1. For Question 1, we are asked to find the exact value of $$Q=\frac{(\sin 2x + b)(2 \sin x - 1)}{a^2 - 4 \tan x}$$ given $x=45^\circ$, $a=18$, and $b=\sqrt{2}$. 2. Calculate the trigonometric values: $$\sin 2x = \sin 90^\circ = 1$$ $$\sin x = \sin 45^\circ = \frac{\sqrt{2}}{2}$$ $$\tan x = \tan 45^\circ = 1$$ 3. Substitute these into numerator and denominator: Numerator: $$(1 + \sqrt{2})(2 \times \frac{\sqrt{2}}{2} - 1) = (1 + \sqrt{2})(\sqrt{2} - 1)$$ 4. Expand numerator: $$(1)(\sqrt{2} - 1) + \sqrt{2}(\sqrt{2} - 1) = (\sqrt{2} - 1) + (2 - \sqrt{2}) = (\sqrt{2} - 1 + 2 - \sqrt{2}) = 1$$ 5. Compute denominator: $$a^2 - 4 \tan x = 18^2 - 4 \times 1 = 324 - 4 = 320$$ 6. So, $$Q = \frac{1}{320}$$ (exact value). 7. Convert exact value to decimal: $$Q \approx 0.003125$$ 8. Part 2(a): to three decimal places, $$Q=0.003$$ 9. Part 2(b): to three significant figures, $$Q=0.00313$$ 10. Part 3: percentage error from rounding to three decimal places: $$\text{Percentage error} = \frac{|0.003125 - 0.003|}{0.003125} \times 100\% = \frac{0.000125}{0.003125} \times 100\% = 4\%$$ --- 11. For Question 2, calculate distance between $A(40,-100)$ and $B(1,-2)$: $$d = \sqrt{(1-40)^2 + (-2+100)^2} = \sqrt{(-39)^2 + 98^2} = \sqrt{1521 + 9604} = \sqrt{11125}$$ 12. Calculate decimal: $$\sqrt{11125} \approx 105.461$$ 13. Part 1: correct to three significant figures: $$105$$ 14. Part 2: correct to one decimal place: $$105.5$$ 15. Part 3: write in form $a \times 10^k$: $$105.5 = 1.055 \times 10^{2}$$ --- 16. Question 3: Given $$F = \frac{(4 \sin 2z - 1)(2 \tan 3z + 1)}{x^2 - y^2}$$ with $x=12$, $y=8$, $z=15^\circ$. 17. Calculate trigonometric values: $$\sin 30^\circ = \frac{1}{2}$$ $$\tan 45^\circ = 1$$ 18. Numerator: $$(4 \times \frac{1}{2} - 1)(2 \times 1 + 1) = (2 - 1)(2 + 1) = 1 \times 3 = 3$$ 19. Denominator: $$12^2 - 8^2 = 144 - 64 = 80$$ 20. Thus, $$F = \frac{3}{80}$$ exact value. 21. Decimal value: $$F = 0.0375$$ 22. Part 2(a): two significant figures, $$0.038$$ 23. Part 2(b): two decimal places, $$0.04$$ 24. Estimate by Sasha: 0.03 25. Percentage error: $$\frac{|0.0375 - 0.03|}{0.0375} \times 100\% = \frac{0.0075}{0.0375} \times 100\% = 20\%$$ --- 26. Question 4: Compute $$A=\sqrt{\frac{\sin \alpha - \sin \beta}{x^2 + 2y}}$$ with $\alpha=54^\circ$, $\beta=18^\circ$, $x=24$, $y=18.25$. 27. Calculate numerator: $$\sin 54^\circ \approx 0.8090$$ $$\sin 18^\circ \approx 0.3090$$ $$0.8090 - 0.3090 = 0.5$$ 28. Denominator: $$24^2 + 2 \times 18.25 = 576 + 36.5 = 612.5$$ 29. Compute expression under square root: $$\frac{0.5}{612.5} \approx 0.0008163$$ 30. Square root: $$A = \sqrt{0.0008163} \approx 0.02858$$ 31. Full calculator display: 0.028582... 