Cambridge IGCSE Mathematics 0580 — 2015 Oct/Nov Paper 4 · Variant 2

1 A film company uses 512 actors in a film. The actors are in the ratio men : women : children = 7 : 11 : 14. (a) (i) Show that there are 224 children in the film. Answer(a)(i) [2] (ii) Find the number of men in the film. Answer(a)(ii) ............................................... [1] (b) Every working day, each child is given $1 to spend. Each child works for 45 days. Calculate the total amount that the film company gives the children to spend. Give your answer correct to the nearest $100. Answer(b) $ ................................................ [2] (c) The children have lessons every day in groups of no more than 12. Calculate the smallest possible number of groups. Answer(c) ............................................... [2] (d) The film costs four million and ninety three thousand dollars to make. (i) Write this number in figures. Answer(d)(i) ............................................... [1] (ii) Write your answer to part (d)(i) in standard form. Answer(d)(ii) ............................................... [1] (e) A DVD copy of the film costs $2.75 to make. The selling price is $8.20 . Calculate the percentage profit. Answer(e) ............................................% [3]

Mark scheme: Question Answer Mark Part marks 512512 5125121 (a) (i) × 14 M2 or M1 for 7 + 11 + 14 7 + 11 + 14 (ii) 112 1 (b) 10 100 2 M1 for 224 × 45 soi by 10080 (c) 19 2 M1 for 224 ÷ 12 soi by 18.66 to 18.67 or 18.7 or 2 18 3 (d) (i) 4 093 000 1 (ii) 4.093 × 106 1FT FT their (d)(i) 2.8 − .275 (e) 198 or 198.1 to 198.2 3 M2 for × 100 oe .275 2.8 2.8 − .275 or M1 for × 100 or .275 .275

More questions on Ratio and proportion

2 The table shows some values for y = x 3 - 3x + 2 . x –2 –1.5 –1 –0.5 0 0.5 1 1.5 2 y 3.125 3.375 2 0 4 (a) Complete the table of values. [4] (b) On the grid, draw the graph of y = x 3 - 3x + 2 for –2  x  2. y 5 4 3 2 1 x –2 –1 0 1 2 –1 [4] (c) By drawing a suitable line, solve the equation x 3 - 3x + 2 = x + 1 for –2  x  2. Answer(c) x = ....................... or x = ...................... [3] (d) By drawing a suitable tangent, find an estimate of the gradient of the curve at the point where x = -1.5 . Answer(d) ............................................... [3]

Mark scheme: 2 (a) 0 4 0.625 0.875 1,1,1,1 (b) Fully correct smooth curve 4 B3 FT for 8 or 9 points or B2 FT for 6 or 7 points or B1 FT for 4 or 5 points (c) line y = x + 1 ruled 3 Line must be fit for purpose ie at least from x = 0 and to x = 2 0.2 to 0.3 B2 for correct line and 1 correct value and 1.8 to 1.95 or B1 for correct line or SC1 for no/wrong line and 2 correct values (d) Tangent ruled at x = −1.5 B1 No daylight between tangent and curve at point of contact. Consider point of contact as midpoint between two vertices of daylight, the midpoint must be between x = –1.6 and x = –1.4 2 dep on B1 2.2 to 5 rise M1 for also dep on any tangent drawn or run close attempt at tangent at any point Must see correct or implied calculation from a drawn tangent

More questions on Graphs of functions

3 Leo measured the rainfall each day, in millimetres, for 120 days. The cumulative frequency table shows the results. Rainfall (r mm) r  20 r  25 r  35 r  40 r  60 r  70 Cumulative 5 13 72 90 117 120 frequency (a) On the grid below, draw a cumulative frequency diagram to show these results. 120 100 80 Cumulative frequency 60 40 20 r 0 10 20 30 40 50 60 70 Rainfall (mm) [3] (b) (i) Find the median. Answer(b)(i) .........................................mm [1] (ii) Use your diagram to find the number of days when the rainfall was more than 50 mm. Answer(b)(ii) ............................................... [2] (c) Use the information in the cumulative frequency table to complete the frequency table below. Rainfall (r mm) 0  r  20 20  r  25 25  r  35 35  r  40 40  r  60 60  r  70 Frequency 5 59 3 [2] (d) Use your frequency table to calculate an estimate of the mean. You must show all your working. Answer(d) ........................................mm [4] (e) In a histogram drawn to show the information in the table in part (c), the frequency density for the interval 25  r  35 is 5.9 . Calculate the frequency density for the intervals 20  r  25 , 40  r  60 and 60  r  70 . Answer(e) 20  r  25 ............................................... 40  r  60 ............................................... 60  r  70 ............................................... [4]

