Cambridge IGCSE Mathematics 0580 — 2004 Oct/Nov Paper 3 · Variant 1

1 (a) Two friends, Hatab and Yasin, went on a cycle ride. For Part of the distance-time graph for their journey is shown below. Examiner's Use 18 16 14 12 Distance 10 from home Yasin (km) 8 Hatab 6 Hatab and Yasin 4 2 0 10 00 11 00 12 00 13 00 14 00 15 00 Time of day For the first part of the journey they cycled at the same speed. (i) Find their speed for the first part of the journey. Answer(a)(i) km/h [1] (ii) At 11 00 they stopped for half an hour. Show this on the graph. [1] (iii) They continued on their ride and at 12 45 they were 16 kilometres from home. Show this part of the journey on the graph. [1] (iv) They stopped again and then had a race going home. (a) For how long did they stop? Answer(a)(iv)(a) min [1] (b) Who won the race? Answer(a)(iv)(b) [1] (v) What was the total length of their journey? Answer(a)(v) km [1] (b) On a certain day the conversion rate between dollars ($) and Indian rupees was For Examiner's $1 = 45 rupees. Use (i) How many rupees were equivalent to $10? Answer(b)(i) rupees [1] (ii) Use this information to draw a conversion graph on the axes below. 500 400 300 Rupees 200 100 0 1 2 3 4 5 6 7 8 9 10 11 Dollars ($) [2] (iii) Use your graph to find (a) how many rupees were equivalent to $6.80, Answer(b)(iii)(a) rupees [1] (b) how many dollars were equivalent to 480 rupees. Answer(b)(iii)(b) $ [1]

Mark scheme: number Marks Total 1 a) i) 10 1 ii) straight line from 1 (11,10) to (11 30,10) iii) straight line from 1√ allow +2 mm in length by (11 30,10) to (12 45,16) eye but must go through the correct points. f.t. from their (1130,10) iv) a) 15 1 allow ¼ hour b) Hatab 1 v) 32 1 b) i) 450 1 ii) straight line ruled from 2 SC1 for freehand or (1,45) to (10,450) broken line or any straight line through the origin ± ½ small square at both points iii) a) 306 ± 4 1 b) 10 60 to 10.80 1 allow 10.6 etc. 11

More questions on Rates

2 For y Examiner's Use 6 5 4 D 3 A 2 1 _7 _6 _5 _4 _3 _2 _1 0 1 2 3 4 5 6 7 x _1 _2 C _3 _4 B _5 _6 (a) Describe fully the single transformation that maps triangle A onto triangle B. Answer(a) [3] (b) Describe fully the single transformation that maps triangle A onto triangle C. Answer(b) [3] (c) Find the centre and the scale factor of the enlargement that maps triangle A onto triangle D. Answer(c) centre ( , ) scale factor [2] (d) On the grid (i) draw the image of triangle A under a reflection in the line x = −1, [2] (ii) draw the image of triangle B under a rotation of 180° about (−4, −3). [2]

Mark scheme: 2 a) translation 1 must be single transformation  − 6  1 SC1 for correct vector   inverted, or   − 7 1  − 12    , or for correct row  − 14  vector, or co-ordinates. Condone missing brackets b) rotation M1 must be single transformation -90 or 90 clockwise o.e. A1 about (0, 0) o.e. A1 IGCSE EXAMINATIONS – NOVEMBER 2004 0580/0581 3 c) (0, 0) 1 1.5 o.e. 1 not 3:2 etc. d) i) correct triangle drawn 2 SC1 for reflection of A in any vertical line or in y = -1 ii) correct triangle drawn 2 SC1 for 180o rotation about any point or SC1 for rotation ± 90o about (-4,-3) 12

More questions on Transformations

3 For A Examiner's Use NOT TO SCALE 2 cm B 10 cm 6 cm 40o C E D On the above diagram, AB = 2 cm, BD = 6 cm, AE = 10 cm, angle BCD = 40° and angle BDE = 90°. (a) Write down the length of AD. Answer(a) AD = cm [1] (b) Calculate the length of DE. Answer(b) DE = cm [2] (c) Calculate the size of angle AED. Answer(c) angle AED = [2] (d) Calculate the length of CD. Answer(d) CD = cm [3] (e) Find the length of CE. Answer(e) CE = cm [1]

Mark scheme: 3 In this question alternative methods must be complete a) 8 1 b) 6 2 M1 for 100 − 64 o.e. must show square root c) art 53.1 2 M1 for sin and 8/10 seen o.e. d) art 7.15 3 M1 for tan 40 and 6 seen +M1 for 6/tan 40 o.e. e) 13.15 or 13.2 1√ f.t. for their b) + d) to 3 s.f. or better 9

