Cambridge IGCSE Mathematics 0580 — 2018 May/June Paper 4 · Variant 1

1 Adele, Barbara and Collette share $680 in the ratio 9 : 7 : 4. (a) Show that Adele receives $306. [1] (b) Calculate the amount that Barbara and Collette each receives. Barbara $ ............................................... Collette $ ............................................... [3] (c) Adele changes her $306 into euros (€) when the exchange rate is €1 = $1.125 . Calculate the number of euros she receives. € ............................................... [2] (d) Barbara spends a total of $17.56 on 5 kg of apples and 3 kg of bananas. Apples cost $2.69 per kilogram. Calculate the cost per kilogram of bananas. $ ............................................... [3] 1 (e) Collette spends half of her share on clothes and of her share on books. 5 Calculate the amount she has left. $ ............................................... [3]

Mark scheme: Question Answer Marks Partial Marks 1(a) 9 1 × 680 9 + 7 + 4 1(b) 238 136 3 B2 for 238 or 136 7 or M1 for × 680 oe or 9 + 7 + 4 4 × 680 oe seen 9 + 7 + 4 1(c) 272 2 M1 for 306 ÷ 1.125 1(d) 1.37 3 M2 for (17.56 −×5 2.69 ) ÷ 3 or M1 for 17.56 − 5 × 2.69 or B1 for 13.45 [cost of apples] 1(e) 40.8[0] 3 3FT for 0.3 × their 136 from part (b) 1 1 or M2 for their 136( + ) or better 2 5 1 1 or M1 for their 136 × or their 136 × 2 5 3 or B1 for 68 or 27.2 or or 0.3 seen 10

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2 The scale drawing shows two boundaries, AB and BC, of a field ABCD. The scale of the drawing is 1 cm represents 8 m. C B A Scale: 1 cm to 8 m (a) The boundaries CD and AD of the field are each 72 m long. (i) Work out the length of CD and AD on the scale drawing. .......................................... cm [1] (ii) Using a ruler and compasses only, complete accurately the scale drawing of the field. [2] (b) A tree in the field is • equidistant from A and B and • equidistant from AB and BC. On the scale drawing, construct two lines to find the position of the tree. Use a straight edge and compasses only and leave in your construction arcs. [4]

Mark scheme: 2(a)(i) 9 1 2(a)(ii) ABCD completed accurately with arcs 2 M1 for intersecting arcs radius their 9 cm or for ABCD completed accurately with no arcs 2(b) Correct ruled perpendicular bisector of 4 B2 for correct ruled perpendicular AB with 2 correct pairs of arcs bisector of AB with 2 correct pairs of Correct ruled bisector of angle ABC with arcs 2 correct pairs of arcs or B1 for correct perpendicular bisector Lines intersecting without/wrong arcs and B2 for correct ruled bisector of angle ABC with 2 correct pairs of arcs or B1 for correct bisector of angle ABC without/wrong arcs If lines do not intersect, max B3

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3 (a) The price of a house decreased from $82 500 to $77 500. Calculate the percentage decrease. ............................................ % [3] (b) Roland invests $12 000 in an account that pays compound interest at a rate of 2.2% per year. Calculate the value of his investment at the end of 6 years. Give your answer correct to the nearest dollar. $ ............................................... [3]

Mark scheme: 3(a) 6.06 or 6.060 to 6.061 3 82500 − 77500[× 100] oe M2 for 82500 or M1 for 77500[× 100] soi 82500 3(b) 13 674 cao 3 6  2.2  M1 for 12000  1 +   100  A1 for 13673.7...

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4 y 8 7 6 5 B 4 3 A 2 C 1 x –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 –1 –2 –3 –4 D –5 –6 (a) Describe fully the single transformation that maps (i) triangle A onto triangle B, ..................................................................................................................................................... ..................................................................................................................................................... [2] (ii) triangle A onto triangle C, ..................................................................................................................................................... ..................................................................................................................................................... [3] (iii) triangle A onto triangle D. ..................................................................................................................................................... ..................................................................................................................................................... [3] (b) On the grid, draw the image of triangle A after an enlargement by scale factor 2, centre ,7 3 [2] ^ h.

