Cambridge A Level Mathematics - Further 9231 — 2025 Oct/Nov Paper 1 · Variant 2

This document has 20 pages. Any blank pages are indicated. [Turn over Cambridge International AS & A Level DC (CE) 340758/4 © UCLES 2025 FURTHER MATHEMATICS 9231/12 Paper 1 Further Pure Mathematics 1 October/November 2025 2 hours You must answer on the question paper. You will need: List of formulae (MF19) INSTRUCTIONS ● Answer all questions. ● Use a black or dark blue pen. You may use an HB pencil for any diagrams or graphs. ● Write your name, centre number and candidate number in the boxes at the top of the page. ● Write your answer to each question in the space provided. ● Do not use an erasable pen or correction fluid. ● Do not write on any bar codes. ● If additional space is needed, you should use the lined page at the end of this booklet; the question number or numbers must be clearly shown. ● You should use a calculator where appropriate. ● You must show all necessary working clearly; no marks will be given for unsupported answers from a calculator. ● Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place for angles in degrees, unless a different level of accuracy is specified in the question. INFORMATION ● The total mark for this paper is 75. ● The number of marks for each question or part question is shown in brackets [ ]. * 0 9 5 9 1 6 7 0 2 1 * ,










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2 9231/12/O/N/25 © UCLES 2025 1 (a) Use standard results from the list of formulae (MF19) to find r r r n 3 1 - = ` j / in terms of n, fully factorising your answer. [3] … … … … … … … … … … … … … … … … … … … … … … … … … … * 0000800000002 * ,










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3 9231/12/O/N/25 © UCLES 2025 [Turn over (b) Express r r r 3 3 - + in the form r A r B r C 1 1 - + + + , where A, B and C are constants to be determined, and hence use the method of differences to find r r r 3 r n 3 2 - + =/ . [5] … … … … … … … … … … … … … … … … … … … … … … … (c) Deduce the value of r r r 3 r 3 2 - + 3 =/ . [1] … … * 0000800000003 * ,










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4 9231/12/O/N/25 © UCLES 2025 2 The cubic equation x bx cx d 0 3 2 + + + = , where b, c and d are constants, has roots a, b and c. It is given that ,2 a b c + + = ,3 2 2 2 a b c + + = .5 4 4 4 a b c + + = (a) Find the values of b and c. [3] … … … … … … … … … … … … … … … … … … … … … … … * 0000800000004 * ,










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5 9231/12/O/N/25 © UCLES 2025 [Turn over (b) Find the value of d. [5] … … … … … … … … … … … … … … … … … … … … … … … … … … … * 0000800000005 * ,










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6 9231/12/O/N/25 © UCLES 2025 3 The sequence of positive numbers , , , u u u 1 2 3 … is such that u 5 1 1 and, for n 1 H , u u u 2 6 5 n n n 1 = + + + . (a) By considering , u 5 n 1 - + prove by mathematical induction that u 5 n 1 for all positive integers n. [5] … … … … … … … … … … … … … … … … … … … … … … … … * 0000800000006 * ,










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7 9231/12/O/N/25 © UCLES 2025 [Turn over (b) Show that u u n n 1 2 + for n 1 H . [3] … … … … … … … … … … … … … … … … … … … … … … … … … … … * 0000800000007 * ,










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8 9231/12/O/N/25 © UCLES 2025 4 Let k and m be non-zero constants. The matrices A, B and C are given by A 0 1 1 1 1 1 = - f p, k m B 0 0 = e o and C 2 1 1 1 1 2 = - e o. (a) Give full details of the geometrical transformation in the x-y plane represented by the matrix B in each of the following cases. (i) m = 1 [2] … … (ii) m = k [2] … … (b) Show that the matrix ABC is singular. [6] … … … … … … … … … … … … … … … … … * 0000800000008 * ,










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10 9231/12/O/N/25 © UCLES 2025 5 The curve C has polar equation tan r 2 2 i = , where r 0 8 1 G G i . (a) Sketch C and state the greatest distance of a point on C from the pole. [2] … (b) Find the exact value of the area of the region bounded by C and the half-line r 8 1 i = . [4] … … … … … … … … … … … … … * 0000800000010 * ,










