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AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 SPS SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
AQA Further AS Paper 2 Statistics 2023 June Q4
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
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
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
    1. 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
  5. (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
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 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 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
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 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
4 marks
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
1 marks
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
  3. (ii) Use your distribution from part 5
    1. to find the probability that the spinner lands on a number greater than 5
      [0pt] [1 mark] 5
  5. 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
  7. (ii) Describe how Clare's results could be used to adjust the model.
AQA Further AS Paper 2 Statistics 2024 June Q6
2 marks
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
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
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
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
2 marks
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
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
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
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
    1. Find the value of \(k\).
  4. (ii) Determine the maximum magnitude of the force predicted by the model. 6
  5. (ii) Determine the maximum magnitude of the fored bed bed at
AQA Further AS Paper 2 Mechanics 2018 June Q7
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 2019 June Q1
1 A turntable rotates at a constant speed of \(33 \frac { 1 } { 3 }\) revolutions per minute.
Find the angular speed in radians per second.
Circle your answer.
\(\frac { 5 \pi } { 9 }\)
\(\frac { 10 \pi } { 9 }\)
\(\frac { 5 \pi } { 3 }\)
\(\frac { 20 \pi } { 9 }\)
AQA Further AS Paper 2 Mechanics 2019 June Q2
1 marks
2 The graph shows the resistance force experienced by a cyclist over the first 20 metres of a bicycle ride.
\includegraphics[max width=\textwidth, alt={}, center]{86bfb16f-4df3-4105-8343-e8c4ae862f27-02_572_1381_1320_328} Find the work done by the resistance force over the 20 metres of the bicycle ride. Circle your answer.
[0pt] [1 mark]
1600 J
3000 J
3200 J
4000 J A formula for the elastic potential energy, \(E\), stored in a stretched spring is given by $$E = \frac { k x ^ { 2 } } { 2 }$$ where \(x\) is the extension of the spring and \(k\) is a constant.
Use dimensional analysis to find the dimensions of \(k\).
AQA Further AS Paper 2 Mechanics 2019 June Q4
2 marks
4
  1. Explain, with the aid of a force diagram, why the magnitude of the frictional force acting on Stephi is 490 newtons. 4 In this question use \(g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\)
AQA Further AS Paper 2 Mechanics 2019 June Q5
5 A car of mass 1000 kg has a maximum speed of \(40 \mathrm {~ms} ^ { - 1 }\) when travelling on a straight horizontal race track. The maximum power output of the car's engine is 48 kW
The total resistance force experienced by the car can be modelled as being proportional to the car's speed. Find the maximum possible acceleration of the car when it is travelling at \(25 \mathrm {~ms} ^ { - 1 }\) on the straight horizontal race track. Fully justify your answer.
AQA Further AS Paper 2 Mechanics 2019 June Q6
6 In this question use \(g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) Martin, who is of mass 40 kg , is using a slide.
The slide is made of two straight sections \(A B\) and \(B C\).
The section \(A B\) has length 15 metres and is at an angle of \(50 ^ { \circ }\) to the horizontal.
The section \(B C\) has length 2 metres and is horizontal.
\includegraphics[max width=\textwidth, alt={}, center]{86bfb16f-4df3-4105-8343-e8c4ae862f27-08_389_702_630_667} Martin pushes himself from \(A\) down the slide with initial speed \(1 \mathrm {~ms} ^ { - 1 }\) He reaches \(B\) with speed \(5 \mathrm {~ms} ^ { - 1 }\) Model Martin as a particle.
6
  1. Find the energy lost as Martin slides from \(A\) to \(B\).
    6
  2. Assume that a resistance force of constant magnitude acts on Martin while he is moving on the slide. 6
    1. Show that the magnitude of this resistance force is approximately 270 N
      6
  4. (ii) Determine if Martin reaches the point \(C\).
AQA Further AS Paper 2 Mechanics 2019 June Q7
7 Two smooth spheres, \(P\) and \(Q\), of equal radius are free to move on a smooth horizontal surface. The masses of \(P\) and \(Q\) are \(3 m\) and \(m\) respectively.
\(P\) is set in motion with speed \(u\) directly towards \(Q\), which is initially at rest.
\(P\) subsequently collides with \(Q\).
\includegraphics[max width=\textwidth, alt={}, center]{86bfb16f-4df3-4105-8343-e8c4ae862f27-10_273_864_685_589} Immediately after the collision, \(P\) moves with speed \(v\) and \(Q\) moves with speed \(w\).
The coefficient of restitution between the spheres is \(e\).
7
    1. Show that $$v = \frac { u ( 3 - e ) } { 4 }$$ 7
  2. (ii) Find \(w\), in terms of \(e\) and \(u\), simplifying your answer.
    7
  3. Deduce that $$\frac { u } { 2 } \leq v \leq \frac { 3 u } { 4 }$$ 7
    1. Find, in terms of \(m\) and \(u\), the maximum magnitude of the impulse that \(P\) exerts on \(Q\).
      7
  5. (ii) Describe the impulse that \(Q\) exerts on \(P\).
    \includegraphics[max width=\textwidth, alt={}, center]{86bfb16f-4df3-4105-8343-e8c4ae862f27-13_2488_1719_219_150} Question number Additional page, if required.
    Write the question numbers in the left-hand margin. Additional page, if required.
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