Cambridge IGCSE Mathematics - International 0607 — 2011 May/June Paper 4 · Variant 1

This document consists of 20 printed pages. IB11 06_0607_41/4RP © UCLES 2011 [Turn over *6342947190* For Examiner's Use UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/41 Paper 4 (Extended) May/June 2011 2 hours 15 minutes Candidates answer on the Question Paper Additional Materials: Geometrical Instruments Graphics 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. Do not use staples, paper clips, highlighters, glue or correction fluid. You may use a pencil for any diagrams or graphs. DO NOT WRITE IN ANY BARCODES. Answer all the questions. Unless instructed otherwise, give your answers exactly or correct to three significant figures as appropriate. Answers in degrees should be given to one decimal place. For π, use your calculator value. You must show all the relevant working to gain full marks and you will be given marks for correct methods, including sketches, even if your answer is incorrect. 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 120. www.XtremePapers.com

2 © UCLES 2011 0607/41/M/J/11 Formula List For the equation ax2 + bx + c = 0 x = 2 _ ± 4 2 _ b b ac a Curved surface area, A, of cylinder of radius r, height h. A = 2πrh Curved surface area, A, of cone of radius r, sloping edge l. A = πrl Curved surface area, A, of sphere of radius r. A = 4πr2 Volume, V, of pyramid, base area A, height h. V= 1 3 Ah Volume, V, of cylinder of radius r, height h. V = πr2h Volume, V, of cone of radius r, height h. V = 1 3 πr2h Volume, V, of sphere of radius r. V = 4 3 πr3 = = sin sin sin a b c A B C a2 = b2 + c2 – 2bc cos A Area = 1 2 bc sin A A a B C b c

3 © UCLES 2011 0607/41/M/J/11 [Turn over For Examiner's Use Answer all the questions. 1 (a) In 2009 the height of a tree was 25.2 m. A year later, the height was 28 m. Calculate the percentage increase in height. Answer(a) % [3] (b) The height of 25.2 m was a 20% increase of the height in 2008. Calculate the height in 2008. Answer(b) m [3] (c) The height of the tree is expected to increase by 5% of its value each year. The height is now 30 m. (i) Calculate the expected height in 3 years time. Answer(c)(i) m [3] (ii) Calculate the number of years it will take for the tree to reach a height of 40 m. Answer(c)(ii) [2]

4 © UCLES 2011 0607/41/M/J/11 For Examiner's Use 2 20 –20 0 –2 4 y x (a) On the axes, sketch the graph of y = x3 – 3x2. [3] (b) Write down the zeros of y = x3 – 3x2. Answer(b) x = x = [1] (c) Write down the co-ordinates of any local maximum or local minimum points. Answer(c) [2]

5 © UCLES 2011 0607/41/M/J/11 [Turn over For Examiner's Use 3 6 5 4 3 2 1 –1 –2 –3 –4 –5 –6 0 –6 –5 –4 –3 –2 –1 1 2 3 4 5 6 y x V P U (a) Describe fully the single transformation that maps triangle P onto (i) triangle U, Answer(a)(i) [3] (ii) triangle V. Answer(a)(ii) [2] (b) On the grid above, draw the following transformations of triangle P. (i) Translation by 6 2 −       . [2] (ii) Stretch by a scale factor 1.5 with the y-axis invariant. [2]

6 © UCLES 2011 0607/41/M/J/11 For Examiner's Use 4 (a) ABCDE is a pentagon with angle EAB = 120°, angle ABC = 140° and angle BCD = 85°. AB is parallel to ED and EA = AB. (i) Calculate the size of angle AED. Answer(a)(i) [1] (ii) Calculate the size of angle EDC. Answer(a)(ii) [2] (iii) On the diagram above, draw the line EB and calculate the size of angle EBC. Answer(a)(iii) [1] (b) 150° O P Q R S NOT TO SCALE P, Q, R and S lie on a circle, centre O. Angle POR = 150°. Calculate the size of (i) angle PSR, Answer(b)(i) [1] (ii) angle PQR. Answer(b)(ii) [1] E A B C D 120° 140° 85° NOT TO SCALE

