Cambridge A Level Mathematics 9709 — 2015 Oct/Nov Paper 6 · Variant 1

*6529685396* Cambridge International Examinations Cambridge International Advanced Subsidiary and Advanced Level MATHEMATICS 9709/61 Paper 6 Probability & Statistics 1 (S1) October/November 2015 1 hour 15 minutes Additional Materials: List of Formulae (MF9) READ THESE INSTRUCTIONS FIRST An answer booklet and a graph paper booklet are provided inside this question paper. You should follow the instructions on the front cover of both booklets. If you need additional answer paper or graph paper ask the invigilator for a continuation booklet or graph paper booklet. Write in dark blue or black pen. You may use an HB pencil for any diagrams or graphs. Do not use staples, paper clips, glue or correction fluid. DO NOT WRITE IN ANY BARCODES. Answer all the questions. Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question. The use of an electronic calculator is expected, where appropriate. You are reminded of the need for clear presentation in your answers. 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. The total number of marks for this paper is 50. This document consists of 3 printed pages, 1 blank page and 2 inserts. JC15 11_9709_61/3R © UCLES 2015 [Turn over

2 1 In a certain town, 76% of cars are fitted with satellite navigation equipment. A random sample of 11 cars from this town is chosen. Find the probability that fewer than 10 of these cars are fitted with this equipment. [4] 2 The random variable X has the distribution N-, 32. It is given that PX < 54.1 = 0.5 and PX > 50.9 = 0.8665. Find the values of - and 3. [4] 3 Robert has a part-time job delivering newspapers. On a number of days he noted the time, correct to the nearest minute, that it took him to do his job. Robert used his results to draw up the following table; two of the values in the table are denoted by a and b. Time (t minutes) 60 −62 63 −64 65 −67 68 −71 Frequency (number of days) 3 9 6 b Frequency density 1 a 2 1.5 (i) Find the values of a and b. [3] (ii) On graph paper, draw a histogram to represent Robert’s times. [3] 4 (a) Amy measured her pulse rate while resting, x beats per minute, at the same time each day on 30 days. The results are summarised below. Σx −80 = −147 Σx −802 = 952 Find the mean and standard deviation of Amy’s pulse rate. [4] (b) Amy’s friend Marok measured her pulse rate every day after running for half an hour. Marok’s pulse rate, in beats per minute, was found to have a mean of 148.6 and a standard deviation of 18.5. Assuming that pulse rates have a normal distribution, find what proportion of Marok’s pulse rates, after running for half an hour, were above 160 beats per minute. [3] 5 (a) Find the number of ways in which all nine letters of the word TENNESSEE can be arranged (i) if all the letters E are together, [3] (ii) if the T is at one end and there is an S at the other end. [3] (b) Four letters are selected from the nine letters of the word VENEZUELA. Find the number of possible selections which contain exactly one E. [3] © UCLES 2015 9709/61/O/N/15

3 6 Nadia is very forgetful. Every time she logs in to her online bank she only has a 40% chance of remembering her password correctly. She is allowed 3 unsuccessful attempts on any one day and then the bank will not let her try again until the next day. (i) Draw a fully labelled tree diagram to illustrate this situation. [3] (ii) Let X be the number of unsuccessful attempts Nadia makes on any day that she tries to log in to her bank. Copy and complete the following table to show the probability distribution of X. [4] x 0 1 2 3 PX = x 0.24 (iii) Calculate the expected number of unsuccessful attempts made by Nadia on any day that she tries to log in. [2] 7 The faces of a biased die are numbered 1, 2, 3, 4, 5 and 6. The probabilities of throwing odd numbers are all the same. The probabilities of throwing even numbers are all the same. The probability of throwing an odd number is twice the probability of throwing an even number. (i) Find the probability of throwing a 3. [3] (ii) The die is thrown three times. Find the probability of throwing two 5s and one 4. [3] (iii) The die is thrown 100 times. Use an approximation to find the probability that an even number is thrown at most 37 times. [5] © UCLES 2015 9709/61/O/N/15

4 BLANK PAGE Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge International Examinations Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at www.cie.org.uk after the live examination series. Cambridge International Examinations 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. © UCLES 2015 9709/61/O/N/15

® IGCSE is the registered trademark of Cambridge International Examinations. CAMBRIDGE INTERNATIONAL EXAMINATIONS Cambridge International Advanced Subsidiary and Advanced Level MARK SCHEME for the October/November 2015 series 9709 MATHEMATICS 9709/61 Paper 6, 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 should be read in conjunction with the question paper and the Principal Examiner Report for Teachers. Cambridge will not enter into discussions about these mark schemes. Cambridge is publishing the mark schemes for the October/November 2015 series for most Cambridge IGCSE®, Cambridge International A and AS Level components and some Cambridge O Level components.

