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

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1 For Edexcel Name GCSE Mathematics Paper 3H (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 20 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 EH3H 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 EH3H Page 2

3 Answer ALL TWENTY 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. Sam presents a radio show each week. The table gives information about the number of songs he played in ten of his shows. Number of songs Number of shows Work out the mean number of songs he played. Q1 (Total 3 marks) EH3H Page 3

4 2. Write each group of numbers in order of size. Start with the smallest number. (a) (b) (Total 4 marks) Q2 3. (a) Simplify q 3 q 6 (1) (b) Factorise x 2 5xy (1) (c) Simplify 4pq 6p 2 q (d) Solve 4(x + 6) = 9 2x x = (Total 7 marks) Q3 EH3H Page 4

5 4. Diagram NOT y accurately drawn B C A O x The diagram is a sketch. A is the point (4, 3). B is the point (3, 6). C is the point (10, 7). (a) Write down AB as a column vector x y. (1) D is the midpoint of AC. (b) Find BD as a column vector x y. (Total 4 marks) Q4 EH3H Page 5

6 5. The stem and leaf diagram shows how long, after their appointment time, each of 28 patients had to wait to see their doctor Key: 2 4 represents 24 minutes (a) Write down the number of patients who waited less than 8 minutes. (1) (b) Write down the modal waiting time. minutes (1) (c) Find the median waiting time. minutes (1) (Total 3 marks) Q5 EH3H Page 6

7 6. y A O x B 4 Describe fully the single transformation that maps triangle A onto triangle B. Q6 (Total 3 marks) EH3H Page 7

8 7. Diagram NOT Tin Box accurately drawn 5 cm 10 cm 10 cm 20 cm 30 cm 40 cm Coffee is sold in tins measuring 10 cm by 10 cm by 5 cm. The tins are packed into boxes measuring 40 cm by 30 cm by 20 cm. (a) Work out the number of tins in a box when it is completely full. A shop pays 2.35 for each tin. (b) Work out how much the shop pays for a box full of tins. (Total 6 marks) Q7 EH3H Page 8

9 8. (a) Estimate the value of (b) Work out the exact value of (Total 6 marks) Q8 EH3H Page 9

10 9. Q R P S The diagram shows trapezium PQRS. (a) Use ruler and compasses to construct the bisector of angle QPS. You must show all your construction lines. (b) Shade the region inside the trapezium which is nearer to PQ than PS and less than 4 cm from R. (Total 4 marks) Q9 EH3H Page 10

11 10. The table shows some expressions. a, b and c represent lengths. π and 2 are numbers which have no dimensions. Tick ( ) a box in each row to show whether each expression could represent a length, an area, a volume or none of these. Expression Length Area Volume None of these 2ab b 2 (a + c) πa 2 + bc Q10 (Total 3 marks) 11. v = T m Make T the subject of the formula. Q11 (Total 2 marks) EH3H Page 11

12 12. (a) Calculate the sum of the interior angles of a pentagon. 2x + 30 x + 30 Diagram NOT accurately drawn 2x 5 2x x + 45 The diagram shows the size, in degrees, of the angles in a pentagon. (b) Use your answer to part (a) to write down an equation in terms of x. (c) Solve your equation to find the value of x. x = (Total 6 marks) Q12 EH3H Page 12

13 13. y Diagram NOT accurately drawn L P B A O x The diagram shows the straight line L with equation y = 1 2 x + 2. The point P lies on L and has y-coordinate 5. (a) Find the x-coordinate of P. The line L crosses the x-axis at the point A and the y-axis at the point B. (b) Find the length of AB in the form k 5, where k is an integer. (5) (Total 7 marks) Q13 EH3H Page 13

14 14. (a) Write down the value of (i) 6 0 (ii) ( 5) 2 (b) Solve 5x 3 = 320 x = (c) Show that = k, where k is an integer, and write down the value of k. k = (Total 7 marks) Q14 EH3H Page 14

15 15. Solve the simultaneous equations x 2 + y 2 = 13 y = 2x + 1 x = y = or x = y = Q15 (Total 6 marks) EH3H Page 15

16 16. Prove that (2n + 1) 2 + (n + 3) 2 is a multiple of 5 for all positive integer values of n. Q16 (Total 4 marks) EH3H Page 16

17 17. Diagram NOT 6 cm P Q accurately drawn 9 cm T x S R The shaded shape was formed by removing triangle RST from rectangle PQRS. PQ = 6 cm. PS = 9 cm. The area of the shaded shape is 42 cm 2. (a) Find the height, x, of triangle RST. x = cm RT = ST. (b) Find the length of RT. cm Angle RTS = y. (c) Find the value of sin y. (Total 7 marks) Q17 EH3H Page 17

18 18. Two boxes each contain some balls. Box P contains 3 white balls and 1 black ball. Box Q contains 2 white balls and 2 black balls. Box P Box Q Jess picks one ball at random from box P and notes its colour. She then puts the ball in box Q. Jess then picks one ball at random from box Q and notes its colour. (a) Complete the tree diagram. Ball from box P Ball from box Q... black black... white... black... white... white (b) Work out the probability that both balls picked out are black. EH3H Page 18

19 (c) Work out the probability that the ball picked from box Q is white. (Total 8 marks) Q y 1 O x 1 The diagram is a sketch of the curve y = sin x for 0 x 360. You are given that sin 20 = (a) Solve the equation for 0 x 360. sin x = (b) Solve the equation for 0 x 180. sin 2x = (Total 4 marks) Q19 EH3H Page 19

20 20. (a) Find the value of a and the value of b such that for all values of x, x 2 + 6x + 10 = (x + a) 2 + b a = b = The curve with equation y = x 2 + 6x + 10 is reflected in the y-axis. (b) Find the equation of the curve following this transformation. y = (1) The curve with equation y = x 2 + 6x + 10 is translated by 3 units in the positive x-direction. (c) Find the equation of the curve following this transformation. y = (Total 6 marks) Q20 TOTAL FOR PAPER: 100 MARKS END EH3H Page 20