It takes 20 minutes for meals to be served when 12 waiters are working

Transcription

1 Q1. The time in minutes (T) for meals to be served at a busy restaurant is inversely proportional to the square of the number of waiters (W) working at that time. It takes 20 minutes for meals to be served when 12 waiters are working. (a) Find an equation connecting T and W. Answer... (3) (b) What is the minimum number of waiters that must be working for a meal to be served within 30 minutes? Answer... (3) (Total 6 marks) Q2. In the diagram, O is the centre of the circle. A, B, C and D are points on the circumference. Angle AOC = 130 (a) Calculate the value of x. Give a reason for your answer. Answer x =...degrees Reason... (2) Page 1 of 227

2 (b) Calculate the value of y. Give a reason for your answer. Answer y =...degrees Reason... (2) (Total 4 marks) Q3. (a) Enlarge the shaded shape by a scale factor of 3. (2) Page 2 of 227

3 (b) How many times bigger is the area of the enlarged shape than the area of the small shape? Answer... (2) (Total 4 marks) Page 3 of 227

4 Q4. ABCD is a quadrilateral. AB = 7 cm, AD = 6 cm and BC = 9 cm. Angle ABC = 75 and angle ADC = 90 Calculate the perimeter of ABCD Answer... cm (Total 5 marks) Q5. OACB is a parallelogram and M is the mid-point of BC. = a and = b Page 4 of 227

5 (a) Express the following vectors in terms of a and b (i) Answer... (1) (ii) Answer... (1) (b) AM is extended to N, where. Show that = b.... (2) (c) What does this tell you about the position of N?... (1) (Total 5 marks) Q6. (a) Calculate the size of an interior angle of a regular octagon..... Answer... degrees (3) Page 5 of 227

6 (b) Part of a tiled floor is shown. The tiles labelled P, Q, R and S are regular octagons. Explain why the tile labelled X is a square..... (3) (Total 6 marks) Page 6 of 227

7 Q7. The diagrams show a rectangle and an L shape All the angles are right angles. All lengths are in centimetres. The shapes are equal in area. Calculate the value of y Answer... cm (Total 6 marks) Q8. (a) Complete the table of values for y = 3x 2 6 x y (1) Page 7 of 227

8 (b) On the grid below, draw the graph of y = 3x 2 6 for values of x between 3 and +4. (2) (c) Use your graph to write down the solutions of 3x 2 6 = 0. Answer... and... (1) Page 8 of 227

9 (d) By drawing an appropriate linear graph, write down the solutions of 3x 2 5x 6 = Answer... (3) (Total 7 marks) Q9. The area of the screen of a television set is A square inches. The length of the diagonal of the screen is d inches. A is directly proportional to the square of d. A television set with an area of 90 square inches has a diagonal of length 15 inches. (a) Find an equation connecting A and d Answer... (3) (b) Find the area of the screen of a television set with a diagonal of length 20 inches... Answer... square inches (1) Page 9 of 227

10 (c) Another television set has a screen with an area of 250 square inches. Find the length of its diagonal..... Answer... inches (3) (Total 7 marks) Q10. The graph shows the function (a) Write down the coordinates of the point where the graph intersects with the y-axis. Answer (...,... ) (1) Page 10 of 227

11 (b) Find the value of a. Answer... (2) (Total 3 marks) Q11. Two spheres of radius 5 cm just fit inside a tube. Calculate the volume inside the tube not filled by the spheres Answer... cm 2 (Total 5 marks) Page 11 of 227

12 Q12. A sign maker designs a letter L. All arcs are quarter circles of radius 2 cm. Not drawn accurately Calculate the area of the L Answer... cm 2 (Total 4 marks) Page 12 of 227

13 Q13. (a) P is inversely proportional to Q. When P = 100, Q = 32 Express P in terms of Q. Answer... (3) (b) P and Q are positive quantities. Sketch a graph of the relationship between P and Q on this diagram. (1) (c) Calculate the value of Q when P is twice as big as Q. Answer... (2) (Total 6 marks) Page 13 of 227

14 Q14. A(1, 1) and B( 2, 4) are two points on the graph of y = x 2 Here are three transformations of the graph y = x 2. On each diagram the graph of y = x 2 is shown dotted. The images A and B of A and B are shown. Write down the equation of the transformed graph in each case. Page 14 of 227

15 (a) y =... (1) (b) y =... (1) Page 15 of 227

16 (c) y =... (1) (Total 3 marks) Q15. The diagram shows a solid made from a cone and a hemisphere. The radius of both shapes is r. The slant height of the cone is l. The perpendicular height of the cone is h. The curved surface area of the cone and the curved surface area of the hemisphere are equal. Page 16 of 227

17 (a) Show that l = 2r (2) (b) Find the perpendicular height, h, of the cone in terms of r. Answer h =... (2) (c) Find the ratio of the volumes of the cone and the hemisphere. Give your answer in surd form. Answer... (2) (Total 6 marks) Q16. (a) Show that can be written as 2x 2 9x + 4 = 0 (2) Page 17 of 227

18 (b) Part of the graph of y = is shown on the grid below. Draw a straight line on the grid which will enable you to solve the equation 2x 2 9x + 4 = 0 (3) (c) Hence, or otherwise, solve the equation 2x 2 9x + 4 = 0 Answer... (2) Page 18 of 227

19 (Total 7 marks) Page 19 of 227

20 Q17. A circle fits exactly inside a semi-circle of diameter 20 cm. Not drawn accurately The shaded area is a π square centimetres. Work out the value of a. You must show your working Answer a =... (Total 4 marks) Q18. This is the graph of y = cos x for 0 x 360 Page 20 of 227

21 Write the equation of each of the transformed graphs. In each case the graph of y = cos x is shown dotted to help you. (a) Equation y =... (1) (b) Equation y =... (1) (c) Equation y =... (1) Page 21 of 227

22 (d) Equation y =... (1) (Total 4 marks) Q19. ABCD is a cyclic quadrilateral. PAQ is a tangent to the circle at A. BC = CD Angle QAB = 38 and angle BAD = 76 Not drawn accurately Page 22 of 227

23 Show that AD is parallel to BC. Give reasons to justify any values you write down or calculate (Total 4 marks) Q20. (a) A calculator displays a number in standard form as Which of the following numbers does the display show? Circle the correct answer (1) (b) Use your calculator to work out cos (tan ) (i) Give all the figures in your calculator display. Answer... (1) Page 23 of 227

24 (ii) Write your answer to an appropriate degree of accuracy. Answer... (1) (c) Use your calculator to work out Answer... (1) (Total 4 marks) Q21. A sphere has radius r. A cone has base radius r and perpendicular height x. The volume of the sphere is double the volume of the cone. Not drawn accurately (a) Show that x =2r (2) Page 24 of 227

25 (b) Calculate the ratio of the surface area of the sphere to the curved surface area of the cone. Give your answer in surd form. Answer... (4) (Total 6 marks) Q22. The diagram shows a cuboid. AB = 3 cm, AE = 4 cm, BC = 12 cm. Not drawn accurately (a) Find the length of BH. Answer... cm (2) Page 25 of 227

26 (b) The angle between BH and BD is x and the angle between BH and BC is y. Which angle is bigger, x or y? You must show your working. Answer... (3) (Total 5 marks) Q23. XYZ is an isosceles triangle in which XZ = XY M and N are points on XZ and XY such that angle MYZ = angle NZY. Page 26 of 227

27 Prove that triangles YMZ and ZNY are congruent (Total 4 marks) Q24. In the diagram SR is parallel to PT. SQT and RQP are straight lines. SR = 20 cm and PT = 30 cm The total height of the two triangles is 40 cm. Not drawn accurately Page 27 of 227

28 Use similar triangles to calculate the height, h cm, of triangle PQT Answer h =... cm (Total 3 marks) Q25. (a) A circle has a radius of 6 cm. A sector has an arc length of 8.4 cm. The angle at the centre of the sector is θ. Not drawn accurately Calculate the value of θ. Answer... degrees (3) Page 28 of 227

29 (b) A cone has base radius 6 cm and height h cm. A smaller cone of base radius 2 cm and height 3 cm is cut from the top. The remaining frustum has dimensions as shown. Not drawn accurately Calculate the volume of the frustum. Answer... cm 3 (5) (Total 8 marks) Page 29 of 227