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11 9231/12/O/N/25 © UCLES 2025 [Turn over (c) Show that C has Cartesian equation x xy y 2 0 4 4 - - = given that r cos x 0 8 1 G G b l and r sin y 0 8 1 G G b l. [4] … … … … … … … … … … … … … … … (d) Using your answer to (b), deduce the exact value of the area bounded by C, the x-axis and the line r cos x 8 1 = b l. [2] … … … … … … … … … … … * 0000800000011 * ,









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12 9231/12/O/N/25 © UCLES 2025 6 The plane P has equation x y z 3 2 1 + + = . (a) Find the perpendicular distance from the origin O to the plane P. [2] … … … … Relative to O, the points A, B, C have position vectors , , , j k i k i j k 2 2 2 - + - - - respectively. (b) Find the acute angle between the planes OAB and P. [4] … … … … … … … … … … … … … … … … … … * 0000800000012 * ,









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13 9231/12/O/N/25 © UCLES 2025 [Turn over (c) Find an equation for the common perpendicular to the lines OC and AB. [8] … … … … … … … … … … … … … … … … … … … … … … … … … … … * 0000800000013 * ,









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14 9231/12/O/N/25 © UCLES 2025 7 The curve C has equation y x x x 1 1 2 = + + + . (a) Find the equations of the asymptotes of C. [3] … … … … … … … … … … … … … (b) Find the coordinates of any stationary points on C. [3] … … … … … … … … … … … … * 0000800000014 * ,









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15 9231/12/O/N/25 © UCLES 2025 [Turn over (c) Sketch C. [3] (d) Sketch the curve with equation y x x x 1 1 2 = + + + . [2] * 0000800000015 * ,









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16 9231/12/O/N/25 © UCLES 2025 (e) Find, in exact form, the set of values of x for which x x x 1 1 3 2 1 + + + . [3] … … … … … … … … … … … … … … … … … … … … … … … … … … … * 0000800000016 * ,









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17 9231/12/O/N/25 © UCLES 2025 Additional page If you use the following page to complete the answer to any question, the question number must be clearly shown. … … … … … … … … … … … … … … … … … … … … … … … … … … * 0000800000017 * ,









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20 9231/12/O/N/25 © UCLES 2025 Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge Assessment International Education Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at www.cambridgeinternational.org after the live examination series. Cambridge Assessment International Education is part of Cambridge Assessment. Cambridge Assessment is the brand name of the University of Cambridge Local Examinations Syndicate (UCLES), which is a department of the University of Cambridge. BLANK PAGE * 0000800000020 * ,










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This document consists of 19 printed pages. © Cambridge University Press & Assessment 2025 [Turn over Cambridge International AS & A Level FURTHER MATHEMATICS 9231/12 Paper 1 Further Pure Mathematics 1 October/November 2025 MARK SCHEME Maximum Mark: 75 Published This mark scheme is published as an aid to teachers and candidates, to indicate the requirements of the examination. It shows the basis on which Examiners were instructed to award marks. It does not indicate the details of the discussions that took place at an Examiners’ meeting before marking began, which would have considered the acceptability of alternative answers. Mark schemes should be read in conjunction with the question paper and the Principal Examiner Report for Teachers. Cambridge International will not enter into discussions about these mark schemes. Cambridge International is publishing the mark schemes for the October/November 2025 series for most Cambridge IGCSE, Cambridge International A and AS Level components, and some Cambridge O Level components.

