Circles, arcs and sectors — Paper 3 · IGCSE Mathematics 0580

5 A bag contains 24 discs. For 10 discs are red, 9 discs are green and 5 discs are yellow. Examiner's Use (a) The number of discs of each colour can be shown by three sectors on a pie chart. The sector angle for the red discs is 150°. Work out the sector angle for (i) the green discs, Answer(a)(i) [1] (ii) the yellow discs. Answer(a)(ii) [1] (iii) Complete the pie chart below and label the sectors. [2] (b) A disc is chosen at random. For Examiner's Find, as a fraction, the probability of each of the following events. Use (i) Event A: the disc is red. Answer(b)(i) [1] (ii) Event B: the disc is red or yellow. Answer(b)(ii) [1] (iii) Event C: the disc is not yellow. Answer(b)(iii) [1] (c) Probability Scale Impossible Certain (c)(i) … (c)(ii) … The diagram shows a horizontal probability scale. Write on the dotted lines in the diagram, the probability of (i) an impossible event, [1] (ii) a certain event. [1] (d) Using the notation, A, B and C, mark the positions of your three answers in part (b) on the Probability Scale diagram in part (c). [3]

12 marks

Mark scheme: 5 (a) (i) 135 (green) B1 (ii) 75 (yellow) B1 (iii) Ruled lines correct to 2° B1ft Only if (a)(i) + (a)(ii) = 210°. 3 correctly labelled sectors B1 Independent of previous marks 10 (b) (i) oe B1 Accept decimals, percentages 24 15 (ii) oe B1 24 19 (iii) oe B1 24 0 12 0 24 (c) (i) 0 B1 SC1 for and or and 12 12 24 24 (ii) 1 B1 (d) Labelled arrows correctly B3ft 1 mark for each. positioned by eye ft their probabilities from (b). [12]

This question in 0580/31 May/June 2007

11 (a) Calculate the area of a circle of radius 6 cm. For Examiner′s Use Answer(a) … cm2 [2] (b) 6 cm NOT TO SCALE Each circle in this rectangle has a radius of 6 cm. The circles fi t exactly in the rectangle. Calculate the shaded area. Answer(b) … cm2 [4]

6 marks

Mark scheme: 11 (a) 113 or 113.09 to 113.112 2 M1 for π × 62 or better (b) 185 or 186 or 185.76 4 or 185.328 to 185.42 M1 for their (a) × 6 M1 for 24 × 36 soi, imp by 864 M1 for their (24 × 36) – their (their (a) × 6) ft their (a) for M3

This question in 0580/31 May/June 2013

9 A family of 2 adults and 3 children are on holiday. For Examiner′s They each hire a mountain bike from the hotel. Use Large mountain bike Small mountain bike First hour Each extra hour First hour Each extra hour $6 $2 $3.60 $1.20 (a) The family hire 2 large and 3 small mountain bikes for 5 hours. (i) Work out the total cost. Answer(a)(i) $ … [3] (ii) The hotel gives the family a discount of 15% on the total cost. Work out how much the family pays. Answer(a)(ii) $ … [2] (b) A wheel of a large bike has a radius of 32 cm. (i) Calculate the circumference of a wheel of a large bike. Answer(b)(i) … cm [2] (ii) The family cross a bridge which is 24 m long. For Examiner′s Use Calculate how many complete turns a wheel of a large bike makes to cross the bridge. Answer(b)(ii) … [2] (c) The diagram shows part of a wheel of a large bike. There is an angle of 9° between two metal spokes. Each spoke is 29 cm long. 9° NOT TO SCALE Calculate the total length of metal, in metres, needed to make the spokes for one wheel. Answer(c) … m [3] _____________________________________________________________________________________ Question 10 is printed on the next page.

12 marks

Mark scheme: 9 (a) (i) 53.2[0] 3 SC2 for 60.80 M2 for 2 × (6 + 4 × 2) + 3 × (3.60 + 4 × 1.20) or better or for 2 × 6 + 3 × 3.60 + 4(2 × 2 + 3 × 1.20) or better if M0 then B1 for 28 or 25.20 or 22.80 or 22.40 or 30.40 or 12 and 10.80 or 16 and 14.40 or 14 and 8.40 seen (ii) 45.22 2ft M1ft for ‘their ai’ × 0.85 oe (b) (i) 201 or 201.06 to 201.1 or 2.01m 2 M1 for 2 × π × 32 oe 2400 (ii) 11 final answer 2 M1ft for both in cm their bi 24 or both in m their bi or SC1 for figs ‘119……’ 360 (c) 11.6 3 M1 for × 29 or better, implied by 9 1160 and M1 indep for ‘their 1160’ / 100 soi or 0.29 seen

