9231 · June 2024
Mathematics - Further
Most candidates demonstrated good knowledge across the whole syllabus. They showed their working clearly and were accurate in their handling of algebra and calculus.
Source: Cambridge International
Cohort performance
Session statistics from official examination reports
No data available in official reports
Key examiner messages
Top priorities from the principal examiner before you revise
Candidates should read each question carefully so that they use all the information given and answer all aspects in adequate depth. They should make clear the method being used.
All sketch graphs need to be fully labelled and carefully drawn to show significant points and behaviour at limits.
Candidates should show all the steps in their solutions, particularly when proving a given result.
Both algebra and arithmetic can often be simplified using common factors and brackets.
Both algebra and arithmetic can often be simplified by the use of common factors and brackets.
Candidates should read questions carefully so that they answer all aspects in adequate depth, particularly when an answer is required in a certain form or in terms of a given variable.
Candidates should make use of results derived or given in earlier parts of a question or given in the list of formulae (MF19).
Candidates should make use of results derived in earlier parts of a question or given in the list of formulae (MF19).
Question difficulty map
How candidates performed on each question in this series
9231/11
Further Pure Mathematics 11
9231/12
Further Pure Mathematics 12
9231/13
Further Pure Mathematics 13
9231/21
Further Pure Mathematics 21
9231/22
Further Pure Mathematics 22
9231/23
Further Pure Mathematics 23
9231/31
Further Mechanics 31
9231/32
Further Mechanics 32
Assessment objectives
Skill and AO weighting from official examiner commentary
No data available in official reports
Method marks watchlist
Where working, steps, or method marks were commonly lost
Method marks · 9231/31 · Q4(a)
Many candidates scored the first method mark as they correctly resolved forces parallel to the inclined plane, but then were not able to find a second suitable equation to eliminate the friction…
Recurring mistakes across years
Themes examiners flag in multiple recent sessions for this subject
No data available in official reports
Question choice intelligence
Mean scores and popularity for optional questions (HKDSE electives)
No data available in official reports
Level exemplars
What candidate scripts at each grade level looked like
No data available in official reports
Grade & admission context
How marks relate to grade thresholds and entry standards
No data available in official reports
Deep insights
What top candidates did
Techniques and approaches examiners rewarded in this series
Cambridge International Advanced Subsidiary and Advanced Level 9231 Further Mathematics Ju…
Cambridge International Advanced Subsidiary and Advanced Level 9231 Further Mathematics June 2024 Principal Examiner Report for Teachers © 2024 FURTHER MATHEMATICS Paper 9231/11 Further Pure Mathematics 11 Key messages • Candidates should read each question carefully so that they use all the information given and answer all aspects in adequate depth. They should make clear the method being used. • All sketch graphs need to be fully labelled and carefully drawn to show significant points and behaviour at limits. • Candidates should show all the steps in their solutions, particularly when proving a given result. • Both algebra and arithmetic can often be simplified using common factors and brackets. General comments Most candidates demonstrated good knowledge across the whole syllabus. They showed their working clearly and were accurate in their handling of algebra and calculus. They also showed understanding of transformations. It seemed that almost all were able to complete the paper in the time allowed. Comments on specific questions Question 1 Candidates were able to employ standard methods of dealing with the roots of equations well. (a) This was almost always correct. (b) Most candidates factorised the expression correctly, however some made sign errors when substituting so did not achieve full credit. (c) A minority of candidates realised that the previous two question parts had given them two of the coefficients needed and could write down the required equation immediately. Some candidates attempted to use a substitution of w = z2 however this was unsuccessful. Many candidates did not attempt the question. (d) Most candidates were able to form and solve the equation for p correctly. Question 2 The general structure of a proof by induction was well understood by most candidates and there were some excellent solutions. There is however a need for more care in stating the hypothesis. In this case candidates needed to write down the function for k and also make the assumption that it can be divided by 74. The inductive step was attempted by rearrangement or by considering the difference between f(k+1) and a multiple of f(k). Many candidates did not show that one of the expressions is a multiple of 74. The best solutions took out a factor of 2 from 64k and from 38k. Candidates are reminded that when not considering f(k+1) directly they need to state why the result they have found implies that f(k+1) is divisible by 74. The final statement usually contained the required reference to 64k + 38k – 2 being divisible by 74 for all positive integers n.
