Circle theorems — Paper 3 · IGCSE Mathematics 0580

9 (a) Calculate the size of one exterior angle of a regular heptagon (seven-sided polygon). For Give your answer correct to 1 decimal place. Examiner's Use Answer(a) [3] (b) D A E so to ro NOT TO SCALE 130o po qo F B C G In the diagram above, DAE and FBCG are parallel lines. AC = BC and angle FBA = 130°. (i) What is the special name given to triangle ABC? Answer(b)(i) [1] (ii) Work out the values of p, q, r, s and t. Answer (b)(ii) p = q = r = s = t = [5] (c) J J, K and L lie on a circle centre O. yo L KOL is a straight line and angle JKL = 65°. NOT TO Find the value of y. 65o O SCALE K Answer(c) y = [2]

11 marks

Mark scheme: 9 (a) 51.4 3 2 for 51 or M1 for any complete method (b) (i) Isosceles 1 (ii) p = 50 1 q = 80 1√ ft for 180 – 2p r = 50 1√ ft for = p s = 50 1√ ft for = p t = 80 1√` ft for = q or 180 – 2p (c) 25 2 M1 for 90 – 65 oe [11]

This question in 0580/31 Oct/Nov 2005

8 For Examiner's Use O A R P B The diagram shows a circular garden, centre O. A straight path AB touches the circle at P. (a) (i) Draw on the diagram the diameter PQ and label the point Q. [1] (ii) Without measuring, write down the size of angle APQ. Answer(a)(ii) Angle APQ= [1] (iii) The point R is marked on the circumference of the circle. Draw the lines PR and QR. [1] (iv) Write down the reason why the angle PRQ is 90°. Answer(a)(iv) [1] (b) Showing all your construction lines, use a straight edge and compasses only to construct (i) the perpendicular bisector of QR, [2] (ii) the bisector of angle PRQ. [2] (c) Shade the region of the garden between PQ and QR which is closer to R than to Q and closer to RQ than to RP. [2] Question 9 is on the next page.

10 marks

Mark scheme: 8 (a) (i) Diameter from P through O to Q B1 (ii) 90 B1cao (iii) P to R and Q to R ruled. B1 (iv) (angle in a ) semi-circle B1 Angle on a diameter. Half the angle at the centre. (b) (i) Bisector of QR with arcs. B2 SC1 if accurate without arcs. Maximum errors 2mm from mid-point and 2° from perpendicular. (ii) Bisector of PRQ with arcs. B2 SC1 if accurate without arcs. Maximum error 2° in line from R. If wrong line and/or angle used treat as misread each time. (c) Correct Shading 2 Dep. on B2 in (b)(i) and (b)(ii). SC1 for ‘correct’ shading but dependent on at least SC1 in (b)(i) and (b)(ii). [10] IGCSE – May/June 2007 0580/0581 03

This question in 0580/31 May/June 2007

4 G D C NOT TO SCALE B 68° E A EG is a diameter of the circle through E,C and G. The tangent AEB is parallel to CD and angle AEC = 68°. Calculate the size of the following angles and give a reason for each answer. (a) Angle CEG = because [2] (b) Angle ECG = because [2] (c) Angle CGE = because [2] (d) Angle ECD = because [2]

8 marks

Mark scheme: ( ) g 4 (a) 22° W1cao Degree symbol not essential throughout question. Tangent (and) radius/ W1 Allow perpendicular for 90° diameter (meet at) 90° (b) 90° W1cao (Angle in a) semi-circle W1 (c) 68° W1ft Ft is180 −( their (a) + their (b)) (Angles in a )triangle W1 or alternate segment (theorem) (=)180° (d) 68° W1cao Alternate or Z (angles) W1 Allow Z correctly placed on the diagram.

This question in 0580/31 Oct/Nov 2008

2 For S T Examiner's 36° Use NOT TO O SCALE R P The points P, R and S lie on a circle, centre O. ROT is a straight line and TS is a tangent to the circle at S. Angle STO = 36°. (a) Write down the size of angle TSO, giving a reason for your answer. Answer(a) Angle TSO = because [2] (b) (i) Calculate the size of angle TOS. Answer(b)(i) Angle TOS = [1] (ii) Show that angle OPR = 63°. Answer(b)(ii) [2] (c) (i) Write down the size of angle PRS. Answer(c)(i) Angle PRS = [1] (ii) Calculate the size of angle PSR. Answer(c)(ii) Angle PSR = [1]

7 marks

Mark scheme: 2 (a) 90° 1 (Angle between) tangent and radius/ diameter 1 dep (b) (i) 54° cao 1 (ii) 1 2 × (180 − 54 ) 2 M1 for using isosceles triangle POR or 180 – 90 – 1/2 (180 – 126) or M1 for using isosceles triangle ROS then or 54/2 followed by triangle PRS (180 – 90 – 27 oe) (c) (i) 90° cao 1 (ii) 27° cao 1

