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← JEE Advanced 2015 · Paper 2
MathematicsMultiple correct+4 / -2Question 49 of 60

JEE Advanced 2015 · Paper 2, Mathematics question 49

Let SS be the set of all non-zero real numbers α\alpha such that the quadratic equation αx2−x+α=0\alpha x^2-x+\alpha=0 has two distinct real roots x1x_1 and x2x_2 satisfying the inequality ∣x1−x2∣<1|x_1-x_2|<1. Which of the following intervals is(are) a subset(s) of SS?

Choose one or more options
Answer and worked solution

Official answer: A, D

Distinct real roots. 1−4α2>01-4\alpha^2>0, so ∣α∣<12|\alpha|<\frac12.

Gap between the roots.
∣x1−x2∣=1−4α2∣α∣<1  ⟹  1−4α2<α2  ⟹  ∣α∣>15.|x_1-x_2|=\frac{\sqrt{1-4\alpha^2}}{|\alpha|}<1\implies1-4\alpha^2<\alpha^2\implies|\alpha|>\frac1{\sqrt5}.

The set. S=(−12,−15)∪(15,12)S=\left(-\frac12,-\frac1{\sqrt5}\right)\cup\left(\frac1{\sqrt5},\frac12\right).

(A), (D)

Topics:Nature of rootsInequalities and modulus
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