Question

If a, b, g are roots of the equation x³ + px² + q = 0 where q ¹ 0 then D = $\left| \begin{matrix} \frac{1}{\alpha} & \frac{1}{\beta} & \frac{1}{\gamma} \\ \frac{1}{\beta} & \frac{1}{\gamma} & \frac{1}{\alpha} \\ \frac{1}{\gamma} & \frac{1}{\alpha} & \frac{1}{\beta} \end{matrix} \right|$

• A. –$\frac{p}{q}$
• B. $\frac{1}{q}$
• C. $\frac{p^{2}}{q}$
• D. None

Answer 1

Answer: (D)

None

Answer 2

S a = – p, S a b = 0, a b g = – q

D=$\left( \frac { 1 } { \alpha } + \frac { 1 } { \beta } + \frac { 1 } { \gamma } \right)$ $\left| \begin{matrix} 1 & 1/\beta & 1/\gamma \\ 1 & 1/\gamma & 1/\alpha \\ 1 & 1/\alpha & 1/\beta \end{matrix} \right|$= $\frac { \Sigma \alpha \beta } { \alpha \beta \gamma }$ $\left| \begin{matrix} 1 & 1/\beta & 1/\gamma \\ 1 & 1/\gamma & 1/\alpha \\ 1 & 1/\alpha & 1/\beta \end{matrix} \right|$= 0