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Ma KEY STAGE 3 Mathematics test Paper 2 Calculator allowed TIER 6–8 2003 Please read this page, but do not open your booklet until your teacher tells you to start. Write your name and the name of your school in the spaces below. First name Last name School Remember ■ The test is 1 hour long. ■ You may use a calculator for any question in this test. ■ You will need: pen, pencil, rubber, ruler and a scientific or graphic calculator. ■ Some formulae you might need are on page 2. ■ This test starts with easier questions. ■ Try to answer all the questions. ■ Write all your answers and working on the test paper – do not use any rough paper. Marks may be awarded for working. ■ Check your work carefully. ■ Ask your teacher if you are not sure what to do. For marker’s use only QCA/03/971 Sourced from SATs-Papers.co.uk Total marks Borderline check http://www.SATs-Papers.co.uk Instructions Answers This means write down your answer or show your working and write down your answer. Calculators You may use a calculator to answer any question in this test. Formulae You might need to use these formulae Trapezium Area = 1 (a + b)h 2 Prism Volume = area of cross-section t length KS3/03/Ma/Tier 6–8/P2 Sourced from SATs-Papers.co.uk 2 http://www.SATs-Papers.co.uk Ratio of ages 1. Paul is 14 years old. His sister is exactly 6 years younger, so this year she is 8 years old. This year, the ratio of Paul’s age to his sister’s age is 14 : 8 14 : 8 written as simply as possible is 7 : 4 (a) When Paul is 21, what will be the ratio of Paul’s age to his sister’s age? Write the ratio as simply as possible. : 1 mark (b) When his sister is 36, what will be the ratio of Paul’s age to his sister’s age? Write the ratio as simply as possible. : 1 mark (c) Could the ratio of their ages ever be 7 : 7? Tick ( ) Yes or No. Yes No Explain how you know. 1 mark KS3/03/Ma/Tier 6–8/P2 Sourced from SATs-Papers.co.uk 3 http://www.SATs-Papers.co.uk Sizing 2. The information in the box describes three different squares, A, B and C. The area of square A is 36cm2 The side length of square B is 36 cm The perimeter of square C is 36cm Put squares A, B and C in order of size, starting with the smallest. You must show calculations to explain how you work out your answer. smallest KS3/03/Ma/Tier 6–8/P2 Sourced from SATs-Papers.co.uk 2 marks largest 4 http://www.SATs-Papers.co.uk Nets 3. The squared paper shows the nets of cuboid A and cuboid B. (a) Do the cuboids have the same surface area? Show calculations to explain how you know. 1 mark (b) Do the cuboids have the same volume? Show calculations to explain how you know. 2 marks KS3/03/Ma/Tier 6–8/P2 Sourced from SATs-Papers.co.uk 5 http://www.SATs-Papers.co.uk Beaches 4. Two beaches are very similar. A survey compared the number of animals found in one square metre on each beach. One beach had not been cleaned. The other beach had been cleaned. (a) The data for the beach that had not been cleaned represent 1620 animals. Complete the table to show how many of each animal were found. 2 marks KS3/03/Ma/Tier 6–8/P2 Sourced from SATs-Papers.co.uk 6 http://www.SATs-Papers.co.uk (b) The data for the beach that had been cleaned represent 15 animals. Complete the table to show how many of each animal were found on the cleaned beach. 2 marks (c) Cleaning the beach changes the numbers of animals and the proportions of animals. Write a sentence to describe both these changes. 1 mark KS3/03/Ma/Tier 6–8/P2 Sourced from SATs-Papers.co.uk 7 http://www.SATs-Papers.co.uk Equations 5. Find the values of t and r 2 t = 3 6 t= 1 mark 2 5 = r 3 r= KS3/03/Ma/Tier 6–8/P2 Sourced from SATs-Papers.co.uk 8 1 mark http://www.SATs-Papers.co.uk Star design 6. This pattern has rotation symmetry of order 6 (a) What is the size of angle w ? Show your working. Not drawn accurately ° 2 marks (b) Each quadrilateral in the pattern is made from two congruent isosceles triangles. What is the size of angle y ? Show your working. Not drawn accurately ° 2 marks KS3/03/Ma/Tier 6–8/P2 Sourced from SATs-Papers.co.uk 9 http://www.SATs-Papers.co.uk Ks and ms 7. On the square grids below you can join dots with two different length lines. Length m is greater than length k (a) Draw a shape with each perimeter shown below. The first one is done for you. Perimeter 4k + 3 m Perimeter 4k + m 1 mark Perimeter 2 ( 2 k + m ) 1 mark KS3/03/Ma/Tier 6–8/P2 Sourced from SATs-Papers.co.uk 10 http://www.SATs-Papers.co.uk (b) What is the area of this triangle? Write it in terms of m 1 mark (c) This is the same triangle and grid. What is the area of the triangle? Write it in terms of k 1 mark (d) Using your answers to (b) and (c), explain why m 2 = 2 k 2 1 mark KS3/03/Ma/Tier 6–8/P2 Sourced from SATs-Papers.co.uk 11 http://www.SATs-Papers.co.uk Giraffe 8. A book gives this information: A baby giraffe was born that was 1.58 metres high. It grew at a rate of 1.3 centimetres every hour. Suppose the baby giraffe continued to grow at this rate. About how many days old would it be when it was 6 metres high? Show your working. days old 3 marks KS3/03/Ma/Tier 6–8/P2 Sourced from SATs-Papers.co.uk 12 http://www.SATs-Papers.co.uk Long-eared owls 9. Owls eat small mammals. They regurgitate the bones and fur in balls called pellets. The table shows the contents of 62 pellets from long-eared owls. Number of mammals found in the pellet 1 2 3 4 5 6 Frequency 9 17 24 6 5 1 (a) Show that the total number of mammals found is 170 1 mark (b) Calculate the mean number of mammals found in each pellet. Show your working and give your answer correct to 1 decimal place. 