Cambridge IGCSE Mathematics 0580 — 2015 May/June Paper 4 · Variant 1

1 12 000 vehicles drive through a road toll on one day. The ratio cars : trucks : motorcycles = 13 : 8 : 3. (a) (i) Show that 6500 cars drive through the road toll on that day. Answer(a)(i) [1] (ii) Calculate the number of trucks that drive through the road toll on that day. Answer(a)(ii) ................................................. [1] (b) The toll charges in 2014 are shown in the table. Vehicle Charge Cars $2 Trucks $5 Motorcycles $1 Show that the total amount paid in tolls on that day is $34 500. Answer(b) [2] (c) This total amount is a decrease of 8% on the total amount paid on the same day in 2013. Calculate the total amount paid on that day in 2013. Answer(c) $ .................................................. [3] (d) 2750 of the 6500 car drivers pay their toll using a credit card. Write down, in its simplest terms, the fraction of car drivers who pay using a credit card. Answer(d) ................................................. [2] (e) To the nearest thousand, 90 000 cars drive through the road toll in one week. Write down the lower bound for this number of cars. Answer(e) ................................................. [1]

Mark scheme: Question Answers Mark Part Marks 1313 1 (a) (i) × 12000 with no 1 13 + 8 + 3 subsequent errors (ii) 4000 1 (b) 2 × 6500 + 5 × their (a)(ii) + 2 B1 for any two of (12000 − 6500 − their (a)(ii) ) 2 × 6500, 5 × their (a)(ii) , (12000 – 6500 – their(a)(ii)) seen or or (13 × 2 + 8 × 5 + 3 × 1) × 500 13 × 2 + 8 × 5 + 3 × 1 34500 (c) 37 500 3 M2 for × 100 oe 100 − 8 or M1 for 34500 associated with (100 – 8)% 11 (d) cao 2 M1 for any correct simplified version of 26 2750 6500 (e) 89 500 1

More questions on Ratio and proportion

2 1 2 The table shows some values for y = x - 2 , x ! 0 . x x –2 –1.5 –1 –0.5 –0.25 –0.2 0.2 0.25 0.5 1 1.5 2 y 4.25 2.58 2.06 2.54 –2.46 –1.94 1.92 3.75 (a) Complete the table of values. [4] 1  x  – 0.2 and 0.2  x  2. (b) On the grid, draw the graph of y = x2 – 2 for – 2 x y 5 4 3 2 1 x –2 –1 0 1 2 –1 –2 –3 [5] 2 1 (c) By drawing a suitable line, use your graph to solve the equation x - 2 = 2 . x Answer(c) x = ..................... or x = ..................... or x = ..................... [3] 2 1 (d) The equation x - 2 = k has only one solution. x Write down the range of values of k for which this is possible. Answer(d) ................................................. [2] (e) By drawing a suitable tangent, find an estimate of the gradient of the curve at the point where x = –1. Answer(e) ................................................. [3]

Mark scheme: 2 (a) 1.5 1.25 −0.75 0.5 4 B1 for each (b) Fully correct curve 5 B5 for correct curve over full domain or B3 FT for 11 or 12 points or B2 FT for 9 or 10 points or B1 FT for 7 or 8 points and B1 independent for one complete branch on each side of the y-axis and not touching or crossing the y-axis SC4 for correct curve with branches joined

More questions on Graphs of functions

3 y 7 6 5 4 3 C 2 A 1 x –12 –11 –10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 –1 –2 –3 –4 B –5 –6 –7 (a) Draw the image of - 1 (i) shape A after a translation by , [2] 3 f p (ii) shape A after a rotation through 180° about the point (0, 0), [2] 1 0 (iii) shape A after the transformation represented by the matrix [3] 0 - 1 f p. (b) Describe fully the single transformation that maps shape A onto shape B. Answer(b) ................................................................................................................................................. .............................................................................................................................................................. [3] (c) Find the matrix which represents the transformation that maps shape A onto shape C. Answer(c) [2] f p

