Questions — Edexcel (10514 questions)

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Edexcel FM2 AS 2020 June Q2
13 marks Standard +0.3
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0567d068-e23c-446e-9e11-f0c292972093-06_531_837_258_632} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} One end of a string of length \(3 a\) is attached to a point \(A\) and the other end is attached to a point \(B\) on a smooth horizontal table. The point \(B\) is vertically below \(A\) with \(A B = a \sqrt { 3 }\) A small smooth bead, \(P\), of mass \(m\) is threaded on to the string. The bead \(P\) moves on the table in a horizontal circle, with centre \(B\), with constant speed \(U\). Both portions, \(A P\) and \(B P\), of the string are taut, as shown in Figure 2. The string is modelled as being light and inextensible and the bead is modelled as a particle.
  1. Show that \(A P = 2 a\)
  2. Find, in terms of \(m , U\) and \(a\), the tension in the string.
  3. Show that \(U ^ { 2 } < a g \sqrt { 3 }\)
  4. Describe what would happen if \(U ^ { 2 } > a g \sqrt { 3 }\)
  5. State briefly how the tension in the string would be affected if the string were not modelled as being light.
Edexcel FM2 AS 2020 June Q3
12 marks Challenging +1.2
  1. At time \(t = 0\), a toy electric car is at rest at a fixed point \(O\). The car then moves in a horizontal straight line so that at time \(t\) seconds \(( t > 0 )\) after leaving \(O\), the velocity of the car is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the acceleration of the car is modelled as \(( p + q v ) \mathrm { ms } ^ { - 2 }\), where \(p\) and \(q\) are constants.
When \(t = 0\), the acceleration of the car is \(3 \mathrm {~ms} ^ { - 2 }\) When \(t = T\), the acceleration of the car is \(\frac { 1 } { 2 } \mathrm {~ms} ^ { - 2 }\) and \(v = 4\)
  1. Show that $$8 \frac { \mathrm {~d} v } { \mathrm {~d} t } = ( 24 - 5 v )$$
  2. Find the exact value of \(T\), simplifying your answer.
Edexcel FM2 AS 2021 June Q1
6 marks Standard +0.8
1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a7901165-1679-4d30-9444-0c27020e32ea-02_744_805_246_632} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A uniform rod of length \(72 a\) is cut into pieces. The pieces are used to make two rigid squares, \(A B C D\) and \(P Q R S\), with sides of length \(10 a\) and \(8 a\) respectively. The two squares are joined to form the rigid framework shown in Figure 1. The squares both lie in the same plane with the rod \(A B\) parallel to the rod \(P Q\).
Given that
  • \(A D\) cuts \(P Q\) in the ratio \(3 : 5\)
  • \(D C\) cuts \(Q R\) in the ratio 5:3
    1. explain why the centre of mass of square \(A B C D\) is at \(Q\).
    2. Find the distance of the centre of mass of the framework from \(B\).
Edexcel FM2 AS 2021 June Q2
10 marks Challenging +1.2
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a7901165-1679-4d30-9444-0c27020e32ea-04_572_889_246_589} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A small smooth ring \(P\), of mass \(m\), is threaded onto a light inextensible string of length 4a. One end of the string is attached to a fixed point \(A\) on a smooth horizontal table. The other end of the string is attached to a fixed point \(B\) which is vertically above \(A\). The ring moves in a horizontal circle with centre \(A\) and radius \(a\), as shown in Figure 2. The ring moves with constant angular speed \(\sqrt { \frac { 2 g } { 3 a } }\) about \(A B\).
The string remains taut throughout the motion.
  1. Find, in terms of \(m\) and \(g\), the magnitude of the normal reaction between \(P\) and the table. The angular speed of \(P\) is now gradually increased.
  2. Find, in terms of \(a\) and \(g\), the angular speed of \(P\) at the instant when it loses contact with the table.
  3. Explain how you have used the fact that \(P\) is smooth.
Edexcel FM2 AS 2021 June Q3
13 marks Standard +0.3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a7901165-1679-4d30-9444-0c27020e32ea-08_547_410_246_829} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} The uniform lamina \(A B C D E F G H I J\) is shown in Figure 3.
The lamina has \(A J = 8 a , A B = 5 a\) and \(B C = D E = E F = F G = G H = H I = I J = 2 a\).
