Questions - AQA - A-Level Maths

AQA Further Paper 3 Mechanics 2022 June Q7

7 Two snooker balls, one white and one red, have equal mass. The balls are on a horizontal table $A B C D$
The white ball is struck so that it moves at a speed of $2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ parallel to $A B$
The white ball hits a stationary red ball.
After the collision, the white ball moves at a speed of $0.8 \mathrm {~ms} ^ { - 1 }$ and at an angle of $30 ^ { \circ }$ to $A B$ After the collision, the red ball moves at a speed $v \mathrm {~ms} ^ { - 1 }$ and at an angle $\theta$ to $C D$
When the collision takes place, the white ball is the same distance from $A B$ as the distance the red ball is from CD The diagram below shows the table and the velocities of the balls after the collision.
\includegraphics[max width=\textwidth, alt={}, center]{0afe3ff2-0af5-4aeb-98c5-1346fa803388-08_595_1370_1121_335} Not to scale After the collision, the white ball hits $A B$ and the red ball hits $C D$
Model the balls as particles that do not experience any air resistance.
7

Explain why the two balls hit the sides of the table at the same time.
7
Show that $\theta = 17.0 ^ { \circ }$ correct to one decimal place.
7
$\quad$ Find $v$
7
Determine which ball travels the greater distance after the collision and before hitting the side of the table. Fully justify your answer.
7
State one possible refinement to the model that you have used.
$8 \quad$ In this question use $g$ as $9.8 \mathrm {~ms} ^ { - 2 }$ A rope is used to pull a crate, of mass 60 kg , along a rough horizontal surface.
The coefficient of friction between the crate and the surface is 0.4 The crate is at rest when the rope starts to pull on it.
The tension in the rope is 240 N and the rope makes an angle of $30 ^ { \circ }$ to the horizontal.
When the crate has moved 5 metres, the rope becomes detached from the crate.

AQA Further Paper 3 Mechanics 2022 June Q8

8

Use an energy method to find the maximum speed of the crate.
8
Use an energy method to find the total distance travelled by the crate.
8
A student claims that in reality the crate is unlikely to travel more than 5.3 metres in total. Comment on the validity of this claim.
\includegraphics[max width=\textwidth, alt={}, center]{0afe3ff2-0af5-4aeb-98c5-1346fa803388-12_2488_1732_219_139}

AQA Further Paper 3 Mechanics 2022 June Q9

3 marks

9 Two blocks have square cross sections. One block has mass 9 kg and its cross section has sides of length 20 cm
The other block has mass 1 kg and its cross section has sides of length 4 cm
The blocks are fixed together to form the composite body shown in Figure 1. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{0afe3ff2-0af5-4aeb-98c5-1346fa803388-13_570_492_717_776}

\end{figure} 9

Find the distance of the centre of mass of the composite body from $A F$
[0pt] [2 marks]
Question 9 continues on the next page 9
A uniform rod has mass 12 kg and length 1 metre. One end of the rod rests against a smooth vertical wall.
The other end of the rod rests on the composite body at point $B$
The composite body is on a horizontal surface.
The coefficient of friction between the composite body and the horizontal surface is 0.3 The angle between the rod and $A B$ is $60 ^ { \circ }$
A particle of mass $m \mathrm {~kg}$ is fixed to the rod at a distance of 75 cm from $B$
The rod, particle and composite body are shown in Figure 2. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{0afe3ff2-0af5-4aeb-98c5-1346fa803388-14_939_1020_1133_511}
\end{figure} 9
1. Write down the magnitude of the vertical reaction force acting on the rod at $B$ in terms of $m$ and $g$
  [0pt] [1 mark] 9
(ii) Show that the magnitude of the horizontal reaction force acting on the rod at $B$ is $$\frac { g ( 6 + 0.75 m ) } { \sqrt { 3 } }$$ 9
(iii) Find the maximum value of $m$ for which the composite body does not slide or topple. Fully justify your answer.

