Circles, arcs and sectors — Paper 2 · IGCSE Mathematics 0580

19 For Examiner's Use A F NOT TO SCALE B E 12 cm 6 cm 40° O C H D G The diagram shows part of a fan. OFG and OAD are sectors, centre O, with radius 18 cm and sector angle 40°. B, C, H and E lie on a circle, centre O and radius 6 cm. Calculate the shaded area. Answer cm2 [4]

4 marks

Mark scheme: 19 314 4 M1 π. 182 . 40/360 or OAD = 113 identified M1 π. 62 (or π. 62 . 40/360) or OBC …”……. M1 2 × (OAD – OBC) + circle oe OR M1 π. 182 . 40/360 M1 π. 62 . 140/360 M1 2 × OAD + 2 × BOE oe

This question in 0580/21 May/June 2009

17 Examiner's Use NOT TO Q R SCALE P 8 m S O W T V U The diagram shows the junction of four paths. In the junction there is a circular area covered in grass. This circle has centre O and radius 8 m. (a) Calculate the area of grass. Answer(a) m2 [2] (b) Q NOT TO SCALE 12 m 45° P O The arc PQ and the other three identical arcs, RS, TU and VW are each part of a circle, centre O, radius 12m. The angle POQ is 45°. The arcs PQ, RS, TU, VW and the circumference of the circle in part(a) are painted white. Calculate the total length painted white. Answer(b) m [4]

6 marks

Mark scheme: 17 (a) 201 2 M1 π × 82 45 (b) 87.9 or 88.0 4 M1 × 2 × π × 12 ….. d 360 M1 2 × π × 8 ……………..e M1 ft for their (4d + e) which must come from multiples of π SC2 43.9 or 44.0

This question in 0580/22 May/June 2010

17 K NOT TO SCALE 40° O 5.6 cm L OKL is a sector of a circle, centre O, radius 5.6 cm. Angle KOL = 40°. Calculate (a) the area of the sector, Answer(a) cm2 [2] (b) the perimeter of the sector. Answer(b) cm [2]

4 marks

Mark scheme: 40 2 17 (a) 10.9 2 M1 for × π × 6.5 360 40 (b) 15.1 2 M1 for × π × 2 × 6.5 (= 3.91..) 360

This question in 0580/23 May/June 2010

18 For Examiner's NOT TO Use SCALE x° 8 cm The diagram shows a sector of a circle of radius 8 cm. The angle of the sector is x°. The perimeter of the sector is (16 + 14π) cm. Find the value of x. Answer x = [3]

3 marks

Mark scheme: x 18 315 3 M1 × 2 × π × 8 oe 360 x M1 × 2 × π × 8 (+ 16) = (16 +) 14π 360 3

This question in 0580/23 Oct/Nov 2010

15 A cylinder has a height of 12 cm and a volume of 920 cm3. Calculate the radius of the base of the cylinder. Answer cm [3]

3 marks

Mark scheme: 15 4.94 3 M1 π r2 × 12 = 920 920 M1 (r2) = their ( π × 12)

This question in 0580/23 May/June 2011

19 A NOT TO SCALE 50° O 9 cm B The diagram shows a sector AOB of a circle, centre O, radius 9 cm with angle AOB = 50°. Calculate the area of the segment shaded in the diagram. Answer cm2 [4]

4 marks

Mark scheme: 50 2 19 4.32 4 M1 for × π × 9 360 M1 for 0.5 × 92 × sin 50 M1 for subtracting their triangle from their sector (dependent on at least M1)

This question in 0580/23 Oct/Nov 2011

20 For R Examiner's Use NOT TO O SCALE 78° 5 cm P T R and T are points on a circle, centre O, with radius 5 cm. PR and PT are tangents to the circle and angle POT = 78°. A thin rope goes from P to R, around the major arc RT and then from T to P. Calculate the length of the rope. Answer cm [6] Question 21 is printed on the next page.

