If the roots of the quadratic equation $x^{2} - a x + b = 0$ are real and differ by a quantity less than 1, then

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none of these

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Solution

The correct option is C
$x^{2} - a x + b = 0$
$α + β = a$
$α β = b$
$0 < | α - β | < 1$
$0 < \sqrt{(α + β)^{2} - 4 α β} < 1$
$0 < \sqrt{a^{2} - 4 b} < 1$
$0 < a^{2} - 4 b < 1$
$- 1 < 4 b - a^{2} < 0$
$∴ a^{2} - 1 < 4 b < a^{2}$
$\frac{a^{2} - 1}{4} < b < \frac{a^{2}}{4}$

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