Cambridge IGCSE Mathematics (with coursework) 0581 — 2011 Oct/Nov Paper 4 · Variant 3

This document consists of 20 printed pages. IB11 11_0581_43/FP © UCLES 2011 [Turn over *3302860416* UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education MATHEMATICS 0581/43 Paper 4 (Extended) October/November 2011 2 hours 30 minutes Candidates answer on the Question Paper. Additional Materials: Electronic calculator Geometrical instruments Mathematical tables (optional) Tracing paper (optional) 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 a pencil for any diagrams or graphs. Do not use staples, paper clips, highlighters, glue or correction fluid. DO NOT WRITE IN ANY BARCODES. Answer all questions. If working is needed for any question it must be shown below that question. 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 2011 0581/43/O/N/11 For Examiner's Use 1 1.2 m 0.8 m 0.5 m d 1.2 m 0.4 m NOT TO SCALE A rectangular tank measures 1.2 m by 0.8 m by 0.5 m. (a) Water flows from the full tank into a cylinder at a rate of 0.3 m3/min. Calculate the time it takes for the full tank to empty. Give your answer in minutes and seconds. Answer(a) min s [3]

3 © UCLES 2011 0581/43/O/N/11 [Turn over For Examiner's Use (b) The radius of the cylinder is 0.4 m. Calculate the depth of water, d, when all the water from the rectangular tank is in the cylinder. Answer(b) d = m [3] (c) The cylinder has a height of 1.2 m and is open at the top. The inside surface is painted at a cost of $2.30 per m2. Calculate the cost of painting the inside surface. Answer(c) $ [4]

4 © UCLES 2011 0581/43/O/N/11 For Examiner's Use 2 (a) Complete the table of values for y = 2x. x –2 –1 0 1 2 3 y 0.25 1 2 8 [2] (b) On the grid, draw the graph of y = 2x for O2 Y x Y 3. y x 10 9 8 7 6 5 4 3 2 1 –1 0 –1 1 2 3 –2 [3]

5 © UCLES 2011 0581/43/O/N/11 [Turn over For Examiner's Use (c) (i) On the grid, draw the straight line which passes through the points (0, 2) and (3, 8). [1] (ii) The equation of this line is y = mx + 2. Show that the value of m is 2. Answer(c)(ii) [1] (iii) One answer to the equation 2x =2x + 2 is x = 3. Use your graph to find the other answer. Answer(c)(iii) x = [1] (d) Draw the tangent to the curve at the point where x = 1. Use this tangent to calculate an estimate of the gradient of y = 2x when x = 1. Answer(d) [3]

6 © UCLES 2011 0581/43/O/N/11 For Examiner's Use 3 (a) 40° 30° D C E B A NOT TO SCALE ABCD is a quadrilateral with angle BAD = 40°. AB is extended to E and angle EBC = 30°. AB = AD and BD = BC. (i) Calculate angle BCD. Answer(a)(i) Angle BCD = [3] (ii) Give a reason why DC is not parallel to AE. Answer(a)(ii) [1] (b) A regular polygon has n sides. Each exterior angle is 2 5n degrees. Find the value of n. Answer(b) n = [3]

7 © UCLES 2011 0581/43/O/N/11 [Turn over For Examiner's Use (c) 25° O C B A NOT TO SCALE The diagram shows a circle centre O. A, B and C are points on the circumference. OC is parallel to AB. Angle OCA = 25°. Calculate angle OBC. Answer(c)Angle OBC = [3]

8 © UCLES 2011 0581/43/O/N/11 For Examiner's Use 4 6 4 2 –2 –4 –6 0 –2 2 4 6 –4 –6 y x P W (a) Draw the reflection of shape P in the line y = x. [2] (b) Draw the translation of shape P by the vector      − 1 2 . [2] (c) (i) Describe fully the single transformation that maps shape P onto shape W. Answer(c)(i) [3] (ii) Find the 2 by 2 matrix which represents this transformation. Answer(c)(ii)           [2] (d) Describe fully the single transformation represented by the matrix       2 0 0 1 . Answer(d) [3]