6 marks

Mark scheme: PT 20 64.8 to 64.9 6 M2 5 tan 78 soi by 23.5 or M1 tan 78 = or 5 5 5 sin 78 or tan 12 sin 12 360 − 2 × 78 M2 × 2 × π × 5 soi by 17.8 360 or M1 for 2π5 seen used M1 for their arc + 2 (their PT) 1 3 2

This question in 0580/21 May/June 2012

14 C 4 cm 4 cm NOT TO x° SCALE A B ABC is a sector of a circle, radius 4 cm and centre C. The length of the arc AB is 8 cm and angle ACB = x°. Calculate the value of x . Answer x = [3]

3 marks

Mark scheme: 14 114.6 or 114.57 (67027..) to 114.59 (1155..) 3 M2 2 × π × 4 × x / M2 x / 360 = 8 / 2π4 360 = 8 or M1 2 × π × 4 × x / or B1 8 / 2π4 or 2π4 / 360 8 seen

This question in 0580/21 Oct/Nov 2012

17 For Examiner's Use A B C AB is the diameter of a circle. C is a point on AB such that AC = 4 cm. (a) Using a straight edge and compasses only, construct (i) the locus of points which are equidistant from A and from B, [2] (ii) the locus of points which are 4 cm from C. [1] (b) Shade the region in the diagram which is • nearer to B than to A and • less than 4 cm from C. [1]

4 marks

Mark scheme: 17 (a) 2 B1 for correct line, on each side of AB (longer than dash at C) B1 for 2 pairs of intersecting arcs R 1 Intention to draw a full correct circle (b) 1 R shaded must be a closed region 7 84

This question in 0580/21 Oct/Nov 2012

17 For B Examiner's Use 5r NOT TO SCALE 4r O A The diagram shows a sector of a circle, centre O, radius 5r. The length of the arc AB is 4r. Find the area of the sector in terms of r, giving your answer in its simplest form. Answer [3]

3 marks

Mark scheme: θ 4r 17 10r2 cao www 3 B1 for ( =) 360 2 × π × 5r 4 r M1 for × ( 5 r 2) π 2 × π × 5 r

This question in 0580/23 Oct/Nov 2012

21 For Examiner′s Use NOT TO SCALE 12 cm 135° 12 cm The diagram shows a sector of a circle of radius 12 cm with an angle of 135°. Calculate the perimeter of the sector. Answer … cm [3] _____________________________________________________________________________________

3 marks

Mark scheme: 21 52.3 or 52.27 to 52.28 3 SC2 for 28.3 or 28.7 to 28.8 135 If 0, M2 for × π × 24 + 2 × 12 360 135 or M1 for ×π × 24 360

This question in 0580/21 May/June 2013

18 For Examiner′s Use B A NOT TO SCALE 5 cm 120° O A and B lie on a circle centre O, radius 5 cm. Angle AOB = 120°. Find the area of the shaded segment. Answer … cm2 [4] _____________________________________________________________________________________

4 marks

Mark scheme: 18 15.4 or 15.35 to 15.36 4 120 2 M1 for ×π × 5 oe 360 1 2 M1 for × 5 × sin 120 oe 2 120 2 1 2 M1 for ×π × 5 − × 5 × sin 120 oe 360 2

This question in 0580/23 May/June 2013

21 For Examiner′s B Use NOT TO 6 cm 4 cm SCALE 65° C A In triangle ABC, AB = 6 cm, BC = 4 cm and angle BCA = 65°. Calculate (a) angle CAB, Answer(a) Angle CAB = … [3] (b) the area of triangle ABC. Answer(b) … cm2 [3]

6 marks

Mark scheme: 4× sin 65 21 (a) 37.2 or 37.17 to 37.19 3 M2 for sin[ ] = 6 4 6 or M1 for = oe sin[] sin 65 (b) 11.7 or 11.72 to 11.74 3 M1 for [B =] 160 – 65 – their (a) 1 M1 for × 4 × 6 × sin their 77.8 2

This question in 0580/22 Oct/Nov 2013

19 E D C NOT TO SCALE 8 cm 60° 30° A B 8 cm The diagram shows a rectangle ABCE. D lies on EC. DAB is a sector of a circle radius 8 cm and sector angle 30°. Calculate the area of the shaded region. Answer … cm2 [7]

7 marks

Mark scheme: 1 19 1.38 or 1.39 or 1.384 to 1.389 7 M3 [Area ∆ =] × 8 cos 60 × 8 sin 60 2 or M1 for [ AE =] 8cos 60 and M1 for [ ED] = 8sin 60 and 30 2 M1 for Area sector × π × 8 360 and M1 for Area rectangle = 8 × 8cos60 or 8 × 4 M1 for their 32 – (their 13.86 + their 16.76) or better

This question in 0580/21 May/June 2014

21 NOT TO SCALE The diagram shows two concentric circles and three radii. The diagram has rotational symmetry of order 3. A club uses the diagram for its badge with some sections shaded. The radius of the large circle is 6 cm and the radius of the small circle is 4 cm. NOT TO SCALE Calculate the total perimeter of the shaded area. Answer … cm [5] __________________________________________________________________________________________ Question 22 is printed on the next page.