32. Part 2(a): three significant figures: $$0.0286$$ 33. Part 2(b): three decimal places: $$0.029$$ 34. Part 3: scientific notation for part (2a): $$2.86 \times 10^{-2}$$ --- 35. Question 5: Volume of cuboid: $$V = l \times w \times h = 9.6 \times 7.4 \times 5.2$$ 36. Multiply: $$9.6 \times 7.4 = 71.04$$ $$71.04 \times 5.2 = 369.9968$$ 37. Exact volume: $$369.9968 \text{ cm}^3$$ 38. Part 2(a): Two decimals: $$370.00$$ 39. Part 2(b): Three significant figures: $$370$$ 40. Part 3: scientific notation: $$3.70 \times 10^{2}$$ --- 41. Question 6: Four weights: 4.92, 4.95, 5.02, 4.95 42. Mean: $$(4.92 + 4.95 + 5.02 + 4.95) / 4 = 19.84 / 4 = 4.96$$ 43. Percentage error from 5 kg: $$(|4.96 - 5| / 5) \times 100\% = (0.04 / 5) \times 100 = 0.8\%$$ 44. Calculate: $$\sqrt{2.15^8} - 5.12^{-0.8}$$ 45. Simplify: $$2.15^8 = (2.15^4)^2$$ $$\sqrt{2.15^8} = 2.15^4$$ 46. Compute $2.15^4$: $$2.15^2 = 4.6225$$ $$2.15^4 = 4.6225^2 \approx 21.37$$ 47. Compute $5.12^{-0.8}$: $$5.12^{-0.8} = \frac{1}{5.12^{0.8}}$$ 48. Approximate $5.12^{0.8}$: $$5.12^{0.8} \approx e^{0.8 \ln 5.12} = e^{0.8 \times 1.634} = e^{1.307} \approx 3.697$$ 49. So, $$5.12^{-0.8} \approx \frac{1}{3.697} = 0.2706$$ 50. Final value: $$21.37 - 0.2706 = 21.1$$ 51. Part 1: nearest integer: $$21$$ 52. Part 2: scientific notation: $$2.11 \times 10^{1}$$ --- 53. Question 7: Exact value $z=0.1475$ 54. Part 1: Express in scientific notation: $$z = 1.475 \times 10^{-1}$$ 55. Part 2: Two significant figures: $$0.15$$ 56. Part 3: Percentage error: $$(|0.1475 - 0.15|/0.1475) \times 100\% = (0.0025 / 0.1475) \times 100\% = 1.69\%$$ --- 57. Question 8: Given $$z=\frac{10 \sin \alpha}{3x + y}, \alpha=30^\circ, x=6, y=46$$ 58. Calculate $\sin 30^\circ = 0.5$ 59. Substitute $$z=\frac{10 \times 0.5}{3 \times 6 + 46} = \frac{5}{18 + 46} = \frac{5}{64}$$ 60. Exact value: $$z=\frac{5}{64}$$ 61. Decimal value: $$0.078125$$ 62. Part 2(a): two decimal places: $$0.08$$ 63. Part 2(b): three significant figures: $$0.0781$$ 64. Part 2(c): scientific notation: $$7.81 \times 10^{-2}$$ --- 65. Question 9: Rectangle dimensions: $$7.6 \times 10^2 \text{ cm} = 760 \text{ cm}$$ $$1.5 \times 10^3 \text{ cm} = 1500 \text{ cm}$$ 66. Area: $$760 \times 1500 = 1,140,000 \text{ cm}^2$$ 67. Scientific notation: $$1.14 \times 10^{6}$$ 68. Natalie's estimate: 1,200,000 69. Percentage error: $$(|1,200,000 - 1,140,000| / 1,140,000) \times 100\% = \frac{60,000}{1,140,000} \times 100\% = 5.26\%$$ --- 70. Question 10: Given $$V = \sqrt{\frac{4 S^3}{243 \pi}}$$ with $S = 529$ cm$^2$. 71. Calculate $S^3$: $$529^3 = 529 \times 529 \times 529$$ $$529^2 = 279,841$$ $$279,841 \times 529 \approx 147,949,289$$ 72. Compute numerator: $$4 \times 147,949,289 = 591,797,156$$ 73. Compute denominator: $$243 \times \pi \approx 243 \times 3.1416 = 762.89$$ 74. Fraction: $$\frac{591,797,156}{762.89} \approx 775,799.7$$ 75. Square root: $$V = \sqrt{775,799.7} \approx 880.78$$ 76. Part 1: To one decimal place: $$880.8$$ 77. Part 2: Nearest integer: $$881$$ 78. Part 3: Scientific notation: $$8.81 \times 10^{2}$$