7 © UCLES 2011 0607/41/M/J/11 [Turn over For Examiner's Use 5 y B x A L 6 0 8 NOT TO SCALE The diagram shows the line, L, which passes through the points A (8, 0) and B (0, 6). (a) The line L also passes through the point (h, 9). Find the value of h. Answer(a) h = [2] (b) Find the equation of the line which is perpendicular to L and which passes through the mid-point of the line AB. Answer(b) [4]

8 © UCLES 2011 0607/41/M/J/11 For Examiner's Use 6 (a) North North A B C 7 cm 9 cm 110° NOT TO SCALE In triangle ABC, AB = 9 cm, AC = 7 cm and angle BAC = 110°. (i) Calculate the area of triangle ABC. Answer(a)(i) cm2 [2] (ii) Calculate the length of BC. Answer(a)(ii) cm [3] (iii) The bearing of A from B is 050°. Find the bearing of C from A. Answer(a)(iii) [2]

9 © UCLES 2011 0607/41/M/J/11 [Turn over For Examiner's Use (b) P Q S R 7 cm 4 cm 11 cm 75° NOT TO SCALE In the quadrilateral PQRS, PQ = 4 cm, PS = 7 cm and QR = 11 cm. Angle QPS = 90° and angle QSR = 75°. Calculate the size of angle QRS. Answer(b) [5]

10 © UCLES 2011 0607/41/M/J/11 For Examiner's Use 7 Nyali sells cakes and ice creams. She records the number of cakes (c) and the number of ice creams (i) she sells each day for 10 days. The results are shown in the table. Number of cakes (c) 48 60 52 40 60 36 70 20 44 50 Number of ice creams (i) 50 18 38 50 30 54 14 70 46 50 (a) Complete the scatter diagram. The first 6 points have been plotted for you. 80 70 60 50 40 30 20 10 10 20 30 40 50 60 70 80 0 Number of ice creams Number of cakes i c [2]

11 © UCLES 2011 0607/41/M/J/11 [Turn over For Examiner's Use (b) Write down one word to describe the correlation between c and i. Answer(b) [1] (c) Find the equation of the line of regression, writing i in terms of c. Answer(c) i = [2] (d) Use your equation to estimate the number of ice creams Nyali sells on a day when she sells 67 cakes. Answer(d) [1]

12 © UCLES 2011 0607/41/M/J/11 For Examiner's Use 8 8 cm 6 cm 12 cm NOT TO SCALE The diagram shows two similar containers. The top of the large container has diameter 8 cm. The top of the small container has diameter 6 cm. (a) The height of the large container is 12 cm. Calculate the height of the small container. Answer(a) cm [2] (b) The volume of the large container is 550 cm3. Calculate the volume of the small container. Answer(b) cm3 [3] (c) Write 550 cm3 in litres. Answer(c) litres [1]

13 © UCLES 2011 0607/41/M/J/11 [Turn over For Examiner's Use 9 y x 1.5 –1 –10 10 0 (a) (i) On the axes, sketch the graph of y = f(x), where f(x) = log x for 0 I x Y 10. [2] (ii) Write down the co-ordinates of the point where the graph of y = f(x) crosses the x-axis. Answer(a)(ii) ( , ) [1] (iii) Write down the equation of the asymptote of the graph of y = f(x). Answer(a)(iii) [1] (iv) Find the range of f(x) for the domain 0 I x Y 10. Answer(a)(iv) [2] (b) Solve the equation 10 log 20 x x − = . Answer(b) x = [2] (c) 10 g( ) 20 x x − = Find the range of g(x) for the domain –10 Y x Y 10.= Answer(c) [2] (d) On the axes, sketch the graph of y = f(x + 1). [2]

14 © UCLES 2011 0607/41/M/J/11 For Examiner's Use 10 40 cm 70 cm NOT TO SCALE The diagram shows a water tank which is open at the top. The tank is in the shape of a cylinder with radius 40 cm and height 70 cm. (a) (i) Calculate the total surface area of the cylinder. Answer(a)(i) cm2 [2] (ii) The cylinder is made of metal which costs $2.40 per square metre. Calculate the cost of the metal. Answer(a)(ii) $ [3]

15 © UCLES 2011 0607/41/M/J/11 [Turn over For Examiner's Use (b) (i) Calculate the volume of the cylinder. Answer(b)(i) cm3 [2] (ii) Water is poured into the empty cylinder at a rate of 8 cm3 per second. Calculate the time taken to fill the cylinder. Give your answer in hours and minutes, correct to the nearest minute. Answer(b)(ii) h min [4]

