OXFORD CAMBRIDGE AND RSA EXAMINATIONS
Advanced Subsidiary General Certificate of Education
Advanced General Certificate of Education
MEI STRUCTURED MATHEMATICS
4773
Decision Mathematics Computation
Thursday
15 JUNE 2006
Afternoon
2 hours 30 minutes
Additional materials:
8 page answer booklet
Graph paper
MEI Examination Formulae and Tables (MF2)
TIME
2 hours 30 minutes
INSTRUCTIONS TO CANDIDATES
•
Write your name, centre number and candidate number in the spaces provided on the answer
booklet.
•
Answer all the questions.
•
Additional sheets, including computer print-outs, should be fastened securely to the answer
booklet.
•
You are permitted to use a graphical calculator in this paper.
COMPUTING RESOURCES
•
Candidates will require access to a computer with a spreadsheet program, a linear programming
package and suitable printing facilities throughout the examination.
INFORMATION FOR CANDIDATES
•
The number of marks is given in brackets [ ] at the end of each question or part question.
•
In each of the questions you are required to write spreadsheet or other routines to carry out
various processes.
•
For each question you attempt, you should submit print-outs showing the routine you have written
and the output it generates.
•
You are not expected to print out and submit everything your routine produces, but you are
required to submit sufficient evidence to convince the examiner that a correct procedure has
been used.
•
The total number of marks for this paper is 72.
This question paper consists of 5 printed pages and 3 blank pages.
HN/3
© OCR 2006 [Y/102/2662]
Registered Charity 1066969
[Turn over
2
1
An investor is considering three investment opportunities over the next five years. He wishes to
maximise the amount of money he has at the end of those five years.
Investment A is a one-year investment. It is available in each of the years and may be started at the
beginning of any year. At the end of a year it will return £1.15 for every £1 invested.
Investment B is a three-year investment. It may be started at the beginning of year 1, year 2, or
year 3. It will return £1.55 for every £1 invested.
Investment C is another one-year investment, but it is not available until the start of year 3. It will
return £1.20 per annum for every £1 invested.
The investor has £50000 to invest.
(i) Define appropriate variables and formulate the investor’s problem as an LP.
[10]
(ii) Solve your LP using your LP package, and interpret your solution.
You should enclose printouts of your formulation and your output with your solution.
[4]
(iii) You should have found that it is not worth investing in B. By experimenting with your LP, or
otherwise, find what the return on B would have to be to make it worthwhile investing in it.
[3]
4773 June 2006
3
2
The Fibonacci recurrence relation is un2 un1 un, with u0 1 and u1 1.
(i) Build a spreadsheet with two columns, the first giving the numbers 0, 1, 2, …, and the second
giving the corresponding Fibonacci numbers.
Print out the first 20 Fibonacci numbers.
[3]
(ii) Write down and solve the auxiliary equation for the Fibonacci recurrence relation. Hence find
an expression for the nth Fibonacci number, and show that it can be expressed in the form
1 Ê Ê1 +
un =
ÁÁ
5 ËË 2
5ˆ
˜
¯
n +1
Ê1 -Á
Ë 2
5ˆ
˜
¯
n +1ˆ
˜.
¯
[9]
(iii) Verify that the formula is correct by coding the formula into the third column of your
spreadsheet.
Print out your spreadsheet formula and print out your spreadsheet.
[2]
(iv) In the fourth column of your spreadsheet compute the Fibonacci ratios Rn, where Rn1 is the
( n 1 ) th Fibonacci number divided by the nth Fibonacci number.
Describe what happens.
Find the exact value of the limit (which is known as the Golden Ratio).
4773 June 2006
[5]
[Turn over
4
3
Four shops, S1, S2, S3 and S4 are to be supplied with crates of material from three warehouses,
W1, W2 and W3. The requirements at the shops are 10 crates at S1, 15 at S2, 12 at S3 and 20 at
S4. There are 20 crates available at each warehouse.
The costs of delivering a single case from each warehouse to each shop are shown in Table 3.1.
cost (£)
S1
S2
S3
S4
W1
2
2
1
5
W2
3
2
2
4
W3
5
5
1
2
Table 3.1
(i) Formulate an LP to solve the problem of moving crates from warehouses to shops at minimum
total cost. Produce a printout of your formulation.
[7]
(ii) Use your LP package to solve your LP. Produce a printout of your solution.
Interpret your solution.
[4]
Two customers, C1 and C2, require 30 and 27 crates respectively. The costs per crate of supplying
each of them from each of the shops is shown in Table 3.2.
cost (£)
S1
S2
S3
S4
C1
4
6
3
2
C2
1
4
2
5
Table 3.2
(iii) Formulate, solve and interpret an LP to find the cheapest way of supplying the two customers
from the warehouses via the shops. There are still 20 crates available at each warehouse, but
the shop requirements no longer apply.
[7]
4773 June 2006
5
4
The weather in Brighting is either wet, showery or dry. On the day following a wet day there is a
20% chance that it will be wet and a 30% chance that it will be showery. On the day following a
showery day there is a 40% chance that it will be wet and a 15% chance that it will be showery.