Cambridge International Advanced Subsidiary and Advanced Level 9231 Further Mathematics Ju…
Cambridge International Advanced Subsidiary and Advanced Level 9231 Further Mathematics June 2024 Principal Examiner Report for Teachers © 2024 Question 3 (a) The first two marks were almost universally achieved. The most elegant solutions kept the expressions fully factorised, factoring out the one twelfth and the initial N(N+1). Those who multiplied out the brackets found themselves with a cubic or quartic requiring further factorisation. (b) Nearly all candidates found the correct partial fractions. Those who set out the telescoping clearly did best. Candidates who wrote out the first few terms to show the effect of the powers of1 4 where generally successful when writing down the answer with the correct form for the final term. (c) This was usually correct if part (b) had been answered correctly. Question 4 (a) Most candidates correctly identified the transformations as stretch and rotation and gave them in the correct order. Candidates usually correctly described both the direction and scale factor for the one-way stretch. Some did not gain full credit as they did not describe both the angle and the centre of rotation for the second transformation. (b) Most candidates clearly know how to find the inverse of a 2 2 matrix and could write the two matrices in the appropriate order. A minority of candidates did not divide by 14, however the most common error was to find M-1 as a single matrix. (c) Most candidates calculated M accurately and made it clear that they were looking for invariant lines rather than points. There were many fully correct solutions. (d) Responses to this question were almost always correct, with the method appearing to be well known to candidates. Question 5 The majority of candidates demonstrated good knowledge and application of the required vector formulae. Several different methods were used to great effect, and it was evident that most students were comfortable with this topic. Candidates are advised to check cross products carefully; accuracy was often lost because of wrong signs. (a) The cross product method was usually applied correctly. (b) There were two methods which were most efficient in this question. The first was to find a vector joining D to a point of the plane and project it on to the normal direction. The second was to substitute the coordinates of D into the modified equation of the plane. Candidates who tried to find the base of the perpendicular from D to the plane often made errors in their working. (c) There were many efficient and accurate solutions using the standard method. A handful of candidates tried first to find a point of intersection. Question 6 (a) Almost all candidates wrote down the correct vertical asymptote. The equation of the oblique asymptote was often given correctly, although errors in the remainder were common when long division was used. Those who used the method of finding coefficients also commonly made errors, some of which were caused by the unknown a in the equation of the curve. (b) Most candidates correctly differentiated the equation for C by using the quotient rule. Candidates generally then used the discriminant to explain that there were no real roots, however not all used the necessary condition 2a > 5 to make their explanation convincing. An elegant solution was to rearrange to give− y a x x 2 d 2 5 = 1 + d ( + 2) and to explain that this meansy x d d 1.
Cambridge International Advanced Subsidiary and Advanced Level 9231 Further Mathematics Ju…
Cambridge International Advanced Subsidiary and Advanced Level 9231 Further Mathematics June 2024 Principal Examiner Report for Teachers © 2024 (c) Most candidates remembered to label the asymptotes. Those who used the fact that there are no turning points drew two branches on the correct side of the asymptotes. A number of graphs showed one or two turning points because they were in the incorrect section of the x-y plane. (d) (i) The idea of reflecting the graph in the x axis is well understood and many graphs correctly showed a cusp or “sharp bounce” off the x-axis and correct behaviour at the vertical asymptote. (ii) Of the candidates who attempted this part, most were correct. However, many candidates did not attempt to draw the line. . (iii) Most of those who got part d(ii) correct used this to write down equations to find critical points. Candidates connected these equations to the information given and therefore found the solution quickly by substituting x = 3 or x = -3 into their equation. Those who tried to work with inequalities were less successful. Question 7 (a) Most candidates were able to produce an acceptable graph. The biggest problem was with the coordinate of the point furthest from the pole: many candidates forgot to take the square root or did not give correct polar form. (b) The majority of candidates obtained the first three marks by writing down the correct integral and using integration by parts. When using a substitution, candidates are strongly advised to change all parts of the function, the limits and the dθ at the same time to avoid problems with constants and signs. When faced withu u u 2 2 d 1+ many tried a logarithmic expression or reversed their integration by parts, without success. Those who formed the equationu u 2 2 1+ = 1 –u2 1 1+ produced the best solutions. Another effective method was to use a second substitution of U = tan w. (c) Most candidates gained the first mark for using the correct function, and the last mark for establishing the change of sign. The differentiation was challenging and required both the chain rule and the product rule for three terms. The most common problems were the omission of one of the terms and sign errors.
