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Unformatted text preview: SUBJECT REPORTS NOVEMBER 2002 MATHEMATICS HIGHER LEVEL Overall grade boundaries
Grade: Mark range: 1 016 2 1731 3 3243 4 4455 5 5665 6 6677 7 78100 Introduction
This subject report is written by the principal examiners. Each of the authors provides general comments on performance, taking into account the comments of the assistant examiners and team leaders. This report is the only means of communication between the senior examiners and the classroom teachers and should therefore be read by all teachers of Mathematics HL. The grade award team studied the responses in the G2 forms, the assistant examiners' reports and the grade descriptors (a description of the criteria to be satisfied for each of the individual grade levels) before determining the grade boundaries. Internal assessment the portfolio
Component grade boundaries
Grade: Mark range: 1 04 2 56 3 79 4 1012 5 1314 6 1517 7 1820 Range and Suitability of Work Submitted
Overall, the range of marks submitted covered a wide spectrum, ranging from 8 to 20 marks. However, the assessment standards also varied considerably. The majority of the assignments were taken from the Teacher Support Material (TSM) for Mathematics HL, including some from the earlier November 1998 edition. There was also a commendable selection of activities designed by teachers themselves, more so than was the case a year ago. However, some teacherdesigned tasks were lacking in suitable components to satisfy Criterion E. Furthermore, Criterion B was often assessed with little attention being paid to the expected care and detail of presentation. It appears that some teachers are also submitting activities designed for Mathematical Methods as part of the Mathematics HL portfolio. This may be expedient for teachers who teach a combined Mathematical Methods/Mathematics HL class, but the activities are not at the level of the HL programme and result in a significant loss of marks to the students. Candidate Performance against each Criterion
Candidates generally performed well against Criterion A (Use of notation and terminology), Criterion C (Mathematical content) and Criterion D (Results or conclusions). It appears that some candidates were given insufficient guidance in meeting Criterion B (Communication), as noted above. Criterion E (Making conjectures) was treated inconsistently by teachers and candidates. In several instances, candidates were haphazardly awarded full marks for merely noting a pattern. Many
Group 5 Mathematics Higher Level 1 IBO 2002 SUBJECT REPORTS NOVEMBER 2002 candidates were not required to engage in formulating a conjecture or presenting a deductive generalization with formal arguments. Criterion F (Use of technology) varied considerably. The full capabilities of a GDC were generally not realized in the limited scope of the activities given. Full marks were often given for appropriate but not resourceful use of technology. Recommendations for the Teaching of future Candidates Assignments should be spaced out over the entire time available in the HL programme, that is two years typically. This will help to avoid the problem of having too many assignments based on too narrow a segment of the syllabus over too short a period of time. Plan to incorporate the use of the GDC to analyse statistical data, calculate vector solutions, model graphs with matrices, generate pseudorandom data and determine or simulate probabilities, in addition to general graphing purposes in class and for portfolio assignments. Spreadsheet data can now also be processed on a GDC Some portfolios contained very poorly presented work. Candidates must be directed to acquire some skill in technical writing. Many candidates have merely shown the steps to the solutions of problems that were severely lacking in explanation, annotation or justification. Provide more feedback to candidates on the assignments. Very few assignments contained actual teacher comments for the candidates. Some assignments were entirely devoid of marks or comments. Moderation is extremely difficult when it is not possible to determine the basis on which the achievement levels were awarded by the teacher. Paper 1
Component grade boundaries
Grade: Mark range: 1 018 2 1937 3 3853 4 5467 5 6880 6 8194 7 95120 No of G2s received: 23 Comparison with last year's paper:
Much easier A little easier Similar standard A little more difficult Much more difficult 0 2 11 2 1 Suitability of question paper:
Too easy Level of difficulty Syllabus coverage Clarity of wording Presentation of paper Appropriate Too Difficult 0
Poor 17
Satisfactory 3
Good 0 0 0 15 6 5 7 15 14 Group 5 Mathematics Higher Level 2 IBO 2002 SUBJECT REPORTS NOVEMBER 2002 General Comments
The teacher responses on the G2 form were very favourable. It was felt that the syllabus coverage was good, that opportunity for GDC usage was very good, that the questions were clearly stated and that the paper was of a good standard. There was some comment that the same style of calculator technique was required in several questions and that candidates who are strong in algebra often have no way of showing their skills now with the GDC usage. It was suggested that there should be more of a blend of questions  one or two conceptual questions, but also one or two questions that allow the candidate to demonstrate excellent mathematical skills rather than technological skills. The assistant examiners were also pleased with the paper but found that candidates had difficulty with Question 2 and Question 20. The examiners found that the examination allowed the candidates to demonstrate a wellrounded knowledge of the syllabus. The candidates demonstrated strengths in algebra, differentiation and progressions. However, there was overwhelming evidence that the areas of the syllabus that continue to present problems for the candidates are probability, trigonometry, rates of change, differential equations and transformations. Examiners felt that the use of the GDC was good and that it seems now that its use has become fairly routine. However, candidates need to be aware that when asked for an exact answer they cannot give an approximate answer obtained by using the GDC. Poor notation and terminology is still common in the work of some candidates. It is inappropriate at this level for candidates to give the equation of the asymptotes as " x is 1 and 4" or " x 1 and 4". Performance on Individual Questions
QUESTION 1 Remainder Theorem Answers: a=4 Most candidates who used the remainder theorem did so successfully. Those who used long division or synthetic division usually made algebraic errors.
