Question

What is the argument of the complex number \(\frac{(1 + i)(2 + i)}{3 - i}\), where \( i = \sqrt{-1} \) ?

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

Hint:

To find the argument of a complex number, express it in polar form and use the angle with the positive real axis.

Answer 1

The Correct Option is D

We are asked to find the argument of the complex number: \[ z = \frac{(1 + i)(2 + i)}{3 - i} \]

Step 1: Multiply the numerator First, multiply the numerator: \[ (1 + i)(2 + i) = 2 + i + 2i + i^2 = 2 + 3i - 1 = 1 + 3i \] Thus, the numerator becomes \( 1 + 3i \).

Step 2: Multiply by the conjugate of the denominator Next, multiply both the numerator and the denominator by the conjugate of \( 3 - i \), which is \( 3 + i \): \[ \frac{(1 + 3i)}{3 - i} \times \frac{3 + i}{3 + i} = \frac{(1 + 3i)(3 + i)}{(3 - i)(3 + i)} = \frac{(1 + 3i)(3 + i)}{3^2 + 1^2} = \frac{(1 + 3i)(3 + i)}{10} \] Step 3: Multiply the numerator Now, multiply the numerator: \[ (1 + 3i)(3 + i) = 3 + i + 9i + 3i^2 = 3 + 10i - 3 = 10i \] Thus, the expression simplifies to: \[ \frac{10i}{10} = i \] Step 4: Find the argument The complex number is \( i \), which lies on the imaginary axis with an argument of \( \frac{\pi}{2} \). Thus, the correct answer is \( \frac{\pi}{2} \).