If z2 + z + 1 = 0,
\(z ∈ C\), then \(\left| \sum_{n=1}^{15} \left( z_n + (-1)^n \frac{1}{z_n} \right)^2 \right|\)
is equal to ________.
If z2 + z + 1 = 0,
\(z ∈ C\), then \(\left| \sum_{n=1}^{15} \left( z_n + (-1)^n \frac{1}{z_n} \right)^2 \right|\)
is equal to ________.
Correct Answer: 2
Solution and Explanation
The correct answer is 2
∵ z2 + z + 1 = 0
⇒ ω or ω2
\(∵\)\(\left| \sum_{n=1}^{15} \left( z_n + (-1)^n \frac{1}{z_n} \right)^2 \right|\)
=\(\left| \sum_{n=1}^{15} z^{2n} + \sum_{n=1}^{15} z^{-2n} + 2.\sum_{n=1}^{15} (-1)^n \right|\)
= |0 + 0 – 2|
= 2
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Concepts Used:
Complex Number
A Complex Number is written in the form
a + ib
where,
- “a” is a real number
- “b” is an imaginary number
The Complex Number consists of a symbol “i” which satisfies the condition i^2 = −1. Complex Numbers are mentioned as the extension of one-dimensional number lines. In a complex plane, a Complex Number indicated as a + bi is usually represented in the form of the point (a, b). We have to pay attention that a Complex Number with absolutely no real part, such as – i, -5i, etc, is called purely imaginary. Also, a Complex Number with perfectly no imaginary part is known as a real number.