Cambridge A Level Thinking Skills 9694 — 2011 Oct/Nov Paper 3 · Variant 1

This document consists of 8 printed pages and 4 blank pages. IB11 11_9694_31/2RP © UCLES 2011 [Turn over *9740711855* UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS General Certificate of Education Advanced Level THINKING SKILLS 9694/31 Paper 3 Problem Analysis and Solution October/November 2011 1 hour and 30 minutes Additional Materials: Answer Booklet/Paper Electronic Calculator READ THESE INSTRUCTIONS FIRST If you have been given an Answer Booklet, follow the instructions on the front cover of the booklet. Write your Centre number, candidate number and name on all the work you hand in. Write in dark blue or black pen. Do not use staples, paper clips, highlighters, glue or correction fluid. DO NOT WRITE ON ANY BARCODES Calculators should be used where appropriate. Answer all the questions. Start each question on a new answer sheet. At the end of the examination, fasten all your work securely together. The number of marks is given in brackets [ ] at the end of each question or part question.

2 © UCLES 2011 9694/31/O/N/11 1 Study the information below and answer the questions. Show your working. The layout of the letters on Sophie’s standard French keyboard is: A Q W Z S X E D C R F V T G B Y H N U J I K O L P M 4 3 2 1 1 1 1 2 3 4 She is a touch-typist (meaning she doesn’t need to look at the keys) and uses the correct eight fingers for the letters, as shown above. The thumbs are not used. She was asked to construct an eight-letter password by using a single letter from each finger, e.g. COMQXKVY. (a) Sophie selected VASOPHIE as her password, and used it until she was forced to change it after 90 days. She decides to use the same eight letters, and to keep VASO as the first four, in that order, but to type P, H, I and E in a different order, e.g. VASOPEIH. How many passwords could she create in this way, excluding the one she has already used? [2] Agnès also has a standard French keyboard, and selects a letter for each finger, going from left to right. She tried what she thought was the simplest one: AZERUIOP, and was surprised to find that it was rejected because someone else had already used it. (b) How many possible passwords could Agnès make (including any already used by others)? [2]

3 © UCLES 2011 9694/31/O/N/11 [Turn over Fred has a standard English keyboard: Q A Z W S X E D C R F V T G B Y H N U J M I K O L P 4 3 2 1 1 1 1 2 3 4 (c) Fred also makes up eight-letter passwords in the same way as Agnès (one per finger and left to right). (i) How many possible passwords could Fred make? [1] (ii) Give an example of a password that could be used by Agnès but not Fred. [1] (iii) How many passwords could be made by Fred but not Agnès? [2] Sophie’s employer is worried that people will use passwords with simple patterns on the keyboard, and decides to give them eight-letter passwords randomly generated by a computer. These passwords use letters which require alternate use of the two hands. Many users have trouble remembering and typing these random passwords. A particularly common error is a transposition of adjacent letters: i.e. to type two letters next to each other in the wrong order. For example, UDWKLGPZ instead of UDKWLGPZ. This has become so common that the system manager has decided to allow passwords with a single error of this type as being ‘near enough’. (d) (i) How many times larger is the collection of acceptable passwords which include such an error than the collection of those that could come from the random generator? [1] (ii) An attacker knows the rules for generating and checking the passwords. What difference does allowing a transposition make to the (very small) chance of guessing the password of a particular user? [1]

