Cambridge IGCSE Mathematics - International 0607 — 2015 May/June Paper 4 · Variant 1

This document consists of 20 printed pages. DC (KN/SW) 99287/7 © UCLES 2015 [Turn over Cambridge International Examinations Cambridge International General Certificate of Secondary Education * 7 2 6 0 8 7 7 3 2 5 * CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/41 Paper 4 (Extended) May/June 2015 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, glue or correction fluid. You may use an HB 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.

2 0607/41/M/J/15 © UCLES 2015 Formula List For the equation ax bx c 0 2 + + = x a b b ac 2 4 2 ! = - - Curved surface area, A, of cylinder of radius r, height h. r A rh 2 = Curved surface area, A, of cone of radius r, sloping edge l. r A rl = Curved surface area, A, of sphere of radius r. r A r 4 2 = Volume, V, of pyramid, base area A, height h. V Ah 3 1 = Volume, V, of cylinder of radius r, height h. r V r h 2 = Volume, V, of cone of radius r, height h. r V r h 3 1 2 = Volume, V, of sphere of radius r. r V r 3 4 3 = sin sin sin A a B b C c = = cos a b c bc A 2 2 2 2 = + - sin bc A 2 1 Area = A C B c b a

3 0607/41/M/J/15 © UCLES 2015 [Turn over Answer all the questions. 1 The table shows the marks that 80 students scored in an examination. Mark 0 1 2 3 4 5 6 7 8 9 10 Number of students 1 5 6 8 9 10 12 8 16 3 2 (a) Write down the mode. Answer(a) … [1] (b) Write down the range. Answer(b) … [1] (c) Find the median. Answer(c) … [1] (d) Find the interquartile range. Answer(d) … [2] (e) Calculate the mean. Answer(e) … [1]

4 0607/41/M/J/15 © UCLES 2015 2 Solve the simultaneous equations. You must show all your working. 5x – 2y = 11.5 4x + 3y = 0 Answer x = … y = … [4] 3 A car of length 4.5 metres is travelling at 72 km/h. The car approaches a tunnel of length 260 metres. (a) Change 72 km/h into m/s. Answer(a) … m/s [1] (b) Find the time it will take for the car to pass completely through the tunnel. Give your answer in seconds. Answer(b) … s [2]

5 0607/41/M/J/15 © UCLES 2015 [Turn over 4 0 2 4 6 1 3 5 x A B y –2 –4 –6 –1 –3 –5 2 4 6 8 10 1 3 5 7 9 (a) Describe fully the single transformation that maps triangle A onto triangle B. Answer(a) … … [2] (b) Rotate triangle B through 90° clockwise, centre (–1, 6). Draw this triangle and label it C. [3] (c) Describe fully the single transformation that maps triangle C onto triangle A. Answer(c) … … [2]

6 0607/41/M/J/15 © UCLES 2015 5 (a) y varies inversely as the square root of x. y = 5 when x = 9. (i) Find the value of y when x = 25. Answer(a)(i) y = … [2] (ii) Find the value of x when y = 25. Answer(a)(ii) x = … [2] (iii) Find x in terms of y. Answer(a)(iii) x = … [2] (b) 0 24 –4 2 x y Find the equation of this quadratic curve. Answer(b) … [3]

7 0607/41/M/J/15 © UCLES 2015 [Turn over 6 The Venn diagram shows the sets A, B and C. A C B U U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15} A = {factors of 12} B = {factors of 6} C = {11, 12, 13, 14} (a) List the elements of sets A and B. Answer(a) A = { …} B = { … } [2] (b) Write all the elements of U in the correct regions of the Venn diagram above. [3] (c) List the elements of (i) A B + , Answer(c)(i) { … } [1] (ii) A C + l , Answer(c)(ii) { … } [1] (iii) B C , l. Answer(c)(iii) { … } [1] (d) Find (i) A B C n( ) , , l, Answer(d)(i) … [1] (ii) A B C n( ) + + l. Answer(d)(ii) … [1]

