Questions — Edexcel (9685 questions)

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Edexcel M1 Specimen Q7
10 marks Standard +0.3
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ab3c0d61-3cab-4050-8288-6052e8404eb1-20_264_684_319_639} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A particle of mass 0.4 kg is held at rest on a fixed rough plane by a horizontal force of magnitude \(P\) newtons. The force acts in the vertical plane containing the line of greatest slope of the inclined plane which passes through the particle. The plane is inclined to the horizontal at an angle \(\alpha\), where \(\tan \alpha = \frac { 3 } { 4 }\), as shown in Figure 2. The coefficient of friction between the particle and the plane is \(\frac { 1 } { 3 }\).
Given that the particle is on the point of sliding up the plane, find
  1. the magnitude of the normal reaction between the particle and the plane,
  2. the value of \(P\).
Edexcel M1 Specimen Q8
17 marks Standard +0.3
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ab3c0d61-3cab-4050-8288-6052e8404eb1-24_862_412_310_774} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Two particles \(A\) and \(B\) have mass 0.4 kg and 0.3 kg respectively. The particles are attached to the ends of a light inextensible string. The string passes over a small smooth pulley which is fixed above a horizontal floor. Both particles are held, with the string taut, at a height of 1 m above the floor, as shown in Figure 3. The particles are released from rest and in the subsequent motion \(B\) does not reach the pulley.
  1. Find the tension in the string immediately after the particles are released.
  2. Find the acceleration of \(A\) immediately after the particles are released. When the particles have been moving for 0.5 s , the string breaks.
  3. Find the further time that elapses until \(B\) hits the floor.
Edexcel M2 2016 June Q1
13 marks Standard +0.3
  1. A particle \(P\) moves along a straight line. The speed of \(P\) at time \(t\) seconds ( \(t \geqslant 0\) ) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(v = \left( p t ^ { 2 } + q t + r \right)\) and \(p , q\) and \(r\) are constants. When \(t = 2\) the speed of \(P\) has its minimum value. When \(t = 0 , v = 11\) and when \(t = 2 , v = 3\)
Find
  1. the acceleration of \(P\) when \(t = 3\)
  2. the distance travelled by \(P\) in the third second of the motion.
Edexcel M2 2016 June Q2
10 marks Standard +0.3
2. A car of mass 800 kg is moving on a straight road which is inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 20 }\). The resistance to the motion of the car from non-gravitational forces is modelled as a constant force of magnitude \(R\) newtons. When the car is moving up the road at a constant speed of \(12.5 \mathrm {~ms} ^ { - 1 }\), the engine of the car is working at a constant rate of \(3 P\) watts. When the car is moving down the road at a constant speed of \(12.5 \mathrm {~ms} ^ { - 1 }\), the engine of the car is working at a constant rate of \(P\) watts.
  1. Find
    1. the value of \(P\),
    2. the value of \(R\).
      (6) When the car is moving up the road at \(12.5 \mathrm {~ms} ^ { - 1 }\) the engine is switched off and the car comes to rest, without braking, in a distance \(d\) metres. The resistance to the motion of the car from non-gravitational forces is still modelled as a constant force of magnitude \(R\) newtons.
  2. Use the work-energy principle to find the value of \(d\).
Edexcel M2 2016 June Q3
6 marks Standard +0.3
3. A particle of mass 0.6 kg is moving with constant velocity ( \(c \mathbf { i } + 2 c \mathbf { j }\) ) \(\mathrm { ms } ^ { - 1 }\), where \(c\) is a positive constant. The particle receives an impulse of magnitude \(2 \sqrt { 10 } \mathrm {~N} \mathrm {~s}\). Immediately after receiving the impulse the particle has velocity ( \(2 c \mathbf { i } - c \mathbf { j }\) ) \(\mathrm { ms } ^ { - 1 }\). Find the value of \(c\).
