Question

Arg \[ \frac{4 + 2i}{1 - 2i} + \frac{3 + 4i}{2 + 3i} \] lies in the interval \[ \text{Arg} \left( \frac{4 + 2i}{1 - 2i} + \frac{3 + 4i}{2 + 3i} \right) \]

• A. \( \left( \frac{\pi}{4}, \frac{\pi}{2} \right) \)
• B. \( \left( -{\pi}, -\frac{\pi}{2} \right) \)
• C. \( \left( -\frac{\pi}{2}, 0 \right) \)
• D. \( \left( 0, \frac{\pi}{4} \right) \)

Hint:

To find the argument of a complex number, multiply by the conjugate of the denominator to eliminate the imaginary part in the denominator. The argument is the angle of the resulting complex number in polar form.

Answer 1

The Correct Option is A

Step 1: First, compute the argument of the complex fraction. The complex number is in the form \( \frac{a + bi}{c + di} \).
To find the argument, multiply both numerator and denominator by the conjugate of the denominator. For \( \frac{4 + 2i}{1 - 2i} \), multiply both by \( 1 + 2i \), and for \( \frac{3 + 4i}{2 + 3i} \), multiply both by \( 2 - 3i \). \[ \frac{4 + 2i}{1 - 2i} = \frac{(4 + 2i)(1 + 2i)}{(1 - 2i)(1 + 2i)} \] \[ \frac{3 + 4i}{2 + 3i} = \frac{(3 + 4i)(2 - 3i)}{(2 + 3i)(2 - 3i)} \] Step 2: After simplifying the expressions, calculate the resulting arguments of both fractions.
Step 3: Adding the two arguments results in the range of values between \( \frac{\pi}{4} \) and \( \frac{\pi}{2} \), confirming the argument lies within this interval.