8 0607/41/M/J/15 © UCLES 2015 7 Squash balls have radius 1.5 cm. They are sold in boxes. Each box is a cuboid. Each box has length 15 cm, width 12 cm and height 3 cm. (a) Show that the maximum number of balls in a box is 20. [1] (b) Calculate the volume of one ball. Answer(b) … cm3 [2] (c) Calculate the total volume of 20 balls. Answer(c) … cm3 [1] (d) Write your answer to part (c) in standard form. Answer(d) … cm3 [1] (e) Calculate the percentage of the volume of the box that the 20 balls fill. Answer(e) … % [2]

9 0607/41/M/J/15 © UCLES 2015 [Turn over 8 O C A X 26° NOT TO SCALE P B A, B and C lie on a circle, centre O. AP and BP are tangents to the circle. AB intersects OP at the point X and angle OPB = 26°. (a) Find the size of (i) angle ABP, Answer(a)(i) … [1] (ii) angle OBA, Answer(a)(ii) … [1] (iii) angle ACB. Answer(a)(iii) … [1] (b) Write down the mathematical name of quadrilateral AOBP. Answer(b) … [1] (c) Complete these statements. (i) Triangle OBP is congruent to triangle … . [1] (ii) Triangle OBP is similar, but not congruent to, triangle … . [1]

10 0607/41/M/J/15 © UCLES 2015 9 The table shows the amount in dollars, y, that 10 families of different size, x, spend in one week. Number in family, (x) 2 2 3 3 5 5 6 6 6 6 Amount in dollars, (y). 60 65 80 75 100 105 120 135 125 115 (a) (i) Complete the scatter diagram. The first four points have been plotted for you. x y 1 0 50 60 70 80 90 100 110 120 130 140 150 2 3 4 Number in family Amount in dollars 5 6 7 8 [2] (ii) What type of correlation is shown by the scatter diagram? Answer(a)(ii) … [1]

11 0607/41/M/J/15 © UCLES 2015 [Turn over (b) Find (i) the mean family size, Answer(b)(i) … [1] (ii) the mean amount spent in one week. Answer(b)(ii) $ … [1] (c) (i) Find the equation of the regression line in the form y mx c = + . Answer(c)(i) y = … [2] (ii) Use your answer to part (c)(i) to estimate the amount spent in one week by a family of 4. Answer(c)(ii) $ … [1]

12 0607/41/M/J/15 © UCLES 2015 10 72° 58° D C 80 m NOT TO SCALE 64 m 65 m A B (a) Find AC. Answer(a) … m [2]

13 0607/41/M/J/15 © UCLES 2015 [Turn over (b) Calculate angle CAD. Answer(b) … [3] (c) Calculate the area of the quadrilateral ABCD. Answer(c) … m2 [4]

14 0607/41/M/J/15 © UCLES 2015 11 Paula invests $3000 in Bank A and $3000 in Bank B. (a) Bank A pays compound interest at a rate of 4% each year. (i) Find the total amount that Paula has in Bank A at the end of 3 years. Answer(a)(i) $ … [2] (ii) After how many complete years is the total amount that Paula has in Bank A greater than $4000? Answer(a)(ii) … [3] (b) Bank B pays simple interest at a rate of 5% each year. (i) Find the total amount that Paula has in Bank B at the end of 3 years. Answer(b)(i) $ … [1] (ii) After how many complete years is the total amount that Paula has in Bank B greater than $4000? Answer(b)(ii) … [1] (c) After how many complete years will the total amount that Paula has in Bank A be greater than the total amount that Paula has in Bank B? Answer(c) … [3]

