Question

The number of non – zero integral solutions of the equation ${\left| {1 - 2i} \right|^x} = {5^x}$is:
A.Zero (no solution)
B.One
C.Two
D.Three

Answer 1

Hint: We will first find the value of the term $\left| {1 - 2i} \right|$by the formula $\left| z \right|$=$\left| {a + ib} \right| = \sqrt {{a^2} + {b^2}} $ when z = a + ib. After that, we will put $\left| {1 - 2i} \right|$as $\sqrt 5 $. Upon simplification, we will compare the powers for the value of x using the formula ${(a^{m})}^{n} = {a}^{mn}$. The number of values of x will be the number of non – zero solutions of the given equation.

Complete step-by-step answer:
We are given the equation: ${\left| {1 - 2i} \right|^x} = {5^x}$
We are required to find the values of x and it should be an integer and not 0 as it is stated non – zero integral solution.
We will first find the value of $\left| {1 - 2i} \right|$.
We know that for a complex number of the form z = a + i b, the modulus of z is given by the formula:$\left| z \right|$=$\left| {a + ib} \right| = \sqrt {{a^2} + {b^2}} $
Here, a = 1 and b = – 2.
Therefore, $\left| {1 - 2i} \right|$= $\sqrt {{1^2} + {{( - 2)}^2}} = \sqrt {1 + 4} = \sqrt 5 $
We are given${\left| {1 - 2i} \right|^x} = {5^x}$. Putting the value of $\left| {1 - 2i} \right|$as $\sqrt 5 $, we get
$
   \Rightarrow {\left( {\sqrt 5 } \right)^x} = {5^x} \\
   \Rightarrow {\left( {{5^{\dfrac{1}{2}}}} \right)^x} = {5^x} \\
 $
Using the formula ${(a^{m})}^{n} = {a}^{mn}$, we get
$ \Rightarrow {5^{\dfrac{x}{2}}} = {5^x}$
Now, we have the same base on both sides , which means the powers must be equal in order that this statement holds true. Therefore, upon comparing powers of 5 both sides, we get
$
   \Rightarrow \dfrac{x}{2} = x \\
   \Rightarrow x = 2x \\
 $
Solving for x, we get
$ \Rightarrow $x – 2x = 0
$ \Rightarrow $ – x = 0
$ \Rightarrow $x = 0
Hence, we get no non – zero integral value of x i.e., no solution.
Hence option(A) is correct.

Note: In such questions, you may get confused about how to solve the question once you have calculated the modulus value of the $\left| {1 - 2i} \right|$. All you need to do is get handy with the concepts. You can also solve this question by putting values of x and then generalising it for all integral values.