If $\;\alpha ,\beta \neq o\;$ , and $\;f(n)=\alpha^{n} + \beta^{n}\;$ and $\;\begin{vmatrix} 3 & 1+f(1) & 1+f(2) \\ 1+f(1) & 1+f(2) & 1+f(3) \\ 1+f(2) & 1+f(3) & 1+f(4)\end{vmatrix} = K (1-\alpha)^{2}(1-\beta)^{2}(\alpha-\beta)^{2}\;$, then$\;K\;$is equal to :

Question

$(a)\;1\qquad(b)\;-1\qquad(c)\;\alpha\beta\qquad(d)\;\large\frac{1}{\alpha \beta}$