1. Peter cuts a square out of a rectangular piece of metal. accurately drawn. x + 2. x + 4. x + 2
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1 1. Peter cuts a square out of a rectangular piece of metal. 2 x + 3 Diagram NOT accurately drawn x + 2 x + 4 x + 2 The length of the rectangle is 2x + 3. The width of the rectangle is x + 4. The length of the side of the square is x + 2. All measurements are in centimetres. The shaded shape in the diagram shows the metal remaining. The area of the shaded shape is 20 cm 2. (a) Show that x 2 + 7x 12 = 0 (4)
2 (b) (i) Solve the equation x 2 + 7x 12 = 0 Give your answers correct to 4 significant figures... (ii) Hence, find the perimeter of the square. Give your answer correct to 3 significant figures... cm (1) (Total 8 marks) 2. The graph of y = f(x) is sketched in the diagrams below. x Diagram A (2, 6) 0 4 y (a) On Diagram A sketch the graph of y = f(x)
3 Diagram B x (2, 6) 0 4 y (b) On Diagram B sketch the graph of y = f(x 1) (Total 4 marks) 3. The expression 8x x 2 can be written in the form p (x q) 2, for all values of x. (a) Find the value of p and the value of q. p =.. q =..
4 (b) The expression 8x x 2 has a maximum value. (i) Find the maximum value of 8x x 2. (ii) State the value of x for which this maximum value occurs. (Total 6 marks) 4. Diagram NOT accurately drawn A A 10 cm 5 cm 5 cm 5 cm 10 cm B 20 cm B 20 cm 15 cm 15 cm The diagram represents a large cone of height 30 cm and base diameter 15 cm. The large cone is made by placing a small cone A of height 10 cm and base diameter 5 cm on top of a frustum B.
5 (a) Calculate the volume of the frustum B. Give your answer correct to 3 significant figures....cm 3 d cm h cm Diagram NOT accurately drawn 3 d cm The diagram shows a frustum. The diameter of the base is 3d cm and the diameter of the top is d cm. The height of the frustum is h cm. The formula for the curved surface area, S cm 2, of the frustum is S = 2πd 2 h d 2 (b) Rearrange the formula to make h the subject. h =...
6 Two mathematically similar frustums have heights of 20 cm and 30 cm. The surface area of the smaller frustum is 450 cm 2. (c) Calculate the surface area of the larger frustum....cm 2 (Total 8 marks) 5. x Diagram NOT accurately drawn 2x The diagram shows a trapezium. The measurements on the diagram are in centimetres. The lengths of the parallel sides are x cm and 20 cm. The height of the trapezium is 2x cm. 20
7 The area of the trapezium is 400 cm 2. (a) Show that x x = 400 (b) Find the value of x. Give your answer correct to 3 decimal places.... (Total 5 marks) 6. (a) Show that (2a 1) 2 (2b 1) 2 = 4(a b)(a + b 1)
8 (b) Prove that the difference between the squares of any two odd numbers is a multiple of 8. (You may assume that any odd number can be written in the form 2r 1, where r is an integer). (Total 6 marks) 7. For all values of x and m, x 2 2mx = (x m) 2 k (a) Express k in terms of m....
9 The expression x 2 2mx has a minimum value as x varies. (b) (i) Find the minimum value of x 2 2mx. Give your answer in terms of m.... (ii) State the value of x for which this minimum value occurs. Give your answer in terms of m.... (Total 5 marks) 8. (a) Expand and simplify 3(2x 1) 2(2x 3) (b) Factorise y 2 + y (1) (Total 3 marks) 9. (a) Complete the table for y = x 2 3x + 1 x y
10 (b) On the grid, draw the graph of y = x 2 3x + 1 y O x (c) Use your graph to find an estimate for the minimum value of y. y = (1)
11 (d) Use a graphical method to find estimates of the solutions to the equation x 2 3x + 1 = 2x 4 x = or x = (Total 8 marks) 10. Two numbers have a difference of 15 and a product of The larger of the two numbers is x. (a) Show that x 2 15x = 0 (b) Solve the equation x 2 15x = 0 (Total 6 marks)
12 11. (a) Factorise 9x 2 6x + 1 (b) Simplify 6x 9x 2 2 7x 3 6x 1 (Total 5 marks)
13 12. Solve the simultaneous equations x 2 + y 2 = 29 y x = 3 (Total 7 marks)
14 13. 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 (Total 7 marks)
15 14. Bill said that the line y = 6 cuts the curve x 2 + y 2 = 25 at two points. (a) By eliminating y show that Bill is incorrect. (b) By eliminating y, find the solutions to the simultaneous equations x 2 + y 2 = 25 y = 2x 2 x =... y =... or x =... y =... (6) (Total 8 marks)
16 15. (a) Solve 40 x = 4 + x 3 x =... (b) Simplify fully 4x 2 4x 2 6x 9... (Total 6 marks)
17 16. y = r t sin x r t sin x r = 8.8 t = 7.2 x = 40 (a) Calculate the value of y. Give your answer correct to 3 significant figures. y =... y = 2 t = 10 x = 30 (b) Find the value of r. r =... (Total 6 marks)
18 17. The straight line L 1 has equation y = 2x + 3 The straight line L 2 is parallel to the straight line L 1. The straight line L 2 passes through the point (3, 2). Find an equation of the straight line L (Total 3 marks) 18. (a) Solve x 2 + x + 11 = 14 Give your solutions correct to 3 significant figures....
