Questions — Edexcel M1 (663 questions)

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Edexcel M1 2015 June Q3
7 marks Moderate -0.3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{de3245a7-cf6e-423e-8689-9a074bdbc23b-04_540_958_116_482} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A particle of mass 2 kg is suspended from a horizontal ceiling by two light inextensible strings, \(P R\) and \(Q R\). The particle hangs at \(R\) in equilibrium, with the strings in a vertical plane. The string \(P R\) is inclined at \(55 ^ { \circ }\) to the horizontal and the string \(Q R\) is inclined at \(35 ^ { \circ }\) to the horizontal, as shown in Figure 1. \section*{Find}
  1. the tension in the string \(P R\),
  2. the tension in the string \(Q R\).
Edexcel M1 2015 June Q4
7 marks Standard +0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{de3245a7-cf6e-423e-8689-9a074bdbc23b-06_428_373_246_788} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A lift of mass 200 kg is being lowered into a mineshaft by a vertical cable attached to the top of the lift. A crate of mass 55 kg is on the floor inside the lift, as shown in Figure 2. The lift descends vertically with constant acceleration. There is a constant upwards resistance of magnitude 150 N on the lift. The crate experiences a constant normal reaction of magnitude 473 N from the floor of the lift.
  1. Find the acceleration of the lift.
  2. Find the magnitude of the force exerted on the lift by the cable.
Edexcel M1 2015 June Q5
12 marks Moderate -0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{de3245a7-cf6e-423e-8689-9a074bdbc23b-08_582_1230_271_374} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} A beam \(A B\) has length 5 m and mass 25 kg . The beam is suspended in equilibrium in a horizontal position by two vertical ropes. One rope is attached to the beam at \(A\) and the other rope is attached to the point \(C\) on the beam where \(C B = 0.5 \mathrm {~m}\), as shown in Figure 3. A particle \(P\) of mass 60 kg is attached to the beam at \(B\) and the beam remains in equilibrium in a horizontal position. The beam is modelled as a uniform rod and the ropes are modelled as light strings.
  1. Find
    1. the tension in the rope attached to the beam at \(A\),
    2. the tension in the rope attached to the beam at \(C\). Particle \(P\) is removed and replaced by a particle \(Q\) of mass \(M \mathrm {~kg}\) at \(B\). Given that the beam remains in equilibrium in a horizontal position,
  2. find
    1. the greatest possible value of \(M\),
    2. the greatest possible tension in the rope attached to the beam at \(C\).
Edexcel M1 2015 June Q6
8 marks Easy -1.3
  1. A particle \(P\) is moving with constant velocity. The position vector of \(P\) at time \(t\) seconds \(( t \geqslant 0 )\) is \(\mathbf { r }\) metres, relative to a fixed origin \(O\), and is given by
$$\mathbf { r } = ( 2 t - 3 ) \mathbf { i } + ( 4 - 5 t ) \mathbf { j }$$
  1. Find the initial position vector of \(P\). The particle \(P\) passes through the point with position vector \(( 3.4 \mathbf { i } - 12 \mathbf { j } )\) m at time \(T\) seconds.
  2. Find the value of \(T\).
  3. Find the speed of \(P\).
Edexcel M1 2015 June Q7
13 marks Standard +0.3
7. A train travels along a straight horizontal track between two stations, \(A\) and \(B\). The train starts from rest at \(A\) and moves with constant acceleration \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) until it reaches a speed of \(V \mathrm {~ms} ^ { - 1 } , ( V < 50 )\). The train then travels at this constant speed before it moves with constant deceleration \(0.25 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) until it comes to rest at \(B\).
  1. Sketch in the space below a speed-time graph for the motion of the train between the two stations \(A\) and \(B\). The total time for the journey from \(A\) to \(B\) is 5 minutes.
  2. Find, in terms of \(V\), the length of time, in seconds, for which the train is
    1. accelerating,
    2. decelerating,
    3. moving with constant speed. Given that the distance between the two stations \(A\) and \(B\) is 6.3 km ,
  3. find the value of \(V\).