Page 2 Mark Scheme Syllabus Paper Cambridge International AS/A Level – October/November 2015 9709 61 © Cambridge International Examinations 2015 Mark Scheme Notes Marks are of the following three types: M Method mark, awarded for a valid method applied to the problem. Method marks are not lost for numerical errors, algebraic slips or errors in units. However, it is not usually sufficient for a candidate just to indicate an intention of using some method or just to quote a formula; the formula or idea must be applied to the specific problem in hand, e.g. by substituting the relevant quantities into the formula. Correct application of a formula without the formula being quoted obviously earns the M mark and in some cases an M mark can be implied from a correct answer. A Accuracy mark, awarded for a correct answer or intermediate step correctly obtained. Accuracy marks cannot be given unless the associated method mark is earned (or implied). B Mark for a correct result or statement independent of method marks. • When a part of a question has two or more “method” steps, the M marks are generally independent unless the scheme specifically says otherwise; and similarly when there are several B marks allocated. The notation DM or DB (or dep*) is used to indicate that a particular M or B mark is dependent on an earlier M or B (asterisked) mark in the scheme. When two or more steps are run together by the candidate, the earlier marks are implied and full credit is given. • The symbol implies that the A or B mark indicated is allowed for work correctly following on from previously incorrect results. Otherwise, A or B marks are given for correct work only. A and B marks are not given for fortuitously “correct” answers or results obtained from incorrect working. • Note: B2 or A2 means that the candidate can earn 2 or 0. B2/1/0 means that the candidate can earn anything from 0 to 2. The marks indicated in the scheme may not be subdivided. If there is genuine doubt whether a candidate has earned a mark, allow the candidate the benefit of the doubt. Unless otherwise indicated, marks once gained cannot subsequently be lost, e.g. wrong working following a correct form of answer is ignored. • Wrong or missing units in an answer should not lead to the loss of a mark unless the scheme specifically indicates otherwise. • For a numerical answer, allow the A or B mark if a value is obtained which is correct to 3 s.f., or which would be correct to 3 s.f. if rounded (1 d.p. in the case of an angle). As stated above, an A or B mark is not given if a correct numerical answer arises fortuitously from incorrect working. For Mechanics questions, allow A or B marks for correct answers which arise from taking g equal to 9.8 or 9.81 instead of 10.

Page 3 Mark Scheme Syllabus Paper Cambridge International AS/A Level – October/November 2015 9709 61 © Cambridge International Examinations 2015 The following abbreviations may be used in a mark scheme or used on the scripts: AEF Any Equivalent Form (of answer is equally acceptable) AG Answer Given on the question paper (so extra checking is needed to ensure that the detailed working leading to the result is valid) BOD Benefit of Doubt (allowed when the validity of a solution may not be absolutely clear) CAO Correct Answer Only (emphasising that no “follow through” from a previous error is allowed) CWO Correct Working Only – often written by a ‘fortuitous’ answer ISW Ignore Subsequent Working MR Misread PA Premature Approximation (resulting in basically correct work that is insufficiently accurate) SOS See Other Solution (the candidate makes a better attempt at the same question) SR Special Ruling (detailing the mark to be given for a specific wrong solution, or a case where some standard marking practice is to be varied in the light of a particular circumstance) Penalties MR –1 A penalty of MR –1 is deducted from A or B marks when the data of a question or part question are genuinely misread and the object and difficulty of the question remain unaltered. In this case all A and B marks then become “follow through ” marks. MR is not applied when the candidate misreads his own figures – this is regarded as an error in accuracy. An MR –2 penalty may be applied in particular cases if agreed at the coordination meeting. PA –1 This is deducted from A or B marks in the case of premature approximation. The PA –1 penalty is usually discussed at the meeting.

