Question

The imaginary part of \(z=\frac{2+i}{3-i}\) is

• A. \(\frac{5}{8}\)
• B. \(\frac{-5}{8}\)
• C. \(\frac{1}{2}\)
• D. \(\frac{3}{4}\)
• E. \(\frac{3}{8}\)

Answer 1

The Correct Option is C

We are given \(z = \frac{2 + i}{3 - i}\), and we need to find the imaginary part of \(z\). 

To simplify the expression for \(z\), we multiply both the numerator and the denominator by the conjugate of the denominator, \(3 + i\):

\(z = \frac{2 + i}{3 - i} \times \frac{3 + i}{3 + i} = \frac{(2 + i)(3 + i)}{(3 - i)(3 + i)}\)

First, simplify the denominator:

\((3 - i)(3 + i) = 3^2 - i^2 = 9 - (-1) = 9 + 1 = 10\)

Now, simplify the numerator:

\((2 + i)(3 + i) = 2(3) + 2(i) + i(3) + i(i) = 6 + 2i + 3i + i^2 = 6 + 5i - 1 = 5 + 5i\)

Thus, we have:

\(z = \frac{5 + 5i}{10} = \frac{5}{10} + \frac{5i}{10} = \frac{1}{2} + \frac{i}{2}\)

The imaginary part of \(z\) is \(\frac{1}{2}\).

The answer is \( \frac{1}{2} \).