30 Q26. The grid below shows graphs of a curve y = x 2 + 2x 3 and 3 straight lines y = x + 1 y = x 2 and y = x + 2 You must use the graphs to answer the following questions. (a) Write down a pair of simultaneous linear equations that have a solution x =, y = Page 30 of 227

31 Answer... (1) (b) Write down and simplify a quadratic equation whose solutions are approximately 3.3 or 0.3. You must show clearly how you obtain your answer. Answer... (2) (c) Write down the approximate solutions to the equation x 2 + x 4 = 0. You must show clearly how you obtain your answer. Answer... (2) (Total 5 marks) Q27. A square of side x and a quarter-circle of radius r have the same area. Not to scale Page 31 of 227

32 Express r in terms of x. Simplify your answer Answer r =... (Total 3 marks) Q28. (a) ABC is a triangle. AC = 19 cm, BC = 17 cm and angle BAC = 60 Not to scale Calculate the size of angle ABC. Answer... degrees (3) Page 32 of 227

33 (b) PQR is a triangle. PR = 23 cm, PQ = 22 cm and angle QPR = 48 Not to scale Calculate the length of QR. Give your answer to an appropriate degree of accuracy. Answer... cm (4) (Total 7 marks) Q29. ABC is an isosceles triangle. The lengths, in cm, of the sides are AB = 4a + 3, BC = 2b + 5 and AC = 2a + b Not to scale Page 33 of 227

34 (a) AB = BC Show that 2a b = 1 (2) (b) The perimeter of the triangle is 32 cm. Find the values of a and b. Answer a =... cm, b =... cm (4) (Total 6 marks) Q30. For a ladder to be safe it must be inclined at between 70 and 80 to the ground. (a) The diagram shows a ladder resting against a wall. Not to scale Page 34 of 227

35 Is it safe? You must show your working. (3) (b) Another ladder rests against a wall. Not to scale Work out the closest distance that the bottom of the ladder can be from the wall so that it is safe. Answer... m (3) (Total 6 marks) Page 35 of 227

36 Q31. A hemispherical bowl of radius 6 cm has the same volume as a cone of perpendicular height 27 cm. Not drawn accurately Calculate the base radius, r, of the cone Answer... cm (Total 4 marks) Page 36 of 227

37 Q32. In the diagram below points Q and S lie on a circle centre O. SR is a tangent to the circle at S. Angle QRS = 40 and angle SOQ = 80 Not drawn accurately Prove that triangle QSR is isosceles (Total 3 marks) Page 37 of 227

38 Q33. Match each of the sketch graphs to one of these equations. A y = 2 2x B y = 2x + 2 C y = 3 x 2 D y = x E y = Graph 1 represents equation... Graph 2 represents equation... Graph 3 represents equation... Graph 4 represents equation... (Total 4 marks) Page 38 of 227

39 Q34. The grid below shows the graph of y = x 2 + 3x 2 (a) By drawing an appropriate straight line on the graph solve the equation x 2 + 3x 3 = 0 Answer... (2) Page 39 of 227

40 (b) By drawing an appropriate straight line on the graph solve the equation x 2 + 2x 1 = 0 Answer... (3) (Total 5 marks) Q35. (a) Explain why the volume of a cube increases by a factor of 8 when the side length is doubled. (2) Page 40 of 227

41 (b) June recently bought a small toy in the local shop. ALIEN Place in water and it becomes 6 times bigger! It was originally 8 cm tall. After she placed it in water it grew to a similarly shaped alien. The height was then 14.5 cm. Is the claim on the pack justified? (3) (Total 5 marks) Q36. A marble paperweight consists of a cuboid and a hemisphere as shown in the diagram. The hemisphere has a radius of 4 cm. Not to scale Page 41 of 227

42 Calculate the volume of the paperweight Answer... (Total 4 marks) Q37. A circle fits inside a semicircle of diameter 10 cm as shown. Not drawn accurately Page 42 of 227

43 Calculate the shaded area Answer... cm 2 (Total 3 marks) Q38. y is directly proportional to the square of x. When y = 5, x = 4. Find the value of y when x = Answer... (Total 3 marks) Page 43 of 227

44 Q39. A giant paper clip is placed alongside a centimetre ruler. The curved ends are semicircles. Calculate the length of wire used to make the clip Answer... cm (Total 5 marks) Q40. (a) ABC is a right-angled triangle. AC = 19 cm and AB = 9 cm. Page 44 of 227

45 Calculate the length of BC. Answer... cm (3) (b) PQR is a right-angled triangle. PQ = 11 cm and QR = 24 cm. Calculate the size of angle PRQ. Answer... degrees (3) (Total 6 marks) Q41. Two towns, A and B, are connected by a motorway of length 100 miles and a dual carriageway of length 80 miles as shown. Page 45 of 227

46 Jack travels from A to B along the motorway at an average speed of 60 mph. Fred travels from A to B along the dual carriageway at an average speed of 50 mph. What is the difference in time between the two journeys? Give your answer in minutes Answer... minutes (Total 4 marks) Q42. A straight line has the equation y = 2x 3 A curve has the equation y 2 = 8x 16 (a) Solve these simultaneous equations to find any points of intersection of the line and the curve. Do not use trial and improvement. You must show all your working. Answer... (5) Page 46 of 227

47 (b) Here are three sketches showing the curve y 2 = 8x 16 and three possible positions of the line y = 2x 3 Sketch 1 Sketch 2 Sketch 3 Which is the correct sketch? You must explain your answer. (2) (Total 7 marks) Page 47 of 227

48 Q43. The sketch shows the graph of y = sin x for 0 x 360 You are given that sin 70 = (a) Write down another solution of the equation sin x = Answer... degrees (1) (b) Solve the equation sin x = for 0 x 360 Answer... degrees... degrees (2) Page 48 of 227

49 (c) On the axes below sketch the graph of y = sin 2x for 0 x 360 (2) (d) Hence write down the four solutions of the equation sin 2x = Answer... degrees... degrees... degrees... degrees (3) (Total 8 marks) Page 49 of 227

50 Q44. The diagram shows the graph of the equation y = x 2 + px + q The graph crosses the x-axis at A and B (2,0). C ( 3, 5) also lies on the graph. (a) Find the values of p and q. Answer p =... q =... (4) (b) Hence work out the coordinates of A. Answer (...,... ) (2) (Total 6 marks) Page 50 of 227

51 Q45. The diagram shows a cylinder. The diameter of the cylinder is 10 cm. The height of the cylinder is 10 cm. (a) Work out the volume of the cylinder. Give your answer in terms of π. Answer... cm 3 (3) Page 51 of 227

52 (b) Twenty of the cylinders are packed in a box of height 10 cm. The diagram shows how the cylinders are arranged inside the box. The shaded area is the space between the cylinders. Work out the volume inside the box that is not filled by the cylinders. Give your answer in terms of π. Answer... cm 3 (4) (Total 7 marks) Q46. In the diagram OACD, OADB and ODEB are parallelograms. Page 52 of 227

53 (a) Express, in terms of a and b, the following vectors. Give your answers in their simplest form. (i)... Answer... (1) (ii)... Answer... (1) (iii) Answer... (1) (b) The point F is such that OCFE is a parallelogram. Write the vector in terms of a and b. Answer... (2) (c) What geometrical relationship is there between the points O, D and F? Justify your answer. (2) (Total 7 marks) Page 53 of 227

54 Q47. A square-based pyramid with a base of side 2 cm has a volume of 2.75 cm 3. Not to scale What is the volume of a similar square-based pyramid with a base of side 6 cm? Answer... cm 3 (Total 2 marks) Q48. A ruined tower is fenced off for safety reasons. To find the height of the tower Rashid stands at a point A and measures the angle of elevation as 18. He then walks 20 metres directly towards the base of the tower to point B where the angle of elevation is 31. Page 54 of 227

55 Calculate the height, h, of the tower Answer... m (Total 6 marks) Q49. The sketch below is of the graph of y = x 2 Page 55 of 227

56 On the axes provided, sketch the following graphs. The graph of y = x 2 is shown dotted on each set of axes to act as a guide. (a) y = x (1) (b) y = (x 2) 2 (1) Page 56 of 227

