Example Candidate Responses (Standards Booklet) IGCSE... · 2017-01-24 · Example Candidate Responses (Standards Booklet) ... The main aim of this booklet is to exemplify standards

Cambridge Secondary 2

Example Candidate Responses (Standards Booklet)Cambridge IGCSE®

International Mathematics

0607

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© Cambridge International Examinations 2013

Contents

Introduction ...........................................................................................................................2

Assessment at a glance ........................................................................................................3

Paper 1 (Core) .......................................................................................................................4

Paper 2 (Extended) .............................................................................................................33

Paper 3 (Core) ..................................................................................................................... 74

Paper 4 (Extended) ............................................................................................................. 91

Paper 5 (Core) ....................................................................................................................110

Paper 6 (Extended) ........................................................................................................... 121

Introduction

2 Cambridge IGCSE International Mathematics 0607

Introduction

The main aim of this booklet is to exemplify standards for those teaching Cambridge IGCSE International Mathematics (0607), and to show how different levels of candidates’ performance relate to the subject’s curriculum and assessment objectives.

In this booklet a range of candidate responses has been chosen to exemplify grades C and E for Papers 1, 3 and 5, and grades A, C and E for Papers 2, 4 and 6. Each response is accompanied by a brief commentary explaining the strengths and weaknesses of the answers.

For ease of reference the following format for Papers 1, 2, 3 and 4 has been adopted:

Question

Mark scheme

Example candidate

response

Examiner comment

Each question is followed by an extract of the mark scheme used by examiners. This, in turn, is followed by examples of marked candidate responses, each with an examiner comment on performance. Comments are given to indicate where and why marks were awarded, and how additional marks could have been obtained. In this way, it is possible to understand what candidates have done to gain their marks and what they still have to do to improve their grades.

For Papers 5 and 6 the mark scheme for the whole paper is followed by examples of whole candidate scripts/papers.

Mark scheme

Example candidate

response

Examiner comment

Pa st papers, Principal Examiner Reports for Teachers and other teacher support materials are available on our Teacher Support website at http://teachers.cie.org.uk

Assessment at a glance


Assessment at a glance

Candidates may follow either the Core Curriculum or the Extended Curriculum. Candidates should attempt to answer all questions on each paper.

Core curriculum Extended curriculum

Paper 1 45 minutes10–12 short response questions.No calculators are permitted.Designed to assess knowledge and use of basic skills and methods.Any part of the syllabus content may be present in this paper but questions will focus on concepts which can be assessed without access to a calculator.40 marks: 25% of assessment

Paper 2 45 minutes10–12 short response questions.No calculators are permitted.Designed to assess knowledge and use of basic skills and methods.Any part of the syllabus content may be present in this paper but questions will focus on concepts which can be assessed without access to a calculator.40 marks: 20% of assessment

Paper 3 1 hour 45 minutes11–15 medium to extended response questions.A graphics calculator is required. Any area of the syllabus may be assessed.Some of the questions will particularly assess the use of the graphics calculator functions described in the syllabus.96 marks: 60% of assessment

Paper 4 2 hours 15 minutes .11–15 medium to extended response questions.A graphics calculator is required.Any area of the syllabus may be assessed.Some of the questions will particularly assess the use of the graphics calculator functions described in the syllabus.120 marks: 60% of assessment

Paper 5 1 hourOne investigation question.A graphics calculator is required. Any area of the syllabus may be assessed.Candidates are assessed on their ability to investigate and solve a more open-ended problem. Clear communication and full reasoning are especially important and mark schemes reflect this.An extended time allowance is given for this paper to allow students to explore and communicate their ideas fully.

24 marks: 15% of assessment

Paper 6 1 hour 30 minutesOne investigation and one modelling question.A graphics calculator is required. Any area of the syllabus may be assessed.Candidates are assessed on their ability to investigate, model, and solve more open-ended problems. Clear communication and full reasoning are especially important and mark schemes reflect this.An extended time allowance is given for this paper to allow students to explore and communicate their ideas fully.40 marks: 20% of assessment

Teachers are reminded that a full syllabus is available on www.cie.org.uk

Paper 1 (Core)


Paper 1 (Core)

Question 1

Mark scheme

Example candidate response – grade C

Examiner comment – grade C

(a) This part was answered correctly indicating a good understanding of rounding to a given number of decimal places. Candidates often find the concept of rounding to a given number of significant figures rather more difficult.

(b) Here is an example that requires careful consideration as to where the decimal point should be placed. In this case the decimal point was incorrectly inserted after the second significant figure. An appreciation that the correct answer is an approximation to the original number would help prevent this type of error.

(c) Here is an example of the slightly more difficult case of converting a number to standard form for which the power of ten is negative. Some candidates at this level may make an error with this case of standard form but this candidate gave the correct answer.

Paper 1 (Core)


Example candidate response – grade E

Examiner comment – grade E

(a) The candidate showed some knowledge of rounding to a given number of decimal places but gave the answer correct to two decimal places. Candidates should always read the question carefully in order to avoid an error of this type.

(b) Candidates at this level often find rounding to a given number of significant figures more difficult than rounding to a given number of decimal places. The first two significant figures (the hundreds and tens digits) were correct but the candidate did not put a zero in the unit’s column. An appreciation that the correct answer is an approximation to the original number would help prevent this type of error.

(c) Candidates sometimes find converting a number to standard form difficult, particularly if a negative power of 10 is required. This candidate did not appear to understand the concept of standard form.

Paper 1 (Core)


Question 2

Mark scheme



Most candidates at this level find a question on factors of this level of difficulty quite straightforward.

(a) The four factors of 15 have been written down correctly.