This question in 0580/32 May/June 2013

1 For 9 (a) The formula for the volume, V, of a cone with radius r, and height h, is V = 3 πr2h . Examiner′s Use (i) To make r the subject of this formula, the fi rst step is 3V = πr2h. Show the remaining steps to make r the subject of this formula. Answer(a)(i) r = … [2] (ii) An ice-cream cone has a volume of 141 cm3 and height 15 cm. Show that the radius of the cone is 3 cm, correct to the nearest whole number. Answer(a)(ii) [2] (b) The open end of an ice-cream cone is a circle of radius 3 cm. Calculate the circumference of this circle. Answer(b) … cm [2] (c) The volume of a ball of ice-cream is 113 cm3. The ball of ice-cream costs $2.15 . Calculate the cost of 1 cm3 of the ice-cream. Give your answer in cents, correct to 1 decimal place. Answer(c) … cents [3]

9 marks

Mark scheme: 3V 3V 3V 9 (a) (i) [r =] 2 B1 for [r2 =] or seen or better πh π h 3 x141 (ii) [r =] M1FT their formula πx15 [r =] 2.99… A1 (b) 18.9 or 18.8 or 18.849 to 18.852 2 M1 for 2 × π × 3 oe (c) 1.9 [cents] cao 3 M1 for 2,15 (or 215) ÷ 113 A1 for 0.019 (0…) or 1.9 (0…) soi

This question in 0580/33 Oct/Nov 2013

3 A NOT TO SCALE O C B D E The diagram shows a circle, centre O and diameter AD. B is on the circumference of the circle and the line CDE touches the circle at D. AD = 21 cm and CD = 16 cm. (a) Calculate (i) the circumference of the circle, Answer(a)(i) … cm [2] (ii) the area of the circle. Answer(a)(ii) … cm2 [2] (b) (i) Write down the size of angle ABD. Answer(b)(i) Angle ABD = … [1] (ii) BD = 9 cm. Show that AB = 19.0 cm, correct to 3 significant figures. Answer(b)(ii) [3] (c) (i) Calculate the area of triangle ABD. Answer(c)(i) … cm2 [2] (ii) Work out the total area of the shaded segments of the circle. Answer(c)(ii) … cm2 [2] (d) (i) Write down the mathematical name of the line CDE. Answer(d)(i) … [1] (ii) Write down the mathematical name of the line OD. Answer(d)(ii) … [1] (iii) Use trigonometry to calculate the size of angle OCD. Answer(d)(iii) Angle OCD = … [2] __________________________________________________________________________________________

16 marks

Mark scheme: 3 (a) (i) 66.0 or 65.97 to 65.98 … 2 M1 for π × 21 (ii) 346 or 346.3 to 346.4 … 2 M1 for π × (21 ÷ 2)2 (b) (i) 90 1 (ii) ( 212 − 9 2 ) M2 M1 for 212 = AB2 + 92 or [AB2] = 212 – 92 18.97(……) A1 (c) (i) 85.5 2 M1 for 0.5 × 19 × 9 (ii) 87.5 or 87.65 to 87.823 2FT M1FT for 0.5 × their (a)(ii) (d) (i) Tangent 1 (ii) Radius 1 ( 21 ÷ 2 ) (iii) 33.3 or 33.27(……) 2 M1 for tan[ ] = or better 16

This question in 0580/32 May/June 2015

6 Swimming pool Shop Distance from home (km) Cinema Home 0 10 00 10 10 10 20 10 30 10 40 10 50 11 00 Time Abjit cycles from his home to the swimming pool. The travel graph for his journey is drawn on the grid. On his journey he passes the cinema and the shop. (a) Write down where Abjit stops on his journey to the swimming pool. … [1] (b) Abjit is cycling fastest between the shop and the swimming pool. Explain how you know this from looking at the graph. … [1] (c) Abjit cycles at 20 km/h from his home to the cinema. This part of the journey takes 12 minutes. (i) Show that the distance from Abjit’s home to the cinema is 4 km. [2] (ii) Complete the scale on the vertical axis of the grid by showing at least two other values. [1] (d) Calculate the speed, in km/h, that Abjit cycles from the cinema to the shop. … km/h [2] (e) When Abjit arrives at the swimming pool it is closed. Without stopping at the swimming pool he cycles home at a constant speed. It takes him 24 minutes to cycle home. Complete the travel graph for his journey home. [1] (f) Calculate the average speed, in km/h, for the whole journey. … km/h [3] (g) Abjit’s bicycle wheel has a radius of 29 cm. (i) Calculate the circumference of the wheel. Give your answer correct to 1 decimal place. … cm [3] (ii) Calculate the number of complete turns the wheel makes when travelling 500 m. … [2]

