2009 part a reportpublic - mathematical institute part a... · scheme which was strictly adhered...

Examiners’ Report: Final Honour School of

Mathematics Part A Trinity Term 2009

October 29, 2009

Part I

A. STATISTICS

• Numbers and percentages in each class.

See Table 1, page 1.

Table 1: Numbers in each class

Range Numbers Percentages %(2009) (2008) (2007) (2006) (2005) (2009) (2008) (2007) (2006) (2005)

70–100 (45) (44) (47) (70) (59) (29.61) (25.58) (29.4) (40.2) (36.6)60–69 (69) (91) (82) (74) (71) (45.39) (52.91) (51.2) (42.5) (44.1)50–59 (29) (25) (23) (24) (25) (19.08) (14.53) (14.4) (13.8) (15.5)40–49 (9) (8) (8) (3) (4) (5.92) (4.65) (5.0) (1.7) (2.5)30–39 (0) (4) (0) (3) (1) (0) (2.33) (0) (1.7) (0.6)0–29 (0) (0) (0) (0) (1) (0) (0) (0) (0) (0.6)

Total (152) (172) (160) (174) (161) (100) (100) (100) (100) (100)

• Numbers of vivas and effects of vivas on classes of result.

Not applicable.

• Marking of scripts.

All scripts were all single marked according to a pre-agreed markingscheme which was strictly adhered to. For details of the extensivechecking process, see Part II, Section A.

• Numbers taking each paper.

All 152 take the set of four papers AC1, AC2, AO1 and AO2. Statisticsfor these papers are shown in Table 2 on page 2.

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Table 2: Numbers taking each paper

Paper Number of Avg StDev Avg StDevCandidates RAW RAW USM USM

AC1 152 43.82 13.64 65.45 9.9AC2 152 57.57 17.17 65.6 10.34AO1 151 50.63 15.79 64.77 11.17AO2 152 63.3 16.1 65.64 10.54

B. New examining methods and procedures

There are new papers of the form of AO1, the short option papers, for alljoint schools now, so AO1(P) and AO1(MC) for Maths & Philosophy andMaths & Computer Science candidates respectively. Maths & Philosophycandidates were also allowed to answer questions on Graph Theory andMods Probability (discrete) on papers AO1(P) and AO2(P).

The upper limit on the number of questions candidates could answer wasremoved for all papers. Rubrics were adjusted to read “the best n questionscount for the total raw marks for this paper.” The rubric on AC2 was alsochanged. Candidates now need to answer one question from each sectionwith a fourth question from any section. Where candidates did not answera question which should have been answered, a zero was recorded.

C. Changes in examining methods and procedures currently

under discussion or contemplated for the future

There is a review of the Structures in Part A and Mods underway at present.This would be likely to be implemented in 2011/12.

D. Notice of examination conventions for candidates

The first Notice to Candidates was issued on 4th December 2008 and thesecond notice on the 12th May.

These can be found at http://www.maths.ox.ac.uk/node/8667, and con-tain details of the examinations and assessments. The course Handbookcontains the full examination conventions and all candidates are issued withthis at Induction in their first year. All notices and examination conventionsare on-line at http://www.maths.ox.ac.uk/notices/undergrad.

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Part II

A. General Comments on the Examination

The examiners would like to express their gratitude to

• Sandy Patel for expertly overseeing Part A examinations during 2008/09.

• Also Waldemar Schlackow for continuing to develop the examinationsdatabase, responding to examiner requests and providing such a goodframework for the examinations data.

• We would also like to thank Helen Lowe, Charlotte Rigdon, and Mar-garet Sloper for all their sterling work in keeping track of the scriptsand marks and everything else they do during the busy examinationperiod.

• We also thank the assessors for their prompt setting of questions andfor the care in checking, responding to comments from examiners andmarking them.

• All the assessors and the internal examiners would like to thank theexternal examiner Professor Tom Korner for his careful reading ofthe draft papers, scrutiny of the examination scripts and insightfulcomments throughout the year.

This year we had agreed protocols with Statistics in advance of the meet-ing for how we would deal with the scaling of shared and related papers.The aim here was to keep a common scaling for all papers but if there wasa case made for having a different scaling on, for example AS1 and AS2, ex-aminers would be open to this. In any eventuality, this did not happen anda common scaling was used across all the papers. We thank the Statisticsdepartment, particularly Dr Neil Laws for working with us on this.

We instructed invigilators on two issues as noted to be problematic lastyear. Namely, about making note of queries and contacting the examinerseven if a query seemed unimportant (these are usually significant in mathe-matics papers), and to note if candidates leave the room or are ill during apaper. There were no incidents to note.

Finally we are pleased to note that no errors in the papers have come tolight.

Timetable

The examinations began on Monday 22nd June at 9.30am and ended onThursday 25th June at 12.30pm. We are pleased to report that examinationtimetables were prepared early this year and thank the Examination Schoolsfor their work.

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Medical certificates and other special circumstances

Examiners received 6 medical notes from the Proctors. Two notes wouldbe passed on to Part B examiners; no action could be taken by Part Aexaminers. The remaining candidates received special consideration andwhere it was deemed appropriate mark(s) were adjusted (in one case only).There were two candidates for special consideration. These are noted inSection F.

Setting and checking of papers and marks processing

As is our usual practice, the questions for AC1 and AC2 are set and markedby examiners, whilst AO1 and AO2 are st by the course lecturer but checkedby examiners for correctness and comparability of standard. This had beena problem last year.

The course lecturers also acted as assessors, marking the questions ontheir course(s).

In the instructions to setters, we gave feedback on last year’s questionsand recorded the proportion of candidates who attained a notional first ineach question to guide as to the expected standard. We asked that questionsbe set and marking schemes established so that a mark of 70 per cent or moreon a question would correspond to a first class performance. The purposewas to reduce the need for any scaling and so that an overall mark of 70 percent should then correspond to a USM of 70, and then one would not be aposition where a candidate might feel that “(s)he had done far better thanthe USM indicated.” Overall, and somewhat fortunately, this has workedbetter this year, although the input from examiners has been constant overboth years.

