1. **Problem:** Determine the answers for each of the given subproblems. 2. **Set notation and intervals:** (a) $(3,7)$ means all numbers strictly between 3 and 7. (b) $]3,7]$ means numbers greater than 3 and up to 7 including 7. (c) $]3,7[$ means numbers strictly between 3 and 7 (same as (a)). 3. **Set difference:** $[2,7] - \{2,7\}$ means the interval from 2 to 7 including endpoints, excluding the points 2 and 7. So the result is $(2,7)$ which corresponds to $]2,7[$. 4. **Given options:** (a) $[1,6]$ is from 1 to 6 including endpoints. (b) $\emptyset$ is the empty set. (d) $\{0\}$ is the set containing 0. None matches $(2,7)$ exactly, so the answer is not listed here. 5. **Next number in the pattern:** $\sqrt{3}, \sqrt{12}, \sqrt{27}, \sqrt{48}$ Rewrite under the root: $3 = 1 \times 3$ $12 = 4 \times 3$ $27 = 9 \times 3$ $48 = 16 \times 3$ The pattern is $\sqrt{n^2 \times 3} = n \sqrt{3}$ with $n=1,2,3,4$. Next is $n=5$, so $\sqrt{25 \times 3} = 5 \sqrt{3} = \sqrt{75}$. Among options: (a) $\sqrt{50}$ (c) $\sqrt{60}$ (d) $\sqrt{90}$ None is $\sqrt{75}$, so none matches exactly. 6. **Expression:** $2^{2017} = 2^{2016} + ...$ Rewrite: $2^{2017} = 2 \times 2^{2016} = 2^{2016} + 2^{2016}$ So the missing term is $2^{2016}$. Answer: (d) $2^{2016}$. 7. **Interval intersection:** Given $[y-1, x] \cap [y,5] = [2,3]$. The intersection of intervals is $[\max(y-1,y), \min(x,5)] = [y, \min(x,5)] = [2,3]$. So $y=2$ and $\min(x,5) = 3$ which means $x=3$ (since $x \leq 5$). Calculate $x^y = 3^2 = 9$. Answer: (c) 9. 8. **Square side length increase:** Side length increases by 10%, so new side = $1.1 \times$ original. Area increases by factor $(1.1)^2 = 1.21$. Percentage increase in area = $(1.21 - 1) \times 100 = 21\%$. Answer: (d) 21. 9. **Solve:** $\frac{x}{2} \neq \frac{8}{x}$ Cross multiply: $x^2 \neq 16$ So $x \neq \pm 4$. Options: (a) -4 (b) $\pm 4$ (c) 4 (d) 16 Since $x$ cannot be $\pm 4$, none of these are solutions, but if the question is to find $x$ such that equality holds, then $x=\pm 4$. 10. **Next odd number after odd $F$:** Odd numbers differ by 2. So next odd number is $F + 2$. Answer: (d) $F + 2$. **Final answers:** - Set difference: $(2,7)$ (not listed) - Next number in pattern: $\sqrt{75}$ (not listed) - $2^{2017} = 2^{2016} + 2^{2016}$ answer (d) - $x^y = 9$ answer (c) - Area increase: 21% answer (d) - $x = \pm 4$ for equality answer (b) - Next odd number: $F + 2$ answer (d)