16 marks

Mark scheme: 6 (a) shop 1 (b) [graph] steepest oe 1 1 (c) (i) 0.2 × 20 or 12 × oe M2 M1 for 12×20 3 (ii) distance axis numbered correctly 1 with at least 2 more numbers 3 3 (d) 12 2 M1 for or [ 60 ] 0.25 15× (e) ruled line from (1034, 8) to 1 (1058, 0) their swimming pool distance × 2 (f) 16.6 or 16.55… 3 M2 for ×60 their 1058 − 1000 dist or M1 for a timeinterval (g) (i) 182.2 3 M1 for 2π × 29 A1 for 182.2 to 182.24 A1FT for their A1 rounded correctly to 1dp 50000 500 (ii) 274 2FT M1FT or their ( g )( i ) their ( g )( i ) ÷ 100 If zero scored SC1 for figs 27[44…] 1732

This question in 0580/32 Feb/March 2016

3 The diagram shows a cylindrical flower vase with radius, r, and height, h. The volume, V, of the vase is V = r r 2 h . NOT TO SCALE h The surface area, A, of the vase is A = 2 r rh + r r 2 . (a) The vase has radius 4 cm and height 15 cm. r (i) Calculate the volume of the vase. Write down the units of your answer. … … [3] (ii) Calculate the surface area of the vase. … cm2 [2] (b) Make h the subject of the formula A = 2 r rh + r r 2 . h = … [2] (c) Factorise completely. 2r rh + r r 2 … [2] (d) Another cylindrical flower vase has radius 6 cm and height 22.5 cm. (i) For this vase and the vase in part (a) the ratio of the radii is 4 : 6 and the ratio of the heights is 15 : 22.5 . Write these ratios in their simplest form. 4 : 6 = … : … 15 : 22.5 = … : … [2] (ii) Write down a mathematical word to complete the statement. The ratios show that the two vases are … [1]

12 marks

Mark scheme: 3 (a) (i) 754 or 753.9 to 754.1 2 M1 for π × 42 × 15 or better cm3 or cubic centimetres 1 Independent mark (ii) 427 or 427.2 to 427.312 2 M1 for 2 × π × 4 × 15 + π × 42 or better A − πr 2 (b) oe final answer 2 B1 for A – πr2 = 2πrh or better 2πr or A π r 2 = h + or better 2π r 2π r (c) πr(2h + r) final answer 2 B1 for π(2rh + r2) or r(2πh + πr) (d) (i) 2 : 3 1 2 2 : 3 1 Accept 1 : 1.5 or : 1 3 (ii) Similar 1

This question in 0580/33 May/June 2016

1 (a) The table shows the temperature at Lexford Station at 10 00 each day for a week. Day Mon Tue Wed Thu Fri Sat Sun Temperature - 3 4 - 1 0 - 5 2 1 (°C) (i) Write down the day which had the coldest temperature. … [1] (ii) Work out the difference in the temperature between Monday and Tuesday. … °C [1] (iii) The temperature falls 6°C from 10 00 to midnight on Sunday. Work out the temperature at midnight. … °C [1] (b) The distance between Lexford Station and Crowton Station is 6.5 km. (i) A train travels between these stations at an average speed of 39 km/h. Work out how long, in minutes, it takes the train to travel between these stations. … min [3] (ii) Each wheel on the train has a diameter of 1.8 m. Work out the number of complete turns each wheel makes in travelling the 6.5 km. … [4] (c) A northbound train leaves Lexford Station every 30 minutes. A bus leaves Lexford Station every 45 minutes. At 11 40 a northbound train and a bus leave the station together. Find the next time when this happens. … [3] (d) Here is part of a timetable for trains going east to west from Lexford Station. Lexford 09 14 09 47 10 21 11 15 11 48 Crowton 09 26 09 59 10 33 11 27 12 00 Doniton Halt 09 42 10 15 10 49 11 43 12 16 Mosshead 10 01 10 34 11 08 12 02 12 35 (i) Work out the number of minutes the 09 14 train takes to travel from Lexford to Mosshead. … min [1] (ii) Freda must arrive at Mosshead by 11 30. Write down the latest time she can catch a train from Lexford. … [1] (e) 437 people go on a coach trip. Each coach seats 62 people. How many coaches are needed? … [2]

17 marks

Mark scheme: Question Answer Marks Partial Marks 1(a)(i) Fri[day] 1 1(a)(ii) 7 1 1(a)(iii) –5 1 1(b)(i) 10 cao 3 6.5 × 60 M2 for oe 39 or M1 for distance ÷ speed 1(b)(ii) 1149 4 M2 for (6.5 × 1000) ÷ (π × 1.8) oe or M1 for π × 1.8 oe A1 for 1149.3 to 1149.5 B1 for their answer to at least 1dp truncated to the integer 1(c) 13 10 3 M2 for [LCM=] 2 × 3 × 3 × 5 or 90 or M1 for [30=] 2 × 3 × 5 or [45=] 3 × 3 × 5 OR M2 for listing times or multiples to at least 13 10 or 90 or M1 for adding times i.e. one correct addition e.g. 12 10 1(d)(i) 47 1 1(d)(ii) 10 21 1 1(e) 8 2 M1 for 437 ÷ 62 oe implied by 7.04… or 7.05