AQA Further Paper 3 Mechanics 2023 June Q1

1 State the dimensions of power.
Circle your answer.
$M L ^ { 2 } T ^ { - 3 }$
$M L ^ { 3 } T ^ { - 3 }$
$M L ^ { 3 } T ^ { - 2 }$
$M L ^ { 2 } T ^ { - 2 }$

AQA Further Paper 3 Mechanics 2023 June Q2

1 marks

2 The force $( 3 \mathbf { i } + 4 \mathbf { j } ) \mathrm { N }$ acts at the point with coordinates $( 0,2 )$
The unit vectors $\mathbf { i }$ and $\mathbf { j }$ are directed along the $x$-axis and the $y$-axis respectively.
Calculate the magnitude of the moment of this force about the origin.
Circle your answer.
[0pt] [1 mark]
6 Nm
8 Nm
10 Nm
14 Nm

AQA Further Paper 3 Mechanics 2023 June Q3

3 A uniform disc has mass 6 kg and diameter 8 cm A uniform rectangular lamina, $A B C D$, has mass 4 kg , width 8 cm and length 20 cm
The disc is fixed to the lamina to form a composite body as shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{cd0d239b-ab92-4d17-9cb8-45722e2894cb-03_448_881_587_577} The sides $A B , A D$ and $C D$ are tangents to the disc.
Calculate the distance of the centre of mass of the composite body from $A D$
Circle your answer.
4 cm
5.6 cm
6.4 cm
8.8 cm

AQA Further Paper 3 Mechanics 2023 June Q4

2 marks

4 A car of mass 1400 kg drives around a horizontal circular bend of radius 60 metres.
The car has a constant speed of $12 \mathrm {~ms} ^ { - 1 }$ on the bend.
Calculate the magnitude of the resultant force acting on the car.
[0pt] [2 marks]
$5 \quad$ A region bounded by the curve with equation $y = 4 - x ^ { 2 }$, the $x$-axis and the $y$-axis is shown below.
\includegraphics[max width=\textwidth, alt={}, center]{cd0d239b-ab92-4d17-9cb8-45722e2894cb-04_641_380_408_831} The region is rotated through $360 ^ { \circ }$ around the $x$-axis to create a uniform solid.

AQA Further Paper 3 Mechanics 2023 June Q6

7 marks

6 Nm
8 Nm
10 Nm
14 Nm 3 A uniform disc has mass 6 kg and diameter 8 cm A uniform rectangular lamina, $A B C D$, has mass 4 kg , width 8 cm and length 20 cm
The disc is fixed to the lamina to form a composite body as shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{cd0d239b-ab92-4d17-9cb8-45722e2894cb-03_448_881_587_577} The sides $A B , A D$ and $C D$ are tangents to the disc.
Calculate the distance of the centre of mass of the composite body from $A D$
Circle your answer.
4 cm
5.6 cm
6.4 cm
8.8 cm 4 A car of mass 1400 kg drives around a horizontal circular bend of radius 60 metres.
The car has a constant speed of $12 \mathrm {~ms} ^ { - 1 }$ on the bend.
Calculate the magnitude of the resultant force acting on the car.
[0pt] [2 marks]
$5 \quad$ A region bounded by the curve with equation $y = 4 - x ^ { 2 }$, the $x$-axis and the $y$-axis is shown below.
\includegraphics[max width=\textwidth, alt={}, center]{cd0d239b-ab92-4d17-9cb8-45722e2894cb-04_641_380_408_831} The region is rotated through $360 ^ { \circ }$ around the $x$-axis to create a uniform solid.
5

Show that the distance of the centre of mass of the solid from the circular face is $\frac { 5 } { 8 }$
[0pt] [5 marks]
5
The solid is suspended in equilibrium from a point on the edge of the circular face.
Find the angle between the circular face and the horizontal, giving your answer to the nearest degree.
6 In this question use $g = 10 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ A sphere of mass 0.8 kg is attached to one end of a string of length 2 metres.
The other end of the string is attached to a fixed point $O$
The sphere is released from rest with the string taut and at an angle of $30 ^ { \circ }$ to the vertical, as shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{cd0d239b-ab92-4d17-9cb8-45722e2894cb-06_464_218_676_909} 6
Find the speed of the sphere when it is directly below $O$
6
State one assumption that you made about the string.
6
As the sphere moves, the string makes an angle $\theta$ with the downward vertical. By finding an expression for the tension in the string in terms of $\theta$, show that the tension is a maximum when the sphere is directly below $O$ 6
A physics student conducts an experiment and uses a device to measure the tension in the string when the sphere is directly below $O$ They find that the tension is 9.5 newtons.
Explain why this result is reasonable, showing any calculations that you make.

