Pythagoras’ theorem — Paper 3 · IGCSE Mathematics 0580

5 For A X B Examiner's Use NOT TO 10 cm SCALE 55º D 18 cm C The diagram shows a rectangular tile ABCD which has a shaded triangle DXB. DC = 18 centimetres, BC = 10 centimetres and angle ADX = 55°. (a) Calculate the area of triangle BDC. Answer(a) cm2 [2] (b) Calculate the length of AX. Answer(b) cm [2] (c) Calculate the shaded area. Answer(c) cm2 [3] (d) Calculate the length of BD. Answer(d) cm [2]

9 marks

Mark scheme: 5 (a) 90 2 M1 for 0.5 × 18 × 10 (b) 14.3 art 2 M1 for 10 × tan 55oe (c) 18.5 to 18.6 3 M1 for 0.5 × 10 × their (b) or M1 18 – their (b) 1 M1 x 10 x their BX 2 M1 for Their (a) – (0.5 × 10 × their (b)) (d) 20.6 art 2 M1 for √( 182 + 102) oe 9

This question in 0580/31 Oct/Nov 2006

6 (a) Write down the name of a polygon with 8 sides. For Examiner's Use Answer(a) [1] (b) Find the size of the interior angle of a regular polygon with 8 sides. Answer(b) [2] (c) A regular 8-sided polygon, centre O, and side 8 cm, is shown below. M is the mid-point of the side AB. F E NOT TO SCALE G D O H C A M B 8 cm (i) Show that OM = 9.66 cm correct to 3 significant figures. Answer (c)(i) [3] (ii) Calculate the area of the triangle AOB. For Examiner's Use Answer(c)(ii) cm2 [2] (iii) Calculate the area of the polygon. Answer(c)(iii) cm2 [1] (d) The polygon forms the cross-section of a box. The box is a prism of height 12 cm. Calculate the volume of the box. Answer(d) cm3 [1] (e) The box contains 200 toffees in the shape of cuboids, 3 cm by 2 cm by 2 cm. Calculate (i) the total volume of the 200 toffees, Answer(e)(i) cm3 [2] (ii) the percentage of the volume of the box not filled by the toffees. Answer(e)(ii) % [3]

15 marks

Mark scheme: 6 (a) Octagon 1 (b) 135 2 M1 for 180 − (360 ÷ 8) oe (c) (i) Angle OAB = their (b)/2 or W1ft 67.5 or 22.5 correct values, angle AOM = 90 − their (b)/2 4 × tan ‘67.5’ or 4 ÷ tan ‘22.5’ M1 9.656… or 9.66 A1cao Dep on W1 and M1 (ii) 38.6 to 38.64 2 M1 for 0.5 × 8 × 9.66 (iii) 308.8 to 309.12 1ft Their (c) (ii) × 8 (d) 3705.6 to 3709.44 or 3710 1ft Their (c) (iii) × 12 (e) (i) 2400 2cao M1 for 3 × 2 × 2 × 200 (ii) 35.2(3…) to 35.3(0…) 3cao M1 for their ((d) − (e) (i)) soi. (d)− (e)(i) M1 for (d) × 100 (e)(i) Or M2 for (1 (d) ) × 100 SC1 for Answer 64.7 to 64.77

This question in 0580/31 May/June 2009

4 For D C Examiner's 7 cm NOT TO Use SCALE M X L A 7 cm B In the diagram, ABCD is a square of side 7 cm. BLC and DMA are equilateral triangles. (a) Find the perimeter of the shape ABLCDM. Answer(a) cm [1] (b) (i) Write down the size of angle CBL. Answer(b)(i) Angle CBL = [1] (ii) Calculate the length of LX. Answer(b)(ii) LX = cm [2] (c) (i) Calculate the area of triangle BLC. Answer(c)(i) cm2 [2] (ii) Calculate the area of the shape ABLCDM. Answer(c)(ii) cm2 [2]

8 marks

Mark scheme: 4 (a) 42 1 (b) (i) 60° 1 x x (ii) 6.06(217…) 2 M1 ft for = cos 30 or = sin 60 or 7 7 x 5.3 = tan 60 or = tan 30 or better 5.3 x (c) (i) 21.2 to 21.4 ft 2ft M1 for 12 × 7 × their (b)(ii) oe (ii) 91.4 to 91.7 ft 2ft M1 ft 7 × 7 + 2 (their (c)(i)) or B1 for 49 5 1 3 75

