Cambridge A Level Mathematics 9709 — 2022 May/June Paper 2 · Variant 3

*3281101778* Cambridge International AS & A Level CANDIDATE NAME CENTRE NUMBER CANDIDATE NUMBER MATHEMATICS 9709/23 Paper 2 Pure Mathematics 2 May/June 2022 1 hour 15 minutes You must answer on the question paper. You will need: List of formulae (MF19) INSTRUCTIONS ³ Answer all questions. ³ Use a black or dark blue pen. You may use an HB pencil for any diagrams or graphs. ³ Write your name, centre number and candidate number in the boxes at the top of the page. ³ Write your answer to each question in the space provided. ³ Do not use an erasable pen or correction fluid. ³ Do not write on any bar codes. ³ 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. ³ You should use a calculator where appropriate. ³ You must show all necessary working clearly; no marks will be given for unsupported answers from a calculator. ³ 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 ³ The total mark for this paper is 50. ³ The number of marks for each question or part question is shown in brackets [ ]. This document has 16 pages. Any blank pages are indicated. JC22 06_9709_23/FP © UCLES 2022 [Turn over

2 1 Given that y = ln x x2 , find the exact value of dy dx when x = e. [3] … … … … … … … … … … … … … … … … … … … … … … … … … © UCLES 2022 9709/23/M/J/22

3 2 (a) Sketch, on the same diagram, the graphs of y = 2x −9 and y = 5x −3. [2] (b) Solve the equation 2x −9 = 5x −3. [2] … … … … … … … … … … … © UCLES 2022 9709/23/M/J/22 [Turn over

4 3 A curve has equation e2x cos 2y + sin y = 1. Find the exact gradient of the curve at the point 0, 1 6π. [5] … … … … … … … … … … … … … … … … … … … … … … … … © UCLES 2022 9709/23/M/J/22

5 4 (a) Use the trapezium rule with three intervals to show that the value of Ó 4 1 ln x dx is approximately ln 12. [4] … … … … … … … … … … … … (b) Use a graph of y = ln x to show that ln 12 is an under-estimate of the true value of Ó 4 1 ln x dx. [2] … … … © UCLES 2022 9709/23/M/J/22 [Turn over

6 5 The polynomial px is defined by px = 2x3 + ax2 −3x −4, where a is a constant. It is given that x −4 is a factor of px. (a) Find the value of a and hence factorise px. [4] … … … … … … … … … … … … … … … … … … … … … … © UCLES 2022 9709/23/M/J/22

7 (b) Show that the equation pe3y = 0 has only one real root and find its exact value. [3] … … … … … … … … … … … … … … … … … … … … … … … … … © UCLES 2022 9709/23/M/J/22 [Turn over

8 6 x y O a a+1 The diagram shows the curve y = 3e2x−1. The shaded region is bounded by the curve and the lines x = a, x = a + 1 and y = 0, where a is a constant. It is given that the area of the shaded region is 120 square units. (a) Show that a = 1 2 ln80 + e2a−1 −1 2. [5] … … … … … … … … … … … … … … … … © UCLES 2022 9709/23/M/J/22

9 … … … … … … … … … … (b) Use an iterative formula, based on the equation in part (a), to find the value of a correct to 3 significant figures. Give the result of each iteration to 5 significant figures. [3] … … … … … … … … … … … … … … © UCLES 2022 9709/23/M/J/22 [Turn over

10 7 x y O π The diagram shows the curves y = 2π −2x and y = sin2x for 0 ≤x ≤π. The shaded region is bounded by the two curves and the line x = 0. Find the exact area of the shaded region. [8] … … … … … … … … … … … … … … … … © UCLES 2022 9709/23/M/J/22

11 … … … … … … … … … … … … … … … … … … … … … … … … … © UCLES 2022 9709/23/M/J/22 [Turn over

12 8 (a) Express 3 sin 21 sec 1 + 10 cos1 −30Å in the form R sin1 + ! where R > 0 and 0Å < ! < 90Å. Give the value of ! correct to 2 decimal places. [6] … … … … … … … … … … … … … … … … … … … … … … … … … © UCLES 2022 9709/23/M/J/22

13 (b) Hence solve the equation 3 sin 4" sec 2" + 10 cos2" −30Å = 2 for 0Å < " < 90Å. [3] … … … … … … … … … … … … … … … … … … … … … … … … … © UCLES 2022 9709/23/M/J/22

14 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. … … … … … … … … … … … … … … … … … … … … … … … … © UCLES 2022 9709/23/M/J/22

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16 BLANK PAGE 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. © UCLES 2022 9709/23/M/J/22

This document consists of 11 printed pages. © UCLES 2022 [Turn over Cambridge International AS Level MATHEMATICS 9709/23 Paper 2 Pure Mathematics 2 May/June 2022 MARK SCHEME Maximum Mark: 50 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 May/June 2022 series for most Cambridge IGCSE, Cambridge International A and AS Level and Cambridge Pre-U components, and some Cambridge O Level components.

9709/23 Cambridge International AS Level – Mark Scheme PUBLISHED May/June 2022 © UCLES 2022 Page 2 of 11 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.

9709/23 Cambridge International AS Level – Mark Scheme PUBLISHED May/June 2022 © UCLES 2022 Page 3 of 11 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.

9709/23 Cambridge International AS Level – Mark Scheme PUBLISHED May/June 2022 © UCLES 2022 Page 4 of 11 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.

