Question

If α ,β are roots of the equation x2+ px - 1/2p2 = 0 ,∀p ∈ R - {0}, hen the minimum value of α4 + β4 is

(a) 2√2

(b) 2 –√ 2

(c) 2

(d) 2 + √2

Answer 1

Correct option (d) 2 + √2

Explanation:

α4 + β4 = (α2 + β2)2 - 2α2β2 = ((α + β)2 - 2α β)2 - 2 (α β)2

= (p2 + 1/p2)2 - 1/2p4 = p4 + 1/2p4 + 2

(p2 + 1/√2p2)2 + 2 + √2

∴ Min value is 2 + √2