5 marks

Mark scheme: 2 21 62.3 or 62.26 to 62.272 5 M1 for × 2π × 6 3 2 1 and M2 for ( + ) × 2π × 4 oe 3 3 2 1 or M1 for × 2π × 4 or × 2π × 4 3 3 and M1 for 2 × (2 + 4 ) + kπ , k ≠ 0 ( ) ( ) ( ) 1 f l i b k  3 

This question in 0580/23 May/June 2014

19 C NOT TO SCALE A B D Two circles, centres A and B, are each of radius 8 cm and intersect at C and D. Each circle passes through the centre of the other circle. (a) Explain why angle CBD is 120°. Answer(a) [1] C (b) For the circle, centre B, fi nd the area of the sector BCD. NOT TO 8 cm SCALE A 120° B 8 cm D Answer(b) … cm2 [2] (c) (i) Find the area of the shaded segment CAD. C NOT TO SCALE A B D Answer(c)(i) … cm2 [3] (ii) Find the area of overlap of the two circles. Answer(c)(ii) … cm2 [1] __________________________________________________________________________________________ Question 20 is printed on the next page.

7 marks

Mark scheme: 19 (a) CBA and BDA are equilateral oe 1 (b) 67[.0] or 67.02 to 67.03 2 M1 for 120360 × π × 8 2 oe (c) (i) 3 M2FT for their (b ) − 12 × 82 × sin 120 oe 39.3 or 39.28 to 39.33 or M1 for 12 × 82 × sin 120 oe (ii) 1FT FT 2 × their(c)(i) correctly evaluated 78.6 or 78.7 or 78.56 to 78.66

This question in 0580/21 Oct/Nov 2014

18 6 cm NOT TO SCALE 15 cm The diagram shows a glass, in the shape of a cone, for drinking milk. The cone has a radius of 6 cm and height 15 cm. A bottle of milk holds 2 litres. (a) How many times can the glass be completely fi lled from the bottle? 1 [The volume, V, of a cone with radius r and height h is V = πr 2h.] 3 Answer(a) … [4] (b) Calculate the volume of milk left in the bottle. Give your answer in cm3. Answer(b) … cm3 [3] __________________________________________________________________________________________ Question 19 is printed on the next page.

7 marks

Mark scheme: 18 (a) 3 4 B3 for 3.536 to 3.54 as an answer or 1 2 M2 for 2000 ÷ π × 6 × 15 3 1 2 or M1 for π × 6 × 15 3 and SC1 for truncating their 3.54 to a whole number (b) 303 to 304 3 M2 for 2000 – their 3 × their volume or M1 for their 3 × their volume

This question in 0580/23 Oct/Nov 2014

19 The diagram shows the positions of three points A, B and C. C B A (a) Draw the locus of points which are 4 cm from C. [1] (b) Using a straight edge and compasses only, construct the locus of points which are equidistant from A and B. [2] (c) Shade the region which is • less than 4 cm from C and • nearer to B than to A. [1] __________________________________________________________________________________________

4 marks

Mark scheme: 19 (a) 1 Correct circle, radius 4 cm centre C (b) 2 B2 for correct bisector with 2 pairs of correct arcs or B1 for correct bisector with no/wrong arcs C (c) 1 Correct complete boundary and correct shading. A B Dep on at least B1 in (b)

This question in 0580/22 May/June 2015

15 The circumference of a circle is 30 cm. (a) Calculate the radius of the circle. Answer(a) … cm [2] (b) The length of the arc of the semi-circle is 15 cm. Calculate the area of the semi-circle. Answer(b) … cm2 [2]

4 marks

Mark scheme: 15 (a) 4.77 or 4.774 to 4.775 2 M1 for 30 ÷ [2]π (b) 35.7 or 35.8 or 35.74 to 35.82 2 M1 for 0.5 × π × (their (a))2 or 0.5 × π × (30 ÷ 2π)2

