Questions - Edexcel - A-Level Maths

Show that the distance of the centre of mass of the dome from the centre of its plane face is $\left( 4 \ln 2 - \frac { 5 } { 2 } \right) a$ metres. A solid uniform right circular cone has base radius $\sqrt { 3 } a$ metres and perpendicular height $4 a$ metres. A toy is formed by attaching the plane surface of the dome to the plane surface of the cone, as shown in Figure 6. The weight of the cone is $k W$ and the weight of the dome is $2 W$
The centre of mass of the toy is a distance $d$ metres from the plane face of the dome.
Show that $d = \frac { | k + 5 - 8 \ln 2 | } { 2 + k } a$ The toy is suspended from a point on the circumference of the plane face of the dome and hangs freely in equilibrium with the plane face of the dome at an angle $\alpha$ to the downward vertical.
Given that $\tan \alpha = \frac { 1 } { 2 \sqrt { 3 } }$
find the exact value of $k$.

Edexcel FM2 2023 June Q8

14 marks Challenging +1.2

8. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{3b070338-1de4-4c33-be29-d37ac06c9fed-28_200_1086_214_552} \captionsetup{labelformat=empty} \caption{Figure 7}

\end{figure} The fixed points $A$ and $B$ lie on a smooth horizontal surface with $A B = 6 \mathrm {~m}$.
A particle $P$ has mass 0.3 kg .
One end of a light elastic string, of natural length 2 m and modulus of elasticity 20 N , is attached to $P$, and the other end is attached to $A$. One end of another light elastic string, of natural length 2 m and modulus of elasticity 40 N , is attached to $P$ and the other end is attached to $B$. The particle $P$ is at rest in equilibrium at the point $E$ on the surface, as shown in Figure 7.

Show that $E B = \frac { 8 } { 3 } \mathrm {~m}$. The particle $P$ is now held at the midpoint of $A B$ and released from rest.
Show that $P$ oscillates with simple harmonic motion about the point $E$. The time between the instant when $P$ is released and the instant when it first returns to the point $E$ is $S$ seconds.
Find the exact value of $S$.
Find the length of time during one oscillation for which the speed of $P$ is more than $2 \mathrm {~ms} ^ { - 1 }$

Edexcel FM2 2024 June Q1

9 marks Standard +0.3

In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

A particle $P$ moves along a straight line. Initially $P$ is at rest at the point $O$ on the line. At time $t$ seconds, where $t \geqslant 0$

the displacement of $P$ from $O$ is $x$ metres
the velocity of $P$ is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ in the positive $x$ direction
the acceleration of $P$ is $\frac { 96 } { ( 3 t + 5 ) ^ { 3 } } \mathrm {~ms} ^ { - 2 }$ in the positive $x$ direction
1. Show that, at time $t$ seconds, $v = p - \frac { q } { ( 3 t + 5 ) ^ { 2 } }$, where $p$ and $q$ are constants to be determined.
2. Find the limiting value of $v$ as $t$ increases.
3. Find the value of $x$ when $t = 2$

Edexcel FM2 2024 June Q2

7 marks Standard +0.8

2. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{c14975b7-6afa-44ce-beab-1cba2e82b249-06_373_847_251_609} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} A uniform rod of length $28 a$ is cut into seven identical rods each of length $4 a$. These rods are joined together to form the rigid framework $A B C D E A$ shown in Figure 1. All seven rods lie in the same plane.
The distance of the centre of mass of the framework from $E D$ is $d$.

Show that $d = \frac { 8 \sqrt { 3 } } { 7 } a$ The weight of the framework is $W$.
The framework is freely pivoted about a horizontal axis through $C$.
The framework is held in equilibrium in a vertical plane, with $A C$ vertical and $A$ below $C$, by a horizontal force that is applied to the framework at $A$. The force acts in the same vertical plane as the framework and has magnitude $F$.
Find $F$ in terms of $W$.