Cambridge International Advanced Subsidiary and Advanced Level 9231 Further Mathematics Ju…
Cambridge International Advanced Subsidiary and Advanced Level 9231 Further Mathematics June 2024 Principal Examiner Report for Teachers © 2024 FURTHER MATHEMATICS Paper 9231/12 Further Pure Mathematics 12 Key messages • Candidates should read each question carefully so that they use all the information given and answer all aspects in adequate depth. They should make clear the method being used. • All sketch graphs need to be fully labelled and carefully drawn to show significant points and behaviour at limits. • Candidates should show all the steps in their solutions, particularly when proving a given result. • Both algebra and arithmetic can often be simplified using common factors and brackets. General comments Most candidates demonstrated good knowledge across the whole syllabus. They showed their working clearly and were accurate in their handling of algebra and calculus. They also showed understanding of transformations. It seemed that almost all were able to complete the paper in the time allowed. Comments on specific questions Question 1 Candidates were able to employ standard methods of dealing with the roots of equations well. (a) This was almost always correct. (b) Most candidates factorised the expression correctly, however some made sign errors when substituting so did not achieve full credit. (c) A minority of candidates realised that the previous two question parts had given them two of the coefficients needed and could write down the required equation immediately. Some candidates attempted to use a substitution of w = z2 however this was unsuccessful. Many candidates did not attempt the question. (d) Most candidates were able to form and solve the equation for p correctly. Question 2 The general structure of a proof by induction was well understood by most candidates and there were some excellent solutions. There is however a need for more care in stating the hypothesis. In this case candidates needed to write down the function for k and also make the assumption that it can be divided by 74. The inductive step was attempted by rearrangement or by considering the difference between f(k+1) and a multiple of f(k). Many candidates did not show that one of the expressions is a multiple of 74. The best solutions took out a factor of 2 from 64k and from 38k. Candidates are reminded that when not considering f(k+1) directly they need to state why the result they have found implies that f(k+1) is divisible by 74. The final statement usually contained the required reference to 64k + 38k – 2 being divisible by 74 for all positive integers n.
Cambridge International Advanced Subsidiary and Advanced Level 9231 Further Mathematics Ju…
Cambridge International Advanced Subsidiary and Advanced Level 9231 Further Mathematics June 2024 Principal Examiner Report for Teachers © 2024 Question 3 (a) The first two marks were almost universally achieved. The most elegant solutions kept the expressions fully factorised, factoring out the one twelfth and the initial N(N+1). Those who multiplied out the brackets found themselves with a cubic or quartic requiring further factorisation. (b) Nearly all candidates found the correct partial fractions. Those who set out the telescoping clearly did best. Candidates who wrote out the first few terms to show the effect of the powers of1 4 where generally successful when writing down the answer with the correct form for the final term. (c) This was usually correct if part (b) had been answered correctly. Question 4 (a) Most candidates correctly identified the transformations as stretch and rotation and gave them in the correct order. Candidates usually correctly described both the direction and scale factor for the one-way stretch. Some did not gain full credit as they did not describe both the angle and the centre of rotation for the second transformation. (b) Most candidates clearly know how to find the inverse of a 2 2 matrix and could write the two matrices in the appropriate order. A minority of candidates did not divide by 14, however the most common error was to find M-1 as a single matrix. (c) Most candidates calculated M accurately and made it clear that they were looking for invariant lines rather than points. There were many fully correct solutions. (d) Responses to this question were almost always correct, with the method appearing to be well known to candidates. Question 5 The majority of candidates demonstrated good knowledge and application of the required vector formulae. Several different methods were used to great effect, and it was evident that most students were comfortable with this topic. Candidates are advised to check cross products carefully; accuracy was often lost because of wrong signs. (a) The cross product method was usually applied correctly. (b) There were two methods which were most efficient in this question. The first was to find a vector joining D to a point of the plane and project it on to the normal direction. The second was to substitute the coordinates of D into the modified equation of the plane. Candidates who tried to find the base of the perpendicular from D to the plane often made errors in their working. (c) There were many efficient and accurate solutions using the standard method. A handful of candidates tried first to find a point of intersection. Question 6 (a) Almost all candidates wrote down the correct vertical asymptote. The equation of the oblique asymptote was often given correctly, although errors in the remainder were common when long division was used. Those who used the method of finding coefficients also commonly made errors, some of which were caused by the unknown a in the equation of the curve. (b) Most candidates correctly differentiated the equation for C by using the quotient rule. Candidates generally then used the discriminant to explain that there were no real roots, however not all used the necessary condition 2a > 5 to make their explanation convincing. An elegant solution was to rearrange to give− y a x x 2 d 2 5 = 1 + d ( + 2) and to explain that this meansy x d d 1.