Mark scheme: 4(a)(i) Translation 2 B1 for each  − 8    oe  2  4(a)(ii) Enlargement 3 B1 for each 1 [sf = ] oe 2 ( −4, 0 ) 4(a)(iii) Rotation 3 B1 for each 90° clockwise oe (1, − 1) 4(b) Triangle with (1, –1), (5, –1), (1,7) 2 B1 for correct size and orientation in wrong position or for 3 correct points not joined

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5 (a) Factorise. (i) 2 mn + m 2 - 6 n - 3m ................................................ [2] (ii) 4y 2 - 81 ................................................ [1] (iii) t 2 - t6 + 8 ................................................ [2] (b) Rearrange the formula to make x the subject. 2m - x k = x x = ............................................... [4] (c) Solve the simultaneous equations. You must show all your working. 1 2 x - 3y = 9 5x + y = 28 x = .............................................. y = ............................................... [3] 3 4(d) - = 6 m + 4 m (i) Show that this equation can be written as 6m 2 + 25m + 16 = 0 . [3] (ii) Solve the equation 6m 2 + 25m + 16 = 0 . Show all your working and give your answers correct to 2 decimal places. m = ..................... or m = ..................... [4]

Mark scheme: 5(a)(i) ( 2 n + m )( m − 3 ) final answer 2 M1 for m ( 2 n + m ) − 3 ( 2 n + m ) or 2 n ( m − 3 ) + m ( m − 3 ) 5(a)(ii) ( 2 y − 9 )( 2 y + 9 ) final answer 1 5(a)(iii) ( t − 4 )( t − 2 ) final answer 2 B1 for ( t − 4 )( t − 2 ) seen and spoiled or M1 for t(t – 2) – 4(t – 2) or t(t – 4) – 2(t – 4) or (t + a)(t + b) where a + b = – 6 or ab = +8 5(b) 2 m 4 2 m [ x = ] M1 for xk = 2 m − x or k = − 1 k + 1 x 2 m M1 for xk + x = 2 m or k + 1 = x M1 for x ( k + )1 = 2 m 5(c) correctly eliminating one variable M1 [x = ] 6 A1 [y = ] −2 A1 If 0 scored SC1 for 2 values satisfying one of the original equations or SC1 if no working shown, but 2 correct answers given 5(d)(i) 3m − 4 ( m + 4 ) = 6 m ( m + 4 ) M1 3m − 4( m + or 4)[ = 6] oe m ( m + 4) 3m − 4 m − 16 = 6 m 2 + 24 m M1 removes brackets correctly 6 m 2 + 25 m + 16 = 0 A1 with no errors or omissions 5(d)(ii) 2 2 2 −25 ± ( 25 ) − 4 ( 6 )(16 ) B1 for ( 25 ) − 4 ( 6) (16 ) ) or better 2 × 6 2  25  or or B1 for  m +  2  12  −25  25  16 ±   − p + q p − q 12  12  6 and if in form or r r B1 for p = −25 and r = 2(6) −0.79 and −3.38 2 B1 for each final ans cao SC1 for −0.8 and −3.4 or for − 0.78 and − 3.37 or −0.789... and −3.377... or 0.79 and 3.38 or −0.79 and −3.38 seen in working

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6 A solid hemisphere has volume 230 cm3. (a) Calculate the radius of the hemisphere. 4 3 [The volume, V, of a sphere with radius r is V = r r .] 3 .......................................... cm [3] (b) A solid cylinder with radius 1.6 cm is attached to the hemisphere to make a toy. NOT TO SCALE The total volume of the toy is 300 cm3. (i) Calculate the height of the cylinder. .......................................... cm [3] (ii) A mathematically similar toy has volume 19 200 cm3. Calculate the radius of the cylinder for this toy. .......................................... cm [3]

Mark scheme: 6(a) 4.79 or 4.788 to 4.789 3 230 × 3 M2 for 3 oe 2 × π 2 3 or M1 for 230 = × π ×r oe 3 If 0 scored SC1 for answer 3.8[0…] 6(b)(i) 8.7[0] or 8.702 to 8.704 3 M2 for (300 − 230) ÷ (1.6 2 π) or M1 for π × 1.6 2 × h 6(b)(ii) 6.4 3 19200 M2 for 1.6 × 3 oe 300 19200 300 or M1 for sf 3 or 3 oe 300 19200  1.6 3 300 or for =    r  19200

