Questions — Edexcel M1 (663 questions)

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Edexcel M1 2008 June Q7
11 marks Standard +0.3
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9dbbbc01-fb66-460d-a42e-2c37ec8b451a-10_291_726_265_607} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} A package of mass 4 kg lies on a rough plane inclined at \(30 ^ { \circ }\) to the horizontal. The package is held in equilibrium by a force of magnitude 45 N acting at an angle of \(50 ^ { \circ }\) to the plane, as shown in Figure 3. The force is acting in a vertical plane through a line of greatest slope of the plane. The package is in equilibrium on the point of moving up the plane. The package is modelled as a particle. Find
  1. the magnitude of the normal reaction of the plane on the package,
  2. the coefficient of friction between the plane and the package.
Edexcel M1 2008 June Q8
15 marks Standard +0.3
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9dbbbc01-fb66-460d-a42e-2c37ec8b451a-12_131_940_269_498} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Two particles \(P\) and \(Q\), of mass 2 kg and 3 kg respectively, are joined by a light inextensible string. Initially the particles are at rest on a rough horizontal plane with the string taut. A constant force \(\mathbf { F }\) of magnitude 30 N is applied to \(Q\) in the direction \(P Q\), as shown in Figure 4. The force is applied for 3 s and during this time \(Q\) travels a distance of 6 m . The coefficient of friction between each particle and the plane is \(\mu\). Find
  1. the acceleration of \(Q\),
  2. the value of \(\mu\),
  3. the tension in the string.
  4. State how in your calculation you have used the information that the string is inextensible. When the particles have moved for 3 s , the force \(\mathbf { F }\) is removed.
  5. Find the time between the instant that the force is removed and the instant that \(Q\) comes to rest.
Edexcel M1 2012 June Q1
6 marks Moderate -0.8
  1. Two particles \(A\) and \(B\), of mass \(5 m \mathrm {~kg}\) and \(2 m \mathrm {~kg}\) respectively, are moving in opposite directions along the same straight horizontal line. The particles collide directly. Immediately before the collision, the speeds of \(A\) and \(B\) are \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. The direction of motion of \(A\) is unchanged by the collision. Immediately after the collision, the speed of \(A\) is \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
    1. Find the speed of \(B\) immediately after the collision.
    In the collision, the magnitude of the impulse exerted on \(A\) by \(B\) is 3.3 N s .
  2. Find the value of \(m\).
Edexcel M1 2012 June Q2
7 marks Standard +0.3
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5c908e75-73df-46be-93bb-09dba2cb3b7e-03_215_716_233_614} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A non-uniform rod \(A B\) has length 3 m and mass 4.5 kg . The rod rests in equilibrium, in a horizontal position, on two smooth supports at \(P\) and at \(Q\), where \(A P = 0.8 \mathrm {~m}\) and \(Q B = 0.6 \mathrm {~m}\), as shown in Figure 1. The centre of mass of the rod is at \(G\). Given that the magnitude of the reaction of the support at \(P\) on the rod is twice the magnitude of the reaction of the support at \(Q\) on the rod, find
  1. the magnitude of the reaction of the support at \(Q\) on the rod,
  2. the distance \(A G\).
Edexcel M1 2012 June Q3
9 marks Standard +0.3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5c908e75-73df-46be-93bb-09dba2cb3b7e-04_432_780_210_584} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A box of mass 5 kg lies on a rough plane inclined at \(30 ^ { \circ }\) to the horizontal. The box is held in equilibrium by a horizontal force of magnitude 20 N , as shown in Figure 2. The force acts in a vertical plane containing a line of greatest slope of the inclined plane.
The box is in equilibrium and on the point of moving down the plane. The box is modelled as a particle. Find
  1. the magnitude of the normal reaction of the plane on the box,
  2. the coefficient of friction between the box and the plane.
Edexcel M1 2012 June Q4
13 marks Moderate -0.8
  1. A car is moving on a straight horizontal road. At time \(t = 0\), the car is moving with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is at the point \(A\). The car maintains the speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for 25 s . The car then moves with constant deceleration \(0.4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), reducing its speed from \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The car then moves with constant speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for 60 s . The car then moves with constant acceleration until it is moving with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the point \(B\).
