For Edexcel Name GCSE Mathematics Paper 3D (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 23 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 EH3D Page 1
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 EH3D Page 2
Answer ALL TWENTY THREE 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. Here is a list of ingredients for making 12 pancakes. Ingredients 2 eggs 250 ml of milk 50 g of butter 110 g of plain flour Work out how much of each ingredient is needed to make 18 pancakes.. eggs ml of milk g of butter. g of plain flour Q1 (Total 3 marks) EH3D Page 3
2. Lisa records how long it takes her to read each book that she buys. The scatter graph shows these times plotted against the number of pages in each book. 10 8 Time to read (hours) 6 4 2 0 100 150 200 250 300 350 400 Number of pages (a) Describe the correlation between the time it takes Lisa to read a book and the number of pages the book has. (b) Draw a line of best fit on the diagram. (1) (1) Lisa buys a book with 260 pages. (c) Use your line of best fit to estimate how long it will take Lisa to read it. hours (1) (Total 3 marks) Q2 EH3D Page 4
3. A is the point ( 1, 7) B is the point (2, 5) B is the midpoint of AC. Find the coordinates of C. (, ) Q3 (Total 2 marks) 4. Here are the first five terms of a number sequence. 5, 8, 11, 14, 17 (a) Write down an expression, in terms of n, for the nth term of this sequence. (b) Explain why 81 will not be a term in this sequence. (Total 4 marks) Q4 EH3D Page 5
5. Sox is a cat. When he was five weeks old he weighed 400 grams. Over the following week his weight increased by 20%. (a) Work out Sox's weight when he was six weeks old. grams As an adult cat, Sox weighs 3.2 kg. (b) Work out 400 grams as a percentage of 3.2 kg. % (3) (Total 5 marks) Q5 EH3D Page 6
6. The box plot gives information about the distribution of the weights of fish in a pond. 0 10 20 30 40 50 60 Weight in grams (a) Write down the median weight of the fish. g (1) (b) Work out the interquartile range of the weights of the fish. g (Total 3 marks) Q6 7. v = 2t 2 7 t = 3 (a) Work out the value of v. v = R = 4x + 3y R = 9 y = 5 (b) Work out the value of x. x = (3) (Total 5 marks) Q7 EH3D Page 7
8. y C 5 4 3 2 B 1 A 6 5 4 3 2 1 O 1 2 3 x 1 2 (a) The points A ( 2, 0), B ( 6, 1) and C ( 5, 5) are plotted on the grid. The quadrilateral ABCD is a square. Write down the coordinates of the point D. (, ) (1) (b) (i) Plot the points E ( 1, 1), F (0, 1), G (2, 2) and H (3, 2) on the grid. (ii) Write down the name of quadrilateral EFGH. (3) (Total 4 marks) Q8 EH3D Page 8
9. Dale makes his own pizzas. He always has a topping of ham, beef, chicken or pepperoni. The table shows the probability that he will have a ham, beef or chicken topping. Topping Ham Beef Chicken Pepperoni Probability 0.1 0.25 0.45 (a) Work out the probability that Dale has a pepperoni topping. In a year, Dale will make 60 pizzas. (b) Work out an estimate for the number of times in a year that Dale will have a pizza with a beef topping. (Total 4 marks) Q9 EH3D Page 9
10. y 9 8 7 6 5 4 3 2 P 1 2 1 O 1 2 3 4 5 6 7 8 9 x 1 2 3 (a) Reflect triangle P in the line y = 4. Label this image Q. (b) Rotate triangle P 90 clockwise about O. Label this image R. (c) Enlarge triangle P by scale factor 3, centre O. Label this image S. (Total 6 marks) Q10 EH3D Page 10
11. O The diagram shows a shaded semicircle of radius 4 cm, centre O. Draw the locus of all points that are 2 cm from the edge of the semicircle. Q11 (Total 3 marks) 12. Solve the equation x 3 4 + x 2 = 3 x = Q12 (Total 4 marks) EH3D Page 11
