Cambridge IGCSE Mathematics - Additional 0606 — 2019 Oct/Nov Paper 1 · Variant 1

This document consists of 16 printed pages. DC (CE/SW) 172295/2 © UCLES 2019 [Turn over * 3 0 6 0 2 5 3 8 4 8 * ADDITIONAL MATHEMATICS 0606/11 Paper 1 October/November 2019 2 hours Candidates answer on the Question Paper. Additional Materials: Electronic calculator READ THESE INSTRUCTIONS FIRST Write your centre number, candidate number and name on all the work you hand in. Write in dark blue or black pen. You may use an HB pencil for any diagrams or graphs. Do not use staples, paper clips, glue or correction fluid. DO NOT WRITE IN ANY BARCODES. Answer all the questions. Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question. The use of an electronic calculator is expected, where appropriate. You are reminded of the need for clear presentation in your answers. At the end of the examination, fasten all your work securely together. The number of marks is given in brackets [ ] at the end of each question or part question. The total number of marks for this paper is 80. Cambridge Assessment International Education Cambridge International General Certificate of Secondary Education

2 0606/11/O/N/19 © UCLES 2019 Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation ax2 + bx + c = 0, x a b b ac 2 4 2 ! = - - . Binomial Theorem (a + b)n = an + ( n 1)an–1 b + ( n 2)an–2 b2 + … + ( n r)an–r br + … + bn, where n is a positive integer and ( n r) = n! (n – r)!r! . 2. TRIGONOMETRY Identities sin2 A + cos2 A = 1 sec2 A = 1 + tan2 A cosec2 A = 1 + cot2 A Formulae for ∆ABC a sin A = b sin B = c sin C a2 = b2 + c2 – 2bc cos A ∆ = 1 2 bc sin A

3 0606/11/O/N/19 © UCLES 2019 [Turn over 1 Using set notation, describe the regions shaded on the Venn diagrams below. A B Y X Z … … [2] 2 Find the values of k for which the line y kx 3 = - and the curve y x x k 2 3 2 = + + do not intersect. [5]

4 0606/11/O/N/19 © UCLES 2019 3 Given that 7 49 1 x y # = and 5 125 25 1 x 5 y 3 2 # = , calculate the value of x and of y. [5]

5 0606/11/O/N/19 © UCLES 2019 [Turn over 4 It is given that ( ) ln y x x 2 3 4 1 2 = - + . (i) Find x y d d . [3] (ii) Find the approximate change in y as x increases from 2 to h 2+ , where h is small. [2]

6 0606/11/O/N/19 © UCLES 2019 5 ( )x 3 1 f e x 2 = + for x R d ( )x x 1 g = + for x R d (i) Write down the range of f and of g. [2] (ii) Evaluate ( ) 0 fg2 . [2] (iii) On the axes below, sketch the graphs of ( ) y x f = and ( ) y x f 1 = - , stating the coordinates of the points where the graphs meet the coordinate axes. [3] y x O

7 0606/11/O/N/19 © UCLES 2019 [Turn over 6 Find the equation of the normal to the curve y x 8 5 = + at the point where x 2 1 = , giving your answer in the form , ax by c 0 + + = where a, b and c are integers. [5]

8 0606/11/O/N/19 © UCLES 2019 7 When lg y is plotted against x, a straight line graph passing through the points (2.2, 3.6) and (3.4, 6) is obtained. (i) Given that , y Abx = find the value of each of the constants A and b. [5] (ii) Find x when . y 900 = [2]

9 0606/11/O/N/19 © UCLES 2019 [Turn over 8 Do not use a calculator in this question. In this question, all lengths are in centimetres. 7 5 + 5 11 2 + 5 4 3 + A B D C The diagram shows the trapezium ABCD in which angle ADC is 90° and AB is parallel to DC. It is given that , AB 4 3 5 = + DC 11 2 5 = + and . AD 7 5 = + (i) Find the perimeter of the trapezium, giving your answer in simplest surd form. [3] (ii) Find the area of the trapezium, giving your answer in simplest surd form. [3]