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7 The graph of y = 10 - 8x 2 for - 1.5 G x G 1.5 is drawn on the grid. y 12 10 8 6 4 2 x – 1.5 – 1 – 0.5 0 0.5 1 1.5 – 2 – 4 – 6 – 8 (a) Write down the equation of the line of symmetry of the graph. ................................................ [1] (b) On the grid opposite, draw the tangent to the curve at the point where x = 0.5 . Find the gradient of this tangent. ................................................ [3] (c) The table shows some values for y = x 3 + 3x + 4 . x - .15 - 1 - .05 0 0.5 1 1.5 y - .39 5.6 8 11.9 (i) Complete the table. [3] (ii) On the grid opposite, draw the graph of y = x 3 + 3x + 4 for - 1.5 G x G 1.5 . [4] (d) Show that the values of x where the two curves intersect are the solutions to the equation x 3 + 8x 2 + 3x - 6 = 0 . [1] (e) By drawing a suitable straight line, solve the equation x 3 + 5x + 2 = 0 for - 1.5 G x G 1.5 . x = ............................................... [3]

Mark scheme: 7(a) x = 0 1 7(b) Tangent ruled at x = 0.5 B1 No daylight between tangent and curve at point of contact −9 to −6.5 2 dep on ruled tangent or close attempt at tangent at x = 0.5 M1 for rise/run also dep on tangent or close attempt at tangent at x = 0.5 7(c)(i) 0 2.4 or better 4 3 B1 for each 7(c)(ii) Correct smooth curve 4 B3FT for 6 or 7 correct plots or B2 FT for 4 or 5 correct plots or B1 FT for 2 or 3 correct plots FT their table 7(d) x 3 + 3 x + 4 = 10 − 8 x 2 and correctly 1 completed 7(e) line y = −2 x + 2 drawn and 3 B2 for ruled y = −2 x + 2 −0.45 to −0.35 nfww or B1 for − 2 x + 2 seen or for line y = –2x + c drawn or for y = cx + 2 (c ≠ 0) drawn and B1 for −0.45 to − 0.35 nfww

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8 (a) The exterior angle of a regular polygon is x° and the interior angle is 8x°. Calculate the number of sides of the polygon. ................................................ [3] (b) C NOT TO SCALE O D B 58° A A, B, C and D are points on the circumference of the circle, centre O. DOB is a straight line and angle DAC = 58°. Find angle CDB. Angle CDB = ............................................... [3] (c) R O NOT TO SCALE 48° P Q P, Q and R are points on the circumference of the circle, centre O. PO is parallel to QR and angle POQ = 48°. (i) Find angle OPR. Angle OPR = ............................................... [2] (ii) The radius of the circle is 5.4 cm. Calculate the length of the major arc PQ. .......................................... cm [3]

Mark scheme: 8(a) 18 3 B2 for 20 nfww or M1 for 8 x + x = 180 or better 8(b) 32 3 B1 for angle DBC = 58 B1 for angle BCD = 90 8(c)(i) 24 2 B1 for angle PRQ = 24 8(c)(ii) 29.4 or 29.40 to 29.41 3 360 − 48 M2 for × 2 × π × 5.4 360 or B2 for answer (minor arc) 4.52 or 4.523 to 4.524… 48 or M1 for × 2 × π × 5.4 360

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59 The probability that it will rain tomorrow is . 8 1 If it rains, the probability that Rafael walks to school is . 6 7 If it does not rain, the probability that Rafael walks to school is . 10 (a) Complete the tree diagram. Walks ........ Rains ........ ........ Does not walk Walks ........ ........ Does not rain ........ Does not walk [3] (b) Calculate the probability that it will rain tomorrow and Rafael walks to school. ................................................ [2] (c) Calculate the probability that Rafael does not walk to school. ................................................ [3]

Mark scheme: 9(a) 5 3 3 B1 for each pair 8 8 1 5 6 6 7 3 10 10 9(b) 5 2 5 1 oe M1FT for their × their 48 8 6 9(c) 304 3 M2 for oe 5 5 3 3 480 their × their + their × their oe 8 6 8 10 or M1 for 5 5 3 3 their × their or their × their 8 6 8 10

