Questions — Edexcel M1 (599 questions)

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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
Edexcel M1 2013 June Q8
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{278c8424-38aa-48c2-bc82-af4be9234f71-13_259_1367_228_294} \captionsetup{labelformat=empty} \caption{Figure 5}
\end{figure} A uniform rod \(A B\) has length 2 m and mass 50 kg . The rod is in equilibrium in a horizontal position, resting on two smooth supports at \(C\) and \(D\), where \(A C = 0.2\) metres and \(D B = x\) metres, as shown in Figure 5. Given that the magnitude of the reaction on the rod at \(D\) is twice the magnitude of the reaction on the rod at \(C\),
  1. find the value of \(x\). The support at \(D\) is now moved to the point \(E\) on the rod, where \(E B = 0.4\) metres. A particle of mass \(m \mathrm {~kg}\) is placed on the rod at \(B\), and the rod remains in equilibrium in a horizontal position. Given that the magnitude of the reaction on the rod at \(E\) is four times the magnitude of the reaction on the rod at \(C\),
  2. find the value of \(m\). \includegraphics[max width=\textwidth, alt={}, center]{278c8424-38aa-48c2-bc82-af4be9234f71-14_77_74_2480_1836}
Edexcel M1 2013 June Q1
\begin{enumerate} \item Particle \(P\) has mass 3 kg and particle \(Q\) has mass \(m \mathrm {~kg}\). The particles are moving in opposite directions along a smooth horizontal plane when they collide directly. Immediately before the collision, the speed of \(P\) is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the speed of \(Q\) is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). In the collision the direction of motion of \(P\) is unchanged and the direction of motion of \(Q\) is reversed. Immediately after the collision, the speed of \(P\) is \(1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the speed of \(Q\) is \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Find the magnitude of the impulse exerted on \(P\) in the collision.
  2. Find the value of \(m\).
    \item A woman travels in a lift. The mass of the woman is 50 kg and the mass of the lift is 950 kg . The lift is being raised vertically by a vertical cable which is attached to the top of the lift. The lift is moving upwards and has constant deceleration of \(2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). By modelling the cable as being light and inextensible, find
Edexcel M1 2013 June Q4
  1. A lorry is moving along a straight horizontal road with constant acceleration. The lorry passes a point \(A\) with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 } , ( u < 34 )\), and 10 seconds later passes a point \(B\) with speed \(34 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Given that \(A B = 240 \mathrm {~m}\), find
    1. the value of \(u\),
    2. the time taken for the lorry to move from \(A\) to the mid-point of \(A B\).
    3. A car is travelling along a straight horizontal road. The car takes 120 s to travel between two sets of traffic lights which are 2145 m apart. The car starts from rest at the first set of traffic lights and moves with constant acceleration for 30 s until its speed is \(22 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The car maintains this speed for \(T\) seconds. The car then moves with constant deceleration, coming to rest at the second set of traffic lights.
    4. Sketch, in the space below, a speed-time graph for the motion of the car between the two sets of traffic lights.
    5. Find the value of \(T\).
    A motorcycle leaves the first set of traffic lights 10 s after the car has left the first set of traffic lights. The motorcycle moves from rest with constant acceleration, a \(\mathrm { m } \mathrm { s } ^ { - 2 }\), and passes the car at the point \(A\) which is 990 m from the first set of traffic lights. When the motorcycle passes the car, the car is moving with speed \(22 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  2. Find the time it takes for the motorcycle to move from the first set of traffic lights to the point \(A\).
  3. Find the value of \(a\).
Edexcel M1 2013 June Q6
6. A beam \(A B\) has length 15 m . The beam rests horizontally in equilibrium on two smooth supports at the points \(P\) and \(Q\), where \(A P = 2 \mathrm {~m}\) and \(Q B = 3 \mathrm {~m}\). When a child of mass 50 kg stands on the beam at \(A\), the beam remains in equilibrium and is on the point of tilting about \(P\). When the same child of mass 50 kg stands on the beam at \(B\), the beam remains in equilibrium and is on the point of tilting about \(Q\). The child is modelled as a particle and the beam is modelled as a non-uniform rod.
    1. Find the mass of the beam.
    2. Find the distance of the centre of mass of the beam from \(A\). When the child stands at the point \(X\) on the beam, it remains horizontal and in equilibrium. Given that the reactions at the two supports are equal in magnitude,
  2. find \(A X\).
Edexcel M1 2013 June Q7
  1. \hspace{0pt} [In this question, the horizontal unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed due east and due north respectively.]
The velocity, \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\), of a particle \(P\) at time \(t\) seconds is given by $$\mathbf { v } = ( 1 - 2 t ) \mathbf { i } + ( 3 t - 3 ) \mathbf { j }$$
  1. Find the speed of \(P\) when \(t = 0\)
  2. Find the bearing on which \(P\) is moving when \(t = 2\)
  3. Find the value of \(t\) when \(P\) is moving
    1. parallel to \(\mathbf { j }\),
    2. parallel to \(( - \mathbf { i } - 3 \mathbf { j } )\).
Edexcel M1 2013 June Q8
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3c8dce6f-367a-42bb-be60-d03d0a23664f-13_526_945_258_502} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Two particles \(A\) and \(B\) have masses \(2 m\) and \(3 m\) respectively. The particles are attached to the ends of a light inextensible string. Particle \(A\) is held at rest on a smooth horizontal table. The string passes over a small smooth pulley which is fixed at the edge of the table. Particle \(B\) hangs at rest vertically below the pulley with the string taut, as shown in Figure 2. Particle \(A\) is released from rest. Assuming that \(A\) has not reached the pulley, find
  1. the acceleration of \(B\),
  2. the tension in the string,
  3. the magnitude and direction of the force exerted on the pulley by the string.
Edexcel M1 2014 June Q1
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
  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
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
  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
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
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
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
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
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
  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
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
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
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
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
  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
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 \(O\).
Edexcel M1 2015 June Q3
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
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
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 \(C\).