(A) The vertices $z_1, z_2, \ldots, z_7$ are the roots of the equation $z^7 - 1 = 0$.By Vieta's formulas for the polynomial $P(z) = z^7 + 0z^6 + 0z^5

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(A) The vertices $z_1, z_2, \ldots, z_7$ are the roots of the equation $z^7 - 1 = 0$.By Vieta's formulas for the polynomial $P(z) = z^7 + 0z^6 + 0z^5 + 0z^4 + 0z^3 + 0z^2 + 0z - 1 = 0$,the sum of the roots taken two at a time is given by the coefficient of $z^5$ divided by the coefficient of $z^7$.Thus,$w = \sum_{1 \leq i Therefore,$|w| = |0| = 0$.

Class 11 Mathematics 4-1.Complex numbers De Moivre's theorem and Roots of unity

Let $z_1, z_2, \ldots, z_7$ be the vertices of a regular heptagon inscribed in the unit circle with the center at the origin in the complex plane. If $w = \sum_{1 \leq i < j \leq 7} z_i z_j$,then $|w|$ is equal to:

KVPY 2017 All KVPY

A
$0$
B
$1$
C
$2$
D
$3$

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4-1.Complex numbers

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Let $z_1, z_2, \ldots, z_7$ be the vertices of a regular heptagon inscribed in the unit circle with the center at the origin in the complex plane. If $w = \sum_{1 \leq i < j \leq 7} z_i z_j$,then $|w|$ is equal to:

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