Cambridge IGCSE Mathematics - Additional 0606 — 2016 May/June Paper 2 · Variant 2

This document consists of 12 printed pages. DC (KN/SW) 106860/4 © UCLES 2016 [Turn over * 9 6 6 2 4 0 1 7 0 8 * ADDITIONAL MATHEMATICS 0606/22 Paper 2 May/June 2016 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 International Examinations Cambridge International General Certificate of Secondary Education

2 0606/22/M/J/16 © UCLES 2016 Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation ax2 + bx + c = 0, x b b ac a = − − 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 0606/22/M/J/16 © UCLES 2016 [Turn over 1 (i) Given that x kx k 2 4 3 0 2 + + - = has no real roots, show that k satisfies k k 4 3 0 2 1 - + . [2] (ii) Solve the inequality k k 4 3 0 2 1 - + . [2] 2 Variables x and y are related by the equation y x x 3 5 1 = - - . (i) Find x y d d , simplifying your answer. [2] (ii) Hence find the approximate change in x when y increases from 9 by the small amount 0.07. [3]

4 0606/22/M/J/16 © UCLES 2016 3 A team of 3 people is to be selected from 7 women and 6 men. Find the number of different teams that could be selected if there must be more women than men on the team. [3] 4 Do not use a calculator in this question. The polynomial p( )x x x qx 2 3 56 3 2 = - + + has a factor x – 2. (i) Show that q = −30. [1] (ii) Factorise p( )x completely and hence state all the solutions of p( )x 0 = . [4]

5 0606/22/M/J/16 © UCLES 2016 [Turn over 5 The coordinates of three points are A(−2, 6), B(6, 10) and C(p, 0). (i) Find the coordinates of M, the mid-point of AB. [2] (ii) Given that CM is perpendicular to AB, find the value of the constant p. [2] (iii) Find angle MCB. [3]

6 0606/22/M/J/16 © UCLES 2016 6 O A B 5 cm rad i The diagram shows a sector of a circle with centre O and radius 5 cm. The length of the arc AB is 7 cm. Angle AOB is θ radians. (i) Explain why θ must be greater than 1 radian. [1] (ii) Find the value of θ. [2] (iii) Calculate the area of the sector AOB. [2] (iv) Calculate the area of the shaded segment. [2]

7 0606/22/M/J/16 © UCLES 2016 [Turn over 7 The matrix A is 4 3 5 2 J L KK N P OO and the matrix B is 4 1 2 3 J L KK N P OO. (i) Find the matrix C such that 3A B C = + . [2] (ii) Show that ( ) det det det AB A B # = . [4] (iii) Find the matrix ( ) AB 1 - . [2]

8 0606/22/M/J/16 © UCLES 2016 8 Find the coordinates of the points of intersection of the curve y x 4 5 3 0 + + = and the line y x 15 10 = + . [6]

9 0606/22/M/J/16 © UCLES 2016 [Turn over 9 (a) Find x x x x 1 d 2 3 2 + + y . [3] (b) (i) Find ( ) sin x x 5 d r + y . [2] (ii) Hence evaluate ( ) sin x x 5 d 0 5 r + - ry . [2]

10 0606/22/M/J/16 © UCLES 2016 10 (a) The graph of the curve ( ) ( ) y p q 4 4 x x 2 = - passes through the points (0, 2) and (0.5, 14). Find the value of p and of q. [3] (b) The variables x and y are connected by the equation ( ) y 10 2 10 x x 2 = - . Using the substitution u 10x = , or otherwise, find the exact value of x when y = 24. [3] (c) Solve ( ) log log x x 1 3 2 2 + - = . [3]

11 0606/22/M/J/16 © UCLES 2016 [Turn over 11 (a) A function f is defined, for all real x, by ( )x x x f 2 = - . Find the greatest value of f (x) and the value of x for which this occurs. [3] (b) The domain of g( )x x x2 = - is such that ( )x g 1 - exists. Explain why x 1 H is a suitable domain for g(x). [1] (c) The functions h and k are defined by : ( ) lg x x 2 h 7 + for x 2 2- , :x x 5 1 k 7 + - for x 1 101 1 1 . (i) Find hk(10). [2] (ii) Find ( )x k 1 - , stating its domain and range. [5] Question 12 is printed on the next page.

