1. Q1 a) Statement: Solve the equation $m/2 + m/3 + 3 = 2 + m/6$. Multiply both sides by 6 to eliminate denominators. $6(m/2 + m/3 + 3) = 6(2 + m/6)$ $3m + 2m + 18 = 12 + m$ $5m + 18 = 12 + m$ $4m = -6$ $m = -3/2 = -1.5$ Answer: $m = -3/2$. 2. Q1 b) Statement: Evaluate $\sqrt{(15.62)^2/(29.21\times\sqrt{10.52})}$. Compute the numerator: $(15.62)^2 = 243.9844$. Compute the inner square root: $\sqrt{10.52} \approx 3.243303$. Compute the denominator: $29.21\times 3.243303 \approx 94.72691$. Divide: $243.9844/94.72691 \approx 2.57566$. Take the square root: $\sqrt{2.57566} \approx 1.6049$. Answer: $\approx 1.605$. 3. Q1 c) Statement: Solve the system $y = x + 2$ and $x^2 + y^2 = 28$. Substitute $y = x + 2$ into the circle equation. $x^2 + (x+2)^2 = 28$ $x^2 + x^2 + 4x + 4 = 28$ $2x^2 + 4x - 24 = 0$ Divide by 2: $x^2 + 2x -12 = 0$ Factor: $(x+4)(x-3) = 0$ So $x = -4$ or $x = 3$. If $x = -4$ then $y = -2$. If $x = 3$ then $y = 5$. Answer: $(-4,-2)$ and $(3,5)$. 4. Q2 a) Statement: Solve $\sqrt{(y+2)/(3-y)} = -15 + 10$. Simplify the right-hand side: $-15 + 10 = -5$. The principal square root $\sqrt{(y+2)/(3-y)}$ is by definition nonnegative, so it cannot equal $-5$. Therefore there is no real solution. 5. Q2 b) Statement: Compute $F = G m_1 m_2 / d^2$ with $G = 6.67\times10^{-11}$, $m_1 = 7.36$, $m_2 = 15.5$, $d = 22.6$ and give standard form to 3 decimals. Compute the product of masses: $7.36\times15.5 = 114.08$. Multiply by G: $6.67\times10^{-11}\times114.08 = 7.608936\times10^{-9}$. Compute $d^2$: $22.6^2 = 510.76$. Divide: $F = 7.608936\times10^{-9}/510.76 \approx 1.48938\times10^{-11}$. Round to three decimal places in standard form: $F \approx 1.489\times10^{-11}$. Answer: $F \approx 1.489\times10^{-11}$. 6. Q3 a) Statement: Use Heron's formula $A = \sqrt{s(s-a)(s-b)(s-c)}$ with $a=3.60$, $b=4.00$, $c=5.20$. Compute semiperimeter: $s = (3.60 + 4.00 + 5.20)/2 = 6.40$. Compute differences: $s-a = 2.80$, $s-b = 2.40$, $s-c = 1.20$. Compute product: $6.40\times2.80\times2.40\times1.20 = 51.6096$. Area: $A = \sqrt{51.6096} \approx 7.1856$. Answer: $A \approx 7.186\text{ cm}^2$. 7. Q3 b) Statement: Given $F = aL + b$ and points $(L,F)=(8.0,5.6)$ and $(2.0,4.4)$ find $a$, $b$ and $F$ at $L=6.5$. Slope: $a = (5.6 - 4.4)/(8.0 - 2.0) = 1.2/6.0 = 0.2$. Intercept: $b = 5.6 - 0.2\times8.0 = 4.0$. Evaluate at $L = 6.5$: $F = 0.2\times6.5 + 4.0 = 1.3 + 4.0 = 5.3$. Answer: $a = 0.2$, $b = 4.0$, $F(6.5) = 5.3$.