Question

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

1. Infinite
2. $1$
3. $2$
4. None of these

Answer 1

In this type of question we have to use the concept of complex numbers. We know that the general representation of complex numbers is given by $z=x+iy$ where $x$ is called the real part and $y$ is known as the imaginary part of the complex number $z$. The value of $i$ is defined to be $i=\sqrt{-1}$. Also we know that the modulus of a complex number is defined as the square root of the sum of real part’s square and imaginary part’s square i.e. $\left| z \right|=\sqrt{{{x}^{2}}+{{y}^{2}}}$.

Complete step-by-step solution:
Now we have to find the number of non-zero integral solutions of ${{\left| 1-i \right|}^{x}}={{2}^{x}}$
Let us consider
$\Rightarrow {{\left| 1-i \right|}^{x}}={{2}^{x}}$
Now we know that if $z=x+iy$ then we can define its modulus as $\left| z \right|=\sqrt{{{x}^{2}}+{{y}^{2}}}$

& \Rightarrow {{\left[ \sqrt{{{1}^{2}}+{{\left( -1 \right)}^{2}}} \right]}^{x}}={{2}^{x}} \\\ & \Rightarrow {{\left( \sqrt{2} \right)}^{x}}={{2}^{x}} \\\ \end{aligned}$$ We will express $$\sqrt{2}$$ as $${{2}^{\dfrac{1}{2}}}$$, so that we can write $$\Rightarrow {{\left( {{2}^{\dfrac{1}{2}}} \right)}^{x}}={{2}^{x}}$$ Now by using the rule of indices $${{\left( {{a}^{m}} \right)}^{n}}={{a}^{mn}}$$ we get, $$\Rightarrow {{2}^{\dfrac{x}{2}}}={{2}^{x}}$$ As we know that if the bases of both the sides are same then we can compare their indices $$\begin{aligned} & \Rightarrow \dfrac{x}{2}=x \\\ & \Rightarrow x-\dfrac{x}{2}=0 \\\ & \Rightarrow \dfrac{x}{2}=0 \\\ & \Rightarrow x=0 \\\ \end{aligned}$$ Hence, the equation $${{\left| 1-i \right|}^{x}}={{2}^{x}}$$ does not have a non-zero integral solution. Thus the number of non-zero integral solutions of the equation $${{\left| 1-i \right|}^{x}}={{2}^{x}}$$ is $$0$$ **Therefore, option (4) is the correct option.** **Note:** In this question students have to remember the definition of modulus of a complex number. Students have to note some of the rules of indices such as $$\sqrt{a}={{a}^{\dfrac{1}{2}}}$$, $${{\left( {{a}^{m}} \right)}^{n}}={{a}^{mn}}$$. Also students have to note that if the bases of both the sides of an equation are the same then they can compare the indices of both the sides.