GCSE Mathematics. Higher Tier. Paper 4G (Calculator) Time: 1 hour and 45 minutes. For Edexcel. Name

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1 For Edexcel Name GCSE Mathematics Paper 4G (Calculator) Higher Tier Time: 1 hour and 45 minutes Materials required Ruler, protractor, compasses, pen, pencil, eraser. Tracing paper may be used. Instructions and Information for Candidates Write your name in the box at the top of the page. Answer all the questions in the spaces provided in this question paper. The marks for each question and for each part of a question are shown in brackets. The total number of marks for this paper is 100. There are 26 questions in this paper. Calculators may be used. If your calculator does not have a π button, take the value of π to be unless the question instructs otherwise. Advice to Candidates Show all stages in any calculation. Work steadily through the paper. Do not spend too long on one question. If you cannot answer a question, leave it and attempt the next one. Return at the end to those you have left out. Written by Shaun Armstrong Only to be copied for use in the purchaser's school or college EH4G Page 1

2 GCSE Mathematics Formulae: Higher Tier Volume of a prism = area of cross section length cross section length Volume of sphere = 4 3 πr3 Surface area of sphere = 4πr 2 Volume of cone = 1 3 πr2 h Curved surface area of cone = πrl r l h r In any triangle ABC b C a The Quadratic Equation The solutions of ax 2 + bx + c = 0 where a 0, are given by x = b± b2 4ac 2a A c B Sine Rule a sin A = b sin B = c sin C Cosine Rule a 2 = b 2 + c 2 2bc cos A Area of triangle = 1 2 ab sin C EH4G Page 2

3 Answer ALL TWENTY SIX questions. Write your answers in the spaces provided. You must write down all the stages in your working. 1. On the grid, draw an enlargement of the shaded shape by a scale factor of 2. Q1 (Total 2 marks) EH4G Page 3

4 2. (a) Use your calculator to work out the value of ( ) 2 Write down all the figures on your calulator display. (b) Write your answer to part (a) correct to 2 significant figures. (1) (Total 3 marks) Q2 3. Diagram NOT accurately drawn cm cm The diagram shows two triangles. (a) State whether or not the triangles are congruent. (1) (b) Explain your answer. (1) (Total 2 marks) Q3 EH4G Page 4

5 4. (a) Multiply out and simplify 7(a + b) 3(2a 3b) (b) Solve 7x 18 = 24 x = (Total 4 marks) Q4 5. Work out the sum of the interior angles of a hexagon. Q5 (Total 2 marks) EH4G Page 5

6 6. Here are some patterns made from sticks. Pattern number 1 Pattern number 2 Pattern number 3 Pattern number 4 Find an expression, in terms of n, for the number of sticks in pattern number n. Q6 (Total 2 marks) 7. Diagram NOT accurately drawn 15.2 cm 6.7 cm Work out the length of the side x in this triangle. Give your answer correct to an appropriate degree of accuracy. x cm Q7 (Total 4 marks) EH4G Page 6

7 8. The diagram shows a 3-D shape. Front (a) Sketch the front elevation of the shape from the direction shown. (b) Sketch the plan of the shape. (Total 4 marks) Q8 EH4G Page 7

8 9. Solve 25x = 20 x = or x = Q9 (Total 4 marks) 10. (a) (i) The sum of two prime numbers is 12 Write down the two numbers. and (ii) The product of two prime numbers is 38 Write down the two numbers. and (b) Daria says The difference between two consecutive prime numbers is never more than 4 Write down an example to show that Daria is not correct. (Total 4 marks) Q10 EH4G Page 8

9 11. A racing car is 4.2 metres long. Diagram NOT accurately drawn A model is made of the car using a scale of 1 : 30 (a) Work out the length, in centimetres, of the model. cm The wheels of the model have a radius of 1.1 centimetres. (b) Work out the circumference of a wheel of the racing car. Give your answer correct to the nearest centimetre. cm (3) (Total 5 marks) Q11 EH4G Page 9

10 12. B C A D ABCD is a rectangle. Shade the region inside the rectangle which is nearer to BC than AD and more than 5 cm from the point B. Q12 (Total 3 marks) 13. The table shows the annual rainfall recorded at a weather station over 6 years. Year Annual rainfall (cm) Work out the three-year moving averages for these data. Q13 (Total 3 marks) EH4G Page 10

