JEE Advanced · Mathematics · 2. Quadratic Equations

Let S be the set of all non - zero real numbers $α$ such that the quadratic equation $α x^{2} - x + α = 0$ has two distinct real roots $x_{1} a n d x_{2}$ satisfying the inequality $|x_{1} - x_{2}| < 1 .$ Which of the following intervals is(are) a subset(s) of S?

A $(- \frac{1}{2}, - \frac{1}{\sqrt{5}})$
B $(- \frac{1}{\sqrt{5}}, 0)$
C $(0, \frac{1}{\sqrt{5}})$
D $(\frac{1}{\sqrt{5}}, \frac{1}{2})$

Verified Solution

Answer & Solution

Correct Answer

(D) $(\frac{1}{\sqrt{5}}, \frac{1}{2})$

Step-by-step Solution

Detailed explanation

α x^{2} - x + α = 0

|x_{1} - x_{2}| < 0

|\frac{\sqrt{1 - 4 α^{2}}}{α}| < 1

\sqrt{1 - 4 α^{2}} < | α |

1 - 4 α^{2} < α^{2}

α^{2} - \frac{1}{5} > 0

α \in (- \infty, - \frac{1}{\sqrt{5}}) \cup (\frac{1}{\sqrt{5}}, \infty)

Also,

1 - 4 α^{2} > 0

α^{2} - \frac{1}{4} < 0

(α - \frac{1}{2}) (α + \frac{1}{2}) < 0

α \in (- \frac{1}{2}, \frac{1}{2})

Taking intersection

α \in (- \frac{1}{2}, - \frac{1}{\sqrt{5}}) \cup (\frac{1}{\sqrt{5}}, \frac{1}{2})

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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(S) \(\|z+1\|^2\) is equal to	(4) 10
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Let S be the set of all non - zero real numbers α such that the quadratic equation αx2-x+α=0 has two distinct real roots x1 and x2 satisfying the inequality x1-x2<1. Which of the following intervals is(are) a subset(s) of S?

Step-by-step Solution

Let S be the set of all non - zero real numbers $α$ such that the quadratic equation $α x^{2} - x + α = 0$ has two distinct real roots $x_{1} a n d x_{2}$ satisfying the inequality $|x_{1} - x_{2}| < 1 .$ Which of the following intervals is(are) a subset(s) of S?