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AQA Further AS Paper 2 Mechanics 2022 June Q6

7 marks Standard +0.3

6 An ice hockey puck, of mass 0.2 kg , is moving in a straight line on a horizontal ice rink under the action of a single force which acts in the direction of motion. At time $t$ seconds, the force has magnitude ( $2 t + 3$ ) newtons.
The force acts on the puck from $t = 0$ to $t = T$ 6

Show that the magnitude of the impulse of the force is $a T ^ { 2 } + b T$, where $a$ and $b$ are integers to be found.
[0pt] [3 marks]
6
While the force acts on the puck, its speed increases from $1 \mathrm {~ms} ^ { - 1 }$ to $4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ Use your answer from part (a) to find $T$, giving your answer to three significant figures.
Fully justify your answer.

AQA Further AS Paper 2 Mechanics 2022 June Q7

9 marks Standard +0.3

7 The particles $A$ and $B$ are moving on a smooth horizontal surface directly towards each other. Particle $A$ has mass 0.4 kg and particle $B$ has mass 0.2 kg
Particle $A$ has speed $4 \mathrm {~ms} ^ { - 1 }$ and particle $B$ has speed $2 \mathrm {~ms} ^ { - 1 }$ when they collide, as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{ec39a757-5867-4798-b26c-73cd5746581c-08_392_1064_625_488} The coefficient of restitution between the particles is $e$ 7

Find the magnitude of the total momentum of the particles before the collision.
[0pt] [2 marks] 7
(i) Show that the speed of $B$ immediately after the collision is $( 4 e + 2 ) \mathrm { ms } ^ { - 1 }$ [0pt] [3 marks]
7 (b) (ii) Find an expression, in terms of $e$, for the speed of $A$ immediately after the collision.
7
Explain what happens to particle $A$ when the collision is perfectly elastic.

AQA Further AS Paper 2 Mechanics 2022 June Q20

1 marks Easy -1.8

20 J
25 J
50 J
100 J 2 Two particles of equal mass are moving on a horizontal surface when they collide.
Immediately before the collision, their velocities are $\left[ \begin{array} { l } 2 \\ 4 \end{array} \right] \mathrm { ms } ^ { - 1 }$ and $\left[ \begin{array} { c } 6 \\ - 2 \end{array} \right] \mathrm { ms } ^ { - 1 }$ As a result of the collision the particles coalesce to become a single particle.
Find the velocity of the single particle, immediately after the collision.
Circle your answer.
[0pt] [1 mark] $\left[ \begin{array} { l } 4 \\ 1 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }$ $\left[ \begin{array} { l } 4 \\ 3 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }$ $\left[ \begin{array} { l } 8 \\ 2 \end{array} \right] \mathrm { ms } ^ { - 1 }$ $\left[ \begin{array} { l } 8 \\ 6 \end{array} \right] \mathrm { ms } ^ { - 1 }$ 3 In this question use $g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ A ball of mass of 0.75 kg is thrown vertically upwards with an initial speed of $12 \mathrm {~ms} ^ { - 1 }$ The ball is thrown from ground level. 3

Calculate the initial kinetic energy of the ball. 3
The maximum height of the ball above the ground is $h$ metres.
Jeff and Gurjas use an energy method to find $h$ Jeff concludes that $h = 7.3$ Gurjas concludes that $h < 7.3$ Explain the reasoning that they have used, showing any calculations that you make.
4 Wavelength is defined as the distance from the highest point on one wave to the highest point on the next wave. Surfers classify waves into one of several types related to their wavelengths.
Two of these classifications are deep water waves and shallow water waves.
4
1. The wavelength $w$ of a deep water wave is given by $$w = \frac { g t ^ { 2 } } { k }$$ where $g$ is the acceleration due to gravity and $t$ is the time period between consecutive waves. Given that the formula for a deep water wave is dimensionally consistent, show that $k$ is a dimensionless constant. 4
2. The wavelength $w$ of a shallow water wave is given by $$w = ( g d ) ^ { \alpha } t ^ { \beta }$$ where $g$ is the acceleration due to gravity, $d$ is the depth of water and $t$ is the time period between consecutive waves. Use dimensional analysis to find the values of $\alpha$ and $\beta$ 5 A car, of mass 1000 kg , is travelling on a straight horizontal road. When the car travels at a speed of $v \mathrm {~ms} ^ { - 1 }$, it experiences a resistance force of magnitude $25 v$ newtons. The car has a maximum speed of $72 \mathrm {~km} \mathrm {~h} ^ { - 1 }$ on the straight road.
  Find the maximum power output of the car.
  Fully justify your answer.

