Question

The number of non-zero integral solutions of the equation ${\left|1-i\right|}^x=2^x$ is

• A. Infinite
• B. $1$
• C. $2$
• D. None of these

Answer 1

The correct option is D

None of these

Explanation for the correct option:

Find the solutions to the given equation.

An equation ${\left|1-i\right|}^x=2^x$ is given.

Find the solutions of the given equation as follows:

$$\begin{gathered} {\left(\sqrt{1^2+1^2}\right)}^x=2^x \\ \Rightarrow {\left(\sqrt{2}\right)}^x=2^x \\ \Rightarrow 2^{\frac{x}{2}}=2^x \end{gathered}$$

On comparing both sides, we get

$$\begin{gathered} \frac{x}{2}=x \\ \Rightarrow \frac{x}{2}-x=0 \\ \Rightarrow x\left(\frac{1}{2}-1\right)=0 \\ \Rightarrow x=0 \end{gathered}$$

Since, the only solution of the given equation is $x=0$.

Therefore, The number of non-zero integral solutions of the given equation is zero.

Hence, option $D$ is the correct answer.