9231/12 Cambridge International AS & A Level – Mark Scheme PUBLISHED October/November 2025 © Cambridge University Press & Assessment 2025 Page 2 of 19 Generic Marking Principles These general marking principles must be applied by all examiners when marking candidate answers. They should be applied alongside the specific content of the mark scheme or generic level descriptions for a question. Each question paper and mark scheme will also comply with these marking principles. GENERIC MARKING PRINCIPLE 1: Marks must be awarded in line with: • the specific content of the mark scheme or the generic level descriptors for the question • the specific skills defined in the mark scheme or in the generic level descriptors for the question • the standard of response required by a candidate as exemplified by the standardisation scripts. GENERIC MARKING PRINCIPLE 2: Marks awarded are always whole marks (not half marks, or other fractions). GENERIC MARKING PRINCIPLE 3: Marks must be awarded positively: • marks are awarded for correct/valid answers, as defined in the mark scheme. However, credit is given for valid answers which go beyond the scope of the syllabus and mark scheme, referring to your Team Leader as appropriate • marks are awarded when candidates clearly demonstrate what they know and can do • marks are not deducted for errors • marks are not deducted for omissions • answers should only be judged on the quality of spelling, punctuation and grammar when these features are specifically assessed by the question as indicated by the mark scheme. The meaning, however, should be unambiguous. GENERIC MARKING PRINCIPLE 4: Rules must be applied consistently, e.g. in situations where candidates have not followed instructions or in the application of generic level descriptors. GENERIC MARKING PRINCIPLE 5: Marks should be awarded using the full range of marks defined in the mark scheme for the question (however; the use of the full mark range may be limited according to the quality of the candidate responses seen).

9231/12 Cambridge International AS & A Level – Mark Scheme PUBLISHED October/November 2025 © Cambridge University Press & Assessment 2025 Page 3 of 19 GENERIC MARKING PRINCIPLE 6: Marks awarded are based solely on the requirements as defined in the mark scheme. Marks should not be awarded with grade thresholds or grade descriptors in mind. Mathematics-Specific Marking Principles 1 Unless a particular method has been specified in the question, full marks may be awarded for any correct method. However, if a calculation is required then no marks will be awarded for a scale drawing. 2 Unless specified in the question, non-integer answers may be given as fractions, decimals or in standard form. Ignore superfluous zeros, provided that the degree of accuracy is not affected. 3 Allow alternative conventions for notation if used consistently throughout the paper, e.g. commas being used as decimal points. 4 Unless otherwise indicated, marks once gained cannot subsequently be lost, e.g. wrong working following a correct form of answer is ignored (isw). 5 Where a candidate has misread a number or sign in the question and used that value consistently throughout, provided that number does not alter the difficulty or the method required, award all marks earned and deduct just 1 A or B mark for the misread. 6 Recovery within working is allowed, e.g. a notation error in the working where the following line of working makes the candidate’s intent clear.

9231/12 Cambridge International AS & A Level – Mark Scheme PUBLISHED October/November 2025 © Cambridge University Press & Assessment 2025 Page 4 of 19 Annotations guidance for centres Examiners use a system of annotations as a shorthand for communicating their marking decisions to one another. Examiners are trained during the standardisation process on how and when to use annotations. The purpose of annotations is to inform the standardisation and monitoring processes and guide the supervising examiners when they are checking the work of examiners within their team. The meaning of annotations and how they are used is specific to each component and is understood by all examiners who mark the component. We publish annotations in our mark schemes to help centres understand the annotations they may see on copies of scripts. Note that there may not be a direct correlation between the number of annotations on a script and the mark awarded. Similarly, the use of an annotation may not be an indication of the quality of the response. The annotations listed below were available to examiners marking this component in this series. Annotations Annotation Meaning More information required Accuracy mark awarded zero Accuracy mark awarded one Independent accuracy mark awarded zero Independent accuracy mark awarded one Independent accuracy mark awarded two Benefit of the doubt Blank Page Incorrect Dep Used to indicate DM0 or DM1

9231/12 Cambridge International AS & A Level – Mark Scheme PUBLISHED October/November 2025 © Cambridge University Press & Assessment 2025 Page 5 of 19 Annotation Meaning DM1 Dependent on the previous M1 mark(s) Follow through Indicate working that is right or wrong Highlighter Highlight a key point in the working Ignore subsequent work Judgement Judgement Method mark awarded zero Method mark awarded one Method mark awarded two Misread Omission or Other solution Off-page comment Allows comments to be entered at the bottom of the RM marking window and then displayed when the associated question item is navigated to. On-page comment Allows comments to be entered in speech bubbles on the candidate response. Judgment made by the PE Premature approximation Special case Indicates that work/page has been seen

9231/12 Cambridge International AS & A Level – Mark Scheme PUBLISHED October/November 2025 © Cambridge University Press & Assessment 2025 Page 6 of 19 Annotation Meaning Error in number of significant figures Correct Transcription error Correct answer from incorrect working