(b) The candidate wrote down the factors of 21 which, when used with the answer to part (a), made it straightforward to give the highest common factor.


Paper 1 (Core)



Most candidates find this type of question quite straightforward. (a) The correct factors were given and as it was not necessary to give them in a particular order the mark

was awarded.

(b) The candidate was able to write down the correct answer without writing down the factors of 21.

Question 3

Mark scheme



(a) The candidate was unable to recall the number of lines of symmetry for a regular pentagon and so produced a sketch with the intention of drawing the lines of symmetry. The number of sides is correct and they are approximately equal but the angles are not equal and so the sketch does not approximate to a regular pentagon. The two lines drawn are diagonals rather than lines of symmetry.

(b) The candidate gave the name of a quadrilateral with rotational symmetry of order 2 but as the rectangle also has two lines of symmetry this is incorrect.

Paper 1 (Core)




(a) Some candidates can recall the number of lines of symmetry for common regular polygons and give the answer without working. In this case the candidate was not able to do this and so drew three sketches, which was a sensible strategy. It did appear that the sketch of the regular pentagon was the one that was used. This was a good sketch with both the sides and angles approximately equal and one correct line of symmetry drawn. The other four lines of symmetry were omitted and so the answer was incorrect.

(b) In order to identify the quadrilateral with the required properties the candidate drew a number of sketches which, as in part (a), was a sensible approach. The sketch of a parallelogram did appear to include an attempt at drawing two lines of symmetry but the parallelogram was selected and the mark earned.

Paper 1 (Core)


Question 4

Mark scheme


Paper 1 (Core)



(a) Candidates often find solving an equation such as this quite straightforward and this candidate showed excellent algebraic skills in solving this equation. All the steps were shown to produce an exemplary solution that scored full marks.

(b) The correct notation was used to identify each end of the inequality to earn one mark. The end points were not joined up and so the second mark, allocated for indicating the correct region, was not awarded. If candidates check their response then sometimes errors such as this can be corrected.



(a) Many candidates at this level are able to solve linear equations. In this case there were a number of errors in the working. The brackets were multiplied out incorrectly and the attempt to collect ‘like terms’ included sign errors with both x and –10.

(b) The end points were shown correctly on the diagram but no attempt was made to use the standard notation to distinguish between them. Also the end points were not joined up and therefore neither of the two available marks was awarded. With more practice at this type of question the candidate could have made a better attempt.

Paper 1 (Core)


Question 5

Mark scheme

Paper 1 (Core)




Candidates at this level usually show a good understanding of this method of defining a sequence.

(a) and (b) The candidate substituted the appropriate values to give the correct answers.

(c) It is important for candidates to read the instructions for this type of question very carefully and to appreciate that ‘show your working’ implies that there must be some explanation for each part of the

answer. The candidate did not write down the equation 2n − 3 = 44 but gave the solution 44 + 32

as the

first step. This was just acceptable and earned the first mark. Although the final conclusion is correct, in

order to earn the second mark it was necessary to give an explanation for this. Having written down 472

,

if the candidate had stated, for example, that ‘n must be a whole number’ or ‘it cannot be 23.5’ and then

given the conclusion this would have earned the second mark.

Paper 1 (Core)




This question was difficult for candidates at this level.

(a) The first term was written down correctly but not the second term.

(b) There was no attempt to substitute n = 100 into 2n − 3.

(c) The candidate correctly assumed that the sequence is generated by adding two each time but without a correct term this did not lead to anything useful.

Paper 1 (Core)


Question 6

Mark scheme

Paper 1 (Core)




Candidates are usually able to answer simple questions involving the symbols for the intersection and union of sets.

(a) The region was correctly identified showing an understanding of the ∪ symbol.

(b) The candidate did not appear to understand the use of the complement of P, (P ’), as the region identified is P rather than P ’ ∩ Q.



Some candidates find the use of set notation quite difficult for all but the simplest cases.

(a) The area representing the intersection was shaded, suggesting that this candidate mistook the symbol ∪ for ∩.

(b) The candidate did well to identify the correct area.

Paper 1 (Core)


Question 7

Mark scheme

Paper 1 (Core)




(a) Candidates at this level normally find this type of question straightforward and in this case the candidate carried out the first step by giving 95° + 35° = 125°. The second step in applying ‘the sum of the angles of a triangle equals 180°’ was written down incorrectly as 125° − 180°. However the intention was clear as the final answer was given correctly as 55° and so the mark was awarded.

(b) The candidate was unable to recall the word ‘similar’ and so did not complete the sentence correctly.

(c) The candidate demonstrated a sound knowledge of similar triangles by writing down 5 × 3 = 15, thus identifying the scale factor as 3 and followed this by correctly calculating 3 × 2.5.

Paper 1 (Core)




(a) Candidates at this level should be able to use ‘the sum of the angles of a triangle equals 180°’ to calculate the third angle form two given angles. Here the candidate carried out the first step by giving 95° + 35° = 125°. The second step was written down and 125 subtracted from 180 correctly.

(b) The statements that ‘the one on the left is enlarged’ and that they have the ‘same degrees’ are correct but the mark is not available in this case since the candidate also stated that the triangles are ‘the same’, which is incorrect. Reading through the answer a second time may have helped the candidate to realise that parts of their statement were contradictory.

(c) There is some working shown for this part above the diagram. The method is correct indicating that the candidate has identified the scale factor and used it to give 2.5 × 3 and this was awarded one mark. The candidate made an error in carrying out the multiplication, which is not typical even for this level, and so was not awarded the accuracy mark. If this part had been checked the mistake may have been corrected although, in this case, it is an error in the method for multiplication.