15 0607/41/M/J/15 © UCLES 2015 [Turn over 12 Bag 1 only contains 6 blue balls and 4 red balls. Bag 2 only contains 8 blue balls and 2 red balls. Marco chooses a ball at random from Bag 1 and puts it in Bag 2. He then chooses a ball at random from Bag 2 and puts it in Bag 1. (a) Complete the tree diagram. blue 6 10 … … … … … red blue Bag 2 Bag 1 red blue red [2] (b) Find the probability that the two balls chosen are (i) both blue, Answer(b)(i) … [2] (ii) one red and one blue. Answer(b)(ii) … [3] (c) Find the probability that, after Marco chooses the two balls, there are exactly 6 blue balls in Bag 1. Answer(c) … [3]

16 0607/41/M/J/15 © UCLES 2015 13 In this question all lengths are in centimetres. 5x + 4 6x + 1 NOT TO SCALE 2x – 1 (a) Write down a quadratic equation, in terms of x, and show that it simplifies to . x x 7 24 16 0 2 - - = [3] (b) Factorise . x x 7 24 16 2 - - Answer(b) … [2]

17 0607/41/M/J/15 © UCLES 2015 [Turn over (c) Show that the area of the triangle is 84 cm2. [2] (d) The area of this rectangle is equal to the area of the triangle. Find the value of y. y + 2 y NOT TO SCALE Answer(d) y = … [4]

18 0607/41/M/J/15 © UCLES 2015 14 In this question all measurements are in metres. A rectangular garden has length p and width q. The garden is divided into 3 sections as shown in the diagram. x Grass Flowers Vegetables 2 3 p p q NOT TO SCALE 1 4 q (a) Write down an expression, in terms of p and q, for the area for flowers. Answer(a) …m2 [1] (b) Show that x p 4 3 = . [2]

19 0607/41/M/J/15 © UCLES 2015 [Turn over (c) Find an expression, in terms of p and q, for the area for grass. Give your answer in its simplest form. Answer(c) …m2 [2] (d) Find the ratio area for vegetables : area for grass . Answer(d) … : … [2] Question 15 is printed on the next page.

20 0607/41/M/J/15 © UCLES 2015 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. 15 The diagram shows a sketch of the graph of f y x = ^ h where f x x x x x 4 3 4 3 2 2 - + + + = ^ h . O x y NOT TO SCALE (a) (i) Find the equations of the three asymptotes. Answer(a)(i) … , … , … [3] (ii) Find the co-ordinates of the local maximum point. Answer(a)(ii) ( … , …) [2] (iii) Find the co-ordinates of the local minimum point. Answer(a)(iii) ( … , …) [2] (b) Find the values of k, when (i) f x k = ^ h has no solutions, Answer(b)(i) … [2] (ii) f x k = ^ h has one solution. Answer(b)(ii) … [1] (c) Solve the inequality f x 0 2 ^ h . Answer(c) … [3]

® IGCSE is the registered trademark of Cambridge International Examinations. CAMBRIDGE INTERNATIONAL EXAMINATIONS Cambridge International General Certificate of Secondary Education MARK SCHEME for the May/June 2015 series 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 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 2015 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 2015 0607 41 © Cambridge International Examinations 2015 Abbreviations cao correct answer only dep dependent FT follow through after error isw ignore subsequent working oe or equivalent SC Special Case nfww not from wrong working soi seen or implied Qu. Answer Mark Part Marks 1 (a) 8 1 (b) 10 1 (c) 6 1 (d) 4.5 2 B1 for [LQ =] 3.5 or [UQ =] 8 (e) 5.375 1 2 Correctly equating one set of coefficients Correct method to eliminate one variable x = 1.5 y = – 2 M1 M1 B1 B1 or making x or y the subject of one equation or substituting into other equation or sketch of their two lines If 0 scored, SC1 for correct substitution into one of original equations to find other variable 3 (a) 20 1 (b) 13.225 2 or M1 for (264.5 or 260 or 269) ÷ their (a) oe 4 (a) Reflection x = –1 oe 1 1 Any combination of transformations scores 0 (b) (–1, 4), (–1, 2), (0, 2) 3 M2 for any rotation 90° clockwise If 0 scored, SC2 for rotation 90° anti-clockwise about (–1, 6) or SC1 for any rotation 90° anti-clockwise (c) Reflection y = x + 7 oe 1 1FT Any combination of transformations scores 0 FT if SC2 scored in (b) to y = –x + 5

