Cambridge IGCSE Mathematics 0580 — 2015 Feb/March Paper 4 · Variant 2

5 (a) X NOT TO 5.4 cm SCALE 62° Y 16 cm Z Show that the area of triangle XYZ is 38.1 cm2, correct to 1 decimal place. Answer(a) [2] (b) NOT TO 48° SCALE 6.7 cm x° 8.4 cm Calculate the value of x. Answer(b) x = ................................................ [4] (c) North A NOT TO SCALE P B Ship A is 180 kilometres from port P on a bearing of 063°. Ship B is 245 kilometres from P on a bearing of 146°. Calculate AB, the distance between the two ships. Answer(c) .......................................... km [5] __________________________________________________________________________________________

Mark scheme: 5 (a) 1 × 16 × 4.5 × sin 62 oe M1 2 A1 38.14... 7.6 × sin 48 (b) 95.6 or 95.64 to 95.65 4 M2 for 4.8 or M1 for implicit form and M1dep for 180 − 48 − their 36.4 (c) 286 or 285.7 to 285.8 5 B1 for [Angle APB=] 83° M2 for 180 2 + 245 2 − 2 × 180 × 245 × cos their 83 or M1 for implicit form and A1 for [AB2 =] 81676[.1...] After 0 scored, SC2 for ans 406.87 to 406.88 or 406.9 or 407 if 146° used in cos rule Or SC1 for 180 2 + 245 2 − 2 × 180 × 245 × cos 146 4

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6 In this question write any probability as a fraction. Navpreet has 15 cards with a shape drawn on each card. 5 cards have a square, 6 cards have a triangle and 4 cards have a circle drawn on them. (a) Navpreet selects a card at random. Write down the probability that the card has a circle drawn on it. Answer(a) ................................................ [1] (b) Navpreet selects a card at random and replaces it. She does this 300 times. Calculate the number of times she expects to select a card with a circle drawn on it. Answer(b) ................................................ [1] (c) Navpreet selects a card at random, replaces it and then selects another card. Calculate the probability that (i) one card has a square drawn on it and the other has a circle drawn on it, Answer(c)(i) ................................................ [3] (ii) neither card has a circle drawn on it. Answer(c)(ii) ................................................ [3] (d) Navpreet selects two cards at random, without replacement. Calculate the probability that (i) only one card has a triangle drawn on it, Answer(d)(i) ................................................ [3] (ii) the two cards have different shapes drawn on them. Answer(d)(ii) ................................................ [4] __________________________________________________________________________________________

Mark scheme: 46 (a) 1 15 (b) 80 1FT FT 300 × their (a) 40  8  5 4 (c) (i) oe 3 M2 for × × 2 oe 225  45  15 15 5 4 or M1 for × 15 15 121 11 (ii) 3 M2 for 11× oe 225 15 15 11 4 or M1 for or 1− seen 15 15 108  18  6 9 9 6 (d) (i) oe 3 M2 for × + × oe 210  35  15 14 15 14 6 9 9 6 or M1 for × oe or × oe 15 14 15 14 6 5 6 4 or × oe or × oe 15 14 15 14 148  74  5 10 6 9 4 11 (ii) oe 4 M3 for × + × + × oe 210  105  15 14 15 14 15 4 5 4 6 5 4 3 or 1 − × − × − × 15 14 15 14 15 14 or M2 for equivalent of 2 of above products added together oe or M1 for one correct relevant product oe

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7 y 7 6 5 A 4 3 2 1 x –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 –1 –2 –3 –4 B –5 –6 C –7 –8 (a) Describe fully the single transformation that maps (i) flag A onto flag B, Answer(a)(i) ................................................................................................................................ ..................................................................................................................................................... [3] (ii) flag A onto flag C. Answer(a)(ii) ............................................................................................................................... ..................................................................................................................................................... [3] 2 (b) Draw the image of flag A after a translation by the vector [2] e 1 o. (c) Draw the image of flag A after a reflection in the line x = 1. [2] 1 0 (d) Describe fully the single transformation represented by the matrix e 0 - 1 o. Answer(d) ........................................................................................................................................... ............................................................................................................................................................. [2] __________________________________________________________________________________________

Mark scheme: 7 (a) (i) Rotation 1 [centre] (0, 0) or origin 1 90° [anticlockwise] oe 1 (ii) Enlargement 1 1 [centre] (−2, 1) 1 [s.f.] − 2  2   k  (b) vertices at (−3, 4) (−3, 5) (−3, 6) 2 SC1 for translation by   or    k   1  (−2, 6) (c) vertices at (7, 3) (7, 4) (7, 5) (6, 5) 2 SC1 for reflection in y = 1 or reflection in any vertical line (d) reflection 1 x-axis oe 1