11 14. Solve x 7 4 = x 5 x = Q14 (Total 3 marks) 15. Here are the equations of eight straight lines. A y = 4 x B y = 4x 1 C y = x + 4 D y = 1 4 x + 1 E y = 4 4x F y = x G y = 1 4x H y = x 1 4 Write down the letter of an equation of (i) a line parallel to the line with equation C, (ii) a line perpendicular to the line with equation C, (iii) a line perpendicular to the line with equation F. equation (1) equation (1) equation (1) (Total 3 marks) Q15 EH4G Page 11

12 16. The table gives information on the distance travelled to work by the employees of a company. Distance (d km) Frequency 0 < d < d < d < d < d 50 5 (a) Draw a histogram to represent this information Distance (d km) (4) EH4G Page 12

13 (b) Work out an estimate for the mean distance travelled to work by these employees. km (4) (Total 8 marks) Q B Diagram NOT accurately drawn 40 cm 16 cm A C In triangle ABC, AB = 40 cm and BC = 16 cm. Angle ACB is a right-angle. Calculate the size of angle BAC. Give your answer correct to 1 decimal place. Q17 (Total 3 marks) EH4G Page 13

14 18. Ron invests a sum of money at 4% compound interest per annum. After 1 year his investment is worth 2600 (a) Calculate the amount of money Ron invested. (3) (b) Calculate the value of his investment after another 2 years. (3) (Total 6 marks) Q Convert 4 pints into litres. litres Q19 (Total 2 marks) EH4G Page 14

15 20. y M y = (1 + x)(4 x) O x The diagram shows a sketch of the curve with equation y = (1 + x)(4 x) (a) Write down the x-coordinates of the points where the curve crosses the x-axis. x = and x = (b) (i) Using your answers to part (a), or otherwise, find the x-coordinate of the maximum point, M, of the curve. (ii) Hence, find the y-coordinate of the point M. x = The curve y = (1 + x)(4 x) is reflected in the x-axis. y = (c) Find the equation of the curve following this transformation. Give your answer in the form y = ax 2 + bx + c. (Total 6 marks) Q20 EH4G Page 15

16 21. Diagram NOT accurately drawn r The radius of the base of a cone is r cm. The height of the cone is 2 cm more than the radius of the base. The volume of the cone is 20π cm 3. (a) Show that r 3 + 2r 2 = 60 The equation r 3 + 2r 2 = 60 has a solution between 3 and 4. (b) Use a trial and improvement method to find this solution. Give your answer correct to 1 decimal place. You must show all your working. r = (4) (Total 6 marks) Q21 EH4G Page 16

17 22. Diagram NOT accurately drawn 21 cm 33 cm A shoebox is in the shape of a cuboid and has a volume of 8300 cm 3 correct to 2 significant figures. The length of the box is 33 cm correct to the nearest cm. The width of the box is 21 cm correct to the nearest cm. Calculate the upper bound for the height of the box. Write down all the figures on your calculator display. cm Q22 (Total 4 marks) EH4G Page 17

18 23. Diagram NOT B accurately drawn cm 5.9 cm A C ABC is a triangle. AB = 4.2 cm. BC = 5.9 cm. Angle ABC = 78. Calculate the area of triangle ABC. Give the units with your answer. Q23 (Total 3 marks) EH4G Page 18

19 24. Diagram NOT accurately drawn 4 cm 3 cm 8 cm 9 cm A frustum is formed by removing a cone of height 4 cm from a larger cone. The frustum has a height of 8 cm. The frustum has a lower radius of 9 cm and an upper radius of 3 cm. Find the volume of the frustum. Give your answer correct to 3 significant figures. cm 3 Q24 (Total 3 marks) EH4G Page 19

20 25. The force, F newtons, on a satellite is inversely proportional to the square of its distance, d km, from the centre of the Earth. When d = 8000, F = 1250 (a) Find a formula for F in terms of d. F = (3) (b) Calculate the value of d when F = 800 d = (Total 5 marks) Q25 EH4G Page 20

21 26. A bag contains n coloured counters. Olga picks a counter at random from the bag. The probability that the counter is green is 1 2. Olga does not replace the first counter. She then picks another counter at random from the bag. The probability that both counters are green is (a) Show that n 2 4 n 1 = (3) (b) Hence, or otherwise, find the value of n. n = (3) (Total 6 marks) Q26 TOTAL FOR PAPER: 100 MARKS END EH4G Page 21