Edexcel FM2 2024 June Q3

9 marks Standard +0.3

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{c14975b7-6afa-44ce-beab-1cba2e82b249-10_433_753_246_657} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 2 shows a hemispherical bowl of internal radius $10 d$ that is fixed with its circular rim horizontal. The centre of the circular rim is at the point $O$.
A particle $P$ moves with constant angular speed on the smooth inner surface of the bowl. The particle $P$ moves in a horizontal circle with radius $8 d$ and centre $C$.

Find, in terms of $g$, the exact magnitude of the acceleration of $P$. The time for $P$ to complete one revolution is $T$.
Find $T$ in terms of $d$ and $g$.

Edexcel FM2 2024 June Q4

12 marks Standard +0.8

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{c14975b7-6afa-44ce-beab-1cba2e82b249-14_675_528_242_772} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} A uniform lamina $O A B$ is in the shape of the region $R$.
Region $R$ lies in the first quadrant and is bounded by the curve with equation $\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 36 } = 1$, the $x$-axis, and the $y$-axis, as shown shaded in Figure 3. The point $A$ is the point of intersection of the curve and the $x$-axis.
The point $B$ is the point of intersection of the curve and the $y$-axis.
One unit on each axis represents 1 m .
The area of $R$ is $6 \pi$
The centre of mass of $R$ lies at the point with coordinates $( \bar { x } , \bar { y } )$

Use algebraic integration to show that $\bar { x } = \frac { 16 } { 3 \pi }$
Use algebraic integration to find the exact value of $\bar { y }$ The lamina is freely suspended from $A$ and hangs in equilibrium with $O A$ at angle $\theta ^ { \circ }$ to the downward vertical.
Find the value of $\theta$

Edexcel FM2 2024 June Q5

11 marks Standard +0.8

A particle $P$ moves in a straight line with simple harmonic motion about a fixed point $O$. The magnitude of the greatest acceleration of $P$ is $18 \mathrm {~m} \mathrm {~s} ^ { - 2 }$

When $P$ is 0.3 m from $O$, the speed of $P$ is $2.4 \mathrm {~ms} ^ { - 1 }$
The amplitude of the motion is $a$ metres.

Show that $a = 0.5$
Find the greatest speed of $P$. During one oscillation, the speed of $P$ is at least $2 \mathrm {~ms} ^ { - 1 }$ for $S$ seconds.
Find the value of $S$.

Edexcel FM2 2024 June Q6

13 marks Challenging +1.2

6. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{c14975b7-6afa-44ce-beab-1cba2e82b249-20_245_435_356_817} \captionsetup{labelformat=empty} \caption{Figure 4}

\end{figure} The shaded region, shown in Figure 4, is bounded by the $x$-axis, the line with equation $x = 6$, the line with equation $y = 2$ and the $y$-axis. This region is rotated through $360 ^ { \circ }$ about the $\boldsymbol { x }$-axis to form a solid of revolution. This solid is used to model a non-uniform cylinder of height 6 cm and radius 2 cm . The mass per unit volume of the cylinder at the point $( x , y , z )$ is $\lambda ( x + 2 ) \mathrm { kg } \mathrm { cm } ^ { - 3 }$, where $0 \leqslant x \leqslant 6$ and $\lambda$ is a constant.

Show that the mass of the cylinder is $120 \lambda \pi \mathrm {~kg}$.
Show that the centre of mass of the cylinder is 3.6 cm from $O$. The point $O$ is the centre of one end of the cylinder. The point $A$ is the centre of the other end of the cylinder. A uniform solid hemisphere of radius 3 cm has density $\lambda \mathrm { kg } \mathrm { cm } ^ { - 3 }$. The hemisphere is attached to the cylinder with the centre of its circular face in contact with the point $A$ on the cylinder to form the model shown in Figure 5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c14975b7-6afa-44ce-beab-1cba2e82b249-20_309_673_1713_696} \captionsetup{labelformat=empty} \caption{Figure 5}
\end{figure} The model is placed with the end containing $O$ on a rough inclined plane which is inclined at angle $\alpha ^ { \circ }$ to the horizontal. The plane is sufficiently rough to prevent the model from sliding. The model is on the point of toppling.
Find the value of $\alpha$.