Mark scheme: 3 (a) Correct diagram 3 B1 for correct vertical plots and B1 for correct horizontal plots and B1 dep on at least B1 for reasonable increasing curve or polygon through their 6 points If zero scored, SC1 for 5 out of 6 correct plots (b) (i) 32 to 34 1 (ii) 120 – reading at r = 50 2FT B1FT for reading at r = 50 seen (c) 8 18 27 2 B1 for 2 correct 1 (d) 4 M1 for mid-values soi 35.2 or 35 or 35.16 to 35.17 6 with x in the correct interval M1 FT for ∑fx nfww including boundaries 120 M1dep for ∑fx ÷ dependent on second M1 earned (e) 1.6 4FT FT from (c) their 8 ÷ 5 and their 27 ÷ 20 1.35 B3FT for any 2 correct 0.3 or B2FT for first or second answer correct or B1 for 0.3 only 0 6

More questions on Averages and measures of spread

4 A NOT TO SCALE B D C The diagram shows a tent ABCD. The front of the tent is an isosceles triangle ABC, with AB = AC. The sides of the tent are congruent triangles ABD and ACD. (a) BC = 1.2 m and angle ABC = 68°. Find AC. Answer(a) AC = ............................................m [3] (b) CD = 2.3 m and AD = 1.9 m. Find angle ADC. Answer(b) Angle ADC = ............................................... [4] (c) The floor of the tent, triangle BCD, is also an isosceles triangle with BD = CD. Calculate the area of the floor of the tent. Answer(c) ...........................................m2 [4] (d) When the tent is on horizontal ground, A is a vertical distance 1.25 m above the ground. Calculate the angle between AD and the ground. Answer(d) ............................................... [3]

Mark scheme: 6.0 4 (a) 1.6[0] or 1.601 to 1.602 3 M2 for oe cos 68 6.0 or M1 for cos 68 = AC 9.1 2 + 3.2 2 − their6.1 2 (b) 43.5 or 43.6 or 43.49 to 43.56 4 M2 for 2 × 9.1 × 3.2 or M1 for implicit statement A1 for [cos = ] 0.724 to 0.726 1 (c) 1.33 or 1.332...nfww 4 M2 for 3.22 − ( × 2.1 )2 2 or M1 for 3.22 = h 2 + (5.0 × 2.1 )2 and M1 for 12 × 2.1 × their .222 (their 2.22 must come from attempt at Pythag or from trig in triangle BCD) .125 (d) oe 41.1 or 41.13 to 41.14 3 M2 for sin = 9.1 or M1 for correct angle identified

More questions on Right-angled triangles

5 (a) The area of shape ABCDEF is 24 cm2. All lengths are in centimetres. 3x – 9 F E 2x NOT TO C SCALE D 3x + 13 A 4x B (i) Show that 5x 2 + 17x - 12 = 0 . Answer(a)(i) [3] (ii) Solve, by factorising, the equation 5x 2 + 17x - 12 = 0 . You must show all your working. Answer(a)(ii) x = .................... or x = .................... [3] (b) Solve the simultaneous equations. You must show all your working. 3x – 2y = 23 –4x – y = –5 Answer(b) x = ............................................... y = ............................................... [3] (c) Solve the equation. 2 ^t + 3h t - = 1 t t + 3 Answer(c) t = ............................................... [5]

