JEE-Advanced 2015 (29/40)

Algebra Level 3

Let SS be the set of all non-zero real numbers α\alpha such that the quadratic equation αx2x+α=0\alpha x^2-x+\alpha=0 has two distinct real roots x1x_1 and x2x_2 satisfying the inequality x1x2<1|x_1-x_2|<1. Which of the following intervals is/are a subset of S?S?

(1)(12,15)(2)(15,0)(3)(0,15)(4)(15,12)\begin{array}{l} (1) \, \left( -\frac{1}{2}, -\frac{1}{\sqrt5} \right) \quad \quad \quad \quad & (2) \, \left( -\frac{1}{\sqrt5},0 \right) \\\\ (3) \, \left( 0, \frac{1}{\sqrt5} \right) & (4) \, \left( \frac{1}{\sqrt5}, \frac{1}{2} \right) \end{array}
Note:

  • Submit your answer as the increasing order of the serial numbers of all the correct options.
  • For example, if your answer is (1),(2),(1), (2), then submit 12 as the correct answer; if your answer is (2),(3),(4),(2),(3),(4), then submit 234 as the correct answer.
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