Cambridge IGCSE Mathematics - Additional 0606 — 2016 Feb/March Paper 1 · Variant 2

This document consists of 16 printed pages. DC (LEG/FD) 106873/3 © UCLES 2016 [Turn over Cambridge International Examinations Cambridge International General Certificate of Secondary Education * 9 2 4 2 8 1 9 0 7 2 * ADDITIONAL MATHEMATICS 0606/12 Paper 1 February/March 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.

2 0606/12/F/M/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 0606/12/F/M/16 © UCLES 2016 [Turn over 1 Find the values of a for which the line y x a 9 = + intersects the curve y x x 2 3 1 2 =- + + at 2 distinct points. [4] 2 Given that p q r p qr p q r a b c 3 1 2 3 2 2 1 = - - - , find the values of a, b and c. [3]

4 0606/12/F/M/16 © UCLES 2016 3 Solve log log x x 3 5 25 + = . [3]

5 0606/12/F/M/16 © UCLES 2016 [Turn over 4 (i) On the axes below, sketch the graphs of y x 2 = - and y x 3 2 = + . [4] –9 9 –6 6 x y 0 (ii) Solve x x 3 2 2 + = - . [3]

6 0606/12/F/M/16 © UCLES 2016 5 (a) A 6-character password is to be chosen from the following 9 characters. letters A B E F numbers 5 8 9 symbols * $ Each character may be used only once in any password. Find the number of different 6-character passwords that may be chosen if (i) there are no restrictions, [1] (ii) the password must consist of 2 letters, 2 numbers and 2 symbols, in that order, [2] (iii) the password must start and finish with a symbol. [2]

7 0606/12/F/M/16 © UCLES 2016 [Turn over (b) An examination consists of a section A, containing 10 short questions, and a section B, containing 5 long questions. Candidates are required to answer 6 questions from section A and 3 questions from section B. Find the number of different selections of questions that can be made if (i) there are no further restrictions, [2] (ii) candidates must answer the first 2 questions in section A and the first question in section B. [2]

8 0606/12/F/M/16 © UCLES 2016 6 A function f is such that f e x 6 x 4 = + ^ h for x R ! . (i) Write down the range of f. [1] (ii) Find f x 1 - ^ h and state its domain and range. [4] (iii) Find f x l^ h. [1] (iv) Hence find the exact solution of f f x x = l ^ ^ h h. [2]

9 0606/12/F/M/16 © UCLES 2016 [Turn over 7 The polynomial f x ax x x b 7 9 3 2 + - + = ^ h is divisible by x 2 1 - . The remainder when f x ^ h is divided by x 2 - is 5 times the remainder when f x ^ h is divided by x 1 + . (i) Show that a 6 = and find the value of b. [4] (ii) Using the values from part (i), show that f x x cx dx e 2 1 2 = - + + ^ ^ ^ h h h, where c, d and e are integers to be found. [2] (iii) Hence factorise f x ^ h completely. [2]

10 0606/12/F/M/16 © UCLES 2016 8 lg y lg x 0 (1.3, 3) (2.55, 4.5) The variables x and y are such that when lgy is plotted against lgx the straight line graph shown above is obtained. (i) Given that y Axb = , find the value of A and of b. [5]

11 0606/12/F/M/16 © UCLES 2016 [Turn over (ii) Find the value of lg y when x 100 = . [2] (iii) Find the value of x when y = 8000. [2]

12 0606/12/F/M/16 © UCLES 2016 9 A C O B r cm i rad The diagram shows a circle, centre O, radius r cm. Points A, B and C are such that A and B lie on the circle and the tangents at A and B meet at C. Angle AOB = i radians. (i) Given that the area of the major sector AOB is 7 times the area of the minor sector AOB, find the value of i. [2] (ii) Given also that the perimeter of the minor sector AOB is 20 cm, show that the value of r, correct to 2 decimal places, is 7.18. [2]

13 0606/12/F/M/16 © UCLES 2016 [Turn over (iii) Using the values of i and r from parts (i) and (ii), find the perimeter of the shaded region ABC. [3] (iv) Find the area of the shaded region ABC. [3]

14 0606/12/F/M/16 © UCLES 2016 10 (i) Find d d ( ) x x x 2 1 2 3 1- 1 J L KK N P OO. [3] (ii) Hence, show that d x x x x px q c 2 1 15 2 1 2 1 2 3 - = - + + ^ ^ ^ h h h y , where c is a constant of integration, and p and q are integers to be found. [6]

15 0606/12/F/M/16 © UCLES 2016 [Turn over (iii) Hence find d x x x 2 1 .0 5 1 2 1 - ^ h y . [2] Question 11 is printed on the next page.

