Cambridge IGCSE Mathematics (with coursework) 0581 — 2008 May/June Paper 4 · Variant 1

This document consists of 13 printed pages and 3 blank pages. IB08 06_0580_04_RP © UCLES 2008 [Turn over *2879260860* UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education MATHEMATICS 0580/04, 0581/04 Paper 4 (Extended) May/June 2008 2 hours 30 minutes Additional Materials: Answer Booklet/Paper Electronic calculator Geometrical instruments Graph paper (1 sheet) Mathematical tables (optional) Tracing paper (optional) READ THESE INSTRUCTIONS FIRST If you have been given an Answer Booklet, follow the instructions on the front cover of the Booklet. 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 a soft pencil for any diagrams or graphs. Do not use staples, paper clips, highlighters, glue or correction fluid. Answer all questions. All working must be clearly shown. It should be done on the same sheet as the rest of the answer. Marks will be given for working which shows that you know how to solve the problem even if you get the answer wrong. Electronic calculators should be used. If the degree of accuracy is not specified in the question, and if the answer is not exact, give the answer to three significant figures. Give answers in degrees to one decimal place. For π use either your calculator value or 3.142. 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 of the marks for this paper is 130. www.XtremePapers.com

2 © UCLES 2008 0580/04/M/J/08 DO NOT WRITE YOUR ANSWERS ON THIS QUESTION PAPER. WRITE ALL YOUR WORKING AND ANSWERS ON THE SEPARATE ANSWER BOOK OR PAPER PROVIDED. 1 Vreni took part in a charity walk. She walked a distance of 20 kilometres. (a) She raised money at a rate of $12.50 for each kilometre. (i) How much money did she raise by walking the 20 kilometres? [1] (ii) The money she raised in part (a)(i) was 5 52 of the total money raised. Work out the total money raised. [2] (iii) In the previous year the total money raised was $2450. Calculate the percentage increase on the previous year’s total. [2] (b) Part of the 20 kilometres was on a road and the rest was on a footpath. The ratio road distance : footpath distance was 3:2. (i) Work out the road distance. [2] (ii) Vreni walked along the road at 3 km / h and along the footpath at 2.5 km / h. How long, in hours and minutes, did Vreni take to walk the 20 kilometres? [2] (iii) Work out Vreni’s average speed. [1] (iv) Vreni started at 08 55. At what time did she finish? [1] (c) On a map, the distance of 20 kilometres was represented by a length of 80 centimetres. The scale of the map was 1 : n. Calculate the value of n. [2] 2 (a) (i) Factorise x2 − x − 20. [2] (ii) Solve the equation x2 − x − 20 = 0. [1] (b) Solve the equation 3x2 − 2x − 2 = 0. Show all your working and give your answers correct to 2 decimal places. [4] (c) y = m2 − 4n2. (i) Factorise m2 − 4n2. [1] (ii) Find the value of y when m = 4.4 and n = 2.8. [1] (iii) m = 2x + 3 and n = x − 1. Find y in terms of x, in its simplest form. [2] (iv) Make n the subject of the formula y = m2 − 4n2. [3]

3 © UCLES 2008 0580/04/M/J/2008 [Turn over (d) (i) m4 − 16n4 can be written as (m2 − kn2)(m2 + kn2). Write down the value of k. [1] (ii) Factorise completely m4n − 16n5. [2] 3 (a) Bag A Bag B Nadia must choose a ball from Bag A or from Bag B. The probability that she chooses Bag A is 2 3 . Bag A contains 5 white and 3 black balls. Bag B contains 6 white and 2 black balls. The tree diagram below shows some of this information. white ball black ball white ball black ball Bag A Bag B q r s p 2 3 5 8 (i) Find the values of p, q, r and s. [3] (ii) Find the probability that Nadia chooses Bag A and then a white ball. [2] (iii) Find the probability that Nadia chooses a white ball. [2] (b) Another bag contains 7 green balls and 3 yellow balls. Sani takes three balls out of the bag, without replacement. (i) Find the probability that all three balls he chooses are yellow. [2] (ii) Find the probability that at least one of the three balls he chooses is green. [1]

4 © UCLES 2008 0580/04/M/J/08 4 200 180 160 140 120 100 80 60 40 20 0 Cumulative frequency ( h) 0 10 20 30 40 Number of hours worked (h) 50 60 70 80 200 people record the number of hours they work in a week. The cumulative frequency graph shows this information.

