(a)Which expression gives the length of the main diagonal of the cuboid?(1)
A) √a + b + c
B) √a2 + b2 + c2
C) a2 + b2 + c2
D) √a * b * c
(b)A cuboid has a = 3 cm, b = 4 cm and c = 12 cm. Calculate the length of the main diagonal.(2)
(Total for Question 1 is 3 marks)
2
The diagram shows a cuboid ABCDEFGH, with diagonal AG drawn together with AC, the diagonal of the base ABCD. E, F, G and H are directly above A, B, C and D respectively.
(a)Give a reason why triangle ACG is right-angled at C.(1)
(b)State which line segment represents the projection of AG onto the base ABCD.(1)
(Total for Question 2 is 2 marks)
3
A cuboid has dimensions 3 cm by 4 cm by 5 cm. Calculate the length of the diagonal of the cuboid, giving your answer to 3 significant figures.
Diagram NOT accurately drawn
(Total for Question 3 is 3 marks)
4
The diagram shows a cuboid ABCDEFGH with AB = 12 cm, BC = 9 cm and CG = 8 cm. E, F, G and H are directly above A, B, C and D respectively.
Diagram NOT accurately drawn
(a)Calculate the length of the diagonal AC of the base ABCD.(2)
(b)Hence calculate the length of the diagonal AG of the cuboid.(2)
(Total for Question 4 is 4 marks)
5
A cuboid has dimensions 5 cm by 6 cm by 7 cm. Calculate the length of the main diagonal of the cuboid, giving your answer to 3 significant figures.
Diagram NOT accurately drawn
(Total for Question 5 is 3 marks)
6
The diagram shows a cuboid ABCDEFGH with AB = 4 cm, BC = 6 cm and CG = 12 cm. Show that the length of the diagonal AG is 14 cm.
Diagram NOT accurately drawn
(Total for Question 6 is 3 marks)
7
The diagram shows a tent in the shape of a triangular prism of length 4 m. The triangular cross-section is isosceles, with base 2.4 m and vertical height 1.6 m.
Diagram NOT accurately drawn
(a)Calculate the slant length of one sloping side of the triangular cross-section.(2)
(b)The rectangular sloping side panel of the tent has width equal to the slant length found in part (a) and length 4 m (the length of the tent). Calculate the length of the diagonal of this rectangular panel, giving your answer to 3 significant figures.(3)
(Total for Question 7 is 5 marks)
8
The diagram shows a cube ABCDEFGH with edge length 6 cm. E, F, G and H are directly above A, B, C and D respectively.
Diagram NOT accurately drawn
(a)Show that the length of the diagonal AG is 6sqrt(3) cm, giving your answer as a simplified surd.(2)
(b)Calculate the angle that the diagonal AG makes with the base ABCD, giving your answer correct to 1 decimal place.(3)
(Total for Question 8 is 5 marks)
9
The diagram shows a square-based pyramid VABCD with square base ABCD of side 8 cm. Vertex V is directly above the centre O of the base, at a height VO = 5 cm.
Diagram NOT accurately drawn
(a)Calculate the distance OA from the centre O of the base to the corner A, giving your answer to 3 significant figures.(2)
(b)Calculate the length of the slant edge VA, giving your answer to 3 significant figures.(3)
(c)Calculate the size of angle VAO, the angle between the slant edge VA and the base, giving your answer correct to 1 decimal place.(3)
(Total for Question 9 is 8 marks)
10
The diagram shows a cuboid ABCDEFGH with AB = 10 cm, BC = 6 cm and CG = 7 cm. E, F, G and H are directly above A, B, C and D respectively.
Diagram NOT accurately drawn
(a)Calculate the length of the diagonal AC of the base ABCD, giving your answer to 3 significant figures.(2)
(b)Calculate the size of angle GAC, the angle between the diagonal AG and the base ABCD, giving your answer correct to 1 decimal place.(3)
(Total for Question 10 is 5 marks)
11
The diagram shows a cuboid ABCDEFGH with AB = 6 cm, BC = 9 cm and CG = 7 cm. E, F, G and H are directly above A, B, C and D respectively.
Diagram NOT accurately drawn
(a)Calculate the length of the diagonal AG, giving your answer to 3 significant figures.(2)
(b)Calculate the size of angle GAB, the angle between the diagonal AG and the edge AB, giving your answer correct to 1 decimal place.(3)
(Total for Question 11 is 5 marks)
12
A gardener, Amara, is laying out a horizontal rectangular field ABCD with AB = 24 m and AD = 7 m. A vertical flagpole stands at corner A, with its top at point T, where AT = 6 m. A wire connects the top of the flagpole, T, to the opposite corner C of the field.