16 0606/12/F/M/16 © UCLES 2016 11 (i) Show that cosec cosec tan 1 1 1 1 2 2 i i i - - + = . [4] (ii) Hence solve cosec cosec tan 1 1 1 1 6 i i i - - + = + for 0 360 1 1 i c c. [4] 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.

® IGCSE is the registered trademark of Cambridge International Examinations. CAMBRIDGE INTERNATIONAL EXAMINATIONS Cambridge International General Certificate of Secondary Education MARK SCHEME for the March 2016 series 0606 ADDITIONAL MATHEMATICS 0606/12 Paper 12, maximum raw mark 80 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 March 2016 series for most Cambridge IGCSE® and Cambridge International A and AS Level components.

Page 2 Mark Scheme Syllabus Paper Cambridge IGCSE – March 2016 0606 12 © 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 2 9 2 3 1 + = − + + ax x x ( ) 2 2 3 8 0 + − + = x a x For 2 distinct roots, ( ) 2 3 64 − > a Critical values 5 − and 11 11, 5 > < − a a M1 M1 A1 A1 for attempt to equate the line and the curve and obtain a 3 term quadratic equation for use of the discriminant for critical values for correct range 2 13, 0, 1 6 = − = = a b c B3 B1 for each 3 5 25 log log 3 + = x x 5 5 5 log 1 log 3 2 log 25 + = x x 5 log 3 = x 125 = x cao Alternative scheme: 25 25 25 log log 3 log 5 + = x x 25 25 25 1 log 2 log 3 log 5 + = x x 25 3 log 2 = x 125 = x cao B1,B1 B1 B1 B1 B1 B1 for 5 1 log 2 x B1 for 5 5 log log 25 x for final answer for change of base for 25 1 log 2 x (must be from correct work) for final answer

Page 3 Mark Scheme Syllabus Paper Cambridge IGCSE – March 2016 0606 12 © Cambridge International Examinations 2016 Question Answer Marks Guidance 4 (i) B1 B1 B1 B1 for a line in correct position for ( ) ( ) 0, 2 , 2, 0 for correct shape for 3 2 = + y x , touching the x-axis for ( ) ( ) 1.5, 0 , 0, 3 − (ii) 2 3 2 − = + x x leading to 1 3 = − x 2 3 2 − = −− x x leading to 5 = − x Alternative: ( ) ( ) 2 2 2 3 4 − = + x x leading to 2 15 28 5 0 + + = x x 1 3 = − x , 5 = − x B1 M1 A1 M1 A1,A1 for 1 3 = − x for correct attempt to deal with ‘negative’ branch. for 5 = − x for equating and squaring to obtain a 3 term quadratic equation A1 for each. 5 (a) (i) 9 6 60480 = P B1 Must be evaluated (ii) 4 3 2 2 2 × × P P =144 M1,A1 M1 for attempt a product of 3 perms (iii) 840 2 × 1680 B1,B1 B1 for either 840, or realising that there are 2 possible positions for the symbols (b) (i) 10 5 6 3 × C C 2100 M1 A1 for unsimplified form (ii) 8 4 4 2 × C C 420 M1 A1 for unsimplified form 6 (i) ( ) f 6 > x B1 Allow B1 for 6 > y (ii) ( ) ( ) 1 1 f ln 6 4 − = − x x Domain: 6 > x Range: ( ) 1 f − ∈
x M1 A1 B1 B1 for a complete method must be ( ) 1 f − = x or y = … must be using the correct variable in both (iii) ( ) 4 f 4e ′ = x x B1 (iv) 4 4 6 e 4e + = x x leading to 1 ln 2 4 = x M1 A1 for a complete, correct method −4 −3 −2 −1 1 2 −4 −2 2 4 6 8 x y