    1. Sketch a speed-time graph to represent the motion of the car from \(A\) to \(B\).
    2. Find the time for which the car is decelerating.
    Given that the distance from \(A\) to \(B\) is 1960 m ,
  2. find the time taken for the car to move from \(A\) to \(B\).
Edexcel M1 2012 June Q5
12 marks Standard +0.3
  1. A particle \(P\) is projected vertically upwards from a point \(A\) with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The point \(A\) is 17.5 m above horizontal ground. The particle \(P\) moves freely under gravity until it reaches the ground with speed \(28 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
    1. Show that \(u = 21\)
    At time \(t\) seconds after projection, \(P\) is 19 m above \(A\).
  2. Find the possible values of \(t\). The ground is soft and, after \(P\) reaches the ground, \(P\) sinks vertically downwards into the ground before coming to rest. The mass of \(P\) is 4 kg and the ground is assumed to exert a constant resistive force of magnitude 5000 N on \(P\).
  3. Find the vertical distance that \(P\) sinks into the ground before coming to rest.
Edexcel M1 2012 June Q6
13 marks Moderate -0.8
6. [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal unit vectors due east and due north respectively and position vectors are given with respect to a fixed origin.] A ship \(S\) is moving with constant velocity \(( - 12 \mathbf { i } + 7.5 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\).
  1. Find the direction in which \(S\) is moving, giving your answer as a bearing. At time \(t\) hours after noon, the position vector of \(S\) is \(\mathbf { s } \mathrm { km }\). When \(t = 0 , \mathbf { s } = 40 \mathbf { i } - 6 \mathbf { j }\).
  2. Write down \(\mathbf { s }\) in terms of \(t\). A fixed beacon \(B\) is at the point with position vector \(( 7 \mathbf { i } + 12.5 \mathbf { j } ) \mathrm { km }\).
  3. Find the distance of \(S\) from \(B\) when \(t = 3\)
  4. Find the distance of \(S\) from \(B\) when \(S\) is due north of \(B\).
Edexcel M1 2012 June Q7
15 marks Moderate -0.3
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5c908e75-73df-46be-93bb-09dba2cb3b7e-12_150_1104_255_422} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Two particles \(P\) and \(Q\), of mass 0.3 kg and 0.5 kg respectively, are joined by a light horizontal rod. The system of the particles and the rod is at rest on a horizontal plane. At time \(t = 0\), a constant force \(\mathbf { F }\) of magnitude 4 N is applied to \(Q\) in the direction \(P Q\), as shown in Figure 3. The system moves under the action of this force until \(t = 6 \mathrm {~s}\). During the motion, the resistance to the motion of \(P\) has constant magnitude 1 N and the resistance to the motion of \(Q\) has constant magnitude 2 N . Find
  1. the acceleration of the particles as the system moves under the action of \(\mathbf { F }\),
  2. the speed of the particles at \(t = 6 \mathrm {~s}\),
  3. the tension in the rod as the system moves under the action of \(\mathbf { F }\). At \(t = 6 \mathrm {~s} , \mathbf { F }\) is removed and the system decelerates to rest. The resistances to motion are unchanged. Find
  4. the distance moved by \(P\) as the system decelerates,
  5. the thrust in the rod as the system decelerates.
Edexcel M1 2014 June Q1
6 marks Moderate -0.8
1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b896c631-00a0-46c5-bce9-16d65f6e3095-02_586_506_285_708} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A particle \(P\) of weight \(W\) newtons is attached to one end of a light inextensible string. The other end of the string is attached to a fixed point \(O\). A horizontal force of magnitude 5 N is applied to \(P\). The particle \(P\) is in equilibrium with the string taut and with \(O P\) making an angle of \(25 ^ { \circ }\) to the downward vertical, as shown in Figure 1. Find
  1. the tension in the string,
  2. the value of \(W\).
