JEE-Advanced 2015 (29/40)

Algebra Level 3

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

\[\begin{array}{} (1) \, \left( -\frac{1}{2}, -\frac{1}{\sqrt5} \right) \quad \quad \quad \quad & (2) \, \left( -\frac{1}{\sqrt5},0 \right) \\\\ (3) \, \left( 0, \frac{1}{\sqrt5} \right) & (4) \, \left( \frac{1}{\sqrt5}, \frac{1}{2} \right) \end{array}\]

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

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