Paper 1 (Core)


Question 8

Mark scheme

Paper 1 (Core)




(a) Candidates generally found part (a) difficult as the equation was given with 2y as the subject. This candidate was not able to make any real progress, such as giving y = 1

2 x + 3

2.

(b) The candidate understood that the gradients of parallel lines are equal, copied down the answer to part (a), and so earned the mark.



Although candidates at this level are sometimes able to find the gradient of a line for which the equation of the line is given in the form ‘y = mx + c’ the fact that this equation had 2y as the subject made it difficult.

(a) A table showing values of y from x = −1 to 3 was given but the candidate was not able to make any further progress.

(b) There was no evidence to suggest that the candidate knew that the gradients of parallel lines are equal.

Paper 1 (Core)


Question 9

Mark scheme

Paper 1 (Core)




Candidates at this level are generally able to answer this type of question well. In this case the candidate wrote down the fraction with the area of the sector as numerator and the total area of the circle as the

denominator. This was then reduced to 15

and by recalling the total number of degrees in a circle, and

correctly finding one fifth of 360°, full marks were scored.

Paper 1 (Core)




This was quite a difficult question but this candidate did well and scored both marks. The diagram appeared to be used to help the candidate work out that there are 5 equal sectors, each with an area of 20 cm2 The candidate was able to recall the total number of degrees in a circle, and correctly dividing 360° by 5, full marks were scored. A check for the division was also carried out by correctly multiplying 72 by 5 and, if time allows, this is well worth doing.

Paper 1 (Core)


Question 10

Mark scheme



(a) Although writing down 3 × 9 = 27 which, if followed by 3 × 3 × 3 = 27, would have been a useful first step the answer of 9 was incorrect.

(b) Candidates at this level can usually factorise an expression with common factors and in this case 3 and y were identified as the factors, the correct expression written down and both marks earned.

Paper 1 (Core)




(a) The candidate gave 9 is the cube root of 27 possibly by incorrectly assuming that the answer follows from 3 × 9 = 27.

(b) Both 3 and y were identified as the factors but when the expression inside the brackets was written down the candidate used 3 instead of 5. The mark scheme required at least a correct partial factorization to earn any marks so no marks were scored. Candidates should always check their work and this is case where the mistake may have been seen and corrected.

Paper 1 (Core)


Question 11

Mark scheme

Paper 1 (Core)




Candidates are usually able to read off the relevant numbers from a Venn diagram.

(a) and (b) The candidate answered both parts correctly.

(c) A fraction was given for the probability as required. The denominator was given correctly as 15, the total number of students, but the candidate gave the numerator as the number of students that only played tennis, rather than the total number that played tennis. This type of error can sometimes be avoided if the question is read carefully.

Paper 1 (Core)




Although this was the penultimate question many candidates scored well.

(a) and (b) The correct numbers were read off from the Venn diagram.

(c) The candidate also identified the relevant numbers and showed a good understanding of simple probability to give the correct answer to score full marks on all three parts.

Paper 1 (Core)


Question 12

Mark scheme

Paper 1 (Core)




Candidates at this level found this question difficult.

(a) The candidate may have had some understanding but used a multiplication sign rather than a plus sign.

(b) The candidate gave y = x2 − 3 rather than y = (x − 3)2 possibly indicating some understanding of transformations.

Paper 1 (Core)




At this level candidates are likely to find it difficult to earn marks.

(a) and (b) The candidate showed that the transformation is a translation and did well to give the correct translations as vectors. However in neither case was the correct equation written down.

Paper 1 (Core)


General comment

Example candidate response – grade C (whole paper)

The candidate showed working where appropriate and answered all the questions.

On this non-calculator paper the candidate showed good computational skills when carrying out calculations. It is not always possible to know if a candidate has checked their work but in this case there are no careless errors but there is the possibility that looking through the work for a second time may have revealed an error such as that in question 4(b).

The candidate showed quite a good understanding of number in questions 1 and 2 but was unable to give the correct cube root in question 10(a). A good understanding of algebra was shown in questions 4, 5 and 10(b) but the concept of transformations within the functions content proved to be too difficult. The use of appropriate mathematical language should be encouraged and in question 7(b) the candidate clearly showed a good understanding of the calculation of lengths of similar figures but was unable to recall the word ‘similar’. The candidate showed some geometrical knowledge when answering questions 3, 7 and 9.

This is an above average script and is one mark above the grade C borderline.

Maximum mark: 40 Script total: 25

Example candidate response – grade E (whole paper)

This is a below average script and is at the grade E level.

The candidate did attempt to show working and attempted all the questions. Candidates should be advised to read each question very carefully before attempting to give a response. It is possible that the error in question 1(a) was caused by a failure to do this. Assuming that there is sufficient time candidates should always check their working carefully and the error in question 10(b) may well have been identified, and corrected, if this had been done.

On this non-calculator paper the candidate showed quite good computational skills, with just one error made when carrying out the multiplication in question 7. There was a mixed response to the number questions with the answers to question 1 indicating a lack of understanding of some topics but question 2 was answered completely correctly. Some knowledge of sets was shown but the candidate was not able to earn any marks on question 4(a), an algebra question requiring the solution to an equation. Many candidates at this level did answer this question correctly. No marks were scored on questions 8 and 12 which were difficult graphical questions for this level. The candidate showed some geometrical knowledge when answering questions 3, 7 and 9.


Paper 2 (Extended)


Paper 2 (Extended)

Question 1

Mark scheme

Example candidate response – grade A

Examiner comment – grade A

(a) The candidate used the most efficient method by writing down 25 3× and went on to give the correct answer.

(b) The correct method was also used and it was unfortunate that an error was made in removing the brackets in the numerator. It is not clear whether the candidate simply wrote down a 2 and a 5 or used 52 but it is an error that could possibly have been rectified if the solution had been checked carefully.

