Questions — Edexcel (10514 questions)

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Edexcel FS2 AS 2024 June Q1
8 marks Standard +0.3
  1. A continuous random variable \(X\) has cumulative distribution function \(\mathrm { F } ( x )\) given by
$$\mathrm { F } ( x ) = \left\{ \begin{array} { c r } 0 & x < - 1 \\ \frac { 1 } { 5 } ( x + 1 ) ^ { 2 } & - 1 \leqslant x \leqslant 0 \\ 1 - \frac { 1 } { 20 } ( 4 - x ) ^ { 2 } & 0 < x \leqslant 4 \\ 1 & x > 4 \end{array} \right.$$
  1. Find the probability density function, \(\mathrm { f } ( x )\)
    1. Sketch \(\mathrm { f } ( x )\)
    2. Hence describe the skewness of the distribution.
  3. Find, to 3 significant figures, the value of \(c\) such that $$\mathrm { P } ( 1 < X < c ) = \mathrm { P } ( c < X < 2 )$$
Edexcel FS2 AS 2024 June Q2
7 marks Standard +0.3
  1. A random sample of size \(n = 8\) of paired data is taken from a population. The data are plotted below. \includegraphics[max width=\textwidth, alt={}, center]{ba41c616-0805-4466-81b8-b985b0bdd94b-06_572_983_335_541}
Test, at the \(1 \%\) level of significance, whether or not there is evidence of a negative rank correlation between the two variables. You should state your hypotheses and critical value and show your working clearly.
Edexcel FS2 AS 2024 June Q3
9 marks Standard +0.3
  1. The continuous random variable \(Y\) has probability density function
$$f ( y ) = \left\{ \begin{array} { c c } \frac { 1 } { 24 } ( y + 2 ) ( 4 - y ) & 0 \leqslant y \leqslant 3 \\ 0 & \text { otherwise } \end{array} \right.$$
  1. Show that the mode of \(Y\) is 1 , justifying your reasoning. Given that \(\mathrm { P } ( Y < 1 ) = \frac { 13 } { 36 }\)
  2. determine whether the median of \(Y\) is less than, equal to, or greater than 2 Give a reason for your answer. Given that \(\mathrm { E } \left( Y ^ { 2 } \right) = \frac { 213 } { 80 }\)
  3. find, using algebraic integration, \(\operatorname { Var } ( 2 Y )\)
Edexcel FS2 AS 2024 June Q4
8 marks Standard +0.3
The continuous random variable \(X\) is uniformly distributed over the interval [2, 7]
  1. Write down the value of \(\mathrm { E } ( X )\)
  2. Find \(\mathrm { P } ( 1 < X < 4 )\)
  3. Find \(\mathrm { P } \left( 2 X ^ { 2 } - 15 X + 27 > 0 \right)\)
  4. Find \(\mathrm { E } \left( \frac { 3 } { X ^ { 2 } } \right)\)
Edexcel FS2 AS 2024 June Q5
8 marks Standard +0.3
  1. A random sample of 24 adults is taken. The height, \(h\) metres, and the arm span, \(s\) metres, for each adult are recorded.
These data are summarised below. $$\mathrm { S } _ { h h } = 0.377 \quad \mathrm {~S} _ { s h } = 0.352 \quad \bar { s } = 1.70 \quad \bar { h } = 1.68$$ The least squares regression line of \(h\) on \(s\) is $$h = a + 0.919 s$$ where \(a\) is a constant.
  1. Calculate the product moment correlation coefficient. A doctor uses the least squares regression line of \(h\) on \(s\) as a model to predict a person's height based on their arm span.
  2. Use the model to predict the height of an adult with arm span 1.79 metres. Ewan has an arm span of 1.70 metres and a height of 1.75 metres. His information is added to the sample as the 25th adult.
  3. Explain how the gradient of the regression line for the sample of 25 adults compares with the gradient of the regression line for the original sample of 24 adults.
    Give a reason for your answer.
Edexcel FS2 AS Specimen Q1
10 marks Standard +0.3
  1. In a gymnastics competition, two judges scored each of 8 competitors on the vault.
CompetitorABCDEFGH
J udge 1's scores4.69.18.48.89.09.59.29.4
J udge 2's scores7.88.88.68.59.19.69.09.3
  1. Calculate Spearman's rank correlation coefficient for these data.
  2. Stating your hypotheses clearly, test at the \(1 \%\) level of significance, whether or not the two judges are generally in agreement.