All the corners are right angles.
  1. Show that the distance of the centre of mass of the lamina from \(A J\) is \(\frac { 49 } { 26 } a\) A light inextensible rope is attached to the lamina at \(A\) and another light inextensible rope is attached to the lamina at \(B\). The lamina hangs in equilibrium with both ropes vertical and \(A B\) horizontal. The weight of the lamina is \(W\).
  2. Find, in terms of \(W\), the tension in the rope attached to the lamina at \(B\). The rope attached to \(B\) breaks and subsequently the lamina hangs freely in equilibrium, suspended from \(A\).
  3. Find the size of the angle between \(A J\) and the downward vertical.
Edexcel FM2 AS 2021 June Q4
11 marks Standard +0.8
  1. A particle \(P\) moves on the \(x\)-axis. At time \(t\) seconds, \(t \geqslant 0 , P\) is \(x\) metres from the origin \(O\) and moving with velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the direction of \(x\) increasing, where
$$v = 5 \sin 2 t$$ When \(t = 0 , x = 1\) and \(P\) is at rest.
  1. Find the magnitude and direction of the acceleration of \(P\) at the instant when \(P\) is next at rest.
  2. Show that \(1 \leqslant x \leqslant 6\)
  3. Find the total time, in the first \(4 \pi\) seconds of the motion, for which \(P\) is more than 3 metres from \(O\)
    \includegraphics[max width=\textwidth, alt={}]{a7901165-1679-4d30-9444-0c27020e32ea-16_2260_52_309_1982}
Edexcel FM2 AS 2022 June Q1
7 marks Standard +0.3
1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{99e1d643-7408-4793-9ebc-b33c91bc5fab-02_474_716_246_676} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A uniform plane lamina is in the shape of an isosceles trapezium \(A B C D E F\), as shown shaded in Figure 1.
  • \(B C E F\) is a square
  • \(A B = C D = a\)
  • \(B C = 3 a\)
    1. Show that the distance of the centre of mass of the lamina from \(A D\) is \(\frac { 11 a } { 8 }\)
The mass of the lamina is \(M\) The lamina is suspended by two light vertical strings, one attached to the lamina at \(A\) and the other attached to the lamina at \(F\) The lamina hangs freely in equilibrium, with \(B F\) horizontal.
  • Find, in terms of \(M\) and \(g\), the tension in the string attached at \(A\)
    •
    Edexcel FM2 AS 2022 June Q2
    12 marks Standard +0.8
    2. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{99e1d643-7408-4793-9ebc-b33c91bc5fab-06_554_547_246_758} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} Uniform wire is used to form the framework shown in Figure 2.
    In the framework
    • \(A B C D\) is a rectangle with \(A D = 2 a\) and \(D C = a\)
    • \(B E C\) is a semicircular arc of radius \(a\) and centre \(O\), where \(O\) lies on \(B C\)
    The diameter of the semicircle is \(B C\) and the point \(E\) is such that \(O E\) is perpendicular to \(B C\). The points \(A , B , C , D\) and \(E\) all lie in the same plane.
    1. Show that the distance of the centre of mass of the framework from \(B C\) is $$\frac { a } { 6 + \pi }$$ The framework is freely suspended from \(A\) and hangs in equilibrium with \(A E\) at an angle \(\theta ^ { \circ }\) to the downward vertical.
    2. Find the value of \(\theta\). The mass of the framework is \(M\).
      A particle of mass \(k M\) is attached to the framework at \(B\).
      The centre of mass of the loaded framework lies on \(O A\).
    3. Find the value of \(k\).
    Edexcel FM2 AS 2022 June Q3
    11 marks Standard +0.8
    1. A cyclist is travelling around a circular track which is banked at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\)
    The cyclist moves with constant speed in a horizontal circle of radius \(r\).
    In an initial model,
    • the cyclist and her cycle are modelled as a particle
    • the track is modelled as being rough so that there is sideways friction between the tyres of the cycle and the track, with coefficient of friction \(\mu\), where \(\mu < \frac { 4 } { 3 }\) Using this model, the maximum speed that the cyclist can travel around the track in a horizontal circle of radius \(r\), without slipping sideways, is \(V\).
