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If $z^{2}+z +1=0$ Where $z$ is a complex number, then the value of

$\left ( z+\frac{1}{z} \right )^{2}+\left ( z^{2}+\frac{1}{z^{2}} \right )^{2}+\left ( z^{3} +\frac{1}{z}\right )^{2}......$ $......+\left ( z^{6}+\frac{1}{z^{6}} \right )^{2}$ is

Option 1)
$18$

Option 2)
$54$

Option 3)
$6$

Option 4)
$12$

Answers (2)

best_answer

As we learnt in

Property of Modulus of z(Complex Number) -

$|z_{1}+z_{2}|^{2}=\left | z_{1} \right |^{2}+\left | z_{2} \right |^{2}+z_{1}.\bar{z_{2}}+z_{2}.\bar{z_{1}}$

- wherein

|.| denotes modulus of z

$\bar{z}$ denotes conjugate of z

$z^{2}+z+1=0$

$\therefore\ \;z=\omega\ \; and\ \; \omega^{2}$

Now take $z=\omega$ , Where $\omega^{3}=1\ \; and\ \frac{1}{z}=\omega^{2}$

Now $z+\frac{1}{z}=\omega+\omega^{2}=-1$

$z^{2}+\frac{1}{z^{2}}=\omega^{2}+\omega^{4}=\omega^{2}+\omega=-1$

$z^{3}+\frac{1}{z^{3}}=\omega^{3}+\omega^{6}=1+1=2$

$z^{4}+\frac{1}{z^{4}}=\omega^{4}+\omega^{8}=\omega+\omega^{2}=-1$

$z^{5}+\frac{1}{z^{5}}=\omega^{5}+\omega^{10}=\omega^{2}+\omega=-1$

$z^{6}+\frac{1}{z^{6}}=\omega^{6}+\omega^{12}=1+1=2$

$\therefore\ \; 1+1+4+1+1+4=12$

Correct option is 4.

Option 1)

$18$

This is an incorrect option.

Option 2)

$54$

This is an incorrect option.

Option 3)

$6$

This is an incorrect option.

Option 4)

$12$

This is the correct option.

Posted by

prateek

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