GENERAL CERTIFICATE OF EDUCATION TYSTYSGRIF ADDYSG GYFFREDINOL

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1 GENERAL CERTIFICATE OF EDUCATION TYSTYSGRIF ADDYSG GYFFREDINOL EXAMINERS' REPORTS MATHEMATICS AS/Advanced JANUARY 008 Unit Page C1 1 C C3 M1 FP1 S1

2 Statistical Information This booklet contains summary details for each unit: number entered; maximum mark available; mean mark achieved; grade ranges. N.B. These refer to 'raw marks' used in the initial assessment, rather than to the uniform marks reported when results are issued. Annual Statistical Report The annual Statistical Report (issued in the second half of the Autumn Term) gives overall outcomes of all examinations administered by WJEC. 1

3 MATHEMATICS General Certificate of Education January 008 Advanced Subsidiary/Advanced Principal Examiner: Unit Statistics The following statistics include all candidates entered for the unit, whether or not they 'cashed in' for an award. The attention of centres is drawn to the fact that the statistics listed should be viewed strictly within the context of this unit and that differences will undoubtedly occur between one year and the next and also between subjects in the same year. Unit Entry Max Mark Mean Mark Grade Ranges A B C D E N.B. The marks given above are raw marks and not uniform marks.

4 Principal Examiner s Report Mathematics C1 January 008 General Comments Most candidates were able to pick up marks on this paper. There did, however, seem to be fewer high marks this time and this may be because candidates found some types of questions in particular questions 4(b)(ii), 7 and 10(c), less accessible than has been the case in recent papers. Individual questions 1. Generally well answered although some candidates still use incorrect formulae for both the mid-point and length of a line.. In part (a), most candidates were able to express 0 as 5 but the other two expressions caused more difficulty. Part (b) seemed to cause a few more problems than is usually the case. 3. Very well answered. 4. Part (a) and, to a lesser extent, part (b)(i) seemed to cause few problems. However, in part (b)(ii), only a minority of candidates realised that the required value of x was 0 1 and only a handful were able to get the final answer of In part (a), most candidates knew the condition for a quadratic equation to have two distinct real roots but not all were able to find the correct range of values for k. Part (b) was generally well answered. 6. In part (a), there seemed to be better and more consistent use of notation than is sometimes the case in this kind of question. In part (b), most candidates showed that they were able to differentiate powers of x correctly. 7. Although almost all candidates were able to show that p = 0 9, not all went on to verify that the value 4 was correct. Only a minority were able to use the rearranged expression to solve the quadratic equation. 8. Most candidates did well on this question although some thought that x + was a factor of the given cubic expression. 9. Candidates found part (a) more difficult than part (b), which was generally well answered. 10. Parts (a) and (b) were generally well answered, as is usually the case for this type of question. However, in part (c), many candidates tried to find the required range of values for k by using a discriminant type of argument for the cubic equation rather than using their sketch for C. 3

5 MATHEMATICS General Certificate of Education January 008 Advanced Subsidiary/Advanced Principal Examiner: Unit Statistics The following statistics include all candidates entered for the unit, whether or not they 'cashed in' for an award. The attention of centres is drawn to the fact that the statistics listed should be viewed strictly within the context of this unit and that differences will undoubtedly occur between one year and the next and also between subjects in the same year. Unit Entry Max Mark Mean Mark Grade Ranges A B C D E N.B. The marks given above are raw marks and not uniform marks. 4

