Processing math: 26%


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

     x_{1}+x_{2}=\frac{1}{\alpha}   and    x_{1}x_{2}=1

 Also,  |x_{1}-x_{2}|<1

 \Rightarrow    |x_{1}-x_{2}|^{2}<1\Rightarrow(x_{1}-x_{2})^{2}<1

 or         (x_{1}-x_{2})^{2}-4x_{1}x_{2}<1

 \Rightarrow    \frac{1}{\alpha^{2}}-4<1    or     \frac{1}{\alpha^{2}}<5

 \Rightarrow     5\alpha^{2}-1>0

or         (\sqrt{5}\alpha-1)(\sqrt{5}\alpha+1)>0

 1232021950_m3.JPG

\therefore     \alpha \epsilon \left(-\infty, -\frac{1}{\sqrt{5}}\right)\cup\left(\frac{1}{\sqrt{5}},\infty\right)            ......(i)

 Also,            D>0

           \Rightarrow      1-4\alpha^{2}>0    or    \alpha \epsilon \left(-\frac{1}{2},\frac{1}{2}\right) .....(ii)

 From Eqs. (i) and (ii) , we get

  \alpha \epsilon \left(-\frac{1}{2}, -\frac{1}{\sqrt{5}}\right)\cup\left(\frac{1}{\sqrt{5}},\frac{1}{2}\right)