Question
If $$f\left( x \right) = a + bx + c{x^2}$$ and $$\alpha ,\beta ,\lambda $$ are roots of the equation $$x^3 = 1,$$ then \[\left| {\begin{array}{*{20}{c}}
a&b&c\\
b&c&a\\
c&a&b
\end{array}} \right|\] is equal to
A.
$$ f\left( \alpha \right) + f\left( \beta \right) + f\left( \lambda \right)$$
B.
$$f\left( \alpha \right)f\left( \beta \right) + f\left( \beta \right)f\left( \lambda \right) + f\left( \gamma \right) + f\left( \alpha \right)$$
C.
$$ f\left( \alpha \right)f\left( \beta \right)f\left( \lambda \right)$$
D.
$$ - f\left( \alpha \right)f\left( \beta \right)f\left( \lambda \right)$$
Answer :
$$ - f\left( \alpha \right)f\left( \beta \right)f\left( \lambda \right)$$
Solution :
\[\left| {\begin{array}{*{20}{c}}
a&b&c\\
b&c&a\\
c&a&b
\end{array}} \right| = - \left( {{a^3} + {b^3} + {c^3} - 3abc} \right)\]
$$\eqalign{
& = - \left( {a + b + c} \right)\left( {a + b{\omega ^2} + c\omega } \right)\left( {a + b\omega + c{\omega ^2}} \right) \cr
& = - f\left( \alpha \right)f\left( \beta \right)f\left( \lambda \right)\,\,\,\,\,\left[ {\because \alpha = 1,\beta = \omega ,\lambda = {\omega ^2}} \right] \cr} $$