The internal examiners met in December to consider the questions forAC1 and AC2. Changes and corrections were agreed. As soon as the lecturecourses were finished, we invited the Michaelmas term course lecturers tocomment on the appropriateness of the questions, in light of their coursesand the notation used. This seems to have worked well and we recommendthat this continues. The revised questions were then sent to the externalexaminer.

Internal examiners met a second time to consider the external exam-iner’s comments and made further changes as necessary before finalising thequestions. The same cycle was repeated towards the latter half of Hilaryterm for the Hilary term courses, and at the end of Hilary term for theTrinity courses, although the schedule here was much tighter. At both ofthese meetings we also looked at AC1 and AC2. Examiners also scrutinisedthe final form of AC1 and AC2 again in their final meeting in early Trinityterm. The Camera Ready Copy was prepared for AC1 and AC2, and all thejoint schools, which examiners signed off as correct. Each assessor signed

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off their questions in AO1 and AO2 in time for submission to Examinationschools at the end of week 3 of Trinity term.

This year all examination scripts were collected from the MathematicalInstitute rather than Examination Schools, which all worked smoothly. Thisis no small task and we commend the administrative staff here. Assessorshad a short time period to return the marks on standardised mark sheets.More time was allowed this year as the deadlines had been too tight lastyear which has mostly worked well. However two sets of marks did come invery late and staff had to stay to complete the data entry before the scriptchecking. Deadlines will need to be adjusted for next year.

A team of graduate checkers under the supervision of Sandy Patel, as-sisted by Charlotte Rigdon, sorted all the scripts for each paper of thisexamination, carefully cross checking against the marks scheme to spot anyunmarked questions or part of questions, addition errors or wrongly recordedmarks. Also sub-totals for each part were checked against the marks scheme,noting correct addition. In this way a number of errors were corrected, eachchange signed by one of the examiners who were present throughout theprocess. A check-sum is also carried out to ensure that marks entered intothe database are correctly read and transposed from the marks sheets.

Determination of University Standardised Marks

This year the Mathematics Teaching Committee issued each examinationboard with broad guidelines on the proportion of candidates that might beexpected in each class. This was based on the average proportion in eachclass over the last five years, together with recent historic data for PartA, and the MPLS Divisional averages. We also followed advice from theTeaching Committee about having the default upper corner closer to 70than 74 (as used last year).

Moderators had drawn the attention of Part A examiners to their aca-demic judgement of the candidates : this cohort was a fairly strong cohortand the proportion in each class awarded in Honour Moderations was areflection of this judgement.

Examiners may recalibrate the raw marks to arrive at university stan-dardised marks reported to candidates, adopting the procedures outlinedbelow, similarly to previous years. Examiners also take into account reportson each question from the examiner/assessor who marked this work, tak-ing into account the standard of work, comparison with previous years, theoverall level of work presented for each question in each examination.

For the construction of a piecewise linear map we let N denote the num-ber of candidates used for classification purposes of the algorithm. For PartA, this is the population of candidates taking Mathematics and Mathematicsand Statistics for AC1 and AC2 (which is 185 candidates) and Mathematicscandidates only for AO1 and AO2, which is 152 candidates.

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Candidates’ raw marks for a given paper are ranked in descending order.The default proportion p1 of notional firsts, p2 of notional upper secondsand p3 notional lower seconds or below in this population is entered into thedatabase, these proportions were set as in 2007 (but slightly higher than in2008).

We count down through the top p1 proportion of candidates on a givenpaper which gives us the candidate at the (100− p1)-th percentile. The rawmark for the last candidate in this percentile in the ranked list is assigned aUSM of 70. Let this raw mark be denoted by R1. Continuing to count downthe list of ranked candidates until another p2 proportion of candidates haselapsed, the last candidate here is assigned a USM of 60. Denote this rawmark by R2. The line segment between (R1, 70) and (R2, 60) is extendedlinearly to the USMs of 72 and 57 respectively (in this way each the non-linearity is away from a class boundary).

Denote the raw marks corresponding to USMs of 72 and 57 by C1 andC2 respectively. A line segment is drawn connecting (C1, 72) to (90, 100) forAC1 and AO1, and (100, 100) for AC2 and AO2. Thus two segments of thepiecewise linear graph are constructed.

Finally, the line segment through the corner at (C2, 57) is extended downtowards the vertical axis as if it were to join the axis at (0, 20), but is brokenat the corner (C3, 37) and joined to the Origin. This gives us the last segmentin this model.

The co-ordinates in the form of (Raw mark,USM) are the corners whichare given in the table below corresponding to the points (C1, 72), (C2, 57)and (C3, 37). In the table we record C1, C2, C3 are the raw marks for eachpaper which were mapped to USMs of 72, 57 and 37 respectively. Thepositions of the finally agreed maps are given in table 3. It should be notedthat the total raw marks for papers AC1 and AO1 are marks out of 90,whilst for AC2 and AO2 they are out of 100.

Table 3: Position of corners of piecewise linear function

Paper C1 C2 C3

AC1 54.4 28.9 13.28AC2 72.6 38.1 17.5AO1 65 35 16.1AO2 78.2 46.7 21.46

The default maps were used for each paper to highlight candidates neareach nominal class boundary and we invited the external examiner to scru-tinise these scripts, for each of the papers.

The default maps are the starting point of the discussions at the exam-ination meetings. We undertook a careful analysis of the performance ofcandidates across the various forms of AC1 and AC2. This can be found in

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the final page of this report, in the Appendix.We also carried out a careful comparative analysis of subsets of questions

on AO1 and AO2. Last year we had seen some quite wide variability.

Performance on Probability and Statistics short questions.

