Cambridge IGCSE Mathematics 0580 — 2012 Oct/Nov Paper 4 · Variant 1

4 (a) Solve the equations. For Examiner's Use (i) 4x – 7 = 8 – 2x Answer(a)(i) x = [2] x − 7 (ii) = 2 3 Answer(a)(ii) x = [2] (b) Simplify the expressions. (i) (3xy 4)3 Answer(b)(i) [2] 1 (ii) (16a6b2) 2 Answer(b)(ii) [2] x 2 − 7 x − 8 (iii) x 2 − 64 Answer(b)(iii) [4]

Mark scheme: 4 5 (a) (i) 2.5 or 2 M1 for one correct step collected 2 i.e 6x = k or ax = 15 or for 4x + 2x = 8 + 7 (ii) 13 2 M1 for x – 7 = 2 × 3 or better (b) (i) 27x3y12 final answer 2 B1 for 2 correct elements (ii) 4a3b[1] final answer 2 B1 for 2 correct elements x + 1 4 M2 for (x – 8)(x + 1) seen (iii) www final answer x + 8 or SC1 for (x + a)(x + b) where a + b = –7 or ab = –8 and B1 for (x + 8)(x – 8) seen [ ]

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5 (a) For NOT TO Examiner's Use SCALE 20 cm 24 cm 46 cm Jose has a fish tank in the shape of a cuboid measuring 46 cm by 24 cm by 20 cm. Calculate the length of the diagonal shown in the diagram. Answer(a) cm [3] (b) Maria has a fish tank with a volume of 20 000 cm3. Write the volume of Maria’s fish tank as a percentage of the volume of Jose’s fish tank. Answer(b) % [3] (c) Lorenzo’s fish tank is mathematically similar to Jose’s and double the volume. Calculate the dimensions of Lorenzo’s fish tank. Answer(c) cm by cm by cm [3] (d) A sphere has a volume of 20 000 cm3. Calculate its radius. 4 [The volume, V, of a sphere with radius r is V = πr3.] 3 Answer(d) cm [3]

Mark scheme: 5 (a) 55.6 to 55.61 www 3 M2 for 46 2 + 24 2 + 20 2 oe [ 3092 ] or M1 for 46² + 24² oe [soi by 2692 or art 51.9] or 46² + 20² oe [soi by 2516 or art 50.2] or 24² + 20² oe [soi by 976 or art 31.2] (b) 90.6 or 90.57 to 90.58 3 20000 M2 for × 100 oe (20 × 24 × 46 ) or M1 for 20 × 24 × 46 [22080] (c) 25.19 to 25.21, 30.23 to 30.246 or 3 M2 for 20 × 3 2 or 24 × 3 2 or 46 × 3 2 30.2, 57.95 to 57.97 or 58[.0] 3 2 oe seen [1.259 to 1.261] M1 for 3 20000 (d) 16.8 to 16.842 3 M2 for oe or answer figs 168 to 4 3π 16842 3 20000 or M1 for [4770 – 4780] seen 4 3π IGCSE – October/November 2012 0580 41

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− 2   2  − 10  For 6 (a) a =   b =   c =   Examiner's  3   −7   21 Use (i) Find 2a + b.     Answer(a)(i)   [1]     (ii) Find ö=b ö. Answer(a)(ii) [2] (iii) ma + nb = c Find the values of m and n. Show all your working. Answer(a)(iii) m = n = [6] (b) For P Examiner's X Use NOT TO SCALE O Y Q In the diagram, OX : XP = 3 : 2 and OY : YQ = 3 : 2 . = p and = q. (i) Write in terms of p and q. Answer(b)(i) = [1] (ii) Write in terms of p and q. Answer(b)(ii) = [1] (iii) Complete the following sentences. The lines XY and PQ are The triangles OXY and OPQ are The ratio of the area of triangle OXY to the area of triangle OPQ is : [3]

Mark scheme: 6  − 2  1 (a) (i)    − 1  M1 for 2 2 + ( − 7 2 ) oe (ii) 7.28 [0] or 2 ± 53 as final answer (iii) [m = ] 3.5 oe and 6 B1 for –2m + 2n = –10 oe [n = ] –1.5 oe and B1 for 3m – 7n = 21 oe and M1 for correct attempt to equate one set of coefficients and M1dep for elimination allow 1 arithmetic error overall ft their sim eqns for both m’s or M1 for correct rearrangement (allow 1 slip) and M1dep for correct substitution ft their sim eqns for both m’s and A1 for 3.5 or –1.5 (b) (i) –p + q 1 Condone column vector used 3 3 3 (ii) – p + q oe 1 FT Correct or ft 5 (their (b)(i)) dep on ap + bq, 5 5 [a ≠ 0, b ≠ 0] Condone column vector used (iii) Parallel 1 similar 1 Accept enlargement 9 : 25 oe 1 e.g 1 : 2.77 [7] or 0.36 : 1

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7 For W A X Examiner's Use NOT TO SCALE E B O 7 cm Z D C Y The vertices A, B, C, D and E of a regular pentagon lie on the circumference of a circle, centre O, radius 7 cm. They also lie on the sides of a rectangle WXYZ. (a) Show that (i) angle DOC = 72°, Answer(a)(i) [1] (ii) angle DCB = 108°, Answer(a)(ii) [2] (iii) angle CBY = 18°. Answer(a)(iii) [1] (b) Show that the length CD of one side of the pentagon is 8.23 cm correct to three significant For figures. Examiner's Use Answer(b) [3] (c) Calculate (i) the area of the triangle DOC, Answer(c)(i) cm2 [2] (ii) the area of the pentagon ABCDE, Answer(c)(ii) cm2 [1] (iii) the area of the sector ODC, Answer(c)(iii) cm2 [2] (iv) the length XY. Answer(c)(iv) cm [2] (d) Calculate the ratio area of the pentagon ABCDE : area of the rectangle WXYZ. Give your answer in the form 1 : n. Answer(d) 1 : [5]