12 0606/22/M/J/16 © UCLES 2016 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 International Examinations Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at www.cie.org.uk after the live examination series. Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. 12 Solve the equation (i) sin cos A A 8 2 7 2 + = for A 0 180 G G c c, [4] (ii) ( ) . cosec B 3 1 2 5 + = for B 0 G G r radians. [4]

® IGCSE is the registered trademark of Cambridge International Examinations. This document consists of 8 printed pages. © UCLES 2016 [Turn over Cambridge International Examinations Cambridge International General Certificate of Secondary Education ADDITIONAL MATHEMATICS 0606/22 Paper 2 May/June 2016 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 will not enter into discussions about these mark schemes. Cambridge is publishing the mark schemes for the May/June 2016 series for most Cambridge IGCSE®, Cambridge International A and AS Level components and some Cambridge O Level components.

Page 2 Mark Scheme Syllabus Paper Cambridge IGCSE – May/June 2016 0606 22 © Cambridge International Examinations 2016 Abbreviations awrt answers which round to cao correct answer only dep dependent FT follow through after error isw ignore subsequent working oe or equivalent rot rounded or truncated SC Special Case soi seen or implied www without wrong working Question Answer Marks Guidance 1 (i) (2k)2 – 4 (1)(4k – 3) [< 0] Correct completion to given inequality 2 4 3 0 k k − + < isw M1 A1 clear attempt at b2 – 4ac (ii) Critical values 1 and 3 soi 1 < k < 3 as final answer M1 A1 May be implied by incorrect inequalities 2 (i) Clear attempt at quotient rule or equivalent product rule ( ) 2 d 14 d 3 y x x   =     − or 2 d 14 d 6 9 y x x x   =     − + cao or correct simplified equivalent M1 A1 condone omission of brackets allow recovery from bracketing errors or omissions if implied in correct work to the correct answer (ii) [y = 9] x = 2 2 0.07 d δ d x y their x x =   ≈    oe 0.005 oe B1 M1 A1 condone 2 0.07 d δ d x y their x x =   =     not from wrong working; answer only does not score 3 Any one of: 6 7 6 7 0 3 1 2 C C C C   × + ×   or 35 + 126 or 13 6 7 6 3 2 1 3 C C C C − × − or 286 – 105 – 20 161 M2 A1 M1 for 6 7 0 3 C C   ×   or 6 7 1 2 C C × or 13 6 7 3 2 1 C C C − × or 13 6 3 3 C C − or 6 7 6 2 1 3 C C C × + or for the numerical equivalent of one of these calculations If M0 then B3 for answer only of 161

Page 3 Mark Scheme Syllabus Paper Cambridge IGCSE – May/June 2016 0606 22 © Cambridge International Examinations 2016 Question Answer Marks Guidance 4 (i) 3 2 2(2) 3(2) 2 56 0 q − + + = with one correct interim step leading to q = −30 B1 allow for only 16 – 12 + 2q + 56 = 0 q = −30 NB = 0 must be seen or may be implied by e.g. −60 = 2q or 60 = −2q; or convincingly showing 3 2 2(2) 3(2) 30(2) 56 0 ; − − + = allow for only 16 – 12 + 2(−30) + 56 = 0 or correct synthetic division at least as far as 2 2 3 56 4 2 2 4 2 1 2 0 q q q − + + then q = −30 (ii) 2 2 28 x x + − ( )( )( ) 2 2 7 4 x x x − − + x = 2, x = −4, x = 3.5 oe B2 M1 A1 B1 for any two terms correct For factorising the correct equation; condone = 0; condone ( )( ) 2 7 4 x x − + only for M1 but for A1 must see all 3 factors in this part; do not allow 7 2 x   −     not from wrong working; answers only do not score 5 (i) (2, 8) B1, B1 (ii) 8 0 2 2 their their p − = − − or better [p =] 6 M1 A1 Condone 8 0 1 2 gradient their their p their AB − − = − oe