Find the value of $\omega$ 4
(i) Find the magnitude of the horizontal resultant force acting on Reena.
4 (b) (ii) Show the direction of this horizontal resultant force on the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{78120346-4a16-4545-925a-d6fab4b750e9-03_380_442_2017_861} 5 An impulse of $\left[ \begin{array} { r } - 5 \\ 12 \end{array} \right] \mathrm { N } \mathrm { s }$ is applied to a particle of mass 5 kg which is moving with velocity $\left[ \begin{array} { l } 6 \\ 2 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }$ 5 (a) Calculate the magnitude of the impulse. 5 (b) Find the speed of the particle immediately after the impulse is applied.

AQA Further AS Paper 2 Mechanics 2023 June Q6

4 marks Moderate -0.5

6 A ball is thrown with speed $u$ at an angle of $45 ^ { \circ }$ to the horizontal from a point $O$ When the horizontal displacement of the ball is $x$, the vertical displacement of the ball above $O$ is $y$ where $$y = x - \frac { k x ^ { 2 } } { u ^ { 2 } }$$ 6

Use dimensional analysis to find the dimensions of $k$ 6
State what can be deduced about $k$ from the dimensions that you found in part (a).

AQA Further AS Paper 2 Mechanics 2023 June Q7

6 marks Standard +0.3

7 Two smooth, equally sized spheres, $A$ and $B$, are moving in the same direction along a straight line on a smooth horizontal surface, as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{78120346-4a16-4545-925a-d6fab4b750e9-06_314_465_420_849} The spheres subsequently collide.
Immediately after the collision, $A$ has speed $2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and $B$ has speed $3.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ The coefficient of restitution between the spheres is $e$ 7

1. Show that $A$ does not change its direction of motion as a result of the collision.
  7
  1. (ii) Find the value of $e$ 7
2. Given that the mass of $B$ is 0.6 kg , find the mass of $A$

AQA Further AS Paper 2 Mechanics 2023 June Q8

7 marks Challenging +1.2

8 In this question use $g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ Omar, a bungee jumper of mass 70 kg , has his ankles attached to one end of an elastic cord. The other end of the cord is attached to a bridge which is 80 metres above the surface of a river. Omar steps off the bridge at the point where the cord is attached and falls vertically downwards. The cord can be modelled as a light elastic string of natural length $L$ metres and modulus of elasticity 2800 N Model Omar as a particle. 8

Given that Omar just reaches the surface of the river before being pulled back up, find the value of $L$ Fully justify your answer.
8
If Omar is not modelled as a particle, explain the effect of revising this assumption on your answer to part (a).

AQA Further AS Paper 2 Mechanics 2023 June Q18

1 marks Easy -1.8

18 J
34 J 2 Two particles are moving directly towards each other when they collide.
Given that the collision is perfectly elastic, state the value of the coefficient of restitution. Circle your answer. $e = - 1$ $e = 0$ $e = \frac { 1 } { 2 }$ $e = 1$ 3 A stone of mass 0.2 kg is thrown vertically upwards with a speed of $10 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ Find the initial kinetic energy of the stone.
Circle your answer.
[0pt] [1 mark]
1 J
5 J
10 J

AQA Further AS Paper 2 Mechanics 2023 June Q20

Easy -1.2

20 J 4 Reena is skating on an ice rink, which has a horizontal surface. She follows a circular path of radius 5 metres and centre $O$ She completes 10 full revolutions in 1 minute, moving with a constant angular speed of $\omega$ radians per second. The mass of Reena is 40 kg
4