      1. Show that \(V = \sqrt { \frac { ( 3 + 4 \mu ) r g } { 4 - 3 \mu } }\)
    In a new simplified model,
    • the cyclist and her cycle are modelled as a particle
    • the motion is now modelled so that there is no sideways friction between the tyres of the cycle and the track
    Using this new model, the speed that the cyclist can travel around the track in a horizontal circle of radius \(r\), without slipping sideways, is \(U\).
  • Find \(U\) in terms of \(r\) and \(g\).
  • Show that \(U < V\).
    •
    Edexcel FM2 AS 2022 June Q4
    10 marks Standard +0.3
    1. A particle \(P\) moves on the \(x\)-axis. At time \(t\) seconds the velocity of \(P\) is \(v \mathrm {~ms} ^ { - 1 }\) in the direction of \(x\) increasing, where
    $$v = \frac { 1 } { 2 } \left( 3 \mathrm { e } ^ { 2 t } - 1 \right) \quad t \geqslant 0$$ The acceleration of \(P\) at time \(t\) seconds is \(a \mathrm {~ms} ^ { - 2 }\)
    1. Show that \(a = 2 v + 1\)
    2. Find the acceleration of \(P\) when \(t = 0\)
    3. Find the exact distance travelled by \(P\) in accelerating from a speed of \(1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to a speed of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
    Edexcel FM2 AS 2023 June Q1
    9 marks Moderate -0.3
    1. Three particles of masses \(4 m , 2 m\) and \(k m\) are placed at the points with coordinates \(( - 3 , - 1 ) , ( 6,1 )\) and \(( - 1,5 )\) respectively.
    Given that the centre of mass of the three particles is at the point with coordinates \(( \bar { x } , \bar { y } )\)
    1. show that \(\bar { x } = \frac { - k } { k + 6 }\)
    2. find \(\bar { y }\) in terms of \(k\). Given that the centre of mass of the three particles lies on the line with equation \(y = 2 x + 3\)
    3. find the value of \(k\). A fourth particle is placed at the point with coordinates \(( \lambda , 4 )\).
      Given that the centre of mass of the four particles also lies on the line with equation \(y = 2 x + 3\)
    4. find the value of \(\lambda\).
    Edexcel FM2 AS 2023 June Q2
    8 marks Standard +0.3
    1. A particle \(P\) is moving along the \(x\)-axis.
    At time \(t\) seconds, \(t \geqslant 0 , P\) has acceleration \(a \mathrm {~ms} ^ { - 2 }\) and velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the direction of \(x\) increasing, where $$v = \mathrm { e } ^ { 2 t } + 6 \mathrm { e } ^ { t } - k t$$ and \(k\) is a positive constant.
    When \(t = \ln 2\), \(a = 0\)
    1. Find the value of \(k\). When \(t = 0\), the particle passes through the fixed point \(A\).
      When \(t = \ln 2\), the particle is \(d\) metres from \(A\).
    2. Showing all stages of your working, find the value of \(d\) correct to 2 significant figures.
      [0pt] [Solutions relying entirely on calculator technology are not acceptable.]
    Edexcel FM2 AS 2023 June Q3
    9 marks Standard +0.8
    1. A girl is cycling round a circular track.
    The girl and her bicycle have a combined mass of 55 kg .
    The coefficient of friction between the track surface and the tyres of the bicycle is \(\mu\).
    The track is banked at an angle of \(15 ^ { \circ }\) to the horizontal.
    The girl and her bicycle are modelled as a particle moving in a horizontal circle of radius 50 m
    The minimum speed at which the girl can cycle round this circle without slipping is \(4.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) Using the model, find the value of \(\mu\).
    Edexcel FM2 AS 2023 June Q4
    14 marks Standard +0.8
    4. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{8fcae18f-6588-4b71-8b7f-c8408de591f4-12_819_853_255_607} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} A uniform triangular lamina \(A B C\) is isosceles, with \(A C = B C\). The midpoint of \(A B\) is \(M\). The length of \(A B\) is \(18 a\) and the length of \(C M\) is \(18 a\). The triangular lamina \(C D E\), with \(D E = 6 a\) and \(C D = 12 a\), has \(E D\) parallel to \(A B\) and \(M D C\) is a straight line. Triangle \(C D E\) is removed from triangle \(A B C\) to form the lamina \(L\), shown shaded in Figure 1. The distance of the centre of mass of \(L\) from \(M C\) is \(d\).
