Question

A real number ‘a’ is called a good number if the inequality (2x^2 – 2x – 3) / (x^2 + x + 1) ≤ a is satisfied for all real x. What is the set of all real numbers?

(a) (-∞, 10/3]

(b) (10/3, ∞)

(c) [10/3, ∞)

(d) [-10/3, ∞)

This question was posed to me during a job interview.

Question is from Applications of Quadratic Equations in division Complex Numbers and Quadratic Equations of Mathematics – Class 11

Answer 1

Right choice is (c) [10/3, ∞)

To explain: We have, (2x^2 – 2x – 3) / (x^2 + x + 1) ≤ a ᵾ x € R

=> 2x^2 – 2x – 3 ≤ a(x^2 + x + 1) ᵾ x € R

=> (2 – a)x^2 – (2 – a)x – (3 – a) ᵾ x € R

2 – a < 0 and (2 – a)x^2 – 4(2 – a)(3 – a) ≤ 0 ᵾ x € R

So, a > 2 and a ≤ 2 or a ≥ 10/3

=> a ≥ 10/3

Therefore, a € [10/3, ∞)