This question in 0580/31 Oct/Nov 2010

5 (a) For C Examiner's D 92° NOT TO Use 140° SCALE 52° A B X In the quadrilateral ABCD, angle BAD = 52°, angle ADC = 140° and angle DCB = 92°. AB is extended to X. (i) Calculate angle CBX. Answer(a)(i) Angle CBX = [2] (ii) The line BY bisects angle CBX. Complete the statement. The lines BY and AD are because [2] (b) T NOT TO O 4x° x° P SCALE U The diagram shows a circle, centre O. PT and PU are tangents to the circle at T and U. Angle TPU = x° and angle TOU = 4x°. Calculate the value of x. Answer(b) x = [3] (c) The exterior angle of a regular polygon is 20°. Calculate the number of sides of the polygon. Answer(c) [2]

9 marks

Mark scheme: 5 (a) (i) 104 2 M1 for 360 – (52 + 140 + 92) implied by 76 (ii) Parallel 1 Dependent on (i) correct Angle YBX = 52° oe 1 Dependent on word parallel already given (b) 36 3 M2 for 360 = 90 + 90 + x + 4x oe (B1 if angle T or U = 90° soi) (c) 18 2 M1 if angle sum = 360 soi or long method

This question in 0580/33 Oct/Nov 2011

7 For H Examiner's Use NOT TO SCALE F C K G A B 117° D E I J The points F, G, H and I lie on a circle, centre C. FG is a diameter and DE is a tangent to the circle at I. DE is parallel to AB and angle GKI = 117°. Complete the following statements. (a) Angle FKI = because [2] (b) Angle FHG = because [2] (c) Angle EIJ = because [2] (d) Angle CIE = because [2]

8 marks

Mark scheme: 7 (a) 63 1 (Angles on a straight) line (add to) 180 1 (b) 90 1 (Angle in a) semi circle 1 (c) 117 1 Corresponding (angles) 1 (d) 90 1 Tangent and radius 1

This question in 0580/33 Oct/Nov 2012

2 (a) For Examiner′s NOT TO Use SCALE 108° 43° p° A B AB is a straight line. Find the value of p. Answer(a) p = … [1] (b) NOT TO 123° SCALE 88° 107° q° Find the value of q. Answer(b) q = … [1] (c) A 48° NOT TO SCALE s° r° D C B DCB is a straight line and AB = AC. Find the values of r and s. Answer(c) r = … s = … [2] (d) For Examiner′s B Use NOT TO 130° SCALE t° A The straight line AB crosses two parallel lines. Find the value of t. Answer(d) t = … [1] (e) B NOT TO SCALE O 124° u° C A A and B lie on a circle, centre O. AC and BC are tangents to the circle. Find the value of u. Answer(e) u = … [2] _____________________________________________________________________________________

7 marks

Mark scheme: 2 (a) 29 1 (b) 42 1 (c) [r =] 66 and [s =] 114 1,1ft Ft is s = 180 – their r (d) 50 1 (e) 56 2 M1 for either angle at A or B indicated as 90 soi IGCSE – May/June 2013 0580 31

This question in 0580/31 May/June 2013

9 For Examiner′s G Use E B C 24° NOT TO SCALE x° O 78° A y° D H F A, B, C and D are points on the circumference of a circle, centre O. EF is a tangent to the circle at A. GH is a straight line through the point A. Angle CBD = 24° and angle OAG = 78°. (a) (i) Write down the mathematical names of lines BC and OA. Answer(a)(i) BC is a … OA is a … [2] (ii) Find the value of x, giving a reason for your answer. Answer(a)(ii) x = … because … … [2] (iii) Find the value of y, giving a reason for your answer. Answer(a)(iii) y = … because … … [3] (b) The diagram shows a regular polygon, centre O. For Examiner′s Use NOT TO SCALE O w° (i) Write down the name of this polygon. Answer(b)(i) … [1] (ii) Find the value of w. Show all your working. Answer(b)(ii) w = … [3] (c) The exterior angle of another regular polygon is 24°. Calculate the number of sides this polygon has. Answer(c) … [2] _____________________________________________________________________________________

13 marks

Mark scheme: 9 (a) (i) Chord 1 Radius 1 (ii) 12 1 Tangent [meets] radius [at] 90 [°] 1 (iii) 66 2 M1 for BCD identified as 90 or 180–24–90 Angles [in] triangle 180 or 1 Angle [in a] semi–circle [= 90] (b) (i) Octagon 1 alternative method (ii) 360 ÷ 8 [= 45] M1 M1 for (8–2) × 180 [=1080] or 6 × 180 [=1080] (180 – their 45) ÷ 2 M1FT M1FT for (their 1080 ÷ 8) ÷ 2 or their 1080 ÷ 16 67.5 A1 A1 for 67.5 (c) 15 2 M1 for 360 / 24

This question in 0580/31 Oct/Nov 2013

4 (a) A regular polygon has 9 sides. For Examiner′s For this polygon, calculate Use (i) the size of one exterior angle, Answer(a)(i) … [2] (ii) the size of one interior angle. Answer(a)(ii) … [1] (b) C w° B 24° y° NOT TO SCALE D O E x° z° A F In the diagram, A, B, C and D are points on the circumference of a circle, centre O. AB is the diameter and EF is a tangent to the circle at A. AB is parallel to DC and angle ACD = 24°. Find (i) w, Answer(b)(i) w = … [1] (ii) x, Answer(b)(ii) x = … [1] (iii) y. Answer(b)(iii) y = … [1] (c) Complete the statement. z = … because … … [2] _____________________________________________________________________________________

