Cambridge A Level Mathematics - Further 9231 — 2023 Oct/Nov Paper 1 · Variant 2

This document has 16 pages. Any blank pages are indicated. [Turn over * 6 2 8 9 7 0 9 4 4 3 * DC (PQ) 311424/2 © UCLES 2023 FURTHER MATHEMATICS 9231/12 Paper 1 Further Pure Mathematics 1 October/November 2023 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 [ ]. Cambridge International AS & A Level

2 9231/12/O/N/23 © UCLES 2023 1 (a) Use standard results from the list of formulae (MF19) to find ( ) r r 3 3 1 r n 2 1 + + =/ in terms of n, simplifying your answer. [3] … … … … … … … … … … … … … … … … … … … … … … … … … …

3 9231/12/O/N/23 © UCLES 2023 [Turn over (b) Show that ( ) ( ) r r r r r r 1 1 1 1 3 3 1 3 3 3 3 2 - + = + + + and hence use the method of differences to find ( ) r r r r 1 3 3 1 r n 3 3 2 1 + + + =/ . [5] … … … … … … … … … … … … … … … … … … … (c) Deduce the value of ( ) r r r r 1 3 3 1 r 3 3 2 1 + + + 3 =/ . [1] … … …

4 9231/12/O/N/23 © UCLES 2023 2 Prove by mathematical induction that, for all positive integers n, ( ) x x x nx n n 2 1 d d e e n n x x 2 2 = + + - ` ` j j . [6] … … … … … … … … … … … … … … … … … … … … … … … … … …

5 9231/12/O/N/23 © UCLES 2023 [Turn over 3 The matrix M is given by M k 0 0 1 0 1 1 1 = e eo o, where k is a constant and k 0 ! and k 1 ! . (a) The matrix M represents a sequence of two geometrical transformations. State the type of each transformation, and make clear the order in which they are applied. [2] … … … The unit square in the x – y plane is transformed by M onto parallelogram OPQR. (b) Find, in terms of k, the area of parallelogram OPQR and the matrix which transforms OPQR onto the unit square. [3] … … … … … (c) Show that the line through the origin with gradient k 1 1 - is invariant under the transformation in the x – y plane represented by M. [3] … … … … … … … … … … … … …

6 9231/12/O/N/23 © UCLES 2023 4 The cubic equation x x x 27 18 6 1 0 3 2 + + - = has roots a, b, c. (a) Show that a cubic equation with roots 3 1 a+ , 3 1 b+ , 3 1 c+ is y y y 2 0 3 2 - + - = . [3] … … … … … … … … … … … … … … … … … … … … … … … … …

7 9231/12/O/N/23 © UCLES 2023 [Turn over The sum ( ) ( ) ( ) 3 1 3 1 3 1 n n n a b c + + + + + is denoted by Sn. (b) Find the values of S2 and S3. [4] … … … … … … … … … … … … … (c) Find the values of S 1 - and S 2 - . [3] … … … … … … … … … … … …

8 9231/12/O/N/23 © UCLES 2023 5 The plane 1 P has equation ( ) ( ) r i j k i j k i k 2 2 3 3 m n = - - + - - + - . (a) Find an equation for 1 P in the form ax by cz d + + = . [4] … … … … … … … … … … … … … … … … The line l, which does not lie in 1 P , has equation ( ) t r i k i j k 3 = - + + + + . (b) Show that l is parallel to 1 P . [2] … … … … … … … …

9 9231/12/O/N/23 © UCLES 2023 [Turn over (c) Find the distance between l and 1 P . [3] … … … … … … … … … … … … (d) The plane 2 P has equation x y z 3 3 2 1 + + = . Find a vector equation of the line of intersection of 1 P and 2 P . [4] … … … … … … … … … … … … …

10 9231/12/O/N/23 © UCLES 2023 6 The curve C has polar equation r e e r 2 1 = - i - - , where r 0 2 1 G G i . (a) Sketch C and state, in exact form, the greatest distance of a point on C from the pole. [3] … (b) Find the exact value of the area of the region bounded by C and the initial line. [5] … … … … … … … … … … … … … …

11 9231/12/O/N/23 © UCLES 2023 [Turn over … … … … … (c) Show that, at the point on C furthest from the initial line, tan 1 0 e r 2 1 i - - = i- and verify that this equation has a root between 0.56 and 0.57. [5] … … … … … … … … … … … … … … … … … … … …

12 9231/12/O/N/23 © UCLES 2023 BLANK PAGE

13 9231/12/O/N/23 © UCLES 2023 [Turn over 7 The curve C has equation ( ) y x f = , where ( )x x x 1 f 2 = + . (a) Find the equations of the asymptotes of C. [3] … … … … … … … … … … … … … … (b) Find the coordinates of any stationary points on C. [2] … … … … … … … … … … …

14 9231/12/O/N/23 © UCLES 2023 (c) Sketch C. [3] (d) Find the coordinates of any stationary points on the curve with equation ( ) y x 1 f = . [2] … … … … … … … …

