Q36 / 90 · KVPY 2017 (19 Nov SB/SX) · Mathematics

Let $z_{1}, z_{2}, \ldots \ldots, z_{7}$ be the vertices of a regular heptagon that is inscribed in the unit circle with centre at the origin in the complex plane. Let $\mathrm{w}=\sum_{1 \leq \mathrm{i} < \mathrm{j} \leq 7} \mathrm{z}_{\mathrm{i}} \mathrm{z}_{\mathrm{j}}$, then $|\mathrm{w}|$ is equal to

  • Correct answer
    A
    0
  • B
    1
  • C
    2
  • D
    3

Solution

given expression can be written as $$ \frac{\left(z_{1}+z_{2}+\ldots+z_{7}\right)^{2}-\sum_{i=1}^{7} z_{i}^{2}}{2}=\sum_{1 \leq i < j \leq-7} z_{i} z_{j} $$ Sum of all seven root of unity $=0$ (By property) $z_{1}^{2}+z_{2}^{2}+z_{3}^{2}+\ldots .+z_{7}^{2}=0$ (By property)

Hence, the correct answer is option: 'A'.

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