Cambridge A Level Mathematics - Further 9231 — 2020 May/June Paper 1 · Variant 2

Cambridge International AS & A Level This document has 16 pages. Blank pages are indicated. DC (LK) 187468/2 © UCLES 2020 [Turn over FURTHER MATHEMATICS 9231/12 Paper 1 Further Pure Mathematics 1 May/June 2020 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 [ ]. * 7 6 3 9 9 5 8 1 4 0 *

2 9231/12/M/J/20 © UCLES 2020 1 Let a be a positive constant. (a) Sketch the curve with equation y x ax 7 = + . [2]

3 9231/12/M/J/20 © UCLES 2020 [Turn over (b) Sketch the curve with equation y x ax 7 = + and find the set of values of x for which x ax a 7 2 2 + . [4] … … … … … … … … … … … … … …

4 9231/12/M/J/20 © UCLES 2020 2 The cubic equation x px x 6 3 5 0 3 2 + - - = , where p is a constant, has roots a, b, c. (a) Find a cubic equation whose roots are 2 a , 2 b , 2 c . [3] … … … … … … … … … … … … … … … … (b) It is given that ( ) 2 2 2 2 a b c a b c + + = + + . (i) Find the value of p. [3] … … … … … … … …

5 9231/12/M/J/20 © UCLES 2020 [Turn over … … … … … … … … … … … … (ii) Find the value of 3 3 3 a b c + + . [2] … … … … … … … … … … … … … … …

6 9231/12/M/J/20 © UCLES 2020 3 The curve C has equation y x x 2 1 2 = + . (a) Find the equations of the asymptotes of C. [3] … … … … … … … … … … … … … (b) Find the coordinates of the stationary points on C. [3] … … … … … … … … … … … …

7 9231/12/M/J/20 © UCLES 2020 [Turn over (c) Sketch C. [3]

8 9231/12/M/J/20 © UCLES 2020 4 (a) By first expressing r 1 1 2 - in partial fractions, show that ( ) r n n an b 1 1 4 3 2 1 r n 2 2 - = - + + =/ , where a and b are integers to be found. [5] … … … … … … … … … … … … … … … … … … … … … … … …

9 9231/12/M/J/20 © UCLES 2020 [Turn over (b) Deduce the value of r 1 1 r 2 2 - 3 =/ . [1] … … … … … … … … … … (c) Find the limit, as n " 3, of r n 1 r n n 2 1 2 - = +/ . [4] … … … … … … … … … … … … … … …

10 9231/12/M/J/20 © UCLES 2020 5 The lines l1 and l2 have equations ( ) r i k i j k 3 3 4 4 m = + + + + and ( ) r i j k j k 3 5 6 5 6 n = - - + + respectively. (a) Find the shortest distance between l1 and l2. [5] … … … … … … … … … … … … … … … … … … … … … … … … … …

11 9231/12/M/J/20 © UCLES 2020 [Turn over The plane P contains l1 and is parallel to the vector i k + . (b) Find the equation of P, giving your answer in the form ax by cz d + + = . [4] … … … … … … … … … … … … … … (c) Find the acute angle between l2 and P. [3] … … … … … … … … … … …

12 9231/12/M/J/20 © UCLES 2020 6 Let A 2 1 0 1 = e o. (a) The transformation in the x-y plane represented by A 1 - transforms a triangle of area cm 30 2 into a triangle of area cm d 2. Find the value of d. [3] … … … … … (b) Prove by mathematical induction that, for all positive integers n, A 2 2 1 0 1 n n n = - e o. [5] … … … … … … … … … … … … … … … … …

13 9231/12/M/J/20 © UCLES 2020 [Turn over (c) The line y x 2 = is invariant under the transformation in the x-y plane represented by A B n , where B 1 33 0 0 = e o. Find the value of n. [5] … … … … … … … … … … … … … … … … … … … … … … … … …

14 9231/12/M/J/20 © UCLES 2020 7 The curve C1 has polar equation cos r i i = , for r 0 2 1 G G i . (a) The point on C1 furthest from the line r 2 1 i = is denoted by P. Show that, at P, tan 2 1 0 i i- = and verify that this equation has a root between 0.6 and 0.7. [5] … … … … … … … … … … … … … … … The curve C2 has polar equation sin r i i = , for r 0 2 1 G G i . The curves C1 and C2 intersect at the pole, denoted by O, and at another point Q. (b) Find the polar coordinates of Q, giving your answers in exact form. [2] … … … … … …