This question in 0580/31 Oct/Nov 2010

9 For B D Examiner's Use 2.7 cm NOT TO cm 2.7 SCALE C A E (a) In the diagram above, AB and ED are vertical. The diagram is symmetrical about a line through C parallel to AB. Angle BCD = 90° and BC = CD = 2.7 cm. (i) Calculate BD. Answer(a)(i) BD = cm [2] (ii) Complete the statement. Triangle BCD is right-angled and [1] (iii) Find the size of angle ABC. Answer(a)(iii) Angle ABC = [1] For Examiner's Use Diagram 1 Diagram 2 Diagram 3 Diagram 4 (b) The pattern of diagrams above is continued by adding more lines and dots. (i) On the grid, draw diagram 4. [1] (ii) Complete the table below. Diagram 1 2 3 4 5 Number of lines 4 7 [2] (c) How many lines will there be in (i) Diagram 9, Answer(c)(i) [1] (ii) Diagram n? Answer(c)(ii) [2] (d) The number of lines in Diagram r is 76. Find the value of r. Answer(d) r = [2] (e) Write down an expression, in terms of n, for the number of dots in Diagram n. Answer(e) [1]

13 marks

Mark scheme: 9 (a) (i) 3.82 art 2 M1 for 2.72 + 2.72 or better 27 or sin 45 = or better BD 27 or cos 45 = or better BD (ii) Isosceles 1 (iii) 45 cao 1 (b) (i) Diagram 4 1 (ii) 10, 13, 16 2 B1 for 2 correct or difference of 3 seen between diagram 4 and diagram 5 in table (c) (i) 28 1 (ii) 3n + 1 oe 2 B1 for pn + 1 (p ≠ 0) or 3n + q (d) 25 2ft M1 for 76 = their (c)(ii) (if linear) (e) 3n + 2 oe 1ft ft their (c)(ii) + 1 (must be a linear expression)

This question in 0580/32 Oct/Nov 2010

4 For North Examiner's Use North NOT TO C SCALE 75 m B 200 m North A Dariella walks 200 m from A to B. She then turns through 90° and walks 75 m from B to C. (a) Calculate (i) the distance AC, Answer(a)(i) m [2] (ii) angle CAB. Answer(a)(ii) Angle CAB = [2] (b) The bearing of B from A is 065°. Find the bearing of (i) C from A, Answer(b)(i) [1] (ii) A from C, Answer(b)(ii) [1] (iii) C from B. Answer(b)(iii) [2]

8 marks

Mark scheme: 4 (a) (i) 214 (213.6…) 2 M1 for 752 + 2002 (ii) 20.6 or (20.55 – 20.56) 2 M1 for tan = 75/200 or sin = 75/their (i) or cos = 200/their (i) (b) (i) (0)44 ((0)44.4…) 1ft B1 65 – their (a)(ii) if < 65 (ii) 224 (224.4…) 1ft 180 + their (b)(i) (iii) 335 2 B1 for 65 below B or 25 above B, may be on diagram

This question in 0580/33 Oct/Nov 2010

10 For Examiner's Use x m 2 m 10 m NOT TO SCALE 5 m The diagram shows a ramp in the form of a triangular prism. The cross-section is a right-angled triangle of length 5 m and height 2 m. (a) Find the value of x. Give your answer correct to 1 decimal place. Answer(a) x = [3] (b) Find the area of the cross-section. Answer(b) m2 [2] (c) The ramp is 10 m long. Calculate the volume of the ramp. Answer(c) m3 [1] (d) Calculate the total surface area of all five faces of the ramp. For Examiner's Use Answer(d) m2 [3] (e) Each face of the ramp is painted. Paint costs $2.25 per square metre. Calculate the total cost of the paint. Answer(e) $ [1] Question 11 is printed on the next page.

10 marks

Mark scheme: 10 (a) 5.4 cao 3 M1 for 22 + 52 (= x2) implied by 29 A1 5.38(51..) or 29 or 5.39 B1 indep for rounding their answer to 1 decimal place (b) 5 2 M1 for 0.5 × 5 × 2 oe (c) 50 1ft 10 × their (b) (d) 134 3ft M2 for 2 × their (b) + 10 × their (a) + 2 × 10 + 5 × 10 or better M1 for any 3 faces correct (e) 301.5(0) 1ft Their (d) × 2.25

This question in 0580/31 Oct/Nov 2011

6 For E NOT TO Examiner's SCALE Use F 16 cm D C 24 cm A 30 cm B The diagram shows a wedge in the shape of a triangular prism. AB = 30 cm, AF = 16 cm and BC = 24 cm. Angle BAF = 90°. (a) Calculate (i) the area of triangle ABF, Answer(a)(i) cm2 [2] (ii) the volume of the wedge. Answer(a)(ii) cm3 [1] (b) (i) Calculate BF. Answer(b)(i) cm [2] (ii) 1.6 cm NOT TO SCALE A coin with diameter 1.6 cm is rolled down the sloping surface of the wedge. It travels in a straight line parallel to BF, starting on FE and ending on BC. Calculate the number of complete turns it makes. Answer(b)(ii) [3] (c) On the grid, complete the net of the wedge. For The base and one of the triangles have been drawn for you. Examiner's Use Each square on the grid represents a square of side 4 centimetres. [3] (d) Calculate the surface area of the wedge. Answer(d) cm2 [3]