AQA Further Paper 3 Mechanics 2023 June Q14

12 marks

14 Nm 3 A uniform disc has mass 6 kg and diameter 8 cm A uniform rectangular lamina, $A B C D$, has mass 4 kg , width 8 cm and length 20 cm
The disc is fixed to the lamina to form a composite body as shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{cd0d239b-ab92-4d17-9cb8-45722e2894cb-03_448_881_587_577} The sides $A B , A D$ and $C D$ are tangents to the disc.
Calculate the distance of the centre of mass of the composite body from $A D$
Circle your answer.
4 cm
5.6 cm
6.4 cm
8.8 cm 4 A car of mass 1400 kg drives around a horizontal circular bend of radius 60 metres.
The car has a constant speed of $12 \mathrm {~ms} ^ { - 1 }$ on the bend.
Calculate the magnitude of the resultant force acting on the car.
[0pt] [2 marks]
$5 \quad$ A region bounded by the curve with equation $y = 4 - x ^ { 2 }$, the $x$-axis and the $y$-axis is shown below.
\includegraphics[max width=\textwidth, alt={}, center]{cd0d239b-ab92-4d17-9cb8-45722e2894cb-04_641_380_408_831} The region is rotated through $360 ^ { \circ }$ around the $x$-axis to create a uniform solid.
5

Show that the distance of the centre of mass of the solid from the circular face is $\frac { 5 } { 8 }$
[0pt] [5 marks]
5
The solid is suspended in equilibrium from a point on the edge of the circular face.
Find the angle between the circular face and the horizontal, giving your answer to the nearest degree.
6 In this question use $g = 10 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ A sphere of mass 0.8 kg is attached to one end of a string of length 2 metres.
The other end of the string is attached to a fixed point $O$
The sphere is released from rest with the string taut and at an angle of $30 ^ { \circ }$ to the vertical, as shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{cd0d239b-ab92-4d17-9cb8-45722e2894cb-06_464_218_676_909} 6
Find the speed of the sphere when it is directly below $O$
6
State one assumption that you made about the string.
6
As the sphere moves, the string makes an angle $\theta$ with the downward vertical. By finding an expression for the tension in the string in terms of $\theta$, show that the tension is a maximum when the sphere is directly below $O$ 6
A physics student conducts an experiment and uses a device to measure the tension in the string when the sphere is directly below $O$ They find that the tension is 9.5 newtons.
Explain why this result is reasonable, showing any calculations that you make.
7 Two particles, $A$ and $B$, are moving on a smooth horizontal surface. A straight line has been marked on the surface and the particles are on opposite sides of the line. Particle $A$ has mass 2 kg and moves with velocity $4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at an angle of $30 ^ { \circ }$ to the line. Particle $B$ has mass 3 kg and moves with velocity $6 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at an angle of $45 ^ { \circ }$ to the line. The particles and their velocities are shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{cd0d239b-ab92-4d17-9cb8-45722e2894cb-08_451_739_858_653} The particles collide when they reach the line and then move together as a single combined particle. 7
Show that the magnitude of the impulse on particle $A$ during the collision is 7.55 Ns correct to three significant figures.
7
State the magnitude of the impulse on $B$ during the collision, giving a reason for your answer. 7
Find the size of the angle between the straight line and the impulse acting on $B$, giving your answer to the nearest degree. 7
During the collision, one particle crosses the straight line.
State which particle crosses the line, giving a reason for your answer.
[0pt] [1 mark] 8 In this question use $g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ A block has mass 10 kg and is at rest 1 metre from a fixed point $O$ on a horizontal surface. One end of an elastic string is attached to the block and the other end of the elastic string is attached to the point $O$ The elastic string has modulus of elasticity 40 newtons and natural length 2 metres.
The coefficient of friction between the block and the surface is 0.6 A force is applied to the block so that it starts to move towards a vertical wall.
The block moves on a line that is perpendicular to the wall.
The force has magnitude 100 newtons and acts at an angle of $30 ^ { \circ }$ to the horizontal.
The situation is shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{cd0d239b-ab92-4d17-9cb8-45722e2894cb-10_239_1339_1176_354} 8
Show that the distance that the block has moved, when the forces acting on it are in equilibrium, is 3.9 metres correct to two significant figures.
[0pt] [4 marks]
8
State one limitation of the model that you have used. 8
Find the maximum speed of the block.
8
The vertical wall is 8.7 metres from $O$ Determine whether the block reaches the wall, showing any calculations that you make.
\includegraphics[max width=\textwidth, alt={}, center]{cd0d239b-ab92-4d17-9cb8-45722e2894cb-13_2492_1721_217_150}

