1. Problem 13: Points A(-4,7), B(4,1), and C(16,-8) lie on a straight line. Find the ratio in which B divides AC. Formula: If point B divides segment AC in ratio $m:n$, then coordinates of B are given by $$x_B = \frac{mx_C + nx_A}{m+n}, \quad y_B = \frac{my_C + ny_A}{m+n}$$ Step 1: Let B divide AC in ratio $m:n$. Step 2: Using the x-coordinates, $$4 = \frac{m \times 16 + n \times (-4)}{m+n} = \frac{16m - 4n}{m+n}$$ Step 3: Multiply both sides by $m+n$: $$4(m+n) = 16m - 4n$$ $$4m + 4n = 16m - 4n$$ Step 4: Rearranging, $$4m + 4n - 16m + 4n = 0$$ $$-12m + 8n = 0$$ $$8n = 12m$$ $$\frac{m}{n} = \frac{8}{12} = \frac{2}{3}$$ Step 5: Check with y-coordinates: $$1 = \frac{m \times (-8) + n \times 7}{m+n} = \frac{-8m + 7n}{m+n}$$ Multiply both sides by $m+n$: $$1(m+n) = -8m + 7n$$ $$m + n = -8m + 7n$$ $$m + n + 8m - 7n = 0$$ $$9m - 6n = 0$$ $$\frac{m}{n} = \frac{6}{9} = \frac{2}{3}$$ Step 6: Both coordinates confirm the ratio $m:n = 2:3$. So, B divides AC in the ratio 2:3. --- 2. Problem 14: Transformation matrix $$T = \begin{bmatrix} a+1 & 4 \\ 4 & a+1 \end{bmatrix}$$ maps triangle PQR of area 0.5 to triangle P'Q'R' of area 4.5. Formula: Area scale factor = $|\det(T)|$ Step 1: Calculate determinant: $$\det(T) = (a+1)(a+1) - 4 \times 4 = (a+1)^2 - 16$$ Step 2: Area scale factor = $\frac{4.5}{0.5} = 9$ Step 3: So, $$| (a+1)^2 - 16 | = 9$$ Step 4: Solve for $(a+1)^2$: Case 1: $$(a+1)^2 - 16 = 9 \Rightarrow (a+1)^2 = 25$$ $$a+1 = \pm 5$$ $$a = 4 \text{ or } -6$$ Case 2: $$(a+1)^2 - 16 = -9 \Rightarrow (a+1)^2 = 7$$ $$a+1 = \pm \sqrt{7}$$ $$a = -1 + \sqrt{7} \text{ or } -1 - \sqrt{7}$$ Step 5: Possible values of $a$ are $4, -6, -1 + \sqrt{7}, -1 - \sqrt{7}$. --- 3. Problem 15: Particle velocity $V = 4 - t$ m/s, distance $S$ at $t=2$ is 7 m. Find $S$ at $t=4$. Step 1: Velocity is derivative of distance: $$V = \frac{dS}{dt} = 4 - t$$ Step 2: Integrate velocity to find distance: $$S = \int (4 - t) dt = 4t - \frac{t^2}{2} + C$$ Step 3: Use initial condition at $t=2$, $S=7$: $$7 = 4(2) - \frac{2^2}{2} + C = 8 - 2 + C = 6 + C$$ $$C = 1$$ Step 4: Distance function: $$S = 4t - \frac{t^2}{2} + 1$$ Step 5: Calculate $S$ at $t=4$: $$S = 4(4) - \frac{4^2}{2} + 1 = 16 - 8 + 1 = 9$$ Final answer: $S = 9$ meters at $t=4$ seconds.