9231/12 Cambridge International AS & A Level – Mark Scheme PUBLISHED October/November 2025 © Cambridge University Press & Assessment 2025 Page 7 of 19 Mark Scheme Notes The following notes are intended to aid interpretation of mark schemes in general, but individual mark schemes may include marks awarded for specific reasons outside the scope of these notes. Types of mark M Method mark, awarded for a valid method applied to the problem. Method marks are not lost for numerical errors, algebraic slips or errors in units. However, it is not usually sufficient for a candidate just to indicate an intention of using some method or just to quote a formula; the formula or idea must be applied to the specific problem in hand, e.g. by substituting the relevant quantities into the formula. Correct application of a formula without the formula being quoted obviously earns the M mark and in some cases an M mark can be implied from a correct answer. A Accuracy mark, awarded for a correct answer or intermediate step correctly obtained. Accuracy marks cannot be given unless the associated method mark is earned (or implied). B Mark for a correct result or statement independent of method marks. DM or DB When a part of a question has two or more ‘method’ steps, the M marks are generally independent unless the scheme specifically says otherwise; and similarly, when there are several B marks allocated. The notation DM or DB is used to indicate that a particular M or B mark is dependent on an earlier M or B (asterisked) mark in the scheme. When two or more steps are run together by the candidate, the earlier marks are implied and full credit is given. FT Implies that the A or B mark indicated is allowed for work correctly following on from previously incorrect results. Otherwise, A or B marks are given for correct work only. • A or B marks are given for correct work only (not for results obtained from incorrect working) unless follow through is allowed (see abbreviation FT above). • For a numerical answer, allow the A or B mark if the answer is correct to 3 significant figures or would be correct to 3 significant figures if rounded (1 decimal place for angles in degrees). • The total number of marks available for each question is shown at the bottom of the Marks column. • Wrong or missing units in an answer should not result in loss of marks unless the guidance indicates otherwise. • Square brackets [ ] around text or numbers show extra information not needed for the mark to be awarded.

9231/12 Cambridge International AS & A Level – Mark Scheme PUBLISHED October/November 2025 © Cambridge University Press & Assessment 2025 Page 8 of 19 Abbreviations AEF/OE Any Equivalent Form (of answer is equally acceptable) / Or Equivalent AG Answer Given on the question paper (so extra checking is needed to ensure that the detailed working leading to the result is valid) CAO Correct Answer Only (emphasising that no ‘follow through’ from a previous error is allowed) CWO Correct Working Only ISW Ignore Subsequent Working SOI Seen Or Implied SC Special Case (detailing the mark to be given for a specific wrong solution, or a case where some standard marking practice is to be varied in the light of a particular circumstance) WWW Without Wrong Working AWRT Answer Which Rounds To

9231/12 Cambridge International AS & A Level – Mark Scheme PUBLISHED October/November 2025 © Cambridge University Press & Assessment 2025 Page 9 of 19 Question Answer Marks Guidance 1(a) ( ) 3 2 2 1 1 4 2 1 ( 1) ( 1) = − = + − +  n r r r n n n n M1 A1 Substitutes formulae. For the 1st A mark allow any letter for n. 1 4 ( 1)( 1)( 2) = + − + n n n n A1 Factorises. 3 1(b) 3 3 2 3 1 1 1 + = − + − + − r r r r r r M1 A1 Finds partial fractions. M mark is for an equation that would lead to values for A, B and C. 2 3 2 (1) 3 (2) (3) ( 1) ( 1) = + = − + − +  n r r f f f r r r 2 (2) 3 (3) (4) + − + f f f 2 (3) 3 (4) (5) + − + f f f  2 ( 2) 3 ( 1) ( ) + − − − + f n f n f n 2 ( 1) 3 ( ) ( 1) + − − + + f n f n f n M1 A1 Shows five complete terms including first two and last two. 3 2 1 2 1 = − + + n n A1 SCB1 if insufficient terms to show all of the cancelling, but must start at r = 2 5 1(c) 3 2 B1FT FT from their part (b). Needs at least two terms in part (b) with one involving a number and a term in n. 1