  3. Give a reason to support the use of Spearman's rank correlation coefficient in this case. The judges also scored the competitors on the beam.
    Spearman's rank correlation coefficient for their ranks on the beam was found to be 0.952
  4. Compare the judges' ranks on the vault with their ranks on the beam.
Edexcel FS2 AS Specimen Q2
11 marks Moderate -0.3
  1. The continuous random variable \(X\) has probability density function
$$f ( x ) = \begin{cases} \frac { 1 } { 18 } ( 11 - 2 x ) & 1 \leqslant x \leqslant 4 \\ 0 & \text { otherwise } \end{cases}$$
  1. Find \(\mathrm { P } ( \mathrm { X } < 3 )\)
  2. State, giving a reason, whether the upper quartile of \(X\) is greater than 3, less than 3 or equal to 3 Given that \(\mathrm { E } ( \mathrm { X } ) = \frac { 9 } { 4 }\)
  3. use algebraic integration to find \(\operatorname { Var } ( \mathrm { X } )\) The cumulative distribution function of \(X\) is given by $$F ( x ) = \left\{ \begin{array} { l r } 0 & x < 1 \\ \frac { 1 } { 18 } \left( 11 x - x ^ { 2 } + c \right) & 1 \leqslant x \leqslant 4 \\ 1 & x > 4 \end{array} \right.$$
  4. Show that \(\mathrm { c } = - 10\)
  5. Find the median of \(X\), giving your answer to 3 significant figures. \section*{Q uestion 2 continued}
Edexcel FS2 AS Specimen Q3
11 marks Standard +0.3
  1. A scientist wants to develop a model to describe the relationship between the average daily temperature, \(\mathrm { x } ^ { \circ } \mathrm { C }\), and a household's daily energy consumption, ykWh , in winter.
A random sample of the average temperature and energy consumption are taken from 10 winter days and are summarised below. $$\begin{gathered} \sum x = 12 \quad \sum x ^ { 2 } = 24.76 \quad \sum y = 251 \quad \sum y ^ { 2 } = 6341 \quad \sum x y = 284.8 \\ S _ { x x } = 10.36 \quad S _ { y y } = 40.9 \end{gathered}$$
  1. Find the product moment correlation coefficient between y and x .
  2. Find the equation of the regression line of \(y\) on \(x\) in the form \(y = a + b x\)
  3. Use your equation to estimate the daily energy consumption when the average daily temperature is \(2 ^ { \circ } \mathrm { C }\)
  4. Calculate the residual sum of squares (RSS). The table shows the residual for each value of x .
    \(\mathbf { x }\)- 0.4- 0.20.30.81.11.41.82.12.52.6
    R esidual- 0.63- 0.32- 0.52- 0.730.742.221.840.32\(f\)- 1.88
  5. Find the value of f.
  6. By considering the signs of the residuals, explain whether or not the linear regression model is a suitable model for these data.
Edexcel FS2 AS Specimen Q4
8 marks Standard +0.3
The continuous random variable \(X\) is uniformly distributed over the interval \([ - 3,5 ]\).
  1. Sketch the probability density function \(\mathrm { f } ( \mathrm { x } )\) of X .
  2. Find the value of k such that \(\mathrm { P } ( \mathrm { X } < 2 [ \mathrm { k } - \mathrm { X } ] ) = 0.25\)
  3. Use algebraic integration to show that \(\mathrm { E } \left( \mathrm { X } ^ { 3 } \right) = 17\)
Edexcel FM1 AS 2018 June Q1
8 marks Moderate -0.5
  1. A small ball of mass 0.3 kg is released from rest from a point 3.6 m above horizontal ground. The ball falls freely under gravity, hits the ground and rebounds vertically upwards.
In the first impact with the ground, the ball receives an impulse of magnitude 4.2 Ns . The ball is modelled as a particle.
  1. Find the speed of the ball immediately after it first hits the ground.
  2. Find the kinetic energy lost by the ball as a result of the impact with the ground.
Edexcel FM1 AS 2018 June Q2
9 marks Standard +0.3
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{cfa9b998-d57d-4980-9316-1bddeac55b90-04_267_891_346_687} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a ramp inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 2 } { 7 }\) A parcel of mass 4 kg is projected, with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), from a point \(A\) on the ramp.