19 y = x 2 + x + 11 The value of y is a prime number when x = 0, 1, 2 and 3 The following statement is not true. y = x 2 + x + 11 is always a prime number when x is an integer (b) Show that the statement is not true (Total 5 marks) 19. Simplify 2 4x 9 2 2x 5x 3... (Total 3marks) 20. (a) On the grid below, draw the graphs of x 2 + y 2 = 100 and 2y = 3x 4
20 (b) Use the graphs to estimate the solutions of the simultaneous equations x 2 + y 2 = 100 and 2y = 3x For all the values of x x 2 + 6x = (x + 3) 2 q (c) Find the value of q. q =... One pair of integer values which satisfy the equation is x = 6 and y = 8 x 2 + y 2 = 100 (d) Find one pair of integer values which satisfy x 2 + 6x + y 2 4y 87 = 0 x =..., y =...
21 y x (Total 10 marks) 21. (a) Make t the subject of the formula v = u + 5t t =..
22 (b) Solve x 3 5 = x 5 x =.. (Total 5 marks) 22. (a) Simplify a 3 a 4 (1) (b) Simplify 3x 2 y 5xy 3 (c) Simplify ( x 1) x 1 2 (1) (d) Factorise a 2 9b 2 (Total 6 marks)
23 23. P = πr + 2r + 2a P = 84 r = 6.7 (a) Work out the value of a. Give your answer correct to 3 significant figures. a =.. (b) Make r the subject of the formula P = πr + 2r + 2a r = (Total 6 marks)
24 24. The diagram below shows a 6-sided shape. All the corners are right angles. All measurements are given in centimetres. The area of the shape is 25 cm 2. (a) Show that 6x x 39 = 0 (b) (i) Solve the equation 6x x 39 = 0 x = or x = (ii) Hence work out the length of the longest side of the shape...cm
25 (4) (Total 7 marks) 25. Simplify fully (a) ( 3xy 2 ) 4. (b) 2 x 3x 2 x 8x 15. (Total 5 marks)
26 26. The graph of y = f (x) is shown on the grids. (a) On this grid, sketch the graph of y = f (x 1)
27 (b) On this grid, sketch the graph of y = 2 f (x) (Total 4 marks) 27. Solve 3x 4 < 16. (Total 2 marks)
28 28. Solve the simultaneous equations 2x + y = 4 5x y = 17 x =. y =. (Total 2 marks) 29. A straight line, L, has equation 3y = 5x 6 Find (i) the gradient of L, (ii) the y-co-ordinate of the point where L cuts the y-axis. (0, ) (Total 2 marks)
29 30. Simplify x 2x x. (Total 3 marks) 31. Simplify fully (i) 4 m m 5.. (ii) 6 p p 2 (iii) 2 q q q 3 6 (iv) 4 k 8 k (Total 4 marks)
30 32. Peter cuts a square out of a rectangular piece of metal. The length of the rectangle is 2x + 3. The width of the rectangle is x + 4. The length of the side of the square is x + 2. All measurements are in centimetres. Diagram NOT accurately drawn The shaded shape in the diagram shows the metal remaining. The area of the shaded shape is 20 cm 2. 2 Show that x 7x 12 0 (Total 4 marks)
31 33. The expression 2 8 x x can be written in the form 2 p x q, for all values of x. (a) Find the value of p and the value of q. p =.. q =.. The expression 2 8x x has a maximum value. (b) State the value of x for which this maximum value occurs. (1) (Total 4 marks)
32 34. A ( x + 4) cm D (2 x 1) cm B C ABCD is a parallelogram. AD = (x + 4) cm, CD = (2x 1) cm. The perimeter of the parallelogram is 24 cm. Diagram NOT accurately drawn (i) Use this information to write down an equation, in terms of x.. (ii) Solve your equation. x = (Total 3 marks) 35. (a) Simplify 12y 3 3y 5
33 (b) Simplify 2w 3 x 2 3w 4 x (Total 4 marks) 36. Make m the subject of the formula 2(2p + m) = 3 5m m = (Total 3 marks)
34 37. Solve the simultaneous equations 3x 4y = 11 5x + 6y = 12 x = y = (Total 4 marks)
35 38. C ( x + 2) metres A (2 x + 1) metres 30º B Diagram NOT accurately drawn AB = (2x + 1) metres. BC = (x + 2) metres. Angle ABC = 30. The area of the triangle ABC is 3 m 2. Calculate the value of x. Give your answer correct to 3 significant figures. (Total 5 marks)