57 (c) (1) (Total 3 marks) Page 57 of 227

58 Q50. The graph of y = x 2 4x + 8 is shown below. (a) (i) By drawing the graph of an appropriate straight line, solve the equation x 2 4x + 8 = 3x Answer... (3) (ii) Hence, or otherwise, solve x 2 7x + 10 = Answer... (1) Page 58 of 227

59 (b) The graph of y = x 2 4x + 8 is to be used to solve the equation x 2 5x + 4 = 0 What straight line graph would need to be drawn? (You do not need to draw it, just state its equation.) Answer y =... (2) (Total 6 marks) Q51. In the diagram, the sides of triangle ABC are tangents to the circle. D, E and F are the points of contact. AE = 5 cm and EC = 4 cm Not to scale (a) Write down the length of CD. Answer...cm (1) (b) The perimeter of the triangle is 32 cm. Calculate the length of DB. Answer...cm (2) (Total 3 marks) Page 59 of 227

60 Q52. Tom is investigating the equation y = x 2 x + 5 He starts to complete a table of values of y for some integer values of x. x y Tom says, "When x is an integer, y is always a prime number". Find a counter-example to show that Tom is wrong. Explain your answer Answer... (Total 2 marks) Q53. A water tank is 50 cm long, 34 cm wide and 24 cm high. It contains water to a depth of 18 cm. Four identical spheres are placed in the tank and are fully submerged. The water level rises by 4.5 cm. Page 60 of 227

61 Calculate the radius of the spheres Answer... cm (Total 5 marks) Q54. ABCD is a quadrilateral. AB = 7 cm, AD = 6 cm and BC = 9 cm. Angle ABC = 75 and angle ADC = 90 Page 61 of 227

62 Calculate the perimeter of ABCD Answer... cm (Total 5 marks) Q55. AB is a chord of a circle, centre O, radius 6 cm. AB = 7 cm Page 62 of 227

63 Calculate the area of the shaded segment Answer... cm 2 (Total 6 marks) Q56. (a) Complete the table of values for y = (0.8) x x y (1) Page 63 of 227

64 (b) On the grid below, draw the graph of y = (0.8) x for values of x from 0 to 4. (2) (c) Use your graph to solve the equation (0.8) x = 0.76 Answer... (1) (Total 4 marks) Page 64 of 227

65 Q57. Enlarge the shaded shape by scale factor with centre of enlargement ( 1, 0). (Total 2 marks) Page 65 of 227

66 Q58. A tin of diameter 7 cm and height 12 cm has a label around it. The label is glued together using a 1 cm overlap. There is a 1 cm gap between the label and the top and the bottom of the tin. Find the length and the height of the label Answer Length =... cm Height =... cm (Total 4 marks) Page 66 of 227

67 Q59. Dario is using trial and improvement to find a solution to the equation The table shows his first trial. x + = 5 x x + Comment Too low Continue the table to find a solution to the equation. Give your answer to 1 decimal place. Answer x =... (Total 4 marks) Q60. (a) Points P, Q, R and S lie on a circle. PQ = QR Angle PQR = 116 Page 67 of 227

68 Explain why angle QSR = (2) (b) The diagram shows a circle, centre O. TA is a tangent to the circle at A. Angle BAC = 58 and angle BAT = 74. (i) Calculate angle BOC Answer Angle BOC =... degrees (1) (ii) Calculate angle OCA Answer Angle OCA =... degrees (3) (Total 6 marks) Page 68 of 227

69 Q61. Which one of the following kites is a cyclic quadrilateral? Give a reason for your answer. Answer... Reason (Total 2 marks) Q62. A square-based pyramid has a base of edge 5 cm. The vertex of the pyramid is directly over the midpoint of the base. The volume of the pyramid is 100cm 3. Page 69 of 227

70 Find the length of the slant edge of the pyramid (marked x in the diagram) Answer... cm (Total 5 marks) Q63. A solid cube has a square hole cut through horizontally and a circular hole cut through vertically. Both holes are cut centrally in the appropriate faces. The dimensions of the cube and the holes are as shown in the diagram. Page 70 of 227

71 Calculate the volume remaining after the holes have been cut Answer... (Total 5 marks) Q64. In triangle ABC, AB = 11 cm, BC = 9 cm and CA = 10 cm. Page 71 of 227

72 Find the area of triangle ABC Answer... cm 2 (Total 5 marks) Q65. ABCD is a rectangle with length 25 cm and width 10 cm. The length of the rectangle is increased by 10%. The width of the rectangle is increased by 20%. Find the percentage increase in the area of the rectangle Answer... % (Total 3 marks) Page 72 of 227

73 Q66. Solve the equation x 2 10x 5 = 0 Give your answers to 2 decimal places Answer... (Total 3 marks) Q67. (a) ABC is a right-angled triangle. AB = 5.1 cm CAB = 48 Find the length of BC (marked x in the diagram). Give your answer to a suitable degree of accuracy. Answer... cm (4) Page 73 of 227

74 (b) PQRS is a parallelogram. PQ = 5.1 cm PS = 6.8 cm QPS = 48 Calculate the area of PQRS. Answer... cm 2 (2) (Total 6 marks) Q68. The diagram shows a circle with centre O and radius 2.5 cm. TA is a tangent to the circle, of length 6 cm. The line from A to the centre O of the circle cuts the circumference at B. Page 74 of 227

75 Calculate the length of AB Answer... cm (Total 4 marks) Q69. (a) Liquid is poured at a steady rate into the bottle shown in the diagram. As the bottle is filled, the height, h, of the liquid in the bottle changes. Which of the five graphs below shows this change? Page 75 of 227

76 Give a reason for your choice. Graph... Reason... (2) (b) Liquid is poured at a steady rate into another container. The graph shows how the height, h, of the liquid in this container changes. Page 76 of 227

77 Sketch a picture of this container. (1) (Total 3 marks) Q70. The diagrams, which are not drawn to scale, show the graph of y = x 2 and four other graphs A, B, C and D. A, B, C and D represent four different transformations of y = x 2. Find the equation of each of the graphs A, B, C and D. Page 77 of 227

78 (Total 4 marks) Q71. The diagram shows two right-angled triangles. AD = 15 cm. CD = 6 cm. Page 78 of 227

79 (a) Given that cos x =, calculate the length BD. Answer BD =... (2) (b) Find the value of sin y. Answer sin y =... (3) (Total 5 marks) Q72. ABC is a triangle. ACD is a straight line. Page 79 of 227

80 Using a ruler and compasses only, make an accurate construction of this diagram. You must show clearly all your construction arcs. The line AD has been drawn for you. (Total 6 marks) Q73. (a) The diagram shows a circle with centre O. Work out the size of the angle marked x. Answer... degrees (1) Page 80 of 227

81 (b) The diagram shows a different circle with centre O. Work out the size of the angle marked y. Answer... degrees (1) (c) A, B and C are points on the circumference of a circle with centre O. BOC is a straight line. Angle ABC = 20 Work out the size of the angle marked z. Explain your answer. Answer... degrees (2) (Total 4 marks) Page 81 of 227

82 Q74. Two similar bottles are shown below. The smaller bottle is 20 cm tall and holds 480 ml of water. The larger bottle is 30 cm tall. How much water does the larger bottle hold? Answer... (Total 3 marks) Q75. A thin-walled glass paperweight consists of a hollow cylinder with a hollow cone on top as shown. The paperweight contains just enough sand to fill the cylinder. The paperweight is now turned upside down. Page 82 of 227

83 Calculate the depth of the sand, (marked x in the diagram) Answer... cm (Total 5 marks) Q76. Two ships, A and B, leave port at hours. Ship A travels at a constant speed of 18 km per hour on a bearing of 070. Ship B travels at a constant speed of 25 km per hour on a bearing of 152. Page 83 of 227

84 Calculate the distance between A and B at hours Answer... km (Total 4 marks) Q77. ABCDEFGH is a cuboid with sides of 5 cm, 5 cm and l2 cm as shown. Page 84 of 227

85 Calculate angle DFH Answer... degrees (Total 5 marks) Q78. In the diagram, the lines AC and BD intersect at E. AB and DC are parallel and AB = DC. Prove that triangles ABE and CDE are congruent (Total 4 marks) Page 85 of 227

86 Q79. ABC is a right-angled triangle. BC = 125 m. Angle CAB = 33. Find the length of AC (marked x in the diagram). Give your answer to an appropriate degree of accuracy Answer... m (Total 4 marks) Q80. Triangles ADE and ABC are similar. DE is parallel to BC. AD = 4 cm, DE = 6 cm and BC = 9 cm. Page 86 of 227