9709/23 Cambridge International AS Level – Mark Scheme PUBLISHED May/June 2022 © UCLES 2022 Page 5 of 11 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

9709/23 Cambridge International AS Level – Mark Scheme PUBLISHED May/June 2022 © UCLES 2022 Page 6 of 11 Question Answer Marks Guidance 1 Differentiate using quotient rule (or product rule) M1 Obtain 4 2 ln  x x x x A1 OE Substitute e  x to obtain 3 1 e  or exact equivalent A1 3 Question Answer Marks Guidance 2(a) Draw V-shaped graph with vertex on positive x-axis *B1 Draw (more or less) correct graph of 5 3   y x with greater gradient DB1 crossing x-axis between origin and vertex of first graph 2 2(b) Attempt solution of linear equation where signs of 2x and 5x are different M1 Solve 2 9 5 3     x x to obtain 12 7 , 1.71 or better A1 and no second answer Alternative method for question 2(b) Attempt solution of 3-term quadratic equation 2 2 (2 9) (5 3)    x x to obtain at least one value of x M1 2 7 2 24 0    x x Obtain 12 7 , 1,71 or better A1 and no second answer 2

9709/23 Cambridge International AS Level – Mark Scheme PUBLISHED May/June 2022 © UCLES 2022 Page 7 of 11 Question Answer Marks Guidance 3 Use product rule to differentiate 2e cos2 x y M1 Must be evidence of implicit differentiation Obtain 2 2 d 2e cos2 2e sin2 d  x x y y y x A1 Obtain 2 2 d d 2e cos2 2e sin2 cos 0 d d          x x y y y y y x x B1 Substitute x- and y-values to find value of first derivative M1 Dependent at least two terms, with at least one involving d d y x Obtain 2 3 or 2 3 3 or exact equivalent A1 5 Question Answer Marks Guidance 4(a) Use y-values   ln1 , ln 2, ln3, ln 4 B1 but not their decimal equivalents Use correct formula, or equivalent, with 1  h M1 allow with decimal equivalents Use both relevant logarithm properties correctly M1 Obtain 1 2[ln1 2ln 2 2ln3 ln 4]    and hence ln12 A1 AG – necessary detail needed 4

9709/23 Cambridge International AS Level – Mark Scheme PUBLISHED May/June 2022 © UCLES 2022 Page 8 of 11 Question Answer Marks Guidance 4(b) Sketch correct graph of y = ln x for at least y ⩾ 0 *B1 Indicate that top of each trapezium is below curve or clear equivalent DB1 but B0 if only one chord shown from (1, 0) to (4, ln 4) allow ‘concave down’ 2 Question Answer Marks Guidance 5(a) Substitute 4  x , equate to zero and attempt solution M1 Obtain 7  a A1 Divide by 4  x at least as far as the x term M1 or use of identity or by inspection Obtain 2 2 1   x x and conclude 2 ( 4)(2 1)    x x x A1 Alternative method for question 5(a) Divide by 4  x at least as far as the x term M1 Equate the remainder to zero M1 Obtain 7  a A1 Obtain 2 2 1   x x and conclude 2 ( 4)(2 1)    x x x A1 4

9709/23 Cambridge International AS Level – Mark Scheme PUBLISHED May/June 2022 © UCLES 2022 Page 9 of 11 Question Answer Marks Guidance 5(b) Apply logarithms and use power law for 3e 4  y M1 Obtain 1 3 ln4 or exact equivalent A1 Use discriminant   1 8 7   or equivalent to show no other roots B1 AG – necessary detail needed 3 Question Answer Marks Guidance 6(a) Integrate to obtain form 2 1 e  x k M1 any constant k, 6  k Obtain correct 2 1 3 2 e  x A1 Apply limits correctly to 2 1 e  x k and equate to 120 M1 Allow one slip Rearrange as far as ...  a M1 Obtain 2 1 1 1 2 2 ln(80 e )     a a A1 AG – necessary detail needed 5 6(b) Use iteration process correctly at least once M1 Obtain final answer 1.76 A1 Answer required to exactly 3 s.f. Show sufficient iterations to 5 s.f. to justify answer or show sign change in the interval [1.755, 1.765] A1 3

9709/23 Cambridge International AS Level – Mark Scheme PUBLISHED May/June 2022 © UCLES 2022 Page 10 of 11 Question Answer Marks Guidance 7 Integrate 2 2 x to obtain form 3 2 1(2 2 )  k x M1 any constant 1k Obtain correct 3 2 1 3 (2 2 )  x A1 OE Apply limits to obtain 3 2 1 3 (2 )  or exact equivalent A1 at this stage or later Use identity 2 1 1 2 2 sin cos2    x x B1 OE Integrate to obtain form 2 3 sin 2  k x k x M1 any constants 2 3 , k k Obtain correct 1 1 2 4 sin 2   x x A1 OE if area taken as positive Substituting correct limits into their 1 1 2 4 sin 2  x x M1 Can be SOI by 1 2  Allow one slip Obtain 3 2 1 1 3 2 (2 )   or exact equivalent A1 CWO 8

9709/23 Cambridge International AS Level – Mark Scheme PUBLISHED May/June 2022 © UCLES 2022 Page 11 of 11 Question Answer Marks Guidance 8(a) Use sin 2 2sin cos     and 1 sec cos    to obtain 6sin B1 Expand second term to obtain 5 3cos 5sin    B1 Simplify to obtain 11sin 5 3cos    B1 State 14  R B1 FT FT their 1 2 cos sin    k k Use appropriate trigonometry to find  M1 Obtain 38.21  A1 AWRT 6 8(b) State or imply 14sin(2 38.21) 2   B1 FT FT their R and  Carry out correct process to find value of  between 0 and 90 M1 Obtain 66.8 A1 AWRT 3