Question

The number of integral values of m for which the equation (1 + m²)x² – 2(1 + 3m)x + (1 + 8m) = 0 has no real root is ______.

Options

• 1

• 2

• infinitely many

• 3

Answer 1

The number of integral values of m for which the equation (1 + m²)x² – 2(1 + 3m)x + (1 + 8m) = 0 has no real root is infinitely many.

Explanation:

Given equation is

(1 + m²)x² – 2(1 + 3m)x + (1 + 8m) = 0

∵ Equation has no real solution

∴ D < 0

⇒ 4 (1 + 3m)² < 4 (1 + m²) (1 + 8m)

⇒ 1 + 9m² + 6m < 1 + 8m + m² + 8m³

⇒ 8m³ – 8m² + 2m > 0

⇒ 2m (4m² – 4m + 1) > 0

⇒ 2m (2m – 1)² > 0

⇒ m > 0 and `"m" ≠ 1/2`  ...`[∵ 1/2  "is not an integer"]`

⇒ Number of integral values of m are infinitely many.