QUESTION 2 Transformations of graphs Answers: 2 x 3  9 x 2 + 13x  6 Solutions were often disappointing with few good answers. Many candidates were quite unaware how to begin. Others defined g ( x ) = f ( x + 1) + 1 .
QUESTION 3 Binomial expansion Answers: The coefficient of x3 is  7 Most candidates knew what had to be done. Often parentheses were omitted, the minus sign was forgotten, or the entire term, rather than just the coefficient, was given.
QUESTION 4 Graphs, asymptotes Answers: y = 1, x = 4, x = 1 Most candidates identified the asymptotes x = 1 and x = 4 but few gave y = 1 . Group 5 Mathematics Higher Level 3 IBO 2002 SUBJECT REPORTS NOVEMBER 2002
QUESTION 5 Probability Answers: (a) (b) 0.25 0.083 Most candidates solved part (a) correctly but then tried to calculate the answer to part (b) as a product of two probabilities.
QUESTION 6 Logarithms Answers: 1275ln 2 This question was answered well by the majority of candidates. The most common mistake was to assume that the expression was a geometric progression.
QUESTION 7 Composite functions Answers: (a) (b) a = 2, b = 1 range is y 0 Part (a) was reasonably well answered with only a few candidates combining the function the wrong way round. In part (b), the wrong answer y > 0 was often seen. QUESTION 8 Means, variance Answers:
a = 5, b = 11 Most candidates were able to show that a + b = 16 but few were able to use the variance to obtain a second equation. QUESTION 9 Solving inequalities Answers:
2.30 < x < 0 or 1 < x < 1.30 or  1 2 ( 13 + 1 < x < 0 or 1 < x < ) 1 2 ( 13  1 ) Many candidates ignored the fact that multiplying an inequality does not preserve the inequality unless the factor is positive. This mistake led them to plot x 3  4 x + 3 . Candidates who did plot x 2  4 + 3 often missed part of the solution because they had an x inappropriate window. Those who tried to solve the problem algebraically were usually unsuccessful. Group 5 Mathematics Higher Level 4 IBO 2002 SUBJECT REPORTS NOVEMBER 2002
QUESTION 10 Lines of intersection of planes Answers: x = t , y = t + 1, z = t (or equivalent) 0 1 or r = 1 + t 1 or equivalent 0 1 Most candidates knew what to do but algebraic errors were not uncommon. Too many candidates gave an equation of a plane as an answer.
QUESTION 11 Distance, velocity, integration Answers: 6m Most candidates simply integrated without realising that the integrand changed sign and therefore obtained 4 m as their answer.
QUESTION 12 Sine and cosine rule, are of triangles Answers:
Area ABC = 2.98(cm 2 ) (accept 2.97) Most candidates failed to realise that there were two possible triangles and simply chose the acute value of C and therefore calculated the larger area. QUESTION 13 Integration by parts Answers: ( cos  )d = sin + cos  2
2 +c Most candidates realised that integration by parts was needed although not all went about it the right way. The constant of integration was often omitted.
QUESTION 14 Points of inflexion Answers: x = 2 Many candidates solved this question correctly. The most common error was thinking that inflexion points occur where the first and not the second derivative is zero.