This question in 0580/23 May/June 2015

11 NOT TO SCALE 6.1 cm A protractor is a semi-circle of radius 6.1 cm. Calculate the perimeter of the protractor. Answer … cm [3] __________________________________________________________________________________________

3 marks

Mark scheme:  2  11 31.4 or 31.36 to 31.37 3 M2 for × 1.6 × π + 2 × 1.6 oe  2  or B2 for 19.16 to 19.17 or 19.2 or M1 for 1.6 × π or for 12 2. × π 3

This question in 0580/21 Oct/Nov 2015

5 Calculate the volume of a hemisphere with radius 5 cm. 4 3 [The volume, V, of a sphere with radius r is V = 3 r r .] Answer … cm3 [2]

2 marks

Mark scheme: 1 4 3 5 262 or 261.7 to 261.83… 2 M1 for × π × 5 2 3 If zero scored SC1 for final answer 524 or 523.5 to 523.7

This question in 0580/22 Oct/Nov 2015

16 NOT TO SCALE 26° 15 cm The diagram shows a sector of a circle with radius 15 cm. Calculate the perimeter of this sector. Answer … cm [3]

3 marks

Mark scheme: 26 16 36.8 or 36.80 to 36.81 3 M1 for × 2 × π × 15 360 M1 for 2 × 15 + a term involving π ( ) 2

This question in 0580/22 Oct/Nov 2015

25 B NOT TO SCALE 8 cm 30° O A C OAB is the sector of a circle, centre O, with radius 8 cm and sector angle 30°. BC is perpendicular to OA. Calculate the area of the region shaded on the diagram. Answer … cm2 [5] __________________________________________________________________________________________ Question 26 is printed on the next page.

5 marks

Mark scheme: 30 × π × 8 2 – 0.5×8cos30×8sin3025 2.9[0] or 2.898 to 2.901 5 M4 for 360 or 30 2 M1 for × π × 8 360 and M2 for [area of triangle =] 0.5×8cos30×8sin30 oe OC BC or M1 for = cos30 oe or = sin30 oe 8 8

This question in 0580/23 Oct/Nov 2015

11 A NOT TO SCALE O 38° 25 cm B The diagram shows a sector of a circle, centre O, radius 25 cm. The sector angle is 38°. Calculate the length of the arc AB. Give your answer correct to 4 significant figures. AB = … cm [3]

3 marks

Mark scheme: 11 16.58 cao 3 B2 for 16.6 or 16.580 to 16.583 final answer or 16.58 not as final answer or 38 M1 for ×2×π×25 360 and B1 for rounding their more accurate answer correctly to 4sf

This question in 0580/22 Feb/March 2016

20 AB is an arc of a circle, centre O, radius 9 cm. A The length of the arc AB is 6 r cm. The area of the sector AOB is k r cm2. Find the value of k. NOT TO SCALE O 9 cm B k = … [3]

3 marks

Mark scheme: 6π 220 27 3 M2 for × π × 9 oe π × 2 × 9 6π or M1 for oe π × 2 × 9

This question in 0580/21 May/June 2016

19 ABCD is a rhombus with side length 10 cm. A NOT TO 10 cm SCALE D 40° B C Angle ADC = 40°. DAC is a sector of a circle with centre D. BAC is a sector of a circle with centre B. Calculate the shaded area. … cm2 [4]

4 marks

Mark scheme: 19 5.53 or 5.54 or 5.534 to 5.543… 4 M3 for 40 1 2 × {( × π × 102)– ( × 102 × sin 40)} 360 2 or M2 for  1  40 × 102 × sin 40 and [2 ×] × π × 102    2  360 or M1 for  1  40 × 102 × sin 40 or [2 ×] × π × 102    2  360

This question in 0580/21 May/June 2017

23 6 cm O 30° NOT TO SCALE The diagram shows a sector of a circle, centre O and radius 6 cm. The sector angle is 30°. The area of the shaded segment is (kr – c) cm2, where k and c are integers. Find the value of k and the value of c. k = … c = … [3]

3 marks

Mark scheme: 23 [k =] 3 3 30 2 M1 for × π × 6 [c =] 9 360 1 M1 for × 6 × 6 × sin30 2

