Right-angled triangles — Paper 2 · IGCSE Mathematics 0580

10 A mountain railway AB is of length 864 m and rises at an angle of 12o to the horizontal. For A train is 586 m above sea level when it is at A. Examiner's Calculate the height above sea level of the train when it reaches B. Use B 864 m NOT TO SCALE 12o A Answer m [3]

3 marks

Mark scheme: 10 766 3* M1 sin 12 = h/864 M1 586 + their “180” cos78 or sine rule

This question in 0580/21 Oct/Nov 2004

15 T NOT TO SCALE h 25° B A 80 m 18° C Mahmoud is working out the height, h metres, of a tower BT which stands on level ground. He measures the angle TAB as 25°. He cannot measure the distance AB and so he walks 80 m from A to C, where angle ACB = 18° and angle ABC = 90°. Calculate (a) the distance AB, Answer(a) m [2] (b) the height of the tower, BT. Answer(b) m [2]

4 marks

Mark scheme: 15 (a) 24.7 2 M1 sin18 = AB/80 or cos72 = AB/80 Allow AB/sin18 = 80/sin90 (b) 11.5 2 M1 tan25 = h/(a) or h/sin25 = (a)/sin65

This question in 0580/21 May/June 2009

17 A NOT TO E SCALE O D 38° C B AB is the diameter of a circle, centre O. C, D and E lie on the circle. EC is parallel to AB and perpendicular to OD. Angle DOC is 38°. Work out (a) angle BOC , Answer(a) Angle BOC = [1] (b) angle CBO, Answer(b) Angle CBO = [1] (c) angle EDO . Answer(c) Angle EDO = [2]

4 marks

Mark scheme: 17 (a) 52 1 (b) 64 1 (c) 71 2 M1 angle CED = 19

This question in 0580/21 May/June 2010

12 The diagram represents the ski lift in Queenstown New Zealand. Examiner's Use T NOT TO SCALE 730 m h 37.1° B (a) The length of the cable from the bottom, B, to the top, T, is 730 metres. The angle of elevation of T from B is 37.1°. Calculate the change in altitude, h metres, from the bottom to the top. Answer(a) m [2] (b) The lift travels along the cable at 3.65 metres per second. Calculate how long it takes to travel from B to T. Give your answer in minutes and seconds. Answer(b) min s [2]

4 marks

Mark scheme: h 12 (a) 440 2 M1 sin 37.1 or cos 52.9 = oe 730 730 (b) 3 min 20 sec 2 M1 .365  6x 3   6x 3 

This question in 0580/22 May/June 2010

11 P cliff NOT TO SCALE beach A F 55 m The diagram shows a point P at the top of a cliff. The point F is on the beach and vertically below P. The point A is 55m from F, along the horizontal beach. The angle of elevation of P from A is 17°. Calculate PF, the height of the cliff. Answer PF = m [3]

3 marks

Mark scheme: h 55 11 16.8 3 M2 tan17 = or tan73 = 55 h 55 h or M1 tan17 = or tan73 = if angle seen in h 55 wrong place at P 2 2 2

This question in 0580/23 May/June 2010

5 J NOT TO 50 m SCALE 20 m G R JGR is a right-angled triangle. JR = 50m and JG = 20m. Calculate angle JRG. Answer Angle JRG = [2]

2 marks

Mark scheme: 20 50 5 23.6 2 M1 sin R = 20/50 or = sin R sin 90 3

This question in 0580/22 Oct/Nov 2010

20 For y Examiner's Use 9 8 7 6 5 4 M 3 2 1 K L x –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 The triangle KLM is shown on the grid. (a) Calculate angle KML. Answer(a) Angle KML = [2] (b) On the grid, draw the shear of triangle KLM, with a shear factor of 3 and the x-axis invariant. [2]

4 marks

Mark scheme: 4 20 (a) 63.4 2 M1 tan(M) = oe 2 (b) Vertices at (4, 1), (8, 1) and (10, 3) 2 B1 two vertices correct

This question in 0580/23 Oct/Nov 2010

5 For Examiner's NOT TO Use 29 cm SCALE x cm 53.2° Calculate the value of x. Answer x = [2]

2 marks

Mark scheme: x 5 23.2 2 M1 for sin 53.2 = implicit form or better 29 8 + 4 + 8 + 9 + y

