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1 For Edexcel Name GCSE Mathematics Paper 4B (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 22 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 EH4B 09 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 EH4B 09 Page 2

3 Answer ALL TWENTY TWO questions. Write your answers in the spaces provided. You must write down all the stages in your working. 1. (a) Find the cube root of (1) (b) Work out (Total 3 marks) Q1 2. Brian thinks of a number. He subtracts 5 from the number and then divides the result by 2. His answer is 19. What number did Brian first think of? Q2 (Total 2 marks) EH4B 09 Page 3

4 3. The scatter graph shows the age, in years, and the annual salary, in thousands of pounds, of nine employees of a company. Salary ( '000s) Age (years) (a) What is the median age of these nine employees? years (1) (b) What is the median annual salary of these nine employees? (1) (c) Describe the relationship between the age of the employees and their salary. (1) EH4B 09 Page 4

5 Another employee of the company is 50 years old. (d) Using the graph, estimate her annual salary. (Total 5 marks) Q3 4. Diagram NOT C accurately drawn B D A E x F In the diagram, ABCDE is a regular pentagon and AEF and CDF are straight lines. Work out the size of angle x. Q4 (Total 3 marks) EH4B 09 Page 5

6 5. (a) The number 135 can be written in the form 3 x y, where x and y are prime numbers. Find the value of x and the value of y. x = y = (b) Find the Highest Common Factor (HCF) of 60 and 135. (c) Find the Lowest Common Multiple (LCM) of 60 and 135. (Total 6 marks) Q5 6. The scale of a map is 1 : (a) The distance between 2 bridges is 1.8 km. Work out the distance in cm between the bridges on the map. cm (b) The area of a car park on the map is 0.5 cm 2. Work out the real area of the car park in m 2. m 2 (Total 4 marks) Q6 EH4B 09 Page 6

7 7. (a) Simplify x 4 x 4 (1) (b) Simplify x + y + 3x 3y (c) Solve x 2 9x + 14 = 0 x = or x = (3) (Total 6 marks) Q7 8. The United Kingdom has a population of approximately (a) Write in standard form. (1) Each person in the United Kingdom drinks an average of 415 litres of tap water per year. (b) Calculate the total amount of tap water drunk in the United Kingdom each year. Give your answer in standard form correct to 3 significant figures. litres (Total 3 marks) Q8 EH4B 09 Page 7

8 9. Diagram NOT y accurately drawn D E z G C 1 1 O F A x B The diagram shows a cuboid on a 3-dimensional grid. The point B has coordinates (4, 0, 2) and the point D has coordinates (0, 2, 0). (a) Write down the coordinates of the points (i) A, (,, ) (ii) G. (,, ) (b) Find the coordinates of the midpoint of BD. (,, ) (Total 4 marks) Q9 EH4B 09 Page 8

9 10. Here is a net of a prism. Draw a sketch of the prism that can be formed from this net. Q10 (Total 2 marks) EH4B 09 Page 9

10 11. The table shows information about how long it took 80 girls from Runfast School to run 100 m. Time in seconds (t) Frequency 12 t < t < t < t < t < t < 24 3 (a) (b) Complete the cumulative frequency table. Time in seconds (t) Cumulative Frequency 12 t < t < t < t < t < t < 24 On the grid, draw a cumulative frequency graph for your table. (1) Cumulative Frequency Time in seconds (t) EH4B 09 Page 10

11 The fastest time was 12.8 s and the slowest time was 23.2 s. (c) Use this information and your graph to draw a box plot for the girls' times Time in seconds (t) (3) Here is a box plot showing information about how long it took 80 girls from Quickstep School to run 100 m Time in seconds (t) (d) Write down two differences between these two distributions. First difference.. Second difference.. (Total 8 marks) Q11 EH4B 09 Page 11

