74 Use the cosine rule. M1 3 M1 for use of cosine rule H AB2 = 2.12 + 1.82 2 × 2.1 × 1.8 × cos 70° A1 A1 for correct substitution into cosine rule – = 5.064 328 A1 A1 for 5.06 rounded to 3 or more sf AB = = 2.250 406 A1 A1 for 2.25 rounded to 2 or 3 sf Extra distance = (2.1 + 1.8) 2.25 km 1.65 km A1 A1 1.65 rounded to 2 or 3 sf – 5 75 3 H Use theorem to find BY. C1 C1 for clear diagram showing all data Pythagoras’ BY = = 3.605 551 3 Use the right-angled triangle XYB to find XY. = tan 6° XY = BY × tan 6° = 0.378 958 7 km M1 M1 for use of theorem Pythagoras’ = 378.9587 m A1 A1 for BY unrounded M1 M1 for use of tangent A1 A1 for XY unrounded 379 m A1 A1 for 379 in metres and rounded to 2 or 3 sf 6 76 a 3 H PM = = PM = 8.944 271 9 cm C1 C1 for use of clear diagram VM = = VM = 10.770 33 cos P = = 0.415 227 4 Angle VPM = 65.466 362° M1 M1 for use of theorem Pythagoras’ Vertical height of V above face PRQ is 8.9 cm A1 A1 for 8.9 rounded to 2 or 3 sf b given by VT in diagram above. VT = = C1 C1 for clear diagram =
74 Use the cosine rule. AB2 = 2.12 + 1.82 – 2 × 2.1 × 1.8 × cos 70° = 5.064 328 AB = = 2.250 406 Extra distance = (2.1 + 1.8) – 2.25 km 1.65 km M1 A1 A1 A1 A1 3 M1 for use of cosine rule A1 for correct substitution into cosine rule A1 for 5.06 rounded to 3 or more sf A1 for 2.25 rounded to 2 or 3 sf A1 1.65 rounded to 2 or 3 sf H 5 75 Use Pythagoras’ theorem to find BY. BY = = 3.605 551 3 Use the right-angled triangle XYB to find XY. = tan 6° XY = BY × tan 6° = 0.378 958 7 km = 378.9587 m 379 m C1 M1 A1 M1 A1 A1 3 C1 for clear diagram showing all data M1 for use of Pythagoras’ theorem A1 for BY unrounded M1 for use of tangent A1 for XY unrounded A1 for 379 in metres and rounded to 2 or 3 sf H 6 76 a b PM = = PM = 8.944 271 9 cm VM = = VM = 10.770 33 cos P = = 0.415 227 4 Angle VPM = 65.466 362° Vertical height of V above face PRQ is given by VT in diagram above. VT = = = 8.9 cm C1 M1 A1 C1 3 C1 for use of clear diagram M1 for use of Pythagoras’ theorem A1 for 8.9 rounded to 2 or 3 sf C1 for clear diagram H