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AQA Further Paper 3 Statistics 2023 June Q8
14 marks Standard +0.3
8 The continuous random variable \(X\) has probability density function $$f ( x ) = \begin{cases} k \sin 2 x & 0 \leq x \leq \frac { \pi } { 6 } \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant. 8
  1. Show that \(k = 4\) 8
  2. Find the cumulative distribution function \(\mathrm { F } ( x )\) 8
  3. Find the median of \(X\), giving your answer to three significant figures. 8
  4. Find the mean of \(X\) giving your answer in the form \(\frac { 1 } { a } ( b \sqrt { 3 } - \pi )\) where \(a\) and \(b\) are integers. \includegraphics[max width=\textwidth, alt={}, center]{1e2fdd33-afa4-486f-a9e2-1d425ed14eee-14_2492_1721_217_150}
AQA Further Paper 3 Statistics 2024 June Q1
1 marks Easy -1.8
1 The random variable \(X\) has a Poisson distribution with mean 16 Find the standard deviation of \(X\) Circle your answer.
4
8
16
256
AQA Further Paper 3 Statistics 2024 June Q4
6 marks Moderate -0.3
4
8
16
256 2 The random variable \(T\) has an exponential distribution with mean 2 Find \(\mathrm { P } ( T \leq 1.4 )\) Circle your answer. \(\mathrm { e } ^ { - 2.8 }\) \(\mathrm { e } ^ { - 0.7 }\) \(1 - e ^ { - 0.7 }\) \(1 - \mathrm { e } ^ { - 2.8 }\) The continuous random variable \(Y\) has cumulative distribution function $$\mathrm { F } ( y ) = \left\{ \begin{array} { l r } 0 & y < 2 \\ - \frac { 1 } { 9 } y ^ { 2 } + \frac { 10 } { 9 } y - \frac { 16 } { 9 } & 2 \leq y < 5 \\ 1 & y \geq 5 \end{array} \right.$$ Find the median of \(Y\) Circle your answer. 2 \(\frac { 10 - 3 \sqrt { 2 } } { 2 }\) \(\frac { 7 } { 2 }\) \(\frac { 10 + 3 \sqrt { 2 } } { 2 }\) Turn over for the next question 4 Research has shown that the mean number of volcanic eruptions on Earth each day is 20 Sandra records 162 volcanic eruptions during a period of one week. Sandra claims that there has been an increase in the mean number of volcanic eruptions per week. Test Sandra's claim at the \(5 \%\) level of significance.
AQA Further Paper 3 Statistics 2024 June Q5
5 marks Standard +0.3
5 The continuous random variable \(X\) has probability density function $$f ( x ) = \begin{cases} \frac { 1 } { 6 } e ^ { \frac { x } { 3 } } & 0 \leq x \leq \ln 27 \\ 0 & \text { otherwise } \end{cases}$$ Show that the mean of \(X\) is \(\frac { 3 } { 2 } ( \ln 27 - 2 )\)
AQA Further Paper 3 Statistics 2024 June Q6
9 marks Standard +0.3
6 Over time it has been accepted that the mean retirement age for professional baseball players is 29.5 years old. Imran claims that the mean retirement age is no longer 29.5 years old.
He takes a random sample of 5 recently retired professional baseball players and records their retirement ages, \(x\). The results are $$\sum x = 152.1 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 7.81$$ 6
  1. State an assumption that you should make about the distribution of the retirement ages to investigate Imran's claim. 6