16 © UCLES 2011 0607/41/M/J/11 For Examiner's Use 11 1 7 8 12 15 19 19 The diagram shows seven numbered cards. (a) A card is chosen at random. (i) Write down the probability that it is numbered 12 or 15. Answer(a)(i) [1] (ii) The probability that the number on the card is greater than x is 5 7 . Write down the value of x. Answer(a)(ii) [1] (b) Two cards are chosen at random, without replacement, from the seven numbered cards. (i) When the first card chosen is numbered 19, write down the probability that the second card is also numbered 19. Answer(b)(i) [1] (ii) First card Second card numbered 19 not numbered 19 numbered 19 not numbered 19 numbered 19 not numbered 19 2 7 Complete the tree diagram, by writing the probabilities in the spaces. [2]

17 © UCLES 2011 0607/41/M/J/11 [Turn over For Examiner's Use (iii) Find the probability that both cards are numbered 19. Answer(b)(iii) [2] (iv) Find the probability that exactly one card is numbered 19. Answer(b)(iv) [3] (c) Cards are chosen at random, without replacement, from the seven numbered cards, until a card that is numbered 19 is chosen. Find the probability that this happens with the third card. Answer(c) [2]

18 © UCLES 2011 0607/41/M/J/11 For Examiner's Use 12 m 12 10 8 6 4 2 10 20 30 40 50 60 70 80 90 100 0 Frequency density Mass (m grams) The histogram shows numbers of apples and their masses (m grams). (a) Complete the frequency table using the information in the histogram. Mass (m grams) 0 Y m I 30 30 Y m I 50 50 Y m I 60 60 Y m I 70 70 Y m I 100 Frequency 60 110 [3] (b) Calculate an estimate of the mean mass of the apples. Answer(b) g [2] (c) Complete the cumulative frequency table using the information in your frequency table. Mass (m grams) m I 30 m I 50 m I 60 m I 70 m I 100 Cumulative frequency 60 560 [2]

19 © UCLES 2011 0607/41/M/J/11 [Turn over For Examiner's Use (d) On the grid below, complete the cumulative frequency curve. 600 500 400 300 200 100 10 20 30 40 50 60 70 80 90 100 0 Cumulative frequency Mass (m grams) [3] (e) Use your cumulative frequency curve to find (i) the median, Answer(e)(i) g [1] (ii) the lower quartile, Answer(e)(ii) g [1] (iii) the inter-quartile range, Answer(e)(iii) g [1] (iv) the number of apples with a mass of at least 40 g. Answer(e)(iv) [2]

20 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. University of 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. © UCLES 2011 0607/41/M/J/11 For Examiner's Use 13 w° x + 2 x x x + 1 NOT TO SCALE The area of the right-angled triangle is equal to the area of the square. (a) Show that x2 – 3x – 2 = 0. [3] (b) Solve the equation x2 – 3x – 2 = 0. Give your answers correct to 2 decimal places. Answer(b) x = or x = [3] (c) Calculate the value of w. Answer(c) w = [2]

UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education MARK SCHEME for the May/June 2011 question paper for the guidance of teachers 0607 CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/41 Paper 4 (Extended), maximum raw mark 120 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 must be read in conjunction with the question papers and the report on the examination. • Cambridge will not enter into discussions or correspondence in connection with these mark schemes. Cambridge is publishing the mark schemes for the May/June 2011 question papers for most IGCSE, GCE Advanced Level and Advanced Subsidiary Level syllabuses and some Ordinary Level syllabuses. www.XtremePapers.com