13. (a) Factorise 6a 9 (1) (b) Simplify 3xy 5x (1) (c) Expand and simplify r(r 2 2) + 2r (d) Find the value of p and the value of q such that for all values of x, x 2 + 4x 1 = (x + p) 2 + q p = q = (3) (Total 7 marks) Q13 EH3D Page 12
14. A Diagram NOT accurately drawn a O D E B c C OABC is a rhombus. OA = a. OC = c. D is the midpoint of AC. E is the midpoint of BD. Express, in terms of a and c, the vectors (a) OB (1) (b) OE (Total 3 marks) Q14 EH3D Page 13
15. The shaded region on the grid below shows all the points which satisfy all three of these inequalities. x a y b y cx + d y 8 7 6 5 4 3 2 1 4 3 2 1 O 1 2 3 4 5 6 x 1 2 Write down the values of the integers a, b, c and d. a = b = c = d = Q15 (Total 3 marks) EH3D Page 14
16. (a) Evaluate (i) 2 3 1 3 (ii) 64 (iii) 4 9 3 2 (4) (b) (i) Express 12 in the form m 3, where m is an integer. (ii) Rationalise the denominator of 1 12 Give your answer in the form 3 n, where n is an integer. (4) (Total 8 marks) Q16 EH3D Page 15
17. Diagram NOT accurately drawn 9 cm 6 cm x cm x cm The diagram shows two candles which have the same volume. One is a hemisphere of radius 6 cm. The other is a cuboid of height 9 cm which has a square base of side x cm. Find the value of x in terms of π. Give your answer in its simplest form. x = Q17 (Total 4 marks) EH3D Page 16
18. Diagram NOT accurately drawn 10 cm 12 cm A B Two cylinders, A and B, are mathematically similar. The height of cylinder A is 10 cm. The height of cylinder B is 12 cm. The total surface area of cylinder A is 200 cm 2. Work out the total surface area of cylinder B. cm 2 Q18 (Total 3 marks) EH3D Page 17
19. Here is a sketch of the graph of y = cos x for 0 x 360. y 1 O 1 90 180 270 360 x On each copy of this diagram, add a sketch of the graph indicated for 0 x 360. (a) y = 2 cos x y 1 O 1 90 180 270 360 x (1) (b) y = cos (x + 60) y 1 O 1 90 180 270 360 x (1) EH3D Page 18
(c) y = 1 + cos ( 1 2 x) y 1 O 1 90 180 270 360 x (Total 4 marks) Q19 20. (a) Change 5 12 to a decimal. (b) Convert the recurring decimal 0. 5 4 to a fraction. Give your answer in its simplest form. (3) (Total 5 marks) Q20 EH3D Page 19
21. A company has 400 employees. The table gives some information about the employees. Children Female 156 No children 47 Male 129 68 The manager of the company wants to carry out a survey of employees' views about working hours. She decides to take a sample of 40 employees, stratified by both gender and whether or not they have children. Find the number of employees in each sub-group that should be in the sample. Female, children Female, no children Male, children Male, no children Q21 (Total 4 marks) EH3D Page 20
22. y Diagram NOT accurately drawn B (1, 7) L 2 C O x A ( 2, 2) L 1 The diagram shows the straight lines L 1 and L 2. The line L 1 passes through the points A ( 2, 2) and B (1, 7) and crosses the y-axis at the point C. The line L 2 is perpendicular to L 1 and also crosses the y-axis at the point C. Find the equation of line L 2. Q22 (Total 5 marks) EH3D Page 21
23. E Diagram NOT accurately drawn B 2 3 x + 2 x 2 7 A C Triangles ABC and DEF are mathematically similar. Angle ABC = angle DEF. Angle ACB = angle DFE. All measurements are in centimetres. (a) Show that 2x 2 3x 20 = 0 D x F (3) (b) Solve the equation 2x 2 3x 20 = 0 x = or x = (3) (c) Find the length of AC. cm (Total 8 marks) Q23 TOTAL FOR PAPER: 100 MARKS END EH3D Page 22