Question

If \( z_1 = \sqrt{3} + i\sqrt{3} \) and \( z_2 = \sqrt{3} + i \), then the complex number
\( \left( \dfrac{z_1}{z_2} \right)^{50} \)
lies in the

• A. first quadrant
• B. second quadrant
• C. third quadrant
• D. fourth quadrant

Hint:

Use arguments to locate powers of complex numbers.

Answer 1

The Correct Option is A

Step 1: Arguments: \( \arg z_1 = 45^\circ \), \( \arg z_2 = 30^\circ \)

Step 2: \( \arg\left(\dfrac{z_1}{z_2}\right) = 15^\circ \Rightarrow \arg\left(\dfrac{z_1}{z_2}\right)^{50} = 750^\circ \equiv 30^\circ \)

Step 3: Hence, the result lies in the first quadrant.