10 0606/11/O/N/19 © UCLES 2019 9 A D B O 15 cm 10 cm 10 cm C The diagram shows a circle with centre O and radius 10 cm. The points A, B, C and D lie on the circle such that the chord AB = 15 cm and the chord CD = 10 cm. The chord AB is parallel to the chord DC. (i) Show that the angle AOB is 1.70 radians correct to 2 decimal places. [2] (ii) Find the perimeter of the shaded region. [4]

11 0606/11/O/N/19 © UCLES 2019 [Turn over (iii) Find the area of the shaded region. [4]

12 0606/11/O/N/19 © UCLES 2019 10 y x O y = 2 + cos 3x y = 1.5 The diagram shows part of the graph of cos y x 2 3 = + and the straight line . . y 1 5 = Find the exact area of the shaded region bounded by the curve and the straight line. You must show all your working. [9]

13 0606/11/O/N/19 © UCLES 2019 [Turn over Continuation of working space for Question 10

14 0606/11/O/N/19 © UCLES 2019 11 (a) Jess wants to arrange 9 different books on a shelf. There are 4 mathematics books, 3 physics books and 2 chemistry books. Find the number of different possible arrangements of the books if (i) there are no restrictions, [1] (ii) a chemistry book is at each end of the shelf, [2] (iii) all the mathematics books are kept together and all the physics books are kept together. [3]

15 0606/11/O/N/19 © UCLES 2019 [Turn over (b) A quiz team of 6 children is to be chosen from a class of 8 boys and 10 girls. Find the number of ways of choosing the team if (i) there are no restrictions, [1] (ii) there are more boys than girls in the team. [4] Question 12 is printed on the next page.

16 0606/11/O/N/19 © UCLES 2019 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. 12 A curve is such that sin x y x 2 3 d d 2 2 r = + b l. Given that the curve has a gradient of 5 at the point , 3 3 5 r r b l, find the equation of the curve. [8]

This document consists of 10 printed pages. © UCLES 2019 [Turn over Cambridge Assessment International Education Cambridge International General Certificate of Secondary Education ADDITIONAL MATHEMATICS 0606/11 Paper 1 October/November 2019 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 October/November 2019 series for most Cambridge IGCSE™, Cambridge International A and AS Level components and some Cambridge O Level components.

0606/11 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2019 © UCLES 2019 Page 2 of 10 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.

0606/11 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2019 © UCLES 2019 Page 3 of 10 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

0606/11 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2019 © UCLES 2019 Page 4 of 10 Question Answer Marks Guidance 1 ′ ∩ A B oe B1 ( ) ( ) ∩ ∪ ∩ X Y X Z or ( ) ∩ ∪ X Y Z B1 2 2 2 3 3 + + = − x x k kx M1 For an attempt to equate and simplify to a 3 term quadratic equation, allow an error in one term ( ) ( ) 2 2 3 3 0 + − + + = x k x k A1 ( ) ( ) 2 3 4 2 3 − −× × + k k M1 For attempt to use the discriminant, allow previous error, leading to a quadratic equation in terms of k 2 14 15 0 − − = k k giving critical values of 1 − and 15 A1 For critical values 1 15 −< < k A1 3 Either 2 7 7 × x y or 2 49 49 × x y or 5 2 5 5 × x y or 5 2 25 25 × x y M1 For expressing the terms on the left hand side of either one of the 2 equations in terms of powers of 7, 49, 5 or 25 2 0 7 7 7 × = x y or 0 2 49 49 49 × = x y A1 5 2 2 5 5 5− × = x y or 5 1 2 25 25 25− × = x y A1 leading to 2 0 + = x y and 5 2 2 + = − x y M1 For attempt to solve two linear equations, with integer coefficients and constants, in terms of x and y 1 1 , 2 4 = − = x y A1 4(i) ( ) ( ) 2 2 d 8 ln 4 1 d 4 1 + = + x x x x B1 ( )( ) ( ) ( ) 2 2 2 8 2 3 2ln 4 1 4 1 d d 2 3 − − + + = − x x x x y x x M1 For attempt to differentiate a quotient A1 For all other terms, not including 2 8 4 1 + x x , correct 4(ii) When d 16 2, 2ln17 d 17 = = − y x x 4.73 = − M1 For attempt to find value of d d y x when 2 = x and multiply by h Change in 4.73 = − y h A1

