QUESTION 1 1.1 Given: \( Z_{1}=5-4 i, Z_{2}=3-i \) and \( Z_{3}=-3 i \) Calculate the following and leave your answers in the form \( a+b i \) : 1.1.1 \( Z_{1}+Z_{2} \) 1.1.2 \( Z_{1} \times Z_{2} \) 1.1.3 \( \frac{Z_{3}}{Z_{1}} \)

To solve the given problems, we will perform the operations on the complex numbers \( Z_{1} = 5 - 4i \), \( Z_{2} = 3 - i \), and \( Z_{3} = -3i \). ### 1.1.1 \( Z_{1} + Z_{2} \) To add the complex numbers, we add their real parts and their imaginary parts separately: \[ Z_{1} + Z_{2} = (5 - 4i) + (3 - i) = (5 + 3) + (-4 - 1)i = 8 - 5i \] So, \[ Z_{1} + Z_{2} = 8 - 5i \] ### 1.1.2 \( Z_{1} \times Z_{2} \) To multiply the complex numbers, we use the distributive property (FOIL method): \[ Z_{1} \times Z_{2} = (5 - 4i)(3 - i) \] Calculating this step-by-step: \[ = 5 \cdot 3 + 5 \cdot (-i) - 4i \cdot 3 - 4i \cdot (-i) \] \[ = 15 - 5i - 12i + 4i^2 \] Since \( i^2 = -1 \), we have: \[ 4i^2 = 4(-1) = -4 \] Now substituting this back in: \[ = 15 - 5i - 12i - 4 = 15 - 4 - 17i = 11 - 17i \] So, \[ Z_{1} \times Z_{2} = 11 - 17i \] ### 1.1.3 \( \frac{Z_{3}}{Z_{1}} \) To divide the complex numbers, we multiply the numerator and the denominator by the conjugate of the denominator: \[ \frac{Z_{3}}{Z_{1}} = \frac{-3i}{5 - 4i} \cdot \frac{5 + 4i}{5 + 4i} = \frac{-3i(5 + 4i)}{(5 - 4i)(5 + 4i)} \] Calculating the denominator: \[ (5 - 4i)(5 + 4i) = 5^2 - (4i)^2 = 25 - 16(-1) = 25 + 16 = 41 \] Now calculating the numerator: \[ -3i(5 + 4i) = -15i - 12i^2 = -15i + 12 = 12 - 15i \] Putting it all together: \[ \frac{Z_{3}}{Z_{1}} = \frac{12 - 15i}{41} = \frac{12}{41} - \frac{15}{41}i \] So, \[ \frac{Z_{3}}{Z_{1}} = \frac{12}{41} - \frac{15}{41}i \] ### Final Answers 1.1.1 \( Z_{1} + Z_{2} = 8 - 5i \) 1.1.2 \( Z_{1} \times Z_{2} = 11 - 17i \) 1.1.3 \( \frac{Z_{3}}{Z_{1}} = \frac{12}{41} - \frac{15}{41}i \)