1. Problem: Find the value of $m$ if $3-3i = (m-2)i + 3$. Rewrite the right side: $(m-2)i + 3 = 3 + (m-2)i$. Equate real parts: $3 = 3$ (true). Equate imaginary parts: $-3 = m-2$. Solve for $m$: $m = -3 + 2 = -1$. 2. Problem: Find the product of roots $\alpha \beta$ of $2x^2 - 4x - 8 = 0$. Using formula for quadratic $ax^2 + bx + c=0$, product of roots is $\frac{c}{a}$. Here, $a=2$, $c=-8$. So, $\alpha \beta = \frac{-8}{2} = -4$. 3. Problem: Find $y$ if $\begin{bmatrix}3 & 5 \\ 8 & y+9\end{bmatrix} \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix} = \begin{bmatrix}3 & 5 \\ 8 & -2y\end{bmatrix}$. Multiplying by identity matrix leaves the first matrix unchanged. So, $\begin{bmatrix}3 & 5 \\ 8 & y+9\end{bmatrix} = \begin{bmatrix}3 & 5 \\ 8 & -2y\end{bmatrix}$. Equate bottom right elements: $y + 9 = -2y$. Solve: $3y = -9 \Rightarrow y = -3$. 4. Problem: Find $f^2(x)$ if $f(x) = \frac{1}{2}x$. $f^2(x) = f(f(x)) = f\left(\frac{1}{2}x\right) = \frac{1}{2} \times \frac{1}{2}x = \frac{1}{4}x$. 5. Problem: Solve $|x-5| = -3$. Absolute value cannot be negative. No solution, so answer is $\phi$ (empty set). 6. Problem: Find discriminant of $x^2 - 3x - 4 = 0$. Discriminant $D = b^2 - 4ac$ with $a=1$, $b=-3$, $c=-4$. Calculate: $D = (-3)^2 - 4(1)(-4) = 9 + 16 = 25$. 7. Problem: Find horizontal component of velocity for projectile launched at $30^\circ$ with initial velocity $20$ m/s. Horizontal component $= 20 \cos 30^\circ = 20 \times \frac{\sqrt{3}}{2} = 10\sqrt{3} \approx 17.3$ m/s. 8. Problem: In triangle ABC, $a=10$, $b=15$, $\alpha=32^\circ$, find $\beta$. Use Law of Sines: $\frac{a}{\sin \alpha} = \frac{b}{\sin \beta}$. Calculate $\sin \beta = \frac{b \sin \alpha}{a} = \frac{15 \times \sin 32^\circ}{10} \approx \frac{15 \times 0.5299}{10} = 0.7949$. Find $\beta = \sin^{-1}(0.7949) \approx 52.6^\circ$. 9. Problem: Which curve is most appropriate for arcs in rainbow effect? Answer: Circular arc. 10. Problem: Distance between centers of two circles with radii 4 cm and 5 cm touching externally. Distance $= 4 + 5 = 9$ cm. 11. Problem: Probability of getting two tails when two coins are tossed. Total outcomes = 4, only one is two tails. Probability $= \frac{1}{4}$. 12. Problem: Equation of line of best fit. Answer: $y = mx + c$. 13. Problem: Angle between tangent line and radius at point P on circle. Answer: $90^\circ$. 14. Problem: Simplify $\left(\frac{2x^2 - 2x}{x+1}\right) \times \left(\frac{2x^2 + 2x}{x-1}\right)$. Factor numerators: $2x(x-1)$ and $2x(x+1)$. Expression becomes $\frac{2x(x-1)}{x+1} \times \frac{2x(x+1)}{x-1} = \frac{2x \cancel{(x-1)}}{x+1} \times \frac{2x (x+1)}{\cancel{(x-1)}}$. Simplify: $2x \times 2x = 4x^2$. 15. Problem: Find y-intercept of $2x + 3y = 2$. Set $x=0$, then $3y=2$, so $y=\frac{2}{3}$. Final answers: i) $-1$, ii) $-4$, iii) $-3$, iv) $\frac{1}{4}x$, v) $\phi$, vi) $25$, vii) $17.3$ m/s, viii) $52.6^\circ$, ix) Circular arc, x) $9$ cm, xi) $\frac{1}{4}$, xii) $y=mx+c$, xiii) $90^\circ$, xiv) $4x^2$, xv) $\frac{2}{3}$.