Assertion :Let the fourth roots of unity be $z_{1}, z_{2}, z_{3}$ and $z_{4}$ , respectively.
$z_{1}^{2} + z_{2}^{2} + z_{3}^{2} + z_{4}^{2} = 0$
Reason: $z_{1} + z_{2} + z_{3} + z_{4} = 0.$

Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

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Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

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Assertion is correct but Reason is incorrect

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Assertion is incorrect and Reason is correct

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Solution

The correct option is B Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
$x^{4} = 1$
$x^{2} = \pm 1$
$x = \pm 1$ and $x = \pm i$
Hence
$z_{1} = 1, z_{2} = - 1, z_{3} = i, z_{4} = - i$
Thus
$\sum z^{2}$
$= 1 + 1 - 1 - 1$
$= 0$
$\sum z$
$= 1 - 1 + i - i$
$= 0$
However Statement 2, is not the correct explanation of statement 1.
$z_{1}^{2} + z_{2}^{2} + z_{3}^{2} + z_{4}^{2}$ is 0 since the roots appear in conjugate pairs.

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