87 Calculate the length of BD Answer... cm (Total 3 marks) Q81. The graph of y = sin x for 0 x 360 is shown on the grid below. The point P(90, 1) lies on the curve. On both of the grids that follow, sketch the graph of the transformed function. In both cases write down the coordinates of the transformed point P. Page 87 of 227

88 (a) y = sin (x 45) P (...,...) (2) (b) y = 2sinx P (...,...) (2) (Total 4 marks) Page 88 of 227

89 Q82. A firm makes cone shaped containers out of card. The card is in the shape of a sector of a circle of radius 12 cm. The angle of the sector is 270. The straight edges are brought together to make the cone. (a) Find the arc length of the card used to make the cone. Give your answer in terms of π... Answer... cm (2) (b) Calculate the radius of the base of the cone... Answer... cm (2) (Total 4 marks) Page 89 of 227

90 Q83. OAB is a triangle where M is the mid-point of OB. P and Q are points on AB such that AP = PQ = QB. = a and = 2b (a) Find, in terms of a and b, expressions for (i) Answer... (1) (ii) Answer... (2) (iii) Answer... (2) Page 90 of 227

91 (b) What can you deduce about quadrilateral OMQP? Give a reason for your answer.... (2) (Total 7 marks) Q84. (a) (i) Write down the value of x. Answer... degrees (1) (ii) Calculate the value of y Answer... degrees (1) Page 91 of 227

92 (b) A and C are points on the circumference of a circle centre B. AD and CD are tangents. Angle ADB = 40. Explain why angle ABC is (2) (c) ABCD is a cyclic quadrilateral. PAQ is a tangent to the circle at A. BC = CD. AD is parallel to BC. Angle BAQ = 32. Page 92 of 227

93 Find the size of angle BAD. You must show all your working Answer Angle BAD =... degrees (5) (Total 9 marks) Q85. The map below shows three boats, A, B and C, on a lake. Along one edge of the lake there is a straight path. Treasure lies at the bottom of the lake. The treasure is: between 150 m and 250 m from B, nearer to A than C, more than 100 m from the path. Page 93 of 227

94 Using a ruler and compasses only, shade the region in which the treasure lies. You must show clearly all your construction arcs. (Total 5 marks) Q86. (a) (i) Write down the value of x. Answer... degrees (1) (ii) Calculate the value of y Answer... degrees (1) Page 94 of 227

95 (b) A and C are points on the circumference of a circle centre B. AD and CD are tangents. Angle ADB = 40. Explain why angle ABC is (2) Page 95 of 227

96 (c) P is a point on the circumference of a circle with centre O. PQ is a tangent of length 8 cm. The area of triangle OPQ is 24 cm 2. Calculate the area of the circle. Give your answer in terms of π Answer... cm 2 (3) (Total 7 marks) Page 96 of 227

97 Q87. Three circles fit inside a rectangle as shown. Two of the circles are identical and the third is larger. The circles have radii 9 cm, 9 cm and 25 cm. Not drawn accurately Calculate the length, l, of the rectangle Answer... cm (Total 6 marks) Page 97 of 227

98 Q88. This diagram is made from 25 small squares and 16 large squares. What percentage of the diagram is shaded? Answer... % (Total 6 marks) Page 98 of 227

99 Q89. The diagram shows a regular pentagon and a regular decagon joined at side XY. Not drawn accurately Show that the points A, B and C lie on a straight line (Total 5 marks) Page 99 of 227

100 Q90. In the figure, AC = 9 cm, AE = 6 cm, BD = 8.5 cm, BE = 4.5 cm and DF = 5 cm, BEDF and AEC are straight lines. Not drawn accurately (a) Show that triangles BEC and AED are similar. (3) (b) By considering triangles BEC and FEA show that AF is parallel to BC. (5) (Total 8 marks) Page 100 of 227

101 Q91. Four points, A( 4, 1), B(5, 11), C(20, 11) and D(11, 1) are joined to form a quadrilateral. Not drawn accurately Prove by calculation that ABCD is a rhombus (Total 4 marks) Page 101 of 227

102 Q92. Find the reflection of the point A(1, 4) in the line y = 2x 3 Use the grid below to help you Answer (...,... ) (Total 3 marks) Page 102 of 227

103 Q93. The diagram shows four points A, B, C and D on the circumference of a circle, centre O. PAQ is the tangent to the circle at A. PBD is a straight line. Angle QAD = 75 Angle APB = 30 Not drawn accurately Work out angle BCD. You must show your working Answer... degrees (Total 3 marks) Page 103 of 227

104 Q94. A manufacturer designs a set of three similar containers to fit inside each other. The diagram shows a sketch of the containers and their oval cross-sections. Not drawn accurately Some information about the containers is shown in this table. Base length Height Area of card used in manufacture Large 30 cm Medium 20 cm 12 cm 1080 cm 2 Small 10 cm (a) Work out the height of the large container. Answer... cm (3) Page 104 of 227

105 (b) cm of card is used to manufacture the medium container. Work out the area of card used to manufacture the small container. Answer... cm 2 (2) (Total 5 marks) Page 105 of 227

106 Q95. The shaded shape is made from two different right-angled triangles. 2 The area of each of the triangles is 6cm. Each of lengths a, b, c and d are a whole number of centimetres. a + c = 10 cm a > b and c > d Not drawn accurately Work out the perimeter of the shaded shape Answer... cm (Total 5 marks) Page 106 of 227

107 Q96. The sector AOB of a circle is shown below. The length of its arc AB is 10π cm. Not drawn accurately The sector is folded so that the straight edges meet and form a cone as shown. (a) Calculate the radius of the base of the cone. Answer... cm (3) 3 (b) The volume of the cone is 80π cm. Work out the perpendicular height of the cone. Answer... cm (3) (Total 6 marks) Page 107 of 227

108 Q97. OABC is a quadrilateral. D, E, F and G are midpoints of OA, AB, BC and OC respectively. = 2a, = 2b and = 2c Not drawn accurately Find the following vectors in terms of a, b and c. For example = c a (a) Answer... (1) (b) Answer... (1) (c) Use your answers to parts (a) and (b) to show that = c a (1) Page 108 of 227

109 (d) Explain how you can tell that DEFG is a parallelogram. (1) (Total 4 marks) Q98. The diagrams show a trapezium and a parallelogram. Not drawn accurately (a) Use the trapezium to explain why 2x + y = 180 (1) (b) The parallelogram can be used to form another equation connecting x and y. Tick a box to show the correct equation. 3x + y = 130 3x + y = 230 3x = y 50 3x + y = 410 (1) Page 109 of 227

110 (c) Hence, or otherwise, work out the values of x and y. Answer x =..., y =... (3) (Total 5 marks) Q99. Airport runways have a two-digit number painted on them. These numbers are used to work out the direction of the runway. To work out the three-figure bearing, multiply the runway number by 10. Here is a diagram of a runway on a three-figure bearing of 280 and a runway on a three-figure bearing of 040. (a) (i) Write down the three-figure bearing for a runway pointing due South. Answer... (1) (ii) Write down the runway number for a runway pointing due South. Answer... (1) Page 110 of 227

111 (iii) A runway has a three-figure bearing of 060. Write down the runway number. Answer... (1) (b) A runway is being painted. By measuring the three-figure bearing, work out the runway number. Answer... (2) (c) Runways are used in both directions. This means that they have two different runway numbers, one at each end. A runway has the number 30 at one end. What runway number is at the other end? Answer... (3) (Total 8 marks) Page 111 of 227

112 Q100. This is the graph of y = sin x for values of x from 0 to 360 On each of the following grids the solid line shows a transformation of the graph of y = sin x. Write down the equation of each of the transformed graphs. On each grid, the graph y = sin x is shown dotted to help you. (a) Answer y =... (1) Page 112 of 227

113 (b) Answer y =... (1) (c) Answer y =... (1) (Total 3 marks) Page 113 of 227

114 Q101. In triangle ABC the length of AB is 13.2 cm. Angle BAC = 40 Angle BCA = 114 Not drawn accuratelly Work out the length of BC. Give your answer to an appropriate degree of accuracy Answer... cm (Total 4 marks) Q102. Advice about wheelchair ramps is that for a ramp that rises more than 7.5 cm the gradient should be maximum For example Not drawn accuratelly Page 114 of 227

