QJEEM03Maa197: JEE Main 2003 Mathematics - Class 11 JEEM03Maa197 Solution: If then :

Question

Verified Solution • 2025–26 Session moderate

If

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

then :

Accepted Answer

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

\Rightarrow

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

\Rightarrow

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

\Rightarrow

{\left[ {{{1 + 2i + {i^2}} \over {1 + 1}}} \right]^x} = 1

\Rightarrow

{\left[ {{{1 + 2i - 1} \over 2}} \right]^x} = 1

\Rightarrow {\left( i \right)^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} = - \left( { - 1} \right) = 1

So when power of

i

is 4 or multiple of 4 then it's value is = 1

\therefore

{\left( i \right)^x} = 1

= {\left( i \right)^{4n}}

where n is a positive integer.

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