Page 4 Mark Scheme Syllabus Paper Cambridge IGCSE – May/June 2016 0606 22 © Cambridge International Examinations 2016 Question Answer Marks Guidance (iii) [ ] ( ) ( ) 2 2 6 2 10 8 MB their their = − + − soi or ( ) ( ) 2 2 1 1 6 2 10 6 2 2 AB   = −− + −     soi or [ ] ( ) ( ) 2 2 2 8 0 MC their their p their = − + − soi or 8 tan[...] 4 = soi or 2 2 2 4.47 8.94 10 2(8.94)(10)cos[...] = + − or 2 2 2 8.94 10 10 2(10)(10)cos[...] = + − 1 20 sin 10 −        oe soi 26.56 to 26.6° or 0.4636 to 0.464 rads cao M1 M1 A1 implied by[ ] 20 MB = or 1 1 80 2 2 AB   =     e.g. 4.47, or[ ] 80 MC = or e.g. 8.94 or 63.4° or equivalents or 1 80 cos 10 −        or 1 20 tan 80 −        or 1 4 tan 8 −      or 90 − 1 8 tan 4 −      or equivalent complete correct method; implies first M1 Not from wrong working 6 (i) Valid explanation B1 e.g. arc length is greater than the radius or 7 is greater than 5 (ii) 7 5θ = 1.4 θ = oe M1 A1 implies M1 (iii) 2 1 5 1.4 2 their × × oe 17.5oe M1 A1

Page 5 Mark Scheme Syllabus Paper Cambridge IGCSE – May/June 2016 0606 22 © Cambridge International Examinations 2016 Question Answer Marks Guidance (iv) [ ] 2 1 triangle area 5 sin 1.4 2 their = × × or 12.3 to 12.32 or for [ 1 2 × base × height=] 1 6.4[4...] 3.8[2...] 2 × × oe 5.18 to 5.2 inclusive M1 A1 may be embedded in a difference calculation implies M1 7 (i) 12 15 4 2 9 6 1 3     +         soi 16 17 10 9       M1 A1 if no method shown, may be implied by their answer with at least 2 correct elements (ii) detA = 4 × 2 – 3 × 5 = −7 or detB = 4 × 3 – 2 × 1= 10 AB = 21 23 14 12       det(AB) = 21 × 12 – 23 × 14 = −70 B1 B2 B1 allow for e.g. (4 × 2 – 3 × 5) × ( 4 × 3 – 2 × 1) = −70 or detA = 8 – 15 = −7 or detB = 12 – 2= 10 or B1 for two elements correct allow for det(AB) = 252 – 322 = −70 For full marks must conclude that detAB = detA × detB or show the product –7 × 10 = –70 otherwise max 3 marks (iii) 12 23 1 14 21 det their their −   ×   −   AB isw B2 correct or correct FT; FT their AB and their non-zero det AB; their AB must be an attempt at a matrix product e.g. 16 10 3 6       B1 for 1 det their their   ×     AB or for 12 23 14 21 k their −   ×   −  