14 marks

Mark scheme: 6 (a) (i) 240 2 M1 for 0.5 × 30 × 16 (ii) 5760 1ft ft is (a)(i) × 24 (b) (i) 34 2 M1 for (FB 2) = 16 2 + 30 2 (ii) 6 3 M1 for (circumference) = 1.6 × π M1 dep their (b)(i) ÷ their 1.6π (6.76 implies M1, M1) If 0 scored either SC1 for their (b)(i) ÷ 3.2 × π and then SC1 for truncating correctly If M1 or still 0 scored then SC1 for truncating correctly any number with at least 1 decimal place (c) 6 by 4 rectangle above 1 6 by their 8.5 rectangle below 1ft ft (b)(i) ÷ 4 Correct triangle on AB 1 1 1 (d) 2400 3cao M2 for × 30 × 16 + × 30 × 16 + 16 × 24 + 2 2 30 × 24 + their 34 × 24 (M1 for any 3 areas) If 0, SC2 for 150 or SC1 for 120 (3 rectangles) or SC1 for 30 (2 triangles)

This question in 0580/32 Oct/Nov 2011

5 For B 20 m C Examiner's Use NOT TO SCALE 15 m A 32 m D The diagram shows a plot of land, ABCD, in the shape of a trapezium. (a) Show that CD = 19.2 m, correct to 1 decimal place. Answer(a) [2] (b) A fence is built around the perimeter of the plot of land. The cost of the fence is $35 for each metre. Calculate the total cost of the fence. Answer(b) $ [2] (c) Calculate the area of the plot of land. Give your answer in square metres. Answer(c) m2 [2] (d) A house is built on the plot of land. For The area of the plot is divided in the ratio house : grounds = 3 : 7 . Examiner's Use Calculate the area of the grounds. Answer(d) m2 [2] (e) (i) In the space below, make a scale drawing of the plot of land. Use a scale of 1 centimetre to represent 4 metres. The side AB has been drawn for you. B A [2] (ii) Measure angle ADC. Answer(e)(ii) Angle ADC = [1] (iii) Use your diagram to find the actual length BD in metres. Answer(e)(iii) BD = m [1]

12 marks

Mark scheme: 5 (a) (CD2 =) (32 – 20)2 + 152 oe M1 (CD =) 369 = 19.20 to 19.21 A1 A0 for 19.2 alone. 2 M1 for 20 + 15 + 32 + 19.2(1) [implied by (b) 3017 86.2(1)] Or M1 for (20 × 35) + (15 × 35) + (32 × 35) + (19.2(1) × 35) 2 M1 for (20 + 32) × 15 ÷ 2 oe (c) 390 2ft M1 for ‘their (c)’ × 7 ÷ 10 (d) 273 2 B1 for C or D correctly positioned (e) (i) trapezium constructed BC = 5 cm, AD = 8 cm Both 90o to AB 1ft (ii) 49 – 53° 1ft (iii) 34.4 – 36.4 m

This question in 0580/31 Oct/Nov 2012

8 For D Examiner's Use NOT TO SCALE 42° E C F H 8.5 m 6 m G 2 m A 12 m B The diagram shows a house, built on level ground. ABCE is a rectangle with AB = 12 m and BC = 8.5 m. CDE is an isosceles triangle. (a) Use trigonometry to calculate DF. Answer(a) DF = m [2] (b) Calculate the area of triangle CDE. Answer(b) m2 [2] (c) A ladder, GH, of length 6 m, leans against the house wall. The foot of the ladder is 2 m from this wall. Calculate AH. Answer(c) AH = m [3] (d) This diagram shows the plan of the driveway to the house. For Examiner's Use HOUSE 12 m NOT TO SCALE 18 m 3 m 14 m Work out the perimeter of the driveway. Answer(d) m [2] (e) The driveway is made from concrete. The concrete is 15 cm thick. Calculate the volume of concrete used for the driveway. Give your answer in cubic metres. Answer(e) m3 [4]

13 marks

Mark scheme: 8 (a) 5.4(0) 2 M1 tan 42 = DF/6 or better (b) 32.4 2ft M1 12 × their 5.4 ft 2 (c) 5.66 3 M2 6 2 − 2 2 or better (accept √32 or 5.65) or M1 62 − 22 or better (accept 32 ) (d) 64 2 M1 12 + 18 + 14 + 3 + 2 + 15 (e) 33.3 cao 4 M1 (12 × 18) + (their (2) × 3) oe and A1 222 and M1 their 222 ft × 0.15

This question in 0580/33 Oct/Nov 2012

7 For Examiner′s North Use B NOT TO SCALE 27 km A 82 km C The diagram shows the positions of three towns A, B and C. B is 27 km north of A and the distance between A and C is 82 km. (a) Calculate BC. Answer(a) BC = … km [2] (b) Write down the three fi gure bearing of C from A. Answer(b) … [1] (c) (i) Use trigonometry to calculate angle ABC. Answer(c)(i) Angle ABC = … [2] (ii) Work out the bearing of C from B. Answer(c)(ii) … [1] (d) (i) Calculate the area of triangle ABC. For Examiner′s Use Answer(d)(i) … km2 [2] (ii) The land forming the triangle ABC is valued at $8400 for each square kilometre. Calculate the value of this land. Answer(d)(ii) $ … [1] _____________________________________________________________________________________