5 © UCLES 2008 0580/04/M/J/2008 [Turn over (a) Use the graph to find (i) the median, [1] (ii) the upper quartile, [1] (iii) the inter-quartile range, [1] (iv) the number of people who work more than 60 hours in a week. [2] (b) Omar uses the graph to make the following frequency table. Hours worked (h) 0IhY10 10IhY20 20IhY30 30IhY40 40IhY50 50IhY60 60IhY70 70IhY80 Frequency 12 34 36 30 38 30 p q (i) Use the graph to find the values of p and q. [2] (ii) Calculate an estimate of the mean number of hours worked in a week. [4] (c) Shalini uses the graph to make a different frequency table. Hours worked (h) 0IhY30 30IhY40 40IhY50 50IhY80 Frequency 82 30 38 50 When she draws a histogram, the height of the column for the interval 30IhY40 is 9 cm. Calculate the height of each of the other three columns. [4]

6 © UCLES 2008 0580/04/M/J/08 5 O A B C D E F Q G H P 12 cm NOT TO SCALE A circle, centre O, touches all the sides of the regular octagon ABCDEFGH shaded in the diagram. The sides of the octagon are of length 12 cm. BA and GH are extended to meet at P. HG and EF are extended to meet at Q. (a) (i) Show that angle BAH is 135°. [2] (ii) Show that angle APH is 90°. [1] (b) Calculate (i) the length of PH, [2] (ii) the length of PQ, [2] (iii) the area of triangle APH, [2] (iv) the area of the octagon. [3] (c) Calculate (i) the radius of the circle, [2] (ii) the area of the circle as a percentage of the area of the octagon. [3]

7 © UCLES 2008 0580/04/M/J/08 [Turn over 6 A B C D 5 4 3 2 1 y x –1 O –4 –3 –2 –1 1 2 3 4 The pentagon OABCD is shown on the grid above. (a) Write as column vectors (i) , [1] (ii) BC . [1] (b) Describe fully the single transformation which maps the side BC onto the side OD. [2] (c) The shaded area inside the pentagon is defined by 5 inequalities. One of these inequalities is y Y 1 2 x + 4. Find the other 4 inequalities. [5]

8 © UCLES 2008 0580/04/M/J/08 7 (a) O A B C D T S 130° 40° NOT TO SCALE z° y° x° A, B, C and D lie on a circle, centre O. SCT is the tangent at C and is parallel to OB. Angle AOB = 130°, and angle BCT = 40°. Angle OBC = x°, angle OBA = y° and angle ADC = z°. (i) Write down the geometrical word which completes the following statement. “ABCD is a quadrilateral.” [1] (ii) Find the values of x, y and z. [3] (iii) Write down the value of angle OCT. [1] (iv) Find the value of the reflex angle AOC. [1]

9 © UCLES 2008 0580/04/M/J/08 [Turn over (b) X P Q R S 7 cm 10 cm NOT TO SCALE P, Q, R and S lie on a circle. PQ = 7 cm and SR = 10 cm. PR and QS intersect at X. The area of triangle SRX = 20 cm2. (i) Write down the geometrical word which completes the following statement. “Triangle PQX is to triangle SRX.” [1] (ii) Calculate the area of triangle PQX. [2] (iii) Calculate the length of the perpendicular height from X to RS. [2]

10 © UCLES 2008 0580/04/M/J/08 8 Answer the whole of this question on a sheet of graph paper. Use one side for your working and one side for your graphs. Alaric invests $100 at 4% per year compound interest. (a) How many dollars will Alaric have after 2 years? [2] (b) After x years, Alaric will have y dollars. He knows a formula to calculate y. The formula is y = 100 × 1.04x x (Years) 0 10 20 30 40 y (Dollars) 100 p 219 q 480 Use this formula to calculate the values of p and q in the table. [2] (c) Using a scale of 2 cm to represent 5 years on the x-axis and 2 cm to represent $50 on the y-axis, draw an x-axis for 0 Y x Y 40 and a y-axis for 0 Y y Y 500. Plot the five points in the table and draw a smooth curve through them. [5] (d) Use your graph to estimate (i) how many dollars Alaric will have after 25 years, [1] (ii) how many years, to the nearest year, it takes for Alaric to have $200. [1] (e) Beatrice invests $100 at 7% per year simple interest. (i) Show that after 20 years Beatrice has $240. [2] (ii) How many dollars will Beatrice have after 40 years? [1] (iii) On the same grid, draw a graph to show how the $100 which Beatrice invests will increase during the 40 years. [2] (f) Alaric first has more than Beatrice after n years. Use your graphs to find the value of n. [1]