Find the value of $\omega$ 4
(i) Find the magnitude of the horizontal resultant force acting on Reena.
4 (b) (ii) Show the direction of this horizontal resultant force on the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{78120346-4a16-4545-925a-d6fab4b750e9-03_380_442_2017_861} 5 An impulse of $\left[ \begin{array} { r } - 5 \\ 12 \end{array} \right] \mathrm { N } \mathrm { s }$ is applied to a particle of mass 5 kg which is moving with velocity $\left[ \begin{array} { l } 6 \\ 2 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }$ 5 (a) Calculate the magnitude of the impulse. 5 (b) Find the speed of the particle immediately after the impulse is applied.
6 A ball is thrown with speed $u$ at an angle of $45 ^ { \circ }$ to the horizontal from a point $O$ When the horizontal displacement of the ball is $x$, the vertical displacement of the ball above $O$ is $y$ where $$y = x - \frac { k x ^ { 2 } } { u ^ { 2 } }$$ 6 (a) Use dimensional analysis to find the dimensions of $k$ 6 (b) State what can be deduced about $k$ from the dimensions that you found in part (a).
7 Two smooth, equally sized spheres, $A$ and $B$, are moving in the same direction along a straight line on a smooth horizontal surface, as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{78120346-4a16-4545-925a-d6fab4b750e9-06_314_465_420_849} The spheres subsequently collide.
Immediately after the collision, $A$ has speed $2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and $B$ has speed $3.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ The coefficient of restitution between the spheres is $e$ 7 (a)
1. Show that $A$ does not change its direction of motion as a result of the collision.
  7
2. Find the value of $e$
7 (b) Given that the mass of $B$ is 0.6 kg , find the mass of $A$

AQA Further AS Paper 2 Discrete 2018 June Q1

1 marks Easy -1.2

1 The table shows some of the outcomes of performing a modular arithmetic operation.

\cline { 2 - 3 } \multicolumn{1}{c\|}{}	2	3
2		1
3	1

Which pair are operations that could each be represented by the table?
Tick ( ✓ ) one box. \includegraphics[max width=\textwidth, alt={}, center]{5a826f8b-4751-4589-ad0a-109fc5c821f2-02_109_111_1338_497} Addition $\bmod 6$ and multiplication $\bmod 5$ \includegraphics[max width=\textwidth, alt={}, center]{5a826f8b-4751-4589-ad0a-109fc5c821f2-02_108_109_1471_497} Addition mod 6 and multiplication $\bmod 6$ \includegraphics[max width=\textwidth, alt={}, center]{5a826f8b-4751-4589-ad0a-109fc5c821f2-02_113_109_1603_497} Addition mod 4 and multiplication $\bmod 5$ \includegraphics[max width=\textwidth, alt={}, center]{5a826f8b-4751-4589-ad0a-109fc5c821f2-02_107_109_1742_497} Addition mod 4 and multiplication mod 6

AQA Further AS Paper 2 Discrete 2018 June Q2

1 marks Standard +0.3

2 The binary operation ⊗ is given by $a \otimes b = 3 a ( 5 + b ) ( \bmod 8 )$ where $a , b \in \mathbb { Z }$ Given that $2 \otimes x = 6$, which of the integers below is a possible value of $x$ ?
Circle your answer.
[0pt] [1 mark]
0123

AQA Further AS Paper 2 Discrete 2018 June Q3

4 marks Moderate -0.5

3 Alex and Sam are playing a zero-sum game. The game is represented by the pay-off matrix for Alex.

Sam
\cline { 2 - 5 }	Strategy
\cline { 2 - 5 }	$\mathbf { S } _ { \mathbf { 1 } }$	$\mathbf { S } _ { \mathbf { 2 } }$	$\mathbf { S } _ { \mathbf { 3 } }$
$\mathbf { A } _ { \mathbf { 1 } }$	2	2	3
\cline { 2 - 5 }	$\mathbf { A } _ { \mathbf { 2 } }$	0	3	5
$\mathbf { A } _ { \mathbf { 3 } }$	- 1	2	- 2

Explain why the value of the game is 2
3
Identify the play-safe strategy for each player.
Each pipe is labelled with its upper capacity in $\mathrm { cm } ^ { 3 } \mathrm {~s} ^ { - 1 }$ \includegraphics[max width=\textwidth, alt={}, center]{5a826f8b-4751-4589-ad0a-109fc5c821f2-04_620_940_450_550}

AQA Further AS Paper 2 Discrete 2018 June Q4

6 marks Moderate -0.5

1. Find the value of the cut given by $\{ A , B , C , D , F , J \} \{ E , G , H \}$.
  4
  1. (ii) State what can be deduced about the maximum flow through the network.
    4
2. 1. List the nodes which are sources of the network. 4
3. (ii) Add a supersource $S$ to the network. 4
4. 1. List the nodes which are sinks of the network. 4
5. (ii) Add a supersink $T$ to the network.