More questions on Transformations

8 (a) The diagram shows a sector of a circle A with centre O and radius 24 cm. NOT TO SCALE x° O 24 cm (i) The total perimeter of the sector is 68 cm. B Calculate the value of x. Answer(a)(i) x = ................................................ [3] (ii) The points A and B of the sector are joined together to O make a hollow cone. The arc AB becomes the circumference of the base of the cone. NOT TO SCALE AB Calculate the volume of the cone. 1 [The volume, V, of a cone with radius r and height h is V = πr2h.] 3 Answer(a)(ii) ......................................... cm3 [6] (b) Q NOT TO SCALE M P O X Y 8 cm The diagram shows a shape made from a square, a quarter circle and a semi-circle. OPXY is a square of side 8 cm. OPQ is a quarter circle, centre O. The line OMQ is the diameter of the semi-circle. Calculate the area of the shape. Answer(b) ......................................... cm2 [5] __________________________________________________________________________________________

Mark scheme: 8 (a) (i) 47.7 or 47.74 to 47.75 3 M1 for [arc =] 68 − 2 × 24 x or 24 + 24 + × 2 π × 24 = 68 360 M1 for [x =] their arc × 360 ÷ (2 × π × 24) 20 (ii) 252 or 252.3 to 252.4.... 6 M1 for r = or 2 π  their 47 7.   × 2 × π × 24  ÷ (2 π )  360  10 A1 for r = 3.18 or 3.182 to 3.183... or π M1 for h 2 = 24 2 − their r 2 A1 for h = 23.8 or 23.78... to 23.79 M1dep on M1 earned for 1 2 V = π × their h × their r 3 2 2 1 2 1  8  (b) 139 or 139.3 to 139.4... nfww 5 M4 for 8 + π × 8 + π ×   4 2  2  1 2 or M1 for π × 8 4 2 1  8  and M1 for π ×   2  2  and M1 for 82 added to at least one term with π

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9 The table shows the height, h cm, of 40 children in a class. Height (h cm) 120 < h  130 130 < h  140 140 < h  144 144 < h  150 150 < h  170 Frequency 3 14 4 6 13 (a) Write down the class interval containing the median. Answer(a) ................................. < h  ................................. [1] (b) Calculate an estimate of the mean height. Answer(b) .......................................... cm [4] (c) Complete the histogram. 2 1.5 Frequency density 1 0.5 0 h 120 130 140 150 160 170 Height (cm) [4] __________________________________________________________________________________________

Mark scheme: 9 (a) 140 < h ≤ 144 1 (b) 144.875 nfww 4 M1 for at least 4 correct mid-values soi M1 for ∑fx where x is in the correct interval, allow one further error/omission M1 dep for ÷ 40 dependent on second method mark (c) 4 correct blocks 4 B3 for 3 correct blocks B2 for 2 correct blocks B1 for 1 correct block or at least 3 correct frequency densities (1.4, 1, 1, 0.65 )

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10 The school cook buys potatoes in small sacks, each of mass 4 kg, and large sacks, each of mass 10 kg. He buys x small sacks and y large sacks. Today, he buys less than 80 kg of potatoes. (a) Show that 2x + 5y < 40. Answer(a) [1] (b) He buys more large sacks than small sacks. He buys no more than 6 large sacks. Write down two inequalities to show this information. Answer(b) ................................................ ................................................ [2] (c) On the grid, show the information in part (a) and part (b) by drawing three straight lines and shading the unwanted regions. y 9 8 7 6 5 4 3 2 1 x 0 5 10 15 20 25 [5] (d) Find the greatest mass of potatoes the cook can buy today. Answer(d) ........................................... kg [2] __________________________________________________________________________________________ Question 11 is printed on the next page.

Mark scheme: 10 (a) 4 x + 10 y < 80 1 With no errors seen (b) y > x 1 y ≤ 6 or y < 7 1 Accept 0 ≤ y ≤ 6 or 0 < y ≤ 6 or 0 ≤ y < 7 or 0 < y < 7 (c) ruled broken line through (5, 6) to B2 SC1 for correct only at (5, 6) or (10, 4) (10,4) ruled broken line y = x B1 ruled solid line y = 6 or broken y = 7 B1 Must be consistent with their (b) correct region indicated B1 (d) 76 2 SC1 for ( 4, 6 ) indicated or 4 x + 10 y evaluated for ( x, y ) in their region, x, y integers

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11 Diagram 1 Diagram 2 Diagram 3 Diagram 4 Diagram 5 Diagram 1 shows two lines of length 1 unit at right angles forming an . Two s are added to Diagram 1 to make Diagram 2. This forms one small square. Three s are added to Diagram 2 to make Diagram 3. This forms three small squares. The sequence of Diagrams continues. (a) Draw Diagram 5. [1] (b) Complete the table. Diagram 1 Diagram 2 Diagram 3 Diagram 4 Diagram 5 Number of lines of length 1 unit 2 6 12 20 Number of small squares 0 1 3 6 [2] (c) Find an expression, in terms of n, for the number of lines of length 1 unit in Diagram n. Answer(c) ................................................ [2] (d) Find an expression, in terms of n, for the number of small squares in Diagram n. Answer(d) ................................................ [2]

Mark scheme: 11 (a) 1 (b) 30 1 10 1 2 + bn + c a, b, c numeric a ≠ 0 (c) n (n + )1 oe 2 B1 for an 1 (d) n (n − )1 oe 2 1 2 B1 for using oe in expression of form 2 1 (an 2 + bn + c ) a ≠ 0 or kn(n − )1 k ≠ 0 2

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