On the day following a dry day there is a 15% chance that it will be wet and a 25% chance that it
will be showery.
Today the weather in Brighting is dry.
(i) Find the probabilities of it being wet, showery or dry in Brighting on the day after tomorrow.
[4]
(ii) Build a spreadsheet to simulate the weather in Brighting tomorrow, and the day after
tomorrow.
(You may wish to set up lookup tables to model the probabilities, and to use “= IF(…,… ,…)”
statements to branch to the appropriate lookup columns.)
Print out the formulae which you use in your spreadsheet.
[7]
(iii) Run your simulation 10 times, putting your results into a table.
Estimate the probabilities of it being wet, showery or dry the day after tomorrow.
[3]
(iv) Extend your spreadsheet to investigate what happens after 20 days.
Print out your spreadsheet.
Run your simulation 10 times, putting your results into a table.
Estimate the probabilities of it being wet, showery or dry after 20 days.
4773 June 2006
[4]
OXFORD CAMBRIDGE AND RSA EXAMINATIONS
Advanced Subsidiary General Certificate of Education
Advanced General Certificate of Education
MEI STRUCTURED MATHEMATICS
4773
Decision Mathematics Computation
Thursday
15 JUNE 2006
Afternoon
2 hours 30 minutes
Additional materials:
8 page answer booklet
Graph paper
MEI Examination Formulae and Tables (MF2)
TIME
2 hours 30 minutes
INSTRUCTIONS TO CANDIDATES
•
Write your name, centre number and candidate number in the spaces provided on the answer
booklet.
•
Answer all the questions.
•
Additional sheets, including computer print-outs, should be fastened securely to the answer
booklet.
•
You are permitted to use a graphical calculator in this paper.
COMPUTING RESOURCES
•
Candidates will require access to a computer with a spreadsheet program, a linear programming
package and suitable printing facilities throughout the examination.
INFORMATION FOR CANDIDATES
•
The number of marks is given in brackets [ ] at the end of each question or part question.
•
In each of the questions you are required to write spreadsheet or other routines to carry out
various processes.
•
For each question you attempt, you should submit print-outs showing the routine you have written
and the output it generates.
•
You are not expected to print out and submit everything your routine produces, but you are
required to submit sufficient evidence to convince the examiner that a correct procedure has
been used.
•
The total number of marks for this paper is 72.
This question paper consists of 5 printed pages and 3 blank pages.
HN/3
© OCR 2006 [Y/102/2662]
Registered Charity 1066969
[Turn over
2
1
An investor is considering three investment opportunities over the next five years. He wishes to
maximise the amount of money he has at the end of those five years.
Investment A is a one-year investment. It is available in each of the years and may be started at the
beginning of any year. At the end of a year it will return £1.15 for every £1 invested.
Investment B is a three-year investment. It may be started at the beginning of year 1, year 2, or
year 3. It will return £1.55 for every £1 invested.
Investment C is another one-year investment, but it is not available until the start of year 3. It will
return £1.20 per annum for every £1 invested.
The investor has £50000 to invest.
(i) Define appropriate variables and formulate the investor’s problem as an LP.
[10]
(ii) Solve your LP using your LP package, and interpret your solution.
You should enclose printouts of your formulation and your output with your solution.
[4]
(iii) You should have found that it is not worth investing in B. By experimenting with your LP, or
otherwise, find what the return on B would have to be to make it worthwhile investing in it.
[3]
4773 June 2006
3
2
The Fibonacci recurrence relation is un2 un1 un, with u0 1 and u1 1.
(i) Build a spreadsheet with two columns, the first giving the numbers 0, 1, 2, …, and the second
giving the corresponding Fibonacci numbers.
Print out the first 20 Fibonacci numbers.
[3]
(ii) Write down and solve the auxiliary equation for the Fibonacci recurrence relation. Hence find
an expression for the nth Fibonacci number, and show that it can be expressed in the form
1 Ê Ê1 +
un =
ÁÁ
5 ËË 2
5ˆ
˜
¯
n +1
Ê1 -Á
Ë 2
5ˆ
˜
¯
n +1ˆ
˜.
¯
[9]
(iii) Verify that the formula is correct by coding the formula into the third column of your
spreadsheet.
Print out your spreadsheet formula and print out your spreadsheet.
[2]
(iv) In the fourth column of your spreadsheet compute the Fibonacci ratios Rn, where Rn1 is the
( n 1 ) th Fibonacci number divided by the nth Fibonacci number.
Describe what happens.
Find the exact value of the limit (which is known as the Golden Ratio).
4773 June 2006
[5]
[Turn over
4
3
Four shops, S1, S2, S3 and S4 are to be supplied with crates of material from three warehouses,
W1, W2 and W3. The requirements at the shops are 10 crates at S1, 15 at S2, 12 at S3 and 20 at
S4. There are 20 crates available at each warehouse.
The costs of delivering a single case from each warehouse to each shop are shown in Table 3.1.
cost (£)
S1
S2
S3
S4
W1
2
2
1
5
W2
3
2
2
4
W3
5
5
1
2
Table 3.1
(i) Formulate an LP to solve the problem of moving crates from warehouses to shops at minimum
total cost. Produce a printout of your formulation.