8 marks

Mark scheme: 4 (a) (i) 40 2 M1 for 360 ÷ 9 (ii) 140 1FT 180 – their (a)(i) (b) (i) [w =] 90 1 (ii) [x =] 24 1 (iii) [y =] 66 1FT 180 – (their w + their x) (c) [z =] 66 1FT (90 – their x) or their y [Angle between] tangent [and] 1 diameter/radius [=] 90°

This question in 0580/33 Oct/Nov 2013

3 (a) Draw the line of symmetry on the shape below. [1] (b) Write down the order of rotational symmetry of the shape below. Answer(b) … [1] (c) (i) NOT TO 72° SCALE 157° x° Work out the value of x. Answer(c)(i) x = … [1] (ii) 49° NOT TO SCALE y° 54° Work out the value of y. Answer(c)(ii) y = … [2] (d) A NOT TO SCALE 34° O B C AC is a diameter of the circle, centre O. Calculate angle ACB. Answer(d) Angle ACB = … [2] (e) The diagram below shows parts of shape P and shape Q. Shape P is a regular hexagon and shape Q is another regular polygon. The two shapes have one side in common. 100° NOT TO SCALE P Q 100° Find the number of sides in shape Q. Show each step of your working. Answer(e) … [5] __________________________________________________________________________________________

12 marks

Mark scheme: 3 (a) correct mirror line 1 (b) 2 1 (c) (i) 131 1 (ii) 103 2 M1 for 180 – 49 – 54 or 49 + 54 or 77 seen or fully correct method (d) 56 2 M1 for 180 – 90 – 34 or better or indication of angle B = 90 (e) 9 with supporting working 5 M2 for internal angle of P =120 or M1 for 180 – (360 ÷ 6) or (6 – 2) × 180 ÷ 6 M1FT for 360 – their ‘120’ – 100 [= 140] M1FT for 360 ÷ (180 – their ‘140’) if M0 then answer of 9 scores SC2 IGCSE – May/June 2014 0580 32

This question in 0580/32 May/June 2014

9 C NOT TO A SCALE O B The diagram shows a circle with diameter AB and centre O. C is a point on the circumference of the circle. (a) Explain how you know that angle ACB is 90° without having to measure it. Answer(a) … [1] (b) AB = 13 cm and AC = 5 cm. Calculate the length BC. Answer(b) BC = … cm [3] (c) Calculate angle ABC. Answer(c) Angle ABC = … [2]

6 marks

Mark scheme: 9 (a) Angle [in the] semi-circle [equals 1 90U] (b) 12 3 M2 for [BC == √(13 2 – 5 2) or better or M1 for 5 2 + BC 2 = 13 2 or better –1 5 (c) 22.6 2 M1FT for tan their 12 or –1 5 M1 for sin 13 or –1 their 12 M1FT for cos 13

This question in 0580/33 May/June 2014

8 (a) D B C NOT TO 63° SCALE A A, B and C lie on a circle with diameter AC. AC is extended to D and angle BAC = 63°. Work out angle BCD. Give reasons to explain your answer. Answer(a) Angle BCD = … because … … … [4] (b) NOT TO SCALE 6 cm 3cm The diagram shows a circle with radius 3 cm inside a square of side 6 cm. Calculate the shaded area. Answer(b) … cm2 [5] (c) F NOT TO SCALE 45 cm 27 cm H G FGH is a right-angled triangle. Calculate (i) GH, Answer(c)(i) GH = … cm [3] (ii) the perimeter of the triangle, Answer(c)(ii) … cm [1] (iii) the area of the triangle. Answer(c)(iii) … cm2 [2] __________________________________________________________________________________________

15 marks

Mark scheme: 8 (a) 153 2 M1 for 90 + 63 or 180 − (90 + 63) oe or [angle BCA =]27 two correct geometrical reasons 2 B1 for angle [in] semi-circle [is 90] B1 for angles [in a] triangle [sum to] 180 or angles [on a] straight line [sum to] 180 3 2 (b) 14.8 5 M2 for × π × 3 or M1 for π × 32 4 or 14.79 to 14.80 M1 for 6 × 6 or 36 M1 dep for their 6 × 6 – their k × π × 32 (c) (i) 36 3 M2 for 45 2 − 27 2 or better or M1 for 452 = GH2 + 272 or better (ii) 108 1FT (iii) 486 2FT M1FT for 0.5 × 27 × their (c)(i)

This question in 0580/31 May/June 2015

2 (a) Write the mathematical name under each of these triangles. 8 cm 8 cm 8 cm 8 cm 8 cm 12 cm NOT TO SCALE 8 cm 12 cm … … … [3] (b) NOT TO 2 cm SCALE 5 cm 8 cm 12 cm (i) Find the perimeter of this shape. Answer(b)(i) … cm [1] (ii) Find the area of this shape. Give the units of your answer. Answer(b)(ii) … … [3] (c) C NOT TO 6 cm SCALE B 16 cm A In the diagram AB is the diameter of the circle and C is a point on the circumference. AB = 16 cm and BC = 6 cm. (i) Give a reason why angle ACB = 90°. Answer(c)(i) … … [1] (ii) Calculate AC. Answer(c)(ii) AC = … cm [3] (iii) Calculate the shaded area. Answer(c)(iii) … cm2 [5]