Edexcel M1 2014 June Q2
10 marks Moderate -0.8
  1. Two forces \(( 4 \mathbf { i } - 2 \mathbf { j } ) \mathrm { N }\) and \(( 2 \mathbf { i } + q \mathbf { j } ) \mathrm { N }\) act on a particle \(P\) of mass 1.5 kg . The resultant of these two forces is parallel to the vector \(( 2 \mathbf { i } + \mathbf { j } )\).
    1. Find the value of \(q\).
    At time \(t = 0 , P\) is moving with velocity \(( - 2 \mathbf { i } + 4 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).
  2. Find the speed of \(P\) at time \(t = 2\) seconds.
Edexcel M1 2014 June Q3
13 marks Moderate -0.3
3. A car starts from rest and moves with constant acceleration along a straight horizontal road. The car reaches a speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in 20 seconds. It moves at constant speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for the next 30 seconds, then moves with constant deceleration \(\frac { 1 } { 2 } \mathrm {~m} \mathrm {~s} ^ { - 2 }\) until it has speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It moves at speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for the next 15 seconds and then moves with constant deceleration \(\frac { 1 } { 3 } \mathrm {~m} \mathrm {~s} ^ { - 2 }\) until it comes to rest.
  1. Sketch, in the space below, a speed-time graph for this journey. In the first 20 seconds of this journey the car travels 140 m . Find
  2. the value of \(V\),
  3. the total time for this journey,
  4. the total distance travelled by the car.
Edexcel M1 2014 June Q4
8 marks Moderate -0.8
  1. At time \(t = 0\), a particle is projected vertically upwards with speed \(u\) from a point \(A\). The particle moves freely under gravity. At time \(T\) the particle is at its maximum height \(H\) above \(A\).
    1. Find \(T\) in terms of \(u\) and \(g\).
    2. Show that \(H = \frac { u ^ { 2 } } { 2 g }\)
    The point \(A\) is at a height \(3 H\) above the ground.
  2. Find, in terms of \(T\), the total time from the instant of projection to the instant when the particle hits the ground.
Edexcel M1 2014 June Q5
14 marks Standard +0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b896c631-00a0-46c5-bce9-16d65f6e3095-09_364_422_269_753} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Two particles \(A\) and \(B\) have masses \(2 m\) and \(3 m\) respectively. The particles are connected by a light inextensible string which passes over a smooth light fixed pulley. The system is held at rest with the string taut. The hanging parts of the string are vertical and \(A\) and \(B\) are above a horizontal plane, as shown in Figure 2. The system is released from rest.
  1. Show that the tension in the string immediately after the particles are released is \(\frac { 12 } { 5 } m g\). After descending \(1.5 \mathrm {~m} , B\) strikes the plane and is immediately brought to rest. In the subsequent motion, \(A\) does not reach the pulley.
  2. Find the distance travelled by \(A\) between the instant when \(B\) strikes the plane and the instant when the string next becomes taut. Given that \(m = 0.5 \mathrm {~kg}\),
  3. find the magnitude of the impulse on \(B\) due to the impact with the plane.
Edexcel M1 2014 June Q6
11 marks Standard +0.3
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b896c631-00a0-46c5-bce9-16d65f6e3095-11_600_969_127_491} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} A non-uniform beam \(A D\) has weight \(W\) newtons and length 4 m . It is held in equilibrium in a horizontal position by two vertical ropes attached to the beam. The ropes are attached to two points \(B\) and \(C\) on the beam, where \(A B = 1 \mathrm {~m}\) and \(C D = 1 \mathrm {~m}\), as shown in Figure 3. The tension in the rope attached to \(C\) is double the tension in the rope attached to \(B\). The beam is modelled as a rod and the ropes are modelled as light inextensible strings.
  1. Find the distance of the centre of mass of the beam from \(A\). A small load of weight \(k W\) newtons is attached to the beam at \(D\). The beam remains in equilibrium in a horizontal position. The load is modelled as a particle. Find
  2. an expression for the tension in the rope attached to \(B\), giving your answer in terms of \(k\) and \(W\),
  3. the set of possible values of \(k\) for which both ropes remain taut.
