1)

Let   π6<θ<π12 . Suppose  α1 and β1 are the roots of the equation x22xsecθ+1=0,  and α2  and β2 are roots of the equation x2+2xtanθ1=0. If  α1>β1  and α2>β2  , α1+β2 equals


A) 2(secθtanθ)

B) 2secθ

C) 2tanθ

D) 0

Answer:

Option C

Explanation:

Here,  x22xsecθ+1=0 has roots  α1 and β1.

      α1,β1=2secθ±4sec2θ42×1

=2secθ±2|tanθ|2

Since, θϵ(π6,π12),

i.e,    Θ ε IV   quadrant = 2secθ±2tanθ2

          α1=secθtanθ      and    β1=secθ+tanθ

 (as α1 > β1 )

 and     x2+2xtanθ1=0 has roots  α2  and β2.

 i.e. α2,β2=2tanθ±4tan2θ+42

       α2=tanθ+secθ

  and   β2=tanθsecθ  [as   α2>β2 ]

 Thus,  α1+β2=2tanθ