QJEEM06Ma8ae1: JEE Main 2006 Mathematics - Class 11 JEEM06Ma8ae1 Solution: If , where z is complex number, then value of is :

Question

Verified Solution • 2025–26 Session moderate

If

{z^2} + z + 1 = 0

, where z is complex number, then value of

{\left( {z + {1 \over z}} \right)^2} + {\left( {{z^2} + {1 \over {{z^2}}}} \right)^2} + {\left( {{z^3} + {1 \over {{z^3}}}} \right)^2} + .......... + {\left( {{z^6} + {1 \over {{z^6}}}} \right)^2}

is :

Accepted Answer

{z^2} + z + 1 = 0 \Rightarrow z = \omega \,\,\,

or

\,\,\,{\omega ^2}

So,

z + {1 \over z} = \omega + {\omega ^2} = - 1

{z^2} + {1 \over {{z^2}}} = {\omega ^2} + \omega = - 1,

{z^3} + {1 \over {{z^3}}} = {\omega ^3} + {\omega ^3} = 2

{z^4} + {1 \over {{z^4}}} = - 1,

{z^5} + {1 \over {{z^5}}} = - 1

and

\,\,\,\,{z^6} + {1 \over {{z^6}}} = 2

\therefore

The given sum

= 1 + 1 + 4 + 1 + 1 + 4 = 12

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