If \( α,β,γ\) are the cube roots of \(-2\) ,then the value of \(xα+yβ+zγ/xβ+yγ+zα\) is (\(x,y,z\) are variables)
- \(\)\(e^{ iπ/3}\)
\(e^{2πi/3}\)
\(1\)
\(-1\)
\(e^{ 4iπ/3}\)
The Correct Option is
Solution and Explanation
\(\dfrac{ xα + yβ + zγ }{xβ + yγ + zα }\)
\(=\dfrac{x(−2)^{1/3} + y (−2^{1/3}ω ) + z ( −2^{1/3}ω 2 )} {x (−2^{1/3}ω) + y (−2^{1/3}ω^2 ) + z(−2)^{1/3} }\)
\(= \dfrac{(−2)^{1/3}(x + yω + zω^2 )ω}{ −2^{1/3}(xω+ yω^2 +z) ω }\)
\(= 1 ω = \dfrac{ω ^3}{ω} = ω^2\)
\(= e^{4πi/3}\)
So, the correct option is (E) : \(e^{4πi/3}\)
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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.