Edexcel FM2 2024 June Q7

14 marks Challenging +1.2

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{c14975b7-6afa-44ce-beab-1cba2e82b249-24_419_935_251_566} \captionsetup{labelformat=empty} \caption{Figure 6}

\end{figure} A smooth solid hemisphere has radius $r$ and the centre of its plane face is $O$.
The hemisphere is fixed with its plane face in contact with horizontal ground, as shown in Figure 6.
A small stone is at the point $A$, the highest point on the surface of the hemisphere. The stone is projected horizontally from $A$ with speed $U$.
The stone is still in contact with the hemisphere at the point $B$, where $O B$ makes an angle $\theta$ with the upward vertical.
The speed of the stone at the instant it reaches $B$ is $v$.
The stone is modelled as a particle $P$ and air resistance is modelled as being negligible.

Use the model to find $v ^ { 2 }$ in terms of $U , r , g$ and $\theta$ When $P$ leaves the surface of the hemisphere, the speed of $P$ is $W$.
Given that $U = \sqrt { \frac { 2 r g } { 3 } }$
show that $W ^ { 2 } = \frac { 8 } { 9 } r g$ After leaving the surface of the hemisphere, $P$ moves freely under gravity until it hits the ground.
Find the speed of $P$ as it hits the ground, giving your answer in terms of $r$ and $g$. At the instant when $P$ hits the ground it is travelling at $\alpha ^ { \circ }$ to the horizontal.
Find the value of $\alpha$.

Edexcel FM2 Specimen Q1

7 marks Standard +0.3

A flag pole is 15 m long.

The flag pole is non-uniform so that, at a distance $x$ metres from its base, the mass per unit length of the flag pole, $m \mathrm {~kg} \mathrm {~m} ^ { - 1 }$ is given by the formula $m = 10 \left( 1 - \frac { x } { 25 } \right)$. The flag pole is modelled as a rod.

Show that the mass of the flag pole is 105 kg .
Find the distance of the centre of mass of the flag pole from its base.

Edexcel FM2 Specimen Q2

8 marks Challenging +1.2

2. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{f06704e5-454c-41c1-9577-b1210f60480d-04_655_643_207_639} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} A hollow right circular cone, of base diameter $4 a$ and height $4 a$ is fixed with its axis vertical and vertex $V$ downwards, as shown in Figure 1. A particle of mass $m$ moves in a horizontal circle with centre $C$ on the rough inner surface of the cone with constant angular speed $\omega$. The height of $C$ above $V$ is $3 a$.
The coefficient of friction between the particle and the inner surface of the cone is $\frac { 1 } { 4 }$. Find, in terms of $a$ and $g$, the greatest possible value of $\omega$.

Edexcel FM2 Specimen Q3

9 marks Challenging +1.2

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{f06704e5-454c-41c1-9577-b1210f60480d-06_608_924_226_541} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} A uniform solid cylinder has radius $2 a$ and height $h ( h > a )$.
A solid hemisphere of radius $a$ is removed from the cylinder to form the vessel $V$.
The plane face of the hemisphere coincides with the upper plane face of the cylinder.
The centre $O$ of the hemisphere is also the centre of the upper plane face of the cylinder, as shown in Figure 2.

Show that the centre of mass of $V$ is $\frac { 3 \left( 8 h ^ { 2 } - a ^ { 2 } \right) } { 8 ( 6 h - a ) }$ from $O$. The vessel $V$ is placed on a rough plane which is inclined at an angle $\phi$ to the horizontal. The lower plane circular face of $V$ is in contact with the inclined plane. Given that $h = 5 a$, the plane is sufficiently rough to prevent $V$ from slipping and $V$ is on the point of toppling,
find, to three significant figures, the size of the angle $\phi$.

Edexcel FM2 Specimen Q4

11 marks Challenging +1.2

A car of mass 500 kg moves along a straight horizontal road.

The engine of the car produces a constant driving force of 1800 N .
The car accelerates from rest from the fixed point $O$ at time $t = 0$ and at time $t$ seconds the car is $x$ metres from $O$, moving with speed $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$. When the speed of the car is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$, the resistance to the motion of the car has magnitude $2 v ^ { 2 } \mathrm {~N}$. At time $T$ seconds, the car is at the point $A$, moving with speed $10 \mathrm {~ms} ^ { - 1 }$.