[7]
(ii) Use your LP package to solve your LP. Produce a printout of your solution.
Interpret your solution.
[4]
Two customers, C1 and C2, require 30 and 27 crates respectively. The costs per crate of supplying
each of them from each of the shops is shown in Table 3.2.
cost (£)
S1
S2
S3
S4
C1
4
6
3
2
C2
1
4
2
5
Table 3.2
(iii) Formulate, solve and interpret an LP to find the cheapest way of supplying the two customers
from the warehouses via the shops. There are still 20 crates available at each warehouse, but
the shop requirements no longer apply.
[7]
4773 June 2006
5
4
The weather in Brighting is either wet, showery or dry. On the day following a wet day there is a
20% chance that it will be wet and a 30% chance that it will be showery. On the day following a
showery day there is a 40% chance that it will be wet and a 15% chance that it will be showery.
On the day following a dry day there is a 15% chance that it will be wet and a 25% chance that it
will be showery.
Today the weather in Brighting is dry.
(i) Find the probabilities of it being wet, showery or dry in Brighting on the day after tomorrow.
[4]
(ii) Build a spreadsheet to simulate the weather in Brighting tomorrow, and the day after
tomorrow.
(You may wish to set up lookup tables to model the probabilities, and to use “= IF(…,… ,…)”
statements to branch to the appropriate lookup columns.)
Print out the formulae which you use in your spreadsheet.
[7]
(iii) Run your simulation 10 times, putting your results into a table.
Estimate the probabilities of it being wet, showery or dry the day after tomorrow.
[3]
(iv) Extend your spreadsheet to investigate what happens after 20 days.
Print out your spreadsheet.
Run your simulation 10 times, putting your results into a table.
Estimate the probabilities of it being wet, showery or dry after 20 days.
4773 June 2006
[4]
4773/01
ADVANCED GCE
MATHEMATICS (MEI)
Decision Mathematics Computation
THURSDAY 12 JUNE 2008
*CUP/T44379*
Additional materials (enclosed):
Morning
Time: 2 hours 30 minutes
None
Additional materials (required):
Answer Booklet (8 pages)
Graph paper
MEI Examination Formulae and Tables (MF2)
INSTRUCTIONS TO CANDIDATES
•
Write your name in capital letters, your Centre Number and Candidate Number in the spaces
provided on the Answer Booklet.
•
Read each question carefully and make sure you know what you have to do before starting
your answer.
•
Answer all the questions.
•
•
•
You are permitted to use a graphical calculator in this paper.
Final answers should be given to a degree of accuracy appropriate to the context.
Additional sheets, including computer print-outs, should be fastened securely to the Answer
Booklet.
COMPUTING RESOURCES
•
Candidates will require access to a computer with a spreadsheet program, a linear
programming package and suitable printing facilities throughout the examination.
INFORMATION FOR CANDIDATES
•
The number of marks is given in brackets [ ] at the end of each question or part question.
•
The total number of marks for this paper is 72.
•
In each of the questions you are required to write spreadsheet or other routines to carry out
various processes.
•
For each question you attempt, you should submit print-outs showing the routine you have
written and the output it generates.
•
You are not expected to print out and submit everything your routine produces, but you are
required to submit sufficient evidence to convince the examiner that a correct procedure has
been used.
This document consists of 6 printed pages and 2 blank pages.
SP (KN) T44379/4
© OCR 2008 [L/102/2662]
OCR is an exempt Charity
[Turn over
2
1
The vertices of the network represent six villages in a rural area in a developing country. The arc
weights represent journey times (hours) along connecting tracks.
B
5.5
A
1.5
1
7.5
F
2
4.5
6
5.5
C
7.5
E
4.5
D
Health centres are to be built in some villages, so that all villages are within 5 hours journey time of a
health centre.
The matrix shows the complete set of least journey times between the villages.
A
B
C
D
E
F
A
0
2.5
5.5
6.5
2
1
B
2.5
0
6
7
4.5
1.5
C
5.5
6
0
7.5
7.5
4.5
D
6.5
7
7.5
0
4.5
5.5
E
2
4.5
7.5
4.5
0
3
F
1
1.5
4.5
5.5
3
0
(i) Let XA, XB, … be indicator variables representing whether or not a health centre is built in village
A, B, …
Write down an inequality using these variables which, when satisfied, will ensure that village A
is within 5 hours journey of a health centre. Explain your inequality by reference to the matrix
above.
[5]
(ii) Formulate an LP, the solution to which will give the minimum number of health centres required
and their locations.
[5]
(iii) Run your LP and interpret the solution.
[3]
(iv) Modify your LP so as to find a second optimal solution to the problem.
[2]
(v) Find a third optimal solution.
[1]
(vi) Explain why it is not necessary to force your variables to be indicator variables, and why it is
preferable not to do so.
[2]
© OCR 2008
4773/01 Jun08
3
2
For a small population of organisms it is approximately the case that in any one-second interval either
a birth occurs, or a death, or both, or neither. There is never more than one birth or one death in a onesecond interval.
