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AQA Further AS Paper 2 Statistics 2023 June Q3
3 marks Easy -1.2
3 The discrete random variable \(X\) has probability distribution
\(x\)- 438
\(\mathrm { P } ( X = x )\)0.20.70.1
Show that \(\mathrm { E } ( 5 X - 7 ) = 3.5\)
AQA Further AS Paper 2 Statistics 2023 June Q4
4 marks Standard +0.3
4 The proportion, \(p\), of people in a particular town who use the local supermarket is unknown. A random sample of 30 people in the town is taken and each person is asked if they use the local supermarket. The manager of the supermarket claims that 35\% of the people in the town use the local supermarket. The random sample is used to conduct a hypothesis test at the \(5 \%\) level of significance with the hypotheses $$\begin{aligned} & \mathrm { H } _ { 0 } : p = 0.35 \\ & \mathrm { H } _ { 1 } : p \neq 0.35 \end{aligned}$$ Show that the probability that a Type I error is made is 0.0356 , correct to four decimal places.
AQA Further AS Paper 2 Statistics 2023 June Q5
6 marks Moderate -0.3
5 Rebekah is investigating the distances, \(X\) light years, between the Earth and visible stars in the night sky. She determines the distance between the Earth and a star for a random sample of 100 visible stars. The summarised results are as follows: $$\sum x = 35522 \quad \text { and } \quad \sum x ^ { 2 } = 32902257$$ 5
  1. Calculate a 97\% confidence interval for the population mean of \(X\), giving your values to the nearest light year.
    5
  2. Mike claims that the population mean is 267 light years. Rebekah says that the confidence interval supports Mike's claim. State, with a reason, whether Rebekah is correct.
AQA Further AS Paper 2 Statistics 2023 June Q6
8 marks Standard +0.3
6 An insurance company models the number of motor claims received in 1 day using a Poisson distribution with mean 65 6
  1. Find the probability that the company receives at most 60 motor claims in 1 day. Give your answer to three decimal places. 6
  2. The company receives motor claims using a telephone line which is open 24 hours a day. Find the probability that the company receives exactly 2 motor claims in 1 hour. Give your answer to three decimal places.
    6
  3. The company models the number of property claims received in 1 day using a Poisson distribution with mean 23 Assume that the number of property claims received is independent of the number of motor claims received. 6 (c) (i) Find the standard deviation of the variable that represents the total number of motor claims and property claims received in 1 day. Give your answer to three significant figures.
    6 (c) (ii) Find the probability that the company receives a total of more than 90 motor claims and property claims in 1 day. Give your answer to three significant figures.
AQA Further AS Paper 2 Statistics 2023 June Q7
10 marks Standard +0.3
7 A theatre has morning, afternoon and evening shows. On one particular day, the theatre asks all of its customers to state whether they enjoyed or did not enjoy the show. The results are summarised in the table.
Morning showAfternoon showEvening showTotal
Enjoyed6291172325
Not enjoyed2535115175
Total87126287500
The theatre claims that there is no association between the show that a customer attends and whether they enjoyed the show. 7
  1. Investigate the theatre's claim, using a \(2.5 \%\) level of significance.
    7
  2. By considering observed and expected frequencies, interpret in context the association between the show that a customer attends and whether they enjoyed the show.
AQA Further AS Paper 2 Statistics 2023 June Q8
8 marks Challenging +1.2
8 The continuous random variable \(X\) has probability density function \(\mathrm { f } ( x )\) It is given that \(\mathrm { f } ( x ) = x ^ { 2 }\) for \(0 \leq x \leq 1\) It is also given that \(\mathrm { f } ( x )\) is a linear function for \(1 < x \leq \frac { 3 } { 2 }\) For all other values of \(x , \mathrm { f } ( x ) = 0\) A sketch of the graph of \(y = \mathrm { f } ( x )\) is shown below. \includegraphics[max width=\textwidth, alt={}, center]{c309e27b-5618-4f94-aecd-a55d8756ef03-12_821_1077_758_543} Show that \(\operatorname { Var } ( X ) = 0.0864\) correct to three significant figures. \includegraphics[max width=\textwidth, alt={}, center]{c309e27b-5618-4f94-aecd-a55d8756ef03-14_2491_1755_173_123} Additional page, if required. number Write the question numbers in the left-hand margin.
AQA Further AS Paper 2 Statistics 2024 June Q1
1 marks Easy -2.0
1 The discrete random variable \(X\) has probability distribution function $$\mathrm { P } ( X = x ) = \begin{cases} 0.45 & x = 1 \\ 0.25 & x = 2 \\ 0.25 & x = 3 \\ 0.05 & x = 4 \\ 0 & \text { otherwise } \end{cases}$$ State the mode of \(X\) Circle your answer.