This question in 0580/33 May/June 2018

10 (a) Using a straight edge and compasses only, construct the equilateral triangle ABC. The base AB has been drawn for you. A B [2] (b) 16 m NOT TO SCALE 14 m 24 m Calculate the area of this trapezium. … m2 [2] (c) Each interior angle of a regular polygon is 162°. Calculate the number of sides of the polygon. … [3] (d) NOT TO SCALE h cm 6h cm The area of this triangle is 363 cm2. Calculate the value of h. h = … [3] (e) NOT TO SCALE This shape is drawn using two semicircles that have the same centre. The large semicircle has radius 7 cm. The small semicircle has radius 3 cm. Calculate the area of the shape. … cm2 [3]

13 marks

Mark scheme: 10(a) correct triangle drawn with arcs 2 B1 for correct triangle without arcs or for correct arcs 10(b) 280 2 1 M1 for ( 24 + 16 ) × 14 oe 2 10(c) 20 3 360 M2 for or better 180 − 162 or M1 for 180 − 162 or ( n − 2 ) × 180 = 162 n or better 10(d) 11 3 2 363 M2 for h = or better 3 1 or M1 for × h × 6 h = 363 oe 2 10(e) 62.8 or 62.83 to 62.84 3 1 2 1 2 M2 for π × 7 − π × 3 oe 2 2  1  2  1  2 or M1 for × π × 7 or × π × 3      2   2 

This question in 0580/32 Feb/March 2019

8 (a) A cylinder has a radius of 6 cm and a height of 17 cm. Show that the volume of this cylinder is 1923 cm3, correct to 4 significant figures. [2] (b) Q NOT TO SCALE P R O Points P, Q and R are on the circumference of a semicircle, centre O and radius 8 cm. Angle POQ = 90°. Calculate the shaded area. … cm2 [5]

7 marks

Mark scheme: 8(a) π × 62 × 17 M1 1922.6 to 1922.91 A1 8(b) 36.5 or 36.53 to 36.54… 5 B2 for 100.53 to 100.54… or 32π or M1 for [0.5 ×] π × 82 oe and B2 for 64 or M1 for [0.5 ×] 16 × 8 oe

This question in 0580/31 May/June 2019

2 80 students each record the name of their mathematics teacher. The number of these students taught by Mr House and by Miss Patel are shown in the bar chart. 24 20 16 Frequency 12 8 4 0 Mr Mrs Mr Miss Mr Jones Brown House Patel Smith (a) How many more students are taught by Miss Patel than by Mr House? … [1] (b) 15 students are taught by Mr Smith. Twice as many students are taught by Mrs Brown than by Mr Jones. Use this information to complete the bar chart. [4] (c) Write down the mode. … [1] (d) One of these students is chosen at random. Work out the probability that this student (i) is taught by Mr House, … [1] (ii) is not taught by either Mr House or Miss Patel. … [2] (e) This information is also to be shown in a pie chart. Work out the sector angle for Miss Patel. … [2]

11 marks

Mark scheme: 2(a) 4 1 2(b) 3 correct bars drawn on bar chart 4 B1 for Mr Smith bar drawn height 15 M2 for their ( 80 − (18 + 14 + 15 ) ) ÷ 3 [× 2 ] or M1 for 80 − (18 + 14 + 15 ) oe 2(c) Mrs Brown 1 FT their bar chart provided 5 bars drawn 2(d)(i) 14 1 oe 80 2(d)(ii) 48 2 FT their bar chart oe 80 18 + 14 M1 for 80 − (18 + 14 ) or oe 80 OR M1FT for adding heights of bars for (Mr Jones, Mrs Brown and Mr Smith) 2(e) 81 2 360 18 M1 for [× 18 ] or [× 360 ] 80 80

This question in 0580/32 May/June 2019

5 The scale drawing shows a play area, ABCDE. The scale is 1 centimetre represents 3 metres. D h E C A B Scale: 1 cm to 3 m (a) Find the actual distance h in metres. h = … m [2] (b) Find the actual area of triangle CDE. … m2 [3] (c) A straight path crosses the play area from C to AB. It is equidistant from CB and CD. Using a straight edge and compasses only, construct the path. Show all your construction arcs. [2] (d) There is a circular pool in the play area. The pool has a diameter of 8 m. Calculate (i) the circumference of the pool, … m [2] (ii) the area of the pool. … m2 [2]

11 marks

Mark scheme: 5(a) 10.8 to 12 2 B1 for 3.6 cm to 4.0 cm measured 5(b) 191 to 220 3 B1 for 11.8 to 12.2 measured or 35.4 to 36.6 M1 for 0.5 × their (a) × their actual EC oe 5(c) Correct ruled bisector of angle BCD 2 B1 for correct angle bisector with with correct arcs no/incorrect arcs, or two pairs of supporting arcs or correct line short of AB with or without arcs 5(d)(i) 25.1 or 25.13 to 25.14 2 M1 for 8 × π oe 5(d)(ii) 50.3 or 50.26 to 50.272 2 M1 for π (0.5 × 8)2 oe