Mark scheme: 7 (a) (i) 360 Q 5 1 Accept longer correct methods (ii) (180 – 72) 2 M1 Accept [(5 – 2) × 180] or 360 / 5 M1 54 × 2 E1 Then Q 5 180 – 72 E1 (iii) 180 – 90 – 72 1 Accept other methods provided they are fully explained (b) 2 × 7 × sin(72/2) oe M2 M1 for 7 × sin (72/2) oe Alt methods M2 for [DC² =] 7² + 7² – 2.7.7 cos 72 or M1 for implicit version or M2 for (7 sin 72)/sin 54 or M1 for DC/sin 72 = 7/sin 54 oe 8.228 to 8.229 E1 Dep on M2 and with no errors seen IGCSE – October/November 2012 0580 41 (c) (i) 23.3[0..] 2 M1 for ½ × 7 × 7 × sin 72 oe (ii) 116.5 to 116.52 or 117 1 ft ft their (c)(i) × 5 (iii) 30.78 to 30.8 2 M1 for 72/360 × π 7 2 (iv) 12.66 to 12.67 or 12.7 2 M1 for 7 + 7 cos 36 oe [7 + 5.66…] e.g 8.23 cos54 + 8.23 sin72 oe [4.84 + 7.83] (d) 1.43 or 1.432 to 1.453 cao 5 B4 for area of rectangle = 168.3 to 169.2 www or area of triangular corners = 51.6 to 52.5 www or B3 for 13.3 to 13.32 seen or M2 for [ZY =] 8.23 + 2(8.23sin18) oe or 2 (8.23 sin 54) or 2 × 7 sin 72 oe or B1 for [CY =] 2.54[3] or 5.08 to 5.09 seen or [AX =] 6.65 to 6.66 seen

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8 A rectangular piece of card has a square of side 2 cm removed from each corner. For Examiner's Use 2 cm 2 cm NOT TO SCALE (2x + 3) cm (x + 5) cm (a) Write expressions, in terms of x, for the dimensions of the rectangular card before the squares are removed from the corners. Answer(a) cm by cm [2] (b) The diagram shows a net for an open box. Show that the volume, V cm3, of the open box is given by the formula V = 4x2 + 26x + 30 . Answer(b) [3] (c) (i) Calculate the values of x when V = 75. For Show all your working and give your answers correct to two decimal places. Examiner's Use Answer(c)(i) x = or x = [5] (ii) Write down the length of the longest edge of the box. Answer(c)(ii) cm [1] Question 9 is printed on the next page.

Mark scheme: 8 (a) 2x + 7 final answer 2 B1 for each, accept in either order x + 9 final answer After 0 scored allow SC1 mark for both correct but unsimplified (b) 2(2x + 3)(x + 5) at any stage M1 The × 2 could be embedded within one of the brackets e.g. (4x + 6)(x + 5) 2x2 + 3x + 10x + 15 or better B1 Expands brackets correctly 4x2 + 26x + 30 E1 No errors seen and two previous stages shown (c) (i) 4x² + 26x – 45 [= 0] soi B1 − 26 ± ( 26 ) 2 − 4( 4 )( −45) B1 ft ft their 4x² + 26x ± k [k ≠ 0] oe 2 ( 4 ) B1 ft In square root B1 ft for ( 26) 2 − 4( 4)( −45) or better (1396) p+ q p− q If in form or ; r r B1 ft for –26 and 2(4) or better –7.92, 1.42 final answers B1 B1 If B0, SC1 for –7.9 and 1.4 or both answers – 7.920…., 1.420….. or for–7.92 , 1.42 seen (ii) 6.42 [0…] 1 ft ft their greatest positive root If their x ≤ 2 then ft x + 5 If their x > 2 then ft 2x + 3 IGCSE – October/November 2012 0580 41 7

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9 Distances from the Sun can be measured in astronomical units, AU. For Earth is a distance of 1 AU from the Sun. Examiner's Use One AU is approximately 1.496 × 108 km. The table shows distances from the Sun. Name Distance from the Sun in AU Distance from the Sun in kilometres Earth 1 1.496 × 108 Mercury 0.387 Jupiter 7.79 × 108 Pluto 5.91 × 109 (a) Complete the table. [3] (b) Light travels at approximately 300 000 kilometres per second. (i) How long does it take light to travel from the Sun to Earth? Give your answer in seconds. Answer(b)(i) s [2] (ii) How long does it take light to travel from the Sun to Pluto? Give your answer in minutes. Answer(b)(ii) min [2] (c) One light year is the distance that light travels in one year (365 days). How far is one light year in kilometres? Give your answer in standard form. Answer(c) km [3] (d) How many astronomical units (AU) are equal to one light year? Answer(d) AU [2]

Mark scheme: 9 (a) 5.79 × 107 oe 1 Accept ans in range 57890000 to 57900000 5.21 1 5.207 39.5 1 39.50…. or 39.51 Accept answers to greater than 3sf (b) (i) 498.6... to 499 2 M1 for 1.496 × 10 8 ÷ 300 000 (ii) 328 or 328.3... 2 M1 for figs 197 or figs 328[3.. ] seen Or their 39.5 × their (b)(i) (c) 9.46[0] to 9.461 × 1012 3 B2 for any correct equivalent or M1 for 300 000 × 3600 × 24 × 365 oe or for answer figs 946 to 9461 (d) 63200 or 63235 to 63242 oe 2 M1 for figs (their (c) ÷ 1496). Implied by first 3 figs correct

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