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Algebra Higher Revision – Paper 2 1. Solve the equation x2 – 10x – 5 = 0 Give your answers to 2 decimal places. .........................…………………………………………………………………………….. .........................…………………………………………………………………………….. .........................…………………………………………………………………………….. .........................…………………………………………………………………………….. .........................…………………………………………………………………………….. .........................…………………………………………………………………………….. .........................…………………………………………………………………………….. Answer ............................................ (Total 3 marks) 2. Solve the equation x2 + 4x – 10 = 0 Give your answers to 2 decimal places. You must show your working. ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ Answer ................................................................. (Total 3 marks) 3. (a) Find the values of a and b such that x2 + 6x – 3 = (x + a)2 + b ...................................................................................................................................... ...................................................................................................................................... ...................................................................................................................................... St Anthony's Catholic Girls' School 1 ...................................................................................................................................... ...................................................................................................................................... ...................................................................................................................................... Answer a = ........................., b = ......................... (2) (b) Hence, or otherwise, solve the equation x2 + 6x – 3 = 0 giving your answers in surd form. ...................................................................................................................................... ...................................................................................................................................... ...................................................................................................................................... ...................................................................................................................................... ...................................................................................................................................... ...................................................................................................................................... ...................................................................................................................................... ...................................................................................................................................... Answer .................................................................. (3) (Total 5 marks) 4. Find the values of a and b such that x2 – 10x + 18 = (x – a)2 + b ............................................................................................................................................... ............................................................................................................................................... ............................................................................................................................................... ............................................................................................................................................... ............................................................................................................................................... ............................................................................................................................................... Answer a = ..........................., b = ........................... (Total 2 marks) St Anthony's Catholic Girls' School 2 5. (a) Find the values of a and b such that x2 + 10x + 40 = (x + a)2 + b ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... Answer a = ......................... , b = ............................... (2) (b) Hence, or otherwise, write down the minimum value of x2 + 10x + 40 ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... Answer ......................................................................... (1) (Total 3 marks) 6. (a) Simplify fully the expression 8 x 2 24 x 2 x 2 5x 3 ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... Answer .................................................... (3) (b) You are given that (x + a)2 + b = x2 – 6x + 13. Find the values of a and b. ..................................................................................................................................... St Anthony's Catholic Girls' School 3 ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... Answer .................................................... (3) (Total 6 marks) 7. Simplify 5 x 2 14 x – 3 x2 – 9 ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ Answer ........................................................ (Total 4 marks) 8. Simplify fully x 2 – 16 3 x 2 10 x – 8 .........................…………………………………………………………………………….. .........................…………………………………………………………………………….. .........................…………………………………………………………………………….. .........................…………………………………………………………………………….. .........................…………………………………………………………………………….. .........................…………………………………………………………………………….. Answer ............................................ (Total 4 marks) St Anthony's Catholic Girls' School 4 9. Solve the equation x 2 – 1 x 1 x –1 ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ Answer ............................................................. (Total 5 marks) 10. Solve the equations (a) 12 – y 5 3 ...............…………………………………………………………………………….. ...............…………………………………………………………………………….. ...............…………………………………………………………………………….. ...............…………………………………………………………………………….. Answer y = ............................................ (3) (b) 2x 1 4x 1 1 4 6 ...............…………………………………………………………………………….. ...............…………………………………………………………………………….. ...............…………………………………………………………………………….. ...............…………………………………………………………………………….. ...............…………………………………………………………………………….. ...............…………………………………………………………………………….. ...............…………………………………………………………………………….. St Anthony's Catholic Girls' School 5 Answer x = ............................................ (4) (Total 7 marks) 11. Solve the equation 1 x 1 5x 3 x–2 .........................…………………………………………………………………………….. .........................…………………………………………………………………………….. .........................…………………………………………………………………………….. .........................…………………………………………………………………………….. .........................…………………………………………………………………………….. .........................…………………………………………………………………………….. .........................…………………………………………………………………………….. .........................…………………………………………………………………………….. .........................…………………………………………………………………………….. .........................…………………………………………………………………………….. .........................