Questions M1, M2, O1, O2

• AO1- Mathematics candidates

Mean =1831.41

366= 5.004

• AS1- Mathematics and Statistics candidates

Mean =719.89

131= 5.495

Performance on Probability and Statistics long questions

Questions M4, M5, O4, O5


Mean =2378.9

161= 14.78


Mean =1459.18

93= 15.69

Performance on Pure Options- Short Questions

A1, B1, C1, D1, D2, E1, E2


Mean =2579.70

422= 6.113


Mean =166.99

26= 6.423

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• AO1(P)- Mathematics and Philosophy candidates

Mean =372.01

61= 6.098

• AO1(MC)- Mathematics and Computing candidates

Mean =177.03

30= 5.901

Performance on Pure Options- Long Questions

A2, B2, C2, D3, D4, E3, E4


Overall average =2830.33

176= 16.081

(exclude C2: the average is 2514.3159

= 15.81)

• AO2(P)- Mathematics and Philosophy candidates


31= 15.354

(exclude C2 and the average is 375

25= 15)

• AO2(MC)- Mathematics and Computing candidates


16= 16.313

(only 1 M &C candidate took C2; if we exclude C2, the average is16.134).


Very few pure answers, C2, D3, D4, E3.

From the sum of all these elements, examiners concluded that therewas a fairly consistent performance across the various schools and that thedefault maps were giving us the level in each class as indicted by TeachingCommittee. Thus we set the default maps as the final maps and our USMswere determined.

The table 4 gives the final Rank and percentage of candidates with thisoverall average USM (or greater).

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Table 4: Rank and percentage of candidates with this overallaverage USM (or greater)

Av USM Rank Candidates with this USM or above %

85 1 1 0.6684 2 3 1.9783 4 8 5.2682 9 9 5.9281 10 11 7.2480 12 15 9.8778 16 18 11.8477 19 22 14.4776 23 23 15.1375 24 26 17.1174 27 28 18.4273 29 32 21.0572 33 38 2571 39 43 28.2970 44 46 30.2669 47 53 34.8768 54 62 40.7967 63 69 45.3966 70 78 51.3265 79 91 59.8764 92 98 64.4763 99 103 67.7662 104 107 70.3961 108 111 73.0360 112 115 75.6659 116 123 80.9258 124 125 82.2457 126 126 82.8956 127 131 86.1855 132 133 87.554 134 136 89.4753 137 139 91.4552 140 140 92.1151 141 142 93.4250 143 144 94.7449 145 146 96.0547 148 150 98.6845 150 151 99.3442 152 152 100

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B. Equal opportunities issues and breakdown of the results

by gender

Table 5, page 10 shows the performances of candidates broken down bygender.

Table 5: Breakdown of results by gender

Range Total Male FemaleNumber % Number % Number %

70–100 45 57.83 33 30.56 12 27.2760–69 69 86.79 52 48.15 17 38.6450–59 29 43.01 17 15.74 12 27.2740–49 9 12.38 6 5.56 3 6.8230–39 0 0 0 0 0 00–29 0 0 0 0 0 0

Total 152 100 108 100 44 100

See 1 page 27 for the overall quartiles for each paper.

C. Detailed numbers on candidates’ performance in each part

of the exam

The Table 6 (and continued in Tables 7 and ??) give the detailed perfor-mance of candidates’ on each question of the exam.

Table 6: Analysis by question

Paper Question rawAvg Avg Std Used Unused Percentage ofNumber Used Dev Marks ≥ 70%

AC1 Q1 5.31 5.31 2.38 151 0 30.05AC1 Q2 5.32 5.32 2.62 151 0 38.34AC1 Q3 2.81 2.81 2.54 151 0 11.40AC1 Q4 5.52 5.52 2.49 152 0 27.46AC1 Q5 4.63 4.63 2.19 152 0 20.10AC1 Q6 3.10 3.10 2.73 150 0 13.02AC1 Q7 4.53 4.53 2.80 152 0 24.74AC1 Q8 7.26 7.26 2.48 152 0 59.79AC1 Q9 5.47 5.47 2.32 152 0 28.87

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Table 7: Analysis by question

Paper Question rawAvg Avg Std Used Unused Percentage ofNumber Used Dev Marks ≥ 70%

AC2 Q1 9.18 11.80 7.43 35 15 13.70AC2 Q2 9.25 9.34 4.67 106 7 5.63AC2 Q3 16.85 16.85 5.71 13 0 42.86AC2 Q4 13.25 14.44 6.51 41 7 21.67AC2 Q5 14.95 15.41 5.43 122 7 33.94AC2 Q6 11.59 11.67 6.90 15 2 26.32AC2 Q7 12.11 13.32 5.53 76 13 13.33AC2 Q8 15.03 15.82 6.42 115 8 41.36AC2 Q9 17.89 20.37 7.84 81 15 53.78

AO1 A1 6.33 6.37 2.38 38 1 46.15AO1 B1 3.82 3.73 2.86 15 2 12.50AO1 C1 6.32 6.37 2.72 54 2 46.43AO1 D1 5.82 5.90 2.12 88 2 36.67AO1 D2 6.59 6.59 2.59 87 0 55.17AO1 E1 5.52 5.60 2.41 70 1 42.25AO1 E2 6.79 6.79 2.50 71 0 56.34AO1 F1 6.09 6.27 3.32 41 2 53.49AO1 G1 6.91 6.91 2.22 129 0 65.89AO1 H1 3.62 3.75 2.39 48 2 14.00AO1 J1 3.71 3.76 3.16 33 1 23.53AO1 K1 6.61 6.61 2.19 79 0 63.29AO1 K2 5.20 5.24 2.23 78 1 22.78AO1 M1 6.02 6.02 2.84 131 0 40.46AO1 M2 3.63 3.74 2.39 120 6 11.90AO1 O1 4.49 4.57 3.11 56 1 29.82AO1 O2 5.73 5.73 2.63 59 0 40.68AO1 P1 7.33 7.33 2.49 57 0 75.44AO1 P2 8.08 8.08 2.80 50 0 76.00

