Cambridge A Level Mathematics (US) 9280 — 2014 May/June Paper 6 · Variant 1

*1772524693* Cambridge International Examinations Cambridge International Advanced Subsidiary and Advanced Level MATHEMATICS (US) 9280/61 Paper 6 Probability & Statistics 1 (S1) May/June 2014 1 hour 15 minutes Additional Materials: Answer Booklet/Paper Graph Paper List of Formulas (MF9) (US) READ THESE INSTRUCTIONS FIRST If you have been given an Answer Booklet, follow the instructions on the front cover of the Booklet. Write your Center number, candidate number, and name on the work you hand in. 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. 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. JC14 06_9280_61/FP © UCLES 2014 [Turn over

2 1 The gasoline consumption of a certain type of car has a normal distribution with mean 24 kilometres per liter and standard deviation 4.7 kilometres per liter. Find the probability that the gasoline consumption of a randomly chosen car of this type is between 21.6 kilometres per liter and 28.7 kilometres per liter. [4] 2 Lengths of a certain type of white radish are normally distributed with mean - cm and standard deviation 3 cm. 4% of these radishes are longer than 12 cm and 32% are longer than 9 cm. Find - and 3. [5] 3 (i) State three conditions which must be satisfied for a situation to be modeled by a binomial distribution. [2] George wants to invest some of his monthly salary. He invests a certain amount of this every month for 18 months. For each month there is a probability of 0.25 that he will buy shares in a large company, there is a probability of 0.15 that he will buy shares in a small company and there is a probability of 0.6 that he will invest in a savings account. (ii) Find the probability that George will buy shares in a small company in at least 3 of these 18 months. [3] 4 A book club sends 6 paperback and 2 hardback books to Mrs Hunt. She chooses 4 of these books at random to take with her on holiday. The random variable X represents the number of paperback books she chooses. (i) Show that the probability that she chooses exactly 2 paperback books is 3 14. [2] (ii) Draw up the probability distribution table for X. [3] (iii) You are given that EX = 3. Find VarX. [2] 5 Playground equipment consists of swings (S), roundabouts (R), climbing frames (C) and play-houses (P). The numbers of pieces of equipment in each of 3 playgrounds are as follows. Playground X Playground Y Playground Z 3S, 2R, 4P 6S, 3R, 1C, 2P 8S, 3R, 4C, 1P Each day Nur takes her child to one of the playgrounds. The probability that she chooses playground X is 1 4. The probability that she chooses playground Y is 1 4. The probability that she chooses playground Z is 1 2. When she arrives at the playground, she chooses one piece of equipment at random. (i) Find the probability that Nur chooses a play-house. [4] (ii) Given that Nur chooses a climbing frame, find the probability that she chose playground Y. [4] © UCLES 2014 9280/61/M/J/14

3 6 Find the number of different ways in which all 8 letters of the word TANZANIA can be arranged so that (i) all the letters A are together, [2] (ii) the first letter is a consonant (T, N, Z), the second letter is a vowel (A, I), the third letter is a consonant, the fourth letter is a vowel, and so on alternately. [3] 4 of the 8 letters of the word TANZANIA are selected. How many possible selections contain (iii) exactly 1 N and 1 A, [2] (iv) exactly 1 N? [3] 7 A typing test is taken by 111 people. The numbers of typing errors they make in the test are summarized in the table below. Number of typing errors 1 −5 6 −20 21 −35 36 −60 61 −80 Frequency 24 9 21 15 42 (i) Draw a histogram on graph paper to represent this information. [5] (ii) Calculate an estimate of the mean number of typing errors for these 111 people. [3] (iii) State which class contains the lower quartile and which class contains the upper quartile. Hence find the least possible value of the interquartile range. [3] © UCLES 2014 9280/61/M/J/14

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. 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 2014 9280/61/M/J/14

CAMBRIDGE INTERNATIONAL EXAMINATIONS GCE Advanced Subsidiary Level and GCE Advanced Level MARK SCHEME for the May/June 2014 series 9280 MATHEMATICS 9280/61 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 May/June 2014 series for most IGCSE, GCE Advanced Level and Advanced Subsidiary Level components and some Ordinary Level components.

