MATHEMATICS 4736 Decision Mathematics 1

Transcription

1 ADVANCED SUBSIDIARY GCE MATHEMATICS 4736 Decision Mathematics 1 QUESTION PAPER Candidates answer on the Printed Answer Book OCR Supplied Materials: Printed Answer Book 4736 List of Formulae (MF1) Other Materials Required: Scientific or graphical calculator Tuesday 22 June 2010 Afternoon Duration: 1 hour 30 minutes INSTRUCTIONS TO CANDIDATES These instructions are the same on the Printed Answer Book and the Question Paper. Write your name clearly in capital letters, your Centre Number and Candidate Number in the spaces provided on the Printed Answer Book. The questions are on the inserted Question Paper. Write your answer to each question in the space provided in the Printed Answer Book. Additional paper may be used if necessary but you must clearly show your Candidate Number, Centre Number and question number(s). Use black ink. Pencil may be used for graphs and diagrams only. Read each question carefully and make sure that you know what you have to do before starting your answer. Answer all the questions. Do not write in the bar codes. You are permitted to use a graphical calculator in this paper. Give non-exact numerical answers correct to 3 significant figures unless a different degree of accuracy is specified in the question or is clearly appropriate. INFORMATION FOR CANDIDATES This information is the same on the Printed Answer Book and the Question Paper. The number of marks is given in brackets [ ] at the end of each question or part question on the Question Paper. You are reminded of the need for clear presentation in your answers. The total number of marks for this paper is 72. The Printed Answer Book consists of 12 pages. The Question Paper consists of 8 pages. Any blank pages are indicated. INSTRUCTION TO EXAMS OFFICER / INVIGILATOR Do not send this Question Paper for marking; it should be retained in the centre or destroyed. OCR 2010 [M/102/2697] OCR is an exempt Charity 2R 0B10 Turn over

2 1 Owen and Hari each want to sort the following list of marks into decreasing order (i) Owen uses bubble sort, starting from the left-hand end of the list. (a) Show the result of the first pass through the list. Record the number of comparisons and the number of swaps used in this first pass. Which marks, if any, are guaranteed to be in their correct final positions after the first pass? [4] (b) Write down the list at the end of the second pass of bubble sort. [1] (c) How many more passes are needed to get the value 95 to the start of the list? [1] (ii) Hari uses shuttle sort, starting from the left-hand end of the list. Show the results of the first and the second pass through the list. Record the number of comparisons and the number of swaps used in each of these passes. [4] (iii) Explain why, for this particular list, the total number of comparisons will be greater using bubble sort than using shuttle sort. [2] Shuttle sort is a quadratic order algorithm. (iv) If it takes Hari 20 seconds to sort a list of ten marks using shuttle sort, approximately how long will it take Hari to sort a list of fifty marks? [2] 2 A simple graph is one in which any two vertices are directly connected by at most one arc and no vertex is directly connected to itself. A connected graph is one in which every vertex is joined, directly or indirectly, to every other vertex. A simply connected graph is one that is both simple and connected. (i) Explain why it is impossible to draw a graph with exactly three vertices in which the vertex orders are 2, 3 and 4. [1] (ii) Draw a graph with exactly four vertices of orders 1, 2, 3 and 4 that is neither simple nor connected. [2] (iii) Explain why there is no simply connected graph with exactly four vertices of orders 1, 2, 3 and 4. State which of the properties simple and connected cannot be achieved. [2] (iv) A simply connected Eulerian graph has exactly five vertices. (a) Explain why there cannot be exactly three vertices of order 4. [1] (b) By considering the vertex orders, explain why there are only four such graphs. Draw an example of each. [3] OCR Jun10

3 3 3 The constraints of a linear programming problem are represented by the graph below. The feasible region is the unshaded region, including its boundaries. y x (i) Write down the inequalities that define the feasible region. [3] The objective is to maximise P 1 = x + 6y. (ii) Find the values of x and y at the optimal point, and the corresponding value of P 1. [3] The objective is changed to maximise P k = kx + 6y, where k is positive. (iii) Calculate the coordinates of the optimal point, and the corresponding value of P k when the optimal point is not the same as in part (ii). [2] (iv) Find the range of values of k for which the point identified in part (ii) is still optimal. [2] OCR Jun10 Turn over

4 4 4 The network below represents a small village. The arcs represent the streets and the weights on the arcs represent distances in km. E 0.3 A 0.5 B 0.1 D C 0.35 F G (i) Use Dijkstra s algorithm to find the shortest path from A to G. You must show your working, including temporary labels, permanent labels and the order in which permanent labels are assigned. Write down the route of the shortest path from A to G. [5] Hannah wants to deliver newsletters along every street; she will start and end at A. (ii) Which standard network problem does Hannah need to solve to find the shortest route that uses every arc? [1] The total weight of all the arcs is 3.7 km. (iii) Hannah knows that she will need to travel AB and EF twice, once in each direction. With this information, use an appropriate algorithm to find the length of the shortest route that Hannah can use. Show all your working. (You may find the lengths of shortest paths between vertices by inspection.) [5] There are street name signs at each vertex except for A and E. Hannah s friend Peter wants to check that the signs have not been vandalised. He will start and end at B. The table below shows the complete set of shortest distances between vertices B, C, D, F and G. B C D F G B C D F G (iv) Apply the nearest neighbour method to this table, starting from B, to find an upper bound for the distance that Peter must travel. [2] (v) Apply Prim s algorithm to the matrix formed by deleting the row and column for vertex G from the table. Start building your tree at vertex B. Draw your tree. Give the order in which vertices are built into your tree and calculate the total weight of your tree. Hence find a lower bound for the distance that Peter must travel. [4] OCR Jun10