Page 2 Mark Scheme: Teachers’ version Syllabus Paper IGCSE – May/June 2011 0607 41 © University of Cambridge International Examinations 2011 1 (a) 11.1(1….) B3 If B0, M2 for (28 – 25.2) ÷ 25.2 (× 100) oe M1 for 28 – 25.2 or 2. 25 28 soi (b) 21 B3 If B0, M2 for 25.2 ÷ 1.2 oe M1 for 120% = 25.2 (c) (i) 34.7 (34.72 to 34.76) final answer B3 If B0, M2 for 30 × 1.053 oe M1 for 30 × 1.05n, n > 1 o.e. SC2 for 4.7 or 4.72 to 4.76 or 34.7 or 34.72 to 34.76 seen (ii) 5.8 to 5.9 or 6 www 2 B2 If B0, M1 for multiplying 30 by 1.05 more than 3 times or dividing 40 by 1.05 more than 3 times or 30 × 1.05n = 40 oe SC1 for 5 [11] 2 (a) B3 B1 for cubic with max then min B1 for x-intercept 3 , (between 2 and 4) B1 for max at origin, 2mm accuracy (b) 0, 3 B1 (c) (0, 0) or (2, –4) B2 B1 for one correct and one incorrect [6] 3 (a) (i) Rotation, (0, 0) 90° clockwise oe B1 B1 B1 All independent (ii) Reflection, y = – x oe B1 B1 Independent (b) (i) Triangle vertices (–5, 3), (–2, 3), (–2, 5) B2 SC1 translation      − k 6 or       2 k (ii) Triangle vertices (1.5, 1), (6, 1), (6, 3) B2 2mm accuracy for 1.5 SC1 for stretch of P wth s.f 1.5 and invariant line x = k or stretch of P s.f. 1.5 with x-axis invariant. [9] 4 (a) (i) 60 B1 (ii) 135 B2 If B0, M1 for (5 – 2) × 180 – (120 + their 60 + 140 + 85) soi by 195 – their (i) oe (iii) 110 B1 (b) (i) 75 B1 (ii) 105 ft B1 ft ft 180 – their (b)(i) only if +ve. Can recover to correct answer [6]

Page 3 Mark Scheme: Teachers’ version Syllabus Paper IGCSE – May/June 2011 0607 41 © University of Cambridge International Examinations 2011 5 (a) – 4 B2 Condone (– 4, 9) M1 for 8 6 3 h = oe (with + or – ) or for 6 4 3 + − = x y implied by answer 4 oe (b) 3 7 3 4 − = x y or other simplified equation B4 isw if equation seen and spoiled If B0, B1 for gradient = 3 4 , B1 for (4, 3) seen and M1 for correct use of any linear equation form with their 3 4 or their (4, 3) [6] 6 (a) (i) 29.6 (0….) M1 A1 M1 for 0.5 × 9 × 7 × sin110 SC1 for 29.6… ww (ii) 92 + 72 – 2 × 9 × 7 cos110 13.2 (13.15 to 13.16) M1 A2 If A0, A1 for 173 or 173.1 or 173.09….. SC2 for 13.2 (13.15 to 13.16) ww (iii) 120 B2 B1 for 60 or 130 or 50 and 70 (with BA extended) seen at A in correct positions (b) 2 2 7 4 + (sin R) = 65 their 11 75 sin × 45.1 (45.05 – 45.07) cao www 5 M2 M2 A1 soi by 65 or 8.06 or 8.062…. (M1 for 42 + 72 soi by 65) (M1 for 11 75 sin 65 their sin = R ) [12] 7 (a) Points at (70, 14), (20, 70), (44, 46) and (50, 50) P2 Points touching correct corners. P1 for 3 points (b) Negative B1 (c) –1.14c + 96.8 (–1.142… and 96.82 to 96.83) B2 B1 –1.14c + k or mc + 96.8 (–1.142… and 96.82 to 96.83) If B0, SC1 if –1.1 and 97 Allow x for c (d) 20 or 21 ft B1ft ft their equation only if answer is positive integer [6] 8 (a) 9 B2 If B0, M1 for 12 × 6 ÷ 8 oe (b) 232.(0……..) B3 If B0, M2 for 550 × (6 ÷ 8)3 oe (M1 for ( ) ( ) 3 6 8 3 8 6 oe or oe ) (c) 0.55 B1 [6]

