Cambridge IGCSE Mathematics - Additional 0606 — 2021 May/June Paper 1 · Variant 3

This document has 16 pages. Any blank pages are indicated. Cambridge IGCSE™ * 2 4 4 5 2 6 9 7 9 6 * ADDITIONAL MATHEMATICS 0606/13 Paper 1 May/June 2021 2 hours You must answer on the question paper. No additional materials are needed. 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. ● 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 80. ● The number of marks for each question or part question is shown in brackets [ ]. DC (LK/FC) 202076/2 © UCLES 2021 [Turn over

2 0606/13/M/J/21 © UCLES 2021 Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation ax bx c 0 2 + + = , x a b b ac 2 4 2 ! = - - Binomial Theorem ( ) a b a a b a b a b n n n r b 1 2 n n n n n r r n 1 2 2 f f + = + + + + + + - - - e e e o o o where n is a positive integer and ( )! ! ! n r n r r n = - e o Arithmetic series ( ) u a n d 1 n = + - ( ) { ( ) } S n a l n a n d 2 1 2 1 2 1 n = + = + - Geometric series u ar n n 1 = - ( ) ( ) S r a r r 1 1 1 n n ! = - - ( ) S r a r 1 1 1 = - 3 2. TRIGONOMETRY Identities sin cos A A 1 2 2 + = sec tan A A 1 2 2 = + ec cos cot A A 1 2 2 = + Formulae for ∆ABC sin sin sin A a B b C c = = cos a b c bc A 2 2 2 2 = + - sin bc A 2 1 T =

3 0606/13/M/J/21 © UCLES 2021 [Turn over 1 Find the possible values of the constant k such that the equation kx kx k 4 3 1 0 2 + + + = has two different real roots. [4]

4 0606/13/M/J/21 © UCLES 2021 2 (a) Find d d e x x x 2 3 ` j. [3] (b) (i) Find d d x x 3 4 2 3 1 + ` j . [2] (ii) Hence find dx x x 3 4 2 0 2 3 2 + - ` j y . [3]

5 0606/13/M/J/21 © UCLES 2021 [Turn over 3 Solve the equation cosec cot cot 2 2 9 2 2 i i i + = + , where i is in radians and r r 2 2 1 1 i - . [5]

6 0606/13/M/J/21 © UCLES 2021 4 (a) Find the first three non-zero terms in the expansion of x 2 4 2 6 - e o in ascending powers of x. Simplify each term. [3] (b) Hence find the term independent of x in the expansion of x x 2 4 3 1 2 6 2 2 - - e e o o . [3]

7 0606/13/M/J/21 © UCLES 2021 [Turn over 5 When ey is plotted against x2 a straight line graph passing through the points (2.24, 5) and (4.74, 10) is obtained. Find y in terms of x. [5]

8 0606/13/M/J/21 © UCLES 2021 6 O D C A 2a a a B The diagram shows a circle, centre O, radius 2a. The points A and B lie on the circumference of the circle. The points C and D are the mid-points of the lines OB and OA respectively. The arc DC is part of a circle centre O. The chord AB is of length 2a. (a) Find angle AOB, giving your answer in radians in terms of r. [1] (b) Find, in terms of a and r, the perimeter of the shaded region ABCD. [2] (c) Find, in terms of a and r, the area of the shaded region ABCD. [3]

9 0606/13/M/J/21 © UCLES 2021 [Turn over 7 (a) A committee of 8 people is to be formed from 5 teachers, 4 doctors and 3 police officers. Find the number of different committees that could be chosen if (i) all 4 doctors are on the committee, [2] (ii) there are at least 2 teachers on the committee. [3] (b) Given that P 6 P n n 5 1 4 # = - , find the value of n. [3]

10 0606/13/M/J/21 © UCLES 2021 8 y x 0 3 P Q R cos y a bx c = + 6 5r The graph shows the curve cos y a bx c = + , for . x 0 2 8 G G , where a, b and c are constants and x is in radians. The curve meets the y-axis at (0, 3) and the x-axis at the point P and point , R 6 5 0 r b l. The curve has a minimum at point Q. The period of cos a bx c + is r radians. (a) Find the value of each of a, b and c. [4] (b) Find the coordinates of P. [1] (c) Find the coordinates of Q. [2]

11 0606/13/M/J/21 © UCLES 2021 [Turn over 9 (a) Show that the equation of the curve y x x 4 2 2 = - - ` `j j can be written as y x ax bx 8 3 2 = + + + , where a and b are integers. Hence find the exact coordinates of the stationary points on the curve. [4] (b) On the axes, sketch the graph of y x x 4 2 2 = - - ` `j j , stating the intercepts with the coordinate axes. [4] y O x (c) Find the possible values of the constant k for which x x k 4 2 2 - - = ` `j j has exactly 4 different solutions. [2]

