Question

Argument of the complex number \(\left(\frac{-1-3i}{2+i}\right)\) is

• A. \(45^\circ\)
• B. \(135^\circ\)
• C. \(225^\circ\)
• D. \(240^\circ\)

Hint:

If complex number lies in 3rd quadrant, argument = \(180^\circ +\) reference angle.

Answer 1

The Correct Option is C

Step 1: Simplify the complex number.
\[ z = \frac{-1-3i}{2+i} \]
Multiply numerator and denominator by conjugate \((2-i)\):
\[ z = \frac{(-1-3i)(2-i)}{(2+i)(2-i)} \]
Step 2: Compute denominator.
\[ (2+i)(2-i)=4+1=5 \]
Step 3: Compute numerator.
\[ (-1-3i)(2-i)=(-1)(2) + (-1)(-i) + (-3i)(2) + (-3i)(-i) \]
\[ = -2 + i -6i + 3i^2 \]
\[ = -2 -5i + 3(-1) = -5 - 5i \]
So:
\[ z = \frac{-5-5i}{5} = -1 - i \]
Step 4: Find argument of \(-1-i\).
Point \((-1,-1)\) lies in third quadrant.
Reference angle = \(45^\circ\).
So argument =
\[ 180^\circ + 45^\circ = 225^\circ \]
Final Answer:
\[ \boxed{225^\circ} \]