This question in 0580/33 May/June 2019

2 Henry decorates a room. (a) Complete Henry’s shopping bill. Item Cost ($) 3 tins of paint at $15.95 each 2 brushes at $7.50 each 1 roll of tape at $2.90 2.90 Total [2] (b) 5.3 m 1.8 m NOT TO 3.2 m SCALE 2.4 m The diagram shows the floor of the room. (i) Calculate the area of the floor. … m2 [2] (ii) Henry buys varnish for the floor of the room. 500 ml of varnish covers 8 m2 of floor. Calculate the amount of varnish Henry needs. … ml [2] (c) This scale drawing shows the window in the room. The scale is 1 centimetre represents 40 centimetres. Scale: 1 cm to 40 cm Work out the actual length and height of the window. Length = … cm Height = … cm [2] (d) NOT TO SCALE 2.6 m 1.9 m 1.8 m The diagram shows one wall of the room. Calculate the area of the wall. … m2 [2] (e) Henry buys a circular mirror for the room. The diameter of the mirror is 80 cm. Calculate the circumference of the mirror. … cm [2]

12 marks

Mark scheme: 2(a) 47.85 2 B1 for one of first two values correct 15[.00] 65.75 2(b)(i) 12.9 2 M1 for 1.8 × 5.3 + 2.4 × (3.2 – 1.8) oe or 3.2 × 2.4 + 1.8 × (5.3 – 2.4) oe or 5.3 × 3.2 − (5.3 – 2.4) × (3.2 – 1.8) oe 2(b)(ii) 806.25 2 FT their (b)(i) × 62.5 M1 for their (b)(i) ÷ 8 × 500 oe 2(c) 160 2 B1 for each 100 or for 4 and 2.5 seen 2(d) 4.05 2 1 M1 for × (9.1 + 6.2 ) × 8.1 oe 2 2(e) 251 or 251.3 to 251.4 2 M1 for π × 80 oe

This question in 0580/32 Oct/Nov 2019

6 NOT TO SCALE G O F 11 cm E D H C 140° B A The diagram shows a circle, centre O, radius 11 cm. C, F, G and H are points on the circumference of the circle. The line AD touches the circle at C and is parallel to the line EG. B is a point on AD and angle ABO = 140°. (a) Write down the mathematical name of the straight line AD. … [1] (b) (i) Find, in terms of r, the circumference of the circle. … cm [2] (ii) Work out angle FOH. Angle FOH = … [2] (iii) Calculate the length of the minor arc FH. … cm [2] (c) (i) Give a reason why angle BCO is 90°. … [1] (ii) Show that BC = 13.11 cm, correct to 2 decimal places. [3] (iii) Calculate BH. BH = … cm [3]

14 marks

Mark scheme: 6(a) Tangent 1 6(b)(i) 22π final answer 2 M1 for 2 × 11 × π 6(b)(ii) 40 2 B1 for angle OBC = 40˚ or angle BOG = 140˚ 6(b)(iii) 7.68 or 7.679 to 7.680 … 2 FT their (b)(ii) and (b)(i) their (b)(ii) M1 for × their (b)(i) 360 6(c)(i) Angle [between] tangent [and] radius 1 6(c)(ii) 180 – 140 or 90 – their (b)(ii) B1 11 M1 tan (180 – 140) = oe BC [BC =] 13.109[...] A1 6(c)(iii) 6.11 or 6.112 to 6.114 3 M1 for [OB2 =] 13.112 + 112 A1 for 17.1 or 17.11 or 17.112 to 17.114 OR 11 M1 for oe sin40 A1 for 17.1 or 17.11... or 17.112 to 17.113

This question in 0580/31 May/June 2020

2 (a) Measure the length of this line in millimetres. … mm [1] (b) x (i) Measure the size of angle x. … [1] (ii) Write down the mathematical name of this type of angle. … [1] (c) A B C x° NOT TO SCALE 26° D ABC is a straight line and BCD is an isosceles triangle. Find the value of x. x = … [2] (d) Work out the size of one interior angle of a regular 16-sided polygon. … [2] (e) X NOT TO SCALE O Y Z (i) Complete this statement. X, Y and Z are points on the … of the circle, centre O. [1] (ii) Give a reason why angle XYZ is 90°. … [1] (f) A circle has diameter 6 cm. Calculate the area of the circle. Give the units of your answer. … … [3]

12 marks

Mark scheme: 2(a) 46 to 50 1 2(b)(i) 221 to 225 1 2(b)(ii) Reflex 1 2(c) 103 2 M1 for (180 – 26) ÷ 2 oe 2(d) 157.5 2 M1 for 180 – 360 ÷ 16 oe or (16 – 2) × 180 ÷ 16 oe 2(e)(i) Circumference 1 2(e)(ii) Angle [in a] semicircle [is] 90° 1 2(f) 28.3 or 28.27 to 28.28 2 M1 for 32 × π oe cm2 1 indep