Let n be the number of organisms in existence at the beginning of a one-second interval.
The probability of a birth occurring in that interval is α × n, where α is a small positive number.
Similarly, the probability of a death occurring in that interval is β × n, where β is a small positive
number.
(i) Build a spreadsheet to simulate the development of a population of size 10 over a
one-minute period.
Print out your spreadsheet and show the formulae which you use.
[6]
(ii) Using α = 0.01 and β = 0.04, run your simulation 10 times.
Tabulate your results and estimate the probability of extinction over the one-minute period.
[3]
(iii) Investigate how the probability of extinction varies for different βs, given that α remains at 0.01.
Tabulate your results.
[3]
You are now to investigate what happens if the population is increased by immigration. In each
one-second period there may be either 0 or 1 immigrants arrive into the population. The probability of
an arrival is γ.
(iv) Modify your spreadsheet to incorporate immigration.
Print out your spreadsheet and show the formulae which you use.
(v) Investigate how immigration affects your results in part (iii), given that γ = 0.05.
© OCR 2008
4773/01 Jun08
[4]
[2]
[Turn over
4
3
(a)
A
B
C
D
E
F
G
A
–
8
–
–
3
–
–
B
8
–
5
7
–
–
–
C
–
5
–
1
–
–
8
D
–
7
1
–
2
–
–
E
3
–
–
2
–
4
–
F
–
–
–
–
4
–
6
G
–
–
8
–
–
6
–
Table 3.1
Table 3.1 represents the capacities of arcs in a flow network.
Formulate and solve an LP to find the maximum flow from A to G.
[9]
(b)
S1
S2
S3
S4
S5
S6
S7
W1
–
8
–
–
3
–
–
W2
8
–
5
7
–
–
–
W3
–
5
–
1
–
–
8
W4
–
7
1
–
2
–
–
W5
3
–
–
2
–
4
–
W6
–
–
–
–
4
–
6
W7
–
–
8
–
–
6
–
Table 3.2
Table 3.2 has the same numerical entries as Table 3.1, but its rows are labelled W1 to W7,
representing warehouses, and its columns are labelled S1 to S7, representing shops supplied from
those warehouses. Each numerical entry represents the cost of moving a single unit from the
corresponding warehouse to the corresponding shop.
Each warehouse has 10 units in stock. Each shop requires 10 units.
Formulate and solve an LP to find a minimum cost distribution pattern.
© OCR 2008
4773/01 Jun08
[9]
5
4
(a) Solve the recurrence relation un + 2 = 3–2 un + 1 – 1–2 un, given that u0 = 5 and u1 = 3.
Give the values of u2, u3, u10 and u1 000 000.
[13]
(b) Successive waves arriving on a beach are found to have heights which follow approximately the
recurrence relation wn + 2 = 3–2 wn + 1 – wn + 5, with w0 = 5 and w1 = 3.
© OCR 2008
(i) Construct a spreadsheet to model wave height.
[2]
(ii) Give the values of w2, w3, w10 and w20.
[2]
(iii) State the limitation of using a spreadsheet to model a recurrence relation.
[1]
4773/01 Jun08
6
BLANK PAGE
© OCR 2008
4773/01 Jun08
7
BLANK PAGE
© OCR 2008
4773/01 Jun08
ADVANCED GCE
4773
MATHEMATICS (MEI)
Decision Mathematics Computation
*OCE/T67869*
Friday 19 June 2009
Afternoon
Candidates answer on the Answer Booklet
OCR Supplied Materials:
•
8 page Answer Booklet
•
Graph paper
•
MEI Examination Formulae and Tables (MF2)
Duration: 2 hours 30 minutes
Other Materials Required:
None
*
4
7
7
3
*
INSTRUCTIONS TO CANDIDATES
•
•
•
•
•
•
•
•
Write your name clearly in capital letters, your Centre Number and Candidate Number in the spaces
provided on the Answer Booklet.
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.
You are permitted to use a graphical calculator in this paper.
Final answers should be given to a degree of accuracy appropriate to the context.
Additional sheets, including computer print-outs, should be fastened securely to the Answer Booklet.
Do not write in the bar codes.
INFORMATION FOR CANDIDATES
•
•
•
•
•
•
The number of marks is given in brackets [ ] at the end of each question or part question.
In each of the questions you are required to write spreadsheet or other routines to carry out various
processes.
For each question you attempt, you should submit print-outs showing the routine you have written and the
output it generates.
You are not expected to print out and submit everything your routine produces, but you are required to
submit sufficient evidence to convince the examiner that a correct procedure has been used.
The total number of marks for this paper is 72.
This document consists of 8 pages. Any blank pages are indicated.
COMPUTING RESOURCES
•
Candidates will require access to a computer with a spreadsheet program, a linear programming package
and suitable printing facilities throughout the examination.
© OCR 2009 [Y/102/2662]
SP (SLM/CGW) T67869/4
OCR is an exempt Charity
Turn over
2
1
At one-minute intervals the automatic steering on a boat finds the bearing which gives the boat’s
actual direction of motion. This bearing is compared with the desired bearing. A correction equal to
the difference between the two bearings is applied. However, this correction takes two minutes to take
effect.