Edexcel M1 2015 June Q8
15 marks Standard +0.3
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{de3245a7-cf6e-423e-8689-9a074bdbc23b-14_643_931_118_534} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Two particles \(P\) and \(Q\) have mass 4 kg and 0.5 kg respectively. The particles are attached to the ends of a light inextensible string. Particle \(P\) is held at rest on a fixed rough plane, which is inclined to the horizontal at an angle \(\alpha\) where \(\tan \alpha = \frac { 4 } { 3 }\). The coefficient of friction between \(P\) and the plane is 0.5 . The string lies along the plane and passes over a small smooth light pulley which is fixed at the top of the plane. Particle \(Q\) hangs freely at rest vertically below the pulley. The string lies in the vertical plane which contains the pulley and a line of greatest slope of the inclined plane, as shown in Figure 4. Particle \(P\) is released from rest with the string taut and slides down the plane. Given that \(Q\) has not hit the pulley, find
  1. the tension in the string during the motion,
  2. the magnitude of the resultant force exerted by the string on the pulley.
Edexcel M1 2016 June Q1
10 marks Moderate -0.3
  1. \hspace{0pt} [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal unit vectors due east and due north respectively and position vectors are given relative to a fixed origin \(O\).]
Two cars \(P\) and \(Q\) are moving on straight horizontal roads with constant velocities. The velocity of \(P\) is \(( 15 \mathbf { i } + 20 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and the velocity of \(Q\) is \(( 20 \mathbf { i } - 5 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\)
  1. Find the direction of motion of \(Q\), giving your answer as a bearing to the nearest degree. At time \(t = 0\), the position vector of \(P\) is \(400 \mathbf { i }\) metres and the position vector of \(Q\) is 800j metres. At time \(t\) seconds, the position vectors of \(P\) and \(Q\) are \(\mathbf { p }\) metres and \(\mathbf { q }\) metres respectively.
  2. Find an expression for
    1. \(\mathbf { p }\) in terms of \(t\),
    2. \(\mathbf { q }\) in terms of \(t\).
  3. Find the position vector of \(Q\) when \(Q\) is due west of \(P\).
Edexcel M1 2016 June Q2
6 marks Moderate -0.8
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d408dd83-c5b2-4e55-b5c1-3e7f3faadbcb-04_327_255_283_847} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A vertical rope \(A B\) has its end \(B\) attached to the top of a scale pan. The scale pan has mass 0.5 kg and carries a brick of mass 1.5 kg , as shown in Figure 1. The scale pan is raised vertically upwards with constant acceleration \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) using the rope \(A B\). The rope is modelled as a light inextensible string.
  1. Find the tension in the rope \(A B\).
  2. Find the magnitude of the force exerted on the scale pan by the brick.
Edexcel M1 2016 June Q3
7 marks Standard +0.3
3. A particle \(P\) of mass 0.4 kg is moving on rough horizontal ground when it hits a fixed vertical plane wall. Immediately before hitting the wall, \(P\) is moving with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a direction perpendicular to the wall. The particle rebounds from the wall and comes to rest at a distance of 5 m from the wall. The coefficient of friction between \(P\) and the ground is \(\frac { 1 } { 8 }\). Find the magnitude of the impulse exerted on \(P\) by the wall.
Edexcel M1 2016 June Q4
12 marks Moderate -0.3
4. Two trains \(M\) and \(N\) are moving in the same direction along parallel straight horizontal tracks. At time \(t = 0 , M\) overtakes \(N\) whilst they are travelling with speeds \(40 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. Train \(M\) overtakes train \(N\) as they pass a point \(X\) at the side of the tracks. After overtaking \(N\), train \(M\) maintains its speed of \(40 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for \(T\) seconds and then decelerates uniformly, coming to rest next to a point \(Y\) at the side of the tracks. After being overtaken, train \(N\) maintains its speed of \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for 25 s and then decelerates uniformly, also coming to rest next to the point \(Y\). The times taken by the trains to travel between \(X\) and \(Y\) are the same.
  1. Sketch, on the same diagram, the speed-time graphs for the motions of the two trains between \(X\) and \(Y\). Given that \(X Y = 975 \mathrm {~m}\),
  2. find the value of \(T\).