Cambridge International Advanced Subsidiary and Advanced Level 9231 Further Mathematics Ju…
Cambridge International Advanced Subsidiary and Advanced Level 9231 Further Mathematics June 2024 Principal Examiner Report for Teachers © 2024 (c) Most candidates remembered to label the asymptotes. Those who used the fact that there are no turning points drew two branches on the correct side of the asymptotes. A number of graphs showed one or two turning points because they were in the incorrect section of the x-y plane. (d) (i) The idea of reflecting the graph in the x axis is well understood and many graphs correctly showed a cusp or “sharp bounce” off the x-axis and correct behaviour at the vertical asymptote. (ii) Of the candidates who attempted this part, most were correct. However, many candidates did not attempt to draw the line. . (iii) Most of those who got part d(ii) correct used this to write down equations to find critical points. Candidates connected these equations to the information given and therefore found the solution quickly by substituting x = 3 or x = -3 into their equation. Those who tried to work with inequalities were less successful. Question 7 (a) Most candidates were able to produce an acceptable graph. The biggest problem was with the coordinate of the point furthest from the pole: many candidates forgot to take the square root or did not give correct polar form. (b) The majority of candidates obtained the first three marks by writing down the correct integral and using integration by parts. When using a substitution, candidates are strongly advised to change all parts of the function, the limits and the dθ at the same time to avoid problems with constants and signs. When faced withu u u 2 2 d 1+ many tried a logarithmic expression or reversed their integration by parts, without success. Those who formed the equationu u 2 2 1+ = 1 –u2 1 1+ produced the best solutions. Another effective method was to use a second substitution of U = tan w. (c) Most candidates gained the first mark for using the correct function, and the last mark for establishing the change of sign. The differentiation was challenging and required both the chain rule and the product rule for three terms. The most common problems were the omission of one of the terms and sign errors.
Command word playbook
How to match each command word to the expected response style
No data available in official reports
Time traps
Sections where candidates spent disproportionate time relative to marks
No data available in official reports
Syllabus traceability
Topics linked to questions and mark weighting in this session
No data available in official reports
MCQ trap analytics
Commonly chosen wrong options from examiner commentary
No data available in official reports
Topic heatmap across years
Mark concentration by topic and exam year for this subject
Mark intensity
Differential equations (Further Pure Mathematics 2)
Integration (Further Pure Mathematics 2)
Matrices (Further Pure Mathematics 2)
Differential equations
Integration
Vectors
Rational functions and graphs
Hyperbolic functions
Difficulty trend
How session difficulty has shifted across recent years
Paper comparison
Marks and duration breakdown across papers in this session
No data available in official reports
Marks you can still earn
Where valid approaches outside the mark scheme may still gain credit
No data available in official reports
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
Candidates should read each question carefully so that they use all the information given and answer all aspects in adequate depth. They should make clear the method being used.
- 2Message
All sketch graphs need to be fully labelled and carefully drawn to show significant points and behaviour at limits.
- 3Message
Candidates should show all the steps in their solutions, particularly when proving a given result.
- 4Message
Both algebra and arithmetic can often be simplified using common factors and brackets.
- 5Message
Both algebra and arithmetic can often be simplified by the use of common factors and brackets.
- 6Message
Candidates should read questions carefully so that they answer all aspects in adequate depth, particularly when an answer is required in a certain form or in terms of a given variable.
- 7Message
Candidates should make use of results derived or given in earlier parts of a question or given in the list of formulae (MF19).
- 8Message
Candidates should make use of results derived in earlier parts of a question or given in the list of formulae (MF19).
- 9Method
Many candidates scored the first method mark as they correctly resolved forces parallel to the inclined plane, but then were not able to find a second suitable
- 10Strength
Cambridge International Advanced Subsidiary and Advanced Level 9231 Further Mathematics Ju…: Cambridge International Advanced Subsidiary and Advanced Level 9231 Further Mathematics June 2024 Pr
- 11Strength
Cambridge International Advanced Subsidiary and Advanced Level 9231 Further Mathematics Ju…: Cambridge International Advanced Subsidiary and Advanced Level 9231 Further Mathematics June 2024 Pr
- 12Strength
Cambridge International Advanced Subsidiary and Advanced Level 9231 Further Mathematics Ju…: Cambridge International Advanced Subsidiary and Advanced Level 9231 Further Mathematics June 2024 Pr
Teacher briefing pack
One-page session summary for tutors and classroom review
June 2024 2024
Mathematics - Further
Cambridge International Advanced Subsidiary and Advanced Level 9231 Further Mathematics June 2024 Principal Examiner Report for Teachers © 2024 FURTHER MATHEMATICS Paper 9231/11 Further Pure Mathematics 11 Key messages • Candidates should read each question carefully so that they
Candidates should read each question carefully so that they use all the information given and answer all aspects in adequate depth. They should make clear the method being used.
All sketch graphs need to be fully labelled and carefully drawn to show significant points and behaviour at limits.
Candidates should show all the steps in their solutions, particularly when proving a given result.
Examiner insights
General comments
- •Most candidates demonstrated good knowledge across the whole syllabus.
- •They showed their working clearly and were accurate in their handling of algebra and calculus.
- •They also showed understanding of transformations.
- •It seemed that almost all were able to complete the paper in the time allowed.