Mark scheme: 5 (a) (i) 4 x (3 x + 13 ) − 2 x (4 x − {3 x − 9}) = 24 M1 oe 12 x 2 + 52 x − 2 x 2 − 18 x M1 Correct removal of all their brackets Dep on two areas added or subtracted 5 x 2 + 17 x − 12 = 0 A1 with no errors or omissions seen and at least one more line of working showing collection of like terms or division by 2 (ii) (5 x − 3 )( x + 4 ) [= 0] M2 M1 for (5 x + a )( x + b ) where ab = −12 or 5b + a = 17 [a, b integers] 3 A1 If zero scored SC1 for correct answers with no oe , − 4 5 working or from other methods. (b) For correctly eliminating one M1 variable x = 3 A1 SC1 if no working shown, but 2 correct answers y = − 7 A1 given If zero scored SC1 for 2 values satisfying one of the original equations (c) t = − 2 nfww 5 M1 for 2(t + 3)(t + 3) − t 2 or better seen M1 for denominator[s] t (t + 3 ) isw or for t (t + 3 ) isw on RHS M1dep for 2t 2 + 12 t + 18 − t 2 = t 2 + 3t oe dependent on both numerators and denominator expanding to give quadratics A1 for 9t + 18 = 0 oe

More questions on Area and perimeter

6 (a) (i) A, B, C and D lie on the circumference of the circle. D C 43° t° NOT TO SCALE A B Find the value of t. Answer(a)(i) t = ............................................... [1] (ii) X, Y and Z lie on the circumference of the circle, centre O. Z X NOT TO w° O SCALE 28° Y Find the value of w, giving reasons for your answer. Answer(a)(ii) w = ......................................... because .............................................................. ..................................................................................................................................................... ..................................................................................................................................................... [3] (iii) E, F, G and H lie on the circumference of the circle. H G 5p° NOT TO SCALE p° F E Find the value of p, giving a reason for your answer. Answer(a)(iii) p = ......................................... because ............................................................. ..................................................................................................................................................... [3] (b) R NOT TO O SCALE P N M Q The diagram shows a circle, centre O. PQ and QR are chords. OM is the perpendicular from O to PQ. (i) Complete the statement. PM : PQ = .................... : .................... [1] (ii) ON is the perpendicular from O to QR and PQ = QR. Complete the statements to show that triangle OMQ is congruent to triangle ONQ. ......................... is a common side. ............... = ............... because M is the midpoint of PQ and N is the midpoint of RQ. ............... = ............... because equal chords are equidistant from .................... [4]

Mark scheme: 6 (a) (i) 43 1 (ii) 62 1 Isosceles triangle or OYZ is 1 isosceles Angle at centre is twice angle at 1 circumference (iii) 30 2 M1 for p + 5p = 180 oe [Opposite angles of a]cyclic 1 quadrilateral [add up to 180°] (b) (i) 1 : 2 oe 1 (ii) OQ 1 MQ = NQ 1 OM = ON 1 Centre or O 1 Not origin

More questions on Circle theorems I

7 y 9 8 7 6 B A 5 4 3 2 1 x 0 –7 –6 –5 –4 –3 –2 –1 1 2 3 4 5 6 7 8 9 –1 –2 D –3 C –4 –5 –6 –7 (a) Describe fully the single transformation that maps (i) shape A onto shape B, Answer(a)(i) ............................................................................................................................... ..................................................................................................................................................... [3] (ii) shape A onto shape C, Answer(a)(ii) .............................................................................................................................. ..................................................................................................................................................... [2] (iii) shape A onto shape D. Answer(a)(iii) ............................................................................................................................. ..................................................................................................................................................... [3] (b) Find the 2 × 2 matrix that represents the transformation in part (a)(iii). Answer(b) [2] f p 2 (c) On the grid, draw the image of shape A after a translation by the vector . [2] e- 3 o 0 1 (d) Describe fully the single transformation represented by the matrix . e 1 0 o Answer(d) ........................................................................................................................................... ............................................................................................................................................................. [2]

Mark scheme: 7 (a) (i) Rotation 1 [+]90 or 90 anticlockwise oe 1 ( 0 , 2 ) 1 Not as column vector (ii) Reflection 1 y = 1 oe 1 (iii) Enlargement 1 1 1 [s f ] −2 oe 1 Origin oe (b)  1  2FT FT their s f from (a)(iii)  − 0   2  oe SC1 for  k 0  , k ≠ 1 or 0 1    0 −   0 k   2  (c) Image at (4 , 1) (6, 1) (6 , 5) 2 ruled or good freehand (4 , 3)  2  SC1 for translation    k   k  or   or for 4 correct vertices not joined  −3  (d) Reflection 1 y = x oe 1