9 marks

Mark scheme: 7 (a) 86.3 or 86.33075….. 2 M1 for [BC =] 27 2 + 82 2 or 729+ 6724 or 7453 (b) 090 cao 1 (c) (i) 71.8 or 71.77492….. 2 M1 for tan [x=] (82÷27) or better oe (ii) 108.2 or 108 1ft (d) (i) 1107 2 M1 for 27×82÷2 or better, imp by 1110 (ii) 9 298 800 1ft

This question in 0580/31 May/June 2013

6 Finn is going camping. For Examiner′s The diagram shows his tent. Use 2.5 m B NOT TO SCALE 1.5 m 1.2 m A M C ABC is an isosceles triangle. M is the midpoint of AC. AB = 1.5 m and BM = 1.2 m. (a) Show that AM = 0.9 m. Answer(a) [2] (b) Use trigonometry to calculate angle ABM. Answer(b) Angle ABM = … [2] (c) The tent is a prism of length 2.5 m. For Examiner′s The area of triangle ABC is 1.08 m2. Use Calculate the volume of the tent. Give the units of your answer. Answer(c) … … [2] (d) Calculate the surface area of the tent, including the base. Answer(d) … m2 [3] _____________________________________________________________________________________

9 marks

Mark scheme: 6 (a) AM2 + 1.22 = 1.52 or [AM2] = 1.52 – 1.22 M1 [AM=] √ (1.52 – 1.22) or √( 2.25 – 1.44) or √0.81 M1dep 2.1 (b) 36.9 or 36.87 or 36.8[6 … ] 2 M1 for cos[ABM] = oe or better 5.1 (c) 2.7 1 m3 1 indep (d) 14.2 or 14.16 3 M2 for 2 × 0.5 × 2 × 0.9 × 1.2 + 2.5 × 2 × 0.9 + 2 × 2.5 × 1.5 or better or M1 for 2.5 × 2 × 0.9 or 2 × 2.5 × 1.5 or better if M0 then SC1 for 13.41 IGCSE – May/June 2013 0580 32

This question in 0580/32 May/June 2013

6 For Examiner′s 30 m A B Use NOT TO SCALE HOUSE 17 m D C The rectangle ABCD shows Mr Liu’s garden. (a) Mr Liu puts a fence around three sides of his garden, AB, BC and CD. The fence costs $3.28 per metre. Calculate the cost of the fence. Answer(a) $ … [2] (b) (i) Calculate the area of Mr Liu’s garden. Answer(b)(i) … m2 [2] (ii) Mr Liu uses an area of 408 m2 in his garden for a lawn, fl owers and vegetables. He divides this area into three parts, in the ratio lawn : fl owers : vegetables = 5 : 3 : 4 . Calculate the area used for each part. Answer(b)(ii) Lawn … m2 Flowers … m2 Vegetables … m2 [3] (c) Mr Liu walks in a straight line across his garden from A to C. For Examiner′s Use Calculate the distance Mr Liu walks. Answer(c) … m [3] (d) Mr Liu has a circular pond, radius 4.5 m, in his garden. (i) Calculate the area of the pond. Answer(d)(i) … m2 [2] (ii) The pond is fi lled with water to a depth of 2 metres. Calculate the volume of water in the pond. Answer(d)(ii) … m3 [1] _____________________________________________________________________________________

13 marks

Mark scheme: 6 (a) 252.56 2 M1 for (30 + 30 + 17) × 3.28 or better oe (b) (i) 510 2 M1 for 30 × 17 (ii) 170 3 M2 for 2 correct areas clearly identified 102 or M1 for 408 ÷ (5 + 3 + 4) soi by 34 or 136 one correct area clearly identified SC2 for three correct answers in incorrect places (c) 34.5 3 M2 for 30 2 + 17 2 soi by 1189 or M1 for 302 + 172 soi by 1189 (d) (i) 63.6 or 63.61 – 63.63 2 M1 for 4.52 × π or 20.25 π (ii) 127 or 127.2… 1FT FT for their (d)(i) × 2 IGCSE – October/November 2013 0580 31

This question in 0580/31 Oct/Nov 2013

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

6 Irina has some solid building blocks. (a) Write down the mathematical name of this solid. Answer(a) … [1] (b) Irina describes the shape of a different block. She says: It has 12 edges and 8 vertices. All the faces are the same shape. Write down the mathematical name of this solid. Answer(b) … [1] (c) The diagram shows the end face of another block. A NOT TO 6 cm SCALE 3 cm B C (i) Show that BC = 5.2 cm, correct to 1 decimal place. Answer(c)(i) [3] (ii) Find the area of triangle ABC. Answer(c)(ii) … cm2 [2] (iii) This block is a triangular prism with length 8 cm. Calculate the volume of the block. Answer(c)(iii) … cm3 [1] (d) The diagram shows another building block. NOT TO SCALE 6 cm 4 cm x cm 8 cm (i) Calculate the area of the end face of this block. Answer(d)(i) … cm2 [2] (ii) The volume of this block is 336 cm3. Find the value of x. Answer(d)(ii) x = … [1] __________________________________________________________________________________________