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Table 8: Analysis by question, continued

Paper Question rawAvg Avg Std Used Unused PercentageofNumber Used Dev Marks ≥ 70%

AO2 A2 16.38 16.38 4.40 16 0 37.50AO2 B2 8.67 11.50 6.66 2 1 0.00AO2 C2 15.41 18.59 8.26 17 5 54.55AO2 D3 13.70 13.98 5.74 49 4 30.19AO2 D4 17.35 18.09 6.14 46 3 59.18AO2 E3 14.57 15.06 4.03 34 3 24.32AO2 E4 15.47 15.93 5.44 14 1 46.67AO2 F2 12.67 13.80 4.72 5 1 33.33AO2 G2 15.92 15.84 4.64 103 2 39.05AO2 H2 17.32 17.71 4.67 24 1 52.00AO2 J2 19.64 19.64 3.99 14 0 78.57AO2 K3 13.19 14.41 4.29 51 8 15.25AO2 K4 11.33 14.63 6.93 8 4 25.00AO2 M4 8.78 10.06 6.44 48 16 14.06AO2 M5 14.30 15.91 6.31 67 10 35.06AO2 O4 17.97 18.85 5.62 33 3 66.67AO2 O5 12.35 16.00 7.33 13 4 29.41AO2 P3 18.11 18.40 5.28 35 1 61.11AO2 P4 17.31 17.57 6.31 28 1 55.17

See the figure 1 on 27 for the quartiles for raw marks for each paper acrossthe various schools. The final table on the last page of this report gives theoverall summary of average marks and standard deviation per question, oncore papers AC1 and AC2, by school.

D. Recommendations for Next Year’s Examiners and Teach-

ing Committee

Candidates are not performing well in the core algebra on papers AC1 andAC2. It may be that the syllabus here requires some simplification.

Prof Korner also requested that when the examination board considersthe draft rankings, examiners have access to historic data here to compareprevious years. In the light of this, possible undesirable results might beavoided.

The mechanism for giving the data to the Mathematics and ComputerScience Board does need to be improved and the way these scripts are dealtwith in the checking and data entry process. (This needs to come under theaegis of the Mathematics examiners in order to allow us to determine theUSMs; this was not clear this year).

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E. Comments on sections and on individual questions

The following comments were submitted by the assessors.

AC1

Overall comments on Algebra

Questions 1 and 2 were reasonably well answered by the majority of candi-dates, question 3 was answered quite well by those who attempted it.

AC1 qu1 Almost all candidates could prove that Iv was an ideal and some can-didates noticed that in K[X] this was a principal ideal. Strongercandidates showed that a (monic) polynomial mv of degree at most nexisted and generated Iv. There were few perfect answers to show thefinal part.

AC1 qu2 All candidates could define a Self-adjoint transformation some ne-glected to say that its eigenvectors were orthonormal (some did notread the question and merely stated that their eigenvalues were realwhich received no marks). Almost all candidates could find the char-acteristic and minimal polynomial and understood their differences.Some seemed to think that if an eigenvalue had multiplicity 2 thenyou could never find 2 o.n. eigenvectors. Sadly for A2 a sizable pro-portion of candidates displayed that they had understood Mods buthad not progressed to master Part A material.

AC1 qu3 The first 2 marks for using the remainder theorem were lost by somecandidates who tried to find an algebraic factorisation for f(x) (orat least show that this failed which wasted a fair amount of theirenergies). Most candidates who progressed to the polynomialX2 − 2008X2 − 2009 noticed that modulo 5 this was f(x) and sousing the homomorphism π you could deduce that this polynomialwas irreducible over Z[X]. Few went on to state Gauss’ Lemma andconclude that this polynomial was irreducible in Q[X].

Analysis

The questions worked well in distinguishing the candidates. The marks wereon average a bit disappointing although question 4 (on open/closed subsetsof R and continuity of C-valued functions) was quite well answered. Question6 on Mobius transforms was less well-answered, with far fewer candidatesattaining over 7/10.

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Differential Equations

Question 7. (Construction of Green’s function).The following were common mistakes:

1. assertion that uniqueness of solution for the boundary - value problemis settled by appeal to Picard’s Theorem;

2. failure to satisfy the boundary condition at the right hand end of theinterval.

3. in evaluating the derivative of the solution, failure to take into accountthat the Green’s function is only piecewise differentiable.

Question 8 (First-order partial differential equation).This material was well-understood, and many candidates correctly identi-fied, and sketched, the region in which the solution is uniquely determined.Question 9 (Plane autonomous system). Candidates frequently overlookedthe fact that critical points must lie in the (x, y)-plane and thus producedspurious points which they attempted to classify. While the phase-planepaths near the origin are approximately elliptical, the axes do not coincidewith the coordinate axes.

AC2

Algebra

The work submitted this year seemed of a lower quality than in recent years,with a distinct lack of basic understanding evident in a number of cases.This of course, may have been accentuated by the fact that candidates are“required” to answer at least one question from each section of this paper.Almost all did, though a number of attempts were cursory in the extreme,and it would have been this last class that may have submitted no algebrain the past.

Question 1

The very first part was “mods”- that a root of the characteristic polynomialis an eigenvalue- but it often produced long answers (including those basedon using the fact that every root of a characteristic polynomial is a root ofthe minimal polynomial). It also gave candidates the opportunity to showthat they didn’t know what “singular” or “nonsingular” meant. Unfortu-nately, far too many candidates assumed that if the multiplicity of a rootof the characteristic polynomial is r then there are r linearly independenteigenvectors. But then, somewhat surprisingly, those candidates did not de-duce in one line that every linear transformation over C is diagonalisable! Ofcourse, if this should have been the case, then the problem, which leads to

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a proof of the Cayley-Hamilton theorem, would have been trivial (as wouldthe whole of algebra...), but this didn’t stop those candidates from gettinginto even deeper water.

