question_answer

If \[\alpha ,\]\[\beta \ne 0,\] and \[f(n)={{\alpha }^{n}}+{{\beta }^{n}}\] \[\left| \begin{matrix}    3 & 1+f(1) & 1+f(2)  \\    1+f(1) & 1+f(2) & 1+f(3)  \\    1+f(2) & 1+f(3) & 1+f(4)  \\ \end{matrix} \right|\]\[=k{{(1-\alpha )}^{2}}\,\,{{(1-\beta )}^{2}}\,\,{{(\alpha -\beta )}^{2}},\]  then k is equal to

A)  \[1\]    

B)  \[-1\]

C)  \[\alpha \beta \]                        

D)  \[\frac{1}{\alpha \beta }\]

E)  None of these