16 marks

Mark scheme: 2 (a) equilateral 3 B1 for each isosceles right-angled or scalene (b) (i) 40 1 (ii) 86 2 M1 for 8 × 12 – 2 × 5 oe cm2 1 B1indep for cm2 (c) (i) angle [in a] semi-circle [=90] 1 accept any correct equivalent statement 2 − 6 2 oe or better (ii) 14.8 3 M2 for 16 or M1 for AC2 + 62 = 162 or better (iii) 56.0 to 56.144 5 M2 for π × 82 ÷ 2 oe or M1 for π × 82 M1 for 6 × their (c)(ii) ÷ 2 oe or 44.4[…] M1dep for the area of their semi-circle – the area of their triangle

This question in 0580/33 May/June 2015

7 (a) 25° 98° NOT TO SCALE y° x° The diagram shows three straight lines crossing at a point. (i) Find the value of x. x = … [1] (ii) Work out the value of y. y = … [1] (b) C A 49° NOT TO SCALE 41° B A, B and C are points on the circumference of a circle. Explain why AB must be a diameter of the circle. … … [2] (c) Q 17.8 cm NOT TO SCALE 35° P R PQR is a right-angled triangle. Use trigonometry to calculate PR. PR = … cm [2] (d) K NOT TO 28.9 cm SCALE M L 21.5 cm KLM is a right-angled triangle. Calculate KL. KL = … cm [3]

9 marks

Mark scheme: 7 (a) (i) 25 1 (ii) 57 1 (b) [∠BCA =] 180 – 49 – 41 = 90° B1 B1 Angle [in a ] semicircle PR (c) 14.6 or 14.58… 2 M1 for cos35 = or better 17.8 (d) 19.3 or 19.31… 3 M2 for [KL =] 28.9 2 − 21.5 2 or better or M1 for 28.92 = KL2 + 21.52 or better

This question in 0580/32 May/June 2016

8 B 73° NOT TO SCALE C F d° O 19° e° b° a° A D c° G E A, B, C, D and E are points on the circumference of a circle, centre O. GAF is a tangent to the circle at A. AB is parallel to EC and AB = AD. (a) Write down the mathematical name of triangle ABD. … [1] (b) Find the value of (i) a, a = … [1] (ii) b, b = … [1] (iii) c, c = … [1] (iv) d, d = … [1] (v) e. e = … [2] (c) The diameter, AC, of the circle is 13 cm. Calculate the circumference of the circle. Give your answer correct to 1 decimal place. … cm [3] Question 9 is printed on the next page.

10 marks

Mark scheme: 8 (a) Isosceles 1 (b) (i) 73 1 (ii) 15 1FT FT is 180 – (73 + 19 + their (b)(i)) (iii) 90 1 (iv) 19 1 (v) 71 2 M1 for [angle CAF = ] 90 – 19 or B1 for angle CAF = 90˚ soi (c) 40.8 cao 3 B2 for 40.84….. or M1 for 13π oe seen in the working B1 independent for rounding their circumference correctly if to more than 1 d p

This question in 0580/33 May/June 2016

4 (a) NOT TO SCALE D O A 49° 5.4 cm B C The diagram shows a circle, centre O, with points B and D on the circumference. The line AC touches the circle at B. OB is parallel to DC and angle OAB = 49°. (i) Write down the mathematical name of the line OB. … [1] (ii) Write down the reason why angle ABO is 90°. … … [1] (iii) Find angle AOB. Angle AOB = … [1] (iv) Write down the reason why angle ADC = angle AOB. … [1] (v) Complete the statement using a mathematical word. Triangle AOB is … to triangle ADC. [1] (vi) AB = 5.4 cm Calculate (a) OB, OB = … cm [2] (b) OA, OA = … cm [2] (c) the area of triangle AOB. … cm2 [2] (b) Here is a polygon with 7 sides. Show that the sum of the interior angles of this polygon is 900°. [1]

12 marks

Mark scheme: 4(a)(i) Radius 1 4(a)(ii) [Angle between] tangent [and] 1 radius 4(a)(iii) 41 1 4(a)(iv) Corresponding [angles] 1 4(a)(v) Similar 1 4(a)(vi)(a) 6.21 or 6.211 to 6.212 2 OB M1 for tan 49 = or better 5.4 4(a)(vi)(b) 8.23 or 8.229 to 8.231 2FT 5.4 M1 for cos 49 = or better OA or for 5.42 + their (vi)(a)2 or better 4(a)(vi)(c) 16.8 or 16.76 to 16.77 2FT M1 for their (vi)(a) × 5.4 ÷ 2 4(b) 5 × 180 1