11 © UCLES 2008 0580/04/M/J/08 [Turn over 9 R O P Q M L NOT TO SCALE S OPQR is a parallelogram. O is the origin. OP= p and OR= r. M is the mid-point of PQ and L is on OR such that OL : LR = 2 : 1. The line PL is extended to the point S. (a) Find, in terms of p and r, in their simplest forms, (i) OQ, [1] (ii) PR , [1] (iii) PL , [1] (iv) the position vector of M. [1] (b) PLS is a straight line and PS = 3 2 PL. Find, in terms of p and / or r, in their simplest forms, (i) PS , [1] (ii) QS . [2] (c) What can you say about the points Q, R and S? [1]

12 © UCLES 2008 0580/04/M/J/08 10 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 A 3 by 3 square x b c can be chosen from the 6 by 6 grid above. d e f g h i (a) One of these squares is 8 9 10 . 14 15 16 20 21 22 In this square, x = 8, c = 10, g = 20 and i = 22. For this square, calculate the value of (i) (i − x) − (g − c), [1] (ii) cg − xi. [1] (b) x b c d e f g h i (i) c = x + 2. Write down g and i in terms of x. [2] (ii) Use your answers to part(b)(i) to show that (i − x) − (g − c) is constant. [1] (iii) Use your answers to part(b)(i) to show that cg − xi is constant. [2]

13 © UCLES 2008 0580/04/M/J/08 (c) The 6 by 6 grid is replaced by a 5 by 5 grid as shown. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 A 3 by 3 square x b c can be chosen from the 5 by 5 grid. d e f g h i For any 3 by 3 square chosen from this 5 by 5 grid, calculate the value of (i) (i − x) − (g − c), [1] (ii) cg − xi. [1] (d) A 3 by 3 square is chosen from an n by n grid. (i) Write down the value of (i − x) − (g − c). [1] (ii) Find g and i in terms of x and n. [2] (iii) Find cg − xi in its simplest form. [1]

16 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. 0580/04/M/J/08 BLANK PAGE

UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education MARK SCHEME for the May/June 2008 question paper 0580, 0581 MATHEMATICS 0580/04, 0581/04 Paper 4 (Extended), maximum raw mark 130 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. All Examiners are instructed that alternative correct answers and unexpected approaches in candidates’ scripts must be given marks that fairly reflect the relevant knowledge and skills demonstrated. Mark schemes must be read in conjunction with the question papers and the report on the examination. • CIE will not enter into discussions or correspondence in connection with these mark schemes. CIE is publishing the mark schemes for the May/June 2008 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 Syllabus Paper IGCSE – May/June 2008 0580, 0581 04 © UCLES 2008 1 (a) (i) 250 B1 (ii) their (a)(i) ÷ 5× 52 o.e. 2600 ft www2 M1 A1 ft SC1 for 12.5 ÷ 5× 52, implied by 130 (iii) 100 2450 2450 × − (a)(ii) their o.e. 6.1(22……..) ft www2 M1 A1ft 100 100 2450 − × (a)(ii) their , x 150 100 2450 = ft M & A only if their (a)(ii) > 2450 (b) (i) 20 ÷ 5× 3 12 www2 M1 A1 Accept 12, 8 or 8, 12 (ii) their (b)(i) ÷ 3 and (20 – their (b)(i)) ÷ 2.5 7 hours 12 mins cao www2 M1 A1 4 and 3.2 or 7.2 or 7h 20 mins seen imply M1 Condone poor notation e.g. 7-12 (iii) 2.78 (2.777–2.778) o.e. cao o.e. in other units B1 o.e. must have units stated e.g. 0.7716..m/s, 46.29 – 46.30 m/min (iv) 16 07 o.e. ft B1 ft ft their (b)(ii) + 08 55 iff finishes on same day and (b)(ii) has hours and mins (c) 20× 100000 ÷ 80 o.e. 25 000 or 2.5× 104 www2 M1 A1 25 000 seen in final ans. After M0, SC1 for figs 25 or 0.00004 final answer [13] 2 (a) (i) (ii) (x + 4)(x – 5) –4, 5 ft B2 B1 ft If B0, SC1 if of form (x ± 4)(x ± 5), Only ft the SC –4, and 5 not from (x – 4)(x + 5). (b) 2.3 2 4.3 2) ( 2) ( 2 − − − ± − − B1,B1 B1 for (–2)2–4(3)(–2) (or better) seen inside a square root. The expression must be in the form r q p ) (or− + then B1 for p = –(–2) and r = 2.3 or better Allow recoveries from incomplete lines –0.55, 1.22 cao B1,B1 If B0, SC1 for –0.5 and 1.2 or both answers correct to 2 or more decimal places (rounded or truncated). –0.54858, 1.21525… (c) (i) (m – 2n)(m + 2n) B1 (ii) –12 B1 (iii) 20x + 5 o.e. cao final ans B2 B1 for (4x2 + 6x + 6x + 9) or (x2 – x – x + 1) or (2x + 3 – 2(x – 1))(2x + 3 + 2(x – 1)) (iv) 4n2 = m2 – y o.e. 4 2 2 y m n − = o.e. 4 ) ( 2 y m n − = o.e. www3 Mark final answer M1 M1 M1 M1 for correct re-arrangement for n2 term (may be –n2) M1 for correct division by 4 or – 4 M1 for correctly taking square root of n² term SC2 for 4 2 m y ± or 4 2 y m − o.e. ww (d) (i) (ii) 4 or –4 or ±4 n(m4 – 16n4) or (m2n – 4n3)(m2 + 4n2) or (m2n + 4n3)(m2 – 4n2) or n(m – 2n)(m + 2n)(m2 + 4n2) B1 M1 A1 Correctly taking out n or a correct factor with n still in one bracket Must be final answer [17]