Mark scheme:  1  k  3 (a) (i) image at (1, 4) (1, 5) (2, 5) (4, 4) 2 SC1 for translation by   or    k   3  or 4 correct vertices plotted but not joined (ii) image at (−2, −1 ) (−5, −1) (−2, −2) 2 SC1 for correct size and orientation, wrong (−3, −2) position or 4 correct vertices plotted but not joined (iii) image at (2, −1) (2, −2) (3 ,−2) 3 B2 for 3 correct vertices plotted (5, −1) or if no / wrong plots allow SC2 for 4 correct coordinates in column matrix or shown in working or SC1 for any 3 correct coordinates or  1 0  2 2 3 5  M1 for    oe  0 − 1 1 2 2 1  (b) enlargement B1 [centre] (1, 0) B1 not as column vector [scale factor] − 3 B1 (c)  0 − 1 2 B1 for one correct row or column or    0 1   1 0     − 1 0 

More questions on Transformations

4 30 students were asked if they had a bicycle (B), a mobile phone (M) and a computer (C). The results are shown in the Venn diagram. B M 2 4 x 7 1 6 3 2 C (a) Work out the value of x. Answer(a) x = ................................................. [1] (b) Use set notation to describe the shaded region in the Venn diagram. Answer(b) ................................................. [1] (c) Find n(C (M B)). Answer(c) ................................................. [1] (d) A student is chosen at random. (i) Write down the probability that the student is a member of the set M  . Answer(d)(i) ................................................. [1] (ii) Write down the probability that the student has a bicycle. Answer(d)(ii) ................................................. [1] (e) Two students are chosen at random from the students who have computers. Find the probability that each of these students has a mobile phone but no bicycle. Answer(e) ................................................ [3]

Mark scheme: 4 (a) 5 1 (b) C ∩ M oe 1 Allow e.g. (B ∩ C ∩ M) ∪ (C ∩ M) (c) 3 1 8 (d) (i) oe 1 0.267 or better 30 14 (ii) oe 1 0.467 or better 30 30 6 5 (e) oe 3 M2 for × 272 17 16 6 or M1 for seen 17 0.110[2…] or better 55

More questions on Introduction to probability

5 (a) Andrei stands on level horizontal ground, 294 m from the foot of a vertical tower which is 55 m high. (i) Calculate the angle of elevation of the top of the tower. Answer(a)(i) ................................................. [2] (ii) Andrei walks a distance x metres directly towards the tower. The angle of elevation of the top of the tower is now 24.8°. Calculate the value of x. Answer(a)(ii) x = ................................................. [4] (b) The diagram shows a pyramid with a horizontal rectangular base. NOT TO y SCALE 4 m 3 m 4.8 m The rectangular base has length 4.8 m and width 3 m and the height of the pyramid is 4 m. Calculate (i) y, the length of a sloping edge of the pyramid, Answer(b)(i) y = ............................................. m [4] (ii) the angle between a sloping edge and the rectangular base of the pyramid. Answer(b)(ii) ................................................ [2]

Mark scheme: 55 5 (a) (i) 10.6 or 10.59... 2 M1 for tan = oe 294 55 (ii) 175 or 174.9[…] to 175.[1…] 4 M2 for [adj =] oe tan 24 8. or M1 for implicit version and M1 dep on at least M1 for 294 − their adj 2 2  1 2 2  2 8.4 + 3  (b) (i) 4.9 or 4.89 to 4.9 4 M3 for 4 +  or M2 for 12 8.4 2 + 32 or M1 for 8.4 2 + 3 2 or 4.2 2 + 5.1 2 4 (ii) 54.7 or 54.71 to 54.722 2 M1 for sin = their 9.4

More questions on Right-angled triangles

6 The table shows the time, t minutes, that 400 people take to complete a test. Time taken 0  t  10 10  t  24 24  t  30 30  t  40 40  t  60 60  t  70 (t mins) Frequency 10 90 135 85 70 10 (a) (i) Write down the modal time interval. Answer(a)(i) .......................................... min [1] (ii) Calculate an estimate of the mean time taken to complete the test. Answer(a)(ii) .......................................... min [4] (b) (i) Complete the table of cumulative frequencies. Time taken t  10 t  24 t  30 t  40 t  60 t  70 (t mins) Cumulative 10 100 400 frequency [2] (ii) On the grid opposite, draw a cumulative frequency diagram to show this information. 400 350 300 250 Cumulative frequency 200 150 100 50 t 0 10 20 30 40 50 60 70 Time taken (minutes) [3] (c) Use your graph to estimate (i) the median time, Answer(c)(i) .......................................... min [1] (ii) the inter-quartile range, Answer(c)(ii) .......................................... min [2] (iii) the 15th percentile, Answer(c)(iii) .......................................... min [2] (iv) the number of people who took more than 50 minutes. Answer(c)(iv) ................................................. [2]