Edexcel M1 2014 June Q7
13 marks Standard +0.3
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b896c631-00a0-46c5-bce9-16d65f6e3095-13_364_833_269_561} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} A particle \(P\) of mass 2.7 kg lies on a rough plane inclined at \(40 ^ { \circ }\) to the horizontal. The particle is held in equilibrium by a force of magnitude 15 N acting at an angle of \(50 ^ { \circ }\) to the plane, as shown in Figure 4. The force acts in a vertical plane containing a line of greatest slope of the plane. The particle is in equilibrium and is on the point of sliding down the plane. Find
  1. the magnitude of the normal reaction of the plane on \(P\),
  2. the coefficient of friction between \(P\) and the plane. The force of magnitude 15 N is removed.
  3. Determine whether \(P\) moves, justifying your answer.
Edexcel M1 2014 June Q1
6 marks Moderate -0.8
1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ed659098-c1cf-4ee1-a12a-bf8b6c42db95-02_332_921_260_516} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A particle of weight \(W\) newtons is attached at \(C\) to two light inextensible strings \(A C\) and \(B C\). The other ends of the strings are attached to fixed points \(A\) and \(B\) on a horizontal ceiling. The particle hangs in equilibrium with \(A C\) and \(B C\) inclined to the horizontal at \(30 ^ { \circ }\) and \(50 ^ { \circ }\) respectively, as shown in Figure 1. Given that the tension in \(B C\) is 6 N , find
  1. the tension in \(A C\),
  2. the value of \(W\).
Edexcel M1 2014 June Q2
7 marks Moderate -0.3
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ed659098-c1cf-4ee1-a12a-bf8b6c42db95-03_435_840_269_561} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A rough plane is inclined at \(40 ^ { \circ }\) to the horizontal. Two points \(A\) and \(B\) are 3 metres apart and lie on a line of greatest slope of the inclined plane, with \(A\) above \(B\), as shown in Figure 2. A particle \(P\) of mass \(m \mathrm {~kg}\) is held at rest on the plane at \(A\). The coefficient of friction between \(P\) and the plane is \(\frac { 1 } { 2 }\). The particle is released.
  1. Find the acceleration of \(P\) down the plane.
  2. Find the speed of \(P\) at \(B\).
Edexcel M1 2014 June Q3
13 marks Moderate -0.8
  1. A ball of mass 0.3 kg is released from rest at a point which is 2 m above horizontal ground. The ball moves freely under gravity. After striking the ground, the ball rebounds vertically and rises to a maximum height of 1.5 m above the ground, before falling to the ground again. The ball is modelled as a particle.
    1. Find the speed of the ball at the instant before it strikes the ground for the first time.
    2. Find the speed of the ball at the instant after it rebounds from the ground for the first time.
    3. Find the magnitude of the impulse on the ball in the first impact with the ground.
    4. Sketch, in the space provided, a velocity-time graph for the motion of the ball from the instant when it is released until the instant when it strikes the ground for the second time.
    5. Find the time between the instant when the ball is released and the instant when it strikes the ground for the second time.
Edexcel M1 2014 June Q4
12 marks Moderate -0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ed659098-c1cf-4ee1-a12a-bf8b6c42db95-07_513_993_276_479} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} A beam \(A B\) has weight \(W\) newtons and length 4 m . The beam is held in equilibrium in a horizontal position by two vertical ropes attached to the beam. One rope is attached to \(A\) and the other rope is attached to the point \(C\) on the beam, where \(A C = d\) metres, as shown in Figure 3. The beam is modelled as a uniform rod and the ropes as light inextensible strings. The tension in the rope attached at \(C\) is double the tension in the rope attached at \(A\).
  1. Find the value of \(d\). A small load of weight \(k W\) newtons is attached to the beam at \(B\). The beam remains in equilibrium in a horizontal position. The load is modelled as a particle. The tension in the rope attached at \(C\) is now four times the tension in the rope attached at \(A\).