This question in 0580/21 May/June 2012

9 P NOT TO SCALE 78.3° A B 58.4 m 170 m The line AB represents the glass walkway between the Petronas Towers in Kuala Lumpur. The walkway is 58.4 metres long and is 170 metres above the ground. The angle of elevation of the point P from A is 78.3°. Calculate the height of P above the ground. Answer m [3]

3 marks

Mark scheme: 9 452 3 x SC2 282 in answer M1 tan 78.3 = space 584. M1 “282” + 170

This question in 0580/22 May/June 2012

21 For P Examiner's Use NOT TO SCALE 5 cm D C 8 cm M A 8 cm B The diagram shows a pyramid on a square base ABCD. The diagonals of the base, AC and BD, intersect at M. The sides of the square are 8 cm and the vertical height of the pyramid, PM, is 5 cm. Calculate (a) the length of the edge PB, Answer(a) PB = cm [3] (b) the angle between PB and the base ABCD. Answer(b) [3]

6 marks

Mark scheme: 1 21 (a) 7.55 www 3 M2 ( √(82 + 82))2 + 52 or 42 + 52 + 42 seen 2 or M1 82 + 82 or 52 + 42 or 42 + 42 or 52 + (their MB)2 seen 5 5 (b) 41.5 www 3 M2 sin(B) = or tan(B) = or a( ) their MB their MB cos(B) = (a) or M1 recognition of angle PBM

This question in 0580/23 May/June 2012

21 For Examiner′s C Use NOT TO SCALE D B X A ABCD is a kite. The diagonals AC and BD intersect at X. AC = 12 cm, BD = 20 cm and DX : XB = 3 : 2 . (a) Calculate angle ABC. Answer(a) Angle ABC = … [3] (b) Calculate the area of the kite. Answer(b) … cm2 [2] _____________________________________________________________________________________

5 marks

Mark scheme: 20 21 (a) 73.7 or 73.73 to 73.74 3 M1 for × 2 or B1 for BX = 8 3 + 2 6 M1 for tan [ ] = or better their 8 (b) 120 2 M1 for 12 × 20 × 12 oe 5

This question in 0580/21 Oct/Nov 2013

21 P 6 cm NOT TO SCALE D C 4 cm M 4 cm A B The diagram shows a pyramid on a square base ABCD with diagonals, AC and BD, of length 8 cm. AC and BD meet at M and the vertex, P, of the pyramid is vertically above M. The sloping edges of the pyramid are of length 6 cm. Calculate (a) the perpendicular height, PM, of the pyramid, Answer(a) PM = … cm [3] (b) the angle between a sloping edge and the base of the pyramid. Answer(b) … [3] __________________________________________________________________________________________ Question 22 is printed on the next page.

6 marks

Mark scheme: 21 (a) 4.47 or 4.472[…] 3 M2 for 6 2 − 4 2 or M1 for [PM ]2 + 4 2 = 6 2 or 6 2 − 4 2 4 (b) 48.2 or 48.18 to 48.19 3 M2 for cos[correct angle] = oe 6 or M1 for recognising a correct angle Page 5 MMarrk SSccheemee Sylllabbuss Papeer IGC SEE – M ay//Juunee 2014 055800 222

This question in 0580/22 May/June 2014

18 F 6 m NOT TO SCALE D C 15 m A B 18 m The diagram shows a rectangular playground ABCD on horizontal ground. A vertical flagpole CF, 6 metres high, stands in corner C. AB = 18 m and BC = 15 m. Calculate the angle of elevation of F from A. Answer … [4] __________________________________________________________________________________________

4 marks

Mark scheme: 6 18 14.4 or 14.36... 4 M3 for tan = 2 2 oe or better their 15 + 18 or M1 for AC = 15 2 + 18 2 and M1 for identifying required angle 18

This question in 0580/22 Feb/March 2015

3 NOT TO SCALE 5 cm x° 2 cm Calculate the value of x. Answer x = … [2]

2 marks

Mark scheme: 2 3 66.4[2…] 2 M1 for cos […= ] oe 5

This question in 0580/23 May/June 2015

8 P 11 cm NOT TO SCALE 37° A O In the diagram, AP is a tangent to the circle at P. O is the centre of the circle, angle PAO = 37° and AP = 11 cm. (a) Write down the size of angle OPA. Answer(a) Angle OPA = … [1] (b) Work out the radius of the circle. Answer(b) … cm [2] __________________________________________________________________________________________

3 marks

Mark scheme: 8 (a) 90 1 OP o (b) 8.29 or 8.289… to 8.29 2 M1 for = tan 37 oe 11

This question in 0580/21 Oct/Nov 2015

9 NOT TO 5 cm SCALE 2 cm x° Calculate the value of x. Answer x = … [2] __________________________________________________________________________________________