As triangular form was set last year and, presumably, every candidatewould have done that question as part of their revision, it is disappointingthat so few tackled this “tweak” successfully. All that is needed to get theform of the matrix is a minor improvement in the inductive “bookkeeping”;some candidates gave a standard proof of upper triangular form while othersdiced with death by pretending they were in an inner product space (where,unfortunately, orthogonal complements are only invariant under the adjointtransformation!). A few used the dual space, but failed to see the delicateaspects of that approach, and became muddled over what should be upperand what lower triangular.

When one came to the “weight space” decomposition, some candidatestried to apply the primary decomposition theorem with the characteristicpolynomial in the place of the minimal polynomial. Unfortunately, this givesrise to a circular argument since it required Cayley-Hamilton theorem (al-beit in the case of a single eigenvalue) for the approach to work even if oneassume that the distinct roots are the same.

Question 2

This question, unwittingly, gave many candidates the opportunity to showlack of basic understanding. Leaving aside those who define

W⊥ = {w ∈ W⊥ : ......}or worse, many seemed to that that the (finite) orthonormal basis given

for W could be extended to an othonormal basis of V - even countable!-by the Gram-Schmidt process. The “projection” part of the question wasdone quite reasonably, modulo error about bases. For the next part, thefew who realised that some geometry was going on fared best, and one evenused that fact that the set {w ∈ W :‖ w ‖= 1} is compact while anotherobserved, correctly, that the entire calculation can be carried out in anyfinite-dimensional space < W, v >, but most seemed to have forgotten themods concepts on inf and sup - in the first case merely showing the relevantinequality, and in the second failing to show that the supremum was actuallyachieved. Sadly, too, there was clear evidence that a very substantial numberof candidates believe that, when you have direct sum decomposition of avector space, then the space is the union of the two subspaces - this cameout from attempts to prove these results for the two cases v ∈ W andv ∈ W⊥”.

Most guessed ”correctly that the result need not be true for W infinitedimensional (though some thought it universally false), but the brief justi-fication was invariably of the type “my proof won’t work” rather than an

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example. A few gave an example for V , but only one candidate offered asubspace W . My concern here is that so few realised that “justify briefly”does require a concrete example (if the answer is “no”), not simply givedetails.

One candidate bravely wrote “in the lectures we were told that [if W isinfinite dimensional, then] W⊥⊥ 6= W ′′ - unfortunately, this is not univer-sally true, for example when W = V !Question 3 This received fewer attempts, and the general standard washigher than for the linear algebra questions. This may not be surprising sincethose who prepared for the ring theory may have been “self-selecting”. Thatsaid, there were some careless errors in computing characteristic polynomialsthat immediately led to wrong answers, while many of those attempting thelast parts of the question seemed ill at ease with the set of ideals in thequotient ring of a polynomial ring.

Analysis

The questions worked well in distinguishing the candidates. Question 5 oncontour integrals was the most popular answered by almost all candidates.Questions 4 (on Laurent expansions) and 6 (on Cauchy’s integral formula)were also answered well, although very few attempted the latter, probablybecause it looked longer and more difficult. In fact the question carefullyguided those candidates who tackled it.

Differential Equations

Question 7 (Picard’s Theorem)The proof of Picard’s theorem, of which a detailed acount was required, wasfrequently done in a highly abbreviated manner and important steps i theargument were omitted. In consequence the average mark was lower thanthe examiner had aimed for. Only a few candidates were able to completethe last part, which involved determining an explicit interval of existence ina special case.Question 8 (Reduction of hyperbolic equation to canonical form)The derivation of canonical form and the solution of an initial-value problemwere well handled for the most part. Few candidates, though, were ableto relate the domain of determinacy to the geometry of the characteristiccurves.Question 9 ( Laplace transform)There were many complete, or almost complete, answers to this question onthe use of the transform to solve an initial - boundary value problem for theheat equation.

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AO1 and AO2

A: Introduction to Fields

The work here, by a relatively small number of candidates, was quite pleas-ing.

The short question was fairly well done by many, though some thoughtthat the splitting field of x6 − 1is Q(

√3, i) [often because x4 + x2 + 1 “is

irreducible”]. But cyclotomic fields had been the subject of an entire lecture.However, one common misunderstanding was that the splitting field is Q(α)where α is a root without showing that this field contains all roots, as wasthe case here.

The long question, although largely covered in lectures, caused somedifficulty and the proofs of things not asked were often proffered as a sub-stitute.

B: Group theory

Given the level of understanding of mods algebra show in AC2, it is perhapsnot surprising that this course proved unpopular. Many made quite a decentattempt at the short question, though there were some howlers, for example,that a finite group is soluble if [the converse of Lagrange’s theorem holds],and that the symmetric group S4 is soluble because it is abelian. Thesolubility of S4 and insolubility of S5 were both carefully shown in lectures,as was the solubility of subgroups of a finite soluble group, though a numberof candidates thought that any subgroup has to sit between two subgroupsof a given composition series. There were only a few attempts at the longquestion that went successfully beyond the initial definitions.

C: Number Theory

(C1) A very large number of candidates omitted important hypotheses inthe theorems they were asked to state. The most common mistake inpart (iii) was forgetting to treat the case in which p divides a. In part(iv), the many candidates tried in vain to use parts (ii) & (iii), whenonly part (i) was needed.

(C2) During marking, it became apparent that the marking allocationsshould be changed. Under the new allocation, there are 5 marks forpart (iv) and 8 marks for part (vi).

Surprisingly many candidates could not accurately state the theoremsand / or give the definitions they were asked to.

In part (vi), many forgot to exclude the possibility of the integer N2+2being even.