9231/12 Cambridge International AS & A Level – Mark Scheme PUBLISHED October/November 2025 © Cambridge University Press & Assessment 2025 Page 10 of 19 Question Answer Marks Guidance 2(a) ( ) 2    − + = + = − b B1 2 3 ( 2) 2 = − −c M1 Uses formula for sum of squares. 1 2 = c A1 3 2(b) 3 2 1 3 0 + + + = S bS cS d M1 Uses original equation or formula for sum of cubes. 3 3 2 3 5 3 = − − − = − S b c d d A1FT Allow FT from part (a) for their value. 2 4 3 1 0 + + + = S bS cS dS M1 Uses original equation. ( )( ) 7 1 2 2 5 ( 2)(5 3 ) 3 2 0 8 0 + − − + + =  − = d d d A1FT Allow FT from part (a) for their value and for their value of 3 S . 7 16 = d A1 CAO 5

9231/12 Cambridge International AS & A Level – Mark Scheme PUBLISHED October/November 2025 © Cambridge University Press & Assessment 2025 Page 11 of 19 Question Answer Marks Guidance 3(a) 1 5  u (given) B1 States base case. Assume that 5  ku for some positive integer k. B1 States inductive hypothesis. Then 1 6 5 5 10 6 5 5 5 2 2 + + + − − − = − = + + k k k k k k u u u u u u M1 Considers 1 5 + − ku , puts over common denominator. 5 0 2 − =  + k k u u . A1 Hence, by induction, 5  n u for all positive integers n. A1 5 3(b) 2 2 1 6 5 6 5 2 4 5 2 2 2 + + + − − − + + − = − = = + + + n n n n n n n n n n n n u u u u u u u u u u u u M1 A1 Considers 1 + − n n u u or uses 1 5 5 . 2 + − − = + n n n u u u ( )( ) 5 1 . 2 − + = + n n n u u u So 1 5 0 +   −  n n n u u u A1 Uses 5.  n u 3 Question Answer Marks Guidance 4(a)(i) Stretch B1 parallel to the x-axis, scale factor k. B1 Allow in the x direction, x axis 2

9231/12 Cambridge International AS & A Level – Mark Scheme PUBLISHED October/November 2025 © Cambridge University Press & Assessment 2025 Page 12 of 19 Question Answer Marks Guidance 4(a)(ii) Enlargement, B1 centre the origin, scale factor k. B1 2 4(b) 0 1 0 2 1 1 2 1 1 1 1 0 1 1 2 1 1 2 1 0 1   − −        − =           −                 m k m m k k m M1 A1 Multiplies two matrices correctly. Allow one error for the M mark. 2 2 2 2 2     − + + −+     + −+ +   m k m k m k m m m k m k m m k A1 2 2 2 2 2 2 2 2 2 + −+ − + −+ − + + − + −+ + + + + −+ k m k m k m k m k m k m m m m k m k m k m k m k m k m M1 A1 Finds determinant. Allow two errors for the M mark ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) 2 2 2 2 2 2 2 2 2 2 2 2 2 6 6 12 0 = + + − −+ −+ − − + + + + −+ + − + −+ − + + = + − = m k m k m m k m k m m k m k m m k m k m m k m k m m k m k m m k m k m k A1 AG. There must be evidence to show that the determinant is equal to zero 6

9231/12 Cambridge International AS & A Level – Mark Scheme PUBLISHED October/November 2025 © Cambridge University Press & Assessment 2025 Page 13 of 19 Question Answer Marks Guidance 5(a) B1 Initial line drawn and correct shape. Maximum distance of C from the pole is 1. B1 2 5(b) 1 8 1 2 π 0 d tan2  M1 Uses 1 2 2 d r with correct limits.   1 1 8 8π 0 π 1 1 2 4 0 sin o 2 d cos2 lnc s2     = = −  M1 A1 Integrates a correct function to get   1 2 lncos2 − condoning the lack of limits and factor of 1 2 for M1A1. 2 1 1 4 2 8 ln ln2 = − = A1 OE, must be exact. ISW 4