6 Principal Examiner s Report Mathematics C January 008 General Comments Candidates, in general, seemed to perform well on this paper and most of the paper seemed to be accessible. The questions which caused most trouble were, somewhat surprisingly, finding the sum of the arithmetic series in 3(b) and calculating the area of the triangle as a surd in 6(b). In 8(c), not all candidates knew how to go about showing that the given circle and line did not intersect. Individual questions 1. Most candidates were able to get full marks for this question.. Part (a) was well answered. In part (b), many candidates who had shown that 3x + 15 = 30, 150, 390, 510, then made algebraic errors when trying to calculate the corresponding values for x. Others started by writing down sin (3x + 15 ) = sin 3x + sin Parts (a) and (c) were well answered but part (b) was extremely disappointing. Some candidates tried to write down the n th term of the series while others, having n obtained an expression for the sum of n terms, then gave their final answer as (n) 4. Generally well answered, in particular the final 4 marks. 5. Many earned full marks on part (a) but there are still some candidates who do not know the correct form of the cosine rule. In part (b), hardly anybody was able to calculate sin BÂC as a surd and then proceed to express the area in the given form. 6. Most candidates found this a very accessible question. The manipulation of logarithms in part (b)(ii) was particularly good. 7. As usual, both parts of this question were generally well answered. 8. Part (a) was well answered, as is usually the case, but in part (b), some candidates did not realise that the tangent was perpendicular to the radius. In part (c), most of the candidates who tried to solve the equations of the line and circle simultaneously were able to derive a correct quadratic in x which they realised they could not factorise. Many, however, thought that this was a sufficient reason to ensure that the circle and line did not intersect. 9. This was quite a straightforward question on which many candidates obtained full marks. Some candidates, however, made heavy weather of solving their simultaneous equations. 5

7 MATHEMATICS General Certificate of Education January 008 Advanced Subsidiary/Advanced Principal Examiner: Unit Statistics The following statistics include all candidates entered for the unit, whether or not they 'cashed in' for an award. The attention of centres is drawn to the fact that the statistics listed should be viewed strictly within the context of this unit and that differences will undoubtedly occur between one year and the next and also between subjects in the same year. Unit Entry Max Mark Mean Mark Grade Ranges A B C D E N.B. The marks given above are raw marks and not uniform marks. 6

8 Report on Mathematics Paper C3 Examinations By R.H. Thomas, Principal Examiner Whilst the overall performance of candidate was similar to those of previous years, there was a market difference in the way candidates responded to parts of the paper. Thus, whilst candidates demonstrated an improved performance on question 1 to 8 (with the exception of 6(a)), this was offset by a generally weaker performance in questions 8, 9 and 10. Some deficiencies in algebraic skills were again evidence, particularly question (b) and 10. The questions are considered in more detail below. Q.1. A favourite question, evoking a good response from most candidates. It should be noted that all working should be shown. Q. (a) As in previous years, candidates were aware of what was required. However as before, many candidates were unable to carry out two correct computations to demonstrate the counter-example. (b) Well answered by many candidates. The following slips/errors occurred:- (i) the omission of brackets in 1 Sec θ tan θ as Sin θ and (iii) the writing of Sec 1 θ as Sin θ Q.3.(a), (b) Well attempted by most candidates. Q.4. Q.5. These were many correct complete answers. Some candidates, albeit relatively few, did not justify their claim that there was a root between 8 and 9: a statement that there is a change of sign between the two expression values (or equivalent) is required. The differentiation was generally well carried out. Marks were mainly lost due to the lack of simplification of answers in (a) and (c). Q.6. (a) Part (ii) posed difficulties for many candidates. These appeared to be a general lack of understanding of 1 1 (i.e. modulus) when applied to other than linear expressions. Thus, relatively few candidates were able to provide a reasonable sketch of y [ l n x] ever when y l n x was sketched correctly. (b) Whilst 3x 4 was considered correctly, there was difficulty in writing down 3x 4. It should be noted that the last mark was only awarded for a clear statement that x satisfies the simultaneous candidates x and x. 3 7

9 Q.7. Candidates encountered difficulty in providing correct answers to the integrals. Typically, it was generally realised that x 3 dx k (x 3) ( C), the then un- 1 simplified value of k ( i. e. 3 not being given correctly. 3 Q.8. (a) There were many correct answers. (b) This proved difficult for most candidates. There was a general realisation that the presence of +1 indicated a y translation and the point (0,0) was transformed correctly. However the x-scaling was not well handled; even when it was realised that an x scaling as involved a magnification of (rather than a compression of ½) was the popular interpretation. Q.9. The writing of correct un-simplified correct expressions for fg(x) and gf (x) posed no difficulty for all but the weakest candidates. However, whilst correctly as 4 x, there was a general failure to simplify e 4 ln x. Q.10. (a) Many candidates were unable to write down the correct range of f. l ne 4 x was simplified (b) The expression for f 1 ( x ) was correctly derived by the majority of candidates. Full marks were obtained by most candidates as a result of the follow through of the result in (a) being allowed. (c) 1 3 The solution of was not often correctly obtained because of the x x occurrence of errors during the clearing of the denominator when the equation was solved, candidates showed good understanding that the x- values obtained were required to be part of the domain. A substantial number of candidates attempted to proceed as follows:- L.H.S of equation > 0. also, R.H.S of equation < 0, since x>0 it is impossible to find a value of x satisfying the equation. The attempts, whilst being credit worthy, were not completed correct in most cases. 8

