1. **Problem:** Find the domain of the function $$d(x) = \frac{3 - x}{x - 3}$$. Since the denominator cannot be zero, set $$x - 3 \neq 0$$ which gives $$x \neq 3$$. **Answer:** The domain is all real numbers except $$x = 3$$. 2. **Problem:** Identify the even function among the given options. An even function satisfies $$f(-x) = f(x)$$. (As the options are unclear, this step is theoretical.) 3. **Problem:** Determine the interval where $$d(x) = (x - 1)^2 + 1$$ is decreasing. Calculate the derivative: $$d'(x) = 2(x - 1)$$. Set $$d'(x) < 0$$ for decreasing: $$2(x - 1) < 0 \Rightarrow x < 1$$. **Answer:** The function is decreasing on $$(-\infty, 1)$$. 4. **Problem:** Solve the equation $$|x - 2| - 3 = 1$$. Rewrite: $$|x - 2| = 4$$. So, $$x - 2 = 4 \Rightarrow x = 6$$ or $$x - 2 = -4 \Rightarrow x = -2$$. **Answer:** The solution set is $$\{ -2, 6 \}$$. 5. **Problem:** Given $$d(x) = 3x + 2$$ and $$d(x) = 5 - x$$, find $$d(5d(3))$$. First, find $$d(3)$$ using the first function: $$d(3) = 3(3) + 2 = 9 + 2 = 11$$. Then find $$d(5d(3)) = d(5 \times 11) = d(55)$$. Using the second function: $$d(55) = 5 - 55 = -50$$. **Answer:** $$d(5d(3)) = -50$$. 6. **Problem:** In triangle ABC, given $$AH = 6$$ cm, $$\angle CHB = 70^\circ$$, and $$\angle CHC = 80^\circ$$, find the length of the diameter of the circumscribed circle. (Since the problem is incomplete or ambiguous, assuming the diameter equals the length of side or related to given data.) **Answer:** The diameter is 12 cm (assuming from options). 7. **Problem:** Given $$3 \sin A = 2$$ and $$\sin B = 4$$ (likely a typo, assuming $$\sin B = \frac{4}{5}$$), find $$\sin \angle ABC$$. (Since data is unclear, cannot solve precisely.) 8. **Problem:** Calculate the limit as $$x \to 0$$ of $$\frac{x^2 - 10}{x - 5}$$. Substitute $$x = 0$$: Numerator: $$0^2 - 10 = -10$$ Denominator: $$0 - 5 = -5$$ Limit = $$\frac{-10}{-5} = 2$$. 9. **Problem:** Calculate the limit as $$x \to 3$$ of $$\frac{|x + 3 - 3|}{3 - x} = \frac{|x|}{3 - x}$$. At $$x = 3$$ numerator is $$|3| = 3$$, denominator is $$3 - 3 = 0$$. Check left and right limits: - From left $$x \to 3^-$$, denominator $$> 0$$, limit tends to $$+\infty$$. - From right $$x \to 3^+$$, denominator $$< 0$$, limit tends to $$-\infty$$. Limit does not exist. 10. **Problem:** Calculate the limit as $$x \to 20$$ of $$\frac{x^4 - 16}{4x + 32}$$. Substitute $$x = 20$$: Numerator: $$20^4 - 16 = 160000 - 16 = 159984$$ Denominator: $$4(20) + 32 = 80 + 32 = 112$$ Limit = $$\frac{159984}{112} = 1428$$. **Summary:** - Domain of $$d(x)$$ excludes $$x=3$$. - $$d(x) = (x-1)^2 + 1$$ decreases on $$(-\infty, 1)$$. - Solution to $$|x-2|-3=1$$ is $$\{-2,6\}$$. - $$d(5d(3)) = -50$$. - Limit at $$x \to 0$$ is 2. - Limit at $$x \to 3$$ does not exist. - Limit at $$x \to 20$$ is 1428.