…………………………………………………………………………….. (Total 5 marks) 12. Solve the simultaneous equations 2x + 3y = 9 3x + 2y = 1 You must show your working. Do not use trial and improvement. .................................................................................................................................................. .................................................................................................................................................. .................................................................................................................................................. .................................................................................................................................................. .................................................................................................................................................. .................................................................................................................................................. .................................................................................................................................................. .................................................................................................................................................. .................................................................................................................................................. St Anthony's Catholic Girls' School 6 Answer x = ..................., y = ............................. (Total 4 marks) 13. Prove that x 2 x – 1 2(2 x 1) – x x 1 x( x 1) ............................................................................................................................................... ............................................................................................................................................... ............................................................................................................................................... ............................................................................................................................................... ............................................................................................................................................... ............................................................................................................................................... ............................................................................................................................................... ............................................................................................................................................... ............................................................................................................................................... ............................................................................................................................................... ............................................................................................................................................... ............................................................................................................................................... (Total 4 marks) 14. Solve the simultaneous equations 5x + 3y = 13 3x + 5y = 3 You must show your working. Do not use trial and improvement. .........................…………………………………………………………………………….. .........................…………………………………………………………………………….. .........................…………………………………………………………………………….. .........................…………………………………………………………………………….. .........................…………………………………………………………………………….. .........................…………………………………………………………………………….. .........................…………………………………………………………………………….. .........................…………………………………………………………………………….. St Anthony's Catholic Girls' School 7 Answer x = ..................., y = ........................ (Total 4 marks) 15. SUPERGROW GARDEN CENTRE ROSES £x each (a) SHRUBS £y each Megan buys 4 roses and 3 shrubs. She pays £33. Use this information to write down an equation in x and y. ..................................................................................................................................... (1) (b) Josh buys 6 roses and 6 shrubs. He pays £57. Use this information to write down another equation in x and y. ..................................................................................................................................... (1) (c) Solve your equations simultaneously to find the values of x and y. You must show your working. Do not use trial and improvement. ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... Answer x = .......................... , y = ............................... (3) (Total 5 marks) 16. (a) Solve the simultaneous equations y = 2x – 5 x2 + y2 = 25 St Anthony's Catholic Girls' School 8 You must show your working. Do not use trial and improvement. ...................................................................................................................................... ...................................................................................................................................... ...................................................................................................................................... ...................................................................................................................................... ...................................................................................................................................... ...................................................................................................................................... ...................................................................................................................................... ...................................................................................................................................... ...................................................................................................................................... ...................................................................................................................................... ...................................................................................................................................... Answer …………………………………………… (6) (b) The graph of y = 2x – 5 is shown below. y 7 y = 2x – 5 6 5 4 3 2 1 –7 –6 –5 –4 –3 –2 –1 1 2 3 4 5 6 7 x –1 –2 –3 –4 –5 –6 –7 (i) On the same axes, draw the graph of x2 + y2 = 25 (2) St Anthony's Catholic Girls' School 9 (ii) Explain the connection between the two graphs and the answers you obtained in part (a). ........................................................................................................................... ........................................................................................................................... (1) (Total 9 marks) 17. Solve the simultaneous equations x2 + y2 = 16 y = 3x 1 Give your answers to an accuracy of 2 decimal places. ............................................................................................................................................... ............................................................................................................................................... ............................................................................................................................................... ............................................................................................................................................... ............................................................................................................................................... ............................................................................................................................................... ............................................................................................................................................... ............................................................................................................................................... ............................................................................................................................................... ............................................................................................................................................... ............................................................................................................................................... ............................................................................................................................................... Answer x = .................................y = ................................ (Total 7 marks) 18. Solve the simultaneous equations y=x+2 y = 3x2 You must show your working. Do not use trial and improvement. ........................................................................................................................................... St Anthony's Catholic Girls' School 10 ........................................................................................................................................... ........................................................................................................................................... ........................................................................................................................................... ........................................................................................................................................... ........................................................................................................................................... ........................................................................................................................................... ........................................................................................................................................... ........................................................................................................................................... ........................................................................................................................................... ........................................................................................................................................... ........................................................................................................................................... Answer .............................................................................................................................. (Total 5 marks) 19. The diagram shows the circle x2 + y2 = 2 and the line y = 2x – 1 The line and the circle intersect at the points A and B. y y = 2x – 1 B O x x 2 + y2 = 2 A Not drawn accurately (a) Show that the x-coordinates of A and B satisfy the equation 5x2 – 4x – 1 = 0 ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... St Anthony's Catholic Girls' School 11 ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... (3) (b) Hence find the coordinates of A and B. ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... Answer A (........., .........) B (........., .........) (3) (Total 6 marks) 20. (a) Expand and simplify (x + 4)2 ...................................................................................................................................... ...................................................................................................................................... ...................................................................................................................................... Answer ........................................................ (2) (b) The diagram shows the circle x2 + y2 = 36 and the line y = x + 4. The line and the circle intersect at the points A and B. y y=x+4 B Not drawn accurately x 2 + y 2 = 36 O x A Show that the x-coordinates of A and B are given by the solutions to the equation x2 + 4x – 10 = 0 St Anthony's Catholic Girls' School 12 ...................................................................................................................................... ...................................................................................................................................... ...................................................................................................................................... ...................................................................................................................................... ...................................................................................................................................... (2) (c) Solve the equation x2 + 4x – 10 = 0. Give your answers to 2 decimal places. You must show your working. ...................................................................................................................................... ...................................................................................................................................... ...................................................................................................................................... ...................................................................................................................................... ...................................................................................................................................... ...................................................................................................................................... Answer ........................................................ (3) (Total 7 marks) 21. (a) On the grid below, draw the graph of x2 + y2 = 9 y 4 3 2 1 –4 –3 –2 –1 O 1 2 3 4 x –1 –2 –3 –4 (1) St Anthony's Catholic Girls' School 13 (b) Write down the equation of the tangent to the curve at the point (0,3). Answer ......................................................................... (1) (Total 2 marks) 22. y is inversely proportional to the square of x. When y = 3, x = 2 Find the value of y when x = 4 ……..…................................................................................................................................ ……..…................................................................................................................................ ……..…................................................................................................................................ ……..…................................................................................................................................ ……..…................................................................................................................................ Answer y = .......................................................... (Total 3 marks) 23. y is directly proportional to the square of x. When y = 5, x = 4. Find the value of y when x = 8. ……….............…............……..............................…………….....…………....................... ……….............…............……..............................…………….....…………....................... ……….............…............……..............................…………….....…………....................... ……….............…............……..............................…………….....…………....................... ……….............…............……..............................…………….....…………....................... ……….............…............……..............................…………….....…………....................... Answer ………………….…………….. (Total 3 marks) 24. y is inversely proportional to the square root of x. When x = 16, y = 2 What is the value of y when x = 0.25? ............................................................................................................................................... ............................................................................................................................................... ............................................................................................................................................... ............................................................................................................................................... St Anthony's Catholic Girls' School 14 ............................................................................................................................................... ............................................................................................................................................... Answer ............................................................. (Total 3 marks) 25. The illumination, L, provided by a torch is inversely proportional to the square of the distance, d, from the torch. When L = 2, d = 10. (a) Find an equation expressing L in terms of d. ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... Answer L = .................................................... (3) (b) Find the value of L when d = 2. ..................................................................................................................................... ..................................................................................................................................... Answer .................................................... (1) (c) Find the value of d when L = 8. ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... Answer .................................................... (2) (Total 6 marks) 26. W and P are both positive quantities. W is directly proportional to the square root of P. When W = 12, P = 16. (a) Express W in terms of P. ...................................................................................................................................... ...................................................................................................................................... St Anthony's Catholic Girls' School 15 ...................................................................................................................................... ...................................................................................................................................... ...................................................................................................................................... Answer .......................................................... (3) (b) What is the value of W when P = 25? ...................................................................................................................................... ...................................................................................................................................... ...................................................................................................................................... Answer .......................................................... (1) (c) What is the value of P when W = 21? ...................................................................................................................................... ...................................................................................................................................... ...................................................................................................................................... Answer .......................................................... (2) (Total 6 marks) 27. The number of days, D, to complete a project is inversely proportional to the number of people, P, who work on the project. (a) The project takes 18 days to complete if 150 people work on it. (i) Find an equation connecting D and P. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... Answer D = ......................................................... (3) (ii) How many people are needed to complete the project in 10 days? ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... St Anthony's Catholic Girls' School 16 Answer ................................................................. (2) (b) Sketch a graph which shows that D is inversely proportional to P. D 0 P (2) (Total 7 marks) 28. Laura is using trial and improvement to find a solution to the equation x3+ 2x = 60 The table shows her first try. Continue the table to find a solution to the equation. x x3 + 2x 3 33 Comment too small Give your answer correct to 1 decimal place. Answer x = .................................................... (Total 4 marks) 29. Parveen is using trial and improvement to find a solution to the equation x3 + 7x = 30 This table shows her first two trials. x St Anthony's Catholic Girls' School x3 + 7x Comment 17 2 22 Too small 3 48 Too big Continue the table to find a solution to the equation. Give your answer to 1 decimal place. Answer ............................................................. (Total 3 marks) 30. In the diagram OP = 4a, PA = a, OB = 5b, BR = 3b and AQ = 2 AB 5 A a P Q 4a O B 5b R 3b Not drawn accurately (a) Find, in terms of a and b, simplifying your answers, (i) AB ........................................................................................................................... ........................................................................................................................... Answer …………………………………………… (1) St Anthony's Catholic Girls' School 18 (ii) PQ ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... Answer …………………………………………… (2) (b) Show clearly that points P, Q and R lie on a straight line. ...................................................................................................................................... ...................................................................................................................................... ...................................................................................................................................... ...................................................................................................................................... ...................................................................................................................................... (3) (Total 6 marks) 31. B Not drawn accurately b Q M P b A a O OAB is a triangle where M is the mid-point of OB. P and Q are points on AB such that AP = PQ = QB. OA = a and OB = 2b (a) Find, in terms of a and b, expressions for (i) BA ........................................................................................................................... ........................................................................................................................... St Anthony's Catholic Girls' School 19 Answer .......................................................... (1) (ii) MQ ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... Answer .......................................................... (2) (iii) OP ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... Answer .......................................................... (2) (b) What can you deduce about quadrilateral OMQP? Give a reason for your answer. ...................................................................................................................................... ...................................................................................................................................... ...................................................................................................................................... (2) (Total 7 marks) 32. OACB is a parallelogram and M is the mid-point of BC. OA = a and OB = b C A Not drawn accurately a M O b St Anthony's Catholic Girls' School B N 20 (a) Express the following vectors in terms of a and b (i) BA Answer .................................................................. (1) (ii) AM Answer .................................................................. (1) (b) AM is extended to N, where AN 2 AM . Show that BN = b ...................................................................................................................................... ...................................................................................................................................... ...................................................................................................................................... ...................................................................................................................................... (2) (c) What does this tell you about the position of N? ...................................................................................................................................... ...................................................................................................................................... ...................................................................................................................................... (1) (Total 5 marks) 33. In triangle ABC, M is the mid-point of BC. AB = s and AC = t B M s A (a) t C Find AM in terms of s and t. Give your answer in its simplest form. ..................................................................................................................................... St Anthony's Catholic Girls' School 21 ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... Answer ......................................................................... (3) (b) AD = s + t The length of AB is not equal to the length of AC. (i) Write down the name of the shape ABDC. Answer ......................................................................... (1) (ii) Write down one fact about the points A, M and D. Explain your answer. Fact ................................................................................................................... Explanation ....................................................................................................... ........................................................................................................................... (2) (Total 6 marks) 34. Show that the sum of any three consecutive integers is always a multiple of 3. .........................