If $1 , \omega , \omega^2$ are the cube roots of unity, then $\frac{1}{1+2\omega} + \frac{1}{2+\omega } - \frac{1}{1+\omega } = $
- 1
- $\omega$
- $\omega^2$
- 0
The Correct Option is D
Solution and Explanation
Cube roots of unity mean \(\omega^3=1\) and \(1+\omega+\omega^2 = 0 →\)
\(\frac{1}{1+2\omega}+\frac{1}{2+\omega}-\frac{1}{1+\omega}\)
\(=\frac{(2+\omega)+(1+2\omega)}{(1+2\omega)(2+\omega)}-\frac{1}{1+\omega}\)
\(=\frac{3+3\omega}{\omega+2\omega^{2}+2+4\omega}-\frac{1}{1+\omega}\)
\(=\frac{3(1+\omega)}{2\omega^{2}+5\omega+2}-\frac{1}{1+\omega}\)
\(=\frac{(3+3\omega)(1+\omega)-(2\omega^2+5\omega+2}{(2\omega^{2}+5\omega+2)(1+\omega)}\)
\(=\frac{3+3\omega+3\omega+3\omega^2-2\omega^2-5\omega-2}{2\omega^2+5\omega+2+2\omega^3+5\omega^2+2\omega}\)
\(=\frac{\omega^2+\omega+1}{2\omega^3+7\omega^2+7\omega+2}\)
\(=\frac{0}{2\omega^{3}+7\omega^{2}+7\omega+2}\)
\(=0\)
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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.