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Question: Answered & Verified by Expert
If \(\alpha\) and \(\beta\) are the roots of the equation \(x^2+x+1=0\), then the equation whose roots are \(\alpha^{19}\) and \(\beta^7\) is
MathematicsQuadratic EquationWBJEEWBJEE 2011
Options:
  • A \(x^2-x-1=0\)
  • B \(x^2-x+1=0\)
  • C \(x^2+x-1=0\)
  • D \(x^2+x+1=0\)
Solution:
1862 Upvotes Verified Answer
The correct answer is: \(x^2+x+1=0\)
Hints: \(\alpha\) and \(\beta\) are the roots of \(x^2+x+1=0\)
\(\begin{array}{ll}
\alpha=\omega & \beta=\omega^2 \\
\alpha^{19}=\omega & \beta^7=\omega^2 \\
x^2-\left(\alpha^{19}+\beta^7\right) x+\alpha^{19} \beta^7=0
\end{array}\)
Thou,
\(\begin{aligned}
& x^2-\left(\omega+\omega^2\right) x+\omega \cdot \omega^2=0 \\
& x^2+x+1=0
\end{aligned}\)

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