Page 6 Mark Scheme Syllabus Paper Cambridge IGCSE – May/June 2016 0606 22 © Cambridge International Examinations 2016 Question Answer Marks Guidance 8 Eliminates y e.g. 5 3 4 0 15 10 x x + + = + or eliminates x e.g. 5 3 4 0 ( 10) 15 y y + + = − Rearrange to a 3-term quadratic 2 60 90 30 0 x x + + = oe or 2 4 10 50 0 y y + − = oe Factorise or solve 3-term quadratic x = 1 2 − , x = −1 isw y = 1 2 2 , y = −5 isw M1 M1 A1 M1 A1 A1 allow even after incorrect rearrangement of the equation of the curve (dependent on resulting equation still in terms of x and y); condone substitution of e.g. 10 15 y + condone sign slips / arithmetic slips or y = 1 2 2 , y = −5 or x = 1 2 − , x = −1 If final A marks not awarded then A1 for a correct x, y pair 9 (a) 2 1 ( ) 2 x x c x + − + isw B3 B1 for each term allow 2 1 ( ) 2 1 x x x c − + + + − isw for B3 (b) (i) cos(5 π) k x + where k < 0 or cos(5 π) 5 x + cos(5 π) ( ) 5 x c − + + M1 A1 (ii) π5 cos(5(0) π) cos(5( ) π) 5 5 − − + − + − or cos(π) cos(0) 5 5 − −   −    0.4 oe M1 A1 correct substitution of the given limits into their expression of the form ( ) cos 5 π k x + , dep on M1 in (b)(i) answer only does not score 10 (a) 2 p q = − and 14 4 2 p q = − oe p = 5 q = 3 M1 A1 A1 (b) Factorise 2 10 2(10 ) 24[ 0] x x − − = or factorise 2 2 24[ 0] u u − − = 10x = 6 x = lg6 cao as final answer M1 A1 A1 or applies the formula or completes the square ignore 10x = −4 for this mark or exact equivalent

Page 7 Mark Scheme Syllabus Paper Cambridge IGCSE – May/June 2016 0606 22 © Cambridge International Examinations 2016 Question Answer Marks Guidance (c) 3 1 2 x x + = oe www x = 1 7 or 0.143 or 0.1428 to 0.1429 M2 A1 combines logs and anti-logs or B1 for one correct log move e.g. 2 1 log 3 x x +  =     or ( ) ( ) 2 2 2 log 1 log log 8 x x + − = or ( ) ( ) 2 2 2 log 1 log 3log 2 x x + − = 11 (a) Valid method when x = 1 2 [greatest value =] 1 4 M1 A1 B1 Completing the square as far as e.g. 2 1 constant 2 x   − −     or calculus as far as 1 − 2x = 0 or finding roots x = 0 and x = 1and using symmetry soi Implies M1 if not clearly from wrong working (b) Valid comment e.g. when x ≥ 1, f ′ is always decreasing B1 Allow e.g. a sketch with a comment such as the curve is one-one [when x ≥ 1] or e.g. the curve is one-one when x > 1 2 (c) (i) k(10) = 8 or5 10 1 8 + −= or stating h(8) h(8) = 1 orlg(8 2) 1 + = cao M1 A1 or[hk( ) ] lg(7 1) x x = + − [hk(10) ] lg(7 10 1) 1 = + − = (ii) ( ) 2 5 1 y x − = − ( ) 2 1 k ( ) 5 1 x x − = − + isw or 1 2 k ( ) 10 26 x x x − = − + isw 5 15 x < < 1 1 k ( ) 101 x − < < M1 A1 B1, B1 B1 or( ) 2 5 1 x y − = − B1for 5 < x oe and B1for x < 15 oe allow (5, 15); one mark for each limit of the interval; if B0 then SC1 for 5 15 x

or ‘5 to 15’ or [5, 15] etc. allow (1, 101)

Page 8 Mark Scheme Syllabus Paper Cambridge IGCSE – May/June 2016 0606 22 © Cambridge International Examinations 2016 Question Answer Marks Guidance 12 (i) 2 8(1 cos ) 2cos 7 A A − + = or better Solves or factorises their 3-term quadratic in cosA 60, 104.477... rounded or truncated to 1 dp or more; B1 M1 A2 with no extras in range; not from clearly wrong working but allow recovery from minor slips or A1 for either, ignoring extras (ii) sin(3 1) 0.4 B + = soi [3B + 1 =] 0.41 or better 0.577, 1.9[0], 2.67 or 0.57669..., 1.89823... , 2.67108... rounded or truncated to 4 or more sf B1 M1 A2 may be implied by 1 2.5 sin(3 1) B = + implies B1 with no extras in range; or A1 for any one correct ignoring extras If M0 then B2 for all 3 correct angles found or B1 for 1 or 2 correct angles found