question_answer

Let \[-\frac{\pi }{6}<\theta <-\frac{\pi }{12}\] Suppose \[{{\alpha }_{1}}\] and \[{{\beta }_{1}}\] are the roots of the equation \[{{x}^{2}}-2x\sec \theta +1=0\] and \[{{\alpha }_{2}}\] and \[{{\beta }_{2}}\] are the roots of the equation\[{{x}^{2}}+2x\tan \theta -1=0\]. If \[{{\alpha }_{1}}>{{\beta }_{1}}\] and \[{{\alpha }_{2}}>{{\beta }_{2}},\] then \[{{\alpha }_{1}}+{{\beta }_{2}}\] equals

A)  \[2(\sec \theta -\tan \theta )\]       

B)  \[2\sec \theta \]

C)  \[-2\tan \theta \]            

D)  \[cosec\theta +\tan \theta \]

E)  None of these