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1)

Let S be the set if all non-zero real numbers  α such that the quadratic equation αx2x+α=0 has two distinct real roots  x1  and x2 satisfying the inequality |x1-x2|<1. Which of the following interval(s) is /are a subset of S?


A) (12,15)

B) (15,0)

C) (0,15)

D) (15,12)

Answer:

Option A,B,C

Explanation:

 Given , x1  and x2 are roots of   αx2x+α=0

     x1+x2=1α   and    x1x2=1

 Also,  |x1x2|<1

     |x1x2|2<1(x1x2)2<1

 or         (x1x2)24x1x2<1

     1α24<1    or     1α2<5

      5α21>0

or         (5α1)(5α+1)>0

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     αϵ(,15)(15,)            ......(i)

 Also,            D>0

                 14α2>0    or    αϵ(12,12) .....(ii)

 From Eqs. (i) and (ii) , we get

  αϵ(12,15)(15,12)