This question in 0580/31 May/June 2017

9 B A NOT TO O SCALE C A, B and C are points on the circumference of a circle, centre O. (a) Write down the mathematical name for (i) the straight line AC, … [1] (ii) the straight line AB. … [1] (b) Give a geometrical reason why angle ABC = 90°. … [1] (c) AB = 20 cm and AC = 52 cm. (i) Use trigonometry to calculate angle BAC. Angle BAC = … [2] (ii) Show that BC = 48 cm. [2] (iii) Work out the area of triangle ABC. … cm2 [2] (iv) Work out the total shaded area. … cm2 [3]

12 marks

Mark scheme: 9(a)(i) Diameter 1 9(a)(ii) Chord 1 9(b) Angle [in] semi-circle [is 90] 1 9(c)(i) 67.4 or 67.38….. 2 20 M1 for cos [ A = ] or better 52 M2 2 − 20 2 M1 for 20 2 + ( BC ) 2 = 52 29(c)(ii)  ( BC ) 2 = 52   9(c)(iii) 480 2 M1 for 0.5 × 20 × 48 or better 9(c)(iv) 582 or 581.8 to 582.0 3 2  1   52  M1 for × π ×  or better    2   2  M1 for their 338π – their (c)(iii)

This question in 0580/31 May/June 2018

2 (a) Draw all the lines of symmetry on each shape. [4] (b) The diagram shows an isosceles triangle and a straight line AB. NOT TO SCALE 48° x° y° A B Find the value of x and the value of y. x = … y = … [2] (c) Find the size of one interior angle of a regular decagon. … [3] (d) P C k° NOT TO B SCALE j° O 37° R A The points A, B and C lie on the circumference of a circle, centre O. PBR is a tangent to the circle and angle BAC = 37°. Find the value of j and the value of k. j = … k = … [3] (e) A NOT TO SCALE 18 cm B D E C ABC and ADE are isosceles triangles, each with perpendicular height 18 cm. BC = 35 cm and DE = 27 cm. Find the total area of the two shaded parts of the diagram. … cm2 [3]

15 marks

Mark scheme: 2(a) [star] 6 correct lines only 2 B1 for 3 correct lines [rectangle] 2 correct lines only 2 B1 for only 1 correct line or 2 correct lines and 1 wrong 2(b) [x = ] 66 2 B1 for one correct angle [y = ] 114 or for both angles adding to 180 2(c) 144 3 M1 for 360 ÷ 10 soi by 36 M1 for [y = ] 180 – their 36 If 0 scored SC2 for a correct interior angle of a regular polygon (greater than 90), providing not from wrong working 2(d) [j = ] 53 3 B2 for one correct angle [k = ] 37 or B1 for 90 seen, marked on drawing in the correct place or for both angles adding to 90 2(e) 72 3 M1 for (18 × 35) ÷ 2 implied by 315 M1 for (18 × 27) ÷ 2 implied by 243

This question in 0580/33 May/June 2018

9 A NOT TO 35° SCALE O E B C D A, B and C are points on the circumference of the circle, centre O. The straight line DE touches the circle at B. (a) Write down the mathematical name for the line DE. … [1] (b) On the circle, draw a radius. [1] (c) Complete the following statements. (i) Angle ABD = … because … … [2] (ii) Angle ACB = … because … … [2] (d) AB = 9 cm. (i) Calculate the area of the circle. Give the units of your answer. … … [3] (ii) Calculate BC. BC = … cm [2]

11 marks

Mark scheme: 9(a) Tangent 1 9(b) Radius drawn on circle 1 9(c)(i) 90 2 B1 for each radius [and] tangent [at 90] 9(c)(ii) 90 2 B1 for each angle [in a] semicircle [= 90] 9(d)(i) 63.6 or 63.61 to 63.63 2 M1 for 4.52 × π cm2 1 9(d)(ii) 5.16 or 5.162 … 2 BC M1 for sin 35 = or better 9

This question in 0580/33 Oct/Nov 2018

4 (a) A NOT TO a° SCALE 118° B C D ABC is an isosceles triangle. BCD is a straight line. Find the value of a. a = … [2] (b) Find the size of one interior angle of a regular 10-sided polygon. … [3] (c) E NOT TO y° SCALE F O x° 58° J G H The points E, F and G lie on the circumference of a circle, centre O. JGH is a tangent to the circle. Find the value of x and the value of y. x = … y = … [2] (d) G E 28° C A NOT TO B 67° SCALE D F In the diagram AG and AF are straight lines. Lines BC and DE are parallel. Find angle CED and give a reason for your answer. Angle CED = … because … [2] (e) R NOT TO SCALE 28 cm P Q 21 cm Calculate PR. PR = … cm [2]

11 marks

Mark scheme: 4(a) 56 2 M1 for 180 – 118 soi by 62 4(b) 144 3 M2 for 180 – (360 ÷ 10) oe M1 for 360 ÷ 10 soi by 36 4(c) 32 2 B1 for each 58 or for their x + their y = 90 or angle F marked as 90 4(d) 28 alternate 2 B1 for each 4(e) 35 2 M1 for 212 + 282 or better

