Question

The argument of the complex number \[ \left( \frac{i}{2} - \frac{2}{i} \right) \] is equal to:

• A. \( \frac{\pi}{4} \)
• B. \( \frac{3\pi}{4} \)
• C. \( \frac{\pi}{12} \)
• D. \( \frac{\pi}{2} \)

Hint:

The argument of a purely imaginary number \( bi \) is \( \frac{\pi}{2} \) if \( b>0 \) and \( -\frac{\pi}{2} \) if \( b<0 \).

Answer 1

The Correct Option is D

Step 1: Simplifying the given complex number.
\[ z = \frac{i}{2} - \frac{2}{i} \] Rewriting the second term: \[ \frac{2}{i} = \frac{2 \times (-i)}{i \times (-i)} = -2i \] Thus, \[ z = \frac{i}{2} - 2i = -\frac{4i}{2} + \frac{i}{2} = -\frac{3i}{2} \] Step 2: Finding the argument.
The given complex number \( z = -\frac{3i}{2} \) is purely imaginary and negative, meaning it lies on the negative imaginary axis. The argument of a purely imaginary number \( bi \) is given by: \[ \theta = \frac{\pi}{2} \quad {if } b>0, \quad {or} \quad -\frac{\pi}{2} { if } b<0. \] Since \( z \) is negative imaginary, \[ {Arg}(z) = \frac{\pi}{2} \] Thus, the correct answer is (D) \( \frac{\pi}{2} \).