(6)
Edexcel M2 2016 June Q4
9 marks Standard +0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{deb9e495-3bfb-4a46-9ee7-3eb421c33499-07_606_883_260_532} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The uniform lamina \(O B C\) is one quarter of a circular disc with centre \(O\) and radius 4 m . The points \(A\) and \(D\), on \(O B\) and \(O C\) respectively, are 3 m from \(O\). The uniform lamina \(A B C D\), shown shaded in Figure 1, is formed by removing the triangle \(O A D\) from \(O B C\). Given that the centre of mass of one quarter of a uniform circular disc of radius \(r\) is at a distance \(\frac { 4 \sqrt { 2 } } { 3 \pi } r\) from the centre of the disc,
  1. find the distance of the centre of mass of the lamina \(A B C D\) from \(A D\). The lamina is freely suspended from \(D\) and hangs in equilibrium.
  2. Find, to the nearest degree, the angle between \(D C\) and the downward vertical.
    \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{deb9e495-3bfb-4a46-9ee7-3eb421c33499-09_915_1269_118_356} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure}
Edexcel M2 2016 June Q5
12 marks Standard +0.3
5. A non-uniform rod \(A B\), of mass 5 kg and length 4 m , rests with one end \(A\) on rough horizontal ground. The centre of mass of the rod is \(d\) metres from \(A\). The rod is held in limiting equilibrium at an angle \(\theta\) to the horizontal by a force \(\mathbf { P }\), which acts in a direction perpendicular to the rod at \(B\), as shown in Figure 2. The line of action of \(\mathbf { P }\) lies in the same vertical plane as the rod.
  1. Find, in terms of \(d , g\) and \(\theta\),
    1. the magnitude of the vertical component of the force exerted on the rod by the ground,
    2. the magnitude of the friction force acting on the rod at \(A\). Given that \(\tan \theta = \frac { 5 } { 12 }\) and that the coefficient of friction between the rod and the ground is \(\frac { 1 } { 2 }\),
  2. find the value of \(d\).
Edexcel M2 2016 June Q6
13 marks Standard +0.3
6. [In this question, \(\mathbf { i }\) is a horizontal unit vector and \(\mathbf { j }\) is an upward vertical unit vector.] A particle \(P\) is projected from a fixed origin \(O\) with velocity ( \(3 \mathbf { i } + 4 \mathbf { j }\) ) \(\mathrm { m } \mathrm { s } ^ { - 1 }\). The particle moves freely under gravity and passes through the point \(A\) with position vector \(\lambda ( \mathbf { i } - \mathbf { j } ) \mathrm { m }\), where \(\lambda\) is a positive constant.
  1. Find the value of \(\lambda\).
  2. Find
    1. the speed of \(P\) at the instant when it passes through \(A\),
    2. the direction of motion of \(P\) at the instant when it passes through \(A\).
      HMAV SIHI NITIIIUM ION OC
      VILV SIHI NI JAHM ION OC
      VJ4V SIHI NI JIIYM ION OC
Edexcel M2 2016 June Q7
12 marks Standard +0.3
7. Two particles \(A\) and \(B\), of mass \(2 m\) and \(3 m\) respectively, are initially at rest on a smooth horizontal surface. Particle \(A\) is projected with speed \(3 u\) towards \(B\). Particle \(A\) collides directly with particle \(B\). The coefficient of restitution between \(A\) and \(B\) is \(\frac { 3 } { 4 }\)
  1. Find
    1. the speed of \(A\) immediately after the collision,
    2. the speed of \(B\) immediately after the collision. After the collision \(B\) hits a fixed smooth vertical wall and rebounds. The wall is perpendicular to the direction of motion of \(B\). The coefficient of restitution between \(B\) and the wall is \(e\). The magnitude of the impulse received by \(B\) when it hits the wall is \(\frac { 27 } { 4 } m u\).
  2. Find the value of \(e\).
  3. Determine whether there is a further collision between \(A\) and \(B\) after \(B\) rebounds from the wall.
Edexcel M3 Specimen Q1
7 marks Moderate -0.8
1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{689d4bd3-db24-4159-986b-40496213321a-02_438_492_328_735} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A garden game is played with a small ball \(B\) of mass \(m\) attached to one end of a light inextensible string of length 13l. The other end of the string is fixed to a point \(A\) on a vertical pole as shown in Figure 1. The ball is hit and moves with constant speed in a horizontal circle of radius \(5 l\) and centre \(C\), where \(C\) is vertically below \(A\). Modelling the ball as a particle, find
  1. the tension in the string,
  2. the speed of the ball.