This question in 0580/22 Oct/Nov 2017

6 Calculate the area of a circle with radius 5.1 cm. … cm2 [2]

2 marks

Mark scheme: 6 81.7 or 81.71 to 81.72… 2 M1 for π × 5.12

This question in 0580/21 May/June 2018

19 The diagram shows a pentagon ABCDE. A E B D C (a) Using a straight edge and compasses only, construct the bisector of angle BCD. [2] (b) Draw the locus of the points inside the pentagon that are 3 cm from E. [1] (c) Shade the region inside the pentagon that is • less than 3 cm from E and • nearer to DC than to BC. [1]

4 marks

Mark scheme: 19(a) Correct rulled bisector withw two 2 B1B for correcct ruled bisecctor with no//wrong arcs pairs of arccs 19(b) Correct arcc centre E raadius 3 cm 1 inside penttagon 19(c) Correct reggion shaded 1 DependentD ono at least B11 in part (a) anda 1 mark in partp (b) and a closed regiion

This question in 0580/22 Oct/Nov 2018

21 A NOT TO M N SCALE C B 10 cm The diagram shows an equilateral triangle ABC with sides of length 10 cm. AMN is a sector of a circle, centre A. M is the mid-point of AC. Work out the area of the shaded region. … cm2 [4]

4 marks

Mark scheme: 21 30.2 or 30.20 to 30.21… 4 2 1 60  10  M3 for × 10 × 10 × sin60 − × π ×  2 360  2  k  10  2 or M1 for × π ×  oe 360  2  1 and M1 for × 10 × 10 × sin c oe 2

This question in 0580/22 Oct/Nov 2018

19 NOT TO SCALE 72° 6 cm The diagram shows a sector of a circle with radius 6 cm and sector angle 72°. The perimeter of this sector is (p + qr ) cm. Find the value of p and the value of q. p = … q = … [3]

3 marks

Mark scheme: 19 [p =] 12 3 B1 for [p =] 12 12 and [q = ] oe 12 5 B2 for [q = ] 5 72 or M1 for [× π ] × 2 × 6 oe 360

This question in 0580/23 Oct/Nov 2018

12 NOT TO SCALE 217° 6.2 cm The diagram shows a sector of a circle with radius 6.2 cm and sector angle 217°. Calculate the area of this sector. … cm2 [2]

2 marks

Mark scheme: 12 72.8 or 72.79 to 72.80… 2 217 2 M1 for × π × 6.2 360

This question in 0580/23 May/June 2019

17 5 cm NOT TO SCALE 45° 3 cm The diagram shows two sectors of circles with the same centre. Calculate the shaded area. … cm2 [3]

3 marks

Mark scheme: 17 6.28 or 6.283 to 6.284 3 45 2 45 2 M2 for × π × 5 oe and × π × 3 360 360 oe 45 2 or M1 for × π × 5 oe 360 45 2 or × π × 3 oe 360 or π × 52 − π × 32 oe

This question in 0580/21 Oct/Nov 2019

6 The table shows how children in Ivan’s class travel to school. Travel to school Number of children Walk 12 Car 7 Bicycle 9 Bus 4 Ivan wants to draw a pie chart to show this information. Find the sector angle for children who walk to school. … [2]

2 marks

Mark scheme: 6 135 2 12 M1 for [× 360 ] 12 + 7 + 9 + 4 360 or [× 12 ] oe 12 + 7 + 9 + 4

This question in 0580/22 Oct/Nov 2019

24 P NOT TO 8 cm SCALE 6.4 cm Q The diagram shows a sector of a circle of radius 8 cm. The length of the arc PQ is 6.4 cm. Find the area of the sector. … cm2 [4]

4 marks

Mark scheme: 24 25.6 or 25.59 to 25.60… 4 6.4 2 M3 for × π × 8 2 × π × 8 x 6.4 or M2 for = oe 360 2 × π × 8 x or M1 for × 2 × π × 8 = 6.4 oe 360

This question in 0580/22 May/June 2020

9 Calculate the area of the sector of a circle with radius 65 mm and sector angle 42°. Give your answer in square centimetres. … cm2 [3]

3 marks

Mark scheme: 9 15.5 or 15.48 to 15.49 3 B2 for 1550 or 1548 to 1549 42 2 or M2 for × π × 6.5 360 42 2 or M1 for × π × 65 360

This question in 0580/23 May/June 2020

12 The total perimeter of a semicircle is 19.02 cm. Calculate the radius of the semicircle. … cm [3]

3 marks

Mark scheme: 12 3.7[0] or 3.689 to 3.699… 3 19.02 M2 for 2 + π or M1 for 2 r + π r [ = 19.02] oe

