9709 · June 2024
Mathematics
Some very good responses were seen but the paper proved very challenging for many candidates. In AS and A Level Mathematics papers the knowledge and use of basic algebraic and trigonometric methods from IGCSE or O Level is expected, as stated in the syllabus.
Source: Cambridge International
Cohort performance
Session statistics from official examination reports
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Key examiner messages
Top priorities from the principal examiner before you revise
The question paper contains a statement in the rubric on the front cover that ‘no marks will be given for unsupported answers from a calculator.’ This means that clear working must be shown to justify solutions, particularly in syllabus items such as quadratic equations and trigonometric equations.
In the case of quadratic equations, for example, it would be necessary to show factorisation, use of the quadratic formula or completing the square as stated in the syllabus.
Using calculators to solve equations and writing down only the solution is not sufficient for certain marks to be awarded.
It is also insufficient to quote only the formula: candidates need to show values substituted into it.
When factorising, candidates should ensure that the factors always expand to give the coefficients of the quadratic equation.
Candidates should take care to read each question carefully and to note the number of marks attached to each question.
Candidates would benefit from spending time to ensure they understand the terminology required in responses and the structure of examination papers at Advanced Level.
For example, the correct terminology for transformation questions is ‘translation’, followed by a column vector, or ‘stretch’, with the word ‘factor’ and both its size and direction clearly indicated.
Question difficulty map
How candidates performed on each question in this series
9709/11
Pure Mathematics 1 (11)
9709/12
Pure Mathematics 1 (12)
9709/13
Pure Mathematics 1 (13)
9709/21
Pure Mathematics 2 (21)
9709/22
Pure Mathematics 2 (22)
9709/23
Pure Mathematics 2 (23)
9709/31
Pure Mathematics 3 (31)
Assessment objectives
Skill and AO weighting from official examiner commentary
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Method marks watchlist
Where working, steps, or method marks were commonly lost
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Recurring mistakes across years
Themes examiners flag in multiple recent sessions for this subject
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Question choice intelligence
Mean scores and popularity for optional questions (HKDSE electives)
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Level exemplars
What candidate scripts at each grade level looked like
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Grade & admission context
How marks relate to grade thresholds and entry standards
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Deep insights
What top candidates did
Techniques and approaches examiners rewarded in this series
Cambridge International Advanced Subsidiary and Advanced Level 9709 Mathematics June 2024…
Cambridge International Advanced Subsidiary and Advanced Level 9709 Mathematics June 2024 Principal Examiner Report for Teachers © 2024 MATHEMATICS Paper 9709/11 Pure Mathematics 1 (11) Key messages The question paper contains a statement in the rubric on the front cover that ‘no marks will be given for unsupported answers from a calculator.’ This means that clear working must be shown to justify solutions, particularly in syllabus items such as quadratic equations and trigonometric equations. In the case of quadratic equations, for example, it would be necessary to show factorisation, use of the quadratic formula or completing the square as stated in the syllabus. Using calculators to solve equations and writing down only the solution is not sufficient for certain marks to be awarded. It is also insufficient to quote only the formula: candidates need to show values substituted into it. When factorising, candidates should ensure that the factors always expand to give the coefficients of the quadratic equation. Candidates should take care to read each question carefully and to note the number of marks attached to each question. General comments Some very good responses were seen but the paper proved very challenging for many candidates. In AS and A Level Mathematics papers the knowledge and use of basic algebraic and trigonometric methods from IGCSE or O Level is expected, as stated in the syllabus. Comments on specific questions Question 1 (a) A well answered question with many candidates getting full marks. Successful candidates factorised the whole expression or factorised the first two terms before completing the square. Candidates should check that their final expression is equivalent to the original expression. (b) A generally well answered question with many achieving full marks. Most candidates did not use their answer