Show that $T = \frac { 25 } { 6 } \ln 2$
Show that the distance from $O$ to $A$ is $125 \ln \frac { 9 } { 8 } \mathrm {~m}$.

Edexcel FM2 Specimen Q5

12 marks Challenging +1.2

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{f06704e5-454c-41c1-9577-b1210f60480d-12_693_515_210_781} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} A shop sign is modelled as a uniform rectangular lamina $A B C D$ with a semicircular lamina removed. The semicircle has radius $a , B C = 4 a$ and $C D = 2 a$.
The centre of the semicircle is at the point $E$ on $A D$ such that $A E = d$, as shown in Figure 3.

Show that the centre of mass of the sign is $\frac { 44 a } { 3 ( 16 - \pi ) }$ from $A D$. The sign is suspended using vertical ropes attached to the sign at $A$ and at $B$ and hangs in equilibrium with $A B$ horizontal. The weight of the sign is $W$ and the ropes are modelled as light inextensible strings.
Find, in terms of $W$ and $\pi$, the tension in the rope attached at $B$. The rope attached at $B$ breaks and the sign hangs freely in equilibrium suspended from $A$, with $A D$ at an angle $\alpha$ to the downward vertical. Given that $\tan \alpha = \frac { 11 } { 18 }$
find $d$ in terms of $a$ and $\pi$.

Edexcel FM2 Specimen Q6

14 marks Standard +0.8

A small bead $B$ of mass $m$ is threaded on a circular hoop.

The hoop has centre $O$ and radius $a$ and is fixed in a vertical plane.
The bead is projected with speed $\sqrt { \frac { 7 } { 2 } g a }$ from the lowest point of the hoop.
The hoop is modelled as being smooth.
When the angle between $O B$ and the downward vertical is $\theta$, the speed of $B$ is $v$.

Show that $v ^ { 2 } = g a \left( \frac { 3 } { 2 } + 2 \cos \theta \right)$
Find the size of $\theta$ at the instant when the contact force between $B$ and the hoop is first zero.
Give a reason why your answer to part (b) is not likely to be the actual value of $\theta$.
Find the magnitude and direction of the acceleration of $B$ at the instant when $B$ is first at instantaneous rest.

Edexcel FM2 Specimen Q7

14 marks Challenging +1.8

Two points $A$ and $B$ are 6 m apart on a smooth horizontal surface.

A light elastic string of natural length 2 m and modulus of elasticity 20 N , has one end attached to the point $A$. A second light elastic string of natural length 2 m and modulus of elasticity 50 N , has one end attached to the point $B$. A particle $P$ of mass 3.5 kg is attached to the free end of each string.
The particle $P$ is held at the point on $A B$ which is 2 m from $B$ and then released from rest.
In the subsequent motion both strings remain taut.

Show that $P$ moves with simple harmonic motion about its equilibrium position.
Find the maximum speed of $P$.
Find the length of time within each oscillation for which $P$ is closer to $A$ than to $B$.

Edexcel FD1 2019 June Q1

8 marks Easy -1.2

1. 0.2
1.7
1.9
1.2
1.4
1.5
2.1
3.0
3.2
3.3

Use the first-fit bin packing algorithm to determine how the numbers listed above can be packed into bins of size 5 The list of numbers is now to be sorted into descending order.
Perform a quick sort on the original list to obtain the sorted list. You should show the result of each pass and identify your pivots clearly. For a list of $n$ numbers, the quick sort algorithm has, on average, order $n \log n$.
Given that it takes 2.32 seconds to run the algorithm when $n = 450$
calculate approximately how long it will take, to the nearest tenth of a second, to run the algorithm when $n = 11250$. You should make your method and working clear.