Edexcel M3 Specimen Q3
9 marks Challenging +1.2
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{689d4bd3-db24-4159-986b-40496213321a-08_325_684_306_639} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A particle of mass 0.5 kg is attached to one end of a light elastic spring of natural length 0.9 m and modulus of elasticity \(\lambda\) newtons. The other end of the spring is attached to a fixed point \(O\) on a rough plane which is inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 3 } { 5 }\). The coefficient of friction between the particle and the plane is 0.15 . The particle is held on the plane at a point which is 1.5 m down the line of greatest slope from \(O\), as shown in Figure 2. The particle is released from rest and first comes to rest again after moving 0.7 m up the plane. Find the value of \(\lambda\).
Edexcel M3 Specimen Q4
10 marks Standard +0.8
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{689d4bd3-db24-4159-986b-40496213321a-12_410_579_312_689} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} A container is formed by removing a right circular solid cone of height \(4 l\) from a uniform solid right circular cylinder of height \(6 l\). The centre \(O\) of the plane face of the cone coincides with the centre of a plane face of the cylinder and the axis of the cone coincides with the axis of the cylinder, as shown in Figure 3. The cylinder has radius \(2 l\) and the base of the cone has radius \(l\).
  1. Find the distance of the centre of mass of the container from \(O\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{689d4bd3-db24-4159-986b-40496213321a-12_435_560_1274_699} \captionsetup{labelformat=empty} \caption{Figure 4}
    \end{figure} The container is placed on a plane which is inclined at an angle \(\theta ^ { \circ }\) to the horizontal. The open face is uppermost, as shown in Figure 4. The plane is sufficiently rough to prevent the container from sliding. The container is on the point of toppling.
  2. Find the value of \(\theta\). \includegraphics[max width=\textwidth, alt={}, center]{689d4bd3-db24-4159-986b-40496213321a-15_78_28_2588_1889}
Edexcel M3 Specimen Q5
12 marks Standard +0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{689d4bd3-db24-4159-986b-40496213321a-16_446_437_324_758} \captionsetup{labelformat=empty} \caption{Figure 5}
\end{figure} A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is fixed at the point \(O\). The particle is initially held with \(O P\) horizontal and the string taut. It is then projected vertically upwards with speed \(u\), where \(u ^ { 2 } = 5 a g\). When \(O P\) has turned through an angle \(\theta\) the speed of \(P\) is \(v\) and the tension in the string is \(T\), as shown in Figure 5.
  1. Find, in terms of \(a , g\) and \(\theta\), an expression for \(v ^ { 2 }\).
  2. Find, in terms of \(m , g\) and \(\theta\), an expression for \(T\).
  3. Prove that \(P\) moves in a complete circle.
  4. Find the maximum speed of \(P\).
Edexcel M1 Q2
6 marks Moderate -0.3
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a9f91ceb-662a-40cd-956b-815052b8f1a0-01_173_520_360_1891} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} A uniform \(\operatorname { rod } A B\) has weight 70 N and length 3 m . It rests in a horizontal position on two smooth supports placed at \(P\) and \(Q\), where \(A P = 0.5 \mathrm {~m}\), as shown in Fig. 1 . The reaction on the rod at \(P\) has magnitude 20 N . Find
  1. the magnitude of the reaction on the rod at \(Q\),
  2. the distance \(A Q\).
    . \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{a9f91ceb-662a-40cd-956b-815052b8f1a0-01_190_476_964_1905} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure} A particle \(P\) of mass 2 kg is held in equilibrium under gravity by two light inextensible strings. One string is horizontal and the other is inclined at an angle \(\alpha\) to the horizontal, as shown in Fig. 2. The tension in the horizontal string is 15 N . The tension in the other string is \(T\) newtons.