2 marks

Mark scheme: 2 9 23.6 or 23.57 to 23.58 2 M1 for sin[= ] oe 5

This question in 0580/23 Oct/Nov 2015

3 C NOT TO SCALE 3.5 m B A 0.9 m Calculate angle BAC. Angle BAC = … [2]

2 marks

Mark scheme: 0.9 3 75.1 or 75.09 to 75.10 2 M1 for cos […= ] 3.5

This question in 0580/22 Feb/March 2016

23 7 cm E F 5 cm B NOT TO A SCALE H G 3 cm D C The diagram shows a cuboid. HD = 3 cm, EH = 5 cm and EF = 7 cm. Calculate (a) the length CE, CE = … cm [4] (b) the angle between CE and the base CDHG. … [3]

7 marks

Mark scheme: 23 (a) 9.11 or 9.110… 4 M3 for 5 2 + 32 + 7 2 or M2 for 5 2 + 32 or 32 + 7 2 or 5 2 + 7 2 or M1 for 5² + 3² or 3² + 7² or 5² + 7² 5 (b) 33.3 or 33.28 to 33.29 3 M2 for sin = oe their ( a ) or B1 for identifying angle ECH

This question in 0580/23 May/June 2016

24 S R NOT TO P Q 8 cm SCALE D C 8 cm A 8 cm B The diagram shows a cube of side length 8 cm. (a) Calculate the length of the diagonal BS. BS = … cm [3] (b) Calculate angle SBD. Angle SBD = … [2]

5 marks

Mark scheme: 24 (a) 13.9 or 13.85 to 13.86 3 M2 for 8 2 + 8 2 + 8 2 oe or M1 for 82 + 82 or better for one face 8 8 2 + 8 2 (b) 35.1 to 35.5[4…] 2 M1 for sin = or cos = their(a) their(a) 8 or tan = oe 8 2 + 8 2

This question in 0580/21 Oct/Nov 2016

9 From the top of a building, 300 metres high, the angle of depression of a car, C, is 52°. NOT TO SCALE 300 m C Calculate the horizontal distance from the car to the base of the building. … m [3]

3 marks

Mark scheme: 300 9 234 or 234.3 to 234.4 3 M2 for [dist = ] oe tan52 or M1 for correct implicit trig statement allow M1 if they use their 52 or their 38 provided it is marked on the diagram or B1 for 52 or 38 correctly placed If zero scored, SC1 for final answer 384

This question in 0580/22 Oct/Nov 2016

12 7 cm NOT TO 4 cm SCALE x° Calculate the value of x. x = … [2]

2 marks

Mark scheme: 12 34.8 or 34.84 to 34.85 2 4 M1 for sin [=] 7

This question in 0580/22 May/June 2017

22 The diagram shows a cube ABCDEFGH of side length 26 cm. H G E NOT TO F SCALE D C A B Calculate the angle between AG and the base of the cube. … [4]

4 marks

Mark scheme: 22 35.3 or 35.26… 4 26 M3 for [tan =] oe 26 2 + 26 2 or M1 for [ AC 2 = ] 26 2 + 26 2 oe and M1 for [tan =] 26 ÷ their AC oe or for angle CAG indicated

This question in 0580/23 May/June 2017

21 E NOT TO SCALE 9 cm D C 12 cm M A 12 cm B The diagram shows a square-based pyramid ABCDE. The diagonals of the square meet at M. E is vertically above M. AB = BC = 12 cm and EM = 9 cm. Calculate the angle between the edge EC and the base, ABCD, of the pyramid. … [4]

4 marks

Mark scheme: 21 46.7 or 46.68 to 46.69 4 9 M3 for tan […=] oe 1 2 2 12 + 12 2 or  1  2 2 12 2 M1 for × 12 + 12 oe e.g.    2  2 and M1 for identifying angle MCE

This question in 0580/21 Oct/Nov 2017

22 B NOT TO 8 cm SCALE 5.5 cm A 16.2 cm The diagram shows a cuboid with dimensions 5.5 cm, 8 cm and 16.2 cm. Calculate the angle between the line AB and the horizontal base of the cuboid. … [4]

4 marks

Mark scheme: 22 25.1 or 25.06… 4 8 M3 for tan = oe 16.2 2 + 5.5 2 or M2 for 16.2 2 + 5.5 2 or M1 for 16.2 2 + 5.5 2 or B1 for identifying correct angle