(i) Letting the actual bearing at time n (minutes) be Bn, explain why Bn+2 – Bn+1 + Bn = 0 when the
boat’s desired bearing is 0° (due North).
[2]
(ii) Given that B0 = 2 and B1 = 4, build a spreadsheet model to show Bn for 0 ⭐ n ⭐ 20.
Describe what happens.
[3]
The performance of the automatic steering is improved by making the correction equal in magnitude to
half the difference between the actual bearing and the desired bearing.
(iii) Given that the desired bearing is 0°, give the recurrence relation for Bn with this improvement.
Build a spreadsheet model to show Bn for 0 ⭐ n ⭐ 20 given that B0 = 2 and B1 = 4.
Describe what happens.
[3]
The performance is further improved by making the correction equal in magnitude to a quarter of the
difference between the actual bearing and the desired bearing.
(iv) The desired bearing is still 0°.
Build a spreadsheet model incorporating this improvement to show Bn for 0 ⭐ n ⭐ 20 given that
B0 = 2 and B1 = 4.
Describe what happens.
[3]
(v) Solve the recurrence relation Bn+2 – Bn+1 + 14 Bn = 0, with B0 = 2 and B1 = 4.
Compare the values for B2, B3, …, B20 given by your solution to those you found in part (iv). [7]
Print out your spreadsheets.
© OCR 2009
4773 Jun09
3
2
The diagram shows a pipe network with a flow of 10 units established through it. Except for pipes
leaving the source (S), or entering the sink (T), flows can be in either direction.
key
capacity
flow
forward
potential
backward
potential
direction of
flow
A
12
10
10 10
0 10
2
20 20
0
10 10
0 10
10 0
10 20
S
20
0
B
C
T
10 0
10 20
20
E
18 18
0 18
10
10
0
D
(i) Use flow augmentation to establish a flow of 30 through the network. Redraw the network after
each augmentation, showing updated labels.
[7]
(ii) Give a cut to prove that the flow of 30 units is maximal.
[2]
(iii) Formulate an LP to find the maximum flow through the network.
[7]
(iv) Run your LP and interpret the results.
[2]
© OCR 2009
4773 Jun09
Turn over
4
3
Bill and Fred are playing each other at cards. Bill is a better player than Fred and has a probability of
0.55 of winning each hand. The loser of a hand pays £1 to the winner.
Bill currently has £5 and Fred has £5.
(i) Build a spreadsheet to simulate what happens.
[3]
(ii) Run your simulation ten times, each time continuing until one of Bill and Fred is ruined (i.e. runs
out of money). Hence estimate the probability that Bill is ruined.
[4]
Bill plays the same game in a casino, again starting with £5. His probability of winning a hand is now
0.45, and this time his opponent, the casino, has a limitless amount of money and cannot be ruined.
When the probability of winning a hand is less than 0.5, Bill will eventually be ruined.
(iii) Use simulation with ten repetitions to estimate how many hands Bill can expect to play before he
is ruined.
[5]
Tim is playing a singles tennis game, and the score is “deuce”. This means that the score is level, and
that the first player to be two points ahead wins the game. The probability of Tim winning each point is
0.55.
(iv) Build a spreadsheet to simulate the remainder of the game.
Use ten repetitions to produce an estimate of the probability of Tim winning the game.
[4]
Minal has a large pot into which she sometimes puts a £1 coin. She also sometimes takes a £1 coin out.
Deposits and withdrawals are in the ratio 55:45.
(v) Build a spreadsheet to simulate this situation, given that there are currently five £1 coins in the
pot.
What difficulty would you encounter in trying to use simulation to estimate the probability of the
pot being emptied?
[2]
© OCR 2009
4773 Jun09
5
4
J. R. Wing has £4 million to invest in three oil well sites. The revenue from each site will depend on
the investment in that site, as shown in the table. The amount invested must be an exact multiple of
£1 million.
Revenue (£ million)
Amount invested
(£ million)
Site 1
Site 2
Site 3
0
4
3
3
1
7
6
7
2
8
10
8
3
9
12
13
4
11
14
15
(i) Let simj be an indicator variable which takes the value 1 if £j million is invested in site i, and
0 otherwise (i = 1, 2 or 3 and j = 0, 1, 2, 3 or 4). Thus, for instance, s2m3 takes the value 1 if
£3 million is invested in site 2.
Formulate the investment problem as an LP involving 15 such indicator variables.
(The instruction “INT 15” after “END” will make LINDO regard all 15 variables as indicator
variables.)
[8]
(ii) Run your LP and interpret the results.
[6]
(iii) Adapt your LP and solve the problem if the amount that Mr Wing has to invest is
(A) £3 million,
(B) £8 million.
Interpret your solution in each case.