0.25
0.45
1
2.5
AQA Further AS Paper 2 Statistics 2024 June Q2
1 marks Easy -1.2
2 A test for association is to be carried out. The tables below show the observed frequencies and the expected frequencies that are to be used for the test.
ObservedXYZ
A28666
B884
C541610
Expected\(\mathbf { X }\)\(\mathbf { Y }\)\(\mathbf { Z }\)
\(\mathbf { A }\)451540
\(\mathbf { B }\)938
\(\mathbf { C }\)361232
It is necessary to merge some rows or columns before the test can be carried out.
Find the entry in the tables that provides evidence for this.
Circle your answer.
[0pt] [1 mark]
Observed A-Z
Observed B-Z
Expected A-X
Expected B-Y
AQA Further AS Paper 2 Statistics 2024 June Q3
3 marks Moderate -0.8
3 The random variable \(X\) has a normal distribution with known variance 15.7 A random sample of size 120 is taken from \(X\) The sample mean is 68.2 Find a 94\% confidence interval for the population mean of \(X\) Give your limits to three significant figures.
AQA Further AS Paper 2 Statistics 2024 June Q4
7 marks Easy -1.2
4 The discrete random variable \(Y\) has probability distribution
\(y\)15213643
\(\mathrm { P } ( Y = y )\)0.160.320.290.23
The standard deviation of \(Y\) is \(s\) 4
  1. Show that \(s = 10.53\) correct to two decimal places.
    [0pt] [4 marks]
    4
  2. The median of \(Y\) is \(m\) Find \(\mathrm { P } ( Y > m - 1.5 s )\)
AQA Further AS Paper 2 Statistics 2024 June Q5
6 marks Easy -1.8
5 A spinner has 8 equal areas numbered 1 to 8, as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{de9f0107-38de-4d0d-8391-4d29b98fa601-06_383_390_319_810} The spinner is spun and lands with one of its edges on the ground. 5
  1. Assume that the spinner lands on each number with equal probability. 5
      1. State a distribution that could be used to model the number that the spinner lands on. 5
      2. Use your distribution from part 5
      (a) (i) to find the probability that the spinner lands on a number greater than 5
      [0pt] [1 mark] 5
    2. Clare spins the spinner 1000 times and records the results in the following table.
      Number
      landed on
      		12345678
      Frequency376411216130815610953
      5
      1. Explain how the data shows that the model used in part (a) may not be valid.
        5
    4. (ii) Describe how Clare's results could be used to adjust the model.
AQA Further AS Paper 2 Statistics 2024 June Q6
11 marks Challenging +1.2
6 The continuous random variable \(X\) has probability density function $$f ( x ) = \begin{cases} \frac { 3 x } { 44 } + \frac { 1 } { 22 } & 1 \leq x \leq 5 \\ 0 & \text { otherwise } \end{cases}$$ 6
  1. Find \(\mathrm { P } ( X > 2 )\) [0pt] [2 marks]
    6
  2. Find the upper quartile of \(X\) Give your answer to two decimal places.
    6
  3. Find \(\operatorname { Var } \left( 44 X ^ { - 3 } \right)\) Give your answer to three decimal places.
AQA Further AS Paper 2 Statistics 2024 June Q7
11 marks Standard +0.3
7 Over a period of time, it has been shown that the mean number of customers entering a small store is 6 per hour. The store runs a promotion, selling many products at lower prices. 7
  1. Luke randomly selects an hour during the promotion and counts 11 customers entering the store. He claims that the promotion has changed the mean number of customers per hour entering the store. Investigate Luke's claim, using the \(5 \%\) level of significance.
    7
  2. Luke randomly selects another hour and carries out the same investigation as in part (a). Find the probability of a Type I error, giving your answer to four decimal places.
    Fully justify your answer.
    7
  3. When observing the store, Luke notices that some customers enter the store together as a group. Explain why the model used in parts (a) and (b) might not be valid.
    DO NOT WRITE/ON THIS PAGE ANSWER IN THE/SPACES PROVIDED number Additional page, if required. Write the question numbers in the left-hand margin.
    Additional page, if required. number Additional page, if required.
    Write the question numbers in the left-hand margin. Additional page, if required. number Additional page, if required.
    Write the question numbers in the left-hand margin.
    Write the question n
AQA Further AS Paper 2 Mechanics 2018 June Q1
1 marks Easy -1.8
1 A particle \(A\), of mass 0.2 kg , collides with a particle \(B\), of mass 0.3 kg Immediately before the collision, the velocity of \(A\) is \(\left[ \begin{array} { c } 4 \\ 12 \end{array} \right] \mathrm { ms } ^ { - 1 }\) and the velocity of \(B\) is \(\left[ \begin{array} { l } - 1 \\ - 3 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\) As a result of the collision the particles coalesce to become a single particle.