2 marks (c) There are about 10 000 long-eared owls in Britain. On average, a long-eared owl regurgitates 1.4 pellets per day. Altogether, how many mammals do the 10 000 long-eared owls eat in one day? Show your working and give your answer to the nearest thousand. 2 marks KS3/03/Ma/Tier 6–8/P2 Sourced from SATs-Papers.co.uk 13 http://www.SATs-Papers.co.uk Pictures 10. Here are four pictures, A, B, C and D. They are not to scale. (a) Picture A can be stretched horizontally to make picture B. Show that the horizontal factor of enlargement is 1.5 1 mark (b) Picture A can be stretched vertically to make picture C. The vertical factor of enlargement is 1.25 What is the height, h, of picture C? cm KS3/03/Ma/Tier 6–8/P2 Sourced from SATs-Papers.co.uk 14 http://www.SATs-Papers.co.uk 1 mark (c) Show that pictures A and D are not mathematically similar. 1 mark (d) Picture E (not shown) is mathematically similar to picture A. The width of picture E is 3cm. What is the height of picture E? cm 1 mark KS3/03/Ma/Tier 6–8/P2 Sourced from SATs-Papers.co.uk 15 http://www.SATs-Papers.co.uk Coffee 11. A cup of coffee costs £ 1.75 The diagram shows how much money different people get when you buy a cup of coffee. Not drawn accurately (a) Complete the table to show what percentage of the cost of a cup of coffee goes to retailers, growers and others. Show your working. Retailers % Growers % Others % 2 marks KS3/03/Ma/Tier 6–8/P2 Sourced from SATs-Papers.co.uk 16 http://www.SATs-Papers.co.uk (b) Some people think the growers should get more. Suppose the percentages changed to: Retailers 23% Growers 10% Others 67% Suppose the retailers still got 44p from each cup of coffee sold. How much would a cup of coffee cost? Show your working. £ 2 marks KS3/03/Ma/Tier 6–8/P2 Sourced from SATs-Papers.co.uk 17 http://www.SATs-Papers.co.uk Cissoid of Diocles 12. The equation of the curve shown is When x = 2.5 calculate the positive value of y Show all the digits on your calculator display. 1 mark When x = 2.5 give both values of y correct to 3 significant figures. and KS3/03/Ma/Tier 6–8/P2 Sourced from SATs-Papers.co.uk 18 http://www.SATs-Papers.co.uk 2 marks Populations 13. The table shows information about some countries. Country Population Area (km2) Canada 3.1 t 107 1.0 t 10 7 France 6.0 t 107 5.5 t 10 5 Gambia 1.4 t 106 1.1 t 10 4 India 1.0 t 109 3.3 t 106 United Kingdom 6.0 t 107 2.4 t 105 United States 2.8 t 10 8 9.3 t 106 (a) Use the table to fill in the gaps: The country with the largest population is 1 mark The country with the smallest area is 1 mark (b) On average, how many more people per km2 are there in the United Kingdom than in the United States? Show your working. 3 marks KS3/03/Ma/Tier 6–8/P2 Sourced from SATs-Papers.co.uk 19 http://www.SATs-Papers.co.uk Joining 14. Two right-angled triangles are joined together to make a larger triangle ACD. Not drawn accurately (a) Show that the perimeter of triangle ACD is 78cm. 2 marks (b) Show that triangle ACD is also a right-angled triangle. 2 marks KS3/03/Ma/Tier 6–8/P2 Sourced from SATs-Papers.co.uk 20 http://www.SATs-Papers.co.uk Squares X 15. y 2 represents a square number; y is an integer. (a) Think about the expression 9 + y 2 Explain how you know there are values of y for which this expression does not represent a square number. 1 mark (b) Explain why the expression 16 y 2 must represent a square number. 1 mark KS3/03/Ma/Tier 6–8/P2 Sourced from SATs-Papers.co.uk 21 http://www.SATs-Papers.co.uk Cylinder 16. A cylinder has a radius of 2.5cm. The volume of the cylinder, in cm3, is 4.5 What is the height of the cylinder? Show your working. cm 3 marks KS3/03/Ma/Tier 6–8/P2 Sourced from SATs-Papers.co.uk 22 http://www.SATs-Papers.co.uk Triangle 17. The diagram shows a triangle. Side XY is of length 11 b Side XZ is of length 2 a + 3b Side YZ is of length a The triangle is isosceles, with XY = XZ The perimeter of the triangle is 91 Use algebra to find the values of a and b 1 mark 1 mark a = b = 1 mark KS3/03/Ma/Tier 6–8/P2 Sourced from SATs-Papers.co.uk 23 http://www.SATs-Papers.co.uk END OF TEST Sourced from SATs-Papers.co.uk © Qualifications and Curriculum Authority 2003 QCA, Key Stage 3 Team, 83 Piccadilly, London W1J 8QA http://www.SATs-Papers.co.uk 254654