This question in 0580/22 Oct/Nov 2018

23 Q P 4 cm NOT TO SCALE D C 6 cm A 12 cm B The diagram shows a triangular prism. AB = 12 cm, BC = 6 cm, PC = 4 cm, angle BCP = 90° and angle QDC = 90°. Calculate the angle between AP and the rectangular base ABCD. … [4]

4 marks

Mark scheme: 23 16.6 or 16.60… 4 4 M3 for tan = oe 12 2 + 6 2 or M2 for 12 2 + 6 2 or M1 for 122 + 62 oe or B1 for recognising angle PAC is required

This question in 0580/23 Oct/Nov 2018

24 R S 97° Q 7.4 cm NOT TO SCALE 8.5 cm 26° 53° P Calculate (a) SR, SR = … cm [3] (b) RQ. RQ = … cm [4]

7 marks

Mark scheme: 24(a) 5.95 or 5.954... 3 7.4 M2 for × sin53 sin97 sin97 sin53 or M1 for = oe 7.4 SR 24(b) 3.73 or 3.733 to 3.734 4 M2 for 8.5 2 + 7.4 2 − 2 × 8.5 × 7.4 × cos26 or M1 for implicit form A1 for 13.9[4...]

This question in 0580/22 Feb/March 2019

7 NOT TO SCALE 12 cm x cm 35° The diagram shows a right-angled triangle. Calculate the value of x. x = … [2]

2 marks

Mark scheme: 7 6.88 or 6.882 to 6.883 2 x M1 for sin 35 [=] oe or better 12

This question in 0580/21 May/June 2019

21 V NOT TO SCALE 10 cm D C M A 8 cm B The diagram shows a pyramid with a square base ABCD of side length 8 cm. The diagonals of the square, AC and BD, intersect at M. V is vertically above M and VM = 10 cm. Calculate the angle between VA and the base. … [4]

4 marks

Mark scheme: 21 60.5 or 60.50… 4 10 M3 for tan = oe 1 2 2 8 + 8 2 1 2 2 or M2 for [ × ] 8 + 8 2 or M1 for 82 + 82 or 4 2 + 4 2 or B1 for recognising the angle required

This question in 0580/21 May/June 2019

24 H G E F 12 cm NOT TO SCALE D C 7 cm A 18 cm B ABCDEFGH is a cuboid. AB = 18 cm, BC = 7 cm and CG = 12 cm. Calculate the angle that the diagonal AG makes with the base ABCD. … [4] Question 25 is printed on the next page.

4 marks

Mark scheme: 24 31.9 or 31.85... 4 12 M3 for tan = oe 18 2 + 7 2 or M2 for 18 2 + 7 2 or M1 for 18 2 + 7 2 or B1 for identifying correct angle CAG

This question in 0580/22 May/June 2019

20 D C 20° NOT TO SCALE O 131° B T A A, B, C and D lie on the circle, centre O. TA is a tangent to the circle at A. Angle ABC = 131° and angle ADB = 20°. Find (a) angle ADC, Angle ADC = … [1] (b) angle AOC, Angle AOC = … [1] (c) angle BAT, Angle BAT = … [1] (d) angle OAB. Angle OAB = … [1]

4 marks

Mark scheme: 20(a) 49 1 20(b) 98 1 FT 2 × their (a) 20(c) 20 1 20(d) 70 1 FT 90 – their (c)

This question in 0580/22 May/June 2020

20 V NOT TO SCALE 18.6 cm D C M A 11 cm B The diagram shows a pyramid with a square base ABCD. The diagonals AC and BD intersect at M. The vertex V is vertically above M. AB = 11 cm and AV = 18.6 cm. Calculate the angle that AV makes with the base. … [4] Question 21 is printed on the next page.

4 marks

Mark scheme: 20 65.3 or 65.28.. 4 1 2 2 11 + 11 2 M3 for cos = or better 186. 1 2 2 or M2 for AM = 11 + 11 oe 2 or M1 for AC2 = 112 + 112 If 0 scored, SC1 for identifying angle VAM

This question in 0580/23 May/June 2020

21 P Q R NOT TO S SCALE U T W V The diagram shows a cuboid PQRSTUVW. PV = 17.2 cm The angle between the line PV and the base TUVW of the cuboid is 43°. Calculate PT. PT = … cm [3]

3 marks

Mark scheme: 21 11.7 or 11.73… 3 PT M2 for sin43 = oe 17.2 or M1 for identifying angle PVT

This question in 0580/21 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

23 V NOT TO SCALE 14 cm D C 10 cm M A 12 cm B The diagram shows a pyramid VABCD with a rectangular base. V is vertically above M, the intersection of the diagonals AC and BD. AB = 12 cm, BC = 10 cm and VC = 14 cm. Calculate the angle that VC makes with the base ABCD. … [4] Question 24 is printed on the next page.