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D: Integration

The scripts generally showed a good understanding of the principles ofLebesgue integration. Most candidates understood how to apply the theo-rems (although there were a lot of invalid applications of the MCT in Seriesin D3) and they had good ideas of how to estimate functions in order toestablish integrability (although quite a few believed in D2 that x−n is inte-grable over (0,∞) for n > 1 or gave false inequalities for logarithms in D3).They showed good judgement of the level of detail that was expected insolutions, and there was a decent standard of presentation in general. Var-ied nomenclature and formulation of standard theorems showed that somecandidates had used the Etheridge notes and some had used notes from thelive lectures. Students who used both sources in combination would havelearned the most.

Of the total of 70 marks available, 21 were for reproduction of mate-rial from lectures (including nullity of the Cantor set and integrability ofxα), while the rest were divided fairly equally between calculations and jus-tifications in unseen examples which were not too similar to examples inlectures and problem sheets. Not much guidance was offered about how toproceed with the examples, and some candidates got stuck, but the absenceof guidance produced a great variety of successful methods on some parts ofquestions.

The marks came out with sensible distributions, although D2 and D4exceeded the Examiners guidelines for high marks (received by the setterafter the questions had been submitted to the Examiners) while it was notas easy to gain half the marks on D3 as the Examiners had requested. Outof about 330 marks on individual questions, there were only 13 cases of fullmarks, but marks of 9 on D2 and 24 on D4 were fairly common. D1 hadvery few marks above 8, because few candidates saw how to simplify theproduct in the last part.

There were very few marks above 22 on D3, and there were a significantnumber of low marks from candidates who did part (i) and then bailed out,probably because there were no signposts pointing to solutions of (ii) and(iii). In this respect, D3 may have been a bit too hard for lesser candidatesand the average mark was on the low side. On the other hand, the medianmark was 16 so the question was accessible to average candidates. More-over, the candidates with low marks may not have spent much time on thequestion and they may well have got higher marks on four other questions.

Candidates in the three joint schools did particularly well on D3. On theother hand, a large proportion of the joint school candidates attempting D2believed that x−n is integrable over (0,∞). Maths & Philosophy candidatesstruggled on D4—not surprisingly since they had not encountered doubleintegrals and polar coordinates in their first year.

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E: Topology

The answers to the Topology questions this year display a good under-standing of the theoretical material. Most candidates answered well to thequestions requiring the mere knowledge of the theory. Many provided alsopartial or complete answer to the second parts of the questions (which con-tained entirely new material, for all of the four questions). Question E4 wasthe least popular; this might point out that the candidates are in the badhabit of revising the very last chapter less than the others. The candidatesfrom the Joint School of Mathematics & Philosophy did particularly well.The Mathematics & Computer Science candidates were less numerous butaltogether did good work. Very few candidates in Mathematics & Statisticsattempted the Topology questions.

Detailed comments on the questions

E1 Almost all candidates answered very well question (a). The topol-ogy appearing in question (b) was rather counter-intuitive and unlikeanything that the candidates might have seen before, nevertheless alarge majority of candidates did part (i), many also solved part (ii).Part (iii) was the less straightforward of all, and it was attemptedsuccessfully by fewer candidates.

The most frequent mistake was to consider only countable intersectionsin the list of properties of closed sets. A few candidates wrote the logicquantifiers in the wrong order in the definition of convergence in (a),(iii). Some candidates confused countable with finite.

E2 This question, on compact sets and their properties, was the mostpopular of all, and had a large number of good answers. Most can-didates answered questions (a) and (b), (i), and attempted to proveat least half of the equivalence in question (ii). An important numberof candidates confused smallest element of a set with infimum in (b),(ii).

E3 Almost all the candidates answered well the questions (a) and (b),(i). Some candidates mistook the closed ball for the open ball in (b),(ii). In (b), (iii), when attempting to prove that [γ] implies [α], somecandidates tried to deduce continuity in an arbitrary fixed point x

by adding two copies of the inequalities provided by the continuity inthe basepoint a, ignoring thus the fact that the distance from x toa is a fixed constant while the constant δ in the continuity propertydecreases with ǫ.

The answers to (b), (iv) and (v), displayed a good knowledge of opensets in metric spaces, and of closures of sets. Still, since question

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(v) was the most difficult, there were only few correct and completeanswers to it.

E4 The candidates attempting this question were considerably fewer thanfor the other questions. Nevertheless, half of those attempting it hadvery good marks. The theoretical questions on connected subsets andquotient topology were well answered. Some candidates did not un-derstand that the property of connectedness of a subset must be con-sidered with respect to the induced topology; others only explained,in (a) (iii), what the quotient space is, without describing its topology.In question (c) some candidates described the equivalence classes inC × C instead of in the subspace X.

F: Multivariable Calculus

AO1: This question was done well by most students who attempted it. Still,there were slightly more errors in the computation of extremals than onewould have expected. Several students lost a mark by not arguing carefullythat Jacobi matrix of the function defining the manifold has maximal rank

AO2: This question was attempted by few students only. The first Parta was done well by all of them. In the first part of (b) the offered solutionswere not completely correct as there fine for m = 1, but not for m > 1. Thelast part was not attempted by anybody.

G: Calculus of Variations

G1: The bookwork part, a derivation of the Euler-Lagrange equation withone dependent variable and three independent variables, caused little diffi-culty but quite a few failed to solve the Laplace equation. Still, quite a lotof attempts with about a quarter scoring 9 or 10.

G2: The bookwork part asked just for a description of the method of La-grange multipliers and then for the derivation of the familiar first integral.This was pretty well done. In the problem part, most attempts simplychecked that the displayed formula worked rather than solving the equa-tion. This loses marks because the displayed formula is the general solutionwith some of the boundary conditions imposed. Surprisingly few attemptscould say how λ was determined. Quite a lot of attempts, with about 20

H: Classical Mechanics

The results of the short paper were very disappointing. Several candidatesfailed to understand that the question dealt with nine degrees of freedom,and considered three degrees of freedom instead. Partial marks were given to

20

candidates who handled three degrees of freedom correctly, or approximatelycorrectly.