The parcel moves up a line of greatest slope of the ramp and first comes to instantaneous rest at the point \(B\), where \(A B = 2.5 \mathrm {~m}\).
The parcel is modelled as a particle.
The total resistance to the motion of the parcel from non-gravitational forces is modelled as a constant force of magnitude \(R\) newtons.
  1. Use the work-energy principle to show that \(R = 8.8\) After coming to instantaneous rest at \(B\), the parcel slides back down the ramp. The total resistance to the motion of the particle is modelled as a constant force of magnitude 8.8N.
  2. Find the speed of the parcel at the instant it returns to \(A\).
  3. Suggest two improvements that could be made to the model.
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Edexcel FM1 AS 2018 June Q3
9 marks Standard +0.3
  1. A van of mass 750 kg is moving along a straight horizontal road. At the instant when the van is moving at \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the resistance to the motion of the van is modelled as a force of magnitude \(\lambda \nu \mathrm { N }\), where \(\lambda\) is a constant.
The engine of the van is working at a constant rate of 18 kW .
At the instant when \(v = 15\), the acceleration of the van is \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 2 }\)
  1. Show that \(\lambda = 50\) The van now moves up a straight road inclined at an angle to the horizontal, where \(\sin \alpha = \frac { 1 } { 15 }\) At the instant when the van is moving at \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the resistance to the motion of the van from non-gravitational forces is modelled as a force of magnitude 50 v . When the engine of the van is working at a constant rate of 12 kW , the van is moving at a constant speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
  2. Find the value of \(V\).
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Edexcel FM1 AS 2018 June Q4
14 marks Standard +0.8
  1. A particle \(P\) of mass \(3 m\) is moving in a straight line on a smooth horizontal floor. A particle \(Q\) of mass \(5 m\) is moving in the opposite direction to \(P\) along the same straight line.
The particles collide directly.
Immediately before the collision, the speed of \(P\) is \(2 u\) and the speed of \(Q\) is \(u\). The coefficient of restitution between \(P\) and \(Q\) is \(e\).
  1. Show that the speed of \(Q\) immediately after the collision is \(\frac { u } { 8 } ( 9 e + 1 )\)
  2. Find the range of values of \(e\) for which the direction of motion of \(P\) is not changed as a result of the collision. When \(P\) and \(Q\) collide they are at a distance \(d\) from a smooth fixed vertical wall, which is perpendicular to their direction of motion. After the collision with \(P\), particle \(Q\) collides directly with the wall and rebounds so that there is a second collision between \(P\) and \(Q\). This second collision takes place at a distance \(x\) from the wall. Given that \(e = \frac { 1 } { 18 }\) and the coefficient of restitution between \(Q\) and the wall is \(\frac { 1 } { 3 }\)
  3. find \(x\) in terms of \(d\).
Edexcel FM1 AS 2019 June Q1
10 marks Standard +0.3
  1. A lorry of mass 16000 kg moves along a straight horizontal road.
The lorry moves at a constant speed of \(25 \mathrm {~ms} ^ { - 1 }\) In an initial model for the motion of the lorry, the resistance to the motion of the lorry is modelled as having constant magnitude 16000 N .
  1. Show that the engine of the lorry is working at a rate of 400 kW . The model for the motion of the lorry along the same road is now refined so that when the speed of the lorry along the same road is \(V \mathrm {~ms} ^ { - 1 }\), the resistance to the motion of the lorry is modelled as having magnitude 640 V newtons. Assuming that the engine of the lorry is working at the same rate of 400 kW
  2. use the refined model to find the speed of the lorry when it is accelerating at \(2.1 \mathrm {~ms} ^ { - 2 }\)
Edexcel FM1 AS 2019 June Q2
13 marks Standard +0.3
Two particles, \(A\) and \(B\), of masses \(2 m\) and \(3 m\) respectively, are moving on a smooth horizontal plane. The particles are moving in opposite directions towards each other along the same straight line when they collide directly. Immediately before the collision the speed of \(A\) is \(2 u\) and the speed of \(B\) is \(u\). In the collision the impulse of \(A\) on \(B\) has magnitude 5 mu .