AQA Further Paper 3 Discrete 2019 June Q1

1 Deanna and Will play a zero-sum game.
The game is represented by the following pay-off matrix for Deanna.

\multirow{6}{*}{Deanna}	Will
	Strategy	X	Y	Z
	A	-1	0	2
	B	-2	-1	3
	C	5	-2	-3
	D	6	-2	0

Which strategy is Deanna's play-safe strategy?
Circle your answer.
A
B
C
D

AQA Further Paper 3 Discrete 2019 June Q2

2 The graph $D$ is shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{22f11ce2-8d07-4f51-9326-b578d1e454f9-02_115_150_1756_945} Which of the graphs below is a subdivision of $D$ ?
Circle your answer.
\includegraphics[max width=\textwidth, alt={}, center]{22f11ce2-8d07-4f51-9326-b578d1e454f9-02_218_154_2124_534}
\includegraphics[max width=\textwidth, alt={}, center]{22f11ce2-8d07-4f51-9326-b578d1e454f9-02_208_150_2129_808}
\includegraphics[max width=\textwidth, alt={}, center]{22f11ce2-8d07-4f51-9326-b578d1e454f9-02_208_154_2129_1082}
\includegraphics[max width=\textwidth, alt={}, center]{22f11ce2-8d07-4f51-9326-b578d1e454f9-02_208_149_2129_1361}

AQA Further Paper 3 Discrete 2019 June Q3

3 The Simplex tableau below is optimal.

$\boldsymbol { P }$	$\boldsymbol { x }$	$\boldsymbol { y }$	$\boldsymbol { z }$	$\boldsymbol { r }$	$\boldsymbol { s }$	value
1	$k ^ { 2 } + k - 6$	0	0	$k - 1$	1	20
0	0	0	1	1.5	0	6
0	0	1	0	0	0.5	86

3

Deduce the range of values that $k$ must satisfy.
3
Write down the value of the variable $s$ which corresponds to the optimal value of $P$.

AQA Further Paper 3 Discrete 2019 June Q4

4 The connected planar graph $P$ has the adjacency matrix

\cline { 2 - 6 } \multicolumn{1}{c\|}{}	$A$	$B$	$C$	$D$	$E$
$A$	0	1	1	0	1
$B$	1	0	1	0	1
$C$	1	1	0	1	1
$D$	0	0	1	0	1
$E$	1	1	1	1	0

4

Draw $P$ 4
Using Euler's formula for connected planar graphs, show that $P$ has exactly 5 faces. 4
Ore's theorem states that a simple graph with $n$ vertices is Hamiltonian if, for every pair of vertices $X$ and $Y$ which are not adjacent, $$\text { degree of } X + \text { degree of } Y \geq n$$ where $n \geq 3$
Using Ore's theorem, prove that the graph $P$ is Hamiltonian.
Fully justify your answer.

AQA Further Paper 3 Discrete 2019 June Q5

5 The set $S$ is defined as $$S = \{ A , B , C , D \}$$ where
$A = \left[ \begin{array} { l l } 1 & 0
0 & 1 \end{array} \right] \quad B = \left[ \begin{array} { c c } 0 & - 1
1 & 0 \end{array} \right] \quad C = \left[ \begin{array} { c c } - 1 & 0
0 & - 1 \end{array} \right] \quad D = \left[ \begin{array} { c c } 0 & 1
- 1 & 0 \end{array} \right]$ The group $G$ is formed by $S$ under matrix multiplication.
The group $H$ is defined as $H = ( \langle \mathrm { i } \rangle , \times )$, where $\mathrm { i } ^ { 2 } = - 1$
5

1. Prove that $B$ is a generator of $G$.
  Fully justify your answer.
  5
(ii) Show that $G \cong H$.
Fully justify your answer.
5
1. Explain why $H$ has no subgroups of order 3
  Fully justify your answer.
  5
(ii) Find all of the subgroups of $H$.

