BC is parallel to DE. AB is twice as long as BD. AD = 36 cm and AC = 27 cm. (a) Work out the length of AB. AB =... cm (2 marks)

Transcription

1 shape and space 2 higher Question 1 BC is parallel to DE. AB is twice as long as BD. AD = 36 cm and AC = 27 cm. (a) Work out the length of AB. (b) Work out the length of AE. AB =... cm AE =... cm

2 Question 2 BE is parallel to CD. ABC and AED are straight lines. AB = 6 cm, BC = 24 cm, CD = 20 cm, AE = 3 cm. (a) Calculate the length of BE. (b) Calculate the length of DE.... cm... cm

3 Question 3 Diagram NOT accurately drawn. ABC is a right-angled triangle. ED is parallel to BC. BC = 7.5 cm, ED = 2.5 cm, EB = 2.6 cm. (a) Calculate the area of trapezium BCDE. AE = x cm. (b) Calculate the value of x. The smallest angle of the trapezium is, The lengths shown in the diagram are correct to the nearest millimetre. (c) Calculate the least possible value for tan.

4 Question 4 (a) Factorise 2x x 33 A cone fits exactly on top of a hemisphere to form a solid toy. The radius, CA, of the base of the cone is 3 cm. AB = 5 cm. (b) Show that the total surface area of the toy is 33 cm 2. The radius of the base of a cylinder is x cm. The height of the cylinder is 9.5 cm longer than the radius of its base. The area of the curved surface of the cylinder is equal to the total surface area, 33 cm 2, of the toy. (c) Calculate the height of the cylinder.... cm (6 marks)

5 Question 5 The diagram shows a trapezium ABCD. AB is parallel to DC. AB = 4.8 m, DC = 5.2 m, AD = 1.6 m, angle BAD = 90, angle ADC = 90. Calculate the area of trapezium ABCD. State the units of your answer. Question 6 The expressions in the table below can be used to calculate lengths, areas, or volumes of some shapes. 2 is a number which has no dimensions. The letters a and b represent lengths. Put a tick in the box underneath the three expressions which can be used to calculate an area. 2a + 2b a 2 + b 2 2ab a 2 b a 3 + b 3 (a + b)(a - b)

6 Question 7 A skip is in the shape of a prism with cross-section ABCD. AD = 2.3 m, DC = 1.3 m and BC = 1.7 m. The width of the skip is 1.5 m. (a) Calculate the area of the shape ABCD. (b) Calculate the volume of the skip. The weight of an empty skip is 650 kg. The skip is full to the top with sand. 1 m 3 of sand weighs 4300 kg. (c) Calculate the total weight of the skip and the sand. Question 8 Here are some expressions. r 2 l 2 r 2 4 r 3 abrl abl r 3(a 2 +b 2 )r rl The letters r, l, a and b represent lengths., 2, 3 and 4 are numbers that have no dimensions. Three of the expressions represent volumes Tick the boxes ( ) underneath these three expressions.

7 Question 9 The diagram shows a triangular prism. BC = 4 cm, CF = 12 cm and angle ABC = 90º. The volume of the triangular prism is 84 cm 3. Work out the length of the side AB of the prism. Question 10 The shape below is the cross section of a prism 10 cm long. Calculate the volume of the prism.

8 Question 11 The diagram shows a cylinder. The height of the cylinder is 26.3 cm. The diameter of the base of the cylinder is 8.6 cm. Calculate the volume of the cylinder. Give your answer correct to 3 significant figures. Question 12 The expressions shown below can be used to calculate lengths, areas or volumes of various shapes. The letters r and h represent lengths., 2, 3, 4, 5 and 10 are numbers which have no dimensions. r( 2 ) 4 2 r h r(r + 4h) rh 4 4r r 3 ( r 2 h) 3r 3 h r 2 ( h r) Draw a circle around each of the expressions that can be used to calculate an area.

9 Question 13 The diagram represents a solid shape. From the expressions below, choose the one that represents the volume of the solid shape. 1 and 3 are numbers which have no dimensions. a, b and h are lengths (b 2 ab + a 2 ), 3 1 h(b 2 + ab + a 2 ), 3 h 2 (b 2 a 2 ), (a 2 + b 2 ), 3 h 2 (b 2 ab + a 2 ). Write down the correct expression.

10 Question 14 This container is made from a cylinder and a cube. The cylinder has a height of 20 cm. It has a base radius of 8 cm. The cube has sides of edges 16 cm. (a) Calculate the total volume, in cm 3, of the cylinder. Give your answer to the nearest cm 3. (b) Calculate the total volume, in cm 3, of the container. Give your answer to the nearest cm 3. When the radius of 8 cm was measured, this measurement was rounded to the nearest centimetre. (c) i) Write down the minimum length, in cm, it could be. ii) Write down the maximum length, in cm, it could be.

11 Question 15 Jomo is going to design a circular roundabout. The roundabout will have a circumference of 7 metres. Jomo is given three estimates for the length of the diameter of the roundabout. The estimates are: metres 2 metres 2.23 metres (a) Give a reason why 2.23 metres is the most reasonable estimate to use. (b) Explain why metres and 2 metres are not appropriate to use. Question 16 A tent has a groundsheet as its horizontal base. The shape of the tent is a triangular prism of length 8 metres, with two identical half right-circular cones, one at each end. The vertical cross-section of the prism is an isosceles triangle of height 2.4 metres and base 3.6 metres. (a) Calculate the area of the groundsheet. Give your answer, in m 2, correct to one decimal place. (b) Calculate the total volume of the tent. Give your answer, in m 2, correct to one decimal place. (4 marks)

12 Question 17 The diagram shows a prism. The cross-section of the prism is a trapezium. The lengths of the parallel sides of the trapezium are 8 cm and 6 cm. The distance between the parallel sides of the trapezium is 5 cm. The length of the prism is 20 cm. (a) Work out the volume of the prism.... cm 3 The prism is made out of gold. Gold has a density of 19.3 grams per cm 3. (b) Work out the mass of the prism. Give your answer in kilograms.... kilograms Question 18 A sphere has a radius of 5.4 cm. A cone has a height of 8 cm. The volume of the sphere is equal to the volume of the cone. Calculate the radius of the base of the cone. Give your answer, in centimetres, correct to 2 significant figures.

13 Question 19 The diagram shows a water tank in the shape of a cuboid. The measurements of the cuboid are 20 cm by 50 cm by 20 cm. (a) Work out the volume of the water tank.... cm 3 Water is poured into the tank at a rate of 5 litres per minute. 1 litre = 1000 cm 3. (b) Work out the time it takes to fill the water tank completely. Give your answer in minutes.... minutes Question 20 The heaviest stick of rock ever made was in the shape of a cylinder. The cylinder had a length of 503 cm and a radius of 21.6 cm. (a) Work out the volume of the cylinder. Give your answer correct to 3 significant figures.... cm 3

14 A small stick of rock, in the shape of a cylinder, has a length of 25 cm and a radius of 2.5 cm. It is made using the same recipe as the heaviest stick of rock. The weight of the heaviest stick of rock ever made was kg. (b) Calculate the weight of the small stick of rock. Give your answer, in grams, correct to 3 significant figures.... g