GCSE Mathematics. Higher Tier. Paper 3J (Non-Calculator) Time: 1 hour and 45 minutes. For Edexcel. Name

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1 For Edexcel Name GCSE Mathematics Paper 3J (Non-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 24 questions in this paper. Calculators must not be used. 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 EH3J 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 EH3J Page 2

3 Answer ALL TWENTY FOUR questions. Write your answers in the spaces provided. You must write down all the stages in your working. You must NOT use a calculator. 1. Diagram NOT D accurately drawn A B 24 C In the diagram, ABC is a straight line and AB = AD = BD. Angle BCD = 24. Find the size of angle BDC. Q1 (Total 3 marks) EH3J Page 3

4 2. Input Output n (n + 4) The diagram shows that if you are given a number, n, add 4 to it and multiply the result by 3, the number obtained is given by the expression 3(n + 4). Write in expressions for the output in each of these diagrams. (a) Input Output n 2 5 Input Output n (Total 4 marks) Q2 EH3J Page 4

5 3. Ms. Tate wants to buy a packet of crisps. The probability that she will choose cheese flavoured crisps is 0.4 (a) Write down the probability that she will not choose cheese flavoured crisps. (1) Mr. Wells has 40 hot drinks in a week. Each time he has a hot drink, the probability that he chooses to have coffee is 0.3 Work out an estimate for the number of drinks of coffee Mr. Wells has each week. (Total 3 marks) Q3 4. (a) Factorise completely 3ab + 6b 2 Expand 4(7y + 2) (1) (c) Expand and simplify x(x 5) + 4(x 1) (Total 5 marks) Q4 EH3J Page 5

6 5. The scatter graph shows the number of hours of sunshine and the maximum temperature for eight cities on the same day Maximum temperature ( C) Hours of sunshine (a) Describe the relationship between the maximum temperature and the number of hours of sunshine. (1) Draw a line of best fit on the diagram. (1) Another city had 3.5 hours of sunshine on the same day. (c) Use your line of best fit to estimate the maximum temperature in the city. C (1) (Total 3 marks) Q5 EH3J Page 6

7 6. Find an expression, in terms of n, for the nth term of each sequence. (a) 1, 5, 9, 13, 17,... 18, 16, 14, 12, 10,... (Total 4 marks) Q6 7. On one day, 7 1 of the workers at an office arrive on time and arrive late The rest of the workers are absent. Calculate the fraction of workers who are absent. Give your answer as a fraction in its simplest form. Q7 (Total 3 marks) EH3J Page 7

8 8. A shop starts selling a new camera. The shop is open 6 days a week. The table gives information about daily sales of the camera over the first 2 weeks. Number of sales Number of days (a) For these data. write down (i) (ii) (iii) the mode, the median, the range. (3) After 3 weeks, the shop has a total of 90 sales of the camera. Work out the mean number of sales per day of the camera in the third week. (4) (Total 7 marks) Q8 EH3J Page 8

9 9. Diagram NOT accurately drawn B A 28 C A, B and C are points on a circle. AB is a diameter of the circle. Angle BAC = 28. (a) Find the size of angle ABC. (1) Give reasons for your answer. (Total 3 marks) Q9 EH3J Page 9

10 10. (a) Use the information that to find the value of (i) = 209 (ii) Use the information that = 209 to find the Highest Common Factor (HCF) of 66 and 418. (Total 4 marks) Q10 EH3J Page 10

11 11. y Diagram NOT accurately drawn A O x B The diagram is a sketch. A is the point (10, 4). B is the point (4, 6). (a) Find the coordinates of the midpoint of AB. (, ) Find the distance of the midpoint of AB from O. Give your answer in the form k 2, where k is an integer. (3) (Total 5 marks) Q11 EH3J Page 11

12 12. Diagram NOT Q M accurately drawn 8.1 cm 6 cm L 9 cm N P 12 cm R Triangles LMN and PQR are similar. Angle LMN = Angle PQR. Angle LNM = Angle PRQ. LM = 8.1 cm. LN = 9 cm. MN = 6 cm. PR = 12 cm. Work out the length of (a) QR, cm PQ. cm (Total 4 marks) Q12 EH3J Page 12

13 13. The table shows some expressions. p, q, and r represent lengths. p + q pr pqr p + qr p(q + r) pqr r Tick ( ) the boxes underneath the two expressions which could represent areas. Q13 (Total 2 marks) 14. (a) Solve 5x + 7 = 11 x = Solve y y 1 2 = 3 y = (4) (Total 6 marks) Q14 EH3J Page 13

14 15. y O x Find the equation of the straight line drawn on the grid. (Total 3 marks) Q15 EH3J Page 14

15 16. (a) Solve the inequality 2x + 5 > 12 Solve the simultaneous equations 3a + 4b = 3 a 2b = 21 a = b = (3) (Total 5 marks) Q16 EH3J Page 15

16 17. (a) Calculate 5% of Work out ( ) ( ) (3) (Total 5 marks) Q17 EH3J Page 16

17 18. D Diagram NOT accurately drawn A C B O ABCD is a parallelogram. O is the point such that OA = 3p, OB = 2p + q, and OC = 4p + 3q. Find, in terms of p and q, the vectors (a) BA OD (Total 4 marks) Q18 EH3J Page 17

18 19. (a) Factorise 9 x 2 Simplify fully x 2 3 x 2 x 2 4 x 5 (3) (Total 5 marks) Q19 EH3J Page 18

19 20. A tin of spagetti weighs 250 grams correct to the nearest gram. A cardboard tray weighs 120 grams correct to the nearest 10 grams. A full tray carries 12 tins of spagetti. Two full trays are weighed. Find the maximum difference between the weights of the two full trays. g Q20 (Total 4 marks) EH3J Page 19

20 21. (a) Find the value of 1 2 (i) 16 (ii) (3) Expand and simplify (2 + 3 )(4 3 ) Give your answer in the form p + q 3, where p and q are integers. (Total 5 marks) Q21 EH3J Page 20

21 22. Graph A Graph B Graph C y y y O x O x O x Graph D Graph E Graph F y y y O x O x O x Each of these equations is represented by one of the sketch graphs above. Write down the letter of the correct graph for each equation. (i) y = x 3 2 Graph (1) (ii) y = 2 x Graph (1) (iii) x 2 + y 2 = 4 Graph (1) (Total 3 marks) Q22 EH3J Page 21

22 23. Diagram NOT accurately drawn 3 cm 4 cm The radius of the base of a cone is 4 cm. The height of the cone is 3 cm. (a) Work out the curved surface area of the cone. Give your answer as a multiple of π. cm 2 The surface area of a sphere is 5 times the curved surface area of the cone. (3) Find the radius of the sphere. cm (3) (Total 6 marks) Q23 EH3J Page 22

23 24. Diagram NOT B accurately drawn A O D C E A, B and C are points on a circle, centre O. DCE is the tangent to the circle at C. Prove that angle ABC = angle ACD. Q24 (Total 4 marks) TOTAL FOR PAPER: 100 MARKS END EH3J Page 23