This question in 0580/32 Oct/Nov 2020

4 (a) The diagram shows the plan of part of Rachel’s garden. 14.6 m 3.2 m NOT TO SCALE 10.9 m 8.5 m Calculate the area. … m2 [3] (b) Rachel has a pond in her garden in the shape of a circle. The circumference of the pond is 4.25 m. Calculate the diameter of the pond. Give your answer in centimetres. … cm [3] (c) A plant pot is a cylinder with radius 15 cm and height 24 cm. Calculate the volume of the pot. … cm3 [2] (d) The diagram shows two mathematically similar plant pots. h NOT TO 21.6 cm SCALE 27 cm 33 cm The smaller pot has height 21.6 cm and diameter 27 cm. The larger pot has diameter 33 cm. Find the height, h, of the larger pot. h = … cm [2] (e) A shop sells bags of compost in three different sizes. Small Medium Large 30 litres 50 litres 75 litres $5.82 $9.45 $14.50 Work out which size of bag gives the best value. Show how you decide. … [3]

13 marks

Mark scheme: 4(a) 112.17 3 M2 for 14.6 × 10.9 – (14.6 – 8.5) × (10.9 – 3.2) oe or M1 for a correct method to find one of the six areas or B1 for 7.7 or 6.1 soi 4(b) 135 or 135.26 to 135.3 3 4.25 M1 for oe π B1 for their answer in metres seen correctly converted to cm or 425 [cm] seen 4(c) 17 000 or 16 960 to 16 970 2 M1 for π × 152 × 24 oe 4(d) 26.4 2 21.6 27 M1 for = oe or better h 33 4(e) Medium 3 M2 for 3 correct consistent divisions shown With correct comparisons made of the but either not evaluated to enough accuracy 3 bags with suitable accuracy shown or wrong bag selected or M1 for 2 correct consistent divisions shown for 2 bags

This question in 0580/33 Oct/Nov 2020

7 (a) Martin, Suki and Pierre make clocks. In one week • Martin makes x clocks. • Suki makes 3 fewer clocks than Martin. • Pierre makes twice as many clocks as Suki. (i) Write an expression for the total number of clocks they make in one week. Give your expression in its simplest form. … [3] (ii) The total number of clocks they make in one week is 35. (a) Work out the value of x. x = … [3] (b) Work out how many more clocks Pierre makes than Martin. … [2] (b) 12 11 1 10 2 9 3 8 4 7 5 6 (i) Complete the clock diagram to show the time 2.30 pm. [1] (ii) Calculate the obtuse angle between the hands of the clock at 2.30 pm. … [2] (c) Work out the number of seconds in 10 days. Give your answer in standard form. … seconds [2] (d) A clock is started at 15 00. The clock is not working correctly and is slow. The clock loses 8 minutes every hour so after one hour the clock shows 15 52. What time will the clock show 3 12 hours after it is started? … [2] (e) The times on two clocks are checked regularly. One clock is checked every 6 days. The other clock is checked every 8 days. Both clocks are checked on 1st January 2021. Find the number of days during 2021 when both clocks will be checked on the same day. [There are 365 days in 2021.] … [4]

19 marks

Mark scheme: 7(a)(i) 4x− 9 cao 3 B2 for x + ( x − 3) + 2( x − 3) oe or B1 for k ( x− )3 seen k = 1,2 or 3 oe 7(a)(ii)(a) 11 nfww 3 M1 for their (a)(i) = 35 M1 for rearranging their( ax + b) = 35 b 35 to ax = 35 − b or x + = or better a a 7(a)(ii)(b) 5 2 FT their x M1 for (their x − 3) × 2 soi or B1 for [Pierre makes] 16 7(b)(i) Half-past two shown correctly on clock 1 face 7(b)(ii) 105 2 3.5 M1 for [× 360 ] oe 12 7(c) 8.64 × 105 2 M1 for 60 × 60 × 24 ×10 or B1 for figs 864 If 0 scored, SC1 for correctly changing their answer into standard form provided their answer >10 000 7(d) 18 02 2 1 M1 for 3 2× 8 or B1 for 1736 seen 7(e) 16 cao 4 B1 for LCM=24 soi 365 364 365 364 M1 for or or or 24 24 48 48 A1 for 15 or 15.2 or 15.16 to 15.17 or 15.20 to 15.21 If A0 scored, SC1 for 7.60[4..] or 7.58[3..] and 8 final answer

This question in 0580/31 May/June 2021

4 (a) A D E 73° z° NOT TO SCALE x° 58° y° F G B C In the diagram, ABC is a triangle. Line DAE is parallel to line FBCG. Find the value of x, the value of y and the value of z. x = … y = … z = … [3] (b) NOT TO SCALE Q R 32° O u° P Points P, Q and R lie on a circle, centre O. Find the value of u. u = … [2] (c) 6.42 cm NOT TO SCALE 72° The diagram shows a sector of a circle with radius 6.42 cm and sector angle 72°. Calculate the perimeter of this sector. … cm [3]