  1. Find the coefficient of restitution between \(A\) and \(B\).
  2. Find the total loss in kinetic energy due to the collision.
Edexcel FM1 AS 2019 June Q3
7 marks Standard +0.3
  1. A particle, \(P\), of mass \(m \mathrm {~kg}\) is projected with speed \(5 \mathrm {~ms} ^ { - 1 }\) down a line of greatest slope of a rough plane. The plane is inclined to the horizontal at an angle \(\alpha\), where \(\sin \alpha = \frac { 3 } { 5 }\) The total resistance to the motion of \(P\) is a force of magnitude \(\frac { 1 } { 5 } m g\) Use the work-energy principle to find the speed of \(P\) at the instant when it has moved a distance 8 m down the plane from the point of projection.
Edexcel FM1 AS 2019 June Q4
10 marks Challenging +1.8
  1. Three particles, \(P , Q\) and \(R\), are at rest on a smooth horizontal plane. The particles lie along a straight line with \(Q\) between \(P\) and \(R\). The particles \(Q\) and \(R\) have masses \(m\) and \(k m\) respectively, where \(k\) is a constant.
Particle \(Q\) is projected towards \(R\) with speed \(u\) and the particles collide directly.
The coefficient of restitution between each pair of particles is \(e\).
  1. Find, in terms of \(e\), the range of values of \(k\) for which there is a second collision. Given that the mass of \(P\) is \(k m\) and that there is a second collision,
  2. write down, in terms of \(u , k\) and \(e\), the speed of \(Q\) after this second collision.
Edexcel FM1 AS 2020 June Q1
5 marks Standard +0.3
Two particles \(P\) and \(Q\) have masses \(m\) and \(4 m\) respectively. The particles are at rest on a smooth horizontal plane. Particle \(P\) is given a horizontal impulse, of magnitude \(I\), in the direction \(P Q\). Particle \(P\) then collides directly with \(Q\). Immediately after this collision, \(P\) is at rest and \(Q\) has speed \(w\). The coefficient of restitution between the particles is \(e\).
  1. Find \(I\) in terms of \(m\) and \(w\).
  2. Show that \(e = \frac { 1 } { 4 }\)
  3. Find, in terms of \(m\) and \(w\), the total kinetic energy lost in the collision between \(P\) and \(Q\).
Edexcel FM1 AS 2020 June Q2
12 marks Standard +0.3
  1. A car of mass 1000 kg moves along a straight horizontal road.
In all circumstances, when the speed of the car is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the resistance to the motion of the car is modelled as a force of magnitude \(c v ^ { 2 } \mathrm {~N}\), where \(c\) is a constant. The maximum power that can be developed by the engine of the car is 50 kW .
At the instant when the speed of the car is \(72 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) and the engine is working at its maximum power, the acceleration of the car is \(2.25 \mathrm {~m} \mathrm {~s} ^ { - 2 }\)
  1. Convert \(72 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) into \(\mathrm { m } \mathrm { s } ^ { - 1 }\)
  2. Find the acceleration of the car at the instant when the speed of the car is \(144 \mathrm { kmh } ^ { - 1 }\) and the engine is working at its maximum power. The maximum speed of the car when the engine is working at its maximum power is \(V \mathrm {~km} \mathrm {~h} ^ { - 1 }\).
  3. Find, to the nearest whole number, the value of \(V\).
Edexcel FM1 AS 2020 June Q3
12 marks Standard +0.8
  1. Three particles \(A , B\) and \(C\) are at rest on a smooth horizontal plane. The particles lie along a straight line with \(B\) between \(A\) and \(C\).
Particle \(B\) has mass \(4 m\) and particle \(C\) has mass \(k m\), where \(k\) is a positive constant. Particle \(B\) is projected with speed \(u\) along the plane towards \(C\) and they collide directly. The coefficient of restitution between \(B\) and \(C\) is \(\frac { 1 } { 4 }\)
  1. Find the range of values of \(k\) for which there would be no further collisions. The magnitude of the impulse on \(B\) in the collision between \(B\) and \(C\) is \(3 m u\)
  2. Find the value of \(k\).
Edexcel FM1 AS 2020 June Q4
11 marks Standard +0.3
A small ball, of mass \(m\), is thrown vertically upwards with speed \(\sqrt { 8 g H }\) from a point \(O\) on a smooth horizontal floor. The ball moves towards a smooth horizontal ceiling that is a vertical distance \(H\) above \(O\). The coefficient of restitution between the ball and the ceiling is \(\frac { 1 } { 2 }\) In a model of the motion of the ball, it is assumed that the ball, as it moves up or down, is subject to air resistance of constant magnitude \(\frac { 1 } { 2 } \mathrm { mg }\).
