Cambridge A Level Mathematics 9709 — 2007 May/June Paper 6 · Variant 1

UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS General Certificate of Education Advanced Subsidiary Level and Advanced Level MATHEMATICS 9709/06 Paper 6 Probability & Statistics 1 (S1) May/June 2007 1 hour 15 minutes Additional Materials: Answer Booklet/Paper Graph Paper List of Formulae (MF9) 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. You may use a soft pencil for any diagrams or graphs. Do not use staples, paper clips, highlighters, glue or correction fluid. 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. Questions carrying smaller numbers of marks are printed earlier in the paper, and questions carrying larger numbers of marks later in the paper. This document consists of 3 printed pages and 1 blank page. ©UCLES 2007 [Turn over *4070068018*

2 1 The length of time, t minutes, taken to do the crossword in a certain newspaper was observed on 12 occasions. The results are summarised below. Σ(t −35) = −15 Σ(t −35)2 = 82.23 Calculate the mean and standard deviation of these times taken to do the crossword. [4] 2 Jamie is equally likely to attend or not to attend a training session before a football match. If he attends, he is certain to be chosen for the team which plays in the match. If he does not attend, there is a probability of 0.6 that he is chosen for the team. (i) Find the probability that Jamie is chosen for the team. [3] (ii) Find the conditional probability that Jamie attended the training session, given that he was chosen for the team. [3] 3 (a) The random variable X is normally distributed. The mean is twice the standard deviation. It is given that P(X > 5.2) = 0.9. Find the standard deviation. [4] (b) A normal distribution has mean µ and standard deviation σ. If 800 observations are taken from this distribution, how many would you expect to be between µ −σ and µ + σ? [3] 4 The lengths of time in minutes to swim a certain distance by the members of a class of twelve 9-year-olds and by the members of a class of eight 16-year-olds are shown below. 9-year-olds: 13.0 16.1 16.0 14.4 15.9 15.1 14.2 13.7 16.7 16.4 15.0 13.2 16-year-olds: 14.8 13.0 11.4 11.7 16.5 13.7 12.8 12.9 (i) Draw a back-to-back stem-and-leaf diagram to represent the information above. [4] (ii) A new pupil joined the 16-year-old class and swam the distance. The mean time for the class of nine pupils was now 13.6 minutes. Find the new pupil’s time to swim the distance. [3] 5 (i) Find the number of ways in which all twelve letters of the word REFRIGERATOR can be arranged (a) if there are no restrictions, [2] (b) if the Rs must all be together. [2] (ii) How many different selections of four letters fromthe twelve letters of the word REFRIGERATOR contain no Rs and two Es? [3] 6 The probability that New Year’s Day is on a Saturday in a randomly chosen year is 1 7. (i) 15 years are chosen randomly. Find the probability that at least 3 of these years have New Year’s Day on a Saturday. [4] (ii) 56 years are chosen randomly. Use a suitable approximation to find the probability that more than 7 of these years have New Year’s Day on a Saturday. [5] © UCLES 2007 9709/06/M/J/07

3 7 A vegetable basket contains 12 peppers, of which 3 are red, 4 are green and 5 are yellow. Three peppers are taken, at random and without replacement, from the basket. (i) Find the probability that the three peppers are all different colours. [3] (ii) Show that the probability that exactly 2 of the peppers taken are green is 12 55. [2] (iii) The number of green peppers taken is denoted by the discrete random variable X. Draw up a probability distribution table for X. [5] © UCLES 2007 9709/06/M/J/07

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. University of 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. 9709/06/M/J/07

UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS GCE Advanced Subsidiary Level and GCE Advanced Level MARK SCHEME for the May/June 2007 question paper 9709 MATHEMATICS 9709/06 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. All Examiners are instructed that alternative correct answers and unexpected approaches in candidates’ scripts must be given marks that fairly reflect the relevant knowledge and skills demonstrated. Mark schemes must be read in conjunction with the question papers and the report on the examination. • CIE will not enter into discussions or correspondence in connection with these mark schemes. CIE is publishing the mark schemes for the May/June 2007 question papers for most IGCSE, GCE Advanced Level and Advanced Subsidiary Level syllabuses and some Ordinary Level syllabuses.

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.