Edexcel FD1 2019 June Q2

14 marks Standard +0.3

2. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{162f9d72-84a4-4b1a-93cf-b7eeb7f957ae-03_663_1421_203_322} \captionsetup{labelformat=empty} \caption{Figure 1
[0pt] [The total weight of the network is 370]}

\end{figure} Figure 1 represents a network of corridors in a building. The number on each arc represents the length, in metres, of the corresponding corridor.

Use Dijkstra's algorithm to find the shortest path from A to D, stating the path and its length. On a particular day, Naasir needs to check the paintwork along each corridor. Naasir must find a route of minimum length. It must traverse each corridor at least once, starting at B and finishing at G .
Use an appropriate algorithm to find the arcs that will need to be traversed twice. You must make your method and working clear.
Find the length of Naasir's route. On a different day, all the corridors that start or finish at B are closed for redecorating. Naasir needs to check all the remaining corridors and may now start at any vertex and finish at any vertex. A route is required that excludes all those corridors that start or finish at B .
1. Determine the possible starting and finishing points so that the length of Naasir's route is minimised. You must give reasons for your answer.
2. Find the length of Naasir's new route.

Edexcel FD1 2019 June Q3

14 marks Moderate -0.5

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{162f9d72-84a4-4b1a-93cf-b7eeb7f957ae-04_666_940_173_534} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} The network in Figure 2 shows the direct roads linking five villages, $\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D }$ and E .
The number on each arc represents the length, in miles, of the corresponding road.
The roads from A to E and from C to B are one-way, as indicated by the arrows.

Complete the initial distance and route tables for the network provided in the answer book.
(2)
Perform the first three iterations of Floyd's algorithm. You should show the distance table and the route table after each of the three iterations. After five iterations of Floyd's algorithm the final distance table and partially completed final route table are shown below. \begin{table}[h]
\captionsetup{labelformat=empty} \caption{Distance table}
\cline { 2 - 6 } \multicolumn{1}{c|}{} A B C D E
A - 12 7 6 3
B 15 - 22 21 18
C 7 5 - 4 7
D 11 9 4 - 3
E 14 12 7 3 -
\end{table} \begin{table}[h]
\captionsetup{labelformat=empty} \caption{Route table}
\cline { 2 - 6 } \multicolumn{1}{c|}{} A B C D E
A A
B A B
C A B C
D C C C D
E D D D D E
\end{table}
1. Explain how the partially completed final route table can be used to find the shortest route from E to A.
2. State this route. Mabintou decides to use the distance table to try to find the shortest cycle that passes through each vertex. Starting at D, she applies the nearest neighbour algorithm to the final distance table.
1. State the cycle obtained using the nearest neighbour algorithm.
2. State the length of this cycle.
3. Interpret the cycle in terms of the actual villages visited.
4. Prove that Mabintou's cycle is not optimal.

Edexcel FD1 2019 June Q4

9 marks Standard +0.3

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{162f9d72-84a4-4b1a-93cf-b7eeb7f957ae-05_1004_1797_205_134} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} The network in Figure 3 shows the activities that need to be undertaken to complete a project. Each activity is represented by an arc and the duration of the activity, in days, is shown in brackets. The early event times and late event times are to be shown at each vertex and one late event time has been completed for you. The total float of activity H is 7 days.

Explain, with detailed reasoning, why $x = 11$
Determine the missing early event times and late event times, and hence complete Diagram 1 in your answer book. Each activity requires one worker and the project must be completed in the shortest possible time using as few workers as possible.
Calculate a lower bound for the number of workers needed to complete the project in the shortest possible time.
Schedule the activities using Grid 1 in the answer book.

Edexcel FD1 2019 June Q5

6 marks Moderate -0.5

Activity	Immediately preceding activities
A	-
B	-
C	-
D	A
E	C
F	B, C, D
G	A
H	B, C, D
I	B, C, D, G
J	B, C, D, G
K	E, H

Draw the activity network described in the precedence table above, using activity on arc. Your activity network must contain only the minimum number of dummies. Given that all the activities shown in the precedence table have the same duration, (b) state the critical path for the network.

