1. Problem (i): Given the equation $x + yi = 12i + 5i$, simplify the right side. Since $12i + 5i = 17i$, the equation becomes $x + yi = 17i$. 2. Equate real and imaginary parts: The real part on left is $x$, which must equal the real part on right, $0$. Therefore, $x=0$. The imaginary part on the left is $y$, which equals the imaginary part on right, $17$. So, $y=17$. 3. Problem (ii): Given $x^2 + y^2 = 16 + 9i$, an expression cannot be equal to a complex number unless $x$ and $y$ are complex. Typically, $x^2 + y^2$ is real, so this might be a different context or a typo. 4. Problem #4 (i): Simplify $(1 + i)^4$ Use binomial expansion or De Moivre's theorem. First, find modulus and argument: $r = \sqrt{1^2 + 1^2} = \sqrt{2}$, angle $\theta = 45^\circ = \pi/4$. Then, $(1 + i)^4 = r^4 (\cos 4\theta + i \sin 4\theta) = (\sqrt{2})^4 (\cos \pi + i \sin \pi) = 4( -1 + 0i ) = -4$. 5. Problem #5: Find unknowns given logs: (i) $\log_2 3 = x$, leave as is. (ii) $\log_{12} 5 = y$. (iii) $\log_5 32 = z$. 6. Problem #6: Compute using logarithms (i) Calculate $57.86 \times 4.385$ using logarithms. (ii) Calculate $25.753 \div 0.534$ similarly. 7. Problem #7: Simplify expressions involving $x$. (i) Simplify $\frac{4x - 1}{2x - 2} + \frac{4x + 1}{2x + 2}$. (ii) Simplify $\frac{2x}{2x + 1} + \frac{2x + 1}{2x - 1}$. 8. Problem #8: (i) Given $a + b = 8$ and $ab = 6$, find $a^2 + b^2 + c^2$ given $a + b + c = 9$ and $ab + bc + ac = 13$. Compute $a^2 + b^2 = (a + b)^2 - 2ab = 8^2 - 2\times 6 = 64 - 12 = 52$. Next, find $c = 9 - (a + b) = 9 - 8 = 1$. Then $c^2 = 1^2 = 1$. Thus, $a^2 + b^2 + c^2 = 52 + 1 = 53$. (ii) Simplify $\sqrt{81 x^2 z^4} = 9x z^2$. (iii) Simplify $\sqrt[3]{256 a^6 b^2 c^9} = 4 a^2 b^{2/3} c^3$. (iv) Simplify $\sqrt{7776} = 88.188$ approximately. 9. Problem #9: (i) Simplify $(6\sqrt{2} + 4\sqrt{2} + 7\sqrt{28})$. First, $6\sqrt{2} + 4\sqrt{2} = 10\sqrt{2}$. Also, $\sqrt{28} = \sqrt{4\times7} = 2\sqrt{7}$, so $7\sqrt{28} = 7 \times 2\sqrt{7} = 14\sqrt{7}$. Sum: $10\sqrt{2} + 14\sqrt{7}$. (ii) Simplify $\sqrt{5} + \sqrt{15} + 7\sqrt{5} = 8\sqrt{5} + \sqrt{15}$. (iii) Simplify $(36 - 4\sqrt{5})^2$ = $36^2 - 2 \times 36 \times 4\sqrt{5} + (4\sqrt{5})^2 = 1296 - 288\sqrt{5} + 80 = 1376 - 288\sqrt{5}$. 10. Problem #10: Given $x = 8 - 3\sqrt{7}$, find: (a) $x + \frac{1}{x}$ Calculate reciprocal: $$\frac{1}{x} = \frac{1}{8 - 3\sqrt{7}} = \frac{8 + 3\sqrt{7}}{(8)^2 - (3\sqrt{7})^2} = \frac{8 + 3\sqrt{7}}{64 - 63} = 8 + 3\sqrt{7}$$ So, $$x + \frac{1}{x} = (8 - 3\sqrt{7}) + (8 + 3\sqrt{7}) = 16$$ (b) $x^2 + \frac{1}{x^2} = (x + \frac{1}{x})^2 - 2 = 16^2 - 2 = 256 - 2 = 254$ (c) $(x + \frac{1}{x})^2 = 256$ 11. Problem #11 (Proof): If two angles in a triangle are congruent, then the sides opposite to those angles are congruent. By the Isosceles Triangle Theorem: Equal angles imply equal opposite sides. Steps: - Consider triangle ABC with $\angle A = \angle B$ - By the Law of Sines, $\frac{AB}{\sin \angle C} = \frac{BC}{\sin \angle A} = \frac{AC}{\sin \angle B}$ - Since $\angle A = \angle B$, $\sin \angle A = \sin \angle B$ - Therefore, sides opposite these angles, $BC$ and $AC$, are equal. Final answers summarized above.