Question

If a, b, g are roots of x³ + ax² + b = 0, then the value of

$\left| \begin{matrix} \alpha & \beta & \gamma \\ \beta & \gamma & \alpha \\ \gamma & \alpha & \beta \end{matrix} \right|$ is –

• A. a³
• B. – a³
• C. a² – 3b
• D. a³ – 3b

Answer 1

Answer: (A)

a³

Answer 2

a + b + g (–1)¹ $\frac{a}{1}$ = – a

ab + bg + ga = (–1)² . $\frac{0}{1}$ = 0 abg = (–1)³ . $\frac{b}{1}$ = – b

D = (a + b + g) $\left| \mspace{6mu}\begin{matrix} 1 & 1 & 1 \\ \beta & \gamma & \alpha \\ \gamma & \alpha & \beta \end{matrix}\mspace{6mu} \right|$

D = – (a + b + g) [a² + b² + g² – ab – bg – ga]

D = – (–a) × [(a + b + g)² – 3 (ab + bg + ga)]

D = a × [(–a)² – 3 (0)]

D = a³