  2. Investigate Imran's claim, using the 10\% level of significance.
AQA Further Paper 3 Statistics 2024 June Q8
5 marks Moderate -0.3
8
16
256 2 The random variable \(T\) has an exponential distribution with mean 2 Find \(\mathrm { P } ( T \leq 1.4 )\) Circle your answer. \(\mathrm { e } ^ { - 2.8 }\) \(\mathrm { e } ^ { - 0.7 }\) \(1 - e ^ { - 0.7 }\) \(1 - \mathrm { e } ^ { - 2.8 }\) The continuous random variable \(Y\) has cumulative distribution function $$\mathrm { F } ( y ) = \left\{ \begin{array} { l r } 0 & y < 2 \\ - \frac { 1 } { 9 } y ^ { 2 } + \frac { 10 } { 9 } y - \frac { 16 } { 9 } & 2 \leq y < 5 \\ 1 & y \geq 5 \end{array} \right.$$ Find the median of \(Y\) Circle your answer. 2 \(\frac { 10 - 3 \sqrt { 2 } } { 2 }\) \(\frac { 7 } { 2 }\) \(\frac { 10 + 3 \sqrt { 2 } } { 2 }\) Turn over for the next question 4 Research has shown that the mean number of volcanic eruptions on Earth each day is 20 Sandra records 162 volcanic eruptions during a period of one week. Sandra claims that there has been an increase in the mean number of volcanic eruptions per week. Test Sandra's claim at the \(5 \%\) level of significance.
5 The continuous random variable \(X\) has probability density function $$f ( x ) = \begin{cases} \frac { 1 } { 6 } e ^ { \frac { x } { 3 } } & 0 \leq x \leq \ln 27 \\ 0 & \text { otherwise } \end{cases}$$ Show that the mean of \(X\) is \(\frac { 3 } { 2 } ( \ln 27 - 2 )\) 6 Over time it has been accepted that the mean retirement age for professional baseball players is 29.5 years old. Imran claims that the mean retirement age is no longer 29.5 years old.
He takes a random sample of 5 recently retired professional baseball players and records their retirement ages, \(x\). The results are $$\sum x = 152.1 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 7.81$$ 6
  1. State an assumption that you should make about the distribution of the retirement ages to investigate Imran's claim. 6
  2. Investigate Imran's claim, using the 10\% level of significance.
AQA Further Paper 3 Statistics 2024 June Q9
11 marks Standard +0.3
9 A company owns three shops, A, B and C, which are based in different towns. Each shop gives a questionnaire to 250 of their customers, and every customer completes the questionnaire. One of the questions asks whether the customer rates the shop as good, satisfactory or poor. For shop A, 26\% of customers rate the shop as good and 38\% of customers rate the shop as poor. For shop B, 32\% of customers rate the shop as good and 40\% of customers rate the shop as satisfactory. Altogether, there are 210 good ratings and 261 satisfactory ratings. 9
  1. Complete the following table with the observed frequencies.
    \multirow{2}{*}{}Rating
    GoodSatisfactoryPoor
    \multirow{3}{*}{Shop}A
    B
    C
    9
  2. Carry out a test for association between shop and rating, using the 1\% level of significance.
AQA Further Paper 3 Statistics 2024 June Q16
Moderate -0.8
16
256 2 The random variable \(T\) has an exponential distribution with mean 2 Find \(\mathrm { P } ( T \leq 1.4 )\) Circle your answer. \(\mathrm { e } ^ { - 2.8 }\) \(\mathrm { e } ^ { - 0.7 }\) \(1 - e ^ { - 0.7 }\) \(1 - \mathrm { e } ^ { - 2.8 }\) The continuous random variable \(Y\) has cumulative distribution function $$\mathrm { F } ( y ) = \left\{ \begin{array} { l r } 0 & y < 2 \\ - \frac { 1 } { 9 } y ^ { 2 } + \frac { 10 } { 9 } y - \frac { 16 } { 9 } & 2 \leq y < 5 \\ 1 & y \geq 5 \end{array} \right.$$ Find the median of \(Y\) Circle your answer. 2 \(\frac { 10 - 3 \sqrt { 2 } } { 2 }\) \(\frac { 7 } { 2 }\) \(\frac { 10 + 3 \sqrt { 2 } } { 2 }\) Turn over for the next question 4 Research has shown that the mean number of volcanic eruptions on Earth each day is 20 Sandra records 162 volcanic eruptions during a period of one week. Sandra claims that there has been an increase in the mean number of volcanic eruptions per week. Test Sandra's claim at the \(5 \%\) level of significance.
