1. **Problem:** Determine whether the elements belong to set $A = \{2,4,5,6,7,10,12\}$. Use $\in$ for "belongs to" and $\notin$ for "does not belong to". 2. **Solution:** - a) $2 \in A$ because 2 is in the set. - b) $3 \notin A$ because 3 is not in the set. - c) $8 \notin A$ because 8 is not in the set. - d) $10 \in A$ because 10 is in the set. **Final answers:** - a) $c$ (correct) - b) $e$ (error/not in set) - c) $e$ - d) $c$ 2. **Problem:** Calculate $23.97 \times 2(11.4 - 6.03) - 7.8$ without a calculator. 3. **Steps:** 1. Calculate inside the parentheses: $11.4 - 6.03 = 5.37$ 2. Multiply by 2: $2 \times 5.37 = 10.74$ 3. Multiply by 23.97: $23.97 \times 10.74$ Calculate $23.97 \times 10.74$: $23.97 \times 10 = 239.7$ $23.97 \times 0.74 = 23.97 \times (0.7 + 0.04) = 23.97 \times 0.7 + 23.97 \times 0.04 = 16.779 + 0.9588 = 17.7378$ Sum: $239.7 + 17.7378 = 257.4378$ 4. Subtract 7.8: $257.4378 - 7.8 = 249.6378$ **Final answer:** $249.6378$ 3. **Problem:** Add and simplify algebraic fractions: $$\frac{2}{a^2 - 4a + 4} + \frac{5}{a - 2}$$ 4. **Steps:** 1. Factor denominator $a^2 - 4a + 4 = (a - 2)^2$ 2. Rewrite first fraction: $\frac{2}{(a - 2)^2}$ 3. Rewrite second fraction with common denominator: $\frac{5}{a - 2} = \frac{5(a - 2)}{(a - 2)^2}$ 4. Add fractions: $$\frac{2}{(a - 2)^2} + \frac{5(a - 2)}{(a - 2)^2} = \frac{2 + 5(a - 2)}{(a - 2)^2}$$ 5. Simplify numerator: $$2 + 5a - 10 = 5a - 8$$ **Final answer:** $$\frac{5a - 8}{(a - 2)^2}$$ 4. **Problem:** Convert $514_8$ (base 8) to base 9 and find the missing digit. 5. **Steps:** 1. Convert $514_8$ to decimal: $$5 \times 8^2 + 1 \times 8^1 + 4 \times 8^0 = 5 \times 64 + 1 \times 8 + 4 = 320 + 8 + 4 = 332$$ 2. Convert decimal 332 to base 9: Divide 332 by 9: $$332 \div 9 = 36 \text{ remainder } 8$$ Divide 36 by 9: $$36 \div 9 = 4 \text{ remainder } 0$$ Divide 4 by 9: $$4 \div 9 = 0 \text{ remainder } 4$$ Digits from last remainder to first: $4 0 8$ **Final answer:** $$514_8 = 408_9$$ 5. **Problem:** Rationalize the denominator: $$\frac{\sqrt{2}}{\sqrt{5} + \sqrt{5}}$$ 6. **Steps:** 1. Note denominator $\sqrt{5} + \sqrt{5} = 2\sqrt{5}$ 2. Rewrite expression: $$\frac{\sqrt{2}}{2\sqrt{5}} = \frac{\sqrt{2}}{2\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}} = \frac{\sqrt{2} \times \sqrt{5}}{2 \times 5} = \frac{\sqrt{10}}{10}$$ **Final answer:** $$\frac{\sqrt{10}}{10}$$ 6. **Problem:** Calculate the volume of a cylinder with radius 2 m and height 6 m. 7. **Steps:** 1. Volume formula for cylinder: $$V = \pi r^2 h$$ 2. Substitute values: $$V = \pi \times 2^2 \times 6 = \pi \times 4 \times 6 = 24\pi$$ **Final answer:** $$24\pi \text{ cubic meters}$$ 7. **Problem:** Find $x$ in the trapezoid figure with parallel lines and given segments. 8. **Steps:** 1. The top base is $x$, the top right segment is 5. 2. The bottom base segments are 16 and 10. 3. Since the trapezoids share parallel lines and transversals, the sum of the top base and the top right segment equals the sum of the bottom base segments: $$x + 5 = 16 + 10$$ 4. Simplify right side: $$x + 5 = 26$$ 5. Solve for $x$: $$x = 26 - 5 = 21$$ **Final answer:** $$x = 21$$