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 GCE A/AS LEVEL – May/June 2007 9709 06 © UCLES 2007 1 mean = 35 – 15/12 M1 For –15/12 seen = 33.75 (33.8) minutes A1 Correct answer sd = 2) 12 / 15 ( 12 / 23 . 82 − − M1 82.23/12 – ( ± their coded mean)2 = 2.3 minutes A1 4 Correct answer 2 (i) P(team) = 0.5 6.0 5.0 × + B1 One correct product M1 Summing two 2-factor products = 0.8 A1 3 Correct answer (ii) P(training session | team) = 6.0 5.0 5.0 5.0 × + M1 Selecting correct term from (i) as their numerator M1 Dividing by their (i) (must be < 1) = 0.625 (5/8) A1 3 Correct answer 3 (a) 282 .1 2 2.5 − = − s s M1 Equation with ± correct LHS seen here or later, can be µ or s, no cc B1 282 .1 ± seen accept ± 1.28 or anything in between M1 solving their equation with recognisable z-value and only 1 unknown occurring twice s = 7.24 or 7.23 A1 4 correct final answer (b)       − + Φ σ µ σ µ = 0.8413 B1 0.8413 (p) seen or implied (can use their own numbers) P( 1 < z ) = 0.3413 6826 .0 2 = × M1 finding the correct area i.e. 2p – 1 0.6826 800 × = 546 ( accept 547) A1 3 correct answer, must be a positive integer OR SR 3 / 2 800× = 533 or 534 SR B1 for 2/3 B1 for 533 or 534 or B2 if 533 or 534 and no working

Page 5 Mark Scheme Syllabus Paper GCE A/AS LEVEL – May/June 2007 9709 06 © UCLES 2007 4 (i) 16 yr olds 9 year olds 7, 4 11 9, 8, 12 7, 0 13 0, 2, 7, 8 14 2, 4, 15 0, 1, 9, 5 16 0, 1, 4, 7, B1 B1 B1 3 columns including an integer stem in the middle, single digits in leaves. Can go downwards One leaf column correct, ordering not necessary Other leaf column correct (ordering not nec) and both leaves labelled correctly (could be in key) Key 7 2 13 means 13.7 minutes and 13.2 minutes B1 4 Key correct both ways or two keys one each way, must have minutes (ii) ∑(8 pupils) = 106.8 B1 106.8 seen or implied ∑(9 pupils) = 13.6 × 9 (= 122.4) B1 for 13.6 × 9 New pupil’s time = 15.6 min B1ft 3 Ft on 122.4 – their ∑8 5 (i) (a) 9979200 !2!4 ! 12 = (9980000) B1 B1 2 Dividing by 4! and 2! only Correct answer (b) !2 !9 = 181440 (181000) B1 9! or 9 × 8! seen not in denom B1 2 correct answer (ii) 6C2 or 4C0× 2C2× 6C2 or 6C4 or 6P2/2! M1 for seeing 6 C something or 6Psomething in a product (could be with 1) M1 for seeing somethingC2 or 6C4 = 15 A1 3 correct answer 15 with no working scores full marks

Page 6 Mark Scheme Syllabus Paper GCE A/AS LEVEL – May/June 2007 9709 06 © UCLES 2007 6 (i) P([3) = 1 – P(0, 1, 2 ) M1 For attempt at 1 – P(0, 1, 2) or 1 – P(0, 1, 2, 3) or P(3...15) or P(4...15) = 1 – (6/7) 15 – 15C1 (1/7) (6/7)14 – 15C2 (1/7)2 (6/7)13 M1 For 1 or more terms with 1/7 and 6/7 to powers which sum to 15 and 15Csomething ( = 1 – 0.0990 –0.2476 – 0.2889) A1 Completely correct unsimplified form = 0.365 (accept 0.364) A1 4 Correct final answer (ii) µ = 56 x 1/7 (= 8) ë2= 56 x 1/7 x 6/7 (= 6.857) B1 8 and 6.857 or 6.86 or 2.618 seen or implied P(more than 7)       − Φ − = 857 .6 8 5.7 1 M1 Standardising attempt with or without cc, must have square root ( ) 1909 .0 857 .6 5.7 8 Φ =       − Φ = M1 Continuity correction either 7.5 or 6.5 M1 Final answer > 0.5 (award this if the long way is used and the final answer is > 0.5) = 0.576 A1 5 Correct final answer 7 (i) P (all different) = × × = 3 12 1 5 1 4 1 3 C C C C M1 Attempt using combinations, with 12C3 denom, or P(RGY) in any order, i.e. 12 x 11 x 10 in denom M1 Correct numerator, or multiplying by 6 = 3/11 ( = 0.273) A1 3 Correct answer (ii) P(exactly 2 G) 3 12 1 8 2 4 C C C × = M1 Attempt using combinations, or mult any P( ) G GG x 3 Or P(GGY) x 3+P(GGR) x 3 = 12/55 AG A1 2 Correct answer AG (iii) x 0 1 2 3 P(X=x) 14/55 28/55 12/55 1/55 decimal 0.255 0.509 0.218 0.018 M1 M1 A1 For seeing P(0, 1, 2, 3) only and 1 or more probs For reasonable attempt at P(X = 0 or 1 or 3) For one correct probability seen other than P(X=2) A1 For a second probability correct other than P(X=2) A1 5 All correct