Find the velocity of the single particle.
Circle your answer.
[0pt] [1 mark] $$\left[ \begin{array} { l } 0.5 \\ 1.5 \end{array} \right] \mathrm { m } \mathrm {~s} ^ { - 1 } \quad \left[ \begin{array} { l } 2 \\ 6 \end{array} \right] \mathrm { m } \mathrm {~s} ^ { - 1 } \quad \left[ \begin{array} { l } 1 \\ 3 \end{array} \right] \mathrm { ms } ^ { - 1 } \quad \left[ \begin{array} { l } 3 \\ 9 \end{array} \right] \mathrm { m } \mathrm {~s} ^ { - 1 }$$
AQA Further AS Paper 2 Mechanics 2018 June Q2
1 marks Easy -1.2
2 A train is travelling at maximum speed with its engine using its maximum power of 1800 kW When travelling at this speed the train experiences a total resistive force of 40000 N Find the maximum speed of the train. Circle your answer.
[0pt] [1 mark] \(22 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) \(45 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) \(54 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) \(90 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
AQA Further AS Paper 2 Mechanics 2018 June Q3
5 marks Standard +0.3
3 The kinetic energy, \(E\), of a compound pendulum is given by $$E = \frac { 1 } { 2 } I \omega ^ { 2 }$$ where \(\omega\) is the angular speed and \(I\) is a quantity called the moment of inertia.
3
  1. Show that for this formula to be dimensionally consistent then \(I\) must have dimensions \(M L ^ { 2 }\), where \(M\) represents mass and \(L\) represents length.
    [0pt] [2 marks]
    3
  2. The time, \(T\), taken for one complete swing of a pendulum is thought to depend on its moment of inertia, \(I\), its weight, \(W\), and the distance, \(h\), of the centre of mass of the pendulum from the point of suspension. The formula being proposed is $$T = k I ^ { \alpha } W ^ { \beta } h ^ { \gamma }$$ where \(k\) is a dimensionless constant. Determine the values of \(\alpha , \beta\) and \(\gamma\).
AQA Further AS Paper 2 Mechanics 2018 June Q4
11 marks Standard +0.8
4 Two smooth spheres \(A\) and \(B\) of equal radius are free to move on a smooth horizontal surface. The masses of \(A\) and \(B\) are \(m\) and \(4 m\) respectively.
The coefficient of restitution between the spheres is \(e\).
The spheres are projected directly towards each other, each with speed \(u\), and subsequently collide. 4
  1. Show that the speed of \(B\) immediately after the impact with \(A\) is $$\frac { u ( 3 - 2 e ) } { 5 }$$ 4
  2. Find the speed of \(A\) in terms of \(u\) and \(e\).
    4
  3. Comment on the direction of motion of the spheres after the collision, justifying your answer.
    4
  4. The magnitude of the impulse on \(B\) due to the collision is \(I\).
    Deduce that $$\frac { 8 m u } { 5 } \leq I \leq \frac { 16 m u } { 5 }$$
AQA Further AS Paper 2 Mechanics 2018 June Q5
6 marks Standard +0.3
5 A car travels around a roundabout at a constant speed. The surface of the roundabout is horizontal. The car has mass 990 kg and the path of the car is a circular arc of radius 48 metres.
A simple model assumes that the car is a particle and the only horizontal force acting on it as it travels around the roundabout is friction. On a dry day typical values of friction, \(F\), between the surface of the roundabout and the tyres of the car are $$7300 \mathrm {~N} \leq F \leq 9200 \mathrm {~N}$$ 5
  1. Using this model calculate a safe speed limit, in miles per hour, for the car as it travels around the roundabout. Explain your reasoning fully.
    Note that there are 1600 metres in one mile.
    5
  2. Gary assumes that on a wet day typical values for friction, \(F\), are $$5400 \mathrm {~N} \leq F \leq 10000 \mathrm {~N}$$ Comment on the validity of Gary's revised assumption.