4 marks

Mark scheme: 23 56.1 or 56.09… 4 1 2 2 10 + 1 2 2 M3 for cos[…] = oe 14 1 2 2 or M2 for [MC =] 10 + 12 oe 2 or M1 for [AC 2 =] 10 2 + 122 oe or B1 for indicating required angle

This question in 0580/22 Feb/March 2021

15 Complete the table showing information about the congruence of pairs of triangles. The first two rows have been completed for you. All diagrams are not to scale. Congruent or Congruence Pair of triangles not congruent criterion Congruent ASA 60° 25° 25° 60° 6 cm 6 cm 3.4 cm 4.8 cm 3 cm Not congruent None 4 cm 3 cm 3.4 cm 7 cm 35° 6.5 cm 6.5 cm 35° 7 cm 4.5 cm 5 cm 5 cm 4 cm 4.5 cm 4 cm 5.2 cm 5.2 cm 35° 65° [3]

3 marks

Mark scheme: 15 Congruent SAS 3 B1 for each correct row Congruent SSS Not congruent None

This question in 0580/22 May/June 2021

21 A NOT TO SCALE 65° B C The shortest distance from B to AC is 12.8 cm. Calculate BC. BC = … cm [3]

3 marks

Mark scheme: 21 14.1 or 14.12... 3 12.8 M2 for sin 65 = oe or better BC or M1 for recognition that the line from B is perpendicular to AC

This question in 0580/23 May/June 2021

9 NOT TO 12 cm SCALE x° 7 cm Calculate the value of x. x = … [2]

2 marks

Mark scheme: 9 54.3 or 54.31… 2 7 M1 for cos [x] = oe 12

This question in 0580/21 Oct/Nov 2021

21 E NOT TO SCALE 9 cm D C O 5 cm A B 5 cm The diagram shows a pyramid ABCDE. The pyramid has a square horizontal base ABCD with side 5 cm. The vertex E is vertically above the centre O of the base. The height OE of the pyramid is 9 cm. Calculate the angle that EC makes with the base ABCD. … [4] Question 22 is printed on the next page.

4 marks

Mark scheme: 21 68.6 or 68.55 to 68.56 4 9 M3 for tan[..] = oe 1 2 2 5 + 5 2 1 2 2 or M2 for 5 + 5 oe 2 or M1 for 52 + 52 oe or 2.52 + 2.52 oe or x2 + x2 = 52 oe or B1 for indicating required angle

This question in 0580/21 Oct/Nov 2021

21 M H G E F NOT TO 14.5 cm SCALE D C 9 cm A B 18.6 cm The diagram shows an open rectangular box ABCDEFGH. AB = 18.6 cm, BC = 9 cm and CG = 14.5 cm. A straight stick AGM rests against A and G and extends outside the box to M. (a) Calculate the angle between the stick and the base of the box. … [4] (b) AM = 30 cm. Show that GM = 4.8 cm, correct to 1 decimal place. [3]

7 marks

Mark scheme: 21(a) 35.1 or 35.05 to 35.06 4 14.5 M3 for tan = oe 18.6 2 + 9 2 or M2 for  AC 2 =  1 8.6 2 + 9 2 oe or better   or  AG 2 =  1 8.6 2 + 9 2 + 14.5 2   or M1 for recognising the angle GAC 21(b) M2 M1 for AG 2 = 18.6 2 + 9 2 + 14.5 2 oe or better 30 − 18.6 2 + 9 2 + 14.5 2 14.5 14.5 30 – or sin ( their(a) ) = sin ( their(a) ) AG or 18.6 2 + 9 2 or cos ( their(a) ) = 2 2 AG 18.6 + 9 30 – cos ( their(a) ) 4.75 to 4.78… A1

This question in 0580/22 Feb/March 2022

18 C NOT TO SCALE 115° 35° A B 7 cm Calculate the length BC. BC = … cm [4]

4 marks

Mark scheme: 18 12.7 or 12.68 to 12.69 4 7sin115 M3 for sin(180 − 115 − 35) or B2 for 8.03… seen OR B1 for [angle C =] 30 7sin115 M2 for sin(their angle C ) sin115 sin(their angle C ) or M1 for = oe BC 7

This question in 0580/21 Oct/Nov 2022

21 H M G NOT TO E F SCALE 8 cm D C 5 cm A B 14 cm The diagram shows a cuboid ABCDEFGH. AB = 14 cm, BC = 5 cm and CG = 8 cm. M is the midpoint of HG. (a) Calculate BM. … cm [3] (b) Calculate the angle that BM makes with the base ABCD. … [3] Question 22 is printed on the next page.