© OCR 2009
[4]
4773 Jun09
6
BLANK PAGE
© OCR 2009
4773 Jun09
7
BLANK PAGE
© OCR 2009
4773 Jun09
8
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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
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© OCR 2009
4773 Jun09
ADVANCED GCE
4773
MATHEMATICS (MEI)
Decision Mathematics Computation
* O C E / 2 3 4 0 0 *
Thursday 24 June 2010
Morning
Candidates answer on the Answer Booklet
OCR Supplied Materials:
•
8 page Answer Booklet
•
MEI Examination Formulae and Tables (MF2)
•
Graph paper
Other Materials Required:
•
Scientific or graphical calculator
•
Computer with appropriate software and
printing facilities
Duration: 2 hours 30 minutes
*
4
7
7
3
*
INSTRUCTIONS TO CANDIDATES
•
•
•
•
•
•
•
•
Write your name clearly in capital letters, your Centre Number and Candidate Number in the spaces
provided on the Answer Booklet.
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.
Final answers should be given to a degree of accuracy appropriate to the context.
Additional sheets, including computer print-outs, should be fastened securely to the Answer Booklet.
INFORMATION FOR CANDIDATES
•
•
•
•
•
•
The number of marks is given in brackets [ ] at the end of each question or part question.
In each of the questions you are required to write spreadsheet or other routines to carry out various
processes.
For each question you attempt, you should submit print-outs showing the routine you have written and the
output it generates.
You are not expected to print out and submit everything your routine produces, but you are required to
submit sufficient evidence to convince the examiner that a correct procedure has been used.
The total number of marks for this paper is 72.
This document consists of 4 pages. Any blank pages are indicated.
COMPUTING RESOURCES
•
Candidates will require access to a computer with a spreadsheet program, a linear programming package
and suitable printing facilities throughout the examination.
© OCR 2010 [Y/102/2662]
DC (LEO) 23400/5
OCR is an exempt Charity
Turn over
2
1
Athos puts £1000 into a deposit account. After a year, interest is added to the account, the amount of
interest being 5% of the balance during the year. Athos then draws out £60. After each subsequent year,
interest at 5% of the balance is added on, and Athos then withdraws £60.
(i) Let un be the amount (in £) in the account after n years, with u0 = 1000. Construct a recurrence
relation for un in terms of un-1.
[3]
(ii) Solve your recurrence relation from part (i), simplifying your answer as far as is possible.
[4]
(iii) Use your answer to part (ii) to find how long Athos can continue to operate the account in this
way.
[2]
Porthos puts £1000 into a deposit account. Every 6 months, interest is added to the account, the amount
of interest being 2 12 % of the balance over those 6 months. He draws out £60 at the end of 12 months
and after each subsequent 12 months.
(iv) Construct a spreadsheet to show how the amount Porthos has in his account varies over time. [3]
(v) Use your spreadsheet to find for how long Porthos can operate his account in this way.
[1]
Aramis puts £1000 into a deposit account. He draws out £30 every 6 months. Every 12 months, interest
is added to the account, the amount of interest being 5% of the average balance over those 12 months.
(vi) Construct a spreadsheet to show how the amount Aramis has in his account varies over time. [4]
(vii) Use your spreadsheet to find for how long Aramis can operate his account in this way.
© OCR 2010
4773 Jun10
[1]
3
2
The distance of the point ( p, q) from the line ax + by = c is given by
c − ap − bq
a2 + b2
.
For example, when a = b = 1 and c = 10, the distance of ( p, q) from x + y = 10 is given by
10 − p − q
.
2
(i) Find the distance of (0, 0) from the line x + y = 10.
Find the distance of (10, 10) from the line x + y = 10.
[1]
Consider the (minimax) LP:
Minimise
subject to
m
m⭓p
m ⭓ –p
m⭓q
m ⭓ –q
m⭓
10 − p − q
2
m⭓
p + q − 10
2
(ii) Rewrite the LP in a form in which it can be submitted to LINDO.
(Approximate 2 by 1.414 214.)
[4]
(iii) Run the LP and draw a diagram to explain what it achieves.
[5]
(iv) Formulate an LP to find the point which is equidistant from the lines y = 0, x + y = 1 and x – y = –1.
(Approximate 2 by 1.414 214.)
[4]
(v) Run your LP.
[1]
(vi) Prove by drawing a diagram and calculating distances that your LP has achieved what was required.
[3]
[Questions 3 and 4 are printed overleaf]
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 (www.ocr.org.uk) 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 2010
4773 Jun10
Turn over
4
3
A logistics company has three depots, D1, D2 and D3. On a particular day it has 10 identical containers
to collect from each of two supply locations, S1 and S2. These containers then have to be shipped from
the depots to four customers, C1, C2, C3 and C4. C1 requires 7 containers, C2 requires 4 containers, C3
requires 6 containers and C4 requires 3 containers. Each depot can handle up to 7 containers.
The transportation costs are shown in the two tables.
S1
S2
D1
2
1
D2
3
8
D3
7
4
D1
D2
D3
C1
2
4
1
C2
3
7
5
C3
9
2
3
C4
1
5
6
(i) Formulate an LP to find the cheapest way to transport the containers from the supply locations to
the depots.
[3]
(ii) Run your LP and interpret the results.