On the other hand, the results of the long paper were better than inprevious years. The marks were assigned generously at the beginning ofthe question, with 10 points in easy reach. In the subsequent part of thequestion it became clear that the marking scheme requires modification,with 3 points for finding a constant of motion instead of the previous 4, and5 points for the equations of motion and the equilibria (previously 4).

J: Electromagnetism

Question 1Although there were a fair number of reasonably complete answers to

this question, many attempts were disappointingly weak. The first part ofthe question was intended to be straightforward: one applies the fact thatthe vector cross product of two vectors is perpendicular to both vectorsto the Biot-Savart law, the latter formula being stated in the question. Anumber of candidates didn’t seem to know how to argue this, and then didn’tproceed to the main part of the question.

The main part of the question involved an application of the Biot-Savartformula to a square loop of wire. After setting up appropriate coordinates,this involves evaluating a simple cross product, and then a (straight) lineintegral which requires a standard change of variable to perform the integralexplicitly. The “trick” was to notice that the integral consists of four equalcontributions, and most candidates realized this (one or two instead stated,incorrectly, that they cancelled pairwise). Common errors were in calculat-ing the cross product or in evaluating the integral (either not knowing howto, or due to algebraic errors).

Question 2Many attempts at this question, which involved a large portion of book-

work, were largely well-done and complete. All candidates correctly statedMaxwell’s equations, which was very pleasing to see, and (the exceptionbeing an occasional algebraic slip) almost all correctly derived Poynting’sTheorem. A common error was to miss the part of the question that askedcandidates to interpret each term in Poynting’s Theorem. I believe thosethat missed this part simply didn’t read the question carefully; those thatdid answer largely did so accurately (all this was bookwork).

The second part of the question was on plane wave solutions to Maxwell’sequations, which again is essentially bookwork, although the starting pointwas a little different to that in lectures. The most common error in answeringthis part of the question (apart from sign errors in computing cross productsor other algebraic slips) was not to show that all four of Maxwell’s equationswere satisfied. Sometimes candidates would check only two of the four, orcheck that the wave equation was satisfied and nothing else.

21

The third part of the question was an application of the formulae in thefirst part of the question to the second part, and many candidates answeredthis correctly. The errors were algebraic slips.

K: Fluid Dynamics and Waves

K3 59 attempts average 12.95Prove Blasius result, given the complex potential, show given values for

square root (z2 − a2) on a flat plate y = 0,−a <= x <= a, demonstratethat there is no force on body, determine stagnation points and sketch theflow.

Blasius result proven uniformly well, virtually no one could explain thegiven values either side of plate (despite this example being given in lec-tures and as part of a tutorial question, sheet 3 q5), consequently force onbody part, stagnation point determination and flow sketch poor. Despitethe indication in the question that the upper and lower surfaces of the plateare different, there appeared little appreciation of this in almost any solution.

K4 12 attempts, average 11.38One space dimension wave problem, derive linearised version of conser-

vation of mass equation, surface equation (both equations given in question)combine two equations to obtain wave equation, substitute in a given geom-etry and velocity, surface solution form to obtain a second order, constantcoefficient linear differential equation, determine when solutions are wavelike.

Very few attempts. Virtually no satisfactory understanding of applica-tion of conservation of mass (or volume as density constant), a few goodsolutions of last part but others let down by basic calculus or algebra.

M: Probability

M1 This was a reasonably standard question and was generally done well.Most candidates made good attempts at (a) and (b), and many did (c) wellalso; for candidates who did have difficulties, their difficulties were usuallywith (c). In (a) quite a few candidates wanted E(Z|Z < a) to be greaterthan zero, which isn’t necessarily true here.

M2 Unfortunately, this question turned out to be a bit too hard. Iexpected candidates to find (c) quite hard, but I was surprised by the extentof the difficulties candidates had with (b) which was covered in lectures, and(a) which is standard bookwork (and was in lectures).

The bookwork in (a) takes just a few lines and is from one of the firstlectures on Markov chains. The majority of candidates did this well, but notall: about a quarter of attempts got fewer marks in total than were availablein (a).

22

I was surprised by the trouble that (b) caused. There are (at least)two ways to do it, at least one of these was in lectures (either method is afew lines, both are in the notes), and an early example on the first Markovchains problem sheet gave practice in calculating pij(n) for a two-state chain.However most candidates could not do (b). As it had been in lectures, I wasdisappointed by the overall quality of attempts at (b).

Part (c) was intended to be harder. Of those who reached (c)(i), mostdrew a diagram (a good idea): several then realised that conditioning on thefirst step allowed them to use (b). Although (c)(ii) was perhaps too hard,there were several excellent answers.

M4 The proportion of candidates getting high marks on this questionmay be in the region of what the examiners would like to see (or it may be alittle lower), but I suspect the average mark will be a bit low. Parts (a)–(c)were done well, but many candidates had difficulties as soon as they reached(d). Part (d)(i) is a thinly-disguised version of an early problem question,however most candidates who attempted it were not able to do it correctly.The most common mistake was to consider a sum of a fixed number of iidexponentials, when in fact the number of terms in the sum should have ageometric distribution. Although I had expected this mistake to crop up, Ihad not expected to see it as frequently as I did. For candidates who tried(d)(ii) and later parts of the question, there was a good spread of marks anda good number of excellent/very good answers, but many attempts stoppedafter (d)(i).

M5 This question was popular, it produced plenty of high marks, butsome low marks as well. Given the difficulties candidates had with M2 onMarkov chains, I was pleased to see the successes of many candidates on thisquestion. In (a), the greatest difficulty was in solving a recurrence relationin (a)(ii): although solving second-order recurrences is in Mods, plenty ofcandidates seem to have forgotten – even though this example was also inthe Part A notes.