5 5 Jenny is making three speciality smoothies for a party: fruit salad, ginger inger and high C. 5 Each litre of fruit salad contains 600 calories and has 120 mg of sugar and 100 mg of vitamin C. Each litre of ginger inger contains 800 calories and has 80 mg of sugar and 40 mg of vitamin C. Each litre of high C contains 500 calories and has 120 mg of sugar and 120 mg of vitamin C. Jenny has enough milk to make 5 litres of fruit salad or 3 litres of ginger inger or 4 litres of high C. This leads to the constraint 12x + 20y in which x represents the number of litres of fruit salad, y represents the number of litres of ginger inger and represents the number of litres of high C. Jenny wants there to be no more than 5000 calories and no more than 800 mg of sugar in total in the smoothies that she makes. (i) Use this information to write down and simplify two more constraints on the values of x, y and, other than that they are non-negative. [4] Jenny wants to maximise the total amount of vitamin C in the smoothies. This gives the following objective. Maximise P = 100x + 40y (ii) Represent Jenny s problem as an initial Simplex tableau. Use the Simplex algorithm, choosing the first pivot from the column and showing all your working, to find the optimum. How much of each type of smoothie should Jenny make? [13] (iii) Show that if the first pivot had been chosen from the x column then the optimum would have been achieved in one iteration instead of two. [5] OCR Jun10

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8 8 Copyright Information OCR is committed to seeking permission to reproduce all third-party content that it uses in its assessment materials. OCR has attempted to identify and contact all copyright holders whose work is used in this paper. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced in the OCR Copyright Acknowledgements Booklet. This is produced for each series of examinations, is given to all schools that receive assessment material and is freely available to download from our public website ( after the live examination series. If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible opportunity. For queries or further information please contact the Copyright Team, First Floor, 9 Hills Road, Cambridge CB2 1GE. OCR is part of the Cambridge Assessment Group; Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. OCR Jun10

9 POST-EXAMINATION CORRECTION ADVANCED SUBSIDIARY GCE MATHEMATICS 4736 Decision Mathematics 1 Thursday 26 May 2011 Morning In response to centre feedback we have provided a correction to the published examination paper to make it more suitable for practice material. Details are provided below: Question 6(ii) should read Now suppose that Richard wants to find the length of the shortest route that he could use to walk along every track, starting and ending at A. Show that for x 1.8 this route has length ( x) km, and for x 1. 8 it has length ( x) km. Any enquiry about this correction should be referred to the Customer Contact Centre on or general.qualifications@ocr.org.uk Ref: JUN11/post-examination correction

10 ADVANCED SUBSIDIARY GCE MATHEMATICS 4736 Decision Mathematics 1 QUESTION PAPER Candidates answer on the printed answer book. OCR supplied materials: Printed answer book 4736 List of Formulae (MF1) Other materials required: Scientific or graphical calculator Thursday 26 May 2011 Morning Duration: 1 hour 30 minutes INSTRUCTIONS TO CANDIDATES These instructions are the same on the printed answer book and the question paper. The question paper will be found in the centre of the printed answer book. Write your name, centre number and candidate number in the spaces provided on the printed answer book. Please write clearly and in capital letters. Write your answer to each question in the space provided in the printed answer book. Additional paper may be used if necessary but you must clearly show your candidate number, centre number and question number(s). Use black ink. Pencil may be used for graphs and diagrams only. Read each question carefully. Make sure you know what you have to do before starting your answer. Answer all the questions. Do not write in the bar codes. You are permitted to use a scientific or graphical calculator in this paper. Give non-exact numerical answers correct to 3 significant figures unless a different degree of accuracy is specified in the question or is clearly appropriate. INFORMATION FOR CANDIDATES This information is the same on the printed answer book and the question paper. The number of marks is given in brackets [ ] at the end of each question or part question on the question paper. You are reminded of the need for clear presentation in your answers. The total number of marks for this paper is 72. The printed answer book consists of 12 pages. The question paper consists of 8 pages. Any blank pages are indicated. INSTRUCTION TO EXAMS OFFICER / INVIGILATOR Do not send this question paper for marking; it should be retained in the centre or destroyed. OCR 2011 [M/102/2697] OCR is an exempt Charity 2R 1A15 Turn over

11 2 1 The constraints of a linear programming problem are represented by the graph below. The feasible region is the unshaded region, including its boundaries. y 6 A 4 B x (i) Write down the inequalities that define the feasible region. [2] The objective is to maximise P m = x + my, where m is a positive, real-valued constant. (ii) In the case when m = 2, calculate the values of x and y at the optimal point, and the corresponding value of P 2. [2] (iii) (a) Write down the values of m for which point A is optimal. (b) Write down the values of m for which point B is optimal. [2] OCR Jun11