12 12. In a sudoku competition, competitors are awarded points for puzzles completed correctly in a fixed period of time. 3 points are awarded for each small puzzle completed and 8 points for each large puzzle. (a) Write down a formula for the number of points obtained, P, when S small and L large puzzles are completed. Siddra gets 43 points in the competition. She completes more small puzzles than large ones. (b) Find how many small puzzles Siddra completed. (3) (Total 5 marks) Q12 EH4B 09 Page 12

13 13. (a) On the grid, draw the graph of y = x 2 2x 5 for values of x from 2 to 4. y O x (4) (b) Use a graphical method to find estimates of the solutions to the equation x 2 2x 5 = x 3 x = or x = (3) (Total 7 marks) Q13 EH4B 09 Page 13

14 14. (a) B Diagram NOT accurately drawn 6.8 cm A cm C The diagram shows a sketch of triangle ABC. AB = 6.8 cm. AC = 5.3 cm. Angle BAC = 103. Make an accurate drawing of triangle ABC. The line AC has been drawn for you. A C EH4B 09 Page 14

15 (b) Using ruler and compasses, construct the perpendicular to the line PQ that passes through the point X. You must show all your construction lines. P X Q (Total 4 marks) Q14 EH4B 09 Page 15

16 15. C D Diagram NOT accurately drawn 10 cm B 22 cm A AB = 22 cm. AC = 10 cm. Angle ACB = angle ADC = angle BAD = 90. (a) Calculate the size of angle BAC. Give your answer correct to 1 decimal place. (3) (b) Calculate the length of CD. Give your answer correct to 3 significant figures. cm (3) (Total 6 marks) Q15 EH4B 09 Page 16

17 16. The table shows the number of sales each quarter of a new product. It also shows four-point moving averages. Year Quarter Number of sales c Four-point moving average a b 57 (a) Work out the four-point moving averages labelled a and b. (b) Work out the number of sales in the 2 nd quarter of year 3, labelled c. a = b = c = (Total 4 marks) Q16 EH4B 09 Page 17

18 17. Diagram NOT accurately drawn 21 m 35 m 35 m The diagram is a sketch of the Louvre Pyramid in Paris. The base is a square of side length 35 m. The height of the pyramid is 21 m. The vertex is directly above the middle of the base. Calculate the volume of the pyramid. m 3 Q17 (Total 3 marks) EH4B 09 Page 18

19 18. Diagram NOT y accurately drawn Q (6, 5) R O P (0, 2) x The diagram shows the straight line which passes through the point P (0, 2) and the point Q (6, 5). (a) Find the equation of this straight line. (3) (b) Find the coordinates of the point R where this line crosses the x-axis. (, ) (c) Write down the equation of the straight line perpendicular to PQ which passes through P. (Total 7 marks) Q18 EH4B 09 Page 19

20 19. The kinetic energy of a ball, E joules, is directly proportional to the square of its speed, v metres per second. E = 126 when v = 2. (a) Express E in terms of v. (3) (b) Work out the value of E when v = 5. E = (1) (c) Work out the value of v when E = 420. Give your answer correct to 3 significant figures. v = (Total 6 marks) Q19 EH4B 09 Page 20

21 20. 2 x = 4 8 Find the value of x. Q20 (Total 3 marks) 21. The probability that Eric is late for a Maths lesson is 0.2 The probability that Greg is late for a Maths lesson is 0.14 The probability that both Eric and Greg are late for a Maths lesson is 0.03 Show that the events Eric being late and Greg being late are not independent. Q21 (Total 2 marks) EH4B 09 Page 21

22 22. Diagram NOT A accurately drawn B 21.8 cm 71 The diagram shows a sector of a circle, centre O. The radius of the circle is 21.8 cm correct to 3 significant figures. The size of angle AOB is 71 correct to 2 significant figures. O (a) Calculate the upper bound and lower bound of the arc length AB. Give your answers correct to 5 significant figures. upper bound = cm lower bound = cm (5) (b) (i) Write down the arc length AB to an appropriate degree of accuracy. cm (ii) Explain why the degree of accuracy you chose is appropriate. (Total 7 marks) Q22 TOTAL FOR PAPER: 100 MARKS END EH4B 09 Page 22

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