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Question: If a, b, g are the roots of x<sup>3</sup> + px<sup>2</sup> + q = 0, where q ¹ 0, then D = \(\left| \...

If a, b, g are the roots of x3 + px2 + q = 0, where q ¹ 0, then D = ∣1/α1/β1/γ1/β1/γ1/α1/γ1/α1/β∣\left| \begin{matrix} 1/\alpha & 1/\beta & 1/\gamma \\ 1/\beta & 1/\gamma & 1/\alpha \\ 1/\gamma & 1/\alpha & 1/\beta \end{matrix} \right| equals-

A

–p/q

B

1/q

C

p2/q

D

None of these

Answer

None of these

Explanation

Solution

We have bg + ga + ab = 0 We can write D as

D= 1α3β3γ3\frac { 1 } { \alpha ^ { 3 } \beta ^ { 3 } \gamma ^ { 3 } } ∣βγγααβγααββγαββγγα∣\left| \begin{matrix} \beta\gamma & \gamma\alpha & \alpha\beta \\ \gamma\alpha & \alpha\beta & \beta\gamma \\ \alpha\beta & \beta\gamma & \gamma\alpha \end{matrix} \right| = 1α3β3γ3\frac{1}{\alpha^{3}\beta^{3}\gamma^{3}}

\beta\gamma + \gamma\alpha + \alpha\beta & \gamma\alpha & \alpha\beta \\ \gamma\alpha + \alpha\beta + \beta\gamma & \alpha\beta & \beta\gamma \\ \alpha\beta + \beta\gamma + \gamma\alpha & \beta\gamma & \gamma\alpha \end{matrix} \right|$$ [using C<sub>1</sub> ® C<sub>1</sub> + C<sub>2</sub> + C<sub>3</sub>]=$\frac{1}{\alpha^{3}\beta^{3}\gamma^{3}}$ $\left| \begin{matrix} 0 & \gamma\alpha & \alpha\beta \\ 0 & \alpha\beta & \beta\gamma \\ 0 & \beta\gamma & \gamma\alpha \end{matrix} \right|$= 0 [all zero property].