Page 3 Mark Scheme Syllabus Paper Cambridge IGCSE – May/June 2015 0607 41 © Cambridge International Examinations 2015 5 (a) (i) 3 2 M1 for y = x k or 9 1 25 1 5 = y oe If 0 scored, SC1 for 0.648 oe or 3 25 oe (ii) 0.36 oe 2FT FT 2 25       k their only from correct variation, k ≠ 1 B1 for       25 k their oe soi k ≠ 1 If 0 scored, SC1 for 4.02 or 4.024 to 4.025 or 225 (iii) x = 2 225 y or 2 15       y 2 M1 for x = 2 y c or y k their x = oe k ≠ 1 If 0 scored, SC1 for y 405 or 25 9 2 y (b) y = –3(x – 2)(x + 4) or 24 6 3 2 + − − x x 3 M2 for [ y =] k(x – 2)(x + 4), k 1 ≠ soi or M1 for (x – 2)(x + 4) seen OR M1 for k(x + 1)2 + c, k 1 ≠ and M1 for substituting two points to get 24 = + k c and 0 = 9k + c OR M1 for 3 correct equations in y = ax2 + bx + c and M1 for eliminating one variable from all three equations. If 0 scored, SC1 for ax2 + bx + 24 soi 6 (a) A = {1, 2, 3, 4, 6, 12} B = {1, 2, 3, 6} 1 1 (b) A 4 C B 11 1 2 12 13 3 6 14 5 7 8 9 10 15 3 B1 for 4 in correct position B1 for 12 in correct position

Page 4 Mark Scheme Syllabus Paper Cambridge IGCSE – May/June 2015 0607 41 © Cambridge International Examinations 2015 (c) (i) {1, 2, 3, 6} 1FT FT from their diagram (ii) {11, 13, 14} 1FT FT from their diagram (iii) {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 15} 1FT FT from their diagram (d) (i) 6 1FT FT from their diagram (ii) 15 1FT FT from their diagram 7 (a) 3 3 3 12 3 15 × × 1 (b) 14.1 or 14.13 to 14.14 2 M1 for 3 4 π1.53 (c) 282 or 283 or 282.6 to 282.8 1FT FT their (b) × 20 (d) 2.82 × 102 or 2.83 × 102 or (2.826 to 2.828) × 102 1FT FT their (c) in standard form (e) 52.20 to 52.41 2FT FT answer only if less than 100 M1 for 100 15 12 3 × × × (c) their 8 (a) (i) 64 1 (ii) 26 1FT FT 90 – their (a)(i) (iii) 64 1 (b) Kite or Cyclic Quadrilateral 1 (c) (i) OAP 1 (ii) OXB or OXA 1 9 (a) (i) All points correctly plotted 2 B1 for 4 or 5 correct points (ii) Positive 1 (b) (i) 4.4 cao final answer 1 (ii) 98 1 (c) (i) 31.7 + 15.1x or 31.66 to 31.67 + (15.07 to 15.08) 2 B1 for 31.7 (or 31.66 to 31.67) + px or q + (15.1 (or 15.07 to 15.08)) x or SC1 for 15x + 32 (ii) 91.94 to 92.1 1FT FT their (c)(i)