15 9231/12/M/J/20 © UCLES 2020 (c) Sketch C1 and C2 on the same diagram. [3] (d) Find, in terms of r, the area of the region bounded by the arc OQ of C1 and the arc OQ of C2. [7] … … … … … … … … … … … … … … … … … …

16 9231/12/M/J/20 © UCLES 2020 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 the Cambridge Assessment Group. Cambridge Assessment is the brand name of the University of Cambridge Local Examinations Syndicate (UCLES), which itself is a department of the University of Cambridge. Additional Page If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown. … … … … … … … … … … … … … … … … … … … … … …

This document consists of 15 printed pages. © UCLES 2020 [Turn over Cambridge International AS & A Level FURTHER MATHEMATICS 9231/12 Paper 1 Further Pure Mathematics 1 May/June 2020 MARK SCHEME Maximum Mark: 75 Published Students did not sit exam papers in the June 2020 series due to the Covid-19 global pandemic. This mark scheme is published to support teachers and students and should be read together with the question paper. It shows the requirements of the exam. The answer column of the mark scheme shows the proposed basis on which Examiners would award marks for this exam. Where appropriate, this column also provides the most likely acceptable alternative responses expected from students. Examiners usually review the mark scheme after they have seen student responses and update the mark scheme if appropriate. In the June series, Examiners were unable to consider the acceptability of alternative responses, as there were no student responses to consider. Mark schemes should usually be read together with the Principal Examiner Report for Teachers. However, because students did not sit exam papers, there is no Principal Examiner Report for Teachers for the June 2020 series. Cambridge International will not enter into discussions about these mark schemes. Cambridge International is publishing the mark schemes for the June 2020 series for most Cambridge IGCSE™ and Cambridge International A & AS Level components, and some Cambridge O Level components.

9231/12 Cambridge International AS & A Level – Mark Scheme PUBLISHED May/June 2020 © UCLES 2020 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 May/June 2020 © UCLES 2020 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, 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 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 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 May/June 2020 © UCLES 2020 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.

9231/12 Cambridge International AS & A Level – Mark Scheme PUBLISHED May/June 2020 © UCLES 2020 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 May/June 2020 © UCLES 2020 Page 6 of 15 Question Answer Marks 1(a) (B1 for axes and asymptotes correct, B1 for branches correct) B1 B1 2

9231/12 Cambridge International AS & A Level – Mark Scheme PUBLISHED May/June 2020 © UCLES 2020 Page 7 of 15 Question Answer Marks 1(b) (B1 FT from sketch in part (a)) B1 FT 7 2 = + ax x a or 2 7 = − + x x a a M1 7 = x and 7 3 = − x A1 7 < − x , 3 7 7 −< < − x , 7 > x A1 4

9231/12 Cambridge International AS & A Level – Mark Scheme PUBLISHED May/June 2020 © UCLES 2020 Page 8 of 15 Question Answer Marks 2(a) 2 = y x B1 ( ) 3 1 2 2 1 2 2 2 2 2 2 6 3 5 0 5 (6 3) ( 5) 3 5 6 6 3 (36 9) 10 2 + − − = = − + − = +  − − − + = − +  y py y py y y y py y y y py y y p M1 3 2 2 36 ( 36) (10 9) 25 0 − + + + − = y p y p y A1 3 2(b)(i) 2 2 2 2 36 36 α β γ = + + + p B1 2 2 1 36 2 36 0 36 6 2 + = −  + + = p p p p M1 6 = − p A1 3 2(b)(ii) 3 3 3 2 2 2 5 6( ) 6( ) 3( ) 1 α β γ α β γ α β γ + + + + + + + = + M1 3 3 3 5 α β γ + = + A1 2

9231/12 Cambridge International AS & A Level – Mark Scheme PUBLISHED May/June 2020 © UCLES 2020 Page 9 of 15 Question Answer Marks 3(a) 1 2 = − x B1 ( ) 1 1 1 2 4 4 1 1 2 4 1 2 1 4( ) 2 1 ( 2 ) 1 + − + = − = + + + x x x x x y M1 1 1 2 4 = − y x A1 3 3(b) ( ) 2 2 d 1 1 0 (2 1) 1 d 2 2 2 1 = − =  + = + y x x x M1 0, 1 = − x A1 (0,0),( 1, 1) −− A1 3

9231/12 Cambridge International AS & A Level – Mark Scheme PUBLISHED May/June 2020 © UCLES 2020 Page 10 of 15 Question Answer Marks 3(c) (B1 for axes and asymptotes correct, B1 for upper branch correct, B1 for lower branch correct) B1 B1 B1 3