12 3 2 Consider the following algorithm. STEP 1 Input a number N STEP 2 Calculate R = N 2 STEP 3 Calculate S = ((N R) + R) 2 STEP 4 If R and S are the same when rounded to 2 decimal places, go to STEP 7 STEP 5 Replace R with the value of S STEP 6 Go to STEP 3 STEP 7 Output the value of R correct to 2 decimal places (i) Work through the algorithm starting with N = 16. Record the values of R and S each time they change and show the value of the output. [2] (ii) Work through the algorithm starting with N = 2. Record the values of R and S each time they change and show the value of the output. [2] (iii) What does the algorithm achieve for positive inputs? [1] (iv) Show that the algorithm fails when it is applied to N = 4. [1] (v) Describe what happens when the algorithm is applied to N = 2. Suggest how the algorithm could be improved to avoid this problem, without imposing a restriction on the allowable input values. [2] OCR Jun11 Turn over

13 4 3 A simple graph is one in which any two vertices are directly connected by at most one arc and no vertex is directly connected to itself. A connected graph is one in which every vertex is joined, directly or indirectly, to every other vertex. A simply connected graph is one that is both simple and connected. (i) Explain why it is impossible to draw a graph with exactly five vertices of orders 1, 2, 3, 4 and 5. [1] (ii) Explain why there is no simply connected graph with exactly five vertices of orders 2, 2, 3, 4 and 5. State which of the properties simple and connected cannot be achieved. [2] (iii) Calculate the number of arcs in a simply connected graph with exactly five vertices of orders 1, 1, 2, 2 and 4. Hence explain why such a graph cannot be a tree. [2] (iv) Draw a simply connected semi-eulerian graph with exactly five vertices that is also a tree. By considering the orders of the vertices, explain why it is impossible to draw a simply connected Eulerian graph with exactly five vertices that is also a tree. [2] In the graph below the vertices represent buildings and the arcs represent pathways between those buildings. A D C F B E (v) By considering the orders of the vertices, explain why it is impossible to walk along these pathways in a continuous route that uses every arc once and only once. Write down the minimum number of arcs that would need to be travelled twice to walk in a continuous route that uses every arc at least once. [2] OCR Jun11

14 5 4 Consider the following LP problem. Maximise P = 3w + 5x 7y + 2, subject to w + 2x 2y 10, 2w + 3y 4 12, 4w + 5x + y 30, and w 0, x 0, y 0, 0. (i) Represent the problem as an initial Simplex tableau. Explain why the pivot can only be chosen from the x column. [4] (ii) Perform one iteration of the Simplex algorithm. Show how each row was obtained and write down the values of w, x, y, and P at this stage. [4] (iii) Perform a second iteration of the Simplex algorithm. Write down the values of w, x, y, and P at this stage and explain how you can tell from this tableau that P can be increased without limit. How could you have known from the LP formulation above that P could be increased without limit? [5] OCR Jun11 Turn over

15 6 5 The arcs in the network below represent the tracks in a forest and the weights on the arcs represent distances in km. 3.2 C A 0.9 B 5.0 D 2.5 F 1.0 G E 6.0 Dijkstra s algorithm is to be used to find the shortest path from A to G. (i) Apply Dijkstra s algorithm to find the shortest path from A to G. Show your working, including temporary labels, permanent labels and the order in which permanent labels are assigned. Do not cross out your working values. Write down the route of the shortest path from A to G and give its length. [6] The track joining B and D is washed away in a flood. It is replaced by a new track of unknown length, x km. 3.2 C A 0.9 B x D 2.5 F 1.0 G E 6.0 (ii) What is the smallest value that x can take so that the route found in part (i) is still a shortest path? If the value of x is smaller than this, what is the weight of the shortest path from A to G? [2] (iii) (a) (b) For what values of x will vertex E have two temporary labels? Write down the values of these temporary labels. [2] For what values of x will vertex C have two temporary labels? Write down the values of these temporary labels. [2] Dijkstra s algorithm has quadratic order. (iv) If a computer takes 20 seconds to apply Dijkstra s algorithm to a complete network with 50 vertices, approximately how long will it take for a complete network with 100 vertices? [2] OCR Jun11

16 7 6 The arcs in the network represent the tracks in a forest. The weights on the arcs represent distances in km. 3.2 C A 0.9 B x D 2.5 F 1.0 G E 6.0 Richard wants to walk along every track in the forest. The total weight of the arcs is x. (i) Find, in terms of x, the length of the shortest route that Richard could use to walk along every track, starting at A and ending at G. Show all of your working. [3] (ii) Now suppose that Richard wants to find the length of the shortest route that he could use to walk along every track, starting and ending at A. Show that for x 1.8 this route has length ( x) km, and for x 1.8 it has length ( x) km. [8] Whenever two tracks join there is an information board for visitors to the forest. Shauna wants to check that the information boards have not been vandalised. She wants to find the length of the shortest possible route that starts and ends at A, passing through every vertex at least once. Consider first the case when x is less than 3.2. (iii) (a) (b) Apply Prim s algorithm to the network, starting from vertex A, to find a minimum spanning tree. Draw the minimum spanning tree and state its total weight. Explain why the solution to Shauna s problem must be longer than this. [3] Use the nearest neighbour strategy, starting from vertex A, and show that it stalls before it has visited every vertex. [2] Now consider the case when x is greater than 3.2 but less than 4.6. (iv) (a) Draw the minimum spanning tree and state its total weight. [2] (b) Use the nearest neighbour strategy, starting from vertex A, to find a route from A to G passing through each vertex once. Write down the route obtained and its total weight. Show how a shortcut can give a shorter route from A to G passing through each vertex. Hence, explaining your method, find an upper bound for Shauna s problem. [4] OCR Jun11