AQA Further Paper 3 Discrete 2019 June Q6

6 A council wants to monitor how long cars are being parked for in short-stay parking bays in a town centre. They employ a traffic warden to walk along the streets in the town centre and issue fines to drivers who park for longer than the stated time. The network below shows streets in the town centre which have short-stay parking bays. Each node represents a street corner and the weight of each arc represents the length, in metres, of the street. The short-stay parking bays are positioned along only one side of each street.
\includegraphics[max width=\textwidth, alt={}, center]{22f11ce2-8d07-4f51-9326-b578d1e454f9-08_483_1108_733_466} The council assumes that the traffic warden will walk at an average speed of 4.8 kilometres per hour when not issuing fines. 6

To monitor all of the parking bays, the traffic warden needs to walk along every street in the town centre at least once, starting and finishing at the same street corner. Find the shortest possible time, to the nearest minute, it can take the traffic warden to monitor all of the parking bays. Fully justify your answer.
6
Explain why the actual time for the traffic warden to walk along every street in the town centre at least once may be different to the value found in part (a).

AQA Further Paper 3 Discrete 2019 June Q7

7 Figure 1 shows a system of water pipes in a manufacturing complex. The number on each arc represents the upper capacity for each pipe in litres per second. The numbers in the circles represent an initial feasible flow of 38 litres of water per second. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{22f11ce2-8d07-4f51-9326-b578d1e454f9-10_874_1360_609_338}

\end{figure} 7

1. Calculate the value of the cut $\{ S , A , B , C \} \{ D , E , F , G , H , T \}$. 7
(ii) Explain, in the context of the question, what can be deduced from your answer to part (a)(i). 7
1. Using the initial feasible flow shown in Figure 1, indicate on Figure 2 potential increases and decreases in the flow along each arc. \begin{figure}[h]
  \captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{22f11ce2-8d07-4f51-9326-b578d1e454f9-11_997_1554_475_242}
  \end{figure} 7
(ii) Use flow augmentation on Figure 2 to find the maximum flow through the manufacturing complex. You must indicate any flow augmenting paths clearly in the table and modify the potential increases and decreases of the flow on Figure 2.
Augmenting Path Flow
Maximum Flow $=$ $\_\_\_\_$ 7
The management of the manufacturing complex want to increase the maximum amount of water which can flow through the system of pipes. To do this they decide to upgrade one of the water pipes by replacing it with a larger capacity pipe. Explain which pipe should be upgraded.
Deduce what effect this upgrade will have on the maximum amount of water which can flow through the system of pipes.
\includegraphics[max width=\textwidth, alt={}, center]{22f11ce2-8d07-4f51-9326-b578d1e454f9-13_2488_1716_219_153}

AQA Further Paper 3 Discrete 2019 June Q8

8 A motor racing team is undertaking a project to build next season's racing car. The project is broken down into 12 separate activities $A , B , \ldots , L$, as shown in the precedence table below. Each activity requires one member of the racing team.

Activity	Duration (days)	Immediate Predecessors
$A$	7	-
B	6	-
C	15	-
D	9	$A , B$
$E$	8	D
$F$	6	C, D
G	7	C
H	14	$E$
$I$	17	$F , G$
$J$	9	H, I
K	8	$I$
L	12	J, K

8

1. Complete the activity network for the project on Figure 3. 8
(ii) Find the earliest start time and the latest finish time for each activity and show these values on Figure 3. 8
Write down the critical path(s).
\section*{Figure 3} Figure 3
\includegraphics[max width=\textwidth, alt={}, center]{22f11ce2-8d07-4f51-9326-b578d1e454f9-15_469_1360_356_338} 8
1. Using Figure 4, draw a resource histogram for the project to show how the project can be completed in the shortest possible time. Assume that each activity is to start as early as possible. \begin{figure}[h]
  \captionsetup{labelformat=empty} \caption{Figure 4} \includegraphics[alt={},max width=\textwidth]{22f11ce2-8d07-4f51-9326-b578d1e454f9-16_698_1534_541_251}
  \end{figure} 8
(ii) The racing team's boss assigns two members of the racing team to work on the project. Explain the effect this has on the minimum completion time for the project.
You may use Figure 5 in your answer. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 5} \includegraphics[alt={},max width=\textwidth]{22f11ce2-8d07-4f51-9326-b578d1e454f9-16_704_1539_1695_248}
\end{figure}

AQA Further Paper 3 Discrete 2020 June Q1

1 The diagram below shows a network of pipes with their capacities.
\includegraphics[max width=\textwidth, alt={}, center]{c297a67f-65fd-47e0-a60c-d38fd86c6081-02_734_1275_630_386} A supersource and a supersink will be added to the network.
To which nodes should the supersource and supersink be connected?
Tick $( \checkmark )$ one box.