Page 3 Mark Scheme Syllabus Paper IGCSE – May/June 2008 0580, 0581 04 © UCLES 2008 3 Accept all probability answers as fractions (non-reduced or reduced), decimals or percentages. –1 once for 2 sf answers or correct words. Condone numerical errors in simplifying or converting after correct answers seen. Ratio answers score zero throughout. (a) (i) 3 1 , 8 3 , 8 6 , 8 2 o.e. B3 –1 each error bod if no letters given (ii) 8 5 3 2 × 12 5 o.e. www2 M1 A1 24 10 , etc., 0.416(6…) (iii) their 8 6 3 1 12 5 × + 3 2 o.e. cao www2 M1 A1 24 16 , 12 8 , etc., 0.666(6….) (b) (i) 8 1 9 2 10 3 × × 120 1 o.e. www2 M1 A1 720 6 , etc., 0.00833(3…) (ii) 120 119 o.e. B1ft 720 714 , etc., 0.991(6…) ft 1 – their (i) not for 7/10 Could start again and have a correct answer independently [10] 4 (a) (i) 36 (36.0–36.4) B1 (ii) 50 (50.0–50.4) B1 (iii) 29 (28.6–29.4) B1 (iv) 20 B2 If B0, SC1 for 19 or 21 or 180 seen (b) (i) p = 16, q = 4 B1,B1 If B0, SC1 if p and q add up to 20 (ii)       200 7220 = 36.1 cso www4 B4 Answer 36 scores 4 marks after some correct working shown with no incorrect working seen M1 for using mid-values at least four correct from 5, 15, 25, 35, 45, 55, 65, 75 M1 (dep on correct mid values or mid- values ±0.5) for ∑fx (at least four correct products) M1 (dependent on 2nd M1) for dividing sum by 200 or 180 + their p + their q (c) 8.2 (8.19–8.20), 11.4, 5 (5.00–5.01) B4 B3 for 2 correct or B2 for 1 correct After B0, SC2 for fd’s 2.7(3…) o.e., 3.8 o.e, 1.6(6...) o.e. or SC1 for 2 of fd’s correct (15) 5 (a) (i) 360 ÷ 8 or (8 – 2)× 180 M1 allow 6× 180