O: Statistics

01

This was a question on distribution transformation and probability plots,with the end part involving applying these ideas to a new distribution. Over-all the question saw a lot of attempts, with a quite wide spread of marks,and some though relatively few perfect answers. The bookwork was gener-ally done well. In the later part of the question, some students mixed uprandom variables and their expectation, or did not allow for the fact thatparameters in the distribution were unknown.02

Here students were tested on information theory, and obtaining approximateconfidence intervals for the negative binomial distribution, using standard

23

approaches from the course. The marks gained were generally solid, withquite a number of perfect answers, though there were some incomplete so-lutions with a steady drop off through the question. The negative binomiallikelihood was generally well handled. One issue noticed with solutions wasa large number of slips with basic algebra (differentiation, expectation) orwith common sense checking (e.g. that information comes out to be pos-itive). Several students got confused with the basics of expectation whencalculating Fisher’s information - writing X instead of E(X), or trying tosubstitute in p for p at the expectation stage. Some marks were also lost forfailing to substitute in the known p for the unknown p in the informationterm to obtain a (theoretically) calculable confidence interval in the lastpart.Paper A02

04

Based on linear models, this long question involved essentially showing thatthe least-squares estimator is the BLUE for the standard linear model. Asthis was new to the students, they were given some direction for how toproceed in part (ii). Overall, this question was answered very well – mostcandidates gaining a beta at least, and many alphas. The first bookworkparts were answered generally nicely, though some candidates had problemswith differentiation of the multivariate function to find the least-squaresestimator or gave insufficient details. Credit was given for the derivationapproach based on vector spaces, rather than via differentiation, if this wasdone carefully. More marks were lost on later parts (i) and (ii). Most candi-dates stated DXθ = θ, but few went on to argue that this implied DX = Ip

because it holds for all θ (rather, candidates simply wrote down DX = Ip).For part (ii), mistakes were made with sign slips, and misunderstanding ofmatrix algebra (e.g. falsely assuming commutability of matrix multiplica-tion). Further, some candidates did not correctly apply the results of (i)(or adapt these to obtain D∗X = 0) to obtain the required result. For thelast part, many candidates lost one or two marks by not concentrating onthe matrix diagonal and instead simply stated “D∗D∗T ≥ 0”, or forgot tomention both mean and variance in arguing for one estimator over the other.05

The question was based on hypothesis testing of categorical data, usingstandard approaches, but with a new distribution, the log-series distribution.This question proved reasonably tough, with fewer attempts than O4, manypartial answers, and relatively few alphas gained though many betas. Ofstudents who attempted both this question and question O4, the markswere often similar, but slightly lower for this question. The first, bookworkparts were well done in general (though with a few errors). One thing thatcame up here was a number of candidates leaving the likelihood ratio teststatistic in terms of theta, rather than the estimator θ. Although candidatesusually suggested replacing θ with θ0 for the second null model, many did

24

not modify the degrees of freedom of the test appropriately. Candidatesstruggled most with the later applied part of the question, sometimes missingthe idea that this involved applying the results of the first part. In particular,many seemed to find it hard to work with the log-series distribution and didnot write down the correct likelihood, leading to not getting the requiredequation for the mle (some credit was still given in this case where analyseswere followed through carefully). The most common mistakes were eitherno inclusion of multiplicities ni of observations in the likelihood (perhapsconfused by the fact that the distribution includes i itself in the expressionfor πi), or including only a single term for one particular πi in the likelihood.The end parts proved easier, and most candidates got at least some marks,though a few did not get the correct estimate for the mle, and many madeerrors in the number of degrees of freedom of the chi-squared test, especiallygetting these the wrong way around between (i) and (ii).

P: Numerical Analysis

It was pleasing to see that, with a very small number of exceptions, thecandidates had excellent understanding of the material in the NumericalAnalysis course and were very well-prepared for the written examination.

P1. The question was concerned with Gershgorin’s Theorem, and its ap-plication to a certain one-parameter family of 3 × 3 matrices. Thequestion was attempted by 56 candidates, and the solutions were gen-erally very good, with roughly half of the candidates handing in perfector almost perfect answers.

P2. The question was concerned with the characterization of best least-squares approximation in terms of orthogonality and the constructionof the best least-squares polynomial approximation to a given contin-uous function. The question was attempted by 50 candidates. Thispart of the syllabus was clearly very well prepared by the candidates,and there were many excellent solutions.

P3. The question was concerned with the definition, existence and unique-ness of the Lagrange interpolation polynomial, the estimation of the in-terpolation error, and the application of the interpolation error boundto a given function. The question was attempted by 36 candidates.There were 16 alpha quality answers, and another 5 high beta qualityanswers.

P4. The question was concerned with Householder matrices, and the re-duction of a symmetric matrix to tridiagonal form using a successionof Householder transformations. There were 29 attempts at the ques-tion; 11 of these were of alpha quality. There were another 5 high betaquality attempts.

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G. Names of members of the Board of Examiners

• Examiners:

Prof M Collins Dr A Curnock (chair)Dr A DayProf R HaydonProf T Korner

• Assessors:

Dr H JohnstonProf C BattyDr C DrutuProf N NiethammerProf P TodProf P ChruscielDr J SparksDr I SobeyDr N LawsDr M WinkelDr S MyersProf E SuliProf B GrifithsDr G Nicholls

Appendix

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Figure 1: Table of quartiles for raw marks of Examination papers

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Figure 2: Comparison of AC1 and AC2 raw marks across all schools

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AC1-Q2

5.290

2.661

5.296

2.627

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AC1-Q3

2.756

2.466

2.808

2.537

3.706

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3.412

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AC1-Q6

3.141

2.804

3.047

2.738

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AC1-Q8

7.165

2.467

7.138

2.444

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AC2-Q2

8.901

4.647

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AC2-Q3

16.214

5.977

10.571

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13.417

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AC2-Q6

12.000

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AC2-Q8

15.252

6.433

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2009 part a reportpublic - mathematical institute part a... · scheme which was strictly adhered...

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