Maths — Decision 1

Practice Paper 2 (Edexcel IAL Maths)

Exam code:YMA01

7 questions · 82 marks · ≈ 1 hour · spans 7 of 9 topics

Assembled from the topic question banks — not an official Edexcel paper

Guided practice

113 marksmedium

3 marks

(a)

1.8 1.4 2.6 1.6 2.8 0.9 3.1 0.8 1.2 2.4 0.6

Use the first-fit bin packing algorithm to determine how the numbers listed above can be packed into bins of size 5

Your answer

3 marks

The list is to be sorted into descending order.

(i) Perform one pass of a bubble sort, starting at the left-hand end of the list. You must write down the list that results at the end of the first pass.

(1 mark)

(ii) Write down the number of comparisons and the number of swaps performed during the first pass.

(2 marks)

Your answer

4 marks

(c)

After a second pass using this bubble sort, the updated list is

2.6 1.8 2.8 1.6 3.1 1.4 1.2 2.4 0.9 0.8 0.6

Use a quick sort on this updated list to obtain the fully sorted list in descending order. You must make your pivots clear.

Your answer

3 marks

(d)

Apply the first-fit decreasing bin packing algorithm to the fully sorted list to pack the numbers into bins of size 5

Your answer

How did you do?/ 13

28 marksmedium

3 marks

(a)

Geometric diagram with nine labelled points (A-J) interconnected by lines, each marked with distances. Points form a network of triangles.

Define the terms

(i) tree,

(ii) minimum spanning tree.

Your answer

3 marks

(b)

Use Kruskal’s algorithm to find the minimum spanning tree for the network shown in Figure 1. You must clearly show the order in which you consider the edges. For each edge, state whether or not you are including it in the minimum spanning tree.

Your answer

2 marks

(c)

Draw the minimum spanning tree using the vertices given in Diagram 1 below and state the weight of the minimum spanning tree.

Diagram 1 with nine labelled dots A to J in a grid-like pattern, each dot evenly spaced apart on the page.

Your answer

How did you do?/ 8

315 marksmedium

6 marks

(a)

Graph with vertices A to J, connected by edges labelled with weights. The total network weight is 383 plus x.

Figure 4 models a network of roads. The number on each edge gives the time, in minutes, to travel along the corresponding road. The vertices, A, B, C, D, E, F, G, H and J represent nine towns. Ezra wishes to travel from A to H as fast as possible.

The time taken to travel between towns G and J is unknown and is denoted by $x$ minutes.

Dijkstra’s algorithm is to be used to find the fastest time to travel from A to H. On Diagram 1 below the “Order of labelling” and “Final value” at A and J, and the “Working values” at J, have already been completed.

Flowchart diagram with nodes A to J showing paths with numeric values. A key explains vertex, order of labelling, final and working values. Labels for fastest time and quickest route.

Use Dijkstra’s algorithm to find the fastest time to travel from A to H. State the quickest route.

Your answer

6 marks

(b)

Ezra needs to travel along each road to check it is in good repair. He wishes to minimise the total time required to traverse the network. Ezra plans to start and finish his inspection route at A. It is given that his route will take at least 440 minutes.

Use the route inspection algorithm and the completed Diagram 1 to find the range of possible values of $x$ .

Your answer

1 mark

(c)

Write down a possible route for Ezra.

Your answer

2 marks

(d)

A new direct road from D to H is under construction and will take 25 minutes to travel along. Ezra will include this new road in a minimum length inspection route starting and finishing at A. It is given that this inspection route takes exactly 488 minutes.

Determine the value of $x$ . You must give reasons for your answer.

Your answer

How did you do?/ 15

410 marksmedium

3 marks

(a)

Diagram of a network graph with vertices labelled A to J, connected by weighted edges. Total network weight is 82. Figure 4 depicted below the graph.

Figure 4 represents a network of 16 roads in a city. The number on each arc represents the time taken, in minutes, to travel along the corresponding road.

Chan needs to check that the roads are in good repair. He must travel along each road at least once. Chan will start and finish at his office at G and must minimise the total time taken for his inspection route.

For this inspection route,

find the time taken and state a possible route. You must make your method and reasoning clear.

Your answer

5 marks

(b)

Chan wonders if he can reduce his travel time by starting from his home at B, travelling along each road at least once and finishing at his office at G.

By considering the pairings of all relevant nodes, find any arcs that would need to be traversed twice in the minimum inspection route from B to G. You must make your method clear, showing your working.

Your answer

2 marks

(c)

Determine which of the two routes ending at G is quicker, the one starting at G or the one starting at B. You must justify your answer.

Your answer

How did you do?/ 10

57 marksmedium

2 marks

(a)

The table below shows the least distances, in km, between six towns, A, B, C, D, E and F.

	A	B	C	D	E	F
A	-	57	76	59	72	65
B	57	-	67	80	66	76
C	76	67	-	71	83	80
D	59	80	71	-	77	78
E	72	66	83	77	-	69
F	65	76	80	78	69	-

Mei must visit each town at least once. She will start and finish at A and wishes her route to minimise the total distance she will travel.