Paper 2 (Extended)




(a) The candidate checked that 75 is not a perfect square and appeared to be aware that it is necessary to write 75 as the product of two factors. However 3 × 25 was not written down, possibly because the candidate did not know that one of the factors must be a perfect square in order to simplify 75.

(b) Some candidates at this level would understand the method of multiplying numerator and denominator by 5 + 3 for this example but in this case the candidate did not.



Candidates at this level usually find it hard to score any marks on a question of this type and level of difficulty.

(a) The answer 3 was given rather than 5 but with no working shown, for example 3 25# , it is not possible to judge if the candidate had any understanding of the method.

(b) The candidate correctly worked out (5 ) (5 )3 3+ + but this is irrelevant to the simplification.

Paper 2 (Extended)


Question 2

Mark scheme



(a) The given terms were used to calculate successive differences and the candidate was then able to write down the next two terms.

(b) The candidate clearly understood that a quadratic in ‘n’ is required and was thus able to write down the correct answer to earn full marks for this question.

Paper 2 (Extended)




(a) Having used the given terms to calculate successive differences the candidate was able to write down the next two terms.

(b) The candidate used the method of differences and correctly identified the second differences as 2 and appeared to understand that the required expression is a quadratic in ‘n’ by writing down n2 − 2 which although incorrect, may have given a clue as to the correct answer. 32 = 9 was also given. However, all this was rejected when the answer of n was written on the answer line.

Paper 2 (Extended)




(a) Candidates at this level will normally be able to identify the pattern for the way in which successive terms are generated for a sequence of this type. Although this candidate did not show the differences between successive terms (these were given in part (b)) the correct values were found.

(b) The candidate did not show anything to indicate that a quadratic expression is required.

Paper 2 (Extended)


Question 3

Mark scheme



Candidates at this level would normally find a linear equation of this type easy to solve and this candidate showed all the relevant steps and gave the correct answer to score full marks.

Paper 2 (Extended)




Candidates usually find a linear equation of this order of difficulty straightforward. This candidate collected the terms in x correctly on the left hand side to earn one mark but made an error when dealing with the numbers by attempting to add 17 to both sides of the equation rather than subtracting 17.



Candidates usually find a linear equation of this order of difficulty straightforward and this candidate gave a good solution showing all the necessary working to earn the two marks.

Paper 2 (Extended)


Question 4

Mark scheme

Paper 2 (Extended)




(a) This was an exemplary solution given to a question that many candidates at this level would find straightforward.

(b) Although candidates sometimes have difficulty with the method, this candidate again produced a good solution but gave 16 + 9 as 27.

Paper 2 (Extended)




The solutions to this question were typical for a candidate at this level.

(a) The working was set out well and the correct answer given.

(b) The candidate appeared to have no knowledge of how to find |q |. Finding the magnitude of a vector is quite routine and something that, with practice, could be improved.

Paper 2 (Extended)




(a) The candidate wrote down the first step correctly but then squared each component of the first vector, rather than multiplying by 2 and attempted to cube each component of the second vector, rather than multiplying by 3. This suggests a lack of practice at this type of question.

(b) This was not attempted.

Paper 2 (Extended)


Question 5

Mark scheme

Paper 2 (Extended)




Most candidates at this level would be able to factorise a quadratic of this difficulty and also to be able to make at least some progress with an inequality such as this.

(a) The candidate was able to write down the correct answer.

(b) The candidate gave a solution that showed all the working and reached the correct answer, so full marks were scored.


Paper 2 (Extended)



Candidates at this level often do very well with algebra questions of this type. Most are able to factorise a simple quadratic expression, although sometimes a sign error is made in examples involving negative signs. Some find the solution of linear inequalities quite difficult, particularly with the inequality sign in the final answer. This candidate produced well presented solutions to both parts (a) and (b) and so scored full marks.



(a) The candidate did not appear to know the method for factorising a quadratic expression.

(b) The first step of the solution was carried out correctly and so the method mark was scored and the next step leading to –2 < –2x was also carried out correctly but two errors were then made. The inequality sign was replaced with an equality sign and 2 given instead of 1.

Paper 2 (Extended)


Question 6

Mark scheme

Paper 2 (Extended)




(a) A straightforward question on sets that was answered correctly.

(b) This is a little more difficult than part (a) but is an example that most candidates will have practised and, again, the correct answer was written down.

Paper 2 (Extended)




Candidates would normally score at least one mark on a relatively straightforward Venn diagram question such as this. This candidate had some knowledge of the notation for the intersection of sets and the complement of a set but was not able to use them correctly in either of the parts (a) or (b).

Paper 2 (Extended)




Some candidates find the use of set notation quite difficult for all but the simplest cases.

(a) The candidate used the intersection sign correctly to give the right answer.

(b) The candidate appeared to make up a sign involving the union sign and the complement sign.

Paper 2 (Extended)


Question 7

Mark scheme

Paper 2 (Extended)




(a) Even candidates at this level often have difficulty with variation questions, particularly with an inverse variation such as this. It is one area of the syllabus that, with practice, candidates could improve on. In this case the candidate assumed that the variation was direct, although not in the form F = kd2, and so did not score any marks.

(b) No marks were available in this case as a ‘follow through’ mark was only available if the answer to part (a) was of a particular form and some attempt had been made to calculate k with k ≠ 1.

Paper 2 (Extended)




All but the most able candidates do find variation questions of this type difficult.

The candidate did not write down anything resembling y x1

2\ and did not use a constant, so the ‘follow

through’ mark in part (b) was not available.

