1. Evaluate the following complex number operations: i) \((2 + 3i) + (7 - 2i)\) Step 1: Add the real parts and the imaginary parts separately. $$2 + 7 = 9$$ $$3i - 2i = i$$ Step 2: Combine the results: $$9 + i$$ \(\therefore (2 + 3i) + (7 - 2i) = 9 + i\) ii) \((3 + 2i) - (7 + 3i)\) Step 1: Subtract the real parts and the imaginary parts separately. $$3 - 7 = -4$$ $$2i - 3i = -i$$ Step 2: Combine the results: $$-4 - i$$ \(\therefore (3 + 2i) - (7 + 3i) = -4 - i\) 2. Find the value of \(\frac{z_1 - z_2}{z_1 + z_2}\) where \(z_1 = -2 + i\) and \(z_2 = 1 - 2i\). Step 1: Compute \(z_1 - z_2\): $$(-2 + i) - (1 - 2i) = -2 + i - 1 + 2i = -3 + 3i$$ Step 2: Compute \(z_1 + z_2\): $$(-2 + i) + (1 - 2i) = -2 + i + 1 - 2i = -1 - i$$ Step 3: Write the expression: $$\frac{-3 + 3i}{-1 - i}$$ Step 4: Multiply numerator and denominator by the conjugate of the denominator \(-1 + i\) to simplify: $$\frac{(-3 + 3i)(-1 + i)}{(-1 - i)(-1 + i)}$$ Step 5: Expand the numerator: $$(-3)(-1) + (-3)(i) + (3i)(-1) + (3i)(i) = 3 - 3i - 3i + 3i^2$$ Knowing \(i^2 = -1\), this becomes: $$3 - 3i - 3i - 3 = -6i$$ Step 6: Expand the denominator: $$(-1)^2 - (i)^2 = 1 - (-1) = 2$$ Step 7: Combine: $$\frac{-6i}{2} = -3i$$ \(\therefore \frac{z_1 - z_2}{z_1 + z_2} = -3i\)