KVPY SX SB Exam 19-Nov-2017 Question Paper

Question : 17 of 120

Marks: +1, -0

Let

z_1, z_2, ......,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

w = ∑↙{1≤i< j≤7} \z_i \z_j

, then |w| is equal to

Solution:

given expression can be written as

{(z_1+ z_2+ ....+ z_7 )-∑↖{7}↙{i=1}z_i^2} /2

=∑↙{1≤i< j≤7} \z_i \z_j

Sum of all seven root of unity = 0 (By property)

z_1^2+ z_2^2+z_3^2 ....+ z_7^2=0

(By property)

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