4 © UCLES 2011 9694/31/O/N/11 2 Study the information below and answer the questions. Show your working. An annuity is an arrangement where a person (‘the policyholder’) makes a single initial payment and then receives a fixed sum of money each year until he or she dies. The table below gives the annual sum received per year for life in return for an initial payment of $100. Those starting younger receive less per year because they are expected to live longer. Annual sum received Age when $100 paid Male Female 50 $4.02 $3.96 55 $4.36 $4.27 60 $4.83 $4.69 (Assume that the proportion of people who die between the ages of 50 and 60 is so small that it can be ignored.) (a) If a 50-year-old man and a 60-year-old man both make a $100 payment at the same time, how much longer will the 50-year-old have to wait than the 60-year-old before getting at least $100 back? [2] (b) Simon believes that if he dies at the average age for a male, he will have received the same amount in total whether he were to purchase an annuity at 50 or 60. What would the average age of death have to be for Simon to be right? [2] Since inflation changes the value of a sum of money over time, the calculations are very complex, but we are assured that the system is such that the expected return ‘in real terms’ is calculated fairly. (c) The figures in the table are lower for females than for males, and yet are fair, in that they have the same total expected payment. What must be the reason for this? [1] Married couples also have the option to get a ‘joint life policy’, which continues to be paid while either the policyholder or their spouse is still alive. Annual sum received per year for life in return for an initial payment of $100: Annual sum received Policyholder’s age when $100 paid Male and wife Female and husband 50 $3.87 $3.90 55 $4.16 $4.19 60 $4.55 $4.59 (d) The rate is less for males than females for joint life policies, although it was greater for males than females for policies only covering the policyholder. Suggest a reason for this. [2]

5 © UCLES 2011 9694/31/O/N/11 [Turn over (e) Janet and John are married, and both aged 55. They have $200 to invest. They can either invest the $200 in one annuity, or they could invest $100 in each of two different annuities. (i) Which option gives them the highest return next year? [1] (ii) Of all the many different options for annuities available to them, the one which will give Janet and John the highest expected total return is the ‘female and husband’ annuity. Explain why this is the case. [2]

6 © UCLES 2011 9694/31/O/N/11 3 Study the information below and answer the questions. Show your working. A newspaper report said that the average oarsman in a winning Olympic ‘coxed four’ (a rowing crew which involves five athletes) had altered dramatically between 1968 and 1988. In particular it said, “…the average weight had increased from 64 kg in 1968 to 72 kg in 1988, and the range of weights in a crew had exactly doubled over those years.” Another report, by a different newspaper, said, “…the average weight remained at 64 kg from 1968 to 1988, but the range of weights exactly doubled over those years.” You can assume that both newspapers had the same raw data (the weights of the athletes involved in 1968 and in 1988). The apparent disagreement in the reports came about because of the different meanings of the word “average”: the first newspaper calculated the means of the weights, while the second calculated the medians. The rules of the competition state that no participant may be less than 55 kg. For the purposes of this investigation, you should only consider weights which are whole numbers of kilograms, and you should not consider weights greater than 100 kg. (a) Give an example of five weights which have a mean of 64 kg. [1] (b) What is the largest range that the weights from 1968 could have had, given that the claims made by the first newspaper are correct (and assuming that it used the mean in its calculation of the average)? [3] (c) Give an example of five weights which have a mean of 72 kg, and a median of 64 kg. [2] (d) Give an example of five weights for the year 1968 and five weights for the year 1988 which could have supported the claims by both of the newspapers (assuming the first newspaper had used the mean in its calculations, and the second had used the median). [3] A third newspaper gave the following response to the two newspaper reports above: “The conflicting reports of the weight differences in the Olympic rowing crews were meaningless, because they treated all five athletes as equals; they are not. One is a coxswain who steers the boat, and is always as light as possible (within the rules of the competition), while the other four are the oarsmen who power the boat. It is the weights of these four oarsmen which might reveal a trend in the way athletes prepare for the sport. The average weight of the four oarsmen actually increased by more than 11 kg from 1968 to 1988, and the range of their weights more than tripled.” (e) Show that the third newspaper could not have been referring to the mean, when it refers to the “average”. [2] (f) Find five weights for the year 1968 and five weights for the year 1988 which are consistent with all three newspaper reports. [4]