11 marks

Mark scheme: 6 (a) Cylinder 1 (b) Cube or cuboid 1 (c) (i) 6 2 − 3 2 M2 M1 for 62 = 32 + BC2 or (BC2 = ) 62 – 32 5.19… A1 (ii) 7.79 to 7.8 2 M1 for 0.5 × 5.2 × 3 (iii) 62.4 1FT FT 8 × their (c)(ii) (d) (i) 28 2 M1 for 0.5 × (6 + 8) × 4 oe (ii) 12 1FT FT 336 ÷ their (d)(i)

This question in 0580/31 Oct/Nov 2015

5 E F D A NOT TO 7 cm SCALE 5 cm B 4 cm C The diagram shows a solid in the shape of a triangular prism. AC = 5 cm, BC = 4 cm and CD = 7 cm. Angle ABC = 90°. (a) What does the word prism tell you about the solid in the diagram? … [1] (b) Show that AB = 3 cm. [2] (c) Calculate the volume of the prism. Give the units of your answer. … … [4] (d) On the 1 cm2 grid, complete the net of the prism. Two faces have been drawn for you. [3] (e) Calculate the surface area of the prism. … cm2 [2]

12 marks

Mark scheme: 5 (a) constant cross-sectional area oe 1 (b) [AB2] + 42 = 52 M1 [AB] = 5 2 − 4 2 = 9 M1 3 × 4 (c) 42 3 M2 for × 7 2 3 × 4 or M1 for 2 If zero scored SC1 for answer 84 cm3 1 B1 independent (d) correct net drawn 3 M1 for 7 × 4 rectangle drawn in correct place M1 for one 3,4,5 triangle drawn correctly

This question in 0580/32 Feb/March 2016

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 (a) A cuboid measures 6 cm by 3 cm by 2 cm. (i) On this 1 cm2 grid, complete the net of the cuboid. [3] (ii) Calculate the volume of the cuboid. … cm3 [2] (b) Write down the mathematical name of this shape. … [1] (c) Mark an obtuse angle on this trapezium. [1] (d) A regular polygon has an exterior angle of 22.5°. Work out how many sides this polygon has. … [2] (e) NOT TO SCALE 20 cm 6 cm x cm The diagram shows a shape made from a semi-circle, radius 6 cm, and a right-angled triangle. (i) Show that x = 16. [2] (ii) Calculate the area of the shape. … cm2 [5]

16 marks

Mark scheme: 8 (a) (i) Correct net 3 B2 for 3 or 4 correct faces in correct position or B1 for 1 or 2 correct faces in correct position (ii) 36 2 M1 for 6 × 3 × 2 oe (b) Hexagon 1 (c) Obtuse angle indicated 1 360 360 (d) 16 2 M1 for or = 22.5 22.5 n 180 ( n − 2 ) or = 157.5 oe n (e) (i) 20 2 − 12 2 M2 M1 for 202 = 122 + x2 or [x2 =] 202 – 122 π6 2 (ii) 153 or 152.5 to 152.6 5 M2 for soi by 56.5… . or 18 π 2 or M1 for π62 soi by 113 or 113.0…. or 113.1…. or 36 π M1 for 0.5 × 12 × 16 soi by 96 M1dep for their 56.5... + their 96 dep on at least M1 earned soi

This question in 0580/31 Oct/Nov 2016

3 (a) x° NOT TO SCALE 74° 71° y° Work out the value of (i) x, x = … [1] (ii) y. y = … [1] (b) 85° NOT TO SCALE w° 128° Work out the value of w. Give reasons for your answer. w = … because … … [3] (c) NOT TO SCALE 15 8 p° Use trigonometry to calculate the value of p. p = … [2] (d) B NOT TO SCALE 225 km A 300 km The diagram shows the path of a plane from airport A to airport B. (i) Show that the distance between A and B is 375 km. [2] (ii) The plane flies at an average speed of 450 km/h. It leaves A at 14 45 and flies directly to B. Work out the time it arrives at B. … [4]

13 marks

Mark scheme: 3 (a) (i) 35 1 (ii) 74 1 (b) 43 and valid reasons 3 reasons include exterior angle [of a triangle] equals the sum of the interior opposite angles or angles on a straight line [sum to 180] and angles in a triangle [sum to 180] B2 for 43 or M1 for 180 – 128 soi by 52 or 128 – 85 B1 for valid reasons (c) 32.2 or 32.23… 2 M1 for sin [… =] 8 ÷ 15 oe (d) (i) [AB] = 300 2 + 225 2 2 M1 for 3002 + 2252 (ii) 15 35 4 M1 for 375 ÷ 450 or [0].833[…] M1 for their [0].833 × 60 or soi by 50 M1 for 14 45 + their 50 soi