10 Mathematics A/AS Level - Jan 008 Paper FP1 General Comments The standard of the scripts was generally good with some excellent candidates but also some who were clearly not suitably prepared for an examination at this level. Solutions to Ques 4 and Ques 7 were often disappointing. It was evident that some candidates were unfamiliar with partial fractions and solutions to problems on induction continue to be poorly presented in general. Comments on Individual Questions Q1 Most candidates knew the method of reduction to echelon form but arithmetic errors were not uncommon. Q Most candidates were able to show that 5. While many candidates realised that this indicated the presence of at least one complex root, some failed to explain why exactly one root must be real. In (b), some candidates gave the third root, incorrectly, as (x + 1). Q3 Q4 It was pleasing to note that most candidates inverted the matrix correctly, without the presence of arithmetic errors noted in Ques 1. Some candidates appeared to be unfamiliar with the term adjugate matrix, giving the cofactor matrix as their answer to (b)(i). It was evident from the scripts that some candidates were unfamiliar with partial fractions which meant that the whole of this question was inaccessible to them. Even candidates who found the partial fractions correctly sometimes went on to sum the terms in the denominator so that expressions involving n( n 1) in the denominator were seen. Q5 Most candidates derived the matrix representing T correctly. However, the determination of the fixed points caused problems for many who failed to realise that a correct solution should include a discussion of the fact that the two equations obtained by putting TX = X are consistent. This fact ensures that the fixed points exist and lie on a line. Q6 Both parts of this question were reasonably well answered with arithmetic errors being the main source of error. Q7 Solutions to questions on induction continue to be generally poor. It was often unclear what was being assumed and what was proved. The expected layout is along the following lines: Proof for n = 1 Assume true for n = k, ie. Consider n = k + 1. Then appropriate algebra. Hence true for n = k true for n = k + 1 and since true for n = 1 the result is proved by induction. A perfect solution of this form was given by only a minority of candidates. 9

11 Q8 Q9 The most common error here was to leave the i in the expressions for the modulus. The effect of this was that the value of the radius was sometimes given as a multiple of i. It is somewhat disturbing to note that this did not appear to worry the candidates. Most candidates managed to take logs successfully at the beginning of the question and many then continued to obtain a correct expression for f (x). Most candidates then obtained, correctly, the x-coordinate of the stationary point but many candidates, however, were unable to classify it as a maximum or minimum, sometimes through an unwise choice of method. Although use of the second derivative can be regarded as the mathematical way to classify stationary points, other methods are often much simpler. In this case, it is evident that f (x) is positive to the left and negative to the right of the stationary point which indicates a maximum. 10

12 Mathematics M1 (Jan 008) Examiner's Report General comments This paper is of comparable standard and length to previous papers on this syllabus. Candidates found all questions assessable and some which are standard fare at this level; the only challenging question being 7(b). There were very many excellent scripts gaining over 70 marks and only a handful of candidates in single figures. Comments on individual question Q1. Most candidates found this question easy and gained high marks. Some very competent candidates lost the last mark of the question in part (e) by failing to specify the units of the axes. This was an excellent start to the paper. Q. This question was generally well done though surprisingly a significant minority of candidates were unable to start the question. Q3. In calculating the normal reaction of the plane on the sledge and the component of weight down the plane, there were numerous sin/cos errors. In using NL down the slope, many candidates omitted either the component of weight or friction. Most candidates thought that the friction acted down the plane, which was obviously incorrect, as this was the direction of motion. Q4. In part (a), almost all candidates used NL with by far the most common errors being sign errors. However, this question was generally well done. In part (b), many candidates brought the weight of the lift into the equation, failing to isolate the forces acting on the man. Again, sign errors were common. Q5. Part (a) was generally well done. The usual mistakes with signs or mixing up of the speeds were evident. Many candidates misquoted the restitution equation. Candidates who had incorrect answers in part (a) often obtained values for the coefficient of restitution which were either negative or greater than 1 without realising that these were obviously incorrect. In part (c), many candidates gave incorrect units for the impulse. Q6. Candidates correctly attempted to apply NLto each particle. The equations that apply to particle B were standard and were mostly correct. For particle A, the component of weight down the slope assisted motion and needed to be added to the tension rather than subtracted. Only a handful of candidates did this correctly. Q7. As usual, this question was not well done with the usual plethora of numerical errors in calculating perpendicular distances. There were also many sign errors in the moment equations and a few candidates left out g, the acceleration due to gravity in an erratic manner. Part (b) proved challenging and only a few candidates realised that when equilibrium was on the point of being lost, the reaction at pivot Q is zero. Many candidates inappropriately tried to use the values of the reactions found in part (a) in this part of the question. 11