Paper 2 (Extended)




Candidates at this level would not normally score marks on all but a very simple proportionality question and in this case no attempt was made with either part.

Paper 2 (Extended)


Question 8

Mark scheme

Paper 2 (Extended)




(a) This question, involving the combination of logarithmic terms, is quite a difficult example and candidates at this level might be expected to make some progress but not necessarily score full marks. This candidate showed a real understanding when giving 3log2 = log23 and 2log6 = log62 and then evaluating 23 and 62 correctly. The terms log9 and log8 were correctly added to give log72 but an incorrect method was used for the last step. This error may have been avoided if the expression log72 − log36 had been written down as a separate step.

(b) This is a difficult question that tests both negative and fractional indices. The candidate appeared to

understand that 1681

416814

3 3=` `j j but then gave 4

9 either from 1681 24 or from

1681

and did not

attempt to allow for the fact that the power is negative.

Paper 2 (Extended)




(a) This question, involving the combination of logarithmic terms, is quite a difficult example but candidates at this level might be expected to show some understanding of how to combine logs. This candidate did well by giving 3log2 = log23 and 2log6 = log62 and then writing down 9 × 8 = 72 but did not use log72 or log 36 for the next step.

(b) Many candidates do not find it easy to make a correct first step with indices questions of this level of difficulty and this was the case with this candidate, who wrote down nothing of any worth.

Paper 2 (Extended)




(a) Not attempted.

(b) The candidate was able to calculate 81 43

correctly but could make no further progress.

Paper 2 (Extended)


Question 9

Mark scheme

Paper 2 (Extended)




Candidates at this level generally have the necessary algebraic skills to tackle a question of this type and this candidate was able to carry out all the necessary steps to find x in terms of c and d but, having reached

the correct answer x d c2 2= + , then made the common error of assuming that d c d c2 2+ = + .



This is not an easy question for a grade C candidate and this candidate did well to write down the correct

first step. Multiplying out (x + c)(x − c) was partially correct but, rather than adding c 2, the candidate divided

x 2 − xc + cx by c 2. However the final method mark was earned by using x cd2

2

2

= , correctly, to give x in

terms of c and d.

Paper 2 (Extended)




Many candidates at this level would either have omitted this question or not have been able to give the correct first step. This candidate did well to eliminate the fractions as well as multiply out the brackets

correctly thus earning the method mark. Unfortunately +xc – xc was incorrectly simplified to +2xc so any further progress was not possible.

Paper 2 (Extended)


Question 10

Mark scheme

Paper 2 (Extended)




The candidate showed a real understanding of the method needed to solve both parts of this question by allocating letters to the separate regions in the diagram and then using the data given in the table to write down five equations. Following these equations the candidate wrote down c = 2 which only follows if the information given in part (b) is also used. It was therefore possible to work the numerical values of all the regions by substituting back into the equations. This was of course fine for part (b) which earned the two marks but did not follow the instructions for part (a) which required ‘3 expressions in terms of x’. It is important that candidates read questions very carefully in order to avoid this type of error. It is possible that this candidate did do this but chose to ignore them or was unable to give the expressions without using c = 2.

Paper 2 (Extended)




The candidate misinterpreted the data in the table by apparently assuming that the Chemistry set is a subset of the Mathematics set. This fundamental error meant that it was not possible for any marks to be scored but in any case in part (a) there was no attempt to give expressions in terms of x. This is an example of a question that requires the instructions to be read particularly carefully by the candidate before starting the solution.

Paper 2 (Extended)




Nothing of any worth was shown in either part of this question.

Paper 2 (Extended)


Question 11

Mark scheme



On this non-calculator paper it is necessary for candidates to be able to recall the exact values for the basic trigonometric ratios as indicated in the syllabus and many candidates at this level are able to do this. In this case a good start was made by drawing a sketch of the cosine function from 0° to 360° but the candidate either did not know that cos60° = 0.5 or did not appreciate how it could be used with the symmetry of the graph to obtain the two answers.

Paper 2 (Extended)




Candidates at this level are sometimes able to give one correct answer to this type of question so this candidate did particularly well to earn two marks. The method was clearly set out with the negative sign ringed and used, together with the diagram at the top right, to identify the second and third quadrants. The

candidate knew that cos−121

` j = 60° and used this to draw the diagram at the left hand side to help calculate the two answers.


This question was omitted.

Paper 2 (Extended)


Question 12

Mark scheme

Paper 2 (Extended)




(a) Most candidates at this level would appreciate that the graph shown in the diagram is related to the graph of y = sinx. This candidate was able to do this and also understood that as the maximum and minimum values are +3 and −3 respectively then the function is of the form y = 3sin(f(x)). The candidate

also attempted to find the value of f(x) but chose 12

x rather than 2x. Checking a particular point such as

x = 90° would have helped to identify the error.

(b) This part was a bit easier than part (a) and the candidate appeared to have no difficulty in producing a well-drawn correct curve.

Paper 2 (Extended)




This was a difficult question and this response was typical for a grade C candidate.

(a) The candidate did not understand that the graph was that of a trigonometric function and assumed that the equation was a quadratic.

(b) The candidate drew a graph with maximum and minimum values of +2 and −2 respectively, to earn one mark, but then drew the graph with the same period as the given graph, not understanding that it is the same as that for y = sinx.

Paper 2 (Extended)




(a) The candidate did not understand that the equation of the graph is a trigonometric function.

(b) Omitted.

Paper 2 (Extended)


General comment

Example candidate response – grade A (whole paper)

This is an above average script and is at the grade A borderline.

As expected for a candidate at this level all the questions were attempted, working was provided and full marks were scored on several questions.

