Question

The number of non-zero integral solutions of the equation ${|1-2i|}^x=5^x$ is

• A. Zero (no solution)
• B. One
• C. Two
• D. Three

Answer 1

Answer: (A)

Zero (no solution)

Giving $|1-2i{|}^x=5^x$

We need to find value of ${}^{\prime }{x}^{\prime }$ for which should be an integer but not zero.

Let first find value of $|1-2i|$

$\therefore |1-2i|=\sqrt{(1{)}^2+(2{)}^2}=\sqrt{5}$ $[z=x+iy$ $\Rightarrow |z|=\sqrt{x^2+y^2}]$

Hence $|1-2i{|}^x=5^x\Rightarrow (\sqrt{5}{)}^x=5^x$

$\therefore (\sqrt{5}{)}^x+(\sqrt{5}{)}^{2x}$

Comparing powers

$\to x=2x$

$\to x-2x=0$

$-x=0$

$x=0$

Hence no non-zero integral solutions.