12 0606/13/M/J/21 © UCLES 2021 10 O a c A B D C E The diagram shows the parallelogram OABC, such that a OA = and c OC = . The point D lies on CB such that : : CD DB 3 1 = . When extended, the lines AB and OD meet at the point E. It is given that OE hOD = and BE kAB = , where h and k are constants. (a) Find DE in terms of a, c and h. [4]

13 0606/13/M/J/21 © UCLES 2021 [Turn over (b) Find DE in terms of a, c and k. [1] (c) Hence find the value of h and of k. [4]

14 0606/13/M/J/21 © UCLES 2021 11 The line x y 2 10 + = intersects the two lines satisfying the equation x y 2 + = at the points A and B. (a) Show that the point ( , ) C 5 20 - lies on the perpendicular bisector of the line AB. [8]

15 0606/13/M/J/21 © UCLES 2021 (b) The point D also lies on this perpendicular bisector. M is the mid-point of AB. The distance CD is three times the distance of CM. Find the possible coordinates of D. [4]

16 0606/13/M/J/21 © UCLES 2021 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. BLANK PAGE

This document consists of 9 printed pages. © UCLES 2021 [Turn over Cambridge IGCSE™ ADDITIONAL MATHEMATICS 0606/13 Paper 1 May/June 2021 MARK SCHEME Maximum Mark: 80 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 2021 series for most Cambridge IGCSE™, Cambridge International A and AS Level components and some Cambridge O Level components.

0606/13 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2021 © UCLES 2021 Page 2 of 9 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.

0606/13 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2021 © UCLES 2021 Page 3 of 9 Maths-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. 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 marks, awarded for a valid method applied to the problem. A Accuracy mark, awarded for a correct answer or intermediate step correctly obtained. For accuracy marks to be given, the associated Method mark must be earned or implied. B Mark for a correct result or statement independent of Method marks. When a part of a question has two or more ‘method’ steps, the M marks are in principle independent unless the scheme specifically says otherwise; and similarly where there are several B marks allocated. The notation ‘dep’ is used to indicate that a particular M or B mark is dependent on an earlier mark in the scheme. Abbreviations awrt answers which round to cao correct answer only dep dependent FT follow through after error isw ignore subsequent working nfww not from wrong working oe or equivalent rot rounded or truncated SC Special Case soi seen or implied

0606/13 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2021 © UCLES 2021 Page 4 of 9 Question Answer Marks Guidance 1 ( ) ( ) 2 4 4 3 1 − + k k k M1 For use of the discriminant to obtain a two term quadratic expression. 2 4 4 0 − = k k M1 Dep to find critical values, allow if only one is found 0, 1 = = k k A1 For both critical values 0 < k 1 > k A1 2(a) ( ) 2 3 3 3e 2 e + x x x x 3 M1 for differentiation of a product A1 for ( ) 2 3 3e x x A1 for 3 2 e + x x 2(b)(i) ( ) 2 2 3 2 3 4 − + x x 2 M1 for ( ) 2 2 3 3 4 − + kx x 2(b)(ii) ( ) 2 1 2 3 0 1 3 4 2   +       x M1 For ( ) 1 2 3 3 4 + k x 1 1 3 3 1 1 16 4 2 2       −                   M1 Dep for correct substitution of limits into their integral 0.466 A1 3 ( ) 2 2 cot 1 2cot 2cot 9 θ θ θ + + = + B1 For use of correct identity ( )( ) 3cot 4 cot 2 0 θ θ + − = 4 cot , cot 2 3 θ θ = − = M1 For attempt to solve their quadratic in cotθ to obtain cotθ = k 3 1 tan , tan 4 2 θ θ = − = M1 For dealing with cotθ = k correctly to get 1 tanθ = k 0.644 θ = − A1 0.464 θ = A1 4(a) 2 4 64 48 15 − + x x 3 B1 for 64 B1 for 2 48 − x B1 for 4 15x

0606/13 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2021 © UCLES 2021 Page 5 of 9 Question Answer Marks Guidance 4(b) 2 4 6 1 9 − + x x B1 ( ) ( ) ( ) 64 9 48 6 15 × + − × − + their their their M1 For considering terms independent of x, must have 3 terms 879 A1 5 2 e = + y mx c B1 May be implied by later work 10 4.74 = + m c 5 2.24 = + m c M1 For at least one correct equation 5 2.5 = m M1 Dep for attempt to solve for m 2, 0.52 = = m c A1 For both ( ) 2 ln 2 0.52 = + y x A1 Alternative 2 e = + y mx c (B1) May be implied by later work Gradient = 10 5 4.74 2.24 − = − m (M1) 10 4.74( ) = + their m c or 5 2.24( ) = + their m c (M1) 2, 0.52 = = m c (A1) For both ( ) 2 ln 2 0.52 = + y x (A1) 6(a) π 3 B1 6(b) π 4 3 + a a 2 B2 FT for π 4 3   × +     their a a or B1 FT for π 3 × their a