AQA Further AS Paper 2 Discrete 2018 June Q5

9 marks Moderate -0.5

5 A group of friends want to prepare a meal. They start preparing the meal at 6:30 pm Activities to prepare the meal are shown in Figure 1 below. \begin{table}[h]

\captionsetup{labelformat=empty} \caption{Figure 1}

Label	Activity	Duration (mins)	Immediate predecessors
A	Weigh rice	1	-
$B$	Cook rice	18	$A$
C	Drain rice	1	B
D	Chop vegetables	10	-
$E$	Fry vegetables	12
$F$	Combine fried vegetables and drained rice	1
G	Prepare sauce ingredients	4	-
$H$	Boil sauce	12
$I$	Serve meal on plates	2

\end{table} 5

1. Use Figure 2 shown below to complete Figure 1 above. 5
  1. (ii) Complete Figure 2 showing the earliest start time and latest finishing time for each activity. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{5a826f8b-4751-4589-ad0a-109fc5c821f2-06_700_1650_1781_194}
    \end{figure} 5
2. 1. State the activity which must be started first so that the meal is served in the shortest possible time. Fully justify your answer.
    5
3. (ii) Determine the earliest possible time at which the preparation of the meal can be completed.
  Question 5 continues on the next page 5
4. The group of friends want to cook spring rolls so that they are served at the same time as the rest of the meal. This requires the additional activities shown in Figure 3. \begin{table}[h]
  \captionsetup{labelformat=empty} \caption{Figure 3}
  Label Activity Duration Immediate predecessors
  J Switch on and heat oven -
  K Put spring rolls in oven and cook
  $L$ Transfer spring rolls to serving dish
  \end{table} It takes 15 seconds to switch on the oven. The oven must be allowed to heat up for 10 minutes before the spring rolls are put in the oven. It takes 15 seconds to put the spring rolls in the oven.
  The spring rolls must cook in the hot oven for 8 minutes.
  It takes 30 seconds to transfer the spring rolls to a serving dish.
  5
5. 1. Complete Figure 3 above. 5
6. (ii) Determine the latest time at which the oven can be switched on in order for the spring rolls to be served at the same time as the rest of the meal.
  [0pt] [2 marks] \includegraphics[max width=\textwidth, alt={}, center]{5a826f8b-4751-4589-ad0a-109fc5c821f2-09_2488_1716_219_153}

AQA Further AS Paper 2 Discrete 2018 June Q6

5 marks Moderate -0.8

6 An animal sanctuary has a rainwater collection site. The manager of the sanctuary is installing a pipe system to connect the rainwater collection site to five other sites in the sanctuary. Each site does not need to be connected directly to the rainwater collection site. There are nine possible routes between the sites that are suitable for water pipes. The distances, in metres, of the nine possible routes are given in the table below.

From/To	Henhouse (H)	Goatshed (G)	Kennels (K)	Cattery (C)
Rainwater collection site (R)	840	810	520	370
Cattery (C)	-	680	610	\multirow{3}{*}{}
Duckpond (D)	480	310
Goatshed (G)	150

Water pipe costs 60 pence per metre. Find the minimum cost of connecting all the sites to the rainwater collection site. Fully justify your answer. $7 \quad$ A linear programming problem has the constraints $$\begin{aligned} 1 \leq x & \leq 6 \\ 1 \leq y & \leq 6 \\ y & \geq x \\ x + y & \leq 11 \end{aligned}$$

