Q93 / 160 · WBJEE 2011 · Mathematics

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

  • A
    \(x^2-x-1=0\)
  • B
    \(x^2-x+1=0\)
  • C
    \(x^2+x-1=0\)
  • Correct answer
    D
    \(x^2+x+1=0\)

Solution

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, \(\begin{aligned} & x^2-\left(\omega+\omega^2\right) x+\omega \cdot \omega^2=0 \\ & x^2+x+1=0 \end{aligned}\)

Hence, the correct answer is option: 'D'.

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