Starting with the minimum spanning tree below, use the shortcut method to find an upper bound below 520 km for Mei’s route. You must state the shortcut(s) you use and the length of your upper bound.

Graph showing a minimum spanning tree with vertices E, B, A, F horizontally and C, D vertically; edge weights are given in kilometres. Total weight is 314 km.

Your answer

2 marks

(b)

Use the nearest neighbour algorithm, starting at A, to find another upper bound for the length of Mei’s route.

Your answer

3 marks

(c)

Starting by deleting E, and all of its arcs, find a lower bound for the length of Mei’s route. Make your method clear.

Your answer

How did you do?/ 7

616 marksmedium

2 marks

(a)

Diagram of a directed graph with nodes numbered 1 to 9, connected by labelled edges; weights in parentheses. Node 5 to 6 has a dashed edge.

A project is modelled by the activity network shown in Figure 2. The activities are represented by the arcs. The number in brackets on each arc gives the time required, in hours, to complete the activity. The numbers in circles are the event numbers. Each activity requires one worker.

Explain the significance of the dummy activity

(i) from event 5 to event 6

(1 mark)

(ii) from event 7 to event 9

(1 mark)

Your answer

4 marks

(b)

Complete Diagram 3 to show the early event times and the late event times.

Flowchart with nodes A to K, numbers in brackets. Arrows depict task flow; solid for direct, dashed for alternate routes. Key shows early and late event times.

Your answer

1 mark

(c)

State the minimum project completion time.

Your answer

2 marks

(d)

Calculate a lower bound for the minimum number of workers required to complete the project in the minimum time. You must show your working.

Your answer

4 marks

(e)

On Grid 1, draw a cascade (Gantt) chart for this project.

Your answer

3 marks

(f)

On Grid 2 , construct a scheduling diagram to show that this project can be completed with three workers in just one more hour than the minimum project completion time.

Numbered grid from 0 to 26 at the top, with vertical dashed lines; labelled "Grid 2" at the bottom.

Your answer

How did you do?/ 16

713 marksmedium

4 marks

(a)

A linear programming problem in $x$ and $y$ is described as follows.

Maximise $P=kx+y$ , where $k$ is a constant

subject to: $3y\geq x$

$x+2y\leq130$

$4x+y\geq100$

$4x+3y\leq300$

Add lines and shading to Diagram 1 below to represent these constraints. Hence determine the feasible region and label it $R$ .

Your answer

5 marks

For the case when $k$ = 0.8

(i) use the objective line method to find the optimal vertex, $V$ , of the feasible region. You must draw and label your objective line and label vertex $V$ clearly.

(2 marks)

(ii) calculate the coordinates of $V$ and hence calculate the corresponding value of $P$ at $V$ .

(3 marks)

Your answer

4 marks

(c)

Given that for a different value of $k$ , $V$ is not the optimal vertex of $R$ ,

determine the range of possible values for $k$ . You must make your method and working clear.

Your answer

How did you do?/ 13

315 marksmedium

6 marks

(a)

Graph with vertices A to J, connected by edges labelled with weights. The total network weight is 383 plus x.

The time taken to travel between towns G and J is unknown and is denoted by $x$ minutes.

Flowchart diagram with nodes A to J showing paths with numeric values. A key explains vertex, order of labelling, final and working values. Labels for fastest time and quickest route.

Use Dijkstra’s algorithm to find the fastest time to travel from A to H. State the quickest route.

Your answer

6 marks

(b)

Use the route inspection algorithm and the completed Diagram 1 to find the range of possible values of $x$ .

Your answer

1 mark

(c)

Write down a possible route for Ezra.

Your answer

2 marks

(d)

Determine the value of $x$ . You must give reasons for your answer.

Your answer

How did you do?/ 15

	A	B	C	D	E	F
A	-	57	76	59	72	65
B	57	-	67	80	66	76
C	76	67	-	71	83	80
D	59	80	71	-	77	78
E	72	66	83	77	-	69
F	65	76	80	78	69	-

	A	B	C	D	E	F
A	-	57	76	59	72	65
B	57	-	67	80	66	76
C	76	67	-	71	83	80
D	59	80	71	-	77	78
E	72	66	83	77	-	69
F	65	76	80	78	69	-

	A	B	C	D	E	F
A	-	57	76	59	72	65
B	57	-	67	80	66	76
C	76	67	-	71	83	80
D	59	80	71	-	77	78
E	72	66	83	77	-	69
F	65	76	80	78	69	-

	A	B	C	D	E	F
A	-	57	76	59	72	65
B	57	-	67	80	66	76
C	76	67	-	71	83	80
D	59	80	71	-	77	78
E	72	66	83	77	-	69
F	65	76	80	78	69	-

	A	B	C	D	E	F
A	-	57	76	59	72	65
B	57	-	67	80	66	76
C	76	67	-	71	83	80
D	59	80	71	-	77	78
E	72	66	83	77	-	69
F	65	76	80	78	69	-

	A	B	C	D	E	F
A	-	57	76	59	72	65
B	57	-	67	80	66	76
C	76	67	-	71	83	80
D	59	80	71	-	77	78
E	72	66	83	77	-	69
F	65	76	80	78	69	-