11 cm. A rectangular container is 12 cm long, 11 cm wide and 10 cm high. The container is filled with water to a depth of 8 cm.

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1 Diagram NOT accurately drawn 10 cm 11 cm 12 cm 3.5 cm A rectangular container is 12 cm long, 11 cm wide and 10 cm high. The container is filled with water to a depth of 8 cm. A metal sphere of radius 3.5 cm is placed in the water. It sinks to the bottom. Calculate the rise in the water level. Give your answer correct to 3 significant figures. Bemrose Community School 1

2 ...cm (Total 4 marks) 2. B A C Diagram NOT accurately drawn 10.4 cm 10.4 cm 120 O The diagram shows a sector OABC of a circle with centre O. OA = OC = 10.4 cm. Angle AOC = 120. Calculate the area of the shaded segment ABC. Give your answer correct to 3 significant figures..cm 2 (Total 4 marks) Bemrose Community School 2

3 3. 40 O Diagram NOT accurately drawn 9 cm The diagram shows a sector of a circle, centre O. The radius of the circle is 9 cm. The angle at the centre of the circle is 40. Find the perimeter of the sector. Leave your answer in terms of π....cm (Total 4 marks) Bemrose Community School 3

4 4. The diagram below shows a 6-sided shape. All the corners are right angles. All measurements are given in centimetres. Diagram NOT accurately drawn The area of the shape is 25 cm 2. Show that 6x x 39 = 0 (Total 3 marks) Bemrose Community School 4

5 5. Diagram NOT accurately drawn The diagram represents a large cone of height 6 cm and base diameter 18 cm. The large cone is made by placing a small cone A of height 2 cm and base diameter 6 cm on top of a frustum B. Calculate the volume of the frustum B. Give your answer in terms of π.... (Total 4 marks) Bemrose Community School 5

6 6. The radius of the base of a cone is x cm and its height is h cm. The radius of a sphere is 2x cm. The volume of the cone and the volume of the sphere are equal. Express h in terms of x. Give your answer in its simplest form. h =.. (Total 3 marks) Bemrose Community School 6

7 7. Diagram NOT accurately drawn B A C 10.4 cm 120 O 10.4 cm The diagram shows a sector OABC of a circle with centre O. OA = OC = 10.4 cm. Angle AOC = 120. (a) Calculate the length of the arc ABC of the sector. Give your answer correct to 3 significant figures....cm (3) Bemrose Community School 7

8 (b) Calculate the area of the shaded segment ABC. Give your answer correct to 3 significant figures....cm 2 (4) (Total 7 marks) 8. The diagram shows a sector of a circle with a radius of x cm and centre O. PQ is an arc of the circle. Angle POQ = 120. O 120 x cm P Diagram NOT accurately drawn Q (a) Write down an expression in terms of and x for (i) the area of this sector,... (ii) the arc length of this sector.... (2) Bemrose Community School 8

9 The sector is the net of the curved surface of this cone. Arc PQ forms the circumference of the circle that makes the base of the cone. h cm x cm The curved surface area of the cone is A cm 2. The volume of the cone is V cm 3. The height of the cone is h cm. Given that V = 3A, (b) find the value of h.... (3) (Total 5 marks) Bemrose Community School 9

10 y y = a sin x Diagram NOT accurately drawn y = cos x + b x The diagram shows part of two graphs. The equation of one graph is The equation of the other graph is y = a sin x y = cos x + b (a) Use the graphs to find the value of a and the value of b. a =... b =... (2) (b) Use the graphs to find the values of x in the range 0 x 720 when a sin x = cos x + b. x =... (2) Bemrose Community School 10

11 (c) Use the graphs to find the value of a sin x (cos x + b) when x = (2) (Total 6 marks) Bemrose Community School 11

12 10. F E Diagram NOT accurately drawn D C 60 cm A cm B The diagram represents a prism. AEFD is a rectangle. ABCD is a square. EB and FC are perpendicular to plane ABCD. AB = 60 cm. AD = 60 cm. Angle ABE = 90. Angle BAE = 30. Calculate the size of the angle that the line DE makes with the plane ABCD. Give your answer correct to 1 decimal place.... Bemrose Community School 12

13 (Total 4 marks) Bemrose Community School 13

14 11. Diagram NOT accurately drawn D A B 25 m C Bemrose Community School 14

15 The diagram shows a vertical tower DC on horizontal ground ABC. ABC is a straight line. The angle of elevation of D from A is 28. The angle of elevation of D from B is 54. AB = 25 m. Calculate the height of the tower. Give your answer correct to 3 significant figures.... m (Total 5 marks) Bemrose Community School 15

16 12. x + 2 Diagram NOT accurately drawn x 5 x + 6 The diagram shows a trapezium. The lengths of three of the sides of the trapezium are x 5, x + 2 and x + 6. All measurements are given in centimetres. The area of the trapezium is 36 cm 2. (a) Show that x 2 x 56 = 0 (4) Bemrose Community School 16