Page 4 Mark Scheme: Teachers’ version Syllabus Paper IGCSE – May/June 2011 0607 41 © University of Cambridge International Examinations 2011 9 (a) (i) B2 B1 for shape B1 for approaching y-axis in 4th quadrant (ii) (1, 0) B1 (iii) x = 0 B1 (iv) (f(x)) 1 ≤ B2 If k to 1, allow B1. Allow in words. Allow y or x f(x) (b) 2.4(0) (2.399…) B2 If B0, M1 for line added to sketch (negative gradient, positive y-intercept, may be freehand) must cross curve. Pen – 1 if y-coord given (c) 1 ) ( 0 ≤ ≤ x g B1B1 Allow in words. Allow y or f(x) or x for g(x). SC1 for 0 < g(x) < 1 (d) B1 ft B1 Shape correct translated to left of their original curve ft B1 for passing through origin Condone if stops at origin [12] 10 (a) (i) 22 600 (22608 to 22623) B2 If B0, M1 for π × 80 × 70 (17584 to 17595.2) Allow 7200π (ii) 5.43 (5.425 to 5.429…) ft B3 ft If B0, M1 for ÷ 1002 and M1 for × 2.40 (M’s independent) ft their (i) ÷ 1002 × 2.40 (b) (i) 351 600 to 352 000 B2 If B0, M1 for π × 402 × 70 Allow 112000π (ii) ÷ 8 ÷ 60 ÷ 60 12 h 12 or 13 min cao final answer www 4 M1 M1 M1 A1 (43 950 to 44 000) (732.5 to 733.3…) (12.20 to 12.22…) or 12 remainder 12.48 to 13.33. (8 × 60 × 60 = 28800) [11]

Page 5 Mark Scheme: Teachers’ version Syllabus Paper IGCSE – May/June 2011 0607 41 © University of Cambridge International Examinations 2011 Throughout question 11, do not allow ratios or words. If decimals or percentages used, usual accuracy applies except penalise two sf by – 1 only once. isw any cancelling/converting to decimal or % 11 (a) (i) 7 2 oe B1 0.286 (0.2857….) (ii) 7 B1 Ignore embellishments (b) (i) 6 1 B1 (ii) 6 4 , 6 2 , 6 5 , 6 1 , 7 5 oe B2 B1 for 3 or 4 correct (iii) 42 2 oe B2 0.0476 (0.04761 to 0.04762) If B0, M1 for their 7 2 × their 6 1 (iv) 42 20 oe B3 0.476 (0.4761 to 0.4762) If B0, M2 for their 7 2 × their 6 5 + their 7 5 × their 6 2 M1 for one of the products (c) 210 40 oe B2 0.19(0) (0.1904 to 0.1905) If B0, M1 for 7 5 × 6 4 × 5 2 [12] 12 (a) 120, 90, 180 B1B1B1 (b) 58.75 ft B2 ft Accept 58.7 or 58.8 ft their frequencies with correct mid-values If B0, M1 for at least two correct mid- values seen (c) 180, 290, 380 ft B2 ft B1 for 2 correct ft their (a) (d) (30, 60), (50, 180), (60, 290), (70, 380), (100, 560) ft P2 ft C1 Points touching lines P1 for 3 or 4 correct ft Smooth curve through these 5 points (0.5 square accuracy) and correct shape (e) (i) 58 60 < ≤med ft B1 ft In all parts of (e) ft their graph but only if it is cumulative (ii) 43 to 46 ft B1 ft (iii) 29 to 36 ft B1 ft ft is upper quartile – their (ii) (iv) 440 to 460 ft B2 ft If B0, SC1ft for 100 to 120 (may be on graph) If use 600 in all 4 parts mark on ft basis but deduct 2 marks [15]

Page 6 Mark Scheme: Teachers’ version Syllabus Paper IGCSE – May/June 2011 0607 41 © University of Cambridge International Examinations 2011 13 (a) 2 )1 )( 2 ( 5.0 x x x = + + or 2 2 )1 )( 2 ( x x x = + + 2 2 2 + + + x x x oe seen 0 2 3 2 = − −x x M1 B1 E1 Must be one of these two forms Independent oe includes 1 2 1 2 2 1 + + + x x x At least one intermediate line after the M1 and no errors or omissions (b) M1 M1 for sketch of parabola with a zero either side of origin or formula correctly used )1( 2 ) 2 )( 1( 4 ) 3 ( ) 3 ( 2 − − − ± − − or better –0.56, 3.56 A1A1 If M0 or other GDC applications, SC2 for –0.56 and 3.56 If M0 or M1, SC1 for –0.6 and 3.6 or –0.561 to –0.562 and 3.561 to 3.562 (c) 39. 15 to 39.60.. ft B2 ft ft a positive root. If B0, M1 for tan = ((their root + 1) / (their root + 2)) [8]