17 8 Copyright Information OCR is committed to seeking permission to reproduce all third-party content that it uses in its assessment materials. OCR has attempted to identify and contact all copyright holders whose work is used in this paper. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced in the OCR Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download from our public website ( after the live examination series. If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible opportunity. For queries or further information please contact the Copyright Team, First Floor, 9 Hills Road, Cambridge CB2 1GE. OCR is part of the Cambridge Assessment Group; Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. OCR Jun11

18 ADVANCED SUBSIDIARY GCE MATHEMATICS 4736 Decision Mathematics 1 PRINTED ANSWER BOOK Candidates answer on this printed answer book. OCR supplied materials: Question paper 4736 (inserted) List of Formulae (MF1) Other materials required: Scientific or graphical calculator Thursday 26 May 2011 Morning Duration: 1 hour 30 minutes * * * * Candidate forename Candidate surname Centre number Candidate number INSTRUCTIONS TO CANDIDATES These instructions are the same on the printed answer book and the question paper. The question paper will be found in the centre of the printed answer book. Write your name, centre number and candidate number in the spaces provided on the printed answer book. Please write clearly and in capital letters. Write your answer to each question in the space provided in the printed answer book. Additional paper may be used if necessary but you must clearly show your candidate number, centre number and question number(s). Use black ink. Pencil may be used for graphs and diagrams only. Read each question carefully. Make sure you know what you have to do before starting your answer. Answer all the questions. Do not write in the bar codes. You are permitted to use a scientific or graphical calculator in this paper. Give non-exact numerical answers correct to 3 significant figures unless a different degree of accuracy is specified in the question or is clearly appropriate. INFORMATION FOR CANDIDATES This information is the same on the printed answer book and the question paper. The number of marks is given in brackets [ ] at the end of each question or part question on the question paper. You are reminded of the need for clear presentation in your answers. The total number of marks for this paper is 72. The printed answer book consists of 12 pages. The question paper consists of 8 pages. Any blank pages are indicated. OCR 2011 [M/102/2697] OCR is an exempt Charity 2R 1A15 Turn over

19 2 1 (i) 1 (ii) 1(iii)(a) 1(iii)(b) OCR 2011

20 3 2 (i) 2 (ii) 2 (iii) OCR 2011 Turn over

21 4 2 (iv) 2 (v) OCR 2011

22 5 3 (i) 3 (ii) 3 (iii) 3 (iv) OCR 2011 Turn over

23 6 3 (v) 4 (i) OCR 2011

24 7 4 (ii) 4 (iii) OCR 2011 Turn over

25 8 5 (i) Key: Order of becoming permanent Permanent label Temporary labels Do not cross out your working values (temporary labels) 3.2 C A 0.9 B 5.0 D 2.5 F 1.0 G E (ii) OCR 2011

26 9 5(iii)(a) 5(iii)(b) 5 (iv) 6 (i) OCR 2011 Turn over

27 10 6 (ii) 6(iii)(a) C A 0.9 B x D 2.5 F 1.0 G E 6.0 6(iii)(b) OCR 2011

28 11 6(iv)(a) C A 0.9 B x D 2.5 F 1.0 G E 6.0 6(iv)(b) OCR 2011

29 12 PLEASE DO NOT WRITE ON THIS PAGE Copyright Information OCR is committed to seeking permission to reproduce all third-party content that it uses in its assessment materials. OCR has attempted to identify and contact all copyright holders whose work is used in this paper. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced in the OCR Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download from our public website ( after the live examination series. If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible opportunity. For queries or further information please contact the Copyright Team, First Floor, 9 Hills Road, Cambridge CB2 1GE. OCR is part of the Cambridge Assessment Group; Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. OCR 2011

30 Thursday 24 May 2012 Morning AS GCE MATHEMATICS 4736 Decision Mathematics 1 QUESTION PAPER * * Candidates answer on the Printed Answer Book. OCR supplied materials: Printed Answer Book 4736 List of Formulae (MF1) Other materials required: Scientific or graphical calculator Duration: 1 hour 30 minutes INSTRUCTIONS TO CANDIDATES These instructions are the same on the Printed Answer Book and the Question Paper. The Question Paper will be found in the centre of the Printed Answer Book. Write your name, centre number and candidate number in the spaces provided on the Printed Answer Book. Please write clearly and in capital letters. Write your answer to each question in the space provided in the Printed Answer Book. Additional paper may be used if necessary but you must clearly show your candidate number, centre number and question number(s). Use black ink. HB pencil may be used for graphs and diagrams only. Answer all the questions. Read each question carefully. Make sure you know what you have to do before starting your answer. Do not write in the bar codes. You are permitted to use a scientific or graphical calculator in this paper. Give non-exact numerical answers correct to 3 significant figures unless a different degree of accuracy is specified in the question or is clearly appropriate. INFORMATION FOR CANDIDATES This information is the same on the Printed Answer Book and the Question Paper. The number of marks is given in brackets [ ] at the end of each question or part question on the Question Paper. You are reminded of the need for clear presentation in your answers. The total number of marks for this paper is 72. The Printed Answer Book consists of 12 pages. The Question Paper consists of 8 pages. Any blank pages are indicated. INSTRUCTION TO EXAMS OFFICER / INVIGILATOR Do not send this Question Paper for marking; it should be retained in the centre or recycled. Please contact OCR Copyright should you wish to re-use this document. OCR 2012 [M/102/2697] DC (NH/CGW) 51124/1 OCR is an exempt Charity Turn over

