1. **Solve the equation:** \(\frac{m}{2} + \frac{m}{3} + 3 = 2 + \frac{m}{6}\) Step 1: Get common denominator for the fractions on the left and right. The denominators are 2,3,6; common denominator is 6. Step 2: Rewrite fractions with denominator 6: $$\frac{3m}{6} + \frac{2m}{6} + 3 = 2 + \frac{m}{6}$$ Step 3: Combine like terms on left: $$\frac{3m + 2m}{6} + 3 = 2 + \frac{m}{6} \Rightarrow \frac{5m}{6} + 3 = 2 + \frac{m}{6}$$ Step 4: Subtract \(\frac{m}{6}\) and 2 from both sides: $$\frac{5m}{6} - \frac{m}{6} + 3 - 2 = 0 \Rightarrow \frac{4m}{6} + 1 = 0$$ Step 5: Simplify \(\frac{4m}{6} = \frac{2m}{3}\) and isolate m: $$\frac{2m}{3} = -1 \Rightarrow m = -\frac{3}{2} = -1.5$$ 2. **Find the value of:** \(\sqrt{\frac{(15.62)^2}{29.21 \times \sqrt{10.52}}}\) Step 1: Square 15.62: $$15.62^2 = 243.9844$$ Step 2: Calculate \(\sqrt{10.52} \approx 3.2439$$ Step 3: Multiply denominator: $$29.21 \times 3.2439 \approx 94.77$$ Step 4: Divide: $$\frac{243.9844}{94.77} \approx 2.575$$ Step 5: Square root: $$\sqrt{2.575} \approx 1.605$$ 3. **Solve the system:** \(y = x + 2\) \(x^2 + y^2 = 28\) Step 1: Substitute \(y = x + 2\) into second equation: $$x^2 + (x + 2)^2 = 28$$ Step 2: Expand binomial: $$x^2 + (x^2 + 4x + 4) = 28 \Rightarrow 2x^2 + 4x + 4 = 28$$ Step 3: Simplify: $$2x^2 + 4x + 4 - 28 = 0 \Rightarrow 2x^2 + 4x - 24 = 0$$ Step 4: Divide entire equation by 2: $$x^2 + 2x - 12 = 0$$ Step 5: Factor quadratic: $$(x + 4)(x - 3) = 0$$ Step 6: Solve for x: \(x = -4 \text{ or } x = 3\) Step 7: Find corresponding y values: For \(x = -4\), \(y = -4 + 2 = -2\) For \(x = 3\), \(y = 3 + 2 = 5\) 4. **Find y in:** \(\sqrt{\frac{y+2}{3-y}} = -15 + 10\) Step 1: Simplify right side: $$-15 + 10 = -5$$ Step 2: Note that square roots are always \(\geq 0\) so \(\sqrt{\frac{y+2}{3-y}} = -5\) has no real solution. Step 3: Therefore, no real value of y satisfies the equation. 5. **Calculate force F:** \(F = G * \frac{m_1 m_2}{d^2}\) Given: $$G = 6.67 \times 10^{-11}, m_1 = 7.36, m_2 = 15.5, d = 22.6$$ Step 1: Calculate \(d^2 = 22.6^2 = 510.76$$ Step 2: Calculate numerator: $$m_1 m_2 = 7.36 \times 15.5 = 114.08$$ Step 3: Calculate fraction: $$\frac{114.08}{510.76} \approx 0.2233$$ Step 4: Calculate force: $$F = 6.67 \times 10^{-11} \times 0.2233 = 1.49 \times 10^{-11}$$ (rounded to 3 decimals in standard form) 6. **Calculate the area A of a triangle:** \(A = \sqrt{s(s - a)(s - b)(s - c)}\) 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 root: $$6.40 \times 2.80 \times 2.40 \times 1.20 = 51.658$$ Step 4: Calculate square root: $$\sqrt{51.658} \approx 7.19$$ Area \(A \approx 7.19\) cm\(^2\) 7. **Find constants \(a\) and \(b\) and value of F when \(L = 6.5\):** \(F = aL + b\) Given points: (i) \(F=5.6, L=8.0\) (ii) \(F=4.4, L=2.0\) Step 1: Write system: $$5.6 = 8a + b$$ $$4.4 = 2a + b$$ Step 2: Subtract the second equation from the first: $$(5.6 - 4.4) = (8a - 2a) + (b - b) \Rightarrow 1.2 = 6a \Rightarrow a = \frac{1.2}{6} = 0.2$$ Step 3: Substitute \(a=0.2\) into second equation: $$4.4 = 2(0.2) + b \Rightarrow 4.4 = 0.4 + b \Rightarrow b = 4.0$$ Step 4: Find \(F\) when \(L=6.5\): $$F = 0.2 \times 6.5 + 4.0 = 1.3 + 4.0 = 5.3$$