QUESTION 15 Calculating median Answers: m= 4 8 ( 42 2 ) Few candidates knew what the median was. Those who did, solved the question well either algebraically or by using their GDC. Group 5 Mathematics Higher Level 5 IBO 2002 SUBJECT REPORTS NOVEMBER 2002
QUESTION 16 Volume, differentation Answers: 1 (cms1 ) (do not accept 0.159) 2 Many candidates were unable to do this question, failing to realise that the chain rule had to be used. Some candidates tried to evaluate usually unsuccessful.
QUESTION 17 Graphs, minimum values Answers: dr dV instead of the much easier , but were dV dr a = 1.33 Few candidates knew how to begin this question. Many of those who did used the fact that the line they were looking for was orthogonal to the curve.
QUESTION 18 Vectors, dot product Answers: a b = 0
2 2 Few candidates gave a correct solution to this question although some simply guessed the correct answer. Others incorrectly started with ( a + b ) = ( a  b ) and expanded this as if a and b were scalars and ended up with the correct answer.
QUESTION 19 Transformations, matrices Answers: reflection in y = x tan 5 12 Many candidates found matrices R and M incorrectly. Those who successfully found them often combined them in the wrong order. Whether they did this correctly or incorrectly, many failed then to recognise the result correctly.
QUESTION 20 Tangents, curves, calculus Answers: y = 2e x Few candidates were able even to show that dy = y and very few of those who did realised dx that they then had to solve a differential equation. Paper 2
Component Grade Boundaries
Grade: Mark range: 1 016 2 1732 3 3341 4 4252 5 5363 6 6474 7 75100 Group 5 Mathematics Higher Level 6 IBO 2002 SUBJECT REPORTS NOVEMBER 2002
No of G2s received: 24 Comparison with last year's paper:
Much easier A little easier Similar standard A little more difficult Much more difficult 0 3 11 1 0 Suitability of question paper:
Too easy Level of difficulty Syllabus coverage Clarity of wording Presentation of paper Appropriate Too Difficult 0
Poor 20
Satisfactory 1
Good 0 0 0 13 8 6 8 13 15 General Comments
This year there was a sense amongst the senior examiners, which was most definitely supported by the comments received from assistant examiners and teachers, that the balance between the option questions of section B was about right. It will always be impossible to write questions of exactly equal difficulty on the current options but the examiners are paying close attention to this issue. All examiners noted that overall there was a general improvement in the way candidates followed instructions with regard to the accuracy of their answers. Examiners are still concerned about the writing of good mathematics in solutions. For example an equation needs a left and right hand side. In this paper too many candidates stated that the equation of 1 1 a line in vector form is just 3 + 2 . 17 5 x 1 1 Candidates need to write this as y = 3 + 2 . z 17 5 The writing of an integral sign without the dx (or dt or whatever variable is needed) is incorrect notation and, in any case, is ambiguous. Again it is necessary to remind teachers that candidates need to be taught certain examination techniques. Candidates should always start each question on a fresh page. They should not start answering parts of Question 2 in the middle of the answer to Question 1. At the end of the examination the examination books or pages can then be organized in the examination order, not necessarily the order in which the questions were answered. If a graph is required for a question then it should be attached in that part of the answer book. If another question needs a graph then use an additional piece of paper and attach it to the appropriate section of examination book. Graphs should always be reasonable in size and legible. Performance on Individual Questions
Section A
Group 5 Mathematics Higher Level 7 IBO 2002 SUBJECT REPORTS NOVEMBER 2002
QUESTION 1 Sequences, proof by induction Answers: (a) u2 = 4 u3 = 8 u4 = 16 (i) (ii) un = 2 n (a)
3 M2 = 2 4 M3 = 3 2 1 3 2 (b) 5 4 M4 = 4 3 (b) n + 1 n Mn = n 1 n This was the question on which nearly all candidates did well. In part (i)(a), most candidates found the first three terms of the given sequence, though some (even good candidates) made minor errors in calculations. In part (b)(i), recognizing the powers of 2 and then forming a conjecture for the nth term was done well. The most common error was conjecturing that un = 2n 1 . Part (b)(ii) was done very well and even candidates who stated wrongly that un = 2n 1 were able to earn follow through marks.