0606/11 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2019 © UCLES 2019 Page 5 of 10 Question Answer Marks Guidance 5(i) f > 1 B1 Must be using correct notation g∈\ B1 Must be using correct notation 5(ii) ( ) ( ) g 0 1, g 1 2 = = and attempt at ( ) f 2 M1 For attempt at 2 g and correct order ( ) f 2 164.8 = awrt 165 A1 5(iii) B3 B1 for correct f and ( ) 0,4 , must be in first and second quadrant B1 for correct f-1 and ( ) 4,0 , must be in first and fourth quadrant B1 for = y x and/or symmetry implied, by ‘matching intercepts’. No intersection. 6 d d y x = k(8x + 5) 1 2 − M1 For attempt to differentiate, must be in the form k(8x + 5) 1 2 − d d y x = 4(8x + 5) 1 2 − A1 When x = 1 2 , y = 3 B1 Normal: 3 1 3 4 2   − = − −     y x M1 For attempt at the normal when x = 1 2 , using correct process for their d d y x and their y. 6x + 8y – 27 = 0 A1

0606/11 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2019 © UCLES 2019 Page 6 of 10 Question Answer Marks Guidance 7(i) lg y = lg A + x lg b B1 For statement, may be implied by subsequent work Either 6 = lg A + 3.4 lg b or 3.6 = lg A + 2.2 lg b M1 For one correct equation M1 For another correct equation and attempt to solve simultaneously lg b = 2, b = 100 A1 lg A = –0.8, A = 10–0.8 or 0.158 A1 Or Gradient = lg b = 2 M1 equating gradient to lg b and attempt to evaluate b = 100 A1 Must be identified as b 6 = lg A + 3.4 lg b or 3.6 = lg A + 2.2 lg b M1 For a correct equation and attempt to find lg A lg A = –0.8, A = 10–0.8 or 0.158 A1 Must be identified as A 7(ii) lg 900 = –0.8 + 2x oe M1 For correct use of y = 900 x = 1.88 A1 8(i) BC2 = ( ) ( ) 2 2 7 5 7 5 + + − = 49 + 14 5 + 5 + 49 – 14 5 + 5 = 108 M1 For use of Pythagoras’ theorem and attempt to expand and simplify BC = 6 3 A1 Perimeter = 22 + 6 5 + 6 3 A1

0606/11 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2019 © UCLES 2019 Page 7 of 10 Question Answer Marks Guidance 8(ii) Either ( )( ) 1 4 3 5 11 2 5 7 5 2 + + + + ( )( ) 1 15 5 5 7 5 2 = + + ( ) 1 105 35 5 15 5 25 2 = + + + M1 Either For a valid method and attempt to expand out and simplify Or ( )( ) ( )( ) 1 4 3 5 7 5 7 5 7 5 2 + + + + − = 28 + 21 5 + 4 5 + 15 + 1 2 (49 – 5) M1 Or For a valid method and attempt to expand out and simplify Area = 65 25 5 + A2 A1for each term 9(i) Either 2 2 2 15 10 10 200cos = + − AOB cos 0.125 = − AOB M1 For use of cosine rule 1.696 = AOB so 1.70 to 2 dp A1 Must have justification to 2 dp Or 7.5 sin 2 10  =     AOB 0.8481 2 = AOB M1 For use of basic trig 1.696 = AOB so 1.70 to 2 dp A1