31 2 1 Satellite navigation systems (sat navs) use a version of Dijkstra s algorithm to find the shortest route between two places. A simplified map is shown below. The values marked represent road distances, in km, for that section of road (from a place to a road junction, or between two places). Ayton (A) Beetown (B) Ceeville (C) 3 Deeham (D) (E) Eborne 6 25 Fort Effleigh (F) (i) Use the map to construct a network with exactly 10 arcs to show the direct distances between these places, with no road junctions shown. For example, there will need to be an arc connecting A to B of weight 22, and also arcs connecting A to C, D, and E. There is no arc connecting A to F (because there is no route from A to F that does not pass through another place). [2] (ii) Apply Dijkstra s algorithm, starting at A, to find the shortest route from A to F. [5] Dijkstra s algorithm has quadratic order (order n 2 ). (iii) If it takes 3 seconds for a certain sat nav to find the shortest route between two places when it has to process 200 places, calculate approximately how many minutes it will take when it has to process 4000 places. [2] OCR Jun12

32 3 2 A simple graph is one in which any two vertices are directly connected by at most one arc and no vertex is directly connected to itself. A connected graph is one in which every vertex is joined, directly or indirectly, to every other vertex. A simply connected graph is one that is both simple and connected. (i) (a) Draw a simply connected Eulerian graph with exactly five vertices and five arcs. [1] (b) Draw a simply connected semi-eulerian graph with exactly five vertices and five arcs, in which one of the vertices has order 4. [1] (c) Draw a simply connected semi-eulerian graph with exactly five vertices and five arcs, in which none of the vertices have order 4. [1] A teacher is organising revision classes for her students. There will be ten revision classes scheduled into a number of sessions. Each class will run in one session only. Each student has chosen two classes to attend. The table shows which classes each student has chosen. Revision classes Student number C1 C2 C3 C4 M1 M2 S1 S2 D1 D (ii) (a) Draw a graph to show this information. Each vertex represents a class. Each arc links the two classes chosen by a student. [2] (b) Show how the teacher can arrange the classes in just two sessions, which satisfy all student choices. For example, C1 and C2 cannot be in the same session. [2] An extra student joins the group. This student chooses to attend the revision classes in M1 and D1. (c) Explain why the teacher cannot now arrange the classes in just two sessions. Do not amend your graph from part (ii)(a). [2] OCR Jun12 Turn over

33 4 3 The constraints of a linear programming problem are represented by the graph below. The feasible region is the unshaded region, including its boundaries. y x (i) Obtain the four inequalities that define the feasible region. [4] (ii) Calculate the coordinates of the vertices of the feasible region, giving your values as fractions. [4] The objective is to maximise P = x + 4y. (iii) Calculate the value of P at each vertex of the feasible region. Hence write down the coordinates of the optimal point, and the corresponding value of P. [3] Suppose that the solution must have integer values for both x and y. (iv) Find the coordinates of the optimal point with integer-valued x and y, and the corresponding value of P. Explain how you know that this is the optimal solution. [2] OCR Jun12

34 4 Consider the following linear programming problem. Maximise P = 5x 6y + 4z, subject to 3x 4y + z 12, 6x + 2z 20, 10x 5y + 5z 30, x 0, y 0, z 0. 5 (i) Use slack variables s, t and u to rewrite the first three constraints as equations. What restrictions are there on the values of s, t and u? [2] (ii) Represent the problem as an initial Simplex tableau. [2] (iii) Show why the pivot for the first iteration of the Simplex algorithm must be the coefficient of z in the third constraint. [2] (iv) Perform one iteration of the Simplex algorithm, showing how the elements of the pivot row were calculated and how this was used to calculate the other rows. [3] (v) Perform a second iteration of the Simplex algorithm and record the values of x, y, z and P at the end of this iteration. [3] (vi) Write down the values of s, t and u from your final tableau and explain what they mean in terms of the original constraints. [2] OCR Jun12 Turn over

35 6 5 Jess and Henry are out shopping. The network represents the main routes between shops in a shopping arcade. The arcs represent pathways and escalators, the vertices represent some of the shops and the weights on the arcs represent distances in metres. M 70 N 40 P R S T V 210 W The total weight of all the arcs is 1200 metres. The table below shows the shortest distances between vertices; some of these are indirect distances. M N P R S T V W M N P R S T V W (i) Use a standard algorithm to find the shortest distance that Jess must travel to cover every arc in the original network, starting and ending at M. [3] (ii) Find the shortest distance that Jess must travel if she just wants to cover every arc, but does not mind where she starts and where she finishes. Which two points are her start and finish? [2] OCR Jun12