    1. Show that \(d = \frac { 4 } { 7 } a\) The lamina \(L\) is suspended by two light inextensible strings. One string is attached to \(L\) at \(A\) and the other string is attached to \(L\) at \(B\).
      The lamina hangs in equilibrium in a vertical plane with the strings vertical and \(A B\) horizontal.
      The weight of \(L\) is \(W\)
    2. Find, in terms of \(W\), the tension in the string attached to \(L\) at \(B\) The string attached to \(L\) at \(B\) breaks, so that \(L\) is now suspended from \(A\). When \(L\) is hanging in equilibrium in a vertical plane, the angle between \(A B\) and the downward vertical through \(A\) is \(\theta ^ { \circ }\)
    3. Find the value of \(\theta\)
    Edexcel FM2 AS 2024 June Q1
    7 marks Standard +0.3
    1. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{fd8bc7b5-adee-4d67-b15d-571255b00b83-02_586_824_244_623} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} A uniform rod of length \(24 a\) is cut into seven pieces which are used to form the framework \(A B C D E F\) shown in Figure 1. It is given that
    • \(A F = B E = C D = A B = F E = 4 a\)
    • \(B C = E D = 2 a\)
    • the rods \(A F , B E\) and \(C D\) are parallel
    • the rods \(A B , B C , F E\) and \(E D\) are parallel
    • \(A F\) is perpendicular to \(A B\)
    • the rods all lie in the same plane
    The distance of the centre of mass of the framework from \(A F\) is \(d\).
    1. Show that \(d = \frac { 19 } { 6 } a\)
    2. Find the distance of the centre of mass of the framework from \(A\).
    Edexcel FM2 AS 2024 June Q2
    10 marks Standard +0.3
    2. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{fd8bc7b5-adee-4d67-b15d-571255b00b83-04_529_794_246_639} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} A thin hollow hemisphere, with centre \(O\) and radius \(a\), is fixed with its axis vertical, as shown in Figure 2. A small ball \(B\) of mass \(m\) moves in a horizontal circle on the inner surface of the hemisphere. The circle has centre \(C\) and radius \(r\). The point \(C\) is vertically below \(O\) such that \(O C = h\). The ball moves with constant angular speed \(\omega\) The inner surface of the hemisphere is modelled as being smooth and \(B\) is modelled as a particle. Air resistance is modelled as being negligible.
    1. Show that \(\omega ^ { 2 } = \frac { g } { h }\) Given that the magnitude of the normal reaction between \(B\) and the surface of the hemisphere is \(3 m g\)
    2. find \(\omega\) in terms of \(g\) and \(a\).
    3. State how, apart from ignoring air resistance, you have used the fact that \(B\) is modelled as a particle.
    Edexcel FM2 AS 2024 June Q3
    11 marks Standard +0.3
    1. A particle \(P\) is moving along the \(x\)-axis. At time \(t\) seconds, \(P\) has velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the positive \(x\) direction and acceleration \(a \mathrm {~ms} ^ { - 2 }\) in the positive \(x\) direction.
    In a model of the motion of \(P\) $$a = 4 - 3 v$$ When \(t = 0 , v = 0\)
    1. Use integration to show that \(v = k \left( 1 - \mathrm { e } ^ { - 3 t } \right)\), where \(k\) is a constant to be found. When \(t = 0 , P\) is at the origin \(O\)
    2. Find, in terms of \(t\) only, the distance of \(P\) from \(O\) at time \(t\) seconds.
    Edexcel FM2 AS 2024 June Q4
    12 marks Challenging +1.2
    4. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{fd8bc7b5-adee-4d67-b15d-571255b00b83-12_351_597_246_735} \captionsetup{labelformat=empty} \caption{Figure 3}
    \end{figure} The uniform triangular lamina \(A B C\) has \(A B\) perpendicular to \(A C\), \(A B = 9 a\) and \(A C = 6 a\). The point \(D\) on \(A B\) is such that \(A D = a\). The rectangle \(D E F G\), with \(D E = 2 a\) and \(E F = 3 a\), is removed from the lamina to form the template shown shaded in Figure 3. The distance of the centre of mass of the template from \(A C\) is \(d\).
