Non-right-angled triangles — Paper 2 · IGCSE Mathematics 0580

21 Examiner's North Use North Q NOT TO 140° SCALE 50 m North R 100 m P The diagram shows three points P, Q and R on horizontal ground. PQ = 50 m, PR = 100 m and angle PQR = 140°. (a) Calculate angle PRQ. Answer(a) Angle PRQ = [3] (b) The bearing of R from Q is 100°. Find the bearing of P from R. Answer(b) [2]

5 marks

Mark scheme: sin 140 21 (a) 18.7 3 M2 for sin R = 50 × (= 0.3219…) 100 sin R sin 140 or M1 for = oe 50 100 (b) 261(.3) 2ft M1 360 – 80 – their (a)

This question in 0580/23 May/June 2010

6 C NOT TO 9 cm SCALE B 28° 15 cm A Calculate the area of triangle ABC. Answer cm2 [2]

2 marks

Mark scheme: 6 31.7 2 M1 0.5 × 9 × 15 × sin 28

This question in 0580/23 May/June 2012

18 C 23° A NOT TO SCALE 13 cm 6 cm B In triangle ABC, AB = 6 cm, BC = 13 cm and angle ACB = 23°. Calculate angle BAC, which is obtuse. Answer Angle BAC = [4]

4 marks

Mark scheme: 18 122.2 4 M2 for 13sin23/6 A1 57.8 or M1 for sin23 = sin A 6 13

This question in 0580/23 Oct/Nov 2012

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

11 A triangle has sides of length 2 cm, 8 cm and 9 cm. Calculate the value of the largest angle in this triangle. Answer … [4] __________________________________________________________________________________________

4 marks

Mark scheme: 8 + 2 9 11 113.9 to 114.0 4 M2 for [cos =] 2 × 8 × 2 or M1 for 92 = 82 + 22 – 2 × 8 × 2 × cos x 13 A1 for −0.406 or −0.4063 to −0.4062 or − 32 If 0 scored SC2 for 54.3[1…] or 11.7 or 11.71 to 11.72 9 2 + 2 2 − 8 2 SC1 for [cos =] or 2 × 9 × 2 9 2 + 8 2 − 2 2 [cos =] 2 × 9 × 8 10 9

This question in 0580/21 May/June 2014

16 A helicopter fl ies from its base B to deliver supplies to two oil rigs at C and D. C is 6 km due east of B and the distance from C to D is 8 km. D is on a bearing of 120° from B. North North NOT TO 120° 6 km SCALE B C 8 km D Find the bearing of D from C. Answer … [5] __________________________________________________________________________________________

5 marks

Mark scheme: 16 142 or 142.0… 5 B1 for CBD = 30 6 × sin theirB M2 for [sin D = ] oe 8 6 8 or M1 for = oe sin D sin(their 30) A1 for [D =] 22 or 22.0 or 22.02… B1FT for 90 + (their30 + their22) evaluated correctly for their final answer or for 360 – 90 – theirBCD evaluated correctly for their final answer 1  4 

This question in 0580/21 Oct/Nov 2014

11 C NOT TO 100° SCALE 30° A 24 cm B Use the sine rule to calculate BC. Answer BC = … cm [3]

3 marks

Mark scheme: 24 sin 30 11 12.2 or 12.18 to 12.19 3 M2 for sin 100 or M1 for correct implicit equation sin 100 sin 30 e.g. = 24 BC u × 10

This question in 0580/23 May/June 2015

13 74° NOT TO SCALE y cm 39° 12.4 cm Calculate the value of y. Answer y = … [3]

3 marks

Mark scheme: 124. 13 8.12 or 8.118… 3 M2 for × sin 39 sin 74 sin 39 sin 74 or M1 for implicit version = oe y 124.

This question in 0580/22 Oct/Nov 2015

15 p° 2.8 cm 3.6 cm NOT TO SCALE 5.3 cm Find the value of p. p = … [4]

4 marks

Mark scheme: 2.8 + 3.6 5.3 15 111.2 or 111.1 to 111.2 4 M2 for [cos =] 2 × 2.8 × 3.6 or M1 for implicit form A1 for [cos =] −0.362 to −0.361

This question in 0580/22 Feb/March 2016

26 A 7 cm NOT TO SCALE 35° B 10 cm C (a) Calculate the area of triangle ABC. … cm2 [2] (b) Calculate the length of AC. AC = … cm [4]

6 marks

Mark scheme: 26 (a) 1 20.1 or 20.07 to 20.08 2 M1 for × 7 × 10 × sin35 oe 2 (b) 5.86 or 5.858….. 4 M2 for 7 2 + 10 2 − 2 × 7 × 10 × cos35 A1 for 34.3 .. or 7 2 + 10 2 − AC 2 M1 for cos35 = 2 × 7 × 10

This question in 0580/21 May/June 2016

17 8.15 m C B 30° NOT TO SCALE 110° A Calculate AC. AC = … m [3]

3 marks

Mark scheme: 17 4.34 or 4.336 to 4.337 3 .815 sin 30 M2 for sin 110 sin 110 sin 30 or M1 for = oe .815 AC

This question in 0580/22 May/June 2017

19 In a triangle PQR, PQ = 8 cm and QR = 7 cm. The area of this triangle is 17 cm2. Calculate the two possible values of angle PQR. Angle PQR = … or … [3]

3 marks

Mark scheme: 19 37.4 or 37.38… 3 B2 for one correct and or M1 for 0.5 × 8 × 7 sin = 17 oe 142.6 or 142.6… If zero or M1 only scored, SC1 for two answers with a sum of 180