36 Henry suggests that Jess only needs to visit each shop. 7 (iii) Apply the nearest neighbour method to the network, starting at M, to write down a closed tour through all the vertices. Calculate the weight of this tour. What does this value tell you about the length of the shortest closed route that passes through every vertex? [4] Henry thinks that Jess does not need to visit shop W. He uses the table of shortest distances to list all the possible connections between M, N, P, R, S, T and V by increasing order of weight. Henry s list is given in your answer book. (iv) Use Kruskal s algorithm on Henry s list to find a minimum spanning tree for M, N, P, R, S, T and V. Draw the tree and calculate its total weight. [2] Jess insists that they must include shop W. (v) Use the weight of the minimum spanning tree for M, N, P, R, S, T and V, and the table of shortest distances, to find a lower bound for the length of the shortest closed route that passes through all eight vertices. [2] [Question 6 is printed overleaf.] OCR Jun12 Turn over

37 8 6 The following flow chart has been written to find a root of the cubic equation x 3 + Ax 2 + Bx + C = 0, given a starting value X that is thought to be near the root. Input the values of A, B and C Input X Calculate Y = X 3 + AX 2 + BX + C Calculate Z = 3X 2 + 2AX + B NO Does Z = 0? YES Let X = W Let W = X (Y Z) Reduce X by 1 NO Is W X between 0.05 and 0.05? YES Output X (i) Work through the algorithm, recording the values of X, Y, Z and W each time they change, for the equation x 3 4x 2 + 5x + 1 = 0, with a starting value of X = 0. [6] (ii) Show what happens when the algorithm is used for the equation x 3 4x 2 + 5x + 1 = 0, with a starting value of X = 1. [2] (iii) Show what happens when the algorithm is used for the equation x 3 4x 2 + 5x + 1 = 0, with a starting value of X = 1. [5] (iv) Identify a possible problem with using this algorithm. [1] Copyright Information OCR is committed to seeking permission to reproduce all third-party content that it uses in its assessment materials. OCR has attempted to identify and contact all copyright holders whose work is used in this paper. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced in the OCR Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download from our public website ( after the live examination series. If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible opportunity. For queries or further information please contact the Copyright Team, First Floor, 9 Hills Road, Cambridge CB2 1GE. OCR is part of the Cambridge Assessment Group; Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. OCR Jun12

38 Friday 25 January 2013 Afternoon AS GCE MATHEMATICS 4736/01 Decision Mathematics 1 QUESTION PAPER * * Candidates answer on the Printed Answer Book. OCR supplied materials: Printed Answer Book 4736 List of Formulae (MF1) Other materials required: Scientific or graphical calculator Duration: 1 hour 30 minutes INSTRUCTIONS TO CANDIDATES These instructions are the same on the Printed Answer Book and the Question Paper. The Question Paper will be found in the centre of the Printed Answer Book. Write your name, centre number and candidate number in the spaces provided on the Printed Answer Book. Please write clearly and in capital letters. Write your answer to each question in the space provided in the Printed Answer Book. Additional paper may be used if necessary but you must clearly show your candidate number, centre number and question number(s). Use black ink. HB pencil may be used for graphs and diagrams only. Answer all the questions. Read each question carefully. Make sure you know what you have to do before starting your answer. Do not write in the bar codes. You are permitted to use a scientific or graphical calculator in this paper. Give non-exact numerical answers correct to 3 significant figures unless a different degree of accuracy is specified in the question or is clearly appropriate. INFORMATION FOR CANDIDATES This information is the same on the Printed Answer Book and the Question Paper. The number of marks is given in brackets [ ] at the end of each question or part question on the Question Paper. You are reminded of the need for clear presentation in your answers. The total number of marks for this paper is 72. The Printed Answer Book consists of 12 pages. The Question Paper consists of 8 pages. Any blank pages are indicated. INSTRUCTION TO EXAMS OFFICER / INVIGILATOR Do not send this Question Paper for marking; it should be retained in the centre or recycled. Please contact OCR Copyright should you wish to re-use this document. OCR 2013 [M/102/2697] DC (KN/SW) 64518/2 OCR is an exempt Charity Turn over

39 1 (i) Use shuttle sort to put this list of values into decreasing order (from largest to smallest) Show the result at the end of each pass through the algorithm and write down the number of comparisons and the number of swaps used in each pass. [5] (ii) The values give the weights, in kg, of sacks of grain. The sacks are to be loaded into boxes, each of which can hold at most 30 kg. Use the first-fit decreasing method to show which sacks should be put into which box. [2] (iii) Suppose that some stronger boxes are available, each of which can hold at most W kg. Find the least value of W for which only four boxes are needed. Show a packing using four of these stronger boxes. [2] 2 A tetromino is a two-dimensional shape made by joining four squares edge-to-edge. Joins are along complete edges. (i) Represent each of the tetrominoes below by a graph in which the nodes represent the squares and two nodes are joined by an arc if the squares share a common edge. [2] (A) (B) (C) (D) (ii) Six simply connected graphs with four nodes are shown below. For each graph, either draw a tetromino that can be represented by the graph, as in part (i), or explain why this is not possible. [3] (1) (2) (3) (4) (5) (6) Two tetrominoes are regarded as being the same if one can be rotated or reflected to form the other. Derek claims that each tetromino corresponds to a unique tree with four nodes, and each tree with four nodes corresponds to a unique tetromino. Derek s claim is wrong. (iii) From the diagrams above, find: (a) a tetromino whose graph does not correspond to a tree; [1] (b) two different tetrominoes whose graphs correspond to the same tree. [1] A pentomino is a two-dimensional shape made by joining five squares edge-to-edge. Joins are along complete edges. Two pentominoes are regarded as being the same if one can be rotated or reflected to form the other. There are twelve distinct pentominoes. (iv) When the pentominoes are represented by graphs, as in part (i), there are only four distinct graphs. Draw these four graphs. [3] OCR /01 Jan13