Rather unusually for a candidate of this ability, two arithmetic errors were made on this non-calculator paper. The candidate did complete the paper and, although it is not known whether there was time for the work to be checked, there is a good chance that, if the working had been looked at again, these errors might have been detected and corrected.

This candidate showed a good knowledge of most sections of the syllabus covered by this paper. As is normal for a grade A candidate the algebra questions were answered well and the questions on surds (radicals), sets, logarithms and functions were all attempted with confidence. In common with many candidates no marks were scored on the questions involving inverse variation and the calculation on powers and roots.


Example candidate response – grade C (whole paper)

This is an average script and is at the grade C borderline.

In order to provide helpful comments on each question this script contains the work of more than one candidate.

Working was shown where appropriate and all the questions were answered.

On this non-calculator paper, as is usual for a grade C candidate, good computational skills were shown when carrying out the numerical calculations. It is always sensible for candidates to check their work and there is the possibility that looking through the work for a second time may have revealed the errors in questions 3 and 9. Apart from question 10 there was no evidence of marks being lost as a result of not reading the question carefully.

A good understanding of algebra was shown and 8 out of the 14 marks were scored on this area of the syllabus. Some knowledge of trigonometry was demonstrated when answering questions 11 and 12 and two marks were earned in question 4, the question on vectors. Other areas of the syllabus such as sets, surds (radicals) and calculation of powers and roots appeared to be less familiar. In order to score as many marks as possible all parts of the syllabus should be covered as there is always the possibility of a straightforward question on one particular topic. This is the case for question 6 in this paper which is a standard question on sets.


Example candidate response – grade E (whole paper)

This is a below average script and is at the grade E level.

Inevitably with a script at this level there are questions which are not attempted or a response given that does not score any marks. Centres with candidates who are working at the grade E level, or below, should consider very carefully whether it is appropriate to enter them for the extended papers. Such a candidate may well earn a higher grade if entered for the core papers and possibly have a better mathematical experience.

Paper 2 (Extended)


This candidate attempted most of the questions and showed working for some of these. Of the six marks scored a method mark was earned for one of the questions on sets and the other five marks were earned on the algebra questions. Little knowledge was shown of some other topics such as surds (radicals), powers and roots, vectors and logarithms. On this non-calculator paper the small number of calculations carried out were done correctly.


Paper 3 (Core)


Paper 3 (Core)

Two questions that typify the type of response expected from a grade C candidate (questions 9 and 12) and a grade E candidate (questions 5 and 7) are given, together with a general comment.

Question 9

Paper 3 (Core)


Mark scheme

Paper 3 (Core)



Paper 3 (Core)



It is worth mentioning that candidates should ensure that their calculators are set to work in degrees and not any other angle measure.

(a) As in this script, many candidates at this level find the reverse bearing hard to calculate with 40° (180 − 140) or 220° (360 − 140) being common answers.

(b) The candidate correctly identified the size of one of the angles in the triangle PSQ. The use of trigonometry was now required, but without showing any further working, and with an incorrect answer, the candidate could not obtain further marks.

(c) (i) The candidate correctly located the position of R although she might have been helped in the next part if she had also drawn the line RS.

(ii) It is not clear how this answer was obtained. Using the symmetry of the completed diagram, it should have been a simple matter to work out the required bearing.

Paper 3 (Core)


Question 12

Paper 3 (Core)


Mark scheme

Paper 3 (Core)



Paper 3 (Core)



(a) The two sketches were correct but should have extended across the complete diagram. If candidates set up the ranges of x and y correctly on their graphics calculators, this will make the sketching a relatively easy process.

(b) Understanding of the term ‘zero’ is essential and it is not clear that the candidate fully understood it. Many candidates at this level who knew how to obtain the answers only gave them to two significant figures instead of three.

(c) The graphics calculator should have been used to obtain the result, which again was required correct to 3 significant figures, as indicated on the front of the question paper. The candidate’s answer of −1 was not sufficiently accurate.

(d) Once again, it was not clear from the candidate’s answer that the terms ‘domain’ and ‘range’ were fully understood.

This grade C script showed typical weaknesses in algebra, co-ordinate geometry and probability. In general there was sufficient working shown and although full marks were obtained to two of the questions, three other questions scored only one mark each. A specific weakness in algebra was the inability of this candidate to add fractions correctly.

Paper 3 (Core)


Question 5

Paper 3 (Core)


Mark scheme

Paper 3 (Core)



Paper 3 (Core)



(a) (i) In common with many candidates at grade E level, the candidate began this algebra question successfully, substituting the value –5 for x in the given expression.

(ii) The correct intention of adding 8 to both sides of the equation was not achieved when the candidate used 8y instead of y + 8. Many candidates at this level used their answer to the previous part to obtain a numeric value for x.

(b) Again the intention was correct when the candidate added the algebraic expressions on the right hand side of the two equations, but the candidate then neglected to do the same to the two ys on the left.

There were a number of possible methods to use in answering this question, the simplest being to equate the two right hand sides, forming a straightforward equation in only one variable.

Paper 3 (Core)


Question 7

Paper 3 (Core)


Mark scheme

Paper 3 (Core)



Paper 3 (Core)


Paper 3 (Core)



(a) The candidate selected the correct formula from page 2 of the question paper, but did not apply it correctly, obtaining the base area by adding two sides of the square instead of multiplying.

(b) (i) This time the correct formula was selected and also correctly used.

(ii) The candidate made no attempt to answer this part of the question. Many candidates at this level found this part difficult, with typical answers being 3 × their previous answer or omitting the base in an otherwise correct attempt.

This grade E script shows a typical lack of understanding of a number of topics, with the questions in statistics and most areas of geometry being either not attempted or else very poorly answered. There was also very little evidence of a graphics calculator being used to any great extent.

