1. **Question 1: Calculate the exact value of** $$Q = \frac{(\sin 2x + b)(2 \sin x - 1)}{a^2 - 4 \tan x}$$ where $x=45^\circ$, $a=18$ and $b=\sqrt{2}$. Step 1: Calculate $\sin 2x$ using the double angle formula. $$\sin 2(45^\circ) = \sin 90^\circ = 1$$ Step 2: Calculate $\sin x$. $$\sin 45^\circ = \frac{\sqrt{2}}{2}$$ Step 3: Evaluate numerator: $$ (\sin 2x + b)(2 \sin x - 1) = \left(1 + \sqrt{2}\right) \left(2 \times \frac{\sqrt{2}}{2} - 1\right) = (1 + \sqrt{2})(\sqrt{2} - 1) $$ Step 4: Simplify $ (1 + \sqrt{2})(\sqrt{2} - 1)$: $$= 1 \times \sqrt{2} - 1 + \sqrt{2} \times \sqrt{2} - \sqrt{2}$$ $$= \sqrt{2} - 1 + 2 - \sqrt{2} = 2 - 1 = 1$$ Step 5: Calculate denominator: $$a^2 - 4 \tan x = 18^2 - 4 \tan 45^\circ = 324 - 4 \times 1 = 324 - 4 = 320$$ Step 6: Calculate $Q$: $$Q= \frac{1}{320}$$ --- 2. **Convert $Q$ to decimal forms:** 2.1: Three decimal places: $$Q = \frac{1}{320} = 0.003125 \approx 0.003$$ 2.2: Three significant figures: $$0.003125 \approx 0.00313$$ --- 3. **Calculate percentage error between exact and 3 decimal places:** Percentage error = $$\frac{|Exact - Approx|}{Exact} \times 100 = \frac{|0.003125 - 0.003|}{0.003125} \times 100 = \frac{0.000125}{0.003125} \times 100 = 4\%$$ --- 4. **Question 2: Distance between points $A(40, -100)$ and $B(1, -2)$** Step 1: Use distance formula: $$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = \sqrt{(1-40)^2 + (-2+100)^2} = \sqrt{(-39)^2 + 98^2}$$ $$= \sqrt{1521 + 9604} = \sqrt{11125}$$ Step 2: Calculate the distance numerically: $$\sqrt{11125} \approx 105.488$$ Step 3: Round answers: - Three significant figures: 105 - One decimal place: 105.5 Step 4: Convert to scientific notation: $$105.5 = 1.055 \times 10^2$$ --- 5. **Question 3: Calculate exact value of** $$F = \frac{(4 \sin 2x - 1)(2 \tan 3z + 1)}{x^2 - y^2}, \quad x=12, y=8, z=15^\circ$$ Step 1: Calculate $\sin 2x$ for $x=12$ degrees: $$\sin 24^\circ \approx 0.406736$$ Step 2: Calculate numerator part 1: $$4 \sin 24^\circ - 1 = 4 \times 0.406736 - 1 = 1.626944 - 1 = 0.626944$$ Step 3: Calculate $\tan 3z$ where $z=15^\circ$: $$\tan 45^\circ = 1$$ Step 4: Numerator part 2: $$2 \times 1 + 1 = 2 + 1 = 3$$ Step 5: Calculate denominator: $$x^2 - y^2 = 12^2 - 8^2 = 144 - 64 = 80$$ Step 6: Calculate $F$: $$F = \frac{0.626944 \times 3}{80} = \frac{1.880832}{80} = 0.02351$$ 6. Round answers: - Two significant figures: 0.024 - Two decimal places: 0.02 7. Percentage error if $F_{estimate} = 0.03$: $$\text{Percent error} = \frac{|0.03 - 0.02351|}{0.02351} \times 100 = \frac{0.00649}{0.02351} \times 100 \approx 27.6\%$$ --- 8. **Question 4: Calculate** $$A = \sqrt{\frac{\sin \alpha - \sin \beta}{x^2 + 2y}}$$ where $\alpha = 54^\circ$, $\beta = 18^\circ$, $x=24$, $y=18.25$. Step 1: Calculate numerator inside root: $$\sin 54^\circ \approx 0.8090, \quad \sin 18^\circ \approx 0.3090$$ $$0.8090 - 0.3090 = 0.5$$ Step 2: Calculate denominator: $$24^2 + 2 \times 18.25 = 576 + 36.5 = 612.5$$ Step 3: Calculate inside root: $$\frac{0.5}{612.5} \approx 0.000816$$ Step 4: Calculate $A$: $$A = \sqrt{0.000816} \approx 0.02857$$ Step 5: Rounding: - Three significant figures: 0.0286 - Three decimal places: 0.029 Step 6: Scientific notation: $$0.0286 = 2.86 \times 10^{-2}$$ --- 9. **Question 5: Volume of cuboid** Length = 9.6 cm, Width = 7.4 cm, Height = 5.2 cm. Step 1: Calculate volume: $$V = l \times w \times h = 9.6 \times 7.4 \times 5.2$$ $$= 369.408$$ Step 2: Round to two decimal places: $$369.41$$ Step 3: Round to three significant figures: $$369$$ Step 4: Express in scientific notation: $$369 = 3.69 \times 10^2$$ --- 10. **Question 6: Cement bags weights: 4.92, 4.95, 5.02, 4.95** Step 1: Calculate mean: $$\frac{4.92 + 4.95 + 5.02 + 4.95}{4} = \frac{19.84}{4} = 4.96$$ Step 2: Calculate percentage error: $$\frac{|5 - 4.96|}{5} \times 100 = \frac{0.04}{5} \times 100 = 0.8\%$$ Step 3: Calculate: $$\sqrt{2.15^8 - 5.12^{-0.8}}$$ Calculate inside: $$2.15^8 \approx 72031.93$$ $$5.12^{-0.8} = \frac{1}{5.12^{0.8}} \approx 1 / 3.536 = 0.283$$ Inside root: $$72031.93 - 0.283 = 72031.65$$ Square root: $$\sqrt{72031.65} \approx 268.41$$ Step 4: Round nearest integer: $$268$$ Step 5: Scientific notation: $$2.68 \times 10^2$$ --- 11. **Question 7: Exact value $z=0.1475$** Step 1: Scientific notation: $$z = 1.475 \times 10^{-1}$$ Step 2: Two significant figures: $$0.15$$ Step 3: Percentage error if $z$ is $0.15$: $$\frac{|0.15 - 0.1475|}{0.1475} \times 100 = \frac{0.0025}{0.1475} \times 100 \approx 1.7\%$$ --- 12. **Question 8: Calculate** $$z = \frac{10 \sin \alpha}{3x + y}, \alpha=30^\circ, x=6, y=46$$ Step 1: Calculate $\sin 30^\circ = 0.5$ Step 2: Calculate denominator: $$3 \times 6 + 46 = 18 + 46 = 64$$ Step 3: Calculate numerator: $$10 \times 0.5 = 5$$ Step 4: Calculate $z$: $$\frac{5}{64} = 0.078125$$ Step 5: Rounding: - Two decimals: 0.08 - Three significant figures: 0.0781 - Scientific notation: $$7.81 \times 10^{-2}$$ --- 13. **Question 9: Area of rectangle with sides $7.6 \times 10^2$ cm and $1.5 \times 10^3$ cm** Step 1: Calculate area: $$(7.6 \times 10^2) \times (1.5 \times 10^3) = 7.6 \times 1.5 \times 10^{2+3} = 11.4 \times 10^5$$ Step 2: Adjust to proper scientific notation: $$1.14 \times 10^6$$ Step 3: Percentage error in estimate $1,200,000$: $$\frac{|1,200,000 - 1,140,000|}{1,140,000} \times 100 = \frac{60,000}{1,140,000} \times 100 \approx 5.26\%$$ --- 14. **Question 10: Volume of hemisphere given** $$V = \sqrt{\frac{4 S^3}{243 \pi}}, S=529$$ Step 1: Calculate numerator inside root: $$4 S^3 = 4 \times 529^3 = 4 \times 148,035,689 = 592,142,756$$ Step 2: Calculate denominator: $$243 \pi \approx 243 \times 3.1416 = 763.41$$ Step 3: Calculate fraction: $$\frac{592,142,756}{763.41} \approx 775,665.22$$ Step 4: Calculate volume: $$\sqrt{775,665.22} \approx 880.78$$ Step 5: Round to one decimal place: $$880.8$$ Step 6: Round to nearest integer: $$881$$ Step 7: Scientific notation: $$8.81 \times 10^2$$