Diagram NOT accurately drawn
(a)Calculate the distance AC, the diagonal of the field.(2)
(b)Calculate the length of the wire TC, giving your answer to 3 significant figures.(3)
(Total for Question 12 is 5 marks)
13
The diagram shows a cuboid ABCDEFGH with AB = 8 cm and BC = 9 cm. The length of the diagonal AG is 17 cm. E, F, G and H are directly above A, B, C and D respectively.
Diagram NOT accurately drawn
(a)Calculate the length of AC, the diagonal of the base ABCD, giving your answer to 3 significant figures.(2)
(b)Calculate the height CG of the cuboid, using the exact value of AC2 = 145.(3)
(Total for Question 13 is 5 marks)
14
A warehouse manager, Ravi, has a storage room in the shape of a cuboid with length 8 m, width 6 m and height 3 m. He wants to store a metal pole of length 10.5 m by laying it along the main diagonal of the room. Determine, showing your working, whether the pole will fit in the room.
Diagram NOT accurately drawn
(Total for Question 14 is 4 marks)
15
The diagram shows a cone with base radius 5 cm and slant height 13 cm.
Diagram NOT accurately drawn
(a)Show that the vertical height of the cone is 12 cm.(3)
(b)Calculate the angle between the slant height and the base of the cone, giving your answer correct to 1 decimal place.(3)
(Total for Question 15 is 6 marks)
16
The diagram shows a cuboid ABCDEFGH with AB = 11 cm, BC = 5 cm and CG = 9 cm. E, F, G and H are directly above A, B, C and D respectively.
Diagram NOT accurately drawn
(a)Calculate the length of BG, the diagonal of the face BCGF, giving your answer to 3 significant figures.(2)
(b)Calculate the size of angle AGB, the angle between the diagonal AG and the plane BCGF, giving your answer correct to 1 decimal place.(3)
(Total for Question 16 is 5 marks)
17
A surveyor, Fatima, is measuring a vertical mast standing at point A on horizontal ground. From point B on the ground, where AB = 40 m, the angle of elevation of the top of the mast, M, is 22 degrees. Point C is also on the ground, such that angle BAC = 90 degrees and AC = 55 m.
Diagram NOT accurately drawn
(a)Calculate the height of the mast, AM, giving your answer to 3 significant figures.(3)
(b)Calculate the angle of elevation of the top of the mast, M, from point C, giving your answer correct to 1 decimal place.(3)
(Total for Question 17 is 6 marks)
18
The diagram shows a cuboid ABCDEFGH with AB = 6 cm, AD = 8 cm and AE = 10 cm. E, F, G and H are directly above A, B, C and D respectively. Diagonals AC, AF and CF are drawn.
Diagram NOT accurately drawn
(a)Calculate the length of AC, the diagonal of the base ABCD.(2)
(b)Calculate the length of AF, the diagonal of the face ABFE, giving your answer to 3 significant figures.(2)
(c)Calculate the length of CF, giving your answer to 3 significant figures.(3)
(d)Use the cosine rule in triangle ACF to calculate the size of angle CAF, the angle between the diagonals AC and AF, giving your answer correct to 1 decimal place. Use the exact values AC2 = 100, AF2 = 136 and CF2 = 164.(4)
(Total for Question 18 is 11 marks)
Mark scheme · 7.15 3D Pythagoras and Trigonometry
Question 1
(a) B1 correct option identified
(a) Answer: B
(b) M1√32 + 42 + 122
(b) A1 13 cm cao
(b) Answer: 13 cm
Question 2
(a) B1 CG is a vertical edge, so CG is perpendicular to every line in the horizontal base, including AC (oe)
(a) Answer: CG is perpendicular to the base ABCD, so angle ACG = 90 degrees.
(b) B1 AC oe
(b) Answer: AC
Question 3
M1√32 + 42 + 52
M1√50 evaluated
A1 awrt 7.07 cm
Answer: 7.07 cm (3 s.f.)
Question 4
(a) M1√122 + 92
(a) A1 15 cm cao
(a) Answer: 15 cm
(b) M1√152 + 82 using their (a) ft
(b) A1 17 cm cao
(b) Answer: 17 cm
Question 5
M1√52 + 62 + 72
M1√110 evaluated
A1 awrt 10.5 cm
Answer: 10.5 cm (3 s.f.)
Question 6
M1√42 + 62 + 122 set up
M1 = √196
A1 cso, = 14 cm shown fully
Answer: 14 cm (given, shown)
Question 7
(a) M1√1.22 + 1.62
(a) A1 2 m cao
(a) Answer: 2 m
(b) M1√42 + (their 2)2 ft
(b) M1√20 evaluated
(b) A1 awrt 4.47 m
(b) Answer: 4.47 m (3 s.f.)