In the better answered questions, a sensible amount of working was shown and this helps to highlight the

candidate’s weakness in some areas. For example in question 1 (a)(ii) the correct fraction 2824

is visible, but

has then been inverted, probably because of a reluctance to accept an answer of over 100%.

Paper 4 (Extended)


Paper 4 (Extended)

Questions 4 and 6 have been chosen for comments.

Question 4 covers a range of topics and grade levels. A grade A candidate would be expected to answer all parts whilst a grade C candidate would be expected to answer parts (a) and (d)(i) and (ii) and possibly attempt parts (b), (c) and (d)(iii). A grade E candidate would be expected to attempt parts (a) and (d)(i), (ii).

Question 6 is a question involving the use of a graphics calculator and so is of particular interest in this new syllabus. A grade A candidate would be expected to successfully answer parts (a), (b) and (c) and attempt the part (d) on range and solutions of f(x) = k. A grade C candidate would be expected to sketch the graph successfully and to find a turning point. A grade E candidate should be able to produce a sketch of a graph.

It must be pointed out that a grade E candidate would not be expected to achieve many marks at this level and the core examination is much more appropriate for this level of candidate to have a more positive experience.

Paper 4 (Extended)


Question 4

Paper 4 (Extended)


Paper 4 (Extended)


Mark scheme

Paper 4 (Extended)



Paper 4 (Extended)


Paper 4 (Extended)



This candidate has completed parts (a), (b) and (c) correctly, showing clear working as expected at grade A. In part (d) the line in part (i) is correctly drawn but in part (ii) the candidate attempts to use a right-angled triangle, which does not exist and overlooks the fact that this part only carries one mark. In part (iii), the triangle PSQ has again been used as right-angled when the sine rule should have been used. A grade A candidate would normally be expected to be successful in part (d).

Paper 4 (Extended)



Paper 4 (Extended)


Paper 4 (Extended)



This script shows how a candidate can score a typical grade C score by gaining marks on some high level parts but losing marks on lower level parts of the question.

(a) (i) Answered correctly.

(ii) The candidate has been unable to add times correctly because of treating seconds as hundredths of minutes.

(b) and (c) Trigonometry in a general triangle has been successfully carried out apart from an accuracy mark being lost. This would be expected of a grade A or B candidate.

(d) (i) A simple line has not been correctly drawn and a grade E candidate could be expected to do this.

(ii) Although the diagram is incorrect, the calculation of an angle is correct.

(iii) The incorrect diagram has resulted in an incorrect sine rule statement and no marks are awarded here.

Paper 4 (Extended)



Paper 4 (Extended)


Paper 4 (Extended)



The candidate has used an incorrect answer to part (a)(i) to find a total time and arrival time in part (a)(ii) and has been awarded two marks for a correct follow through. The line in part (d)(i) has been correctly drawn. The trigonometry in parts (b), (c) and (d)(iii) have not been attempted and this shows a typical grade E response.

Overall, this script shows how a candidate may just reach a grade E by attempting and succeeding in a few parts of questions. It also shows how the majority of the questions are above a grade E level and the examination is not a positive experience for this level of candidate and there is a risk that the candidate may end up without a grade.

Paper 4 (Extended)


Question 6

Paper 4 (Extended)


Mark scheme

Paper 4 (Extended)



Paper 4 (Extended)



Question 6 often reflected experience of use of the graphics calculator. This candidate has drawn an excellent sketch of the graph, has found two of the three asymptotes and successfully written down the co-ordinates of the maximum point. The answer given for the horizontal asymptote shows a range of y values not seen on the sketch, indicating a possible misunderstanding of asymptote. The very discriminating part (d) has been incorrectly answered by giving values of x when both parts require values or ranges of values of f(x). This response is quite typical of a grade A candidate.

Paper 4 (Extended)



Paper 4 (Extended)



Question 6 shows a very common answer from grade C candidates – a good sketch but nothing else meeting with success. The attempts made in the other parts are indicative of a candidate who finds the interpretation of a graph difficult or has received insufficient preparation in this type of question.

Paper 5 (Core)


Paper 5 (Core)

Mark scheme

Paper 5 (Core)


Paper 5 (Core)



Paper 5 (Core)


Paper 5 (Core)


Paper 5 (Core)


Paper 5 (Core)



This candidate gives a strong response to this paper and in particular shows the ability to tackle an investigation by exploring possibilities. Communication skills are good and the exploration in question 7 has been showed clearly.

In question 3, in common with many others, the candidate has not understood what is required and has omitted to demonstrate that Pick’s Equation gives the same result as the area worked out using the usual area formulae. In this question and in the subsequent question it is clear that Pick’s Equation itself has been understood and the investigation in question 5 has been tackled well. The abstraction in question 6 caused difficulty and the candidate appears to have confused p for the minimum the number of dots (vertices) with the number of sides. More clarity was required here.

The investigative skills of this candidate are seen most clearly in question 7 where a tabular setting allowed the candidate to find three possibilities. However the values chosen are not ordered and with a more structured exploration the candidate might have found all six pairs.

One mark was lost here because the idea of the previous question (that p > 2) was ignored.

In this paper the questions follow a logical progression and so this candidate should have taken notice of what had been stated previously.

Paper 5 (Core)



Paper 5 (Core)


Paper 5 (Core)


Paper 5 (Core)



This candidate shows what was necessary to gain a grade E. The connection between area and the values of p and i has been found but its significance was not properly understood. Thus no meaningful progress was made with the exploration of different possibilities in question 5. The quadrilateral fits the erroneous values given in part 5(a) but no credit could be given as the area is obviously far more than 4. Reflection on the result would have helped this candidate to realise that such a shape could not have an area of 4.

