Q50 / 90 · KVPY 2010 (SB/SX) · Mathematics

Let $I, \omega$ and $\omega^{2}$ be the cube roots of unity. The least possible degree of a polynomial, with real coefficients, having $2 \mathrm{~m}^{2}, 3+4 \mathrm{c}, 3+4 \mathrm{c}^{2}$ and $5-\omega-\omega^{2}$ as roots is-

  • A
    4
  • Correct answer
    B
    5
  • C
    6
  • D
    8

Solution

roots $\rightarrow 2 \omega^{2}, 3+4 \omega, \quad 3+4 \omega^{2}, \quad 5-\omega-\omega^{2}$ $$ \begin{array}{c} \alpha \quad \beta \quad \gamma \quad \delta \\ \delta=5-\left(\omega+\omega^{2}\right)=5-(-1)=6 \end{array} $$ If $\alpha=2 \omega^{2}$ is a root then $2 \omega$ has to be a root too. total $\rightarrow \min .5$ roots, hence min. degree $\rightarrow 5$

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

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