Supersource	Supersink
$P , Q$	$U , V , W$
$U , V , W$	$P , Q$
$V , X$	$U , W$
$U , W$	$V , X$

□
□
□
\includegraphics[max width=\textwidth, alt={}, center]{c297a67f-65fd-47e0-a60c-d38fd86c6081-02_118_113_2261_1324}

AQA Further Paper 3 Discrete 2020 June Q2

2 Which of the following statements is true about the operation of matrix multiplication on the set of all $2 \times 2$ real matrices? Tick ( $\checkmark$ ) one box. Matrix multiplication is associative and commutative.
\includegraphics[max width=\textwidth, alt={}, center]{c297a67f-65fd-47e0-a60c-d38fd86c6081-03_109_112_552_1599} Matrix multiplication is associative but not commutative. □ Matrix multiplication is commutative but not associative. □ Matrix multiplication is not commutative and not associative. □

AQA Further Paper 3 Discrete 2020 June Q3

4 marks

3 A company is installing an internal telephone network between the offices in a council building. Each office is required to be connected with telephone cables, either directly or indirectly, to every other office in the building. The lengths of cable, in metres, needed to connect the offices are shown in the table below.

	Education	Housing	Refuse Collection	Payroll	Social Care	Transport
Education	-	27	13	35	16	24
Housing	27	-	29	30	22	24
Refuse Collection	13	29	-	26	23	17
Payroll	35	30	26	-	20	40
Social Care	16	22	23	20	-	21
Transport	24	24	17	40	21	-

The council wants the total length of cable that is used to be as small as possible.
The cost to the council to install one metre of cable is $\pounds 8$
3

1. Find the minimum total cost to the council to install the cable required for the internal telephone network.
  [0pt] [4 marks]
  3
(ii) Suggest a reason why the total cost to the council for installing the internal telephone network is likely to be different from your answer to part (a)(i). 3
Before the company starts installing the cable, it is told that the Education office cannot be connected directly to the Transport office due to issues with the building. Explain the possible impact of this on your answer to part (a)(i).

AQA Further Paper 3 Discrete 2020 June Q4

4 Joe, a courier, is required to deliver parcels to six different locations, $A , B , C , D , E$ and $F$. Joe needs to start and finish his journey at the depot.
The distances, in miles, between the depot and the six different locations are shown in the table below.

	Depot	$\boldsymbol { A }$	$\boldsymbol { B }$	C	$\boldsymbol { D }$	$E$	$F$
Depot	-	18	17	15	16	19	30
$\boldsymbol { A }$	18	-	29	20	25	35	21
B	17	29	-	26	30	16	14
C	15	20	26	-	28	31	27
D	16	25	30	28	-	34	24
E	19	35	16	31	34	-	28
F	30	21	14	27	24	28	-

The minimum total distance that Joe can travel in order to make all six deliveries, starting and finishing at the depot, is $L$ miles. 4

Using the nearest neighbour algorithm starting from the depot, find an upper bound for $L$.
4
By deleting the depot, find a lower bound for $L$.
4
Joe starts from the depot, delivers parcels to all six different locations and arrives back at the depot, covering 134 miles in the process. Joe claims that this is the minimum total distance that is possible for the journey. Comment on Joe's claim.

AQA Further Paper 3 Discrete 2020 June Q5

5 The planar graph $P$ is shown below.
\includegraphics[max width=\textwidth, alt={}, center]{c297a67f-65fd-47e0-a60c-d38fd86c6081-08_410_406_360_817} 5

Determine the number of faces of $P$.
5
Akwasi claims that $P$ is semi-Eulerian as it is connected and it has exactly two vertices with even degree. Comment on the validity of Akwasi's claim.
\includegraphics[max width=\textwidth, alt={}, center]{c297a67f-65fd-47e0-a60c-d38fd86c6081-09_2488_1716_219_153}

AQA Further Paper 3 Discrete 2020 June Q6

6 The group $( G , \boldsymbol { A } )$ has the elements $e , r , r ^ { 2 } , q , q r$ and $q r ^ { 2 }$, where $r ^ { 2 } = r \boldsymbol { \Delta } r , q r = q \boldsymbol { \Delta } r , q r ^ { 2 } = q \boldsymbol { \Delta } r ^ { 2 }$ and $e$ is the identity element of $G$. The elements $q$ and $r$ have the following properties: $$\begin{aligned} & r \boldsymbol { \Delta } r \boldsymbol { \Delta } r = e
& q \boldsymbol { \Delta } q = e
& r ^ { 2 } \boldsymbol { \Delta } q = q \boldsymbol { \Delta } r \end{aligned}$$ 6

1. State the order of $G$. 6
(ii) Prove that the inverse of $q r$ is $q r$.
6
Complete the Cayley table for elements of $G$. 6

Complete the Cayley table for elements of $G$.