Edexcel M1 2016 June Q5
10 marks Standard +0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d408dd83-c5b2-4e55-b5c1-3e7f3faadbcb-08_321_917_285_518} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A particle \(P\) of mass 2 kg is held at rest in equilibrium on a rough plane by a constant force of magnitude 40 N . The direction of the force is inclined to the plane at an angle of \(30 ^ { \circ }\). The plane is inclined to the horizontal at an angle of \(20 ^ { \circ }\), as shown in Figure 2. The line of action of the force lies in the vertical plane containing \(P\) and a line of greatest slope of the plane. The coefficient of friction between \(P\) and the plane is \(\mu\). Given that \(P\) is on the point of sliding up the plane, find the value of \(\mu\).
Edexcel M1 2016 June Q6
7 marks Standard +0.3
6. A non-uniform plank \(A B\) has length 6 m and mass 30 kg . The plank rests in equilibrium in a horizontal position on supports at the points \(S\) and \(T\) of the plank where \(A S = 0.5 \mathrm {~m}\) and \(T B = 2 \mathrm {~m}\). When a block of mass \(M \mathrm {~kg}\) is placed on the plank at \(A\), the plank remains horizontal and in equilibrium and the plank is on the point of tilting about \(S\). When the block is moved to \(B\), the plank remains horizontal and in equilibrium and the plank is on the point of tilting about \(T\). The distance of the centre of mass of the plank from \(A\) is \(d\) metres. The block is modelled as a particle and the plank is modelled as a non-uniform rod. Find
  1. the value of \(d\),
  2. the value of \(M\).
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Edexcel M1 2016 June Q7
11 marks Moderate -0.3
7. Two forces \(\mathbf { F } _ { 1 }\) and \(\mathbf { F } _ { 2 }\) act on a particle \(P\). The force \(\mathbf { F } _ { 1 }\) is given by \(\mathbf { F } _ { 1 } = ( - \mathbf { i } + 2 \mathbf { j } ) \mathrm { N }\) and \(\mathbf { F } _ { 2 }\) acts in the direction of the vector \(( \mathbf { i } + \mathbf { j } )\).
Given that the resultant of \(\mathbf { F } _ { 1 }\) and \(\mathbf { F } _ { 2 }\) acts in the direction of the vector ( \(\mathbf { i } + 3 \mathbf { j }\) ),
  1. find \(\mathbf { F } _ { 2 }\) (7) The acceleration of \(P\) is \(( 3 \mathbf { i } + 9 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }\). At time \(t = 0\), the velocity of \(P\) is \(( 3 \mathbf { i } - 22 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\)
  2. Find the speed of \(P\) when \(t = 3\) seconds.
Edexcel M1 2016 June Q8
12 marks Standard +0.3
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d408dd83-c5b2-4e55-b5c1-3e7f3faadbcb-14_460_981_274_475} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Two particles \(P\) and \(Q\) have masses 1.5 kg and 3 kg respectively. The particles are attached to the ends of a light inextensible string. Particle \(P\) is held at rest on a fixed rough horizontal table. The coefficient of friction between \(P\) and the table is \(\frac { 1 } { 5 }\). The string is parallel to the table and passes over a small smooth light pulley which is fixed at the edge of the table. Particle \(Q\) hangs freely at rest vertically below the pulley, as shown in Figure 3. Particle \(P\) is released from rest with the string taut and slides along the table. Assuming that \(P\) has not reached the pulley, find
  1. the tension in the string during the motion,
  2. the magnitude and direction of the resultant force exerted on the pulley by the string.
Edexcel M1 2017 June Q1
6 marks Moderate -0.8
  1. Three forces, \(( 15 \mathbf { i } + \mathbf { j } ) \mathrm { N } , ( 5 q \mathbf { i } - p \mathbf { j } ) \mathrm { N }\) and \(( - 3 p \mathbf { i } - q \mathbf { j } ) \mathrm { N }\), where \(p\) and \(q\) are constants, act on a particle. Given that the particle is in equilibrium, find the value of \(p\) and the value of \(q\).
    (6)
Edexcel M1 2017 June Q2
7 marks Moderate -0.8
2. Two particles, \(P\) and \(Q\), have masses \(2 m\) and \(3 m\) respectively. They are moving towards each other in opposite directions on a smooth horizontal plane when they collide directly. Immediately before they collide the speed of \(P\) is \(4 u\) and the speed of \(Q\) is \(3 u\). As a result of the collision, \(Q\) has its direction of motion reversed and is moving with speed \(u\).