Mark scheme: 6 (a) (i) 24 < t Y 30 1 (ii) 30.9 or 30.875 nfww 4 M1 for midpoints soi (condone 1 error or omission) 5, 17, 27, 35, 50, 65 soi M1 for use of ∑fx with x in correct interval including both boundaries (condone 1 further error or omission) (50, 1530, 3645, 2975, 3500, 650) and M1 (dep on 2nd M1) for ∑fx ÷ 400 (b) (i) [10 100] 235 320 390 [400] 2 B1 for any two correct SC1 for 235, n, n + 70 n > 235 (ii) Correct curve or polygon 3 B1 for correct horizontal placement B1FT for correct vertical placement B1FT dep on at least B1 for reasonable increasing curve or polygon through their 6 points If zero scored SC1 for 5 out of 6 points correctly plotted (c) (i) 27.5 to 29 1 (ii) 12 to 14 2 B1 for 36 to 38 or 24 seen (iii) 18 to 20 2 B1 for 60 seen or marked on grid (iv) 30 to 45 2 B1 for 355 to 370 seen

More questions on Averages and measures of spread

7 (a) P NOT TO SCALE 8.4 cm Q 7.6 cm 62° R In the triangle PQR, QR = 7.6 cm and PR = 8.4 cm. Angle QRP = 62°. Calculate (i) PQ, Answer(a)(i) PQ = ........................................... cm [4] (ii) the area of triangle PQR. Answer(a)(ii) .......................................... cm2 [2] (b) North H NOT TO North SCALE 63 km G J The diagram shows the positions of three small islands G, H and J. The bearing of H from G is 045°. The bearing of J from G is 126°. The bearing of J from H is 164°. The distance HJ is 63 km. Calculate the distance GJ. Answer(b) GJ = .......................................... km [5]

Mark scheme: 7 (a) (i) 8.27 or 8.269... nfww 4 M2 for 6.7 2 + 4.8 2 − 2 × 6.7 × 4.8 × cos ( 62) oe or M1 for implicit form A1 for [PQ2 =] 68.3 to 68.5 (ii) 28.2 or 28.18.. 2 M1 for 0.5 × 7.6 × 8.4 × sin 62 oe (b) 55.8 or 55.78 to 55.79 nfww 5 B1 for [HGJ] = 81 B1 for [GHJ] = 61 63 M2 for [GJ = ] × sin (their 61) sin (their 81) or M1 for implicit form After M0, SC1 for final answer of 68.1…

More questions on Right-angled triangles

8 (a) Jamil, Kiera and Luther collect badges. Jamil has x badges. Kiera has 12 badges more than Jamil. Luther has 3 times as many badges as Kiera. Altogether they have 123 badges. Form an equation and solve it to find the value of x. Answer(a) x = ................................................. [3] (b) Find the integer values of t which satisfy the inequalities. 4t + 7 < 39  7t + 2 Answer(b) ................................................. [3] (c) Solve the following equations. 21 - x = 4 (i) 3 x + Answer(c)(i) x = ................................................. [3] (ii) 3x2 + 7x – 5 = 0 Show all your working and give your answers correct to 2 decimal places. Answer(c)(ii) x = ......................... or x = ......................... [4] __________________________________________________________________________________________

Mark scheme: 8 (a) 5x = 75 or 5x + 48 = 123 B2 M1 for x + (x + 12) + 3(x + 12) = 123 oe 15 B1 (b) 6, 7 3 B2 for answer of 6 or 7 OR M1 for t < 8 37 M1 for t [ 7 OR SC2 for final answer of 5, 6, 7 or 6, 7, 8 or SC1 for final answer of 5, 6, 7, 8 (c) (i) 1.8 oe 3 M1 for 21 – x = 4(x + 3) or better B1 for [±]5x = k or kx = [±]9 2  7  (ii) 72 − 4 × 3 × ( −5) or better nfww B1 or for  x +   6  and − 7 + q − 7 − q 7 5  7  2 or oe B1 or for – ± +   2()3 2 (3) 6 3  6  − 2.91 and 0.57 final ans cao B1B1 SC1 for 0.6 or 0.573… and − 2.9 or – 2.907 or –2.906… or − 0.57 and 2.91 or 0.57 and − 2.91 seen in working