115 (a) Access to a Village Hall is by two steps each 16 cm high. It is proposed to build a ramp alongside the steps as shown in the diagram. Not drawn accurately Does the proposed ramp follow the advice given? You must show your working (2) Page 115 of 227

116 (b) For a wheelchair ramp that rises less than 7.5 cm the maximum gradient should be For example Not drawn accurately A ramp is designed for a step that rises 6 cm. Not drawn accurately Work out the length of a ramp that uses the maximum gradient Answer... cm (4) (Total 6 marks) Page 116 of 227

117 Q103. The diagram shows a kite ABCD. AB = x + 1 and CD = 3x 2 Not drawn accurately (a) Show that, in terms of x, the perimeter of the kite is 8x 2 (1) (b) The perimeter of the kite is 16 cm. Write down and solve an equation to work out the value of x Answer x =... cm (3) (Total 4 marks) Q104. A, B and C are three points such that = 5a 3b and = 7.5a 4.5b (a) Write down a fact about the points A, B and C. (1) Page 117 of 227

118 (b) Write down the ratio of the lengths AB : BC in its simplest form. (1) (Total 2 marks) 2 2 Q105. The diagram shows the graph of x + y = 25 (a) By drawing a linear graph, write down one solution to the simultaneous equations 2 2 x + y = 25 and y = x Answer x =..., y =... (2) Page 118 of 227

119 (b) Explain why the x-coordinate in your answer to part (a) is an approximation of (2) (Total 4 marks) Q106. (a) The diagram shows a circle with centre O. TS is a tangent. Not drawn accuratelly Work out the value of x. Answer... degrees (1) Page 119 of 227

120 (b) The diagram shows another circle. Not drawn accuratelly Write down the value of y. Answer... degrees (1) (Total 2 marks) Q107. A baker is weighing out amounts of bread dough using a machine. The machine is set to weigh 400 grams of dough. The amounts of dough are each within 10% of the weight set. 300 amounts of dough are produced by the machine. What is the maximum total amount of bread dough that could be produced? Give your answer in kilograms Answer... kg (Total 3 marks) Page 120 of 227

121 Q108. The diagram shows an accurate scale drawing of the plan view of a house and garage. The drawing is on a centimetre grid. The front of the house is 12.5 metres long. (a) Explain why the scale is 1 centimetre represents 2.5 metres. (1) (b) The owner of the house is designing an extension to be joined to the back wall of the house. The extension is rectangular and is 7.5 metres long and 5 metres wide. It must be at least 2.5 metres from the garage. Make an accurate sketch of the possible extension on the grid. (3) (Total 4 marks) Page 121 of 227

122 Q109. The diagram shows a rhombus made of two triangles X and Y. M is the midpoint of diagonal AC. Not drawn accuratelly (a) Describe fully a single transformation that maps triangle X onto triangle Y. (2) (b) Describe fully a different single transformation that maps triangle X onto triangle Y. (3) (Total 5 marks) Q110. AB is a diameter of the circle, centre O. ABC is a straight line. DTC is a tangent to the circle at T. Angle BCT = 32 and angle TAB = x Not drawn accurately Page 122 of 227

123 Find the value of x. Give reasons for all angles you write down or calculate Answer... degrees (Total 4 marks) Q111. This right-angled triangle has sides of lengths (x 2) cm, (x + 5) cm and 10 cm. Not drawn accurately Page 123 of 227

124 Calculate the value of x. Give your answer to an appropriate degree of accuracy Answer... cm (Total 5 marks) Q112. An old windmill is the shape of a truncated cone. The mill is 12 metres high and has 4 floors, equally spaced. The diameter of the ground floor is 8 metres and the diameter of the roof is 6 metres. Not drawn accurately The mill is for sale. This is the advert. DEVELOPMENT OPPORTUNITY OLD MILL FOR SALE Over 150 square metres of floor space Page 124 of 227

125 Is the claim about floor space justified? You must show your working (Total 5 marks) Q is a prime number. (a) (i) Write down 251 Give your answer to 1 decimal place. Answer... (1) (ii) Explain how to test that 251 is a prime number (2) (b) (i) Express 2008 as a product of its prime factors Answer... (2) Page 125 of 227

126 (ii) Write down all the factors of Answer... (1) 2 2 (c) (i) Show that (x + y)(x y) = x y (1) (ii) Hence find both pairs of integers x and y such that 2 2 x y = Answer x =..., y =... x =..., y =... (4) (Total 11 marks) Page 126 of 227

127 Q114. The diagram shows two right-angled triangles. AD = 40 cm CD = 7 cm cos x = Not drawn accurately Find the value of sin y Answer... (Total 6 marks) Page 127 of 227

128 Q115. OABC is a quadrilateral. P, Q, R and S are the mid-points of OA, AB, BC and CO respectively. = 2a, = 2b and = 2c Not drawn accurately (a) Write down, in terms of a and b, the vector. Answer... (1) (b) Write down, in terms of c and b, the vector. Answer... (1) (c) Show that = = b (2) (d) Using your answer to part (c) write down a geometrical fact about the line joining the midpoints of two sides of a triangle. (1) Page 128 of 227

129 (e) What type of quadrilateral is formed by joining the mid-points of the four sides of a quadrilateral? Give a reason for your answer. Type of quadrilateral... Reason... (2) (Total 7 marks) Q116. The rule for this sequence is that each term is the mean of the two previous terms. a x y b (a) Find an expression for a in terms of x and y. Answer... (2) (b) Find an expression for b in terms of x and y. Simplify your answer. Answer... (2) (Total 4 marks) Q117. Four identical circular discs fit into a rectangle 10 cm long. Not drawn accurately Page 129 of 227

130 Ten of the same discs fit into a rectangle 22 cm long. Not drawn accurately 24 discs are placed together in the same way. How long is the rectangle? Answer...cm (Total 3 marks) Q118. Joe uses a ruler and compasses to find the centre of the circle drawn below. He starts by drawing a chord on the circle. Complete Joe s construction to find the centre of the circle. (Total 3 marks) Page 130 of 227

131 Q119. The rule for continuing a Fibonacci sequence is to add the last two terms to make the next term. For example, the sequence that starts 1, 1, continues as 1, 1, 2, 3, 5, 8, Two other Fibonacci sequences start a, 2a, and b, 4b, The fifth terms of these two sequences are equal. Given that a + b = 11, work out the values of a and b Answer a =... b =... (Total 4 marks) Q120. The diagram shows a cone. The diameter of the base of the cone is x cm. The height of the cone is also x cm. 3 The volume of the cone is V cm. Page 131 of 227

132 Find a formula for x in terms of V and π Answer... (Total 4 marks) Q121. ABCDEF is a regular hexagon. AFGH and AJKB are squares. Not drawn accurately Page 132 of 227

133 Show that triangle AHJ is equilateral (Total 4 marks) Q122. Solve the equation 3x 5x 7 = 0 2 Give your answers to 2 decimal places Answer... (Total 3 marks) Page 133 of 227

134 Q123. Triangle ABC has a right angle at B. Angle BAC = 38 AB = 7.21 cm Not drawn accurately Calculate the length of BC. Give your answer to an appropriate degree of accuracy Answer... cm (Total 4 marks) Q124. The diagram shows the graph of y = sin x for 0 x 360 Page 134 of 227

135 (a) Write down a possible equation of the following graph. Answer... (1) (b) Write down a possible equation of the following graph. Answer... (1) Page 135 of 227

136 (c) Write down a possible equation of the following graph. Answer... (1) (Total 3 marks) Q125. The diagram shows a hollow cylinder and a solid sphere. The radius of the cylinder = 3 cm The radius of the sphere = 3 cm The height of the cylinder = 6 cm Not drawn accurately Page 136 of 227

137 The sphere just fits inside the cylinder as shown. Not drawn accurately Work out the volume of the space left inside the cylinder. Give your answer in terms of π as simply as possible Answer... cm 3 (Total 5 marks) Page 137 of 227

138 Q126. (a) In the diagram O is the centre of the circle. Not drawn accurately What is the value of x? Answer... degrees (1) (b) Not drawn accurately What is the value of y? Answer... degrees (1) (Total 2 marks) Page 138 of 227

