Question

If x² − x + a − 3 < 0 for at least one negative value of x, then complete set of values of a is

(A)  (−∞, 4)

(B)  (−∞, 2)

(C)  (−∞, 3) 

(D)  (−∞, 1)

Answer 1

Correct option (D) (−∞, 1)

The equation x² − x + a −3 = 0 must have at least one negative root. For real roots, D ≥ 0

⇒ 1 − 4(a − 3) ≥ 0

⇒ a ≤ 13/4

Both roots will be non-negative if

D ≥ 0, a − 3 ≥ 0, 1 ≥ 0

⇒ a ≤ 13 4 , a ≥ 3

⇒ a ∈ [3,13/4]

Thus equation will at least one negative root if