8 marks

Mark scheme: 4(a) 122 3 B1 for each 73 49 FT 122 – their 73 or their 122 – their 73 or (their 122) – 73 4(b) 58 2 M1 for 180 – 90 – 32 or 90 – 32 or angle PQR identified as 90 4(c) 20.9 or 20.90 to 20.91 3 72 M2 for × 2 × π × 6.42 + 2 × 6.42 oe 360 72 or M1 for × 2 × π × 6.42 oe 360

This question in 0580/32 Oct/Nov 2021

4 (a) The diagram shows the net of a cuboid. NOT TO SCALE 5 cm 7.8 cm A (i) Work out the area of the shaded rectangle, A. … cm2 [2] (ii) The volume of the cuboid is 468 cm 3. Complete the statement. The dimensions of the cuboid are … cm by … cm by … cm [2] (b) A cylinder has a radius of 8 cm and a height of 12 cm. Calculate, in terms of r, the volume of the cylinder. … cm3 [2] (c) NOT TO 7 cm SCALE 12 cm The diagram shows a circle with a diameter of 7 cm and a parallelogram with a base of 12 cm. The circle touches two of the sides of the parallelogram. Calculate the shaded area. … cm2 [3]

9 marks

Mark scheme: 4(a)(i) 39 2 M1 for 7.8  5 or B1 for 7.8 and 5 marked on two correct sides of rectangle A 4(a)(ii) 5 7.8 12 2 468 468 M1FT for oe or for 7.8  5 their(i) 4(b) 768π final answer 2 M1 for π  82  12 oe 4(c) 45.5 or 45.51 to 45.52 3 M2 for 12  7  π  3.52 oe or M1 for 12  7 oe or M1 for π  3.52 oe

This question in 0580/32 May/June 2022

7 (a) NOT TO 2 cm A SCALE 4 cm B 12 cm The area of rectangle A is equal to the area of rectangle B. Work out which rectangle has the greater perimeter and by how much. Rectangle … has the greater perimeter by … cm [4] (b) A circle has an area of 150 cm 2. Calculate the radius of this circle. … cm [3] (c) NOT TO SCALE 16 cm 16 cm 12 cm An isosceles triangle has base 12 cm and sides 16 cm. Find the area of this triangle. … cm2 [5]

12 marks

Mark scheme: 7(a) A 8 4 B2 for [length =] 6 or M1 for [area A =] 2 × 12 M1 for 2 × (2 + 12) oe or 2 × (4 + 24 ÷ 4) oe 7(b) 6.91 or 6.909… 3 150 M2 for π or M1 for [r2 = ] 150 ÷ π 7(c) 89.0 or 88.98 to 89 5 2 1 2  12  M4 for 2 × 12 × 16 −  oe  2  OR B3 for [height =] 14.8 or 14.83… or 220 or 2 55 or M2 for 162 − ( 122 ) 2 oe or better or M1 for [...]2 + ( 122 ) 2 = 162 oe M1dep for 12 × 12 × their 14.8 oe

This question in 0580/32 Oct/Nov 2022

5 (a) NOT TO SCALE This rectangle has an area of 12 cm2 and a perimeter of 16 cm. NOT TO SCALE This shape is made from six of these rectangles. Find the area and perimeter of this shape. Area = … cm2 Perimeter = … cm [4] (b) NOT TO SCALE 11.7 cm 8.4 cm 16 cm Find the area of this triangle. … cm2 [2] (c) A circle has a circumference of 28 cm. Work out the radius of the circle. … cm [2] (d) A cube has a volume of 125 m3. Work out the surface area of the cube. … m2 [3]

11 marks

Mark scheme: 5(a) [a =] 72 4 B1 for [a =] 72 [p =] 52 AND B3 for [p =] 52 or B1 for lengths of 6 and 2 M1 for use of 6l + 8w 5(b) 67.2 2 16  8.4 M1 for oe 2 If M0 scored, SC1 for 101 to 101.4 …. 5(c) 4.46 2 M1 for 28 ÷ 2π oe 5(d) 150 3 M2 for 6 × ( 3 125 )2 oe or M1 for 3 125 oe

This question in 0580/32 May/June 2023

2 (a) The diagram shows a circle. NOT TO SCALE (i) The diameter of this circle is 168 mm. Write down the radius of this circle. … mm [1] (ii) On the diagram, draw a chord of this circle. [1] (b) The scale drawing shows the position of ship A and the position of ship B. The scale is 1 cm represents 6 km. North B North A Scale : 1 cm to 6 km Another ship, C, is 45 km from ship B on a bearing of 124°. (i) On the scale drawing, mark the position of ship C. [2] (ii) Find the actual distance of ship C from ship A. … km [2] (c) (i) Show that the interior angle of a regular octagon is 135°. [1] (ii) NOT TO SCALE Show that two regular octagons and a square meet at a point without any gaps. [1] (d) E F NOT TO 49° SCALE D The diagram shows points D, E and F on the circumference of a circle. DF is a diameter of the circle. Find angle EDF. Angle EDF = … [2]