AQA Further AS Paper 2 Discrete 2018 June Q7

14 marks Standard +0.8

1. Complete Figure 4 to identify the feasible region for the problem. \begin{figure}[h]
  \captionsetup{labelformat=empty} \caption{Figure 4} \includegraphics[alt={},max width=\textwidth]{5a826f8b-4751-4589-ad0a-109fc5c821f2-12_922_940_849_552}
  \end{figure} 7
  1. (ii) Determine the maximum value of $5 x + 4 y$ subject to the constraints.
    7
2. The simple-connected graph $G$ has seven vertices. The vertices of $G$ have degree $1,2,3 , v , w , x$ and $y$ 7
3. 1. Explain why $x \geq 1$ and $y \geq 1$ 7
  2. Explain why $x \leq 6$ and $y \leq 6$ 7
  3. Explain why $x + y \leq 11$ 7
  4. State an additional constraint that applies to the values of $x$ and $y$ in this context.
    7
  5. The graph $G$ also has eight edges. The inequalities used in part (a)(i) apply to the graph $G$.
  7
4. 1. Given that $v + w = 4$, find all the feasible values of $x$ and $y$.
    7
5. (ii) It is also given that the graph $G$ is semi-Eulerian. On Figure 5, draw $G$. Figure 5

AQA Further AS Paper 2 Discrete 2019 June Q1

1 marks Easy -1.2

1 The network represents a system of pipes.
The number on each arc represents the upper capacity for each pipe in $\mathrm { cm } ^ { 3 } \mathrm {~s} ^ { - 1 }$ \includegraphics[max width=\textwidth, alt={}, center]{dcf97b92-d067-41d4-89a6-ea5bab9ea4ff-03_691_1067_721_482} The value of the cut $\{ S , A , B \} \{ C , D , E , T \}$ is $V \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }$ Find $V$. Circle your answer.
[0pt] [1 mark]
25303137

AQA Further AS Paper 2 Discrete 2019 June Q2

1 marks Moderate -0.5

2 Part of an activity network is shown in the diagram below. $A B C$ is part of the critical path of the activity network. \includegraphics[max width=\textwidth, alt={}, center]{dcf97b92-d067-41d4-89a6-ea5bab9ea4ff-04_264_908_447_566} The duration of activity $B$ is $d$.
Which of the following statements about $d$ is correct? Circle your answer. $$0 < d < 10 \quad d = 10 \quad 10 < d < 20 \quad d = 20$$

AQA Further AS Paper 2 Discrete 2019 June Q3

4 marks Moderate -0.8

3 Manon makes apple cakes and banana cakes. Each apple cake is made with 3 eggs and 100 grams of flour. Each banana cake is made with 2 eggs and 150 grams of flour. Manon has 36 eggs and 1500 grams of flour.
Manon wants to make as many cakes as possible.
Formulate Manon's situation as a linear programming problem, clearly defining any variables you introduce.

AQA Further AS Paper 2 Discrete 2019 June Q4

6 marks Easy -1.2

State the definition of a bipartite graph. 4
A jazz quintet has five musical instruments: bassoon, clarinet, flute, oboe and violin. Jay, Kay, Lee, Mel and Nish are musicians and each plays a musical instrument in the jazz quintet. Jay knows how to play the bassoon and the clarinet.
Kay knows how to play the bassoon, the oboe and the violin.
Lee knows how to play the clarinet and the flute.
Mel knows how to play the clarinet, the oboe and the violin.
Nish knows how to play the flute, the oboe and the violin. 4 (b) (i) Draw a graph to show which musicians know how to play which instruments. 4 (b) (ii) Nish arrives late to a jazz quintet rehearsal. Each of the other four musicians is already playing an instrument: \begin{displayquote} Jay is playing the clarinet
Kay is playing the oboe
Lee is playing the flute
Mel is playing the violin. \end{displayquote} Explain how the graph in part (b)(i) shows that there is no instrument available that Nish knows how to play. 4 (b) (iii) When Nish arrives the rehearsal stops. When they restart the rehearsal, Nish is playing the flute. Draw all possible subgraphs of the graph in part (b)(i) that show how Jay, Kay, Lee and Mel can each be assigned a unique musical instrument they know how to play.
[0pt] [2 marks]