Edexcel FD1 2019 June Q6

12 marks Standard +0.8

6. A linear programming problem in $x , y$ and $z$ is described as follows. Maximise $\quad P = 2 x + 2 y - z$
subject to $\quad 3 x + y + 2 z \leqslant 30$ $$\begin{aligned} x - y + z & \geqslant 8 \\ 4 y + 2 z & \geqslant 15 \\ x , y , z & \geqslant 0 \end{aligned}$$

Explain why the Simplex algorithm cannot be used to solve this linear programming problem.

Set up the initial tableau for solving this linear programming problem using the big-M method. After a first iteration of the big-M method, the tableau is

b.v.	$x$	$y$	$z$	$s _ { 1 }$	$S _ { 2 }$	$S _ { 3 }$	$a _ { 1 }$	$a _ { 2 }$	Value
$s _ { 1 }$	3	0	1.5	1	0	0.25	0	-0.25	26.25
$a _ { 1 }$	1	0	1.5	0	-1	-0.25	1	0.25	11.75
$y$	0	1	0.5	0	0	-0.25	0	0.25	3.75
$P$	$- ( 2 + M )$	0	2-1.5M	0	M	$- 0.5 + 0.25 M$	0	$0.5 + 0.75 M$	7.5-11.75M

State the value of each variable after the first iteration.
Explain why the solution given by the first iteration is not feasible. Taking the most negative entry in the profit row to indicate the pivot column,
obtain the most efficient pivot for a second iteration. You must give reasons for your answer.

Edexcel FD1 2020 June Q1

6 marks Moderate -0.8

The table below shows the lengths, in km , of the roads in a network connecting seven towns, $\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D } , \mathrm { E } , \mathrm { F }$ and G .

	A	B	C	D	E	F	G
A	-	24	-	22	35	-	-
B	24	-	25	27	-	-	-
C	-	25	-	33	31	36	26
D	22	27	33	-	-	42	-
E	35	-	31	-	-	37	29
F	-	-	36	42	37	-	40
G	-	-	26	-	29	40	-

By adding the arcs from vertex D along with their weights, complete the drawing of the network on Diagram 1 in the answer book.
Use Kruskal's algorithm to find a minimum spanning tree for the network. You should list the arcs in the order that you consider them. In each case, state whether you are adding the arc to your minimum spanning tree.
State the weight of the minimum spanning tree.

Edexcel FD1 2020 June Q2

15 marks Moderate -0.3

2. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{bd357978-6464-43fd-854f-4188b5408e91-03_688_1102_267_482} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} The network in Figure 1 shows the activities that need to be undertaken to complete a project. Each activity is represented by an arc and the duration, in hours, of the corresponding activity is shown in brackets.

Explain why each of the dummy activities is required.
Complete the table in the answer book to show the immediately preceding activities for each activity.
1. Complete Diagram 1 in the answer book to show the early event times and the late event times.
2. State the minimum completion time for the project.
3. State the critical activities. Each activity requires one worker. Each worker is able to do any of the activities. Once an activity is started it must be completed without interruption.
On Grid 1 in the answer book, draw a resource histogram to show the number of workers required at each time when each activity begins at its earliest possible start time.
Determine whether or not the project can be completed in the minimum possible time using fewer workers than the number indicated by the resource histogram in (d). You must justify your answer with reference to the resource histogram and the completed Diagram 1.

Edexcel FD1 2020 June Q3

9 marks Moderate -0.5

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{bd357978-6464-43fd-854f-4188b5408e91-04_387_519_214_774} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Direct roads between five villages, A, B, C, D and E, are shown in Figure 2. The weight on each arc is the time, in minutes, it takes to travel along the corresponding road. The road from D to C is one-way as indicated by the arrow on the corresponding arc. Floyd's algorithm is to be used to find the complete network of shortest times between the five villages.