[3]
(iii) Using your answer to part (ii), formulate an LP to find the cheapest way to transport these
containers from the depots to the customers.
[3]
(iv) Run your LP and interpret the results.
[2]
(v) Formulate an LP to find the cheapest way to transport the containers from the supply locations to
the customers via the depots.
[4]
(vi) Run your LP and interpret and comment on your results.
4
[3]
Each individual in a population produces either 0, 1 or 2 offspring, the probabilities being 0.1, 0.5 and
0.4 respectively.
The population starts from a single individual (generation 0), whose offspring form generation 1. In
turn the offspring from members of the population in generation 1 form generation 2, etc.
(i) Show how to use a spreadsheet command to simulate the offspring produced by a member of the
population.
[4]
(ii) Build a spreadsheet to simulate generations 1 and 2.
Print out the formulae which you used in your spreadsheet.
[9]
(iii) Run your simulation 20 times and hence estimate the probabilities of there being 0, 1, 2, 3 or 4
individuals in generation 2.
[2]
A second population has individuals which reproduce according to the same rule, but this population
starts with 2 individuals in generation 0.
(iv) Use your simulation model to produce one simulation of the number in this population at
generation 2.
Explain how you produced your result. List all possible results.
[3]
© OCR 2010
4773 Jun10
Monday 24 June 2013 – Afternoon
A2 GCE MATHEMATICS (MEI)
4773/01
Decision Mathematics Computation
Candidates answer on the Answer Booklet.
* 4 7 1 5 9 3 0 6 1 3 *
OCR supplied materials:
•
12 page Answer Booklet (OCR12)
(sent with general stationery)
•
Graph paper
•
MEI Examination Formulae and Tables (MF2)
Other materials required:
•
Scientific or graphical calculator
•
Computer with appropriate software and
printing facilities
Duration: 2 hours 30 minutes
*
4
7
7
3
0
1
*
INSTRUCTIONS TO CANDIDATES
•
•
•
•
•
•
•
•
Write your name, centre number and candidate number in the spaces provided on the
Answer Booklet. Please write clearly and in capital letters.
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.
You are permitted to use a graphical calculator in this paper.
Final answers should be given to a degree of accuracy appropriate to the context.
Additional sheets, including computer print-outs, should be fastened securely to the
Answer Booklet.
Do not write in the bar codes.
INFORMATION FOR CANDIDATES
•
•
•
•
•
•
The number of marks is given in brackets [ ] at the end of each question or part
question.
In each of the questions you are required to write spreadsheet or other routines to carry
out various processes.
For each question you attempt, you should submit print-outs showing the routine you
have written and the output it generates.
You are not expected to print out and submit everything your routine produces, but
you are required to submit sufficient evidence to convince the examiner that a correct
procedure has been used.
The total number of marks for this paper is 72.
This document consists of 8 pages. Any blank pages are indicated.
COMPUTING RESOURCES
•
Candidates will require access to a computer with a spreadsheet program, a linear
programming package and suitable printing facilities throughout the examination.
© OCR 2013 [Y/102/2662]
DC (CW) 51942/3
OCR is an exempt Charity
Turn over
2
1
The bread man calls early at a remote mountain village on every third day, including weekends. Ioanna
always buys either one or two loaves, randomly and each with probability 0.5.
The following random variable is a good model of Ioanna’s daily bread requirements.
Daily requirements (loaves)
1
4
1
2
3
4
1
Probability
3
7
2
7
1
7
1
7
(i) Build a spreadsheet simulation of this system and run it for 100 days, starting with a day on which
Ioanna starts with half a loaf in stock and on which the bread man calls.
[7]
(ii) Define two measures of system performance. Add these measures to your simulation.
[2]
(iii) Repeat your simulation a number of times and report on the behaviour of your two measures.
[4]
Ioanna’s friend advises her that it would be better if she took account of her stock level when purchasing
bread. She suggests that Ioanna should purchase one loaf if her stock is 0.75 loaves or more, and two loaves
otherwise.
(iv) Investigate and report on how this system performs, comparing it with the original system.
© OCR 2013
4773/01 Jun13
[5]
3
2
(This question is concerned only with working days, and ignores weekends.)
Ioanna likes to keep €500 in her current account. Her bank is a small agricultural bank in a rural region,
which does not offer internet banking. Ioanna has a separate savings account, and she either transfers money
from this to her current account, or vice-versa, to keep the current account balance at €500. However, her
instructions to move money take 3 days to be put into effect.
At the close of banking on day 1, Ioanna’s current account has a balance of €450, and she issues an
instruction to move €50 in from her savings account.
At the end of day 2, the current account contains €520, and she instructs that €20 be moved to her savings
account.
At the end of day 3, the current account contains €410, and she instructs that €90 be moved in to it from her
savings account.
She continues to operate this strategy in subsequent days.
During day 4, the €50 from her day 1 instruction arrives in the current account. Assume that this is the only
change to the current account balance during day 4.
(i) Assuming that there are no changes to the current account in subsequent days other than those following
Ioanna’s instructions, give a recurrence relation for e n , the amount in the current account on day n, in
[1]
terms of e n - 1 and e n - 3 (n H 4 ).