  3. Find the size of the angle \(\alpha\).
    (6 marks)
  4. Find the value of \(T\). You must ensure that your answers to parts of questions are clearly labelled.
    You must show sufficient working to make your methods clear to the Examiner. Answers without working may gain no credit.
Edexcel M1 Q3
9 marks Moderate -0.3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a9f91ceb-662a-40cd-956b-815052b8f1a0-02_280_428_340_516} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure} Two particles \(A\) and \(B\) have masses \(3 m\) and \(k m\) respectively, where \(k > 3\). They are connected by a light inextensible string which passes over a smooth fixed pulley. The system is released from rest with the string taut and the hanging parts of the string vertical, as shown in Fig. 3. While the particles are moving freely, \(A\) has an acceleration of magnitude \(\frac { 2 } { 5 } g\).
  1. Find, in terms of \(m\) and \(g\), the tension in the string.
    (3 marks)
  2. State why \(B\) also has an acceleration of magnitude \(\frac { 2 } { 5 } g\).
  3. Find the value of \(k\).
  4. State how you have used the fact that the string is light.
    (1 mark)
Edexcel M1 Q4
7 marks Easy -1.2
4. A particle \(P\) moves in a straight line with constant velocity. Initially \(P\) is at the point \(A\) with position vector \(( 2 \mathbf { i } - \mathbf { j } ) \mathrm { m }\) relative to a fixed origin \(O\), and 2 s later it is at the point \(B\) with position vector \(( 6 \mathbf { i } + \mathbf { j } ) \mathrm { m }\).
  1. Find the velocity of \(P\).
  2. Find, in degrees to one decimal place, the size of the angle between the direction of motion of \(P\) and the vector \(\mathbf { i }\). Three seconds after it passes \(B\) the particle \(P\) reaches the point \(C\).
  3. Find, in m to one decimal place, the distance \(O C\).
Edexcel M1 Q5
12 marks Standard +0.3
5. Two small balls \(A\) and \(B\) have masses 0.6 kg and 0.2 kg respectively. They are moving towards each other in opposite directions on a horizontal table when they collide directly. Immediately before the collision, the speed of \(A\) is \(4.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the speed of \(B\) is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Immediately after the collision, \(A\) and \(B\) move in the same direction and the speed of \(B\) is twice the speed of \(A\).
By modelling the balls as particles, find
  1. the speed of \(B\) immediately after the collision,
  2. the magnitude of the impulse exerted on \(B\) in the collision, stating the units in which your answer is given.
    (3 marks)
    The table is rough. After the collision, \(B\) moves a distance of 2 m on the table before coming to rest.
  3. Find the coefficient of friction between \(B\) and the table.
    (6 marks)
Edexcel M1 Q6
9 marks Moderate -0.3
6. A parachutist drops from a helicopter \(H\) and falls vertically from rest towards the ground. Her parachute opens 2 s after she leaves \(H\) and her speed then reduces to \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). For the first 2 s her motion is modelled as that of a particle falling freely under gravity. For the next 5 s the model is motion with constant deceleration, so that her speed is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the end of this period. For the rest of the time before she reaches the ground, the model is motion with constant speed of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Sketch a speed-time graph to illustrate her motion from \(H\) to the ground.
    (3 marks)
  2. Find her speed when the parachute opens.
    (2 marks)
    A safety rule states that the helicopter must be high enough to allow the parachute to open and for the speed of a parachutist to reduce to \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) before reaching the ground. Using the assumptions made in the above model,
  3. find the minimum height of \(H\) for which the woman can make a drop without breaking this safety rule.
    (5 marks)
    Given that \(H\) is 125 m above the ground when the woman starts her drop,
  4. find the total time taken for her to reach the ground.
  5. State one way in which the model could be refined to make it more realistic.
    (1 mark)
Edexcel M3 Q5
12 marks Standard +0.8
5. A cyclist is travelling around a circular track which is banked at \(25 ^ { \circ }\) to the horizontal. The coefficient of friction between the cycle's tyres and the track is 0.6 . The cyclist moves with constant speed in a horizontal circle of radius 40 m , without the tyres slipping. Find the maximum speed of the cyclist.