More questions on Right-angled triangles

4 (a) For A Examiner's Use NOT TO SCALE 5 cm 6 cm B C 4 cm (i) In the space below, using a ruler and compasses only, construct the above triangle accurately. [3] (ii) Using the triangle you have drawn, measure and write down the size of angle ACB. Answer(a)(ii) angle ACB = [1] (b) In the diagram below two points, P and Q, are joined by a straight line. For Examiner's Use P Q (i) On the diagram draw the locus of all the points that are 4 centimetres from the line PQ. [3] (ii) On the same diagram, using a straight edge and compasses only, construct the locus of the points that are equidistant from P and Q. Show all your construction lines. [2] (iii) Shade the region which contains the points that are closer to P than to Q and are less than 4 centimetres from the line PQ. [2]

Mark scheme: 4 a) i) triangle drawn with three 3 2 for two sides correct, sides the correct length with arcs ± 0.1 cm 1 for two sides correct without arcs ii) 56 ± 2 c.a.o. 1 b) in this part of the question deduct 1 once for broken lines i) complete locus drawn 3 1 for a line correct distance from PQ 1 for a semicircle IGCSE EXAMINATIONS – NOVEMBER 2004 0580/0581 3 ii) correct line drawn B1 ± 1 mm, ± 1o correct arcs, radius > 4 cm B1 iii) correct area shaded 2 SC1 for shading on left hand side of their ‘mediator’ or inside lines drawn for their b) i) 11

More questions on Right-angled triangles

5 (a) For A Examiner's Use NOT TO SCALE D 140o 80o B yo C In the diagram above AB=BC and AD=DC. (i) What is the special name of the quadrilateral ABCD? Answer(a)(i) [1] (ii) On the diagram draw the line of symmetry. [1] (iii) Calculate the value of y. Answer(a)(iii) y = [2] (b) N po 40o M ro qo O NOT TO SCALE K L In the diagram above, the points K,L,M and N lie on the circle centre O. KN is parallel to LM. Find the values of p,q and r. Answer(b) p = , q = , r = [3] (c) For xo Examiner's Use NOT TO SCALE The diagram above shows a regular seven-sided polygon. Each of the interior angles measures x°. One of the angles is marked in the diagram. Calculate the value of x, giving your answer correct to 1 decimal place. Show all your working. Answer(c) x = [4]

Mark scheme: 5 a) i) kite 1 ii) correct line BD drawn 1 Allow broken line, one line only iii) 70 2 360 − 140 − 80 M1 for o.e. 2 b) (p =) 90 1 (q =) 50 1 (r =) 50 1√ f.t. from their q, not strict f.t. c) 128.6 c.a.o. 4 360 M2 for 180 - or 7 5× 180 o.e. 7 (may be implied by art 129) +A1 for 128.57 11

More questions on Geometrical terms

6 (a) Complete the table below for y = x2 − 2x. For Examiner's x −2 −1 0 1 2 3 4 Use y 8 −1 3 8 [3] (b) On the grid below, draw the graph of y = x2 − 2x for −2 x 4. y 8 7 6 5 4 3 2 y = 2 1 _4 _3 _2 _1 0 1 2 3 4 x _1 _2 [4] (c) The line y = 2 is drawn on the diagram. Use your graph to find the values of x that solve the equation x2 − 2x = 2. Answer(c) x = or x = [2] (d) Complete the table below for y = 4 − x. x −4 0 4 y 8 [2] (e) On the grid above, draw the line y = 4 − x for −4 x 4. [1] (f) Write down the x coordinates of the points of intersection of the graphs of y = x2 − 2x and y = 4 − x. Answer(f) x = or x = [2]

Mark scheme: 6 a) 3 0 0 1,1,1 b) 7 correct points plotted P3√ P2√ for 5 or 6 points ± ½ sm. sq. P1√ for 4 points. not strict f.t. smooth curve through all correct points C1 incorrectly plotted points should be ignored for C1. Minimum curved, not pointed c) -0.8 to -0.7 c.a.o. 1 ignore any y values 2.7 to 2.8 c.a.o. 1 IGCSE EXAMINATIONS – NOVEMBER 2004 0580/0581 3 d) 4 0 1,1 e) correct line drawn through 1 complete line (-4,8) and (4,0) f) -1.7 to -1.4 c.a.o. 1 ignore any y values 2.4 to 2.7 c.a.o. 1 14