AQA Further AS Paper 2 Discrete 2019 June Q5

5 marks Standard +0.3

Complete the Cayley table in Figure 1 for multiplication modulo 4 \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{dcf97b92-d067-41d4-89a6-ea5bab9ea4ff-08_761_1017_434_493}
\end{figure} 5
The set $S$ is defined as $$S = \{ a , b , c , d \}$$ Figure 2 shows an incomplete Cayley table for $S$ under the commutative binary operation • \begin{table}[h]
\captionsetup{labelformat=empty} \caption{Figure 2}
• $a$ $b$ $c$ $d$
$a$ $b$ $a$ $a$ $c$
$b$ $c$ $c$
$c$ $d$ $d$
$d$ $d$ $d$
\end{table} 5 (b) (i) Complete the Cayley table in Figure 2. 5 (b) (ii) Determine whether the binary operation • is associative when acting on the elements of $S$. Fully justify your answer.

AQA Further AS Paper 2 Discrete 2019 June Q6

13 marks Standard +0.8

6 The diagram shows a nature reserve which has its entrance at $A$, eight information signs at $B , C , \ldots , I$, and fifteen grass paths. The length of each grass path is given in metres.
The total length of the grass paths is 1465 metres. \includegraphics[max width=\textwidth, alt={}, center]{dcf97b92-d067-41d4-89a6-ea5bab9ea4ff-10_812_1192_584_424} To cut the grass, Ashley starts at the entrance and drives a mower along every grass path in the nature reserve. The mower moves at 7 kilometres per hour. 6

Find the least possible time that it takes for Ashley to cut the grass on all fifteen paths in the nature reserve and return to the entrance. Fully justify your answer.
6

Brook visits every information sign in the nature reserve to update them, starting and finishing at the entrance. For the eight information signs, the minimum connecting distance of the grass paths is 510 metres. 6 (b)

Determine a lower bound for the distance Brook walks to visit every information sign.
Fully justify your answer.
[0pt] [2 marks]
6
Using the nearest neighbour algorithm starting from the entrance, determine an upper bound for the distance Brook walks to visit every information sign.
[0pt] [2 marks]
6
Brook takes one minute to update the information at one information sign. Brook walks on the grass paths at an average speed of 5 kilometres per hour. Ashley and Brook start from the entrance at the same time.

6 (c) (i) Use your answers from parts (a) and (b) to show that Ashley and Brook will return to the entrance at approximately the same time. Fully justify your answer.
6 (c) (ii) State an assumption that you have used in part (c)(i). \includegraphics[max width=\textwidth, alt={}, center]{dcf97b92-d067-41d4-89a6-ea5bab9ea4ff-13_2488_1716_219_153} $7 \quad$ Ali and Bex play a zero-sum game. The game is represented by the following pay-off matrix for Ali.

\multirow{2}{*}{}	Bex
	Strategy	$\mathbf { B } _ { \mathbf { 1 } }$	$\mathbf { B } _ { \mathbf { 2 } }$	$\mathbf { B } _ { \mathbf { 3 } }$
\multirow{4}{*}{Ali}	$\mathbf { A } _ { \mathbf { 1 } }$	2	-1	3
	$\mathbf { A } _ { \mathbf { 2 } }$	-4	-2	2
	$\mathbf { A } _ { \mathbf { 3 } }$	0	1	1
	$\mathrm { A } _ { 4 }$	-3	2	-2

•	\(a\)	\(b\)	\(c\)	\(d\)
\(a\)	\(b\)	\(a\)	\(a\)	\(c\)
\(b\)		\(c\)		\(c\)
\(c\)		\(d\)	\(d\)
\(d\)			\(d\)	\(d\)

Label	Activity	Duration	Immediate predecessors
J	Switch on and heat oven		-
K	Put spring rolls in oven and cook
\(L\)	Transfer spring rolls to serving dish

\multirow{2}{*}{}	Bex
	Strategy	\(\mathbf { B } _ { \mathbf { 1 } }\)	\(\mathbf { B } _ { \mathbf { 2 } }\)	\(\mathbf { B } _ { \mathbf { 3 } }\)
\multirow{4}{*}{Ali}	\(\mathbf { A } _ { \mathbf { 1 } }\)	2	-1	3
	\(\mathbf { A } _ { \mathbf { 2 } }\)	-4	-2	2
	\(\mathbf { A } _ { \mathbf { 3 } }\)	0	1	1
	\(\mathrm { A } _ { 4 }\)	-3	2	-2

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