139 Q127. The diagram shows two identical shapes A and B. Describe fully the single transformation which takes shape A to shape B (Total 2 marks) Page 139 of 227

140 Q128. In the diagram AB and CD are parallel. Not drawn accurately (a) Write down the value of x. Answer... degrees (1) (b) Work out the value of y. Answer... degrees (2) (Total 3 marks) 2 Q129. This is the graph of y = x 4x + 1 Page 140 of 227

141 By drawing an appropriate linear graph, solve the equation x 5x + 3 = Answer... (Total 4 marks) Q130. (a) A test tube is formed from a cylinder and a hemisphere as shown. Work out the total volume of the test tube. Answer... cm 3 (4) Page 141 of 227

142 (b) The test tube is filled with water to a depth of d cm, as shown in the next diagram. The water occupies exactly half the full capacity of the test tube. Work out the value of d. Answer... cm (4) (Total 8 marks) Page 142 of 227

143 Q131. O is the centre of the circle. Angle PRS = 134 Not drawn accurately Work out the size of the reflex angle POQ. You must show your working Answer... degrees (Total 3 marks) Page 143 of 227

144 Q132. Triangle ABC has AB = 6 cm, AC = 10 cm, BC = 14 cm Not drawn accurately Calculate the largest angle in the triangle Answer... degrees (Total 3 marks) Q133. Katy is using the quadratic formula to solve a quadratic equation. After correctly substituting the values, she writes (a) What is the quadratic equation Katy is trying to solve? Answer... (3) Page 144 of 227

145 (b) Explain why Katy will not be able to find any solutions to the equation. (1) (Total 4 marks) Q134. (a) The right-angled triangle has sides shown. Not drawn accurately Show that x = 9 cm (2) Page 145 of 227

146 (b) This right-angled triangle has sides n, m and n + 1. m and n are integers. Prove that m must be an odd number. (5) (Total 7 marks) Q135. Here are four equations of graphs. 2 3 A y = 3x + 2 B 2x + 3y = 6 C y = 3x D y = x (a) Here are three sketch graphs. Match each graph to its equation. Equation... Equation... Equation... (3) Page 146 of 227

147 (b) On the axes below, sketch the graph of the other equation. (1) (Total 4 marks) Q136. A gold bar has a trapezium cross-sectional area. The dimensions are shown in the diagram. Not drawn accurately (a) Calculate the cross-sectional area of the gold bar. Answer... cm 2 (2) Page 147 of 227

148 (b) Gold has a density of 19.3 grams per cm. Work out the mass of the gold bar. Give your answer in kilograms. 3 Answer... kg (4) (Total 6 marks) Q137. (a) Factorise x + 10x 2 Answer... (1) (b) Factorise y 36 2 Answer... (1) (c) Solve the equation 5w + 6 = 9 w Answer w =... (3) Page 148 of 227

149 (d) Solve the equation Answer x =... (4) (Total 9 marks) Q138. A restaurant serves garlic bread. All the garlic breads are circular and the same thickness. They can be made with different diameters as shown. Robert is going to order a 14-inch garlic bread. The restaurant has a special offer. Special Offer Get one 7-inch garlic bread and one 10-inch garlic bread for the same price as a 14-inch garlic bread. Page 149 of 227

150 Robert says that if he has the special offer he will get less garlic bread. Is Robert correct? You must show your working (Total 4 marks) Q139. You have a square piece of paper which is folded in half to form a rectangle as shown. The perimeter of the rectangle is 39 centimetres. What is the area of the square you started with? Answer... cm 2 (Total 4 marks) Page 150 of 227

151 Q140. Is the statement below always true, sometimes true or never true? Tick the correct box. The circumference of a circle of diameter 10 cm is greater than the perimeter of a triangle with a base 10 cm. Always true Sometimes true Never true Explain your answer (Total 2 marks) Q141. Triangle T is drawn on the grid. (a) Draw the image of T after a rotation of 90 anticlockwise about O. (3) Page 151 of 227

152 (b) The triangle T is reflected to form a new triangle S. The coordinates of S are ( 4, 4), ( 3, 3), and ( 4, 1). Work out the equation of the mirror line. Answer... (2) (Total 5 marks) Q142. The radius of the Earth and the radius of Jupiter are in the approximate ratio 1 : 11. The mass of the Earth and the mass of Jupiter are in the approximate ratio 1 : 320. You will need the following information. The Earth and Jupiter are spherical The volume of a sphere of radius r is (a) Show that the approximate ratio of the volume of the Earth to the volume of Jupiter is 1 : (1) Page 152 of 227

153 (b) You are given density = Work out the approximate ratio of the average density of the Earth to the average density of Jupiter in the form 1 : n Answer 1:... (2) (Total 3 marks) Page 153 of 227

154 . (a) 2880 implies M2 or k = 2880 and equation seen using k (b) (W = their =) their so need 10 waiters. ft answer rounded up if awarded. ft ft [6] M2. (a) 65 angle at centre (b) 115 ft 180 their 65 provided reason given is not contradictory ft Opposite angles (of cyclic quad) or other valid explanation eg x + y = 180 [4] M3. (a) Correct enlargement for enlargement any scale factor (not 1) Accept any orientation B2 Page 154 of 227

155 (b) or 3 3 or 54 6 SC1 for their (SF in (a)) 2 Accept ratio 1 : 9 or 9 : 1 [4] M4. AC 2 = cos 75 AC 2 = Accept 97.4 DC 2 = (their AC) DC = 7.8(35 ) 29.8(35 ) or 7.84 or or 30 with correct working ft ft [5] M5. (a) (i) b + a or a b (ii) b a oe (b) oe = a + b a Page 155 of 227

156 (c) = 2 or OBN a straight line or BN = OB or B is midpoint of ON [5] M6. (a) or 45 seen or or 1080 or (2 8 4) right angles 180 (their 45) (their 1080) 8 dep (b) 360 (their ) or in X Sides of X are equal or (regular) octagons so sides are equal 4 lines of symmetry or rotational symmetry of order 4 scores 3 marks Other symmetry scores [6] Page 156 of 227

157 M7. (3x + 2)(x + 1) Rectangle 3x 2 + 5x + 2 Rectangle x 3x + 5(x + y) or x 3x + x 5 + y 5 or x(3x + 5) + y 5 or (3x + 5)(x + y) 3x y 3x 2 + 5x + 5y 5y = L shape oe dependent on a previous and a term in y oe dep [6] M8. (a) 6 (b) Plot points Draw curve (c) x = 1.4 and 1.4 (d) (3x 2 6) (3x 2 5x 6) Sight of (±) 5x (+ k) = 5x Draw y = 5x x = 2.5, 0.8 Accept 2.4 to 2.55 and 0.75 to 0.85 ft [7] Page 157 of 227

158 M9. (a) A d 2 or A = kd 2 When d = 15, A = = 225k k = 0.4 or A = 0.4d 2 Accept A = oe (b) d = 20 => A = A = unsupported SC1 (c) A = = 0.4d 2 d 2 = = 625 Dep on M2 in (a) Accept dep d = 25 [7] 0. (a) (0, 1) Generally marked (b) Matching any (non zero for x) Values eg, a 1 = 3, a 2 = 9, etc Must show as power a = 3 [3] Page 158 of 227

159 1. Volume of one (or two) spheres Allow 10 for r for = (1047.2) {524} {1048, 1050} oe 500π/3 or 1000π/3 (523.3 to 523.7) or ( to ) Volume of cylinder = {1570, 1571} Allow 10 for h or 10 for r for (not both) oe 500π (1570 to 1571) Volume remaining ( =) 523.6, 524 oe Due to different values of π an answer between and gets full marks ft If both Ms awarded and one value is correct. ft [5] 2. Breaks down into areas of rectangles and areas of (quarter) circles Any combination of rectangles and circles or or is enough evidence for area of circles NB from 2 π 2, if seen is M0 NB 3.14 on its own is not evidence of the area of a quarter circle as it is π Page 159 of 227