9231/12 Cambridge International AS & A Level – Mark Scheme PUBLISHED May/June 2020 © UCLES 2020 Page 11 of 15 Question Answer Marks 4(a) ( )( ) 2 1 1 1 1 1 1 1 2 1 1 1   = = −   − + − + −   r r r r r M1 A1 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 3 2 4 3 5 4 6 2 1 1 1 =   = − + − + − + − + + − + −   − − + −     r n r n n n n M1 1 1 1 1 1 2 2 1   = + − −   +   n n A1 ( ) 3 2 1 4 2 1 + = − + n n n A1 5 4(b) 3 4 B1 1 4(c) 2 2 2 2 2 1 2 2 1 1 1 = + = = = − − − −    n n n r n r r n n n r r r M1 ( ) ( ) ( ) ( ) ( ) ( ) 2 1 4 1 3 4 1 3 2 1 4 4 2 1 4 2 1 2 1 4 2 1     =  + + + + − − + − =       −   +    + + n n n n n n n n n n n n n n n n M1A1 1 2 → as →∞ n A1 4

9231/12 Cambridge International AS & A Level – Mark Scheme PUBLISHED May/June 2020 © UCLES 2020 Page 12 of 15 Question Answer Marks 5(a) 3 3 0 5 0 5 6 3 9           − − = −           − −      B1 4 1 4 4 6 0 5 6 5     = −       i j k M1 A1 0 4 1 15 5 6 1.71 77 77 9 5       − − = =       −     M1 A1 5 5(b) 4 1 4 4 3 1 0 1 4     =     −   i j k M1 A1 4(3) 3(0) 4(3) 0 4 3 4 0 + − =  + − = x y z M1 A1 4 5(c) 0 4 5 3 61 41cos cos 61 41 6 4 9 α α    =     − −  =    (A1 FT their normal) M1 A1FT Acute angle between 2l and Π is 90 10.4 α ° − = A1 3

9231/12 Cambridge International AS & A Level – Mark Scheme PUBLISHED May/June 2020 © UCLES 2020 Page 13 of 15 Question Answer Marks 6(a) ( ) 1 1 2 1 det det − − = = A A M1 A1 15 = d A1 3 6(b) 1 1 2 0 2 0 1 1 2 1 1     = =       −     A so true when 1. = n B1 Assume that it is true for = n k , so 2 0 2 1 1   =     −   k k k A . B1 Then ( ) 1 1 1 1 2 0 2 0 2 0 2 0 1 1 2 2 1 1 1 2 1 1 2 1 1 + + + +           = = =             − + − −         k k k k k k k A M1A1 So, it is also true for 1 = + n k . Hence, by induction, true for all positive integers. A1 5 6(c) 2 0 1 0 2 0 33 0 2 1 1 2 32 0       = =           − +       n n n n n A B M1A1 2 0 2 2 32 0 (2 32)     =          + +       n n n n x x y x B1 1 (2 32) 2 32 5 2 +   + = = = n n n x x n M1 A1 5

9231/12 Cambridge International AS & A Level – Mark Scheme PUBLISHED May/June 2020 © UCLES 2020 Page 14 of 15 Question Answer Marks 7(a) 2 cos θ θ = x B1 2 d 2 cos sin cos 0 d θ θ θ θ θ = − + = x M1 A1 cos 0 2 tan 1 0 θ θ θ ≠  −= A1 2(0.6)tan0.6 1 0.179 −= − and 2(0.7)tan0.7 1 0.179 −= B1 5 7(b) cos sin tan 1 θ θ θ θ θ =  = M1 ( ) 1 1 8 4 π 2, π √ A1 2 7(c) (B1 for initial line drawn and C1 correct, B1 for C2 correct, B1 for intersection correct) B1 B1 B1 3

9231/12 Cambridge International AS & A Level – Mark Scheme PUBLISHED May/June 2020 © UCLES 2020 Page 15 of 15 Question Answer Marks 7(d) ( ) 1 4π 2 2 2 1 2 0 cos sin d θ θ θ θ −  M1 1 4π 2 1 2 0 cos2 d θ θ θ =  M1 1 1 4 4 π π 2 1 1 2 2 0 0 sin 2 sin 2 d θ θ θ θ θ     = −        M1 A1 1 1 1 4 4 4 π π π 2 1 1 1 1 2 2 2 2 0 0 0 sin 2 cos2 cos2 d θ θ θ θ θ θ     = −− −            M1 1 4π 2 1 1 4 2 0 sin 2 cos2 sin 2 θ θ θ θ θ   = + −   A1 2 1 1 64 8 π = − A1 7