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10 (a) In 2017, the membership fee for a sports club was $79.50 . This was an increase of 6% on the fee in 2016. Calculate the fee in 2016. $ ............................................... [3] (b) On one day, the number of members using the exercise machines was 40, correct to the nearest 10. Each member used a machine for 30 minutes, correct to the nearest 5 minutes. Calculate the lower bound for the number of minutes the exercise machines were used on this day. ......................................... min [2] (c) On another day, the number of members using the exercise machines (E), the swimming pool (S) and the tennis courts (T) is shown on the Venn diagram. E S 33 20 5 4 7 8 16 T (i) Find the number of members using only the tennis courts. ................................................ [1] (ii) Find the number of members using the swimming pool. ................................................ [1] (iii) A member using the swimming pool is chosen at random. Find the probability that this member also uses the tennis courts and the exercise machines. ................................................ [2] (iv) Find n T + E , S ^ ^ hh. ................................................ [1]

Mark scheme: 10(a) 75 3 M2 for 79.5 ÷ 1.06 oe or M1 for 79.5 associated with 106 [%] 10(b) 962.5 cao 2 B1 for 35 or 27.5 seen 10(c)(i) 16 1 10(c)(ii) 50 1 10(c)(iii) 4 2 FT their (c)(ii) for 1 or 2 marks oe 4 k 50 B1 for , k > 4 or , k < 50 k their 50 10(c)(iv) 19 1

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4 8 - 3 11 (a) OA = AB = AC = c3m c- 7m c 6m Find (i) OB , OB = ............................................... [3] (ii) BC. BC = [2] f p (b) S R NOT TO SCALE b X P Q a PQRS is a parallelogram with diagonals PR and SQ intersecting at X. PQ = a and PS = b . Find QX in terms of a and b. Give your answer in its simplest form. QX = ............................................... [2] 2 5 (c) M = c 1 8 m Calculate (i) M2, M2 = [2] f p (ii) M -1 . M -1 = [2] f p

Mark scheme: 11(a)(i) 12.6 or 12.64 to 12.65 3 2 2 M2 for 12 + ( − 4 ) OR  12  B1 for    − 4  M1 for (their12)2 + ( their − 4 ) 2 11(a)(ii)  −11  2  −11   k    B1 for   or   or for  13   k   13  JJJG  − 8  [ BA = ]    7  11(b) 1 2 M1 for correct route or correct (b − a) oe unsimplified answer 2 JJJG or B1 for QS = b – a oe 11(c)(i)  9 50  2 B1 for 2 correct elements    10 69  11(c)(ii) 1  8 − 5  2  8 −5  1  a b    oe isw B1 for k   or   11  − 1 2   − 1 2  11  c d  or det = 11 soi

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12 Marco is making patterns with grey and white circular mats. Pattern 1 Pattern 2 Pattern 3 Pattern 4 The patterns form a sequence. Marco makes a table to show some information about the patterns. Pattern number 1 2 3 4 5 Number of grey mats 6 9 12 15 Total number of mats 6 10 15 21 (a) Complete the table for Pattern 5. [2] (b) Find an expression, in terms of n, for the number of grey mats in Pattern n. ................................................ [2] (c) Marco makes a pattern with 24 grey mats. Find the total number of mats in this pattern. ................................................ [2] (d) Marco needs a total of 6 mats to make the first pattern. He needs a total of 16 mats to make the first two patterns. 1 3 2 He needs a total of n + an + bn mats to make the first n patterns. 6 Find the value of a and the value of b. a = ............................................... b = ............................................... [6]

Mark scheme: 12(a) 18 2 B1 for each 28 12(b) 3n + 3 oe 2 B1 for 3n + k oe or cn + 3 oe c ≠ 0 12(c) 45 2 M1 for identifying 7th pattern or M1 for their ( 3n + 3 ) = 24 12(d) 3 13 6 M1 for any correct substitution [ a = ] oe [b = ] oe 1 2 3 e.g. (2)³ + 2²a + 2b 6 A1 for one of e.g. 1 + a + b = 6 oe 6 8 + 4a + 2b = 16 oe 6 27 + 9a + 3b = 31 oe 6 64 + 16 a + 4b = 52 oe 6 A1 for another of the above M1 for correctly eliminating one variable from their equations 3 A1 for a = 2 13 A1 for b = oe 3

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