Using this model,
  1. use the work-energy principle to find, in terms of \(g\) and \(H\), the speed of the ball immediately before it strikes the ceiling,
  2. find, in terms of \(g\) and \(H\), the speed of the ball immediately before it strikes the floor at \(O\) for the first time. In a simplified model of the motion of the ball, it is assumed that the ball, as it moves up or down, is subject to no air resistance. Using this simplified model,
  3. explain, without any detailed calculation, why the speed of the ball, immediately before it strikes the floor at \(O\) for the first time, would still be less than \(\sqrt { 8 g H }\)
Edexcel FM1 AS 2021 June Q1
7 marks Standard +0.3
1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{05f6f654-05e5-41d5-a6e4-11cd91a6df83-02_826_700_244_550} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A small book of mass \(m\) is held on a rough straight desk lid which is inclined at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\). The book is released from rest at a distance of 0.5 m from the edge of the desk lid, as shown in Figure 1. The book slides down the desk lid and then hits the floor that is 0.8 m below the edge of the desk lid. The coefficient of friction between the book and the desk lid is 0.4 The book is modelled as a particle which, after leaving the desk lid, is assumed to move freely under gravity.
  1. Find, in terms of \(m\) and \(g\), the magnitude of the normal reaction on the book as it slides down the desk lid.
  2. Use the work-energy principle to find the speed of the book as it hits the floor.
Edexcel FM1 AS 2021 June Q2
9 marks Standard +0.8
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{05f6f654-05e5-41d5-a6e4-11cd91a6df83-06_458_278_248_986} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A particle of mass em is at rest on a smooth horizontal plane between two smooth fixed parallel vertical walls, as shown in the plan view in Figure 2. The particle is projected along the plane with speed \(u\) towards one of the walls and strikes the wall at right angles. The coefficient of restitution between the particle and each wall is \(e\) and air resistance is modelled as being negligible. Using the model,
  1. find, in terms of \(m , u\) and \(e\), an expression for the total loss in the kinetic energy of the particle as a result of the first two impacts. Given that \(e\) can vary such that \(0 < e < 1\) and using the model,
  2. find the value of \(e\) for which the total loss in the kinetic energy of the particle as a result of the first two impacts is a maximum,
  3. describe the subsequent motion of the particle.
Edexcel FM1 AS 2021 June Q3
11 marks Standard +0.3
  1. The total mass of a cyclist and his bicycle is 100 kg .
In all circumstances, the magnitude of the resistance to the motion of the cyclist from non-gravitational forces is modelled as being \(k v ^ { 2 } \mathrm {~N}\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of the cyclist. The cyclist can freewheel, without pedalling, down a slope that is inclined to the horizontal at an angle \(\alpha\), where \(\sin \alpha = \frac { 1 } { 35 }\), at a constant speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) When he is pedalling up a slope that is inclined to the horizontal at an angle \(\beta\), where \(\sin \beta = \frac { 1 } { 70 }\), and he is moving at the same constant speed \(V \mathrm {~ms} ^ { - 1 }\), he is working at a constant rate of \(P\) watts.
  1. Find \(P\) in terms of \(V\). If he pedals and works at a rate of 35 V watts on a horizontal road, he moves at a constant speed of \(U \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
  2. Find \(U\) in terms of \(V\).
Edexcel FM1 AS 2021 June Q4
13 marks Standard +0.8
  1. Two particles, \(P\) and \(Q\), have masses \(m\) and \(e m\) respectively. The particles are moving on a smooth horizontal plane in the same direction along the same straight line when they collide directly. The coefficient of restitution between \(P\) and \(Q\) is \(e\), where \(0 < e < 1\)
Immediately before the collision the speed of \(P\) is \(u\) and the speed of \(Q\) is \(e u\).
  1. Show that the speed of \(Q\) immediately after the collision is \(u\).
  2. Show that the direction of motion of \(P\) is unchanged by the collision. The magnitude of the impulse on \(Q\) in the collision is \(\frac { 2 } { 9 } m u\)
  3. Find the possible values of \(e\).