More questions on Equations

9 (a) The points A, B, C and D lie on a circle. AC is a diameter of the circle. ST is the tangent to the circle at A. S A 48° T 27° NOT TO B D y° x° SCALE C Find the value of (i) x, Answer(a)(i) x = ................................................. [2] (ii) y. Answer(a)(ii) y = ................................................. [2] (b) The diagram shows a shaded shape formed by three semi-circular arcs. The radius of each semi-circle is shown in the diagram. NOT TO SCALE 6 cm 4 cm 2 cm (i) Calculate the perimeter of the shaded shape. Answer(b)(i) ........................................... cm [2] (ii) The shaded shape is made from metal 1.6 mm thick. Calculate the volume of metal used to make this shape. Give your answer in cubic millimetres. Answer(b)(ii) ........................................ mm3 [5]

Mark scheme: 9 (a) (i) 42 2 B1 for BAC = 90 − 48 (ii) 111 2 B1 for 111 or 69 or ACD = 27 correctly placed on diagram or indicated (b) (i) 37.7 or 37.69 to 37.704 nfww 2 M1 for 6π + 4π ± 2π oe (ii) 12100, 12060, 12070, 5 SC4 for answer with figs 121 12062.4 to 12065.6 nfww or 1206 to 1207 OR M2 for total area = 12 π 6 2 + 12 π 4 2 − 12 π 2 2 or 12 π 60 2 + 12 π 40 2 − 12 π 20 2 or M1 for 12 π 6 2 or 12 π 4 2 or 12 π 2 2 or 12 π 60 2 or 12 π 40 2 or 12 π 20 2 A1 for area = 75.39 to 75.41 or 7539 to 7541 and M1 dep for volume = their area × thickness

More questions on Angles

10 The diagram is a scale drawing of three straight roads, AB, BC and CD. The scale is 1 : 5000. C D A B Scale 1 : 5000 (a) Find the actual length of the road BC. Give your answer in metres. Answer(a) ............................................. m [2] (b) Another straight road starts at M, the midpoint of AB. This road is perpendicular to AB and it meets the road CD at X. Using a straight edge and compasses only, construct MX. [2] (c) There is a park in the area enclosed by the four roads. The park is • less than 290 m from B and • nearer to CD than to CB. Using a ruler and compasses only, construct the boundaries of the park. Leave in all your construction arcs and label the park P. [5] Question 11 is printed on the next page.

Mark scheme: 10 (a) 475 or 465 to 485 2 B1 for 9.3 to 9.7 [cm] seen (b) Correct perpendicular bisector with 2 B1 for accurate with no/wrong arcs two pairs of intersecting arcs or M1 for correct intersecting arcs (c) Compass drawn arc centre B radius 5.8 2 M1 for compass drawn arc centre B or B1 for 5.8 cm stated or used Accurate angle bisector at C with 2 B1 for accurate with no/wrong arcs correct intersecting arcs or M1 for correct intersecting arcs P 1 cao At

More questions on Geometrical constructions

11 (a) Make x the subject of the formula. xr A - x = t Answer(a) x = ................................................. [4] (b) Find the value of a and the value of b when x2 – 16x + a = (x + b)2. Answer(b) a = ................................................ b = ................................................. [3] (c) Write as a single fraction in its simplest form. 6 5 - x - 4 3x - 2 Answer(c) ................................................. [3]

Mark scheme: At 11 (a) final answer oe nfww 4 B1 for t (A – x) = xr t + r or tA – tx = xr xr or A = + x t M1 for correctly completing multiplication by t (eliminating any bracket) and x terms isolated M1 for correct factorisation M1 dep for correct division (b) [a = ] 64 3 B1 for 2b = –16 or (x – 8)2 [b = ] −8 B1 for a = (their b)2 If 0 scored, SC1 for x2 +2bx + b2 soi 13 x + 8 (c) final answer nfww 3 B1 for 6(3x – 2) – 5(x – 4) or better seen ( x − 4 )(3 x − 2 ) B1 for (x – 4)(3x – 2) oe seen as denom 13 x − 32 or SC2 for final answer ( x − 4 )(3 x − 2 )

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