6 marks

Mark scheme: 21(a) 11.7 or 11.74 to 11.75 3 2  14  M2 for   + 52 + 82 oe  2   14  2  14  2 or M1 for   + 52, 52 + 82 or   + 82  2   2  21(b) 42.9 to 43.14 3 8 M2 for sin [….] = oe their (a) or M1 for recognising angle MBX where X is the midpoint of DC

This question in 0580/23 Oct/Nov 2023

12 NOT TO SCALE 14 cm x° 8.5 cm The diagram shows a right-angled triangle. Calculate the value of x. x = … [2]

2 marks

Mark scheme: 12 52.6 or 52.61 to 52.62 2 8.5 M1 for cos[...  ] oe 14

This question in 0580/22 May/June 2024

9 C NOT TO SCALE 8 cm 37° A B The diagram shows a right-angled triangle. Calculate AB. AB = … cm [2]

2 marks

Mark scheme: 9 6.39 or 6.389... 2 M1 for cos37 AB oe 8

This question in 0580/23 May/June 2024

20 S NOT TO 4 cm SCALE 18 cm T 28° P Q R 9 cm The diagram shows two right-angled triangles PQS and RQT. PQR and QTS are straight lines. Calculate angle QTR. Angle QTR = … [5]

5 marks

Mark scheme: 20 63.7 or 63.68 to 63.69 5 9 M4 for tan [QTR] = oe 18sin28 − 4 OR M3 for 18sin28 – 4 or M2 for 18sin28 QS or M1 for = sin28 oe 18 and 9 M1 for tan [QTR] = oe theirQT

This question in 0580/21 Oct/Nov 2024

20 NOT TO 6 cm SCALE 30° x cm Find the exact value of x. x = … [4]

4 marks

Mark scheme: 20 4 1 3 6 3 oe B1 for tan30 = or 3 3 3 1 or for sin60 = and sin30 = 2 2 6 6sin ( 60 ) M2 for or tan30 sin ( 30 ) 6 sin60 sin30 or M1 for = tan30 or = x x 6

This question in 0580/21 May/June 2025

23 (a) Write down the value of cos 90°. … [1] (b) x NOT TO SCALE n 30° 30° The diagram shows two different right-angled triangles joined by a common side. Find x in terms of n. x = … [5] Question 24 is on the next page.

6 marks

Mark scheme: 23(a) 0 1 23(b) 4 n 4 n 3 5 B4 for correct explicit expression to find [x =] or oe x 3 3 2 n 2 n e.g. x = or x = oe cos30 3 2 OR M3 for correct explicit method to find x n  sin 30 with 2 trig functions e.g. x = cos30 oe or a correct implicit method with one trig 2n function e.g. cos 30 = oe x OR B2 for hypotenuse of lower triangle = 2n or M2 for any correct implicit method with 2 trig functions e.g. cos 30 = n  sin 30 oe x OR M1 for seeing any correct trig value from sin 30 or cos 60 = 0.5, cos 30 or 3 3 sin 60 = or tan 30 = oe 2 3 or for hypotenuse of lower triangle = n oe sin30

This question in 0580/23 May/June 2025

1 NOT TO x° SCALE 25° The diagram shows an isosceles triangle. Find the value of x. x = … [2]

2 marks

Mark scheme: Question Answer Marks Partial Marks 1 130 2 M1 for 180 – 2 × 25

This question in 0580/23 Oct/Nov 2025

10 C 11 cm NOT TO SCALE x° B A 8 cm The diagram shows a right-angled triangle ABC. (a) Work out the exact length of AC. … cm [3] (b) cos x = k Write down the value of k. k = … [1]

4 marks

Mark scheme: 10(a) 57 3 M2 for 112 – 82 or M1 for 112 = AC2 + 82 10(b) 8 1 11

This question in 0580/23 Oct/Nov 2025