UPSEE Solved Model Paper 6

we have

λx2+3x+4

x2+2x+2

< 5 for all x ∊ R
clearly x2+2x+2 > 0 for all x
∴ λx2 + 3x + 4 < 5 (x2+2x+2)
⇒ λx2 + 3x + 4 < 5x2 + 10x + 10
⇒ λx2−5x2 + 3x - 10x + 4 - 10 < 0
⇒ x2(λ−5) - 7x - 6 < 0
⇒ λ - 5 < 0 and the discriminent is less than zero
⇒ λ - 5 < 0 and (−7)2 - 4 (λ - 5) - 6 < 0
⇒ λ < 5 and 49 + 24λ - 120 < 0
⇒ λ < 5 and 24λ - 71 < 0
⇒ λ < 5 and 24λ < 71
⇒ λ <

71

24