This question in 0580/31 May/June 2019

7 (a) A triangle is isosceles. One of its angles is 96°. Find the other two angles. … and … [1] (b) NOT TO SCALE 45° 6x° 5x° 3x° Find the value of x. x = … [4] (c) Work out the size of one interior angle of a regular polygon with 20 sides. … [3] (d) C 7.4 m 2.3 m NOT TO SCALE A B The diagram shows a right-angled triangle ABC. Calculate the length of AB. AB = … m [2] (e) The diagram shows the vertices of a triangle lying on the circumference of a circle with centre O. 61° NOT TO SCALE O b° Find the value of b. Give a reason for your answer. b = … because … [2]

12 marks

Mark scheme: 7(a) 42, 42 1 7(b) 22.5 4 B3 for 14 x = 315 or M2 for 45 + 3 x + 5 x + 6 x = 360 oe or M1 for 45 + 3 x + 5 x + 6 x oe or 14x If 0 scored and 45 + bx = 360 or better seen then 360 − 45 SC1 for x = oe b OR 360 − 45 B3 for 14 or B1 for 14 and B1 for 360 − 45 oe 7(c) 162 3 360 ( 20 − 2 )180 M2 for 180 − oe or oe 20 20 360 or M1 for or ( 20 − 2 ) 180 20 7(d) 7.75 or 7.74[9…] 2 M1 for  x 2 =  7.4 2 + 2.3 2 or better   7(e) 29 2 B1 for each angle [in a] semicircle [is] 90°

This question in 0580/32 May/June 2019

2 (a) B A O C In the diagram, A, B and C are points on the circle, centre O. (i) On the diagram, draw a chord. [1] (ii) Explain why angle ABC is 90°. … [1] (b) The length of the edge of a cube is 8 cm. Calculate the surface area of this cube. … cm2 [2] (c) A cuboid measures 5 cm by 4 cm by 2 cm. (i) Calculate the volume of this cuboid. Give the units of your answer. … … [3] (ii) On the 1 cm2 grid, draw an accurate net of this cuboid. One face has been drawn for you. [3]

10 marks

Mark scheme: 2(a)(i) Chord correctly drawn 1 2(a)(ii) Angle [in a] semicircle [is 90º] 1 2(b) 384 2 M1 for 8 × 8[× 6] 2(c)(i) 40 2 M1 for 5 × 4 × 2 cm3 1 2(c)(ii) Correct net 3 B2 for 4 more correct faces in correct position B1 for 2 or 3 more correct faces in correct position

This question in 0580/31 Oct/Nov 2019

7 (a) NOT TO w° SCALE 118° The diagram shows an isosceles triangle and a straight line. Work out the value of w. w = … [2] (b) E F NOT TO SCALE A 31° x° B y° D C ABCD is a rectangle. AE is parallel to DBF. Find the value of x and the value of y. x = … y = … [2] (c) B NOT TO SCALE a° 53° A C A, B and C are points on a circle. AC is a diameter of the circle. Find the value of a. a = … [2] (d) NOT TO SCALE P Two regular octagons and a square meet at point P. Show, by calculation, that the three interior angles at P add up to 360°. [3]

9 marks

Mark scheme: 7(a) 56 2 M1 for 180 – 118 oe or 180 – 2 × their 62 oe 7(b) [x =] 31 2 B1 for each [y =] 121 or M1 for their y = 90 + their x 7(c) 37 2 B1 for the angle ABC marked as 90 or M1 for 180 – (90 + 53) oe 7(d) 360 M2 360 180 – or (8 – 2) × 180 ÷ 8 M1 for or (8 – 2) × 180 8 8 135 + 135 + 90 [= 360] A1

This question in 0580/31 Oct/Nov 2020

8 (a) A P NOT TO SCALE 53° O x° B Q P and Q are points on the circle, centre O. APB is a tangent to the circle at P. (i) Write down the mathematical name for the line PQ. … [1] (ii) Explain why angle OPB is 90°. … [1] (iii) Find the value of x. x = … [3] (b) a° b° 65° 48° NOT TO SCALE c° The diagram shows two parallel lines and two straight lines. (i) Find the value of a. Give a reason for your answer. a = … because … [2] (ii) Find the value of b. Give a reason for your answer. b = … because … [2] (iii) Find the value of c. c = … [2] (c) NOT TO 11.8 cm SCALE 34° x cm Calculate the value of x. x = … [3] Question 9 is printed on the next page.

14 marks

Mark scheme: 8(a)(i) Chord 1 8(a)(ii) Angle [between] tangent [and] radius [is] 1 90° 8(a)(iii) 106 3 M1 for 90 – 53 soi by 37 M1 for 180 – 2 × their angle OPQ 8(b)(i) 48 2 B1 for 48 corresponding 8(b)(ii) 67 2 B1 for 67 angles [on a straight] line [add to] 180 8(b)(iii) 115 2 B1 for 65 or 115 seen in correct position 8(c) 17.5 or 17.49… 3 11.8 M2 for [x =] or [x =] 11.8 tan 56 tan34 11.8 or M1 for tan [34] = x x or tan 56 = 11.8