from part (a). The most common method where a majority of candidates gained full marks was one of substitution for x2 before factorising. Of those who did not get full marks the main reasons were not showing a factor of 3 before factorising or not showing their substitution. Question 2 (a) Successful students scored all 4 marks by correctly stating the transformations, beginning with a stretch followed by a translation. Candidates who started with a translation invariably used the vector− 0 2 rather than− 0 2 3 . Where marks were lost it was due to a failure to fully describe the transformation e.g. translation down 2, rather than translation down 2 vertically. (b) Most candidates gave the answer in the form f (x) = a sin x + b and many of these gave the correct answer. Question 3
Cambridge International Advanced Subsidiary and Advanced Level 9709 Mathematics June 2024…
Cambridge International Advanced Subsidiary and Advanced Level 9709 Mathematics June 2024 Principal Examiner Report for Teachers © 2024 Question 8 (a) Successful candidates gained full marks on this question whilst many gained some marks. Some candidates set up two simultaneous equations in d and p, solved them to find d and p and then found the 10th term. Other successful candidates set up one equation in p, solved it to find p and then found the 10th term. Some candidates didn’t gain credit for their answers because of sign errors, poor algebra, confusing p and d or using sum of the series formula instead of the nth term formula. (b) Overall, candidates were more successful in solving part (b) than part (a) with some candidates gaining full marks. Candidates were familiar with setting up a quadratic equation from a geometric sequence and many then solved the equation to find q or r and hence the sum to infinity. Some candidates did not show a complete method for solving the quadratic and were unable to gain full marks. Question 9 Many candidates were able to integrate, and more successful candidates stated that the volume of revolution = π − 2 3 1 d (5 4) x x and integrated appropriately. Other candidates found the area under the curve rather than the volume of revolution and they are to be reminded to read each question carefully to gain maximum credit. Few candidates found the volume of the cylinder but those that did used integration or the formula for the volume of a cylinder. Question 10 Successful candidates used substitution to find (x – 3)2 + (mx – 9)2 = 18, simplified this to (m2 + 1)x2 – (6 + 18m)x + 72 = 0 and then used b2 – 4ac = 0 to set up a quadratic in m; m2 + 6m – 7 = 0. Candidates who reached this point were generally then able to find the two points correctly. Some candidates used the substitution y = mx + c and were unable to make any significant progress. A few candidates mistakenly used m =1 9 . Question 11 (a) Many candidates differentiated successfully and solved –4 12 x +2 3 x = 0 to find x = 2. A few candidates stated an incorrect solution of x = 0. A fully correct solution was found by a small number of candidates using a variety of notations. (b) Candidates that were successful found the equation of the tangent and normal and equated them to find the point of intersection. The area of the triangle was found by a variety of methods including integration and Pythagoras with the area of a triangle. However, most candidates were unable to make much progress.
Cambridge International Advanced Subsidiary and Advanced Level 9709 Mathematics June 2024…
Cambridge International Advanced Subsidiary and Advanced Level 9709 Mathematics June 2024 Principal Examiner Report for Teachers © 2024 MATHEMATICS Paper 9709/12 Pure Mathematics 1 (12) Key messages Candidates would benefit from spending time to ensure they understand the terminology required in responses and the structure of examination papers at Advanced Level. For example, the correct terminology for transformation questions is ‘translation’, followed by a column vector, or ‘stretch’, with the word ‘factor’ and both its size and direction clearly indicated. If information is given in the introduction to the question, then it is true and relevant for the whole question. However, if information is given in part (a), then it is only true and relevant for that part of the question. The phrase ‘it is given instead’ is often used to emphasise this in a subsequent part of a question. Many candidates failed to appreciate this distinction in Question 8. Where exact answers are requested, such as in Question 5(a), then a rounded decimal is not acceptable. Exact answers should not contain fractions divided by fractions but should be simplified. Premature rounding often means that the final answer is inaccurate. If a final answer is requested to three significant figures, then intermediate working should be carried out to at least four significant figures. Care needs to be taken to ensure that the whole of a given function is used and not just part of it. This was a particular issue in both Questions 9 and 