Page 4 Mark Scheme Syllabus Paper Cambridge IGCSE – March 2016 0606 12 © Cambridge International Examinations 2016 Question Answer Marks Guidance 7 (i) ( ) 1 7 9 f 0 2 8 4 2  = + − + =     a b 8 22 + = a b ( ) 8 28 18 5 7 9 + − + = −+ + + a b a b 13 4 70 − = a b leading to 6, 2 = = a b M1 M1 DM1 A1 for attempt at 1 f 2       for attempt at ( ) ( ) f 2 5f 1 = − Allow if the ‘wrong way’ round for attempt to solve simultaneous equations A1 for both (ii) ( )( ) 2 2 1 3 5 2 − + − x x x B2,1,0 –1 each error (iii) ( )( )( ) 2 1 3 1 2 − − + x x x M1 A1FT for attempt to factorise their quadratic factor must be 3 linear factors 8 (i) lg lg lg = + y A b x Gradient = 1.2 so 1.2 = b Intercept 1.44 = 27.5 = A B1 M1 A1 M1 A1 may be implied by later work for attempt at gradient for 1.2 = b for attempt to find y-intercept for , allow awrt 28 (ii) when 100 = x , lg 2 = x lg 3.84 = y ( allow 3.8 to 3.9) M1 A1 for correct use of graph or equation (iii) when 8000, lg8000 3.9 = = y , lg 2.05 = x leading to 113, = x 2.05 10 or 112 M1 A1 for correct use of graph or equation

Page 5 Mark Scheme Syllabus Paper Cambridge IGCSE – March 2016 0606 12 © Cambridge International Examinations 2016 Question Answer Marks Guidance 9 (i) ( ) 2 2 7 1 2π 2 2 θ θ = − r r π 4 θ = oe M1 A1 for a valid method allow in degrees (ii) π 20 4 + + = r r r , leading to 7.180(3..) = r M1 A1 for valid method Must show enough accuracy to get A1 (iii) Perimeter π π 2 tan 4 8 = + r r 5.6394 5.9484 11.6 = + = B1,B1 B1 B1 for arc length, B1 for twice AC for 11.6 (iv) Area 2 1 π ( ) 2 4 = × − r AC r 21.356 20.246 = − or equivalent method using triangles 1.08 Area 1.11

B1,B1 B1 B1 for area of quadrilateral, allow unsimplified, B1 for sector area for area in given range 10 (i) ( ) ( ) 1 3 2 2 3 2 2 1 2 1 2 × × − + − x x x B1 M1 A1 for ( ) 1 2 3 2 2 1 2 × − x for attempt at differentiation of a product for all else correct (ii) ( ) ( ) ( ) 1 3 3 2 2 2 3 2 1 d 2 1 2 1 d − = − − − ∫ ∫ x x x x x x x ( ) ( ) 3 5 2 2 1 2 2 1 2 1 2 5 = − − × − x x x ( ) ( ) ( ) 1 3 2 2 1 1 2 1 d 2 1 2 1 3 5   − = − − −     ∫x x x x x x ( ) ( ) 3 2 2 1 3 1 15 − = + x x M1 B1,B1 M1 DM1 A1 for attempt to use part (i) B1 for ( ) 3 2 2 1 − x x , allow if divided by 3 B1 for ( ) 5 2 1 2 2 1 2 5 × − x , allow if divided by 3 for taking out a common factor of ( ) 3 2 2 1 − x for attempt to obtain a linear factor (iii) 1 4 0 15   × −     M1 A1FT for attempt to use limits correctly FT on their 15 + px q

Page 6 Mark Scheme Syllabus Paper Cambridge IGCSE – March 2016 0606 12 © Cambridge International Examinations 2016 Question Answer Marks Guidance 11 (i) 2 1 1 cosec 1 cosec 1 cosec 1 cosec 1 cosec 1 θ θ θ θ θ + − + − = − + − 2 2 cot θ = 2 2tan θ = Alternative scheme: 1 1 sin sin cosec 1 cosec 1 1 sin 1 cos θ θ θ θ θ θ − = − − + − + ( ) ( ) 2 2 2 sin sin sin sin 1 sin θ θ θ θ θ + − − = − 2 2 2sin cos θ θ = 2 2tan θ = M1 A1 M1 A1 M1 A1 M1 A1 for attempt to obtain a single fraction all correct as shown for use of correct identity for ‘finishing off’ for attempt to obtain a single fraction in terms of sinθ only all correct as shown for use of correct identity for ‘finishing off’ (ii) 2 2tan 6 tan θ θ = + ( )( ) 2tan 3 tan 2 0 θ θ + − = 3 tan 2 θ = − , tan 2 θ = o o o o 63.4 , 123.7 , 243.4 , 303.7 θ = M1 DM1 A1,A1 for attempt to use (i), to obtain a quadratic equation and valid attempt to solve for attempt to solve trig equation for each ‘pair’