Question 1 was successfully answered although the calculations of the areas did not seem to be that easy for the candidate. This led to a correct observation in question 2 but it was not formulated as an equation as required: no connection between area A and the values of p and i is seen and this candidate assumed a slightly different question to what was written. In question 3, in common with many others, the candidate has not understood what is required and omitted to demonstrate that Pick’s Equation gives the same result as the area worked out using the usual formulae. The candidate is not so careful about communication: for instance one notices the unusual idea of taking (p+2i−2)/vertices. This idea is considered, discarded, but not erased from the final answer resulting in a confusion of communication. In question 4, as in question 3, the candidate evaluates p+2i−2 but remains unclear about what to do with the answer. One mark was awarded for communication for showing how the answer was found. Little progress was made with the rest of the paper and this candidate reverts to using areas of rectangles and triangles in question 7 although the question cannot be solved in this way and the instruction says to use Pick’s Equation.

Paper 6 (Extended)


Paper 6 (Extended)

Mark scheme

Paper 6 (Extended)


Paper 6 (Extended)


Paper 6 (Extended)


Paper 6 (Extended)


Paper 6 (Extended)



Paper 6 (Extended)


Paper 6 (Extended)


Paper 6 (Extended)


Paper 6 (Extended)


Paper 6 (Extended)


Paper 6 (Extended)


Paper 6 (Extended)



This is at the lower end for a grade A script.

The general standard of algebra is very high throughout, although a little more precision in writing the mathematics would have allowed more marks to be gained. Throughout the whole paper this candidate has shown impressive communication skills and so was awarded the maximum marks for communication.

Part A

The candidate shows good reasoning in the first question and gives a full and correct answer. In question 1(d) and question 3 the candidate lucidly shows that Pick’s Theorem gives the same result as when working with areas calculated in the usual manner. In question 2 the observation that an increase in i increases the area is insufficient and here there was a need to supply a more precise connection between i and A. The answer in part (c) is close – the difference between > and ≥ being overlooked. In question 5(a) the candidate quickly establishes the key equation (2p + i = 5) but then misinterprets this as inverse proportion and so loses the marks for this question. The geometrical interpretation of what was found in part (a) was not seen in part (b) and one feels more perseverance might have yielded a correct diagram for p = 4, i = 3.

Part B

The candidate is comfortable using formulae. However, the explanations in questions 1(a) and 1(d) do not address the facts that allow one to produce the correct formulae in the first place. The candidate successfully recognises the model from the graph and then is able to answer all of questions 2, 3 and 4 correctly. A typical candidate at this level uses the graphics calculator in question 5 to produce a clear graph such as is seen here. A communication mark was gained by this candidate for choosing a sensible scale for the sketch.

In the final question the candidate did not consider the difference in the suitability of the model for x > 2 and x < 2. A closer reading of the domains given might have allowed more success, although this question proved the most difficult for all candidates.

Paper 6 (Extended)



Paper 6 (Extended)


Paper 6 (Extended)


Paper 6 (Extended)


Paper 6 (Extended)


Paper 6 (Extended)


Paper 6 (Extended)


Paper 6 (Extended)



This is an example of a typical grade C script. The candidate has shown good application and ability in investigating all the possibilities and part A in fact shows a very strong set of responses. This may have been at the expense of the modelling question which was answered below the standard expected of a candidate at this level. Indeed part A gained about three times as many marks as part B.

Part A

The algebra is efficiently tackled and one sees that this candidate is even able to go directly to the formula for area. The candidate did not understand what was implied by the word “Show” and assumes that Pick’s Equation gives the correct answer. The explanations regarding Pick’s Equation (question 2) were often too difficult for candidates at this level where more precision was required. The candidate shows good communication in questions 4 and 5 and produces a helpful table to organise findings. Such tables are useful in identifying patterns and so allowing candidates to find all the possibilities. This candidate explores the geometrical possibilities in question 5(b) but has not read the question carefully enough and not all shapes are quadrilaterals as required by the question.

Part B

The candidate shows clear understanding of the formulae presented but fails to explain how these formulae arise. After question 1 there is very little accurate work. Subsequently the work becomes untidy and confused (question 2(b)) and there is a costly careless error in swapping question 4(a)(i) and 4(a)(ii). There was no attempt at the questions on the final page.

This candidate might have benefited from better time management and a more careful reading of some questions.

Paper 6 (Extended)


Example candidate response - grade E

Paper 6 (Extended)


Paper 6 (Extended)


Paper 6 (Extended)


Paper 6 (Extended)


Paper 6 (Extended)


Paper 6 (Extended)


Paper 6 (Extended)



This candidate shows the minimum requirements to gain grade E.

The use of algebra is therefore at a low level. However the idea of investigation has been understood and a sensible method has been employed.

Part A

The candidate knows of the difference method but can make nothing of the calculations that appear. In questions 3 and 4 the candidate is able to demonstrate understanding of how Pick’s Equation is applied. Having been given Pick’s Equation the candidate makes a good effort in identifying all the possibilities in question 5(a) setting out the results logically. In the geometrical follow-up in question 5(b) a more careful reading of the instructions is required as the candidate overlooks the requirement that the shapes are quadrilaterals.

Part B

In question 1 the candidate is able to substitute a variable into the formula given but is not able to explain from where results come. The graph is in essence correct but there has been carelessness is copying from the graphics calculator. Also, in spite of identifying the correct model, too many arithmetic mistakes have been made.

Lack of clear communication is a weakness of this candidate. With less strong algebraic skills and difficulty in explaining algebraic expressions this candidate is typical of a grade E candidate. A lack of care in some questions caused this candidate to lose marks.



























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