36 39. Make t the subject of the formula v = 5t + u. t = (Total 2 marks) 40. Find the gradient of the straight line with equation 5y = 3 2x. (Total 2 marks)
37 41. y y y y y y y y y x x x x x x x x x O O O O O O O O O A D G B E H C F I Write down the letter of the graph which could have the equation (i) y = 3x 2 (ii) y = 2x 2 + 5x 3 (iii) y = x 3 (Total 3 marks)
38 42. 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.... (b) Rearrange y = 2 1 x + 1 to make x the subject.... (Total 4 marks)
39 43. Solve the simultaneous equations x + y = 4 x 2 + y 2 = 40 x =..., y =... or x =..., y =... (Total 7 marks) 44. In the table below, put a tick ( ) next to the identity. 4a + 3 = 8a 4 C = 2 r 2(c + 1) = 2c + 2 4m + 3 (Total 1 mark)
40 45. Simplify fully 5x 3 y 4 7xy 2 (Total 2 marks) 46. (a) Complete the table of values for the graph of y = 4x(11 2x) x y (b) On the grid, draw the graph of y = 4x(11 2x) 80 y x 20 40
41 (c) Use your graph to find the maximum value of y... (1) (Total 5 marks) 47. (a) Complete the table of values for y = x 3 3x + 1 x y (1) (b) On the grid, draw the graph of y = x 3 3x + 1 for 2 x 2 y O 1 2 x 1 2 (Total 3 marks)
42 48. Make a the subject of the formula 2(3a c) = 5c (Total 3 marks) 49. Solve x 2 + 3x 5 = 0 Give your solutions correct to 4 significant figures.. (Total 3 marks) 50. (a) Expand (x + 5)(x + 8)
43 (b) Factorise x 2 5x 14 (Total 4 marks) 51. Make y the subject of the formula x = 3y + 2 (Total 2 marks)
44 52. Simplify x 2 x (5 x) 2 25 (Total 2 marks) 53. 3( x 3) 2 x x 1 Diagram NOT accurately drawn The lengths, in cm, of the sides of the triangle are 3(x 3), 4x 1 and 2x + 5 (a) Write down, in terms of x, an expression for the perimeter of the triangle.... cm (1)
45 The perimeter of the triangle is 49 cm. (b) Work out the value of x. x =... (Total 3 marks) 54. (a) Complete the table of values for y = x 2 6x + 10 x y 10 10
46 (b) On the grid draw the graph of y = x 2 6x + 10 y O x (Total 4 marks) 55. The first four terms of an arithmetic sequence are Find, in terms of n, an expression for the nth term of this sequence.... (Total 2 marks)
47 56. Expand and simplify (y + 5)(y + 3)... (Total 2 marks) 57. (a) Factorise completely 2(x 5) 2 + 3(x 5)... 3( y 4) (b) Simplify 2 ( y 4)... (1) (Total 3 marks)
48 58. Here are five graphs labelled A, B, C, D and E. A y B y x x C y D y x x E y x Each of the equations in the table represents one of the graphs A to E. Write the letter of each graph in the correct place in the table. Equation x + y = 5 y = x 5 y = 5 x y = 5 x = 5 Graph (Total 3 marks)
49 59. Expand and simplify (x 9)(x + 4)... (Total 2 marks) 60. P = πr + 2r + 2a P = 84 r = 6.7 Work out the value of a. Give your answer correct to 3 significant figures. a = (Total 3 marks)
50 61. 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) 62. Simplify fully (3xy 2 ) 4. (Total 2 marks)
51 63. By eliminating x, find the solutions to the simultaneous equations x 2y = 1 x 2 + y 2 = 13 x =.., y =. or x =.., y =. (Total 7 marks)
Questions Q1. x =... (Total for Question is 4 marks) Q2. Write down the value of (i) 7. (ii) 5 1. (iii) 9 ½. (Total for Question is 3 marks) Q3.
Questions Q1. The equation x 3 6x = 72 has a solution between 4 and 5 Use a trial and improvement method to find this solution. Give your answer correct to one decimal place. You must show all your working.
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