More questions on Graphs of functions

7 (a) Rajeesh thought of a number. For He multiplied this number by 2. Examiner's He then added 10. Use The answer was 42. (i) What was the number Rajeesh first thought of? Answer(a)(i) [1] (ii) Simon thought of a number x. He multiplied this number by 3 and then added 8. Write down an expression in x for his answer. Answer(a)(ii) [2] (b) Simplify − 8a + 7b − a − 2b. Answer(b) [2] (c) Factorise fully 6a − 9a2 . Answer(c) [2] (d) Make t the subject of the formula v = u + at. Answer(d) t= [2] (e) Solve the simultaneous equations 8x + 2y = 13, 3x + y = 4. Answer(e) x = , y = [4]

Mark scheme: 7 a) i) 16 1 ii) 3x + 8 o.e. 2 M1 for 3x. allow n instead of x. deduct 1 for ‘= x’ or ‘= 0’ or = any number, but allow a different letter b) -9a 1 +5b 1 c) 3a(2 – 3a) 2 M1 for any correct partial factorisation d) v - u 2 M1 for v – u seen o.e. a e) (x=) 2.5 2 M1 for correct multiplication of LHS of one or both equations to equalise coefficients or for a recognisable attempt to eliminate one variable (y=) -3.5 2 M1 for correct substitution of their other value or M2 correct matrix method 13

More questions on Equations

8 (a) The list shows the rainfall in millimetres in Prestbury for the 12 months of 2002. For Examiner's 61 146 22 54 67 94 141 22 37 167 87 170 Use (i) Write down the mode. Answer(a)(i) mm [1] (ii) Find the median. Answer(a)(ii) mm [2] (iii) Calculate the mean. Answer(a)(iii) mm [2] (b) During the years 1996 - 2000 the total rainfall in Prestbury was 5400 millimetres. The pie chart shows how this was spread over the five years. 1996 2000 1997 1999 1998 (i) Measure the angles of the sectors for 1998, 1999 and 2000. For Write your answers in the table below. [3] Examiner's Use (ii) Work out the annual rainfall, in millimetres, for each of the years 1998, 1999 and 2000. Write your answers in the table below. [3] Answers (b)(i) and (ii) Year Angle (degrees) Rainfall (mm) 1996 54 810 1997 60 900 1998 1999 2000 Total 360 5400 (iii) What do you notice about the trend in the rainfall from 1996 to 2000? Answer(b)(iii) [1]

Mark scheme: 8 a) i) 22 1 ii) 67 + 87 2 M1 for evidence of 77 or 2 ranking seen anywhere. e.g. 67,87 iii) 89 2 ∑x M1 for their 12 IGCSE EXAMINATIONS – NOVEMBER 2004 0580/0581 3 b) i) 72 ± 1 1 80 ± 1 1 94 ± 1 1 ii) 1080 ± 5 1√ strict f.t.s for their angle x 15 ± 5 1200 ± 5 1√ 1410 ± 5 1√ iii) appropriate observation 1 12

More questions on Averages and range

9 (a) A pattern of numbers is shown below. For Examiner's row Use 1 1 2 2 3 4 3 5 6 7 8 9 4 10 11 12 13 14 15 16 5 17 18 19 20 21 22 23 24 25 6 26 ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... (i) On the diagram complete row 6. [1] (ii) The last numbers in each row form a sequence. 1, 4, 9, 16, 25, …………… (a) What is the special name given to these numbers? Answer(a)(ii)(a) [1] (b) Write down the last number in the 10th row. Answer(a)(ii)(b) [1] (c) Write down an expression for the last number in the nth row. Answer(a)(ii)(c) [1] (iii) The numbers in the middle column of the pattern form a sequence. 1, 3, 7, 13, 21, 31, ………….. (a) Write down the next number in this sequence. Answer(a)(iii)(a) [1] (b) The expression for the nth number in this sequence is n2 − n + 1. Work out the 30th number. Answer(a)(iii)(b) [2] (b) Another pattern of numbers is shown below. For Examiner's Use row 1 1 2 3 4 5 6 7 8 9 10 2 11 12 13 14 15 16 17 18 19 20 3 21 22 23 24 25 26 27 28 29 30 4 31 32 33 34 35 36 37 38 39 40 (i) What is the last number in the 10th row? Answer(b)(i) [1] (ii) Find an expression for the last number in the nth row. Answer(b)(ii) [1] (iii) What is the first number in the 10th row? Answer(b)(iii) [1] (iv) Find an expression for the first number in the nth row. Answer(b)(iv) [1]

Mark scheme: 9 a) i) 27 to 36 entered correctly 1 ii) a) square 1 b) 100 1 c) n2 c.a.o. 1 allow n x n iii) a) 43 c.a.o. 1 b) 871 2 M1 for 900 – 30 + 1 o.e. b) i) 100 1 ii) 10n c.a.o. 1 allow 10 x n iii) 91 1 vi) 10n – 9 o.e. 1 11 Total 104

More questions on Sequences