Set up initial time and route matrices. The matrices after two iterations of Floyd's algorithm are shown below. \begin{table}[h]
\captionsetup{labelformat=empty} \caption{Time matrix}
\cline { 2 - 6 } \multicolumn{1}{c|}{} A B C D E
A - 8 4 7 18
B 8 - 3 15 10
C 4 3 - 11 6
D 7 15 1 - 1
E 18 10 6 1 -
\end{table} \begin{table}[h]
\captionsetup{labelformat=empty} \caption{Route matrix}
\cline { 2 - 6 } \multicolumn{1}{c|}{} A B C D E
A A B C D B
B A B C A E
C A B C A E
D A A C D E
E B B C D E
\end{table}
Perform the next two iterations of Floyd's algorithm that follow from the tables above. You should show the time and route matrices after each iteration. The final time matrix after completion of Floyd's algorithm is shown below. \begin{table}[h]
\captionsetup{labelformat=empty} \caption{Final time matrix}
\cline { 2 - 6 } \multicolumn{1}{c|}{} A B C D E
A - 7 4 7 8
B 7 - 3 10 9
C 4 3 - 7 6
D 5 4 1 - 1
E 6 5 2 1 -
\end{table}
1. Use the nearest neighbour algorithm, starting at A , to find a Hamiltonian cycle in the complete network of shortest times.
2. Find the time taken for this cycle.
3. Interpret the cycle in terms of the actual villages visited.

\cline { 2 - 6 } \multicolumn{1}{c\|}{}	A	B	C	D	E
A	-	12	7	6	3
B	15	-	22	21	18
C	7	5	-	4	7
D	11	9	4	-	3
E	14	12	7	3	-

b.v.	\(x\)	\(y\)	\(z\)	\(s _ { 1 }\)	\(S _ { 2 }\)	\(S _ { 3 }\)	\(a _ { 1 }\)	\(a _ { 2 }\)	Value
\(s _ { 1 }\)	3	0	1.5	1	0	0.25	0	-0.25	26.25
\(a _ { 1 }\)	1	0	1.5	0	-1	-0.25	1	0.25	11.75
\(y\)	0	1	0.5	0	0	-0.25	0	0.25	3.75
\(P\)	\(- ( 2 + M )\)	0	2-1.5M	0	M	\(- 0.5 + 0.25 M\)	0	\(0.5 + 0.75 M\)	7.5-11.75M

	A	B	C	D	E	F	G
A	-	24	-	22	35	-	-
B	24	-	25	27	-	-	-
C	-	25	-	33	31	36	26
D	22	27	33	-	-	42	-
E	35	-	31	-	-	37	29
F	-	-	36	42	37	-	40
G	-	-	26	-	29	40	-

	A	B	C	D	E	F	G
A	-	24	-	22	35	-	-
B	24	-	25	27	-	-	-
C	-	25	-	33	31	36	26
D	22	27	33	-	-	42	-
E	35	-	31	-	-	37	29
F	-	-	36	42	37	-	40
G	-	-	26	-	29	40	-

Questions — Edexcel (9670 questions)

Browse by module

Browse by board

Edexcel FM2 2023 June Q7

Edexcel FM2 2023 June Q8

Edexcel FM2 2024 June Q1

Edexcel FM2 2024 June Q2

Edexcel FM2 2024 June Q3

Edexcel FM2 2024 June Q4

Edexcel FM2 2024 June Q5

Edexcel FM2 2024 June Q6

Edexcel FM2 2024 June Q7

Edexcel FM2 Specimen Q1

Edexcel FM2 Specimen Q2

Edexcel FM2 Specimen Q3

Edexcel FM2 Specimen Q4

Edexcel FM2 Specimen Q5

Edexcel FM2 Specimen Q6

Edexcel FM2 Specimen Q7

Edexcel FD1 2019 June Q1

Edexcel FD1 2019 June Q2

Edexcel FD1 2019 June Q3

Edexcel FD1 2019 June Q4

Edexcel FD1 2019 June Q5

Edexcel FD1 2019 June Q6

Edexcel FD1 2020 June Q1

Edexcel FD1 2020 June Q2

Edexcel FD1 2020 June Q3

	A	B	C	D	E	F	G
A	-	24	-	22	35	-	-
B	24	-	25	27	-	-	-
C	-	25	-	33	31	36	26
D	22	27	33	-	-	42	-
E	35	-	31	-	-	37	29
F	-	-	36	42	37	-	40
G	-	-	26	-	29	40	-