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8 © UCLES 2011 9694/31/O/N/11 4 Study the information below and answer the questions. Show your working. This is the film schedule for the Red Carpet multi-screen cinema from this Wednesday to next Tuesday. Screen Film Daily at: Extra Shows 1 Colossal (124 minutes) 14:15 17:25 20:35 11:05 (Sat only) 2 Wolf Gang (98 minutes) 13:45 (not Mon) 16:00 18:15 20:30 11:30 (Sat & Sun) 3 Illinois (102 minutes) 13:20 (not Mon) 15:40 18:00 20:20 11:00 (Sat only) 13:50 (not Mon) 15:45 17:40 19:35 4 Mahatma (83 minutes) 21:30 12:05 (Sat & Sun) 5 Lupine Jives (109 minutes) 14:10 16:30 18:50 21:10 11:50 (Sat & Sun) All cinema tickets are valid throughout the cinema for the whole of the day of purchase. The Red Carpet is in Ivystone. It runs a courtesy bus service for the benefit of customers living in neighbouring villages. Two buses operate, travelling a circular route in opposite directions. Bus Timetable Red Carpet (Ivystone) Paltrow Winslet Red Carpet Sat only Not Mon Red Carpet 09:40 11:50 13:30 15:10 16:50 18:30 20:10 22:10 Eastwood 09:48 11:58 13:38 15:18 16:58 18:38 20:18 22:18 Sarandon 09:59 12:09 13:49 15:29 17:09 18:49 20:29 22:29 Paltrow 10:05 12:15 13:55 15:35 17:15 18:55 20:35 22:35 Heston 10:14 12:24 14:04 15:44 17:24 19:04 20:44 22:44 Winslet 10:21 12:31 14:11 15:51 17:31 19:11 20:51 22:51 Redford 10:32 12:42 14:22 16:02 17:42 19:22 21:02 23:02 Hepburn 10:40 12:50 14:30 16:10 17:50 19:30 21:10 23:10 Red Carpet 10:47 12:57 14:37 16:17 17:57 19:37 21:17 23:17

9 © UCLES 2011 9694/31/O/N/11 Red Carpet (Ivystone) Winslet Paltrow Red Carpet Sat & Sun only Red Carpet 10:10 12:40 14:20 16:00 17:40 19:20 21:00 23:20 Hepburn 10:17 12:47 14:27 16:07 17:47 19:27 21:07 23:27 Redford 10:25 12:55 14:35 16:15 17:55 19:35 21:15 23:35 Winslet 10:36 13:06 14:46 16:26 18:06 19:46 21:26 23:46 Heston 10:43 13:13 14:53 16:33 18:13 19:53 21:33 23:53 Paltrow 10:52 13:22 15:02 16:42 18:22 20:02 21:42 00:02 Sarandon 10:58 13:28 15:08 16:48 18:28 20:08 21:48 00:08 Eastwood 11:09 13:39 15:19 16:59 18:39 20:19 21:59 00:19 Red Carpet 11:17 13:47 15:27 17:07 18:47 20:27 22:07 00:27 (a) At what time will the last film finish each day? [2] (b) How many times will each of the five films be shown during this seven-day period? [3] (c) Dustin lives in Winslet but works in Ivystone. Every Thursday, after work, he goes to the Red Carpet, arriving between 17:30 and 17:35. His routine never varies: he watches the first film that starts after his arrival, then the next different film that starts after his first one has finished. (i) Which two films will Dustin watch this Thursday? [2] (ii) How long after the second film finishes will he have to wait for a bus to take him home? [2] (d) Nicole lives in Redford. She has arranged to meet her friend Meryl outside the Red Carpet at 15:30 on Saturday. (i) What is the time of the latest bus that Nicole can catch from Redford? [1] Nicole and Meryl want to see Colossal, Illinois and Mahatma. They don’t mind which order they see them in, but they do want to watch each of the three films all the way through. (ii) What is the earliest time that they will be able to leave the Red Carpet? [3] The two buses travelling in opposite directions pass each other a number of times each day. On Saturday, for example, the first time they pass is between Winslet and Redford at approximately 10:28. (e) Between which two villages and at approximately what time do the buses pass for the last time each day? [2]

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UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS GCE Advanced Level MARK SCHEME for the October/November 2011 question paper for the guidance of teachers 9694 THINKING SKILLS 9694/31 Paper 3 (Problem Analysis and Solution), maximum raw mark 50 This mark scheme is published as an aid to teachers and candidates, to indicate the requirements of the examination. It shows the basis on which Examiners were instructed to award marks. It does not indicate the details of the discussions that took place at an Examiners’ meeting before marking began, which would have considered the acceptability of alternative answers. Mark schemes must be read in conjunction with the question papers and the report on the examination. • Cambridge will not enter into discussions or correspondence in connection with these mark schemes. Cambridge is publishing the mark schemes for the October/November 2011 question papers for most IGCSE, GCE Advanced Level and Advanced Subsidiary Level syllabuses and some Ordinary Level syllabuses.