0606/11 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2019 © UCLES 2019 Page 8 of 10 Question Answer Marks Guidance 9(ii) Angle π 3 = DOC B1 Either π 0.5 2π 1.696 3   = = − −     AOD BOC 1.77 = = AOD BOC M1 For attempt to get AOD or BOC Arc lengths = 17.7 M1 For attempt at arc length using their previous answer Perimeter = ( ) 15 10 2 17.7 60.4 + + × = A1 Or Arc AB = 17 or Arc CD = 10π 3 M1 For either arc length (20π – arc AB – arc CD) M1 Perimeter = 60.4 A1 9(iii) Either Area of each sector = ( ) 2 110 1.770 2 M1 For area of sector using their BOC Area of triangles = 1 π 1 100 sin 100sin1.70 2 3 2     × × + ×         M1 For area of one triangle using the sine rule oe Total area = 177 + 43.3 + 49.6 M1 For plan Area = awrt 270 A1 Or Area of upper segment = ( ) 2 110 1.696 sin1.696 2 − M1 For area of a sector or area of a triangle using the sine rule oe Area of lower segment = 2 1 π π 10 sin 2 3 3   −     M1 For whichever has not been obtained in previous part Shaded area = 100π – are of the 2 segments Area = 314.2 – 35.2 – 9.06 M1 For plan Area = awrt 270 A1

0606/11 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2019 © UCLES 2019 Page 9 of 10 Question Answer Marks Guidance 10 1.5 2 cos3 = + x cos3 0.5 = − x M1 For correct attempt to find points of intersection 2π 4π 3 , 3 3 = x M1 For dealing with 3x correctly 2π 9 = x or o 40 A1 4π 9 = x or o 80 A1 Either ( ) 4π 9 2π 9 1.5 2 cos3 d − + ∫ x x M1 For subtraction method – condone omission of or incorrect limits [ ] 4π 9 2π 9 0.5 sin3 − − x k x M1 For attempt to integrate – condone omission of or incorrect limits 4π 9 2π 9 1 0.5 sin3 3   − −     x x A1 All correct – condone omission of or incorrect limits 2π 3 π 3 9 6 9 6     − + −− −             M1 Dep for application of limits, must be in radians Area = 3 π 3 9 − A1 Or 2π 1.5 9   ×     M1 For attempt at rectangle (must include subtraction subsequently) [ ] 4π 9 2π 9 2 sin3 + x k x M1 For attempt to integrate – condone omission of or incorrect limits 4π 9 2π 9 1 2 sin3 3   +     x x A1 All correct – condone omission of or incorrect limits 8π 3 4π 3 9 6 9 6       − − +                   M1 Dep for application of limits, must be in radians Area = 3 π 3 9 − A1

0606/11 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2019 © UCLES 2019 Page 10 of 10 Question Answer Marks Guidance 11(a)(i) 362 880 B1 11(a)(ii) 7! 2 × B1 For 7! 10 080 B1 For 7! 2 × leading to 10080 11(a)(iii) Total = 4! 4! 3! × × = 3456 B3 B1 for treating as 4 separate units 4! B1 for either number of ways of arranging the maths books amongst themselves 4! or the number of ways of arranging the physics books amongst themselves 3! 11(b)(i) 18 564 B1 11(b)(ii) Total 3738 B4 B1 4 boys 3150 B1 5 boys 560 B1 6 boys 28 12 d π cos d 3   = + +     y k x c x M1 For attempt to integrate d π 2cos d 3   = − + +     y x c x A1 All correct, condone omission of +c 2π 5 2cos 3 = − + c M1 Dep for attempt to find c d π 2cos 4 d 3   = − + +     y x x A1 ( ) π sin 3   = + + +     y p x qx d M1 attempt to integrate a second time to obtain π sin 3   = +     y p x π 2sin 4 3   = − + + +     y x x d A1 All correct, condone omission of +d 5π 2π 4π 2sin 3 3 3 = − + + d M1 Dep for attempt to find a second arbitrary constant π π 2sin 4 3 3 3   = − + + + +     y x x or π 2sin 4 2.78 3   = − + + +     y x x A1