In part (ii)(a), nearly all candidates used a calculator to find the powers of the matrix M. Most candidates made a correct conjecture in part (ii)(b)(i). It is pleasing to report that candidates have made great improvements in writing a proof by mathematical induction. The only part in which some candidates still appear to be weak is in putting together a complete conclusion that links the various parts of the proof into a final statement. QUESTION 2 Complex Numbers
Answers: (a) (i) (ii) z =1 or arg z = 2 3 (c) 3 3 2 + i 2 2 Although this question was not intended to be particularly difficult, candidates certainly found it to be very challenging. Many weaker candidates did not even attempt it. In trying to find the modulus and argument of z in parts (a)(i) and (ii), candidates generally tried to evaluate z in either Cartesian form, or using de Moivre's theorem. There were many errors made in both calculations. Few candidates realized that the modulus could have been stated immediately and those trying to use the polar form often ignored the minus sign in two of the terms. Some candidates who did find z correctly were unable to state the correct argument. These candidates would have benefited from making a sketch to first illustrate the position of z in the complex plane. Group 5 Mathematics Higher Level 8 IBO 2002 SUBJECT REPORTS NOVEMBER 2002 In part (b), showing that z was a cube root of one was impossible for candidates who had an incorrect value of z from part (a). Some candidates found the cube roots of one but did not relate their answers to part (a). Part (c) was poorly done with many calculation errors in the evaluation of the given expression. QUESTION 3 Graphs, asymptotes, calculus
Answers: (b) (i) (ii)
x = 3 x = 4.39 (= e2  3) y = 0.901 (= ln 3  2) (c) (d) x = 1.34 or x = 3.05 (ii)
(iii) Area of A = 3.05 0 Area of A = 10.6 ( 4  (1  x ) )  (ln( x + 3)  2)dx
2 (e) The maximum value is 4.63. This question was well answered by a very large number of candidates. Candidates in general made very good use of their graphical display calculators. In part (a), the sketches were well done but many candidates failed to show the vertical asymptote at x = 3, even though they stated it later in part (b). Some candidates had a point on x = 3. In part (b), most candidates used their calculators well to find the intercepts. Some marks were lost by failure to follow the rules on accuracy of answers. In part (c), there was again good use of the calculator to solve f(x) = g(x). In part (d)(i), the most common error in sketching the required area was to ignore the x 0 condition. In part (d)(ii), there were many correct integrals, the most common error involving how to deal with the region below the x axis. In part (d)(iii), some candidates tried to integrate algebraically, but most used the definite integral capability on their calculator to find the area. Part (e) was the hardest part of the question with only a few correct solutions. It was very common to see candidates simply assume that the maximum vertical distance occurred at the maximum point of the quadratic. QUESTION 4 Vectors, parametric form, calculus
Answers:
(i) (a) (b) (ii) (b) x = + 1, y = 2 + 3, z = 5  17 P(4, 3, 2) y2 ln x + ln 5 = ln 2 + 1 or y = x 5 x  1 x This question covered two separate topics. The vector geometry was done well but solving a differential equation seems to be an area of the programme that is not well understood. In part (i)(a), most candidates found the equation of the line but not necessarily in the required parametric form. In part (i)(b), most candidates did an excellent job using the zero scalar product as the necessary condition for point P. Group 5 Mathematics Higher Level 9 IBO 2002 SUBJECT REPORTS NOVEMBER 2002 In part (ii)(a),(b) many candidates used the given substitution to obtain the differential equation in terms of v. However the left hand side was not always dealt with as well as the right hand side. Many candidates were unable to separate the variables to solve the equation. It was not uncommon to see candidates actually substituting numbers straight into the differential equation leading to meaningless numerical values. QUESTION 5 Probability
Answers:
(a) 6 (= 0.353) 17 5 (= 0.238) 21 0.442 (Accept 0.443) (b) (c) Question 5, along with Question 2, turned out to be the hardest questions on this examination paper. Probability is still an area of the program that appears to give candidates many problems. In part (a)(i), many candidates successfully showed that A and B were not independent but then continued in part (ii) as if the events were independent. In part (b), many candidates considered correctly the various combinations of two electricians and one plumber but forgot that two others had to be chosen from the remaining five. There were many candidates who made no attempt at part (c), although candidates familiar with the normal distribution often gave completely correct solutions. Section B
QUESTION 6 Statistics Answers:
(i) (ii) (iii) 0.1847 (accept 0.1848) [2.703, 2.707] (a) (i) (ii) At the 5% level, we must accept H1 At the 1% level, there is not enough evidence to conclude that the die is not fair (and hence we accept H 0 ) (iv) accept H 0 Part (i) involving the Poisson distribution was successfully completed by most candidates. In part (ii), many candidates did not realize that the tdistribution was required and instead used the normal distribution.. The chisquared test in part (iii) was generally well done but some candidates did not state the hypotheses and explanations of "level of significance" were often very weak. Part (iv) was done very badly with only a few candidates recognizing the contingency table required. Group 5 Mathematics Higher Level 10 IBO 2002 SUBJECT REPORTS NOVEMBER 2002 QUESTION 7 Sets, Relations and Groups
Answers:
(i) (a) + 0 1 2 3 (b) * a b c d a b a d c b a b c d c d c a b d c d b a 0 0 1 2 3 1 1 2 3 0 2 2 3 0 1 3 3 0 1 2 (ii) (b) (ii) {5,10} {1, 4,6,9} {2,3,7,8}
c d a c (iv) (a) a b b d (b) a b c d a b c d ; a b c d b a c d a b c d a b c d a b c d a b ; b c d a c d (c) c d a ; a b d b c d a b c This was the most popular Section B question. Parts (i)(a) and (i)(b) were done very well. In part (ii)(a), there were many excellent solutions proving that R was an equivalence relation. In part (b), the definitions of what is meant by "the equivalence class containing a" were often very poor and few candidates were able to list the equivalence classes. There were very few attempts at part (iii). Confusion between subsets and subgroups seemed to be a problem with some candidates even quoting Lagrange's Theorem. Many candidates scored well in part (iv) although attempts to show in part (c) that the elements form a subgroup was often incomplete. Group 5 Mathematics Higher Level 11 IBO 2002 SUBJECT REPORTS NOVEMBER 2002 QUESTION 8 Discrete Mathematics
Answers:
(i) (b) d = 14 x = 3 and y = 7 (ii) n +1 n +1 3 1+ 3  1 3 12 (or yn = 0.144(2.73n +1  (0.732) n +1 ) ) yn = (( ) ( ) ) (iii) (b) (c) yes yes PTWXYUR S Q (v) (b) There were very few attempts at this question. In part (i), explaining and using Euclid's algorithm for finding the greatest common divisor was well done by many candidates. Candidates found the algebra required to solve the recurrence relation in part (ii) to be a challenge. In part (iii), candidates showed good understanding of graph theory. Proofs offered in part (iv) often lacked sufficient justification. In part (v), descriptions of the process were often weak but there were many completely correct solutions using the depthfirst algorithm for finding a spanning tree. QUESTION 9 Analysis, approximation
Answers:
(i) (ii) (b) (c) (ii) (i) (iii) (iv) 3.14159 k = 1.318311 3.1415926146 < < 3.141637847 = 3.142 This was the second most popular option question. However candidates sometimes struggled through the various parts, not following the way the question was trying to lead them. Many candidates made part (i)(a) much harder than necessary. Candidates need to be made aware that the first part of a question with only a 1 mark allocation is meant to be quick and easy. In part (b)(i), no candidate was able to give a complete justification as to why the method converged. In part (ii) there were many reasonable attempts to apply the NewtonRaphson method. In part (ii)(a), showing that the fourth derivative was bounded was done quite well. Most candidates found part (ii)(b) straightforward. In part (c)(i), although the method was known, many candidates had great difficulty calculating a correct approximation to A. In part (c)(ii), only a few candidates were able to refer to the error term for Simpson's rule.
Group 5 Mathematics Higher Level 12 IBO 2002 SUBJECT REPORTS NOVEMBER 2002 Few candidates had any idea how to proceed in parts (iii) and (iv). In part (iii), the work on the sequence of partial sums was not at all well done. In part (b) some candidates scored marks with alternative proofs with regard to the divergence of the harmonic series. QUESTION 10 Euclidean Geometry, conic sections
Answers:
(i) (a) (b) (iii) (a) (b) parabola m 2  18m  16c + 161 = 0 n= ( a 2  b 2 ) cos a PF1 = a + c cos = a + c cos PF2 = a  c cos = a  c cos (c) c NF1 = (a + c cos ) a c NF2 = (a  c cos ) a Only a very small number of candidates attempted this question. It was either done very well or very poorly. Most candidates recognized the conic in part (i)(a) as a parabola. In part (i)(b), only a few good candidates were able to find the equation of the tangent and then the relationship between m and c. In part (ii), amongst the good candidates, there were some excellent solutions using Ceva's Theorem. Some good answers were seen to part (iii). Some candidates found the equations in parts (b) and (c) rather than the required distances. Group 5 Mathematics Higher Level 13 IBO 2002 ...
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 Winter '10
 Chang
 Math, Calculus

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