A	$e$	$r$	$r ^ { 2 }$	$q$	$q r$	$q r ^ { 2 }$
$e$	$e$	$r$	$r ^ { 2 }$	$q$	$q r$	$q r ^ { 2 }$
$r$	$r$	$r ^ { 2 }$	$e$
$r ^ { 2 }$	$r ^ { 2 }$	$e$	$r$
$q$	$q$	$q r$	$q r ^ { 2 }$	$e$
$q r$	$q r$	$q r ^ { 2 }$	$q$	$r ^ { 2 }$
$q r ^ { 2 }$	$q r ^ { 2 }$	$q$	$q r$	$r$	$r ^ { 2 }$	$e$

6

State the name of a group which is isomorphic to $G$.

AQA Further Paper 3 Discrete 2020 June Q7

7 An engineering company makes brake kits and clutch kits to sell to motorsport teams. The table below summarises the time taken and costs involved in making the two different types of kit.

Type of kit	Time taken to make a kit (hours)	Cost to engineering company per kit (£)	Profit to engineering company per kit (£)
Brake kit	5	500	2000
Clutch kit	3	200	1000

The workers at the engineering company have a combined 2500 hours available to make the kits every month. The engineering company has $\pounds 200000$ available to cover the costs of making the kits every month. To meet the minimum demands of the motorsport teams, the engineering company must make at least 100 of each type of kit every month. 7

Using a graphical method on the grid opposite, find the number of each type of kit that the engineering company should make every month, in order to maximise its total monthly profit. Show clearly how you obtain your answer.
\includegraphics[max width=\textwidth, alt={}, center]{c297a67f-65fd-47e0-a60c-d38fd86c6081-13_2486_1709_221_153} Do not write outside the box 7
Give a reason why the engineering company may not be able to make the number of each kit that you found in part (a). 7
During one particular month the engineering company removes the need to make at least 100 of each type of kit. Explain whether or not this has an effect on your answer to part (a).

AQA Further Paper 3 Discrete 2020 June Q8

8 Daryl and Clare play a zero-sum game. The game is represented by the following pay-off matrix for Daryl. Clare

\(\boldsymbol { P }\)	\(\boldsymbol { x }\)	\(\boldsymbol { y }\)	\(\boldsymbol { z }\)	\(\boldsymbol { r }\)	\(\boldsymbol { s }\)	value
1	\(k ^ { 2 } + k - 6\)	0	0	\(k - 1\)	1	20
0	0	0	1	1.5	0	6
0	0	1	0	0	0.5	86

\cline { 2 - 6 } \multicolumn{1}{c\|}{}	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)
\(A\)	0	1	1	0	1
\(B\)	1	0	1	0	1
\(C\)	1	1	0	1	1
\(D\)	0	0	1	0	1
\(E\)	1	1	1	1	0

Supersource	Supersink
\(P , Q\)	\(U , V , W\)
\(U , V , W\)	\(P , Q\)
\(V , X\)	\(U , W\)
\(U , W\)	\(V , X\)

A	\(e\)	\(r\)	\(r ^ { 2 }\)	\(q\)	\(q r\)	\(q r ^ { 2 }\)
\(e\)	\(e\)	\(r\)	\(r ^ { 2 }\)	\(q\)	\(q r\)	\(q r ^ { 2 }\)
\(r\)	\(r\)	\(r ^ { 2 }\)	\(e\)
\(r ^ { 2 }\)	\(r ^ { 2 }\)	\(e\)	\(r\)
\(q\)	\(q\)	\(q r\)	\(q r ^ { 2 }\)	\(e\)
\(q r\)	\(q r\)	\(q r ^ { 2 }\)	\(q\)	\(r ^ { 2 }\)
\(q r ^ { 2 }\)	\(q r ^ { 2 }\)	\(q\)	\(q r\)	\(r\)	\(r ^ { 2 }\)	\(e\)

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