    1. Show that \(d = \frac { 23 } { 7 } a\) The template is freely suspended from \(A\) and hangs in equilibrium with \(A B\) at an angle \(\theta ^ { \circ }\) to the downward vertical through \(A\).
    2. Find the value of \(\theta\) A new piece, of exactly the same size and shape as the template, is cut from a lamina of a different uniform material. The template and the new piece are joined together to form the model shown in Figure 4. Both parts of the model lie in the same plane. \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{fd8bc7b5-adee-4d67-b15d-571255b00b83-12_369_1185_1667_440} \captionsetup{labelformat=empty} \caption{Figure 4}
      \end{figure} The weight of \(C P Q R S T A\) is \(W\) The weight of \(A D G F E B C\) is \(4 W\) The model is freely suspended from \(A\).
      A horizontal force of magnitude \(X\), acting in the same vertical plane as the model, is now applied to the model at \(T\) so that \(A C\) is vertical, as shown in Figure 4.
    3. Find \(X\) in terms of \(W\).
    Edexcel FM2 AS Specimen Q1
    8 marks Standard +0.3
    1. A particle \(P\) moves on the \(x\)-axis. At time \(t\) seconds the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the direction of \(x\) increasing, where
    $$v = ( t - 2 ) ( 3 t - 10 ) , \quad t \geqslant 0$$ When \(t = 0 , P\) is at the origin \(O\).
    1. Find the acceleration of \(P\) at time \(t\) seconds.
    2. Find the total distance travelled by \(P\) in the first 2 seconds of its motion.
    3. Show that \(P\) never returns to \(O\), explaining your reasoning.
    Edexcel FM2 AS Specimen Q2
    16 marks Challenging +1.2
    A light inextensible string has length 7a. One end of the string is attached to a fixed point \(A\) and the other end of the string is attached to a fixed point \(B\), with \(A\) vertically above \(B\) and \(A B = 5 a\). A particle of mass \(m\) is attached to a point \(P\) on the string where \(A P = 4 a\). The particle moves in a horizontal circle with constant angular speed \(\omega\), with both \(A P\) and \(B P\) taut.
    1. Show that
      1. the tension in \(A P\) is \(\frac { 4 m } { 25 } \left( 9 a \omega ^ { 2 } + 5 g \right)\)
      2. the tension in \(B P\) is \(\frac { 3 m } { 25 } \left( 16 a \omega ^ { 2 } - 5 g \right)\). The string will break if the tension in it reaches a magnitude of \(4 m g\).
        The time for the particle to make one revolution is \(S\).
    2. Show that $$3 \pi \sqrt { \frac { a } { 5 g } } < S < 8 \pi \sqrt { \frac { a } { 5 g } }$$
    3. State how in your calculations you have used the assumption that the string is light.
    Edexcel FM2 AS Specimen Q3
    16 marks Challenging +1.2
    3. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{5bfd2018-ea46-4ea5-9cf7-4210d125a91c-07_611_1146_280_456} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} Figure 1 shows the shape and dimensions of a template \(O P Q R S T U V\) made from thin uniform metal. \(O P = 5 \mathrm {~m} , P Q = 2 \mathrm {~m} , Q R = 1 \mathrm {~m} , R S = 1 \mathrm {~m} , T U = 2 \mathrm {~m} , U V = 1 \mathrm {~m} , V O = 3 \mathrm {~m}\).
    Figure 1 also shows a coordinate system with \(O\) as origin and the \(x\)-axis and \(y\)-axis along \(O P\) and \(O V\) respectively. The unit of length on both axes is the metre. The centre of mass of the template has coordinates \(( \bar { x } , \bar { y } )\).
      1. Show that \(\bar { y } = 1\)
      2. Find the value of \(\bar { x }\). A new design requires the template to have its centre of mass at the point (2.5,1). In order to achieve this, two circular discs, each of radius \(r\) metres, are removed from the template which is shown in Figure 1, to form a new template \(L\). The centre of the first disc is ( \(0.5,0.5\) ) and the centre of the second disc is ( \(0.5 , a\) ) where \(a\) is a constant.