AQA Further AS Paper 2 Mechanics 2018 June Q6
7 marks Standard +0.3
6 At a fairground a dodgem car is moving in a straight horizontal line towards a side wall that is perpendicular to the velocity of the car. The speed of the car is \(1.8 \mathrm {~ms} ^ { - 1 }\) It collides with the side wall and rebounds along its original path with a speed of \(1.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) The total mass of the dodgem car and the passengers is 250 kg
6
  1. Find the magnitude of the impulse on the car during the collision with the side wall.
    6
  2. A possible model for the magnitude of the force, \(F\) newtons, acting on the dodgem car due to its collision with the side wall is given by $$F = k t ( 4 - 5 t ) \quad \text { for } 0 \leq t \leq 0.8$$ 6 (b)
    1. Find the value of \(k\).
    2. Determine the maximum magnitude of the force predicted by the model.
    6 (b) (ii) Determine the maximum magnitude of the fored bed bed at
AQA Further AS Paper 2 Mechanics 2018 June Q7
9 marks Standard +0.3
7
  1. Find Dominic's speed at the point when the cord initially becomes taut.
    7
  2. Determine whether or not Dominic enters the river and gets wet.
    7
  3. One limitation of this model is that Dominic is not a particle.
    Explain the effect of revising this assumption on your answer to part (b). \includegraphics[max width=\textwidth, alt={}, center]{1b79a789-c003-46c9-9235-254c1d8a0501-12_2492_1721_217_150} Question number Additional page, if required.
    Write the question numbers in the left-hand margin. Question number Additional page, if required.
    Write the question numbers in the left-hand margin. Additional page, if required.
    Write the question numbers in the left-hand margin.
AQA Further AS Paper 2 Mechanics 2022 June Q1
1 marks Easy -1.8
1 A box is being pushed in a straight line along horizontal ground by a force.
The force is applied in the direction of motion and has magnitude 10 newtons. The box moves 5 metres in 2 seconds. Calculate the work done by the force.
Circle your answer.
20 J
25 J
50 J
100 J
AQA Further AS Paper 2 Mechanics 2022 June Q2
1 marks Easy -1.8
2 Two particles of equal mass are moving on a horizontal surface when they collide.
Immediately before the collision, their velocities are \(\left[ \begin{array} { l } 2 \\ 4 \end{array} \right] \mathrm { ms } ^ { - 1 }\) and \(\left[ \begin{array} { c } 6 \\ - 2 \end{array} \right] \mathrm { ms } ^ { - 1 }\) As a result of the collision the particles coalesce to become a single particle.
Find the velocity of the single particle, immediately after the collision.
Circle your answer.
[0pt] [1 mark] \(\left[ \begin{array} { l } 4 \\ 1 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\) \(\left[ \begin{array} { l } 4 \\ 3 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\) \(\left[ \begin{array} { l } 8 \\ 2 \end{array} \right] \mathrm { ms } ^ { - 1 }\) \(\left[ \begin{array} { l } 8 \\ 6 \end{array} \right] \mathrm { ms } ^ { - 1 }\)
AQA Further AS Paper 2 Mechanics 2022 June Q3
4 marks Moderate -0.3
3 In this question use \(g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) A ball of mass of 0.75 kg is thrown vertically upwards with an initial speed of \(12 \mathrm {~ms} ^ { - 1 }\) The ball is thrown from ground level. 3
  1. Calculate the initial kinetic energy of the ball. 3
  2. The maximum height of the ball above the ground is \(h\) metres.
    Jeff and Gurjas use an energy method to find \(h\) Jeff concludes that \(h = 7.3\) Gurjas concludes that \(h < 7.3\) Explain the reasoning that they have used, showing any calculations that you make.
AQA Further AS Paper 2 Mechanics 2022 June Q4
5 marks Moderate -0.8
4 Wavelength is defined as the distance from the highest point on one wave to the highest point on the next wave. Surfers classify waves into one of several types related to their wavelengths.
Two of these classifications are deep water waves and shallow water waves.
4
  1. The wavelength \(w\) of a deep water wave is given by $$w = \frac { g t ^ { 2 } } { k }$$ where \(g\) is the acceleration due to gravity and \(t\) is the time period between consecutive waves. Given that the formula for a deep water wave is dimensionally consistent, show that \(k\) is a dimensionless constant. 4
  2. The wavelength \(w\) of a shallow water wave is given by $$w = ( g d ) ^ { \alpha } t ^ { \beta }$$ where \(g\) is the acceleration due to gravity, \(d\) is the depth of water and \(t\) is the time period between consecutive waves. Use dimensional analysis to find the values of \(\alpha\) and \(\beta\)
AQA Further AS Paper 2 Mechanics 2022 June Q5
5 marks Moderate -0.3
5 A car, of mass 1000 kg , is travelling on a straight horizontal road. When the car travels at a speed of \(v \mathrm {~ms} ^ { - 1 }\), it experiences a resistance force of magnitude \(25 v\) newtons. The car has a maximum speed of \(72 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) on the straight road.
Find the maximum power output of the car.
Fully justify your answer.