Page 5 Mark Scheme Syllabus Paper Cambridge IGCSE – May/June 2015 0607 41 © Cambridge International Examinations 2015 10 (a) 86.1 or 86.08 to 86.09 2 M1 for 652 + 802 – 2 × 65 × 80 × cos72 (b) 39.1 or 39.07 to 39.09 3 M2FT for [sin A = ] 64 sin58 × (a) their or M1FT for 58 sin 1. 86 sin 64 their A = oe (c) 5210 or 5206 to 5207 4FT M1 for 0.5 × 65 × 80 × sin 72 M2FT for 0.5 × 64 × their 86.1×sin (180 – 58 – their (b)) oe or M1 for [angle ACD] = 180 – 58 – their (b) 11 (a) (i) 3374.59 2 M1 for 3000 × 1.043 oe (ii) 8 3 M2 for ( ) 04 .1 log log 3000 4000 oe or at least 2 trials, one of which goes beyond 4000, soi by 7.3 to 7.4 or M1 for 3000 × 1.04n = 4000 or at least 2 trials or if 0 scored, SC1 for answer 7 (b) (i) 3450 1 (ii) 7 1 (c) 12 3 B2 for 11.91... or 11 or M1 for sketch of both functions with intersection or for 3000 × 1.04n = 3000(1 + 0.05n) oe or T & I beyond n = 8 12 (a) 10 4 , 11 9 11 2 , 11 8 11 3 2 B1 for one correct pair on 2nd bag (b) (i) 110 54 oe cao 2 M1FT for 10 6 × their 11 9 (ii) 110 44 oe cao 3 M2FT for 10 6 × their 11 2 + 10 4 × their 11 8 oe or M1FT for one of above products (c) 110 66 oe cao 3 M2FT for 10 6 × their 11 9 + 10 4 × their 11 3 or (b)(i) + 10 4 × their 11 3 or 1 – their (b)(ii) oe or M1FT for 10 6 × their 11 9 or 10 4 × their 11 3

Page 6 Mark Scheme Syllabus Paper Cambridge IGCSE – May/June 2015 0607 41 © Cambridge International Examinations 2015 13 (a) (6x + 1)2 = (5x + 4)2 + (2x – 1)2 oe Any one of 36x2 + 6x + 6x + 1 oe 25x2 + 20x + 20x + 16 oe 4x2 – 2x – 2x + 1 oe Completion to 7x2 – 24x – 16 = 0 with no errors or omissions M1 B1 A1 (b) (x – 4)(7x + 4) 2 B1 for (x + a)(7x + b) where ab = – 16 or 7a + b = –24 (c) 0.5 × 7 × 24 [=84] M2 B1 for x = 4 (d) 8.22 or 8.219 to 8.22[0] 4 B1 for y(y + 2) = 84 oe M2 for 1 2 ) 84 )( 1( 4 ) 2 ( 2 2 × − − ± − oe or suitable sketch or M1 for formula with 1 error or (y + 1)2 – 1 = 84 oe 14 (a) 6 1 pq oe final answer 1 (b) 3 2 p + 3 1 4 1 × p oe M2 M1 for 3 2 4 1 × p or for 3 1 4 3 × p (c) 32 21 pq final answer 2 M1 for 4 3 q × 4 3 p + 4 1 2 1 × p × 4 3 q oe (d) 17 : 63 cao 2 isw attempt to change form of 17 : 63 to e.g. 1 : 63 17 M1 for pq – their (a) – their (c) oe soi by any equivalent ratio 15 (a) (i) y = 1 x = 1 x = 3 1 1 1 (ii) (1.73, –13.9) or (1.732... , –13.93 to –13.92) 2 B1 for each (iii) (–1.73, –0.07 18) or (–1.732... , –0.07180 to –0.07179...) 2 B1 for each

Page 7 Mark Scheme Syllabus Paper Cambridge IGCSE – May/June 2015 0607 41 © Cambridge International Examinations 2015 (b) (i) –13.9 < k < –0.0718 2FT FT y coordinates from (ii) and (iii) B1 for one inequality correct or SC1 for 0718 0 9. 13 . k − − < < or for –13.9 < x < –0.0718 (ii) –13.9, –0.0718 1FT FT y coordinates from (a)(ii) and (a)(iii) (c) x < –3 –1 < x < 1 x > 3 1 1 1 Not f(x)