Page 2 Mark Scheme Syllabus Paper GCE AS/A LEVEL – May/June 2014 9280 61 © Cambridge International Examinations 2014 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 GCE AS/A LEVEL – May/June 2014 9280 61 © Cambridge International Examinations 2014 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 AS/A LEVEL – May/June 2014 9280 61 © Cambridge International Examinations 2014 1 P (21.6 < x < 28.7) = P                         − < < − 7.4 24 7. 28 7.4 24 6. 21 z = P (–0.5106 < z < 1) = Φ (1) – Φ (–0.5106) = 0.8413 – (1 – 0.6953) = 0.537 (0.5366) M1 A1 M1 A1 4 Standardising; no cc, no sq rt One rounding to Φ (0.841 or 0.695) Φ1 + Φ2 – 1 Correct answer 2 1.751 = σ µ − 12 0.468 = σ µ − 9 σ = 2.34 µ = 7.91 B1 B1 M1 M1 A1 5 Rounding to ±1.75 seen ±0.468 seen An eqn with a z-value, µ and σ no √σ, no σ2 Sensible attempt to eliminate µ or σ by substitution or subtraction, need a value correct answers 3 (i) constant / given p, independent trials, fixed / given no. of trials, only two outcomes B1 B1 2 Any one correct Any 3 correct (ii) P (x [ 3) = 1 – P (0, 1, 2) = 1 – [(0.85)18 + (0.85)17(0.15) × 18 + (0.85)16(0.15)2 × 18C2] = 0.520 M1 M1 A1 3 Any binomial expression pr (1 – p) 18–r 18Cr seen 1 – P (0, 1, 2 ), any n,p,q Correct answer 4 (i) P (exactly 2) = 4 8 2 6 C C = 14 3 70 15 = AG OR P (2) = 2 4C 5 1 6 2 7 5 8 6 × × × × = 14 3 AG M1 A1 2 6Cx / 8Cx seen or 4C2 mult by 4 fractions (last 2 can be implied) Answer legit obtained (ii) x 2 3 4 Prob 3/14 8/14 3/14 B1 B1 B1 3 2, 3, 4 only in top line one correct prob other than P (2) third correct prob ft Σ = 1 (iii) Var (X) = 14 48 14 72 14 12 + + – 32 = 7 3 (0.429) M1 A1 2 using Σx2p – 32 (or their {E(X)}2) must be evaluated correct answer

Page 5 Mark Scheme Syllabus Paper GCE AS/A LEVEL – May/June 2014 9280 61 © Cambridge International Examinations 2014 5 (i) P(X and P) = 4 1 × 9 4 = 9 1 P(Y and P) = 4 1 × 12 2 = 24 1 P(Z and P) = 2 1 × 16 1 = 32 1 P(P) = 288 53 = 0.184 M1 A1 M1 A1 4 Mult a playground prob with a P prob One correct prob Summing at least two 2-factor probs Correct answer (ii) P(Y | C) = ) ( P ) ( P C C Y ∩ 16 4 2 1 12 1 4 1 12 1 4 1 × + × × = 48 7 48 1 = 7 1 M1 M1 A1 A1 4 Attempt at ) ( P C Y ∩ as numerator of a fraction Attempt at P(C) in form of summing two 2- factor products, seen anywhere Correct unsimplified P(C) seen anywhere Correct answer 6 (i) !2 !6 = 360 B1 B1 2 6! Seen alone Dividing by 2! only (ii) !3 !4 !2 !4 × = 48 B1 B1 B1 3 4! seen mult Dividing by 2! or 3! (Mult by 4 implied B1B1) Correct answer (iii) 1N and 1A: N A xx in 3C2 = 3 ways M1 A1 2 3Cx or xC2 seen alone Correct answer (iv) 0 A : Nxxx = 1 way 2 As: NAAx in 3C1 = 3 ways 3 As: NAAA in 1 way Total = 8 ways M1 M1 A1 3 Finding ways with 0 or 2 or 3 As Summing 3 or 4 options Correct answer

Page 6 Mark Scheme Syllabus Paper GCE AS/A LEVEL – May/June 2014 9280 61 © Cambridge International Examinations 2014 7 (i) class widths 5, 15, 15, 25, 20 fd = 20 42 , 25 15 , 15 21 , 15 9 , 5 24 = 4.8, 0.6, 1.4, 0.6, 2.1 fd 5 4 3 2 1 10.5 20.5 30.5 40.5 50.5 60.5 70.5 80.5 errors M1 B1 M1 A1 B1 5 Attempt at class widths Correct widths of bars, with or without halves, seen on diagram Attempt at fd or scaled freq Correct heights seen on graph ft their fd Correct labels, scales and halves (ii) mean = 111 42 70.5 15 48 21 28 9 13 24 3     × + × + × + × + × = 40.2 errors M1 M1 A1 3 Using mid points using (Σ their fx) / their 111 correct answer (iii) LQ in 6 – 20 UQ in 61 – 80 Least value of IQ range is 61 – 20 = 41 B1 B1 B1 3 ft any or both wrong quartile ranges if sensible