Page 4 Mark Scheme Syllabus Paper IGCSE – May/June 2008 0580, 0581 04 © UCLES 2008 180 – their (360 ÷ 8) o.e. ÷ 8 M1 dependent (ii) 45° used or use implied o.e. E1 Accept sketch with values (b) (i) 12 l = cos45 o.e. (PH = ) 8.49 (8.485….) www2 M1 A1 For o.e. allow implicit expression Accept 72 , 18 2 , 8 3 , 2 6 (ii) (PQ =) 2×their PH + 12 o.e. (PQ =) 29.(0) (28.96–29.00) ft www2 M1 A1 ft ft their PH accept surd form (iii) their PH×their PH ÷2 o.e. (Area APH =) 36 (35.95–36.1) ft www2 M1 A1 ft ft their PH (iv) (their PQ)2 – 4 × their area of triangle o.e. (Area octagon = ) 695 (694.0–697.1) cao www3 M2 A1 If M0, M1 for a clear collection of areas leading to the octagon possibly without any calculation shown (c) (i) 0.5 of their PQ o.e. 14.5 (14.47–14.53) cao www2 M1 A1 e.g. 6 + PH, 6tan67.5° accept surd form (ii) 2 ( r) their × π 100 × area octagon their area circle their 94.8 (94.35 to 95.60) cao www3 M1 M1 A1 (660.5…) Dependent on first M1 and circle smaller than the octagon [17] 6 (a) (i)       1 2 B1 Allow (2 1), condone omission of brackets (ii)       1 2 ft B1ft Allow (2 1), condone omission of brackets ft their (i) if a vector (b) Translation       −4 0 o.e. B1, B1 Allow (0 –4), condone omission of brackets, allow in words Any extra transformation spoils both marks (c) y > 0 o.e. x < 2 o.e. y > 2 1 x o.e. y < 2x + 4 o.e. B1 B1 B1 B2 For all four, condone strict inequalities and only penalise first incorrect sign, which may be = or an inequality sign If B0, B1 for 2x or for 4 if other co-efficient is not zero y < 2 1 x + 4 gets zero [9] 7 (a) (i) cyclic B1 Condone concyclic (ii) Any one of 40, 45, 50 Any one of 20, 25, 30 Any one of 105, 110, 115 B1 B1 B1 Angle BCT = 40° is inconsistent with ST parallel to OB. So different values of angles x, y, z, OCT and AOC can be arrived at, depending on route taken. (iii) Any one of 80, 85, 90 B1 (iv) Any one of 210, 215, 220, 225, 230 B1 (b) (i) Similar (or enlargement) B1 (ii) 2       10 7 or 2       7 10 o.e. seen 9.8 (9.79 to 9.81) www2 M1 A1 (0.49), (2.04) It is possible to do (iii) then (ii) and full marks can still be scored (iii) 2 1 ×10×height = 20 4 www2 M1 A1 [11]

Page 5 Mark Scheme Syllabus Paper IGCSE – May/June 2008 0580, 0581 04 © UCLES 2008 8 (a) 108(.16) (allow 108.2(0)) www2 M1 A1 M1 for method of compound interest used (b) 148(.02…) 324(.3…) B1 B1 (c) Correct axes full domains 5 correct pts 100, 148 ft, 219, 324ft, 480 Smooth exponential curve, correct shape through 5 points S1 P3ft C1 Condone absence of labels P2ft for 4 correct, P1ft for 3 correct Points must be in correct square vertically, including on line Scale error – remove that part and try to mark the rest (d) (i) 265 – 270 B1ft If out of range, then ft their graph at 25 years (ii) 17 or 18 cao B1 (e) (i) (100) 20 7 (100) × × o.e. 100 + 7 × 20 or better M1 E1 No errors (ii) 380 B1 (iii) Correct straight ruled line for x – range 0 to 35 L2 P1ft for 2 of (0,100), (20,240) (40,380)ft correctly plotted (f) 27 – 29 cao B1 [17] 9 (a) (i) p + r B1 Answers in bracketed column form penalise only once throughout (ii) –p + r B1 (iii) –p + 3 2 r B1 (iv) p + 2 1 r B1 (b) (i) 2 3 × (–p + 3 2 r) or – 2 3 p + r isw after correct answer seen B1 ft ft only 2 3 × their (a)(iii) (ii) PS QP + o.e. – 2 3 p www 2 M1 A1 ft o.e. is any correct route of at least 2 vectors ft their (b)(i) – r (c) lie on a straight line B1 dependent on their (b)(ii) being a multiple of p [8] 10(a) (i) 4 B1 (ii) 24 B1 (b) (i) x + 12, x + 14 o.e. B1,B1 Any order ignore ref to g and i (ii) (x + 14 – x) and (x + 12 – (x + 2)) 14 – 10 or 14 – 12 + 2 or 4 E1 x + 12 and x + 14 must be seen to be used No errors seen (iii) (x + 2)(x + 12) – x(x + 14) 24 B1 E1 Subtraction can be implied later Dep on B1 and no errors anywhere for the E mark (c) (i) 4 B1 (ii) 20 B1 (d) (i) 4 B1 (ii) x + 2n o.e., x + 2+ 2n o.e. B1,B1 (iii) 4n B1 Allow 4×n, n×4, n4 [13]