This question in 0580/23 May/June 2020

15 NOT TO SCALE 60° 7.5 cm Calculate the area of this sector of a circle. … cm2 [2]

2 marks

Mark scheme: 15 29.5 or 29.45 to 29.46 2 60 2 M1 for × π× 7.5 oe 360

This question in 0580/22 Oct/Nov 2020

21 B NOT TO SCALE 8 cm 30° O P A OAB is the sector of a circle, centre O. OB = 8 cm and angle AOB = 30°. BP is perpendicular to OA. (a) Calculate AP. AP = … cm [3] (b) Work out the area of the shaded region APB. … cm2 [3]

6 marks

Mark scheme: 21(a) 1.07 or 1.071 to 1.072 3 M2 for [8 −] 8 cos 30 oe OP or M1 for = cos30 oe 8 21(b) 2.9[0] or 2.895 to 2.901 3 30 2 M1 for × π× 8 oe 360 1 M1 for × 8 × their 6.93 × sin30 oe 2 1 or × 8cos30 × 4 oe 2

This question in 0580/23 Oct/Nov 2020

19 (a) NOT TO A SCALE B C The diagram shows a shape made from an equilateral triangle ABC and a sector of a circle. Points B and C lie on the circle, centre A. The side length of the equilateral triangle is 12.4 cm. Work out the perimeter of the shape. … cm [3] (b) NOT TO SCALE 41° The diagram shows two sectors of a circle. The major sector is shaded. The area of the major sector is 74.5 cm2. Calculate the radius of the circle. … cm [3]

6 marks

Mark scheme: 19(a) 77.3 or 77.32 to 77.33… 3 360 − 60 M2 for × π× 12.4 × 2 oe 360 [± n × 12.4] or M1 for angle 60° or 300° soi k or for × π× 12.4 × 2 oe [± n × 12.4] 360 19(b) 5.17 or 5.172 to 5.173… 3 74.5 360 2 M2 for × = r oe or better π 360 − 41 360 − 41 2 or M1 for 74.5 = × πr oe 360 74.5 360 or for × oe π k

This question in 0580/22 May/June 2021

19 12.6 cm NOT TO SCALE 80° O The diagram shows a sector of a circle, centre O, radius 12.6 cm. Calculate the perimeter of the shaded segment. … cm [4]

4 marks

Mark scheme: 19 33.8 or 33.78 to 33.80 4 M2 for 2 × 12.6 × sin 40 oe ( ... ) or M1 for sin 40 = oe 12.6 80 M1 for × 2 × π × 12.6 oe 360

This question in 0580/23 Oct/Nov 2021

14 Calculate the circumference of a circle with radius 4.7 cm. … cm [2]

2 marks

Mark scheme: 14 29.5 or 29.53… 2 M1 for 2 × π× 4.7 oe

This question in 0580/22 Feb/March 2022

15 The perimeter of a sector of a circle with radius 8 cm is 26 cm. Calculate the angle of this sector. … [3]

3 marks

Mark scheme: 15 71.6 or 71.61 to 71.62 3 angle 26 −−8 8 M2 for = or better 360 2π  8 angle or M1 for  2π  8 oe 360

This question in 0580/21 Oct/Nov 2022

14 P 9 cm NOT TO SCALE 6 cm O Q The diagram shows a sector of a circle with centre O and radius 9 cm. The length of the chord PQ is 6 cm. Calculate the length of the arc PQ. … cm [3]

3 marks

Mark scheme: 14 6.12 or 6.116… to 6.118 3 3 9 2 + 9 2 − 6 2 M1 for sin = oe or cos = oe 9 2 9 9 their angle M1 dep for  2 9 dependent 360 on use of trig for their angle

This question in 0580/22 Feb/March 2023

11 B 2.6 cm C NOT TO 3.2 cm SCALE O D A The diagram shows a shape, ABCD, formed by the sectors of two circles with the same centre O. Both sector angles are 140°, OC = 3.2 cm and CB = 2.6 cm . The area of the shape is k rcm2 . Find the value of k. k = … [3]