40 3 3 The total weight of the arcs in the network below is 230. A 13 C 14 E 14 G B 14 D 15 F 12 H (i) Apply Dijkstra s algorithm to the copy of the network in the answer book to find the least weight path from A to H. Give the path and its weight. [6] In the remainder of this question, any least weight paths required may be found without using a formal algorithm. (ii) The arc AD is removed. Apply the route inspection algorithm, showing your working, to find the weight of the least weight closed route that uses every arc (except AD) at least once. [4] (iii) Suppose, instead, that the arc AD is available, but arcs AC and CD are both removed. Apply the route inspection algorithm, showing your working, to find the weight of the least weight closed route that uses every arc (except AC and CD) at least once. [4] OCR /01 Jan13 Turn over

41 4 Pam has seven employees. When it snows they all need to be contacted by telephone. 4 The table shows the expected time, in minutes, that it will take Pam and her employees to contact each other. Pam Alan Bob Caz Dan Ella Fred Gita Pam Alan Bob Caz Dan Ella Fred Gita (i) Use the nearest neighbour method, starting from Pam, to find a cycle through all the employees and Pam. If there is a choice of names choose the one that occurs first alphabetically. Calculate the total weight of this cycle. [5] (ii) Apply Prim s algorithm to the copy of the table in the answer book, starting by crossing out the row for Pam and looking down the column for Pam. List the arcs in the order in which they were chosen. Draw the resulting minimum spanning tree and calculate its total weight. [6] (iii) Find a lower bound for the minimum weight cycle through Pam and her seven employees by initially removing Gita from the minimum spanning tree. [3] Pam realises that it takes less time if she splits the employees into teams. (iv) Use the minimum spanning tree to suggest how to split the employees into two teams, so that Pam contacts the two team leaders and they each contact the members of their team. Using this solution, find the minimum elapsed time by which all the employees can be contacted. [3] OCR /01 Jan13

42 5 5 Roland Neede, the baker, is making cupcakes. He makes three sizes of cupcake: miniature, small and standard. Miniature cupcakes are sold in boxes of 24 and each cupcake uses 3 units of topping and 2 decorations. Small cupcakes are sold in boxes of 20 and each cupcake uses 5 units of topping and 3 decorations. Standard cupcakes are sold in boxes of 12 and each cupcake uses 7 units of topping and 4 decorations. Roland has no restriction on the amount of cake mix that he uses but he only has 5000 units of topping and 3000 decorations available. Cupcakes are only sold in complete boxes, and Roland assumes that he can sell all the boxes of cupcakes that he makes. Irrespective of size, each box of cupcakes sold will give Roland a profit of 1. Roland wants to maximise his total profit. Let x denote the number of boxes of miniature cupcakes, y denote the number of boxes of small cupcakes and z denote the number of boxes of standard cupcakes that Roland makes. (i) Construct an objective function, P, to be maximised. [1] (ii) By considering the number of units of topping used, show that 18x + 25y + 21z G [2] (iii) Construct a similar constraint by considering the number of decorations used, simplifying the coefficients so that they are integers with no common factor. [2] (iv) Set up an initial Simplex tableau to represent Roland s problem. [3] (v) Perform one iteration of the Simplex algorithm, choosing a pivot from the x column. Explain how the choice of pivot row was made and show how each row was calculated. [5] (vi) Write down the values of x, y and z from the first iteration of the Simplex algorithm. Hence find the maximum profit that Roland can make, remembering that cupcakes can only be sold in complete boxes. Calculate the number of units of topping and the number of decorations that are left over with this solution. [4] (vii) The constraint from the number of units of topping can be rewritten as 18P + 7y + 3z G Form a similar expression for the constraint from the number of decorations. Use this to find the number of boxes of small cupcakes which maximises the profit when there are no decorations left over. Find the solution which gives the maximum profit using all the topping and all the decorations, and find the values of x, y and z for this solution. [5] OCR /01 Jan13

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45 8 Copyright Information OCR is committed to seeking permission to reproduce all third-party content that it uses in its assessment materials. OCR has attempted to identify and contact all copyright holders whose work is used in this paper. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced in the OCR Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download from our public website ( after the live examination series. If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible opportunity. For queries or further information please contact the Copyright Team, First Floor, 9 Hills Road, Cambridge CB2 1GE. OCR is part of the Cambridge Assessment Group; Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. OCR /01 Jan13

