constant velocity before it moves with a constant deceleration of \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), coming to rest at \(B\).
  1. For this model of the motion of the train between \(A\) and \(B\),
    1. state the value of the constant velocity of the train,
    2. state the time for which the train is decelerating,
    3. sketch a velocity-time graph. The total distance between the two stations is 4800 m .
  2. Using the model, find the total time taken by the train to travel from \(A\) to \(B\).
  3. Suggest one improvement that could be made to the model of the motion of the train from \(A\) to \(B\) in order to make the model more realistic.
Edexcel AS Paper 2 2018 June Q8
10 marks Standard +0.3
  1. A particle, \(P\), moves along the \(x\)-axis. At time \(t\) seconds, \(t \geqslant 0\), the displacement, \(x\) metres, of \(P\) from the origin \(O\), is given by \(x = \frac { 1 } { 2 } t ^ { 2 } \left( t ^ { 2 } - 2 t + 1 \right)\)
    1. Find the times when \(P\) is instantaneously at rest.
    2. Find the total distance travelled by \(P\) in the time interval \(0 \leqslant t \leqslant 2\)
    3. Show that \(P\) will never move along the negative \(x\)-axis.
Edexcel AS Paper 2 2018 June Q9
9 marks Standard +0.3
9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2edcf965-9c93-4a9b-9395-2d3c023801af-26_551_276_210_890} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Two small balls, \(P\) and \(Q\), have masses \(2 m\) and \(k m\) respectively, where \(k < 2\).
The balls are attached to the ends of a string that passes over a fixed pulley.
The system is held at rest with the string taut and the hanging parts of the string vertical, as shown in Figure 1. The system is released from rest and, in the subsequent motion, \(P\) moves downwards with an acceleration of magnitude \(\frac { 5 g } { 7 }\) The balls are modelled as particles moving freely.
The string is modelled as being light and inextensible.
The pulley is modelled as being small and smooth.
Using the model,
  1. find, in terms of \(m\) and \(g\), the tension in the string,
  2. explain why the acceleration of \(Q\) also has magnitude \(\frac { 5 g } { 7 }\)
  3. find the value of \(k\).
  4. Identify one limitation of the model that will affect the accuracy of your answer to part (c).
Edexcel AS Paper 2 Specimen Q1
4 marks Easy -1.8
  1. Sara is investigating the variation in daily maximum gust, \(t \mathrm { kn }\), for Camborne in June and July 1987.
She used the large data set to select a sample of size 20 from the June and July data for 1987. Sara selected the first value using a random number from 1 to 4 and then selected every third value after that.
  1. State the sampling technique Sara used.
  2. From your knowledge of the large data set explain why this process may not generate a sample of size 20 . The data Sara collected are summarised as follows $$n = 20 \quad \sum t = 374 \quad \sum t ^ { 2 } = 7600$$
  3. Calculate the standard deviation.
Edexcel AS Paper 2 Specimen Q2
5 marks Moderate -0.8
  1. The partially completed histogram and the partially completed table show the time, to the nearest minute, that a random sample of motorists was delayed by roadworks on a stretch of motorway. \includegraphics[max width=\textwidth, alt={}, center]{8f3dbcb4-3260-4493-a230-12577b4ed691-04_1227_1465_354_301}
Delay (minutes)Number of motorists
4-66
7-8
917
10-1245
13-159
16-20
Estimate the percentage of these motorists who were delayed by the roadworks for between 8.5 and 13.5 minutes.
Edexcel AS Paper 2 Specimen Q3
5 marks Easy -1.2
  1. The Venn diagram shows the probabilities for students at a college taking part in various sports. \(A\) represents the event that a student takes part in Athletics. \(T\) represents the event that a student takes part in Tennis. \(C\) represents the event that a student takes part in Cricket. \(p\) and \(q\) are probabilities. \includegraphics[max width=\textwidth, alt={}, center]{8f3dbcb4-3260-4493-a230-12577b4ed691-06_668_935_596_566}
The probability that a student selected at random takes part in Athletics or Tennis is 0.75
  1. Find the value of \(p\).
  2. State, giving a reason, whether or not the events \(A\) and \(T\) are statistically independent. Show your working clearly.
  3. Find the probability that a student selected at random does not take part in Athletics or Cricket.
Edexcel AS Paper 2 Specimen Q4
7 marks Moderate -0.8
  1. Sara was studying the relationship between rainfall, \(r \mathrm {~mm}\), and humidity, \(h \%\), in the UK. She takes a random sample of 11 days from May 1987 for Leuchars from the large data set.