This question in 0580/33 Oct/Nov 2016

9 120 m D C NOT TO 90 m SCALE A B 150 m The diagram shows a field in the shape of a trapezium. AB = 150 m, BC = 90 m and CD = 120 m. Angle ABC = angle BCD = 90°. (a) Calculate the area of the field. … m2 [2] (b) (i) Show that AD = 95 m, correct to the nearest metre. [3] (ii) A fence is built around the perimeter of the field. It costs $48 to build each 5-metre section of the fence. Calculate the cost of building this fence. $ … [3]

8 marks

Mark scheme: 9(a) 12 150 2 1 M1 for × (120 + 150 ) × 90 oe 2 or M1 for 90 2 + (150 − 120 ) 2 2 + (150 − 120 ) 2 M29(b)(i) [ AD = ] 90 = 94.9 or 94.8[…] A1 9(b)(ii) 4368 3 95 + 120 + 150 + 90 M2 for × 48 oe 5 or M1 for 95 + 120 + 150 + 90 soi or 455 95 120 150 90 or and and and 5 5 5 5

This question in 0580/31 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

7 (a) NOT TO 70° SCALE x° The diagram shows an isosceles triangle. Find the value of x. x = … [2] (b) b° 32° NOT TO 50° SCALE a° c° The diagram shows two pairs of parallel lines. Find the value of a, the value of b and the value of c. a = … b = … c = … [3] (c) 23 cm NOT TO w cm SCALE 14 cm The diagram shows a rectangle 14 cm by w cm. The diagonal is 23 cm. Calculate the value of w. w = … [3] (d) NOT TO SCALE O The diagram shows a square with vertices on the circumference of a circle, centre O. The radius of the circle is 6 cm. Work out the shaded area. … cm2 [5]

13 marks

Mark scheme: 7(a) 55 2 M1 for 180 − 70 7(b) [a =] 32 3 B1 for each [b =] 98 [c =] 82 7(c) 18.2 or 18.24 to 18.25 3 M2 for 232 − 14 2 or better or M1 for 142 + […]2 = 232 7(d) 41.1 or 41.09 to 41.112 5 M1 for π × 62 M2 for 21 × 6 × 6 × 4 or M1 for 21 × 6 × 6 M1 for π × 62 − 21 × 6 × 6 × 4

This question in 0580/32 May/June 2020

9 (a) A speedboat travels at 84 kilometres per hour. Change this speed into metres per minute. … m/min [2] (b) North X 39 km North NOT TO 21 km SCALE Y Z The speedboat starts at X and travels to Y, then to Z and then back to X. Z is due south of X and Y is due west of Z. XY = 39 km and XZ = 21 km. (i) Calculate YZ. YZ = … km [3] (ii) Calculate angle YXZ. Angle YXZ = … [2] (iii) Find the bearing of Y from X. … [1]

8 marks

Mark scheme: 9(a) 1400 2 84 × 1000 M1 for 60 or B1 for final answer figs 14 9(b)(i) 32.9 or 32.86… 3 M2 for 39 2 − 212 or better or M1 for x 2 + 212 = 39 2 9(b)(ii) 57.4 or 57.42 2 FT their (b)(i) if used in tan or sin for angle at X 21 M1 for cos = oe 39 If 0 scored, SC1 for finding other angle in triangle, possibly with their 32.9 9(b)(iii) 237.4 1 FT their (b)(ii) + 180

This question in 0580/33 May/June 2020

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

3 (a) The diagram shows a scale drawing of Joel’s rectangular garden. The scale is 1 centimetre represents 8 metres. Scale: 1 cm to 8 m Find the actual area of his garden. … m2 [3] (b) The diagram shows a rectangular gate, FGHJ, in Joel’s garden. F J NOT TO SCALE G H GJ = 2.1 m and FG = 0.85 m. Find FJ. FJ = … m [3] (c) 85 m NOT TO SCALE patio grass 24 m The diagram shows Brenda’s rectangular garden. There is a patio in the shape of a quarter-circle. She wants to grow grass in the shaded part of the garden. She needs 40 g of grass seed per square metre. Grass seed is sold in 1 kg bags which cost $6.80 per bag. Calculate the cost of the grass seed she needs to buy. $ … [6]

12 marks

Mark scheme: 3(a) 3840 3 B1 for 5.8 to 6.2 or 9.8 to 10.2 or 4050 or 3637.76 to 4047.36 M1 for their length × their width or for use of 82 3(b) 1.92 or 1.920… 3 M2 for 2.12 − 0.852 or better or M1 for 2.12 = FJ2 + 0.852 or better 3(c) 435.2[0] cao 6 M1 for 85 × 24 M1 for π × 242 ÷ 4 oe M1 for their 2040 – their 452 M1 for their area × 100040 oe M1 for their 64 × 6.8[0]