10. General comments The paper was generally found to be accessible for candidates and many excellent scripts were seen. Candidates seemed to have sufficient time to finish the paper. Presentation of work was mostly good, although some answers still seem to be written in pencil and then overwritten with ink. This practice produces a very unclear image when the script is scanned and makes it difficult to mark. Consequently, appropriate marks may not be awarded. Centres should strongly advise candidates not to do this. Comments on specific questions Question 1 This question was a good start to the paper for most candidates. Many were able to obtain the required coefficients, form a correct equation and solve it. Common errors included obtaining80a instead of2 80a and multiplying the wrong coefficient by 12. Question 2 Many fully correct descriptions were given, and full marks were obtained for answers as succinct as: ‘A stretch of factor 4 in the y-direction followed by a translation of − 3 .' 8 It is very important that the correct terminology is used and that the required order is clear. Question 3 Many fully complete answers were seen for part (a) although it is important that ‘= 0’ is retained from the question and not simply inserted at the end of the answer. In part (b), many candidates were unable to gain a mark as they used a calculator to solve the quadratic equation rather than factorisation or another valid
Cambridge International Advanced Subsidiary and Advanced Level 9709 Mathematics June 2024…
Cambridge International Advanced Subsidiary and Advanced Level 9709 Mathematics June 2024 Principal Examiner Report for Teachers © 2024 method. Candidates need to be aware that simply quoting the quadratic formula is insufficient: values need to be substituted into it. A significant number of candidates seemed unable to obtain the two required answers for− − 1 3 sin 4 in the given range. The value of− 48.6 from a calculator was commonly given as part of the solution, as was 131.4. Question 4 Part (a) proved to be straightforward for most candidates, but the other two parts were much more challenging for many of them. In part (b), it was common for only one part of the required inequality for the range to be given, as was confusion between and . It was often unclear what candidates meant when they referred to ‘it’ being many-to-one, as this could have applied to the given function or the inverse. Candidates are encouraged to simply state that “ −1 g does not exist because g is not one to one”. In part (c), most candidates were able to find 25 16 f , ( ) hf x and equate them, but many were unable to solve the resulting equation. Those who did manage this often failed to discount − 3 2 2 as a possible solution. Question 5 In part (a) many candidates were able to find a correct answer but not in an exact form. In part (b)(i), most candidates could correctly find the sum to infinity, but many were unsure how to present it: − tan 1 cos was acceptable as there are no embedded fractions (fractions divided by fractions), whereas − sin cos 1 cos and − tan sin 1 tan were not acceptable. Part (b)(ii) was generally well done, but some weaker candidates found the tenth term instead of the required sum. Question 6 Part (a) was very well done by most candidates with the vast majority using differentiation to find the minimum point. Those who attempted to complete the square sometimes became confused and thought = 2 x rather than = 2. x A significant number of candidates failed to see the connection between part (a) and part (b) and wasted time finding the co-ordinates of A and B again. The integration to find the areas was generally well done, but candidates need to remember to clearly show the limits substituted into the functions. Questions like 6(b) generally involve subtracting one area from another; most candidates are more likely to score full marks if they work out the areas separately and then subtract. This avoids having to deal with applying the minus sign to all the terms of the second integral. Question 7 Several different methods were used to find the two possible values in part (a), but the most common and most successful method was to replace y in the circle equation with 2a – x and then use the discriminant. Errors often occurred when candidates were expanding the brackets, and they are encouraged to take care when performing this process and collecting like terms. The 18 on the right-hand side of the equation was sometimes forgotten or ignored. Part (b) proved to be challenging and was omitted by about 30% of candidates. Many who did attempt it unnecessarily spent time finding the point of intersection of the tangent and the circle rather than using the centre of the circle. Using a diagram may well have avoided this waste and should always be encouraged for this type of question. Others seem confused about what was being asked for and were unable to make any progress. Again, a diagram may well have made clear what was required.