3 marks

Mark scheme: 11 9.1 3 M2 for 140 2 140 2  π  3.2  2.6  –  π  3.2 oe 360 360 140 2 or M1 for  π  3.2 oe 360 140 2 or  π  3.2  2.6  oe 360 2 or π  3.2  2.6  2 – π  3.2

This question in 0580/22 May/June 2023

14 NOT TO 48° SCALE 9 cm The diagram shows a circle with radius 9 cm. Calculate the area of the shaded major sector. … cm2 [3]

3 marks

Mark scheme: 14 221 or 220.5 to 220.6 3 360  48 2 M2 for   9 360 k 2 or M1 for   9 where k < 360 360 or B1 for 312

This question in 0580/21 May/June 2024

9 A B NOT TO SCALE D C Points A, B, C and D lie on a circle. ABCD is a square with area 72 cm 2. Calculate the area of the circle. Give your answer as a multiple of r. … cm2 [3]

3 marks

Mark scheme: 9 36π cao 3 B2 for answer 113 or 113.0 to 113.1… or an answer in terms of π which rounds to 113 or M1 for correct first step for finding d or r e.g. 72 + 72 = d 2 oe  72  2  72  2 72 = r2 + r2 oe or   +   = r2 oe      2   2  r 72 = oe sin 45 1 72 × r × r = oe 2 4

This question in 0580/22 Oct/Nov 2024

10 18 cm NOT TO SCALE 45° O The diagram shows a sector of a circle, centre O. The length of the arc is nr cm . Find the value of n. n = … [2]

2 marks

Mark scheme: 10 9 2 45 oe M1 for 2 π  18 oe 2 360

This question in 0580/22 May/June 2025

20 A NOT TO 40° B SCALE O The diagram shows a sector of a circle, centre O. The radius of the circle is 6 cm. Calculate the length of the major arc AB. Give your answer in its simplest form in terms of r. … cm [3]

3 marks

Mark scheme: 20 32 2 3 360 − 40 π or 10 π cao M2 for π 6 2 3 3 360 360 − 40 22 or  6 2 oe 360 7 40 or M1 for π 6 2 oe 360 or B1 for major sector angle = 320 soi or 12π

This question in 0580/23 May/June 2025

12 The diagram shows a sector of a circle with centre O and radius 9 cm. 9 cm NOT TO SCALE O 9 cm The perimeter of the sector is ( 18 + 2 r ) cm. Find the area of the sector. Give your answer in terms of r. … cm2 [4]

4 marks

Mark scheme: 12 9π 4 M3 for a fully correct method 2 π 2 e.g. π 9 oe 2 π  9 OR 1 B2 for 40 or oe 9 or M1 for 18 + k =2 π 9 18 + 2π oe M1 for ( their k )  π  9 2 oe OR SC2 for answer 81 + 9π oe

This question in 0580/21 Oct/Nov 2025

23 A NOT TO SCALE O 12 cm B 240° The diagram shows a major sector, AOB, of a circle. The sector angle is 240° and the radius is 12 cm. (a) Show that the length of the major arc AB is 16rcm . [1] (b) OA is joined to OB to form a cone. Work out the volume of the cone. `a bj r Give your answer in the form where a is an integer and b is a prime number. 3 … cm3 [6] Question 24 is printed on the next page.

7 marks

Mark scheme: 23(a) 240 M1 2 12 oe 360 23(b) 6 256 5  256 5 256 5  ( ) ( ) ( ) cao B5 for answer or 3 3 3 seen then spoilt 256 5  ( ) or an answer equivalent to 3 but not in required form OR B2 for radius of cone = 8 oe or M1 for 2πr = 16π B2 for [height of cone =] 80 oe or M1 for 122 = their 82 + h2 or better 1 2 M1dep for  their 8  their 80 oe 3 dep on attempt at Pythagoras to find height of cone

This question in 0580/22 Oct/Nov 2025

19 A NOT TO SCALE 60° B O The diagram shows a circle, centre O, radius 12 cm. (a) Work out the area of the minor sector AOB. Give your answer in terms of r in its simplest form. … cm2 [2] (b) Calculate the length of the major arc AB. Give your answer in terms of r in its simplest form. … cm [3]

5 marks

Mark scheme: 19(a) 24π cao 2 60 2 M1 for  12 oe 360 19(b) 20π cao 3 360 − 60 M2 for  12  2 oe 360 k or M1 for  12  2 oe 360 where k < 360

This question in 0580/23 Oct/Nov 2025