    2. Find the value of \(r\).
      1. Explain how symmetry can be used to find the value of \(a\).
      2. Find the value of \(a\). The template \(L\) is now freely suspended from the point \(U\) and hangs in equilibrium.
    4. Find the size of the angle between the line \(T U\) and the horizontal.
    Edexcel FD1 AS 2018 June Q1
    9 marks Moderate -0.5
    1. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{3e853c6d-e90e-4a09-b990-1c2c146b54e1-2_1105_1459_463_402} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} Figure 1 represents a network of roads.
    The number on each arc represents the time taken, in minutes, to drive along the corresponding road.
      1. Use Dijkstra's algorithm to find the shortest time needed to travel from A to H .
      2. State the quickest route. For a network with \(n\) vertices, Dijkstra's algorithm has order \(n ^ { 2 }\)
    2. If it takes 1.5 seconds to run the algorithm when \(n = 250\), calculate approximately how long it will take, in seconds, to run the algorithm when \(n = 9500\). You should make your method and working clear.
    3. Explain why your answer to part (b) is only an approximation.
    Edexcel FD1 AS 2018 June Q2
    10 marks Moderate -0.8
    2. A simply connected graph is a connected graph in which any two vertices are directly connected by at most one arc and no vertex is directly connected to itself.
    1. Given that a simply connected graph has exactly four vertices,
      1. write down the minimum number of arcs it can have,
      2. write down the maximum number of arcs it can have.
      1. Draw a simply connected graph that has exactly four vertices and exactly five arcs.
      2. State, with justification, whether your graph is Eulerian, semi-Eulerian or neither.
    3. By considering the orders of the vertices, explain why there is only one simply connected graph with exactly four vertices and exactly five arcs.
    Edexcel FD1 AS 2018 June Q3
    10 marks Moderate -0.8
    3.
    ActivityTime taken (days)Immediately preceding activities
    A5-
    B8-
    C4-
    D14A
    E10A
    F3B, C, E
    G7C
    H5D, F, G
    I7H
    J9H
    The table above shows the activities required for the completion of a building project. For each activity, the table shows the time it takes, in days, and the immediately preceding activities. Each activity requires one worker. The project is to be completed in the shortest possible time. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{3e853c6d-e90e-4a09-b990-1c2c146b54e1-4_486_1161_1194_551} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} Figure 2 shows a partially completed activity network used to model the project. The activities are represented by the arcs and the number in brackets on each arc is the time taken, in days, to complete the corresponding activity.
    1. Add the missing activities and necessary dummies to Diagram 1 in the answer book.
    2. Complete Diagram 1 in the answer book to show the early event times and the late event times.
    3. State the critical activities. At the beginning of the project it is decided that activity G is no longer required.
    4. Explain what effect, if any, this will have on
      1. the shortest completion time of the project if activity G is no longer required,
      2. the timing of the remaining activities.
    Edexcel FD1 AS 2018 June Q4
    11 marks Standard +0.3
    4. The manager of a factory is planning the production schedule for the next three weeks for a range of cabinets. The following constraints apply to the production schedule.
    • The total number of cabinets produced in week 3 cannot be fewer than the total number produced in weeks 1 and 2
    • At most twice as many cabinets must be produced in week 3 as in week 2
    • The number of cabinets produced in weeks 2 and 3 must, in total, be at most 125
    The production cost for each cabinet produced in weeks 1,2 and 3 is \(\pounds 250 , \pounds 275\) and \(\pounds 200\) respectively.
    The factory manager decides to formulate a linear programming problem to find a production schedule that minimises the total cost of production. The objective is to minimise \(250 x + 275 y + 200 z\)
    1. Explain what the variables \(x , y\) and \(z\) represent.
    2. Write down the constraints of the linear programming problem in terms of \(x , y\) and \(z\). Due to demand, exactly 150 cabinets must be produced during these three weeks. This reduces the constraints to $$\begin{gathered} x + y \leqslant 75 \\ x + 3 y \geqslant 150 \\ x \geqslant 25 \\ y \geqslant 0 \end{gathered}$$ which are shown in Diagram 1 in the answer book.
      Given that the manager does not want any cabinets left unfinished at the end of a week,
      1. use a graphical approach to solve the linear programming problem and hence determine the production schedule which minimises the cost of production. You should make your method and working clear.
      2. Find the minimum total cost of the production schedule.