JEE Math Practice Question

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

1. A.$$1$$
2. B.$$-1$$
3. C.$$\alpha \beta $$
4. D.$${1 \over {\alpha \beta }}$$