Edexcel D1 Q1
7 marks Standard +0.3
1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6c6b7934-ab46-4a87-8a11-f99bf9a5d743-02_629_700_196_443} \captionsetup{labelformat=empty} \caption{Fig 1}
\end{figure}
  1. Find a Hamiltonian cycle for the graph shown in Figure 1.
  2. Starting with your cycle, construct a plane drawing of the graph, showing your method clearly.
    (5 marks)
Edexcel D1 Q2
8 marks Easy -1.3
2. (a) The following list of numbers is to be sorted into descending order. $$\begin{array} { l l l l l l } 35 & 23 & 10 & 46 & 24 & 11 \end{array}$$ Use the Bubble sort algorithm to obtain a sorted list, giving the state of the list at each stage where two values could be interchanged.
(b) Find the maximum number of interchanges needed when 8 values are sorted into descending order using the Bubble sort algorithm.
(c) Use the first-fit decreasing algorithm to fit the data in part (a) into bins of size 50. Explain how you decided in which bin to place the number 11.
Edexcel D1 Q3
8 marks Moderate -0.8
3. This question should be answered on the sheet provided. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6c6b7934-ab46-4a87-8a11-f99bf9a5d743-03_744_1524_319_315} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} Figure 2 shows an activity network. The nodes represent events and the arcs represent the activities. The number in each bracket gives the time, in days, needed to complete the activity.
  1. Calculate the early and late times for each event using appropriate forward and backward scanning.
    (5 marks)
  2. Hence, determine the activities which lie on the critical path.
  3. State the minimum number of days needed to complete the entire project.
Edexcel D1 Q4
8 marks Moderate -0.8
4. This question should be answered on the sheet provided. The manager of an outdoor centre must staff each activity offered by the centre with an appropriately qualified instructor. The table below shows the sports for which each member of staff is qualified to supervise.
NameActivities
FatimaWindsurfing, Sailing
GavinClimbing, Orienteering
HassanWindsurfing, Climbing
IainSailing, Diving
JaneDiving, Sailing, Orienteering
  1. Draw a bipartite graph to model this situation. Initially the manager allocates Fatima, Gavin, Iain and Jane to supervise the first sport listed against their names in the table.
  2. Starting from this matching, use the maximum matching algorithm to find a complete matching. Indicate clearly how you have applied the algorithm.
    (6 marks)
Edexcel D1 Q5
12 marks Moderate -0.3
5. This question should be answered on the sheet provided. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6c6b7934-ab46-4a87-8a11-f99bf9a5d743-05_834_1436_306_280} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure} Figure 3 shows a weighted network. The number on each arc indicates the weight of that arc.
  1. Use Dijkstra's algorithm to find a path of least weight from \(A\) to \(J\) and state the weight of the path. Your solution must show clearly how you have applied the algorithm including:
    1. the order in which the vertices were labelled,
    2. how you determined the path of least weight.
  2. Find if there are any other paths of least weight and explain your answer.
  3. Describe a practical problem that could be modelled by the above network.
Edexcel D1 Q6
15 marks Standard +0.3
6. This question should be answered on the sheet provided. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6c6b7934-ab46-4a87-8a11-f99bf9a5d743-06_723_1292_276_349} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure} Figure 4 above shows a capacitated, directed network. The number on each arc indicates the capacity of that arc.
  1. Calculate the values of cut \(C _ { 1 }\) and \(C _ { 2 }\).
  2. Find the minimum cut and state its value.
    (2 marks) \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{6c6b7934-ab46-4a87-8a11-f99bf9a5d743-06_647_1303_1430_347} \captionsetup{labelformat=empty} \caption{Fig. 5}
    \end{figure} Figure 5 shows a feasible flow through the same network.
  3. State the values of \(x , y\) and \(z\).
  4. Using this as your initial flow pattern, use the labelling procedure to find a maximal flow. You should list each flow-augmenting route you use together with its flow. State how you know that you have found a maximal flow.
    (8 marks)