9231/12 Cambridge International AS & A Level – Mark Scheme PUBLISHED October/November 2025 © Cambridge University Press & Assessment 2025 Page 14 of 19 Question Answer Marks Guidance 5(c) 2 2 2tan 1 tan   = − r M1 Uses relevant identity for tan2. ( ) 2 2 2 2 2 2 2 2 1 + = = − − y x y x xy x y x y M1 For the M mark uses either 2 2 = + r x y or tan= y x . A1 Use both 2 2 = + r x y and tan= y x correctly. ( )( ) 2 2 2 2 2 + − = x y x y xy 4 4 2 − = x y xy 4 4 2 0. − − = x xy y A1 At least one intermediate step before the answer given. Needs to be in this form as answer given. 4 5(d) ( )( ) 1 1 1 1 2 8 8 8 cos π sin π ln2 − M1 Area of triangle minus their (b). 1 1 8 8 2 ln2 − A1 By double angle formula or otherwise. 2 Question Answer Marks Guidance 6(a) 2 2 2 1 1 14 1 3 2 = + + M1 A1 Allow 0.267 or better 2

9231/12 Cambridge International AS & A Level – Mark Scheme PUBLISHED October/November 2025 © Cambridge University Press & Assessment 2025 Page 15 of 19 Question Answer Marks Guidance 6(b) 1 0 1 2 4 2 0 1 2   − =   −  i j k M1 A1 Finds normal to OAB. Allow 1 error for the M mark to be awarded. 1 1 4 3 21 14 cos cos 21 14 2 2 17    =     =   M1 Takes dot product of normal vectors. 7.5° A1 Accept 0.131 radians. Mark final answer of 7.5°or better. 4

9231/12 Cambridge International AS & A Level – Mark Scheme PUBLISHED October/November 2025 © Cambridge University Press & Assessment 2025 Page 16 of 19 Question Answer Marks Guidance 6(c) 2 2 2 1 , 1 1 2 2 2 1 3 3                       = − = −+  =             − −  − −  + + −  +     OP OQ PQ M1 A1 Finds  PQ or direction of common perpendicular. 2 2 1 0 2 1 1 2 3             − =       − − +    − −+ +  M1 Uses that dot product of  PQ with line direction is zero, or, alternatively,  PQ is a multiple of the common perpendicular 1 1 1      . 6 6 1   − = A1 Deduces one equation. 7 2 2 1 0 14 2 3 2 1 6 3         −       =        − −    −+ + − = +  A1 Deduces second equation. 3 2 1 1 4 4 1 4 6 3 1 . 4 1          =  = − = −         − −      OQ M1 A1 Solves for  and substitutes into  OQ and  PQ or 7 12 2 7 1 . 12 1      =  = −     −    OP 1 4 6 1 1 1 1 1       = − +       −    k r B1 FT OE. FT using their common perpendicular. Must have r = 8

9231/12 Cambridge International AS & A Level – Mark Scheme PUBLISHED October/November 2025 © Cambridge University Press & Assessment 2025 Page 17 of 19 Question Answer Marks Guidance 7(a) 1 = − x B1 Vertical asymptote. ( 1) 1 1 + + = + x x y x M1 Oblique asymptote. = y x A1 3 7(b) ( ) 2 2 d 2 0 d 1 + = = + y x x x x M1 Sets d 0. d = y x ( ) 0,1 , ( ) 2, 3 − − A1 A1 1 mark for each correct coordinate. SCB1 if just the correct x values listed 3

9231/12 Cambridge International AS & A Level – Mark Scheme PUBLISHED October/November 2025 © Cambridge University Press & Assessment 2025 Page 18 of 19 Question Answer Marks Guidance 7(c) B1FT Axes and asymptotes labelled. B1 Left branch correct with all approaches to the asymptotes correct. B1 Right branch correct with a smooth minimum point. 3

9231/12 Cambridge International AS & A Level – Mark Scheme PUBLISHED October/November 2025 © Cambridge University Press & Assessment 2025 Page 19 of 19 Question Answer Marks Guidance 7(d) B1FT Symmetrical about x = 0 and follow through from part (c) right branch B1 Correct shape of a curve and smooth at (0,1) from a correct graph in part (c) 2 7(e) 2 1 3 1 + + = + x x x 2 2 2 0 − − = x x M1 Finds critical point. 1 3 = + x A1 For selecting just the positive value of x. 1 3 1 3 −   − + x A1 3