Cambridge International Advanced Subsidiary and Advanced Level 9709 Mathematics June 2024…
Cambridge International Advanced Subsidiary and Advanced Level 9709 Mathematics June 2024 Principal Examiner Report for Teachers © 2024 Question 8 About 40% of candidates scored no marks on part (a)(i), with a number attempting to use the arc length to show the given result. Many candidates showed this successfully using right angled triangle trigonometry. Part (a)(ii) produced a mixed response. Many candidates could find the areas of the sectors, but the height of the trapezium was more challenging, with some simply using 0.2 or 0.4 instead of calculating it. When working out several areas that are to be added together, clear labelling of those areas can enable candidates to be awarded method marks even if their areas are wrong. This should always be encouraged. Part (b) again proved challenging and was omitted by almost 20% of candidates. Of those who did attempt it, many failed to appreciate that because the radius had changed EF was no longer 2.4. Question 9 A small but significant number of candidates missed out the – 6x and only considered the first part of the function. This significantly changed the nature of the question and therefore no marks could be awarded in part (a). Care needs to be taken by candidates to avoid this type of mistake. Those who considered the whole of the derivative usually realised what was required, although weaker candidates sometimes differentiated or integrated unnecessarily. It is important to note that if the candidate gave their final answer as two separate inequalities, then the word ‘and’ rather than ‘or’ needed to be used. In part (b), those candidates who realised that integration was required were almost always able to do so correctly, but there was confusion over the implication of f(1) = –1. Some candidates failed to include a constant of integration and others set f(x) = 0. Question 10 This question proved challenging for many candidates, especially part (b) which was omitted by 20% of them. In part (a), most candidates knew how to start their response and the differentiation was generally done well. Many seemed unsure how to find the required x value and were, therefore, unable to make any further progress. In part (b), many failed to see the link with the first part of the question and spent time finding the x-coordinate which had been previously found. Others assumed that the gradient of the perpendicular bisector would be1 3 because the gradient of the curve was –3, or failed to find the midpoint of AB. About half of the candidates were able to find the correct equation but about half of these lost the final mark as they did not give their answer in the required form.
Cambridge International Advanced Subsidiary and Advanced Level 9709 Mathematics June 2024…
Cambridge International Advanced Subsidiary and Advanced Level 9709 Mathematics June 2024 Principal Examiner Report for Teachers © 2024 MATHEMATICS Paper 9709/13 Pure Mathematics 1 (13) Key messages it is important that centres remind their students to note the ninth bullet point of the examination instructions on the front page of the examination paper: ‘You must show your working clearly; no marks will be given for unsupported answers from a calculator.’ Examiners are unable to give credit when, for example, quadratic equation solutions, definite integration results, gradients, simultaneous equation solutions, intersection points and sums of series arrive from little or no working. Even when these results are correct they are not acceptable. Examiners have made allowances when these results are used in later stages of a solution but this is not automatic. Centres should not allow calculators with equation solvers, programmable calculators or graphical calculators to be used in this examination. General comments Most candidates appeared well prepared for this exam and the vast majority attempted most questions. With a few exceptions questions were interpreted correctly and an appropriate method of solution selected. The questions involving series and calculus were particularly well answered whilst those involving transformation geometry caused problems for many candidates. Comments on specific questions Question 1 This question was generally well answered by most candidates. Many candidates demonstrated that the application of the binomial expansion was well understood and the multiplication necessary to obtain the required term was carried out successfully. Question 2 (a) This part was well answered by a minority of candidates. Those who used an algebraic method were occasionally successful as were those who used calculus to find the coordinates of B. Where candidates identified the graph was the result of a simple translation of y = k cos x they usually reached the correct answers by considering where the graph is zero and where it is a minimum. This question part was often omitted by some candidates. (b) In contrast to part (a) this part was usually completed correctly. Some candidates chose to work in degrees and some mixed the units but managed to find the correct result. Finding the inverse trigonometric function was generally done accurately and most answers were given in the required exact form. Question 3 (a) The formulae for arc length and sector area were used effectively by most candidates to produce two equations in r and θ. Although elimination of θ was the efficient route to a quadratic equation there were numerous correct equations in θ found through elimination of r. Sight of clear method of solution for these quadratics was the only way full marks could be awarded. At this stage it was sufficient to find both pairs of solutions or just the appropriate pair. Those candidates who showed no working after obtaining the two initial equations in r and θ, yet still managed to reach correct solutions, could gain no more than half marks.
