1. Problem: Find the results for each set and algebra question. (a) $Q \cup Q = Q$ because the union of a set with itself is the set. (b) $Q \cap Q^{c} = \varnothing$ because the intersection of a set and its complement is empty. (c) $N \cup Z = Z$ since natural numbers $N$ are a subset of integers $Z$. (d) $Z \cup Q^{c}$ is the union of integers and the complement of rationals. Since $Z \subset Q$, $Q^{c}$ contains irrationals, so $Z \cup Q^{c}$ is all real numbers $\mathbb{R}$. 2. Given $2^{x} = 3$, find $2^{x+1}$. Use the property $2^{x+1} = 2^{x} \times 2^{1} = 3 \times 2 = 6$. Answer: (c) 6. 3. If $(x-1)$ is a factor of $x^{2} - 4x + 3$, find the other factor. Factor the trinomial: $x^{2} - 4x + 3 = (x-1)(x-3)$. Other factor is $x-3$. Answer: (d) $x - 3$. 4. For $k x^{2} - 12x + 9$ to be a perfect square, it must be $(ax + b)^{2} = a^{2}x^{2} + 2abx + b^{2}$. Matching coefficients: $a^{2} = k$, $2ab = -12$, $b^{2} = 9$. From $b^{2} = 9$, $b = \pm 3$. From $2ab = -12$, $2a(\pm 3) = -12 \Rightarrow a = \frac{-12}{2b} = \frac{-12}{\pm 6} = \mp 2$. Then $k = a^{2} = (\mp 2)^{2} = 4$. Answer: (a) 4. 5. Solve $x^{2} + 16 = 0$ in $\mathbb{R}$. Rewrite: $x^{2} = -16$. No real solution since square of real number cannot be negative. Solution set is empty $\varnothing$. Answer: (c) $\varnothing$. 6. In triangle ABC, if $(AC)^{2} + (BC)^{2} = (AB)^{2} - 5$, determine $\angle C$. By Pythagoras, if $(AC)^{2} + (BC)^{2} < (AB)^{2}$, then $\angle C$ is obtuse. Since right side is $(AB)^{2} - 5 < (AB)^{2}$, $\angle C$ is obtuse. Answer: (c) obtuse. 7. Given triangle ABC with $\angle B = 42^{\circ}$, $\angle C = x^{\circ}$, and $AB = AC$ (isosceles), find $x$. Sum of angles in triangle: $42 + x + x = 180$. $2x = 180 - 42 = 138$. $x = 69^{\circ}$. Answer: (c) 69. 8. Length opposite $30^{\circ}$ in right triangle is half the hypotenuse. Answer: (a) $\frac{1}{2}$. 9. Upper bound = 18, center = 16, find lower bound. Center = $\frac{upper + lower}{2} \Rightarrow 16 = \frac{18 + lower}{2}$. Multiply both sides by 2: $32 = 18 + lower$. $lower = 32 - 18 = 14$. Answer: (b) 14.