A cone has base radius r and vertical height h. Which of the following gives the slant height l of the cone?
A) √r2 + h2
B) √r2 - h2
C) √h2 - r2
D) r + h
(Total for Question 1 is 1 mark)
2
State Pythagoras' theorem for a right-angled triangle with hypotenuse h and shorter sides x and y.
(Total for Question 2 is 1 mark)
3
Calculate the length of the diagonal of a rectangle with length 9 cm and width 12 cm.
(Total for Question 3 is 2 marks)
4
A cone has base radius 5 cm and slant height 13 cm. Find h2, the square of the vertical height, without evaluating the square root.
(Total for Question 4 is 1 mark)
5
Hence write down the vertical height of the cone described in the previous question.
(Total for Question 5 is 1 mark)
6
A cuboid has a square base of side 7 cm and height 24 cm. State the length of the diagonal of the square base, in exact surd form.
(Total for Question 6 is 1 mark)
7
To find the angle between the space diagonal of a cuboid and its base, which trigonometric ratio directly relates the vertical height, the base diagonal and the angle?
A) Sine
B) Cosine
C) Tangent
D) Pythagoras' theorem
(Total for Question 7 is 1 mark)
8
A cuboid has length 2 cm, width 10 cm and height 11 cm. Calculate the length of the space diagonal connecting two opposite vertices of the cuboid.
(Total for Question 8 is 2 marks)
9
A cuboid has length 4 cm, width 13 cm and height 16 cm. Calculate the length of the space diagonal connecting two opposite vertices of the cuboid.
(Total for Question 9 is 2 marks)
10
A square-based pyramid has base ABCD with side 8 cm and centre M; each sloping edge has length 12 cm. Given that AM = 4sqrt(2) cm, find the vertical height of the pyramid, giving your answer correct to 3 significant figures.
(Total for Question 10 is 2 marks)
11
A vertical pole stands at corner A of a rectangular horizontal field ABCD, with AB = 9 m and AD = 12 m. Given that the diagonal AC = 15 m, and the angle of elevation of the top of the pole from corner C is 20 degrees, calculate the height of the pole, giving your answer correct to 3 significant figures.
(Total for Question 11 is 2 marks)
12
A cone has base radius 9 cm and vertical height 12 cm. State the value of tan(θ), where θ is the semi-vertical angle of the cone (the angle between the slant height and the vertical axis).
(Total for Question 12 is 1 mark)
13
Hence find the semi-vertical angle θ described in the previous question, giving your answer correct to 1 decimal place.
(Total for Question 13 is 2 marks)
14
A cuboid has a rectangular base with length 20 cm and width 15 cm, and height 4 cm. Calculate the length of the diagonal connecting two opposite vertices of the cuboid, giving your answer correct to 3 significant figures.
(Total for Question 14 is 3 marks)
15
A cone has base diameter 20 cm and slant height 26 cm. Calculate the vertical height of the cone, giving your answer correct to 3 significant figures.
(Total for Question 15 is 3 marks)
16
A room is in the shape of a cuboid, 6 m long, 8 m wide and 3 m high. Calculate the angle between the diagonal of the floor and the diagonal connecting two opposite corners of the room (one on the floor, one on the ceiling), giving your answer correct to 1 decimal place.
(Total for Question 16 is 3 marks)
17
A square-based pyramid has base side 10 cm and vertical height 12 cm, with apex V directly above the centre M of the base. Calculate the length of a sloping edge VA, giving your answer correct to 3 significant figures.
(Total for Question 17 is 3 marks)
18
A vertical mast MT stands at corner M of a horizontal rectangular sports pitch MNPQ, with MN = 40 m and MQ = 30 m. The angle of elevation of the top T of the mast from corner P (diagonally opposite M) is 15 degrees.
(a)Find the length of the diagonal MP.(1)
(b)Hence calculate the height of the mast, MT, giving your answer correct to 3 significant figures.(3)
(Total for Question 18 is 4 marks)
19
In a storage container, three edges OA, OB and OC meet at corner O and are mutually perpendicular, with OA = 12 cm, OB = 9 cm and OC = 8 cm.
(a)Find the length AB.(1)
(b)Find the length AC, giving your answer correct to 3 significant figures.(1)
(c)Find the length BC, giving your answer correct to 3 significant figures.(1)
(d)Using exact (unrounded) values, calculate the size of angle BAC, giving your answer correct to 1 decimal place.(2)
(Total for Question 19 is 5 marks)
Mark scheme · G6D 3D Pythagoras and Trigonometry: Fluency and Exam Drill
Question 1
B1 A cao
Answer: A) √r2 + h2
Question 2
B1 h2 = x2 + y2 oe
Answer: h2 = x2 + y2
Question 3
M1 d = √92 + 122
A1 d = 15 cm cao
Answer: 15 cm
Question 4
B1 144 cao
Answer: 144
Question 5
B1 12 cm, ft from h2 = 144
Answer: 12 cm
Question 6
B1 7sqrt(2) cm (exact) cao
Answer: 7sqrt(2) cm
Question 7
B1 C cao
Answer: C) Tangent
Question 8
M1 d = √22 + 102 + 112
A1 d = 15 cm cao
Answer: 15 cm
Question 9
M1 d = √42 + 132 + 162
A1 d = 21 cm cao
Answer: 21 cm
Question 10
M1 height = √122 - (4sqrt(2))2 = √144 - 32
A1 10.6 cm awrt (3sf)
Answer: 10.6 cm
Question 11
M1 height = 15 * tan(20)
A1 5.46 m awrt (3sf)
Answer: 5.46 m
Question 12
B1 9/12 oe (0.75 or 3/4) cao
Answer: 0.75
Question 13
M1 θ = arctan(0.75), ft from part above
A1 36.9 degrees awrt (1dp)
Answer: 36.9 degrees
Question 14
M1 finds the base diagonal, √202 + 152 = √625 = 25
M1 combines with the height: d = √252 + 42
A1 25.3 cm awrt (3sf)
Answer: 25.3 cm
Question 15
M1 radius = 20/2 = 10 (half the diameter)
M1 h = √262 - 102
A1 h = 24 cm cao (exact)
Answer: 24 cm
Question 16
M1 floor diagonal = √62 + 82 = √100 = 10
M1 tan(angle) = 3/10, using the right-angled triangle formed by the height, floor diagonal and space diagonal
A1 16.7 degrees awrt (1dp)
Answer: 16.7 degrees
Question 17
M1 finds AM2 = (10/√2)2 = 50, the square of half the base diagonal
M1 VA = √122 + 50
A1 13.9 cm awrt (3sf)
Answer: 13.9 cm
Question 18
(a) B1 MP = 50 m cao
(a) Answer: 50 m
(b) M1 recognises triangle MPT is right-angled at M, since the mast is vertical