More questions on Transformations

8 A line AB joins the points A (3, 4) and B (5, 8). (a) Write down the co-ordinates of the midpoint of the line AB. Answer(a) ( .................. , .................. ) [2] (b) Calculate the distance AB. Answer(b) AB = ............................................... [3] (c) Find the equation of the line AB. Answer(c) ............................................... [3] (d) A line perpendicular to AB passes through the origin and through the point (6, r). Find the value of r. Answer(d) r = ............................................... [3]

Mark scheme: 8 (a) ( 4 , 6 ) 1, 1 (b) 4.47 or 4.472 3 M2 for (8 − 4 )2 + (5 − 3 )2 or better or M1 for (8 − 4 )2 + (5 − 3)2 or better (c) y = 2 x − 2 oe 3 B2 for 2 x − 2 or y = 2 x + c oe 8 − 4 or M1 for [m =] oe soi by 2x 5 − 3 and M1 for (3, 4) or (5, 8) or their midpoint substituted into their y = mx + c with m numerical (d) − 3 3 M1 for use of gradient × their m = − 1 soi by 1 – 2 M1 for r = their gradient × 6 [+0]

More questions on Equations of linear graphs

9 f(x) = 2x + 5 g(x) = 2x h(x) = 7 - 3x (a) Find (i) f(3), Answer(a)(i) ............................................... [1] (ii) gg(3). Answer(a)(ii) ............................................... [2] (b) Find f –1(x). Answer(b) f –1(x) = ............................................... [2] (c) Find fh(x), giving your answer in its simplest form. Answer(c) ............................................... [2] (d) Find the integer values of x which satisfy this inequality. 1  f(x)  9 Answer(d) ............................................... [3] Question 10 is printed on the next page.

Mark scheme: 9 (a) (i) 11 1 (ii) 256 2 M1 for [g(3) =] 8 or 23 or 2 2 x x − 5 (b) 2 M1 for x = 2 y + 5 or 2 x = y − 5 or better oe final answer 2 y 5 or = x + 2 2 (c) 19 − 6 x final answer 2 M1 for 2 (7 −x3 ) + 5 (d) − 1, 0, 1, 2 3 Additional values count as errors B2 for one error /omission or B1 for two errors/omissions or M2 for –2 < x ⩽ 2oe seen or M1 for –2 < x or x ⩽ 2 or x = −2 and x = 2 or − 4 < 2 x ⩽ 4

More questions on Introduction to algebra

10 The table shows the first five terms of sequences A, B and C. Sequence 1st term 2nd term 3rd term 4th term 5th term 6th term A 3 4 5 6 7 B 0 1 4 9 16 C –3 –3 –1 3 9 (a) Complete the table for the 6th term of each sequence. [2] (b) Write down the nth term of sequence A. Answer(b) ............................................... [1] (c) (i) Find the nth term of sequence B. Answer(c)(i) ............................................... [2] (ii) Find the value of n when the nth term of sequence B is 8281. Answer(c)(ii) n = ............................................... [2] (d) (i) Find the nth term of sequence C in its simplest form. Answer(d)(i) ............................................... [2] (ii) Find the 8th term of sequence C. Answer(d)(ii) ............................................... [1] n - 1 1 (e) The nth term of another sequence D is - . e 2 o Complete the table for the first four terms of sequence D. Sequence 1st term 2nd term 3rd term 4th term D [3]

Mark scheme: 10 (a) 8 2 B1 for 2 correct 25 17 (b) n + 2 oe 1 (c) (i) (n − 1)2 oe 2 M1 for (n + k)2 for integer k (ii) 92 2 M1 for 8281 or 91 seen (d) (i) n2 − 3n − 1 final answer 2 M1 for their (n − 1)2 − their (n + 2) soi (ii) 39 1 (e) 1 1 and −2 oe 1 1 oe 1 4 1 oe 1 −8

More questions on Sequences