This question in 0580/22 May/June 2017

15 39 cm NOT TO SCALE x° 24 cm 71.8° Find the value of x. x = … [3]

3 marks

Mark scheme: 15 35.8 or 35.77…. 3 24 × sin71.8 M2 for [sin =] 39 39 24 or M1 for = oe sin71.8 sin x

This question in 0580/23 May/June 2017

14 C 84.6° NOT TO 5.9 cm SCALE B 17.8 cm A Use the sine rule to find angle ABC. Angle ABC = … [3]

3 marks

Mark scheme: 14 19.3 or 19.26 to 19.27 nfww 3 sin84.6 M2 for [sin = ]5.9 × 17.8 5.9 17.8 or M1 for = oe sin B sin84.6

This question in 0580/21 May/June 2018

19 N 16 cm 14 cm NOT TO SCALE L M 19 cm Calculate angle LMN. Angle LMN = … [4]

4 marks

Mark scheme: 19 46.2 or 46.17 to 46.18 4 16 2 + 19 2 − 14 2 M2 for [cos =] 2 × 16 × 19 or M1 for 142 = 192 + 162 – 2 × 19 × 16cosM 421 A1 for 0.692… or 608

This question in 0580/21 May/June 2018

23 x° NOT TO 11 cm 5 cm SCALE 13 cm Calculate the value of x. x = … [4]

4 marks

Mark scheme: 23 102.1 or 102.06 to 102.07 4 112 + 5 2 − 132 M2 for [cos x =] 2 × 11 × 5 or M1 for 13 2 = 112 + 5 2 − 2 × 11 × 5cos x 23 A1 for – 0.209…. or − 110

This question in 0580/23 May/June 2018

16 12 cm NOT TO 8 cm SCALE x 39° Calculate the obtuse angle x in this triangle. x = … [3]

3 marks

Mark scheme: 16 109.3 or 109.26 to 109.27 3 12 sin 39 M2 for 8 8 12 or M1 for = oe sin 39 sin(...)

This question in 0580/21 Oct/Nov 2019

23 A triangle has sides of length 11 cm, 10 cm and 9 cm. Calculate the largest angle in the triangle. … [4]

4 marks

Mark scheme: 23 70.5 or 70.52 to 70.53 4 B3 for 59(.0) or 58.99… or 50.5 or 50.47 to 50.48 OR 10 2 + 9 2 − 112 M2 for oe or equivalent 2 × 10 × 9 expression for smaller angle or M1 for 112 = 10 2 + 9 2 − 2 × 10 × 9cos(...) oe or equivalent expression for smaller angle 1 A1 for oe 3

This question in 0580/23 May/June 2021

21 North 112° B 12.8 km NOT TO SCALE C 25° A 17.6 km The diagram shows the positions of three ships A, B and C. AC = 17.6 km, BC = 12.8 km and angle BAC = 25°. The bearing of C from B is 112° and angle ABC is obtuse. Calculate the bearing of B from A. … [5] Question 22 is printed on the next page.

5 marks

Mark scheme: 21 076 or 076.4 to 076.5 5 B3 for [angle ABC =] 144 or 144.4 to 144.5 OR 17.6sin 25 M2 for [sin ABC =] oe 12.8 17.6 12.8 or M1 for = oe sin B sin25 M1 for 180 – their 35.5 AND M1 for their angle ABC – (180 – 112) oe

This question in 0580/23 Oct/Nov 2022

18 North W NOT TO SCALE 62.4 km U 27.3 km V The diagram shows the position of three towns, U, V and W. U is due west of V and angle UVW = 125°. Calculate the bearing of U from W. … [4]

4 marks

Mark scheme: 18 236[.0…] 4 27.3  sin125 M2 for 62.4 27.3 62.4 or M1 for  sin UWV sin125 M1 for 180 + (125 – 90 ) + their 21 oe or 180   90 their 34  oe If 0 scored SC1 for the correct bearing marked at W

This question in 0580/21 May/June 2023

17 NOT TO 92.5 cm SCALE x° 71 cm The diagram shows a triangle with an acute angle marked x°. The area of the triangle is 2143 cm 2. Work out the value of x. x = … [2]

2 marks

Mark scheme: 17 40.7 or 40.73 to 40.74 2 1 M1 for  92.5  71 sinx = 2143 oe 2

This question in 0580/22 May/June 2023

18 105° 8.3 cm NOT TO SCALE y° 16.2 cm Calculate the value of y. y = … [3]

3 marks

Mark scheme: 18 29.7 or 29.66[...] 3 8.3sin105 M2 for [sin y = ] 16.2 16.2 8.3 or M1 for = oe sin105 sin y

This question in 0580/21 Oct/Nov 2023

19 R NOT TO 13.5 cm SCALE Q 42° 18 cm P Calculate the obtuse angle PRQ. Angle PRQ = … [4]

4 marks

Mark scheme: 19 116.9 or 116.85… 4 −1 18sin 42  M3 for 180 – sin    13.5  or B3 for 63.1 or 63.14 to 63.15 18sin 42 or M2 for [sin PRQ = ] 13.5 18 13.5 or M1 for = oe sin PRQ sin 42

This question in 0580/23 Oct/Nov 2023

23 B NOT TO 6 cm SCALE 30° A 10 cm C Work out the area of triangle ABC. … cm2 [3]

3 marks

Mark scheme: 23 15 3 1 M1 for 6 10  sin30 oe 2 1 B1 for sin 30 = soi 2

This question in 0580/23 Oct/Nov 2025