If \;≤ft( 1+i/1-i )^x = 1 then [AIEEE 2003]

Correct Answer is : x=4nx = 4nx=4n, where nnn is any positive integer

x = 2n+1, where n is any positive integer

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.

Correct Answer is : x=4nx = 4nx=4n, where nnn is any positive integer

Test Index

Complex Numbers Part 1

Section: Mathematics

Question : 97 of 100

Marks: +1, -0

\;\left( \frac{1+i}{1-i} \right)^x = 1

Solution:

\;\left( \frac{1+i}{1-i} \right)^x = 1

\Rightarrow \left[ \; \frac{(1+i)(1+i)}{(1-i)(1+i)} \right] ^x = 1

\Rightarrow \left[ \; \frac{(1+i)^2}{1-i^2} \right] ^x = 1

\Rightarrow \left[ \; \frac{1+2i+i^2}{1+1} \right] ^x = 1

\Rightarrow \left[ \; \frac{1+2i-1}{2} \right] ^x = 1

\Rightarrow (i)^x = 1

We know

i = \sqrt{-1}

\therefore i^2 = -1

\Rightarrow i^3 = -1 \times i = -i

\Rightarrow i^4 = -i \times i = -i^2 = -(-1) = 1

So when power of

i

is 4 or multiple of 4 then it's value is

=1

\therefore (i)^x = 1 = (i)^{4n}

where

n

is a positive integer.

Go to Question:

Prev Question

Next Question