Command word playbook
How to match each command word to the expected response style
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Time traps
Sections where candidates spent disproportionate time relative to marks
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Syllabus traceability
Topics linked to questions and mark weighting in this session
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MCQ trap analytics
Commonly chosen wrong options from examiner commentary
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Topic heatmap across years
Mark concentration by topic and exam year for this subject
Mark intensity
Kinematics of motion in a straight line (Mechanics (for Paper 4))
Differentiation
Differentiation (Pure Mathematics 1 (for Paper 1))
Differentiation (Pure Mathematics 2 (for Paper 2))
The Poisson distribution (Probability & Statistics 2 (for Paper 6))
Integration (Pure Mathematics 2 (for Paper 2))
Series (Pure Mathematics 1 (for Paper 1))
Integration (Pure Mathematics 3 (for Paper 3))
Difficulty trend
How session difficulty has shifted across recent years
Paper comparison
Marks and duration breakdown across papers in this session
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Marks you can still earn
Where valid approaches outside the mark scheme may still gain credit
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Practise what examiners flagged
Target weak topics from this report inside the Revui app
Self-diagnostic checklist
Key actions before you sit this paper — copy and tick off as you revise
- 1Message
The question paper contains a statement in the rubric on the front cover that ‘no marks will be given for unsupported answers from a calculator.’ This means that clear working must be shown to justify solutions, particularly in syllabus items such as quadratic equations and trigonometric equations.
- 2Message
In the case of quadratic equations, for example, it would be necessary to show factorisation, use of the quadratic formula or completing the square as stated in the syllabus.
- 3Message
Using calculators to solve equations and writing down only the solution is not sufficient for certain marks to be awarded.
- 4Message
It is also insufficient to quote only the formula: candidates need to show values substituted into it.
- 5Message
When factorising, candidates should ensure that the factors always expand to give the coefficients of the quadratic equation.
- 6Message
Candidates should take care to read each question carefully and to note the number of marks attached to each question.
- 7Message
Candidates would benefit from spending time to ensure they understand the terminology required in responses and the structure of examination papers at Advanced Level.
- 8Message
For example, the correct terminology for transformation questions is ‘translation’, followed by a column vector, or ‘stretch’, with the word ‘factor’ and both its size and direction clearly indicated.
- 9Strength
Cambridge International Advanced Subsidiary and Advanced Level 9709 Mathematics June 2024…: Cambridge International Advanced Subsidiary and Advanced Level 9709 Mathematics June 2024 Principal
- 10Strength
Cambridge International Advanced Subsidiary and Advanced Level 9709 Mathematics June 2024…: Cambridge International Advanced Subsidiary and Advanced Level 9709 Mathematics June 2024 Principal
- 11Strength
Cambridge International Advanced Subsidiary and Advanced Level 9709 Mathematics June 2024…: Cambridge International Advanced Subsidiary and Advanced Level 9709 Mathematics June 2024 Principal
Teacher briefing pack
One-page session summary for tutors and classroom review
June 2024 2024
Mathematics
Cambridge International Advanced Subsidiary and Advanced Level 9709 Mathematics June 2024 Principal Examiner Report for Teachers © 2024 MATHEMATICS Paper 9709/11 Pure Mathematics 1 (11) Key messages The question paper contains a statement in the rubric on the front cover that ‘no
The question paper contains a statement in the rubric on the front cover that ‘no marks will be given for unsupported answers from a calculator.’ This means that clear working must be shown to justify solutions, particularly in syllabus items such as quadratic equations and trigonometric equations.
In the case of quadratic equations, for example, it would be necessary to show factorisation, use of the quadratic formula or completing the square as stated in the syllabus.
Using calculators to solve equations and writing down only the solution is not sufficient for certain marks to be awarded.
Examiner insights
General comments
- •Some very good responses were seen but the paper proved very challenging for many candidates.
- •In AS and A Level Mathematics papers the knowledge and use of basic algebraic and trigonometric methods from IGCSE or O Level is expected, as stated in the syllabus.