She obtained the following results.
\(h\)9386959786949797879786
\(r\)1.10.33.720.6002.41.10.10.90.1
Sara examined the rainfall figures and found $$Q _ { 1 } = 0.1 \quad Q _ { 2 } = 0.9 \quad Q _ { 3 } = 2.4$$ A value that is more than 1.5 times the interquartile range (IQR) above \(Q _ { 3 }\) is called an outlier.
  1. Show that \(r = 20.6\) is an outlier.
  2. Give a reason why Sara might:
    1. include
    2. exclude
      this day's reading. Sara decided to exclude this day's reading and drew the following scatter diagram for the remaining 10 days' values of \(r\) and \(h\). \includegraphics[max width=\textwidth, alt={}, center]{8f3dbcb4-3260-4493-a230-12577b4ed691-08_988_1081_1555_420}
  3. Give an interpretation of the correlation between rainfall and humidity. The equation of the regression line of \(r\) on \(h\) for these 10 days is \(r = - 12.8 + 0.15 h\)
  4. Give an interpretation of the gradient of this regression line.
    1. Comment on the suitability of Sara's sampling method for this study.
    2. Suggest how Sara could make better use of the large data set for her study.
Edexcel AS Paper 2 Specimen Q5
9 marks Easy -1.2
5. (a) The discrete random variable \(X \sim \mathrm {~B} ( 40,0.27 )\) $$\text { Find } \quad \mathrm { P } ( X \geqslant 16 )$$ Past records suggest that \(30 \%\) of customers who buy baked beans from a large supermarket buy them in single tins. A new manager suspects that there has been a change in the proportion of customers who buy baked beans in single tins. A random sample of 20 customers who had bought baked beans was taken.
(b) Write down the hypotheses that should be used to test the manager's suspicion.
(c) Using a \(10 \%\) level of significance, find the critical region for a two-tailed test to answer the manager's suspicion. You should state the probability of rejection in each tail, which should be less than 0.05
(d) Find the actual significance level of a test based on your critical region from part (c). One afternoon the manager observes that 12 of the 20 customers who bought baked beans, bought their beans in single tins.
(e) Comment on the manager's suspicion in the light of this observation. Later it was discovered that the local scout group visited the supermarket that afternoon to buy food for their camping trip.
(f) Comment on the validity of the model used to obtain the answer to part (e), giving a reason for your answer.
Edexcel AS Paper 2 Specimen Q6
4 marks Easy -1.2
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8f3dbcb4-3260-4493-a230-12577b4ed691-12_520_1072_616_388} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A car moves along a straight horizontal road. At time \(t = 0\), the velocity of the car is \(U \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The car then accelerates with constant acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for \(T\) seconds. The car travels a distance \(D\) metres during these \(T\) seconds. Figure 1 shows the velocity-time graph for the motion of the car for \(0 \leqslant t \leqslant T\).
Using the graph, show that \(D = U T + 1 / 2 a T ^ { 2 }\).
(No credit will be given for answers which use any of the kinematics (suvat) formulae listed under Mechanics in the AS Mathematics section of the formulae booklet.)
Edexcel AS Paper 2 Specimen Q7
7 marks Standard +0.3
  1. A car is moving along a straight horizontal road with constant acceleration. There are three points \(A , B\) and \(C\), in that order, on the road, where \(A B = 22 \mathrm {~m}\) and \(B C = 104 \mathrm {~m}\). The car takes 2 s to travel from \(A\) to \(B\) and 4 s to travel from \(B\) to \(C\).
Find
  1. the acceleration of the car,
  2. the speed of the car at the instant it passes \(A\).
Edexcel AS Paper 2 Specimen Q8
9 marks Standard +0.3
  1. A bird leaves its nest at time \(t = 0\) for a short flight along a straight line.
The bird then returns to its nest.
The bird is modelled as a particle moving in a straight horizontal line.
The distance, \(s\) metres, of the bird from its nest at time \(t\) seconds is given by $$s = \frac { 1 } { 10 } \left( t ^ { 4 } - 20 t ^ { 3 } + 100 t ^ { 2 } \right) , \quad \text { where } 0 \leqslant t \leqslant 10$$
  1. Explain the restriction, \(0 \leqslant t \leqslant 10\)
  2. Find the distance of the bird from the nest when the bird first comes to instantaneous rest.