Question:
If $\left(\frac{1+i}{1-i}\right)^{x}=1,$ then
If $\left(\frac{1+i}{1-i}\right)^{x}=1,$ then
Updated On: Apr 2, 2024
- $x = 4n$, where $n$ is any positive integer
- $x = 2n$, where $n$ is any positive integer
- $x = 4n+1$, where $n$ is any positive integer
- $x = 2n+1$, where $n$ is any positive integer
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The Correct Option is A
Solution and Explanation
$\frac{1+i}{1-i}=\frac{\left(1+i\right)^{2}}{2}=i$
$\left(\frac{1+i}{1-i}\right)^{x}=i^{x}$
$\Rightarrow x = 4n.$
Hence, (A) is the correct answer.
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Concepts Used:
Complex Numbers and Quadratic Equations
Complex Number: Any number that is formed as a+ib is called a complex number. For example: 9+3i,7+8i are complex numbers. Here i = -1. With this we can say that i² = 1. So, for every equation which does not have a real solution we can use i = -1.
Quadratic equation: A polynomial that has two roots or is of the degree 2 is called a quadratic equation. The general form of a quadratic equation is y=ax²+bx+c. Here a≠0, b and c are the real numbers.