This question in 0580/33 Oct/Nov 2020

9 (a) On the 1cm2 grid, draw one rectangle that has • a perimeter of 22 cm and • an area of 24cm2. [2] (b) 94° 127° NOT TO SCALE x° 298° Work out the value of x. Write down the two geometrical properties needed to find x. 1 … 2 … x = … [4] (c) P Draw a tangent to the circle at point P. [1] (d) The exterior angle of a regular polygon is 24°. Work out the number of sides of this polygon. … [1] (e) D 13.6 cm x cm NOT TO SCALE 41° A B C 7.4 cm Calculate the value of x. x = … [5]

13 marks

Mark scheme: 9(a) 8 cm by 3 cm rectangle drawn 2 B1 for rectangle with perimeter 22 or for rectangle with area 24 If no rectangle drawn, SC1 for showing calculations that go together and satisfy either area=24 or perimeter=22 9(b) 77 with two correct properties 4 B2 for 77 or M1 for 360 − 298 B1 for angles [at a] point [add to] 360 B1 for angles [in a] quadrilateral [add to] 360 9(c) Ruled tangent drawn 1 9(d) 15 1 9(e) 17.4 or 17.39… 5 M2 for 13.6 2 − 7.4 2 oe or better 2 2 2 or M1 for 7.4 + ( BD ) = 13.6 oe and theirBD M2FT for x = sin 41 theirBD or M1FT for sin41 = oe or better x BD or B1 for stating sin41 = or better x

This question in 0580/32 Feb/March 2021

8 (a) B NOT TO 6 cm SCALE A 10 cm O C A, B and C lie on a circle, centre O, diameter AC. (i) Complete this statement. Angle ABC is 90° because … [1] (ii) Work out the area of triangle ABC. … cm2 [2] (iii) Work out AC. AC = … cm [2] (b) Make r the subject of the formula A = rr2 . r = … [2] (c) NOT TO SCALE The diagram shows a circle inside a square. The circle touches the four sides of the square. The area of the square is 81 cm 2. Calculate the shaded area. … cm2 [4] Question 9 is printed on the next page.

11 marks

Mark scheme: 8(a)(i) Angle [in a] semicircle 1 8(a)(ii) 30 2 6 × 10 M1 for 2 8(a)(iii) 11.7 or 11.66… 2 2 2 2 M1 for [x =] 6 +10 or better 8(b) A 2 A 2 [ r = ] M1 for = r or A = π × r π π 8(c) 17.4 or 17.37 to 17.38… 4 2  81  M3 for 81 −π  oe    2  OR M1 for 81  their 81 2 and M1 for π     2   81 2 and M1 for 81 − their π     2 

This question in 0580/31 May/June 2021

6 (a) NOT TO SCALE B O A 52° C AB is the diameter of a circle, centre O. C is a point on the circle and angle BAC = 52°. Find angle ABC. Angle ABC = … [2] (b) The diagram shows the positions of town A, town B and town C. North B NOT TO SCALE North A C The bearing of town B from town A is 042°. The bearing of town C from town A is 146°. (i) Find angle BAC. Angle BAC = … [2] (ii) Find the bearing of town A from town B. … [2] (c) A NOT TO x° SCALE 117° B C D Triangle ABC is isosceles with AB = AC. BCD is a straight line and angle ACD = 117°. Find the value of x. x = … [3]

9 marks

Mark scheme: 6(a) 38 2 B1 for angle ACB marked as 90° or M1 for 90 – 52 oe 6(b)(i) 104 2 M1 for 146 – 42 oe or B1 for 42 or 146 correctly marked on diagram 6(b)(ii) 222 2 M1 for 180 + 42 oe 6(c) 54 3 M1 for 180 – 117 or 63 M1 for 180 – 2 × their 63 or 117 – 63

This question in 0580/33 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

8 (a) P NOT TO SCALE 114° Q R S In the diagram, PQ = PR and QRS is a straight line. (i) Write down the mathematical name of triangle PQR. … [1] (ii) Work out angle QPR. Angle QPR = … [3] (b) C F NOT TO SCALE D O A 68° E B In the diagram, D, E and F are points on a circle, centre O. AB is a tangent to the circle at E. Lines AB and CD are parallel and angle BED = 68° . (i) Find angle CDE and give a reason for your answer. Angle CDE = … because … … [2] (ii) Find angle DEF and give a reason for your answer. Angle DEF = … because … … [2] (iii) Work out angle EFD. Write down the two further geometrical properties needed to find angle EFD. Angle EFD = … 1. … 2. … [3] (c) O NOT TO 60° SCALE 7.5 cm 7.5 cm Q P POQ is a sector of a circle, centre O and radius 7.5 cm. The sector angle is 60°. Calculate the length of the arc PQ. PQ = … cm [2]

13 marks

Mark scheme: 8(a)(i) Isosceles 1 8(a)(ii) 48 3 M2 for 180 – 2 × (180 – 114) oe or M1 for 180 – 114 or B1 for PQR = 66 or PRQ = 66 8(b)(i) 68 2 B1 for each Alternate [angles] 8(b)(ii) 22 2 B1 for each Angle [between] tangent [and] radius [=] 90° 8(b)(iii) 68 with two correct reasons 3 B1 for each Angle [in a] semicircle [=] 90° Angles [in a] triangle add to 180° 8(c) 7.85 or 7.86 or 7.853 to 7.855 2 60 M1 for × 2π × 7.5 oe 360

