Question

The number of integer solutions of the equation $|1 - i|^x = 2^x$ is:

• A. 1
• B. 0
• C. 2
• D. 3

Hint:

Use modulus rules and properties of exponents carefully with complex numbers.

Answer 1

The Correct Option is A

We know: $|1 - i| = \sqrt{1^2 + 1^2} = \sqrt{2}$
So equation becomes: \[ (\sqrt{2})^x = 2^x \Rightarrow (2^{1/2})^x = 2^x \Rightarrow 2^{x/2} = 2^x \Rightarrow \frac{x}{2} = x \Rightarrow x = 0 \] Only integer solution: $x = 0$