…………………………………………………………………………….. .........................…………………………………………………………………………….. .........................…………………………………………………………………………….. .........................…………………………………………………………………………….. .........................…………………………………………………………………………….. .........................…………………………………………………………………………….. (Total 3 marks) 35. Prove that, for any positive integer a, a3 + a is always even. ................................………………………………………………………………………… ................................………………………………………………………………………… ................................………………………………………………………………………… St Anthony's Catholic Girls' School 22 ................................………………………………………………………………………… (Total 2 marks) 36. Zoe states that the product of two consecutive integers is never divisible by 50. By means of an example, show that Zoe is not correct. ................................………………………………………………………………………… ................................………………………………………………………………………… ................................………………………………………………………………………… Answer ................................................ (Total 2 marks) 37. Explain why the sum of three consecutive integers is divisible by three. ............................................................................................................................................... ............................................................................................................................................... ............................................................................................................................................... ............................................................................................................................................... (Total 2 marks) 38. Prove that the product of two odd numbers is always an odd number. ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ (Total 2 marks) 39. (a) n is a positive integer. (i) Explain why n(n + 1) must be an even number. .................………............................................................................................... .................………............................................................................................... (1) (ii) Explain why 2n + 1 must be an odd number. St Anthony's Catholic Girls' School 23 .................………............................................................................................... .................………............................................................................................... (1) (b) Expand and simplify (2n + 1)2 .................………............................................................................................... .................………............................................................................................... Answer ......................................................................... (2) (c) Prove that the square of any odd number is always 1 more than a multiple of 8. ……..................………............................................................................................... ……..................………............................................................................................... ……..................………............................................................................................... ……..................………............................................................................................... ……..................………............................................................................................... ……..................………............................................................................................... (3) (Total 7 marks) 40. (a) This is a page from Zoe’s exercise book. 23 – 13 = 7 (prime) 33 – 23 = 19 (prime) 43 – 33 = 37 (prime) The difference between consecutive cube numbers is always a prime number. Give a counter example to show that Zoe is wrong. Justify your answer . ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... St Anthony's Catholic Girls' School 24 (2) (b) Prove that (n + 5)2 – (n + 3)2 = 4(n + 4) ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... (3) (Total 5 marks) 41. The base of a triangle is 7 cm longer than its height. The area of the triangle is 32 cm2. (a) Taking the height to be h cm, show that h2+ 7h 64 = 0 ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... (3) (b) Solve this equation to find the height of the triangle. Give your answer to 2 decimal places. ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... Answer....................................................................................................................cm (3) (Total 6 marks) St Anthony's Catholic Girls' School 25 42. (a) On the grid draw the graph of 4. y = 3x + 1 for values of x from 0 to ...................................................................................................................................... ...................................................................................................................................... y 13 12 11 10 9 8 7 6 5 4 3 2 1 0 0 1 2 3 4 x (3) (b) Use your graph to solve 5.5 = 3x + 1 ...................................................................................................................................... ...................................................................................................................................... Answer x = ………..………………………………… (2) (Total 5 marks) St Anthony's Catholic Girls' School 26 43. The graph of y = x2 – 4x + 8 is shown below. y 14 13 12 11 10 9 8 7 6 5 4 3 2 1 –1 O 1 2 3 4 5 x –1 –2 (a) (i) By drawing the graph of an appropriate straight line, solve the equation x2 – 4x + 8 = 3x – 2 ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... Answer .......................................................................... (3) (ii) Hence, or otherwise, solve x2 – 7x + 10 = 0 ........................................................................................................................... St Anthony's Catholic Girls' School 27 ........................................................................................................................... Answer .......................................................................... (1) (b) The graph of y = x2 – 4x + 8 is to be used to solve the equation x2 – 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) 44. The graph y = x2 – 2x – 4 is drawn below for values of x between –3 and +4. y 12 10 8 6 4 2 –3 –2 –1 0 1 2 3 4 x –2 –4 –6 St Anthony's Catholic Girls' School 28 (a) Using the graph, find the solutions of x2 – 2x – 4 = 0, giving your answers to 1 decimal place. .....................………………………………………………………………………… Answer ............................................ (1) (b) By drawing an appropriate linear graph, write down the solutions of x2 – 3x – 2 = 0 .....................………………………………………………………………………… .....................………………………………………………………………………… .....................………………………………………………………………………… Answer ............................................ (3) (Total 4 marks) 45. (a) Complete the table of values for y = x2 – 3x – 4. x –3 y 14 –2 –1 0 1 2 0 –4 –6 –6 3 4 5 6 0 6 14 (2) (b) On the grid below, draw the graph y = x2 – 3x – 4 for values of x between –3 and +6 . y 14 12 10 8 6 4 2 –4 –2 0 2 4 6 x –2 –4 –6 –8 St Anthony's Catholic Girls' School 29 (2) (c) 2 Write down the solutions of x – 3x – 4 = 0 .....................………………………………………………………………………… Answer ................................................ (1) (d) By drawing an appropriate linear graph, write down the solutions of x2 – 4x – 1 = 0 .....................………………………………………………………………………… .....................………………………………………………………………………… .....................………………………………………………………………………… .....................………………………………………………………………………… .....................………………………………………………………………………… Answer ................................................ (3) (Total 8 marks) 46. The line PQ is shown on the grid. y 6 5 P 4 3 2 1 –5 –4 –3 –2 –1 O 1 2 3 4 5 6 x –1 –2 –3 Q –4 –5 St Anthony's Catholic Girls' School 30 (a) Find the gradient of a line which is perpendicular to PQ. ...................................................................................................................................... ...................................................................................................................................... ...................................................................................................................................... Answer .................................................................. (3) (b) Hence find the equation of the perpendicular bisector of the line PQ. ...................................................................................................................................... ...................................................................................................................................... ...................................................................................................................................... Answer .................................................................. (2) (Total 5 marks) 47. The graph of 4y + 3x = 12 has been drawn on the grid below. Draw another line on the grid to solve the simultaneous equations 4y + 3x = 12 y = 2x – 4 ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ Answer St Anthony's Catholic Girls' School x = ......................., y = ………………. 31 y 4 3 2 1 1 O 2 3 4 x –1 –2 –3 –4 (Total 3 marks) 48. (a) Complete the table of values for y = x3 – 4 x y –2 –1 –5 0 1 2 4 (2) St Anthony's Catholic Girls' School 32 (b) On the grid, draw the graph of y = x3 – 4 for values of x from –2 to +2. y 4 2 –2 –1 1 O 2 x –2 –4 –6 –8 –10 –12 (2) (Total 4 marks) 49. The line l on the graph passes through the points A (0, 3) and B (–4, 11). y 15 l B 10 5 A –10 (a) –5 O 5 x Calculate the gradient of the line l. St Anthony's Catholic Girls' School 33 ...................................................................................................................................... ...................................................................................................................................... Answer ...................................................... (2) (b) Write down the equation of the line l. ...................................................................................................................................... Answer ...................................................... (1) (c) Write down the equation of the line which also passes through the point (0, 3) but is perpendicular to line l. ...................................................................................................................................... ...................................................................................................................................... Answer ...................................................... (2) (Total 5 marks) 50. Below are three graphs. Match each graph with one of the following equations. Equation A: Equation B: Equation C: Equation D: y = 3x – p y = x2 + p 3x + 4y = p y = px3 In each case p is a positive number. (i) (ii) (iii) y y x Answer y x Graph x (i) Equation ................................................ Graph (ii) Equation ................................................ Graph (iii) Equation ................................................ (Total 3 marks) St Anthony's Catholic Girls' School 34 51. Match each of the sketch graphs to one of these equations. A y = 2 – 2x C y = 3 – x2 B y = 2x + 2 D y = x3 + 4 y E y= 2 x y 1 2 x x y y 3 4 x x Graph 1 represents equation ……................ Graph 2 represents equation ……................ Graph 3 represents equation ……................ Graph 4 represents equation ……................ (Total 4 marks) 52. (a) This L-shape is made of rectangles. 2.5 cm Not to scale 5.6 cm 1.8 cm 7.2 cm Calculate the area of the L-shape. ..................................................................................................................................... St Anthony's Catholic Girls' School 35 ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... Answer ……………………............................ cm2 (3) (b) This T-shape is also made of rectangles. 5x 3 cm 7 cm 3 cm 2x 2x Not to scale x The perimeter of the T-shape is 29 cm. Work out the value of x. ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... Answer ……………………............................ cm (4) (Total 7 marks) 53. A triangle has angles of 63°, 2x and x. 2x Not drawn accurately x St Anthony's Catholic Girls' School 63° 36 Work out the value of x. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. Answer ................................................... degrees (Total 3 marks) 54. (a) Write down the first three terms of the sequence whose nth term is given by 5n 4n 7 ..................................................................................................................................... ..................................................................................................................................... Answer .................................................... (2) (b) Which term of the sequence has a value of 1? ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... Answer .................................................... (2) (Total 4 marks) 55. A sequence of numbers is shown. 2 (a) 5 8 11 14 Find an expression for the nth term of the sequence. ...................................................................................................................................... ...................................................................................................................................... Answer ......................................................................... (2) St Anthony's Catholic Girls' School 37 (b) Explain why 99 will not be a term in this sequence. ...................................................................................................................................... ...................................................................................................................................... ...................................................................................................................................... (2) (Total 4 marks) 56. Find the nth term of the following sequences. (a) 1, 4, 9, 16, 25, ... ..................................................................................................................................... Answer............................................................................... (1) (b) –2, 1, 6, 13, 22, ... ..................................................................................................................................... ..................................................................................................................................... Answer............................................................................... (1) (Total 2 marks) 57. A sequence of numbers is shown. 5 (a) 9 13 17 21 Find an expression for the nth term of the sequence. ...................................................................................................................................... ...................................................................................................................................... Answer ................................................................ (2) (b) Explain why 83 will not be a term in this sequence. ...................................................................................................................................... ...................................................................................................................................... (2) (Total 4 marks) St Anthony's Catholic Girls' School 38