Page 2 Mark Scheme: Teachers’ version Syllabus Paper GCE A LEVEL – October/November 2011 9694 31 © University of Cambridge International Examinations 2011 1 (a) How many passwords could she create in this way, excluding the one she has already used? [2] There are 4 ways to select the fifth letter, 3 the sixth, etc. So (4 × 3 × 2 × 1) – 1 ways. 23. 1 mark for 4! = 24 and 1 mark for – 1. OR 1 mark for a systematic attempt to list the passwords [i.e. if solution involves one letter held while the others are permuted, and then another letter given a similar treatment. Not necessary for the list of permutations to be complete]. (b) How many possible passwords could Agnès make (including any already used by others)? [2] Agnès has 3 choices for the first, second and third, 6 for the fourth, 5 for the fifth, 2 for 6th, 7th, and 8th. 3 × 3 × 3 × 6 × 5 × 2 × 2 × 2 = 6480. 1 mark for 3 × 3 × 3 × 6 × 5 × 2 × 2 × 2 with missing or incorrect answer. OR 1 mark for the correct eight numbers combined according to an incorrect process (for example 3 + 3 + 3 + 6 + 5 + 2 + 2 + 2 = 26). (c) (i) How many possible passwords could Fred make? [1] Fred has 3 × 3 × 3 × 6 × 6 × 2 × 2 × 1 = 3888. (ii) Give an example of a password that could be used by Agnès but not Fred. [1] A valid password starting with W, ending in M or with second letter Z is necessary and sufficient. E.g. WSCBJIOP, AXCFJKOM, QZCRHILP. (iii) How many passwords could be made by Fred but not Agnès? [2] To determine the number that only Fred can produce, it is simplest to calculate the number that are common. The differences between keyboards are the position of W and Z, the location of M, and the positions of Q and A. As Q and A use the same finger, they can be ignored. The common set is 2 × 2 × 3 × 6 × 5 × 2 × 2 × 1 = 1440, so there are 3888 – 1440 = 2448 unique to Fred. 1 mark for 1440. OR 1 mark for logic but faulty arithmetic. Creditworthy answers must delineate the correct collection of cases for the problem to be answered. (d) (i) How many times larger is the collection of acceptable passwords which include such an error than the collection of those that could come from the random generator? [1] There are seven possible switches of adjacent letters, (and each one cannot be produced from any other switch) so there are 7 times as many. Allow 8 [for identifying the total acceptable password increase, error or no error].

Page 3 Mark Scheme: Teachers’ version Syllabus Paper GCE A LEVEL – October/November 2011 9694 31 © University of Cambridge International Examinations 2011 (ii) An attacker knows the rules for generating and checking the passwords. What difference does allowing a transposition make to the (very small) chance of guessing the password of a particular user? [1] It makes no difference since all variants are treated as equivalent. 2 (a) If a 50-year-old man and a 60-year-old man both make a $100 payment at the same time, how much longer will the 50-year-old have to wait than the 60-year-old before getting at least $100 back? [2] Answer: 4 years. Allow sensible rounding of 4.17. If 2 marks can not be awarded, award 1 mark for seeing: 100/4.02 → 25 years or 100/4.83 → 21 years or 100/4.02 – 100/4.83 → 24.87 – 20.70 → 4.17 → rounded to 5 years. (b) Simon believes that if he dies at the average age for a male, he will have received the same amount in total whether he were to purchase an annuity at 50 or 60. What would the average age of death have to be for Simon to be right? [2] 4.83(x – 60) = 4.02(x – 50) 0.81(x – 60) = 40.2 x = 109.63 Allow sensible rounding (including 110). If 2 marks can not be awarded, award 1 mark for attempting to formulate the problem algebraically or by trial and improvement. (c) The figures in the table are lower for females than for males, and yet are fair, in that they have the same total expected payment. What must be the reason for this? [1] (With all other parameters being the same,) the average age of death must be greater for females. (d) The rate is less for males than females for joint life policies, although it was greater for males than females for policies only covering the policyholder. Suggest a reason for this. [2] If the person taking out the annuity and partner were always of the same age then the figures should be equal; as they are not it must be because there is an expected age difference. On average, the male in a pair is older. SC: It may be the case that couples in which the man takes out the policy tend to live longer. If 2 marks cannot awarded, award 1 mark for answers which show some appreciation that the age difference between the husband and wife is the (likely) cause. SC: award 1 mark for an answer which offers an explanation relating to possible incentives that might be offered to attract females to take out the policies.