  1. Find the speed of \(P\) immediately after the collision.
  2. State whether or not the direction of motion of \(P\) has been reversed by the collision.
  3. Find the magnitude of the impulse exerted on \(P\) by \(Q\) in the collision.
Edexcel M1 2017 June Q3
9 marks Standard +0.3
3. A plank \(A B\) has length 6 m and mass 30 kg . The point \(C\) is on the plank with \(C B = 2 \mathrm {~m}\). The plank rests in equilibrium in a horizontal position on supports at \(A\) and \(C\). Two people, each of mass 75 kg , stand on the plank. One person stands at the point \(P\) of the plank, where \(A P = x\) metres, and the other person stands at the point \(Q\) of the plank, where \(A Q = 2 x\) metres. The plank remains horizontal and in equilibrium with the magnitude of the reaction at \(C\) five times the magnitude of the reaction at \(A\). The plank is modelled as a uniform rod and each person is modelled as a particle.
  1. Find the value of \(x\).
  2. State two ways in which you have used the assumptions made in modelling the plank as a uniform rod.
Edexcel M1 2017 June Q4
9 marks Standard +0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c809d34e-83db-4a16-a831-001f9f36b1c3-10_291_926_251_516} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A particle \(P\) of mass 5 kg is held at rest in equilibrium on a rough inclined plane by a horizontal force of magnitude 10 N . The plane is inclined to the horizontal at an angle \(\alpha\) where \(\tan \alpha = \frac { 3 } { 4 }\), as shown in Figure 1. The line of action of the force lies in the vertical plane containing \(P\) and a line of greatest slope of the plane. The coefficient of friction between \(P\) and the plane is \(\mu\). Given that \(P\) is on the point of sliding down the plane, find the value of \(\mu\). \includegraphics[max width=\textwidth, alt={}, center]{c809d34e-83db-4a16-a831-001f9f36b1c3-13_2460_72_311_27}
Edexcel M1 2017 June Q5
6 marks Moderate -0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c809d34e-83db-4a16-a831-001f9f36b1c3-14_346_241_262_845} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A vertical light rod \(P Q\) has a particle of mass 0.5 kg attached to it at \(P\) and a particle of mass 0.75 kg attached to it at \(Q\), to form a system, as shown in Figure 2. The system is accelerated vertically upwards by a vertical force of magnitude 15 N applied to the particle at \(Q\). Find the thrust in the rod.
Edexcel M1 2017 June Q6
9 marks Moderate -0.8
6. A cyclist is moving along a straight horizontal road and passes a point \(A\). Five seconds later, at the instant when she is moving with speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), she passes the point \(B\). She moves with constant acceleration from \(A\) to \(B\). Given that \(A B = 40 \mathrm {~m}\), find
  1. the acceleration of the cyclist as she moves from \(A\) to \(B\),
  2. the time it takes her to travel from \(A\) to the midpoint of \(A B\).
Edexcel M1 2017 June Q7
14 marks Standard +0.3
7. [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal unit vectors due east and due north respectively and position vectors are given relative to a fixed origin \(O\).] Two ships, \(P\) and \(Q\), are moving with constant velocities.
The velocity of \(P\) is \(( 9 \mathbf { i } - 2 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\) and the velocity of \(Q\) is \(( 4 \mathbf { i } + 8 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\)
  1. Find the direction of motion of \(P\), giving your answer as a bearing to the nearest degree. When \(t = 0\), the position vector of \(P\) is \(( 9 \mathbf { i } + 10 \mathbf { j } ) \mathrm { km }\) and the position vector of \(Q\) is \(( \mathbf { i } + 4 \mathbf { j } ) \mathrm { km }\). At time \(t\) hours, the position vectors of \(P\) and \(Q\) are \(\mathbf { p } \mathrm { km }\) and \(\mathbf { q } \mathrm { km }\) respectively.
  2. Find an expression for
    1. \(\mathbf { p }\) in terms of \(t\),
    2. \(\mathbf { q }\) in terms of \(t\).
