Given Z1 = 2 - 5i and Z2 = 2 + 3i.

Here's the solution for question 1, part (i) and (ii) verification. Given $Z_1 = 2 - 5i$ and $Z_2 = 2 + 3i$. i) Verify $\overline{Z_1 + Z_2} = \overline{Z_1} + \overline{Z_2}$ Step 1: Calculate $Z_1 + Z_2$. $$Z_1 + Z_2 = (2 - 5i) + (2 + 3i) = (2 + 2) + (-5 + 3)i = 4 - 2i$$ Step 2: Find the conjugate of $Z_1 + Z_2$. $$\overline{Z_1 + Z_2} = \overline{4 - 2i} = 4 + 2i$$ Step 3: Find $\overline{Z_1}$ and $\overline{Z_2}$. $$\overline{Z_1} = \overline{2 - 5i} = 2 + 5i$$ $$\overline{Z_2} = \overline{2 + 3i} = 2 - 3i$$ Step 4: Calculate $\overline{Z_1} + \overline{Z_2}$. $$\overline{Z_1} + \overline{Z_2} = (2 + 5i) + (2 - 3i) = (2 + 2) + (5 - 3)i = 4 + 2i$$ Step 5: Compare the results. Since $\overline{Z_1 + Z_2} = 4 + 2i$ and $\overline{Z_1} + \overline{Z_2} = 4 + 2i$, $$\overline{Z_1 + Z_2} = \overline{Z_1} + \overline{Z_2}$$ This property is verified. ii) Verify $\overline{Z_1 \cdot Z_2} = \overline{Z_1} \cdot \overline{Z_2}$ Step 1: Calculate $Z_1 \cdot Z_2$. $$Z_1 \cdot Z_2 = (2 - 5i)(2 + 3i)$$ $$Z_1 \cdot Z_2 = 2(2) + 2(3i) - 5i(2) - 5i(3i)$$ $$Z_1 \cdot Z_2 = 4 + 6i - 10i - 15i^2$$ Since $i^2 = -1$: $$Z_1 \cdot Z_2 = 4 - 4i - 15(-1)$$ $$Z_1 \cdot Z_2 = 4 - 4i + 15 = 19 - 4i$$ Step 2: Find the conjugate of $Z_1 \cdot Z_2$. $$\overline{Z_1 \cdot Z_2} = \overline{19 - 4i} = 19 + 4i$$ Step 3: Use $\overline{Z_1}$ and $\overline{Z_2}$ from the previous part. $$\overline{Z_1} = 2 + 5i$$ $$\overline{Z_2} = 2 - 3i$$ Step 4: Calculate $\overline{Z_1} \cdot \overline{Z_2}$. $$\overline{Z_1} \cdot \overline{Z_2} = (2 + 5i)(2 - 3i)$$ $$\overline{Z_1} \cdot \overline{Z_2} = 2(2) + 2(-3i) + 5i(2) + 5i(-3i)$$ $$\overline{Z_1} \cdot \overline{Z_2} = 4 - 6i + 10i - 15i^2$$ Since $i^2 = -1$: $$\overline{Z_1} \cdot \overline{Z_2} = 4 + 4i - 15(-1)$$ $$\overline{Z_1} \cdot \overline{Z_2} = 4 + 4i + 15 = 19 + 4i$$ Step 5: Compare the results. Since $\overline{Z_1 \cdot Z_2} = 19 + 4i$ and $\overline{Z_1} \cdot \overline{Z_2} = 19 + 4i$, $$\overline{Z_1 \cdot Z_2} = \overline{Z_1} \cdot \overline{Z_2}$$ This property is verified. Both properties are verified. Send me the next one 📸