Page 4 Mark Scheme: Teachers’ version Syllabus Paper GCE A LEVEL – October/November 2011 9694 31 © University of Cambridge International Examinations 2011 (e) (i) Which option gives them the highest return next year? [1] The highest immediate return would be for John to put all the money into a single life policy for him. (2 x $4.36). (ii) Of all the many different options for annuities available to them, the one which will give Janet and John the highest expected total return is the ‘female and husband’ annuity. Explain why this is the case. [2] The only reason that one can be better than the others is if there is some extra information known to them but not those setting the rates. In this case we (only) know that they are the same age, which means that John is younger than would be expected for a spouse and can thus expect to live for longer. This means that she should put all the money into the female + spouse option: $4.19 If 2 marks can not be awarded, award 1 mark for simply pointing out that the assumed age difference between man and wife does not apply to Janet and John / any reference to the non-averageness of John and Janet as causes of the anomaly. Individual life expectancies in a marriage are not independent of each other. 3 (a) Give an example of five weights which have a mean of 64kg. [1] Any five numbers which add up to 320, all of which are between 55 and 100 inclusive (units not needed). (b) What is the largest range that the weights from 1968 could have had, given that the claims made by the first newspaper are correct (and assuming that it used the mean in its calculation of the average)? [3] Largest = 22. 45 is the largest range achievable if the mean is to be 64 [55/55/55/55/100], but it must half this (or less) if the range is to double by 1988. 3 marks for the correct answer. If 3 marks cannot be awarded, 2 marks for an answer of 22.5. 1 mark for appreciating that the greatest possible range is 45. (c) Give an example of five weights which have a mean of 72 kg, and a median of 64 kg. [2] Five weights between 55 and 100 which add up to 360, and a median of 64. E.g. 64/64/64/84/84. 1 mark for weights with a mean of 72. 1 mark for weights with a median of 64.

Page 5 Mark Scheme: Teachers’ version Syllabus Paper GCE A LEVEL – October/November 2011 9694 31 © University of Cambridge International Examinations 2011 (d) Give an example of five weights for the year 1968 and five weights for the year 1988 which could have supported the claims by both of the newspapers (assuming the first newspaper had used the mean in its calculations, and the second had used the median). [3] Possible answer: 1968 = 60, 61, 64, 64, 71 and 1988 = 64, 64, 64, 82, 86. 1 mark for answers in which the range is doubled. If this first mark is awarded, award the two further marks if the medians are 64 (1 mark), and the means increase from 64 to 72 (1 mark). (e) Show that the third newspaper could not have been referring to the mean, when it refers to the average above. [2] Mean of the five athletes goes from 64 to 72, and cox is 55. So mean of oarsmen is ((5 x 64) – 55)/4 = 66.25 (or 66) in 1968. And is ((5 x 72) – 55)/4 = 76.25 (or 76) in 1988. Which is not more than 11kg Alternative explanation: the mean of the five athletes increasing from 64 to 72 entails a total increase of (360 – 320) 40kg. If the mean weight of the four oarsmen were to increase by 11, this would entail an increase of (4 x 11) 44 kg. So the 3rd newspaper report’s use of the mean would conflict with the previous two. 2 marks if the candidate shows the underlined figures. If 2 marks cannot be awarded, award 1 mark for a correct calculation of the means of the rowers for a particular valid collection of weights, OR an attempt to calculate the change in total weight implied by the oarsmen’s increase referring to the mean. (f) Find five weights for year 1968 and five weights for the year 1988 which are consistent with all three newspaper reports. [4] correct answers: 1968: 55 [cox], 64, 64, 65 & 72. 1988: 55 [cox], 64, 64, 88 & 89. Award 4 marks for these weights, given unambiguously. If 4 marks cannot be awarded, give 1 mark for each (awarded independently of each other) of the following four requirements being fulfilled: (i) two sets of five weights which include a cox (55/56kg), have the right median (64kg), and have a mean which increases from 64 to 72. (ii) two sets of 5 weights whose range doubles. (iii) two sets or weights for the four oarsmen which increase by (more than) 11. (iv) two sets or weights for the four oarsmen whose range (more than) triples.

