Questions & Answers

Question

Answers

A.Zero (no solution)

B.One

C.Two

D.Three

Answer
Verified

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.