15 9231/12/O/N/23 © UCLES 2023 (e) Sketch the curve with equation ( ) y x 1 f = and find, in exact form, the set of values for which ( ) ( ) x x 1 f f 2 . [6] … … … … … …

16 9231/12/O/N/23 © UCLES 2023 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. Additional page If you use the following page to complete the answer to any question, the question number must be clearly shown. … … … … … … … … … … … … … … … … … … … … … …

This document consists of 15 printed pages. © UCLES 2023 [Turn over Cambridge International AS & A Level FURTHER MATHEMATICS 9231/12 Paper 1 Further Pure Mathematics 1 October/November 2023 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 2023 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 2023 © UCLES 2023 Page 2 of 15 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 descriptors 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). 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.

9231/12 Cambridge International AS & A Level – Mark Scheme PUBLISHED October/November 2023 © UCLES 2023 Page 3 of 15 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 2023 © UCLES 2023 Page 4 of 15 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 2023 © UCLES 2023 Page 5 of 15 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 2023 © UCLES 2023 Page 6 of 15 Question Answer Marks Guidance 1(a) 3 1 2 2 ( 1)(2 1) ( 1) + + + + + n n n n n n M1 A1 Substitutes correct formulae from MF19. 3 2 3 3 n n n + + A1 Simplifies 3 1(b) 3 2 3 2 3 3 3 3 3 3 3 3 3 3 1 3 1 3 1 ( 1) ( 1) ( 1) ( 1 ( 1 ) 3 1 ) 3 + − + = − + + + − + = + + + + = r r r r r r r r r r r r r r r r M1 A1 Puts over a common denominator and expands, AG. ( ) 3 3 1 1 2 3 3 3 3 3 3 3 3 3 1 ( 1) 1 1 1 1 1 1 1 2 1 1 2 3 ( 1) = =   −     + + = + + = − + − + + − +   n n r r r r r n n r r r M1 A1 Shows three complete terms, including last. ( ) 3 1 1 1 n − + A1 5 1(c) 1 B1FT FT from their answer to part (b). 1

9231/12 Cambridge International AS & A Level – Mark Scheme PUBLISHED October/November 2023 © UCLES 2023 Page 7 of 15 Question Answer Marks Guidance 2 ( ) ( ) 2 2 2 d e e 2 e 2 e d = + = + x x x x x x x x x x so true when 1. = n M1 A1 Differentiates once using the product rule. Assume that ( ) ( ) 2 2 d e 2 ( 1) e d = + + − x x k k x x kx k k x [for some value of k ]. B1 States inductive hypothesis. ( ) ( ) ( ) 2 1 2 1 d e 2 ( 1) e e 2 2 d + + = + + − + + x x x k k x x kx k k x k x M1 Differentiates kth derivative. ( ) 2 2( 1) ( 1) ex x k x k k + + + + A1 So true when 1. = + n k By induction, true for all positive integers n. A1 States conclusion. 6 Question Answer Marks Guidance 3(a) Shear followed by a stretch. B2 Award B1 if given in the wrong order. 2 3(b) det = = OPQR k M B1 1 1 0 1 1 −   =   −   k k M M1 A1 3

9231/12 Cambridge International AS & A Level – Mark Scheme PUBLISHED October/November 2023 © UCLES 2023 Page 8 of 15 Question Answer Marks Guidance 3(c) 1 1 0 1 1 −            k x k x B1 Sets 1 1 . − = k y x 1 1 1 1 1 0 1 1 − − −         = =         +        k k k k kx x kx k x x x x M1 Shows that 1 1 . − = k Y X 1 1 k x k x −       A1 Alternative method for 3(c) 0 1 1     =     +     k x kx y x y B1 Transforms       x y to       X Y = X kx and = + mX x y = + mkx x mx M1 Uses y = mx and Y = mX 1 1 = − m k 1 1 = − y x k A1 AG 3

9231/12 Cambridge International AS & A Level – Mark Scheme PUBLISHED October/November 2023 © UCLES 2023 Page 9 of 15 Question Answer Marks Guidance 4(a) ( ) 1 3 1 3 1 − + =  = y x x y 3 2 1 1 1 27 18 6 1 0 3 3 3 − − −        + + −=             y y y B1 Substitutes. ( ) ( ) 3 2 1 2 1 2( 1) 1 0  − + − + − −= y y y 3 2 2 3 3 1 2 4 2 2 2 1 0  − + −+ − + + − −= y y y y y y M1 Expands. 3 2 2 0 + − − = y y y A1 AG. 3 4(b) 2 2 1 2(1) 1 = − = − S M1 A1 Uses formula for sum of squares, AG. ( ) 3 3 3 3 1) 1) 6 1 (3 (3 (3 1 1 )    + + + = + + = −− + S M1 Uses 3 2 2 + = − y y y or expands and uses original equation. 4 A1 4 4(c) ( ) ( )( ) ( ) ( )( ) 1 (3 (3 1 (3 1 2 1) 3 1 3 1 3 1 3 1 ) 1) 3 1 3 1          − + + + + + + + + + + = = + S B1 1 2 2 1 1 2 3 1 3 − − = − + = − + S S S M1 Uses 2 1 2 1 . − − = −+ y y y 3 4 2 −= − S A1 CAO 3