This question in 0580/33 May/June 2021

8 (a) B NOT TO 125 m SCALE A C 100 m The diagram shows a right-angled triangle, ABC. (i) Show that BC = 75 m . [2] (ii) Calculate angle BAC. Angle BAC = … [2] (b) x cm NOT TO SCALE 20 cm 12 cm 16 cm The diagram shows a shape made from two right-angled triangles. The total area of this shape is 246 cm 2. Work out the value of x. x = … [3]

7 marks

Mark scheme: 8(a)(i) 2 2 M2 M1 for BC2 + 1002 = 1252 125 − 100 or for [BC2] = 1252 – 1002 8(a)(ii) 36.9 or 36.86 to 36.87 2 100 M1 for cos [BAC =] oe or better 125 75 or for tan [BAC =] oe or better 100 75 or for sin [BAC =] oe or better 125 8(b) 15 nfww 3 M2 for ½ × 16 × 12 + ½ × 20 × x = 246 or better or M1 for ½ × 16 × 12

This question in 0580/31 Oct/Nov 2021

10 B NOT TO 13.2 cm SCALE 4.54.5 cmcm C 8.9 cm A The diagram shows a right-angled triangle, ABC, and a semicircle. The radius of the semicircle is 4.5 cm. AC = 8.9 cm and BC = 13.2 cm. (a) Calculate the shaded area. Give the units of your answer. … … [5] (b) Calculate AB. AB = … cm [2]

7 marks

Mark scheme: 10(a) 26.9 or 26.92 to 26.93….. 4 M3 for 0.5 × 8.9 × 13.2 – 0.5 × π × 4.52 oe or M1 for 0.5 × 8.9 ×13.2 and M1 for 0.5 × π × 4.52 and M1 for subtraction of their areas cm2 1 10(b) 15.9 or 15.92[……] 2 M1 for 8.92 + 13.22 or better

This question in 0580/31 May/June 2023

10 (a) U P 21.4 cm 14.4 cm NOT TO 9.6 cm SCALE Q R V W 12.8 cm Triangle PQR is mathematically similar to triangle UVW. Calculate VW. VW = … cm [2] (b) ABC is a right-angled triangle. A NOT TO 9.1 cm SCALE 3.5 cm C B Calculate BC. BC = … cm [3] (c) DEF is a right-angled triangle. D 8.4 cm NOT TO SCALE 35° F E Calculate EF. EF = … cm [2] (d) JKL is a right-angled triangle. 10 cm J L NOT TO 8 cm SCALE K Calculate angle JKL. Angle JKL = … [2]

9 marks

Mark scheme: 10(a) 19.2 2 14.4 VW M1 for = oe or better 9.6 12.8 10(b) 8.4 3 M2 for 9.12 – 3.52 oe or better or M1 for […]2 + 3.52 = 9.12 oe or better 10(c) 6.88 to 6.881 2 [...] M1 for cos35 = oe or better 8.4  or sin 55 = oe or better 8.4 10(d) 51.3[4…] 2 10 M1 for tan[… =] oe or better 8

This question in 0580/31 Oct/Nov 2023

8 (a) In triangle RST, RT = 7 cm and ST = 4 cm. (i) Using a ruler and compasses only, construct triangle RST. Leave in your construction arcs. The line RS has been drawn for you. R S [2] (ii) Measure the distance from S to the midpoint of RT. Give your answer in millimetres. … mm [1] (b) Town A is 8.5 cm from town B on a map. The scale of the map is 1 : 50 000. Calculate the actual distance from town A to town B. Give your answer in kilometres. … km [2] (c) E x° NOT TO SCALE 118° A B C D The diagram shows triangle BCE and a straight line ABCD. BE = CE and angle DCE = 118°. Find the value of x. x = … [2] (d) A NOT TO 8.9 cm SCALE 4.8 cm C B The diagram shows a right-angled triangle ABC. Show that BC is 7.5 cm, correct to 2 significant figures. [3] Question 9 is printed on the next page.

10 marks

Mark scheme: 8(a)(i) Correct triangle with correct arcs 2 B1 for correct triangle with incorrect or no arcs or for two correct arcs If 0 scored, SC1 for triangle with arcs but lines interchanged 8(a)(ii) 52 1 FT their complete triangle 8(b) 4.25 2 B1 for figs 425 as answer or 1 cm = 0.5 km seen or M1 for 8.5  50 000 oe 8(c) 56 2 M1 for 180 – 2  (180 – 118) oe or B1 for [angle] ECB or EBC = 62 8(d) 8.92 – 4.82 M2 M1 for 4.82 + BC2 = 8.92 2 2 A1 8.9 − 4.8 = 7.49… or 56.17 = 7.49…

This question in 0580/32 Oct/Nov 2023

9 (a) C NOT TO SCALE 18 cm 9.6 cm A O B The diagram shows a semicircle with diameter AB. O is the midpoint of AB and C is a point on the circumference. (i) Calculate the area of triangle ABC. … cm2 [2] (ii) Show that AB = 20.4 cm. [2] (iii) Calculate the shaded area. … cm2 [2] (b) The diagram shows two right-angled triangles, PQR and XYZ. Y NOT TO 46° SCALE Q 26 cm 51° P R X 34 cm Z Calculate the difference in the heights RQ and XY. … cm [4]