  2. Find the value of \(k\).
Edexcel M1 2014 June Q5
12 marks Moderate -0.3
5. A particle \(P\) of mass 0.5 kg is moving under the action of a single force \(( 3 \mathbf { i } - 2 \mathbf { j } ) \mathrm { N }\).
  1. Show that the magnitude of the acceleration of \(P\) is \(2 \sqrt { 13 } \mathrm {~m} \mathrm {~s} ^ { - 2 }\). At time \(t = 0\), the velocity of \(P\) is \(( \mathbf { i } + 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).
  2. Find the velocity of \(P\) at time \(t = 2\) seconds. Another particle \(Q\) moves with constant velocity \(\mathbf { v } = ( 2 \mathbf { i } - \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).
  3. Find the distance moved by \(Q\) in 2 seconds.
  4. Show that at time \(t = 3.5\) seconds both particles are moving in the same direction.
Edexcel M1 2014 June Q6
9 marks Moderate -0.3
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ed659098-c1cf-4ee1-a12a-bf8b6c42db95-11_472_908_285_520} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Two forces \(\mathbf { P }\) and \(\mathbf { Q }\) act on a particle at \(O\). The angle between the lines of action of \(\mathbf { P }\) and \(\mathbf { Q }\) is \(120 ^ { \circ }\) as shown in Figure 4. The force \(\mathbf { P }\) has magnitude 20 N and the force \(\mathbf { Q }\) has magnitude \(X\) newtons. The resultant of \(\mathbf { P }\) and \(\mathbf { Q }\) is the force \(\mathbf { R }\). Given that the magnitude of \(\mathbf { R }\) is \(3 X\) newtons, find, giving your answers to 3 significant figures
  1. the value of \(X\),
  2. the magnitude of \(( \mathbf { P } - \mathbf { Q } )\).
Edexcel M1 2014 June Q7
16 marks Standard +0.3
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ed659098-c1cf-4ee1-a12a-bf8b6c42db95-13_490_316_267_815} \captionsetup{labelformat=empty} \caption{Figure 5}
\end{figure} Three particles \(A , B\) and \(C\) have masses \(3 m , 2 m\) and \(2 m\) respectively. Particle \(C\) is attached to particle \(B\). Particles \(A\) and \(B\) are connected by a light inextensible string which passes over a smooth light fixed pulley. The system is held at rest with the string taut and the hanging parts of the string vertical, as shown in Figure 5. The system is released from rest and \(A\) moves upwards.
    1. Show that the acceleration of \(A\) is \(\frac { g } { 7 }\)
    2. Find the tension in the string as \(A\) ascends. At the instant when \(A\) is 0.7 m above its original position, \(C\) separates from \(B\) and falls away. In the subsequent motion, \(A\) does not reach the pulley.
  2. Find the speed of \(A\) at the instant when it is 0.7 m above its original position.
  3. Find the acceleration of \(A\) at the instant after \(C\) separates from \(B\).
  4. Find the greatest height reached by \(A\) above its original position. \includegraphics[max width=\textwidth, alt={}, center]{ed659098-c1cf-4ee1-a12a-bf8b6c42db95-14_115_161_2455_1784}
Edexcel M1 2015 June Q1
6 marks Moderate -0.5
  1. Particle \(P\) of mass \(m\) and particle \(Q\) of mass \(k m\) are moving in opposite directions on a smooth horizontal plane when they collide directly. Immediately before the collision the speed of \(P\) is \(5 u\) and the speed of \(Q\) is \(u\). Immediately after the collision the speed of each particle is halved and the direction of motion of each particle is reversed.
Find
  1. the value of \(k\),
  2. the magnitude of the impulse exerted on \(P\) by \(Q\) in the collision.
Edexcel M1 2015 June Q2
7 marks Moderate -0.8
2. A small stone is projected vertically upwards from a point \(O\) with a speed of \(19.6 \mathrm {~ms} ^ { - 1 }\). Modelling the stone as a particle moving freely under gravity,
  1. find the greatest height above \(O\) reached by the stone,
  2. find the length of time for which the stone is more than 14.7 m above \(O\).