9 © UCLES 2011 0581/43/O/N/11 [Turn over For Examiner's Use 5 (a) The cost of a bottle of juice is 5 cents more than the cost of a bottle of water. Mohini buys 3 bottles of water and 6 bottles of juice. The total cost is $5.25. Find the cost of a bottle of water. Give your answer in cents. Answer(a) cents [4] (b) The cost of a biscuit is x cents. The cost of a cake is (x + 3) cents. The number of biscuits Roshni can buy for 72 cents is 2 more than the number of cakes she can buy for 72 cents. (i) Show that x2 + 3x O 108 = 0. Answer(b)(i) [3] (ii) Solve the equation x2 + 3x O 108 = 0. Answer(b)(ii) x = or x = [3] (iii) Find the total cost of 2 biscuits and 1 cake. Answer(b)(iii) cents [1]

10 © UCLES 2011 0581/43/O/N/11 For Examiner's Use 6 Q P C B A D 3 cm 4 cm 12 cm NOT TO SCALE The diagram shows a triangular prism of length 12 cm. The rectangle ABCD is horizontal and the rectangle DCPQ is vertical. The cross-section is triangle PBC in which angle BCP = 90°, BC = 4 cm and CP = 3 cm. (a) (i) Calculate the length of AP. Answer(a)(i) AP = cm [3] (ii) Calculate the angle of elevation of P from A. Answer(a)(ii) [2]

11 © UCLES 2011 0581/43/O/N/11 [Turn over For Examiner's Use (b) (i) Calculate angle PBC. Answer(b)(i) Angle PBC = [2] (ii) X is on BP so that angle BXC = 120°. Calculate the length of XC. Answer(b)(ii) XC = cm [3]

12 © UCLES 2011 0581/43/O/N/11 For Examiner's Use 7 The times, t minutes, taken for 200 students to cycle one kilometre are shown in the table. Time (t minutes) 0 I=t=Y=2 2 I=t=Y=3 3 I=t=Y=4 4 I=t=Y=8 Frequency 24 68 72 36 (a) Write down the class interval that contains the median. Answer(a) [1] (b) Calculate an estimate of the mean. Show all your working. Answer(b) min [4]

13 © UCLES 2011 0581/43/O/N/11 [Turn over For Examiner's Use (c) (i) Use the information in the table opposite to complete the cumulative frequency table. Time (t minutes) t Y 2 t Y 3 t Y 4 t Y 8 Cumulative frequency 24 200 [1] (ii) On the grid, draw a cumulative frequency diagram. 200 180 160 140 120 100 80 60 40 20 Cumulative frequency t 0 1 2 3 4 Time (minutes) 5 6 7 8 [3] (iii) Use your diagram to find the median, the lower quartile and the inter-quartile range. Answer(c)(iii) Median = min Lower quartile = min Inter-quartile range = min [3]

14 © UCLES 2011 0581/43/O/N/11 For Examiner's Use 8 f(x) = x2 + x O1 g(x) = 1 O 2x h(x) = 3x (a) Find the value of hg(–2). Answer(a) [2] (b) Find g –1(x). Answer(b) g O1(x) = [2] (c) Solve the equation f(x) = 0. Show all your working and give your answers correct to 2 decimal places. Answer(c) x = or x = [4] (d) Find fg(x). Give your answer in its simplest form. Answer(d) fg(x) = [3] (e) Solve the equation h – 1(x) = 2. Answer(e) x = [1]

15 © UCLES 2011 0581/43/O/N/11 [Turn over For Examiner's Use 9 Set A S U M S Set B M I N U S The diagram shows two sets of cards. (a) One card is chosen at random from Set A and replaced. (i) Write down the probability that the card chosen shows the letter M. Answer(a)(i) [1] (ii) If this is carried out 100 times, write down the expected number of times the card chosen shows the letter M. Answer(a)(ii) [1] (b) Two cards are chosen at random, without replacement, from Set A. Find the probability that both cards show the letter S. Answer(b) [2] (c) One card is chosen at random from Set A and one card is chosen at random from Set B. Find the probability that exactly one of the two cards shows the letter U. Answer(c) [3] (d) A card is chosen at random, without replacement, from Set B until the letter shown is either I or U. Find the probability that this does not happen until the 4th card is chosen. Answer(d) [2]