10 marks

Mark scheme: 9(a)(i) 86.4 2 M1 for ½ × 9.6 × 18 oe 9(a)(ii) 2 2 M2 M1 for 9.62 + 182 9.6 + 18 leading to 20.4 9(a)(iii) 77[.0] or 77.02 to 77.05 2 2  20.4  M1 for [½ ×] π ×   soi  2  9(b) 12.6 or 12.62 to 12.63 4 34 M3 for – 26 × sin 51 oe tan 46 or RQ M1 for sin 51 = or better oe 26 34 M1 for tan 46 = or better oe XY

This question in 0580/33 Oct/Nov 2023

8 B 3.6 m A NOT TO SCALE 4.7 m E F 1.7 m C D 5.5 m The diagram shows a plan, ABCDE, of the floor of a room in Jo’s house. F is a point inside the room. (a) (i) Show that EF = 1.9 m . [1] (ii) Work out AF. AF = … m [1] (b) Calculate the area of the floor. … m2 [3] (c) A cupboard in the room is in the shape of a cuboid. The area of the base of the cupboard is 1.2 m2 and the height of the cupboard is 2.3 m. Calculate the volume of the cupboard. Give the units of your answer. … … [2] (d) Jo buys 275 floor tiles which cost $1.64 each. Calculate the total cost of the floor tiles. $ … [1] (e) Jo builds a patio in the shape of a semicircle with radius 2.3 m. Calculate the area of the patio. … m2 [2] Question 9 is printed on the next page.

10 marks

Mark scheme: 8(a)(i) 5.5 – 3.6 [= 1.9] 1 8(a)(ii) 3 1 8(b) 23 3 M2 for 4.7 × 3.6 + 1.9 × 1.7 + 0.5 × 1.9 × their (a)(ii) or 3.6 ×their(a)(ii) + 5.5 × 1.7 + 0.5 × 1.9 × their(a)(ii) or 5.5 × 4.7 − 0.5 × 1.9 × their (a)(ii) or 0.5  (3.6 + 5.5)  their (a)(ii) + 1.7  5.5 or 0.5  (1.7 + 4.7)  1.9 + 3.6  4.7 or M1 for 4.7 × 3.6 + 1.9 × 1.7 or 3.6 × their (a)(ii) + 5.5 × 1.7 or 0.5 × 1.9 × their (a)(ii) or 5.5 × 4.7 or 0.5  (3.6 + 5.5)  their (a)(ii) or 0.5  (1.7 + 4.7)  1.9 8(c) 2.76 1 m3 1 8(d) 451 1 8(e) 8.31 or 8.309 to 8.311 2 M1 for [0.5]π × 2.32 oe

This question in 0580/32 May/June 2024

5 (a) k (i) Measure angle k. … [1] (ii) Write down the mathematical name for this type of angle. … [1] (b) The diagram shows a pair of parallel lines and a straight line. Angles a, b, c, d and x are labelled. a NOT TO x SCALE b c d Complete the statements. Angle … is alternate to angle x. Angle … is corresponding to angle x. [2] (c) The diagram shows a parallelogram. y° NOT TO SCALE 42° Find the value of y. y = … [1] (d) B C NOT TO 17° SCALE O A A, B and C lie on a circle, centre O. Find angle ACB. Angle ACB = … [2] (e) The interior angle of a regular polygon is 171°. Work out the number of sides of this polygon. … [2] (f) NOT TO 18.5 cm SCALE 7.4 cm x° Calculate the value of x. x = … [2]

11 marks

Mark scheme: 5(a)(i) 326 1 5(a)(ii) reflex 1 5(b) c 2 B1 for each d 5(c) 138 1 5(d) 73° 2 M1 for 180 – 90 – 17 oe or 90 – 17 or angle B marked as 90° 5(e) 40 2 M1 for 360 ÷ (180 – 171) oe 180  n  2  or  171 oe n 5(f) 23.6 or 23.57 to 23.58 2 7.4 M1 for sin […. = ] oe 18.5

This question in 0580/33 May/June 2024

21 The diagram shows a right-angled triangle ABC. A NOT TO SCALE 52.5 cm 59.5 cm B C Calculate BC. BC = … cm [3]

3 marks

Mark scheme: 21 28 nfww 3 M2 for 59.52 – 52.52 oe or M1 for 52.52 + […]2 = 59.52 oe

This question in 0580/31 May/June 2025

23 The diagram shows a right-angled triangle. 35 cm NOT TO SCALE 22 cm h cm Calculate the value of h. h = … [2]

2 marks

Mark scheme: 23 41.3 or 41.34… 2 M1 for 352 + 222 oe or better

This question in 0580/33 May/June 2025