17 (b) (i) Solve the equation x 2 x 56 = 0 (ii) Hence find the length of the shortest side of the trapezium. cm (4) (Total 8 marks) Bemrose Community School 17

18 13. y 100 y = a bcos( kt) O t The graph of y = a bcos(kt), for values of t between 0 and 120, is drawn on the grid. Use the graph to find an estimate for the value of (i) a, (ii) b, (iii) k (Total 3 marks) Bemrose Community School 18

19 14. H E 3 cm A 5 cm F B D 7 cm G C Diagram NOT accurately drawn The diagram represents a cuboid ABCDEFGH. AB = 5 cm. BC = 7 cm. AE = 3 cm. (a) Calculate the length of AG. Give your answer correct to 3 significant figures.... cm (2) Bemrose Community School 19

20 (b) Calculate the size of the angle between AG and the face ABCD. Give your answer correct to 1 decimal place.... (2) (Total 4 marks) Bemrose Community School 20

21 15. T Diagram NOT accurately drawn x x + 5 O x + 8 A AT is a tangent at T to a circle, centre O. OT = x cm, AT = (x + 5) cm, OA = (x + 8) cm. (a) Show that x 2 6x 39 = 0 (4) (b) Solve the equation x 2 6x 39 = 0 to find the radius of the circle. Give your answer correct to 3 significant figures.... cm (3) Bemrose Community School 21

22 (Total 7 marks) Bemrose Community School 22

23 16. Diagram NOT accurately drawn 3 cm 3 cm The radius of a sphere is 3 cm. The radius of the base of a cone is also 3 cm. The volume of the sphere is 3 times the volume of the cone. Work out the curved surface area of the cone. Give your answer as a multiple of. Bemrose Community School 23

24 cm 2 (Total 7 marks) Bemrose Community School 24

25 17. 4 cm 6 cm Diagram NOT accurately drawn A B Cylinder A and cylinder B are mathematically similar. The length of cylinder A is 4 cm and the length of cylinder B is 6 cm. The volume of cylinder A is 80 cm 3. Calculate the volume of cylinder B. cm 3 (Total 3 marks) Bemrose Community School 25

26 18. L m x Elliot did an experiment to find the value of g m/s 2, the acceleration due to gravity. He measured the time, T seconds, that a block took to slide L m down a smooth slope of angle x. He then used the formula g = T 2 2L sin x to calculate an estimate for g. T = 1.3 correct to 1 decimal place. L = 4.50 correct to 2 decimal places. x = 30 correct to the nearest integer. (a) Calculate the lower bound and the upper bound for the value of g. Give your answers correct to 3 decimal places. Lower bound... Upper bound... (4) Bemrose Community School 26

27 (b) Use your answers to part (a) to write down the value of g to a suitable degree of accuracy. Explain your reasoning (1) (Total 5 marks) 19. This is a sketch of the curve with equation y =f(x). It passes through the origin O. y y = f( x) O x A (2, 4) The only vertex of the curve is at A (2, 4) (a) Write down the coordinates of the vertex of the curve with equation (i) y = f(x 3), (...,...) (ii) y = f(x) 5, (...,...) (iii) y = f(x), (...,...) (iv) y = f(2x). (...,...) (4) Bemrose Community School 27

28 The curve with equation y = x 2 has been translated to give the curve y = f(x). (b) Find f(x) in terms of x. f(x) =... (4) (Total 8 marks) 20. x x A cuboid has a square base of side x cm. The height of the cuboid is 1 cm more than the length x cm. The volume of the cuboid is 230 cm 3. (a) Show that x 3 + x 2 = 230 (2) Bemrose Community School 28

29 The equation x 3 + x 2 = 230 has a solution between x = 5 and x = 6. (b) Use a trial and improvement method to find this solution. Give your answer correct to 1 decimal place. You must show all your working. x =... (4) (Total 6 marks) 21. A straight line has equation y = 2 1 x +1 The point P lies on the straight line. P has a y-coordinate of 5. (a) Find the x-coordinate of P.... (2) Bemrose Community School 29

30 (b) Write down the equation of a different straight line that is parallel to y = 2 1 x (1) (c) Rearrange y = 2 1 x + 1 to make x the subject.... (2) (Total 5 marks) 22. A X Diagram NOT accurately drawn 8 cm C cm B In triangle ABC, AC = 8 cm, CB = 15 cm, Angle ACB = 70. Bemrose Community School 30

31 (a) Calculate the area of triangle ABC. Give your answer correct to 3 significant figures.... cm 2 (2) X is the point on AB such that angle CXB = 90. (b) Calculate the length of CX. Give your answer correct to 3 significant figures.... cm (4) (Total 6 marks) Bemrose Community School 31

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