(ii) Construct a spreadsheet to show how Ioanna’s current account balance varies over a period of 25
working days.
[2]
Ioanna is promised an improved service in which her instructions are put into effect after 2 working days.
(iii) Give a recurrence relation for e n , the amount in the current account on day n, in terms of e n - 1 and e n - 2
[1]
(n H 3) under this new service.
(iv) Construct a spreadsheet to show how Ioanna’s current account balance would vary under this new
service over a period of 25 working days, starting with €450 in the account on day 1, and €520 on day
2. During day 3, €110 leaves the current account and €50 arrives following Ioanna’s day 1 instruction.
Subsequently the only changes are due to Ioanna’s instructions.
[2]
Ioanna is still unhappy with the fluctuations in the level of her current account, and complains to her bank
manager. He cannot improve the bank’s service level any further, but he advises her to make her daily
adjustments equal to a proportion of the difference between €500 and the amount in her current account, ie
so that the daily change is € p(500 – balance), where 0 G p G 1.
(v) Give the new recurrence relation for e n when Ioanna implements this advice.
(vi) Solve the recurrence relation when p =
(vii) Construct a spreadsheet to check your answer to part (vi).
© OCR 2013
2
, e = 450 and e 2 = 520 .
9 1
4773/01 Jun13
[1]
[9]
[2]
Turnover
4
3
The manager of an athletics club has 8 runners to allocate to positions 1, 2, 3 and 4 in two sprint relay teams.
The table shows historical information giving the past mean times (in seconds) of the athletes when running
in each of the four positions. She wants to minimise the expected total running time.
position
1
2
3
4
A
11.12
11.34
11.74
11.63
B
12.01
12.23
11.89
12.17
C
11.24
11.09
11.56
11.65
D
13.34
12.95
12.67
13.01
E
12.54
12.37
12.21
12.45
F
11.87
11.74
11.35
11.21
G
11.52
11.42
11.37
11.74
H
12.08
12.43
12.32
12.57
athlete
The manager sets up the problem as an allocation problem. There are 8 athletes to be allocated to 8
positions. Numbers 1, 2, 3 and 4 represent the positions in one team, and numbers 5, 6, 7 and 8 represent the
corresponding positions in the other team.
(i) Set up an LP to solve this allocation problem. Solve it and interpret your solution.
(ii) Athlete C, one of the fastest, complains that this method of team selection will not maximise his
chances of winning a medal. Why might he argue thus?
[1]
(iii) Set up LPs to choose the best team out of the 8 athletes, and their best positions, and the best positions
for the athletes in the second team. Solve your LPs and interpret the solutions.
[10]
© OCR 2013
4773/01 Jun13
[7]
5
4
Each of the customers at a restaurant orders a main meal. In addition, some have a starter, some have a
dessert, and some have both starter and dessert.
The individual dishes vary in price, but the management is encouraging custom by making two offers: any
starter and main for £15; any main and dessert for £10.
In addition the management is encouraging custom by not restricting the offers to orders placed by
individuals. So, for instance, if two people share a meal in which one has a starter, a main and a dessert, and
the other has just a main, then they could choose to pay £25, ie £15 for a starter and a main, and £10 for a
main and a dessert.
A party of 8 diners orders the following dishes.
5 starters priced at £8.50, £7.65, £4.32, £5.67 and £5.67
8 mains priced at £12.42, £9.85, £13.36, £21.25, £12.42, £17.85, £13.63 and £13.63
4 desserts priced at £6.85, £5.32, £3.42 and £10.18
The following LP computes the minimum price payable by the party.
min 8.50s1+7.65s2+4.32s3+5.67s4+5.67s5
+12.42m1+9.85m2+13.36m3+21.25m4+12.42m5+17.85m6+13.63m7+13.63m8
+6.85d1+5.32d2+3.42d3+10.18d4
+15sm+10md
st m1+m2+m3+m4+m5+m6+m7+m8+sm+md=8
s1+s2+s3+s4+s5+sm=5
d1+d2+d3+d4+md=4
end
int 17
(i) Run the LP, and interpret the output.
[3]
(ii) Explain what the variables represent, and the meaning of “int 17”.
[6]
(iii) The optimal solution involves the party paying separately for the cheapest starter. Explain why this is
so, when the cheapest starter costs more than the cheapest dessert.
[1]
To encourage even more custom the restaurant’s management considers introducing a special price of £17.50
for any 3-course meal, starter, main and dessert, with the same rules as before, ie which dishes individual
party members eat is ignored in the pricing.
(iv) Modify the LP to find the minimum price now payable by the party of 8.
[5]
(v) Run your modified LP and show that the new offer is of no worth to the party.
[1]
(vi) Verify that, to be of any worth at all to the party, the 3-course meal would have to be priced at £15.14.
Interpret the corresponding solution.
[2]
© OCR 2013
4773/01 Jun13
6
BLANKPAGE
© OCR 2013
4773/01 Jun13
7
BLANKPAGE
© OCR 2013
4773/01 Jun13
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 (www.ocr.org.uk) 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 2013
4773/01 Jun13