160 Uses an addition method (method 1) and finds Area of one (or 5) external quadrants or or Uses a subtraction method (methods 2 and 3) and finds 5 area one quadrant or dep = , {0.9, 0.86, 0.858} or = ( 5) {4.3, 4.29} 15.71, or π Allow 52 if 52.3 or or a full method seen [4] 3. (a) or or k = 3200 or P = 3200/Q or PQ = 3200 or Q = 3200/P (b) Correct sketch graph (c) (Their 3200) or 2Q = (Their 3200) Q or Q = (Their 3200) 2Q (Q =) 40 ft Their value of k ft [6] Page 160 of 227

161 4. (a) (b) (c) oe [3] 5. (a) Or l = 2r Clearly shown since answer given (b) h 2 = 4r 2 r 2 Attempt to use Pythagoras theorem correctly h = 3r h 2 = 3r 2 is sufficient or h = (3r 2 ) (c) 3r : ft with Their h if 1 st earned dep 3 : 2 [6] 6. (a) Multiply through by x, 4 = x(9 2x) is enough Expanding and rearranging must be seen (answer given) Page 161 of 227

162 (b) Attempt at identified as being required line Points worked out eg, table of values (2 points minimum) Correct line plotted With ruler, must intersect the curve twice dep (c) Solutions can come from factorising ie, x = 4 No ft from incorrect factors [7] 7. or Condone use of π = 3.(14...) and (Their 50π) (Their 25π) 25 dep [4] Page 162 of 227

163 8. (a) y = cosx + 1 y = 1 + cosx (b) (c) y = 2cosx y = cos2x (d) y = cos(90 x), y = cos (x +270) y = cos (x 90) or y = sin x [4] 9. ABD = 66 (Alt segment) or angles in triangle if ADB found first DCB = 104 (opposite in cyclic) In all alternatives, for first 3 B marks do not award the first time no reason or wrong reason given, otherwise accept angles identified in answer or on diagram. NB Mark positively ie, ignore wrong values or reasons unless totally contradictory. DBC = 38 (isosceles) CBA = 104 CBA + BAD = 180 (interior) In all alternatives, reason must be given for final Accept allied or angles between parallel lines. Dependent on correct angles. Alt. 1 ADB = 38 (Alt segment) DCB = 104 (opposite in cyclic) CBD = 38 (isosceles) CBD = ADB (alternate) Use of Z angles is not acceptable Dependent on correct angles Page 163 of 227

164 Alt. 2 ADB = 38 (Alt segment) DCB = 104 (opposite in cyclic) BDC = 38 (isosceles) ADC = 76 BDC + BCD = 180 (interior) Dependent on correct angles Alt. 3 ADB = 38 and ABD = 66 (Alt segment) DCB = 104 (opposite in cyclic) CBD = CBD = 38 (isosceles) DCB = CBA and CDA and BAD = (isosceles trapezium) [4] M20. (a) (b) (i) (052) (ii) 0.9, 0.91, 0.912, 9 or 9.1 or ft their answer for (b)(i) to 1, 2 or 3sf eg Gradians (b)(i) B0 (b)(ii) 0.02, , etc ft ft (c) or [4] Page 164 of 227

165 M21. (a) π r 3 = 2 π r 2 x Must include the factor of 2 Allow use of h instead of x Simplified to give x = 2r Alternatively Allow substitution of 2r for height of cone and verification of result ie 2 Vol cone = 2 π r 2 2r = π r 3 (must be seen) (b) (l) 2 = r 2 + 4r 2 (l) 2 = r 2 + (2r) 2 is (l) 2 = r 2 + 2r 2 is M0 (l) = 5 r Surface area cone = π r 5 r Using their l if from an attempt at Pythagoras 4 : 5 Allow 5 : 4 SC2 for a complete numerical solution [6] M22. (a) Correct Pythagoras in two appropriate right-angled triangles 13 or simply BH ² = 12² + 3² + 4² Page 165 of 227

166 (b) HB = 13, HC = 5 or DB = 153 with attempt at trig. Ratio Explanations may not involve any calculations eg BC < BD or HC > HD together with some comparison such as BH is common (diagrams drawn, to illustrate, are appropriate) Two correct, comparable trig. ratios eg sin x = and sin y = For example: BH is common and triangles BHD and BHC are right-angled, so y must be bigger because the height is greater y Good explanation and correct conclusion this earns all 3 marks [5] M23. YZ = ZY Angle MZY = angle NYZ base angles of (Isosceles) XYZ Note Reason necessary eg you might see If XZ = XY then angle XZY = angle XYZ Angle MYZ = angle NZY Triangles congruent, ASA Note Dependent on earning first 3 marks Must give correct reason for congruence (ASA) Only allow AAS if complete argument stating third angles equal dep [4] Page 166 of 227

167 M24. Sight of correct ratio or scale factor ie 20 : 30, 2 : 3,, 1 oe sight of or earns this mark 40 oe eg might work out then subtract 24 Note 2 : 3 ratio might be scaled up to give ratio of 16 : 24 (, ) Must state h = 24 for Alternatively h/30 = (40 h)/20 20h = 30(40 h) 20h = h 50h = 1200 h = 24 [3] M25. (a) 80.2(1...) r= 12 giving 40.1 is Ml, Al, A0 r = 3 giving is Ml, Al, A0 (b) h = 9 (cm) h = 12 gives Ml, A0 Ml for difference of two cone volumes Al if all correct, Page 167 of 227

168 (V) = 327 or (cm 3 ) Accept 330 if working seen, ft their h if both M's awarded. ft [8] ALTERNATIVE linear scale factor 1:3 Must be used. Just writing it down does not qualify as a method unless progress is made. Volume scale factor 1:27 Volume small cone Volume large cone 27 (their ) D (V) = 327 or (cm 3 ) Accept 330 if working seen. Scs only only,,, [8] M26. (a) y = x+ 1, y = x 2 x + 1= x 2 x 2 = x 2 + 2x 3 Page 168 of 227

169 (b) x 2 + 3x 1 = 0 Simplified to 3 terms in x 2, x and constant e.g. x 2 = 1 3x x 2 + 2x 3 = + 1 (c) 1.6, 2.6 Accept 1.5 to 1.6, 2.5 to 2.6 [5] M27. oe e.g. [3] M28. (a) Accept Sin B = (1...) B = 75.4(...) Page 169 of 227

170 (b) x 2 = cos 48 x 2 = 335.8(...) x = 18.32(...) 18 or 18.3 ft only if an error made in calculation of x 2 but not on ( ( = 1)) cos 48 (= = 0.818) Independent mark. Award if value > 3sf seen or calculation seen. ft ft [7] M29. (a) 4a + 3 = 2b + 5 (b) 4a 2b = 2 (-2) Must indicate division by 2 4a+3+2b+5+2a+b=32 6a + 3b = 24 2a + b = 8 Bl for any version (1) 3: 6a 3b = 3 12a = 27 For attempt to eliminate AB or 4a + 3 =12 and BC or 2b + 5 = 12 a = 2.25 [6] Page 170 of 227

171 M30. (a) Sight of tan unless alternative method used Tan 1 (5.59/1.5) 90 tan 1 (l.5/5.59), 1.5tan70 and 1.5tan80 74.(98) or 75 so safe 4.1(2) and 8.5(1) D (b) Sight of cos 4 cos80 D with working [6] M31. Vol Hemisphere = Ml, 144π ( π) ( π) = r 2 or... their 144π...(4 6 3 ) (2 27) = r 2 (r =) 4 (r =) 4 [4] Page 171 of 227

172 M32. angle OSQ = angle OQS = 50 Isosceles triangle OQS Penalise no reason angle OSR = 90 angle QSR = 40 Tangent-radius property first time only angle QSR = angle QRS (Isosceles) [3] M33. (Graph 1) D (Graph 2) A (Graph 3) E (Graph 4) C [4] M34. (a) Line y= 1 drawn or points on curve 0.8, 3.8 (±0.1) Accept y = 1 written in body of script. (b) Attempt to split equation into x 2 + 3x 2 = ax + b Or x 2 + 3x 2 -(x 2 + 2x 1) Or x 2 + 3x 2 + ax + b = x 2 + 2x 1 Line (y = x 1) drawn 0.4, 2.4 (±0.1) f.t. on their line if Ml awarded, e.g. y = x + 1(1, 3), y = 1 x(0.6 (0.7), 4.6 ( 4.7)),y = 1 x(0.2, 4.2) ft [5] Page 172 of 227

