1. a) Solve the equation: $$\frac{m}{2} + \frac{m}{3} + 3 = 2 + \frac{m}{6}$$ Step 1: Find a common denominator for the terms involving $m$, which is 6. Rewrite: $$\frac{3m}{6} + \frac{2m}{6} + 3 = 2 + \frac{m}{6}$$ Step 2: Combine terms on the left: $$\frac{3m + 2m}{6} + 3 = 2 + \frac{m}{6} \implies \frac{5m}{6} + 3 = 2 + \frac{m}{6}$$ Step 3: Subtract $\frac{m}{6}$ from both sides: $$\frac{5m}{6} - \frac{m}{6} + 3 = 2 \implies \frac{4m}{6} + 3 = 2$$ Step 4: Simplify $\frac{4m}{6}$ to $\frac{2m}{3}$: $$\frac{2m}{3} + 3 = 2$$ Step 5: Subtract 3 from both sides: $$\frac{2m}{3} = 2 - 3 = -1$$ Step 6: Multiply both sides by $\frac{3}{2}$: $$m = -1 \times \frac{3}{2} = -\frac{3}{2} = -1.5$$ Answer: $m = -1.5$ 1. b) Calculate: $$\sqrt{\frac{15.62^2}{29.21 \times \sqrt{10.52}}}$$ Step 1: Calculate $15.62^2$: $$15.62^2 = 243.9844$$ Step 2: Calculate $\sqrt{10.52}$: $$\sqrt{10.52} \approx 3.243$$ Step 3: Multiply denominator: $$29.21 \times 3.243 \approx 94.739$$ Step 4: Divide numerator by denominator: $$\frac{243.9844}{94.739} \approx 2.576$$ Step 5: Take the square root: $$\sqrt{2.576} \approx 1.605$$ Answer: Approximately $1.605$ 1. c) Solve system: $$y = x + 2$$ $$x^2 + y^2 = 28$$ Step 1: Substitute $y = x + 2$ into the second equation: $$x^2 + (x+2)^2 = 28$$ Step 2: Expand: $$x^2 + x^2 + 4x + 4 = 28$$ Step 3: Combine like terms: $$2x^2 + 4x + 4 = 28$$ Step 4: Subtract 28 from both sides: $$2x^2 + 4x + 4 - 28 = 0 \implies 2x^2 + 4x - 24 = 0$$ Step 5: Divide entire equation by 2: $$x^2 + 2x - 12 = 0$$ Step 6: Factor quadratic: $$(x + 4)(x - 3) = 0$$ Step 7: Solve for $x$: $$x = -4 \text{ or } x = 3$$ Step 8: Find corresponding $y$ values: - If $x = -4$, $y = -4 + 2 = -2$ - If $x = 3$, $y = 3 + 2 = 5$ Answer: Solutions are $(x, y) = (-4, -2)$ and $(3, 5)$ 2. a) Solve: $$\sqrt{\frac{y + 2}{3 - y}} = -15 + 10$$ Step 1: Simplify the right side: $$-15 + 10 = -5$$ Step 2: Note that the square root of a real number can't be negative, so no real solution. Step 3: To check, if we assume the expression equals $-5$, there's no real $y$ satisfying this. Answer: No real solution since the left side is always non-negative but right side is negative. 2. b) Calculate force $F$ using: $$F = G \times \frac{m_1 m_2}{d^2}$$ Given: $$G = 6.67 \times 10^{-11}$$ $$m_1 = 7.36$$ $$m_2 = 15.5$$ $$d = 22.6$$ Step 1: Calculate $m_1 m_2$: $$7.36 \times 15.5 = 114.08$$ Step 2: Square the distance: $$22.6^2 = 510.76$$ Step 3: Compute fraction: $$\frac{114.08}{510.76} \approx 0.2233$$ Step 4: Multiply by $G$: $$6.67 \times 10^{-11} \times 0.2233 = 1.49 \times 10^{-11}$$ Answer: $F \approx 1.49 \times 10^{-11}$ (Newtons), correct to 3 decimal places in standard form. 3. a) Calculate area $A$ of triangle using Heron's formula: $$A = \sqrt{s(s-a)(s-b)(s-c)}$$ where $$s = \frac{a + b + c}{2}$$ Given: $$a = 3.60, b = 4.00, c = 5.20$$ Step 1: Calculate $s$: $$s = \frac{3.60 + 4.00 + 5.20}{2} = \frac{12.80}{2} = 6.40$$ Step 2: Calculate each term: $$s - a = 6.40 - 3.60 = 2.80$$ $$s - b = 6.40 - 4.00 = 2.40$$ $$s - c = 6.40 - 5.20 = 1.20$$ Step 3: Multiply terms inside the root: $$6.40 \times 2.80 \times 2.40 \times 1.20 = 51.6864$$ Step 4: Take square root for area: $$A = \sqrt{51.6864} \approx 7.19$$ Answer: Area $A \approx 7.19$ square cm 3. b) Given: $$F = aL + b$$ Two points: $$F=5.6, L=8.0$$ $$F=4.4, L=2.0$$ Step 1: Write equations: $$5.6 = 8a + b$$ $$4.4 = 2a + b$$ Step 2: Subtract second from first to eliminate $b$: $$(5.6 - 4.4) = (8a - 2a) \, \Rightarrow \, 1.2 = 6a$$ Step 3: Solve for $a$: $$a = \frac{1.2}{6} = 0.2$$ Step 4: Substitute $a=0.2$ into second equation to find $b$: $$4.4 = 2(0.2) + b \implies b = 4.4 - 0.4 = 4.0$$ Step 5: Find $F$ when $L=6.5$: $$F = 0.2 \times 6.5 + 4.0 = 1.3 + 4.0 = 5.3$$ Answer: $a=0.2$, $b=4.0$, and at $L=6.5$, $F=5.3$