Question

If a and b are integers in the below mentioned equations then find minimum possible value of a + b²

a²x² - 4x + 9 < 0

x² + bx + 12 ≥ 0

1. 38/ 3
2. 41/ 3
3. 47/ 3
4. 51/ 3

Answer 1

Correct Answer - Option 1 : 38/ 3

Calculation:

a²x² -4x + 9 < 0

Dividing above equation by a²

⇒ x² – (4x/ a²) + (9/ a²) < 0

⇒ (x – (3/ a)) ² + (6x/ a) – (4x/ a²) < 0

⇒ (x – (3/ a)) ² < (4x/ a²) – (6x/ a)

⇒ (4/ a²) – (6/ a) < 0

⇒ 4 – 6a < 0

⇒ 4 > 6a

⇒ a < 2/ 3

x² + bx + 12 ≥ 0

⇒ x² + bx + b² ≥ b² – 12

⇒ (x + b)² ≥ b² – 12

⇒ 12 ≥ b²

⇒ b ≤ 2√ 3

a + b²

⇒ (2/ 3) + (12)

⇒ (2 + 36)/ 3

⇒ 38/ 3

∴ required answer is 38/ 3