  3. Hence show that, at time \(t\) hours, $$\overrightarrow { Q P } = ( 8 + 5 t ) \mathbf { i } + ( 6 - 10 t ) \mathbf { j }$$
  4. Find the values of \(t\) when the ships are 10 km apart.
Edexcel M1 2017 June Q8
15 marks Standard +0.3
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c809d34e-83db-4a16-a831-001f9f36b1c3-24_638_951_242_500} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Two particles, \(A\) and \(B\), have masses \(2 m\) and \(m\) respectively. The particles are attached to the ends of a light inextensible string. Particle \(A\) is held at rest on a fixed rough horizontal table at a distance \(d\) from a small smooth light pulley which is fixed at the edge of the table at the point \(P\). The coefficient of friction between \(A\) and the table is \(\mu\), where \(\mu < \frac { 1 } { 2 }\). The string is parallel to the table from \(A\) to \(P\) and passes over the pulley. Particle \(B\) hangs freely at rest vertically below \(P\) with the string taut and at a height \(h\), ( \(h < d\) ), above a horizontal floor, as shown in Figure 3. Particle \(A\) is released from rest with the string taut and slides along the table.
    1. Write down an equation of motion for \(A\).
    2. Write down an equation of motion for \(B\).
  2. Hence show that, until \(B\) hits the floor, the acceleration of \(A\) is \(\frac { g } { 3 } ( 1 - 2 \mu )\).
  3. Find, in terms of \(g , h\) and \(\mu\), the speed of \(A\) at the instant when \(B\) hits the floor. After \(B\) hits the floor, \(A\) continues to slide along the table. Given that \(\mu = \frac { 1 } { 3 }\) and that \(A\) comes to rest at \(P\),
  4. find \(d\) in terms of \(h\).
  5. Describe what would happen if \(\mu = \frac { 1 } { 2 }\)
    (Total 15 marks)
    Leave blank
    Q8
Edexcel M1 2018 June Q1
6 marks Moderate -0.8
  1. Two particles, \(P\) and \(Q\), have masses \(3 m\) and \(m\) respectively. They are moving in opposite directions towards each other along the same straight line on a smooth horizontal plane and collide directly. The speeds of \(P\) and \(Q\) immediately before the collision are \(2 u\) and \(4 u\) respectively. The magnitude of the impulse received by each particle in the collision is \(\frac { 21 m u } { 4 }\).
    1. Find the speed of \(P\) after the collision.
    2. Find the speed of \(Q\) after the collision.
Edexcel M1 2018 June Q2
10 marks Standard +0.8
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4fd21e83-0bdf-4bb1-8a3f-76beada511ae-04_333_976_287_550} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A particle of mass 2 kg lies on a rough plane. The plane is inclined to the horizontal at \(30 ^ { \circ }\). The coefficient of friction between the particle and the plane is \(\frac { 1 } { 4 }\). The particle is held in equilibrium by a force of magnitude \(P\) newtons. The force makes an angle of \(20 ^ { \circ }\) with the horizontal and acts in a vertical plane containing a line of greatest slope of the plane, as shown in Figure 1. Find the least possible value of \(P\).
Edexcel M1 2018 June Q3
7 marks Standard +0.3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4fd21e83-0bdf-4bb1-8a3f-76beada511ae-08_426_1226_221_360} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A wooden beam \(A B\), of mass 150 kg and length 9 m , rests in a horizontal position supported by two vertical ropes. The ropes are attached to the beam at \(C\) and \(D\), where \(A C = 1.5 \mathrm {~m}\) and \(B D = 3.5 \mathrm {~m}\). A gymnast of mass 60 kg stands on the beam at the point \(P\), where \(A P = 3 \mathrm {~m}\), as shown in Figure 2. The beam remains horizontal and in equilibrium. By modelling the gymnast as a particle, the beam as a uniform rod and the ropes as light inextensible strings,
  1. find the tension in the rope attached to the beam at \(C\). The gymnast at \(P\) remains on the beam at \(P\) and another gymnast, who is also modelled as a particle, stands on the beam at \(B\). The beam remains horizontal and in equilibrium. The mass of the gymnast at \(B\) is the largest possible for which the beam remains horizontal and in equilibrium.
  2. Find the tension in the rope attached to the beam at \(D\).