13 Q8. Most candidates resolved in two perpendicular directions, usually parallel and perpendicular to the force of magnitude18 N, with varying degrees of success in obtaining the two perpendicular components of the resultant. A substantial number of candidates were not able to combine these components to calculate the magnitude and direction of the resultant. Q9. The main error in this question is in finding the centre of mass of the triangular lamina ABC and relating it to the origin D. Otherwise the question is well done. 1

14 Mathematics A/AS Level - Jan 008 Paper S1 General Comments The standard of the candidature was generally satisfactory. Comments on Individual Questions Q1 Most candidates were able to solve (a) correctly. In part (b), although most candidates investigated independence by comparing P( A B) with P( A) P( B), some chose the longer method of trying to compare P(A) with P( A B) which sometimes led to arithmetic errors. The most common errors in (c) were either to assume that P( A B ) 1 P( A B) or that P( A B ) P( A B ) P( B ) P( A) P( B ) P( B ). Q Part (a) was solved correctly by many candidates. Part (b), however, caused problems for some candidates and a variety of incorrect methods were seen. Q3 Q4 Most candidates solved (a)(i) correctly but in (a)(ii), some candidates evaluated the probabilities of 3 and 4 errors separately but then failed to add them. Candidates solved (b)(i) either by finding the probability of no errors on one page and raising this to the fourth power or by noting that the number of errors on 4 pages has a Poisson distribution with mean 3.8 although this latter result is an S topic, it is of course an acceptable method. A common error in (b)(ii) was to evaluate the required probability by wrongly assuming that there is just one error on the third page as opposed to at least one error. Parts (a) and (b) were well done in general but (c) caused problems for many candidates with some stopping after showing that X = 4, in the apparent belief that this was all that was required. Q5 This question was well done by most candidates although some translated 0.5% incorrectly as As noted in previous reports, it is advisable to draw an appropriate tree diagram as opposed to a purely algebraic solution. Q6 Q7 Many candidates were unable to give a correct solution to (a) with the following incorrect solutions seen not infrequently, [0,1], [0,0.3] and {0,0.1,0.,0.3}. In (b)(i), candidates who merely verified that 0. were given only partial credit. While problems such as (b)(iii) have caused difficulties in the past, it was pleasing this time to see most candidates solving this one correctly. Part (a) was well solved in general, the most common error being a misinterpretation of the term more than. Candidates who used a Poisson approximation in (a) were given no credit. Again in (b), the most common error was a misinterpretation of the term less than. 13

15 Q8 Solutions to this question were often disappointing. In (a), some candidates who correctly wrote the mean as x(4 x) dx went on to integrate this as 1 x (4x x ). In (b), it was pleasing to note that attempts to find F(x) from f(x) were better than in recent examinations. In (c), many candidates evaluated P(X > 1.) as F(1.) instead of 1 F(1.). In (d), most candidates showed that the median m satisfies the quadratic equation m 8m 7 0. Attempts at solving this equation, however, were often extremely disappointing with the following incorrect solution not uncommon: m ( m 4) 7 m 7 or m 4 7 etc It continues to be a curious fact that some candidates appear not to take their pure mathematical knowledge into the examination room when sitting a paper in statistics. GCE Examiners Report Mathematics (January 008)/LG 14

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