5 The continuous random variable \(X\) has probability density function $$f ( x ) = \begin{cases} \frac { 1 } { 6 } e ^ { \frac { x } { 3 } } & 0 \leq x \leq \ln 27 \\ 0 & \text { otherwise } \end{cases}$$ Show that the mean of \(X\) is \(\frac { 3 } { 2 } ( \ln 27 - 2 )\) 6 Over time it has been accepted that the mean retirement age for professional baseball players is 29.5 years old. Imran claims that the mean retirement age is no longer 29.5 years old.
He takes a random sample of 5 recently retired professional baseball players and records their retirement ages, \(x\). The results are $$\sum x = 152.1 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 7.81$$ 6
  1. State an assumption that you should make about the distribution of the retirement ages to investigate Imran's claim. 6
  2. Investigate Imran's claim, using the 10\% level of significance.
AQA Further Paper 3 Mechanics 2019 June Q1
1 marks Moderate -0.8
1 A spring has natural length 0.4 metres and modulus of elasticity 55 N
Calculate the elastic potential energy stored in the spring when the extension of the spring is 0.08 metres. Circle your answer. \(0.176 \mathrm {~J} \quad 0.44 \mathrm {~J} \quad 0.88 \mathrm {~J} \quad 1.76 \mathrm {~J}\)
AQA Further Paper 3 Mechanics 2019 June Q2
1 marks Easy -1.8
2 A particle has an angular speed of 72 revolutions per minute.
Find the angular speed in radians per second.
Circle your answer.
[0pt] [1 mark] \(\frac { 6 \pi } { 5 } \quad \frac { 12 \pi } { 5 } \quad 12 \pi \quad 24 \pi\)
AQA Further Paper 3 Mechanics 2019 June Q3
3 marks Standard +0.3
3 A disc, of mass \(m\) and radius \(r\), rotates about an axis through its centre, perpendicular to the plane face of the disc. The angular speed of the disc is \(\omega\).
A possible model for the kinetic energy \(E\) of the disc is $$E = k m ^ { a } r ^ { b } \omega ^ { c }$$ where \(a , b\) and \(c\) are constants and \(k\) is a dimensionless constant.
Find the values of \(a , b\) and \(c\).
AQA Further Paper 3 Mechanics 2019 June Q4
8 marks Moderate -0.3
4 In this question use \(g = 10 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) An inelastic string has length 1.2 metres.
One end of the string is attached to a fixed point \(O\).
A sphere, of mass 500 grams, is attached to the other end of the string.
The sphere is held, with the string taut and at an angle of \(20 ^ { \circ }\) to the vertical, touching the chin of a student, as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{f2470caa-0f73-4ec1-b08f-525c02ed2e67-04_739_511_676_762} The sphere is released from rest.
Assume that the student stays perfectly still once the sphere has been released.
4
  1. Calculate the maximum speed of the sphere.
    4
AQA Further Paper 3 Mechanics 2019 June Q5
11 marks Standard +0.8
5 The triangular region shown below is rotated through \(360 ^ { \circ }\) around the \(x\)-axis, to form a solid cone. \includegraphics[max width=\textwidth, alt={}, center]{f2470caa-0f73-4ec1-b08f-525c02ed2e67-06_328_755_415_644} The coordinates of the vertices of the triangle are \(( 0,0 ) , ( 8,0 )\) and \(( 0,4 )\).
All units are in centimetres. 5
  1. State an assumption that you should make about the cone in order to find the position of its centre of mass. 5
  2. Using integration, prove that the centre of mass of the cone is 2 cm from its plane face.
    5
  3. The cone is placed with its plane face on a rough board. One end of the board is lifted so that the angle between the board and the horizontal is gradually increased. Eventually the cone topples without sliding. 5 (c) (i) Find the angle between the board and the horizontal when the cone topples, giving your answer to the nearest degree. 5 (c) (ii) Find the range of possible values for the coefficient of friction between the cone and the board.
AQA Further Paper 3 Mechanics 2019 June Q6
6 marks Standard +0.3
6 A ball moving on a smooth horizontal surface collides with a fixed vertical wall. Before the collision, the ball moves with speed \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and at an angle of \(40 ^ { \circ }\) to the wall. After the collision, the ball moves with speed \(5 \mathrm {~ms} ^ { - 1 }\) and at an angle of \(26 ^ { \circ }\) to the wall. Model the ball as a particle.