16 © UCLES 2011 0581/43/O/N/11 For Examiner's Use 10 Hassan stores books in large boxes and small boxes. Each large box holds 20 books and each small box holds 10 books. He has x large boxes and y small boxes. (a) Hassan must store at least 200 books. Show that 2x + y [ 20. Answer(a) [1] (b) Hassan must not use more than 15 boxes. He must use at least 3 small boxes. The number of small boxes must be less than or equal to the number of large boxes. Write down three inequalities to show this information. Answer(b) [3] (c) On the grid, show the information in part (a) and part (b) by drawing four straight lines and shading the unwanted regions. y x 20 18 16 14 12 10 8 6 4 2 0 2 4 6 8 10 12 14 16 18 20 [6]

17 © UCLES 2011 0581/43/O/N/11 [Turn over For Examiner's Use (d) A large box costs $5 and a small box costs $2. (i) Find the least possible total cost of the boxes. Answer(d)(i) $ [1] (ii) Find the number of large boxes and the number of small boxes which give this least possible cost. Answer(d)(ii) Number of large boxes = Number of small boxes = [2] Question 11 is printed on the next page.

18 © UCLES 2011 0581/43/O/N/11 For Examiner's Use 11 (a) y x –3 –2 –1 0 1 2 3 4 5 5 4 3 2 1 P Q The points P and Q have co-ordinates (–3, 1) and (5, 2). (i) Write as a column vector. Answer(a)(i) =           [1] (ii) = 2      − 1 1 Mark the point R on the grid. [1] (iii) Write down the position vector of the point P. Answer(a)(iii)           [1]

19 © UCLES 2011 0581/43/O/N/11 [Turn over For Examiner's Use (b) O L U K V M u v NOT TO SCALE In the diagram, = u and = v. K is on UV so that = 3 2 and L is on OU so that = 4 3 . M is the midpoint of KL. Find the following in terms of u and v, giving your answers in their simplest form. (i) Answer(b)(i) = [4] (ii) Answer(b)(ii) = [2] Question 12 is printed on the next page.

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 0581/43/O/N/11 For Examiner's Use 12 (a) The nth term of a sequence is n(n+1). (i) Write the two missing terms in the spaces. 2, 6, , 20, [2] (ii) Write down an expression in terms of n for the (n + 1)th term. Answer(a)(ii) [1] (iii) The difference between the nth term and the (n + 1)th term is pn + q. Find the values of p and q. Answer(a)(iii) p = q = [2] (iv) Find the positions of the two consecutive terms which have a difference of 140. Answer(a)(iv) and [2] (b) A sequence u1, u2, u3, u4, …………. is given by the following rules. u1 = 2, u2 = 3 and un = 2u 2 − n + u 1 − n for n [ 3. For example, the third term is u3 and u3 = 2u1 + u2 = 2 × 2 + 3 = 7. So, the sequence is 2, 3, 7, u4, u5, ….. (i) Show that u4 = 13. Answer(b)(i) [1] (ii) Find the value of u5 . Answer(b)(ii) u5 = [1] (iii) Two consecutive terms of the sequence are 3413 and 6827 . Find the term before and the term after these two given terms. Answer(b)(iii) , 3413, 6827, [2]

UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education MARK SCHEME for the October/November 2011 question paper for the guidance of teachers 0581 MATHEMATICS 0581/43 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, 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 October/November 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 – October/November 2011 0581 43 © University of Cambridge International Examinations 2011 Abbreviations cao correct answer only cso correct solution only dep dependent ft follow through after error isw ignore subsequent working oe or equivalent SC Special Case www without wrong working art anything rounding to soi seen or implied Qu. Answers Mark Part Marks 1 (a) 1 min 36 s www 3 M1 for 1.2 × 0.8 × 0.5 (= 0.48) A1 1.6 or 96 If A0, B1 for correctly converting to min and sec Dep on M1 (b) 0.954 to 0.956 www 3 M2 for 2 0.4 0.48 π their × or M1 for π × 0.42 × d = ‘0.48’ (c) 8.09 to 8.10 www 4 M1 for π × 0.42 (0.503) condone × 2 and M1 for π × 0.8 × 1.2 (3.02) M1 for their area × 2.3 (dep M1 M1) 2 (a) 0.5, 4 1+1 (b) 6 points plotted ft P2 P1 for 5 points Correct shaped curve through 6 points (exponential) C1 Ignore to left of x = −2 (c) (i) Correct ruled line reaching both points L1 (ii) 6 ÷ 3 oe 1 Allow ‘test’ with a coordinate on the line (not 0, 2) (iii) –0.8 to –0.6 1 Dep on L1 (d) Tangent drawn at (1, 2) T1 Not chord, allow up to 1 mm daylight Rise/run attempt using correct scales M1 Dep on T1 1.2 to 1.6 cao A1 3 (a) (i) 50 www3 3 B1 for angle ADB or ABD = 70 B1 for angle DBC = 80 (ii) Angle DCB ≠ angle CBE oe 1 Accept angle CDB ≠ angle ABD (b) 12 B3 M2 for n n 360 2 5 = oe or M1 for 360 soi (c) 65 www 3 OAC = 25, CAB = 25, OBA = 50, BOC = 50, AOB = 80, AOC = 130 B1 each, max 2

Page 3 Mark Scheme: Teachers’ version Syllabus Paper IGCSE – October/November 2011 0581 43 © University of Cambridge International Examinations 2011 4 (a) Image (1, –1), (1, –2), (4, –2), (3, –1) 2 B1 if vertices plotted only or reflects in y = –x (b) Image (–3, 2), (–4, 2), (–4, 5), (–3, 4) 2 B1 for translation by      − k 2 or       1 k (c) (i) Rotation only, 1 Spoilt if extras 90 clockwise oe, 1 (Centre) (0, 0) oe 1 (ii)       − 0 1 1 0 2 B1 for one row or one column correct (d) Stretch only, 1 Spoilt if extras (Factor) 2, 1 x-axis oe invariant 1 5 (a) 55 www B4 M3 for 3w + 6(w + 5) = 525 oe in $ or (3j – 5) + 6j = 525 oe in $ or M2 for j = w + figs5 oe and 3w + 6j = figs525 or M1 for w and w + figs5 or j and j – figs5 (b) (i) 2 3 72 72 = + −x x oe M2 M1 for x 72 or 3 72 + x 72(x + 3) – 72x = 2x(x + 3) oe M1 Dep on 3 terms above Fractions removed, isw (ii) –12, 9 www 3 M2 for (x + 12)(x – 9) or 2 441 3 ± − or SC1 for ) )( ( b x a x + + where ab = –108 or a + b = 3 or 2 108 1 4 3 3 2 − × × − ± − (iii) 30 1 ft 3 × a positive root + 3 6 (a) (i) 13 or 13.0 www 3 M1 for 32 + 42 oe Equiv if find AC first and M1 for ) 4 3 ( their 12 2 2 2 + + (ii) 13.32 to 13.35 or 13.3 2 M1 for sin = AP their 3 or tan = AC their 3 oe (b) (i) 36.86 to 36.87 or 36.9 2 M1 for tan (PBC) = 4 3 oe (ii) 2.770 to 2.774 or 2.77 3 M2 for 120 sin ) )( ( their sin 4 i b or M1 for correct implicit eqn