This question in 0580/33 May/June 2022

6 (a) Write down the mathematical name of this solid. … [1] (b) B C A D 104° NOT TO SCALE x° E The diagram shows triangle BCE and a straight line ABCD. BE = CE and angle ABE = 104°. Find the value of x. x = … [2] (c) Work out the size of one interior angle of a regular polygon with 15 sides. … [2] (d) B y° O NOT TO A 38° SCALE C A, B and C are points on a circle, centre O. (i) Write down the mathematical name of the line BC. … [1] (ii) Draw a tangent to the circle at point B. [1] (iii) The area of the circle is 245.5 cm 2. Calculate AB. AB = … cm [3] (iv) Find the value of y. y = … [2]

12 marks

Mark scheme: 6(a) Cylinder 1 6(b) 28 2 M1 for 180 – 104 oe 6(c) 156 2 360 (15 − 2 )180 M1 for 180 – oe or oe 15 15 6(d)(i) Chord 1 6(d)(ii) Tangent drawn at point B 1 6(d)(iii) 17.7 or 17.67 to 17.68 3 M2 for [2] 245.5  π oe or M1 for 245.5 ÷ π oe 6(d)(iv) 52 2 M1 for 180 – 90 – 38 oe or B1 for [angle ACB =] 90 correctly identified

This question in 0580/31 Oct/Nov 2022

7 (a) D F NOT TO SCALE E O C B 28° A The diagram shows a circle, centre O, with points B, D and E on the circumference. AOEF is a straight line. The straight line AC touches the circle at B. (i) Write down the mathematical name for (a) line BOD … [1] (b) line ABC. … [1] (ii) Write down the two geometrical reasons why angle AOB is 62°. … and … [2] (iii) Give the geometrical reason why angle DOE is also 62°. … [1] (iv) (a) Find angle DEB. Angle DEB = … [1] (b) Find angle ODE. Angle ODE = … [2] (c) Find angle BEF. Angle BEF = … [2] (b) Write down two geometrical properties that show that a polygon is regular. … and … [2] (c) Work out the interior angle of a regular 10-sided polygon. … [2]

14 marks

Mark scheme: 7(a)(i)(a) Diameter 1 7(a)(i)(b) Tangent 1 7(a)(ii) Angle between tangent and radius = 90 2 B1 for each Angles in a triangle add to 180 7(a)(iii) Opposite angles are equal 1 7(a)(iv)(a) 90 1 7(a)(iv)(b) 59 2 M1 for (180 – 62) ÷ 2 oe 7(a)(iv)(c) 149 2 B1 for OEB = 31 or B1FT for 180 – their a + their b or 298 – (their a + their b) 7(b) Equal sides 2 B1 for each Equal angles 7(c) 144 2 M1 for 180 – (360 ÷ 10) oe (10  2)  180 or oe 10

This question in 0580/31 May/June 2023

7 (a) Write down the mathematical name for this solid. … [1] (b) x (i) Measure the size of angle x. … [1] (ii) Write down the mathematical name for this type of angle. … [1] (c) A NOT TO SCALE O C B Points A, B and C lie on the circle, centre O. AB = 11 cm and BC = 5 cm . (i) Give a geometrical reason why angle ACB is 90°. … [1] (ii) Calculate the circumference of the circle. … cm [2] (iii) Show that AC is 9.8 cm, correct to 2 significant figures. [3] (d) The surface area of a sphere is 250 cm2. Calculate the radius of the sphere. [The surface area, A, of a sphere with radius r is A = 4 r r 2 .] … cm [3]

12 marks

Mark scheme: 7(a) Cylinder 1 7(b)(i) 137 1 7(b)(ii) Obtuse 1 7(c)(i) Angle in a semicircle is 90º 1 7(c)(ii) 34. 6 or 34.55 to 34.562 2 M1 for 11 × π 7(c)(iii) 112 – 52 = AC2 M2 M1 for 52 +(…)2 = 112 96 = 9.79…or 9.80 A1 7(d) 4.46 or 4.460 … 3 250 M2 for 4 250 or M1 for 4

This question in 0580/33 Oct/Nov 2024

18 (a) D C Points C and D lie on the circle. Write down the mathematical name for the line CD. … [1] (b) The diagram shows a circle with centre O. O Points K, L and M lie on the circumference of the circle. Draw triangle KLM so that angle KLM = 90° . [1]

2 marks

Mark scheme: 18(a) Chord 1 18(b) Correct triangle drawn 1

This question in 0580/33 May/June 2025

20 (a) Work out the size of one interior angle of a regular 8-sided polygon. … [2] (b) D NOT TO C SCALE x° B 71° O E A A, B and C lie on the circumference of the circle, centre O. AB is a diameter. DBE is a tangent to the circle at B. Find the value of x. Give a geometrical reason for your answer. … because … … [2]

4 marks

Mark scheme: 20(a) 135 2 ( 8 − 2 )180 M1 for 180 – 360 ÷ 8 or oe 8 20(b) 19 2 B1 for each Angle between tangent and radius = 90

This question in 0580/31 Oct/Nov 2025