Page 6 Mark Scheme: Teachers’ version Syllabus Paper GCE A LEVEL – October/November 2011 9694 31 © University of Cambridge International Examinations 2011 4 (a) At what time will the last film finish each day? [2] Answer: 22:59 award 2 marks If 2 marks cannot be awarded, award 1 mark for evidence that Lupine Jives will be the last film to finish. (b) How many times will each of the five films be shown during this seven-day period? [3] Answer: Colossal 22 times Wolf Gang 29 times Illinois 28 times award 3 marks Mahatma 36 times Lupine Jives 30 times Deduct 1 mark for each incorrect figure. (Minimum 1 mark if candidates get at least one film correct.) Penalise oversight of ‘not Mon’s and ‘extra shows’ once only. (c) (i) Which two films will Dustin watch this Thursday? [2] Answer: Mahatma (17:40 to 19:03) award 1 mark and Illinois (20:20 to 22:02) award 1 mark Times need not be given. Alternatively, award 1 mark for a second film that correctly follows an incorrect first film; e.g. Illinois (18:00 to 19:42) and Wolf Gang (20:30 to 22:08) OR: Colossal (17.25 start) and Mahatma (19.35 start) OR Wolf Gang (18:15 to 19:53) and Illinois. (ii) How long after the second film finishes will he have to wait for a bus to take him home? [2] Answer: 8 minutes (22:02 to 22:10) award 2 marks 2 marks for a fully correct follow through from part (c)(i). If 2 marks cannot be awarded, award 1 mark for sight of correct finish time for second film (may be seen in part (c)(i)) OR for a correct waiting time/finishing time consistent with an incorrect finishing time (seen) of a film. (d) (i) What is the time of the latest bus that Nicole can catch from Redford? [1] Answer: 14:35

Page 7 Mark Scheme: Teachers’ version Syllabus Paper GCE A LEVEL – October/November 2011 9694 31 © University of Cambridge International Examinations 2011 (ii) What is the earliest time that they will be able to leave the Red Carpet? [3] Answer: 20:58 or 21.00 (the bus departure time) award 3 marks If 3 marks cannot be awarded, award 1 mark each for evidence of the following (max 2): • If they see Colossal last they won’t leave until 22:39. • It is possible to see either Illinois or Mahatma before the 17:25 showing of Colossal and the other one afterwards. • The 19:35 showing of Mahatma will finish before the 20:20 showing of Illinois. Alternatively, award up to 2 marks for the second-best solution (such as Mahatma, followed by Colossal followed by Illinois: finishes at 22.02, first bus leaving at 22.10). Award 1 mark for a finish time consistent with a correct alternative ordering (the two remaining alternatives (I – M – C and M – I – C) both finish at 22:39, catching the 23:20 bus. (e) Between which two villages and at approximately what time do the buses pass for the last time each day? [2] Answer: Between Redford and Hepburn. Accept any time or time interval between 21:07 and 21:10 (inclusive).