46 Tuesday 10 June 2014 Morning AS GCE MATHEMATICS 4736/01 Decision Mathematics 1 QUESTION PAPER * * Candidates answer on the Printed Answer Book. OCR supplied materials: Printed Answer Book 4736/01 List of Formulae (MF1) Other materials required: Scientific or graphical calculator Duration: 1 hour 30 minutes INSTRUCTIONS TO CANDIDATES These instructions are the same on the Printed Answer Book and the Question Paper. The Question Paper will be found inside the Printed Answer Book. Write your name, centre number and candidate number in the spaces provided on the Printed Answer Book. Please write clearly and in capital letters. Write your answer to each question in the space provided in the Printed Answer Book. Additional paper may be used if necessary but you must clearly show your candidate number, centre number and question number(s). Use black ink. HB pencil may be used for graphs and diagrams only. Answer all the questions. Read each question carefully. Make sure you know what you have to do before starting your answer. Do not write in the bar codes. You are permitted to use a scientific or graphical calculator in this paper. Give non-exact numerical answers correct to 3 significant figures unless a different degree of accuracy is specified in the question or is clearly appropriate. INFORMATION FOR CANDIDATES This information is the same on the Printed Answer Book and the Question Paper. The number of marks is given in brackets [ ] at the end of each question or part question on the Question Paper. You are reminded of the need for clear presentation in your answers. The total number of marks for this paper is 72. The Printed Answer Book consists of 12 pages. The Question Paper consists of 8 pages. Any blank pages are indicated. INSTRUCTION TO EXAMS OFFICER / INVIGILATOR Do not send this Question Paper for marking; it should be retained in the centre or recycled. Please contact OCR Copyright should you wish to re-use this document. OCR 2014 [M/102/2697] DC (NF/SW) 80661/3 OCR is an exempt Charity Turn over

47 2 1 Sangita needs to move some heavy boxes to her new house. She has borrowed a van that can carry at most 600 kg. She will have to make several deliveries to her new house. The masses of the boxes have been recorded in kg as: (i) Use the first-fit method to show how Sangita could pack the boxes into the van. How many deliveries does this solution require? [3] (ii) Use the first-fit decreasing method to show how Sangita could pack the boxes into the van. There is no need to use a sorting algorithm, but you should write down the sorted list before showing the packing. How many deliveries does this solution require? [4] Sangita then realises that she cannot fit more than four boxes in the van at a time. (iii) Find a way to pack the boxes into the van so that she makes as few deliveries as possible. [2] 2 A simple graph is one in which any two vertices are directly connected by at most one arc and no vertex is directly connected to itself. A connected graph is one in which every vertex is joined, directly or indirectly, to every other vertex. A simply connected graph is one that is both simple and connected. (i) (a) Draw a simply connected graph that has exactly four vertices and exactly five arcs. Is your graph Eulerian, semi-eulerian or neither? Explain how you know. [3] (b) By considering the sum of the vertex orders, show that there is only one possible simply connected graph with exactly four vertices and exactly five arcs. [5] (ii) Draw five distinct simply connected graphs each with exactly five vertices and exactly five arcs. [3] OCR /01 Jun14

48 3 3 The following algorithm finds two positive integers for which the sum of their squares equals a given input, when this is possible. The function INT(X ) gives the largest integer that is less than or equal to X. For example: INT ^6. 9h = 6, INT ^7h = 7, INT ^7. 1h = 7. Line 10 Input a positive integer, N Line 20 Let C = 1 Line 30 If C H N jump to line 110 Line 40 Let X = ^N - C 2 h [you may record your answer as a surd or a decimal] Line 50 Let Y = INT( X) Line 60 If X = Y jump to line 100 Line 70 If C 2 Y jump to line 110 Line 80 Add 1 to C Line 90 Go back to line 30 Line 100 Print C, X and stop Line 110 Print FAIL and stop (i) Apply the algorithm to the input N = 500. You only need to write down values when they change and there is no need to record the use of lines 30, 60, 70 or 90. [4] (ii) Apply the algorithm to the input N = 7. [2] (iii) Explain why lines 70 and 110 are needed. [1] The algorithm has order N. (iv) If it takes 0.7 seconds to run the algorithm when N = 3000, roughly how long will it take when N = ? [2] OCR /01 Jun14 Turn over

49 4 4 The network below represents a treasure trail. The arcs represent paths and the weights show distances in units of 100 metres. The total length of the paths shown is 4200 metres. B 1 E A 1 C 4 F 3 H D 4 G (i) Apply Dijkstra s algorithm to the network, starting at A, to find the shortest distance (in metres) from A to each of the other vertices. [5] Alex wants to hunt for the treasure. His current location is marked on the network as A. The clues to the location of the treasure are located on the paths. Every path has at least one clue and some paths have more than one. This means that Alex will need to search along the full length of every path to find all the clues. (ii) Showing your working, find the length of the shortest route that Alex can take, starting and ending at A, to find every clue. [3] The clues tell Alex that the treasure is located at the point marked as H on the network. (iii) Write down the shortest route from A to H. [1] Zac also starts at A and searches along every path to find the clues. He also uses a shortest route to do this, but without returning to A. Instead he proceeds directly to the treasure at H. (iv) Calculate the length of the shortest route that Zac can take to search for all the clues and reach the treasure. [2] OCR /01 Jun14