6
  1. Find the coefficient of restitution between the ball and the wall, giving your answer correct to two significant figures.
    6
  2. Determine whether or not the wall is smooth. Fully justify your answer.
AQA Further Paper 3 Mechanics 2019 June Q7
9 marks Challenging +1.8
7 A particle of mass 2.5 kilograms is attached to one end of a light, inextensible string of length 75 cm . The other end of this string is attached to a point \(A\). The particle is also attached to one end of an elastic string of natural length 30 cm and modulus of elasticity \(\lambda \mathrm { N }\). The other end of this string is attached to a point \(B\), which is 60 cm vertically below \(A\). The particle is set in motion so that it describes a horizontal circle with centre \(B\). The angular speed of the particle is \(8 \mathrm { rad } \mathrm { s } { } ^ { - 1 }\) Find \(\lambda\), giving your answer in terms of \(g\).
AQA Further Paper 3 Mechanics 2019 June Q8
11 marks Challenging +1.8
8 In this question use \(g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) A 'reverse' bungee jump consists of two identical elastic ropes. One end of each elastic rope is attached to either side of the top of a gorge. The other ends are both attached to Hannah, who has mass 84 kg
Hannah is modelled as a particle, as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{f2470caa-0f73-4ec1-b08f-525c02ed2e67-12_467_844_678_598} The depth of the gorge is 50 metres and the width of the gorge is 40 metres.
Each elastic rope has natural length 30 metres and modulus of elasticity 3150 N
Hannah is released from rest at the centre of the bottom of the gorge.
8
  1. Show that the speed of Hannah when the ropes become slack is \(30 \mathrm {~ms} ^ { - 1 }\) correct to two significant figures.
    8
  2. Determine whether Hannah is moving up or down when the ropes become taut again. [5 marks] \includegraphics[max width=\textwidth, alt={}, center]{f2470caa-0f73-4ec1-b08f-525c02ed2e67-14_2492_1721_217_150} 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. 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.
AQA Further Paper 3 Mechanics 2020 June Q1
1 marks Moderate -0.8
1 A rigid rod, \(A B\), has mass 2 kg and length 4 metres.
Two particles of masses 5 kg and 3 kg are fixed to \(A\) and \(B\) respectively to create a composite body, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{b0d0c552-71cb-4e5a-b545-de8a9052def0-02_120_730_769_653} Find the distance of the centre of mass of the composite body from \(B\). Circle your answer.
1.5 metres
1.6 metres
2.4 metres
2.5 metres
AQA Further Paper 3 Mechanics 2020 June Q2
1 marks Standard +0.3
2 The tension, \(T\) newtons, in a spring is given by \(T = 20 e\), where \(e\) metres is the extension of the spring. Calculate the work done when the extension is increased from 0.2 metres to 0.4 metres. Circle your answer.
[0pt] [1 mark]
0.4 J 0.9 J 1.2 J 1.6 J
AQA Further Paper 3 Mechanics 2020 June Q3
2 marks Easy -1.8
3 The speed, \(v\), of a particle moving in a horizontal circle is given by the formula \(v = r \omega\) where: \(v =\) speed \(r =\) radius \(\omega =\) angular speed.
Show that the dimensions of angular speed are \(T ^ { - 1 }\) [0pt] [2 marks]
AQA Further Paper 3 Mechanics 2020 June Q4
8 marks Standard +0.3
4 A car has mass 1000 kg and travels on a straight horizontal road. The maximum speed of the car on this road is \(48 \mathrm {~ms} ^ { - 1 }\) In a simple model, it is assumed that the car experiences a resistance force that is proportional to its speed. When the car travels at \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the magnitude of the resistance force is 600 newtons. 4
  1. Show that the maximum power of the car is 69120 W
    4
  2. Find the maximum acceleration of the car when it is travelling at \(25 \mathrm {~ms} ^ { - 1 }\) 4
  3. Find the maximum acceleration of the car when it is travelling at \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) 4
  4. Comment on the validity of the model in the context of your answers to parts (b) and (c).
AQA Further Paper 3 Mechanics 2020 June Q5
17 marks Standard +0.8
5 A ball, of mass 0.3 kg , is moving on a smooth horizontal surface. The ball collides with a smooth fixed vertical wall and rebounds.
