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Question

Question: If the roots of equation x<sup>2</sup> – ax + b = 0 are real and differ by a quantity less than 1. T...

If the roots of equation x2 – ax + b = 0 are real and differ by a quantity less than 1. Then –

A

a214\frac{a^{2} - 1}{4}< b <a24\frac{a^{2}}{4}

B

a215\frac{a^{2} - 1}{5}< b <a25\frac{a^{2}}{5}

C

a216\frac{a^{2} - 1}{6}< b <a22\frac{a^{2}}{2}

D

None of these

Answer

a214\frac{a^{2} - 1}{4}< b <a24\frac{a^{2}}{4}

Explanation

Solution

Roots are real and unequal a2 – 4b > 0

⇒ b < a24\frac{a^{2}}{4}

From question | α – β | < 1, α, β are roots

(α – β)2 = (α + β)2 – 4αβ = a2 – 4b

⇒ | α – β |2 = a2 – 4b

But | α – β | < 1 ⇒  | α – β |2 < 1

⇒ a2 – 4b < 1

⇒ a2 – 1 < 4b ⇒ a214\frac{a^{2} - 1}{4}< b

a214\frac{a^{2} - 1}{4} < b < a24\frac{a^{2}}{4}