173 M35. (a) Linear scale factor is 2 Allow numerical examples but must be complete 2 3 = 8, 4 3 = 64, 64 = 8 8, but the increase by a factor of 8 must be shown and not assumed B2 Volume scale factor is lsf 3 Allow algebra (2x) 3 = 8x 3 (b) (14.5 8) 3 or = , =5.95(4) Volume increases by about 6 so claim justified. Allow Almost but not quite [5] M36. Use of Must use 4 or 8 as radius. (Volume hemisphere =) to (inclusive) if π = 3.14 used. (Volume paperweight =) 500+(their 134) (=634) If Ml awarded. ft cm 3 This mark is independent [4] Page 173 of 227

174 M37. Area semicircle area circle Accept π and/or π 5 2 for π5 2 2 π(2.5) 2 (= 12.5π 6.25π = ) Accept fractions, decimals or in terms of π 19.6(...)(= 6.25π) ft on one error only, e.g. Accept fractions, decimals or in terms of π. Use of π as 3.14 gives Al Common errors e.g. π π 5 2 = = 78.6 = Ml,A0,A0ft. π l0 2 2 π = = =,AO, ft π π 5 2 = = 39.3 Ml, AO, AO (non-sensible answer) ft [3] M38. y = kx 2 or y a x 2 oe 5 = k 16 k = oe 20 [3] Page 174 of 227

175 M39. Breaks problem down into sum of lines and (semi-)circles Length of lines (= 17.6) Sc 17.6 only Use of 2 πr 2 or πd 2 but must use with numbers. D Length of semi-circles 0.9π + 0.6π + 0.7π (= 6.9(11..)) 2.8, 1.9,2.2 Total = 24.5(...) ft on 1 arithmetical or reading from scale error and both M's awarded. 4.1 = , 5.9 = , 4.7 = , 2.9 = 2.9 ft [5] M40. (a) BC 2 = (= 280) x = 19 2 BC = 280 For squaring, subtracting and evidence of square rooting BC = 17 or 16.7(...) 17 with no working gets 3 D Page 175 of 227

176 (b) Sight of tangent or Angle = tan l (1 24) tan 1 (0.458) M2 for any complete correct method Sin = 11/ 697 or 11/26.4 Cos = 24/ 697 or 24/26.4 D 25 or 24.6(...) 25 with no working gets 3 Radians 0.43 gradians Penalise on first occurrence only. [6] M or or /60 60 or 80/ min or 96 min Their(100 96) or reversed 4 D D [4] Page 176 of 227

177 M42. (a) (2x 3) 2 = 4x 2 6x 6x + 9 condone one error 4x is two errors 4x 2 12x + 9 = 8x 16 or 4x 2 20x +25 (= 0) for equating expressions and/or simplifying this must lead to a quadratic equation (2x 5)(2x 5) ( 0) ft from their quadratic equation (if 'formula' used, substitution must be completely correct) x = 2.5 y = 2 (b) Only one solution so straight line must be a tangent to the curve Hence sketch 2 ft from their solution(s) to (a) clear solution(s) in (a) correct sketch in (b) can earn (no explanation) or B2 (with explanation) B2 ft ALTERNATIVE (a) oe for setting up attempt to eliminate x y 2 4y + 4 (= 0) oe condone one error this must lead to a quadratic equation (y 2)(y 2)( 0) ft from their quadratic equation (if formula used, substitution must be completely correct) y = 2 x = 2.5 Page 177 of 227

178 (b) Only one solution so straight line must be a tangent to the curve Hence sketch 2 ft from their solution(s) to (a) clear solution(s) in (a) correct sketch in (b) can earn (no explanation) or B2 (with explanation) B2 ft [7] M43. (a) x = scores (b) x = 250, x = 290 for each B2 (c) correct sketch of double cycle for sketch only as far as 180 or slight inaccuracy B2 (d) x = 35, x = 55 must have 35 and ½ of their (a) x = 215 ft x = 235 for each of (their ), (their ) ft [8] M44. (a) 0 = 4 + 2p + q 5 = 9 3p + q for substitution of both sets of coordinates allow one error 5 = 5 + 5p oe for correct attempt at elimination of p or q p = 2 D q = 8 p = 2 and q = 8 from no obvious working scores 4 Page 178 of 227

179 (b) Solving their x 2 + px + q = 0 if formula used substitution must be completely correct ( 4, 0) [6] M45. (a) π( )5 2 condone π ( ) or (their area) 10π ( ) or (their area) 10 condone their area must contain π 250π or 250 π or π to 790 scores M2 A0 do not accept π250 ignore fw 250π can be recovered in (b) (b) their their 1000 their 250π 20 their (π 5 2 ) 20 their 250π 20 their ( π) their 2000 their 500π π 20( π) 10( π) 4290 to 4500 scores M3 A0 ignore fw except 15000π [7] Page 179 of 227

180 M46. (a) (i) a + b b + a (ii) (iii) 2a + b b + 2a b a a + b (b) CF = OE a + 2b a + b + b oe (c) Straight line because OD = a + b 3 times bigger because OF = 3a + 3b [7] M for Accept 74 or 74.3 Height first pyramid is Height second is Volume is = This line,. [2] Page 180 of 227

181 M48. Angle ATB = 13 for use of sine rule, for correct substitution. =, BT = ( ) AT = H = BT sin31 H=14.2 or 14.15(...) Ft only if both Ms awarded. NB 14.2 can come from BT = 27.5 or AT = 46 Deduct 1 for pa if seen. ft [6] M49. (a) Parallel curve translated up y axis 2 need not be marked, needs to look symmetrical (b) Parallel curve translated in positive direction along x axis Must 'sit on' x axis and look symmetrical (c) Curve through (0,0) nearer to x axis than original Must look symmetrical [3] M50. (a) (i) y = 3x 2 plotted must draw correct line x = 2, x = 5 for each, must be correct answers...no ft. coordinates given... lose 1 mark A2 Page 181 of 227

182 (ii) x = 2, x = 5 must have both solutions (ft answers from part (a) earns 1 mark) (b) x 2 4x +8 = x + 4 allow one slip in manipulation y = x + 4 Straight line to be clearly stated [6] M51. (a) 4 (b) ( ) ( 2) or 14 or or equivalent 7 [3] M52. Any correctly evaluated counter example with non-prime conclusion. Examples 4 and 5 => 25 and not prime 5 and 6 => 35 and not prime 8 and 9 => 77 and not prime accept any indication of "not prime" any correctly evaluated trial with no conclusion Examples 3 and 4 => 17 4 and 5 => 25 5 and 6 => 35 6 and 7 => 47 7 and 8 => 61 8 and 9 => 77 or incorrectly evaluated trial that gives a counter example with non-prime conclusion B2 [2] Page 182 of 227

183 M53. Extra volume = = 7650 r 3 = their Dependent on their coming from a volume calculation. r 3 = (3 their ) 4π Allow (3 7650) 4π R = 7.7, 7.70, D [5] M54. AC 2 = cos75 AC 2 = 97., AC= 9.9, 9.86 Their AC AC 2 must be > 36 = if correct DC = 7.8(3 ) Answer must be accurate to 2 sf or better ft Perimeter = 29.8(. ) ft their DC + 22 but both Ms must be awarded. ft [5] Page 183 of 227

184 M55. Angle at centre = 2 sin 1 ( ) Half angle sin 1 ( ) gets = 71.(...) Area sector = their π 6 2 Area sector = 22.4(.)_ M for use of area sector formula not for πr 2 4 for example. Their sector their triangle area Must make a valid attempt at calculating the area of the triangle. (17.06 ) and at least one of the previous M marks must be awarded. Area segment = D [6] M56. (a) 0.51(2) (b) Correct plots Smooth curve ±0.5 square Use of ruler or double lines or discontinuities B0 ft ft (c) 1.2 ft their graph. If double line at y = 0.76 then B0. Within tolerance of their graph ft [4] Page 184 of 227

185 M57. B2 fully correct for any translation of correct answer. Alternative scheme. for rays from at least 3 corners through ( 1, 0) and attempt at drawing a reduced shape in 3rd quadrant. if correct shape B2 [2] M58. C = π 7 = C = 2π 3.5 Must substitute numbers. C = πd or 2πr is M0 until used. NB π 3.5 is M0 as wrong method (πr) = 21.98, = 22 Length = to 23 ft their if awarded. ft Height = 10 cm Allow answers transposed. [4] Page 185 of 227