Before the ball hits the wall, the ball is moving at \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(30 ^ { \circ }\) to the wall as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{b0d0c552-71cb-4e5a-b545-de8a9052def0-06_634_268_584_886} The magnitude of the force, \(F\) newtons, exerted on the ball by the wall at time \(t\) seconds is modelled by $$F = k t ^ { 2 } ( 0.1 - t ) ^ { 2 } \quad \text { for } \quad 0 \leq t \leq 0.1$$ where \(k\) is a constant. The ball is in contact with the wall for 0.1 seconds.
\includegraphics[max width=\textwidth, alt={}]{b0d0c552-71cb-4e5a-b545-de8a9052def0-07_2484_1709_219_153}
5 (b) Explain why \(1800000 < k \leq 3600000\) Fully justify your answer.
5 (c) Given that \(k = 2400000\) Find the speed of the ball after the collision with the wall.
[0pt] [4 marks]
AQA Further Paper 3 Mechanics 2020 June Q6
9 marks Standard +0.3
6 A particle moves with constant speed on a circular path of radius 2 metres. The centre of the circle has position vector \(2 \mathbf { j }\) metres.
At time \(t = 0\), the particle is at the origin and is moving in the positive \(\mathbf { i }\) direction.
The particle returns to the origin every 4 seconds.
The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular.
6
  1. Calculate the angular speed of the particle.
    6
  2. Write down an expression for the position vector of the particle at time \(t\) seconds.
    6
  3. Find an expression for the acceleration of the particle at time \(t\) seconds.
    6
  4. State the magnitude of the acceleration of the particle.
    6
  5. State the time when the acceleration is first directed towards the origin.
AQA Further Paper 3 Mechanics 2020 June Q7
8 marks Standard +0.3
7 In this question use \(g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) A box, of mass 8 kg , is on a rough horizontal surface.
A string attached to the box is used to pull it along the surface.
The string is inclined at an angle of \(40 ^ { \circ }\) above the horizontal.
The tension in the string is 50 newtons.
As the box moves a distance of \(x\) metres, its speed increases from \(2 \mathrm {~ms} ^ { - 1 }\) to \(5 \mathrm {~ms} ^ { - 1 }\) The coefficient of friction between the box and the surface is 0.4
7
  1. By using an energy method, find \(x\).
    7
  2. Describe how the model could be refined to obtain a more realistic value of \(x\) and use an energy argument to explain whether this would increase or decrease the value of \(x\).
AQA Further Paper 3 Mechanics 2020 June Q8
8 marks Challenging +1.8
8 A ladder has length 4 metres and mass 20 kg The ladder rests in equilibrium with one end on a horizontal surface and the ladder resting on the top of a vertical wall. In this position the ladder is on the point of slipping.
The top of the wall is 1.5 metres above the horizontal surface.
The angle between the ladder and the horizontal surface is \(\alpha\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{b0d0c552-71cb-4e5a-b545-de8a9052def0-14_362_863_804_593} The coefficient of friction between the ladder and the wall is 0.5
The coefficient of friction between the ladder and the ground is also 0.5
Show that $$\cos \alpha \sin ^ { 2 } \alpha = \frac { 3 } { 10 }$$ stating clearly any assumptions you make. \includegraphics[max width=\textwidth, alt={}, center]{b0d0c552-71cb-4e5a-b545-de8a9052def0-16_2490_1735_219_139}
AQA Further Paper 3 Mechanics 2022 June Q1
1 marks Easy -1.2
1 The graph shows how a force, \(F\) newtons, varies during a 5 second period of time. \includegraphics[max width=\textwidth, alt={}, center]{0afe3ff2-0af5-4aeb-98c5-1346fa803388-02_575_1182_680_429} Calculate the magnitude of the impulse of the force.
Circle your answer.
[0pt] [1 mark]
17.5 N s
25 Ns
35 Ns
70 Ns