9231/12 Cambridge International AS & A Level – Mark Scheme PUBLISHED October/November 2023 © UCLES 2023 Page 10 of 15 Question Answer Marks Guidance 5(a) 2 1 1 2 3 8 ~ 4 3 0 1 6 3         − − = − −         −     i j k M1 A1 Finds perpendicular to 1 . 1(1) 4( 1) 3( 2) 1 − − + − = − M1 Uses point on 1.  4 3 1 − + = − x y z A1 4 5(b) 1 1 4 1 1 4 3 0 3 1     − = − + =       M1 A1 Shows dot product with direction of line is 0. 2 5(c) 2 2 2 4 1 1 1 4 1 4 3 3 3 −       −    + +       or 2 2 2 3 1 1 0 4 1 1 4 3 1 3   −         − +      + +           M1 A1 Uses correct formula for distance from point on l to 1 . ( ) 2 2 2 1 3.1 0. 4 1.3 1 1 4 3 − + − + + + + 1 ( 0.196) 26 = A1 3

9231/12 Cambridge International AS & A Level – Mark Scheme PUBLISHED October/November 2023 © UCLES 2023 Page 11 of 15 Question Answer Marks Guidance 5(d) States point common to both planes e.g. 1 15 4 15 . 0           B1 5 7 0 4 7           −     or 0 5 17 1 17                   or alternative. 17 1 4 3 7 3 3 2 15 −     − =       i j k M1 A1 Finds direction of line. 5 7 4 7 17 0 7 15    −       = +         −     r A1 OE. 4

9231/12 Cambridge International AS & A Level – Mark Scheme PUBLISHED October/November 2023 © UCLES 2023 Page 12 of 15 Question Answer Marks Guidance 6(a) B1 Initial line drawn. Correct shape, r strictly decreasing. B1 Correct shape at extremities. 1 2 π 1 e − − B1 May be seen on their diagram. 3 6(b) ( ) 1 1 2 2 2 π π 1 2 0 e e d   − −−  M1 Uses correct formula with correct limits. 1 1 2 2 π π 2 π 1 2 0 e d 2e e    −− − − − +  A1 1 1 2 2 π 2 π 2 π 1 1 2 0 e 2e e    −− − −   − + +   M1 A1 Integrates. ( ) 1 1 2 2 3 1 1 1 1 1 1 2 2 2 2 4 4 4 π π π π π π π πe e πe e 2e e e 2 − − − − − − − − + + + − = + − + A1 5 0 =

9231/12 Cambridge International AS & A Level – Mark Scheme PUBLISHED October/November 2023 © UCLES 2023 Page 13 of 15 Question Answer Marks Guidance 6(c) ( ) 1 2π s e i e n   − − = − y B1 Uses sin = y r ( ) ( ) 1 2π cos sin e 0 d e e d      − − − = + − − = y M1 A1 Sets derivative equal to zero. 1 2 1 2 π 1 2 π e 1 tan 0 1 0 e e tan e        − − − −   −   + = −  − − =        A1 AG. 1 2 0.56 π 1 e tan0.56 0.00912 − − − = and 1 2 0.57 π 1 e tan0.57 0.00856 − − − = − B1 Shows sign change. 5 Question Answer Marks Guidance 7(a) 1 = − x B1 Vertical asymptote. ( )( ) 1 1 1 1 + − + = + x x y x M1 Oblique asymptote. 1 = − 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 ( ) 0,0 , ( ) 2, 4 − − A1 2

9231/12 Cambridge International AS & A Level – Mark Scheme PUBLISHED October/November 2023 © UCLES 2023 Page 14 of 15 Question Answer Marks Guidance 7(c) B1 Axes and asymptotes. B1 Left branch correct. B1 Right branch correct. 3 7(d) ( ) 1 4 2, −− B1 B1 B1 for each correct coordinate. SC B1 for ( ) 1 4 2, −− and ( ) 0,0 . 2 x y

9231/12 Cambridge International AS & A Level – Mark Scheme PUBLISHED October/November 2023 © UCLES 2023 Page 15 of 15 Question Answer Marks Guidance 7(e) B1 Left branch correct. B1 Right branch correct. 2 1 1 = + x x or 2 1 1 = − + x x 2 1 0 − −= x x M2 Finds critical points, award M1 for each case. 1 1 2 2 5 = − x or 1 1 2 2 5 = + x A1 1, − x 1 1 1 1 2 2 2 2 5 5, +   − x 0  x B1 Condone missing 0  x . 6 x y