JEE Math Practice Question

Let $$f(x) = \left| {\begin{matrix} {1 + {{\sin }^2}x} & {{{\cos }^2}x} & {\sin 2x} \\ {{{\sin }^2}x} & {1 + {{\cos }^2}x} & {\sin 2x} \\ {{{\sin }^2}x} & {{{\cos }^2}x} & {1 + \sin 2x} \\ \end{matrix} } \right|,\,x \in \left[ {{\pi \over 6},{\pi \over 3}} \right]$$. If $$\alpha$$ and $$\beta$$ respectively are the maximum and the minimum values of $$f$$, then

1. A.$${\alpha ^2} - {\beta ^2} = 4\sqrt 3 $$
2. B.$${\beta ^2} - 2\sqrt \alpha = {{19} \over 4}$$
3. C.$${\beta ^2} + 2\sqrt \alpha = {{19} \over 4}$$
4. D.$${\alpha ^2} + {\beta ^2} = {9 \over 2}$$