Question 8
(a) M1√62 + 62 + 62 = √108 seen
(a) A1 cso, simplified to 6sqrt(3) cm
(a) Answer: 6sqrt(3) cm (given, shown)
(b) M1 base diagonal AC = 6sqrt(2) (awrt 8.49) found
(b) M1 tan(angle) = 6 / 8.49 ft
(b) A1 awrt 35.3 degrees
(b) Answer: 35.3 degrees (1 d.p.)
Question 9
(a) M1 OA = (8 x √2) / 2 oe, or √42 + 42
(a) A1 awrt 5.66 cm
(a) Answer: 5.66 cm (3 s.f.)
(b) M1√52 + (their OA)2 ft
(b) M1√57 evaluated
(b) A1 awrt 7.55 cm
(b) Answer: 7.55 cm (3 s.f.)
(c) M1 tan(angle) = 5 / (their OA) ft
(c) M1 arctan evaluated
(c) A1 awrt 41.5 degrees
(c) Answer: 41.5 degrees (1 d.p.)
Question 10
(a) M1√102 + 62
(a) A1 awrt 11.7 cm
(a) Answer: 11.7 cm (3 s.f.)
(b) M1 tan(angle) = 7 / (their AC) ft, using unrounded AC where possible
(b) M1 arctan evaluated
(b) A1 awrt 31.0 degrees
(b) Answer: 31.0 degrees (1 d.p.)
Question 11
(a) M1√62 + 92 + 72
(a) A1 awrt 12.9 cm
(a) Answer: 12.9 cm (3 s.f.)
(b) M1 BG = √92 + 72 found, and angle ABG identified as 90 degrees, giving tan(angle GAB) = BG / AB (or equivalent using cos(angle GAB) = AB / AG)
(b) M1 correct value substituted and inverse trig applied
(b) A1 awrt 62.2 degrees
(b) Answer: 62.2 degrees (1 d.p.)
Question 12
(a) M1√242 + 72
(a) A1 25 m cao
(a) Answer: 25 m
(b) M1√62 + (their AC)2 ft
(b) M1√661 evaluated
(b) A1 awrt 25.7 m
(b) Answer: 25.7 m (3 s.f.)
Question 13
(a) M1√82 + 92
(a) A1 awrt 12.0 cm
(a) Answer: 12.0 cm (3 s.f.)
(b) M1 CG2 = 172 - 145
(b) M1 CG2 = 144
(b) A1 12 cm cao
(b) Answer: 12 cm
Question 14
M1√82 + 62 + 32
M1√109 evaluated
A1 awrt 10.4 m
C1 dependent on A1, correct conclusion: No, the pole will not fit, since 10.4 m < 10.5 m (oe)
Answer: Diagonal = 10.4 m (3 s.f.); No, the pole does not fit since 10.4 m < 10.5 m
Question 15
(a) M1√132 - 52 set up
(a) M1 = √144
(a) A1 cso, = 12 cm shown fully
(a) Answer: 12 cm (given, shown)
(b) M1 tan(angle) = 12 / 5
(b) M1 arctan(2.4) evaluated
(b) A1 awrt 67.4 degrees
(b) Answer: 67.4 degrees (1 d.p.)
Question 16
(a) M1√52 + 92
(a) A1 awrt 10.3 cm
(a) Answer: 10.3 cm (3 s.f.)
(b) M1 recognise AB is perpendicular to plane BCGF, so tan(angle AGB) = AB / BG
(b) M1 tan(angle) = 11 / (their BG) evaluated
(b) A1 awrt 46.9 degrees
(b) Answer: 46.9 degrees (1 d.p.)
Question 17
(a) M1 tan(22) = AM / 40
(a) M1 AM = 40 x tan(22) evaluated
(a) A1 awrt 16.2 m
(a) Answer: 16.2 m (3 s.f.)
(b) M1 tan(angle) = (their AM) / 55, using AC as the horizontal distance since angle BAC = 90 degrees
(b) M1 arctan evaluated
(b) A1 awrt 16.4 degrees, ft their AM
(b) Answer: 16.4 degrees (1 d.p.)
Question 18
(a) M1√62 + 82
(a) A1 10 cm cao
(a) Answer: 10 cm
(b) M1√62 + 102
(b) A1 awrt 11.7 cm
(b) Answer: 11.7 cm (3 s.f.)
(c) M1 identify CG = 10 cm (height) and GF = 8 cm (equal to AD), with angle CGF = 90 degrees
(c) M1√102 + 82 = √164
(c) A1 awrt 12.8 cm
(c) Answer: 12.8 cm (3 s.f.)
(d) M1 cos(A) = (AC2 + AF2 - CF2) / (2 x AC x AF) correctly rearranged
(d) M1 correct substitution: (100 + 136 - 164) / (2 x 10 x 11.6619)