Surds: 30. Simplify each of the following:(a) 27,(b) 243,(c) 18 + 98,(d) 28 175 + 112,(e) 5 125 8,(f) 273 27,(g) 112 63 + 224 28 ,(h) 2 + 1 + 1 37 ,(i) 1 3+7

+

1 37

31. Express the following in the form of a + b:(a) (3 2)2 (b) (3 23)2 (c) (3 + 1)2(d) 32+23 3223(e) 2332 23+32 (f) 7+2 72

32. Rationalize the denominators of the following, giving your answer in the simplest form possible:(a) 1+2 5+3 + 12 53(b) 73+25 3523(c) 14 7+2

33. Find the value of each of the following:(a) 1 3+5 + 1 35(b) 50+18 328(c) 1 (15) 2 + 1 (1+5) 2 Indices:

34. Evaluate each of the following without using a calculator: (a) 71(b) 170 (c) 49 3 2(d) 8 2 3 (e) 243 3 5 (f) 81 1 4 (g) ( 1 27 ) 4 3(h) (1 4 ) 2 (i) ( 1 573 ) 1 2 (j) 512 4 3 (k) 8 2 3 41(k) ( 1 625 ) 1 4 43

35. Simplify each of the following giving your answer in index form: (a) 1 2 1 3 1 6(b) 3 4 2(c) 124 46 (d) 16 5 2 4 3 2(e) ( 1 3 2 5) 15(f) ( 1 4 3 8) 24

(g) 3 (h) 7 8 (i) 3 4 3

(j) +2 6 +3 9(k) 83 643 3(l) ( 2 924

) 3

36. Solve the following equations:(a) 3x = 81(b) 32x = 8(c) 5x = 125(d) 2x = 1 8

(e) 16x = 1 2

(f) 7x = 1 49(g) 5x = 1(h) 34x = 27x+3 (i) 4x 32x = 6 (j) 2x 4x-1 = 82x (k) 3x 92x = 272x (l) 53x 4x+1 = 1 125

37. By using appropriate substitution, or otherwise, solve the following equations: (a) 22x + 2x+2 = 12(b) 32t + 3t+2 = 31 3(c) 22x-1 - 9 2x-2 +1 = 0 (d) 32x+1 + 9 = 3x+3 + 3x(e) 9x + 3 = 4(3x)(f) 5 2x = 2 4x + 2 (g) 9x+1 - 28(3x) + 3 = 0(h) 32x+2 + 81 = 246(3x)

38. Solve the following pairs of simultaneous equations:(a) 7x-y = 49, 7x+y = 343(b) 75 = ab2, 375 = ab3 (c) 3x+y = 243, 22x-5y = 8(d) 52x+y = 625, 24x-2y = 1 16(e) 3x 81y = 27, 2x 8y = 1 16Logarithms:

39. Write the following in logarithmic from:(a) 52 = 25 (b) 120 = 1(c) 73 = 343(d) 3-1 = 1 3(e) 1 8 = 2-3(f) 216 = 63

40. Write the following in index form: (a) log2 8 = 3(b) log5 625 = 4(c) log2 1 2 = 1 (d) log9 1 = 0(e) log6 1 36 = 2(f) log10 1000 = 3

41. Solve the following equations: (a) log3 1 = (b) log2 25 = (c) log5 1 5 = (d) log4 0.25 = (e) log2 128 = (f) log7 7 = (g) log5 5 = (h) log4 = 1 2(i) log5 = 3

42. Simplify the following logarithms: (a) log10 5(b) log8 64(c) log5 5 3(d) log5 1255 (e) log16 1 4(f) log5 5(g) log3 12 + log3 4 (h) log7 5 + log7 15 (i) 3log6 5 log6 25 (j) log2 21 + log2 3 log2 5(k) 2log3 5 log3 10 + 3log3 4(l) 1 2 log10 81 17 log10 17 4 + 2log10 5 3 + 3 2 log10 17

43. If log3 2 = 0.6309 and log3 5 = 1.456, evaluate the following without the use of calculators or logarithm tables: (a) log3 10(b) log3 15(c) log3 25(d) log3 5 (e) log3 2.5(f) log3 31 3(g) log3 1 8(h) log3 100 (i) log3 12(j) log3 0.12(k) log3 0.08(l) log3 52

44. Evaluate the following without using calculations:(a) 2log10 4 log10 2 + 3log10 5(b) log10 27 + log10 14 9log10 9 log10 7 log10 6 (c) 2 3 log2 8 3 2 log2 16 + 1 2 log2 32 (d) log4 10 9 log4 24 25 3log4 5 6

(e) log10 175 log10 91 + log10 52 (f) log6 4 15 + log6 32 7 + log6 9 4 log6 8 105

45. Solve the following equations, giving your answer correct to 3 significant figures, where necessary: (a) 75-3 = 3x+2(b) 4 5x = 0.74(c) 52x + 100 = 5x+2 (d) 72x - 7 7x + 12 = 0

46. Simplify log27+log8log125 log6log5

47. Given that = 3, (a) Find the value of (1 3 ) in terms of a. (b) Find the value of a and b if 32 5 32 = 14 Rectangular Coordinates:

48. Find the length of the line joining the following pairs of points: (a) (1, 2), (4, 6)(b) (3, 1), (2, 0)(c) (4, 2), (2, 5) (d) (-1, 4), (2, 6)(e) (0, 0), (-1, -2)(f) (-1, -4), (-3, -2)

49. Find the coordinates of the midpoints of the lines joining the pairs of points given in 48.

50. Find the length of the line from the origin to the point (7, 4).

51. Show, using Pythagoras' Theorem, that the lines joining A(1, 6), B(-1, 4) and C(2, 1) from the right angled triangle.

52. Show that ABC is isosceles where A, B, and C are the points (7, 3), (- 4, 1), (-3, -2). 53. Find the midpoint of the base of ABC in No. 52. Hence find the area of ABC.

54. Prove that the lines OA and OB are perpendicular where A, B are the points (4, 3) and(3, -4) respectively.

55. In the triangle ABC, A, B, and, C are points (0, 2), (1, 5) and (-1, 4). Find the coordinates of the point D such that AD is a median and find the length of this median.

56. A, B and M are three points such that M is the midpoint of AB. The coordinates of A and M are (5, 7) and (0, 2) respectively. Find the coordinates of B. Equation of the straight line:

57. Write down the equation of the line through the origin and with gradient (a) 2(b)-1(c) 1/3(d) -1/4(e) 0(f) Draw a sketch showing these lines on the same pair of axes.

58. Write down the equation of the line passing through the given point and with the given gradient. (a) (0,1), (b) (0,0), (c) (-1,-4),

59. Write down the equation of the line passing through the given points. (a) (0,0), (2,1)(b) (1,4), (3,0)(c) (2,0), (0,4)(e) (-1,-3), (-4,-3)

60. Write down the inequality which defines the region: (a) Above the line through the origin with gradient 1. (b) Below the line through (1,2) and (0,4)

(c) Above the line x + y - 2 = 0(d) Below the line 2x - y + 4 = 0

61. Write down the equation of the line passing through the origin and perpendicular to: (a) (2,1), 3x + y - 2 = 0(b) (-1,-2), 2x - 3y + 6 = 0

62. Write down the equation of the line passing through the given point and perpendicular to the given line. 63. Write down the equation of the line passing through (3,-2) and parallel to: (a) 5x - y + 3 = 0(b) x + 7y - 5 = 0