Page 4 Mark Scheme: Teachers’ version Syllabus Paper IGCSE – October/November 2011 0581 43 © University of Cambridge International Examinations 2011 7 (a) 3 < t ≤ 4 1 Condone alt. notation used for class (b) 1 2.5 3.5 6 M1 Mid-interval values soi ∑ fx with x in correct interval M1 Allow 1 slip (24 170 252 216) 662 ÷ 200 M1 M1 dep on second M1 3.31 cso A1 (c) (i) 92, 164 1 (ii) (2, 24), (3, 92), (4, 164), (8, 200) ft P2ft P1ft for 3 points Curve/polygon through the 4 points 1ft ft increasing curve/polygon (iii) 3 ≤ med ≤ 3.2 B1 2.4 ≤ lq ≤ 2.7 B1 0.9 ≤ iqr ≤ 1.5 B1 8 (a) 243 2 B1 for (g(–2) =) 5 seen or 3(1–2x) (b) 2 1 x − or 2 1 − − x final ans 2 M1 for x = 1 − 2y or x = (1 − y)/2 (c) )1( 2 )1 )( 1( 4 1 1 2 − − ± − B2 B1 for )1 )( 1( 4 12 − − or better ( 5 ) seen anywhere If in form r q p + or r q p − B1 for p = –1 and r = 2(1) –1.62, 0.62 B1B1 SC1 for –1.62 and 0.62 seen or –1.6 or –1.618.. and 0.6 or 0.618… (d) 4x2 – 6x + 1 final ans www3 3 M1 for (1 – 2x)2 + (1 – 2x) – 1 or better and B1 for (1 – 2x)2 = 1 – 2x – 2x + 4x2 or better (e) 9 1

Page 5 Mark Scheme: Teachers’ version Syllabus Paper IGCSE – October/November 2011 0581 43 © University of Cambridge International Examinations 2011 9 Accept fraction, %, dec equivalents (3sf or better when not exact) throughout but not ratio or words isw incorrect cancelling/conversion to other forms (a) (i) 4 1 oe 1 (ii) 25 cao 1ft ft their 4 1 × 100 to 3sf or better or rounding or truncating to integer Not 25/100 (b) 12 2 oe cao 2 M1 for 4 2 × 3 1 0.167, 16.7% (c) 20 7 oe cao 3 M2 for 4 1 × 5 4 + 4 3 × 5 1 or M1 for 4 1 × 5 4 or 4 3 × 5 1 After 0, SC1 for 7 correct in list (condone UU in addition) (d) 60 6 oe cao 2 M1 for 5 3 × 4 2 × 3 1 ×       2 2 10 (a) 20x + 10y ≥ 200 1 In (a), (b) −1 once for wrong symbol (b) x + y ≤ 15, y ≥ 3, y ≤ x 3 B1 for each (c) All lines long enough to make full boundary of region, accept dashed or solid lines, 2 mm acc at intercepts 2x + y = 20 ruled B2 B1 for ruled line through (10, 0) or (0, 20) x + y = 15 ruled B1 y = x ruled B1 y = 3 ruled B1 −1 once, freehand Quadrilateral identified R1 Allow if slight inaccuracy(s) in diagonal lines Allow any clear indication of region (d) (i) 47 cao 1 (ii) 7, 6 cao 2 M1 for any 5x + 2y in their region evaluated to equal their 47

Page 6 Mark Scheme: Teachers’ version Syllabus Paper IGCSE – October/November 2011 0581 43 © University of Cambridge International Examinations 2011 11 (a) (i)       1 8 1 (ii) Point (3, 4) indicated 1 (iii)      − 1 3 1 (b) (i) 12 5 − u + 3 2 v oe 2 terms 4 M1 for any correct route L to K e.g. LU + UK and B1 for LU = u/4 oe or OL = ¾u oe and B1 for UK = 2/3 (v – u) oe or VK = 1/3(u – v) oe all Bs are soi (ii) 24 13 u + 3 1 v oe 2 terms 2 M1 for correct route from O to M e.g. OL + LM (can be in terms of u, v) 12 (a) (i) 12, …, 30 2 B1 each (ii) (n + 1)(n + 2) oe 1 isw if expand incorrectly (iii) p = 2 1 q = 2 1 (iv) 69(th), 70(th) 2 M1 for their 2n + 2 = 140 soi (b) (i) 2 × 3 + 7 1 Accept 2 × 3 + 2 × 2 + 3 (ii) 27 1 (iii) 1707, …, 13 653 1,1