0606/13 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2021 © UCLES 2021 Page 6 of 9 Question Answer Marks Guidance 6(c) ( ) 2 1 π 2 sin 2 3 a B1 FT their π 3 2 1 π 2 3 a B1 FT their π 3 2 2 π 3 6 − a a B1 FT their π 3 7(a)(i) 8 4 C M1 For realisation that there are 4 places left and 8 people available to fill them 70 A1 7(a)(ii) 1 teacher on committee: 5 ways B1 12 8 C 5 − M1 490 A1 Alternative 2 teachers: 70 3 teachers: 210 4 teachers: 175 5 teachers: 35 (2) B1 for 2 correct cases 490 (B1) 7(b) ( ) ( ) ( ) 1 ! ! 6 5 ! 1 4 ! − = − −− n n n n B1 ( ) ( ) 6 5 ! 5 ! = − − n n n M1 For simplification of either ! n and ( )1 ! − n or ‘cancelling out’ of the terms of ( ) 5 ! − n 6 = n A1 nfww 8(a) 2 = b B1 At ( ) 0, 3 : 3 = + a c B1 At 5π , 0 6       : 5π 0 cos 3 = + a c 0 2 = + a c M1 For use of their b and 5π , 0 6       6 = a 3 = − c A1 For both

0606/13 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2021 © UCLES 2021 Page 7 of 9 Question Answer Marks Guidance 8(b) π , 0 6       B1 Allow for π 6 = x 8(c) π , 9 2   −     2 B1 for π 2 B1 for 9 − 9(a) 3 2 2 4 8 = − − + y x x x B1 2 d 3 4 4 d = − − y x x x ( )( ) 3 2 2 0 + − = x x M1 For attempt to differentiate, allow one slip and for equating their d d y x to zero and attempt to solve to obtain x = k 2 256 , 3 27   −     A1 ( ) 2,0 A1 9(b) 4 B1 for curve with maximum in the second quadrant B1 for 8 = y either on the curve or stated B1 for 2 = ± x either on the curve or stated B1 for a cusp at 2 = − x and a min at 2 = x 9(c) 256 0 27 < < k 2 FT on their 256 27 B1 for either 0 < k or 256 27 < k 10(a) 3 4 =  CD a B1 3 4 = +  OD c a B1 3 4   = +      OE h c a B1 3 3 4 4     = + − +          DE h c a c a oe cao B1 10(b) 1 4 = +  DE k a c B1

0606/13 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2021 © UCLES 2021 Page 8 of 9 Question Answer Marks Guidance 10(c) ( ) 3 3 1 1 4 4 4   − + + = +     h h k c a a c M1 For equating their answer to (a) to their answer to (b) ( ) 3 3 1 1 4 4 4   − + + = +     h h k c a a c 1 −= h k M1 For attempt to equate like vectors once. 4 3 = h A1 1 3 = k A1 11(a) 2 10 + = x y 2 + = x y M1 For attempt to solve simultaneously ( ) 6, 8 − A1 2 10 + = x y 2 + = − x y M1 For attempt to solve simultaneously ( ) 14, 12 − A1 Alternative ( ) ( ) 2 2 10 10 4 4 − + − + = x x x x or ( ) ( ) 2 2 10 2 2 10 2 4 − + − + = y y y y (M1) For attempt to eliminate one of the variables using ( ) 2 4 + = x y 2 20 84 0 + + = x x or 2 20 96 0 − + = y y (M1) Dep for attempt to obtain a 3 term quadratic equation = 0 and solve to obtain at least one solution, allow 1 arithmetic error ( ) 14, 12 − (A1) ( ) 6, 8 − (A1) Mid-point of AB: ( ) 10, 10 − M1 For attempt to obtain the mid-point using their coordinates for A and B. Gradient of line perpendicular to AB = 2 M1 For attempt to obtain the perpendicular gradient using their coordinates for A and B. ( ) ( ) 10 2 10 − = − − y their their x their M1 ( ) 20 10 2 5 10 − = −+ oe A1 For verification

0606/13 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2021 © UCLES 2021 Page 9 of 9 Question Answer Marks Guidance 11(b) ( ) 10, 50 2 FT on their midpoint B1 for each coordinate ( ) 20, 10 − − 2 FT on their midpoint B1 for each coordinate