Page 4 Mark Scheme Syllabus Paper Cambridge International AS/A Level – October/November 2015 9709 61 © Cambridge International Examinations 2015 1 p = 0.76 P(fewer than 10) = 1 – P(10, 11) = 1 – (0.76)10(0.24)11C10 – (0.76)11 = 1 – 0.219 = 0.781 M1 M1 M1 A1 [4] Any binomial term 11Cxpx(1-p)11-x, 0 < p < 1 Any binomial term nCx(0.76)x(0.24)n-x 1 – P(10, 11) oe binomial expression Correct answer 2 µ = 54.1 z = -1.11 –1.11 = σ 1. 54 9. 50 − σ = 2.88 B1 B1 M1 A1 [4] Stated or evaluated Accept rounding to ± 1.1 Standardising no cc no sq rt Correct answer 3 (i) a = 9/cw a = 9/2 = 4.5 1.5 = b/4 so b = 6 M1 A1 A1 [3] Using fd = f/cw Correct a Correct b (ii) B1 B1 B1 [3] Correct heights ft their b Correct widths, ie 3, 2, 3, 4 starting either 60 or 59.5 Labels fd, time or minutes and squiggle and bars from 59.5 to 71.5 4 (i) x = 80 – 147/30 = 80 – 4.9 = 75.1 sd = √               − 2 30 147 30 952 = √ 7.72… sd = 2.78 M1 A1 M1 A1 [4] For –147/30 oe seen Correct answer 952/30 – (± their coded mean)2 Correct answer (ii) P(x > 160) = P       − > 5. 18 6. 148 160 z = P(z > 0.616) = 1 – 0.7310 = 0.269 M1 M1 A1 [3] Standardising no cc no sq rt 1 – Φ Correct answer fd 6 4 2 60 62 64 66 68 70 72 Time in minutes

Page 5 Mark Scheme Syllabus Paper Cambridge International AS/A Level – October/November 2015 9709 61 © Cambridge International Examinations 2015 5 (i) 5 (i) eg **(EEEE)*** Number of ways = !2!2 !6 = 180 M1 M1 A1 [3] Mult by 6! oe Dividing by 2!2! oe Correct answer (ii) S*******T or T*******S Number of ways = !2!4 !7 × 2 = 210 M1 M1 A1 [3] Mult by 7! Or dividing by one of 2! or 4! Mult by 2 Correct answer (iii) exactly one E in 6C3 ways = 20 M1 M1 A1 [3] 6Cx as a single answer xC3 as a single answer correct answer 6 (i) M1 A1 A1 [3] 3 pairs S (bank, log in, success oe) and F oe seen no extra bits. Exactly 3 pairs, must be labelled Correct diagram with all probs correct (ii) x 0 1 2 3 Prob 0.4 0.144 0.216 B1 M1 A1 B1 [4] P(0) correct Multiplying two of more factors of 0.4 and 0.6 One more correct prob One more correct prob (iii) E(X) = 0.24+ 2× 0.144 + 3× 0.216 = 1.176 (1.18) M1 A1 [2] Using Σpixi Correct answer 7 (i) let P(2, 4, 6) all = p then P(1, 3, 5) all = 2p 3p + 6p = 1 p = 1/9 so prob (3) = 2/9 (0.222) M1 M1 A1 [3] Using P(even) = 2P(odd) or vice versa oe Summing P(odd+ even) or P(1, 2, 3, 4, 5, 6) = 1 Correct answer (ii) P(5, 5, 6) = 2/9 × 2/9 × 1/9 × 3C2 P(5, 5, 6) = 4/243 (0.0165) M1 M1 A1 [3] Mult three probs together Mult by 3 oe ie summing 3 options Correct answer (iii) µ = 100×1/3 = 33.3, σ = 100×1/3×2/3 = 22.2 P(x ⩽ 37)= P       z ⩽       − 9 200 3 100 5. 37 =P(z ⩽ 0.8839) = 0.812 B1 M1 M1 M1 A1 [5] Unsimplified 100/3 and 200/9 seen Standardising need sq rt 36.5 or 37.5 seen correct area using their mean Correct answer S 0.4 0.4 S 0.6 F S 0.6 F 0.4 0.6 F