Q94: The value of $\frac{5+5i}{3-4i} + \frac{20}{4+3i}$ is given by : - Mathematics - ICAR AIEEA PG Statistical Sciences 09 July 2023 (Shift 1) Paper | Agrigyan

Simplifying each term by multiplying numerator and denominator by the complex conjugate:
First term: $\frac{5+5i}{3-4i} = \frac{(5+5i)(3+4i)}{3^2 + (-4)^2} = \frac{15 + 20i + 15i + 20i^2}{25} = \frac{15 - 20 + 35i}{25} = \frac{-5 + 35i}{25} = -\frac{1}{5} + \frac{7}{5}i$.
Second term: $\frac{20}{4+3i} = \frac{20(4-3i)}{4^2 + 3^2} = \frac{80 - 60i}{25} = \frac{16}{5} - \frac{12}{5}i$.
Adding the two terms:
$\left(-\frac{1}{5} + \frac{7}{5}i\right) + \left(\frac{16}{5} - \frac{12}{5}i\right) = \frac{-1+16}{5} + \frac{7-12}{5}i = \frac{15}{5} - \frac{5}{5}i = 3 - i$.

Detailed analysis of options:

• $3 - i$ is correct.
• $1 + i$ is incorrect.
• $\frac{5}{2} - \frac{i}{2}$ is incorrect.
• $-2 + 9i$ is incorrect.