10 marks

Mark scheme: 2(a)(i) 84 1 2(a)(ii) Any chord 1 2(b)(i) Accurate position marked 2 B1 for accurate distance or accurate angle 2(b)(ii) 57 2 FT their diagram for 1 and 2 marks B1 for 9.2 to 9.6 seen or M1 for their length  6 2(c)(i) 360 ( 8 − 2 )  180 M1 180 – or 8 8 2(c)(ii) 135 + 135 + 90 = 360 M1 2(d) 41 2 M1 for 180 – 90 – 49 oe or angle DEF identified as 90°

This question in 0580/31 Oct/Nov 2023

8 (a) 8.2 cm NOT TO 5.4 cm SCALE 12.6 cm Find the area of this trapezium. … cm2 [2] (b) NOT TO SCALE 4.5 cm b cm The area of this triangle is 15.3 cm 2. Find the value of b. b = … [2] (c) A circle has a circumference of 58.6 cm . Find the radius of this circle. … cm [2] (d) 28 cm NOT TO SCALE 12 cm The diagram shows a rectangle with two semicircles removed. Calculate the shaded area. … cm2 [4]

10 marks

Mark scheme: 8(a) 56.2 or 56.16 2 8.2  12.6 M1 for  5.4 oe 2 8(b) 6.8 2 M1 for 12 × 4.5 × b = 15.3 or better 8(c) 9.33 or 9.325 to 9.327 2 M1 for 58.6 ÷ [2]π 8(d) 223 or 222.8 to 222.91 4 M1 for 28 × 12 or 336 M1 for π × 62 oe M1 for their 336 − their 113

This question in 0580/33 May/June 2024

29 10 cm NOT TO SCALE 9 cm 16 cm This shape is made from a rectangle and a semicircle. The diameter of the semicircle is 10 cm. Calculate the perimeter of this shape. … cm [3] Question 30 is printed on the next page.

3 marks

Mark scheme: 29 55.7 or 55.70 to 55.71 3 M2 for 16 + 9 + 9 + (16 – 10) 1 + their( ×π×10) oe 2 1 or M1 for π × 10 [× ] 2 1 or 2× π × 5 [× ] oe 2 or 16 + 9 + 9 + (16 – 10) oe

This question in 0580/31 May/June 2025

22 The diagram shows a solid cylinder with height 10 cm. NOT TO SCALE 10 cm The volume of the cylinder is 478 cm3. (a) Find the radius of the cylinder. … cm [3] (b) The cylinder is made from gold. The density of the gold is 19.3 g/cm3. Calculate the mass of the cylinder. mass :Density = D volume … g [1]

4 marks

Mark scheme: 22(a) 3.9[0] or 3.900 to 3.901 3 2 478 M2 for  r =  oe   10π or M1 for π × r2 × 10 = 478 22(b) 9225.4 1

This question in 0580/33 May/June 2025

22 NOT TO SCALE 16 m 10 m The diagram shows a garden. The garden has a circular pond and the shaded area is grass. The width of the grass area is equal to the diameter of the pond. (a) Find the area of the pond. … m2 [2] (b) Find the area of the grass. … m2 [2] (c) Find the percentage of the garden that is grass. … % [2]

6 marks

Mark scheme: 22(a) 78.5 or 78.6 or 78.53 to 78.55 2 2  10  M1 for π ×   oe  2  22(b) 121 or 120.7 to 120.8 2 FT 1 M1FT for 10 × 16 − × their (a) oe 2 22(c) 60.4 to 60.8 2 their ( b ) M1FT for [100] oe 1 10  16 + their ( a ) 2 their ( b ) or [100] oe their ( b ) + their ( a )

This question in 0580/32 Oct/Nov 2025

23 (a) Jo buys a candle for $3.20 . She sells the candle for $4.64 . Find the percentage profit on the candle. … % [2] (b) The candle is in the shape of a cone. The volume of the candle is 900 cm3. The height of the candle is 8 cm. Calculate the radius of the candle. … cm [3]

5 marks

Mark scheme: 23(a) 45 2 4.64 −3.2 M1 for [×100] oe 3.2 4.64 or ×100[–100] oe 3.20 4.64 or – 1 [×100] oe 3.20 23(b) 10.4 or 10.36… 3 900  3 M2 for oe 8π or 900  3 M1 for [r2 =] oe 8π 1 2 or for 900 = πr  8 oe and a correct first 3 step

This question in 0580/33 Oct/Nov 2025